# Nearly-universal quantification

[polymorphism] [subtyping]

This issue is about an interaction between polymorphism and constrained type constructors. These are a feature of several type systems allowing the definition of a type constructor $T[-]$, where $T[A]$ is a valid type only for certain $A$.

This feature is different from GADTs (indexed types), which allow the definition of a type constructor $T[-]$ where $T[A]$ is always a valid type, but one which is only inhabited for certain $A$.

To explain this distinction, here's a GADT $\n{G}$ and a constrained type $\n{C}$:

type 'a g = G_int : { name : string } -> int g

type 'a c = { name: string } constraint 'a = int

sealed trait G[T]
case class G_int(i: Int) extends G[Int]

class C[A >: Int <: Int](name : String)


The type $\n{C}[A]$ is only legal when $A = \n{Int}$, whereas $\n{G}[A]$ is always valid, but is an empty type unless $A = \n{Int}$:

type ok = string g

type bad = string c
(* Error: This type string should be an instance of type int *)

type Ok = G[String]

type Bad = C[String]
// Error: type arguments [A] do not conform to
// class C's type parameter bounds [A >: Int <: Int]


With constrained types, there are two different ways to interpret a polymorphic type such as $∀α. \n{C}[α] → \n{String}$:

1. $∀α.;\dots$ really means "for all $α$", so the type above is invalid since $\n{C}[α]$ is not a valid type for all $α$.

2. $∀α.;\dots$ means "for all $α$ for which the right-hand side is valid", so the type above is valid, but can only be used with $α = \n{Int}$.

Both OCaml and Scala choose option (1):

let poly : 'a . 'a c -> string =
fun _ -> "hello"
(* Error: The universal type variable 'a cannot be generalized:
it is bound to int. *)

def poly[A](x: C[A]): String = "hello"
// Error: type arguments [A] do not conform to
// class C's type parameter bounds [A >: Int <: Int]


Option (2) requires slightly less annotation by the programmer, but it can be easy to lose track of the implicit constraints on $α$. Rust chooses this option, which led to a tricky bug 1.

In Rust, the type &'a T means a reference to type T which is valid for lifetime 'a. Lifetimes are ordered by a subtyping relation 'a: 'b (pronounced "'a outlives 'b"), if the lifetime 'a contains all of 'b. References to references &'a &'b T are valid, but only if 'b: 'a as the reference's lifetime must not outlive its contents.

There is also a longest lifetime 'static, such that 'static: 'a for any lifetime 'a. The bug allows any reference to be converted to a 'static reference, breaking soundness guarantees:

// Counterexample by Aaron Turon
static UNIT: &'static &'static () = &&();

fn foo<'a, 'b, T>(_: &'a &'b (), v: &'b T) -> &'a T { v }

fn bad<'c, T>(x: &'c T) -> &'static T {
let f: fn(&'static &'c (), &'c T) -> &'static T = foo;
f(UNIT, x)
}


The issue is that the type of foo uses &'a &'b (), which is a well-formed type only if 'b: 'a, as a reference cannot outlive its contents. This constraint 'b: 'a is relied upon, to allow v to be returned.

Since Rust uses option (2) above, this constraint does not appear explicitly in the type of foo:

fn foo<'a, 'b, T> (_: &'a &'b (), v: &'b T) -> &'a T


Contravariance allows arguments to be passed with a longer lifetime than required by the function, allowing foo to be used at the following type, where one argument lifetime has been extended to 'static:

fn foo<'a, 'b, T> (_: &'a &'static (), v: &'b T) -> &'a T


However, the implicit constraint 'b: 'a has been lost, as validity now requires only that 'static: 'a, which is always true. This allows the type to be instantiated with 'a := 'static, 'b := 'c:

fn foo<T> (_: &'static &'static (), v: &'c T) -> &'static T


which is unsound.