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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>Index - mathlib3 docs</title>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<meta name="description" content="The Lean 3 mathematical library, mathlib, is a community-driven effort to build a unified library of mathematics formalized in the Lean proof assistant." />
<link rel="canonical" href="https://leanprover-community.github.io/mathlib_docs/index.html" />
<meta property="og:title" content="Index - mathlib3 docs">
<meta property="og:site_name" content="mathlib for Lean 3 - API documentation">
<meta property="og:description" content="The Lean 3 mathematical library, mathlib, is a community-driven effort to build a unified library of mathematics formalized in the Lean proof assistant.">
<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">Index</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"></nav>
<main>
<a id="top"></a>
<h1>Welcome to mathlib 3's documentation page</h1>
<p>This is the API reference for mathlib 3, the library of mathematics developed for Lean 3.
If you need information about installing Lean or mathlib, or getting started with a project, please
visit our <a href="https://leanprover-community.github.io">community website</a>.<p>
<p>For documentation for the Lean 4 version, see <a href="https://leanprover-community.github.io/mathlib4_docs">mathlib4's documentation</a>.</p>
<p>This documentation was automatically generated using
<a href="https://github.com/leanprover-community/doc-gen">doc-gen</a> on the following source commits:</p>
<ul>
<li>mathlib <a href="https://github.com/leanprover-community/mathlib/tree/bf2428c9486c407ca38b5b3fb10b87dad0bc99fa">bf2428c9486c407ca38b5b3fb10b87dad0bc99fa</a></li>
<li>Lean <a href="https://github.com/leanprover-community/lean/blob/cce7990ea86a78bdb383e38ed7f9b5ba93c60ce0/library">cce7990ea86a78bdb383e38ed7f9b5ba93c60ce0</a></li>
<li>doc-gen <a href="https://github.com/leanprover-community/doc-gen/tree/0ab85f2ad359ecb52311f31bbf61ea9cf60ffb4c">0ab85f2ad359ecb52311f31bbf61ea9cf60ffb4c</a></li>
</ul>
<p>On the left menu, the "General" and "Additional documentation" sections provide links with extra
information about mathlib and related topics from our community, respectively. The "Library" section
contains dropdown/expandable subsections according to mathlib's tree structure.</p>
</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 ">
<a href="https://pygae.github.io/lean-ga-docs/for_mathlib/algebra/ring_quot.html">ring_quot</a>
</div>
</details>
<details class="nav_sect" data-path="for_mathlib/linear_algebra" >
<summary>linear_algebra</summary>
<details class="nav_sect" data-path="for_mathlib/linear_algebra/bilinear_form" >
<summary>bilinear_form</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/for_mathlib/linear_algebra/bilinear_form/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/for_mathlib/linear_algebra/bilinear_form/tensor_product.html">tensor_product</a>
</div>
</details>
<details class="nav_sect" data-path="for_mathlib/linear_algebra/quadratic_form" >
<summary>quadratic_form</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/for_mathlib/linear_algebra/quadratic_form/isometric_map.html">isometric_map</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/for_mathlib/linear_algebra/quadratic_form/prod.html">prod</a>
</div>
</details>
<details class="nav_sect" data-path="for_mathlib/linear_algebra/tensor_product" >
<summary>tensor_product</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/for_mathlib/linear_algebra/tensor_product/inner_product_space.html">inner_product_space</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/for_mathlib/linear_algebra/tensor_product/opposite.html">opposite</a>
</div>
</details>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/for_mathlib/linear_algebra/tensor_product.html">tensor_product</a>
</div>
</details>
<details class="nav_sect" data-path="for_mathlib/ring_theory" >
<summary>ring_theory</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/for_mathlib/ring_theory/tensor_product.html">tensor_product</a>
</div>
</details>
</details>
<details class="nav_sect" data-path="geometric_algebra" >
<summary>geometric_algebra</summary>
<details class="nav_sect" data-path="geometric_algebra/from_mathlib" >
<summary>from_mathlib</summary>
<details class="nav_sect" data-path="geometric_algebra/from_mathlib/concrete" >
<summary>concrete</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/geometric_algebra/from_mathlib/concrete/cga.html">cga</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/geometric_algebra/from_mathlib/concrete/default.html">default</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/geometric_algebra/from_mathlib/concrete/pga.html">pga</a>
</div>
</details>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/geometric_algebra/from_mathlib/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/geometric_algebra/from_mathlib/category_theory.html">category_theory</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/geometric_algebra/from_mathlib/complexification.html">complexification</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/geometric_algebra/from_mathlib/conjugation.html">conjugation</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/geometric_algebra/from_mathlib/contract.html">contract</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/geometric_algebra/from_mathlib/default.html">default</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/geometric_algebra/from_mathlib/even_odd.html">even_odd</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/geometric_algebra/from_mathlib/even_odd_equiv.html">even_odd_equiv</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/geometric_algebra/from_mathlib/fold.html">fold</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/geometric_algebra/from_mathlib/mathoverflow.html">mathoverflow</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/geometric_algebra/from_mathlib/versors.html">versors</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/geometric_algebra/from_mathlib/zmod_grading.html">zmod_grading</a>
</div>
</details>
<details class="nav_sect" data-path="geometric_algebra/nursery" >
<summary>nursery</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/geometric_algebra/nursery/chisolm.html">chisolm</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/geometric_algebra/nursery/graded.html">graded</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/geometric_algebra/nursery/instances.html">instances</a>
