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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
g = self: super: { foo = super.foo + " + "; }
let
f = final: { # attributes };
overlay = final: prev: { # attributes };
g = extends overlay f;
in fix g
```
:::{.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; }
```
that has access to the unmodified input (`super`) as well as the final
non-recursive representation of the attribute set (`self`). `extends`
differs from the native `//` operator insofar as that it's applied *before*
references to `self` are resolved:
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
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`.