Division in Numbers: factorisation of signatures

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12624 85f007b7-540e-0410-9357-904b9bb8a0f7

5 files changed, 17 insertions, 54 deletions

diff --git a/theories/Numbers/Integer/Abstract/ZDivEucl.v b/theories/Numbers/Integer/Abstract/ZDivEucl.v
index 2ec1cc9b04..e6af6f16f7 100644
--- a/theories/Numbers/Integer/Abstract/ZDivEucl.v
+++ b/theories/Numbers/Integer/Abstract/ZDivEucl.v
@@ -26,21 +26,9 @@ Require Import ZAxioms ZProperties NZDiv.
 Open Scope NumScope.
 
 Module Type ZDiv (Import Z : ZAxiomsSig).
-
- Parameter Inline div : t -> t -> t.
- Parameter Inline modulo : t -> t -> t.
-
- Infix "/" := div : NumScope.
- Infix "mod" := modulo (at level 40, no associativity) : NumScope.
-
- Declare Instance div_wd : Proper (eq==>eq==>eq) div.
- Declare Instance mod_wd : Proper (eq==>eq==>eq) modulo.
-
+ Include Type NZDivCommon Z. (** div, mod, compat with eq, equation a=bq+r *)
  Definition abs z := max z (-z).
-
- Axiom div_mod : forall a b, b ~= 0 -> a == b*(a/b) + (a mod b).
  Axiom mod_always_pos : forall a b, 0 <= a mod b < abs b.
-
 End ZDiv.
 
 Module Type ZDivSig := ZAxiomsSig <+ ZDiv.
diff --git a/theories/Numbers/Integer/Abstract/ZDivFloor.v b/theories/Numbers/Integer/Abstract/ZDivFloor.v
index 4736312939..c152fa83ea 100644
--- a/theories/Numbers/Integer/Abstract/ZDivFloor.v
+++ b/theories/Numbers/Integer/Abstract/ZDivFloor.v
@@ -29,20 +29,9 @@ Require Import ZAxioms ZProperties NZDiv.
 Open Scope NumScope.
 
 Module Type ZDiv (Import Z : ZAxiomsSig).
-
- Parameter Inline div : t -> t -> t.
- Parameter Inline modulo : t -> t -> t.
-
- Infix "/" := div : NumScope.
- Infix "mod" := modulo (at level 40, no associativity) : NumScope.
-
- Declare Instance div_wd : Proper (eq==>eq==>eq) div.
- Declare Instance mod_wd : Proper (eq==>eq==>eq) modulo.
-
- Axiom div_mod : forall a b, b ~= 0 -> a == b*(a/b) + (a mod b).
+ Include Type NZDivCommon Z. (** div, mod, compat with eq, equation a=bq+r *)
  Axiom mod_pos_bound : forall a b, 0 < b -> 0 <= a mod b < b.
  Axiom mod_neg_bound : forall a b, b < 0 -> b < a mod b <= 0.
-
 End ZDiv.
 
 Module Type ZDivSig := ZAxiomsSig <+ ZDiv.
diff --git a/theories/Numbers/Integer/Abstract/ZDivTrunc.v b/theories/Numbers/Integer/Abstract/ZDivTrunc.v
index 2522f70c12..eca1e2ba17 100644
--- a/theories/Numbers/Integer/Abstract/ZDivTrunc.v
+++ b/theories/Numbers/Integer/Abstract/ZDivTrunc.v
@@ -29,21 +29,10 @@ Require Import ZAxioms ZProperties NZDiv.
 Open Scope NumScope.
 
