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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).A:Type
R:A -> A -> Type
complement (flip R) = flip (complement R)
  Proof.A:Type
R:A -> A -> Type
complement (flip R) = flip (complement R) 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

  Instance StrictOrder_Asymmetric `(StrictOrder R) : Asymmetric R.A:Type
R:crelation A
H:StrictOrder R
Asymmetric R
  Proof.A:Type
R:crelation A
H:StrictOrder R
Asymmetric R 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.

  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.A:Type
R:crelation A
H:Symmetric R
subrelation (flip R) R
  Proof.A:Type
R:crelation A
H:Symmetric R
subrelation (flip R) R hnf.A:Type
R:crelation A
H:Symmetric R
forall x y : A, flip R x y -> R x y intros x y H'.A:Type
R:crelation A
H:Symmetric R
x, y:A
H':flip R x y
R x y red in H'.A:Type
R:crelation A
H:Symmetric R
x, y:A
H':R y x
R x y apply symmetry.A:Type
R:crelation A
H:Symmetric R
x, y:A
H':R y x
R y x assumption. Qed.

  Section flip.
  
    Lemma flip_Reflexive `{Reflexive R} : Reflexive (flip R).A:Type
R:crelation A
H:Reflexive R
Reflexive (flip R)
    Proof.A:Type
R:crelation A
H:Reflexive R
Reflexive (flip R) tauto. Qed.
    
    Definition flip_Irreflexive `(Irreflexive R) : Irreflexive (flip R) :=
      irreflexivity (R:=R).
    
    Definition flip_Symmetric `(Symmetric R) : Symmetric (flip R) :=
      fun x y H => symmetry (R:=R) H.
    
    Definition flip_Asymmetric `(Asymmetric R) : Asymmetric (flip R) :=
      fun x y H H' => asymmetry (R:=R) H H'.
    
    Definition flip_Transitive `(Transitive R) : Transitive (flip R) :=
      fun x y z H H' => transitivity (R:=R) H' H.

    Definition flip_Antisymmetric `(Antisymmetric eqA R) :
      Antisymmetric eqA (flip R).A:Type
eqA:crelation A
equ:Equivalence eqA
R:crelation A
H:Antisymmetric eqA R
Antisymmetric eqA (flip R)
    Proof.A:Type
eqA:crelation A
equ:Equivalence eqA
R:crelation A
H:Antisymmetric eqA R
Antisymmetric eqA (flip R) firstorder. Qed.

Inversing the larger structures

    Lemma flip_PreOrder `(PreOrder R) : PreOrder (flip R).A:Type
R:crelation A
H:PreOrder R
PreOrder (flip R)
    Proof.A:Type
R:crelation A
H:PreOrder R
PreOrder (flip R) firstorder. Qed.

    Lemma flip_StrictOrder `(StrictOrder R) : StrictOrder (flip R).A:Type
R:crelation A
H:StrictOrder R
StrictOrder (flip R)
    Proof.A:Type
R:crelation A
H:StrictOrder R
StrictOrder (flip R) firstorder. Qed.

    Lemma flip_PER `(PER R) : PER (flip R).A:Type
R:crelation A
H:PER R
PER (flip R)
    Proof.A:Type
R:crelation A
H:PER R
PER (flip R) firstorder. Qed.

    Lemma flip_Equivalence `(Equivalence R) : Equivalence (flip R).A:Type
R:crelation A
H:Equivalence R
Equivalence (flip R)
    Proof.A:Type
R:crelation A
H:Equivalence R
Equivalence (flip R) firstorder. Qed.

  End flip.

  Section complement.

    Definition complement_Irreflexive `(Reflexive R)
      : Irreflexive (complement R).A:Type
R:crelation A
H:Reflexive R
Irreflexive (complement R)
    Proof.A:Type
R:crelation A
H:Reflexive R
Irreflexive (complement R) firstorder. Qed.

    Definition complement_Symmetric `(Symmetric R) : Symmetric (complement R).A:Type
R:crelation A
H:Symmetric R
Symmetric (complement R)
    Proof.A:Type
R:crelation A
H:Symmetric R
Symmetric (complement R) 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.

  Instance equivalence_rewrite_crelation `(Equivalence eqA) : RewriteRelation eqA.
  Defined.

