# Subtyping and Variance

Subtyping is implicit and can occur at any stage in type checking or inference. Subtyping in Rust is very restricted and occurs only due to variance with respect to lifetimes and between types with higher ranked lifetimes. If we were to erase lifetimes from types, then the only subtyping would be due to type equality.

Consider the following example: string literals always have `'static` lifetime. Nevertheless, we can assign `s` to `t`:

    #![allow(unused_variables)]
    fn main() {
    fn bar<'a>() {
        let s: &'static str = "hi";
        let t: &'a str = s;
    }
    }

Since `'static` outlives the lifetime parameter `'a`, `&'static str` is a subtype of `&'a str`.

Higher-ranked function pointers and trait objects have another subtype relation. They are subtypes of types that are given by substitutions of the higher-ranked lifetimes. Some examples:

    #![allow(unused_variables)]
    fn main() {
    // Here 'a is substituted for 'static
    let subtype: &(for<'a> fn(&'a i32) -> &'a i32) = &((|x| x) as fn(&_) -> &_);
    let supertype: &(fn(&'static i32) -> &'static i32) = subtype;

    // This works similarly for trait objects
    let subtype: &(for<'a> Fn(&'a i32) -> &'a i32) = &|x| x;
    let supertype: &(Fn(&'static i32) -> &'static i32) = subtype;

    // We can also substitute one higher-ranked lifetime for another
    let subtype: &(for<'a, 'b> fn(&'a i32, &'b i32))= &((|x, y| {}) as fn(&_, &_));
    let supertype: &for<'c> fn(&'c i32, &'c i32) = subtype;
    }

## Variance

Variance is a property that generic types have with respect to their arguments. A generic type's variance in a parameter is how the subtyping of the parameter affects the subtyping of the type.

- `F<T>` is covariant over `T` if `T` being a subtype of `U` implies that `F<T>` is a subtype of `F<U>` (subtyping "passes through")
- `F<T>` is contravariant over `T` if `T` being a subtype of `U` implies that `F<U>` is a subtype of `F<T>`
- `F<T>` is invariant over `T` otherwise (no subtyping relation can be derived)

Variance of types is automatically determined as follows

Type | Variance in `'a` | Variance in `T`
--- | --- | ---
`&'a T` | covariant | covariant
`&'a mut T` | covariant | invariant
`*const T` |  | covariant
`*mut T` |  | invariant
`[T]` and `[T; n]` |  | covariant
`fn() -> T` |  | covariant
`fn(T) -> ()` |  | contravariant
`std::cell::UnsafeCell<T>` |  | invariant
`std::marker::PhantomData<T>` |  | covariant
`Trait<T> + 'a` | covariant | invariant

The variance of other `struct`, `enum`, `union`, and tuple types is decided by looking at the variance of the types of their fields. If the parameter is used in positions with different variances then the parameter is invariant. For example the following struct is covariant in `'a` and `T` and invariant in `'b` and `U`.

    #![allow(unused_variables)]
    fn main() {
    use std::cell::UnsafeCell;
    struct Variance<'a, 'b, T, U: 'a> {
        x: &'a U,               // This makes `Variance` covariant in 'a, and would
                                // make it covariant in U, but U is used later
        y: *const T,            // Covariant in T
        z: UnsafeCell<&'b f64>, // Invariant in 'b
        w: *mut U,              // Invariant in U, makes the whole struct invariant
    }
    }