Typed closure conversion

Typed closure conversion is a really lovely explanation of the structure of a closure-converted function. Every closure has its own particular environment structure that binds the particular set of free variables that occur in its body. But this means that if you represent a closure directly as a pair of an environment record and a closed procedure (which is explicitly parameterized over the environment record), then the representations of functions that are supposed to have the same type can end up with incompatible environment types. For example, the program

let val y = 1
in
  if true then
      λx.x+y
  else
      λz.z
end

converts naively to the transparent representation

let val y = 1
in
  if true then
      (λ(env, x).x+#y(env), {y=y})
  else
      (λ(env,z).z, {})
end

The problem here is the type of the environment has leaked information about the internals of the procedure into the external interface. So Minamide et al use existential types as a nice, lightweight type abstraction for hiding the environment layout as an abstract type. So you instead get the opaque representation

let val y = 1
in
  if true then
      pack {y:int} with (λ(env,x).x+#y(env), {y=y})
      as ∃t.((t × int) → int) × t
  else
      pack {} with (λ(env,z).z, {})
      as ∃t.((t × int) → int) × t
end

Update: fixed the types in the existentials-- since I'm using an uncurried representation the argument type is a pair, not an arrow type.