-
This commit is contained in:
Jeff Hykin
parent
11c40a8dcf
commit
9cb4ed1ffc
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@ -1,309 +1,319 @@
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{ lib, ... }:
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rec {
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/*
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`fix f` computes the fixed point of the given function `f`. In other words, the return value is `x` in `x = f x`.
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`f` must be a lazy function.
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This means that `x` must be a value that can be partially evaluated,
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such as an attribute set, a list, or a function.
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This way, `f` can use one part of `x` to compute another part.
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**Relation to syntactic recursion**
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This section explains `fix` by refactoring from syntactic recursion to a call of `fix` instead.
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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).
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```nix
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nix-repl> rec {
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foo = "foo";
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bar = "bar";
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foobar = foo + bar;
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}
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{ bar = "bar"; foo = "foo"; foobar = "foobar"; }
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```
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This is convenient when constructing a value to pass to a function for example,
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but an equivalent effect can be achieved with the `let` binding syntax:
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```nix
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nix-repl> let self = {
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foo = "foo";
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bar = "bar";
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foobar = self.foo + self.bar;
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}; in self
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{ bar = "bar"; foo = "foo"; foobar = "foobar"; }
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```
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But in general you can get more reuse out of `let` bindings by refactoring them to a function.
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```nix
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nix-repl> f = self: {
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foo = "foo";
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bar = "bar";
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foobar = self.foo + self.bar;
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}
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```
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This is where `fix` comes in, it contains the syntactic recursion that's not in `f` anymore.
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```nix
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nix-repl> fix = f:
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let self = f self; in self;
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```
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By applying `fix` we get the final result.
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```nix
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nix-repl> fix f
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{ bar = "bar"; foo = "foo"; foobar = "foobar"; }
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```
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Such a refactored `f` using `fix` is not useful by itself.
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See [`extends`](#function-library-lib.fixedPoints.extends) for an example use case.
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There `self` is also often called `final`.
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Type: fix :: (a -> a) -> a
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Example:
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fix (self: { foo = "foo"; bar = "bar"; foobar = self.foo + self.bar; })
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=> { bar = "bar"; foo = "foo"; foobar = "foobar"; }
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fix (self: [ 1 2 (elemAt self 0 + elemAt self 1) ])
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=> [ 1 2 3 ]
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*/
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fix = f: let x = f x; in x;
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/*
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A variant of `fix` that records the original recursive attribute set in the
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result, in an attribute named `__unfix__`.
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This is useful in combination with the `extends` function to
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implement deep overriding.
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*/
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fix' = f: let x = f x // { __unfix__ = f; }; in x;
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/*
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Return the fixpoint that `f` converges to when called iteratively, starting
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with the input `x`.
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```
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nix-repl> converge (x: x / 2) 16
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0
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```
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Type: (a -> a) -> a -> a
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*/
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converge = f: x:
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let
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x' = f x;
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in
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if x' == x
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then x
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else converge f x';
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/*
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Extend a function using an overlay.
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Overlays allow modifying and extending fixed-point functions, specifically ones returning attribute sets.
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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.
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This is possible due to Nix's lazy evaluation.
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A fixed-point function returning an attribute set has the form
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```nix
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final: { # attributes }
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```
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where `final` refers to the lazily evaluated attribute set returned by the fixed-point function.
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An overlay to such a fixed-point function has the form
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```nix
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final: prev: { # attributes }
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```
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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.
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Applying an overlay is done with `extends`:
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```nix
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let
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f = final: { # attributes };
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overlay = final: prev: { # attributes };
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in extends overlay f;
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```
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To get the value of `final`, use `lib.fix`:
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```nix
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let
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f = final: { # attributes };
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overlay = final: prev: { # attributes };
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g = extends overlay f;
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in fix g
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```
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:::{.note}
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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.
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The given fixed-point function is called with a separate argument than if it was evaluated with `lib.fix`.
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:::
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:::{.example}
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# Extend a fixed-point function with an overlay
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Define a fixed-point function `f` that expects its own output as the argument `final`:
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```nix-repl
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f = final: {
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# Constant value a
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a = 1;
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# b depends on the final value of a, available as final.a
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b = final.a + 2;
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}
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```
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Evaluate this using [`lib.fix`](#function-library-lib.fixedPoints.fix) to get the final result:
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```nix-repl
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fix f
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=> { a = 1; b = 3; }
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```
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An overlay represents a modification or extension of such a fixed-point function.
