# Variance: PEP 695 syntax

```toml
[environment]
python-version = "3.12"
```

Type variables have a property called _variance_ that affects the subtyping and assignability
relations. Much more detail can be found in the [spec]. To summarize, each typevar is either
**covariant**, **contravariant**, **invariant**, or **bivariant**. (Note that bivariance is not
currently mentioned in the typing spec, but is a fourth case that we must consider.)

For all of the examples below, we will consider typevars `T` and `U`, two generic classes using
those typevars `C[T]` and `D[U]`, and two types `A` and `B`.

(Note that dynamic types like `Any` never participate in subtyping, so `C[Any]` is neither a subtype
nor supertype of any other specialization of `C`, regardless of `T`'s variance. It is, however,
assignable to any specialization of `C`, regardless of variance, via materialization.)

## Covariance

With a covariant typevar, subtyping and assignability are in "alignment": if `A <: B` and `C <: D`,
then `C[A] <: C[B]` and `C[A] <: D[B]`.

Types that "produce" data on demand are covariant in their typevar. If you expect a sequence of
`int`s, someone can safely provide a sequence of `bool`s, since each `bool` element that you would
get from the sequence is a valid `int`.

```py
from ty_extensions import is_assignable_to, is_equivalent_to, is_subtype_of, static_assert, Unknown
from typing import Any, Never

class A: ...
class B(A): ...

class C[T]:
    def receive(self) -> T:
        raise ValueError

class D[U](C[U]):
    pass

static_assert(is_assignable_to(C[B], C[A]))
static_assert(not is_assignable_to(C[A], C[B]))
static_assert(is_assignable_to(C[A], C[Any]))
static_assert(is_assignable_to(C[B], C[Any]))
static_assert(is_assignable_to(C[Any], C[A]))
static_assert(is_assignable_to(C[Any], C[B]))

static_assert(is_assignable_to(D[B], C[A]))
static_assert(not is_assignable_to(D[A], C[B]))
static_assert(is_assignable_to(D[A], C[Any]))
static_assert(is_assignable_to(D[B], C[Any]))
static_assert(is_assignable_to(D[Any], C[A]))
static_assert(is_assignable_to(D[Any], C[B]))

static_assert(is_subtype_of(C[B], C[A]))
static_assert(not is_subtype_of(C[A], C[B]))
static_assert(not is_subtype_of(C[A], C[Any]))
static_assert(not is_subtype_of(C[B], C[Any]))
static_assert(not is_subtype_of(C[Any], C[A]))
static_assert(not is_subtype_of(C[Any], C[B]))
static_assert(is_subtype_of(C[Any], C[object]))
static_assert(is_subtype_of(C[Never], C[Any]))

static_assert(is_subtype_of(D[B], C[A]))
static_assert(not is_subtype_of(D[A], C[B]))
static_assert(not is_subtype_of(D[A], C[Any]))
static_assert(not is_subtype_of(D[B], C[Any]))
static_assert(not is_subtype_of(D[Any], C[A]))
static_assert(not is_subtype_of(D[Any], C[B]))

static_assert(is_equivalent_to(C[A], C[A]))
static_assert(is_equivalent_to(C[B], C[B]))
static_assert(not is_equivalent_to(C[B], C[A]))
static_assert(not is_equivalent_to(C[A], C[B]))
static_assert(not is_equivalent_to(C[A], C[Any]))
static_assert(not is_equivalent_to(C[B], C[Any]))
static_assert(not is_equivalent_to(C[Any], C[A]))
static_assert(not is_equivalent_to(C[Any], C[B]))

static_assert(not is_equivalent_to(D[A], C[A]))
static_assert(not is_equivalent_to(D[B], C[B]))
static_assert(not is_equivalent_to(D[B], C[A]))
static_assert(not is_equivalent_to(D[A], C[B]))
static_assert(not is_equivalent_to(D[A], C[Any]))
static_assert(not is_equivalent_to(D[B], C[Any]))
static_assert(not is_equivalent_to(D[Any], C[A]))
static_assert(not is_equivalent_to(D[Any], C[B]))

static_assert(is_equivalent_to(C[Any], C[Any]))
static_assert(is_equivalent_to(C[Any], C[Unknown]))

static_assert(not is_equivalent_to(D[Any], C[Any]))
static_assert(not is_equivalent_to(D[Any], C[Unknown]))
```

## Contravariance

With a contravariant typevar, subtyping and assignability are in "opposition": if `A <: B` and
`C <: D`, then `C[B] <: C[A]` and `D[B] <: C[A]`.