Leibniz equality.

  Section Leibniz.
    Instance eq_Reflexive : Reflexive (@eq A) := @eq_refl A.
    Instance eq_Symmetric : Symmetric (@eq A) := @eq_sym A.
    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.

    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.

Instance impl_Reflexive : Reflexive impl.
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.

Instance iff_equivalence : Equivalence iff. 
Instance arrow_Reflexive : Reflexive arrow.
Instance arrow_Transitive : Transitive arrow.

Instance iffT_Reflexive : Reflexive iffT.Reflexive iffT 
Proof.Reflexive iffT firstorder. Defined.
Instance iffT_Symmetric : Symmetric iffT.Symmetric iffT 
Proof.Symmetric iffT firstorder. Defined. 
Instance iffT_Transitive : Transitive iffT.Transitive iffT
Proof.Transitive iffT 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.

  Instance relation_equivalence_equivalence :
    Equivalence relation_equivalence.A:Type
Equivalence relation_equivalence
  Proof.A:Type
Equivalence relation_equivalence
    split; red; unfold relation_equivalence, iffT.A:Type
forall (x : crelation A) (x0 y : A),
(x x0 y -> x x0 y) * (x x0 y -> x x0 y)
A:Type
forall x y : crelation A,
(forall x0 y0 : A,
 (x x0 y0 -> y x0 y0) * (y x0 y0 -> x x0 y0)) ->
forall x0 y0 : A,
(y x0 y0 -> x x0 y0) * (x x0 y0 -> y x0 y0)
A:Type
forall x y z : crelation A,
(forall x0 y0 : A,
 (x x0 y0 -> y x0 y0) * (y x0 y0 -> x x0 y0)) ->
(forall x0 y0 : A,
 (y x0 y0 -> z x0 y0) * (z x0 y0 -> y x0 y0)) ->
forall x0 y0 : A,
(x x0 y0 -> z x0 y0) * (z x0 y0 -> x x0 y0)
    -A:Type
forall (x : crelation A) (x0 y : A),
(x x0 y -> x x0 y) * (x x0 y -> x x0 y) firstorder.
    -A:Type
forall x y : crelation A,
(forall x0 y0 : A,
 (x x0 y0 -> y x0 y0) * (y x0 y0 -> x x0 y0)) ->
forall x0 y0 : A,
(y x0 y0 -> x x0 y0) * (x x0 y0 -> y x0 y0) firstorder.
    -A:Type
forall x y z : crelation A,
(forall x0 y0 : A,
 (x x0 y0 -> y x0 y0) * (y x0 y0 -> x x0 y0)) ->
(forall x0 y0 : A,
 (y x0 y0 -> z x0 y0) * (z x0 y0 -> y x0 y0)) ->
forall x0 y0 : A,
(x x0 y0 -> z x0 y0) * (z x0 y0 -> x x0 y0) intros.A:Type
x, y, z:crelation A
X:forall x1 y1 : A,
(x x1 y1 -> y x1 y1) * (y x1 y1 -> x x1 y1)
X0:forall x1 y1 : A,
(y x1 y1 -> z x1 y1) * (z x1 y1 -> y x1 y1)
x0, y0:A
((x x0 y0 -> z x0 y0) * (z x0 y0 -> x x0 y0))%type specialize (X x0 y0).A:Type
x, y, z:crelation A
x0, y0:A
X:((x x0 y0 -> y x0 y0) * (y x0 y0 -> x x0 y0))%type
X0:forall x1 y1 : A,
(y x1 y1 -> z x1 y1) * (z x1 y1 -> y x1 y1)
((x x0 y0 -> z x0 y0) * (z x0 y0 -> x x0 y0))%type specialize (X0 x0 y0).A:Type
x, y, z:crelation A
x0, y0:A
X:((x x0 y0 -> y x0 y0) * (y x0 y0 -> x x0 y0))%type
X0:((y x0 y0 -> z x0 y0) * (z x0 y0 -> y x0 y0))%type
((x x0 y0 -> z x0 y0) * (z x0 y0 -> x x0 y0))%type firstorder.
  Qed.
    
