Numbers: BigN and BigZ get instantiations of all properties about div and mod

 NB: for declaring div and mod as a morphism, even when divisor is zero,
 I've slightly changed the definition of div_eucl: it now starts by a
 check of whether the divisor is zero. Not very nice, but this way
 we can say that BigN.div and BigZ.div _always_ answer like Zdiv.Zdiv.

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12646 85f007b7-540e-0410-9357-904b9bb8a0f7

4 files changed, 47 insertions, 25 deletions

diff --git a/theories/Numbers/Natural/BigN/BigN.v b/theories/Numbers/Natural/BigN/BigN.v
index bc2f3ecdf..b87056a63 100644
--- a/theories/Numbers/Natural/BigN/BigN.v
+++ b/theories/Numbers/Natural/BigN/BigN.v
@@ -15,12 +15,7 @@ Author: Arnaud Spiwack
 *)
 
 Require Export Int31.
-Require Import CyclicAxioms.
-Require Import Cyclic31.
-Require Import NSig.
-Require Import NSigNAxioms.
-Require Import NMake.
-Require Import NProperties.
+Require Import CyclicAxioms Cyclic31 NSig NSigNAxioms NMake NProperties NDiv.
 
 Module BigN <: NType := NMake.Make Int31Cyclic.
 
@@ -28,6 +23,7 @@ Module BigN <: NType := NMake.Make Int31Cyclic.
 
 Module Export BigNAxiomsMod := NSig_NAxioms BigN.
 Module Export BigNPropMod := NPropFunct BigNAxiomsMod.
+Module Export BigDivModProp := NDivPropFunct BigNAxiomsMod BigNPropMod.
 
 (** Notations about [BigN] *)
 
@@ -73,6 +69,7 @@ Infix "<=" := BigN.le : bigN_scope.
 Notation "x > y" := (BigN.lt y x)(only parsing) : bigN_scope.
 Notation "x >= y" := (BigN.le y x)(only parsing) : bigN_scope.
 Notation "[ i ]" := (BigN.to_Z i) : bigN_scope.
+Infix "mod" := modulo (at level 40, no associativity) : bigN_scope.
 
 Local Open Scope bigN_scope.
 
diff --git a/theories/Numbers/Natural/BigN/NMake_gen.ml b/theories/Numbers/Natural/BigN/NMake_gen.ml
index 769d804d7..b8e879c66 100644
--- a/theories/Numbers/Natural/BigN/NMake_gen.ml
+++ b/theories/Numbers/Natural/BigN/NMake_gen.ml
@@ -1956,6 +1956,7 @@ let _ =
   pr "";
 
   pr " Definition div_eucl x y :=";
+  pr "  if eq_bool y zero then (zero,zero) else";
   pr "  match compare x y with";
   pr "  | Eq => (one, zero)";
   pr "  | Lt => (zero, x)";
@@ -1964,7 +1965,6 @@ let _ =
   pr "";
 
   pr " Theorem spec_div_eucl: forall x y,";
-  pr "      0 < [y] ->";
   pr "      let (q,r) := div_eucl x y in";
   pr "      ([q], [r]) = Zdiv_eucl [x] [y].";
   pa " Admitted.";
@@ -1973,8 +1973,13 @@ let _ =
   pp "   exact (spec_0 w0_spec).";
   pp " assert (F1: [one] = 1).";
   pp "   exact (spec_1 w0_spec).";
-  pp " intros x y H; generalize (spec_compare x y);";
-  pp "   unfold div_eucl; case compare; try rewrite F0;";
+  pp " intros x y. unfold div_eucl.";
+  pp " generalize (spec_eq_bool y zero). destruct eq_bool; rewrite F0.";
+  pp " intro H. rewrite H. destruct [x]; auto.";
+  pp " intro H'.";
+  pp " assert (0 < [y]) by (generalize (spec_pos y); auto with zarith).";
+  pp " clear H'.";
+  pp " generalize (spec_compare x y); case compare; try rewrite F0;";
   pp "   try rewrite F1; intros; auto with zarith.";
   pp " rewrite H0; generalize (Z_div_same [y] (Zlt_gt _ _ H))";
   pp "                        (Z_mod_same [y] (Zlt_gt _ _ H));";
@@ -1994,10 +1999,10 @@ let _ =
   pr "";
 
   pr " Theorem spec_div:";
-  pr "   forall x y, 0 < [y] -> [div x y] = [x] / [y].";
+  pr "   forall x y, [div x y] = [x] / [y].";
   pa " Admitted.";
   pp " Proof.";
-  pp " intros x y H1; unfold div; generalize (spec_div_eucl x y H1);";
+  pp " intros x y; unfold div; generalize (spec_div_eucl x y);";
   pp "   case div_eucl; simpl fst.";
   pp " intros xx yy; unfold Zdiv; case Zdiv_eucl; intros qq rr H; ";
   pp "  injection H; auto.";
@@ -2099,6 +2104,7 @@ let _ =
   pr "";
 
   pr " Definition modulo x y := ";
+  pr "  if eq_bool y zero then zero else";
   pr "  match compare x y with";
   pr "  | Eq => zero";
   pr "  | Lt => x";
@@ -2107,15 +2113,20 @@ let _ =
   pr "";
 