Types that "consume" data are contravariant in their typevar. If you expect a consumer that receives
`bool`s, someone can safely provide a consumer that expects to receive `int`s, since each `bool`
that you pass into the consumer is a valid `int`.

```py
from ty_extensions import is_assignable_to, is_equivalent_to, is_subtype_of, static_assert, Unknown
from typing import Any, Never

class A: ...
class B(A): ...

class C[T]:
    def send(self, value: T): ...

class D[U](C[U]):
    pass

static_assert(not is_assignable_to(C[B], C[A]))
static_assert(is_assignable_to(C[A], C[B]))
static_assert(is_assignable_to(C[A], C[Any]))
static_assert(is_assignable_to(C[B], C[Any]))
static_assert(is_assignable_to(C[Any], C[A]))
static_assert(is_assignable_to(C[Any], C[B]))

static_assert(not is_assignable_to(D[B], C[A]))
static_assert(is_assignable_to(D[A], C[B]))
static_assert(is_assignable_to(D[A], C[Any]))
static_assert(is_assignable_to(D[B], C[Any]))
static_assert(is_assignable_to(D[Any], C[A]))
static_assert(is_assignable_to(D[Any], C[B]))

static_assert(not is_subtype_of(C[B], C[A]))
static_assert(is_subtype_of(C[A], C[B]))
static_assert(not is_subtype_of(C[A], C[Any]))
static_assert(not is_subtype_of(C[B], C[Any]))
static_assert(not is_subtype_of(C[Any], C[A]))
static_assert(not is_subtype_of(C[Any], C[B]))
static_assert(is_subtype_of(C[object], C[Any]))
static_assert(is_subtype_of(C[Any], C[Never]))

static_assert(not is_subtype_of(D[B], C[A]))
static_assert(is_subtype_of(D[A], C[B]))
static_assert(not is_subtype_of(D[A], C[Any]))
static_assert(not is_subtype_of(D[B], C[Any]))
static_assert(not is_subtype_of(D[Any], C[A]))
static_assert(not is_subtype_of(D[Any], C[B]))

static_assert(is_equivalent_to(C[A], C[A]))
static_assert(is_equivalent_to(C[B], C[B]))
static_assert(not is_equivalent_to(C[B], C[A]))
static_assert(not is_equivalent_to(C[A], C[B]))
static_assert(not is_equivalent_to(C[A], C[Any]))
static_assert(not is_equivalent_to(C[B], C[Any]))
static_assert(not is_equivalent_to(C[Any], C[A]))
static_assert(not is_equivalent_to(C[Any], C[B]))

static_assert(not is_equivalent_to(D[A], C[A]))
static_assert(not is_equivalent_to(D[B], C[B]))
static_assert(not is_equivalent_to(D[B], C[A]))
static_assert(not is_equivalent_to(D[A], C[B]))
static_assert(not is_equivalent_to(D[A], C[Any]))
static_assert(not is_equivalent_to(D[B], C[Any]))
static_assert(not is_equivalent_to(D[Any], C[A]))
static_assert(not is_equivalent_to(D[Any], C[B]))

static_assert(is_equivalent_to(C[Any], C[Any]))
static_assert(is_equivalent_to(C[Any], C[Unknown]))

static_assert(not is_equivalent_to(D[Any], C[Any]))
static_assert(not is_equivalent_to(D[Any], C[Unknown]))
```

## Bounded typevars in contravariant positions

When a bounded typevar appears in a contravariant position, the actual type doesn't need to satisfy
the bound directly. The typevar can be solved to the intersection of the actual type and the bound
(e.g., `Never` when disjoint).

```py
class Contra[T]:
    def append(self, x: T): ...

def f[T: int](x: Contra[T]) -> T:
    raise NotImplementedError

def _(x: Contra[str]):
    reveal_type(f(x))  # revealed: Never
```

## Invariance

With an invariant typevar, only equivalent specializations of the generic class are subtypes of or
assignable to each other.

This often occurs for types that are both producers _and_ consumers, like a mutable `list`.
Iterating over the elements in a list would work with a covariant typevar, just like with the
"producer" type above. Appending elements to a list would work with a contravariant typevar, just
like with the "consumer" type above. However, a typevar cannot be both covariant and contravariant
at the same time!

If you expect a mutable list of `int`s, it's not safe for someone to provide you with a mutable list
of `bool`s, since you might try to add an element to the list: if you try to add an `int`, the list
would no longer only contain elements that are subtypes of `bool`.

Conversely, if you expect a mutable list of `bool`s, it's not safe for someone to provide you with a
mutable list of `int`s, since you might try to extract elements from the list: you expect every
element that you extract to be a subtype of `bool`, but the list can contain any `int`.

