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<h1 id="sec-functions-library-fixedPoints">lib.fixedPoints: explicit recursion functions</h1>
<h2 id="function-library-lib.fixedPoints.fix"><code>lib.fixedPoints.fix</code></h2>
<p><code>fix f</code> computes the fixed point of the given function <code>f</code>. In other words, the return value is <code>x</code> in <code>x = f x</code>.</p>
<p><code>f</code> must be a lazy function.
This means that <code>x</code> must be a value that can be partially evaluated,
such as an attribute set, a list, or a function.
This way, <code>f</code> can use one part of <code>x</code> to compute another part.</p>
<p><strong>Relation to syntactic recursion</strong></p>
<p>This section explains <code>fix</code> by refactoring from syntactic recursion to a call of <code>fix</code> instead.</p>
<p>For context, Nix lets you define attributes in terms of other attributes syntactically using the <a href="https://nixos.org/manual/nix/stable/language/constructs.html#recursive-sets"><code>rec { }</code> syntax</a>.</p>
<div class="highlight"><pre><span></span><code>nix-repl<span class="o">&gt;</span> <span class="k">rec</span> <span class="p">{</span>
<span class="ss">foo</span> <span class="o">=</span> <span class="s2">&quot;foo&quot;</span><span class="p">;</span>
<span class="ss">bar</span> <span class="o">=</span> <span class="s2">&quot;bar&quot;</span><span class="p">;</span>
<span class="ss">foobar</span> <span class="o">=</span> foo <span class="o">+</span> bar<span class="p">;</span>
<span class="p">}</span>
<span class="p">{</span> <span class="ss">bar</span> <span class="o">=</span> <span class="s2">&quot;bar&quot;</span><span class="p">;</span> <span class="ss">foo</span> <span class="o">=</span> <span class="s2">&quot;foo&quot;</span><span class="p">;</span> <span class="ss">foobar</span> <span class="o">=</span> <span class="s2">&quot;foobar&quot;</span><span class="p">;</span> <span class="p">}</span>
</code></pre></div>
<p>This is convenient when constructing a value to pass to a function for example,
but an equivalent effect can be achieved with the <code>let</code> binding syntax:</p>
<div class="highlight"><pre><span></span><code>nix-repl<span class="o">&gt;</span> <span class="k">let</span> <span class="ss">self</span> <span class="o">=</span> <span class="p">{</span>
<span class="ss">foo</span> <span class="o">=</span> <span class="s2">&quot;foo&quot;</span><span class="p">;</span>
<span class="ss">bar</span> <span class="o">=</span> <span class="s2">&quot;bar&quot;</span><span class="p">;</span>
<span class="ss">foobar</span> <span class="o">=</span> self<span class="o">.</span>foo <span class="o">+</span> self<span class="o">.</span>bar<span class="p">;</span>
<span class="p">};</span> <span class="k">in</span> self
<span class="p">{</span> <span class="ss">bar</span> <span class="o">=</span> <span class="s2">&quot;bar&quot;</span><span class="p">;</span> <span class="ss">foo</span> <span class="o">=</span> <span class="s2">&quot;foo&quot;</span><span class="p">;</span> <span class="ss">foobar</span> <span class="o">=</span> <span class="s2">&quot;foobar&quot;</span><span class="p">;</span> <span class="p">}</span>
</code></pre></div>
<p>But in general you can get more reuse out of <code>let</code> bindings by refactoring them to a function.</p>
<div class="highlight"><pre><span></span><code>nix-repl<span class="o">&gt;</span> <span class="ss">f</span> <span class="o">=</span> self<span class="p">:</span> <span class="p">{</span>
<span class="ss">foo</span> <span class="o">=</span> <span class="s2">&quot;foo&quot;</span><span class="p">;</span>
<span class="ss">bar</span> <span class="o">=</span> <span class="s2">&quot;bar&quot;</span><span class="p">;</span>
<span class="ss">foobar</span> <span class="o">=</span> self<span class="o">.</span>foo <span class="o">+</span> self<span class="o">.</span>bar<span class="p">;</span>
<span class="p">}</span>
</code></pre></div>
<p>This is where <code>fix</code> comes in, it contains the syntactic recursion that's not in <code>f</code> anymore.</p>
<div class="highlight"><pre><span></span><code>nix-repl<span class="o">&gt;</span> <span class="ss">fix</span> <span class="o">=</span> f<span class="p">:</span>
<span class="k">let</span> <span class="ss">self</span> <span class="o">=</span> f self<span class="p">;</span> <span class="k">in</span> self<span class="p">;</span>
</code></pre></div>
<p>By applying <code>fix</code> we get the final result.</p>
<div class="highlight"><pre><span></span><code>nix-repl<span class="o">&gt;</span> fix f
<span class="p">{</span> <span class="ss">bar</span> <span class="o">=</span> <span class="s2">&quot;bar&quot;</span><span class="p">;</span> <span class="ss">foo</span> <span class="o">=</span> <span class="s2">&quot;foo&quot;</span><span class="p">;</span> <span class="ss">foobar</span> <span class="o">=</span> <span class="s2">&quot;foobar&quot;</span><span class="p">;</span> <span class="p">}</span>
</code></pre></div>
<p>Such a refactored <code>f</code> using <code>fix</code> is not useful by itself.
