The Coq Proof Assistant

# Library Coq.Classes.Morphisms

# Typeclass-based morphism definition and standard, minimal instances

Author: Matthieu Sozeau Institution: LRI, CNRS UMR 8623 - University Paris Sud
Require Import Coq.Program.Basics.
Require Import Coq.Program.Tactics.
Require Import Coq.Relations.Relation_Definitions.
Require Export Coq.Classes.RelationClasses.
Generalizable Variables A eqA B C D R RA RB RC m f x y.

# Morphisms.

We now turn to the definition of Proper and declare standard instances. These will be used by the setoid_rewrite tactic later.
A morphism for a relation R is a proper element of the relation. The relation R will be instantiated by respectful and A by an arrow type for usual morphisms.
Section Proper.
Let U := Type.
Context {A B : U}.
Class Proper (R : relation A) (m : A) : Prop :=
proper_prf : R m m.
Every element in the carrier of a reflexive relation is a morphism for this relation. We use a proxy class for this case which is used internally to discharge reflexivity constraints. The Reflexive instance will almost always be used, but it won't apply in general to any kind of Proper (A -> B) _ _ goal, making proof-search much slower. A cleaner solution would be to be able to set different priorities in different hint bases and select a particular hint database for resolution of a type class constraint.
Class ProperProxy (R : relation A) (m : A) : Prop :=
proper_proxy : R m m.
Class ReflexiveProxy (R : relation A) : Prop :=
reflexive_proxy : forall x, R x x.
Lemma eq_proper_proxy (x : A) : ProperProxy (@eq A) x.
Every reflexive relation gives rise to a morphism. If the relation is not determined (is an evar), then we restrict the solutions to predefined ones (equality, or iff on Prop), using ground instances. If the relation is determined then ReflexiveProxy calls back to Reflexive.
Lemma reflexive_proper `{ReflexiveProxy R} (x : A) : Proper R x.
Lemma reflexive_proper_proxy `(ReflexiveProxy R) (x : A) : ProperProxy R x.
Lemma proper_proper_proxy x `(Proper R x) : ProperProxy R x.
Lemma reflexive_reflexive_proxy `(Reflexive A R) : ReflexiveProxy R.
Respectful morphisms.
The fully dependent version, not used yet.
Definition respectful_hetero
(A B : Type)
(C : A -> Type) (D : B -> Type)
(R : A -> B -> Prop)
(R' : forall (x : A) (y : B), C x -> D y -> Prop) :
(forall x : A, C x) -> (forall x : B, D x) -> Prop :=
fun f g => forall x y, R x y -> R' x y (f x) (g y).
The non-dependent version is an instance where we forget dependencies.
Definition respectful (R : relation A) (R' : relation B) : relation (A -> B) :=
Eval compute in @respectful_hetero A A (fun _ => B) (fun _ => B) R (fun _ _ => R').
End Proper.
Non-dependent pointwise lifting
Definition pointwise_relation A {B} (R : relation B) : relation (A -> B) :=
fun f g => forall a, R (f a) (g a).
We let Coq infer these relations when a default relation should be found on the function space.
Lemma rewrite_relation_pointwise {A B R} `{RewriteRelation B R}:
RewriteRelation (@pointwise_relation A B R).
Lemma rewrite_relation_eq_dom {A B R} `{RewriteRelation B R}:
RewriteRelation (respectful (@Logic.eq A) R).
Pointwise reflexive
Ltac rewrite_relation_fun :=
class_apply @rewrite_relation_pointwise ||
class_apply @rewrite_relation_eq_dom.
Global Hint Extern 2 (@RewriteRelation (_ -> _) _) =>
rewrite_relation_fun : typeclass_instances.
Lemma eq_rewrite_relation {A} : RewriteRelation (@eq A).
Ltac eq_rewrite_relation A :=
solve [unshelve class_apply @eq_rewrite_relation].
Global Hint Extern 100 (@RewriteRelation ?A _) => eq_rewrite_relation A : typeclass_instances.
We favor the use of Leibniz equality or a declared reflexive relation when resolving ProperProxy, otherwise, if the relation is given (not an evar), we fall back to Proper.
#[global]
Hint Extern 1 (ProperProxy _ _) =>
class_apply @eq_proper_proxy || class_apply @reflexive_proper_proxy : typeclass_instances.
#[global]
Hint Extern 2 (ProperProxy ?R _) =>
not_evar R; class_apply @proper_proper_proxy : typeclass_instances.
Ltac find_rewrite_relation A R kont :=
assert (@RewriteRelation A R); [solve [unshelve typeclasses eauto]|]; kont R.
This hint helps infer "generic" reflexive relations, based only on the type of the carrier, when the relation is only partially defined (contains evars).
