More improvements of BigN, BigZ, BigQ:

 - ring/field: detection of constants for ring/field,
   detection of power, potential use of euclidean division.
 - for BigN and BigZ, x^n now takes a N as 2nd arg instead of a positive
 - mention that we can use (r)omega thanks to (ugly) BigN.zify, BigZ.zify.
   By the way, BigN.zify could still be improved (no insertion of positivity
   hyps yet, unlike the original zify).
 - debug of BigQ.qify (autorewrite was looping on spec_0).
 - for BigQ, start of a generic functor of properties QProperties.
 - BigQ now implements OrderedType, TotalOrder, and contains facts
   about min and max.

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12681 85f007b7-540e-0410-9357-904b9bb8a0f7

2 files changed, 215 insertions, 105 deletions

diff --git a/theories/Numbers/Rational/BigQ/BigQ.v b/theories/Numbers/Rational/BigQ/BigQ.v
index 15abaaa42b..0bc71166d9 100644
--- a/theories/Numbers/Rational/BigQ/BigQ.v
+++ b/theories/Numbers/Rational/BigQ/BigQ.v
@@ -11,7 +11,7 @@
 (** Initial authors: Benjamin Gregoire, Laurent Thery, INRIA, 2007 *)
 
 Require Export BigZ.
-Require Import Field Qfield QSig QMake.
+Require Import Field Qfield QSig QMake Orders GenericMinMax.
 
 (** We choose for BigQ an implemention with
     multiple representation of 0: 0, 1/0, 2/0 etc.
@@ -35,7 +35,9 @@ End BigN_BigZ.
 
 (** This allows to build [BigQ] out of [BigN] and [BigQ] via [QMake] *)
 
-Module BigQ <: QSig.QType := QMake.Make BigN BigZ BigN_BigZ.
+Module BigQ <: QType <: OrderedTypeFull <: TotalOrder :=
+ QMake.Make BigN BigZ BigN_BigZ <+ !QProperties <+ HasEqBool2Dec
+ <+ !MinMaxLogicalProperties <+ !MinMaxDecProperties.
 
 (** Notations about [BigQ] *)
 
@@ -76,6 +78,8 @@ Arguments Scope BigQ.power_norm [bigQ_scope bigQ_scope].
 
 (** As in QArith, we use [#] to denote fractions *)
 Notation "p # q" := (BigQ.Qq p q) (at level 55, no associativity) : bigQ_scope.
+Local Notation "0" := BigQ.zero : bigQ_scope.
+Local Notation "1" := BigQ.one : bigQ_scope.
 Infix "+" := BigQ.add : bigQ_scope.
 Infix "-" := BigQ.sub : bigQ_scope.
 Notation "- x" := (BigQ.opp x) : bigQ_scope.
@@ -84,146 +88,102 @@ Infix "/" := BigQ.div : bigQ_scope.
 Infix "^" := BigQ.power : bigQ_scope.
 Infix "?=" := BigQ.compare : bigQ_scope.
 Infix "==" := BigQ.eq : bigQ_scope.
+Notation "x != y" := (~x==y)%bigQ (at level 70, no associativity) : bigQ_scope.
 Infix "<" := BigQ.lt : bigQ_scope.
 Infix "<=" := BigQ.le : bigQ_scope.
 Notation "x > y" := (BigQ.lt y x)(only parsing) : bigQ_scope.
 Notation "x >= y" := (BigQ.le y x)(only parsing) : bigQ_scope.
+Notation "x < y < z" := (x<y /\ y<z)%bigQ : bigQ_scope.
+Notation "x < y <= z" := (x<y /\ y<=z)%bigQ : bigQ_scope.
+Notation "x <= y < z" := (x<=y /\ y<z)%bigQ : bigQ_scope.
+Notation "x <= y <= z" := (x<=y /\ y<=z)%bigQ : bigQ_scope.
 Notation "[ q ]" := (BigQ.to_Q q) : bigQ_scope.
 
 Local Open Scope bigQ_scope.
 
