core/lib/fixed-points.nix

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{ lib, ... }:
rec {
/*
`fix f` computes the fixed point of the given function `f`. In other words, the return value is `x` in `x = f x`.
`f` must be a lazy function.
This means that `x` must be a value that can be partially evaluated,
such as an attribute set, a list, or a function.
This way, `f` can use one part of `x` to compute another part.
**Relation to syntactic recursion**
This section explains `fix` by refactoring from syntactic recursion to a call of `fix` instead.
For context, Nix lets you define attributes in terms of other attributes syntactically using the [`rec { }` syntax](https://nixos.org/manual/nix/stable/language/constructs.html#recursive-sets).
```nix
nix-repl> rec {
foo = "foo";
bar = "bar";
foobar = foo + bar;
}
{ bar = "bar"; foo = "foo"; foobar = "foobar"; }
```
This is convenient when constructing a value to pass to a function for example,
but an equivalent effect can be achieved with the `let` binding syntax:
```nix
nix-repl> let self = {
foo = "foo";
bar = "bar";
foobar = self.foo + self.bar;
}; in self
{ bar = "bar"; foo = "foo"; foobar = "foobar"; }
```
But in general you can get more reuse out of `let` bindings by refactoring them to a function.
```nix
nix-repl> f = self: {
foo = "foo";
bar = "bar";
foobar = self.foo + self.bar;
}
```
This is where `fix` comes in, it contains the syntactic recursion that's not in `f` anymore.
```nix
nix-repl> fix = f:
let self = f self; in self;
```
By applying `fix` we get the final result.
```nix
nix-repl> fix f
{ bar = "bar"; foo = "foo"; foobar = "foobar"; }
```
Such a refactored `f` using `fix` is not useful by itself.
See [`extends`](#function-library-lib.fixedPoints.extends) for an example use case.
There `self` is also often called `final`.
Type: fix :: (a -> a) -> a
Example:
fix (self: { foo = "foo"; bar = "bar"; foobar = self.foo + self.bar; })
=> { bar = "bar"; foo = "foo"; foobar = "foobar"; }
fix (self: [ 1 2 (elemAt self 0 + elemAt self 1) ])
=> [ 1 2 3 ]
*/
fix = f: let x = f x; in x;
/*
A variant of `fix` that records the original recursive attribute set in the
result, in an attribute named `__unfix__`.
This is useful in combination with the `extends` function to
implement deep overriding.
*/
fix' = f: let x = f x // { __unfix__ = f; }; in x;
/*
Return the fixpoint that `f` converges to when called iteratively, starting
with the input `x`.
```
nix-repl> converge (x: x / 2) 16
0
```
Type: (a -> a) -> a -> a
*/
converge = f: x:
let
x' = f x;
in
if x' == x
then x
else converge f x';
/*
Extend a function using an overlay.
Overlays allow modifying and extending fixed-point functions, specifically ones returning attribute sets.
A fixed-point function is a function which is intended to be evaluated by passing the result of itself as the argument.
This is possible due to Nix's lazy evaluation.
A fixed-point function returning an attribute set has the form
```nix
final: { # attributes }
```
where `final` refers to the lazily evaluated attribute set returned by the fixed-point function.
An overlay to such a fixed-point function has the form
```nix
final: prev: { # attributes }
```
where `prev` refers to the result of the original function to `final`, and `final` is the result of the composition of the overlay and the original function.
Applying an overlay is done with `extends`:
```nix
let
f = final: { # attributes };
overlay = final: prev: { # attributes };
in extends overlay f;
```
To get the value of `final`, use `lib.fix`:
```nix
let
f = final: { # attributes };
overlay = final: prev: { # attributes };
g = extends overlay f;
in fix g
```
:::{.note}
The argument to the given fixed-point function after applying an overlay will *not* refer to its own return value, but rather to the value after evaluating the overlay function.
The given fixed-point function is called with a separate argument than if it was evaluated with `lib.fix`.
