Require Export RelDefinitions.
(** * Relators *)
(** With this infrastructure in place, we can define actual relators that cover the commonly used type constructors. There are two broad categories: those related to function types, and those derived from inductive types. *)
(** As a convention, we name relators and the associated monotonicity theorems by appending the suffix [_rel] to the name of original type and those of its constructors. Likewise, we use the suffix [_subrel] for monotonicity theorems that characterize the variance of corresponding relators with respect to [subrel], and the suffix [_relim] for an associated instance of [RElim]. *)
(** ** A note about universes *)
(** Our use of [subrel] illustrates the fact that we consider relations over relations, and use relators to build up relations over the types of our relators to characterize their variance. In other words: we relate all kind of higher-order things as a matter of course and make sure our library eats its own dog food. This makes it relatively easy to run into universe inconsistency issues. This note is to explain why those happen, fix the terminology, and formulate best practices to avoid such problems.  The critical universe level in Coqrel is the universe [U] from which the arguments and return type of [rel] are taken (so that conceptually we will write [rel : U -> U -> U]). We will call "small types" the inhabitants of [U], and "large types" those taken from a universe strictly larger than [U]. At the moment, [U] is implicit, but at some point we may want declare an explicit [Universe] to aid debugging and enhance clarity.  Generally speaking, [U] is a pretty large universe: for the most part, the world of relational things sits well above the world of the things we want relate, no matter how high-order those are. Even [rel A B] is still a small type, so that [@subrel : forall A B : U, rel (rel A B) (rel A B)] above didn't cause any issue, although the type of [rel] itself, namely [U -> U -> U : U+1], is of course a large type.  With this in mind, consider the relational operator [rel_compose : forall A B C, rel A B -> rel B C -> rel A C]. We wish to register a monotonicity property for [rel_compose], establishing that composing larger relations yields a larger composite relation. As usual, we express this property by establishing that [rel_compose] is related to itself by an appropriate relation. However, the type of [rel_compose] may involve [U : U+1] as the type of the arguments [A B C], so that the type of [rel_compose] may itself be forced into sort [U+1]. Consequently, attempting to define that relation will yield a universe inconsistency as soon as, say, [subrel] is used as an argument of [rel_compose].  Fortunately, even if [A B C : U], the specialized form [rel_compose A B C] is still in the small type [rel A B -> rel B C -> rel A C : U]. Moreover, there is no relational structure that we want to capture regarding the type parameters [A B C], so that we don't lose anything by defining our monotonicity property for the specialized [rel_compose A B C], as long as the property itself is generic in the specific values of [A B C].  This is why, as a general rule, we define the [foo_subrel] properties of our relators on partially applied versions thereof, and use a [foo_subrel_params] hint to make sure our monotonicity tactic can resolve them. *)
(** ** Relators for function types *)
(** *** Pointwise extension of a relation *)
(** One useful special case is the pointwise extension of a relation on the domain to the function type. This is equivalent to [eq ==> R], however with the formulation below we don't have consider two equal elements of the domain. *)
Definition arrow_pointwise_rel A {B1 B2}:
rel B1 B2 -> rel (A -> B1) (A -> B2) :=
fun RB f g => forall a, RB (f a) (g a).
Arguments arrow_pointwise_rel A%type {B1%type B2%type} RB%rel _ _.
Notation "- ==> R" := (arrow_pointwise_rel _ R)
(at level 55, right associativity) : rel_scope.
Global Instance arrow_pointwise_subrel {A B1 B2}:
Monotonic (@arrow_pointwise_rel A B1 B2) (subrel ++> subrel).
Global Instance arrow_pointwise_subrel_params:
Params (@arrow_pointwise_rel) 3.
Defined.
Hint Extern 0 (RIntro _ (- ==> _) _ _) =>
eapply arrow_pointwise_rintro : typeclass_instances.