</div>
</details>
<details class="nav_sect" data-path="geometric_algebra/translations" >
<summary>translations</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/geometric_algebra/translations/hol_light.html">hol_light</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/geometric_algebra/translations/ida.html">ida</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/geometric_algebra/translations/ida_derived.html">ida_derived</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/geometric_algebra/translations/laurents.html">laurents</a>
</div>
</details>
</details>
<h4>mathlib</h4>
<details class="nav_sect" data-path="algebra" >
<summary>algebra</summary>
<details class="nav_sect" data-path="algebra/algebra" >
<summary>algebra</summary>
<details class="nav_sect" data-path="algebra/algebra/subalgebra" >
<summary>subalgebra</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/algebra/subalgebra/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/algebra/subalgebra/tower.html">tower</a>
</div>
</details>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/algebra/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/algebra/bilinear.html">bilinear</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/algebra/equiv.html">equiv</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/algebra/hom.html">hom</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/algebra/operations.html">operations</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/algebra/pi.html">pi</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/algebra/prod.html">prod</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/algebra/restrict_scalars.html">restrict_scalars</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/algebra/tower.html">tower</a>
</div>
</details>
<details class="nav_sect" data-path="algebra/big_operators" >
<summary>big_operators</summary>
<details class="nav_sect" data-path="algebra/big_operators/multiset" >
<summary>multiset</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/big_operators/multiset/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/big_operators/multiset/lemmas.html">lemmas</a>
</div>
</details>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/big_operators/associated.html">associated</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/big_operators/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/big_operators/fin.html">fin</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/big_operators/finprod.html">finprod</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/big_operators/finsupp.html">finsupp</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/big_operators/intervals.html">intervals</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/big_operators/nat_antidiagonal.html">nat_antidiagonal</a>
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<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/big_operators/option.html">option</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/big_operators/order.html">order</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/big_operators/pi.html">pi</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/big_operators/ring.html">ring</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/big_operators/ring_equiv.html">ring_equiv</a>
</div>
</details>
<details class="nav_sect" data-path="algebra/category" >
<summary>category</summary>
<details class="nav_sect" data-path="algebra/category/Algebra" >
<summary>Algebra</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/category/Algebra/basic.html">basic</a>
</div>
</details>
<details class="nav_sect" data-path="algebra/category/Group" >
<summary>Group</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/category/Group/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/category/Group/preadditive.html">preadditive</a>
</div>
</details>
<details class="nav_sect" data-path="algebra/category/Module" >
<summary>Module</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/category/Module/basic.html">basic</a>
</div>
</details>
<details class="nav_sect" data-path="algebra/category/Mon" >
<summary>Mon</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/category/Mon/basic.html">basic</a>
</div>
</details>
<details class="nav_sect" data-path="algebra/category/Ring" >
<summary>Ring</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/category/Ring/basic.html">basic</a>
</div>
</details>
</details>
<details class="nav_sect" data-path="algebra/char_p" >
<summary>char_p</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/char_p/algebra.html">algebra</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/char_p/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/char_p/invertible.html">invertible</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/char_p/pi.html">pi</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/char_p/quotient.html">quotient</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/char_p/two.html">two</a>
</div>
</details>
<details class="nav_sect" data-path="algebra/char_zero" >
<summary>char_zero</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/char_zero/defs.html">defs</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/char_zero/infinite.html">infinite</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/char_zero/lemmas.html">lemmas</a>
</div>
</details>
<details class="nav_sect" data-path="algebra/direct_sum" >
<summary>direct_sum</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/direct_sum/algebra.html">algebra</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/direct_sum/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/direct_sum/decomposition.html">decomposition</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/direct_sum/finsupp.html">finsupp</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/direct_sum/internal.html">internal</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/direct_sum/module.html">module</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/direct_sum/ring.html">ring</a>
</div>
</details>
<details class="nav_sect" data-path="algebra/divisibility" >
<summary>divisibility</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/divisibility/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/divisibility/units.html">units</a>
</div>
</details>
<details class="nav_sect" data-path="algebra/euclidean_domain" >
<summary>euclidean_domain</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/algebra/euclidean_domain/basic.html">basic</a>
</div>