-(** [BigQ] is a setoid *)
-
-Instance BigQeq_rel : Equivalence BigQ.eq.
-Proof. unfold BigQ.eq. split; red; eauto with qarith. Qed.
-
-Instance BigQadd_wd : Proper (BigQ.eq==>BigQ.eq==>BigQ.eq) BigQ.add.
-Proof.
- do 3 red. unfold BigQ.eq; intros.
- rewrite !BigQ.spec_add, H, H0. reflexivity.
-Qed.
-
-Instance BigQopp_wd : Proper (BigQ.eq==>BigQ.eq) BigQ.opp.
-Proof.
- do 2 red. unfold BigQ.eq; intros.
- rewrite !BigQ.spec_opp, H; reflexivity.
-Qed.
-
-Instance BigQsub_wd : Proper (BigQ.eq==>BigQ.eq==>BigQ.eq) BigQ.sub.
-Proof.
- do 3 red. unfold BigQ.eq; intros.
- rewrite !BigQ.spec_sub, H, H0; reflexivity.
-Qed.
-
-Instance BigQmul_wd : Proper (BigQ.eq==>BigQ.eq==>BigQ.eq) BigQ.mul.
-Proof.
- do 3 red. unfold BigQ.eq; intros.
- rewrite !BigQ.spec_mul, H, H0; reflexivity.
-Qed.
-
-Instance BigQinv_wd : Proper (BigQ.eq==>BigQ.eq) BigQ.inv.
-Proof.
- do 2 red; unfold BigQ.eq; intros.
- rewrite !BigQ.spec_inv, H; reflexivity.
-Qed.
-
-Instance BigQdiv_wd : Proper (BigQ.eq==>BigQ.eq==>BigQ.eq) BigQ.div.
-Proof.
- do 3 red; unfold BigQ.eq; intros.
- rewrite !BigQ.spec_div, H, H0; reflexivity.
-Qed.
-
-(* TODO : fix this. For the moment it's useless (horribly slow)
-Hint Rewrite
- BigQ.spec_0 BigQ.spec_1 BigQ.spec_m1 BigQ.spec_compare
- BigQ.spec_red BigQ.spec_add BigQ.spec_sub BigQ.spec_opp
- BigQ.spec_mul BigQ.spec_inv BigQ.spec_div BigQ.spec_power_pos
- BigQ.spec_square : bigq. *)
-
-
 (** [BigQ] is a field *)
 
 Lemma BigQfieldth :
- field_theory BigQ.zero BigQ.one BigQ.add BigQ.mul BigQ.sub BigQ.opp BigQ.div BigQ.inv BigQ.eq.
+ field_theory 0 1 BigQ.add BigQ.mul BigQ.sub BigQ.opp
+  BigQ.div BigQ.inv BigQ.eq.
 Proof.
 constructor.
-constructor; intros; red.
-rewrite BigQ.spec_add, BigQ.spec_0; ring.
-rewrite ! BigQ.spec_add; ring.
-rewrite ! BigQ.spec_add; ring.
-rewrite BigQ.spec_mul, BigQ.spec_1; ring.
-rewrite ! BigQ.spec_mul; ring.
-rewrite ! BigQ.spec_mul; ring.
-rewrite BigQ.spec_add, ! BigQ.spec_mul, BigQ.spec_add; ring.
-unfold BigQ.sub; apply Qeq_refl.
-rewrite BigQ.spec_add, BigQ.spec_0, BigQ.spec_opp; ring.
-compute; discriminate.
-intros; red.
-unfold BigQ.div; apply Qeq_refl.
-intros; red.
-rewrite BigQ.spec_mul, BigQ.spec_inv, BigQ.spec_1; field.
-rewrite <- BigQ.spec_0; auto.
-Qed.
-
-Lemma BigQpowerth :
- power_theory BigQ.one BigQ.mul BigQ.eq Z_of_N BigQ.power.
-Proof.
 constructor.
-intros; red.
-rewrite BigQ.spec_power.
-replace ([r] ^ Z_of_N n)%Q with (pow_N 1 Qmult [r] n)%Q.
-destruct n.
-simpl; compute; auto.
-induction p; simpl; auto; try rewrite !BigQ.spec_mul, !IHp; apply Qeq_refl.
-destruct n; reflexivity.
+exact BigQ.add_0_l. exact BigQ.add_comm. exact BigQ.add_assoc.
+exact BigQ.mul_1_l. exact BigQ.mul_comm. exact BigQ.mul_assoc.
+exact BigQ.mul_add_distr_r. exact BigQ.sub_add_opp.
+exact BigQ.add_opp_diag_r. exact BigQ.neq_1_0.
+exact BigQ.div_mul_inv. exact BigQ.mul_inv_diag_l.
 Qed.
 