(** Note that although the elimination rule could use a single variable and dispense with the equality premise, it actually uses two distinct variables [m] and [n], and a premise in [P] that [m = n]. Hence the effect is similar to the elimination rule for the equivalent relation [eq ==> R]. We do this because in contexts where we have a single term [t] against which both [m] and [n] unify, it is easy and cheap to automatically prove that [t = t], but in contexts where we have two disctinct variables and a proof of equality, it is more involved to massage the goal into a form where the single-variable version of the rule could apply. *)
Hint Extern 1 (RElim (- ==> _) _ _ _ _) =>
eapply arrow_pointwise_relim : typeclass_instances.
Hint Extern 1 (Reflexive (- ==> _)) =>
eapply arrow_pointwise_refl : typeclass_instances.
Global Instance arrow_pointwise_rel_compose {T} `(RCompose) :
RCompose (A := T -> A) (- ==> RAB) (- ==> RBC) (- ==> RAC).
(** *** Dependent pointwise extension *)
(** Like we did for non-dependent functions, we can provide a simpler definition for the special case where [E] is [eq]. *)
Definition forall_pointwise_rel {V: Type} {FV1 FV2: V -> Type}:
(forall v, rel (FV1 v) (FV2 v)) ->
rel (forall v, FV1 v) (forall v, FV2 v) :=
fun FE f g =>
forall v, FE v (f v) (g v).
Arguments forall_pointwise_rel {V%type FV1%type FV2%type} FE%rel _ _.
Notation "'forallr' - @ v : V , FE" :=
(forall_pointwise_rel (V := V) (fun v => FE))
(v ident, at level 200).
Notation "'forallr' - @ v , FE" :=
(forall_pointwise_rel (fun v => FE))
(v ident, at level 200).
Notation "'forallr' - : V , FE" :=
(forall_pointwise_rel (V := V) (fun _ => FE))
(at level 200).
Notation "'forallr' - , FE" :=
(forall_pointwise_rel (fun _ => FE))
(at level 200).
Hint Extern 0 (RIntro _ (forall_pointwise_rel _) _ _) =>
eapply forall_pointwise_rintro : typeclass_instances.
Hint Extern 1 (RElim (forall_pointwise_rel _) _ _ _ _) =>
eapply forall_pointwise_relim : typeclass_instances.
(** *** Dependent products (restricted version) *)
(** This is a restricted version of [forall_rel] with [E] in [Prop], when the codomain relation only depends on the arguments. *)
Definition forallp_rel {V1 V2} (E: rel V1 V2) {FV1: V1->Type} {FV2: V2->Type}:
(forall v1 v2, rel (FV1 v1) (FV2 v2)) ->
rel (forall v1, FV1 v1) (forall v2, FV2 v2) :=
fun FE f g =>
forall v1 v2, E v1 v2 -> FE v1 v2 (f v1) (g v2).
Arguments forallp_rel {V1%type V2%type} E%rel {FV1%type FV2%type} FE%rel _ _.
Notation "'forallr' v1 v2 : E , R" :=
(forallp_rel E (fun v1 v2 => R))
(at level 200, v1 ident, v2 ident, right associativity)
: rel_scope.
Hint Extern 0 (RIntro _ (forallp_rel _ _) _ _) =>
eapply forallp_rintro : typeclass_instances.
(** Since [e : E v1 v2] cannot be unified in [Q], the elimination rule adds an [E v1 v2] premise to [P] instead. *)
Hint Extern 1 (RElim (forallp_rel _ _) _ _ _ _) =>
eapply forallp_relim : typeclass_instances.
(** TODO: A further specialization could be [foralln_rel], which does not even require that [v1], [v2] be related (let alone care about the proof). In other words, a [forallr v1 v2, R] which would essentially boil down to [forallr v1 v2 : ⊤, R]. *)
(** *** Sets ([A -> Prop]) *)
(** Sets of type [A -> Prop] can be related using the regular arrow relator, as in [R ++> impl]. This states that for any two elements related by [R], if the first is in the first set, then the second must be in the second set.  However, very often we need the following relator instead, which states that for any element of the first set, there exists an element of the second set which is related to it. This is useful for example when defining simulation diagrams. *)
Definition set_le {A B} (R: rel A B): rel (A -> Prop) (B -> Prop) :=
fun sA sB => forall a, sA a -> exists b, sB b /\ R a b.