See <a href="#function-library-lib.fixedPoints.extends"><code>extends</code></a> for an example use case.
There <code>self</code> is also often called <code>final</code>.</p>
<h3 id="inputs">Inputs</h3>
<p><code>f</code></p>
<p>: 1. Function argument</p>
<h3 id="type">Type</h3>
<div class="highlight"><pre><span></span><code>fix :: (a -&gt; a) -&gt; a
</code></pre></div>
<h3 id="examples">Examples</h3>
<div class="admonition example">
<p class="admonition-title"><code>lib.fixedPoints.fix</code> usage example</p>
<div class="highlight"><pre><span></span><code>fix <span class="p">(</span>self<span class="p">:</span> <span class="p">{</span> <span class="ss">foo</span> <span class="o">=</span> <span class="s2">&quot;foo&quot;</span><span class="p">;</span> <span class="ss">bar</span> <span class="o">=</span> <span class="s2">&quot;bar&quot;</span><span class="p">;</span> <span class="ss">foobar</span> <span class="o">=</span> self<span class="o">.</span>foo <span class="o">+</span> self<span class="o">.</span>bar<span class="p">;</span> <span class="p">})</span>
<span class="o">=&gt;</span> <span class="p">{</span> <span class="ss">bar</span> <span class="o">=</span> <span class="s2">&quot;bar&quot;</span><span class="p">;</span> <span class="ss">foo</span> <span class="o">=</span> <span class="s2">&quot;foo&quot;</span><span class="p">;</span> <span class="ss">foobar</span> <span class="o">=</span> <span class="s2">&quot;foobar&quot;</span><span class="p">;</span> <span class="p">}</span>
fix <span class="p">(</span>self<span class="p">:</span> <span class="p">[</span> <span class="mi">1</span> <span class="mi">2</span> <span class="p">(</span>elemAt self <span class="mi">0</span> <span class="o">+</span> elemAt self <span class="mi">1</span><span class="p">)</span> <span class="p">])</span>
<span class="o">=&gt;</span> <span class="p">[</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="p">]</span>
</code></pre></div>
</div>
<p>Located at <a href="https://github.com/NixOS/nixpkgs/blob/master/lib/fixed-points.nix#L93">lib/fixed-points.nix:93</a> in <code>&lt;nixpkgs&gt;</code>.</p>
<h2 id="function-library-lib.fixedPoints.fix-prime"><code>lib.fixedPoints.fix'</code></h2>
<p>A variant of <code>fix</code> that records the original recursive attribute set in the
result, in an attribute named <code>__unfix__</code>.</p>
<p>This is useful in combination with the <code>extends</code> function to
implement deep overriding.</p>
<h3 id="inputs_1">Inputs</h3>
<p><code>f</code></p>
<p>: 1. Function argument</p>
<p>Located at <a href="https://github.com/NixOS/nixpkgs/blob/master/lib/fixed-points.nix#L109">lib/fixed-points.nix:109</a> in <code>&lt;nixpkgs&gt;</code>.</p>
<h2 id="function-library-lib.fixedPoints.converge"><code>lib.fixedPoints.converge</code></h2>
<p>Return the fixpoint that <code>f</code> converges to when called iteratively, starting
with the input <code>x</code>.</p>
<div class="highlight"><pre><span></span><code>nix-repl&gt; converge (x: x / 2) 16
0
</code></pre></div>
<h3 id="inputs_2">Inputs</h3>
<p><code>f</code></p>
<p>: 1. Function argument</p>
<p><code>x</code></p>
<p>: 2. Function argument</p>
<h3 id="type_1">Type</h3>
<div class="highlight"><pre><span></span><code>(a -&gt; a) -&gt; a -&gt; a
</code></pre></div>
<p>Located at <a href="https://github.com/NixOS/nixpkgs/blob/master/lib/fixed-points.nix#L137">lib/fixed-points.nix:137</a> in <code>&lt;nixpkgs&gt;</code>.</p>
