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. *)

Lemma arrow_pointwise_relim {A B1 B2} (R: rel B1 B2) f g (m n: A) P Q:
RElim R (f m) (g n) P Q ->
RElim (- ==> R) f g (m = n /\ P) Q.