148a4031 - mathematics_in

Authored by: Jeremy Avigad
Authored at: 2023-05-29 21:48:42 -0400
Committed by: Jeremy Avigad
Committed at: 2023-05-29 21:48:42 -0400
Object name: 148a4031356727652363039f18eed5671002c902

initial commit

Changes

126 changed files (+34728/-0)

.docker/gitpod/Dockerfile (new)
- @@ -0,0 +1,37 @@
  + # This is the Dockerfile for `leanprovercommunity/mathlib:gitpod`. + + # gitpod doesn't support multiple FROM statements, (or rather, you can't copy from one to another) + # so we just install everything in one go + FROM ubuntu:jammy + + USER root + + RUN apt-get update && apt-get install sudo git curl git bash-completion python3 -y && apt-get clean + + RUN useradd -l -u 33333 -G sudo -md /home/gitpod -s /bin/bash -p gitpod gitpod \ + # passwordless sudo for users in the 'sudo' group + && sed -i.bkp -e 's/%sudo\s\+ALL=(ALL$:ALL$\?)\s\+ALL/%sudo ALL=NOPASSWD:ALL/g' /etc/sudoers + USER gitpod + WORKDIR /home/gitpod + + SHELL ["/bin/bash", "-c"] + + # gitpod bash prompt + RUN { echo && echo "PS1='\[\033[01;32m\]\u\[\033[00m\] \[\033[01;34m\]\w\[\033[00m\]\$(__git_ps1 \" (%s)\") $ '" ; } >> .bashrc + + # install elan + RUN curl https://raw.githubusercontent.com/leanprover/elan/master/elan-init.sh -sSf | sh -s -- -y --default-toolchain none + + # install whichever toolchain mathlib is currently using + RUN . ~/.profile && elan toolchain install $(curl https://raw.githubusercontent.com/leanprover-community/mathlib4/master/lean-toolchain) + + ENV PATH="/home/gitpod/.local/bin:/home/gitpod/.elan/bin:${PATH}" + + # fix the infoview when the container is used on gitpod: + ENV VSCODE_API_VERSION="1.50.0" + + # ssh to github once to bypass the unknown fingerprint warning + RUN ssh -o StrictHostKeyChecking=no github.com || true + + # run sudo once to suppress usage info + RUN sudo echo finished
.gitignore (new)
- @@ -0,0 +1,4 @@
  + /.vscode + *.olean + /_target + /leanpkg.path
.gitpod.yml (new)
- @@ -0,0 +1,9 @@
  + image: + file: .docker/gitpod/Dockerfile + + vscode: + extensions: + - leanprover.lean4 + + tasks: + - init: lake exe cache get
README.md (new)
- @@ -0,0 +1,85 @@
  + # Mathematics in Lean + + This tutorial depends on Lean 4, VS Code, and Mathlib. + You can find the textbook both online and in this repository + in + [html format](https://leanprover-community.github.io/mathematics_in_lean/) + or as a + [pdf document](https://leanprover-community.github.io/mathematics_in_lean/mathematics_in_lean.pdf). + The book is designed to be read as you + work through examples and exercises, + using a copy of this repository on your computer. + Alternatively, you can use Gitpod to run Lean and VS Code in the cloud. + + This version of *Mathematics in Lean* is designed for [Lean 4](https://leanprover.github.io/) and + [Mathlib](https://github.com/leanprover-community/mathlib4). + For the Lean 3 version, see [github.com/leanprover-community/mathematics_in_lean3](github.com/leanprover-community/mathematics_in_lean3). + + + ## To use this repository on your computer + + Do the following: + + 1. Install Lean 4 and VS Code following + these [instructions](https://github.com/leanprover/lean4/blob/master/doc/quickstart.md). + + 2. In a terminal, navigate to the folder where you want to put a copy of the + repository, and type `git clone git@github.com:leanprover-community/mathematics_in_lean.git` + to fetch it from github. + + 3. Navigate to `mathematics_in_lean`, and execute `lake exe cache get` to fetch a compiled + version of the library, `Mathlib`. + + 4. Type `code .` to open the folder in `VS Code`. Alternatively, you can run `VS Code` and + choose `Open Folder` from the `File` menu. Be sure to open the folder `mathematics_in_lean`, + not any other folder. + + You can open the book in a side panel in VS Code as follows: + + 1. Type ``ctrl-shift-P``. + + 2. Type ``Lean 4: Open Documentation View`` in the bar that appears, and then + press return. (You can press return to select it as soon as it is highlighted + in the menu.) + + 3. In the window that opens, click on ``Open documentation of current project``. + + Each section in the book has an associated Lean file + with examples and exercises. + You can find them in the folder `src`, organized by chapter. + We recommend making a copy of that folder, + naming it something like `my_files`. + That way you can experiment with the files as you go + while leaving the originals intact. + + If you have an HTML previewer installed as an extension, you can open `html/index.html` + and read the book in VS Code while you work on the exercises. + + You can update to a newer version of this repository + by typing ``git pull`` followed by ``lake exe cache get`` + inside the ``mathematics_in_lean`` folder. + This will update the `src` folder, but will not change `my_files`. + + ## To use this repository with Gitpod + + If you have a Gitpod account or are willing to sign up for one, + just point your browser to [https://gitpod.io/#/https://github.com/leanprover-community/mathematics_in_lean](https://gitpod.io/#/https://github.com/leanprover-community/mathematics_in_lean). + This creates a virtual machine in the cloud, + and installs Lean and mathlib. + It then presents you with a VS Code window, running in a virtual + copy of the repository. + You can then make a copy of the `src` directory, and so on, + following the instructions above. + + Gitpod gives you 50 free hours every month. + When you are done working, choose `Stop workspace` from the menu on the left. + The workspace should also stop automatically + 30 minutes after the last interaction or 3 minutes after closing the tab. + + To restart a previous workspace, go to [https://gitpod.io/workspaces/](https://gitpod.io/workspaces/). + If you change the filter from Active to All, you will see all your recent workspaces. You can pin a workspace to keep it on the list of active ones. + + ## Contributing + + PRs and issues should be opened at the upstream + [source repository](https://github.com/avigad/mathematics_in_lean_source).
html/.buildinfo (new)
- @@ -0,0 +1,4 @@
  + # Sphinx build info version 1 + # This file hashes the configuration used when building these files. When it is not found, a full rebuild will be done. + config: 88166e69df2da4dbd33869bb72fe3995 + tags: 645f666f9bcd5a90fca523b33c5a78b7
html/C01_Introduction.html (new)
- @@ -0,0 +1,363 @@
 + <!DOCTYPE html> + <html class="writer-html5" lang="en" > + <head> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + + <meta name="viewport" content="width=device-width, initial-scale=1.0" /> + <title>1. Introduction — Mathematics in Lean 0.1 documentation</title> + <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="shortcut icon" href="_static/favicon.ico"/> +  + + <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> + <script src="_static/jquery.js"></script> + <script src="_static/underscore.js"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> + <script src="_static/doctools.js"></script> + <script src="_static/js/theme.js"></script> + <link rel="index" title="Index" href="genindex.html" /> + <link rel="search" title="Search" href="search.html" /> + <link rel="next" title="2. Basics" href="C02_Basics.html" /> + <link rel="prev" title="Mathematics in Lean" href="index.html" /> + </head> + + <body class="wy-body-for-nav"> + <div class="wy-grid-for-nav"> + <nav data-toggle="wy-nav-shift" class="wy-nav-side"> + <div class="wy-side-scroll"> + <div class="wy-side-nav-search" > + <a href="index.html" class="icon icon-home"> Mathematics in Lean + </a> + <div role="search"> + <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> + <input type="text" name="q" placeholder="Search docs" /> + <input type="hidden" name="check_keywords" value="yes" /> + <input type="hidden" name="area" value="default" /> + </form> + </div> + </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> + <ul class="current"> + <li class="toctree-l1 current"><a class="current reference internal" href="#">1. Introduction</a><ul> + <li class="toctree-l2"><a class="reference internal" href="#getting-started">1.1. Getting Started</a></li> + <li class="toctree-l2"><a class="reference internal" href="#overview">1.2. Overview</a></li> + </ul> + </li> + <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> + <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> + <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> + <li class="toctree-l1"><a class="reference internal" href="C06_Abstract_Algebra.html">6. Abstract Algebra</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Topology.html">7. Topology</a></li> + </ul> + <ul> + <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li> + </ul> + + </div> + </div> + </nav> + + <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > + + <a href="index.html">Mathematics in Lean</a> + </nav> + + <div class="wy-nav-content"> + <div class="rst-content"> + <div role="navigation" aria-label="Page navigation"> + <ul class="wy-breadcrumbs"> + <li><a href="index.html" class="icon icon-home"></a> »</li> + <li>1. Introduction</li> + <li class="wy-breadcrumbs-aside"> + <a href="_sources/C01_Introduction.rst.txt" rel="nofollow"> View page source</a> + </li> + </ul> + <hr/> + </div> + <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> + <div itemprop="articleBody"> + + <section id="introduction"> + <h1>1. Introduction<a class="headerlink" href="#introduction" title="Permalink to this heading"></a></h1> + <section id="getting-started"> + <h2>1.1. Getting Started<a class="headerlink" href="#getting-started" title="Permalink to this heading"></a></h2> + The goal of this book is to teach you to formalize mathematics using the + Lean 4 interactive proof assistant. + It assumes that you know some mathematics, but it does not require much. + Although we will cover examples ranging from number theory + to measure theory and analysis, + we will focus on elementary aspects of those fields, + in the hopes that if they are not familiar to you, + you can pick them up as you go. + We also don’t presuppose any background in formalization. + Formalization can be seen as a kind of computer programming: + we will write mathematical definitions, theorems, and proofs in + a regimented language, like a programming language, + that Lean can understand. + In return, Lean provides feedback and information, + interprets expressions and guarantees that they are well-formed, + and ultimately certifies the correctness of our proofs. + You can learn more about Lean from the + <a class="reference external" href="https://leanprover.github.io">Lean project page</a> + and the + <a class="reference external" href="https://leanprover-community.github.io/">Lean community web pages</a>. + This tutorial is based on Lean’s large and ever-growing library, mathlib. + We also strongly recommend taking a look at the + <a class="reference external" href="https://leanprover.zulipchat.com/">Lean Zulip online chat group</a> + if you haven’t already. + You’ll find a lively and welcoming community of Lean enthusiasts there, + happy to answer questions and offer moral support. + Although you can read a pdf or html version of this book online, + it designed to be read interactively, + running Lean from inside the VS Code editor. + To get started: + <ol class="arabic simple"> + <li>Install Lean 4 and VS Code following + these <cite>instructions <https://github.com/leanprover/lean4/blob/master/doc/quickstart.md>_</cite>.</li> + <li>In a terminal, navigate to the folder where you want to put a copy of the + repository, and type <code class="docutils literal notranslate">git clone git@github.com:leanprover-community/mathematics_in_lean.git</code> + to fetch it from github.</li> + <li>Navigate to <code class="docutils literal notranslate">mathematics_in_lean</code>, and execute <code class="docutils literal notranslate">lake exe cache get</code> to fetch a compiled + version of the library, <code class="docutils literal notranslate">mathlib</code>.</li> + <li>Type <code class="docutils literal notranslate">code .</code> to open the folder in <code class="docutils literal notranslate">VS Code</code>. Alternatively, you can run <code class="docutils literal notranslate">VS Code</code> and + choose <code class="docutils literal notranslate">Open Folder</code> from the <code class="docutils literal notranslate">File</code> menu. Be sure to open the folder <code class="docutils literal notranslate">mathematics_in_lean</code>, + not any other folder.</li> + </ol> + Opening any Lean file will simultaneously open this + book in a VS Code window. You can update to a newer version by tying + <code class="docutils literal notranslate">git pull</code> followed by <code class="docutils literal notranslate">lake exe cache get</code> inside + the <code class="docutils literal notranslate">mathematics_in_lean</code> folder. + Alternatively, you can run Lean and VS Code in the cloud, + using <a class="reference external" href="https://gitpod.io/">Gitpod</a>. + You can find instructions as to how to do that on the Mathematics in Lean + <a class="reference external" href="https://github.com/leanprover-community/mathematics_in_lean">project page</a> + on Github. + Each section in this book has an associated Lean file with examples + and exercises. You can find them in the folder <cite>src</cite>, + organized by chapter. We recommend making a copy of that folder + so that you can experiment with the files as you go, + while leaving the originals intact. + The text will often include examples, like this one: + <div class="highlight-lean notranslate"><div class="highlight"><pre>#eval "Hello, World!" + </pre></div> + </div> + You should be able to find the corresponding example in the associated + Lean file. + If you click on the line, VS Code will show you Lean’s feedback in + the <code class="docutils literal notranslate">Lean Goal</code> window, and if you hover + your cursor over the <code class="docutils literal notranslate">#eval</code> command VS Code will show you Lean’s response + to this command in a pop-up window. + You are encouraged to edit the file and try examples of your own. + This book moreover provides lots of challenging exercises for you to try. + Don’t rush past these! + Lean is about doing mathematics interactively, not just reading about it. + Working through the exercises is central to the experience. + You can always compare your solutions to the ones in the <code class="docutils literal notranslate">solutions</code> + folder associated with each section. + </section> + <section id="overview"> + <h2>1.2. Overview<a class="headerlink" href="#overview" title="Permalink to this heading"></a></h2> + Put simply, Lean is a tool for building complex expressions in a formal language + known as dependent type theory. + Every expression has a type, and you can use the <cite>#check</cite> command to + print it. + Some expressions have types like <cite>ℕ</cite> or <cite>ℕ → ℕ</cite>. + These are mathematical objects. + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check 2 + 2 + + def f (x : ℕ) := + x + 3 + + #check f + </pre></div> + </div> + Some expressions have type <cite>Prop</cite>. + These are mathematical statements. + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check 2 + 2 = 4 + + def FermatLastTheorem := + ∀ x y z n : ℕ, n > 2 ∧ x * y * z ≠ 0 → x ^ n + y ^ n ≠ z ^ n + + #check FermatLastTheorem + </pre></div> + </div> + Some expressions have a type, <cite>P</cite>, where <cite>P</cite> itself has type <cite>Prop</cite>. + Such an expression is a proof of the proposition <cite>P</cite>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem easy : 2 + 2 = 4 := + rfl + + #check easy + + theorem hard : FermatLastTheorem := + sorry + + #check hard + </pre></div> + </div> + If you manage to construct an expression of type <cite>fermat_last_theorem</cite> and + Lean accepts it as a term of that type, + you have done something very impressive. + (Using <code class="docutils literal notranslate">sorry</code> is cheating, and Lean knows it.) + So now you know the game. + All that is left to learn are the rules. + This book is complementary to a companion tutorial, + <a class="reference external" href="https://leanprover.github.io/theorem_proving_in_lean/">Theorem Proving in Lean</a>, + which provides a more thorough introduction to the underlying logical framework + and core syntax of Lean. + Theorem Proving in Lean is for people who prefer to read a user manual cover to cover before + using a new dishwasher. + If you are the kind of person who prefers to hit the start button and + figure out how to activate the potscrubber feature later, + it makes more sense to start here and refer back to + Theorem Proving in Lean as necessary. + Another thing that distinguishes Mathematics in Lean from + Theorem Proving in Lean is that here we place a much greater + emphasis on the use of tactics. + Given that we are trying to build complex expressions, + Lean offers two ways of going about it: + we can write down the expressions themselves + (that is, suitable text descriptions thereof), + or we can provide Lean with instructions as to how to construct them. + For example, the following expression represents a proof of the fact that + if <code class="docutils literal notranslate">n</code> is even then so is <code class="docutils literal notranslate">m * n</code>: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : ∀ m n : Nat, Even n → Even (m * n) := fun m n ⟨k, (hk : n = k + k)⟩ => + have hmn : m * n = m * k + m * k := by rw [hk, mul_add] + show ∃ l, m * n = l + l from ⟨_, hmn⟩ + </pre></div> + </div> + The proof term can be compressed to a single line: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : ∀ m n : Nat, Even n → Even (m * n) := fun m n ⟨k, hk⟩ => ⟨m * k, by rw [hk, mul_add]⟩ + </pre></div> + </div> + The following is, instead, a tactic-style proof of the same theorem: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : ∀ m n : Nat, Even n → Even (m * n) := by + -- say m and n are natural numbers, and assume n=2*k + rintro m n ⟨k, hk⟩ + -- We need to prove m*n is twice a natural number. Let's show it's twice m*k. + use m * k + -- substitute in for n + rw [hk] + -- and now it's obvious + ring + </pre></div> + </div> + As you enter each line of such a proof in VS Code, + Lean displays the proof state in a separate window, + telling you what facts you have already established and what + tasks remain to prove your theorem. + You can replay the proof by stepping through the lines, + since Lean will continue to show you the state of the proof + at the point where the cursor is. + In this example, you will then see that + the first line of the proof introduces <code class="docutils literal notranslate">m</code> and <code class="docutils literal notranslate">n</code> + (we could have renamed them at that point, if we wanted to), + and also decomposes the hypothesis <code class="docutils literal notranslate">even n</code> to + a <code class="docutils literal notranslate">k</code> and the assumption that <code class="docutils literal notranslate">n = 2 * k</code>. + The second line, <code class="docutils literal notranslate">use m * k</code>, + declares that we are going to show that <code class="docutils literal notranslate">m * n</code> is even by + showing <code class="docutils literal notranslate">m * n = 2 * (m * k)</code>. + The next line uses the <code class="docutils literal notranslate">rewrite</code> tactic + to replace <code class="docutils literal notranslate">n</code> by <code class="docutils literal notranslate">2 * k</code> in the goal, + and the <cite>ring</cite> tactic solves the resulting goal <code class="docutils literal notranslate">m * (2 * k) = 2 * (m * k)</code>. + The ability to build a proof in small steps with incremental feedback + is extremely powerful. For that reason, + tactic proofs are often easier and quicker to write than + proof terms. + There isn’t a sharp distinction between the two: + tactic proofs can be inserted in proof terms, + as we did with the phrase <code class="docutils literal notranslate">by rw [hk, mul_left_comm]</code> in the example above. + We will also see that, conversely, + it is often useful to insert a short proof term in the middle of a tactic proof. + That said, in this book, our emphasis will be on the use of tactics. + In our example, the tactic proof can also be reduced to a one-liner: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : ∀ m n : Nat, Even n → Even (m * n) := by + rintro m n ⟨k, hk⟩; use m * k; rw [hk]; ring + </pre></div> + </div> + Here we have used tactics to carry out small proof steps. + But they can also provide substantial automation, + and justify longer calculations and bigger inferential steps. + For example, we can invoke Lean’s simplifier with + specific rules for simplifying statements about parity to + prove our theorem automatically. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : ∀ m n : Nat, Even n → Even (m * n) := by + intros; simp [*, parity_simps] + </pre></div> + </div> + Another big difference between the two introductions is that + Theorem Proving in Lean depends only on core Lean and its built-in + tactics, whereas Mathematics in Lean is built on top of Lean’s + powerful and ever-growing library, mathlib. + As a result, we can show you how to use some of the mathematical + objects and theorems in the library, + and some of the very useful tactics. + This book is not meant to be used as an overview of the library; + the <a class="reference external" href="https://leanprover-community.github.io/">community</a> + web pages contain extensive documentation. + Rather, our goal is to introduce you to the style of thinking that + underlies that formalization, + so that you are comfortable browsing the library and + finding things on your own. + Interactive theorem proving can be frustrating, + and the learning curve is steep. + But the Lean community is very welcoming to newcomers, + and people are available on the + <a class="reference external" href="https://leanprover.zulipchat.com/">Lean Zulip chat group</a> round the clock + to answer questions. + We hope to see you there, and have no doubt that + soon enough you, too, will be able to answer such questions + and contribute to the development of mathlib. + So here is your mission, should you choose to accept it: + dive in, try the exercises, come to Zulip with questions, and have fun. + But be forewarned: + interactive theorem proving will challenge you to think about + mathematics and mathematical reasoning in fundamentally new ways. + Your life may never be the same. + Acknowledgments. We are grateful to Gabriel Ebner for setting up the + infrastructure for running this tutorial in VS Code, + and to Scott Morrison and Mario Carneiro for help porting it from Lean 3. + We are also grateful for help and corrections from + Bryan Gin-ge Chen, Johan Commelin, Mathieu Guay-Paquet, Julian Külshammer, + Giovanni Mascellani, Hunter Monroe, Pietro Monticone, Bartosz Piotrowski, + and Guilherme Silva. + Our work has been partially supported by the Hoskinson Center for + Formal Mathematics. + </section> + </section> + + + </div> + </div> + <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> + <a href="index.html" class="btn btn-neutral float-left" title="Mathematics in Lean" accesskey="p" rel="prev"> Previous</a> + <a href="C02_Basics.html" class="btn btn-neutral float-right" title="2. Basics" accesskey="n" rel="next">Next </a> + </div> + + <hr/> + + <div role="contentinfo"> + © Copyright 2020, Jeremy Avigad, Kevin Buzzard, Robert Y. Lewis, Patrick Massot. + </div> + + Built with <a href="https://www.sphinx-doc.org/">Sphinx</a> using a + <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a> + provided by <a href="https://readthedocs.org">Read the Docs</a>. + + + </footer> + </div> + </div> + </section> + </div> + <script> + jQuery(function () { + SphinxRtdTheme.Navigation.enable(true); + }); + </script> + + </body> + </html>
html/C02_Basics.html (new)
- @@ -0,0 +1,1438 @@
 + <!DOCTYPE html> + <html class="writer-html5" lang="en" > + <head> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + + <meta name="viewport" content="width=device-width, initial-scale=1.0" /> + <title>2. Basics — Mathematics in Lean 0.1 documentation</title> + <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="shortcut icon" href="_static/favicon.ico"/> +  + + <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> + <script src="_static/jquery.js"></script> + <script src="_static/underscore.js"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> + <script src="_static/doctools.js"></script> + <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> + <script src="_static/js/theme.js"></script> + <link rel="index" title="Index" href="genindex.html" /> + <link rel="search" title="Search" href="search.html" /> + <link rel="next" title="3. Logic" href="C03_Logic.html" /> + <link rel="prev" title="1. Introduction" href="C01_Introduction.html" /> + </head> + + <body class="wy-body-for-nav"> + <div class="wy-grid-for-nav"> + <nav data-toggle="wy-nav-shift" class="wy-nav-side"> + <div class="wy-side-scroll"> + <div class="wy-side-nav-search" > + <a href="index.html" class="icon icon-home"> Mathematics in Lean + </a> + <div role="search"> + <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> + <input type="text" name="q" placeholder="Search docs" /> + <input type="hidden" name="check_keywords" value="yes" /> + <input type="hidden" name="area" value="default" /> + </form> + </div> + </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> + <ul class="current"> + <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a></li> + <li class="toctree-l1 current"><a class="current reference internal" href="#">2. Basics</a><ul> + <li class="toctree-l2"><a class="reference internal" href="#calculating">2.1. Calculating</a></li> + <li class="toctree-l2"><a class="reference internal" href="#proving-identities-in-algebraic-structures">2.2. Proving Identities in Algebraic Structures</a></li> + <li class="toctree-l2"><a class="reference internal" href="#using-theorems-and-lemmas">2.3. Using Theorems and Lemmas</a></li> + <li class="toctree-l2"><a class="reference internal" href="#more-on-order-and-divisibility">2.4. More on Order and Divisibility</a></li> + <li class="toctree-l2"><a class="reference internal" href="#proving-facts-about-algebraic-structures">2.5. Proving Facts about Algebraic Structures</a></li> + </ul> + </li> + <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> + <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> + <li class="toctree-l1"><a class="reference internal" href="C06_Abstract_Algebra.html">6. Abstract Algebra</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Topology.html">7. Topology</a></li> + </ul> + <ul> + <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li> + </ul> + + </div> + </div> + </nav> + + <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > + + <a href="index.html">Mathematics in Lean</a> + </nav> + + <div class="wy-nav-content"> + <div class="rst-content"> + <div role="navigation" aria-label="Page navigation"> + <ul class="wy-breadcrumbs"> + <li><a href="index.html" class="icon icon-home"></a> »</li> + <li>2. Basics</li> + <li class="wy-breadcrumbs-aside"> + <a href="_sources/C02_Basics.rst.txt" rel="nofollow"> View page source</a> + </li> + </ul> + <hr/> + </div> + <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> + <div itemprop="articleBody"> + + <section id="basics"> + <h1>2. Basics<a class="headerlink" href="#basics" title="Permalink to this heading"></a></h1> + This chapter is designed to introduce you to the nuts and + bolts of mathematical reasoning in Lean: calculating, + applying lemmas and theorems, + and reasoning about generic structures. + <section id="calculating"> + <h2>2.1. Calculating<a class="headerlink" href="#calculating" title="Permalink to this heading"></a></h2> + We generally learn to carry out mathematical calculations + without thinking of them as proofs. + But when we justify each step in a calculation, + as Lean requires us to do, + the net result is a proof that the left-hand side of the calculation + is equal to the right-hand side. + In Lean, stating a theorem is tantamount to stating a goal, + namely, the goal of proving the theorem. + Lean provides the <code class="docutils literal notranslate">rewrite</code> tactic, abbreviated <code class="docutils literal notranslate">rw</code>, + to replace the left-hand side of an identity by the right-hand side + in the goal. If <code class="docutils literal notranslate">a</code>, <code class="docutils literal notranslate">b</code>, and <code class="docutils literal notranslate">c</code> are real numbers, + <code class="docutils literal notranslate">mul_assoc a b c</code> is the identity <code class="docutils literal notranslate">a * b * c = a * (b * c)</code> + and <code class="docutils literal notranslate">mul_comm a b</code> is the identity <code class="docutils literal notranslate">a * b = b * a</code>. + Lean provides automation that generally eliminates the need + to refer the facts like these explicitly, + but they are useful for the purposes of illustration. + In Lean, multiplication associates to the left, + so the left-hand side of <code class="docutils literal notranslate">mul_assoc</code> could also be written <code class="docutils literal notranslate">(a * b) * c</code>. + However, it is generally good style to be mindful of Lean’s + notational conventions and leave out parentheses when Lean does as well. + Let’s try out <code class="docutils literal notranslate">rw</code>. + <div class="highlight-lean notranslate" id="index-1"><div class="highlight"><pre>example (a b c : ℝ) : a * b * c = b * (a * c) := by + rw [mul_comm a b] + rw [mul_assoc b a c] + </pre></div> + </div> + The <code class="docutils literal notranslate">import</code> line at the beginning of the example + imports the theory of the real numbers from <code class="docutils literal notranslate">mathlib</code>. + For the sake of brevity, + we generally suppress information like this when it + is repeated from example to example. + Clicking the <code class="docutils literal notranslate">try it!</code> button displays the full + example as it is meant to be processed and checked by Lean. + You are welcome to make changes to see what happens. + You can type the <code class="docutils literal notranslate">ℝ</code> character as <code class="docutils literal notranslate">\R</code> or <code class="docutils literal notranslate">\real</code> + in VS Code. + The symbol doesn’t appear until you hit space or the tab key. + If you hover over a symbol when reading a Lean file, + VS Code will show you the syntax that can be used to enter it. + If your keyboard does not have an easily accessible backslash, + you can change the leading character by changing the + <code class="docutils literal notranslate">lean.input.leader</code> setting. + When a cursor is in the middle of a tactic proof, + Lean reports on the current proof state in the + Lean infoview window. + As you move your cursor past each step of the proof, + you can see the state change. + A typical proof state in Lean might look as follows: + <div class="highlight-lean notranslate"><div class="highlight"><pre>1 goal + x y : ℕ, + h₁ : Prime x, + h₂ : ¬Even x, + h₃ : y > x + ⊢ y ≥ 4 + </pre></div> + </div> + The lines before the one that begins with <code class="docutils literal notranslate">⊢</code> denote the context: + they are the objects and assumptions currently at play. + In this example, these include two objects, <code class="docutils literal notranslate">x</code> and <code class="docutils literal notranslate">y</code>, + each a natural number. + They also include three assumptions, + labelled <code class="docutils literal notranslate">h₁</code>, <code class="docutils literal notranslate">h₂</code>, and <code class="docutils literal notranslate">h₃</code>. + In Lean, everything in a context is labelled with an identifier. + You can type these subscripted labels as <code class="docutils literal notranslate">h\1</code>, <code class="docutils literal notranslate">h\2</code>, and <code class="docutils literal notranslate">h\3</code>, + but any legal identifiers would do: + you can use <code class="docutils literal notranslate">h1</code>, <code class="docutils literal notranslate">h2</code>, <code class="docutils literal notranslate">h3</code> instead, + or <code class="docutils literal notranslate">foo</code>, <code class="docutils literal notranslate">bar</code>, and <code class="docutils literal notranslate">baz</code>. + The last line represents the goal, + that is, the fact to be proved. + Sometimes people use target for the fact to be proved, + and goal for the combination of the context and the target. + In practice, the intended meaning is usually clear. + Try proving these identities, + in each case replacing <code class="docutils literal notranslate">sorry</code> by a tactic proof. + With the <code class="docutils literal notranslate">rw</code> tactic, you can use a left arrow (<code class="docutils literal notranslate">\l</code>) + to reverse an identity. + For example, <code class="docutils literal notranslate">rw ← mul_assoc a b c</code> + replaces <code class="docutils literal notranslate">a * (b * c)</code> by <code class="docutils literal notranslate">a * b * c</code> in the current goal. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a b c : ℝ) : c * b * a = b * (a * c) := by + sorry + + example (a b c : ℝ) : a * (b * c) = b * (a * c) := by + sorry + </pre></div> + </div> + You can also use identities like <code class="docutils literal notranslate">mul_assoc</code> and <code class="docutils literal notranslate">mul_comm</code> without arguments. + In this case, the rewrite tactic tries to match the left-hand side with + an expression in the goal, + using the first pattern it finds. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a b c : ℝ) : a * b * c = b * c * a := by + rw [mul_assoc] + rw [mul_comm] + </pre></div> + </div> + You can also provide partial information. + For example, <code class="docutils literal notranslate">mul_comm a</code> matches any pattern of the form + <code class="docutils literal notranslate">a * ?</code> and rewrites it to <code class="docutils literal notranslate">? * a</code>. + Try doing the first of these examples without + providing any arguments at all, + and the second with only one argument. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a b c : ℝ) : a * (b * c) = b * (c * a) := by + sorry + + example (a b c : ℝ) : a * (b * c) = b * (a * c) := by + sorry + </pre></div> + </div> + You an also use <code class="docutils literal notranslate">rw</code> with facts from the local context. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by + rw [h'] + rw [← mul_assoc] + rw [h] + rw [mul_assoc] + </pre></div> + </div> + Try these: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a b c d e f : ℝ) (h : b * c = e * f) : a * b * c * d = a * e * f * d := by + sorry + + example (a b c d : ℝ) (hyp : c = b * a - d) (hyp' : d = a * b) : c = 0 := by + sorry + </pre></div> + </div> + For the second one, you can use the theorem <code class="docutils literal notranslate">sub_self</code>, + where <code class="docutils literal notranslate">sub_self a</code> is the identity <code class="docutils literal notranslate">a - a = 0</code>. + We now introduce some useful features of Lean. + First, multiple rewrite commands can be carried out + with a single command, + by listing the relevant identities within square brackets. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by + rw [h', ← mul_assoc, h, mul_assoc] + </pre></div> + </div> + You still see the incremental progress by placing the cursor after + a comma in any list of rewrites. + Another trick is that we can declare variables once and for all outside + an example or theorem. + When Lean sees them mentioned in the statement of the theorem, + it includes them automatically. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (a b c d e f g : ℝ) + + example (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by + rw [h', ← mul_assoc, h, mul_assoc] + </pre></div> + </div> + Inspection of the tactic state at the beginning of the above proof + reveals that Lean indeed included the relevant variables, leaving out + <cite>g</cite> that doesn’t feature in the statement. + We can delimit the scope of the declaration by putting it + in a <code class="docutils literal notranslate">section ... end</code> block. + Finally, recall from the introduction that Lean provides us with a + command to determine the type of an expression: + <div class="highlight-lean notranslate"><div class="highlight"><pre>section + variable (a b c : ℝ) + + #check a + #check a + b + #check (a : ℝ) + #check mul_comm a b + #check (mul_comm a b : a * b = b * a) + #check mul_assoc c a b + #check mul_comm a + #check mul_comm + #check @mul_comm + + end + </pre></div> + </div> + The <code class="docutils literal notranslate">#check</code> command works for both objects and facts. + In response to the command <code class="docutils literal notranslate">#check a</code>, Lean reports that <code class="docutils literal notranslate">a</code> has type <code class="docutils literal notranslate">ℝ</code>. + In response to the command <code class="docutils literal notranslate">#check mul_comm a b</code>, + Lean reports that <code class="docutils literal notranslate">mul_comm a b</code> is a proof of the fact <code class="docutils literal notranslate">a * b = b * a</code>. + The command <code class="docutils literal notranslate">#check (a : ℝ)</code> states our expectation that the + type of <code class="docutils literal notranslate">a</code> is <code class="docutils literal notranslate">ℝ</code>, + and Lean will raise an error if that is not the case. + We will explain the output of the last three <code class="docutils literal notranslate">#check</code> commands later, + but in the meanwhile, you can take a look at them, + and experiment with some <code class="docutils literal notranslate">#check</code> commands of your own. + Let’s try some more examples. The theorem <code class="docutils literal notranslate">two_mul a</code> says + that <code class="docutils literal notranslate">2 * a = a + a</code>. The theorems <code class="docutils literal notranslate">add_mul</code> and <code class="docutils literal notranslate">mul_add</code> + express the distributivity of multiplication over addition, + and the theorem <code class="docutils literal notranslate">add_assoc</code> expresses the associativity of addition. + Use the <code class="docutils literal notranslate">#check</code> command to see the precise statements. + <div class="highlight-lean notranslate" id="index-3"><div class="highlight"><pre>example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by + rw [mul_add, add_mul, add_mul] + rw [← add_assoc, add_assoc (a * a)] + rw [mul_comm b a, ← two_mul] + </pre></div> + </div> + Whereas it is possible to figure out what it going on in this proof + by stepping through it in the editor, + it is hard to read on its own. + Lean provides a more structured way of writing proofs like this + using the <code class="docutils literal notranslate">calc</code> keyword. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := + calc + (a + b) * (a + b) = a * a + b * a + (a * b + b * b) := by + rw [mul_add, add_mul, add_mul] + _ = a * a + (b * a + a * b) + b * b := by + rw [← add_assoc, add_assoc (a * a)] + _ = a * a + 2 * (a * b) + b * b := by + rw [mul_comm b a, ← two_mul] + </pre></div> + </div> + Notice that the proof does not begin with <code class="docutils literal notranslate">by</code>: + an expression that begins with <code class="docutils literal notranslate">calc</code> is a proof term. + A <code class="docutils literal notranslate">calc</code> expression can also be used inside a tactic proof, + but Lean interprets it as the instruction to use the resulting + proof term to solve the goal. + The <code class="docutils literal notranslate">calc</code> syntax is finicky: the dots and underscires and justification + have to be in the format indicated above. + Lean uses indentation to determine things like where a block + of tactics or a <code class="docutils literal notranslate">calc</code> block begins and ends; + try changing the indentation in the proof above to see what happens. + One way to write a <code class="docutils literal notranslate">calc</code> proof is to outline it first + using the <code class="docutils literal notranslate">sorry</code> tactic for justification, + make sure Lean accepts the expression modulo these, + and then justify the individual steps using tactics. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := + calc + (a + b) * (a + b) = a * a + b * a + (a * b + b * b) := by + sorry + _ = a * a + (b * a + a * b) + b * b := by + sorry + _ = a * a + 2 * (a * b) + b * b := by + sorry + </pre></div> + </div> + Try proving the following identity using both a pure <code class="docutils literal notranslate">rw</code> proof + and a more structured <code class="docutils literal notranslate">calc</code> proof: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : (a + b) * (c + d) = a * c + a * d + b * c + b * d := by + sorry + </pre></div> + </div> + The following exercise is a little more challenging. + You can use the theorems listed underneath. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a b : ℝ) : (a + b) * (a - b) = a ^ 2 - b ^ 2 := by + sorry + + #check pow_two a + #check mul_sub a b c + #check add_mul a b c + #check add_sub a b c + #check sub_sub a b c + #check add_zero a + </pre></div> + </div> + We can also perform rewriting in an assumption in the context. + For example, <code class="docutils literal notranslate">rw [mul_comm a b] at hyp</code> replaces <code class="docutils literal notranslate">a * b</code> by <code class="docutils literal notranslate">b * a</code> + in the assumption <code class="docutils literal notranslate">hyp</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a b c d : ℝ) (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by + rw [hyp'] at hyp + rw [mul_comm d a] at hyp + rw [← two_mul (a * d)] at hyp + rw [← mul_assoc 2 a d] at hyp + exact hyp + </pre></div> + </div> + In the last step, the <code class="docutils literal notranslate">exact</code> tactic can use <code class="docutils literal notranslate">hyp</code> to solve the goal + because at that point <code class="docutils literal notranslate">hyp</code> matches the goal exactly. + We close this section by noting that <code class="docutils literal notranslate">mathlib</code> provides a + useful bit of automation with a <code class="docutils literal notranslate">ring</code> tactic, + which is designed to prove identities in any commutative ring. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : c * b * a = b * (a * c) := by + ring + + example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by + ring + + example : (a + b) * (a - b) = a ^ 2 - b ^ 2 := by + ring + + example (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by + rw [hyp, hyp'] + ring + </pre></div> + </div> + The <code class="docutils literal notranslate">ring</code> tactic is imported indirectly when we + import <code class="docutils literal notranslate">Data.Real.Basic</code>, + but we will see in the next section that it can be used + for calculations on structures other than the real numbers. + It can be imported explicitly with the command + <code class="docutils literal notranslate">import tactic</code>. + We will see there are similar tactics for other common kind of algebraic + structures. + There is a variation of <code class="docutils literal notranslate">rw</code> called <code class="docutils literal notranslate">nth_rewrite</code> that allows you to replace only particular instances of an expression in the goal. + Possible matches are enumerated starting with zero, + so in the following example, <code class="docutils literal notranslate">nth_rewrite 1 h</code> replaces the second + occurrence of <code class="docutils literal notranslate">a + b</code> with <code class="docutils literal notranslate">c</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a b c : ℕ) (h : a + b = c) : (a + b) * (a + b) = a * c + b * c := by + nth_rw 2 [h] + rw [add_mul] + </pre></div> + </div> + See also <code class="docutils literal notranslate">nth_rewrite_lhs</code> and <code class="docutils literal notranslate">nth_rewrite_rhs</code>. + For a more sophisticated means of rewriting particular subexpressions, + see the <a class="reference external" href="https://leanprover-community.github.io/extras/conv.html">documentation for the conversion tactic</a>. + </section> + <section id="proving-identities-in-algebraic-structures"> + <h2>2.2. Proving Identities in Algebraic Structures<a class="headerlink" href="#proving-identities-in-algebraic-structures" title="Permalink to this heading"></a></h2> + Mathematically, a ring consists of a collection of objects, + $R$, operations $+$ $\times$, and constants $0$ + and $1$, and an operation $x \mapsto -x$ such that: + <ul class="simple"> + <li>$R$ with $+$ is an abelian group, with $0$ + as the additive identity and negation as inverse.</li> + <li>Multiplication is associative with identity $1$, + and multiplication distributes over addition.</li> + </ul> + In Lean, the collection of objects is represented as a type, <code class="docutils literal notranslate">R</code>. + The ring axioms are as follows: + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (R : Type _) [Ring R] + + #check (add_assoc : ∀ a b c : R, a + b + c = a + (b + c)) + #check (add_comm : ∀ a b : R, a + b = b + a) + #check (zero_add : ∀ a : R, 0 + a = a) + #check (add_left_neg : ∀ a : R, -a + a = 0) + #check (mul_assoc : ∀ a b c : R, a * b * c = a * (b * c)) + #check (mul_one : ∀ a : R, a * 1 = a) + #check (one_mul : ∀ a : R, 1 * a = a) + #check (mul_add : ∀ a b c : R, a * (b + c) = a * b + a * c) + #check (add_mul : ∀ a b c : R, (a + b) * c = a * c + b * c) + </pre></div> + </div> + You will learn more about the square brackets in the first line later, + but for the time being, + suffice it to say that the declaration gives us a type, <code class="docutils literal notranslate">R</code>, + and a ring structure on <code class="docutils literal notranslate">R</code>. + Lean then allows us to use generic ring notation with elements of <code class="docutils literal notranslate">R</code>, + and to make use of a library of theorems about rings. + The names of some of the theorems should look familiar: + they are exactly the ones we used to calculate with the real numbers + in the last section. + Lean is good not only for proving things about concrete mathematical + structures like the natural numbers and the integers, + but also for proving things about abstract structures, + characterized axiomatically, like rings. + Moreover, Lean supports generic reasoning about + both abstract and concrete structures, + and can be trained to recognized appropriate instances. + So any theorem about rings can be applied to concrete rings + like the integers, <code class="docutils literal notranslate">ℤ</code>, the rational numbers, <code class="docutils literal notranslate">ℚ</code>, + and the complex numbers <code class="docutils literal notranslate">ℂ</code>. + It can also be applied to any instance of an abstract + structure that extends rings, + such as any ordered ring or any field. + Not all important properties of the real numbers hold in an + arbitrary ring, however. + For example, multiplication on the real numbers + is commutative, + but that does not hold in general. + If you have taken a course in linear algebra, + you will recognize that, for every $n$, + the $n$ by $n$ matrices of real numbers + form a ring in which commutativity usually fails. If we declare <code class="docutils literal notranslate">R</code> to be a + commutative ring, in fact, all the theorems + in the last section continue to hold when we replace + <code class="docutils literal notranslate">ℝ</code> by <code class="docutils literal notranslate">R</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (R : Type _) [CommRing R] + variable (a b c d : R) + + example : c * b * a = b * (a * c) := by ring + + example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by ring + + example : (a + b) * (a - b) = a ^ 2 - b ^ 2 := by ring + + example (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by + rw [hyp, hyp'] + ring + </pre></div> + </div> + We leave it to you to check that all the other proofs go through unchanged. + Notice that when a proof is short, like <code class="docutils literal notranslate">by ring</code> or <code class="docutils literal notranslate">by linarith</code> + or <code class="docutils literal notranslate">by sorry</code>, + it is common (and permissible) to put it on the same line as + the <code class="docutils literal notranslate">by</code>. + Good proof-writing style should strike a balance between concision and readability. + The goal of this section is to strengthen the skills + you have developed in the last section + and apply them to reasoning axiomatically about rings. + We will start with the axioms listed above, + and use them to derive other facts. + Most of the facts we prove are already in <code class="docutils literal notranslate">mathlib</code>. + We will give the versions we prove the same names + to help you learn the contents of the library + as well as the naming conventions. + Lean provides an organizational mechanism similar + to those used in programming languages: + when a definition or theorem <code class="docutils literal notranslate">foo</code> is introduced in a namespace + <code class="docutils literal notranslate">bar</code>, its full name is <code class="docutils literal notranslate">bar.foo</code>. + The command <code class="docutils literal notranslate">open bar</code> later opens the namespace, + which allows us to use the shorter name <code class="docutils literal notranslate">foo</code>. + To avoid errors due to name clashes, + in the next example we put our versions of the library + theorems in a new namespace called <code class="docutils literal notranslate">MyRing.</code> + The next example shows that we do not need <code class="docutils literal notranslate">add_zero</code> or <code class="docutils literal notranslate">add_right_neg</code> + as ring axioms, because they follow from the other axioms. + <div class="highlight-lean notranslate"><div class="highlight"><pre>namespace MyRing + variable {R : Type _} [Ring R] + + theorem add_zero (a : R) : a + 0 = a := by rw [add_comm, zero_add] + + theorem add_right_neg (a : R) : a + -a = 0 := by rw [add_comm, add_left_neg] + + #check @MyRing.add_zero + #check @add_zero + + end MyRing + </pre></div> + </div> + The net effect is that we can temporarily reprove a theorem in the library, + and then go on using the library version after that. + But don’t cheat! + In the exercises that follow, take care to use only the + general facts about rings that we have proved earlier in this section. + (If you are paying careful attention, you may have noticed that we + changed the round brackets in <code class="docutils literal notranslate">(R : Type _)</code> for + curly brackets in <code class="docutils literal notranslate">{R : Type _}</code>. + This declares <code class="docutils literal notranslate">R</code> to be an implicit argument. + We will explain what this means in a moment, + but don’t worry about it in the meanwhile.) + Here is a useful theorem: + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem neg_add_cancel_left (a b : R) : -a + (a + b) = b := by + rw [← add_assoc, add_left_neg, zero_add] + </pre></div> + </div> + Prove the companion version: + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem add_neg_cancel_right (a b : R) : a + b + -b = a := by + sorry + </pre></div> + </div> + Use these to prove the following: + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem add_left_cancel {a b c : R} (h : a + b = a + c) : b = c := by + sorry + + theorem add_right_cancel {a b c : R} (h : a + b = c + b) : a = c := by + sorry + </pre></div> + </div> + With enough planning, you can do each of them with three rewrites. + We can now explain the use of the curly braces. + Imagine you are in a situation where you have <code class="docutils literal notranslate">a</code>, <code class="docutils literal notranslate">b</code>, and <code class="docutils literal notranslate">c</code> + in your context, + as well as a hypothesis <code class="docutils literal notranslate">h : a + b = a + c</code>, + and you would like to draw the conclusion <code class="docutils literal notranslate">b = c</code>. + In Lean, you can apply a theorem to hypotheses and facts just + the same way that you can apply them to objects, + so you might think that <code class="docutils literal notranslate">add_left_cancel a b c h</code> is a + proof of the fact <code class="docutils literal notranslate">b = c</code>. + But notice that explicitly writing <code class="docutils literal notranslate">a</code>, <code class="docutils literal notranslate">b</code>, and <code class="docutils literal notranslate">c</code> + is redundant, because the hypothesis <code class="docutils literal notranslate">h</code> makes it clear that + those are the objects we have in mind. + In this case, typing a few extra characters is not onerous, + but if we wanted to apply <code class="docutils literal notranslate">add_left_cancel</code> to more complicated expressions, + writing them would be tedious. + In cases like these, + Lean allows us to mark arguments as implicit, + meaning that they are supposed to be left out and inferred by other means, + such as later arguments and hypotheses. + The curly brackets in <code class="docutils literal notranslate">{a b c : R}</code> do exactly that. + So, given the statement of the theorem above, + the correct expression is simply <code class="docutils literal notranslate">add_left_cancel h</code>. + To illustrate, let us show that <code class="docutils literal notranslate">a * 0 = 0</code> + follows from the ring axioms. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem mul_zero (a : R) : a * 0 = 0 := by + have h : a * 0 + a * 0 = a * 0 + 0 := by + rw [← mul_add, add_zero, add_zero] + rw [add_left_cancel h] + </pre></div> + </div> + We have used a new trick! + If you step through the proof, + you can see what is going on. + The <code class="docutils literal notranslate">have</code> tactic introduces a new goal, + <code class="docutils literal notranslate">a * 0 + a * 0 = a * 0 + 0</code>, + with the same context as the original goal. + The fact that the next line is indented indicates that Lean + is expecting a block of tactics that serves to prove this + new goal. + The indentation therefore promotes a modular style of proof: + the indented subproof establishes the goal + that was introduced by the <code class="docutils literal notranslate">have</code>. + After that, we are back to proving the original goal, + except a new hypothesis <code class="docutils literal notranslate">h</code> has been added: + having proved it, we are now free to use it. + At this point, the goal is exactly the result of <code class="docutils literal notranslate">add_left_cancel h</code>. + We could equally well have closed the proof with + <code class="docutils literal notranslate">apply add_left_cancel h</code> or <code class="docutils literal notranslate">exact add_left_cancel h</code>. + Remember that multiplication is not assumed to be commutative, + so the following theorem also requires some work. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem zero_mul (a : R) : 0 * a = 0 := by + sorry + </pre></div> + </div> + By now, you should also be able replace each <code class="docutils literal notranslate">sorry</code> in the next + exercise with a proof, + still using only facts about rings that we have + established in this section. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem neg_eq_of_add_eq_zero {a b : R} (h : a + b = 0) : -a = b := by + sorry + + theorem eq_neg_of_add_eq_zero {a b : R} (h : a + b = 0) : a = -b := by + sorry + + theorem neg_zero : (-0 : R) = 0 := by + apply neg_eq_of_add_eq_zero + rw [add_zero] + + theorem neg_neg (a : R) : - -a = a := by + sorry + </pre></div> + </div> + We had to use the annotation <code class="docutils literal notranslate">(-0 : R)</code> instead of <code class="docutils literal notranslate">0</code> in the third theorem + because without specifying <code class="docutils literal notranslate">R</code> + it is impossible for Lean to infer which <code class="docutils literal notranslate">0</code> we have in mind, + and by default it would be interpreted as a natural number. + In Lean, subtraction in a ring is provably equal to + addition of the additive inverse. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a b : R) : a - b = a + -b := + sub_eq_add_neg a b + </pre></div> + </div> + On the real numbers, it is defined that way: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a b : ℝ) : a - b = a + -b := + rfl + + example (a b : ℝ) : a - b = a + -b := by + rfl + </pre></div> + </div> + The proof term <code class="docutils literal notranslate">rfl</code> is short for <code class="docutils literal notranslate">reflexivity</code>. + Presenting it as a proof of <code class="docutils literal notranslate">a - b = a + -b</code> forces Lean + to unfold the definition and recognize both sides as being the same. + The <code class="docutils literal notranslate">reflexivity</code> tactic, which can be abbreviated as <code class="docutils literal notranslate">refl</code>, + does the same. + This is an instance of what is known as a definitional equality + in Lean’s underlying logic. + This means that not only can one rewrite with <code class="docutils literal notranslate">sub_eq_add_neg</code> + to replace <code class="docutils literal notranslate">a - b = a + -b</code>, + but in some contexts, when dealing with the real numbers, + you can use the two sides of the equation interchangeably. + For example, you now have enough information to prove the theorem + <code class="docutils literal notranslate">self_sub</code> from the last section: + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem self_sub (a : R) : a - a = 0 := + sorry + </pre></div> + </div> + Show that you can prove this using <code class="docutils literal notranslate">rw</code>, + but if you replace the arbitrary ring <code class="docutils literal notranslate">R</code> by + the real numbers, you can also prove it + using either <code class="docutils literal notranslate">apply</code> or <code class="docutils literal notranslate">exact</code>. + Lean knows that <code class="docutils literal notranslate">1 + 1 = 2</code> holds in any ring. + With a bit of effort, + you can use that to prove the theorem <code class="docutils literal notranslate">two_mul</code> from + the last section: + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem one_add_one_eq_two : 1 + 1 = (2 : R) := by + norm_num + + theorem two_mul (a : R) : 2 * a = a + a := + sorry + </pre></div> + </div> + We close this section by noting that some of the facts about + addition and negation that we established above do not + need the full strength of the ring axioms, or even + commutativity of addition. The weaker notion of a group + can be axiomatized as follows: + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (A : Type _) [AddGroup A] + + #check (add_assoc : ∀ a b c : A, a + b + c = a + (b + c)) + #check (zero_add : ∀ a : A, 0 + a = a) + #check (add_left_neg : ∀ a : A, -a + a = 0) + </pre></div> + </div> + It is conventional to use additive notation when + the group operation is commutative, + and multiplicative notation otherwise. + So Lean defines a multiplicative version as well as the + additive version (and also their abelian variants, + <code class="docutils literal notranslate">AddCommGroup</code> and <code class="docutils literal notranslate">CommGroup</code>). + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {G : Type _} [Group G] + + #check (mul_assoc : ∀ a b c : G, a * b * c = a * (b * c)) + #check (one_mul : ∀ a : G, 1 * a = a) + #check (mul_left_inv : ∀ a : G, a⁻¹ * a = 1) + </pre></div> + </div> + If you are feeling cocky, try proving the following facts about + groups, using only these axioms. + You will need to prove a number of helper lemmas along the way. + The proofs we have carried out in this section provide some hints. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem mul_right_inv (a : G) : a * a⁻¹ = 1 := by + sorry + + theorem mul_one (a : G) : a * 1 = a := by + sorry + + theorem mul_inv_rev (a b : G) : (a * b)⁻¹ = b⁻¹ * a⁻¹ := by + sorry + </pre></div> + </div> + Explicitly invoking those lemmas is tedious, so mathlib provides + tactics similar to <cite>ring</cite> in order to cover most uses: <cite>group</cite> + is for non-commutative multiplicative groups, <cite>abel</cite> for abelian + additive groups, and <cite>noncomm_ring</cite> for non-commutative groups. + It may seem odd that the algebraic structures are called + <cite>Ring</cite> and <cite>CommRing</cite> while the tactics are named + <cite>noncomm_ring</cite> and <cite>ring</cite>. This is partly for historical reasons, + but also for the convenience of using a shorter name for the + tactic that deals with commutative rings, since it is used more often. + </section> + <section id="using-theorems-and-lemmas"> + <h2>2.3. Using Theorems and Lemmas<a class="headerlink" href="#using-theorems-and-lemmas" title="Permalink to this heading"></a></h2> + Rewriting is great for proving equations, + but what about other sorts of theorems? + For example, how can we prove an inequality, + like the fact that $a + e^b \le a + e^c$ holds whenever $b \le c$? + We have already seen that theorems can be applied to arguments and hypotheses, + and that the <code class="docutils literal notranslate">apply</code> and <code class="docutils literal notranslate">exact</code> tactics can be used to solve goals. + In this section, we will make good use of these tools. + Consider the library theorems <code class="docutils literal notranslate">le_refl</code> and <code class="docutils literal notranslate">le_trans</code>: + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (le_refl : ∀ a : ℝ, a ≤ a) + #check (le_trans : a ≤ b → b ≤ c → a ≤ c) + </pre></div> + </div> + As we explain in more detail in <a class="reference internal" href="C03_Logic.html#implication-and-the-universal-quantifier">Section 3.1</a>, + the implicit parentheses in the statement of <code class="docutils literal notranslate">le_trans</code> + associate to the right, so it should be interpreted as <code class="docutils literal notranslate">a ≤ b → (b ≤ c → a ≤ c)</code>. + The library designers have set the arguments to <code class="docutils literal notranslate">le_trans</code> implicit, + so that Lean will not let you provide them explicitly (unless you + really insist, as we will discuss later). + Rather, it expects to infer them from the context in which they are used. + For example, when hypotheses <code class="docutils literal notranslate">h : a ≤ b</code> and <code class="docutils literal notranslate">h' : b ≤ c</code> + are in the context, + all the following work: + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (h : a ≤ b) (h' : b ≤ c) + + #check (le_refl : ∀ a : Real, a ≤ a) + #check (le_refl a : a ≤ a) + #check (le_trans : a ≤ b → b ≤ c → a ≤ c) + #check (le_trans h : b ≤ c → a ≤ c) + #check (le_trans h h' : a ≤ c) + </pre></div> + </div> + The <code class="docutils literal notranslate">apply</code> tactic takes a proof of a general statement or implication, + tries to match the conclusion with the current goal, + and leaves the hypotheses, if any, as new goals. + If the given proof matches the goal exactly + (modulo definitional equality), + you can use the <code class="docutils literal notranslate">exact</code> tactic instead of <code class="docutils literal notranslate">apply</code>. + So, all of these work: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := by + apply le_trans + · apply h₀ + . apply h₁ + + example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := by + apply le_trans h₀ + apply h₁ + + example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := + le_trans h₀ h₁ + + example (x : ℝ) : x ≤ x := by + apply le_refl + + example (x : ℝ) : x ≤ x := + le_refl x + </pre></div> + </div> + In the first example, applying <code class="docutils literal notranslate">le_trans</code> + creates two goals, + and we use the dots to indicate where the proof of each begins. + The dots are optional, but they serve to focus the goal: + within the block introduced by the dot, only one goal is visible, + and it must be completed before the end of the block. + Here we end the first block by starting a new one with another dot. + We could just as well have decreased the indentation. + In the fourth example and in the last example, + we avoid going into tactic mode entirely: + <code class="docutils literal notranslate">le_trans h₀ h₁</code> and <code class="docutils literal notranslate">le_refl x</code> are the proof terms we need. + Here are a few more library theorems: + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (le_refl : ∀ a, a ≤ a) + #check (le_trans : a ≤ b → b ≤ c → a ≤ c) + #check (lt_of_le_of_lt : a ≤ b → b < c → a < c) + #check (lt_of_lt_of_le : a < b → b ≤ c → a < c) + #check (lt_trans : a < b → b < c → a < c) + </pre></div> + </div> + Use them together with <code class="docutils literal notranslate">apply</code> and <code class="docutils literal notranslate">exact</code> to prove the following: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) (h₃ : d < e) : a < e := by + sorry + </pre></div> + </div> + In fact, Lean has a tactic that does this sort of thing automatically: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) (h₃ : d < e) : a < e := by + linarith + </pre></div> + </div> + The <code class="docutils literal notranslate">linarith</code> tactic is designed to handle linear arithmetic. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : 2 * a ≤ 3 * b) (h' : 1 ≤ a) (h'' : d = 2) : d + a ≤ 5 * b := by + linarith + </pre></div> + </div> + In addition to equations and inequalities in the context, + <code class="docutils literal notranslate">linarith</code> will use additional inequalities that you pass as arguments. + In the next example, <code class="docutils literal notranslate">exp_le_exp.mpr h'</code> is a proof of + <code class="docutils literal notranslate">exp b ≤ exp c</code>, as we will explain in a moment. + Notice that, in Lean, we write <code class="docutils literal notranslate">f x</code> to denote the application + of a function <code class="docutils literal notranslate">f</code> to the argument <code class="docutils literal notranslate">x</code>, + exactly the same way we write <code class="docutils literal notranslate">h x</code> to denote the result of + applying a fact or theorem <code class="docutils literal notranslate">h</code> to the argument <code class="docutils literal notranslate">x</code>. + Parentheses are only needed for compound arguments, + as in <code class="docutils literal notranslate">f (x + y)</code>. Without the parentheses, <code class="docutils literal notranslate">f x + y</code> + would be parsed as <code class="docutils literal notranslate">(f x) + y</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : 1 ≤ a) (h' : b ≤ c) : 2 + a + exp b ≤ 3 * a + exp c := by + linarith [exp_le_exp.mpr h'] + </pre></div> + </div> + Here are some more theorems in the library that can be used to establish + inequalities on the real numbers. + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (exp_le_exp : exp a ≤ exp b ↔ a ≤ b) + #check (exp_lt_exp : exp a < exp b ↔ a < b) + #check (log_le_log : 0 < a → 0 < b → (log a ≤ log b ↔ a ≤ b)) + #check (log_lt_log : 0 < a → a < b → log a < log b) + #check (add_le_add : a ≤ b → c ≤ d → a + c ≤ b + d) + #check (add_le_add_left : a ≤ b → ∀ c, c + a ≤ c + b) + #check (add_le_add_right : a ≤ b → ∀ c, a + c ≤ b + c) + #check (add_lt_add_of_le_of_lt : a ≤ b → c < d → a + c < b + d) + #check (add_lt_add_of_lt_of_le : a < b → c ≤ d → a + c < b + d) + #check (add_lt_add_left : a < b → ∀ c, c + a < c + b) + #check (add_lt_add_right : a < b → ∀ c, a + c < b + c) + #check (add_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a + b) + #check (add_pos : 0 < a → 0 < b → 0 < a + b) + #check (add_pos_of_pos_of_nonneg : 0 < a → 0 ≤ b → 0 < a + b) + #check (exp_pos : ∀ a, 0 < exp a) + #check @add_le_add_left + </pre></div> + </div> + Some of the theorems, <code class="docutils literal notranslate">exp_le_exp</code>, <code class="docutils literal notranslate">exp_lt_exp</code>, and <code class="docutils literal notranslate">log_le_log</code> + use a bi-implication, which represents the + phrase “if and only if.” + (You can type it in VS Code with <code class="docutils literal notranslate">\lr</code> of <code class="docutils literal notranslate">\iff</code>). + We will discuss this connective in greater detail in the next chapter. + Such a theorem can be used with <code class="docutils literal notranslate">rw</code> to rewrite a goal to + an equivalent one: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : a ≤ b) : exp a ≤ exp b := by + rw [exp_le_exp] + exact h + </pre></div> + </div> + In this section, however, we will use the fact that if <code class="docutils literal notranslate">h : A ↔ B</code> + is such an equivalence, + then <code class="docutils literal notranslate">h.mp</code> establishes the forward direction, <code class="docutils literal notranslate">A → B</code>, + and <code class="docutils literal notranslate">h.mpr</code> establishes the reverse direction, <code class="docutils literal notranslate">B → A</code>. + Here, <code class="docutils literal notranslate">mp</code> stands for “modus ponens” and + <code class="docutils literal notranslate">mpr</code> stands for “modus ponens reverse.” + You can also use <code class="docutils literal notranslate">h.1</code> and <code class="docutils literal notranslate">h.2</code> for <code class="docutils literal notranslate">h.mp</code> and <code class="docutils literal notranslate">h.mpr</code>, + respectively, if you prefer. + Thus the following proof works: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h₀ : a ≤ b) (h₁ : c < d) : a + exp c + e < b + exp d + e := by + apply add_lt_add_of_lt_of_le + · apply add_lt_add_of_le_of_lt h₀ + apply exp_lt_exp.mpr h₁ + apply le_refl + </pre></div> + </div> + The first line, <code class="docutils literal notranslate">apply add_lt_add_of_lt_of_le</code>, + creates two goals, + and once again we use a dot to separate the + proof of the first from the proof of the second. + Try the following examples on your own. + The example in the middle shows you that the <code class="docutils literal notranslate">norm_num</code> + tactic can be used to solve concrete numeric goals. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := by sorry + + example : (0 : ℝ) < 1 := by norm_num + + example (h : a ≤ b) : log (1 + exp a) ≤ log (1 + exp b) := by + have h₀ : 0 < 1 + exp a := by sorry + have h₁ : 0 < 1 + exp b := by sorry + apply (log_le_log h₀ h₁).mpr + sorry + </pre></div> + </div> + From these examples, it should be clear that being able to + find the library theorems you need constitutes an important + part of formalization. + There are a number of strategies you can use: + <ul class="simple"> + <li>You can browse mathlib in its + <a class="reference external" href="https://github.com/leanprover-community/mathlib">GitHub repository</a>.</li> + <li>You can use the API documentation on the mathlib + <a class="reference external" href="https://leanprover-community.github.io/mathlib_docs/">web pages</a>.</li> + <li>You can rely on mathlib naming conventions and tab completion in + the editor to guess a theorem name. + In Lean, a theorem named <code class="docutils literal notranslate">A_of_B_of_C</code> establishes + something of the form <code class="docutils literal notranslate">A</code> from hypotheses of the form <code class="docutils literal notranslate">B</code> and <code class="docutils literal notranslate">C</code>, + where <code class="docutils literal notranslate">A</code>, <code class="docutils literal notranslate">B</code>, and <code class="docutils literal notranslate">C</code> + approximate the way we might read the goals out loud. + So a theorem establishing something like <code class="docutils literal notranslate">x + y ≤ ...</code> will probably + start with <code class="docutils literal notranslate">add_le</code>. + Typing <code class="docutils literal notranslate">add_le</code> and hitting tab will give you some helpful choices.</li> + <li>If you right-click on an existing theorem name in VS Code, + the editor will show a menu with the option to + jump to the file where the theorem is defined, + and you can find similar theorems nearby.</li> + <li>You can use the <code class="docutils literal notranslate">library_search</code> tactic, + which tries to find the relevant theorem in the library.</li> + </ul> + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : 0 ≤ a ^ 2 := by + -- library_search + exact sq_nonneg a + </pre></div> + </div> + To try out <code class="docutils literal notranslate">library_search</code> in this example, + delete the <code class="docutils literal notranslate">exact</code> command and uncomment the previous line. + If you replace <code class="docutils literal notranslate">library_search</code> with <code class="docutils literal notranslate">suggest</code>, + you’ll see a long list of suggestions. + In this case, the suggestions are not helpful, but in other cases + it does better. + Using these tricks, + see if you can find what you need to do the + next example: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : a ≤ b) : c - exp b ≤ c - exp a := by + sorry + </pre></div> + </div> + Using the same tricks, confirm that <code class="docutils literal notranslate">linarith</code> instead of <code class="docutils literal notranslate">library_search</code> + can also finish the job. + Here is another example of an inequality: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : 2 * a * b ≤ a ^ 2 + b ^ 2 := by + have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 + calc + a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring + _ ≥ 0 := by apply pow_two_nonneg + + calc + 2 * a * b = 2 * a * b + 0 := by ring + _ ≤ 2 * a * b + (a ^ 2 - 2 * a * b + b ^ 2) := + add_le_add (le_refl _) h + _ = a ^ 2 + b ^ 2 := by ring + </pre></div> + </div> + Mathlib tends to put spaces around binary operations like <code class="docutils literal notranslate">*</code> and <code class="docutils literal notranslate">^</code>, + but in this example, the more compressed format increases readability. + There are a number of things worth noticing. + First, an expression <code class="docutils literal notranslate">s ≥ t</code> is definitionally equivalent to <code class="docutils literal notranslate">t ≤ s</code>. + In principle, this means one should be able to use them interchangeably. + But some of Lean’s automation does not recognize the equivalence, + so mathlib tends to favor <code class="docutils literal notranslate">≤</code> over <code class="docutils literal notranslate">≥</code>. + Second, we have used the <code class="docutils literal notranslate">ring</code> tactic extensively. + It is a real timesaver! + Finally, notice that in the second line of the + second <code class="docutils literal notranslate">calc</code> proof, + instead of writing <code class="docutils literal notranslate">by exact add_le_add (le_refl _) h</code>, + we can simply write the proof term <code class="docutils literal notranslate">add_le_add (le_refl _) h</code>. + In fact, the only cleverness in the proof above is figuring + out the hypothesis <code class="docutils literal notranslate">h</code>. + Once we have it, the second calculation involves only + linear arithmetic, and <code class="docutils literal notranslate">linarith</code> can handle it: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : 2 * a * b ≤ a ^ 2 + b ^ 2 := by + have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 + calc + a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring + _ ≥ 0 := by apply pow_two_nonneg + linarith + </pre></div> + </div> + How nice! We challenge you to use these ideas to prove the + following theorem. You can use the theorem <code class="docutils literal notranslate">abs_le'.mpr</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : abs (a * b) ≤ (a ^ 2 + b ^ 2) / 2 := by + sorry + + #check abs_le'.mpr + </pre></div> + </div> + If you managed to solve this, congratulations! + You are well on your way to becoming a master formalizer. + </section> + <section id="more-on-order-and-divisibility"> + <h2>2.4. More on Order and Divisibility<a class="headerlink" href="#more-on-order-and-divisibility" title="Permalink to this heading"></a></h2> + The <code class="docutils literal notranslate">min</code> function on the real numbers is uniquely characterized + by the following three facts: + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (min_le_left a b : min a b ≤ a) + #check (min_le_right a b : min a b ≤ b) + #check (le_min : c ≤ a → c ≤ b → c ≤ min a b) + </pre></div> + </div> + Can you guess the names of the theorems that characterize + <code class="docutils literal notranslate">max</code> in a similar way? + Notice that we have to apply <code class="docutils literal notranslate">min</code> to a pair of arguments <code class="docutils literal notranslate">a</code> and <code class="docutils literal notranslate">b</code> + by writing <code class="docutils literal notranslate">min a b</code> rather than <code class="docutils literal notranslate">min (a, b)</code>. + Formally, <code class="docutils literal notranslate">min</code> is a function of type <code class="docutils literal notranslate">ℝ → ℝ → ℝ</code>. + When we write a type like this with multiple arrows, + the convention is that the implicit parentheses associate + to the right, so the type is interpreted as <code class="docutils literal notranslate">ℝ → (ℝ → ℝ)</code>. + The net effect is that if <code class="docutils literal notranslate">a</code> and <code class="docutils literal notranslate">b</code> have type <code class="docutils literal notranslate">ℝ</code> + then <code class="docutils literal notranslate">min a</code> has type <code class="docutils literal notranslate">ℝ → ℝ</code> and + <code class="docutils literal notranslate">min a b</code> has type <code class="docutils literal notranslate">ℝ</code>, so <code class="docutils literal notranslate">min</code> acts like a function + of two arguments, as we expect. Handling multiple + arguments in this way is known as currying, + after the logician Haskell Curry. + The order of operations in Lean can also take some getting used to. + Function application binds tighter than infix operations, so the + expression <code class="docutils literal notranslate">min a b + c</code> is interpreted as <code class="docutils literal notranslate">(min a b) + c</code>. + With time, these conventions will become second nature. + Using the theorem <code class="docutils literal notranslate">le_antisymm</code>, we can show that two + real numbers are equal if each is less than or equal to the other. + Using this and the facts above, + we can show that <code class="docutils literal notranslate">min</code> is commutative: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : min a b = min b a := by + apply le_antisymm + · show min a b ≤ min b a + apply le_min + · apply min_le_right + apply min_le_left + · show min b a ≤ min a b + apply le_min + · apply min_le_right + apply min_le_left + </pre></div> + </div> + Here we have used dots to separate proofs of + different goals. + Our usage is inconsistent: + at the outer level, + we use dots and indentation for both goals, + whereas for the nested proofs, + we use dots only until a single goal remains. + Both conventions are reasonable and useful. + We also use the <code class="docutils literal notranslate">show</code> tactic to structure + the proof + and indicate what is being proved in each block. + The proof still works without the <code class="docutils literal notranslate">show</code> commands, + but using them makes the proof easier to read and maintain. + It may bother you that the the proof is repetitive. + To foreshadow skills you will learn later on, + we note that one way to avoid the repetition + is to state a local lemma and then use it: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : min a b = min b a := by + have h : ∀ x y : ℝ, min x y ≤ min y x := by + intro x y + apply le_min + apply min_le_right + apply min_le_left + apply le_antisymm + apply h + apply h + </pre></div> + </div> + We will say more about the universal quantifier in + <a class="reference internal" href="C03_Logic.html#implication-and-the-universal-quantifier">Section 3.1</a>, + but suffice it to say here that the hypothesis + <code class="docutils literal notranslate">h</code> says that the desired inequality holds for + any <code class="docutils literal notranslate">x</code> and <code class="docutils literal notranslate">y</code>, + and the <code class="docutils literal notranslate">intro</code> tactic introduces an arbitrary + <code class="docutils literal notranslate">x</code> and <code class="docutils literal notranslate">y</code> to establish the conclusion. + The first <code class="docutils literal notranslate">apply</code> after <code class="docutils literal notranslate">le_antisymm</code> implicitly + uses <code class="docutils literal notranslate">h a b</code>, whereas the second one uses <code class="docutils literal notranslate">h b a</code>. + Another solution is to use the <code class="docutils literal notranslate">repeat</code> tactic, + which applies a tactic (or a block) as many times + as it can. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : min a b = min b a := by + apply le_antisymm + repeat + apply le_min + apply min_le_right + apply min_le_left + </pre></div> + </div> + In any case, + whether or not you use these tricks, + we encourage you to prove the following: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : max a b = max b a := by + sorry + + example : min (min a b) c = min a (min b c) := by + sorry + </pre></div> + </div> + Of course, you are welcome to prove the associativity of <code class="docutils literal notranslate">max</code> as well. + It is an interesting fact that <code class="docutils literal notranslate">min</code> distributes over <code class="docutils literal notranslate">max</code> + the way that multiplication distributes over addition, + and vice-versa. + In other words, on the real numbers, we have the identity + <code class="docutils literal notranslate">min a (max b c) ≤ max (min a b) (min a c)</code> + as well as the corresponding version with <code class="docutils literal notranslate">max</code> and <code class="docutils literal notranslate">min</code> + switched. + But in the next section we will see that this does not follow + from the transitivity and reflexivity of <code class="docutils literal notranslate">≤</code> and + the characterizing properties of <code class="docutils literal notranslate">min</code> and <code class="docutils literal notranslate">max</code> enumerated above. + We need to use the fact that <code class="docutils literal notranslate">≤</code> on the real numbers is a total order, + which is to say, + it satisfies <code class="docutils literal notranslate">∀ x y, x ≤ y ∨ y ≤ x</code>. + Here the disjunction symbol, <code class="docutils literal notranslate">∨</code>, represents “or”. + In the first case, we have <code class="docutils literal notranslate">min x y = x</code>, + and in the second case, we have <code class="docutils literal notranslate">min x y = y</code>. + We will learn how to reason by cases in <a class="reference internal" href="C03_Logic.html#disjunction">Section 3.5</a>, + but for now we will stick to examples that don’t require the case split. + Here is one such example: + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem aux : min a b + c ≤ min (a + c) (b + c) := by + sorry + + example : min a b + c = min (a + c) (b + c) := by + sorry + </pre></div> + </div> + It is clear that <code class="docutils literal notranslate">aux</code> provides one of the two inequalities + needed to prove the equality, + but applying it to suitable values yields the other direction + as well. + As a hint, you can use the theorem <code class="docutils literal notranslate">add_neg_cancel_right</code> + and the <code class="docutils literal notranslate">linarith</code> tactic. + Lean’s naming convention is made manifest + in the library’s name for the triangle inequality: + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (abs_add : ∀ a b : ℝ, abs (a + b) ≤ abs a + abs b) + </pre></div> + </div> + Use it to prove the following variant: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : abs a - abs b ≤ abs (a - b) := + sorry + </pre></div> + </div> + See if you can do this in three lines or less. + You can use the theorem <code class="docutils literal notranslate">sub_add_cancel</code>. + Another important relation that we will make use of + in the sections to come is the divisibility relation + on the natural numbers, <code class="docutils literal notranslate">x ∣ y</code>. + Be careful: the divisibility symbol is not the + ordinary bar on your keyboard. + Rather, it is a unicode character obtained by + typing <code class="docutils literal notranslate">\|</code> in VS Code. + By convention, mathlib uses <code class="docutils literal notranslate">dvd</code> + to refer to it in theorem names. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h₀ : x ∣ y) (h₁ : y ∣ z) : x ∣ z := + dvd_trans h₀ h₁ + + example : x ∣ y * x * z := by + apply dvd_mul_of_dvd_left + apply dvd_mul_left + + example : x ∣ x ^ 2 := by + apply dvd_mul_left + </pre></div> + </div> + In the last example, the exponent is a natural + number, and applying <code class="docutils literal notranslate">dvd_mul_left</code> + forces Lean to expand the definition of <code class="docutils literal notranslate">x^2</code> to + <code class="docutils literal notranslate">x * x^1</code>. + See if you can guess the names of the theorems + you need to prove the following: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : x ∣ w) : x ∣ y * (x * z) + x ^ 2 + w ^ 2 := by + sorry + </pre></div> + </div> + With respect to divisibility, the greatest common divisor, + <code class="docutils literal notranslate">gcd</code>, and least common multiple, <code class="docutils literal notranslate">lcm</code>, + are analogous to <code class="docutils literal notranslate">min</code> and <code class="docutils literal notranslate">max</code>. + Since every number divides <code class="docutils literal notranslate">0</code>, + <code class="docutils literal notranslate">0</code> is really the greatest element with respect to divisibility: + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (m n : ℕ) + + open Nat + + #check (gcd_zero_right n : gcd n 0 = n) + + #check (gcd_zero_left n : gcd 0 n = n) + + #check (lcm_zero_right n : lcm n 0 = 0) + + #check (lcm_zero_left n : lcm 0 n = 0) + </pre></div> + </div> + The functions <code class="docutils literal notranslate">gcd</code> and <code class="docutils literal notranslate">lcm</code> for natural numbers are in the + <code class="docutils literal notranslate">Nat</code> namespace, + which means that the full identifiers are <code class="docutils literal notranslate">Nat.gcd</code> and <code class="docutils literal notranslate">Nat.lcm</code>. + Similarly, the names of the theorems listed are prefixed by <code class="docutils literal notranslate">Nat</code>. + The command <code class="docutils literal notranslate">open Nat</code> opens the namespace, + allowing us to use the shorter names. + See if you can guess the names of the theorems you will need to + prove the following: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : gcd m n = gcd n m := by + sorry + </pre></div> + </div> + Hint: you can use <code class="docutils literal notranslate">dvd_antisymm</code>, but if you do, Lean will + complain that the expression is ambiguous between the generic + theorem and the version <code class="docutils literal notranslate">Nat.dvd_antisymm</code>, + the one specifically for the natural numbers. + You can use <code class="docutils literal notranslate">_root_.dvd_antisymm</code> to specify the generic one; + either one will work. + </section> + <section id="proving-facts-about-algebraic-structures"> + <h2>2.5. Proving Facts about Algebraic Structures<a class="headerlink" href="#proving-facts-about-algebraic-structures" title="Permalink to this heading"></a></h2> + In <a class="reference internal" href="#proving-identities-in-algebraic-structures">Section 2.2</a>, + we saw that many common identities governing the real numbers hold + in more general classes of algebraic structures, + such as commutative rings. + We can use any axioms we want to describe an algebraic structure, + not just equations. + For example, a partial order consists of a set with a + binary relation that is reflexive and transitive, + like <code class="docutils literal notranslate">≤</code> on the real numbers. + Lean knows about partial orders: + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} [PartialOrder α] + variable (x y z : α) + + #check x ≤ y + #check (le_refl x : x ≤ x) + #check (le_trans : x ≤ y → y ≤ z → x ≤ z) + </pre></div> + </div> + Here we are adopting the mathlib convention of using + letters like <code class="docutils literal notranslate">α</code>, <code class="docutils literal notranslate">β</code>, and <code class="docutils literal notranslate">γ</code> + (entered as <code class="docutils literal notranslate">\a</code>, <code class="docutils literal notranslate">\b</code>, and <code class="docutils literal notranslate">\g</code>) + for arbitrary types. + The library often uses letters like <code class="docutils literal notranslate">R</code> and <code class="docutils literal notranslate">G</code> + for the carries of algebraic structures like rings and groups, + respectively, + but in general Greek letters are used for types, + especially when there is little or no structure + associated with them. + Associated to any partial order, <code class="docutils literal notranslate">≤</code>, + there is also a strict partial order, <code class="docutils literal notranslate"><</code>, + which acts somewhat like <code class="docutils literal notranslate"><</code> on the real numbers. + Saying that <code class="docutils literal notranslate">x</code> is less than <code class="docutils literal notranslate">y</code> in this order + is equivalent to saying that it is less-than-or-equal to <code class="docutils literal notranslate">y</code> + and not equal to <code class="docutils literal notranslate">y</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check x < y + #check (lt_irrefl x : ¬x < x) + #check (lt_trans : x < y → y < z → x < z) + #check (lt_of_le_of_lt : x ≤ y → y < z → x < z) + #check (lt_of_lt_of_le : x < y → y ≤ z → x < z) + + example : x < y ↔ x ≤ y ∧ x ≠ y := + lt_iff_le_and_ne + </pre></div> + </div> + In this example, the symbol <code class="docutils literal notranslate">∧</code> stands for “and,” + the symbol <code class="docutils literal notranslate">¬</code> stands for “not,” and + <code class="docutils literal notranslate">x ≠ y</code> abbreviates <code class="docutils literal notranslate">¬ (x = y)</code>. + In <a class="reference internal" href="C03_Logic.html#logic">Chapter 3</a>, you will learn how to use + these logical connectives to prove that <code class="docutils literal notranslate"><</code> + has the properties indicated. + A lattice is a structure that extends a partial + order with operations <code class="docutils literal notranslate">⊓</code> and <code class="docutils literal notranslate">⊔</code> that are + analogous to <code class="docutils literal notranslate">min</code> and <code class="docutils literal notranslate">max</code> on the real numbers: + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} [Lattice α] + variable (x y z : α) + + #check x ⊓ y + #check (inf_le_left : x ⊓ y ≤ x) + #check (inf_le_right : x ⊓ y ≤ y) + #check (le_inf : z ≤ x → z ≤ y → z ≤ x ⊓ y) + #check x ⊔ y + #check (le_sup_left : x ≤ x ⊔ y) + #check (le_sup_right : y ≤ x ⊔ y) + #check (sup_le : x ≤ z → y ≤ z → x ⊔ y ≤ z) + </pre></div> + </div> + The characterizations of <code class="docutils literal notranslate">⊓</code> and <code class="docutils literal notranslate">⊔</code> justify calling them + the greatest lower bound and least upper bound, respectively. + You can type them in VS code using <code class="docutils literal notranslate">\glb</code> and <code class="docutils literal notranslate">\lub</code>. + The symbols are also often called then infimum and + the supremum, + and mathlib refers to them as <code class="docutils literal notranslate">inf</code> and <code class="docutils literal notranslate">sup</code> in + theorem names. + To further complicate matters, + they are also often called meet and join. + Therefore, if you work with lattices, + you have to keep the following dictionary in mind: + <ul class="simple"> + <li><code class="docutils literal notranslate">⊓</code> is the greatest lower bound, infimum, or meet.</li> + <li><code class="docutils literal notranslate">⊔</code> is the least upper bound, supremum, or join.</li> + </ul> + Some instances of lattices include: + <ul class="simple"> + <li><code class="docutils literal notranslate">min</code> and <code class="docutils literal notranslate">max</code> on any total order, such as the integers or real numbers with <code class="docutils literal notranslate">≤</code></li> + <li><code class="docutils literal notranslate">∩</code> and <code class="docutils literal notranslate">∪</code> on the collection of subsets of some domain, with the ordering <code class="docutils literal notranslate">⊆</code></li> + <li><code class="docutils literal notranslate">∧</code> and <code class="docutils literal notranslate">∨</code> on boolean truth values, with ordering <code class="docutils literal notranslate">x ≤ y</code> if either <code class="docutils literal notranslate">x</code> is false or <code class="docutils literal notranslate">y</code> is true</li> + <li><code class="docutils literal notranslate">gcd</code> and <code class="docutils literal notranslate">lcm</code> on the natural numbers (or positive natural numbers), with the divisibility ordering, <code class="docutils literal notranslate">∣</code></li> + <li>the collection of linear subspaces of a vector space, + where the greatest lower bound is given by the intersection, + the least upper bound is given by the sum of the two spaces, + and the ordering is inclusion</li> + <li>the collection of topologies on a set (or, in Lean, a type), + where the greatest lower bound of two topologies consists of + the topology that is generated by their union, + the least upper bound is their intersection, + and the ordering is reverse inclusion</li> + </ul> + You can check that, as with <code class="docutils literal notranslate">min</code> / <code class="docutils literal notranslate">max</code> and <code class="docutils literal notranslate">gcd</code> / <code class="docutils literal notranslate">lcm</code>, + you can prove the commutativity and associativity of the infimum and supremum + using only their characterizing axioms, + together with <code class="docutils literal notranslate">le_refl</code> and <code class="docutils literal notranslate">le_trans</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : x ⊓ y = y ⊓ x := by + sorry + + example : x ⊓ y ⊓ z = x ⊓ (y ⊓ z) := by + sorry + + example : x ⊔ y = y ⊔ x := by + sorry + + example : x ⊔ y ⊔ z = x ⊔ (y ⊔ z) := by + sorry + </pre></div> + </div> + You can find these theorems in the mathlib as <code class="docutils literal notranslate">inf_comm</code>, <code class="docutils literal notranslate">inf_assoc</code>, + <code class="docutils literal notranslate">sup_comm</code>, and <code class="docutils literal notranslate">sup_assoc</code>, respectively. + Another good exercise is to prove the absorption laws + using only those axioms: + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem absorb1 : x ⊓ (x ⊔ y) = x := by + sorry + + theorem absorb2 : x ⊔ x ⊓ y = x := by + sorry + </pre></div> + </div> + These can be found in mathlib with the names <code class="docutils literal notranslate">inf_sup_self</code> and <code class="docutils literal notranslate">sup_inf_self</code>. + A lattice that satisfies the additional identities + <code class="docutils literal notranslate">x ⊓ (y ⊔ z) = (x ⊓ y) ⊔ (x ⊓ z)</code> and + <code class="docutils literal notranslate">x ⊔ (y ⊓ z) = (x ⊔ y) ⊓ (x ⊔ z)</code> + is called a distributive lattice. Lean knows about these too: + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} [DistribLattice α] + variable (x y z : α) + + #check (inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) + #check (inf_sup_right : (x ⊔ y) ⊓ z = x ⊓ z ⊔ y ⊓ z) + #check (sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) + #check (sup_inf_right : x ⊓ y ⊔ z = (x ⊔ z) ⊓ (y ⊔ z)) + </pre></div> + </div> + The left and right versions are easily shown to be + equivalent, given the commutativity of <code class="docutils literal notranslate">⊓</code> and <code class="docutils literal notranslate">⊔</code>. + It is a good exercise to show that not every lattice + is distributive + by providing an explicit description of a + nondistributive lattice with finitely many elements. + It is also a good exercise to show that in any lattice, + either distributivity law implies the other: + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} [Lattice α] + variable (a b c : α) + + example (h : ∀ x y z : α, x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) : a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c) := by + sorry + + example (h : ∀ x y z : α, x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) : a ⊓ (b ⊔ c) = a ⊓ b ⊔ a ⊓ c := by + sorry + </pre></div> + </div> + It is possible to combine axiomatic structures into larger ones. + For example, a strict ordered ring consists of a commutative ring together + with a partial order on the carrier + satisfying additional axioms that say that the ring operations + are compatible with the order: + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {R : Type _} [StrictOrderedRing R] + variable (a b c : R) + + #check (add_le_add_left : a ≤ b → ∀ c, c + a ≤ c + b) + #check (mul_pos : 0 < a → 0 < b → 0 < a * b) + </pre></div> + </div> + <a class="reference internal" href="C03_Logic.html#logic">Chapter 3</a> will provide the means to derive the following from <code class="docutils literal notranslate">mul_pos</code> + and the definition of <code class="docutils literal notranslate"><</code>: + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (mul_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a * b) + </pre></div> + </div> + It is then an extended exercise to show that many common facts + used to reason about arithmetic and the ordering on the real + numbers hold generically for any ordered ring. + Here are a couple of examples you can try, + using only properties of rings, partial orders, and the facts + enumerated in the last two examples: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : a ≤ b → 0 ≤ b - a := by + sorry + + example : 0 ≤ b - a → a ≤ b := by + sorry + + example (h : a ≤ b) (h' : 0 ≤ c) : a * c ≤ b * c := by + sorry + </pre></div> + </div> + Finally, here is one last example. + A metric space consists of a set equipped with a notion of + distance, <code class="docutils literal notranslate">dist x y</code>, + mapping any pair of elements to a real number. + The distance function is assumed to satisfy the following axioms: + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {X : Type _} [MetricSpace X] + variable (x y z : X) + + #check (dist_self x : dist x x = 0) + #check (dist_comm x y : dist x y = dist y x) + #check (dist_triangle x y z : dist x z ≤ dist x y + dist y z) + </pre></div> + </div> + Having mastered this section, + you can show that it follows from these axioms that distances are + always nonnegative: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x y : X) : 0 ≤ dist x y := by + sorry + </pre></div> + </div> + We recommend making use of the theorem <code class="docutils literal notranslate">nonneg_of_mul_nonneg_left</code>. + As you may have guessed, this theorem is called <code class="docutils literal notranslate">dist_nonneg</code> in mathlib. + </section> + </section> + + + </div> + </div> + <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> + <a href="C01_Introduction.html" class="btn btn-neutral float-left" title="1. Introduction" accesskey="p" rel="prev"> Previous</a> + <a href="C03_Logic.html" class="btn btn-neutral float-right" title="3. Logic" accesskey="n" rel="next">Next </a> + </div> + + <hr/> + + <div role="contentinfo"> + © Copyright 2020, Jeremy Avigad, Kevin Buzzard, Robert Y. Lewis, Patrick Massot. + </div> + + Built with <a href="https://www.sphinx-doc.org/">Sphinx</a> using a + <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a> + provided by <a href="https://readthedocs.org">Read the Docs</a>. + + + </footer> + </div> + </div> + </section> + </div> + <script> + jQuery(function () { + SphinxRtdTheme.Navigation.enable(true); + }); + </script> + + </body> + </html>
html/C03_Logic.html (new)
- @@ -0,0 +1,1703 @@
 + <!DOCTYPE html> + <html class="writer-html5" lang="en" > + <head> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + + <meta name="viewport" content="width=device-width, initial-scale=1.0" /> + <title>3. Logic — Mathematics in Lean 0.1 documentation</title> + <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="shortcut icon" href="_static/favicon.ico"/> +  + + <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> + <script src="_static/jquery.js"></script> + <script src="_static/underscore.js"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> + <script src="_static/doctools.js"></script> + <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> + <script src="_static/js/theme.js"></script> + <link rel="index" title="Index" href="genindex.html" /> + <link rel="search" title="Search" href="search.html" /> + <link rel="next" title="4. Sets and Functions" href="C04_Sets_and_Functions.html" /> + <link rel="prev" title="2. Basics" href="C02_Basics.html" /> + </head> + + <body class="wy-body-for-nav"> + <div class="wy-grid-for-nav"> + <nav data-toggle="wy-nav-shift" class="wy-nav-side"> + <div class="wy-side-scroll"> + <div class="wy-side-nav-search" > + <a href="index.html" class="icon icon-home"> Mathematics in Lean + </a> + <div role="search"> + <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> + <input type="text" name="q" placeholder="Search docs" /> + <input type="hidden" name="check_keywords" value="yes" /> + <input type="hidden" name="area" value="default" /> + </form> + </div> + </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> + <ul class="current"> + <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a></li> + <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> + <li class="toctree-l1 current"><a class="current reference internal" href="#">3. Logic</a><ul> + <li class="toctree-l2"><a class="reference internal" href="#implication-and-the-universal-quantifier">3.1. Implication and the Universal Quantifier</a></li> + <li class="toctree-l2"><a class="reference internal" href="#the-existential-quantifier">3.2. The Existential Quantifier</a></li> + <li class="toctree-l2"><a class="reference internal" href="#negation">3.3. Negation</a></li> + <li class="toctree-l2"><a class="reference internal" href="#conjunction-and-bi-implication">3.4. Conjunction and Bi-implication</a></li> + <li class="toctree-l2"><a class="reference internal" href="#disjunction">3.5. Disjunction</a></li> + <li class="toctree-l2"><a class="reference internal" href="#sequences-and-convergence">3.6. Sequences and Convergence</a></li> + </ul> + </li> + <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> + <li class="toctree-l1"><a class="reference internal" href="C06_Abstract_Algebra.html">6. Abstract Algebra</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Topology.html">7. Topology</a></li> + </ul> + <ul> + <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li> + </ul> + + </div> + </div> + </nav> + + <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > + + <a href="index.html">Mathematics in Lean</a> + </nav> + + <div class="wy-nav-content"> + <div class="rst-content"> + <div role="navigation" aria-label="Page navigation"> + <ul class="wy-breadcrumbs"> + <li><a href="index.html" class="icon icon-home"></a> »</li> + <li>3. Logic</li> + <li class="wy-breadcrumbs-aside"> + <a href="_sources/C03_Logic.rst.txt" rel="nofollow"> View page source</a> + </li> + </ul> + <hr/> + </div> + <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> + <div itemprop="articleBody"> + + <section id="logic"> + <h1>3. Logic<a class="headerlink" href="#logic" title="Permalink to this heading"></a></h1> + In the last chapter, we dealt with equations, inequalities, + and basic mathematical statements like + “$x$ divides $y$.” + Complex mathematical statements are built up from + simple ones like these + using logical terms like “and,” “or,” “not,” and + “if … then,” “every,” and “some.” + In this chapter, we show you how to work with statements + that are built up in this way. + <section id="implication-and-the-universal-quantifier"> + <h2>3.1. Implication and the Universal Quantifier<a class="headerlink" href="#implication-and-the-universal-quantifier" title="Permalink to this heading"></a></h2> + Consider the statement after the <code class="docutils literal notranslate">#check</code>: + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check ∀ x : ℝ, 0 ≤ x → abs x = x + </pre></div> + </div> + In words, we would say “for every real number <code class="docutils literal notranslate">x</code>, if <code class="docutils literal notranslate">0 ≤ x</code> then + the absolute value of <code class="docutils literal notranslate">x</code> equals <code class="docutils literal notranslate">x</code>”. + We can also have more complicated statements like: + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε + </pre></div> + </div> + In words, we would say “for every <code class="docutils literal notranslate">x</code>, <code class="docutils literal notranslate">y</code>, and <code class="docutils literal notranslate">ε</code>, + if <code class="docutils literal notranslate">0 < ε ≤ 1</code>, the absolute value of <code class="docutils literal notranslate">x</code> is less than <code class="docutils literal notranslate">ε</code>, + and the absolute value of <code class="docutils literal notranslate">y</code> is less than <code class="docutils literal notranslate">ε</code>, + then the absolute value of <code class="docutils literal notranslate">x * y</code> is less than <code class="docutils literal notranslate">ε</code>.” + In Lean, in a sequence of implications there are + implicit parentheses grouped to the right. + So the expression above means + “if <code class="docutils literal notranslate">0 < ε</code> then if <code class="docutils literal notranslate">ε ≤ 1</code> then if <code class="docutils literal notranslate">abs x < ε</code> …” + As a result, the expression says that all the + assumptions together imply the conclusion. + You have already seen that even though the universal quantifier + in this statement + ranges over objects and the implication arrows introduce hypotheses, + Lean treats the two in very similar ways. + In particular, if you have proved a theorem of that form, + you can apply it to objects and hypotheses in the same way: + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem my_lemma : ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := + sorry + + section + variable (a b δ : ℝ) + variable (h₀ : 0 < δ) (h₁ : δ ≤ 1) + variable (ha : abs a < δ) (hb : abs b < δ) + + #check my_lemma a b δ + #check my_lemma a b δ h₀ h₁ + #check my_lemma a b δ h₀ h₁ ha hb + + end + </pre></div> + </div> + You have also already seen that it is common in Lean + to use curly brackets to make quantified variables implicit + when they can be inferred from subsequent hypotheses. + When we do that, we can just apply a lemma to the hypotheses without + mentioning the objects. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem my_lemma2 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := + sorry + + section + variable (a b δ : ℝ) + variable (h₀ : 0 < δ) (h₁ : δ ≤ 1) + variable (ha : abs a < δ) (hb : abs b < δ) + + #check my_lemma2 h₀ h₁ ha hb + + end + </pre></div> + </div> + At this stage, you also know that if you use + the <code class="docutils literal notranslate">apply</code> tactic to apply <code class="docutils literal notranslate">my_lemma</code> + to a goal of the form <code class="docutils literal notranslate">abs (a * b) < δ</code>, + you are left with new goals that require you to prove + each of the hypotheses. + To prove a statement like this, use the <code class="docutils literal notranslate">intro</code> tactic. + Take a look at what it does in this example: + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem my_lemma3 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := by + intro x y ε epos ele1 xlt ylt + sorry + </pre></div> + </div> + We can use any names we want for the universally quantified variables; + they do not have to be <code class="docutils literal notranslate">x</code>, <code class="docutils literal notranslate">y</code>, and <code class="docutils literal notranslate">ε</code>. + Notice that we have to introduce the variables + even though they are marked implicit: + making them implicit means that we leave them out when + we write an expression using <code class="docutils literal notranslate">my_lemma</code>, + but they are still an essential part of the statement + that we are proving. + After the <code class="docutils literal notranslate">intro</code> command, + the goal is what it would have been at the start if we + listed all the variables and hypotheses before the colon, + as we did in the last section. + In a moment, we will see why it is sometimes necessary to + introduce variables and hypotheses after the proof begins. + To help you prove the lemma, we will start you off: + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem my_lemma4 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := by + intro x y ε epos ele1 xlt ylt + calc + abs (x * y) = abs x * abs y := sorry + _ ≤ abs x * ε := sorry + _ < 1 * ε := sorry + _ = ε := sorry + </pre></div> + </div> + Finish the proof using the theorems + <code class="docutils literal notranslate">abs_mul</code>, <code class="docutils literal notranslate">mul_le_mul</code>, <code class="docutils literal notranslate">abs_nonneg</code>, + <code class="docutils literal notranslate">mul_lt_mul_right</code>, and <code class="docutils literal notranslate">one_mul</code>. + Remember that you can find theorems like these using + tab completion. + Remember also that you can use <code class="docutils literal notranslate">.mp</code> and <code class="docutils literal notranslate">.mpr</code> + or <code class="docutils literal notranslate">.1</code> and <code class="docutils literal notranslate">.2</code> to extract the two directions + of an if-and-only-if statement. + Universal quantifiers are often hidden in definitions, + and Lean will unfold definitions to expose them when necessary. + For example, let’s define two predicates, + <code class="docutils literal notranslate">fn_ub f a</code> and <code class="docutils literal notranslate">fn_lb f a</code>, + where <code class="docutils literal notranslate">f</code> is a function from the real numbers to the real + numbers and <code class="docutils literal notranslate">a</code> is a real number. + The first says that <code class="docutils literal notranslate">a</code> is an upper bound on the + values of <code class="docutils literal notranslate">f</code>, + and the second says that <code class="docutils literal notranslate">a</code> is a lower bound + on the values of <code class="docutils literal notranslate">f</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, f x ≤ a + + def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, a ≤ f x + </pre></div> + </div> + In the next example, <code class="docutils literal notranslate">fun x ↦ f x + g x</code> is a name for the + function that maps <code class="docutils literal notranslate">x</code> to <code class="docutils literal notranslate">f x + g x</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (hfa : FnUb f a) (hgb : FnUb g b) : FnUb (fun x ↦ f x + g x) (a + b) := by + intro x + dsimp + apply add_le_add + apply hfa + apply hgb + </pre></div> + </div> + Applying <code class="docutils literal notranslate">intro</code> to the goal <code class="docutils literal notranslate">fn_ub (fun x ↦ f x + g x) (a + b)</code> + forces Lean to unfold the definition of <code class="docutils literal notranslate">fn_ub</code> + and introduce <code class="docutils literal notranslate">x</code> for the universal quantifier. + The goal is then <code class="docutils literal notranslate">(fun (x : ℝ) ↦ f x + g x) x ≤ a + b</code>. + But applying <code class="docutils literal notranslate">(fun x ↦ f x + g x)</code> to <code class="docutils literal notranslate">x</code> should result in <code class="docutils literal notranslate">f x + g x</code>, + and the <code class="docutils literal notranslate">dsimp</code> command performs that simplification. + (The “d” stands for “definitional.”) + You can delete that command and the proof still works; + Lean would have to perform that contraction anyhow to make + sense of the next <code class="docutils literal notranslate">apply</code>. + The <code class="docutils literal notranslate">dsimp</code> command simply makes the goal more readable + and helps us figure out what to do next. + Another option is to use the <code class="docutils literal notranslate">change</code> tactic + by writing <code class="docutils literal notranslate">change f x + g x ≤ a + b</code>. + This helps make the proof more readable, + and gives you more control over how the goal is transformed. + The rest of the proof is routine. + The last two <code class="docutils literal notranslate">apply</code> commands force Lean to unfold the definitions + of <code class="docutils literal notranslate">fn_ub</code> in the hypotheses. + Try carrying out similar proofs of these: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (hfa : FnLb f a) (hgb : FnLb g b) : FnLb (fun x => f x + g x) (a + b) := + sorry + + example (nnf : FnLb f 0) (nng : FnLb g 0) : FnLb (fun x => f x * g x) 0 := + sorry + + example (hfa : FnUb f a) (hfb : FnUb g b) (nng : FnLb g 0) (nna : 0 ≤ a) : + FnUb (fun x => f x * g x) (a * b) := + sorry + </pre></div> + </div> + Even though we have defined <code class="docutils literal notranslate">fn_ub</code> and <code class="docutils literal notranslate">fn_lb</code> for functions + from the reals to the reals, + you should recognize that the definitions and proofs are much + more general. + The definitions make sense for functions between any two types + for which there is a notion of order on the codomain. + Checking the type of the theorem <code class="docutils literal notranslate">add_le_add</code> shows that it holds + of any structure that is an “ordered additive commutative monoid”; + the details of what that means don’t matter now, + but it is worth knowing that the natural numbers, integers, rationals, + and real numbers are all instances. + So if we prove the theorem <code class="docutils literal notranslate">fn_ub_add</code> at that level of generality, + it will apply in all these instances. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} {R : Type _} [OrderedCancelAddCommMonoid R] + + #check @add_le_add + + def FnUb' (f : α → R) (a : R) : Prop := + ∀ x, f x ≤ a + + theorem fn_ub_add {f g : α → R} {a b : R} (hfa : FnUb' f a) (hgb : FnUb' g b) : + FnUb' (fun x => f x + g x) (a + b) := fun x => add_le_add (hfa x) (hgb x) + </pre></div> + </div> + You have already seen square brackets like these in + Section <a class="reference internal" href="C02_Basics.html#proving-identities-in-algebraic-structures">Section 2.2</a>, + though we still haven’t explained what they mean. + For concreteness, we will stick to the real numbers + for most of our examples, + but it is worth knowing that mathlib contains definitions and theorems + that work at a high level of generality. + For another example of a hidden universal quantifier, + mathlib defines a predicate <code class="docutils literal notranslate">monotone</code>, + which says that a function is nondecreasing in its arguments: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (f : ℝ → ℝ) (h : Monotone f) : ∀ {a b}, a ≤ b → f a ≤ f b := + @h + </pre></div> + </div> + The property <code class="docutils literal notranslate">Monotone f</code> is defined to be exactly the expression + after the colon. We need to put the <code class="docutils literal notranslate">@</code> symbol before <code class="docutils literal notranslate">h</code> because + if we don’t, + Lean expands the implicit arguments to <code class="docutils literal notranslate">h</code> and inserts placeholders. + Proving statements about monotonicity + involves using <code class="docutils literal notranslate">intro</code> to introduce two variables, + say, <code class="docutils literal notranslate">a</code> and <code class="docutils literal notranslate">b</code>, and the hypothesis <code class="docutils literal notranslate">a ≤ b</code>. + To use a monotonicity hypothesis, + you can apply it to suitable arguments and hypotheses, + and then apply the resulting expression to the goal. + Or you can apply it to the goal and let Lean help you + work backwards by displaying the remaining hypotheses + as new subgoals. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f x + g x := by + intro a b aleb + apply add_le_add + apply mf aleb + apply mg aleb + </pre></div> + </div> + When a proof is this short, it is often convenient + to give a proof term instead. + To describe a proof that temporarily introduces objects + <code class="docutils literal notranslate">a</code> and <code class="docutils literal notranslate">b</code> and a hypothesis <code class="docutils literal notranslate">aleb</code>, + Lean uses the notation <code class="docutils literal notranslate">fun a b aleb => ...</code>. + This is analogous to the way that an expression + like <code class="docutils literal notranslate">fun x => x^2</code> describes a function + by temporarily naming an object, <code class="docutils literal notranslate">x</code>, + and then using it to describe a value. + So the <code class="docutils literal notranslate">intro</code> command in the previous proof + corresponds to the lambda abstraction in the next proof term. + The <code class="docutils literal notranslate">apply</code> commands then correspond to building + the application of the theorem to its arguments. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f x + g x := + fun a b aleb => add_le_add (mf aleb) (mg aleb) + </pre></div> + </div> + Here is a useful trick: if you start writing + the proof term <code class="docutils literal notranslate">fun a b aleb => _</code> using + an underscore where the rest of the + expression should go, + Lean will flag an error, + indicating that it can’t guess the value of that expression. + If you check the Lean Goal window in VS Code or + hover over the squiggly error marker, + Lean will show you the goal that the remaining + expression has to solve. + Try proving these, with either tactics or proof terms: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {c : ℝ} (mf : Monotone f) (nnc : 0 ≤ c) : Monotone fun x => c * f x := + sorry + + example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f (g x) := + sorry + </pre></div> + </div> + Here are some more examples. + A function $f$ from $\Bbb R$ to + $\Bbb R$ is said to be even if + $f(-x) = f(x)$ for every $x$, + and odd if $f(-x) = -f(x)$ for every $x$. + The following example defines these two notions formally + and establishes one fact about them. + You can complete the proofs of the others. + <div class="highlight-lean notranslate"><div class="highlight"><pre>def FnEven (f : ℝ → ℝ) : Prop := + ∀ x, f x = f (-x) + + def FnOdd (f : ℝ → ℝ) : Prop := + ∀ x, f x = -f (-x) + + example (ef : FnEven f) (eg : FnEven g) : FnEven fun x => f x + g x := by + intro x + calc + (fun x => f x + g x) x = f x + g x := rfl + _ = f (-x) + g (-x) := by rw [ef, eg] + + + example (of : FnOdd f) (og : FnOdd g) : FnEven fun x => f x * g x := by + sorry + + example (ef : FnEven f) (og : FnOdd g) : FnOdd fun x => f x * g x := by + sorry + + example (ef : FnEven f) (og : FnOdd g) : FnEven fun x => f (g x) := by + sorry + </pre></div> + </div> + The first proof can be shortened using <code class="docutils literal notranslate">dsimp</code> or <code class="docutils literal notranslate">change</code> + to get rid of the lambda. + But you can check that the subsequent <code class="docutils literal notranslate">rw</code> won’t work + unless we get rid of the lambda explicitly, + because otherwise it cannot find the patterns <code class="docutils literal notranslate">f x</code> and <code class="docutils literal notranslate">g x</code> + in the expression. + Contrary to some other tactics, <code class="docutils literal notranslate">rw</code> operates on the syntactic level, + it won’t unfold definitions or apply reductions for you + (it has a variant called <code class="docutils literal notranslate">erw</code> that tries a little harder in this + direction, but not much harder). + You can find implicit universal quantifiers all over the place, + once you know how to spot them. + Mathlib includes a good library for rudimentary set theory. + Lean’s logical foundation imposes the restriction that when + we talk about sets, we are always talking about sets of + elements of some type. If <code class="docutils literal notranslate">x</code> has type <code class="docutils literal notranslate">α</code> and <code class="docutils literal notranslate">s</code> has + type <code class="docutils literal notranslate">Set α</code>, then <code class="docutils literal notranslate">x ∈ s</code> is a proposition that + asserts that <code class="docutils literal notranslate">x</code> is an element of <code class="docutils literal notranslate">s</code>. + If <code class="docutils literal notranslate">s</code> and <code class="docutils literal notranslate">t</code> are of type <code class="docutils literal notranslate">Set α</code>, + then the subset relation <code class="docutils literal notranslate">s ⊆ t</code> is defined to mean + <code class="docutils literal notranslate">∀ {x : α}, x ∈ s → x ∈ t</code>. + The variable in the quantifier is marked implicit so that + given <code class="docutils literal notranslate">h : s ⊆ t</code> and <code class="docutils literal notranslate">h' : x ∈ s</code>, + we can write <code class="docutils literal notranslate">h h'</code> as justification for <code class="docutils literal notranslate">x ∈ t</code>. + The following example provides a tactic proof and a proof term + justifying the reflexivity of the subset relation, + and asks you to do the same for transitivity. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} (r s t : Set α) + + example : s ⊆ s := by + intro x xs + exact xs + + theorem Subset.refl : s ⊆ s := fun x xs => xs + + theorem Subset.trans : r ⊆ s → s ⊆ t → r ⊆ t := by + sorry + </pre></div> + </div> + Just as we defined <code class="docutils literal notranslate">FnUb</code> for functions, + we can define <code class="docutils literal notranslate">SetUb s a</code> to mean that <code class="docutils literal notranslate">a</code> + is an upper bound on the set <code class="docutils literal notranslate">s</code>, + assuming <code class="docutils literal notranslate">s</code> is a set of elements of some type that + has an order associated with it. + In the next example, we ask you to prove that + if <code class="docutils literal notranslate">a</code> is a bound on <code class="docutils literal notranslate">s</code> and <code class="docutils literal notranslate">a ≤ b</code>, + then <code class="docutils literal notranslate">b</code> is a bound on <code class="docutils literal notranslate">s</code> as well. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} [PartialOrder α] + variable (s : Set α) (a b : α) + + def SetUb (s : Set α) (a : α) := + ∀ x, x ∈ s → x ≤ a + + example (h : SetUb s a) (h' : a ≤ b) : SetUb s b := + sorry + </pre></div> + </div> + We close this section with one last important example. + A function $f$ is said to be injective if for + every $x_1$ and $x_2$, + if $f(x_1) = f(x_2)$ then $x_1 = x_2$. + Mathlib defines <code class="docutils literal notranslate">function.injective f</code> with + <code class="docutils literal notranslate">x₁</code> and <code class="docutils literal notranslate">x₂</code> implicit. + The next example shows that, on the real numbers, + any function that adds a constant is injective. + We then ask you to show that multiplication by a nonzero + constant is also injective. + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Function + + example (c : ℝ) : Injective fun x => x + c := by + intro x₁ x₂ h' + exact (add_left_inj c).mp h' + + example {c : ℝ} (h : c ≠ 0) : Injective fun x => c * x := by + sorry + </pre></div> + </div> + Finally, show that the composition of two injective functions is injective: + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} {β : Type _} {γ : Type _} + variable {g : β → γ} {f : α → β} + + example (injg : Injective g) (injf : Injective f) : Injective fun x => g (f x) := by + sorry + </pre></div> + </div> + </section> + <section id="the-existential-quantifier"> + <h2>3.2. The Existential Quantifier<a class="headerlink" href="#the-existential-quantifier" title="Permalink to this heading"></a></h2> + The existential quantifier, which can be entered as <code class="docutils literal notranslate">\ex</code> in VS Code, + is used to represent the phrase “there exists.” + The formal expression <code class="docutils literal notranslate">∃ x : ℝ, 2 < x ∧ x < 3</code> in Lean says + that there is a real number between 2 and 3. + (We will discuss the conjunction symbol, <code class="docutils literal notranslate">∧</code>, in <a class="reference internal" href="#conjunction-and-biimplication">Section 3.4</a>.) + The canonical way to prove such a statement is to exhibit a real number + and show that it has the stated property. + The number 2.5, which we can enter as <code class="docutils literal notranslate">5 / 2</code> + or <code class="docutils literal notranslate">(5 : ℝ) / 2</code> when Lean cannot infer from context that we have + the real numbers in mind, has the required property, + and the <code class="docutils literal notranslate">norm_num</code> tactic can prove that it meets the description. + There are a few ways we can put the information together. + Given a goal that begins with an existential quantifier, + the <code class="docutils literal notranslate">use</code> tactic is used to provide the object, + leaving the goal of proving the property. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : ∃ x : ℝ, 2 < x ∧ x < 3 := by + use 5 / 2 + norm_num + </pre></div> + </div> + Alternatively, we can use Lean’s anonymous constructor notation + to construct the proof. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : ∃ x : ℝ, 2 < x ∧ x < 3 := + have h : 2 < (5 : ℝ) / 2 ∧ (5 : ℝ) / 2 < 3 := by norm_num + ⟨5 / 2, h⟩ + </pre></div> + </div> + The left and right angle brackets, + which can be entered as <code class="docutils literal notranslate">\<</code> and <code class="docutils literal notranslate">\></code> respectively, + tell Lean to put together the given data using + whatever construction is appropriate + for the current goal. + We can use the notation without going first into tactic mode: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : ∃ x : ℝ, 2 < x ∧ x < 3 := + ⟨5 / 2, by norm_num⟩ + </pre></div> + </div> + So now we know how to prove an exists statement. + But how do we use one? + If we know that there exists an object with a certain property, + we should be able to give a name to an arbitrary one + and reason about it. + For example, remember the predicates <code class="docutils literal notranslate">FnUb f a</code> and <code class="docutils literal notranslate">FnLb f a</code> + from the last section, + which say that <code class="docutils literal notranslate">a</code> is an upper bound or lower bound on <code class="docutils literal notranslate">f</code>, + respectively. + We can use the existential quantifier to say that “<code class="docutils literal notranslate">f</code> is bounded” + without specifying the bound: + <div class="highlight-lean notranslate"><div class="highlight"><pre>def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, f x ≤ a + + def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, a ≤ f x + + def FnHasUb (f : ℝ → ℝ) := + ∃ a, FnUb f a + + def FnHasLb (f : ℝ → ℝ) := + ∃ a, FnLb f a + </pre></div> + </div> + We can use the theorem <code class="docutils literal notranslate">FnUb_add</code> from the last section + to prove that if <code class="docutils literal notranslate">f</code> and <code class="docutils literal notranslate">g</code> have upper bounds, + then so does `` gun x => f x + g x``. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {f g : ℝ → ℝ} + + example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x => f x + g x := by + cases' ubf with a ubfa + cases' ubg with b ubfb + use a + b + apply fnUb_add ubfa ubfb + </pre></div> + </div> + The <code class="docutils literal notranslate">cases</code> tactic unpacks the information + in the existential quantifier. + Given the hypothesis <code class="docutils literal notranslate">ubf</code> that there is an upper bound + for <code class="docutils literal notranslate">f</code>, + <code class="docutils literal notranslate">cases</code> adds a new variable for an upper + bound to the context, + together with the hypothesis that it has the given property. + The <code class="docutils literal notranslate">with</code> clause allows us to specify the names + we want Lean to use. + The goal is left unchanged; + what has changed is that we can now use + the new object and the new hypothesis + to prove the goal. + This is a common pattern in mathematics: + we unpack objects whose existence is asserted or implied + by some hypothesis, and then use it to establish the existence + of something else. + Try using this pattern to establish the following. + You might find it useful to turn some of the examples + from the last section into named theorems, + as we did with <code class="docutils literal notranslate">fn_ub_add</code>, + or you can insert the arguments directly + into the proofs. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (lbf : FnHasLb f) (lbg : FnHasLb g) : FnHasLb fun x => f x + g x := by + sorry + + example {c : ℝ} (ubf : FnHasUb f) (h : c ≥ 0) : FnHasUb fun x => c * f x := by + sorry + </pre></div> + </div> + The task of unpacking information in a hypothesis is + so important that Lean and mathlib provide a number of + ways to do it. + A cousin of the <code class="docutils literal notranslate">cases</code> tactic, <code class="docutils literal notranslate">rcases</code>, is more + flexible in that it allows us to unpack nested data. + (The “r” stands for “recursive.”) + In the <code class="docutils literal notranslate">with</code> clause for unpacking an existential quantifier, + we name the object and the hypothesis by presenting + them as a pattern <code class="docutils literal notranslate">⟨a, h⟩</code> that <code class="docutils literal notranslate">rcases</code> then tries to match. + The <code class="docutils literal notranslate">rintro</code> tactic (which can also be written <code class="docutils literal notranslate">rintro</code>) + is a combination of <code class="docutils literal notranslate">intro</code> and <code class="docutils literal notranslate">rcases</code>. + These examples illustrate their use: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x => f x + g x := by + rcases ubf with ⟨a, ubfa⟩ + rcases ubg with ⟨b, ubfb⟩ + exact ⟨a + b, fnUb_add ubfa ubfb⟩ + + example : FnHasUb f → FnHasUb g → FnHasUb fun x => f x + g x := by + rintro ⟨a, ubfa⟩ ⟨b, ubfb⟩ + exact ⟨a + b, fnUb_add ubfa ubfb⟩ + </pre></div> + </div> + In fact, Lean also supports a pattern-matching lambda + in expressions and proof terms: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : FnHasUb f → FnHasUb g → FnHasUb fun x => f x + g x := + fun ⟨a, ubfa⟩ ⟨b, ubfb⟩ => ⟨a + b, fnUb_add ubfa ubfb⟩ + </pre></div> + </div> + These are power-user moves, and there is no harm + in favoring the use of <code class="docutils literal notranslate">cases</code> until you are more comfortable + with the existential quantifier. + But we will come to learn that all of these tools, + including <code class="docutils literal notranslate">cases</code>, <code class="docutils literal notranslate">use</code>, and the anonymous constructors, + are like Swiss army knives when it comes to theorem proving. + They can be used for a wide range of purposes, + not just for unpacking exists statements. + To illustrate one way that <code class="docutils literal notranslate">rcases</code> can be used, + we prove an old mathematical chestnut: + if two integers <code class="docutils literal notranslate">x</code> and <code class="docutils literal notranslate">y</code> can each be written as + a sum of two squares, + then so can their product, <code class="docutils literal notranslate">x * y</code>. + In fact, the statement is true for any commutative + ring, not just the integers. + In the next example, <code class="docutils literal notranslate">rcases</code> unpacks two existential + quantifiers at once. + We then provide the magic values needed to express <code class="docutils literal notranslate">x * y</code> + as a sum of squares as a list to the <code class="docutils literal notranslate">use</code> statement, + and we use <code class="docutils literal notranslate">ring</code> to verify that they work. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} [CommRing α] + + def SumOfSquares (x : α) := + ∃ a b, x = a ^ 2 + b ^ 2 + + theorem sumOfSquares_mul {x y : α} (sosx : SumOfSquares x) (sosy : SumOfSquares y) : + SumOfSquares (x * y) := by + rcases sosx with ⟨a, b, xeq⟩ + rcases sosy with ⟨c, d, yeq⟩ + rw [xeq, yeq] + use a * c - b * d, a * d + b * c + ring + </pre></div> + </div> + This proof doesn’t provide much insight, + but here is one way to motivate it. + A Gaussian integer is a number of the form $a + bi$ + where $a$ and $b$ are integers and $i = \sqrt{-1}$. + The norm of the Gaussian integer $a + bi$ is, by definition, + $a^2 + b^2$. + So the norm of a Gaussian integer is a sum of squares, + and any sum of squares can be expressed in this way. + The theorem above reflects the fact that norm of a product of + Gaussian integers is the product of their norms: + if $x$ is the norm of $a + bi$ and + $y$ in the norm of $c + di$, + then $xy$ is the norm of $(a + bi) (c + di)$. + Our cryptic proof illustrates the fact that + the proof that is easiest to formalize isn’t always + the most perspicuous one. + In the chapters to come, + we will provide you with the means to define the Gaussian + integers and use them to provide an alternative proof. + The pattern of unpacking an equation inside an existential quantifier + and then using it to rewrite an expression in the goal + comes up often, + so much so that the <code class="docutils literal notranslate">rcases</code> tactic provides + an abbreviation: + if you use the keyword <code class="docutils literal notranslate">rfl</code> in place of a new identifier, + <code class="docutils literal notranslate">rcases</code> does the rewriting automatically (this trick doesn’t work + with pattern-matching lambdas). + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem sumOfSquares_mul' {x y : α} (sosx : SumOfSquares x) (sosy : SumOfSquares y) : + SumOfSquares (x * y) := by + rcases sosx with ⟨a, b, rfl⟩ + rcases sosy with ⟨c, d, rfl⟩ + use a * c - b * d, a * d + b * c + ring + </pre></div> + </div> + As with the universal quantifier, + you can find existential quantifiers hidden all over + if you know how to spot them. + For example, divisibility is implicitly an “exists” statement. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (divab : a ∣ b) (divbc : b ∣ c) : a ∣ c := by + cases' divab with d beq + cases' divbc with e ceq + rw [ceq, beq] + use d * e; ring + </pre></div> + </div> + And once again, this provides a nice setting for using + <code class="docutils literal notranslate">rcases</code> with <code class="docutils literal notranslate">rfl</code>. + Try it out in the proof above. + It feels pretty good! + Then try proving the following: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (divab : a ∣ b) (divac : a ∣ c) : a ∣ b + c := by + sorry + </pre></div> + </div> + For another important example, a function $f : \alpha \to \beta$ + is said to be surjective if for every $y$ in the + codomain, $\beta$, + there is an $x$ in the domain, $\alpha$, + such that $f(x) = y$. + Notice that this statement includes both a universal + and an existential quantifier, which explains + why the next example makes use of both <code class="docutils literal notranslate">intro</code> and <code class="docutils literal notranslate">use</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {c : ℝ} : Surjective fun x => x + c := by + intro x + use x - c + dsimp; ring + </pre></div> + </div> + Try this example yourself: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {c : ℝ} (h : c ≠ 0) : Surjective fun x => c * x := by + sorry + </pre></div> + </div> + At this point, it is worth mentioning that there is a tactic, <cite>field_simp</cite>, + that will often clear denominators in a useful way. + It can be used in conjunction with the <code class="docutils literal notranslate">ring</code> tactic. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x y : ℝ) (h : x - y ≠ 0) : (x ^ 2 - y ^ 2) / (x - y) = x + y := by + field_simp [h] + ring + </pre></div> + </div> + You can use the theorem <code class="docutils literal notranslate">div_mul_cancel</code>. + The next example uses a surjectivity hypothesis + by applying it to a suitable value. + Note that you can use <code class="docutils literal notranslate">cases</code> with any expression, + not just a hypothesis. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {f : ℝ → ℝ} (h : Surjective f) : ∃ x, f x ^ 2 = 4 := by + cases' h 2 with x hx + use x + rw [hx] + norm_num + </pre></div> + </div> + See if you can use these methods to show that + the composition of surjective functions is surjective. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} {β : Type _} {γ : Type _} + variable {g : β → γ} {f : α → β} + + example (surjg : Surjective g) (surjf : Surjective f) : Surjective fun x => g (f x) := by + sorry + </pre></div> + </div> + </section> + <section id="negation"> + <h2>3.3. Negation<a class="headerlink" href="#negation" title="Permalink to this heading"></a></h2> + The symbol <code class="docutils literal notranslate">¬</code> is meant to express negation, + so <code class="docutils literal notranslate">¬ x < y</code> says that <code class="docutils literal notranslate">x</code> is not less than <code class="docutils literal notranslate">y</code>, + <code class="docutils literal notranslate">¬ x = y</code> (or, equivalently, <code class="docutils literal notranslate">x ≠ y</code>) says that + <code class="docutils literal notranslate">x</code> is not equal to <code class="docutils literal notranslate">y</code>, + and <code class="docutils literal notranslate">¬ ∃ z, x < z ∧ z < y</code> says that there does not exist a <code class="docutils literal notranslate">z</code> + strictly between <code class="docutils literal notranslate">x</code> and <code class="docutils literal notranslate">y</code>. + In Lean, the notation <code class="docutils literal notranslate">¬ A</code> abbreviates <code class="docutils literal notranslate">A → False</code>, + which you can think of as saying that <code class="docutils literal notranslate">A</code> implies a contradiction. + Practically speaking, this means that you already know something + about how to work with negations: + you can prove <code class="docutils literal notranslate">¬ A</code> by introducing a hypothesis <code class="docutils literal notranslate">h : A</code> + and proving <code class="docutils literal notranslate">False</code>, + and if you have <code class="docutils literal notranslate">h : ¬ A</code> and <code class="docutils literal notranslate">h' : A</code>, + then applying <code class="docutils literal notranslate">h</code> to <code class="docutils literal notranslate">h'</code> yields <code class="docutils literal notranslate">False</code>. + To illustrate, consider the irreflexivity principle <code class="docutils literal notranslate">lt_irrefl</code> + for a strict order, + which says that we have <code class="docutils literal notranslate">¬ a < a</code> for every <code class="docutils literal notranslate">a</code>. + The asymmetry principle <code class="docutils literal notranslate">lt_asymm</code> says that we have + <code class="docutils literal notranslate">a < b → ¬ b < a</code>. Let’s show that <code class="docutils literal notranslate">lt_asymm</code> follows + from <code class="docutils literal notranslate">lt_irrefl</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : a < b) : ¬b < a := by + intro h' + have : a < a := lt_trans h h' + apply lt_irrefl a this + </pre></div> + </div> + This example introduces a couple of new tricks. + First, when you use <code class="docutils literal notranslate">have</code> without providing + a label, + Lean uses the name <code class="docutils literal notranslate">this</code>, + providing a convenient way to refer back to it. + Also, the <code class="docutils literal notranslate">from</code> tactic is syntactic sugar for <code class="docutils literal notranslate">exact</code>, + providing a nice way to justify a <code class="docutils literal notranslate">have</code> with an explicit + proof term. + But what you should really be paying attention to in this + proof is the result of the <code class="docutils literal notranslate">intro</code> tactic, + which leaves a goal of <code class="docutils literal notranslate">False</code>, + and the fact that we eventually prove <code class="docutils literal notranslate">False</code> + by applying <code class="docutils literal notranslate">lt_irrefl</code> to a proof of <code class="docutils literal notranslate">a < a</code>. + Here is another example, which uses the + predicate <code class="docutils literal notranslate">FnHasUb</code> defined in the last section, + which says that a function has an upper bound. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : ∀ a, ∃ x, f x > a) : ¬FnHasUb f := by + intro fnub + cases' fnub with a fnuba + cases' h a with x hx + have : f x ≤ a := fnuba x + linarith + </pre></div> + </div> + See if you can prove these in a similar way: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : ∀ a, ∃ x, f x < a) : ¬FnHasLb f := + sorry + + example : ¬FnHasUb fun x => x := + sorry + </pre></div> + </div> + Mathlib offers a number of useful theorems for relating orders + and negations: + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (not_le_of_gt : a > b → ¬a ≤ b) + #check (not_lt_of_ge : a ≥ b → ¬a < b) + #check (lt_of_not_ge : ¬a ≥ b → a < b) + #check (le_of_not_gt : ¬a > b → a ≤ b) + </pre></div> + </div> + Recall the predicate <code class="docutils literal notranslate">Monotone f</code>, + which says that <code class="docutils literal notranslate">f</code> is nondecreasing. + Use some of the theorems just enumerated to prove the following: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : Monotone f) (h' : f a < f b) : a < b := by + sorry + + example (h : a ≤ b) (h' : f b < f a) : ¬Monotone f := by + sorry + </pre></div> + </div> + Remember that it is often convenient to use <code class="docutils literal notranslate">linarith</code> + when a goal follows from linear equations and + inequalities that in the context. + We can show that the first example in the last snippet + cannot be proved if we replace <code class="docutils literal notranslate"><</code> by <code class="docutils literal notranslate">≤</code>. + Notice that we can prove the negation of a universally + quantified statement by giving a counterexample. + Complete the proof. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : ¬∀ {f : ℝ → ℝ}, Monotone f → ∀ {a b}, f a ≤ f b → a ≤ b := by + intro h + let f := fun x : ℝ => (0 : ℝ) + have monof : Monotone f := by sorry + have h' : f 1 ≤ f 0 := le_refl _ + sorry + </pre></div> + </div> + This example introduces the <code class="docutils literal notranslate">let</code> tactic, + which adds a local definition to the context. + If you put the cursor after the <code class="docutils literal notranslate">let</code> command, + in the goal window you will see that the definition + <code class="docutils literal notranslate">f : ℝ → ℝ := fun x => 0</code> has been added to the context. + Lean will unfold the definition of <code class="docutils literal notranslate">f</code> when it has to. + In particular, when we prove <code class="docutils literal notranslate">f 1 ≤ f 0</code> with <code class="docutils literal notranslate">le_refl</code>, + Lean reduces <code class="docutils literal notranslate">f 1</code> and <code class="docutils literal notranslate">f 0</code> to <code class="docutils literal notranslate">0</code>. + Use <code class="docutils literal notranslate">le_of_not_gt</code> to prove the following: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x : ℝ) (h : ∀ ε > 0, x < ε) : x ≤ 0 := by + sorry + </pre></div> + </div> + Implicit in many of the proofs we have just done + is the fact that if <code class="docutils literal notranslate">P</code> is any property, + saying that there is nothing with property <code class="docutils literal notranslate">P</code> + is the same as saying that everything fails to have + property <code class="docutils literal notranslate">P</code>, + and saying that not everything has property <code class="docutils literal notranslate">P</code> + is equivalent to saying that something fails to have property <code class="docutils literal notranslate">P</code>. + In other words, all four of the following implications + are valid (but one of them cannot be proved with what we explained so + far): + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} (P : α → Prop) (Q : Prop) + + example (h : ¬∃ x, P x) : ∀ x, ¬P x := by + sorry + + example (h : ∀ x, ¬P x) : ¬∃ x, P x := by + sorry + + example (h : ¬∀ x, P x) : ∃ x, ¬P x := by + sorry + + example (h : ∃ x, ¬P x) : ¬∀ x, P x := by + sorry + </pre></div> + </div> + The first, second, and fourth are straightforward to + prove using the methods you have already seen. + We encourage you to try it. + The third is more difficult, however, + because it concludes that an object exists + from the fact that its nonexistence is contradictory. + This is an instance of classical mathematical reasoning. + We can use proof by contradiction + to prove the third implication as follows. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : ¬∀ x, P x) : ∃ x, ¬P x := by + by_contra h' + apply h + intro x + show P x + by_contra h'' + exact h' ⟨x, h''⟩ + </pre></div> + </div> + Make sure you understand how this works. + The <code class="docutils literal notranslate">by_contra</code> tactic + allows us to prove a goal <code class="docutils literal notranslate">Q</code> by assuming <code class="docutils literal notranslate">¬ Q</code> + and deriving a contradiction. + In fact, it is equivalent to using the + equivalence <code class="docutils literal notranslate">not_not : ¬ ¬ Q ↔ Q</code>. + Confirm that you can prove the forward direction + of this equivalence using <code class="docutils literal notranslate">by_contra</code>, + while the reverse direction follows from the + ordinary rules for negation. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : ¬¬Q) : Q := by + sorry + + example (h : Q) : ¬¬Q := by + sorry + </pre></div> + </div> + Use proof by contradiction to establish the following, + which is the converse of one of the implications we proved above. + (Hint: use <code class="docutils literal notranslate">intro</code> first.) + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by + sorry + </pre></div> + </div> + It is often tedious to work with compound statements with + a negation in front, + and it is a common mathematical pattern to replace such + statements with equivalent forms in which the negation + has been pushed inward. + To facilitate this, mathlib offers a <code class="docutils literal notranslate">push_neg</code> tactic, + which restates the goal in this way. + The command <code class="docutils literal notranslate">push_neg at h</code> restates the hypothesis <code class="docutils literal notranslate">h</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : ¬∀ a, ∃ x, f x > a) : FnHasUb f := by + push_neg at h + exact h + + example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by + simp only [FnHasUb, FnUb] at h + push_neg at h + exact h + </pre></div> + </div> + In the second example, we use Lean’s simplifier to + expand the definitions of <code class="docutils literal notranslate">FnHasUb</code> and <code class="docutils literal notranslate">FnUb</code>. + (We need to use <code class="docutils literal notranslate">simp</code> rather than <code class="docutils literal notranslate">rw</code> + to expand <code class="docutils literal notranslate">FnUb</code>, + because it appears in the scope of a quantifier.) + You can verify that in the examples above + with <code class="docutils literal notranslate">¬ ∃ x, P x</code> and <code class="docutils literal notranslate">¬ ∀ x, P x</code>, + the <code class="docutils literal notranslate">push_neg</code> tactic does the expected thing. + Without even knowing how to use the conjunction + symbol, + you should be able to use <code class="docutils literal notranslate">push_neg</code> + to prove the following: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : ¬Monotone f) : ∃ x y, x ≤ y ∧ f y < f x := by + sorry + </pre></div> + </div> + Mathlib also has a tactic, <code class="docutils literal notranslate">contrapose</code>, + which transforms a goal <code class="docutils literal notranslate">A → B</code> to <code class="docutils literal notranslate">¬ B → ¬ A</code>. + Similarly, given a goal of proving <code class="docutils literal notranslate">B</code> from + hypothesis <code class="docutils literal notranslate">h : A</code>, + <code class="docutils literal notranslate">contrapose h</code> leaves you with a goal of proving + <code class="docutils literal notranslate">¬ A</code> from hypothesis <code class="docutils literal notranslate">¬ B</code>. + Using <code class="docutils literal notranslate">contrapose!</code> instead of <code class="docutils literal notranslate">contrapose</code> + applies <code class="docutils literal notranslate">push_neg</code> to the goal and the relevant + hypothesis as well. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by + contrapose! h + exact h + + example (x : ℝ) (h : ∀ ε > 0, x ≤ ε) : x ≤ 0 := by + contrapose! h + use x / 2 + constructor <;> linarith + </pre></div> + </div> + We have not yet explained the <code class="docutils literal notranslate">constructor</code> command + or the use of the semicolon after it, + but we will do that in the next section. + We close this section with + the principle of ex falso, + which says that anything follows from a contradiction. + In Lean, this is represented by <code class="docutils literal notranslate">False.elim</code>, + which establishes <code class="docutils literal notranslate">False → P</code> for any proposition <code class="docutils literal notranslate">P</code>. + This may seem like a strange principle, + but it comes up fairly often. + We often prove a theorem by splitting on cases, + and sometimes we can show that one of + the cases is contradictory. + In that case, we need to assert that the contradiction + establishes the goal so we can move on to the next one. + (We will see instances of reasoning by cases in + <a class="reference internal" href="#disjunction">Section 3.5</a>.) + Lean provides a number of ways of closing + a goal once a contradiction has been reached. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : 0 < 0) : a > 37 := by + exfalso + apply lt_irrefl 0 h + + example (h : 0 < 0) : a > 37 := + absurd h (lt_irrefl 0) + + example (h : 0 < 0) : a > 37 := by + have h' : ¬0 < 0 := lt_irrefl 0 + contradiction + </pre></div> + </div> + The <code class="docutils literal notranslate">exfalso</code> tactic replaces the current goal with + the goal of proving <code class="docutils literal notranslate">False</code>. + Given <code class="docutils literal notranslate">h : P</code> and <code class="docutils literal notranslate">h' : ¬ P</code>, + the term <code class="docutils literal notranslate">absurd h h'</code> establishes any proposition. + Finally, the <code class="docutils literal notranslate">contradiction</code> tactic tries to close a goal + by finding a contradiction in the hypotheses, + such as a pair of the form <code class="docutils literal notranslate">h : P</code> and <code class="docutils literal notranslate">h' : ¬ P</code>. + Of course, in this example, <code class="docutils literal notranslate">linarith</code> also works. + </section> + <section id="conjunction-and-bi-implication"> + <h2>3.4. Conjunction and Bi-implication<a class="headerlink" href="#conjunction-and-bi-implication" title="Permalink to this heading"></a></h2> + You have already seen that the conjunction symbol, <code class="docutils literal notranslate">∧</code>, + is used to express “and.” + The <code class="docutils literal notranslate">constructor</code> tactic allows you to prove a statement of + the form <code class="docutils literal notranslate">A ∧ B</code> + by proving <code class="docutils literal notranslate">A</code> and then proving <code class="docutils literal notranslate">B</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y := by + constructor + · assumption + intro h + apply h₁ + rw [h] + </pre></div> + </div> + In this example, the <code class="docutils literal notranslate">assumption</code> tactic + tells Lean to find an assumption that will solve the goal. + Notice that the final <code class="docutils literal notranslate">rw</code> finishes the goal by + applying the reflexivity of <code class="docutils literal notranslate">≤</code>. + The following are alternative ways of carrying out + the previous examples using the anonymous constructor + angle brackets. + The first is a slick proof-term version of the + previous proof, + which drops into tactic mode at the keyword <code class="docutils literal notranslate">by</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y := + ⟨h₀, fun h => h₁ (by rw [h])⟩ + + example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y := + have h : x ≠ y := by + contrapose! h₁ + rw [h₁] + ⟨h₀, h⟩ + </pre></div> + </div> + Using a conjunction instead of proving one involves unpacking the proofs of the + two parts. + You can use the <code class="docutils literal notranslate">cases</code> tactic for that, + as well as <code class="docutils literal notranslate">rcases</code>, <code class="docutils literal notranslate">rintro</code>, or a pattern-matching lambda, + all in a manner similar to the way they are used with + the existential quantifier. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by + cases' h with h₀ h₁ + contrapose! h₁ + exact le_antisymm h₀ h₁ + + example {x y : ℝ} : x ≤ y ∧ x ≠ y → ¬y ≤ x := by + rintro ⟨h₀, h₁⟩ h' + exact h₁ (le_antisymm h₀ h') + + example {x y : ℝ} : x ≤ y ∧ x ≠ y → ¬y ≤ x := + fun ⟨h₀, h₁⟩ h' => h₁ (le_antisymm h₀ h') + </pre></div> + </div> + In contrast to using an existential quantifier, + you can also extract proofs of the two components + of a hypothesis <code class="docutils literal notranslate">h : A ∧ B</code> + by writing <code class="docutils literal notranslate">h.left</code> and <code class="docutils literal notranslate">h.right</code>, + or, equivalently, <code class="docutils literal notranslate">h.1</code> and <code class="docutils literal notranslate">h.2</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by + intro h' + apply h.right + exact le_antisymm h.left h' + + example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := + fun h' => h.right (le_antisymm h.left h') + </pre></div> + </div> + Try using these techniques to come up with various ways of proving of the following: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {m n : ℕ} (h : m ∣ n ∧ m ≠ n) : m ∣ n ∧ ¬n ∣ m := + sorry + </pre></div> + </div> + You can nest uses of <code class="docutils literal notranslate">∃</code> and <code class="docutils literal notranslate">∧</code> + with anonymous constructors, <code class="docutils literal notranslate">rintro</code>, and <code class="docutils literal notranslate">rcases</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : ∃ x : ℝ, 2 < x ∧ x < 4 := + ⟨5 / 2, by norm_num, by norm_num⟩ + + example (x y : ℝ) : (∃ z : ℝ, x < z ∧ z < y) → x < y := by + rintro ⟨z, xltz, zlty⟩ + exact lt_trans xltz zlty + + example (x y : ℝ) : (∃ z : ℝ, x < z ∧ z < y) → x < y := + fun ⟨z, xltz, zlty⟩ => lt_trans xltz zlty + </pre></div> + </div> + You can also use the <code class="docutils literal notranslate">use</code> tactic: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : ∃ x : ℝ, 2 < x ∧ x < 4 := by + use 5 / 2 + constructor <;> norm_num + + example : ∃ m n : ℕ, 4 < m ∧ m < n ∧ n < 10 ∧ Nat.Prime m ∧ Nat.Prime n := by + use 5 + use 7 + norm_num + sorry + + example {x y : ℝ} : x ≤ y ∧ x ≠ y → x ≤ y ∧ ¬y ≤ x := by + rintro ⟨h₀, h₁⟩ + use h₀ + exact fun h' => h₁ (le_antisymm h₀ h') + </pre></div> + </div> + In the first example, the semicolon after the <code class="docutils literal notranslate">constructor</code> command tells Lean to use the + <code class="docutils literal notranslate">norm_num</code> tactic on both of the goals that result. + In Lean, <code class="docutils literal notranslate">A ↔ B</code> is not defined to be <code class="docutils literal notranslate">(A → B) ∧ (B → A)</code>, + but it could have been, + and it behaves roughly the same way. + You have already seen that you can write <code class="docutils literal notranslate">h.mp</code> and <code class="docutils literal notranslate">h.mpr</code> + or <code class="docutils literal notranslate">h.1</code> and <code class="docutils literal notranslate">h.2</code> for the two directions of <code class="docutils literal notranslate">h : A ↔ B</code>. + You can also use <code class="docutils literal notranslate">cases</code> and friends. + To prove an if-and-only-if statement, + you can uses <code class="docutils literal notranslate">constructor</code> or angle brackets, + just as you would if you were proving a conjunction. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {x y : ℝ} (h : x ≤ y) : ¬y ≤ x ↔ x ≠ y := by + constructor + · contrapose! + rintro rfl + rfl + contrapose! + exact le_antisymm h + + example {x y : ℝ} (h : x ≤ y) : ¬y ≤ x ↔ x ≠ y := + ⟨fun h₀ h₁ => h₀ (by rw [h₁]), fun h₀ h₁ => h₀ (le_antisymm h h₁)⟩ + </pre></div> + </div> + The last proof term is inscrutable. Remember that you can + use underscores while writing an expression like that to + see what Lean expects. + Try out the various techniques and gadgets you have just seen + in order to prove the following: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {x y : ℝ} : x ≤ y ∧ ¬y ≤ x ↔ x ≤ y ∧ x ≠ y := + sorry + </pre></div> + </div> + For a more interesting exercise, show that for any + two real numbers <code class="docutils literal notranslate">x</code> and <code class="docutils literal notranslate">y</code>, + <code class="docutils literal notranslate">x^2 + y^2 = 0</code> if and only if <code class="docutils literal notranslate">x = 0</code> and <code class="docutils literal notranslate">y = 0</code>. + We suggest proving an auxiliary lemma using + <code class="docutils literal notranslate">linarith</code>, <code class="docutils literal notranslate">pow_two_nonneg</code>, and <code class="docutils literal notranslate">pow_eq_zero</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem aux {x y : ℝ} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := + have h' : x ^ 2 = 0 := by sorry + pow_eq_zero h' + + example (x y : ℝ) : x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := + sorry + </pre></div> + </div> + In Lean, bi-implication leads a double-life. + You can treat it like a conjunction and use its two + parts separately. + But Lean also knows that it is a reflexive, symmetric, + and transitive relation between propositions, + and you can also use it with <code class="docutils literal notranslate">calc</code> and <code class="docutils literal notranslate">rw</code>. + It is often convenient to rewrite a statement to + an equivalent one. + In the next example, we use <code class="docutils literal notranslate">abs_lt</code> to + replace an expression of the form <code class="docutils literal notranslate">abs x < y</code> + by the equivalent expression <code class="docutils literal notranslate">- y < x ∧ x < y</code>, + and in the one after that we use <code class="docutils literal notranslate">Nat.dvd_gcd_iff</code> + to replace an expression of the form <code class="docutils literal notranslate">m ∣ Nat.gcd n k</code> by the equivalent expression <code class="docutils literal notranslate">m ∣ n ∧ m ∣ k</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x : ℝ) : abs (x + 3) < 5 → -8 < x ∧ x < 2 := by + rw [abs_lt] + intro h + constructor <;> linarith + + example : 3 ∣ Nat.gcd 6 15 := by + rw [Nat.dvd_gcd_iff] + constructor <;> norm_num + </pre></div> + </div> + See if you can use <code class="docutils literal notranslate">rw</code> with the theorem below + to provide a short proof that negation is not a + nondecreasing function. (Note that <code class="docutils literal notranslate">push_neg</code> won’t + unfold definitions for you, so the <code class="docutils literal notranslate">rw monotone</code> in + the proof of the theorem is needed.) + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem not_monotone_iff {f : ℝ → ℝ} : ¬Monotone f ↔ ∃ x y, x ≤ y ∧ f x > f y := by + rw [Monotone] + push_neg + rfl + + example : ¬Monotone fun x : ℝ => -x := by + sorry + </pre></div> + </div> + The remaining exercises in this section are designed + to give you some more practice with conjunction and + bi-implication. Remember that a partial order is a + binary relation that is transitive, reflexive, and + antisymmetric. + An even weaker notion sometimes arises: + a preorder is just a reflexive, transitive relation. + For any pre-order <code class="docutils literal notranslate">≤</code>, + Lean axiomatizes the associated strict pre-order by + <code class="docutils literal notranslate">a < b ↔ a ≤ b ∧ ¬ b ≤ a</code>. + Show that if <code class="docutils literal notranslate">≤</code> is a partial order, + then <code class="docutils literal notranslate">a < b</code> is equivalent to <code class="docutils literal notranslate">a ≤ b ∧ a ≠ b</code>: + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} [PartialOrder α] + variable (a b : α) + + example : a < b ↔ a ≤ b ∧ a ≠ b := by + rw [lt_iff_le_not_le] + sorry + </pre></div> + </div> + Beyond logical operations, you should not need + anything more than <code class="docutils literal notranslate">le_refl</code> and <code class="docutils literal notranslate">le_antisymm</code>. + Then show that even in the case where <code class="docutils literal notranslate">≤</code> + is only assumed to be a preorder, + we can prove that the strict order is irreflexive + and transitive. + You do not need anything more than <code class="docutils literal notranslate">le_refl</code> and <code class="docutils literal notranslate">le_trans</code>. + In the second example, + for convenience, we use the simplifier rather than <code class="docutils literal notranslate">rw</code> + to express <code class="docutils literal notranslate"><</code> in terms of <code class="docutils literal notranslate">≤</code> and <code class="docutils literal notranslate">¬</code>. + We will come back to the simplifier later, + but here we are only relying on the fact that it will + use the indicated lemma repeatedly, even if it needs + to be instantiated to different values. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} [Preorder α] + variable (a b c : α) + + example : ¬a < a := by + rw [lt_iff_le_not_le] + sorry + + example : a < b → b < c → a < c := by + simp only [lt_iff_le_not_le] + sorry + </pre></div> + </div> + </section> + <section id="disjunction"> + <h2>3.5. Disjunction<a class="headerlink" href="#disjunction" title="Permalink to this heading"></a></h2> + The canonical way to prove a disjunction <code class="docutils literal notranslate">A ∨ B</code> is to prove + <code class="docutils literal notranslate">A</code> or to prove <code class="docutils literal notranslate">B</code>. + The <code class="docutils literal notranslate">left</code> tactic chooses <code class="docutils literal notranslate">A</code>, + and the <code class="docutils literal notranslate">right</code> tactic chooses <code class="docutils literal notranslate">B</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {x y : ℝ} + + example (h : y > x ^ 2) : y > 0 ∨ y < -1 := by + left + linarith [pow_two_nonneg x] + + example (h : -y > x ^ 2 + 1) : y > 0 ∨ y < -1 := by + right + linarith [pow_two_nonneg x] + </pre></div> + </div> + We cannot use an anonymous constructor to construct a proof + of an “or” because Lean would have to guess + which disjunct we are trying to prove. + When we write proof terms we can use + <code class="docutils literal notranslate">Or.inl</code> and <code class="docutils literal notranslate">Or.inr</code> instead + to make the choice explicitly. + Here, <code class="docutils literal notranslate">inl</code> is short for “introduction left” and + <code class="docutils literal notranslate">inr</code> is short for “introduction right.” + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : y > 0) : y > 0 ∨ y < -1 := + Or.inl h + + example (h : y < -1) : y > 0 ∨ y < -1 := + Or.inr h + </pre></div> + </div> + It may seem strange to prove a disjunction by proving one side + or the other. + In practice, which case holds usually depends a case distinction + that is implicit or explicit in the assumptions and the data. + The <code class="docutils literal notranslate">cases</code> tactic allows us to make use of a hypothesis + of the form <code class="docutils literal notranslate">A ∨ B</code>. + In contrast to the use of <code class="docutils literal notranslate">cases</code> with conjunction or an + existential quantifier, + here the <code class="docutils literal notranslate">cases</code> tactic produces two goals. + Both have the same conclusion, but in the first case, + <code class="docutils literal notranslate">A</code> is assumed to be true, + and in the second case, + <code class="docutils literal notranslate">B</code> is assumed to be true. + In other words, as the name suggests, + the <code class="docutils literal notranslate">cases</code> tactic carries out a proof by cases. + As usual, we can tell Lean what names to use for the hypotheses. + In the next example, we tell Lean + to use the name <code class="docutils literal notranslate">h</code> on each branch. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : x < abs y → x < y ∨ x < -y := by + cases' le_or_gt 0 y with h h + · rw [abs_of_nonneg h] + intro h + left + exact h + rw [abs_of_neg h] + intro h; right; exact h + </pre></div> + </div> + The absolute value function is defined in such a way + that we can immediately prove that + <code class="docutils literal notranslate">x ≥ 0</code> implies <code class="docutils literal notranslate">abs x = x</code> + (this is the theorem <code class="docutils literal notranslate">abs_of_nonneg</code>) + and <code class="docutils literal notranslate">x < 0</code> implies <code class="docutils literal notranslate">abs x = -x</code> (this is <code class="docutils literal notranslate">abs_of_neg</code>). + The expression <code class="docutils literal notranslate">le_or_gt 0 x</code> establishes <code class="docutils literal notranslate">0 ≤ x ∨ x < 0</code>, + allowing us to split on those two cases. + Try proving the triangle inequality using the two + first two theorems in the next snippet. + They are given the same names they have in mathlib. + <div class="highlight-lean notranslate"><div class="highlight"><pre>namespace MyAbs + + theorem le_abs_self (x : ℝ) : x ≤ abs x := by + sorry + + theorem neg_le_abs_self (x : ℝ) : -x ≤ abs x := by + sorry + + theorem abs_add (x y : ℝ) : abs (x + y) ≤ abs x + abs y := by + sorry + </pre></div> + </div> + In case you enjoyed these (pun intended) and + you want more practice with disjunction, + try these. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem lt_abs : x < abs y ↔ x < y ∨ x < -y := by + sorry + + theorem abs_lt : abs x < y ↔ -y < x ∧ x < y := by + sorry + </pre></div> + </div> + You can also use <code class="docutils literal notranslate">rcases</code> and <code class="docutils literal notranslate">rintro</code> with disjunctions. + When these result in a genuine case split with multiple goals, + the patterns for each new goal are separated by a vertical bar. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {x : ℝ} (h : x ≠ 0) : x < 0 ∨ x > 0 := by + rcases lt_trichotomy x 0 with (xlt | xeq | xgt) + · left + exact xlt + · contradiction + right; exact xgt + </pre></div> + </div> + You can still nest patterns and use the <code class="docutils literal notranslate">rfl</code> keyword + to substitute equations: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {m n k : ℕ} (h : m ∣ n ∨ m ∣ k) : m ∣ n * k := by + rcases h with (⟨a, rfl⟩ | ⟨b, rfl⟩) + · rw [mul_assoc] + apply dvd_mul_right + rw [mul_comm, mul_assoc] + apply dvd_mul_right + </pre></div> + </div> + See if you can prove the following with a single (long) line. + Use <code class="docutils literal notranslate">rcases</code> to unpack the hypotheses and split on cases, + and use a semicolon and <code class="docutils literal notranslate">linarith</code> to solve each branch. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {z : ℝ} (h : ∃ x y, z = x ^ 2 + y ^ 2 ∨ z = x ^ 2 + y ^ 2 + 1) : z ≥ 0 := by + sorry + </pre></div> + </div> + On the real numbers, an equation <code class="docutils literal notranslate">x * y = 0</code> + tells us that <code class="docutils literal notranslate">x = 0</code> or <code class="docutils literal notranslate">y = 0</code>. + In mathlib, this fact is known as <code class="docutils literal notranslate">eq_zero_or_eq_zero_of_mul_eq_zero</code>, + and it is another nice example of how a disjunction can arise. + See if you can use it to prove the following: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {x : ℝ} (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by + sorry + + example {x y : ℝ} (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by + sorry + </pre></div> + </div> + Remember that you can use the <code class="docutils literal notranslate">ring</code> tactic to help + with calculations. + In an arbitrary ring $R$, an element $x$ such + that $x y = 0$ for some nonzero $y$ is called + a left zero divisor, + an element $x$ such + that $y x = 0$ for some nonzero $y$ is called + a right zero divisor, + and an element that is either a left or right zero divisor + is called simply a zero divisor. + The theorem <code class="docutils literal notranslate">eq_zero_or_eq_zero_of_mul_eq_zero</code> + says that the real numbers have no nontrivial zero divisors. + A commutative ring with this property is called an integral domain. + Your proofs of the two theorems above should work equally well + in any integral domain: + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {R : Type _} [CommRing R] [IsDomain R] + variable (x y : R) + + example (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by + sorry + + example (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by + sorry + </pre></div> + </div> + In fact, if you are careful, you can prove the first + theorem without using commutativity of multiplication. + In that case, it suffices to assume that <code class="docutils literal notranslate">R</code> is + a <code class="docutils literal notranslate">Ring</code> instead of an <code class="docutils literal notranslate">CommRing</code>. + Sometimes in a proof we want to split on cases + depending on whether some statement is true or not. + For any proposition <code class="docutils literal notranslate">P</code>, we can use + <code class="docutils literal notranslate">em P : P ∨ ¬ P</code>. + The name <code class="docutils literal notranslate">em</code> is short for “excluded middle.” + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (P : Prop) : ¬¬P → P := by + intro h + cases em P + · assumption + contradiction + </pre></div> + </div> + Alternatively, you can use the <code class="docutils literal notranslate">by_cases</code> tactic. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (P : Prop) : ¬¬P → P := by + intro h + by_cases h' : P + · assumption + contradiction + </pre></div> + </div> + Notice that the <code class="docutils literal notranslate">by_cases</code> tactic lets you + specify a label for the hypothesis that is + introduced in each branch, + in this case, <code class="docutils literal notranslate">h' : P</code> in one and <code class="docutils literal notranslate">h' : ¬ P</code> + in the other. + If you leave out the label, + Lean uses <code class="docutils literal notranslate">h</code> by default. + Try proving the following equivalence, + using <code class="docutils literal notranslate">by_cases</code> to establish one direction. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (P Q : Prop) : P → Q ↔ ¬P ∨ Q := by + sorry + </pre></div> + </div> + </section> + <section id="sequences-and-convergence"> + <h2>3.6. Sequences and Convergence<a class="headerlink" href="#sequences-and-convergence" title="Permalink to this heading"></a></h2> + We now have enough skills at our disposal to do some real mathematics. + In Lean, we can represent a sequence $s_0, s_1, s_2, \ldots$ of + real numbers as a function <code class="docutils literal notranslate">s : ℕ → ℝ</code>. + Such a sequence is said to converge to a number $a$ if for every + $\varepsilon > 0$ there is a point beyond which the sequence + remains within $\varepsilon$ of $a$, + that is, there is a number $N$ such that for every + $n \ge N$, $| s_n - a | < \varepsilon$. + In Lean, we can render this as follows: + <div class="highlight-lean notranslate"><div class="highlight"><pre>def ConvergesTo (s : ℕ → ℝ) (a : ℝ) := + ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε + </pre></div> + </div> + The notation <code class="docutils literal notranslate">∀ ε > 0, ...</code> is a convenient abbreviation + for <code class="docutils literal notranslate">∀ ε, ε > 0 → ...</code>, and, similarly, + <code class="docutils literal notranslate">∀ n ≥ N, ...</code> abbreviates <code class="docutils literal notranslate">∀ n, n ≥ N →  ...</code>. + And remember that <code class="docutils literal notranslate">ε > 0</code>, in turn, is defined as <code class="docutils literal notranslate">0 < ε</code>, + and <code class="docutils literal notranslate">n ≥ N</code> is defined as <code class="docutils literal notranslate">N ≤ n</code>. + In this section, we’ll establish some properties of convergence. + But first, we will discuss three tactics for working with equality + that will prove useful. + The first, the <code class="docutils literal notranslate">ext</code> tactic, + gives us a way of proving that two functions are equal. + Let $f(x) = x + 1$ and $g(x) = 1 + x$ + be functions from reals to reals. + Then, of course, $f = g$, because they return the same + value for every $x$. + The <code class="docutils literal notranslate">ext</code> tactic enables us to prove an equation between functions + by proving that their values are the same + at all the values of their arguments. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : (fun x y : ℝ => (x + y) ^ 2) = fun x y : ℝ => x ^ 2 + 2 * x * y + y ^ 2 := by + ext + ring + </pre></div> + </div> + We’ll see later that <code class="docutils literal notranslate">ext</code> is actually more general, and also one can + specify the name of the variables that appear. + For instance you can try to replace <code class="docutils literal notranslate">ext</code> with <code class="docutils literal notranslate">ext u v</code> in the + above proof. + The second tactic, the <code class="docutils literal notranslate">congr</code> tactic, + allows us to prove an equation between two expressions + by reconciling the parts that are different: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a b : ℝ) : abs a = abs (a - b + b) := by + congr + ring + </pre></div> + </div> + Here the <code class="docutils literal notranslate">congr</code> tactic peels off the <code class="docutils literal notranslate">abs</code> on each side, + leaving us to prove <code class="docutils literal notranslate">a = a - b + b</code>. + Finally, the <code class="docutils literal notranslate">convert</code> tactic is used to apply a theorem + to a goal when the conclusion of the theorem doesn’t quite match. + For example, suppose we want to prove <code class="docutils literal notranslate">a < a * a</code> from <code class="docutils literal notranslate">1 < a</code>. + A theorem in the library, <code class="docutils literal notranslate">mul_lt_mul_right</code>, + will let us prove <code class="docutils literal notranslate">1 * a < a * a</code>. + One possibility is to work backwards and rewrite the goal + so that it has that form. + Instead, the <code class="docutils literal notranslate">convert</code> tactic lets us apply the theorem + as it is, + and leaves us with the task of proving the equations that + are needed to make the goal match. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {a : ℝ} (h : 1 < a) : a < a * a := by + convert(mul_lt_mul_right _).2 h + · rw [one_mul] + exact lt_trans zero_lt_one h + </pre></div> + </div> + This example illustrates another useful trick: when we apply an + expression with an underscore + and Lean can’t fill it in for us automatically, + it simply leaves it for us as another goal. + The following shows that any constant sequence $a, a, a, \ldots$ + converges. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem convergesTo_const (a : ℝ) : ConvergesTo (fun x : ℕ => a) a := by + intro ε εpos + use 0 + intro n nge; dsimp + rw [sub_self, abs_zero] + apply εpos + </pre></div> + </div> + Lean has a tactic, <code class="docutils literal notranslate">simp</code>, which can often save you the + trouble of carrying out steps like <code class="docutils literal notranslate">rw [sub_self, abs_zero]</code> + by hand. + We will tell you more about it soon. + For a more interesting theorem, let’s show that if <code class="docutils literal notranslate">s</code> + converges to <code class="docutils literal notranslate">a</code> and <code class="docutils literal notranslate">t</code> converges to <code class="docutils literal notranslate">b</code>, then + <code class="docutils literal notranslate">fun n ↦ s n + t n</code> converges to <code class="docutils literal notranslate">a + b</code>. + It is helpful to have a clear pen-and-paper + proof in mind before you start writing a formal one. + Given <code class="docutils literal notranslate">ε</code> greater than <code class="docutils literal notranslate">0</code>, + the idea is to use the hypotheses to obtain an <code class="docutils literal notranslate">Ns</code> + such that beyond that point, <code class="docutils literal notranslate">s</code> is within <code class="docutils literal notranslate">ε / 2</code> + of <code class="docutils literal notranslate">a</code>, + and an <code class="docutils literal notranslate">Nt</code> such that beyond that point, <code class="docutils literal notranslate">t</code> is within + <code class="docutils literal notranslate">ε / 2</code> of <code class="docutils literal notranslate">b</code>. + Then, whenever <code class="docutils literal notranslate">n</code> is greater than or equal to the + maximum of <code class="docutils literal notranslate">Ns</code> and <code class="docutils literal notranslate">Nt</code>, + the sequence <code class="docutils literal notranslate">fun n ↦ s n + t n</code> should be within <code class="docutils literal notranslate">ε</code> + of <code class="docutils literal notranslate">a + b</code>. + The following example begins to implement this strategy. + See if you can finish it off. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem convergesTo_add {s t : ℕ → ℝ} {a b : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t b) : + ConvergesTo (fun n => s n + t n) (a + b) := by + intro ε εpos + dsimp + have ε2pos : 0 < ε / 2 := by linarith + cases' cs (ε / 2) ε2pos with Ns hs + cases' ct (ε / 2) ε2pos with Nt ht + use max Ns Nt + sorry + </pre></div> + </div> + As hints, you can use <code class="docutils literal notranslate">le_of_max_le_left</code> and <code class="docutils literal notranslate">le_of_max_le_right</code>, + and <code class="docutils literal notranslate">norm_num</code> can prove <code class="docutils literal notranslate">ε / 2 + ε / 2 = ε</code>. + Also, it is helpful to use the <code class="docutils literal notranslate">congr</code> tactic to + show that <code class="docutils literal notranslate">abs (s n + t n - (a + b))</code> is equal to + <code class="docutils literal notranslate">abs ((s n - a) + (t n - b)),</code> + since then you can use the triangle inequality. + Notice that we marked all the variables <code class="docutils literal notranslate">s</code>, <code class="docutils literal notranslate">t</code>, <code class="docutils literal notranslate">a</code>, and <code class="docutils literal notranslate">b</code> + implicit because they can be inferred from the hypotheses. + Proving the same theorem with multiplication in place + of addition is tricky. + We will get there by proving some auxiliary statements first. + See if you can also finish off the next proof, + which shows that if <code class="docutils literal notranslate">s</code> converges to <code class="docutils literal notranslate">a</code>, + then <code class="docutils literal notranslate">fun n => c * s n</code> converges to <code class="docutils literal notranslate">c * a</code>. + It is helpful to split into cases depending on whether <code class="docutils literal notranslate">c</code> + is equal to zero or not. + We have taken care of the zero case, + and we have left you to prove the result with + the extra assumption that <code class="docutils literal notranslate">c</code> is nonzero. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem convergesTo_mul_const {s : ℕ → ℝ} {a : ℝ} (c : ℝ) (cs : ConvergesTo s a) : + ConvergesTo (fun n => c * s n) (c * a) := by + by_cases h : c = 0 + · convert convergesTo_const 0 + · rw [h, MulZeroClass.zero_mul] + rw [h, MulZeroClass.zero_mul] + have acpos : 0 < abs c := abs_pos.mpr h + sorry + </pre></div> + </div> + The next theorem is also independently interesting: + it shows that a convergent sequence is eventually bounded + in absolute value. + We have started you off; see if you can finish it. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem exists_abs_le_of_convergesTo {s : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) : + ∃ N b, ∀ n, N ≤ n → abs (s n) < b := by + cases' cs 1 zero_lt_one with N h + use N, abs a + 1 + sorry + </pre></div> + </div> + In fact, the theorem could be strengthened to assert + that there is a bound <code class="docutils literal notranslate">b</code> that holds for all values of <code class="docutils literal notranslate">n</code>. + But this version is strong enough for our purposes, + and we will see at the end of this section that it + holds more generally. + The next lemma is auxiliary: we prove that if + <code class="docutils literal notranslate">s</code> converges to <code class="docutils literal notranslate">a</code> and <code class="docutils literal notranslate">t</code> converges to <code class="docutils literal notranslate">0</code>, + then <code class="docutils literal notranslate">fun n => s n * t n</code> converges to <code class="docutils literal notranslate">0</code>. + To do so, we use the previous theorem to find a <code class="docutils literal notranslate">B</code> + that bounds <code class="docutils literal notranslate">s</code> beyond some point <code class="docutils literal notranslate">N₀</code>. + See if you can understand the strategy we have outlined + and finish the proof. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem aux {s t : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t 0) : + ConvergesTo (fun n => s n * t n) 0 := by + intro ε εpos + dsimp + rcases exists_abs_le_of_convergesTo cs with ⟨N₀, B, h₀⟩ + have Bpos : 0 < B := lt_of_le_of_lt (abs_nonneg _) (h₀ N₀ (le_refl _)) + have pos₀ : ε / B > 0 := div_pos εpos Bpos + cases' ct _ pos₀ with N₁ h₁ + sorry + </pre></div> + </div> + If you have made it this far, congratulations! + We are now within striking distance of our theorem. + The following proof finishes it off. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem convergesTo_mul {s t : ℕ → ℝ} {a b : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t b) : + ConvergesTo (fun n => s n * t n) (a * b) := by + have h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 := by + apply aux cs + convert convergesTo_add ct (convergesTo_const (-b)) + ring + have := convergesTo_add h₁ (convergesTo_mul_const b cs) + convert convergesTo_add h₁ (convergesTo_mul_const b cs) using 1 + · ext; ring + ring + </pre></div> + </div> + For another challenging exercise, + try filling out the following sketch of a proof that limits + are unique. + (If you are feeling bold, + you can delete the proof sketch and try proving it from scratch.) + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem convergesTo_unique {s : ℕ → ℝ} {a b : ℝ} (sa : ConvergesTo s a) (sb : ConvergesTo s b) : + a = b := by + by_contra abne + have : abs (a - b) > 0 := by sorry + let ε := abs (a - b) / 2 + have εpos : ε > 0 := by + change abs (a - b) / 2 > 0 + linarith + cases' sa ε εpos with Na hNa + cases' sb ε εpos with Nb hNb + let N := max Na Nb + have absa : abs (s N - a) < ε := by sorry + have absb : abs (s N - b) < ε := by sorry + have : abs (a - b) < abs (a - b) := by sorry + exact lt_irrefl _ this + </pre></div> + </div> + We close the section with the observation that our proofs can be generalized. + For example, the only properties that we have used of the + natural numbers is that their structure carries a partial order + with <code class="docutils literal notranslate">min</code> and <code class="docutils literal notranslate">max</code>. + You can check that everything still works if you replace <code class="docutils literal notranslate">ℕ</code> + everywhere by any linear order <code class="docutils literal notranslate">α</code>: + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} [LinearOrder α] + + def ConvergesTo' (s : α → ℝ) (a : ℝ) := + ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε + </pre></div> + </div> + In <a class="reference internal" href="C07_Topology.html#filters">Section 7.1</a>, we will see that mathlib has mechanisms + for dealing with convergence in vastly more general terms, + not only abstracting away particular features of the domain + and codomain, + but also abstracting over different types of convergence. + </section> + </section> + + + </div> + </div> + <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> + <a href="C02_Basics.html" class="btn btn-neutral float-left" title="2. Basics" accesskey="p" rel="prev"> Previous</a> + <a href="C04_Sets_and_Functions.html" class="btn btn-neutral float-right" title="4. Sets and Functions" accesskey="n" rel="next">Next </a> + </div> + + <hr/> + + <div role="contentinfo"> + © Copyright 2020, Jeremy Avigad, Kevin Buzzard, Robert Y. Lewis, Patrick Massot. + </div> + + Built with <a href="https://www.sphinx-doc.org/">Sphinx</a> using a + <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a> + provided by <a href="https://readthedocs.org">Read the Docs</a>. + + + </footer> + </div> + </div> + </section> + </div> + <script> + jQuery(function () { + SphinxRtdTheme.Navigation.enable(true); + }); + </script> + + </body> + </html>
html/C04_Sets_and_Functions.html (new)
- @@ -0,0 +1,1188 @@
 + <!DOCTYPE html> + <html class="writer-html5" lang="en" > + <head> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + + <meta name="viewport" content="width=device-width, initial-scale=1.0" /> + <title>4. Sets and Functions — Mathematics in Lean 0.1 documentation</title> + <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="shortcut icon" href="_static/favicon.ico"/> +  + + <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> + <script src="_static/jquery.js"></script> + <script src="_static/underscore.js"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> + <script src="_static/doctools.js"></script> + <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> + <script src="_static/js/theme.js"></script> + <link rel="index" title="Index" href="genindex.html" /> + <link rel="search" title="Search" href="search.html" /> + <link rel="next" title="5. Number Theory" href="C05_Number_Theory.html" /> + <link rel="prev" title="3. Logic" href="C03_Logic.html" /> + </head> + + <body class="wy-body-for-nav"> + <div class="wy-grid-for-nav"> + <nav data-toggle="wy-nav-shift" class="wy-nav-side"> + <div class="wy-side-scroll"> + <div class="wy-side-nav-search" > + <a href="index.html" class="icon icon-home"> Mathematics in Lean + </a> + <div role="search"> + <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> + <input type="text" name="q" placeholder="Search docs" /> + <input type="hidden" name="check_keywords" value="yes" /> + <input type="hidden" name="area" value="default" /> + </form> + </div> + </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> + <ul class="current"> + <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a></li> + <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> + <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> + <li class="toctree-l1 current"><a class="current reference internal" href="#">4. Sets and Functions</a><ul> + <li class="toctree-l2"><a class="reference internal" href="#sets">4.1. Sets</a></li> + <li class="toctree-l2"><a class="reference internal" href="#functions">4.2. Functions</a></li> + <li class="toctree-l2"><a class="reference internal" href="#the-schroder-bernstein-theorem">4.3. The Schröder-Bernstein Theorem</a></li> + </ul> + </li> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> + <li class="toctree-l1"><a class="reference internal" href="C06_Abstract_Algebra.html">6. Abstract Algebra</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Topology.html">7. Topology</a></li> + </ul> + <ul> + <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li> + </ul> + + </div> + </div> + </nav> + + <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > + + <a href="index.html">Mathematics in Lean</a> + </nav> + + <div class="wy-nav-content"> + <div class="rst-content"> + <div role="navigation" aria-label="Page navigation"> + <ul class="wy-breadcrumbs"> + <li><a href="index.html" class="icon icon-home"></a> »</li> + <li>4. Sets and Functions</li> + <li class="wy-breadcrumbs-aside"> + <a href="_sources/C04_Sets_and_Functions.rst.txt" rel="nofollow"> View page source</a> + </li> + </ul> + <hr/> + </div> + <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> + <div itemprop="articleBody"> + + <section id="sets-and-functions"> + <h1>4. Sets and Functions<a class="headerlink" href="#sets-and-functions" title="Permalink to this heading"></a></h1> + The vocabulary of sets, relations, and functions provides a uniform + language for carrying out constructions in all the branches of + mathematics. + Since functions and relations can be defined in terms of sets, + axiomatic set theory can be used as a foundation for mathematics. + Lean’s foundation is based instead on the primitive notion of a type, + and it includes ways of defining functions between types. + Every expression in Lean has a type: + there are natural numbers, real numbers, functions from reals to reals, + groups, vector spaces, and so on. + Some expressions are types, + which is to say, + their type is <code class="docutils literal notranslate">Type</code>. + Lean and mathlib provide ways of defining new types, + and ways of defining objects of those types. + Conceptually, you can think of a type as just a set of objects. + Requiring every object to have a type has some advantages. + For example, it makes it possible to overload notation like <code class="docutils literal notranslate">+</code>, + and it sometimes makes input less verbose + because Lean can infer a lot of information from + an object’s type. + The type system also enables Lean to flag errors when you + apply a function to the wrong number of arguments, + or apply a function to arguments of the wrong type. + Lean’s library does define elementary set-theoretic notions. + In contrast to set theory, + in Lean a set is always a set of objects of some type, + such as a set natural numbers or a set of functions + from real numbers to real numbers. + The distinction between types and set takes some getting used to, + but this chapter will take you through the essentials. + <section id="sets"> + <h2>4.1. Sets<a class="headerlink" href="#sets" title="Permalink to this heading"></a></h2> + If <code class="docutils literal notranslate">α</code> is any type, the type <code class="docutils literal notranslate">Set α</code> consists of sets + of elements of <code class="docutils literal notranslate">α</code>. + This type supports the usual set-theoretic operations and relations. + For example, <code class="docutils literal notranslate">s ⊆ t</code> says that <code class="docutils literal notranslate">s</code> is a subset of <code class="docutils literal notranslate">t</code>, + <code class="docutils literal notranslate">s ∩ t</code> denotes the intersection of <code class="docutils literal notranslate">s</code> and <code class="docutils literal notranslate">t</code>, + and <code class="docutils literal notranslate">s ∪ t</code> denotes their union. + The subset relation can be typed with <code class="docutils literal notranslate">\ss</code> or <code class="docutils literal notranslate">\sub</code>, + intersection can be typed with <code class="docutils literal notranslate">\i</code> or <code class="docutils literal notranslate">\cap</code>, + and union can be typed with <code class="docutils literal notranslate">\un</code> or <code class="docutils literal notranslate">\cup</code>. + The library also defines the set <code class="docutils literal notranslate">univ</code>, + which consists of all the elements of type <code class="docutils literal notranslate">α</code>, + and the empty set, <code class="docutils literal notranslate">∅</code>, which can be typed as <code class="docutils literal notranslate">\empty</code>. + Given <code class="docutils literal notranslate">x : α</code> and <code class="docutils literal notranslate">s : Set α</code>, + the expression <code class="docutils literal notranslate">x ∈ s</code> says that <code class="docutils literal notranslate">x</code> is a member of <code class="docutils literal notranslate">s</code>. + Theorems that mention set membership often include <code class="docutils literal notranslate">mem</code> + in their name. + The expression <code class="docutils literal notranslate">x ∉ s</code> abbreviates <code class="docutils literal notranslate">¬ x ∈ s</code>. + You can type <code class="docutils literal notranslate">∈</code> as <code class="docutils literal notranslate">\in</code> or <code class="docutils literal notranslate">\mem</code> and <code class="docutils literal notranslate">∉</code> as <code class="docutils literal notranslate">\notin</code>. + One way to prove things about sets is to use <code class="docutils literal notranslate">rw</code> + or the simplifier to expand the definitions. + In the second example below, we use <code class="docutils literal notranslate">simp only</code> + to tell the simplifier to use only the list + of identities we give it, + and not its full database of identities. + Unlike <code class="docutils literal notranslate">rw</code>, <code class="docutils literal notranslate">simp</code> can perform simplifications + inside a universal or existential quantifier. + If you step through the proof, + you can see the effects of these commands. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} + variable (s t u : Set α) + open Set + + example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := by + rw [subset_def, inter_def, inter_def] + rw [subset_def] at h + dsimp + rintro x ⟨xs, xu⟩ + exact ⟨h _ xs, xu⟩ + + example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := by + simp only [subset_def, mem_inter_iff] at * + rintro x ⟨xs, xu⟩ + exact ⟨h _ xs, xu⟩ + </pre></div> + </div> + In this example, we open the <code class="docutils literal notranslate">set</code> namespace to have + access to the shorter names for the theorems. + But, in fact, we can delete the calls to <code class="docutils literal notranslate">rw</code> and <code class="docutils literal notranslate">simp</code> + entirely: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := by + intro x xsu + exact ⟨h xsu.1, xsu.2⟩ + </pre></div> + </div> + What is going on here is known as definitional reduction: + to make sense of the <code class="docutils literal notranslate">intro</code> command and the anonymous constructors + Lean is forced to expand the definitions. + The following examples also illustrate the phenomenon: + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem foo (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := + fun x ⟨xs, xu⟩ => ⟨h xs, xu⟩ + + example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := + fun x ⟨xs, xu⟩ => ⟨h xs, xu⟩ + </pre></div> + </div> + Due to a quirk of how Lean processes its input, + the first example fails if we replace <code class="docutils literal notranslate">theorem foo</code> with <code class="docutils literal notranslate">example</code>. + This illustrates the pitfalls of relying on definitional reduction + too heavily. + It is often convenient, + but sometimes we have to fall back on unfolding definitions manually. + To deal with unions, we can use <code class="docutils literal notranslate">Set.union_def</code> and <code class="docutils literal notranslate">Set.mem_union</code>. + Since <code class="docutils literal notranslate">x ∈ s ∪ t</code> unfolds to <code class="docutils literal notranslate">x ∈ s ∨ x ∈ t</code>, + we can also use the <code class="docutils literal notranslate">cases</code> tactic to force a definitional reduction. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : s ∩ (t ∪ u) ⊆ s ∩ t ∪ s ∩ u := by + intro x hx + have xs : x ∈ s := hx.1 + have xtu : x ∈ t ∪ u := hx.2 + cases' xtu with xt xu + · left + show x ∈ s ∩ t + exact ⟨xs, xt⟩ + right + show x ∈ s ∩ u + exact ⟨xs, xu⟩ + </pre></div> + </div> + Since intersection binds tighter than union, + the use of parentheses in the expression <code class="docutils literal notranslate">(s ∩ t) ∪ (s ∩ u)</code> + is unnecessary, but they make the meaning of the expression clearer. + The following is a shorter proof of the same fact: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : s ∩ (t ∪ u) ⊆ s ∩ t ∪ s ∩ u := by + rintro x ⟨xs, xt | xu⟩ + · left + exact ⟨xs, xt⟩ + right; exact ⟨xs, xu⟩ + </pre></div> + </div> + As an exercise, try proving the other inclusion: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : s ∩ t ∪ s ∩ u ⊆ s ∩ (t ∪ u) := by + sorry + </pre></div> + </div> + It might help to know that when using <code class="docutils literal notranslate">rintro</code>, + sometimes we need to use parentheses around a disjunctive pattern + <code class="docutils literal notranslate">h1 | h2</code> to get Lean to parse it correctly. + The library also defines set difference, <code class="docutils literal notranslate">s \ t</code>, + where the backslash is a special unicode character + entered as <code class="docutils literal notranslate">\\</code>. + The expression <code class="docutils literal notranslate">x ∈ s \ t</code> expands to <code class="docutils literal notranslate">x ∈ s ∧ x ∉ t</code>. + (The <code class="docutils literal notranslate">∉</code> can be entered as <code class="docutils literal notranslate">\notin</code>.) + It can be rewritten manually using <code class="docutils literal notranslate">Set.diff_eq</code> and <code class="docutils literal notranslate">dsimp</code> + or <code class="docutils literal notranslate">Set.mem_diff</code>, + but the following two proofs of the same inclusion + show how to avoid using them. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : (s \ t) \ u ⊆ s \ (t ∪ u) := by + intro x xstu + have xs : x ∈ s := xstu.1.1 + have xnt : x ∉ t := xstu.1.2 + have xnu : x ∉ u := xstu.2 + constructor + · exact xs + intro xtu + -- x ∈ t ∨ x ∈ u + cases' xtu with xt xu + · show False + exact xnt xt + show False; exact xnu xu + + example : (s \ t) \ u ⊆ s \ (t ∪ u) := by + rintro x ⟨⟨xs, xnt⟩, xnu⟩ + use xs + rintro (xt | xu) <;> contradiction + </pre></div> + </div> + As an exercise, prove the reverse inclusion: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : s \ (t ∪ u) ⊆ (s \ t) \ u := by + sorry + </pre></div> + </div> + To prove that two sets are equal, + it suffices to show that every element of one is an element + of the other. + This principle is known as “extensionality,” + and, unsurprisingly, + the <code class="docutils literal notranslate">ext</code> tactic is equipped to handle it. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : s ∩ t = t ∩ s := by + ext x + simp only [mem_inter_iff] + constructor + · rintro ⟨xs, xt⟩ + exact ⟨xt, xs⟩ + rintro ⟨xt, xs⟩; exact ⟨xs, xt⟩ + </pre></div> + </div> + Once again, deleting the line <code class="docutils literal notranslate">simp only [mem_inter_iff]</code> + does not harm the proof. + In fact, if you like inscrutable proof terms, + the following one-line proof is for you: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : s ∩ t = t ∩ s := + Set.ext fun x => ⟨fun ⟨xs, xt⟩ => ⟨xt, xs⟩, fun ⟨xt, xs⟩ => ⟨xs, xt⟩⟩ + </pre></div> + </div> + The dollar sign is a useful syntax: + writing <code class="docutils literal notranslate">f $ ...</code> + is essentially the same as writing <code class="docutils literal notranslate">f (...)</code>, + but it saves us the trouble of having to close + a set of parentheses at the end of a long expression. + Here is an even shorter proof, + using the simplifier: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : s ∩ t = t ∩ s := by ext x; simp [and_comm] + </pre></div> + </div> + An alternative to using <code class="docutils literal notranslate">ext</code> is to use + the theorem <code class="docutils literal notranslate">Subset.antisymm</code> + which allows us to prove an equation <code class="docutils literal notranslate">s = t</code> + between sets by proving <code class="docutils literal notranslate">s ⊆ t</code> and <code class="docutils literal notranslate">t ⊆ s</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : s ∩ t = t ∩ s := by + apply Subset.antisymm + · rintro x ⟨xs, xt⟩ + exact ⟨xt, xs⟩ + rintro x ⟨xt, xs⟩; exact ⟨xs, xt⟩ + </pre></div> + </div> + Try finishing this proof term: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : s ∩ t = t ∩ s := + Subset.antisymm sorry sorry + </pre></div> + </div> + Remember that you can replace <cite>sorry</cite> by an underscore, + and when you hover over it, + Lean will show you what it expects at that point. + Here are some set-theoretic identities you might enjoy proving: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : s ∩ (s ∪ t) = s := by + sorry + + example : s ∪ s ∩ t = s := by + sorry + + example : s \ t ∪ t = s ∪ t := by + sorry + + example : s \ t ∪ t \ s = (s ∪ t) \ (s ∩ t) := by + sorry + </pre></div> + </div> + When it comes to representing sets, + here is what is going on underneath the hood. + In type theory, a property or predicate on a type <code class="docutils literal notranslate">α</code> + is just a function <code class="docutils literal notranslate">P : α → Prop</code>. + This makes sense: + given <code class="docutils literal notranslate">a : α</code>, <code class="docutils literal notranslate">P a</code> is just the proposition + that <code class="docutils literal notranslate">P</code> holds of <code class="docutils literal notranslate">a</code>. + In the library, <code class="docutils literal notranslate">Set α</code> is defined to be <code class="docutils literal notranslate">α → Prop</code> and <code class="docutils literal notranslate">x ∈ s</code> is defined to be <code class="docutils literal notranslate">s x</code>. + In other words, sets are really properties, treated as objects. + The library also defines set-builder notation. + The expression <code class="docutils literal notranslate">{ y | P y }</code> unfolds to <code class="docutils literal notranslate">(fun y ↦ P y)</code>, + so <code class="docutils literal notranslate">x ∈ { y | P y }</code> reduces to <code class="docutils literal notranslate">P x</code>. + So we can turn the property of being even into the set of even numbers: + <div class="highlight-lean notranslate"><div class="highlight"><pre>def evens : Set ℕ := + { n | Even n } + + def odds : Set ℕ := + { n | ¬Even n } + + example : evens ∪ odds = univ := by + rw [evens, odds] + ext n + simp + apply Classical.em + </pre></div> + </div> + You should step through this proof and make sure + you understand what is going on. + Try deleting the line <code class="docutils literal notranslate">rw [evens, odds]</code> + and confirm that the proof still works. + In fact, set-builder notation is used to define + <ul class="simple"> + <li><code class="docutils literal notranslate">s ∩ t</code> as <code class="docutils literal notranslate">{x | x ∈ s ∧ x ∈ t}</code>,</li> + <li><code class="docutils literal notranslate">s ∪ t</code> as <code class="docutils literal notranslate">{x | x ∈ s ∨ x ∈ t}</code>,</li> + <li><code class="docutils literal notranslate">∅</code> as <code class="docutils literal notranslate">{x | False}</code>, and</li> + <li><code class="docutils literal notranslate">univ</code> as <code class="docutils literal notranslate">{x | True}</code>.</li> + </ul> + We often need to indicate the type of <code class="docutils literal notranslate">∅</code> and <code class="docutils literal notranslate">univ</code> + explicitly, + because Lean has trouble guessing which ones we mean. + The following examples show how Lean unfolds the last + two definitions when needed. In the second one, + <code class="docutils literal notranslate">trivial</code> is the canonical proof of <code class="docutils literal notranslate">True</code> in the library. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x : ℕ) (h : x ∈ (∅ : Set ℕ)) : False := + h + + example (x : ℕ) : x ∈ (univ : Set ℕ) := + trivial + </pre></div> + </div> + As an exercise, prove the following inclusion. + Use <code class="docutils literal notranslate">intro n</code> to unfold the definition of subset, + and use the simplifier to reduce the + set-theoretic constructions to logic. + We also recommend using the theorems + <code class="docutils literal notranslate">Nat.Prime.eq_two_or_odd</code> and <code class="docutils literal notranslate">Nat.even_iff</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : { n | Nat.Prime n } ∩ { n | n > 2 } ⊆ { n | ¬Even n } := by + sorry + </pre></div> + </div> + Be careful: it is somewhat confusing that the library has multiple versions + of the predicate <code class="docutils literal notranslate">Prime</code>. + The most general one makes sense in any commutative monoid with a zero element. + The predicate <code class="docutils literal notranslate">Nat.Prime</code> is specific to the natural numbers. + Fortunately, there is a theorem that says that in the specific case, + the two notions agree, so you can always rewrite one to the other. + <div class="highlight-lean notranslate"><div class="highlight"><pre>#print Prime + + #print Nat.Prime + + example (n : ℕ) : Prime n ↔ Nat.Prime n := + Nat.prime_iff.symm + + example (n : ℕ) (h : Prime n) : Nat.Prime n := by + rw [Nat.prime_iff] + exact h + </pre></div> + </div> + The <cite>rwa</cite> tactic follows a rewrite with the assumption tactic. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (n : ℕ) (h : Prime n) : Nat.Prime n := by + rwa [Nat.prime_iff] + </pre></div> + </div> + Lean introduces the notation <code class="docutils literal notranslate">∀ x ∈ s, ...</code>, + “for every <code class="docutils literal notranslate">x</code> in <code class="docutils literal notranslate">s</code> .,” + as an abbreviation for <code class="docutils literal notranslate">∀ x, x ∈ s → ...</code>. + It also introduces the notation <code class="docutils literal notranslate">∃ x ∈ s, ...,</code> + “there exists an <code class="docutils literal notranslate">x</code> in <code class="docutils literal notranslate">s</code> such that ..” + These are sometimes known as bounded quantifiers, + because the construction serves to restrict their significance + to the set <code class="docutils literal notranslate">s</code>. + As a result, theorems in the library that make use of them + often contain <code class="docutils literal notranslate">ball</code> or <code class="docutils literal notranslate">bex</code> in the name. + The theorem <code class="docutils literal notranslate">bex_def</code> asserts that <code class="docutils literal notranslate">∃ x ∈ s, ...</code> is equivalent + to <code class="docutils literal notranslate">∃ x, x ∈ s ∧ ...,</code> + but when they are used with <code class="docutils literal notranslate">rintro</code>, <code class="docutils literal notranslate">use</code>, + and anonymous constructors, + these two expressions behave roughly the same. + As a result, we usually don’t need to use <code class="docutils literal notranslate">bex_def</code> + to transform them explicitly. + Here is are some examples of how they are used: + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (s t : Set ℕ) + + example (h₀ : ∀ x ∈ s, ¬Even x) (h₁ : ∀ x ∈ s, Prime x) : ∀ x ∈ s, ¬Even x ∧ Prime x := by + intro x xs + constructor + · apply h₀ x xs + apply h₁ x xs + + example (h : ∃ x ∈ s, ¬Even x ∧ Prime x) : ∃ x ∈ s, Prime x := by + rcases h with ⟨x, xs, _, prime_x⟩ + use x, xs + exact prime_x + </pre></div> + </div> + See if you can prove these slight variations: + <div class="highlight-lean notranslate"><div class="highlight"><pre>section + variable (ssubt : s ⊆ t) + + example (h₀ : ∀ x ∈ t, ¬Even x) (h₁ : ∀ x ∈ t, Prime x) : ∀ x ∈ s, ¬Even x ∧ Prime x := by + sorry + + example (h : ∃ x ∈ s, ¬Even x ∧ Prime x) : ∃ x ∈ t, Prime x := by + sorry + + end + </pre></div> + </div> + Indexed unions and intersections are + another important set-theoretic construction. + We can model a sequence $A_0, A_1, A_2, \ldots$ of sets of + elements of <code class="docutils literal notranslate">α</code> + as a function <code class="docutils literal notranslate">A : ℕ → Set α</code>, + in which case <code class="docutils literal notranslate">⋃ i, A i</code> denotes their union, + and <code class="docutils literal notranslate">⋂ i, A i</code> denotes their intersection. + There is nothing special about the natural numbers here, + so <code class="docutils literal notranslate">ℕ</code> can be replaced by any type <code class="docutils literal notranslate">I</code> + used to index the sets. + The following illustrates their use. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α I : Type _} + variable (A B : I → Set α) + variable (s : Set α) + + open Set + + example : (s ∩ ⋃ i, A i) = ⋃ i, A i ∩ s := by + ext x + simp only [mem_inter_iff, mem_iUnion] + constructor + · rintro ⟨xs, ⟨i, xAi⟩⟩ + exact ⟨i, xAi, xs⟩ + rintro ⟨i, xAi, xs⟩ + exact ⟨xs, ⟨i, xAi⟩⟩ + + example : (⋂ i, A i ∩ B i) = (⋂ i, A i) ∩ ⋂ i, B i := by + ext x + simp only [mem_inter_iff, mem_iInter] + constructor + · intro h + constructor + · intro i + exact (h i).1 + intro i + exact (h i).2 + rintro ⟨h1, h2⟩ i + constructor + · exact h1 i + exact h2 i + </pre></div> + </div> + Parentheses are often needed with an + indexed union or intersection because, + as with the quantifiers, + the scope of the bound variable extends as far as it can. + Try proving the following identity. + One direction requires classical logic! + We recommend using <code class="docutils literal notranslate">by_cases xs : x ∈ s</code> + at an appropriate point in the proof. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : (s ∪ ⋂ i, A i) = ⋂ i, A i ∪ s := by + sorry + </pre></div> + </div> + Mathlib also has bounded unions and intersections, + which are analogous to the bounded quantifiers. + You can unpack their meaning with <code class="docutils literal notranslate">mem_Union₂</code> + and <code class="docutils literal notranslate">mem_Inter₂</code>. + As the following examples show, + Lean’s simplifier carries out these replacements as well. + <div class="highlight-lean notranslate"><div class="highlight"><pre>def primes : Set ℕ := + { x | Nat.Prime x } + + example : (⋃ p ∈ primes, { x | p ^ 2 ∣ x }) = { x | ∃ p ∈ primes, p ^ 2 ∣ x } :=by + ext + rw [mem_iUnion₂] + simp + + example : (⋃ p ∈ primes, { x | p ^ 2 ∣ x }) = { x | ∃ p ∈ primes, p ^ 2 ∣ x } := by + ext + simp + + example : (⋂ p ∈ primes, { x | ¬p ∣ x }) ⊆ { x | x = 1 } := by + intro x + contrapose! + simp + apply Nat.exists_prime_and_dvd + </pre></div> + </div> + Try solving the following example, which is similar. + If you start typing <code class="docutils literal notranslate">eq_univ</code>, + tab completion will tell you that <code class="docutils literal notranslate">apply eq_univ_of_forall</code> + is a good way to start the proof. + We also recommend using the theorem <code class="docutils literal notranslate">Nat.exists_infinite_primes</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : (⋃ p ∈ primes, { x | x ≤ p }) = univ := by + sorry + </pre></div> + </div> + Give a collection of sets, <code class="docutils literal notranslate">s : Set (Set α)</code>, + their union, <code class="docutils literal notranslate">⋃₀ s</code>, has type <code class="docutils literal notranslate">Set α</code> + and is defined as <code class="docutils literal notranslate">{x | ∃ t ∈ s, x ∈ t}</code>. + Similarly, their intersection, <code class="docutils literal notranslate">⋂₀ s</code>, is defined as + <code class="docutils literal notranslate">{x | ∀ t ∈ s, x ∈ t}</code>. + These operations are called <code class="docutils literal notranslate">sUnion</code> and <code class="docutils literal notranslate">sInter</code>, respectively. + The following examples show their relationship to bounded union + and intersection. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} (s : Set (Set α)) + + example : ⋃₀ s = ⋃ t ∈ s, t := by + ext x + rw [mem_iUnion₂] + simp + + example : ⋂₀ s = ⋂ t ∈ s, t := by + ext x + rw [mem_iInter₂] + rfl + </pre></div> + </div> + In the library, these identities are called + <code class="docutils literal notranslate">sUnion_eq_biUnion</code> and <code class="docutils literal notranslate">sInter_eq_biInter</code>. + </section> + <section id="functions"> + <h2>4.2. Functions<a class="headerlink" href="#functions" title="Permalink to this heading"></a></h2> + If <code class="docutils literal notranslate">f : α → β</code> is a function and <code class="docutils literal notranslate">p</code> is a set of + elements of type <code class="docutils literal notranslate">β</code>, + the library defines <code class="docutils literal notranslate">preimage f p</code>, written <code class="docutils literal notranslate">f ⁻¹' p</code>, + to be <code class="docutils literal notranslate">{x | f x ∈ p}</code>. + The expression <code class="docutils literal notranslate">x ∈ f ⁻¹' p</code> reduces to <code class="docutils literal notranslate">f x ∈ p</code>. + This is often convenient, as in the following example: + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α β : Type _} + variable (f : α → β) + variable (s t : Set α) + variable (u v : Set β) + + open Function + open Set + + example : f ⁻¹' (u ∩ v) = f ⁻¹' u ∩ f ⁻¹' v := by + ext + rfl + </pre></div> + </div> + If <code class="docutils literal notranslate">s</code> is a set of elements of type <code class="docutils literal notranslate">α</code>, + the library also defines <code class="docutils literal notranslate">image f s</code>, + written <code class="docutils literal notranslate">f '' s</code>, + to be <code class="docutils literal notranslate">{y | ∃ x, x ∈ s ∧ f x = y}</code>. + So a hypothesis <code class="docutils literal notranslate">y ∈ f '' s</code> decomposes to a triple + <code class="docutils literal notranslate">⟨x, xs, xeq⟩</code> with <code class="docutils literal notranslate">x : α</code> satisfying the hypotheses <code class="docutils literal notranslate">xs : x ∈ s</code> + and <code class="docutils literal notranslate">xeq : f x = y</code>. + The <code class="docutils literal notranslate">rfl</code> tag in the <code class="docutils literal notranslate">rintro</code> tactic (see <a class="reference internal" href="C03_Logic.html#the-existential-quantifier">Section 3.2</a>) was made precisely + for this sort of situation. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : f '' (s ∪ t) = f '' s ∪ f '' t := by + ext y; constructor + · rintro ⟨x, xs | xt, rfl⟩ + · left + use x, xs + right + use x, xt + rintro (⟨x, xs, rfl⟩ | ⟨x, xt, rfl⟩) + · use x, Or.inl xs + use x, Or.inr xt + </pre></div> + </div> + Notice also that the <code class="docutils literal notranslate">use</code> tactic applies <code class="docutils literal notranslate">refl</code> + to close goals when it can. + Here is another example: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : s ⊆ f ⁻¹' (f '' s) := by + intro x xs + show f x ∈ f '' s + use x, xs + </pre></div> + </div> + We can replace the line <code class="docutils literal notranslate">use x, xs</code> by + <code class="docutils literal notranslate">apply mem_image_of_mem f xs</code> if we want to + use a theorem specifically designed for that purpose. + But knowing that the image is defined in terms + of an existential quantifier is often convenient. + The following equivalence is a good exercise: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : f '' s ⊆ v ↔ s ⊆ f ⁻¹' v := by + sorry + </pre></div> + </div> + It shows that <code class="docutils literal notranslate">image f</code> and <code class="docutils literal notranslate">preimage f</code> are + an instance of what is known as a Galois connection + between <code class="docutils literal notranslate">Set α</code> and <code class="docutils literal notranslate">Set β</code>, + each partially ordered by the subset relation. + In the library, this equivalence is named + <code class="docutils literal notranslate">image_subset_iff</code>. + In practice, the right-hand side is often the + more useful representation, + because <code class="docutils literal notranslate">y ∈ f ⁻¹' t</code> unfolds to <code class="docutils literal notranslate">f y ∈ t</code> + whereas working with <code class="docutils literal notranslate">x ∈ f '' s</code> requires + decomposing an existential quantifier. + Here is a long list of set-theoretic identities for + you to enjoy. + You don’t have to do all of them at once; + do a few of them, + and set the rest aside for a rainy day. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : Injective f) : f ⁻¹' (f '' s) ⊆ s := by + sorry + + example : f '' (f ⁻¹' u) ⊆ u := by + sorry + + example (h : Surjective f) : u ⊆ f '' (f ⁻¹' u) := by + sorry + + example (h : s ⊆ t) : f '' s ⊆ f '' t := by + sorry + + example (h : u ⊆ v) : f ⁻¹' u ⊆ f ⁻¹' v := by + sorry + + example : f ⁻¹' (u ∪ v) = f ⁻¹' u ∪ f ⁻¹' v := by + sorry + + example : f '' (s ∩ t) ⊆ f '' s ∩ f '' t := by + sorry + + example (h : Injective f) : f '' s ∩ f '' t ⊆ f '' (s ∩ t) := by + sorry + + example : f '' s \ f '' t ⊆ f '' (s \ t) := by + sorry + + example : f ⁻¹' u \ f ⁻¹' v ⊆ f ⁻¹' (u \ v) := by + sorry + + example : f '' s ∩ v = f '' (s ∩ f ⁻¹' v) := by + sorry + + example : f '' (s ∩ f ⁻¹' u) ⊆ f '' s ∪ u := by + sorry + + example : s ∩ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∩ u) := by + sorry + + example : s ∪ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∪ u) := by + sorry + </pre></div> + </div> + You can also try your hand at the next group of exercises, + which characterize the behavior of images and preimages + with respect to indexed unions and intersections. + In the third exercise, the argument <code class="docutils literal notranslate">i : I</code> is needed + to guarantee that the index set is nonempty. + To prove any of these, we recommend using <code class="docutils literal notranslate">ext</code> or <code class="docutils literal notranslate">intro</code> + to unfold the meaning of an equation or inclusion between sets, + and then calling <code class="docutils literal notranslate">simp</code> to unpack the conditions for membership. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {I : Type _} (A : I → Set α) (B : I → Set β) + + example : (f '' ⋃ i, A i) = ⋃ i, f '' A i := by + ext y; simp + constructor + · rintro ⟨x, ⟨i, xAi⟩, fxeq⟩ + use i, x + exact ⟨xAi, fxeq⟩ + rintro ⟨i, x, xAi, fxeq⟩ + exact ⟨x, ⟨i, xAi⟩, fxeq⟩ + + example : (f '' ⋂ i, A i) ⊆ ⋂ i, f '' A i := by + intro y; simp + intro x h fxeq i + use x + exact ⟨h i, fxeq⟩ + + example (i : I) (injf : Injective f) : (⋂ i, f '' A i) ⊆ f '' ⋂ i, A i := by + intro y; simp + intro h + rcases h i with ⟨x, xAi, fxeq⟩ + use x; constructor + · intro i' + rcases h i' with ⟨x', x'Ai, fx'eq⟩ + have : f x = f x' := by rw [fxeq, fx'eq] + have : x = x' := injf this + rw [this] + exact x'Ai + exact fxeq + + example : (f ⁻¹' ⋃ i, B i) = ⋃ i, f ⁻¹' B i := by + ext x + simp + + example : (f ⁻¹' ⋂ i, B i) = ⋂ i, f ⁻¹' B i := by + ext x + simp + </pre></div> + </div> + The library defines a predicate <code class="docutils literal notranslate">InjOn f s</code> to say that + <code class="docutils literal notranslate">f</code> is injective on <code class="docutils literal notranslate">s</code>. + It is defined as follows: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : InjOn f s ↔ ∀ x₁ ∈ s, ∀ x₂ ∈ s, f x₁ = f x₂ → x₁ = x₂ := + Iff.refl _ + </pre></div> + </div> + The statement <code class="docutils literal notranslate">Injective f</code> is provably equivalent + to <code class="docutils literal notranslate">InjOn f univ</code>. + Similarly, the library defines <code class="docutils literal notranslate">range f</code> to be + <code class="docutils literal notranslate">{x | ∃y, f y = x}</code>, + so <code class="docutils literal notranslate">range f</code> is provably equal to <code class="docutils literal notranslate">f '' univ</code>. + This is a common theme in mathlib: + although many properties of functions are defined relative + to their full domain, + there are often relativized versions that restrict + the statements to a subset of the domain type. + Here is are some examples of <code class="docutils literal notranslate">InjOn</code> and <code class="docutils literal notranslate">range</code> in use: + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Set Real + + example : InjOn log { x | x > 0 } := by + intro x xpos y ypos + intro e + -- log x = log y + calc + x = exp (log x) := by rw [exp_log xpos] + _ = exp (log y) := by rw [e] + _ = y := by rw [exp_log ypos] + + + example : range exp = { y | y > 0 } := by + ext y; constructor + · rintro ⟨x, rfl⟩ + apply exp_pos + intro ypos + use log y + rw [exp_log ypos] + </pre></div> + </div> + Try proving these: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : InjOn sqrt { x | x ≥ 0 } := by + sorry + + example : InjOn (fun x => x ^ 2) { x : ℝ | x ≥ 0 } := by + sorry + + example : sqrt '' { x | x ≥ 0 } = { y | y ≥ 0 } := by + sorry + + example : (range fun x => x ^ 2) = { y : ℝ | y ≥ 0 } := by + sorry + </pre></div> + </div> + To define the inverse of a function <code class="docutils literal notranslate">f : α → β</code>, + we will use two new ingredients. + First, we need to deal with the fact that + an arbitrary type in Lean may be empty. + To define the inverse to <code class="docutils literal notranslate">f</code> at <code class="docutils literal notranslate">y</code> when there is + no <code class="docutils literal notranslate">x</code> satisfying <code class="docutils literal notranslate">f x = y</code>, + we want to assign a default value in <code class="docutils literal notranslate">α</code>. + Adding the annotation <code class="docutils literal notranslate">[Inhabited α]</code> as a variable + is tantamount to assuming that <code class="docutils literal notranslate">α</code> has a + preferred element, which is denoted <code class="docutils literal notranslate">default</code>. + Second, in the case where there is more than one <code class="docutils literal notranslate">x</code> + such that <code class="docutils literal notranslate">f x = y</code>, + the inverse function needs to choose one of them. + This requires an appeal to the axiom of choice. + Lean allows various ways of accessing it; + one convenient method is to use the classical <code class="docutils literal notranslate">some</code> + operator, illustrated below. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α β : Type _} [Inhabited α] + + #check (default : α) + + variable (P : α → Prop) (h : ∃ x, P x) + + #check Classical.choose h + + example : P (Classical.choose h) := + Classical.choose_spec h + </pre></div> + </div> + Given <code class="docutils literal notranslate">h : ∃ x, P x</code>, the value of <code class="docutils literal notranslate">Classical.some h</code> + is some <code class="docutils literal notranslate">x</code> satisfying <code class="docutils literal notranslate">P x</code>. + The theorem <code class="docutils literal notranslate">Classical.some_spec h</code> says that <code class="docutils literal notranslate">Classical.some h</code> + meets this specification. + With these in hand, we can define the inverse function + as follows: + <div class="highlight-lean notranslate"><div class="highlight"><pre>noncomputable section + + open Classical + + def inverse (f : α → β) : β → α := fun y : β => + if h : ∃ x, f x = y then Classical.choose h else default + + theorem inverse_spec {f : α → β} (y : β) (h : ∃ x, f x = y) : f (inverse f y) = y := by + rw [inverse]; dsimp; rw [dif_pos h] + exact Classical.choose_spec h + </pre></div> + </div> + The lines <code class="docutils literal notranslate">noncomputable theory</code> and <code class="docutils literal notranslate">open Classical</code> + are needed because we are using classical logic in an essential way. + On input <code class="docutils literal notranslate">y</code>, the function <code class="docutils literal notranslate">inverse f</code> + returns some value of <code class="docutils literal notranslate">x</code> satisfying <code class="docutils literal notranslate">f x = y</code> if there is one, + and a default element of <code class="docutils literal notranslate">α</code> otherwise. + This is an instance of a dependent if construction, + since in the positive case, the value returned, + <code class="docutils literal notranslate">Classical.choose h</code>, depends on the assumption <code class="docutils literal notranslate">h</code>. + The identity <code class="docutils literal notranslate">dif_pos h</code> rewrites <code class="docutils literal notranslate">if h : e then a else b</code> + to <code class="docutils literal notranslate">a</code> given <code class="docutils literal notranslate">h : e</code>, + and, similarly, <code class="docutils literal notranslate">dif_neg h</code> rewrites it to <code class="docutils literal notranslate">b</code> given <code class="docutils literal notranslate">h : ¬ e</code>. + The theorem <code class="docutils literal notranslate">inverse_spec</code> says that <code class="docutils literal notranslate">inverse f</code> + meets the first part of this specification. + Don’t worry if you do not fully understand how these work. + The theorem <code class="docutils literal notranslate">inverse_spec</code> alone should be enough to show + that <code class="docutils literal notranslate">inverse f</code> is a left inverse if and only if <code class="docutils literal notranslate">f</code> is injective + and a right inverse if and only if <code class="docutils literal notranslate">f</code> is surjective. + Look up the definition of <code class="docutils literal notranslate">LeftInverse</code> and <code class="docutils literal notranslate">RightInverse</code> + by double-clicking or right-clicking on them in VS Code, + or using the commands <code class="docutils literal notranslate">#print LeftInverse</code> and <code class="docutils literal notranslate">#print RightInverse</code>. + Then try to prove the two theorems. + They are tricky! + It helps to do the proofs on paper before + you start hacking through the details. + You should be able to prove each of them with about a half-dozen + short lines. + If you are looking for an extra challenge, + try to condense each proof to a single-line proof term. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (f : α → β) + + open Function + + example : Injective f ↔ LeftInverse (inverse f) f := + sorry + + example : Surjective f ↔ RightInverse (inverse f) f := + sorry + </pre></div> + </div> + We close this section with a type-theoretic statement of Cantor’s + famous theorem that there is no surjective function from a set + to its power set. + See if you can understand the proof, + and then fill in the two lines that are missing. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem Cantor : ∀ f : α → Set α, ¬Surjective f := by + intro f surjf + let S := { i | i ∉ f i } + rcases surjf S with ⟨j, h⟩ + have h₁ : j ∉ f j := by + intro h' + have : j ∉ f j := by rwa [h] at h' + contradiction + have h₂ : j ∈ S + sorry + have h₃ : j ∉ S + sorry + contradiction + </pre></div> + </div> + </section> + <section id="the-schroder-bernstein-theorem"> + <h2>4.3. The Schröder-Bernstein Theorem<a class="headerlink" href="#the-schroder-bernstein-theorem" title="Permalink to this heading"></a></h2> + We close this chapter with an elementary but nontrivial theorem of set theory. + Let $\alpha$ and $\beta$ be sets. + (In our formalization, they will actually be types.) + Suppose $f : \alpha → \beta$ and $g : \beta → \alpha$ + are both injective. + Intuitively, this means that $\alpha$ is no bigger than $\beta$ and vice-versa. + If $\alpha$ and $\beta$ are finite, this implies that + they have the same cardinality, which is equivalent to saying that there + is a bijection between them. + In the nineteenth century, Cantor stated that same result holds even in the + case where $\alpha$ and $\beta$ are infinite. + This was eventually established by Dedekind, Schröder, and Bernstein + independently. + Our formalization will introduce some new methods that we will explain + in greater detail in chapters to come. + Don’t worry if they go by too quickly here. + Our goal is to show you that you already have the skills to contribute + to the formal proof of a real mathematical result. + To understand the idea behind the proof, consider the image of the map + $g$ in $\alpha$. + On that image, the inverse of $g$ is defined and is a bijection + with $\beta$. + <a class="reference internal image-reference" href="_images/schroeder_bernstein1.png"><img alt="the Schröder Bernstein theorem" class="align-center" src="_images/schroeder_bernstein1.png" style="height: 150px;" /></a> + The problem is that the bijection does not include the shaded region + in the diagram, which is nonempty if $g$ is not surjective. + Alternatively, we can use $f$ to map all of + $\alpha$ to $\beta$, + but in that case the problem is that if $f$ is not surjective, + it will miss some elements of $\beta$. + <a class="reference internal image-reference" href="_images/schroeder_bernstein2.png"><img alt="the Schröder Bernstein theorem" class="align-center" src="_images/schroeder_bernstein2.png" style="height: 150px;" /></a> + But now consider the composition $g \circ f$ from $\alpha$ to + itself. Because the composition is injective, it forms a bijection between + $\alpha$ and its image, yielding a scaled-down copy of $\alpha$ + inside itself. + <a class="reference internal image-reference" href="_images/schroeder_bernstein3.png"><img alt="the Schröder Bernstein theorem" class="align-center" src="_images/schroeder_bernstein3.png" style="height: 150px;" /></a> + This composition maps the inner shaded ring to yet another such + set, which we can think of as an even smaller concentric shaded ring, + and so on. + This yields a + concentric sequence of shaded rings, each of which is in + bijective correspondence with the next. + If we map each ring to the next and leave the unshaded + parts of $\alpha$ alone, + we have a bijection of $\alpha$ with the image of $g$. + Composing with $g^{-1}$, this yields the desired + bijection between $\alpha$ and $\beta$. + We can describe this bijection more simply. + Let $A$ be the union of the sequence of shaded regions, and + define $h : \alpha \to \beta$ as follows: + <div class="math notranslate nohighlight"> + \[\begin{split}h(x) = \begin{cases} + f(x) & \text{if $x \in A$} \\ + g^{-1}(x) & \text{otherwise.} + \end{cases}\end{split}\]</div> + In other words, we use $f$ on the shaded parts, + and we use the inverse of $g$ everywhere else. + The resulting map $h$ is injective + because each component is injective + and the images of the two components are disjoint. + To see that it is surjective, + suppose we are given a $y$ in $\beta$, and + consider $g(y)$. + If $g(y)$ is in one of the shaded regions, + it cannot be in the first ring, so we have $g(y) = g(f(x))$ + for some $x$ is in the previous ring. + By the injectivity of $g$, we have $h(x) = f(x) = y$. + If $g(y)$ is not in the shaded region, + then by the definition of $h$, we have $h(g(y))= y$. + Either way, $y$ is in the image of $h$. + This argument should sound plausible, but the details are delicate. + Formalizing the proof will not only improve our confidence in the + result, but also help us understand it better. + Because the proof uses classical logic, we tell Lean that our definitions + will generally not be computable. + <div class="highlight-lean notranslate"><div class="highlight"><pre>noncomputable section + open Classical + variable {α β : Type _} [Nonempty β] + </pre></div> + </div> + The annotation <code class="docutils literal notranslate">[Nonempty β]</code> specifies that <code class="docutils literal notranslate">β</code> is nonempty. + We use it because the mathlib primitive that we will use to + construct $g^{-1}$ requires it. + The case of the theorem where $\beta$ is empty is trivial, + and even though it would not be hard to generalize the formalization to cover + that case as well, we will not bother. + Specifically, we need the hypothesis <code class="docutils literal notranslate">[Nonempty β]</code> for the operation + <code class="docutils literal notranslate">invFun</code> that is defined in mathlib. + Given <code class="docutils literal notranslate">x : α</code>, <code class="docutils literal notranslate">invFun g x</code> chooses a preimage of <code class="docutils literal notranslate">x</code> + in <code class="docutils literal notranslate">β</code> if there is one, + and returns an arbitrary element of <code class="docutils literal notranslate">β</code> otherwise. + The function <code class="docutils literal notranslate">invFun g</code> is always a left inverse if <code class="docutils literal notranslate">g</code> is injective + and a right inverse if <code class="docutils literal notranslate">g</code> is surjective. + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (invFun g : α → β) + #check (leftInverse_invFun : Injective g → LeftInverse (invFun g) g) + #check (leftInverse_invFun : Injective g → ∀ y, invFun g (g y) = y) + #check (invFun_eq : (∃ y, g y = x) → g (invFun g x) = x) + </pre></div> + </div> + We define the set corresponding to the union of the shaded regions as follows. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (f : α → β) (g : β → α) + + def sbAux : ℕ → Set α + | 0 => univ \ g '' univ + | n + 1 => g '' (f '' sbAux n) + + def sbSet := + ⋃ n, sbAux f g n + </pre></div> + </div> + The definition <code class="docutils literal notranslate">sb_aux</code> is an example of a recursive definition, + which we will explain in the next chapter. + It defines a sequence of sets + <div class="math notranslate nohighlight"> + \[\begin{split}S_0 &= \alpha ∖ g(\beta) \\ + S_{n+1} &= g(f(S_n)).\end{split}\]</div> + The definition <code class="docutils literal notranslate">sb_set</code> corresponds to the set + $A = \bigcup_{n \in \mathbb{N}} S_n$ in our proof sketch. + The function $h$ described above is now defined as follows: + <div class="highlight-lean notranslate"><div class="highlight"><pre>def sbFun (x : α) : β := + if x ∈ sbSet f g then f x else invFun g x + </pre></div> + </div> + We will need the fact that our definition of $g^{-1}$ is a + right inverse on the complement of $A$, + which is to say, on the non-shaded regions of $\alpha$. + This is so because the outermost ring, $S_0$, is equal to + $\alpha \setminus g(\beta)$, so the complement of $A$ is + contained in $g(\beta)$. + As a result, for every $x$ in the complement of $A$, + there is a $y$ such that $g(y) = x$. + (By the injectivity of $g$, this $y$ is unique, + but next theorem says only that <code class="docutils literal notranslate">inv_fun g x</code> returns some <code class="docutils literal notranslate">y</code> + such that <code class="docutils literal notranslate">g y = x</code>.) + Step through the proof below, make sure you understand what is going on, + and fill in the remaining parts. + You will need to use <code class="docutils literal notranslate">inv_fun_eq</code> at the end. + Notice that rewriting with <code class="docutils literal notranslate">sb_aux</code> here replaces <code class="docutils literal notranslate">sb_aux f g 0</code> + with the right-hand side of the corresponding defining equation. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem sb_right_inv {x : α} (hx : x ∉ sbSet f g) : g (invFun g x) = x := by + have : x ∈ g '' univ := by + contrapose! hx + rw [sbSet, mem_iUnion] + use 0 + rw [sbAux, mem_diff] + sorry + have : ∃ y, g y = x := by + sorry + sorry + </pre></div> + </div> + We now turn to the proof that $h$ is injective. + Informally, the proof goes as follows. + First, suppose $h(x_1) = h(x_2)$. + If $x_1$ is in $A$, then $h(x_1) = f(x_1)$, + and we can show that $x_2$ is in $A$ as follows. + If it isn’t, then we have $h(x_2) = g^{-1}(x_2)$. + From $f(x_1) = h(x_1) = h(x_2)$ we have $g(f(x_1)) = x_2$. + From the definition of $A$, since $x_1$ is in $A$, + $x_2$ is in $A$ as well, a contradiction. + Hence, if $x_1$ is in $A$, so is $x_2$, + in which case we have $f(x_1) = h(x_1) = h(x_2) = f(x_2)$. + The injectivity of $f$ then implies $x_1 = x_2$. + The symmetric argument shows that if $x_2$ is in $A$, + then so is $x_1$, which again implies $x_1 = x_2$. + The only remaining possibility is that neither $x_1$ nor $x_2$ + is in $A$. In that case, we have + $g^{-1}(x_1) = h(x_1) = h(x_2) = g^{-1}(x_2)$. + Applying $g$ to both sides yields $x_1 = x_2$. + Once again, we encourage you to step through the following proof + to see how the argument plays out in Lean. + See if you can finish off the proof using <code class="docutils literal notranslate">sb_right_inv</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem sb_injective (hf : Injective f) (hg : Injective g) : Injective (sbFun f g) := by + set A := sbSet f g with A_def + set h := sbFun f g with h_def + intro x₁ x₂ + intro (hxeq : h x₁ = h x₂) + show x₁ = x₂ + simp only [h_def, sbFun, ← A_def] at hxeq + by_cases xA : x₁ ∈ A ∨ x₂ ∈ A + · wlog x₁A : x₁ ∈ A generalizing x₁ x₂ hxeq xA + · symm + apply this hxeq.symm xA.symm (xA.resolve_left x₁A) + have x₂A : x₂ ∈ A := by + apply not_imp_self.mp + intro (x₂nA : x₂ ∉ A) + rw [if_pos x₁A, if_neg x₂nA] at hxeq + rw [A_def, sbSet, mem_iUnion] at x₁A + have x₂eq : x₂ = g (f x₁) := by + sorry + rcases x₁A with ⟨n, hn⟩ + rw [A_def, sbSet, mem_iUnion] + use n + 1 + simp [sbAux] + exact ⟨x₁, hn, x₂eq.symm⟩ + sorry + push_neg at xA + sorry + </pre></div> + </div> + The proof introduces some new tactics. + To start with, notice the <code class="docutils literal notranslate">set</code> tactic, which introduces abbreviations + <code class="docutils literal notranslate">A</code> and <code class="docutils literal notranslate">h</code> for <code class="docutils literal notranslate">sb_set f g</code> and <code class="docutils literal notranslate">sb_fun f g</code> respectively. + We name the corresponding defining equations <code class="docutils literal notranslate">A_def</code> and <code class="docutils literal notranslate">h_def</code>. + The abbreviations are definitional, which is to say, Lean will sometimes + unfold them automatically when needed. + But not always; for example, when using <code class="docutils literal notranslate">rw</code>, we generally need to + use <code class="docutils literal notranslate">A_def</code> and <code class="docutils literal notranslate">h_def</code> explicitly. + So the definitions bring a tradeoff: they can make expressions shorter + and more readable, but they sometimes require us to do more work. + A more interesting tactic is the <code class="docutils literal notranslate">wlog</code> tactic, which encapsulates + the symmetry argument in the informal proof above. + We will not dwell on it now, but notice that it does exactly what we want. + If you hover over the tactic you can take a look at its documentation. + The argument for surjectivity is even easier. + Given $y$ in $\beta$, + we consider two cases, depending on whether $g(y)$ is in $A$. + If it is, it can’t be in $S_0$, the outermost ring, + because by definition that is disjoint from the image of $g$. + Thus it is an element of $S_{n+1}$ for some $n$. + This means that it is of the form $g(f(x))$ for some + $x$ in $S_n$. + By the injectivity of $g$, we have $f(x) = y$. + In the case where $g(y)$ is in the complement of $A$, + we immediately have $h(g(y))= y$, and we are done. + Once again, we encourage you to step through the proof and fill in + the missing parts. + The tactic <code class="docutils literal notranslate">cases n with n</code> splits on the cases <code class="docutils literal notranslate">g y ∈ sb_aux f g 0</code> + and <code class="docutils literal notranslate">g y ∈ sb_aux f g n.succ</code>. + In both cases, calling the simplifier with <code class="docutils literal notranslate">simp [sb_aux]</code> + applies the corresponding defining equation of <code class="docutils literal notranslate">sb_aux</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem sb_surjective (hf : Injective f) (hg : Injective g) : Surjective (sbFun f g) := by + set A := sbSet f g with A_def + set h := sbFun f g with h_def + intro y + by_cases gyA : g y ∈ A + · rw [A_def, sbSet, mem_iUnion] at gyA + rcases gyA with ⟨n, hn⟩ + cases' n with n + · simp [sbAux] at hn + simp [sbAux] at hn + rcases hn with ⟨x, xmem, hx⟩ + use x + have : x ∈ A := by + rw [A_def, sbSet, mem_iUnion] + exact ⟨n, xmem⟩ + simp only [h_def, sbFun, if_pos this] + exact hg hx + sorry + </pre></div> + </div> + We can now put it all together. The final statement is short and sweet, + and the proof uses the fact that <code class="docutils literal notranslate">Bijective h</code> unfolds to + <code class="docutils literal notranslate">Injective h ∧ Surjective h</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem schroeder_bernstein {f : α → β} {g : β → α} (hf : Injective f) (hg : Injective g) : + ∃ h : α → β, Bijective h := + ⟨sbFun f g, sb_injective f g hf hg, sb_surjective f g hf hg⟩ + </pre></div> + </div> + </section> + </section> + + + </div> + </div> + <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> + <a href="C03_Logic.html" class="btn btn-neutral float-left" title="3. Logic" accesskey="p" rel="prev"> Previous</a> + <a href="C05_Number_Theory.html" class="btn btn-neutral float-right" title="5. Number Theory" accesskey="n" rel="next">Next </a> + </div> + + <hr/> + + <div role="contentinfo"> + © Copyright 2020, Jeremy Avigad, Kevin Buzzard, Robert Y. Lewis, Patrick Massot. + </div> + + Built with <a href="https://www.sphinx-doc.org/">Sphinx</a> using a + <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a> + provided by <a href="https://readthedocs.org">Read the Docs</a>. + + + </footer> + </div> + </div> + </section> + </div> + <script> + jQuery(function () { + SphinxRtdTheme.Navigation.enable(true); + }); + </script> + + </body> + </html>
html/C05_Number_Theory.html (new)
- @@ -0,0 +1,1181 @@
 + <!DOCTYPE html> + <html class="writer-html5" lang="en" > + <head> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + + <meta name="viewport" content="width=device-width, initial-scale=1.0" /> + <title>5. Number Theory — Mathematics in Lean 0.1 documentation</title> + <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="shortcut icon" href="_static/favicon.ico"/> +  + + <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> + <script src="_static/jquery.js"></script> + <script src="_static/underscore.js"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> + <script src="_static/doctools.js"></script> + <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> + <script src="_static/js/theme.js"></script> + <link rel="index" title="Index" href="genindex.html" /> + <link rel="search" title="Search" href="search.html" /> + <link rel="next" title="6. Abstract Algebra" href="C06_Abstract_Algebra.html" /> + <link rel="prev" title="4. Sets and Functions" href="C04_Sets_and_Functions.html" /> + </head> + + <body class="wy-body-for-nav"> + <div class="wy-grid-for-nav"> + <nav data-toggle="wy-nav-shift" class="wy-nav-side"> + <div class="wy-side-scroll"> + <div class="wy-side-nav-search" > + <a href="index.html" class="icon icon-home"> Mathematics in Lean + </a> + <div role="search"> + <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> + <input type="text" name="q" placeholder="Search docs" /> + <input type="hidden" name="check_keywords" value="yes" /> + <input type="hidden" name="area" value="default" /> + </form> + </div> + </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> + <ul class="current"> + <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a></li> + <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> + <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> + <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> + <li class="toctree-l1 current"><a class="current reference internal" href="#">5. Number Theory</a><ul> + <li class="toctree-l2"><a class="reference internal" href="#irrational-roots">5.1. Irrational Roots</a></li> + <li class="toctree-l2"><a class="reference internal" href="#induction-and-recursion">5.2. Induction and Recursion</a></li> + <li class="toctree-l2"><a class="reference internal" href="#id2">5.3. Induction and Recursion</a></li> + <li class="toctree-l2"><a class="reference internal" href="#infinitely-many-primes">5.4. Infinitely Many Primes</a></li> + </ul> + </li> + <li class="toctree-l1"><a class="reference internal" href="C06_Abstract_Algebra.html">6. Abstract Algebra</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Topology.html">7. Topology</a></li> + </ul> + <ul> + <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li> + </ul> + + </div> + </div> + </nav> + + <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > + + <a href="index.html">Mathematics in Lean</a> + </nav> + + <div class="wy-nav-content"> + <div class="rst-content"> + <div role="navigation" aria-label="Page navigation"> + <ul class="wy-breadcrumbs"> + <li><a href="index.html" class="icon icon-home"></a> »</li> + <li>5. Number Theory</li> + <li class="wy-breadcrumbs-aside"> + <a href="_sources/C05_Number_Theory.rst.txt" rel="nofollow"> View page source</a> + </li> + </ul> + <hr/> + </div> + <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> + <div itemprop="articleBody"> + + <section id="number-theory"> + <h1>5. Number Theory<a class="headerlink" href="#number-theory" title="Permalink to this heading"></a></h1> + In this chapter, we show you how to formalize some elementary + results in number theory. + As we deal with more substantive mathematical content, + the proofs will get longer and more involved, + building on the skills you have already mastered. + <section id="irrational-roots"> + <h2>5.1. Irrational Roots<a class="headerlink" href="#irrational-roots" title="Permalink to this heading"></a></h2> + Let’s start with a fact known to the ancient greeks, namely, + that the square root of 2 is irrational. + If we suppose otherwise, + we can write $\sqrt{2} = a / b$ as a fraction + in lowest terms. Squaring both sides yields $a^2 = 2 b^2$, + which implies that $a$ is even. + If we write $a = 2c$, then we get $4c^2 = 2 b^2$ + and hence $b^2 = 2 c^2$. + This implies that $b$ is also even, contradicting + the fact that we have assumed that $a / b$ has been + reduced to lowest terms. + Saying that $a / b$ is a fraction in lowest terms means + that $a$ and $b$ do not have any factors in common, + which is to say, they are coprime. + Mathlib defines the predicate <code class="docutils literal notranslate">Nat.coprime m n</code> to be <code class="docutils literal notranslate">Nat.gcd m n = 1</code>. + Using Lean’s anonymous projection notation, if <code class="docutils literal notranslate">s</code> and <code class="docutils literal notranslate">t</code> are + expressions of type <code class="docutils literal notranslate">Nat</code>, we can write <code class="docutils literal notranslate">s.coprime t</code> instead of + <code class="docutils literal notranslate">Nat.coprime s t</code>, and similarly for <code class="docutils literal notranslate">Nat.gcd</code>. + As usual, Lean will often unfold the definition of <code class="docutils literal notranslate">Nat.coprime</code> automatically + when necessary, + but we can also do it manually by rewriting or simplifying with + the identifier <code class="docutils literal notranslate">Nat.coprime</code>. + The <code class="docutils literal notranslate">norm_num</code> tactic is smart enough to compute concrete values. + <div class="highlight-lean notranslate"><div class="highlight"><pre>#print Nat.coprime + + example (m n : Nat) (h : m.coprime n) : m.gcd n = 1 := + h + + example (m n : Nat) (h : m.coprime n) : m.gcd n = 1 := by + rw [Nat.coprime] at h + exact h + + example : Nat.coprime 12 7 := by norm_num + + example : Nat.gcd 12 8 = 4 := by norm_num + </pre></div> + </div> + We have already encountered the <code class="docutils literal notranslate">gcd</code> function in + <a class="reference internal" href="C02_Basics.html#more-on-order-and-divisibility">Section 2.4</a>. + There is also a version of <code class="docutils literal notranslate">gcd</code> for the integers; + we will return to a discussion of the relationship between + different number systems below. + There are even a generic <code class="docutils literal notranslate">gcd</code> function and generic + notions of <code class="docutils literal notranslate">Prime</code> and <code class="docutils literal notranslate">coprime</code> + that make sense in general classes of algebraic structures. + We will come to understand how Lean manages this generality + in the next chapter. + In the meanwhile, in this section, we will restrict attention + to the natural numbers. + We also need the notion of a prime number, <code class="docutils literal notranslate">Nat.Prime</code>. + The theorem <code class="docutils literal notranslate">Nat.prime_def_lt</code> provides one familiar characterization, + and <code class="docutils literal notranslate">Nat.Prime.eq_one_or_self_of_dvd</code> provides another. + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check @Nat.prime_def_lt + + example (p : ℕ) (prime_p : Nat.Prime p) : 2 ≤ p ∧ ∀ m : ℕ, m < p → m ∣ p → m = 1 := by + rwa [Nat.prime_def_lt] at prime_p + + #check Nat.Prime.eq_one_or_self_of_dvd + + example (p : ℕ) (prime_p : Nat.Prime p) : ∀ m : ℕ, m ∣ p → m = 1 ∨ m = p := + prime_p.eq_one_or_self_of_dvd + + example : Nat.Prime 17 := by norm_num + + -- commonly used + example : Nat.Prime 2 := + Nat.prime_two + + example : Nat.Prime 3 := + Nat.prime_three + </pre></div> + </div> + In the natural numbers, a prime number has the property that it cannot + be written as a product of nontrivial factors. + In a broader mathematical context, an element of a ring that has this property + is said to be irreducible. + An element of a ring is said to be prime if whenever it divides a product, + it divides one of the factors. + It is an important property of the natural numbers + that in that setting the two notions coincide, + giving rise to the theorem <code class="docutils literal notranslate">Nat.Prime.dvd_mul</code>. + We can use this fact to establish a key property in the argument + above: + if the square of a number is even, then that number is even as well. + Mathlib defines the predicate <code class="docutils literal notranslate">Even</code> in <code class="docutils literal notranslate">Data.Nat.Parity</code>, + but for reasons that will become clear below, + we will simply use <code class="docutils literal notranslate">2 ∣ m</code> to express that <code class="docutils literal notranslate">m</code> is even. + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check @Nat.Prime.dvd_mul + #check Nat.Prime.dvd_mul Nat.prime_two + #check Nat.prime_two.dvd_mul + + theorem even_of_even_sqr {m : ℕ} (h : 2 ∣ m ^ 2) : 2 ∣ m := by + rw [pow_two, Nat.prime_two.dvd_mul] at h + cases h <;> assumption + + example {m : ℕ} (h : 2 ∣ m ^ 2) : 2 ∣ m := + Nat.Prime.dvd_of_dvd_pow Nat.prime_two h + </pre></div> + </div> + As we proceed, you will need to become proficient at finding the facts you + need. + Remember that if you can guess the prefix of the name and + you have imported the relevant library, + you can use tab completion (sometimes with <code class="docutils literal notranslate">ctrl-tab</code>) to find + what you are looking for. + You can use <code class="docutils literal notranslate">ctrl-click</code> on any identifier to jump to the file + where it is defined, which enables you to browse definitions and theorems + nearby. + You can also use the search engine on the + <a class="reference external" href="https://leanprover-community.github.io/">Lean community web pages</a>, + and if all else fails, + don’t hesitate to ask on + <a class="reference external" href="https://leanprover.zulipchat.com/">Zulip</a>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a b c : Nat) (h : a * b = a * c) (h' : a ≠ 0) : b = c := + -- library_search suggests the following: + (mul_right_inj' h').mp h + </pre></div> + </div> + The heart of our proof of the irrationality of the square root of two + is contained in the following theorem. + See if you can fill out the proof sketch, using + <code class="docutils literal notranslate">even_of_even_sqr</code> and the theorem <code class="docutils literal notranslate">Nat.dvd_gcd</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {m n : ℕ} (coprime_mn : m.coprime n) : m ^ 2 ≠ 2 * n ^ 2 := by + intro sqr_eq + have : 2 ∣ m := by + sorry + obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this + have : 2 * (2 * k ^ 2) = 2 * n ^ 2 := by + rw [← sqr_eq, meq] + ring + have : 2 * k ^ 2 = n ^ 2 := + sorry + have : 2 ∣ n := by + sorry + have : 2 ∣ m.gcd n := by + sorry + have : 2 ∣ 1 := by + sorry + norm_num at this + </pre></div> + </div> + In fact, with very few changes, we can replace <code class="docutils literal notranslate">2</code> by an arbitrary prime. + Give it a try in the next example. + At the end of the proof, you’ll need to derive a contradiction from + <code class="docutils literal notranslate">p ∣ 1</code>. + You can use <code class="docutils literal notranslate">Nat.Prime.two_le</code>, which says that + any prime number is greater than or equal to two, + and <code class="docutils literal notranslate">Nat.le_of_dvd</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {m n p : ℕ} (coprime_mn : m.coprime n) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by + sorry + </pre></div> + </div> + Let us consider another approach. + Here is a quick proof that if $p$ is prime, then + $m^2 \ne p n^2$: if we assume $m^2 = p n^2$ + and consider the factorization of $m$ and $n$ into primes, + then $p$ occurs an even number of times on the left side of the equation + and an odd number of times on the right, a contradiction. + Note that this argument requires that $n$ and hence $m$ + are not equal to zero. + The formalization below confirms that this assumption is sufficient. + The unique factorization theorem says that any natural number other + than zero can be written as the product of primes in a unique way. + Mathlib contains a formal version of this, expressed in terms of a function + <code class="docutils literal notranslate">Nat.factors</code>, which returns the list of + prime factors of a number in nondecreasing order. + The library proves that all the elements of <code class="docutils literal notranslate">Nat.factors n</code> + are prime, that any <code class="docutils literal notranslate">n</code> greater than zero is equal to the + product of its factors, + and that if <code class="docutils literal notranslate">n</code> is equal to the product of another list of prime numbers, + then that list is a permutation of <code class="docutils literal notranslate">Nat.factors n</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check Nat.factors + #check Nat.prime_of_mem_factors + #check Nat.prod_factors + #check Nat.factors_unique + </pre></div> + </div> + You can browse these theorems and others nearby, even though we have not + talked about list membership, products, or permutations yet. + We won’t need any of that for the task at hand. + We will instead use the fact that Mathlib has a function <code class="docutils literal notranslate">Nat.factorization</code>, + that represents the same data as a function. + Specifically, <code class="docutils literal notranslate">Nat.factorization n p</code>, which we can also write + <code class="docutils literal notranslate">n.factorization p</code>, returns the multiplicity of <code class="docutils literal notranslate">p</code> in the prime + factorization of <code class="docutils literal notranslate">n</code>. We will use the following three facts. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem factorization_mul' {m n : ℕ} (mnez : m ≠ 0) (nnez : n ≠ 0) (p : ℕ) : + (m * n).factorization p = m.factorization p + n.factorization p := by + rw [Nat.factorization_mul mnez nnez] + rfl + + theorem factorization_pow' (n k p : ℕ) : (n ^ k).factorization p = k * n.factorization p := by + rw [Nat.factorization_pow] + rfl + + theorem Nat.Prime.factorization' {p : ℕ} (prime_p : p.Prime) : p.factorization p = 1 := by + rw [prime_p.factorization] + simp + </pre></div> + </div> + In fact, <code class="docutils literal notranslate">n.factorization</code> is defined in Lean as a function of finite support, + which explains the strange notation you will see as you step through the + proofs above. Don’t worry about this now. For our purposes here, we can use + the three theorems above as a black box. + The next example shows that the simplifier is smart enough to replace + <code class="docutils literal notranslate">n^2 ≠ 0</code> by <code class="docutils literal notranslate">n ≠ 0</code>. The tactic <code class="docutils literal notranslate">simpa</code> just calls <code class="docutils literal notranslate">simp</code> + followed by <code class="docutils literal notranslate">assumption</code>. + See if you can use the identities above to fill in the missing parts + of the proof. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {m n p : ℕ} (nnz : n ≠ 0) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by + intro sqr_eq + have nsqr_nez : n ^ 2 ≠ 0 := by simpa + have eq1 : Nat.factorization (m ^ 2) p = 2 * m.factorization p := by + sorry + have eq2 : (p * n ^ 2).factorization p = 2 * n.factorization p + 1 := by + sorry + have : 2 * m.factorization p % 2 = (2 * n.factorization p + 1) % 2 := by rw [← eq1, sqr_eq, eq2] + rw [add_comm, Nat.add_mul_mod_self_left, Nat.mul_mod_right] at this + norm_num at this + </pre></div> + </div> + A nice thing about this proof is that it also generalizes. There is + nothing special about <code class="docutils literal notranslate">2</code>; with small changes, the proof shows that + whenever we write <code class="docutils literal notranslate">m^k = r * n^k</code>, the multiplicity of any prime <code class="docutils literal notranslate">p</code> + in <code class="docutils literal notranslate">r</code> has to be a multiple of <code class="docutils literal notranslate">k</code>. + To use <code class="docutils literal notranslate">Nat.count_factors_mul_of_pos</code> with <code class="docutils literal notranslate">r * n^k</code>, + we need to know that <code class="docutils literal notranslate">r</code> is positive. + But when <code class="docutils literal notranslate">r</code> is zero, the theorem below is trivial, and easily + proved by the simplifier. + So the proof is carried out in cases. + The line <code class="docutils literal notranslate">cases r with r</code> replaces the goal with two versions: + one in which <code class="docutils literal notranslate">r</code> is replaced by <code class="docutils literal notranslate">0</code>, + and the other in which <code class="docutils literal notranslate">r</code> is replaces by <code class="docutils literal notranslate">r.succ</code>, + the successor of <code class="docutils literal notranslate">r</code>. + In the second case, we can use the theorem <code class="docutils literal notranslate">r.succ_ne_zero</code>, which + establishes <code class="docutils literal notranslate">r.succ ≠ 0</code>. + Notice also that the line that begins <code class="docutils literal notranslate">have : npow_nz</code> provides a + short proof-term proof of <code class="docutils literal notranslate">n^k ≠ 0</code>. + To understand how it works, try replacing it with a tactic proof, + and then think about how the tactics describe the proof term. + See if you can fill in the missing parts of the proof below. + At the very end, you can use <code class="docutils literal notranslate">Nat.dvd_sub'</code> and <code class="docutils literal notranslate">Nat.dvd_mul_right</code> + to finish it off. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {m n k r : ℕ} (nnz : n ≠ 0) (pow_eq : m ^ k = r * n ^ k) {p : ℕ} (prime_p : p.Prime) : + k ∣ r.factorization p := by + cases' r with r + · simp + have npow_nz : n ^ k ≠ 0 := fun npowz => nnz (pow_eq_zero npowz) + have eq1 : (m ^ k).factorization p = k * m.factorization p := by + sorry + have eq2 : (r.succ * n ^ k).factorization p = k * n.factorization p + r.succ.factorization p := by + sorry + have : r.succ.factorization p = k * m.factorization p - k * n.factorization p := by + rw [← eq1, pow_eq, eq2, add_comm, Nat.add_sub_cancel] + rw [this] + sorry + </pre></div> + </div> + There are a number of ways in which we might want to improve on these results. + To start with, a proof that the square root of two is irrational + should say something about the square root of two, + which can be understood as an element of the + real or complex numbers. + And stating that it is irrational should say something about the + rational numbers, namely, that no rational number is equal to it. + Moreover, we should extend the theorems in this section to the integers. + Although it is mathematically obvious that if we could write the square root of + two as a quotient of two integers then we could write it as a quotient + of two natural numbers, + proving this formally requires some effort. + In Mathlib, the natural numbers, the integers, the rationals, the reals, + and the complex numbers are represented by separate data types. + Restricting attention to the separate domains is often helpful: + we will see that it is easy to do induction on the natural numbers, + and it is easiest to reason about divisibility of integers when the + real numbers are not part of the picture. + But having to mediate between the different domains is a headache, + one we will have to contend with. + We will return to this issue later in this chapter. + We should also expect to be able to strengthen the conclusion of the + last theorem to say that the number <code class="docutils literal notranslate">r</code> is a <code class="docutils literal notranslate">k</code>-th power, + since its <code class="docutils literal notranslate">k</code>-th root is just the product of each prime dividing <code class="docutils literal notranslate">r</code> + raised to its multiplicity in <code class="docutils literal notranslate">r</code> divided by <code class="docutils literal notranslate">k</code>. + To be able to do that we will need better means for reasoning about + products and sums over a finite set, + which is also a topic we will return to. + In fact, the results in this section are all established in much + greater generality in mathlib, + in <code class="docutils literal notranslate">Data.Real.Irrational</code>. + The notion of <code class="docutils literal notranslate">multiplicity</code> is defined for an + arbitrary commutative monoid, + and that it takes values in the extended natural numbers <code class="docutils literal notranslate">enat</code>, + which adds the value infinity to the natural numbers. + In the next chapter, we will begin to develop the means to + appreciate the way that Lean supports this sort of generality. + </section> + <section id="induction-and-recursion"> + <h2>5.2. Induction and Recursion<a class="headerlink" href="#induction-and-recursion" title="Permalink to this heading"></a></h2> + The set of natural numbers $\mathbb{N} = \{ 0, 1, 2, \ldots \}$ + is not only fundamentally important in its own right, + but also a plays a central role in the construction of new mathematical objects. + Lean’s foundation allows us to declare inductive types, + which are types generated inductively by a given list of + constructors. + In Lean, the natural numbers are declared as follows. + .. _section_induction_and_recursion: + </section> + <section id="id2"> + <h2>5.3. Induction and Recursion<a class="headerlink" href="#id2" title="Permalink to this heading"></a></h2> + The set of natural numbers $\mathbb{N} = \{ 0, 1, 2, \ldots \}$ + is not only fundamentally important in its own right, + but also a plays a central role in the construction of new mathematical objects. + Lean’s foundation allows us to declare inductive types, + which are types generated inductively by a given list of + constructors. + In Lean, the natural numbers are declared as follows. + <div class="highlight-lean notranslate"><div class="highlight"><pre>inductive Nat + | zero : Nat + | succ (n : Nat) : Nat + </pre></div> + </div> + You can find this in the library by writting <code class="docutils literal notranslate">#check Nat</code> and + then using <code class="docutils literal notranslate">ctrl-click</code> on the identifier <code class="docutils literal notranslate">Nat</code>. + The command specifies that <code class="docutils literal notranslate">Nat</code> is the datatype generated + freely and inductively by the two constructors <code class="docutils literal notranslate">zero : Nat</code> and + <code class="docutils literal notranslate">succ : Nat → Nat</code>. + Of course, the library introduces notation <code class="docutils literal notranslate">ℕ</code> and <code class="docutils literal notranslate">0</code> for + <code class="docutils literal notranslate">nat</code> and <code class="docutils literal notranslate">zero</code> respectively. (Numerals are translated to binary + representations, but we don’t have to worry about the details of that now.) + What “freely” means for the working mathematician is that the type + <code class="docutils literal notranslate">Nat</code> has an element <code class="docutils literal notranslate">zero</code> and an injective successor function + <code class="docutils literal notranslate">succ</code> whose image does not include <code class="docutils literal notranslate">zero</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (n : Nat) : n.succ ≠ Nat.zero := + Nat.succ_ne_zero n + + example (m n : Nat) (h : m.succ = n.succ) : m = n := + Nat.succ.inj h + </pre></div> + </div> + What the word “inductively” means for the working mathematician is that + the natural numbers comes with a principle of proof by induction + and a principle of definition by recursion. + This section will show you how to use these. + Here is an example of a recursive definition of the factorial + function. + <div class="highlight-lean notranslate"><div class="highlight"><pre>def fac : ℕ → ℕ + | 0 => 1 + | n + 1 => (n + 1) * fac n + </pre></div> + </div> + The syntax takes some getting used to. + Notice that there is no <code class="docutils literal notranslate">:=</code> on the first line. + The next two lines provide the base case and inductive step + for a recursive definition. + These equations hold definitionally, but they can also + be used manually by giving the name <code class="docutils literal notranslate">fac</code> to <code class="docutils literal notranslate">simp</code> or <code class="docutils literal notranslate">rw</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : fac 0 = 1 := + rfl + + example : fac 0 = 1 := by + rw [fac] + + example : fac 0 = 1 := by + simp [fac] + + example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := + rfl + + example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := by + rw [fac] + + example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := by + simp [fac] + </pre></div> + </div> + The factorial function is actually already defined in mathlib as + <code class="docutils literal notranslate">Nat.factorial</code>. Once again, you can jump to it by typing + <code class="docutils literal notranslate">#check Nat.factorial</code> and using <code class="docutils literal notranslate">ctrl-click.</code> + For illustrative purposes, we will continue using <code class="docutils literal notranslate">fac</code> in the examples. + The annotation <code class="docutils literal notranslate">@[simp]</code> before the definition + of <code class="docutils literal notranslate">Nat.factorial</code> specifies that + the defining equation should be added to the database of identities + that the simplifier uses by default. + The principle of induction says that we can prove a general statement + about the natural numbers by proving that the statement holds of 0 + and that whenever it holds of a natural number $n$, + it also holds of $n + 1$. + The line <code class="docutils literal notranslate">induction n with n ih</code> in the proof + below therefore results in two goals: + in the first we need to prove <code class="docutils literal notranslate">0 < fac 0</code>, + and in the second we have the added assumption <code class="docutils literal notranslate">ih : 0 < fac n</code> + and a required to prove <code class="docutils literal notranslate">0 < fac (n + 1)</code>. + The phrase <code class="docutils literal notranslate">with n ih</code> serves to name the variable and + the assumption for the inductive hypothesis, + and you can choose whatever names you want for them. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem fac_pos (n : ℕ) : 0 < fac n := by + induction' n with n ih + · rw [fac] + exact zero_lt_one + rw [fac] + exact mul_pos n.succ_pos ih + </pre></div> + </div> + The <code class="docutils literal notranslate">induction</code> tactic is smart enough to include hypotheses + that depend on the induction variable as part of the + induction hypothesis. + Step through the next example to see what is going on. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem dvd_fac {i n : ℕ} (ipos : 0 < i) (ile : i ≤ n) : i ∣ fac n := by + induction' n with n ih + · exact absurd ipos (not_lt_of_ge ile) + rw [fac] + cases' Nat.of_le_succ ile with h h + · apply dvd_mul_of_dvd_right (ih h) + rw [h] + apply dvd_mul_right + </pre></div> + </div> + The following example provides a crude lower bound for the factorial + function. + It turns out to be easier to start with a proof by cases, + so that the remainder of the proof starts with the case + $n = 1$. + See if you can complete the argument with a proof by induction. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem pow_two_le_fac (n : ℕ) : 2 ^ (n - 1) ≤ fac n := by + cases' n with n + · simp [fac] + sorry + </pre></div> + </div> + Induction is often used to prove identities involving finite sums and + products. + Mathlib defines the expressions <code class="docutils literal notranslate">Finset.sum s f</code> where + <code class="docutils literal notranslate">s : Finset α</code> if a finite set of elements of the type <code class="docutils literal notranslate">α</code> and + <code class="docutils literal notranslate">f</code> is a function defined on <code class="docutils literal notranslate">α</code>. + The codomain of <code class="docutils literal notranslate">f</code> can be any type that supports a commutative, + associative addition operation with a zero element. + If you import <code class="docutils literal notranslate">Algebra.BigOperators.Basic</code> and issue the command + <code class="docutils literal notranslate">open BigOperators</code>, you can use the more suggestive notation + <code class="docutils literal notranslate">∑ x in s, f x</code>. Of course, there is are an analogous operation and + notation for finite products. + We will talk about the <code class="docutils literal notranslate">Finset</code> type and the operations + it supports in the next section, and again in a later chapter. + For now, we will only make use + of <code class="docutils literal notranslate">Finset.range n</code>, which is the finite set of natural numbers + less than <code class="docutils literal notranslate">n</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} (s : Finset ℕ) (f : ℕ → ℕ) (n : ℕ) + + #check Finset.sum s f + #check Finset.prod s f + + open BigOperators + open Finset + + example : s.sum f = ∑ x in s, f x := + rfl + + example : s.prod f = ∏ x in s, f x := + rfl + + example : (range n).sum f = ∑ x in range n, f x := + rfl + + example : (range n).prod f = ∏ x in range n, f x := + rfl + </pre></div> + </div> + The facts <code class="docutils literal notranslate">Finset.sum_range_zero</code> and <code class="docutils literal notranslate">Finset.sum_range_succ</code> + provide a recursive description summation up to $n$, + and similarly for products. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (f : ℕ → ℕ) : (∑ x in range 0, f x) = 0 := + Finset.sum_range_zero f + + example (f : ℕ → ℕ) (n : ℕ) : (∑ x in range n.succ, f x) = (∑ x in range n, f x) + f n := + Finset.sum_range_succ f n + + example (f : ℕ → ℕ) : (∏ x in range 0, f x) = 1 := + Finset.prod_range_zero f + + example (f : ℕ → ℕ) (n : ℕ) : (∏ x in range n.succ, f x) = (∏ x in range n, f x) * f n := + Finset.prod_range_succ f n + </pre></div> + </div> + The first identity in each pair holds definitionally, which is to say, + you can replace the proofs by <code class="docutils literal notranslate">rfl</code>. + The following expresses the factorial function that we defined as a product. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (n : ℕ) : fac n = ∏ i in range n, (i + 1) := by + induction' n with n ih + · rw [fac, prod_range_zero] + rw [fac, ih, prod_range_succ, mul_comm] + </pre></div> + </div> + The fact that we include <code class="docutils literal notranslate">mul_comm</code> as a simplification rule deserves + comment. + It should seem dangerous to simplify with the identity <code class="docutils literal notranslate">x * y = y * x</code>, + which would ordinarily loop indefinitely. + Lean’s simplifier is smart enough to recognize that, and applies the rule + only in the case where the resulting term has a smaller value in some + fixed but arbitrary ordering of the terms. + The following example shows that simplifying using the three rules + <code class="docutils literal notranslate">mul_assoc</code>, <code class="docutils literal notranslate">mul_comm</code>, and <code class="docutils literal notranslate">mul_left_comm</code> + manages to identify products that are the same up to the + placement of parentheses and ordering of variables. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a b c d e f : ℕ) : a * (b * c * f * (d * e)) = d * (a * f * e) * (c * b) := by + simp [mul_assoc, mul_comm, mul_left_comm] + </pre></div> + </div> + Roughly, the rules work by pushing parentheses to the right + and then re-ordering the expressions on both sides until they + both follow the same canonical order. Simplifying with these + rules, and the corresponding rules for addition, is a handy trick. + Returning to summation identities, we suggest stepping through the following proof + that the sum of the natural numbers up to an including $n$ is + $n (n + 1) / 2$. + The first step of the proof clears the denominator. + This is generally useful when formalizing identities, + because calculations with division generally have side conditions. + (It is similarly useful to avoid using subtraction on the natural numbers when possible.) + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem sum_id (n : ℕ) : (∑ i in range (n + 1), i) = n * (n + 1) / 2 := by + symm; apply Nat.div_eq_of_eq_mul_right (by norm_num : 0 < 2) + induction' n with n ih + · simp + rw [Finset.sum_range_succ, mul_add 2, ← ih, Nat.succ_eq_add_one] + ring + </pre></div> + </div> + We encourage you to prove the analogous identity for sums of squares, + and other identities you can find on the web. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem sum_sqr (n : ℕ) : (∑ i in range (n + 1), i ^ 2) = n * (n + 1) * (2 * n + 1) / 6 := by + sorry + </pre></div> + </div> + In Lean’s core library, addition and multiplication are themselves defined + using recursive definitions, + and their fundamental properties are established using induction. + If you like thinking about foundational topics like that, + you might enjoy working through proofs + of the commutativity and associativity of multiplication and addition + and the distributivity of multiplication over addition. + You can do this on a copy of the natural numbers + following the outline below. + Notice that we can use the <code class="docutils literal notranslate">induction</code> tactic with <code class="docutils literal notranslate">MyNat</code>; + Lean is smart enough to know to + use the relevant induction principle (which is, of course, + the same as that for <code class="docutils literal notranslate">Nat</code>). + We start you off with the commutativity of addition. + A good rule of thumb is that because addition and multiplication + are defined by recursion on the second argument, + it is generally advantageous to do proofs by induction on a variable + that occurs in that position. + It is a bit tricky to decide which variable to use in the proof + of associativity. + It can be confusing to write things without the usual notation + for zero, one, addition, and multiplication. + We will learn how to define such notation later. + Working in the namespace <code class="docutils literal notranslate">MyNat</code> means that we can write + <code class="docutils literal notranslate">zero</code> and <code class="docutils literal notranslate">succ</code> rather than <code class="docutils literal notranslate">MyNat.zero</code> and <code class="docutils literal notranslate">MyNat.succ</code>, + and that these interpretations of the names take precedence over + others. + Outside the namespace, the full name of the <code class="docutils literal notranslate">add</code> defined below, + for example, is <code class="docutils literal notranslate">MyNat.add</code>. + If you find that you really enjoy this sort of thing, try defining + truncated subtraction and exponentiation and proving some of their + properties as well. + Remember that truncated subtraction cuts off at zero. + To define that, it is useful to define a predecessor function, <code class="docutils literal notranslate">pred</code>, + that subtracts one from any nonzero number and fixes zero. + The function <code class="docutils literal notranslate">pred</code> can be defined by a simple instance of recursion. + <div class="highlight-lean notranslate"><div class="highlight"><pre>inductive MyNat + | zero : MyNat + | succ : MyNat → MyNat + + namespace MyNat + + def add : MyNat → MyNat → MyNat + | x, zero => x + | x, succ y => succ (add x y) + + def mul : MyNat → MyNat → MyNat + | x, zero => zero + | x, succ y => add (mul x y) x + + theorem zero_add (n : MyNat) : add zero n = n := by + induction' n with n ih + · rfl + rw [add, ih] + + theorem succ_add (m n : MyNat) : add (succ m) n = succ (add m n) := by + induction' n with n ih + · rfl + rw [add, ih] + rfl + + theorem add_comm (m n : MyNat) : add m n = add n m := by + induction' n with n ih + · rw [zero_add] + rfl + rw [add, succ_add, ih] + + theorem add_assoc (m n k : MyNat) : add (add m n) k = add m (add n k) := by + sorry + theorem mul_add (m n k : MyNat) : mul m (add n k) = add (mul m n) (mul m k) := by + sorry + theorem zero_mul (n : MyNat) : mul zero n = zero := by + sorry + theorem succ_mul (m n : MyNat) : mul (succ m) n = add (mul m n) n := by + sorry + theorem mul_comm (m n : MyNat) : mul m n = mul n m := by + sorry + end MyNat + </pre></div> + </div> + </section> + <section id="infinitely-many-primes"> + <h2>5.4. Infinitely Many Primes<a class="headerlink" href="#infinitely-many-primes" title="Permalink to this heading"></a></h2> + Let us continue our exploration of induction and recursion with another + mathematical standard: a proof that there are infinitely many primes. + One way to formulate this is as the statement that + for every natural number + $n$, there is a prime number greater than $n$. + To prove this, let $p$ be any prime factor of $n! + 1$. + If $p$ is less than $n$, it divides $n!$. + Since it also divides $n! + 1$, it divides 1, a contradiction. + Hence $p$ is greater than $n$. + To formalize that proof, we need to show that any number greater than or equal + to 2 has a prime factor. + To do that, we will need to show that any natural number that is + not equal to 0 or 1 is greater-than or equal to 2. + And this brings us to a quirky feature of formalization: + it is often trivial statements like this that are among the most + annoying to formalize. + Here we consider a few ways to do it. + To start with, we can use the <code class="docutils literal notranslate">cases</code> tactic and the fact that the + successor function respects the ordering on the natural numbers. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem two_le {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by + cases m; contradiction + case succ m => + cases m; contradiction + repeat' apply Nat.succ_le_succ + apply zero_le + </pre></div> + </div> + Another strategy is to use the tactic <code class="docutils literal notranslate">interval_cases</code>, + which automatically splits the goal into cases when + the variable in question is contained in an interval + of natural numbers or integers. + Remember that you can hover over it to see its documentation. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by + by_contra h + push_neg at h + interval_cases m <;> contradiction + </pre></div> + </div> + Recall that the semicolon after <code class="docutils literal notranslate">interval_cases m</code> means + that the next tactic is applied to each of the cases that it generates. + Yet another option is to use the tactic, <code class="docutils literal notranslate">decide</code>, which tries + to find a decision procedure to solve the problem. + Lean knows that you can decide the truth value of a statement that + begins with a bounded quantifier <code class="docutils literal notranslate">∀ x, x < n → ...</code> or <code class="docutils literal notranslate">∃ x, x < n ∧ ...</code> + by deciding each of the finitely many instances. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by + by_contra h + push_neg at h + revert h0 h1 + revert h m + decide + </pre></div> + </div> + With the theorem <code class="docutils literal notranslate">two_le</code> in hand, let’s start by showing that every + natural number greater than two has a prime divisor. + Mathlib contains a function <code class="docutils literal notranslate">Nat.minFac</code> that + returns the smallest prime divisor, + but for the sake of learning new parts of the library, + we’ll avoid using it and prove the theorem directly. + Here, ordinary induction isn’t enough. + We want to use strong induction, which allows us to prove + that every natural number $n$ has a property $P$ + by showing that for every number $n$, if $P$ holds + of all values less than $n$, it holds at $n$ as well. + In Lean, this principle is called <code class="docutils literal notranslate">Nat.strong_induction_on</code>, + and we can use the <code class="docutils literal notranslate">with</code> keyword to tell the induction tactic + to use it. + Notice that when we do that, there is no base case; it is subsumed + by the general induction step. + The argument is simply as follows. Assuming $n ≥ 2$, + if $n$ is prime, we’re done. If it isn’t, + then by one of the characterizations of what it means to be a prime number, + it has a nontrivial factor, $m$, + and we can apply the inductive hypothesis to that. + Step through the next proof to see how that plays out. + The line <code class="docutils literal notranslate">dsimp at ih</code> simplifies the expression of the + inductive hypothesis to make it more readable. + The proof still works if you delete that line. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem exists_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ p : Nat, p.Prime ∧ p ∣ n := by + by_cases np : n.Prime + · use n, np + induction' n using Nat.strong_induction_on with n ih + dsimp at ih + rw [Nat.prime_def_lt] at np + push_neg at np + rcases np h with ⟨m, mltn, mdvdn, mne1⟩ + have : m ≠ 0 := by + intro mz + rw [mz, zero_dvd_iff] at mdvdn + linarith + have mgt2 : 2 ≤ m := two_le this mne1 + by_cases mp : m.Prime + · use m, mp + exact mdvdn + . rcases ih m mltn mgt2 mp with ⟨p, pp, pdvd⟩ + use p, pp + apply pdvd.trans mdvdn + </pre></div> + </div> + We can now prove the following formulation of our theorem. + See if you can fill out the sketch. + You can use <code class="docutils literal notranslate">Nat.factorial_pos</code>, <code class="docutils literal notranslate">Nat.dvd_factorial</code>, + and <code class="docutils literal notranslate">Nat.dvd_sub</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem primes_infinite : ∀ n, ∃ p > n, Nat.Prime p := by + intro n + have : 2 ≤ Nat.factorial (n + 1) + 1 := by + sorry + rcases exists_prime_factor this with ⟨p, pp, pdvd⟩ + refine' ⟨p, _, pp⟩ + show p > n + by_contra ple + push_neg at ple + have : p ∣ Nat.factorial (n + 1) := by + sorry + have : p ∣ 1 := by + sorry + show False + sorry + </pre></div> + </div> + Let’s consider a variation of the proof above, where instead + of using the factorial function, + we suppose that we are given by a finite set + $\{ p_1, \ldots, p_n \}$ and we consider a prime factor of + $\prod_{i = 1}^n p_i + 1$. + That prime factor has to be distinct from each + $p_i$, showing that there is no finite set that contains + all the prime numbers. + Formalizing this argument requires us to reason about finite + sets. In Lean, for any type <code class="docutils literal notranslate">α</code>, the type <code class="docutils literal notranslate">Finset α</code> + represents finite sets of elements of type <code class="docutils literal notranslate">α</code>. + Reasoning about finite sets computationally requires having + a procedure to test equality on <code class="docutils literal notranslate">α</code>, which is why the snippet + below includes the assumption <code class="docutils literal notranslate">[DecidableEq α]</code>. + For concrete data types like <code class="docutils literal notranslate">ℕ</code>, <code class="docutils literal notranslate">ℤ</code>, and <code class="docutils literal notranslate">ℚ</code>, + the assumption is satisfied automatically. When reasoning about + the real numbers, it can be satisfied using classical logic + and abandoning the computational interpretation. + We use the command <code class="docutils literal notranslate">open Finset</code> to avail ourselves of shorter names + for the relevant theorems. Unlike the case with sets, + most equivalences involving finsets do not hold definitionally, + so they need to be expanded manually using equivalances like + <code class="docutils literal notranslate">Finset.subset_iff</code>, <code class="docutils literal notranslate">Finset.mem_union</code>, <code class="docutils literal notranslate">Finset.mem_inter</code>, + and <code class="docutils literal notranslate">Finset.mem_sdiff</code>. The <code class="docutils literal notranslate">ext</code> tactic can still be used + to reduce show that two finite sets are equal by showing + that every element of one is an element of the other. + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Finset + + section + variable {α : Type _} [DecidableEq α] (r s t : Finset α) + + example : r ∩ (s ∪ t) ⊆ r ∩ s ∪ r ∩ t := by + rw [subset_iff] + intro x + rw [mem_inter, mem_union, mem_union, mem_inter, mem_inter] + tauto + + example : r ∩ (s ∪ t) ⊆ r ∩ s ∪ r ∩ t := by + simp [subset_iff] + intro x + tauto + + example : r ∩ s ∪ r ∩ t ⊆ r ∩ (s ∪ t) := by + simp [subset_iff] + intro x + tauto + + example : r ∩ s ∪ r ∩ t = r ∩ (s ∪ t) := by + ext x + simp + tauto + + end + </pre></div> + </div> + We have used a new trick: the <code class="docutils literal notranslate">tauto</code> tactic (and a strengthened + version, <code class="docutils literal notranslate">tauto!</code>, which uses classical logic) can be used to + dispense with propositional tautologies. See if you can use + these methods to prove the two examples below. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : (r ∪ s) ∩ (r ∪ t) = r ∪ s ∩ t := by + sorry + example : (r \ s) \ t = r \ (s ∪ t) := by + sorry + </pre></div> + </div> + The theorem <code class="docutils literal notranslate">Finset.dvd_prod_of_mem</code> tells us that if an + <code class="docutils literal notranslate">n</code> is an element of a finite set <code class="docutils literal notranslate">s</code>, then <code class="docutils literal notranslate">n</code> divides + <code class="docutils literal notranslate">∏ i in s, i</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (s : Finset ℕ) (n : ℕ) (h : n ∈ s) : n ∣ ∏ i in s, i := + Finset.dvd_prod_of_mem _ h + </pre></div> + </div> + We also need to know that the converse holds in the case where + <code class="docutils literal notranslate">n</code> is prime and <code class="docutils literal notranslate">s</code> is a set of primes. + To show that, we need the following lemma, which you should + be able to prove using the theorem <code class="docutils literal notranslate">Nat.Prime.eq_one_or_self_of_dvd</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem Nat.Prime.eq_of_dvd_of_prime {p q : ℕ} (prime_p : Nat.Prime p) (prime_q : Nat.Prime q) + (h : p ∣ q) : p = q := by + sorry + </pre></div> + </div> + We can use this lemma to show that if a prime <code class="docutils literal notranslate">p</code> divides a product of a finite + set of primes, then it divides one of them. + Mathlib provides a useful principle of induction on finite sets: + to show that a property holds of an arbitrary finite set <code class="docutils literal notranslate">s</code>, + show that it holds of the empty set, and show that it is preserved + when we add a single new element <code class="docutils literal notranslate">a ∉ s</code>. + The principle is known as <code class="docutils literal notranslate">Finset.induction_on</code>. + When we tell the induction tactic to use it, we can also specify the names + <code class="docutils literal notranslate">a</code> and <code class="docutils literal notranslate">s</code>, the name for the assumption <code class="docutils literal notranslate">a ∉ s</code> in the inductive step, + and the name of the inductive hypothesis. + The expression <code class="docutils literal notranslate">Finset.insert a s</code> denotes the union of <code class="docutils literal notranslate">s</code> with the singleton <code class="docutils literal notranslate">a</code>. + The identities <code class="docutils literal notranslate">Finset.prod_empty</code> and <code class="docutils literal notranslate">Finset.prod_insert</code> then provide + the relevant rewrite rules for the product. + In the proof below, the first <code class="docutils literal notranslate">simp</code> applies <code class="docutils literal notranslate">Finset.prod_empty</code>. + Step through the beginning of the proof to see the induction unfold, + and then finish it off. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem mem_of_dvd_prod_primes {s : Finset ℕ} {p : ℕ} (prime_p : p.Prime) : + (∀ n ∈ s, Nat.Prime n) → (p ∣ ∏ n in s, n) → p ∈ s := by + intro h₀ h₁ + induction' s using Finset.induction_on with a s ans ih + · simp at h₁ + linarith [prime_p.two_le] + simp [Finset.prod_insert ans, prime_p.dvd_mul] at h₀ h₁ + rw [mem_insert] + sorry + </pre></div> + </div> + We need one last property of finite sets. + Given an element <code class="docutils literal notranslate">s : Set α</code> and a predicate + <code class="docutils literal notranslate">P</code> on <code class="docutils literal notranslate">α</code>, in <a class="reference internal" href="C04_Sets_and_Functions.html#sets-and-functions">Chapter 4</a> + we wrote <code class="docutils literal notranslate">{ x ∈ s | P x }</code> for the set of + elements of <code class="docutils literal notranslate">s</code> that satisfy <code class="docutils literal notranslate">P</code>. + Given <code class="docutils literal notranslate">s : Finset α</code>, + the analogous notion is written <code class="docutils literal notranslate">s.filter P</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (s : Finset ℕ) (x : ℕ) : x ∈ s.filter Nat.Prime ↔ x ∈ s ∧ x.Prime := + mem_filter + </pre></div> + </div> + We now prove an alternative formulation of the statement that there are infinitely many + primes, namely, that given any <code class="docutils literal notranslate">s : Finset ℕ</code>, there is a prime <code class="docutils literal notranslate">p</code> that is not + an element of <code class="docutils literal notranslate">s</code>. + Aiming for a contradiction, we assume that all the primes are in <code class="docutils literal notranslate">s</code>, and then + cut down to a set <code class="docutils literal notranslate">s'</code> that contains all and only the primes. + Taking the product of that set, adding one, and finding a prime factor + of the result + leads to the contradiction we are looking for. + See if you can complete the sketch below. + You can use <code class="docutils literal notranslate">Finset.prod_pos</code> in the proof of the first <code class="docutils literal notranslate">have</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem primes_infinite' : ∀ s : Finset Nat, ∃ p, Nat.Prime p ∧ p ∉ s := by + intro s + by_contra h + push_neg at h + set s' := s.filter Nat.Prime with s'_def + have mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ n.Prime := by + intro n + simp [s'_def] + apply h + have : 2 ≤ (∏ i in s', i) + 1 := by + sorry + rcases exists_prime_factor this with ⟨p, pp, pdvd⟩ + have : p ∣ ∏ i in s', i := by + sorry + have : p ∣ 1 := by + convert Nat.dvd_sub' pdvd this + simp + show False + sorry + </pre></div> + </div> + We have thus seen two ways of saying that there are infinitely many primes: + saying that they are not bounded by any <code class="docutils literal notranslate">n</code>, and saying that they are + not contained in any finite set <code class="docutils literal notranslate">s</code>. + The two proofs below show that these formulations are equivalent. + In the second, in order to form <code class="docutils literal notranslate">s.filter Q</code>, we have to assume that there + is a procedure for deciding whether or not <code class="docutils literal notranslate">Q</code> holds. Lean knows that there + is a procedure for <code class="docutils literal notranslate">Nat.Prime</code>. In general, if we use classical logic + by writing <code class="docutils literal notranslate">open Classical</code>, + we can dispense with the assumption. + In mathlib, <code class="docutils literal notranslate">Finset.sup s f</code> denotes the supremum of the values of <code class="docutils literal notranslate">f x</code> as <code class="docutils literal notranslate">x</code> + ranges over <code class="docutils literal notranslate">s</code>, returning <code class="docutils literal notranslate">0</code> in the case where <code class="docutils literal notranslate">s</code> is empty and + the codomain of <code class="docutils literal notranslate">f</code> is <code class="docutils literal notranslate">ℕ</code>. In the first proof, we use <code class="docutils literal notranslate">s.sup id</code>, + where <code class="docutils literal notranslate">id</code> is the identity function, to refer to the maximum value in <code class="docutils literal notranslate">s</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem bounded_of_ex_finset (Q : ℕ → Prop) : + (∃ s : Finset ℕ, ∀ k, Q k → k ∈ s) → ∃ n, ∀ k, Q k → k < n := by + rintro ⟨s, hs⟩ + use s.sup id + 1 + intro k Qk + apply Nat.lt_succ_of_le + show id k ≤ s.sup id + apply le_sup (hs k Qk) + + theorem ex_finset_of_bounded (Q : ℕ → Prop) [DecidablePred Q] : + (∃ n, ∀ k, Q k → k ≤ n) → ∃ s : Finset ℕ, ∀ k, Q k ↔ k ∈ s := by + rintro ⟨n, hn⟩ + use (range (n + 1)).filter Q + intro k + simp [Nat.lt_succ_iff] + exact hn k + </pre></div> + </div> + A small variation on our second proof that there are infinitely many primes + shows that there are infinitely many primes congruent to 3 modulo 4. + The argument goes as follows. + First, notice that if the product of two numbers $m$ and $n$ + is equal to 3 modulo 4, then one of the two numbers is congruent to three modulo 4. + After all, both have to be odd, and if they are both congruent to 1 modulo 4, + so is their product. + We can use this observation to show that if some number + greater than 2 is congruent to 3 modulo 4, + then that number has a prime divisor that is also congruent to 3 modulo 4. + Now suppose there are only finitely many prime numbers congruent to 3 + modulo 4, say, $p_1, \ldots, p_k$. + Without loss of generality, we can assume that $p_1 = 3$. + Consider the product $4 \prod_{i = 2}^k p_i + 3$. + It is easy to see that this is congruent to 3 modulo 4, so it has + a prime factor $p$ congruent to 3 modulo 4. + It can’t be the case that $p = 3$; since $p$ + divides $4 \prod_{i = 2}^k p_i + 3$, if $p$ + were equal to 3 then it would also divide $\prod_{i = 2}^k p_i$, + which implies that $p$ is equal to + one of the $p_i$ for $i = 2, \ldots, k$; + and we have excluded 3 from this list. + So $p$ has to be one of the other elements $p_i$. + But in that case, $p$ divides $4 \prod_{i = 2}^k p_i$ + and hence 3, which contradicts the fact that it is not 3. + In Lean, the notation <code class="docutils literal notranslate">n % m</code>, read “<code class="docutils literal notranslate">n</code> modulo <code class="docutils literal notranslate">m</code>,” + denotes the remainder of the division of <code class="docutils literal notranslate">n</code> by <code class="docutils literal notranslate">m</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : 27 % 4 = 3 := by norm_num + </pre></div> + </div> + We can then render the statement “<code class="docutils literal notranslate">n</code> is congruent to 3 modulo 4” + as <code class="docutils literal notranslate">n % 4 = 3</code>. The following example and theorems sum up + the facts about this function that we will need to use below. + The first named theorem is another illustration of reasoning by + a small number of cases. + In the second named theorem, remember that the semicolon means that + the subsequent tactic block is applied to both of the goals + that result from the application of <code class="docutils literal notranslate">two_le</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (n : ℕ) : (4 * n + 3) % 4 = 3 := by + rw [add_comm, Nat.add_mul_mod_self_left] + norm_num + + theorem mod_4_eq_3_or_mod_4_eq_3 {m n : ℕ} (h : m * n % 4 = 3) : m % 4 = 3 ∨ n % 4 = 3 := by + revert h + rw [Nat.mul_mod] + have : m % 4 < 4 := Nat.mod_lt m (by norm_num) + interval_cases hm : m % 4 <;> simp [hm] + have : n % 4 < 4 := Nat.mod_lt n (by norm_num) + interval_cases hn : n % 4 <;> simp [hn] + + theorem two_le_of_mod_4_eq_3 {n : ℕ} (h : n % 4 = 3) : 2 ≤ n := by + apply two_le <;> + · intro neq + rw [neq] at h + norm_num at h + </pre></div> + </div> + We will also need the following fact, which says that if + <code class="docutils literal notranslate">m</code> is a nontrivial divisor of <code class="docutils literal notranslate">n</code>, then so is <code class="docutils literal notranslate">n / m</code>. + See if you can complete the proof using <code class="docutils literal notranslate">Nat.div_dvd_of_dvd</code> + and <code class="docutils literal notranslate">Nat.div_lt_self</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem aux {m n : ℕ} (h₀ : m ∣ n) (h₁ : 2 ≤ m) (h₂ : m < n) : n / m ∣ n ∧ n / m < n := by + sorry + </pre></div> + </div> + Now put all the pieces together to prove that any + number congruent to 3 modulo 4 has a prime divisor with that + same property. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem exists_prime_factor_mod_4_eq_3 {n : Nat} (h : n % 4 = 3) : + ∃ p : Nat, p.Prime ∧ p ∣ n ∧ p % 4 = 3 := by + by_cases np : n.Prime + · use n + exact ⟨np, dvd_rfl, h⟩ + induction' n using Nat.strong_induction_on with n ih + dsimp at ih + rw [Nat.prime_def_lt] at np + push_neg at np + rcases np (two_le_of_mod_4_eq_3 h) with ⟨m, mltn, mdvdn, mne1⟩ + have mge2 : 2 ≤ m := by + apply two_le _ mne1 + intro mz + rw [mz, zero_dvd_iff] at mdvdn + linarith + have neq : m * (n / m) = n := Nat.mul_div_cancel' mdvdn + have : m % 4 = 3 ∨ n / m % 4 = 3 := by + apply mod_4_eq_3_or_mod_4_eq_3 + rw [neq, h] + cases' this with h1 h1 + . sorry + sorry + </pre></div> + </div> + We are in the home stretch. Given a set <code class="docutils literal notranslate">s</code> of prime + numbers, we need to talk about the result of removing 3 from that + set, if it is present. The function <code class="docutils literal notranslate">Finset.erase</code> handles that. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (m n : ℕ) (s : Finset ℕ) (h : m ∈ erase s n) : m ≠ n ∧ m ∈ s := by + rwa [mem_erase] at h + + example (m n : ℕ) (s : Finset ℕ) (h : m ∈ erase s n) : m ≠ n ∧ m ∈ s := by + simp at h + assumption + </pre></div> + </div> + We are now ready to prove that there are infinitely many primes + congruent to 3 modulo 4. + Fill in the missing parts below. + Our solution uses <code class="docutils literal notranslate">Nat.dvd_add_iff_left</code> and <code class="docutils literal notranslate">Nat.dvd_sub'</code> + along the way. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem primes_mod_4_eq_3_infinite : ∀ n, ∃ p > n, Nat.Prime p ∧ p % 4 = 3 := by + by_contra h + push_neg at h + cases' h with n hn + have : ∃ s : Finset Nat, ∀ p : ℕ, p.Prime ∧ p % 4 = 3 ↔ p ∈ s := by + apply ex_finset_of_bounded + use n + contrapose! hn + rcases hn with ⟨p, ⟨pp, p4⟩, pltn⟩ + exact ⟨p, pltn, pp, p4⟩ + cases' this with s hs + have h₁ : ((4 * ∏ i in erase s 3, i) + 3) % 4 = 3 := by + sorry + rcases exists_prime_factor_mod_4_eq_3 h₁ with ⟨p, pp, pdvd, p4eq⟩ + have ps : p ∈ s := by + sorry + have pne3 : p ≠ 3 := by + sorry + have : p ∣ 4 * ∏ i in erase s 3, i := by + sorry + have : p ∣ 3 := by + sorry + have : p = 3 := by + sorry + contradiction + </pre></div> + </div> + If you managed to complete the proof, congratulations! This has been a serious + feat of formalization. + </section> + </section> + + + </div> + </div> + <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> + <a href="C04_Sets_and_Functions.html" class="btn btn-neutral float-left" title="4. Sets and Functions" accesskey="p" rel="prev"> Previous</a> + <a href="C06_Abstract_Algebra.html" class="btn btn-neutral float-right" title="6. Abstract Algebra" accesskey="n" rel="next">Next </a> + </div> + + <hr/> + + <div role="contentinfo"> + © Copyright 2020, Jeremy Avigad, Kevin Buzzard, Robert Y. Lewis, Patrick Massot. + </div> + + Built with <a href="https://www.sphinx-doc.org/">Sphinx</a> using a + <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a> + provided by <a href="https://readthedocs.org">Read the Docs</a>. + + + </footer> + </div> + </div> + </section> + </div> + <script> + jQuery(function () { + SphinxRtdTheme.Navigation.enable(true); + }); + </script> + + </body> + </html>
html/C06_Abstract_Algebra.html (new)
- @@ -0,0 +1,1565 @@