:::
:::{.example}
# Extend a fixed-point function with an overlay
Define a fixed-point function `f` that expects its own output as the argument `final`:
```nix-repl
f = final: {
# Constant value a
a = 1;
# b depends on the final value of a, available as final.a
b = final.a + 2;
}
```
Evaluate this using [`lib.fix`](#function-library-lib.fixedPoints.fix) to get the final result:
```nix-repl
fix f
=> { a = 1; b = 3; }
```
An overlay represents a modification or extension of such a fixed-point function.
Here's an example of an overlay:
```nix-repl
overlay = final: prev: {
# Modify the previous value of a, available as prev.a
a = prev.a + 10;
# Extend the attribute set with c, letting it depend on the final values of a and b
c = final.a + final.b;
}
```
Use `extends overlay f` to apply the overlay to the fixed-point function `f`.
This produces a new fixed-point function `g` with the combined behavior of `f` and `overlay`:
```nix-repl
g = extends overlay f
```
The result is a function, so we can't print it directly, but it's the same as:
```nix-repl
g' = final: {
# The constant from f, but changed with the overlay
a = 1 + 10;
# Unchanged from f
b = final.a + 2;
# Extended in the overlay
c = final.a + final.b;
}
```
Evaluate this using [`lib.fix`](#function-library-lib.fixedPoints.fix) again to get the final result:
```nix-repl
fix g
=> { a = 11; b = 13; c = 24; }
```
:::
Type:
extends :: (Attrs -> Attrs -> Attrs) # The overlay to apply to the fixed-point function
-> (Attrs -> Attrs) # A fixed-point function
-> (Attrs -> Attrs) # The resulting fixed-point function
Example:
f = final: { a = 1; b = final.a + 2; }
fix f
=> { a = 1; b = 3; }
fix (extends (final: prev: { a = prev.a + 10; }) f)
=> { a = 11; b = 13; }
fix (extends (final: prev: { b = final.a + 5; }) f)
=> { a = 1; b = 6; }
fix (extends (final: prev: { c = final.a + final.b; }) f)
=> { a = 1; b = 3; c = 4; }
*/
extends =
# The overlay to apply to the fixed-point function
overlay:
# The fixed-point function
f:
# Wrap with parenthesis to prevent nixdoc from rendering the `final` argument in the documentation
# The result should be thought of as a function, the argument of that function is not an argument to `extends` itself
(
final:
let
prev = f final;
in
prev // overlay final prev
);
/*
Compose two extending functions of the type expected by 'extends'
into one where changes made in the first are available in the
'super' of the second
*/
composeExtensions =
f: g: final: prev:
let fApplied = f final prev;
prev' = prev // fApplied;
in fApplied // g final prev';
/*
Compose several extending functions of the type expected by 'extends' into
one where changes made in preceding functions are made available to
subsequent ones.
```
composeManyExtensions : [packageSet -> packageSet -> packageSet] -> packageSet -> packageSet -> packageSet
^final ^prev ^overrides ^final ^prev ^overrides
```
*/
composeManyExtensions =
lib.foldr (x: y: composeExtensions x y) (final: prev: {});
/*
Create an overridable, recursive attribute set. For example:
```
nix-repl> obj = makeExtensible (self: { })
nix-repl> obj
{ __unfix__ = «lambda»; extend = «lambda»; }
nix-repl> obj = obj.extend (self: super: { foo = "foo"; })
nix-repl> obj
{ __unfix__ = «lambda»; extend = «lambda»; foo = "foo"; }
nix-repl> obj = obj.extend (self: super: { foo = super.foo + " + "; bar = "bar"; foobar = self.foo + self.bar; })
nix-repl> obj
{ __unfix__ = «lambda»; bar = "bar"; extend = «lambda»; foo = "foo + "; foobar = "foo + bar"; }
```
*/
makeExtensible = makeExtensibleWithCustomName "extend";
/*
Same as `makeExtensible` but the name of the extending attribute is
customized.
*/
makeExtensibleWithCustomName = extenderName: rattrs:
fix' (self: (rattrs self) // {
${extenderName} = f: makeExtensibleWithCustomName extenderName (extends f rattrs);
});
}