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hole_commands.html
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<!DOCTYPE html>
<html lang="en">
<head>
<link rel="stylesheet" href="https://pygae.github.io/lean-ga-docs/style.css">
<link rel="stylesheet" href="https://pygae.github.io/lean-ga-docs/pygments.css">
<link rel="stylesheet" href="https://pygae.github.io/lean-ga-docs/pygments-dark.css">
<link rel="shortcut icon" href="https://pygae.github.io/lean-ga-docs/favicon.ico">
<title>Hole commands - mathlib3 docs</title>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<meta name="description" content="Both VS Code and emacs support integration for 'hole commands'. In VS Code, if you enter `{! !}`, a small light bulb symbol will appear, and clicking on it gives a drop down menu of available hole…" />
<link rel="canonical" href="https://leanprover-community.github.io/mathlib_docs/hole_commands.html" />
<meta property="og:title" content="Hole commands - mathlib3 docs">
<meta property="og:site_name" content="mathlib for Lean 3 - API documentation">
<meta property="og:description" content="Both VS Code and emacs support integration for 'hole commands'. In VS Code, if you enter `{! !}`, a small light bulb symbol will appear, and clicking on it gives a drop down menu of available hole…">
<meta property="og:image" content="https://pygae.github.io/lean-ga-docs/meta-og.png">
<meta name="twitter:card" content="summary">
<script src="https://pygae.github.io/lean-ga-docs/color_scheme.js"></script>
</head>
<body>
<input id="nav_toggle" type="checkbox">
<header>
<h1><label for="nav_toggle"></label><a href="https://leanprover-community.github.io/lean3">mathlib3</a>
<span>documentation</span></h1>
<p class="header_filename break_within">Hole commands</p>
<form action="https://google.com/search" method="get" id="search_form">
<input type="hidden" name="sitesearch" value="https://leanprover-community.github.io/mathlib_docs">
<input type="text" name="q" autocomplete="off">
<button>Google site search</button>
</form>
</header>
<nav class="internal_nav">
<h3><a href="#top">Hole commands</a></h3>
<details class="tagfilter-div">
<summary>Filter by tag</summary>
<label><input type="checkbox" id="tagfilter-selectall" name="tagfilter-selectall">Select/deselect all</label>
<br><hr>
<label><input type="checkbox" class="tagfilter" name="Try-this" value="Try-this">Try this</label>
<br>
<label><input type="checkbox" class="tagfilter" name="goal-information" value="goal-information">goal information</label>
<br>
<label><input type="checkbox" class="tagfilter" name="instances" value="instances">instances</label>
<br>
<label><input type="checkbox" class="tagfilter" name="linting" value="linting">linting</label>
<br>
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<br>
<label><input type="checkbox" class="tagfilter" name="search" value="search">search</label>
<br>
</details>
<div class="taclink pattern-matching"><a href="#Equations Stub">Equations Stub</a></div>
<div class="taclink linting"><a href="#Lint">Lint</a></div>
<div class="taclink goal-information"><a href="#List Constructors">List Constructors</a></div>
<div class="taclink pattern-matching"><a href="#Match Stub">Match Stub</a></div>
<div class="taclink instances"><a href="#instance_stub">instance_stub</a></div>
<div class="taclink search Try-this"><a href="#library_search">library_search</a></div>
<div class="taclink search"><a href="#tidy">tidy</a></div>
</nav>
<main>
<a id="top"></a>
<div class="docfile">
<h1>Hole commands</h1>
<p>Both VS Code and emacs support integration for 'hole commands'.</p>
<p>In VS Code, if you enter <code>{! !}</code>, a small light bulb symbol will appear, and
clicking on it gives a drop down menu of available hole commands. Running one
of these will replace the <code>{! !}</code> with whatever text that hole command provides.</p>
<p>As of Lean 3.16.0c, hole commands can also be activated at underscores <code>_</code>.</p>
<p>In emacs, you can do something similar with <code>C-c SPC</code>.</p>
<p>Many of these commands are available whenever <code>tactic.core</code> is imported.