-Lemma BigQ_eq_bool_iff :
- forall x y, BigQ.eq_bool x y = true <-> x==y.
+Lemma BigQpowerth :
+ power_theory 1 BigQ.mul BigQ.eq Z_of_N BigQ.power.
 Proof.
-intros. rewrite BigQ.spec_eq_bool. apply Qeq_bool_iff.
+constructor. intros. BigQ.qify.
+replace ([r] ^ Z_of_N n)%Q with (pow_N 1 Qmult [r] n)%Q by (now destruct n).
+destruct n. reflexivity.
+induction p; simpl; auto; rewrite ?BigQ.spec_mul, ?IHp; reflexivity.
 Qed.
 
-Lemma BigQ_eq_bool_correct :
- forall x y, BigQ.eq_bool x y = true -> x==y.
-Proof. now apply BigQ_eq_bool_iff. Qed.
-
-Lemma BigQ_eq_bool_complete :
- forall x y, x==y -> BigQ.eq_bool x y = true.
-Proof. now apply BigQ_eq_bool_iff. Qed.
-
-(* TODO : improve later the detection of constants ... *)
+Ltac isBigQcst t :=
+ match t with
+ | BigQ.Qz ?t => isBigZcst t
+ | BigQ.Qq ?n ?d => match isBigZcst n with
+             | true => isBigNcst d
+             | false => constr:false
+             end
+ | BigQ.zero => constr:true
+ | BigQ.one => constr:true
+ | BigQ.minus_one => constr:true
+ | _ => constr:false
+ end.
 
 Ltac BigQcst t :=
- match t with
-   | BigQ.zero => BigQ.zero
-   | BigQ.one => BigQ.one
-   | BigQ.minus_one => BigQ.minus_one
-   | _ => NotConstant
+ match isBigQcst t with
+ | true => constr:t
+ | false => constr:NotConstant
  end.
 
 Add Field BigQfield : BigQfieldth
- (decidable BigQ_eq_bool_correct,
-  completeness BigQ_eq_bool_complete,
+ (decidable BigQ.eqb_correct,
+  completeness BigQ.eqb_complete,
   constants [BigQcst],
   power_tac BigQpowerth [Qpow_tac]).
 
-Section Examples.
+Section TestField.
 
 Let ex1 : forall x y z, (x+y)*z ==  (x*z)+(y*z).
   intros.
   ring.
 Qed.
 
-Let ex8 : forall x, x ^ 1 == x.
+Let ex8 : forall x, x ^ 2 == x*x.
   intro.
   ring.
 Qed.
 
-Let ex10 : forall x y, ~(y==BigQ.zero) -> (x/y)*y == x.
+Let ex10 : forall x y, y!=0 -> (x/y)*y == x.
 intros.
 field.
 auto.
 Qed.
 
-End Examples.
\ No newline at end of file
+End TestField.
+
+(** [BigQ] can also benefit from an "order" tactic *)
+
+Module BigQ_Order := !OrdersTac.MakeOrderTac BigQ.
+Ltac bigQ_order := BigQ_Order.order.
+
+Section TestOrder.
+Let test : forall x y : bigQ, x<=y -> y<=x -> x==y.
+Proof. bigQ_order. Qed.
+End TestOrder.
+
+(** We can also reason by switching to QArith thanks to tactic
+    BigQ.qify. *)
+
+Section TestQify.
+Let test : forall x : bigQ, 0+x == 1*x.
+Proof. intro x. BigQ.qify. ring. Qed.
+End TestQify.
diff --git a/theories/Numbers/Rational/SpecViaQ/QSig.v b/theories/Numbers/Rational/SpecViaQ/QSig.v
index 1959f4ad69..10d0189a32 100644
--- a/theories/Numbers/Rational/SpecViaQ/QSig.v
+++ b/theories/Numbers/Rational/SpecViaQ/QSig.v
@@ -8,7 +8,7 @@
 
