# Type Lambdas - More Details

## Syntax

```
Type ::= ... | TypeParamClause ‘=>>’ Type
TypeParamClause ::= ‘[’ TypeParam {‘,’ TypeParam} ‘]’
TypeParam ::= {Annotation} (id [HkTypeParamClause] | ‘_’) TypeBounds
TypeBounds ::= [‘>:’ Type] [‘<:’ Type]
```

### Type Checking

A type lambda such as `[X] =>> F[X]`

defines a function from types to types. The parameter(s) may carry bounds. If a parameter is bounded, as in `[X >: L <: U] =>> F[X]`

it is checked that arguments to the parameters conform to the bounds `L`

and `U`

. Only the upper bound `U`

can be F-bounded, i.e. `X`

can appear in it.

## Subtyping Rules

Assume two type lambdas

```
type TL1 = [X >: L1 <: U1] =>> R1
type TL2 = [X >: L2 <: U2] =>> R2
```

Then `TL1 <: TL2`

, if

- the type interval
`L2..U2`

is contained in the type interval`L1..U1`

(i.e.`L1 <: L2`

and`U2 <: U1`

), `R1 <: R2`

Here we have relied on alpha renaming to match the two bound types `X`

.

A partially applied type constructor such as `List`

is assumed to be equivalent to its eta expansion. I.e, `List = [X] =>> List[X]`

. This allows type constructors to be compared with type lambdas.

## Relationship with Parameterized Type Definitions

A parameterized type definition

```
type T[X] = R
```

is regarded as a shorthand for an unparameterized definition with a type lambda as right-hand side:

```
type T = [X] =>> R
```

If the type definition carries `+`

or `-`

variance annotations, it is checked that the variance annotations are satisfied by the type lambda. For instance,

```
type F2[A, +B] = A => B
```

expands to

```
type F2 = [A, B] =>> A => B
```

and at the same time it is checked that the parameter `B`

appears covariantly in `A => B`

.

A parameterized abstract type

```
type T[X] >: L <: U
```

is regarded as shorthand for an unparameterized abstract type with type lambdas as bounds.

```
type T >: ([X] =>> L) <: ([X] =>> U)
```

However, if `L`

is `Nothing`

it is not parameterized, since `Nothing`

is treated as a bottom type for all kinds. For instance,

```
type T[X] <: X => X
```

is expanded to

```
type T >: Nothing <: ([X] =>> X => X)
```

instead of

```
type T >: ([X] =>> Nothing) <: ([X] =>> X => X)
```

The same expansions apply to type parameters. For instance,

```
[F[X] <: Coll[X]]
```

is treated as a shorthand for

```
[F >: Nothing <: [X] =>> Coll[X]]
```

Abstract types and opaque type aliases remember the variances they were created with. So the type

```
type F2[-A, +B]
```

is known to be contravariant in `A`

and covariant in `B`

and can be instantiated only with types that satisfy these constraints. Likewise

```
opaque type O[X] = List[X]
```

`O`

is known to be invariant (and not covariant, as its right-hand side would suggest). On the other hand, a transparent alias

```
type O2[X] = List[X]
```

would be treated as covariant, `X`

is used covariantly on its right-hand side.

**Note**: The decision to treat `Nothing`

as universal bottom type is provisional, and might be changed after further discussion.

**Note**: Scala 2 and 3 differ in that Scala 2 also treats `Any`

as universal top-type. This is not done in Scala 3. See also the discussion on kind polymorphism

## Curried Type Parameters

The body of a type lambda can again be a type lambda. Example:

```
type TL = [X] =>> [Y] =>> (X, Y)
```

Currently, no special provision is made to infer type arguments to such curried type lambdas. This is left for future work.