Commands that require additional imports are noted below.
All should be available with <code>import <a href="https://pygae.github.io/lean-ga-docs/init/meta/tactic.html#tactic">tactic</a></code>.</p>
<div class="tactic pattern-matching">
<h2 id="Equations Stub"><a href="#Equations Stub">Equations Stub</a></h2>
<p>Invoking hole command "Equations Stub" ("Generate a list of equations for a recursive definition")
in the following:</p>
<div class="codehilite"><pre><span></span><code><span class="kd">meta</span> <span class="kd">def</span> <span class="n">foo</span> <span class="o">:</span> <span class="o">{</span><span class="bp">!</span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr">expr</a></span> <span class="bp">→</span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/tactic.html#tactic">tactic</a></span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/core.html#unit">unit</a></span> <span class="bp">!</span><span class="o">}</span> <span class="c1">-- `:=` is omitted</span>
</code></pre></div>
<p>produces:</p>
<div class="codehilite"><pre><span></span><code><span class="kd">meta</span> <span class="kd">def</span> <span class="n">foo</span> <span class="o">:</span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr">expr</a></span> <span class="bp">→</span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/tactic.html#tactic">tactic</a></span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/core.html#unit">unit</a></span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.var">expr.var</a></span> <span class="n">a</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.sort">expr.sort</a></span> <span class="n">a</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.const">expr.const</a></span> <span class="n">a</span> <span class="n">a_1</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.mvar">expr.mvar</a></span> <span class="n">a</span> <span class="n">a_1</span> <span class="n">a_2</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.local_const">expr.local_const</a></span> <span class="n">a</span> <span class="n">a_1</span> <span class="n">a_2</span> <span class="n">a_3</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.app">expr.app</a></span> <span class="n">a</span> <span class="n">a_1</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.lam">expr.lam</a></span> <span class="n">a</span> <span class="n">a_1</span> <span class="n">a_2</span> <span class="n">a_3</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.pi">expr.pi</a></span> <span class="n">a</span> <span class="n">a_1</span> <span class="n">a_2</span> <span class="n">a_3</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.elet">expr.elet</a></span> <span class="n">a</span> <span class="n">a_1</span> <span class="n">a_2</span> <span class="n">a_3</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.macro">expr.macro</a></span> <span class="n">a</span> <span class="n">a_1</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
</code></pre></div>
<p>A similar result can be obtained by invoking "Equations Stub" on the following:</p>
<div class="codehilite"><pre><span></span><code><span class="kd">meta</span> <span class="kd">def</span> <span class="n">foo</span> <span class="o">:</span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr">expr</a></span> <span class="bp">→</span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/tactic.html#tactic">tactic</a></span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/core.html#unit">unit</a></span> <span class="o">:=</span> <span class="c1">-- do not forget to write `:=`!!</span>
<span class="o">{</span><span class="bp">!</span> <span class="bp">!</span><span class="o">}</span>
</code></pre></div>
<div class="codehilite"><pre><span></span><code><span class="kd">meta</span> <span class="kd">def</span> <span class="n">foo</span> <span class="o">:</span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr">expr</a></span> <span class="bp">→</span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/tactic.html#tactic">tactic</a></span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/core.html#unit">unit</a></span> <span class="o">:=</span> <span class="c1">-- don't forget to erase `:=`!!</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.var">expr.var</a></span> <span class="n">a</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.sort">expr.sort</a></span> <span class="n">a</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.const">expr.const</a></span> <span class="n">a</span> <span class="n">a_1</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.mvar">expr.mvar</a></span> <span class="n">a</span> <span class="n">a_1</span> <span class="n">a_2</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.local_const">expr.local_const</a></span> <span class="n">a</span> <span class="n">a_1</span> <span class="n">a_2</span> <span class="n">a_3</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.app">expr.app</a></span> <span class="n">a</span> <span class="n">a_1</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.lam">expr.lam</a></span> <span class="n">a</span> <span class="n">a_1</span> <span class="n">a_2</span> <span class="n">a_3</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.pi">expr.pi</a></span> <span class="n">a</span> <span class="n">a_1</span> <span class="n">a_2</span> <span class="n">a_3</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.elet">expr.elet</a></span> <span class="n">a</span> <span class="n">a_1</span> <span class="n">a_2</span> <span class="n">a_3</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.macro">expr.macro</a></span> <span class="n">a</span> <span class="n">a_1</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