<h2 id="function-library-lib.fixedPoints.extends"><code>lib.fixedPoints.extends</code></h2>
<p>Extend a function using an overlay.</p>
<p>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.</p>
<p>A fixed-point function returning an attribute set has the form</p>
<div class="highlight"><pre><span></span><code>final<span class="p">:</span> <span class="p">{</span> <span class="c1"># attributes }</span>
</code></pre></div>
<p>where <code>final</code> refers to the lazily evaluated attribute set returned by the fixed-point function.</p>
<p>An overlay to such a fixed-point function has the form</p>
<div class="highlight"><pre><span></span><code>final<span class="p">:</span> prev<span class="p">:</span> <span class="p">{</span> <span class="c1"># attributes }</span>
</code></pre></div>
<p>where <code>prev</code> refers to the result of the original function to <code>final</code>, and <code>final</code> is the result of the composition of the overlay and the original function.</p>
<p>Applying an overlay is done with <code>extends</code>:</p>
<div class="highlight"><pre><span></span><code><span class="k">let</span>
<span class="ss">f</span> <span class="o">=</span> final<span class="p">:</span> <span class="p">{</span> <span class="c1"># attributes };</span>
<span class="ss">overlay</span> <span class="o">=</span> final<span class="p">:</span> prev<span class="p">:</span> <span class="p">{</span> <span class="c1"># attributes };</span>
<span class="k">in</span> extends overlay f<span class="p">;</span>
</code></pre></div>
<p>To get the value of <code>final</code>, use <code>lib.fix</code>:</p>
<div class="highlight"><pre><span></span><code><span class="k">let</span>
<span class="ss">f</span> <span class="o">=</span> final<span class="p">:</span> <span class="p">{</span> <span class="c1"># attributes };</span>
<span class="ss">overlay</span> <span class="o">=</span> final<span class="p">:</span> prev<span class="p">:</span> <span class="p">{</span> <span class="c1"># attributes };</span>
<span class="ss">g</span> <span class="o">=</span> extends overlay f<span class="p">;</span>
<span class="k">in</span> fix g
</code></pre></div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The argument to the given fixed-point function after applying an overlay will <em>not</em> refer to its own return value, but rather to the value after evaluating the overlay function.</p>
</div>
<p>The given fixed-point function is called with a separate argument than if it was evaluated with <code>lib.fix</code>.</p>
<div class="admonition example">
<p class="admonition-title">Example</p>
</div>
<h3 id="extend-a-fixed-point-function-with-an-overlay">Extend a fixed-point function with an overlay</h3>
<p>Define a fixed-point function <code>f</code> that expects its own output as the argument <code>final</code>:</p>
<div class="highlight"><pre><span></span><code>f = final: {
# Constant value a
a = 1;
# b depends on the final value of a, available as final.a
b = final.a + 2;
}
</code></pre></div>
<p>Evaluate this using <a href="#function-library-lib.fixedPoints.fix"><code>lib.fix</code></a> to get the final result:</p>
<div class="highlight"><pre><span></span><code>fix f
=&gt; { a = 1; b = 3; }
</code></pre></div>
<p>An overlay represents a modification or extension of such a fixed-point function.
