Map preserves uniqueness of labels.

Ap preserves uniqueness of labels.

ApSingle preserves uniqueness of labels.

Zip preserves uniqueness of labels.

If we know that r has been extended with l .== t, then we know that this extension at the label l must be t.

This allows us to derive Map f r .! l ≈ f t from r .! l ≈ t

This allows us to derive Ap ϕ ρ .! l ≈ f t from ϕ .! l ≈ f and ρ .! l ≈ t

This allows us to derive ApSingle r x .! l ≈ f x from r .! l ≈ f

Proof that the Map type family preserves labels and their ordering.

Proof that the Ap type family preserves labels and their ordering.

Proof that the ApSingle type family preserves labels and their ordering.

Proof that the Ap type family preserves labels and their ordering.

Map distributes over MinJoin

ApSingle distributes over MinJoin

FreeForall can be used when a Forall constraint is necessary but there is no particular constraint we care about.

FreeForall can be used when a BiForall constraint is necessary but there is no particular constraint we care about.

Allow any Forall over a row-type, be usable for Unconstrained1.

This allows us to derive a Forall (Map f r) .. from a Forall r ...

This allows us to derive a Forall (ApSingle f r) .. from a Forall f ...

Two rows are subsets of a third if and only if their disjoint union is a subset of that third.

If two rows are each subsets of a third, their join is a subset of the third

If a row is a subset of another, then its restriction is also a subset of the other

Subset is transitive

Map distributes over Difference

ApSingle distributes over Difference

A class to capture the idea of As so that it can be partially applied in a context.

This data type is used to for its ability to existentially bind a type variable. Particularly, it says that for the type a, there exists a t such that a ~ f t and c t holds.

A class to capture the idea of As' so that it can be partially applied in a context.

Like As, but here we know the underlying value is some f applied to the given type a.

Values of type Dict p capture a dictionary for a constraint of type p.

e.g.

Dict :: Dict (Eq Int)

captures a dictionary that proves we have an:

instance Eq Int

Pattern matching on the Dict constructor will bring this instance into scope.

This is the type of entailment.

a :- b is read as a "entails" b.

With this we can actually build a category for Constraint resolution.

e.g.

Because Eq a is a superclass of Ord a, we can show that Ord a entails Eq a.

Because instance Ord a => Ord [a] exists, we can show that Ord a entails Ord [a] as well.

This relationship is captured in the :- entailment type here.

Since p :- p and entailment composes, :- forms the arrows of a Category of constraints. However, Category only became sufficiently general to support this instance in GHC 7.8, so prior to 7.8 this instance is unavailable.

But due to the coherence of instance resolution in Haskell, this Category has some very interesting properties. Notably, in the absence of IncoherentInstances, this category is "thin", which is to say that between any two objects (constraints) there is at most one distinguishable arrow.

This means that for instance, even though there are two ways to derive Ord a :- Eq [a], the answers from these two paths _must_ by construction be equal. This is a property that Haskell offers that is pretty much unique in the space of languages with things they call "type classes".

What are the two ways?

Well, we can go from Ord a :- Eq a via the superclass relationship, and then from Eq a :- Eq [a] via the instance, or we can go from Ord a :- Ord [a] via the instance then from Ord [a] :- Eq [a] through the superclass relationship and this diagram by definition must "commute".

Diagrammatically,

                   Ord a
               ins /     \ cls
                  v       v
            Ord [a]     Eq a
               cls \     / ins
                    v   v
                   Eq [a]

This safety net ensures that pretty much anything you can write with this library is sensible and can't break any assumptions on the behalf of library authors.

Witnesses that a value of type e contains evidence of the constraint c.

Mainly intended to allow (\\) to be overloaded, since it's a useful operator.

Operator version of withDict, with the arguments flipped

From a Dict, takes a value in an environment where the instance witnessed by the Dict is in scope, and evaluates it.

Essentially a deconstruction of a Dict into its continuation-style form.

Can also be used to deconstruct an entailment, a :- b, using a context a.

withDict :: Dict c -> (c => r) -> r
withDict :: a => (a :- c) -> (c => r) -> r

A null constraint

A null constraint of one argument

A null constraint of two arguments