Global Instance set_le_subrel {A B}:
Monotonic (@set_le A B) (subrel ++> subrel).
Global Instance set_le_subrel_params:
Params (@set_le) 3.
Defined.
Hint Extern 1 (Reflexive (set_le _)) =>
eapply set_le_refl : typeclass_instances.
Global Instance set_le_compose `(RCompose) :
RCompose (set_le RAB) (set_le RBC) (set_le RAC).
(** We define [set_ge] as well. *)
Definition set_ge {A B} (R: rel A B): rel (A -> Prop) (B -> Prop) :=
fun sA sB => forall b, sB b -> exists a, sA a /\ R a b.
Global Instance set_ge_subrel {A B}:
Monotonic (@set_ge A B) (subrel ++> subrel).
Global Instance set_ge_subrel_params:
Params (@set_ge) 3.
Defined.
Hint Extern 1 (Reflexive (set_ge _)) =>
eapply set_ge_refl : typeclass_instances.
Global Instance set_ge_compose `(RCompose) :
RCompose (set_ge RAB) (set_ge RBC) (set_ge RAC).
(** ** Inductive types *)
(** For inductive types, there is a systematic way of converting their definition into that of the corresponding relator. Where the original inductive definition quantifies over types, the corresponding relator will quantify over pairs of types and relations between them. Then, the constructors of the relator will essentially be [Proper] instances for the original constructors. In other words, the resulting relation will be the smallest one such that the constructors are order-preserving. *)
(** *** Nullary type constructors *)
(** As a proof-of-concept, we start with the most elementary types [Empty_set] and [unit], which can be considered as nullary type constructors related to [sum] and [prod] below. *)
Inductive Empty_set_rel: rel Empty_set Empty_set := .
Inductive unit_rel: rel unit unit :=
tt_rel: Proper unit_rel tt.
Global Existing Instance tt_rel.
(** *** Sum types *)
(** The definition of [sum_rel] could look something like this: << Inductive sum_rel: forall {A1 A2 B1 B2}, rel A1 A2 -> rel B1 B2 -> rel (A1+B1) (A2+B2):= | inl_rel: Proper (∀ RA : rel, ∀ RB : rel, RA ++> sum_rel RA RB) (@inl) | inr_rel: Proper (∀ RA : rel, ∀ RB : rel, RB ++> sum_rel RA RB) (@inr). >> However, to minimize the need for [inversion]s we want to keep as many arguments as possible as parameters of the inductive type. *)
Inductive sum_rel {A1 A2 B1 B2} RA RB: rel (A1 + B1)%type (A2 + B2)%type :=
| inl_rel_def: (RA ++> sum_rel RA RB)%rel (@inl A1 B1) (@inl A2 B2)
| inr_rel_def: (RB ++> sum_rel RA RB)%rel (@inr A1 B1) (@inr A2 B2).
Infix "+" := sum_rel : rel_scope.
(** Since it is not possible to retype the constructors after the fact, we use the [_def] suffix when defining them, then redeclare a corresponding, full-blown [Proper] instance. *)
Global Instance inl_rel:
Monotonic (@inl) (forallr RA, forallr RB, RA ++> RA + RB).
Global Instance inr_rel:
Monotonic (@inr) (forallr RA, forallr RB, RB ++> RA + RB).
Global Instance sum_subrel {A1 A2 B1 B2}:
Monotonic (@sum_rel A1 A2 B1 B2) (subrel ++> subrel ++> subrel).
Global Instance sum_subrel_params:
Params (@sum_rel) 4.
Defined.
Hint Extern 2 (Reflexive (_ + _)) =>
eapply sum_rel_refl : typeclass_instances.
Hint Extern 2 (Coreflexive (_ + _)) =>
eapply sum_rel_corefl : typeclass_instances.

Lemma sum_rel_trans {A B} (R1: rel A A) (R2: rel B B):
Transitive R1 -> Transitive R2 -> Transitive (R1 + R2).