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(* -*- coding: utf-8 -*- *)
(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** * Typeclass-based relations, tactics and standard instances
This is the basic theory needed to formalize morphisms and setoids.
Author: Matthieu Sozeau
Institution: LRI, CNRS UMR 8623 - University Paris Sud
*)
Require Export Coq.Classes.Init.
Require Import Coq.Program.Basics.
Require Import Coq.Program.Tactics.
Generalizable Variables A B C D R S T U l eqA eqB eqC eqD.
Set Universe Polymorphism.
Definition crelation (A : Type) := A -> A -> Type.
Definition arrow (A B : Type) := A -> B.
Definition flip {A B C : Type} (f : A -> B -> C) := fun x y => f y x.
Definition iffT (A B : Type) := ((A -> B) * (B -> A))%type.
(** We allow to unfold the [crelation] definition while doing morphism search. *)
Section Defs.
Context {A : Type}.
(** We rebind crelational properties in separate classes to be able to overload each proof. *)
Class Reflexive (R : crelation A) :=
reflexivity : forall x : A, R x x.
Definition complement (R : crelation A) : crelation A :=
fun x y => R x y -> False.
(** Opaque for proof-search. *)
Typeclasses Opaque complement iffT.
(** These are convertible. *)
Lemma complement_inverse R : complement (flip R) = flip (complement R).
Proof. reflexivity. Qed.
Class Irreflexive (R : crelation A) :=
irreflexivity : Reflexive (complement R).
Class Symmetric (R : crelation A) :=
symmetry : forall {x y}, R x y -> R y x.
Class Asymmetric (R : crelation A) :=
asymmetry : forall {x y}, R x y -> (complement R y x : Type).
Class Transitive (R : crelation A) :=
transitivity : forall {x y z}, R x y -> R y z -> R x z.
(** Various combinations of reflexivity, symmetry and transitivity. *)
(** A [PreOrder] is both Reflexive and Transitive. *)
Class PreOrder (R : crelation A) := {
PreOrder_Reflexive :> Reflexive R | 2 ;
PreOrder_Transitive :> Transitive R | 2 }.
(** A [StrictOrder] is both Irreflexive and Transitive. *)
Class StrictOrder (R : crelation A) := {
StrictOrder_Irreflexive :> Irreflexive R ;
StrictOrder_Transitive :> Transitive R }.
(** By definition, a strict order is also asymmetric *)
Global Instance StrictOrder_Asymmetric `(StrictOrder R) : Asymmetric R.
Proof. firstorder. Qed.
(** A partial equivalence crelation is Symmetric and Transitive. *)
Class PER (R : crelation A) := {
PER_Symmetric :> Symmetric R | 3 ;
PER_Transitive :> Transitive R | 3 }.
(** Equivalence crelations. *)
Class Equivalence (R : crelation A) := {
Equivalence_Reflexive :> Reflexive R ;
Equivalence_Symmetric :> Symmetric R ;
Equivalence_Transitive :> Transitive R }.
(** An Equivalence is a PER plus reflexivity. *)
Global Instance Equivalence_PER {R} `(Equivalence R) : PER R | 10 :=
{ PER_Symmetric := Equivalence_Symmetric ;
PER_Transitive := Equivalence_Transitive }.
(** We can now define antisymmetry w.r.t. an equivalence crelation on the carrier. *)
Class Antisymmetric eqA `{equ : Equivalence eqA} (R : crelation A) :=
antisymmetry : forall {x y}, R x y -> R y x -> eqA x y.
Class subrelation (R R' : crelation A) :=
is_subrelation : forall {x y}, R x y -> R' x y.
(** Any symmetric crelation is equal to its inverse. *)
Lemma subrelation_symmetric R `(Symmetric R) : subrelation (flip R) R.
Proof. hnf. intros x y H'. red in H'. apply symmetry. assumption. Qed.
Section flip.
Lemma flip_Reflexive `{Reflexive R} : Reflexive (flip R).
Proof. tauto. Qed.
Program Definition flip_Irreflexive `(Irreflexive R) : Irreflexive (flip R) :=
irreflexivity (R:=R).
Program Definition flip_Symmetric `(Symmetric R) : Symmetric (flip R) :=
fun x y H => symmetry (R:=R) H.
Program Definition flip_Asymmetric `(Asymmetric R) : Asymmetric (flip R) :=
fun x y H H' => asymmetry (R:=R) H H'.
Program Definition flip_Transitive `(Transitive R) : Transitive (flip R) :=
fun x y z H H' => transitivity (R:=R) H' H.