Here's an example of an overlay:</p>
<div class="highlight"><pre><span></span><code>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;
}
</code></pre></div>
<p>Use <code>extends overlay f</code> to apply the overlay to the fixed-point function <code>f</code>.
This produces a new fixed-point function <code>g</code> with the combined behavior of <code>f</code> and <code>overlay</code>:</p>
<div class="highlight"><pre><span></span><code>g = extends overlay f
</code></pre></div>
<p>The result is a function, so we can't print it directly, but it's the same as:</p>
<div class="highlight"><pre><span></span><code>g&#39; = 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;
}
</code></pre></div>
<p>Evaluate this using <a href="#function-library-lib.fixedPoints.fix"><code>lib.fix</code></a> again to get the final result:</p>
<div class="highlight"><pre><span></span><code>fix g
=&gt; { a = 11; b = 13; c = 24; }
</code></pre></div>
<h3 id="inputs_3">Inputs</h3>
<p><code>overlay</code></p>
<p>: The overlay to apply to the fixed-point function</p>
<p><code>f</code></p>
<p>: The fixed-point function</p>
<h3 id="type_2">Type</h3>
<div class="highlight"><pre><span></span><code>extends :: (Attrs -&gt; Attrs -&gt; Attrs) # The overlay to apply to the fixed-point function
-&gt; (Attrs -&gt; Attrs) # A fixed-point function
-&gt; (Attrs -&gt; Attrs) # The resulting fixed-point function
</code></pre></div>
<h3 id="examples_1">Examples</h3>
<div class="admonition example">
<p class="admonition-title"><code>lib.fixedPoints.extends</code> usage example</p>
<div class="highlight"><pre><span></span><code><span class="ss">f</span> <span class="o">=</span> final<span class="p">:</span> <span class="p">{</span> <span class="ss">a</span> <span class="o">=</span> <span class="mi">1</span><span class="p">;</span> <span class="ss">b</span> <span class="o">=</span> final<span class="o">.</span>a <span class="o">+</span> <span class="mi">2</span><span class="p">;</span> <span class="p">}</span>
fix <span class="ss">f</span>
<span class="o">=&gt;</span> <span class="p">{</span> <span class="ss">a</span> <span class="o">=</span> <span class="mi">1</span><span class="p">;</span> <span class="ss">b</span> <span class="o">=</span> <span class="mi">3</span><span class="p">;</span> <span class="p">}</span>
fix <span class="p">(</span>extends <span class="p">(</span>final<span class="p">:</span> prev<span class="p">:</span> <span class="p">{</span> <span class="ss">a</span> <span class="o">=</span> prev<span class="o">.</span>a <span class="o">+</span> <span class="mi">10</span><span class="p">;</span> <span class="p">})</span> f<span class="p">)</span>
<span class="o">=&gt;</span> <span class="p">{</span> <span class="ss">a</span> <span class="o">=</span> <span class="mi">11</span><span class="p">;</span> <span class="ss">b</span> <span class="o">=</span> <span class="mi">13</span><span class="p">;</span> <span class="p">}</span>
fix <span class="p">(</span>extends <span class="p">(</span>final<span class="p">:</span> prev<span class="p">:</span> <span class="p">{</span> <span class="ss">b</span> <span class="o">=</span> final<span class="o">.</span>a <span class="o">+</span> <span class="mi">5</span><span class="p">;</span> <span class="p">})</span> f<span class="p">)</span>
<span class="o">=&gt;</span> <span class="p">{</span> <span class="ss">a</span> <span class="o">=</span> <span class="mi">1</span><span class="p">;</span> <span class="ss">b</span> <span class="o">=</span> <span class="mi">6</span><span class="p">;</span> <span class="p">}</span>