 (*i $Id$ i*)
 
-Require Import QArith Qpower Qminmax.
+Require Import QArith Qpower Qminmax Orders RelationPairs GenericMinMax.
 
 Open Scope Q_scope.
 
@@ -23,7 +23,7 @@ Module Type QType.
  Parameter t : Type.
 
  Parameter to_Q : t -> Q.
- Notation "[ x ]" := (to_Q x).
+ Local Notation "[ x ]" := (to_Q x).
 
  Definition eq x y := [x] == [y].
  Definition lt x y := [x] < [y].
@@ -73,3 +73,153 @@ End QType.
      that expect reduced arguments and return reduced results. *)
 
 (** TODO : also speak of specifications via Qcanon ... *)
+
+Module Type QType_Notation (Import Q : QType).
+ Notation "[ x ]" := (to_Q x).
+ Infix "=="  := eq (at level 70).
+ Notation "x != y" := (~x==y) (at level 70).
+ Infix "<=" := le.
+ Infix "<" := lt.
+ Notation "0" := zero.
+ Notation "1" := one.
+ Infix "+" := add.
+ Infix "-" := sub.
+ Infix "*" := mul.
+ Notation "- x" := (opp x).
+ Infix "/" := div.
+ Notation "/ x" := (inv x).
+ Infix "^" := power.
+End QType_Notation.
+
+Module Type QType' := QType <+ QType_Notation.
+
+
+Module QProperties (Import Q : QType').
+
+(** Conversion to Q *)
+
+Hint Rewrite
+ spec_red spec_compare spec_eq_bool spec_min spec_max
+ spec_add spec_sub spec_opp spec_mul spec_square spec_inv spec_div
+ spec_power : qsimpl.
+Ltac qify := unfold eq, lt, le in *; autorewrite with qsimpl;
+ try rewrite spec_0 in *; try rewrite spec_1 in *; try rewrite spec_m1 in *.
+
+(** NB: do not add [spec_0] in the autorewrite database. Otherwise,
+    after instanciation in BigQ, this lemma become convertible to 0=0,
+    and autorewrite loops. Idem for [spec_1] and [spec_m1] *)
+
+(** Morphisms *)
+
+Ltac solve_wd1 := intros x x' Hx; qify; now rewrite Hx.
+Ltac solve_wd2 := intros x x' Hx y y' Hy; qify; now rewrite Hx, Hy.
+
+Local Obligation Tactic := solve_wd2 || solve_wd1.
+
+Instance : Measure to_Q.
+Instance eq_equiv : Equivalence eq.
+
+Program Instance lt_wd : Proper (eq==>eq==>iff) lt.
+Program Instance le_wd : Proper (eq==>eq==>iff) le.
+Program Instance red_wd : Proper (eq==>eq) red.
+Program Instance compare_wd : Proper (eq==>eq==>Logic.eq) compare.
+Program Instance eq_bool_wd : Proper (eq==>eq==>Logic.eq) eq_bool.
+Program Instance min_wd : Proper (eq==>eq==>eq) min.
+Program Instance max_wd : Proper (eq==>eq==>eq) max.
+Program Instance add_wd : Proper (eq==>eq==>eq) add.
+Program Instance sub_wd : Proper (eq==>eq==>eq) sub.
+Program Instance opp_wd : Proper (eq==>eq) opp.
+Program Instance mul_wd : Proper (eq==>eq==>eq) mul.
+Program Instance square_wd : Proper (eq==>eq) square.
+Program Instance inv_wd : Proper (eq==>eq) inv.
+Program Instance div_wd : Proper (eq==>eq==>eq) div.
+Program Instance power_wd : Proper (eq==>Logic.eq==>eq) power.