In the end, if you expect a mutable list, you must always be given a list of exactly that type,
since we can't know in advance which of the allowed methods you'll want to use.

```py
from ty_extensions import is_assignable_to, is_equivalent_to, is_subtype_of, static_assert, Unknown
from typing import Any, Never

class A: ...
class B(A): ...

class C[T]:
    def send(self, value: T): ...
    def receive(self) -> T:
        raise ValueError

class D[U](C[U]):
    pass

static_assert(not is_assignable_to(C[B], C[A]))
static_assert(not is_assignable_to(C[A], C[B]))
static_assert(is_assignable_to(C[A], C[Any]))
static_assert(is_assignable_to(C[B], C[Any]))
static_assert(is_assignable_to(C[Any], C[A]))
static_assert(is_assignable_to(C[Any], C[B]))

static_assert(not is_assignable_to(D[B], C[A]))
static_assert(not is_assignable_to(D[A], C[B]))
static_assert(is_assignable_to(D[A], C[Any]))
static_assert(is_assignable_to(D[B], C[Any]))
static_assert(is_assignable_to(D[Any], C[A]))
static_assert(is_assignable_to(D[Any], C[B]))

static_assert(not is_subtype_of(C[B], C[A]))
static_assert(not is_subtype_of(C[A], C[B]))
static_assert(not is_subtype_of(C[A], C[Any]))
static_assert(not is_subtype_of(C[B], C[Any]))
static_assert(not is_subtype_of(C[Any], C[A]))
static_assert(not is_subtype_of(C[Any], C[B]))
static_assert(not is_subtype_of(C[object], C[Any]))
static_assert(not is_subtype_of(C[Any], C[Never]))

static_assert(not is_subtype_of(D[B], C[A]))
static_assert(not is_subtype_of(D[A], C[B]))
static_assert(not is_subtype_of(D[A], C[Any]))
static_assert(not is_subtype_of(D[B], C[Any]))
static_assert(not is_subtype_of(D[Any], C[A]))
static_assert(not is_subtype_of(D[Any], C[B]))

static_assert(is_equivalent_to(C[A], C[A]))
static_assert(is_equivalent_to(C[B], C[B]))
static_assert(not is_equivalent_to(C[B], C[A]))
static_assert(not is_equivalent_to(C[A], C[B]))
static_assert(not is_equivalent_to(C[A], C[Any]))
static_assert(not is_equivalent_to(C[B], C[Any]))
static_assert(not is_equivalent_to(C[Any], C[A]))
static_assert(not is_equivalent_to(C[Any], C[B]))

static_assert(not is_equivalent_to(D[A], C[A]))
static_assert(not is_equivalent_to(D[B], C[B]))
static_assert(not is_equivalent_to(D[B], C[A]))
static_assert(not is_equivalent_to(D[A], C[B]))
static_assert(not is_equivalent_to(D[A], C[Any]))
static_assert(not is_equivalent_to(D[B], C[Any]))
static_assert(not is_equivalent_to(D[Any], C[A]))
static_assert(not is_equivalent_to(D[Any], C[B]))

static_assert(is_equivalent_to(C[Any], C[Any]))
static_assert(is_equivalent_to(C[Any], C[Unknown]))

static_assert(not is_equivalent_to(D[Any], C[Any]))
static_assert(not is_equivalent_to(D[Any], C[Unknown]))
```

## Bivariance

With a bivariant typevar, _all_ specializations of the generic class are assignable to (and in fact,
gradually equivalent to) each other, and all specializations are subtypes of (and equivalent to)
each other.

