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# Closure Typing

## Region sharing

Consider the following function:

f () = do x = 5 g () = x g

Without closure information this function would have the following extended type:

f :: forall %r. () -> () -> Int %r

Remember that the `forall %r` at the front of the type is supposed to indicate that the return value is freshly allocated. This is certainly true if we apply both arguments:

twoSeparateInts :: Tuple2 %r1 (Int %r2, Int %r3) twoSeparateInts = (f () (), f () ())

In the type of `twoSeparateInts`, the different regions variables on each of the `Int` constructors means that they do not alias. This in turn means that it is safe to treat one as `Const` and the other as `Mutable`.

But what happens if we partially apply `f`? The standard type system will re-generalize the type for the new binding and we're left with:

f_unit :: forall %r1. () -> Int %r1 f_unit = f ()

We've now got a function which returns the *same* `Int` every time we call it, but the type says it's supposed to be fresh! The problem here is that `x` was free in our original definition of `g` so is shared between applications of it.

## Closure typing for region sharing

Closure typing is used to track the sharing of regions between function calls like this one.

The type inferred for `f` is actually:

f :: forall %r0 . () -> () -($c0)> Int %r0 :- $c0 = x : %r0

The `$c0` annotation tells us that this function has a object free in its closure, and that the object contains region `%r0`. The `x : %r0` syntax gives a name to this term, and is pronounced "x *of* region zero".

Similarly to variables in the expression language, the exact names given to closure terms is not significant to the type system. However, to make inferred types easier to read, DDC names the closure terms in a function's type after the free variables that gave rise to those terms. The user is free to use whatever names they wish in type signatures. Two closure terms with different names but identical right hand sides are taken as being equivalent.

If we use this new type and apply the first argument we have:

f_unit :: () -($c0)> Int %r0 :- $c0 = x : %r0

This type says that `f_unit` is a function that takes a unit value and returns an <hask>Int</hask>, but the <hask>Int</hask> is free in its closure so is shared by all calls to it. The type system does not generalise regions which are free in these closures.