</code></pre></div>
<div class="tags">Tags:
<ul>
<li>pattern matching</li>
</ul>
</div>
<details class="rel_decls"><summary>Related declarations</summary>
<ul>
<li><a href="https://pygae.github.io/lean-ga-docs/tactic/core.html#tactic.eqn_stub">tactic.eqn_stub</a></li>
</ul>
</details>
<details class="imports"><summary>Import using</summary><ul><li>import tactic.core</li>
<li>import tactic.basic</li></ul></details>
</div>
<div class="tactic linting">
<h2 id="Lint"><a href="#Lint">Lint</a></h2>
<p>Invoking the hole command <code><a href="https://pygae.github.io/lean-ga-docs/tactic/lint/frontend.html#lint">lint</a></code> ("Find common mistakes in current file") will print text that
indicates mistakes made in the file above the command. It is equivalent to copying and pasting the
output of <code>#lint</code>. On large files, it may take some time before the output appears.</p>
<div class="tags">Tags:
<ul>
<li>linting</li>
</ul>
</div>
<details class="rel_decls"><summary>Related declarations</summary>
<ul>
<li><a href="https://pygae.github.io/lean-ga-docs/tactic/lint/frontend.html#lint_hole_cmd">lint_hole_cmd</a></li>
</ul>
</details>
<details class="imports"><summary>Import using</summary><ul><li>import tactic.lint.frontend</li>
<li>import tactic.basic</li></ul></details>
</div>
<div class="tactic goal-information">
<h2 id="List Constructors"><a href="#List Constructors">List Constructors</a></h2>
<p>This command lists the constructors that can be used to satisfy the expected type.</p>
<p>Invoking "List Constructors" ("Show the list of constructors of the expected type")
in the following hole:</p>
<div class="codehilite"><pre><span></span><code><span class="kd">def</span> <span class="n">foo</span> <span class="o">:</span> <span class="n">ℤ</span> <span class="bp">⊕</span> <span class="n">ℕ</span> <span class="o">:=</span>
<span class="o">{</span><span class="bp">!</span> <span class="bp">!</span><span class="o">}</span>
</code></pre></div>
<p>produces:</p>
<div class="codehilite"><pre><span></span><code><span class="kd">def</span> <span class="n">foo</span> <span class="o">:</span> <span class="n">ℤ</span> <span class="bp">⊕</span> <span class="n">ℕ</span> <span class="o">:=</span>
<span class="o">{</span><span class="bp">!</span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/core.html#sum.inl">sum.inl</a></span><span class="o">,</span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/core.html#sum.inr">sum.inr</a></span> <span class="bp">!</span><span class="o">}</span>
</code></pre></div>
<p>and will display:</p>
<div class="codehilite"><pre><span></span><code><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/core.html#sum.inl">sum.inl</a></span> <span class="o">:</span> <span class="n">ℤ</span> <span class="bp">→</span> <span class="n">ℤ</span> <span class="bp">⊕</span> <span class="n">ℕ</span>
<span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/core.html#sum.inr">sum.inr</a></span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">ℤ</span> <span class="bp">⊕</span> <span class="n">ℕ</span>
<span class="bp">```</span>
</code></pre></div>
<div class="tags">Tags:
<ul>
<li>goal information</li>
</ul>
</div>
<details class="rel_decls"><summary>Related declarations</summary>
<ul>
<li><a href="https://pygae.github.io/lean-ga-docs/tactic/core.html#tactic.list_constructors_hole">tactic.list_constructors_hole</a></li>
</ul>
</details>
<details class="imports"><summary>Import using</summary><ul><li>import tactic.core</li>
<li>import tactic.basic</li></ul></details>
</div>
<div class="tactic pattern-matching">
<h2 id="Match Stub"><a href="#Match Stub">Match Stub</a></h2>
<p>Hole command used to generate a <code>match</code> expression.</p>
<p>In the following:</p>
<div class="codehilite"><pre><span></span><code><span class="kd">meta</span> <span class="kd">def</span> <span class="n">foo</span> <span class="o">(</span><span class="n">e</span> <span class="o">:</span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr">expr</a></span><span class="o">)</span> <span class="o">:</span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/tactic.html#tactic">tactic</a></span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/core.html#unit">unit</a></span> <span class="o">:=</span>
<span class="o">{</span><span class="bp">!</span> <span class="n">e</span> <span class="bp">!</span><span class="o">}</span>
</code></pre></div>
<p>invoking hole command "Match Stub" ("Generate a list of equations for a <code>match</code> expression")
produces:</p>
<div class="codehilite"><pre><span></span><code><span class="kd">meta</span> <span class="kd">def</span> <span class="n">foo</span> <span class="o">(</span><span class="n">e</span> <span class="o">:</span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr">expr</a></span><span class="o">)</span> <span class="o">:</span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/tactic.html#tactic">tactic</a></span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/core.html#unit">unit</a></span> <span class="o">:=</span>