Program Definition flip_Antisymmetric `(Antisymmetric eqA R) :
Antisymmetric eqA (flip R).
Proof. firstorder. Qed.
(** Inversing the larger structures *)
Lemma flip_PreOrder `(PreOrder R) : PreOrder (flip R).
Proof. firstorder. Qed.
Lemma flip_StrictOrder `(StrictOrder R) : StrictOrder (flip R).
Proof. firstorder. Qed.
Lemma flip_PER `(PER R) : PER (flip R).
Proof. firstorder. Qed.
Lemma flip_Equivalence `(Equivalence R) : Equivalence (flip R).
Proof. firstorder. Qed.
End flip.
Section complement.
Definition complement_Irreflexive `(Reflexive R)
: Irreflexive (complement R).
Proof. firstorder. Qed.
Definition complement_Symmetric `(Symmetric R) : Symmetric (complement R).
Proof. firstorder. Qed.
End complement.
(** Rewrite crelation on a given support: declares a crelation as a rewrite
crelation for use by the generalized rewriting tactic.
It helps choosing if a rewrite should be handled
by the generalized or the regular rewriting tactic using leibniz equality.
Users can declare an [RewriteRelation A RA] anywhere to declare default
crelations. This is also done automatically by the [Declare Relation A RA]
commands. *)
Class RewriteRelation (RA : crelation A).
(** Any [Equivalence] declared in the context is automatically considered
a rewrite crelation. *)
Global Instance equivalence_rewrite_crelation `(Equivalence eqA) : RewriteRelation eqA.
Defined.
(** Leibniz equality. *)
Section Leibniz.
Global Instance eq_Reflexive : Reflexive (@eq A) := @eq_refl A.
Global Instance eq_Symmetric : Symmetric (@eq A) := @eq_sym A.
Global Instance eq_Transitive : Transitive (@eq A) := @eq_trans A.
(** Leibinz equality [eq] is an equivalence crelation.
The instance has low priority as it is always applicable
if only the type is constrained. *)
Global Program Instance eq_equivalence : Equivalence (@eq A) | 10.
End Leibniz.
End Defs.
(** Default rewrite crelations handled by [setoid_rewrite]. *)
Instance: RewriteRelation impl.
Defined.
Instance: RewriteRelation iff.
Defined.
(** Hints to drive the typeclass resolution avoiding loops
due to the use of full unification. *)
Hint Extern 1 (Reflexive (complement _)) => class_apply @irreflexivity : typeclass_instances.
Hint Extern 3 (Symmetric (complement _)) => class_apply complement_Symmetric : typeclass_instances.
Hint Extern 3 (Irreflexive (complement _)) => class_apply complement_Irreflexive : typeclass_instances.
Hint Extern 3 (Reflexive (flip _)) => apply flip_Reflexive : typeclass_instances.
Hint Extern 3 (Irreflexive (flip _)) => class_apply flip_Irreflexive : typeclass_instances.
Hint Extern 3 (Symmetric (flip _)) => class_apply flip_Symmetric : typeclass_instances.
Hint Extern 3 (Asymmetric (flip _)) => class_apply flip_Asymmetric : typeclass_instances.
Hint Extern 3 (Antisymmetric (flip _)) => class_apply flip_Antisymmetric : typeclass_instances.
Hint Extern 3 (Transitive (flip _)) => class_apply flip_Transitive : typeclass_instances.
Hint Extern 3 (StrictOrder (flip _)) => class_apply flip_StrictOrder : typeclass_instances.
Hint Extern 3 (PreOrder (flip _)) => class_apply flip_PreOrder : typeclass_instances.
Hint Extern 4 (subrelation (flip _) _) =>
class_apply @subrelation_symmetric : typeclass_instances.
Hint Resolve irreflexivity : ord.
Unset Implicit Arguments.
(** A HintDb for crelations. *)
Ltac solve_crelation :=
match goal with
| [ |- ?R ?x ?x ] => reflexivity
| [ H : ?R ?x ?y |- ?R ?y ?x ] => symmetry ; exact H
end.
Hint Extern 4 => solve_crelation : crelations.
(** We can already dualize all these properties. *)
(** * Standard instances. *)
Ltac reduce_hyp H :=
match type of H with
| context [ _ <-> _ ] => fail 1
| _ => red in H ; try reduce_hyp H
end.
Ltac reduce_goal :=
match goal with
| [ |- _ <-> _ ] => fail 1
| _ => red ; intros ; try reduce_goal
end.
