# 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.
Require Import Coq.Relations.Relation_Definitions.

Generalizable Variables A B C D R S T U l eqA eqB eqC eqD.

We allow to unfold the relation definition while doing morphism search.

Section Defs.
Context {A : Type}.

We rebind relational properties in separate classes to be able to overload each proof.

Class Reflexive (R : relation A) :=
reflexivity : forall x : A, R x x.

Definition complement (R : relation A) : relation A := fun x y => R x y -> False.

Opaque for proof-search.
Typeclasses Opaque complement.

These are convertible.
Lemma complement_inverse R : complement (flip R) = flip (complement R).

Class Irreflexive (R : relation A) :=
irreflexivity : Reflexive (complement R).

Class Symmetric (R : relation A) :=
symmetry : forall {x y}, R x y -> R y x.

Class Asymmetric (R : relation A) :=
asymmetry : forall {x y}, R x y -> R y x -> False.

Class Transitive (R : relation 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 : relation A) : Prop := {
#[global] PreOrder_Reflexive :: Reflexive R | 2 ;
#[global] PreOrder_Transitive :: Transitive R | 2 }.

A StrictOrder is both Irreflexive and Transitive.

Class StrictOrder (R : relation A) : Prop := {
#[global] StrictOrder_Irreflexive :: Irreflexive R ;
#[global] StrictOrder_Transitive :: Transitive R }.

By definition, a strict order is also asymmetric
Global Instance StrictOrder_Asymmetric `(StrictOrder R) : Asymmetric R.

A partial equivalence relation is Symmetric and Transitive.

Class PER (R : relation A) : Prop := {
#[global] PER_Symmetric :: Symmetric R | 3 ;
#[global] PER_Transitive :: Transitive R | 3 }.

Equivalence relations.

Class Equivalence (R : relation A) : Prop := {
#[global] Equivalence_Reflexive :: Reflexive R ;
#[global] Equivalence_Symmetric :: Symmetric R ;
#[global] Equivalence_Transitive :: Transitive R }.

An Equivalence is a PER plus reflexivity.

Global Instance Equivalence_PER {R} `(E:Equivalence R) : PER R | 10 :=
{ }.

An Equivalence is a PreOrder plus symmetry.

Global Instance Equivalence_PreOrder {R} `(E:Equivalence R) : PreOrder R | 10 :=
{ }.

We can now define antisymmetry w.r.t. an equivalence relation on the carrier.

Class Antisymmetric eqA `{equ : Equivalence eqA} (R : relation A) :=
antisymmetry : forall {x y}, R x y -> R y x -> eqA x y.

Class subrelation (R R' : relation A) : Prop :=
is_subrelation : forall {x y}, R x y -> R' x y.

Any symmetric relation is equal to its inverse.

Lemma subrelation_symmetric R `(Symmetric R) : subrelation (flip R) R.

Section flip.

Lemma flip_Reflexive `{Reflexive R} : Reflexive (flip R).

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).

Inversing the larger structures

Lemma flip_PreOrder `(PreOrder R) : PreOrder (flip R).

Lemma flip_StrictOrder `(StrictOrder R) : StrictOrder (flip R).

Lemma flip_PER `(PER R) : PER (flip R).

Lemma flip_Equivalence `(Equivalence R) : Equivalence (flip R).

End flip.

Section complement.

Definition complement_Irreflexive `(Reflexive R)
: Irreflexive (complement R).

Definition complement_Symmetric `(Symmetric R) : Symmetric (complement R).
End complement.

Rewrite relation on a given support: declares a relation as a rewrite relation 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 relations on a given type `A`. This is also done automatically by the Declare Relation A RA commands. It has no mode declaration: it will assign `?A := Prop, ?R := iff` on an entirely unspecified query `RewriteRelation ?A ?R`, or any prefered rewrite relation of priority < 2.

Class RewriteRelation (RA : relation A).

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 relation. The instance has low priority as it is always applicable if only the type is constrained.

End Leibniz.

Leibniz disequality.
Section LeibnizNot.
Disequality is symmetric.
Global Instance neq_Symmetric : Symmetric (fun x y : A => x <> y) := (@not_eq_sym A).
End LeibnizNot.
End Defs.