fix <span class="p">(</span>extends <span class="p">(</span>final<span class="p">:</span> prev<span class="p">:</span> <span class="p">{</span> <span class="ss">c</span> <span class="o">=</span> final<span class="o">.</span>a <span class="o">+</span> final<span class="o">.</span>b<span class="p">;</span> <span class="p">})</span> f<span class="p">)</span>
<span class="o">=&gt;</span> <span class="p">{</span> <span class="ss">a</span> <span class="o">=</span> <span class="mi">1</span><span class="p">;</span> <span class="ss">b</span> <span class="o">=</span> <span class="mi">3</span><span class="p">;</span> <span class="ss">c</span> <span class="o">=</span> <span class="mi">4</span><span class="p">;</span> <span class="p">}</span>
</code></pre></div>
</div>
<p>Located at <a href="https://github.com/NixOS/nixpkgs/blob/master/lib/fixed-points.nix#L301">lib/fixed-points.nix:301</a> in <code>&lt;nixpkgs&gt;</code>.</p>
<h2 id="function-library-lib.fixedPoints.composeExtensions"><code>lib.fixedPoints.composeExtensions</code></h2>
<p>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</p>
<h3 id="inputs_4">Inputs</h3>
<p><code>f</code></p>
<p>: 1. Function argument</p>
<p><code>g</code></p>
<p>: 2. Function argument</p>
<p><code>final</code></p>
<p>: 3. Function argument</p>
<p><code>prev</code></p>
<p>: 4. Function argument</p>
<p>Located at <a href="https://github.com/NixOS/nixpkgs/blob/master/lib/fixed-points.nix#L337">lib/fixed-points.nix:337</a> in <code>&lt;nixpkgs&gt;</code>.</p>
<h2 id="function-library-lib.fixedPoints.composeManyExtensions"><code>lib.fixedPoints.composeManyExtensions</code></h2>
<p>Compose several extending functions of the type expected by 'extends' into
one where changes made in preceding functions are made available to
subsequent ones.</p>
<div class="highlight"><pre><span></span><code>composeManyExtensions : [packageSet -&gt; packageSet -&gt; packageSet] -&gt; packageSet -&gt; packageSet -&gt; packageSet
^final ^prev ^overrides ^final ^prev ^overrides
</code></pre></div>
<p>Located at <a href="https://github.com/NixOS/nixpkgs/blob/master/lib/fixed-points.nix#L353">lib/fixed-points.nix:353</a> in <code>&lt;nixpkgs&gt;</code>.</p>
<h2 id="function-library-lib.fixedPoints.makeExtensible"><code>lib.fixedPoints.makeExtensible</code></h2>
<p>Create an overridable, recursive attribute set. For example:</p>
<div class="highlight"><pre><span></span><code>nix-repl&gt; obj = makeExtensible (self: { })
nix-repl&gt; obj
{ __unfix__ = «lambda»; extend = «lambda»; }
nix-repl&gt; obj = obj.extend (self: super: { foo = &quot;foo&quot;; })
nix-repl&gt; obj
{ __unfix__ = «lambda»; extend = «lambda»; foo = &quot;foo&quot;; }
nix-repl&gt; obj = obj.extend (self: super: { foo = super.foo + &quot; + &quot;; bar = &quot;bar&quot;; foobar = self.foo + self.bar; })
nix-repl&gt; obj
{ __unfix__ = «lambda»; bar = &quot;bar&quot;; extend = «lambda»; foo = &quot;foo + &quot;; foobar = &quot;foo + bar&quot;; }
</code></pre></div>
<p>Located at <a href="https://github.com/NixOS/nixpkgs/blob/master/lib/fixed-points.nix#L376">lib/fixed-points.nix:376</a> in <code>&lt;nixpkgs&gt;</code>.</p>
<h2 id="function-library-lib.fixedPoints.makeExtensibleWithCustomName"><code>lib.fixedPoints.makeExtensibleWithCustomName</code></h2>
<p>Same as <code>makeExtensible</code> but the name of the extending attribute is
customized.</p>
<h3 id="inputs_5">Inputs</h3>
<p><code>extenderName</code></p>
<p>: 1. Function argument</p>
<p><code>rattrs</code></p>
<p>: 2. Function argument</p>
<p>Located at <a href="https://github.com/NixOS/nixpkgs/blob/master/lib/fixed-points.nix#L393">lib/fixed-points.nix:393</a> in <code>&lt;nixpkgs&gt;</code>.</p>
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