<span class="k">match</span> <span class="n">e</span> <span class="k">with</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.var">expr.var</a></span> <span class="n">a</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.sort">expr.sort</a></span> <span class="n">a</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.const">expr.const</a></span> <span class="n">a</span> <span class="n">a_1</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.mvar">expr.mvar</a></span> <span class="n">a</span> <span class="n">a_1</span> <span class="n">a_2</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.local_const">expr.local_const</a></span> <span class="n">a</span> <span class="n">a_1</span> <span class="n">a_2</span> <span class="n">a_3</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.app">expr.app</a></span> <span class="n">a</span> <span class="n">a_1</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.lam">expr.lam</a></span> <span class="n">a</span> <span class="n">a_1</span> <span class="n">a_2</span> <span class="n">a_3</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.pi">expr.pi</a></span> <span class="n">a</span> <span class="n">a_1</span> <span class="n">a_2</span> <span class="n">a_3</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.elet">expr.elet</a></span> <span class="n">a</span> <span class="n">a_1</span> <span class="n">a_2</span> <span class="n">a_3</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="bp">|</span> <span class="o">(</span><span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html#expr.macro">expr.macro</a></span> <span class="n">a</span> <span class="n">a_1</span><span class="o">)</span> <span class="o">:=</span> <span class="n">_</span>
<span class="kd">end</span>
</code></pre></div>
<div class="tags">Tags:
<ul>
<li>pattern matching</li>
</ul>
</div>
<details class="rel_decls"><summary>Related declarations</summary>
<ul>
<li><a href="https://pygae.github.io/lean-ga-docs/tactic/core.html#tactic.match_stub">tactic.match_stub</a></li>
</ul>
</details>
<details class="imports"><summary>Import using</summary><ul><li>import tactic.core</li>
<li>import tactic.basic</li></ul></details>
</div>
<div class="tactic instances">
<h2 id="instance_stub"><a href="#instance_stub">instance_stub</a></h2>
<p>Hole command used to fill in a structure's field when specifying an instance.</p>
<p>In the following:</p>
<div class="codehilite"><pre><span></span><code><span class="kd">instance</span> <span class="o">:</span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/control/monad.html#monad">monad</a></span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/logic.html#id">id</a></span> <span class="o">:=</span>
<span class="o">{</span><span class="bp">!</span> <span class="bp">!</span><span class="o">}</span>
</code></pre></div>
<p>invoking the hole command "Instance Stub" ("Generate a skeleton for the structure under
construction.") produces:</p>
<div class="codehilite"><pre><span></span><code><span class="kd">instance</span> <span class="o">:</span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/control/monad.html#monad">monad</a></span> <span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/logic.html#id">id</a></span> <span class="o">:=</span>
<span class="o">{</span> <span class="n">map</span> <span class="o">:=</span> <span class="n">_</span><span class="o">,</span>
<span class="n">map_const</span> <span class="o">:=</span> <span class="n">_</span><span class="o">,</span>
<span class="n">pure</span> <span class="o">:=</span> <span class="n">_</span><span class="o">,</span>
<span class="n">seq</span> <span class="o">:=</span> <span class="n">_</span><span class="o">,</span>
<span class="n">seq_left</span> <span class="o">:=</span> <span class="n">_</span><span class="o">,</span>
<span class="n">seq_right</span> <span class="o">:=</span> <span class="n">_</span><span class="o">,</span>
<span class="n">bind</span> <span class="o">:=</span> <span class="n">_</span> <span class="o">}</span>
</code></pre></div>
<div class="tags">Tags:
<ul>
<li>instances</li>
</ul>
</div>
<details class="rel_decls"><summary>Related declarations</summary>
<ul>
<li><a href="https://pygae.github.io/lean-ga-docs/tactic/core.html#tactic.instance_stub">tactic.instance_stub</a></li>
</ul>
</details>
<details class="imports"><summary>Import using</summary><ul><li>import tactic.core</li>
<li>import tactic.basic</li></ul></details>
</div>
<div class="tactic search Try-this">
<h2 id="library_search"><a href="#library_search">library_search</a></h2>
<p>Invoking the hole command <code>library_search</code> ("Use <code>library_search</code> to complete the goal") calls
the tactic <code>library_search</code> to produce a proof term with the type of the hole.</p>
<p>Running it on</p>
<div class="codehilite"><pre><span></span><code><span class="kd">example</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="mi">1</span> <span class="o">:=</span>
<span class="o">{</span><span class="bp">!!</span><span class="o">}</span>
</code></pre></div>
<p>produces</p>
<div class="codehilite"><pre><span></span><code><span class="kd">example</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="mi">1</span> <span class="o">:=</span>
<span class="n"><a href="https://pygae.github.io/lean-ga-docs/init/data/nat/lemmas.html#nat.one_pos">nat.one_pos</a></span>
</code></pre></div>
<div class="tags">Tags:
<ul>
<li>search</li>
<li>Try this</li>
</ul>
</div>
<details class="rel_decls"><summary>Related declarations</summary>
<ul>
<li><a href="https://pygae.github.io/lean-ga-docs/tactic/suggest.html#tactic.library_search_hole_cmd">tactic.library_search_hole_cmd</a></li>
</ul>
</details>