This is a bit of pathological case, which really only happens when the class doesn't use the typevar
at all. (If it did, it would have to be covariant, contravariant, or invariant, depending on _how_
the typevar was used.)

```py
from ty_extensions import is_assignable_to, is_equivalent_to, is_subtype_of, static_assert, Unknown
from typing import Any, Never

class A: ...
class B(A): ...

class C[T]:
    pass

class D[U](C[U]):
    pass

static_assert(is_assignable_to(C[B], C[A]))
static_assert(is_assignable_to(C[A], C[B]))
static_assert(is_assignable_to(C[A], C[Any]))
static_assert(is_assignable_to(C[B], C[Any]))
static_assert(is_assignable_to(C[Any], C[A]))
static_assert(is_assignable_to(C[Any], C[B]))

static_assert(is_assignable_to(D[B], C[A]))
static_assert(is_subtype_of(C[A], C[A]))
static_assert(is_assignable_to(D[A], C[B]))
static_assert(is_assignable_to(D[A], C[Any]))
static_assert(is_assignable_to(D[B], C[Any]))
static_assert(is_assignable_to(D[Any], C[A]))
static_assert(is_assignable_to(D[Any], C[B]))

static_assert(is_subtype_of(C[B], C[A]))
static_assert(is_subtype_of(C[A], C[B]))
static_assert(is_subtype_of(C[A], C[Any]))
static_assert(is_subtype_of(C[B], C[Any]))
static_assert(is_subtype_of(C[Any], C[A]))
static_assert(is_subtype_of(C[Any], C[B]))
static_assert(is_subtype_of(C[Any], C[Any]))
static_assert(is_subtype_of(C[object], C[Any]))
static_assert(is_subtype_of(C[Any], C[Never]))

static_assert(is_subtype_of(D[B], C[A]))
static_assert(is_subtype_of(D[A], C[B]))
static_assert(is_subtype_of(D[A], C[Any]))
static_assert(is_subtype_of(D[B], C[Any]))
static_assert(is_subtype_of(D[Any], C[A]))
static_assert(is_subtype_of(D[Any], C[B]))

static_assert(is_equivalent_to(C[A], C[A]))
static_assert(is_equivalent_to(C[B], C[B]))
static_assert(is_equivalent_to(C[B], C[A]))
static_assert(is_equivalent_to(C[A], C[B]))
static_assert(is_equivalent_to(C[A], C[Any]))
static_assert(is_equivalent_to(C[B], C[Any]))
static_assert(is_equivalent_to(C[Any], C[A]))
static_assert(is_equivalent_to(C[Any], C[B]))

static_assert(not is_equivalent_to(D[A], C[A]))
static_assert(not is_equivalent_to(D[B], C[B]))
static_assert(not is_equivalent_to(D[B], C[A]))
static_assert(not is_equivalent_to(D[A], C[B]))
static_assert(not is_equivalent_to(D[A], C[Any]))
static_assert(not is_equivalent_to(D[B], C[Any]))
static_assert(not is_equivalent_to(D[Any], C[A]))
static_assert(not is_equivalent_to(D[Any], C[B]))

static_assert(is_equivalent_to(C[Any], C[Any]))
static_assert(is_equivalent_to(C[Any], C[Unknown]))

static_assert(not is_equivalent_to(D[Any], C[Any]))
static_assert(not is_equivalent_to(D[Any], C[Unknown]))
```

## Mutual Recursion

This example due to Martin Huschenbett's PyCon 2025 talk,
[Linear Time variance Inference for PEP 695][linear-time-variance-talk]

```py
from ty_extensions import is_subtype_of, static_assert
from typing import Any

class A: ...
class B(A): ...

class C[X]:
    def f(self) -> "D[X]":
        return D()

    def g(self, x: X) -> None: ...

class D[Y]:
    def h(self) -> C[Y]:
        return C()
```

`C` is contravariant in `X`, and `D` in `Y`:

- `C` has two occurrences of `X`
    - `X` occurs in the return type of `f` as `D[X]` (`X` is substituted in for `Y`)
        - `D` has one occurrence of `Y`
            - `Y` occurs in the return type of `h` as `C[Y]`
    - `X` occurs contravariantly as a parameter in `g`

Thus the variance of `X` in `C` depends on itself. We want to infer the least restrictive possible
variance, so in such cases we begin by assuming that the point where we detect the cycle is
bivariant.

If we thus assume `X` is bivariant in `C`, then `Y` will be bivariant in `D`, as `D`'s only
occurrence of `Y` is in `C`. Then we consider `X` in `C` once more. We have two occurrences: `D[X]`
covariantly in a return type, and `X` contravariantly in an argument type. With one bivariant and
one contravariant occurrence, we update our inference of `X` in `C` to contravariant---the supremum
of contravariant and bivariant in the lattice.

Now that we've updated the variance of `X` in `C`, we re-evaluate `Y` in `D`. It only has the one
occurrence `C[Y]`, which we now infer is contravariant, and so we infer contravariance for `Y` in
`D` as well.

Because the variance of `X` in `C` depends on that of `Y` in `D`, we have to re-evaluate now that
we've updated the latter to contravariant. The variance of `X` in `C` is now the supremum of
contravariant and contravariant---giving us contravariant---and so remains unchanged.

