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, 88 insertions, 31 deletions

diff --git a/theories/Numbers/Integer/BigZ/BigZ.v b/theories/Numbers/Integer/BigZ/BigZ.v
index 1ec7960ae..fc94f693a 100644
--- a/theories/Numbers/Integer/BigZ/BigZ.v
+++ b/theories/Numbers/Integer/BigZ/BigZ.v
@@ -11,10 +11,7 @@
 (*i $Id$ i*)
 
 Require Export BigN.
-Require Import ZProperties.
-Require Import ZSig.
-Require Import ZSigZAxioms.
-Require Import ZMake.
+Require Import ZProperties ZDivFloor ZSig ZSigZAxioms ZMake.
 
 Module BigZ <: ZType := ZMake.Make BigN.
 
@@ -22,6 +19,7 @@ Module BigZ <: ZType := ZMake.Make BigN.
 
 Module Export BigZAxiomsMod := ZSig_ZAxioms BigZ.
 Module Export BigZPropMod := ZPropFunct BigZAxiomsMod.
+Module Export BigZDivPropMod := ZDivPropFunct BigZAxiomsMod BigZPropMod.
 
 (** Notations about [BigZ] *)
 
@@ -71,6 +69,7 @@ Infix "<=" := BigZ.le : bigZ_scope.
 Notation "x > y" := (BigZ.lt y x)(only parsing) : bigZ_scope.
 Notation "x >= y" := (BigZ.le y x)(only parsing) : bigZ_scope.
 Notation "[ i ]" := (BigZ.to_Z i) : bigZ_scope.
+Infix "mod" := modulo (at level 40, no associativity) : bigN_scope.
 
 Local Open Scope bigZ_scope.
 
diff --git a/theories/Numbers/Integer/BigZ/ZMake.v b/theories/Numbers/Integer/BigZ/ZMake.v
index 827877fc5..0ab509650 100644
--- a/theories/Numbers/Integer/BigZ/ZMake.v
+++ b/theories/Numbers/Integer/BigZ/ZMake.v
@@ -369,17 +369,17 @@ Module Make (N:NType) <: ZType.
   end.
 
 
- Theorem spec_div_eucl: forall x y,
+ Theorem spec_div_eucl_nz: forall x y,
       to_Z y <> 0 ->
       let (q,r) := div_eucl x y in
       (to_Z q, to_Z r) = Zdiv_eucl (to_Z x) (to_Z y).
  unfold div_eucl, to_Z; intros [x | x] [y | y] H.
  assert (H1: 0 < N.to_Z y).
    generalize (N.spec_pos y); auto with zarith.
- generalize (N.spec_div_eucl x y H1); case N.div_eucl; auto.
+ generalize (N.spec_div_eucl x y); case N.div_eucl; auto.
  assert (HH: 0 < N.to_Z y).
    generalize (N.spec_pos y); auto with zarith.
- generalize (N.spec_div_eucl x y HH); case N.div_eucl; auto.
+ generalize (N.spec_div_eucl x y); case N.div_eucl; auto.
  intros q r; generalize (N.spec_pos x) HH; unfold Zdiv_eucl;
    case_eq (N.to_Z x); case_eq (N.to_Z y);
      try (intros; apply False_ind; auto with zarith; fail).
@@ -411,7 +411,7 @@ Module Make (N:NType) <: ZType.
    intros; apply False_ind; auto with zarith.
  assert (HH: 0 < N.to_Z y).
    generalize (N.spec_pos y); auto with zarith.
- generalize (N.spec_div_eucl x y HH); case N.div_eucl; auto.
+ generalize (N.spec_div_eucl x y); case N.div_eucl; auto.
  intros q r; generalize (N.spec_pos x) HH; unfold Zdiv_eucl;
    case_eq (N.to_Z x); case_eq (N.to_Z y);
      try (intros; apply False_ind; auto with zarith; fail).
@@ -441,7 +441,7 @@ Module Make (N:NType) <: ZType.
  rewrite N.spec_0; generalize (N.spec_pos r); intros; apply False_ind; auto with zarith.
  assert (H1: 0 < N.to_Z y).
    generalize (N.spec_pos y); auto with zarith.
- generalize (N.spec_div_eucl x y H1); case N.div_eucl; auto.
+ generalize (N.spec_div_eucl x y); case N.div_eucl; auto.
  intros q r; generalize (N.spec_pos x) H1; unfold Zdiv_eucl;
    case_eq (N.to_Z x); case_eq (N.to_Z y);
      try (intros; apply False_ind; auto with zarith; fail).
@@ -455,11 +455,43 @@ Module Make (N:NType) <: ZType.
  rewrite <- H2; auto.
  Qed.
 