 + <!DOCTYPE html> + <html class="writer-html5" lang="en" > + <head> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + + <meta name="viewport" content="width=device-width, initial-scale=1.0" /> + <title>6. Abstract Algebra — Mathematics in Lean 0.1 documentation</title> + <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="shortcut icon" href="_static/favicon.ico"/> +  + + <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> + <script src="_static/jquery.js"></script> + <script src="_static/underscore.js"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> + <script src="_static/doctools.js"></script> + <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> + <script src="_static/js/theme.js"></script> + <link rel="index" title="Index" href="genindex.html" /> + <link rel="search" title="Search" href="search.html" /> + <link rel="next" title="7. Topology" href="C07_Topology.html" /> + <link rel="prev" title="5. Number Theory" href="C05_Number_Theory.html" /> + </head> + + <body class="wy-body-for-nav"> + <div class="wy-grid-for-nav"> + <nav data-toggle="wy-nav-shift" class="wy-nav-side"> + <div class="wy-side-scroll"> + <div class="wy-side-nav-search" > + <a href="index.html" class="icon icon-home"> Mathematics in Lean + </a> + <div role="search"> + <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> + <input type="text" name="q" placeholder="Search docs" /> + <input type="hidden" name="check_keywords" value="yes" /> + <input type="hidden" name="area" value="default" /> + </form> + </div> + </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> + <ul class="current"> + <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a></li> + <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> + <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> + <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> + <li class="toctree-l1 current"><a class="current reference internal" href="#">6. Abstract Algebra</a><ul> + <li class="toctree-l2"><a class="reference internal" href="#structures">6.1. Structures</a></li> + <li class="toctree-l2"><a class="reference internal" href="#algebraic-structures">6.2. Algebraic Structures</a></li> + <li class="toctree-l2"><a class="reference internal" href="#building-the-gaussian-integers">6.3. Building the Gaussian Integers</a></li> + </ul> + </li> + <li class="toctree-l1"><a class="reference internal" href="C07_Topology.html">7. Topology</a></li> + </ul> + <ul> + <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li> + </ul> + + </div> + </div> + </nav> + + <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > + + <a href="index.html">Mathematics in Lean</a> + </nav> + + <div class="wy-nav-content"> + <div class="rst-content"> + <div role="navigation" aria-label="Page navigation"> + <ul class="wy-breadcrumbs"> + <li><a href="index.html" class="icon icon-home"></a> »</li> + <li>6. Abstract Algebra</li> + <li class="wy-breadcrumbs-aside"> + <a href="_sources/C06_Abstract_Algebra.rst.txt" rel="nofollow"> View page source</a> + </li> + </ul> + <hr/> + </div> + <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> + <div itemprop="articleBody"> + + <section id="abstract-algebra"> + <h1>6. Abstract Algebra<a class="headerlink" href="#abstract-algebra" title="Permalink to this heading"></a></h1> + Modern mathematics makes essential use of algebraic + structures, + which encapsulate patterns that can be instantiated in + multiple settings. + The subject provides various ways of defining such structures and + constructing particular instances. + Lean therefore provides corresponding ways of + defining structures formally and working with them. + You have already seen examples of algebraic structures in Lean, + such as rings and lattices, which were discussed in + <a class="reference internal" href="C02_Basics.html#basics">Chapter 2</a>. + This chapter will explain the mysterious square bracket annotations + that you saw there, + <code class="docutils literal notranslate">[Ring α]</code> and <code class="docutils literal notranslate">[Lattice α]</code>. + It will also show you how to define and use + algebraic structures on your own. + For more technical detail, you can consult <a class="reference external" href="https://leanprover.github.io/theorem_proving_in_lean/">Theorem Proving in Lean</a>, + and a paper by Anne Baanen, <a class="reference external" href="https://arxiv.org/abs/2202.01629">Use and abuse of instance parameters in the Lean mathematical library</a>. + <section id="structures"> + <h2>6.1. Structures<a class="headerlink" href="#structures" title="Permalink to this heading"></a></h2> + In the broadest sense of the term, a structure is a specification + of a collection of data, possibly with constraints that the + data is required to satisfy. + An instance of the structure is a particular bundle of data satisfying + the constraints. For example, we can specify that a point is + a tuple of three real numbers: + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] + structure Point where + x : ℝ + y : ℝ + z : ℝ + </pre></div> + </div> + The <code class="docutils literal notranslate">@[ext]</code> annotation tells Lean to automatically generate theorems + that can be used to prove that two instances of a structure are equal + when their components are equal, a property known as extensionality. + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check Point.ext + + example (a b : Point) (hx : a.x = b.x) (hy : a.y = b.y) (hz : a.z = b.z) : a = b := by + ext + repeat' assumption + </pre></div> + </div> + We can then define particular instances of the <code class="docutils literal notranslate">Point</code> structure. + Lean provides multiple ways of doing that. + <div class="highlight-lean notranslate"><div class="highlight"><pre>def myPoint1 : Point where + x := 2 + y := -1 + z := 4 + + def myPoint2 : Point := + ⟨2, -1, 4⟩ + + def myPoint3 := + Point.mk 2 (-1) 4 + </pre></div> + </div> + In the first example, the fields of the structure are named + explicitly. + The function <code class="docutils literal notranslate">Point.mk</code> referred to in the definition of <code class="docutils literal notranslate">myPoint3</code> + is known as the constructor for the <code class="docutils literal notranslate">Point</code> structure, because + it serves to construct elements. + You can specify a different name if you want, like <code class="docutils literal notranslate">build</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>structure Point' where build :: + x : ℝ + y : ℝ + z : ℝ + + #check Point'.build 2 (-1) 4 + </pre></div> + </div> + The next two examples show how to define functions on structures. + Whereas the second example makes the <code class="docutils literal notranslate">Point.mk</code> + constructor explicit, the first example uses an anonymous constructor + for brevity. + Lean can infer the relevant constructor from the indicated type of + <code class="docutils literal notranslate">add</code>. + It is conventional to put definitions and theorems associated + with a structure like <code class="docutils literal notranslate">Point</code> in a namespace with the same name. + In the example below, because we have opened the <code class="docutils literal notranslate">Point</code> + namespace, the full name of <code class="docutils literal notranslate">add</code> is <code class="docutils literal notranslate">Point.add</code>. + When the namespace is not open, we have to use the full name. + But remember that it is often convenient to use + anonymous projection notation, + which allows us to write <code class="docutils literal notranslate">a.add b</code> instead of <code class="docutils literal notranslate">Point.add a b</code>. + Lean interprets the former as the latter because <code class="docutils literal notranslate">a</code> has type <code class="docutils literal notranslate">Point</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>namespace Point + + def add (a b : Point) : Point := + ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ + + def add' (a b : Point) : Point where + x := a.x + b.x + y := a.y + b.y + z := a.z + b.z + + #check add myPoint1 myPoint2 + #check myPoint1.add myPoint2 + + end Point + + #check Point.add myPoint1 myPoint2 + #check myPoint1.add myPoint2 + </pre></div> + </div> + Below we will continue to put definitions in the relevant + namespace, but we will leave the namespacing commands out of the quoted + snippets. To prove properties of the addition function, + we can use <code class="docutils literal notranslate">rw</code> to expand the definition and <code class="docutils literal notranslate">ext</code> to + reduce an equation between two elements of the structure to equations + between the components. + Below we use the <code class="docutils literal notranslate">protected</code> keyword so that the name of the + theorem is <code class="docutils literal notranslate">Point.add_comm</code>, even when the namespace is open. + This is helpful when we want to avoid ambiguity with a generic + theorem like <code class="docutils literal notranslate">add_comm</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>protected theorem add_comm (a b : Point) : add a b = add b a := by + rw [add, add] + ext <;> dsimp + repeat' apply add_comm + + example (a b : Point) : add a b = add b a := by simp [add, add_comm] + </pre></div> + </div> + Because Lean can unfold definitions and simplify projections + internally, sometimes the equations we want hold definitionally. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem add_x (a b : Point) : (a.add b).x = a.x + b.x := + rfl + </pre></div> + </div> + It is also possible to define functions on structures using + pattern matching, + in a manner similar to the way we defined recursive functions in + <a class="reference internal" href="C05_Number_Theory.html#section-induction-and-recursion">Section 5.2</a>. + The definitions <code class="docutils literal notranslate">addAlt</code> and <code class="docutils literal notranslate">addAlt'</code> below are essentially the + same; the only difference is that we use anonymous constructor notation + in the second. + Although it is sometimes convenient to define functions this way, + the definitional properties are not as convenient. + For example, the expressions <code class="docutils literal notranslate">addAlt a b</code> and <code class="docutils literal notranslate">addAlt' a b</code> + cannot be simplified until we decompose <code class="docutils literal notranslate">a</code> and <code class="docutils literal notranslate">b</code> into + components, which we can do with <code class="docutils literal notranslate">cases</code>, <code class="docutils literal notranslate">rcases</code>, etc. + <div class="highlight-lean notranslate"><div class="highlight"><pre>def addAlt : Point → Point → Point + | Point.mk x₁ y₁ z₁, Point.mk x₂ y₂ z₂ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ + + def addAlt' : Point → Point → Point + | ⟨x₁, y₁, z₁⟩, ⟨x₂, y₂, z₂⟩ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ + + theorem addAlt_x (a b : Point) : (a.addAlt b).x = a.x + b.x := by + cases a + cases b + rfl + + theorem addAlt_comm (a b : Point) : addAlt a b = addAlt b a := by + rcases a with ⟨xa, ya, za⟩ + rcases b with ⟨xb, yb, zb⟩ + rw [addAlt, addAlt] + ext <;> dsimp + apply add_comm + repeat' apply add_comm + + example (a b : Point) : addAlt a b = addAlt b a := by + rcases a with ⟨xa, ya, za⟩ + rcases b with ⟨xb, yb, zb⟩ + simp [addAlt, add_comm] + + example : ∀ a b : Point, addAlt a b = addAlt b a := by + rintro ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ + simp [addAlt, add_comm] + + example : ∀ a b : Point, add a b = add b a := fun ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ => by + simp [add, add_comm] + </pre></div> + </div> + Mathematical constructions often involve taking apart bundled information and + putting it together again in different ways. + It therefore makes sense that Lean and mathlib offer so many ways + of doing this efficiently. + As an exercise, try proving that <code class="docutils literal notranslate">Point.add</code> is associative. + Then define scalar multiplication for a point and show that it + distributes over addition. + <div class="highlight-lean notranslate"><div class="highlight"><pre>protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by + sorry + def smul (r : ℝ) (a : Point) : Point := + sorry + theorem smul_distrib (r : ℝ) (a b : Point) : (smul r a).add (smul r b) = smul r (a.add b) := by + sorry + </pre></div> + </div> + Using structures is only the first step on the road to + algebraic abstraction. + We don’t yet have a way to link <code class="docutils literal notranslate">Point.add</code> to the generic <code class="docutils literal notranslate">+</code> symbol, + or to connect <code class="docutils literal notranslate">Point.add_comm</code> and <code class="docutils literal notranslate">Point.add_assoc</code> to + the generic <code class="docutils literal notranslate">add_comm</code> and <code class="docutils literal notranslate">add_assoc</code> theorems. + These tasks belong to the algebraic aspect of using structures, + and we will explain how to carry them out in the next section. + For now, just think of a structure as a way of bundling together objects + and information. + It is especially useful that a structure can specify not only + data types but also constraints that the data must satisfy. + In Lean, the latter are represented as fields of type <code class="docutils literal notranslate">Prop</code>. + For example, the standard 2-simplex is defined to be the set of + points $(x, y, z)$ satisfying $x ≥ 0$, $y ≥ 0$, $z ≥ 0$, + and $x + y + z = 1$. + If you are not familiar with the notion, you should draw a picture, + and convince yourself that this set is + the equilateral triangle in three-space with vertices + $(1, 0, 0)$, $(0, 1, 0)$, and $(0, 0, 1)$, + together with its interior. + We can represent it in Lean as follows: + <div class="highlight-lean notranslate"><div class="highlight"><pre>structure StandardTwoSimplex where + x : ℝ + y : ℝ + z : ℝ + x_nonneg : 0 ≤ x + y_nonneg : 0 ≤ y + z_nonneg : 0 ≤ z + sum_eq : x + y + z = 1 + </pre></div> + </div> + Notice that the last four fields refer to <code class="docutils literal notranslate">x</code>, <code class="docutils literal notranslate">y</code>, and <code class="docutils literal notranslate">z</code>, + that is, the first three fields. + We can define a map from the two-simplex to itself that swaps <code class="docutils literal notranslate">x</code> and <code class="docutils literal notranslate">y</code>: + <div class="highlight-lean notranslate"><div class="highlight"><pre>def swapXy (a : StandardTwoSimplex) : StandardTwoSimplex + where + x := a.y + y := a.x + z := a.z + x_nonneg := a.y_nonneg + y_nonneg := a.x_nonneg + z_nonneg := a.z_nonneg + sum_eq := by rw [add_comm a.y a.x, a.sum_eq] + </pre></div> + </div> + More interestingly, we can compute the midpoint of two points on + the simplex. We have added the phrase <code class="docutils literal notranslate">noncomputable section</code> + at the beginning of this file in order to use division on the real numbers. + <div class="highlight-lean notranslate"><div class="highlight"><pre>noncomputable section + + def midpoint (a b : StandardTwoSimplex) : StandardTwoSimplex + where + x := (a.x + b.x) / 2 + y := (a.y + b.y) / 2 + z := (a.z + b.z) / 2 + x_nonneg := div_nonneg (add_nonneg a.x_nonneg b.x_nonneg) (by norm_num) + y_nonneg := div_nonneg (add_nonneg a.y_nonneg b.y_nonneg) (by norm_num) + z_nonneg := div_nonneg (add_nonneg a.z_nonneg b.z_nonneg) (by norm_num) + sum_eq := by field_simp; linarith [a.sum_eq, b.sum_eq] + </pre></div> + </div> + Here we have established <code class="docutils literal notranslate">x_nonneg</code>, <code class="docutils literal notranslate">y_nonneg</code>, and <code class="docutils literal notranslate">z_nonneg</code> + with concise proof terms, but establish <code class="docutils literal notranslate">sum_eq</code> in tactic mode, + using <code class="docutils literal notranslate">by</code>. + Given a parameter $\lambda$ satisfying $0 \le \lambda \le 1$, + we can take the weighted average $\lambda a + (1 - \lambda) b$ + of two points $a$ and $b$ in the standard 2-simplex. + We challenge you to define that function, in analogy to the <code class="docutils literal notranslate">midpoint</code> + function above. + <div class="highlight-lean notranslate"><div class="highlight"><pre>def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) + (a b : StandardTwoSimplex) : StandardTwoSimplex + where + sorry + </pre></div> + </div> + Structures can depend on parameters. + For example, we can generalize the standard 2-simplex to the standard + $n$-simplex for any $n$. + At this stage, you don’t have to know anything about the type <cite>Fin n</cite> + except that it has $n$ elements, and that Lean knows + how to sum over it. + <div class="highlight-lean notranslate"><div class="highlight"><pre>open BigOperators + + structure StandardSimplex (n : ℕ) where + V : Fin n → ℝ + NonNeg : ∀ i : Fin n, 0 ≤ V i + sum_eq_one : (∑ i, V i) = 1 + + namespace StandardSimplex + + def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n + where + V i := (a.V i + b.V i) / 2 + NonNeg := by + intro i + apply div_nonneg + · linarith [a.NonNeg i, b.NonNeg i] + norm_num + sum_eq_one := by + simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one] + field_simp + + end StandardSimplex + </pre></div> + </div> + As an exercise, see if you can define the weighted average of + two points in the standard $n$-simplex. + You can use <code class="docutils literal notranslate">Finset.sum_add_distrib</code> + and <code class="docutils literal notranslate">Finset.mul_sum</code> to manipulate the relevant sums. + We have seen that structures can be used to bundle together data + and properties. + Interestingly, they can also be used to bundle together properties + without the data. + For example, the next structure, <code class="docutils literal notranslate">IsLinear</code>, bundles together + the two components of linearity. + <div class="highlight-lean notranslate"><div class="highlight"><pre>structure IsLinear (f : ℝ → ℝ) where + is_additive : ∀ x y, f (x + y) = f x + f y + preserves_mul : ∀ x c, f (c * x) = c * f x + + section + variable (f : ℝ → ℝ) (linf : IsLinear f) + + #check linf.is_additive + #check linf.preserves_mul + + end + </pre></div> + </div> + It is worth pointing out that structures are not the only way to bundle + together data. + The <code class="docutils literal notranslate">Point</code> data structure can be defined using the generic type product, + and <code class="docutils literal notranslate">IsLinear</code> can be defined with a simple <code class="docutils literal notranslate">and</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>def Point'' := + ℝ × ℝ × ℝ + + def IsLinear' (f : ℝ → ℝ) := + (∀ x y, f (x + y) = f x + f y) ∧ ∀ x c, f (c * x) = c * f x + </pre></div> + </div> + Generic type constructions can even be used in place of structures + with dependencies between their components. + For example, the subtype construction combines a piece of data with + a property. + You can think of the type <code class="docutils literal notranslate">PReal</code> in the next example as being + the type of positive real numbers. + Any <code class="docutils literal notranslate">x : PReal</code> has two components: the value, and the property of being + positive. + You can access these components as <code class="docutils literal notranslate">x.val</code>, which has type <code class="docutils literal notranslate">ℝ</code>, + and <code class="docutils literal notranslate">x.property</code>, which represents the fact <code class="docutils literal notranslate">0 < x.val</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>def PReal := + { y : ℝ // 0 < y } + + section + variable (x : PReal) + + #check x.val + #check x.property + #check x.1 + #check x.2 + + end + </pre></div> + </div> + We could have used subtypes to define the standard 2-simplex, + as well as the standard $n$-simplex for an arbitrary $n$. + <div class="highlight-lean notranslate"><div class="highlight"><pre>def StandardTwoSimplex' := + { p : ℝ × ℝ × ℝ // 0 ≤ p.1 ∧ 0 ≤ p.2.1 ∧ 0 ≤ p.2.2 ∧ p.1 + p.2.1 + p.2.2 = 1 } + + def StandardSimplex' (n : ℕ) := + { v : Fin n → ℝ // (∀ i : Fin n, 0 ≤ v i) ∧ (∑ i, v i) = 1 } + </pre></div> + </div> + Similarly, Sigma types are generalizations of ordered pairs, + whereby the type of the second component depends on the type of + the first. + <div class="highlight-lean notranslate"><div class="highlight"><pre>def StdSimplex := Σ n : ℕ, StandardSimplex n + + section + variable (s : StdSimplex) + + #check s.fst + #check s.snd + + #check s.1 + #check s.2 + + end + </pre></div> + </div> + Given <code class="docutils literal notranslate">s : StdSimplex</code>, the first component <code class="docutils literal notranslate">s.fst</code> is a natural + number, and the second component is an element of the corresponding + simplex <code class="docutils literal notranslate">StandardSimplex s.fst</code>. + The difference between a Sigma type and a subtype is that + the second component of a Sigma type is data rather than a proposition. + But even though we can use products, subtypes, and Sigma types + instead of structures, using structures has a number of advantages. + Defining a structure abstracts away the underlying representation + and provides custom names for the functions that access the components. + This makes proofs more robust: + proofs that rely only on the interface to a structure + will generally continue to work when we change the definition, + as long as we redefine the old accessors in terms of the new definition. + Moreover, as we are about to see, Lean provides support for + weaving structures together into a rich, interconnected hierarchy, + and for managing the interactions between them. + </section> + <section id="algebraic-structures"> + <h2>6.2. Algebraic Structures<a class="headerlink" href="#algebraic-structures" title="Permalink to this heading"></a></h2> + To clarify what we mean by the phrase algebraic structure, + it will help to consider some examples. + <ol class="arabic simple"> + <li>A partially ordered set consists of a set $P$ and + a binary relation $\le$ on $P$ that is transitive + and antireflexive.</li> + <li>A group consists of a set $G$ with an associative + binary operation, an identity element + $1$, and a function $g \mapsto g^{-1}$ that returns + an inverse for each $g$ in $G$. + A group is abelian or commutative if the operation is commutative.</li> + <li>A lattice is a partially ordered set with meets and joins.</li> + <li>A ring consists of an (additively written) abelian group + $(R, +, 0, x \mapsto -x)$ + together with an associative multiplication operation + $\cdot$ and an identity $1$, + such that multiplication distributes over addition. + A ring is commutative if the multiplication is commutative.</li> + <li>An ordered ring $(R, +, 0, -, \cdot, 1, \le)$ consists of a ring + together with a partial order on its elements, such that $a \le b$ implies + $a + c \le b + c$ for every $a$, $b$, and $c$ in $R$, + and $0 \le a$ and $0 \le b$ implies $0 \le a b$ for + every $a$ and $b$ in $R$.</li> + <li>A metric space consists of a set $X$ and a function + $d : X \times X \to \mathbb{R}$ such that the following hold: + <ul class="simple"> + <li>$d(x, y) \ge 0$ for every $x$ and $y$ in $X$.</li> + <li>$d(x, y) = 0$ if and only if $x = y$.</li> + <li>$d(x, y) = d(y, x)$ for every $x$ and $y$ in $X$.</li> + <li>$d(x, z) \le d(x, y) + d(y, z)$ for every $x$, $y$, and + $z$ in $X$.</li> + </ul> + </li> + <li>A topological space consists of a set $X$ and a collection $\mathcal T$ + of subsets of $X$, called the open subsets of $X$, such that + the following hold: + <ul class="simple"> + <li>The empty set and $X$ are open.</li> + <li>The intersection of two open sets is open.</li> + <li>An arbitrary union of open sets is open.</li> + </ul> + </li> + </ol> + In each of these examples, the elements of the structure belong to a + set, the carrier set, + that sometimes stands proxy for the entire structure. + For example, when we say “let $G$ be a group” and then + “let $g \in G$,” we are using $G$ to stand for both + the structure and its carrier. + Not every algebraic structure is associated with a single carrier set in this way. + For example, a bipartite graph involves a relation between two sets, + as does a Galois connection, + A category also involves two sets of interest, commonly called the objects + and the morphisms. + The examples indicate some of the things that a proof assistant has to do + in order to support algebraic reasoning. + First, it needs to recognize concrete instances of structures. + The number systems $\mathbb{Z}$, $\mathbb{Q}$, + and $\mathbb{R}$ are all ordered rings, + and we should be able to apply a generic theorem about ordered rings + in any of these instances. + Sometimes a concrete set may be an instance of a structure in more than one way. + For example, in addition to the usual topology on $\mathbb{R}$, + which forms the basis for real analysis, + we can also consider the discrete topology on $\mathbb{R}$, + in which every set is open. + Second, a proof assistant needs to support generic notation on structures. + In Lean, the notation <code class="docutils literal notranslate">*</code> + is used for multiplication in all the usual number systems, + as well as for multiplication in generic groups and rings. + When we use an expression like <code class="docutils literal notranslate">f x * y</code>, + Lean has to use information about the types of <code class="docutils literal notranslate">f</code>, <code class="docutils literal notranslate">x</code>, and <code class="docutils literal notranslate">y</code> + to determine which multiplication we have in mind. + Third, it needs to deal with the fact that structures can inherit + definitions, theorems, and notation from other structures in various ways. + Some structures extend others by adding more axioms. + A commutative ring is still a ring, so any definition + that makes sense in a ring also makes sense in a commutative ring, + and any theorem that holds in a ring also holds in a commutative ring. + Some structures extend others by adding more data. + For example, the additive part of any ring is an additive group. + The ring structure adds a multiplication and an identity, + as well as axioms that govern them and relate them to the additive part. + Sometimes we can define one structure in terms of another. + Any metric space has a canonical topology associated with it, + the metric space topology, and there are various topologies that can be + associated with any linear ordering. + Finally, it is important to keep in mind that mathematics allows us to + use functions and operations to define structures in the same way we + use functions and operations to define numbers. + Products and powers of groups are again groups. + For every $n$, the integers modulo $n$ form a ring, + and for every $k > 0$, the $k \times k$ matrices of polynomials + with coefficients in that ring again form a ring. + Thus we can calculate with structures just as easily as we can calculate + with their elements. + This means that algebraic structures lead dual lives in mathematics, + as containers for collections of objects and as objects in their own right. + A proof assistant has to accommodate this dual role. + When dealing with elements of a type that has an algebraic structure + associated with it, + a proof assistant needs to recognize the structure and find the relevant + definitions, theorems, and notation. + All this should sound like a lot of work, and it is. + But Lean uses a small collection of fundamental mechanisms to + carry out these tasks. + The goal of this section is to explain these mechanisms and show you + how to use them. + The first ingredient is almost too obvious to mention: + formally speaking, algebraic structures are structures in the sense + of <a class="reference internal" href="#section-structures">Section 6.1</a>. + An algebraic structure is a specification of a bundle of data satisfying + some axiomatic hypotheses, and we saw in <a class="reference internal" href="#section-structures">Section 6.1</a> that + this is exactly what the <code class="docutils literal notranslate">structure</code> command is designed to accommodate. + It’s a marriage made in heaven! + Given a data type <code class="docutils literal notranslate">α</code>, we can define the group structure on <code class="docutils literal notranslate">α</code> + as follows. + <div class="highlight-lean notranslate"><div class="highlight"><pre>structure Group₁ (α : Type _) where + mul : α → α → α + one : α + inv : α → α + mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) + mul_one : ∀ x : α, mul x one = x + one_mul : ∀ x : α, mul one x = x + mul_left_inv : ∀ x : α, mul (inv x) x = one + </pre></div> + </div> + Notice that the type <code class="docutils literal notranslate">α</code> is a parameter in the definition of <code class="docutils literal notranslate">group₁</code>. + So you should think of an object <code class="docutils literal notranslate">struc : Group₁ α</code> as being + a group structure on <code class="docutils literal notranslate">α</code>. + We saw in <a class="reference internal" href="C02_Basics.html#proving-identities-in-algebraic-structures">Section 2.2</a> + that the counterpart <code class="docutils literal notranslate">mul_right_inv</code> to <code class="docutils literal notranslate">mul_left_inv</code> + follows from the other group axioms, so there is no need + to add it to the definition. + This definition of a group is similar to the definition of <code class="docutils literal notranslate">Group</code> in + mathlib, + and we have chosen the name <code class="docutils literal notranslate">Group₁</code> to distinguish our version. + If you write <code class="docutils literal notranslate">#check Group</code> and ctrl-click on the definition, + you will see that the mathlib version of <code class="docutils literal notranslate">Group</code> is defined to + extend another structure; we will explain how to do that later. + If you type <code class="docutils literal notranslate">#print Group</code> you will also see that the mathlib + version of <code class="docutils literal notranslate">Group</code> has a number of extra fields. + For reasons we will explain later, sometimes it is useful to add + redundant information to a structure, + so that there are additional fields for objects and functions + that can be defined from the core + data. Don’t worry about that for now. + Rest assured that our simplified version <code class="docutils literal notranslate">Group₁</code> is + morally the same as the definition of a group that mathlib uses. + It is sometimes useful to bundle + the type together with the structure, and mathlib also + contains a definition of a <code class="docutils literal notranslate">GroupCat</code> structure that is equivalent to + the following: + <div class="highlight-lean notranslate"><div class="highlight"><pre>structure Group₁Cat where + α : Type _ + str : Group₁ α + </pre></div> + </div> + The mathlib version is found in <code class="docutils literal notranslate">Algebra.Category.Group.Basic</code>, + and you can <code class="docutils literal notranslate">#check</code> it if you add this to the imports at the + beginning of the examples file. + For reasons that will become clearer below, it is more often + useful to keep the type <code class="docutils literal notranslate">α</code> separate from the structure <code class="docutils literal notranslate">Group α</code>. + We refer to the two objects together as a partially bundled structure, + since the representation combines most, but not all, of the components + into one structure. It is common in mathlib + to use capital roman letters like <code class="docutils literal notranslate">G</code> for a type + when it is used as the carrier type for a group. + Let’s construct a group, which is to say, an element of the <code class="docutils literal notranslate">Group₁</code> type. + For any pair of types <code class="docutils literal notranslate">α</code> and <code class="docutils literal notranslate">β</code>, Mathlib defines the type <code class="docutils literal notranslate">Equiv α β</code> + of equivalences between <code class="docutils literal notranslate">α</code> and <code class="docutils literal notranslate">β</code>. + Mathlib also defines the suggestive notation <code class="docutils literal notranslate">α ≃ β</code> for this type. + An element <code class="docutils literal notranslate">f : α ≃ β</code> is a bijection between <code class="docutils literal notranslate">α</code> and <code class="docutils literal notranslate">β</code> + represented by four components: + a function <code class="docutils literal notranslate">f.toFun</code> from <code class="docutils literal notranslate">α</code> to <code class="docutils literal notranslate">β</code>, + the inverse function <code class="docutils literal notranslate">f.invFun</code> from <code class="docutils literal notranslate">β</code> to <code class="docutils literal notranslate">α</code>, + and two properties that specify these functions are indeed inverse + to one another. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (α β γ : Type _) + variable (f : α ≃ β) (g : β ≃ γ) + + #check Equiv α β + #check (f.toFun : α → β) + #check (f.invFun : β → α) + #check (f.right_inv : ∀ x : β, f (f.invFun x) = x) + #check (f.left_inv : ∀ x : α, f.invFun (f x) = x) + #check (Equiv.refl α : α ≃ α) + #check (f.symm : β ≃ α) + #check (f.trans g : α ≃ γ) + </pre></div> + </div> + Notice the creative naming of the last three constructions. We think of the + identity function <code class="docutils literal notranslate">Equiv.refl</code>, the inverse operation <code class="docutils literal notranslate">Equiv.symm</code>, + and the composition operation <code class="docutils literal notranslate">Equiv.trans</code> as explicit evidence + that the property of being in bijective correspondence is an equivalence relation. + Notice also that <code class="docutils literal notranslate">f.trans g</code> requires composing the forward functions + in reverse order. Mathlib has declared a coercion from <code class="docutils literal notranslate">Equiv α β</code> + to the function type <code class="docutils literal notranslate">α → β</code>, so we can omit writing <code class="docutils literal notranslate">.toFun</code> + and have Lean insert it for us. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x : α) : (f.trans g).toFun x = g.toFun (f.toFun x) := + rfl + + example (x : α) : (f.trans g) x = g (f x) := + rfl + + example : (f.trans g : α → γ) = g ∘ f := + rfl + </pre></div> + </div> + Mathlib also defines the type <code class="docutils literal notranslate">perm α</code> of equivalences between + <code class="docutils literal notranslate">α</code> and itself. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (α : Type _) : Equiv.Perm α = (α ≃ α) := + rfl + </pre></div> + </div> + It should be clear that <code class="docutils literal notranslate">Equiv.Perm α</code> forms a group under composition + of equivalences. We orient things so that <code class="docutils literal notranslate">mul f g</code> is + equal to <code class="docutils literal notranslate">g.trans f</code>, whose forward function is <code class="docutils literal notranslate">f ∘ g</code>. + In other words, multiplication is what we ordinarily think of as + composition of the bijections. Here we define this group: + <div class="highlight-lean notranslate"><div class="highlight"><pre>def permGroup {α : Type _} : Group₁ (Equiv.Perm α) + where + mul f g := Equiv.trans g f + one := Equiv.refl α + inv := Equiv.symm + mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm + one_mul := Equiv.trans_refl + mul_one := Equiv.refl_trans + mul_left_inv := Equiv.self_trans_symm + </pre></div> + </div> + In fact, mathlib defines exactly this <code class="docutils literal notranslate">Group</code> structure on <code class="docutils literal notranslate">Equiv.Perm α</code> + in the file <code class="docutils literal notranslate">GroupTheory.Perm.Basic</code>. + As always, you can hover over the theorems used in the definition of + <code class="docutils literal notranslate">permGroup</code> to see their statements, + and you can jump to their definitions in the original file to learn + more about how they are implemented. + In ordinary mathematics, we generally think of notation as + independent of structure. + For example, we can consider groups $(G_1, \cdot, 1, \cdot^{-1})$, + $(G_2, \circ, e, i(\cdot))$, and $(G_3, +, 0, -)$. + In the first case, we write the binary operation as $\cdot$, + the identity at $1$, and the inverse function as $x \mapsto x^{-1}$. + In the second and third cases, we use the notational alternatives shown. + When we formalize the notion of a group in Lean, however, + the notation is more tightly linked to the structure. + In Lean, the components of any <code class="docutils literal notranslate">Group</code> are named + <code class="docutils literal notranslate">mul</code>, <code class="docutils literal notranslate">one</code>, and <code class="docutils literal notranslate">inv</code>, + and in a moment we will see how multiplicative notation is + set up to refer to them. + If we want to use additive notation, we instead use an isomorphic structure + <code class="docutils literal notranslate">AdditiveGroup</code>. Its components are named <code class="docutils literal notranslate">add</code>, <code class="docutils literal notranslate">zero</code>, + and <code class="docutils literal notranslate">neg</code>, and the associated notation is what you would expect it to be. + Recall the type <code class="docutils literal notranslate">Point</code> that we defined in <a class="reference internal" href="#section-structures">Section 6.1</a>, + and the addition function that we defined there. + These definitions are reproduced in the examples file that accompanies + this section. + As an exercise, define an <code class="docutils literal notranslate">AddGroup₁</code> structure that is similar + to the <code class="docutils literal notranslate">Group₁</code> structure we defined above, except that it uses the + additive naming scheme just described. + Define negation and a zero on the <code class="docutils literal notranslate">Point</code> data type, + and define the <code class="docutils literal notranslate">AddGroup₁</code> structure on <code class="docutils literal notranslate">Point</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>structure AddGroup₁ (α : Type _) where + (add : α → α → α) + -- fill in the rest + @[ext] + structure Point where + x : ℝ + y : ℝ + z : ℝ + + namespace Point + + def add (a b : Point) : Point := + ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ + + def neg (a : point) : point := sorry + + def zero : point := sorry + + def add_group_point : add_group₁ point := sorry + end Point + </pre></div> + </div> + We are making progress. + Now we know how to define algebraic structures in Lean, + and we know how to define instances of those structures. + But we also want to associate notation with structures + so that we can use it with each instance. + Moreover, we want to arrange it so that we can define an operation + on a structure and use it with any particular instance, + and we want to arrange it so that we can prove a theorem about + a structure and use it with any instance. + In fact, mathlib is already set up to use generic group notation, + definitions, and theorems for <code class="docutils literal notranslate">Equiv.Perm α</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} (f g : Equiv.Perm α) (n : ℕ) + + #check f * g + #check mul_assoc f g g⁻¹ + + -- group power, defined for any group + #check g ^ n + + example : f * g * g⁻¹ = f := by rw [mul_assoc, mul_right_inv, mul_one] + + example : f * g * g⁻¹ = f := + mul_inv_cancel_right f g + + example {α : Type _} (f g : Equiv.Perm α) : g.symm.trans (g.trans f) = f := + mul_inv_cancel_right f g + </pre></div> + </div> + You can check that this is not the case for the additive group structure + on <code class="docutils literal notranslate">Point</code> that we asked you to define above. + Our task now is to understand that magic that goes on under the hood + in order to make the examples for <code class="docutils literal notranslate">Equiv.Perm α</code> work the way they do. + The issue is that Lean needs to be able to find the relevant + notation and the implicit group structure, + using the information that is found in the expressions that we type. + Similarly, when we write <code class="docutils literal notranslate">x + y</code> with expressions <code class="docutils literal notranslate">x</code> and <code class="docutils literal notranslate">y</code> + that have type <code class="docutils literal notranslate">ℝ</code>, Lean needs to interpret the <code class="docutils literal notranslate">+</code> + symbol as the relevant addition function on the reals. + It also has to recognize the type <code class="docutils literal notranslate">ℝ</code> as an instance of a commutative ring, + so that all the definitions and theorems for a commutative ring are available. + For another example, + continuity is defined in Lean relative to any two topological spaces. + When we have <code class="docutils literal notranslate">f : ℝ → ℂ</code> and we write <code class="docutils literal notranslate">Continuous f</code>, Lean has to find the + relevant topologies on <code class="docutils literal notranslate">ℝ</code> and <code class="docutils literal notranslate">ℂ</code>. + The magic is achieved with a combination of three things. + <ol class="arabic simple"> + <li>Logic. A definition that should be interpreted in any group takes, as + arguments, the type of the group and the group structure as arguments. + Similarly, a theorem about the elements of an arbitrary group + begins with universal quantifiers over + the type of the group and the group structure.</li> + <li>Implicit arguments. The arguments for the type and the structure + are generally left implicit, so that we do not have to write them + or see them in the Lean information window. Lean fills the + information in for us silently.</li> + <li>Type class inference. Also known as class inference, + this is a simple but powerful mechanism + that enables us to register information for Lean to use later on. + When Lean is called on to fill in implicit arguments to a + definition, theorem, or piece of notation, + it can make use of information that has been registered.</li> + </ol> + Whereas an annotation <code class="docutils literal notranslate">(grp : Group G)</code> tells Lean that it should + expect to be given that argument explicitly and the annotation + <code class="docutils literal notranslate">{grp : Group G}</code> tells Lean that it should try to figure it out + from contextual cues in the expression, + the annotation <code class="docutils literal notranslate">[grp : Group G]</code> tells Lean that the corresponding + argument should be synthesized using type class inference. + Since the whole point to the use of such arguments is that + we generally do not need to refer to them explicitly, + Lean allows us to write <code class="docutils literal notranslate">[Group G]</code> and leave the name anonymous. + You have probably already noticed that Lean chooses names like <code class="docutils literal notranslate">_inst_1</code> + automatically. + When we use the anonymous square-bracket annotation with the <code class="docutils literal notranslate">variables</code> command, + then as long as the variables are still in scope, + Lean automatically adds the argument <code class="docutils literal notranslate">[Group G]</code> to any definition or + theorem that mentions <code class="docutils literal notranslate">G</code>. + How do we register the information that Lean needs to use to carry + out the search? + Returning to our group example, we need only make two changes. + First, instead of using the <code class="docutils literal notranslate">structure</code> command to define the group structure, + we use the keyword <code class="docutils literal notranslate">class</code> to indicate that it is a candidate + for class inference. + Second, instead of defining particular instances with <code class="docutils literal notranslate">def</code>, + we use the keyword <code class="docutils literal notranslate">instance</code> to register the particular instance with + Lean. As with the names of class variables, we are allowed to leave the + name of an instance definition anonymous, + since in general we intend Lean to find it and put it to use + without troubling us with the details. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class Group₂ (α : Type _) where + mul : α → α → α + one : α + inv : α → α + mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) + mul_one : ∀ x : α, mul x one = x + one_mul : ∀ x : α, mul one x = x + mul_left_inv : ∀ x : α, mul (inv x) x = one + + instance {α : Type _} : Group₂ (Equiv.Perm α) where + mul f g := Equiv.trans g f + one := Equiv.refl α + inv := Equiv.symm + mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm + one_mul := Equiv.trans_refl + mul_one := Equiv.refl_trans + mul_left_inv := Equiv.self_trans_symm + </pre></div> + </div> + The following illustrates their use. + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check @Group₂.mul + + def mySquare {α : Type _} [Group₂ α] (x : α) := + Group₂.mul x x + + #check @mySquare + + section + variable {β : Type _} (f g : Equiv.Perm β) + + example : Group₂.mul f g = g.trans f := + rfl + + example : mySquare f = f.trans f := + rfl + + end + </pre></div> + </div> + The <code class="docutils literal notranslate">#check</code> command shows that <code class="docutils literal notranslate">Group₂.mul</code> has an implicit argument + <code class="docutils literal notranslate">[Group₂ α]</code> that we expect to be found by class inference, + where <code class="docutils literal notranslate">α</code> is the type of the arguments to <code class="docutils literal notranslate">Group₂.mul</code>. + In other words, <code class="docutils literal notranslate">{α : Type*}</code> is the implicit argument for the type + of the group elements and <code class="docutils literal notranslate">[Group₂ α]</code> is the implicit argument for the + group structure on <code class="docutils literal notranslate">α</code>. + Similarly, when we define a generic squaring function <code class="docutils literal notranslate">my_square</code> + for <code class="docutils literal notranslate">Group₂</code>, we use an implicit argument <code class="docutils literal notranslate">{α : Type*}</code> for + the type of the elements and an implicit argument <code class="docutils literal notranslate">[Group₂ α]</code> for + the <code class="docutils literal notranslate">Group₂</code> structure. + In the first example, + when we write <code class="docutils literal notranslate">Group₂.mul f g</code>, the type of <code class="docutils literal notranslate">f</code> and <code class="docutils literal notranslate">g</code> + tells Lean that in the argument <code class="docutils literal notranslate">α</code> to <code class="docutils literal notranslate">Group₂.mul</code> + has to be instantiated to <code class="docutils literal notranslate">Equiv.Perm β</code>. + That means that Lean has to find an element of <code class="docutils literal notranslate">Group₂ (Equiv.Perm β)</code>. + The previous <code class="docutils literal notranslate">instance</code> declaration tells Lean exactly how to do that. + Problem solved! + This simple mechanism for registering information so that Lean can find it + when it needs it is remarkably useful. + Here is one way it comes up. + In Lean’s foundation, a data type <code class="docutils literal notranslate">α</code> may be empty. + In a number of applications, however, it is useful to know that a + type has at least one element. + For example, the function <code class="docutils literal notranslate">List.head</code>, which returns the first + element of a list, can return the default value when the list is empty. + To make that work, the Lean library defines a class <code class="docutils literal notranslate">Inhabited α</code>, + which does nothing more than store a default value. + We can show that the <code class="docutils literal notranslate">Point</code> type is an instance: + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance : Inhabited Point where default := ⟨0, 0, 0⟩ + + #check (default : Point) + + example : ([] : List Point).headI = default := + rfl + </pre></div> + </div> + The class inference mechanism is also used for generic notation. + The expression <code class="docutils literal notranslate">x + y</code> is an abbreviation for <code class="docutils literal notranslate">Add.add x y</code> + where—you guessed it—<code class="docutils literal notranslate">Add α</code> is a class that stores + a binary function on <code class="docutils literal notranslate">α</code>. + Writing <code class="docutils literal notranslate">x + y</code> tells Lean to find a registered instance of <code class="docutils literal notranslate">[Add.add α]</code> + and use the corresponding function. + Below, we register the addition function for <code class="docutils literal notranslate">Point</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance : Add Point where add := Point.add + + section + variable (x y : Point) + + #check x + y + + example : x + y = Point.add x y := + rfl + + end + </pre></div> + </div> + In this way, we can assign the notation <code class="docutils literal notranslate">+</code> to binary operations on other + types as well. + But we can do even better. We have seen that <code class="docutils literal notranslate">*</code> can be used in any + group, <code class="docutils literal notranslate">+</code> can be used in any additive group, and both can be used in + any ring. + When we define a new instance of a ring in Lean, + we don’t have to define <code class="docutils literal notranslate">+</code> and <code class="docutils literal notranslate">*</code> for that instance, + because Lean knows that these are defined for every ring. + We can use this method to specify notation for our <code class="docutils literal notranslate">Group₂</code> class: + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance hasMulGroup₂ {α : Type _} [Group₂ α] : Mul α := + ⟨Group₂.mul⟩ + + instance hasOneGroup₂ {α : Type _} [Group₂ α] : One α := + ⟨Group₂.one⟩ + + instance hasInvGroup₂ {α : Type _} [Group₂ α] : Inv α := + ⟨Group₂.inv⟩ + + section + variable {α : Type _} (f g : Equiv.Perm α) + + #check f * 1 * g⁻¹ + + def foo : f * 1 * g⁻¹ = g.symm.trans ((Equiv.refl α).trans f) := + rfl + + end + </pre></div> + </div> + In this case, we have to supply names for the instances, because + Lean has a hard time coming up with good defaults. + What makes this approach work is that Lean carries out a recursive search. + According to the instances we have declared, Lean can find an instance of + <code class="docutils literal notranslate">Mul (Equiv.Perm α)</code> by finding an + instance of <code class="docutils literal notranslate">Group₂ (Equiv.Perm α)</code>, and it can find an instance of + <code class="docutils literal notranslate">Group₂ (Equiv.Perm α)</code> because we have provided one. + Lean is capable of finding these two facts and chaining them together. + The example we have just given is dangerous, because Lean’s + library also has an instance of <code class="docutils literal notranslate">Group (Equiv.Perm α)</code>, and + multiplication is defined on any group. + So it is ambiguous as to which instance is found. + In fact, Lean favors more recent declarations unless you explicitly + specify a different priority. + Also, there is another way to tell Lean that one structure is an + instance of another, using the <code class="docutils literal notranslate">extends</code> keyword. + This is how <code class="docutils literal notranslate">mathlib</code> specifies that, for example, + every commutative ring is a ring. + You can find more information in a + <a class="reference external" href="https://leanprover.github.io/theorem_proving_in_lean4/type_classes.html#managing-type-class-inference">section on class inference</a> in Theorem Proving in Lean. + In general, it is a bad idea to specify a value of + <code class="docutils literal notranslate">*</code> for an instance of an algebraic structure that already has + the notation defined. + Redefining the notion of <code class="docutils literal notranslate">Group</code> in Lean is an artificial example. + In this case, however, both interpretations of the group notation unfold to + <code class="docutils literal notranslate">Equiv.trans</code>, <code class="docutils literal notranslate">Equiv.refl</code>, and <code class="docutils literal notranslate">Equiv.symm</code>, in the same way. + As a similarly artificial exercise, + define a class <code class="docutils literal notranslate">AddGroup₂</code> in analogy to <code class="docutils literal notranslate">Group₂</code>. + Define the usual notation for addition, negation, and zero + on any <code class="docutils literal notranslate">AddGroup₂</code> + using the classes <code class="docutils literal notranslate">Add</code>, <code class="docutils literal notranslate">Neg</code>, and <code class="docutils literal notranslate">Zero</code>. + Then show <code class="docutils literal notranslate">Point</code> is an instance of <code class="docutils literal notranslate">AddGroup₂</code>. + Try it out and make sure that the additive group notation works for + elements of <code class="docutils literal notranslate">Point</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class AddGroup₂ (α : Type _) where + add : α → α → α + -- fill in the rest + </pre></div> + </div> + It is not a big problem that we have already declared instances + <code class="docutils literal notranslate">Add</code>, <code class="docutils literal notranslate">Neg</code>, and <code class="docutils literal notranslate">Zero</code> for <code class="docutils literal notranslate">Point</code> above. + Once again, the two ways of synthesizing the notation should come up + with the same answer. + Class inference is subtle, and you have to be careful when using it, + because it configures automation that invisibly governs the interpretation of + the expressions we type. + When used wisely, however, class inference is a powerful tool. + It is what makes algebraic reasoning possible in Lean. + </section> + <section id="building-the-gaussian-integers"> + <h2>6.3. Building the Gaussian Integers<a class="headerlink" href="#building-the-gaussian-integers" title="Permalink to this heading"></a></h2> + We will now illustrate the use of the algebraic hierarchy in Lean by + building an important mathematical object, the Gaussian integers, + and showing that it is a Euclidean domain. In other words, according to + the terminology we have been using, we will define the Gaussian integers + and show that they are an instance of the Euclidean domain structure. + In ordinary mathematical terms, the set of Gaussian integers $\Bbb{Z}[i]$ + is the set of complex numbers $\{ a + b i \mid a, b \in \Bbb{Z}\}$. + But rather than define them as a subset of the complex numbers, our goal + here is to define them as a data type in their own right. We do this by + representing a Gaussian integer as a pair of integers, which we think of as the + real and imaginary parts. + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] + structure gaussInt where + re : ℤ + im : ℤ + </pre></div> + </div> + We first show that the Gaussian integers have the structure of a ring, + with <code class="docutils literal notranslate">0</code> defined to be <code class="docutils literal notranslate">⟨0, 0⟩</code>, <code class="docutils literal notranslate">1</code> defined to be <code class="docutils literal notranslate">⟨1, 0⟩</code>, and + addition defined pointwise. To work out the definition of multiplication, + remember that we want the element $i$, represented by <code class="docutils literal notranslate">⟨0, 1⟩</code>, to + be a square root of $-1$. Thus we want + <div class="math notranslate nohighlight"> + \[\begin{split}(a + bi) (c + di) & = ac + bci + adi + bd i^2 \\ + & = (ac - bd) + (bc + ad)i.\end{split}\]</div> + This explains the definition of <code class="docutils literal notranslate">hasMul</code> below. + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance : Zero gaussInt := + ⟨⟨0, 0⟩⟩ + + instance : One gaussInt := + ⟨⟨1, 0⟩⟩ + + instance : Add gaussInt := + ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ + + instance : Neg gaussInt := + ⟨fun x => ⟨-x.re, -x.im⟩⟩ + + instance : Mul gaussInt := + ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ + </pre></div> + </div> + As noted in <a class="reference internal" href="#section-structures">Section 6.1</a>, it is a good idea to put all the definitions + related to a data type in a namespace with the same name. Thus in the Lean + files associated with this chapter, these definitions are made in the + <code class="docutils literal notranslate">GaussInt</code> namespace. + Notice that here we are defining the interpretations of the notation <code class="docutils literal notranslate">0</code>, + <code class="docutils literal notranslate">1</code>, <code class="docutils literal notranslate">+</code>, <code class="docutils literal notranslate">-</code>, and <code class="docutils literal notranslate">*</code> directly, rather than naming them + <code class="docutils literal notranslate">GaussInt.zero</code> and the like and assigning the notation to those. + It is often useful to have an explicit name for the definitions, for example, + to use with <code class="docutils literal notranslate">simp</code> and <code class="docutils literal notranslate">rewrite</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := + rfl + + theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := + rfl + + theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := + rfl + + theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := + rfl + + theorem mul_def (x y : gaussInt) : x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := + rfl + </pre></div> + </div> + It is also useful to name the rules that compute the real and imaginary + parts, and to declare them to the simplifier. + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[simp] + theorem zero_re : (0 : gaussInt).re = 0 := + rfl + + @[simp] + theorem zero_im : (0 : gaussInt).im = 0 := + rfl + + @[simp] + theorem one_re : (1 : gaussInt).re = 1 := + rfl + + @[simp] + theorem one_im : (1 : gaussInt).im = 0 := + rfl + + @[simp] + theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := + rfl + + @[simp] + theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := + rfl + + @[simp] + theorem neg_re (x : gaussInt) : (-x).re = -x.re := + rfl + + @[simp] + theorem neg_im (x : gaussInt) : (-x).im = -x.im := + rfl + + @[simp] + theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := + rfl + + @[simp] + theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := + rfl + </pre></div> + </div> + It is now surprisingly easy to show that the Gaussian integers are an instance + of a commutative ring. We are putting the structure concept to good use. + Each particular Gaussian integer is an instance of the <code class="docutils literal notranslate">gaussInt</code> structure, + whereas the type <code class="docutils literal notranslate">gaussInt</code> itself, together with the relevant operations, is an + instance of the <code class="docutils literal notranslate">CommRing</code> structure. The <code class="docutils literal notranslate">CommRing</code> structure, in turn, + extends the notational structures <code class="docutils literal notranslate">Zero</code>, <code class="docutils literal notranslate">One</code>, <code class="docutils literal notranslate">Add</code>, + <code class="docutils literal notranslate">Neg</code>, and <code class="docutils literal notranslate">Mul</code>. + If you type <code class="docutils literal notranslate">instance : CommRing gaussInt := _</code>, click on the light bulb + that appears in VS Code, and then ask Lean to fill in a skeleton for the + structure definition, you will see a scary number of entries. + Jumping to the definition of the structure, however, shows that many of the + fields have default definitions that Lean will fill in for you automatically. + The essential ones appear in the definition below. In each case, the relevant + identity is proved by unfolding definitions, using the <code class="docutils literal notranslate">ext</code> tactic + to reduce the identities to their real and imaginary components, + simplifying, and, if necessary, carrying out the relevant ring calculation in + the integers. + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance instCommRing : CommRing gaussInt where + zero := 0 + one := 1 + add := (· + ·) + neg x := -x + mul := (· * ·) + add_assoc := by + intros + ext <;> simp <;> ring + zero_add := by + intro + ext <;> simp + add_zero := by + intro + ext <;> simp + add_left_neg := by + intro + ext <;> simp + add_comm := by + intros + ext <;> simp <;> ring + mul_assoc := by + intros + ext <;> simp <;> ring + one_mul := by + intro + ext <;> simp + mul_one := by + intro + ext <;> simp + left_distrib := by + intros + ext <;> simp <;> ring + right_distrib := by + intros + ext <;> simp <;> ring + mul_comm := by + intros + ext <;> simp <;> ring + zero_mul := sorry + mul_zero := sorry + </pre></div> + </div> + Lean’s library defines the class of nontrivial types to be types with at + least two distinct elements. In the context of a ring, this is equivalent + to saying the the zero is not equal to the one. Since some common theorems + depend on that fact, we may as well establish it now. + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance : Nontrivial gaussInt := by + use 0, 1 + rw [Ne, gaussInt.ext_iff] + simp + </pre></div> + </div> + We will now show that the Gaussian integers have an important additional + property. A Euclidean domain is a ring $R$ equipped with a norm + function $N : R \to \mathbb{N}$ with the following two properties: + <ul class="simple"> + <li>For every $a$ and $b \ne 0$ in $R$, there are + $q$ and $r$ in $R$ such that $a = bq + r$ and + either $r = 0$ or <cite>N(r) < N(b)</cite>.</li> + <li>For every $a$ and $b \ne 0$, $N(a) \le N(ab)$.</li> + </ul> + The ring of integers $\Bbb{Z}$ with $N(a) = |a|$ is an + archetypal example of a Euclidean domain. + In that case, we can take $q$ to be the + result of integer division of $a$ by $b$ and $r$ + to be the remainder. These functions are defined in Lean so that the + satisfy the following: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a b : ℤ) : a = b * (a / b) + a % b := + Eq.symm <| Int.ediv_add_emod a b + + example (a b : ℤ) : b ≠ 0 → 0 ≤ a % b := + Int.emod_nonneg a + + example (a b : ℤ) : b ≠ 0 → a % b < abs b := + Int.emod_lt a + </pre></div> + </div> + In an arbitrary ring, an element $a$ is said to be a unit if it divides + $1$. A nonzero element $a$ is said to be irreducible if it cannot + be written in the form $a = bc$ + where neither $b$ nor $c$ is a unit. In the integers, every + irreducible element $a$ is prime, which is to say, whenever $a$ + divides a product $bc$, it divides either $b$ or $c$. But + in other rings this property can fail. In the ring + $\Bbb{Z}[\sqrt{-5}]$, we have + <div class="math notranslate nohighlight"> + \[6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5}),\]</div> + and the elements $2$, $3$, $1 + \sqrt{-5}$, and + $1 - \sqrt{-5}$ are all irreducible, but they are not prime. For example, + $2$ divides the product $(1 + \sqrt{-5})(1 - \sqrt{-5})$, + but it does not divide either factor. In particular, we no longer have + unique factorization: the number $6$ can be factored into irreducible + elements in more than one way. + In contrast, every Euclidean domain is a unique factorization domain, which + implies that every irreducible element is prime. + The axioms for a Euclidean domain imply that one can write any nonzero element + as a finite product of irreducible elements. They also imply that one can use + the Euclidean algorithm to find a greatest common divisor of any two + nonzero elements <code class="docutils literal notranslate">a</code> and <code class="docutils literal notranslate">b</code>, i.e.~an element that is divisible by any + other common divisor. This, in turn, implies that factorization + into irreducible elements is unique up to multiplication by units. + We now show that the Gaussian integers are a Euclidean domain with + the norm defined by $N(a + bi) = (a + bi)(a - bi) = a^2 + b^2$. + The Gaussian integer $a - bi$ is called the conjugate of $a + bi$. + It is not hard to check that for any complex numbers $x$ and $y$, + we have $N(xy) = N(x)N(y)$. + To see that this definition of the norm makes the complex numbers a Euclidean + domain, only the first property is challenging. Suppose + we want to write $a + bi = (c + di) q + r$ for suitable $q$ + and $r$. Treating $a + bi$ and $c + di$ are complex + numbers, carry out the division + <div class="math notranslate nohighlight"> + \[\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c-di)} = + \frac{ac + bd}{c^2 + d^2} + \frac{bc -ad}{c^2+d^2} i.\]</div> + The real and imaginary parts might not be integers, but we can round + them to the nearest integers $u$ and $v$. We can then express the + right-hand size as $(u + vi) + (u' + v'i)$, where + $u' + v'i$ is the part left over. Note that we have + $|u'| \le 1/2$ and $|v'| \le 1/2$, and hence + <div class="math notranslate nohighlight"> + \[N(u' + v' i) = (u')^2 + (v')^2 \le 1/4 + 1/4 \le 1/2.\]</div> + Multiplying through by $c + di$, we have + <div class="math notranslate nohighlight"> + \[a + bi = (c + di) (u + vi) + (c + di) (u' + v'i).\]</div> + Setting $q = u + vi$ and $r = (c + di) (u' + v'i)$, we have + $a + bi = (c + di) q + r$, and we only need to + bound $N(r)$: + <div class="math notranslate nohighlight"> + \[N(r) = N(c + di)N(u' + v'i) \le N(c + di) \cdot 1/2 < N(c + di).\]</div> + The argument we just carried out requires viewing the Gaussian integers + as a subset of the complex numbers. One option for formalizing it in Lean + is therefore to embed the Gaussian integers in the complex numbers, embed + the integers in the Gaussian integers, define the rounding function from the + real numbers to the integers, and take great care to pass back and forth + between these number systems appropriately. + In fact, this is exactly the approach that is followed in mathlib, + where the Gaussian integers themselves are constructed as a special case + of a ring of quadratic integers. + See the file <a class="reference external" href="https://github.com/leanprover-community/mathlib/blob/master/src/number_theory/zsqrtd/gaussian_int.lean">gaussian_int.lean</a>. + Here we will instead carry out an argument that stays in the integers. + This illustrates an choice one commonly faces when formalizing mathematics. + Given an argument that requires concepts or machinery that is not already + in the library, one has two choices: either formalizes the concepts or machinery + needed, or adapt the argument to make use of concepts and machinery you + already have. + The first choice is generally a good investment of time when the results + can be used in other contexts. + Pragmatically speaking, however, sometimes seeking a more elementary proof + is more efficient. + The usual quotient-remainder theorem for the integers says that for + every $a$ and nonzero $b$, there are $q$ and $r$ + such that $a = b q + r$ and $0 \le r < b$. + Here we will make use of the following variation, which says that there + are $q'$ and $r'$ such that + $a = b q' + r'$ and $|r'| \le b/2$. + You can check that if the value of $r$ in the first statement + satisfies $r \le b/2$, we can take $q' = q$ and $r' = r$, + and otherwise we can take $q' = q + 1$ and $r' = r - b$. + We are grateful to Heather Macbeth for suggesting the following more + elegant approach, which avoids definition by cases. + We simply add <code class="docutils literal notranslate">b / 2</code> to <code class="docutils literal notranslate">a</code> before dividing and then subtract it + from the remainder. + <div class="highlight-lean notranslate"><div class="highlight"><pre>def div' (a b : ℤ) := + (a + b / 2) / b + + def mod' (a b : ℤ) := + (a + b / 2) % b - b / 2 + + theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by + rw [div', mod'] + linarith [Int.ediv_add_emod (a + b / 2) b] + + theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by + rw [mod', abs_le] + constructor + · linarith [Int.emod_nonneg (a + b / 2) h.ne'] + have := Int.emod_lt_of_pos (a + b / 2) h + have := Int.ediv_add_emod b 2 + have := Int.emod_lt_of_pos b zero_lt_two + revert this; intro this -- FIXME, this should not be needed + linarith + </pre></div> + </div> + Note the use of our old friend, <code class="docutils literal notranslate">linarith</code>. We will also need to express + <code class="docutils literal notranslate">mod'</code> in terms of <code class="docutils literal notranslate">div'</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] + </pre></div> + </div> + We will use the fact that $x^2 + y^2$ is equal to zero if and only if + $x$ and $y$ are both zero. As an exercise, we ask you to prove + that this holds in any ordered ring. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : + x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by + sorry + </pre></div> + </div> + We will put all the remaining definitions and theorems in this section + in the <code class="docutils literal notranslate">gaussInt</code> namespace. + First, we define the <code class="docutils literal notranslate">norm</code> function and ask you to establish + some of its properties. + The proofs are all short. + <div class="highlight-lean notranslate"><div class="highlight"><pre>def norm (x : gaussInt) := + x.re ^ 2 + x.im ^ 2 + + @[simp] + theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by + sorry + theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by + sorry + theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by + sorry + theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by + sorry + </pre></div> + </div> + Next we define the conjugate function: + <div class="highlight-lean notranslate"><div class="highlight"><pre>def conj (x : gaussInt) : gaussInt := + ⟨x.re, -x.im⟩ + + @[simp] + theorem conj_re (x : gaussInt) : (conj x).re = x.re := + rfl + + @[simp] + theorem conj_im (x : gaussInt) : (conj x).im = -x.im := + rfl + + theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] + </pre></div> + </div> + Finally, we define division for the Gaussian integers + with the notation <code class="docutils literal notranslate">x / y</code>, that rounds the complex quotient to the nearest + Gaussian integer. We use our bespoke <code class="docutils literal notranslate">Int.div'</code> for that purpose. + As we calculated above, if <code class="docutils literal notranslate">x</code> is $a + bi$ and <code class="docutils literal notranslate">y</code> is $c + di$, + then the real and imaginary parts of <code class="docutils literal notranslate">x / y</code> are the nearest integers to + <div class="math notranslate nohighlight"> + \[\frac{ac + bd}{c^2 + d^2} \quad \text{and} \quad \frac{bc -ad}{c^2+d^2},\]</div> + respectively. Here the numerators are the real and imaginary parts of + $(a + bi) (c - di)$, and the denominators are both equal to the norm + of $c + di$. + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance : Div gaussInt := + ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ + </pre></div> + </div> + Having defined <code class="docutils literal notranslate">x / y</code>, We define <code class="docutils literal notranslate">x % y</code> to be the remainder, + <code class="docutils literal notranslate">x - (x / y) * y</code>. As above, we record the definitions in the + theorems <code class="docutils literal notranslate">div_def</code> and + <code class="docutils literal notranslate">mod_def</code> so that we can use them with <code class="docutils literal notranslate">simp</code> and <code class="docutils literal notranslate">rewrite</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance : Mod gaussInt := + ⟨fun x y => x - y * (x / y)⟩ + + theorem div_def (x y : gaussInt) : + x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := + rfl + + theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := + rfl + </pre></div> + </div> + These definitions immediately yield <code class="docutils literal notranslate">x = y * (x / y) + x % y</code> for every + <code class="docutils literal notranslate">x</code> and <code class="docutils literal notranslate">y</code>, so all we need to do is show that the norm of <code class="docutils literal notranslate">x % y</code> is + less than the norm of <code class="docutils literal notranslate">y</code> when <code class="docutils literal notranslate">y</code> is not zero. + We just defined the real and imaginary parts of <code class="docutils literal notranslate">x / y</code> to be + <code class="docutils literal notranslate">div' (x * conj y).re (norm y)</code> and <code class="docutils literal notranslate">div' (x * conj y).im (norm y)</code>, + respectively. + Calculating, we have + <blockquote> + <div><code class="docutils literal notranslate">(x % y) * conj y = (x - x / y * y) * conj y = x * conj y - x * (y * conj j)</code> + </div></blockquote> + The real and imaginary parts of the right-hand side are exactly <code class="docutils literal notranslate">mod' (x * conj y).re (norm y)</code> and <code class="docutils literal notranslate">mod' (x * conj y).im (norm y)</code>. + By the properties of <code class="docutils literal notranslate">div'</code> and <code class="docutils literal notranslate">mod'</code>, + these are guaranteed to be less than or equal to <code class="docutils literal notranslate">norm y / 2</code>. + So we have + <blockquote> + <div><code class="docutils literal notranslate">norm ((x % y) * conj y) ≤ (norm y / 2)^2 + (norm y / 2)^2 ≤ (norm y / 2) * norm y</code>. + </div></blockquote> + On the other hand, we have + <blockquote> + <div><code class="docutils literal notranslate">norm ((x % y) * conj y) = norm (x % y) * norm (conj y) = norm (x % y) * norm y</code>. + </div></blockquote> + Dividing through by <code class="docutils literal notranslate">norm y</code> we have <code class="docutils literal notranslate">norm (x % y) ≤ (norm y) / 2 < norm y</code>, + as required. + This messy calculation is carried out in the next proof. We encourage you + to step through the details and see if you can find a nicer argument. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : (x % y).norm < y.norm := by + have norm_y_pos : 0 < norm y := by rwa [norm_pos] + have : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by + rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] + ext <;> simp <;> ring + have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by + conv => + lhs + rw [← norm_conj y, ← norm_mul, this, norm] + simp + trans 2 * (y.norm / 2) ^ 2 + · rw [two_mul] + apply add_le_add <;> + · rw [sq_le_sq] + apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) + apply le_abs_self + rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] + apply mul_le_mul_of_nonneg_left + · apply Int.ediv_mul_le + norm_num + apply Int.ediv_nonneg (norm_nonneg y) + norm_num + have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos + apply lt_of_le_of_lt this + apply Int.ediv_lt_of_lt_mul + · norm_num + linarith + </pre></div> + </div> + We are in the home stretch. Our <code class="docutils literal notranslate">norm</code> function maps Gaussian integers to + nonnegative integers. We need a function that maps Gaussian integers to natural + numbers, and we obtain that by composing <code class="docutils literal notranslate">norm</code> with the function + <code class="docutils literal notranslate">Int.natAbs</code>, which maps integers to the natural numbers. + The first of the next two lemmas establishes that mapping the norm to the + natural numbers and back to the integers does not change the value. + The second one re-expresses the fact that the norm is decreasing. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := + Int.natAbs_of_nonneg (norm_nonneg _) + + theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : (x % y).norm.natAbs < y.norm.natAbs := by + apply Int.ofNat_lt.1 + simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] + apply norm_mod_lt x hy + </pre></div> + </div> + We also need to establish the second key property of the norm function + on a Euclidean domain. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : + ¬(norm (x * y)).natAbs < (norm x).natAbs := by + apply not_lt_of_ge + rw [norm_mul, Int.natAbs_mul] + apply le_mul_of_one_le_right (Nat.zero_le _) + apply Int.ofNat_le.1 + rw [coe_natAbs_norm] + exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) + </pre></div> + </div> + We can now put it together to show that the Gaussian integers are an + instance of a Euclidean domain. We use the quotient and remainder function we + have defined. + The mathlib definition of a Euclidean domain is more general than the one + above in that it allows us to show that remainder decreases with respect + to any well-founded measure. + Comparing the values of a norm function that returns natural numbers is + just one instance of such a measure, + and in that case, the required properties are the theorems + <code class="docutils literal notranslate">natAbs_norm_mod_lt</code> and <code class="docutils literal notranslate">not_norm_mul_left_lt_norm</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance : EuclideanDomain gaussInt := + { gaussInt.instCommRing with + quotient := (· / ·) + remainder := (· % ·) + quotient_mul_add_remainder_eq := fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] + quotient_zero := fun x => by + simp [div_def, norm, Int.div'] + rfl + r := Measure (Int.natAbs ∘ norm) + r_wellFounded := (measure (Int.natAbs ∘ norm)).2 + remainder_lt := natAbs_norm_mod_lt + mul_left_not_lt := not_norm_mul_left_lt_norm } + </pre></div> + </div> + An immediate payoff is that we now know that, in the Gaussian integers, + the notions of being prime and being irreducible coincide. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x : gaussInt) : Irreducible x ↔ Prime x := + PrincipalIdealRing.irreducible_iff_prime + </pre></div> + </div> + </section> + </section> + + + </div> + </div> + <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> + <a href="C05_Number_Theory.html" class="btn btn-neutral float-left" title="5. Number Theory" accesskey="p" rel="prev"> Previous</a> + <a href="C07_Topology.html" class="btn btn-neutral float-right" title="7. Topology" accesskey="n" rel="next">Next </a> + </div> + + <hr/> + + <div role="contentinfo"> + © Copyright 2020, Jeremy Avigad, Kevin Buzzard, Robert Y. Lewis, Patrick Massot. + </div> + + Built with <a href="https://www.sphinx-doc.org/">Sphinx</a> using a + <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a> + provided by <a href="https://readthedocs.org">Read the Docs</a>. + + + </footer> + </div> + </div> + </section> + </div> + <script> + jQuery(function () { + SphinxRtdTheme.Navigation.enable(true); + }); + </script> + + </body> + </html>
html/C07_Topology.html (new)
- @@ -0,0 +1,1217 @@
 + <!DOCTYPE html> + <html class="writer-html5" lang="en" > + <head> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + + <meta name="viewport" content="width=device-width, initial-scale=1.0" /> + <title>7. Topology — Mathematics in Lean 0.1 documentation</title> + <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="shortcut icon" href="_static/favicon.ico"/> +  + + <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> + <script src="_static/jquery.js"></script> + <script src="_static/underscore.js"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> + <script src="_static/doctools.js"></script> + <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> + <script src="_static/js/theme.js"></script> + <link rel="index" title="Index" href="genindex.html" /> + <link rel="search" title="Search" href="search.html" /> + <link rel="next" title="Index" href="genindex.html" /> + <link rel="prev" title="6. Abstract Algebra" href="C06_Abstract_Algebra.html" /> + </head> + + <body class="wy-body-for-nav"> + <div class="wy-grid-for-nav"> + <nav data-toggle="wy-nav-shift" class="wy-nav-side"> + <div class="wy-side-scroll"> + <div class="wy-side-nav-search" > + <a href="index.html" class="icon icon-home"> Mathematics in Lean + </a> + <div role="search"> + <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> + <input type="text" name="q" placeholder="Search docs" /> + <input type="hidden" name="check_keywords" value="yes" /> + <input type="hidden" name="area" value="default" /> + </form> + </div> + </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> + <ul class="current"> + <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a></li> + <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> + <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> + <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> + <li class="toctree-l1"><a class="reference internal" href="C06_Abstract_Algebra.html">6. Abstract Algebra</a></li> + <li class="toctree-l1 current"><a class="current reference internal" href="#">7. Topology</a><ul> + <li class="toctree-l2"><a class="reference internal" href="#filters">7.1. Filters</a></li> + <li class="toctree-l2"><a class="reference internal" href="#metric-spaces">7.2. Metric spaces</a><ul> + <li class="toctree-l3"><a class="reference internal" href="#convergence-and-continuity">7.2.1. Convergence and continuity</a></li> + <li class="toctree-l3"><a class="reference internal" href="#balls-open-sets-and-closed-sets">7.2.2. Balls, open sets and closed sets</a></li> + <li class="toctree-l3"><a class="reference internal" href="#compactness">7.2.3. Compactness</a></li> + <li class="toctree-l3"><a class="reference internal" href="#uniformly-continuous-functions">7.2.4. Uniformly continuous functions</a></li> + <li class="toctree-l3"><a class="reference internal" href="#completeness">7.2.5. Completeness</a></li> + </ul> + </li> + <li class="toctree-l2"><a class="reference internal" href="#topological-spaces">7.3. Topological spaces</a><ul> + <li class="toctree-l3"><a class="reference internal" href="#fundamentals">7.3.1. Fundamentals</a></li> + <li class="toctree-l3"><a class="reference internal" href="#separation-and-countability">7.3.2. Separation and countability</a></li> + <li class="toctree-l3"><a class="reference internal" href="#id5">7.3.3. Compactness</a></li> + </ul> + </li> + </ul> + </li> + </ul> + <ul> + <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li> + </ul> + + </div> + </div> + </nav> + + <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > + + <a href="index.html">Mathematics in Lean</a> + </nav> + + <div class="wy-nav-content"> + <div class="rst-content"> + <div role="navigation" aria-label="Page navigation"> + <ul class="wy-breadcrumbs"> + <li><a href="index.html" class="icon icon-home"></a> »</li> + <li>7. Topology</li> + <li class="wy-breadcrumbs-aside"> + <a href="_sources/C07_Topology.rst.txt" rel="nofollow"> View page source</a> + </li> + </ul> + <hr/> + </div> + <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> + <div itemprop="articleBody"> + + <section id="index-0"> + <h1>7. Topology<a class="headerlink" href="#index-0" title="Permalink to this heading"></a></h1> + Calculus is based on the concept of a function, which is used to model + quantities that depend on one another. + For example, it is common to study quantities that change over time. + The notion of a limit is also fundamental. + We may say that the limit of a function $f(x)$ is a value $b$ + as $x$ approaches a value $a$, + or that $f(x)$ converges to $b$ as $x$ approaches $a$. + Equivalently, we may say that a $f(x)$ approaches $a$ as $x$ + approaches a value $b$, or that it tends to $b$ + as $x$ tends to $a$. + We have already begun to consider such notions in <a class="reference internal" href="C03_Logic.html#sequences-and-convergence">Section 3.6</a>. + Topology is the abstract study of limits and continuity. + Having covered the essentials of formalization in Chapters <a class="reference internal" href="C02_Basics.html#basics">2</a> + to <a class="reference internal" href="C06_Abstract_Algebra.html#abstract-algebra">6</a>, + in this chapter, we will explain how topological notions are formalized in mathlib. + Not only do topological abstractions apply in much greater generality, + but that also, somewhat paradoxically, make it easier to reason about limits + and continuity in concrete instances. + Topological notions build on quite a few layers of mathematical structure. + The first layer is naive set theory, + as described in <a class="reference internal" href="C04_Sets_and_Functions.html#sets-and-functions">Chapter 4</a>. + The next layer is the theory of filters, which we will describe in <a class="reference internal" href="#filters">Section 7.1</a>. + On top of that, we layer + the theories of topological spaces, metric spaces, and a slightly more exotic + intermediate notion called a uniform space. + Whereas previous chapters relied on mathematical notions that were likely + familiar to you, + the notion of a filter less well known, + even to many working mathematicians. + The notion is essential, however, for formalizing mathematics effectively. + Let us explain why. + Let <code class="docutils literal notranslate">f : ℝ → ℝ</code> be any function. We can consider + the limit of <code class="docutils literal notranslate">f x</code> as <code class="docutils literal notranslate">x</code> approaches some value <code class="docutils literal notranslate">x₀</code>, + but we can also consider the limit of <code class="docutils literal notranslate">f x</code> as <code class="docutils literal notranslate">x</code> approaches infinity + or negative infinity. + We can moreover consider the limit of <code class="docutils literal notranslate">f x</code> as <code class="docutils literal notranslate">x</code> approaches <code class="docutils literal notranslate">x₀</code> from + the right, conventionally written <code class="docutils literal notranslate">x₀⁺</code>, or from the left, + written <code class="docutils literal notranslate">x₀⁻</code>. There are variations where <code class="docutils literal notranslate">x</code> approaches <code class="docutils literal notranslate">x₀</code> or <code class="docutils literal notranslate">x₀⁺</code> + or <code class="docutils literal notranslate">x₀⁻</code> but + is not allowed to take on the value <code class="docutils literal notranslate">x₀</code> itself. + This results in at least eight ways that <code class="docutils literal notranslate">x</code> can approach something. + We can also restrict to rational values of <code class="docutils literal notranslate">x</code> + or place other constraints on the domain, but let’s stick to those 8 cases. + We have a similar variety of options on the codomain: + we can specify that <code class="docutils literal notranslate">f x</code> approaches a value from the left or right, + or that it approaches positive or negative infinity, and so on. + For example, we may wish to say that <code class="docutils literal notranslate">f x</code> tends to <code class="docutils literal notranslate">+∞</code> + when <code class="docutils literal notranslate">x</code> tends to <code class="docutils literal notranslate">x₀</code> from the right without + being equal to <code class="docutils literal notranslate">x₀</code>. + This results in 64 different kinds of limit statements, + and we haven’t even begun to deal with limits of sequences, + as we did in <a class="reference internal" href="C03_Logic.html#sequences-and-convergence">Section 3.6</a>. + The problem is compounded even further when it comes to the supporting lemmas. + For instance, limits compose: if + <code class="docutils literal notranslate">f x</code> tends to <code class="docutils literal notranslate">y₀</code> when <code class="docutils literal notranslate">x</code> tends to <code class="docutils literal notranslate">x₀</code> and + <code class="docutils literal notranslate">g y</code> tends to <code class="docutils literal notranslate">z₀</code> when <code class="docutils literal notranslate">y</code> tends to <code class="docutils literal notranslate">y₀</code> then + <code class="docutils literal notranslate">g ∘ f x</code> tends to <code class="docutils literal notranslate">z₀</code> when <code class="docutils literal notranslate">x</code> tends to <code class="docutils literal notranslate">x₀</code>. + There are three notions of “tends to” at play here, + each of which can be instantiated in any of the eight ways described + in the previous paragraph. + This results in 512 lemmas, a lot to have to add to a library! + Informally, mathematicians generally prove two or three of these + and simply note that the rest can be proved “in the same way.” + Formalizing mathematics requires making the relevant notion of “sameness” + fully explicit, and that is exactly what Bourbaki’s theory of filters + manages to do. + <section id="filters"> + <h2>7.1. Filters<a class="headerlink" href="#filters" title="Permalink to this heading"></a></h2> + A filter on a type <code class="docutils literal notranslate">X</code> is a collection of sets of <code class="docutils literal notranslate">X</code> that satisfies three + conditions that we will spell out below. The notion + supports two related ideas: + <ul class="simple"> + <li>limits, including all the kinds of limits discussed above: finite and infinite limits of sequences, finite and infinite limits of functions at a point or at infinity, and so on.</li> + <li>things happening eventually, including things happening for large enough <code class="docutils literal notranslate">n : ℕ</code>, or sufficiently near a point <code class="docutils literal notranslate">x</code>, or for sufficiently close pairs of points, or almost everywhere in the sense of measure theory. Dually, filters can also express the idea of things happening often: for arbitrarily large <code class="docutils literal notranslate">n</code>, at a point in any neighborhood of given a point, etc.</li> + </ul> + The filters that correspond to these descriptions will be defined later in this section, but we can already name them: + <ul class="simple"> + <li><code class="docutils literal notranslate">(at_top : Filter ℕ)</code>, made of sets of <code class="docutils literal notranslate">ℕ</code> containing <code class="docutils literal notranslate">{n | n ≥ N}</code> for some <code class="docutils literal notranslate">N</code></li> + <li><code class="docutils literal notranslate">𝓝 x</code>, made of neighborhoods of <code class="docutils literal notranslate">x</code> in a topological space</li> + <li><code class="docutils literal notranslate">𝓤 X</code>, made of entourages of a uniform space (uniform spaces generalize metric spaces and topological groups)</li> + <li><code class="docutils literal notranslate">μ.a_e</code> , made of sets whose complement has zero measure with respect to a measure <code class="docutils literal notranslate">μ</code>.</li> + </ul> + The general definition is as follows: a filter <code class="docutils literal notranslate">F : Filter X</code> is a + collection of sets <code class="docutils literal notranslate">F.sets : Set (Set X)</code> satisfying the following: + <ul class="simple"> + <li><code class="docutils literal notranslate">F.univ_sets : univ ∈ F.sets</code></li> + <li><code class="docutils literal notranslate">F.sets_of_superset : ∀ {U V}, U ∈ F.sets → U ⊆ V → V ∈ F.sets</code></li> + <li><code class="docutils literal notranslate">F.inter_sets : ∀ {U V}, U ∈ F.sets → V ∈ F.sets → U ∩ V ∈ F.sets</code>.</li> + </ul> + The first condition says that the set of all elements of <code class="docutils literal notranslate">X</code> belongs to <code class="docutils literal notranslate">F.sets</code>. + The second condition says that if <code class="docutils literal notranslate">U</code> belongs to <code class="docutils literal notranslate">F.sets</code> then anything + containing <code class="docutils literal notranslate">U</code> also belongs to <code class="docutils literal notranslate">F.sets</code>. + The third condition says that <code class="docutils literal notranslate">F.sets</code> is closed under finite intersections. + In mathlib, a filter <code class="docutils literal notranslate">F</code> is defined to be a structure bundling <code class="docutils literal notranslate">F.sets</code> and its + three properties, but the properties carry no additional data, + and it is convenient to blur the distinction between <code class="docutils literal notranslate">F</code> and <code class="docutils literal notranslate">F.sets</code>. We + therefore define <code class="docutils literal notranslate">U ∈ F</code> to mean <code class="docutils literal notranslate">U ∈ F.sets</code>. + This explains why the word <code class="docutils literal notranslate">sets</code> appears in the names of some lemmas that + that mention <code class="docutils literal notranslate">U ∈ F</code>. + It may help to think of a filter as defining a notion of a “sufficiently large” set. The first + condition then says that <code class="docutils literal notranslate">univ</code> is sufficiently large, the second one says that a set containing a sufficiently + large set is sufficiently large and the third one says that the intersection of two sufficiently large sets + is sufficiently large. + It may be even more useful to think of a filter on a type <code class="docutils literal notranslate">X</code> + as a generalized element of <code class="docutils literal notranslate">Set X</code>. For instance, <code class="docutils literal notranslate">at_top</code> is the + “set of very large numbers” and <code class="docutils literal notranslate">𝓝 x₀</code> is the “set of points very close to <code class="docutils literal notranslate">x₀</code>.” + One manifestation of this view is that we can associate to any <code class="docutils literal notranslate">s : Set X</code> the so-called principal filter + consisting of all sets that contain <code class="docutils literal notranslate">s</code>. + This definition is already in mathlib and has a notation <code class="docutils literal notranslate">𝓟</code> (localized in the <code class="docutils literal notranslate">Filter</code> namespace). + For the purpose of demonstration, we ask you to take this opportunity to work out the definition here. + <div class="highlight-lean notranslate"><div class="highlight"><pre>def principal {α : Type _} (s : Set α) : Filter α + where + sets := { t | s ⊆ t } + univ_sets := sorry + sets_of_superset := sorry + inter_sets := sorry + </pre></div> + </div> + For our second example, we ask you to define the filter <code class="docutils literal notranslate">at_top : Filter ℕ</code>. + (We could use any type with a preorder instead of <code class="docutils literal notranslate">ℕ</code>.) + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : Filter ℕ := + { sets := { s | ∃ a, ∀ b, a ≤ b → b ∈ s } + univ_sets := sorry + sets_of_superset := sorry + inter_sets := sorry } + </pre></div> + </div> + We can also directly define the filter <code class="docutils literal notranslate">𝓝 x</code> of neighborhoods of any <code class="docutils literal notranslate">x : ℝ</code>. + In the real numbers, a neighborhood of <code class="docutils literal notranslate">x</code> is a set containing an open interval + $(x_0 - \varepsilon, x_0 + \varepsilon)$, + defined in mathlib as <code class="docutils literal notranslate">Ioo (x₀ - ε) (x₀ + ε)</code>. + (This is notion of a neighborhood is only a special case of a more general construction in mathlib.) + With these examples, we can already define what is means for a function <code class="docutils literal notranslate">f : X → Y</code> + to converge to some <code class="docutils literal notranslate">G : Filter Y</code> along some <code class="docutils literal notranslate">F : Filter X</code>, + as follows: + <div class="highlight-lean notranslate"><div class="highlight"><pre>def Tendsto₁ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := + ∀ V ∈ G, f ⁻¹' V ∈ F + </pre></div> + </div> + When <code class="docutils literal notranslate">X</code> is <code class="docutils literal notranslate">ℕ</code> and <code class="docutils literal notranslate">Y</code> is <code class="docutils literal notranslate">ℝ</code>, <code class="docutils literal notranslate">Tendsto₁ u at_top (𝓝 x)</code> is equivalent to saying that the sequence <code class="docutils literal notranslate">u : ℕ → ℝ</code> + converges to the real number <code class="docutils literal notranslate">x</code>. When both <code class="docutils literal notranslate">X</code> and <code class="docutils literal notranslate">Y</code> are <code class="docutils literal notranslate">ℝ</code>, <code class="docutils literal notranslate">Tendsto f (𝓝 x₀) (𝓝 y₀)</code> + is equivalent to the familiar notion $\lim_{x \to x₀} f(x) = y₀$. + All of the other kinds of limits mentioned in the introduction are + also equivalent to instances of <code class="docutils literal notranslate">Tendsto₁</code> for suitable choices of filters on the source and target. + The notion <code class="docutils literal notranslate">Tendsto₁</code> above is definitionally equivalent to the notion <code class="docutils literal notranslate">Tendsto</code> that is defined in mathlib, + but the latter is defined more abstractly. + The problem with the definition of <code class="docutils literal notranslate">Tendsto₁</code> is that it exposes a quantifier and elements of <code class="docutils literal notranslate">G</code>, + and it hides the intuition that we get by viewing filters as generalized sets. We can + hide the quantifier <code class="docutils literal notranslate">∀ V</code> and make the intuition more salient by using more algebraic and set-theoretic machinery. + The first ingredient is the pushforward operation $f_*$ associated to any map <code class="docutils literal notranslate">f : X → Y</code>, + denoted <code class="docutils literal notranslate">Filter.map f</code> in mathlib. Given a filter <code class="docutils literal notranslate">F</code> on <code class="docutils literal notranslate">X</code>, <code class="docutils literal notranslate">Filter.map f F : Filter Y</code> is defined so that + <code class="docutils literal notranslate">V ∈ Filter.map f F ↔ f ⁻¹' V ∈ F</code> holds definitionally. + In this examples file we’ve opened the <code class="docutils literal notranslate">Filter</code> namespace so that + <code class="docutils literal notranslate">Filter.map</code> can be written as <code class="docutils literal notranslate">map</code>. This means that we can rewrite the definition of <code class="docutils literal notranslate">Tendsto</code> using + the order relation on <code class="docutils literal notranslate">Filter Y</code>, which is reversed inclusion of the set of members. + In other words, given <code class="docutils literal notranslate">G H : Filter Y</code>, we have <code class="docutils literal notranslate">G ≤ H ↔ ∀ V : Set Y, V ∈ H → V ∈ G</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>def Tendsto₂ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := + map f F ≤ G + + example {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) : + Tendsto₂ f F G ↔ Tendsto₁ f F G := + Iff.rfl + </pre></div> + </div> + It may seem that the order relation on filters is backward. But recall that we can view filters on <code class="docutils literal notranslate">X</code> as + generalized elements of <code class="docutils literal notranslate">Set X</code>, via the inclusion of <code class="docutils literal notranslate">𝓟 : Set X → Filter X</code> which maps any set <code class="docutils literal notranslate">s</code> to the corresponding principal filter. + This inclusion is order preserving, so the order relation on <code class="docutils literal notranslate">Filter</code> can indeed be seen as the natural inclusion relation + between generalized sets. In this analogy, pushforward is analogous to the direct image. + And, indeed, <code class="docutils literal notranslate">map f (𝓟 s) = 𝓟 (f '' s)</code>. + We can now understand intuitively why a sequence <code class="docutils literal notranslate">u : ℕ → ℝ</code> converges to + a point <code class="docutils literal notranslate">x₀</code> if and only if we have <code class="docutils literal notranslate">map u at_top ≤ 𝓝 x₀</code>. + The inequality means the “direct image under <code class="docutils literal notranslate">u</code>” of + “the set of very big natural numbers” is “included” in “the set of points very close to <code class="docutils literal notranslate">x₀</code>.” + As promised, the definition of <code class="docutils literal notranslate">Tendsto₂</code> does not exhibit any quantifiers or sets. + It also leverages the algebraic properties of the pushforward operation. + First, each <code class="docutils literal notranslate">Filter.map f</code> is monotone. And, second, <code class="docutils literal notranslate">Filter.map</code> is compatible with + composition. + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (@Filter.map_mono : ∀ {α β} {m : α → β}, Monotone (map m)) + + #check + (@Filter.map_map : + ∀ {α β γ} {f : Filter α} {m : α → β} {m' : β → γ}, map m' (map m f) = map (m' ∘ m) f) + </pre></div> + </div> + Together these two properties allow us to prove that limits compose, yielding in one shot all 256 variants + of the composition lemma described in the introduction, and lots more. + You can practice proving the following statement using either the definition + of <code class="docutils literal notranslate">Tendsto₁</code> in terms of the + universal quantifier or the algebraic definition, + together with the two lemmas above. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} + (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := + sorry + </pre></div> + </div> + The pushforward construction uses a map to push filters from the map source to the map target. + There also a pullback operation, <code class="docutils literal notranslate">Filter.comap</code>, going in the other direction. + This generalizes the + preimage operation on sets. For any map <code class="docutils literal notranslate">f</code>, + <code class="docutils literal notranslate">Filter.map f</code> and <code class="docutils literal notranslate">Filter.comap f</code> form what is known as a Galois connection, + which is to say, they satisfy + <blockquote> + <div><code class="docutils literal notranslate">Filter.map_le_iff_le_comap : Filter.map f F ≤ G ↔ F ≤ Filter.comap f G</code> + </div></blockquote> + for every <code class="docutils literal notranslate">F</code> and <code class="docutils literal notranslate">G</code>. + This operation could be used to provided another formulation of <code class="docutils literal notranslate">Tendsto</code> that would be provably + (but not definitionally) equivalent to the one in mathlib. + The <code class="docutils literal notranslate">comap</code> operation can be used to restrict filters to a subtype. For instance, suppose we have <code class="docutils literal notranslate">f : ℝ → ℝ</code>, + <code class="docutils literal notranslate">x₀ : ℝ</code> and <code class="docutils literal notranslate">y₀ : ℝ</code>, and suppose we want to state that <code class="docutils literal notranslate">f x</code> approaches <code class="docutils literal notranslate">y₀</code> when <code class="docutils literal notranslate">x</code> approaches <code class="docutils literal notranslate">x₀</code> within the rational numbers. + We can pull the filter <code class="docutils literal notranslate">𝓝 x₀</code> back to <code class="docutils literal notranslate">ℚ</code> using the coercion map + <code class="docutils literal notranslate">(↑) : ℚ → ℝ</code> and state <code class="docutils literal notranslate">Tendsto (f ∘ (↑) : ℚ → ℝ) (comap (↑) (𝓝 x₀)) (𝓝 y₀)</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (f : ℝ → ℝ) (x₀ y₀ : ℝ) + + #check comap ((↑) : ℚ → ℝ) (𝓝 x₀) + + #check Tendsto (f ∘ (↑)) (comap ((↑) : ℚ → ℝ) (𝓝 x₀)) (𝓝 y₀) + </pre></div> + </div> + The pullback operation is also compatible with composition, but it contravariant, + which is to say, it reverses the order of the arguments. + <div class="highlight-lean notranslate"><div class="highlight"><pre>section + variable {α β γ : Type _} (F : Filter α) {m : γ → β} {n : β → α} + + #check (comap_comap : comap m (comap n F) = comap (n ∘ m) F) + + end + </pre></div> + </div> + Let’s now shift attention to the plane <code class="docutils literal notranslate">ℝ × ℝ</code> and try to understand how the neighborhoods of a point + <code class="docutils literal notranslate">(x₀, y₀)</code> are related to <code class="docutils literal notranslate">𝓝 x₀</code> and <code class="docutils literal notranslate">𝓝 y₀</code>. There is a product operation + <code class="docutils literal notranslate">Filter.prod : Filter X → Filter Y → Filter (X × Y)</code>, denoted by <code class="docutils literal notranslate">×ᶠ</code>, which answers this question: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : 𝓝 (x₀, y₀) = 𝓝 x₀ ×ᶠ 𝓝 y₀ := + nhds_prod_eq + </pre></div> + </div> + The product operation is defined in terms of the pullback operation and the <code class="docutils literal notranslate">inf</code> operation: + <blockquote> + <div><code class="docutils literal notranslate">F ×ᶠ G = (comap prod.fst F) ⊓ (comap prod.snd G)</code>. + </div></blockquote> + Here the <code class="docutils literal notranslate">inf</code> operation refers to the lattice structure on <code class="docutils literal notranslate">filter X</code> for any type <code class="docutils literal notranslate">X</code>, whereby + <code class="docutils literal notranslate">F ⊓ G</code> is the greatest filter that is smaller than both <code class="docutils literal notranslate">F</code> and <code class="docutils literal notranslate">G</code>. + Thus the <code class="docutils literal notranslate">inf</code> operation generalizes the notion of the intersection of sets. + A lot of proofs in mathlib use all of the aforementioned structure (<code class="docutils literal notranslate">map</code>, <code class="docutils literal notranslate">comap</code>, <code class="docutils literal notranslate">inf</code>, <code class="docutils literal notranslate">sup</code>, and <code class="docutils literal notranslate">prod</code>) + to give algebraic proofs about convergence without ever referring to members of filters. + You can practice doing this in a proof of the following lemma, unfolding the definition of <code class="docutils literal notranslate">Tendsto</code> + and <code class="docutils literal notranslate">Filter.prod</code> if needed. + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check le_inf_iff + + example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : + Tendsto f atTop (𝓝 (x₀, y₀)) ↔ + Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := + sorry + </pre></div> + </div> + The ordered type <code class="docutils literal notranslate">filter X</code> is actually a complete lattice, + which is to say, there is a bottom element, there is a top element, and + every set of filters on <code class="docutils literal notranslate">X</code> has an <code class="docutils literal notranslate">Inf</code> and a <code class="docutils literal notranslate">Sup</code>. + Note that given the second property in the definition of a filter + (if <code class="docutils literal notranslate">U</code> belongs to <code class="docutils literal notranslate">F</code> then anything larger than <code class="docutils literal notranslate">U</code> also belongs to <code class="docutils literal notranslate">F</code>), + the first property + (the set of all inhabitants of <code class="docutils literal notranslate">X</code> belongs to <code class="docutils literal notranslate">F</code>) is + equivalent to the property that <code class="docutils literal notranslate">F</code> is not the empty collection of sets. + This shouldn’t be confused with the more subtle question as to whether + the empty set is an element of <code class="docutils literal notranslate">F</code>. The + definition of a filter does not prohibit <code class="docutils literal notranslate">∅ ∈ F</code>, + but if the empty set is in <code class="docutils literal notranslate">F</code> then + every set is in <code class="docutils literal notranslate">F</code>, which is to say, <code class="docutils literal notranslate">∀ U : Set X, U ∈ F</code>. + In this case, <code class="docutils literal notranslate">F</code> is a rather trivial filter, which is precisely the + bottom element of the complete lattice <code class="docutils literal notranslate">filter X</code>. + This contrasts with the definition of filters in + Bourbaki, which doesn’t allow filters containing the empty set. + Because we include the trivial filter in our definition, we sometimes need to explicitly assume + nontriviality in some lemmas. + In return, however, the theory has nicer global properties. + We have already seen that including the trivial filter gives us a + bottom element. It also allows us to define <code class="docutils literal notranslate">principal : Set X → Filter X</code>, + which maps <code class="docutils literal notranslate">∅</code> to <code class="docutils literal notranslate">⊥</code>, without adding a precondition to rule out the empty set. + And it allows us to define the pullback operation without a precondition as well. + Indeed, it can happen that <code class="docutils literal notranslate">comap f F = ⊥</code> although <code class="docutils literal notranslate">F ≠ ⊥</code>. For instance, + given <code class="docutils literal notranslate">x₀ : ℝ</code> and <code class="docutils literal notranslate">s : Set ℝ</code>, the pullback of <code class="docutils literal notranslate">𝓝 x₀</code> under the coercion + from the subtype corresponding to <code class="docutils literal notranslate">s</code> is nontrivial if and only if <code class="docutils literal notranslate">x₀</code> belongs to the + closure of <code class="docutils literal notranslate">s</code>. + In order to manage lemmas that do need to assume some filter is nontrivial, mathlib has + a type class <code class="docutils literal notranslate">Filter.ne_bot</code>, and the library has lemmas that assume + <code class="docutils literal notranslate">(F : Filter X) [F.ne_bot]</code>. The instance database knows, for example, that <code class="docutils literal notranslate">(at_top : Filter ℕ).ne_bot</code>, + and it knows that pushing forward a nontrivial filter gives a nontrivial filter. + As a result, a lemma assuming <code class="docutils literal notranslate">[F.ne_bot]</code> will automatically apply to <code class="docutils literal notranslate">map u at_top</code> for any sequence <code class="docutils literal notranslate">u</code>. + Our tour of the algebraic properties of filters and their relation to limits is essentially done, + but we have not yet justified our claim to have recaptured the usual limit notions. + Superficially, it may seem that <code class="docutils literal notranslate">Tendsto u at_top (𝓝 x₀)</code> + is stronger than the notion of convergence defined in <a class="reference internal" href="C03_Logic.html#sequences-and-convergence">Section 3.6</a> because we ask that every neighborhood of <code class="docutils literal notranslate">x₀</code> + has a preimage belonging to <code class="docutils literal notranslate">at_top</code>, whereas the usual definition only requires + this for the standard neighborhoods <code class="docutils literal notranslate">Ioo (x₀ - ε) (x₀ + ε)</code>. + The key is that, by definition, every neighborhood contains such a standard one. + This observation leads to the notion of a filter basis. + Given <code class="docutils literal notranslate">F : Filter X</code>, + a family of sets <cite>s : ι → Set X</cite> is a basis for <code class="docutils literal notranslate">F</code> if for every set <code class="docutils literal notranslate">U</code>, + we have <code class="docutils literal notranslate">U ∈ F</code> if and only if it contains some <code class="docutils literal notranslate">s i</code>. In other words, formally speaking, + <code class="docutils literal notranslate">s</code> is a basis if it satisfies + <code class="docutils literal notranslate">∀ U : Set X, U ∈ F ↔ ∃ i, s i ⊆ U</code>. It is even more flexible to consider + a predicate on <code class="docutils literal notranslate">ι</code> that selects only some of the values <code class="docutils literal notranslate">i</code> in the indexing type. + In the case of <code class="docutils literal notranslate">𝓝 x₀</code>, we want <code class="docutils literal notranslate">ι</code> to be <code class="docutils literal notranslate">ℝ</code>, we write <code class="docutils literal notranslate">ε</code> for <code class="docutils literal notranslate">i</code>, and the predicate should select the positive values of <code class="docutils literal notranslate">ε</code>. + So the fact that the sets <code class="docutils literal notranslate">Ioo  (x₀ - ε) (x₀ + ε)</code> form a basis for the + neighborhood topology on <code class="docutils literal notranslate">ℝ</code> is stated as follows: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x₀ : ℝ) : HasBasis (𝓝 x₀) (fun ε : ℝ => 0 < ε) fun ε => Ioo (x₀ - ε) (x₀ + ε) := + nhds_basis_Ioo_pos x₀ + </pre></div> + </div> + There is also a nice basis for the filter <code class="docutils literal notranslate">at_top</code>. The lemma + <code class="docutils literal notranslate">Filter.has_basis.tendsto_iff</code> allows + us to reformulate a statement of the form <code class="docutils literal notranslate">Tendsto f F G</code> + given bases for <code class="docutils literal notranslate">F</code> and <code class="docutils literal notranslate">G</code>. + Putting these pieces together gives us essentially the notion of convergence + that we used in <a class="reference internal" href="C03_Logic.html#sequences-and-convergence">Section 3.6</a>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (u : ℕ → ℝ) (x₀ : ℝ) : + Tendsto u atTop (𝓝 x₀) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, u n ∈ Ioo (x₀ - ε) (x₀ + ε) := by + have : atTop.HasBasis (fun _ : ℕ => True) Ici := atTop_basis + rw [this.tendsto_iff (nhds_basis_Ioo_pos x₀)] + simp + </pre></div> + </div> + We now show how filters facilitate working with properties that hold for sufficiently large numbers + or for points that are sufficiently close to a given point. In <a class="reference internal" href="C03_Logic.html#sequences-and-convergence">Section 3.6</a>, we were often faced with the situation where + we knew that some property <code class="docutils literal notranslate">P n</code> holds for sufficiently large <code class="docutils literal notranslate">n</code> and that some + other property <code class="docutils literal notranslate">Q n</code> holds for sufficiently large <code class="docutils literal notranslate">n</code>. + Using <code class="docutils literal notranslate">cases</code> twice gave us <code class="docutils literal notranslate">N_P</code> and <code class="docutils literal notranslate">N_Q</code> satisfying + <code class="docutils literal notranslate">∀ n ≥ N_P, P n</code> and <code class="docutils literal notranslate">∀ n ≥ N_Q, Q n</code>. Using <code class="docutils literal notranslate">set N := max N_P N_Q</code>, we could + eventually prove <code class="docutils literal notranslate">∀ n ≥ N, P n ∧ Q n</code>. + Doing this repeatedly becomes tiresome. + We can do better by noting that the statement “<code class="docutils literal notranslate">P n</code> and <code class="docutils literal notranslate">Q n</code> hold for large enough <code class="docutils literal notranslate">n</code>” means + that we have <code class="docutils literal notranslate">{n | P n} ∈ at_top</code> and <code class="docutils literal notranslate">{n | Q n} ∈ at_top</code>. + The fact that <code class="docutils literal notranslate">at_top</code> is a filter implies that the intersection of two elements of <code class="docutils literal notranslate">at_top</code> + is again in <code class="docutils literal notranslate">at_top</code>, so we have <code class="docutils literal notranslate">{n | P n ∧ Q n} ∈ at_top</code>. + Writing <code class="docutils literal notranslate">{n | P n} ∈ at_top</code> is unpleasant, + but we can use the more suggestive notation <code class="docutils literal notranslate">∀ᶠ n in at_top, P n</code>. + Here the superscripted <code class="docutils literal notranslate">f</code> stands for “Filter.” + You can think of the notation as saying that for all <code class="docutils literal notranslate">n</code> in the “set of very large numbers,” <code class="docutils literal notranslate">P n</code> holds. The <code class="docutils literal notranslate">∀ᶠ</code> + notation stands for <code class="docutils literal notranslate">Filter.Eventually</code>, and the lemma <code class="docutils literal notranslate">Filter.Eventually.and</code> uses the intersection property of filters to do what we just described: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (P Q : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in atTop, Q n) : + ∀ᶠ n in atTop, P n ∧ Q n := + hP.and hQ + </pre></div> + </div> + This notation is so convenient and intuitive that we also have specializations + when <code class="docutils literal notranslate">P</code> is an equality or inequality statement. For example, let <code class="docutils literal notranslate">u</code> and <code class="docutils literal notranslate">v</code> be + two sequences of real numbers, and let us show that if + <code class="docutils literal notranslate">u n</code> and <code class="docutils literal notranslate">v n</code> coincide for sufficiently large <code class="docutils literal notranslate">n</code> then + <code class="docutils literal notranslate">u</code> tends to <code class="docutils literal notranslate">x₀</code> if and only if <code class="docutils literal notranslate">v</code> tends to <code class="docutils literal notranslate">x₀</code>. + First we’ll use the generic <code class="docutils literal notranslate">Eventually</code> and then the one + specialized for the equality predicate, <code class="docutils literal notranslate">Eventually_eq</code>. The two statements are + definitionally equivalent so the same proof work in both cases. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (u v : ℕ → ℝ) (h : ∀ᶠ n in atTop, u n = v n) (x₀ : ℝ) : + Tendsto u atTop (𝓝 x₀) ↔ Tendsto v atTop (𝓝 x₀) := + tendsto_congr' h + + example (u v : ℕ → ℝ) (h : u =ᶠ[atTop] v) (x₀ : ℝ) : + Tendsto u atTop (𝓝 x₀) ↔ Tendsto v atTop (𝓝 x₀) := + tendsto_congr' h + </pre></div> + </div> + It is instructive to review the definition of filters in terms of <code class="docutils literal notranslate">Eventually</code>. + Given <code class="docutils literal notranslate">F : Filter X</code>, for any predicates <code class="docutils literal notranslate">P</code> and <code class="docutils literal notranslate">Q</code> on <code class="docutils literal notranslate">X</code>, + <ul class="simple"> + <li>the condition <code class="docutils literal notranslate">univ ∈ F</code> ensures <code class="docutils literal notranslate">(∀ x, P x) → ∀ᶠ x in F, P x</code>,</li> + <li>the condition <code class="docutils literal notranslate">U ∈ F → U ⊆ V → V ∈ F</code> ensures <code class="docutils literal notranslate">(∀ᶠ x in F, P x) → (∀ x, P x → Q x) → ∀ᶠ x in F, Q x</code>, and</li> + <li>the condition <code class="docutils literal notranslate">U ∈ F → V ∈ F → U ∩ V ∈ F</code> ensures <code class="docutils literal notranslate">(∀ᶠ x in F, P x) → (∀ᶠ x in F, Q x) → ∀ᶠ x in F, P x ∧ Q x</code>.</li> + </ul> + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check @eventually_of_forall + + #check @Eventually.mono + + #check @Eventually.and + </pre></div> + </div> + The second item, corresponding to <code class="docutils literal notranslate">eventually.mono</code>, supports nice ways + of using filters, especially when combined + with <code class="docutils literal notranslate">eventually.and</code>. The <code class="docutils literal notranslate">filter_upwards</code> tactic allows us to combine them. + Compare: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (P Q R : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in atTop, Q n) + (hR : ∀ᶠ n in atTop, P n ∧ Q n → R n) : ∀ᶠ n in atTop, R n := by + apply (hP.and (hQ.and hR)).mono + rintro n ⟨h, h', h''⟩ + exact h'' ⟨h, h'⟩ + + example (P Q R : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in atTop, Q n) + (hR : ∀ᶠ n in atTop, P n ∧ Q n → R n) : ∀ᶠ n in atTop, R n := by + filter_upwards [hP, hQ, hR] + intro n h h' h'' + exact h'' ⟨h, h'⟩ + </pre></div> + </div> + Readers who know about measure theory will note that the filter <code class="docutils literal notranslate">μ.ae</code> of sets whose complement has measure zero + (aka “the set consisting of almost every point”) is not very useful as the source or target of <code class="docutils literal notranslate">Tendsto</code>, but it can be conveniently + used with <code class="docutils literal notranslate">eventually</code> to say that a property holds for almost every point. + There is a dual version of <code class="docutils literal notranslate">∀ᶠ x in F, P x</code>, which is occasionally useful: + <code class="docutils literal notranslate">∃ᶠ x in F, P x</code> means + <code class="docutils literal notranslate">{x | ¬P x} ∉ F</code>. For example, <code class="docutils literal notranslate">∃ᶠ n in at_top, P n</code> means there are arbitrarily large <code class="docutils literal notranslate">n</code> such that <code class="docutils literal notranslate">P n</code> holds. + The <code class="docutils literal notranslate">∃ᶠ</code> notation stands for <code class="docutils literal notranslate">Filter.frequently</code>. + For a more sophisticated example, consider the following statement about a sequence + <code class="docutils literal notranslate">u</code>, a set <code class="docutils literal notranslate">M</code>, and a value <code class="docutils literal notranslate">x</code>: + <blockquote> + <div>If <code class="docutils literal notranslate">u</code> converges to <code class="docutils literal notranslate">x</code> and <code class="docutils literal notranslate">u n</code> belongs to <code class="docutils literal notranslate">M</code> for + sufficiently large <code class="docutils literal notranslate">n</code> then <code class="docutils literal notranslate">x</code> is in the closure of <code class="docutils literal notranslate">M</code>. + </div></blockquote> + This can be formalized as follows: + <blockquote> + <div><code class="docutils literal notranslate">Tendsto u at_top (𝓝 x) → (∀ᶠ n in at_top, u n ∈ M) → x ∈ closure M</code>. + </div></blockquote> + This is a special case of the theorem <code class="docutils literal notranslate">mem_closure_of_tendsto</code> from the + topology library. + See if you can prove it using the quoted lemmas, + using the fact that <code class="docutils literal notranslate">cluster_pt x F</code> means <code class="docutils literal notranslate">(𝓝 x ⊓ F).ne_bot</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check mem_closure_iff_clusterPt + + #check le_principal_iff + + #check neBot_of_le + + example (u : ℕ → ℝ) (M : Set ℝ) (x : ℝ) (hux : Tendsto u atTop (𝓝 x)) + (huM : ∀ᶠ n in atTop, u n ∈ M) : x ∈ closure M := + sorry + </pre></div> + </div> + </section> + <section id="metric-spaces"> + <h2>7.2. Metric spaces<a class="headerlink" href="#metric-spaces" title="Permalink to this heading"></a></h2> + Examples in the previous section focus on sequences of real numbers. In this section we will go up a bit in generality and focus on + metric spaces. A metric space is a type <code class="docutils literal notranslate">X</code> equipped with a distance function <code class="docutils literal notranslate">dist : X → X → ℝ</code> which is a generalization of + the function <code class="docutils literal notranslate">fun x y ↦ |x - y|</code> from the case where <code class="docutils literal notranslate">X = ℝ</code>. + Introducing such a space is easy and we will check all properties required from the distance function. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {X : Type _} [MetricSpace X] (a b c : X) + + #check (dist a b : ℝ) + + #check (dist_nonneg : 0 ≤ dist a b) + + #check (dist_eq_zero : dist a b = 0 ↔ a = b) + + #check (dist_comm a b : dist a b = dist b a) + + #check (dist_triangle a b c : dist a c ≤ dist a b + dist b c) + </pre></div> + </div> + Note we also have variants where the distance can be infinite or where <code class="docutils literal notranslate">dist a b</code> can be zero without having <code class="docutils literal notranslate">a = b</code> or both. + They are called <code class="docutils literal notranslate">EMetricSpace</code>, <code class="docutils literal notranslate">PseudoMetricSpace</code> and <code class="docutils literal notranslate">PseudoEMetricSpace</code> respectively (here “e” stands for “extended”). + Note that our journey from <code class="docutils literal notranslate">ℝ</code> to metric spaces jumped over the special case of normed spaces that also require linear algebra and + will be explained as part of the calculus chapter. + <section id="convergence-and-continuity"> + <h3>7.2.1. Convergence and continuity<a class="headerlink" href="#convergence-and-continuity" title="Permalink to this heading"></a></h3> + Using distance functions, we can already define convergent sequences and continuous functions between metric spaces. + They are actually defined in a more general setting covered in the next section, + but we have lemmas recasting the definition is terms of distances. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {u : ℕ → X} {a : X} : Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := + Metric.tendsto_atTop + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} : + Continuous f ↔ ∀ x : X, ∀ ε > 0, ∃ δ > 0, ∀ x', dist x' x < δ → dist (f x') (f x) < ε := + Metric.continuous_iff + </pre></div> + </div> + A lot of lemmas have some continuity assumptions, no we end up proving a lot of continuity results and there + is a <code class="docutils literal notranslate">continuity</code> tactic devoted to this task. Let’s prove a continuity statement that will be needed + in an exercise below. Notice that Lean knows how to treat a product of two metric spaces as a metric space, so + it makes sense to consider continuous functions from <code class="docutils literal notranslate">X × X</code> to <code class="docutils literal notranslate">ℝ</code>. + In particular the (uncurried version of the) distance function is such a function. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := by continuity + </pre></div> + </div> + This tactic is a bit slow, so it is also useful to know + how to do it by hand. We first need to use that <code class="docutils literal notranslate">fun p : X × X ↦ f p.1</code> is continuous because it + is the composition of <code class="docutils literal notranslate">f</code>, which is continuous by assumption <code class="docutils literal notranslate">hf</code>, and the projection <code class="docutils literal notranslate">prod.fst</code> whose continuity + is the content of the lemma <code class="docutils literal notranslate">continuous_fst</code>. The composition property is <code class="docutils literal notranslate">Continuous.comp</code> which is + in the <code class="docutils literal notranslate">Continuous</code> namespace so we can use dot notation to compress + <code class="docutils literal notranslate">Continuous.comp hf continuous_fst</code> into <code class="docutils literal notranslate">hf.comp continuous_fst</code> which is actually more readable + since it really reads as composing our assumption and our lemma. + We can do the same for the second component to get continuity of <code class="docutils literal notranslate">fun p : X × X ↦ f p.2</code>. We then assemble + those two continuities using <code class="docutils literal notranslate">Continuous.prod_mk</code> to get + <code class="docutils literal notranslate">(hf.comp continuous_fst).prod_mk (hf.comp continuous_snd) : Continuous (fun p : X × X ↦ (f p.1, f p.2))</code> + and compose once more to get our full proof. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := + continuous_dist.comp ((hf.comp continuous_fst).prod_mk (hf.comp continuous_snd)) + </pre></div> + </div> + The combination of <code class="docutils literal notranslate">Continuous.prod_mk</code> and <code class="docutils literal notranslate">continuous_dist</code> via <code class="docutils literal notranslate">Continuous.comp</code> feels clunky, + even when heavily using dot notation as above. A more serious issue is that this nice proof requires a lot of + planning. Lean accepts the above proof term because it is a full term proving a statement which is + definitionally equivalent to our goal, the crucial definition to unfold being that of a composition of functions. + Indeed our target function <code class="docutils literal notranslate">fun p : X × X ↦ dist (f p.1) (f p.2)</code> is not presented as a composition. + The proof term we provided proves continuity of <code class="docutils literal notranslate">dist ∘ (fun p : X × X ↦ (f p.1, f p.2))</code> which happens + to be definitionally equal to our target function. But if we try to build this proof gradually using + tactics starting with <code class="docutils literal notranslate">apply continuous_dist.comp</code> then Lean’s elaborator will fail to recognize a + composition and refuse to apply this lemma. It is especially bad at this when products of types are involved. + A better lemma to apply here is + <code class="docutils literal notranslate">Continuous.dist {f g : X → Y} : Continuous f → Continuous g → Continuous (fun x ↦ dist (f x) (g x))</code> + which is nicer to Lean’s elaborator and also provides a shorter proof when directly providing a full + proof term, as can be seen from the following two new proofs of the above statement: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := by + apply Continuous.dist + exact hf.comp continuous_fst + exact hf.comp continuous_snd + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := + (hf.comp continuous_fst).dist (hf.comp continuous_snd) + </pre></div> + </div> + Note that, without the elaboration issue coming from composition, another way to compress + our proof would be to use <code class="docutils literal notranslate">Continuous.prod_map</code> which is sometimes useful and gives + as an alternate proof term <code class="docutils literal notranslate">continuous_dist.comp (hf.prod_map hf)</code> which even shorter to type. + Since it is sad to decide between a version which is better for elaboration and a version which is shorter + to type, let us wrap this discussion with a last bit of compression offered + by <code class="docutils literal notranslate">Continuous.fst'</code> which allows to compress <code class="docutils literal notranslate">hf.comp continuous_fst</code> to <code class="docutils literal notranslate">hf.fst'</code> (and the same with <code class="docutils literal notranslate">snd</code>) + and get our final proof, now bordering obfuscation. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := + hf.fst'.dist hf.snd' + </pre></div> + </div> + It’s your turn now to prove some continuity lemma. After trying the continuity tactic, you will need + <code class="docutils literal notranslate">Continuous.add</code>, <code class="docutils literal notranslate">continuous_pow</code> and <code class="docutils literal notranslate">continuous_id</code> to do it by hand. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {f : ℝ → X} (hf : Continuous f) : Continuous fun x : ℝ => f (x ^ 2 + x) := + sorry + </pre></div> + </div> + So far we saw continuity as a global notion, but one can also define continuity at a point. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {X Y : Type _} [MetricSpace X] [MetricSpace Y] (f : X → Y) (a : X) : + ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x}, dist x a < δ → dist (f x) (f a) < ε := + Metric.continuousAt_iff + </pre></div> + </div> + </section> + <section id="balls-open-sets-and-closed-sets"> + <h3>7.2.2. Balls, open sets and closed sets<a class="headerlink" href="#balls-open-sets-and-closed-sets" title="Permalink to this heading"></a></h3> + Once we have a distance function, the most important geometric definitions are (open) balls and closed balls. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (r : ℝ) + + example : Metric.ball a r = { b | dist b a < r } := + rfl + + example : Metric.closedBall a r = { b | dist b a ≤ r } := + rfl + </pre></div> + </div> + Note that <cite>r</cite> is any real number here, there is no sign restriction. Of course some statements do require a radius condition. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (hr : 0 < r) : a ∈ Metric.ball a r := + Metric.mem_ball_self hr + + example (hr : 0 ≤ r) : a ∈ Metric.closedBall a r := + Metric.mem_closedBall_self hr + </pre></div> + </div> + Once we have balls, we can define open sets. They are actually defined in a more general setting covered in the next section, + but we have lemmas recasting the definition is terms of balls. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (s : Set X) : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s := + Metric.isOpen_iff + </pre></div> + </div> + Then closed sets are sets whose complement is open. Their important property is they are closed under limits. The closure of a set is the smallest subset containing it. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {s : Set X} : IsClosed s ↔ IsOpen (sᶜ) := + isOpen_compl_iff.symm + + example {s : Set X} (hs : IsClosed s) {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) + (hus : ∀ n, u n ∈ s) : a ∈ s := + hs.mem_of_tendsto hu (eventually_of_forall hus) + + example {s : Set X} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, a ∈ Metric.ball b ε := + Metric.mem_closure_iff + </pre></div> + </div> + Do the next exercise without using <cite>mem_closure_iff_seq_limit</cite> + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := + sorry + </pre></div> + </div> + Remember from the filters sections that neighborhood filters play a big role in mathlib. + In the metric space context, the crucial point is that balls provide bases for those filters. + The main lemmas here are <code class="docutils literal notranslate">Metric.nhds_basis_ball</code> and <code class="docutils literal notranslate">Metric.nhds_basis_closedBall</code> + that claim this for open and closed balls with positive radius. The center point is an implicit + argument so we can invoke <code class="docutils literal notranslate">Filter.HasBasis.mem_iff</code> as in the following example. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.ball x ε ⊆ s := + Metric.nhds_basis_ball.mem_iff + + example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.closedBall x ε ⊆ s := + Metric.nhds_basis_closedBall.mem_iff + </pre></div> + </div> + </section> + <section id="compactness"> + <h3>7.2.3. Compactness<a class="headerlink" href="#compactness" title="Permalink to this heading"></a></h3> + Compactness is an important topological notion. It distinguishes subsets of a metric space + that enjoy the same kind of properties as segments in reals compared to other intervals: + <ul class="simple"> + <li>Any sequence taking value in a compact set has a subsequence that converges in this set</li> + <li>Any continuous function on a nonempty compact set with values in real numbers is bounded and + achieves its bounds somewhere (this is called the extreme values theorem).</li> + <li>Compact sets are closed sets.</li> + </ul> + Let us first check that the unit interval in reals is indeed a compact set, and then check the above + claims for compact sets in general metric spaces. In the second statement we only + need continuity on the given set so we will use <code class="docutils literal notranslate">ContinuousOn</code> instead of <code class="docutils literal notranslate">Continuous</code>, and + we will give separate statements for the minimum and the maximum. Of course all these results + are deduced from more general versions, some of which will be discussed in later sections. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : IsCompact (Set.Icc 0 1 : Set ℝ) := + isCompact_Icc + + example {s : Set X} (hs : IsCompact s) {u : ℕ → X} (hu : ∀ n, u n ∈ s) : + ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := + hs.tendsto_subseq hu + + example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : + ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y := + hs.exists_forall_le hs' hfs + + example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : + ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x := + hs.exists_forall_ge hs' hfs + + example {s : Set X} (hs : IsCompact s) : IsClosed s := + hs.isClosed + </pre></div> + </div> + We can also metric spaces which are globally compact, using an extra <code class="docutils literal notranslate">Prop</code>-valued type class: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := + isCompact_univ + </pre></div> + </div> + In a compact metric space any closed set is compact, this is <code class="docutils literal notranslate">IsCompact.isClosed</code>. + </section> + <section id="uniformly-continuous-functions"> + <h3>7.2.4. Uniformly continuous functions<a class="headerlink" href="#uniformly-continuous-functions" title="Permalink to this heading"></a></h3> + We now turn to uniformity notions on metric spaces : uniformly continuous functions, Cauchy sequences and completeness. + Again those are defined in a more general context but we have lemmas in the metric name space to access their elementary definitions. + We start with uniform continuity. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {X : Type _} [MetricSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} : + UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : X}, dist a b < δ → dist (f a) (f b) < ε := + Metric.uniformContinuous_iff + </pre></div> + </div> + In order to practice manipulating all those definitions, we will prove that continuous + functions from a compact metric space to a metric space are uniformly continuous + (we will see a more general version in a later section). + We will first give an informal sketch. Let <code class="docutils literal notranslate">f : X → Y</code> be a continuous function from + a compact metric space to a metric space. + We fix <code class="docutils literal notranslate">ε > 0</code> and start looking for some <code class="docutils literal notranslate">δ</code>. + Let <code class="docutils literal notranslate">φ : X × X → ℝ := fun p ↦ dist (f p.1) (f p.2)</code> and let <code class="docutils literal notranslate">K := { p : X × X | ε ≤ φ p }</code>. + Observe <code class="docutils literal notranslate">φ</code> is continuous since <code class="docutils literal notranslate">f</code> and distance are continuous. + And <code class="docutils literal notranslate">K</code> is clearly closed (use <code class="docutils literal notranslate">isClosed_le</code>) hence compact since <code class="docutils literal notranslate">X</code> is compact. + Then we discuss two possibilities using <code class="docutils literal notranslate">eq_empty_or_nonempty</code>. + If <code class="docutils literal notranslate">K</code> is empty then we are clearly done (we can set <code class="docutils literal notranslate">δ = 1</code> for instance). + So let’s assume <code class="docutils literal notranslate">K</code> is not empty, and use the extreme value theorem to choose <code class="docutils literal notranslate">(x₀, x₁)</code> attaining the infimum + of the distance function on <code class="docutils literal notranslate">K</code>. We can then set <code class="docutils literal notranslate">δ = dist x₀ x₁</code> and check everything works. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} + (hf : Continuous f) : UniformContinuous f := + sorry + </pre></div> + </div> + </section> + <section id="completeness"> + <h3>7.2.5. Completeness<a class="headerlink" href="#completeness" title="Permalink to this heading"></a></h3> + A Cauchy sequence in a metric space is a sequence whose terms get closer and closer to each other. + There are a couple of equivalent ways to state that idea. + In particular converging sequences are Cauchy. The converse is true only in so-called complete + spaces. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε := + Metric.cauchySeq_iff + + example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, dist (u n) (u N) < ε := + Metric.cauchySeq_iff' + + example [CompleteSpace X] (u : ℕ → X) (hu : CauchySeq u) : ∃ x, Tendsto u atTop (𝓝 x) := + cauchySeq_tendsto_of_complete hu + </pre></div> + </div> + We’ll practice using this definition by proving a convenient criterion which is a special case of a + criterion appearing in mathlib. This is also a good opportunity to practice using big sums in + a geometric context. In addition to the explanations from the filters section, you will probably need + <code class="docutils literal notranslate">tendsto_pow_atTop_nhds_0_of_lt_1</code>, <code class="docutils literal notranslate">Tendsto.mul</code> and <code class="docutils literal notranslate">dist_le_range_sum_dist</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem cauchySeq_of_le_geometric_two' {u : ℕ → X} + (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by + rw [Metric.cauchySeq_iff'] + intro ε ε_pos + obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by sorry + use N + intro n hn + obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn + calc + dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := sorry + _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := sorry + _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := sorry + _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2) ^ i := sorry + _ ≤ 1 / 2 ^ N * 2 := sorry + _ < ε := sorry + </pre></div> + </div> + We are ready for the final boss of this section: Baire’s theorem for complete metric spaces! + The proof skeleton below shows interesting techniques. It uses the <code class="docutils literal notranslate">choose</code> tactic in its exclamation + mark variant (you should experiment with removing this exclamation mark) and it shows how to + define something inductively in the middle of a proof using <code class="docutils literal notranslate">Nat.rec_on</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Metric + + example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) : + Dense (⋂ n, f n) := by + let B : ℕ → ℝ := fun n => (1 / 2) ^ n + have Bpos : ∀ n, 0 < B n + sorry + /- Translate the density assumption into two functions `center` and `radius` associating + to any n, x, δ, δpos a center and a positive radius such that + `closedBall center radius` is included both in `f n` and in `closedBall x δ`. + We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ + have : + ∀ (n : ℕ) (x : X), + ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := + by sorry + choose! center radius Hpos HB Hball using this + intro x + rw [mem_closure_iff_nhds_basis nhds_basis_closedBall] + intro ε εpos + /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all + `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball + `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that + `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a + limit which belongs to all the `f n`. -/ + let F : ℕ → X × ℝ := fun n => + Nat.recOn n (Prod.mk x (min ε (B 0))) fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2) + let c : ℕ → X := fun n => (F n).1 + let r : ℕ → ℝ := fun n => (F n).2 + have rpos : ∀ n, 0 < r n := by sorry + have rB : ∀ n, r n ≤ B n := by sorry + have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := by sorry + have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by sorry + have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist + -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. + rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩ + -- this point `y` will be the desired point. We will check that it belongs to all + -- `f n` and to `ball x ε`. + use y + have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry + have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry + sorry + </pre></div> + </div> + </section> + </section> + <section id="topological-spaces"> + <h2>7.3. Topological spaces<a class="headerlink" href="#topological-spaces" title="Permalink to this heading"></a></h2> + <section id="fundamentals"> + <h3>7.3.1. Fundamentals<a class="headerlink" href="#fundamentals" title="Permalink to this heading"></a></h3> + We now go up in generality and introduce topological spaces. We will review the two main ways to define + topological spaces and then explain how the category of topological spaces is much better behaved than + the category of metric spaces. Note that we won’t be using mathlib category theory here, only having + a somewhat categorical point of view. + The first way to think about the transition from metric spaces to topological spaces is that we only + remember the notion of open sets (or equivalently the notion of closed sets). From this point of view, + a topological space is a type equipped with a collection of sets that are called open sets. This collection + has to satisfy a number of axioms presented below (this collection is slightly redundant but we will ignore that). + <div class="highlight-lean notranslate"><div class="highlight"><pre>section + variable {X : Type _} [TopologicalSpace X] + + example : IsOpen (univ : Set X) := + isOpen_univ + + example : IsOpen (∅ : Set X) := + isOpen_empty + + example {ι : Type _} {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋃ i, s i) := + isOpen_iUnion hs + + example {ι : Type _} [Fintype ι] {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋂ i, s i) := + isOpen_iInter hs + </pre></div> + </div> + Closed sets are then defined as sets whose complement is open. A function between topological spaces + is (globally) continuous if all preimages of open sets are open. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {Y : Type _} [TopologicalSpace Y] + + example {f : X → Y} : Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) := + continuous_def + </pre></div> + </div> + With this definition we already see that, compared to metric spaces, topological spaces only remember + enough information to talk about continuous functions: two topological structures on a type are + the same if and only if they have the same continuous functions (indeed the identity function will + be continuous in both direction if and only if the two structures have the same open sets). + However as soon as we move on to continuity at a point we see the limitations of the approach based + on open sets. In mathlib it is much more frequent to think of topological spaces as types equipped + with a neighborhood filter <code class="docutils literal notranslate">𝓝 x</code> attached to each point <code class="docutils literal notranslate">x</code> (the corresponding function + <code class="docutils literal notranslate">X → Filter X</code> satisfies certain conditions explained further down). Remember from the filters section that + these gadget play two related roles. First <code class="docutils literal notranslate">𝓝 x</code> is seen as the generalized set of points of <code class="docutils literal notranslate">X</code> + that are close to <code class="docutils literal notranslate">x</code>. And then it is seen as giving a way to say, for any predicate <code class="docutils literal notranslate">P : X → Prop</code>, + that this predicates holds for points that are close enough to <code class="docutils literal notranslate">x</code>. Let us state + that <code class="docutils literal notranslate">f : X → Y</code> is continuous at <code class="docutils literal notranslate">x</code>. The purely filtery way is to say that the direct image under + <code class="docutils literal notranslate">f</code> of the generalized set of points that are close to <code class="docutils literal notranslate">x</code> is contained in the generalized set of + points that are close to <code class="docutils literal notranslate">f x</code>. Recall this spelled either <code class="docutils literal notranslate">map f (𝓝 x) ≤ 𝓝 (f x)</code> + or <code class="docutils literal notranslate">Tendsto f (𝓝 x) (𝓝 (f x))</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {f : X → Y} {x : X} : ContinuousAt f x ↔ map f (𝓝 x) ≤ 𝓝 (f x) := + Iff.rfl + </pre></div> + </div> + One can also spell it using both neighborhoods seen as ordinary sets and a neighborhood filter + seen as a generalized set: “for any neighborhood <code class="docutils literal notranslate">U</code> of <code class="docutils literal notranslate">f x</code>, all points close to <code class="docutils literal notranslate">x</code> + are sent to <code class="docutils literal notranslate">U</code>”. Note that the proof is again <code class="docutils literal notranslate">iff.rfl</code>, this point of view is definitionally + equivalent to the previous one. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {f : X → Y} {x : X} : ContinuousAt f x ↔ ∀ U ∈ 𝓝 (f x), ∀ᶠ x in 𝓝 x, f x ∈ U := + Iff.rfl + </pre></div> + </div> + We now explain how to go from one point of view to the other. In terms of open sets, we can + simply define members of <code class="docutils literal notranslate">𝓝 x</code> as sets that contain an open set containing <code class="docutils literal notranslate">x</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := + mem_nhds_iff + </pre></div> + </div> + To go in the other direction we need to discuss the condition that <code class="docutils literal notranslate">𝓝 : X → Filter X</code> must satisfy + in order to be the neighborhood function of a topology. + The first constraint is that <code class="docutils literal notranslate">𝓝 x</code>, seen as a generalized set, contains the set <code class="docutils literal notranslate">{x}</code> seen as the generalized set + <code class="docutils literal notranslate">pure x</code> (explaining this weird name would be too much of a digression, so we simply accept it for now). + Another way to say it is that if a predicate holds for points close to <code class="docutils literal notranslate">x</code> then it holds at <code class="docutils literal notranslate">x</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x : X) : pure x ≤ 𝓝 x := + pure_le_nhds x + + example (x : X) (P : X → Prop) (h : ∀ᶠ y in 𝓝 x, P y) : P x := + pure_le_nhds x h + </pre></div> + </div> + Then a more subtle requirement is that, for any predicate <code class="docutils literal notranslate">P : X → Prop</code> and any <code class="docutils literal notranslate">x</code>, if <code class="docutils literal notranslate">P y</code> holds for <code class="docutils literal notranslate">y</code> close + to <code class="docutils literal notranslate">x</code> then for <code class="docutils literal notranslate">y</code> close to <code class="docutils literal notranslate">x</code> and <code class="docutils literal notranslate">z</code> close to <code class="docutils literal notranslate">y</code>, <code class="docutils literal notranslate">P z</code> holds. More precisely we have: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {P : X → Prop} {x : X} (h : ∀ᶠ y in 𝓝 x, P y) : ∀ᶠ y in 𝓝 x, ∀ᶠ z in 𝓝 y, P z := + eventually_eventually_nhds.mpr h + </pre></div> + </div> + Those two results characterize the functions <code class="docutils literal notranslate">X → Filter X</code> that are neighborhood functions for a topological space + structure on <code class="docutils literal notranslate">X</code>. There is a still a function <code class="docutils literal notranslate">topological_space.mk_of_nhds : (X → Filter X) → topological_space X</code> + but it will give back its input as a neighborhood function only if it satisfies the above two constraints. + More precisely we have a lemma <code class="docutils literal notranslate">topological_space.nhds_mk_of_nhds</code> saying that in a different way and our + next exercise deduces this different way from how we stated it above. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) + (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → ∀ᶠ y in n a, ∀ᶠ x in n y, p x) : + ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := + sorry + </pre></div> + </div> + Note that <code class="docutils literal notranslate">topological_space.mk_of_nhds</code> is not so frequently used, but it still good to know in what + precise sense the neighborhood filters is all there is in a topological space structure. + The next thing to know in order to efficiently use topological spaces in mathlib is that we use a lot + of formal properties of <code class="docutils literal notranslate">topological_space : Type u → Type u</code>. From a purely mathematical point of view, + those formal properties are a very clean way to explain how topological spaces solve issues that metric spaces + have. From this point of view, the issues solved by topological spaces is that metric spaces enjoy very + little fonctoriality, and have very bad categorical properties in general. This comes on top of the fact + already discussed that metric spaces contain a lot of geometrical information that is not topologically relevant. + Let us focus on fonctoriality first. A metric space structure can be induced on a subset or, + equivalently, it can be pulled back by an injective map. But that’s pretty much everything. + They cannot be pulled back by general map or pushed forward, even by surjective maps. + In particular there is no sensible distance to put on a quotient of a metric space or on an uncountable + products of metric spaces. Consider for instance the type <code class="docutils literal notranslate">ℝ → ℝ</code>, seen as + a product of copies of <code class="docutils literal notranslate">ℝ</code> indexed by <code class="docutils literal notranslate">ℝ</code>. We would like to say that pointwise convergence of + sequences of functions is a respectable notion of convergence. But there is no distance on + <code class="docutils literal notranslate">ℝ → ℝ</code> that gives this notion of convergence. Relatedly, there is no distance ensuring that + a map <code class="docutils literal notranslate">f : X → (ℝ → ℝ)</code> is continuous if and only <code class="docutils literal notranslate">fun x ↦ f x t</code> is continuous for every <code class="docutils literal notranslate">t : ℝ</code>. + We now review the data used to solve all those issues. First we can use any map <code class="docutils literal notranslate">f : X → Y</code> to + push or pull topologies from one side to the other. Those two operations form a Galois connection. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {X Y : Type _} + + example (f : X → Y) : TopologicalSpace X → TopologicalSpace Y := + TopologicalSpace.coinduced f + + example (f : X → Y) : TopologicalSpace Y → TopologicalSpace X := + TopologicalSpace.induced f + + example (f : X → Y) (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) : + TopologicalSpace.coinduced f T_X ≤ T_Y ↔ T_X ≤ TopologicalSpace.induced f T_Y := + coinduced_le_iff_le_induced + </pre></div> + </div> + Those operations are compactible with composition of functions. + As usual, pushing forward is covariant and pulling back is contravariant, see <code class="docutils literal notranslate">coinduced_compose</code> and <code class="docutils literal notranslate">induced_compose</code>. + On paper we will use notations $f_*T$ for <code class="docutils literal notranslate">topological_space.coinduced f T</code> and + $f^*T$ for <code class="docutils literal notranslate">topological_space.induced f T</code>. + Then the next big piece is a complete lattice structure on <code class="docutils literal notranslate">topological_structure X</code> + for any given structure. If you think of topologies are being primarily the data of open sets then you expect + the order relation on <code class="docutils literal notranslate">topological_structure X</code> to come from <code class="docutils literal notranslate">Set (Set X)</code>, ie you expect <code class="docutils literal notranslate">t ≤ t'</code> + if a set <code class="docutils literal notranslate">u</code> is open for <code class="docutils literal notranslate">t'</code> as soon as it is open for <code class="docutils literal notranslate">t</code>. However we already know that mathlib focuses + on neighborhoods more than open sets so, for any <code class="docutils literal notranslate">x : X</code> we want <code class="docutils literal notranslate">fun T : topological_space X ↦ @nhds X T x</code> + to be order preserving. And we know the order relation on <code class="docutils literal notranslate">filter X</code> is designed to ensure an order + preserving <code class="docutils literal notranslate">principal : Set X → Filter X</code>, allowing to see filters as generalized sets. + So the order relation we do use on <code class="docutils literal notranslate">topological_structure X</code> is opposite to the one coming from <code class="docutils literal notranslate">Set (Set X)</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {T T' : TopologicalSpace X} : T ≤ T' ↔ ∀ s, T'.IsOpen s → T.IsOpen s := + Iff.rfl + </pre></div> + </div> + Now we can recover continuity by combining the push-foward (or pull-back) operation with the order relation. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) (f : X → Y) : + Continuous f ↔ TopologicalSpace.coinduced f T_X ≤ T_Y := + continuous_iff_coinduced_le + </pre></div> + </div> + With this definition and the compatibility of push-forward and composition, we + get for free the universal property that, for any topological space $Z$, + a function $g : Y → Z$ is continuous for the topology $f_*T_X$ if and only if + $g ∘ f$ is continuous. + <div class="math notranslate nohighlight"> + \[\begin{split}g \text{ continuous } &⇔ g_*(f_*T_X) ≤ T_Z \\ + &⇔ (g ∘ f)_* T_X ≤ T_Z \\ + &⇔ g ∘ f \text{ continuous}\end{split}\]</div> + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {Z : Type _} (f : X → Y) (T_X : TopologicalSpace X) (T_Z : TopologicalSpace Z) (g : Y → Z) : + @Continuous Y Z (TopologicalSpace.coinduced f T_X) T_Z g ↔ @Continuous X Z T_X T_Z (g ∘ f) := by + rw [continuous_iff_coinduced_le, coinduced_compose, continuous_iff_coinduced_le] + </pre></div> + </div> + So we already get quotient topologies (using the projection map as <code class="docutils literal notranslate">f</code>). This wasn’t using that + <code class="docutils literal notranslate">topological_space X</code> is a complete lattice for all <code class="docutils literal notranslate">X</code>. Let’s now see how all this structure + proves the existence of the product topology by abstract non-sense. + We considered the case of <code class="docutils literal notranslate">ℝ → ℝ</code> above, but let’s now consider the general case of <code class="docutils literal notranslate">Π i, X i</code> for + some <code class="docutils literal notranslate">ι : Type*</code> and <code class="docutils literal notranslate">X : ι → Type*</code>. We want, for any topological space <code class="docutils literal notranslate">Z</code> and any function + <code class="docutils literal notranslate">f : Z → Π i, X i</code>, that <code class="docutils literal notranslate">f</code> is continuous if and only if <code class="docutils literal notranslate">(fun x ↦ x i) ∘ f</code> is continuous. + Let us explore that constraint “on papar” using notation $p_i$ for the projection + <code class="docutils literal notranslate">(fun (x : Π i, X i) ↦ x i)</code>: + <div class="math notranslate nohighlight"> + \[\begin{split}(∀ i, p_i ∘ f \text{ continuous}) &⇔ ∀ i, (p_i ∘ f)_* T_Z ≤ T_{X_i} \\ + &⇔ ∀ i, (p_i)_* f_* T_Z ≤ T_{X_i}\\ + &⇔ ∀ i, f_* T_Z ≤ (p_i)^*T_{X_i}\\ + &⇔ f_* T_Z ≤ \inf \left[(p_i)^*T_{X_i}\right]\end{split}\]</div> + So we see that what is the topology we want on <code class="docutils literal notranslate">Π i, X i</code>: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (ι : Type _) (X : ι → Type _) (T_X : ∀ i, TopologicalSpace <| X i) : + (Pi.topologicalSpace : TopologicalSpace (∀ i, X i)) = + ⨅ i, TopologicalSpace.induced (fun x => x i) (T_X i) := + rfl + </pre></div> + </div> + This ends our tour of how mathlib thinks that topological spaces fix defects of the theory of metric spaces + by being a more functorial theory and having a complete lattice structure for any fixed type. + </section> + <section id="separation-and-countability"> + <h3>7.3.2. Separation and countability<a class="headerlink" href="#separation-and-countability" title="Permalink to this heading"></a></h3> + We saw that the category of topological spaces have very nice properties. The price to pay for + this is existence of rather pathological topological spaces. + There are a number of assumptions you can make on a topological space to ensure its behavior + is closer to what metric spaces do. The most important is <code class="docutils literal notranslate">t2_space</code>, also called “Hausdorff”, + that will ensure that limits are unique. + A stronger separation property is regularity that ensure that each point has a basis of closed + neighborhood. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [TopologicalSpace X] [T2Space X] {u : ℕ → X} {a b : X} (ha : Tendsto u atTop (𝓝 a)) + (hb : Tendsto u atTop (𝓝 b)) : a = b := + tendsto_nhds_unique ha hb + + example [TopologicalSpace X] [RegularSpace X] (a : X) : + (𝓝 a).HasBasis (fun s : Set X => s ∈ 𝓝 a ∧ IsClosed s) id := + closed_nhds_basis a + </pre></div> + </div> + Note that, in every topological space, each point has a basis of open neighborhood, by definition. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [TopologicalSpace X] {x : X} : (𝓝 x).HasBasis (fun t : Set X => t ∈ 𝓝 x ∧ IsOpen t) id := + nhds_basis_opens' x + </pre></div> + </div> + Our main goal is now to prove the basic theorem which allows extension by continuity. + From Bourbaki’s general topology book, I.8.5, Theorem 1 (taking only the non-trivial implication): + Let $X$ be a topological space, $A$ a dense subset of $X$, $f : A → Y$ + a continuous mapping of $A$ into a regular space $Y$. If, for each $x$ in $X$, + $f(y)$ tends to a limit in $Y$ when $y$ tends to $x$ + while remaining in $A$ then there exists a continuous extension $φ$ of $f$ to + $X$. + Actually <code class="docutils literal notranslate">mathlib</code> contains a more general version of the above lemma, <code class="docutils literal notranslate">DenseInducing.continuousAt_extend</code>, + but we’ll stick to Bourbaki’s version here. + Remember that, given <code class="docutils literal notranslate">A : Set X</code>, <code class="docutils literal notranslate">↥A</code> is the subtype associated to <code class="docutils literal notranslate">A</code>, and Lean will automatically + insert that funny up arrow when needed. And the (inclusion) coercion map is <code class="docutils literal notranslate">(↑) : A → X</code>. + The assumption “tends to $x$ while remaining in $A$” corresponds to the pull-back filter + <code class="docutils literal notranslate">comap (↑) (𝓝 x)</code>. + Let’s prove first an auxiliary lemma, extracted to simplify the context + (in particular we don’t need Y to be a topological space here). + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem aux {X Y A : Type _} [TopologicalSpace X] {c : A → X} {f : A → Y} {x : X} {F : Filter Y} + (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : + ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := + sorry + </pre></div> + </div> + Let’s now turn to the main proof of the extension by continuity theorem. + When Lean needs a topology on <code class="docutils literal notranslate">↥A</code> it will use the induced topology, thanks to the instance + <code class="docutils literal notranslate">subtype.topological_space</code>. + This all happens automatically. The only relevant lemma is + <code class="docutils literal notranslate">nhds_induced (↑) : ∀ a : ↥A, 𝓝 a = comap (↑) (𝓝 ↑a)</code> + (this is actually a general lemma about induced topologies). + The proof outline is: + The main assumption and the axiom of choice give a function <code class="docutils literal notranslate">φ</code> such that + <code class="docutils literal notranslate">∀ x, Tendsto f (comap (↑) $ 𝓝 x) (𝓝 (φ x))</code> + (because <code class="docutils literal notranslate">Y</code> is Hausdorff, <code class="docutils literal notranslate">φ</code> is entirely determined, but we won’t need that until we try to + prove that <code class="docutils literal notranslate">φ</code> indeed extends <code class="docutils literal notranslate">f</code>). + Let’s first prove <code class="docutils literal notranslate">φ</code> is continuous. Fix any <code class="docutils literal notranslate">x : X</code>. + Since <code class="docutils literal notranslate">Y</code> is regular, it suffices to check that for every closed neighborhood + <code class="docutils literal notranslate">V'</code> of <code class="docutils literal notranslate">φ x</code>, <code class="docutils literal notranslate">φ ⁻¹' V' ∈ 𝓝 x</code>. + The limit assumption gives (through the auxiliary lemma above) + some <code class="docutils literal notranslate">V ∈ 𝓝 x</code> such <code class="docutils literal notranslate">is_open V ∧ (↑) ⁻¹' V ⊆ f ⁻¹' V'</code>. + Since <code class="docutils literal notranslate">V ∈ 𝓝 x</code>, it suffices to prove <code class="docutils literal notranslate">V ⊆ φ ⁻¹' V'</code>, ie <code class="docutils literal notranslate">∀ y ∈ V, φ y ∈ V'</code>. + Let’s fix <code class="docutils literal notranslate">y</code> in <code class="docutils literal notranslate">V</code>. Because <code class="docutils literal notranslate">V</code> is open, it is a neighborhood of <code class="docutils literal notranslate">y</code>. + In particular <code class="docutils literal notranslate">(↑) ⁻¹' V ∈ comap (↑) (𝓝 y)</code> and a fortiori <code class="docutils literal notranslate">f ⁻¹' V' ∈ comap (↑) (𝓝 y)</code>. + In addition <code class="docutils literal notranslate">comap (↑)$ 𝓝 y ≠ ⊥</code> because <code class="docutils literal notranslate">A</code> is dense. + Because we know <code class="docutils literal notranslate">Tendsto f (comap (↑) $ 𝓝 y) (𝓝 (φ y))</code> this implies + <code class="docutils literal notranslate">φ y ∈ closure V'</code> and, since <code class="docutils literal notranslate">V'</code> is closed, we have proved <code class="docutils literal notranslate">φ y ∈ V'</code>. + It remains to prove that <code class="docutils literal notranslate">φ</code> extends <code class="docutils literal notranslate">f</code>. This is were continuity of <code class="docutils literal notranslate">f</code> enters the discussion, + together with the fact that <code class="docutils literal notranslate">Y</code> is Hausdorff. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace Y] {A : Set X} + (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f) + (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) <| 𝓝 x) <| 𝓝 c) : + ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := + sorry + + #check @HasBasis.tendsto_right_iff + </pre></div> + </div> + In addition to separation property, the main kind of assumption you can make on a topological + space to bring it closer to metric spaces is countability assumption. The main one is first countability + asking that every point has a countable neighborhood basic. In particular this ensures that closure + of sets can be understood using sequences. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] {s : Set X} {a : X} : + a ∈ closure s ↔ ∃ u : ℕ → X, (∀ n, u n ∈ s) ∧ Tendsto u atTop (𝓝 a) := + mem_closure_iff_seq_limit + </pre></div> + </div> + </section> + <section id="id5"> + <h3>7.3.3. Compactness<a class="headerlink" href="#id5" title="Permalink to this heading"></a></h3> + Let us now discuss how compactness is defined for topological spaces. As usual there are several ways + to think about it and mathlib goes for the filter version. + We first need to define cluster points of filters. Given a filter <code class="docutils literal notranslate">F</code> on a topological space <code class="docutils literal notranslate">X</code>, + a point <code class="docutils literal notranslate">x : X</code> is a cluster point of <code class="docutils literal notranslate">F</code> if <code class="docutils literal notranslate">F</code>, seen as a generalized set, has non-empty intersection + with the generalized set of points that are close to <code class="docutils literal notranslate">x</code>. + Then we can say that a set <code class="docutils literal notranslate">s</code> is compact if every nonempty generalized set <code class="docutils literal notranslate">F</code> contained in <code class="docutils literal notranslate">s</code>, + ie such that <code class="docutils literal notranslate">F ≤ 𝓟 s</code>, has a cluster point in <code class="docutils literal notranslate">s</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable [TopologicalSpace X] + + example {F : Filter X} {x : X} : ClusterPt x F ↔ NeBot (𝓝 x ⊓ F) := + Iff.rfl + + example {s : Set X} : IsCompact s ↔ ∀ (F : Filter X) [NeBot F], F ≤ 𝓟 s → ∃ a ∈ s, ClusterPt a F := + Iff.rfl + </pre></div> + </div> + For instance if <code class="docutils literal notranslate">F</code> is <code class="docutils literal notranslate">map u at_top</code>, the image under <code class="docutils literal notranslate">u : ℕ → X</code> of <code class="docutils literal notranslate">at_top</code>, the generalized set + of very large natural numbers, then the assumption <code class="docutils literal notranslate">F ≤ 𝓟 s</code> means that <code class="docutils literal notranslate">u n</code> belongs to <code class="docutils literal notranslate">s</code> for <code class="docutils literal notranslate">n</code> + large enough. Saying that <code class="docutils literal notranslate">x</code> is a cluster point of <code class="docutils literal notranslate">map u at_top</code> says the image of very large numbers + intersects the set of points that are close to <code class="docutils literal notranslate">x</code>. In case <code class="docutils literal notranslate">𝓝 x</code> has a countable basis, we can + interpret this as saying that <code class="docutils literal notranslate">u</code> has a subsequence converging to <code class="docutils literal notranslate">x</code>, and we get back what compactness + looks like in metric spaces. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [TopologicalSpace.FirstCountableTopology X] {s : Set X} {u : ℕ → X} (hs : IsCompact s) + (hu : ∀ n, u n ∈ s) : ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := + hs.tendsto_subseq hu + </pre></div> + </div> + Cluster points behave nicely with continuous functions. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable [TopologicalSpace Y] + + example {x : X} {F : Filter X} {G : Filter Y} (H : ClusterPt x F) {f : X → Y} + (hfx : ContinuousAt f x) (hf : Tendsto f F G) : ClusterPt (f x) G := + ClusterPt.map H hfx hf + </pre></div> + </div> + As an exercise, we will prove that the image of a compact set under a continuous map is + compact. In addition to what we saw already, you should use <code class="docutils literal notranslate">Filter.push_pull</code> and + <code class="docutils literal notranslate">ne_bot.of_map</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) : + IsCompact (f '' s) := by + intro F F_ne F_le + have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by sorry + have Hne : (𝓟 s ⊓ comap f F).NeBot := by sorry + have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left + sorry + </pre></div> + </div> + One can also express compactness in terms of open covers: <code class="docutils literal notranslate">s</code> is compact if every family of open sets that + cover <code class="docutils literal notranslate">s</code> has a finite covering sub-family. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {ι : Type _} {s : Set X} (hs : IsCompact s) (U : ι → Set X) (hUo : ∀ i, IsOpen (U i)) + (hsU : s ⊆ ⋃ i, U i) : ∃ t : Finset ι, s ⊆ ⋃ i ∈ t, U i := + hs.elim_finite_subcover U hUo hsU + </pre></div> + </div> + A topological space <code class="docutils literal notranslate">X</code> is compact if <code class="docutils literal notranslate">(univ : Set X)</code> is compact. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [CompactSpace X] : IsCompact (univ : Set X) := + isCompact_univ + </pre></div> + </div> + </section> + </section> + </section> + + + </div> + </div> + <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> + <a href="C06_Abstract_Algebra.html" class="btn btn-neutral float-left" title="6. Abstract Algebra" accesskey="p" rel="prev"> Previous</a> + <a href="genindex.html" class="btn btn-neutral float-right" title="Index" accesskey="n" rel="next">Next </a> + </div> + + <hr/> + + <div role="contentinfo"> + © Copyright 2020, Jeremy Avigad, Kevin Buzzard, Robert Y. Lewis, Patrick Massot. + </div> + + Built with <a href="https://www.sphinx-doc.org/">Sphinx</a> using a + <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a> + provided by <a href="https://readthedocs.org">Read the Docs</a>. + + + </footer> + </div> + </div> + </section> + </div> + <script> + jQuery(function () { + SphinxRtdTheme.Navigation.enable(true); + }); + </script> + + </body> + </html>
html/C08_Differential_Calculus.html (new)
- @@ -0,0 +1,118 @@
 + <!DOCTYPE html> + <html class="writer-html5" lang="en" > + <head> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + + <meta name="viewport" content="width=device-width, initial-scale=1.0" /> + <title>Differential Calculus — Mathematics in Lean 0.1 documentation</title> + <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="shortcut icon" href="_static/favicon.ico"/> +  + + <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> + <script src="_static/jquery.js"></script> + <script src="_static/underscore.js"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> + <script src="_static/doctools.js"></script> + <script src="_static/js/theme.js"></script> + <link rel="index" title="Index" href="genindex.html" /> + <link rel="search" title="Search" href="search.html" /> + </head> + + <body class="wy-body-for-nav"> + <div class="wy-grid-for-nav"> + <nav data-toggle="wy-nav-shift" class="wy-nav-side"> + <div class="wy-side-scroll"> + <div class="wy-side-nav-search" > + <a href="index.html" class="icon icon-home"> Mathematics in Lean + </a> + <div role="search"> + <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> + <input type="text" name="q" placeholder="Search docs" /> + <input type="hidden" name="check_keywords" value="yes" /> + <input type="hidden" name="area" value="default" /> + </form> + </div> + </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> + <ul> + <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a></li> + <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> + <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> + <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> + <li class="toctree-l1"><a class="reference internal" href="C06_Abstract_Algebra.html">6. Abstract Algebra</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Topology.html">7. Topology</a></li> + </ul> + <ul> + <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li> + </ul> + + </div> + </div> + </nav> + + <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > + + <a href="index.html">Mathematics in Lean</a> + </nav> + + <div class="wy-nav-content"> + <div class="rst-content"> + <div role="navigation" aria-label="Page navigation"> + <ul class="wy-breadcrumbs"> + <li><a href="index.html" class="icon icon-home"></a> »</li> + <li>Differential Calculus</li> + <li class="wy-breadcrumbs-aside"> + <a href="_sources/C08_Differential_Calculus.rst.txt" rel="nofollow"> View page source</a> + </li> + </ul> + <hr/> + </div> + <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> + <div itemprop="articleBody"> + + <section id="index-0"> + <h1>Differential Calculus<a class="headerlink" href="#index-0" title="Permalink to this heading"></a></h1> + We now consider the formalization of notions from analysis, + starting with differentiation in this chapter + and turning integration and measure theory in the next. + In <code class="xref std std-numref docutils literal notranslate">elementary_differential_calculus</code>, we stick with the + setting of functions from the real numbers to the real numbers, + which is familiar from any introductory calculus class. + In <code class="xref std std-numref docutils literal notranslate">normed_spaces</code>, we then consider the notion of a derivative in + a much broader setting. + </section> + + + </div> + </div> + <footer> + + <hr/> + + <div role="contentinfo"> + © Copyright 2020, Jeremy Avigad, Kevin Buzzard, Robert Y. Lewis, Patrick Massot. + </div> + + Built with <a href="https://www.sphinx-doc.org/">Sphinx</a> using a + <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a> + provided by <a href="https://readthedocs.org">Read the Docs</a>. + + + </footer> + </div> + </div> + </section> + </div> + <script> + jQuery(function () { + SphinxRtdTheme.Navigation.enable(true); + }); + </script> + + </body> + </html>
html/C09_Integration_and_Measure_Theory.html (new)
- @@ -0,0 +1,110 @@
 + <!DOCTYPE html> + <html class="writer-html5" lang="en" > + <head> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + + <meta name="viewport" content="width=device-width, initial-scale=1.0" /> + <title>Integration and Measure Theory — Mathematics in Lean 0.1 documentation</title> + <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="shortcut icon" href="_static/favicon.ico"/> +  + + <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> + <script src="_static/jquery.js"></script> + <script src="_static/underscore.js"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> + <script src="_static/doctools.js"></script> + <script src="_static/js/theme.js"></script> + <link rel="index" title="Index" href="genindex.html" /> + <link rel="search" title="Search" href="search.html" /> + </head> + + <body class="wy-body-for-nav"> + <div class="wy-grid-for-nav"> + <nav data-toggle="wy-nav-shift" class="wy-nav-side"> + <div class="wy-side-scroll"> + <div class="wy-side-nav-search" > + <a href="index.html" class="icon icon-home"> Mathematics in Lean + </a> + <div role="search"> + <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> + <input type="text" name="q" placeholder="Search docs" /> + <input type="hidden" name="check_keywords" value="yes" /> + <input type="hidden" name="area" value="default" /> + </form> + </div> + </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> + <ul> + <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a></li> + <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> + <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> + <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> + <li class="toctree-l1"><a class="reference internal" href="C06_Abstract_Algebra.html">6. Abstract Algebra</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Topology.html">7. Topology</a></li> + </ul> + <ul> + <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li> + </ul> + + </div> + </div> + </nav> + + <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > + + <a href="index.html">Mathematics in Lean</a> + </nav> + + <div class="wy-nav-content"> + <div class="rst-content"> + <div role="navigation" aria-label="Page navigation"> + <ul class="wy-breadcrumbs"> + <li><a href="index.html" class="icon icon-home"></a> »</li> + <li>Integration and Measure Theory</li> + <li class="wy-breadcrumbs-aside"> + <a href="_sources/C09_Integration_and_Measure_Theory.rst.txt" rel="nofollow"> View page source</a> + </li> + </ul> + <hr/> + </div> + <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> + <div itemprop="articleBody"> + + <section id="index-0"> + <h1>Integration and Measure Theory<a class="headerlink" href="#index-0" title="Permalink to this heading"></a></h1> + </section> + + + </div> + </div> + <footer> + + <hr/> + + <div role="contentinfo"> + © Copyright 2020, Jeremy Avigad, Kevin Buzzard, Robert Y. Lewis, Patrick Massot. + </div> + + Built with <a href="https://www.sphinx-doc.org/">Sphinx</a> using a + <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a> + provided by <a href="https://readthedocs.org">Read the Docs</a>. + + + </footer> + </div> + </div> + </section> + </div> + <script> + jQuery(function () { + SphinxRtdTheme.Navigation.enable(true); + }); + </script> + + </body> + </html>
html/_images/schroeder_bernstein1.png (new)
html/_images/schroeder_bernstein2.png (new)
html/_images/schroeder_bernstein3.png (new)
html/_sources/C01_Introduction.rst.txt (new)
- @@ -0,0 +1,7 @@
  + .. _introduction: + + Introduction + ============ + + .. include:: C01_Introduction/S01_Getting_Started.inc + .. include:: C01_Introduction/S02_Overview.inc
html/_sources/C02_Basics.rst.txt (new)
- @@ -0,0 +1,19 @@
  + .. _basics: + + Basics + ====== + + This chapter is designed to introduce you to the nuts and + bolts of mathematical reasoning in Lean: calculating, + applying lemmas and theorems, + and reasoning about generic structures. + + .. include:: C02_Basics/S01_Calculating.inc + + .. include:: C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.inc + + .. include:: C02_Basics/S03_Using_Theorems_and_Lemmas.inc + + .. include:: C02_Basics/S04_More_on_Order_and_Divisibility.inc + + .. include:: C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.inc
html/_sources/C03_Logic.rst.txt (new)
- @@ -0,0 +1,37 @@
  + .. _logic: + + Logic + ===== + + In the last chapter, we dealt with equations, inequalities, + and basic mathematical statements like + ":math:`x` divides :math:`y`." + Complex mathematical statements are built up from + simple ones like these + using logical terms like "and," "or," "not," and + "if ... then," "every," and "some." + In this chapter, we show you how to work with statements + that are built up in this way. + + .. include:: C03_Logic/S01_Implication_and_the_Universal_Quantifier.inc + + .. include:: C03_Logic/S02_The_Existential_Quantifier.inc + + .. include:: C03_Logic/S03_Negation.inc + + .. include:: C03_Logic/S04_Conjunction_and_Bi-implication.inc + + .. include:: C03_Logic/S05_Disjunction.inc + + .. include:: C03_Logic/S06_Sequences_and_Convergence.inc + + + + + + + + + + +
html/_sources/C04_Sets_and_Functions.rst.txt (new)
- @@ -0,0 +1,45 @@
  + .. _sets_and_functions: + + Sets and Functions + ================== + + The vocabulary of sets, relations, and functions provides a uniform + language for carrying out constructions in all the branches of + mathematics. + Since functions and relations can be defined in terms of sets, + axiomatic set theory can be used as a foundation for mathematics. + + Lean's foundation is based instead on the primitive notion of a *type*, + and it includes ways of defining functions between types. + Every expression in Lean has a type: + there are natural numbers, real numbers, functions from reals to reals, + groups, vector spaces, and so on. + Some expressions *are* types, + which is to say, + their type is ``Type``. + Lean and mathlib provide ways of defining new types, + and ways of defining objects of those types. + + Conceptually, you can think of a type as just a set of objects. + Requiring every object to have a type has some advantages. + For example, it makes it possible to overload notation like ``+``, + and it sometimes makes input less verbose + because Lean can infer a lot of information from + an object's type. + The type system also enables Lean to flag errors when you + apply a function to the wrong number of arguments, + or apply a function to arguments of the wrong type. + + Lean's library does define elementary set-theoretic notions. + In contrast to set theory, + in Lean a set is always a set of objects of some type, + such as a set natural numbers or a set of functions + from real numbers to real numbers. + The distinction between types and set takes some getting used to, + but this chapter will take you through the essentials. + + .. include:: C04_Sets_and_Functions/S01_Sets.inc + + .. include:: C04_Sets_and_Functions/S02_Functions.inc + + .. include:: C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.inc
html/_sources/C05_Number_Theory.rst.txt (new)
- @@ -0,0 +1,14 @@
  + .. _number_theory: + + Number Theory + ============= + + In this chapter, we show you how to formalize some elementary + results in number theory. + As we deal with more substantive mathematical content, + the proofs will get longer and more involved, + building on the skills you have already mastered. + + .. include:: C05_Number_Theory/S01_Irrational_Roots.inc + .. include:: C05_Number_Theory/S02_Induction_and_Recursion.inc + .. include:: C05_Number_Theory/S03_Infinitely_Many_Primes.inc
html/_sources/C06_Abstract_Algebra.rst.txt (new)
- @@ -0,0 +1,29 @@