+
+(** Let's implement [HasCompare] *)
+
+Lemma compare_spec : forall x y, CompSpec eq lt x y (compare x y).
+Proof. intros. qify. destruct (Qcompare_spec [x] [y]); auto. Qed.
+
+(** Let's implement [TotalOrder] *)
+
+Definition lt_compat := lt_wd.
+Instance lt_strorder : StrictOrder lt.
+
+Lemma le_lteq : forall x y, x<=y <-> x<y \/ x==y.
+Proof. intros. qify. apply Qle_lteq. Qed.
+
+Lemma lt_total : forall x y, x<y \/ x==y \/ y<x.
+Proof. intros. destruct (compare_spec x y); auto. Qed.
+
+(** Let's implement [HasEqBool] *)
+
+Definition eqb := eq_bool.
+
+Lemma eqb_eq : forall x y, eq_bool x y = true <-> x == y.
+Proof. intros. qify. apply Qeq_bool_iff. Qed.
+
+Lemma eqb_correct : forall x y, eq_bool x y = true -> x == y.
+Proof. now apply eqb_eq. Qed.
+
+Lemma eqb_complete : forall x y, x == y -> eq_bool x y = true.
+Proof. now apply eqb_eq. Qed.
+
+(** Let's implement [HasMinMax] *)
+
+Lemma max_l : forall x y, y<=x -> max x y == x.
+Proof. intros x y. qify. apply Qminmax.Q.max_l. Qed.
+
+Lemma max_r : forall x y, x<=y -> max x y == y.
+Proof. intros x y. qify. apply Qminmax.Q.max_r. Qed.
+
+Lemma min_l : forall x y, x<=y -> min x y == x.
+Proof. intros x y. qify. apply Qminmax.Q.min_l. Qed.
+
+Lemma min_r : forall x y, y<=x -> min x y == y.
+Proof. intros x y. qify. apply Qminmax.Q.min_r. Qed.
+
+(** Q is a ring *)
+
+Lemma add_0_l : forall x, 0+x == x.
+Proof. intros. qify. apply Qplus_0_l. Qed.
+
+Lemma add_comm : forall x y, x+y == y+x.
+Proof. intros. qify. apply Qplus_comm. Qed.
+
+Lemma add_assoc : forall x y z, x+(y+z) == x+y+z.
+Proof. intros. qify. apply Qplus_assoc. Qed.
+
+Lemma mul_1_l : forall x, 1*x == x.
+Proof. intros. qify. apply Qmult_1_l. Qed.
+
+Lemma mul_comm : forall x y, x*y == y*x.
+Proof. intros. qify. apply Qmult_comm. Qed.
+
+Lemma mul_assoc : forall x y z, x*(y*z) == x*y*z.
+Proof. intros. qify. apply Qmult_assoc. Qed.
+
+Lemma mul_add_distr_r : forall x y z, (x+y)*z == x*z + y*z.
+Proof. intros. qify. apply Qmult_plus_distr_l. Qed.
+
+Lemma sub_add_opp : forall x y, x-y == x+(-y).
+Proof. intros. qify. now unfold Qminus. Qed.
+
+Lemma add_opp_diag_r : forall x, x+(-x) == 0.
+Proof. intros. qify. apply Qplus_opp_r. Qed.
+
+(** Q is a field *)
+
+Lemma neq_1_0 : 1!=0.
+Proof. intros. qify. apply Q_apart_0_1. Qed.
+
+Lemma div_mul_inv : forall x y, x/y == x*(/y).
+Proof. intros. qify. now unfold Qdiv. Qed.
+
+Lemma mul_inv_diag_l : forall x, x!=0 -> /x * x == 1.
+Proof. intros x. qify. rewrite Qmult_comm. apply Qmult_inv_r. Qed.
+
+End QProperties.
+
+Module QTypeExt (Q : QType)
+ <: QType <: TotalOrder <: HasCompare Q <: HasMinMax Q <: HasEqBool Q
+ := Q <+ QProperties.
\ No newline at end of file