Default rewrite relations handled by setoid_rewrite on Prop.
#[global]
Instance inverse_impl_rewrite_relation : RewriteRelation (flip impl) | 3 := {}.
#[global]
Instance impl_rewrite_relation : RewriteRelation impl | 3 := {}.
#[global]
Instance iff_rewrite_relation : RewriteRelation iff | 2 := {}.

Any Equivalence declared in the context is automatically considered a rewrite relation. This only applies if the relation is at least partially defined: setoid_rewrite won't try to infer arbitrary user rewrite relations.

Definition equivalence_rewrite_relation `(eqa : Equivalence A eqA) : RewriteRelation eqA :=
Build_RewriteRelation _.

Ltac equiv_rewrite_relation R :=
tryif is_evar R then fail
else class_apply equivalence_rewrite_relation.

#[global]
Hint Extern 10 (@RewriteRelation ?A ?R) => equiv_rewrite_relation R : typeclass_instances.

Hints to drive the typeclass resolution avoiding loops due to the use of full unification.
#[global]
Hint Extern 1 (Reflexive (complement _)) => class_apply @irreflexivity : typeclass_instances.
#[global]
Hint Extern 3 (Symmetric (complement _)) => class_apply complement_Symmetric : typeclass_instances.
#[global]
Hint Extern 3 (Irreflexive (complement _)) => class_apply complement_Irreflexive : typeclass_instances.

#[global]
Hint Extern 3 (Reflexive (flip _)) => apply flip_Reflexive : typeclass_instances.
#[global]
Hint Extern 3 (Irreflexive (flip _)) => class_apply flip_Irreflexive : typeclass_instances.
#[global]
Hint Extern 3 (Symmetric (flip _)) => class_apply flip_Symmetric : typeclass_instances.
#[global]
Hint Extern 3 (Asymmetric (flip _)) => class_apply flip_Asymmetric : typeclass_instances.
#[global]
Hint Extern 3 (Antisymmetric (flip _)) => class_apply flip_Antisymmetric : typeclass_instances.
#[global]
Hint Extern 3 (Transitive (flip _)) => class_apply flip_Transitive : typeclass_instances.
#[global]
Hint Extern 3 (StrictOrder (flip _)) => class_apply flip_StrictOrder : typeclass_instances.
#[global]
Hint Extern 3 (PreOrder (flip _)) => class_apply flip_PreOrder : typeclass_instances.

#[global]
Hint Extern 4 (subrelation (flip _) _) =>
class_apply @subrelation_symmetric : typeclass_instances.

Arguments irreflexivity {A R Irreflexive} [x] _ : rename.
Arguments symmetry {A} {R} {_} [x] [y] _.
Arguments asymmetry {A} {R} {_} [x] [y] _ _.
Arguments transitivity {A} {R} {_} [x] [y] [z] _ _.
Arguments Antisymmetric A eqA {_} _.

#[global]
Hint Resolve irreflexivity : ord.

Unset Implicit Arguments.

Ltac solve_relation :=
match goal with
| [ |- ?R ?x ?x ] => reflexivity
| [ H : ?R ?x ?y |- ?R ?y ?x ] => symmetry ; exact H
end.

#[global]
Hint Extern 4 => solve_relation : relations.

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_relation :=
unfold flip, impl, arrow ; try reduce ; program_simpl ;
try ( solve [ dintuition auto with relations ]).

Logical implication.

#[global]
Program Instance impl_Reflexive : Reflexive impl.
#[global]
Program Instance impl_Transitive : Transitive impl.

Logical equivalence.

#[global]
Instance iff_Reflexive : Reflexive iff := iff_refl.
#[global]
Instance iff_Symmetric : Symmetric iff := iff_sym.
#[global]
Instance iff_Transitive : Transitive iff := iff_trans.

Logical equivalence iff is an equivalence relation.

#[global]
Program Instance iff_equivalence : Equivalence iff.

We now develop a generalization of results on relations for arbitrary predicates. The resulting theory can be applied to homogeneous binary relations but also to arbitrary n-ary predicates.

Local Open Scope list_scope.

A compact representation of non-dependent arities, with the codomain singled-out.

Inductive Tlist : Type := Tnil : Tlist | Tcons : Type -> Tlist -> Tlist.

Fixpoint arrows (l : Tlist) (r : Type) : Type :=
match l with
| Tnil => r
| A :: l' => A -> arrows l' r
end.