Once we've completed a turn around the cycle with nothing changed, we've reached a fixed-point---the
variance inference will not change any further---and so we finally conclude that both `X` in `C` and
`Y` in `D` are contravariant.

```py
static_assert(not is_subtype_of(C[B], C[A]))
static_assert(is_subtype_of(C[A], C[B]))
static_assert(not is_subtype_of(C[A], C[Any]))
static_assert(not is_subtype_of(C[B], C[Any]))
static_assert(not is_subtype_of(C[Any], C[A]))
static_assert(not is_subtype_of(C[Any], C[B]))

static_assert(not is_subtype_of(D[B], D[A]))
static_assert(is_subtype_of(D[A], D[B]))
static_assert(not is_subtype_of(D[A], D[Any]))
static_assert(not is_subtype_of(D[B], D[Any]))
static_assert(not is_subtype_of(D[Any], D[A]))
static_assert(not is_subtype_of(D[Any], D[B]))
```

## Class Attributes

### Mutable Attributes

Normal attributes are mutable, and so make the enclosing class invariant in this typevar (see
[inv]).

```py
from ty_extensions import is_subtype_of, static_assert

class A: ...
class B(A): ...

class C[T]:
    x: T

static_assert(not is_subtype_of(C[B], C[A]))
static_assert(not is_subtype_of(C[A], C[B]))
```

One might think that occurrences in the types of normal attributes are covariant, but they are
mutable, and thus the occurrences are invariant.

### Immutable Attributes

Immutable attributes can't be written to, and thus constrain the typevar to covariance, not
invariance.

#### Final attributes

```py
from typing import Final
from ty_extensions import is_subtype_of, static_assert

class A: ...
class B(A): ...

class C[T]:
    x: Final[T]  # error: [final-without-value]

static_assert(is_subtype_of(C[B], C[A]))
static_assert(not is_subtype_of(C[A], C[B]))
```

#### Underscore-prefixed attributes

Underscore-prefixed instance attributes are considered private, and thus are assumed not externally
mutated.

```py
from ty_extensions import is_subtype_of, static_assert

class A: ...
class B(A): ...

class C[T]:
    _x: T

    @property
    def x(self) -> T:
        return self._x

static_assert(is_subtype_of(C[B], C[A]))
static_assert(not is_subtype_of(C[A], C[B]))

class D[T]:
    def __init__(self, x: T):
        self._x = x

    @property
    def x(self) -> T:
        return self._x

static_assert(is_subtype_of(D[B], D[A]))
static_assert(not is_subtype_of(D[A], D[B]))
```

#### Frozen dataclasses in Python 3.12 and earlier

```py
from dataclasses import dataclass, field
from ty_extensions import is_subtype_of, static_assert

class A: ...
class B(A): ...

@dataclass(frozen=True)
class D[U]:
    y: U

static_assert(is_subtype_of(D[B], D[A]))
static_assert(not is_subtype_of(D[A], D[B]))

@dataclass(frozen=True)
class E[U]:
    y: U = field()

static_assert(is_subtype_of(E[B], E[A]))
static_assert(not is_subtype_of(E[A], E[B]))
```

This also works for dataclass-transformers:

```py
from typing import dataclass_transform

@dataclass_transform(frozen_default=False)
class ModelBase:
    def __init_subclass__(cls, frozen: bool = False) -> None:
        pass

class NonFrozenModel[T](ModelBase):
    value: T

static_assert(not is_subtype_of(NonFrozenModel[B], NonFrozenModel[A]))
static_assert(not is_subtype_of(NonFrozenModel[A], NonFrozenModel[B]))

class FrozenModel[T](ModelBase, frozen=True):
    value: T

static_assert(is_subtype_of(FrozenModel[B], FrozenModel[A]))
static_assert(not is_subtype_of(FrozenModel[A], FrozenModel[B]))
```

#### Frozen dataclasses in Python 3.13 and later

```toml
[environment]
python-version = "3.13"
```

Python 3.13 introduced a new synthesized `__replace__` method on dataclasses, which uses every field
type in a contravariant position (as a parameter to `__replace__`). This means that frozen
dataclasses on Python 3.13+ can't be covariant in their field types.

```py
from dataclasses import dataclass
from ty_extensions import is_subtype_of, static_assert

class A: ...
class B(A): ...

@dataclass(frozen=True)
class D[U]:
    y: U

static_assert(not is_subtype_of(D[B], D[A]))
static_assert(not is_subtype_of(D[A], D[B]))
```

The same holds for dataclass-transformers:

```py
from typing import dataclass_transform

@dataclass_transform(frozen_default=True)
class ModelBase:
    def __init_subclass__(cls, frozen: bool = True) -> None:
        pass

class DefaultFrozenModel[T](ModelBase):
    value: T

static_assert(not is_subtype_of(DefaultFrozenModel[B], DefaultFrozenModel[A]))
static_assert(not is_subtype_of(DefaultFrozenModel[A], DefaultFrozenModel[B]))

class ExplicitFrozenModel[T](ModelBase, frozen=True):
    value: T

static_assert(not is_subtype_of(ExplicitFrozenModel[B], ExplicitFrozenModel[A]))
static_assert(not is_subtype_of(ExplicitFrozenModel[A], ExplicitFrozenModel[B]))
```