Ltac reflexive_proxy_tac A R :=
tryif has_evar R then
find_rewrite_relation A R ltac:(fun RA =>
class_apply (@reflexive_reflexive_proxy A RA))
else
class_apply @reflexive_reflexive_proxy.
#[global]
Hint Extern 1 (@ReflexiveProxy ?A ?R) => reflexive_proxy_tac A R : typeclass_instances.
Notations reminiscent of the old syntax for declaring morphisms.
Declare Scope signature_scope.
Delimit Scope signature_scope with signature.
Module ProperNotations.
Notation " R ++> R' " := (@respectful _ _ (R%signature) (R'%signature))
(right associativity, at level 55) : signature_scope.
Notation " R ==> R' " := (@respectful _ _ (R%signature) (R'%signature))
(right associativity, at level 55) : signature_scope.
Notation " R --> R' " := (@respectful _ _ (flip (R%signature)) (R'%signature))
(right associativity, at level 55) : signature_scope.
End ProperNotations.
Arguments Proper {A}%type R%signature m.
Arguments respectful {A B}%type (R R')%signature _ _.
Export ProperNotations.
Local Open Scope signature_scope.
solve_proper try to solve the goal Proper (?==> ... ==>?) f by repeated introductions and setoid rewrites. It should work fine when f is a combination of already known morphisms and quantifiers.
Ltac solve_respectful t :=
match goal with
| |- respectful _ _ _ _ =>
let H := fresh "H" in
intros ? ? H; solve_respectful ltac:(setoid_rewrite H; t)
| _ => t; reflexivity
end.
Ltac solve_proper := unfold Proper; solve_respectful ltac:(idtac).
f_equiv is a clone of f_equal that handles setoid equivalences. For example, if we know that f is a morphism for E1==>E2==>E, then the goal E (f x y) (f x' y') will be transformed by f_equiv into the subgoals E1 x x' and E2 y y'.
Ltac f_equiv :=
match goal with
| |- ?R (?f ?x) (?f' _) =>
let T := type of x in
let Rx := fresh "R" in
evar (Rx : relation T);
let H := fresh in
assert (H : (Rx==>R)%signature f f');
unfold Rx in *; clear Rx; [ f_equiv | apply H; clear H; try reflexivity ]
| |- ?R ?f ?f' =>
solve [change (Proper R f); eauto with typeclass_instances | reflexivity ]
| _ => idtac
end.
Section Relations.
Let U := Type.
Context {A B : U} (P : A -> U).
forall_def reifies the dependent product as a definition.
Definition forall_def : Type := forall x : A, P x.
Dependent pointwise lifting of a relation on the range.
Definition forall_relation
(sig : forall a, relation (P a)) : relation (forall x, P x) :=
fun f g => forall a, sig a (f a) (g a).
Lemma pointwise_pointwise (R : relation B) :
relation_equivalence (pointwise_relation A R) (@eq A ==> R).
Subrelations induce a morphism on the identity.
Global Instance subrelation_id_proper `(subrelation A RA RA') : Proper (RA ==> RA') id.
The subrelation property goes through products as usual.
Lemma subrelation_respectful `(subl : subrelation A RA' RA, subr : subrelation B RB RB') :
subrelation (RA ==> RB) (RA' ==> RB').
And of course it is reflexive.
Lemma subrelation_refl R : @subrelation A R R.
Proper is itself a covariant morphism for subrelation. We use an unconvertible premise to avoid looping.
Lemma subrelation_proper `(mor : Proper A R' m)
`(unc : Unconvertible (relation A) R R')
`(sub : subrelation A R' R) : Proper R m.
Global Instance proper_subrelation_proper :
Proper (subrelation ++> eq ==> impl) (@Proper A).
Global Instance pointwise_subrelation `(sub : subrelation B R R') :
subrelation (pointwise_relation A R) (pointwise_relation A R') | 4.
For dependent function types.
Lemma forall_subrelation (R S : forall x : A, relation (P x)) :
(forall a, subrelation (R a) (S a)) -> subrelation (forall_relation R) (forall_relation S).
End Relations.
Arguments forall_relation {A P}%type sig%signature _ _.
Arguments pointwise_relation A%type {B}%type R%signature _ _.
#[global]
Hint Unfold Reflexive : core.
#[global]
Hint Unfold Symmetric : core.
#[global]
Hint Unfold Transitive : core.
Resolution with subrelation: favor decomposing products over applying reflexivity for unconstrained goals.
Ltac subrelation_tac T U :=
(is_ground T ; is_ground U ; class_apply @subrelation_refl) ||
class_apply @subrelation_respectful || class_apply @subrelation_refl.
#[global]
Hint Extern 3 (@subrelation _ ?T ?U) => subrelation_tac T U : typeclass_instances.
CoInductive apply_subrelation : Prop := do_subrelation.
