(* Copyright (c) 2012-2015, Robbert Krebbers. *)
(* This file is distributed under the terms of the BSD license. *)
(** 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.
Set Default Proof Using "Type*".
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 _). 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.
Proof.
intros [? HXY] ?. split; [by trans Y|].
contradict HXY. by trans Z.
Qed.
Lemma strict_transitive_r `{!Transitive R} X Y Z : X ⊆ Y → Y ⊂ Z → X ⊂ Z.
Proof.
intros ? [? HYZ]. split; [by trans Y|].
contradict HYZ. by trans X.
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.
Global Instance preorder_subset_dec_slow `{∀ X Y, Decision (X ⊆ Y)}
(X Y : A) : Decision (X ⊂ Y) | 100 := _.
Lemma strict_spec_alt `{!AntiSymm (=) R} X Y : X ⊂ Y ↔ X ⊆ Y ∧ X ≠ Y.
Proof.
split.
- intros [? HYX]. split. done. by intros <-.
- intros [? HXY]. split. done. by contradict HXY; apply (anti_symm R).
Qed.
Lemma po_eq_dec `{!PartialOrder R, ∀ X Y, Decision (X ⊆ Y)} : EqDecision A.
Proof.
refine (λ X Y, cast_if_and (decide (X ⊆ Y)) (decide (Y ⊆ X)));
abstract (rewrite anti_symm_iff; tauto).
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.
Global Instance trichotomy_total
`{!Trichotomy (strict R), !Reflexive R} : Total R.
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.
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} X Y :
Decision (X ⊂ Y) :=
match trichotomyT R X Y with
| inleft (left H) => left H
| inleft (right H) => right (irreflexive_eq _ _ H)
| inright H => right (strict_anti_symm _ _ H)
end.
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)
end
end.