+ Lemma Zdiv_eucl_0 : forall a, Zdiv_eucl a 0 = (0,0).
+ Proof. destruct a; auto. Qed.
+
+ Theorem spec_div_eucl: forall x y,
+      let (q,r) := div_eucl x y in
+      (to_Z q, to_Z r) = Zdiv_eucl (to_Z x) (to_Z y).
+ Proof.
+ intros. destruct (Z_eq_dec (to_Z y) 0) as [EQ|NEQ];
+  [|apply spec_div_eucl_nz; auto].
+ unfold div_eucl.
+ destruct x; destruct y; simpl in *.
+ generalize (N.spec_div_eucl t0 t1). destruct N.div_eucl; simpl; auto.
+ generalize (N.spec_div_eucl t0 t1). destruct N.div_eucl; simpl; auto.
+ assert (EQ' : N.to_Z t1 = 0) by auto with zarith.
+ rewrite EQ'. simpl. rewrite Zdiv_eucl_0. injection 1; intros.
+ generalize (N.spec_compare N.zero t3); destruct N.compare.
+ simpl. intros. f_equal; auto with zarith.
+ rewrite N.spec_0; intro; exfalso; auto with zarith.
+ rewrite N.spec_0; intro; exfalso; auto with zarith.
+ generalize (N.spec_div_eucl t0 t1). destruct N.div_eucl; simpl; auto.
+ assert (EQ' : N.to_Z t1 = 0) by auto with zarith.
+ rewrite EQ'. simpl. rewrite 2 Zdiv_eucl_0. injection 1; intros.
+ generalize (N.spec_compare N.zero t3); destruct N.compare.
+ simpl. intros. f_equal; auto with zarith.
+ rewrite N.spec_0; intro; exfalso; auto with zarith.
+ rewrite N.spec_0; intro; exfalso; auto with zarith.
+ generalize (N.spec_div_eucl t0 t1). destruct N.div_eucl; simpl; auto.
+ assert (EQ' : N.to_Z t1 = 0) by auto with zarith.
+ rewrite EQ'. simpl. rewrite 2 Zdiv_eucl_0. injection 1; intros.
+ f_equal; auto with zarith.
+ Qed.
+
  Definition div x y := fst (div_eucl x y).
 
  Definition spec_div: forall x y,
-     to_Z y <> 0 -> to_Z (div x y) = to_Z x / to_Z y.
- intros x y H1; generalize (spec_div_eucl x y H1); unfold div, Zdiv.
+     to_Z (div x y) = to_Z x / to_Z y.
+ intros x y; generalize (spec_div_eucl x y); unfold div, Zdiv.
  case div_eucl; case Zdiv_eucl; simpl; auto.
  intros q r q11 r1 H; injection H; auto.
  Qed.
@@ -467,8 +499,8 @@ Module Make (N:NType) <: ZType.
  Definition modulo x y := snd (div_eucl x y).
 
  Theorem spec_modulo:
-   forall x y, to_Z y <> 0  -> to_Z (modulo x y) = to_Z x mod to_Z y.
- intros x y H1; generalize (spec_div_eucl x y H1); unfold modulo, Zmod.
+   forall x y, to_Z (modulo x y) = to_Z x mod to_Z y.
+ intros x y; generalize (spec_div_eucl x y); unfold modulo, Zmod.
  case div_eucl; case Zdiv_eucl; simpl; auto.
  intros q r q11 r1 H; injection H; auto.
  Qed.
diff --git a/theories/Numbers/Integer/SpecViaZ/ZSig.v b/theories/Numbers/Integer/SpecViaZ/ZSig.v
index ef3cd5c34..a7c5473aa 100644
--- a/theories/Numbers/Integer/SpecViaZ/ZSig.v
+++ b/theories/Numbers/Integer/SpecViaZ/ZSig.v
@@ -98,17 +98,16 @@ Module Type ZType.
 
  Parameter div_eucl : t -> t -> t * t.
 
- Parameter spec_div_eucl: forall x y, [y] <> 0 ->
+ Parameter spec_div_eucl: forall 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, [y] <> 0 -> [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, [y] <> 0 ->
-   [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/Integer/SpecViaZ/ZSigZAxioms.v b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v
index 80166d5bf..cb5a771da 100644
--- a/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v
+++ b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v
@@ -8,11 +8,15 @@
 
 (*i $Id$ i*)
 
-Require Import ZArith ZAxioms ZSgnAbs ZSig.
+Require Import ZArith ZAxioms ZDivFloor ZSig.
 
