Library stdpp.orders

Properties about arbitrary pre-, partial, and total orders. We do not use the relation because we often have multiple orders on the same structure
From stdpp Require Export tactics.
From stdpp Require Import options.

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.
  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 R). Qed.
  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.
    intros [? HXY] ?. split; [by trans Y|].
    contradict HXY. by trans Z.
  Lemma strict_transitive_r `{!Transitive R} X Y Z : X Y Y Z X Z.
    intros ? [? HYZ]. split; [by trans Y|].
    contradict HYZ. by trans X.
  Global Instance: Irreflexive (strict R).
  Proof. firstorder. Qed.
  Global Instance: Transitive R StrictOrder (strict R).
    split; try apply _.
    eauto using strict_transitive_r, strict_include.
  Global Instance preorder_subset_dec_slow `{!RelDecision R} :
    RelDecision (strict R) | 100.
  Proof. solve_decision. Defined.
  Lemma strict_spec_alt `{!AntiSymm (=) R} X Y : X Y X Y X Y.
    - intros [? HYX]. split; [ done | by intros <- ].
    - intros [? HXY]. split; [ done | by contradict HXY; apply (anti_symm R) ].
  Lemma po_eq_dec `{!PartialOrder R, !RelDecision R} : EqDecision A.
    refine (λ X Y, cast_if_and (decide (X Y)) (decide (Y X)));
     abstract (rewrite anti_symm_iff; tauto).
  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.
  Global Instance trichotomy_total
    `{!Trichotomy (strict R), !Reflexive R} : Total R.
    intros X Y.
    destruct (trichotomy (strict R) X Y) as [[??]|[<-|[??]]]; intuition.
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.
  Lemma strict_anti_symm `{!StrictOrder R} X Y :
    X Y Y X False.
  Proof. intros. apply (irreflexivity R X). by trans Y. Qed.
  Global Instance trichotomyT_dec `{!TrichotomyT R, !StrictOrder R} :
      RelDecision R := λ X Y,
    match trichotomyT R X Y with
    | inleft (left H) ⇒ left H
    | inleft (right H) ⇒ right (irreflexive_eq _ _ H)
    | inright Hright (strict_anti_symm _ _ H)
  Global Instance trichotomyT_trichotomy `{!TrichotomyT R} : Trichotomy R.
  Proof. intros X Y. destruct (trichotomyT R X Y) as [[|]|]; tauto. Qed.
End strict_orders.

Ltac simplify_order := repeat
  match goal with
  | _progress simplify_eq/=
  | H : ?R ?x ?x |- _by destruct (irreflexivity _ _ H)
  | H1 : ?R ?x ?y |- _
    match goal with
    | H2 : R y x |- _
      assert (x = y) by (by apply (anti_symm R)); clear H1 H2
    | H2 : R y ?z |- _
      unless (R x z) by done;
      assert (R x z) by (by trans y)