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(***********************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *)
(* \VV/ *************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
Require Import Basics OrdersTac.
Require Export Orders.
Set Implicit Arguments.
Unset Strict Implicit.
(** * Properties of [OrderedTypeFull] *)
Module OrderedTypeFullFacts (Import O:OrderedTypeFull').
Module OrderTac := OTF_to_OrderTac O.
Ltac order := OrderTac.order.
Ltac iorder := intuition order.
Instance le_compat : Proper (eq==>eq==>iff) le.
Proof. repeat red; iorder. Qed.
Instance le_preorder : PreOrder le.
Proof. split; red; order. Qed.
Instance le_order : PartialOrder eq le.
Proof. compute; iorder. Qed.
Instance le_antisym : Antisymmetric _ eq le.
Proof. apply partial_order_antisym; auto with *. Qed.
Lemma le_not_gt_iff : forall x y, x<=y <-> ~y<x.
Proof. iorder. Qed.
Lemma lt_not_ge_iff : forall x y, x<y <-> ~y<=x.
Proof. iorder. Qed.
Lemma le_or_gt : forall x y, x<=y \/ y<x.
Proof. intros. rewrite le_lteq; destruct (O.compare_spec x y); auto. Qed.
Lemma lt_or_ge : forall x y, x<y \/ y<=x.
Proof. intros. rewrite le_lteq; destruct (O.compare_spec x y); iorder. Qed.
Lemma eq_is_le_ge : forall x y, x==y <-> x<=y /\ y<=x.
Proof. iorder. Qed.
End OrderedTypeFullFacts.
(** * Properties of [OrderedType] *)
Module OrderedTypeFacts (Import O: OrderedType').
Module OrderTac := OT_to_OrderTac O.
Ltac order := OrderTac.order.
Notation "x <= y" := (~lt y x) : order.
Infix "?=" := compare (at level 70, no associativity) : order.
Local Open Scope order.
Tactic Notation "elim_compare" constr(x) constr(y) :=
destruct (compare_spec x y).
Tactic Notation "elim_compare" constr(x) constr(y) "as" ident(h) :=
destruct (compare_spec x y) as [h|h|h].
(** The following lemmas are either re-phrasing of [eq_equiv] and
[lt_strorder] or immediately provable by [order]. Interest:
compatibility, test of order, etc *)
Definition eq_refl (x:t) : x==x := Equivalence_Reflexive x.
Definition eq_sym (x y:t) : x==y -> y==x := Equivalence_Symmetric x y.
Definition eq_trans (x y z:t) : x==y -> y==z -> x==z :=
Equivalence_Transitive x y z.
Definition lt_trans (x y z:t) : x<y -> y<z -> x<z :=
StrictOrder_Transitive x y z.
Definition lt_irrefl (x:t) : ~x<x := StrictOrder_Irreflexive x.
(** Some more about [compare] *)
Lemma compare_eq_iff : forall x y, (x ?= y) = Eq <-> x==y.
Proof.
intros; elim_compare x y; intuition; try discriminate; order.
Qed.
Lemma compare_lt_iff : forall x y, (x ?= y) = Lt <-> x<y.
Proof.
intros; elim_compare x y; intuition; try discriminate; order.
Qed.
Lemma compare_gt_iff : forall x y, (x ?= y) = Gt <-> y<x.
Proof.
intros; elim_compare x y; intuition; try discriminate; order.
Qed.
Lemma compare_ge_iff : forall x y, (x ?= y) <> Lt <-> y<=x.
Proof.
intros; rewrite compare_lt_iff; intuition.
Qed.
Lemma compare_le_iff : forall x y, (x ?= y) <> Gt <-> x<=y.
Proof.
intros; rewrite compare_gt_iff; intuition.
Qed.
Hint Rewrite compare_eq_iff compare_lt_iff compare_gt_iff : order.
Instance compare_compat : Proper (eq==>eq==>Logic.eq) compare.
Proof.
intros x x' Hxx' y y' Hyy'.
elim_compare x' y'; autorewrite with order; order.
Qed.
Lemma compare_refl : forall x, (x ?= x) = Eq.
Proof.
intros; elim_compare x x; auto; order.
Qed.
Lemma compare_antisym : forall x y, (y ?= x) = CompOpp (x ?= y).
Proof.
intros; elim_compare x y; simpl; autorewrite with order; order.