-(** * The interface [ZSig.ZType] implies the interface [ZAxiomsSig] *)
+(** * The interface [ZSig.ZType] implies the interface [ZAxiomsSig]
 
-Module ZSig_ZAxioms (Z:ZType) <: ZAxiomsExtSig.
+    It also provides [sgn], [abs], [div], [mod]
+*)
+
+
+Module ZSig_ZAxioms (Z:ZType) <: ZAxiomsSig <: ZDivSig.
 
 Local Notation "[ x ]" := (Z.to_Z x).
 Local Infix "=="  := Z.eq (at level 70).
@@ -26,7 +30,7 @@ Local Infix "<" := Z.lt.
 
 Hint Rewrite
  Z.spec_0 Z.spec_1 Z.spec_add Z.spec_sub Z.spec_pred Z.spec_succ
- Z.spec_mul Z.spec_opp Z.spec_of_Z : zspec.
+ Z.spec_mul Z.spec_opp Z.spec_of_Z Z.spec_div Z.spec_modulo: zspec.
 
 Ltac zsimpl := unfold Z.eq in *; autorewrite with zspec.
 Ltac zcongruence := repeat red; intros; zsimpl; congruence.
@@ -235,32 +239,53 @@ Qed.
 
 Theorem abs_eq : forall n, 0 <= n -> Z.abs n == n.
 Proof.
-intros. red. rewrite Z.spec_abs. apply Zabs_eq.
- rewrite <- Z.spec_0, <- spec_le; auto.
+intros n. red. rewrite spec_le, Z.spec_0, Z.spec_abs. apply Zabs_eq.
 Qed.
 
 Theorem abs_neq : forall n, n <= 0 -> Z.abs n == -n.
 Proof.
-intros. red. rewrite Z.spec_abs, Z.spec_opp. apply Zabs_non_eq.
- rewrite <- Z.spec_0, <- spec_le; auto.
+intros n. red. rewrite spec_le, Z.spec_0, Z.spec_abs, Z.spec_opp.
+ apply Zabs_non_eq.
 Qed.
 
 Theorem sgn_null : forall n, n==0 -> Z.sgn n == 0.
 Proof.
-intros. red. rewrite Z.spec_sgn, Z.spec_0. rewrite Zsgn_null.
- rewrite <- Z.spec_0; auto.
+intros n. unfold Z.eq. rewrite Z.spec_sgn, Z.spec_0. rewrite Zsgn_null; auto.
 Qed.
 
 Theorem sgn_pos : forall n, 0<n -> Z.sgn n == Z.succ 0.
 Proof.
-intros. red. rewrite Z.spec_sgn. zsimpl. rewrite Zsgn_pos.
- rewrite <- Z.spec_0, <- spec_lt; auto.
+intros n. red. rewrite spec_lt, Z.spec_sgn. zsimpl. rewrite Zsgn_pos; auto.
 Qed.
 
 Theorem sgn_neg : forall n, n<0 -> Z.sgn n == Z.opp (Z.succ 0).
 Proof.
-intros. red. rewrite Z.spec_sgn. zsimpl. rewrite Zsgn_neg.
- rewrite <- Z.spec_0, <- spec_lt; auto.
+intros n. red. rewrite spec_lt, Z.spec_sgn. zsimpl. rewrite Zsgn_neg; auto.
+Qed.
+
+Program Instance div_wd : Proper (Z.eq==>Z.eq==>Z.eq) Z.div.
+Program Instance mod_wd : Proper (Z.eq==>Z.eq==>Z.eq) Z.modulo.
+
+Theorem div_mod : forall a b, ~b==0 -> a == b*(Z.div a b) + (Z.modulo a b).
+Proof.
+intros a b. unfold Z.eq; zsimpl. intros.
+apply Z_div_mod_eq_full; auto.
+Qed.
+
+Theorem mod_pos_bound :
+ forall a b, 0 < b -> 0 <= Z.modulo a b /\ Z.modulo a b < b.
+Proof.
+intros a b. rewrite 2 spec_lt, spec_le, Z.spec_0. intros.
+rewrite Z.spec_modulo; auto with zarith.
+apply Z_mod_lt; auto with zarith.
+Qed.
+
+Theorem mod_neg_bound :
+ forall a b, b < 0 -> b < Z.modulo a b /\ Z.modulo a b <= 0.
+Proof.
+intros a b. rewrite 2 spec_lt, spec_le, Z.spec_0. intros.
+rewrite Z.spec_modulo; auto with zarith.
+apply Z_mod_neg; auto with zarith.
 Qed.
 
 (** Aliases *)
@@ -280,5 +305,7 @@ Definition min := Z.min.
 Definition max := Z.max.
 Definition abs := Z.abs.
 Definition sgn := Z.sgn.
+Definition div := Z.div.
+Definition modulo := Z.modulo.
 
 End ZSig_ZAxioms.