 + .. _abstract_algebra: + + Abstract Algebra + ================ + + Modern mathematics makes essential use of algebraic + structures, + which encapsulate patterns that can be instantiated in + multiple settings. + The subject provides various ways of defining such structures and + constructing particular instances. + + Lean therefore provides corresponding ways of + defining structures formally and working with them. + You have already seen examples of algebraic structures in Lean, + such as rings and lattices, which were discussed in + :numref:`Chapter %s <basics>`. + This chapter will explain the mysterious square bracket annotations + that you saw there, + ``[Ring α]`` and ``[Lattice α]``. + It will also show you how to define and use + algebraic structures on your own. + + For more technical detail, you can consult `Theorem Proving in Lean <https://leanprover.github.io/theorem_proving_in_lean/>`_, + and a paper by Anne Baanen, `Use and abuse of instance parameters in the Lean mathematical library <https://arxiv.org/abs/2202.01629>`_. + + .. include:: C06_Abstract_Algebra/S01_Structures.inc + .. include:: C06_Abstract_Algebra/S02_Algebraic_Structures.inc + .. include:: C06_Abstract_Algebra/S03_Building_the_Gaussian_Integers.inc
html/_sources/C07_Topology.rst.txt (new)
- @@ -0,0 +1,84 @@
 + .. _topology: + + .. index:: topology + + Topology + ======== + + Calculus is based on the concept of a function, which is used to model + quantities that depend on one another. + For example, it is common to study quantities that change over time. + The notion of a *limit* is also fundamental. + We may say that the limit of a function :math:`f(x)` is a value :math:`b` + as :math:`x` approaches a value :math:`a`, + or that :math:`f(x)` *converges to* :math:`b` as :math:`x` approaches :math:`a`. + Equivalently, we may say that a :math:`f(x)` approaches :math:`a` as :math:`x` + approaches a value :math:`b`, or that it *tends to* :math:`b` + as :math:`x` tends to :math:`a`. + We have already begun to consider such notions in :numref:`sequences_and_convergence`. + + *Topology* is the abstract study of limits and continuity. + Having covered the essentials of formalization in Chapters :numref:`%s <basics>` + to :numref:`%s <abstract_algebra>`, + in this chapter, we will explain how topological notions are formalized in mathlib. + Not only do topological abstractions apply in much greater generality, + but that also, somewhat paradoxically, make it easier to reason about limits + and continuity in concrete instances. + + Topological notions build on quite a few layers of mathematical structure. + The first layer is naive set theory, + as described in :numref:`Chapter %s <sets_and_functions>`. + The next layer is the theory of *filters*, which we will describe in :numref:`filters`. + On top of that, we layer + the theories of *topological spaces*, *metric spaces*, and a slightly more exotic + intermediate notion called a *uniform space*. + + Whereas previous chapters relied on mathematical notions that were likely + familiar to you, + the notion of a filter less well known, + even to many working mathematicians. + The notion is essential, however, for formalizing mathematics effectively. + Let us explain why. + Let ``f : ℝ → ℝ`` be any function. We can consider + the limit of ``f x`` as ``x`` approaches some value ``x₀``, + but we can also consider the limit of ``f x`` as ``x`` approaches infinity + or negative infinity. + We can moreover consider the limit of ``f x`` as ``x`` approaches ``x₀`` from + the right, conventionally written ``x₀⁺``, or from the left, + written ``x₀⁻``. There are variations where ``x`` approaches ``x₀`` or ``x₀⁺`` + or ``x₀⁻`` but + is not allowed to take on the value ``x₀`` itself. + This results in at least eight ways that ``x`` can approach something. + We can also restrict to rational values of ``x`` + or place other constraints on the domain, but let's stick to those 8 cases. + + We have a similar variety of options on the codomain: + we can specify that ``f x`` approaches a value from the left or right, + or that it approaches positive or negative infinity, and so on. + For example, we may wish to say that ``f x`` tends to ``+∞`` + when ``x`` tends to ``x₀`` from the right without + being equal to ``x₀``. + This results in 64 different kinds of limit statements, + and we haven't even begun to deal with limits of sequences, + as we did in :numref:`sequences_and_convergence`. + + The problem is compounded even further when it comes to the supporting lemmas. + For instance, limits compose: if + ``f x`` tends to ``y₀`` when ``x`` tends to ``x₀`` and + ``g y`` tends to ``z₀`` when ``y`` tends to ``y₀`` then + ``g ∘ f x`` tends to ``z₀`` when ``x`` tends to ``x₀``. + There are three notions of "tends to" at play here, + each of which can be instantiated in any of the eight ways described + in the previous paragraph. + This results in 512 lemmas, a lot to have to add to a library! + Informally, mathematicians generally prove two or three of these + and simply note that the rest can be proved "in the same way." + Formalizing mathematics requires making the relevant notion of "sameness" + fully explicit, and that is exactly what Bourbaki's theory of filters + manages to do. + + .. include:: C07_Topology/S01_Filters.inc + + .. include:: C07_Topology/S02_Metric_Spaces.inc + + .. include:: C07_Topology/S03_Topological_Spaces.inc
html/_sources/C08_Differential_Calculus.rst.txt (new)
- @@ -0,0 +1,19 @@
  + .. _differential_calculus: + + .. index:: differential calculus + + Differential Calculus + ===================== + + We now consider the formalization of notions from *analysis*, + starting with differentiation in this chapter + and turning integration and measure theory in the next. + In :numref:`elementary_differential_calculus`, we stick with the + setting of functions from the real numbers to the real numbers, + which is familiar from any introductory calculus class. + In :numref:`normed_spaces`, we then consider the notion of a derivative in + a much broader setting. + + .. + .. include:: C08_Differential_Calculus/S01_Elementary_Differential_Calculus.inc + .. include:: C08_Differential_Calculus/S02_Differential_Calculus_in_Normed_Spaces.inc
html/_sources/C09_Integration_and_Measure_Theory.rst.txt (new)
- @@ -0,0 +1,11 @@
  + .. _integration_and_measure_theory: + + .. index:: integration + + Integration and Measure Theory + ============================== + + .. + .. include:: C09_Integration_and_Measure_Theory/S01_Elementary_Integration.inc + .. include:: C09_Integration_and_Measure_Theory/S02_Measure_Theory.inc + .. include:: C09_Integration_and_Measure_Theory/S03_Integration.inc
html/_sources/genindex.rst.txt (new)
- @@ -0,0 +1,2 @@
  + Index + =====
html/_sources/index.rst.txt (new)
- @@ -0,0 +1,21 @@
  + + + Mathematics in Lean + =================== + + .. toctree:: + :numbered: + :maxdepth: 2 + + C01_Introduction + C02_Basics + C03_Logic + C04_Sets_and_Functions + C05_Number_Theory + C06_Abstract_Algebra + C07_Topology + + .. toctree:: + :hidden: + + genindex
html/_static/_sphinx_javascript_frameworks_compat.js (new)
- @@ -0,0 +1,134 @@
 + /* + * _sphinx_javascript_frameworks_compat.js + * ~~~~~~~~~~ + * + * Compatability shim for jQuery and underscores.js. + * + * WILL BE REMOVED IN Sphinx 6.0 + * xref RemovedInSphinx60Warning + * + */ + + /** + * select a different prefix for underscore + */ + $u = _.noConflict(); + + + /** + * small helper function to urldecode strings + * + * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/decodeURIComponent#Decoding_query_parameters_from_a_URL + */ + jQuery.urldecode = function(x) { + if (!x) { + return x + } + return decodeURIComponent(x.replace(/\+/g, ' ')); + }; + + /** + * small helper function to urlencode strings + */ + jQuery.urlencode = encodeURIComponent; + + /** + * This function returns the parsed url parameters of the + * current request. Multiple values per key are supported, + * it will always return arrays of strings for the value parts. + */ + jQuery.getQueryParameters = function(s) { + if (typeof s === 'undefined') + s = document.location.search; + var parts = s.substr(s.indexOf('?') + 1).split('&'); + var result = {}; + for (var i = 0; i < parts.length; i++) { + var tmp = parts[i].split('=', 2); + var key = jQuery.urldecode(tmp[0]); + var value = jQuery.urldecode(tmp[1]); + if (key in result) + result[key].push(value); + else + result[key] = [value]; + } + return result; + }; + + /** + * highlight a given string on a jquery object by wrapping it in + * span elements with the given class name. + */ + jQuery.fn.highlightText = function(text, className) { + function highlight(node, addItems) { + if (node.nodeType === 3) { + var val = node.nodeValue; + var pos = val.toLowerCase().indexOf(text); + if (pos >= 0 && + !jQuery(node.parentNode).hasClass(className) && + !jQuery(node.parentNode).hasClass("nohighlight")) { + var span; + var isInSVG = jQuery(node).closest("body, svg, foreignObject").is("svg"); + if (isInSVG) { + span = document.createElementNS("http://www.w3.org/2000/svg", "tspan"); + } else { + span = document.createElement("span"); + span.className = className; + } + span.appendChild(document.createTextNode(val.substr(pos, text.length))); + node.parentNode.insertBefore(span, node.parentNode.insertBefore( + document.createTextNode(val.substr(pos + text.length)), + node.nextSibling)); + node.nodeValue = val.substr(0, pos); + if (isInSVG) { + var rect = document.createElementNS("http://www.w3.org/2000/svg", "rect"); + var bbox = node.parentElement.getBBox(); + rect.x.baseVal.value = bbox.x; + rect.y.baseVal.value = bbox.y; + rect.width.baseVal.value = bbox.width; + rect.height.baseVal.value = bbox.height; + rect.setAttribute('class', className); + addItems.push({ + "parent": node.parentNode, + "target": rect}); + } + } + } + else if (!jQuery(node).is("button, select, textarea")) { + jQuery.each(node.childNodes, function() { + highlight(this, addItems); + }); + } + } + var addItems = []; + var result = this.each(function() { + highlight(this, addItems); + }); + for (var i = 0; i < addItems.length; ++i) { + jQuery(addItems[i].parent).before(addItems[i].target); + } + return result; + }; + + /* + * backward compatibility for jQuery.browser + * This will be supported until firefox bug is fixed. + */ + if (!jQuery.browser) { + jQuery.uaMatch = function(ua) { + ua = ua.toLowerCase(); + + var match = /(chrome)[ \/]([\w.]+)/.exec(ua) || + /(webkit)[ \/]([\w.]+)/.exec(ua) || + /(opera)(?:.*version|)[ \/]([\w.]+)/.exec(ua) || + /(msie) ([\w.]+)/.exec(ua) || + ua.indexOf("compatible") < 0 && /(mozilla)(?:.*? rv:([\w.]+)|)/.exec(ua) || + []; + + return { + browser: match[ 1 ] || "", + version: match[ 2 ] || "0" + }; + }; + jQuery.browser = {}; + jQuery.browser[jQuery.uaMatch(navigator.userAgent).browser] = true; + }
html/_static/basic.css (new)
- @@ -0,0 +1,932 @@
  + /* + * basic.css + * ~~~~~~~~~ + * + * Sphinx stylesheet -- basic theme. + * + * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS. + * :license: BSD, see LICENSE for details. + * + */ + + /* -- main layout ----------------------------------------------------------- */ + + div.clearer { + clear: both; + } + + div.section::after { + display: block; + content: ''; + clear: left; + } + + /* -- relbar ---------------------------------------------------------------- */ + + div.related { + width: 100%; + font-size: 90%; + } + + div.related h3 { + display: none; + } + + div.related ul { + margin: 0; + padding: 0 0 0 10px; + list-style: none; + } + + div.related li { + display: inline; + } + + div.related li.right { + float: right; + margin-right: 5px; + } + + /* -- sidebar --------------------------------------------------------------- */ + + div.sphinxsidebarwrapper { + padding: 10px 5px 0 10px; + } + + div.sphinxsidebar { + float: left; + width: 230px; + margin-left: -100%; + font-size: 90%; + word-wrap: break-word; + overflow-wrap : break-word; + } + + div.sphinxsidebar ul { + list-style: none; + } + + div.sphinxsidebar ul ul, + div.sphinxsidebar ul.want-points { + margin-left: 20px; + list-style: square; + } + + div.sphinxsidebar ul ul { + margin-top: 0; + margin-bottom: 0; + } + + div.sphinxsidebar form { + margin-top: 10px; + } + + div.sphinxsidebar input { + border: 1px solid #98dbcc; + font-family: sans-serif; + font-size: 1em; + } + + div.sphinxsidebar #searchbox form.search { + overflow: hidden; + } + + div.sphinxsidebar #searchbox input[type="text"] { + float: left; + width: 80%; + padding: 0.25em; + box-sizing: border-box; + } + + div.sphinxsidebar #searchbox input[type="submit"] { + float: left; + width: 20%; + border-left: none; + padding: 0.25em; + box-sizing: border-box; + } + + + img { + border: 0; + max-width: 100%; + } + + /* -- search page ----------------------------------------------------------- */ + + ul.search { + margin: 10px 0 0 20px; + padding: 0; + } + + ul.search li { + padding: 5px 0 5px 20px; + background-image: url(file.png); + background-repeat: no-repeat; + background-position: 0 7px; + } + + ul.search li a { + font-weight: bold; + } + + ul.search li p.context { + color: #888; + margin: 2px 0 0 30px; + text-align: left; + } + + ul.keywordmatches li.goodmatch a { + font-weight: bold; + } + + /* -- index page ------------------------------------------------------------ */ + + table.contentstable { + width: 90%; + margin-left: auto; + margin-right: auto; + } + + table.contentstable p.biglink { + line-height: 150%; + } + + a.biglink { + font-size: 1.3em; + } + + span.linkdescr { + font-style: italic; + padding-top: 5px; + font-size: 90%; + } + + /* -- general index --------------------------------------------------------- */ + + table.indextable { + width: 100%; + } + + table.indextable td { + text-align: left; + vertical-align: top; + } + + table.indextable ul { + margin-top: 0; + margin-bottom: 0; + list-style-type: none; + } + + table.indextable > tbody > tr > td > ul { + padding-left: 0em; + } + + table.indextable tr.pcap { + height: 10px; + } + + table.indextable tr.cap { + margin-top: 10px; + background-color: #f2f2f2; + } + + img.toggler { + margin-right: 3px; + margin-top: 3px; + cursor: pointer; + } + + div.modindex-jumpbox { + border-top: 1px solid #ddd; + border-bottom: 1px solid #ddd; + margin: 1em 0 1em 0; + padding: 0.4em; + } + + div.genindex-jumpbox { + border-top: 1px solid #ddd; + border-bottom: 1px solid #ddd; + margin: 1em 0 1em 0; + padding: 0.4em; + } + + /* -- domain module index --------------------------------------------------- */ + + table.modindextable td { + padding: 2px; + border-collapse: collapse; + } + + /* -- general body styles --------------------------------------------------- */ + + div.body { + min-width: 360px; + max-width: 800px; + } + + div.body p, div.body dd, div.body li, div.body blockquote { + -moz-hyphens: auto; + -ms-hyphens: auto; + -webkit-hyphens: auto; + hyphens: auto; + } + + a.headerlink { + visibility: hidden; + } + + a.brackets:before, + span.brackets > a:before{ + content: "["; + } + + a.brackets:after, + span.brackets > a:after { + content: "]"; + } + + h1:hover > a.headerlink, + h2:hover > a.headerlink, + h3:hover > a.headerlink, + h4:hover > a.headerlink, + h5:hover > a.headerlink, + h6:hover > a.headerlink, + dt:hover > a.headerlink, + caption:hover > a.headerlink, + p.caption:hover > a.headerlink, + div.code-block-caption:hover > a.headerlink { + visibility: visible; + } + + div.body p.caption { + text-align: inherit; + } + + div.body td { + text-align: left; + } + + .first { + margin-top: 0 !important; + } + + p.rubric { + margin-top: 30px; + font-weight: bold; + } + + img.align-left, figure.align-left, .figure.align-left, object.align-left { + clear: left; + float: left; + margin-right: 1em; + } + + img.align-right, figure.align-right, .figure.align-right, object.align-right { + clear: right; + float: right; + margin-left: 1em; + } + + img.align-center, figure.align-center, .figure.align-center, object.align-center { + display: block; + margin-left: auto; + margin-right: auto; + } + + img.align-default, figure.align-default, .figure.align-default { + display: block; + margin-left: auto; + margin-right: auto; + } + + .align-left { + text-align: left; + } + + .align-center { + text-align: center; + } + + .align-default { + text-align: center; + } + + .align-right { + text-align: right; + } + + /* -- sidebars -------------------------------------------------------------- */ + + div.sidebar, + aside.sidebar { + margin: 0 0 0.5em 1em; + border: 1px solid #ddb; + padding: 7px; + background-color: #ffe; + width: 40%; + float: right; + clear: right; + overflow-x: auto; + } + + p.sidebar-title { + font-weight: bold; + } + + div.admonition, div.topic, aside.topic, blockquote { + clear: left; + } + + /* -- topics ---------------------------------------------------------------- */ + + div.topic, aside.topic { + border: 1px solid #ccc; + padding: 7px; + margin: 10px 0 10px 0; + } + + p.topic-title { + font-size: 1.1em; + font-weight: bold; + margin-top: 10px; + } + + /* -- admonitions ----------------------------------------------------------- */ + + div.admonition { + margin-top: 10px; + margin-bottom: 10px; + padding: 7px; + } + + div.admonition dt { + font-weight: bold; + } + + p.admonition-title { + margin: 0px 10px 5px 0px; + font-weight: bold; + } + + div.body p.centered { + text-align: center; + margin-top: 25px; + } + + /* -- content of sidebars/topics/admonitions -------------------------------- */ + + div.sidebar > :last-child, + aside.sidebar > :last-child, + div.topic > :last-child, + aside.topic > :last-child, + div.admonition > :last-child { + margin-bottom: 0; + } + + div.sidebar::after, + aside.sidebar::after, + div.topic::after, + aside.topic::after, + div.admonition::after, + blockquote::after { + display: block; + content: ''; + clear: both; + } + + /* -- tables ---------------------------------------------------------------- */ + + table.docutils { + margin-top: 10px; + margin-bottom: 10px; + border: 0; + border-collapse: collapse; + } + + table.align-center { + margin-left: auto; + margin-right: auto; + } + + table.align-default { + margin-left: auto; + margin-right: auto; + } + + table caption span.caption-number { + font-style: italic; + } + + table caption span.caption-text { + } + + table.docutils td, table.docutils th { + padding: 1px 8px 1px 5px; + border-top: 0; + border-left: 0; + border-right: 0; + border-bottom: 1px solid #aaa; + } + + th { + text-align: left; + padding-right: 5px; + } + + table.citation { + border-left: solid 1px gray; + margin-left: 1px; + } + + table.citation td { + border-bottom: none; + } + + th > :first-child, + td > :first-child { + margin-top: 0px; + } + + th > :last-child, + td > :last-child { + margin-bottom: 0px; + } + + /* -- figures --------------------------------------------------------------- */ + + div.figure, figure { + margin: 0.5em; + padding: 0.5em; + } + + div.figure p.caption, figcaption { + padding: 0.3em; + } + + div.figure p.caption span.caption-number, + figcaption span.caption-number { + font-style: italic; + } + + div.figure p.caption span.caption-text, + figcaption span.caption-text { + } + + /* -- field list styles ----------------------------------------------------- */ + + table.field-list td, table.field-list th { + border: 0 !important; + } + + .field-list ul { + margin: 0; + padding-left: 1em; + } + + .field-list p { + margin: 0; + } + + .field-name { + -moz-hyphens: manual; + -ms-hyphens: manual; + -webkit-hyphens: manual; + hyphens: manual; + } + + /* -- hlist styles ---------------------------------------------------------- */ + + table.hlist { + margin: 1em 0; + } + + table.hlist td { + vertical-align: top; + } + + /* -- object description styles --------------------------------------------- */ + + .sig { + font-family: 'Consolas', 'Menlo', 'DejaVu Sans Mono', 'Bitstream Vera Sans Mono', monospace; + } + + .sig-name, code.descname { + background-color: transparent; + font-weight: bold; + } + + .sig-name { + font-size: 1.1em; + } + + code.descname { + font-size: 1.2em; + } + + .sig-prename, code.descclassname { + background-color: transparent; + } + + .optional { + font-size: 1.3em; + } + + .sig-paren { + font-size: larger; + } + + .sig-param.n { + font-style: italic; + } + + /* C++ specific styling */ + + .sig-inline.c-texpr, + .sig-inline.cpp-texpr { + font-family: unset; + } + + .sig.c .k, .sig.c .kt, + .sig.cpp .k, .sig.cpp .kt { + color: #0033B3; + } + + .sig.c .m, + .sig.cpp .m { + color: #1750EB; + } + + .sig.c .s, .sig.c .sc, + .sig.cpp .s, .sig.cpp .sc { + color: #067D17; + } + + + /* -- other body styles ----------------------------------------------------- */ + + ol.arabic { + list-style: decimal; + } + + ol.loweralpha { + list-style: lower-alpha; + } + + ol.upperalpha { + list-style: upper-alpha; + } + + ol.lowerroman { + list-style: lower-roman; + } + + ol.upperroman { + list-style: upper-roman; + } + + :not(li) > ol > li:first-child > :first-child, + :not(li) > ul > li:first-child > :first-child { + margin-top: 0px; + } + + :not(li) > ol > li:last-child > :last-child, + :not(li) > ul > li:last-child > :last-child { + margin-bottom: 0px; + } + + ol.simple ol p, + ol.simple ul p, + ul.simple ol p, + ul.simple ul p { + margin-top: 0; + } + + ol.simple > li:not(:first-child) > p, + ul.simple > li:not(:first-child) > p { + margin-top: 0; + } + + ol.simple p, + ul.simple p { + margin-bottom: 0; + } + + /* Docutils 0.17 and older (footnotes & citations) */ + dl.footnote > dt, + dl.citation > dt { + float: left; + margin-right: 0.5em; + } + + dl.footnote > dd, + dl.citation > dd { + margin-bottom: 0em; + } + + dl.footnote > dd:after, + dl.citation > dd:after { + content: ""; + clear: both; + } + + /* Docutils 0.18+ (footnotes & citations) */ + aside.footnote > span, + div.citation > span { + float: left; + } + aside.footnote > span:last-of-type, + div.citation > span:last-of-type { + padding-right: 0.5em; + } + aside.footnote > p { + margin-left: 2em; + } + div.citation > p { + margin-left: 4em; + } + aside.footnote > p:last-of-type, + div.citation > p:last-of-type { + margin-bottom: 0em; + } + aside.footnote > p:last-of-type:after, + div.citation > p:last-of-type:after { + content: ""; + clear: both; + } + + /* Footnotes & citations ends */ + + dl.field-list { + display: grid; + grid-template-columns: fit-content(30%) auto; + } + + dl.field-list > dt { + font-weight: bold; + word-break: break-word; + padding-left: 0.5em; + padding-right: 5px; + } + + dl.field-list > dt:after { + content: ":"; + } + + dl.field-list > dd { + padding-left: 0.5em; + margin-top: 0em; + margin-left: 0em; + margin-bottom: 0em; + } + + dl { + margin-bottom: 15px; + } + + dd > :first-child { + margin-top: 0px; + } + + dd ul, dd table { + margin-bottom: 10px; + } + + dd { + margin-top: 3px; + margin-bottom: 10px; + margin-left: 30px; + } + + dl > dd:last-child, + dl > dd:last-child > :last-child { + margin-bottom: 0; + } + + dt:target, span.highlighted { + background-color: #fbe54e; + } + + rect.highlighted { + fill: #fbe54e; + } + + dl.glossary dt { + font-weight: bold; + font-size: 1.1em; + } + + .versionmodified { + font-style: italic; + } + + .system-message { + background-color: #fda; + padding: 5px; + border: 3px solid red; + } + + .footnote:target { + background-color: #ffa; + } + + .line-block { + display: block; + margin-top: 1em; + margin-bottom: 1em; + } + + .line-block .line-block { + margin-top: 0; + margin-bottom: 0; + margin-left: 1.5em; + } + + .guilabel, .menuselection { + font-family: sans-serif; + } + + .accelerator { + text-decoration: underline; + } + + .classifier { + font-style: oblique; + } + + .classifier:before { + font-style: normal; + margin: 0 0.5em; + content: ":"; + display: inline-block; + } + + abbr, acronym { + border-bottom: dotted 1px; + cursor: help; + } + + /* -- code displays --------------------------------------------------------- */ + + pre { + overflow: auto; + overflow-y: hidden; /* fixes display issues on Chrome browsers */ + } + + pre, div[class*="highlight-"] { + clear: both; + } + + span.pre { + -moz-hyphens: none; + -ms-hyphens: none; + -webkit-hyphens: none; + hyphens: none; + white-space: nowrap; + } + + div[class*="highlight-"] { + margin: 1em 0; + } + + td.linenos pre { + border: 0; + background-color: transparent; + color: #aaa; + } + + table.highlighttable { + display: block; + } + + table.highlighttable tbody { + display: block; + } + + table.highlighttable tr { + display: flex; + } + + table.highlighttable td { + margin: 0; + padding: 0; + } + + table.highlighttable td.linenos { + padding-right: 0.5em; + } + + table.highlighttable td.code { + flex: 1; + overflow: hidden; + } + + .highlight .hll { + display: block; + } + + div.highlight pre, + table.highlighttable pre { + margin: 0; + } + + div.code-block-caption + div { + margin-top: 0; + } + + div.code-block-caption { + margin-top: 1em; + padding: 2px 5px; + font-size: small; + } + + div.code-block-caption code { + background-color: transparent; + } + + table.highlighttable td.linenos, + span.linenos, + div.highlight span.gp { /* gp: Generic.Prompt */ + user-select: none; + -webkit-user-select: text; /* Safari fallback only */ + -webkit-user-select: none; /* Chrome/Safari */ + -moz-user-select: none; /* Firefox */ + -ms-user-select: none; /* IE10+ */ + } + + div.code-block-caption span.caption-number { + padding: 0.1em 0.3em; + font-style: italic; + } + + div.code-block-caption span.caption-text { + } + + div.literal-block-wrapper { + margin: 1em 0; + } + + code.xref, a code { + background-color: transparent; + font-weight: bold; + } + + h1 code, h2 code, h3 code, h4 code, h5 code, h6 code { + background-color: transparent; + } + + .viewcode-link { + float: right; + } + + .viewcode-back { + float: right; + font-family: sans-serif; + } + + div.viewcode-block:target { + margin: -1px -10px; + padding: 0 10px; + } + + /* -- math display ---------------------------------------------------------- */ + + img.math { + vertical-align: middle; + } + + div.body div.math p { + text-align: center; + } + + span.eqno { + float: right; + } + + span.eqno a.headerlink { + position: absolute; + z-index: 1; + } + + div.math:hover a.headerlink { + visibility: visible; + } + + /* -- printout stylesheet --------------------------------------------------- */ + + @media print { + div.document, + div.documentwrapper, + div.bodywrapper { + margin: 0 !important; + width: 100%; + } + + div.sphinxsidebar, + div.related, + div.footer, + #top-link { + display: none; + } + }
html/_static/css/badge_only.css (new)
- @@ -0,0 +1,1 @@
  + .fa:before{-webkit-font-smoothing:antialiased}.clearfix{*zoom:1}.clearfix:after,.clearfix:before{display:table;content:""}.clearfix:after{clear:both}@font-face{font-family:FontAwesome;font-style:normal;font-weight:400;src:url(fonts/fontawesome-webfont.eot?674f50d287a8c48dc19ba404d20fe713?#iefix) format("embedded-opentype"),url(fonts/fontawesome-webfont.woff2?af7ae505a9eed503f8b8e6982036873e) format("woff2"),url(fonts/fontawesome-webfont.woff?fee66e712a8a08eef5805a46892932ad) format("woff"),url(fonts/fontawesome-webfont.ttf?b06871f281fee6b241d60582ae9369b9) format("truetype"),url(fonts/fontawesome-webfont.svg?912ec66d7572ff821749319396470bde#FontAwesome) format("svg")}.fa:before{font-family:FontAwesome;font-style:normal;font-weight:400;line-height:1}.fa:before,a .fa{text-decoration:inherit}.fa:before,a .fa,li .fa{display:inline-block}li .fa-large:before{width:1.875em}ul.fas{list-style-type:none;margin-left:2em;text-indent:-.8em}ul.fas li .fa{width:.8em}ul.fas li .fa-large:before{vertical-align:baseline}.fa-book:before,.icon-book:before{content:"\f02d"}.fa-caret-down:before,.icon-caret-down:before{content:"\f0d7"}.fa-caret-up:before,.icon-caret-up:before{content:"\f0d8"}.fa-caret-left:before,.icon-caret-left:before{content:"\f0d9"}.fa-caret-right:before,.icon-caret-right:before{content:"\f0da"}.rst-versions{position:fixed;bottom:0;left:0;width:300px;color:#fcfcfc;background:#1f1d1d;font-family:Lato,proxima-nova,Helvetica Neue,Arial,sans-serif;z-index:400}.rst-versions a{color:#2980b9;text-decoration:none}.rst-versions .rst-badge-small{display:none}.rst-versions .rst-current-version{padding:12px;background-color:#272525;display:block;text-align:right;font-size:90%;cursor:pointer;color:#27ae60}.rst-versions .rst-current-version:after{clear:both;content:"";display:block}.rst-versions .rst-current-version .fa{color:#fcfcfc}.rst-versions .rst-current-version .fa-book,.rst-versions .rst-current-version .icon-book{float:left}.rst-versions .rst-current-version.rst-out-of-date{background-color:#e74c3c;color:#fff}.rst-versions .rst-current-version.rst-active-old-version{background-color:#f1c40f;color:#000}.rst-versions.shift-up{height:auto;max-height:100%;overflow-y:scroll}.rst-versions.shift-up .rst-other-versions{display:block}.rst-versions .rst-other-versions{font-size:90%;padding:12px;color:grey;display:none}.rst-versions .rst-other-versions hr{display:block;height:1px;border:0;margin:20px 0;padding:0;border-top:1px solid #413d3d}.rst-versions .rst-other-versions dd{display:inline-block;margin:0}.rst-versions .rst-other-versions dd a{display:inline-block;padding:6px;color:#fcfcfc}.rst-versions.rst-badge{width:auto;bottom:20px;right:20px;left:auto;border:none;max-width:300px;max-height:90%}.rst-versions.rst-badge .fa-book,.rst-versions.rst-badge .icon-book{float:none;line-height:30px}.rst-versions.rst-badge.shift-up .rst-current-version{text-align:right}.rst-versions.rst-badge.shift-up .rst-current-version .fa-book,.rst-versions.rst-badge.shift-up .rst-current-version .icon-book{float:left}.rst-versions.rst-badge>.rst-current-version{width:auto;height:30px;line-height:30px;padding:0 6px;display:block;text-align:center}@media screen and (max-width:768px){.rst-versions{width:85%;display:none}.rst-versions.shift{display:block}}
html/_static/css/custom.css (new)
- @@ -0,0 +1,16 @@
  + .rst-content .admonition .admonition-title:before { + content: none; + } + + .rst-content .admonition-theorem .admonition-title:before { + content: none; + } + + .rst-content .admonition-proof .admonition-title:before { + content: none; + } + + code.literal { + color: #404040 !important; + background-color: #fbfbfb !important; + }
html/_static/css/fonts/Roboto-Slab-Bold.woff (new)
html/_static/css/fonts/Roboto-Slab-Bold.woff2 (new)
html/_static/css/fonts/Roboto-Slab-Regular.woff (new)
html/_static/css/fonts/Roboto-Slab-Regular.woff2 (new)
html/_static/css/fonts/fontawesome-webfont.eot (new)
html/_static/css/fonts/fontawesome-webfont.svg (new)
- @@ -0,0 +1,2671 @@
 + <?xml version="1.0" standalone="no"?> + <!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd" > + <svg> + <metadata> + Created by FontForge 20120731 at Mon Oct 24 17:37:40 2016 + By ,,, + Copyright Dave Gandy 2016. All rights reserved. + </metadata> + <defs> + + <font-face + font-family="FontAwesome" + font-weight="400" + font-stretch="normal" + units-per-em="1792" + panose-1="0 0 0 0 0 0 0 0 0 0" + ascent="1536" + descent="-256" + bbox="-1.02083 -256.962 2304.6 1537.02" + underline-thickness="0" + underline-position="0" + unicode-range="U+0020-F500" + /> + <missing-glyph horiz-adv-x="896" + d="M224 112h448v1312h-448v-1312zM112 0v1536h672v-1536h-672z" /> + <glyph glyph-name=".notdef" horiz-adv-x="896" + d="M224 112h448v1312h-448v-1312zM112 0v1536h672v-1536h-672z" /> + <glyph glyph-name=".null" horiz-adv-x="0" + /> + <glyph glyph-name="nonmarkingreturn" horiz-adv-x="597" + /> + <glyph glyph-name="space" unicode=" " horiz-adv-x="448" + /> + <glyph glyph-name="dieresis" unicode="¨" horiz-adv-x="1792" + /> + <glyph glyph-name="copyright" unicode="©" horiz-adv-x="1792" + /> + <glyph glyph-name="registered" unicode="®" horiz-adv-x="1792" + /> + <glyph glyph-name="acute" unicode="´" horiz-adv-x="1792" + /> + <glyph glyph-name="AE" unicode="Æ" horiz-adv-x="1792" + /> + <glyph glyph-name="Oslash" unicode="Ø" horiz-adv-x="1792" + /> + <glyph glyph-name="trademark" unicode="™" horiz-adv-x="1792" + /> + <glyph glyph-name="infinity" unicode="∞" horiz-adv-x="1792" + /> + <glyph glyph-name="notequal" unicode="≠" horiz-adv-x="1792" + /> + <glyph glyph-name="glass" unicode="" horiz-adv-x="1792" + d="M1699 1350q0 -35 -43 -78l-632 -632v-768h320q26 0 45 -19t19 -45t-19 -45t-45 -19h-896q-26 0 -45 19t-19 45t19 45t45 19h320v768l-632 632q-43 43 -43 78q0 23 18 36.5t38 17.5t43 4h1408q23 0 43 -4t38 -17.5t18 -36.5z" /> + <glyph glyph-name="music" unicode="" + d="M1536 1312v-1120q0 -50 -34 -89t-86 -60.5t-103.5 -32t-96.5 -10.5t-96.5 10.5t-103.5 32t-86 60.5t-34 89t34 89t86 60.5t103.5 32t96.5 10.5q105 0 192 -39v537l-768 -237v-709q0 -50 -34 -89t-86 -60.5t-103.5 -32t-96.5 -10.5t-96.5 10.5t-103.5 32t-86 60.5t-34 89 + t34 89t86 60.5t103.5 32t96.5 10.5q105 0 192 -39v967q0 31 19 56.5t49 35.5l832 256q12 4 28 4q40 0 68 -28t28 -68z" /> + <glyph glyph-name="search" unicode="" horiz-adv-x="1664" + d="M1152 704q0 185 -131.5 316.5t-316.5 131.5t-316.5 -131.5t-131.5 -316.5t131.5 -316.5t316.5 -131.5t316.5 131.5t131.5 316.5zM1664 -128q0 -52 -38 -90t-90 -38q-54 0 -90 38l-343 342q-179 -124 -399 -124q-143 0 -273.5 55.5t-225 150t-150 225t-55.5 273.5 + t55.5 273.5t150 225t225 150t273.5 55.5t273.5 -55.5t225 -150t150 -225t55.5 -273.5q0 -220 -124 -399l343 -343q37 -37 37 -90z" /> + <glyph glyph-name="envelope" unicode="" horiz-adv-x="1792" + d="M1664 32v768q-32 -36 -69 -66q-268 -206 -426 -338q-51 -43 -83 -67t-86.5 -48.5t-102.5 -24.5h-1h-1q-48 0 -102.5 24.5t-86.5 48.5t-83 67q-158 132 -426 338q-37 30 -69 66v-768q0 -13 9.5 -22.5t22.5 -9.5h1472q13 0 22.5 9.5t9.5 22.5zM1664 1083v11v13.5t-0.5 13 + t-3 12.5t-5.5 9t-9 7.5t-14 2.5h-1472q-13 0 -22.5 -9.5t-9.5 -22.5q0 -168 147 -284q193 -152 401 -317q6 -5 35 -29.5t46 -37.5t44.5 -31.5t50.5 -27.5t43 -9h1h1q20 0 43 9t50.5 27.5t44.5 31.5t46 37.5t35 29.5q208 165 401 317q54 43 100.5 115.5t46.5 131.5z + M1792 1120v-1088q0 -66 -47 -113t-113 -47h-1472q-66 0 -113 47t-47 113v1088q0 66 47 113t113 47h1472q66 0 113 -47t47 -113z" /> + <glyph glyph-name="heart" unicode="" horiz-adv-x="1792" + d="M896 -128q-26 0 -44 18l-624 602q-10 8 -27.5 26t-55.5 65.5t-68 97.5t-53.5 121t-23.5 138q0 220 127 344t351 124q62 0 126.5 -21.5t120 -58t95.5 -68.5t76 -68q36 36 76 68t95.5 68.5t120 58t126.5 21.5q224 0 351 -124t127 -344q0 -221 -229 -450l-623 -600 + q-18 -18 -44 -18z" /> + <glyph glyph-name="star" unicode="" horiz-adv-x="1664" + d="M1664 889q0 -22 -26 -48l-363 -354l86 -500q1 -7 1 -20q0 -21 -10.5 -35.5t-30.5 -14.5q-19 0 -40 12l-449 236l-449 -236q-22 -12 -40 -12q-21 0 -31.5 14.5t-10.5 35.5q0 6 2 20l86 500l-364 354q-25 27 -25 48q0 37 56 46l502 73l225 455q19 41 49 41t49 -41l225 -455 + l502 -73q56 -9 56 -46z" /> + <glyph glyph-name="star_empty" unicode="" horiz-adv-x="1664" + d="M1137 532l306 297l-422 62l-189 382l-189 -382l-422 -62l306 -297l-73 -421l378 199l377 -199zM1664 889q0 -22 -26 -48l-363 -354l86 -500q1 -7 1 -20q0 -50 -41 -50q-19 0 -40 12l-449 236l-449 -236q-22 -12 -40 -12q-21 0 -31.5 14.5t-10.5 35.5q0 6 2 20l86 500 + l-364 354q-25 27 -25 48q0 37 56 46l502 73l225 455q19 41 49 41t49 -41l225 -455l502 -73q56 -9 56 -46z" /> + <glyph glyph-name="user" unicode="" horiz-adv-x="1280" + d="M1280 137q0 -109 -62.5 -187t-150.5 -78h-854q-88 0 -150.5 78t-62.5 187q0 85 8.5 160.5t31.5 152t58.5 131t94 89t134.5 34.5q131 -128 313 -128t313 128q76 0 134.5 -34.5t94 -89t58.5 -131t31.5 -152t8.5 -160.5zM1024 1024q0 -159 -112.5 -271.5t-271.5 -112.5 + t-271.5 112.5t-112.5 271.5t112.5 271.5t271.5 112.5t271.5 -112.5t112.5 -271.5z" /> + <glyph glyph-name="film" unicode="" horiz-adv-x="1920" + d="M384 -64v128q0 26 -19 45t-45 19h-128q-26 0 -45 -19t-19 -45v-128q0 -26 19 -45t45 -19h128q26 0 45 19t19 45zM384 320v128q0 26 -19 45t-45 19h-128q-26 0 -45 -19t-19 -45v-128q0 -26 19 -45t45 -19h128q26 0 45 19t19 45zM384 704v128q0 26 -19 45t-45 19h-128 + q-26 0 -45 -19t-19 -45v-128q0 -26 19 -45t45 -19h128q26 0 45 19t19 45zM1408 -64v512q0 26 -19 45t-45 19h-768q-26 0 -45 -19t-19 -45v-512q0 -26 19 -45t45 -19h768q26 0 45 19t19 45zM384 1088v128q0 26 -19 45t-45 19h-128q-26 0 -45 -19t-19 -45v-128q0 -26 19 -45 + t45 -19h128q26 0 45 19t19 45zM1792 -64v128q0 26 -19 45t-45 19h-128q-26 0 -45 -19t-19 -45v-128q0 -26 19 -45t45 -19h128q26 0 45 19t19 45zM1408 704v512q0 26 -19 45t-45 19h-768q-26 0 -45 -19t-19 -45v-512q0 -26 19 -45t45 -19h768q26 0 45 19t19 45zM1792 320v128 + q0 26 -19 45t-45 19h-128q-26 0 -45 -19t-19 -45v-128q0 -26 19 -45t45 -19h128q26 0 45 19t19 45zM1792 704v128q0 26 -19 45t-45 19h-128q-26 0 -45 -19t-19 -45v-128q0 -26 19 -45t45 -19h128q26 0 45 19t19 45zM1792 1088v128q0 26 -19 45t-45 19h-128q-26 0 -45 -19 + t-19 -45v-128q0 -26 19 -45t45 -19h128q26 0 45 19t19 45zM1920 1248v-1344q0 -66 -47 -113t-113 -47h-1600q-66 0 -113 47t-47 113v1344q0 66 47 113t113 47h1600q66 0 113 -47t47 -113z" /> + <glyph glyph-name="th_large" unicode="" horiz-adv-x="1664" + d="M768 512v-384q0 -52 -38 -90t-90 -38h-512q-52 0 -90 38t-38 90v384q0 52 38 90t90 38h512q52 0 90 -38t38 -90zM768 1280v-384q0 -52 -38 -90t-90 -38h-512q-52 0 -90 38t-38 90v384q0 52 38 90t90 38h512q52 0 90 -38t38 -90zM1664 512v-384q0 -52 -38 -90t-90 -38 + h-512q-52 0 -90 38t-38 90v384q0 52 38 90t90 38h512q52 0 90 -38t38 -90zM1664 1280v-384q0 -52 -38 -90t-90 -38h-512q-52 0 -90 38t-38 90v384q0 52 38 90t90 38h512q52 0 90 -38t38 -90z" /> + <glyph glyph-name="th" unicode="" horiz-adv-x="1792" + d="M512 288v-192q0 -40 -28 -68t-68 -28h-320q-40 0 -68 28t-28 68v192q0 40 28 68t68 28h320q40 0 68 -28t28 -68zM512 800v-192q0 -40 -28 -68t-68 -28h-320q-40 0 -68 28t-28 68v192q0 40 28 68t68 28h320q40 0 68 -28t28 -68zM1152 288v-192q0 -40 -28 -68t-68 -28h-320 + q-40 0 -68 28t-28 68v192q0 40 28 68t68 28h320q40 0 68 -28t28 -68zM512 1312v-192q0 -40 -28 -68t-68 -28h-320q-40 0 -68 28t-28 68v192q0 40 28 68t68 28h320q40 0 68 -28t28 -68zM1152 800v-192q0 -40 -28 -68t-68 -28h-320q-40 0 -68 28t-28 68v192q0 40 28 68t68 28 + h320q40 0 68 -28t28 -68zM1792 288v-192q0 -40 -28 -68t-68 -28h-320q-40 0 -68 28t-28 68v192q0 40 28 68t68 28h320q40 0 68 -28t28 -68zM1152 1312v-192q0 -40 -28 -68t-68 -28h-320q-40 0 -68 28t-28 68v192q0 40 28 68t68 28h320q40 0 68 -28t28 -68zM1792 800v-192 + q0 -40 -28 -68t-68 -28h-320q-40 0 -68 28t-28 68v192q0 40 28 68t68 28h320q40 0 68 -28t28 -68zM1792 1312v-192q0 -40 -28 -68t-68 -28h-320q-40 0 -68 28t-28 68v192q0 40 28 68t68 28h320q40 0 68 -28t28 -68z" /> + <glyph glyph-name="th_list" unicode="" horiz-adv-x="1792" + d="M512 288v-192q0 -40 -28 -68t-68 -28h-320q-40 0 -68 28t-28 68v192q0 40 28 68t68 28h320q40 0 68 -28t28 -68zM512 800v-192q0 -40 -28 -68t-68 -28h-320q-40 0 -68 28t-28 68v192q0 40 28 68t68 28h320q40 0 68 -28t28 -68zM1792 288v-192q0 -40 -28 -68t-68 -28h-960 + q-40 0 -68 28t-28 68v192q0 40 28 68t68 28h960q40 0 68 -28t28 -68zM512 1312v-192q0 -40 -28 -68t-68 -28h-320q-40 0 -68 28t-28 68v192q0 40 28 68t68 28h320q40 0 68 -28t28 -68zM1792 800v-192q0 -40 -28 -68t-68 -28h-960q-40 0 -68 28t-28 68v192q0 40 28 68t68 28 + h960q40 0 68 -28t28 -68zM1792 1312v-192q0 -40 -28 -68t-68 -28h-960q-40 0 -68 28t-28 68v192q0 40 28 68t68 28h960q40 0 68 -28t28 -68z" /> + <glyph glyph-name="ok" unicode="" horiz-adv-x="1792" + d="M1671 970q0 -40 -28 -68l-724 -724l-136 -136q-28 -28 -68 -28t-68 28l-136 136l-362 362q-28 28 -28 68t28 68l136 136q28 28 68 28t68 -28l294 -295l656 657q28 28 68 28t68 -28l136 -136q28 -28 28 -68z" /> + <glyph glyph-name="remove" unicode="" horiz-adv-x="1408" + d="M1298 214q0 -40 -28 -68l-136 -136q-28 -28 -68 -28t-68 28l-294 294l-294 -294q-28 -28 -68 -28t-68 28l-136 136q-28 28 -28 68t28 68l294 294l-294 294q-28 28 -28 68t28 68l136 136q28 28 68 28t68 -28l294 -294l294 294q28 28 68 28t68 -28l136 -136q28 -28 28 -68 + t-28 -68l-294 -294l294 -294q28 -28 28 -68z" /> + <glyph glyph-name="zoom_in" unicode="" horiz-adv-x="1664" + d="M1024 736v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-224v-224q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v224h-224q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h224v224q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5v-224h224 + q13 0 22.5 -9.5t9.5 -22.5zM1152 704q0 185 -131.5 316.5t-316.5 131.5t-316.5 -131.5t-131.5 -316.5t131.5 -316.5t316.5 -131.5t316.5 131.5t131.5 316.5zM1664 -128q0 -53 -37.5 -90.5t-90.5 -37.5q-54 0 -90 38l-343 342q-179 -124 -399 -124q-143 0 -273.5 55.5 + t-225 150t-150 225t-55.5 273.5t55.5 273.5t150 225t225 150t273.5 55.5t273.5 -55.5t225 -150t150 -225t55.5 -273.5q0 -220 -124 -399l343 -343q37 -37 37 -90z" /> + <glyph glyph-name="zoom_out" unicode="" horiz-adv-x="1664" + d="M1024 736v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-576q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h576q13 0 22.5 -9.5t9.5 -22.5zM1152 704q0 185 -131.5 316.5t-316.5 131.5t-316.5 -131.5t-131.5 -316.5t131.5 -316.5t316.5 -131.5t316.5 131.5t131.5 316.5z + M1664 -128q0 -53 -37.5 -90.5t-90.5 -37.5q-54 0 -90 38l-343 342q-179 -124 -399 -124q-143 0 -273.5 55.5t-225 150t-150 225t-55.5 273.5t55.5 273.5t150 225t225 150t273.5 55.5t273.5 -55.5t225 -150t150 -225t55.5 -273.5q0 -220 -124 -399l343 -343q37 -37 37 -90z + " /> + <glyph glyph-name="off" unicode="" + d="M1536 640q0 -156 -61 -298t-164 -245t-245 -164t-298 -61t-298 61t-245 164t-164 245t-61 298q0 182 80.5 343t226.5 270q43 32 95.5 25t83.5 -50q32 -42 24.5 -94.5t-49.5 -84.5q-98 -74 -151.5 -181t-53.5 -228q0 -104 40.5 -198.5t109.5 -163.5t163.5 -109.5 + t198.5 -40.5t198.5 40.5t163.5 109.5t109.5 163.5t40.5 198.5q0 121 -53.5 228t-151.5 181q-42 32 -49.5 84.5t24.5 94.5q31 43 84 50t95 -25q146 -109 226.5 -270t80.5 -343zM896 1408v-640q0 -52 -38 -90t-90 -38t-90 38t-38 90v640q0 52 38 90t90 38t90 -38t38 -90z" /> + <glyph glyph-name="signal" unicode="" horiz-adv-x="1792" + d="M256 96v-192q0 -14 -9 -23t-23 -9h-192q-14 0 -23 9t-9 23v192q0 14 9 23t23 9h192q14 0 23 -9t9 -23zM640 224v-320q0 -14 -9 -23t-23 -9h-192q-14 0 -23 9t-9 23v320q0 14 9 23t23 9h192q14 0 23 -9t9 -23zM1024 480v-576q0 -14 -9 -23t-23 -9h-192q-14 0 -23 9t-9 23 + v576q0 14 9 23t23 9h192q14 0 23 -9t9 -23zM1408 864v-960q0 -14 -9 -23t-23 -9h-192q-14 0 -23 9t-9 23v960q0 14 9 23t23 9h192q14 0 23 -9t9 -23zM1792 1376v-1472q0 -14 -9 -23t-23 -9h-192q-14 0 -23 9t-9 23v1472q0 14 9 23t23 9h192q14 0 23 -9t9 -23z" /> + <glyph glyph-name="cog" unicode="" + d="M1024 640q0 106 -75 181t-181 75t-181 -75t-75 -181t75 -181t181 -75t181 75t75 181zM1536 749v-222q0 -12 -8 -23t-20 -13l-185 -28q-19 -54 -39 -91q35 -50 107 -138q10 -12 10 -25t-9 -23q-27 -37 -99 -108t-94 -71q-12 0 -26 9l-138 108q-44 -23 -91 -38 + q-16 -136 -29 -186q-7 -28 -36 -28h-222q-14 0 -24.5 8.5t-11.5 21.5l-28 184q-49 16 -90 37l-141 -107q-10 -9 -25 -9q-14 0 -25 11q-126 114 -165 168q-7 10 -7 23q0 12 8 23q15 21 51 66.5t54 70.5q-27 50 -41 99l-183 27q-13 2 -21 12.5t-8 23.5v222q0 12 8 23t19 13 + l186 28q14 46 39 92q-40 57 -107 138q-10 12 -10 24q0 10 9 23q26 36 98.5 107.5t94.5 71.5q13 0 26 -10l138 -107q44 23 91 38q16 136 29 186q7 28 36 28h222q14 0 24.5 -8.5t11.5 -21.5l28 -184q49 -16 90 -37l142 107q9 9 24 9q13 0 25 -10q129 -119 165 -170q7 -8 7 -22 + q0 -12 -8 -23q-15 -21 -51 -66.5t-54 -70.5q26 -50 41 -98l183 -28q13 -2 21 -12.5t8 -23.5z" /> + <glyph glyph-name="trash" unicode="" horiz-adv-x="1408" + d="M512 800v-576q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9t-9 23v576q0 14 9 23t23 9h64q14 0 23 -9t9 -23zM768 800v-576q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9t-9 23v576q0 14 9 23t23 9h64q14 0 23 -9t9 -23zM1024 800v-576q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9t-9 23v576 + q0 14 9 23t23 9h64q14 0 23 -9t9 -23zM1152 76v948h-896v-948q0 -22 7 -40.5t14.5 -27t10.5 -8.5h832q3 0 10.5 8.5t14.5 27t7 40.5zM480 1152h448l-48 117q-7 9 -17 11h-317q-10 -2 -17 -11zM1408 1120v-64q0 -14 -9 -23t-23 -9h-96v-948q0 -83 -47 -143.5t-113 -60.5h-832 + q-66 0 -113 58.5t-47 141.5v952h-96q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h309l70 167q15 37 54 63t79 26h320q40 0 79 -26t54 -63l70 -167h309q14 0 23 -9t9 -23z" /> + <glyph glyph-name="home" unicode="" horiz-adv-x="1664" + d="M1408 544v-480q0 -26 -19 -45t-45 -19h-384v384h-256v-384h-384q-26 0 -45 19t-19 45v480q0 1 0.5 3t0.5 3l575 474l575 -474q1 -2 1 -6zM1631 613l-62 -74q-8 -9 -21 -11h-3q-13 0 -21 7l-692 577l-692 -577q-12 -8 -24 -7q-13 2 -21 11l-62 74q-8 10 -7 23.5t11 21.5 + l719 599q32 26 76 26t76 -26l244 -204v195q0 14 9 23t23 9h192q14 0 23 -9t9 -23v-408l219 -182q10 -8 11 -21.5t-7 -23.5z" /> + <glyph glyph-name="file_alt" unicode="" + d="M1468 1156q28 -28 48 -76t20 -88v-1152q0 -40 -28 -68t-68 -28h-1344q-40 0 -68 28t-28 68v1600q0 40 28 68t68 28h896q40 0 88 -20t76 -48zM1024 1400v-376h376q-10 29 -22 41l-313 313q-12 12 -41 22zM1408 -128v1024h-416q-40 0 -68 28t-28 68v416h-768v-1536h1280z + " /> + <glyph glyph-name="time" unicode="" + d="M896 992v-448q0 -14 -9 -23t-23 -9h-320q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h224v352q0 14 9 23t23 9h64q14 0 23 -9t9 -23zM1312 640q0 148 -73 273t-198 198t-273 73t-273 -73t-198 -198t-73 -273t73 -273t198 -198t273 -73t273 73t198 198t73 273zM1536 640 + q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="road" unicode="" horiz-adv-x="1920" + d="M1111 540v4l-24 320q-1 13 -11 22.5t-23 9.5h-186q-13 0 -23 -9.5t-11 -22.5l-24 -320v-4q-1 -12 8 -20t21 -8h244q12 0 21 8t8 20zM1870 73q0 -73 -46 -73h-704q13 0 22 9.5t8 22.5l-20 256q-1 13 -11 22.5t-23 9.5h-272q-13 0 -23 -9.5t-11 -22.5l-20 -256 + q-1 -13 8 -22.5t22 -9.5h-704q-46 0 -46 73q0 54 26 116l417 1044q8 19 26 33t38 14h339q-13 0 -23 -9.5t-11 -22.5l-15 -192q-1 -14 8 -23t22 -9h166q13 0 22 9t8 23l-15 192q-1 13 -11 22.5t-23 9.5h339q20 0 38 -14t26 -33l417 -1044q26 -62 26 -116z" /> + <glyph glyph-name="download_alt" unicode="" horiz-adv-x="1664" + d="M1280 192q0 26 -19 45t-45 19t-45 -19t-19 -45t19 -45t45 -19t45 19t19 45zM1536 192q0 26 -19 45t-45 19t-45 -19t-19 -45t19 -45t45 -19t45 19t19 45zM1664 416v-320q0 -40 -28 -68t-68 -28h-1472q-40 0 -68 28t-28 68v320q0 40 28 68t68 28h465l135 -136 + q58 -56 136 -56t136 56l136 136h464q40 0 68 -28t28 -68zM1339 985q17 -41 -14 -70l-448 -448q-18 -19 -45 -19t-45 19l-448 448q-31 29 -14 70q17 39 59 39h256v448q0 26 19 45t45 19h256q26 0 45 -19t19 -45v-448h256q42 0 59 -39z" /> + <glyph glyph-name="download" unicode="" + d="M1120 608q0 -12 -10 -24l-319 -319q-11 -9 -23 -9t-23 9l-320 320q-15 16 -7 35q8 20 30 20h192v352q0 14 9 23t23 9h192q14 0 23 -9t9 -23v-352h192q14 0 23 -9t9 -23zM768 1184q-148 0 -273 -73t-198 -198t-73 -273t73 -273t198 -198t273 -73t273 73t198 198t73 273 + t-73 273t-198 198t-273 73zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="upload" unicode="" + d="M1118 660q-8 -20 -30 -20h-192v-352q0 -14 -9 -23t-23 -9h-192q-14 0 -23 9t-9 23v352h-192q-14 0 -23 9t-9 23q0 12 10 24l319 319q11 9 23 9t23 -9l320 -320q15 -16 7 -35zM768 1184q-148 0 -273 -73t-198 -198t-73 -273t73 -273t198 -198t273 -73t273 73t198 198 + t73 273t-73 273t-198 198t-273 73zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="inbox" unicode="" + d="M1023 576h316q-1 3 -2.5 8.5t-2.5 7.5l-212 496h-708l-212 -496q-1 -3 -2.5 -8.5t-2.5 -7.5h316l95 -192h320zM1536 546v-482q0 -26 -19 -45t-45 -19h-1408q-26 0 -45 19t-19 45v482q0 62 25 123l238 552q10 25 36.5 42t52.5 17h832q26 0 52.5 -17t36.5 -42l238 -552 + q25 -61 25 -123z" /> + <glyph glyph-name="play_circle" unicode="" + d="M1184 640q0 -37 -32 -55l-544 -320q-15 -9 -32 -9q-16 0 -32 8q-32 19 -32 56v640q0 37 32 56q33 18 64 -1l544 -320q32 -18 32 -55zM1312 640q0 148 -73 273t-198 198t-273 73t-273 -73t-198 -198t-73 -273t73 -273t198 -198t273 -73t273 73t198 198t73 273zM1536 640 + q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="repeat" unicode="" + d="M1536 1280v-448q0 -26 -19 -45t-45 -19h-448q-42 0 -59 40q-17 39 14 69l138 138q-148 137 -349 137q-104 0 -198.5 -40.5t-163.5 -109.5t-109.5 -163.5t-40.5 -198.5t40.5 -198.5t109.5 -163.5t163.5 -109.5t198.5 -40.5q119 0 225 52t179 147q7 10 23 12q15 0 25 -9 + l137 -138q9 -8 9.5 -20.5t-7.5 -22.5q-109 -132 -264 -204.5t-327 -72.5q-156 0 -298 61t-245 164t-164 245t-61 298t61 298t164 245t245 164t298 61q147 0 284.5 -55.5t244.5 -156.5l130 129q29 31 70 14q39 -17 39 -59z" /> + <glyph glyph-name="refresh" unicode="" + d="M1511 480q0 -5 -1 -7q-64 -268 -268 -434.5t-478 -166.5q-146 0 -282.5 55t-243.5 157l-129 -129q-19 -19 -45 -19t-45 19t-19 45v448q0 26 19 45t45 19h448q26 0 45 -19t19 -45t-19 -45l-137 -137q71 -66 161 -102t187 -36q134 0 250 65t186 179q11 17 53 117 + q8 23 30 23h192q13 0 22.5 -9.5t9.5 -22.5zM1536 1280v-448q0 -26 -19 -45t-45 -19h-448q-26 0 -45 19t-19 45t19 45l138 138q-148 137 -349 137q-134 0 -250 -65t-186 -179q-11 -17 -53 -117q-8 -23 -30 -23h-199q-13 0 -22.5 9.5t-9.5 22.5v7q65 268 270 434.5t480 166.5 + q146 0 284 -55.5t245 -156.5l130 129q19 19 45 19t45 -19t19 -45z" /> + <glyph glyph-name="list_alt" unicode="" horiz-adv-x="1792" + d="M384 352v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5zM384 608v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5z + M384 864v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5zM1536 352v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-960q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h960q13 0 22.5 -9.5t9.5 -22.5z + M1536 608v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-960q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h960q13 0 22.5 -9.5t9.5 -22.5zM1536 864v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-960q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h960q13 0 22.5 -9.5 + t9.5 -22.5zM1664 160v832q0 13 -9.5 22.5t-22.5 9.5h-1472q-13 0 -22.5 -9.5t-9.5 -22.5v-832q0 -13 9.5 -22.5t22.5 -9.5h1472q13 0 22.5 9.5t9.5 22.5zM1792 1248v-1088q0 -66 -47 -113t-113 -47h-1472q-66 0 -113 47t-47 113v1088q0 66 47 113t113 47h1472q66 0 113 -47 + t47 -113z" /> + <glyph glyph-name="lock" unicode="" horiz-adv-x="1152" + d="M320 768h512v192q0 106 -75 181t-181 75t-181 -75t-75 -181v-192zM1152 672v-576q0 -40 -28 -68t-68 -28h-960q-40 0 -68 28t-28 68v576q0 40 28 68t68 28h32v192q0 184 132 316t316 132t316 -132t132 -316v-192h32q40 0 68 -28t28 -68z" /> + <glyph glyph-name="flag" unicode="" horiz-adv-x="1792" + d="M320 1280q0 -72 -64 -110v-1266q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v1266q-64 38 -64 110q0 53 37.5 90.5t90.5 37.5t90.5 -37.5t37.5 -90.5zM1792 1216v-763q0 -25 -12.5 -38.5t-39.5 -27.5q-215 -116 -369 -116q-61 0 -123.5 22t-108.5 48 + t-115.5 48t-142.5 22q-192 0 -464 -146q-17 -9 -33 -9q-26 0 -45 19t-19 45v742q0 32 31 55q21 14 79 43q236 120 421 120q107 0 200 -29t219 -88q38 -19 88 -19q54 0 117.5 21t110 47t88 47t54.5 21q26 0 45 -19t19 -45z" /> + <glyph glyph-name="headphones" unicode="" horiz-adv-x="1664" + d="M1664 650q0 -166 -60 -314l-20 -49l-185 -33q-22 -83 -90.5 -136.5t-156.5 -53.5v-32q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9t-9 23v576q0 14 9 23t23 9h64q14 0 23 -9t9 -23v-32q71 0 130 -35.5t93 -95.5l68 12q29 95 29 193q0 148 -88 279t-236.5 209t-315.5 78 + t-315.5 -78t-236.5 -209t-88 -279q0 -98 29 -193l68 -12q34 60 93 95.5t130 35.5v32q0 14 9 23t23 9h64q14 0 23 -9t9 -23v-576q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9t-9 23v32q-88 0 -156.5 53.5t-90.5 136.5l-185 33l-20 49q-60 148 -60 314q0 151 67 291t179 242.5 + t266 163.5t320 61t320 -61t266 -163.5t179 -242.5t67 -291z" /> + <glyph glyph-name="volume_off" unicode="" horiz-adv-x="768" + d="M768 1184v-1088q0 -26 -19 -45t-45 -19t-45 19l-333 333h-262q-26 0 -45 19t-19 45v384q0 26 19 45t45 19h262l333 333q19 19 45 19t45 -19t19 -45z" /> + <glyph glyph-name="volume_down" unicode="" horiz-adv-x="1152" + d="M768 1184v-1088q0 -26 -19 -45t-45 -19t-45 19l-333 333h-262q-26 0 -45 19t-19 45v384q0 26 19 45t45 19h262l333 333q19 19 45 19t45 -19t19 -45zM1152 640q0 -76 -42.5 -141.5t-112.5 -93.5q-10 -5 -25 -5q-26 0 -45 18.5t-19 45.5q0 21 12 35.5t29 25t34 23t29 36 + t12 56.5t-12 56.5t-29 36t-34 23t-29 25t-12 35.5q0 27 19 45.5t45 18.5q15 0 25 -5q70 -27 112.5 -93t42.5 -142z" /> + <glyph glyph-name="volume_up" unicode="" horiz-adv-x="1664" + d="M768 1184v-1088q0 -26 -19 -45t-45 -19t-45 19l-333 333h-262q-26 0 -45 19t-19 45v384q0 26 19 45t45 19h262l333 333q19 19 45 19t45 -19t19 -45zM1152 640q0 -76 -42.5 -141.5t-112.5 -93.5q-10 -5 -25 -5q-26 0 -45 18.5t-19 45.5q0 21 12 35.5t29 25t34 23t29 36 + t12 56.5t-12 56.5t-29 36t-34 23t-29 25t-12 35.5q0 27 19 45.5t45 18.5q15 0 25 -5q70 -27 112.5 -93t42.5 -142zM1408 640q0 -153 -85 -282.5t-225 -188.5q-13 -5 -25 -5q-27 0 -46 19t-19 45q0 39 39 59q56 29 76 44q74 54 115.5 135.5t41.5 173.5t-41.5 173.5 + t-115.5 135.5q-20 15 -76 44q-39 20 -39 59q0 26 19 45t45 19q13 0 26 -5q140 -59 225 -188.5t85 -282.5zM1664 640q0 -230 -127 -422.5t-338 -283.5q-13 -5 -26 -5q-26 0 -45 19t-19 45q0 36 39 59q7 4 22.5 10.5t22.5 10.5q46 25 82 51q123 91 192 227t69 289t-69 289 + t-192 227q-36 26 -82 51q-7 4 -22.5 10.5t-22.5 10.5q-39 23 -39 59q0 26 19 45t45 19q13 0 26 -5q211 -91 338 -283.5t127 -422.5z" /> + <glyph glyph-name="qrcode" unicode="" horiz-adv-x="1408" + d="M384 384v-128h-128v128h128zM384 1152v-128h-128v128h128zM1152 1152v-128h-128v128h128zM128 129h384v383h-384v-383zM128 896h384v384h-384v-384zM896 896h384v384h-384v-384zM640 640v-640h-640v640h640zM1152 128v-128h-128v128h128zM1408 128v-128h-128v128h128z + M1408 640v-384h-384v128h-128v-384h-128v640h384v-128h128v128h128zM640 1408v-640h-640v640h640zM1408 1408v-640h-640v640h640z" /> + <glyph glyph-name="barcode" unicode="" horiz-adv-x="1792" + d="M63 0h-63v1408h63v-1408zM126 1h-32v1407h32v-1407zM220 1h-31v1407h31v-1407zM377 1h-31v1407h31v-1407zM534 1h-62v1407h62v-1407zM660 1h-31v1407h31v-1407zM723 1h-31v1407h31v-1407zM786 1h-31v1407h31v-1407zM943 1h-63v1407h63v-1407zM1100 1h-63v1407h63v-1407z + M1226 1h-63v1407h63v-1407zM1352 1h-63v1407h63v-1407zM1446 1h-63v1407h63v-1407zM1635 1h-94v1407h94v-1407zM1698 1h-32v1407h32v-1407zM1792 0h-63v1408h63v-1408z" /> + <glyph glyph-name="tag" unicode="" + d="M448 1088q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM1515 512q0 -53 -37 -90l-491 -492q-39 -37 -91 -37q-53 0 -90 37l-715 716q-38 37 -64.5 101t-26.5 117v416q0 52 38 90t90 38h416q53 0 117 -26.5t102 -64.5 + l715 -714q37 -39 37 -91z" /> + <glyph glyph-name="tags" unicode="" horiz-adv-x="1920" + d="M448 1088q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM1515 512q0 -53 -37 -90l-491 -492q-39 -37 -91 -37q-53 0 -90 37l-715 716q-38 37 -64.5 101t-26.5 117v416q0 52 38 90t90 38h416q53 0 117 -26.5t102 -64.5 + l715 -714q37 -39 37 -91zM1899 512q0 -53 -37 -90l-491 -492q-39 -37 -91 -37q-36 0 -59 14t-53 45l470 470q37 37 37 90q0 52 -37 91l-715 714q-38 38 -102 64.5t-117 26.5h224q53 0 117 -26.5t102 -64.5l715 -714q37 -39 37 -91z" /> + <glyph glyph-name="book" unicode="" horiz-adv-x="1664" + d="M1639 1058q40 -57 18 -129l-275 -906q-19 -64 -76.5 -107.5t-122.5 -43.5h-923q-77 0 -148.5 53.5t-99.5 131.5q-24 67 -2 127q0 4 3 27t4 37q1 8 -3 21.5t-3 19.5q2 11 8 21t16.5 23.5t16.5 23.5q23 38 45 91.5t30 91.5q3 10 0.5 30t-0.5 28q3 11 17 28t17 23 + q21 36 42 92t25 90q1 9 -2.5 32t0.5 28q4 13 22 30.5t22 22.5q19 26 42.5 84.5t27.5 96.5q1 8 -3 25.5t-2 26.5q2 8 9 18t18 23t17 21q8 12 16.5 30.5t15 35t16 36t19.5 32t26.5 23.5t36 11.5t47.5 -5.5l-1 -3q38 9 51 9h761q74 0 114 -56t18 -130l-274 -906 + q-36 -119 -71.5 -153.5t-128.5 -34.5h-869q-27 0 -38 -15q-11 -16 -1 -43q24 -70 144 -70h923q29 0 56 15.5t35 41.5l300 987q7 22 5 57q38 -15 59 -43zM575 1056q-4 -13 2 -22.5t20 -9.5h608q13 0 25.5 9.5t16.5 22.5l21 64q4 13 -2 22.5t-20 9.5h-608q-13 0 -25.5 -9.5 + t-16.5 -22.5zM492 800q-4 -13 2 -22.5t20 -9.5h608q13 0 25.5 9.5t16.5 22.5l21 64q4 13 -2 22.5t-20 9.5h-608q-13 0 -25.5 -9.5t-16.5 -22.5z" /> + <glyph glyph-name="bookmark" unicode="" horiz-adv-x="1280" + d="M1164 1408q23 0 44 -9q33 -13 52.5 -41t19.5 -62v-1289q0 -34 -19.5 -62t-52.5 -41q-19 -8 -44 -8q-48 0 -83 32l-441 424l-441 -424q-36 -33 -83 -33q-23 0 -44 9q-33 13 -52.5 41t-19.5 62v1289q0 34 19.5 62t52.5 41q21 9 44 9h1048z" /> + <glyph glyph-name="print" unicode="" horiz-adv-x="1664" + d="M384 0h896v256h-896v-256zM384 640h896v384h-160q-40 0 -68 28t-28 68v160h-640v-640zM1536 576q0 26 -19 45t-45 19t-45 -19t-19 -45t19 -45t45 -19t45 19t19 45zM1664 576v-416q0 -13 -9.5 -22.5t-22.5 -9.5h-224v-160q0 -40 -28 -68t-68 -28h-960q-40 0 -68 28t-28 68 + v160h-224q-13 0 -22.5 9.5t-9.5 22.5v416q0 79 56.5 135.5t135.5 56.5h64v544q0 40 28 68t68 28h672q40 0 88 -20t76 -48l152 -152q28 -28 48 -76t20 -88v-256h64q79 0 135.5 -56.5t56.5 -135.5z" /> + <glyph glyph-name="camera" unicode="" horiz-adv-x="1920" + d="M960 864q119 0 203.5 -84.5t84.5 -203.5t-84.5 -203.5t-203.5 -84.5t-203.5 84.5t-84.5 203.5t84.5 203.5t203.5 84.5zM1664 1280q106 0 181 -75t75 -181v-896q0 -106 -75 -181t-181 -75h-1408q-106 0 -181 75t-75 181v896q0 106 75 181t181 75h224l51 136 + q19 49 69.5 84.5t103.5 35.5h512q53 0 103.5 -35.5t69.5 -84.5l51 -136h224zM960 128q185 0 316.5 131.5t131.5 316.5t-131.5 316.5t-316.5 131.5t-316.5 -131.5t-131.5 -316.5t131.5 -316.5t316.5 -131.5z" /> + <glyph glyph-name="font" unicode="" horiz-adv-x="1664" + d="M725 977l-170 -450q33 0 136.5 -2t160.5 -2q19 0 57 2q-87 253 -184 452zM0 -128l2 79q23 7 56 12.5t57 10.5t49.5 14.5t44.5 29t31 50.5l237 616l280 724h75h53q8 -14 11 -21l205 -480q33 -78 106 -257.5t114 -274.5q15 -34 58 -144.5t72 -168.5q20 -45 35 -57 + q19 -15 88 -29.5t84 -20.5q6 -38 6 -57q0 -5 -0.5 -13.5t-0.5 -12.5q-63 0 -190 8t-191 8q-76 0 -215 -7t-178 -8q0 43 4 78l131 28q1 0 12.5 2.5t15.5 3.5t14.5 4.5t15 6.5t11 8t9 11t2.5 14q0 16 -31 96.5t-72 177.5t-42 100l-450 2q-26 -58 -76.5 -195.5t-50.5 -162.5 + q0 -22 14 -37.5t43.5 -24.5t48.5 -13.5t57 -8.5t41 -4q1 -19 1 -58q0 -9 -2 -27q-58 0 -174.5 10t-174.5 10q-8 0 -26.5 -4t-21.5 -4q-80 -14 -188 -14z" /> + <glyph glyph-name="bold" unicode="" horiz-adv-x="1408" + d="M555 15q74 -32 140 -32q376 0 376 335q0 114 -41 180q-27 44 -61.5 74t-67.5 46.5t-80.5 25t-84 10.5t-94.5 2q-73 0 -101 -10q0 -53 -0.5 -159t-0.5 -158q0 -8 -1 -67.5t-0.5 -96.5t4.5 -83.5t12 -66.5zM541 761q42 -7 109 -7q82 0 143 13t110 44.5t74.5 89.5t25.5 142 + q0 70 -29 122.5t-79 82t-108 43.5t-124 14q-50 0 -130 -13q0 -50 4 -151t4 -152q0 -27 -0.5 -80t-0.5 -79q0 -46 1 -69zM0 -128l2 94q15 4 85 16t106 27q7 12 12.5 27t8.5 33.5t5.5 32.5t3 37.5t0.5 34v35.5v30q0 982 -22 1025q-4 8 -22 14.5t-44.5 11t-49.5 7t-48.5 4.5 + t-30.5 3l-4 83q98 2 340 11.5t373 9.5q23 0 68 -0.5t68 -0.5q70 0 136.5 -13t128.5 -42t108 -71t74 -104.5t28 -137.5q0 -52 -16.5 -95.5t-39 -72t-64.5 -57.5t-73 -45t-84 -40q154 -35 256.5 -134t102.5 -248q0 -100 -35 -179.5t-93.5 -130.5t-138 -85.5t-163.5 -48.5 + t-176 -14q-44 0 -132 3t-132 3q-106 0 -307 -11t-231 -12z" /> + <glyph glyph-name="italic" unicode="" horiz-adv-x="1024" + d="M0 -126l17 85q22 7 61.5 16.5t72 19t59.5 23.5q28 35 41 101q1 7 62 289t114 543.5t52 296.5v25q-24 13 -54.5 18.5t-69.5 8t-58 5.5l19 103q33 -2 120 -6.5t149.5 -7t120.5 -2.5q48 0 98.5 2.5t121 7t98.5 6.5q-5 -39 -19 -89q-30 -10 -101.5 -28.5t-108.5 -33.5 + q-8 -19 -14 -42.5t-9 -40t-7.5 -45.5t-6.5 -42q-27 -148 -87.5 -419.5t-77.5 -355.5q-2 -9 -13 -58t-20 -90t-16 -83.5t-6 -57.5l1 -18q17 -4 185 -31q-3 -44 -16 -99q-11 0 -32.5 -1.5t-32.5 -1.5q-29 0 -87 10t-86 10q-138 2 -206 2q-51 0 -143 -9t-121 -11z" /> + <glyph glyph-name="text_height" unicode="" horiz-adv-x="1792" + d="M1744 128q33 0 42 -18.5t-11 -44.5l-126 -162q-20 -26 -49 -26t-49 26l-126 162q-20 26 -11 44.5t42 18.5h80v1024h-80q-33 0 -42 18.5t11 44.5l126 162q20 26 49 26t49 -26l126 -162q20 -26 11 -44.5t-42 -18.5h-80v-1024h80zM81 1407l54 -27q12 -5 211 -5q44 0 132 2 + t132 2q36 0 107.5 -0.5t107.5 -0.5h293q6 0 21 -0.5t20.5 0t16 3t17.5 9t15 17.5l42 1q4 0 14 -0.5t14 -0.5q2 -112 2 -336q0 -80 -5 -109q-39 -14 -68 -18q-25 44 -54 128q-3 9 -11 48t-14.5 73.5t-7.5 35.5q-6 8 -12 12.5t-15.5 6t-13 2.5t-18 0.5t-16.5 -0.5 + q-17 0 -66.5 0.5t-74.5 0.5t-64 -2t-71 -6q-9 -81 -8 -136q0 -94 2 -388t2 -455q0 -16 -2.5 -71.5t0 -91.5t12.5 -69q40 -21 124 -42.5t120 -37.5q5 -40 5 -50q0 -14 -3 -29l-34 -1q-76 -2 -218 8t-207 10q-50 0 -151 -9t-152 -9q-3 51 -3 52v9q17 27 61.5 43t98.5 29t78 27 + q19 42 19 383q0 101 -3 303t-3 303v117q0 2 0.5 15.5t0.5 25t-1 25.5t-3 24t-5 14q-11 12 -162 12q-33 0 -93 -12t-80 -26q-19 -13 -34 -72.5t-31.5 -111t-42.5 -53.5q-42 26 -56 44v383z" /> + <glyph glyph-name="text_width" unicode="" + d="M81 1407l54 -27q12 -5 211 -5q44 0 132 2t132 2q70 0 246.5 1t304.5 0.5t247 -4.5q33 -1 56 31l42 1q4 0 14 -0.5t14 -0.5q2 -112 2 -336q0 -80 -5 -109q-39 -14 -68 -18q-25 44 -54 128q-3 9 -11 47.5t-15 73.5t-7 36q-10 13 -27 19q-5 2 -66 2q-30 0 -93 1t-103 1 + t-94 -2t-96 -7q-9 -81 -8 -136l1 -152v52q0 -55 1 -154t1.5 -180t0.5 -153q0 -16 -2.5 -71.5t0 -91.5t12.5 -69q40 -21 124 -42.5t120 -37.5q5 -40 5 -50q0 -14 -3 -29l-34 -1q-76 -2 -218 8t-207 10q-50 0 -151 -9t-152 -9q-3 51 -3 52v9q17 27 61.5 43t98.5 29t78 27 + q7 16 11.5 74t6 145.5t1.5 155t-0.5 153.5t-0.5 89q0 7 -2.5 21.5t-2.5 22.5q0 7 0.5 44t1 73t0 76.5t-3 67.5t-6.5 32q-11 12 -162 12q-41 0 -163 -13.5t-138 -24.5q-19 -12 -34 -71.5t-31.5 -111.5t-42.5 -54q-42 26 -56 44v383zM1310 125q12 0 42 -19.5t57.5 -41.5 + t59.5 -49t36 -30q26 -21 26 -49t-26 -49q-4 -3 -36 -30t-59.5 -49t-57.5 -41.5t-42 -19.5q-13 0 -20.5 10.5t-10 28.5t-2.5 33.5t1.5 33t1.5 19.5h-1024q0 -2 1.5 -19.5t1.5 -33t-2.5 -33.5t-10 -28.5t-20.5 -10.5q-12 0 -42 19.5t-57.5 41.5t-59.5 49t-36 30q-26 21 -26 49 + t26 49q4 3 36 30t59.5 49t57.5 41.5t42 19.5q13 0 20.5 -10.5t10 -28.5t2.5 -33.5t-1.5 -33t-1.5 -19.5h1024q0 2 -1.5 19.5t-1.5 33t2.5 33.5t10 28.5t20.5 10.5z" /> + <glyph glyph-name="align_left" unicode="" horiz-adv-x="1792" + d="M1792 192v-128q0 -26 -19 -45t-45 -19h-1664q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h1664q26 0 45 -19t19 -45zM1408 576v-128q0 -26 -19 -45t-45 -19h-1280q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h1280q26 0 45 -19t19 -45zM1664 960v-128q0 -26 -19 -45 + t-45 -19h-1536q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h1536q26 0 45 -19t19 -45zM1280 1344v-128q0 -26 -19 -45t-45 -19h-1152q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h1152q26 0 45 -19t19 -45z" /> + <glyph glyph-name="align_center" unicode="" horiz-adv-x="1792" + d="M1792 192v-128q0 -26 -19 -45t-45 -19h-1664q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h1664q26 0 45 -19t19 -45zM1408 576v-128q0 -26 -19 -45t-45 -19h-896q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h896q26 0 45 -19t19 -45zM1664 960v-128q0 -26 -19 -45t-45 -19 + h-1408q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h1408q26 0 45 -19t19 -45zM1280 1344v-128q0 -26 -19 -45t-45 -19h-640q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h640q26 0 45 -19t19 -45z" /> + <glyph glyph-name="align_right" unicode="" horiz-adv-x="1792" + d="M1792 192v-128q0 -26 -19 -45t-45 -19h-1664q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h1664q26 0 45 -19t19 -45zM1792 576v-128q0 -26 -19 -45t-45 -19h-1280q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h1280q26 0 45 -19t19 -45zM1792 960v-128q0 -26 -19 -45 + t-45 -19h-1536q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h1536q26 0 45 -19t19 -45zM1792 1344v-128q0 -26 -19 -45t-45 -19h-1152q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h1152q26 0 45 -19t19 -45z" /> + <glyph glyph-name="align_justify" unicode="" horiz-adv-x="1792" + d="M1792 192v-128q0 -26 -19 -45t-45 -19h-1664q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h1664q26 0 45 -19t19 -45zM1792 576v-128q0 -26 -19 -45t-45 -19h-1664q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h1664q26 0 45 -19t19 -45zM1792 960v-128q0 -26 -19 -45 + t-45 -19h-1664q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h1664q26 0 45 -19t19 -45zM1792 1344v-128q0 -26 -19 -45t-45 -19h-1664q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h1664q26 0 45 -19t19 -45z" /> + <glyph glyph-name="list" unicode="" horiz-adv-x="1792" + d="M256 224v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-192q-13 0 -22.5 9.5t-9.5 22.5v192q0 13 9.5 22.5t22.5 9.5h192q13 0 22.5 -9.5t9.5 -22.5zM256 608v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-192q-13 0 -22.5 9.5t-9.5 22.5v192q0 13 9.5 22.5t22.5 9.5h192q13 0 22.5 -9.5 + t9.5 -22.5zM256 992v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-192q-13 0 -22.5 9.5t-9.5 22.5v192q0 13 9.5 22.5t22.5 9.5h192q13 0 22.5 -9.5t9.5 -22.5zM1792 224v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-1344q-13 0 -22.5 9.5t-9.5 22.5v192q0 13 9.5 22.5t22.5 9.5h1344 + q13 0 22.5 -9.5t9.5 -22.5zM256 1376v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-192q-13 0 -22.5 9.5t-9.5 22.5v192q0 13 9.5 22.5t22.5 9.5h192q13 0 22.5 -9.5t9.5 -22.5zM1792 608v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-1344q-13 0 -22.5 9.5t-9.5 22.5v192q0 13 9.5 22.5 + t22.5 9.5h1344q13 0 22.5 -9.5t9.5 -22.5zM1792 992v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-1344q-13 0 -22.5 9.5t-9.5 22.5v192q0 13 9.5 22.5t22.5 9.5h1344q13 0 22.5 -9.5t9.5 -22.5zM1792 1376v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-1344q-13 0 -22.5 9.5t-9.5 22.5v192 + q0 13 9.5 22.5t22.5 9.5h1344q13 0 22.5 -9.5t9.5 -22.5z" /> + <glyph glyph-name="indent_left" unicode="" horiz-adv-x="1792" + d="M384 992v-576q0 -13 -9.5 -22.5t-22.5 -9.5q-14 0 -23 9l-288 288q-9 9 -9 23t9 23l288 288q9 9 23 9q13 0 22.5 -9.5t9.5 -22.5zM1792 224v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-1728q-13 0 -22.5 9.5t-9.5 22.5v192q0 13 9.5 22.5t22.5 9.5h1728q13 0 22.5 -9.5 + t9.5 -22.5zM1792 608v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-1088q-13 0 -22.5 9.5t-9.5 22.5v192q0 13 9.5 22.5t22.5 9.5h1088q13 0 22.5 -9.5t9.5 -22.5zM1792 992v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-1088q-13 0 -22.5 9.5t-9.5 22.5v192q0 13 9.5 22.5t22.5 9.5h1088 + q13 0 22.5 -9.5t9.5 -22.5zM1792 1376v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-1728q-13 0 -22.5 9.5t-9.5 22.5v192q0 13 9.5 22.5t22.5 9.5h1728q13 0 22.5 -9.5t9.5 -22.5z" /> + <glyph glyph-name="indent_right" unicode="" horiz-adv-x="1792" + d="M352 704q0 -14 -9 -23l-288 -288q-9 -9 -23 -9q-13 0 -22.5 9.5t-9.5 22.5v576q0 13 9.5 22.5t22.5 9.5q14 0 23 -9l288 -288q9 -9 9 -23zM1792 224v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-1728q-13 0 -22.5 9.5t-9.5 22.5v192q0 13 9.5 22.5t22.5 9.5h1728q13 0 22.5 -9.5 + t9.5 -22.5zM1792 608v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-1088q-13 0 -22.5 9.5t-9.5 22.5v192q0 13 9.5 22.5t22.5 9.5h1088q13 0 22.5 -9.5t9.5 -22.5zM1792 992v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-1088q-13 0 -22.5 9.5t-9.5 22.5v192q0 13 9.5 22.5t22.5 9.5h1088 + q13 0 22.5 -9.5t9.5 -22.5zM1792 1376v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-1728q-13 0 -22.5 9.5t-9.5 22.5v192q0 13 9.5 22.5t22.5 9.5h1728q13 0 22.5 -9.5t9.5 -22.5z" /> + <glyph glyph-name="facetime_video" unicode="" horiz-adv-x="1792" + d="M1792 1184v-1088q0 -42 -39 -59q-13 -5 -25 -5q-27 0 -45 19l-403 403v-166q0 -119 -84.5 -203.5t-203.5 -84.5h-704q-119 0 -203.5 84.5t-84.5 203.5v704q0 119 84.5 203.5t203.5 84.5h704q119 0 203.5 -84.5t84.5 -203.5v-165l403 402q18 19 45 19q12 0 25 -5 + q39 -17 39 -59z" /> + <glyph glyph-name="picture" unicode="" horiz-adv-x="1920" + d="M640 960q0 -80 -56 -136t-136 -56t-136 56t-56 136t56 136t136 56t136 -56t56 -136zM1664 576v-448h-1408v192l320 320l160 -160l512 512zM1760 1280h-1600q-13 0 -22.5 -9.5t-9.5 -22.5v-1216q0 -13 9.5 -22.5t22.5 -9.5h1600q13 0 22.5 9.5t9.5 22.5v1216 + q0 13 -9.5 22.5t-22.5 9.5zM1920 1248v-1216q0 -66 -47 -113t-113 -47h-1600q-66 0 -113 47t-47 113v1216q0 66 47 113t113 47h1600q66 0 113 -47t47 -113z" /> + <glyph glyph-name="pencil" unicode="" + d="M363 0l91 91l-235 235l-91 -91v-107h128v-128h107zM886 928q0 22 -22 22q-10 0 -17 -7l-542 -542q-7 -7 -7 -17q0 -22 22 -22q10 0 17 7l542 542q7 7 7 17zM832 1120l416 -416l-832 -832h-416v416zM1515 1024q0 -53 -37 -90l-166 -166l-416 416l166 165q36 38 90 38 + q53 0 91 -38l235 -234q37 -39 37 -91z" /> + <glyph glyph-name="map_marker" unicode="" horiz-adv-x="1024" + d="M768 896q0 106 -75 181t-181 75t-181 -75t-75 -181t75 -181t181 -75t181 75t75 181zM1024 896q0 -109 -33 -179l-364 -774q-16 -33 -47.5 -52t-67.5 -19t-67.5 19t-46.5 52l-365 774q-33 70 -33 179q0 212 150 362t362 150t362 -150t150 -362z" /> + <glyph glyph-name="adjust" unicode="" + d="M768 96v1088q-148 0 -273 -73t-198 -198t-73 -273t73 -273t198 -198t273 -73zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="tint" unicode="" horiz-adv-x="1024" + d="M512 384q0 36 -20 69q-1 1 -15.5 22.5t-25.5 38t-25 44t-21 50.5q-4 16 -21 16t-21 -16q-7 -23 -21 -50.5t-25 -44t-25.5 -38t-15.5 -22.5q-20 -33 -20 -69q0 -53 37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM1024 512q0 -212 -150 -362t-362 -150t-362 150t-150 362 + q0 145 81 275q6 9 62.5 90.5t101 151t99.5 178t83 201.5q9 30 34 47t51 17t51.5 -17t33.5 -47q28 -93 83 -201.5t99.5 -178t101 -151t62.5 -90.5q81 -127 81 -275z" /> + <glyph glyph-name="edit" unicode="" horiz-adv-x="1792" + d="M888 352l116 116l-152 152l-116 -116v-56h96v-96h56zM1328 1072q-16 16 -33 -1l-350 -350q-17 -17 -1 -33t33 1l350 350q17 17 1 33zM1408 478v-190q0 -119 -84.5 -203.5t-203.5 -84.5h-832q-119 0 -203.5 84.5t-84.5 203.5v832q0 119 84.5 203.5t203.5 84.5h832 + q63 0 117 -25q15 -7 18 -23q3 -17 -9 -29l-49 -49q-14 -14 -32 -8q-23 6 -45 6h-832q-66 0 -113 -47t-47 -113v-832q0 -66 47 -113t113 -47h832q66 0 113 47t47 113v126q0 13 9 22l64 64q15 15 35 7t20 -29zM1312 1216l288 -288l-672 -672h-288v288zM1756 1084l-92 -92 + l-288 288l92 92q28 28 68 28t68 -28l152 -152q28 -28 28 -68t-28 -68z" /> + <glyph glyph-name="share" unicode="" horiz-adv-x="1664" + d="M1408 547v-259q0 -119 -84.5 -203.5t-203.5 -84.5h-832q-119 0 -203.5 84.5t-84.5 203.5v832q0 119 84.5 203.5t203.5 84.5h255v0q13 0 22.5 -9.5t9.5 -22.5q0 -27 -26 -32q-77 -26 -133 -60q-10 -4 -16 -4h-112q-66 0 -113 -47t-47 -113v-832q0 -66 47 -113t113 -47h832 + q66 0 113 47t47 113v214q0 19 18 29q28 13 54 37q16 16 35 8q21 -9 21 -29zM1645 1043l-384 -384q-18 -19 -45 -19q-12 0 -25 5q-39 17 -39 59v192h-160q-323 0 -438 -131q-119 -137 -74 -473q3 -23 -20 -34q-8 -2 -12 -2q-16 0 -26 13q-10 14 -21 31t-39.5 68.5t-49.5 99.5 + t-38.5 114t-17.5 122q0 49 3.5 91t14 90t28 88t47 81.5t68.5 74t94.5 61.5t124.5 48.5t159.5 30.5t196.5 11h160v192q0 42 39 59q13 5 25 5q26 0 45 -19l384 -384q19 -19 19 -45t-19 -45z" /> + <glyph glyph-name="check" unicode="" horiz-adv-x="1664" + d="M1408 606v-318q0 -119 -84.5 -203.5t-203.5 -84.5h-832q-119 0 -203.5 84.5t-84.5 203.5v832q0 119 84.5 203.5t203.5 84.5h832q63 0 117 -25q15 -7 18 -23q3 -17 -9 -29l-49 -49q-10 -10 -23 -10q-3 0 -9 2q-23 6 -45 6h-832q-66 0 -113 -47t-47 -113v-832 + q0 -66 47 -113t113 -47h832q66 0 113 47t47 113v254q0 13 9 22l64 64q10 10 23 10q6 0 12 -3q20 -8 20 -29zM1639 1095l-814 -814q-24 -24 -57 -24t-57 24l-430 430q-24 24 -24 57t24 57l110 110q24 24 57 24t57 -24l263 -263l647 647q24 24 57 24t57 -24l110 -110 + q24 -24 24 -57t-24 -57z" /> + <glyph glyph-name="move" unicode="" horiz-adv-x="1792" + d="M1792 640q0 -26 -19 -45l-256 -256q-19 -19 -45 -19t-45 19t-19 45v128h-384v-384h128q26 0 45 -19t19 -45t-19 -45l-256 -256q-19 -19 -45 -19t-45 19l-256 256q-19 19 -19 45t19 45t45 19h128v384h-384v-128q0 -26 -19 -45t-45 -19t-45 19l-256 256q-19 19 -19 45 + t19 45l256 256q19 19 45 19t45 -19t19 -45v-128h384v384h-128q-26 0 -45 19t-19 45t19 45l256 256q19 19 45 19t45 -19l256 -256q19 -19 19 -45t-19 -45t-45 -19h-128v-384h384v128q0 26 19 45t45 19t45 -19l256 -256q19 -19 19 -45z" /> + <glyph glyph-name="step_backward" unicode="" horiz-adv-x="1024" + d="M979 1395q19 19 32 13t13 -32v-1472q0 -26 -13 -32t-32 13l-710 710q-9 9 -13 19v-678q0 -26 -19 -45t-45 -19h-128q-26 0 -45 19t-19 45v1408q0 26 19 45t45 19h128q26 0 45 -19t19 -45v-678q4 10 13 19z" /> + <glyph glyph-name="fast_backward" unicode="" horiz-adv-x="1792" + d="M1747 1395q19 19 32 13t13 -32v-1472q0 -26 -13 -32t-32 13l-710 710q-9 9 -13 19v-710q0 -26 -13 -32t-32 13l-710 710q-9 9 -13 19v-678q0 -26 -19 -45t-45 -19h-128q-26 0 -45 19t-19 45v1408q0 26 19 45t45 19h128q26 0 45 -19t19 -45v-678q4 10 13 19l710 710 + q19 19 32 13t13 -32v-710q4 10 13 19z" /> + <glyph glyph-name="backward" unicode="" horiz-adv-x="1664" + d="M1619 1395q19 19 32 13t13 -32v-1472q0 -26 -13 -32t-32 13l-710 710q-9 9 -13 19v-710q0 -26 -13 -32t-32 13l-710 710q-19 19 -19 45t19 45l710 710q19 19 32 13t13 -32v-710q4 10 13 19z" /> + <glyph glyph-name="play" unicode="" horiz-adv-x="1408" + d="M1384 609l-1328 -738q-23 -13 -39.5 -3t-16.5 36v1472q0 26 16.5 36t39.5 -3l1328 -738q23 -13 23 -31t-23 -31z" /> + <glyph glyph-name="pause" unicode="" + d="M1536 1344v-1408q0 -26 -19 -45t-45 -19h-512q-26 0 -45 19t-19 45v1408q0 26 19 45t45 19h512q26 0 45 -19t19 -45zM640 1344v-1408q0 -26 -19 -45t-45 -19h-512q-26 0 -45 19t-19 45v1408q0 26 19 45t45 19h512q26 0 45 -19t19 -45z" /> + <glyph glyph-name="stop" unicode="" + d="M1536 1344v-1408q0 -26 -19 -45t-45 -19h-1408q-26 0 -45 19t-19 45v1408q0 26 19 45t45 19h1408q26 0 45 -19t19 -45z" /> + <glyph glyph-name="forward" unicode="" horiz-adv-x="1664" + d="M45 -115q-19 -19 -32 -13t-13 32v1472q0 26 13 32t32 -13l710 -710q9 -9 13 -19v710q0 26 13 32t32 -13l710 -710q19 -19 19 -45t-19 -45l-710 -710q-19 -19 -32 -13t-13 32v710q-4 -10 -13 -19z" /> + <glyph glyph-name="fast_forward" unicode="" horiz-adv-x="1792" + d="M45 -115q-19 -19 -32 -13t-13 32v1472q0 26 13 32t32 -13l710 -710q9 -9 13 -19v710q0 26 13 32t32 -13l710 -710q9 -9 13 -19v678q0 26 19 45t45 19h128q26 0 45 -19t19 -45v-1408q0 -26 -19 -45t-45 -19h-128q-26 0 -45 19t-19 45v678q-4 -10 -13 -19l-710 -710 + q-19 -19 -32 -13t-13 32v710q-4 -10 -13 -19z" /> + <glyph glyph-name="step_forward" unicode="" horiz-adv-x="1024" + d="M45 -115q-19 -19 -32 -13t-13 32v1472q0 26 13 32t32 -13l710 -710q9 -9 13 -19v678q0 26 19 45t45 19h128q26 0 45 -19t19 -45v-1408q0 -26 -19 -45t-45 -19h-128q-26 0 -45 19t-19 45v678q-4 -10 -13 -19z" /> + <glyph glyph-name="eject" unicode="" horiz-adv-x="1538" + d="M14 557l710 710q19 19 45 19t45 -19l710 -710q19 -19 13 -32t-32 -13h-1472q-26 0 -32 13t13 32zM1473 0h-1408q-26 0 -45 19t-19 45v256q0 26 19 45t45 19h1408q26 0 45 -19t19 -45v-256q0 -26 -19 -45t-45 -19z" /> + <glyph glyph-name="chevron_left" unicode="" horiz-adv-x="1280" + d="M1171 1235l-531 -531l531 -531q19 -19 19 -45t-19 -45l-166 -166q-19 -19 -45 -19t-45 19l-742 742q-19 19 -19 45t19 45l742 742q19 19 45 19t45 -19l166 -166q19 -19 19 -45t-19 -45z" /> + <glyph glyph-name="chevron_right" unicode="" horiz-adv-x="1280" + d="M1107 659l-742 -742q-19 -19 -45 -19t-45 19l-166 166q-19 19 -19 45t19 45l531 531l-531 531q-19 19 -19 45t19 45l166 166q19 19 45 19t45 -19l742 -742q19 -19 19 -45t-19 -45z" /> + <glyph glyph-name="plus_sign" unicode="" + d="M1216 576v128q0 26 -19 45t-45 19h-256v256q0 26 -19 45t-45 19h-128q-26 0 -45 -19t-19 -45v-256h-256q-26 0 -45 -19t-19 -45v-128q0 -26 19 -45t45 -19h256v-256q0 -26 19 -45t45 -19h128q26 0 45 19t19 45v256h256q26 0 45 19t19 45zM1536 640q0 -209 -103 -385.5 + t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="minus_sign" unicode="" + d="M1216 576v128q0 26 -19 45t-45 19h-768q-26 0 -45 -19t-19 -45v-128q0 -26 19 -45t45 -19h768q26 0 45 19t19 45zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5 + t103 -385.5z" /> + <glyph glyph-name="remove_sign" unicode="" + d="M1149 414q0 26 -19 45l-181 181l181 181q19 19 19 45q0 27 -19 46l-90 90q-19 19 -46 19q-26 0 -45 -19l-181 -181l-181 181q-19 19 -45 19q-27 0 -46 -19l-90 -90q-19 -19 -19 -46q0 -26 19 -45l181 -181l-181 -181q-19 -19 -19 -45q0 -27 19 -46l90 -90q19 -19 46 -19 + q26 0 45 19l181 181l181 -181q19 -19 45 -19q27 0 46 19l90 90q19 19 19 46zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="ok_sign" unicode="" + d="M1284 802q0 28 -18 46l-91 90q-19 19 -45 19t-45 -19l-408 -407l-226 226q-19 19 -45 19t-45 -19l-91 -90q-18 -18 -18 -46q0 -27 18 -45l362 -362q19 -19 45 -19q27 0 46 19l543 543q18 18 18 45zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103 + t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="question_sign" unicode="" + d="M896 160v192q0 14 -9 23t-23 9h-192q-14 0 -23 -9t-9 -23v-192q0 -14 9 -23t23 -9h192q14 0 23 9t9 23zM1152 832q0 88 -55.5 163t-138.5 116t-170 41q-243 0 -371 -213q-15 -24 8 -42l132 -100q7 -6 19 -6q16 0 25 12q53 68 86 92q34 24 86 24q48 0 85.5 -26t37.5 -59 + q0 -38 -20 -61t-68 -45q-63 -28 -115.5 -86.5t-52.5 -125.5v-36q0 -14 9 -23t23 -9h192q14 0 23 9t9 23q0 19 21.5 49.5t54.5 49.5q32 18 49 28.5t46 35t44.5 48t28 60.5t12.5 81zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5 + t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="info_sign" unicode="" + d="M1024 160v160q0 14 -9 23t-23 9h-96v512q0 14 -9 23t-23 9h-320q-14 0 -23 -9t-9 -23v-160q0 -14 9 -23t23 -9h96v-320h-96q-14 0 -23 -9t-9 -23v-160q0 -14 9 -23t23 -9h448q14 0 23 9t9 23zM896 1056v160q0 14 -9 23t-23 9h-192q-14 0 -23 -9t-9 -23v-160q0 -14 9 -23 + t23 -9h192q14 0 23 9t9 23zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="screenshot" unicode="" + d="M1197 512h-109q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h109q-32 108 -112.5 188.5t-188.5 112.5v-109q0 -26 -19 -45t-45 -19h-128q-26 0 -45 19t-19 45v109q-108 -32 -188.5 -112.5t-112.5 -188.5h109q26 0 45 -19t19 -45v-128q0 -26 -19 -45t-45 -19h-109 + q32 -108 112.5 -188.5t188.5 -112.5v109q0 26 19 45t45 19h128q26 0 45 -19t19 -45v-109q108 32 188.5 112.5t112.5 188.5zM1536 704v-128q0 -26 -19 -45t-45 -19h-143q-37 -161 -154.5 -278.5t-278.5 -154.5v-143q0 -26 -19 -45t-45 -19h-128q-26 0 -45 19t-19 45v143 + q-161 37 -278.5 154.5t-154.5 278.5h-143q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h143q37 161 154.5 278.5t278.5 154.5v143q0 26 19 45t45 19h128q26 0 45 -19t19 -45v-143q161 -37 278.5 -154.5t154.5 -278.5h143q26 0 45 -19t19 -45z" /> + <glyph glyph-name="remove_circle" unicode="" + d="M1097 457l-146 -146q-10 -10 -23 -10t-23 10l-137 137l-137 -137q-10 -10 -23 -10t-23 10l-146 146q-10 10 -10 23t10 23l137 137l-137 137q-10 10 -10 23t10 23l146 146q10 10 23 10t23 -10l137 -137l137 137q10 10 23 10t23 -10l146 -146q10 -10 10 -23t-10 -23 + l-137 -137l137 -137q10 -10 10 -23t-10 -23zM1312 640q0 148 -73 273t-198 198t-273 73t-273 -73t-198 -198t-73 -273t73 -273t198 -198t273 -73t273 73t198 198t73 273zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5 + t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="ok_circle" unicode="" + d="M1171 723l-422 -422q-19 -19 -45 -19t-45 19l-294 294q-19 19 -19 45t19 45l102 102q19 19 45 19t45 -19l147 -147l275 275q19 19 45 19t45 -19l102 -102q19 -19 19 -45t-19 -45zM1312 640q0 148 -73 273t-198 198t-273 73t-273 -73t-198 -198t-73 -273t73 -273t198 -198 + t273 -73t273 73t198 198t73 273zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="ban_circle" unicode="" + d="M1312 643q0 161 -87 295l-754 -753q137 -89 297 -89q111 0 211.5 43.5t173.5 116.5t116 174.5t43 212.5zM313 344l755 754q-135 91 -300 91q-148 0 -273 -73t-198 -199t-73 -274q0 -162 89 -299zM1536 643q0 -157 -61 -300t-163.5 -246t-245 -164t-298.5 -61t-298.5 61 + t-245 164t-163.5 246t-61 300t61 299.5t163.5 245.5t245 164t298.5 61t298.5 -61t245 -164t163.5 -245.5t61 -299.5z" /> + <glyph glyph-name="arrow_left" unicode="" + d="M1536 640v-128q0 -53 -32.5 -90.5t-84.5 -37.5h-704l293 -294q38 -36 38 -90t-38 -90l-75 -76q-37 -37 -90 -37q-52 0 -91 37l-651 652q-37 37 -37 90q0 52 37 91l651 650q38 38 91 38q52 0 90 -38l75 -74q38 -38 38 -91t-38 -91l-293 -293h704q52 0 84.5 -37.5 + t32.5 -90.5z" /> + <glyph glyph-name="arrow_right" unicode="" + d="M1472 576q0 -54 -37 -91l-651 -651q-39 -37 -91 -37q-51 0 -90 37l-75 75q-38 38 -38 91t38 91l293 293h-704q-52 0 -84.5 37.5t-32.5 90.5v128q0 53 32.5 90.5t84.5 37.5h704l-293 294q-38 36 -38 90t38 90l75 75q38 38 90 38q53 0 91 -38l651 -651q37 -35 37 -90z" /> + <glyph glyph-name="arrow_up" unicode="" horiz-adv-x="1664" + d="M1611 565q0 -51 -37 -90l-75 -75q-38 -38 -91 -38q-54 0 -90 38l-294 293v-704q0 -52 -37.5 -84.5t-90.5 -32.5h-128q-53 0 -90.5 32.5t-37.5 84.5v704l-294 -293q-36 -38 -90 -38t-90 38l-75 75q-38 38 -38 90q0 53 38 91l651 651q35 37 90 37q54 0 91 -37l651 -651 + q37 -39 37 -91z" /> + <glyph glyph-name="arrow_down" unicode="" horiz-adv-x="1664" + d="M1611 704q0 -53 -37 -90l-651 -652q-39 -37 -91 -37q-53 0 -90 37l-651 652q-38 36 -38 90q0 53 38 91l74 75q39 37 91 37q53 0 90 -37l294 -294v704q0 52 38 90t90 38h128q52 0 90 -38t38 -90v-704l294 294q37 37 90 37q52 0 91 -37l75 -75q37 -39 37 -91z" /> + <glyph glyph-name="share_alt" unicode="" horiz-adv-x="1792" + d="M1792 896q0 -26 -19 -45l-512 -512q-19 -19 -45 -19t-45 19t-19 45v256h-224q-98 0 -175.5 -6t-154 -21.5t-133 -42.5t-105.5 -69.5t-80 -101t-48.5 -138.5t-17.5 -181q0 -55 5 -123q0 -6 2.5 -23.5t2.5 -26.5q0 -15 -8.5 -25t-23.5 -10q-16 0 -28 17q-7 9 -13 22 + t-13.5 30t-10.5 24q-127 285 -127 451q0 199 53 333q162 403 875 403h224v256q0 26 19 45t45 19t45 -19l512 -512q19 -19 19 -45z" /> + <glyph glyph-name="resize_full" unicode="" + d="M755 480q0 -13 -10 -23l-332 -332l144 -144q19 -19 19 -45t-19 -45t-45 -19h-448q-26 0 -45 19t-19 45v448q0 26 19 45t45 19t45 -19l144 -144l332 332q10 10 23 10t23 -10l114 -114q10 -10 10 -23zM1536 1344v-448q0 -26 -19 -45t-45 -19t-45 19l-144 144l-332 -332 + q-10 -10 -23 -10t-23 10l-114 114q-10 10 -10 23t10 23l332 332l-144 144q-19 19 -19 45t19 45t45 19h448q26 0 45 -19t19 -45z" /> + <glyph glyph-name="resize_small" unicode="" + d="M768 576v-448q0 -26 -19 -45t-45 -19t-45 19l-144 144l-332 -332q-10 -10 -23 -10t-23 10l-114 114q-10 10 -10 23t10 23l332 332l-144 144q-19 19 -19 45t19 45t45 19h448q26 0 45 -19t19 -45zM1523 1248q0 -13 -10 -23l-332 -332l144 -144q19 -19 19 -45t-19 -45 + t-45 -19h-448q-26 0 -45 19t-19 45v448q0 26 19 45t45 19t45 -19l144 -144l332 332q10 10 23 10t23 -10l114 -114q10 -10 10 -23z" /> + <glyph glyph-name="plus" unicode="" horiz-adv-x="1408" + d="M1408 800v-192q0 -40 -28 -68t-68 -28h-416v-416q0 -40 -28 -68t-68 -28h-192q-40 0 -68 28t-28 68v416h-416q-40 0 -68 28t-28 68v192q0 40 28 68t68 28h416v416q0 40 28 68t68 28h192q40 0 68 -28t28 -68v-416h416q40 0 68 -28t28 -68z" /> + <glyph glyph-name="minus" unicode="" horiz-adv-x="1408" + d="M1408 800v-192q0 -40 -28 -68t-68 -28h-1216q-40 0 -68 28t-28 68v192q0 40 28 68t68 28h1216q40 0 68 -28t28 -68z" /> + <glyph glyph-name="asterisk" unicode="" horiz-adv-x="1664" + d="M1482 486q46 -26 59.5 -77.5t-12.5 -97.5l-64 -110q-26 -46 -77.5 -59.5t-97.5 12.5l-266 153v-307q0 -52 -38 -90t-90 -38h-128q-52 0 -90 38t-38 90v307l-266 -153q-46 -26 -97.5 -12.5t-77.5 59.5l-64 110q-26 46 -12.5 97.5t59.5 77.5l266 154l-266 154 + q-46 26 -59.5 77.5t12.5 97.5l64 110q26 46 77.5 59.5t97.5 -12.5l266 -153v307q0 52 38 90t90 38h128q52 0 90 -38t38 -90v-307l266 153q46 26 97.5 12.5t77.5 -59.5l64 -110q26 -46 12.5 -97.5t-59.5 -77.5l-266 -154z" /> + <glyph glyph-name="exclamation_sign" unicode="" + d="M768 1408q209 0 385.5 -103t279.5 -279.5t103 -385.5t-103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103zM896 161v190q0 14 -9 23.5t-22 9.5h-192q-13 0 -23 -10t-10 -23v-190q0 -13 10 -23t23 -10h192 + q13 0 22 9.5t9 23.5zM894 505l18 621q0 12 -10 18q-10 8 -24 8h-220q-14 0 -24 -8q-10 -6 -10 -18l17 -621q0 -10 10 -17.5t24 -7.5h185q14 0 23.5 7.5t10.5 17.5z" /> + <glyph glyph-name="gift" unicode="" + d="M928 180v56v468v192h-320v-192v-468v-56q0 -25 18 -38.5t46 -13.5h192q28 0 46 13.5t18 38.5zM472 1024h195l-126 161q-26 31 -69 31q-40 0 -68 -28t-28 -68t28 -68t68 -28zM1160 1120q0 40 -28 68t-68 28q-43 0 -69 -31l-125 -161h194q40 0 68 28t28 68zM1536 864v-320 + q0 -14 -9 -23t-23 -9h-96v-416q0 -40 -28 -68t-68 -28h-1088q-40 0 -68 28t-28 68v416h-96q-14 0 -23 9t-9 23v320q0 14 9 23t23 9h440q-93 0 -158.5 65.5t-65.5 158.5t65.5 158.5t158.5 65.5q107 0 168 -77l128 -165l128 165q61 77 168 77q93 0 158.5 -65.5t65.5 -158.5 + t-65.5 -158.5t-158.5 -65.5h440q14 0 23 -9t9 -23z" /> + <glyph glyph-name="leaf" unicode="" horiz-adv-x="1792" + d="M1280 832q0 26 -19 45t-45 19q-172 0 -318 -49.5t-259.5 -134t-235.5 -219.5q-19 -21 -19 -45q0 -26 19 -45t45 -19q24 0 45 19q27 24 74 71t67 66q137 124 268.5 176t313.5 52q26 0 45 19t19 45zM1792 1030q0 -95 -20 -193q-46 -224 -184.5 -383t-357.5 -268 + q-214 -108 -438 -108q-148 0 -286 47q-15 5 -88 42t-96 37q-16 0 -39.5 -32t-45 -70t-52.5 -70t-60 -32q-43 0 -63.5 17.5t-45.5 59.5q-2 4 -6 11t-5.5 10t-3 9.5t-1.5 13.5q0 35 31 73.5t68 65.5t68 56t31 48q0 4 -14 38t-16 44q-9 51 -9 104q0 115 43.5 220t119 184.5 + t170.5 139t204 95.5q55 18 145 25.5t179.5 9t178.5 6t163.5 24t113.5 56.5l29.5 29.5t29.5 28t27 20t36.5 16t43.5 4.5q39 0 70.5 -46t47.5 -112t24 -124t8 -96z" /> + <glyph glyph-name="fire" unicode="" horiz-adv-x="1408" + d="M1408 -160v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-1344q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h1344q13 0 22.5 -9.5t9.5 -22.5zM1152 896q0 -78 -24.5 -144t-64 -112.5t-87.5 -88t-96 -77.5t-87.5 -72t-64 -81.5t-24.5 -96.5q0 -96 67 -224l-4 1l1 -1 + q-90 41 -160 83t-138.5 100t-113.5 122.5t-72.5 150.5t-27.5 184q0 78 24.5 144t64 112.5t87.5 88t96 77.5t87.5 72t64 81.5t24.5 96.5q0 94 -66 224l3 -1l-1 1q90 -41 160 -83t138.5 -100t113.5 -122.5t72.5 -150.5t27.5 -184z" /> + <glyph glyph-name="eye_open" unicode="" horiz-adv-x="1792" + d="M1664 576q-152 236 -381 353q61 -104 61 -225q0 -185 -131.5 -316.5t-316.5 -131.5t-316.5 131.5t-131.5 316.5q0 121 61 225q-229 -117 -381 -353q133 -205 333.5 -326.5t434.5 -121.5t434.5 121.5t333.5 326.5zM944 960q0 20 -14 34t-34 14q-125 0 -214.5 -89.5 + t-89.5 -214.5q0 -20 14 -34t34 -14t34 14t14 34q0 86 61 147t147 61q20 0 34 14t14 34zM1792 576q0 -34 -20 -69q-140 -230 -376.5 -368.5t-499.5 -138.5t-499.5 139t-376.5 368q-20 35 -20 69t20 69q140 229 376.5 368t499.5 139t499.5 -139t376.5 -368q20 -35 20 -69z" /> + <glyph glyph-name="eye_close" unicode="" horiz-adv-x="1792" + d="M555 201l78 141q-87 63 -136 159t-49 203q0 121 61 225q-229 -117 -381 -353q167 -258 427 -375zM944 960q0 20 -14 34t-34 14q-125 0 -214.5 -89.5t-89.5 -214.5q0 -20 14 -34t34 -14t34 14t14 34q0 86 61 147t147 61q20 0 34 14t14 34zM1307 1151q0 -7 -1 -9 + q-106 -189 -316 -567t-315 -566l-49 -89q-10 -16 -28 -16q-12 0 -134 70q-16 10 -16 28q0 12 44 87q-143 65 -263.5 173t-208.5 245q-20 31 -20 69t20 69q153 235 380 371t496 136q89 0 180 -17l54 97q10 16 28 16q5 0 18 -6t31 -15.5t33 -18.5t31.5 -18.5t19.5 -11.5 + q16 -10 16 -27zM1344 704q0 -139 -79 -253.5t-209 -164.5l280 502q8 -45 8 -84zM1792 576q0 -35 -20 -69q-39 -64 -109 -145q-150 -172 -347.5 -267t-419.5 -95l74 132q212 18 392.5 137t301.5 307q-115 179 -282 294l63 112q95 -64 182.5 -153t144.5 -184q20 -34 20 -69z + " /> + <glyph glyph-name="warning_sign" unicode="" horiz-adv-x="1792" + d="M1024 161v190q0 14 -9.5 23.5t-22.5 9.5h-192q-13 0 -22.5 -9.5t-9.5 -23.5v-190q0 -14 9.5 -23.5t22.5 -9.5h192q13 0 22.5 9.5t9.5 23.5zM1022 535l18 459q0 12 -10 19q-13 11 -24 11h-220q-11 0 -24 -11q-10 -7 -10 -21l17 -457q0 -10 10 -16.5t24 -6.5h185 + q14 0 23.5 6.5t10.5 16.5zM1008 1469l768 -1408q35 -63 -2 -126q-17 -29 -46.5 -46t-63.5 -17h-1536q-34 0 -63.5 17t-46.5 46q-37 63 -2 126l768 1408q17 31 47 49t65 18t65 -18t47 -49z" /> + <glyph glyph-name="plane" unicode="" horiz-adv-x="1408" + d="M1376 1376q44 -52 12 -148t-108 -172l-161 -161l160 -696q5 -19 -12 -33l-128 -96q-7 -6 -19 -6q-4 0 -7 1q-15 3 -21 16l-279 508l-259 -259l53 -194q5 -17 -8 -31l-96 -96q-9 -9 -23 -9h-2q-15 2 -24 13l-189 252l-252 189q-11 7 -13 23q-1 13 9 25l96 97q9 9 23 9 + q6 0 8 -1l194 -53l259 259l-508 279q-14 8 -17 24q-2 16 9 27l128 128q14 13 30 8l665 -159l160 160q76 76 172 108t148 -12z" /> + <glyph glyph-name="calendar" unicode="" horiz-adv-x="1664" + d="M128 -128h288v288h-288v-288zM480 -128h320v288h-320v-288zM128 224h288v320h-288v-320zM480 224h320v320h-320v-320zM128 608h288v288h-288v-288zM864 -128h320v288h-320v-288zM480 608h320v288h-320v-288zM1248 -128h288v288h-288v-288zM864 224h320v320h-320v-320z + M512 1088v288q0 13 -9.5 22.5t-22.5 9.5h-64q-13 0 -22.5 -9.5t-9.5 -22.5v-288q0 -13 9.5 -22.5t22.5 -9.5h64q13 0 22.5 9.5t9.5 22.5zM1248 224h288v320h-288v-320zM864 608h320v288h-320v-288zM1248 608h288v288h-288v-288zM1280 1088v288q0 13 -9.5 22.5t-22.5 9.5h-64 + q-13 0 -22.5 -9.5t-9.5 -22.5v-288q0 -13 9.5 -22.5t22.5 -9.5h64q13 0 22.5 9.5t9.5 22.5zM1664 1152v-1280q0 -52 -38 -90t-90 -38h-1408q-52 0 -90 38t-38 90v1280q0 52 38 90t90 38h128v96q0 66 47 113t113 47h64q66 0 113 -47t47 -113v-96h384v96q0 66 47 113t113 47 + h64q66 0 113 -47t47 -113v-96h128q52 0 90 -38t38 -90z" /> + <glyph glyph-name="random" unicode="" horiz-adv-x="1792" + d="M666 1055q-60 -92 -137 -273q-22 45 -37 72.5t-40.5 63.5t-51 56.5t-63 35t-81.5 14.5h-224q-14 0 -23 9t-9 23v192q0 14 9 23t23 9h224q250 0 410 -225zM1792 256q0 -14 -9 -23l-320 -320q-9 -9 -23 -9q-13 0 -22.5 9.5t-9.5 22.5v192q-32 0 -85 -0.5t-81 -1t-73 1 + t-71 5t-64 10.5t-63 18.5t-58 28.5t-59 40t-55 53.5t-56 69.5q59 93 136 273q22 -45 37 -72.5t40.5 -63.5t51 -56.5t63 -35t81.5 -14.5h256v192q0 14 9 23t23 9q12 0 24 -10l319 -319q9 -9 9 -23zM1792 1152q0 -14 -9 -23l-320 -320q-9 -9 -23 -9q-13 0 -22.5 9.5t-9.5 22.5 + v192h-256q-48 0 -87 -15t-69 -45t-51 -61.5t-45 -77.5q-32 -62 -78 -171q-29 -66 -49.5 -111t-54 -105t-64 -100t-74 -83t-90 -68.5t-106.5 -42t-128 -16.5h-224q-14 0 -23 9t-9 23v192q0 14 9 23t23 9h224q48 0 87 15t69 45t51 61.5t45 77.5q32 62 78 171q29 66 49.5 111 + t54 105t64 100t74 83t90 68.5t106.5 42t128 16.5h256v192q0 14 9 23t23 9q12 0 24 -10l319 -319q9 -9 9 -23z" /> + <glyph glyph-name="comment" unicode="" horiz-adv-x="1792" + d="M1792 640q0 -174 -120 -321.5t-326 -233t-450 -85.5q-70 0 -145 8q-198 -175 -460 -242q-49 -14 -114 -22q-17 -2 -30.5 9t-17.5 29v1q-3 4 -0.5 12t2 10t4.5 9.5l6 9t7 8.5t8 9q7 8 31 34.5t34.5 38t31 39.5t32.5 51t27 59t26 76q-157 89 -247.5 220t-90.5 281 + q0 130 71 248.5t191 204.5t286 136.5t348 50.5q244 0 450 -85.5t326 -233t120 -321.5z" /> + <glyph glyph-name="magnet" unicode="" + d="M1536 704v-128q0 -201 -98.5 -362t-274 -251.5t-395.5 -90.5t-395.5 90.5t-274 251.5t-98.5 362v128q0 26 19 45t45 19h384q26 0 45 -19t19 -45v-128q0 -52 23.5 -90t53.5 -57t71 -30t64 -13t44 -2t44 2t64 13t71 30t53.5 57t23.5 90v128q0 26 19 45t45 19h384 + q26 0 45 -19t19 -45zM512 1344v-384q0 -26 -19 -45t-45 -19h-384q-26 0 -45 19t-19 45v384q0 26 19 45t45 19h384q26 0 45 -19t19 -45zM1536 1344v-384q0 -26 -19 -45t-45 -19h-384q-26 0 -45 19t-19 45v384q0 26 19 45t45 19h384q26 0 45 -19t19 -45z" /> + <glyph glyph-name="chevron_up" unicode="" horiz-adv-x="1792" + d="M1683 205l-166 -165q-19 -19 -45 -19t-45 19l-531 531l-531 -531q-19 -19 -45 -19t-45 19l-166 165q-19 19 -19 45.5t19 45.5l742 741q19 19 45 19t45 -19l742 -741q19 -19 19 -45.5t-19 -45.5z" /> + <glyph glyph-name="chevron_down" unicode="" horiz-adv-x="1792" + d="M1683 728l-742 -741q-19 -19 -45 -19t-45 19l-742 741q-19 19 -19 45.5t19 45.5l166 165q19 19 45 19t45 -19l531 -531l531 531q19 19 45 19t45 -19l166 -165q19 -19 19 -45.5t-19 -45.5z" /> + <glyph glyph-name="retweet" unicode="" horiz-adv-x="1920" + d="M1280 32q0 -13 -9.5 -22.5t-22.5 -9.5h-960q-8 0 -13.5 2t-9 7t-5.5 8t-3 11.5t-1 11.5v13v11v160v416h-192q-26 0 -45 19t-19 45q0 24 15 41l320 384q19 22 49 22t49 -22l320 -384q15 -17 15 -41q0 -26 -19 -45t-45 -19h-192v-384h576q16 0 25 -11l160 -192q7 -10 7 -21 + zM1920 448q0 -24 -15 -41l-320 -384q-20 -23 -49 -23t-49 23l-320 384q-15 17 -15 41q0 26 19 45t45 19h192v384h-576q-16 0 -25 12l-160 192q-7 9 -7 20q0 13 9.5 22.5t22.5 9.5h960q8 0 13.5 -2t9 -7t5.5 -8t3 -11.5t1 -11.5v-13v-11v-160v-416h192q26 0 45 -19t19 -45z + " /> + <glyph glyph-name="shopping_cart" unicode="" horiz-adv-x="1664" + d="M640 0q0 -52 -38 -90t-90 -38t-90 38t-38 90t38 90t90 38t90 -38t38 -90zM1536 0q0 -52 -38 -90t-90 -38t-90 38t-38 90t38 90t90 38t90 -38t38 -90zM1664 1088v-512q0 -24 -16.5 -42.5t-40.5 -21.5l-1044 -122q13 -60 13 -70q0 -16 -24 -64h920q26 0 45 -19t19 -45 + t-19 -45t-45 -19h-1024q-26 0 -45 19t-19 45q0 11 8 31.5t16 36t21.5 40t15.5 29.5l-177 823h-204q-26 0 -45 19t-19 45t19 45t45 19h256q16 0 28.5 -6.5t19.5 -15.5t13 -24.5t8 -26t5.5 -29.5t4.5 -26h1201q26 0 45 -19t19 -45z" /> + <glyph glyph-name="folder_close" unicode="" horiz-adv-x="1664" + d="M1664 928v-704q0 -92 -66 -158t-158 -66h-1216q-92 0 -158 66t-66 158v960q0 92 66 158t158 66h320q92 0 158 -66t66 -158v-32h672q92 0 158 -66t66 -158z" /> + <glyph glyph-name="folder_open" unicode="" horiz-adv-x="1920" + d="M1879 584q0 -31 -31 -66l-336 -396q-43 -51 -120.5 -86.5t-143.5 -35.5h-1088q-34 0 -60.5 13t-26.5 43q0 31 31 66l336 396q43 51 120.5 86.5t143.5 35.5h1088q34 0 60.5 -13t26.5 -43zM1536 928v-160h-832q-94 0 -197 -47.5t-164 -119.5l-337 -396l-5 -6q0 4 -0.5 12.5 + t-0.5 12.5v960q0 92 66 158t158 66h320q92 0 158 -66t66 -158v-32h544q92 0 158 -66t66 -158z" /> + <glyph glyph-name="resize_vertical" unicode="" horiz-adv-x="768" + d="M704 1216q0 -26 -19 -45t-45 -19h-128v-1024h128q26 0 45 -19t19 -45t-19 -45l-256 -256q-19 -19 -45 -19t-45 19l-256 256q-19 19 -19 45t19 45t45 19h128v1024h-128q-26 0 -45 19t-19 45t19 45l256 256q19 19 45 19t45 -19l256 -256q19 -19 19 -45z" /> + <glyph glyph-name="resize_horizontal" unicode="" horiz-adv-x="1792" + d="M1792 640q0 -26 -19 -45l-256 -256q-19 -19 -45 -19t-45 19t-19 45v128h-1024v-128q0 -26 -19 -45t-45 -19t-45 19l-256 256q-19 19 -19 45t19 45l256 256q19 19 45 19t45 -19t19 -45v-128h1024v128q0 26 19 45t45 19t45 -19l256 -256q19 -19 19 -45z" /> + <glyph glyph-name="bar_chart" unicode="" horiz-adv-x="2048" + d="M640 640v-512h-256v512h256zM1024 1152v-1024h-256v1024h256zM2048 0v-128h-2048v1536h128v-1408h1920zM1408 896v-768h-256v768h256zM1792 1280v-1152h-256v1152h256z" /> + <glyph glyph-name="twitter_sign" unicode="" + d="M1280 926q-56 -25 -121 -34q68 40 93 117q-65 -38 -134 -51q-61 66 -153 66q-87 0 -148.5 -61.5t-61.5 -148.5q0 -29 5 -48q-129 7 -242 65t-192 155q-29 -50 -29 -106q0 -114 91 -175q-47 1 -100 26v-2q0 -75 50 -133.5t123 -72.5q-29 -8 -51 -8q-13 0 -39 4 + q21 -63 74.5 -104t121.5 -42q-116 -90 -261 -90q-26 0 -50 3q148 -94 322 -94q112 0 210 35.5t168 95t120.5 137t75 162t24.5 168.5q0 18 -1 27q63 45 105 109zM1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5 + t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="facebook_sign" unicode="" + d="M1248 1408q119 0 203.5 -84.5t84.5 -203.5v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-188v595h199l30 232h-229v148q0 56 23.5 84t91.5 28l122 1v207q-63 9 -178 9q-136 0 -217.5 -80t-81.5 -226v-171h-200v-232h200v-595h-532q-119 0 -203.5 84.5t-84.5 203.5v960 + q0 119 84.5 203.5t203.5 84.5h960z" /> + <glyph glyph-name="camera_retro" unicode="" horiz-adv-x="1792" + d="M928 704q0 14 -9 23t-23 9q-66 0 -113 -47t-47 -113q0 -14 9 -23t23 -9t23 9t9 23q0 40 28 68t68 28q14 0 23 9t9 23zM1152 574q0 -106 -75 -181t-181 -75t-181 75t-75 181t75 181t181 75t181 -75t75 -181zM128 0h1536v128h-1536v-128zM1280 574q0 159 -112.5 271.5 + t-271.5 112.5t-271.5 -112.5t-112.5 -271.5t112.5 -271.5t271.5 -112.5t271.5 112.5t112.5 271.5zM256 1216h384v128h-384v-128zM128 1024h1536v118v138h-828l-64 -128h-644v-128zM1792 1280v-1280q0 -53 -37.5 -90.5t-90.5 -37.5h-1536q-53 0 -90.5 37.5t-37.5 90.5v1280 + q0 53 37.5 90.5t90.5 37.5h1536q53 0 90.5 -37.5t37.5 -90.5z" /> + <glyph glyph-name="key" unicode="" horiz-adv-x="1792" + d="M832 1024q0 80 -56 136t-136 56t-136 -56t-56 -136q0 -42 19 -83q-41 19 -83 19q-80 0 -136 -56t-56 -136t56 -136t136 -56t136 56t56 136q0 42 -19 83q41 -19 83 -19q80 0 136 56t56 136zM1683 320q0 -17 -49 -66t-66 -49q-9 0 -28.5 16t-36.5 33t-38.5 40t-24.5 26 + l-96 -96l220 -220q28 -28 28 -68q0 -42 -39 -81t-81 -39q-40 0 -68 28l-671 671q-176 -131 -365 -131q-163 0 -265.5 102.5t-102.5 265.5q0 160 95 313t248 248t313 95q163 0 265.5 -102.5t102.5 -265.5q0 -189 -131 -365l355 -355l96 96q-3 3 -26 24.5t-40 38.5t-33 36.5 + t-16 28.5q0 17 49 66t66 49q13 0 23 -10q6 -6 46 -44.5t82 -79.5t86.5 -86t73 -78t28.5 -41z" /> + <glyph glyph-name="cogs" unicode="" horiz-adv-x="1920" + d="M896 640q0 106 -75 181t-181 75t-181 -75t-75 -181t75 -181t181 -75t181 75t75 181zM1664 128q0 52 -38 90t-90 38t-90 -38t-38 -90q0 -53 37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM1664 1152q0 52 -38 90t-90 38t-90 -38t-38 -90q0 -53 37.5 -90.5t90.5 -37.5 + t90.5 37.5t37.5 90.5zM1280 731v-185q0 -10 -7 -19.5t-16 -10.5l-155 -24q-11 -35 -32 -76q34 -48 90 -115q7 -11 7 -20q0 -12 -7 -19q-23 -30 -82.5 -89.5t-78.5 -59.5q-11 0 -21 7l-115 90q-37 -19 -77 -31q-11 -108 -23 -155q-7 -24 -30 -24h-186q-11 0 -20 7.5t-10 17.5 + l-23 153q-34 10 -75 31l-118 -89q-7 -7 -20 -7q-11 0 -21 8q-144 133 -144 160q0 9 7 19q10 14 41 53t47 61q-23 44 -35 82l-152 24q-10 1 -17 9.5t-7 19.5v185q0 10 7 19.5t16 10.5l155 24q11 35 32 76q-34 48 -90 115q-7 11 -7 20q0 12 7 20q22 30 82 89t79 59q11 0 21 -7 + l115 -90q34 18 77 32q11 108 23 154q7 24 30 24h186q11 0 20 -7.5t10 -17.5l23 -153q34 -10 75 -31l118 89q8 7 20 7q11 0 21 -8q144 -133 144 -160q0 -8 -7 -19q-12 -16 -42 -54t-45 -60q23 -48 34 -82l152 -23q10 -2 17 -10.5t7 -19.5zM1920 198v-140q0 -16 -149 -31 + q-12 -27 -30 -52q51 -113 51 -138q0 -4 -4 -7q-122 -71 -124 -71q-8 0 -46 47t-52 68q-20 -2 -30 -2t-30 2q-14 -21 -52 -68t-46 -47q-2 0 -124 71q-4 3 -4 7q0 25 51 138q-18 25 -30 52q-149 15 -149 31v140q0 16 149 31q13 29 30 52q-51 113 -51 138q0 4 4 7q4 2 35 20 + t59 34t30 16q8 0 46 -46.5t52 -67.5q20 2 30 2t30 -2q51 71 92 112l6 2q4 0 124 -70q4 -3 4 -7q0 -25 -51 -138q17 -23 30 -52q149 -15 149 -31zM1920 1222v-140q0 -16 -149 -31q-12 -27 -30 -52q51 -113 51 -138q0 -4 -4 -7q-122 -71 -124 -71q-8 0 -46 47t-52 68 + q-20 -2 -30 -2t-30 2q-14 -21 -52 -68t-46 -47q-2 0 -124 71q-4 3 -4 7q0 25 51 138q-18 25 -30 52q-149 15 -149 31v140q0 16 149 31q13 29 30 52q-51 113 -51 138q0 4 4 7q4 2 35 20t59 34t30 16q8 0 46 -46.5t52 -67.5q20 2 30 2t30 -2q51 71 92 112l6 2q4 0 124 -70 + q4 -3 4 -7q0 -25 -51 -138q17 -23 30 -52q149 -15 149 -31z" /> + <glyph glyph-name="comments" unicode="" horiz-adv-x="1792" + d="M1408 768q0 -139 -94 -257t-256.5 -186.5t-353.5 -68.5q-86 0 -176 16q-124 -88 -278 -128q-36 -9 -86 -16h-3q-11 0 -20.5 8t-11.5 21q-1 3 -1 6.5t0.5 6.5t2 6l2.5 5t3.5 5.5t4 5t4.5 5t4 4.5q5 6 23 25t26 29.5t22.5 29t25 38.5t20.5 44q-124 72 -195 177t-71 224 + q0 139 94 257t256.5 186.5t353.5 68.5t353.5 -68.5t256.5 -186.5t94 -257zM1792 512q0 -120 -71 -224.5t-195 -176.5q10 -24 20.5 -44t25 -38.5t22.5 -29t26 -29.5t23 -25q1 -1 4 -4.5t4.5 -5t4 -5t3.5 -5.5l2.5 -5t2 -6t0.5 -6.5t-1 -6.5q-3 -14 -13 -22t-22 -7 + q-50 7 -86 16q-154 40 -278 128q-90 -16 -176 -16q-271 0 -472 132q58 -4 88 -4q161 0 309 45t264 129q125 92 192 212t67 254q0 77 -23 152q129 -71 204 -178t75 -230z" /> + <glyph glyph-name="thumbs_up_alt" unicode="" + d="M256 192q0 26 -19 45t-45 19t-45 -19t-19 -45t19 -45t45 -19t45 19t19 45zM1408 768q0 51 -39 89.5t-89 38.5h-352q0 58 48 159.5t48 160.5q0 98 -32 145t-128 47q-26 -26 -38 -85t-30.5 -125.5t-59.5 -109.5q-22 -23 -77 -91q-4 -5 -23 -30t-31.5 -41t-34.5 -42.5 + t-40 -44t-38.5 -35.5t-40 -27t-35.5 -9h-32v-640h32q13 0 31.5 -3t33 -6.5t38 -11t35 -11.5t35.5 -12.5t29 -10.5q211 -73 342 -73h121q192 0 192 167q0 26 -5 56q30 16 47.5 52.5t17.5 73.5t-18 69q53 50 53 119q0 25 -10 55.5t-25 47.5q32 1 53.5 47t21.5 81zM1536 769 + q0 -89 -49 -163q9 -33 9 -69q0 -77 -38 -144q3 -21 3 -43q0 -101 -60 -178q1 -139 -85 -219.5t-227 -80.5h-36h-93q-96 0 -189.5 22.5t-216.5 65.5q-116 40 -138 40h-288q-53 0 -90.5 37.5t-37.5 90.5v640q0 53 37.5 90.5t90.5 37.5h274q36 24 137 155q58 75 107 128 + q24 25 35.5 85.5t30.5 126.5t62 108q39 37 90 37q84 0 151 -32.5t102 -101.5t35 -186q0 -93 -48 -192h176q104 0 180 -76t76 -179z" /> + <glyph glyph-name="thumbs_down_alt" unicode="" + d="M256 1088q0 26 -19 45t-45 19t-45 -19t-19 -45t19 -45t45 -19t45 19t19 45zM1408 512q0 35 -21.5 81t-53.5 47q15 17 25 47.5t10 55.5q0 69 -53 119q18 31 18 69q0 37 -17.5 73.5t-47.5 52.5q5 30 5 56q0 85 -49 126t-136 41h-128q-131 0 -342 -73q-5 -2 -29 -10.5 + t-35.5 -12.5t-35 -11.5t-38 -11t-33 -6.5t-31.5 -3h-32v-640h32q16 0 35.5 -9t40 -27t38.5 -35.5t40 -44t34.5 -42.5t31.5 -41t23 -30q55 -68 77 -91q41 -43 59.5 -109.5t30.5 -125.5t38 -85q96 0 128 47t32 145q0 59 -48 160.5t-48 159.5h352q50 0 89 38.5t39 89.5z + M1536 511q0 -103 -76 -179t-180 -76h-176q48 -99 48 -192q0 -118 -35 -186q-35 -69 -102 -101.5t-151 -32.5q-51 0 -90 37q-34 33 -54 82t-25.5 90.5t-17.5 84.5t-31 64q-48 50 -107 127q-101 131 -137 155h-274q-53 0 -90.5 37.5t-37.5 90.5v640q0 53 37.5 90.5t90.5 37.5 + h288q22 0 138 40q128 44 223 66t200 22h112q140 0 226.5 -79t85.5 -216v-5q60 -77 60 -178q0 -22 -3 -43q38 -67 38 -144q0 -36 -9 -69q49 -73 49 -163z" /> + <glyph glyph-name="star_half" unicode="" horiz-adv-x="896" + d="M832 1504v-1339l-449 -236q-22 -12 -40 -12q-21 0 -31.5 14.5t-10.5 35.5q0 6 2 20l86 500l-364 354q-25 27 -25 48q0 37 56 46l502 73l225 455q19 41 49 41z" /> + <glyph glyph-name="heart_empty" unicode="" horiz-adv-x="1792" + d="M1664 940q0 81 -21.5 143t-55 98.5t-81.5 59.5t-94 31t-98 8t-112 -25.5t-110.5 -64t-86.5 -72t-60 -61.5q-18 -22 -49 -22t-49 22q-24 28 -60 61.5t-86.5 72t-110.5 64t-112 25.5t-98 -8t-94 -31t-81.5 -59.5t-55 -98.5t-21.5 -143q0 -168 187 -355l581 -560l580 559 + q188 188 188 356zM1792 940q0 -221 -229 -450l-623 -600q-18 -18 -44 -18t-44 18l-624 602q-10 8 -27.5 26t-55.5 65.5t-68 97.5t-53.5 121t-23.5 138q0 220 127 344t351 124q62 0 126.5 -21.5t120 -58t95.5 -68.5t76 -68q36 36 76 68t95.5 68.5t120 58t126.5 21.5 + q224 0 351 -124t127 -344z" /> + <glyph glyph-name="signout" unicode="" horiz-adv-x="1664" + d="M640 96q0 -4 1 -20t0.5 -26.5t-3 -23.5t-10 -19.5t-20.5 -6.5h-320q-119 0 -203.5 84.5t-84.5 203.5v704q0 119 84.5 203.5t203.5 84.5h320q13 0 22.5 -9.5t9.5 -22.5q0 -4 1 -20t0.5 -26.5t-3 -23.5t-10 -19.5t-20.5 -6.5h-320q-66 0 -113 -47t-47 -113v-704 + q0 -66 47 -113t113 -47h288h11h13t11.5 -1t11.5 -3t8 -5.5t7 -9t2 -13.5zM1568 640q0 -26 -19 -45l-544 -544q-19 -19 -45 -19t-45 19t-19 45v288h-448q-26 0 -45 19t-19 45v384q0 26 19 45t45 19h448v288q0 26 19 45t45 19t45 -19l544 -544q19 -19 19 -45z" /> + <glyph glyph-name="linkedin_sign" unicode="" + d="M237 122h231v694h-231v-694zM483 1030q-1 52 -36 86t-93 34t-94.5 -34t-36.5 -86q0 -51 35.5 -85.5t92.5 -34.5h1q59 0 95 34.5t36 85.5zM1068 122h231v398q0 154 -73 233t-193 79q-136 0 -209 -117h2v101h-231q3 -66 0 -694h231v388q0 38 7 56q15 35 45 59.5t74 24.5 + q116 0 116 -157v-371zM1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="pushpin" unicode="" horiz-adv-x="1152" + d="M480 672v448q0 14 -9 23t-23 9t-23 -9t-9 -23v-448q0 -14 9 -23t23 -9t23 9t9 23zM1152 320q0 -26 -19 -45t-45 -19h-429l-51 -483q-2 -12 -10.5 -20.5t-20.5 -8.5h-1q-27 0 -32 27l-76 485h-404q-26 0 -45 19t-19 45q0 123 78.5 221.5t177.5 98.5v512q-52 0 -90 38 + t-38 90t38 90t90 38h640q52 0 90 -38t38 -90t-38 -90t-90 -38v-512q99 0 177.5 -98.5t78.5 -221.5z" /> + <glyph glyph-name="external_link" unicode="" horiz-adv-x="1792" + d="M1408 608v-320q0 -119 -84.5 -203.5t-203.5 -84.5h-832q-119 0 -203.5 84.5t-84.5 203.5v832q0 119 84.5 203.5t203.5 84.5h704q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-704q-66 0 -113 -47t-47 -113v-832q0 -66 47 -113t113 -47h832q66 0 113 47t47 113v320 + q0 14 9 23t23 9h64q14 0 23 -9t9 -23zM1792 1472v-512q0 -26 -19 -45t-45 -19t-45 19l-176 176l-652 -652q-10 -10 -23 -10t-23 10l-114 114q-10 10 -10 23t10 23l652 652l-176 176q-19 19 -19 45t19 45t45 19h512q26 0 45 -19t19 -45z" /> + <glyph glyph-name="signin" unicode="" + d="M1184 640q0 -26 -19 -45l-544 -544q-19 -19 -45 -19t-45 19t-19 45v288h-448q-26 0 -45 19t-19 45v384q0 26 19 45t45 19h448v288q0 26 19 45t45 19t45 -19l544 -544q19 -19 19 -45zM1536 992v-704q0 -119 -84.5 -203.5t-203.5 -84.5h-320q-13 0 -22.5 9.5t-9.5 22.5 + q0 4 -1 20t-0.5 26.5t3 23.5t10 19.5t20.5 6.5h320q66 0 113 47t47 113v704q0 66 -47 113t-113 47h-288h-11h-13t-11.5 1t-11.5 3t-8 5.5t-7 9t-2 13.5q0 4 -1 20t-0.5 26.5t3 23.5t10 19.5t20.5 6.5h320q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="trophy" unicode="" horiz-adv-x="1664" + d="M458 653q-74 162 -74 371h-256v-96q0 -78 94.5 -162t235.5 -113zM1536 928v96h-256q0 -209 -74 -371q141 29 235.5 113t94.5 162zM1664 1056v-128q0 -71 -41.5 -143t-112 -130t-173 -97.5t-215.5 -44.5q-42 -54 -95 -95q-38 -34 -52.5 -72.5t-14.5 -89.5q0 -54 30.5 -91 + t97.5 -37q75 0 133.5 -45.5t58.5 -114.5v-64q0 -14 -9 -23t-23 -9h-832q-14 0 -23 9t-9 23v64q0 69 58.5 114.5t133.5 45.5q67 0 97.5 37t30.5 91q0 51 -14.5 89.5t-52.5 72.5q-53 41 -95 95q-113 5 -215.5 44.5t-173 97.5t-112 130t-41.5 143v128q0 40 28 68t68 28h288v96 + q0 66 47 113t113 47h576q66 0 113 -47t47 -113v-96h288q40 0 68 -28t28 -68z" /> + <glyph glyph-name="github_sign" unicode="" + d="M519 336q4 6 -3 13q-9 7 -14 2q-4 -6 3 -13q9 -7 14 -2zM491 377q-5 7 -12 4q-6 -4 0 -12q7 -8 12 -5q6 4 0 13zM450 417q2 4 -5 8q-7 2 -8 -2q-3 -5 4 -8q8 -2 9 2zM471 394q2 1 1.5 4.5t-3.5 5.5q-6 7 -10 3t1 -11q6 -6 11 -2zM557 319q2 7 -9 11q-9 3 -13 -4 + q-2 -7 9 -11q9 -3 13 4zM599 316q0 8 -12 8q-10 0 -10 -8t11 -8t11 8zM638 323q-2 7 -13 5t-9 -9q2 -8 12 -6t10 10zM1280 640q0 212 -150 362t-362 150t-362 -150t-150 -362q0 -167 98 -300.5t252 -185.5q18 -3 26.5 5t8.5 20q0 52 -1 95q-6 -1 -15.5 -2.5t-35.5 -2t-48 4 + t-43.5 20t-29.5 41.5q-23 59 -57 74q-2 1 -4.5 3.5l-8 8t-7 9.5t4 7.5t19.5 3.5q6 0 15 -2t30 -15.5t33 -35.5q16 -28 37.5 -42t43.5 -14t38 3.5t30 9.5q7 47 33 69q-49 6 -86 18.5t-73 39t-55.5 76t-19.5 119.5q0 79 53 137q-24 62 5 136q19 6 54.5 -7.5t60.5 -29.5l26 -16 + q58 17 128 17t128 -17q11 7 28.5 18t55.5 26t57 9q29 -74 5 -136q53 -58 53 -137q0 -57 -14 -100.5t-35.5 -70t-53.5 -44.5t-62.5 -26t-68.5 -12q35 -31 35 -95q0 -40 -0.5 -89t-0.5 -51q0 -12 8.5 -20t26.5 -5q154 52 252 185.5t98 300.5zM1536 1120v-960 + q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="upload_alt" unicode="" horiz-adv-x="1664" + d="M1280 64q0 26 -19 45t-45 19t-45 -19t-19 -45t19 -45t45 -19t45 19t19 45zM1536 64q0 26 -19 45t-45 19t-45 -19t-19 -45t19 -45t45 -19t45 19t19 45zM1664 288v-320q0 -40 -28 -68t-68 -28h-1472q-40 0 -68 28t-28 68v320q0 40 28 68t68 28h427q21 -56 70.5 -92 + t110.5 -36h256q61 0 110.5 36t70.5 92h427q40 0 68 -28t28 -68zM1339 936q-17 -40 -59 -40h-256v-448q0 -26 -19 -45t-45 -19h-256q-26 0 -45 19t-19 45v448h-256q-42 0 -59 40q-17 39 14 69l448 448q18 19 45 19t45 -19l448 -448q31 -30 14 -69z" /> + <glyph glyph-name="lemon" unicode="" + d="M1407 710q0 44 -7 113.5t-18 96.5q-12 30 -17 44t-9 36.5t-4 48.5q0 23 5 68.5t5 67.5q0 37 -10 55q-4 1 -13 1q-19 0 -58 -4.5t-59 -4.5q-60 0 -176 24t-175 24q-43 0 -94.5 -11.5t-85 -23.5t-89.5 -34q-137 -54 -202 -103q-96 -73 -159.5 -189.5t-88 -236t-24.5 -248.5 + q0 -40 12.5 -120t12.5 -121q0 -23 -11 -66.5t-11 -65.5t12 -36.5t34 -14.5q24 0 72.5 11t73.5 11q57 0 169.5 -15.5t169.5 -15.5q181 0 284 36q129 45 235.5 152.5t166 245.5t59.5 275zM1535 712q0 -165 -70 -327.5t-196 -288t-281 -180.5q-124 -44 -326 -44 + q-57 0 -170 14.5t-169 14.5q-24 0 -72.5 -14.5t-73.5 -14.5q-73 0 -123.5 55.5t-50.5 128.5q0 24 11 68t11 67q0 40 -12.5 120.5t-12.5 121.5q0 111 18 217.5t54.5 209.5t100.5 194t150 156q78 59 232 120q194 78 316 78q60 0 175.5 -24t173.5 -24q19 0 57 5t58 5 + q81 0 118 -50.5t37 -134.5q0 -23 -5 -68t-5 -68q0 -13 2 -25t3.5 -16.5t7.5 -20.5t8 -20q16 -40 25 -118.5t9 -136.5z" /> + <glyph glyph-name="phone" unicode="" horiz-adv-x="1408" + d="M1408 296q0 -27 -10 -70.5t-21 -68.5q-21 -50 -122 -106q-94 -51 -186 -51q-27 0 -53 3.5t-57.5 12.5t-47 14.5t-55.5 20.5t-49 18q-98 35 -175 83q-127 79 -264 216t-216 264q-48 77 -83 175q-3 9 -18 49t-20.5 55.5t-14.5 47t-12.5 57.5t-3.5 53q0 92 51 186 + q56 101 106 122q25 11 68.5 21t70.5 10q14 0 21 -3q18 -6 53 -76q11 -19 30 -54t35 -63.5t31 -53.5q3 -4 17.5 -25t21.5 -35.5t7 -28.5q0 -20 -28.5 -50t-62 -55t-62 -53t-28.5 -46q0 -9 5 -22.5t8.5 -20.5t14 -24t11.5 -19q76 -137 174 -235t235 -174q2 -1 19 -11.5t24 -14 + t20.5 -8.5t22.5 -5q18 0 46 28.5t53 62t55 62t50 28.5q14 0 28.5 -7t35.5 -21.5t25 -17.5q25 -15 53.5 -31t63.5 -35t54 -30q70 -35 76 -53q3 -7 3 -21z" /> + <glyph glyph-name="check_empty" unicode="" horiz-adv-x="1408" + d="M1120 1280h-832q-66 0 -113 -47t-47 -113v-832q0 -66 47 -113t113 -47h832q66 0 113 47t47 113v832q0 66 -47 113t-113 47zM1408 1120v-832q0 -119 -84.5 -203.5t-203.5 -84.5h-832q-119 0 -203.5 84.5t-84.5 203.5v832q0 119 84.5 203.5t203.5 84.5h832 + q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="bookmark_empty" unicode="" horiz-adv-x="1280" + d="M1152 1280h-1024v-1242l423 406l89 85l89 -85l423 -406v1242zM1164 1408q23 0 44 -9q33 -13 52.5 -41t19.5 -62v-1289q0 -34 -19.5 -62t-52.5 -41q-19 -8 -44 -8q-48 0 -83 32l-441 424l-441 -424q-36 -33 -83 -33q-23 0 -44 9q-33 13 -52.5 41t-19.5 62v1289 + q0 34 19.5 62t52.5 41q21 9 44 9h1048z" /> + <glyph glyph-name="phone_sign" unicode="" + d="M1280 343q0 11 -2 16t-18 16.5t-40.5 25t-47.5 26.5t-45.5 25t-28.5 15q-5 3 -19 13t-25 15t-21 5q-15 0 -36.5 -20.5t-39.5 -45t-38.5 -45t-33.5 -20.5q-7 0 -16.5 3.5t-15.5 6.5t-17 9.5t-14 8.5q-99 55 -170 126.5t-127 170.5q-2 3 -8.5 14t-9.5 17t-6.5 15.5 + t-3.5 16.5q0 13 20.5 33.5t45 38.5t45 39.5t20.5 36.5q0 10 -5 21t-15 25t-13 19q-3 6 -15 28.5t-25 45.5t-26.5 47.5t-25 40.5t-16.5 18t-16 2q-48 0 -101 -22q-46 -21 -80 -94.5t-34 -130.5q0 -16 2.5 -34t5 -30.5t9 -33t10 -29.5t12.5 -33t11 -30q60 -164 216.5 -320.5 + t320.5 -216.5q6 -2 30 -11t33 -12.5t29.5 -10t33 -9t30.5 -5t34 -2.5q57 0 130.5 34t94.5 80q22 53 22 101zM1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z + " /> + <glyph glyph-name="twitter" unicode="" horiz-adv-x="1664" + d="M1620 1128q-67 -98 -162 -167q1 -14 1 -42q0 -130 -38 -259.5t-115.5 -248.5t-184.5 -210.5t-258 -146t-323 -54.5q-271 0 -496 145q35 -4 78 -4q225 0 401 138q-105 2 -188 64.5t-114 159.5q33 -5 61 -5q43 0 85 11q-112 23 -185.5 111.5t-73.5 205.5v4q68 -38 146 -41 + q-66 44 -105 115t-39 154q0 88 44 163q121 -149 294.5 -238.5t371.5 -99.5q-8 38 -8 74q0 134 94.5 228.5t228.5 94.5q140 0 236 -102q109 21 205 78q-37 -115 -142 -178q93 10 186 50z" /> + <glyph glyph-name="facebook" unicode="" horiz-adv-x="1024" + d="M959 1524v-264h-157q-86 0 -116 -36t-30 -108v-189h293l-39 -296h-254v-759h-306v759h-255v296h255v218q0 186 104 288.5t277 102.5q147 0 228 -12z" /> + <glyph glyph-name="github" unicode="" + d="M768 1408q209 0 385.5 -103t279.5 -279.5t103 -385.5q0 -251 -146.5 -451.5t-378.5 -277.5q-27 -5 -40 7t-13 30q0 3 0.5 76.5t0.5 134.5q0 97 -52 142q57 6 102.5 18t94 39t81 66.5t53 105t20.5 150.5q0 119 -79 206q37 91 -8 204q-28 9 -81 -11t-92 -44l-38 -24 + q-93 26 -192 26t-192 -26q-16 11 -42.5 27t-83.5 38.5t-85 13.5q-45 -113 -8 -204q-79 -87 -79 -206q0 -85 20.5 -150t52.5 -105t80.5 -67t94 -39t102.5 -18q-39 -36 -49 -103q-21 -10 -45 -15t-57 -5t-65.5 21.5t-55.5 62.5q-19 32 -48.5 52t-49.5 24l-20 3q-21 0 -29 -4.5 + t-5 -11.5t9 -14t13 -12l7 -5q22 -10 43.5 -38t31.5 -51l10 -23q13 -38 44 -61.5t67 -30t69.5 -7t55.5 3.5l23 4q0 -38 0.5 -88.5t0.5 -54.5q0 -18 -13 -30t-40 -7q-232 77 -378.5 277.5t-146.5 451.5q0 209 103 385.5t279.5 279.5t385.5 103zM291 305q3 7 -7 12 + q-10 3 -13 -2q-3 -7 7 -12q9 -6 13 2zM322 271q7 5 -2 16q-10 9 -16 3q-7 -5 2 -16q10 -10 16 -3zM352 226q9 7 0 19q-8 13 -17 6q-9 -5 0 -18t17 -7zM394 184q8 8 -4 19q-12 12 -20 3q-9 -8 4 -19q12 -12 20 -3zM451 159q3 11 -13 16q-15 4 -19 -7t13 -15q15 -6 19 6z + M514 154q0 13 -17 11q-16 0 -16 -11q0 -13 17 -11q16 0 16 11zM572 164q-2 11 -18 9q-16 -3 -14 -15t18 -8t14 14z" /> + <glyph glyph-name="unlock" unicode="" horiz-adv-x="1664" + d="M1664 960v-256q0 -26 -19 -45t-45 -19h-64q-26 0 -45 19t-19 45v256q0 106 -75 181t-181 75t-181 -75t-75 -181v-192h96q40 0 68 -28t28 -68v-576q0 -40 -28 -68t-68 -28h-960q-40 0 -68 28t-28 68v576q0 40 28 68t68 28h672v192q0 185 131.5 316.5t316.5 131.5 + t316.5 -131.5t131.5 -316.5z" /> + <glyph glyph-name="credit_card" unicode="" horiz-adv-x="1920" + d="M1760 1408q66 0 113 -47t47 -113v-1216q0 -66 -47 -113t-113 -47h-1600q-66 0 -113 47t-47 113v1216q0 66 47 113t113 47h1600zM160 1280q-13 0 -22.5 -9.5t-9.5 -22.5v-224h1664v224q0 13 -9.5 22.5t-22.5 9.5h-1600zM1760 0q13 0 22.5 9.5t9.5 22.5v608h-1664v-608 + q0 -13 9.5 -22.5t22.5 -9.5h1600zM256 128v128h256v-128h-256zM640 128v128h384v-128h-384z" /> + <glyph glyph-name="rss" unicode="" horiz-adv-x="1408" + d="M384 192q0 -80 -56 -136t-136 -56t-136 56t-56 136t56 136t136 56t136 -56t56 -136zM896 69q2 -28 -17 -48q-18 -21 -47 -21h-135q-25 0 -43 16.5t-20 41.5q-22 229 -184.5 391.5t-391.5 184.5q-25 2 -41.5 20t-16.5 43v135q0 29 21 47q17 17 43 17h5q160 -13 306 -80.5 + t259 -181.5q114 -113 181.5 -259t80.5 -306zM1408 67q2 -27 -18 -47q-18 -20 -46 -20h-143q-26 0 -44.5 17.5t-19.5 42.5q-12 215 -101 408.5t-231.5 336t-336 231.5t-408.5 102q-25 1 -42.5 19.5t-17.5 43.5v143q0 28 20 46q18 18 44 18h3q262 -13 501.5 -120t425.5 -294 + q187 -186 294 -425.5t120 -501.5z" /> + <glyph glyph-name="hdd" unicode="" + d="M1040 320q0 -33 -23.5 -56.5t-56.5 -23.5t-56.5 23.5t-23.5 56.5t23.5 56.5t56.5 23.5t56.5 -23.5t23.5 -56.5zM1296 320q0 -33 -23.5 -56.5t-56.5 -23.5t-56.5 23.5t-23.5 56.5t23.5 56.5t56.5 23.5t56.5 -23.5t23.5 -56.5zM1408 160v320q0 13 -9.5 22.5t-22.5 9.5 + h-1216q-13 0 -22.5 -9.5t-9.5 -22.5v-320q0 -13 9.5 -22.5t22.5 -9.5h1216q13 0 22.5 9.5t9.5 22.5zM178 640h1180l-157 482q-4 13 -16 21.5t-26 8.5h-782q-14 0 -26 -8.5t-16 -21.5zM1536 480v-320q0 -66 -47 -113t-113 -47h-1216q-66 0 -113 47t-47 113v320q0 25 16 75 + l197 606q17 53 63 86t101 33h782q55 0 101 -33t63 -86l197 -606q16 -50 16 -75z" /> + <glyph glyph-name="bullhorn" unicode="" horiz-adv-x="1792" + d="M1664 896q53 0 90.5 -37.5t37.5 -90.5t-37.5 -90.5t-90.5 -37.5v-384q0 -52 -38 -90t-90 -38q-417 347 -812 380q-58 -19 -91 -66t-31 -100.5t40 -92.5q-20 -33 -23 -65.5t6 -58t33.5 -55t48 -50t61.5 -50.5q-29 -58 -111.5 -83t-168.5 -11.5t-132 55.5q-7 23 -29.5 87.5 + t-32 94.5t-23 89t-15 101t3.5 98.5t22 110.5h-122q-66 0 -113 47t-47 113v192q0 66 47 113t113 47h480q435 0 896 384q52 0 90 -38t38 -90v-384zM1536 292v954q-394 -302 -768 -343v-270q377 -42 768 -341z" /> + <glyph glyph-name="bell" unicode="" horiz-adv-x="1792" + d="M912 -160q0 16 -16 16q-59 0 -101.5 42.5t-42.5 101.5q0 16 -16 16t-16 -16q0 -73 51.5 -124.5t124.5 -51.5q16 0 16 16zM246 128h1300q-266 300 -266 832q0 51 -24 105t-69 103t-121.5 80.5t-169.5 31.5t-169.5 -31.5t-121.5 -80.5t-69 -103t-24 -105q0 -532 -266 -832z + M1728 128q0 -52 -38 -90t-90 -38h-448q0 -106 -75 -181t-181 -75t-181 75t-75 181h-448q-52 0 -90 38t-38 90q50 42 91 88t85 119.5t74.5 158.5t50 206t19.5 260q0 152 117 282.5t307 158.5q-8 19 -8 39q0 40 28 68t68 28t68 -28t28 -68q0 -20 -8 -39q190 -28 307 -158.5 + t117 -282.5q0 -139 19.5 -260t50 -206t74.5 -158.5t85 -119.5t91 -88z" /> + <glyph glyph-name="certificate" unicode="" + d="M1376 640l138 -135q30 -28 20 -70q-12 -41 -52 -51l-188 -48l53 -186q12 -41 -19 -70q-29 -31 -70 -19l-186 53l-48 -188q-10 -40 -51 -52q-12 -2 -19 -2q-31 0 -51 22l-135 138l-135 -138q-28 -30 -70 -20q-41 11 -51 52l-48 188l-186 -53q-41 -12 -70 19q-31 29 -19 70 + l53 186l-188 48q-40 10 -52 51q-10 42 20 70l138 135l-138 135q-30 28 -20 70q12 41 52 51l188 48l-53 186q-12 41 19 70q29 31 70 19l186 -53l48 188q10 41 51 51q41 12 70 -19l135 -139l135 139q29 30 70 19q41 -10 51 -51l48 -188l186 53q41 12 70 -19q31 -29 19 -70 + l-53 -186l188 -48q40 -10 52 -51q10 -42 -20 -70z" /> + <glyph glyph-name="hand_right" unicode="" horiz-adv-x="1792" + d="M256 192q0 26 -19 45t-45 19t-45 -19t-19 -45t19 -45t45 -19t45 19t19 45zM1664 768q0 51 -39 89.5t-89 38.5h-576q0 20 15 48.5t33 55t33 68t15 84.5q0 67 -44.5 97.5t-115.5 30.5q-24 0 -90 -139q-24 -44 -37 -65q-40 -64 -112 -145q-71 -81 -101 -106 + q-69 -57 -140 -57h-32v-640h32q72 0 167 -32t193.5 -64t179.5 -32q189 0 189 167q0 26 -5 56q30 16 47.5 52.5t17.5 73.5t-18 69q53 50 53 119q0 25 -10 55.5t-25 47.5h331q52 0 90 38t38 90zM1792 769q0 -105 -75.5 -181t-180.5 -76h-169q-4 -62 -37 -119q3 -21 3 -43 + q0 -101 -60 -178q1 -139 -85 -219.5t-227 -80.5q-133 0 -322 69q-164 59 -223 59h-288q-53 0 -90.5 37.5t-37.5 90.5v640q0 53 37.5 90.5t90.5 37.5h288q10 0 21.5 4.5t23.5 14t22.5 18t24 22.5t20.5 21.5t19 21.5t14 17q65 74 100 129q13 21 33 62t37 72t40.5 63t55 49.5 + t69.5 17.5q125 0 206.5 -67t81.5 -189q0 -68 -22 -128h374q104 0 180 -76t76 -179z" /> + <glyph glyph-name="hand_left" unicode="" horiz-adv-x="1792" + d="M1376 128h32v640h-32q-35 0 -67.5 12t-62.5 37t-50 46t-49 54q-8 9 -12 14q-72 81 -112 145q-14 22 -38 68q-1 3 -10.5 22.5t-18.5 36t-20 35.5t-21.5 30.5t-18.5 11.5q-71 0 -115.5 -30.5t-44.5 -97.5q0 -43 15 -84.5t33 -68t33 -55t15 -48.5h-576q-50 0 -89 -38.5 + t-39 -89.5q0 -52 38 -90t90 -38h331q-15 -17 -25 -47.5t-10 -55.5q0 -69 53 -119q-18 -32 -18 -69t17.5 -73.5t47.5 -52.5q-4 -24 -4 -56q0 -85 48.5 -126t135.5 -41q84 0 183 32t194 64t167 32zM1664 192q0 26 -19 45t-45 19t-45 -19t-19 -45t19 -45t45 -19t45 19t19 45z + M1792 768v-640q0 -53 -37.5 -90.5t-90.5 -37.5h-288q-59 0 -223 -59q-190 -69 -317 -69q-142 0 -230 77.5t-87 217.5l1 5q-61 76 -61 178q0 22 3 43q-33 57 -37 119h-169q-105 0 -180.5 76t-75.5 181q0 103 76 179t180 76h374q-22 60 -22 128q0 122 81.5 189t206.5 67 + q38 0 69.5 -17.5t55 -49.5t40.5 -63t37 -72t33 -62q35 -55 100 -129q2 -3 14 -17t19 -21.5t20.5 -21.5t24 -22.5t22.5 -18t23.5 -14t21.5 -4.5h288q53 0 90.5 -37.5t37.5 -90.5z" /> + <glyph glyph-name="hand_up" unicode="" + d="M1280 -64q0 26 -19 45t-45 19t-45 -19t-19 -45t19 -45t45 -19t45 19t19 45zM1408 700q0 189 -167 189q-26 0 -56 -5q-16 30 -52.5 47.5t-73.5 17.5t-69 -18q-50 53 -119 53q-25 0 -55.5 -10t-47.5 -25v331q0 52 -38 90t-90 38q-51 0 -89.5 -39t-38.5 -89v-576 + q-20 0 -48.5 15t-55 33t-68 33t-84.5 15q-67 0 -97.5 -44.5t-30.5 -115.5q0 -24 139 -90q44 -24 65 -37q64 -40 145 -112q81 -71 106 -101q57 -69 57 -140v-32h640v32q0 72 32 167t64 193.5t32 179.5zM1536 705q0 -133 -69 -322q-59 -164 -59 -223v-288q0 -53 -37.5 -90.5 + t-90.5 -37.5h-640q-53 0 -90.5 37.5t-37.5 90.5v288q0 10 -4.5 21.5t-14 23.5t-18 22.5t-22.5 24t-21.5 20.5t-21.5 19t-17 14q-74 65 -129 100q-21 13 -62 33t-72 37t-63 40.5t-49.5 55t-17.5 69.5q0 125 67 206.5t189 81.5q68 0 128 -22v374q0 104 76 180t179 76 + q105 0 181 -75.5t76 -180.5v-169q62 -4 119 -37q21 3 43 3q101 0 178 -60q139 1 219.5 -85t80.5 -227z" /> + <glyph glyph-name="hand_down" unicode="" + d="M1408 576q0 84 -32 183t-64 194t-32 167v32h-640v-32q0 -35 -12 -67.5t-37 -62.5t-46 -50t-54 -49q-9 -8 -14 -12q-81 -72 -145 -112q-22 -14 -68 -38q-3 -1 -22.5 -10.5t-36 -18.5t-35.5 -20t-30.5 -21.5t-11.5 -18.5q0 -71 30.5 -115.5t97.5 -44.5q43 0 84.5 15t68 33 + t55 33t48.5 15v-576q0 -50 38.5 -89t89.5 -39q52 0 90 38t38 90v331q46 -35 103 -35q69 0 119 53q32 -18 69 -18t73.5 17.5t52.5 47.5q24 -4 56 -4q85 0 126 48.5t41 135.5zM1280 1344q0 26 -19 45t-45 19t-45 -19t-19 -45t19 -45t45 -19t45 19t19 45zM1536 580 + q0 -142 -77.5 -230t-217.5 -87l-5 1q-76 -61 -178 -61q-22 0 -43 3q-54 -30 -119 -37v-169q0 -105 -76 -180.5t-181 -75.5q-103 0 -179 76t-76 180v374q-54 -22 -128 -22q-121 0 -188.5 81.5t-67.5 206.5q0 38 17.5 69.5t49.5 55t63 40.5t72 37t62 33q55 35 129 100 + q3 2 17 14t21.5 19t21.5 20.5t22.5 24t18 22.5t14 23.5t4.5 21.5v288q0 53 37.5 90.5t90.5 37.5h640q53 0 90.5 -37.5t37.5 -90.5v-288q0 -59 59 -223q69 -190 69 -317z" /> + <glyph glyph-name="circle_arrow_left" unicode="" + d="M1280 576v128q0 26 -19 45t-45 19h-502l189 189q19 19 19 45t-19 45l-91 91q-18 18 -45 18t-45 -18l-362 -362l-91 -91q-18 -18 -18 -45t18 -45l91 -91l362 -362q18 -18 45 -18t45 18l91 91q18 18 18 45t-18 45l-189 189h502q26 0 45 19t19 45zM1536 640 + q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="circle_arrow_right" unicode="" + d="M1285 640q0 27 -18 45l-91 91l-362 362q-18 18 -45 18t-45 -18l-91 -91q-18 -18 -18 -45t18 -45l189 -189h-502q-26 0 -45 -19t-19 -45v-128q0 -26 19 -45t45 -19h502l-189 -189q-19 -19 -19 -45t19 -45l91 -91q18 -18 45 -18t45 18l362 362l91 91q18 18 18 45zM1536 640 + q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="circle_arrow_up" unicode="" + d="M1284 641q0 27 -18 45l-362 362l-91 91q-18 18 -45 18t-45 -18l-91 -91l-362 -362q-18 -18 -18 -45t18 -45l91 -91q18 -18 45 -18t45 18l189 189v-502q0 -26 19 -45t45 -19h128q26 0 45 19t19 45v502l189 -189q19 -19 45 -19t45 19l91 91q18 18 18 45zM1536 640 + q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="circle_arrow_down" unicode="" + d="M1284 639q0 27 -18 45l-91 91q-18 18 -45 18t-45 -18l-189 -189v502q0 26 -19 45t-45 19h-128q-26 0 -45 -19t-19 -45v-502l-189 189q-19 19 -45 19t-45 -19l-91 -91q-18 -18 -18 -45t18 -45l362 -362l91 -91q18 -18 45 -18t45 18l91 91l362 362q18 18 18 45zM1536 640 + q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="globe" unicode="" + d="M768 1408q209 0 385.5 -103t279.5 -279.5t103 -385.5t-103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103zM1042 887q-2 -1 -9.5 -9.5t-13.5 -9.5q2 0 4.5 5t5 11t3.5 7q6 7 22 15q14 6 52 12q34 8 51 -11 + q-2 2 9.5 13t14.5 12q3 2 15 4.5t15 7.5l2 22q-12 -1 -17.5 7t-6.5 21q0 -2 -6 -8q0 7 -4.5 8t-11.5 -1t-9 -1q-10 3 -15 7.5t-8 16.5t-4 15q-2 5 -9.5 11t-9.5 10q-1 2 -2.5 5.5t-3 6.5t-4 5.5t-5.5 2.5t-7 -5t-7.5 -10t-4.5 -5q-3 2 -6 1.5t-4.5 -1t-4.5 -3t-5 -3.5 + q-3 -2 -8.5 -3t-8.5 -2q15 5 -1 11q-10 4 -16 3q9 4 7.5 12t-8.5 14h5q-1 4 -8.5 8.5t-17.5 8.5t-13 6q-8 5 -34 9.5t-33 0.5q-5 -6 -4.5 -10.5t4 -14t3.5 -12.5q1 -6 -5.5 -13t-6.5 -12q0 -7 14 -15.5t10 -21.5q-3 -8 -16 -16t-16 -12q-5 -8 -1.5 -18.5t10.5 -16.5 + q2 -2 1.5 -4t-3.5 -4.5t-5.5 -4t-6.5 -3.5l-3 -2q-11 -5 -20.5 6t-13.5 26q-7 25 -16 30q-23 8 -29 -1q-5 13 -41 26q-25 9 -58 4q6 1 0 15q-7 15 -19 12q3 6 4 17.5t1 13.5q3 13 12 23q1 1 7 8.5t9.5 13.5t0.5 6q35 -4 50 11q5 5 11.5 17t10.5 17q9 6 14 5.5t14.5 -5.5 + t14.5 -5q14 -1 15.5 11t-7.5 20q12 -1 3 17q-4 7 -8 9q-12 4 -27 -5q-8 -4 2 -8q-1 1 -9.5 -10.5t-16.5 -17.5t-16 5q-1 1 -5.5 13.5t-9.5 13.5q-8 0 -16 -15q3 8 -11 15t-24 8q19 12 -8 27q-7 4 -20.5 5t-19.5 -4q-5 -7 -5.5 -11.5t5 -8t10.5 -5.5t11.5 -4t8.5 -3 + q14 -10 8 -14q-2 -1 -8.5 -3.5t-11.5 -4.5t-6 -4q-3 -4 0 -14t-2 -14q-5 5 -9 17.5t-7 16.5q7 -9 -25 -6l-10 1q-4 0 -16 -2t-20.5 -1t-13.5 8q-4 8 0 20q1 4 4 2q-4 3 -11 9.5t-10 8.5q-46 -15 -94 -41q6 -1 12 1q5 2 13 6.5t10 5.5q34 14 42 7l5 5q14 -16 20 -25 + q-7 4 -30 1q-20 -6 -22 -12q7 -12 5 -18q-4 3 -11.5 10t-14.5 11t-15 5q-16 0 -22 -1q-146 -80 -235 -222q7 -7 12 -8q4 -1 5 -9t2.5 -11t11.5 3q9 -8 3 -19q1 1 44 -27q19 -17 21 -21q3 -11 -10 -18q-1 2 -9 9t-9 4q-3 -5 0.5 -18.5t10.5 -12.5q-7 0 -9.5 -16t-2.5 -35.5 + t-1 -23.5l2 -1q-3 -12 5.5 -34.5t21.5 -19.5q-13 -3 20 -43q6 -8 8 -9q3 -2 12 -7.5t15 -10t10 -10.5q4 -5 10 -22.5t14 -23.5q-2 -6 9.5 -20t10.5 -23q-1 0 -2.5 -1t-2.5 -1q3 -7 15.5 -14t15.5 -13q1 -3 2 -10t3 -11t8 -2q2 20 -24 62q-15 25 -17 29q-3 5 -5.5 15.5 + t-4.5 14.5q2 0 6 -1.5t8.5 -3.5t7.5 -4t2 -3q-3 -7 2 -17.5t12 -18.5t17 -19t12 -13q6 -6 14 -19.5t0 -13.5q9 0 20 -10.5t17 -19.5q5 -8 8 -26t5 -24q2 -7 8.5 -13.5t12.5 -9.5l16 -8t13 -7q5 -2 18.5 -10.5t21.5 -11.5q10 -4 16 -4t14.5 2.5t13.5 3.5q15 2 29 -15t21 -21 + q36 -19 55 -11q-2 -1 0.5 -7.5t8 -15.5t9 -14.5t5.5 -8.5q5 -6 18 -15t18 -15q6 4 7 9q-3 -8 7 -20t18 -10q14 3 14 32q-31 -15 -49 18q0 1 -2.5 5.5t-4 8.5t-2.5 8.5t0 7.5t5 3q9 0 10 3.5t-2 12.5t-4 13q-1 8 -11 20t-12 15q-5 -9 -16 -8t-16 9q0 -1 -1.5 -5.5t-1.5 -6.5 + q-13 0 -15 1q1 3 2.5 17.5t3.5 22.5q1 4 5.5 12t7.5 14.5t4 12.5t-4.5 9.5t-17.5 2.5q-19 -1 -26 -20q-1 -3 -3 -10.5t-5 -11.5t-9 -7q-7 -3 -24 -2t-24 5q-13 8 -22.5 29t-9.5 37q0 10 2.5 26.5t3 25t-5.5 24.5q3 2 9 9.5t10 10.5q2 1 4.5 1.5t4.5 0t4 1.5t3 6q-1 1 -4 3 + q-3 3 -4 3q7 -3 28.5 1.5t27.5 -1.5q15 -11 22 2q0 1 -2.5 9.5t-0.5 13.5q5 -27 29 -9q3 -3 15.5 -5t17.5 -5q3 -2 7 -5.5t5.5 -4.5t5 0.5t8.5 6.5q10 -14 12 -24q11 -40 19 -44q7 -3 11 -2t4.5 9.5t0 14t-1.5 12.5l-1 8v18l-1 8q-15 3 -18.5 12t1.5 18.5t15 18.5q1 1 8 3.5 + t15.5 6.5t12.5 8q21 19 15 35q7 0 11 9q-1 0 -5 3t-7.5 5t-4.5 2q9 5 2 16q5 3 7.5 11t7.5 10q9 -12 21 -2q8 8 1 16q5 7 20.5 10.5t18.5 9.5q7 -2 8 2t1 12t3 12q4 5 15 9t13 5l17 11q3 4 0 4q18 -2 31 11q10 11 -6 20q3 6 -3 9.5t-15 5.5q3 1 11.5 0.5t10.5 1.5 + q15 10 -7 16q-17 5 -43 -12zM879 10q206 36 351 189q-3 3 -12.5 4.5t-12.5 3.5q-18 7 -24 8q1 7 -2.5 13t-8 9t-12.5 8t-11 7q-2 2 -7 6t-7 5.5t-7.5 4.5t-8.5 2t-10 -1l-3 -1q-3 -1 -5.5 -2.5t-5.5 -3t-4 -3t0 -2.5q-21 17 -36 22q-5 1 -11 5.5t-10.5 7t-10 1.5t-11.5 -7 + q-5 -5 -6 -15t-2 -13q-7 5 0 17.5t2 18.5q-3 6 -10.5 4.5t-12 -4.5t-11.5 -8.5t-9 -6.5t-8.5 -5.5t-8.5 -7.5q-3 -4 -6 -12t-5 -11q-2 4 -11.5 6.5t-9.5 5.5q2 -10 4 -35t5 -38q7 -31 -12 -48q-27 -25 -29 -40q-4 -22 12 -26q0 -7 -8 -20.5t-7 -21.5q0 -6 2 -16z" /> + <glyph glyph-name="wrench" unicode="" horiz-adv-x="1664" + d="M384 64q0 26 -19 45t-45 19t-45 -19t-19 -45t19 -45t45 -19t45 19t19 45zM1028 484l-682 -682q-37 -37 -90 -37q-52 0 -91 37l-106 108q-38 36 -38 90q0 53 38 91l681 681q39 -98 114.5 -173.5t173.5 -114.5zM1662 919q0 -39 -23 -106q-47 -134 -164.5 -217.5 + t-258.5 -83.5q-185 0 -316.5 131.5t-131.5 316.5t131.5 316.5t316.5 131.5q58 0 121.5 -16.5t107.5 -46.5q16 -11 16 -28t-16 -28l-293 -169v-224l193 -107q5 3 79 48.5t135.5 81t70.5 35.5q15 0 23.5 -10t8.5 -25z" /> + <glyph glyph-name="tasks" unicode="" horiz-adv-x="1792" + d="M1024 128h640v128h-640v-128zM640 640h1024v128h-1024v-128zM1280 1152h384v128h-384v-128zM1792 320v-256q0 -26 -19 -45t-45 -19h-1664q-26 0 -45 19t-19 45v256q0 26 19 45t45 19h1664q26 0 45 -19t19 -45zM1792 832v-256q0 -26 -19 -45t-45 -19h-1664q-26 0 -45 19 + t-19 45v256q0 26 19 45t45 19h1664q26 0 45 -19t19 -45zM1792 1344v-256q0 -26 -19 -45t-45 -19h-1664q-26 0 -45 19t-19 45v256q0 26 19 45t45 19h1664q26 0 45 -19t19 -45z" /> + <glyph glyph-name="filter" unicode="" horiz-adv-x="1408" + d="M1403 1241q17 -41 -14 -70l-493 -493v-742q0 -42 -39 -59q-13 -5 -25 -5q-27 0 -45 19l-256 256q-19 19 -19 45v486l-493 493q-31 29 -14 70q17 39 59 39h1280q42 0 59 -39z" /> + <glyph glyph-name="briefcase" unicode="" horiz-adv-x="1792" + d="M640 1280h512v128h-512v-128zM1792 640v-480q0 -66 -47 -113t-113 -47h-1472q-66 0 -113 47t-47 113v480h672v-160q0 -26 19 -45t45 -19h320q26 0 45 19t19 45v160h672zM1024 640v-128h-256v128h256zM1792 1120v-384h-1792v384q0 66 47 113t113 47h352v160q0 40 28 68 + t68 28h576q40 0 68 -28t28 -68v-160h352q66 0 113 -47t47 -113z" /> + <glyph glyph-name="fullscreen" unicode="" + d="M1283 995l-355 -355l355 -355l144 144q29 31 70 14q39 -17 39 -59v-448q0 -26 -19 -45t-45 -19h-448q-42 0 -59 40q-17 39 14 69l144 144l-355 355l-355 -355l144 -144q31 -30 14 -69q-17 -40 -59 -40h-448q-26 0 -45 19t-19 45v448q0 42 40 59q39 17 69 -14l144 -144 + l355 355l-355 355l-144 -144q-19 -19 -45 -19q-12 0 -24 5q-40 17 -40 59v448q0 26 19 45t45 19h448q42 0 59 -40q17 -39 -14 -69l-144 -144l355 -355l355 355l-144 144q-31 30 -14 69q17 40 59 40h448q26 0 45 -19t19 -45v-448q0 -42 -39 -59q-13 -5 -25 -5q-26 0 -45 19z + " /> + <glyph glyph-name="group" unicode="" horiz-adv-x="1920" + d="M593 640q-162 -5 -265 -128h-134q-82 0 -138 40.5t-56 118.5q0 353 124 353q6 0 43.5 -21t97.5 -42.5t119 -21.5q67 0 133 23q-5 -37 -5 -66q0 -139 81 -256zM1664 3q0 -120 -73 -189.5t-194 -69.5h-874q-121 0 -194 69.5t-73 189.5q0 53 3.5 103.5t14 109t26.5 108.5 + t43 97.5t62 81t85.5 53.5t111.5 20q10 0 43 -21.5t73 -48t107 -48t135 -21.5t135 21.5t107 48t73 48t43 21.5q61 0 111.5 -20t85.5 -53.5t62 -81t43 -97.5t26.5 -108.5t14 -109t3.5 -103.5zM640 1280q0 -106 -75 -181t-181 -75t-181 75t-75 181t75 181t181 75t181 -75 + t75 -181zM1344 896q0 -159 -112.5 -271.5t-271.5 -112.5t-271.5 112.5t-112.5 271.5t112.5 271.5t271.5 112.5t271.5 -112.5t112.5 -271.5zM1920 671q0 -78 -56 -118.5t-138 -40.5h-134q-103 123 -265 128q81 117 81 256q0 29 -5 66q66 -23 133 -23q59 0 119 21.5t97.5 42.5 + t43.5 21q124 0 124 -353zM1792 1280q0 -106 -75 -181t-181 -75t-181 75t-75 181t75 181t181 75t181 -75t75 -181z" /> + <glyph glyph-name="link" unicode="" horiz-adv-x="1664" + d="M1456 320q0 40 -28 68l-208 208q-28 28 -68 28q-42 0 -72 -32q3 -3 19 -18.5t21.5 -21.5t15 -19t13 -25.5t3.5 -27.5q0 -40 -28 -68t-68 -28q-15 0 -27.5 3.5t-25.5 13t-19 15t-21.5 21.5t-18.5 19q-33 -31 -33 -73q0 -40 28 -68l206 -207q27 -27 68 -27q40 0 68 26 + l147 146q28 28 28 67zM753 1025q0 40 -28 68l-206 207q-28 28 -68 28q-39 0 -68 -27l-147 -146q-28 -28 -28 -67q0 -40 28 -68l208 -208q27 -27 68 -27q42 0 72 31q-3 3 -19 18.5t-21.5 21.5t-15 19t-13 25.5t-3.5 27.5q0 40 28 68t68 28q15 0 27.5 -3.5t25.5 -13t19 -15 + t21.5 -21.5t18.5 -19q33 31 33 73zM1648 320q0 -120 -85 -203l-147 -146q-83 -83 -203 -83q-121 0 -204 85l-206 207q-83 83 -83 203q0 123 88 209l-88 88q-86 -88 -208 -88q-120 0 -204 84l-208 208q-84 84 -84 204t85 203l147 146q83 83 203 83q121 0 204 -85l206 -207 + q83 -83 83 -203q0 -123 -88 -209l88 -88q86 88 208 88q120 0 204 -84l208 -208q84 -84 84 -204z" /> + <glyph glyph-name="cloud" unicode="" horiz-adv-x="1920" + d="M1920 384q0 -159 -112.5 -271.5t-271.5 -112.5h-1088q-185 0 -316.5 131.5t-131.5 316.5q0 132 71 241.5t187 163.5q-2 28 -2 43q0 212 150 362t362 150q158 0 286.5 -88t187.5 -230q70 62 166 62q106 0 181 -75t75 -181q0 -75 -41 -138q129 -30 213 -134.5t84 -239.5z + " /> + <glyph glyph-name="beaker" unicode="" horiz-adv-x="1664" + d="M1527 88q56 -89 21.5 -152.5t-140.5 -63.5h-1152q-106 0 -140.5 63.5t21.5 152.5l503 793v399h-64q-26 0 -45 19t-19 45t19 45t45 19h512q26 0 45 -19t19 -45t-19 -45t-45 -19h-64v-399zM748 813l-272 -429h712l-272 429l-20 31v37v399h-128v-399v-37z" /> + <glyph glyph-name="cut" unicode="" horiz-adv-x="1792" + d="M960 640q26 0 45 -19t19 -45t-19 -45t-45 -19t-45 19t-19 45t19 45t45 19zM1260 576l507 -398q28 -20 25 -56q-5 -35 -35 -51l-128 -64q-13 -7 -29 -7q-17 0 -31 8l-690 387l-110 -66q-8 -4 -12 -5q14 -49 10 -97q-7 -77 -56 -147.5t-132 -123.5q-132 -84 -277 -84 + q-136 0 -222 78q-90 84 -79 207q7 76 56 147t131 124q132 84 278 84q83 0 151 -31q9 13 22 22l122 73l-122 73q-13 9 -22 22q-68 -31 -151 -31q-146 0 -278 84q-82 53 -131 124t-56 147q-5 59 15.5 113t63.5 93q85 79 222 79q145 0 277 -84q83 -52 132 -123t56 -148 + q4 -48 -10 -97q4 -1 12 -5l110 -66l690 387q14 8 31 8q16 0 29 -7l128 -64q30 -16 35 -51q3 -36 -25 -56zM579 836q46 42 21 108t-106 117q-92 59 -192 59q-74 0 -113 -36q-46 -42 -21 -108t106 -117q92 -59 192 -59q74 0 113 36zM494 91q81 51 106 117t-21 108 + q-39 36 -113 36q-100 0 -192 -59q-81 -51 -106 -117t21 -108q39 -36 113 -36q100 0 192 59zM672 704l96 -58v11q0 36 33 56l14 8l-79 47l-26 -26q-3 -3 -10 -11t-12 -12q-2 -2 -4 -3.5t-3 -2.5zM896 480l96 -32l736 576l-128 64l-768 -431v-113l-160 -96l9 -8q2 -2 7 -6 + q4 -4 11 -12t11 -12l26 -26zM1600 64l128 64l-520 408l-177 -138q-2 -3 -13 -7z" /> + <glyph glyph-name="copy" unicode="" horiz-adv-x="1792" + d="M1696 1152q40 0 68 -28t28 -68v-1216q0 -40 -28 -68t-68 -28h-960q-40 0 -68 28t-28 68v288h-544q-40 0 -68 28t-28 68v672q0 40 20 88t48 76l408 408q28 28 76 48t88 20h416q40 0 68 -28t28 -68v-328q68 40 128 40h416zM1152 939l-299 -299h299v299zM512 1323l-299 -299 + h299v299zM708 676l316 316v416h-384v-416q0 -40 -28 -68t-68 -28h-416v-640h512v256q0 40 20 88t48 76zM1664 -128v1152h-384v-416q0 -40 -28 -68t-68 -28h-416v-640h896z" /> + <glyph glyph-name="paper_clip" unicode="" horiz-adv-x="1408" + d="M1404 151q0 -117 -79 -196t-196 -79q-135 0 -235 100l-777 776q-113 115 -113 271q0 159 110 270t269 111q158 0 273 -113l605 -606q10 -10 10 -22q0 -16 -30.5 -46.5t-46.5 -30.5q-13 0 -23 10l-606 607q-79 77 -181 77q-106 0 -179 -75t-73 -181q0 -105 76 -181 + l776 -777q63 -63 145 -63q64 0 106 42t42 106q0 82 -63 145l-581 581q-26 24 -60 24q-29 0 -48 -19t-19 -48q0 -32 25 -59l410 -410q10 -10 10 -22q0 -16 -31 -47t-47 -31q-12 0 -22 10l-410 410q-63 61 -63 149q0 82 57 139t139 57q88 0 149 -63l581 -581q100 -98 100 -235 + z" /> + <glyph glyph-name="save" unicode="" + d="M384 0h768v384h-768v-384zM1280 0h128v896q0 14 -10 38.5t-20 34.5l-281 281q-10 10 -34 20t-39 10v-416q0 -40 -28 -68t-68 -28h-576q-40 0 -68 28t-28 68v416h-128v-1280h128v416q0 40 28 68t68 28h832q40 0 68 -28t28 -68v-416zM896 928v320q0 13 -9.5 22.5t-22.5 9.5 + h-192q-13 0 -22.5 -9.5t-9.5 -22.5v-320q0 -13 9.5 -22.5t22.5 -9.5h192q13 0 22.5 9.5t9.5 22.5zM1536 896v-928q0 -40 -28 -68t-68 -28h-1344q-40 0 -68 28t-28 68v1344q0 40 28 68t68 28h928q40 0 88 -20t76 -48l280 -280q28 -28 48 -76t20 -88z" /> + <glyph glyph-name="sign_blank" unicode="" + d="M1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="reorder" unicode="" + d="M1536 192v-128q0 -26 -19 -45t-45 -19h-1408q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h1408q26 0 45 -19t19 -45zM1536 704v-128q0 -26 -19 -45t-45 -19h-1408q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h1408q26 0 45 -19t19 -45zM1536 1216v-128q0 -26 -19 -45 + t-45 -19h-1408q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h1408q26 0 45 -19t19 -45z" /> + <glyph glyph-name="ul" unicode="" horiz-adv-x="1792" + d="M384 128q0 -80 -56 -136t-136 -56t-136 56t-56 136t56 136t136 56t136 -56t56 -136zM384 640q0 -80 -56 -136t-136 -56t-136 56t-56 136t56 136t136 56t136 -56t56 -136zM1792 224v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-1216q-13 0 -22.5 9.5t-9.5 22.5v192q0 13 9.5 22.5 + t22.5 9.5h1216q13 0 22.5 -9.5t9.5 -22.5zM384 1152q0 -80 -56 -136t-136 -56t-136 56t-56 136t56 136t136 56t136 -56t56 -136zM1792 736v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-1216q-13 0 -22.5 9.5t-9.5 22.5v192q0 13 9.5 22.5t22.5 9.5h1216q13 0 22.5 -9.5t9.5 -22.5z + M1792 1248v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-1216q-13 0 -22.5 9.5t-9.5 22.5v192q0 13 9.5 22.5t22.5 9.5h1216q13 0 22.5 -9.5t9.5 -22.5z" /> + <glyph glyph-name="ol" unicode="" horiz-adv-x="1792" + d="M381 -84q0 -80 -54.5 -126t-135.5 -46q-106 0 -172 66l57 88q49 -45 106 -45q29 0 50.5 14.5t21.5 42.5q0 64 -105 56l-26 56q8 10 32.5 43.5t42.5 54t37 38.5v1q-16 0 -48.5 -1t-48.5 -1v-53h-106v152h333v-88l-95 -115q51 -12 81 -49t30 -88zM383 543v-159h-362 + q-6 36 -6 54q0 51 23.5 93t56.5 68t66 47.5t56.5 43.5t23.5 45q0 25 -14.5 38.5t-39.5 13.5q-46 0 -81 -58l-85 59q24 51 71.5 79.5t105.5 28.5q73 0 123 -41.5t50 -112.5q0 -50 -34 -91.5t-75 -64.5t-75.5 -50.5t-35.5 -52.5h127v60h105zM1792 224v-192q0 -13 -9.5 -22.5 + t-22.5 -9.5h-1216q-13 0 -22.5 9.5t-9.5 22.5v192q0 14 9 23t23 9h1216q13 0 22.5 -9.5t9.5 -22.5zM384 1123v-99h-335v99h107q0 41 0.5 121.5t0.5 121.5v12h-2q-8 -17 -50 -54l-71 76l136 127h106v-404h108zM1792 736v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-1216 + q-13 0 -22.5 9.5t-9.5 22.5v192q0 14 9 23t23 9h1216q13 0 22.5 -9.5t9.5 -22.5zM1792 1248v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-1216q-13 0 -22.5 9.5t-9.5 22.5v192q0 13 9.5 22.5t22.5 9.5h1216q13 0 22.5 -9.5t9.5 -22.5z" /> + <glyph glyph-name="strikethrough" unicode="" horiz-adv-x="1792" + d="M1760 640q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-1728q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h1728zM483 704q-28 35 -51 80q-48 98 -48 188q0 181 134 309q133 127 393 127q50 0 167 -19q66 -12 177 -48q10 -38 21 -118q14 -123 14 -183q0 -18 -5 -45l-12 -3l-84 6 + l-14 2q-50 149 -103 205q-88 91 -210 91q-114 0 -182 -59q-67 -58 -67 -146q0 -73 66 -140t279 -129q69 -20 173 -66q58 -28 95 -52h-743zM990 448h411q7 -39 7 -92q0 -111 -41 -212q-23 -56 -71 -104q-37 -35 -109 -81q-80 -48 -153 -66q-80 -21 -203 -21q-114 0 -195 23 + l-140 40q-57 16 -72 28q-8 8 -8 22v13q0 108 -2 156q-1 30 0 68l2 37v44l102 2q15 -34 30 -71t22.5 -56t12.5 -27q35 -57 80 -94q43 -36 105 -57q59 -22 132 -22q64 0 139 27q77 26 122 86q47 61 47 129q0 84 -81 157q-34 29 -137 71z" /> + <glyph glyph-name="underline" unicode="" + d="M48 1313q-37 2 -45 4l-3 88q13 1 40 1q60 0 112 -4q132 -7 166 -7q86 0 168 3q116 4 146 5q56 0 86 2l-1 -14l2 -64v-9q-60 -9 -124 -9q-60 0 -79 -25q-13 -14 -13 -132q0 -13 0.5 -32.5t0.5 -25.5l1 -229l14 -280q6 -124 51 -202q35 -59 96 -92q88 -47 177 -47 + q104 0 191 28q56 18 99 51q48 36 65 64q36 56 53 114q21 73 21 229q0 79 -3.5 128t-11 122.5t-13.5 159.5l-4 59q-5 67 -24 88q-34 35 -77 34l-100 -2l-14 3l2 86h84l205 -10q76 -3 196 10l18 -2q6 -38 6 -51q0 -7 -4 -31q-45 -12 -84 -13q-73 -11 -79 -17q-15 -15 -15 -41 + q0 -7 1.5 -27t1.5 -31q8 -19 22 -396q6 -195 -15 -304q-15 -76 -41 -122q-38 -65 -112 -123q-75 -57 -182 -89q-109 -33 -255 -33q-167 0 -284 46q-119 47 -179 122q-61 76 -83 195q-16 80 -16 237v333q0 188 -17 213q-25 36 -147 39zM1536 -96v64q0 14 -9 23t-23 9h-1472 + q-14 0 -23 -9t-9 -23v-64q0 -14 9 -23t23 -9h1472q14 0 23 9t9 23z" /> + <glyph glyph-name="table" unicode="" horiz-adv-x="1664" + d="M512 160v192q0 14 -9 23t-23 9h-320q-14 0 -23 -9t-9 -23v-192q0 -14 9 -23t23 -9h320q14 0 23 9t9 23zM512 544v192q0 14 -9 23t-23 9h-320q-14 0 -23 -9t-9 -23v-192q0 -14 9 -23t23 -9h320q14 0 23 9t9 23zM1024 160v192q0 14 -9 23t-23 9h-320q-14 0 -23 -9t-9 -23 + v-192q0 -14 9 -23t23 -9h320q14 0 23 9t9 23zM512 928v192q0 14 -9 23t-23 9h-320q-14 0 -23 -9t-9 -23v-192q0 -14 9 -23t23 -9h320q14 0 23 9t9 23zM1024 544v192q0 14 -9 23t-23 9h-320q-14 0 -23 -9t-9 -23v-192q0 -14 9 -23t23 -9h320q14 0 23 9t9 23zM1536 160v192 + q0 14 -9 23t-23 9h-320q-14 0 -23 -9t-9 -23v-192q0 -14 9 -23t23 -9h320q14 0 23 9t9 23zM1024 928v192q0 14 -9 23t-23 9h-320q-14 0 -23 -9t-9 -23v-192q0 -14 9 -23t23 -9h320q14 0 23 9t9 23zM1536 544v192q0 14 -9 23t-23 9h-320q-14 0 -23 -9t-9 -23v-192 + q0 -14 9 -23t23 -9h320q14 0 23 9t9 23zM1536 928v192q0 14 -9 23t-23 9h-320q-14 0 -23 -9t-9 -23v-192q0 -14 9 -23t23 -9h320q14 0 23 9t9 23zM1664 1248v-1088q0 -66 -47 -113t-113 -47h-1344q-66 0 -113 47t-47 113v1088q0 66 47 113t113 47h1344q66 0 113 -47t47 -113 + z" /> + <glyph glyph-name="magic" unicode="" horiz-adv-x="1664" + d="M1190 955l293 293l-107 107l-293 -293zM1637 1248q0 -27 -18 -45l-1286 -1286q-18 -18 -45 -18t-45 18l-198 198q-18 18 -18 45t18 45l1286 1286q18 18 45 18t45 -18l198 -198q18 -18 18 -45zM286 1438l98 -30l-98 -30l-30 -98l-30 98l-98 30l98 30l30 98zM636 1276 + l196 -60l-196 -60l-60 -196l-60 196l-196 60l196 60l60 196zM1566 798l98 -30l-98 -30l-30 -98l-30 98l-98 30l98 30l30 98zM926 1438l98 -30l-98 -30l-30 -98l-30 98l-98 30l98 30l30 98z" /> + <glyph glyph-name="truck" unicode="" horiz-adv-x="1792" + d="M640 128q0 52 -38 90t-90 38t-90 -38t-38 -90t38 -90t90 -38t90 38t38 90zM256 640h384v256h-158q-13 0 -22 -9l-195 -195q-9 -9 -9 -22v-30zM1536 128q0 52 -38 90t-90 38t-90 -38t-38 -90t38 -90t90 -38t90 38t38 90zM1792 1216v-1024q0 -15 -4 -26.5t-13.5 -18.5 + t-16.5 -11.5t-23.5 -6t-22.5 -2t-25.5 0t-22.5 0.5q0 -106 -75 -181t-181 -75t-181 75t-75 181h-384q0 -106 -75 -181t-181 -75t-181 75t-75 181h-64q-3 0 -22.5 -0.5t-25.5 0t-22.5 2t-23.5 6t-16.5 11.5t-13.5 18.5t-4 26.5q0 26 19 45t45 19v320q0 8 -0.5 35t0 38 + t2.5 34.5t6.5 37t14 30.5t22.5 30l198 198q19 19 50.5 32t58.5 13h160v192q0 26 19 45t45 19h1024q26 0 45 -19t19 -45z" /> + <glyph glyph-name="pinterest" unicode="" + d="M1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103q-111 0 -218 32q59 93 78 164q9 34 54 211q20 -39 73 -67.5t114 -28.5q121 0 216 68.5t147 188.5t52 270q0 114 -59.5 214t-172.5 163t-255 63q-105 0 -196 -29t-154.5 -77t-109 -110.5t-67 -129.5t-21.5 -134 + q0 -104 40 -183t117 -111q30 -12 38 20q2 7 8 31t8 30q6 23 -11 43q-51 61 -51 151q0 151 104.5 259.5t273.5 108.5q151 0 235.5 -82t84.5 -213q0 -170 -68.5 -289t-175.5 -119q-61 0 -98 43.5t-23 104.5q8 35 26.5 93.5t30 103t11.5 75.5q0 50 -27 83t-77 33 + q-62 0 -105 -57t-43 -142q0 -73 25 -122l-99 -418q-17 -70 -13 -177q-206 91 -333 281t-127 423q0 209 103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="pinterest_sign" unicode="" + d="M1248 1408q119 0 203.5 -84.5t84.5 -203.5v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-725q85 122 108 210q9 34 53 209q21 -39 73.5 -67t112.5 -28q181 0 295.5 147.5t114.5 373.5q0 84 -35 162.5t-96.5 139t-152.5 97t-197 36.5q-104 0 -194.5 -28.5t-153 -76.5 + t-107.5 -109.5t-66.5 -128t-21.5 -132.5q0 -102 39.5 -180t116.5 -110q13 -5 23.5 0t14.5 19q10 44 15 61q6 23 -11 42q-50 62 -50 150q0 150 103.5 256.5t270.5 106.5q149 0 232.5 -81t83.5 -210q0 -168 -67.5 -286t-173.5 -118q-60 0 -97 43.5t-23 103.5q8 34 26.5 92.5 + t29.5 102t11 74.5q0 49 -26.5 81.5t-75.5 32.5q-61 0 -103.5 -56.5t-42.5 -139.5q0 -72 24 -121l-98 -414q-24 -100 -7 -254h-183q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960z" /> + <glyph glyph-name="google_plus_sign" unicode="" + d="M917 631q0 26 -6 64h-362v-132h217q-3 -24 -16.5 -50t-37.5 -53t-66.5 -44.5t-96.5 -17.5q-99 0 -169 71t-70 171t70 171t169 71q92 0 153 -59l104 101q-108 100 -257 100q-160 0 -272 -112.5t-112 -271.5t112 -271.5t272 -112.5q165 0 266.5 105t101.5 270zM1262 585 + h109v110h-109v110h-110v-110h-110v-110h110v-110h110v110zM1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="google_plus" unicode="" horiz-adv-x="2304" + d="M1437 623q0 -208 -87 -370.5t-248 -254t-369 -91.5q-149 0 -285 58t-234 156t-156 234t-58 285t58 285t156 234t234 156t285 58q286 0 491 -192l-199 -191q-117 113 -292 113q-123 0 -227.5 -62t-165.5 -168.5t-61 -232.5t61 -232.5t165.5 -168.5t227.5 -62 + q83 0 152.5 23t114.5 57.5t78.5 78.5t49 83t21.5 74h-416v252h692q12 -63 12 -122zM2304 745v-210h-209v-209h-210v209h-209v210h209v209h210v-209h209z" /> + <glyph glyph-name="money" unicode="" horiz-adv-x="1920" + d="M768 384h384v96h-128v448h-114l-148 -137l77 -80q42 37 55 57h2v-288h-128v-96zM1280 640q0 -70 -21 -142t-59.5 -134t-101.5 -101t-138 -39t-138 39t-101.5 101t-59.5 134t-21 142t21 142t59.5 134t101.5 101t138 39t138 -39t101.5 -101t59.5 -134t21 -142zM1792 384 + v512q-106 0 -181 75t-75 181h-1152q0 -106 -75 -181t-181 -75v-512q106 0 181 -75t75 -181h1152q0 106 75 181t181 75zM1920 1216v-1152q0 -26 -19 -45t-45 -19h-1792q-26 0 -45 19t-19 45v1152q0 26 19 45t45 19h1792q26 0 45 -19t19 -45z" /> + <glyph glyph-name="caret_down" unicode="" horiz-adv-x="1024" + d="M1024 832q0 -26 -19 -45l-448 -448q-19 -19 -45 -19t-45 19l-448 448q-19 19 -19 45t19 45t45 19h896q26 0 45 -19t19 -45z" /> + <glyph glyph-name="caret_up" unicode="" horiz-adv-x="1024" + d="M1024 320q0 -26 -19 -45t-45 -19h-896q-26 0 -45 19t-19 45t19 45l448 448q19 19 45 19t45 -19l448 -448q19 -19 19 -45z" /> + <glyph glyph-name="caret_left" unicode="" horiz-adv-x="640" + d="M640 1088v-896q0 -26 -19 -45t-45 -19t-45 19l-448 448q-19 19 -19 45t19 45l448 448q19 19 45 19t45 -19t19 -45z" /> + <glyph glyph-name="caret_right" unicode="" horiz-adv-x="640" + d="M576 640q0 -26 -19 -45l-448 -448q-19 -19 -45 -19t-45 19t-19 45v896q0 26 19 45t45 19t45 -19l448 -448q19 -19 19 -45z" /> + <glyph glyph-name="columns" unicode="" horiz-adv-x="1664" + d="M160 0h608v1152h-640v-1120q0 -13 9.5 -22.5t22.5 -9.5zM1536 32v1120h-640v-1152h608q13 0 22.5 9.5t9.5 22.5zM1664 1248v-1216q0 -66 -47 -113t-113 -47h-1344q-66 0 -113 47t-47 113v1216q0 66 47 113t113 47h1344q66 0 113 -47t47 -113z" /> + <glyph glyph-name="sort" unicode="" horiz-adv-x="1024" + d="M1024 448q0 -26 -19 -45l-448 -448q-19 -19 -45 -19t-45 19l-448 448q-19 19 -19 45t19 45t45 19h896q26 0 45 -19t19 -45zM1024 832q0 -26 -19 -45t-45 -19h-896q-26 0 -45 19t-19 45t19 45l448 448q19 19 45 19t45 -19l448 -448q19 -19 19 -45z" /> + <glyph glyph-name="sort_down" unicode="" horiz-adv-x="1024" + d="M1024 448q0 -26 -19 -45l-448 -448q-19 -19 -45 -19t-45 19l-448 448q-19 19 -19 45t19 45t45 19h896q26 0 45 -19t19 -45z" /> + <glyph glyph-name="sort_up" unicode="" horiz-adv-x="1024" + d="M1024 832q0 -26 -19 -45t-45 -19h-896q-26 0 -45 19t-19 45t19 45l448 448q19 19 45 19t45 -19l448 -448q19 -19 19 -45z" /> + <glyph glyph-name="envelope_alt" unicode="" horiz-adv-x="1792" + d="M1792 826v-794q0 -66 -47 -113t-113 -47h-1472q-66 0 -113 47t-47 113v794q44 -49 101 -87q362 -246 497 -345q57 -42 92.5 -65.5t94.5 -48t110 -24.5h1h1q51 0 110 24.5t94.5 48t92.5 65.5q170 123 498 345q57 39 100 87zM1792 1120q0 -79 -49 -151t-122 -123 + q-376 -261 -468 -325q-10 -7 -42.5 -30.5t-54 -38t-52 -32.5t-57.5 -27t-50 -9h-1h-1q-23 0 -50 9t-57.5 27t-52 32.5t-54 38t-42.5 30.5q-91 64 -262 182.5t-205 142.5q-62 42 -117 115.5t-55 136.5q0 78 41.5 130t118.5 52h1472q65 0 112.5 -47t47.5 -113z" /> + <glyph glyph-name="linkedin" unicode="" + d="M349 911v-991h-330v991h330zM370 1217q1 -73 -50.5 -122t-135.5 -49h-2q-82 0 -132 49t-50 122q0 74 51.5 122.5t134.5 48.5t133 -48.5t51 -122.5zM1536 488v-568h-329v530q0 105 -40.5 164.5t-126.5 59.5q-63 0 -105.5 -34.5t-63.5 -85.5q-11 -30 -11 -81v-553h-329 + q2 399 2 647t-1 296l-1 48h329v-144h-2q20 32 41 56t56.5 52t87 43.5t114.5 15.5q171 0 275 -113.5t104 -332.5z" /> + <glyph glyph-name="undo" unicode="" + d="M1536 640q0 -156 -61 -298t-164 -245t-245 -164t-298 -61q-172 0 -327 72.5t-264 204.5q-7 10 -6.5 22.5t8.5 20.5l137 138q10 9 25 9q16 -2 23 -12q73 -95 179 -147t225 -52q104 0 198.5 40.5t163.5 109.5t109.5 163.5t40.5 198.5t-40.5 198.5t-109.5 163.5 + t-163.5 109.5t-198.5 40.5q-98 0 -188 -35.5t-160 -101.5l137 -138q31 -30 14 -69q-17 -40 -59 -40h-448q-26 0 -45 19t-19 45v448q0 42 40 59q39 17 69 -14l130 -129q107 101 244.5 156.5t284.5 55.5q156 0 298 -61t245 -164t164 -245t61 -298z" /> + <glyph glyph-name="legal" unicode="" horiz-adv-x="1792" + d="M1771 0q0 -53 -37 -90l-107 -108q-39 -37 -91 -37q-53 0 -90 37l-363 364q-38 36 -38 90q0 53 43 96l-256 256l-126 -126q-14 -14 -34 -14t-34 14q2 -2 12.5 -12t12.5 -13t10 -11.5t10 -13.5t6 -13.5t5.5 -16.5t1.5 -18q0 -38 -28 -68q-3 -3 -16.5 -18t-19 -20.5 + t-18.5 -16.5t-22 -15.5t-22 -9t-26 -4.5q-40 0 -68 28l-408 408q-28 28 -28 68q0 13 4.5 26t9 22t15.5 22t16.5 18.5t20.5 19t18 16.5q30 28 68 28q10 0 18 -1.5t16.5 -5.5t13.5 -6t13.5 -10t11.5 -10t13 -12.5t12 -12.5q-14 14 -14 34t14 34l348 348q14 14 34 14t34 -14 + q-2 2 -12.5 12t-12.5 13t-10 11.5t-10 13.5t-6 13.5t-5.5 16.5t-1.5 18q0 38 28 68q3 3 16.5 18t19 20.5t18.5 16.5t22 15.5t22 9t26 4.5q40 0 68 -28l408 -408q28 -28 28 -68q0 -13 -4.5 -26t-9 -22t-15.5 -22t-16.5 -18.5t-20.5 -19t-18 -16.5q-30 -28 -68 -28 + q-10 0 -18 1.5t-16.5 5.5t-13.5 6t-13.5 10t-11.5 10t-13 12.5t-12 12.5q14 -14 14 -34t-14 -34l-126 -126l256 -256q43 43 96 43q52 0 91 -37l363 -363q37 -39 37 -91z" /> + <glyph glyph-name="dashboard" unicode="" horiz-adv-x="1792" + d="M384 384q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM576 832q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM1004 351l101 382q6 26 -7.5 48.5t-38.5 29.5 + t-48 -6.5t-30 -39.5l-101 -382q-60 -5 -107 -43.5t-63 -98.5q-20 -77 20 -146t117 -89t146 20t89 117q16 60 -6 117t-72 91zM1664 384q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM1024 1024q0 53 -37.5 90.5 + t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM1472 832q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM1792 384q0 -261 -141 -483q-19 -29 -54 -29h-1402q-35 0 -54 29 + q-141 221 -141 483q0 182 71 348t191 286t286 191t348 71t348 -71t286 -191t191 -286t71 -348z" /> + <glyph glyph-name="comment_alt" unicode="" horiz-adv-x="1792" + d="M896 1152q-204 0 -381.5 -69.5t-282 -187.5t-104.5 -255q0 -112 71.5 -213.5t201.5 -175.5l87 -50l-27 -96q-24 -91 -70 -172q152 63 275 171l43 38l57 -6q69 -8 130 -8q204 0 381.5 69.5t282 187.5t104.5 255t-104.5 255t-282 187.5t-381.5 69.5zM1792 640 + q0 -174 -120 -321.5t-326 -233t-450 -85.5q-70 0 -145 8q-198 -175 -460 -242q-49 -14 -114 -22h-5q-15 0 -27 10.5t-16 27.5v1q-3 4 -0.5 12t2 10t4.5 9.5l6 9t7 8.5t8 9q7 8 31 34.5t34.5 38t31 39.5t32.5 51t27 59t26 76q-157 89 -247.5 220t-90.5 281q0 174 120 321.5 + t326 233t450 85.5t450 -85.5t326 -233t120 -321.5z" /> + <glyph glyph-name="comments_alt" unicode="" horiz-adv-x="1792" + d="M704 1152q-153 0 -286 -52t-211.5 -141t-78.5 -191q0 -82 53 -158t149 -132l97 -56l-35 -84q34 20 62 39l44 31l53 -10q78 -14 153 -14q153 0 286 52t211.5 141t78.5 191t-78.5 191t-211.5 141t-286 52zM704 1280q191 0 353.5 -68.5t256.5 -186.5t94 -257t-94 -257 + t-256.5 -186.5t-353.5 -68.5q-86 0 -176 16q-124 -88 -278 -128q-36 -9 -86 -16h-3q-11 0 -20.5 8t-11.5 21q-1 3 -1 6.5t0.5 6.5t2 6l2.5 5t3.5 5.5t4 5t4.5 5t4 4.5q5 6 23 25t26 29.5t22.5 29t25 38.5t20.5 44q-124 72 -195 177t-71 224q0 139 94 257t256.5 186.5 + t353.5 68.5zM1526 111q10 -24 20.5 -44t25 -38.5t22.5 -29t26 -29.5t23 -25q1 -1 4 -4.5t4.5 -5t4 -5t3.5 -5.5l2.5 -5t2 -6t0.5 -6.5t-1 -6.5q-3 -14 -13 -22t-22 -7q-50 7 -86 16q-154 40 -278 128q-90 -16 -176 -16q-271 0 -472 132q58 -4 88 -4q161 0 309 45t264 129 + q125 92 192 212t67 254q0 77 -23 152q129 -71 204 -178t75 -230q0 -120 -71 -224.5t-195 -176.5z" /> + <glyph glyph-name="bolt" unicode="" horiz-adv-x="896" + d="M885 970q18 -20 7 -44l-540 -1157q-13 -25 -42 -25q-4 0 -14 2q-17 5 -25.5 19t-4.5 30l197 808l-406 -101q-4 -1 -12 -1q-18 0 -31 11q-18 15 -13 39l201 825q4 14 16 23t28 9h328q19 0 32 -12.5t13 -29.5q0 -8 -5 -18l-171 -463l396 98q8 2 12 2q19 0 34 -15z" /> + <glyph glyph-name="sitemap" unicode="" horiz-adv-x="1792" + d="M1792 288v-320q0 -40 -28 -68t-68 -28h-320q-40 0 -68 28t-28 68v320q0 40 28 68t68 28h96v192h-512v-192h96q40 0 68 -28t28 -68v-320q0 -40 -28 -68t-68 -28h-320q-40 0 -68 28t-28 68v320q0 40 28 68t68 28h96v192h-512v-192h96q40 0 68 -28t28 -68v-320 + q0 -40 -28 -68t-68 -28h-320q-40 0 -68 28t-28 68v320q0 40 28 68t68 28h96v192q0 52 38 90t90 38h512v192h-96q-40 0 -68 28t-28 68v320q0 40 28 68t68 28h320q40 0 68 -28t28 -68v-320q0 -40 -28 -68t-68 -28h-96v-192h512q52 0 90 -38t38 -90v-192h96q40 0 68 -28t28 -68 + z" /> + <glyph glyph-name="umbrella" unicode="" horiz-adv-x="1664" + d="M896 708v-580q0 -104 -76 -180t-180 -76t-180 76t-76 180q0 26 19 45t45 19t45 -19t19 -45q0 -50 39 -89t89 -39t89 39t39 89v580q33 11 64 11t64 -11zM1664 681q0 -13 -9.5 -22.5t-22.5 -9.5q-11 0 -23 10q-49 46 -93 69t-102 23q-68 0 -128 -37t-103 -97 + q-7 -10 -17.5 -28t-14.5 -24q-11 -17 -28 -17q-18 0 -29 17q-4 6 -14.5 24t-17.5 28q-43 60 -102.5 97t-127.5 37t-127.5 -37t-102.5 -97q-7 -10 -17.5 -28t-14.5 -24q-11 -17 -29 -17q-17 0 -28 17q-4 6 -14.5 24t-17.5 28q-43 60 -103 97t-128 37q-58 0 -102 -23t-93 -69 + q-12 -10 -23 -10q-13 0 -22.5 9.5t-9.5 22.5q0 5 1 7q45 183 172.5 319.5t298 204.5t360.5 68q140 0 274.5 -40t246.5 -113.5t194.5 -187t115.5 -251.5q1 -2 1 -7zM896 1408v-98q-42 2 -64 2t-64 -2v98q0 26 19 45t45 19t45 -19t19 -45z" /> + <glyph glyph-name="paste" unicode="" horiz-adv-x="1792" + d="M768 -128h896v640h-416q-40 0 -68 28t-28 68v416h-384v-1152zM1024 1312v64q0 13 -9.5 22.5t-22.5 9.5h-704q-13 0 -22.5 -9.5t-9.5 -22.5v-64q0 -13 9.5 -22.5t22.5 -9.5h704q13 0 22.5 9.5t9.5 22.5zM1280 640h299l-299 299v-299zM1792 512v-672q0 -40 -28 -68t-68 -28 + h-960q-40 0 -68 28t-28 68v160h-544q-40 0 -68 28t-28 68v1344q0 40 28 68t68 28h1088q40 0 68 -28t28 -68v-328q21 -13 36 -28l408 -408q28 -28 48 -76t20 -88z" /> + <glyph glyph-name="light_bulb" unicode="" horiz-adv-x="1024" + d="M736 960q0 -13 -9.5 -22.5t-22.5 -9.5t-22.5 9.5t-9.5 22.5q0 46 -54 71t-106 25q-13 0 -22.5 9.5t-9.5 22.5t9.5 22.5t22.5 9.5q50 0 99.5 -16t87 -54t37.5 -90zM896 960q0 72 -34.5 134t-90 101.5t-123 62t-136.5 22.5t-136.5 -22.5t-123 -62t-90 -101.5t-34.5 -134 + q0 -101 68 -180q10 -11 30.5 -33t30.5 -33q128 -153 141 -298h228q13 145 141 298q10 11 30.5 33t30.5 33q68 79 68 180zM1024 960q0 -155 -103 -268q-45 -49 -74.5 -87t-59.5 -95.5t-34 -107.5q47 -28 47 -82q0 -37 -25 -64q25 -27 25 -64q0 -52 -45 -81q13 -23 13 -47 + q0 -46 -31.5 -71t-77.5 -25q-20 -44 -60 -70t-87 -26t-87 26t-60 70q-46 0 -77.5 25t-31.5 71q0 24 13 47q-45 29 -45 81q0 37 25 64q-25 27 -25 64q0 54 47 82q-4 50 -34 107.5t-59.5 95.5t-74.5 87q-103 113 -103 268q0 99 44.5 184.5t117 142t164 89t186.5 32.5 + t186.5 -32.5t164 -89t117 -142t44.5 -184.5z" /> + <glyph glyph-name="exchange" unicode="" horiz-adv-x="1792" + d="M1792 352v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-1376v-192q0 -13 -9.5 -22.5t-22.5 -9.5q-12 0 -24 10l-319 320q-9 9 -9 22q0 14 9 23l320 320q9 9 23 9q13 0 22.5 -9.5t9.5 -22.5v-192h1376q13 0 22.5 -9.5t9.5 -22.5zM1792 896q0 -14 -9 -23l-320 -320q-9 -9 -23 -9 + q-13 0 -22.5 9.5t-9.5 22.5v192h-1376q-13 0 -22.5 9.5t-9.5 22.5v192q0 13 9.5 22.5t22.5 9.5h1376v192q0 14 9 23t23 9q12 0 24 -10l319 -319q9 -9 9 -23z" /> + <glyph glyph-name="cloud_download" unicode="" horiz-adv-x="1920" + d="M1280 608q0 14 -9 23t-23 9h-224v352q0 13 -9.5 22.5t-22.5 9.5h-192q-13 0 -22.5 -9.5t-9.5 -22.5v-352h-224q-13 0 -22.5 -9.5t-9.5 -22.5q0 -14 9 -23l352 -352q9 -9 23 -9t23 9l351 351q10 12 10 24zM1920 384q0 -159 -112.5 -271.5t-271.5 -112.5h-1088 + q-185 0 -316.5 131.5t-131.5 316.5q0 130 70 240t188 165q-2 30 -2 43q0 212 150 362t362 150q156 0 285.5 -87t188.5 -231q71 62 166 62q106 0 181 -75t75 -181q0 -76 -41 -138q130 -31 213.5 -135.5t83.5 -238.5z" /> + <glyph glyph-name="cloud_upload" unicode="" horiz-adv-x="1920" + d="M1280 672q0 14 -9 23l-352 352q-9 9 -23 9t-23 -9l-351 -351q-10 -12 -10 -24q0 -14 9 -23t23 -9h224v-352q0 -13 9.5 -22.5t22.5 -9.5h192q13 0 22.5 9.5t9.5 22.5v352h224q13 0 22.5 9.5t9.5 22.5zM1920 384q0 -159 -112.5 -271.5t-271.5 -112.5h-1088 + q-185 0 -316.5 131.5t-131.5 316.5q0 130 70 240t188 165q-2 30 -2 43q0 212 150 362t362 150q156 0 285.5 -87t188.5 -231q71 62 166 62q106 0 181 -75t75 -181q0 -76 -41 -138q130 -31 213.5 -135.5t83.5 -238.5z" /> + <glyph glyph-name="user_md" unicode="" horiz-adv-x="1408" + d="M384 192q0 -26 -19 -45t-45 -19t-45 19t-19 45t19 45t45 19t45 -19t19 -45zM1408 131q0 -121 -73 -190t-194 -69h-874q-121 0 -194 69t-73 190q0 68 5.5 131t24 138t47.5 132.5t81 103t120 60.5q-22 -52 -22 -120v-203q-58 -20 -93 -70t-35 -111q0 -80 56 -136t136 -56 + t136 56t56 136q0 61 -35.5 111t-92.5 70v203q0 62 25 93q132 -104 295 -104t295 104q25 -31 25 -93v-64q-106 0 -181 -75t-75 -181v-89q-32 -29 -32 -71q0 -40 28 -68t68 -28t68 28t28 68q0 42 -32 71v89q0 52 38 90t90 38t90 -38t38 -90v-89q-32 -29 -32 -71q0 -40 28 -68 + t68 -28t68 28t28 68q0 42 -32 71v89q0 68 -34.5 127.5t-93.5 93.5q0 10 0.5 42.5t0 48t-2.5 41.5t-7 47t-13 40q68 -15 120 -60.5t81 -103t47.5 -132.5t24 -138t5.5 -131zM1088 1024q0 -159 -112.5 -271.5t-271.5 -112.5t-271.5 112.5t-112.5 271.5t112.5 271.5t271.5 112.5 + t271.5 -112.5t112.5 -271.5z" /> + <glyph glyph-name="stethoscope" unicode="" horiz-adv-x="1408" + d="M1280 832q0 26 -19 45t-45 19t-45 -19t-19 -45t19 -45t45 -19t45 19t19 45zM1408 832q0 -62 -35.5 -111t-92.5 -70v-395q0 -159 -131.5 -271.5t-316.5 -112.5t-316.5 112.5t-131.5 271.5v132q-164 20 -274 128t-110 252v512q0 26 19 45t45 19q6 0 16 -2q17 30 47 48 + t65 18q53 0 90.5 -37.5t37.5 -90.5t-37.5 -90.5t-90.5 -37.5q-33 0 -64 18v-402q0 -106 94 -181t226 -75t226 75t94 181v402q-31 -18 -64 -18q-53 0 -90.5 37.5t-37.5 90.5t37.5 90.5t90.5 37.5q35 0 65 -18t47 -48q10 2 16 2q26 0 45 -19t19 -45v-512q0 -144 -110 -252 + t-274 -128v-132q0 -106 94 -181t226 -75t226 75t94 181v395q-57 21 -92.5 70t-35.5 111q0 80 56 136t136 56t136 -56t56 -136z" /> + <glyph glyph-name="suitcase" unicode="" horiz-adv-x="1792" + d="M640 1152h512v128h-512v-128zM288 1152v-1280h-64q-92 0 -158 66t-66 158v832q0 92 66 158t158 66h64zM1408 1152v-1280h-1024v1280h128v160q0 40 28 68t68 28h576q40 0 68 -28t28 -68v-160h128zM1792 928v-832q0 -92 -66 -158t-158 -66h-64v1280h64q92 0 158 -66 + t66 -158z" /> + <glyph glyph-name="bell_alt" unicode="" horiz-adv-x="1792" + d="M912 -160q0 16 -16 16q-59 0 -101.5 42.5t-42.5 101.5q0 16 -16 16t-16 -16q0 -73 51.5 -124.5t124.5 -51.5q16 0 16 16zM1728 128q0 -52 -38 -90t-90 -38h-448q0 -106 -75 -181t-181 -75t-181 75t-75 181h-448q-52 0 -90 38t-38 90q50 42 91 88t85 119.5t74.5 158.5 + t50 206t19.5 260q0 152 117 282.5t307 158.5q-8 19 -8 39q0 40 28 68t68 28t68 -28t28 -68q0 -20 -8 -39q190 -28 307 -158.5t117 -282.5q0 -139 19.5 -260t50 -206t74.5 -158.5t85 -119.5t91 -88z" /> + <glyph glyph-name="coffee" unicode="" horiz-adv-x="1920" + d="M1664 896q0 80 -56 136t-136 56h-64v-384h64q80 0 136 56t56 136zM0 128h1792q0 -106 -75 -181t-181 -75h-1280q-106 0 -181 75t-75 181zM1856 896q0 -159 -112.5 -271.5t-271.5 -112.5h-64v-32q0 -92 -66 -158t-158 -66h-704q-92 0 -158 66t-66 158v736q0 26 19 45 + t45 19h1152q159 0 271.5 -112.5t112.5 -271.5z" /> + <glyph glyph-name="food" unicode="" horiz-adv-x="1408" + d="M640 1472v-640q0 -61 -35.5 -111t-92.5 -70v-779q0 -52 -38 -90t-90 -38h-128q-52 0 -90 38t-38 90v779q-57 20 -92.5 70t-35.5 111v640q0 26 19 45t45 19t45 -19t19 -45v-416q0 -26 19 -45t45 -19t45 19t19 45v416q0 26 19 45t45 19t45 -19t19 -45v-416q0 -26 19 -45 + t45 -19t45 19t19 45v416q0 26 19 45t45 19t45 -19t19 -45zM1408 1472v-1600q0 -52 -38 -90t-90 -38h-128q-52 0 -90 38t-38 90v512h-224q-13 0 -22.5 9.5t-9.5 22.5v800q0 132 94 226t226 94h256q26 0 45 -19t19 -45z" /> + <glyph glyph-name="file_text_alt" unicode="" + d="M1468 1156q28 -28 48 -76t20 -88v-1152q0 -40 -28 -68t-68 -28h-1344q-40 0 -68 28t-28 68v1600q0 40 28 68t68 28h896q40 0 88 -20t76 -48zM1024 1400v-376h376q-10 29 -22 41l-313 313q-12 12 -41 22zM1408 -128v1024h-416q-40 0 -68 28t-28 68v416h-768v-1536h1280z + M384 736q0 14 9 23t23 9h704q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-704q-14 0 -23 9t-9 23v64zM1120 512q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-704q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h704zM1120 256q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-704 + q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h704z" /> + <glyph glyph-name="building" unicode="" horiz-adv-x="1408" + d="M384 224v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5zM384 480v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5z + M640 480v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5zM384 736v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5z + M1152 224v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5zM896 480v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5z + M640 736v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5zM384 992v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5z + M1152 480v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5zM896 736v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5z + M640 992v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5zM384 1248v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5z + M1152 736v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5zM896 992v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5z + M640 1248v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5zM1152 992v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5z + M896 1248v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5zM1152 1248v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5z + M896 -128h384v1536h-1152v-1536h384v224q0 13 9.5 22.5t22.5 9.5h320q13 0 22.5 -9.5t9.5 -22.5v-224zM1408 1472v-1664q0 -26 -19 -45t-45 -19h-1280q-26 0 -45 19t-19 45v1664q0 26 19 45t45 19h1280q26 0 45 -19t19 -45z" /> + <glyph glyph-name="hospital" unicode="" horiz-adv-x="1408" + d="M384 224v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5zM384 480v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5z + M640 480v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5zM384 736v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5z + M1152 224v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5zM896 480v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5z + M640 736v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5zM1152 480v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5z + M896 736v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5zM1152 736v-64q0 -13 -9.5 -22.5t-22.5 -9.5h-64q-13 0 -22.5 9.5t-9.5 22.5v64q0 13 9.5 22.5t22.5 9.5h64q13 0 22.5 -9.5t9.5 -22.5z + M896 -128h384v1152h-256v-32q0 -40 -28 -68t-68 -28h-448q-40 0 -68 28t-28 68v32h-256v-1152h384v224q0 13 9.5 22.5t22.5 9.5h320q13 0 22.5 -9.5t9.5 -22.5v-224zM896 1056v320q0 13 -9.5 22.5t-22.5 9.5h-64q-13 0 -22.5 -9.5t-9.5 -22.5v-96h-128v96q0 13 -9.5 22.5 + t-22.5 9.5h-64q-13 0 -22.5 -9.5t-9.5 -22.5v-320q0 -13 9.5 -22.5t22.5 -9.5h64q13 0 22.5 9.5t9.5 22.5v96h128v-96q0 -13 9.5 -22.5t22.5 -9.5h64q13 0 22.5 9.5t9.5 22.5zM1408 1088v-1280q0 -26 -19 -45t-45 -19h-1280q-26 0 -45 19t-19 45v1280q0 26 19 45t45 19h320 + v288q0 40 28 68t68 28h448q40 0 68 -28t28 -68v-288h320q26 0 45 -19t19 -45z" /> + <glyph glyph-name="ambulance" unicode="" horiz-adv-x="1920" + d="M640 128q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM256 640h384v256h-158q-14 -2 -22 -9l-195 -195q-7 -12 -9 -22v-30zM1536 128q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5 + t90.5 37.5t37.5 90.5zM1664 800v192q0 14 -9 23t-23 9h-224v224q0 14 -9 23t-23 9h-192q-14 0 -23 -9t-9 -23v-224h-224q-14 0 -23 -9t-9 -23v-192q0 -14 9 -23t23 -9h224v-224q0 -14 9 -23t23 -9h192q14 0 23 9t9 23v224h224q14 0 23 9t9 23zM1920 1344v-1152 + q0 -26 -19 -45t-45 -19h-192q0 -106 -75 -181t-181 -75t-181 75t-75 181h-384q0 -106 -75 -181t-181 -75t-181 75t-75 181h-128q-26 0 -45 19t-19 45t19 45t45 19v416q0 26 13 58t32 51l198 198q19 19 51 32t58 13h160v320q0 26 19 45t45 19h1152q26 0 45 -19t19 -45z" /> + <glyph glyph-name="medkit" unicode="" horiz-adv-x="1792" + d="M1280 416v192q0 14 -9 23t-23 9h-224v224q0 14 -9 23t-23 9h-192q-14 0 -23 -9t-9 -23v-224h-224q-14 0 -23 -9t-9 -23v-192q0 -14 9 -23t23 -9h224v-224q0 -14 9 -23t23 -9h192q14 0 23 9t9 23v224h224q14 0 23 9t9 23zM640 1152h512v128h-512v-128zM256 1152v-1280h-32 + q-92 0 -158 66t-66 158v832q0 92 66 158t158 66h32zM1440 1152v-1280h-1088v1280h160v160q0 40 28 68t68 28h576q40 0 68 -28t28 -68v-160h160zM1792 928v-832q0 -92 -66 -158t-158 -66h-32v1280h32q92 0 158 -66t66 -158z" /> + <glyph glyph-name="fighter_jet" unicode="" horiz-adv-x="1920" + d="M1920 576q-1 -32 -288 -96l-352 -32l-224 -64h-64l-293 -352h69q26 0 45 -4.5t19 -11.5t-19 -11.5t-45 -4.5h-96h-160h-64v32h64v416h-160l-192 -224h-96l-32 32v192h32v32h128v8l-192 24v128l192 24v8h-128v32h-32v192l32 32h96l192 -224h160v416h-64v32h64h160h96 + q26 0 45 -4.5t19 -11.5t-19 -11.5t-45 -4.5h-69l293 -352h64l224 -64l352 -32q128 -28 200 -52t80 -34z" /> + <glyph glyph-name="beer" unicode="" horiz-adv-x="1664" + d="M640 640v384h-256v-256q0 -53 37.5 -90.5t90.5 -37.5h128zM1664 192v-192h-1152v192l128 192h-128q-159 0 -271.5 112.5t-112.5 271.5v320l-64 64l32 128h480l32 128h960l32 -192l-64 -32v-800z" /> + <glyph glyph-name="h_sign" unicode="" + d="M1280 192v896q0 26 -19 45t-45 19h-128q-26 0 -45 -19t-19 -45v-320h-512v320q0 26 -19 45t-45 19h-128q-26 0 -45 -19t-19 -45v-896q0 -26 19 -45t45 -19h128q26 0 45 19t19 45v320h512v-320q0 -26 19 -45t45 -19h128q26 0 45 19t19 45zM1536 1120v-960 + q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="f0fe" unicode="" + d="M1280 576v128q0 26 -19 45t-45 19h-320v320q0 26 -19 45t-45 19h-128q-26 0 -45 -19t-19 -45v-320h-320q-26 0 -45 -19t-19 -45v-128q0 -26 19 -45t45 -19h320v-320q0 -26 19 -45t45 -19h128q26 0 45 19t19 45v320h320q26 0 45 19t19 45zM1536 1120v-960 + q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="double_angle_left" unicode="" horiz-adv-x="1024" + d="M627 160q0 -13 -10 -23l-50 -50q-10 -10 -23 -10t-23 10l-466 466q-10 10 -10 23t10 23l466 466q10 10 23 10t23 -10l50 -50q10 -10 10 -23t-10 -23l-393 -393l393 -393q10 -10 10 -23zM1011 160q0 -13 -10 -23l-50 -50q-10 -10 -23 -10t-23 10l-466 466q-10 10 -10 23 + t10 23l466 466q10 10 23 10t23 -10l50 -50q10 -10 10 -23t-10 -23l-393 -393l393 -393q10 -10 10 -23z" /> + <glyph glyph-name="double_angle_right" unicode="" horiz-adv-x="1024" + d="M595 576q0 -13 -10 -23l-466 -466q-10 -10 -23 -10t-23 10l-50 50q-10 10 -10 23t10 23l393 393l-393 393q-10 10 -10 23t10 23l50 50q10 10 23 10t23 -10l466 -466q10 -10 10 -23zM979 576q0 -13 -10 -23l-466 -466q-10 -10 -23 -10t-23 10l-50 50q-10 10 -10 23t10 23 + l393 393l-393 393q-10 10 -10 23t10 23l50 50q10 10 23 10t23 -10l466 -466q10 -10 10 -23z" /> + <glyph glyph-name="double_angle_up" unicode="" horiz-adv-x="1152" + d="M1075 224q0 -13 -10 -23l-50 -50q-10 -10 -23 -10t-23 10l-393 393l-393 -393q-10 -10 -23 -10t-23 10l-50 50q-10 10 -10 23t10 23l466 466q10 10 23 10t23 -10l466 -466q10 -10 10 -23zM1075 608q0 -13 -10 -23l-50 -50q-10 -10 -23 -10t-23 10l-393 393l-393 -393 + q-10 -10 -23 -10t-23 10l-50 50q-10 10 -10 23t10 23l466 466q10 10 23 10t23 -10l466 -466q10 -10 10 -23z" /> + <glyph glyph-name="double_angle_down" unicode="" horiz-adv-x="1152" + d="M1075 672q0 -13 -10 -23l-466 -466q-10 -10 -23 -10t-23 10l-466 466q-10 10 -10 23t10 23l50 50q10 10 23 10t23 -10l393 -393l393 393q10 10 23 10t23 -10l50 -50q10 -10 10 -23zM1075 1056q0 -13 -10 -23l-466 -466q-10 -10 -23 -10t-23 10l-466 466q-10 10 -10 23 + t10 23l50 50q10 10 23 10t23 -10l393 -393l393 393q10 10 23 10t23 -10l50 -50q10 -10 10 -23z" /> + <glyph glyph-name="angle_left" unicode="" horiz-adv-x="640" + d="M627 992q0 -13 -10 -23l-393 -393l393 -393q10 -10 10 -23t-10 -23l-50 -50q-10 -10 -23 -10t-23 10l-466 466q-10 10 -10 23t10 23l466 466q10 10 23 10t23 -10l50 -50q10 -10 10 -23z" /> + <glyph glyph-name="angle_right" unicode="" horiz-adv-x="640" + d="M595 576q0 -13 -10 -23l-466 -466q-10 -10 -23 -10t-23 10l-50 50q-10 10 -10 23t10 23l393 393l-393 393q-10 10 -10 23t10 23l50 50q10 10 23 10t23 -10l466 -466q10 -10 10 -23z" /> + <glyph glyph-name="angle_up" unicode="" horiz-adv-x="1152" + d="M1075 352q0 -13 -10 -23l-50 -50q-10 -10 -23 -10t-23 10l-393 393l-393 -393q-10 -10 -23 -10t-23 10l-50 50q-10 10 -10 23t10 23l466 466q10 10 23 10t23 -10l466 -466q10 -10 10 -23z" /> + <glyph glyph-name="angle_down" unicode="" horiz-adv-x="1152" + d="M1075 800q0 -13 -10 -23l-466 -466q-10 -10 -23 -10t-23 10l-466 466q-10 10 -10 23t10 23l50 50q10 10 23 10t23 -10l393 -393l393 393q10 10 23 10t23 -10l50 -50q10 -10 10 -23z" /> + <glyph glyph-name="desktop" unicode="" horiz-adv-x="1920" + d="M1792 544v832q0 13 -9.5 22.5t-22.5 9.5h-1600q-13 0 -22.5 -9.5t-9.5 -22.5v-832q0 -13 9.5 -22.5t22.5 -9.5h1600q13 0 22.5 9.5t9.5 22.5zM1920 1376v-1088q0 -66 -47 -113t-113 -47h-544q0 -37 16 -77.5t32 -71t16 -43.5q0 -26 -19 -45t-45 -19h-512q-26 0 -45 19 + t-19 45q0 14 16 44t32 70t16 78h-544q-66 0 -113 47t-47 113v1088q0 66 47 113t113 47h1600q66 0 113 -47t47 -113z" /> + <glyph glyph-name="laptop" unicode="" horiz-adv-x="1920" + d="M416 256q-66 0 -113 47t-47 113v704q0 66 47 113t113 47h1088q66 0 113 -47t47 -113v-704q0 -66 -47 -113t-113 -47h-1088zM384 1120v-704q0 -13 9.5 -22.5t22.5 -9.5h1088q13 0 22.5 9.5t9.5 22.5v704q0 13 -9.5 22.5t-22.5 9.5h-1088q-13 0 -22.5 -9.5t-9.5 -22.5z + M1760 192h160v-96q0 -40 -47 -68t-113 -28h-1600q-66 0 -113 28t-47 68v96h160h1600zM1040 96q16 0 16 16t-16 16h-160q-16 0 -16 -16t16 -16h160z" /> + <glyph glyph-name="tablet" unicode="" horiz-adv-x="1152" + d="M640 128q0 26 -19 45t-45 19t-45 -19t-19 -45t19 -45t45 -19t45 19t19 45zM1024 288v960q0 13 -9.5 22.5t-22.5 9.5h-832q-13 0 -22.5 -9.5t-9.5 -22.5v-960q0 -13 9.5 -22.5t22.5 -9.5h832q13 0 22.5 9.5t9.5 22.5zM1152 1248v-1088q0 -66 -47 -113t-113 -47h-832 + q-66 0 -113 47t-47 113v1088q0 66 47 113t113 47h832q66 0 113 -47t47 -113z" /> + <glyph glyph-name="mobile_phone" unicode="" horiz-adv-x="768" + d="M464 128q0 33 -23.5 56.5t-56.5 23.5t-56.5 -23.5t-23.5 -56.5t23.5 -56.5t56.5 -23.5t56.5 23.5t23.5 56.5zM672 288v704q0 13 -9.5 22.5t-22.5 9.5h-512q-13 0 -22.5 -9.5t-9.5 -22.5v-704q0 -13 9.5 -22.5t22.5 -9.5h512q13 0 22.5 9.5t9.5 22.5zM480 1136 + q0 16 -16 16h-160q-16 0 -16 -16t16 -16h160q16 0 16 16zM768 1152v-1024q0 -52 -38 -90t-90 -38h-512q-52 0 -90 38t-38 90v1024q0 52 38 90t90 38h512q52 0 90 -38t38 -90z" /> + <glyph glyph-name="circle_blank" unicode="" + d="M768 1184q-148 0 -273 -73t-198 -198t-73 -273t73 -273t198 -198t273 -73t273 73t198 198t73 273t-73 273t-198 198t-273 73zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103 + t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="quote_left" unicode="" horiz-adv-x="1664" + d="M768 576v-384q0 -80 -56 -136t-136 -56h-384q-80 0 -136 56t-56 136v704q0 104 40.5 198.5t109.5 163.5t163.5 109.5t198.5 40.5h64q26 0 45 -19t19 -45v-128q0 -26 -19 -45t-45 -19h-64q-106 0 -181 -75t-75 -181v-32q0 -40 28 -68t68 -28h224q80 0 136 -56t56 -136z + M1664 576v-384q0 -80 -56 -136t-136 -56h-384q-80 0 -136 56t-56 136v704q0 104 40.5 198.5t109.5 163.5t163.5 109.5t198.5 40.5h64q26 0 45 -19t19 -45v-128q0 -26 -19 -45t-45 -19h-64q-106 0 -181 -75t-75 -181v-32q0 -40 28 -68t68 -28h224q80 0 136 -56t56 -136z" /> + <glyph glyph-name="quote_right" unicode="" horiz-adv-x="1664" + d="M768 1216v-704q0 -104 -40.5 -198.5t-109.5 -163.5t-163.5 -109.5t-198.5 -40.5h-64q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h64q106 0 181 75t75 181v32q0 40 -28 68t-68 28h-224q-80 0 -136 56t-56 136v384q0 80 56 136t136 56h384q80 0 136 -56t56 -136zM1664 1216 + v-704q0 -104 -40.5 -198.5t-109.5 -163.5t-163.5 -109.5t-198.5 -40.5h-64q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h64q106 0 181 75t75 181v32q0 40 -28 68t-68 28h-224q-80 0 -136 56t-56 136v384q0 80 56 136t136 56h384q80 0 136 -56t56 -136z" /> + <glyph glyph-name="spinner" unicode="" horiz-adv-x="1792" + d="M526 142q0 -53 -37.5 -90.5t-90.5 -37.5q-52 0 -90 38t-38 90q0 53 37.5 90.5t90.5 37.5t90.5 -37.5t37.5 -90.5zM1024 -64q0 -53 -37.5 -90.5t-90.5 -37.5t-90.5 37.5t-37.5 90.5t37.5 90.5t90.5 37.5t90.5 -37.5t37.5 -90.5zM320 640q0 -53 -37.5 -90.5t-90.5 -37.5 + t-90.5 37.5t-37.5 90.5t37.5 90.5t90.5 37.5t90.5 -37.5t37.5 -90.5zM1522 142q0 -52 -38 -90t-90 -38q-53 0 -90.5 37.5t-37.5 90.5t37.5 90.5t90.5 37.5t90.5 -37.5t37.5 -90.5zM558 1138q0 -66 -47 -113t-113 -47t-113 47t-47 113t47 113t113 47t113 -47t47 -113z + M1728 640q0 -53 -37.5 -90.5t-90.5 -37.5t-90.5 37.5t-37.5 90.5t37.5 90.5t90.5 37.5t90.5 -37.5t37.5 -90.5zM1088 1344q0 -80 -56 -136t-136 -56t-136 56t-56 136t56 136t136 56t136 -56t56 -136zM1618 1138q0 -93 -66 -158.5t-158 -65.5q-93 0 -158.5 65.5t-65.5 158.5 + q0 92 65.5 158t158.5 66q92 0 158 -66t66 -158z" /> + <glyph glyph-name="circle" unicode="" + d="M1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="reply" unicode="" horiz-adv-x="1792" + d="M1792 416q0 -166 -127 -451q-3 -7 -10.5 -24t-13.5 -30t-13 -22q-12 -17 -28 -17q-15 0 -23.5 10t-8.5 25q0 9 2.5 26.5t2.5 23.5q5 68 5 123q0 101 -17.5 181t-48.5 138.5t-80 101t-105.5 69.5t-133 42.5t-154 21.5t-175.5 6h-224v-256q0 -26 -19 -45t-45 -19t-45 19 + l-512 512q-19 19 -19 45t19 45l512 512q19 19 45 19t45 -19t19 -45v-256h224q713 0 875 -403q53 -134 53 -333z" /> + <glyph glyph-name="github_alt" unicode="" horiz-adv-x="1664" + d="M640 320q0 -40 -12.5 -82t-43 -76t-72.5 -34t-72.5 34t-43 76t-12.5 82t12.5 82t43 76t72.5 34t72.5 -34t43 -76t12.5 -82zM1280 320q0 -40 -12.5 -82t-43 -76t-72.5 -34t-72.5 34t-43 76t-12.5 82t12.5 82t43 76t72.5 34t72.5 -34t43 -76t12.5 -82zM1440 320 + q0 120 -69 204t-187 84q-41 0 -195 -21q-71 -11 -157 -11t-157 11q-152 21 -195 21q-118 0 -187 -84t-69 -204q0 -88 32 -153.5t81 -103t122 -60t140 -29.5t149 -7h168q82 0 149 7t140 29.5t122 60t81 103t32 153.5zM1664 496q0 -207 -61 -331q-38 -77 -105.5 -133t-141 -86 + t-170 -47.5t-171.5 -22t-167 -4.5q-78 0 -142 3t-147.5 12.5t-152.5 30t-137 51.5t-121 81t-86 115q-62 123 -62 331q0 237 136 396q-27 82 -27 170q0 116 51 218q108 0 190 -39.5t189 -123.5q147 35 309 35q148 0 280 -32q105 82 187 121t189 39q51 -102 51 -218 + q0 -87 -27 -168q136 -160 136 -398z" /> + <glyph glyph-name="folder_close_alt" unicode="" horiz-adv-x="1664" + d="M1536 224v704q0 40 -28 68t-68 28h-704q-40 0 -68 28t-28 68v64q0 40 -28 68t-68 28h-320q-40 0 -68 -28t-28 -68v-960q0 -40 28 -68t68 -28h1216q40 0 68 28t28 68zM1664 928v-704q0 -92 -66 -158t-158 -66h-1216q-92 0 -158 66t-66 158v960q0 92 66 158t158 66h320 + q92 0 158 -66t66 -158v-32h672q92 0 158 -66t66 -158z" /> + <glyph glyph-name="folder_open_alt" unicode="" horiz-adv-x="1920" + d="M1781 605q0 35 -53 35h-1088q-40 0 -85.5 -21.5t-71.5 -52.5l-294 -363q-18 -24 -18 -40q0 -35 53 -35h1088q40 0 86 22t71 53l294 363q18 22 18 39zM640 768h768v160q0 40 -28 68t-68 28h-576q-40 0 -68 28t-28 68v64q0 40 -28 68t-68 28h-320q-40 0 -68 -28t-28 -68 + v-853l256 315q44 53 116 87.5t140 34.5zM1909 605q0 -62 -46 -120l-295 -363q-43 -53 -116 -87.5t-140 -34.5h-1088q-92 0 -158 66t-66 158v960q0 92 66 158t158 66h320q92 0 158 -66t66 -158v-32h544q92 0 158 -66t66 -158v-160h192q54 0 99 -24.5t67 -70.5q15 -32 15 -68z + " /> + <glyph glyph-name="expand_alt" unicode="" horiz-adv-x="1792" + /> + <glyph glyph-name="collapse_alt" unicode="" horiz-adv-x="1792" + /> + <glyph glyph-name="smile" unicode="" + d="M1134 461q-37 -121 -138 -195t-228 -74t-228 74t-138 195q-8 25 4 48.5t38 31.5q25 8 48.5 -4t31.5 -38q25 -80 92.5 -129.5t151.5 -49.5t151.5 49.5t92.5 129.5q8 26 32 38t49 4t37 -31.5t4 -48.5zM640 896q0 -53 -37.5 -90.5t-90.5 -37.5t-90.5 37.5t-37.5 90.5 + t37.5 90.5t90.5 37.5t90.5 -37.5t37.5 -90.5zM1152 896q0 -53 -37.5 -90.5t-90.5 -37.5t-90.5 37.5t-37.5 90.5t37.5 90.5t90.5 37.5t90.5 -37.5t37.5 -90.5zM1408 640q0 130 -51 248.5t-136.5 204t-204 136.5t-248.5 51t-248.5 -51t-204 -136.5t-136.5 -204t-51 -248.5 + t51 -248.5t136.5 -204t204 -136.5t248.5 -51t248.5 51t204 136.5t136.5 204t51 248.5zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="frown" unicode="" + d="M1134 307q8 -25 -4 -48.5t-37 -31.5t-49 4t-32 38q-25 80 -92.5 129.5t-151.5 49.5t-151.5 -49.5t-92.5 -129.5q-8 -26 -31.5 -38t-48.5 -4q-26 8 -38 31.5t-4 48.5q37 121 138 195t228 74t228 -74t138 -195zM640 896q0 -53 -37.5 -90.5t-90.5 -37.5t-90.5 37.5 + t-37.5 90.5t37.5 90.5t90.5 37.5t90.5 -37.5t37.5 -90.5zM1152 896q0 -53 -37.5 -90.5t-90.5 -37.5t-90.5 37.5t-37.5 90.5t37.5 90.5t90.5 37.5t90.5 -37.5t37.5 -90.5zM1408 640q0 130 -51 248.5t-136.5 204t-204 136.5t-248.5 51t-248.5 -51t-204 -136.5t-136.5 -204 + t-51 -248.5t51 -248.5t136.5 -204t204 -136.5t248.5 -51t248.5 51t204 136.5t136.5 204t51 248.5zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="meh" unicode="" + d="M1152 448q0 -26 -19 -45t-45 -19h-640q-26 0 -45 19t-19 45t19 45t45 19h640q26 0 45 -19t19 -45zM640 896q0 -53 -37.5 -90.5t-90.5 -37.5t-90.5 37.5t-37.5 90.5t37.5 90.5t90.5 37.5t90.5 -37.5t37.5 -90.5zM1152 896q0 -53 -37.5 -90.5t-90.5 -37.5t-90.5 37.5 + t-37.5 90.5t37.5 90.5t90.5 37.5t90.5 -37.5t37.5 -90.5zM1408 640q0 130 -51 248.5t-136.5 204t-204 136.5t-248.5 51t-248.5 -51t-204 -136.5t-136.5 -204t-51 -248.5t51 -248.5t136.5 -204t204 -136.5t248.5 -51t248.5 51t204 136.5t136.5 204t51 248.5zM1536 640 + q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="gamepad" unicode="" horiz-adv-x="1920" + d="M832 448v128q0 14 -9 23t-23 9h-192v192q0 14 -9 23t-23 9h-128q-14 0 -23 -9t-9 -23v-192h-192q-14 0 -23 -9t-9 -23v-128q0 -14 9 -23t23 -9h192v-192q0 -14 9 -23t23 -9h128q14 0 23 9t9 23v192h192q14 0 23 9t9 23zM1408 384q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5 + t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM1664 640q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM1920 512q0 -212 -150 -362t-362 -150q-192 0 -338 128h-220q-146 -128 -338 -128q-212 0 -362 150 + t-150 362t150 362t362 150h896q212 0 362 -150t150 -362z" /> + <glyph glyph-name="keyboard" unicode="" horiz-adv-x="1920" + d="M384 368v-96q0 -16 -16 -16h-96q-16 0 -16 16v96q0 16 16 16h96q16 0 16 -16zM512 624v-96q0 -16 -16 -16h-224q-16 0 -16 16v96q0 16 16 16h224q16 0 16 -16zM384 880v-96q0 -16 -16 -16h-96q-16 0 -16 16v96q0 16 16 16h96q16 0 16 -16zM1408 368v-96q0 -16 -16 -16 + h-864q-16 0 -16 16v96q0 16 16 16h864q16 0 16 -16zM768 624v-96q0 -16 -16 -16h-96q-16 0 -16 16v96q0 16 16 16h96q16 0 16 -16zM640 880v-96q0 -16 -16 -16h-96q-16 0 -16 16v96q0 16 16 16h96q16 0 16 -16zM1024 624v-96q0 -16 -16 -16h-96q-16 0 -16 16v96q0 16 16 16 + h96q16 0 16 -16zM896 880v-96q0 -16 -16 -16h-96q-16 0 -16 16v96q0 16 16 16h96q16 0 16 -16zM1280 624v-96q0 -16 -16 -16h-96q-16 0 -16 16v96q0 16 16 16h96q16 0 16 -16zM1664 368v-96q0 -16 -16 -16h-96q-16 0 -16 16v96q0 16 16 16h96q16 0 16 -16zM1152 880v-96 + q0 -16 -16 -16h-96q-16 0 -16 16v96q0 16 16 16h96q16 0 16 -16zM1408 880v-96q0 -16 -16 -16h-96q-16 0 -16 16v96q0 16 16 16h96q16 0 16 -16zM1664 880v-352q0 -16 -16 -16h-224q-16 0 -16 16v96q0 16 16 16h112v240q0 16 16 16h96q16 0 16 -16zM1792 128v896h-1664v-896 + h1664zM1920 1024v-896q0 -53 -37.5 -90.5t-90.5 -37.5h-1664q-53 0 -90.5 37.5t-37.5 90.5v896q0 53 37.5 90.5t90.5 37.5h1664q53 0 90.5 -37.5t37.5 -90.5z" /> + <glyph glyph-name="flag_alt" unicode="" horiz-adv-x="1792" + d="M1664 491v616q-169 -91 -306 -91q-82 0 -145 32q-100 49 -184 76.5t-178 27.5q-173 0 -403 -127v-599q245 113 433 113q55 0 103.5 -7.5t98 -26t77 -31t82.5 -39.5l28 -14q44 -22 101 -22q120 0 293 92zM320 1280q0 -35 -17.5 -64t-46.5 -46v-1266q0 -14 -9 -23t-23 -9 + h-64q-14 0 -23 9t-9 23v1266q-29 17 -46.5 46t-17.5 64q0 53 37.5 90.5t90.5 37.5t90.5 -37.5t37.5 -90.5zM1792 1216v-763q0 -39 -35 -57q-10 -5 -17 -9q-218 -116 -369 -116q-88 0 -158 35l-28 14q-64 33 -99 48t-91 29t-114 14q-102 0 -235.5 -44t-228.5 -102 + q-15 -9 -33 -9q-16 0 -32 8q-32 19 -32 56v742q0 35 31 55q35 21 78.5 42.5t114 52t152.5 49.5t155 19q112 0 209 -31t209 -86q38 -19 89 -19q122 0 310 112q22 12 31 17q31 16 62 -2q31 -20 31 -55z" /> + <glyph glyph-name="flag_checkered" unicode="" horiz-adv-x="1792" + d="M832 536v192q-181 -16 -384 -117v-185q205 96 384 110zM832 954v197q-172 -8 -384 -126v-189q215 111 384 118zM1664 491v184q-235 -116 -384 -71v224q-20 6 -39 15q-5 3 -33 17t-34.5 17t-31.5 15t-34.5 15.5t-32.5 13t-36 12.5t-35 8.5t-39.5 7.5t-39.5 4t-44 2 + q-23 0 -49 -3v-222h19q102 0 192.5 -29t197.5 -82q19 -9 39 -15v-188q42 -17 91 -17q120 0 293 92zM1664 918v189q-169 -91 -306 -91q-45 0 -78 8v-196q148 -42 384 90zM320 1280q0 -35 -17.5 -64t-46.5 -46v-1266q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9t-9 23v1266 + q-29 17 -46.5 46t-17.5 64q0 53 37.5 90.5t90.5 37.5t90.5 -37.5t37.5 -90.5zM1792 1216v-763q0 -39 -35 -57q-10 -5 -17 -9q-218 -116 -369 -116q-88 0 -158 35l-28 14q-64 33 -99 48t-91 29t-114 14q-102 0 -235.5 -44t-228.5 -102q-15 -9 -33 -9q-16 0 -32 8 + q-32 19 -32 56v742q0 35 31 55q35 21 78.5 42.5t114 52t152.5 49.5t155 19q112 0 209 -31t209 -86q38 -19 89 -19q122 0 310 112q22 12 31 17q31 16 62 -2q31 -20 31 -55z" /> + <glyph glyph-name="terminal" unicode="" horiz-adv-x="1664" + d="M585 553l-466 -466q-10 -10 -23 -10t-23 10l-50 50q-10 10 -10 23t10 23l393 393l-393 393q-10 10 -10 23t10 23l50 50q10 10 23 10t23 -10l466 -466q10 -10 10 -23t-10 -23zM1664 96v-64q0 -14 -9 -23t-23 -9h-960q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h960q14 0 23 -9 + t9 -23z" /> + <glyph glyph-name="code" unicode="" horiz-adv-x="1920" + d="M617 137l-50 -50q-10 -10 -23 -10t-23 10l-466 466q-10 10 -10 23t10 23l466 466q10 10 23 10t23 -10l50 -50q10 -10 10 -23t-10 -23l-393 -393l393 -393q10 -10 10 -23t-10 -23zM1208 1204l-373 -1291q-4 -13 -15.5 -19.5t-23.5 -2.5l-62 17q-13 4 -19.5 15.5t-2.5 24.5 + l373 1291q4 13 15.5 19.5t23.5 2.5l62 -17q13 -4 19.5 -15.5t2.5 -24.5zM1865 553l-466 -466q-10 -10 -23 -10t-23 10l-50 50q-10 10 -10 23t10 23l393 393l-393 393q-10 10 -10 23t10 23l50 50q10 10 23 10t23 -10l466 -466q10 -10 10 -23t-10 -23z" /> + <glyph glyph-name="reply_all" unicode="" horiz-adv-x="1792" + d="M640 454v-70q0 -42 -39 -59q-13 -5 -25 -5q-27 0 -45 19l-512 512q-19 19 -19 45t19 45l512 512q29 31 70 14q39 -17 39 -59v-69l-397 -398q-19 -19 -19 -45t19 -45zM1792 416q0 -58 -17 -133.5t-38.5 -138t-48 -125t-40.5 -90.5l-20 -40q-8 -17 -28 -17q-6 0 -9 1 + q-25 8 -23 34q43 400 -106 565q-64 71 -170.5 110.5t-267.5 52.5v-251q0 -42 -39 -59q-13 -5 -25 -5q-27 0 -45 19l-512 512q-19 19 -19 45t19 45l512 512q29 31 70 14q39 -17 39 -59v-262q411 -28 599 -221q169 -173 169 -509z" /> + <glyph glyph-name="star_half_empty" unicode="" horiz-adv-x="1664" + d="M1186 579l257 250l-356 52l-66 10l-30 60l-159 322v-963l59 -31l318 -168l-60 355l-12 66zM1638 841l-363 -354l86 -500q5 -33 -6 -51.5t-34 -18.5q-17 0 -40 12l-449 236l-449 -236q-23 -12 -40 -12q-23 0 -34 18.5t-6 51.5l86 500l-364 354q-32 32 -23 59.5t54 34.5 + l502 73l225 455q20 41 49 41q28 0 49 -41l225 -455l502 -73q45 -7 54 -34.5t-24 -59.5z" /> + <glyph glyph-name="location_arrow" unicode="" horiz-adv-x="1408" + d="M1401 1187l-640 -1280q-17 -35 -57 -35q-5 0 -15 2q-22 5 -35.5 22.5t-13.5 39.5v576h-576q-22 0 -39.5 13.5t-22.5 35.5t4 42t29 30l1280 640q13 7 29 7q27 0 45 -19q15 -14 18.5 -34.5t-6.5 -39.5z" /> + <glyph glyph-name="crop" unicode="" horiz-adv-x="1664" + d="M557 256h595v595zM512 301l595 595h-595v-595zM1664 224v-192q0 -14 -9 -23t-23 -9h-224v-224q0 -14 -9 -23t-23 -9h-192q-14 0 -23 9t-9 23v224h-864q-14 0 -23 9t-9 23v864h-224q-14 0 -23 9t-9 23v192q0 14 9 23t23 9h224v224q0 14 9 23t23 9h192q14 0 23 -9t9 -23 + v-224h851l246 247q10 9 23 9t23 -9q9 -10 9 -23t-9 -23l-247 -246v-851h224q14 0 23 -9t9 -23z" /> + <glyph glyph-name="code_fork" unicode="" horiz-adv-x="1024" + d="M288 64q0 40 -28 68t-68 28t-68 -28t-28 -68t28 -68t68 -28t68 28t28 68zM288 1216q0 40 -28 68t-68 28t-68 -28t-28 -68t28 -68t68 -28t68 28t28 68zM928 1088q0 40 -28 68t-68 28t-68 -28t-28 -68t28 -68t68 -28t68 28t28 68zM1024 1088q0 -52 -26 -96.5t-70 -69.5 + q-2 -287 -226 -414q-67 -38 -203 -81q-128 -40 -169.5 -71t-41.5 -100v-26q44 -25 70 -69.5t26 -96.5q0 -80 -56 -136t-136 -56t-136 56t-56 136q0 52 26 96.5t70 69.5v820q-44 25 -70 69.5t-26 96.5q0 80 56 136t136 56t136 -56t56 -136q0 -52 -26 -96.5t-70 -69.5v-497 + q54 26 154 57q55 17 87.5 29.5t70.5 31t59 39.5t40.5 51t28 69.5t8.5 91.5q-44 25 -70 69.5t-26 96.5q0 80 56 136t136 56t136 -56t56 -136z" /> + <glyph glyph-name="unlink" unicode="" horiz-adv-x="1664" + d="M439 265l-256 -256q-11 -9 -23 -9t-23 9q-9 10 -9 23t9 23l256 256q10 9 23 9t23 -9q9 -10 9 -23t-9 -23zM608 224v-320q0 -14 -9 -23t-23 -9t-23 9t-9 23v320q0 14 9 23t23 9t23 -9t9 -23zM384 448q0 -14 -9 -23t-23 -9h-320q-14 0 -23 9t-9 23t9 23t23 9h320 + q14 0 23 -9t9 -23zM1648 320q0 -120 -85 -203l-147 -146q-83 -83 -203 -83q-121 0 -204 85l-334 335q-21 21 -42 56l239 18l273 -274q27 -27 68 -27.5t68 26.5l147 146q28 28 28 67q0 40 -28 68l-274 275l18 239q35 -21 56 -42l336 -336q84 -86 84 -204zM1031 1044l-239 -18 + l-273 274q-28 28 -68 28q-39 0 -68 -27l-147 -146q-28 -28 -28 -67q0 -40 28 -68l274 -274l-18 -240q-35 21 -56 42l-336 336q-84 86 -84 204q0 120 85 203l147 146q83 83 203 83q121 0 204 -85l334 -335q21 -21 42 -56zM1664 960q0 -14 -9 -23t-23 -9h-320q-14 0 -23 9 + t-9 23t9 23t23 9h320q14 0 23 -9t9 -23zM1120 1504v-320q0 -14 -9 -23t-23 -9t-23 9t-9 23v320q0 14 9 23t23 9t23 -9t9 -23zM1527 1353l-256 -256q-11 -9 -23 -9t-23 9q-9 10 -9 23t9 23l256 256q10 9 23 9t23 -9q9 -10 9 -23t-9 -23z" /> + <glyph glyph-name="question" unicode="" horiz-adv-x="1024" + d="M704 280v-240q0 -16 -12 -28t-28 -12h-240q-16 0 -28 12t-12 28v240q0 16 12 28t28 12h240q16 0 28 -12t12 -28zM1020 880q0 -54 -15.5 -101t-35 -76.5t-55 -59.5t-57.5 -43.5t-61 -35.5q-41 -23 -68.5 -65t-27.5 -67q0 -17 -12 -32.5t-28 -15.5h-240q-15 0 -25.5 18.5 + t-10.5 37.5v45q0 83 65 156.5t143 108.5q59 27 84 56t25 76q0 42 -46.5 74t-107.5 32q-65 0 -108 -29q-35 -25 -107 -115q-13 -16 -31 -16q-12 0 -25 8l-164 125q-13 10 -15.5 25t5.5 28q160 266 464 266q80 0 161 -31t146 -83t106 -127.5t41 -158.5z" /> + <glyph glyph-name="_279" unicode="" horiz-adv-x="640" + d="M640 192v-128q0 -26 -19 -45t-45 -19h-512q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h64v384h-64q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h384q26 0 45 -19t19 -45v-576h64q26 0 45 -19t19 -45zM512 1344v-192q0 -26 -19 -45t-45 -19h-256q-26 0 -45 19t-19 45v192 + q0 26 19 45t45 19h256q26 0 45 -19t19 -45z" /> + <glyph glyph-name="exclamation" unicode="" horiz-adv-x="640" + d="M512 288v-224q0 -26 -19 -45t-45 -19h-256q-26 0 -45 19t-19 45v224q0 26 19 45t45 19h256q26 0 45 -19t19 -45zM542 1344l-28 -768q-1 -26 -20.5 -45t-45.5 -19h-256q-26 0 -45.5 19t-20.5 45l-28 768q-1 26 17.5 45t44.5 19h320q26 0 44.5 -19t17.5 -45z" /> + <glyph glyph-name="superscript" unicode="" + d="M897 167v-167h-248l-159 252l-24 42q-8 9 -11 21h-3q-1 -3 -2.5 -6.5t-3.5 -8t-3 -6.5q-10 -20 -25 -44l-155 -250h-258v167h128l197 291l-185 272h-137v168h276l139 -228q2 -4 23 -42q8 -9 11 -21h3q3 9 11 21l25 42l140 228h257v-168h-125l-184 -267l204 -296h109z + M1534 846v-206h-514l-3 27q-4 28 -4 46q0 64 26 117t65 86.5t84 65t84 54.5t65 54t26 64q0 38 -29.5 62.5t-70.5 24.5q-51 0 -97 -39q-14 -11 -36 -38l-105 92q26 37 63 66q83 65 188 65q110 0 178 -59.5t68 -158.5q0 -56 -24.5 -103t-62 -76.5t-81.5 -58.5t-82 -50.5 + t-65.5 -51.5t-30.5 -63h232v80h126z" /> + <glyph glyph-name="subscript" unicode="" + d="M897 167v-167h-248l-159 252l-24 42q-8 9 -11 21h-3q-1 -3 -2.5 -6.5t-3.5 -8t-3 -6.5q-10 -20 -25 -44l-155 -250h-258v167h128l197 291l-185 272h-137v168h276l139 -228q2 -4 23 -42q8 -9 11 -21h3q3 9 11 21l25 42l140 228h257v-168h-125l-184 -267l204 -296h109z + M1536 -50v-206h-514l-4 27q-3 45 -3 46q0 64 26 117t65 86.5t84 65t84 54.5t65 54t26 64q0 38 -29.5 62.5t-70.5 24.5q-51 0 -97 -39q-14 -11 -36 -38l-105 92q26 37 63 66q80 65 188 65q110 0 178 -59.5t68 -158.5q0 -66 -34.5 -118.5t-84 -86t-99.5 -62.5t-87 -63t-41 -73 + h232v80h126z" /> + <glyph glyph-name="_283" unicode="" horiz-adv-x="1920" + d="M896 128l336 384h-768l-336 -384h768zM1909 1205q15 -34 9.5 -71.5t-30.5 -65.5l-896 -1024q-38 -44 -96 -44h-768q-38 0 -69.5 20.5t-47.5 54.5q-15 34 -9.5 71.5t30.5 65.5l896 1024q38 44 96 44h768q38 0 69.5 -20.5t47.5 -54.5z" /> + <glyph glyph-name="puzzle_piece" unicode="" horiz-adv-x="1664" + d="M1664 438q0 -81 -44.5 -135t-123.5 -54q-41 0 -77.5 17.5t-59 38t-56.5 38t-71 17.5q-110 0 -110 -124q0 -39 16 -115t15 -115v-5q-22 0 -33 -1q-34 -3 -97.5 -11.5t-115.5 -13.5t-98 -5q-61 0 -103 26.5t-42 83.5q0 37 17.5 71t38 56.5t38 59t17.5 77.5q0 79 -54 123.5 + t-135 44.5q-84 0 -143 -45.5t-59 -127.5q0 -43 15 -83t33.5 -64.5t33.5 -53t15 -50.5q0 -45 -46 -89q-37 -35 -117 -35q-95 0 -245 24q-9 2 -27.5 4t-27.5 4l-13 2q-1 0 -3 1q-2 0 -2 1v1024q2 -1 17.5 -3.5t34 -5t21.5 -3.5q150 -24 245 -24q80 0 117 35q46 44 46 89 + q0 22 -15 50.5t-33.5 53t-33.5 64.5t-15 83q0 82 59 127.5t144 45.5q80 0 134 -44.5t54 -123.5q0 -41 -17.5 -77.5t-38 -59t-38 -56.5t-17.5 -71q0 -57 42 -83.5t103 -26.5q64 0 180 15t163 17v-2q-1 -2 -3.5 -17.5t-5 -34t-3.5 -21.5q-24 -150 -24 -245q0 -80 35 -117 + q44 -46 89 -46q22 0 50.5 15t53 33.5t64.5 33.5t83 15q82 0 127.5 -59t45.5 -143z" /> + <glyph glyph-name="microphone" unicode="" horiz-adv-x="1152" + d="M1152 832v-128q0 -221 -147.5 -384.5t-364.5 -187.5v-132h256q26 0 45 -19t19 -45t-19 -45t-45 -19h-640q-26 0 -45 19t-19 45t19 45t45 19h256v132q-217 24 -364.5 187.5t-147.5 384.5v128q0 26 19 45t45 19t45 -19t19 -45v-128q0 -185 131.5 -316.5t316.5 -131.5 + t316.5 131.5t131.5 316.5v128q0 26 19 45t45 19t45 -19t19 -45zM896 1216v-512q0 -132 -94 -226t-226 -94t-226 94t-94 226v512q0 132 94 226t226 94t226 -94t94 -226z" /> + <glyph glyph-name="microphone_off" unicode="" horiz-adv-x="1408" + d="M271 591l-101 -101q-42 103 -42 214v128q0 26 19 45t45 19t45 -19t19 -45v-128q0 -53 15 -113zM1385 1193l-361 -361v-128q0 -132 -94 -226t-226 -94q-55 0 -109 19l-96 -96q97 -51 205 -51q185 0 316.5 131.5t131.5 316.5v128q0 26 19 45t45 19t45 -19t19 -45v-128 + q0 -221 -147.5 -384.5t-364.5 -187.5v-132h256q26 0 45 -19t19 -45t-19 -45t-45 -19h-640q-26 0 -45 19t-19 45t19 45t45 19h256v132q-125 13 -235 81l-254 -254q-10 -10 -23 -10t-23 10l-82 82q-10 10 -10 23t10 23l1234 1234q10 10 23 10t23 -10l82 -82q10 -10 10 -23 + t-10 -23zM1005 1325l-621 -621v512q0 132 94 226t226 94q102 0 184.5 -59t116.5 -152z" /> + <glyph glyph-name="shield" unicode="" horiz-adv-x="1280" + d="M1088 576v640h-448v-1137q119 63 213 137q235 184 235 360zM1280 1344v-768q0 -86 -33.5 -170.5t-83 -150t-118 -127.5t-126.5 -103t-121 -77.5t-89.5 -49.5t-42.5 -20q-12 -6 -26 -6t-26 6q-16 7 -42.5 20t-89.5 49.5t-121 77.5t-126.5 103t-118 127.5t-83 150 + t-33.5 170.5v768q0 26 19 45t45 19h1152q26 0 45 -19t19 -45z" /> + <glyph glyph-name="calendar_empty" unicode="" horiz-adv-x="1664" + d="M128 -128h1408v1024h-1408v-1024zM512 1088v288q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-288q0 -14 9 -23t23 -9h64q14 0 23 9t9 23zM1280 1088v288q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-288q0 -14 9 -23t23 -9h64q14 0 23 9t9 23zM1664 1152v-1280 + q0 -52 -38 -90t-90 -38h-1408q-52 0 -90 38t-38 90v1280q0 52 38 90t90 38h128v96q0 66 47 113t113 47h64q66 0 113 -47t47 -113v-96h384v96q0 66 47 113t113 47h64q66 0 113 -47t47 -113v-96h128q52 0 90 -38t38 -90z" /> + <glyph glyph-name="fire_extinguisher" unicode="" horiz-adv-x="1408" + d="M512 1344q0 26 -19 45t-45 19t-45 -19t-19 -45t19 -45t45 -19t45 19t19 45zM1408 1376v-320q0 -16 -12 -25q-8 -7 -20 -7q-4 0 -7 1l-448 96q-11 2 -18 11t-7 20h-256v-102q111 -23 183.5 -111t72.5 -203v-800q0 -26 -19 -45t-45 -19h-512q-26 0 -45 19t-19 45v800 + q0 106 62.5 190.5t161.5 114.5v111h-32q-59 0 -115 -23.5t-91.5 -53t-66 -66.5t-40.5 -53.5t-14 -24.5q-17 -35 -57 -35q-16 0 -29 7q-23 12 -31.5 37t3.5 49q5 10 14.5 26t37.5 53.5t60.5 70t85 67t108.5 52.5q-25 42 -25 86q0 66 47 113t113 47t113 -47t47 -113 + q0 -33 -14 -64h302q0 11 7 20t18 11l448 96q3 1 7 1q12 0 20 -7q12 -9 12 -25z" /> + <glyph glyph-name="rocket" unicode="" horiz-adv-x="1664" + d="M1440 1088q0 40 -28 68t-68 28t-68 -28t-28 -68t28 -68t68 -28t68 28t28 68zM1664 1376q0 -249 -75.5 -430.5t-253.5 -360.5q-81 -80 -195 -176l-20 -379q-2 -16 -16 -26l-384 -224q-7 -4 -16 -4q-12 0 -23 9l-64 64q-13 14 -8 32l85 276l-281 281l-276 -85q-3 -1 -9 -1 + q-14 0 -23 9l-64 64q-17 19 -5 39l224 384q10 14 26 16l379 20q96 114 176 195q188 187 358 258t431 71q14 0 24 -9.5t10 -22.5z" /> + <glyph glyph-name="maxcdn" unicode="" horiz-adv-x="1792" + d="M1745 763l-164 -763h-334l178 832q13 56 -15 88q-27 33 -83 33h-169l-204 -953h-334l204 953h-286l-204 -953h-334l204 953l-153 327h1276q101 0 189.5 -40.5t147.5 -113.5q60 -73 81 -168.5t0 -194.5z" /> + <glyph glyph-name="chevron_sign_left" unicode="" + d="M909 141l102 102q19 19 19 45t-19 45l-307 307l307 307q19 19 19 45t-19 45l-102 102q-19 19 -45 19t-45 -19l-454 -454q-19 -19 -19 -45t19 -45l454 -454q19 -19 45 -19t45 19zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5 + t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="chevron_sign_right" unicode="" + d="M717 141l454 454q19 19 19 45t-19 45l-454 454q-19 19 -45 19t-45 -19l-102 -102q-19 -19 -19 -45t19 -45l307 -307l-307 -307q-19 -19 -19 -45t19 -45l102 -102q19 -19 45 -19t45 19zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5 + t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="chevron_sign_up" unicode="" + d="M1165 397l102 102q19 19 19 45t-19 45l-454 454q-19 19 -45 19t-45 -19l-454 -454q-19 -19 -19 -45t19 -45l102 -102q19 -19 45 -19t45 19l307 307l307 -307q19 -19 45 -19t45 19zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5 + t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="chevron_sign_down" unicode="" + d="M813 237l454 454q19 19 19 45t-19 45l-102 102q-19 19 -45 19t-45 -19l-307 -307l-307 307q-19 19 -45 19t-45 -19l-102 -102q-19 -19 -19 -45t19 -45l454 -454q19 -19 45 -19t45 19zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5 + t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="html5" unicode="" horiz-adv-x="1408" + d="M1130 939l16 175h-884l47 -534h612l-22 -228l-197 -53l-196 53l-13 140h-175l22 -278l362 -100h4v1l359 99l50 544h-644l-15 181h674zM0 1408h1408l-128 -1438l-578 -162l-574 162z" /> + <glyph glyph-name="css3" unicode="" horiz-adv-x="1792" + d="M275 1408h1505l-266 -1333l-804 -267l-698 267l71 356h297l-29 -147l422 -161l486 161l68 339h-1208l58 297h1209l38 191h-1208z" /> + <glyph glyph-name="anchor" unicode="" horiz-adv-x="1792" + d="M960 1280q0 26 -19 45t-45 19t-45 -19t-19 -45t19 -45t45 -19t45 19t19 45zM1792 352v-352q0 -22 -20 -30q-8 -2 -12 -2q-12 0 -23 9l-93 93q-119 -143 -318.5 -226.5t-429.5 -83.5t-429.5 83.5t-318.5 226.5l-93 -93q-9 -9 -23 -9q-4 0 -12 2q-20 8 -20 30v352 + q0 14 9 23t23 9h352q22 0 30 -20q8 -19 -7 -35l-100 -100q67 -91 189.5 -153.5t271.5 -82.5v647h-192q-26 0 -45 19t-19 45v128q0 26 19 45t45 19h192v163q-58 34 -93 92.5t-35 128.5q0 106 75 181t181 75t181 -75t75 -181q0 -70 -35 -128.5t-93 -92.5v-163h192q26 0 45 -19 + t19 -45v-128q0 -26 -19 -45t-45 -19h-192v-647q149 20 271.5 82.5t189.5 153.5l-100 100q-15 16 -7 35q8 20 30 20h352q14 0 23 -9t9 -23z" /> + <glyph glyph-name="unlock_alt" unicode="" horiz-adv-x="1152" + d="M1056 768q40 0 68 -28t28 -68v-576q0 -40 -28 -68t-68 -28h-960q-40 0 -68 28t-28 68v576q0 40 28 68t68 28h32v320q0 185 131.5 316.5t316.5 131.5t316.5 -131.5t131.5 -316.5q0 -26 -19 -45t-45 -19h-64q-26 0 -45 19t-19 45q0 106 -75 181t-181 75t-181 -75t-75 -181 + v-320h736z" /> + <glyph glyph-name="bullseye" unicode="" + d="M1024 640q0 -106 -75 -181t-181 -75t-181 75t-75 181t75 181t181 75t181 -75t75 -181zM1152 640q0 159 -112.5 271.5t-271.5 112.5t-271.5 -112.5t-112.5 -271.5t112.5 -271.5t271.5 -112.5t271.5 112.5t112.5 271.5zM1280 640q0 -212 -150 -362t-362 -150t-362 150 + t-150 362t150 362t362 150t362 -150t150 -362zM1408 640q0 130 -51 248.5t-136.5 204t-204 136.5t-248.5 51t-248.5 -51t-204 -136.5t-136.5 -204t-51 -248.5t51 -248.5t136.5 -204t204 -136.5t248.5 -51t248.5 51t204 136.5t136.5 204t51 248.5zM1536 640 + q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="ellipsis_horizontal" unicode="" horiz-adv-x="1408" + d="M384 800v-192q0 -40 -28 -68t-68 -28h-192q-40 0 -68 28t-28 68v192q0 40 28 68t68 28h192q40 0 68 -28t28 -68zM896 800v-192q0 -40 -28 -68t-68 -28h-192q-40 0 -68 28t-28 68v192q0 40 28 68t68 28h192q40 0 68 -28t28 -68zM1408 800v-192q0 -40 -28 -68t-68 -28h-192 + q-40 0 -68 28t-28 68v192q0 40 28 68t68 28h192q40 0 68 -28t28 -68z" /> + <glyph glyph-name="ellipsis_vertical" unicode="" horiz-adv-x="384" + d="M384 288v-192q0 -40 -28 -68t-68 -28h-192q-40 0 -68 28t-28 68v192q0 40 28 68t68 28h192q40 0 68 -28t28 -68zM384 800v-192q0 -40 -28 -68t-68 -28h-192q-40 0 -68 28t-28 68v192q0 40 28 68t68 28h192q40 0 68 -28t28 -68zM384 1312v-192q0 -40 -28 -68t-68 -28h-192 + q-40 0 -68 28t-28 68v192q0 40 28 68t68 28h192q40 0 68 -28t28 -68z" /> + <glyph glyph-name="_303" unicode="" + d="M512 256q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM863 162q-13 233 -176.5 396.5t-396.5 176.5q-14 1 -24 -9t-10 -23v-128q0 -13 8.5 -22t21.5 -10q154 -11 264 -121t121 -264q1 -13 10 -21.5t22 -8.5h128 + q13 0 23 10t9 24zM1247 161q-5 154 -56 297.5t-139.5 260t-205 205t-260 139.5t-297.5 56q-14 1 -23 -9q-10 -10 -10 -23v-128q0 -13 9 -22t22 -10q204 -7 378 -111.5t278.5 -278.5t111.5 -378q1 -13 10 -22t22 -9h128q13 0 23 10q11 9 9 23zM1536 1120v-960 + q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="play_sign" unicode="" + d="M768 1408q209 0 385.5 -103t279.5 -279.5t103 -385.5t-103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103zM1152 585q32 18 32 55t-32 55l-544 320q-31 19 -64 1q-32 -19 -32 -56v-640q0 -37 32 -56 + q16 -8 32 -8q17 0 32 9z" /> + <glyph glyph-name="ticket" unicode="" horiz-adv-x="1792" + d="M1024 1084l316 -316l-572 -572l-316 316zM813 105l618 618q19 19 19 45t-19 45l-362 362q-18 18 -45 18t-45 -18l-618 -618q-19 -19 -19 -45t19 -45l362 -362q18 -18 45 -18t45 18zM1702 742l-907 -908q-37 -37 -90.5 -37t-90.5 37l-126 126q56 56 56 136t-56 136 + t-136 56t-136 -56l-125 126q-37 37 -37 90.5t37 90.5l907 906q37 37 90.5 37t90.5 -37l125 -125q-56 -56 -56 -136t56 -136t136 -56t136 56l126 -125q37 -37 37 -90.5t-37 -90.5z" /> + <glyph glyph-name="minus_sign_alt" unicode="" + d="M1280 576v128q0 26 -19 45t-45 19h-896q-26 0 -45 -19t-19 -45v-128q0 -26 19 -45t45 -19h896q26 0 45 19t19 45zM1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5 + t84.5 -203.5z" /> + <glyph glyph-name="check_minus" unicode="" horiz-adv-x="1408" + d="M1152 736v-64q0 -14 -9 -23t-23 -9h-832q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h832q14 0 23 -9t9 -23zM1280 288v832q0 66 -47 113t-113 47h-832q-66 0 -113 -47t-47 -113v-832q0 -66 47 -113t113 -47h832q66 0 113 47t47 113zM1408 1120v-832q0 -119 -84.5 -203.5 + t-203.5 -84.5h-832q-119 0 -203.5 84.5t-84.5 203.5v832q0 119 84.5 203.5t203.5 84.5h832q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="level_up" unicode="" horiz-adv-x="1024" + d="M1018 933q-18 -37 -58 -37h-192v-864q0 -14 -9 -23t-23 -9h-704q-21 0 -29 18q-8 20 4 35l160 192q9 11 25 11h320v640h-192q-40 0 -58 37q-17 37 9 68l320 384q18 22 49 22t49 -22l320 -384q27 -32 9 -68z" /> + <glyph glyph-name="level_down" unicode="" horiz-adv-x="1024" + d="M32 1280h704q13 0 22.5 -9.5t9.5 -23.5v-863h192q40 0 58 -37t-9 -69l-320 -384q-18 -22 -49 -22t-49 22l-320 384q-26 31 -9 69q18 37 58 37h192v640h-320q-14 0 -25 11l-160 192q-13 14 -4 34q9 19 29 19z" /> + <glyph glyph-name="check_sign" unicode="" + d="M685 237l614 614q19 19 19 45t-19 45l-102 102q-19 19 -45 19t-45 -19l-467 -467l-211 211q-19 19 -45 19t-45 -19l-102 -102q-19 -19 -19 -45t19 -45l358 -358q19 -19 45 -19t45 19zM1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5 + t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="edit_sign" unicode="" + d="M404 428l152 -152l-52 -52h-56v96h-96v56zM818 818q14 -13 -3 -30l-291 -291q-17 -17 -30 -3q-14 13 3 30l291 291q17 17 30 3zM544 128l544 544l-288 288l-544 -544v-288h288zM1152 736l92 92q28 28 28 68t-28 68l-152 152q-28 28 -68 28t-68 -28l-92 -92zM1536 1120 + v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="_312" unicode="" + d="M1280 608v480q0 26 -19 45t-45 19h-480q-42 0 -59 -39q-17 -41 14 -70l144 -144l-534 -534q-19 -19 -19 -45t19 -45l102 -102q19 -19 45 -19t45 19l534 534l144 -144q18 -19 45 -19q12 0 25 5q39 17 39 59zM1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960 + q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="share_sign" unicode="" + d="M1005 435l352 352q19 19 19 45t-19 45l-352 352q-30 31 -69 14q-40 -17 -40 -59v-160q-119 0 -216 -19.5t-162.5 -51t-114 -79t-76.5 -95.5t-44.5 -109t-21.5 -111.5t-5 -110.5q0 -181 167 -404q11 -12 25 -12q7 0 13 3q22 9 19 33q-44 354 62 473q46 52 130 75.5 + t224 23.5v-160q0 -42 40 -59q12 -5 24 -5q26 0 45 19zM1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="compass" unicode="" + d="M640 448l256 128l-256 128v-256zM1024 1039v-542l-512 -256v542zM1312 640q0 148 -73 273t-198 198t-273 73t-273 -73t-198 -198t-73 -273t73 -273t198 -198t273 -73t273 73t198 198t73 273zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103 + t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="collapse" unicode="" + d="M1145 861q18 -35 -5 -66l-320 -448q-19 -27 -52 -27t-52 27l-320 448q-23 31 -5 66q17 35 57 35h640q40 0 57 -35zM1280 160v960q0 13 -9.5 22.5t-22.5 9.5h-960q-13 0 -22.5 -9.5t-9.5 -22.5v-960q0 -13 9.5 -22.5t22.5 -9.5h960q13 0 22.5 9.5t9.5 22.5zM1536 1120 + v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="collapse_top" unicode="" + d="M1145 419q-17 -35 -57 -35h-640q-40 0 -57 35q-18 35 5 66l320 448q19 27 52 27t52 -27l320 -448q23 -31 5 -66zM1280 160v960q0 13 -9.5 22.5t-22.5 9.5h-960q-13 0 -22.5 -9.5t-9.5 -22.5v-960q0 -13 9.5 -22.5t22.5 -9.5h960q13 0 22.5 9.5t9.5 22.5zM1536 1120v-960 + q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="_317" unicode="" + d="M1088 640q0 -33 -27 -52l-448 -320q-31 -23 -66 -5q-35 17 -35 57v640q0 40 35 57q35 18 66 -5l448 -320q27 -19 27 -52zM1280 160v960q0 14 -9 23t-23 9h-960q-14 0 -23 -9t-9 -23v-960q0 -14 9 -23t23 -9h960q14 0 23 9t9 23zM1536 1120v-960q0 -119 -84.5 -203.5 + t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="eur" unicode="" horiz-adv-x="1024" + d="M976 229l35 -159q3 -12 -3 -22.5t-17 -14.5l-5 -1q-4 -2 -10.5 -3.5t-16 -4.5t-21.5 -5.5t-25.5 -5t-30 -5t-33.5 -4.5t-36.5 -3t-38.5 -1q-234 0 -409 130.5t-238 351.5h-95q-13 0 -22.5 9.5t-9.5 22.5v113q0 13 9.5 22.5t22.5 9.5h66q-2 57 1 105h-67q-14 0 -23 9 + t-9 23v114q0 14 9 23t23 9h98q67 210 243.5 338t400.5 128q102 0 194 -23q11 -3 20 -15q6 -11 3 -24l-43 -159q-3 -13 -14 -19.5t-24 -2.5l-4 1q-4 1 -11.5 2.5l-17.5 3.5t-22.5 3.5t-26 3t-29 2.5t-29.5 1q-126 0 -226 -64t-150 -176h468q16 0 25 -12q10 -12 7 -26 + l-24 -114q-5 -26 -32 -26h-488q-3 -37 0 -105h459q15 0 25 -12q9 -12 6 -27l-24 -112q-2 -11 -11 -18.5t-20 -7.5h-387q48 -117 149.5 -185.5t228.5 -68.5q18 0 36 1.5t33.5 3.5t29.5 4.5t24.5 5t18.5 4.5l12 3l5 2q13 5 26 -2q12 -7 15 -21z" /> + <glyph glyph-name="gbp" unicode="" horiz-adv-x="1024" + d="M1020 399v-367q0 -14 -9 -23t-23 -9h-956q-14 0 -23 9t-9 23v150q0 13 9.5 22.5t22.5 9.5h97v383h-95q-14 0 -23 9.5t-9 22.5v131q0 14 9 23t23 9h95v223q0 171 123.5 282t314.5 111q185 0 335 -125q9 -8 10 -20.5t-7 -22.5l-103 -127q-9 -11 -22 -12q-13 -2 -23 7 + q-5 5 -26 19t-69 32t-93 18q-85 0 -137 -47t-52 -123v-215h305q13 0 22.5 -9t9.5 -23v-131q0 -13 -9.5 -22.5t-22.5 -9.5h-305v-379h414v181q0 13 9 22.5t23 9.5h162q14 0 23 -9.5t9 -22.5z" /> + <glyph glyph-name="usd" unicode="" horiz-adv-x="1024" + d="M978 351q0 -153 -99.5 -263.5t-258.5 -136.5v-175q0 -14 -9 -23t-23 -9h-135q-13 0 -22.5 9.5t-9.5 22.5v175q-66 9 -127.5 31t-101.5 44.5t-74 48t-46.5 37.5t-17.5 18q-17 21 -2 41l103 135q7 10 23 12q15 2 24 -9l2 -2q113 -99 243 -125q37 -8 74 -8q81 0 142.5 43 + t61.5 122q0 28 -15 53t-33.5 42t-58.5 37.5t-66 32t-80 32.5q-39 16 -61.5 25t-61.5 26.5t-62.5 31t-56.5 35.5t-53.5 42.5t-43.5 49t-35.5 58t-21 66.5t-8.5 78q0 138 98 242t255 134v180q0 13 9.5 22.5t22.5 9.5h135q14 0 23 -9t9 -23v-176q57 -6 110.5 -23t87 -33.5 + t63.5 -37.5t39 -29t15 -14q17 -18 5 -38l-81 -146q-8 -15 -23 -16q-14 -3 -27 7q-3 3 -14.5 12t-39 26.5t-58.5 32t-74.5 26t-85.5 11.5q-95 0 -155 -43t-60 -111q0 -26 8.5 -48t29.5 -41.5t39.5 -33t56 -31t60.5 -27t70 -27.5q53 -20 81 -31.5t76 -35t75.5 -42.5t62 -50 + t53 -63.5t31.5 -76.5t13 -94z" /> + <glyph glyph-name="inr" unicode="" horiz-adv-x="898" + d="M898 1066v-102q0 -14 -9 -23t-23 -9h-168q-23 -144 -129 -234t-276 -110q167 -178 459 -536q14 -16 4 -34q-8 -18 -29 -18h-195q-16 0 -25 12q-306 367 -498 571q-9 9 -9 22v127q0 13 9.5 22.5t22.5 9.5h112q132 0 212.5 43t102.5 125h-427q-14 0 -23 9t-9 23v102 + q0 14 9 23t23 9h413q-57 113 -268 113h-145q-13 0 -22.5 9.5t-9.5 22.5v133q0 14 9 23t23 9h832q14 0 23 -9t9 -23v-102q0 -14 -9 -23t-23 -9h-233q47 -61 64 -144h171q14 0 23 -9t9 -23z" /> + <glyph glyph-name="jpy" unicode="" horiz-adv-x="1027" + d="M603 0h-172q-13 0 -22.5 9t-9.5 23v330h-288q-13 0 -22.5 9t-9.5 23v103q0 13 9.5 22.5t22.5 9.5h288v85h-288q-13 0 -22.5 9t-9.5 23v104q0 13 9.5 22.5t22.5 9.5h214l-321 578q-8 16 0 32q10 16 28 16h194q19 0 29 -18l215 -425q19 -38 56 -125q10 24 30.5 68t27.5 61 + l191 420q8 19 29 19h191q17 0 27 -16q9 -14 1 -31l-313 -579h215q13 0 22.5 -9.5t9.5 -22.5v-104q0 -14 -9.5 -23t-22.5 -9h-290v-85h290q13 0 22.5 -9.5t9.5 -22.5v-103q0 -14 -9.5 -23t-22.5 -9h-290v-330q0 -13 -9.5 -22.5t-22.5 -9.5z" /> + <glyph glyph-name="rub" unicode="" horiz-adv-x="1280" + d="M1043 971q0 100 -65 162t-171 62h-320v-448h320q106 0 171 62t65 162zM1280 971q0 -193 -126.5 -315t-326.5 -122h-340v-118h505q14 0 23 -9t9 -23v-128q0 -14 -9 -23t-23 -9h-505v-192q0 -14 -9.5 -23t-22.5 -9h-167q-14 0 -23 9t-9 23v192h-224q-14 0 -23 9t-9 23v128 + q0 14 9 23t23 9h224v118h-224q-14 0 -23 9t-9 23v149q0 13 9 22.5t23 9.5h224v629q0 14 9 23t23 9h539q200 0 326.5 -122t126.5 -315z" /> + <glyph glyph-name="krw" unicode="" horiz-adv-x="1792" + d="M514 341l81 299h-159l75 -300q1 -1 1 -3t1 -3q0 1 0.5 3.5t0.5 3.5zM630 768l35 128h-292l32 -128h225zM822 768h139l-35 128h-70zM1271 340l78 300h-162l81 -299q0 -1 0.5 -3.5t1.5 -3.5q0 1 0.5 3t0.5 3zM1382 768l33 128h-297l34 -128h230zM1792 736v-64q0 -14 -9 -23 + t-23 -9h-213l-164 -616q-7 -24 -31 -24h-159q-24 0 -31 24l-166 616h-209l-167 -616q-7 -24 -31 -24h-159q-11 0 -19.5 7t-10.5 17l-160 616h-208q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h175l-33 128h-142q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h109l-89 344q-5 15 5 28 + q10 12 26 12h137q26 0 31 -24l90 -360h359l97 360q7 24 31 24h126q24 0 31 -24l98 -360h365l93 360q5 24 31 24h137q16 0 26 -12q10 -13 5 -28l-91 -344h111q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-145l-34 -128h179q14 0 23 -9t9 -23z" /> + <glyph glyph-name="btc" unicode="" horiz-adv-x="1280" + d="M1167 896q18 -182 -131 -258q117 -28 175 -103t45 -214q-7 -71 -32.5 -125t-64.5 -89t-97 -58.5t-121.5 -34.5t-145.5 -15v-255h-154v251q-80 0 -122 1v-252h-154v255q-18 0 -54 0.5t-55 0.5h-200l31 183h111q50 0 58 51v402h16q-6 1 -16 1v287q-13 68 -89 68h-111v164 + l212 -1q64 0 97 1v252h154v-247q82 2 122 2v245h154v-252q79 -7 140 -22.5t113 -45t82.5 -78t36.5 -114.5zM952 351q0 36 -15 64t-37 46t-57.5 30.5t-65.5 18.5t-74 9t-69 3t-64.5 -1t-47.5 -1v-338q8 0 37 -0.5t48 -0.5t53 1.5t58.5 4t57 8.5t55.5 14t47.5 21t39.5 30 + t24.5 40t9.5 51zM881 827q0 33 -12.5 58.5t-30.5 42t-48 28t-55 16.5t-61.5 8t-58 2.5t-54 -1t-39.5 -0.5v-307q5 0 34.5 -0.5t46.5 0t50 2t55 5.5t51.5 11t48.5 18.5t37 27t27 38.5t9 51z" /> + <glyph glyph-name="file" unicode="" + d="M1024 1024v472q22 -14 36 -28l408 -408q14 -14 28 -36h-472zM896 992q0 -40 28 -68t68 -28h544v-1056q0 -40 -28 -68t-68 -28h-1344q-40 0 -68 28t-28 68v1600q0 40 28 68t68 28h800v-544z" /> + <glyph glyph-name="file_text" unicode="" + d="M1468 1060q14 -14 28 -36h-472v472q22 -14 36 -28zM992 896h544v-1056q0 -40 -28 -68t-68 -28h-1344q-40 0 -68 28t-28 68v1600q0 40 28 68t68 28h800v-544q0 -40 28 -68t68 -28zM1152 160v64q0 14 -9 23t-23 9h-704q-14 0 -23 -9t-9 -23v-64q0 -14 9 -23t23 -9h704 + q14 0 23 9t9 23zM1152 416v64q0 14 -9 23t-23 9h-704q-14 0 -23 -9t-9 -23v-64q0 -14 9 -23t23 -9h704q14 0 23 9t9 23zM1152 672v64q0 14 -9 23t-23 9h-704q-14 0 -23 -9t-9 -23v-64q0 -14 9 -23t23 -9h704q14 0 23 9t9 23z" /> + <glyph glyph-name="sort_by_alphabet" unicode="" horiz-adv-x="1664" + d="M1191 1128h177l-72 218l-12 47q-2 16 -2 20h-4l-3 -20q0 -1 -3.5 -18t-7.5 -29zM736 96q0 -12 -10 -24l-319 -319q-10 -9 -23 -9q-12 0 -23 9l-320 320q-15 16 -7 35q8 20 30 20h192v1376q0 14 9 23t23 9h192q14 0 23 -9t9 -23v-1376h192q14 0 23 -9t9 -23zM1572 -23 + v-233h-584v90l369 529q12 18 21 27l11 9v3q-2 0 -6.5 -0.5t-7.5 -0.5q-12 -3 -30 -3h-232v-115h-120v229h567v-89l-369 -530q-6 -8 -21 -26l-11 -11v-2l14 2q9 2 30 2h248v119h121zM1661 874v-106h-288v106h75l-47 144h-243l-47 -144h75v-106h-287v106h70l230 662h162 + l230 -662h70z" /> + <glyph glyph-name="_329" unicode="" horiz-adv-x="1664" + d="M1191 104h177l-72 218l-12 47q-2 16 -2 20h-4l-3 -20q0 -1 -3.5 -18t-7.5 -29zM736 96q0 -12 -10 -24l-319 -319q-10 -9 -23 -9q-12 0 -23 9l-320 320q-15 16 -7 35q8 20 30 20h192v1376q0 14 9 23t23 9h192q14 0 23 -9t9 -23v-1376h192q14 0 23 -9t9 -23zM1661 -150 + v-106h-288v106h75l-47 144h-243l-47 -144h75v-106h-287v106h70l230 662h162l230 -662h70zM1572 1001v-233h-584v90l369 529q12 18 21 27l11 9v3q-2 0 -6.5 -0.5t-7.5 -0.5q-12 -3 -30 -3h-232v-115h-120v229h567v-89l-369 -530q-6 -8 -21 -26l-11 -10v-3l14 3q9 1 30 1h248 + v119h121z" /> + <glyph glyph-name="sort_by_attributes" unicode="" horiz-adv-x="1792" + d="M736 96q0 -12 -10 -24l-319 -319q-10 -9 -23 -9q-12 0 -23 9l-320 320q-15 16 -7 35q8 20 30 20h192v1376q0 14 9 23t23 9h192q14 0 23 -9t9 -23v-1376h192q14 0 23 -9t9 -23zM1792 -32v-192q0 -14 -9 -23t-23 -9h-832q-14 0 -23 9t-9 23v192q0 14 9 23t23 9h832 + q14 0 23 -9t9 -23zM1600 480v-192q0 -14 -9 -23t-23 -9h-640q-14 0 -23 9t-9 23v192q0 14 9 23t23 9h640q14 0 23 -9t9 -23zM1408 992v-192q0 -14 -9 -23t-23 -9h-448q-14 0 -23 9t-9 23v192q0 14 9 23t23 9h448q14 0 23 -9t9 -23zM1216 1504v-192q0 -14 -9 -23t-23 -9h-256 + q-14 0 -23 9t-9 23v192q0 14 9 23t23 9h256q14 0 23 -9t9 -23z" /> + <glyph glyph-name="sort_by_attributes_alt" unicode="" horiz-adv-x="1792" + d="M1216 -32v-192q0 -14 -9 -23t-23 -9h-256q-14 0 -23 9t-9 23v192q0 14 9 23t23 9h256q14 0 23 -9t9 -23zM736 96q0 -12 -10 -24l-319 -319q-10 -9 -23 -9q-12 0 -23 9l-320 320q-15 16 -7 35q8 20 30 20h192v1376q0 14 9 23t23 9h192q14 0 23 -9t9 -23v-1376h192 + q14 0 23 -9t9 -23zM1408 480v-192q0 -14 -9 -23t-23 -9h-448q-14 0 -23 9t-9 23v192q0 14 9 23t23 9h448q14 0 23 -9t9 -23zM1600 992v-192q0 -14 -9 -23t-23 -9h-640q-14 0 -23 9t-9 23v192q0 14 9 23t23 9h640q14 0 23 -9t9 -23zM1792 1504v-192q0 -14 -9 -23t-23 -9h-832 + q-14 0 -23 9t-9 23v192q0 14 9 23t23 9h832q14 0 23 -9t9 -23z" /> + <glyph glyph-name="sort_by_order" unicode="" + d="M1346 223q0 63 -44 116t-103 53q-52 0 -83 -37t-31 -94t36.5 -95t104.5 -38q50 0 85 27t35 68zM736 96q0 -12 -10 -24l-319 -319q-10 -9 -23 -9q-12 0 -23 9l-320 320q-15 16 -7 35q8 20 30 20h192v1376q0 14 9 23t23 9h192q14 0 23 -9t9 -23v-1376h192q14 0 23 -9t9 -23 + zM1486 165q0 -62 -13 -121.5t-41 -114t-68 -95.5t-98.5 -65.5t-127.5 -24.5q-62 0 -108 16q-24 8 -42 15l39 113q15 -7 31 -11q37 -13 75 -13q84 0 134.5 58.5t66.5 145.5h-2q-21 -23 -61.5 -37t-84.5 -14q-106 0 -173 71.5t-67 172.5q0 105 72 178t181 73q123 0 205 -94.5 + t82 -252.5zM1456 882v-114h-469v114h167v432q0 7 0.5 19t0.5 17v16h-2l-7 -12q-8 -13 -26 -31l-62 -58l-82 86l192 185h123v-654h165z" /> + <glyph glyph-name="sort_by_order_alt" unicode="" + d="M1346 1247q0 63 -44 116t-103 53q-52 0 -83 -37t-31 -94t36.5 -95t104.5 -38q50 0 85 27t35 68zM736 96q0 -12 -10 -24l-319 -319q-10 -9 -23 -9q-12 0 -23 9l-320 320q-15 16 -7 35q8 20 30 20h192v1376q0 14 9 23t23 9h192q14 0 23 -9t9 -23v-1376h192q14 0 23 -9 + t9 -23zM1456 -142v-114h-469v114h167v432q0 7 0.5 19t0.5 17v16h-2l-7 -12q-8 -13 -26 -31l-62 -58l-82 86l192 185h123v-654h165zM1486 1189q0 -62 -13 -121.5t-41 -114t-68 -95.5t-98.5 -65.5t-127.5 -24.5q-62 0 -108 16q-24 8 -42 15l39 113q15 -7 31 -11q37 -13 75 -13 + q84 0 134.5 58.5t66.5 145.5h-2q-21 -23 -61.5 -37t-84.5 -14q-106 0 -173 71.5t-67 172.5q0 105 72 178t181 73q123 0 205 -94.5t82 -252.5z" /> + <glyph glyph-name="_334" unicode="" horiz-adv-x="1664" + d="M256 192q0 26 -19 45t-45 19q-27 0 -45.5 -19t-18.5 -45q0 -27 18.5 -45.5t45.5 -18.5q26 0 45 18.5t19 45.5zM416 704v-640q0 -26 -19 -45t-45 -19h-288q-26 0 -45 19t-19 45v640q0 26 19 45t45 19h288q26 0 45 -19t19 -45zM1600 704q0 -86 -55 -149q15 -44 15 -76 + q3 -76 -43 -137q17 -56 0 -117q-15 -57 -54 -94q9 -112 -49 -181q-64 -76 -197 -78h-36h-76h-17q-66 0 -144 15.5t-121.5 29t-120.5 39.5q-123 43 -158 44q-26 1 -45 19.5t-19 44.5v641q0 25 18 43.5t43 20.5q24 2 76 59t101 121q68 87 101 120q18 18 31 48t17.5 48.5 + t13.5 60.5q7 39 12.5 61t19.5 52t34 50q19 19 45 19q46 0 82.5 -10.5t60 -26t40 -40.5t24 -45t12 -50t5 -45t0.5 -39q0 -38 -9.5 -76t-19 -60t-27.5 -56q-3 -6 -10 -18t-11 -22t-8 -24h277q78 0 135 -57t57 -135z" /> + <glyph glyph-name="_335" unicode="" horiz-adv-x="1664" + d="M256 960q0 -26 -19 -45t-45 -19q-27 0 -45.5 19t-18.5 45q0 27 18.5 45.5t45.5 18.5q26 0 45 -18.5t19 -45.5zM416 448v640q0 26 -19 45t-45 19h-288q-26 0 -45 -19t-19 -45v-640q0 -26 19 -45t45 -19h288q26 0 45 19t19 45zM1545 597q55 -61 55 -149q-1 -78 -57.5 -135 + t-134.5 -57h-277q4 -14 8 -24t11 -22t10 -18q18 -37 27 -57t19 -58.5t10 -76.5q0 -24 -0.5 -39t-5 -45t-12 -50t-24 -45t-40 -40.5t-60 -26t-82.5 -10.5q-26 0 -45 19q-20 20 -34 50t-19.5 52t-12.5 61q-9 42 -13.5 60.5t-17.5 48.5t-31 48q-33 33 -101 120q-49 64 -101 121 + t-76 59q-25 2 -43 20.5t-18 43.5v641q0 26 19 44.5t45 19.5q35 1 158 44q77 26 120.5 39.5t121.5 29t144 15.5h17h76h36q133 -2 197 -78q58 -69 49 -181q39 -37 54 -94q17 -61 0 -117q46 -61 43 -137q0 -32 -15 -76z" /> + <glyph glyph-name="youtube_sign" unicode="" + d="M919 233v157q0 50 -29 50q-17 0 -33 -16v-224q16 -16 33 -16q29 0 29 49zM1103 355h66v34q0 51 -33 51t-33 -51v-34zM532 621v-70h-80v-423h-74v423h-78v70h232zM733 495v-367h-67v40q-39 -45 -76 -45q-33 0 -42 28q-6 17 -6 54v290h66v-270q0 -24 1 -26q1 -15 15 -15 + q20 0 42 31v280h67zM985 384v-146q0 -52 -7 -73q-12 -42 -53 -42q-35 0 -68 41v-36h-67v493h67v-161q32 40 68 40q41 0 53 -42q7 -21 7 -74zM1236 255v-9q0 -29 -2 -43q-3 -22 -15 -40q-27 -40 -80 -40q-52 0 -81 38q-21 27 -21 86v129q0 59 20 86q29 38 80 38t78 -38 + q21 -29 21 -86v-76h-133v-65q0 -51 34 -51q24 0 30 26q0 1 0.5 7t0.5 16.5v21.5h68zM785 1079v-156q0 -51 -32 -51t-32 51v156q0 52 32 52t32 -52zM1318 366q0 177 -19 260q-10 44 -43 73.5t-76 34.5q-136 15 -412 15q-275 0 -411 -15q-44 -5 -76.5 -34.5t-42.5 -73.5 + q-20 -87 -20 -260q0 -176 20 -260q10 -43 42.5 -73t75.5 -35q137 -15 412 -15t412 15q43 5 75.5 35t42.5 73q20 84 20 260zM563 1017l90 296h-75l-51 -195l-53 195h-78q7 -23 23 -69l24 -69q35 -103 46 -158v-201h74v201zM852 936v130q0 58 -21 87q-29 38 -78 38 + q-51 0 -78 -38q-21 -29 -21 -87v-130q0 -58 21 -87q27 -38 78 -38q49 0 78 38q21 27 21 87zM1033 816h67v370h-67v-283q-22 -31 -42 -31q-15 0 -16 16q-1 2 -1 26v272h-67v-293q0 -37 6 -55q11 -27 43 -27q36 0 77 45v-40zM1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5 + h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="youtube" unicode="" + d="M971 292v-211q0 -67 -39 -67q-23 0 -45 22v301q22 22 45 22q39 0 39 -67zM1309 291v-46h-90v46q0 68 45 68t45 -68zM343 509h107v94h-312v-94h105v-569h100v569zM631 -60h89v494h-89v-378q-30 -42 -57 -42q-18 0 -21 21q-1 3 -1 35v364h-89v-391q0 -49 8 -73 + q12 -37 58 -37q48 0 102 61v-54zM1060 88v197q0 73 -9 99q-17 56 -71 56q-50 0 -93 -54v217h-89v-663h89v48q45 -55 93 -55q54 0 71 55q9 27 9 100zM1398 98v13h-91q0 -51 -2 -61q-7 -36 -40 -36q-46 0 -46 69v87h179v103q0 79 -27 116q-39 51 -106 51q-68 0 -107 -51 + q-28 -37 -28 -116v-173q0 -79 29 -116q39 -51 108 -51q72 0 108 53q18 27 21 54q2 9 2 58zM790 1011v210q0 69 -43 69t-43 -69v-210q0 -70 43 -70t43 70zM1509 260q0 -234 -26 -350q-14 -59 -58 -99t-102 -46q-184 -21 -555 -21t-555 21q-58 6 -102.5 46t-57.5 99 + q-26 112 -26 350q0 234 26 350q14 59 58 99t103 47q183 20 554 20t555 -20q58 -7 102.5 -47t57.5 -99q26 -112 26 -350zM511 1536h102l-121 -399v-271h-100v271q-14 74 -61 212q-37 103 -65 187h106l71 -263zM881 1203v-175q0 -81 -28 -118q-38 -51 -106 -51q-67 0 -105 51 + q-28 38 -28 118v175q0 80 28 117q38 51 105 51q68 0 106 -51q28 -37 28 -117zM1216 1365v-499h-91v55q-53 -62 -103 -62q-46 0 -59 37q-8 24 -8 75v394h91v-367q0 -33 1 -35q3 -22 21 -22q27 0 57 43v381h91z" /> + <glyph glyph-name="xing" unicode="" horiz-adv-x="1408" + d="M597 869q-10 -18 -257 -456q-27 -46 -65 -46h-239q-21 0 -31 17t0 36l253 448q1 0 0 1l-161 279q-12 22 -1 37q9 15 32 15h239q40 0 66 -45zM1403 1511q11 -16 0 -37l-528 -934v-1l336 -615q11 -20 1 -37q-10 -15 -32 -15h-239q-42 0 -66 45l-339 622q18 32 531 942 + q25 45 64 45h241q22 0 31 -15z" /> + <glyph glyph-name="xing_sign" unicode="" + d="M685 771q0 1 -126 222q-21 34 -52 34h-184q-18 0 -26 -11q-7 -12 1 -29l125 -216v-1l-196 -346q-9 -14 0 -28q8 -13 24 -13h185q31 0 50 36zM1309 1268q-7 12 -24 12h-187q-30 0 -49 -35l-411 -729q1 -2 262 -481q20 -35 52 -35h184q18 0 25 12q8 13 -1 28l-260 476v1 + l409 723q8 16 0 28zM1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="youtube_play" unicode="" horiz-adv-x="1792" + d="M711 408l484 250l-484 253v-503zM896 1270q168 0 324.5 -4.5t229.5 -9.5l73 -4q1 0 17 -1.5t23 -3t23.5 -4.5t28.5 -8t28 -13t31 -19.5t29 -26.5q6 -6 15.5 -18.5t29 -58.5t26.5 -101q8 -64 12.5 -136.5t5.5 -113.5v-40v-136q1 -145 -18 -290q-7 -55 -25 -99.5t-32 -61.5 + l-14 -17q-14 -15 -29 -26.5t-31 -19t-28 -12.5t-28.5 -8t-24 -4.5t-23 -3t-16.5 -1.5q-251 -19 -627 -19q-207 2 -359.5 6.5t-200.5 7.5l-49 4l-36 4q-36 5 -54.5 10t-51 21t-56.5 41q-6 6 -15.5 18.5t-29 58.5t-26.5 101q-8 64 -12.5 136.5t-5.5 113.5v40v136 + q-1 145 18 290q7 55 25 99.5t32 61.5l14 17q14 15 29 26.5t31 19.5t28 13t28.5 8t23.5 4.5t23 3t17 1.5q251 18 627 18z" /> + <glyph glyph-name="dropbox" unicode="" horiz-adv-x="1792" + d="M402 829l494 -305l-342 -285l-490 319zM1388 274v-108l-490 -293v-1l-1 1l-1 -1v1l-489 293v108l147 -96l342 284v2l1 -1l1 1v-2l343 -284zM554 1418l342 -285l-494 -304l-338 270zM1390 829l338 -271l-489 -319l-343 285zM1239 1418l489 -319l-338 -270l-494 304z" /> + <glyph glyph-name="stackexchange" unicode="" + d="M1289 -96h-1118v480h-160v-640h1438v640h-160v-480zM347 428l33 157l783 -165l-33 -156zM450 802l67 146l725 -339l-67 -145zM651 1158l102 123l614 -513l-102 -123zM1048 1536l477 -641l-128 -96l-477 641zM330 65v159h800v-159h-800z" /> + <glyph glyph-name="instagram" unicode="" + d="M1024 640q0 106 -75 181t-181 75t-181 -75t-75 -181t75 -181t181 -75t181 75t75 181zM1162 640q0 -164 -115 -279t-279 -115t-279 115t-115 279t115 279t279 115t279 -115t115 -279zM1270 1050q0 -38 -27 -65t-65 -27t-65 27t-27 65t27 65t65 27t65 -27t27 -65zM768 1270 + q-7 0 -76.5 0.5t-105.5 0t-96.5 -3t-103 -10t-71.5 -18.5q-50 -20 -88 -58t-58 -88q-11 -29 -18.5 -71.5t-10 -103t-3 -96.5t0 -105.5t0.5 -76.5t-0.5 -76.5t0 -105.5t3 -96.5t10 -103t18.5 -71.5q20 -50 58 -88t88 -58q29 -11 71.5 -18.5t103 -10t96.5 -3t105.5 0t76.5 0.5 + t76.5 -0.5t105.5 0t96.5 3t103 10t71.5 18.5q50 20 88 58t58 88q11 29 18.5 71.5t10 103t3 96.5t0 105.5t-0.5 76.5t0.5 76.5t0 105.5t-3 96.5t-10 103t-18.5 71.5q-20 50 -58 88t-88 58q-29 11 -71.5 18.5t-103 10t-96.5 3t-105.5 0t-76.5 -0.5zM1536 640q0 -229 -5 -317 + q-10 -208 -124 -322t-322 -124q-88 -5 -317 -5t-317 5q-208 10 -322 124t-124 322q-5 88 -5 317t5 317q10 208 124 322t322 124q88 5 317 5t317 -5q208 -10 322 -124t124 -322q5 -88 5 -317z" /> + <glyph glyph-name="flickr" unicode="" + d="M1248 1408q119 0 203.5 -84.5t84.5 -203.5v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960zM698 640q0 88 -62 150t-150 62t-150 -62t-62 -150t62 -150t150 -62t150 62t62 150zM1262 640q0 88 -62 150 + t-150 62t-150 -62t-62 -150t62 -150t150 -62t150 62t62 150z" /> + <glyph glyph-name="adn" unicode="" + d="M768 914l201 -306h-402zM1133 384h94l-459 691l-459 -691h94l104 160h522zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="f171" unicode="" horiz-adv-x="1408" + d="M815 677q8 -63 -50.5 -101t-111.5 -6q-39 17 -53.5 58t-0.5 82t52 58q36 18 72.5 12t64 -35.5t27.5 -67.5zM926 698q-14 107 -113 164t-197 13q-63 -28 -100.5 -88.5t-34.5 -129.5q4 -91 77.5 -155t165.5 -56q91 8 152 84t50 168zM1165 1240q-20 27 -56 44.5t-58 22 + t-71 12.5q-291 47 -566 -2q-43 -7 -66 -12t-55 -22t-50 -43q30 -28 76 -45.5t73.5 -22t87.5 -11.5q228 -29 448 -1q63 8 89.5 12t72.5 21.5t75 46.5zM1222 205q-8 -26 -15.5 -76.5t-14 -84t-28.5 -70t-58 -56.5q-86 -48 -189.5 -71.5t-202 -22t-201.5 18.5q-46 8 -81.5 18 + t-76.5 27t-73 43.5t-52 61.5q-25 96 -57 292l6 16l18 9q223 -148 506.5 -148t507.5 148q21 -6 24 -23t-5 -45t-8 -37zM1403 1166q-26 -167 -111 -655q-5 -30 -27 -56t-43.5 -40t-54.5 -31q-252 -126 -610 -88q-248 27 -394 139q-15 12 -25.5 26.5t-17 35t-9 34t-6 39.5 + t-5.5 35q-9 50 -26.5 150t-28 161.5t-23.5 147.5t-22 158q3 26 17.5 48.5t31.5 37.5t45 30t46 22.5t48 18.5q125 46 313 64q379 37 676 -50q155 -46 215 -122q16 -20 16.5 -51t-5.5 -54z" /> + <glyph glyph-name="bitbucket_sign" unicode="" + d="M848 666q0 43 -41 66t-77 1q-43 -20 -42.5 -72.5t43.5 -70.5q39 -23 81 4t36 72zM928 682q8 -66 -36 -121t-110 -61t-119 40t-56 113q-2 49 25.5 93t72.5 64q70 31 141.5 -10t81.5 -118zM1100 1073q-20 -21 -53.5 -34t-53 -16t-63.5 -8q-155 -20 -324 0q-44 6 -63 9.5 + t-52.5 16t-54.5 32.5q13 19 36 31t40 15.5t47 8.5q198 35 408 1q33 -5 51 -8.5t43 -16t39 -31.5zM1142 327q0 7 5.5 26.5t3 32t-17.5 16.5q-161 -106 -365 -106t-366 106l-12 -6l-5 -12q26 -154 41 -210q47 -81 204 -108q249 -46 428 53q34 19 49 51.5t22.5 85.5t12.5 71z + M1272 1020q9 53 -8 75q-43 55 -155 88q-216 63 -487 36q-132 -12 -226 -46q-38 -15 -59.5 -25t-47 -34t-29.5 -54q8 -68 19 -138t29 -171t24 -137q1 -5 5 -31t7 -36t12 -27t22 -28q105 -80 284 -100q259 -28 440 63q24 13 39.5 23t31 29t19.5 40q48 267 80 473zM1536 1120 + v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="tumblr" unicode="" horiz-adv-x="1024" + d="M944 207l80 -237q-23 -35 -111 -66t-177 -32q-104 -2 -190.5 26t-142.5 74t-95 106t-55.5 120t-16.5 118v544h-168v215q72 26 129 69.5t91 90t58 102t34 99t15 88.5q1 5 4.5 8.5t7.5 3.5h244v-424h333v-252h-334v-518q0 -30 6.5 -56t22.5 -52.5t49.5 -41.5t81.5 -14 + q78 2 134 29z" /> + <glyph glyph-name="tumblr_sign" unicode="" + d="M1136 75l-62 183q-44 -22 -103 -22q-36 -1 -62 10.5t-38.5 31.5t-17.5 40.5t-5 43.5v398h257v194h-256v326h-188q-8 0 -9 -10q-5 -44 -17.5 -87t-39 -95t-77 -95t-118.5 -68v-165h130v-418q0 -57 21.5 -115t65 -111t121 -85.5t176.5 -30.5q69 1 136.5 25t85.5 50z + M1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="long_arrow_down" unicode="" horiz-adv-x="768" + d="M765 237q8 -19 -5 -35l-350 -384q-10 -10 -23 -10q-14 0 -24 10l-355 384q-13 16 -5 35q9 19 29 19h224v1248q0 14 9 23t23 9h192q14 0 23 -9t9 -23v-1248h224q21 0 29 -19z" /> + <glyph glyph-name="long_arrow_up" unicode="" horiz-adv-x="768" + d="M765 1043q-9 -19 -29 -19h-224v-1248q0 -14 -9 -23t-23 -9h-192q-14 0 -23 9t-9 23v1248h-224q-21 0 -29 19t5 35l350 384q10 10 23 10q14 0 24 -10l355 -384q13 -16 5 -35z" /> + <glyph glyph-name="long_arrow_left" unicode="" horiz-adv-x="1792" + d="M1792 736v-192q0 -14 -9 -23t-23 -9h-1248v-224q0 -21 -19 -29t-35 5l-384 350q-10 10 -10 23q0 14 10 24l384 354q16 14 35 6q19 -9 19 -29v-224h1248q14 0 23 -9t9 -23z" /> + <glyph glyph-name="long_arrow_right" unicode="" horiz-adv-x="1792" + d="M1728 643q0 -14 -10 -24l-384 -354q-16 -14 -35 -6q-19 9 -19 29v224h-1248q-14 0 -23 9t-9 23v192q0 14 9 23t23 9h1248v224q0 21 19 29t35 -5l384 -350q10 -10 10 -23z" /> + <glyph glyph-name="apple" unicode="" horiz-adv-x="1408" + d="M1393 321q-39 -125 -123 -250q-129 -196 -257 -196q-49 0 -140 32q-86 32 -151 32q-61 0 -142 -33q-81 -34 -132 -34q-152 0 -301 259q-147 261 -147 503q0 228 113 374q113 144 284 144q72 0 177 -30q104 -30 138 -30q45 0 143 34q102 34 173 34q119 0 213 -65 + q52 -36 104 -100q-79 -67 -114 -118q-65 -94 -65 -207q0 -124 69 -223t158 -126zM1017 1494q0 -61 -29 -136q-30 -75 -93 -138q-54 -54 -108 -72q-37 -11 -104 -17q3 149 78 257q74 107 250 148q1 -3 2.5 -11t2.5 -11q0 -4 0.5 -10t0.5 -10z" /> + <glyph glyph-name="windows" unicode="" horiz-adv-x="1664" + d="M682 530v-651l-682 94v557h682zM682 1273v-659h-682v565zM1664 530v-786l-907 125v661h907zM1664 1408v-794h-907v669z" /> + <glyph glyph-name="android" unicode="" horiz-adv-x="1408" + d="M493 1053q16 0 27.5 11.5t11.5 27.5t-11.5 27.5t-27.5 11.5t-27 -11.5t-11 -27.5t11 -27.5t27 -11.5zM915 1053q16 0 27 11.5t11 27.5t-11 27.5t-27 11.5t-27.5 -11.5t-11.5 -27.5t11.5 -27.5t27.5 -11.5zM103 869q42 0 72 -30t30 -72v-430q0 -43 -29.5 -73t-72.5 -30 + t-73 30t-30 73v430q0 42 30 72t73 30zM1163 850v-666q0 -46 -32 -78t-77 -32h-75v-227q0 -43 -30 -73t-73 -30t-73 30t-30 73v227h-138v-227q0 -43 -30 -73t-73 -30q-42 0 -72 30t-30 73l-1 227h-74q-46 0 -78 32t-32 78v666h918zM931 1255q107 -55 171 -153.5t64 -215.5 + h-925q0 117 64 215.5t172 153.5l-71 131q-7 13 5 20q13 6 20 -6l72 -132q95 42 201 42t201 -42l72 132q7 12 20 6q12 -7 5 -20zM1408 767v-430q0 -43 -30 -73t-73 -30q-42 0 -72 30t-30 73v430q0 43 30 72.5t72 29.5q43 0 73 -29.5t30 -72.5z" /> + <glyph glyph-name="linux" unicode="" + d="M663 1125q-11 -1 -15.5 -10.5t-8.5 -9.5q-5 -1 -5 5q0 12 19 15h10zM750 1111q-4 -1 -11.5 6.5t-17.5 4.5q24 11 32 -2q3 -6 -3 -9zM399 684q-4 1 -6 -3t-4.5 -12.5t-5.5 -13.5t-10 -13q-10 -11 -1 -12q4 -1 12.5 7t12.5 18q1 3 2 7t2 6t1.5 4.5t0.5 4v3t-1 2.5t-3 2z + M1254 325q0 18 -55 42q4 15 7.5 27.5t5 26t3 21.5t0.5 22.5t-1 19.5t-3.5 22t-4 20.5t-5 25t-5.5 26.5q-10 48 -47 103t-72 75q24 -20 57 -83q87 -162 54 -278q-11 -40 -50 -42q-31 -4 -38.5 18.5t-8 83.5t-11.5 107q-9 39 -19.5 69t-19.5 45.5t-15.5 24.5t-13 15t-7.5 7 + q-14 62 -31 103t-29.5 56t-23.5 33t-15 40q-4 21 6 53.5t4.5 49.5t-44.5 25q-15 3 -44.5 18t-35.5 16q-8 1 -11 26t8 51t36 27q37 3 51 -30t4 -58q-11 -19 -2 -26.5t30 -0.5q13 4 13 36v37q-5 30 -13.5 50t-21 30.5t-23.5 15t-27 7.5q-107 -8 -89 -134q0 -15 -1 -15 + q-9 9 -29.5 10.5t-33 -0.5t-15.5 5q1 57 -16 90t-45 34q-27 1 -41.5 -27.5t-16.5 -59.5q-1 -15 3.5 -37t13 -37.5t15.5 -13.5q10 3 16 14q4 9 -7 8q-7 0 -15.5 14.5t-9.5 33.5q-1 22 9 37t34 14q17 0 27 -21t9.5 -39t-1.5 -22q-22 -15 -31 -29q-8 -12 -27.5 -23.5 + t-20.5 -12.5q-13 -14 -15.5 -27t7.5 -18q14 -8 25 -19.5t16 -19t18.5 -13t35.5 -6.5q47 -2 102 15q2 1 23 7t34.5 10.5t29.5 13t21 17.5q9 14 20 8q5 -3 6.5 -8.5t-3 -12t-16.5 -9.5q-20 -6 -56.5 -21.5t-45.5 -19.5q-44 -19 -70 -23q-25 -5 -79 2q-10 2 -9 -2t17 -19 + q25 -23 67 -22q17 1 36 7t36 14t33.5 17.5t30 17t24.5 12t17.5 2.5t8.5 -11q0 -2 -1 -4.5t-4 -5t-6 -4.5t-8.5 -5t-9 -4.5t-10 -5t-9.5 -4.5q-28 -14 -67.5 -44t-66.5 -43t-49 -1q-21 11 -63 73q-22 31 -25 22q-1 -3 -1 -10q0 -25 -15 -56.5t-29.5 -55.5t-21 -58t11.5 -63 + q-23 -6 -62.5 -90t-47.5 -141q-2 -18 -1.5 -69t-5.5 -59q-8 -24 -29 -3q-32 31 -36 94q-2 28 4 56q4 19 -1 18q-2 -1 -4 -5q-36 -65 10 -166q5 -12 25 -28t24 -20q20 -23 104 -90.5t93 -76.5q16 -15 17.5 -38t-14 -43t-45.5 -23q8 -15 29 -44.5t28 -54t7 -70.5q46 24 7 92 + q-4 8 -10.5 16t-9.5 12t-2 6q3 5 13 9.5t20 -2.5q46 -52 166 -36q133 15 177 87q23 38 34 30q12 -6 10 -52q-1 -25 -23 -92q-9 -23 -6 -37.5t24 -15.5q3 19 14.5 77t13.5 90q2 21 -6.5 73.5t-7.5 97t23 70.5q15 18 51 18q1 37 34.5 53t72.5 10.5t60 -22.5zM626 1152 + q3 17 -2.5 30t-11.5 15q-9 2 -9 -7q2 -5 5 -6q10 0 7 -15q-3 -20 8 -20q3 0 3 3zM1045 955q-2 8 -6.5 11.5t-13 5t-14.5 5.5q-5 3 -9.5 8t-7 8t-5.5 6.5t-4 4t-4 -1.5q-14 -16 7 -43.5t39 -31.5q9 -1 14.5 8t3.5 20zM867 1168q0 11 -5 19.5t-11 12.5t-9 3q-6 0 -8 -2t0 -4 + t5 -3q14 -4 18 -31q0 -3 8 2q2 2 2 3zM921 1401q0 2 -2.5 5t-9 7t-9.5 6q-15 15 -24 15q-9 -1 -11.5 -7.5t-1 -13t-0.5 -12.5q-1 -4 -6 -10.5t-6 -9t3 -8.5q4 -3 8 0t11 9t15 9q1 1 9 1t15 2t9 7zM1486 60q20 -12 31 -24.5t12 -24t-2.5 -22.5t-15.5 -22t-23.5 -19.5 + t-30 -18.5t-31.5 -16.5t-32 -15.5t-27 -13q-38 -19 -85.5 -56t-75.5 -64q-17 -16 -68 -19.5t-89 14.5q-18 9 -29.5 23.5t-16.5 25.5t-22 19.5t-47 9.5q-44 1 -130 1q-19 0 -57 -1.5t-58 -2.5q-44 -1 -79.5 -15t-53.5 -30t-43.5 -28.5t-53.5 -11.5q-29 1 -111 31t-146 43 + q-19 4 -51 9.5t-50 9t-39.5 9.5t-33.5 14.5t-17 19.5q-10 23 7 66.5t18 54.5q1 16 -4 40t-10 42.5t-4.5 36.5t10.5 27q14 12 57 14t60 12q30 18 42 35t12 51q21 -73 -32 -106q-32 -20 -83 -15q-34 3 -43 -10q-13 -15 5 -57q2 -6 8 -18t8.5 -18t4.5 -17t1 -22q0 -15 -17 -49 + t-14 -48q3 -17 37 -26q20 -6 84.5 -18.5t99.5 -20.5q24 -6 74 -22t82.5 -23t55.5 -4q43 6 64.5 28t23 48t-7.5 58.5t-19 52t-20 36.5q-121 190 -169 242q-68 74 -113 40q-11 -9 -15 15q-3 16 -2 38q1 29 10 52t24 47t22 42q8 21 26.5 72t29.5 78t30 61t39 54 + q110 143 124 195q-12 112 -16 310q-2 90 24 151.5t106 104.5q39 21 104 21q53 1 106 -13.5t89 -41.5q57 -42 91.5 -121.5t29.5 -147.5q-5 -95 30 -214q34 -113 133 -218q55 -59 99.5 -163t59.5 -191q8 -49 5 -84.5t-12 -55.5t-20 -22q-10 -2 -23.5 -19t-27 -35.5 + t-40.5 -33.5t-61 -14q-18 1 -31.5 5t-22.5 13.5t-13.5 15.5t-11.5 20.5t-9 19.5q-22 37 -41 30t-28 -49t7 -97q20 -70 1 -195q-10 -65 18 -100.5t73 -33t85 35.5q59 49 89.5 66.5t103.5 42.5q53 18 77 36.5t18.5 34.5t-25 28.5t-51.5 23.5q-33 11 -49.5 48t-15 72.5 + t15.5 47.5q1 -31 8 -56.5t14.5 -40.5t20.5 -28.5t21 -19t21.5 -13t16.5 -9.5z" /> + <glyph glyph-name="dribble" unicode="" + d="M1024 36q-42 241 -140 498h-2l-2 -1q-16 -6 -43 -16.5t-101 -49t-137 -82t-131 -114.5t-103 -148l-15 11q184 -150 418 -150q132 0 256 52zM839 643q-21 49 -53 111q-311 -93 -673 -93q-1 -7 -1 -21q0 -124 44 -236.5t124 -201.5q50 89 123.5 166.5t142.5 124.5t130.5 81 + t99.5 48l37 13q4 1 13 3.5t13 4.5zM732 855q-120 213 -244 378q-138 -65 -234 -186t-128 -272q302 0 606 80zM1416 536q-210 60 -409 29q87 -239 128 -469q111 75 185 189.5t96 250.5zM611 1277q-1 0 -2 -1q1 1 2 1zM1201 1132q-185 164 -433 164q-76 0 -155 -19 + q131 -170 246 -382q69 26 130 60.5t96.5 61.5t65.5 57t37.5 40.5zM1424 647q-3 232 -149 410l-1 -1q-9 -12 -19 -24.5t-43.5 -44.5t-71 -60.5t-100 -65t-131.5 -64.5q25 -53 44 -95q2 -5 6.5 -17t7.5 -17q36 5 74.5 7t73.5 2t69 -1.5t64 -4t56.5 -5.5t48 -6.5t36.5 -6 + t25 -4.5zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="skype" unicode="" + d="M1173 473q0 50 -19.5 91.5t-48.5 68.5t-73 49t-82.5 34t-87.5 23l-104 24q-30 7 -44 10.5t-35 11.5t-30 16t-16.5 21t-7.5 30q0 77 144 77q43 0 77 -12t54 -28.5t38 -33.5t40 -29t48 -12q47 0 75.5 32t28.5 77q0 55 -56 99.5t-142 67.5t-182 23q-68 0 -132 -15.5 + t-119.5 -47t-89 -87t-33.5 -128.5q0 -61 19 -106.5t56 -75.5t80 -48.5t103 -32.5l146 -36q90 -22 112 -36q32 -20 32 -60q0 -39 -40 -64.5t-105 -25.5q-51 0 -91.5 16t-65 38.5t-45.5 45t-46 38.5t-54 16q-50 0 -75.5 -30t-25.5 -75q0 -92 122 -157.5t291 -65.5 + q73 0 140 18.5t122.5 53.5t88.5 93.5t33 131.5zM1536 256q0 -159 -112.5 -271.5t-271.5 -112.5q-130 0 -234 80q-77 -16 -150 -16q-143 0 -273.5 55.5t-225 150t-150 225t-55.5 273.5q0 73 16 150q-80 104 -80 234q0 159 112.5 271.5t271.5 112.5q130 0 234 -80 + q77 16 150 16q143 0 273.5 -55.5t225 -150t150 -225t55.5 -273.5q0 -73 -16 -150q80 -104 80 -234z" /> + <glyph glyph-name="foursquare" unicode="" horiz-adv-x="1280" + d="M1000 1102l37 194q5 23 -9 40t-35 17h-712q-23 0 -38.5 -17t-15.5 -37v-1101q0 -7 6 -1l291 352q23 26 38 33.5t48 7.5h239q22 0 37 14.5t18 29.5q24 130 37 191q4 21 -11.5 40t-36.5 19h-294q-29 0 -48 19t-19 48v42q0 29 19 47.5t48 18.5h346q18 0 35 13.5t20 29.5z + M1227 1324q-15 -73 -53.5 -266.5t-69.5 -350t-35 -173.5q-6 -22 -9 -32.5t-14 -32.5t-24.5 -33t-38.5 -21t-58 -10h-271q-13 0 -22 -10q-8 -9 -426 -494q-22 -25 -58.5 -28.5t-48.5 5.5q-55 22 -55 98v1410q0 55 38 102.5t120 47.5h888q95 0 127 -53t10 -159zM1227 1324 + l-158 -790q4 17 35 173.5t69.5 350t53.5 266.5z" /> + <glyph glyph-name="trello" unicode="" + d="M704 192v1024q0 14 -9 23t-23 9h-480q-14 0 -23 -9t-9 -23v-1024q0 -14 9 -23t23 -9h480q14 0 23 9t9 23zM1376 576v640q0 14 -9 23t-23 9h-480q-14 0 -23 -9t-9 -23v-640q0 -14 9 -23t23 -9h480q14 0 23 9t9 23zM1536 1344v-1408q0 -26 -19 -45t-45 -19h-1408 + q-26 0 -45 19t-19 45v1408q0 26 19 45t45 19h1408q26 0 45 -19t19 -45z" /> + <glyph glyph-name="female" unicode="" horiz-adv-x="1280" + d="M1280 480q0 -40 -28 -68t-68 -28q-51 0 -80 43l-227 341h-45v-132l247 -411q9 -15 9 -33q0 -26 -19 -45t-45 -19h-192v-272q0 -46 -33 -79t-79 -33h-160q-46 0 -79 33t-33 79v272h-192q-26 0 -45 19t-19 45q0 18 9 33l247 411v132h-45l-227 -341q-29 -43 -80 -43 + q-40 0 -68 28t-28 68q0 29 16 53l256 384q73 107 176 107h384q103 0 176 -107l256 -384q16 -24 16 -53zM864 1280q0 -93 -65.5 -158.5t-158.5 -65.5t-158.5 65.5t-65.5 158.5t65.5 158.5t158.5 65.5t158.5 -65.5t65.5 -158.5z" /> + <glyph glyph-name="male" unicode="" horiz-adv-x="1024" + d="M1024 832v-416q0 -40 -28 -68t-68 -28t-68 28t-28 68v352h-64v-912q0 -46 -33 -79t-79 -33t-79 33t-33 79v464h-64v-464q0 -46 -33 -79t-79 -33t-79 33t-33 79v912h-64v-352q0 -40 -28 -68t-68 -28t-68 28t-28 68v416q0 80 56 136t136 56h640q80 0 136 -56t56 -136z + M736 1280q0 -93 -65.5 -158.5t-158.5 -65.5t-158.5 65.5t-65.5 158.5t65.5 158.5t158.5 65.5t158.5 -65.5t65.5 -158.5z" /> + <glyph glyph-name="gittip" unicode="" + d="M773 234l350 473q16 22 24.5 59t-6 85t-61.5 79q-40 26 -83 25.5t-73.5 -17.5t-54.5 -45q-36 -40 -96 -40q-59 0 -95 40q-24 28 -54.5 45t-73.5 17.5t-84 -25.5q-46 -31 -60.5 -79t-6 -85t24.5 -59zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103 + t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="sun" unicode="" horiz-adv-x="1792" + d="M1472 640q0 117 -45.5 223.5t-123 184t-184 123t-223.5 45.5t-223.5 -45.5t-184 -123t-123 -184t-45.5 -223.5t45.5 -223.5t123 -184t184 -123t223.5 -45.5t223.5 45.5t184 123t123 184t45.5 223.5zM1748 363q-4 -15 -20 -20l-292 -96v-306q0 -16 -13 -26q-15 -10 -29 -4 + l-292 94l-180 -248q-10 -13 -26 -13t-26 13l-180 248l-292 -94q-14 -6 -29 4q-13 10 -13 26v306l-292 96q-16 5 -20 20q-5 17 4 29l180 248l-180 248q-9 13 -4 29q4 15 20 20l292 96v306q0 16 13 26q15 10 29 4l292 -94l180 248q9 12 26 12t26 -12l180 -248l292 94 + q14 6 29 -4q13 -10 13 -26v-306l292 -96q16 -5 20 -20q5 -16 -4 -29l-180 -248l180 -248q9 -12 4 -29z" /> + <glyph glyph-name="_366" unicode="" + d="M1262 233q-54 -9 -110 -9q-182 0 -337 90t-245 245t-90 337q0 192 104 357q-201 -60 -328.5 -229t-127.5 -384q0 -130 51 -248.5t136.5 -204t204 -136.5t248.5 -51q144 0 273.5 61.5t220.5 171.5zM1465 318q-94 -203 -283.5 -324.5t-413.5 -121.5q-156 0 -298 61 + t-245 164t-164 245t-61 298q0 153 57.5 292.5t156 241.5t235.5 164.5t290 68.5q44 2 61 -39q18 -41 -15 -72q-86 -78 -131.5 -181.5t-45.5 -218.5q0 -148 73 -273t198 -198t273 -73q118 0 228 51q41 18 72 -13q14 -14 17.5 -34t-4.5 -38z" /> + <glyph glyph-name="archive" unicode="" horiz-adv-x="1792" + d="M1088 704q0 26 -19 45t-45 19h-256q-26 0 -45 -19t-19 -45t19 -45t45 -19h256q26 0 45 19t19 45zM1664 896v-960q0 -26 -19 -45t-45 -19h-1408q-26 0 -45 19t-19 45v960q0 26 19 45t45 19h1408q26 0 45 -19t19 -45zM1728 1344v-256q0 -26 -19 -45t-45 -19h-1536 + q-26 0 -45 19t-19 45v256q0 26 19 45t45 19h1536q26 0 45 -19t19 -45z" /> + <glyph glyph-name="bug" unicode="" horiz-adv-x="1664" + d="M1632 576q0 -26 -19 -45t-45 -19h-224q0 -171 -67 -290l208 -209q19 -19 19 -45t-19 -45q-18 -19 -45 -19t-45 19l-198 197q-5 -5 -15 -13t-42 -28.5t-65 -36.5t-82 -29t-97 -13v896h-128v-896q-51 0 -101.5 13.5t-87 33t-66 39t-43.5 32.5l-15 14l-183 -207 + q-20 -21 -48 -21q-24 0 -43 16q-19 18 -20.5 44.5t15.5 46.5l202 227q-58 114 -58 274h-224q-26 0 -45 19t-19 45t19 45t45 19h224v294l-173 173q-19 19 -19 45t19 45t45 19t45 -19l173 -173h844l173 173q19 19 45 19t45 -19t19 -45t-19 -45l-173 -173v-294h224q26 0 45 -19 + t19 -45zM1152 1152h-640q0 133 93.5 226.5t226.5 93.5t226.5 -93.5t93.5 -226.5z" /> + <glyph glyph-name="vk" unicode="" horiz-adv-x="1920" + d="M1917 1016q23 -64 -150 -294q-24 -32 -65 -85q-40 -51 -55 -72t-30.5 -49.5t-12 -42t13 -34.5t32.5 -43t57 -53q4 -2 5 -4q141 -131 191 -221q3 -5 6.5 -12.5t7 -26.5t-0.5 -34t-25 -27.5t-59 -12.5l-256 -4q-24 -5 -56 5t-52 22l-20 12q-30 21 -70 64t-68.5 77.5t-61 58 + t-56.5 15.5q-3 -1 -8 -3.5t-17 -14.5t-21.5 -29.5t-17 -52t-6.5 -77.5q0 -15 -3.5 -27.5t-7.5 -18.5l-4 -5q-18 -19 -53 -22h-115q-71 -4 -146 16.5t-131.5 53t-103 66t-70.5 57.5l-25 24q-10 10 -27.5 30t-71.5 91t-106 151t-122.5 211t-130.5 272q-6 16 -6 27t3 16l4 6 + q15 19 57 19l274 2q12 -2 23 -6.5t16 -8.5l5 -3q16 -11 24 -32q20 -50 46 -103.5t41 -81.5l16 -29q29 -60 56 -104t48.5 -68.5t41.5 -38.5t34 -14t27 5q2 1 5 5t12 22t13.5 47t9.5 81t0 125q-2 40 -9 73t-14 46l-6 12q-25 34 -85 43q-13 2 5 24q16 19 38 30q53 26 239 24 + q82 -1 135 -13q20 -5 33.5 -13.5t20.5 -24t10.5 -32t3.5 -45.5t-1 -55t-2.5 -70.5t-1.5 -82.5q0 -11 -1 -42t-0.5 -48t3.5 -40.5t11.5 -39t22.5 -24.5q8 -2 17 -4t26 11t38 34.5t52 67t68 107.5q60 104 107 225q4 10 10 17.5t11 10.5l4 3l5 2.5t13 3t20 0.5l288 2 + q39 5 64 -2.5t31 -16.5z" /> + <glyph glyph-name="weibo" unicode="" horiz-adv-x="1792" + d="M675 252q21 34 11 69t-45 50q-34 14 -73 1t-60 -46q-22 -34 -13 -68.5t43 -50.5t74.5 -2.5t62.5 47.5zM769 373q8 13 3.5 26.5t-17.5 18.5q-14 5 -28.5 -0.5t-21.5 -18.5q-17 -31 13 -45q14 -5 29 0.5t22 18.5zM943 266q-45 -102 -158 -150t-224 -12 + q-107 34 -147.5 126.5t6.5 187.5q47 93 151.5 139t210.5 19q111 -29 158.5 -119.5t2.5 -190.5zM1255 426q-9 96 -89 170t-208.5 109t-274.5 21q-223 -23 -369.5 -141.5t-132.5 -264.5q9 -96 89 -170t208.5 -109t274.5 -21q223 23 369.5 141.5t132.5 264.5zM1563 422 + q0 -68 -37 -139.5t-109 -137t-168.5 -117.5t-226 -83t-270.5 -31t-275 33.5t-240.5 93t-171.5 151t-65 199.5q0 115 69.5 245t197.5 258q169 169 341.5 236t246.5 -7q65 -64 20 -209q-4 -14 -1 -20t10 -7t14.5 0.5t13.5 3.5l6 2q139 59 246 59t153 -61q45 -63 0 -178 + q-2 -13 -4.5 -20t4.5 -12.5t12 -7.5t17 -6q57 -18 103 -47t80 -81.5t34 -116.5zM1489 1046q42 -47 54.5 -108.5t-6.5 -117.5q-8 -23 -29.5 -34t-44.5 -4q-23 8 -34 29.5t-4 44.5q20 63 -24 111t-107 35q-24 -5 -45 8t-25 37q-5 24 8 44.5t37 25.5q60 13 119 -5.5t101 -65.5z + M1670 1209q87 -96 112.5 -222.5t-13.5 -241.5q-9 -27 -34 -40t-52 -4t-40 34t-5 52q28 82 10 172t-80 158q-62 69 -148 95.5t-173 8.5q-28 -6 -52 9.5t-30 43.5t9.5 51.5t43.5 29.5q123 26 244 -11.5t208 -134.5z" /> + <glyph glyph-name="renren" unicode="" + d="M1133 -34q-171 -94 -368 -94q-196 0 -367 94q138 87 235.5 211t131.5 268q35 -144 132.5 -268t235.5 -211zM638 1394v-485q0 -252 -126.5 -459.5t-330.5 -306.5q-181 215 -181 495q0 187 83.5 349.5t229.5 269.5t325 137zM1536 638q0 -280 -181 -495 + q-204 99 -330.5 306.5t-126.5 459.5v485q179 -30 325 -137t229.5 -269.5t83.5 -349.5z" /> + <glyph glyph-name="_372" unicode="" horiz-adv-x="1408" + d="M1402 433q-32 -80 -76 -138t-91 -88.5t-99 -46.5t-101.5 -14.5t-96.5 8.5t-86.5 22t-69.5 27.5t-46 22.5l-17 10q-113 -228 -289.5 -359.5t-384.5 -132.5q-19 0 -32 13t-13 32t13 31.5t32 12.5q173 1 322.5 107.5t251.5 294.5q-36 -14 -72 -23t-83 -13t-91 2.5t-93 28.5 + t-92 59t-84.5 100t-74.5 146q114 47 214 57t167.5 -7.5t124.5 -56.5t88.5 -77t56.5 -82q53 131 79 291q-7 -1 -18 -2.5t-46.5 -2.5t-69.5 0.5t-81.5 10t-88.5 23t-84 42.5t-75 65t-54.5 94.5t-28.5 127.5q70 28 133.5 36.5t112.5 -1t92 -30t73.5 -50t56 -61t42 -63t27.5 -56 + t16 -39.5l4 -16q12 122 12 195q-8 6 -21.5 16t-49 44.5t-63.5 71.5t-54 93t-33 112.5t12 127t70 138.5q73 -25 127.5 -61.5t84.5 -76.5t48 -85t20.5 -89t-0.5 -85.5t-13 -76.5t-19 -62t-17 -42l-7 -15q1 -4 1 -50t-1 -72q3 7 10 18.5t30.5 43t50.5 58t71 55.5t91.5 44.5 + t112 14.5t132.5 -24q-2 -78 -21.5 -141.5t-50 -104.5t-69.5 -71.5t-81.5 -45.5t-84.5 -24t-80 -9.5t-67.5 1t-46.5 4.5l-17 3q-23 -147 -73 -283q6 7 18 18.5t49.5 41t77.5 52.5t99.5 42t117.5 20t129 -23.5t137 -77.5z" /> + <glyph glyph-name="stack_exchange" unicode="" horiz-adv-x="1280" + d="M1259 283v-66q0 -85 -57.5 -144.5t-138.5 -59.5h-57l-260 -269v269h-529q-81 0 -138.5 59.5t-57.5 144.5v66h1238zM1259 609v-255h-1238v255h1238zM1259 937v-255h-1238v255h1238zM1259 1077v-67h-1238v67q0 84 57.5 143.5t138.5 59.5h846q81 0 138.5 -59.5t57.5 -143.5z + " /> + <glyph glyph-name="_374" unicode="" + d="M1152 640q0 -14 -9 -23l-320 -320q-9 -9 -23 -9q-13 0 -22.5 9.5t-9.5 22.5v192h-352q-13 0 -22.5 9.5t-9.5 22.5v192q0 13 9.5 22.5t22.5 9.5h352v192q0 14 9 23t23 9q12 0 24 -10l319 -319q9 -9 9 -23zM1312 640q0 148 -73 273t-198 198t-273 73t-273 -73t-198 -198 + t-73 -273t73 -273t198 -198t273 -73t273 73t198 198t73 273zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="arrow_circle_alt_left" unicode="" + d="M1152 736v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-352v-192q0 -14 -9 -23t-23 -9q-12 0 -24 10l-319 319q-9 9 -9 23t9 23l320 320q9 9 23 9q13 0 22.5 -9.5t9.5 -22.5v-192h352q13 0 22.5 -9.5t9.5 -22.5zM1312 640q0 148 -73 273t-198 198t-273 73t-273 -73t-198 -198 + t-73 -273t73 -273t198 -198t273 -73t273 73t198 198t73 273zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="_376" unicode="" + d="M1024 960v-640q0 -26 -19 -45t-45 -19q-20 0 -37 12l-448 320q-27 19 -27 52t27 52l448 320q17 12 37 12q26 0 45 -19t19 -45zM1280 160v960q0 13 -9.5 22.5t-22.5 9.5h-960q-13 0 -22.5 -9.5t-9.5 -22.5v-960q0 -13 9.5 -22.5t22.5 -9.5h960q13 0 22.5 9.5t9.5 22.5z + M1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="dot_circle_alt" unicode="" + d="M1024 640q0 -106 -75 -181t-181 -75t-181 75t-75 181t75 181t181 75t181 -75t75 -181zM768 1184q-148 0 -273 -73t-198 -198t-73 -273t73 -273t198 -198t273 -73t273 73t198 198t73 273t-73 273t-198 198t-273 73zM1536 640q0 -209 -103 -385.5t-279.5 -279.5 + t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="_378" unicode="" horiz-adv-x="1664" + d="M1023 349l102 -204q-58 -179 -210 -290t-339 -111q-156 0 -288.5 77.5t-210 210t-77.5 288.5q0 181 104.5 330t274.5 211l17 -131q-122 -54 -195 -165.5t-73 -244.5q0 -185 131.5 -316.5t316.5 -131.5q126 0 232.5 65t165 175.5t49.5 236.5zM1571 249l58 -114l-256 -128 + q-13 -7 -29 -7q-40 0 -57 35l-239 477h-472q-24 0 -42.5 16.5t-21.5 40.5l-96 779q-2 17 6 42q14 51 57 82.5t97 31.5q66 0 113 -47t47 -113q0 -69 -52 -117.5t-120 -41.5l37 -289h423v-128h-407l16 -128h455q40 0 57 -35l228 -455z" /> + <glyph glyph-name="vimeo_square" unicode="" + d="M1292 898q10 216 -161 222q-231 8 -312 -261q44 19 82 19q85 0 74 -96q-4 -57 -74 -167t-105 -110q-43 0 -82 169q-13 54 -45 255q-30 189 -160 177q-59 -7 -164 -100l-81 -72l-81 -72l52 -67q76 52 87 52q57 0 107 -179q15 -55 45 -164.5t45 -164.5q68 -179 164 -179 + q157 0 383 294q220 283 226 444zM1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="_380" unicode="" horiz-adv-x="1152" + d="M1152 704q0 -191 -94.5 -353t-256.5 -256.5t-353 -94.5h-160q-14 0 -23 9t-9 23v611l-215 -66q-3 -1 -9 -1q-10 0 -19 6q-13 10 -13 26v128q0 23 23 31l233 71v93l-215 -66q-3 -1 -9 -1q-10 0 -19 6q-13 10 -13 26v128q0 23 23 31l233 71v250q0 14 9 23t23 9h160 + q14 0 23 -9t9 -23v-181l375 116q15 5 28 -5t13 -26v-128q0 -23 -23 -31l-393 -121v-93l375 116q15 5 28 -5t13 -26v-128q0 -23 -23 -31l-393 -121v-487q188 13 318 151t130 328q0 14 9 23t23 9h160q14 0 23 -9t9 -23z" /> + <glyph glyph-name="plus_square_o" unicode="" horiz-adv-x="1408" + d="M1152 736v-64q0 -14 -9 -23t-23 -9h-352v-352q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9t-9 23v352h-352q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h352v352q0 14 9 23t23 9h64q14 0 23 -9t9 -23v-352h352q14 0 23 -9t9 -23zM1280 288v832q0 66 -47 113t-113 47h-832 + q-66 0 -113 -47t-47 -113v-832q0 -66 47 -113t113 -47h832q66 0 113 47t47 113zM1408 1120v-832q0 -119 -84.5 -203.5t-203.5 -84.5h-832q-119 0 -203.5 84.5t-84.5 203.5v832q0 119 84.5 203.5t203.5 84.5h832q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="_382" unicode="" horiz-adv-x="2176" + d="M620 416q-110 -64 -268 -64h-128v64h-64q-13 0 -22.5 23.5t-9.5 56.5q0 24 7 49q-58 2 -96.5 10.5t-38.5 20.5t38.5 20.5t96.5 10.5q-7 25 -7 49q0 33 9.5 56.5t22.5 23.5h64v64h128q158 0 268 -64h1113q42 -7 106.5 -18t80.5 -14q89 -15 150 -40.5t83.5 -47.5t22.5 -40 + t-22.5 -40t-83.5 -47.5t-150 -40.5q-16 -3 -80.5 -14t-106.5 -18h-1113zM1739 668q53 -36 53 -92t-53 -92l81 -30q68 48 68 122t-68 122zM625 400h1015q-217 -38 -456 -80q-57 0 -113 -24t-83 -48l-28 -24l-288 -288q-26 -26 -70.5 -45t-89.5 -19h-96l-93 464h29 + q157 0 273 64zM352 816h-29l93 464h96q46 0 90 -19t70 -45l288 -288q4 -4 11 -10.5t30.5 -23t48.5 -29t61.5 -23t72.5 -10.5l456 -80h-1015q-116 64 -273 64z" /> + <glyph glyph-name="_383" unicode="" horiz-adv-x="1664" + d="M1519 760q62 0 103.5 -40.5t41.5 -101.5q0 -97 -93 -130l-172 -59l56 -167q7 -21 7 -47q0 -59 -42 -102t-101 -43q-47 0 -85.5 27t-53.5 72l-55 165l-310 -106l55 -164q8 -24 8 -47q0 -59 -42 -102t-102 -43q-47 0 -85 27t-53 72l-55 163l-153 -53q-29 -9 -50 -9 + q-61 0 -101.5 40t-40.5 101q0 47 27.5 85t71.5 53l156 53l-105 313l-156 -54q-26 -8 -48 -8q-60 0 -101 40.5t-41 100.5q0 47 27.5 85t71.5 53l157 53l-53 159q-8 24 -8 47q0 60 42 102.5t102 42.5q47 0 85 -27t53 -72l54 -160l310 105l-54 160q-8 24 -8 47q0 59 42.5 102 + t101.5 43q47 0 85.5 -27.5t53.5 -71.5l53 -161l162 55q21 6 43 6q60 0 102.5 -39.5t42.5 -98.5q0 -45 -30 -81.5t-74 -51.5l-157 -54l105 -316l164 56q24 8 46 8zM725 498l310 105l-105 315l-310 -107z" /> + <glyph glyph-name="_384" unicode="" + d="M1248 1408q119 0 203.5 -84.5t84.5 -203.5v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960zM1280 352v436q-31 -35 -64 -55q-34 -22 -132.5 -85t-151.5 -99q-98 -69 -164 -69v0v0q-66 0 -164 69 + q-47 32 -142 92.5t-142 92.5q-12 8 -33 27t-31 27v-436q0 -40 28 -68t68 -28h832q40 0 68 28t28 68zM1280 925q0 41 -27.5 70t-68.5 29h-832q-40 0 -68 -28t-28 -68q0 -37 30.5 -76.5t67.5 -64.5q47 -32 137.5 -89t129.5 -83q3 -2 17 -11.5t21 -14t21 -13t23.5 -13 + t21.5 -9.5t22.5 -7.5t20.5 -2.5t20.5 2.5t22.5 7.5t21.5 9.5t23.5 13t21 13t21 14t17 11.5l267 174q35 23 66.5 62.5t31.5 73.5z" /> + <glyph glyph-name="_385" unicode="" horiz-adv-x="1792" + d="M127 640q0 163 67 313l367 -1005q-196 95 -315 281t-119 411zM1415 679q0 -19 -2.5 -38.5t-10 -49.5t-11.5 -44t-17.5 -59t-17.5 -58l-76 -256l-278 826q46 3 88 8q19 2 26 18.5t-2.5 31t-28.5 13.5l-205 -10q-75 1 -202 10q-12 1 -20.5 -5t-11.5 -15t-1.5 -18.5t9 -16.5 + t19.5 -8l80 -8l120 -328l-168 -504l-280 832q46 3 88 8q19 2 26 18.5t-2.5 31t-28.5 13.5l-205 -10q-7 0 -23 0.5t-26 0.5q105 160 274.5 253.5t367.5 93.5q147 0 280.5 -53t238.5 -149h-10q-55 0 -92 -40.5t-37 -95.5q0 -12 2 -24t4 -21.5t8 -23t9 -21t12 -22.5t12.5 -21 + t14.5 -24t14 -23q63 -107 63 -212zM909 573l237 -647q1 -6 5 -11q-126 -44 -255 -44q-112 0 -217 32zM1570 1009q95 -174 95 -369q0 -209 -104 -385.5t-279 -278.5l235 678q59 169 59 276q0 42 -6 79zM896 1536q182 0 348 -71t286 -191t191 -286t71 -348t-71 -348t-191 -286 + t-286 -191t-348 -71t-348 71t-286 191t-191 286t-71 348t71 348t191 286t286 191t348 71zM896 -215q173 0 331.5 68t273 182.5t182.5 273t68 331.5t-68 331.5t-182.5 273t-273 182.5t-331.5 68t-331.5 -68t-273 -182.5t-182.5 -273t-68 -331.5t68 -331.5t182.5 -273 + t273 -182.5t331.5 -68z" /> + <glyph glyph-name="_386" unicode="" horiz-adv-x="1792" + d="M1086 1536v-1536l-272 -128q-228 20 -414 102t-293 208.5t-107 272.5q0 140 100.5 263.5t275 205.5t391.5 108v-172q-217 -38 -356.5 -150t-139.5 -255q0 -152 154.5 -267t388.5 -145v1360zM1755 954l37 -390l-525 114l147 83q-119 70 -280 99v172q277 -33 481 -157z" /> + <glyph glyph-name="_387" unicode="" horiz-adv-x="2048" + d="M960 1536l960 -384v-128h-128q0 -26 -20.5 -45t-48.5 -19h-1526q-28 0 -48.5 19t-20.5 45h-128v128zM256 896h256v-768h128v768h256v-768h128v768h256v-768h128v768h256v-768h59q28 0 48.5 -19t20.5 -45v-64h-1664v64q0 26 20.5 45t48.5 19h59v768zM1851 -64 + q28 0 48.5 -19t20.5 -45v-128h-1920v128q0 26 20.5 45t48.5 19h1782z" /> + <glyph glyph-name="_388" unicode="" horiz-adv-x="2304" + d="M1774 700l18 -316q4 -69 -82 -128t-235 -93.5t-323 -34.5t-323 34.5t-235 93.5t-82 128l18 316l574 -181q22 -7 48 -7t48 7zM2304 1024q0 -23 -22 -31l-1120 -352q-4 -1 -10 -1t-10 1l-652 206q-43 -34 -71 -111.5t-34 -178.5q63 -36 63 -109q0 -69 -58 -107l58 -433 + q2 -14 -8 -25q-9 -11 -24 -11h-192q-15 0 -24 11q-10 11 -8 25l58 433q-58 38 -58 107q0 73 65 111q11 207 98 330l-333 104q-22 8 -22 31t22 31l1120 352q4 1 10 1t10 -1l1120 -352q22 -8 22 -31z" /> + <glyph glyph-name="_389" unicode="" + d="M859 579l13 -707q-62 11 -105 11q-41 0 -105 -11l13 707q-40 69 -168.5 295.5t-216.5 374.5t-181 287q58 -15 108 -15q44 0 111 15q63 -111 133.5 -229.5t167 -276.5t138.5 -227q37 61 109.5 177.5t117.5 190t105 176t107 189.5q54 -14 107 -14q56 0 114 14v0 + q-28 -39 -60 -88.5t-49.5 -78.5t-56.5 -96t-49 -84q-146 -248 -353 -610z" /> + <glyph glyph-name="uniF1A0" unicode="" + d="M768 750h725q12 -67 12 -128q0 -217 -91 -387.5t-259.5 -266.5t-386.5 -96q-157 0 -299 60.5t-245 163.5t-163.5 245t-60.5 299t60.5 299t163.5 245t245 163.5t299 60.5q300 0 515 -201l-209 -201q-123 119 -306 119q-129 0 -238.5 -65t-173.5 -176.5t-64 -243.5 + t64 -243.5t173.5 -176.5t238.5 -65q87 0 160 24t120 60t82 82t51.5 87t22.5 78h-436v264z" /> + <glyph glyph-name="f1a1" unicode="" horiz-adv-x="1792" + d="M1095 369q16 -16 0 -31q-62 -62 -199 -62t-199 62q-16 15 0 31q6 6 15 6t15 -6q48 -49 169 -49q120 0 169 49q6 6 15 6t15 -6zM788 550q0 -37 -26 -63t-63 -26t-63.5 26t-26.5 63q0 38 26.5 64t63.5 26t63 -26.5t26 -63.5zM1183 550q0 -37 -26.5 -63t-63.5 -26t-63 26 + t-26 63t26 63.5t63 26.5t63.5 -26t26.5 -64zM1434 670q0 49 -35 84t-85 35t-86 -36q-130 90 -311 96l63 283l200 -45q0 -37 26 -63t63 -26t63.5 26.5t26.5 63.5t-26.5 63.5t-63.5 26.5q-54 0 -80 -50l-221 49q-19 5 -25 -16l-69 -312q-180 -7 -309 -97q-35 37 -87 37 + q-50 0 -85 -35t-35 -84q0 -35 18.5 -64t49.5 -44q-6 -27 -6 -56q0 -142 140 -243t337 -101q198 0 338 101t140 243q0 32 -7 57q30 15 48 43.5t18 63.5zM1792 640q0 -182 -71 -348t-191 -286t-286 -191t-348 -71t-348 71t-286 191t-191 286t-71 348t71 348t191 286t286 191 + t348 71t348 -71t286 -191t191 -286t71 -348z" /> + <glyph glyph-name="_392" unicode="" + d="M939 407q13 -13 0 -26q-53 -53 -171 -53t-171 53q-13 13 0 26q5 6 13 6t13 -6q42 -42 145 -42t145 42q5 6 13 6t13 -6zM676 563q0 -31 -23 -54t-54 -23t-54 23t-23 54q0 32 22.5 54.5t54.5 22.5t54.5 -22.5t22.5 -54.5zM1014 563q0 -31 -23 -54t-54 -23t-54 23t-23 54 + q0 32 22.5 54.5t54.5 22.5t54.5 -22.5t22.5 -54.5zM1229 666q0 42 -30 72t-73 30q-42 0 -73 -31q-113 78 -267 82l54 243l171 -39q1 -32 23.5 -54t53.5 -22q32 0 54.5 22.5t22.5 54.5t-22.5 54.5t-54.5 22.5q-48 0 -69 -43l-189 42q-17 5 -21 -13l-60 -268q-154 -6 -265 -83 + q-30 32 -74 32q-43 0 -73 -30t-30 -72q0 -30 16 -55t42 -38q-5 -25 -5 -48q0 -122 120 -208.5t289 -86.5q170 0 290 86.5t120 208.5q0 25 -6 49q25 13 40.5 37.5t15.5 54.5zM1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960 + q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="_393" unicode="" + d="M866 697l90 27v62q0 79 -58 135t-138 56t-138 -55.5t-58 -134.5v-283q0 -20 -14 -33.5t-33 -13.5t-32.5 13.5t-13.5 33.5v120h-151v-122q0 -82 57.5 -139t139.5 -57q81 0 138.5 56.5t57.5 136.5v280q0 19 13.5 33t33.5 14q19 0 32.5 -14t13.5 -33v-54zM1199 502v122h-150 + v-126q0 -20 -13.5 -33.5t-33.5 -13.5q-19 0 -32.5 14t-13.5 33v123l-90 -26l-60 28v-123q0 -80 58 -137t139 -57t138.5 57t57.5 139zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103 + t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="f1a4" unicode="" horiz-adv-x="1920" + d="M1062 824v118q0 42 -30 72t-72 30t-72 -30t-30 -72v-612q0 -175 -126 -299t-303 -124q-178 0 -303.5 125.5t-125.5 303.5v266h328v-262q0 -43 30 -72.5t72 -29.5t72 29.5t30 72.5v620q0 171 126.5 292t301.5 121q176 0 302 -122t126 -294v-136l-195 -58zM1592 602h328 + v-266q0 -178 -125.5 -303.5t-303.5 -125.5q-177 0 -303 124.5t-126 300.5v268l131 -61l195 58v-270q0 -42 30 -71.5t72 -29.5t72 29.5t30 71.5v275z" /> + <glyph glyph-name="_395" unicode="" + d="M1472 160v480h-704v704h-480q-93 0 -158.5 -65.5t-65.5 -158.5v-480h704v-704h480q93 0 158.5 65.5t65.5 158.5zM1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5 + t84.5 -203.5z" /> + <glyph glyph-name="_396" unicode="" horiz-adv-x="2048" + d="M328 1254h204v-983h-532v697h328v286zM328 435v369h-123v-369h123zM614 968v-697h205v697h-205zM614 1254v-204h205v204h-205zM901 968h533v-942h-533v163h328v82h-328v697zM1229 435v369h-123v-369h123zM1516 968h532v-942h-532v163h327v82h-327v697zM1843 435v369h-123 + v-369h123z" /> + <glyph glyph-name="_397" unicode="" + d="M1046 516q0 -64 -38 -109t-91 -45q-43 0 -70 15v277q28 17 70 17q53 0 91 -45.5t38 -109.5zM703 944q0 -64 -38 -109.5t-91 -45.5q-43 0 -70 15v277q28 17 70 17q53 0 91 -45t38 -109zM1265 513q0 134 -88 229t-213 95q-20 0 -39 -3q-23 -78 -78 -136q-87 -95 -211 -101 + v-636l211 41v206q51 -19 117 -19q125 0 213 95t88 229zM922 940q0 134 -88.5 229t-213.5 95q-74 0 -141 -36h-186v-840l211 41v206q55 -19 116 -19q125 0 213.5 95t88.5 229zM1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960 + q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="_398" unicode="" horiz-adv-x="2038" + d="M1222 607q75 3 143.5 -20.5t118 -58.5t101 -94.5t84 -108t75.5 -120.5q33 -56 78.5 -109t75.5 -80.5t99 -88.5q-48 -30 -108.5 -57.5t-138.5 -59t-114 -47.5q-44 37 -74 115t-43.5 164.5t-33 180.5t-42.5 168.5t-72.5 123t-122.5 48.5l-10 -2l-6 -4q4 -5 13 -14 + q6 -5 28 -23.5t25.5 -22t19 -18t18 -20.5t11.5 -21t10.5 -27.5t4.5 -31t4 -40.5l1 -33q1 -26 -2.5 -57.5t-7.5 -52t-12.5 -58.5t-11.5 -53q-35 1 -101 -9.5t-98 -10.5q-39 0 -72 10q-2 16 -2 47q0 74 3 96q2 13 31.5 41.5t57 59t26.5 51.5q-24 2 -43 -24 + q-36 -53 -111.5 -99.5t-136.5 -46.5q-25 0 -75.5 63t-106.5 139.5t-84 96.5q-6 4 -27 30q-482 -112 -513 -112q-16 0 -28 11t-12 27q0 15 8.5 26.5t22.5 14.5l486 106q-8 14 -8 25t5.5 17.5t16 11.5t20 7t23 4.5t18.5 4.5q4 1 15.5 7.5t17.5 6.5q15 0 28 -16t20 -33 + q163 37 172 37q17 0 29.5 -11t12.5 -28q0 -15 -8.5 -26t-23.5 -14l-182 -40l-1 -16q-1 -26 81.5 -117.5t104.5 -91.5q47 0 119 80t72 129q0 36 -23.5 53t-51 18.5t-51 11.5t-23.5 34q0 16 10 34l-68 19q43 44 43 117q0 26 -5 58q82 16 144 16q44 0 71.5 -1.5t48.5 -8.5 + t31 -13.5t20.5 -24.5t15.5 -33.5t17 -47.5t24 -60l50 25q-3 -40 -23 -60t-42.5 -21t-40 -6.5t-16.5 -20.5zM1282 842q-5 5 -13.5 15.5t-12 14.5t-10.5 11.5t-10 10.5l-8 8t-8.5 7.5t-8 5t-8.5 4.5q-7 3 -14.5 5t-20.5 2.5t-22 0.5h-32.5h-37.5q-126 0 -217 -43 + q16 30 36 46.5t54 29.5t65.5 36t46 36.5t50 55t43.5 50.5q12 -9 28 -31.5t32 -36.5t38 -13l12 1v-76l22 -1q247 95 371 190q28 21 50 39t42.5 37.5t33 31t29.5 34t24 31t24.5 37t23 38t27 47.5t29.5 53l7 9q-2 -53 -43 -139q-79 -165 -205 -264t-306 -142q-14 -3 -42 -7.5 + t-50 -9.5t-39 -14q3 -19 24.5 -46t21.5 -34q0 -11 -26 -30zM1061 -79q39 26 131.5 47.5t146.5 21.5q9 0 22.5 -15.5t28 -42.5t26 -50t24 -51t14.5 -33q-121 -45 -244 -45q-61 0 -125 11zM822 568l48 12l109 -177l-73 -48zM1323 51q3 -15 3 -16q0 -7 -17.5 -14.5t-46 -13 + t-54 -9.5t-53.5 -7.5t-32 -4.5l-7 43q21 2 60.5 8.5t72 10t60.5 3.5h14zM866 679l-96 -20l-6 17q10 1 32.5 7t34.5 6q19 0 35 -10zM1061 45h31l10 -83l-41 -12v95zM1950 1535v1v-1zM1950 1535l-1 -5l-2 -2l1 3zM1950 1535l1 1z" /> + <glyph glyph-name="_399" unicode="" + d="M1167 -50q-5 19 -24 5q-30 -22 -87 -39t-131 -17q-129 0 -193 49q-5 4 -13 4q-11 0 -26 -12q-7 -6 -7.5 -16t7.5 -20q34 -32 87.5 -46t102.5 -12.5t99 4.5q41 4 84.5 20.5t65 30t28.5 20.5q12 12 7 29zM1128 65q-19 47 -39 61q-23 15 -76 15q-47 0 -71 -10 + q-29 -12 -78 -56q-26 -24 -12 -44q9 -8 17.5 -4.5t31.5 23.5q3 2 10.5 8.5t10.5 8.5t10 7t11.5 7t12.5 5t15 4.5t16.5 2.5t20.5 1q27 0 44.5 -7.5t23 -14.5t13.5 -22q10 -17 12.5 -20t12.5 1q23 12 14 34zM1483 346q0 22 -5 44.5t-16.5 45t-34 36.5t-52.5 14 + q-33 0 -97 -41.5t-129 -83.5t-101 -42q-27 -1 -63.5 19t-76 49t-83.5 58t-100 49t-111 19q-115 -1 -197 -78.5t-84 -178.5q-2 -112 74 -164q29 -20 62.5 -28.5t103.5 -8.5q57 0 132 32.5t134 71t120 70.5t93 31q26 -1 65 -31.5t71.5 -67t68 -67.5t55.5 -32q35 -3 58.5 14 + t55.5 63q28 41 42.5 101t14.5 106zM1536 506q0 -164 -62 -304.5t-166 -236t-242.5 -149.5t-290.5 -54t-293 57.5t-247.5 157t-170.5 241.5t-64 302q0 89 19.5 172.5t49 145.5t70.5 118.5t78.5 94t78.5 69.5t64.5 46.5t42.5 24.5q14 8 51 26.5t54.5 28.5t48 30t60.5 44 + q36 28 58 72.5t30 125.5q129 -155 186 -193q44 -29 130 -68t129 -66q21 -13 39 -25t60.5 -46.5t76 -70.5t75 -95t69 -122t47 -148.5t19.5 -177.5z" /> + <glyph glyph-name="_400" unicode="" + d="M1070 463l-160 -160l-151 -152l-30 -30q-65 -64 -151.5 -87t-171.5 -2q-16 -70 -72 -115t-129 -45q-85 0 -145 60.5t-60 145.5q0 72 44.5 128t113.5 72q-22 86 1 173t88 152l12 12l151 -152l-11 -11q-37 -37 -37 -89t37 -90q37 -37 89 -37t89 37l30 30l151 152l161 160z + M729 1145l12 -12l-152 -152l-12 12q-37 37 -89 37t-89 -37t-37 -89.5t37 -89.5l29 -29l152 -152l160 -160l-151 -152l-161 160l-151 152l-30 30q-68 67 -90 159.5t5 179.5q-70 15 -115 71t-45 129q0 85 60 145.5t145 60.5q76 0 133.5 -49t69.5 -123q84 20 169.5 -3.5 + t149.5 -87.5zM1536 78q0 -85 -60 -145.5t-145 -60.5q-74 0 -131 47t-71 118q-86 -28 -179.5 -6t-161.5 90l-11 12l151 152l12 -12q37 -37 89 -37t89 37t37 89t-37 89l-30 30l-152 152l-160 160l152 152l160 -160l152 -152l29 -30q64 -64 87.5 -150.5t2.5 -171.5 + q76 -11 126.5 -68.5t50.5 -134.5zM1534 1202q0 -77 -51 -135t-127 -69q26 -85 3 -176.5t-90 -158.5l-12 -12l-151 152l12 12q37 37 37 89t-37 89t-89 37t-89 -37l-30 -30l-152 -152l-160 -160l-152 152l161 160l152 152l29 30q67 67 159 89.5t178 -3.5q11 75 68.5 126 + t135.5 51q85 0 145 -60.5t60 -145.5z" /> + <glyph glyph-name="f1ab" unicode="" + d="M654 458q-1 -3 -12.5 0.5t-31.5 11.5l-20 9q-44 20 -87 49q-7 5 -41 31.5t-38 28.5q-67 -103 -134 -181q-81 -95 -105 -110q-4 -2 -19.5 -4t-18.5 0q6 4 82 92q21 24 85.5 115t78.5 118q17 30 51 98.5t36 77.5q-8 1 -110 -33q-8 -2 -27.5 -7.5t-34.5 -9.5t-17 -5 + q-2 -2 -2 -10.5t-1 -9.5q-5 -10 -31 -15q-23 -7 -47 0q-18 4 -28 21q-4 6 -5 23q6 2 24.5 5t29.5 6q58 16 105 32q100 35 102 35q10 2 43 19.5t44 21.5q9 3 21.5 8t14.5 5.5t6 -0.5q2 -12 -1 -33q0 -2 -12.5 -27t-26.5 -53.5t-17 -33.5q-25 -50 -77 -131l64 -28 + q12 -6 74.5 -32t67.5 -28q4 -1 10.5 -25.5t4.5 -30.5zM449 944q3 -15 -4 -28q-12 -23 -50 -38q-30 -12 -60 -12q-26 3 -49 26q-14 15 -18 41l1 3q3 -3 19.5 -5t26.5 0t58 16q36 12 55 14q17 0 21 -17zM1147 815l63 -227l-139 42zM39 15l694 232v1032l-694 -233v-1031z + M1280 332l102 -31l-181 657l-100 31l-216 -536l102 -31l45 110l211 -65zM777 1294l573 -184v380zM1088 -29l158 -13l-54 -160l-40 66q-130 -83 -276 -108q-58 -12 -91 -12h-84q-79 0 -199.5 39t-183.5 85q-8 7 -8 16q0 8 5 13.5t13 5.5q4 0 18 -7.5t30.5 -16.5t20.5 -11 + q73 -37 159.5 -61.5t157.5 -24.5q95 0 167 14.5t157 50.5q15 7 30.5 15.5t34 19t28.5 16.5zM1536 1050v-1079l-774 246q-14 -6 -375 -127.5t-368 -121.5q-13 0 -18 13q0 1 -1 3v1078q3 9 4 10q5 6 20 11q107 36 149 50v384l558 -198q2 0 160.5 55t316 108.5t161.5 53.5 + q20 0 20 -21v-418z" /> + <glyph glyph-name="_402" unicode="" horiz-adv-x="1792" + d="M288 1152q66 0 113 -47t47 -113v-1088q0 -66 -47 -113t-113 -47h-128q-66 0 -113 47t-47 113v1088q0 66 47 113t113 47h128zM1664 989q58 -34 93 -93t35 -128v-768q0 -106 -75 -181t-181 -75h-864q-66 0 -113 47t-47 113v1536q0 40 28 68t68 28h672q40 0 88 -20t76 -48 + l152 -152q28 -28 48 -76t20 -88v-163zM928 0v128q0 14 -9 23t-23 9h-128q-14 0 -23 -9t-9 -23v-128q0 -14 9 -23t23 -9h128q14 0 23 9t9 23zM928 256v128q0 14 -9 23t-23 9h-128q-14 0 -23 -9t-9 -23v-128q0 -14 9 -23t23 -9h128q14 0 23 9t9 23zM928 512v128q0 14 -9 23 + t-23 9h-128q-14 0 -23 -9t-9 -23v-128q0 -14 9 -23t23 -9h128q14 0 23 9t9 23zM1184 0v128q0 14 -9 23t-23 9h-128q-14 0 -23 -9t-9 -23v-128q0 -14 9 -23t23 -9h128q14 0 23 9t9 23zM1184 256v128q0 14 -9 23t-23 9h-128q-14 0 -23 -9t-9 -23v-128q0 -14 9 -23t23 -9h128 + q14 0 23 9t9 23zM1184 512v128q0 14 -9 23t-23 9h-128q-14 0 -23 -9t-9 -23v-128q0 -14 9 -23t23 -9h128q14 0 23 9t9 23zM1440 0v128q0 14 -9 23t-23 9h-128q-14 0 -23 -9t-9 -23v-128q0 -14 9 -23t23 -9h128q14 0 23 9t9 23zM1440 256v128q0 14 -9 23t-23 9h-128 + q-14 0 -23 -9t-9 -23v-128q0 -14 9 -23t23 -9h128q14 0 23 9t9 23zM1440 512v128q0 14 -9 23t-23 9h-128q-14 0 -23 -9t-9 -23v-128q0 -14 9 -23t23 -9h128q14 0 23 9t9 23zM1536 896v256h-160q-40 0 -68 28t-28 68v160h-640v-512h896z" /> + <glyph glyph-name="_403" unicode="" + d="M1344 1536q26 0 45 -19t19 -45v-1664q0 -26 -19 -45t-45 -19h-1280q-26 0 -45 19t-19 45v1664q0 26 19 45t45 19h1280zM512 1248v-64q0 -14 9 -23t23 -9h64q14 0 23 9t9 23v64q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23zM512 992v-64q0 -14 9 -23t23 -9h64q14 0 23 9 + t9 23v64q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23zM512 736v-64q0 -14 9 -23t23 -9h64q14 0 23 9t9 23v64q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23zM512 480v-64q0 -14 9 -23t23 -9h64q14 0 23 9t9 23v64q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23zM384 160v64 + q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-64q0 -14 9 -23t23 -9h64q14 0 23 9t9 23zM384 416v64q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-64q0 -14 9 -23t23 -9h64q14 0 23 9t9 23zM384 672v64q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-64q0 -14 9 -23t23 -9h64 + q14 0 23 9t9 23zM384 928v64q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-64q0 -14 9 -23t23 -9h64q14 0 23 9t9 23zM384 1184v64q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-64q0 -14 9 -23t23 -9h64q14 0 23 9t9 23zM896 -96v192q0 14 -9 23t-23 9h-320q-14 0 -23 -9 + t-9 -23v-192q0 -14 9 -23t23 -9h320q14 0 23 9t9 23zM896 416v64q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-64q0 -14 9 -23t23 -9h64q14 0 23 9t9 23zM896 672v64q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-64q0 -14 9 -23t23 -9h64q14 0 23 9t9 23zM896 928v64 + q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-64q0 -14 9 -23t23 -9h64q14 0 23 9t9 23zM896 1184v64q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-64q0 -14 9 -23t23 -9h64q14 0 23 9t9 23zM1152 160v64q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-64q0 -14 9 -23t23 -9h64 + q14 0 23 9t9 23zM1152 416v64q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-64q0 -14 9 -23t23 -9h64q14 0 23 9t9 23zM1152 672v64q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-64q0 -14 9 -23t23 -9h64q14 0 23 9t9 23zM1152 928v64q0 14 -9 23t-23 9h-64q-14 0 -23 -9 + t-9 -23v-64q0 -14 9 -23t23 -9h64q14 0 23 9t9 23zM1152 1184v64q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-64q0 -14 9 -23t23 -9h64q14 0 23 9t9 23z" /> + <glyph glyph-name="_404" unicode="" horiz-adv-x="1280" + d="M1188 988l-292 -292v-824q0 -46 -33 -79t-79 -33t-79 33t-33 79v384h-64v-384q0 -46 -33 -79t-79 -33t-79 33t-33 79v824l-292 292q-28 28 -28 68t28 68q29 28 68.5 28t67.5 -28l228 -228h368l228 228q28 28 68 28t68 -28q28 -29 28 -68.5t-28 -67.5zM864 1152 + q0 -93 -65.5 -158.5t-158.5 -65.5t-158.5 65.5t-65.5 158.5t65.5 158.5t158.5 65.5t158.5 -65.5t65.5 -158.5z" /> + <glyph glyph-name="uniF1B1" unicode="" horiz-adv-x="1664" + d="M780 1064q0 -60 -19 -113.5t-63 -92.5t-105 -39q-76 0 -138 57.5t-92 135.5t-30 151q0 60 19 113.5t63 92.5t105 39q77 0 138.5 -57.5t91.5 -135t30 -151.5zM438 581q0 -80 -42 -139t-119 -59q-76 0 -141.5 55.5t-100.5 133.5t-35 152q0 80 42 139.5t119 59.5 + q76 0 141.5 -55.5t100.5 -134t35 -152.5zM832 608q118 0 255 -97.5t229 -237t92 -254.5q0 -46 -17 -76.5t-48.5 -45t-64.5 -20t-76 -5.5q-68 0 -187.5 45t-182.5 45q-66 0 -192.5 -44.5t-200.5 -44.5q-183 0 -183 146q0 86 56 191.5t139.5 192.5t187.5 146t193 59zM1071 819 + q-61 0 -105 39t-63 92.5t-19 113.5q0 74 30 151.5t91.5 135t138.5 57.5q61 0 105 -39t63 -92.5t19 -113.5q0 -73 -30 -151t-92 -135.5t-138 -57.5zM1503 923q77 0 119 -59.5t42 -139.5q0 -74 -35 -152t-100.5 -133.5t-141.5 -55.5q-77 0 -119 59t-42 139q0 74 35 152.5 + t100.5 134t141.5 55.5z" /> + <glyph glyph-name="_406" unicode="" horiz-adv-x="768" + d="M704 1008q0 -145 -57 -243.5t-152 -135.5l45 -821q2 -26 -16 -45t-44 -19h-192q-26 0 -44 19t-16 45l45 821q-95 37 -152 135.5t-57 243.5q0 128 42.5 249.5t117.5 200t160 78.5t160 -78.5t117.5 -200t42.5 -249.5z" /> + <glyph glyph-name="_407" unicode="" horiz-adv-x="1792" + d="M896 -93l640 349v636l-640 -233v-752zM832 772l698 254l-698 254l-698 -254zM1664 1024v-768q0 -35 -18 -65t-49 -47l-704 -384q-28 -16 -61 -16t-61 16l-704 384q-31 17 -49 47t-18 65v768q0 40 23 73t61 47l704 256q22 8 44 8t44 -8l704 -256q38 -14 61 -47t23 -73z + " /> + <glyph glyph-name="_408" unicode="" horiz-adv-x="2304" + d="M640 -96l384 192v314l-384 -164v-342zM576 358l404 173l-404 173l-404 -173zM1664 -96l384 192v314l-384 -164v-342zM1600 358l404 173l-404 173l-404 -173zM1152 651l384 165v266l-384 -164v-267zM1088 1030l441 189l-441 189l-441 -189zM2176 512v-416q0 -36 -19 -67 + t-52 -47l-448 -224q-25 -14 -57 -14t-57 14l-448 224q-4 2 -7 4q-2 -2 -7 -4l-448 -224q-25 -14 -57 -14t-57 14l-448 224q-33 16 -52 47t-19 67v416q0 38 21.5 70t56.5 48l434 186v400q0 38 21.5 70t56.5 48l448 192q23 10 50 10t50 -10l448 -192q35 -16 56.5 -48t21.5 -70 + v-400l434 -186q36 -16 57 -48t21 -70z" /> + <glyph glyph-name="_409" unicode="" horiz-adv-x="2048" + d="M1848 1197h-511v-124h511v124zM1596 771q-90 0 -146 -52.5t-62 -142.5h408q-18 195 -200 195zM1612 186q63 0 122 32t76 87h221q-100 -307 -427 -307q-214 0 -340.5 132t-126.5 347q0 208 130.5 345.5t336.5 137.5q138 0 240.5 -68t153 -179t50.5 -248q0 -17 -2 -47h-658 + q0 -111 57.5 -171.5t166.5 -60.5zM277 236h296q205 0 205 167q0 180 -199 180h-302v-347zM277 773h281q78 0 123.5 36.5t45.5 113.5q0 144 -190 144h-260v-294zM0 1282h594q87 0 155 -14t126.5 -47.5t90 -96.5t31.5 -154q0 -181 -172 -263q114 -32 172 -115t58 -204 + q0 -75 -24.5 -136.5t-66 -103.5t-98.5 -71t-121 -42t-134 -13h-611v1260z" /> + <glyph glyph-name="_410" unicode="" + d="M1248 1408q119 0 203.5 -84.5t84.5 -203.5v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960zM499 1041h-371v-787h382q117 0 197 57.5t80 170.5q0 158 -143 200q107 52 107 164q0 57 -19.5 96.5 + t-56.5 60.5t-79 29.5t-97 8.5zM477 723h-176v184h163q119 0 119 -90q0 -94 -106 -94zM486 388h-185v217h189q124 0 124 -113q0 -104 -128 -104zM1136 356q-68 0 -104 38t-36 107h411q1 10 1 30q0 132 -74.5 220.5t-203.5 88.5q-128 0 -210 -86t-82 -216q0 -135 79 -217 + t213 -82q205 0 267 191h-138q-11 -34 -47.5 -54t-75.5 -20zM1126 722q113 0 124 -122h-254q4 56 39 89t91 33zM964 988h319v-77h-319v77z" /> + <glyph glyph-name="_411" unicode="" horiz-adv-x="1792" + d="M1582 954q0 -101 -71.5 -172.5t-172.5 -71.5t-172.5 71.5t-71.5 172.5t71.5 172.5t172.5 71.5t172.5 -71.5t71.5 -172.5zM812 212q0 104 -73 177t-177 73q-27 0 -54 -6l104 -42q77 -31 109.5 -106.5t1.5 -151.5q-31 -77 -107 -109t-152 -1q-21 8 -62 24.5t-61 24.5 + q32 -60 91 -96.5t130 -36.5q104 0 177 73t73 177zM1642 953q0 126 -89.5 215.5t-215.5 89.5q-127 0 -216.5 -89.5t-89.5 -215.5q0 -127 89.5 -216t216.5 -89q126 0 215.5 89t89.5 216zM1792 953q0 -189 -133.5 -322t-321.5 -133l-437 -319q-12 -129 -109 -218t-229 -89 + q-121 0 -214 76t-118 192l-230 92v429l389 -157q79 48 173 48q13 0 35 -2l284 407q2 187 135.5 319t320.5 132q188 0 321.5 -133.5t133.5 -321.5z" /> + <glyph glyph-name="_412" unicode="" + d="M1242 889q0 80 -57 136.5t-137 56.5t-136.5 -57t-56.5 -136q0 -80 56.5 -136.5t136.5 -56.5t137 56.5t57 136.5zM632 301q0 -83 -58 -140.5t-140 -57.5q-56 0 -103 29t-72 77q52 -20 98 -40q60 -24 120 1.5t85 86.5q24 60 -1.5 120t-86.5 84l-82 33q22 5 42 5 + q82 0 140 -57.5t58 -140.5zM1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v153l172 -69q20 -92 93.5 -152t168.5 -60q104 0 181 70t87 173l345 252q150 0 255.5 105.5t105.5 254.5q0 150 -105.5 255.5t-255.5 105.5 + q-148 0 -253 -104.5t-107 -252.5l-225 -322q-9 1 -28 1q-75 0 -137 -37l-297 119v468q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5zM1289 887q0 -100 -71 -170.5t-171 -70.5t-170.5 70.5t-70.5 170.5t70.5 171t170.5 71q101 0 171.5 -70.5t70.5 -171.5z + " /> + <glyph glyph-name="_413" unicode="" horiz-adv-x="1792" + d="M836 367l-15 -368l-2 -22l-420 29q-36 3 -67 31.5t-47 65.5q-11 27 -14.5 55t4 65t12 55t21.5 64t19 53q78 -12 509 -28zM449 953l180 -379l-147 92q-63 -72 -111.5 -144.5t-72.5 -125t-39.5 -94.5t-18.5 -63l-4 -21l-190 357q-17 26 -18 56t6 47l8 18q35 63 114 188 + l-140 86zM1680 436l-188 -359q-12 -29 -36.5 -46.5t-43.5 -20.5l-18 -4q-71 -7 -219 -12l8 -164l-230 367l211 362l7 -173q170 -16 283 -5t170 33zM895 1360q-47 -63 -265 -435l-317 187l-19 12l225 356q20 31 60 45t80 10q24 -2 48.5 -12t42 -21t41.5 -33t36 -34.5 + t36 -39.5t32 -35zM1550 1053l212 -363q18 -37 12.5 -76t-27.5 -74q-13 -20 -33 -37t-38 -28t-48.5 -22t-47 -16t-51.5 -14t-46 -12q-34 72 -265 436l313 195zM1407 1279l142 83l-220 -373l-419 20l151 86q-34 89 -75 166t-75.5 123.5t-64.5 80t-47 46.5l-17 13l405 -1 + q31 3 58 -10.5t39 -28.5l11 -15q39 -61 112 -190z" /> + <glyph glyph-name="_414" unicode="" horiz-adv-x="2048" + d="M480 448q0 66 -47 113t-113 47t-113 -47t-47 -113t47 -113t113 -47t113 47t47 113zM516 768h1016l-89 357q-2 8 -14 17.5t-21 9.5h-768q-9 0 -21 -9.5t-14 -17.5zM1888 448q0 66 -47 113t-113 47t-113 -47t-47 -113t47 -113t113 -47t113 47t47 113zM2048 544v-384 + q0 -14 -9 -23t-23 -9h-96v-128q0 -80 -56 -136t-136 -56t-136 56t-56 136v128h-1024v-128q0 -80 -56 -136t-136 -56t-136 56t-56 136v128h-96q-14 0 -23 9t-9 23v384q0 93 65.5 158.5t158.5 65.5h28l105 419q23 94 104 157.5t179 63.5h768q98 0 179 -63.5t104 -157.5 + l105 -419h28q93 0 158.5 -65.5t65.5 -158.5z" /> + <glyph glyph-name="_415" unicode="" horiz-adv-x="2048" + d="M1824 640q93 0 158.5 -65.5t65.5 -158.5v-384q0 -14 -9 -23t-23 -9h-96v-64q0 -80 -56 -136t-136 -56t-136 56t-56 136v64h-1024v-64q0 -80 -56 -136t-136 -56t-136 56t-56 136v64h-96q-14 0 -23 9t-9 23v384q0 93 65.5 158.5t158.5 65.5h28l105 419q23 94 104 157.5 + t179 63.5h128v224q0 14 9 23t23 9h448q14 0 23 -9t9 -23v-224h128q98 0 179 -63.5t104 -157.5l105 -419h28zM320 160q66 0 113 47t47 113t-47 113t-113 47t-113 -47t-47 -113t47 -113t113 -47zM516 640h1016l-89 357q-2 8 -14 17.5t-21 9.5h-768q-9 0 -21 -9.5t-14 -17.5z + M1728 160q66 0 113 47t47 113t-47 113t-113 47t-113 -47t-47 -113t47 -113t113 -47z" /> + <glyph glyph-name="_416" unicode="" + d="M1504 64q0 -26 -19 -45t-45 -19h-462q1 -17 6 -87.5t5 -108.5q0 -25 -18 -42.5t-43 -17.5h-320q-25 0 -43 17.5t-18 42.5q0 38 5 108.5t6 87.5h-462q-26 0 -45 19t-19 45t19 45l402 403h-229q-26 0 -45 19t-19 45t19 45l402 403h-197q-26 0 -45 19t-19 45t19 45l384 384 + q19 19 45 19t45 -19l384 -384q19 -19 19 -45t-19 -45t-45 -19h-197l402 -403q19 -19 19 -45t-19 -45t-45 -19h-229l402 -403q19 -19 19 -45z" /> + <glyph glyph-name="_417" unicode="" + d="M1127 326q0 32 -30 51q-193 115 -447 115q-133 0 -287 -34q-42 -9 -42 -52q0 -20 13.5 -34.5t35.5 -14.5q5 0 37 8q132 27 243 27q226 0 397 -103q19 -11 33 -11q19 0 33 13.5t14 34.5zM1223 541q0 40 -35 61q-237 141 -548 141q-153 0 -303 -42q-48 -13 -48 -64 + q0 -25 17.5 -42.5t42.5 -17.5q7 0 37 8q122 33 251 33q279 0 488 -124q24 -13 38 -13q25 0 42.5 17.5t17.5 42.5zM1331 789q0 47 -40 70q-126 73 -293 110.5t-343 37.5q-204 0 -364 -47q-23 -7 -38.5 -25.5t-15.5 -48.5q0 -31 20.5 -52t51.5 -21q11 0 40 8q133 37 307 37 + q159 0 309.5 -34t253.5 -95q21 -12 40 -12q29 0 50.5 20.5t21.5 51.5zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="_418" unicode="" horiz-adv-x="1024" + d="M1024 1233l-303 -582l24 -31h279v-415h-507l-44 -30l-142 -273l-30 -30h-301v303l303 583l-24 30h-279v415h507l44 30l142 273l30 30h301v-303z" /> + <glyph glyph-name="_419" unicode="" horiz-adv-x="2304" + d="M784 164l16 241l-16 523q-1 10 -7.5 17t-16.5 7q-9 0 -16 -7t-7 -17l-14 -523l14 -241q1 -10 7.5 -16.5t15.5 -6.5q22 0 24 23zM1080 193l11 211l-12 586q0 16 -13 24q-8 5 -16 5t-16 -5q-13 -8 -13 -24l-1 -6l-10 -579q0 -1 11 -236v-1q0 -10 6 -17q9 -11 23 -11 + q11 0 20 9q9 7 9 20zM35 533l20 -128l-20 -126q-2 -9 -9 -9t-9 9l-17 126l17 128q2 9 9 9t9 -9zM121 612l26 -207l-26 -203q-2 -9 -10 -9q-9 0 -9 10l-23 202l23 207q0 9 9 9q8 0 10 -9zM401 159zM213 650l25 -245l-25 -237q0 -11 -11 -11q-10 0 -12 11l-21 237l21 245 + q2 12 12 12q11 0 11 -12zM307 657l23 -252l-23 -244q-2 -13 -14 -13q-13 0 -13 13l-21 244l21 252q0 13 13 13q12 0 14 -13zM401 639l21 -234l-21 -246q-2 -16 -16 -16q-6 0 -10.5 4.5t-4.5 11.5l-20 246l20 234q0 6 4.5 10.5t10.5 4.5q14 0 16 -15zM784 164zM495 785 + l21 -380l-21 -246q0 -7 -5 -12.5t-12 -5.5q-16 0 -18 18l-18 246l18 380q2 18 18 18q7 0 12 -5.5t5 -12.5zM589 871l19 -468l-19 -244q0 -8 -5.5 -13.5t-13.5 -5.5q-18 0 -20 19l-16 244l16 468q2 19 20 19q8 0 13.5 -5.5t5.5 -13.5zM687 911l18 -506l-18 -242 + q-2 -21 -22 -21q-19 0 -21 21l-16 242l16 506q0 9 6.5 15.5t14.5 6.5q9 0 15 -6.5t7 -15.5zM1079 169v0v0v0zM881 915l15 -510l-15 -239q0 -10 -7.5 -17.5t-17.5 -7.5t-17 7t-8 18l-14 239l14 510q0 11 7.5 18t17.5 7t17.5 -7t7.5 -18zM980 896l14 -492l-14 -236 + q0 -11 -8 -19t-19 -8t-19 8t-9 19l-12 236l12 492q1 12 9 20t19 8t18.5 -8t8.5 -20zM1192 404l-14 -231v0q0 -13 -9 -22t-22 -9t-22 9t-10 22l-6 114l-6 117l12 636v3q2 15 12 24q9 7 20 7q8 0 15 -5q14 -8 16 -26zM2304 423q0 -117 -83 -199.5t-200 -82.5h-786 + q-13 2 -22 11t-9 22v899q0 23 28 33q85 34 181 34q195 0 338 -131.5t160 -323.5q53 22 110 22q117 0 200 -83t83 -201z" /> + <glyph glyph-name="uniF1C0" unicode="" + d="M768 768q237 0 443 43t325 127v-170q0 -69 -103 -128t-280 -93.5t-385 -34.5t-385 34.5t-280 93.5t-103 128v170q119 -84 325 -127t443 -43zM768 0q237 0 443 43t325 127v-170q0 -69 -103 -128t-280 -93.5t-385 -34.5t-385 34.5t-280 93.5t-103 128v170q119 -84 325 -127 + t443 -43zM768 384q237 0 443 43t325 127v-170q0 -69 -103 -128t-280 -93.5t-385 -34.5t-385 34.5t-280 93.5t-103 128v170q119 -84 325 -127t443 -43zM768 1536q208 0 385 -34.5t280 -93.5t103 -128v-128q0 -69 -103 -128t-280 -93.5t-385 -34.5t-385 34.5t-280 93.5 + t-103 128v128q0 69 103 128t280 93.5t385 34.5z" /> + <glyph glyph-name="uniF1C1" unicode="" + d="M1468 1156q28 -28 48 -76t20 -88v-1152q0 -40 -28 -68t-68 -28h-1344q-40 0 -68 28t-28 68v1600q0 40 28 68t68 28h896q40 0 88 -20t76 -48zM1024 1400v-376h376q-10 29 -22 41l-313 313q-12 12 -41 22zM1408 -128v1024h-416q-40 0 -68 28t-28 68v416h-768v-1536h1280z + M894 465q33 -26 84 -56q59 7 117 7q147 0 177 -49q16 -22 2 -52q0 -1 -1 -2l-2 -2v-1q-6 -38 -71 -38q-48 0 -115 20t-130 53q-221 -24 -392 -83q-153 -262 -242 -262q-15 0 -28 7l-24 12q-1 1 -6 5q-10 10 -6 36q9 40 56 91.5t132 96.5q14 9 23 -6q2 -2 2 -4q52 85 107 197 + q68 136 104 262q-24 82 -30.5 159.5t6.5 127.5q11 40 42 40h21h1q23 0 35 -15q18 -21 9 -68q-2 -6 -4 -8q1 -3 1 -8v-30q-2 -123 -14 -192q55 -164 146 -238zM318 54q52 24 137 158q-51 -40 -87.5 -84t-49.5 -74zM716 974q-15 -42 -2 -132q1 7 7 44q0 3 7 43q1 4 4 8 + q-1 1 -1 2q-1 2 -1 3q-1 22 -13 36q0 -1 -1 -2v-2zM592 313q135 54 284 81q-2 1 -13 9.5t-16 13.5q-76 67 -127 176q-27 -86 -83 -197q-30 -56 -45 -83zM1238 329q-24 24 -140 24q76 -28 124 -28q14 0 18 1q0 1 -2 3z" /> + <glyph glyph-name="_422" unicode="" + d="M1468 1156q28 -28 48 -76t20 -88v-1152q0 -40 -28 -68t-68 -28h-1344q-40 0 -68 28t-28 68v1600q0 40 28 68t68 28h896q40 0 88 -20t76 -48zM1024 1400v-376h376q-10 29 -22 41l-313 313q-12 12 -41 22zM1408 -128v1024h-416q-40 0 -68 28t-28 68v416h-768v-1536h1280z + M233 768v-107h70l164 -661h159l128 485q7 20 10 46q2 16 2 24h4l3 -24q1 -3 3.5 -20t5.5 -26l128 -485h159l164 661h70v107h-300v-107h90l-99 -438q-5 -20 -7 -46l-2 -21h-4q0 3 -0.5 6.5t-1.5 8t-1 6.5q-1 5 -4 21t-5 25l-144 545h-114l-144 -545q-2 -9 -4.5 -24.5 + t-3.5 -21.5l-4 -21h-4l-2 21q-2 26 -7 46l-99 438h90v107h-300z" /> + <glyph glyph-name="_423" unicode="" + d="M1468 1156q28 -28 48 -76t20 -88v-1152q0 -40 -28 -68t-68 -28h-1344q-40 0 -68 28t-28 68v1600q0 40 28 68t68 28h896q40 0 88 -20t76 -48zM1024 1400v-376h376q-10 29 -22 41l-313 313q-12 12 -41 22zM1408 -128v1024h-416q-40 0 -68 28t-28 68v416h-768v-1536h1280z + M429 106v-106h281v106h-75l103 161q5 7 10 16.5t7.5 13.5t3.5 4h2q1 -4 5 -10q2 -4 4.5 -7.5t6 -8t6.5 -8.5l107 -161h-76v-106h291v106h-68l-192 273l195 282h67v107h-279v-107h74l-103 -159q-4 -7 -10 -16.5t-9 -13.5l-2 -3h-2q-1 4 -5 10q-6 11 -17 23l-106 159h76v107 + h-290v-107h68l189 -272l-194 -283h-68z" /> + <glyph glyph-name="_424" unicode="" + d="M1468 1156q28 -28 48 -76t20 -88v-1152q0 -40 -28 -68t-68 -28h-1344q-40 0 -68 28t-28 68v1600q0 40 28 68t68 28h896q40 0 88 -20t76 -48zM1024 1400v-376h376q-10 29 -22 41l-313 313q-12 12 -41 22zM1408 -128v1024h-416q-40 0 -68 28t-28 68v416h-768v-1536h1280z + M416 106v-106h327v106h-93v167h137q76 0 118 15q67 23 106.5 87t39.5 146q0 81 -37 141t-100 87q-48 19 -130 19h-368v-107h92v-555h-92zM769 386h-119v268h120q52 0 83 -18q56 -33 56 -115q0 -89 -62 -120q-31 -15 -78 -15z" /> + <glyph glyph-name="_425" unicode="" + d="M1468 1156q28 -28 48 -76t20 -88v-1152q0 -40 -28 -68t-68 -28h-1344q-40 0 -68 28t-28 68v1600q0 40 28 68t68 28h896q40 0 88 -20t76 -48zM1024 1400v-376h376q-10 29 -22 41l-313 313q-12 12 -41 22zM1408 -128v1024h-416q-40 0 -68 28t-28 68v416h-768v-1536h1280z + M1280 320v-320h-1024v192l192 192l128 -128l384 384zM448 512q-80 0 -136 56t-56 136t56 136t136 56t136 -56t56 -136t-56 -136t-136 -56z" /> + <glyph glyph-name="_426" unicode="" + d="M640 1152v128h-128v-128h128zM768 1024v128h-128v-128h128zM640 896v128h-128v-128h128zM768 768v128h-128v-128h128zM1468 1156q28 -28 48 -76t20 -88v-1152q0 -40 -28 -68t-68 -28h-1344q-40 0 -68 28t-28 68v1600q0 40 28 68t68 28h896q40 0 88 -20t76 -48zM1024 1400 + v-376h376q-10 29 -22 41l-313 313q-12 12 -41 22zM1408 -128v1024h-416q-40 0 -68 28t-28 68v416h-128v-128h-128v128h-512v-1536h1280zM781 593l107 -349q8 -27 8 -52q0 -83 -72.5 -137.5t-183.5 -54.5t-183.5 54.5t-72.5 137.5q0 25 8 52q21 63 120 396v128h128v-128h79 + q22 0 39 -13t23 -34zM640 128q53 0 90.5 19t37.5 45t-37.5 45t-90.5 19t-90.5 -19t-37.5 -45t37.5 -45t90.5 -19z" /> + <glyph glyph-name="_427" unicode="" + d="M1468 1156q28 -28 48 -76t20 -88v-1152q0 -40 -28 -68t-68 -28h-1344q-40 0 -68 28t-28 68v1600q0 40 28 68t68 28h896q40 0 88 -20t76 -48zM1024 1400v-376h376q-10 29 -22 41l-313 313q-12 12 -41 22zM1408 -128v1024h-416q-40 0 -68 28t-28 68v416h-768v-1536h1280z + M620 686q20 -8 20 -30v-544q0 -22 -20 -30q-8 -2 -12 -2q-12 0 -23 9l-166 167h-131q-14 0 -23 9t-9 23v192q0 14 9 23t23 9h131l166 167q16 15 35 7zM1037 -3q31 0 50 24q129 159 129 363t-129 363q-16 21 -43 24t-47 -14q-21 -17 -23.5 -43.5t14.5 -47.5 + q100 -123 100 -282t-100 -282q-17 -21 -14.5 -47.5t23.5 -42.5q18 -15 40 -15zM826 145q27 0 47 20q87 93 87 219t-87 219q-18 19 -45 20t-46 -17t-20 -44.5t18 -46.5q52 -57 52 -131t-52 -131q-19 -20 -18 -46.5t20 -44.5q20 -17 44 -17z" /> + <glyph glyph-name="_428" unicode="" + d="M1468 1156q28 -28 48 -76t20 -88v-1152q0 -40 -28 -68t-68 -28h-1344q-40 0 -68 28t-28 68v1600q0 40 28 68t68 28h896q40 0 88 -20t76 -48zM1024 1400v-376h376q-10 29 -22 41l-313 313q-12 12 -41 22zM1408 -128v1024h-416q-40 0 -68 28t-28 68v416h-768v-1536h1280z + M768 768q52 0 90 -38t38 -90v-384q0 -52 -38 -90t-90 -38h-384q-52 0 -90 38t-38 90v384q0 52 38 90t90 38h384zM1260 766q20 -8 20 -30v-576q0 -22 -20 -30q-8 -2 -12 -2q-14 0 -23 9l-265 266v90l265 266q9 9 23 9q4 0 12 -2z" /> + <glyph glyph-name="_429" unicode="" + d="M1468 1156q28 -28 48 -76t20 -88v-1152q0 -40 -28 -68t-68 -28h-1344q-40 0 -68 28t-28 68v1600q0 40 28 68t68 28h896q40 0 88 -20t76 -48zM1024 1400v-376h376q-10 29 -22 41l-313 313q-12 12 -41 22zM1408 -128v1024h-416q-40 0 -68 28t-28 68v416h-768v-1536h1280z + M480 768q8 11 21 12.5t24 -6.5l51 -38q11 -8 12.5 -21t-6.5 -24l-182 -243l182 -243q8 -11 6.5 -24t-12.5 -21l-51 -38q-11 -8 -24 -6.5t-21 12.5l-226 301q-14 19 0 38zM1282 467q14 -19 0 -38l-226 -301q-8 -11 -21 -12.5t-24 6.5l-51 38q-11 8 -12.5 21t6.5 24l182 243 + l-182 243q-8 11 -6.5 24t12.5 21l51 38q11 8 24 6.5t21 -12.5zM662 6q-13 2 -20.5 13t-5.5 24l138 831q2 13 13 20.5t24 5.5l63 -10q13 -2 20.5 -13t5.5 -24l-138 -831q-2 -13 -13 -20.5t-24 -5.5z" /> + <glyph glyph-name="_430" unicode="" + d="M1497 709v-198q-101 -23 -198 -23q-65 -136 -165.5 -271t-181.5 -215.5t-128 -106.5q-80 -45 -162 3q-28 17 -60.5 43.5t-85 83.5t-102.5 128.5t-107.5 184t-105.5 244t-91.5 314.5t-70.5 390h283q26 -218 70 -398.5t104.5 -317t121.5 -235.5t140 -195q169 169 287 406 + q-142 72 -223 220t-81 333q0 192 104 314.5t284 122.5q178 0 273 -105.5t95 -297.5q0 -159 -58 -286q-7 -1 -19.5 -3t-46 -2t-63 6t-62 25.5t-50.5 51.5q31 103 31 184q0 87 -29 132t-79 45q-53 0 -85 -49.5t-32 -140.5q0 -186 105 -293.5t267 -107.5q62 0 121 14z" /> + <glyph glyph-name="_431" unicode="" horiz-adv-x="1792" + d="M216 367l603 -402v359l-334 223zM154 511l193 129l-193 129v-258zM973 -35l603 402l-269 180l-334 -223v-359zM896 458l272 182l-272 182l-272 -182zM485 733l334 223v359l-603 -402zM1445 640l193 -129v258zM1307 733l269 180l-603 402v-359zM1792 913v-546 + q0 -41 -34 -64l-819 -546q-21 -13 -43 -13t-43 13l-819 546q-34 23 -34 64v546q0 41 34 64l819 546q21 13 43 13t43 -13l819 -546q34 -23 34 -64z" /> + <glyph glyph-name="_432" unicode="" horiz-adv-x="2048" + d="M1800 764q111 -46 179.5 -145.5t68.5 -221.5q0 -164 -118 -280.5t-285 -116.5q-4 0 -11.5 0.5t-10.5 0.5h-1209h-1h-2h-5q-170 10 -288 125.5t-118 280.5q0 110 55 203t147 147q-12 39 -12 82q0 115 82 196t199 81q95 0 172 -58q75 154 222.5 248t326.5 94 + q166 0 306 -80.5t221.5 -218.5t81.5 -301q0 -6 -0.5 -18t-0.5 -18zM468 498q0 -122 84 -193t208 -71q137 0 240 99q-16 20 -47.5 56.5t-43.5 50.5q-67 -65 -144 -65q-55 0 -93.5 33.5t-38.5 87.5q0 53 38.5 87t91.5 34q44 0 84.5 -21t73 -55t65 -75t69 -82t77 -75t97 -55 + t121.5 -21q121 0 204.5 71.5t83.5 190.5q0 121 -84 192t-207 71q-143 0 -241 -97l93 -108q66 64 142 64q52 0 92 -33t40 -84q0 -57 -37 -91.5t-94 -34.5q-43 0 -82.5 21t-72 55t-65.5 75t-69.5 82t-77.5 75t-96.5 55t-118.5 21q-122 0 -207 -70.5t-85 -189.5z" /> + <glyph glyph-name="_433" unicode="" horiz-adv-x="1792" + d="M896 1536q182 0 348 -71t286 -191t191 -286t71 -348t-71 -348t-191 -286t-286 -191t-348 -71t-348 71t-286 191t-191 286t-71 348t71 348t191 286t286 191t348 71zM896 1408q-190 0 -361 -90l194 -194q82 28 167 28t167 -28l194 194q-171 90 -361 90zM218 279l194 194 + q-28 82 -28 167t28 167l-194 194q-90 -171 -90 -361t90 -361zM896 -128q190 0 361 90l-194 194q-82 -28 -167 -28t-167 28l-194 -194q171 -90 361 -90zM896 256q159 0 271.5 112.5t112.5 271.5t-112.5 271.5t-271.5 112.5t-271.5 -112.5t-112.5 -271.5t112.5 -271.5 + t271.5 -112.5zM1380 473l194 -194q90 171 90 361t-90 361l-194 -194q28 -82 28 -167t-28 -167z" /> + <glyph glyph-name="_434" unicode="" horiz-adv-x="1792" + d="M1760 640q0 -176 -68.5 -336t-184 -275.5t-275.5 -184t-336 -68.5t-336 68.5t-275.5 184t-184 275.5t-68.5 336q0 213 97 398.5t265 305.5t374 151v-228q-221 -45 -366.5 -221t-145.5 -406q0 -130 51 -248.5t136.5 -204t204 -136.5t248.5 -51t248.5 51t204 136.5 + t136.5 204t51 248.5q0 230 -145.5 406t-366.5 221v228q206 -31 374 -151t265 -305.5t97 -398.5z" /> + <glyph glyph-name="uniF1D0" unicode="" horiz-adv-x="1792" + d="M19 662q8 217 116 406t305 318h5q0 -1 -1 -3q-8 -8 -28 -33.5t-52 -76.5t-60 -110.5t-44.5 -135.5t-14 -150.5t39 -157.5t108.5 -154q50 -50 102 -69.5t90.5 -11.5t69.5 23.5t47 32.5l16 16q39 51 53 116.5t6.5 122.5t-21 107t-26.5 80l-14 29q-10 25 -30.5 49.5t-43 41 + t-43.5 29.5t-35 19l-13 6l104 115q39 -17 78 -52t59 -61l19 -27q1 48 -18.5 103.5t-40.5 87.5l-20 31l161 183l160 -181q-33 -46 -52.5 -102.5t-22.5 -90.5l-4 -33q22 37 61.5 72.5t67.5 52.5l28 17l103 -115q-44 -14 -85 -50t-60 -65l-19 -29q-31 -56 -48 -133.5t-7 -170 + t57 -156.5q33 -45 77.5 -60.5t85 -5.5t76 26.5t57.5 33.5l21 16q60 53 96.5 115t48.5 121.5t10 121.5t-18 118t-37 107.5t-45.5 93t-45 72t-34.5 47.5l-13 17q-14 13 -7 13l10 -3q40 -29 62.5 -46t62 -50t64 -58t58.5 -65t55.5 -77t45.5 -88t38 -103t23.5 -117t10.5 -136 + q3 -259 -108 -465t-312 -321t-456 -115q-185 0 -351 74t-283.5 198t-184 293t-60.5 353z" /> + <glyph glyph-name="uniF1D1" unicode="" horiz-adv-x="1792" + d="M874 -102v-66q-208 6 -385 109.5t-283 275.5l58 34q29 -49 73 -99l65 57q148 -168 368 -212l-17 -86q65 -12 121 -13zM276 428l-83 -28q22 -60 49 -112l-57 -33q-98 180 -98 385t98 385l57 -33q-30 -56 -49 -112l82 -28q-35 -100 -35 -212q0 -109 36 -212zM1528 251 + l58 -34q-106 -172 -283 -275.5t-385 -109.5v66q56 1 121 13l-17 86q220 44 368 212l65 -57q44 50 73 99zM1377 805l-233 -80q14 -42 14 -85t-14 -85l232 -80q-31 -92 -98 -169l-185 162q-57 -67 -147 -85l48 -241q-52 -10 -98 -10t-98 10l48 241q-90 18 -147 85l-185 -162 + q-67 77 -98 169l232 80q-14 42 -14 85t14 85l-233 80q33 93 99 169l185 -162q59 68 147 86l-48 240q44 10 98 10t98 -10l-48 -240q88 -18 147 -86l185 162q66 -76 99 -169zM874 1448v-66q-65 -2 -121 -13l17 -86q-220 -42 -368 -211l-65 56q-38 -42 -73 -98l-57 33 + q106 172 282 275.5t385 109.5zM1705 640q0 -205 -98 -385l-57 33q27 52 49 112l-83 28q36 103 36 212q0 112 -35 212l82 28q-19 56 -49 112l57 33q98 -180 98 -385zM1585 1063l-57 -33q-35 56 -73 98l-65 -56q-148 169 -368 211l17 86q-56 11 -121 13v66q209 -6 385 -109.5 + t282 -275.5zM1748 640q0 173 -67.5 331t-181.5 272t-272 181.5t-331 67.5t-331 -67.5t-272 -181.5t-181.5 -272t-67.5 -331t67.5 -331t181.5 -272t272 -181.5t331 -67.5t331 67.5t272 181.5t181.5 272t67.5 331zM1792 640q0 -182 -71 -348t-191 -286t-286 -191t-348 -71 + t-348 71t-286 191t-191 286t-71 348t71 348t191 286t286 191t348 71t348 -71t286 -191t191 -286t71 -348z" /> + <glyph glyph-name="uniF1D2" unicode="" + d="M582 228q0 -66 -93 -66q-107 0 -107 63q0 64 98 64q102 0 102 -61zM546 694q0 -85 -74 -85q-77 0 -77 84q0 90 77 90q36 0 55 -25.5t19 -63.5zM712 769v125q-78 -29 -135 -29q-50 29 -110 29q-86 0 -145 -57t-59 -143q0 -50 29.5 -102t73.5 -67v-3q-38 -17 -38 -85 + q0 -53 41 -77v-3q-113 -37 -113 -139q0 -45 20 -78.5t54 -51t72 -25.5t81 -8q224 0 224 188q0 67 -48 99t-126 46q-27 5 -51.5 20.5t-24.5 39.5q0 44 49 52q77 15 122 70t45 134q0 24 -10 52q37 9 49 13zM771 350h137q-2 27 -2 82v387q0 46 2 69h-137q3 -23 3 -71v-392 + q0 -50 -3 -75zM1280 366v121q-30 -21 -68 -21q-53 0 -53 82v225h52q9 0 26.5 -1t26.5 -1v117h-105q0 82 3 102h-140q4 -24 4 -55v-47h-60v-117q36 3 37 3q3 0 11 -0.5t12 -0.5v-2h-2v-217q0 -37 2.5 -64t11.5 -56.5t24.5 -48.5t43.5 -31t66 -12q64 0 108 24zM924 1072 + q0 36 -24 63.5t-60 27.5t-60.5 -27t-24.5 -64q0 -36 25 -62.5t60 -26.5t59.5 27t24.5 62zM1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="_438" unicode="" horiz-adv-x="1792" + d="M595 22q0 100 -165 100q-158 0 -158 -104q0 -101 172 -101q151 0 151 105zM536 777q0 61 -30 102t-89 41q-124 0 -124 -145q0 -135 124 -135q119 0 119 137zM805 1101v-202q-36 -12 -79 -22q16 -43 16 -84q0 -127 -73 -216.5t-197 -112.5q-40 -8 -59.5 -27t-19.5 -58 + q0 -31 22.5 -51.5t58 -32t78.5 -22t86 -25.5t78.5 -37.5t58 -64t22.5 -98.5q0 -304 -363 -304q-69 0 -130 12.5t-116 41t-87.5 82t-32.5 127.5q0 165 182 225v4q-67 41 -67 126q0 109 63 137v4q-72 24 -119.5 108.5t-47.5 165.5q0 139 95 231.5t235 92.5q96 0 178 -47 + q98 0 218 47zM1123 220h-222q4 45 4 134v609q0 94 -4 128h222q-4 -33 -4 -124v-613q0 -89 4 -134zM1724 442v-196q-71 -39 -174 -39q-62 0 -107 20t-70 50t-39.5 78t-18.5 92t-4 103v351h2v4q-7 0 -19 1t-18 1q-21 0 -59 -6v190h96v76q0 54 -6 89h227q-6 -41 -6 -165h171 + v-190q-15 0 -43.5 2t-42.5 2h-85v-365q0 -131 87 -131q61 0 109 33zM1148 1389q0 -58 -39 -101.5t-96 -43.5q-58 0 -98 43.5t-40 101.5q0 59 39.5 103t98.5 44q58 0 96.5 -44.5t38.5 -102.5z" /> + <glyph glyph-name="_439" unicode="" + d="M809 532l266 499h-112l-157 -312q-24 -48 -44 -92l-42 92l-155 312h-120l263 -493v-324h101v318zM1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="uniF1D5" unicode="" horiz-adv-x="1280" + d="M842 964q0 -80 -57 -136.5t-136 -56.5q-60 0 -111 35q-62 -67 -115 -146q-247 -371 -202 -859q1 -22 -12.5 -38.5t-34.5 -18.5h-5q-20 0 -35 13.5t-17 33.5q-14 126 -3.5 247.5t29.5 217t54 186t69 155.5t74 125q61 90 132 165q-16 35 -16 77q0 80 56.5 136.5t136.5 56.5 + t136.5 -56.5t56.5 -136.5zM1223 953q0 -158 -78 -292t-212.5 -212t-292.5 -78q-64 0 -131 14q-21 5 -32.5 23.5t-6.5 39.5q5 20 23 31.5t39 7.5q51 -13 108 -13q97 0 186 38t153 102t102 153t38 186t-38 186t-102 153t-153 102t-186 38t-186 -38t-153 -102t-102 -153 + t-38 -186q0 -114 52 -218q10 -20 3.5 -40t-25.5 -30t-39.5 -3t-30.5 26q-64 123 -64 265q0 119 46.5 227t124.5 186t186 124t226 46q158 0 292.5 -78t212.5 -212.5t78 -292.5z" /> + <glyph glyph-name="uniF1D6" unicode="" horiz-adv-x="1792" + d="M270 730q-8 19 -8 52q0 20 11 49t24 45q-1 22 7.5 53t22.5 43q0 139 92.5 288.5t217.5 209.5q139 66 324 66q133 0 266 -55q49 -21 90 -48t71 -56t55 -68t42 -74t32.5 -84.5t25.5 -89.5t22 -98l1 -5q55 -83 55 -150q0 -14 -9 -40t-9 -38q0 -1 1.5 -3.5t3.5 -5t2 -3.5 + q77 -114 120.5 -214.5t43.5 -208.5q0 -43 -19.5 -100t-55.5 -57q-9 0 -19.5 7.5t-19 17.5t-19 26t-16 26.5t-13.5 26t-9 17.5q-1 1 -3 1l-5 -4q-59 -154 -132 -223q20 -20 61.5 -38.5t69 -41.5t35.5 -65q-2 -4 -4 -16t-7 -18q-64 -97 -302 -97q-53 0 -110.5 9t-98 20 + t-104.5 30q-15 5 -23 7q-14 4 -46 4.5t-40 1.5q-41 -45 -127.5 -65t-168.5 -20q-35 0 -69 1.5t-93 9t-101 20.5t-74.5 40t-32.5 64q0 40 10 59.5t41 48.5q11 2 40.5 13t49.5 12q4 0 14 2q2 2 2 4l-2 3q-48 11 -108 105.5t-73 156.5l-5 3q-4 0 -12 -20q-18 -41 -54.5 -74.5 + t-77.5 -37.5h-1q-4 0 -6 4.5t-5 5.5q-23 54 -23 100q0 275 252 466z" /> + <glyph glyph-name="uniF1D7" unicode="" horiz-adv-x="2048" + d="M580 1075q0 41 -25 66t-66 25q-43 0 -76 -25.5t-33 -65.5q0 -39 33 -64.5t76 -25.5q41 0 66 24.5t25 65.5zM1323 568q0 28 -25.5 50t-65.5 22q-27 0 -49.5 -22.5t-22.5 -49.5q0 -28 22.5 -50.5t49.5 -22.5q40 0 65.5 22t25.5 51zM1087 1075q0 41 -24.5 66t-65.5 25 + q-43 0 -76 -25.5t-33 -65.5q0 -39 33 -64.5t76 -25.5q41 0 65.5 24.5t24.5 65.5zM1722 568q0 28 -26 50t-65 22q-27 0 -49.5 -22.5t-22.5 -49.5q0 -28 22.5 -50.5t49.5 -22.5q39 0 65 22t26 51zM1456 965q-31 4 -70 4q-169 0 -311 -77t-223.5 -208.5t-81.5 -287.5 + q0 -78 23 -152q-35 -3 -68 -3q-26 0 -50 1.5t-55 6.5t-44.5 7t-54.5 10.5t-50 10.5l-253 -127l72 218q-290 203 -290 490q0 169 97.5 311t264 223.5t363.5 81.5q176 0 332.5 -66t262 -182.5t136.5 -260.5zM2048 404q0 -117 -68.5 -223.5t-185.5 -193.5l55 -181l-199 109 + q-150 -37 -218 -37q-169 0 -311 70.5t-223.5 191.5t-81.5 264t81.5 264t223.5 191.5t311 70.5q161 0 303 -70.5t227.5 -192t85.5 -263.5z" /> + <glyph glyph-name="_443" unicode="" horiz-adv-x="1792" + d="M1764 1525q33 -24 27 -64l-256 -1536q-5 -29 -32 -45q-14 -8 -31 -8q-11 0 -24 5l-453 185l-242 -295q-18 -23 -49 -23q-13 0 -22 4q-19 7 -30.5 23.5t-11.5 36.5v349l864 1059l-1069 -925l-395 162q-37 14 -40 55q-2 40 32 59l1664 960q15 9 32 9q20 0 36 -11z" /> + <glyph glyph-name="_444" unicode="" horiz-adv-x="1792" + d="M1764 1525q33 -24 27 -64l-256 -1536q-5 -29 -32 -45q-14 -8 -31 -8q-11 0 -24 5l-527 215l-298 -327q-18 -21 -47 -21q-14 0 -23 4q-19 7 -30 23.5t-11 36.5v452l-472 193q-37 14 -40 55q-3 39 32 59l1664 960q35 21 68 -2zM1422 26l221 1323l-1434 -827l336 -137 + l863 639l-478 -797z" /> + <glyph glyph-name="_445" unicode="" + d="M1536 640q0 -156 -61 -298t-164 -245t-245 -164t-298 -61q-172 0 -327 72.5t-264 204.5q-7 10 -6.5 22.5t8.5 20.5l137 138q10 9 25 9q16 -2 23 -12q73 -95 179 -147t225 -52q104 0 198.5 40.5t163.5 109.5t109.5 163.5t40.5 198.5t-40.5 198.5t-109.5 163.5 + t-163.5 109.5t-198.5 40.5q-98 0 -188 -35.5t-160 -101.5l137 -138q31 -30 14 -69q-17 -40 -59 -40h-448q-26 0 -45 19t-19 45v448q0 42 40 59q39 17 69 -14l130 -129q107 101 244.5 156.5t284.5 55.5q156 0 298 -61t245 -164t164 -245t61 -298zM896 928v-448q0 -14 -9 -23 + t-23 -9h-320q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h224v352q0 14 9 23t23 9h64q14 0 23 -9t9 -23z" /> + <glyph glyph-name="_446" unicode="" + d="M768 1280q-130 0 -248.5 -51t-204 -136.5t-136.5 -204t-51 -248.5t51 -248.5t136.5 -204t204 -136.5t248.5 -51t248.5 51t204 136.5t136.5 204t51 248.5t-51 248.5t-136.5 204t-204 136.5t-248.5 51zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103 + t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="_447" unicode="" horiz-adv-x="1792" + d="M1682 -128q-44 0 -132.5 3.5t-133.5 3.5q-44 0 -132 -3.5t-132 -3.5q-24 0 -37 20.5t-13 45.5q0 31 17 46t39 17t51 7t45 15q33 21 33 140l-1 391q0 21 -1 31q-13 4 -50 4h-675q-38 0 -51 -4q-1 -10 -1 -31l-1 -371q0 -142 37 -164q16 -10 48 -13t57 -3.5t45 -15 + t20 -45.5q0 -26 -12.5 -48t-36.5 -22q-47 0 -139.5 3.5t-138.5 3.5q-43 0 -128 -3.5t-127 -3.5q-23 0 -35.5 21t-12.5 45q0 30 15.5 45t36 17.5t47.5 7.5t42 15q33 23 33 143l-1 57v813q0 3 0.5 26t0 36.5t-1.5 38.5t-3.5 42t-6.5 36.5t-11 31.5t-16 18q-15 10 -45 12t-53 2 + t-41 14t-18 45q0 26 12 48t36 22q46 0 138.5 -3.5t138.5 -3.5q42 0 126.5 3.5t126.5 3.5q25 0 37.5 -22t12.5 -48q0 -30 -17 -43.5t-38.5 -14.5t-49.5 -4t-43 -13q-35 -21 -35 -160l1 -320q0 -21 1 -32q13 -3 39 -3h699q25 0 38 3q1 11 1 32l1 320q0 139 -35 160 + q-18 11 -58.5 12.5t-66 13t-25.5 49.5q0 26 12.5 48t37.5 22q44 0 132 -3.5t132 -3.5q43 0 129 3.5t129 3.5q25 0 37.5 -22t12.5 -48q0 -30 -17.5 -44t-40 -14.5t-51.5 -3t-44 -12.5q-35 -23 -35 -161l1 -943q0 -119 34 -140q16 -10 46 -13.5t53.5 -4.5t41.5 -15.5t18 -44.5 + q0 -26 -12 -48t-36 -22z" /> + <glyph glyph-name="_448" unicode="" horiz-adv-x="1280" + d="M1278 1347v-73q0 -29 -18.5 -61t-42.5 -32q-50 0 -54 -1q-26 -6 -32 -31q-3 -11 -3 -64v-1152q0 -25 -18 -43t-43 -18h-108q-25 0 -43 18t-18 43v1218h-143v-1218q0 -25 -17.5 -43t-43.5 -18h-108q-26 0 -43.5 18t-17.5 43v496q-147 12 -245 59q-126 58 -192 179 + q-64 117 -64 259q0 166 88 286q88 118 209 159q111 37 417 37h479q25 0 43 -18t18 -43z" /> + <glyph glyph-name="_449" unicode="" + d="M352 128v-128h-352v128h352zM704 256q26 0 45 -19t19 -45v-256q0 -26 -19 -45t-45 -19h-256q-26 0 -45 19t-19 45v256q0 26 19 45t45 19h256zM864 640v-128h-864v128h864zM224 1152v-128h-224v128h224zM1536 128v-128h-736v128h736zM576 1280q26 0 45 -19t19 -45v-256 + q0 -26 -19 -45t-45 -19h-256q-26 0 -45 19t-19 45v256q0 26 19 45t45 19h256zM1216 768q26 0 45 -19t19 -45v-256q0 -26 -19 -45t-45 -19h-256q-26 0 -45 19t-19 45v256q0 26 19 45t45 19h256zM1536 640v-128h-224v128h224zM1536 1152v-128h-864v128h864z" /> + <glyph glyph-name="uniF1E0" unicode="" + d="M1216 512q133 0 226.5 -93.5t93.5 -226.5t-93.5 -226.5t-226.5 -93.5t-226.5 93.5t-93.5 226.5q0 12 2 34l-360 180q-92 -86 -218 -86q-133 0 -226.5 93.5t-93.5 226.5t93.5 226.5t226.5 93.5q126 0 218 -86l360 180q-2 22 -2 34q0 133 93.5 226.5t226.5 93.5 + t226.5 -93.5t93.5 -226.5t-93.5 -226.5t-226.5 -93.5q-126 0 -218 86l-360 -180q2 -22 2 -34t-2 -34l360 -180q92 86 218 86z" /> + <glyph glyph-name="_451" unicode="" + d="M1280 341q0 88 -62.5 151t-150.5 63q-84 0 -145 -58l-241 120q2 16 2 23t-2 23l241 120q61 -58 145 -58q88 0 150.5 63t62.5 151t-62.5 150.5t-150.5 62.5t-151 -62.5t-63 -150.5q0 -7 2 -23l-241 -120q-62 57 -145 57q-88 0 -150.5 -62.5t-62.5 -150.5t62.5 -150.5 + t150.5 -62.5q83 0 145 57l241 -120q-2 -16 -2 -23q0 -88 63 -150.5t151 -62.5t150.5 62.5t62.5 150.5zM1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="_452" unicode="" horiz-adv-x="1792" + d="M571 947q-10 25 -34 35t-49 0q-108 -44 -191 -127t-127 -191q-10 -25 0 -49t35 -34q13 -5 24 -5q42 0 60 40q34 84 98.5 148.5t148.5 98.5q25 11 35 35t0 49zM1513 1303l46 -46l-244 -243l68 -68q19 -19 19 -45.5t-19 -45.5l-64 -64q89 -161 89 -343q0 -143 -55.5 -273.5 + t-150 -225t-225 -150t-273.5 -55.5t-273.5 55.5t-225 150t-150 225t-55.5 273.5t55.5 273.5t150 225t225 150t273.5 55.5q182 0 343 -89l64 64q19 19 45.5 19t45.5 -19l68 -68zM1521 1359q-10 -10 -22 -10q-13 0 -23 10l-91 90q-9 10 -9 23t9 23q10 9 23 9t23 -9l90 -91 + q10 -9 10 -22.5t-10 -22.5zM1751 1129q-11 -9 -23 -9t-23 9l-90 91q-10 9 -10 22.5t10 22.5q9 10 22.5 10t22.5 -10l91 -90q9 -10 9 -23t-9 -23zM1792 1312q0 -14 -9 -23t-23 -9h-96q-14 0 -23 9t-9 23t9 23t23 9h96q14 0 23 -9t9 -23zM1600 1504v-96q0 -14 -9 -23t-23 -9 + t-23 9t-9 23v96q0 14 9 23t23 9t23 -9t9 -23zM1751 1449l-91 -90q-10 -10 -22 -10q-13 0 -23 10q-10 9 -10 22.5t10 22.5l90 91q10 9 23 9t23 -9q9 -10 9 -23t-9 -23z" /> + <glyph glyph-name="_453" unicode="" horiz-adv-x="1792" + d="M609 720l287 208l287 -208l-109 -336h-355zM896 1536q182 0 348 -71t286 -191t191 -286t71 -348t-71 -348t-191 -286t-286 -191t-348 -71t-348 71t-286 191t-191 286t-71 348t71 348t191 286t286 191t348 71zM1515 186q149 203 149 454v3l-102 -89l-240 224l63 323 + l134 -12q-150 206 -389 282l53 -124l-287 -159l-287 159l53 124q-239 -76 -389 -282l135 12l62 -323l-240 -224l-102 89v-3q0 -251 149 -454l30 132l326 -40l139 -298l-116 -69q117 -39 240 -39t240 39l-116 69l139 298l326 40z" /> + <glyph glyph-name="_454" unicode="" horiz-adv-x="1792" + d="M448 224v-192q0 -14 -9 -23t-23 -9h-192q-14 0 -23 9t-9 23v192q0 14 9 23t23 9h192q14 0 23 -9t9 -23zM256 608v-192q0 -14 -9 -23t-23 -9h-192q-14 0 -23 9t-9 23v192q0 14 9 23t23 9h192q14 0 23 -9t9 -23zM832 224v-192q0 -14 -9 -23t-23 -9h-192q-14 0 -23 9t-9 23 + v192q0 14 9 23t23 9h192q14 0 23 -9t9 -23zM640 608v-192q0 -14 -9 -23t-23 -9h-192q-14 0 -23 9t-9 23v192q0 14 9 23t23 9h192q14 0 23 -9t9 -23zM66 768q-28 0 -47 19t-19 46v129h514v-129q0 -27 -19 -46t-46 -19h-383zM1216 224v-192q0 -14 -9 -23t-23 -9h-192 + q-14 0 -23 9t-9 23v192q0 14 9 23t23 9h192q14 0 23 -9t9 -23zM1024 608v-192q0 -14 -9 -23t-23 -9h-192q-14 0 -23 9t-9 23v192q0 14 9 23t23 9h192q14 0 23 -9t9 -23zM1600 224v-192q0 -14 -9 -23t-23 -9h-192q-14 0 -23 9t-9 23v192q0 14 9 23t23 9h192q14 0 23 -9t9 -23 + zM1408 608v-192q0 -14 -9 -23t-23 -9h-192q-14 0 -23 9t-9 23v192q0 14 9 23t23 9h192q14 0 23 -9t9 -23zM1792 1016v-13h-514v10q0 104 -382 102q-382 -1 -382 -102v-10h-514v13q0 17 8.5 43t34 64t65.5 75.5t110.5 76t160 67.5t224 47.5t293.5 18.5t293 -18.5t224 -47.5 + t160.5 -67.5t110.5 -76t65.5 -75.5t34 -64t8.5 -43zM1792 608v-192q0 -14 -9 -23t-23 -9h-192q-14 0 -23 9t-9 23v192q0 14 9 23t23 9h192q14 0 23 -9t9 -23zM1792 962v-129q0 -27 -19 -46t-46 -19h-384q-27 0 -46 19t-19 46v129h514z" /> + <glyph glyph-name="_455" unicode="" horiz-adv-x="1792" + d="M704 1216v-768q0 -26 -19 -45t-45 -19v-576q0 -26 -19 -45t-45 -19h-512q-26 0 -45 19t-19 45v512l249 873q7 23 31 23h424zM1024 1216v-704h-256v704h256zM1792 320v-512q0 -26 -19 -45t-45 -19h-512q-26 0 -45 19t-19 45v576q-26 0 -45 19t-19 45v768h424q24 0 31 -23z + M736 1504v-224h-352v224q0 14 9 23t23 9h288q14 0 23 -9t9 -23zM1408 1504v-224h-352v224q0 14 9 23t23 9h288q14 0 23 -9t9 -23z" /> + <glyph glyph-name="_456" unicode="" horiz-adv-x="1792" + d="M1755 1083q37 -38 37 -90.5t-37 -90.5l-401 -400l150 -150l-160 -160q-163 -163 -389.5 -186.5t-411.5 100.5l-362 -362h-181v181l362 362q-124 185 -100.5 411.5t186.5 389.5l160 160l150 -150l400 401q38 37 91 37t90 -37t37 -90.5t-37 -90.5l-400 -401l234 -234 + l401 400q38 37 91 37t90 -37z" /> + <glyph glyph-name="_457" unicode="" horiz-adv-x="1792" + d="M873 796q0 -83 -63.5 -142.5t-152.5 -59.5t-152.5 59.5t-63.5 142.5q0 84 63.5 143t152.5 59t152.5 -59t63.5 -143zM1375 796q0 -83 -63 -142.5t-153 -59.5q-89 0 -152.5 59.5t-63.5 142.5q0 84 63.5 143t152.5 59q90 0 153 -59t63 -143zM1600 616v667q0 87 -32 123.5 + t-111 36.5h-1112q-83 0 -112.5 -34t-29.5 -126v-673q43 -23 88.5 -40t81 -28t81 -18.5t71 -11t70 -4t58.5 -0.5t56.5 2t44.5 2q68 1 95 -27q6 -6 10 -9q26 -25 61 -51q7 91 118 87q5 0 36.5 -1.5t43 -2t45.5 -1t53 1t54.5 4.5t61 8.5t62 13.5t67 19.5t67.5 27t72 34.5z + M1763 621q-121 -149 -372 -252q84 -285 -23 -465q-66 -113 -183 -148q-104 -32 -182 15q-86 51 -82 164l-1 326v1q-8 2 -24.5 6t-23.5 5l-1 -338q4 -114 -83 -164q-79 -47 -183 -15q-117 36 -182 150q-105 180 -22 463q-251 103 -372 252q-25 37 -4 63t60 -1q4 -2 11.5 -7 + t10.5 -8v694q0 72 47 123t114 51h1257q67 0 114 -51t47 -123v-694l21 15q39 27 60 1t-4 -63z" /> + <glyph glyph-name="_458" unicode="" horiz-adv-x="1792" + d="M896 1102v-434h-145v434h145zM1294 1102v-434h-145v434h145zM1294 342l253 254v795h-1194v-1049h326v-217l217 217h398zM1692 1536v-1013l-434 -434h-326l-217 -217h-217v217h-398v1158l109 289h1483z" /> + <glyph glyph-name="_459" unicode="" + d="M773 217v-127q-1 -292 -6 -305q-12 -32 -51 -40q-54 -9 -181.5 38t-162.5 89q-13 15 -17 36q-1 12 4 26q4 10 34 47t181 216q1 0 60 70q15 19 39.5 24.5t49.5 -3.5q24 -10 37.5 -29t12.5 -42zM624 468q-3 -55 -52 -70l-120 -39q-275 -88 -292 -88q-35 2 -54 36 + q-12 25 -17 75q-8 76 1 166.5t30 124.5t56 32q13 0 202 -77q71 -29 115 -47l84 -34q23 -9 35.5 -30.5t11.5 -48.5zM1450 171q-7 -54 -91.5 -161t-135.5 -127q-37 -14 -63 7q-14 10 -184 287l-47 77q-14 21 -11.5 46t19.5 46q35 43 83 26q1 -1 119 -40q203 -66 242 -79.5 + t47 -20.5q28 -22 22 -61zM778 803q5 -102 -54 -122q-58 -17 -114 71l-378 598q-8 35 19 62q41 43 207.5 89.5t224.5 31.5q40 -10 49 -45q3 -18 22 -305.5t24 -379.5zM1440 695q3 -39 -26 -59q-15 -10 -329 -86q-67 -15 -91 -23l1 2q-23 -6 -46 4t-37 32q-30 47 0 87 + q1 1 75 102q125 171 150 204t34 39q28 19 65 2q48 -23 123 -133.5t81 -167.5v-3z" /> + <glyph glyph-name="_460" unicode="" horiz-adv-x="2048" + d="M1024 1024h-384v-384h384v384zM1152 384v-128h-640v128h640zM1152 1152v-640h-640v640h640zM1792 384v-128h-512v128h512zM1792 640v-128h-512v128h512zM1792 896v-128h-512v128h512zM1792 1152v-128h-512v128h512zM256 192v960h-128v-960q0 -26 19 -45t45 -19t45 19 + t19 45zM1920 192v1088h-1536v-1088q0 -33 -11 -64h1483q26 0 45 19t19 45zM2048 1408v-1216q0 -80 -56 -136t-136 -56h-1664q-80 0 -136 56t-56 136v1088h256v128h1792z" /> + <glyph glyph-name="_461" unicode="" horiz-adv-x="2048" + d="M1024 13q-20 0 -93 73.5t-73 93.5q0 32 62.5 54t103.5 22t103.5 -22t62.5 -54q0 -20 -73 -93.5t-93 -73.5zM1294 284q-2 0 -40 25t-101.5 50t-128.5 25t-128.5 -25t-101 -50t-40.5 -25q-18 0 -93.5 75t-75.5 93q0 13 10 23q78 77 196 121t233 44t233 -44t196 -121 + q10 -10 10 -23q0 -18 -75.5 -93t-93.5 -75zM1567 556q-11 0 -23 8q-136 105 -252 154.5t-268 49.5q-85 0 -170.5 -22t-149 -53t-113.5 -62t-79 -53t-31 -22q-17 0 -92 75t-75 93q0 12 10 22q132 132 320 205t380 73t380 -73t320 -205q10 -10 10 -22q0 -18 -75 -93t-92 -75z + M1838 827q-11 0 -22 9q-179 157 -371.5 236.5t-420.5 79.5t-420.5 -79.5t-371.5 -236.5q-11 -9 -22 -9q-17 0 -92.5 75t-75.5 93q0 13 10 23q187 186 445 288t527 102t527 -102t445 -288q10 -10 10 -23q0 -18 -75.5 -93t-92.5 -75z" /> + <glyph glyph-name="_462" unicode="" horiz-adv-x="1792" + d="M384 0q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM768 0q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM384 384q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5 + t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM1152 0q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM768 384q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5 + t37.5 90.5zM384 768q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM1152 384q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM768 768q0 53 -37.5 90.5t-90.5 37.5 + t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM1536 0v384q0 52 -38 90t-90 38t-90 -38t-38 -90v-384q0 -52 38 -90t90 -38t90 38t38 90zM1152 768q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5z + M1536 1088v256q0 26 -19 45t-45 19h-1280q-26 0 -45 -19t-19 -45v-256q0 -26 19 -45t45 -19h1280q26 0 45 19t19 45zM1536 768q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM1664 1408v-1536q0 -52 -38 -90t-90 -38 + h-1408q-52 0 -90 38t-38 90v1536q0 52 38 90t90 38h1408q52 0 90 -38t38 -90z" /> + <glyph glyph-name="_463" unicode="" + d="M1519 890q18 -84 -4 -204q-87 -444 -565 -444h-44q-25 0 -44 -16.5t-24 -42.5l-4 -19l-55 -346l-2 -15q-5 -26 -24.5 -42.5t-44.5 -16.5h-251q-21 0 -33 15t-9 36q9 56 26.5 168t26.5 168t27 167.5t27 167.5q5 37 43 37h131q133 -2 236 21q175 39 287 144q102 95 155 246 + q24 70 35 133q1 6 2.5 7.5t3.5 1t6 -3.5q79 -59 98 -162zM1347 1172q0 -107 -46 -236q-80 -233 -302 -315q-113 -40 -252 -42q0 -1 -90 -1l-90 1q-100 0 -118 -96q-2 -8 -85 -530q-1 -10 -12 -10h-295q-22 0 -36.5 16.5t-11.5 38.5l232 1471q5 29 27.5 48t51.5 19h598 + q34 0 97.5 -13t111.5 -32q107 -41 163.5 -123t56.5 -196z" /> + <glyph glyph-name="_464" unicode="" horiz-adv-x="1792" + d="M441 864q33 0 52 -26q266 -364 362 -774h-446q-127 441 -367 749q-12 16 -3 33.5t29 17.5h373zM1000 507q-49 -199 -125 -393q-79 310 -256 594q40 221 44 449q211 -340 337 -650zM1099 1216q235 -324 384.5 -698.5t184.5 -773.5h-451q-41 665 -553 1472h435zM1792 640 + q0 -424 -101 -812q-67 560 -359 1083q-25 301 -106 584q-4 16 5.5 28.5t25.5 12.5h359q21 0 38.5 -13t22.5 -33q115 -409 115 -850z" /> + <glyph glyph-name="uniF1F0" unicode="" horiz-adv-x="2304" + d="M1975 546h-138q14 37 66 179l3 9q4 10 10 26t9 26l12 -55zM531 611l-58 295q-11 54 -75 54h-268l-2 -13q311 -79 403 -336zM710 960l-162 -438l-17 89q-26 70 -85 129.5t-131 88.5l135 -510h175l261 641h-176zM849 318h166l104 642h-166zM1617 944q-69 27 -149 27 + q-123 0 -201 -59t-79 -153q-1 -102 145 -174q48 -23 67 -41t19 -39q0 -30 -30 -46t-69 -16q-86 0 -156 33l-22 11l-23 -144q74 -34 185 -34q130 -1 208.5 59t80.5 160q0 106 -140 174q-49 25 -71 42t-22 38q0 22 24.5 38.5t70.5 16.5q70 1 124 -24l15 -8zM2042 960h-128 + q-65 0 -87 -54l-246 -588h174l35 96h212q5 -22 20 -96h154zM2304 1280v-1280q0 -52 -38 -90t-90 -38h-2048q-52 0 -90 38t-38 90v1280q0 52 38 90t90 38h2048q52 0 90 -38t38 -90z" /> + <glyph glyph-name="_466" unicode="" horiz-adv-x="2304" + d="M1119 1195q-128 85 -281 85q-103 0 -197.5 -40.5t-162.5 -108.5t-108.5 -162t-40.5 -197q0 -104 40.5 -198t108.5 -162t162 -108.5t198 -40.5q153 0 281 85q-131 107 -178 265.5t0.5 316.5t177.5 265zM1152 1171q-126 -99 -172 -249.5t-0.5 -300.5t172.5 -249 + q127 99 172.5 249t-0.5 300.5t-172 249.5zM1185 1195q130 -107 177.5 -265.5t0.5 -317t-178 -264.5q128 -85 281 -85q104 0 198 40.5t162 108.5t108.5 162t40.5 198q0 103 -40.5 197t-108.5 162t-162.5 108.5t-197.5 40.5q-153 0 -281 -85zM1926 473h7v3h-17v-3h7v-17h3v17z + M1955 456h4v20h-5l-6 -13l-6 13h-5v-20h3v15l6 -13h4l5 13v-15zM1947 16v-2h-2h-3v3h3h2v-1zM1947 7h3l-4 5h2l1 1q1 1 1 3t-1 3l-1 1h-3h-6v-13h3v5h1zM685 75q0 19 11 31t30 12q18 0 29 -12.5t11 -30.5q0 -19 -11 -31t-29 -12q-19 0 -30 12t-11 31zM1158 119q30 0 35 -32 + h-70q5 32 35 32zM1514 75q0 19 11 31t29 12t29.5 -12.5t11.5 -30.5q0 -19 -11 -31t-30 -12q-18 0 -29 12t-11 31zM1786 75q0 18 11.5 30.5t29.5 12.5t29.5 -12.5t11.5 -30.5q0 -19 -11.5 -31t-29.5 -12t-29.5 12.5t-11.5 30.5zM1944 3q-2 0 -4 1q-1 0 -3 2t-2 3q-1 2 -1 4 + q0 3 1 4q0 2 2 4l1 1q2 0 2 1q2 1 4 1q3 0 4 -1l4 -2l2 -4v-1q1 -2 1 -3l-1 -1v-3t-1 -1l-1 -2q-2 -2 -4 -2q-1 -1 -4 -1zM599 7h30v85q0 24 -14.5 38.5t-39.5 15.5q-32 0 -47 -24q-14 24 -45 24q-24 0 -39 -20v16h-30v-135h30v75q0 36 33 36q30 0 30 -36v-75h29v75 + q0 36 33 36q30 0 30 -36v-75zM765 7h29v68v67h-29v-16q-17 20 -43 20q-29 0 -48 -20t-19 -51t19 -51t48 -20q28 0 43 20v-17zM943 48q0 34 -47 40l-14 2q-23 4 -23 14q0 15 25 15q23 0 43 -11l12 24q-22 14 -55 14q-26 0 -41 -12t-15 -32q0 -33 47 -39l13 -2q24 -4 24 -14 + q0 -17 -31 -17q-25 0 -45 14l-13 -23q25 -17 58 -17q29 0 45.5 12t16.5 32zM1073 14l-8 25q-13 -7 -26 -7q-19 0 -19 22v61h48v27h-48v41h-30v-41h-28v-27h28v-61q0 -50 47 -50q21 0 36 10zM1159 146q-29 0 -48 -20t-19 -51q0 -32 19.5 -51.5t49.5 -19.5q33 0 55 19l-14 22 + q-18 -15 -39 -15q-34 0 -41 33h101v12q0 32 -18 51.5t-46 19.5zM1318 146q-23 0 -35 -20v16h-30v-135h30v76q0 35 29 35q10 0 18 -4l9 28q-9 4 -21 4zM1348 75q0 -31 19.5 -51t52.5 -20q29 0 48 16l-14 24q-18 -13 -35 -12q-18 0 -29.5 12t-11.5 31t11.5 31t29.5 12 + q19 0 35 -12l14 24q-20 16 -48 16q-33 0 -52.5 -20t-19.5 -51zM1593 7h30v68v67h-30v-16q-15 20 -42 20q-29 0 -48.5 -20t-19.5 -51t19.5 -51t48.5 -20q28 0 42 20v-17zM1726 146q-23 0 -35 -20v16h-29v-135h29v76q0 35 29 35q10 0 18 -4l9 28q-8 4 -21 4zM1866 7h29v68v122 + h-29v-71q-15 20 -43 20t-47.5 -20.5t-19.5 -50.5t19.5 -50.5t47.5 -20.5q29 0 43 20v-17zM1944 27l-2 -1h-3q-2 -1 -4 -3q-3 -1 -3 -4q-1 -2 -1 -6q0 -3 1 -5q0 -2 3 -4q2 -2 4 -3t5 -1q4 0 6 1q0 1 2 2l2 1q1 1 3 4q1 2 1 5q0 4 -1 6q-1 1 -3 4q0 1 -2 2l-2 1q-1 0 -3 0.5 + t-3 0.5zM2304 1280v-1280q0 -52 -38 -90t-90 -38h-2048q-52 0 -90 38t-38 90v1280q0 52 38 90t90 38h2048q52 0 90 -38t38 -90z" /> + <glyph glyph-name="_467" unicode="" horiz-adv-x="2304" + d="M313 759q0 -51 -36 -84q-29 -26 -89 -26h-17v220h17q61 0 89 -27q36 -31 36 -83zM2089 824q0 -52 -64 -52h-19v101h20q63 0 63 -49zM380 759q0 74 -50 120.5t-129 46.5h-95v-333h95q74 0 119 38q60 51 60 128zM410 593h65v333h-65v-333zM730 694q0 40 -20.5 62t-75.5 42 + q-29 10 -39.5 19t-10.5 23q0 16 13.5 26.5t34.5 10.5q29 0 53 -27l34 44q-41 37 -98 37q-44 0 -74 -27.5t-30 -67.5q0 -35 18 -55.5t64 -36.5q37 -13 45 -19q19 -12 19 -34q0 -20 -14 -33.5t-36 -13.5q-48 0 -71 44l-42 -40q44 -64 115 -64q51 0 83 30.5t32 79.5zM1008 604 + v77q-37 -37 -78 -37q-49 0 -80.5 32.5t-31.5 82.5q0 48 31.5 81.5t77.5 33.5q43 0 81 -38v77q-40 20 -80 20q-74 0 -125.5 -50.5t-51.5 -123.5t51 -123.5t125 -50.5q42 0 81 19zM2240 0v527q-65 -40 -144.5 -84t-237.5 -117t-329.5 -137.5t-417.5 -134.5t-504 -118h1569 + q26 0 45 19t19 45zM1389 757q0 75 -53 128t-128 53t-128 -53t-53 -128t53 -128t128 -53t128 53t53 128zM1541 584l144 342h-71l-90 -224l-89 224h-71l142 -342h35zM1714 593h184v56h-119v90h115v56h-115v74h119v57h-184v-333zM2105 593h80l-105 140q76 16 76 94q0 47 -31 73 + t-87 26h-97v-333h65v133h9zM2304 1274v-1268q0 -56 -38.5 -95t-93.5 -39h-2040q-55 0 -93.5 39t-38.5 95v1268q0 56 38.5 95t93.5 39h2040q55 0 93.5 -39t38.5 -95z" /> + <glyph glyph-name="f1f3" unicode="" horiz-adv-x="2304" + d="M119 854h89l-45 108zM740 328l74 79l-70 79h-163v-49h142v-55h-142v-54h159zM898 406l99 -110v217zM1186 453q0 33 -40 33h-84v-69h83q41 0 41 36zM1475 457q0 29 -42 29h-82v-61h81q43 0 43 32zM1197 923q0 29 -42 29h-82v-60h81q43 0 43 31zM1656 854h89l-44 108z + M699 1009v-271h-66v212l-94 -212h-57l-94 212v-212h-132l-25 60h-135l-25 -60h-70l116 271h96l110 -257v257h106l85 -184l77 184h108zM1255 453q0 -20 -5.5 -35t-14 -25t-22.5 -16.5t-26 -10t-31.5 -4.5t-31.5 -1t-32.5 0.5t-29.5 0.5v-91h-126l-80 90l-83 -90h-256v271h260 + l80 -89l82 89h207q109 0 109 -89zM964 794v-56h-217v271h217v-57h-152v-49h148v-55h-148v-54h152zM2304 235v-229q0 -55 -38.5 -94.5t-93.5 -39.5h-2040q-55 0 -93.5 39.5t-38.5 94.5v678h111l25 61h55l25 -61h218v46l19 -46h113l20 47v-47h541v99l10 1q10 0 10 -14v-86h279 + v23q23 -12 55 -18t52.5 -6.5t63 0.5t51.5 1l25 61h56l25 -61h227v58l34 -58h182v378h-180v-44l-25 44h-185v-44l-23 44h-249q-69 0 -109 -22v22h-172v-22q-24 22 -73 22h-628l-43 -97l-43 97h-198v-44l-22 44h-169l-78 -179v391q0 55 38.5 94.5t93.5 39.5h2040 + q55 0 93.5 -39.5t38.5 -94.5v-678h-120q-51 0 -81 -22v22h-177q-55 0 -78 -22v22h-316v-22q-31 22 -87 22h-209v-22q-23 22 -91 22h-234l-54 -58l-50 58h-349v-378h343l55 59l52 -59h211v89h21q59 0 90 13v-102h174v99h8q8 0 10 -2t2 -10v-87h529q57 0 88 24v-24h168 + q60 0 95 17zM1546 469q0 -23 -12 -43t-34 -29q25 -9 34 -26t9 -46v-54h-65v45q0 33 -12 43.5t-46 10.5h-69v-99h-65v271h154q48 0 77 -15t29 -58zM1269 936q0 -24 -12.5 -44t-33.5 -29q26 -9 34.5 -25.5t8.5 -46.5v-53h-65q0 9 0.5 26.5t0 25t-3 18.5t-8.5 16t-17.5 8.5 + t-29.5 3.5h-70v-98h-64v271l153 -1q49 0 78 -14.5t29 -57.5zM1798 327v-56h-216v271h216v-56h-151v-49h148v-55h-148v-54zM1372 1009v-271h-66v271h66zM2065 357q0 -86 -102 -86h-126v58h126q34 0 34 25q0 16 -17 21t-41.5 5t-49.5 3.5t-42 22.5t-17 55q0 39 26 60t66 21 + h130v-57h-119q-36 0 -36 -25q0 -16 17.5 -20.5t42 -4t49 -2.5t42 -21.5t17.5 -54.5zM2304 407v-101q-24 -35 -88 -35h-125v58h125q33 0 33 25q0 13 -12.5 19t-31 5.5t-40 2t-40 8t-31 24t-12.5 48.5q0 39 26.5 60t66.5 21h129v-57h-118q-36 0 -36 -25q0 -20 29 -22t68.5 -5 + t56.5 -26zM2139 1008v-270h-92l-122 203v-203h-132l-26 60h-134l-25 -60h-75q-129 0 -129 133q0 138 133 138h63v-59q-7 0 -28 1t-28.5 0.5t-23 -2t-21.5 -6.5t-14.5 -13.5t-11.5 -23t-3 -33.5q0 -38 13.5 -58t49.5 -20h29l92 213h97l109 -256v256h99l114 -188v188h66z" /> + <glyph glyph-name="_469" unicode="" horiz-adv-x="2304" + d="M745 630q0 -37 -25.5 -61.5t-62.5 -24.5q-29 0 -46.5 16t-17.5 44q0 37 25 62.5t62 25.5q28 0 46.5 -16.5t18.5 -45.5zM1530 779q0 -42 -22 -57t-66 -15l-32 -1l17 107q2 11 13 11h18q22 0 35 -2t25 -12.5t12 -30.5zM1881 630q0 -36 -25.5 -61t-61.5 -25q-29 0 -47 16 + t-18 44q0 37 25 62.5t62 25.5q28 0 46.5 -16.5t18.5 -45.5zM513 801q0 59 -38.5 85.5t-100.5 26.5h-160q-19 0 -21 -19l-65 -408q-1 -6 3 -11t10 -5h76q20 0 22 19l18 110q1 8 7 13t15 6.5t17 1.5t19 -1t14 -1q86 0 135 48.5t49 134.5zM822 489l41 261q1 6 -3 11t-10 5h-76 + q-14 0 -17 -33q-27 40 -95 40q-72 0 -122.5 -54t-50.5 -127q0 -59 34.5 -94t92.5 -35q28 0 58 12t48 32q-4 -12 -4 -21q0 -16 13 -16h69q19 0 22 19zM1269 752q0 5 -4 9.5t-9 4.5h-77q-11 0 -18 -10l-106 -156l-44 150q-5 16 -22 16h-75q-5 0 -9 -4.5t-4 -9.5q0 -2 19.5 -59 + t42 -123t23.5 -70q-82 -112 -82 -120q0 -13 13 -13h77q11 0 18 10l255 368q2 2 2 7zM1649 801q0 59 -38.5 85.5t-100.5 26.5h-159q-20 0 -22 -19l-65 -408q-1 -6 3 -11t10 -5h82q12 0 16 13l18 116q1 8 7 13t15 6.5t17 1.5t19 -1t14 -1q86 0 135 48.5t49 134.5zM1958 489 + l41 261q1 6 -3 11t-10 5h-76q-14 0 -17 -33q-26 40 -95 40q-72 0 -122.5 -54t-50.5 -127q0 -59 34.5 -94t92.5 -35q29 0 59 12t47 32q0 -1 -2 -9t-2 -12q0 -16 13 -16h69q19 0 22 19zM2176 898v1q0 14 -13 14h-74q-11 0 -13 -11l-65 -416l-1 -2q0 -5 4 -9.5t10 -4.5h66 + q19 0 21 19zM392 764q-5 -35 -26 -46t-60 -11l-33 -1l17 107q2 11 13 11h19q40 0 58 -11.5t12 -48.5zM2304 1280v-1280q0 -52 -38 -90t-90 -38h-2048q-52 0 -90 38t-38 90v1280q0 52 38 90t90 38h2048q52 0 90 -38t38 -90z" /> + <glyph glyph-name="_470" unicode="" horiz-adv-x="2304" + d="M1597 633q0 -69 -21 -106q-19 -35 -52 -35q-23 0 -41 9v224q29 30 57 30q57 0 57 -122zM2035 669h-110q6 98 56 98q51 0 54 -98zM476 534q0 59 -33 91.5t-101 57.5q-36 13 -52 24t-16 25q0 26 38 26q58 0 124 -33l18 112q-67 32 -149 32q-77 0 -123 -38q-48 -39 -48 -109 + q0 -58 32.5 -90.5t99.5 -56.5q39 -14 54.5 -25.5t15.5 -27.5q0 -31 -48 -31q-29 0 -70 12.5t-72 30.5l-18 -113q72 -41 168 -41q81 0 129 37q51 41 51 117zM771 749l19 111h-96v135l-129 -21l-18 -114l-46 -8l-17 -103h62v-219q0 -84 44 -120q38 -30 111 -30q32 0 79 11v118 + q-32 -7 -44 -7q-42 0 -42 50v197h77zM1087 724v139q-15 3 -28 3q-32 0 -55.5 -16t-33.5 -46l-10 56h-131v-471h150v306q26 31 82 31q16 0 26 -2zM1124 389h150v471h-150v-471zM1746 638q0 122 -45 179q-40 52 -111 52q-64 0 -117 -56l-8 47h-132v-645l150 25v151 + q36 -11 68 -11q83 0 134 56q61 65 61 202zM1278 986q0 33 -23 56t-56 23t-56 -23t-23 -56t23 -56.5t56 -23.5t56 23.5t23 56.5zM2176 629q0 113 -48 176q-50 64 -144 64q-96 0 -151.5 -66t-55.5 -180q0 -128 63 -188q55 -55 161 -55q101 0 160 40l-16 103q-57 -31 -128 -31 + q-43 0 -63 19q-23 19 -28 66h248q2 14 2 52zM2304 1280v-1280q0 -52 -38 -90t-90 -38h-2048q-52 0 -90 38t-38 90v1280q0 52 38 90t90 38h2048q52 0 90 -38t38 -90z" /> + <glyph glyph-name="_471" unicode="" horiz-adv-x="2048" + d="M1558 684q61 -356 298 -556q0 -52 -38 -90t-90 -38h-448q0 -106 -75 -181t-181 -75t-180.5 74.5t-75.5 180.5zM1024 -176q16 0 16 16t-16 16q-59 0 -101.5 42.5t-42.5 101.5q0 16 -16 16t-16 -16q0 -73 51.5 -124.5t124.5 -51.5zM2026 1424q8 -10 7.5 -23.5t-10.5 -22.5 + l-1872 -1622q-10 -8 -23.5 -7t-21.5 11l-84 96q-8 10 -7.5 23.5t10.5 21.5l186 161q-19 32 -19 66q50 42 91 88t85 119.5t74.5 158.5t50 206t19.5 260q0 152 117 282.5t307 158.5q-8 19 -8 39q0 40 28 68t68 28t68 -28t28 -68q0 -20 -8 -39q124 -18 219 -82.5t148 -157.5 + l418 363q10 8 23.5 7t21.5 -11z" /> + <glyph glyph-name="_472" unicode="" horiz-adv-x="2048" + d="M1040 -160q0 16 -16 16q-59 0 -101.5 42.5t-42.5 101.5q0 16 -16 16t-16 -16q0 -73 51.5 -124.5t124.5 -51.5q16 0 16 16zM503 315l877 760q-42 88 -132.5 146.5t-223.5 58.5q-93 0 -169.5 -31.5t-121.5 -80.5t-69 -103t-24 -105q0 -384 -137 -645zM1856 128 + q0 -52 -38 -90t-90 -38h-448q0 -106 -75 -181t-181 -75t-180.5 74.5t-75.5 180.5l149 129h757q-166 187 -227 459l111 97q61 -356 298 -556zM1942 1520l84 -96q8 -10 7.5 -23.5t-10.5 -22.5l-1872 -1622q-10 -8 -23.5 -7t-21.5 11l-84 96q-8 10 -7.5 23.5t10.5 21.5l186 161 + q-19 32 -19 66q50 42 91 88t85 119.5t74.5 158.5t50 206t19.5 260q0 152 117 282.5t307 158.5q-8 19 -8 39q0 40 28 68t68 28t68 -28t28 -68q0 -20 -8 -39q124 -18 219 -82.5t148 -157.5l418 363q10 8 23.5 7t21.5 -11z" /> + <glyph glyph-name="_473" unicode="" horiz-adv-x="1408" + d="M512 160v704q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-704q0 -14 9 -23t23 -9h64q14 0 23 9t9 23zM768 160v704q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-704q0 -14 9 -23t23 -9h64q14 0 23 9t9 23zM1024 160v704q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-704 + q0 -14 9 -23t23 -9h64q14 0 23 9t9 23zM480 1152h448l-48 117q-7 9 -17 11h-317q-10 -2 -17 -11zM1408 1120v-64q0 -14 -9 -23t-23 -9h-96v-948q0 -83 -47 -143.5t-113 -60.5h-832q-66 0 -113 58.5t-47 141.5v952h-96q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h309l70 167 + q15 37 54 63t79 26h320q40 0 79 -26t54 -63l70 -167h309q14 0 23 -9t9 -23z" /> + <glyph glyph-name="_474" unicode="" + d="M1150 462v-109q0 -50 -36.5 -89t-94 -60.5t-118 -32.5t-117.5 -11q-205 0 -342.5 139t-137.5 346q0 203 136 339t339 136q34 0 75.5 -4.5t93 -18t92.5 -34t69 -56.5t28 -81v-109q0 -16 -16 -16h-118q-16 0 -16 16v70q0 43 -65.5 67.5t-137.5 24.5q-140 0 -228.5 -91.5 + t-88.5 -237.5q0 -151 91.5 -249.5t233.5 -98.5q68 0 138 24t70 66v70q0 7 4.5 11.5t10.5 4.5h119q6 0 11 -4.5t5 -11.5zM768 1280q-130 0 -248.5 -51t-204 -136.5t-136.5 -204t-51 -248.5t51 -248.5t136.5 -204t204 -136.5t248.5 -51t248.5 51t204 136.5t136.5 204t51 248.5 + t-51 248.5t-136.5 204t-204 136.5t-248.5 51zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="_475" unicode="" + d="M972 761q0 108 -53.5 169t-147.5 61q-63 0 -124 -30.5t-110 -84.5t-79.5 -137t-30.5 -180q0 -112 53.5 -173t150.5 -61q96 0 176 66.5t122.5 166t42.5 203.5zM1536 640q0 -111 -37 -197t-98.5 -135t-131.5 -74.5t-145 -27.5q-6 0 -15.5 -0.5t-16.5 -0.5q-95 0 -142 53 + q-28 33 -33 83q-52 -66 -131.5 -110t-173.5 -44q-161 0 -249.5 95.5t-88.5 269.5q0 157 66 290t179 210.5t246 77.5q87 0 155 -35.5t106 -99.5l2 19l11 56q1 6 5.5 12t9.5 6h118q5 0 13 -11q5 -5 3 -16l-120 -614q-5 -24 -5 -48q0 -39 12.5 -52t44.5 -13q28 1 57 5.5t73 24 + t77 50t57 89.5t24 137q0 292 -174 466t-466 174q-130 0 -248.5 -51t-204 -136.5t-136.5 -204t-51 -248.5t51 -248.5t136.5 -204t204 -136.5t248.5 -51q228 0 405 144q11 9 24 8t21 -12l41 -49q8 -12 7 -24q-2 -13 -12 -22q-102 -83 -227.5 -128t-258.5 -45q-156 0 -298 61 + t-245 164t-164 245t-61 298t61 298t164 245t245 164t298 61q344 0 556 -212t212 -556z" /> + <glyph glyph-name="_476" unicode="" horiz-adv-x="1792" + d="M1698 1442q94 -94 94 -226.5t-94 -225.5l-225 -223l104 -104q10 -10 10 -23t-10 -23l-210 -210q-10 -10 -23 -10t-23 10l-105 105l-603 -603q-37 -37 -90 -37h-203l-256 -128l-64 64l128 256v203q0 53 37 90l603 603l-105 105q-10 10 -10 23t10 23l210 210q10 10 23 10 + t23 -10l104 -104l223 225q93 94 225.5 94t226.5 -94zM512 64l576 576l-192 192l-576 -576v-192h192z" /> + <glyph glyph-name="f1fc" unicode="" horiz-adv-x="1792" + d="M1615 1536q70 0 122.5 -46.5t52.5 -116.5q0 -63 -45 -151q-332 -629 -465 -752q-97 -91 -218 -91q-126 0 -216.5 92.5t-90.5 219.5q0 128 92 212l638 579q59 54 130 54zM706 502q39 -76 106.5 -130t150.5 -76l1 -71q4 -213 -129.5 -347t-348.5 -134q-123 0 -218 46.5 + t-152.5 127.5t-86.5 183t-29 220q7 -5 41 -30t62 -44.5t59 -36.5t46 -17q41 0 55 37q25 66 57.5 112.5t69.5 76t88 47.5t103 25.5t125 10.5z" /> + <glyph glyph-name="_478" unicode="" horiz-adv-x="1792" + d="M1792 128v-384h-1792v384q45 0 85 14t59 27.5t47 37.5q30 27 51.5 38t56.5 11q24 0 44 -7t31 -15t33 -27q29 -25 47 -38t58 -27t86 -14q45 0 85 14.5t58 27t48 37.5q21 19 32.5 27t31 15t43.5 7q35 0 56.5 -11t51.5 -38q28 -24 47 -37.5t59 -27.5t85 -14t85 14t59 27.5 + t47 37.5q30 27 51.5 38t56.5 11q34 0 55.5 -11t51.5 -38q28 -24 47 -37.5t59 -27.5t85 -14zM1792 448v-192q-24 0 -44 7t-31 15t-33 27q-29 25 -47 38t-58 27t-85 14q-46 0 -86 -14t-58 -27t-47 -38q-22 -19 -33 -27t-31 -15t-44 -7q-35 0 -56.5 11t-51.5 38q-29 25 -47 38 + t-58 27t-86 14q-45 0 -85 -14.5t-58 -27t-48 -37.5q-21 -19 -32.5 -27t-31 -15t-43.5 -7q-35 0 -56.5 11t-51.5 38q-28 24 -47 37.5t-59 27.5t-85 14q-46 0 -86 -14t-58 -27t-47 -38q-30 -27 -51.5 -38t-56.5 -11v192q0 80 56 136t136 56h64v448h256v-448h256v448h256v-448 + h256v448h256v-448h64q80 0 136 -56t56 -136zM512 1312q0 -77 -36 -118.5t-92 -41.5q-53 0 -90.5 37.5t-37.5 90.5q0 29 9.5 51t23.5 34t31 28t31 31.5t23.5 44.5t9.5 67q38 0 83 -74t45 -150zM1024 1312q0 -77 -36 -118.5t-92 -41.5q-53 0 -90.5 37.5t-37.5 90.5 + q0 29 9.5 51t23.5 34t31 28t31 31.5t23.5 44.5t9.5 67q38 0 83 -74t45 -150zM1536 1312q0 -77 -36 -118.5t-92 -41.5q-53 0 -90.5 37.5t-37.5 90.5q0 29 9.5 51t23.5 34t31 28t31 31.5t23.5 44.5t9.5 67q38 0 83 -74t45 -150z" /> + <glyph glyph-name="_479" unicode="" horiz-adv-x="2048" + d="M2048 0v-128h-2048v1536h128v-1408h1920zM1664 1024l256 -896h-1664v576l448 576l576 -576z" /> + <glyph glyph-name="_480" unicode="" horiz-adv-x="1792" + d="M768 646l546 -546q-106 -108 -247.5 -168t-298.5 -60q-209 0 -385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103v-762zM955 640h773q0 -157 -60 -298.5t-168 -247.5zM1664 768h-768v768q209 0 385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="_481" unicode="" horiz-adv-x="2048" + d="M2048 0v-128h-2048v1536h128v-1408h1920zM1920 1248v-435q0 -21 -19.5 -29.5t-35.5 7.5l-121 121l-633 -633q-10 -10 -23 -10t-23 10l-233 233l-416 -416l-192 192l585 585q10 10 23 10t23 -10l233 -233l464 464l-121 121q-16 16 -7.5 35.5t29.5 19.5h435q14 0 23 -9 + t9 -23z" /> + <glyph glyph-name="_482" unicode="" horiz-adv-x="1792" + d="M1292 832q0 -6 10 -41q10 -29 25 -49.5t41 -34t44 -20t55 -16.5q325 -91 325 -332q0 -146 -105.5 -242.5t-254.5 -96.5q-59 0 -111.5 18.5t-91.5 45.5t-77 74.5t-63 87.5t-53.5 103.5t-43.5 103t-39.5 106.5t-35.5 95q-32 81 -61.5 133.5t-73.5 96.5t-104 64t-142 20 + q-96 0 -183 -55.5t-138 -144.5t-51 -185q0 -160 106.5 -279.5t263.5 -119.5q177 0 258 95q56 63 83 116l84 -152q-15 -34 -44 -70l1 -1q-131 -152 -388 -152q-147 0 -269.5 79t-190.5 207.5t-68 274.5q0 105 43.5 206t116 176.5t172 121.5t204.5 46q87 0 159 -19t123.5 -50 + t95 -80t72.5 -99t58.5 -117t50.5 -124.5t50 -130.5t55 -127q96 -200 233 -200q81 0 138.5 48.5t57.5 128.5q0 42 -19 72t-50.5 46t-72.5 31.5t-84.5 27t-87.5 34t-81 52t-65 82t-39 122.5q-3 16 -3 33q0 110 87.5 192t198.5 78q78 -3 120.5 -14.5t90.5 -53.5h-1 + q12 -11 23 -24.5t26 -36t19 -27.5l-129 -99q-26 49 -54 70v1q-23 21 -97 21q-49 0 -84 -33t-35 -83z" /> + <glyph glyph-name="_483" unicode="" + d="M1432 484q0 173 -234 239q-35 10 -53 16.5t-38 25t-29 46.5q0 2 -2 8.5t-3 12t-1 7.5q0 36 24.5 59.5t60.5 23.5q54 0 71 -15h-1q20 -15 39 -51l93 71q-39 54 -49 64q-33 29 -67.5 39t-85.5 10q-80 0 -142 -57.5t-62 -137.5q0 -7 2 -23q16 -96 64.5 -140t148.5 -73 + q29 -8 49 -15.5t45 -21.5t38.5 -34.5t13.5 -46.5v-5q1 -58 -40.5 -93t-100.5 -35q-97 0 -167 144q-23 47 -51.5 121.5t-48 125.5t-54 110.5t-74 95.5t-103.5 60.5t-147 24.5q-101 0 -192 -56t-144 -148t-50 -192v-1q4 -108 50.5 -199t133.5 -147.5t196 -56.5q186 0 279 110 + q20 27 31 51l-60 109q-42 -80 -99 -116t-146 -36q-115 0 -191 87t-76 204q0 105 82 189t186 84q112 0 170 -53.5t104 -172.5q8 -21 25.5 -68.5t28.5 -76.5t31.5 -74.5t38.5 -74t45.5 -62.5t55.5 -53.5t66 -33t80 -13.5q107 0 183 69.5t76 174.5zM1536 1120v-960 + q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="_484" unicode="" horiz-adv-x="2048" + d="M1152 640q0 104 -40.5 198.5t-109.5 163.5t-163.5 109.5t-198.5 40.5t-198.5 -40.5t-163.5 -109.5t-109.5 -163.5t-40.5 -198.5t40.5 -198.5t109.5 -163.5t163.5 -109.5t198.5 -40.5t198.5 40.5t163.5 109.5t109.5 163.5t40.5 198.5zM1920 640q0 104 -40.5 198.5 + t-109.5 163.5t-163.5 109.5t-198.5 40.5h-386q119 -90 188.5 -224t69.5 -288t-69.5 -288t-188.5 -224h386q104 0 198.5 40.5t163.5 109.5t109.5 163.5t40.5 198.5zM2048 640q0 -130 -51 -248.5t-136.5 -204t-204 -136.5t-248.5 -51h-768q-130 0 -248.5 51t-204 136.5 + t-136.5 204t-51 248.5t51 248.5t136.5 204t204 136.5t248.5 51h768q130 0 248.5 -51t204 -136.5t136.5 -204t51 -248.5z" /> + <glyph glyph-name="_485" unicode="" horiz-adv-x="2048" + d="M0 640q0 130 51 248.5t136.5 204t204 136.5t248.5 51h768q130 0 248.5 -51t204 -136.5t136.5 -204t51 -248.5t-51 -248.5t-136.5 -204t-204 -136.5t-248.5 -51h-768q-130 0 -248.5 51t-204 136.5t-136.5 204t-51 248.5zM1408 128q104 0 198.5 40.5t163.5 109.5 + t109.5 163.5t40.5 198.5t-40.5 198.5t-109.5 163.5t-163.5 109.5t-198.5 40.5t-198.5 -40.5t-163.5 -109.5t-109.5 -163.5t-40.5 -198.5t40.5 -198.5t109.5 -163.5t163.5 -109.5t198.5 -40.5z" /> + <glyph glyph-name="_486" unicode="" horiz-adv-x="2304" + d="M762 384h-314q-40 0 -57.5 35t6.5 67l188 251q-65 31 -137 31q-132 0 -226 -94t-94 -226t94 -226t226 -94q115 0 203 72.5t111 183.5zM576 512h186q-18 85 -75 148zM1056 512l288 384h-480l-99 -132q105 -103 126 -252h165zM2176 448q0 132 -94 226t-226 94 + q-60 0 -121 -24l174 -260q15 -23 10 -49t-27 -40q-15 -11 -36 -11q-35 0 -53 29l-174 260q-93 -95 -93 -225q0 -132 94 -226t226 -94t226 94t94 226zM2304 448q0 -185 -131.5 -316.5t-316.5 -131.5t-316.5 131.5t-131.5 316.5q0 97 39.5 183.5t109.5 149.5l-65 98l-353 -469 + q-18 -26 -51 -26h-197q-23 -164 -149 -274t-294 -110q-185 0 -316.5 131.5t-131.5 316.5t131.5 316.5t316.5 131.5q114 0 215 -55l137 183h-224q-26 0 -45 19t-19 45t19 45t45 19h384v-128h435l-85 128h-222q-26 0 -45 19t-19 45t19 45t45 19h256q33 0 53 -28l267 -400 + q91 44 192 44q185 0 316.5 -131.5t131.5 -316.5z" /> + <glyph glyph-name="_487" unicode="" + d="M384 320q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM1408 320q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM1362 716l-72 384q-5 23 -22.5 37.5t-40.5 14.5 + h-918q-23 0 -40.5 -14.5t-22.5 -37.5l-72 -384q-5 -30 14 -53t49 -23h1062q30 0 49 23t14 53zM1136 1328q0 20 -14 34t-34 14h-640q-20 0 -34 -14t-14 -34t14 -34t34 -14h640q20 0 34 14t14 34zM1536 603v-603h-128v-128q0 -53 -37.5 -90.5t-90.5 -37.5t-90.5 37.5 + t-37.5 90.5v128h-768v-128q0 -53 -37.5 -90.5t-90.5 -37.5t-90.5 37.5t-37.5 90.5v128h-128v603q0 112 25 223l103 454q9 78 97.5 137t230 89t312.5 30t312.5 -30t230 -89t97.5 -137l105 -454q23 -102 23 -223z" /> + <glyph glyph-name="_488" unicode="" horiz-adv-x="2048" + d="M1463 704q0 -35 -25 -60.5t-61 -25.5h-702q-36 0 -61 25.5t-25 60.5t25 60.5t61 25.5h702q36 0 61 -25.5t25 -60.5zM1677 704q0 86 -23 170h-982q-36 0 -61 25t-25 60q0 36 25 61t61 25h908q-88 143 -235 227t-320 84q-177 0 -327.5 -87.5t-238 -237.5t-87.5 -327 + q0 -86 23 -170h982q36 0 61 -25t25 -60q0 -36 -25 -61t-61 -25h-908q88 -143 235.5 -227t320.5 -84q132 0 253 51.5t208 139t139 208t52 253.5zM2048 959q0 -35 -25 -60t-61 -25h-131q17 -85 17 -170q0 -167 -65.5 -319.5t-175.5 -263t-262.5 -176t-319.5 -65.5 + q-246 0 -448.5 133t-301.5 350h-189q-36 0 -61 25t-25 61q0 35 25 60t61 25h132q-17 85 -17 170q0 167 65.5 319.5t175.5 263t262.5 176t320.5 65.5q245 0 447.5 -133t301.5 -350h188q36 0 61 -25t25 -61z" /> + <glyph glyph-name="_489" unicode="" horiz-adv-x="1280" + d="M953 1158l-114 -328l117 -21q165 451 165 518q0 56 -38 56q-57 0 -130 -225zM654 471l33 -88q37 42 71 67l-33 5.5t-38.5 7t-32.5 8.5zM362 1367q0 -98 159 -521q17 10 49 10q15 0 75 -5l-121 351q-75 220 -123 220q-19 0 -29 -17.5t-10 -37.5zM283 608q0 -36 51.5 -119 + t117.5 -153t100 -70q14 0 25.5 13t11.5 27q0 24 -32 102q-13 32 -32 72t-47.5 89t-61.5 81t-62 32q-20 0 -45.5 -27t-25.5 -47zM125 273q0 -41 25 -104q59 -145 183.5 -227t281.5 -82q227 0 382 170q152 169 152 427q0 43 -1 67t-11.5 62t-30.5 56q-56 49 -211.5 75.5 + t-270.5 26.5q-37 0 -49 -11q-12 -5 -12 -35q0 -34 21.5 -60t55.5 -40t77.5 -23.5t87.5 -11.5t85 -4t70 0h23q24 0 40 -19q15 -19 19 -55q-28 -28 -96 -54q-61 -22 -93 -46q-64 -46 -108.5 -114t-44.5 -137q0 -31 18.5 -88.5t18.5 -87.5l-3 -12q-4 -12 -4 -14 + q-137 10 -146 216q-8 -2 -41 -2q2 -7 2 -21q0 -53 -40.5 -89.5t-94.5 -36.5q-82 0 -166.5 78t-84.5 159q0 34 33 67q52 -64 60 -76q77 -104 133 -104q12 0 26.5 8.5t14.5 20.5q0 34 -87.5 145t-116.5 111q-43 0 -70 -44.5t-27 -90.5zM11 264q0 101 42.5 163t136.5 88 + q-28 74 -28 104q0 62 61 123t122 61q29 0 70 -15q-163 462 -163 567q0 80 41 130.5t119 50.5q131 0 325 -581q6 -17 8 -23q6 16 29 79.5t43.5 118.5t54 127.5t64.5 123t70.5 86.5t76.5 36q71 0 112 -49t41 -122q0 -108 -159 -550q61 -15 100.5 -46t58.5 -78t26 -93.5 + t7 -110.5q0 -150 -47 -280t-132 -225t-211 -150t-278 -55q-111 0 -223 42q-149 57 -258 191.5t-109 286.5z" /> + <glyph glyph-name="_490" unicode="" horiz-adv-x="2048" + d="M785 528h207q-14 -158 -98.5 -248.5t-214.5 -90.5q-162 0 -254.5 116t-92.5 316q0 194 93 311.5t233 117.5q148 0 232 -87t97 -247h-203q-5 64 -35.5 99t-81.5 35q-57 0 -88.5 -60.5t-31.5 -177.5q0 -48 5 -84t18 -69.5t40 -51.5t66 -18q95 0 109 139zM1497 528h206 + q-14 -158 -98 -248.5t-214 -90.5q-162 0 -254.5 116t-92.5 316q0 194 93 311.5t233 117.5q148 0 232 -87t97 -247h-204q-4 64 -35 99t-81 35q-57 0 -88.5 -60.5t-31.5 -177.5q0 -48 5 -84t18 -69.5t39.5 -51.5t65.5 -18q49 0 76.5 38t33.5 101zM1856 647q0 207 -15.5 307 + t-60.5 161q-6 8 -13.5 14t-21.5 15t-16 11q-86 63 -697 63q-625 0 -710 -63q-5 -4 -17.5 -11.5t-21 -14t-14.5 -14.5q-45 -60 -60 -159.5t-15 -308.5q0 -208 15 -307.5t60 -160.5q6 -8 15 -15t20.5 -14t17.5 -12q44 -33 239.5 -49t470.5 -16q610 0 697 65q5 4 17 11t20.5 14 + t13.5 16q46 60 61 159t15 309zM2048 1408v-1536h-2048v1536h2048z" /> + <glyph glyph-name="_491" unicode="" + d="M992 912v-496q0 -14 -9 -23t-23 -9h-160q-14 0 -23 9t-9 23v496q0 112 -80 192t-192 80h-272v-1152q0 -14 -9 -23t-23 -9h-160q-14 0 -23 9t-9 23v1344q0 14 9 23t23 9h464q135 0 249 -66.5t180.5 -180.5t66.5 -249zM1376 1376v-880q0 -135 -66.5 -249t-180.5 -180.5 + t-249 -66.5h-464q-14 0 -23 9t-9 23v960q0 14 9 23t23 9h160q14 0 23 -9t9 -23v-768h272q112 0 192 80t80 192v880q0 14 9 23t23 9h160q14 0 23 -9t9 -23z" /> + <glyph glyph-name="_492" unicode="" + d="M1311 694v-114q0 -24 -13.5 -38t-37.5 -14h-202q-24 0 -38 14t-14 38v114q0 24 14 38t38 14h202q24 0 37.5 -14t13.5 -38zM821 464v250q0 53 -32.5 85.5t-85.5 32.5h-133q-68 0 -96 -52q-28 52 -96 52h-130q-53 0 -85.5 -32.5t-32.5 -85.5v-250q0 -22 21 -22h55 + q22 0 22 22v230q0 24 13.5 38t38.5 14h94q24 0 38 -14t14 -38v-230q0 -22 21 -22h54q22 0 22 22v230q0 24 14 38t38 14h97q24 0 37.5 -14t13.5 -38v-230q0 -22 22 -22h55q21 0 21 22zM1410 560v154q0 53 -33 85.5t-86 32.5h-264q-53 0 -86 -32.5t-33 -85.5v-410 + q0 -21 22 -21h55q21 0 21 21v180q31 -42 94 -42h191q53 0 86 32.5t33 85.5zM1536 1176v-1072q0 -96 -68 -164t-164 -68h-1072q-96 0 -164 68t-68 164v1072q0 96 68 164t164 68h1072q96 0 164 -68t68 -164z" /> + <glyph glyph-name="_493" unicode="" + d="M915 450h-294l147 551zM1001 128h311l-324 1024h-440l-324 -1024h311l383 314zM1536 1120v-960q0 -118 -85 -203t-203 -85h-960q-118 0 -203 85t-85 203v960q0 118 85 203t203 85h960q118 0 203 -85t85 -203z" /> + <glyph glyph-name="_494" unicode="" horiz-adv-x="2048" + d="M2048 641q0 -21 -13 -36.5t-33 -19.5l-205 -356q3 -9 3 -18q0 -20 -12.5 -35.5t-32.5 -19.5l-193 -337q3 -8 3 -16q0 -23 -16.5 -40t-40.5 -17q-25 0 -41 18h-400q-17 -20 -43 -20t-43 20h-399q-17 -20 -43 -20q-23 0 -40 16.5t-17 40.5q0 8 4 20l-193 335 + q-20 4 -32.5 19.5t-12.5 35.5q0 9 3 18l-206 356q-20 5 -32.5 20.5t-12.5 35.5q0 21 13.5 36.5t33.5 19.5l199 344q0 1 -0.5 3t-0.5 3q0 36 34 51l209 363q-4 10 -4 18q0 24 17 40.5t40 16.5q26 0 44 -21h396q16 21 43 21t43 -21h398q18 21 44 21q23 0 40 -16.5t17 -40.5 + q0 -6 -4 -18l207 -358q23 -1 39 -17.5t16 -38.5q0 -13 -7 -27l187 -324q19 -4 31.5 -19.5t12.5 -35.5zM1063 -158h389l-342 354h-143l-342 -354h360q18 16 39 16t39 -16zM112 654q1 -4 1 -13q0 -10 -2 -15l208 -360l15 -6l188 199v347l-187 194q-13 -8 -29 -10zM986 1438 + h-388l190 -200l554 200h-280q-16 -16 -38 -16t-38 16zM1689 226q1 6 5 11l-64 68l-17 -79h76zM1583 226l22 105l-252 266l-296 -307l63 -64h463zM1495 -142l16 28l65 310h-427l333 -343q8 4 13 5zM578 -158h5l342 354h-373v-335l4 -6q14 -5 22 -13zM552 226h402l64 66 + l-309 321l-157 -166v-221zM359 226h163v189l-168 -177q4 -8 5 -12zM358 1051q0 -1 0.5 -2t0.5 -2q0 -16 -8 -29l171 -177v269zM552 1121v-311l153 -157l297 314l-223 236zM556 1425l-4 -8v-264l205 74l-191 201q-6 -2 -10 -3zM1447 1438h-16l-621 -224l213 -225zM1023 946 + l-297 -315l311 -319l296 307zM688 634l-136 141v-284zM1038 270l-42 -44h85zM1374 618l238 -251l132 624l-3 5l-1 1zM1718 1018q-8 13 -8 29v2l-216 376q-5 1 -13 5l-437 -463l310 -327zM522 1142v223l-163 -282zM522 196h-163l163 -283v283zM1607 196l-48 -227l130 227h-82 + zM1729 266l207 361q-2 10 -2 14q0 1 3 16l-171 296l-129 -612l77 -82q5 3 15 7z" /> + <glyph glyph-name="f210" unicode="" + d="M0 856q0 131 91.5 226.5t222.5 95.5h742l352 358v-1470q0 -132 -91.5 -227t-222.5 -95h-780q-131 0 -222.5 95t-91.5 227v790zM1232 102l-176 180v425q0 46 -32 79t-78 33h-484q-46 0 -78 -33t-32 -79v-492q0 -46 32.5 -79.5t77.5 -33.5h770z" /> + <glyph glyph-name="_496" unicode="" + d="M934 1386q-317 -121 -556 -362.5t-358 -560.5q-20 89 -20 176q0 208 102.5 384.5t278.5 279t384 102.5q82 0 169 -19zM1203 1267q93 -65 164 -155q-389 -113 -674.5 -400.5t-396.5 -676.5q-93 72 -155 162q112 386 395 671t667 399zM470 -67q115 356 379.5 622t619.5 384 + q40 -92 54 -195q-292 -120 -516 -345t-343 -518q-103 14 -194 52zM1536 -125q-193 50 -367 115q-135 -84 -290 -107q109 205 274 370.5t369 275.5q-21 -152 -101 -284q65 -175 115 -370z" /> + <glyph glyph-name="f212" unicode="" horiz-adv-x="2048" + d="M1893 1144l155 -1272q-131 0 -257 57q-200 91 -393 91q-226 0 -374 -148q-148 148 -374 148q-193 0 -393 -91q-128 -57 -252 -57h-5l155 1272q224 127 482 127q233 0 387 -106q154 106 387 106q258 0 482 -127zM1398 157q129 0 232 -28.5t260 -93.5l-124 1021 + q-171 78 -368 78q-224 0 -374 -141q-150 141 -374 141q-197 0 -368 -78l-124 -1021q105 43 165.5 65t148.5 39.5t178 17.5q202 0 374 -108q172 108 374 108zM1438 191l-55 907q-211 -4 -359 -155q-152 155 -374 155q-176 0 -336 -66l-114 -941q124 51 228.5 76t221.5 25 + q209 0 374 -102q172 107 374 102z" /> + <glyph glyph-name="_498" unicode="" horiz-adv-x="2048" + d="M1500 165v733q0 21 -15 36t-35 15h-93q-20 0 -35 -15t-15 -36v-733q0 -20 15 -35t35 -15h93q20 0 35 15t15 35zM1216 165v531q0 20 -15 35t-35 15h-101q-20 0 -35 -15t-15 -35v-531q0 -20 15 -35t35 -15h101q20 0 35 15t15 35zM924 165v429q0 20 -15 35t-35 15h-101 + q-20 0 -35 -15t-15 -35v-429q0 -20 15 -35t35 -15h101q20 0 35 15t15 35zM632 165v362q0 20 -15 35t-35 15h-101q-20 0 -35 -15t-15 -35v-362q0 -20 15 -35t35 -15h101q20 0 35 15t15 35zM2048 311q0 -166 -118 -284t-284 -118h-1244q-166 0 -284 118t-118 284 + q0 116 63 214.5t168 148.5q-10 34 -10 73q0 113 80.5 193.5t193.5 80.5q102 0 180 -67q45 183 194 300t338 117q149 0 275 -73.5t199.5 -199.5t73.5 -275q0 -66 -14 -122q135 -33 221 -142.5t86 -247.5z" /> + <glyph glyph-name="_499" unicode="" + d="M0 1536h1536v-1392l-776 -338l-760 338v1392zM1436 209v926h-1336v-926l661 -294zM1436 1235v201h-1336v-201h1336zM181 937v-115h-37v115h37zM181 789v-115h-37v115h37zM181 641v-115h-37v115h37zM181 493v-115h-37v115h37zM181 345v-115h-37v115h37zM207 202l15 34 + l105 -47l-15 -33zM343 142l15 34l105 -46l-15 -34zM478 82l15 34l105 -46l-15 -34zM614 23l15 33l104 -46l-15 -34zM797 10l105 46l15 -33l-105 -47zM932 70l105 46l15 -34l-105 -46zM1068 130l105 46l15 -34l-105 -46zM1203 189l105 47l15 -34l-105 -46zM259 1389v-36h-114 + v36h114zM421 1389v-36h-115v36h115zM583 1389v-36h-115v36h115zM744 1389v-36h-114v36h114zM906 1389v-36h-114v36h114zM1068 1389v-36h-115v36h115zM1230 1389v-36h-115v36h115zM1391 1389v-36h-114v36h114zM181 1049v-79h-37v115h115v-36h-78zM421 1085v-36h-115v36h115z + M583 1085v-36h-115v36h115zM744 1085v-36h-114v36h114zM906 1085v-36h-114v36h114zM1068 1085v-36h-115v36h115zM1230 1085v-36h-115v36h115zM1355 970v79h-78v36h115v-115h-37zM1355 822v115h37v-115h-37zM1355 674v115h37v-115h-37zM1355 526v115h37v-115h-37zM1355 378 + v115h37v-115h-37zM1355 230v115h37v-115h-37zM760 265q-129 0 -221 91.5t-92 221.5q0 129 92 221t221 92q130 0 221.5 -92t91.5 -221q0 -130 -91.5 -221.5t-221.5 -91.5zM595 646q0 -36 19.5 -56.5t49.5 -25t64 -7t64 -2t49.5 -9t19.5 -30.5q0 -49 -112 -49q-97 0 -123 51 + h-3l-31 -63q67 -42 162 -42q29 0 56.5 5t55.5 16t45.5 33t17.5 53q0 46 -27.5 69.5t-67.5 27t-79.5 3t-67 5t-27.5 25.5q0 21 20.5 33t40.5 15t41 3q34 0 70.5 -11t51.5 -34h3l30 58q-3 1 -21 8.5t-22.5 9t-19.5 7t-22 7t-20 4.5t-24 4t-23 1q-29 0 -56.5 -5t-54 -16.5 + t-43 -34t-16.5 -53.5z" /> + <glyph glyph-name="_500" unicode="" horiz-adv-x="2048" + d="M863 504q0 112 -79.5 191.5t-191.5 79.5t-191 -79.5t-79 -191.5t79 -191t191 -79t191.5 79t79.5 191zM1726 505q0 112 -79 191t-191 79t-191.5 -79t-79.5 -191q0 -113 79.5 -192t191.5 -79t191 79.5t79 191.5zM2048 1314v-1348q0 -44 -31.5 -75.5t-76.5 -31.5h-1832 + q-45 0 -76.5 31.5t-31.5 75.5v1348q0 44 31.5 75.5t76.5 31.5h431q44 0 76 -31.5t32 -75.5v-161h754v161q0 44 32 75.5t76 31.5h431q45 0 76.5 -31.5t31.5 -75.5z" /> + <glyph glyph-name="_501" unicode="" horiz-adv-x="2048" + d="M1430 953zM1690 749q148 0 253 -98.5t105 -244.5q0 -157 -109 -261.5t-267 -104.5q-85 0 -162 27.5t-138 73.5t-118 106t-109 126t-103.5 132.5t-108.5 126.5t-117 106t-136 73.5t-159 27.5q-154 0 -251.5 -91.5t-97.5 -244.5q0 -157 104 -250t263 -93q100 0 208 37.5 + t193 98.5q5 4 21 18.5t30 24t22 9.5q14 0 24.5 -10.5t10.5 -24.5q0 -24 -60 -77q-101 -88 -234.5 -142t-260.5 -54q-133 0 -245.5 58t-180 165t-67.5 241q0 205 141.5 341t347.5 136q120 0 226.5 -43.5t185.5 -113t151.5 -153t139 -167.5t133.5 -153.5t149.5 -113 + t172.5 -43.5q102 0 168.5 61.5t66.5 162.5q0 95 -64.5 159t-159.5 64q-30 0 -81.5 -18.5t-68.5 -18.5q-20 0 -35.5 15t-15.5 35q0 18 8.5 57t8.5 59q0 159 -107.5 263t-266.5 104q-58 0 -111.5 -18.5t-84 -40.5t-55.5 -40.5t-33 -18.5q-15 0 -25.5 10.5t-10.5 25.5 + q0 19 25 46q59 67 147 103.5t182 36.5q191 0 318 -125.5t127 -315.5q0 -37 -4 -66q57 15 115 15z" /> + <glyph glyph-name="_502" unicode="" horiz-adv-x="1664" + d="M1216 832q0 26 -19 45t-45 19h-128v128q0 26 -19 45t-45 19t-45 -19t-19 -45v-128h-128q-26 0 -45 -19t-19 -45t19 -45t45 -19h128v-128q0 -26 19 -45t45 -19t45 19t19 45v128h128q26 0 45 19t19 45zM640 0q0 -53 -37.5 -90.5t-90.5 -37.5t-90.5 37.5t-37.5 90.5 + t37.5 90.5t90.5 37.5t90.5 -37.5t37.5 -90.5zM1536 0q0 -53 -37.5 -90.5t-90.5 -37.5t-90.5 37.5t-37.5 90.5t37.5 90.5t90.5 37.5t90.5 -37.5t37.5 -90.5zM1664 1088v-512q0 -24 -16 -42.5t-41 -21.5l-1044 -122q1 -7 4.5 -21.5t6 -26.5t2.5 -22q0 -16 -24 -64h920 + q26 0 45 -19t19 -45t-19 -45t-45 -19h-1024q-26 0 -45 19t-19 45q0 14 11 39.5t29.5 59.5t20.5 38l-177 823h-204q-26 0 -45 19t-19 45t19 45t45 19h256q16 0 28.5 -6.5t20 -15.5t13 -24.5t7.5 -26.5t5.5 -29.5t4.5 -25.5h1201q26 0 45 -19t19 -45z" /> + <glyph glyph-name="_503" unicode="" horiz-adv-x="1664" + d="M1280 832q0 26 -19 45t-45 19t-45 -19l-147 -146v293q0 26 -19 45t-45 19t-45 -19t-19 -45v-293l-147 146q-19 19 -45 19t-45 -19t-19 -45t19 -45l256 -256q19 -19 45 -19t45 19l256 256q19 19 19 45zM640 0q0 -53 -37.5 -90.5t-90.5 -37.5t-90.5 37.5t-37.5 90.5 + t37.5 90.5t90.5 37.5t90.5 -37.5t37.5 -90.5zM1536 0q0 -53 -37.5 -90.5t-90.5 -37.5t-90.5 37.5t-37.5 90.5t37.5 90.5t90.5 37.5t90.5 -37.5t37.5 -90.5zM1664 1088v-512q0 -24 -16 -42.5t-41 -21.5l-1044 -122q1 -7 4.5 -21.5t6 -26.5t2.5 -22q0 -16 -24 -64h920 + q26 0 45 -19t19 -45t-19 -45t-45 -19h-1024q-26 0 -45 19t-19 45q0 14 11 39.5t29.5 59.5t20.5 38l-177 823h-204q-26 0 -45 19t-19 45t19 45t45 19h256q16 0 28.5 -6.5t20 -15.5t13 -24.5t7.5 -26.5t5.5 -29.5t4.5 -25.5h1201q26 0 45 -19t19 -45z" /> + <glyph glyph-name="_504" unicode="" horiz-adv-x="2048" + d="M212 768l623 -665l-300 665h-323zM1024 -4l349 772h-698zM538 896l204 384h-262l-288 -384h346zM1213 103l623 665h-323zM683 896h682l-204 384h-274zM1510 896h346l-288 384h-262zM1651 1382l384 -512q14 -18 13 -41.5t-17 -40.5l-960 -1024q-18 -20 -47 -20t-47 20 + l-960 1024q-16 17 -17 40.5t13 41.5l384 512q18 26 51 26h1152q33 0 51 -26z" /> + <glyph glyph-name="_505" unicode="" horiz-adv-x="2048" + d="M1811 -19q19 19 45 19t45 -19l128 -128l-90 -90l-83 83l-83 -83q-18 -19 -45 -19t-45 19l-83 83l-83 -83q-19 -19 -45 -19t-45 19l-83 83l-83 -83q-19 -19 -45 -19t-45 19l-83 83l-83 -83q-19 -19 -45 -19t-45 19l-83 83l-83 -83q-19 -19 -45 -19t-45 19l-83 83l-83 -83 + q-19 -19 -45 -19t-45 19l-83 83l-83 -83q-19 -19 -45 -19t-45 19l-128 128l90 90l83 -83l83 83q19 19 45 19t45 -19l83 -83l83 83q19 19 45 19t45 -19l83 -83l83 83q19 19 45 19t45 -19l83 -83l83 83q19 19 45 19t45 -19l83 -83l83 83q19 19 45 19t45 -19l83 -83l83 83 + q19 19 45 19t45 -19l83 -83zM237 19q-19 -19 -45 -19t-45 19l-128 128l90 90l83 -82l83 82q19 19 45 19t45 -19l83 -82l64 64v293l-210 314q-17 26 -7 56.5t40 40.5l177 58v299h128v128h256v128h256v-128h256v-128h128v-299l177 -58q30 -10 40 -40.5t-7 -56.5l-210 -314 + v-293l19 18q19 19 45 19t45 -19l83 -82l83 82q19 19 45 19t45 -19l128 -128l-90 -90l-83 83l-83 -83q-18 -19 -45 -19t-45 19l-83 83l-83 -83q-19 -19 -45 -19t-45 19l-83 83l-83 -83q-19 -19 -45 -19t-45 19l-83 83l-83 -83q-19 -19 -45 -19t-45 19l-83 83l-83 -83 + q-19 -19 -45 -19t-45 19l-83 83l-83 -83q-19 -19 -45 -19t-45 19l-83 83zM640 1152v-128l384 128l384 -128v128h-128v128h-512v-128h-128z" /> + <glyph glyph-name="_506" unicode="" + d="M576 0l96 448l-96 128l-128 64zM832 0l128 640l-128 -64l-96 -128zM992 1010q-2 4 -4 6q-10 8 -96 8q-70 0 -167 -19q-7 -2 -21 -2t-21 2q-97 19 -167 19q-86 0 -96 -8q-2 -2 -4 -6q2 -18 4 -27q2 -3 7.5 -6.5t7.5 -10.5q2 -4 7.5 -20.5t7 -20.5t7.5 -17t8.5 -17t9 -14 + t12 -13.5t14 -9.5t17.5 -8t20.5 -4t24.5 -2q36 0 59 12.5t32.5 30t14.5 34.5t11.5 29.5t17.5 12.5h12q11 0 17.5 -12.5t11.5 -29.5t14.5 -34.5t32.5 -30t59 -12.5q13 0 24.5 2t20.5 4t17.5 8t14 9.5t12 13.5t9 14t8.5 17t7.5 17t7 20.5t7.5 20.5q2 7 7.5 10.5t7.5 6.5 + q2 9 4 27zM1408 131q0 -121 -73 -190t-194 -69h-874q-121 0 -194 69t-73 190q0 61 4.5 118t19 125.5t37.5 123.5t63.5 103.5t93.5 74.5l-90 220h214q-22 64 -22 128q0 12 2 32q-194 40 -194 96q0 57 210 99q17 62 51.5 134t70.5 114q32 37 76 37q30 0 84 -31t84 -31t84 31 + t84 31q44 0 76 -37q36 -42 70.5 -114t51.5 -134q210 -42 210 -99q0 -56 -194 -96q7 -81 -20 -160h214l-82 -225q63 -33 107.5 -96.5t65.5 -143.5t29 -151.5t8 -148.5z" /> + <glyph glyph-name="_507" unicode="" horiz-adv-x="2304" + d="M2301 500q12 -103 -22 -198.5t-99 -163.5t-158.5 -106t-196.5 -31q-161 11 -279.5 125t-134.5 274q-12 111 27.5 210.5t118.5 170.5l-71 107q-96 -80 -151 -194t-55 -244q0 -27 -18.5 -46.5t-45.5 -19.5h-256h-69q-23 -164 -149 -274t-294 -110q-185 0 -316.5 131.5 + t-131.5 316.5t131.5 316.5t316.5 131.5q76 0 152 -27l24 45q-123 110 -304 110h-64q-26 0 -45 19t-19 45t19 45t45 19h128q78 0 145 -13.5t116.5 -38.5t71.5 -39.5t51 -36.5h512h115l-85 128h-222q-30 0 -49 22.5t-14 52.5q4 23 23 38t43 15h253q33 0 53 -28l70 -105 + l114 114q19 19 46 19h101q26 0 45 -19t19 -45v-128q0 -26 -19 -45t-45 -19h-179l115 -172q131 63 275 36q143 -26 244 -134.5t118 -253.5zM448 128q115 0 203 72.5t111 183.5h-314q-35 0 -55 31q-18 32 -1 63l147 277q-47 13 -91 13q-132 0 -226 -94t-94 -226t94 -226 + t226 -94zM1856 128q132 0 226 94t94 226t-94 226t-226 94q-60 0 -121 -24l174 -260q15 -23 10 -49t-27 -40q-15 -11 -36 -11q-35 0 -53 29l-174 260q-93 -95 -93 -225q0 -132 94 -226t226 -94z" /> + <glyph glyph-name="_508" unicode="" + d="M1408 0q0 -63 -61.5 -113.5t-164 -81t-225 -46t-253.5 -15.5t-253.5 15.5t-225 46t-164 81t-61.5 113.5q0 49 33 88.5t91 66.5t118 44.5t131 29.5q26 5 48 -10.5t26 -41.5q5 -26 -10.5 -48t-41.5 -26q-58 -10 -106 -23.5t-76.5 -25.5t-48.5 -23.5t-27.5 -19.5t-8.5 -12 + q3 -11 27 -26.5t73 -33t114 -32.5t160.5 -25t201.5 -10t201.5 10t160.5 25t114 33t73 33.5t27 27.5q-1 4 -8.5 11t-27.5 19t-48.5 23.5t-76.5 25t-106 23.5q-26 4 -41.5 26t-10.5 48q4 26 26 41.5t48 10.5q71 -12 131 -29.5t118 -44.5t91 -66.5t33 -88.5zM1024 896v-384 + q0 -26 -19 -45t-45 -19h-64v-384q0 -26 -19 -45t-45 -19h-256q-26 0 -45 19t-19 45v384h-64q-26 0 -45 19t-19 45v384q0 53 37.5 90.5t90.5 37.5h384q53 0 90.5 -37.5t37.5 -90.5zM928 1280q0 -93 -65.5 -158.5t-158.5 -65.5t-158.5 65.5t-65.5 158.5t65.5 158.5t158.5 65.5 + t158.5 -65.5t65.5 -158.5z" /> + <glyph glyph-name="_509" unicode="" horiz-adv-x="1792" + d="M1280 512h305q-5 -6 -10 -10.5t-9 -7.5l-3 -4l-623 -600q-18 -18 -44 -18t-44 18l-624 602q-5 2 -21 20h369q22 0 39.5 13.5t22.5 34.5l70 281l190 -667q6 -20 23 -33t39 -13q21 0 38 13t23 33l146 485l56 -112q18 -35 57 -35zM1792 940q0 -145 -103 -300h-369l-111 221 + q-8 17 -25.5 27t-36.5 8q-45 -5 -56 -46l-129 -430l-196 686q-6 20 -23.5 33t-39.5 13t-39 -13.5t-22 -34.5l-116 -464h-423q-103 155 -103 300q0 220 127 344t351 124q62 0 126.5 -21.5t120 -58t95.5 -68.5t76 -68q36 36 76 68t95.5 68.5t120 58t126.5 21.5q224 0 351 -124 + t127 -344z" /> + <glyph glyph-name="venus" unicode="" horiz-adv-x="1280" + d="M1152 960q0 -221 -147.5 -384.5t-364.5 -187.5v-260h224q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-224v-224q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9t-9 23v224h-224q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h224v260q-150 16 -271.5 103t-186 224t-52.5 292 + q11 134 80.5 249t182 188t245.5 88q170 19 319 -54t236 -212t87 -306zM128 960q0 -185 131.5 -316.5t316.5 -131.5t316.5 131.5t131.5 316.5t-131.5 316.5t-316.5 131.5t-316.5 -131.5t-131.5 -316.5z" /> + <glyph glyph-name="_511" unicode="" + d="M1472 1408q26 0 45 -19t19 -45v-416q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9t-9 23v262l-382 -383q126 -156 126 -359q0 -117 -45.5 -223.5t-123 -184t-184 -123t-223.5 -45.5t-223.5 45.5t-184 123t-123 184t-45.5 223.5t45.5 223.5t123 184t184 123t223.5 45.5 + q203 0 359 -126l382 382h-261q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h416zM576 0q185 0 316.5 131.5t131.5 316.5t-131.5 316.5t-316.5 131.5t-316.5 -131.5t-131.5 -316.5t131.5 -316.5t316.5 -131.5z" /> + <glyph