#### NamedTuple

```py
from typing import NamedTuple
from ty_extensions import is_subtype_of, static_assert

class A: ...
class B(A): ...

class E[V](NamedTuple):
    z: V

static_assert(is_subtype_of(E[B], E[A]))
static_assert(not is_subtype_of(E[A], E[B]))
```

A subclass of a `NamedTuple` can still be covariant:

```py
class D[T](E[T]):
    pass

static_assert(is_subtype_of(D[B], D[A]))
static_assert(not is_subtype_of(D[A], D[B]))
```

But adding a new generic attribute on the subclass makes it invariant (the added attribute is not a
`NamedTuple` field, and thus not immutable):

```py
class C[T](E[T]):
    w: T

static_assert(not is_subtype_of(C[B], C[A]))
static_assert(not is_subtype_of(C[A], C[B]))
```

### Properties

Properties constrain to covariance if they are get-only and invariant if they are get-set:

```py
from ty_extensions import static_assert, is_subtype_of

class A: ...
class B(A): ...

class C[T]:
    @property
    def x(self) -> T | None:
        return None

class D[U]:
    @property
    def y(self) -> U | None:
        return None

    @y.setter
    def y(self, value: U): ...

static_assert(is_subtype_of(C[B], C[A]))
static_assert(not is_subtype_of(C[A], C[B]))
static_assert(not is_subtype_of(D[B], D[A]))
static_assert(not is_subtype_of(D[A], D[B]))
```

### Implicit Attributes

Implicit attributes work like normal ones

```py
from ty_extensions import static_assert, is_subtype_of

class A: ...
class B(A): ...

class C[T]:
    def f(self) -> None:
        self.x: T | None = None

static_assert(not is_subtype_of(C[B], C[A]))
static_assert(not is_subtype_of(C[A], C[B]))
```

### Constructors: excluding `__init__` and `__new__`

We consider it invalid to call `__init__` explicitly on an existing object. Likewise, `__new__` is
only used at the beginning of an object's life. As such, we don't need to worry about the variance
impact of these methods.

```py
from ty_extensions import static_assert, is_subtype_of

class A: ...
class B(A): ...

class C[T]:
    def __init__(self, x: T): ...
    def __new__(self, x: T): ...

static_assert(is_subtype_of(C[B], C[A]))
static_assert(is_subtype_of(C[A], C[B]))
```

This example is then bivariant because it doesn't use `T` outside of the two exempted methods.

This holds likewise for dataclasses with synthesized `__init__`:

```py
from dataclasses import dataclass

@dataclass(init=True, frozen=True)
class D[T]:
    x: T

# Covariant due to the read-only T-typed attribute; the `__init__` is ignored and doesn't make it
# invariant:

static_assert(is_subtype_of(D[B], D[A]))
static_assert(not is_subtype_of(D[A], D[B]))
```

## Union Types

Union types are covariant in all their members. If `A <: B`, then `A | C <: B | C` and
`C | A <: C | B`.

```py
from ty_extensions import is_assignable_to, is_subtype_of, static_assert

class A: ...
class B(A): ...
class C: ...

# Union types are covariant in their members
static_assert(is_subtype_of(B | C, A | C))
static_assert(is_subtype_of(C | B, C | A))
static_assert(not is_subtype_of(A | C, B | C))
static_assert(not is_subtype_of(C | A, C | B))

# Assignability follows the same pattern
static_assert(is_assignable_to(B | C, A | C))
static_assert(is_assignable_to(C | B, C | A))
static_assert(not is_assignable_to(A | C, B | C))
static_assert(not is_assignable_to(C | A, C | B))
```