We can define abbreviations for operation and relation types based on arrows.
We define n-ary predicates as functions into Prop.

Notation predicate l := (arrows l Prop).

Unary predicates, or sets.

Definition unary_predicate A := predicate (A::Tnil).

Homogeneous binary relations, equivalent to relation A.
We can close a predicate by universal or existential quantification.

Fixpoint predicate_all (l : Tlist) : predicate l -> Prop :=
match l with
| Tnil => fun f => f
| A :: tl => fun f => forall x : A, predicate_all tl (f x)
end.

Fixpoint predicate_exists (l : Tlist) : predicate l -> Prop :=
match l with
| Tnil => fun f => f
| A :: tl => fun f => exists x : A, predicate_exists tl (f x)
end.

Pointwise extension of a binary operation on T to a binary operation on functions whose codomain is T. For an operator on Prop this lifts the operator to a binary operation.

Fixpoint pointwise_extension {T : Type} (op : binary_operation T)
(l : Tlist) : binary_operation (arrows l T) :=
match l with
| Tnil => fun R R' => op R R'
| A :: tl => fun R R' =>
fun x => pointwise_extension op tl (R x) (R' x)
end.

Pointwise lifting, equivalent to doing pointwise_extension and closing using predicate_all.

Fixpoint pointwise_lifting (op : binary_relation Prop) (l : Tlist) : binary_relation (predicate l) :=
match l with
| Tnil => fun R R' => op R R'
| A :: tl => fun R R' =>
forall x, pointwise_lifting op tl (R x) (R' x)
end.

The n-ary equivalence relation, defined by lifting the 0-ary iff relation.
The n-ary implication relation, defined by lifting the 0-ary impl relation.
Notations for pointwise equivalence and implication of predicates.

Declare Scope predicate_scope.

Infix "<∙>" := predicate_equivalence (at level 95, no associativity) : predicate_scope.
Infix "-∙>" := predicate_implication (at level 70, right associativity) : predicate_scope.

Local Open Scope predicate_scope.

The pointwise liftings of conjunction and disjunctions. Note that these are binary_operations, building new relations out of old ones.

Definition predicate_intersection := pointwise_extension and.
Definition predicate_union := pointwise_extension or.

Infix "/∙\" := predicate_intersection (at level 80, right associativity) : predicate_scope.
Infix "\∙/" := predicate_union (at level 85, right associativity) : predicate_scope.

The always True and always False predicates.

Fixpoint true_predicate {l : Tlist} : predicate l :=
match l with
| Tnil => True
| A :: tl => fun _ => @true_predicate tl
end.

Fixpoint false_predicate {l : Tlist} : predicate l :=
match l with
| Tnil => False
| A :: tl => fun _ => @false_predicate tl
end.

Notation "∙⊤∙" := true_predicate : predicate_scope.
Notation "∙⊥∙" := false_predicate : predicate_scope.

Predicate equivalence is an equivalence, and predicate implication defines a preorder.

#[global]
Program Instance predicate_equivalence_equivalence {l} :
Equivalence (@predicate_equivalence l).

#[global]
Program Instance predicate_implication_preorder {l} :
PreOrder (@predicate_implication l).

We define the various operations which define the algebra on binary relations, from the general ones.

Section Binary.
Context {A : Type}.

Definition relation_equivalence : relation (relation A) :=
@predicate_equivalence (_::_::Tnil).

Global Instance relation_equivalence_rewrite_relation: RewriteRelation relation_equivalence := {}.

Definition relation_conjunction (R : relation A) (R' : relation A) : relation A :=
@predicate_intersection (A::A::Tnil) R R'.

Definition relation_disjunction (R : relation A) (R' : relation A) : relation A :=
@predicate_union (A::A::Tnil) R R'.

Relation equivalence is an equivalence, and subrelation defines a partial order.

### Partial Order.

A partial order is a preorder which is additionally antisymmetric. We give an equivalent definition, up-to an equivalence relation on the carrier.
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.

Lemma PartialOrder_inverse `(PartialOrder eqA R) : PartialOrder eqA (flip R).
End Binary.

#[global]
Hint Extern 3 (PartialOrder (flip _)) => class_apply PartialOrder_inverse : typeclass_instances.

The partial order defined by subrelation and relation equivalence.