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Here's an example of an overlay:
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```nix-repl
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overlay = final: prev: {
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# Modify the previous value of a, available as prev.a
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a = prev.a + 10;
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# Extend the attribute set with c, letting it depend on the final values of a and b
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c = final.a + final.b;
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}
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```
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Use `extends overlay f` to apply the overlay to the fixed-point function `f`.
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This produces a new fixed-point function `g` with the combined behavior of `f` and `overlay`:
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```nix-repl
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g = extends overlay f
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```
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The result is a function, so we can't print it directly, but it's the same as:
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```nix-repl
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g' = final: {
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# The constant from f, but changed with the overlay
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a = 1 + 10;
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# Unchanged from f
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b = final.a + 2;
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# Extended in the overlay
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c = final.a + final.b;
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}
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```
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Evaluate this using [`lib.fix`](#function-library-lib.fixedPoints.fix) again to get the final result:
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```nix-repl
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fix g
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=> { a = 11; b = 13; c = 24; }
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```
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:::
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Type:
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extends :: (Attrs -> Attrs -> Attrs) # The overlay to apply to the fixed-point function
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-> (Attrs -> Attrs) # A fixed-point function
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-> (Attrs -> Attrs) # The resulting fixed-point function
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Example:
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f = final: { a = 1; b = final.a + 2; }
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{ foldr }:
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let
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/*
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`fix f` computes the fixed point of the given function `f`. In other words, the return value is `x` in `x = f x`.
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`f` must be a lazy function.
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This means that `x` must be a value that can be partially evaluated,
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such as an attribute set, a list, or a function.
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This way, `f` can use one part of `x` to compute another part.
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**Relation to syntactic recursion**
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This section explains `fix` by refactoring from syntactic recursion to a call of `fix` instead.
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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).
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```nix
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nix-repl> rec {
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foo = "foo";
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bar = "bar";
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foobar = foo + bar;
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}
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{ bar = "bar"; foo = "foo"; foobar = "foobar"; }
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```
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This is convenient when constructing a value to pass to a function for example,
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but an equivalent effect can be achieved with the `let` binding syntax:
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```nix
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nix-repl> let self = {
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foo = "foo";
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bar = "bar";
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foobar = self.foo + self.bar;
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}; in self
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{ bar = "bar"; foo = "foo"; foobar = "foobar"; }
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```
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But in general you can get more reuse out of `let` bindings by refactoring them to a function.
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```nix
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nix-repl> f = self: {
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foo = "foo";
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bar = "bar";
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foobar = self.foo + self.bar;
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}
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```
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This is where `fix` comes in, it contains the syntactic recursion that's not in `f` anymore.
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```nix
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nix-repl> fix = f:
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let self = f self; in self;
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```
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By applying `fix` we get the final result.
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```nix
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nix-repl> fix f
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{ bar = "bar"; foo = "foo"; foobar = "foobar"; }
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```
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Such a refactored `f` using `fix` is not useful by itself.
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See [`extends`](#function-library-lib.fixedPoints.extends) for an example use case.
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There `self` is also often called `final`.
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Type: fix :: (a -> a) -> a
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Example:
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fix (self: { foo = "foo"; bar = "bar"; foobar = self.foo + self.bar; })
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=> { bar = "bar"; foo = "foo"; foobar = "foobar"; }
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fix (self: [ 1 2 (elemAt self 0 + elemAt self 1) ])
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=> [ 1 2 3 ]
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*/
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fix = f: let x = f x; in x;
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/*
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A variant of `fix` that records the original recursive attribute set in the
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result, in an attribute named `__unfix__`.
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This is useful in combination with the `extends` function to
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implement deep overriding.
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*/
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fix' = f: let x = f x // { __unfix__ = f; }; in x;
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/*
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Return the fixpoint that `f` converges to when called iteratively, starting
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with the input `x`.
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```
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nix-repl> converge (x: x / 2) 16
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0
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```
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Type: (a -> a) -> a -> a
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*/
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converge = f: x:
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let
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x' = f x;
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in
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if x' == x
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then x
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else converge f x';
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/*
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Extend a function using an overlay.
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||||
|
||||
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
|
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|
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```nix
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final: { # attributes }
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||||
```
|
||||
|
||||
where `final` refers to the lazily evaluated attribute set returned by the fixed-point function.
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||||
|
||||
An overlay to such a fixed-point function has the form
|
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|
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```nix
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final: prev: { # attributes }
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||||
```
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||||
|
||||
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.