glyph-name="_512" unicode="" horiz-adv-x="1280" + d="M830 1220q145 -72 233.5 -210.5t88.5 -305.5q0 -221 -147.5 -384.5t-364.5 -187.5v-132h96q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-96v-96q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9t-9 23v96h-96q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h96v132q-217 24 -364.5 187.5 + t-147.5 384.5q0 167 88.5 305.5t233.5 210.5q-165 96 -228 273q-6 16 3.5 29.5t26.5 13.5h69q21 0 29 -20q44 -106 140 -171t214 -65t214 65t140 171q8 20 37 20h61q17 0 26.5 -13.5t3.5 -29.5q-63 -177 -228 -273zM576 256q185 0 316.5 131.5t131.5 316.5t-131.5 316.5 + t-316.5 131.5t-316.5 -131.5t-131.5 -316.5t131.5 -316.5t316.5 -131.5z" /> + <glyph glyph-name="_513" unicode="" + d="M1024 1504q0 14 9 23t23 9h288q26 0 45 -19t19 -45v-288q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9t-9 23v134l-254 -255q126 -158 126 -359q0 -221 -147.5 -384.5t-364.5 -187.5v-132h96q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-96v-96q0 -14 -9 -23t-23 -9h-64 + q-14 0 -23 9t-9 23v96h-96q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h96v132q-149 16 -270.5 103t-186.5 223.5t-53 291.5q16 204 160 353.5t347 172.5q118 14 228 -19t198 -103l255 254h-134q-14 0 -23 9t-9 23v64zM576 256q185 0 316.5 131.5t131.5 316.5t-131.5 316.5 + t-316.5 131.5t-316.5 -131.5t-131.5 -316.5t131.5 -316.5t316.5 -131.5z" /> + <glyph glyph-name="_514" unicode="" horiz-adv-x="1792" + d="M1280 1504q0 14 9 23t23 9h288q26 0 45 -19t19 -45v-288q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9t-9 23v134l-254 -255q126 -158 126 -359q0 -221 -147.5 -384.5t-364.5 -187.5v-132h96q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-96v-96q0 -14 -9 -23t-23 -9h-64 + q-14 0 -23 9t-9 23v96h-96q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h96v132q-217 24 -364.5 187.5t-147.5 384.5q0 201 126 359l-52 53l-101 -111q-9 -10 -22 -10.5t-23 7.5l-48 44q-10 8 -10.5 21.5t8.5 23.5l105 115l-111 112v-134q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9 + t-9 23v288q0 26 19 45t45 19h288q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-133l106 -107l86 94q9 10 22 10.5t23 -7.5l48 -44q10 -8 10.5 -21.5t-8.5 -23.5l-90 -99l57 -56q158 126 359 126t359 -126l255 254h-134q-14 0 -23 9t-9 23v64zM832 256q185 0 316.5 131.5 + t131.5 316.5t-131.5 316.5t-316.5 131.5t-316.5 -131.5t-131.5 -316.5t131.5 -316.5t316.5 -131.5z" /> + <glyph glyph-name="_515" unicode="" horiz-adv-x="1792" + d="M1790 1007q12 -155 -52.5 -292t-186 -224t-271.5 -103v-260h224q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-224v-224q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9t-9 23v224h-512v-224q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9t-9 23v224h-224q-14 0 -23 9t-9 23v64q0 14 9 23 + t23 9h224v260q-150 16 -271.5 103t-186 224t-52.5 292q17 206 164.5 356.5t352.5 169.5q206 21 377 -94q171 115 377 94q205 -19 352.5 -169.5t164.5 -356.5zM896 647q128 131 128 313t-128 313q-128 -131 -128 -313t128 -313zM576 512q115 0 218 57q-154 165 -154 391 + q0 224 154 391q-103 57 -218 57q-185 0 -316.5 -131.5t-131.5 -316.5t131.5 -316.5t316.5 -131.5zM1152 128v260q-137 15 -256 94q-119 -79 -256 -94v-260h512zM1216 512q185 0 316.5 131.5t131.5 316.5t-131.5 316.5t-316.5 131.5q-115 0 -218 -57q154 -167 154 -391 + q0 -226 -154 -391q103 -57 218 -57z" /> + <glyph glyph-name="_516" unicode="" horiz-adv-x="1920" + d="M1536 1120q0 14 9 23t23 9h288q26 0 45 -19t19 -45v-288q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9t-9 23v134l-254 -255q76 -95 107.5 -214t9.5 -247q-31 -182 -166 -312t-318 -156q-210 -29 -384.5 80t-241.5 300q-117 6 -221 57.5t-177.5 133t-113.5 192.5t-32 230 + q9 135 78 252t182 191.5t248 89.5q118 14 227.5 -19t198.5 -103l255 254h-134q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h288q26 0 45 -19t19 -45v-288q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9t-9 23v134l-254 -255q59 -74 93 -169q182 -9 328 -124l255 254h-134q-14 0 -23 9 + t-9 23v64zM1024 704q0 20 -4 58q-162 -25 -271 -150t-109 -292q0 -20 4 -58q162 25 271 150t109 292zM128 704q0 -168 111 -294t276 -149q-3 29 -3 59q0 210 135 369.5t338 196.5q-53 120 -163.5 193t-245.5 73q-185 0 -316.5 -131.5t-131.5 -316.5zM1088 -128 + q185 0 316.5 131.5t131.5 316.5q0 168 -111 294t-276 149q3 -28 3 -59q0 -210 -135 -369.5t-338 -196.5q53 -120 163.5 -193t245.5 -73z" /> + <glyph glyph-name="_517" unicode="" horiz-adv-x="2048" + d="M1664 1504q0 14 9 23t23 9h288q26 0 45 -19t19 -45v-288q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9t-9 23v134l-254 -255q76 -95 107.5 -214t9.5 -247q-32 -180 -164.5 -310t-313.5 -157q-223 -34 -409 90q-117 -78 -256 -93v-132h96q14 0 23 -9t9 -23v-64q0 -14 -9 -23 + t-23 -9h-96v-96q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9t-9 23v96h-96q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h96v132q-155 17 -279.5 109.5t-187 237.5t-39.5 307q25 187 159.5 322.5t320.5 164.5q224 34 410 -90q146 97 320 97q201 0 359 -126l255 254h-134q-14 0 -23 9 + t-9 23v64zM896 391q128 131 128 313t-128 313q-128 -131 -128 -313t128 -313zM128 704q0 -185 131.5 -316.5t316.5 -131.5q117 0 218 57q-154 167 -154 391t154 391q-101 57 -218 57q-185 0 -316.5 -131.5t-131.5 -316.5zM1216 256q185 0 316.5 131.5t131.5 316.5 + t-131.5 316.5t-316.5 131.5q-117 0 -218 -57q154 -167 154 -391t-154 -391q101 -57 218 -57z" /> + <glyph glyph-name="_518" unicode="" + d="M1472 1408q26 0 45 -19t19 -45v-416q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9t-9 23v262l-213 -214l140 -140q9 -10 9 -23t-9 -22l-46 -46q-9 -9 -22 -9t-23 9l-140 141l-78 -79q126 -156 126 -359q0 -117 -45.5 -223.5t-123 -184t-184 -123t-223.5 -45.5t-223.5 45.5 + t-184 123t-123 184t-45.5 223.5t45.5 223.5t123 184t184 123t223.5 45.5q203 0 359 -126l78 78l-172 172q-9 10 -9 23t9 22l46 46q9 9 22 9t23 -9l172 -172l213 213h-261q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h416zM576 0q185 0 316.5 131.5t131.5 316.5t-131.5 316.5 + t-316.5 131.5t-316.5 -131.5t-131.5 -316.5t131.5 -316.5t316.5 -131.5z" /> + <glyph glyph-name="_519" unicode="" horiz-adv-x="1280" + d="M640 892q217 -24 364.5 -187.5t147.5 -384.5q0 -167 -87 -306t-236 -212t-319 -54q-133 15 -245.5 88t-182 188t-80.5 249q-12 155 52.5 292t186 224t271.5 103v132h-160q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h160v165l-92 -92q-10 -9 -23 -9t-22 9l-46 46q-9 9 -9 22 + t9 23l202 201q19 19 45 19t45 -19l202 -201q9 -10 9 -23t-9 -22l-46 -46q-9 -9 -22 -9t-23 9l-92 92v-165h160q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-160v-132zM576 -128q185 0 316.5 131.5t131.5 316.5t-131.5 316.5t-316.5 131.5t-316.5 -131.5t-131.5 -316.5 + t131.5 -316.5t316.5 -131.5z" /> + <glyph glyph-name="_520" unicode="" horiz-adv-x="2048" + d="M1901 621q19 -19 19 -45t-19 -45l-294 -294q-9 -10 -22.5 -10t-22.5 10l-45 45q-10 9 -10 22.5t10 22.5l185 185h-294v-224q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9t-9 23v224h-132q-24 -217 -187.5 -364.5t-384.5 -147.5q-167 0 -306 87t-212 236t-54 319q15 133 88 245.5 + t188 182t249 80.5q155 12 292 -52.5t224 -186t103 -271.5h132v224q0 14 9 23t23 9h64q14 0 23 -9t9 -23v-224h294l-185 185q-10 9 -10 22.5t10 22.5l45 45q9 10 22.5 10t22.5 -10zM576 128q185 0 316.5 131.5t131.5 316.5t-131.5 316.5t-316.5 131.5t-316.5 -131.5 + t-131.5 -316.5t131.5 -316.5t316.5 -131.5z" /> + <glyph glyph-name="_521" unicode="" horiz-adv-x="1280" + d="M1152 960q0 -221 -147.5 -384.5t-364.5 -187.5v-612q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9t-9 23v612q-217 24 -364.5 187.5t-147.5 384.5q0 117 45.5 223.5t123 184t184 123t223.5 45.5t223.5 -45.5t184 -123t123 -184t45.5 -223.5zM576 512q185 0 316.5 131.5 + t131.5 316.5t-131.5 316.5t-316.5 131.5t-316.5 -131.5t-131.5 -316.5t131.5 -316.5t316.5 -131.5z" /> + <glyph glyph-name="_522" unicode="" horiz-adv-x="1280" + d="M1024 576q0 185 -131.5 316.5t-316.5 131.5t-316.5 -131.5t-131.5 -316.5t131.5 -316.5t316.5 -131.5t316.5 131.5t131.5 316.5zM1152 576q0 -117 -45.5 -223.5t-123 -184t-184 -123t-223.5 -45.5t-223.5 45.5t-184 123t-123 184t-45.5 223.5t45.5 223.5t123 184t184 123 + t223.5 45.5t223.5 -45.5t184 -123t123 -184t45.5 -223.5z" /> + <glyph glyph-name="_523" unicode="" horiz-adv-x="1792" + /> + <glyph glyph-name="_524" unicode="" horiz-adv-x="1792" + /> + <glyph glyph-name="_525" unicode="" + d="M1451 1408q35 0 60 -25t25 -60v-1366q0 -35 -25 -60t-60 -25h-391v595h199l30 232h-229v148q0 56 23.5 84t91.5 28l122 1v207q-63 9 -178 9q-136 0 -217.5 -80t-81.5 -226v-171h-200v-232h200v-595h-735q-35 0 -60 25t-25 60v1366q0 35 25 60t60 25h1366z" /> + <glyph glyph-name="_526" unicode="" horiz-adv-x="1280" + d="M0 939q0 108 37.5 203.5t103.5 166.5t152 123t185 78t202 26q158 0 294 -66.5t221 -193.5t85 -287q0 -96 -19 -188t-60 -177t-100 -149.5t-145 -103t-189 -38.5q-68 0 -135 32t-96 88q-10 -39 -28 -112.5t-23.5 -95t-20.5 -71t-26 -71t-32 -62.5t-46 -77.5t-62 -86.5 + l-14 -5l-9 10q-15 157 -15 188q0 92 21.5 206.5t66.5 287.5t52 203q-32 65 -32 169q0 83 52 156t132 73q61 0 95 -40.5t34 -102.5q0 -66 -44 -191t-44 -187q0 -63 45 -104.5t109 -41.5q55 0 102 25t78.5 68t56 95t38 110.5t20 111t6.5 99.5q0 173 -109.5 269.5t-285.5 96.5 + q-200 0 -334 -129.5t-134 -328.5q0 -44 12.5 -85t27 -65t27 -45.5t12.5 -30.5q0 -28 -15 -73t-37 -45q-2 0 -17 3q-51 15 -90.5 56t-61 94.5t-32.5 108t-11 106.5z" /> + <glyph glyph-name="_527" unicode="" + d="M985 562q13 0 97.5 -44t89.5 -53q2 -5 2 -15q0 -33 -17 -76q-16 -39 -71 -65.5t-102 -26.5q-57 0 -190 62q-98 45 -170 118t-148 185q-72 107 -71 194v8q3 91 74 158q24 22 52 22q6 0 18 -1.5t19 -1.5q19 0 26.5 -6.5t15.5 -27.5q8 -20 33 -88t25 -75q0 -21 -34.5 -57.5 + t-34.5 -46.5q0 -7 5 -15q34 -73 102 -137q56 -53 151 -101q12 -7 22 -7q15 0 54 48.5t52 48.5zM782 32q127 0 243.5 50t200.5 134t134 200.5t50 243.5t-50 243.5t-134 200.5t-200.5 134t-243.5 50t-243.5 -50t-200.5 -134t-134 -200.5t-50 -243.5q0 -203 120 -368l-79 -233 + l242 77q158 -104 345 -104zM782 1414q153 0 292.5 -60t240.5 -161t161 -240.5t60 -292.5t-60 -292.5t-161 -240.5t-240.5 -161t-292.5 -60q-195 0 -365 94l-417 -134l136 405q-108 178 -108 389q0 153 60 292.5t161 240.5t240.5 161t292.5 60z" /> + <glyph glyph-name="_528" unicode="" horiz-adv-x="1792" + d="M128 128h1024v128h-1024v-128zM128 640h1024v128h-1024v-128zM1696 192q0 40 -28 68t-68 28t-68 -28t-28 -68t28 -68t68 -28t68 28t28 68zM128 1152h1024v128h-1024v-128zM1696 704q0 40 -28 68t-68 28t-68 -28t-28 -68t28 -68t68 -28t68 28t28 68zM1696 1216 + q0 40 -28 68t-68 28t-68 -28t-28 -68t28 -68t68 -28t68 28t28 68zM1792 384v-384h-1792v384h1792zM1792 896v-384h-1792v384h1792zM1792 1408v-384h-1792v384h1792z" /> + <glyph glyph-name="_529" unicode="" horiz-adv-x="2048" + d="M704 640q-159 0 -271.5 112.5t-112.5 271.5t112.5 271.5t271.5 112.5t271.5 -112.5t112.5 -271.5t-112.5 -271.5t-271.5 -112.5zM1664 512h352q13 0 22.5 -9.5t9.5 -22.5v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-352v-352q0 -13 -9.5 -22.5t-22.5 -9.5h-192q-13 0 -22.5 9.5 + t-9.5 22.5v352h-352q-13 0 -22.5 9.5t-9.5 22.5v192q0 13 9.5 22.5t22.5 9.5h352v352q0 13 9.5 22.5t22.5 9.5h192q13 0 22.5 -9.5t9.5 -22.5v-352zM928 288q0 -52 38 -90t90 -38h256v-238q-68 -50 -171 -50h-874q-121 0 -194 69t-73 190q0 53 3.5 103.5t14 109t26.5 108.5 + t43 97.5t62 81t85.5 53.5t111.5 20q19 0 39 -17q79 -61 154.5 -91.5t164.5 -30.5t164.5 30.5t154.5 91.5q20 17 39 17q132 0 217 -96h-223q-52 0 -90 -38t-38 -90v-192z" /> + <glyph glyph-name="_530" unicode="" horiz-adv-x="2048" + d="M704 640q-159 0 -271.5 112.5t-112.5 271.5t112.5 271.5t271.5 112.5t271.5 -112.5t112.5 -271.5t-112.5 -271.5t-271.5 -112.5zM1781 320l249 -249q9 -9 9 -23q0 -13 -9 -22l-136 -136q-9 -9 -22 -9q-14 0 -23 9l-249 249l-249 -249q-9 -9 -23 -9q-13 0 -22 9l-136 136 + q-9 9 -9 22q0 14 9 23l249 249l-249 249q-9 9 -9 23q0 13 9 22l136 136q9 9 22 9q14 0 23 -9l249 -249l249 249q9 9 23 9q13 0 22 -9l136 -136q9 -9 9 -22q0 -14 -9 -23zM1283 320l-181 -181q-37 -37 -37 -91q0 -53 37 -90l83 -83q-21 -3 -44 -3h-874q-121 0 -194 69 + t-73 190q0 53 3.5 103.5t14 109t26.5 108.5t43 97.5t62 81t85.5 53.5t111.5 20q19 0 39 -17q154 -122 319 -122t319 122q20 17 39 17q28 0 57 -6q-28 -27 -41 -50t-13 -56q0 -54 37 -91z" /> + <glyph glyph-name="_531" unicode="" horiz-adv-x="2048" + d="M256 512h1728q26 0 45 -19t19 -45v-448h-256v256h-1536v-256h-256v1216q0 26 19 45t45 19h128q26 0 45 -19t19 -45v-704zM832 832q0 106 -75 181t-181 75t-181 -75t-75 -181t75 -181t181 -75t181 75t75 181zM2048 576v64q0 159 -112.5 271.5t-271.5 112.5h-704 + q-26 0 -45 -19t-19 -45v-384h1152z" /> + <glyph glyph-name="_532" unicode="" + d="M1536 1536l-192 -448h192v-192h-274l-55 -128h329v-192h-411l-357 -832l-357 832h-411v192h329l-55 128h-274v192h192l-192 448h256l323 -768h378l323 768h256zM768 320l108 256h-216z" /> + <glyph glyph-name="_533" unicode="" + d="M1088 1536q185 0 316.5 -93.5t131.5 -226.5v-896q0 -130 -125.5 -222t-305.5 -97l213 -202q16 -15 8 -35t-30 -20h-1056q-22 0 -30 20t8 35l213 202q-180 5 -305.5 97t-125.5 222v896q0 133 131.5 226.5t316.5 93.5h640zM768 192q80 0 136 56t56 136t-56 136t-136 56 + t-136 -56t-56 -136t56 -136t136 -56zM1344 768v512h-1152v-512h1152z" /> + <glyph glyph-name="_534" unicode="" + d="M1088 1536q185 0 316.5 -93.5t131.5 -226.5v-896q0 -130 -125.5 -222t-305.5 -97l213 -202q16 -15 8 -35t-30 -20h-1056q-22 0 -30 20t8 35l213 202q-180 5 -305.5 97t-125.5 222v896q0 133 131.5 226.5t316.5 93.5h640zM288 224q66 0 113 47t47 113t-47 113t-113 47 + t-113 -47t-47 -113t47 -113t113 -47zM704 768v512h-544v-512h544zM1248 224q66 0 113 47t47 113t-47 113t-113 47t-113 -47t-47 -113t47 -113t113 -47zM1408 768v512h-576v-512h576z" /> + <glyph glyph-name="_535" unicode="" horiz-adv-x="1792" + d="M597 1115v-1173q0 -25 -12.5 -42.5t-36.5 -17.5q-17 0 -33 8l-465 233q-21 10 -35.5 33.5t-14.5 46.5v1140q0 20 10 34t29 14q14 0 44 -15l511 -256q3 -3 3 -5zM661 1014l534 -866l-534 266v600zM1792 996v-1054q0 -25 -14 -40.5t-38 -15.5t-47 13l-441 220zM1789 1116 + q0 -3 -256.5 -419.5t-300.5 -487.5l-390 634l324 527q17 28 52 28q14 0 26 -6l541 -270q4 -2 4 -6z" /> + <glyph glyph-name="_536" unicode="" + d="M809 532l266 499h-112l-157 -312q-24 -48 -44 -92l-42 92l-155 312h-120l263 -493v-324h101v318zM1536 1408v-1536h-1536v1536h1536z" /> + <glyph glyph-name="_537" unicode="" horiz-adv-x="2296" + d="M478 -139q-8 -16 -27 -34.5t-37 -25.5q-25 -9 -51.5 3.5t-28.5 31.5q-1 22 40 55t68 38q23 4 34 -21.5t2 -46.5zM1819 -139q7 -16 26 -34.5t38 -25.5q25 -9 51.5 3.5t27.5 31.5q2 22 -39.5 55t-68.5 38q-22 4 -33 -21.5t-2 -46.5zM1867 -30q13 -27 56.5 -59.5t77.5 -41.5 + q45 -13 82 4.5t37 50.5q0 46 -67.5 100.5t-115.5 59.5q-40 5 -63.5 -37.5t-6.5 -76.5zM428 -30q-13 -27 -56 -59.5t-77 -41.5q-45 -13 -82 4.5t-37 50.5q0 46 67.5 100.5t115.5 59.5q40 5 63 -37.5t6 -76.5zM1158 1094h1q-41 0 -76 -15q27 -8 44 -30.5t17 -49.5 + q0 -35 -27 -60t-65 -25q-52 0 -80 43q-5 -23 -5 -42q0 -74 56 -126.5t135 -52.5q80 0 136 52.5t56 126.5t-56 126.5t-136 52.5zM1462 1312q-99 109 -220.5 131.5t-245.5 -44.5q27 60 82.5 96.5t118 39.5t121.5 -17t99.5 -74.5t44.5 -131.5zM2212 73q8 -11 -11 -42 + q7 -23 7 -40q1 -56 -44.5 -112.5t-109.5 -91.5t-118 -37q-48 -2 -92 21.5t-66 65.5q-687 -25 -1259 0q-23 -41 -66.5 -65t-92.5 -22q-86 3 -179.5 80.5t-92.5 160.5q2 22 7 40q-19 31 -11 42q6 10 31 1q14 22 41 51q-7 29 2 38q11 10 39 -4q29 20 59 34q0 29 13 37 + q23 12 51 -16q35 5 61 -2q18 -4 38 -19v73q-11 0 -18 2q-53 10 -97 44.5t-55 87.5q-9 38 0 81q15 62 93 95q2 17 19 35.5t36 23.5t33 -7.5t19 -30.5h13q46 -5 60 -23q3 -3 5 -7q10 1 30.5 3.5t30.5 3.5q-15 11 -30 17q-23 40 -91 43q0 6 1 10q-62 2 -118.5 18.5t-84.5 47.5 + q-32 36 -42.5 92t-2.5 112q16 126 90 179q23 16 52 4.5t32 -40.5q0 -1 1.5 -14t2.5 -21t3 -20t5.5 -19t8.5 -10q27 -14 76 -12q48 46 98 74q-40 4 -162 -14l47 46q61 58 163 111q145 73 282 86q-20 8 -41 15.5t-47 14t-42.5 10.5t-47.5 11t-43 10q595 126 904 -139 + q98 -84 158 -222q85 -10 121 9h1q5 3 8.5 10t5.5 19t3 19.5t3 21.5l1 14q3 28 32 40t52 -5q73 -52 91 -178q7 -57 -3.5 -113t-42.5 -91q-28 -32 -83.5 -48.5t-115.5 -18.5v-10q-71 -2 -95 -43q-14 -5 -31 -17q11 -1 32 -3.5t30 -3.5q1 5 5 8q16 18 60 23h13q5 18 19 30t33 8 + t36 -23t19 -36q79 -32 93 -95q9 -40 1 -81q-12 -53 -56 -88t-97 -44q-10 -2 -17 -2q0 -49 -1 -73q20 15 38 19q26 7 61 2q28 28 51 16q14 -9 14 -37q33 -16 59 -34q27 13 38 4q10 -10 2 -38q28 -30 41 -51q23 8 31 -1zM1937 1025q0 -29 -9 -54q82 -32 112 -132 + q4 37 -9.5 98.5t-41.5 90.5q-20 19 -36 17t-16 -20zM1859 925q35 -42 47.5 -108.5t-0.5 -124.5q67 13 97 45q13 14 18 28q-3 64 -31 114.5t-79 66.5q-15 -15 -52 -21zM1822 921q-30 0 -44 1q42 -115 53 -239q21 0 43 3q16 68 1 135t-53 100zM258 839q30 100 112 132 + q-9 25 -9 54q0 18 -16.5 20t-35.5 -17q-28 -29 -41.5 -90.5t-9.5 -98.5zM294 737q29 -31 97 -45q-13 58 -0.5 124.5t47.5 108.5v0q-37 6 -52 21q-51 -16 -78.5 -66t-31.5 -115q9 -17 18 -28zM471 683q14 124 73 235q-19 -4 -55 -18l-45 -19v1q-46 -89 -20 -196q25 -3 47 -3z + M1434 644q8 -38 16.5 -108.5t11.5 -89.5q3 -18 9.5 -21.5t23.5 4.5q40 20 62 85.5t23 125.5q-24 2 -146 4zM1152 1285q-116 0 -199 -82.5t-83 -198.5q0 -117 83 -199.5t199 -82.5t199 82.5t83 199.5q0 116 -83 198.5t-199 82.5zM1380 646q-105 2 -211 0v1q-1 -27 2.5 -86 + t13.5 -66q29 -14 93.5 -14.5t95.5 10.5q9 3 11 39t-0.5 69.5t-4.5 46.5zM1112 447q8 4 9.5 48t-0.5 88t-4 63v1q-212 -3 -214 -3q-4 -20 -7 -62t0 -83t14 -46q34 -15 101 -16t101 10zM718 636q-16 -59 4.5 -118.5t77.5 -84.5q15 -8 24 -5t12 21q3 16 8 90t10 103 + q-69 -2 -136 -6zM591 510q3 -23 -34 -36q132 -141 271.5 -240t305.5 -154q172 49 310.5 146t293.5 250q-33 13 -30 34q0 2 0.5 3.5t1.5 3t1 2.5v1v-1q-17 2 -50 5.5t-48 4.5q-26 -90 -82 -132q-51 -38 -82 1q-5 6 -9 14q-7 13 -17 62q-2 -5 -5 -9t-7.5 -7t-8 -5.5t-9.5 -4 + l-10 -2.5t-12 -2l-12 -1.5t-13.5 -1t-13.5 -0.5q-106 -9 -163 11q-4 -17 -10 -26.5t-21 -15t-23 -7t-36 -3.5q-6 -1 -9 -1q-179 -17 -203 40q-2 -63 -56 -54q-47 8 -91 54q-12 13 -20 26q-17 29 -26 65q-58 -6 -87 -10q1 -2 4 -10zM507 -118q3 14 3 30q-17 71 -51 130 + t-73 70q-41 12 -101.5 -14.5t-104.5 -80t-39 -107.5q35 -53 100 -93t119 -42q51 -2 94 28t53 79zM510 53q23 -63 27 -119q195 113 392 174q-98 52 -180.5 120t-179.5 165q-6 -4 -29 -13q0 -1 -1 -4t-1 -5q31 -18 22 -37q-12 -23 -56 -34q-10 -13 -29 -24h-1q-2 -83 1 -150 + q19 -34 35 -73zM579 -113q532 -21 1145 0q-254 147 -428 196q-76 -35 -156 -57q-8 -3 -16 0q-65 21 -129 49q-208 -60 -416 -188h-1v-1q1 0 1 1zM1763 -67q4 54 28 120q14 38 33 71l-1 -1q3 77 3 153q-15 8 -30 25q-42 9 -56 33q-9 20 22 38q-2 4 -2 9q-16 4 -28 12 + q-204 -190 -383 -284q198 -59 414 -176zM2155 -90q5 54 -39 107.5t-104 80t-102 14.5q-38 -11 -72.5 -70.5t-51.5 -129.5q0 -16 3 -30q10 -49 53 -79t94 -28q54 2 119 42t100 93z" /> + <glyph glyph-name="_538" unicode="" horiz-adv-x="2304" + d="M1524 -25q0 -68 -48 -116t-116 -48t-116.5 48t-48.5 116t48.5 116.5t116.5 48.5t116 -48.5t48 -116.5zM775 -25q0 -68 -48.5 -116t-116.5 -48t-116 48t-48 116t48 116.5t116 48.5t116.5 -48.5t48.5 -116.5zM0 1469q57 -60 110.5 -104.5t121 -82t136 -63t166 -45.5 + t200 -31.5t250 -18.5t304 -9.5t372.5 -2.5q139 0 244.5 -5t181 -16.5t124 -27.5t71 -39.5t24 -51.5t-19.5 -64t-56.5 -76.5t-89.5 -91t-116 -104.5t-139 -119q-185 -157 -286 -247q29 51 76.5 109t94 105.5t94.5 98.5t83 91.5t54 80.5t13 70t-45.5 55.5t-116.5 41t-204 23.5 + t-304 5q-168 -2 -314 6t-256 23t-204.5 41t-159.5 51.5t-122.5 62.5t-91.5 66.5t-68 71.5t-50.5 69.5t-40 68t-36.5 59.5z" /> + <glyph glyph-name="_539" unicode="" horiz-adv-x="1792" + d="M896 1472q-169 0 -323 -66t-265.5 -177.5t-177.5 -265.5t-66 -323t66 -323t177.5 -265.5t265.5 -177.5t323 -66t323 66t265.5 177.5t177.5 265.5t66 323t-66 323t-177.5 265.5t-265.5 177.5t-323 66zM896 1536q182 0 348 -71t286 -191t191 -286t71 -348t-71 -348 + t-191 -286t-286 -191t-348 -71t-348 71t-286 191t-191 286t-71 348t71 348t191 286t286 191t348 71zM496 704q16 0 16 -16v-480q0 -16 -16 -16h-32q-16 0 -16 16v480q0 16 16 16h32zM896 640q53 0 90.5 -37.5t37.5 -90.5q0 -35 -17.5 -64t-46.5 -46v-114q0 -14 -9 -23 + t-23 -9h-64q-14 0 -23 9t-9 23v114q-29 17 -46.5 46t-17.5 64q0 53 37.5 90.5t90.5 37.5zM896 1408q209 0 385.5 -103t279.5 -279.5t103 -385.5t-103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103zM544 928v-96 + q0 -14 9 -23t23 -9h64q14 0 23 9t9 23v96q0 93 65.5 158.5t158.5 65.5t158.5 -65.5t65.5 -158.5v-96q0 -14 9 -23t23 -9h64q14 0 23 9t9 23v96q0 146 -103 249t-249 103t-249 -103t-103 -249zM1408 192v512q0 26 -19 45t-45 19h-896q-26 0 -45 -19t-19 -45v-512 + q0 -26 19 -45t45 -19h896q26 0 45 19t19 45z" /> + <glyph glyph-name="_540" unicode="" horiz-adv-x="2304" + d="M1920 1024v-768h-1664v768h1664zM2048 448h128v384h-128v288q0 14 -9 23t-23 9h-1856q-14 0 -23 -9t-9 -23v-960q0 -14 9 -23t23 -9h1856q14 0 23 9t9 23v288zM2304 832v-384q0 -53 -37.5 -90.5t-90.5 -37.5v-160q0 -66 -47 -113t-113 -47h-1856q-66 0 -113 47t-47 113 + v960q0 66 47 113t113 47h1856q66 0 113 -47t47 -113v-160q53 0 90.5 -37.5t37.5 -90.5z" /> + <glyph glyph-name="_541" unicode="" horiz-adv-x="2304" + d="M256 256v768h1280v-768h-1280zM2176 960q53 0 90.5 -37.5t37.5 -90.5v-384q0 -53 -37.5 -90.5t-90.5 -37.5v-160q0 -66 -47 -113t-113 -47h-1856q-66 0 -113 47t-47 113v960q0 66 47 113t113 47h1856q66 0 113 -47t47 -113v-160zM2176 448v384h-128v288q0 14 -9 23t-23 9 + h-1856q-14 0 -23 -9t-9 -23v-960q0 -14 9 -23t23 -9h1856q14 0 23 9t9 23v288h128z" /> + <glyph glyph-name="_542" unicode="" horiz-adv-x="2304" + d="M256 256v768h896v-768h-896zM2176 960q53 0 90.5 -37.5t37.5 -90.5v-384q0 -53 -37.5 -90.5t-90.5 -37.5v-160q0 -66 -47 -113t-113 -47h-1856q-66 0 -113 47t-47 113v960q0 66 47 113t113 47h1856q66 0 113 -47t47 -113v-160zM2176 448v384h-128v288q0 14 -9 23t-23 9 + h-1856q-14 0 -23 -9t-9 -23v-960q0 -14 9 -23t23 -9h1856q14 0 23 9t9 23v288h128z" /> + <glyph glyph-name="_543" unicode="" horiz-adv-x="2304" + d="M256 256v768h512v-768h-512zM2176 960q53 0 90.5 -37.5t37.5 -90.5v-384q0 -53 -37.5 -90.5t-90.5 -37.5v-160q0 -66 -47 -113t-113 -47h-1856q-66 0 -113 47t-47 113v960q0 66 47 113t113 47h1856q66 0 113 -47t47 -113v-160zM2176 448v384h-128v288q0 14 -9 23t-23 9 + h-1856q-14 0 -23 -9t-9 -23v-960q0 -14 9 -23t23 -9h1856q14 0 23 9t9 23v288h128z" /> + <glyph glyph-name="_544" unicode="" horiz-adv-x="2304" + d="M2176 960q53 0 90.5 -37.5t37.5 -90.5v-384q0 -53 -37.5 -90.5t-90.5 -37.5v-160q0 -66 -47 -113t-113 -47h-1856q-66 0 -113 47t-47 113v960q0 66 47 113t113 47h1856q66 0 113 -47t47 -113v-160zM2176 448v384h-128v288q0 14 -9 23t-23 9h-1856q-14 0 -23 -9t-9 -23 + v-960q0 -14 9 -23t23 -9h1856q14 0 23 9t9 23v288h128z" /> + <glyph glyph-name="_545" unicode="" horiz-adv-x="1280" + d="M1133 493q31 -30 14 -69q-17 -40 -59 -40h-382l201 -476q10 -25 0 -49t-34 -35l-177 -75q-25 -10 -49 0t-35 34l-191 452l-312 -312q-19 -19 -45 -19q-12 0 -24 5q-40 17 -40 59v1504q0 42 40 59q12 5 24 5q27 0 45 -19z" /> + <glyph glyph-name="_546" unicode="" horiz-adv-x="1024" + d="M832 1408q-320 0 -320 -224v-416h128v-128h-128v-544q0 -224 320 -224h64v-128h-64q-272 0 -384 146q-112 -146 -384 -146h-64v128h64q320 0 320 224v544h-128v128h128v416q0 224 -320 224h-64v128h64q272 0 384 -146q112 146 384 146h64v-128h-64z" /> + <glyph glyph-name="_547" unicode="" horiz-adv-x="2048" + d="M2048 1152h-128v-1024h128v-384h-384v128h-1280v-128h-384v384h128v1024h-128v384h384v-128h1280v128h384v-384zM1792 1408v-128h128v128h-128zM128 1408v-128h128v128h-128zM256 -128v128h-128v-128h128zM1664 0v128h128v1024h-128v128h-1280v-128h-128v-1024h128v-128 + h1280zM1920 -128v128h-128v-128h128zM1280 896h384v-768h-896v256h-384v768h896v-256zM512 512h640v512h-640v-512zM1536 256v512h-256v-384h-384v-128h640z" /> + <glyph glyph-name="_548" unicode="" horiz-adv-x="2304" + d="M2304 768h-128v-640h128v-384h-384v128h-896v-128h-384v384h128v128h-384v-128h-384v384h128v640h-128v384h384v-128h896v128h384v-384h-128v-128h384v128h384v-384zM2048 1024v-128h128v128h-128zM1408 1408v-128h128v128h-128zM128 1408v-128h128v128h-128zM256 256 + v128h-128v-128h128zM1536 384h-128v-128h128v128zM384 384h896v128h128v640h-128v128h-896v-128h-128v-640h128v-128zM896 -128v128h-128v-128h128zM2176 -128v128h-128v-128h128zM2048 128v640h-128v128h-384v-384h128v-384h-384v128h-384v-128h128v-128h896v128h128z" /> + <glyph glyph-name="_549" unicode="" + d="M1024 288v-416h-928q-40 0 -68 28t-28 68v1344q0 40 28 68t68 28h1344q40 0 68 -28t28 -68v-928h-416q-40 0 -68 -28t-28 -68zM1152 256h381q-15 -82 -65 -132l-184 -184q-50 -50 -132 -65v381z" /> + <glyph glyph-name="_550" unicode="" + d="M1400 256h-248v-248q29 10 41 22l185 185q12 12 22 41zM1120 384h288v896h-1280v-1280h896v288q0 40 28 68t68 28zM1536 1312v-1024q0 -40 -20 -88t-48 -76l-184 -184q-28 -28 -76 -48t-88 -20h-1024q-40 0 -68 28t-28 68v1344q0 40 28 68t68 28h1344q40 0 68 -28t28 -68 + z" /> + <glyph glyph-name="_551" unicode="" horiz-adv-x="2304" + d="M1951 538q0 -26 -15.5 -44.5t-38.5 -23.5q-8 -2 -18 -2h-153v140h153q10 0 18 -2q23 -5 38.5 -23.5t15.5 -44.5zM1933 751q0 -25 -15 -42t-38 -21q-3 -1 -15 -1h-139v129h139q3 0 8.5 -0.5t6.5 -0.5q23 -4 38 -21.5t15 -42.5zM728 587v308h-228v-308q0 -58 -38 -94.5 + t-105 -36.5q-108 0 -229 59v-112q53 -15 121 -23t109 -9l42 -1q328 0 328 217zM1442 403v113q-99 -52 -200 -59q-108 -8 -169 41t-61 142t61 142t169 41q101 -7 200 -58v112q-48 12 -100 19.5t-80 9.5l-28 2q-127 6 -218.5 -14t-140.5 -60t-71 -88t-22 -106t22 -106t71 -88 + t140.5 -60t218.5 -14q101 4 208 31zM2176 518q0 54 -43 88.5t-109 39.5v3q57 8 89 41.5t32 79.5q0 55 -41 88t-107 36q-3 0 -12 0.5t-14 0.5h-455v-510h491q74 0 121.5 36.5t47.5 96.5zM2304 1280v-1280q0 -52 -38 -90t-90 -38h-2048q-52 0 -90 38t-38 90v1280q0 52 38 90 + t90 38h2048q52 0 90 -38t38 -90z" /> + <glyph glyph-name="_552" unicode="" horiz-adv-x="2304" + d="M858 295v693q-106 -41 -172 -135.5t-66 -211.5t66 -211.5t172 -134.5zM1362 641q0 117 -66 211.5t-172 135.5v-694q106 41 172 135.5t66 211.5zM1577 641q0 -159 -78.5 -294t-213.5 -213.5t-294 -78.5q-119 0 -227.5 46.5t-187 125t-125 187t-46.5 227.5q0 159 78.5 294 + t213.5 213.5t294 78.5t294 -78.5t213.5 -213.5t78.5 -294zM1960 634q0 139 -55.5 261.5t-147.5 205.5t-213.5 131t-252.5 48h-301q-176 0 -323.5 -81t-235 -230t-87.5 -335q0 -171 87 -317.5t236 -231.5t323 -85h301q129 0 251.5 50.5t214.5 135t147.5 202.5t55.5 246z + M2304 1280v-1280q0 -52 -38 -90t-90 -38h-2048q-52 0 -90 38t-38 90v1280q0 52 38 90t90 38h2048q52 0 90 -38t38 -90z" /> + <glyph glyph-name="_553" unicode="" horiz-adv-x="1792" + d="M1664 -96v1088q0 13 -9.5 22.5t-22.5 9.5h-1088q-13 0 -22.5 -9.5t-9.5 -22.5v-1088q0 -13 9.5 -22.5t22.5 -9.5h1088q13 0 22.5 9.5t9.5 22.5zM1792 992v-1088q0 -66 -47 -113t-113 -47h-1088q-66 0 -113 47t-47 113v1088q0 66 47 113t113 47h1088q66 0 113 -47t47 -113 + zM1408 1376v-160h-128v160q0 13 -9.5 22.5t-22.5 9.5h-1088q-13 0 -22.5 -9.5t-9.5 -22.5v-1088q0 -13 9.5 -22.5t22.5 -9.5h160v-128h-160q-66 0 -113 47t-47 113v1088q0 66 47 113t113 47h1088q66 0 113 -47t47 -113z" /> + <glyph glyph-name="_554" unicode="" horiz-adv-x="2304" + d="M1728 1088l-384 -704h768zM448 1088l-384 -704h768zM1269 1280q-14 -40 -45.5 -71.5t-71.5 -45.5v-1291h608q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-1344q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h608v1291q-40 14 -71.5 45.5t-45.5 71.5h-491q-14 0 -23 9t-9 23v64 + q0 14 9 23t23 9h491q21 57 70 92.5t111 35.5t111 -35.5t70 -92.5h491q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-491zM1088 1264q33 0 56.5 23.5t23.5 56.5t-23.5 56.5t-56.5 23.5t-56.5 -23.5t-23.5 -56.5t23.5 -56.5t56.5 -23.5zM2176 384q0 -73 -46.5 -131t-117.5 -91 + t-144.5 -49.5t-139.5 -16.5t-139.5 16.5t-144.5 49.5t-117.5 91t-46.5 131q0 11 35 81t92 174.5t107 195.5t102 184t56 100q18 33 56 33t56 -33q4 -7 56 -100t102 -184t107 -195.5t92 -174.5t35 -81zM896 384q0 -73 -46.5 -131t-117.5 -91t-144.5 -49.5t-139.5 -16.5 + t-139.5 16.5t-144.5 49.5t-117.5 91t-46.5 131q0 11 35 81t92 174.5t107 195.5t102 184t56 100q18 33 56 33t56 -33q4 -7 56 -100t102 -184t107 -195.5t92 -174.5t35 -81z" /> + <glyph glyph-name="_555" unicode="" + d="M1408 1408q0 -261 -106.5 -461.5t-266.5 -306.5q160 -106 266.5 -306.5t106.5 -461.5h96q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-1472q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h96q0 261 106.5 461.5t266.5 306.5q-160 106 -266.5 306.5t-106.5 461.5h-96q-14 0 -23 9 + t-9 23v64q0 14 9 23t23 9h1472q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-96zM874 700q77 29 149 92.5t129.5 152.5t92.5 210t35 253h-1024q0 -132 35 -253t92.5 -210t129.5 -152.5t149 -92.5q19 -7 30.5 -23.5t11.5 -36.5t-11.5 -36.5t-30.5 -23.5q-77 -29 -149 -92.5 + t-129.5 -152.5t-92.5 -210t-35 -253h1024q0 132 -35 253t-92.5 210t-129.5 152.5t-149 92.5q-19 7 -30.5 23.5t-11.5 36.5t11.5 36.5t30.5 23.5z" /> + <glyph glyph-name="_556" unicode="" + d="M1408 1408q0 -261 -106.5 -461.5t-266.5 -306.5q160 -106 266.5 -306.5t106.5 -461.5h96q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-1472q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h96q0 261 106.5 461.5t266.5 306.5q-160 106 -266.5 306.5t-106.5 461.5h-96q-14 0 -23 9 + t-9 23v64q0 14 9 23t23 9h1472q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-96zM1280 1408h-1024q0 -66 9 -128h1006q9 61 9 128zM1280 -128q0 130 -34 249.5t-90.5 208t-126.5 152t-146 94.5h-230q-76 -31 -146 -94.5t-126.5 -152t-90.5 -208t-34 -249.5h1024z" /> + <glyph glyph-name="_557" unicode="" + d="M1408 1408q0 -261 -106.5 -461.5t-266.5 -306.5q160 -106 266.5 -306.5t106.5 -461.5h96q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-1472q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h96q0 261 106.5 461.5t266.5 306.5q-160 106 -266.5 306.5t-106.5 461.5h-96q-14 0 -23 9 + t-9 23v64q0 14 9 23t23 9h1472q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-96zM1280 1408h-1024q0 -206 85 -384h854q85 178 85 384zM1223 192q-54 141 -145.5 241.5t-194.5 142.5h-230q-103 -42 -194.5 -142.5t-145.5 -241.5h910z" /> + <glyph glyph-name="_558" unicode="" + d="M1408 1408q0 -261 -106.5 -461.5t-266.5 -306.5q160 -106 266.5 -306.5t106.5 -461.5h96q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-1472q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h96q0 261 106.5 461.5t266.5 306.5q-160 106 -266.5 306.5t-106.5 461.5h-96q-14 0 -23 9 + t-9 23v64q0 14 9 23t23 9h1472q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-96zM874 700q77 29 149 92.5t129.5 152.5t92.5 210t35 253h-1024q0 -132 35 -253t92.5 -210t129.5 -152.5t149 -92.5q19 -7 30.5 -23.5t11.5 -36.5t-11.5 -36.5t-30.5 -23.5q-137 -51 -244 -196 + h700q-107 145 -244 196q-19 7 -30.5 23.5t-11.5 36.5t11.5 36.5t30.5 23.5z" /> + <glyph glyph-name="_559" unicode="" + d="M1504 -64q14 0 23 -9t9 -23v-128q0 -14 -9 -23t-23 -9h-1472q-14 0 -23 9t-9 23v128q0 14 9 23t23 9h1472zM130 0q3 55 16 107t30 95t46 87t53.5 76t64.5 69.5t66 60t70.5 55t66.5 47.5t65 43q-43 28 -65 43t-66.5 47.5t-70.5 55t-66 60t-64.5 69.5t-53.5 76t-46 87 + t-30 95t-16 107h1276q-3 -55 -16 -107t-30 -95t-46 -87t-53.5 -76t-64.5 -69.5t-66 -60t-70.5 -55t-66.5 -47.5t-65 -43q43 -28 65 -43t66.5 -47.5t70.5 -55t66 -60t64.5 -69.5t53.5 -76t46 -87t30 -95t16 -107h-1276zM1504 1536q14 0 23 -9t9 -23v-128q0 -14 -9 -23t-23 -9 + h-1472q-14 0 -23 9t-9 23v128q0 14 9 23t23 9h1472z" /> + <glyph glyph-name="_560" unicode="" + d="M768 1152q-53 0 -90.5 -37.5t-37.5 -90.5v-128h-32v93q0 48 -32 81.5t-80 33.5q-46 0 -79 -33t-33 -79v-429l-32 30v172q0 48 -32 81.5t-80 33.5q-46 0 -79 -33t-33 -79v-224q0 -47 35 -82l310 -296q39 -39 39 -102q0 -26 19 -45t45 -19h640q26 0 45 19t19 45v25 + q0 41 10 77l108 436q10 36 10 77v246q0 48 -32 81.5t-80 33.5q-46 0 -79 -33t-33 -79v-32h-32v125q0 40 -25 72.5t-64 40.5q-14 2 -23 2q-46 0 -79 -33t-33 -79v-128h-32v122q0 51 -32.5 89.5t-82.5 43.5q-5 1 -13 1zM768 1280q84 0 149 -50q57 34 123 34q59 0 111 -27 + t86 -76q27 7 59 7q100 0 170 -71.5t70 -171.5v-246q0 -51 -13 -108l-109 -436q-6 -24 -6 -71q0 -80 -56 -136t-136 -56h-640q-84 0 -138 58.5t-54 142.5l-308 296q-76 73 -76 175v224q0 99 70.5 169.5t169.5 70.5q11 0 16 -1q6 95 75.5 160t164.5 65q52 0 98 -21 + q72 69 174 69z" /> + <glyph glyph-name="_561" unicode="" horiz-adv-x="1792" + d="M880 1408q-46 0 -79 -33t-33 -79v-656h-32v528q0 46 -33 79t-79 33t-79 -33t-33 -79v-528v-256l-154 205q-38 51 -102 51q-53 0 -90.5 -37.5t-37.5 -90.5q0 -43 26 -77l384 -512q38 -51 102 -51h688q34 0 61 22t34 56l76 405q5 32 5 59v498q0 46 -33 79t-79 33t-79 -33 + t-33 -79v-272h-32v528q0 46 -33 79t-79 33t-79 -33t-33 -79v-528h-32v656q0 46 -33 79t-79 33zM880 1536q68 0 125.5 -35.5t88.5 -96.5q19 4 42 4q99 0 169.5 -70.5t70.5 -169.5v-17q105 6 180.5 -64t75.5 -175v-498q0 -40 -8 -83l-76 -404q-14 -79 -76.5 -131t-143.5 -52 + h-688q-60 0 -114.5 27.5t-90.5 74.5l-384 512q-51 68 -51 154q0 106 75 181t181 75q78 0 128 -34v434q0 99 70.5 169.5t169.5 70.5q23 0 42 -4q31 61 88.5 96.5t125.5 35.5z" /> + <glyph glyph-name="_562" unicode="" horiz-adv-x="1792" + d="M1073 -128h-177q-163 0 -226 141q-23 49 -23 102v5q-62 30 -98.5 88.5t-36.5 127.5q0 38 5 48h-261q-106 0 -181 75t-75 181t75 181t181 75h113l-44 17q-74 28 -119.5 93.5t-45.5 145.5q0 106 75 181t181 75q46 0 91 -17l628 -239h401q106 0 181 -75t75 -181v-668 + q0 -88 -54 -157.5t-140 -90.5l-339 -85q-92 -23 -186 -23zM1024 583l-155 -71l-163 -74q-30 -14 -48 -41.5t-18 -60.5q0 -46 33 -79t79 -33q26 0 46 10l338 154q-49 10 -80.5 50t-31.5 90v55zM1344 272q0 46 -33 79t-79 33q-26 0 -46 -10l-290 -132q-28 -13 -37 -17 + t-30.5 -17t-29.5 -23.5t-16 -29t-8 -40.5q0 -50 31.5 -82t81.5 -32q20 0 38 9l352 160q30 14 48 41.5t18 60.5zM1112 1024l-650 248q-24 8 -46 8q-53 0 -90.5 -37.5t-37.5 -90.5q0 -40 22.5 -73t59.5 -47l526 -200v-64h-640q-53 0 -90.5 -37.5t-37.5 -90.5t37.5 -90.5 + t90.5 -37.5h535l233 106v198q0 63 46 106l111 102h-69zM1073 0q82 0 155 19l339 85q43 11 70 45.5t27 78.5v668q0 53 -37.5 90.5t-90.5 37.5h-308l-136 -126q-36 -33 -36 -82v-296q0 -46 33 -77t79 -31t79 35t33 81v208h32v-208q0 -70 -57 -114q52 -8 86.5 -48.5t34.5 -93.5 + q0 -42 -23 -78t-61 -53l-310 -141h91z" /> + <glyph glyph-name="_563" unicode="" horiz-adv-x="2048" + d="M1151 1536q61 0 116 -28t91 -77l572 -781q118 -159 118 -359v-355q0 -80 -56 -136t-136 -56h-384q-80 0 -136 56t-56 136v177l-286 143h-546q-80 0 -136 56t-56 136v32q0 119 84.5 203.5t203.5 84.5h420l42 128h-686q-100 0 -173.5 67.5t-81.5 166.5q-65 79 -65 182v32 + q0 80 56 136t136 56h959zM1920 -64v355q0 157 -93 284l-573 781q-39 52 -103 52h-959q-26 0 -45 -19t-19 -45q0 -32 1.5 -49.5t9.5 -40.5t25 -43q10 31 35.5 50t56.5 19h832v-32h-832q-26 0 -45 -19t-19 -45q0 -44 3 -58q8 -44 44 -73t81 -29h640h91q40 0 68 -28t28 -68 + q0 -15 -5 -30l-64 -192q-10 -29 -35 -47.5t-56 -18.5h-443q-66 0 -113 -47t-47 -113v-32q0 -26 19 -45t45 -19h561q16 0 29 -7l317 -158q24 -13 38.5 -36t14.5 -50v-197q0 -26 19 -45t45 -19h384q26 0 45 19t19 45z" /> + <glyph glyph-name="_564" unicode="" horiz-adv-x="2048" + d="M459 -256q-77 0 -137.5 47.5t-79.5 122.5l-101 401q-13 57 -13 108q0 45 -5 67l-116 477q-7 27 -7 57q0 93 62 161t155 78q17 85 82.5 139t152.5 54q83 0 148 -51.5t85 -132.5l83 -348l103 428q20 81 85 132.5t148 51.5q89 0 155.5 -57.5t80.5 -144.5q92 -10 152 -79 + t60 -162q0 -24 -7 -59l-123 -512q10 7 37.5 28.5t38.5 29.5t35 23t41 20.5t41.5 11t49.5 5.5q105 0 180 -74t75 -179q0 -62 -28.5 -118t-78.5 -94l-507 -380q-68 -51 -153 -51h-694zM1104 1408q-38 0 -68.5 -24t-39.5 -62l-164 -682h-127l-145 602q-9 38 -39.5 62t-68.5 24 + q-48 0 -80 -33t-32 -80q0 -15 3 -28l132 -547h-26l-99 408q-9 37 -40 62.5t-69 25.5q-47 0 -80 -33t-33 -79q0 -14 3 -26l116 -478q7 -28 9 -86t10 -88l100 -401q8 -32 34 -52.5t59 -20.5h694q42 0 76 26l507 379q56 43 56 110q0 52 -37.5 88.5t-89.5 36.5q-43 0 -77 -26 + l-307 -230v227q0 4 32 138t68 282t39 161q4 18 4 29q0 47 -32 81t-79 34q-39 0 -69.5 -24t-39.5 -62l-116 -482h-26l150 624q3 14 3 28q0 48 -31.5 82t-79.5 34z" /> + <glyph glyph-name="_565" unicode="" horiz-adv-x="1792" + d="M640 1408q-53 0 -90.5 -37.5t-37.5 -90.5v-512v-384l-151 202q-41 54 -107 54q-52 0 -89 -38t-37 -90q0 -43 26 -77l384 -512q38 -51 102 -51h718q22 0 39.5 13.5t22.5 34.5l92 368q24 96 24 194v217q0 41 -28 71t-68 30t-68 -28t-28 -68h-32v61q0 48 -32 81.5t-80 33.5 + q-46 0 -79 -33t-33 -79v-64h-32v90q0 55 -37 94.5t-91 39.5q-53 0 -90.5 -37.5t-37.5 -90.5v-96h-32v570q0 55 -37 94.5t-91 39.5zM640 1536q107 0 181.5 -77.5t74.5 -184.5v-220q22 2 32 2q99 0 173 -69q47 21 99 21q113 0 184 -87q27 7 56 7q94 0 159 -67.5t65 -161.5 + v-217q0 -116 -28 -225l-92 -368q-16 -64 -68 -104.5t-118 -40.5h-718q-60 0 -114.5 27.5t-90.5 74.5l-384 512q-51 68 -51 154q0 105 74.5 180.5t179.5 75.5q71 0 130 -35v547q0 106 75 181t181 75zM768 128v384h-32v-384h32zM1024 128v384h-32v-384h32zM1280 128v384h-32 + v-384h32z" /> + <glyph glyph-name="_566" unicode="" + d="M1288 889q60 0 107 -23q141 -63 141 -226v-177q0 -94 -23 -186l-85 -339q-21 -86 -90.5 -140t-157.5 -54h-668q-106 0 -181 75t-75 181v401l-239 628q-17 45 -17 91q0 106 75 181t181 75q80 0 145.5 -45.5t93.5 -119.5l17 -44v113q0 106 75 181t181 75t181 -75t75 -181 + v-261q27 5 48 5q69 0 127.5 -36.5t88.5 -98.5zM1072 896q-33 0 -60.5 -18t-41.5 -48l-74 -163l-71 -155h55q50 0 90 -31.5t50 -80.5l154 338q10 20 10 46q0 46 -33 79t-79 33zM1293 761q-22 0 -40.5 -8t-29 -16t-23.5 -29.5t-17 -30.5t-17 -37l-132 -290q-10 -20 -10 -46 + q0 -46 33 -79t79 -33q33 0 60.5 18t41.5 48l160 352q9 18 9 38q0 50 -32 81.5t-82 31.5zM128 1120q0 -22 8 -46l248 -650v-69l102 111q43 46 106 46h198l106 233v535q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5v-640h-64l-200 526q-14 37 -47 59.5t-73 22.5 + q-53 0 -90.5 -37.5t-37.5 -90.5zM1180 -128q44 0 78.5 27t45.5 70l85 339q19 73 19 155v91l-141 -310q-17 -38 -53 -61t-78 -23q-53 0 -93.5 34.5t-48.5 86.5q-44 -57 -114 -57h-208v32h208q46 0 81 33t35 79t-31 79t-77 33h-296q-49 0 -82 -36l-126 -136v-308 + q0 -53 37.5 -90.5t90.5 -37.5h668z" /> + <glyph glyph-name="_567" unicode="" horiz-adv-x="1973" + d="M857 992v-117q0 -13 -9.5 -22t-22.5 -9h-298v-812q0 -13 -9 -22.5t-22 -9.5h-135q-13 0 -22.5 9t-9.5 23v812h-297q-13 0 -22.5 9t-9.5 22v117q0 14 9 23t23 9h793q13 0 22.5 -9.5t9.5 -22.5zM1895 995l77 -961q1 -13 -8 -24q-10 -10 -23 -10h-134q-12 0 -21 8.5 + t-10 20.5l-46 588l-189 -425q-8 -19 -29 -19h-120q-20 0 -29 19l-188 427l-45 -590q-1 -12 -10 -20.5t-21 -8.5h-135q-13 0 -23 10q-9 10 -9 24l78 961q1 12 10 20.5t21 8.5h142q20 0 29 -19l220 -520q10 -24 20 -51q3 7 9.5 24.5t10.5 26.5l221 520q9 19 29 19h141 + q13 0 22 -8.5t10 -20.5z" /> + <glyph glyph-name="_568" unicode="" horiz-adv-x="1792" + d="M1042 833q0 88 -60 121q-33 18 -117 18h-123v-281h162q66 0 102 37t36 105zM1094 548l205 -373q8 -17 -1 -31q-8 -16 -27 -16h-152q-20 0 -28 17l-194 365h-155v-350q0 -14 -9 -23t-23 -9h-134q-14 0 -23 9t-9 23v960q0 14 9 23t23 9h294q128 0 190 -24q85 -31 134 -109 + t49 -180q0 -92 -42.5 -165.5t-115.5 -109.5q6 -10 9 -16zM896 1376q-150 0 -286 -58.5t-234.5 -157t-157 -234.5t-58.5 -286t58.5 -286t157 -234.5t234.5 -157t286 -58.5t286 58.5t234.5 157t157 234.5t58.5 286t-58.5 286t-157 234.5t-234.5 157t-286 58.5zM1792 640 + q0 -182 -71 -348t-191 -286t-286 -191t-348 -71t-348 71t-286 191t-191 286t-71 348t71 348t191 286t286 191t348 71t348 -71t286 -191t191 -286t71 -348z" /> + <glyph glyph-name="_569" unicode="" horiz-adv-x="1792" + d="M605 303q153 0 257 104q14 18 3 36l-45 82q-6 13 -24 17q-16 2 -27 -11l-4 -3q-4 -4 -11.5 -10t-17.5 -13.5t-23.5 -14.5t-28.5 -13t-33.5 -9.5t-37.5 -3.5q-76 0 -125 50t-49 127q0 76 48 125.5t122 49.5q37 0 71.5 -14t50.5 -28l16 -14q11 -11 26 -10q16 2 24 14l53 78 + q13 20 -2 39q-3 4 -11 12t-30 23.5t-48.5 28t-67.5 22.5t-86 10q-148 0 -246 -96.5t-98 -240.5q0 -146 97 -241.5t247 -95.5zM1235 303q153 0 257 104q14 18 4 36l-45 82q-8 14 -25 17q-16 2 -27 -11l-4 -3q-4 -4 -11.5 -10t-17.5 -13.5t-23.5 -14.5t-28.5 -13t-33.5 -9.5 + t-37.5 -3.5q-76 0 -125 50t-49 127q0 76 48 125.5t122 49.5q37 0 71.5 -14t50.5 -28l16 -14q11 -11 26 -10q16 2 24 14l53 78q13 20 -2 39q-3 4 -11 12t-30 23.5t-48.5 28t-67.5 22.5t-86 10q-147 0 -245.5 -96.5t-98.5 -240.5q0 -146 97 -241.5t247 -95.5zM896 1376 + q-150 0 -286 -58.5t-234.5 -157t-157 -234.5t-58.5 -286t58.5 -286t157 -234.5t234.5 -157t286 -58.5t286 58.5t234.5 157t157 234.5t58.5 286t-58.5 286t-157 234.5t-234.5 157t-286 58.5zM896 1536q182 0 348 -71t286 -191t191 -286t71 -348t-71 -348t-191 -286t-286 -191 + t-348 -71t-348 71t-286 191t-191 286t-71 348t71 348t191 286t286 191t348 71z" /> + <glyph glyph-name="f260" unicode="" horiz-adv-x="2048" + d="M736 736l384 -384l-384 -384l-672 672l672 672l168 -168l-96 -96l-72 72l-480 -480l480 -480l193 193l-289 287zM1312 1312l672 -672l-672 -672l-168 168l96 96l72 -72l480 480l-480 480l-193 -193l289 -287l-96 -96l-384 384z" /> + <glyph glyph-name="f261" unicode="" horiz-adv-x="1792" + d="M717 182l271 271l-279 279l-88 -88l192 -191l-96 -96l-279 279l279 279l40 -40l87 87l-127 128l-454 -454zM1075 190l454 454l-454 454l-271 -271l279 -279l88 88l-192 191l96 96l279 -279l-279 -279l-40 40l-87 -88zM1792 640q0 -182 -71 -348t-191 -286t-286 -191 + t-348 -71t-348 71t-286 191t-191 286t-71 348t71 348t191 286t286 191t348 71t348 -71t286 -191t191 -286t71 -348z" /> + <glyph glyph-name="_572" unicode="" horiz-adv-x="2304" + d="M651 539q0 -39 -27.5 -66.5t-65.5 -27.5q-39 0 -66.5 27.5t-27.5 66.5q0 38 27.5 65.5t66.5 27.5q38 0 65.5 -27.5t27.5 -65.5zM1805 540q0 -39 -27.5 -66.5t-66.5 -27.5t-66.5 27.5t-27.5 66.5t27.5 66t66.5 27t66.5 -27t27.5 -66zM765 539q0 79 -56.5 136t-136.5 57 + t-136.5 -56.5t-56.5 -136.5t56.5 -136.5t136.5 -56.5t136.5 56.5t56.5 136.5zM1918 540q0 80 -56.5 136.5t-136.5 56.5q-79 0 -136 -56.5t-57 -136.5t56.5 -136.5t136.5 -56.5t136.5 56.5t56.5 136.5zM850 539q0 -116 -81.5 -197.5t-196.5 -81.5q-116 0 -197.5 82t-81.5 197 + t82 196.5t197 81.5t196.5 -81.5t81.5 -196.5zM2004 540q0 -115 -81.5 -196.5t-197.5 -81.5q-115 0 -196.5 81.5t-81.5 196.5t81.5 196.5t196.5 81.5q116 0 197.5 -81.5t81.5 -196.5zM1040 537q0 191 -135.5 326.5t-326.5 135.5q-125 0 -231 -62t-168 -168.5t-62 -231.5 + t62 -231.5t168 -168.5t231 -62q191 0 326.5 135.5t135.5 326.5zM1708 1110q-254 111 -556 111q-319 0 -573 -110q117 0 223 -45.5t182.5 -122.5t122 -183t45.5 -223q0 115 43.5 219.5t118 180.5t177.5 123t217 50zM2187 537q0 191 -135 326.5t-326 135.5t-326.5 -135.5 + t-135.5 -326.5t135.5 -326.5t326.5 -135.5t326 135.5t135 326.5zM1921 1103h383q-44 -51 -75 -114.5t-40 -114.5q110 -151 110 -337q0 -156 -77 -288t-209 -208.5t-287 -76.5q-133 0 -249 56t-196 155q-47 -56 -129 -179q-11 22 -53.5 82.5t-74.5 97.5 + q-80 -99 -196.5 -155.5t-249.5 -56.5q-155 0 -287 76.5t-209 208.5t-77 288q0 186 110 337q-9 51 -40 114.5t-75 114.5h365q149 100 355 156.5t432 56.5q224 0 421 -56t348 -157z" /> + <glyph glyph-name="f263" unicode="" horiz-adv-x="1280" + d="M640 629q-188 0 -321 133t-133 320q0 188 133 321t321 133t321 -133t133 -321q0 -187 -133 -320t-321 -133zM640 1306q-92 0 -157.5 -65.5t-65.5 -158.5q0 -92 65.5 -157.5t157.5 -65.5t157.5 65.5t65.5 157.5q0 93 -65.5 158.5t-157.5 65.5zM1163 574q13 -27 15 -49.5 + t-4.5 -40.5t-26.5 -38.5t-42.5 -37t-61.5 -41.5q-115 -73 -315 -94l73 -72l267 -267q30 -31 30 -74t-30 -73l-12 -13q-31 -30 -74 -30t-74 30q-67 68 -267 268l-267 -268q-31 -30 -74 -30t-73 30l-12 13q-31 30 -31 73t31 74l267 267l72 72q-203 21 -317 94 + q-39 25 -61.5 41.5t-42.5 37t-26.5 38.5t-4.5 40.5t15 49.5q10 20 28 35t42 22t56 -2t65 -35q5 -4 15 -11t43 -24.5t69 -30.5t92 -24t113 -11q91 0 174 25.5t120 50.5l38 25q33 26 65 35t56 2t42 -22t28 -35z" /> + <glyph glyph-name="_574" unicode="" + d="M927 956q0 -66 -46.5 -112.5t-112.5 -46.5t-112.5 46.5t-46.5 112.5t46.5 112.5t112.5 46.5t112.5 -46.5t46.5 -112.5zM1141 593q-10 20 -28 32t-47.5 9.5t-60.5 -27.5q-10 -8 -29 -20t-81 -32t-127 -20t-124 18t-86 36l-27 18q-31 25 -60.5 27.5t-47.5 -9.5t-28 -32 + q-22 -45 -2 -74.5t87 -73.5q83 -53 226 -67l-51 -52q-142 -142 -191 -190q-22 -22 -22 -52.5t22 -52.5l9 -9q22 -22 52.5 -22t52.5 22l191 191q114 -115 191 -191q22 -22 52.5 -22t52.5 22l9 9q22 22 22 52.5t-22 52.5l-191 190l-52 52q141 14 225 67q67 44 87 73.5t-2 74.5 + zM1092 956q0 134 -95 229t-229 95t-229 -95t-95 -229t95 -229t229 -95t229 95t95 229zM1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="_575" unicode="" horiz-adv-x="1720" + d="M1565 1408q65 0 110 -45.5t45 -110.5v-519q0 -176 -68 -336t-182.5 -275t-274 -182.5t-334.5 -67.5q-176 0 -335.5 67.5t-274.5 182.5t-183 275t-68 336v519q0 64 46 110t110 46h1409zM861 344q47 0 82 33l404 388q37 35 37 85q0 49 -34.5 83.5t-83.5 34.5q-47 0 -82 -33 + l-323 -310l-323 310q-35 33 -81 33q-49 0 -83.5 -34.5t-34.5 -83.5q0 -51 36 -85l405 -388q33 -33 81 -33z" /> + <glyph glyph-name="_576" unicode="" horiz-adv-x="2304" + d="M1494 -103l-295 695q-25 -49 -158.5 -305.5t-198.5 -389.5q-1 -1 -27.5 -0.5t-26.5 1.5q-82 193 -255.5 587t-259.5 596q-21 50 -66.5 107.5t-103.5 100.5t-102 43q0 5 -0.5 24t-0.5 27h583v-50q-39 -2 -79.5 -16t-66.5 -43t-10 -64q26 -59 216.5 -499t235.5 -540 + q31 61 140 266.5t131 247.5q-19 39 -126 281t-136 295q-38 69 -201 71v50l513 -1v-47q-60 -2 -93.5 -25t-12.5 -69q33 -70 87 -189.5t86 -187.5q110 214 173 363q24 55 -10 79.5t-129 26.5q1 7 1 25v24q64 0 170.5 0.5t180 1t92.5 0.5v-49q-62 -2 -119 -33t-90 -81 + l-213 -442q13 -33 127.5 -290t121.5 -274l441 1017q-14 38 -49.5 62.5t-65 31.5t-55.5 8v50l460 -4l1 -2l-1 -44q-139 -4 -201 -145q-526 -1216 -559 -1291h-49z" /> + <glyph glyph-name="_577" unicode="" horiz-adv-x="1792" + d="M949 643q0 -26 -16.5 -45t-41.5 -19q-26 0 -45 16.5t-19 41.5q0 26 17 45t42 19t44 -16.5t19 -41.5zM964 585l350 581q-9 -8 -67.5 -62.5t-125.5 -116.5t-136.5 -127t-117 -110.5t-50.5 -51.5l-349 -580q7 7 67 62t126 116.5t136 127t117 111t50 50.5zM1611 640 + q0 -201 -104 -371q-3 2 -17 11t-26.5 16.5t-16.5 7.5q-13 0 -13 -13q0 -10 59 -44q-74 -112 -184.5 -190.5t-241.5 -110.5l-16 67q-1 10 -15 10q-5 0 -8 -5.5t-2 -9.5l16 -68q-72 -15 -146 -15q-199 0 -372 105q1 2 13 20.5t21.5 33.5t9.5 19q0 13 -13 13q-6 0 -17 -14.5 + t-22.5 -34.5t-13.5 -23q-113 75 -192 187.5t-110 244.5l69 15q10 3 10 15q0 5 -5.5 8t-10.5 2l-68 -15q-14 72 -14 139q0 206 109 379q2 -1 18.5 -12t30 -19t17.5 -8q13 0 13 12q0 6 -12.5 15.5t-32.5 21.5l-20 12q77 112 189 189t244 107l15 -67q2 -10 15 -10q5 0 8 5.5 + t2 10.5l-15 66q71 13 134 13q204 0 379 -109q-39 -56 -39 -65q0 -13 12 -13q11 0 48 64q111 -75 187.5 -186t107.5 -241l-56 -12q-10 -2 -10 -16q0 -5 5.5 -8t9.5 -2l57 13q14 -72 14 -140zM1696 640q0 163 -63.5 311t-170.5 255t-255 170.5t-311 63.5t-311 -63.5 + t-255 -170.5t-170.5 -255t-63.5 -311t63.5 -311t170.5 -255t255 -170.5t311 -63.5t311 63.5t255 170.5t170.5 255t63.5 311zM1792 640q0 -182 -71 -348t-191 -286t-286 -191t-348 -71t-348 71t-286 191t-191 286t-71 348t71 348t191 286t286 191t348 71t348 -71t286 -191 + t191 -286t71 -348z" /> + <glyph glyph-name="_578" unicode="" horiz-adv-x="1792" + d="M893 1536q240 2 451 -120q232 -134 352 -372l-742 39q-160 9 -294 -74.5t-185 -229.5l-276 424q128 159 311 245.5t383 87.5zM146 1131l337 -663q72 -143 211 -217t293 -45l-230 -451q-212 33 -385 157.5t-272.5 316t-99.5 411.5q0 267 146 491zM1732 962 + q58 -150 59.5 -310.5t-48.5 -306t-153 -272t-246 -209.5q-230 -133 -498 -119l405 623q88 131 82.5 290.5t-106.5 277.5zM896 942q125 0 213.5 -88.5t88.5 -213.5t-88.5 -213.5t-213.5 -88.5t-213.5 88.5t-88.5 213.5t88.5 213.5t213.5 88.5z" /> + <glyph glyph-name="_579" unicode="" horiz-adv-x="1792" + d="M903 -256q-283 0 -504.5 150.5t-329.5 398.5q-58 131 -67 301t26 332.5t111 312t179 242.5l-11 -281q11 14 68 15.5t70 -15.5q42 81 160.5 138t234.5 59q-54 -45 -119.5 -148.5t-58.5 -163.5q25 -8 62.5 -13.5t63 -7.5t68 -4t50.5 -3q15 -5 9.5 -45.5t-30.5 -75.5 + q-5 -7 -16.5 -18.5t-56.5 -35.5t-101 -34l15 -189l-139 67q-18 -43 -7.5 -81.5t36 -66.5t65.5 -41.5t81 -6.5q51 9 98 34.5t83.5 45t73.5 17.5q61 -4 89.5 -33t19.5 -65q-1 -2 -2.5 -5.5t-8.5 -12.5t-18 -15.5t-31.5 -10.5t-46.5 -1q-60 -95 -144.5 -135.5t-209.5 -29.5 + q74 -61 162.5 -82.5t168.5 -6t154.5 52t128 87.5t80.5 104q43 91 39 192.5t-37.5 188.5t-78.5 125q87 -38 137 -79.5t77 -112.5q15 170 -57.5 343t-209.5 284q265 -77 412 -279.5t151 -517.5q2 -127 -40.5 -255t-123.5 -238t-189 -196t-247.5 -135.5t-288.5 -49.5z" /> + <glyph glyph-name="_580" unicode="" horiz-adv-x="1792" + d="M1493 1308q-165 110 -359 110q-155 0 -293 -73t-240 -200q-75 -93 -119.5 -218t-48.5 -266v-42q4 -141 48.5 -266t119.5 -218q102 -127 240 -200t293 -73q194 0 359 110q-121 -108 -274.5 -168t-322.5 -60q-29 0 -43 1q-175 8 -333 82t-272 193t-181 281t-67 339 + q0 182 71 348t191 286t286 191t348 71h3q168 -1 320.5 -60.5t273.5 -167.5zM1792 640q0 -192 -77 -362.5t-213 -296.5q-104 -63 -222 -63q-137 0 -255 84q154 56 253.5 233t99.5 405q0 227 -99 404t-253 234q119 83 254 83q119 0 226 -65q135 -125 210.5 -295t75.5 -361z + " /> + <glyph glyph-name="_581" unicode="" horiz-adv-x="1792" + d="M1792 599q0 -56 -7 -104h-1151q0 -146 109.5 -244.5t257.5 -98.5q99 0 185.5 46.5t136.5 130.5h423q-56 -159 -170.5 -281t-267.5 -188.5t-321 -66.5q-187 0 -356 83q-228 -116 -394 -116q-237 0 -237 263q0 115 45 275q17 60 109 229q199 360 475 606 + q-184 -79 -427 -354q63 274 283.5 449.5t501.5 175.5q30 0 45 -1q255 117 433 117q64 0 116 -13t94.5 -40.5t66.5 -76.5t24 -115q0 -116 -75 -286q101 -182 101 -390zM1722 1239q0 83 -53 132t-137 49q-108 0 -254 -70q121 -47 222.5 -131.5t170.5 -195.5q51 135 51 216z + M128 2q0 -86 48.5 -132.5t134.5 -46.5q115 0 266 83q-122 72 -213.5 183t-137.5 245q-98 -205 -98 -332zM632 715h728q-5 142 -113 237t-251 95q-144 0 -251.5 -95t-112.5 -237z" /> + <glyph glyph-name="_582" unicode="" horiz-adv-x="2048" + d="M1792 288v960q0 13 -9.5 22.5t-22.5 9.5h-1600q-13 0 -22.5 -9.5t-9.5 -22.5v-960q0 -13 9.5 -22.5t22.5 -9.5h1600q13 0 22.5 9.5t9.5 22.5zM1920 1248v-960q0 -66 -47 -113t-113 -47h-736v-128h352q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-832q-14 0 -23 9t-9 23 + v64q0 14 9 23t23 9h352v128h-736q-66 0 -113 47t-47 113v960q0 66 47 113t113 47h1600q66 0 113 -47t47 -113z" /> + <glyph glyph-name="_583" unicode="" horiz-adv-x="1792" + d="M138 1408h197q-70 -64 -126 -149q-36 -56 -59 -115t-30 -125.5t-8.5 -120t10.5 -132t21 -126t28 -136.5q4 -19 6 -28q51 -238 81 -329q57 -171 152 -275h-272q-48 0 -82 34t-34 82v1304q0 48 34 82t82 34zM1346 1408h308q48 0 82 -34t34 -82v-1304q0 -48 -34 -82t-82 -34 + h-178q212 210 196 565l-469 -101q-2 -45 -12 -82t-31 -72t-59.5 -59.5t-93.5 -36.5q-123 -26 -199 40q-32 27 -53 61t-51.5 129t-64.5 258q-35 163 -45.5 263t-5.5 139t23 77q20 41 62.5 73t102.5 45q45 12 83.5 6.5t67 -17t54 -35t43 -48t34.5 -56.5l468 100 + q-68 175 -180 287z" /> + <glyph glyph-name="_584" unicode="" + d="M1401 -11l-6 -6q-113 -113 -259 -175q-154 -64 -317 -64q-165 0 -317 64q-148 63 -259 175q-113 112 -175 258q-42 103 -54 189q-4 28 48 36q51 8 56 -20q1 -1 1 -4q18 -90 46 -159q50 -124 152 -226q98 -98 226 -152q132 -56 276 -56q143 0 276 56q128 55 225 152l6 6 + q10 10 25 6q12 -3 33 -22q36 -37 17 -58zM929 604l-66 -66l63 -63q21 -21 -7 -49q-17 -17 -32 -17q-10 0 -19 10l-62 61l-66 -66q-5 -5 -15 -5q-15 0 -31 16l-2 2q-18 15 -18 29q0 7 8 17l66 65l-66 66q-16 16 14 45q18 18 31 18q6 0 13 -5l65 -66l65 65q18 17 48 -13 + q27 -27 11 -44zM1400 547q0 -118 -46 -228q-45 -105 -126 -186q-80 -80 -187 -126t-228 -46t-228 46t-187 126q-82 82 -125 186q-15 33 -15 40h-1q-9 27 43 44q50 16 60 -12q37 -99 97 -167h1v339v2q3 136 102 232q105 103 253 103q147 0 251 -103t104 -249 + q0 -147 -104.5 -251t-250.5 -104q-58 0 -112 16q-28 11 -13 61q16 51 44 43l14 -3q14 -3 33 -6t30 -3q104 0 176 71.5t72 174.5q0 101 -72 171q-71 71 -175 71q-107 0 -178 -80q-64 -72 -64 -160v-413q110 -67 242 -67q96 0 185 36.5t156 103.5t103.5 155t36.5 183 + q0 198 -141 339q-140 140 -339 140q-200 0 -340 -140q-53 -53 -77 -87l-2 -2q-8 -11 -13 -15.5t-21.5 -9.5t-38.5 3q-21 5 -36.5 16.5t-15.5 26.5v680q0 15 10.5 26.5t27.5 11.5h877q30 0 30 -55t-30 -55h-811v-483h1q40 42 102 84t108 61q109 46 231 46q121 0 228 -46 + t187 -126q81 -81 126 -186q46 -112 46 -229zM1369 1128q9 -8 9 -18t-5.5 -18t-16.5 -21q-26 -26 -39 -26q-9 0 -16 7q-106 91 -207 133q-128 56 -276 56q-133 0 -262 -49q-27 -10 -45 37q-9 25 -8 38q3 16 16 20q130 57 299 57q164 0 316 -64q137 -58 235 -152z" /> + <glyph glyph-name="_585" unicode="" horiz-adv-x="1792" + d="M1551 60q15 6 26 3t11 -17.5t-15 -33.5q-13 -16 -44 -43.5t-95.5 -68t-141 -74t-188 -58t-229.5 -24.5q-119 0 -238 31t-209 76.5t-172.5 104t-132.5 105t-84 87.5q-8 9 -10 16.5t1 12t8 7t11.5 2t11.5 -4.5q192 -117 300 -166q389 -176 799 -90q190 40 391 135z + M1758 175q11 -16 2.5 -69.5t-28.5 -102.5q-34 -83 -85 -124q-17 -14 -26 -9t0 24q21 45 44.5 121.5t6.5 98.5q-5 7 -15.5 11.5t-27 6t-29.5 2.5t-35 0t-31.5 -2t-31 -3t-22.5 -2q-6 -1 -13 -1.5t-11 -1t-8.5 -1t-7 -0.5h-5.5h-4.5t-3 0.5t-2 1.5l-1.5 3q-6 16 47 40t103 30 + q46 7 108 1t76 -24zM1364 618q0 -31 13.5 -64t32 -58t37.5 -46t33 -32l13 -11l-227 -224q-40 37 -79 75.5t-58 58.5l-19 20q-11 11 -25 33q-38 -59 -97.5 -102.5t-127.5 -63.5t-140 -23t-137.5 21t-117.5 65.5t-83 113t-31 162.5q0 84 28 154t72 116.5t106.5 83t122.5 57 + t130 34.5t119.5 18.5t99.5 6.5v127q0 65 -21 97q-34 53 -121 53q-6 0 -16.5 -1t-40.5 -12t-56 -29.5t-56 -59.5t-48 -96l-294 27q0 60 22 119t67 113t108 95t151.5 65.5t190.5 24.5q100 0 181 -25t129.5 -61.5t81 -83t45 -86t12.5 -73.5v-589zM692 597q0 -86 70 -133 + q66 -44 139 -22q84 25 114 123q14 45 14 101v162q-59 -2 -111 -12t-106.5 -33.5t-87 -71t-32.5 -114.5z" /> + <glyph glyph-name="_586" unicode="" horiz-adv-x="1792" + d="M1536 1280q52 0 90 -38t38 -90v-1280q0 -52 -38 -90t-90 -38h-1408q-52 0 -90 38t-38 90v1280q0 52 38 90t90 38h128v96q0 66 47 113t113 47h64q66 0 113 -47t47 -113v-96h384v96q0 66 47 113t113 47h64q66 0 113 -47t47 -113v-96h128zM1152 1376v-288q0 -14 9 -23t23 -9 + h64q14 0 23 9t9 23v288q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23zM384 1376v-288q0 -14 9 -23t23 -9h64q14 0 23 9t9 23v288q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23zM1536 -128v1024h-1408v-1024h1408zM896 448h224q14 0 23 -9t9 -23v-64q0 -14 -9 -23t-23 -9h-224 + v-224q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9t-9 23v224h-224q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h224v224q0 14 9 23t23 9h64q14 0 23 -9t9 -23v-224z" /> + <glyph glyph-name="_587" unicode="" horiz-adv-x="1792" + d="M1152 416v-64q0 -14 -9 -23t-23 -9h-576q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h576q14 0 23 -9t9 -23zM128 -128h1408v1024h-1408v-1024zM512 1088v288q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-288q0 -14 9 -23t23 -9h64q14 0 23 9t9 23zM1280 1088v288q0 14 -9 23 + t-23 9h-64q-14 0 -23 -9t-9 -23v-288q0 -14 9 -23t23 -9h64q14 0 23 9t9 23zM1664 1152v-1280q0 -52 -38 -90t-90 -38h-1408q-52 0 -90 38t-38 90v1280q0 52 38 90t90 38h128v96q0 66 47 113t113 47h64q66 0 113 -47t47 -113v-96h384v96q0 66 47 113t113 47h64q66 0 113 -47 + t47 -113v-96h128q52 0 90 -38t38 -90z" /> + <glyph glyph-name="_588" unicode="" horiz-adv-x="1792" + d="M1111 151l-46 -46q-9 -9 -22 -9t-23 9l-188 189l-188 -189q-10 -9 -23 -9t-22 9l-46 46q-9 9 -9 22t9 23l189 188l-189 188q-9 10 -9 23t9 22l46 46q9 9 22 9t23 -9l188 -188l188 188q10 9 23 9t22 -9l46 -46q9 -9 9 -22t-9 -23l-188 -188l188 -188q9 -10 9 -23t-9 -22z + M128 -128h1408v1024h-1408v-1024zM512 1088v288q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-288q0 -14 9 -23t23 -9h64q14 0 23 9t9 23zM1280 1088v288q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-288q0 -14 9 -23t23 -9h64q14 0 23 9t9 23zM1664 1152v-1280 + q0 -52 -38 -90t-90 -38h-1408q-52 0 -90 38t-38 90v1280q0 52 38 90t90 38h128v96q0 66 47 113t113 47h64q66 0 113 -47t47 -113v-96h384v96q0 66 47 113t113 47h64q66 0 113 -47t47 -113v-96h128q52 0 90 -38t38 -90z" /> + <glyph glyph-name="_589" unicode="" horiz-adv-x="1792" + d="M1303 572l-512 -512q-10 -9 -23 -9t-23 9l-288 288q-9 10 -9 23t9 22l46 46q9 9 22 9t23 -9l220 -220l444 444q10 9 23 9t22 -9l46 -46q9 -9 9 -22t-9 -23zM128 -128h1408v1024h-1408v-1024zM512 1088v288q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-288q0 -14 9 -23 + t23 -9h64q14 0 23 9t9 23zM1280 1088v288q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-288q0 -14 9 -23t23 -9h64q14 0 23 9t9 23zM1664 1152v-1280q0 -52 -38 -90t-90 -38h-1408q-52 0 -90 38t-38 90v1280q0 52 38 90t90 38h128v96q0 66 47 113t113 47h64q66 0 113 -47 + t47 -113v-96h384v96q0 66 47 113t113 47h64q66 0 113 -47t47 -113v-96h128q52 0 90 -38t38 -90z" /> + <glyph glyph-name="_590" unicode="" horiz-adv-x="1792" + d="M448 1536q26 0 45 -19t19 -45v-891l536 429q17 14 40 14q26 0 45 -19t19 -45v-379l536 429q17 14 40 14q26 0 45 -19t19 -45v-1152q0 -26 -19 -45t-45 -19h-1664q-26 0 -45 19t-19 45v1664q0 26 19 45t45 19h384z" /> + <glyph glyph-name="_591" unicode="" horiz-adv-x="1024" + d="M512 448q66 0 128 15v-655q0 -26 -19 -45t-45 -19h-128q-26 0 -45 19t-19 45v655q62 -15 128 -15zM512 1536q212 0 362 -150t150 -362t-150 -362t-362 -150t-362 150t-150 362t150 362t362 150zM512 1312q14 0 23 9t9 23t-9 23t-23 9q-146 0 -249 -103t-103 -249 + q0 -14 9 -23t23 -9t23 9t9 23q0 119 84.5 203.5t203.5 84.5z" /> + <glyph glyph-name="_592" unicode="" horiz-adv-x="1792" + d="M1745 1239q10 -10 10 -23t-10 -23l-141 -141q-28 -28 -68 -28h-1344q-26 0 -45 19t-19 45v256q0 26 19 45t45 19h576v64q0 26 19 45t45 19h128q26 0 45 -19t19 -45v-64h512q40 0 68 -28zM768 320h256v-512q0 -26 -19 -45t-45 -19h-128q-26 0 -45 19t-19 45v512zM1600 768 + q26 0 45 -19t19 -45v-256q0 -26 -19 -45t-45 -19h-1344q-40 0 -68 28l-141 141q-10 10 -10 23t10 23l141 141q28 28 68 28h512v192h256v-192h576z" /> + <glyph glyph-name="_593" unicode="" horiz-adv-x="2048" + d="M2020 1525q28 -20 28 -53v-1408q0 -20 -11 -36t-29 -23l-640 -256q-24 -11 -48 0l-616 246l-616 -246q-10 -5 -24 -5q-19 0 -36 11q-28 20 -28 53v1408q0 20 11 36t29 23l640 256q24 11 48 0l616 -246l616 246q32 13 60 -6zM736 1390v-1270l576 -230v1270zM128 1173 + v-1270l544 217v1270zM1920 107v1270l-544 -217v-1270z" /> + <glyph glyph-name="_594" unicode="" horiz-adv-x="1792" + d="M512 1536q13 0 22.5 -9.5t9.5 -22.5v-1472q0 -20 -17 -28l-480 -256q-7 -4 -15 -4q-13 0 -22.5 9.5t-9.5 22.5v1472q0 20 17 28l480 256q7 4 15 4zM1760 1536q13 0 22.5 -9.5t9.5 -22.5v-1472q0 -20 -17 -28l-480 -256q-7 -4 -15 -4q-13 0 -22.5 9.5t-9.5 22.5v1472 + q0 20 17 28l480 256q7 4 15 4zM640 1536q8 0 14 -3l512 -256q18 -10 18 -29v-1472q0 -13 -9.5 -22.5t-22.5 -9.5q-8 0 -14 3l-512 256q-18 10 -18 29v1472q0 13 9.5 22.5t22.5 9.5z" /> + <glyph glyph-name="_595" unicode="" horiz-adv-x="1792" + d="M640 640q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM1024 640q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM1408 640q0 53 -37.5 90.5t-90.5 37.5 + t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5zM1792 640q0 -174 -120 -321.5t-326 -233t-450 -85.5q-110 0 -211 18q-173 -173 -435 -229q-52 -10 -86 -13q-12 -1 -22 6t-13 18q-4 15 20 37q5 5 23.5 21.5t25.5 23.5t23.5 25.5t24 31.5t20.5 37 + t20 48t14.5 57.5t12.5 72.5q-146 90 -229.5 216.5t-83.5 269.5q0 174 120 321.5t326 233t450 85.5t450 -85.5t326 -233t120 -321.5z" /> + <glyph glyph-name="_596" unicode="" horiz-adv-x="1792" + d="M640 640q0 -53 -37.5 -90.5t-90.5 -37.5t-90.5 37.5t-37.5 90.5t37.5 90.5t90.5 37.5t90.5 -37.5t37.5 -90.5zM1024 640q0 -53 -37.5 -90.5t-90.5 -37.5t-90.5 37.5t-37.5 90.5t37.5 90.5t90.5 37.5t90.5 -37.5t37.5 -90.5zM1408 640q0 -53 -37.5 -90.5t-90.5 -37.5 + t-90.5 37.5t-37.5 90.5t37.5 90.5t90.5 37.5t90.5 -37.5t37.5 -90.5zM896 1152q-204 0 -381.5 -69.5t-282 -187.5t-104.5 -255q0 -112 71.5 -213.5t201.5 -175.5l87 -50l-27 -96q-24 -91 -70 -172q152 63 275 171l43 38l57 -6q69 -8 130 -8q204 0 381.5 69.5t282 187.5 + t104.5 255t-104.5 255t-282 187.5t-381.5 69.5zM1792 640q0 -174 -120 -321.5t-326 -233t-450 -85.5q-70 0 -145 8q-198 -175 -460 -242q-49 -14 -114 -22h-5q-15 0 -27 10.5t-16 27.5v1q-3 4 -0.5 12t2 10t4.5 9.5l6 9t7 8.5t8 9q7 8 31 34.5t34.5 38t31 39.5t32.5 51 + t27 59t26 76q-157 89 -247.5 220t-90.5 281q0 130 71 248.5t191 204.5t286 136.5t348 50.5t348 -50.5t286 -136.5t191 -204.5t71 -248.5z" /> + <glyph glyph-name="_597" unicode="" horiz-adv-x="1024" + d="M512 345l512 295v-591l-512 -296v592zM0 640v-591l512 296zM512 1527v-591l-512 -296v591zM512 936l512 295v-591z" /> + <glyph glyph-name="_598" unicode="" horiz-adv-x="1792" + d="M1709 1018q-10 -236 -332 -651q-333 -431 -562 -431q-142 0 -240 263q-44 160 -132 482q-72 262 -157 262q-18 0 -127 -76l-77 98q24 21 108 96.5t130 115.5q156 138 241 146q95 9 153 -55.5t81 -203.5q44 -287 66 -373q55 -249 120 -249q51 0 154 161q101 161 109 246 + q13 139 -109 139q-57 0 -121 -26q120 393 459 382q251 -8 236 -326z" /> + <glyph glyph-name="f27e" unicode="" + d="M0 1408h1536v-1536h-1536v1536zM1085 293l-221 631l221 297h-634l221 -297l-221 -631l317 -304z" /> + <glyph glyph-name="uniF280" unicode="" + d="M0 1408h1536v-1536h-1536v1536zM908 1088l-12 -33l75 -83l-31 -114l25 -25l107 57l107 -57l25 25l-31 114l75 83l-12 33h-95l-53 96h-32l-53 -96h-95zM641 925q32 0 44.5 -16t11.5 -63l174 21q0 55 -17.5 92.5t-50.5 56t-69 25.5t-85 7q-133 0 -199 -57.5t-66 -182.5v-72 + h-96v-128h76q20 0 20 -8v-382q0 -14 -5 -20t-18 -7l-73 -7v-88h448v86l-149 14q-6 1 -8.5 1.5t-3.5 2.5t-0.5 4t1 7t0.5 10v387h191l38 128h-231q-6 0 -2 6t4 9v80q0 27 1.5 40.5t7.5 28t19.5 20t36.5 5.5zM1248 96v86l-54 9q-7 1 -9.5 2.5t-2.5 3t1 7.5t1 12v520h-275 + l-23 -101l83 -22q23 -7 23 -27v-370q0 -14 -6 -18.5t-20 -6.5l-70 -9v-86h352z" /> + <glyph glyph-name="uniF281" unicode="" horiz-adv-x="1792" + d="M1792 690q0 -58 -29.5 -105.5t-79.5 -72.5q12 -46 12 -96q0 -155 -106.5 -287t-290.5 -208.5t-400 -76.5t-399.5 76.5t-290 208.5t-106.5 287q0 47 11 94q-51 25 -82 73.5t-31 106.5q0 82 58 140.5t141 58.5q85 0 145 -63q218 152 515 162l116 521q3 13 15 21t26 5 + l369 -81q18 37 54 59.5t79 22.5q62 0 106 -43.5t44 -105.5t-44 -106t-106 -44t-105.5 43.5t-43.5 105.5l-334 74l-104 -472q300 -9 519 -160q58 61 143 61q83 0 141 -58.5t58 -140.5zM418 491q0 -62 43.5 -106t105.5 -44t106 44t44 106t-44 105.5t-106 43.5q-61 0 -105 -44 + t-44 -105zM1228 136q11 11 11 26t-11 26q-10 10 -25 10t-26 -10q-41 -42 -121 -62t-160 -20t-160 20t-121 62q-11 10 -26 10t-25 -10q-11 -10 -11 -25.5t11 -26.5q43 -43 118.5 -68t122.5 -29.5t91 -4.5t91 4.5t122.5 29.5t118.5 68zM1225 341q62 0 105.5 44t43.5 106 + q0 61 -44 105t-105 44q-62 0 -106 -43.5t-44 -105.5t44 -106t106 -44z" /> + <glyph glyph-name="_602" unicode="" horiz-adv-x="1792" + d="M69 741h1q16 126 58.5 241.5t115 217t167.5 176t223.5 117.5t276.5 43q231 0 414 -105.5t294 -303.5q104 -187 104 -442v-188h-1125q1 -111 53.5 -192.5t136.5 -122.5t189.5 -57t213 -3t208 46.5t173.5 84.5v-377q-92 -55 -229.5 -92t-312.5 -38t-316 53 + q-189 73 -311.5 249t-124.5 372q-3 242 111 412t325 268q-48 -60 -78 -125.5t-46 -159.5h635q8 77 -8 140t-47 101.5t-70.5 66.5t-80.5 41t-75 20.5t-56 8.5l-22 1q-135 -5 -259.5 -44.5t-223.5 -104.5t-176 -140.5t-138 -163.5z" /> + <glyph glyph-name="_603" unicode="" horiz-adv-x="2304" + d="M0 32v608h2304v-608q0 -66 -47 -113t-113 -47h-1984q-66 0 -113 47t-47 113zM640 256v-128h384v128h-384zM256 256v-128h256v128h-256zM2144 1408q66 0 113 -47t47 -113v-224h-2304v224q0 66 47 113t113 47h1984z" /> + <glyph glyph-name="_604" unicode="" horiz-adv-x="1792" + d="M1584 246l-218 111q-74 -120 -196.5 -189t-263.5 -69q-147 0 -271 72t-196 196t-72 270q0 110 42.5 209.5t115 172t172 115t209.5 42.5q131 0 247.5 -60.5t192.5 -168.5l215 125q-110 169 -286.5 265t-378.5 96q-161 0 -308 -63t-253 -169t-169 -253t-63 -308t63 -308 + t169 -253t253 -169t308 -63q213 0 397.5 107t290.5 292zM1030 643l693 -352q-116 -253 -334.5 -400t-492.5 -147q-182 0 -348 71t-286 191t-191 286t-71 348t71 348t191 286t286 191t348 71q260 0 470.5 -133.5t335.5 -366.5zM1543 640h-39v-160h-96v352h136q32 0 54.5 -20 + t28.5 -48t1 -56t-27.5 -48t-57.5 -20z" /> + <glyph glyph-name="uniF285" unicode="" horiz-adv-x="1792" + d="M1427 827l-614 386l92 151h855zM405 562l-184 116v858l1183 -743zM1424 697l147 -95v-858l-532 335zM1387 718l-500 -802h-855l356 571z" /> + <glyph glyph-name="uniF286" unicode="" horiz-adv-x="1792" + d="M640 528v224q0 16 -16 16h-96q-16 0 -16 -16v-224q0 -16 16 -16h96q16 0 16 16zM1152 528v224q0 16 -16 16h-96q-16 0 -16 -16v-224q0 -16 16 -16h96q16 0 16 16zM1664 496v-752h-640v320q0 80 -56 136t-136 56t-136 -56t-56 -136v-320h-640v752q0 16 16 16h96 + q16 0 16 -16v-112h128v624q0 16 16 16h96q16 0 16 -16v-112h128v112q0 16 16 16h96q16 0 16 -16v-112h128v112q0 6 2.5 9.5t8.5 5t9.5 2t11.5 0t9 -0.5v391q-32 15 -32 50q0 23 16.5 39t38.5 16t38.5 -16t16.5 -39q0 -35 -32 -50v-17q45 10 83 10q21 0 59.5 -7.5t54.5 -7.5 + q17 0 47 7.5t37 7.5q16 0 16 -16v-210q0 -15 -35 -21.5t-62 -6.5q-18 0 -54.5 7.5t-55.5 7.5q-40 0 -90 -12v-133q1 0 9 0.5t11.5 0t9.5 -2t8.5 -5t2.5 -9.5v-112h128v112q0 16 16 16h96q16 0 16 -16v-112h128v112q0 16 16 16h96q16 0 16 -16v-624h128v112q0 16 16 16h96 + q16 0 16 -16z" /> + <glyph glyph-name="_607" unicode="" horiz-adv-x="2304" + d="M2288 731q16 -8 16 -27t-16 -27l-320 -192q-8 -5 -16 -5q-9 0 -16 4q-16 10 -16 28v128h-858q37 -58 83 -165q16 -37 24.5 -55t24 -49t27 -47t27 -34t31.5 -26t33 -8h96v96q0 14 9 23t23 9h320q14 0 23 -9t9 -23v-320q0 -14 -9 -23t-23 -9h-320q-14 0 -23 9t-9 23v96h-96 + q-32 0 -61 10t-51 23.5t-45 40.5t-37 46t-33.5 57t-28.5 57.5t-28 60.5q-23 53 -37 81.5t-36 65t-44.5 53.5t-46.5 17h-360q-22 -84 -91 -138t-157 -54q-106 0 -181 75t-75 181t75 181t181 75q88 0 157 -54t91 -138h104q24 0 46.5 17t44.5 53.5t36 65t37 81.5q19 41 28 60.5 + t28.5 57.5t33.5 57t37 46t45 40.5t51 23.5t61 10h107q21 57 70 92.5t111 35.5q80 0 136 -56t56 -136t-56 -136t-136 -56q-62 0 -111 35.5t-70 92.5h-107q-17 0 -33 -8t-31.5 -26t-27 -34t-27 -47t-24 -49t-24.5 -55q-46 -107 -83 -165h1114v128q0 18 16 28t32 -1z" /> + <glyph glyph-name="_608" unicode="" horiz-adv-x="1792" + d="M1150 774q0 -56 -39.5 -95t-95.5 -39h-253v269h253q56 0 95.5 -39.5t39.5 -95.5zM1329 774q0 130 -91.5 222t-222.5 92h-433v-896h180v269h253q130 0 222 91.5t92 221.5zM1792 640q0 -182 -71 -348t-191 -286t-286 -191t-348 -71t-348 71t-286 191t-191 286t-71 348 + t71 348t191 286t286 191t348 71t348 -71t286 -191t191 -286t71 -348z" /> + <glyph glyph-name="_609" unicode="" horiz-adv-x="2304" + d="M1645 438q0 59 -34 106.5t-87 68.5q-7 -45 -23 -92q-7 -24 -27.5 -38t-44.5 -14q-12 0 -24 3q-31 10 -45 38.5t-4 58.5q23 71 23 143q0 123 -61 227.5t-166 165.5t-228 61q-134 0 -247 -73t-167 -194q108 -28 188 -106q22 -23 22 -55t-22 -54t-54 -22t-55 22 + q-75 75 -180 75q-106 0 -181 -74.5t-75 -180.5t75 -180.5t181 -74.5h1046q79 0 134.5 55.5t55.5 133.5zM1798 438q0 -142 -100.5 -242t-242.5 -100h-1046q-169 0 -289 119.5t-120 288.5q0 153 100 267t249 136q62 184 221 298t354 114q235 0 408.5 -158.5t196.5 -389.5 + q116 -25 192.5 -118.5t76.5 -214.5zM2048 438q0 -175 -97 -319q-23 -33 -64 -33q-24 0 -43 13q-26 17 -32 48.5t12 57.5q71 104 71 233t-71 233q-18 26 -12 57t32 49t57.5 11.5t49.5 -32.5q97 -142 97 -318zM2304 438q0 -244 -134 -443q-23 -34 -64 -34q-23 0 -42 13 + q-26 18 -32.5 49t11.5 57q108 164 108 358q0 195 -108 357q-18 26 -11.5 57.5t32.5 48.5q26 18 57 12t49 -33q134 -198 134 -442z" /> + <glyph glyph-name="_610" unicode="" + d="M1500 -13q0 -89 -63 -152.5t-153 -63.5t-153.5 63.5t-63.5 152.5q0 90 63.5 153.5t153.5 63.5t153 -63.5t63 -153.5zM1267 268q-115 -15 -192.5 -102.5t-77.5 -205.5q0 -74 33 -138q-146 -78 -379 -78q-109 0 -201 21t-153.5 54.5t-110.5 76.5t-76 85t-44.5 83 + t-23.5 66.5t-6 39.5q0 19 4.5 42.5t18.5 56t36.5 58t64 43.5t94.5 18t94 -17.5t63 -41t35.5 -53t17.5 -49t4 -33.5q0 -34 -23 -81q28 -27 82 -42t93 -17l40 -1q115 0 190 51t75 133q0 26 -9 48.5t-31.5 44.5t-49.5 41t-74 44t-93.5 47.5t-119.5 56.5q-28 13 -43 20 + q-116 55 -187 100t-122.5 102t-72 125.5t-20.5 162.5q0 78 20.5 150t66 137.5t112.5 114t166.5 77t221.5 28.5q120 0 220 -26t164.5 -67t109.5 -94t64 -105.5t19 -103.5q0 -46 -15 -82.5t-36.5 -58t-48.5 -36t-49 -19.5t-39 -5h-8h-32t-39 5t-44 14t-41 28t-37 46t-24 70.5 + t-10 97.5q-15 16 -59 25.5t-81 10.5l-37 1q-68 0 -117.5 -31t-70.5 -70t-21 -76q0 -24 5 -43t24 -46t53 -51t97 -53.5t150 -58.5q76 -25 138.5 -53.5t109 -55.5t83 -59t60.5 -59.5t41 -62.5t26.5 -62t14.5 -63.5t6 -62t1 -62.5z" /> + <glyph glyph-name="_611" unicode="" + d="M704 352v576q0 14 -9 23t-23 9h-256q-14 0 -23 -9t-9 -23v-576q0 -14 9 -23t23 -9h256q14 0 23 9t9 23zM1152 352v576q0 14 -9 23t-23 9h-256q-14 0 -23 -9t-9 -23v-576q0 -14 9 -23t23 -9h256q14 0 23 9t9 23zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103 + t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="_612" unicode="" + d="M768 1408q209 0 385.5 -103t279.5 -279.5t103 -385.5t-103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103zM768 96q148 0 273 73t198 198t73 273t-73 273t-198 198t-273 73t-273 -73t-198 -198t-73 -273 + t73 -273t198 -198t273 -73zM864 320q-14 0 -23 9t-9 23v576q0 14 9 23t23 9h192q14 0 23 -9t9 -23v-576q0 -14 -9 -23t-23 -9h-192zM480 320q-14 0 -23 9t-9 23v576q0 14 9 23t23 9h192q14 0 23 -9t9 -23v-576q0 -14 -9 -23t-23 -9h-192z" /> + <glyph glyph-name="_613" unicode="" + d="M1088 352v576q0 14 -9 23t-23 9h-576q-14 0 -23 -9t-9 -23v-576q0 -14 9 -23t23 -9h576q14 0 23 9t9 23zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5 + t103 -385.5z" /> + <glyph glyph-name="_614" unicode="" + d="M768 1408q209 0 385.5 -103t279.5 -279.5t103 -385.5t-103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103zM768 96q148 0 273 73t198 198t73 273t-73 273t-198 198t-273 73t-273 -73t-198 -198t-73 -273 + t73 -273t198 -198t273 -73zM480 320q-14 0 -23 9t-9 23v576q0 14 9 23t23 9h576q14 0 23 -9t9 -23v-576q0 -14 -9 -23t-23 -9h-576z" /> + <glyph glyph-name="_615" unicode="" horiz-adv-x="1792" + d="M1757 128l35 -313q3 -28 -16 -50q-19 -21 -48 -21h-1664q-29 0 -48 21q-19 22 -16 50l35 313h1722zM1664 967l86 -775h-1708l86 775q3 24 21 40.5t43 16.5h256v-128q0 -53 37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5v128h384v-128q0 -53 37.5 -90.5t90.5 -37.5 + t90.5 37.5t37.5 90.5v128h256q25 0 43 -16.5t21 -40.5zM1280 1152v-256q0 -26 -19 -45t-45 -19t-45 19t-19 45v256q0 106 -75 181t-181 75t-181 -75t-75 -181v-256q0 -26 -19 -45t-45 -19t-45 19t-19 45v256q0 159 112.5 271.5t271.5 112.5t271.5 -112.5t112.5 -271.5z" /> + <glyph glyph-name="_616" unicode="" horiz-adv-x="2048" + d="M1920 768q53 0 90.5 -37.5t37.5 -90.5t-37.5 -90.5t-90.5 -37.5h-15l-115 -662q-8 -46 -44 -76t-82 -30h-1280q-46 0 -82 30t-44 76l-115 662h-15q-53 0 -90.5 37.5t-37.5 90.5t37.5 90.5t90.5 37.5h1792zM485 -32q26 2 43.5 22.5t15.5 46.5l-32 416q-2 26 -22.5 43.5 + t-46.5 15.5t-43.5 -22.5t-15.5 -46.5l32 -416q2 -25 20.5 -42t43.5 -17h5zM896 32v416q0 26 -19 45t-45 19t-45 -19t-19 -45v-416q0 -26 19 -45t45 -19t45 19t19 45zM1280 32v416q0 26 -19 45t-45 19t-45 -19t-19 -45v-416q0 -26 19 -45t45 -19t45 19t19 45zM1632 27l32 416 + q2 26 -15.5 46.5t-43.5 22.5t-46.5 -15.5t-22.5 -43.5l-32 -416q-2 -26 15.5 -46.5t43.5 -22.5h5q25 0 43.5 17t20.5 42zM476 1244l-93 -412h-132l101 441q19 88 89 143.5t160 55.5h167q0 26 19 45t45 19h384q26 0 45 -19t19 -45h167q90 0 160 -55.5t89 -143.5l101 -441 + h-132l-93 412q-11 44 -45.5 72t-79.5 28h-167q0 -26 -19 -45t-45 -19h-384q-26 0 -45 19t-19 45h-167q-45 0 -79.5 -28t-45.5 -72z" /> + <glyph glyph-name="_617" unicode="" horiz-adv-x="1792" + d="M991 512l64 256h-254l-64 -256h254zM1759 1016l-56 -224q-7 -24 -31 -24h-327l-64 -256h311q15 0 25 -12q10 -14 6 -28l-56 -224q-5 -24 -31 -24h-327l-81 -328q-7 -24 -31 -24h-224q-16 0 -26 12q-9 12 -6 28l78 312h-254l-81 -328q-7 -24 -31 -24h-225q-15 0 -25 12 + q-9 12 -6 28l78 312h-311q-15 0 -25 12q-9 12 -6 28l56 224q7 24 31 24h327l64 256h-311q-15 0 -25 12q-10 14 -6 28l56 224q5 24 31 24h327l81 328q7 24 32 24h224q15 0 25 -12q9 -12 6 -28l-78 -312h254l81 328q7 24 32 24h224q15 0 25 -12q9 -12 6 -28l-78 -312h311 + q15 0 25 -12q9 -12 6 -28z" /> + <glyph glyph-name="_618" unicode="" + d="M841 483l148 -148l-149 -149zM840 1094l149 -149l-148 -148zM710 -130l464 464l-306 306l306 306l-464 464v-611l-255 255l-93 -93l320 -321l-320 -321l93 -93l255 255v-611zM1429 640q0 -209 -32 -365.5t-87.5 -257t-140.5 -162.5t-181.5 -86.5t-219.5 -24.5 + t-219.5 24.5t-181.5 86.5t-140.5 162.5t-87.5 257t-32 365.5t32 365.5t87.5 257t140.5 162.5t181.5 86.5t219.5 24.5t219.5 -24.5t181.5 -86.5t140.5 -162.5t87.5 -257t32 -365.5z" /> + <glyph glyph-name="_619" unicode="" horiz-adv-x="1024" + d="M596 113l173 172l-173 172v-344zM596 823l173 172l-173 172v-344zM628 640l356 -356l-539 -540v711l-297 -296l-108 108l372 373l-372 373l108 108l297 -296v711l539 -540z" /> + <glyph glyph-name="_620" unicode="" + d="M1280 256q0 52 -38 90t-90 38t-90 -38t-38 -90t38 -90t90 -38t90 38t38 90zM512 1024q0 52 -38 90t-90 38t-90 -38t-38 -90t38 -90t90 -38t90 38t38 90zM1536 256q0 -159 -112.5 -271.5t-271.5 -112.5t-271.5 112.5t-112.5 271.5t112.5 271.5t271.5 112.5t271.5 -112.5 + t112.5 -271.5zM1440 1344q0 -20 -13 -38l-1056 -1408q-19 -26 -51 -26h-160q-26 0 -45 19t-19 45q0 20 13 38l1056 1408q19 26 51 26h160q26 0 45 -19t19 -45zM768 1024q0 -159 -112.5 -271.5t-271.5 -112.5t-271.5 112.5t-112.5 271.5t112.5 271.5t271.5 112.5 + t271.5 -112.5t112.5 -271.5z" /> + <glyph glyph-name="_621" unicode="" horiz-adv-x="1792" + d="M104 830l792 -1015l-868 630q-18 13 -25 34.5t0 42.5l101 308v0zM566 830h660l-330 -1015v0zM368 1442l198 -612h-462l198 612q8 23 33 23t33 -23zM1688 830l101 -308q7 -21 0 -42.5t-25 -34.5l-868 -630l792 1015v0zM1688 830h-462l198 612q8 23 33 23t33 -23z" /> + <glyph glyph-name="_622" unicode="" horiz-adv-x="1792" + d="M384 704h160v224h-160v-224zM1221 372v92q-104 -36 -243 -38q-135 -1 -259.5 46.5t-220.5 122.5l1 -96q88 -80 212 -128.5t272 -47.5q129 0 238 49zM640 704h640v224h-640v-224zM1792 736q0 -187 -99 -352q89 -102 89 -229q0 -157 -129.5 -268t-313.5 -111 + q-122 0 -225 52.5t-161 140.5q-19 -1 -57 -1t-57 1q-58 -88 -161 -140.5t-225 -52.5q-184 0 -313.5 111t-129.5 268q0 127 89 229q-99 165 -99 352q0 209 120 385.5t326.5 279.5t449.5 103t449.5 -103t326.5 -279.5t120 -385.5z" /> + <glyph glyph-name="_623" unicode="" + d="M515 625v-128h-252v128h252zM515 880v-127h-252v127h252zM1273 369v-128h-341v128h341zM1273 625v-128h-672v128h672zM1273 880v-127h-672v127h672zM1408 20v1240q0 8 -6 14t-14 6h-32l-378 -256l-210 171l-210 -171l-378 256h-32q-8 0 -14 -6t-6 -14v-1240q0 -8 6 -14 + t14 -6h1240q8 0 14 6t6 14zM553 1130l185 150h-406zM983 1130l221 150h-406zM1536 1260v-1240q0 -62 -43 -105t-105 -43h-1240q-62 0 -105 43t-43 105v1240q0 62 43 105t105 43h1240q62 0 105 -43t43 -105z" /> + <glyph glyph-name="_624" unicode="" horiz-adv-x="1792" + d="M896 720q-104 196 -160 278q-139 202 -347 318q-34 19 -70 36q-89 40 -94 32t34 -38l39 -31q62 -43 112.5 -93.5t94.5 -116.5t70.5 -113t70.5 -131q9 -17 13 -25q44 -84 84 -153t98 -154t115.5 -150t131 -123.5t148.5 -90.5q153 -66 154 -60q1 3 -49 37q-53 36 -81 57 + q-77 58 -179 211t-185 310zM549 177q-76 60 -132.5 125t-98 143.5t-71 154.5t-58.5 186t-52 209t-60.5 252t-76.5 289q273 0 497.5 -36t379 -92t271 -144.5t185.5 -172.5t110 -198.5t56 -199.5t12.5 -198.5t-9.5 -173t-20 -143.5t-13 -107l323 -327h-104l-281 285 + q-22 -2 -91.5 -14t-121.5 -19t-138 -6t-160.5 17t-167.5 59t-179 111z" /> + <glyph glyph-name="_625" unicode="" horiz-adv-x="1792" + d="M1374 879q-6 26 -28.5 39.5t-48.5 7.5q-261 -62 -401 -62t-401 62q-26 6 -48.5 -7.5t-28.5 -39.5t7.5 -48.5t39.5 -28.5q194 -46 303 -58q-2 -158 -15.5 -269t-26.5 -155.5t-41 -115.5l-9 -21q-10 -25 1 -49t36 -34q9 -4 23 -4q44 0 60 41l8 20q54 139 71 259h42 + q17 -120 71 -259l8 -20q16 -41 60 -41q14 0 23 4q25 10 36 34t1 49l-9 21q-28 71 -41 115.5t-26.5 155.5t-15.5 269q109 12 303 58q26 6 39.5 28.5t7.5 48.5zM1024 1024q0 53 -37.5 90.5t-90.5 37.5t-90.5 -37.5t-37.5 -90.5t37.5 -90.5t90.5 -37.5t90.5 37.5t37.5 90.5z + M1600 640q0 -143 -55.5 -273.5t-150 -225t-225 -150t-273.5 -55.5t-273.5 55.5t-225 150t-150 225t-55.5 273.5t55.5 273.5t150 225t225 150t273.5 55.5t273.5 -55.5t225 -150t150 -225t55.5 -273.5zM896 1408q-156 0 -298 -61t-245 -164t-164 -245t-61 -298t61 -298 + t164 -245t245 -164t298 -61t298 61t245 164t164 245t61 298t-61 298t-164 245t-245 164t-298 61zM1792 640q0 -182 -71 -348t-191 -286t-286 -191t-348 -71t-348 71t-286 191t-191 286t-71 348t71 348t191 286t286 191t348 71t348 -71t286 -191t191 -286t71 -348z" /> + <glyph glyph-name="_626" unicode="" + d="M1438 723q34 -35 29 -82l-44 -551q-4 -42 -34.5 -70t-71.5 -28q-6 0 -9 1q-44 3 -72.5 36.5t-25.5 77.5l35 429l-143 -8q55 -113 55 -240q0 -216 -148 -372l-137 137q91 101 91 235q0 145 -102.5 248t-247.5 103q-134 0 -236 -92l-137 138q120 114 284 141l264 300 + l-149 87l-181 -161q-33 -30 -77 -27.5t-73 35.5t-26.5 77t34.5 73l239 213q26 23 60 26.5t64 -14.5l488 -283q36 -21 48 -68q17 -67 -26 -117l-205 -232l371 20q49 3 83 -32zM1240 1180q-74 0 -126 52t-52 126t52 126t126 52t126.5 -52t52.5 -126t-52.5 -126t-126.5 -52z + M613 -62q106 0 196 61l139 -139q-146 -116 -335 -116q-148 0 -273.5 73t-198.5 198t-73 273q0 188 116 336l139 -139q-60 -88 -60 -197q0 -145 102.5 -247.5t247.5 -102.5z" /> + <glyph glyph-name="_627" unicode="" + d="M880 336v-160q0 -14 -9 -23t-23 -9h-160q-14 0 -23 9t-9 23v160q0 14 9 23t23 9h160q14 0 23 -9t9 -23zM1136 832q0 -50 -15 -90t-45.5 -69t-52 -44t-59.5 -36q-32 -18 -46.5 -28t-26 -24t-11.5 -29v-32q0 -14 -9 -23t-23 -9h-160q-14 0 -23 9t-9 23v68q0 35 10.5 64.5 + t24 47.5t39 35.5t41 25.5t44.5 21q53 25 75 43t22 49q0 42 -43.5 71.5t-95.5 29.5q-56 0 -95 -27q-29 -20 -80 -83q-9 -12 -25 -12q-11 0 -19 6l-108 82q-10 7 -12 20t5 23q122 192 349 192q129 0 238.5 -89.5t109.5 -214.5zM768 1280q-130 0 -248.5 -51t-204 -136.5 + t-136.5 -204t-51 -248.5t51 -248.5t136.5 -204t204 -136.5t248.5 -51t248.5 51t204 136.5t136.5 204t51 248.5t-51 248.5t-136.5 204t-204 136.5t-248.5 51zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5 + t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="_628" unicode="" horiz-adv-x="1408" + d="M366 1225q-64 0 -110 45.5t-46 110.5q0 64 46 109.5t110 45.5t109.5 -45.5t45.5 -109.5q0 -65 -45.5 -110.5t-109.5 -45.5zM917 583q0 -50 -30 -67.5t-63.5 -6.5t-47.5 34l-367 438q-7 12 -14 15.5t-11 1.5l-3 -3q-7 -8 4 -21l122 -139l1 -354l-161 -457 + q-67 -192 -92 -234q-15 -26 -28 -32q-50 -26 -103 -1q-29 13 -41.5 43t-9.5 57q2 17 197 618l5 416l-85 -164l35 -222q4 -24 -1 -42t-14 -27.5t-19 -16t-17 -7.5l-7 -2q-19 -3 -34.5 3t-24 16t-14 22t-7.5 19.5t-2 9.5l-46 299l211 381q23 34 113 34q75 0 107 -40l424 -521 + q7 -5 14 -17l3 -3l-1 -1q7 -13 7 -29zM514 433q43 -113 88.5 -225t69.5 -168l24 -55q36 -93 42 -125q11 -70 -36 -97q-35 -22 -66 -16t-51 22t-29 35h-1q-6 16 -8 25l-124 351zM1338 -159q31 -49 31 -57q0 -5 -3 -7q-9 -5 -14.5 0.5t-15.5 26t-16 30.5q-114 172 -423 661 + q3 -1 7 1t7 4l3 2q11 9 11 17z" /> + <glyph glyph-name="_629" unicode="" horiz-adv-x="2304" + d="M504 542h171l-1 265zM1530 641q0 87 -50.5 140t-146.5 53h-54v-388h52q91 0 145 57t54 138zM956 1018l1 -756q0 -14 -9.5 -24t-23.5 -10h-216q-14 0 -23.5 10t-9.5 24v62h-291l-55 -81q-10 -15 -28 -15h-267q-21 0 -30.5 18t3.5 35l556 757q9 14 27 14h332q14 0 24 -10 + t10 -24zM1783 641q0 -193 -125.5 -303t-324.5 -110h-270q-14 0 -24 10t-10 24v756q0 14 10 24t24 10h268q200 0 326 -109t126 -302zM1939 640q0 -11 -0.5 -29t-8 -71.5t-21.5 -102t-44.5 -108t-73.5 -102.5h-51q38 45 66.5 104.5t41.5 112t21 98t9 72.5l1 27q0 8 -0.5 22.5 + t-7.5 60t-20 91.5t-41 111.5t-66 124.5h43q41 -47 72 -107t45.5 -111.5t23 -96t10.5 -70.5zM2123 640q0 -11 -0.5 -29t-8 -71.5t-21.5 -102t-45 -108t-74 -102.5h-51q38 45 66.5 104.5t41.5 112t21 98t9 72.5l1 27q0 8 -0.5 22.5t-7.5 60t-19.5 91.5t-40.5 111.5t-66 124.5 + h43q41 -47 72 -107t45.5 -111.5t23 -96t10.5 -70.5zM2304 640q0 -11 -0.5 -29t-8 -71.5t-21.5 -102t-44.5 -108t-73.5 -102.5h-51q38 45 66 104.5t41 112t21 98t9 72.5l1 27q0 8 -0.5 22.5t-7.5 60t-19.5 91.5t-40.5 111.5t-66 124.5h43q41 -47 72 -107t45.5 -111.5t23 -96 + t9.5 -70.5z" /> + <glyph glyph-name="uniF2A0" unicode="" horiz-adv-x="1408" + d="M617 -153q0 11 -13 58t-31 107t-20 69q-1 4 -5 26.5t-8.5 36t-13.5 21.5q-15 14 -51 14q-23 0 -70 -5.5t-71 -5.5q-34 0 -47 11q-6 5 -11 15.5t-7.5 20t-6.5 24t-5 18.5q-37 128 -37 255t37 255q1 4 5 18.5t6.5 24t7.5 20t11 15.5q13 11 47 11q24 0 71 -5.5t70 -5.5 + q36 0 51 14q9 8 13.5 21.5t8.5 36t5 26.5q2 9 20 69t31 107t13 58q0 22 -43.5 52.5t-75.5 42.5q-20 8 -45 8q-34 0 -98 -18q-57 -17 -96.5 -40.5t-71 -66t-46 -70t-45.5 -94.5q-6 -12 -9 -19q-49 -107 -68 -216t-19 -244t19 -244t68 -216q56 -122 83 -161q63 -91 179 -127 + l6 -2q64 -18 98 -18q25 0 45 8q32 12 75.5 42.5t43.5 52.5zM776 760q-26 0 -45 19t-19 45.5t19 45.5q37 37 37 90q0 52 -37 91q-19 19 -19 45t19 45t45 19t45 -19q75 -75 75 -181t-75 -181q-21 -19 -45 -19zM957 579q-27 0 -45 19q-19 19 -19 45t19 45q112 114 112 272 + t-112 272q-19 19 -19 45t19 45t45 19t45 -19q150 -150 150 -362t-150 -362q-18 -19 -45 -19zM1138 398q-27 0 -45 19q-19 19 -19 45t19 45q90 91 138.5 208t48.5 245t-48.5 245t-138.5 208q-19 19 -19 45t19 45t45 19t45 -19q109 -109 167 -249t58 -294t-58 -294t-167 -249 + q-18 -19 -45 -19z" /> + <glyph glyph-name="uniF2A1" unicode="" horiz-adv-x="2176" + d="M192 352q-66 0 -113 -47t-47 -113t47 -113t113 -47t113 47t47 113t-47 113t-113 47zM704 352q-66 0 -113 -47t-47 -113t47 -113t113 -47t113 47t47 113t-47 113t-113 47zM704 864q-66 0 -113 -47t-47 -113t47 -113t113 -47t113 47t47 113t-47 113t-113 47zM1472 352 + q-66 0 -113 -47t-47 -113t47 -113t113 -47t113 47t47 113t-47 113t-113 47zM1984 352q-66 0 -113 -47t-47 -113t47 -113t113 -47t113 47t47 113t-47 113t-113 47zM1472 864q-66 0 -113 -47t-47 -113t47 -113t113 -47t113 47t47 113t-47 113t-113 47zM1984 864 + q-66 0 -113 -47t-47 -113t47 -113t113 -47t113 47t47 113t-47 113t-113 47zM1984 1376q-66 0 -113 -47t-47 -113t47 -113t113 -47t113 47t47 113t-47 113t-113 47zM384 192q0 -80 -56 -136t-136 -56t-136 56t-56 136t56 136t136 56t136 -56t56 -136zM896 192q0 -80 -56 -136 + t-136 -56t-136 56t-56 136t56 136t136 56t136 -56t56 -136zM384 704q0 -80 -56 -136t-136 -56t-136 56t-56 136t56 136t136 56t136 -56t56 -136zM896 704q0 -80 -56 -136t-136 -56t-136 56t-56 136t56 136t136 56t136 -56t56 -136zM384 1216q0 -80 -56 -136t-136 -56 + t-136 56t-56 136t56 136t136 56t136 -56t56 -136zM1664 192q0 -80 -56 -136t-136 -56t-136 56t-56 136t56 136t136 56t136 -56t56 -136zM896 1216q0 -80 -56 -136t-136 -56t-136 56t-56 136t56 136t136 56t136 -56t56 -136zM2176 192q0 -80 -56 -136t-136 -56t-136 56 + t-56 136t56 136t136 56t136 -56t56 -136zM1664 704q0 -80 -56 -136t-136 -56t-136 56t-56 136t56 136t136 56t136 -56t56 -136zM2176 704q0 -80 -56 -136t-136 -56t-136 56t-56 136t56 136t136 56t136 -56t56 -136zM1664 1216q0 -80 -56 -136t-136 -56t-136 56t-56 136 + t56 136t136 56t136 -56t56 -136zM2176 1216q0 -80 -56 -136t-136 -56t-136 56t-56 136t56 136t136 56t136 -56t56 -136z" /> + <glyph glyph-name="uniF2A2" unicode="" horiz-adv-x="1792" + d="M128 -192q0 -26 -19 -45t-45 -19t-45 19t-19 45t19 45t45 19t45 -19t19 -45zM320 0q0 -26 -19 -45t-45 -19t-45 19t-19 45t19 45t45 19t45 -19t19 -45zM365 365l256 -256l-90 -90l-256 256zM704 384q0 -26 -19 -45t-45 -19t-45 19t-19 45t19 45t45 19t45 -19t19 -45z + M1411 704q0 -59 -11.5 -108.5t-37.5 -93.5t-44 -67.5t-53 -64.5q-31 -35 -45.5 -54t-33.5 -50t-26.5 -64t-7.5 -74q0 -159 -112.5 -271.5t-271.5 -112.5q-26 0 -45 19t-19 45t19 45t45 19q106 0 181 75t75 181q0 57 11.5 105.5t37 91t43.5 66.5t52 63q40 46 59.5 72 + t37.5 74.5t18 103.5q0 185 -131.5 316.5t-316.5 131.5t-316.5 -131.5t-131.5 -316.5q0 -26 -19 -45t-45 -19t-45 19t-19 45q0 117 45.5 223.5t123 184t184 123t223.5 45.5t223.5 -45.5t184 -123t123 -184t45.5 -223.5zM896 576q0 -26 -19 -45t-45 -19t-45 19t-19 45t19 45 + t45 19t45 -19t19 -45zM1184 704q0 -26 -19 -45t-45 -19t-45 19t-19 45q0 93 -65.5 158.5t-158.5 65.5q-92 0 -158 -65.5t-66 -158.5q0 -26 -19 -45t-45 -19t-45 19t-19 45q0 146 103 249t249 103t249 -103t103 -249zM1578 993q10 -25 -1 -49t-36 -34q-9 -4 -23 -4 + q-19 0 -35.5 11t-23.5 30q-68 178 -224 295q-21 16 -25 42t12 47q17 21 43 25t47 -12q183 -137 266 -351zM1788 1074q9 -25 -1.5 -49t-35.5 -34q-11 -4 -23 -4q-44 0 -60 41q-92 238 -297 393q-22 16 -25.5 42t12.5 47q16 22 42 25.5t47 -12.5q235 -175 341 -449z" /> + <glyph glyph-name="uniF2A3" unicode="" horiz-adv-x="2304" + d="M1032 576q-59 2 -84 55q-17 34 -48 53.5t-68 19.5q-53 0 -90.5 -37.5t-37.5 -90.5q0 -56 36 -89l10 -8q34 -31 82 -31q37 0 68 19.5t48 53.5q25 53 84 55zM1600 704q0 56 -36 89l-10 8q-34 31 -82 31q-37 0 -68 -19.5t-48 -53.5q-25 -53 -84 -55q59 -2 84 -55 + q17 -34 48 -53.5t68 -19.5q53 0 90.5 37.5t37.5 90.5zM1174 925q-17 -35 -55 -48t-73 4q-62 31 -134 31q-51 0 -99 -17q3 0 9.5 0.5t9.5 0.5q92 0 170.5 -50t118.5 -133q17 -36 3.5 -73.5t-49.5 -54.5q-18 -9 -39 -9q21 0 39 -9q36 -17 49.5 -54.5t-3.5 -73.5 + q-40 -83 -118.5 -133t-170.5 -50h-6q-16 2 -44 4l-290 27l-239 -120q-14 -7 -29 -7q-40 0 -57 35l-160 320q-11 23 -4 47.5t29 37.5l209 119l148 267q17 155 91.5 291.5t195.5 236.5q31 25 70.5 21.5t64.5 -34.5t21.5 -70t-34.5 -65q-70 -59 -117 -128q123 84 267 101 + q40 5 71.5 -19t35.5 -64q5 -40 -19 -71.5t-64 -35.5q-84 -10 -159 -55q46 10 99 10q115 0 218 -50q36 -18 49 -55.5t-5 -73.5zM2137 1085l160 -320q11 -23 4 -47.5t-29 -37.5l-209 -119l-148 -267q-17 -155 -91.5 -291.5t-195.5 -236.5q-26 -22 -61 -22q-45 0 -74 35 + q-25 31 -21.5 70t34.5 65q70 59 117 128q-123 -84 -267 -101q-4 -1 -12 -1q-36 0 -63.5 24t-31.5 60q-5 40 19 71.5t64 35.5q84 10 159 55q-46 -10 -99 -10q-115 0 -218 50q-36 18 -49 55.5t5 73.5q17 35 55 48t73 -4q62 -31 134 -31q51 0 99 17q-3 0 -9.5 -0.5t-9.5 -0.5 + q-92 0 -170.5 50t-118.5 133q-17 36 -3.5 73.5t49.5 54.5q18 9 39 9q-21 0 -39 9q-36 17 -49.5 54.5t3.5 73.5q40 83 118.5 133t170.5 50h6h1q14 -2 42 -4l291 -27l239 120q14 7 29 7q40 0 57 -35z" /> + <glyph glyph-name="uniF2A4" unicode="" horiz-adv-x="1792" + d="M1056 704q0 -26 19 -45t45 -19t45 19t19 45q0 146 -103 249t-249 103t-249 -103t-103 -249q0 -26 19 -45t45 -19t45 19t19 45q0 93 66 158.5t158 65.5t158 -65.5t66 -158.5zM835 1280q-117 0 -223.5 -45.5t-184 -123t-123 -184t-45.5 -223.5q0 -26 19 -45t45 -19t45 19 + t19 45q0 185 131.5 316.5t316.5 131.5t316.5 -131.5t131.5 -316.5q0 -55 -18 -103.5t-37.5 -74.5t-59.5 -72q-34 -39 -52 -63t-43.5 -66.5t-37 -91t-11.5 -105.5q0 -106 -75 -181t-181 -75q-26 0 -45 -19t-19 -45t19 -45t45 -19q159 0 271.5 112.5t112.5 271.5q0 41 7.5 74 + t26.5 64t33.5 50t45.5 54q35 41 53 64.5t44 67.5t37.5 93.5t11.5 108.5q0 117 -45.5 223.5t-123 184t-184 123t-223.5 45.5zM591 561l226 -226l-579 -579q-12 -12 -29 -12t-29 12l-168 168q-12 12 -12 29t12 29zM1612 1524l168 -168q12 -12 12 -29t-12 -30l-233 -233 + l-26 -25l-71 -71q-66 153 -195 258l91 91l207 207q13 12 30 12t29 -12z" /> + <glyph glyph-name="uniF2A5" unicode="" + d="M866 1021q0 -27 -13 -94q-11 -50 -31.5 -150t-30.5 -150q-2 -11 -4.5 -12.5t-13.5 -2.5q-20 -2 -31 -2q-58 0 -84 49.5t-26 113.5q0 88 35 174t103 124q28 14 51 14q28 0 36.5 -16.5t8.5 -47.5zM1352 597q0 14 -39 75.5t-52 66.5q-21 8 -34 8q-91 0 -226 -77l-2 2 + q3 22 27.5 135t24.5 178q0 233 -242 233q-24 0 -68 -6q-94 -17 -168.5 -89.5t-111.5 -166.5t-37 -189q0 -146 80.5 -225t227.5 -79q25 0 25 -3t-1 -5q-4 -34 -26 -117q-14 -52 -51.5 -101t-82.5 -49q-42 0 -42 47q0 24 10.5 47.5t25 39.5t29.5 28.5t26 20t11 8.5q0 3 -7 10 + q-24 22 -58.5 36.5t-65.5 14.5q-35 0 -63.5 -34t-41 -75t-12.5 -75q0 -88 51.5 -142t138.5 -54q82 0 155 53t117.5 126t65.5 153q6 22 15.5 66.5t14.5 66.5q3 12 14 18q118 60 227 60q48 0 127 -18q1 -1 4 -1q5 0 9.5 4.5t4.5 8.5zM1536 1120v-960q0 -119 -84.5 -203.5 + t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="uniF2A6" unicode="" horiz-adv-x="1535" + d="M744 1231q0 24 -2 38.5t-8.5 30t-21 23t-37.5 7.5q-39 0 -78 -23q-105 -58 -159 -190.5t-54 -269.5q0 -44 8.5 -85.5t26.5 -80.5t52.5 -62.5t81.5 -23.5q4 0 18 -0.5t20 0t16 3t15 8.5t7 16q16 77 48 231.5t48 231.5q19 91 19 146zM1498 575q0 -7 -7.5 -13.5t-15.5 -6.5 + l-6 1q-22 3 -62 11t-72 12.5t-63 4.5q-167 0 -351 -93q-15 -8 -21 -27q-10 -36 -24.5 -105.5t-22.5 -100.5q-23 -91 -70 -179.5t-112.5 -164.5t-154.5 -123t-185 -47q-135 0 -214.5 83.5t-79.5 219.5q0 53 19.5 117t63 116.5t97.5 52.5q38 0 120 -33.5t83 -61.5 + q0 -1 -16.5 -12.5t-39.5 -31t-46 -44.5t-39 -61t-16 -74q0 -33 16.5 -53t48.5 -20q45 0 85 31.5t66.5 78t48 105.5t32.5 107t16 90v9q0 2 -3.5 3.5t-8.5 1.5h-10t-10 -0.5t-6 -0.5q-227 0 -352 122.5t-125 348.5q0 108 34.5 221t96 210t156 167.5t204.5 89.5q52 9 106 9 + q374 0 374 -360q0 -98 -38 -273t-43 -211l3 -3q101 57 182.5 88t167.5 31q22 0 53 -13q19 -7 80 -102.5t61 -116.5z" /> + <glyph glyph-name="uniF2A7" unicode="" horiz-adv-x="1664" + d="M831 863q32 0 59 -18l222 -148q61 -40 110 -97l146 -170q40 -46 29 -106l-72 -413q-6 -32 -29.5 -53.5t-55.5 -25.5l-527 -56l-352 -32h-9q-39 0 -67.5 28t-28.5 68q0 37 27 64t65 32l260 32h-448q-41 0 -69.5 30t-26.5 71q2 39 32 65t69 26l442 1l-521 64q-41 5 -66 37 + t-19 73q6 35 34.5 57.5t65.5 22.5h10l481 -60l-351 94q-38 10 -62 41.5t-18 68.5q6 36 33 58.5t62 22.5q6 0 20 -2l448 -96l217 -37q1 0 3 -0.5t3 -0.5q23 0 30.5 23t-12.5 36l-186 125q-35 23 -42 63.5t18 73.5q27 38 76 38zM761 661l186 -125l-218 37l-5 2l-36 38 + l-238 262q-1 1 -2.5 3.5t-2.5 3.5q-24 31 -18.5 70t37.5 64q31 23 68 17.5t64 -33.5l142 -147q-2 -1 -5 -3.5t-4 -4.5q-32 -45 -23 -99t55 -85zM1648 1115l15 -266q4 -73 -11 -147l-48 -219q-12 -59 -67 -87l-106 -54q2 62 -39 109l-146 170q-53 61 -117 103l-222 148 + q-34 23 -76 23q-51 0 -88 -37l-235 312q-25 33 -18 73.5t41 63.5q33 22 71.5 14t62.5 -40l266 -352l-262 455q-21 35 -10.5 75t47.5 59q35 18 72.5 6t57.5 -46l241 -420l-136 337q-15 35 -4.5 74t44.5 56q37 19 76 6t56 -51l193 -415l101 -196q8 -15 23 -17.5t27 7.5t11 26 + l-12 224q-2 41 26 71t69 31q39 0 67 -28.5t30 -67.5z" /> + <glyph glyph-name="uniF2A8" unicode="" horiz-adv-x="1792" + d="M335 180q-2 0 -6 2q-86 57 -168.5 145t-139.5 180q-21 30 -21 69q0 9 2 19t4 18t7 18t8.5 16t10.5 17t10 15t12 15.5t11 14.5q184 251 452 365q-110 198 -110 211q0 19 17 29q116 64 128 64q18 0 28 -16l124 -229q92 19 192 19q266 0 497.5 -137.5t378.5 -369.5 + q20 -31 20 -69t-20 -69q-91 -142 -218.5 -253.5t-278.5 -175.5q110 -198 110 -211q0 -20 -17 -29q-116 -64 -127 -64q-19 0 -29 16l-124 229l-64 119l-444 820l7 7q-58 -24 -99 -47q3 -5 127 -234t243 -449t119 -223q0 -7 -9 -9q-13 -3 -72 -3q-57 0 -60 7l-456 841 + q-39 -28 -82 -68q24 -43 214 -393.5t190 -354.5q0 -10 -11 -10q-14 0 -82.5 22t-72.5 28l-106 197l-224 413q-44 -53 -78 -106q2 -3 18 -25t23 -34l176 -327q0 -10 -10 -10zM1165 282l49 -91q273 111 450 385q-180 277 -459 389q67 -64 103 -148.5t36 -176.5 + q0 -106 -47 -200.5t-132 -157.5zM848 896q0 -20 14 -34t34 -14q86 0 147 -61t61 -147q0 -20 14 -34t34 -14t34 14t14 34q0 126 -89 215t-215 89q-20 0 -34 -14t-14 -34zM1214 961l-9 4l7 -7z" /> + <glyph glyph-name="uniF2A9" unicode="" horiz-adv-x="1280" + d="M1050 430q0 -215 -147 -374q-148 -161 -378 -161q-232 0 -378 161q-147 159 -147 374q0 147 68 270.5t189 196.5t268 73q96 0 182 -31q-32 -62 -39 -126q-66 28 -143 28q-167 0 -280.5 -123t-113.5 -291q0 -170 112.5 -288.5t281.5 -118.5t281 118.5t112 288.5 + q0 89 -32 166q66 13 123 49q41 -98 41 -212zM846 619q0 -192 -79.5 -345t-238.5 -253l-14 -1q-29 0 -62 5q83 32 146.5 102.5t99.5 154.5t58.5 189t30 192.5t7.5 178.5q0 69 -3 103q55 -160 55 -326zM791 947v-2q-73 214 -206 440q88 -59 142.5 -186.5t63.5 -251.5z + M1035 744q-83 0 -160 75q218 120 290 247q19 37 21 56q-42 -94 -139.5 -166.5t-204.5 -97.5q-35 54 -35 113q0 37 17 79t43 68q46 44 157 74q59 16 106 58.5t74 100.5q74 -105 74 -253q0 -109 -24 -170q-32 -77 -88.5 -130.5t-130.5 -53.5z" /> + <glyph glyph-name="uniF2AA" unicode="" + d="M1050 495q0 78 -28 147q-41 -25 -85 -34q22 -50 22 -114q0 -117 -77 -198.5t-193 -81.5t-193.5 81.5t-77.5 198.5q0 115 78 199.5t193 84.5q53 0 98 -19q4 43 27 87q-60 21 -125 21q-154 0 -257.5 -108.5t-103.5 -263.5t103.5 -261t257.5 -106t257.5 106.5t103.5 260.5z + M872 850q2 -24 2 -71q0 -63 -5 -123t-20.5 -132.5t-40.5 -130t-68.5 -106t-100.5 -70.5q21 -3 42 -3h10q219 139 219 411q0 116 -38 225zM872 850q-4 80 -44 171.5t-98 130.5q92 -156 142 -302zM1207 955q0 102 -51 174q-41 -86 -124 -109q-69 -19 -109 -53.5t-40 -99.5 + q0 -40 24 -77q74 17 140.5 67t95.5 115q-4 -52 -74.5 -111.5t-138.5 -97.5q52 -52 110 -52q51 0 90 37t60 90q17 42 17 117zM1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960q119 0 203.5 -84.5 + t84.5 -203.5z" /> + <glyph glyph-name="uniF2AB" unicode="" + d="M1279 388q0 22 -22 27q-67 15 -118 59t-80 108q-7 19 -7 25q0 15 19.5 26t43 17t43 20.5t19.5 36.5q0 19 -18.5 31.5t-38.5 12.5q-12 0 -32 -8t-31 -8q-4 0 -12 2q5 95 5 114q0 79 -17 114q-36 78 -103 121.5t-152 43.5q-199 0 -275 -165q-17 -35 -17 -114q0 -19 5 -114 + q-4 -2 -14 -2q-12 0 -32 7.5t-30 7.5q-21 0 -38.5 -12t-17.5 -32q0 -21 19.5 -35.5t43 -20.5t43 -17t19.5 -26q0 -6 -7 -25q-64 -138 -198 -167q-22 -5 -22 -27q0 -46 137 -68q2 -5 6 -26t11.5 -30.5t23.5 -9.5q12 0 37.5 4.5t39.5 4.5q35 0 67 -15t54 -32.5t57.5 -32.5 + t76.5 -15q43 0 79 15t57.5 32.5t53.5 32.5t67 15q14 0 39.5 -4t38.5 -4q16 0 23 10t11 30t6 25q137 22 137 68zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5 + t103 -385.5z" /> + <glyph glyph-name="uniF2AC" unicode="" horiz-adv-x="1664" + d="M848 1408q134 1 240.5 -68.5t163.5 -192.5q27 -58 27 -179q0 -47 -9 -191q14 -7 28 -7q18 0 51 13.5t51 13.5q29 0 56 -18t27 -46q0 -32 -31.5 -54t-69 -31.5t-69 -29t-31.5 -47.5q0 -15 12 -43q37 -82 102.5 -150t144.5 -101q28 -12 80 -23q28 -6 28 -35 + q0 -70 -219 -103q-7 -11 -11 -39t-14 -46.5t-33 -18.5q-20 0 -62 6.5t-64 6.5q-37 0 -62 -5q-32 -5 -63 -22.5t-58 -38t-58 -40.5t-76 -33.5t-99 -13.5q-52 0 -96.5 13.5t-75 33.5t-57.5 40.5t-58 38t-62 22.5q-26 5 -63 5q-24 0 -65.5 -7.5t-58.5 -7.5q-25 0 -35 18.5 + t-14 47.5t-11 40q-219 33 -219 103q0 29 28 35q52 11 80 23q78 32 144.5 101t102.5 150q12 28 12 43q0 28 -31.5 47.5t-69.5 29.5t-69.5 31.5t-31.5 52.5q0 27 26 45.5t55 18.5q15 0 48 -13t53 -13q18 0 32 7q-9 142 -9 190q0 122 27 180q64 137 172 198t264 63z" /> + <glyph glyph-name="uniF2AD" unicode="" + d="M1280 388q0 22 -22 27q-67 14 -118 58t-80 109q-7 14 -7 25q0 15 19.5 26t42.5 17t42.5 20.5t19.5 36.5q0 19 -18.5 31.5t-38.5 12.5q-11 0 -31 -8t-32 -8q-4 0 -12 2q5 63 5 115q0 78 -17 114q-36 78 -102.5 121.5t-152.5 43.5q-198 0 -275 -165q-18 -38 -18 -115 + q0 -38 6 -114q-10 -2 -15 -2q-11 0 -31.5 8t-30.5 8q-20 0 -37.5 -12.5t-17.5 -32.5q0 -21 19.5 -35.5t42.5 -20.5t42.5 -17t19.5 -26q0 -11 -7 -25q-64 -138 -198 -167q-22 -5 -22 -27q0 -47 138 -69q2 -5 6 -26t11 -30.5t23 -9.5q13 0 38.5 5t38.5 5q35 0 67.5 -15 + t54.5 -32.5t57.5 -32.5t76.5 -15q43 0 79 15t57.5 32.5t54 32.5t67.5 15q13 0 39 -4.5t39 -4.5q15 0 22.5 9.5t11.5 31t5 24.5q138 22 138 69zM1536 1120v-960q0 -119 -84.5 -203.5t-203.5 -84.5h-960q-119 0 -203.5 84.5t-84.5 203.5v960q0 119 84.5 203.5t203.5 84.5h960 + q119 0 203.5 -84.5t84.5 -203.5z" /> + <glyph glyph-name="uniF2AE" unicode="" horiz-adv-x="2304" + d="M2304 1536q-69 -46 -125 -92t-89 -81t-59.5 -71.5t-37.5 -57.5t-22 -44.5t-14 -29.5q-10 -18 -35.5 -136.5t-48.5 -164.5q-15 -29 -50 -60.5t-67.5 -50.5t-72.5 -41t-48 -28q-47 -31 -151 -231q-341 14 -630 -158q-92 -53 -303 -179q47 16 86 31t55 22l15 7 + q71 27 163 64.5t133.5 53.5t108 34.5t142.5 31.5q186 31 465 -7q1 0 10 -3q11 -6 14 -17t-3 -22l-194 -345q-15 -29 -47 -22q-128 24 -354 24q-146 0 -402 -44.5t-392 -46.5q-82 -1 -149 13t-107 37t-61 40t-33 34l-1 1v2q0 6 6 6q138 0 371 55q192 366 374.5 524t383.5 158 + q5 0 14.5 -0.5t38 -5t55 -12t61.5 -24.5t63 -39.5t54 -59t40 -82.5l102 177q2 4 21 42.5t44.5 86.5t61 109.5t84 133.5t100.5 137q66 82 128 141.5t121.5 96.5t92.5 53.5t88 39.5z" /> + <glyph glyph-name="uniF2B0" unicode="" + d="M1322 640q0 -45 -5 -76l-236 14l224 -78q-19 -73 -58 -141l-214 103l177 -158q-44 -61 -107 -108l-157 178l103 -215q-61 -37 -140 -59l-79 228l14 -240q-38 -6 -76 -6t-76 6l14 238l-78 -226q-74 19 -140 59l103 215l-157 -178q-59 43 -108 108l178 158l-214 -104 + q-39 69 -58 141l224 79l-237 -14q-5 42 -5 76q0 35 5 77l238 -14l-225 79q19 73 58 140l214 -104l-177 159q46 61 107 108l158 -178l-103 215q67 39 140 58l77 -224l-13 236q36 6 75 6q38 0 76 -6l-14 -237l78 225q74 -19 140 -59l-103 -214l158 178q61 -47 107 -108 + l-177 -159l213 104q37 -62 58 -141l-224 -78l237 14q5 -31 5 -77zM1352 640q0 160 -78.5 295.5t-213 214t-292.5 78.5q-119 0 -227 -46.5t-186.5 -125t-124.5 -187.5t-46 -229q0 -119 46 -228t124.5 -187.5t186.5 -125t227 -46.5q158 0 292.5 78.5t213 214t78.5 294.5z + M1425 1023v-766l-657 -383l-657 383v766l657 383zM768 -183l708 412v823l-708 411l-708 -411v-823zM1536 1088v-896l-768 -448l-768 448v896l768 448z" /> + <glyph glyph-name="uniF2B1" unicode="" horiz-adv-x="1664" + d="M339 1318h691l-26 -72h-665q-110 0 -188.5 -79t-78.5 -189v-771q0 -95 60.5 -169.5t153.5 -93.5q23 -5 98 -5v-72h-45q-140 0 -239.5 100t-99.5 240v771q0 140 99.5 240t239.5 100zM1190 1536h247l-482 -1294q-23 -61 -40.5 -103.5t-45 -98t-54 -93.5t-64.5 -78.5 + t-79.5 -65t-95.5 -41t-116 -18.5v195q163 26 220 182q20 52 20 105q0 54 -20 106l-285 733h228l187 -585zM1664 978v-1111h-795q37 55 45 73h678v1038q0 85 -49.5 155t-129.5 99l25 67q101 -34 163.5 -123.5t62.5 -197.5z" /> + <glyph glyph-name="uniF2B2" unicode="" horiz-adv-x="1792" + d="M852 1227q0 -29 -17 -52.5t-45 -23.5t-45 23.5t-17 52.5t17 52.5t45 23.5t45 -23.5t17 -52.5zM688 -149v114q0 30 -20.5 51.5t-50.5 21.5t-50 -21.5t-20 -51.5v-114q0 -30 20.5 -52t49.5 -22q30 0 50.5 22t20.5 52zM860 -149v114q0 30 -20 51.5t-50 21.5t-50.5 -21.5 + t-20.5 -51.5v-114q0 -30 20.5 -52t50.5 -22q29 0 49.5 22t20.5 52zM1034 -149v114q0 30 -20.5 51.5t-50.5 21.5t-50.5 -21.5t-20.5 -51.5v-114q0 -30 20.5 -52t50.5 -22t50.5 22t20.5 52zM1208 -149v114q0 30 -20.5 51.5t-50.5 21.5t-50.5 -21.5t-20.5 -51.5v-114 + q0 -30 20.5 -52t50.5 -22t50.5 22t20.5 52zM1476 535q-84 -160 -232 -259.5t-323 -99.5q-123 0 -229.5 51.5t-178.5 137t-113 197.5t-41 232q0 88 21 174q-104 -175 -104 -390q0 -162 65 -312t185 -251q30 57 91 57q56 0 86 -50q32 50 87 50q56 0 86 -50q32 50 87 50t87 -50 + q30 50 86 50q28 0 52.5 -15.5t37.5 -40.5q112 94 177 231.5t73 287.5zM1326 564q0 75 -72 75q-17 0 -47 -6q-95 -19 -149 -19q-226 0 -226 243q0 86 30 204q-83 -127 -83 -275q0 -150 89 -260.5t235 -110.5q111 0 210 70q13 48 13 79zM884 1223q0 50 -32 89.5t-81 39.5 + t-81 -39.5t-32 -89.5q0 -51 31.5 -90.5t81.5 -39.5t81.5 39.5t31.5 90.5zM1513 884q0 96 -37.5 179t-113 137t-173.5 54q-77 0 -149 -35t-127 -94q-48 -159 -48 -268q0 -104 45.5 -157t147.5 -53q53 0 142 19q36 6 53 6q51 0 77.5 -28t26.5 -80q0 -26 -4 -46 + q75 68 117.5 165.5t42.5 200.5zM1792 667q0 -111 -33.5 -249.5t-93.5 -204.5q-58 -64 -195 -142.5t-228 -104.5l-4 -1v-114q0 -43 -29.5 -75t-72.5 -32q-56 0 -86 50q-32 -50 -87 -50t-87 50q-30 -50 -86 -50q-55 0 -87 50q-30 -50 -86 -50q-47 0 -75 33.5t-28 81.5 + q-90 -68 -198 -68q-118 0 -211 80q54 1 106 20q-113 31 -182 127q32 -7 71 -7q89 0 164 46q-192 192 -240 306q-24 56 -24 160q0 57 9 125.5t31.5 146.5t55 141t86.5 105t120 42q59 0 81 -52q19 29 42 54q2 3 12 13t13 16q10 15 23 38t25 42t28 39q87 111 211.5 177 + t260.5 66q35 0 62 -4q59 64 146 64q83 0 140 -57q5 -5 5 -12q0 -5 -6 -13.5t-12.5 -16t-16 -17l-10.5 -10.5q17 -6 36 -18t19 -24q0 -6 -16 -25q157 -138 197 -378q25 30 60 30q45 0 100 -49q90 -80 90 -279z" /> + <glyph glyph-name="uniF2B3" unicode="" + d="M917 631q0 33 -6 64h-362v-132h217q-12 -76 -74.5 -120.5t-142.5 -44.5q-99 0 -169 71.5t-70 170.5t70 170.5t169 71.5q93 0 153 -59l104 101q-108 100 -257 100q-160 0 -272 -112.5t-112 -271.5t112 -271.5t272 -112.5q165 0 266.5 105t101.5 270zM1262 585h109v110 + h-109v110h-110v-110h-110v-110h110v-110h110v110zM1536 640q0 -209 -103 -385.5t-279.5 -279.5t-385.5 -103t-385.5 103t-279.5 279.5t-103 385.5t103 385.5t279.5 279.5t385.5 103t385.5 -103t279.5 -279.5t103 -385.5z" /> + <glyph glyph-name="uniF2B4" unicode="" + d="M1536 1024v-839q0 -48 -49 -62q-174 -52 -338 -52q-73 0 -215.5 29.5t-227.5 29.5q-164 0 -370 -48v-338h-160v1368q-63 25 -101 81t-38 124q0 91 64 155t155 64t155 -64t64 -155q0 -68 -38 -124t-101 -81v-68q190 44 343 44q99 0 198 -15q14 -2 111.5 -22.5t149.5 -20.5 + q77 0 165 18q11 2 80 21t89 19q26 0 45 -19t19 -45z" /> + <glyph glyph-name="uniF2B5" unicode="" horiz-adv-x="2304" + d="M192 384q40 0 56 32t0 64t-56 32t-56 -32t0 -64t56 -32zM1665 442q-10 13 -38.5 50t-41.5 54t-38 49t-42.5 53t-40.5 47t-45 49l-125 -140q-83 -94 -208.5 -92t-205.5 98q-57 69 -56.5 158t58.5 157l177 206q-22 11 -51 16.5t-47.5 6t-56.5 -0.5t-49 -1q-92 0 -158 -66 + l-158 -158h-155v-544q5 0 21 0.5t22 0t19.5 -2t20.5 -4.5t17.5 -8.5t18.5 -13.5l297 -292q115 -111 227 -111q78 0 125 47q57 -20 112.5 8t72.5 85q74 -6 127 44q20 18 36 45.5t14 50.5q10 -10 43 -10q43 0 77 21t49.5 53t12 71.5t-30.5 73.5zM1824 384h96v512h-93l-157 180 + q-66 76 -169 76h-167q-89 0 -146 -67l-209 -243q-28 -33 -28 -75t27 -75q43 -51 110 -52t111 49l193 218q25 23 53.5 21.5t47 -27t8.5 -56.5q16 -19 56 -63t60 -68q29 -36 82.5 -105.5t64.5 -84.5q52 -66 60 -140zM2112 384q40 0 56 32t0 64t-56 32t-56 -32t0 -64t56 -32z + M2304 960v-640q0 -26 -19 -45t-45 -19h-434q-27 -65 -82 -106.5t-125 -51.5q-33 -48 -80.5 -81.5t-102.5 -45.5q-42 -53 -104.5 -81.5t-128.5 -24.5q-60 -34 -126 -39.5t-127.5 14t-117 53.5t-103.5 81l-287 282h-358q-26 0 -45 19t-19 45v672q0 26 19 45t45 19h421 + q14 14 47 48t47.5 48t44 40t50.5 37.5t51 25.5t62 19.5t68 5.5h117q99 0 181 -56q82 56 181 56h167q35 0 67 -6t56.5 -14.5t51.5 -26.5t44.5 -31t43 -39.5t39 -42t41 -48t41.5 -48.5h355q26 0 45 -19t19 -45z" /> + <glyph glyph-name="uniF2B6" unicode="" horiz-adv-x="1792" + d="M1792 882v-978q0 -66 -47 -113t-113 -47h-1472q-66 0 -113 47t-47 113v978q0 15 11 24q8 7 39 34.5t41.5 36t45.5 37.5t70 55.5t96 73t143.5 107t192.5 140.5q5 4 52.5 40t71.5 52.5t64 35t69 18.5t69 -18.5t65 -35.5t71 -52t52 -40q110 -80 192.5 -140.5t143.5 -107 + t96 -73t70 -55.5t45.5 -37.5t41.5 -36t39 -34.5q11 -9 11 -24zM1228 297q263 191 345 252q11 8 12.5 20.5t-6.5 23.5l-38 52q-8 11 -21 12.5t-24 -6.5q-231 -169 -343 -250q-5 -3 -52 -39t-71.5 -52.5t-64.5 -35t-69 -18.5t-69 18.5t-64.5 35t-71.5 52.5t-52 39 + q-186 134 -343 250q-11 8 -24 6.5t-21 -12.5l-38 -52q-8 -11 -6.5 -23.5t12.5 -20.5q82 -61 345 -252q10 -8 50 -38t65 -47t64 -39.5t77.5 -33.5t75.5 -11t75.5 11t79 34.5t64.5 39.5t65 47.5t48 36.5z" /> + <glyph glyph-name="uniF2B7" unicode="" horiz-adv-x="1792" + d="M1474 623l39 -51q8 -11 6.5 -23.5t-11.5 -20.5q-43 -34 -126.5 -98.5t-146.5 -113t-67 -51.5q-39 -32 -60 -48t-60.5 -41t-76.5 -36.5t-74 -11.5h-1h-1q-37 0 -74 11.5t-76 36.5t-61 41.5t-60 47.5q-5 4 -65 50.5t-143.5 111t-122.5 94.5q-11 8 -12.5 20.5t6.5 23.5 + l37 52q8 11 21.5 13t24.5 -7q94 -73 306 -236q5 -4 43.5 -35t60.5 -46.5t56.5 -32.5t58.5 -17h1h1q24 0 58.5 17t56.5 32.5t60.5 46.5t43.5 35q258 198 313 242q11 8 24 6.5t21 -12.5zM1664 -96v928q-90 83 -159 139q-91 74 -389 304q-3 2 -43 35t-61 48t-56 32.5t-59 17.5 + h-1h-1q-24 0 -59 -17.5t-56 -32.5t-61 -48t-43 -35q-215 -166 -315.5 -245.5t-129.5 -104t-82 -74.5q-14 -12 -21 -19v-928q0 -13 9.5 -22.5t22.5 -9.5h1472q13 0 22.5 9.5t9.5 22.5zM1792 832v-928q0 -66 -47 -113t-113 -47h-1472q-66 0 -113 47t-47 113v928q0 56 41 94 + q123 114 350 290.5t233 181.5q36 30 59 47.5t61.5 42t76 36.5t74.5 12h1h1q37 0 74.5 -12t76 -36.5t61.5 -42t59 -47.5q43 -36 156 -122t226 -177t201 -173q41 -38 41 -94z" /> + <glyph glyph-name="uniF2B8" unicode="" + d="M330 1l202 -214l-34 236l-216 213zM556 -225l274 218l-11 245l-300 -215zM245 413l227 -213l-48 327l-245 204zM495 189l317 214l-14 324l-352 -200zM843 178l95 -80l-2 239l-103 79q0 -1 1 -8.5t0 -12t-5 -7.5l-78 -52l85 -70q7 -6 7 -88zM138 930l256 -200l-68 465 + l-279 173zM1173 267l15 234l-230 -164l2 -240zM417 722l373 194l-19 441l-423 -163zM1270 357l20 233l-226 142l-2 -105l144 -95q6 -4 4 -9l-7 -119zM1461 496l30 222l-179 -128l-20 -228zM1273 329l-71 49l-8 -117q0 -5 -4 -8l-234 -187q-7 -5 -14 0l-98 83l7 -161 + q0 -5 -4 -8l-293 -234q-4 -2 -6 -2q-8 2 -8 3l-228 242q-4 4 -59 277q-2 7 5 11l61 37q-94 86 -95 92l-72 351q-2 7 6 12l94 45q-133 100 -135 108l-96 466q-2 10 7 13l433 135q5 0 8 -1l317 -153q6 -4 6 -9l20 -463q0 -7 -6 -10l-118 -61l126 -85q5 -2 5 -8l5 -123l121 74 + q5 4 11 0l84 -56l3 110q0 6 5 9l206 126q6 3 11 0l245 -135q4 -4 5 -7t-6.5 -60t-17.5 -124.5t-10 -70.5q0 -5 -4 -7l-191 -153q-6 -5 -13 0z" /> + <glyph glyph-name="uniF2B9" unicode="" horiz-adv-x="1664" + d="M1201 298q0 57 -5.5 107t-21 100.5t-39.5 86t-64 58t-91 22.5q-6 -4 -33.5 -20.5t-42.5 -24.5t-40.5 -20t-49 -17t-46.5 -5t-46.5 5t-49 17t-40.5 20t-42.5 24.5t-33.5 20.5q-51 0 -91 -22.5t-64 -58t-39.5 -86t-21 -100.5t-5.5 -107q0 -73 42 -121.5t103 -48.5h576 + q61 0 103 48.5t42 121.5zM1028 892q0 108 -76.5 184t-183.5 76t-183.5 -76t-76.5 -184q0 -107 76.5 -183t183.5 -76t183.5 76t76.5 183zM1664 352v-192q0 -14 -9 -23t-23 -9h-96v-224q0 -66 -47 -113t-113 -47h-1216q-66 0 -113 47t-47 113v1472q0 66 47 113t113 47h1216 + q66 0 113 -47t47 -113v-224h96q14 0 23 -9t9 -23v-192q0 -14 -9 -23t-23 -9h-96v-128h96q14 0 23 -9t9 -23v-192q0 -14 -9 -23t-23 -9h-96v-128h96q14 0 23 -9t9 -23z" /> + <glyph glyph-name="uniF2BA" unicode="" horiz-adv-x="1664" + d="M1028 892q0 -107 -76.5 -183t-183.5 -76t-183.5 76t-76.5 183q0 108 76.5 184t183.5 76t183.5 -76t76.5 -184zM980 672q46 0 82.5 -17t60 -47.5t39.5 -67t24 -81t11.5 -82.5t3.5 -79q0 -67 -39.5 -118.5t-105.5 -51.5h-576q-66 0 -105.5 51.5t-39.5 118.5q0 48 4.5 93.5 + t18.5 98.5t36.5 91.5t63 64.5t93.5 26h5q7 -4 32 -19.5t35.5 -21t33 -17t37 -16t35 -9t39.5 -4.5t39.5 4.5t35 9t37 16t33 17t35.5 21t32 19.5zM1664 928q0 -13 -9.5 -22.5t-22.5 -9.5h-96v-128h96q13 0 22.5 -9.5t9.5 -22.5v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-96v-128h96 + q13 0 22.5 -9.5t9.5 -22.5v-192q0 -13 -9.5 -22.5t-22.5 -9.5h-96v-224q0 -66 -47 -113t-113 -47h-1216q-66 0 -113 47t-47 113v1472q0 66 47 113t113 47h1216q66 0 113 -47t47 -113v-224h96q13 0 22.5 -9.5t9.5 -22.5v-192zM1408 -96v1472q0 13 -9.5 22.5t-22.5 9.5h-1216 + q-13 0 -22.5 -9.5t-9.5 -22.5v-1472q0 -13 9.5 -22.5t22.5 -9.5h1216q13 0 22.5 9.5t9.5 22.5z" /> + <glyph glyph-name="uniF2BB" unicode="" horiz-adv-x="2048" + d="M1024 405q0 64 -9 117.5t-29.5 103t-60.5 78t-97 28.5q-6 -4 -30 -18t-37.5 -21.5t-35.5 -17.5t-43 -14.5t-42 -4.5t-42 4.5t-43 14.5t-35.5 17.5t-37.5 21.5t-30 18q-57 0 -97 -28.5t-60.5 -78t-29.5 -103t-9 -117.5t37 -106.5t91 -42.5h512q54 0 91 42.5t37 106.5z + M867 925q0 94 -66.5 160.5t-160.5 66.5t-160.5 -66.5t-66.5 -160.5t66.5 -160.5t160.5 -66.5t160.5 66.5t66.5 160.5zM1792 416v64q0 14 -9 23t-23 9h-576q-14 0 -23 -9t-9 -23v-64q0 -14 9 -23t23 -9h576q14 0 23 9t9 23zM1792 676v56q0 15 -10.5 25.5t-25.5 10.5h-568 + q-15 0 -25.5 -10.5t-10.5 -25.5v-56q0 -15 10.5 -25.5t25.5 -10.5h568q15 0 25.5 10.5t10.5 25.5zM1792 928v64q0 14 -9 23t-23 9h-576q-14 0 -23 -9t-9 -23v-64q0 -14 9 -23t23 -9h576q14 0 23 9t9 23zM2048 1248v-1216q0 -66 -47 -113t-113 -47h-352v96q0 14 -9 23t-23 9 + h-64q-14 0 -23 -9t-9 -23v-96h-768v96q0 14 -9 23t-23 9h-64q-14 0 -23 -9t-9 -23v-96h-352q-66 0 -113 47t-47 113v1216q0 66 47 113t113 47h1728q66 0 113 -47t47 -113z" /> + <glyph glyph-name="uniF2BC" unicode="" horiz-adv-x="2048" + d="M1024 405q0 -64 -37 -106.5t-91 -42.5h-512q-54 0 -91 42.5t-37 106.5t9 117.5t29.5 103t60.5 78t97 28.5q6 -4 30 -18t37.5 -21.5t35.5 -17.5t43 -14.5t42 -4.5t42 4.5t43 14.5t35.5 17.5t37.5 21.5t30 18q57 0 97 -28.5t60.5 -78t29.5 -103t9 -117.5zM867 925 + q0 -94 -66.5 -160.5t-160.5 -66.5t-160.5 66.5t-66.5 160.5t66.5 160.5t160.5 66.5t160.5 -66.5t66.5 -160.5zM1792 480v-64q0 -14 -9 -23t-23 -9h-576q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h576q14 0 23 -9t9 -23zM1792 732v-56q0 -15 -10.5 -25.5t-25.5 -10.5h-568 + q-15 0 -25.5 10.5t-10.5 25.5v56q0 15 10.5 25.5t25.5 10.5h568q15 0 25.5 -10.5t10.5 -25.5zM1792 992v-64q0 -14 -9 -23t-23 -9h-576q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h576q14 0 23 -9t9 -23zM1920 32v1216q0 13 -9.5 22.5t-22.5 9.5h-1728q-13 0 -22.5 -9.5 + t-9.5 -22.5v-1216q0 -13 9.5 -22.5t22.5 -9.5h352v96q0 14 9 23t23 9h64q14 0 23 -9t9 -23v-96h768v96q0 14 9 23t23 9h64q14 0 23 -9t9 -23v-96h352q13 0 22.5 9.5t9.5 22.5zM2048 1248v-1216q0 -66 -47 -113t-113 -47h-1728q-66 0 -113 47t-47 113v1216q0 66 47 113 + t113 47h1728q66 0 113 -47t47 -113z" /> + <glyph glyph-name="uniF2BD" unicode="" horiz-adv-x="1792" + d="M1523 197q-22 155 -87.5 257.5t-184.5 118.5q-67 -74 -159.5 -115.5t-195.5 -41.5t-195.5 41.5t-159.5 115.5q-119 -16 -184.5 -118.5t-87.5 -257.5q106 -150 271 -237.5t356 -87.5t356 87.5t271 237.5zM1280 896q0 159 -112.5 271.5t-271.5 112.5t-271.5 -112.5 + t-112.5 -271.5t112.5 -271.5t271.5 -112.5t271.5 112.5t112.5 271.5zM1792 640q0 -182 -71 -347.5t-190.5 -286t-285.5 -191.5t-349 -71q-182 0 -348 71t-286 191t-191 286t-71 348t71 348t191 286t286 191t348 71t348 -71t286 -191t191 -286t71 -348z" /> + <glyph glyph-name="uniF2BE" unicode="" horiz-adv-x="1792" + d="M896 1536q182 0 348 -71t286 -191t191 -286t71 -348q0 -181 -70.5 -347t-190.5 -286t-286 -191.5t-349 -71.5t-349 71t-285.5 191.5t-190.5 286t-71 347.5t71 348t191 286t286 191t348 71zM1515 185q149 205 149 455q0 156 -61 298t-164 245t-245 164t-298 61t-298 -61 + t-245 -164t-164 -245t-61 -298q0 -250 149 -455q66 327 306 327q131 -128 313 -128t313 128q240 0 306 -327zM1280 832q0 159 -112.5 271.5t-271.5 112.5t-271.5 -112.5t-112.5 -271.5t112.5 -271.5t271.5 -112.5t271.5 112.5t112.5 271.5z" /> + <glyph glyph-name="uniF2C0" unicode="" + d="M1201 752q47 -14 89.5 -38t89 -73t79.5 -115.5t55 -172t22 -236.5q0 -154 -100 -263.5t-241 -109.5h-854q-141 0 -241 109.5t-100 263.5q0 131 22 236.5t55 172t79.5 115.5t89 73t89.5 38q-79 125 -79 272q0 104 40.5 198.5t109.5 163.5t163.5 109.5t198.5 40.5 + t198.5 -40.5t163.5 -109.5t109.5 -163.5t40.5 -198.5q0 -147 -79 -272zM768 1408q-159 0 -271.5 -112.5t-112.5 -271.5t112.5 -271.5t271.5 -112.5t271.5 112.5t112.5 271.5t-112.5 271.5t-271.5 112.5zM1195 -128q88 0 150.5 71.5t62.5 173.5q0 239 -78.5 377t-225.5 145 + q-145 -127 -336 -127t-336 127q-147 -7 -225.5 -145t-78.5 -377q0 -102 62.5 -173.5t150.5 -71.5h854z" /> + <glyph glyph-name="uniF2C1" unicode="" horiz-adv-x="1280" + d="M1024 278q0 -64 -37 -107t-91 -43h-512q-54 0 -91 43t-37 107t9 118t29.5 104t61 78.5t96.5 28.5q80 -75 188 -75t188 75q56 0 96.5 -28.5t61 -78.5t29.5 -104t9 -118zM870 797q0 -94 -67.5 -160.5t-162.5 -66.5t-162.5 66.5t-67.5 160.5t67.5 160.5t162.5 66.5 + t162.5 -66.5t67.5 -160.5zM1152 -96v1376h-1024v-1376q0 -13 9.5 -22.5t22.5 -9.5h960q13 0 22.5 9.5t9.5 22.5zM1280 1376v-1472q0 -66 -47 -113t-113 -47h-960q-66 0 -113 47t-47 113v1472q0 66 47 113t113 47h352v-96q0 -14 9 -23t23 -9h192q14 0 23 9t9 23v96h352 + q66 0 113 -47t47 -113z" /> + <glyph glyph-name="uniF2C2" unicode="" horiz-adv-x="2048" + d="M896 324q0 54 -7.5 100.5t-24.5 90t-51 68.5t-81 25q-64 -64 -156 -64t-156 64q-47 0 -81 -25t-51 -68.5t-24.5 -90t-7.5 -100.5q0 -55 31.5 -93.5t75.5 -38.5h426q44 0 75.5 38.5t31.5 93.5zM768 768q0 80 -56 136t-136 56t-136 -56t-56 -136t56 -136t136 -56t136 56 + t56 136zM1792 288v64q0 14 -9 23t-23 9h-704q-14 0 -23 -9t-9 -23v-64q0 -14 9 -23t23 -9h704q14 0 23 9t9 23zM1408 544v64q0 14 -9 23t-23 9h-320q-14 0 -23 -9t-9 -23v-64q0 -14 9 -23t23 -9h320q14 0 23 9t9 23zM1792 544v64q0 14 -9 23t-23 9h-192q-14 0 -23 -9t-9 -23 + v-64q0 -14 9 -23t23 -9h192q14 0 23 9t9 23zM1792 800v64q0 14 -9 23t-23 9h-704q-14 0 -23 -9t-9 -23v-64q0 -14 9 -23t23 -9h704q14 0 23 9t9 23zM128 1152h1792v96q0 14 -9 23t-23 9h-1728q-14 0 -23 -9t-9 -23v-96zM2048 1248v-1216q0 -66 -47 -113t-113 -47h-1728 + q-66 0 -113 47t-47 113v1216q0 66 47 113t113 47h1728q66 0 113 -47t47 -113z" /> + <glyph glyph-name="uniF2C3" unicode="" horiz-adv-x="2048" + d="M896 324q0 -55 -31.5 -93.5t-75.5 -38.5h-426q-44 0 -75.5 38.5t-31.5 93.5q0 54 7.5 100.5t24.5 90t51 68.5t81 25q64 -64 156 -64t156 64q47 0 81 -25t51 -68.5t24.5 -90t7.5 -100.5zM768 768q0 -80 -56 -136t-136 -56t-136 56t-56 136t56 136t136 56t136 -56t56 -136z + M1792 352v-64q0 -14 -9 -23t-23 -9h-704q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h704q14 0 23 -9t9 -23zM1408 608v-64q0 -14 -9 -23t-23 -9h-320q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h320q14 0 23 -9t9 -23zM1792 608v-64q0 -14 -9 -23t-23 -9h-192q-14 0 -23 9t-9 23v64 + q0 14 9 23t23 9h192q14 0 23 -9t9 -23zM1792 864v-64q0 -14 -9 -23t-23 -9h-704q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h704q14 0 23 -9t9 -23zM1920 32v1120h-1792v-1120q0 -13 9.5 -22.5t22.5 -9.5h1728q13 0 22.5 9.5t9.5 22.5zM2048 1248v-1216q0 -66 -47 -113t-113 -47 + h-1728q-66 0 -113 47t-47 113v1216q0 66 47 113t113 47h1728q66 0 113 -47t47 -113z" /> + <glyph glyph-name="uniF2C4" unicode="" horiz-adv-x="1792" + d="M1255 749q0 318 -105 474.5t-330 156.5q-222 0 -326 -157t-104 -474q0 -316 104 -471.5t326 -155.5q74 0 131 17q-22 43 -39 73t-44 65t-53.5 56.5t-63 36t-77.5 14.5q-46 0 -79 -16l-49 97q105 91 276 91q132 0 215.5 -54t150.5 -155q67 149 67 402zM1645 117h117 + q3 -27 -2 -67t-26.5 -95t-58 -100.5t-107 -78t-162.5 -32.5q-71 0 -130.5 19t-105.5 56t-79 78t-66 96q-97 -27 -205 -27q-150 0 -292.5 58t-253 158.5t-178 249t-67.5 317.5q0 170 67.5 319.5t178.5 250.5t253.5 159t291.5 58q121 0 238.5 -36t217 -106t176 -164.5 + t119.5 -219t43 -261.5q0 -190 -80.5 -347.5t-218.5 -264.5q47 -70 93.5 -106.5t104.5 -36.5q61 0 94 37.5t38 85.5z" /> + <glyph glyph-name="uniF2C5" unicode="" horiz-adv-x="2304" + d="M453 -101q0 -21 -16 -37.5t-37 -16.5q-1 0 -13 3q-63 15 -162 140q-225 284 -225 676q0 341 213 614q39 51 95 103.5t94 52.5q19 0 35 -13.5t16 -32.5q0 -27 -63 -90q-98 -102 -147 -184q-119 -199 -119 -449q0 -281 123 -491q50 -85 136 -173q2 -3 14.5 -16t19.5 -21 + t17 -20.5t14.5 -23.5t4.5 -21zM1796 33q0 -29 -17.5 -48.5t-46.5 -19.5h-1081q-26 0 -45 19t-19 45q0 29 17.5 48.5t46.5 19.5h1081q26 0 45 -19t19 -45zM1581 644q0 -134 -67 -233q-25 -38 -69.5 -78.5t-83.5 -60.5q-16 -10 -27 -10q-7 0 -15 6t-8 12q0 9 19 30t42 46 + t42 67.5t19 88.5q0 76 -35 130q-29 42 -46 42q-3 0 -3 -5q0 -12 7.5 -35.5t7.5 -36.5q0 -22 -21.5 -35t-44.5 -13q-66 0 -66 76q0 15 1.5 44t1.5 44q0 25 -10 46q-13 25 -42 53.5t-51 28.5q-5 0 -7 -0.5t-3.5 -2.5t-1.5 -6q0 -2 16 -26t16 -54q0 -37 -19 -68t-46 -54 + t-53.5 -46t-45.5 -54t-19 -68q0 -98 42 -160q29 -43 79 -63q16 -5 17 -10q1 -2 1 -5q0 -16 -18 -16q-6 0 -33 11q-119 43 -195 139.5t-76 218.5q0 55 24.5 115.5t60 115t70.5 108.5t59.5 113.5t24.5 111.5q0 53 -25 94q-29 48 -56 64q-19 9 -19 21q0 20 41 20q50 0 110 -29 + q41 -19 71 -44.5t49.5 -51t33.5 -62.5t22 -69t16 -80q0 -1 3 -17.5t4.5 -25t5.5 -25t9 -27t11 -21.5t14.5 -16.5t18.5 -5.5q23 0 37 14t14 37q0 25 -20 67t-20 52t10 10q27 0 93 -70q72 -76 102.5 -156t30.5 -186zM2304 615q0 -274 -138 -503q-19 -32 -48 -72t-68 -86.5 + t-81 -77t-74 -30.5q-16 0 -31 15.5t-15 31.5q0 15 29 50.5t68.5 77t48.5 52.5q183 230 183 531q0 131 -20.5 235t-72.5 211q-58 119 -163 228q-2 3 -13 13.5t-16.5 16.5t-15 17.5t-15 20t-9.5 18.5t-4 19q0 19 16 35.5t35 16.5q70 0 196 -169q98 -131 146 -273t60 -314 + q2 -42 2 -64z" /> + <glyph glyph-name="uniF2C6" unicode="" horiz-adv-x="1792" + d="M1189 229l147 693q9 44 -10.5 63t-51.5 7l-864 -333q-29 -11 -39.5 -25t-2.5 -26.5t32 -19.5l221 -69l513 323q21 14 32 6q7 -5 -4 -15l-415 -375v0v0l-16 -228q23 0 45 22l108 104l224 -165q64 -36 81 38zM1792 640q0 -182 -71 -348t-191 -286t-286 -191t-348 -71 + t-348 71t-286 191t-191 286t-71 348t71 348t191 286t286 191t348 71t348 -71t286 -191t191 -286t71 -348z" /> + <glyph glyph-name="uniF2C7" unicode="" horiz-adv-x="1024" + d="M640 192q0 -80 -56 -136t-136 -56t-136 56t-56 136q0 60 35 110t93 71v907h128v-907q58 -21 93 -71t35 -110zM768 192q0 77 -34 144t-94 112v768q0 80 -56 136t-136 56t-136 -56t-56 -136v-768q-60 -45 -94 -112t-34 -144q0 -133 93.5 -226.5t226.5 -93.5t226.5 93.5 + t93.5 226.5zM896 192q0 -185 -131.5 -316.5t-316.5 -131.5t-316.5 131.5t-131.5 316.5q0 182 128 313v711q0 133 93.5 226.5t226.5 93.5t226.5 -93.5t93.5 -226.5v-711q128 -131 128 -313zM1024 768v-128h-192v128h192zM1024 1024v-128h-192v128h192zM1024 1280v-128h-192 + v128h192z" /> + <glyph glyph-name="uniF2C8" unicode="" horiz-adv-x="1024" + d="M640 192q0 -80 -56 -136t-136 -56t-136 56t-56 136q0 60 35 110t93 71v651h128v-651q58 -21 93 -71t35 -110zM768 192q0 77 -34 144t-94 112v768q0 80 -56 136t-136 56t-136 -56t-56 -136v-768q-60 -45 -94 -112t-34 -144q0 -133 93.5 -226.5t226.5 -93.5t226.5 93.5 + t93.5 226.5zM896 192q0 -185 -131.5 -316.5t-316.5 -131.5t-316.5 131.5t-131.5 316.5q0 182 128 313v711q0 133 93.5 226.5t226.5 93.5t226.5 -93.5t93.5 -226.5v-711q128 -131 128 -313zM1024 768v-128h-192v128h192zM1024 1024v-128h-192v128h192zM1024 1280v-128h-192 + v128h192z" /> + <glyph glyph-name="uniF2C9" unicode="" horiz-adv-x="1024" + d="M640 192q0 -80 -56 -136t-136 -56t-136 56t-56 136q0 60 35 110t93 71v395h128v-395q58 -21 93 -71t35 -110zM768 192q0 77 -34 144t-94 112v768q0 80 -56 136t-136 56t-136 -56t-56 -136v-768q-60 -45 -94 -112t-34 -144q0 -133 93.5 -226.5t226.5 -93.5t226.5 93.5 + t93.5 226.5zM896 192q0 -185 -131.5 -316.5t-316.5 -131.5t-316.5 131.5t-131.5 316.5q0 182 128 313v711q0 133 93.5 226.5t226.5 93.5t226.5 -93.5t93.5 -226.5v-711q128 -131 128 -313zM1024 768v-128h-192v128h192zM1024 1024v-128h-192v128h192zM1024 1280v-128h-192 + v128h192z" /> + <glyph glyph-name="uniF2CA" unicode="" horiz-adv-x="1024" + d="M640 192q0 -80 -56 -136t-136 -56t-136 56t-56 136q0 60 35 110t93 71v139h128v-139q58 -21 93 -71t35 -110zM768 192q0 77 -34 144t-94 112v768q0 80 -56 136t-136 56t-136 -56t-56 -136v-768q-60 -45 -94 -112t-34 -144q0 -133 93.5 -226.5t226.5 -93.5t226.5 93.5 + t93.5 226.5zM896 192q0 -185 -131.5 -316.5t-316.5 -131.5t-316.5 131.5t-131.5 316.5q0 182 128 313v711q0 133 93.5 226.5t226.5 93.5t226.5 -93.5t93.5 -226.5v-711q128 -131 128 -313zM1024 768v-128h-192v128h192zM1024 1024v-128h-192v128h192zM1024 1280v-128h-192 + v128h192z" /> + <glyph glyph-name="uniF2CB" unicode="" horiz-adv-x="1024" + d="M640 192q0 -80 -56 -136t-136 -56t-136 56t-56 136q0 79 56 135.5t136 56.5t136 -56.5t56 -135.5zM768 192q0 77 -34 144t-94 112v768q0 80 -56 136t-136 56t-136 -56t-56 -136v-768q-60 -45 -94 -112t-34 -144q0 -133 93.5 -226.5t226.5 -93.5t226.5 93.5t93.5 226.5z + M896 192q0 -185 -131.5 -316.5t-316.5 -131.5t-316.5 131.5t-131.5 316.5q0 182 128 313v711q0 133 93.5 226.5t226.5 93.5t226.5 -93.5t93.5 -226.5v-711q128 -131 128 -313zM1024 768v-128h-192v128h192zM1024 1024v-128h-192v128h192zM1024 1280v-128h-192v128h192z" /> + <glyph glyph-name="uniF2CC" unicode="" horiz-adv-x="1920" + d="M1433 1287q10 -10 10 -23t-10 -23l-626 -626q-10 -10 -23 -10t-23 10l-82 82q-10 10 -10 23t10 23l44 44q-72 91 -81.5 207t46.5 215q-74 71 -176 71q-106 0 -181 -75t-75 -181v-1280h-256v1280q0 104 40.5 198.5t109.5 163.5t163.5 109.5t198.5 40.5q106 0 201 -41 + t166 -115q94 39 197 24.5t185 -79.5l44 44q10 10 23 10t23 -10zM1344 1024q26 0 45 -19t19 -45t-19 -45t-45 -19t-45 19t-19 45t19 45t45 19zM1600 896q-26 0 -45 19t-19 45t19 45t45 19t45 -19t19 -45t-19 -45t-45 -19zM1856 1024q26 0 45 -19t19 -45t-19 -45t-45 -19 + t-45 19t-19 45t19 45t45 19zM1216 896q26 0 45 -19t19 -45t-19 -45t-45 -19t-45 19t-19 45t19 45t45 19zM1408 832q0 26 19 45t45 19t45 -19t19 -45t-19 -45t-45 -19t-45 19t-19 45zM1728 896q26 0 45 -19t19 -45t-19 -45t-45 -19t-45 19t-19 45t19 45t45 19zM1088 768 + q26 0 45 -19t19 -45t-19 -45t-45 -19t-45 19t-19 45t19 45t45 19zM1344 640q-26 0 -45 19t-19 45t19 45t45 19t45 -19t19 -45t-19 -45t-45 -19zM1600 768q26 0 45 -19t19 -45t-19 -45t-45 -19t-45 19t-19 45t19 45t45 19zM1216 512q-26 0 -45 19t-19 45t19 45t45 19t45 -19 + t19 -45t-19 -45t-45 -19zM1472 640q26 0 45 -19t19 -45t-19 -45t-45 -19t-45 19t-19 45t19 45t45 19zM1088 512q26 0 45 -19t19 -45t-19 -45t-45 -19t-45 19t-19 45t19 45t45 19zM1344 512q26 0 45 -19t19 -45t-19 -45t-45 -19t-45 19t-19 45t19 45t45 19zM1216 384 + q26 0 45 -19t19 -45t-19 -45t-45 -19t-45 19t-19 45t19 45t45 19zM1088 256q26 0 45 -19t19 -45t-19 -45t-45 -19t-45 19t-19 45t19 45t45 19z" /> + <glyph glyph-name="uniF2CD" unicode="" horiz-adv-x="1792" + d="M1664 448v-192q0 -169 -128 -286v-194q0 -14 -9 -23t-23 -9h-64q-14 0 -23 9t-9 23v118q-63 -22 -128 -22h-768q-65 0 -128 22v-110q0 -17 -9.5 -28.5t-22.5 -11.5h-64q-13 0 -22.5 11.5t-9.5 28.5v186q-128 117 -128 286v192h1536zM704 864q0 -14 -9 -23t-23 -9t-23 9 + t-9 23t9 23t23 9t23 -9t9 -23zM768 928q0 -14 -9 -23t-23 -9t-23 9t-9 23t9 23t23 9t23 -9t9 -23zM704 992q0 -14 -9 -23t-23 -9t-23 9t-9 23t9 23t23 9t23 -9t9 -23zM832 992q0 -14 -9 -23t-23 -9t-23 9t-9 23t9 23t23 9t23 -9t9 -23zM768 1056q0 -14 -9 -23t-23 -9t-23 9 + t-9 23t9 23t23 9t23 -9t9 -23zM704 1120q0 -14 -9 -23t-23 -9t-23 9t-9 23t9 23t23 9t23 -9t9 -23zM1792 608v-64q0 -14 -9 -23t-23 -9h-1728q-14 0 -23 9t-9 23v64q0 14 9 23t23 9h96v640q0 106 75 181t181 75q108 0 184 -78q46 19 98 12t93 -39l22 22q11 11 22 0l42 -42 + q11 -11 0 -22l-314 -314q-11 -11 -22 0l-42 42q-11 11 0 22l22 22q-36 46 -40.5 104t23.5 108q-37 35 -88 35q-53 0 -90.5 -37.5t-37.5 -90.5v-640h1504q14 0 23 -9t9 -23zM896 1056q0 -14 -9 -23t-23 -9t-23 9t-9 23t9 23t23 9t23 -9t9 -23zM832 1120q0 -14 -9 -23t-23 -9 + t-23 9t-9 23t9 23t23 9t23 -9t9 -23zM768 1184q0 -14 -9 -23t-23 -9t-23 9t-9 23t9 23t23 9t23 -9t9 -23zM960 1120q0 -14 -9 -23t-23 -9t-23 9t-9 23t9 23t23 9t23 -9t9 -23zM896 1184q0 -14 -9 -23t-23 -9t-23 9t-9 23t9 23t23 9t23 -9t9 -23zM832 1248q0 -14 -9 -23 + t-23 -9t-23 9t-9 23t9 23t23 9t23 -9t9 -23zM1024 1184q0 -14 -9 -23t-23 -9t-23 9t-9 23t9 23t23 9t23 -9t9 -23zM960 1248q0 -14 -9 -23t-23 -9t-23 9t-9 23t9 23t23 9t23 -9t9 -23zM1088 1248q0 -14 -9 -23t-23 -9t-23 9t-9 23t9 23t23 9t23 -9t9 -23z" /> + <glyph glyph-name="uniF2CE" unicode="" + d="M994 344q0 -86 -17 -197q-31 -215 -55 -313q-22 -90 -152 -90t-152 90q-24 98 -55 313q-17 110 -17 197q0 168 224 168t224 -168zM1536 768q0 -240 -134 -434t-350 -280q-8 -3 -15 3t-6 15q7 48 10 66q4 32 6 47q1 9 9 12q159 81 255.5 234t96.5 337q0 180 -91 330.5 + t-247 234.5t-337 74q-124 -7 -237 -61t-193.5 -140.5t-128 -202t-46.5 -240.5q1 -184 99 -336.5t257 -231.5q7 -3 9 -12q3 -21 6 -45q1 -9 5 -32.5t6 -35.5q1 -9 -6.5 -15t-15.5 -2q-148 58 -261 169.5t-173.5 264t-52.5 319.5q7 143 66 273.5t154.5 227t225 157.5t272.5 70 + q164 10 315.5 -46.5t261 -160.5t175 -250.5t65.5 -308.5zM994 800q0 -93 -65.5 -158.5t-158.5 -65.5t-158.5 65.5t-65.5 158.5t65.5 158.5t158.5 65.5t158.5 -65.5t65.5 -158.5zM1282 768q0 -122 -53.5 -228.5t-146.5 -177.5q-8 -6 -16 -2t-10 14q-6 52 -29 92q-7 10 3 20 + q58 54 91 127t33 155q0 111 -58.5 204t-157.5 141.5t-212 36.5q-133 -15 -229 -113t-109 -231q-10 -92 23.5 -176t98.5 -144q10 -10 3 -20q-24 -41 -29 -93q-2 -9 -10 -13t-16 2q-95 74 -148.5 183t-51.5 234q3 131 69 244t177 181.5t241 74.5q144 7 268 -60t196.5 -187.5 + t72.5 -263.5z" /> + <glyph glyph-name="uniF2D0" unicode="" horiz-adv-x="1792" + d="M256 128h1280v768h-1280v-768zM1792 1248v-1216q0 -66 -47 -113t-113 -47h-1472q-66 0 -113 47t-47 113v1216q0 66 47 113t113 47h1472q66 0 113 -47t47 -113z" /> + <glyph glyph-name="uniF2D1" unicode="" horiz-adv-x="1792" + d="M1792 224v-192q0 -66 -47 -113t-113 -47h-1472q-66 0 -113 47t-47 113v192q0 66 47 113t113 47h1472q66 0 113 -47t47 -113z" /> + <glyph glyph-name="uniF2D2" unicode="" horiz-adv-x="2048" + d="M256 0h768v512h-768v-512zM1280 512h512v768h-768v-256h96q66 0 113 -47t47 -113v-352zM2048 1376v-960q0 -66 -47 -113t-113 -47h-608v-352q0 -66 -47 -113t-113 -47h-960q-66 0 -113 47t-47 113v960q0 66 47 113t113 47h608v352q0 66 47 113t113 47h960q66 0 113 -47 + t47 -113z" /> + <glyph glyph-name="uniF2D3" unicode="" horiz-adv-x="1792" + d="M1175 215l146 146q10 10 10 23t-10 23l-233 233l233 233q10 10 10 23t-10 23l-146 146q-10 10 -23 10t-23 -10l-233 -233l-233 233q-10 10 -23 10t-23 -10l-146 -146q-10 -10 -10 -23t10 -23l233 -233l-233 -233q-10 -10 -10 -23t10 -23l146 -146q10 -10 23 -10t23 10 + l233 233l233 -233q10 -10 23 -10t23 10zM1792 1248v-1216q0 -66 -47 -113t-113 -47h-1472q-66 0 -113 47t-47 113v1216q0 66 47 113t113 47h1472q66 0 113 -47t47 -113z" /> + <glyph glyph-name="uniF2D4" unicode="" horiz-adv-x="1792" + d="M1257 425l-146 -146q-10 -10 -23 -10t-23 10l-169 169l-169 -169q-10 -10 -23 -10t-23 10l-146 146q-10 10 -10 23t10 23l169 169l-169 169q-10 10 -10 23t10 23l146 146q10 10 23 10t23 -10l169 -169l169 169q10 10 23 10t23 -10l146 -146q10 -10 10 -23t-10 -23 + l-169 -169l169 -169q10 -10 10 -23t-10 -23zM256 128h1280v1024h-1280v-1024zM1792 1248v-1216q0 -66 -47 -113t-113 -47h-1472q-66 0 -113 47t-47 113v1216q0 66 47 113t113 47h1472q66 0 113 -47t47 -113z" /> + <glyph glyph-name="uniF2D5" unicode="" horiz-adv-x="1792" + d="M1070 358l306 564h-654l-306 -564h654zM1792 640q0 -182 -71 -348t-191 -286t-286 -191t-348 -71t-348 71t-286 191t-191 286t-71 348t71 348t191 286t286 191t348 71t348 -71t286 -191t191 -286t71 -348z" /> + <glyph glyph-name="uniF2D6" unicode="" horiz-adv-x="1794" + d="M1291 1060q-15 17 -35 8.5t-26 -28.5t5 -38q14 -17 40 -14.5t34 20.5t-18 52zM895 814q-8 -8 -19.5 -8t-18.5 8q-8 8 -8 19t8 18q7 8 18.5 8t19.5 -8q7 -7 7 -18t-7 -19zM1060 740l-35 -35q-12 -13 -29.5 -13t-30.5 13l-38 38q-12 13 -12 30t12 30l35 35q12 12 29.5 12 + t30.5 -12l38 -39q12 -12 12 -29.5t-12 -29.5zM951 870q-7 -8 -18.5 -8t-19.5 8q-7 8 -7 19t7 19q8 8 19 8t19 -8t8 -19t-8 -19zM1354 968q-34 -64 -107.5 -85.5t-127.5 16.5q-38 28 -61 66.5t-21 87.5t39 92t75.5 53t70.5 -5t70 -51q2 -2 13 -12.5t14.5 -13.5t13 -13.5 + t12.5 -15.5t10 -15.5t8.5 -18t4 -18.5t1 -21t-5 -22t-9.5 -24zM1555 486q3 20 -8.5 34.5t-27.5 21.5t-33 17t-23 20q-40 71 -84 98.5t-113 11.5q19 13 40 18.5t33 4.5l12 -1q2 45 -34 90q6 20 6.5 40.5t-2.5 30.5l-3 10q43 24 71 65t34 91q10 84 -43 150.5t-137 76.5 + q-60 7 -114 -18.5t-82 -74.5q-30 -51 -33.5 -101t14.5 -87t43.5 -64t56.5 -42q-45 4 -88 36t-57 88q-28 108 32 222q-16 21 -29 32q-50 0 -89 -19q19 24 42 37t36 14l13 1q0 50 -13 78q-10 21 -32.5 28.5t-47 -3.5t-37.5 -40q2 4 4 7q-7 -28 -6.5 -75.5t19 -117t48.5 -122.5 + q-25 -14 -47 -36q-35 -16 -85.5 -70.5t-84.5 -101.5l-33 -46q-90 -34 -181 -125.5t-75 -162.5q1 -16 11 -27q-15 -12 -30 -30q-21 -25 -21 -54t21.5 -40t63.5 6q41 19 77 49.5t55 60.5q-2 2 -6.5 5t-20.5 7.5t-33 3.5q23 5 51 12.5t40 10t27.5 6t26 4t23.5 0.5q14 -7 22 34 + q7 37 7 90q0 102 -40 150q106 -103 101 -219q-1 -29 -15 -50t-27 -27l-13 -6q-4 -7 -19 -32t-26 -45.5t-26.5 -52t-25 -61t-17 -63t-6.5 -66.5t10 -63q-35 54 -37 80q-22 -24 -34.5 -39t-33.5 -42t-30.5 -46t-16.5 -41t-0.5 -38t25.5 -27q45 -25 144 64t190.5 221.5 + t122.5 228.5q86 52 145 115.5t86 119.5q47 -93 154 -178q104 -83 167 -80q39 2 46 43zM1794 640q0 -182 -71 -348t-191 -286t-286.5 -191t-348.5 -71t-348.5 71t-286.5 191t-191 286t-71 348t71 348t191 286t286.5 191t348.5 71t348.5 -71t286.5 -191t191 -286t71 -348z" /> + <glyph glyph-name="uniF2D7" unicode="" + d="M518 1353v-655q103 -1 191.5 1.5t125.5 5.5l37 3q68 2 90.5 24.5t39.5 94.5l33 142h103l-14 -322l7 -319h-103l-29 127q-15 68 -45 93t-84 26q-87 8 -352 8v-556q0 -78 43.5 -115.5t133.5 -37.5h357q35 0 59.5 2t55 7.5t54 18t48.5 32t46 50.5t39 73l93 216h89 + q-6 -37 -31.5 -252t-30.5 -276q-146 5 -263.5 8t-162.5 4h-44h-628l-376 -12v102l127 25q67 13 91.5 37t25.5 79l8 643q3 402 -8 645q-2 61 -25.5 84t-91.5 36l-127 24v102l376 -12h702q139 0 374 27q-6 -68 -14 -194.5t-12 -219.5l-5 -92h-93l-32 124q-31 121 -74 179.5 + t-113 58.5h-548q-28 0 -35.5 -8.5t-7.5 -30.5z" /> + <glyph glyph-name="uniF2D8" unicode="" + d="M922 739v-182q0 -4 0.5 -15t0 -15l-1.5 -12t-3.5 -11.5t-6.5 -7.5t-11 -5.5t-16 -1.5v309q9 0 16 -1t11 -5t6.5 -5.5t3.5 -9.5t1 -10.5v-13.5v-14zM1238 643v-121q0 -1 0.5 -12.5t0 -15.5t-2.5 -11.5t-7.5 -10.5t-13.5 -3q-9 0 -14 9q-4 10 -4 165v7v8.5v9t1.5 8.5l3.5 7 + t5 5.5t8 1.5q6 0 10 -1.5t6.5 -4.5t4 -6t2 -8.5t0.5 -8v-9.5v-9zM180 407h122v472h-122v-472zM614 407h106v472h-159l-28 -221q-20 148 -32 221h-158v-472h107v312l45 -312h76l43 319v-319zM1039 712q0 67 -5 90q-3 16 -11 28.5t-17 20.5t-25 14t-26.5 8.5t-31 4t-29 1.5 + h-29.5h-12h-91v-472h56q169 -1 197 24.5t25 180.5q-1 62 -1 100zM1356 515v133q0 29 -2 45t-9.5 33.5t-24.5 25t-46 7.5q-46 0 -77 -34v154h-117v-472h110l7 30q30 -36 77 -36q50 0 66 30.5t16 83.5zM1536 1248v-1216q0 -66 -47 -113t-113 -47h-1216q-66 0 -113 47t-47 113 + v1216q0 66 47 113t113 47h1216q66 0 113 -47t47 -113z" /> + <glyph glyph-name="uniF2D9" unicode="" horiz-adv-x="2176" + d="M1143 -197q-6 1 -11 4q-13 8 -36 23t-86 65t-116.5 104.5t-112 140t-89.5 172.5q-17 3 -175 37q66 -213 235 -362t391 -184zM502 409l168 -28q-25 76 -41 167.5t-19 145.5l-4 53q-84 -82 -121 -224q5 -65 17 -114zM612 1018q-43 -64 -77 -148q44 46 74 68zM2049 584 + q0 161 -62 307t-167.5 252t-250.5 168.5t-304 62.5q-147 0 -281 -52.5t-240 -148.5q-30 -58 -45 -160q60 51 143 83.5t158.5 43t143 13.5t108.5 -1l40 -3q33 -1 53 -15.5t24.5 -33t6.5 -37t-1 -28.5q-126 11 -227.5 0.5t-183 -43.5t-142.5 -71.5t-131 -98.5 + q4 -36 11.5 -92.5t35.5 -178t62 -179.5q123 -6 247.5 14.5t214.5 53.5t162.5 67t109.5 59l37 24q22 16 39.5 20.5t30.5 -5t17 -34.5q14 -97 -39 -121q-208 -97 -467 -134q-135 -20 -317 -16q41 -96 110 -176.5t137 -127t130.5 -79t101.5 -43.5l39 -12q143 -23 263 15 + q195 99 314 289t119 418zM2123 621q-14 -135 -40 -212q-70 -208 -181.5 -346.5t-318.5 -253.5q-48 -33 -82 -44q-72 -26 -163 -16q-36 -3 -73 -3q-283 0 -504.5 173t-295.5 442q-1 0 -4 0.5t-5 0.5q-6 -50 2.5 -112.5t26 -115t36 -98t31.5 -71.5l14 -26q8 -12 54 -82 + q-71 38 -124.5 106.5t-78.5 140t-39.5 137t-17.5 107.5l-2 42q-5 2 -33.5 12.5t-48.5 18t-53 20.5t-57.5 25t-50 25.5t-42.5 27t-25 25.5q19 -10 50.5 -25.5t113 -45.5t145.5 -38l2 32q11 149 94 290q41 202 176 365q28 115 81 214q15 28 32 45t49 32q158 74 303.5 104 + t302 11t306.5 -97q220 -115 333 -336t87 -474z" /> + <glyph glyph-name="uniF2DA" unicode="" horiz-adv-x="1792" + d="M1341 752q29 44 -6.5 129.5t-121.5 142.5q-58 39 -125.5 53.5t-118 4.5t-68.5 -37q-12 -23 -4.5 -28t42.5 -10q23 -3 38.5 -5t44.5 -9.5t56 -17.5q36 -13 67.5 -31.5t53 -37t40 -38.5t30.5 -38t22 -34.5t16.5 -28.5t12 -18.5t10.5 -6t11 9.5zM1704 178 + q-52 -127 -148.5 -220t-214.5 -141.5t-253 -60.5t-266 13.5t-251 91t-210 161.5t-141.5 235.5t-46.5 303.5q1 41 8.5 84.5t12.5 64t24 80.5t23 73q-51 -208 1 -397t173 -318t291 -206t346 -83t349 74.5t289 244.5q20 27 18 14q0 -4 -4 -14zM1465 627q0 -104 -40.5 -199 + t-108.5 -164t-162 -109.5t-198 -40.5t-198 40.5t-162 109.5t-108.5 164t-40.5 199t40.5 199t108.5 164t162 109.5t198 40.5t198 -40.5t162 -109.5t108.5 -164t40.5 -199zM1752 915q-65 147 -180.5 251t-253 153.5t-292 53.5t-301 -36.5t-275.5 -129t-220 -211.5t-131 -297 + t-10 -373q-49 161 -51.5 311.5t35.5 272.5t109 227t165.5 180.5t207 126t232 71t242.5 9t236 -54t216 -124.5t178 -197q33 -50 62 -121t31 -112zM1690 573q12 244 -136.5 416t-396.5 240q-8 0 -10 5t24 8q125 -4 230 -50t173 -120t116 -168.5t58.5 -199t-1 -208 + t-61.5 -197.5t-122.5 -167t-185 -117.5t-248.5 -46.5q108 30 201.5 80t174 123t129.5 176.5t55 225.5z" /> + <glyph glyph-name="uniF2DB" unicode="" + d="M192 256v-128h-112q-16 0 -16 16v16h-48q-16 0 -16 16v32q0 16 16 16h48v16q0 16 16 16h112zM192 512v-128h-112q-16 0 -16 16v16h-48q-16 0 -16 16v32q0 16 16 16h48v16q0 16 16 16h112zM192 768v-128h-112q-16 0 -16 16v16h-48q-16 0 -16 16v32q0 16 16 16h48v16 + q0 16 16 16h112zM192 1024v-128h-112q-16 0 -16 16v16h-48q-16 0 -16 16v32q0 16 16 16h48v16q0 16 16 16h112zM192 1280v-128h-112q-16 0 -16 16v16h-48q-16 0 -16 16v32q0 16 16 16h48v16q0 16 16 16h112zM1280 1440v-1472q0 -40 -28 -68t-68 -28h-832q-40 0 -68 28 + t-28 68v1472q0 40 28 68t68 28h832q40 0 68 -28t28 -68zM1536 208v-32q0 -16 -16 -16h-48v-16q0 -16 -16 -16h-112v128h112q16 0 16 -16v-16h48q16 0 16 -16zM1536 464v-32q0 -16 -16 -16h-48v-16q0 -16 -16 -16h-112v128h112q16 0 16 -16v-16h48q16 0 16 -16zM1536 720v-32 + q0 -16 -16 -16h-48v-16q0 -16 -16 -16h-112v128h112q16 0 16 -16v-16h48q16 0 16 -16zM1536 976v-32q0 -16 -16 -16h-48v-16q0 -16 -16 -16h-112v128h112q16 0 16 -16v-16h48q16 0 16 -16zM1536 1232v-32q0 -16 -16 -16h-48v-16q0 -16 -16 -16h-112v128h112q16 0 16 -16v-16 + h48q16 0 16 -16z" /> + <glyph glyph-name="uniF2DC" unicode="" horiz-adv-x="1664" + d="M1566 419l-167 -33l186 -107q23 -13 29.5 -38.5t-6.5 -48.5q-14 -23 -39 -29.5t-48 6.5l-186 106l55 -160q13 -38 -12 -63.5t-60.5 -20.5t-48.5 42l-102 300l-271 156v-313l208 -238q16 -18 17 -39t-11 -36.5t-28.5 -25t-37 -5.5t-36.5 22l-112 128v-214q0 -26 -19 -45 + t-45 -19t-45 19t-19 45v214l-112 -128q-16 -18 -36.5 -22t-37 5.5t-28.5 25t-11 36.5t17 39l208 238v313l-271 -156l-102 -300q-13 -37 -48.5 -42t-60.5 20.5t-12 63.5l55 160l-186 -106q-23 -13 -48 -6.5t-39 29.5q-13 23 -6.5 48.5t29.5 38.5l186 107l-167 33 + q-29 6 -42 29t-8.5 46.5t25.5 40t50 10.5l310 -62l271 157l-271 157l-310 -62q-4 -1 -13 -1q-27 0 -44 18t-19 40t11 43t40 26l167 33l-186 107q-23 13 -29.5 38.5t6.5 48.5t39 30t48 -7l186 -106l-55 160q-13 38 12 63.5t60.5 20.5t48.5 -42l102 -300l271 -156v313 + l-208 238q-16 18 -17 39t11 36.5t28.5 25t37 5.5t36.5 -22l112 -128v214q0 26 19 45t45 19t45 -19t19 -45v-214l112 128q16 18 36.5 22t37 -5.5t28.5 -25t11 -36.5t-17 -39l-208 -238v-313l271 156l102 300q13 37 48.5 42t60.5 -20.5t12 -63.5l-55 -160l186 106 + q23 13 48 6.5t39 -29.5q13 -23 6.5 -48.5t-29.5 -38.5l-186 -107l167 -33q27 -5 40 -26t11 -43t-19 -40t-44 -18q-9 0 -13 1l-310 62l-271 -157l271 -157l310 62q29 6 50 -10.5t25.5 -40t-8.5 -46.5t-42 -29z" /> + <glyph glyph-name="uniF2DD" unicode="" horiz-adv-x="1792" + d="M1473 607q7 118 -33 226.5t-113 189t-177 131t-221 57.5q-116 7 -225.5 -32t-192 -110.5t-135 -175t-59.5 -220.5q-7 -118 33 -226.5t113 -189t177.5 -131t221.5 -57.5q155 -9 293 59t224 195.5t94 283.5zM1792 1536l-349 -348q120 -117 180.5 -272t50.5 -321 + q-11 -183 -102 -339t-241 -255.5t-332 -124.5l-999 -132l347 347q-120 116 -180.5 271.5t-50.5 321.5q11 184 102 340t241.5 255.5t332.5 124.5q167 22 500 66t500 66z" /> + <glyph glyph-name="uniF2DE" unicode="" horiz-adv-x="1792" + d="M948 508l163 -329h-51l-175 350l-171 -350h-49l179 374l-78 33l21 49l240 -102l-21 -50zM563 1100l304 -130l-130 -304l-304 130zM907 915l240 -103l-103 -239l-239 102zM1188 765l191 -81l-82 -190l-190 81zM1680 640q0 159 -62 304t-167.5 250.5t-250.5 167.5t-304 62 + t-304 -62t-250.5 -167.5t-167.5 -250.5t-62 -304t62 -304t167.5 -250.5t250.5 -167.5t304 -62t304 62t250.5 167.5t167.5 250.5t62 304zM1792 640q0 -182 -71 -348t-191 -286t-286 -191t-348 -71t-348 71t-286 191t-191 286t-71 348t71 348t191 286t286 191t348 71t348 -71 + t286 -191t191 -286t71 -348z" /> + <glyph glyph-name="uniF2E0" unicode="" horiz-adv-x="1920" + d="M1334 302q-4 24 -27.5 34t-49.5 10.5t-48.5 12.5t-25.5 38q-5 47 33 139.5t75 181t32 127.5q-14 101 -117 103q-45 1 -75 -16l-3 -2l-5 -2.5t-4.5 -2t-5 -2t-5 -0.5t-6 1.5t-6 3.5t-6.5 5q-3 2 -9 8.5t-9 9t-8.5 7.5t-9.5 7.5t-9.5 5.5t-11 4.5t-11.5 2.5q-30 5 -48 -3 + t-45 -31q-1 -1 -9 -8.5t-12.5 -11t-15 -10t-16.5 -5.5t-17 3q-54 27 -84 40q-41 18 -94 -5t-76 -65q-16 -28 -41 -98.5t-43.5 -132.5t-40 -134t-21.5 -73q-22 -69 18.5 -119t110.5 -46q30 2 50.5 15t38.5 46q7 13 79 199.5t77 194.5q6 11 21.5 18t29.5 0q27 -15 21 -53 + q-2 -18 -51 -139.5t-50 -132.5q-6 -38 19.5 -56.5t60.5 -7t55 49.5q4 8 45.5 92t81.5 163.5t46 88.5q20 29 41 28q29 0 25 -38q-2 -16 -65.5 -147.5t-70.5 -159.5q-12 -53 13 -103t74 -74q17 -9 51 -15.5t71.5 -8t62.5 14t20 48.5zM383 86q3 -15 -5 -27.5t-23 -15.5 + q-14 -3 -26.5 5t-15.5 23q-3 14 5 27t22 16t27 -5t16 -23zM953 -177q12 -17 8.5 -37.5t-20.5 -32.5t-37.5 -8t-32.5 21q-11 17 -7.5 37.5t20.5 32.5t37.5 8t31.5 -21zM177 635q-18 -27 -49.5 -33t-57.5 13q-26 18 -32 50t12 58q18 27 49.5 33t57.5 -12q26 -19 32 -50.5 + t-12 -58.5zM1467 -42q19 -28 13 -61.5t-34 -52.5t-60.5 -13t-51.5 34t-13 61t33 53q28 19 60.5 13t52.5 -34zM1579 562q69 -113 42.5 -244.5t-134.5 -207.5q-90 -63 -199 -60q-20 -80 -84.5 -127t-143.5 -44.5t-140 57.5q-12 -9 -13 -10q-103 -71 -225 -48.5t-193 126.5 + q-50 73 -53 164q-83 14 -142.5 70.5t-80.5 128t-2 152t81 138.5q-36 60 -38 128t24.5 125t79.5 98.5t121 50.5q32 85 99 148t146.5 91.5t168 17t159.5 -66.5q72 21 140 17.5t128.5 -36t104.5 -80t67.5 -115t17.5 -140.5q52 -16 87 -57t45.5 -89t-5.5 -99.5t-58 -87.5z + M455 1222q14 -20 9.5 -44.5t-24.5 -38.5q-19 -14 -43.5 -9.5t-37.5 24.5q-14 20 -9.5 44.5t24.5 38.5q19 14 43.5 9.5t37.5 -24.5zM614 1503q4 -16 -5 -30.5t-26 -18.5t-31 5.5t-18 26.5q-3 17 6.5 31t25.5 18q17 4 31 -5.5t17 -26.5zM1800 555q4 -20 -6.5 -37t-30.5 -21 + q-19 -4 -36 6.5t-21 30.5t6.5 37t30.5 22q20 4 36.5 -7.5t20.5 -30.5zM1136 1448q16 -27 8.5 -58.5t-35.5 -47.5q-27 -16 -57.5 -8.5t-46.5 34.5q-16 28 -8.5 59t34.5 48t58 9t47 -36zM1882 792q4 -15 -4 -27.5t-23 -16.5q-15 -3 -27.5 5.5t-15.5 22.5q-3 15 5 28t23 16 + q14 3 26.5 -5t15.5 -23zM1691 1033q15 -22 10.5 -49t-26.5 -43q-22 -15 -49 -10t-42 27t-10 49t27 43t48.5 11t41.5 -28z" /> + <glyph glyph-name="uniF2E1" unicode="" horiz-adv-x="1792" + /> + <glyph glyph-name="uniF2E2" unicode="" horiz-adv-x="1792" + /> + <glyph glyph-name="uniF2E3" unicode="" horiz-adv-x="1792" + /> + <glyph glyph-name="uniF2E4" unicode="" horiz-adv-x="1792" + /> + <glyph glyph-name="uniF2E5" unicode="" horiz-adv-x="1792" + /> + <glyph glyph-name="uniF2E6" unicode="" horiz-adv-x="1792" + /> + <glyph glyph-name="uniF2E7" unicode="" horiz-adv-x="1792" + /> + <glyph glyph-name="_698" unicode="" horiz-adv-x="1792" + /> + <glyph glyph-name="uniF2E9" unicode="" horiz-adv-x="1792" + /> + <glyph glyph-name="uniF2EA" unicode="" horiz-adv-x="1792" + /> + <glyph glyph-name="uniF2EB" unicode="" horiz-adv-x="1792" + /> + <glyph glyph-name="uniF2EC" unicode="" horiz-adv-x="1792" + /> + <glyph glyph-name="uniF2ED" unicode="" horiz-adv-x="1792" + /> + <glyph glyph-name="uniF2EE" unicode="" horiz-adv-x="1792" + /> + <glyph glyph-name="lessequal" unicode="" horiz-adv-x="1792" + /> + + </defs></svg>
html/_static/css/fonts/fontawesome-webfont.ttf (new)
html/_static/css/fonts/fontawesome-webfont.woff (new)
html/_static/css/fonts/fontawesome-webfont.woff2 (new)
html/_static/css/fonts/lato-bold-italic.woff (new)
html/_static/css/fonts/lato-bold-italic.woff2 (new)
html/_static/css/fonts/lato-bold.woff (new)
html/_static/css/fonts/lato-bold.woff2 (new)
html/_static/css/fonts/lato-normal-italic.woff (new)
html/_static/css/fonts/lato-normal-italic.woff2 (new)
html/_static/css/fonts/lato-normal.woff (new)
html/_static/css/fonts/lato-normal.woff2 (new)
html/_static/css/theme.css (new)
- @@ -0,0 +1,4 @@
  + html{box-sizing:border-box}*,:after,:before{box-sizing:inherit}article,aside,details,figcaption,figure,footer,header,hgroup,nav,section{display:block}audio,canvas,video{display:inline-block;*display:inline;*zoom:1}[hidden],audio:not([controls]){display:none}*{-webkit-box-sizing:border-box;-moz-box-sizing:border-box;box-sizing:border-box}html{font-size:100%;-webkit-text-size-adjust:100%;-ms-text-size-adjust:100%}body{margin:0}a:active,a:hover{outline:0}abbr[title]{border-bottom:1px dotted}b,strong{font-weight:700}blockquote{margin:0}dfn{font-style:italic}ins{background:#ff9;text-decoration:none}ins,mark{color:#000}mark{background:#ff0;font-style:italic;font-weight:700}.rst-content code,.rst-content tt,code,kbd,pre,samp{font-family:monospace,serif;_font-family:courier new,monospace;font-size:1em}pre{white-space:pre}q{quotes:none}q:after,q:before{content:"";content:none}small{font-size:85%}sub,sup{font-size:75%;line-height:0;position:relative;vertical-align:baseline}sup{top:-.5em}sub{bottom:-.25em}dl,ol,ul{margin:0;padding:0;list-style:none;list-style-image:none}li{list-style:none}dd{margin:0}img{border:0;-ms-interpolation-mode:bicubic;vertical-align:middle;max-width:100%}svg:not(:root){overflow:hidden}figure,form{margin:0}label{cursor:pointer}button,input,select,textarea{font-size:100%;margin:0;vertical-align:baseline;*vertical-align:middle}button,input{line-height:normal}button,input[type=button],input[type=reset],input[type=submit]{cursor:pointer;-webkit-appearance:button;*overflow:visible}button[disabled],input[disabled]{cursor:default}input[type=search]{-webkit-appearance:textfield;-moz-box-sizing:content-box;-webkit-box-sizing:content-box;box-sizing:content-box}textarea{resize:vertical}table{border-collapse:collapse;border-spacing:0}td{vertical-align:top}.chromeframe{margin:.2em 0;background:#ccc;color:#000;padding:.2em 0}.ir{display:block;border:0;text-indent:-999em;overflow:hidden;background-color:transparent;background-repeat:no-repeat;text-align:left;direction:ltr;*line-height:0}.ir br{display:none}.hidden{display:none!important;visibility:hidden}.visuallyhidden{border:0;clip:rect(0 0 0 0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}.visuallyhidden.focusable:active,.visuallyhidden.focusable:focus{clip:auto;height:auto;margin:0;overflow:visible;position:static;width:auto}.invisible{visibility:hidden}.relative{position:relative}big,small{font-size:100%}@media print{body,html,section{background:none!important}*{box-shadow:none!important;text-shadow:none!important;filter:none!important;-ms-filter:none!important}a,a:visited{text-decoration:underline}.ir a:after,a[href^="#"]:after,a[href^="javascript:"]:after{content:""}blockquote,pre{page-break-inside:avoid}thead{display:table-header-group}img,tr{page-break-inside:avoid}img{max-width:100%!important}@page{margin:.5cm}.rst-content .toctree-wrapper>p.caption,h2,h3,p{orphans:3;widows:3}.rst-content .toctree-wrapper>p.caption,h2,h3{page-break-after:avoid}}.btn,.fa:before,.icon:before,.rst-content .admonition,.rst-content .admonition-title:before,.rst-content .admonition-todo,.rst-content .attention,.rst-content .caution,.rst-content .code-block-caption .headerlink:before,.rst-content .danger,.rst-content .eqno .headerlink:before,.rst-content .error,.rst-content .hint,.rst-content .important,.rst-content .note,.rst-content .seealso,.rst-content .tip,.rst-content .warning,.rst-content code.download span:first-child:before,.rst-content dl dt .headerlink:before,.rst-content h1 .headerlink:before,.rst-content h2 .headerlink:before,.rst-content h3 .headerlink:before,.rst-content h4 .headerlink:before,.rst-content h5 .headerlink:before,.rst-content h6 .headerlink:before,.rst-content p.caption .headerlink:before,.rst-content p .headerlink:before,.rst-content table>caption .headerlink:before,.rst-content tt.download span:first-child:before,.wy-alert,.wy-dropdown .caret:before,.wy-inline-validate.wy-inline-validate-danger .wy-input-context:before,.wy-inline-validate.wy-inline-validate-info .wy-input-context:before,.wy-inline-validate.wy-inline-validate-success .wy-input-context:before,.wy-inline-validate.wy-inline-validate-warning .wy-input-context:before,.wy-menu-vertical li.current>a,.wy-menu-vertical li.current>a button.toctree-expand:before,.wy-menu-vertical li.on a,.wy-menu-vertical li.on a button.toctree-expand:before,.wy-menu-vertical li button.toctree-expand:before,.wy-nav-top a,.wy-side-nav-search .wy-dropdown>a,.wy-side-nav-search>a,input[type=color],input[type=date],input[type=datetime-local],input[type=datetime],input[type=email],input[type=month],input[type=number],input[type=password],input[type=search],input[type=tel],input[type=text],input[type=time],input[type=url],input[type=week],select,textarea{-webkit-font-smoothing:antialiased}.clearfix{*zoom:1}.clearfix:after,.clearfix:before{display:table;content:""}.clearfix:after{clear:both}/*! + * Font Awesome 4.7.0 by @davegandy - http://fontawesome.io - @fontawesome + * License - http://fontawesome.io/license (Font: SIL OFL 1.1, CSS: MIT License) + */@font-face{font-family:FontAwesome;src:url(fonts/fontawesome-webfont.eot?674f50d287a8c48dc19ba404d20fe713);src:url(fonts/fontawesome-webfont.eot?674f50d287a8c48dc19ba404d20fe713?#iefix&v=4.7.0) format("embedded-opentype"),url(fonts/fontawesome-webfont.woff2?af7ae505a9eed503f8b8e6982036873e) format("woff2"),url(fonts/fontawesome-webfont.woff?fee66e712a8a08eef5805a46892932ad) format("woff"),url(fonts/fontawesome-webfont.ttf?b06871f281fee6b241d60582ae9369b9) format("truetype"),url(fonts/fontawesome-webfont.svg?912ec66d7572ff821749319396470bde#fontawesomeregular) format("svg");font-weight:400;font-style:normal}.fa,.icon,.rst-content .admonition-title,.rst-content .code-block-caption .headerlink,.rst-content .eqno .headerlink,.rst-content code.download span:first-child,.rst-content dl dt .headerlink,.rst-content h1 .headerlink,.rst-content h2 .headerlink,.rst-content h3 .headerlink,.rst-content h4 .headerlink,.rst-content h5 .headerlink,.rst-content h6 .headerlink,.rst-content p.caption .headerlink,.rst-content p .headerlink,.rst-content table>caption .headerlink,.rst-content tt.download span:first-child,.wy-menu-vertical li.current>a button.toctree-expand,.wy-menu-vertical li.on a button.toctree-expand,.wy-menu-vertical li button.toctree-expand{display:inline-block;font:normal normal normal 14px/1 FontAwesome;font-size:inherit;text-rendering:auto;-webkit-font-smoothing:antialiased;-moz-osx-font-smoothing:grayscale}.fa-lg{font-size:1.33333em;line-height:.75em;vertical-align:-15%}.fa-2x{font-size:2em}.fa-3x{font-size:3em}.fa-4x{font-size:4em}.fa-5x{font-size:5em}.fa-fw{width:1.28571em;text-align:center}.fa-ul{padding-left:0;margin-left:2.14286em;list-style-type:none}.fa-ul>li{position:relative}.fa-li{position:absolute;left:-2.14286em;width:2.14286em;top:.14286em;text-align:center}.fa-li.fa-lg{left:-1.85714em}.fa-border{padding:.2em .25em .15em;border:.08em solid #eee;border-radius:.1em}.fa-pull-left{float:left}.fa-pull-right{float:right}.fa-pull-left.icon,.fa.fa-pull-left,.rst-content .code-block-caption .fa-pull-left.headerlink,.rst-content .eqno .fa-pull-left.headerlink,.rst-content .fa-pull-left.admonition-title,.rst-content code.download span.fa-pull-left:first-child,.rst-content dl dt .fa-pull-left.headerlink,.rst-content h1 .fa-pull-left.headerlink,.rst-content h2 .fa-pull-left.headerlink,.rst-content h3 .fa-pull-left.headerlink,.rst-content h4 .fa-pull-left.headerlink,.rst-content h5 .fa-pull-left.headerlink,.rst-content h6 .fa-pull-left.headerlink,.rst-content p .fa-pull-left.headerlink,.rst-content table>caption .fa-pull-left.headerlink,.rst-content tt.download span.fa-pull-left:first-child,.wy-menu-vertical li.current>a button.fa-pull-left.toctree-expand,.wy-menu-vertical li.on a button.fa-pull-left.toctree-expand,.wy-menu-vertical li button.fa-pull-left.toctree-expand{margin-right:.3em}.fa-pull-right.icon,.fa.fa-pull-right,.rst-content .code-block-caption .fa-pull-right.headerlink,.rst-content .eqno .fa-pull-right.headerlink,.rst-content .fa-pull-right.admonition-title,.rst-content code.download span.fa-pull-right:first-child,.rst-content dl dt .fa-pull-right.headerlink,.rst-content h1 .fa-pull-right.headerlink,.rst-content h2 .fa-pull-right.headerlink,.rst-content h3 .fa-pull-right.headerlink,.rst-content h4 .fa-pull-right.headerlink,.rst-content h5 .fa-pull-right.headerlink,.rst-content h6 .fa-pull-right.headerlink,.rst-content p .fa-pull-right.headerlink,.rst-content table>caption .fa-pull-right.headerlink,.rst-content tt.download span.fa-pull-right:first-child,.wy-menu-vertical li.current>a button.fa-pull-right.toctree-expand,.wy-menu-vertical li.on a button.fa-pull-right.toctree-expand,.wy-menu-vertical li button.fa-pull-right.toctree-expand{margin-left:.3em}.pull-right{float:right}.pull-left{float:left}.fa.pull-left,.pull-left.icon,.rst-content .code-block-caption .pull-left.headerlink,.rst-content .eqno .pull-left.headerlink,.rst-content .pull-left.admonition-title,.rst-content code.download span.pull-left:first-child,.rst-content dl dt .pull-left.headerlink,.rst-content h1 .pull-left.headerlink,.rst-content h2 .pull-left.headerlink,.rst-content h3 .pull-left.headerlink,.rst-content h4 .pull-left.headerlink,.rst-content h5 .pull-left.headerlink,.rst-content h6 .pull-left.headerlink,.rst-content p .pull-left.headerlink,.rst-content table>caption .pull-left.headerlink,.rst-content tt.download span.pull-left:first-child,.wy-menu-vertical li.current>a button.pull-left.toctree-expand,.wy-menu-vertical li.on a button.pull-left.toctree-expand,.wy-menu-vertical li button.pull-left.toctree-expand{margin-right:.3em}.fa.pull-right,.pull-right.icon,.rst-content .code-block-caption .pull-right.headerlink,.rst-content .eqno .pull-right.headerlink,.rst-content .pull-right.admonition-title,.rst-content code.download span.pull-right:first-child,.rst-content dl dt .pull-right.headerlink,.rst-content h1 .pull-right.headerlink,.rst-content h2 .pull-right.headerlink,.rst-content h3 .pull-right.headerlink,.rst-content h4 .pull-right.headerlink,.rst-content h5 .pull-right.headerlink,.rst-content h6 .pull-right.headerlink,.rst-content p .pull-right.headerlink,.rst-content table>caption .pull-right.headerlink,.rst-content tt.download span.pull-right:first-child,.wy-menu-vertical li.current>a button.pull-right.toctree-expand,.wy-menu-vertical li.on a button.pull-right.toctree-expand,.wy-menu-vertical li button.pull-right.toctree-expand{margin-left:.3em}.fa-spin{-webkit-animation:fa-spin 2s linear infinite;animation:fa-spin 2s linear infinite}.fa-pulse{-webkit-animation:fa-spin 1s steps(8) infinite;animation:fa-spin 1s steps(8) infinite}@-webkit-keyframes fa-spin{0%{-webkit-transform:rotate(0deg);transform:rotate(0deg)}to{-webkit-transform:rotate(359deg);transform:rotate(359deg)}}@keyframes fa-spin{0%{-webkit-transform:rotate(0deg);transform:rotate(0deg)}to{-webkit-transform:rotate(359deg);transform:rotate(359deg)}}.fa-rotate-90{-ms-filter:"progid:DXImageTransform.Microsoft.BasicImage(rotation=1)";-webkit-transform:rotate(90deg);-ms-transform:rotate(90deg);transform:rotate(90deg)}.fa-rotate-180{-ms-filter:"progid:DXImageTransform.Microsoft.BasicImage(rotation=2)";-webkit-transform:rotate(180deg);-ms-transform:rotate(180deg);transform:rotate(180deg)}.fa-rotate-270{-ms-filter:"progid:DXImageTransform.Microsoft.BasicImage(rotation=3)";-webkit-transform:rotate(270deg);-ms-transform:rotate(270deg);transform:rotate(270deg)}.fa-flip-horizontal{-ms-filter:"progid:DXImageTransform.Microsoft.BasicImage(rotation=0, mirror=1)";-webkit-transform:scaleX(-1);-ms-transform:scaleX(-1);transform:scaleX(-1)}.fa-flip-vertical{-ms-filter:"progid:DXImageTransform.Microsoft.BasicImage(rotation=2, mirror=1)";-webkit-transform:scaleY(-1);-ms-transform:scaleY(-1);transform:scaleY(-1)}:root .fa-flip-horizontal,:root .fa-flip-vertical,:root .fa-rotate-90,:root .fa-rotate-180,:root .fa-rotate-270{filter:none}.fa-stack{position:relative;display:inline-block;width:2em;height:2em;line-height:2em;vertical-align:middle}.fa-stack-1x,.fa-stack-2x{position:absolute;left:0;width:100%;text-align:center}.fa-stack-1x{line-height:inherit}.fa-stack-2x{font-size:2em}.fa-inverse{color:#fff}.fa-glass:before{content:""}.fa-music:before{content:""}.fa-search:before,.icon-search:before{content:""}.fa-envelope-o:before{content:""}.fa-heart:before{content:""}.fa-star:before{content:""}.fa-star-o:before{content:""}.fa-user:before{content:""}.fa-film:before{content:""}.fa-th-large:before{content:""}.fa-th:before{content:""}.fa-th-list:before{content:""}.fa-check:before{content:""}.fa-close:before,.fa-remove:before,.fa-times:before{content:""}.fa-search-plus:before{content:""}.fa-search-minus:before{content:""}.fa-power-off:before{content:""}.fa-signal:before{content:""}.fa-cog:before,.fa-gear:before{content:""}.fa-trash-o:before{content:""}.fa-home:before,.icon-home:before{content:""}.fa-file-o:before{content:""}.fa-clock-o:before{content:""}.fa-road:before{content:""}.fa-download:before,.rst-content code.download span:first-child:before,.rst-content tt.download span:first-child:before{content:""}.fa-arrow-circle-o-down:before{content:""}.fa-arrow-circle-o-up:before{content:""}.fa-inbox:before{content:""}.fa-play-circle-o:before{content:""}.fa-repeat:before,.fa-rotate-right:before{content:""}.fa-refresh:before{content:""}.fa-list-alt:before{content:""}.fa-lock:before{content:""}.fa-flag:before{content:""}.fa-headphones:before{content:""}.fa-volume-off:before{content:""}.fa-volume-down:before{content:""}.fa-volume-up:before{content:""}.fa-qrcode:before{content:""}.fa-barcode:before{content:""}.fa-tag:before{content:""}.fa-tags:before{content:""}.fa-book:before,.icon-book:before{content:""}.fa-bookmark:before{content:""}.fa-print:before{content:""}.fa-camera:before{content:""}.fa-font:before{content:""}.fa-bold:before{content:""}.fa-italic:before{content:""}.fa-text-height:before{content:""}.fa-text-width:before{content:""}.fa-align-left:before{content:""}.fa-align-center:before{content:""}.fa-align-right:before{content:""}.fa-align-justify:before{content:""}.fa-list:before{content:""}.fa-dedent:before,.fa-outdent:before{content:""}.fa-indent:before{content:""}.fa-video-camera:before{content:""}.fa-image:before,.fa-photo:before,.fa-picture-o:before{content:""}.fa-pencil:before{content:""}.fa-map-marker:before{content:""}.fa-adjust:before{content:""}.fa-tint:before{content:""}.fa-edit:before,.fa-pencil-square-o:before{content:""}.fa-share-square-o:before{content:""}.fa-check-square-o:before{content:""}.fa-arrows:before{content:""}.fa-step-backward:before{content:""}.fa-fast-backward:before{content:""}.fa-backward:before{content:""}.fa-play:before{content:""}.fa-pause:before{content:""}.fa-stop:before{content:""}.fa-forward:before{content:""}.fa-fast-forward:before{content:""}.fa-step-forward:before{content:""}.fa-eject:before{content:""}.fa-chevron-left:before{content:""}.fa-chevron-right:before{content:""}.fa-plus-circle:before{content:""}.fa-minus-circle:before{content:""}.fa-times-circle:before,.wy-inline-validate.wy-inline-validate-danger .wy-input-context:before{content:""}.fa-check-circle:before,.wy-inline-validate.wy-inline-validate-success .wy-input-context:before{content:""}.fa-question-circle:before{content:""}.fa-info-circle:before{content:""}.fa-crosshairs:before{content:""}.fa-times-circle-o:before{content:""}.fa-check-circle-o:before{content:""}.fa-ban:before{content:""}.fa-arrow-left:before{content:""}.fa-arrow-right:before{content:""}.fa-arrow-up:before{content:""}.fa-arrow-down:before{content:""}.fa-mail-forward:before,.fa-share:before{content:""}.fa-expand:before{content:""}.fa-compress:before{content:""}.fa-plus:before{content:""}.fa-minus:before{content:""}.fa-asterisk:before{content:""}.fa-exclamation-circle:before,.rst-content .admonition-title:before,.wy-inline-validate.wy-inline-validate-info .wy-input-context:before,.wy-inline-validate.wy-inline-validate-warning .wy-input-context:before{content:""}.fa-gift:before{content:""}.fa-leaf:before{content:""}.fa-fire:before,.icon-fire:before{content:""}.fa-eye:before{content:""}.fa-eye-slash:before{content:""}.fa-exclamation-triangle:before,.fa-warning:before{content:""}.fa-plane:before{content:""}.fa-calendar:before{content:""}.fa-random:before{content:""}.fa-comment:before{content:""}.fa-magnet:before{content:""}.fa-chevron-up:before{content:""}.fa-chevron-down:before{content:""}.fa-retweet:before{content:""}.fa-shopping-cart:before{content:""}.fa-folder:before{content:""}.fa-folder-open:before{content:""}.fa-arrows-v:before{content:""}.fa-arrows-h:before{content:""}.fa-bar-chart-o:before,.fa-bar-chart:before{content:""}.fa-twitter-square:before{content:""}.fa-facebook-square:before{content:""}.fa-camera-retro:before{content:""}.fa-key:before{content:""}.fa-cogs:before,.fa-gears:before{content:""}.fa-comments:before{content:""}.fa-thumbs-o-up:before{content:""}.fa-thumbs-o-down:before{content:""}.fa-star-half:before{content:""}.fa-heart-o:before{content:""}.fa-sign-out:before{content:""}.fa-linkedin-square:before{content:""}.fa-thumb-tack:before{content:""}.fa-external-link:before{content:""}.fa-sign-in:before{content:""}.fa-trophy:before{content:""}.fa-github-square:before{content:""}.fa-upload:before{content:""}.fa-lemon-o:before{content:""}.fa-phone:before{content:""}.fa-square-o:before{content:""}.fa-bookmark-o:before{content:""}.fa-phone-square:before{content:""}.fa-twitter:before{content:""}.fa-facebook-f:before,.fa-facebook:before{content:""}.fa-github:before,.icon-github:before{content:""}.fa-unlock:before{content:""}.fa-credit-card:before{content:""}.fa-feed:before,.fa-rss:before{content:""}.fa-hdd-o:before{content:""}.fa-bullhorn:before{content:""}.fa-bell:before{content:""}.fa-certificate:before{content:""}.fa-hand-o-right:before{content:""}.fa-hand-o-left:before{content:""}.fa-hand-o-up:before{content:""}.fa-hand-o-down:before{content:""}.fa-arrow-circle-left:before,.icon-circle-arrow-left:before{content:""}.fa-arrow-circle-right:before,.icon-circle-arrow-right:before{content:""}.fa-arrow-circle-up:before{content:""}.fa-arrow-circle-down:before{content:""}.fa-globe:before{content:""}.fa-wrench:before{content:""}.fa-tasks:before{content:""}.fa-filter:before{content:""}.fa-briefcase:before{content:""}.fa-arrows-alt:before{content:""}.fa-group:before,.fa-users:before{content:""}.fa-chain:before,.fa-link:before,.icon-link:before{content:""}.fa-cloud:before{content:""}.fa-flask:before{content:""}.fa-cut:before,.fa-scissors:before{content:""}.fa-copy:before,.fa-files-o:before{content:""}.fa-paperclip:before{content:""}.fa-floppy-o:before,.fa-save:before{content:""}.fa-square:before{content:""}.fa-bars:before,.fa-navicon:before,.fa-reorder:before{content:""}.fa-list-ul:before{content:""}.fa-list-ol:before{content:""}.fa-strikethrough:before{content:""}.fa-underline:before{content:""}.fa-table:before{content:""}.fa-magic:before{content:""}.fa-truck:before{content:""}.fa-pinterest:before{content:""}.fa-pinterest-square:before{content:""}.fa-google-plus-square:before{content:""}.fa-google-plus:before{content:""}.fa-money:before{content:""}.fa-caret-down:before,.icon-caret-down:before,.wy-dropdown .caret:before{content:""}.fa-caret-up:before{content:""}.fa-caret-left:before{content:""}.fa-caret-right:before{content:""}.fa-columns:before{content:""}.fa-sort:before,.fa-unsorted:before{content:""}.fa-sort-desc:before,.fa-sort-down:before{content:""}.fa-sort-asc:before,.fa-sort-up:before{content:""}.fa-envelope:before{content:""}.fa-linkedin:before{content:""}.fa-rotate-left:before,.fa-undo:before{content:""}.fa-gavel:before,.fa-legal:before{content:""}.fa-dashboard:before,.fa-tachometer:before{content:""}.fa-comment-o:before{content:""}.fa-comments-o:before{content:""}.fa-bolt:before,.fa-flash:before{content:""}.fa-sitemap:before{content:""}.fa-umbrella:before{content:""}.fa-clipboard:before,.fa-paste:before{content:""}.fa-lightbulb-o:before{content:""}.fa-exchange:before{content:""}.fa-cloud-download:before{content:""}.fa-cloud-upload:before{content:""}.fa-user-md:before{content:""}.fa-stethoscope:before{content:""}.fa-suitcase:before{content:""}.fa-bell-o:before{content:""}.fa-coffee:before{content:""}.fa-cutlery:before{content:""}.fa-file-text-o:before{content:""}.fa-building-o:before{content:""}.fa-hospital-o:before{content:""}.fa-ambulance:before{content:""}.fa-medkit:before{content:""}.fa-fighter-jet:before{content:""}.fa-beer:before{content:""}.fa-h-square:before{content:""}.fa-plus-square:before{content:""}.fa-angle-double-left:before{content:""}.fa-angle-double-right:before{content:""}.fa-angle-double-up:before{content:""}.fa-angle-double-down:before{content:""}.fa-angle-left:before{content:""}.fa-angle-right:before{content:""}.fa-angle-up:before{content:""}.fa-angle-down:before{content:""}.fa-desktop:before{content:""}.fa-laptop:before{content:""}.fa-tablet:before{content:""}.fa-mobile-phone:before,.fa-mobile:before{content:""}.fa-circle-o:before{content:""}.fa-quote-left:before{content:""}.fa-quote-right:before{content:""}.fa-spinner:before{content:""}.fa-circle:before{content:""}.fa-mail-reply:before,.fa-reply:before{content:""}.fa-github-alt:before{content:""}.fa-folder-o:before{content:""}.fa-folder-open-o:before{content:""}.fa-smile-o:before{content:""}.fa-frown-o:before{content:""}.fa-meh-o:before{content:""}.fa-gamepad:before{content:""}.fa-keyboard-o:before{content:""}.fa-flag-o:before{content:""}.fa-flag-checkered:before{content:""}.fa-terminal:before{content:""}.fa-code:before{content:""}.fa-mail-reply-all:before,.fa-reply-all:before{content:""}.fa-star-half-empty:before,.fa-star-half-full:before,.fa-star-half-o:before{content:""}.fa-location-arrow:before{content:""}.fa-crop:before{content:""}.fa-code-fork:before{content:""}.fa-chain-broken:before,.fa-unlink:before{content:""}.fa-question:before{content:""}.fa-info:before{content:""}.fa-exclamation:before{content:""}.fa-superscript:before{content:""}.fa-subscript:before{content:""}.fa-eraser:before{content:""}.fa-puzzle-piece:before{content:""}.fa-microphone:before{content:""}.fa-microphone-slash:before{content:""}.fa-shield:before{content:""}.fa-calendar-o:before{content:""}.fa-fire-extinguisher:before{content:""}.fa-rocket:before{content:""}.fa-maxcdn:before{content:""}.fa-chevron-circle-left:before{content:""}.fa-chevron-circle-right:before{content:""}.fa-chevron-circle-up:before{content:""}.fa-chevron-circle-down:before{content:""}.fa-html5:before{content:""}.fa-css3:before{content:""}.fa-anchor:before{content:""}.fa-unlock-alt:before{content:""}.fa-bullseye:before{content:""}.fa-ellipsis-h:before{content:""}.fa-ellipsis-v:before{content:""}.fa-rss-square:before{content:""}.fa-play-circle:before{content:""}.fa-ticket:before{content:""}.fa-minus-square:before{content:""}.fa-minus-square-o:before,.wy-menu-vertical li.current>a button.toctree-expand:before,.wy-menu-vertical li.on a button.toctree-expand:before{content:""}.fa-level-up:before{content:""}.fa-level-down:before{content:""}.fa-check-square:before{content:""}.fa-pencil-square:before{content:""}.fa-external-link-square:before{content:""}.fa-share-square:before{content:""}.fa-compass:before{content:""}.fa-caret-square-o-down:before,.fa-toggle-down:before{content:""}.fa-caret-square-o-up:before,.fa-toggle-up:before{content:""}.fa-caret-square-o-right:before,.fa-toggle-right:before{content:""}.fa-eur:before,.fa-euro:before{content:""}.fa-gbp:before{content:""}.fa-dollar:before,.fa-usd:before{content:""}.fa-inr:before,.fa-rupee:before{content:""}.fa-cny:before,.fa-jpy:before,.fa-rmb:before,.fa-yen:before{content:""}.fa-rouble:before,.fa-rub:before,.fa-ruble:before{content:""}.fa-krw:before,.fa-won:before{content:""}.fa-bitcoin:before,.fa-btc:before{content:""}.fa-file:before{content:""}.fa-file-text:before{content:""}.fa-sort-alpha-asc:before{content:""}.fa-sort-alpha-desc:before{content:""}.fa-sort-amount-asc:before{content:""}.fa-sort-amount-desc:before{content:""}.fa-sort-numeric-asc:before{content:""}.fa-sort-numeric-desc:before{content:""}.fa-thumbs-up:before{content:""}.fa-thumbs-down:before{content:""}.fa-youtube-square:before{content:""}.fa-youtube:before{content:""}.fa-xing:before{content:""}.fa-xing-square:before{content:""}.fa-youtube-play:before{content:""}.fa-dropbox:before{content:""}.fa-stack-overflow:before{content:""}.fa-instagram:before{content:""}.fa-flickr:before{content:""}.fa-adn:before{content:""}.fa-bitbucket:before,.icon-bitbucket:before{content:""}.fa-bitbucket-square:before{content:""}.fa-tumblr:before{content:""}.fa-tumblr-square:before{content:""}.fa-long-arrow-down:before{content:""}.fa-long-arrow-up:before{content:""}.fa-long-arrow-left:before{content:""}.fa-long-arrow-right:before{content:""}.fa-apple:before{content:""}.fa-windows:before{content:""}.fa-android:before{content:""}.fa-linux:before{content:""}.fa-dribbble:before{content:""}.fa-skype:before{content:""}.fa-foursquare:before{content:""}.fa-trello:before{content:""}.fa-female:before{content:""}.fa-male:before{content:""}.fa-gittip:before,.fa-gratipay:before{content:""}.fa-sun-o:before{content:""}.fa-moon-o:before{content:""}.fa-archive:before{content:""}.fa-bug:before{content:""}.fa-vk:before{content:""}.fa-weibo:before{content:""}.fa-renren:before{content:""}.fa-pagelines:before{content:""}.fa-stack-exchange:before{content:""}.fa-arrow-circle-o-right:before{content:""}.fa-arrow-circle-o-left:before{content:""}.fa-caret-square-o-left:before,.fa-toggle-left:before{content:""}.fa-dot-circle-o:before{content:""}.fa-wheelchair:before{content:""}.fa-vimeo-square:before{content:""}.fa-try:before,.fa-turkish-lira:before{content:""}.fa-plus-square-o:before,.wy-menu-vertical li button.toctree-expand:before{content:""}.fa-space-shuttle:before{content:""}.fa-slack:before{content:""}.fa-envelope-square:before{content:""}.fa-wordpress:before{content:""}.fa-openid:before{content:""}.fa-bank:before,.fa-institution:before,.fa-university:before{content:""}.fa-graduation-cap:before,.fa-mortar-board:before{content:""}.fa-yahoo:before{content:""}.fa-google:before{content:""}.fa-reddit:before{content:""}.fa-reddit-square:before{content:""}.fa-stumbleupon-circle:before{content:""}.fa-stumbleupon:before{content:""}.fa-delicious:before{content:""}.fa-digg:before{content:""}.fa-pied-piper-pp:before{content:""}.fa-pied-piper-alt:before{content:""}.fa-drupal:before{content:""}.fa-joomla:before{content:""}.fa-language:before{content:""}.fa-fax:before{content:""}.fa-building:before{content:""}.fa-child:before{content:""}.fa-paw:before{content:""}.fa-spoon:before{content:""}.fa-cube:before{content:""}.fa-cubes:before{content:""}.fa-behance:before{content:""}.fa-behance-square:before{content:""}.fa-steam:before{content:""}.fa-steam-square:before{content:""}.fa-recycle:before{content:""}.fa-automobile:before,.fa-car:before{content:""}.fa-cab:before,.fa-taxi:before{content:""}.fa-tree:before{content:""}.fa-spotify:before{content:""}.fa-deviantart:before{content:""}.fa-soundcloud:before{content:""}.fa-database:before{content:""}.fa-file-pdf-o:before{content:""}.fa-file-word-o:before{content:""}.fa-file-excel-o:before{content:""}.fa-file-powerpoint-o:before{content:""}.fa-file-image-o:before,.fa-file-photo-o:before,.fa-file-picture-o:before{content:""}.fa-file-archive-o:before,.fa-file-zip-o:before{content:""}.fa-file-audio-o:before,.fa-file-sound-o:before{content:""}.fa-file-movie-o:before,.fa-file-video-o:before{content:""}.fa-file-code-o:before{content:""}.fa-vine:before{content:""}.fa-codepen:before{content:""}.fa-jsfiddle:before{content:""}.fa-life-bouy:before,.fa-life-buoy:before,.fa-life-ring:before,.fa-life-saver:before,.fa-support:before{content:""}.fa-circle-o-notch:before{content:""}.fa-ra:before,.fa-rebel:before,.fa-resistance:before{content:""}.fa-empire:before,.fa-ge:before{content:""}.fa-git-square:before{content:""}.fa-git:before{content:""}.fa-hacker-news:before,.fa-y-combinator-square:before,.fa-yc-square:before{content:""}.fa-tencent-weibo:before{content:""}.fa-qq:before{content:""}.fa-wechat:before,.fa-weixin:before{content:""}.fa-paper-plane:before,.fa-send:before{content:""}.fa-paper-plane-o:before,.fa-send-o:before{content:""}.fa-history:before{content:""}.fa-circle-thin:before{content:""}.fa-header:before{content:""}.fa-paragraph:before{content:""}.fa-sliders:before{content:""}.fa-share-alt:before{content:""}.fa-share-alt-square:before{content:""}.fa-bomb:before{content:""}.fa-futbol-o:before,.fa-soccer-ball-o:before{content:""}.fa-tty:before{content:""}.fa-binoculars:before{content:""}.fa-plug:before{content:""}.fa-slideshare:before{content:""}.fa-twitch:before{content:""}.fa-yelp:before{content:""}.fa-newspaper-o:before{content:""}.fa-wifi:before{content:""}.fa-calculator:before{content:""}.fa-paypal:before{content:""}.fa-google-wallet:before{content:""}.fa-cc-visa:before{content:""}.fa-cc-mastercard:before{content:""}.fa-cc-discover:before{content:""}.fa-cc-amex:before{content:""}.fa-cc-paypal:before{content:""}.fa-cc-stripe:before{content:""}.fa-bell-slash:before{content:""}.fa-bell-slash-o:before{content:""}.fa-trash:before{content:""}.fa-copyright:before{content:""}.fa-at:before{content:""}.fa-eyedropper:before{content:""}.fa-paint-brush:before{content:""}.fa-birthday-cake:before{content:""}.fa-area-chart:before{content:""}.fa-pie-chart:before{content:""}.fa-line-chart:before{content:""}.fa-lastfm:before{content:""}.fa-lastfm-square:before{content:""}.fa-toggle-off:before{content:""}.fa-toggle-on:before{content:""}.fa-bicycle:before{content:""}.fa-bus:before{content:""}.fa-ioxhost:before{content:""}.fa-angellist:before{content:""}.fa-cc:before{content:""}.fa-ils:before,.fa-shekel:before,.fa-sheqel:before{content:""}.fa-meanpath:before{content:""}.fa-buysellads:before{content:""}.fa-connectdevelop:before{content:""}.fa-dashcube:before{content:""}.fa-forumbee:before{content:""}.fa-leanpub:before{content:""}.fa-sellsy:before{content:""}.fa-shirtsinbulk:before{content:""}.fa-simplybuilt:before{content:""}.fa-skyatlas:before{content:""}.fa-cart-plus:before{content:""}.fa-cart-arrow-down:before{content:""}.fa-diamond:before{content:""}.fa-ship:before{content:""}.fa-user-secret:before{content:""}.fa-motorcycle:before{content:""}.fa-street-view:before{content:""}.fa-heartbeat:before{content:""}.fa-venus:before{content:""}.fa-mars:before{content:""}.fa-mercury:before{content:""}.fa-intersex:before,.fa-transgender:before{content:""}.fa-transgender-alt:before{content:""}.fa-venus-double:before{content:""}.fa-mars-double:before{content:""}.fa-venus-mars:before{content:""}.fa-mars-stroke:before{content:""}.fa-mars-stroke-v:before{content:""}.fa-mars-stroke-h:before{content:""}.fa-neuter:before{content:""}.fa-genderless:before{content:""}.fa-facebook-official:before{content:""}.fa-pinterest-p:before{content:""}.fa-whatsapp:before{content:""}.fa-server:before{content:""}.fa-user-plus:before{content:""}.fa-user-times:before{content:""}.fa-bed:before,.fa-hotel:before{content:""}.fa-viacoin:before{content:""}.fa-train:before{content:""}.fa-subway:before{content:""}.fa-medium:before{content:""}.fa-y-combinator:before,.fa-yc:before{content:""}.fa-optin-monster:before{content:""}.fa-opencart:before{content:""}.fa-expeditedssl:before{content:""}.fa-battery-4:before,.fa-battery-full:before,.fa-battery:before{content:""}.fa-battery-3:before,.fa-battery-three-quarters:before{content:""}.fa-battery-2:before,.fa-battery-half:before{content:""}.fa-battery-1:before,.fa-battery-quarter:before{content:""}.fa-battery-0:before,.fa-battery-empty:before{content:""}.fa-mouse-pointer:before{content:""}.fa-i-cursor:before{content:""}.fa-object-group:before{content:""}.fa-object-ungroup:before{content:""}.fa-sticky-note:before{content:""}.fa-sticky-note-o:before{content:""}.fa-cc-jcb:before{content:""}.fa-cc-diners-club:before{content:""}.fa-clone:before{content:""}.fa-balance-scale:before{content:""}.fa-hourglass-o:before{content:""}.fa-hourglass-1:before,.fa-hourglass-start:before{content:""}.fa-hourglass-2:before,.fa-hourglass-half:before{content:""}.fa-hourglass-3:before,.fa-hourglass-end:before{content:""}.fa-hourglass:before{content:""}.fa-hand-grab-o:before,.fa-hand-rock-o:before{content:""}.fa-hand-paper-o:before,.fa-hand-stop-o:before{content:""}.fa-hand-scissors-o:before{content:""}.fa-hand-lizard-o:before{content:""}.fa-hand-spock-o:before{content:""}.fa-hand-pointer-o:before{content:""}.fa-hand-peace-o:before{content:""}.fa-trademark:before{content:""}.fa-registered:before{content:""}.fa-creative-commons:before{content:""}.fa-gg:before{content:""}.fa-gg-circle:before{content:""}.fa-tripadvisor:before{content:""}.fa-odnoklassniki:before{content:""}.fa-odnoklassniki-square:before{content:""}.fa-get-pocket:before{content:""}.fa-wikipedia-w:before{content:""}.fa-safari:before{content:""}.fa-chrome:before{content:""}.fa-firefox:before{content:""}.fa-opera:before{content:""}.fa-internet-explorer:before{content:""}.fa-television:before,.fa-tv:before{content:""}.fa-contao:before{content:""}.fa-500px:before{content:""}.fa-amazon:before{content:""}.fa-calendar-plus-o:before{content:""}.fa-calendar-minus-o:before{content:""}.fa-calendar-times-o:before{content:""}.fa-calendar-check-o:before{content:""}.fa-industry:before{content:""}.fa-map-pin:before{content:""}.fa-map-signs:before{content:""}.fa-map-o:before{content:""}.fa-map:before{content:""}.fa-commenting:before{content:""}.fa-commenting-o:before{content:""}.fa-houzz:before{content:""}.fa-vimeo:before{content:""}.fa-black-tie:before{content:""}.fa-fonticons:before{content:""}.fa-reddit-alien:before{content:""}.fa-edge:before{content:""}.fa-credit-card-alt:before{content:""}.fa-codiepie:before{content:""}.fa-modx:before{content:""}.fa-fort-awesome:before{content:""}.fa-usb:before{content:""}.fa-product-hunt:before{content:""}.fa-mixcloud:before{content:""}.fa-scribd:before{content:""}.fa-pause-circle:before{content:""}.fa-pause-circle-o:before{content:""}.fa-stop-circle:before{content:""}.fa-stop-circle-o:before{content:""}.fa-shopping-bag:before{content:""}.fa-shopping-basket:before{content:""}.fa-hashtag:before{content:""}.fa-bluetooth:before{content:""}.fa-bluetooth-b:before{content:""}.fa-percent:before{content:""}.fa-gitlab:before,.icon-gitlab:before{content:""}.fa-wpbeginner:before{content:""}.fa-wpforms:before{content:""}.fa-envira:before{content:""}.fa-universal-access:before{content:""}.fa-wheelchair-alt:before{content:""}.fa-question-circle-o:before{content:""}.fa-blind:before{content:""}.fa-audio-description:before{content:""}.fa-volume-control-phone:before{content:""}.fa-braille:before{content:""}.fa-assistive-listening-systems:before{content:""}.fa-american-sign-language-interpreting:before,.fa-asl-interpreting:before{content:""}.fa-deaf:before,.fa-deafness:before,.fa-hard-of-hearing:before{content:""}.fa-glide:before{content:""}.fa-glide-g:before{content:""}.fa-sign-language:before,.fa-signing:before{content:""}.fa-low-vision:before{content:""}.fa-viadeo:before{content:""}.fa-viadeo-square:before{content:""}.fa-snapchat:before{content:""}.fa-snapchat-ghost:before{content:""}.fa-snapchat-square:before{content:""}.fa-pied-piper:before{content:""}.fa-first-order:before{content:""}.fa-yoast:before{content:""}.fa-themeisle:before{content:""}.fa-google-plus-circle:before,.fa-google-plus-official:before{content:""}.fa-fa:before,.fa-font-awesome:before{content:""}.fa-handshake-o:before{content:""}.fa-envelope-open:before{content:""}.fa-envelope-open-o:before{content:""}.fa-linode:before{content:""}.fa-address-book:before{content:""}.fa-address-book-o:before{content:""}.fa-address-card:before,.fa-vcard:before{content:""}.fa-address-card-o:before,.fa-vcard-o:before{content:""}.fa-user-circle:before{content:""}.fa-user-circle-o:before{content:""}.fa-user-o:before{content:""}.fa-id-badge:before{content:""}.fa-drivers-license:before,.fa-id-card:before{content:""}.fa-drivers-license-o:before,.fa-id-card-o:before{content:""}.fa-quora:before{content:""}.fa-free-code-camp:before{content:""}.fa-telegram:before{content:""}.fa-thermometer-4:before,.fa-thermometer-full:before,.fa-thermometer:before{content:""}.fa-thermometer-3:before,.fa-thermometer-three-quarters:before{content:""}.fa-thermometer-2:before,.fa-thermometer-half:before{content:""}.fa-thermometer-1:before,.fa-thermometer-quarter:before{content:""}.fa-thermometer-0:before,.fa-thermometer-empty:before{content:""}.fa-shower:before{content:""}.fa-bath:before,.fa-bathtub:before,.fa-s15:before{content:""}.fa-podcast:before{content:""}.fa-window-maximize:before{content:""}.fa-window-minimize:before{content:""}.fa-window-restore:before{content:""}.fa-times-rectangle:before,.fa-window-close:before{content:""}.fa-times-rectangle-o:before,.fa-window-close-o:before{content:""}.fa-bandcamp:before{content:""}.fa-grav:before{content:""}.fa-etsy:before{content:""}.fa-imdb:before{content:""}.fa-ravelry:before{content:""}.fa-eercast:before{content:""}.fa-microchip:before{content:""}.fa-snowflake-o:before{content:""}.fa-superpowers:before{content:""}.fa-wpexplorer:before{content:""}.fa-meetup:before{content:""}.sr-only{position:absolute;width:1px;height:1px;padding:0;margin:-1px;overflow:hidden;clip:rect(0,0,0,0);border:0}.sr-only-focusable:active,.sr-only-focusable:focus{position:static;width:auto;height:auto;margin:0;overflow:visible;clip:auto}.fa,.icon,.rst-content .admonition-title,.rst-content .code-block-caption .headerlink,.rst-content .eqno .headerlink,.rst-content code.download span:first-child,.rst-content dl dt .headerlink,.rst-content h1 .headerlink,.rst-content h2 .headerlink,.rst-content h3 .headerlink,.rst-content h4 .headerlink,.rst-content h5 .headerlink,.rst-content h6 .headerlink,.rst-content p.caption .headerlink,.rst-content p .headerlink,.rst-content table>caption .headerlink,.rst-content tt.download span:first-child,.wy-dropdown .caret,.wy-inline-validate.wy-inline-validate-danger .wy-input-context,.wy-inline-validate.wy-inline-validate-info .wy-input-context,.wy-inline-validate.wy-inline-validate-success .wy-input-context,.wy-inline-validate.wy-inline-validate-warning .wy-input-context,.wy-menu-vertical li.current>a button.toctree-expand,.wy-menu-vertical li.on a button.toctree-expand,.wy-menu-vertical li button.toctree-expand{font-family:inherit}.fa:before,.icon:before,.rst-content .admonition-title:before,.rst-content .code-block-caption .headerlink:before,.rst-content .eqno .headerlink:before,.rst-content code.download span:first-child:before,.rst-content dl dt .headerlink:before,.rst-content h1 .headerlink:before,.rst-content h2 .headerlink:before,.rst-content h3 .headerlink:before,.rst-content h4 .headerlink:before,.rst-content h5 .headerlink:before,.rst-content h6 .headerlink:before,.rst-content p.caption .headerlink:before,.rst-content p .headerlink:before,.rst-content table>caption .headerlink:before,.rst-content tt.download span:first-child:before,.wy-dropdown .caret:before,.wy-inline-validate.wy-inline-validate-danger .wy-input-context:before,.wy-inline-validate.wy-inline-validate-info .wy-input-context:before,.wy-inline-validate.wy-inline-validate-success .wy-input-context:before,.wy-inline-validate.wy-inline-validate-warning .wy-input-context:before,.wy-menu-vertical li.current>a button.toctree-expand:before,.wy-menu-vertical li.on a button.toctree-expand:before,.wy-menu-vertical li button.toctree-expand:before{font-family:FontAwesome;display:inline-block;font-style:normal;font-weight:400;line-height:1;text-decoration:inherit}.rst-content .code-block-caption a .headerlink,.rst-content .eqno a .headerlink,.rst-content a .admonition-title,.rst-content code.download a span:first-child,.rst-content dl dt a .headerlink,.rst-content h1 a .headerlink,.rst-content h2 a .headerlink,.rst-content h3 a .headerlink,.rst-content h4 a .headerlink,.rst-content h5 a .headerlink,.rst-content h6 a .headerlink,.rst-content p.caption a .headerlink,.rst-content p a .headerlink,.rst-content table>caption a .headerlink,.rst-content tt.download a span:first-child,.wy-menu-vertical li.current>a button.toctree-expand,.wy-menu-vertical li.on a button.toctree-expand,.wy-menu-vertical li a button.toctree-expand,a .fa,a .icon,a .rst-content .admonition-title,a .rst-content .code-block-caption .headerlink,a .rst-content .eqno .headerlink,a .rst-content code.download span:first-child,a .rst-content dl dt .headerlink,a .rst-content h1 .headerlink,a .rst-content h2 .headerlink,a .rst-content h3 .headerlink,a .rst-content h4 .headerlink,a .rst-content h5 .headerlink,a .rst-content h6 .headerlink,a .rst-content p.caption .headerlink,a .rst-content p .headerlink,a .rst-content table>caption .headerlink,a .rst-content tt.download span:first-child,a .wy-menu-vertical li button.toctree-expand{display:inline-block;text-decoration:inherit}.btn .fa,.btn .icon,.btn .rst-content .admonition-title,.btn .rst-content .code-block-caption .headerlink,.btn .rst-content .eqno .headerlink,.btn .rst-content code.download span:first-child,.btn .rst-content dl dt .headerlink,.btn .rst-content h1 .headerlink,.btn .rst-content h2 .headerlink,.btn .rst-content h3 .headerlink,.btn .rst-content h4 .headerlink,.btn .rst-content h5 .headerlink,.btn .rst-content h6 .headerlink,.btn .rst-content p .headerlink,.btn .rst-content table>caption .headerlink,.btn .rst-content tt.download span:first-child,.btn .wy-menu-vertical li.current>a button.toctree-expand,.btn .wy-menu-vertical li.on a button.toctree-expand,.btn .wy-menu-vertical li button.toctree-expand,.nav .fa,.nav .icon,.nav .rst-content .admonition-title,.nav .rst-content .code-block-caption .headerlink,.nav .rst-content .eqno .headerlink,.nav .rst-content code.download span:first-child,.nav .rst-content dl dt .headerlink,.nav .rst-content h1 .headerlink,.nav .rst-content h2 .headerlink,.nav .rst-content h3 .headerlink,.nav .rst-content h4 .headerlink,.nav .rst-content h5 .headerlink,.nav .rst-content h6 .headerlink,.nav .rst-content p .headerlink,.nav .rst-content table>caption .headerlink,.nav .rst-content tt.download span:first-child,.nav .wy-menu-vertical li.current>a button.toctree-expand,.nav .wy-menu-vertical li.on a button.toctree-expand,.nav .wy-menu-vertical li button.toctree-expand,.rst-content .btn .admonition-title,.rst-content .code-block-caption .btn .headerlink,.rst-content .code-block-caption .nav .headerlink,.rst-content .eqno .btn .headerlink,.rst-content .eqno .nav .headerlink,.rst-content .nav .admonition-title,.rst-content code.download .btn span:first-child,.rst-content code.download .nav span:first-child,.rst-content dl dt .btn .headerlink,.rst-content dl dt .nav .headerlink,.rst-content h1 .btn .headerlink,.rst-content h1 .nav .headerlink,.rst-content h2 .btn .headerlink,.rst-content h2 .nav .headerlink,.rst-content h3 .btn .headerlink,.rst-content h3 .nav .headerlink,.rst-content h4 .btn .headerlink,.rst-content h4 .nav .headerlink,.rst-content h5 .btn .headerlink,.rst-content h5 .nav .headerlink,.rst-content h6 .btn .headerlink,.rst-content h6 .nav .headerlink,.rst-content p .btn .headerlink,.rst-content p .nav .headerlink,.rst-content table>caption .btn .headerlink,.rst-content table>caption .nav .headerlink,.rst-content tt.download .btn span:first-child,.rst-content tt.download .nav span:first-child,.wy-menu-vertical li .btn button.toctree-expand,.wy-menu-vertical li.current>a .btn button.toctree-expand,.wy-menu-vertical li.current>a .nav button.toctree-expand,.wy-menu-vertical li .nav button.toctree-expand,.wy-menu-vertical li.on a .btn button.toctree-expand,.wy-menu-vertical li.on a .nav button.toctree-expand{display:inline}.btn .fa-large.icon,.btn .fa.fa-large,.btn .rst-content .code-block-caption .fa-large.headerlink,.btn .rst-content .eqno .fa-large.headerlink,.btn .rst-content .fa-large.admonition-title,.btn .rst-content code.download span.fa-large:first-child,.btn .rst-content dl dt .fa-large.headerlink,.btn .rst-content h1 .fa-large.headerlink,.btn .rst-content h2 .fa-large.headerlink,.btn .rst-content h3 .fa-large.headerlink,.btn .rst-content h4 .fa-large.headerlink,.btn .rst-content h5 .fa-large.headerlink,.btn .rst-content h6 .fa-large.headerlink,.btn .rst-content p .fa-large.headerlink,.btn .rst-content table>caption .fa-large.headerlink,.btn .rst-content tt.download span.fa-large:first-child,.btn .wy-menu-vertical li button.fa-large.toctree-expand,.nav .fa-large.icon,.nav .fa.fa-large,.nav .rst-content .code-block-caption .fa-large.headerlink,.nav .rst-content .eqno .fa-large.headerlink,.nav .rst-content .fa-large.admonition-title,.nav .rst-content code.download span.fa-large:first-child,.nav .rst-content dl dt .fa-large.headerlink,.nav .rst-content h1 .fa-large.headerlink,.nav .rst-content h2 .fa-large.headerlink,.nav .rst-content h3 .fa-large.headerlink,.nav .rst-content h4 .fa-large.headerlink,.nav .rst-content h5 .fa-large.headerlink,.nav .rst-content h6 .fa-large.headerlink,.nav .rst-content p .fa-large.headerlink,.nav .rst-content table>caption .fa-large.headerlink,.nav .rst-content tt.download span.fa-large:first-child,.nav .wy-menu-vertical li button.fa-large.toctree-expand,.rst-content .btn .fa-large.admonition-title,.rst-content .code-block-caption .btn .fa-large.headerlink,.rst-content .code-block-caption .nav .fa-large.headerlink,.rst-content .eqno .btn .fa-large.headerlink,.rst-content .eqno .nav .fa-large.headerlink,.rst-content .nav .fa-large.admonition-title,.rst-content code.download .btn span.fa-large:first-child,.rst-content code.download .nav span.fa-large:first-child,.rst-content dl dt .btn .fa-large.headerlink,.rst-content dl dt .nav .fa-large.headerlink,.rst-content h1 .btn .fa-large.headerlink,.rst-content h1 .nav .fa-large.headerlink,.rst-content h2 .btn .fa-large.headerlink,.rst-content h2 .nav .fa-large.headerlink,.rst-content h3 .btn .fa-large.headerlink,.rst-content h3 .nav .fa-large.headerlink,.rst-content h4 .btn .fa-large.headerlink,.rst-content h4 .nav .fa-large.headerlink,.rst-content h5 .btn .fa-large.headerlink,.rst-content h5 .nav .fa-large.headerlink,.rst-content h6 .btn .fa-large.headerlink,.rst-content h6 .nav .fa-large.headerlink,.rst-content p .btn .fa-large.headerlink,.rst-content p .nav .fa-large.headerlink,.rst-content table>caption .btn .fa-large.headerlink,.rst-content table>caption .nav .fa-large.headerlink,.rst-content tt.download .btn span.fa-large:first-child,.rst-content tt.download .nav span.fa-large:first-child,.wy-menu-vertical li .btn button.fa-large.toctree-expand,.wy-menu-vertical li .nav button.fa-large.toctree-expand{line-height:.9em}.btn .fa-spin.icon,.btn .fa.fa-spin,.btn .rst-content .code-block-caption .fa-spin.headerlink,.btn .rst-content .eqno .fa-spin.headerlink,.btn .rst-content .fa-spin.admonition-title,.btn .rst-content code.download span.fa-spin:first-child,.btn .rst-content dl dt .fa-spin.headerlink,.btn .rst-content h1 .fa-spin.headerlink,.btn .rst-content h2 .fa-spin.headerlink,.btn .rst-content h3 .fa-spin.headerlink,.btn .rst-content h4 .fa-spin.headerlink,.btn .rst-content h5 .fa-spin.headerlink,.btn .rst-content h6 .fa-spin.headerlink,.btn .rst-content p .fa-spin.headerlink,.btn .rst-content table>caption .fa-spin.headerlink,.btn .rst-content tt.download span.fa-spin:first-child,.btn .wy-menu-vertical li button.fa-spin.toctree-expand,.nav .fa-spin.icon,.nav .fa.fa-spin,.nav .rst-content .code-block-caption .fa-spin.headerlink,.nav .rst-content .eqno .fa-spin.headerlink,.nav .rst-content .fa-spin.admonition-title,.nav .rst-content code.download span.fa-spin:first-child,.nav .rst-content dl dt .fa-spin.headerlink,.nav .rst-content h1 .fa-spin.headerlink,.nav .rst-content h2 .fa-spin.headerlink,.nav .rst-content h3 .fa-spin.headerlink,.nav .rst-content h4 .fa-spin.headerlink,.nav .rst-content h5 .fa-spin.headerlink,.nav .rst-content h6 .fa-spin.headerlink,.nav .rst-content p .fa-spin.headerlink,.nav .rst-content table>caption .fa-spin.headerlink,.nav .rst-content tt.download span.fa-spin:first-child,.nav .wy-menu-vertical li button.fa-spin.toctree-expand,.rst-content .btn .fa-spin.admonition-title,.rst-content .code-block-caption .btn .fa-spin.headerlink,.rst-content .code-block-caption .nav .fa-spin.headerlink,.rst-content .eqno .btn .fa-spin.headerlink,.rst-content .eqno .nav .fa-spin.headerlink,.rst-content .nav .fa-spin.admonition-title,.rst-content code.download .btn span.fa-spin:first-child,.rst-content code.download .nav span.fa-spin:first-child,.rst-content dl dt .btn .fa-spin.headerlink,.rst-content dl dt .nav .fa-spin.headerlink,.rst-content h1 .btn .fa-spin.headerlink,.rst-content h1 .nav .fa-spin.headerlink,.rst-content h2 .btn .fa-spin.headerlink,.rst-content h2 .nav .fa-spin.headerlink,.rst-content h3 .btn .fa-spin.headerlink,.rst-content h3 .nav .fa-spin.headerlink,.rst-content h4 .btn .fa-spin.headerlink,.rst-content h4 .nav .fa-spin.headerlink,.rst-content h5 .btn .fa-spin.headerlink,.rst-content h5 .nav .fa-spin.headerlink,.rst-content h6 .btn .fa-spin.headerlink,.rst-content h6 .nav .fa-spin.headerlink,.rst-content p .btn .fa-spin.headerlink,.rst-content p .nav .fa-spin.headerlink,.rst-content table>caption .btn .fa-spin.headerlink,.rst-content table>caption .nav .fa-spin.headerlink,.rst-content tt.download .btn span.fa-spin:first-child,.rst-content tt.download .nav span.fa-spin:first-child,.wy-menu-vertical li .btn button.fa-spin.toctree-expand,.wy-menu-vertical li .nav button.fa-spin.toctree-expand{display:inline-block}.btn.fa:before,.btn.icon:before,.rst-content .btn.admonition-title:before,.rst-content .code-block-caption .btn.headerlink:before,.rst-content .eqno .btn.headerlink:before,.rst-content code.download span.btn:first-child:before,.rst-content dl dt .btn.headerlink:before,.rst-content h1 .btn.headerlink:before,.rst-content h2 .btn.headerlink:before,.rst-content h3 .btn.headerlink:before,.rst-content h4 .btn.headerlink:before,.rst-content h5 .btn.headerlink:before,.rst-content h6 .btn.headerlink:before,.rst-content p .btn.headerlink:before,.rst-content table>caption .btn.headerlink:before,.rst-content tt.download span.btn:first-child:before,.wy-menu-vertical li button.btn.toctree-expand:before{opacity:.5;-webkit-transition:opacity .05s ease-in;-moz-transition:opacity .05s ease-in;transition:opacity .05s ease-in}.btn.fa:hover:before,.btn.icon:hover:before,.rst-content .btn.admonition-title:hover:before,.rst-content .code-block-caption .btn.headerlink:hover:before,.rst-content .eqno .btn.headerlink:hover:before,.rst-content code.download span.btn:first-child:hover:before,.rst-content dl dt .btn.headerlink:hover:before,.rst-content h1 .btn.headerlink:hover:before,.rst-content h2 .btn.headerlink:hover:before,.rst-content h3 .btn.headerlink:hover:before,.rst-content h4 .btn.headerlink:hover:before,.rst-content h5 .btn.headerlink:hover:before,.rst-content h6 .btn.headerlink:hover:before,.rst-content p .btn.headerlink:hover:before,.rst-content table>caption .btn.headerlink:hover:before,.rst-content tt.download span.btn:first-child:hover:before,.wy-menu-vertical li button.btn.toctree-expand:hover:before{opacity:1}.btn-mini .fa:before,.btn-mini .icon:before,.btn-mini .rst-content .admonition-title:before,.btn-mini .rst-content .code-block-caption .headerlink:before,.btn-mini .rst-content .eqno .headerlink:before,.btn-mini .rst-content code.download span:first-child:before,.btn-mini .rst-content dl dt .headerlink:before,.btn-mini .rst-content h1 .headerlink:before,.btn-mini .rst-content h2 .headerlink:before,.btn-mini .rst-content h3 .headerlink:before,.btn-mini .rst-content h4 .headerlink:before,.btn-mini .rst-content h5 .headerlink:before,.btn-mini .rst-content h6 .headerlink:before,.btn-mini .rst-content p .headerlink:before,.btn-mini .rst-content table>caption .headerlink:before,.btn-mini .rst-content tt.download span:first-child:before,.btn-mini .wy-menu-vertical li button.toctree-expand:before,.rst-content .btn-mini .admonition-title:before,.rst-content .code-block-caption .btn-mini .headerlink:before,.rst-content .eqno .btn-mini .headerlink:before,.rst-content code.download .btn-mini span:first-child:before,.rst-content dl dt .btn-mini .headerlink:before,.rst-content h1 .btn-mini .headerlink:before,.rst-content h2 .btn-mini .headerlink:before,.rst-content h3 .btn-mini .headerlink:before,.rst-content h4 .btn-mini .headerlink:before,.rst-content h5 .btn-mini .headerlink:before,.rst-content h6 .btn-mini .headerlink:before,.rst-content p .btn-mini .headerlink:before,.rst-content table>caption .btn-mini .headerlink:before,.rst-content tt.download .btn-mini span:first-child:before,.wy-menu-vertical li .btn-mini button.toctree-expand:before{font-size:14px;vertical-align:-15%}.rst-content .admonition,.rst-content .admonition-todo,.rst-content .attention,.rst-content .caution,.rst-content .danger,.rst-content .error,.rst-content .hint,.rst-content .important,.rst-content .note,.rst-content .seealso,.rst-content .tip,.rst-content .warning,.wy-alert{padding:12px;line-height:24px;margin-bottom:24px;background:#e7f2fa}.rst-content .admonition-title,.wy-alert-title{font-weight:700;display:block;color:#fff;background:#6ab0de;padding:6px 12px;margin:-12px -12px 12px}.rst-content .danger,.rst-content .error,.rst-content .wy-alert-danger.admonition,.rst-content .wy-alert-danger.admonition-todo,.rst-content .wy-alert-danger.attention,.rst-content .wy-alert-danger.caution,.rst-content .wy-alert-danger.hint,.rst-content .wy-alert-danger.important,.rst-content .wy-alert-danger.note,.rst-content .wy-alert-danger.seealso,.rst-content .wy-alert-danger.tip,.rst-content .wy-alert-danger.warning,.wy-alert.wy-alert-danger{background:#fdf3f2}.rst-content .danger .admonition-title,.rst-content .danger .wy-alert-title,.rst-content .error .admonition-title,.rst-content .error .wy-alert-title,.rst-content .wy-alert-danger.admonition-todo .admonition-title,.rst-content .wy-alert-danger.admonition-todo .wy-alert-title,.rst-content .wy-alert-danger.admonition .admonition-title,.rst-content .wy-alert-danger.admonition .wy-alert-title,.rst-content .wy-alert-danger.attention .admonition-title,.rst-content .wy-alert-danger.attention .wy-alert-title,.rst-content .wy-alert-danger.caution .admonition-title,.rst-content .wy-alert-danger.caution .wy-alert-title,.rst-content .wy-alert-danger.hint .admonition-title,.rst-content .wy-alert-danger.hint .wy-alert-title,.rst-content .wy-alert-danger.important .admonition-title,.rst-content .wy-alert-danger.important .wy-alert-title,.rst-content .wy-alert-danger.note .admonition-title,.rst-content .wy-alert-danger.note .wy-alert-title,.rst-content .wy-alert-danger.seealso .admonition-title,.rst-content .wy-alert-danger.seealso .wy-alert-title,.rst-content .wy-alert-danger.tip .admonition-title,.rst-content .wy-alert-danger.tip .wy-alert-title,.rst-content .wy-alert-danger.warning .admonition-title,.rst-content .wy-alert-danger.warning .wy-alert-title,.rst-content .wy-alert.wy-alert-danger .admonition-title,.wy-alert.wy-alert-danger .rst-content .admonition-title,.wy-alert.wy-alert-danger .wy-alert-title{background:#f29f97}.rst-content .admonition-todo,.rst-content .attention,.rst-content .caution,.rst-content .warning,.rst-content .wy-alert-warning.admonition,.rst-content .wy-alert-warning.danger,.rst-content .wy-alert-warning.error,.rst-content .wy-alert-warning.hint,.rst-content .wy-alert-warning.important,.rst-content .wy-alert-warning.note,.rst-content .wy-alert-warning.seealso,.rst-content .wy-alert-warning.tip,.wy-alert.wy-alert-warning{background:#ffedcc}.rst-content .admonition-todo .admonition-title,.rst-content .admonition-todo .wy-alert-title,.rst-content .attention .admonition-title,.rst-content .attention .wy-alert-title,.rst-content .caution .admonition-title,.rst-content .caution .wy-alert-title,.rst-content .warning .admonition-title,.rst-content .warning .wy-alert-title,.rst-content .wy-alert-warning.admonition .admonition-title,.rst-content .wy-alert-warning.admonition .wy-alert-title,.rst-content .wy-alert-warning.danger .admonition-title,.rst-content .wy-alert-warning.danger .wy-alert-title,.rst-content .wy-alert-warning.error .admonition-title,.rst-content .wy-alert-warning.error .wy-alert-title,.rst-content .wy-alert-warning.hint .admonition-title,.rst-content .wy-alert-warning.hint .wy-alert-title,.rst-content .wy-alert-warning.important .admonition-title,.rst-content .wy-alert-warning.important .wy-alert-title,.rst-content .wy-alert-warning.note .admonition-title,.rst-content .wy-alert-warning.note .wy-alert-title,.rst-content .wy-alert-warning.seealso .admonition-title,.rst-content .wy-alert-warning.seealso .wy-alert-title,.rst-content .wy-alert-warning.tip .admonition-title,.rst-content .wy-alert-warning.tip .wy-alert-title,.rst-content .wy-alert.wy-alert-warning .admonition-title,.wy-alert.wy-alert-warning .rst-content .admonition-title,.wy-alert.wy-alert-warning .wy-alert-title{background:#f0b37e}.rst-content .note,.rst-content .seealso,.rst-content .wy-alert-info.admonition,.rst-content .wy-alert-info.admonition-todo,.rst-content .wy-alert-info.attention,.rst-content .wy-alert-info.caution,.rst-content .wy-alert-info.danger,.rst-content .wy-alert-info.error,.rst-content .wy-alert-info.hint,.rst-content .wy-alert-info.important,.rst-content .wy-alert-info.tip,.rst-content .wy-alert-info.warning,.wy-alert.wy-alert-info{background:#e7f2fa}.rst-content .note .admonition-title,.rst-content .note .wy-alert-title,.rst-content .seealso .admonition-title,.rst-content .seealso .wy-alert-title,.rst-content .wy-alert-info.admonition-todo .admonition-title,.rst-content .wy-alert-info.admonition-todo .wy-alert-title,.rst-content .wy-alert-info.admonition .admonition-title,.rst-content .wy-alert-info.admonition .wy-alert-title,.rst-content .wy-alert-info.attention .admonition-title,.rst-content .wy-alert-info.attention .wy-alert-title,.rst-content .wy-alert-info.caution .admonition-title,.rst-content .wy-alert-info.caution .wy-alert-title,.rst-content .wy-alert-info.danger .admonition-title,.rst-content .wy-alert-info.danger .wy-alert-title,.rst-content .wy-alert-info.error .admonition-title,.rst-content .wy-alert-info.error .wy-alert-title,.rst-content .wy-alert-info.hint .admonition-title,.rst-content .wy-alert-info.hint .wy-alert-title,.rst-content .wy-alert-info.important .admonition-title,.rst-content .wy-alert-info.important .wy-alert-title,.rst-content .wy-alert-info.tip .admonition-title,.rst-content .wy-alert-info.tip .wy-alert-title,.rst-content .wy-alert-info.warning .admonition-title,.rst-content .wy-alert-info.warning .wy-alert-title,.rst-content .wy-alert.wy-alert-info .admonition-title,.wy-alert.wy-alert-info .rst-content .admonition-title,.wy-alert.wy-alert-info .wy-alert-title{background:#6ab0de}.rst-content .hint,.rst-content .important,.rst-content .tip,.rst-content .wy-alert-success.admonition,.rst-content .wy-alert-success.admonition-todo,.rst-content .wy-alert-success.attention,.rst-content .wy-alert-success.caution,.rst-content .wy-alert-success.danger,.rst-content .wy-alert-success.error,.rst-content .wy-alert-success.note,.rst-content .wy-alert-success.seealso,.rst-content .wy-alert-success.warning,.wy-alert.wy-alert-success{background:#dbfaf4}.rst-content .hint .admonition-title,.rst-content .hint .wy-alert-title,.rst-content .important .admonition-title,.rst-content .important .wy-alert-title,.rst-content .tip .admonition-title,.rst-content .tip .wy-alert-title,.rst-content .wy-alert-success.admonition-todo .admonition-title,.rst-content .wy-alert-success.admonition-todo .wy-alert-title,.rst-content .wy-alert-success.admonition .admonition-title,.rst-content .wy-alert-success.admonition .wy-alert-title,.rst-content .wy-alert-success.attention .admonition-title,.rst-content .wy-alert-success.attention .wy-alert-title,.rst-content .wy-alert-success.caution .admonition-title,.rst-content .wy-alert-success.caution .wy-alert-title,.rst-content .wy-alert-success.danger .admonition-title,.rst-content .wy-alert-success.danger .wy-alert-title,.rst-content .wy-alert-success.error .admonition-title,.rst-content .wy-alert-success.error .wy-alert-title,.rst-content .wy-alert-success.note .admonition-title,.rst-content .wy-alert-success.note .wy-alert-title,.rst-content .wy-alert-success.seealso .admonition-title,.rst-content .wy-alert-success.seealso .wy-alert-title,.rst-content .wy-alert-success.warning .admonition-title,.rst-content .wy-alert-success.warning .wy-alert-title,.rst-content .wy-alert.wy-alert-success .admonition-title,.wy-alert.wy-alert-success .rst-content .admonition-title,.wy-alert.wy-alert-success .wy-alert-title{background:#1abc9c}.rst-content .wy-alert-neutral.admonition,.rst-content .wy-alert-neutral.admonition-todo,.rst-content .wy-alert-neutral.attention,.rst-content .wy-alert-neutral.caution,.rst-content .wy-alert-neutral.danger,.rst-content .wy-alert-neutral.error,.rst-content .wy-alert-neutral.hint,.rst-content .wy-alert-neutral.important,.rst-content .wy-alert-neutral.note,.rst-content .wy-alert-neutral.seealso,.rst-content .wy-alert-neutral.tip,.rst-content .wy-alert-neutral.warning,.wy-alert.wy-alert-neutral{background:#f3f6f6}.rst-content .wy-alert-neutral.admonition-todo .admonition-title,.rst-content .wy-alert-neutral.admonition-todo .wy-alert-title,.rst-content .wy-alert-neutral.admonition .admonition-title,.rst-content .wy-alert-neutral.admonition .wy-alert-title,.rst-content .wy-alert-neutral.attention .admonition-title,.rst-content .wy-alert-neutral.attention .wy-alert-title,.rst-content .wy-alert-neutral.caution .admonition-title,.rst-content .wy-alert-neutral.caution .wy-alert-title,.rst-content .wy-alert-neutral.danger .admonition-title,.rst-content .wy-alert-neutral.danger .wy-alert-title,.rst-content .wy-alert-neutral.error .admonition-title,.rst-content .wy-alert-neutral.error .wy-alert-title,.rst-content .wy-alert-neutral.hint .admonition-title,.rst-content .wy-alert-neutral.hint .wy-alert-title,.rst-content .wy-alert-neutral.important .admonition-title,.rst-content .wy-alert-neutral.important .wy-alert-title,.rst-content .wy-alert-neutral.note .admonition-title,.rst-content .wy-alert-neutral.note .wy-alert-title,.rst-content .wy-alert-neutral.seealso .admonition-title,.rst-content .wy-alert-neutral.seealso .wy-alert-title,.rst-content .wy-alert-neutral.tip .admonition-title,.rst-content .wy-alert-neutral.tip .wy-alert-title,.rst-content .wy-alert-neutral.warning .admonition-title,.rst-content .wy-alert-neutral.warning .wy-alert-title,.rst-content .wy-alert.wy-alert-neutral .admonition-title,.wy-alert.wy-alert-neutral .rst-content .admonition-title,.wy-alert.wy-alert-neutral .wy-alert-title{color:#404040;background:#e1e4e5}.rst-content .wy-alert-neutral.admonition-todo a,.rst-content .wy-alert-neutral.admonition a,.rst-content .wy-alert-neutral.attention a,.rst-content .wy-alert-neutral.caution a,.rst-content .wy-alert-neutral.danger a,.rst-content .wy-alert-neutral.error a,.rst-content .wy-alert-neutral.hint a,.rst-content .wy-alert-neutral.important a,.rst-content .wy-alert-neutral.note a,.rst-content .wy-alert-neutral.seealso a,.rst-content .wy-alert-neutral.tip a,.rst-content .wy-alert-neutral.warning a,.wy-alert.wy-alert-neutral a{color:#2980b9}.rst-content .admonition-todo p:last-child,.rst-content .admonition p:last-child,.rst-content .attention p:last-child,.rst-content .caution p:last-child,.rst-content .danger p:last-child,.rst-content .error p:last-child,.rst-content .hint p:last-child,.rst-content .important p:last-child,.rst-content .note p:last-child,.rst-content .seealso p:last-child,.rst-content .tip p:last-child,.rst-content .warning p:last-child,.wy-alert p:last-child{margin-bottom:0}.wy-tray-container{position:fixed;bottom:0;left:0;z-index:600}.wy-tray-container li{display:block;width:300px;background:transparent;color:#fff;text-align:center;box-shadow:0 5px 5px 0 rgba(0,0,0,.1);padding:0 24px;min-width:20%;opacity:0;height:0;line-height:56px;overflow:hidden;-webkit-transition:all .3s ease-in;-moz-transition:all .3s ease-in;transition:all .3s ease-in}.wy-tray-container li.wy-tray-item-success{background:#27ae60}.wy-tray-container li.wy-tray-item-info{background:#2980b9}.wy-tray-container li.wy-tray-item-warning{background:#e67e22}.wy-tray-container li.wy-tray-item-danger{background:#e74c3c}.wy-tray-container li.on{opacity:1;height:56px}@media screen and (max-width:768px){.wy-tray-container{bottom:auto;top:0;width:100%}.wy-tray-container li{width:100%}}button{font-size:100%;margin:0;vertical-align:baseline;*vertical-align:middle;cursor:pointer;line-height:normal;-webkit-appearance:button;*overflow:visible}button::-moz-focus-inner,input::-moz-focus-inner{border:0;padding:0}button[disabled]{cursor:default}.btn{display:inline-block;border-radius:2px;line-height:normal;white-space:nowrap;text-align:center;cursor:pointer;font-size:100%;padding:6px 12px 8px;color:#fff;border:1px solid rgba(0,0,0,.1);background-color:#27ae60;text-decoration:none;font-weight:400;font-family:Lato,proxima-nova,Helvetica Neue,Arial,sans-serif;box-shadow:inset 0 1px 2px -1px hsla(0,0%,100%,.5),inset 0 -2px 0 0 rgba(0,0,0,.1);outline-none:false;vertical-align:middle;*display:inline;zoom:1;-webkit-user-drag:none;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;-webkit-transition:all .1s linear;-moz-transition:all .1s linear;transition:all .1s linear}.btn-hover{background:#2e8ece;color:#fff}.btn:hover{background:#2cc36b;color:#fff}.btn:focus{background:#2cc36b;outline:0}.btn:active{box-shadow:inset 0 -1px 0 0 rgba(0,0,0,.05),inset 0 2px 0 0 rgba(0,0,0,.1);padding:8px 12px 6px}.btn:visited{color:#fff}.btn-disabled,.btn-disabled:active,.btn-disabled:focus,.btn-disabled:hover,.btn:disabled{background-image:none;filter:progid:DXImageTransform.Microsoft.gradient(enabled = false);filter:alpha(opacity=40);opacity:.4;cursor:not-allowed;box-shadow:none}.btn::-moz-focus-inner{padding:0;border:0}.btn-small{font-size:80%}.btn-info{background-color:#2980b9!important}.btn-info:hover{background-color:#2e8ece!important}.btn-neutral{background-color:#f3f6f6!important;color:#404040!important}.btn-neutral:hover{background-color:#e5ebeb!important;color:#404040}.btn-neutral:visited{color:#404040!important}.btn-success{background-color:#27ae60!important}.btn-success:hover{background-color:#295!important}.btn-danger{background-color:#e74c3c!important}.btn-danger:hover{background-color:#ea6153!important}.btn-warning{background-color:#e67e22!important}.btn-warning:hover{background-color:#e98b39!important}.btn-invert{background-color:#222}.btn-invert:hover{background-color:#2f2f2f!important}.btn-link{background-color:transparent!important;color:#2980b9;box-shadow:none;border-color:transparent!important}.btn-link:active,.btn-link:hover{background-color:transparent!important;color:#409ad5!important;box-shadow:none}.btn-link:visited{color:#9b59b6}.wy-btn-group .btn,.wy-control .btn{vertical-align:middle}.wy-btn-group{margin-bottom:24px;*zoom:1}.wy-btn-group:after,.wy-btn-group:before{display:table;content:""}.wy-btn-group:after{clear:both}.wy-dropdown{position:relative;display:inline-block}.wy-dropdown-active .wy-dropdown-menu{display:block}.wy-dropdown-menu{position:absolute;left:0;display:none;float:left;top:100%;min-width:100%;background:#fcfcfc;z-index:100;border:1px solid #cfd7dd;box-shadow:0 2px 2px 0 rgba(0,0,0,.1);padding:12px}.wy-dropdown-menu>dd>a{display:block;clear:both;color:#404040;white-space:nowrap;font-size:90%;padding:0 12px;cursor:pointer}.wy-dropdown-menu>dd>a:hover{background:#2980b9;color:#fff}.wy-dropdown-menu>dd.divider{border-top:1px solid #cfd7dd;margin:6px 0}.wy-dropdown-menu>dd.search{padding-bottom:12px}.wy-dropdown-menu>dd.search input[type=search]{width:100%}.wy-dropdown-menu>dd.call-to-action{background:#e3e3e3;text-transform:uppercase;font-weight:500;font-size:80%}.wy-dropdown-menu>dd.call-to-action:hover{background:#e3e3e3}.wy-dropdown-menu>dd.call-to-action .btn{color:#fff}.wy-dropdown.wy-dropdown-up .wy-dropdown-menu{bottom:100%;top:auto;left:auto;right:0}.wy-dropdown.wy-dropdown-bubble .wy-dropdown-menu{background:#fcfcfc;margin-top:2px}.wy-dropdown.wy-dropdown-bubble .wy-dropdown-menu a{padding:6px 12px}.wy-dropdown.wy-dropdown-bubble .wy-dropdown-menu a:hover{background:#2980b9;color:#fff}.wy-dropdown.wy-dropdown-left .wy-dropdown-menu{right:0;left:auto;text-align:right}.wy-dropdown-arrow:before{content:" ";border-bottom:5px solid #f5f5f5;border-left:5px solid transparent;border-right:5px solid transparent;position:absolute;display:block;top:-4px;left:50%;margin-left:-3px}.wy-dropdown-arrow.wy-dropdown-arrow-left:before{left:11px}.wy-form-stacked select{display:block}.wy-form-aligned .wy-help-inline,.wy-form-aligned input,.wy-form-aligned label,.wy-form-aligned select,.wy-form-aligned textarea{display:inline-block;*display:inline;*zoom:1;vertical-align:middle}.wy-form-aligned .wy-control-group>label{display:inline-block;vertical-align:middle;width:10em;margin:6px 12px 0 0;float:left}.wy-form-aligned .wy-control{float:left}.wy-form-aligned .wy-control label{display:block}.wy-form-aligned .wy-control select{margin-top:6px}fieldset{margin:0}fieldset,legend{border:0;padding:0}legend{width:100%;white-space:normal;margin-bottom:24px;font-size:150%;*margin-left:-7px}label,legend{display:block}label{margin:0 0 .3125em;color:#333;font-size:90%}input,select,textarea{font-size:100%;margin:0;vertical-align:baseline;*vertical-align:middle}.wy-control-group{margin-bottom:24px;max-width:1200px;margin-left:auto;margin-right:auto;*zoom:1}.wy-control-group:after,.wy-control-group:before{display:table;content:""}.wy-control-group:after{clear:both}.wy-control-group.wy-control-group-required>label:after{content:" *";color:#e74c3c}.wy-control-group .wy-form-full,.wy-control-group .wy-form-halves,.wy-control-group .wy-form-thirds{padding-bottom:12px}.wy-control-group .wy-form-full input[type=color],.wy-control-group .wy-form-full input[type=date],.wy-control-group .wy-form-full input[type=datetime-local],.wy-control-group .wy-form-full input[type=datetime],.wy-control-group .wy-form-full input[type=email],.wy-control-group .wy-form-full input[type=month],.wy-control-group .wy-form-full input[type=number],.wy-control-group .wy-form-full input[type=password],.wy-control-group .wy-form-full input[type=search],.wy-control-group .wy-form-full input[type=tel],.wy-control-group .wy-form-full input[type=text],.wy-control-group .wy-form-full input[type=time],.wy-control-group .wy-form-full input[type=url],.wy-control-group .wy-form-full input[type=week],.wy-control-group .wy-form-full select,.wy-control-group .wy-form-halves input[type=color],.wy-control-group .wy-form-halves input[type=date],.wy-control-group .wy-form-halves input[type=datetime-local],.wy-control-group .wy-form-halves input[type=datetime],.wy-control-group .wy-form-halves input[type=email],.wy-control-group .wy-form-halves input[type=month],.wy-control-group .wy-form-halves input[type=number],.wy-control-group .wy-form-halves input[type=password],.wy-control-group .wy-form-halves input[type=search],.wy-control-group .wy-form-halves input[type=tel],.wy-control-group .wy-form-halves input[type=text],.wy-control-group .wy-form-halves input[type=time],.wy-control-group .wy-form-halves input[type=url],.wy-control-group .wy-form-halves input[type=week],.wy-control-group .wy-form-halves select,.wy-control-group .wy-form-thirds input[type=color],.wy-control-group .wy-form-thirds input[type=date],.wy-control-group .wy-form-thirds input[type=datetime-local],.wy-control-group .wy-form-thirds input[type=datetime],.wy-control-group .wy-form-thirds input[type=email],.wy-control-group .wy-form-thirds input[type=month],.wy-control-group .wy-form-thirds input[type=number],.wy-control-group .wy-form-thirds input[type=password],.wy-control-group .wy-form-thirds input[type=search],.wy-control-group .wy-form-thirds input[type=tel],.wy-control-group .wy-form-thirds input[type=text],.wy-control-group .wy-form-thirds input[type=time],.wy-control-group .wy-form-thirds input[type=url],.wy-control-group .wy-form-thirds input[type=week],.wy-control-group .wy-form-thirds select{width:100%}.wy-control-group .wy-form-full{float:left;display:block;width:100%;margin-right:0}.wy-control-group .wy-form-full:last-child{margin-right:0}.wy-control-group .wy-form-halves{float:left;display:block;margin-right:2.35765%;width:48.82117%}.wy-control-group .wy-form-halves:last-child,.wy-control-group .wy-form-halves:nth-of-type(2n){margin-right:0}.wy-control-group .wy-form-halves:nth-of-type(odd){clear:left}.wy-control-group .wy-form-thirds{float:left;display:block;margin-right:2.35765%;width:31.76157%}.wy-control-group .wy-form-thirds:last-child,.wy-control-group .wy-form-thirds:nth-of-type(3n){margin-right:0}.wy-control-group .wy-form-thirds:nth-of-type(3n+1){clear:left}.wy-control-group.wy-control-group-no-input .wy-control,.wy-control-no-input{margin:6px 0 0;font-size:90%}.wy-control-no-input{display:inline-block}.wy-control-group.fluid-input input[type=color],.wy-control-group.fluid-input input[type=date],.wy-control-group.fluid-input input[type=datetime-local],.wy-control-group.fluid-input input[type=datetime],.wy-control-group.fluid-input input[type=email],.wy-control-group.fluid-input input[type=month],.wy-control-group.fluid-input input[type=number],.wy-control-group.fluid-input input[type=password],.wy-control-group.fluid-input input[type=search],.wy-control-group.fluid-input input[type=tel],.wy-control-group.fluid-input input[type=text],.wy-control-group.fluid-input input[type=time],.wy-control-group.fluid-input input[type=url],.wy-control-group.fluid-input input[type=week]{width:100%}.wy-form-message-inline{padding-left:.3em;color:#666;font-size:90%}.wy-form-message{display:block;color:#999;font-size:70%;margin-top:.3125em;font-style:italic}.wy-form-message p{font-size:inherit;font-style:italic;margin-bottom:6px}.wy-form-message p:last-child{margin-bottom:0}input{line-height:normal}input[type=button],input[type=reset],input[type=submit]{-webkit-appearance:button;cursor:pointer;font-family:Lato,proxima-nova,Helvetica Neue,Arial,sans-serif;*overflow:visible}input[type=color],input[type=date],input[type=datetime-local],input[type=datetime],input[type=email],input[type=month],input[type=number],input[type=password],input[type=search],input[type=tel],input[type=text],input[type=time],input[type=url],input[type=week]{-webkit-appearance:none;padding:6px;display:inline-block;border:1px solid #ccc;font-size:80%;font-family:Lato,proxima-nova,Helvetica Neue,Arial,sans-serif;box-shadow:inset 0 1px 3px #ddd;border-radius:0;-webkit-transition:border .3s linear;-moz-transition:border .3s linear;transition:border .3s linear}input[type=datetime-local]{padding:.34375em .625em}input[disabled]{cursor:default}input[type=checkbox],input[type=radio]{padding:0;margin-right:.3125em;*height:13px;*width:13px}input[type=checkbox],input[type=radio],input[type=search]{-webkit-box-sizing:border-box;-moz-box-sizing:border-box;box-sizing:border-box}input[type=search]::-webkit-search-cancel-button,input[type=search]::-webkit-search-decoration{-webkit-appearance:none}input[type=color]:focus,input[type=date]:focus,input[type=datetime-local]:focus,input[type=datetime]:focus,input[type=email]:focus,input[type=month]:focus,input[type=number]:focus,input[type=password]:focus,input[type=search]:focus,input[type=tel]:focus,input[type=text]:focus,input[type=time]:focus,input[type=url]:focus,input[type=week]:focus{outline:0;outline:thin dotted\9;border-color:#333}input.no-focus:focus{border-color:#ccc!important}input[type=checkbox]:focus,input[type=file]:focus,input[type=radio]:focus{outline:thin dotted #333;outline:1px auto #129fea}input[type=color][disabled],input[type=date][disabled],input[type=datetime-local][disabled],input[type=datetime][disabled],input[type=email][disabled],input[type=month][disabled],input[type=number][disabled],input[type=password][disabled],input[type=search][disabled],input[type=tel][disabled],input[type=text][disabled],input[type=time][disabled],input[type=url][disabled],input[type=week][disabled]{cursor:not-allowed;background-color:#fafafa}input:focus:invalid,select:focus:invalid,textarea:focus:invalid{color:#e74c3c;border:1px solid #e74c3c}input:focus:invalid:focus,select:focus:invalid:focus,textarea:focus:invalid:focus{border-color:#e74c3c}input[type=checkbox]:focus:invalid:focus,input[type=file]:focus:invalid:focus,input[type=radio]:focus:invalid:focus{outline-color:#e74c3c}input.wy-input-large{padding:12px;font-size:100%}textarea{overflow:auto;vertical-align:top;width:100%;font-family:Lato,proxima-nova,Helvetica Neue,Arial,sans-serif}select,textarea{padding:.5em .625em;display:inline-block;border:1px solid #ccc;font-size:80%;box-shadow:inset 0 1px 3px #ddd;-webkit-transition:border .3s linear;-moz-transition:border .3s linear;transition:border .3s linear}select{border:1px solid #ccc;background-color:#fff}select[multiple]{height:auto}select:focus,textarea:focus{outline:0}input[readonly],select[disabled],select[readonly],textarea[disabled],textarea[readonly]{cursor:not-allowed;background-color:#fafafa}input[type=checkbox][disabled],input[type=radio][disabled]{cursor:not-allowed}.wy-checkbox,.wy-radio{margin:6px 0;color:#404040;display:block}.wy-checkbox input,.wy-radio input{vertical-align:baseline}.wy-form-message-inline{display:inline-block;*display:inline;*zoom:1;vertical-align:middle}.wy-input-prefix,.wy-input-suffix{white-space:nowrap;padding:6px}.wy-input-prefix .wy-input-context,.wy-input-suffix .wy-input-context{line-height:27px;padding:0 8px;display:inline-block;font-size:80%;background-color:#f3f6f6;border:1px solid #ccc;color:#999}.wy-input-suffix .wy-input-context{border-left:0}.wy-input-prefix .wy-input-context{border-right:0}.wy-switch{position:relative;display:block;height:24px;margin-top:12px;cursor:pointer}.wy-switch:before{left:0;top:0;width:36px;height:12px;background:#ccc}.wy-switch:after,.wy-switch:before{position:absolute;content:"";display:block;border-radius:4px;-webkit-transition:all .2s ease-in-out;-moz-transition:all .2s ease-in-out;transition:all .2s ease-in-out}.wy-switch:after{width:18px;height:18px;background:#999;left:-3px;top:-3px}.wy-switch span{position:absolute;left:48px;display:block;font-size:12px;color:#ccc;line-height:1}.wy-switch.active:before{background:#1e8449}.wy-switch.active:after{left:24px;background:#27ae60}.wy-switch.disabled{cursor:not-allowed;opacity:.8}.wy-control-group.wy-control-group-error .wy-form-message,.wy-control-group.wy-control-group-error>label{color:#e74c3c}.wy-control-group.wy-control-group-error input[type=color],.wy-control-group.wy-control-group-error input[type=date],.wy-control-group.wy-control-group-error input[type=datetime-local],.wy-control-group.wy-control-group-error input[type=datetime],.wy-control-group.wy-control-group-error input[type=email],.wy-control-group.wy-control-group-error input[type=month],.wy-control-group.wy-control-group-error input[type=number],.wy-control-group.wy-control-group-error input[type=password],.wy-control-group.wy-control-group-error input[type=search],.wy-control-group.wy-control-group-error input[type=tel],.wy-control-group.wy-control-group-error input[type=text],.wy-control-group.wy-control-group-error input[type=time],.wy-control-group.wy-control-group-error input[type=url],.wy-control-group.wy-control-group-error input[type=week],.wy-control-group.wy-control-group-error textarea{border:1px solid #e74c3c}.wy-inline-validate{white-space:nowrap}.wy-inline-validate .wy-input-context{padding:.5em .625em;display:inline-block;font-size:80%}.wy-inline-validate.wy-inline-validate-success .wy-input-context{color:#27ae60}.wy-inline-validate.wy-inline-validate-danger .wy-input-context{color:#e74c3c}.wy-inline-validate.wy-inline-validate-warning .wy-input-context{color:#e67e22}.wy-inline-validate.wy-inline-validate-info .wy-input-context{color:#2980b9}.rotate-90{-webkit-transform:rotate(90deg);-moz-transform:rotate(90deg);-ms-transform:rotate(90deg);-o-transform:rotate(90deg);transform:rotate(90deg)}.rotate-180{-webkit-transform:rotate(180deg);-moz-transform:rotate(180deg);-ms-transform:rotate(180deg);-o-transform:rotate(180deg);transform:rotate(180deg)}.rotate-270{-webkit-transform:rotate(270deg);-moz-transform:rotate(270deg);-ms-transform:rotate(270deg);-o-transform:rotate(270deg);transform:rotate(270deg)}.mirror{-webkit-transform:scaleX(-1);-moz-transform:scaleX(-1);-ms-transform:scaleX(-1);-o-transform:scaleX(-1);transform:scaleX(-1)}.mirror.rotate-90{-webkit-transform:scaleX(-1) rotate(90deg);-moz-transform:scaleX(-1) rotate(90deg);-ms-transform:scaleX(-1) rotate(90deg);-o-transform:scaleX(-1) rotate(90deg);transform:scaleX(-1) rotate(90deg)}.mirror.rotate-180{-webkit-transform:scaleX(-1) rotate(180deg);-moz-transform:scaleX(-1) rotate(180deg);-ms-transform:scaleX(-1) rotate(180deg);-o-transform:scaleX(-1) rotate(180deg);transform:scaleX(-1) rotate(180deg)}.mirror.rotate-270{-webkit-transform:scaleX(-1) rotate(270deg);-moz-transform:scaleX(-1) rotate(270deg);-ms-transform:scaleX(-1) rotate(270deg);-o-transform:scaleX(-1) rotate(270deg);transform:scaleX(-1) rotate(270deg)}@media only screen and (max-width:480px){.wy-form button[type=submit]{margin:.7em 0 0}.wy-form input[type=color],.wy-form input[type=date],.wy-form input[type=datetime-local],.wy-form input[type=datetime],.wy-form input[type=email],.wy-form input[type=month],.wy-form input[type=number],.wy-form input[type=password],.wy-form input[type=search],.wy-form input[type=tel],.wy-form input[type=text],.wy-form input[type=time],.wy-form input[type=url],.wy-form input[type=week],.wy-form label{margin-bottom:.3em;display:block}.wy-form input[type=color],.wy-form input[type=date],.wy-form input[type=datetime-local],.wy-form input[type=datetime],.wy-form input[type=email],.wy-form input[type=month],.wy-form input[type=number],.wy-form input[type=password],.wy-form input[type=search],.wy-form input[type=tel],.wy-form input[type=time],.wy-form input[type=url],.wy-form input[type=week]{margin-bottom:0}.wy-form-aligned .wy-control-group label{margin-bottom:.3em;text-align:left;display:block;width:100%}.wy-form-aligned .wy-control{margin:1.5em 0 0}.wy-form-message,.wy-form-message-inline,.wy-form .wy-help-inline{display:block;font-size:80%;padding:6px 0}}@media screen and (max-width:768px){.tablet-hide{display:none}}@media screen and (max-width:480px){.mobile-hide{display:none}}.float-left{float:left}.float-right{float:right}.full-width{width:100%}.rst-content table.docutils,.rst-content table.field-list,.wy-table{border-collapse:collapse;border-spacing:0;empty-cells:show;margin-bottom:24px}.rst-content table.docutils caption,.rst-content table.field-list caption,.wy-table caption{color:#000;font:italic 85%/1 arial,sans-serif;padding:1em 0;text-align:center}.rst-content table.docutils td,.rst-content table.docutils th,.rst-content table.field-list td,.rst-content table.field-list th,.wy-table td,.wy-table th{font-size:90%;margin:0;overflow:visible;padding:8px 16px}.rst-content table.docutils td:first-child,.rst-content table.docutils th:first-child,.rst-content table.field-list td:first-child,.rst-content table.field-list th:first-child,.wy-table td:first-child,.wy-table th:first-child{border-left-width:0}.rst-content table.docutils thead,.rst-content table.field-list thead,.wy-table thead{color:#000;text-align:left;vertical-align:bottom;white-space:nowrap}.rst-content table.docutils thead th,.rst-content table.field-list thead th,.wy-table thead th{font-weight:700;border-bottom:2px solid #e1e4e5}.rst-content table.docutils td,.rst-content table.field-list td,.wy-table td{background-color:transparent;vertical-align:middle}.rst-content table.docutils td p,.rst-content table.field-list td p,.wy-table td p{line-height:18px}.rst-content table.docutils td p:last-child,.rst-content table.field-list td p:last-child,.wy-table td p:last-child{margin-bottom:0}.rst-content table.docutils .wy-table-cell-min,.rst-content table.field-list .wy-table-cell-min,.wy-table .wy-table-cell-min{width:1%;padding-right:0}.rst-content table.docutils .wy-table-cell-min input[type=checkbox],.rst-content table.field-list .wy-table-cell-min input[type=checkbox],.wy-table .wy-table-cell-min input[type=checkbox]{margin:0}.wy-table-secondary{color:grey;font-size:90%}.wy-table-tertiary{color:grey;font-size:80%}.rst-content table.docutils:not(.field-list) tr:nth-child(2n-1) td,.wy-table-backed,.wy-table-odd td,.wy-table-striped tr:nth-child(2n-1) td{background-color:#f3f6f6}.rst-content table.docutils,.wy-table-bordered-all{border:1px solid #e1e4e5}.rst-content table.docutils td,.wy-table-bordered-all td{border-bottom:1px solid #e1e4e5;border-left:1px solid #e1e4e5}.rst-content table.docutils tbody>tr:last-child td,.wy-table-bordered-all tbody>tr:last-child td{border-bottom-width:0}.wy-table-bordered{border:1px solid #e1e4e5}.wy-table-bordered-rows td{border-bottom:1px solid #e1e4e5}.wy-table-bordered-rows tbody>tr:last-child td{border-bottom-width:0}.wy-table-horizontal td,.wy-table-horizontal th{border-width:0 0 1px;border-bottom:1px solid #e1e4e5}.wy-table-horizontal tbody>tr:last-child td{border-bottom-width:0}.wy-table-responsive{margin-bottom:24px;max-width:100%;overflow:auto}.wy-table-responsive table{margin-bottom:0!important}.wy-table-responsive table td,.wy-table-responsive table th{white-space:nowrap}a{color:#2980b9;text-decoration:none;cursor:pointer}a:hover{color:#3091d1}a:visited{color:#9b59b6}html{height:100%}body,html{overflow-x:hidden}body{font-family:Lato,proxima-nova,Helvetica Neue,Arial,sans-serif;font-weight:400;color:#404040;min-height:100%;background:#edf0f2}.wy-text-left{text-align:left}.wy-text-center{text-align:center}.wy-text-right{text-align:right}.wy-text-large{font-size:120%}.wy-text-normal{font-size:100%}.wy-text-small,small{font-size:80%}.wy-text-strike{text-decoration:line-through}.wy-text-warning{color:#e67e22!important}a.wy-text-warning:hover{color:#eb9950!important}.wy-text-info{color:#2980b9!important}a.wy-text-info:hover{color:#409ad5!important}.wy-text-success{color:#27ae60!important}a.wy-text-success:hover{color:#36d278!important}.wy-text-danger{color:#e74c3c!important}a.wy-text-danger:hover{color:#ed7669!important}.wy-text-neutral{color:#404040!important}a.wy-text-neutral:hover{color:#595959!important}.rst-content .toctree-wrapper>p.caption,h1,h2,h3,h4,h5,h6,legend{margin-top:0;font-weight:700;font-family:Roboto Slab,ff-tisa-web-pro,Georgia,Arial,sans-serif}p{line-height:24px;font-size:16px;margin:0 0 24px}h1{font-size:175%}.rst-content .toctree-wrapper>p.caption,h2{font-size:150%}h3{font-size:125%}h4{font-size:115%}h5{font-size:110%}h6{font-size:100%}hr{display:block;height:1px;border:0;border-top:1px solid #e1e4e5;margin:24px 0;padding:0}.rst-content code,.rst-content tt,code{white-space:nowrap;max-width:100%;background:#fff;border:1px solid #e1e4e5;font-size:75%;padding:0 5px;font-family:SFMono-Regular,Menlo,Monaco,Consolas,Liberation Mono,Courier New,Courier,monospace;color:#e74c3c;overflow-x:auto}.rst-content tt.code-large,code.code-large{font-size:90%}.rst-content .section ul,.rst-content .toctree-wrapper ul,.rst-content section ul,.wy-plain-list-disc,article ul{list-style:disc;line-height:24px;margin-bottom:24px}.rst-content .section ul li,.rst-content .toctree-wrapper ul li,.rst-content section ul li,.wy-plain-list-disc li,article ul li{list-style:disc;margin-left:24px}.rst-content .section ul li p:last-child,.rst-content .section ul li ul,.rst-content .toctree-wrapper ul li p:last-child,.rst-content .toctree-wrapper ul li ul,.rst-content section ul li p:last-child,.rst-content section ul li ul,.wy-plain-list-disc li p:last-child,.wy-plain-list-disc li ul,article ul li p:last-child,article ul li ul{margin-bottom:0}.rst-content .section ul li li,.rst-content .toctree-wrapper ul li li,.rst-content section ul li li,.wy-plain-list-disc li li,article ul li li{list-style:circle}.rst-content .section ul li li li,.rst-content .toctree-wrapper ul li li li,.rst-content section ul li li li,.wy-plain-list-disc li li li,article ul li li li{list-style:square}.rst-content .section ul li ol li,.rst-content .toctree-wrapper ul li ol li,.rst-content section ul li ol li,.wy-plain-list-disc li ol li,article ul li ol li{list-style:decimal}.rst-content .section ol,.rst-content .section ol.arabic,.rst-content .toctree-wrapper ol,.rst-content .toctree-wrapper ol.arabic,.rst-content section ol,.rst-content section ol.arabic,.wy-plain-list-decimal,article ol{list-style:decimal;line-height:24px;margin-bottom:24px}.rst-content .section ol.arabic li,.rst-content .section ol li,.rst-content .toctree-wrapper ol.arabic li,.rst-content .toctree-wrapper ol li,.rst-content section ol.arabic li,.rst-content section ol li,.wy-plain-list-decimal li,article ol li{list-style:decimal;margin-left:24px}.rst-content .section ol.arabic li ul,.rst-content .section ol li p:last-child,.rst-content .section ol li ul,.rst-content .toctree-wrapper ol.arabic li ul,.rst-content .toctree-wrapper ol li p:last-child,.rst-content .toctree-wrapper ol li ul,.rst-content section ol.arabic li ul,.rst-content section ol li p:last-child,.rst-content section ol li ul,.wy-plain-list-decimal li p:last-child,.wy-plain-list-decimal li ul,article ol li p:last-child,article ol li ul{margin-bottom:0}.rst-content .section ol.arabic li ul li,.rst-content .section ol li ul li,.rst-content .toctree-wrapper ol.arabic li ul li,.rst-content .toctree-wrapper ol li ul li,.rst-content section ol.arabic li ul li,.rst-content section ol li ul li,.wy-plain-list-decimal li ul li,article ol li ul li{list-style:disc}.wy-breadcrumbs{*zoom:1}.wy-breadcrumbs:after,.wy-breadcrumbs:before{display:table;content:""}.wy-breadcrumbs:after{clear:both}.wy-breadcrumbs li{display:inline-block}.wy-breadcrumbs li.wy-breadcrumbs-aside{float:right}.wy-breadcrumbs li a{display:inline-block;padding:5px}.wy-breadcrumbs li a:first-child{padding-left:0}.rst-content .wy-breadcrumbs li tt,.wy-breadcrumbs li .rst-content tt,.wy-breadcrumbs li code{padding:5px;border:none;background:none}.rst-content .wy-breadcrumbs li tt.literal,.wy-breadcrumbs li .rst-content tt.literal,.wy-breadcrumbs li code.literal{color:#404040}.wy-breadcrumbs-extra{margin-bottom:0;color:#b3b3b3;font-size:80%;display:inline-block}@media screen and (max-width:480px){.wy-breadcrumbs-extra,.wy-breadcrumbs li.wy-breadcrumbs-aside{display:none}}@media print{.wy-breadcrumbs li.wy-breadcrumbs-aside{display:none}}html{font-size:16px}.wy-affix{position:fixed;top:1.618em}.wy-menu a:hover{text-decoration:none}.wy-menu-horiz{*zoom:1}.wy-menu-horiz:after,.wy-menu-horiz:before{display:table;content:""}.wy-menu-horiz:after{clear:both}.wy-menu-horiz li,.wy-menu-horiz ul{display:inline-block}.wy-menu-horiz li:hover{background:hsla(0,0%,100%,.1)}.wy-menu-horiz li.divide-left{border-left:1px solid #404040}.wy-menu-horiz li.divide-right{border-right:1px solid #404040}.wy-menu-horiz a{height:32px;display:inline-block;line-height:32px;padding:0 16px}.wy-menu-vertical{width:300px}.wy-menu-vertical header,.wy-menu-vertical p.caption{color:#55a5d9;height:32px;line-height:32px;padding:0 1.618em;margin:12px 0 0;display:block;font-weight:700;text-transform:uppercase;font-size:85%;white-space:nowrap}.wy-menu-vertical ul{margin-bottom:0}.wy-menu-vertical li.divide-top{border-top:1px solid #404040}.wy-menu-vertical li.divide-bottom{border-bottom:1px solid #404040}.wy-menu-vertical li.current{background:#e3e3e3}.wy-menu-vertical li.current a{color:grey;border-right:1px solid #c9c9c9;padding:.4045em 2.427em}.wy-menu-vertical li.current a:hover{background:#d6d6d6}.rst-content .wy-menu-vertical li tt,.wy-menu-vertical li .rst-content tt,.wy-menu-vertical li code{border:none;background:inherit;color:inherit;padding-left:0;padding-right:0}.wy-menu-vertical li button.toctree-expand{display:block;float:left;margin-left:-1.2em;line-height:18px;color:#4d4d4d;border:none;background:none;padding:0}.wy-menu-vertical li.current>a,.wy-menu-vertical li.on a{color:#404040;font-weight:700;position:relative;background:#fcfcfc;border:none;padding:.4045em 1.618em}.wy-menu-vertical li.current>a:hover,.wy-menu-vertical li.on a:hover{background:#fcfcfc}.wy-menu-vertical li.current>a:hover button.toctree-expand,.wy-menu-vertical li.on a:hover button.toctree-expand{color:grey}.wy-menu-vertical li.current>a button.toctree-expand,.wy-menu-vertical li.on a button.toctree-expand{display:block;line-height:18px;color:#333}.wy-menu-vertical li.toctree-l1.current>a{border-bottom:1px solid #c9c9c9;border-top:1px solid #c9c9c9}.wy-menu-vertical .toctree-l1.current .toctree-l2>ul,.wy-menu-vertical .toctree-l2.current .toctree-l3>ul,.wy-menu-vertical .toctree-l3.current .toctree-l4>ul,.wy-menu-vertical .toctree-l4.current .toctree-l5>ul,.wy-menu-vertical .toctree-l5.current .toctree-l6>ul,.wy-menu-vertical .toctree-l6.current .toctree-l7>ul,.wy-menu-vertical .toctree-l7.current .toctree-l8>ul,.wy-menu-vertical .toctree-l8.current .toctree-l9>ul,.wy-menu-vertical .toctree-l9.current .toctree-l10>ul,.wy-menu-vertical .toctree-l10.current .toctree-l11>ul{display:none}.wy-menu-vertical .toctree-l1.current .current.toctree-l2>ul,.wy-menu-vertical .toctree-l2.current .current.toctree-l3>ul,.wy-menu-vertical .toctree-l3.current .current.toctree-l4>ul,.wy-menu-vertical .toctree-l4.current .current.toctree-l5>ul,.wy-menu-vertical .toctree-l5.current .current.toctree-l6>ul,.wy-menu-vertical .toctree-l6.current .current.toctree-l7>ul,.wy-menu-vertical .toctree-l7.current .current.toctree-l8>ul,.wy-menu-vertical .toctree-l8.current .current.toctree-l9>ul,.wy-menu-vertical .toctree-l9.current .current.toctree-l10>ul,.wy-menu-vertical .toctree-l10.current .current.toctree-l11>ul{display:block}.wy-menu-vertical li.toctree-l3,.wy-menu-vertical li.toctree-l4{font-size:.9em}.wy-menu-vertical li.toctree-l2 a,.wy-menu-vertical li.toctree-l3 a,.wy-menu-vertical li.toctree-l4 a,.wy-menu-vertical li.toctree-l5 a,.wy-menu-vertical li.toctree-l6 a,.wy-menu-vertical li.toctree-l7 a,.wy-menu-vertical li.toctree-l8 a,.wy-menu-vertical li.toctree-l9 a,.wy-menu-vertical li.toctree-l10 a{color:#404040}.wy-menu-vertical li.toctree-l2 a:hover button.toctree-expand,.wy-menu-vertical li.toctree-l3 a:hover button.toctree-expand,.wy-menu-vertical li.toctree-l4 a:hover button.toctree-expand,.wy-menu-vertical li.toctree-l5 a:hover button.toctree-expand,.wy-menu-vertical li.toctree-l6 a:hover button.toctree-expand,.wy-menu-vertical li.toctree-l7 a:hover button.toctree-expand,.wy-menu-vertical li.toctree-l8 a:hover button.toctree-expand,.wy-menu-vertical li.toctree-l9 a:hover button.toctree-expand,.wy-menu-vertical li.toctree-l10 a:hover button.toctree-expand{color:grey}.wy-menu-vertical li.toctree-l2.current li.toctree-l3>a,.wy-menu-vertical li.toctree-l3.current li.toctree-l4>a,.wy-menu-vertical li.toctree-l4.current li.toctree-l5>a,.wy-menu-vertical li.toctree-l5.current li.toctree-l6>a,.wy-menu-vertical li.toctree-l6.current li.toctree-l7>a,.wy-menu-vertical li.toctree-l7.current li.toctree-l8>a,.wy-menu-vertical li.toctree-l8.current li.toctree-l9>a,.wy-menu-vertical li.toctree-l9.current li.toctree-l10>a,.wy-menu-vertical li.toctree-l10.current li.toctree-l11>a{display:block}.wy-menu-vertical li.toctree-l2.current>a{padding:.4045em 2.427em}.wy-menu-vertical li.toctree-l2.current li.toctree-l3>a{padding:.4045em 1.618em .4045em 4.045em}.wy-menu-vertical li.toctree-l3.current>a{padding:.4045em 4.045em}.wy-menu-vertical li.toctree-l3.current li.toctree-l4>a{padding:.4045em 1.618em .4045em 5.663em}.wy-menu-vertical li.toctree-l4.current>a{padding:.4045em 5.663em}.wy-menu-vertical li.toctree-l4.current li.toctree-l5>a{padding:.4045em 1.618em .4045em 7.281em}.wy-menu-vertical li.toctree-l5.current>a{padding:.4045em 7.281em}.wy-menu-vertical li.toctree-l5.current li.toctree-l6>a{padding:.4045em 1.618em .4045em 8.899em}.wy-menu-vertical li.toctree-l6.current>a{padding:.4045em 8.899em}.wy-menu-vertical li.toctree-l6.current li.toctree-l7>a{padding:.4045em 1.618em .4045em 10.517em}.wy-menu-vertical li.toctree-l7.current>a{padding:.4045em 10.517em}.wy-menu-vertical li.toctree-l7.current li.toctree-l8>a{padding:.4045em 1.618em .4045em 12.135em}.wy-menu-vertical li.toctree-l8.current>a{padding:.4045em 12.135em}.wy-menu-vertical li.toctree-l8.current li.toctree-l9>a{padding:.4045em 1.618em .4045em 13.753em}.wy-menu-vertical li.toctree-l9.current>a{padding:.4045em 13.753em}.wy-menu-vertical li.toctree-l9.current li.toctree-l10>a{padding:.4045em 1.618em .4045em 15.371em}.wy-menu-vertical li.toctree-l10.current>a{padding:.4045em 15.371em}.wy-menu-vertical li.toctree-l10.current li.toctree-l11>a{padding:.4045em 1.618em .4045em 16.989em}.wy-menu-vertical li.toctree-l2.current>a,.wy-menu-vertical li.toctree-l2.current li.toctree-l3>a{background:#c9c9c9}.wy-menu-vertical li.toctree-l2 button.toctree-expand{color:#a3a3a3}.wy-menu-vertical li.toctree-l3.current>a,.wy-menu-vertical li.toctree-l3.current li.toctree-l4>a{background:#bdbdbd}.wy-menu-vertical li.toctree-l3 button.toctree-expand{color:#969696}.wy-menu-vertical li.current ul{display:block}.wy-menu-vertical li ul{margin-bottom:0;display:none}.wy-menu-vertical li ul li a{margin-bottom:0;color:#d9d9d9;font-weight:400}.wy-menu-vertical a{line-height:18px;padding:.4045em 1.618em;display:block;position:relative;font-size:90%;color:#d9d9d9}.wy-menu-vertical a:hover{background-color:#4e4a4a;cursor:pointer}.wy-menu-vertical a:hover button.toctree-expand{color:#d9d9d9}.wy-menu-vertical a:active{background-color:#2980b9;cursor:pointer;color:#fff}.wy-menu-vertical a:active button.toctree-expand{color:#fff}.wy-side-nav-search{display:block;width:300px;padding:.809em;margin-bottom:.809em;z-index:200;background-color:#2980b9;text-align:center;color:#fcfcfc}.wy-side-nav-search input[type=text]{width:100%;border-radius:50px;padding:6px 12px;border-color:#2472a4}.wy-side-nav-search img{display:block;margin:auto auto .809em;height:45px;width:45px;background-color:#2980b9;padding:5px;border-radius:100%}.wy-side-nav-search .wy-dropdown>a,.wy-side-nav-search>a{color:#fcfcfc;font-size:100%;font-weight:700;display:inline-block;padding:4px 6px;margin-bottom:.809em;max-width:100%}.wy-side-nav-search .wy-dropdown>a:hover,.wy-side-nav-search>a:hover{background:hsla(0,0%,100%,.1)}.wy-side-nav-search .wy-dropdown>a img.logo,.wy-side-nav-search>a img.logo{display:block;margin:0 auto;height:auto;width:auto;border-radius:0;max-width:100%;background:transparent}.wy-side-nav-search .wy-dropdown>a.icon img.logo,.wy-side-nav-search>a.icon img.logo{margin-top:.85em}.wy-side-nav-search>div.version{margin-top:-.4045em;margin-bottom:.809em;font-weight:400;color:hsla(0,0%,100%,.3)}.wy-nav .wy-menu-vertical header{color:#2980b9}.wy-nav .wy-menu-vertical a{color:#b3b3b3}.wy-nav .wy-menu-vertical a:hover{background-color:#2980b9;color:#fff}[data-menu-wrap]{-webkit-transition:all .2s ease-in;-moz-transition:all .2s ease-in;transition:all .2s ease-in;position:absolute;opacity:1;width:100%;opacity:0}[data-menu-wrap].move-center{left:0;right:auto;opacity:1}[data-menu-wrap].move-left{right:auto;left:-100%;opacity:0}[data-menu-wrap].move-right{right:-100%;left:auto;opacity:0}.wy-body-for-nav{background:#fcfcfc}.wy-grid-for-nav{position:absolute;width:100%;height:100%}.wy-nav-side{position:fixed;top:0;bottom:0;left:0;padding-bottom:2em;width:300px;overflow-x:hidden;overflow-y:hidden;min-height:100%;color:#9b9b9b;background:#343131;z-index:200}.wy-side-scroll{width:320px;position:relative;overflow-x:hidden;overflow-y:scroll;height:100%}.wy-nav-top{display:none;background:#2980b9;color:#fff;padding:.4045em .809em;position:relative;line-height:50px;text-align:center;font-size:100%;*zoom:1}.wy-nav-top:after,.wy-nav-top:before{display:table;content:""}.wy-nav-top:after{clear:both}.wy-nav-top a{color:#fff;font-weight:700}.wy-nav-top img{margin-right:12px;height:45px;width:45px;background-color:#2980b9;padding:5px;border-radius:100%}.wy-nav-top i{font-size:30px;float:left;cursor:pointer;padding-top:inherit}.wy-nav-content-wrap{margin-left:300px;background:#fcfcfc;min-height:100%}.wy-nav-content{padding:1.618em 3.236em;height:100%;max-width:800px;margin:auto}.wy-body-mask{position:fixed;width:100%;height:100%;background:rgba(0,0,0,.2);display:none;z-index:499}.wy-body-mask.on{display:block}footer{color:grey}footer p{margin-bottom:12px}.rst-content footer span.commit tt,footer span.commit .rst-content tt,footer span.commit code{padding:0;font-family:SFMono-Regular,Menlo,Monaco,Consolas,Liberation Mono,Courier New,Courier,monospace;font-size:1em;background:none;border:none;color:grey}.rst-footer-buttons{*zoom:1}.rst-footer-buttons:after,.rst-footer-buttons:before{width:100%;display:table;content:""}.rst-footer-buttons:after{clear:both}.rst-breadcrumbs-buttons{margin-top:12px;*zoom:1}.rst-breadcrumbs-buttons:after,.rst-breadcrumbs-buttons:before{display:table;content:""}.rst-breadcrumbs-buttons:after{clear:both}#search-results .search li{margin-bottom:24px;border-bottom:1px solid #e1e4e5;padding-bottom:24px}#search-results .search li:first-child{border-top:1px solid #e1e4e5;padding-top:24px}#search-results .search li a{font-size:120%;margin-bottom:12px;display:inline-block}#search-results .context{color:grey;font-size:90%}.genindextable li>ul{margin-left:24px}@media screen and (max-width:768px){.wy-body-for-nav{background:#fcfcfc}.wy-nav-top{display:block}.wy-nav-side{left:-300px}.wy-nav-side.shift{width:85%;left:0}.wy-menu.wy-menu-vertical,.wy-side-nav-search,.wy-side-scroll{width:auto}.wy-nav-content-wrap{margin-left:0}.wy-nav-content-wrap .wy-nav-content{padding:1.618em}.wy-nav-content-wrap.shift{position:fixed;min-width:100%;left:85%;top:0;height:100%;overflow:hidden}}@media screen and (min-width:1100px){.wy-nav-content-wrap{background:rgba(0,0,0,.05)}.wy-nav-content{margin:0;background:#fcfcfc}}@media print{.rst-versions,.wy-nav-side,footer{display:none}.wy-nav-content-wrap{margin-left:0}}.rst-versions{position:fixed;bottom:0;left:0;width:300px;color:#fcfcfc;background:#1f1d1d;font-family:Lato,proxima-nova,Helvetica Neue,Arial,sans-serif;z-index:400}.rst-versions a{color:#2980b9;text-decoration:none}.rst-versions .rst-badge-small{display:none}.rst-versions .rst-current-version{padding:12px;background-color:#272525;display:block;text-align:right;font-size:90%;cursor:pointer;color:#27ae60;*zoom:1}.rst-versions .rst-current-version:after,.rst-versions .rst-current-version:before{display:table;content:""}.rst-versions .rst-current-version:after{clear:both}.rst-content .code-block-caption .rst-versions .rst-current-version .headerlink,.rst-content .eqno .rst-versions .rst-current-version .headerlink,.rst-content .rst-versions .rst-current-version .admonition-title,.rst-content code.download .rst-versions .rst-current-version span:first-child,.rst-content dl dt .rst-versions .rst-current-version .headerlink,.rst-content h1 .rst-versions .rst-current-version .headerlink,.rst-content h2 .rst-versions .rst-current-version .headerlink,.rst-content h3 .rst-versions .rst-current-version .headerlink,.rst-content h4 .rst-versions .rst-current-version .headerlink,.rst-content h5 .rst-versions .rst-current-version .headerlink,.rst-content h6 .rst-versions .rst-current-version .headerlink,.rst-content p .rst-versions .rst-current-version .headerlink,.rst-content table>caption .rst-versions .rst-current-version .headerlink,.rst-content tt.download .rst-versions .rst-current-version span:first-child,.rst-versions .rst-current-version .fa,.rst-versions .rst-current-version .icon,.rst-versions .rst-current-version .rst-content .admonition-title,.rst-versions .rst-current-version .rst-content .code-block-caption .headerlink,.rst-versions .rst-current-version .rst-content .eqno .headerlink,.rst-versions .rst-current-version .rst-content code.download span:first-child,.rst-versions .rst-current-version .rst-content dl dt .headerlink,.rst-versions .rst-current-version .rst-content h1 .headerlink,.rst-versions .rst-current-version .rst-content h2 .headerlink,.rst-versions .rst-current-version .rst-content h3 .headerlink,.rst-versions .rst-current-version .rst-content h4 .headerlink,.rst-versions .rst-current-version .rst-content h5 .headerlink,.rst-versions .rst-current-version .rst-content h6 .headerlink,.rst-versions .rst-current-version .rst-content p .headerlink,.rst-versions .rst-current-version .rst-content table>caption .headerlink,.rst-versions .rst-current-version .rst-content tt.download span:first-child,.rst-versions .rst-current-version .wy-menu-vertical li button.toctree-expand,.wy-menu-vertical li .rst-versions .rst-current-version button.toctree-expand{color:#fcfcfc}.rst-versions .rst-current-version .fa-book,.rst-versions .rst-current-version .icon-book{float:left}.rst-versions .rst-current-version.rst-out-of-date{background-color:#e74c3c;color:#fff}.rst-versions .rst-current-version.rst-active-old-version{background-color:#f1c40f;color:#000}.rst-versions.shift-up{height:auto;max-height:100%;overflow-y:scroll}.rst-versions.shift-up .rst-other-versions{display:block}.rst-versions .rst-other-versions{font-size:90%;padding:12px;color:grey;display:none}.rst-versions .rst-other-versions hr{display:block;height:1px;border:0;margin:20px 0;padding:0;border-top:1px solid #413d3d}.rst-versions .rst-other-versions dd{display:inline-block;margin:0}.rst-versions .rst-other-versions dd a{display:inline-block;padding:6px;color:#fcfcfc}.rst-versions.rst-badge{width:auto;bottom:20px;right:20px;left:auto;border:none;max-width:300px;max-height:90%}.rst-versions.rst-badge .fa-book,.rst-versions.rst-badge .icon-book{float:none;line-height:30px}.rst-versions.rst-badge.shift-up .rst-current-version{text-align:right}.rst-versions.rst-badge.shift-up .rst-current-version .fa-book,.rst-versions.rst-badge.shift-up .rst-current-version .icon-book{float:left}.rst-versions.rst-badge>.rst-current-version{width:auto;height:30px;line-height:30px;padding:0 6px;display:block;text-align:center}@media screen and (max-width:768px){.rst-versions{width:85%;display:none}.rst-versions.shift{display:block}}.rst-content .toctree-wrapper>p.caption,.rst-content h1,.rst-content h2,.rst-content h3,.rst-content h4,.rst-content h5,.rst-content h6{margin-bottom:24px}.rst-content img{max-width:100%;height:auto}.rst-content div.figure,.rst-content figure{margin-bottom:24px}.rst-content div.figure .caption-text,.rst-content figure .caption-text{font-style:italic}.rst-content div.figure p:last-child.caption,.rst-content figure p:last-child.caption{margin-bottom:0}.rst-content div.figure.align-center,.rst-content figure.align-center{text-align:center}.rst-content .section>a>img,.rst-content .section>img,.rst-content section>a>img,.rst-content section>img{margin-bottom:24px}.rst-content abbr[title]{text-decoration:none}.rst-content.style-external-links a.reference.external:after{font-family:FontAwesome;content:"\f08e";color:#b3b3b3;vertical-align:super;font-size:60%;margin:0 .2em}.rst-content blockquote{margin-left:24px;line-height:24px;margin-bottom:24px}.rst-content pre.literal-block{white-space:pre;margin:0;padding:12px;font-family:SFMono-Regular,Menlo,Monaco,Consolas,Liberation Mono,Courier New,Courier,monospace;display:block;overflow:auto}.rst-content div[class^=highlight],.rst-content pre.literal-block{border:1px solid #e1e4e5;overflow-x:auto;margin:1px 0 24px}.rst-content div[class^=highlight] div[class^=highlight],.rst-content pre.literal-block div[class^=highlight]{padding:0;border:none;margin:0}.rst-content div[class^=highlight] td.code{width:100%}.rst-content .linenodiv pre{border-right:1px solid #e6e9ea;margin:0;padding:12px;font-family:SFMono-Regular,Menlo,Monaco,Consolas,Liberation Mono,Courier New,Courier,monospace;user-select:none;pointer-events:none}.rst-content div[class^=highlight] pre{white-space:pre;margin:0;padding:12px;display:block;overflow:auto}.rst-content div[class^=highlight] pre .hll{display:block;margin:0 -12px;padding:0 12px}.rst-content .linenodiv pre,.rst-content div[class^=highlight] pre,.rst-content pre.literal-block{font-family:SFMono-Regular,Menlo,Monaco,Consolas,Liberation Mono,Courier New,Courier,monospace;font-size:12px;line-height:1.4}.rst-content div.highlight .gp,.rst-content div.highlight span.linenos{user-select:none;pointer-events:none}.rst-content div.highlight span.linenos{display:inline-block;padding-left:0;padding-right:12px;margin-right:12px;border-right:1px solid #e6e9ea}.rst-content .code-block-caption{font-style:italic;font-size:85%;line-height:1;padding:1em 0;text-align:center}@media print{.rst-content .codeblock,.rst-content div[class^=highlight],.rst-content div[class^=highlight] pre{white-space:pre-wrap}}.rst-content .admonition,.rst-content .admonition-todo,.rst-content .attention,.rst-content .caution,.rst-content .danger,.rst-content .error,.rst-content .hint,.rst-content .important,.rst-content .note,.rst-content .seealso,.rst-content .tip,.rst-content .warning{clear:both}.rst-content .admonition-todo .last,.rst-content .admonition-todo>:last-child,.rst-content .admonition .last,.rst-content .admonition>:last-child,.rst-content .attention .last,.rst-content .attention>:last-child,.rst-content .caution .last,.rst-content .caution>:last-child,.rst-content .danger .last,.rst-content .danger>:last-child,.rst-content .error .last,.rst-content .error>:last-child,.rst-content .hint .last,.rst-content .hint>:last-child,.rst-content .important .last,.rst-content .important>:last-child,.rst-content .note .last,.rst-content .note>:last-child,.rst-content .seealso .last,.rst-content .seealso>:last-child,.rst-content .tip .last,.rst-content .tip>:last-child,.rst-content .warning .last,.rst-content .warning>:last-child{margin-bottom:0}.rst-content .admonition-title:before{margin-right:4px}.rst-content .admonition table{border-color:rgba(0,0,0,.1)}.rst-content .admonition table td,.rst-content .admonition table th{background:transparent!important;border-color:rgba(0,0,0,.1)!important}.rst-content .section ol.loweralpha,.rst-content .section ol.loweralpha>li,.rst-content .toctree-wrapper ol.loweralpha,.rst-content .toctree-wrapper ol.loweralpha>li,.rst-content section ol.loweralpha,.rst-content section ol.loweralpha>li{list-style:lower-alpha}.rst-content .section ol.upperalpha,.rst-content .section ol.upperalpha>li,.rst-content .toctree-wrapper ol.upperalpha,.rst-content .toctree-wrapper ol.upperalpha>li,.rst-content section ol.upperalpha,.rst-content section ol.upperalpha>li{list-style:upper-alpha}.rst-content .section ol li>*,.rst-content .section ul li>*,.rst-content .toctree-wrapper ol li>*,.rst-content .toctree-wrapper ul li>*,.rst-content section ol li>*,.rst-content section ul li>*{margin-top:12px;margin-bottom:12px}.rst-content .section ol li>:first-child,.rst-content .section ul li>:first-child,.rst-content .toctree-wrapper ol li>:first-child,.rst-content .toctree-wrapper ul li>:first-child,.rst-content section ol li>:first-child,.rst-content section ul li>:first-child{margin-top:0}.rst-content .section ol li>p,.rst-content .section ol li>p:last-child,.rst-content .section ul li>p,.rst-content .section ul li>p:last-child,.rst-content .toctree-wrapper ol li>p,.rst-content .toctree-wrapper ol li>p:last-child,.rst-content .toctree-wrapper ul li>p,.rst-content .toctree-wrapper ul li>p:last-child,.rst-content section ol li>p,.rst-content section ol li>p:last-child,.rst-content section ul li>p,.rst-content section ul li>p:last-child{margin-bottom:12px}.rst-content .section ol li>p:only-child,.rst-content .section ol li>p:only-child:last-child,.rst-content .section ul li>p:only-child,.rst-content .section ul li>p:only-child:last-child,.rst-content .toctree-wrapper ol li>p:only-child,.rst-content .toctree-wrapper ol li>p:only-child:last-child,.rst-content .toctree-wrapper ul li>p:only-child,.rst-content .toctree-wrapper ul li>p:only-child:last-child,.rst-content section ol li>p:only-child,.rst-content section ol li>p:only-child:last-child,.rst-content section ul li>p:only-child,.rst-content section ul li>p:only-child:last-child{margin-bottom:0}.rst-content .section ol li>ol,.rst-content .section ol li>ul,.rst-content .section ul li>ol,.rst-content .section ul li>ul,.rst-content .toctree-wrapper ol li>ol,.rst-content .toctree-wrapper ol li>ul,.rst-content .toctree-wrapper ul li>ol,.rst-content .toctree-wrapper ul li>ul,.rst-content section ol li>ol,.rst-content section ol li>ul,.rst-content section ul li>ol,.rst-content section ul li>ul{margin-bottom:12px}.rst-content .section ol.simple li>*,.rst-content .section ol.simple li ol,.rst-content .section ol.simple li ul,.rst-content .section ul.simple li>*,.rst-content .section ul.simple li ol,.rst-content .section ul.simple li ul,.rst-content .toctree-wrapper ol.simple li>*,.rst-content .toctree-wrapper ol.simple li ol,.rst-content .toctree-wrapper ol.simple li ul,.rst-content .toctree-wrapper ul.simple li>*,.rst-content .toctree-wrapper ul.simple li ol,.rst-content .toctree-wrapper ul.simple li ul,.rst-content section ol.simple li>*,.rst-content section ol.simple li ol,.rst-content section ol.simple li ul,.rst-content section ul.simple li>*,.rst-content section ul.simple li ol,.rst-content section ul.simple li ul{margin-top:0;margin-bottom:0}.rst-content .line-block{margin-left:0;margin-bottom:24px;line-height:24px}.rst-content .line-block .line-block{margin-left:24px;margin-bottom:0}.rst-content .topic-title{font-weight:700;margin-bottom:12px}.rst-content .toc-backref{color:#404040}.rst-content .align-right{float:right;margin:0 0 24px 24px}.rst-content .align-left{float:left;margin:0 24px 24px 0}.rst-content .align-center{margin:auto}.rst-content .align-center:not(table){display:block}.rst-content .code-block-caption .headerlink,.rst-content .eqno .headerlink,.rst-content .toctree-wrapper>p.caption .headerlink,.rst-content dl dt .headerlink,.rst-content h1 .headerlink,.rst-content h2 .headerlink,.rst-content h3 .headerlink,.rst-content h4 .headerlink,.rst-content h5 .headerlink,.rst-content h6 .headerlink,.rst-content p.caption .headerlink,.rst-content p .headerlink,.rst-content table>caption .headerlink{opacity:0;font-size:14px;font-family:FontAwesome;margin-left:.5em}.rst-content .code-block-caption .headerlink:focus,.rst-content .code-block-caption:hover .headerlink,.rst-content .eqno .headerlink:focus,.rst-content .eqno:hover .headerlink,.rst-content .toctree-wrapper>p.caption .headerlink:focus,.rst-content .toctree-wrapper>p.caption:hover .headerlink,.rst-content dl dt .headerlink:focus,.rst-content dl dt:hover .headerlink,.rst-content h1 .headerlink:focus,.rst-content h1:hover .headerlink,.rst-content h2 .headerlink:focus,.rst-content h2:hover .headerlink,.rst-content h3 .headerlink:focus,.rst-content h3:hover .headerlink,.rst-content h4 .headerlink:focus,.rst-content h4:hover .headerlink,.rst-content h5 .headerlink:focus,.rst-content h5:hover .headerlink,.rst-content h6 .headerlink:focus,.rst-content h6:hover .headerlink,.rst-content p.caption .headerlink:focus,.rst-content p.caption:hover .headerlink,.rst-content p .headerlink:focus,.rst-content p:hover .headerlink,.rst-content table>caption .headerlink:focus,.rst-content table>caption:hover .headerlink{opacity:1}.rst-content .btn:focus{outline:2px solid}.rst-content table>caption .headerlink:after{font-size:12px}.rst-content .centered{text-align:center}.rst-content .sidebar{float:right;width:40%;display:block;margin:0 0 24px 24px;padding:24px;background:#f3f6f6;border:1px solid #e1e4e5}.rst-content .sidebar dl,.rst-content .sidebar p,.rst-content .sidebar ul{font-size:90%}.rst-content .sidebar .last,.rst-content .sidebar>:last-child{margin-bottom:0}.rst-content .sidebar .sidebar-title{display:block;font-family:Roboto Slab,ff-tisa-web-pro,Georgia,Arial,sans-serif;font-weight:700;background:#e1e4e5;padding:6px 12px;margin:-24px -24px 24px;font-size:100%}.rst-content .highlighted{background:#f1c40f;box-shadow:0 0 0 2px #f1c40f;display:inline;font-weight:700}.rst-content .citation-reference,.rst-content .footnote-reference{vertical-align:baseline;position:relative;top:-.4em;line-height:0;font-size:90%}.rst-content .hlist{width:100%}.rst-content dl dt span.classifier:before{content:" : "}.rst-content dl dt span.classifier-delimiter{display:none!important}html.writer-html4 .rst-content table.docutils.citation,html.writer-html4 .rst-content table.docutils.footnote{background:none;border:none}html.writer-html4 .rst-content table.docutils.citation td,html.writer-html4 .rst-content table.docutils.citation tr,html.writer-html4 .rst-content table.docutils.footnote td,html.writer-html4 .rst-content table.docutils.footnote tr{border:none;background-color:transparent!important;white-space:normal}html.writer-html4 .rst-content table.docutils.citation td.label,html.writer-html4 .rst-content table.docutils.footnote td.label{padding-left:0;padding-right:0;vertical-align:top}html.writer-html5 .rst-content dl.field-list,html.writer-html5 .rst-content dl.footnote{display:grid;grid-template-columns:max-content auto}html.writer-html5 .rst-content dl.field-list>dt,html.writer-html5 .rst-content dl.footnote>dt{padding-left:1rem}html.writer-html5 .rst-content dl.field-list>dt:after,html.writer-html5 .rst-content dl.footnote>dt:after{content:":"}html.writer-html5 .rst-content dl.field-list>dd,html.writer-html5 .rst-content dl.field-list>dt,html.writer-html5 .rst-content dl.footnote>dd,html.writer-html5 .rst-content dl.footnote>dt{margin-bottom:0}html.writer-html5 .rst-content dl.footnote{font-size:.9rem}html.writer-html5 .rst-content dl.footnote>dt{margin:0 .5rem .5rem 0;line-height:1.2rem;word-break:break-all;font-weight:400}html.writer-html5 .rst-content dl.footnote>dt>span.brackets{margin-right:.5rem}html.writer-html5 .rst-content dl.footnote>dt>span.brackets:before{content:"["}html.writer-html5 .rst-content dl.footnote>dt>span.brackets:after{content:"]"}html.writer-html5 .rst-content dl.footnote>dt>span.fn-backref{font-style:italic}html.writer-html5 .rst-content dl.footnote>dd{margin:0 0 .5rem;line-height:1.2rem}html.writer-html5 .rst-content dl.footnote>dd p,html.writer-html5 .rst-content dl.option-list kbd{font-size:.9rem}.rst-content table.docutils.footnote,html.writer-html4 .rst-content table.docutils.citation,html.writer-html5 .rst-content dl.footnote{color:grey}.rst-content table.docutils.footnote code,.rst-content table.docutils.footnote tt,html.writer-html4 .rst-content table.docutils.citation code,html.writer-html4 .rst-content table.docutils.citation tt,html.writer-html5 .rst-content dl.footnote code,html.writer-html5 .rst-content dl.footnote tt{color:#555}.rst-content .wy-table-responsive.citation,.rst-content .wy-table-responsive.footnote{margin-bottom:0}.rst-content .wy-table-responsive.citation+:not(.citation),.rst-content .wy-table-responsive.footnote+:not(.footnote){margin-top:24px}.rst-content .wy-table-responsive.citation:last-child,.rst-content .wy-table-responsive.footnote:last-child{margin-bottom:24px}.rst-content table.docutils th{border-color:#e1e4e5}html.writer-html5 .rst-content table.docutils th{border:1px solid #e1e4e5}html.writer-html5 .rst-content table.docutils td>p,html.writer-html5 .rst-content table.docutils th>p{line-height:1rem;margin-bottom:0;font-size:.9rem}.rst-content table.docutils td .last,.rst-content table.docutils td .last>:last-child{margin-bottom:0}.rst-content table.field-list,.rst-content table.field-list td{border:none}.rst-content table.field-list td p{font-size:inherit;line-height:inherit}.rst-content table.field-list td>strong{display:inline-block}.rst-content table.field-list .field-name{padding-right:10px;text-align:left;white-space:nowrap}.rst-content table.field-list .field-body{text-align:left}.rst-content code,.rst-content tt{color:#000;font-family:SFMono-Regular,Menlo,Monaco,Consolas,Liberation Mono,Courier New,Courier,monospace;padding:2px 5px}.rst-content code big,.rst-content code em,.rst-content tt big,.rst-content tt em{font-size:100%!important;line-height:normal}.rst-content code.literal,.rst-content tt.literal{color:#e74c3c;white-space:normal}.rst-content code.xref,.rst-content tt.xref,a .rst-content code,a .rst-content tt{font-weight:700;color:#404040}.rst-content kbd,.rst-content pre,.rst-content samp{font-family:SFMono-Regular,Menlo,Monaco,Consolas,Liberation Mono,Courier New,Courier,monospace}.rst-content a code,.rst-content a tt{color:#2980b9}.rst-content dl{margin-bottom:24px}.rst-content dl dt{font-weight:700;margin-bottom:12px}.rst-content dl ol,.rst-content dl p,.rst-content dl table,.rst-content dl ul{margin-bottom:12px}.rst-content dl dd{margin:0 0 12px 24px;line-height:24px}html.writer-html4 .rst-content dl:not(.docutils),html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.glossary):not(.simple){margin-bottom:24px}html.writer-html4 .rst-content dl:not(.docutils)>dt,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.glossary):not(.simple)>dt{display:table;margin:6px 0;font-size:90%;line-height:normal;background:#e7f2fa;color:#2980b9;border-top:3px solid #6ab0de;padding:6px;position:relative}html.writer-html4 .rst-content dl:not(.docutils)>dt:before,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.glossary):not(.simple)>dt:before{color:#6ab0de}html.writer-html4 .rst-content dl:not(.docutils)>dt .headerlink,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.glossary):not(.simple)>dt .headerlink{color:#404040;font-size:100%!important}html.writer-html4 .rst-content dl:not(.docutils) dl:not(.field-list)>dt,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.glossary):not(.simple) dl:not(.field-list)>dt{margin-bottom:6px;border:none;border-left:3px solid #ccc;background:#f0f0f0;color:#555}html.writer-html4 .rst-content dl:not(.docutils) dl:not(.field-list)>dt .headerlink,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.glossary):not(.simple) dl:not(.field-list)>dt .headerlink{color:#404040;font-size:100%!important}html.writer-html4 .rst-content dl:not(.docutils)>dt:first-child,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.glossary):not(.simple)>dt:first-child{margin-top:0}html.writer-html4 .rst-content dl:not(.docutils) code.descclassname,html.writer-html4 .rst-content dl:not(.docutils) code.descname,html.writer-html4 .rst-content dl:not(.docutils) tt.descclassname,html.writer-html4 .rst-content dl:not(.docutils) tt.descname,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.glossary):not(.simple) code.descclassname,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.glossary):not(.simple) code.descname,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.glossary):not(.simple) tt.descclassname,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.glossary):not(.simple) tt.descname{background-color:transparent;border:none;padding:0;font-size:100%!important}html.writer-html4 .rst-content dl:not(.docutils) code.descname,html.writer-html4 .rst-content dl:not(.docutils) tt.descname,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.glossary):not(.simple) code.descname,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.glossary):not(.simple) tt.descname{font-weight:700}html.writer-html4 .rst-content dl:not(.docutils) .optional,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.glossary):not(.simple) .optional{display:inline-block;padding:0 4px;color:#000;font-weight:700}html.writer-html4 .rst-content dl:not(.docutils) .property,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.glossary):not(.simple) .property{display:inline-block;padding-right:8px;max-width:100%}html.writer-html4 .rst-content dl:not(.docutils) .k,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.glossary):not(.simple) .k{font-style:italic}html.writer-html4 .rst-content dl:not(.docutils) .descclassname,html.writer-html4 .rst-content dl:not(.docutils) .descname,html.writer-html4 .rst-content dl:not(.docutils) .sig-name,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.glossary):not(.simple) .descclassname,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.glossary):not(.simple) .descname,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.glossary):not(.simple) .sig-name{font-family:SFMono-Regular,Menlo,Monaco,Consolas,Liberation Mono,Courier New,Courier,monospace;color:#000}.rst-content .viewcode-back,.rst-content .viewcode-link{display:inline-block;color:#27ae60;font-size:80%;padding-left:24px}.rst-content .viewcode-back{display:block;float:right}.rst-content p.rubric{margin-bottom:12px;font-weight:700}.rst-content code.download,.rst-content tt.download{background:inherit;padding:inherit;font-weight:400;font-family:inherit;font-size:inherit;color:inherit;border:inherit;white-space:inherit}.rst-content code.download span:first-child,.rst-content tt.download span:first-child{-webkit-font-smoothing:subpixel-antialiased}.rst-content code.download span:first-child:before,.rst-content tt.download span:first-child:before{margin-right:4px}.rst-content .guilabel{border:1px solid #7fbbe3;background:#e7f2fa;font-size:80%;font-weight:700;border-radius:4px;padding:2.4px 6px;margin:auto 2px}.rst-content .versionmodified{font-style:italic}@media screen and (max-width:480px){.rst-content .sidebar{width:100%}}span[id*=MathJax-Span]{color:#404040}.math{text-align:center}@font-face{font-family:Lato;src:url(fonts/lato-normal.woff2?bd03a2cc277bbbc338d464e679fe9942) format("woff2"),url(fonts/lato-normal.woff?27bd77b9162d388cb8d4c4217c7c5e2a) format("woff");font-weight:400;font-style:normal;font-display:block}@font-face{font-family:Lato;src:url(fonts/lato-bold.woff2?cccb897485813c7c256901dbca54ecf2) format("woff2"),url(fonts/lato-bold.woff?d878b6c29b10beca227e9eef4246111b) format("woff");font-weight:700;font-style:normal;font-display:block}@font-face{font-family:Lato;src:url(fonts/lato-bold-italic.woff2?0b6bb6725576b072c5d0b02ecdd1900d) format("woff2"),url(fonts/lato-bold-italic.woff?9c7e4e9eb485b4a121c760e61bc3707c) format("woff");font-weight:700;font-style:italic;font-display:block}@font-face{font-family:Lato;src:url(fonts/lato-normal-italic.woff2?4eb103b4d12be57cb1d040ed5e162e9d) format("woff2"),url(fonts/lato-normal-italic.woff?f28f2d6482446544ef1ea1ccc6dd5892) format("woff");font-weight:400;font-style:italic;font-display:block}@font-face{font-family:Roboto Slab;font-style:normal;font-weight:400;src:url(fonts/Roboto-Slab-Regular.woff2?7abf5b8d04d26a2cafea937019bca958) format("woff2"),url(fonts/Roboto-Slab-Regular.woff?c1be9284088d487c5e3ff0a10a92e58c) format("woff");font-display:block}@font-face{font-family:Roboto Slab;font-style:normal;font-weight:700;src:url(fonts/Roboto-Slab-Bold.woff2?9984f4a9bda09be08e83f2506954adbe) format("woff2"),url(fonts/Roboto-Slab-Bold.woff?bed5564a116b05148e3b3bea6fb1162a) format("woff");font-display:block}
html/_static/doctools.js (new)
- @@ -0,0 +1,264 @@
 + /* + * doctools.js + * ~~~~~~~~~~~ + * + * Base JavaScript utilities for all Sphinx HTML documentation. + * + * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS. + * :license: BSD, see LICENSE for details. + * + */ + "use strict"; + + const _ready = (callback) => { + if (document.readyState !== "loading") { + callback(); + } else { + document.addEventListener("DOMContentLoaded", callback); + } + }; + + /** + * highlight a given string on a node by wrapping it in + * span elements with the given class name. + */ + const _highlight = (node, addItems, text, className) => { + if (node.nodeType === Node.TEXT_NODE) { + const val = node.nodeValue; + const parent = node.parentNode; + const pos = val.toLowerCase().indexOf(text); + if ( + pos >= 0 && + !parent.classList.contains(className) && + !parent.classList.contains("nohighlight") + ) { + let span; + + const closestNode = parent.closest("body, svg, foreignObject"); + const isInSVG = closestNode && closestNode.matches("svg"); + if (isInSVG) { + span = document.createElementNS("http://www.w3.org/2000/svg", "tspan"); + } else { + span = document.createElement("span"); + span.classList.add(className); + } + + span.appendChild(document.createTextNode(val.substr(pos, text.length))); + parent.insertBefore( + span, + parent.insertBefore( + document.createTextNode(val.substr(pos + text.length)), + node.nextSibling + ) + ); + node.nodeValue = val.substr(0, pos); + + if (isInSVG) { + const rect = document.createElementNS( + "http://www.w3.org/2000/svg", + "rect" + ); + const bbox = parent.getBBox(); + rect.x.baseVal.value = bbox.x; + rect.y.baseVal.value = bbox.y; + rect.width.baseVal.value = bbox.width; + rect.height.baseVal.value = bbox.height; + rect.setAttribute("class", className); + addItems.push({ parent: parent, target: rect }); + } + } + } else if (node.matches && !node.matches("button, select, textarea")) { + node.childNodes.forEach((el) => _highlight(el, addItems, text, className)); + } + }; + const _highlightText = (thisNode, text, className) => { + let addItems = []; + _highlight(thisNode, addItems, text, className); + addItems.forEach((obj) => + obj.parent.insertAdjacentElement("beforebegin", obj.target) + ); + }; + + /** + * Small JavaScript module for the documentation. + */ + const Documentation = { + init: () => { + Documentation.highlightSearchWords(); + Documentation.initDomainIndexTable(); + Documentation.initOnKeyListeners(); + }, + + /** + * i18n support + */ + TRANSLATIONS: {}, + PLURAL_EXPR: (n) => (n === 1 ? 0 : 1), + LOCALE: "unknown", + + // gettext and ngettext don't access this so that the functions + // can safely bound to a different name (_ = Documentation.gettext) + gettext: (string) => { + const translated = Documentation.TRANSLATIONS[string]; + switch (typeof translated) { + case "undefined": + return string; // no translation + case "string": + return translated; // translation exists + default: + return translated[0]; // (singular, plural) translation tuple exists + } + }, + + ngettext: (singular, plural, n) => { + const translated = Documentation.TRANSLATIONS[singular]; + if (typeof translated !== "undefined") + return translated[Documentation.PLURAL_EXPR(n)]; + return n === 1 ? singular : plural; + }, + + addTranslations: (catalog) => { + Object.assign(Documentation.TRANSLATIONS, catalog.messages); + Documentation.PLURAL_EXPR = new Function( + "n", + `return (${catalog.plural_expr})` + ); + Documentation.LOCALE = catalog.locale; + }, + + /** + * highlight the search words provided in the url in the text + */ + highlightSearchWords: () => { + const highlight = + new URLSearchParams(window.location.search).get("highlight") || ""; + const terms = highlight.toLowerCase().split(/\s+/).filter(x => x); + if (terms.length === 0) return; // nothing to do + + // There should never be more than one element matching "div.body" + const divBody = document.querySelectorAll("div.body"); + const body = divBody.length ? divBody[0] : document.querySelector("body"); + window.setTimeout(() => { + terms.forEach((term) => _highlightText(body, term, "highlighted")); + }, 10); + + const searchBox = document.getElementById("searchbox"); + if (searchBox === null) return; + searchBox.appendChild( + document + .createRange() + .createContextualFragment( + '' + + '<a href="javascript:Documentation.hideSearchWords()">' + + Documentation.gettext("Hide Search Matches") + + "</a>" + ) + ); + }, + + /** + * helper function to hide the search marks again + */ + hideSearchWords: () => { + document + .querySelectorAll("#searchbox .highlight-link") + .forEach((el) => el.remove()); + document + .querySelectorAll("span.highlighted") + .forEach((el) => el.classList.remove("highlighted")); + const url = new URL(window.location); + url.searchParams.delete("highlight"); + window.history.replaceState({}, "", url); + }, + + /** + * helper function to focus on search bar + */ + focusSearchBar: () => { + document.querySelectorAll("input[name=q]")[0]?.focus(); + }, + + /** + * Initialise the domain index toggle buttons + */ + initDomainIndexTable: () => { + const toggler = (el) => { + const idNumber = el.id.substr(7); + const toggledRows = document.querySelectorAll(`tr.cg-${idNumber}`); + if (el.src.substr(-9) === "minus.png") { + el.src = `${el.src.substr(0, el.src.length - 9)}plus.png`; + toggledRows.forEach((el) => (el.style.display = "none")); + } else { + el.src = `${el.src.substr(0, el.src.length - 8)}minus.png`; + toggledRows.forEach((el) => (el.style.display = "")); + } + }; + + const togglerElements = document.querySelectorAll("img.toggler"); + togglerElements.forEach((el) => + el.addEventListener("click", (event) => toggler(event.currentTarget)) + ); + togglerElements.forEach((el) => (el.style.display = "")); + if (DOCUMENTATION_OPTIONS.COLLAPSE_INDEX) togglerElements.forEach(toggler); + }, + + initOnKeyListeners: () => { + // only install a listener if it is really needed + if ( + !DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS && + !DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS + ) + return; + + const blacklistedElements = new Set([ + "TEXTAREA", + "INPUT", + "SELECT", + "BUTTON", + ]); + document.addEventListener("keydown", (event) => { + if (blacklistedElements.has(document.activeElement.tagName)) return; // bail for input elements + if (event.altKey || event.ctrlKey || event.metaKey) return; // bail with special keys + + if (!event.shiftKey) { + switch (event.key) { + case "ArrowLeft": + if (!DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS) break; + + const prevLink = document.querySelector('link[rel="prev"]'); + if (prevLink && prevLink.href) { + window.location.href = prevLink.href; + event.preventDefault(); + } + break; + case "ArrowRight": + if (!DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS) break; + + const nextLink = document.querySelector('link[rel="next"]'); + if (nextLink && nextLink.href) { + window.location.href = nextLink.href; + event.preventDefault(); + } + break; + case "Escape": + if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS) break; + Documentation.hideSearchWords(); + event.preventDefault(); + } + } + + // some keyboard layouts may need Shift to get / + switch (event.key) { + case "/": + if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS) break; + Documentation.focusSearchBar(); + event.preventDefault(); + } + }); + }, + }; + + // quick alias for translations + const _ = Documentation.gettext; + + _ready(Documentation.init);
html/_static/documentation_options.js (new)
- @@ -0,0 +1,14 @@
  + var DOCUMENTATION_OPTIONS = { + URL_ROOT: document.getElementById("documentation_options").getAttribute('data-url_root'), + VERSION: '0.1', + LANGUAGE: 'en', + COLLAPSE_INDEX: false, + BUILDER: 'html', + FILE_SUFFIX: '.html', + LINK_SUFFIX: '.html', + HAS_SOURCE: true, + SOURCELINK_SUFFIX: '.txt', + NAVIGATION_WITH_KEYS: false, + SHOW_SEARCH_SUMMARY: true, + ENABLE_SEARCH_SHORTCUTS: false, + };
html/_static/favicon.ico (new)
html/_static/file.png (new)
html/_static/jquery-3.6.0.js (new)
- @@ -0,0 +1,10881 @@
 + /*! + * jQuery JavaScript Library v3.6.0 + * https://jquery.com/ + * + * Includes Sizzle.js + * https://sizzlejs.com/ + * + * Copyright OpenJS Foundation and other contributors + * Released under the MIT license + * https://jquery.org/license + * + * Date: 2021-03-02T17:08Z + */ + ( function( global, factory ) { + + "use strict"; + + if ( typeof module === "object" && typeof module.exports === "object" ) { + + // For CommonJS and CommonJS-like environments where a proper `window` + // is present, execute the factory and get jQuery. + // For environments that do not have a `window` with a `document` + // (such as Node.js), expose a factory as module.exports. + // This accentuates the need for the creation of a real `window`. + // e.g. var jQuery = require("jquery")(window); + // See ticket #14549 for more info. + module.exports = global.document ? + factory( global, true ) : + function( w ) { + if ( !w.document ) { + throw new Error( "jQuery requires a window with a document" ); + } + return factory( w ); + }; + } else { + factory( global ); + } + + // Pass this if window is not defined yet + } )( typeof window !== "undefined" ? window : this, function( window, noGlobal ) { + + // Edge <= 12 - 13+, Firefox <=18 - 45+, IE 10 - 11, Safari 5.1 - 9+, iOS 6 - 9.1 + // throw exceptions when non-strict code (e.g., ASP.NET 4.5) accesses strict mode + // arguments.callee.caller (trac-13335). But as of jQuery 3.0 (2016), strict mode should be common + // enough that all such attempts are guarded in a try block. + "use strict"; + + var arr = []; + + var getProto = Object.getPrototypeOf; + + var slice = arr.slice; + + var flat = arr.flat ? function( array ) { + return arr.flat.call( array ); + } : function( array ) { + return arr.concat.apply( [], array ); + }; + + + var push = arr.push; + + var indexOf = arr.indexOf; + + var class2type = {}; + + var toString = class2type.toString; + + var hasOwn = class2type.hasOwnProperty; + + var fnToString = hasOwn.toString; + + var ObjectFunctionString = fnToString.call( Object ); + + var support = {}; + + var isFunction = function isFunction( obj ) { + + // Support: Chrome <=57, Firefox <=52 + // In some browsers, typeof returns "function" for HTML <object> elements + // (i.e., `typeof document.createElement( "object" ) === "function"`). + // We don't want to classify *any* DOM node as a function. + // Support: QtWeb <=3.8.5, WebKit <=534.34, wkhtmltopdf tool <=0.12.5 + // Plus for old WebKit, typeof returns "function" for HTML collections + // (e.g., `typeof document.getElementsByTagName("div") === "function"`). (gh-4756) + return typeof obj === "function" && typeof obj.nodeType !== "number" && + typeof obj.item !== "function"; + }; + + + var isWindow = function isWindow( obj ) { + return obj != null && obj === obj.window; + }; + + + var document = window.document; + + + + var preservedScriptAttributes = { + type: true, + src: true, + nonce: true, + noModule: true + }; + + function DOMEval( code, node, doc ) { + doc = doc || document; + + var i, val, + script = doc.createElement( "script" ); + + script.text = code; + if ( node ) { + for ( i in preservedScriptAttributes ) { + + // Support: Firefox 64+, Edge 18+ + // Some browsers don't support the "nonce" property on scripts. + // On the other hand, just using `getAttribute` is not enough as + // the `nonce` attribute is reset to an empty string whenever it + // becomes browsing-context connected. + // See https://github.com/whatwg/html/issues/2369 + // See https://html.spec.whatwg.org/#nonce-attributes + // The `node.getAttribute` check was added for the sake of + // `jQuery.globalEval` so that it can fake a nonce-containing node + // via an object. + val = node[ i ] || node.getAttribute && node.getAttribute( i ); + if ( val ) { + script.setAttribute( i, val ); + } + } + } + doc.head.appendChild( script ).parentNode.removeChild( script ); + } + + + function toType( obj ) { + if ( obj == null ) { + return obj + ""; + } + + // Support: Android <=2.3 only (functionish RegExp) + return typeof obj === "object" || typeof obj === "function" ? + class2type[ toString.call( obj ) ] || "object" : + typeof obj; + } + /* global Symbol */ + // Defining this global in .eslintrc.json would create a danger of using the global + // unguarded in another place, it seems safer to define global only for this module + + + + var + version = "3.6.0", + + // Define a local copy of jQuery + jQuery = function( selector, context ) { + + // The jQuery object is actually just the init constructor 'enhanced' + // Need init if jQuery is called (just allow error to be thrown if not included) + return new jQuery.fn.init( selector, context ); + }; + + jQuery.fn = jQuery.prototype = { + + // The current version of jQuery being used + jquery: version, + + constructor: jQuery, + + // The default length of a jQuery object is 0 + length: 0, + + toArray: function() { + return slice.call( this ); + }, + + // Get the Nth element in the matched element set OR + // Get the whole matched element set as a clean array + get: function( num ) { + + // Return all the elements in a clean array + if ( num == null ) { + return slice.call( this ); + } + + // Return just the one element from the set + return num < 0 ? this[ num + this.length ] : this[ num ]; + }, + + // Take an array of elements and push it onto the stack + // (returning the new matched element set) + pushStack: function( elems ) { + + // Build a new jQuery matched element set + var ret = jQuery.merge( this.constructor(), elems ); + + // Add the old object onto the stack (as a reference) + ret.prevObject = this; + + // Return the newly-formed element set + return ret; + }, + + // Execute a callback for every element in the matched set. + each: function( callback ) { + return jQuery.each( this, callback ); + }, + + map: function( callback ) { + return this.pushStack( jQuery.map( this, function( elem, i ) { + return callback.call( elem, i, elem ); + } ) ); + }, + + slice: function() { + return this.pushStack( slice.apply( this, arguments ) ); + }, + + first: function() { + return this.eq( 0 ); + }, + + last: function() { + return this.eq( -1 ); + }, + + even: function() { + return this.pushStack( jQuery.grep( this, function( _elem, i ) { + return ( i + 1 ) % 2; + } ) ); + }, + + odd: function() { + return this.pushStack( jQuery.grep( this, function( _elem, i ) { + return i % 2; + } ) ); + }, + + eq: function( i ) { + var len = this.length, + j = +i + ( i < 0 ? len : 0 ); + return this.pushStack( j >= 0 && j < len ? [ this[ j ] ] : [] ); + }, + + end: function() { + return this.prevObject || this.constructor(); + }, + + // For internal use only. + // Behaves like an Array's method, not like a jQuery method. + push: push, + sort: arr.sort, + splice: arr.splice + }; + + jQuery.extend = jQuery.fn.extend = function() { + var options, name, src, copy, copyIsArray, clone, + target = arguments[ 0 ] || {}, + i = 1, + length = arguments.length, + deep = false; + + // Handle a deep copy situation + if ( typeof target === "boolean" ) { + deep = target; + + // Skip the boolean and the target + target = arguments[ i ] || {}; + i++; + } + + // Handle case when target is a string or something (possible in deep copy) + if ( typeof target !== "object" && !isFunction( target ) ) { + target = {}; + } + + // Extend jQuery itself if only one argument is passed + if ( i === length ) { + target = this; + i--; + } + + for ( ; i < length; i++ ) { + + // Only deal with non-null/undefined values + if ( ( options = arguments[ i ] ) != null ) { + + // Extend the base object + for ( name in options ) { + copy = options[ name ]; + + // Prevent Object.prototype pollution + // Prevent never-ending loop + if ( name === "__proto__" || target === copy ) { + continue; + } + + // Recurse if we're merging plain objects or arrays + if ( deep && copy && ( jQuery.isPlainObject( copy ) || + ( copyIsArray = Array.isArray( copy ) ) ) ) { + src = target[ name ]; + + // Ensure proper type for the source value + if ( copyIsArray && !Array.isArray( src ) ) { + clone = []; + } else if ( !copyIsArray && !jQuery.isPlainObject( src ) ) { + clone = {}; + } else { + clone = src; + } + copyIsArray = false; + + // Never move original objects, clone them + target[ name ] = jQuery.extend( deep, clone, copy ); + + // Don't bring in undefined values + } else if ( copy !== undefined ) { + target[ name ] = copy; + } + } + } + } + + // Return the modified object + return target; + }; + + jQuery.extend( { + + // Unique for each copy of jQuery on the page + expando: "jQuery" + ( version + Math.random() ).replace( /\D/g, "" ), + + // Assume jQuery is ready without the ready module + isReady: true, + + error: function( msg ) { + throw new Error( msg ); + }, + + noop: function() {}, + + isPlainObject: function( obj ) { + var proto, Ctor; + + // Detect obvious negatives + // Use toString instead of jQuery.type to catch host objects + if ( !obj || toString.call( obj ) !== "[object Object]" ) { + return false; + } + + proto = getProto( obj ); + + // Objects with no prototype (e.g., `Object.create( null )`) are plain + if ( !proto ) { + return true; + } + + // Objects with prototype are plain iff they were constructed by a global Object function + Ctor = hasOwn.call( proto, "constructor" ) && proto.constructor; + return typeof Ctor === "function" && fnToString.call( Ctor ) === ObjectFunctionString; + }, + + isEmptyObject: function( obj ) { + var name; + + for ( name in obj ) { + return false; + } + return true; + }, + + // Evaluates a script in a provided context; falls back to the global one + // if not specified. + globalEval: function( code, options, doc ) { + DOMEval( code, { nonce: options && options.nonce }, doc ); + }, + + each: function( obj, callback ) { + var length, i = 0; + + if ( isArrayLike( obj ) ) { + length = obj.length; + for ( ; i < length; i++ ) { + if ( callback.call( obj[ i ], i, obj[ i ] ) === false ) { + break; + } + } + } else { + for ( i in obj ) { + if ( callback.call( obj[ i ], i, obj[ i ] ) === false ) { + break; + } + } + } + + return obj; + }, + + // results is for internal usage only + makeArray: function( arr, results ) { + var ret = results || []; + + if ( arr != null ) { + if ( isArrayLike( Object( arr ) ) ) { + jQuery.merge( ret, + typeof arr === "string" ? + [ arr ] : arr + ); + } else { + push.call( ret, arr ); + } + } + + return ret; + }, + + inArray: function( elem, arr, i ) { + return arr == null ? -1 : indexOf.call( arr, elem, i ); + }, + + // Support: Android <=4.0 only, PhantomJS 1 only + // push.apply(_, arraylike) throws on ancient WebKit + merge: function( first, second ) { + var len = +second.length, + j = 0, + i = first.length; + + for ( ; j < len; j++ ) { + first[ i++ ] = second[ j ]; + } + + first.length = i; + + return first; + }, + + grep: function( elems, callback, invert ) { + var callbackInverse, + matches = [], + i = 0, + length = elems.length, + callbackExpect = !invert; + + // Go through the array, only saving the items + // that pass the validator function + for ( ; i < length; i++ ) { + callbackInverse = !callback( elems[ i ], i ); + if ( callbackInverse !== callbackExpect ) { + matches.push( elems[ i ] ); + } + } + + return matches; + }, + + // arg is for internal usage only + map: function( elems, callback, arg ) { + var length, value, + i = 0, + ret = []; + + // Go through the array, translating each of the items to their new values + if ( isArrayLike( elems ) ) { + length = elems.length; + for ( ; i < length; i++ ) { + value = callback( elems[ i ], i, arg ); + + if ( value != null ) { + ret.push( value ); + } + } + + // Go through every key on the object, + } else { + for ( i in elems ) { + value = callback( elems[ i ], i, arg ); + + if ( value != null ) { + ret.push( value ); + } + } + } + + // Flatten any nested arrays + return flat( ret ); + }, + + // A global GUID counter for objects + guid: 1, + + // jQuery.support is not used in Core but other projects attach their + // properties to it so it needs to exist. + support: support + } ); + + if ( typeof Symbol === "function" ) { + jQuery.fn[ Symbol.iterator ] = arr[ Symbol.iterator ]; + } + + // Populate the class2type map + jQuery.each( "Boolean Number String Function Array Date RegExp Object Error Symbol".split( " " ), + function( _i, name ) { + class2type[ "[object " + name + "]" ] = name.toLowerCase(); + } ); + + function isArrayLike( obj ) { + + // Support: real iOS 8.2 only (not reproducible in simulator) + // `in` check used to prevent JIT error (gh-2145) + // hasOwn isn't used here due to false negatives + // regarding Nodelist length in IE + var length = !!obj && "length" in obj && obj.length, + type = toType( obj ); + + if ( isFunction( obj ) || isWindow( obj ) ) { + return false; + } + + return type === "array" || length === 0 || + typeof length === "number" && length > 0 && ( length - 1 ) in obj; + } + var Sizzle = + /*! + * Sizzle CSS Selector Engine v2.3.6 + * https://sizzlejs.com/ + * + * Copyright JS Foundation and other contributors + * Released under the MIT license + * https://js.foundation/ + * + * Date: 2021-02-16 + */ + ( function( window ) { + var i, + support, + Expr, + getText, + isXML, + tokenize, + compile, + select, + outermostContext, + sortInput, + hasDuplicate, + + // Local document vars + setDocument, + document, + docElem, + documentIsHTML, + rbuggyQSA, + rbuggyMatches, + matches, + contains, + + // Instance-specific data + expando = "sizzle" + 1 * new Date(), + preferredDoc = window.document, + dirruns = 0, + done = 0, + classCache = createCache(), + tokenCache = createCache(), + compilerCache = createCache(), + nonnativeSelectorCache = createCache(), + sortOrder = function( a, b ) { + if ( a === b ) { + hasDuplicate = true; + } + return 0; + }, + + // Instance methods + hasOwn = ( {} ).hasOwnProperty, + arr = [], + pop = arr.pop, + pushNative = arr.push, + push = arr.push, + slice = arr.slice, + + // Use a stripped-down indexOf as it's faster than native + // https://jsperf.com/thor-indexof-vs-for/5 + indexOf = function( list, elem ) { + var i = 0, + len = list.length; + for ( ; i < len; i++ ) { + if ( list[ i ] === elem ) { + return i; + } + } + return -1; + }, + + booleans = "checked|selected|async|autofocus|autoplay|controls|defer|disabled|hidden|" + + "ismap|loop|multiple|open|readonly|required|scoped", + + // Regular expressions + + // http://www.w3.org/TR/css3-selectors/#whitespace + whitespace = "[\\x20\\t\\r\\n\\f]", + + // https://www.w3.org/TR/css-syntax-3/#ident-token-diagram + identifier = "(?:\\\\[\\da-fA-F]{1,6}" + whitespace + + "?|\\\\[^\\r\\n\\f]|[\\w-]|[^\0-\\x7f])+", + + // Attribute selectors: http://www.w3.org/TR/selectors/#attribute-selectors + attributes = "\\[" + whitespace + "*(" + identifier + ")(?:" + whitespace + + + // Operator (capture 2) + "*([*^$|!~]?=)" + whitespace + + + // "Attribute values must be CSS identifiers [capture 5] + // or strings [capture 3 or capture 4]" + "*(?:'((?:\\\\.|[^\\\\'])*)'|\"((?:\\\\.|[^\\\\\"])*)\"|(" + identifier + "))|)" + + whitespace + "*\\]", + + pseudos = ":(" + identifier + ")(?:\$(" + + + // To reduce the number of selectors needing tokenize in the preFilter, prefer arguments: + // 1. quoted (capture 3; capture 4 or capture 5) + "('((?:\\\\.|[^\\\\'])*)'|\"((?:\\\\.|[^\\\\\"])*)\")|" + + + // 2. simple (capture 6) + "((?:\\\\.|[^\\\\()[\\]]|" + attributes + ")*)|" + + + // 3. anything else (capture 2) + ".*" + + ")\$|)", + + // Leading and non-escaped trailing whitespace, capturing some non-whitespace characters preceding the latter + rwhitespace = new RegExp( whitespace + "+", "g" ), + rtrim = new RegExp( "^" + whitespace + "+|((?:^|[^\\\\])(?:\\\\.)*)" + + whitespace + "+$", "g" ), + + rcomma = new RegExp( "^" + whitespace + "*," + whitespace + "*" ), + rcombinators = new RegExp( "^" + whitespace + "*([>+~]|" + whitespace + ")" + whitespace + + "*" ), + rdescend = new RegExp( whitespace + "|>" ), + + rpseudo = new RegExp( pseudos ), + ridentifier = new RegExp( "^" + identifier + "$" ), + + matchExpr = { + "ID": new RegExp( "^#(" + identifier + ")" ), + "CLASS": new RegExp( "^\\.(" + identifier + ")" ), + "TAG": new RegExp( "^(" + identifier + "|[*])" ), + "ATTR": new RegExp( "^" + attributes ), + "PSEUDO": new RegExp( "^" + pseudos ), + "CHILD": new RegExp( "^:(only|first|last|nth|nth-last)-(child|of-type)(?:\$" + + whitespace + "*(even|odd|(([+-]|)(\\d*)n|)" + whitespace + "*(?:([+-]|)" + + whitespace + "*(\\d+)|))" + whitespace + "*\$|)", "i" ), + "bool": new RegExp( "^(?:" + booleans + ")$", "i" ), + + // For use in libraries implementing .is() + // We use this for POS matching in `select` + "needsContext": new RegExp( "^" + whitespace + + "*[>+~]|:(even|odd|eq|gt|lt|nth|first|last)(?:\$" + whitespace + + "*((?:-\\d)?\\d*)" + whitespace + "*\$|)(?=[^-]|$)", "i" ) + }, + + rhtml = /HTML$/i, + rinputs = /^(?:input|select|textarea|button)$/i, + rheader = /^h\d$/i, + + rnative = /^[^{]+\{\s*\[native \w/, + + // Easily-parseable/retrievable ID or TAG or CLASS selectors + rquickExpr = /^(?:#([\w-]+)|(\w+)|\.([\w-]+))$/, + + rsibling = /[+~]/, + + // CSS escapes + // http://www.w3.org/TR/CSS21/syndata.html#escaped-characters + runescape = new RegExp( "\\\\[\\da-fA-F]{1,6}" + whitespace + "?|\\\\([^\\r\\n\\f])", "g" ), + funescape = function( escape, nonHex ) { + var high = "0x" + escape.slice( 1 ) - 0x10000; + + return nonHex ? + + // Strip the backslash prefix from a non-hex escape sequence + nonHex : + + // Replace a hexadecimal escape sequence with the encoded Unicode code point + // Support: IE <=11+ + // For values outside the Basic Multilingual Plane (BMP), manually construct a + // surrogate pair + high < 0 ? + String.fromCharCode( high + 0x10000 ) : + String.fromCharCode( high >> 10 | 0xD800, high & 0x3FF | 0xDC00 ); + }, + + // CSS string/identifier serialization + // https://drafts.csswg.org/cssom/#common-serializing-idioms + rcssescape = /([\0-\x1f\x7f]|^-?\d)|^-$|[^\0-\x1f\x7f-\uFFFF\w-]/g, + fcssescape = function( ch, asCodePoint ) { + if ( asCodePoint ) { + + // U+0000 NULL becomes U+FFFD REPLACEMENT CHARACTER + if ( ch === "\0" ) { + return "\uFFFD"; + } + + // Control characters and (dependent upon position) numbers get escaped as code points + return ch.slice( 0, -1 ) + "\\" + + ch.charCodeAt( ch.length - 1 ).toString( 16 ) + " "; + } + + // Other potentially-special ASCII characters get backslash-escaped + return "\\" + ch; + }, + + // Used for iframes + // See setDocument() + // Removing the function wrapper causes a "Permission Denied" + // error in IE + unloadHandler = function() { + setDocument(); + }, + + inDisabledFieldset = addCombinator( + function( elem ) { + return elem.disabled === true && elem.nodeName.toLowerCase() === "fieldset"; + }, + { dir: "parentNode", next: "legend" } + ); + + // Optimize for push.apply( _, NodeList ) + try { + push.apply( + ( arr = slice.call( preferredDoc.childNodes ) ), + preferredDoc.childNodes + ); + + // Support: Android<4.0 + // Detect silently failing push.apply + // eslint-disable-next-line no-unused-expressions + arr[ preferredDoc.childNodes.length ].nodeType; + } catch ( e ) { + push = { apply: arr.length ? + + // Leverage slice if possible + function( target, els ) { + pushNative.apply( target, slice.call( els ) ); + } : + + // Support: IE<9 + // Otherwise append directly + function( target, els ) { + var j = target.length, + i = 0; + + // Can't trust NodeList.length + while ( ( target[ j++ ] = els[ i++ ] ) ) {} + target.length = j - 1; + } + }; + } + + function Sizzle( selector, context, results, seed ) { + var m, i, elem, nid, match, groups, newSelector, + newContext = context && context.ownerDocument, + + // nodeType defaults to 9, since context defaults to document + nodeType = context ? context.nodeType : 9; + + results = results || []; + + // Return early from calls with invalid selector or context + if ( typeof selector !== "string" || !selector || + nodeType !== 1 && nodeType !== 9 && nodeType !== 11 ) { + + return results; + } + + // Try to shortcut find operations (as opposed to filters) in HTML documents + if ( !seed ) { + setDocument( context ); + context = context || document; + + if ( documentIsHTML ) { + + // If the selector is sufficiently simple, try using a "get*By*" DOM method + // (excepting DocumentFragment context, where the methods don't exist) + if ( nodeType !== 11 && ( match = rquickExpr.exec( selector ) ) ) { + + // ID selector + if ( ( m = match[ 1 ] ) ) { + + // Document context + if ( nodeType === 9 ) { + if ( ( elem = context.getElementById( m ) ) ) { + + // Support: IE, Opera, Webkit + // TODO: identify versions + // getElementById can match elements by name instead of ID + if ( elem.id === m ) { + results.push( elem ); + return results; + } + } else { + return results; + } + + // Element context + } else { + + // Support: IE, Opera, Webkit + // TODO: identify versions + // getElementById can match elements by name instead of ID + if ( newContext && ( elem = newContext.getElementById( m ) ) && + contains( context, elem ) && + elem.id === m ) { + + results.push( elem ); + return results; + } + } + + // Type selector + } else if ( match[ 2 ] ) { + push.apply( results, context.getElementsByTagName( selector ) ); + return results; + + // Class selector + } else if ( ( m = match[ 3 ] ) && support.getElementsByClassName && + context.getElementsByClassName ) { + + push.apply( results, context.getElementsByClassName( m ) ); + return results; + } + } + + // Take advantage of querySelectorAll + if ( support.qsa && + !nonnativeSelectorCache[ selector + " " ] && + ( !rbuggyQSA || !rbuggyQSA.test( selector ) ) && + + // Support: IE 8 only + // Exclude object elements + ( nodeType !== 1 || context.nodeName.toLowerCase() !== "object" ) ) { + + newSelector = selector; + newContext = context; + + // qSA considers elements outside a scoping root when evaluating child or + // descendant combinators, which is not what we want. + // In such cases, we work around the behavior by prefixing every selector in the + // list with an ID selector referencing the scope context. + // The technique has to be used as well when a leading combinator is used + // as such selectors are not recognized by querySelectorAll. + // Thanks to Andrew Dupont for this technique. + if ( nodeType === 1 && + ( rdescend.test( selector ) || rcombinators.test( selector ) ) ) { + + // Expand context for sibling selectors + newContext = rsibling.test( selector ) && testContext( context.parentNode ) || + context; + + // We can use :scope instead of the ID hack if the browser + // supports it & if we're not changing the context. + if ( newContext !== context || !support.scope ) { + + // Capture the context ID, setting it first if necessary + if ( ( nid = context.getAttribute( "id" ) ) ) { + nid = nid.replace( rcssescape, fcssescape ); + } else { + context.setAttribute( "id", ( nid = expando ) ); + } + } + + // Prefix every selector in the list + groups = tokenize( selector ); + i = groups.length; + while ( i-- ) { + groups[ i ] = ( nid ? "#" + nid : ":scope" ) + " " + + toSelector( groups[ i ] ); + } + newSelector = groups.join( "," ); + } + + try { + push.apply( results, + newContext.querySelectorAll( newSelector ) + ); + return results; + } catch ( qsaError ) { + nonnativeSelectorCache( selector, true ); + } finally { + if ( nid === expando ) { + context.removeAttribute( "id" ); + } + } + } + } + } + + // All others + return select( selector.replace( rtrim, "$1" ), context, results, seed ); + } + + /** + * Create key-value caches of limited size + * @returns {function(string, object)} Returns the Object data after storing it on itself with + * property name the (space-suffixed) string and (if the cache is larger than Expr.cacheLength) + * deleting the oldest entry + */ + function createCache() { + var keys = []; + + function cache( key, value ) { + + // Use (key + " ") to avoid collision with native prototype properties (see Issue #157) + if ( keys.push( key + " " ) > Expr.cacheLength ) { + + // Only keep the most recent entries + delete cache[ keys.shift() ]; + } + return ( cache[ key + " " ] = value ); + } + return cache; + } + + /** + * Mark a function for special use by Sizzle + * @param {Function} fn The function to mark + */ + function markFunction( fn ) { + fn[ expando ] = true; + return fn; + } + + /** + * Support testing using an element + * @param {Function} fn Passed the created element and returns a boolean result + */ + function assert( fn ) { + var el = document.createElement( "fieldset" ); + + try { + return !!fn( el ); + } catch ( e ) { + return false; + } finally { + + // Remove from its parent by default + if ( el.parentNode ) { + el.parentNode.removeChild( el ); + } + + // release memory in IE + el = null; + } + } + + /** + * Adds the same handler for all of the specified attrs + * @param {String} attrs Pipe-separated list of attributes + * @param {Function} handler The method that will be applied + */ + function addHandle( attrs, handler ) { + var arr = attrs.split( "|" ), + i = arr.length; + + while ( i-- ) { + Expr.attrHandle[ arr[ i ] ] = handler; + } + } + + /** + * Checks document order of two siblings + * @param {Element} a + * @param {Element} b + * @returns {Number} Returns less than 0 if a precedes b, greater than 0 if a follows b + */ + function siblingCheck( a, b ) { + var cur = b && a, + diff = cur && a.nodeType === 1 && b.nodeType === 1 && + a.sourceIndex - b.sourceIndex; + + // Use IE sourceIndex if available on both nodes + if ( diff ) { + return diff; + } + + // Check if b follows a + if ( cur ) { + while ( ( cur = cur.nextSibling ) ) { + if ( cur === b ) { + return -1; + } + } + } + + return a ? 1 : -1; + } + + /** + * Returns a function to use in pseudos for input types + * @param {String} type + */ + function createInputPseudo( type ) { + return function( elem ) { + var name = elem.nodeName.toLowerCase(); + return name === "input" && elem.type === type; + }; + } + + /** + * Returns a function to use in pseudos for buttons + * @param {String} type + */ + function createButtonPseudo( type ) { + return function( elem ) { + var name = elem.nodeName.toLowerCase(); + return ( name === "input" || name === "button" ) && elem.type === type; + }; + } + + /** + * Returns a function to use in pseudos for :enabled/:disabled + * @param {Boolean} disabled true for :disabled; false for :enabled + */ + function createDisabledPseudo( disabled ) { + + // Known :disabled false positives: fieldset[disabled] > legend:nth-of-type(n+2) :can-disable + return function( elem ) { + + // Only certain elements can match :enabled or :disabled + // https://html.spec.whatwg.org/multipage/scripting.html#selector-enabled + // https://html.spec.whatwg.org/multipage/scripting.html#selector-disabled + if ( "form" in elem ) { + + // Check for inherited disabledness on relevant non-disabled elements: + // * listed form-associated elements in a disabled fieldset + // https://html.spec.whatwg.org/multipage/forms.html#category-listed + // https://html.spec.whatwg.org/multipage/forms.html#concept-fe-disabled + // * option elements in a disabled optgroup + // https://html.spec.whatwg.org/multipage/forms.html#concept-option-disabled + // All such elements have a "form" property. + if ( elem.parentNode && elem.disabled === false ) { + + // Option elements defer to a parent optgroup if present + if ( "label" in elem ) { + if ( "label" in elem.parentNode ) { + return elem.parentNode.disabled === disabled; + } else { + return elem.disabled === disabled; + } + } + + // Support: IE 6 - 11 + // Use the isDisabled shortcut property to check for disabled fieldset ancestors + return elem.isDisabled === disabled || + + // Where there is no isDisabled, check manually + /* jshint -W018 */ + elem.isDisabled !== !disabled && + inDisabledFieldset( elem ) === disabled; + } + + return elem.disabled === disabled; + + // Try to winnow out elements that can't be disabled before trusting the disabled property. + // Some victims get caught in our net (label, legend, menu, track), but it shouldn't + // even exist on them, let alone have a boolean value. + } else if ( "label" in elem ) { + return elem.disabled === disabled; + } + + // Remaining elements are neither :enabled nor :disabled + return false; + }; + } + + /** + * Returns a function to use in pseudos for positionals + * @param {Function} fn + */ + function createPositionalPseudo( fn ) { + return markFunction( function( argument ) { + argument = +argument; + return markFunction( function( seed, matches ) { + var j, + matchIndexes = fn( [], seed.length, argument ), + i = matchIndexes.length; + + // Match elements found at the specified indexes + while ( i-- ) { + if ( seed[ ( j = matchIndexes[ i ] ) ] ) { + seed[ j ] = !( matches[ j ] = seed[ j ] ); + } + } + } ); + } ); + } + + /** + * Checks a node for validity as a Sizzle context + * @param {Element|Object=} context + * @returns {Element|Object|Boolean} The input node if acceptable, otherwise a falsy value + */ + function testContext( context ) { + return context && typeof context.getElementsByTagName !== "undefined" && context; + } + + // Expose support vars for convenience + support = Sizzle.support = {}; + + /** + * Detects XML nodes + * @param {Element|Object} elem An element or a document + * @returns {Boolean} True iff elem is a non-HTML XML node + */ + isXML = Sizzle.isXML = function( elem ) { + var namespace = elem && elem.namespaceURI, + docElem = elem && ( elem.ownerDocument || elem ).documentElement; + + // Support: IE <=8 + // Assume HTML when documentElement doesn't yet exist, such as inside loading iframes + // https://bugs.jquery.com/ticket/4833 + return !rhtml.test( namespace || docElem && docElem.nodeName || "HTML" ); + }; + + /** + * Sets document-related variables once based on the current document + * @param {Element|Object} [doc] An element or document object to use to set the document + * @returns {Object} Returns the current document + */ + setDocument = Sizzle.setDocument = function( node ) { + var hasCompare, subWindow, + doc = node ? node.ownerDocument || node : preferredDoc; + + // Return early if doc is invalid or already selected + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + if ( doc == document || doc.nodeType !== 9 || !doc.documentElement ) { + return document; + } + + // Update global variables + document = doc; + docElem = document.documentElement; + documentIsHTML = !isXML( document ); + + // Support: IE 9 - 11+, Edge 12 - 18+ + // Accessing iframe documents after unload throws "permission denied" errors (jQuery #13936) + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + if ( preferredDoc != document && + ( subWindow = document.defaultView ) && subWindow.top !== subWindow ) { + + // Support: IE 11, Edge + if ( subWindow.addEventListener ) { + subWindow.addEventListener( "unload", unloadHandler, false ); + + // Support: IE 9 - 10 only + } else if ( subWindow.attachEvent ) { + subWindow.attachEvent( "onunload", unloadHandler ); + } + } + + // Support: IE 8 - 11+, Edge 12 - 18+, Chrome <=16 - 25 only, Firefox <=3.6 - 31 only, + // Safari 4 - 5 only, Opera <=11.6 - 12.x only + // IE/Edge & older browsers don't support the :scope pseudo-class. + // Support: Safari 6.0 only + // Safari 6.0 supports :scope but it's an alias of :root there. + support.scope = assert( function( el ) { + docElem.appendChild( el ).appendChild( document.createElement( "div" ) ); + return typeof el.querySelectorAll !== "undefined" && + !el.querySelectorAll( ":scope fieldset div" ).length; + } ); + + /* Attributes + ---------------------------------------------------------------------- */ + + // Support: IE<8 + // Verify that getAttribute really returns attributes and not properties + // (excepting IE8 booleans) + support.attributes = assert( function( el ) { + el.className = "i"; + return !el.getAttribute( "className" ); + } ); + + /* getElement(s)By* + ---------------------------------------------------------------------- */ + + // Check if getElementsByTagName("*") returns only elements + support.getElementsByTagName = assert( function( el ) { + el.appendChild( document.createComment( "" ) ); + return !el.getElementsByTagName( "*" ).length; + } ); + + // Support: IE<9 + support.getElementsByClassName = rnative.test( document.getElementsByClassName ); + + // Support: IE<10 + // Check if getElementById returns elements by name + // The broken getElementById methods don't pick up programmatically-set names, + // so use a roundabout getElementsByName test + support.getById = assert( function( el ) { + docElem.appendChild( el ).id = expando; + return !document.getElementsByName || !document.getElementsByName( expando ).length; + } ); + + // ID filter and find + if ( support.getById ) { + Expr.filter[ "ID" ] = function( id ) { + var attrId = id.replace( runescape, funescape ); + return function( elem ) { + return elem.getAttribute( "id" ) === attrId; + }; + }; + Expr.find[ "ID" ] = function( id, context ) { + if ( typeof context.getElementById !== "undefined" && documentIsHTML ) { + var elem = context.getElementById( id ); + return elem ? [ elem ] : []; + } + }; + } else { + Expr.filter[ "ID" ] = function( id ) { + var attrId = id.replace( runescape, funescape ); + return function( elem ) { + var node = typeof elem.getAttributeNode !== "undefined" && + elem.getAttributeNode( "id" ); + return node && node.value === attrId; + }; + }; + + // Support: IE 6 - 7 only + // getElementById is not reliable as a find shortcut + Expr.find[ "ID" ] = function( id, context ) { + if ( typeof context.getElementById !== "undefined" && documentIsHTML ) { + var node, i, elems, + elem = context.getElementById( id ); + + if ( elem ) { + + // Verify the id attribute + node = elem.getAttributeNode( "id" ); + if ( node && node.value === id ) { + return [ elem ]; + } + + // Fall back on getElementsByName + elems = context.getElementsByName( id ); + i = 0; + while ( ( elem = elems[ i++ ] ) ) { + node = elem.getAttributeNode( "id" ); + if ( node && node.value === id ) { + return [ elem ]; + } + } + } + + return []; + } + }; + } + + // Tag + Expr.find[ "TAG" ] = support.getElementsByTagName ? + function( tag, context ) { + if ( typeof context.getElementsByTagName !== "undefined" ) { + return context.getElementsByTagName( tag ); + + // DocumentFragment nodes don't have gEBTN + } else if ( support.qsa ) { + return context.querySelectorAll( tag ); + } + } : + + function( tag, context ) { + var elem, + tmp = [], + i = 0, + + // By happy coincidence, a (broken) gEBTN appears on DocumentFragment nodes too + results = context.getElementsByTagName( tag ); + + // Filter out possible comments + if ( tag === "*" ) { + while ( ( elem = results[ i++ ] ) ) { + if ( elem.nodeType === 1 ) { + tmp.push( elem ); + } + } + + return tmp; + } + return results; + }; + + // Class + Expr.find[ "CLASS" ] = support.getElementsByClassName && function( className, context ) { + if ( typeof context.getElementsByClassName !== "undefined" && documentIsHTML ) { + return context.getElementsByClassName( className ); + } + }; + + /* QSA/matchesSelector + ---------------------------------------------------------------------- */ + + // QSA and matchesSelector support + + // matchesSelector(:active) reports false when true (IE9/Opera 11.5) + rbuggyMatches = []; + + // qSa(:focus) reports false when true (Chrome 21) + // We allow this because of a bug in IE8/9 that throws an error + // whenever `document.activeElement` is accessed on an iframe + // So, we allow :focus to pass through QSA all the time to avoid the IE error + // See https://bugs.jquery.com/ticket/13378 + rbuggyQSA = []; + + if ( ( support.qsa = rnative.test( document.querySelectorAll ) ) ) { + + // Build QSA regex + // Regex strategy adopted from Diego Perini + assert( function( el ) { + + var input; + + // Select is set to empty string on purpose + // This is to test IE's treatment of not explicitly + // setting a boolean content attribute, + // since its presence should be enough + // https://bugs.jquery.com/ticket/12359 + docElem.appendChild( el ).innerHTML = "<a id='" + expando + "'></a>" + + "<select id='" + expando + "-\r\\' msallowcapture=''>" + + "<option selected=''></option></select>"; + + // Support: IE8, Opera 11-12.16 + // Nothing should be selected when empty strings follow ^= or $= or *= + // The test attribute must be unknown in Opera but "safe" for WinRT + // https://msdn.microsoft.com/en-us/library/ie/hh465388.aspx#attribute_section + if ( el.querySelectorAll( "[msallowcapture^='']" ).length ) { + rbuggyQSA.push( "[*^$]=" + whitespace + "*(?:''|\"\")" ); + } + + // Support: IE8 + // Boolean attributes and "value" are not treated correctly + if ( !el.querySelectorAll( "[selected]" ).length ) { + rbuggyQSA.push( "\\[" + whitespace + "*(?:value|" + booleans + ")" ); + } + + // Support: Chrome<29, Android<4.4, Safari<7.0+, iOS<7.0+, PhantomJS<1.9.8+ + if ( !el.querySelectorAll( "[id~=" + expando + "-]" ).length ) { + rbuggyQSA.push( "~=" ); + } + + // Support: IE 11+, Edge 15 - 18+ + // IE 11/Edge don't find elements on a `[name='']` query in some cases. + // Adding a temporary attribute to the document before the selection works + // around the issue. + // Interestingly, IE 10 & older don't seem to have the issue. + input = document.createElement( "input" ); + input.setAttribute( "name", "" ); + el.appendChild( input ); + if ( !el.querySelectorAll( "[name='']" ).length ) { + rbuggyQSA.push( "\\[" + whitespace + "*name" + whitespace + "*=" + + whitespace + "*(?:''|\"\")" ); + } + + // Webkit/Opera - :checked should return selected option elements + // http://www.w3.org/TR/2011/REC-css3-selectors-20110929/#checked + // IE8 throws error here and will not see later tests + if ( !el.querySelectorAll( ":checked" ).length ) { + rbuggyQSA.push( ":checked" ); + } + + // Support: Safari 8+, iOS 8+ + // https://bugs.webkit.org/show_bug.cgi?id=136851 + // In-page `selector#id sibling-combinator selector` fails + if ( !el.querySelectorAll( "a#" + expando + "+*" ).length ) { + rbuggyQSA.push( ".#.+[+~]" ); + } + + // Support: Firefox <=3.6 - 5 only + // Old Firefox doesn't throw on a badly-escaped identifier. + el.querySelectorAll( "\\\f" ); + rbuggyQSA.push( "[\\r\\n\\f]" ); + } ); + + assert( function( el ) { + el.innerHTML = "<a href='' disabled='disabled'></a>" + + "<select disabled='disabled'><option/></select>"; + + // Support: Windows 8 Native Apps + // The type and name attributes are restricted during .innerHTML assignment + var input = document.createElement( "input" ); + input.setAttribute( "type", "hidden" ); + el.appendChild( input ).setAttribute( "name", "D" ); + + // Support: IE8 + // Enforce case-sensitivity of name attribute + if ( el.querySelectorAll( "[name=d]" ).length ) { + rbuggyQSA.push( "name" + whitespace + "*[*^$|!~]?=" ); + } + + // FF 3.5 - :enabled/:disabled and hidden elements (hidden elements are still enabled) + // IE8 throws error here and will not see later tests + if ( el.querySelectorAll( ":enabled" ).length !== 2 ) { + rbuggyQSA.push( ":enabled", ":disabled" ); + } + + // Support: IE9-11+ + // IE's :disabled selector does not pick up the children of disabled fieldsets + docElem.appendChild( el ).disabled = true; + if ( el.querySelectorAll( ":disabled" ).length !== 2 ) { + rbuggyQSA.push( ":enabled", ":disabled" ); + } + + // Support: Opera 10 - 11 only + // Opera 10-11 does not throw on post-comma invalid pseudos + el.querySelectorAll( "*,:x" ); + rbuggyQSA.push( ",.*:" ); + } ); + } + + if ( ( support.matchesSelector = rnative.test( ( matches = docElem.matches || + docElem.webkitMatchesSelector || + docElem.mozMatchesSelector || + docElem.oMatchesSelector || + docElem.msMatchesSelector ) ) ) ) { + + assert( function( el ) { + + // Check to see if it's possible to do matchesSelector + // on a disconnected node (IE 9) + support.disconnectedMatch = matches.call( el, "*" ); + + // This should fail with an exception + // Gecko does not error, returns false instead + matches.call( el, "[s!='']:x" ); + rbuggyMatches.push( "!=", pseudos ); + } ); + } + + rbuggyQSA = rbuggyQSA.length && new RegExp( rbuggyQSA.join( "|" ) ); + rbuggyMatches = rbuggyMatches.length && new RegExp( rbuggyMatches.join( "|" ) ); + + /* Contains + ---------------------------------------------------------------------- */ + hasCompare = rnative.test( docElem.compareDocumentPosition ); + + // Element contains another + // Purposefully self-exclusive + // As in, an element does not contain itself + contains = hasCompare || rnative.test( docElem.contains ) ? + function( a, b ) { + var adown = a.nodeType === 9 ? a.documentElement : a, + bup = b && b.parentNode; + return a === bup || !!( bup && bup.nodeType === 1 && ( + adown.contains ? + adown.contains( bup ) : + a.compareDocumentPosition && a.compareDocumentPosition( bup ) & 16 + ) ); + } : + function( a, b ) { + if ( b ) { + while ( ( b = b.parentNode ) ) { + if ( b === a ) { + return true; + } + } + } + return false; + }; + + /* Sorting + ---------------------------------------------------------------------- */ + + // Document order sorting + sortOrder = hasCompare ? + function( a, b ) { + + // Flag for duplicate removal + if ( a === b ) { + hasDuplicate = true; + return 0; + } + + // Sort on method existence if only one input has compareDocumentPosition + var compare = !a.compareDocumentPosition - !b.compareDocumentPosition; + if ( compare ) { + return compare; + } + + // Calculate position if both inputs belong to the same document + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + compare = ( a.ownerDocument || a ) == ( b.ownerDocument || b ) ? + a.compareDocumentPosition( b ) : + + // Otherwise we know they are disconnected + 1; + + // Disconnected nodes + if ( compare & 1 || + ( !support.sortDetached && b.compareDocumentPosition( a ) === compare ) ) { + + // Choose the first element that is related to our preferred document + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + if ( a == document || a.ownerDocument == preferredDoc && + contains( preferredDoc, a ) ) { + return -1; + } + + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + if ( b == document || b.ownerDocument == preferredDoc && + contains( preferredDoc, b ) ) { + return 1; + } + + // Maintain original order + return sortInput ? + ( indexOf( sortInput, a ) - indexOf( sortInput, b ) ) : + 0; + } + + return compare & 4 ? -1 : 1; + } : + function( a, b ) { + + // Exit early if the nodes are identical + if ( a === b ) { + hasDuplicate = true; + return 0; + } + + var cur, + i = 0, + aup = a.parentNode, + bup = b.parentNode, + ap = [ a ], + bp = [ b ]; + + // Parentless nodes are either documents or disconnected + if ( !aup || !bup ) { + + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + /* eslint-disable eqeqeq */ + return a == document ? -1 : + b == document ? 1 : + /* eslint-enable eqeqeq */ + aup ? -1 : + bup ? 1 : + sortInput ? + ( indexOf( sortInput, a ) - indexOf( sortInput, b ) ) : + 0; + + // If the nodes are siblings, we can do a quick check + } else if ( aup === bup ) { + return siblingCheck( a, b ); + } + + // Otherwise we need full lists of their ancestors for comparison + cur = a; + while ( ( cur = cur.parentNode ) ) { + ap.unshift( cur ); + } + cur = b; + while ( ( cur = cur.parentNode ) ) { + bp.unshift( cur ); + } + + // Walk down the tree looking for a discrepancy + while ( ap[ i ] === bp[ i ] ) { + i++; + } + + return i ? + + // Do a sibling check if the nodes have a common ancestor + siblingCheck( ap[ i ], bp[ i ] ) : + + // Otherwise nodes in our document sort first + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + /* eslint-disable eqeqeq */ + ap[ i ] == preferredDoc ? -1 : + bp[ i ] == preferredDoc ? 1 : + /* eslint-enable eqeqeq */ + 0; + }; + + return document; + }; + + Sizzle.matches = function( expr, elements ) { + return Sizzle( expr, null, null, elements ); + }; + + Sizzle.matchesSelector = function( elem, expr ) { + setDocument( elem ); + + if ( support.matchesSelector && documentIsHTML && + !nonnativeSelectorCache[ expr + " " ] && + ( !rbuggyMatches || !rbuggyMatches.test( expr ) ) && + ( !rbuggyQSA || !rbuggyQSA.test( expr ) ) ) { + + try { + var ret = matches.call( elem, expr ); + + // IE 9's matchesSelector returns false on disconnected nodes + if ( ret || support.disconnectedMatch || + + // As well, disconnected nodes are said to be in a document + // fragment in IE 9 + elem.document && elem.document.nodeType !== 11 ) { + return ret; + } + } catch ( e ) { + nonnativeSelectorCache( expr, true ); + } + } + + return Sizzle( expr, document, null, [ elem ] ).length > 0; + }; + + Sizzle.contains = function( context, elem ) { + + // Set document vars if needed + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + if ( ( context.ownerDocument || context ) != document ) { + setDocument( context ); + } + return contains( context, elem ); + }; + + Sizzle.attr = function( elem, name ) { + + // Set document vars if needed + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + if ( ( elem.ownerDocument || elem ) != document ) { + setDocument( elem ); + } + + var fn = Expr.attrHandle[ name.toLowerCase() ], + + // Don't get fooled by Object.prototype properties (jQuery #13807) + val = fn && hasOwn.call( Expr.attrHandle, name.toLowerCase() ) ? + fn( elem, name, !documentIsHTML ) : + undefined; + + return val !== undefined ? + val : + support.attributes || !documentIsHTML ? + elem.getAttribute( name ) : + ( val = elem.getAttributeNode( name ) ) && val.specified ? + val.value : + null; + }; + + Sizzle.escape = function( sel ) { + return ( sel + "" ).replace( rcssescape, fcssescape ); + }; + + Sizzle.error = function( msg ) { + throw new Error( "Syntax error, unrecognized expression: " + msg ); + }; + + /** + * Document sorting and removing duplicates + * @param {ArrayLike} results + */ + Sizzle.uniqueSort = function( results ) { + var elem, + duplicates = [], + j = 0, + i = 0; + + // Unless we *know* we can detect duplicates, assume their presence + hasDuplicate = !support.detectDuplicates; + sortInput = !support.sortStable && results.slice( 0 ); + results.sort( sortOrder ); + + if ( hasDuplicate ) { + while ( ( elem = results[ i++ ] ) ) { + if ( elem === results[ i ] ) { + j = duplicates.push( i ); + } + } + while ( j-- ) { + results.splice( duplicates[ j ], 1 ); + } + } + + // Clear input after sorting to release objects + // See https://github.com/jquery/sizzle/pull/225 + sortInput = null; + + return results; + }; + + /** + * Utility function for retrieving the text value of an array of DOM nodes + * @param {Array|Element} elem + */ + getText = Sizzle.getText = function( elem ) { + var node, + ret = "", + i = 0, + nodeType = elem.nodeType; + + if ( !nodeType ) { + + // If no nodeType, this is expected to be an array + while ( ( node = elem[ i++ ] ) ) { + + // Do not traverse comment nodes + ret += getText( node ); + } + } else if ( nodeType === 1 || nodeType === 9 || nodeType === 11 ) { + + // Use textContent for elements + // innerText usage removed for consistency of new lines (jQuery #11153) + if ( typeof elem.textContent === "string" ) { + return elem.textContent; + } else { + + // Traverse its children + for ( elem = elem.firstChild; elem; elem = elem.nextSibling ) { + ret += getText( elem ); + } + } + } else if ( nodeType === 3 || nodeType === 4 ) { + return elem.nodeValue; + } + + // Do not include comment or processing instruction nodes + + return ret; + }; + + Expr = Sizzle.selectors = { + + // Can be adjusted by the user + cacheLength: 50, + + createPseudo: markFunction, + + match: matchExpr, + + attrHandle: {}, + + find: {}, + + relative: { + ">": { dir: "parentNode", first: true }, + " ": { dir: "parentNode" }, + "+": { dir: "previousSibling", first: true }, + "~": { dir: "previousSibling" } + }, + + preFilter: { + "ATTR": function( match ) { + match[ 1 ] = match[ 1 ].replace( runescape, funescape ); + + // Move the given value to match[3] whether quoted or unquoted + match[ 3 ] = ( match[ 3 ] || match[ 4 ] || + match[ 5 ] || "" ).replace( runescape, funescape ); + + if ( match[ 2 ] === "~=" ) { + match[ 3 ] = " " + match[ 3 ] + " "; + } + + return match.slice( 0, 4 ); + }, + + "CHILD": function( match ) { + + /* matches from matchExpr["CHILD"] + 1 type (only|nth|...) + 2 what (child|of-type) + 3 argument (even|odd|\d*|\d*n([+-]\d+)?|...) + 4 xn-component of xn+y argument ([+-]?\d*n|) + 5 sign of xn-component + 6 x of xn-component + 7 sign of y-component + 8 y of y-component + */ + match[ 1 ] = match[ 1 ].toLowerCase(); + + if ( match[ 1 ].slice( 0, 3 ) === "nth" ) { + + // nth-* requires argument + if ( !match[ 3 ] ) { + Sizzle.error( match[ 0 ] ); + } + + // numeric x and y parameters for Expr.filter.CHILD + // remember that false/true cast respectively to 0/1 + match[ 4 ] = +( match[ 4 ] ? + match[ 5 ] + ( match[ 6 ] || 1 ) : + 2 * ( match[ 3 ] === "even" || match[ 3 ] === "odd" ) ); + match[ 5 ] = +( ( match[ 7 ] + match[ 8 ] ) || match[ 3 ] === "odd" ); + + // other types prohibit arguments + } else if ( match[ 3 ] ) { + Sizzle.error( match[ 0 ] ); + } + + return match; + }, + + "PSEUDO": function( match ) { + var excess, + unquoted = !match[ 6 ] && match[ 2 ]; + + if ( matchExpr[ "CHILD" ].test( match[ 0 ] ) ) { + return null; + } + + // Accept quoted arguments as-is + if ( match[ 3 ] ) { + match[ 2 ] = match[ 4 ] || match[ 5 ] || ""; + + // Strip excess characters from unquoted arguments + } else if ( unquoted && rpseudo.test( unquoted ) && + + // Get excess from tokenize (recursively) + ( excess = tokenize( unquoted, true ) ) && + + // advance to the next closing parenthesis + ( excess = unquoted.indexOf( ")", unquoted.length - excess ) - unquoted.length ) ) { + + // excess is a negative index + match[ 0 ] = match[ 0 ].slice( 0, excess ); + match[ 2 ] = unquoted.slice( 0, excess ); + } + + // Return only captures needed by the pseudo filter method (type and argument) + return match.slice( 0, 3 ); + } + }, + + filter: { + + "TAG": function( nodeNameSelector ) { + var nodeName = nodeNameSelector.replace( runescape, funescape ).toLowerCase(); + return nodeNameSelector === "*" ? + function() { + return true; + } : + function( elem ) { + return elem.nodeName && elem.nodeName.toLowerCase() === nodeName; + }; + }, + + "CLASS": function( className ) { + var pattern = classCache[ className + " " ]; + + return pattern || + ( pattern = new RegExp( "(^|" + whitespace + + ")" + className + "(" + whitespace + "|$)" ) ) && classCache( + className, function( elem ) { + return pattern.test( + typeof elem.className === "string" && elem.className || + typeof elem.getAttribute !== "undefined" && + elem.getAttribute( "class" ) || + "" + ); + } ); + }, + + "ATTR": function( name, operator, check ) { + return function( elem ) { + var result = Sizzle.attr( elem, name ); + + if ( result == null ) { + return operator === "!="; + } + if ( !operator ) { + return true; + } + + result += ""; + + /* eslint-disable max-len */ + + return operator === "=" ? result === check : + operator === "!=" ? result !== check : + operator === "^=" ? check && result.indexOf( check ) === 0 : + operator === "*=" ? check && result.indexOf( check ) > -1 : + operator === "$=" ? check && result.slice( -check.length ) === check : + operator === "~=" ? ( " " + result.replace( rwhitespace, " " ) + " " ).indexOf( check ) > -1 : + operator === "|=" ? result === check || result.slice( 0, check.length + 1 ) === check + "-" : + false; + /* eslint-enable max-len */ + + }; + }, + + "CHILD": function( type, what, _argument, first, last ) { + var simple = type.slice( 0, 3 ) !== "nth", + forward = type.slice( -4 ) !== "last", + ofType = what === "of-type"; + + return first === 1 && last === 0 ? + + // Shortcut for :nth-*(n) + function( elem ) { + return !!elem.parentNode; + } : + + function( elem, _context, xml ) { + var cache, uniqueCache, outerCache, node, nodeIndex, start, + dir = simple !== forward ? "nextSibling" : "previousSibling", + parent = elem.parentNode, + name = ofType && elem.nodeName.toLowerCase(), + useCache = !xml && !ofType, + diff = false; + + if ( parent ) { + + // :(first|last|only)-(child|of-type) + if ( simple ) { + while ( dir ) { + node = elem; + while ( ( node = node[ dir ] ) ) { + if ( ofType ? + node.nodeName.toLowerCase() === name : + node.nodeType === 1 ) { + + return false; + } + } + + // Reverse direction for :only-* (if we haven't yet done so) + start = dir = type === "only" && !start && "nextSibling"; + } + return true; + } + + start = [ forward ? parent.firstChild : parent.lastChild ]; + + // non-xml :nth-child(...) stores cache data on `parent` + if ( forward && useCache ) { + + // Seek `elem` from a previously-cached index + + // ...in a gzip-friendly way + node = parent; + outerCache = node[ expando ] || ( node[ expando ] = {} ); + + // Support: IE <9 only + // Defend against cloned attroperties (jQuery gh-1709) + uniqueCache = outerCache[ node.uniqueID ] || + ( outerCache[ node.uniqueID ] = {} ); + + cache = uniqueCache[ type ] || []; + nodeIndex = cache[ 0 ] === dirruns && cache[ 1 ]; + diff = nodeIndex && cache[ 2 ]; + node = nodeIndex && parent.childNodes[ nodeIndex ]; + + while ( ( node = ++nodeIndex && node && node[ dir ] || + + // Fallback to seeking `elem` from the start + ( diff = nodeIndex = 0 ) || start.pop() ) ) { + + // When found, cache indexes on `parent` and break + if ( node.nodeType === 1 && ++diff && node === elem ) { + uniqueCache[ type ] = [ dirruns, nodeIndex, diff ]; + break; + } + } + + } else { + + // Use previously-cached element index if available + if ( useCache ) { + + // ...in a gzip-friendly way + node = elem; + outerCache = node[ expando ] || ( node[ expando ] = {} ); + + // Support: IE <9 only + // Defend against cloned attroperties (jQuery gh-1709) + uniqueCache = outerCache[ node.uniqueID ] || + ( outerCache[ node.uniqueID ] = {} ); + + cache = uniqueCache[ type ] || []; + nodeIndex = cache[ 0 ] === dirruns && cache[ 1 ]; + diff = nodeIndex; + } + + // xml :nth-child(...) + // or :nth-last-child(...) or :nth(-last)?-of-type(...) + if ( diff === false ) { + + // Use the same loop as above to seek `elem` from the start + while ( ( node = ++nodeIndex && node && node[ dir ] || + ( diff = nodeIndex = 0 ) || start.pop() ) ) { + + if ( ( ofType ? + node.nodeName.toLowerCase() === name : + node.nodeType === 1 ) && + ++diff ) { + + // Cache the index of each encountered element + if ( useCache ) { + outerCache = node[ expando ] || + ( node[ expando ] = {} ); + + // Support: IE <9 only + // Defend against cloned attroperties (jQuery gh-1709) + uniqueCache = outerCache[ node.uniqueID ] || + ( outerCache[ node.uniqueID ] = {} ); + + uniqueCache[ type ] = [ dirruns, diff ]; + } + + if ( node === elem ) { + break; + } + } + } + } + } + + // Incorporate the offset, then check against cycle size + diff -= last; + return diff === first || ( diff % first === 0 && diff / first >= 0 ); + } + }; + }, + + "PSEUDO": function( pseudo, argument ) { + + // pseudo-class names are case-insensitive + // http://www.w3.org/TR/selectors/#pseudo-classes + // Prioritize by case sensitivity in case custom pseudos are added with uppercase letters + // Remember that setFilters inherits from pseudos + var args, + fn = Expr.pseudos[ pseudo ] || Expr.setFilters[ pseudo.toLowerCase() ] || + Sizzle.error( "unsupported pseudo: " + pseudo ); + + // The user may use createPseudo to indicate that + // arguments are needed to create the filter function + // just as Sizzle does + if ( fn[ expando ] ) { + return fn( argument ); + } + + // But maintain support for old signatures + if ( fn.length > 1 ) { + args = [ pseudo, pseudo, "", argument ]; + return Expr.setFilters.hasOwnProperty( pseudo.toLowerCase() ) ? + markFunction( function( seed, matches ) { + var idx, + matched = fn( seed, argument ), + i = matched.length; + while ( i-- ) { + idx = indexOf( seed, matched[ i ] ); + seed[ idx ] = !( matches[ idx ] = matched[ i ] ); + } + } ) : + function( elem ) { + return fn( elem, 0, args ); + }; + } + + return fn; + } + }, + + pseudos: { + + // Potentially complex pseudos + "not": markFunction( function( selector ) { + + // Trim the selector passed to compile + // to avoid treating leading and trailing + // spaces as combinators + var input = [], + results = [], + matcher = compile( selector.replace( rtrim, "$1" ) ); + + return matcher[ expando ] ? + markFunction( function( seed, matches, _context, xml ) { + var elem, + unmatched = matcher( seed, null, xml, [] ), + i = seed.length; + + // Match elements unmatched by `matcher` + while ( i-- ) { + if ( ( elem = unmatched[ i ] ) ) { + seed[ i ] = !( matches[ i ] = elem ); + } + } + } ) : + function( elem, _context, xml ) { + input[ 0 ] = elem; + matcher( input, null, xml, results ); + + // Don't keep the element (issue #299) + input[ 0 ] = null; + return !results.pop(); + }; + } ), + + "has": markFunction( function( selector ) { + return function( elem ) { + return Sizzle( selector, elem ).length > 0; + }; + } ), + + "contains": markFunction( function( text ) { + text = text.replace( runescape, funescape ); + return function( elem ) { + return ( elem.textContent || getText( elem ) ).indexOf( text ) > -1; + }; + } ), + + // "Whether an element is represented by a :lang() selector + // is based solely on the element's language value + // being equal to the identifier C, + // or beginning with the identifier C immediately followed by "-". + // The matching of C against the element's language value is performed case-insensitively. + // The identifier C does not have to be a valid language name." + // http://www.w3.org/TR/selectors/#lang-pseudo + "lang": markFunction( function( lang ) { + + // lang value must be a valid identifier + if ( !ridentifier.test( lang || "" ) ) { + Sizzle.error( "unsupported lang: " + lang ); + } + lang = lang.replace( runescape, funescape ).toLowerCase(); + return function( elem ) { + var elemLang; + do { + if ( ( elemLang = documentIsHTML ? + elem.lang : + elem.getAttribute( "xml:lang" ) || elem.getAttribute( "lang" ) ) ) { + + elemLang = elemLang.toLowerCase(); + return elemLang === lang || elemLang.indexOf( lang + "-" ) === 0; + } + } while ( ( elem = elem.parentNode ) && elem.nodeType === 1 ); + return false; + }; + } ), + + // Miscellaneous + "target": function( elem ) { + var hash = window.location && window.location.hash; + return hash && hash.slice( 1 ) === elem.id; + }, + + "root": function( elem ) { + return elem === docElem; + }, + + "focus": function( elem ) { + return elem === document.activeElement && + ( !document.hasFocus || document.hasFocus() ) && + !!( elem.type || elem.href || ~elem.tabIndex ); + }, + + // Boolean properties + "enabled": createDisabledPseudo( false ), + "disabled": createDisabledPseudo( true ), + + "checked": function( elem ) { + + // In CSS3, :checked should return both checked and selected elements + // http://www.w3.org/TR/2011/REC-css3-selectors-20110929/#checked + var nodeName = elem.nodeName.toLowerCase(); + return ( nodeName === "input" && !!elem.checked ) || + ( nodeName === "option" && !!elem.selected ); + }, + + "selected": function( elem ) { + + // Accessing this property makes selected-by-default + // options in Safari work properly + if ( elem.parentNode ) { + // eslint-disable-next-line no-unused-expressions + elem.parentNode.selectedIndex; + } + + return elem.selected === true; + }, + + // Contents + "empty": function( elem ) { + + // http://www.w3.org/TR/selectors/#empty-pseudo + // :empty is negated by element (1) or content nodes (text: 3; cdata: 4; entity ref: 5), + // but not by others (comment: 8; processing instruction: 7; etc.) + // nodeType < 6 works because attributes (2) do not appear as children + for ( elem = elem.firstChild; elem; elem = elem.nextSibling ) { + if ( elem.nodeType < 6 ) { + return false; + } + } + return true; + }, + + "parent": function( elem ) { + return !Expr.pseudos[ "empty" ]( elem ); + }, + + // Element/input types + "header": function( elem ) { + return rheader.test( elem.nodeName ); + }, + + "input": function( elem ) { + return rinputs.test( elem.nodeName ); + }, + + "button": function( elem ) { + var name = elem.nodeName.toLowerCase(); + return name === "input" && elem.type === "button" || name === "button"; + }, + + "text": function( elem ) { + var attr; + return elem.nodeName.toLowerCase() === "input" && + elem.type === "text" && + + // Support: IE<8 + // New HTML5 attribute values (e.g., "search") appear with elem.type === "text" + ( ( attr = elem.getAttribute( "type" ) ) == null || + attr.toLowerCase() === "text" ); + }, + + // Position-in-collection + "first": createPositionalPseudo( function() { + return [ 0 ]; + } ), + + "last": createPositionalPseudo( function( _matchIndexes, length ) { + return [ length - 1 ]; + } ), + + "eq": createPositionalPseudo( function( _matchIndexes, length, argument ) { + return [ argument < 0 ? argument + length : argument ]; + } ), + + "even": createPositionalPseudo( function( matchIndexes, length ) { + var i = 0; + for ( ; i < length; i += 2 ) { + matchIndexes.push( i ); + } + return matchIndexes; + } ), + + "odd": createPositionalPseudo( function( matchIndexes, length ) { + var i = 1; + for ( ; i < length; i += 2 ) { + matchIndexes.push( i ); + } + return matchIndexes; + } ), + + "lt": createPositionalPseudo( function( matchIndexes, length, argument ) { + var i = argument < 0 ? + argument + length : + argument > length ? + length : + argument; + for ( ; --i >= 0; ) { + matchIndexes.push( i ); + } + return matchIndexes; + } ), + + "gt": createPositionalPseudo( function( matchIndexes, length, argument ) { + var i = argument < 0 ? argument + length : argument; + for ( ; ++i < length; ) { + matchIndexes.push( i ); + } + return matchIndexes; + } ) + } + }; + + Expr.pseudos[ "nth" ] = Expr.pseudos[ "eq" ]; + + // Add button/input type pseudos + for ( i in { radio: true, checkbox: true, file: true, password: true, image: true } ) { + Expr.pseudos[ i ] = createInputPseudo( i ); + } + for ( i in { submit: true, reset: true } ) { + Expr.pseudos[ i ] = createButtonPseudo( i ); + } + + // Easy API for creating new setFilters + function setFilters() {} + setFilters.prototype = Expr.filters = Expr.pseudos; + Expr.setFilters = new setFilters(); + + tokenize = Sizzle.tokenize = function( selector, parseOnly ) { + var matched, match, tokens, type, + soFar, groups, preFilters, + cached = tokenCache[ selector + " " ]; + + if ( cached ) { + return parseOnly ? 