Ltac proper_subrelation :=
match goal with
[ H : apply_subrelation |- _ ] => clear H ; class_apply @subrelation_proper
end.
#[global]
Hint Extern 5 (@Proper _ ?H _) => proper_subrelation : typeclass_instances.
Essential subrelation instances for iff, impl and pointwise_relation.
#[global]
Instance iff_impl_subrelation : subrelation iff impl | 2.
#[global]
Instance iff_flip_impl_subrelation : subrelation iff (flip impl) | 2.
We use an extern hint to help unification.
#[global]
Hint Extern 4 (subrelation (@forall_relation ?A ?B ?R) (@forall_relation _ _ ?S)) =>
apply (@forall_subrelation A B R S) ; intro : typeclass_instances.
Section GenericInstances.
Let U := Type.
Context {A B C : U}.
We can build a PER on the Coq function space if we have PERs on the domain and codomain.
Program Instance respectful_per `(PER A R, PER B R') : PER (R ==> R').
The complement of a relation conserves its proper elements.
Program Definition complement_proper
`(mR : Proper (A -> A -> Prop) (RA ==> RA ==> iff) R) :
Proper (RA ==> RA ==> iff) (complement R) := _.
The flip too, actually the flip instance is a bit more general.
Program Definition flip_proper
`(mor : Proper (A -> B -> C) (RA ==> RB ==> RC) f) :
Proper (RB ==> RA ==> RC) (flip f) := _.
Every Transitive relation gives rise to a binary morphism on impl, contravariant in the first argument, covariant in the second.
Proper declarations for partial applications.
Every Transitive relation induces a morphism by "pushing" an R x y on the left of an R x z proof to get an R y z goal.
Every Symmetric and Transitive relation gives rise to an equivariant morphism.
Lemma symmetric_equiv_flip `(Symmetric A R) : relation_equivalence R (flip R).
Global Instance reflexive_eq_dom_reflexive `{Reflexive B R'}:
Reflexive (respectful (@Logic.eq A) R').
respectful is a morphism for relation equivalence.
Global Instance respectful_morphism :
Proper (relation_equivalence ++> relation_equivalence ++> relation_equivalence)
(@respectful A B).
R is Reflexive, hence we can build the needed proof.
Lemma Reflexive_partial_app_morphism `(Proper (A -> B) (R ==> R') m, ProperProxy A R x) :
Proper R' (m x).
Lemma flip_respectful (R : relation A) (R' : relation B) :
relation_equivalence (flip (R ==> R')) (flip R ==> flip R').
Treating flip: can't make them direct instances as we need at least a flip present in the goal.
Lemma flip1 `(subrelation A R' R) : subrelation (flip (flip R')) R.
Lemma flip2 `(subrelation A R R') : subrelation R (flip (flip R')).
That's if and only if
Lemma eq_subrelation `(Reflexive A R) : subrelation (@eq A) R.
Once we have normalized, we will apply this instance to simplify the problem.
Definition proper_flip_proper `(mor : Proper A R m) : Proper (flip R) m := mor.
Lemma proper_eq (x : A) : Proper (@eq A) x.
End GenericInstances.
Class PartialApplication.
CoInductive normalization_done : Prop := did_normalization.
Class Params {A : Type} (of : A) (arity : nat).
Ltac partial_application_tactic :=
let rec do_partial_apps H m cont :=
match m with
| ?m' ?x => class_apply @Reflexive_partial_app_morphism ;
[(do_partial_apps H m' ltac:(idtac))|clear H]
| _ => cont
end
in
let rec do_partial H ar m :=
lazymatch ar with
| 0%nat => do_partial_apps H m ltac:(fail 1)
| S ?n' =>
match m with
?m' ?x => do_partial H n' m'
end
end
in
let params m sk fk :=
(let m' := fresh in head_of_constr m' m ;
let n := fresh in evar (n:nat) ;
let v := eval compute in n in clear n ;
let H := fresh in
assert(H:Params m' v) by (subst m'; once typeclasses eauto) ;
let v' := eval compute in v in subst m';
(sk H v' || fail 1))
|| fk
in
let on_morphism m cont :=
params m ltac:(fun H n => do_partial H n m)
ltac:(cont)
in
match goal with
| [ _ : normalization_done |- _ ] => fail 1
| [ _ : @Params _ _ _ |- _ ] => fail 1
| [ |- @Proper ?T _ (?m ?x) ] =>
match goal with
| [ H : PartialApplication |- _ ] =>
class_apply @Reflexive_partial_app_morphism; [|clear H]
| _ => on_morphism (m x)
ltac:(class_apply @Reflexive_partial_app_morphism)
end
end.
Bootstrap !!!
#[global]
Instance proper_proper {A} : Proper (relation_equivalence ==> eq ==> iff) (@Proper A).