<details class="imports"><summary>Import using</summary><ul><li>import tactic.suggest</li>
<li>import tactic.basic</li></ul></details>
</div>
<div class="tactic search">
<h2 id="tidy"><a href="#tidy">tidy</a></h2>
<p>Invoking the hole command <code>tidy</code> ("Use <code>tidy</code> to complete the goal") runs the tactic of
the same name, replacing the hole with the tactic script <code>tidy</code> produces.</p>
<div class="tags">Tags:
<ul>
<li>search</li>
</ul>
</div>
<details class="rel_decls"><summary>Related declarations</summary>
<ul>
<li><a href="https://pygae.github.io/lean-ga-docs/tactic/tidy.html#tactic.tidy_hole_cmd">tactic.tidy_hole_cmd</a></li>
</ul>
</details>
<details class="imports"><summary>Import using</summary><ul><li>import tactic.tidy</li>
<li>import tactic.basic</li></ul></details>
</div>
</div>
</main>
<nav class="nav">
<h3>General documentation</h3>
<div class="nav_link"><a href="https://pygae.github.io/lean-ga-docs/">index</a></div>
<div class="nav_link"><a href="https://pygae.github.io/lean-ga-docs/foundational_types.html">foundational types</a></div>
<div class="nav_link"><a href="https://pygae.github.io/lean-ga-docs/tactics.html">tactics</a></div>
<div class="nav_link"><a href="https://pygae.github.io/lean-ga-docs/commands.html">commands</a></div>
<div class="nav_link"><a href="https://pygae.github.io/lean-ga-docs/hole_commands.html">hole commands</a></div>
<div class="nav_link"><a href="https://pygae.github.io/lean-ga-docs/attributes.html">attributes</a></div>
<div class="nav_link"><a href="https://pygae.github.io/lean-ga-docs/notes.html">notes</a></div>
<div class="nav_link"><a href="https://pygae.github.io/lean-ga-docs/references.html">references</a></div>
<h3>Additional documentation</h3>
<h3>Library</h3>
<h4>core</h4>
<details class="nav_sect" data-path="data" >
<summary>data</summary>
<details class="nav_sect" data-path="data/buffer" >
<summary>buffer</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/data/buffer/parser.html">parser</a>
</div>
</details>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/data/buffer.html">buffer</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/data/dlist.html">dlist</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/data/vector.html">vector</a>
</div>
</details>
<details class="nav_sect" data-path="init" >
<summary>init</summary>
<details class="nav_sect" data-path="init/algebra" >
<summary>algebra</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/algebra/classes.html">classes</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/algebra/functions.html">functions</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/algebra/order.html">order</a>
</div>
</details>
<details class="nav_sect" data-path="init/control" >
<summary>control</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/alternative.html">alternative</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/applicative.html">applicative</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/combinators.html">combinators</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/except.html">except</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/functor.html">functor</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/id.html">id</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/lawful.html">lawful</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/lift.html">lift</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/monad.html">monad</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/monad_fail.html">monad_fail</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/option.html">option</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/reader.html">reader</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/state.html">state</a>
</div>
</details>
<details class="nav_sect" data-path="init/data" >
<summary>data</summary>
<details class="nav_sect" data-path="init/data/array" >
<summary>array</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/array/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/array/slice.html">slice</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/bool" >
<summary>bool</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/bool/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/bool/lemmas.html">lemmas</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/char" >
<summary>char</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/char/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/char/classes.html">classes</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/char/lemmas.html">lemmas</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/fin" >
<summary>fin</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/fin/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/fin/ops.html">ops</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/int" >
<summary>int</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/int/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/int/bitwise.html">bitwise</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/int/comp_lemmas.html">comp_lemmas</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/int/order.html">order</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/list" >
<summary>list</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/list/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/list/instances.html">instances</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/list/lemmas.html">lemmas</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/list/qsort.html">qsort</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/nat" >
