OrderedType implementation for various numerical datatypes + min/max structures

 - A richer OrderedTypeFull interface : OrderedType + predicate "le"
 - Implementations {Nat,N,P,Z,Q}OrderedType.v, also providing "order" tactics
 - By the way: as suggested by S. Lescuyer, specification of compare is
   now inductive
 - GenericMinMax: axiomatisation + properties of min and max out of
    OrderedTypeFull structures.
 - MinMax.v, {Z,P,N,Q}minmax.v are specialization of GenericMinMax,
    with also some domain-specific results, and compatibility layer
    with already existing results.
 - Some ML code of plugins had to be adapted, otherwise wrong "eq",
   "lt" or simimlar constants were found by functions like coq_constant.
 - Beware of the aliasing problems: for instance eq:=@eq t instead of
   eq:=@eq M.t in Make_UDT made (r)omega stopped working (Z_as_OT.t
   instead of Z in statement of Zmax_spec).
 - Some Morphism declaration are now ambiguous: switch to new syntax
   anyway.
 - Misc adaptations of FSets/MSets
 - Classes/RelationPairs.v: from two relations over A and B, we
   inspect relations over A*B and their properties in terms of classes.

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12461 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 130 insertions, 0 deletions

diff --git a/theories/QArith/Qminmax.v b/theories/QArith/Qminmax.v
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+++ b/theories/QArith/Qminmax.v
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+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+Require Import QArith_base OrderedType2 QOrderedType GenericMinMax.
+
+(** * Maximum and Minimum of two rational numbers *)
+
+Local Open Scope Q_scope.
+
+(** [Qmin] and [Qmax] are obtained the usual way from [Qcompare]. *)
+
+Definition Qmax := gmax Qcompare.
+Definition Qmin := gmin Qcompare.
+
+Module QHasMinMax <: HasMinMax Q_as_OT.
+ Module QMM := GenericMinMax Q_as_OT.
+ Definition max := Qmax.
+ Definition min := Qmin.
+ Definition max_spec := QMM.max_spec.
+ Definition min_spec := QMM.min_spec.
+End QHasMinMax.
+
+(** We obtain hence all the generic properties of max and min. *)
+
+Module Import QMinMaxProps := MinMaxProperties Q_as_OT QHasMinMax.
+
+Definition Qmax_case_strong := max_case_strong.
+Definition Qmax_case := max_case.
+Definition Qmax_monotone := max_monotone.
+Definition Qmax_spec := max_spec.
+Definition Qmax_spec_le := max_spec_le.
+Definition Qmax_dec := max_dec.
+Definition Qmax_unicity := max_unicity.
+Definition Qmax_unicity_ext := max_unicity_ext.
+Definition Qmax_id := max_id.
+Notation Qmax_idempotent := Qmax_id (only parsing).
+Definition Qmax_assoc := max_assoc.
+Definition Qmax_comm := max_comm.
+Definition Qmax_l := max_l.
+Definition Qmax_r := max_r.
+Definition Nle_max_l := le_max_l.
+Definition Nle_max_r := le_max_r.
+Definition Qmax_le := max_le.
+Definition Qmax_le_iff := max_le_iff.
+Definition Qmax_lt_iff := max_lt_iff.
+Definition Qmax_lub_l := max_lub_l.
+Definition Qmax_lub_r := max_lub_r.
+Definition Qmax_lub := max_lub.
+Definition Qmax_lub_iff := max_lub_iff.
+Definition Qmax_lub_lt := max_lub_lt.
+Definition Qmax_lub_lt_iff := max_lub_lt_iff.
+Definition Qmax_le_compat_l := max_le_compat_l.
+Definition Qmax_le_compat_r := max_le_compat_r.
+Definition Qmax_le_compat := max_le_compat.
+
+Definition Qmin_case_strong := min_case_strong.
+Definition Qmin_case := min_case.
+Definition Qmin_monotone := min_monotone.
+Definition Qmin_spec := min_spec.
+Definition Qmin_spec_le := min_spec_le.
+Definition Qmin_dec := min_dec.
+Definition Qmin_unicity := min_unicity.
+Definition Qmin_unicity_ext := min_unicity_ext.
+Definition Qmin_id := min_id.
+Notation Qmin_idempotent := Qmin_id (only parsing).
+Definition Qmin_assoc := min_assoc.
+Definition Qmin_comm := min_comm.
+Definition Qmin_l := min_l.
+Definition Qmin_r := min_r.
+Definition Nle_min_l := le_min_l.
+Definition Nle_min_r := le_min_r.
+Definition Qmin_le := min_le.
+Definition Qmin_le_iff := min_le_iff.
+Definition Qmin_lt_iff := min_lt_iff.
+Definition Qmin_glb_l := min_glb_l.
+Definition Qmin_glb_r := min_glb_r.
+Definition Qmin_glb := min_glb.
+Definition Qmin_glb_iff := min_glb_iff.
+Definition Qmin_glb_lt := min_glb_lt.
+Definition Qmin_glb_lt_iff := min_glb_lt_iff.
+Definition Qmin_le_compat_l := min_le_compat_l.
+Definition Qmin_le_compat_r := min_le_compat_r.
+Definition Qmin_le_compat := min_le_compat.
+
+Definition Qmin_max_absorption := min_max_absorption.
+Definition Qmax_min_absorption := max_min_absorption.
+Definition Qmax_min_distr := max_min_distr.
+Definition Qmin_max_distr := min_max_distr.
+Definition Qmax_min_modular := max_min_modular.
+Definition Qmin_max_modular := min_max_modular.
+Definition Qmax_min_disassoc := max_min_disassoc.
+Definition Qmax_min_antimonotone := max_min_antimonotone.
+Definition Qmin_max_antimonotone := min_max_antimonotone.
+
+
+
+(** * Properties specific to the [Q] domain *)
+
+(** Compatibilities (consequences of monotonicity) *)
+
+Lemma Qplus_max_distr_l : forall n m p, Qmax (p + n) (p + m) == p + Qmax n m.
+Proof.
+ intros. apply Qmax_monotone.
+ intros x x' Hx; rewrite Hx; auto with qarith.
+ intros x x' Hx. apply Qplus_le_compat; q_order.
+Qed.
+
+Lemma Qplus_max_distr_r : forall n m p, Qmax (n + p) (m + p) == Qmax n m + p.
+Proof.
+ intros. rewrite (Qplus_comm n p), (Qplus_comm m p), (Qplus_comm _ p).
+ apply Qplus_max_distr_l.
+Qed.
+
+Lemma Qplus_min_distr_l : forall n m p, Qmin (p + n) (p + m) == p + Qmin n m.
+Proof.
+ intros. apply Qmin_monotone.
+ intros x x' Hx; rewrite Hx; auto with qarith.
+ intros x x' Hx. apply Qplus_le_compat; q_order.
+Qed.
+
+Lemma Qplus_min_distr_r : forall n m p, Qmin (n + p) (m + p) == Qmin n m + p.
+Proof.
+ intros. rewrite (Qplus_comm n p), (Qplus_comm m p), (Qplus_comm _ p).
+ apply Qplus_min_distr_l.
+Qed.