  Instance relation_implication_preorder : PreOrder (@subrelation A).A:Type
PreOrder subrelation
  Proof.A:Type
PreOrder subrelation 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

  Instance partial_order_antisym `(PartialOrder eqA R) : ! Antisymmetric A eqA R.A:Type
eqA:crelation A
equ:Equivalence eqA
R:crelation A
preo:PreOrder R
H:PartialOrder eqA R
Antisymmetric eqA R
  Proof with auto.A:Type
eqA:crelation A
equ:Equivalence eqA
R:crelation A
preo:PreOrder R
H:PartialOrder eqA R
Antisymmetric eqA R
    reduce_goal.A:Type
eqA:crelation A
equ:Equivalence eqA
R:crelation A
preo:PreOrder R
H:PartialOrder eqA R
x, y:A
X:R x y
X0:R y x
eqA x y
    apply H.A:Type
eqA:crelation A
equ:Equivalence eqA
R:crelation A
preo:PreOrder R
H:PartialOrder eqA R
x, y:A
X:R x y
X0:R y x
relation_conjunction R (flip R) x y firstorder.
  Qed.

  Lemma PartialOrder_inverse `(PartialOrder eqA R) : PartialOrder eqA (flip R).A:Type
eqA:crelation A
equ:Equivalence eqA
R:crelation A
preo:PreOrder R
H:PartialOrder eqA R
PartialOrder eqA (flip R)
  Proof.A:Type
eqA:crelation A
equ:Equivalence eqA
R:crelation A
preo:PreOrder R
H:PartialOrder eqA R
PartialOrder eqA (flip R)
    unfold flip; constructor; unfold flip.A:Type
eqA:crelation A
equ:Equivalence eqA
R:crelation A
preo:PreOrder R
H:PartialOrder eqA R
x, y:A
eqA x y ->
relation_conjunction (fun x0 y0 : A => R y0 x0)
  (fun x0 y0 : A => R x0 y0) x y
A:Type
eqA:crelation A
equ:Equivalence eqA
R:crelation A
preo:PreOrder R
H:PartialOrder eqA R
x, y:A
relation_conjunction (fun x0 y0 : A => R y0 x0)
  (fun x0 y0 : A => R x0 y0) x y -> 
eqA x y
    -A:Type
eqA:crelation A
equ:Equivalence eqA
R:crelation A
preo:PreOrder R
H:PartialOrder eqA R
x, y:A
eqA x y ->
relation_conjunction (fun x0 y0 : A => R y0 x0)
  (fun x0 y0 : A => R x0 y0) x y intros.A:Type
eqA:crelation A
equ:Equivalence eqA
R:crelation A
preo:PreOrder R
H:PartialOrder eqA R
x, y:A
X:eqA x y
relation_conjunction (fun x0 y0 : A => R y0 x0)
  (fun x0 y0 : A => R x0 y0) x y apply H.A:Type
eqA:crelation A
equ:Equivalence eqA
R:crelation A
preo:PreOrder R
H:PartialOrder eqA R
x, y:A
X:eqA x y
eqA y x apply symmetry.A:Type
eqA:crelation A
equ:Equivalence eqA
R:crelation A
preo:PreOrder R
H:PartialOrder eqA R
x, y:A
X:eqA x y
eqA x y apply X.
    -A:Type
eqA:crelation A
equ:Equivalence eqA
R:crelation A
preo:PreOrder R
H:PartialOrder eqA R
x, y:A
relation_conjunction (fun x0 y0 : A => R y0 x0)
  (fun x0 y0 : A => R x0 y0) x y -> eqA x y unfold relation_conjunction.A:Type
eqA:crelation A
equ:Equivalence eqA
R:crelation A
preo:PreOrder R
H:PartialOrder eqA R
x, y:A
R y x * R x y -> eqA x y intros [H1 H2].A:Type
eqA:crelation A
equ:Equivalence eqA
R:crelation A
preo:PreOrder R
H:PartialOrder eqA R
x, y:A
H1:R y x
H2:R x y
eqA x y apply H.A:Type
eqA:crelation A
equ:Equivalence eqA
R:crelation A
preo:PreOrder R
H:PartialOrder eqA R
x, y:A
H1:R y x
H2:R x y
relation_conjunction R (flip R) x y 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.