<summary>nat</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/nat/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/nat/bitwise.html">bitwise</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/nat/div.html">div</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/nat/gcd.html">gcd</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/nat/lemmas.html">lemmas</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/option" >
<summary>option</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/option/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/option/instances.html">instances</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/ordering" >
<summary>ordering</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/ordering/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/ordering/lemmas.html">lemmas</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/sigma" >
<summary>sigma</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/sigma/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/sigma/lex.html">lex</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/string" >
<summary>string</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/string/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/string/ops.html">ops</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/subtype" >
<summary>subtype</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/subtype/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/subtype/instances.html">instances</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/sum" >
<summary>sum</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/sum/basic.html">basic</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/unsigned" >
<summary>unsigned</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/unsigned/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/unsigned/ops.html">ops</a>
</div>
</details>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/prod.html">prod</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/punit.html">punit</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/quot.html">quot</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/repr.html">repr</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/set.html">set</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/setoid.html">setoid</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/to_string.html">to_string</a>
</div>
</details>
<details class="nav_sect" data-path="init/meta" >
<summary>meta</summary>
<details class="nav_sect" data-path="init/meta/converter" >
<summary>converter</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/converter/conv.html">conv</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/converter/interactive.html">interactive</a>
</div>
</details>
<details class="nav_sect" data-path="init/meta/lean" >
<summary>lean</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/lean/parser.html">parser</a>
</div>
</details>
<details class="nav_sect" data-path="init/meta/smt" >
<summary>smt</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/smt/congruence_closure.html">congruence_closure</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/smt/ematch.html">ematch</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/smt/interactive.html">interactive</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/smt/rsimp.html">rsimp</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/smt/smt_tactic.html">smt_tactic</a>
</div>
</details>
<details class="nav_sect" data-path="init/meta/widget" >
<summary>widget</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/widget/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/widget/html_cmd.html">html_cmd</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/widget/interactive_expr.html">interactive_expr</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/widget/replace_save_info.html">replace_save_info</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/widget/tactic_component.html">tactic_component</a>
</div>
</details>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/ac_tactics.html">ac_tactics</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/async_tactic.html">async_tactic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/attribute.html">attribute</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/backward.html">backward</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/case_tag.html">case_tag</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/comp_value_tactics.html">comp_value_tactics</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/congr_lemma.html">congr_lemma</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/congr_tactic.html">congr_tactic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/constructor_tactic.html">constructor_tactic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/contradiction_tactic.html">contradiction_tactic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/declaration.html">declaration</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/derive.html">derive</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/environment.html">environment</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/exceptional.html">exceptional</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html">expr</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/expr_address.html">expr_address</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/feature_search