A Reduction is the result of an operation that rewrites a type ty_in. The Reduction includes the rewritten type ty_out and a Coercion co such that co :: ty_in ~ ty_out, where the role of the coercion is determined by the context. That is, the LHS type of the coercion is the original type ty_in, while its RHS type is the rewritten type ty_out.

A Reduction is always homogeneous, unless it is wrapped inside a HetReduction, which separately stores the kind coercion.

See Note [The Reduction type].

A Reduction in which the Coercion has Nominal role.

A Reduction in which the Coercion has Representational role.

Stores a heterogeneous reduction.

The stored kind coercion must relate the kinds of the stored reduction. That is, in HetReduction (Reduction co xi) kco, we must have:

 co :: ty ~ xi
kco :: typeKind ty ~ typeKind xi

A collection of Reductions where the coercions and the types are stored separately.

Use unzipRedns to obtain Reductions from a list of Reductions.

This datatype is used in mkAppRedns, mkClassPredRedns and mkTyConAppRedn, which expect separate types and coercions.

Invariant: the two stored lists are of the same length, and the RHS type of each coercion is the corresponding type.

Create a Reduction from a pair of a Coercion and a 'Type.

Pre-condition: the RHS type of the coercion matches the provided type (perhaps up to zonking).

Use coercionRedn when you only have the coercion.

Create Reductions from individual lists of coercions and types.

The lists should be of the same length, and the RHS type of each coercion should match the specified type in the other list.

Create a heterogeneous reduction.

Pre-condition: the provided kind coercion (second argument) relates the kinds of the stored reduction. That is, if the coercion stored in the Reduction is of the form

co :: ty ~ xi

Then the kind coercion supplied must be of the form:

kco :: typeKind ty ~ typeKind xi

Turn a Coercion into a Reduction by inspecting the RHS type of the coercion.

Prefer using mkReduction when you already know the RHS type of the coercion, to avoid computing it anew.

Get the original, unreduced type corresponding to a Reduction.

This is obtained by computing the LHS kind of the stored coercion, which may be slow.

Downgrade the role of the coercion stored in the Reduction.

Downgrade the role of the coercion stored in the Reduction, from Nominal to Representational.

Compose a reduction with a coercion on the left.

Pre-condition: the provided coercion's RHS type must match the LHS type of the coercion that is stored in the reduction.

Apply a cast to the result of a Reduction.

Given a Reduction ty1 ~co1~> (ty2 :: k2) and a kind coercion kco with LHS kind k2, produce a new Reduction ty1 ~co2~> ( ty2 |> kco ) of the given Role (which must match the role of the coercion stored in the Reduction argument).

Apply a cast to the result of a Reduction, using an MCoercionN.

Given a Reduction ty1 ~co1~> (ty2 :: k2) and a kind coercion mco with LHS kind k2, produce a new Reduction ty1 ~co2~> ( ty2 |> mco ) of the given Role (which must match the role of the coercion stored in the Reduction argument).

Apply a cast to a Reduction, casting both the original and the reduced type.

Given cast_co and Reduction ty ~co~> xi, this function returns the Reduction (ty |> cast_co) ~return_co~> (xi |> cast_co) of the given Role (which must match the role of the coercion stored in the Reduction argument).

Pre-condition: the Type passed in is the same as the LHS type of the coercion stored in the Reduction.

Apply casts on both sides of a Reduction (of the given Role).

Use mkCastRedn1 when you want to cast both the original and reduced types in a Reduction using the same coercion.

Pre-condition: the Type passed in is the same as the LHS type of the coercion stored in the Reduction.

The reflexive reduction.

Create a Reduction from a kind cast, in which the casted type is the rewritten type.

Given ty :: k1, mco :: k1 ~ k2, produces the Reduction ty ~res_co~> (ty |> mco) at the given Role.

Create a Reduction from a kind cast, in which the casted type is the rewritten type.

Given ty :: k1, mco :: k1 ~ k2, produces the Reduction ty ~res_co~> (ty |> mco) at the given Role.

Create a Reduction from a kind cast, in which the casted type is the original (non-rewritten) type.

Given ty :: k1, mco :: k1 ~ k2, produces the Reduction (ty |> mco) ~res_co~> ty at the given Role.

Create a Reduction from a kind cast, in which the casted type is the original (non-rewritten) type.

Given ty :: k1, mco :: k1 ~ k2, produces the Reduction (ty |> mco) ~res_co~> ty at the given Role.

Apply one Reduction to another.

Combines mkAppCo and mkAppTy.

Combines mkAppCos and mkAppTys.

Create a function Reduction.

Combines mkFunCo and mkFunTy.

Create a Reduction associated to a Π type, from a kind Reduction and a body Reduction.

Combines mkForAllCo and mkForAllTy.

Create a Reduction of a quantified type from a Reduction of the body.

Combines mkHomoForAllCos and mkForAllTys.

TyConAppCo for Reductions: combines mkTyConAppCo and mkTyConApp.

Reduce the arguments of a Class TyCon.

Create a Reduction from a coercion between coercions.

Combines mkProofIrrelCo and mkCoercionTy.

Create a reflexive Reduction whose RHS is the given Coercion, with the specified Role.

Homogenise a heterogeneous reduction.

Given HetReduction (Reduction co xi) kco, with

 co :: ty ~ xi
kco :: typeKind(ty) ~ typeKind(xi)

this returns the homogeneous reduction:

hco :: ty ~ ( xi |> sym kco )

Obtain Reductions from a list of Reductions by unzipping.

Stores Reductions as well as a kind coercion.

Used when rewriting arguments to a type function f.

Invariant: when the stored reductions are of the form co_i :: ty_i ~ xi_i, the kind coercion is of the form kco :: typeKind (f ty_1 ... ty_n) ~ typeKind (f xi_1 ... xi_n)

The type function f depends on context.