Qed.
(** For compatibility reasons *)
Definition eq_dec := eq_dec.
Lemma lt_dec : forall x y : t, {lt x y} + {~ lt x y}.
Proof.
intros x y; destruct (CompSpec2Type (compare_spec x y));
[ right | left | right ]; order.
Defined.
Definition eqb x y : bool := if eq_dec x y then true else false.
Lemma if_eq_dec : forall x y (B:Type)(b b':B),
(if eq_dec x y then b else b') =
(match compare x y with Eq => b | _ => b' end).
Proof.
intros; destruct eq_dec; elim_compare x y; auto; order.
Qed.
Lemma eqb_alt :
forall x y, eqb x y = match compare x y with Eq => true | _ => false end.
Proof.
unfold eqb; intros; apply if_eq_dec.
Qed.
Instance eqb_compat : Proper (eq==>eq==>Logic.eq) eqb.
Proof.
intros x x' Hxx' y y' Hyy'.
rewrite 2 eqb_alt, Hxx', Hyy'; auto.
Qed.
End OrderedTypeFacts.
(** * Tests of the order tactic
Is it at least capable of proving some basic properties ? *)
Module OrderedTypeTest (Import O:OrderedType').
Module Import MO := OrderedTypeFacts O.
Local Open Scope order.
Lemma lt_not_eq x y : x<y -> ~x==y. Proof. order. Qed.
Lemma lt_eq x y z : x<y -> y==z -> x<z. Proof. order. Qed.
Lemma eq_lt x y z : x==y -> y<z -> x<z. Proof. order. Qed.
Lemma le_eq x y z : x<=y -> y==z -> x<=z. Proof. order. Qed.
Lemma eq_le x y z : x==y -> y<=z -> x<=z. Proof. order. Qed.
Lemma neq_eq x y z : ~x==y -> y==z -> ~x==z. Proof. order. Qed.
Lemma eq_neq x y z : x==y -> ~y==z -> ~x==z. Proof. order. Qed.
Lemma le_lt_trans x y z : x<=y -> y<z -> x<z. Proof. order. Qed.
Lemma lt_le_trans x y z : x<y -> y<=z -> x<z. Proof. order. Qed.
Lemma le_trans x y z : x<=y -> y<=z -> x<=z. Proof. order. Qed.
Lemma le_antisym x y : x<=y -> y<=x -> x==y. Proof. order. Qed.
Lemma le_neq x y : x<=y -> ~x==y -> x<y. Proof. order. Qed.
Lemma neq_sym x y : ~x==y -> ~y==x. Proof. order. Qed.
Lemma lt_le x y : x<y -> x<=y. Proof. order. Qed.
Lemma gt_not_eq x y : y<x -> ~x==y. Proof. order. Qed.
Lemma eq_not_lt x y : x==y -> ~x<y. Proof. order. Qed.
Lemma eq_not_gt x y : x==y -> ~ y<x. Proof. order. Qed.
Lemma lt_not_gt x y : x<y -> ~ y<x. Proof. order. Qed.
Lemma eq_is_nlt_ngt x y : x==y <-> ~x<y /\ ~y<x.
Proof. intuition; order. Qed.
End OrderedTypeTest.
(** * Reversed OrderedTypeFull.
we can switch the orientation of the order. This is used for
example when deriving properties of [min] out of the ones of [max]
(see [GenericMinMax]).
*)
Module OrderedTypeRev (O:OrderedTypeFull) <: OrderedTypeFull.
Definition t := O.t.
Definition eq := O.eq.
Instance eq_equiv : Equivalence eq.
Definition eq_dec := O.eq_dec.
Definition lt := flip O.lt.
Definition le := flip O.le.
Instance lt_strorder: StrictOrder lt.
Proof. unfold lt; auto with *. Qed.
Instance lt_compat : Proper (eq==>eq==>iff) lt.
Proof. unfold lt; auto with *. Qed.
Lemma le_lteq : forall x y, le x y <-> lt x y \/ eq x y.
Proof. intros; unfold le, lt, flip. rewrite O.le_lteq; intuition. Qed.
Definition compare := flip O.compare.
Lemma compare_spec : forall x y, CompSpec eq lt x y (compare x y).
Proof.
intros; unfold compare, eq, lt, flip.
destruct (O.compare_spec y x); auto with relations.
Qed.
End OrderedTypeRev.
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