.html">feature_search</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/float.html">float</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/format.html">format</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/fun_info.html">fun_info</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/has_reflect.html">has_reflect</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/hole_command.html">hole_command</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/injection_tactic.html">injection_tactic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/instance_cache.html">instance_cache</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/interaction_monad.html">interaction_monad</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/interactive.html">interactive</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/interactive_base.html">interactive_base</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/json.html">json</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/level.html">level</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/local_context.html">local_context</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/match_tactic.html">match_tactic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/mk_dec_eq_instance.html">mk_dec_eq_instance</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/mk_has_reflect_instance.html">mk_has_reflect_instance</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/mk_has_sizeof_instance.html">mk_has_sizeof_instance</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/mk_inhabited_instance.html">mk_inhabited_instance</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/module_info.html">module_info</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/name.html">name</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/occurrences.html">occurrences</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/options.html">options</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/pexpr.html">pexpr</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/rb_map.html">rb_map</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/rec_util.html">rec_util</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/ref.html">ref</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/relation_tactics.html">relation_tactics</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/rewrite_tactic.html">rewrite_tactic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/set_get_option_tactics.html">set_get_option_tactics</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/simp_tactic.html">simp_tactic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/tactic.html">tactic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/tagged_format.html">tagged_format</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/task.html">task</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/type_context.html">type_context</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/vm.html">vm</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/well_founded_tactics.html">well_founded_tactics</a>
</div>
</details>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/cc_lemmas.html">cc_lemmas</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/classical.html">classical</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/coe.html">coe</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/core.html">core</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/default.html">default</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/function.html">function</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/funext.html">funext</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/ite_simp.html">ite_simp</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/logic.html">logic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/propext.html">propext</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/util.html">util</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/version.html">version</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/wf.html">wf</a>
</div>
</details>
<details class="nav_sect" data-path="system" >
<summary>system</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/system/io.html">io</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/system/io_interface.html">io_interface</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/system/random.html">random</a>
</div>
</details>
<h4>lean-ga</h4>
<details class="nav_sect" data-path="examples" >
<summary>examples</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/examples/recursors.html">recursors</a>
</div>
</details>
<details class="nav_sect" data-path="for_mathlib" >
<summary>for_mathlib</summary>
<details class="nav_sect" data-path="for_mathlib/algebra" >
<summary>algebra</summary>
<details class="nav_sect" data-path="for_mathlib/algebra/algebra" >
<summary>algebra</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/for_mathlib/algebra/algebra/opposite.html">opposite</a>
</div>
</details>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/for_mathlib/algebra/algebra_operations.html">algebra_operations</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/for_mathlib/algebra/center_submonoid.html">center_submonoid</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/for_mathlib/algebra/filtration.html">filtration</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/for_mathlib/algebra/monoid_algebra.html">monoid_algebra</a>
</div>
<div class="nav_link ">