 Module Type ZDiv (Import Z : ZAxiomsSig).
-
- Parameter Inline div : t -> t -> t.
- Parameter Inline modulo : t -> t -> t.
-
- Infix "/" := div : NumScope.
- Infix "mod" := modulo (at level 40, no associativity) : NumScope.
-
- Declare Instance div_wd : Proper (eq==>eq==>eq) div.
- Declare Instance mod_wd : Proper (eq==>eq==>eq) modulo.
-
- Axiom div_mod : forall a b, b ~= 0 -> a == b*(a/b) + (a mod b).
+ Include Type NZDivCommon Z. (** div, mod, compat with eq, equation a=bq+r *)
  Axiom mod_bound : forall a b, 0<=a -> 0<b -> 0 <= a mod b < b.
  Axiom mod_opp_l : forall a b, b ~= 0 -> (-a) mod b == - (a mod b).
  Axiom mod_opp_r : forall a b, b ~= 0 -> a mod (-b) == a mod b.
-
 End ZDiv.
 
 Module Type ZDivSig := ZAxiomsSig <+ ZDiv.
diff --git a/theories/Numbers/NatInt/NZDiv.v b/theories/Numbers/NatInt/NZDiv.v
index 1be2f85087..bb63b420b3 100644
--- a/theories/Numbers/NatInt/NZDiv.v
+++ b/theories/Numbers/NatInt/NZDiv.v
@@ -12,20 +12,28 @@ Require Import NZAxioms NZMulOrder.
 
 Open Scope NumScope.
 
-Module Type NZDiv (Import NZ : NZOrdAxiomsSig).
+(** This first signature will be common to all divisions over NZ, N and Z *)
 
+Module Type NZDivCommon (Import NZ : NZAxiomsSig).
  Parameter Inline div : t -> t -> t.
  Parameter Inline modulo : t -> t -> t.
-
  Infix "/" := div : NumScope.
  Infix "mod" := modulo (at level 40, no associativity) : NumScope.
-
  Declare Instance div_wd : Proper (eq==>eq==>eq) div.
  Declare Instance mod_wd : Proper (eq==>eq==>eq) modulo.
-
  Axiom div_mod : forall a b, b ~= 0 -> a == b*(a/b) + (a mod b).
- Axiom mod_bound : forall a b, 0<=a -> 0<b -> 0 <= a mod b < b.
+End NZDivCommon.
 
+(** The different divisions will only differ in the conditions
+    they impose on [modulo]. For NZ, we only describe behavior
+    on positive numbers.
+
+    NB: This axiom would also be true for N and Z, but redundant.
+*)
+
+Module Type NZDiv (Import NZ : NZOrdAxiomsSig).
+ Include Type NZDivCommon NZ.
+ Axiom mod_bound : forall a b, 0<=a -> 0<b -> 0 <= a mod b < b.
 End NZDiv.
 
 Module Type NZDivSig := NZOrdAxiomsSig <+ NZDiv.
diff --git a/theories/Numbers/Natural/Abstract/NDiv.v b/theories/Numbers/Natural/Abstract/NDiv.v
index 1aa10958f0..6c417868a8 100644
--- a/theories/Numbers/Natural/Abstract/NDiv.v
+++ b/theories/Numbers/Natural/Abstract/NDiv.v
@@ -13,19 +13,8 @@ Require Import NAxioms NProperties NZDiv.
 Open Scope NumScope.
 
 Module Type NDiv (Import N : NAxiomsSig).
-
- Parameter Inline div : t -> t -> t.
- Parameter Inline modulo : t -> t -> t.
-
- Infix "/" := div : NumScope.
- Infix "mod" := modulo (at level 40, no associativity) : NumScope.
-
- Declare Instance div_wd : Proper (eq==>eq==>eq) div.
- Declare Instance mod_wd : Proper (eq==>eq==>eq) modulo.
-
- Axiom div_mod : forall a b, b ~= 0 -> a == b*(a/b) + (a mod b).
+ Include Type NZDivCommon N. (** div, mod, compat with eq, equation a=bq+r *)
  Axiom mod_upper_bound : forall a b, b ~= 0 -> a mod b < b.
-
 End NDiv.
 
 Module Type NDivSig := NAxiomsSig <+ NDiv.