orders.v 3.74 KB
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(* Copyright (c) 2012-2019, Coq-std++ developers. *)
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(* This file is distributed under the terms of the BSD license. *)
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(** Properties about arbitrary pre-, partial, and total orders. We do not use
the relation [⊆] because we often have multiple orders on the same structure *)
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From stdpp Require Export tactics.
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Set Default Proof Using "Type".
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Section orders.
  Context {A} {R : relation A}.
  Implicit Types X Y : A.
  Infix "⊆" := R.
  Notation "X ⊈ Y" := (¬X  Y).
  Infix "⊂" := (strict R).

  Lemma reflexive_eq `{!Reflexive R} X Y : X = Y  X  Y.
  Proof. by intros <-. Qed.
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  Lemma anti_symm_iff `{!PartialOrder R} X Y : X = Y  R X Y  R Y X.
  Proof. split. by intros ->. by intros [??]; apply (anti_symm _). Qed.
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  Lemma strict_spec X Y : X  Y  X  Y  Y  X.
  Proof. done. Qed.
  Lemma strict_include X Y : X  Y  X  Y.
  Proof. by intros [? _]. Qed.
  Lemma strict_ne X Y : X  Y  X  Y.
  Proof. by intros [??] <-. Qed.
  Lemma strict_ne_sym X Y : X  Y  Y  X.
  Proof. by intros [??] <-. Qed.
  Lemma strict_transitive_l `{!Transitive R} X Y Z : X  Y  Y  Z  X  Z.
  Proof.
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    intros [? HXY] ?. split; [by trans Y|].
    contradict HXY. by trans Z.
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  Qed.
  Lemma strict_transitive_r `{!Transitive R} X Y Z : X  Y  Y  Z  X  Z.
  Proof.
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    intros ? [? HYZ]. split; [by trans Y|].
    contradict HYZ. by trans X.
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  Qed.
  Global Instance: Irreflexive (strict R).
  Proof. firstorder. Qed.
  Global Instance: Transitive R  StrictOrder (strict R).
  Proof.
    split; try apply _.
    eauto using strict_transitive_r, strict_include.
  Qed.
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  Global Instance preorder_subset_dec_slow `{!RelDecision R} :
    RelDecision (strict R) | 100.
  Proof. solve_decision. Defined.
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  Lemma strict_spec_alt `{!AntiSymm (=) R} X Y : X  Y  X  Y  X  Y.
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  Proof.
    split.
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    - intros [? HYX]. split. done. by intros <-.
    - intros [? HXY]. split. done. by contradict HXY; apply (anti_symm R).
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  Qed.
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  Lemma po_eq_dec `{!PartialOrder R, !RelDecision R} : EqDecision A.
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  Proof.
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    refine (λ X Y, cast_if_and (decide (X  Y)) (decide (Y  X)));
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     abstract (rewrite anti_symm_iff; tauto).
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  Defined.
  Lemma total_not `{!Total R} X Y : X  Y  Y  X.
  Proof. intros. destruct (total R X Y); tauto. Qed.
  Lemma total_not_strict `{!Total R} X Y : X  Y  Y  X.
  Proof. red; auto using total_not. Qed.
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  Global Instance trichotomy_total
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    `{!Trichotomy (strict R), !Reflexive R} : Total R.
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  Proof.
    intros X Y.
    destruct (trichotomy (strict R) X Y) as [[??]|[<-|[??]]]; intuition.
  Qed.
End orders.

Section strict_orders.
  Context {A} {R : relation A}.
  Implicit Types X Y : A.
  Infix "⊂" := R.

  Lemma irreflexive_eq `{!Irreflexive R} X Y : X = Y  ¬X  Y.
  Proof. intros ->. apply (irreflexivity R). Qed.
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  Lemma strict_anti_symm `{!StrictOrder R} X Y :
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    X  Y  Y  X  False.
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  Proof. intros. apply (irreflexivity R X). by trans Y. Qed.
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  Global Instance trichotomyT_dec `{!TrichotomyT R, !StrictOrder R} :
      RelDecision R := λ X Y,
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    match trichotomyT R X Y with
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    | inleft (left H) => left H
    | inleft (right H) => right (irreflexive_eq _ _ H)
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    | inright H => right (strict_anti_symm _ _ H)
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    end.
  Global Instance trichotomyT_trichotomy `{!TrichotomyT R} : Trichotomy R.
  Proof. intros X Y. destruct (trichotomyT R X Y) as [[|]|]; tauto. Qed.
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End strict_orders.
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Ltac simplify_order := repeat
  match goal with
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  | _ => progress simplify_eq/=
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  | H : ?R ?x ?x |- _ => by destruct (irreflexivity _ _ H)
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  | H1 : ?R ?x ?y |- _ =>
    match goal with
    | H2 : R y x |- _ =>
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      assert (x = y) by (by apply (anti_symm R)); clear H1 H2
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    | H2 : R y ?z |- _ =>
      unless (R x z) by done;
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      assert (R x z) by (by trans y)
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    end
  end.