Tactic Notation "reduce" "in" hyp(Hid) := reduce_hyp Hid.
Ltac reduce := reduce_goal.
Tactic Notation "apply" "*" constr(t) :=
first [ refine t | refine (t _) | refine (t _ _) | refine (t _ _ _) | refine (t _ _ _ _) |
refine (t _ _ _ _ _) | refine (t _ _ _ _ _ _) | refine (t _ _ _ _ _ _ _) ].
Ltac simpl_crelation :=
unfold flip, impl, arrow ; try reduce ; program_simpl ;
try ( solve [ dintuition ]).
Local Obligation Tactic := simpl_crelation.
(** Logical implication. *)
Program Instance impl_Reflexive : Reflexive impl.
Program Instance impl_Transitive : Transitive impl.
(** Logical equivalence. *)
Instance iff_Reflexive : Reflexive iff := iff_refl.
Instance iff_Symmetric : Symmetric iff := iff_sym.
Instance iff_Transitive : Transitive iff := iff_trans.
(** Logical equivalence [iff] is an equivalence crelation. *)
Program Instance iff_equivalence : Equivalence iff.
Program Instance arrow_Reflexive : Reflexive arrow.
Program Instance arrow_Transitive : Transitive arrow.
Instance iffT_Reflexive : Reflexive iffT.
Proof. firstorder. Defined.
Instance iffT_Symmetric : Symmetric iffT.
Proof. firstorder. Defined.
Instance iffT_Transitive : Transitive iffT.
Proof. firstorder. Defined.
(** We now develop a generalization of results on crelations for arbitrary predicates.
The resulting theory can be applied to homogeneous binary crelations but also to
arbitrary n-ary predicates. *)
Local Open Scope list_scope.
(** A compact representation of non-dependent arities, with the codomain singled-out. *)
(** We define the various operations which define the algebra on binary crelations *)
Section Binary.
Context {A : Type}.
Definition relation_equivalence : crelation (crelation A) :=
fun R R' => forall x y, iffT (R x y) (R' x y).
Global Instance: RewriteRelation relation_equivalence.
Defined.
Definition relation_conjunction (R : crelation A) (R' : crelation A) : crelation A :=
fun x y => prod (R x y) (R' x y).
Definition relation_disjunction (R : crelation A) (R' : crelation A) : crelation A :=
fun x y => sum (R x y) (R' x y).
(** Relation equivalence is an equivalence, and subrelation defines a partial order. *)
Global Instance relation_equivalence_equivalence :
Equivalence relation_equivalence.
Proof.
split; red; unfold relation_equivalence, iffT.
- firstorder.
- firstorder.
- intros. specialize (X x0 y0). specialize (X0 x0 y0). firstorder.
Qed.
Global Instance relation_implication_preorder : PreOrder (@subrelation A).
Proof. firstorder. Qed.
(** *** Partial Order.
A partial order is a preorder which is additionally antisymmetric.
We give an equivalent definition, up-to an equivalence crelation
on the carrier. *)
Class PartialOrder eqA `{equ : Equivalence A eqA} R `{preo : PreOrder A R} :=
partial_order_equivalence : relation_equivalence eqA (relation_conjunction R (flip R)).
(** The equivalence proof is sufficient for proving that [R] must be a
morphism for equivalence (see Morphisms). It is also sufficient to
show that [R] is antisymmetric w.r.t. [eqA] *)
Global Instance partial_order_antisym `(PartialOrder eqA R) : ! Antisymmetric A eqA R.
Proof with auto.
reduce_goal.
apply H. firstorder.
Qed.
Lemma PartialOrder_inverse `(PartialOrder eqA R) : PartialOrder eqA (flip R).
Proof.
unfold flip; constructor; unfold flip.
- intros. apply H. apply symmetry. apply X.
- unfold relation_conjunction. intros [H1 H2]. apply H. constructor; assumption.
Qed.
End Binary.
Hint Extern 3 (PartialOrder (flip _)) => class_apply PartialOrder_inverse : typeclass_instances.
(** The partial order defined by subrelation and crelation equivalence. *)
(* Program Instance subrelation_partial_order : *)
(* ! PartialOrder (crelation A) relation_equivalence subrelation. *)
(* Obligation Tactic := idtac. *)
(* Next Obligation. *)
(* Proof. *)
(* intros x. refine (fun x => x). *)
(* Qed. *)
Typeclasses Opaque relation_equivalence.
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