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|
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Applying an overlay is done with `extends`:
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|
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```nix
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let
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f = final: { # attributes };
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overlay = final: prev: { # attributes };
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in extends overlay f;
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```
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To get the value of `final`, use `lib.fix`:
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```nix
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let
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f = final: { # attributes };
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overlay = final: prev: { # attributes };
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g = extends overlay f;
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in fix g
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```
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:::{.note}
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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`.
|
||||
:::
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||||
|
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:::{.example}
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||||
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# Extend a fixed-point function with an overlay
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||||
|
||||
Define a fixed-point function `f` that expects its own output as the argument `final`:
|
||||
|
||||
```nix-repl
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f = final: {
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# Constant value a
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a = 1;
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# b depends on the final value of a, available as final.a
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b = final.a + 2;
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}
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```
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Evaluate this using [`lib.fix`](#function-library-lib.fixedPoints.fix) to get the final result:
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```nix-repl
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fix f
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=> { a = 1; b = 3; }
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fix (extends (final: prev: { a = prev.a + 10; }) f)
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=> { a = 11; b = 13; }
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fix (extends (final: prev: { b = final.a + 5; }) f)
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=> { a = 1; b = 6; }
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fix (extends (final: prev: { c = final.a + final.b; }) f)
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=> { a = 1; b = 3; c = 4; }
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*/
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extends =
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# The overlay to apply to the fixed-point function
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overlay:
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# The fixed-point function
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f:
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# Wrap with parenthesis to prevent nixdoc from rendering the `final` argument in the documentation
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# The result should be thought of as a function, the argument of that function is not an argument to `extends` itself
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(
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final:
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let
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prev = f final;
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in
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prev // overlay final prev
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||||
);
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||||
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||||
/*
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Compose two extending functions of the type expected by 'extends'
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into one where changes made in the first are available in the
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'super' of the second
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*/
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composeExtensions =
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f: g: final: prev:
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let fApplied = f final prev;
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prev' = prev // fApplied;
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in fApplied // g final prev';
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||||
|
||||
/*
|
||||
Compose several extending functions of the type expected by 'extends' into
|
||||
one where changes made in preceding functions are made available to
|
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subsequent ones.
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||||
|
||||
```
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composeManyExtensions : [packageSet -> packageSet -> packageSet] -> packageSet -> packageSet -> packageSet
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^final ^prev ^overrides ^final ^prev ^overrides
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||||
```
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||||
*/
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composeManyExtensions =
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lib.foldr (x: y: composeExtensions x y) (final: prev: {});
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||||
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/*
|
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Create an overridable, recursive attribute set. For example:
|
||||
|
||||
```
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||||
nix-repl> obj = makeExtensible (self: { })
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nix-repl> obj
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{ __unfix__ = «lambda»; extend = «lambda»; }
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nix-repl> obj = obj.extend (self: super: { foo = "foo"; })
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||||
nix-repl> obj
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{ __unfix__ = «lambda»; extend = «lambda»; foo = "foo"; }
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|
||||
nix-repl> obj = obj.extend (self: super: { foo = super.foo + " + "; bar = "bar"; foobar = self.foo + self.bar; })
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||||
|
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nix-repl> obj
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{ __unfix__ = «lambda»; bar = "bar"; extend = «lambda»; foo = "foo + "; foobar = "foo + bar"; }
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||||
```
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||||
*/
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||||
makeExtensible = makeExtensibleWithCustomName "extend";
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||||
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||||
/*
|
||||
Same as `makeExtensible` but the name of the extending attribute is
|
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customized.
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||||
*/
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||||
makeExtensibleWithCustomName = extenderName: rattrs:
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fix' (self: (rattrs self) // {
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||||
${extenderName} = f: makeExtensibleWithCustomName extenderName (extends f rattrs);
|
||||
});
|
||||
}
|
||||
```
|
||||
|
||||
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 =
|
||||
foldr (x: y: composeExtensions x y) (final: prev: {});
|
||||
|
||||
/*
|
||||
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);
|
||||
});
|
||||
|
||||
/*
|
||||
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";
|
||||
in
|
||||
{
|
||||
fix = fix;
|
||||
fix' = fix';
|
||||
converge = converge;
|
||||
extends = extends;
|
||||
composeExtensions = composeExtensions;
|
||||
composeManyExtensions = composeManyExtensions;
|
||||
makeExtensibleWithCustomName = makeExtensibleWithCustomName;
|
||||
makeExtensible = makeExtensible;
|
||||
}
|
||||
Loading…
Reference in a new issue