## Intersection Types

Intersection types cannot be expressed directly in Python syntax, but they occur when type narrowing
creates constraints through control flow. In ty's representation, intersection types are covariant
in their positive conjuncts and contravariant in their negative conjuncts.

```py
from ty_extensions import is_assignable_to, is_subtype_of, static_assert, Intersection, Not

class A: ...
class B(A): ...
class C: ...

# Test covariance in positive conjuncts
# If B <: A, then Intersection[X, B] <: Intersection[X, A]
static_assert(is_subtype_of(Intersection[C, B], Intersection[C, A]))
static_assert(not is_subtype_of(Intersection[C, A], Intersection[C, B]))

static_assert(is_assignable_to(Intersection[C, B], Intersection[C, A]))
static_assert(not is_assignable_to(Intersection[C, A], Intersection[C, B]))

# Test contravariance in negative conjuncts
# If B <: A, then Intersection[X, Not[A]] <: Intersection[X, Not[B]]
# (excluding supertype A is more restrictive than excluding subtype B)
static_assert(is_subtype_of(Intersection[C, Not[A]], Intersection[C, Not[B]]))
static_assert(not is_subtype_of(Intersection[C, Not[B]], Intersection[C, Not[A]]))

static_assert(is_assignable_to(Intersection[C, Not[A]], Intersection[C, Not[B]]))
static_assert(not is_assignable_to(Intersection[C, Not[B]], Intersection[C, Not[A]]))
```

## Subclass Types (type[T])

The `type[T]` construct represents the type of classes that are subclasses of `T`. It is covariant
in `T` because if `A <: B`, then `type[A] <: type[B]` holds.

```py
from ty_extensions import is_assignable_to, is_subtype_of, static_assert

class A: ...
class B(A): ...

# type[T] is covariant in T
static_assert(is_subtype_of(type[B], type[A]))
static_assert(not is_subtype_of(type[A], type[B]))

static_assert(is_assignable_to(type[B], type[A]))
static_assert(not is_assignable_to(type[A], type[B]))

# With generic classes using type[T]
class ClassContainer[T]:
    def __init__(self, cls: type[T]) -> None:
        self.cls = cls

    def create_instance(self) -> T:
        return self.cls()

# ClassContainer is covariant in T due to type[T]
static_assert(is_subtype_of(ClassContainer[B], ClassContainer[A]))
static_assert(not is_subtype_of(ClassContainer[A], ClassContainer[B]))

static_assert(is_assignable_to(ClassContainer[B], ClassContainer[A]))
static_assert(not is_assignable_to(ClassContainer[A], ClassContainer[B]))

# Practical example: you can pass a ClassContainer[B] where ClassContainer[A] is expected
# because type[B] can safely be used where type[A] is expected
def use_a_class_container(container: ClassContainer[A]) -> A:
    return container.create_instance()

b_container = ClassContainer[B](B)
a_instance: A = use_a_class_container(b_container)  # This should work
```

## TypeIs

```toml
[environment]
python-version = "3.13"
```

`TypeIs[T]` is invariant in `T`. See the [typing spec][typeis-spec] for a justification.

```py
from typing import TypeIs
from ty_extensions import is_assignable_to, is_subtype_of, static_assert

class A:
    pass

class B(A):
    pass

class C[T]:
    def check(self, x: object) -> TypeIs[T]:
        # this is a bad check, but we only care about it type-checking
        return False

static_assert(not is_subtype_of(C[B], C[A]))
static_assert(not is_subtype_of(C[A], C[B]))
static_assert(not is_assignable_to(C[B], C[A]))
static_assert(not is_assignable_to(C[A], C[B]))
```

## TypeGuard

`TypeGuard[T]` is covariant in `T`. The typing spec doesn't explicitly call this out, but it follows
from similar logic to invariance of `TypeIs` except without the negative case.

Formally, suppose we have types `A` and `B` with `B < A`. Take `x: object` to be the value that all
subsequent `TypeGuard`s are narrowing.

We can assign `p: TypeGuard[A] = q` where `q: TypeGuard[B]` because

- if `q` is `False`, then no constraints were learned on `x` before and none are now learned, so
    nothing changes
- if `q` is `True`, then we know `x: B`. From `B < A`, we conclude `x: A`.

We _cannot_ assign `p: TypeGuard[B] = q` where `q: TypeGuard[A]` because if `q` is `True`, we would
be concluding `x: B` from `x: A`, which is an unsafe downcast.

```py
from typing import TypeGuard
from ty_extensions import is_assignable_to, is_subtype_of, static_assert

class A:
    pass

class B(A):
    pass

class C[T]:
    def check(self, x: object) -> TypeGuard[T]:
        # this is a bad check, but we only care about it type-checking
        return False

static_assert(is_subtype_of(C[B], C[A]))
static_assert(not is_subtype_of(C[A], C[B]))
static_assert(is_assignable_to(C[B], C[A]))
static_assert(not is_assignable_to(C[A], C[B]))
```

