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# Type Functions: Desugaring

## Desugaring indexed data types

The kind signature of an indexed data type

data T (a1::<kind1>) .. (an::<kindn>) :: <kind>

turns into an F_{C} type function declaration

type T_n : <kind1> -> .. -> <kindn> -> <kind>

A member of an indexed data type

data T t1 .. tn b1 .. bm = <constructors>

turns into an equality axiom and a vanilla data declaration

axiom cTinst : (forall c1..cr. T_n t1 .. tn) :=: (forall c1..cr. Tinst c1 .. cr) data Tinst c1 .. cr b1 .. bm = <constructors>

where the `ci` are the free variables of the `tj`. Moreover, we morally replace all occurences of `T` in the rest of the program by `T_n`. No such replacement is required in the actual implementation as the arity index at F_{C} type functions is just a formal device used in the formal development. In the implementation, it is perfectly fine to retain the original name and maintain the arity information separately.

## Inserting coercions

To ensure that the F_{C} code generated by the above desugaring still type checks, we need to introduce cast expressions using `cTinst` to move between the indexed type `T_n` and the representation types, such as `Tinst`, of its members. The simplicity of type checking and desugaring indexed data types - as opposed to general type functions - is due to the locations where these casts need to be added being well defined. More precisely, there are two different kinds of locations corresponding to the introduction and elimination of indexed data types:

- Wrappers for data constructors introduce indexed types.
- Case expressions scrutinising indexed types eliminate them.

## Wrappers for indexed data types

The wrapper of a data constructor acts as an impedance matcher between the source-level signatures of the constructor and its actual representation; in particular, it evaluates strict arguments and unboxes flattened arguments. In the case of a constructor for an indexed data type, it additionally has to apply the coercion between the type function representing the source type and its representation as a vanilla data type. So, for example, if we have (continuing the example from above)

data T t1 .. tn b1 .. bm = C s1 .. sk

then we generate a wrapper

C = /\c1..cr b1..bm -> \x1..xk -> Con C [c1,..,cr,b1,..,bm] [x1,..,xk] |> sym (cTinst@c1..@cr b1 .. bm)

The generation of constructor wrappers is performed by `MkId.mkDataConIds`.

## Case expressions for indexed data types

When we scrutinise an indexed type in a case expression, we need to first cast it to the vanilla data type representing the family member from which the constructors guarding the alternatives are drawn. (This implies that we cannot have any case expression mixing constructors from two or more family members. In fact, if we had that capability, we would have open GADT definitions in the Löh/Hinze sense.)

So, whether we need to cast the scrutinee of a case expression depends on the constructors appearing in the alternatives, which are type checked by `TcPat.tcConPat`. This function uses `TcUnify.boxySplitTyConApp` to match the type of the scrutinee against the result type of the data constructor. In the case of GADTs and indexed types, this is not just a matter of extracting the arguments from the type constructor application, but we need to match against type patterns. This matching is already conveniently performed by the code for GADTs.

If the data constructor is from an indexed type, we need to propagate a coercion (to be applied to the scrutinee) outwards. For this, GHC also already has a mechanism, namely the variant `CoPat` of `HsPat.Pat`. It enables us to attach a coercion function, of type `HsBinds.ExprCoFun`, to a pattern, which the desugarer will pick up in `Match.matchCoercion` and apply to the match variable of the case expression.

`ExprCoFun` represents, besides coercions due to type instantiation, also type equality coercions of type `Coercion.Coercion`. We use them for coercions that are exactly the converse of the coercion used in the wrapper of the data constructor of the current case alternative. (There is also an equivalent of `CoPat` for expressions, namely `HsCoerce` of `HsExpr.HsExpr`.)