Ltac proper_reflexive :=
match goal with
| [ _ : normalization_done |- _ ] => fail 1
| _ => class_apply proper_eq || class_apply @reflexive_proper
end.
#[global]
Hint Extern 1 (subrelation (flip _) _) => class_apply @flip1 : typeclass_instances.
#[global]
Hint Extern 1 (subrelation _ (flip _)) => class_apply @flip2 : typeclass_instances.
#[global]
Hint Extern 1 (Proper _ (complement _)) => apply @complement_proper
: typeclass_instances.
#[global]
Hint Extern 1 (Proper _ (flip _)) => apply @flip_proper
: typeclass_instances.
#[global]
Hint Extern 2 (@Proper _ (flip _) _) => class_apply @proper_flip_proper
: typeclass_instances.
#[global]
Hint Extern 4 (@Proper _ _ _) => partial_application_tactic
: typeclass_instances.
#[global]
Hint Extern 7 (@Proper _ _ _) => proper_reflexive
: typeclass_instances.
Special-purpose class to do normalization of signatures w.r.t. flip.
Section Normalize.
Context (A : Type).
Class Normalizes (m : relation A) (m' : relation A) : Prop :=
normalizes : relation_equivalence m m'.
Current strategy: add flip everywhere and reduce using subrelation afterwards.
Lemma proper_normalizes_proper `(Normalizes R0 R1, Proper A R1 m) : Proper R0 m.
Lemma flip_atom R : Normalizes R (flip (flip R)).
End Normalize.
Lemma flip_arrow {A : Type} {B : Type}
`(NA : Normalizes A R (flip R'''), NB : Normalizes B R' (flip R'')) :
Normalizes (A -> B) (R ==> R') (flip (R''' ==> R'')%signature).
Ltac normalizes :=
match goal with
| [ |- Normalizes _ (respectful _ _) _ ] => class_apply @flip_arrow
| _ => class_apply @flip_atom
end.
Ltac proper_normalization :=
match goal with
| [ _ : normalization_done |- _ ] => fail 1
| [ _ : apply_subrelation |- @Proper _ ?R _ ] =>
let H := fresh "H" in
set(H:=did_normalization) ; class_apply @proper_normalizes_proper
end.
#[global]
Hint Extern 1 (Normalizes _ _ _) => normalizes : typeclass_instances.
#[global]
Hint Extern 6 (@Proper _ _ _) => proper_normalization
: typeclass_instances.
When the relation on the domain is symmetric, we can flip the relation on the codomain. Same for binary functions.
Lemma proper_sym_flip :
forall `(Symmetric A R1)`(Proper (A->B) (R1==>R2) f),
Proper (R1==>flip R2) f.
Lemma proper_sym_flip_2 :
forall `(Symmetric A R1)`(Symmetric B R2)`(Proper (A->B->C) (R1==>R2==>R3) f),
Proper (R1==>R2==>flip R3) f.
When the relation on the domain is symmetric, a predicate is compatible with iff as soon as it is compatible with impl. Same with a binary relation.
Lemma proper_sym_impl_iff : forall `(Symmetric A R)`(Proper _ (R==>impl) f),
Proper (R==>iff) f.
Lemma proper_sym_impl_iff_2 :
forall `(Symmetric A R)`(Symmetric B R')`(Proper _ (R==>R'==>impl) f),
Proper (R==>R'==>iff) f.
A PartialOrder is compatible with its underlying equivalence.
#[global]
Instance PartialOrder_proper `(PartialOrder A eqA R) :
Proper (eqA==>eqA==>iff) R.
From a PartialOrder to the corresponding StrictOrder: lt = le /\ ~eq. If the order is total, we could also say gt = ~le.
Lemma PartialOrder_StrictOrder `(PartialOrder A eqA R) :
StrictOrder (relation_conjunction R (complement eqA)).
From a StrictOrder to the corresponding PartialOrder: le = lt \/ eq. If the order is total, we could also say ge = ~lt.
Lemma StrictOrder_PreOrder
`(Equivalence A eqA, StrictOrder A R, Proper _ (eqA==>eqA==>iff) R) :
PreOrder (relation_disjunction R eqA).
#[global]
Hint Extern 4 (PreOrder (relation_disjunction _ _)) =>
class_apply StrictOrder_PreOrder : typeclass_instances.
Lemma StrictOrder_PartialOrder
`(Equivalence A eqA, StrictOrder A R, Proper _ (eqA==>eqA==>iff) R) :
PartialOrder eqA (relation_disjunction R eqA).
#[global]
Hint Extern 4 (StrictOrder (relation_conjunction _ _)) =>
class_apply PartialOrder_StrictOrder : typeclass_instances.
#[global]
Hint Extern 4 (PartialOrder _ (relation_disjunction _ _)) =>
class_apply StrictOrder_PartialOrder : typeclass_instances.