## Type aliases

The variance of the type alias matches the variance of the value type (RHS type).

```py
from ty_extensions import static_assert, is_subtype_of
from typing import Literal

class Covariant[T]:
    def get(self) -> T:
        raise ValueError

type CovariantLiteral1 = Covariant[Literal[1]]
type CovariantInt = Covariant[int]
type MyCovariant[T] = Covariant[T]

static_assert(is_subtype_of(CovariantLiteral1, CovariantInt))
static_assert(is_subtype_of(MyCovariant[Literal[1]], MyCovariant[int]))

class Contravariant[T]:
    def set(self, value: T):
        pass

type ContravariantLiteral1 = Contravariant[Literal[1]]
type ContravariantInt = Contravariant[int]
type MyContravariant[T] = Contravariant[T]

static_assert(is_subtype_of(ContravariantInt, ContravariantLiteral1))
static_assert(is_subtype_of(MyContravariant[int], MyContravariant[Literal[1]]))

class Invariant[T]:
    def get(self) -> T:
        raise ValueError

    def set(self, value: T):
        pass

type InvariantLiteral1 = Invariant[Literal[1]]
type InvariantInt = Invariant[int]
type MyInvariant[T] = Invariant[T]

static_assert(not is_subtype_of(InvariantInt, InvariantLiteral1))
static_assert(not is_subtype_of(InvariantLiteral1, InvariantInt))
static_assert(not is_subtype_of(MyInvariant[Literal[1]], MyInvariant[int]))
static_assert(not is_subtype_of(MyInvariant[int], MyInvariant[Literal[1]]))

class Bivariant[T]:
    pass

type BivariantLiteral1 = Bivariant[Literal[1]]
type BivariantInt = Bivariant[int]
type MyBivariant[T] = Bivariant[T]

static_assert(is_subtype_of(BivariantInt, BivariantLiteral1))
static_assert(is_subtype_of(BivariantLiteral1, BivariantInt))
static_assert(is_subtype_of(MyBivariant[Literal[1]], MyBivariant[int]))
static_assert(is_subtype_of(MyBivariant[int], MyBivariant[Literal[1]]))
```

## Inheriting from generic classes with inferred variance

When inheriting from a generic class with our type variable substituted in, we count its occurrences
as well. In the following example, `T` is covariant in `C`, and contravariant in the subclass `D` if
you only count its own occurrences. Because we count both then, `T` is invariant in `D`.

```py
from ty_extensions import is_subtype_of, static_assert

class A:
    pass

class B(A):
    pass

class C[T]:
    def f() -> T | None:
        pass

static_assert(is_subtype_of(C[B], C[A]))
static_assert(not is_subtype_of(C[A], C[B]))

class D[T](C[T]):
    def g(x: T) -> None:
        pass

static_assert(not is_subtype_of(D[B], D[A]))
static_assert(not is_subtype_of(D[A], D[B]))
```

## Inheriting from generic classes with explicit variance

```py
from typing import TypeVar, Generic
from ty_extensions import is_subtype_of, static_assert

T = TypeVar("T")
T_co = TypeVar("T_co", covariant=True)
T_contra = TypeVar("T_contra", contravariant=True)

class A:
    pass

class B(A):
    pass

class Invariant(Generic[T]):
    pass

static_assert(not is_subtype_of(Invariant[B], Invariant[A]))
static_assert(not is_subtype_of(Invariant[A], Invariant[B]))

class DerivedInvariant[T](Invariant[T]):
    pass

static_assert(not is_subtype_of(DerivedInvariant[B], DerivedInvariant[A]))
static_assert(not is_subtype_of(DerivedInvariant[A], DerivedInvariant[B]))

class Covariant(Generic[T_co]):
    pass

static_assert(is_subtype_of(Covariant[B], Covariant[A]))
static_assert(not is_subtype_of(Covariant[A], Covariant[B]))

class DerivedCovariant[T](Covariant[T]):
    pass

static_assert(is_subtype_of(DerivedCovariant[B], DerivedCovariant[A]))
static_assert(not is_subtype_of(DerivedCovariant[A], DerivedCovariant[B]))

class Contravariant(Generic[T_contra]):
    pass

static_assert(not is_subtype_of(Contravariant[B], Contravariant[A]))
static_assert(is_subtype_of(Contravariant[A], Contravariant[B]))

class DerivedContravariant[T](Contravariant[T]):
    pass

static_assert(not is_subtype_of(DerivedContravariant[B], DerivedContravariant[A]))
static_assert(is_subtype_of(DerivedContravariant[A], DerivedContravariant[B]))
```

[linear-time-variance-talk]: https://www.youtube.com/watch?v=7uixlNTOY4s&t=9705s
[spec]: https://typing.python.org/en/latest/spec/generics.html#variance
[typeis-spec]: https://typing.python.org/en/latest/spec/narrowing.html#typeis