   pr " Theorem spec_modulo:";
-  pr "   forall x y, 0 < [y] -> [modulo x y] = [x] mod [y].";
+  pr "   forall x y, [modulo x y] = [x] mod [y].";
   pa " Admitted.";
   pp " Proof.";
   pp " assert (F0: [zero] = 0).";
   pp "   exact (spec_0 w0_spec).";
   pp " assert (F1: [one] = 1).";
   pp "   exact (spec_1 w0_spec).";
-  pp " intros x y H; generalize (spec_compare x y);";
-  pp "   unfold modulo; case compare; try rewrite F0;";
+  pp " intros x y. unfold modulo.";
+  pp " generalize (spec_eq_bool y zero). destruct eq_bool; rewrite F0.";
+  pp " intro H; rewrite H. destruct [x]; auto.";
+  pp " intro H'.";
+  pp " assert (H : 0 < [y]) by (generalize (spec_pos y); auto with zarith).";
+  pp " clear H'.";
+  pp " generalize (spec_compare x y); case compare; try rewrite F0;";
   pp "   try rewrite F1; intros; try split; auto with zarith.";
   pp " rewrite H0; apply sym_equal; apply Z_mod_same; auto with zarith.";
   pp " apply sym_equal; apply Zmod_small; auto with zarith.";
diff --git a/theories/Numbers/Natural/SpecViaZ/NSig.v b/theories/Numbers/Natural/SpecViaZ/NSig.v
index 2dfa783aa..ecb49d32d 100644
--- a/theories/Numbers/Natural/SpecViaZ/NSig.v
+++ b/theories/Numbers/Natural/SpecViaZ/NSig.v
@@ -94,17 +94,15 @@ Module Type NType.
  Parameter div_eucl : t -> t -> t * t.
 
  Parameter spec_div_eucl: forall x y,
-      0 < [y] ->
-      let (q,r) := div_eucl x y in ([q], [r]) = Zdiv_eucl [x] [y].
+  let (q,r) := div_eucl x y in ([q], [r]) = Zdiv_eucl [x] [y].
 
  Parameter div : t -> t -> t.
 
- Parameter spec_div: forall x y, 0 < [y] -> [div x y] = [x] / [y].
+ Parameter spec_div: forall x y, [div x y] = [x] / [y].
 
  Parameter modulo : t -> t -> t.
 
- Parameter spec_modulo:
-   forall x y, 0 < [y] -> [modulo x y] = [x] mod [y].
+ Parameter spec_modulo: forall x y, [modulo x y] = [x] mod [y].
 
  Parameter gcd : t -> t -> t.
 
diff --git a/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v b/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v
index 66d3a89c2..8a34185d9 100644
--- a/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v
+++ b/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v
@@ -8,14 +8,11 @@
 
 (*i $Id$ i*)
 
-Require Import ZArith.
-Require Import Nnat.
-Require Import NAxioms.
-Require Import NSig.
+Require Import ZArith Nnat NAxioms NDiv NSig.
 
 (** * The interface [NSig.NType] implies the interface [NAxiomsSig] *)
 
-Module NSig_NAxioms (N:NType) <: NAxiomsSig.
+Module NSig_NAxioms (N:NType) <: NAxiomsSig <: NDivSig.
 
 Delimit Scope NumScope with Num.
 Bind Scope NumScope with N.t.
@@ -28,7 +25,8 @@ Local Infix "-" := N.sub : NumScope.
 Local Infix "*" := N.mul : NumScope.
 
 Hint Rewrite
- N.spec_0 N.spec_succ N.spec_add N.spec_mul N.spec_pred N.spec_sub : num.
+ N.spec_0 N.spec_succ N.spec_add N.spec_mul N.spec_pred N.spec_sub
+ N.spec_div N.spec_modulo : num.
 Ltac nsimpl := autorewrite with num.
 Ltac ncongruence := unfold N.eq; repeat red; intros; nsimpl; congruence.
 
@@ -212,6 +210,22 @@ Proof.
 red; nsimpl; auto.
 Qed.
 
+Program Instance div_wd : Proper (N.eq==>N.eq==>N.eq) N.div.
+Program Instance mod_wd : Proper (N.eq==>N.eq==>N.eq) N.modulo.
+
+Theorem div_mod : forall a b, ~b==0 -> a == b*(N.div a b) + (N.modulo a b).
+Proof.
+intros a b. unfold N.eq. nsimpl. intros.
+apply Z_div_mod_eq_full; auto.
+Qed.
+
+Theorem mod_upper_bound : forall a b, ~b==0 -> N.modulo a b < b.
+Proof.
+intros a b. unfold N.eq. rewrite spec_lt. nsimpl. intros.
+destruct (Z_mod_lt [a] [b]); auto.
+generalize (N.spec_pos b); auto with zarith.
+Qed.
+
 Definition recursion (A : Type) (a : A) (f : N.t -> A -> A) (n : N.t) :=
   Nrect (fun _ => A) a (fun n a => f (N.of_N n) a) (N.to_N n).
 Implicit Arguments recursion [A].
@@ -282,5 +296,7 @@ Definition lt := N.lt.
 Definition le := N.le.
 Definition min := N.min.
 Definition max := N.max.
+Definition div := N.div.
+Definition modulo := N.modulo.
 
 End NSig_NAxioms.