0 : cached.slice( 0 ); + } + + soFar = selector; + groups = []; + preFilters = Expr.preFilter; + + while ( soFar ) { + + // Comma and first run + if ( !matched || ( match = rcomma.exec( soFar ) ) ) { + if ( match ) { + + // Don't consume trailing commas as valid + soFar = soFar.slice( match[ 0 ].length ) || soFar; + } + groups.push( ( tokens = [] ) ); + } + + matched = false; + + // Combinators + if ( ( match = rcombinators.exec( soFar ) ) ) { + matched = match.shift(); + tokens.push( { + value: matched, + + // Cast descendant combinators to space + type: match[ 0 ].replace( rtrim, " " ) + } ); + soFar = soFar.slice( matched.length ); + } + + // Filters + for ( type in Expr.filter ) { + if ( ( match = matchExpr[ type ].exec( soFar ) ) && ( !preFilters[ type ] || + ( match = preFilters[ type ]( match ) ) ) ) { + matched = match.shift(); + tokens.push( { + value: matched, + type: type, + matches: match + } ); + soFar = soFar.slice( matched.length ); + } + } + + if ( !matched ) { + break; + } + } + + // Return the length of the invalid excess + // if we're just parsing + // Otherwise, throw an error or return tokens + return parseOnly ? + soFar.length : + soFar ? + Sizzle.error( selector ) : + + // Cache the tokens + tokenCache( selector, groups ).slice( 0 ); + }; + + function toSelector( tokens ) { + var i = 0, + len = tokens.length, + selector = ""; + for ( ; i < len; i++ ) { + selector += tokens[ i ].value; + } + return selector; + } + + function addCombinator( matcher, combinator, base ) { + var dir = combinator.dir, + skip = combinator.next, + key = skip || dir, + checkNonElements = base && key === "parentNode", + doneName = done++; + + return combinator.first ? + + // Check against closest ancestor/preceding element + function( elem, context, xml ) { + while ( ( elem = elem[ dir ] ) ) { + if ( elem.nodeType === 1 || checkNonElements ) { + return matcher( elem, context, xml ); + } + } + return false; + } : + + // Check against all ancestor/preceding elements + function( elem, context, xml ) { + var oldCache, uniqueCache, outerCache, + newCache = [ dirruns, doneName ]; + + // We can't set arbitrary data on XML nodes, so they don't benefit from combinator caching + if ( xml ) { + while ( ( elem = elem[ dir ] ) ) { + if ( elem.nodeType === 1 || checkNonElements ) { + if ( matcher( elem, context, xml ) ) { + return true; + } + } + } + } else { + while ( ( elem = elem[ dir ] ) ) { + if ( elem.nodeType === 1 || checkNonElements ) { + outerCache = elem[ expando ] || ( elem[ expando ] = {} ); + + // Support: IE <9 only + // Defend against cloned attroperties (jQuery gh-1709) + uniqueCache = outerCache[ elem.uniqueID ] || + ( outerCache[ elem.uniqueID ] = {} ); + + if ( skip && skip === elem.nodeName.toLowerCase() ) { + elem = elem[ dir ] || elem; + } else if ( ( oldCache = uniqueCache[ key ] ) && + oldCache[ 0 ] === dirruns && oldCache[ 1 ] === doneName ) { + + // Assign to newCache so results back-propagate to previous elements + return ( newCache[ 2 ] = oldCache[ 2 ] ); + } else { + + // Reuse newcache so results back-propagate to previous elements + uniqueCache[ key ] = newCache; + + // A match means we're done; a fail means we have to keep checking + if ( ( newCache[ 2 ] = matcher( elem, context, xml ) ) ) { + return true; + } + } + } + } + } + return false; + }; + } + + function elementMatcher( matchers ) { + return matchers.length > 1 ? + function( elem, context, xml ) { + var i = matchers.length; + while ( i-- ) { + if ( !matchers[ i ]( elem, context, xml ) ) { + return false; + } + } + return true; + } : + matchers[ 0 ]; + } + + function multipleContexts( selector, contexts, results ) { + var i = 0, + len = contexts.length; + for ( ; i < len; i++ ) { + Sizzle( selector, contexts[ i ], results ); + } + return results; + } + + function condense( unmatched, map, filter, context, xml ) { + var elem, + newUnmatched = [], + i = 0, + len = unmatched.length, + mapped = map != null; + + for ( ; i < len; i++ ) { + if ( ( elem = unmatched[ i ] ) ) { + if ( !filter || filter( elem, context, xml ) ) { + newUnmatched.push( elem ); + if ( mapped ) { + map.push( i ); + } + } + } + } + + return newUnmatched; + } + + function setMatcher( preFilter, selector, matcher, postFilter, postFinder, postSelector ) { + if ( postFilter && !postFilter[ expando ] ) { + postFilter = setMatcher( postFilter ); + } + if ( postFinder && !postFinder[ expando ] ) { + postFinder = setMatcher( postFinder, postSelector ); + } + return markFunction( function( seed, results, context, xml ) { + var temp, i, elem, + preMap = [], + postMap = [], + preexisting = results.length, + + // Get initial elements from seed or context + elems = seed || multipleContexts( + selector || "*", + context.nodeType ? [ context ] : context, + [] + ), + + // Prefilter to get matcher input, preserving a map for seed-results synchronization + matcherIn = preFilter && ( seed || !selector ) ? + condense( elems, preMap, preFilter, context, xml ) : + elems, + + matcherOut = matcher ? + + // If we have a postFinder, or filtered seed, or non-seed postFilter or preexisting results, + postFinder || ( seed ? preFilter : preexisting || postFilter ) ? + + // ...intermediate processing is necessary + [] : + + // ...otherwise use results directly + results : + matcherIn; + + // Find primary matches + if ( matcher ) { + matcher( matcherIn, matcherOut, context, xml ); + } + + // Apply postFilter + if ( postFilter ) { + temp = condense( matcherOut, postMap ); + postFilter( temp, [], context, xml ); + + // Un-match failing elements by moving them back to matcherIn + i = temp.length; + while ( i-- ) { + if ( ( elem = temp[ i ] ) ) { + matcherOut[ postMap[ i ] ] = !( matcherIn[ postMap[ i ] ] = elem ); + } + } + } + + if ( seed ) { + if ( postFinder || preFilter ) { + if ( postFinder ) { + + // Get the final matcherOut by condensing this intermediate into postFinder contexts + temp = []; + i = matcherOut.length; + while ( i-- ) { + if ( ( elem = matcherOut[ i ] ) ) { + + // Restore matcherIn since elem is not yet a final match + temp.push( ( matcherIn[ i ] = elem ) ); + } + } + postFinder( null, ( matcherOut = [] ), temp, xml ); + } + + // Move matched elements from seed to results to keep them synchronized + i = matcherOut.length; + while ( i-- ) { + if ( ( elem = matcherOut[ i ] ) && + ( temp = postFinder ? indexOf( seed, elem ) : preMap[ i ] ) > -1 ) { + + seed[ temp ] = !( results[ temp ] = elem ); + } + } + } + + // Add elements to results, through postFinder if defined + } else { + matcherOut = condense( + matcherOut === results ? + matcherOut.splice( preexisting, matcherOut.length ) : + matcherOut + ); + if ( postFinder ) { + postFinder( null, results, matcherOut, xml ); + } else { + push.apply( results, matcherOut ); + } + } + } ); + } + + function matcherFromTokens( tokens ) { + var checkContext, matcher, j, + len = tokens.length, + leadingRelative = Expr.relative[ tokens[ 0 ].type ], + implicitRelative = leadingRelative || Expr.relative[ " " ], + i = leadingRelative ? 1 : 0, + + // The foundational matcher ensures that elements are reachable from top-level context(s) + matchContext = addCombinator( function( elem ) { + return elem === checkContext; + }, implicitRelative, true ), + matchAnyContext = addCombinator( function( elem ) { + return indexOf( checkContext, elem ) > -1; + }, implicitRelative, true ), + matchers = [ function( elem, context, xml ) { + var ret = ( !leadingRelative && ( xml || context !== outermostContext ) ) || ( + ( checkContext = context ).nodeType ? + matchContext( elem, context, xml ) : + matchAnyContext( elem, context, xml ) ); + + // Avoid hanging onto element (issue #299) + checkContext = null; + return ret; + } ]; + + for ( ; i < len; i++ ) { + if ( ( matcher = Expr.relative[ tokens[ i ].type ] ) ) { + matchers = [ addCombinator( elementMatcher( matchers ), matcher ) ]; + } else { + matcher = Expr.filter[ tokens[ i ].type ].apply( null, tokens[ i ].matches ); + + // Return special upon seeing a positional matcher + if ( matcher[ expando ] ) { + + // Find the next relative operator (if any) for proper handling + j = ++i; + for ( ; j < len; j++ ) { + if ( Expr.relative[ tokens[ j ].type ] ) { + break; + } + } + return setMatcher( + i > 1 && elementMatcher( matchers ), + i > 1 && toSelector( + + // If the preceding token was a descendant combinator, insert an implicit any-element `*` + tokens + .slice( 0, i - 1 ) + .concat( { value: tokens[ i - 2 ].type === " " ? "*" : "" } ) + ).replace( rtrim, "$1" ), + matcher, + i < j && matcherFromTokens( tokens.slice( i, j ) ), + j < len && matcherFromTokens( ( tokens = tokens.slice( j ) ) ), + j < len && toSelector( tokens ) + ); + } + matchers.push( matcher ); + } + } + + return elementMatcher( matchers ); + } + + function matcherFromGroupMatchers( elementMatchers, setMatchers ) { + var bySet = setMatchers.length > 0, + byElement = elementMatchers.length > 0, + superMatcher = function( seed, context, xml, results, outermost ) { + var elem, j, matcher, + matchedCount = 0, + i = "0", + unmatched = seed && [], + setMatched = [], + contextBackup = outermostContext, + + // We must always have either seed elements or outermost context + elems = seed || byElement && Expr.find[ "TAG" ]( "*", outermost ), + + // Use integer dirruns iff this is the outermost matcher + dirrunsUnique = ( dirruns += contextBackup == null ? 1 : Math.random() || 0.1 ), + len = elems.length; + + if ( outermost ) { + + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + outermostContext = context == document || context || outermost; + } + + // Add elements passing elementMatchers directly to results + // Support: IE<9, Safari + // Tolerate NodeList properties (IE: "length"; Safari: <number>) matching elements by id + for ( ; i !== len && ( elem = elems[ i ] ) != null; i++ ) { + if ( byElement && elem ) { + j = 0; + + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + if ( !context && elem.ownerDocument != document ) { + setDocument( elem ); + xml = !documentIsHTML; + } + while ( ( matcher = elementMatchers[ j++ ] ) ) { + if ( matcher( elem, context || document, xml ) ) { + results.push( elem ); + break; + } + } + if ( outermost ) { + dirruns = dirrunsUnique; + } + } + + // Track unmatched elements for set filters + if ( bySet ) { + + // They will have gone through all possible matchers + if ( ( elem = !matcher && elem ) ) { + matchedCount--; + } + + // Lengthen the array for every element, matched or not + if ( seed ) { + unmatched.push( elem ); + } + } + } + + // `i` is now the count of elements visited above, and adding it to `matchedCount` + // makes the latter nonnegative. + matchedCount += i; + + // Apply set filters to unmatched elements + // NOTE: This can be skipped if there are no unmatched elements (i.e., `matchedCount` + // equals `i`), unless we didn't visit _any_ elements in the above loop because we have + // no element matchers and no seed. + // Incrementing an initially-string "0" `i` allows `i` to remain a string only in that + // case, which will result in a "00" `matchedCount` that differs from `i` but is also + // numerically zero. + if ( bySet && i !== matchedCount ) { + j = 0; + while ( ( matcher = setMatchers[ j++ ] ) ) { + matcher( unmatched, setMatched, context, xml ); + } + + if ( seed ) { + + // Reintegrate element matches to eliminate the need for sorting + if ( matchedCount > 0 ) { + while ( i-- ) { + if ( !( unmatched[ i ] || setMatched[ i ] ) ) { + setMatched[ i ] = pop.call( results ); + } + } + } + + // Discard index placeholder values to get only actual matches + setMatched = condense( setMatched ); + } + + // Add matches to results + push.apply( results, setMatched ); + + // Seedless set matches succeeding multiple successful matchers stipulate sorting + if ( outermost && !seed && setMatched.length > 0 && + ( matchedCount + setMatchers.length ) > 1 ) { + + Sizzle.uniqueSort( results ); + } + } + + // Override manipulation of globals by nested matchers + if ( outermost ) { + dirruns = dirrunsUnique; + outermostContext = contextBackup; + } + + return unmatched; + }; + + return bySet ? + markFunction( superMatcher ) : + superMatcher; + } + + compile = Sizzle.compile = function( selector, match /* Internal Use Only */ ) { + var i, + setMatchers = [], + elementMatchers = [], + cached = compilerCache[ selector + " " ]; + + if ( !cached ) { + + // Generate a function of recursive functions that can be used to check each element + if ( !match ) { + match = tokenize( selector ); + } + i = match.length; + while ( i-- ) { + cached = matcherFromTokens( match[ i ] ); + if ( cached[ expando ] ) { + setMatchers.push( cached ); + } else { + elementMatchers.push( cached ); + } + } + + // Cache the compiled function + cached = compilerCache( + selector, + matcherFromGroupMatchers( elementMatchers, setMatchers ) + ); + + // Save selector and tokenization + cached.selector = selector; + } + return cached; + }; + + /** + * A low-level selection function that works with Sizzle's compiled + * selector functions + * @param {String|Function} selector A selector or a pre-compiled + * selector function built with Sizzle.compile + * @param {Element} context + * @param {Array} [results] + * @param {Array} [seed] A set of elements to match against + */ + select = Sizzle.select = function( selector, context, results, seed ) { + var i, tokens, token, type, find, + compiled = typeof selector === "function" && selector, + match = !seed && tokenize( ( selector = compiled.selector || selector ) ); + + results = results || []; + + // Try to minimize operations if there is only one selector in the list and no seed + // (the latter of which guarantees us context) + if ( match.length === 1 ) { + + // Reduce context if the leading compound selector is an ID + tokens = match[ 0 ] = match[ 0 ].slice( 0 ); + if ( tokens.length > 2 && ( token = tokens[ 0 ] ).type === "ID" && + context.nodeType === 9 && documentIsHTML && Expr.relative[ tokens[ 1 ].type ] ) { + + context = ( Expr.find[ "ID" ]( token.matches[ 0 ] + .replace( runescape, funescape ), context ) || [] )[ 0 ]; + if ( !context ) { + return results; + + // Precompiled matchers will still verify ancestry, so step up a level + } else if ( compiled ) { + context = context.parentNode; + } + + selector = selector.slice( tokens.shift().value.length ); + } + + // Fetch a seed set for right-to-left matching + i = matchExpr[ "needsContext" ].test( selector ) ? 0 : tokens.length; + while ( i-- ) { + token = tokens[ i ]; + + // Abort if we hit a combinator + if ( Expr.relative[ ( type = token.type ) ] ) { + break; + } + if ( ( find = Expr.find[ type ] ) ) { + + // Search, expanding context for leading sibling combinators + if ( ( seed = find( + token.matches[ 0 ].replace( runescape, funescape ), + rsibling.test( tokens[ 0 ].type ) && testContext( context.parentNode ) || + context + ) ) ) { + + // If seed is empty or no tokens remain, we can return early + tokens.splice( i, 1 ); + selector = seed.length && toSelector( tokens ); + if ( !selector ) { + push.apply( results, seed ); + return results; + } + + break; + } + } + } + } + + // Compile and execute a filtering function if one is not provided + // Provide `match` to avoid retokenization if we modified the selector above + ( compiled || compile( selector, match ) )( + seed, + context, + !documentIsHTML, + results, + !context || rsibling.test( selector ) && testContext( context.parentNode ) || context + ); + return results; + }; + + // One-time assignments + + // Sort stability + support.sortStable = expando.split( "" ).sort( sortOrder ).join( "" ) === expando; + + // Support: Chrome 14-35+ + // Always assume duplicates if they aren't passed to the comparison function + support.detectDuplicates = !!hasDuplicate; + + // Initialize against the default document + setDocument(); + + // Support: Webkit<537.32 - Safari 6.0.3/Chrome 25 (fixed in Chrome 27) + // Detached nodes confoundingly follow *each other* + support.sortDetached = assert( function( el ) { + + // Should return 1, but returns 4 (following) + return el.compareDocumentPosition( document.createElement( "fieldset" ) ) & 1; + } ); + + // Support: IE<8 + // Prevent attribute/property "interpolation" + // https://msdn.microsoft.com/en-us/library/ms536429%28VS.85%29.aspx + if ( !assert( function( el ) { + el.innerHTML = "<a href='#'></a>"; + return el.firstChild.getAttribute( "href" ) === "#"; + } ) ) { + addHandle( "type|href|height|width", function( elem, name, isXML ) { + if ( !isXML ) { + return elem.getAttribute( name, name.toLowerCase() === "type" ? 1 : 2 ); + } + } ); + } + + // Support: IE<9 + // Use defaultValue in place of getAttribute("value") + if ( !support.attributes || !assert( function( el ) { + el.innerHTML = "<input/>"; + el.firstChild.setAttribute( "value", "" ); + return el.firstChild.getAttribute( "value" ) === ""; + } ) ) { + addHandle( "value", function( elem, _name, isXML ) { + if ( !isXML && elem.nodeName.toLowerCase() === "input" ) { + return elem.defaultValue; + } + } ); + } + + // Support: IE<9 + // Use getAttributeNode to fetch booleans when getAttribute lies + if ( !assert( function( el ) { + return el.getAttribute( "disabled" ) == null; + } ) ) { + addHandle( booleans, function( elem, name, isXML ) { + var val; + if ( !isXML ) { + return elem[ name ] === true ? name.toLowerCase() : + ( val = elem.getAttributeNode( name ) ) && val.specified ? + val.value : + null; + } + } ); + } + + return Sizzle; + + } )( window ); + + + + jQuery.find = Sizzle; + jQuery.expr = Sizzle.selectors; + + // Deprecated + jQuery.expr[ ":" ] = jQuery.expr.pseudos; + jQuery.uniqueSort = jQuery.unique = Sizzle.uniqueSort; + jQuery.text = Sizzle.getText; + jQuery.isXMLDoc = Sizzle.isXML; + jQuery.contains = Sizzle.contains; + jQuery.escapeSelector = Sizzle.escape; + + + + + var dir = function( elem, dir, until ) { + var matched = [], + truncate = until !== undefined; + + while ( ( elem = elem[ dir ] ) && elem.nodeType !== 9 ) { + if ( elem.nodeType === 1 ) { + if ( truncate && jQuery( elem ).is( until ) ) { + break; + } + matched.push( elem ); + } + } + return matched; + }; + + + var siblings = function( n, elem ) { + var matched = []; + + for ( ; n; n = n.nextSibling ) { + if ( n.nodeType === 1 && n !== elem ) { + matched.push( n ); + } + } + + return matched; + }; + + + var rneedsContext = jQuery.expr.match.needsContext; + + + + function nodeName( elem, name ) { + + return elem.nodeName && elem.nodeName.toLowerCase() === name.toLowerCase(); + + } + var rsingleTag = ( /^<([a-z][^\/\0>:\x20\t\r\n\f]*)[\x20\t\r\n\f]*\/?>(?:<\/\1>|)$/i ); + + + + // Implement the identical functionality for filter and not + function winnow( elements, qualifier, not ) { + if ( isFunction( qualifier ) ) { + return jQuery.grep( elements, function( elem, i ) { + return !!qualifier.call( elem, i, elem ) !== not; + } ); + } + + // Single element + if ( qualifier.nodeType ) { + return jQuery.grep( elements, function( elem ) { + return ( elem === qualifier ) !== not; + } ); + } + + // Arraylike of elements (jQuery, arguments, Array) + if ( typeof qualifier !== "string" ) { + return jQuery.grep( elements, function( elem ) { + return ( indexOf.call( qualifier, elem ) > -1 ) !== not; + } ); + } + + // Filtered directly for both simple and complex selectors + return jQuery.filter( qualifier, elements, not ); + } + + jQuery.filter = function( expr, elems, not ) { + var elem = elems[ 0 ]; + + if ( not ) { + expr = ":not(" + expr + ")"; + } + + if ( elems.length === 1 && elem.nodeType === 1 ) { + return jQuery.find.matchesSelector( elem, expr ) ? [ elem ] : []; + } + + return jQuery.find.matches( expr, jQuery.grep( elems, function( elem ) { + return elem.nodeType === 1; + } ) ); + }; + + jQuery.fn.extend( { + find: function( selector ) { + var i, ret, + len = this.length, + self = this; + + if ( typeof selector !== "string" ) { + return this.pushStack( jQuery( selector ).filter( function() { + for ( i = 0; i < len; i++ ) { + if ( jQuery.contains( self[ i ], this ) ) { + return true; + } + } + } ) ); + } + + ret = this.pushStack( [] ); + + for ( i = 0; i < len; i++ ) { + jQuery.find( selector, self[ i ], ret ); + } + + return len > 1 ? jQuery.uniqueSort( ret ) : ret; + }, + filter: function( selector ) { + return this.pushStack( winnow( this, selector || [], false ) ); + }, + not: function( selector ) { + return this.pushStack( winnow( this, selector || [], true ) ); + }, + is: function( selector ) { + return !!winnow( + this, + + // If this is a positional/relative selector, check membership in the returned set + // so $("p:first").is("p:last") won't return true for a doc with two "p". + typeof selector === "string" && rneedsContext.test( selector ) ? + jQuery( selector ) : + selector || [], + false + ).length; + } + } ); + + + // Initialize a jQuery object + + + // A central reference to the root jQuery(document) + var rootjQuery, + + // A simple way to check for HTML strings + // Prioritize #id over <tag> to avoid XSS via location.hash (#9521) + // Strict HTML recognition (#11290: must start with <) + // Shortcut simple #id case for speed + rquickExpr = /^(?:\s*(<[\w\W]+>)[^>]*|#([\w-]+))$/, + + init = jQuery.fn.init = function( selector, context, root ) { + var match, elem; + + // HANDLE: $(""), $(null), $(undefined), $(false) + if ( !selector ) { + return this; + } + + // Method init() accepts an alternate rootjQuery + // so migrate can support jQuery.sub (gh-2101) + root = root || rootjQuery; + + // Handle HTML strings + if ( typeof selector === "string" ) { + if ( selector[ 0 ] === "<" && + selector[ selector.length - 1 ] === ">" && + selector.length >= 3 ) { + + // Assume that strings that start and end with <> are HTML and skip the regex check + match = [ null, selector, null ]; + + } else { + match = rquickExpr.exec( selector ); + } + + // Match html or make sure no context is specified for #id + if ( match && ( match[ 1 ] || !context ) ) { + + // HANDLE: $(html) -> $(array) + if ( match[ 1 ] ) { + context = context instanceof jQuery ? context[ 0 ] : context; + + // Option to run scripts is true for back-compat + // Intentionally let the error be thrown if parseHTML is not present + jQuery.merge( this, jQuery.parseHTML( + match[ 1 ], + context && context.nodeType ? context.ownerDocument || context : document, + true + ) ); + + // HANDLE: $(html, props) + if ( rsingleTag.test( match[ 1 ] ) && jQuery.isPlainObject( context ) ) { + for ( match in context ) { + + // Properties of context are called as methods if possible + if ( isFunction( this[ match ] ) ) { + this[ match ]( context[ match ] ); + + // ...and otherwise set as attributes + } else { + this.attr( match, context[ match ] ); + } + } + } + + return this; + + // HANDLE: $(#id) + } else { + elem = document.getElementById( match[ 2 ] ); + + if ( elem ) { + + // Inject the element directly into the jQuery object + this[ 0 ] = elem; + this.length = 1; + } + return this; + } + + // HANDLE: $(expr, $(...)) + } else if ( !context || context.jquery ) { + return ( context || root ).find( selector ); + + // HANDLE: $(expr, context) + // (which is just equivalent to: $(context).find(expr) + } else { + return this.constructor( context ).find( selector ); + } + + // HANDLE: $(DOMElement) + } else if ( selector.nodeType ) { + this[ 0 ] = selector; + this.length = 1; + return this; + + // HANDLE: $(function) + // Shortcut for document ready + } else if ( isFunction( selector ) ) { + return root.ready !== undefined ? + root.ready( selector ) : + + // Execute immediately if ready is not present + selector( jQuery ); + } + + return jQuery.makeArray( selector, this ); + }; + + // Give the init function the jQuery prototype for later instantiation + init.prototype = jQuery.fn; + + // Initialize central reference + rootjQuery = jQuery( document ); + + + var rparentsprev = /^(?:parents|prev(?:Until|All))/, + + // Methods guaranteed to produce a unique set when starting from a unique set + guaranteedUnique = { + children: true, + contents: true, + next: true, + prev: true + }; + + jQuery.fn.extend( { + has: function( target ) { + var targets = jQuery( target, this ), + l = targets.length; + + return this.filter( function() { + var i = 0; + for ( ; i < l; i++ ) { + if ( jQuery.contains( this, targets[ i ] ) ) { + return true; + } + } + } ); + }, + + closest: function( selectors, context ) { + var cur, + i = 0, + l = this.length, + matched = [], + targets = typeof selectors !== "string" && jQuery( selectors ); + + // Positional selectors never match, since there's no _selection_ context + if ( !rneedsContext.test( selectors ) ) { + for ( ; i < l; i++ ) { + for ( cur = this[ i ]; cur && cur !== context; cur = cur.parentNode ) { + + // Always skip document fragments + if ( cur.nodeType < 11 && ( targets ? + targets.index( cur ) > -1 : + + // Don't pass non-elements to Sizzle + cur.nodeType === 1 && + jQuery.find.matchesSelector( cur, selectors ) ) ) { + + matched.push( cur ); + break; + } + } + } + } + + return this.pushStack( matched.length > 1 ? jQuery.uniqueSort( matched ) : matched ); + }, + + // Determine the position of an element within the set + index: function( elem ) { + + // No argument, return index in parent + if ( !elem ) { + return ( this[ 0 ] && this[ 0 ].parentNode ) ? this.first().prevAll().length : -1; + } + + // Index in selector + if ( typeof elem === "string" ) { + return indexOf.call( jQuery( elem ), this[ 0 ] ); + } + + // Locate the position of the desired element + return indexOf.call( this, + + // If it receives a jQuery object, the first element is used + elem.jquery ? elem[ 0 ] : elem + ); + }, + + add: function( selector, context ) { + return this.pushStack( + jQuery.uniqueSort( + jQuery.merge( this.get(), jQuery( selector, context ) ) + ) + ); + }, + + addBack: function( selector ) { + return this.add( selector == null ? + this.prevObject : this.prevObject.filter( selector ) + ); + } + } ); + + function sibling( cur, dir ) { + while ( ( cur = cur[ dir ] ) && cur.nodeType !== 1 ) {} + return cur; + } + + jQuery.each( { + parent: function( elem ) { + var parent = elem.parentNode; + return parent && parent.nodeType !== 11 ? parent : null; + }, + parents: function( elem ) { + return dir( elem, "parentNode" ); + }, + parentsUntil: function( elem, _i, until ) { + return dir( elem, "parentNode", until ); + }, + next: function( elem ) { + return sibling( elem, "nextSibling" ); + }, + prev: function( elem ) { + return sibling( elem, "previousSibling" ); + }, + nextAll: function( elem ) { + return dir( elem, "nextSibling" ); + }, + prevAll: function( elem ) { + return dir( elem, "previousSibling" ); + }, + nextUntil: function( elem, _i, until ) { + return dir( elem, "nextSibling", until ); + }, + prevUntil: function( elem, _i, until ) { + return dir( elem, "previousSibling", until ); + }, + siblings: function( elem ) { + return siblings( ( elem.parentNode || {} ).firstChild, elem ); + }, + children: function( elem ) { + return siblings( elem.firstChild ); + }, + contents: function( elem ) { + if ( elem.contentDocument != null && + + // Support: IE 11+ + // <object> elements with no `data` attribute has an object + // `contentDocument` with a `null` prototype. + getProto( elem.contentDocument ) ) { + + return elem.contentDocument; + } + + // Support: IE 9 - 11 only, iOS 7 only, Android Browser <=4.3 only + // Treat the template element as a regular one in browsers that + // don't support it. + if ( nodeName( elem, "template" ) ) { + elem = elem.content || elem; + } + + return jQuery.merge( [], elem.childNodes ); + } + }, function( name, fn ) { + jQuery.fn[ name ] = function( until, selector ) { + var matched = jQuery.map( this, fn, until ); + + if ( name.slice( -5 ) !== "Until" ) { + selector = until; + } + + if ( selector && typeof selector === "string" ) { + matched = jQuery.filter( selector, matched ); + } + + if ( this.length > 1 ) { + + // Remove duplicates + if ( !guaranteedUnique[ name ] ) { + jQuery.uniqueSort( matched ); + } + + // Reverse order for parents* and prev-derivatives + if ( rparentsprev.test( name ) ) { + matched.reverse(); + } + } + + return this.pushStack( matched ); + }; + } ); + var rnothtmlwhite = ( /[^\x20\t\r\n\f]+/g ); + + + + // Convert String-formatted options into Object-formatted ones + function createOptions( options ) { + var object = {}; + jQuery.each( options.match( rnothtmlwhite ) || [], function( _, flag ) { + object[ flag ] = true; + } ); + return object; + } + + /* + * Create a callback list using the following parameters: + * + * options: an optional list of space-separated options that will change how + * the callback list behaves or a more traditional option object + * + * By default a callback list will act like an event callback list and can be + * "fired" multiple times. + * + * Possible options: + * + * once: will ensure the callback list can only be fired once (like a Deferred) + * + * memory: will keep track of previous values and will call any callback added + * after the list has been fired right away with the latest "memorized" + * values (like a Deferred) + * + * unique: will ensure a callback can only be added once (no duplicate in the list) + * + * stopOnFalse: interrupt callings when a callback returns false + * + */ + jQuery.Callbacks = function( options ) { + + // Convert options from String-formatted to Object-formatted if needed + // (we check in cache first) + options = typeof options === "string" ? + createOptions( options ) : + jQuery.extend( {}, options ); + + var // Flag to know if list is currently firing + firing, + + // Last fire value for non-forgettable lists + memory, + + // Flag to know if list was already fired + fired, + + // Flag to prevent firing + locked, + + // Actual callback list + list = [], + + // Queue of execution data for repeatable lists + queue = [], + + // Index of currently firing callback (modified by add/remove as needed) + firingIndex = -1, + + // Fire callbacks + fire = function() { + + // Enforce single-firing + locked = locked || options.once; + + // Execute callbacks for all pending executions, + // respecting firingIndex overrides and runtime changes + fired = firing = true; + for ( ; queue.length; firingIndex = -1 ) { + memory = queue.shift(); + while ( ++firingIndex < list.length ) { + + // Run callback and check for early termination + if ( list[ firingIndex ].apply( memory[ 0 ], memory[ 1 ] ) === false && + options.stopOnFalse ) { + + // Jump to end and forget the data so .add doesn't re-fire + firingIndex = list.length; + memory = false; + } + } + } + + // Forget the data if we're done with it + if ( !options.memory ) { + memory = false; + } + + firing = false; + + // Clean up if we're done firing for good + if ( locked ) { + + // Keep an empty list if we have data for future add calls + if ( memory ) { + list = []; + + // Otherwise, this object is spent + } else { + list = ""; + } + } + }, + + // Actual Callbacks object + self = { + + // Add a callback or a collection of callbacks to the list + add: function() { + if ( list ) { + + // If we have memory from a past run, we should fire after adding + if ( memory && !firing ) { + firingIndex = list.length - 1; + queue.push( memory ); + } + + ( function add( args ) { + jQuery.each( args, function( _, arg ) { + if ( isFunction( arg ) ) { + if ( !options.unique || !self.has( arg ) ) { + list.push( arg ); + } + } else if ( arg && arg.length && toType( arg ) !== "string" ) { + + // Inspect recursively + add( arg ); + } + } ); + } )( arguments ); + + if ( memory && !firing ) { + fire(); + } + } + return this; + }, + + // Remove a callback from the list + remove: function() { + jQuery.each( arguments, function( _, arg ) { + var index; + while ( ( index = jQuery.inArray( arg, list, index ) ) > -1 ) { + list.splice( index, 1 ); + + // Handle firing indexes + if ( index <= firingIndex ) { + firingIndex--; + } + } + } ); + return this; + }, + + // Check if a given callback is in the list. + // If no argument is given, return whether or not list has callbacks attached. + has: function( fn ) { + return fn ? + jQuery.inArray( fn, list ) > -1 : + list.length > 0; + }, + + // Remove all callbacks from the list + empty: function() { + if ( list ) { + list = []; + } + return this; + }, + + // Disable .fire and .add + // Abort any current/pending executions + // Clear all callbacks and values + disable: function() { + locked = queue = []; + list = memory = ""; + return this; + }, + disabled: function() { + return !list; + }, + + // Disable .fire + // Also disable .add unless we have memory (since it would have no effect) + // Abort any pending executions + lock: function() { + locked = queue = []; + if ( !memory && !firing ) { + list = memory = ""; + } + return this; + }, + locked: function() { + return !!locked; + }, + + // Call all callbacks with the given context and arguments + fireWith: function( context, args ) { + if ( !locked ) { + args = args || []; + args = [ context, args.slice ? args.slice() : args ]; + queue.push( args ); + if ( !firing ) { + fire(); + } + } + return this; + }, + + // Call all the callbacks with the given arguments + fire: function() { + self.fireWith( this, arguments ); + return this; + }, + + // To know if the callbacks have already been called at least once + fired: function() { + return !!fired; + } + }; + + return self; + }; + + + function Identity( v ) { + return v; + } + function Thrower( ex ) { + throw ex; + } + + function adoptValue( value, resolve, reject, noValue ) { + var method; + + try { + + // Check for promise aspect first to privilege synchronous behavior + if ( value && isFunction( ( method = value.promise ) ) ) { + method.call( value ).done( resolve ).fail( reject ); + + // Other thenables + } else if ( value && isFunction( ( method = value.then ) ) ) { + method.call( value, resolve, reject ); + + // Other non-thenables + } else { + + // Control `resolve` arguments by letting Array#slice cast boolean `noValue` to integer: + // * false: [ value ].slice( 0 ) => resolve( value ) + // * true: [ value ].slice( 1 ) => resolve() + resolve.apply( undefined, [ value ].slice( noValue ) ); + } + + // For Promises/A+, convert exceptions into rejections + // Since jQuery.when doesn't unwrap thenables, we can skip the extra checks appearing in + // Deferred#then to conditionally suppress rejection. + } catch ( value ) { + + // Support: Android 4.0 only + // Strict mode functions invoked without .call/.apply get global-object context + reject.apply( undefined, [ value ] ); + } + } + + jQuery.extend( { + + Deferred: function( func ) { + var tuples = [ + + // action, add listener, callbacks, + // ... .then handlers, argument index, [final state] + [ "notify", "progress", jQuery.Callbacks( "memory" ), + jQuery.Callbacks( "memory" ), 2 ], + [ "resolve", "done", jQuery.Callbacks( "once memory" ), + jQuery.Callbacks( "once memory" ), 0, "resolved" ], + [ "reject", "fail", jQuery.Callbacks( "once memory" ), + jQuery.Callbacks( "once memory" ), 1, "rejected" ] + ], + state = "pending", + promise = { + state: function() { + return state; + }, + always: function() { + deferred.done( arguments ).fail( arguments ); + return this; + }, + "catch": function( fn ) { + return promise.then( null, fn ); + }, + + // Keep pipe for back-compat + pipe: function( /* fnDone, fnFail, fnProgress */ ) { + var fns = arguments; + + return jQuery.Deferred( function( newDefer ) { + jQuery.each( tuples, function( _i, tuple ) { + + // Map tuples (progress, done, fail) to arguments (done, fail, progress) + var fn = isFunction( fns[ tuple[ 4 ] ] ) && fns[ tuple[ 4 ] ]; + + // deferred.progress(function() { bind to newDefer or newDefer.notify }) + // deferred.done(function() { bind to newDefer or newDefer.resolve }) + // deferred.fail(function() { bind to newDefer or newDefer.reject }) + deferred[ tuple[ 1 ] ]( function() { + var returned = fn && fn.apply( this, arguments ); + if ( returned && isFunction( returned.promise ) ) { + returned.promise() + .progress( newDefer.notify ) + .done( newDefer.resolve ) + .fail( newDefer.reject ); + } else { + newDefer[ tuple[ 0 ] + "With" ]( + this, + fn ? [ returned ] : arguments + ); + } + } ); + } ); + fns = null; + } ).promise(); + }, + then: function( onFulfilled, onRejected, onProgress ) { + var maxDepth = 0; + function resolve( depth, deferred, handler, special ) { + return function() { + var that = this, + args = arguments, + mightThrow = function() { + var returned, then; + + // Support: Promises/A+ section 2.3.3.3.3 + // https://promisesaplus.com/#point-59 + // Ignore double-resolution attempts + if ( depth < maxDepth ) { + return; + } + + returned = handler.apply( that, args ); + + // Support: Promises/A+ section 2.3.1 + // https://promisesaplus.com/#point-48 + if ( returned === deferred.promise() ) { + throw new TypeError( "Thenable self-resolution" ); + } + + // Support: Promises/A+ sections 2.3.3.1, 3.5 + // https://promisesaplus.com/#point-54 + // https://promisesaplus.com/#point-75 + // Retrieve `then` only once + then = returned && + + // Support: Promises/A+ section 2.3.4 + // https://promisesaplus.com/#point-64 + // Only check objects and functions for thenability + ( typeof returned === "object" || + typeof returned === "function" ) && + returned.then; + + // Handle a returned thenable + if ( isFunction( then ) ) { + + // Special processors (notify) just wait for resolution + if ( special ) { + then.call( + returned, + resolve( maxDepth, deferred, Identity, special ), + resolve( maxDepth, deferred, Thrower, special ) + ); + + // Normal processors (resolve) also hook into progress + } else { + + // ...and disregard older resolution values + maxDepth++; + + then.call( + returned, + resolve( maxDepth, deferred, Identity, special ), + resolve( maxDepth, deferred, Thrower, special ), + resolve( maxDepth, deferred, Identity, + deferred.notifyWith ) + ); + } + + // Handle all other returned values + } else { + + // Only substitute handlers pass on context + // and multiple values (non-spec behavior) + if ( handler !== Identity ) { + that = undefined; + args = [ returned ]; + } + + // Process the value(s) + // Default process is resolve + ( special || deferred.resolveWith )( that, args ); + } + }, + + // Only normal processors (resolve) catch and reject exceptions + process = special ? + mightThrow : + function() { + try { + mightThrow(); + } catch ( e ) { + + if ( jQuery.Deferred.exceptionHook ) { + jQuery.Deferred.exceptionHook( e, + process.stackTrace ); + } + + // Support: Promises/A+ section 2.3.3.3.4.1 + // https://promisesaplus.com/#point-61 + // Ignore post-resolution exceptions + if ( depth + 1 >= maxDepth ) { + + // Only substitute handlers pass on context + // and multiple values (non-spec behavior) + if ( handler !== Thrower ) { + that = undefined; + args = [ e ]; + } + + deferred.rejectWith( that, args ); + } + } + }; + + // Support: Promises/A+ section 2.3.3.3.1 + // https://promisesaplus.com/#point-57 + // Re-resolve promises immediately to dodge false rejection from + // subsequent errors + if ( depth ) { + process(); + } else { + + // Call an optional hook to record the stack, in case of exception + // since it's otherwise lost when execution goes async + if ( jQuery.Deferred.getStackHook ) { + process.stackTrace = jQuery.Deferred.getStackHook(); + } + window.setTimeout( process ); + } + }; + } + + return jQuery.Deferred( function( newDefer ) { + + // progress_handlers.add( ... ) + tuples[ 0 ][ 3 ].add( + resolve( + 0, + newDefer, + isFunction( onProgress ) ? + onProgress : + Identity, + newDefer.notifyWith + ) + ); + + // fulfilled_handlers.add( ... ) + tuples[ 1 ][ 3 ].add( + resolve( + 0, + newDefer, + isFunction( onFulfilled ) ? + onFulfilled : + Identity + ) + ); + + // rejected_handlers.add( ... ) + tuples[ 2 ][ 3 ].add( + resolve( + 0, + newDefer, + isFunction( onRejected ) ? + onRejected : + Thrower + ) + ); + } ).promise(); + }, + + // Get a promise for this deferred + // If obj is provided, the promise aspect is added to the object + promise: function( obj ) { + return obj != null ? jQuery.extend( obj, promise ) : promise; + } + }, + deferred = {}; + + // Add list-specific methods + jQuery.each( tuples, function( i, tuple ) { + var list = tuple[ 2 ], + stateString = tuple[ 5 ]; + + // promise.progress = list.add + // promise.done = list.add + // promise.fail = list.add + promise[ tuple[ 1 ] ] = list.add; + + // Handle state + if ( stateString ) { + list.add( + function() { + + // state = "resolved" (i.e., fulfilled) + // state = "rejected" + state = stateString; + }, + + // rejected_callbacks.disable + // fulfilled_callbacks.disable + tuples[ 3 - i ][ 2 ].disable, + + // rejected_handlers.disable + // fulfilled_handlers.disable + tuples[ 3 - i ][ 3 ].disable, + + // progress_callbacks.lock + tuples[ 0 ][ 2 ].lock, + + // progress_handlers.lock + tuples[ 0 ][ 3 ].lock + ); + } + + // progress_handlers.fire + // fulfilled_handlers.fire + // rejected_handlers.fire + list.add( tuple[ 3 ].fire ); + + // deferred.notify = function() { deferred.notifyWith(...) } + // deferred.resolve = function() { deferred.resolveWith(...) } + // deferred.reject = function() { deferred.rejectWith(...) } + deferred[ tuple[ 0 ] ] = function() { + deferred[ tuple[ 0 ] + "With" ]( this === deferred ? undefined : this, arguments ); + return this; + }; + + // deferred.notifyWith = list.fireWith + // deferred.resolveWith = list.fireWith + // deferred.rejectWith = list.fireWith + deferred[ tuple[ 0 ] + "With" ] = list.fireWith; + } ); + + // Make the deferred a promise + promise.promise( deferred ); + + // Call given func if any + if ( func ) { + func.call( deferred, deferred ); + } + + // All done! + return deferred; + }, + + // Deferred helper + when: function( singleValue ) { + var + + // count of uncompleted subordinates + remaining = arguments.length, + + // count of unprocessed arguments + i = remaining, + + // subordinate fulfillment data + resolveContexts = Array( i ), + resolveValues = slice.call( arguments ), + + // the primary Deferred + primary = jQuery.Deferred(), + + // subordinate callback factory + updateFunc = function( i ) { + return function( value ) { + resolveContexts[ i ] = this; + resolveValues[ i ] = arguments.length > 1 ? slice.call( arguments ) : value; + if ( !( --remaining ) ) { + primary.resolveWith( resolveContexts, resolveValues ); + } + }; + }; + + // Single- and empty arguments are adopted like Promise.resolve + if ( remaining <= 1 ) { + adoptValue( singleValue, primary.done( updateFunc( i ) ).resolve, primary.reject, + !remaining ); + + // Use .then() to unwrap secondary thenables (cf. gh-3000) + if ( primary.state() === "pending" || + isFunction( resolveValues[ i ] && resolveValues[ i ].then ) ) { + + return primary.then(); + } + } + + // Multiple arguments are aggregated like Promise.all array elements + while ( i-- ) { + adoptValue( resolveValues[ i ], updateFunc( i ), primary.reject ); + } + + return primary.promise(); + } + } ); + + + // These usually indicate a programmer mistake during development, + // warn about them ASAP rather than swallowing them by default. + var rerrorNames = /^(Eval|Internal|Range|Reference|Syntax|Type|URI)Error$/; + + jQuery.Deferred.exceptionHook = function( error, stack ) { + + // Support: IE 8 - 9 only + // Console exists when dev tools are open, which can happen at any time + if ( window.console && window.console.warn && error && rerrorNames.test( error.name ) ) { + window.console.warn( "jQuery.Deferred exception: " + error.message, error.stack, stack ); + } + }; + + + + + jQuery.readyException = function( error ) { + window.setTimeout( function() { + throw error; + } ); + }; + + + + + // The deferred used on DOM ready + var readyList = jQuery.Deferred(); + + jQuery.fn.ready = function( fn ) { + + readyList + .then( fn ) + + // Wrap jQuery.readyException in a function so that the lookup + // happens at the time of error handling instead of callback + // registration. + .catch( function( error ) { + jQuery.readyException( error ); + } ); + + return this; + }; + + jQuery.extend( { + + // Is the DOM ready to be used? Set to true once it occurs. + isReady: false, + + // A counter to track how many items to wait for before + // the ready event fires. See #6781 + readyWait: 1, + + // Handle when the DOM is ready + ready: function( wait ) { + + // Abort if there are pending holds or we're already ready + if ( wait === true ? --jQuery.readyWait : jQuery.isReady ) { + return; + } + + // Remember that the DOM is ready + jQuery.isReady = true; + + // If a normal DOM Ready event fired, decrement, and wait if need be + if ( wait !== true && --jQuery.readyWait > 0 ) { + return; + } + + // If there are functions bound, to execute + readyList.resolveWith( document, [ jQuery ] ); + } + } ); + + jQuery.ready.then = readyList.then; + + // The ready event handler and self cleanup method + function completed() { + document.removeEventListener( "DOMContentLoaded", completed ); + window.removeEventListener( "load", completed ); + jQuery.ready(); + } + + // Catch cases where $(document).ready() is called + // after the browser event has already occurred. + // Support: IE <=9 - 10 only + // Older IE sometimes signals "interactive" too soon + if ( document.readyState === "complete" || + ( document.readyState !== "loading" && !document.documentElement.doScroll ) ) { + + // Handle it asynchronously to allow scripts the opportunity to delay ready + window.setTimeout( jQuery.ready ); + + } else { + + // Use the handy event callback + document.addEventListener( "DOMContentLoaded", completed ); + + // A fallback to window.onload, that will always work + window.addEventListener( "load", completed ); + } + + + + + // Multifunctional method to get and set values of a collection + // The value/s can optionally be executed if it's a function + var access = function( elems, fn, key, value, chainable, emptyGet, raw ) { + var i = 0, + len = elems.length, + bulk = key == null; + + // Sets many values + if ( toType( key ) === "object" ) { + chainable = true; + for ( i in key ) { + access( elems, fn, i, key[ i ], true, emptyGet, raw ); + } + + // Sets one value + } else if ( value !== undefined ) { + chainable = true; + + if ( !isFunction( value ) ) { + raw = true; + } + + if ( bulk ) { + + // Bulk operations run against the entire set + if ( raw ) { + fn.call( elems, value ); + fn = null; + + // ...except when executing function values + } else { + bulk = fn; + fn = function( elem, _key, value ) { + return bulk.call( jQuery( elem ), value ); + }; + } + } + + if ( fn ) { + for ( ; i < len; i++ ) { + fn( + elems[ i ], key, raw ? + value : + value.call( elems[ i ], i, fn( elems[ i ], key ) ) + ); + } + } + } + + if ( chainable ) { + return elems; + } + + // Gets + if ( bulk ) { + return fn.call( elems ); + } + + return len ? fn( elems[ 0 ], key ) : emptyGet; + }; + + + // Matches dashed string for camelizing + var rmsPrefix = /^-ms-/, + rdashAlpha = /-([a-z])/g; + + // Used by camelCase as callback to replace() + function fcamelCase( _all, letter ) { + return letter.toUpperCase(); + } + + // Convert dashed to camelCase; used by the css and data modules + // Support: IE <=9 - 11, Edge 12 - 15 + // Microsoft forgot to hump their vendor prefix (#9572) + function camelCase( string ) { + return string.replace( rmsPrefix, "ms-" ).replace( rdashAlpha, fcamelCase ); + } + var acceptData = function( owner ) { + + // Accepts only: + // - Node + // - Node.ELEMENT_NODE + // - Node.DOCUMENT_NODE + // - Object + // - Any + return owner.nodeType === 1 || owner.nodeType === 9 || !( +owner.nodeType ); + }; + + + + + function Data() { + this.expando = jQuery.expando + Data.uid++; + } + + Data.uid = 1; + + Data.prototype = { + + cache: function( owner ) { + + // Check if the owner object already has a cache + var value = owner[ this.expando ]; + + // If not, create one + if ( !value ) { + value = {}; + + // We can accept data for non-element nodes in modern browsers, + // but we should not, see #8335. + // Always return an empty object. + if ( acceptData( owner ) ) { + + // If it is a node unlikely to be stringify-ed or looped over + // use plain assignment + if ( owner.nodeType ) { + owner[ this.expando ] = value; + + // Otherwise secure it in a non-enumerable property + // configurable must be true to allow the property to be + // deleted when data is removed + } else { + Object.defineProperty( owner, this.expando, { + value: value, + configurable: true + } ); + } + } + } + + return value; + }, + set: function( owner, data, value ) { + var prop, + cache = this.cache( owner ); + + // Handle: [ owner, key, value ] args + // Always use camelCase key (gh-2257) + if ( typeof data === "string" ) { + cache[ camelCase( data ) ] = value; + + // Handle: [ owner, { properties } ] args + } else { + + // Copy the properties one-by-one to the cache object + for ( prop in data ) { + cache[ camelCase( prop ) ] = data[ prop ]; + } + } + return cache; + }, + get: function( owner, key ) { + return key === undefined ? + this.cache( owner ) : + + // Always use camelCase key (gh-2257) + owner[ this.expando ] && owner[ this.expando ][ camelCase( key ) ]; + }, + access: function( owner, key, value ) { + + // In cases where either: + // + // 1. No key was specified + // 2. A string key was specified, but no value provided + // + // Take the "read" path and allow the get method to determine + // which value to return, respectively either: + // + // 1. The entire cache object + // 2. The data stored at the key + // + if ( key === undefined || + ( ( key && typeof key === "string" ) && value === undefined ) ) { + + return this.get( owner, key ); + } + + // When the key is not a string, or both a key and value + // are specified, set or extend (existing objects) with either: + // + // 1. An object of properties + // 2. A key and value + // + this.set( owner, key, value ); + + // Since the "set" path can have two possible entry points + // return the expected data based on which path was taken[*] + return value !== undefined ? value : key; + }, + remove: function( owner, key ) { + var i, + cache = owner[ this.expando ]; + + if ( cache === undefined ) { + return; + } + + if ( key !== undefined ) { + + // Support array or space separated string of keys + if ( Array.isArray( key ) ) { + + // If key is an array of keys... + // We always set camelCase keys, so remove that. + key = key.map( camelCase ); + } else { + key = camelCase( key ); + + // If a key with the spaces exists, use it. + // Otherwise, create an array by matching non-whitespace + key = key in cache ? + [ key ] : + ( key.match( rnothtmlwhite ) || [] ); + } + + i = key.length; + + while ( i-- ) { + delete cache[ key[ i ] ]; + } + } + + // Remove the expando if there's no more data + if ( key === undefined || jQuery.isEmptyObject( cache ) ) { + + // Support: Chrome <=35 - 45 + // Webkit & Blink performance suffers when deleting properties + // from DOM nodes, so set to undefined instead + // https://bugs.chromium.org/p/chromium/issues/detail?id=378607 (bug restricted) + if ( owner.nodeType ) { + owner[ this.expando ] = undefined; + } else { + delete owner[ this.expando ]; + } + } + }, + hasData: function( owner ) { + var cache = owner[ this.expando ]; + return cache !== undefined && !jQuery.isEmptyObject( cache ); + } + }; + var dataPriv = new Data(); + + var dataUser = new Data(); + + + + // Implementation Summary + // + // 1. Enforce API surface and semantic compatibility with 1.9.x branch + // 2. Improve the module's maintainability by reducing the storage + // paths to a single mechanism. + // 3. Use the same single mechanism to support "private" and "user" data. + // 4. _Never_ expose "private" data to user code (TODO: Drop _data, _removeData) + // 5. Avoid exposing implementation details on user objects (eg. expando properties) + // 6. Provide a clear path for implementation upgrade to WeakMap in 2014 + + var rbrace = /^(?:\{[\w\W]*\}|\[[\w\W]*\])$/, + rmultiDash = /[A-Z]/g; + + function getData( data ) { + if ( data === "true" ) { + return true; + } + + if ( data === "false" ) { + return false; + } + + if ( data === "null" ) { + return null; + } + + // Only convert to a number if it doesn't change the string + if ( data === +data + "" ) { + return +data; + } + + if ( rbrace.test( data ) ) { + return JSON.parse( data ); + } + + return data; + } + + function dataAttr( elem, key, data ) { + var name; + + // If nothing was found internally, try to fetch any + // data from the HTML5 data-* attribute + if ( data === undefined && elem.nodeType === 1 ) { + name = "data-" + key.replace( rmultiDash, "-$&" ).toLowerCase(); + data = elem.getAttribute( name ); + + if ( typeof data === "string" ) { + try { + data = getData( data ); + } catch ( e ) {} + + // Make sure we set the data so it isn't changed later + dataUser.set( elem, key, data ); + } else { + data = undefined; + } + } + return data; + } + + jQuery.extend( { + hasData: function( elem ) { + return dataUser.hasData( elem ) || dataPriv.hasData( elem ); + }, + + data: function( elem, name, data ) { + return dataUser.access( elem, name, data ); + }, + + removeData: function( elem, name ) { + dataUser.remove( elem, name ); + }, + + // TODO: Now that all calls to _data and _removeData have been replaced + // with direct calls to dataPriv methods, these can be deprecated. + _data: function( elem, name, data ) { + return dataPriv.access( elem, name, data ); + }, + + _removeData: function( elem, name ) { + dataPriv.remove( elem, name ); + } + } ); + + jQuery.fn.extend( { + data: function( key, value ) { + var i, name, data, + elem = this[ 0 ], + attrs = elem && elem.attributes; + + // Gets all values + if ( key === undefined ) { + if ( this.length ) { + data = dataUser.get( elem ); + + if ( elem.nodeType === 1 && !dataPriv.get( elem, "hasDataAttrs" ) ) { + i = attrs.length; + while ( i-- ) { + + // Support: IE 11 only + // The attrs elements can be null (#14894) + if ( attrs[ i ] ) { + name = attrs[ i ].name; + if ( name.indexOf( "data-" ) === 0 ) { + name = camelCase( name.slice( 5 ) ); + dataAttr( elem, name, data[ name ] ); + } + } + } + dataPriv.set( elem, "hasDataAttrs", true ); + } + } + + return data; + } + + // Sets multiple values + if ( typeof key === "object" ) { + return this.each( function() { + dataUser.set( this, key ); + } ); + } + + return access( this, function( value ) { + var data; + + // The calling jQuery object (element matches) is not empty + // (and therefore has an element appears at this[ 0 ]) and the + // `value` parameter was not undefined. An empty jQuery object + // will result in `undefined` for elem = this[ 0 ] which will + // throw an exception if an attempt to read a data cache is made. + if ( elem && value === undefined ) { + + // Attempt to get data from the cache + // The key will always be camelCased in Data + data = dataUser.get( elem, key ); + if ( data !== undefined ) { + return data; + } + + // Attempt to "discover" the data in + // HTML5 custom data-* attrs + data = dataAttr( elem, key ); + if ( data !== undefined ) { + return data; + } + + // We tried really hard, but the data doesn't exist. + return; + } + + // Set the data... + this.each( function() { + + // We always store the camelCased key + dataUser.set( this, key, value ); + } ); + }, null, value, arguments.length > 1, null, true ); + }, + + removeData: function( key ) { + return this.each( function() { + dataUser.remove( this, key ); + } ); + } + } ); + + + jQuery.extend( { + queue: function( elem, type, data ) { + var queue; + + if ( elem ) { + type = ( type || "fx" ) + "queue"; + queue = dataPriv.get( elem, type ); + + // Speed up dequeue by getting out quickly if this is just a lookup + if ( data ) { + if ( !queue || Array.isArray( data ) ) { + queue = dataPriv.access( elem, type, jQuery.makeArray( data ) ); + } else { + queue.push( data ); + } + } + return queue || []; + } + }, + + dequeue: function( elem, type ) { + type = type || "fx"; + + var queue = jQuery.queue( elem, type ), + startLength = queue.length, + fn = queue.shift(), + hooks = jQuery._queueHooks( elem, type ), + next = function() { + jQuery.dequeue( elem, type ); + }; + + // If the fx queue is dequeued, always remove the progress sentinel + if ( fn === "inprogress" ) { + fn = queue.shift(); + startLength--; + } + + if ( fn ) { + + // Add a progress sentinel to prevent the fx queue from being + // automatically dequeued + if ( type === "fx" ) { + queue.unshift( "inprogress" ); + } + + // Clear up the last queue stop function + delete hooks.stop; + fn.call( elem, next, hooks ); + } + + if ( !startLength && hooks ) { + hooks.empty.fire(); + } + }, + + // Not public - generate a queueHooks object, or return the current one + _queueHooks: function( elem, type ) { + var key = type + "queueHooks"; + return dataPriv.get( elem, key ) || dataPriv.access( elem, key, { + empty: jQuery.Callbacks( "once memory" ).add( function() { + dataPriv.remove( elem, [ type + "queue", key ] ); + } ) + } ); + } + } ); + + jQuery.fn.extend( { + queue: function( type, data ) { + var setter = 2; + + if ( typeof type !== "string" ) { + data = type; + type = "fx"; + setter--; + } + + if ( arguments.length < setter ) { + return jQuery.queue( this[ 0 ], type ); + } + + return data === undefined ? + this : + this.each( function() { + var queue = jQuery.queue( this, type, data ); + + // Ensure a hooks for this queue + jQuery._queueHooks( this, type ); + + if ( type === "fx" && queue[ 0 ] !== "inprogress" ) { + jQuery.dequeue( this, type ); + } + } ); + }, + dequeue: function( type ) { + return this.each( function() { + jQuery.dequeue( this, type ); + } ); + }, + clearQueue: function( type ) { + return this.queue( type || "fx", [] ); + }, + + // Get a promise resolved when queues of a certain type + // are emptied (fx is the type by default) + promise: function( type, obj ) { + var tmp, + count = 1, + defer = jQuery.Deferred(), + elements = this, + i = this.length, + resolve = function() { + if ( !( --count ) ) { + defer.resolveWith( elements, [ elements ] ); + } + }; + + if ( typeof type !== "string" ) { + obj = type; + type = undefined; + } + type = type || "fx"; + + while ( i-- ) { + tmp = dataPriv.get( elements[ i ], type + "queueHooks" ); + if ( tmp && tmp.empty ) { + count++; + tmp.empty.add( resolve ); + } + } + resolve(); + return defer.promise( obj ); + } + } ); + var pnum = ( /[+-]?(?:\d*\.|)\d+(?:[eE][+-]?\d+|)/ ).source; + + var rcssNum = new RegExp( "^(?:([+-])=|)(" + pnum + ")([a-z%]*)$", "i" ); + + + var cssExpand = [ "Top", "Right", "Bottom", "Left" ]; + + var documentElement = document.documentElement; + + + + var isAttached = function( elem ) { + return jQuery.contains( elem.ownerDocument, elem ); + }, + composed = { composed: true }; + + // Support: IE 9 - 11+, Edge 12 - 18+, iOS 10.0 - 10.2 only + // Check attachment across shadow DOM boundaries when possible (gh-3504) + // Support: iOS 10.0-10.2 only + // Early iOS 10 versions support `attachShadow` but not `getRootNode`, + // leading to errors. We need to check for `getRootNode`. + if ( documentElement.getRootNode ) { + isAttached = function( elem ) { + return jQuery.contains( elem.ownerDocument, elem ) || + elem.getRootNode( composed ) === elem.ownerDocument; + }; + } + var isHiddenWithinTree = function( elem, el ) { + + // isHiddenWithinTree might be called from jQuery#filter function; + // in that case, element will be second argument + elem = el || elem; + + // Inline style trumps all + return elem.style.display === "none" || + elem.style.display === "" && + + // Otherwise, check computed style + // Support: Firefox <=43 - 45 + // Disconnected elements can have computed display: none, so first confirm that elem is + // in the document. + isAttached( elem ) && + + jQuery.css( elem, "display" ) === "none"; + }; + + + + function adjustCSS( elem, prop, valueParts, tween ) { + var adjusted, scale, + maxIterations = 20, + currentValue = tween ? + function() { + return tween.cur(); + } : + function() { + return jQuery.css( elem, prop, "" ); + }, + initial = currentValue(), + unit = valueParts && valueParts[ 3 ] || ( jQuery.cssNumber[ prop ] ? "" : "px" ), + + // Starting value computation is required for potential unit mismatches + initialInUnit = elem.nodeType && + ( jQuery.cssNumber[ prop ] || unit !== "px" && +initial ) && + rcssNum.exec( jQuery.css( elem, prop ) ); + + if ( initialInUnit && initialInUnit[ 3 ] !== unit ) { + + // Support: Firefox <=54 + // Halve the iteration target value to prevent interference from CSS upper bounds (gh-2144) + initial = initial / 2; + + // Trust units reported by jQuery.css + unit = unit || initialInUnit[ 3 ]; + + // Iteratively approximate from a nonzero starting point + initialInUnit = +initial || 1; + + while ( maxIterations-- ) { + + // Evaluate and update our best guess (doubling guesses that zero out). + // Finish if the scale equals or crosses 1 (making the old*new product non-positive). + jQuery.style( elem, prop, initialInUnit + unit ); + if ( ( 1 - scale ) * ( 1 - ( scale = currentValue() / initial || 0.5 ) ) <= 0 ) { + maxIterations = 0; + } + initialInUnit = initialInUnit / scale; + + } + + initialInUnit = initialInUnit * 2; + jQuery.style( elem, prop, initialInUnit + unit ); + + // Make sure we update the tween properties later on + valueParts = valueParts || []; + } + + if ( valueParts ) { + initialInUnit = +initialInUnit || +initial || 0; + + // Apply relative offset (+=/-=) if specified + adjusted = valueParts[ 1 ] ? + initialInUnit + ( valueParts[ 1 ] + 1 ) * valueParts[ 2 ] : + +valueParts[ 2 ]; + if ( tween ) { + tween.unit = unit; + tween.start = initialInUnit; + tween.end = adjusted; + } + } + return adjusted; + } + + + var defaultDisplayMap = {}; + + function getDefaultDisplay( elem ) { + var temp, + doc = elem.ownerDocument, + nodeName = elem.nodeName, + display = defaultDisplayMap[ nodeName ]; + + if ( display ) { + return display; + } + + temp = doc.body.appendChild( doc.createElement( nodeName ) ); + display = jQuery.css( temp, "display" ); + + temp.parentNode.removeChild( temp ); + + if ( display === "none" ) { + display = "block"; + } + defaultDisplayMap[ nodeName ] = display; + + return display; + } + + function showHide( elements, show ) { + var display, elem, + values = [], + index = 0, + length = elements.length; + + // Determine new display value for elements that need to change + for ( ; index < length; index++ ) { + elem = elements[ index ]; + if ( !elem.style ) { + continue; + } + + display = elem.style.display; + if ( show ) { + + // Since we force visibility upon cascade-hidden elements, an immediate (and slow) + // check is required in this first loop unless we have a nonempty display value (either + // inline or about-to-be-restored) + if ( display === "none" ) { + values[ index ] = dataPriv.get( elem, "display" ) || null; + if ( !values[ index ] ) { + elem.style.display = ""; + } + } + if ( elem.style.display === "" && isHiddenWithinTree( elem ) ) { + values[ index ] = getDefaultDisplay( elem ); + } + } else { + if ( display !== "none" ) { + values[ index ] = "none"; + + // Remember what we're overwriting + dataPriv.set( elem, "display", display ); + } + } + } + + // Set the display of the elements in a second loop to avoid constant reflow + for ( index = 0; index < length; index++ ) { + if ( values[ index ] != null ) { + elements[ index ].style.display = values[ index ]; + } + } + + return elements; + } + + jQuery.fn.extend( { + show: function() { + return showHide( this, true ); + }, + hide: function() { + return showHide( this ); + }, + toggle: function( state ) { + if ( typeof state === "boolean" ) { + return state ? this.show() : this.hide(); + } + + return this.each( function() { + if ( isHiddenWithinTree( this ) ) { + jQuery( this ).show(); + } else { + jQuery( this ).hide(); + } + } ); + } + } ); + var rcheckableType = ( /^(?:checkbox|radio)$/i ); + + var rtagName = ( /<([a-z][^\/\0>\x20\t\r\n\f]*)/i ); + + var rscriptType = ( /^$|^module$|\/(?:java|ecma)script/i ); + + + + ( function() { + var fragment = document.createDocumentFragment(), + div = fragment.appendChild( document.createElement( "div" ) ), + input = document.createElement( "input" ); + + // Support: Android 4.0 - 4.3 only + // Check state lost if the name is set (#11217) + // Support: Windows Web Apps (WWA) + // `name` and `type` must use .setAttribute for WWA (#14901) + input.setAttribute( "type", "radio" ); + input.setAttribute( "checked", "checked" ); + input.setAttribute( "name", "t" ); + + div.appendChild( input ); + + // Support: Android <=4.1 only + // Older WebKit doesn't clone checked state correctly in fragments + support.checkClone = div.cloneNode( true ).cloneNode( true ).lastChild.checked; + + // Support: IE <=11 only + // Make sure textarea (and checkbox) defaultValue is properly cloned + div.innerHTML = "<textarea>x</textarea>"; + support.noCloneChecked = !!div.cloneNode( true ).lastChild.defaultValue; + + // Support: IE <=9 only + // IE <=9 replaces <option> tags with their contents when inserted outside of + // the select element. + div.innerHTML = "<option></option>"; + support.option = !!div.lastChild; + } )(); + + + // We have to close these tags to support XHTML (#13200) + var wrapMap = { + + // XHTML parsers do not magically insert elements in the + // same way that tag soup parsers do. So we cannot shorten + // this by omitting <tbody> or other required elements. + thead: [ 1, "<table>", "</table>" ], + col: [ 2, "<table><colgroup>", "</colgroup></table>" ], + tr: [ 2, "<table><tbody>", "</tbody></table>" ], + td: [ 3, "<table><tbody><tr>", "</tr></tbody></table>" ], + + _default: [ 0, "", "" ] + }; + + wrapMap.tbody = wrapMap.tfoot = wrapMap.colgroup = wrapMap.caption = wrapMap.thead; + wrapMap.th = wrapMap.td; + + // Support: IE <=9 only + if ( !support.option ) { + wrapMap.optgroup = wrapMap.option = [ 1, "<select multiple='multiple'>", "</select>" ]; + } + + + function getAll( context, tag ) { + + // Support: IE <=9 - 11 only + // Use typeof to avoid zero-argument method invocation on host objects (#15151) + var ret; + + if ( typeof context.getElementsByTagName !== "undefined" ) { + ret = context.getElementsByTagName( tag || "*" ); + + } else if ( typeof context.querySelectorAll !== "undefined" ) { + ret = context.querySelectorAll( tag || "*" ); + + } else { + ret = []; + } + + if ( tag === undefined || tag && nodeName( context, tag ) ) { + return jQuery.merge( [ context ], ret ); + } + + return ret; + } + + + // Mark scripts as having already been evaluated + function setGlobalEval( elems, refElements ) { + var i = 0, + l = elems.length; + + for ( ; i < l; i++ ) { + dataPriv.set( + elems[ i ], + "globalEval", + !refElements || dataPriv.get( refElements[ i ], "globalEval" ) + ); + } + } + + + var rhtml = /<|&#?\w+;/; + + function buildFragment( elems, context, scripts, selection, ignored ) { + var elem, tmp, tag, wrap, attached, j, + fragment = context.createDocumentFragment(), + nodes = [], + i = 0, + l = elems.length; + + for ( ; i < l; i++ ) { + elem = elems[ i ]; + + if ( elem || elem === 0 ) { + + // Add nodes directly + if ( toType( elem ) === "object" ) { + + // Support: Android <=4.0 only, PhantomJS 1 only + // push.apply(_, arraylike) throws on ancient WebKit + jQuery.merge( nodes, elem.nodeType ? [ elem ] : elem ); + + // Convert non-html into a text node + } else if ( !rhtml.test( elem ) ) { + nodes.push( context.createTextNode( elem ) ); + + // Convert html into DOM nodes + } else { + tmp = tmp || fragment.appendChild( context.createElement( "div" ) ); + + // Deserialize a standard representation + tag = ( rtagName.exec( elem ) || [ "", "" ] )[ 1 ].toLowerCase(); + wrap = wrapMap[ tag ] || wrapMap._default; + tmp.innerHTML = wrap[ 1 ] + jQuery.htmlPrefilter( elem ) + wrap[ 2 ]; + + // Descend through wrappers to the right content + j = wrap[ 0 ]; + while ( j-- ) { + tmp = tmp.lastChild; + } + + // Support: Android <=4.0 only, PhantomJS 1 only + // push.apply(_, arraylike) throws on ancient WebKit + jQuery.merge( nodes, tmp.childNodes ); + + // Remember the top-level container + tmp = fragment.firstChild; + + // Ensure the created nodes are orphaned (#12392) + tmp.textContent = ""; + } + } + } + + // Remove wrapper from fragment + fragment.textContent = ""; + + i = 0; + while ( ( elem = nodes[ i++ ] ) ) { + + // Skip elements already in the context collection (trac-4087) + if ( selection && jQuery.inArray( elem, selection ) > -1 ) { + if ( ignored ) { + ignored.push( elem ); + } + continue; + } + + attached = isAttached( elem ); + + // Append to fragment + tmp = getAll( fragment.appendChild( elem ), "script" ); + + // Preserve script evaluation history + if ( attached ) { + setGlobalEval( tmp ); + } + + // Capture executables + if ( scripts ) { + j = 0; + while ( ( elem = tmp[ j++ ] ) ) { + if ( rscriptType.test( elem.type || "" ) ) { + scripts.push( elem ); + } + } + } + } + + return fragment; + } + + + var rtypenamespace = /^([^.]*)(?:\.(.+)|)/; + + function returnTrue() { + return true; + } + + function returnFalse() { + return false; + } + + // Support: IE <=9 - 11+ + // focus() and blur() are asynchronous, except when they are no-op. + // So expect focus to be synchronous when the element is already active, + // and blur to be synchronous when the element is not already active. + // (focus and blur are always synchronous in other supported browsers, + // this just defines when we can count on it). + function expectSync( elem, type ) { + return ( elem === safeActiveElement() ) === ( type === "focus" ); + } + + // Support: IE <=9 only + // Accessing document.activeElement can throw unexpectedly + // https://bugs.jquery.com/ticket/13393 + function safeActiveElement() { + try { + return document.activeElement; + } catch ( err ) { } + } + + function on( elem, types, selector, data, fn, one ) { + var origFn, type; + + // Types can be a map of types/handlers + if ( typeof types === "object" ) { + + // ( types-Object, selector, data ) + if ( typeof selector !== "string" ) { + + // ( types-Object, data ) + data = data || selector; + selector = undefined; + } + for ( type in types ) { + on( elem, type, selector, data, types[ type ], one ); + } + return elem; + } + + if ( data == null && fn == null ) { + + // ( types, fn ) + fn = selector; + data = selector = undefined; + } else if ( fn == null ) { + if ( typeof selector === "string" ) { + + // ( types, selector, fn ) + fn = data; + data = undefined; + } else { + + // ( types, data, fn ) + fn = data; + data = selector; + selector = undefined; + } + } + if ( fn === false ) { + fn = returnFalse; + } else if ( !fn ) { + return elem; + } + + if ( one === 1 ) { + origFn = fn; + fn = function( event ) { + + // Can use an empty set, since event contains the info + jQuery().off( event ); + return origFn.apply( this, arguments ); + }; + + // Use same guid so caller can remove using origFn + fn.guid = origFn.guid || ( origFn.guid = jQuery.guid++ ); + } + return elem.each( function() { + jQuery.event.add( this, types, fn, data, selector ); + } ); + } + + /* + * Helper functions for managing events -- not part of the public interface. + * Props to Dean Edwards' addEvent library for many of the ideas. + */ + jQuery.event = { + + global: {}, + + add: function( elem, types, handler, data, selector ) { + + var handleObjIn, eventHandle, tmp, + events, t, handleObj, + special, handlers, type, namespaces, origType, + elemData = dataPriv.get( elem ); + + // Only attach events to objects that accept data + if ( !acceptData( elem ) ) { + return; + } + + // Caller can pass in an object of custom data in lieu of the handler + if ( handler.handler ) { + handleObjIn = handler; + handler = handleObjIn.handler; + selector = handleObjIn.selector; + } + + // Ensure that invalid selectors throw exceptions at attach time + // Evaluate against documentElement in case elem is a non-element node (e.g., document) + if ( selector ) { + jQuery.find.matchesSelector( documentElement, selector ); + } + + // Make sure that the handler has a unique ID, used to find/remove it later + if ( !handler.guid ) { + handler.guid = jQuery.guid++; + } + + // Init the element's event structure and main handler, if this is the first + if ( !( events = elemData.events ) ) { + events = elemData.events = Object.create( null ); + } + if ( !( eventHandle = elemData.handle ) ) { + eventHandle = elemData.handle = function( e ) { + + // Discard the second event of a jQuery.event.trigger() and + // when an event is called after a page has unloaded + return typeof jQuery !== "undefined" && jQuery.event.triggered !== e.type ? + jQuery.event.dispatch.apply( elem, arguments ) : undefined; + }; + } + + // Handle multiple events separated by a space + types = ( types || "" ).match( rnothtmlwhite ) || [ "" ]; + t = types.length; + while ( t-- ) { + tmp = rtypenamespace.exec( types[ t ] ) || []; + type = origType = tmp[ 1 ]; + namespaces = ( tmp[ 2 ] || "" ).split( "." ).sort(); + + // There *must* be a type, no attaching namespace-only handlers + if ( !type ) { + continue; + } + + // If event changes its type, use the special event handlers for the changed type + special = jQuery.event.special[ type ] || {}; + + // If selector defined, determine special event api type, otherwise given type + type = ( selector ? special.delegateType : special.bindType ) || type; + + // Update special based on newly reset type + special = jQuery.event.special[ type ] || {}; + + // handleObj is passed to all event handlers + handleObj = jQuery.extend( { + type: type, + origType: origType, + data: data, + handler: handler, + guid: handler.guid, + selector: selector, + needsContext: selector && jQuery.expr.match.needsContext.test( selector ), + namespace: namespaces.join( "." ) + }, handleObjIn ); + + // Init the event handler queue if we're the first + if ( !( handlers = events[ type ] ) ) { + handlers = events[ type ] = []; + handlers.delegateCount = 0; + + // Only use addEventListener if the special events handler returns false + if ( !special.setup || + special.setup.call( elem, data, namespaces, eventHandle ) === false ) { + + if ( elem.addEventListener ) { + elem.addEventListener( type, eventHandle ); + } + } + } + + if ( special.add ) { + special.add.call( elem, handleObj ); + + if ( !handleObj.handler.guid ) { + handleObj.handler.guid = handler.guid; + } + } + + // Add to the element's handler list, delegates in front + if ( selector ) { + handlers.splice( handlers.delegateCount++, 0, handleObj ); + } else { + handlers.push( handleObj ); + } + + // Keep track of which events have ever been used, for event optimization + jQuery.event.global[ type ] = true; + } + + }, + + // Detach an event or set of events from an element + remove: function( elem, types, handler, selector, mappedTypes ) { + + var j, origCount, tmp, + events, t, handleObj, + special, handlers, type, namespaces, origType, + elemData = dataPriv.hasData( elem ) && dataPriv.get( elem ); + + if ( !elemData || !( events = elemData.events ) ) { + return; + } + + // Once for each type.namespace in types; type may be omitted + types = ( types || "" ).match( rnothtmlwhite ) || [ "" ]; + t = types.length; + while ( t-- ) { + tmp = rtypenamespace.exec( types[ t ] ) || []; + type = origType = tmp[ 1 ]; + namespaces = ( tmp[ 2 ] || "" ).split( "." ).sort(); + + // Unbind all events (on this namespace, if provided) for the element + if ( !type ) { + for ( type in events ) { + jQuery.event.remove( elem, type + types[ t ], handler, selector, true ); + } + continue; + } + + special = jQuery.event.special[ type ] || {}; + type = ( selector ? special.delegateType : special.bindType ) || type; + handlers = events[ type ] || []; + tmp = tmp[ 2 ] && + new RegExp( "(^|\\.)" + namespaces.join( "\\.(?:.*\\.|)" ) + "(\\.|$)" ); + + // Remove matching events + origCount = j = handlers.length; + while ( j-- ) { + handleObj = handlers[ j ]; + + if ( ( mappedTypes || origType === handleObj.origType ) && + ( !handler || handler.guid === handleObj.guid ) && + ( !tmp || tmp.test( handleObj.namespace ) ) && + ( !selector || selector === handleObj.selector || + selector === "**" && handleObj.selector ) ) { + handlers.splice( j, 1 ); + + if ( handleObj.selector ) { + handlers.delegateCount--; + } + if ( special.remove ) { + special.remove.call( elem, handleObj ); + } + } + } + + // Remove generic event handler if we removed something and no more handlers exist + // (avoids potential for endless recursion during removal of special event handlers) + if ( origCount && !handlers.length ) { + if ( !special.teardown || + special.teardown.call( elem, namespaces, elemData.handle ) === false ) { + + jQuery.removeEvent( elem, type, elemData.handle ); + } + + delete events[ type ]; + } + } + + // Remove data and the expando if it's no longer used + if ( jQuery.isEmptyObject( events ) ) { + dataPriv.remove( elem, "handle events" ); + } + }, + + dispatch: function( nativeEvent ) { + + var i, j, ret, matched, handleObj, handlerQueue, + args = new Array( arguments.length ), + + // Make a writable jQuery.Event from the native event object + event = jQuery.event.fix( nativeEvent ), + + handlers = ( + dataPriv.get( this, "events" ) || Object.create( null ) + )[ event.type ] || [], + special = jQuery.event.special[ event.type ] || {}; + + // Use the fix-ed jQuery.Event rather than the (read-only) native event + args[ 0 ] = event; + + for ( i = 1; i < arguments.length; i++ ) { + args[ i ] = arguments[ i ]; + } + + event.delegateTarget = this; + + // Call the preDispatch hook for the mapped type, and let it bail if desired + if ( special.preDispatch && special.preDispatch.call( this, event ) === false ) { + return; + } + + // Determine handlers + handlerQueue = jQuery.event.handlers.call( this, event, handlers ); + + // Run delegates first; they may want to stop propagation beneath us + i = 0; + while ( ( matched = handlerQueue[ i++ ] ) && !event.isPropagationStopped() ) { + event.currentTarget = matched.elem; + + j = 0; + while ( ( handleObj = matched.handlers[ j++ ] ) && + !event.isImmediatePropagationStopped() ) { + + // If the event is namespaced, then each handler is only invoked if it is + // specially universal or its namespaces are a superset of the event's. + if ( !event.rnamespace || handleObj.namespace === false || + event.rnamespace.test( handleObj.namespace ) ) { + + event.handleObj = handleObj; + event.data = handleObj.data; + + ret = ( ( jQuery.event.special[ handleObj.origType ] || {} ).handle || + handleObj.handler ).apply( matched.elem, args ); + + if ( ret !== undefined ) { + if ( ( event.result = ret ) === false ) { + event.preventDefault(); + event.stopPropagation(); + } + } + } + } + } + + // Call the postDispatch hook for the mapped type + if ( special.postDispatch ) { + special.postDispatch.call( this, event ); + } + + return event.result; + }, + + handlers: function( event, handlers ) { + var i, handleObj, sel, matchedHandlers, matchedSelectors, + handlerQueue = [], + delegateCount = handlers.delegateCount, + cur = event.target; + + // Find delegate handlers + if ( delegateCount && + + // Support: IE <=9 + // Black-hole SVG <use> instance trees (trac-13180) + cur.nodeType && + + // Support: Firefox <=42 + // Suppress spec-violating clicks indicating a non-primary pointer button (trac-3861) + // https://www.w3.org/TR/DOM-Level-3-Events/#event-type-click + // Support: IE 11 only + // ...but not arrow key "clicks" of radio inputs, which can have `button` -1 (gh-2343) + !( event.type === "click" && event.button >= 1 ) ) { + + for ( ; cur !== this; cur = cur.parentNode || this ) { + + // Don't check non-elements (#13208) + // Don't process clicks on disabled elements (#6911, #8165, #11382, #11764) + if ( cur.nodeType === 1 && !( event.type === "click" && cur.disabled === true ) ) { + matchedHandlers = []; + matchedSelectors = {}; + for ( i = 0; i < delegateCount; i++ ) { + handleObj = handlers[ i ]; + + // Don't conflict with Object.prototype properties (#13203) + sel = handleObj.selector + " "; + + if ( matchedSelectors[ sel ] === undefined ) { + matchedSelectors[ sel ] = handleObj.needsContext ? + jQuery( sel, this ).index( cur ) > -1 : + jQuery.find( sel, this, null, [ cur ] ).length; + } + if ( matchedSelectors[ sel ] ) { + matchedHandlers.push( handleObj ); + } + } + if ( matchedHandlers.length ) { + handlerQueue.push( { elem: cur, handlers: matchedHandlers } ); + } + } + } + } + + // Add the remaining (directly-bound) handlers + cur = this; + if ( delegateCount < handlers.length ) { + handlerQueue.push( { elem: cur, handlers: handlers.slice( delegateCount ) } ); + } + + return handlerQueue; + }, + + addProp: function( name, hook ) { + Object.defineProperty( jQuery.Event.prototype, name, { + enumerable: true, + configurable: true, + + get: isFunction( hook ) ? + function() { + if ( this.originalEvent ) { + return hook( this.originalEvent ); + } + } : + function() { + if ( this.originalEvent ) { + return this.originalEvent[ name ]; + } + }, + + set: function( value ) { + Object.defineProperty( this, name, { + enumerable: true, + configurable: true, + writable: true, + value: value + } ); + } + } ); + }, + + fix: function( originalEvent ) { + return originalEvent[ jQuery.expando ] ? + originalEvent : + new jQuery.Event( originalEvent ); + }, + + special: { + load: { + + // Prevent triggered image.load events from bubbling to window.load + noBubble: true + }, + click: { + + // Utilize native event to ensure correct state for checkable inputs + setup: function( data ) { + + // For mutual compressibility with _default, replace `this` access with a local var. + // `|| data` is dead code meant only to preserve the variable through minification. + var el = this || data; + + // Claim the first handler + if ( rcheckableType.test( el.type ) && + el.click && nodeName( el, "input" ) ) { + + // dataPriv.set( el, "click", ... ) + leverageNative( el, "click", returnTrue ); + } + + // Return false to allow normal processing in the caller + return false; + }, + trigger: function( data ) { + + // For mutual compressibility with _default, replace `this` access with a local var. + // `|| data` is dead code meant only to preserve the variable through minification. + var el = this || data; + + // Force setup before triggering a click + if ( rcheckableType.test( el.type ) && + el.click && nodeName( el, "input" ) ) { + + leverageNative( el, "click" ); + } + + // Return non-false to allow normal event-path propagation + return true; + }, + + // For cross-browser consistency, suppress native .click() on links + // Also prevent it if we're currently inside a leveraged native-event stack + _default: function( event ) { + var target = event.target; + return rcheckableType.test( target.type ) && + target.click && nodeName( target, "input" ) && + dataPriv.get( target, "click" ) || + nodeName( target, "a" ); + } + }, + + beforeunload: { + postDispatch: function( event ) { + + // Support: Firefox 20+ + // Firefox doesn't alert if the returnValue field is not set. + if ( event.result !== undefined && event.originalEvent ) { + event.originalEvent.returnValue = event.result; + } + } + } + } + }; + + // Ensure the presence of an event listener that handles manually-triggered + // synthetic events by interrupting progress until reinvoked in response to + // *native* events that it fires directly, ensuring that state changes have + // already occurred before other listeners are invoked. + function leverageNative( el, type, expectSync ) { + + // Missing expectSync indicates a trigger call, which must force setup through jQuery.event.add + if ( !expectSync ) { + if ( dataPriv.get( el, type ) === undefined ) { + jQuery.event.add( el, type, returnTrue ); + } + return; + } + + // Register the controller as a special universal handler for all event namespaces + dataPriv.set( el, type, false ); + jQuery.event.add( el, type, { + namespace: false, + handler: function( event ) { + var notAsync, result, + saved = dataPriv.get( this, type ); + + if ( ( event.isTrigger & 1 ) && this[ type ] ) { + + // Interrupt processing of the outer synthetic .trigger()ed event + // Saved data should be false in such cases, but might be a leftover capture object + // from an async native handler (gh-4350) + if ( !saved.length ) { + + // Store arguments for use when handling the inner native event + // There will always be at least one argument (an event object), so this array + // will not be confused with a leftover capture object. + saved = slice.call( arguments ); + dataPriv.set( this, type, saved ); + + // Trigger the native event and capture its result + // Support: IE <=9 - 11+ + // focus() and blur() are asynchronous + notAsync = expectSync( this, type ); + this[ type ](); + result = dataPriv.get( this, type ); + if ( saved !== result || notAsync ) { + dataPriv.set( this, type, false ); + } else { + result = {}; + } + if ( saved !== result ) { + + // Cancel the outer synthetic event + event.stopImmediatePropagation(); + event.preventDefault(); + + // Support: Chrome 86+ + // In Chrome, if an element having a focusout handler is blurred by + // clicking outside of it, it invokes the handler synchronously. If + // that handler calls `.remove()` on the element, the data is cleared, + // leaving `result` undefined. We need to guard against this. + return result && result.value; + } + + // If this is an inner synthetic event for an event with a bubbling surrogate + // (focus or blur), assume that the surrogate already propagated from triggering the + // native event and prevent that from happening again here. + // This technically gets the ordering wrong w.r.t. to `.trigger()` (in which the + // bubbling surrogate propagates *after* the non-bubbling base), but that seems + // less bad than duplication. + } else if ( ( jQuery.event.special[ type ] || {} ).delegateType ) { + event.stopPropagation(); + } + + // If this is a native event triggered above, everything is now in order + // Fire an inner synthetic event with the original arguments + } else if ( saved.length ) { + + // ...and capture the result + dataPriv.set( this, type, { + value: jQuery.event.trigger( + + // Support: IE <=9 - 11+ + // Extend with the prototype to reset the above stopImmediatePropagation() + jQuery.extend( saved[ 0 ], jQuery.Event.prototype ), + saved.slice( 1 ), + this + ) + } ); + + // Abort handling of the native event + event.stopImmediatePropagation(); + } + } + } ); + } + + jQuery.removeEvent = function( elem, type, handle ) { + + // This "if" is needed for plain objects + if ( elem.removeEventListener ) { + elem.removeEventListener( type, handle ); + } + }; + + jQuery.Event = function( src, props ) { + + // Allow instantiation without the 'new' keyword + if ( !( this instanceof jQuery.Event ) ) { + return new jQuery.Event( src, props ); + } + + // Event object + if ( src && src.type ) { + this.originalEvent = src; + this.type = src.type; + + // Events bubbling up the document may have been marked as prevented + // by a handler lower down the tree; reflect the correct value. + this.isDefaultPrevented = src.defaultPrevented || + src.defaultPrevented === undefined && + + // Support: Android <=2.3 only + src.returnValue === false ? + returnTrue : + returnFalse; + + // Create target properties + // Support: Safari <=6 - 7 only + // Target should not be a text node (#504, #13143) + this.target = ( src.target && src.target.nodeType === 3 ) ? + src.target.parentNode : + src.target; + + this.currentTarget = src.currentTarget; + this.relatedTarget = src.relatedTarget; + + // Event type + } else { + this.type = src; + } + + // Put explicitly provided properties onto the event object + if ( props ) { + jQuery.extend( this, props ); + } + + // Create a timestamp if incoming event doesn't have one + this.timeStamp = src && src.timeStamp || Date.now(); + + // Mark it as fixed + this[ jQuery.expando ] = true; + }; + + // jQuery.Event is based on DOM3 Events as specified by the ECMAScript Language Binding + // https://www.w3.org/TR/2003/WD-DOM-Level-3-Events-20030331/ecma-script-binding.html + jQuery.Event.prototype = { + constructor: jQuery.Event, + isDefaultPrevented: returnFalse, + isPropagationStopped: returnFalse, + isImmediatePropagationStopped: returnFalse, + isSimulated: false, + + preventDefault: function() { + var e = this.originalEvent; + + this.isDefaultPrevented = returnTrue; + + if ( e && !this.isSimulated ) { + e.preventDefault(); + } + }, + stopPropagation: function() { + var e = this.originalEvent; + + this.isPropagationStopped = returnTrue; + + if ( e && !this.isSimulated ) { + e.stopPropagation(); + } + }, + stopImmediatePropagation: function() { + var e = this.originalEvent; + + this.isImmediatePropagationStopped = returnTrue; + + if ( e && !this.isSimulated ) { + e.stopImmediatePropagation(); + } + + this.stopPropagation(); + } + }; + + // Includes all common event props including KeyEvent and MouseEvent specific props + jQuery.each( { + altKey: true, + bubbles: true, + cancelable: true, + changedTouches: true, + ctrlKey: true, + detail: true, + eventPhase: true, + metaKey: true, + pageX: true, + pageY: true, + shiftKey: true, + view: true, + "char": true, + code: true, + charCode: true, + key: true, + keyCode: true, + button: true, + buttons: true, + clientX: true, + clientY: true, + offsetX: true, + offsetY: true, + pointerId: true, + pointerType: true, + screenX: true, + screenY: true, + targetTouches: true, + toElement: true, + touches: true, + which: true + }, jQuery.event.addProp ); + + jQuery.each( { focus: "focusin", blur: "focusout" }, function( type, delegateType ) { + jQuery.event.special[ type ] = { + + // Utilize native event if possible so blur/focus sequence is correct + setup: function() { + + // Claim the first handler + // dataPriv.set( this, "focus", ... ) + // dataPriv.set( this, "blur", ... ) + leverageNative( this, type, expectSync ); + + // Return false to allow normal processing in the caller + return false; + }, + trigger: function() { + + // Force setup before trigger + leverageNative( this, type ); + + // Return non-false to allow normal event-path propagation + return true; + }, + + // Suppress native focus or blur as it's already being fired + // in leverageNative. + _default: function() { + return true; + }, + + delegateType: delegateType + }; + } ); + + // Create mouseenter/leave events using mouseover/out and event-time checks + // so that event delegation works in jQuery. + // Do the same for pointerenter/pointerleave and pointerover/pointerout + // + // Support: Safari 7 only + // Safari sends mouseenter too often; see: + // https://bugs.chromium.org/p/chromium/issues/detail?id=470258 + // for the description of the bug (it existed in older Chrome versions as well). + jQuery.each( { + mouseenter: "mouseover", + mouseleave: "mouseout", + pointerenter: "pointerover", + pointerleave: "pointerout" + }, function( orig, fix ) { + jQuery.event.special[ orig ] = { + delegateType: fix, + bindType: fix, + + handle: function( event ) { + var ret, + target = this, + related = event.relatedTarget, + handleObj = event.handleObj; + + // For mouseenter/leave call the handler if related is outside the target. + // NB: No relatedTarget if the mouse left/entered the browser window + if ( !related || ( related !== target && !jQuery.contains( target, related ) ) ) { + event.type = handleObj.origType; + ret = handleObj.handler.apply( this, arguments ); + event.type = fix; + } + return ret; + } + }; + } ); + + jQuery.fn.extend( { + + on: function( types, selector, data, fn ) { + return on( this, types, selector, data, fn ); + }, + one: function( types, selector, data, fn ) { + return on( this, types, selector, data, fn, 1 ); + }, + off: function( types, selector, fn ) { + var handleObj, type; + if ( types && types.preventDefault && types.handleObj ) { + + // ( event ) dispatched jQuery.Event + handleObj = types.handleObj; + jQuery( types.delegateTarget ).off( + handleObj.namespace ? + handleObj.origType + "." + handleObj.namespace : + handleObj.origType, + handleObj.selector, + handleObj.handler + ); + return this; + } + if ( typeof types === "object" ) { + + // ( types-object [, selector] ) + for ( type in types ) { + this.off( type, selector, types[ type ] ); + } + return this; + } + if ( selector === false || typeof selector === "function" ) { + + // ( types [, fn] ) + fn = selector; + selector = undefined; + } + if ( fn === false ) { + fn = returnFalse; + } + return this.each( function() { + jQuery.event.remove( this, types, fn, selector ); + } ); + } + } ); + + + var + + // Support: IE <=10 - 11, Edge 12 - 13 only + // In IE/Edge using regex groups here causes severe slowdowns. + // See https://connect.microsoft.com/IE/feedback/details/1736512/ + rnoInnerhtml = /<script|<style|<link/i, + + // checked="checked" or checked + rchecked = /checked\s*(?:[^=]|=\s*.checked.)/i, + rcleanScript = /^\s*<!(?:\[CDATA\[|--)|(?:\]\]|--)>\s*$/g; + + // Prefer a tbody over its parent table for containing new rows + function manipulationTarget( elem, content ) { + if ( nodeName( elem, "table" ) && + nodeName( content.nodeType !== 11 ? content : content.firstChild, "tr" ) ) { + + return jQuery( elem ).children( "tbody" )[ 0 ] || elem; + } + + return elem; + } + + // Replace/restore the type attribute of script elements for safe DOM manipulation + function disableScript( elem ) { + elem.type = ( elem.getAttribute( "type" ) !== null ) + "/" + elem.type; + return elem; + } + function restoreScript( elem ) { + if ( ( elem.type || "" ).slice( 0, 5 ) === "true/" ) { + elem.type = elem.type.slice( 5 ); + } else { + elem.removeAttribute( "type" ); + } + + return elem; + } + + function cloneCopyEvent( src, dest ) { + var i, l, type, pdataOld, udataOld, udataCur, events; + + if ( dest.nodeType !== 1 ) { + return; + } + + // 1. Copy private data: events, handlers, etc. + if ( dataPriv.hasData( src ) ) { + pdataOld = dataPriv.get( src ); + events = pdataOld.events; + + if ( events ) { + dataPriv.remove( dest, "handle events" ); + + for ( type in events ) { + for ( i = 0, l = events[ type ].length; i < l; i++ ) { + jQuery.event.add( dest, type, events[ type ][ i ] ); + } + } + } + } + + // 2. Copy user data + if ( dataUser.hasData( src ) ) { + udataOld = dataUser.access( src ); + udataCur = jQuery.extend( {}, udataOld ); + + dataUser.set( dest, udataCur ); + } + } + + // Fix IE bugs, see support tests + function fixInput( src, dest ) { + var nodeName = dest.nodeName.toLowerCase(); + + // Fails to persist the checked state of a cloned checkbox or radio button. + if ( nodeName === "input" && rcheckableType.test( src.type ) ) { + dest.checked = src.checked; + + // Fails to return the selected option to the default selected state when cloning options + } else if ( nodeName === "input" || nodeName === "textarea" ) { + dest.defaultValue = src.defaultValue; + } + } + + function domManip( collection, args, callback, ignored ) { + + // Flatten any nested arrays + args = flat( args ); + + var fragment, first, scripts, hasScripts, node, doc, + i = 0, + l = collection.length, + iNoClone = l - 1, + value = args[ 0 ], + valueIsFunction = isFunction( value ); + + // We can't cloneNode fragments that contain checked, in WebKit + if ( valueIsFunction || + ( l > 1 && typeof value === "string" && + !support.checkClone && rchecked.test( value ) ) ) { + return collection.each( function( index ) { + var self = collection.eq( index ); + if ( valueIsFunction ) { + args[ 0 ] = value.call( this, index, self.html() ); + } + domManip( self, args, callback, ignored ); + } ); + } + + if ( l ) { + fragment = buildFragment( args, collection[ 0 ].ownerDocument, false, collection, ignored ); + first = fragment.firstChild; + + if ( fragment.childNodes.length === 1 ) { + fragment = first; + } + + // Require either new content or an interest in ignored elements to invoke the callback + if ( first || ignored ) { + scripts = jQuery.map( getAll( fragment, "script" ), disableScript ); + hasScripts = scripts.length; + + // Use the original fragment for the last item + // instead of the first because it can end up + // being emptied incorrectly in certain situations (#8070). + for ( ; i < l; i++ ) { + node = fragment; + + if ( i !== iNoClone ) { + node = jQuery.clone( node, true, true ); + + // Keep references to cloned scripts for later restoration + if ( hasScripts ) { + + // Support: Android <=4.0 only, PhantomJS 1 only + // push.apply(_, arraylike) throws on ancient WebKit + jQuery.merge( scripts, getAll( node, "script" ) ); + } + } + + callback.call( collection[ i ], node, i ); + } + + if ( hasScripts ) { + doc = scripts[ scripts.length - 1 ].ownerDocument; + + // Reenable scripts + jQuery.map( scripts, restoreScript ); + + // Evaluate executable scripts on first document insertion + for ( i = 0; i < hasScripts; i++ ) { + node = scripts[ i ]; + if ( rscriptType.test( node.type || "" ) && + !dataPriv.access( node, "globalEval" ) && + jQuery.contains( doc, node ) ) { + + if ( node.src && ( node.type || "" ).toLowerCase() !== "module" ) { + + // Optional AJAX dependency, but won't run scripts if not present + if ( jQuery._evalUrl && !node.noModule ) { + jQuery._evalUrl( node.src, { + nonce: node.nonce || node.getAttribute( "nonce" ) + }, doc ); + } + } else { + DOMEval( node.textContent.replace( rcleanScript, "" ), node, doc ); + } + } + } + } + } + } + + return collection; + } + + function remove( elem, selector, keepData ) { + var node, + nodes = selector ? jQuery.filter( selector, elem ) : elem, + i = 0; + + for ( ; ( node = nodes[ i ] ) != null; i++ ) { + if ( !keepData && node.nodeType === 1 ) { + jQuery.cleanData( getAll( node ) ); + } + + if ( node.parentNode ) { + if ( keepData && isAttached( node ) ) { + setGlobalEval( getAll( node, "script" ) ); + } + node.parentNode.removeChild( node ); + } + } + + return elem; + } + + jQuery.extend( { + htmlPrefilter: function( html ) { + return html; + }, + + clone: function( elem, dataAndEvents, deepDataAndEvents ) { + var i, l, srcElements, destElements, + clone = elem.cloneNode( true ), + inPage = isAttached( elem ); + + // Fix IE cloning issues + if ( !support.noCloneChecked && ( elem.nodeType === 1 || elem.nodeType === 11 ) && + !jQuery.isXMLDoc( elem ) ) { + + // We eschew Sizzle here for performance reasons: https://jsperf.com/getall-vs-sizzle/2 + destElements = getAll( clone ); + srcElements = getAll( elem ); + + for ( i = 0, l = srcElements.length; i < l; i++ ) { + fixInput( srcElements[ i ], destElements[ i ] ); + } + } + + // Copy the events from the original to the clone + if ( dataAndEvents ) { + if ( deepDataAndEvents ) { + srcElements = srcElements || getAll( elem ); + destElements = destElements || getAll( clone ); + + for ( i = 0, l = srcElements.length; i < l; i++ ) { + cloneCopyEvent( srcElements[ i ], destElements[ i ] ); + } + } else { + cloneCopyEvent( elem, clone ); + } + } + + // Preserve script evaluation history + destElements = getAll( clone, "script" ); + if ( destElements.length > 0 ) { + setGlobalEval( destElements, !inPage && getAll( elem, "script" ) ); + } + + // Return the cloned set + return clone; + }, + + cleanData: function( elems ) { + var data, elem, type, + special = jQuery.event.special, + i = 0; + + for ( ; ( elem = elems[ i ] ) !== undefined; i++ ) { + if ( acceptData( elem ) ) { + if ( ( data = elem[ dataPriv.expando ] ) ) { + if ( data.events ) { + for ( type in data.events ) { + if ( special[ type ] ) { + jQuery.event.remove( elem, type ); + + // This is a shortcut to avoid jQuery.event.remove's overhead + } else { + jQuery.removeEvent( elem, type, data.handle ); + } + } + } + + // Support: Chrome <=35 - 45+ + // Assign undefined instead of using delete, see Data#remove + elem[ dataPriv.expando ] = undefined; + } + if ( elem[ dataUser.expando ] ) { + + // Support: Chrome <=35 - 45+ + // Assign undefined instead of using delete, see Data#remove + elem[ dataUser.expando ] = undefined; + } + } + } + } + } ); + + jQuery.fn.extend( { + detach: function( selector ) { + return remove( this, selector, true ); + }, + + remove: function( selector ) { + return remove( this, selector ); + }, + + text: function( value ) { + return access( this, function( value ) { + return value === undefined ? + jQuery.text( this ) : + this.empty().each( function() { + if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) { + this.textContent = value; + } + } ); + }, null, value, arguments.length ); + }, + + append: function() { + return domManip( this, arguments, function( elem ) { + if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) { + var target = manipulationTarget( this, elem ); + target.appendChild( elem ); + } + } ); + }, + + prepend: function() { + return domManip( this, arguments, function( elem ) { + if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) { + var target = manipulationTarget( this, elem ); + target.insertBefore( elem, target.firstChild ); + } + } ); + }, + + before: function() { + return domManip( this, arguments, function( elem ) { + if ( this.parentNode ) { + this.parentNode.insertBefore( elem, this ); + } + } ); + }, + + after: function() { + return domManip( this, arguments, function( elem ) { + if ( this.parentNode ) { + this.parentNode.insertBefore( elem, this.nextSibling ); + } + } ); + }, + + empty: function() { + var elem, + i = 0; + + for ( ; ( elem = this[ i ] ) != null; i++ ) { + if ( elem.nodeType === 1 ) { + + // Prevent memory leaks + jQuery.cleanData( getAll( elem, false ) ); + + // Remove any remaining nodes + elem.textContent = ""; + } + } + + return this; + }, + + clone: function( dataAndEvents, deepDataAndEvents ) { + dataAndEvents = dataAndEvents == null ? false : dataAndEvents; + deepDataAndEvents = deepDataAndEvents == null ? dataAndEvents : deepDataAndEvents; + + return this.map( function() { + return jQuery.clone( this, dataAndEvents, deepDataAndEvents ); + } ); + }, + + html: function( value ) { + return access( this, function( value ) { + var elem = this[ 0 ] || {}, + i = 0, + l = this.length; + + if ( value === undefined && elem.nodeType === 1 ) { + return elem.innerHTML; + } + + // See if we can take a shortcut and just use innerHTML + if ( typeof value === "string" && !rnoInnerhtml.test( value ) && + !wrapMap[ ( rtagName.exec( value ) || [ "", "" ] )[ 1 ].toLowerCase() ] ) { + + value = jQuery.htmlPrefilter( value ); + + try { + for ( ; i < l; i++ ) { + elem = this[ i ] || {}; + + // Remove element nodes and prevent memory leaks + if ( elem.nodeType === 1 ) { + jQuery.cleanData( getAll( elem, false ) ); + elem.innerHTML = value; + } + } + + elem = 0; + + // If using innerHTML throws an exception, use the fallback method + } catch ( e ) {} + } + + if ( elem ) { + this.empty().append( value ); + } + }, null, value, arguments.length ); + }, + + replaceWith: function() { + var ignored = []; + + // Make the changes, replacing each non-ignored context element with the new content + return domManip( this, arguments, function( elem ) { + var parent = this.parentNode; + + if ( jQuery.inArray( this, ignored ) < 0 ) { + jQuery.cleanData( getAll( this ) ); + if ( parent ) { + parent.replaceChild( elem, this ); + } + } + + // Force callback invocation + }, ignored ); + } + } ); + + jQuery.each( { + appendTo: "append", + prependTo: "prepend", + insertBefore: "before", + insertAfter: "after", + replaceAll: "replaceWith" + }, function( name, original ) { + jQuery.fn[ name ] = function( selector ) { + var elems, + ret = [], + insert = jQuery( selector ), + last = insert.length - 1, + i = 0; + + for ( ; i <= last; i++ ) { + elems = i === last ? this : this.clone( true ); + jQuery( insert[ i ] )[ original ]( elems ); + + // Support: Android <=4.0 only, PhantomJS 1 only + // .get() because push.apply(_, arraylike) throws on ancient WebKit + push.apply( ret, elems.get() ); + } + + return this.pushStack( ret ); + }; + } ); + var rnumnonpx = new RegExp( "^(" + pnum + ")(?!px)[a-z%]+$", "i" ); + + var getStyles = function( elem ) { + + // Support: IE <=11 only, Firefox <=30 (#15098, #14150) + // IE throws on elements created in popups + // FF meanwhile throws on frame elements through "defaultView.getComputedStyle" + var view = elem.ownerDocument.defaultView; + + if ( !view || !view.opener ) { + view = window; + } + + return view.getComputedStyle( elem ); + }; + + var swap = function( elem, options, callback ) { + var ret, name, + old = {}; + + // Remember the old values, and insert the new ones + for ( name in options ) { + old[ name ] = elem.style[ name ]; + elem.style[ name ] = options[ name ]; + } + + ret = callback.call( elem ); + + // Revert the old values + for ( name in options ) { + elem.style[ name ] = old[ name ]; + } + + return ret; + }; + + + var rboxStyle = new RegExp( cssExpand.join( "|" ), "i" ); + + + + ( function() { + + // Executing both pixelPosition & boxSizingReliable tests require only one layout + // so they're executed at the same time to save the second computation. + function computeStyleTests() { + + // This is a singleton, we need to execute it only once + if ( !div ) { + return; + } + + container.style.cssText = "position:absolute;left:-11111px;width:60px;" + + "margin-top:1px;padding:0;border:0"; + div.style.cssText = + "position:relative;display:block;box-sizing:border-box;overflow:scroll;" + + "margin:auto;border:1px;padding:1px;" + + "width:60%;top:1%"; + documentElement.appendChild( container ).appendChild( div ); + + var divStyle = window.getComputedStyle( div ); + pixelPositionVal = divStyle.top !== "1%"; + + // Support: Android 4.0 - 4.3 only, Firefox <=3 - 44 + reliableMarginLeftVal = roundPixelMeasures( divStyle.marginLeft ) === 12; + + // Support: Android 4.0 - 4.3 only, Safari <=9.1 - 10.1, iOS <=7.0 - 9.3 + // Some styles come back with percentage values, even though they shouldn't + div.style.right = "60%"; + pixelBoxStylesVal = roundPixelMeasures( divStyle.right ) === 36; + + // Support: IE 9 - 11 only + // Detect misreporting of content dimensions for box-sizing:border-box elements + boxSizingReliableVal = roundPixelMeasures( divStyle.width ) === 36; + + // Support: IE 9 only + // Detect overflow:scroll screwiness (gh-3699) + // Support: Chrome <=64 + // Don't get tricked when zoom affects offsetWidth (gh-4029) + div.style.position = "absolute"; + scrollboxSizeVal = roundPixelMeasures( div.offsetWidth / 3 ) === 12; + + documentElement.removeChild( container ); + + // Nullify the div so it wouldn't be stored in the memory and + // it will also be a sign that checks already performed + div = null; + } + + function roundPixelMeasures( measure ) { + return Math.round( parseFloat( measure ) ); + } + + var pixelPositionVal, boxSizingReliableVal, scrollboxSizeVal, pixelBoxStylesVal, + reliableTrDimensionsVal, reliableMarginLeftVal, + container = document.createElement( "div" ), + div = document.createElement( "div" ); + + // Finish early in limited (non-browser) environments + if ( !div.style ) { + return; + } + + // Support: IE <=9 - 11 only + // Style of cloned element affects source element cloned (#8908) + div.style.backgroundClip = "content-box"; + div.cloneNode( true ).style.backgroundClip = ""; + support.clearCloneStyle = div.style.backgroundClip === "content-box"; + + jQuery.extend( support, { + boxSizingReliable: function() { + computeStyleTests(); + return boxSizingReliableVal; + }, + pixelBoxStyles: function() { + computeStyleTests(); + return pixelBoxStylesVal; + }, + pixelPosition: function() { + computeStyleTests(); + return pixelPositionVal; + }, + reliableMarginLeft: function() { + computeStyleTests(); + return reliableMarginLeftVal; + }, + scrollboxSize: function() { + computeStyleTests(); + return scrollboxSizeVal; + }, + + // Support: IE 9 - 11+, Edge 15 - 18+ + // IE/Edge misreport `getComputedStyle` of table rows with width/height + // set in CSS while `offset*` properties report correct values. + // Behavior in IE 9 is more subtle than in newer versions & it passes + // some versions of this test; make sure not to make it pass there! + // + // Support: Firefox 70+ + // Only Firefox includes border widths + // in computed dimensions. (gh-4529) + reliableTrDimensions: function() { + var table, tr, trChild, trStyle; + if ( reliableTrDimensionsVal == null ) { + table = document.createElement( "table" ); + tr = document.createElement( "tr" ); + trChild = document.createElement( "div" ); + + table.style.cssText = "position:absolute;left:-11111px;border-collapse:separate"; + tr.style.cssText = "border:1px solid"; + + // Support: Chrome 86+ + // Height set through cssText does not get applied. + // Computed height then comes back as 0. + tr.style.height = "1px"; + trChild.style.height = "9px"; + + // Support: Android 8 Chrome 86+ + // In our bodyBackground.html iframe, + // display for all div elements is set to "inline", + // which causes a problem only in Android 8 Chrome 86. + // Ensuring the div is display: block + // gets around this issue. + trChild.style.display = "block"; + + documentElement + .appendChild( table ) + .appendChild( tr ) + .appendChild( trChild ); + + trStyle = window.getComputedStyle( tr ); + reliableTrDimensionsVal = ( parseInt( trStyle.height, 10 ) + + parseInt( trStyle.borderTopWidth, 10 ) + + parseInt( trStyle.borderBottomWidth, 10 ) ) === tr.offsetHeight; + + documentElement.removeChild( table ); + } + return reliableTrDimensionsVal; + } + } ); + } )(); + + + function curCSS( elem, name, computed ) { + var width, minWidth, maxWidth, ret, + + // Support: Firefox 51+ + // Retrieving style before computed somehow + // fixes an issue with getting wrong values + // on detached elements + style = elem.style; + + computed = computed || getStyles( elem ); + + // getPropertyValue is needed for: + // .css('filter') (IE 9 only, #12537) + // .css('--customProperty) (#3144) + if ( computed ) { + ret = computed.getPropertyValue( name ) || computed[ name ]; + + if ( ret === "" && !isAttached( elem ) ) { + ret = jQuery.style( elem, name ); + } + + // A tribute to the "awesome hack by Dean Edwards" + // Android Browser returns percentage for some values, + // but width seems to be reliably pixels. + // This is against the CSSOM draft spec: + // https://drafts.csswg.org/cssom/#resolved-values + if ( !support.pixelBoxStyles() && rnumnonpx.test( ret ) && rboxStyle.test( name ) ) { + + // Remember the original values + width = style.width; + minWidth = style.minWidth; + maxWidth = style.maxWidth; + + // Put in the new values to get a computed value out + style.minWidth = style.maxWidth = style.width = ret; + ret = computed.width; + + // Revert the changed values + style.width = width; + style.minWidth = minWidth; + style.maxWidth = maxWidth; + } + } + + return ret !== undefined ? + + // Support: IE <=9 - 11 only + // IE returns zIndex value as an integer. + ret + "" : + ret; + } + + + function addGetHookIf( conditionFn, hookFn ) { + + // Define the hook, we'll check on the first run if it's really needed. + return { + get: function() { + if ( conditionFn() ) { + + // Hook not needed (or it's not possible to use it due + // to missing dependency), remove it. + delete this.get; + return; + } + + // Hook needed; redefine it so that the support test is not executed again. + return ( this.get = hookFn ).apply( this, arguments ); + } + }; + } + + + var cssPrefixes = [ "Webkit", "Moz", "ms" ], + emptyStyle = document.createElement( "div" ).style, + vendorProps = {}; + + // Return a vendor-prefixed property or undefined + function vendorPropName( name ) { + + // Check for vendor prefixed names + var capName = name[ 0 ].toUpperCase() + name.slice( 1 ), + i = cssPrefixes.length; + + while ( i-- ) { + name = cssPrefixes[ i ] + capName; + if ( name in emptyStyle ) { + return name; + } + } + } + + // Return a potentially-mapped jQuery.cssProps or vendor prefixed property + function finalPropName( name ) { + var final = jQuery.cssProps[ name ] || vendorProps[ name ]; + + if ( final ) { + return final; + } + if ( name in emptyStyle ) { + return name; + } + return vendorProps[ name ] = vendorPropName( name ) || name; + } + + + var + + // Swappable if display is none or starts with table + // except "table", "table-cell", or "table-caption" + // See here for display values: https://developer.mozilla.org/en-US/docs/CSS/display + rdisplayswap = /^(none|table(?!-c[ea]).+)/, + rcustomProp = /^--/, + cssShow = { position: "absolute", visibility: "hidden", display: "block" }, + cssNormalTransform = { + letterSpacing: "0", + fontWeight: "400" + }; + + function setPositiveNumber( _elem, value, subtract ) { + + // Any relative (+/-) values have already been + // normalized at this point + var matches = rcssNum.exec( value ); + return matches ? + + // Guard against undefined "subtract", e.g., when used as in cssHooks + Math.max( 0, matches[ 2 ] - ( subtract || 0 ) ) + ( matches[ 3 ] || "px" ) : + value; + } + + function boxModelAdjustment( elem, dimension, box, isBorderBox, styles, computedVal ) { + var i = dimension === "width" ? 1 : 0, + extra = 0, + delta = 0; + + // Adjustment may not be necessary + if ( box === ( isBorderBox ? "border" : "content" ) ) { + return 0; + } + + for ( ; i < 4; i += 2 ) { + + // Both box models exclude margin + if ( box === "margin" ) { + delta += jQuery.css( elem, box + cssExpand[ i ], true, styles ); + } + + // If we get here with a content-box, we're seeking "padding" or "border" or "margin" + if ( !isBorderBox ) { + + // Add padding + delta += jQuery.css( elem, "padding" + cssExpand[ i ], true, styles ); + + // For "border" or "margin", add border + if ( box !== "padding" ) { + delta += jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles ); + + // But still keep track of it otherwise + } else { + extra += jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles ); + } + + // If we get here with a border-box (content + padding + border), we're seeking "content" or + // "padding" or "margin" + } else { + + // For "content", subtract padding + if ( box === "content" ) { + delta -= jQuery.css( elem, "padding" + cssExpand[ i ], true, styles ); + } + + // For "content" or "padding", subtract border + if ( box !== "margin" ) { + delta -= jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles ); + } + } + } + + // Account for positive content-box scroll gutter when requested by providing computedVal + if ( !isBorderBox && computedVal >= 0 ) { + + // offsetWidth/offsetHeight is a rounded sum of content, padding, scroll gutter, and border + // Assuming integer scroll gutter, subtract the rest and round down + delta += Math.max( 0, Math.ceil( + elem[ "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 ) ] - + computedVal - + delta - + extra - + 0.5 + + // If offsetWidth/offsetHeight is unknown, then we can't determine content-box scroll gutter + // Use an explicit zero to avoid NaN (gh-3964) + ) ) || 0; + } + + return delta; + } + + function getWidthOrHeight( elem, dimension, extra ) { + + // Start with computed style + var styles = getStyles( elem ), + + // To avoid forcing a reflow, only fetch boxSizing if we need it (gh-4322). + // Fake content-box until we know it's needed to know the true value. + boxSizingNeeded = !support.boxSizingReliable() || extra, + isBorderBox = boxSizingNeeded && + jQuery.css( elem, "boxSizing", false, styles ) === "border-box", + valueIsBorderBox = isBorderBox, + + val = curCSS( elem, dimension, styles ), + offsetProp = "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 ); + + // Support: Firefox <=54 + // Return a confounding non-pixel value or feign ignorance, as appropriate. + if ( rnumnonpx.test( val ) ) { + if ( !extra ) { + return val; + } + val = "auto"; + } + + + // Support: IE 9 - 11 only + // Use offsetWidth/offsetHeight for when box sizing is unreliable. + // In those cases, the computed value can be trusted to be border-box. + if ( ( !support.boxSizingReliable() && isBorderBox || + + // Support: IE 10 - 11+, Edge 15 - 18+ + // IE/Edge misreport `getComputedStyle` of table rows with width/height + // set in CSS while `offset*` properties report correct values. + // Interestingly, in some cases IE 9 doesn't suffer from this issue. + !support.reliableTrDimensions() && nodeName( elem, "tr" ) || + + // Fall back to offsetWidth/offsetHeight when value is "auto" + // This happens for inline elements with no explicit setting (gh-3571) + val === "auto" || + + // Support: Android <=4.1 - 4.3 only + // Also use offsetWidth/offsetHeight for misreported inline dimensions (gh-3602) + !parseFloat( val ) && jQuery.css( elem, "display", false, styles ) === "inline" ) && + + // Make sure the element is visible & connected + elem.getClientRects().length ) { + + isBorderBox = jQuery.css( elem, "boxSizing", false, styles ) === "border-box"; + + // Where available, offsetWidth/offsetHeight approximate border box dimensions. + // Where not available (e.g., SVG), assume unreliable box-sizing and interpret the + // retrieved value as a content box dimension. + valueIsBorderBox = offsetProp in elem; + if ( valueIsBorderBox ) { + val = elem[ offsetProp ]; + } + } + + // Normalize "" and auto + val = parseFloat( val ) || 0; + + // Adjust for the element's box model + return ( val + + boxModelAdjustment( + elem, + dimension, + extra || ( isBorderBox ? "border" : "content" ), + valueIsBorderBox, + styles, + + // Provide the current computed size to request scroll gutter calculation (gh-3589) + val + ) + ) + "px"; + } + + jQuery.extend( { + + // Add in style property hooks for overriding the default + // behavior of getting and setting a style property + cssHooks: { + opacity: { + get: function( elem, computed ) { + if ( computed ) { + + // We should always get a number back from opacity + var ret = curCSS( elem, "opacity" ); + return ret === "" ? "1" : ret; + } + } + } + }, + + // Don't automatically add "px" to these possibly-unitless properties + cssNumber: { + "animationIterationCount": true, + "columnCount": true, + "fillOpacity": true, + "flexGrow": true, + "flexShrink": true, + "fontWeight": true, + "gridArea": true, + "gridColumn": true, + "gridColumnEnd": true, + "gridColumnStart": true, + "gridRow": true, + "gridRowEnd": true, + "gridRowStart": true, + "lineHeight": true, + "opacity": true, + "order": true, + "orphans": true, + "widows": true, + "zIndex": true, + "zoom": true + }, + + // Add in properties whose names you wish to fix before + // setting or getting the value + cssProps: {}, + + // Get and set the style property on a DOM Node + style: function( elem, name, value, extra ) { + + // Don't set styles on text and comment nodes + if ( !elem || elem.nodeType === 3 || elem.nodeType === 8 || !elem.style ) { + return; + } + + // Make sure that we're working with the right name + var ret, type, hooks, + origName = camelCase( name ), + isCustomProp = rcustomProp.test( name ), + style = elem.style; + + // Make sure that we're working with the right name. We don't + // want to query the value if it is a CSS custom property + // since they are user-defined. + if ( !isCustomProp ) { + name = finalPropName( origName ); + } + + // Gets hook for the prefixed version, then unprefixed version + hooks = jQuery.cssHooks[ name ] || jQuery.cssHooks[ origName ]; + + // Check if we're setting a value + if ( value !== undefined ) { + type = typeof value; + + // Convert "+=" or "-=" to relative numbers (#7345) + if ( type === "string" && ( ret = rcssNum.exec( value ) ) && ret[ 1 ] ) { + value = adjustCSS( elem, name, ret ); + + // Fixes bug #9237 + type = "number"; + } + + // Make sure that null and NaN values aren't set (#7116) + if ( value == null || value !== value ) { + return; + } + + // If a number was passed in, add the unit (except for certain CSS properties) + // The isCustomProp check can be removed in jQuery 4.0 when we only auto-append + // "px" to a few hardcoded values. + if ( type === "number" && !isCustomProp ) { + value += ret && ret[ 3 ] || ( jQuery.cssNumber[ origName ] ? "" : "px" ); + } + + // background-* props affect original clone's values + if ( !support.clearCloneStyle && value === "" && name.indexOf( "background" ) === 0 ) { + style[ name ] = "inherit"; + } + + // If a hook was provided, use that value, otherwise just set the specified value + if ( !hooks || !( "set" in hooks ) || + ( value = hooks.set( elem, value, extra ) ) !== undefined ) { + + if ( isCustomProp ) { + style.setProperty( name, value ); + } else { + style[ name ] = value; + } + } + + } else { + + // If a hook was provided get the non-computed value from there + if ( hooks && "get" in hooks && + ( ret = hooks.get( elem, false, extra ) ) !== undefined ) { + + return ret; + } + + // Otherwise just get the value from the style object + return style[ name ]; + } + }, + + css: function( elem, name, extra, styles ) { + var val, num, hooks, + origName = camelCase( name ), + isCustomProp = rcustomProp.test( name ); + + // Make sure that we're working with the right name. We don't + // want to modify the value if it is a CSS custom property + // since they are user-defined. + if ( !isCustomProp ) { + name = finalPropName( origName ); + } + + // Try prefixed name followed by the unprefixed name + hooks = jQuery.cssHooks[ name ] || jQuery.cssHooks[ origName ]; + + // If a hook was provided get the computed value from there + if ( hooks && "get" in hooks ) { + val = hooks.get( elem, true, extra ); + } + + // Otherwise, if a way to get the computed value exists, use that + if ( val === undefined ) { + val = curCSS( elem, name, styles ); + } + + // Convert "normal" to computed value + if ( val === "normal" && name in cssNormalTransform ) { + val = cssNormalTransform[ name ]; + } + + // Make numeric if forced or a qualifier was provided and val looks numeric + if ( extra === "" || extra ) { + num = parseFloat( val ); + return extra === true || isFinite( num ) ? num || 0 : val; + } + + return val; + } + } ); + + jQuery.each( [ "height", "width" ], function( _i, dimension ) { + jQuery.cssHooks[ dimension ] = { + get: function( elem, computed, extra ) { + if ( computed ) { + + // Certain elements can have dimension info if we invisibly show them + // but it must have a current display style that would benefit + return rdisplayswap.test( jQuery.css( elem, "display" ) ) && + + // Support: Safari 8+ + // Table columns in Safari have non-zero offsetWidth & zero + // getBoundingClientRect().width unless display is changed. + // Support: IE <=11 only + // Running getBoundingClientRect on a disconnected node + // in IE throws an error. + ( !elem.getClientRects().length || !elem.getBoundingClientRect().width ) ? + swap( elem, cssShow, function() { + return getWidthOrHeight( elem, dimension, extra ); + } ) : + getWidthOrHeight( elem, dimension, extra ); + } + }, + + set: function( elem, value, extra ) { + var matches, + styles = getStyles( elem ), + + // Only read styles.position if the test has a chance to fail + // to avoid forcing a reflow. + scrollboxSizeBuggy = !support.scrollboxSize() && + styles.position === "absolute", + + // To avoid forcing a reflow, only fetch boxSizing if we need it (gh-3991) + boxSizingNeeded = scrollboxSizeBuggy || extra, + isBorderBox = boxSizingNeeded && + jQuery.css( elem, "boxSizing", false, styles ) === "border-box", + subtract = extra ? + boxModelAdjustment( + elem, + dimension, + extra, + isBorderBox, + styles + ) : + 0; + + // Account for unreliable border-box dimensions by comparing offset* to computed and + // faking a content-box to get border and padding (gh-3699) + if ( isBorderBox && scrollboxSizeBuggy ) { + subtract -= Math.ceil( + elem[ "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 ) ] - + parseFloat( styles[ dimension ] ) - + boxModelAdjustment( elem, dimension, "border", false, styles ) - + 0.5 + ); + } + + // Convert to pixels if value adjustment is needed + if ( subtract && ( matches = rcssNum.exec( value ) ) && + ( matches[ 3 ] || "px" ) !== "px" ) { + + elem.style[ dimension ] = value; + value = jQuery.css( elem, dimension ); + } + + return setPositiveNumber( elem, value, subtract ); + } + }; + } ); + + jQuery.cssHooks.marginLeft = addGetHookIf( support.reliableMarginLeft, + function( elem, computed ) { + if ( computed ) { + return ( parseFloat( curCSS( elem, "marginLeft" ) ) || + elem.getBoundingClientRect().left - + swap( elem, { marginLeft: 0 }, function() { + return elem.getBoundingClientRect().left; + } ) + ) + "px"; + } + } + ); + + // These hooks are used by animate to expand properties + jQuery.each( { + margin: "", + padding: "", + border: "Width" + }, function( prefix, suffix ) { + jQuery.cssHooks[ prefix + suffix ] = { + expand: function( value ) { + var i = 0, + expanded = {}, + + // Assumes a single number if not a string + parts = typeof value === "string" ? value.split( " " ) : [ value ]; + + for ( ; i < 4; i++ ) { + expanded[ prefix + cssExpand[ i ] + suffix ] = + parts[ i ] || parts[ i - 2 ] || parts[ 0 ]; + } + + return expanded; + } + }; + + if ( prefix !== "margin" ) { + jQuery.cssHooks[ prefix + suffix ].set = setPositiveNumber; + } + } ); + + jQuery.fn.extend( { + css: function( name, value ) { + return access( this, function( elem, name, value ) { + var styles, len, + map = {}, + i = 0; + + if ( Array.isArray( name ) ) { + styles = getStyles( elem ); + len = name.length; + + for ( ; i < len; i++ ) { + map[ name[ i ] ] = jQuery.css( elem, name[ i ], false, styles ); + } + + return map; + } + + return value !== undefined ? + jQuery.style( elem, name, value ) : + jQuery.css( elem, name ); + }, name, value, arguments.length > 1 ); + } + } ); + + + function Tween( elem, options, prop, end, easing ) { + return new Tween.prototype.init( elem, options, prop, end, easing ); + } + jQuery.Tween = Tween; + + Tween.prototype = { + constructor: Tween, + init: function( elem, options, prop, end, easing, unit ) { + this.elem = elem; + this.prop = prop; + this.easing = easing || jQuery.easing._default; + this.options = options; + this.start = this.now = this.cur(); + this.end = end; + this.unit = unit || ( jQuery.cssNumber[ prop ] ? "" : "px" ); + }, + cur: function() { + var hooks = Tween.propHooks[ this.prop ]; + + return hooks && hooks.get ? + hooks.get( this ) : + Tween.propHooks._default.get( this ); + }, + run: function( percent ) { + var eased, + hooks = Tween.propHooks[ this.prop ]; + + if ( this.options.duration ) { + this.pos = eased = jQuery.easing[ this.easing ]( + percent, this.options.duration * percent, 0, 1, this.options.duration + ); + } else { + this.pos = eased = percent; + } + this.now = ( this.end - this.start ) * eased + this.start; + + if ( this.options.step ) { + this.options.step.call( this.elem, this.now, this ); + } + + if ( hooks && hooks.set ) { + hooks.set( this ); + } else { + Tween.propHooks._default.set( this ); + } + return this; + } + }; + + Tween.prototype.init.prototype = Tween.prototype; + + Tween.propHooks = { + _default: { + get: function( tween ) { + var result; + + // Use a property on the element directly when it is not a DOM element, + // or when there is no matching style property that exists. + if ( tween.elem.nodeType !== 1 || + tween.elem[ tween.prop ] != null && tween.elem.style[ tween.prop ] == null ) { + return tween.elem[ tween.prop ]; + } + + // Passing an empty string as a 3rd parameter to .css will automatically + // attempt a parseFloat and fallback to a string if the parse fails. + // Simple values such as "10px" are parsed to Float; + // complex values such as "rotate(1rad)" are returned as-is. + result = jQuery.css( tween.elem, tween.prop, "" ); + + // Empty strings, null, undefined and "auto" are converted to 0. + return !result || result === "auto" ? 0 : result; + }, + set: function( tween ) { + + // Use step hook for back compat. + // Use cssHook if its there. + // Use .style if available and use plain properties where available. + if ( jQuery.fx.step[ tween.prop ] ) { + jQuery.fx.step[ tween.prop ]( tween ); + } else if ( tween.elem.nodeType === 1 && ( + jQuery.cssHooks[ tween.prop ] || + tween.elem.style[ finalPropName( tween.prop ) ] != null ) ) { + jQuery.style( tween.elem, tween.prop, tween.now + tween.unit ); + } else { + tween.elem[ tween.prop ] = tween.now; + } + } + } + }; + + // Support: IE <=9 only + // Panic based approach to setting things on disconnected nodes + Tween.propHooks.scrollTop = Tween.propHooks.scrollLeft = { + set: function( tween ) { + if ( tween.elem.nodeType && tween.elem.parentNode ) { + tween.elem[ tween.prop ] = tween.now; + } + } + }; + + jQuery.easing = { + linear: function( p ) { + return p; + }, + swing: function( p ) { + return 0.5 - Math.cos( p * Math.PI ) / 2; + }, + _default: "swing" + }; + + jQuery.fx = Tween.prototype.init; + + // Back compat <1.8 extension point + jQuery.fx.step = {}; + + + + + var + fxNow, inProgress, + rfxtypes = /^(?:toggle|show|hide)$/, + rrun = /queueHooks$/; + + function schedule() { + if ( inProgress ) { + if ( document.hidden === false && window.requestAnimationFrame ) { + window.requestAnimationFrame( schedule ); + } else { + window.setTimeout( schedule, jQuery.fx.interval ); + } + + jQuery.fx.tick(); + } + } + + // Animations created synchronously will run synchronously + function createFxNow() { + window.setTimeout( function() { + fxNow = undefined; + } ); + return ( fxNow = Date.now() ); + } + + // Generate parameters to create a standard animation + function genFx( type, includeWidth ) { + var which, + i = 0, + attrs = { height: type }; + + // If we include width, step value is 1 to do all cssExpand values, + // otherwise step value is 2 to skip over Left and Right + includeWidth = includeWidth ? 1 : 0; + for ( ; i < 4; i += 2 - includeWidth ) { + which = cssExpand[ i ]; + attrs[ "margin" + which ] = attrs[ "padding" + which ] = type; + } + + if ( includeWidth ) { + attrs.opacity = attrs.width = type; + } + + return attrs; + } + + function createTween( value, prop, animation ) { + var tween, + collection = ( Animation.tweeners[ prop ] || [] ).concat( Animation.tweeners[ "*" ] ), + index = 0, + length = collection.length; + for ( ; index < length; index++ ) { + if ( ( tween = collection[ index ].call( animation, prop, value ) ) ) { + + // We're done with this property + return tween; + } + } + } + + function defaultPrefilter( elem, props, opts ) { + var prop, value, toggle, hooks, oldfire, propTween, restoreDisplay, display, + isBox = "width" in props || "height" in props, + anim = this, + orig = {}, + style = elem.style, + hidden = elem.nodeType && isHiddenWithinTree( elem ), + dataShow = dataPriv.get( elem, "fxshow" ); + + // Queue-skipping animations hijack the fx hooks + if ( !opts.queue ) { + hooks = jQuery._queueHooks( elem, "fx" ); + if ( hooks.unqueued == null ) { + hooks.unqueued = 0; + oldfire = hooks.empty.fire; + hooks.empty.fire = function() { + if ( !hooks.unqueued ) { + oldfire(); + } + }; + } + hooks.unqueued++; + + anim.always( function() { + + // Ensure the complete handler is called before this completes + anim.always( function() { + hooks.unqueued--; + if ( !jQuery.queue( elem, "fx" ).length ) { + hooks.empty.fire(); + } + } ); + } ); + } + + // Detect show/hide animations + for ( prop in props ) { + value = props[ prop ]; + if ( rfxtypes.test( value ) ) { + delete props[ prop ]; + toggle = toggle || value === "toggle"; + if ( value === ( hidden ? "hide" : "show" ) ) { + + // Pretend to be hidden if this is a "show" and + // there is still data from a stopped show/hide + if ( value === "show" && dataShow && dataShow[ prop ] !== undefined ) { + hidden = true; + + // Ignore all other no-op show/hide data + } else { + continue; + } + } + orig[ prop ] = dataShow && dataShow[ prop ] || jQuery.style( elem, prop ); + } + } + + // Bail out if this is a no-op like .hide().hide() + propTween = !jQuery.isEmptyObject( props ); + if ( !propTween && jQuery.isEmptyObject( orig ) ) { + return; + } + + // Restrict "overflow" and "display" styles during box animations + if ( isBox && elem.nodeType === 1 ) { + + // Support: IE <=9 - 11, Edge 12 - 15 + // Record all 3 overflow attributes because IE does not infer the shorthand + // from identically-valued overflowX and overflowY and Edge just mirrors + // the overflowX value there. + opts.overflow = [ style.overflow, style.overflowX, style.overflowY ]; + + // Identify a display type, preferring old show/hide data over the CSS cascade + restoreDisplay = dataShow && dataShow.display; + if ( restoreDisplay == null ) { + restoreDisplay = dataPriv.get( elem, "display" ); + } + display = jQuery.css( elem, "display" ); + if ( display === "none" ) { + if ( restoreDisplay ) { + display = restoreDisplay; + } else { + + // Get nonempty value(s) by temporarily forcing visibility + showHide( [ elem ], true ); + restoreDisplay = elem.style.display || restoreDisplay; + display = jQuery.css( elem, "display" ); + showHide( [ elem ] ); + } + } + + // Animate inline elements as inline-block + if ( display === "inline" || display === "inline-block" && restoreDisplay != null ) { + if ( jQuery.css( elem, "float" ) === "none" ) { + + // Restore the original display value at the end of pure show/hide animations + if ( !propTween ) { + anim.done( function() { + style.display = restoreDisplay; + } ); + if ( restoreDisplay == null ) { + display = style.display; + restoreDisplay = display === "none" ? "" : display; + } + } + style.display = "inline-block"; + } + } + } + + if ( opts.overflow ) { + style.overflow = "hidden"; + anim.always( function() { + style.overflow = opts.overflow[ 0 ]; + style.overflowX = opts.overflow[ 1 ]; + style.overflowY = opts.overflow[ 2 ]; + } ); + } + + // Implement show/hide animations + propTween = false; + for ( prop in orig ) { + + // General show/hide setup for this element animation + if ( !propTween ) { + if ( dataShow ) { + if ( "hidden" in dataShow ) { + hidden = dataShow.hidden; + } + } else { + dataShow = dataPriv.access( elem, "fxshow", { display: restoreDisplay } ); + } + + // Store hidden/visible for toggle so `.stop().toggle()` "reverses" + if ( toggle ) { + dataShow.hidden = !hidden; + } + + // Show elements before animating them + if ( hidden ) { + showHide( [ elem ], true ); + } + + /* eslint-disable no-loop-func */ + + anim.done( function() { + + /* eslint-enable no-loop-func */ + + // The final step of a "hide" animation is actually hiding the element + if ( !hidden ) { + showHide( [ elem ] ); + } + dataPriv.remove( elem, "fxshow" ); + for ( prop in orig ) { + jQuery.style( elem, prop, orig[ prop ] ); + } + } ); + } + + // Per-property setup + propTween = createTween( hidden ? dataShow[ prop ] : 0, prop, anim ); + if ( !( prop in dataShow ) ) { + dataShow[ prop ] = propTween.start; + if ( hidden ) { + propTween.end = propTween.start; + propTween.start = 0; + } + } + } + } + + function propFilter( props, specialEasing ) { + var index, name, easing, value, hooks; + + // camelCase, specialEasing and expand cssHook pass + for ( index in props ) { + name = camelCase( index ); + easing = specialEasing[ name ]; + value = props[ index ]; + if ( Array.isArray( value ) ) { + easing = value[ 1 ]; + value = props[ index ] = value[ 0 ]; + } + + if ( index !== name ) { + props[ name ] = value; + delete props[ index ]; + } + + hooks = jQuery.cssHooks[ name ]; + if ( hooks && "expand" in hooks ) { + value = hooks.expand( value ); + delete props[ name ]; + + // Not quite $.extend, this won't overwrite existing keys. + // Reusing 'index' because we have the correct "name" + for ( index in value ) { + if ( !( index in props ) ) { + props[ index ] = value[ index ]; + specialEasing[ index ] = easing; + } + } + } else { + specialEasing[ name ] = easing; + } + } + } + + function Animation( elem, properties, options ) { + var result, + stopped, + index = 0, + length = Animation.prefilters.length, + deferred = jQuery.Deferred().always( function() { + + // Don't match elem in the :animated selector + delete tick.elem; + } ), + tick = function() { + if ( stopped ) { + return false; + } + var currentTime = fxNow || createFxNow(), + remaining = Math.max( 0, animation.startTime + animation.duration - currentTime ), + + // Support: Android 2.3 only + // Archaic crash bug won't allow us to use `1 - ( 0.5 || 0 )` (#12497) + temp = remaining / animation.duration || 0, + percent = 1 - temp, + index = 0, + length = animation.tweens.length; + + for ( ; index < length; index++ ) { + animation.tweens[ index ].run( percent ); + } + + deferred.notifyWith( elem, [ animation, percent, remaining ] ); + + // If there's more to do, yield + if ( percent < 1 && length ) { + return remaining; + } + + // If this was an empty animation, synthesize a final progress notification + if ( !length ) { + deferred.notifyWith( elem, [ animation, 1, 0 ] ); + } + + // Resolve the animation and report its conclusion + deferred.resolveWith( elem, [ animation ] ); + return false; + }, + animation = deferred.promise( { + elem: elem, + props: jQuery.extend( {}, properties ), + opts: jQuery.extend( true, { + specialEasing: {}, + easing: jQuery.easing._default + }, options ), + originalProperties: properties, + originalOptions: options, + startTime: fxNow || createFxNow(), + duration: options.duration, + tweens: [], + createTween: function( prop, end ) { + var tween = jQuery.Tween( elem, animation.opts, prop, end, + animation.opts.specialEasing[ prop ] || animation.opts.easing ); + animation.tweens.push( tween ); + return tween; + }, + stop: function( gotoEnd ) { + var index = 0, + + // If we are going to the end, we want to run all the tweens + // otherwise we skip this part + length = gotoEnd ? animation.tweens.length : 0; + if ( stopped ) { + return this; + } + stopped = true; + for ( ; index < length; index++ ) { + animation.tweens[ index ].run( 1 ); + } + + // Resolve when we played the last frame; otherwise, reject + if ( gotoEnd ) { + deferred.notifyWith( elem, [ animation, 1, 0 ] ); + deferred.resolveWith( elem, [ animation, gotoEnd ] ); + } else { + deferred.rejectWith( elem, [ animation, gotoEnd ] ); + } + return this; + } + } ), + props = animation.props; + + propFilter( props, animation.opts.specialEasing ); + + for ( ; index < length; index++ ) { + result = Animation.prefilters[ index ].call( animation, elem, props, animation.opts ); + if ( result ) { + if ( isFunction( result.stop ) ) { + jQuery._queueHooks( animation.elem, animation.opts.queue ).stop = + result.stop.bind( result ); + } + return result; + } + } + + jQuery.map( props, createTween, animation ); + + if ( isFunction( animation.opts.start ) ) { + animation.opts.start.call( elem, animation ); + } + + // Attach callbacks from options + animation + .progress( animation.opts.progress ) + .done( animation.opts.done, animation.opts.complete ) + .fail( animation.opts.fail ) + .always( animation.opts.always ); + + jQuery.fx.timer( + jQuery.extend( tick, { + elem: elem, + anim: animation, + queue: animation.opts.queue + } ) + ); + + return animation; + } + + jQuery.Animation = jQuery.extend( Animation, { + + tweeners: { + "*": [ function( prop, value ) { + var tween = this.createTween( prop, value ); + adjustCSS( tween.elem, prop, rcssNum.exec( value ), tween ); + return tween; + } ] + }, + + tweener: function( props, callback ) { + if ( isFunction( props ) ) { + callback = props; + props = [ "*" ]; + } else { + props = props.match( rnothtmlwhite ); + } + + var prop, + index = 0, + length = props.length; + + for ( ; index < length; index++ ) { + prop = props[ index ]; + Animation.tweeners[ prop ] = Animation.tweeners[ prop ] || []; + Animation.tweeners[ prop ].unshift( callback ); + } + }, + + prefilters: [ defaultPrefilter ], + + prefilter: function( callback, prepend ) { + if ( prepend ) { + Animation.prefilters.unshift( callback ); + } else { + Animation.prefilters.push( callback ); + } + } + } ); + + jQuery.speed = function( speed, easing, fn ) { + var opt = speed && typeof speed === "object" ? jQuery.extend( {}, speed ) : { + complete: fn || !fn && easing || + isFunction( speed ) && speed, + duration: speed, + easing: fn && easing || easing && !isFunction( easing ) && easing + }; + + // Go to the end state if fx are off + if ( jQuery.fx.off ) { + opt.duration = 0; + + } else { + if ( typeof opt.duration !== "number" ) { + if ( opt.duration in jQuery.fx.speeds ) { + opt.duration = jQuery.fx.speeds[ opt.duration ]; + + } else { + opt.duration = jQuery.fx.speeds._default; + } + } + } + + // Normalize opt.queue - true/undefined/null -> "fx" + if ( opt.queue == null || opt.queue === true ) { + opt.queue = "fx"; + } + + // Queueing + opt.old = opt.complete; + + opt.complete = function() { + if ( isFunction( opt.old ) ) { + opt.old.call( this ); + } + + if ( opt.queue ) { + jQuery.dequeue( this, opt.queue ); + } + }; + + return opt; + }; + + jQuery.fn.extend( { + fadeTo: function( speed, to, easing, callback ) { + + // Show any hidden elements after setting opacity to 0 + return this.filter( isHiddenWithinTree ).css( "opacity", 0 ).show() + + // Animate to the value specified + .end().animate( { opacity: to }, speed, easing, callback ); + }, + animate: function( prop, speed, easing, callback ) { + var empty = jQuery.isEmptyObject( prop ), + optall = jQuery.speed( speed, easing, callback ), + doAnimation = function() { + + // Operate on a copy of prop so per-property easing won't be lost + var anim = Animation( this, jQuery.extend( {}, prop ), optall ); + + // Empty animations, or finishing resolves immediately + if ( empty || dataPriv.get( this, "finish" ) ) { + anim.stop( true ); + } + }; + + doAnimation.finish = doAnimation; + + return empty || optall.queue === false ? + this.each( doAnimation ) : + this.queue( optall.queue, doAnimation ); + }, + stop: function( type, clearQueue, gotoEnd ) { + var stopQueue = function( hooks ) { + var stop = hooks.stop; + delete hooks.stop; + stop( gotoEnd ); + }; + + if ( typeof type !== "string" ) { + gotoEnd = clearQueue; + clearQueue = type; + type = undefined; + } + if ( clearQueue ) { + this.queue( type || "fx", [] ); + } + + return this.each( function() { + var dequeue = true, + index = type != null && type + "queueHooks", + timers = jQuery.timers, + data = dataPriv.get( this ); + + if ( index ) { + if ( data[ index ] && data[ index ].stop ) { + stopQueue( data[ index ] ); + } + } else { + for ( index in data ) { + if ( data[ index ] && data[ index ].stop && rrun.test( index ) ) { + stopQueue( data[ index ] ); + } + } + } + + for ( index = timers.length; index--; ) { + if ( timers[ index ].elem === this && + ( type == null || timers[ index ].queue === type ) ) { + + timers[ index ].anim.stop( gotoEnd ); + dequeue = false; + timers.splice( index, 1 ); + } + } + + // Start the next in the queue if the last step wasn't forced. + // Timers currently will call their complete callbacks, which + // will dequeue but only if they were gotoEnd. + if ( dequeue || !gotoEnd ) { + jQuery.dequeue( this, type ); + } + } ); + }, + finish: function( type ) { + if ( type !== false ) { + type = type || "fx"; + } + return this.each( function() { + var index, + data = dataPriv.get( this ), + queue = data[ type + "queue" ], + hooks = data[ type + "queueHooks" ], + timers = jQuery.timers, + length = queue ? queue.length : 0; + + // Enable finishing flag on private data + data.finish = true; + + // Empty the queue first + jQuery.queue( this, type, [] ); + + if ( hooks && hooks.stop ) { + hooks.stop.call( this, true ); + } + + // Look for any active animations, and finish them + for ( index = timers.length; index--; ) { + if ( timers[ index ].elem === this && timers[ index ].queue === type ) { + timers[ index ].anim.stop( true ); + timers.splice( index, 1 ); + } + } + + // Look for any animations in the old queue and finish them + for ( index = 0; index < length; index++ ) { + if ( queue[ index ] && queue[ index ].finish ) { + queue[ index ].finish.call( this ); + } + } + + // Turn off finishing flag + delete data.finish; + } ); + } + } ); + + jQuery.each( [ "toggle", "show", "hide" ], function( _i, name ) { + var cssFn = jQuery.fn[ name ]; + jQuery.fn[ name ] = function( speed, easing, callback ) { + return speed == null || typeof speed === "boolean" ? + cssFn.apply( this, arguments ) : + this.animate( genFx( name, true ), speed, easing, callback ); + }; + } ); + + // Generate shortcuts for custom animations + jQuery.each( { + slideDown: genFx( "show" ), + slideUp: genFx( "hide" ), + slideToggle: genFx( "toggle" ), + fadeIn: { opacity: "show" }, + fadeOut: { opacity: "hide" }, + fadeToggle: { opacity: "toggle" } + }, function( name, props ) { + jQuery.fn[ name ] = function( speed, easing, callback ) { + return this.animate( props, speed, easing, callback ); + }; + } ); + + jQuery.timers = []; + jQuery.fx.tick = function() { + var timer, + i = 0, + timers = jQuery.timers; + + fxNow = Date.now(); + + for ( ; i < timers.length; i++ ) { + timer = timers[ i ]; + + // Run the timer and safely remove it when done (allowing for external removal) + if ( !timer() && timers[ i ] === timer ) { + timers.splice( i--, 1 ); + } + } + + if ( !timers.length ) { + jQuery.fx.stop(); + } + fxNow = undefined; + }; + + jQuery.fx.timer = function( timer ) { + jQuery.timers.push( timer ); + jQuery.fx.start(); + }; + + jQuery.fx.interval = 13; + jQuery.fx.start = function() { + if ( inProgress ) { + return; + } + + inProgress = true; + schedule(); + }; + + jQuery.fx.stop = function() { + inProgress = null; + }; + + jQuery.fx.speeds = { + slow: 600, + fast: 200, + + // Default speed + _default: 400 + }; + + + // Based off of the plugin by Clint Helfers, with permission. + // https://web.archive.org/web/20100324014747/http://blindsignals.com/index.php/2009/07/jquery-delay/ + jQuery.fn.delay = function( time, type ) { + time = jQuery.fx ? jQuery.fx.speeds[ time ] || time : time; + type = type || "fx"; + + return this.queue( type, function( next, hooks ) { + var timeout = window.setTimeout( next, time ); + hooks.stop = function() { + window.clearTimeout( timeout ); + }; + } ); + }; + + + ( function() { + var input = document.createElement( "input" ), + select = document.createElement( "select" ), + opt = select.appendChild( document.createElement( "option" ) ); + + input.type = "checkbox"; + + // Support: Android <=4.3 only + // Default value for a checkbox should be "on" + support.checkOn = input.value !== ""; + + // Support: IE <=11 only + // Must access selectedIndex to make default options select + support.optSelected = opt.selected; + + // Support: IE <=11 only + // An input loses its value after becoming a radio + input = document.createElement( "input" ); + input.value = "t"; + input.type = "radio"; + support.radioValue = input.value === "t"; + } )(); + + + var boolHook, + attrHandle = jQuery.expr.attrHandle; + + jQuery.fn.extend( { + attr: function( name, value ) { + return access( this, jQuery.attr, name, value, arguments.length > 1 ); + }, + + removeAttr: function( name ) { + return this.each( function() { + jQuery.removeAttr( this, name ); + } ); + } + } ); + + jQuery.extend( { + attr: function( elem, name, value ) { + var ret, hooks, + nType = elem.nodeType; + + // Don't get/set attributes on text, comment and attribute nodes + if ( nType === 3 || nType === 8 || nType === 2 ) { + return; + } + + // Fallback to prop when attributes are not supported + if ( typeof elem.getAttribute === "undefined" ) { + return jQuery.prop( elem, name, value ); + } + + // Attribute hooks are determined by the lowercase version + // Grab necessary hook if one is defined + if ( nType !== 1 || !jQuery.isXMLDoc( elem ) ) { + hooks = jQuery.attrHooks[ name.toLowerCase() ] || + ( jQuery.expr.match.bool.test( name ) ? boolHook : undefined ); + } + + if ( value !== undefined ) { + if ( value === null ) { + jQuery.removeAttr( elem, name ); + return; + } + + if ( hooks && "set" in hooks && + ( ret = hooks.set( elem, value, name ) ) !== undefined ) { + return ret; + } + + elem.setAttribute( name, value + "" ); + return value; + } + + if ( hooks && "get" in hooks && ( ret = hooks.get( elem, name ) ) !== null ) { + return ret; + } + + ret = jQuery.find.attr( elem, name ); + + // Non-existent attributes return null, we normalize to undefined + return ret == null ? undefined : ret; + }, + + attrHooks: { + type: { + set: function( elem, value ) { + if ( !support.radioValue && value === "radio" && + nodeName( elem, "input" ) ) { + var val = elem.value; + elem.setAttribute( "type", value ); + if ( val ) { + elem.value = val; + } + return value; + } + } + } + }, + + removeAttr: function( elem, value ) { + var name, + i = 0, + + // Attribute names can contain non-HTML whitespace characters + // https://html.spec.whatwg.org/multipage/syntax.html#attributes-2 + attrNames = value && value.match( rnothtmlwhite ); + + if ( attrNames && elem.nodeType === 1 ) { + while ( ( name = attrNames[ i++ ] ) ) { + elem.removeAttribute( name ); + } + } + } + } ); + + // Hooks for boolean attributes + boolHook = { + set: function( elem, value, name ) { + if ( value === false ) { + + // Remove boolean attributes when set to false + jQuery.removeAttr( elem, name ); + } else { + elem.setAttribute( name, name ); + } + return name; + } + }; + + jQuery.each( jQuery.expr.match.bool.source.match( /\w+/g ), function( _i, name ) { + var getter = attrHandle[ name ] || jQuery.find.attr; + + attrHandle[ name ] = function( elem, name, isXML ) { + var ret, handle, + lowercaseName = name.toLowerCase(); + + if ( !isXML ) { + + // Avoid an infinite loop by temporarily removing this function from the getter + handle = attrHandle[ lowercaseName ]; + attrHandle[ lowercaseName ] = ret; + ret = getter( elem, name, isXML ) != null ? + lowercaseName : + null; + attrHandle[ lowercaseName ] = handle; + } + return ret; + }; + } ); + + + + + var rfocusable = /^(?:input|select|textarea|button)$/i, + rclickable = /^(?:a|area)$/i; + + jQuery.fn.extend( { + prop: function( name, value ) { + return access( this, jQuery.prop, name, value, arguments.length > 1 ); + }, + + removeProp: function( name ) { + return this.each( function() { + delete this[ jQuery.propFix[ name ] || name ]; + } ); + } + } ); + + jQuery.extend( { + prop: function( elem, name, value ) { + var ret, hooks, + nType = elem.nodeType; + + // Don't get/set properties on text, comment and attribute nodes + if ( nType === 3 || nType === 8 || nType === 2 ) { + return; + } + + if ( nType !== 1 || !jQuery.isXMLDoc( elem ) ) { + + // Fix name and attach hooks + name = jQuery.propFix[ name ] || name; + hooks = jQuery.propHooks[ name ]; + } + + if ( value !== undefined ) { + if ( hooks && "set" in hooks && + ( ret = hooks.set( elem, value, name ) ) !== undefined ) { + return ret; + } + + return ( elem[ name ] = value ); + } + + if ( hooks && "get" in hooks && ( ret = hooks.get( elem, name ) ) !== null ) { + return ret; + } + + return elem[ name ]; + }, + + propHooks: { + tabIndex: { + get: function( elem ) { + + // Support: IE <=9 - 11 only + // elem.tabIndex doesn't always return the + // correct value when it hasn't been explicitly set + // https://web.archive.org/web/20141116233347/http://fluidproject.org/blog/2008/01/09/getting-setting-and-removing-tabindex-values-with-javascript/ + // Use proper attribute retrieval(#12072) + var tabindex = jQuery.find.attr( elem, "tabindex" ); + + if ( tabindex ) { + return parseInt( tabindex, 10 ); + } + + if ( + rfocusable.test( elem.nodeName ) || + rclickable.test( elem.nodeName ) && + elem.href + ) { + return 0; + } + + return -1; + } + } + }, + + propFix: { + "for": "htmlFor", + "class": "className" + } + } ); + + // Support: IE <=11 only + // Accessing the selectedIndex property + // forces the browser to respect setting selected + // on the option + // The getter ensures a default option is selected + // when in an optgroup + // eslint rule "no-unused-expressions" is disabled for this code + // since it considers such accessions noop + if ( !support.optSelected ) { + jQuery.propHooks.selected = { + get: function( elem ) { + + /* eslint no-unused-expressions: "off" */ + + var parent = elem.parentNode; + if ( parent && parent.parentNode ) { + parent.parentNode.selectedIndex; + } + return null; + }, + set: function( elem ) { + + /* eslint no-unused-expressions: "off" */ + + var parent = elem.parentNode; + if ( parent ) { + parent.selectedIndex; + + if ( parent.parentNode ) { + parent.parentNode.selectedIndex; + } + } + } + }; + } + + jQuery.each( [ + "tabIndex", + "readOnly", + "maxLength", + "cellSpacing", + "cellPadding", + "rowSpan", + "colSpan", + "useMap", + "frameBorder", + "contentEditable" + ], function() { + jQuery.propFix[ this.toLowerCase() ] = this; + } ); + + + + + // Strip and collapse whitespace according to HTML spec + // https://infra.spec.whatwg.org/#strip-and-collapse-ascii-whitespace + function stripAndCollapse( value ) { + var tokens = value.match( rnothtmlwhite ) || []; + return tokens.join( " " ); + } + + + function getClass( elem ) { + return elem.getAttribute && elem.getAttribute( "class" ) || ""; + } + + function classesToArray( value ) { + if ( Array.isArray( value ) ) { + return value; + } + if ( typeof value === "string" ) { + return value.match( rnothtmlwhite ) || []; + } + return []; + } + + jQuery.fn.extend( { + addClass: function( value ) { + var classes, elem, cur, curValue, clazz, j, finalValue, + i = 0; + + if ( isFunction( value ) ) { + return this.each( function( j ) { + jQuery( this ).addClass( value.call( this, j, getClass( this ) ) ); + } ); + } + + classes = classesToArray( value ); + + if ( classes.length ) { + while ( ( elem = this[ i++ ] ) ) { + curValue = getClass( elem ); + cur = elem.nodeType === 1 && ( " " + stripAndCollapse( curValue ) + " " ); + + if ( cur ) { + j = 0; + while ( ( clazz = classes[ j++ ] ) ) { + if ( cur.indexOf( " " + clazz + " " ) < 0 ) { + cur += clazz + " "; + } + } + + // Only assign if different to avoid unneeded rendering. + finalValue = stripAndCollapse( cur ); + if ( curValue !== finalValue ) { + elem.setAttribute( "class", finalValue ); + } + } + } + } + + return this; + }, + + removeClass: function( value ) { + var classes, elem, cur, curValue, clazz, j, finalValue, + i = 0; + + if ( isFunction( value ) ) { + return this.each( function( j ) { + jQuery( this ).removeClass( value.call( this, j, getClass( this ) ) ); + } ); + } + + if ( !arguments.length ) { + return this.attr( "class", "" ); + } + + classes = classesToArray( value ); + + if ( classes.length ) { + while ( ( elem = this[ i++ ] ) ) { + curValue = getClass( elem ); + + // This expression is here for better compressibility (see addClass) + cur = elem.nodeType === 1 && ( " " + stripAndCollapse( curValue ) + " " ); + + if ( cur ) { + j = 0; + while ( ( clazz = classes[ j++ ] ) ) { + + // Remove *all* instances + while ( cur.indexOf( " " + clazz + " " ) > -1 ) { + cur = cur.replace( " " + clazz + " ", " " ); + } + } + + // Only assign if different to avoid unneeded rendering. + finalValue = stripAndCollapse( cur ); + if ( curValue !== finalValue ) { + elem.setAttribute( "class", finalValue ); + } + } + } + } + + return this; + }, + + toggleClass: function( value, stateVal ) { + var type = typeof value, + isValidValue = type === "string" || Array.isArray( value ); + + if ( typeof stateVal === "boolean" && isValidValue ) { + return stateVal ? this.addClass( value ) : this.removeClass( value ); + } + + if ( isFunction( value ) ) { + return this.each( function( i ) { + jQuery( this ).toggleClass( + value.call( this, i, getClass( this ), stateVal ), + stateVal + ); + } ); + } + + return this.each( function() { + var className, i, self, classNames; + + if ( isValidValue ) { + + // Toggle individual class names + i = 0; + self = jQuery( this ); + classNames = classesToArray( value ); + + while ( ( className = classNames[ i++ ] ) ) { + + // Check each className given, space separated list + if ( self.hasClass( className ) ) { + self.removeClass( className ); + } else { + self.addClass( className ); + } + } + + // Toggle whole class name + } else if ( value === undefined || type === "boolean" ) { + className = getClass( this ); + if ( className ) { + + // Store className if set + dataPriv.set( this, "__className__", className ); + } + + // If the element has a class name or if we're passed `false`, + // then remove the whole classname (if there was one, the above saved it). + // Otherwise bring back whatever was previously saved (if anything), + // falling back to the empty string if nothing was stored. + if ( this.setAttribute ) { + this.setAttribute( "class", + className || value === false ? + "" : + dataPriv.get( this, "__className__" ) || "" + ); + } + } + } ); + }, + + hasClass: function( selector ) { + var className, elem, + i = 0; + + className = " " + selector + " "; + while ( ( elem = this[ i++ ] ) ) { + if ( elem.nodeType === 1 && + ( " " + stripAndCollapse( getClass( elem ) ) + " " ).indexOf( className ) > -1 ) { + return true; + } + } + + return false; + } + } ); + + + + + var rreturn = /\r/g; + + jQuery.fn.extend( { + val: function( value ) { + var hooks, ret, valueIsFunction, + elem = this[ 0 ]; + + if ( !arguments.length ) { + if ( elem ) { + hooks = jQuery.valHooks[ elem.type ] || + jQuery.valHooks[ elem.nodeName.toLowerCase() ]; + + if ( hooks && + "get" in hooks && + ( ret = hooks.get( elem, "value" ) ) !== undefined + ) { + return ret; + } + + ret = elem.value; + + // Handle most common string cases + if ( typeof ret === "string" ) { + return ret.replace( rreturn, "" ); + } + + // Handle cases where value is null/undef or number + return ret == null ? "" : ret; + } + + return; + } + + valueIsFunction = isFunction( value ); + + return this.each( function( i ) { + var val; + + if ( this.nodeType !== 1 ) { + return; + } + + if ( valueIsFunction ) { + val = value.call( this, i, jQuery( this ).val() ); + } else { + val = value; + } + + // Treat null/undefined as ""; convert numbers to string + if ( val == null ) { + val = ""; + + } else if ( typeof val === "number" ) { + val += ""; + + } else if ( Array.isArray( val ) ) { + val = jQuery.map( val, function( value ) { + return value == null ? "" : value + ""; + } ); + } + + hooks = jQuery.valHooks[ this.type ] || jQuery.valHooks[ this.nodeName.toLowerCase() ]; + + // If set returns undefined, fall back to normal setting + if ( !hooks || !( "set" in hooks ) || hooks.set( this, val, "value" ) === undefined ) { + this.value = val; + } + } ); + } + } ); + + jQuery.extend( { + valHooks: { + option: { + get: function( elem ) { + + var val = jQuery.find.attr( elem, "value" ); + return val != null ? + val : + + // Support: IE <=10 - 11 only + // option.text throws exceptions (#14686, #14858) + // Strip and collapse whitespace + // https://html.spec.whatwg.org/#strip-and-collapse-whitespace + stripAndCollapse( jQuery.text( elem ) ); + } + }, + select: { + get: function( elem ) { + var value, option, i, + options = elem.options, + index = elem.selectedIndex, + one = elem.type === "select-one", + values = one ? null : [], + max = one ? index + 1 : options.length; + + if ( index < 0 ) { + i = max; + + } else { + i = one ? index : 0; + } + + // Loop through all the selected options + for ( ; i < max; i++ ) { + option = options[ i ]; + + // Support: IE <=9 only + // IE8-9 doesn't update selected after form reset (#2551) + if ( ( option.selected || i === index ) && + + // Don't return options that are disabled or in a disabled optgroup + !option.disabled && + ( !option.parentNode.disabled || + !nodeName( option.parentNode, "optgroup" ) ) ) { + + // Get the specific value for the option + value = jQuery( option ).val(); + + // We don't need an array for one selects + if ( one ) { + return value; + } + + // Multi-Selects return an array + values.push( value ); + } + } + + return values; + }, + + set: function( elem, value ) { + var optionSet, option, + options = elem.options, + values = jQuery.makeArray( value ), + i = options.length; + + while ( i-- ) { + option = options[ i ]; + + /* eslint-disable no-cond-assign */ + + if ( option.selected = + jQuery.inArray( jQuery.valHooks.option.get( option ), values ) > -1 + ) { + optionSet = true; + } + + /* eslint-enable no-cond-assign */ + } + + // Force browsers to behave consistently when non-matching value is set + if ( !optionSet ) { + elem.selectedIndex = -1; + } + return values; + } + } + } + } ); + + // Radios and checkboxes getter/setter + jQuery.each( [ "radio", "checkbox" ], function() { + jQuery.valHooks[ this ] = { + set: function( elem, value ) { + if ( Array.isArray( value ) ) { + return ( elem.checked = jQuery.inArray( jQuery( elem ).val(), value ) > -1 ); + } + } + }; + if ( !support.checkOn ) { + jQuery.valHooks[ this ].get = function( elem ) { + return elem.getAttribute( "value" ) === null ? "on" : elem.value; + }; + } + } ); + + + + + // Return jQuery for attributes-only inclusion + + + support.focusin = "onfocusin" in window; + + + var rfocusMorph = /^(?:focusinfocus|focusoutblur)$/, + stopPropagationCallback = function( e ) { + e.stopPropagation(); + }; + + jQuery.extend( jQuery.event, { + + trigger: function( event, data, elem, onlyHandlers ) { + + var i, cur, tmp, bubbleType, ontype, handle, special, lastElement, + eventPath = [ elem || document ], + type = hasOwn.call( event, "type" ) ? event.type : event, + namespaces = hasOwn.call( event, "namespace" ) ? event.namespace.split( "." ) : []; + + cur = lastElement = tmp = elem = elem || document; + + // Don't do events on text and comment nodes + if ( elem.nodeType === 3 || elem.nodeType === 8 ) { + return; + } + + // focus/blur morphs to focusin/out; ensure we're not firing them right now + if ( rfocusMorph.test( type + jQuery.event.triggered ) ) { + return; + } + + if ( type.indexOf( "." ) > -1 ) { + + // Namespaced trigger; create a regexp to match event type in handle() + namespaces = type.split( "." ); + type = namespaces.shift(); + namespaces.sort(); + } + ontype = type.indexOf( ":" ) < 0 && "on" + type; + + // Caller can pass in a jQuery.Event object, Object, or just an event type string + event = event[ jQuery.expando ] ? + event : + new jQuery.Event( type, typeof event === "object" && event ); + + // Trigger bitmask: & 1 for native handlers; & 2 for jQuery (always true) + event.isTrigger = onlyHandlers ? 2 : 3; + event.namespace = namespaces.join( "." ); + event.rnamespace = event.namespace ? + new RegExp( "(^|\\.)" + namespaces.join( "\\.(?:.*\\.|)" ) + "(\\.|$)" ) : + null; + + // Clean up the event in case it is being reused + event.result = undefined; + if ( !event.target ) { + event.target = elem; + } + + // Clone any incoming data and prepend the event, creating the handler arg list + data = data == null ? + [ event ] : + jQuery.makeArray( data, [ event ] ); + + // Allow special events to draw outside the lines + special = jQuery.event.special[ type ] || {}; + if ( !onlyHandlers && special.trigger && special.trigger.apply( elem, data ) === false ) { + return; + } + + // Determine event propagation path in advance, per W3C events spec (#9951) + // Bubble up to document, then to window; watch for a global ownerDocument var (#9724) + if ( !onlyHandlers && !special.noBubble && !isWindow( elem ) ) { + + bubbleType = special.delegateType || type; + if ( !rfocusMorph.test( bubbleType + type ) ) { + cur = cur.parentNode; + } + for ( ; cur; cur = cur.parentNode ) { + eventPath.push( cur ); + tmp = cur; + } + + // Only add window if we got to document (e.g., not plain obj or detached DOM) + if ( tmp === ( elem.ownerDocument || document ) ) { + eventPath.push( tmp.defaultView || tmp.parentWindow || window ); + } + } + + // Fire handlers on the event path + i = 0; + while ( ( cur = eventPath[ i++ ] ) && !event.isPropagationStopped() ) { + lastElement = cur; + event.type = i > 1 ? + bubbleType : + special.bindType || type; + + // jQuery handler + handle = ( dataPriv.get( cur, "events" ) || Object.create( null ) )[ event.type ] && + dataPriv.get( cur, "handle" ); + if ( handle ) { + handle.apply( cur, data ); + } + + // Native handler + handle = ontype && cur[ ontype ]; + if ( handle && handle.apply && acceptData( cur ) ) { + event.result = handle.apply( cur, data ); + if ( event.result === false ) { + event.preventDefault(); + } + } + } + event.type = type; + + // If nobody prevented the default action, do it now + if ( !onlyHandlers && !event.isDefaultPrevented() ) { + + if ( ( !special._default || + special._default.apply( eventPath.pop(), data ) === false ) && + acceptData( elem ) ) { + + // Call a native DOM method on the target with the same name as the event. + // Don't do default actions on window, that's where global variables be (#6170) + if ( ontype && isFunction( elem[ type ] ) && !isWindow( elem ) ) { + + // Don't re-trigger an onFOO event when we call its FOO() method + tmp = elem[ ontype ]; + + if ( tmp ) { + elem[ ontype ] = null; + } + + // Prevent re-triggering of the same event, since we already bubbled it above + jQuery.event.triggered = type; + + if ( event.isPropagationStopped() ) { + lastElement.addEventListener( type, stopPropagationCallback ); + } + + elem[ type ](); + + if ( event.isPropagationStopped() ) { + lastElement.removeEventListener( type, stopPropagationCallback ); + } + + jQuery.event.triggered = undefined; + + if ( tmp ) { + elem[ ontype ] = tmp; + } + } + } + } + + return event.result; + }, + + // Piggyback on a donor event to simulate a different one + // Used only for `focus(in | out)` events + simulate: function( type, elem, event ) { + var e = jQuery.extend( + new jQuery.Event(), + event, + { + type: type, + isSimulated: true + } + ); + + jQuery.event.trigger( e, null, elem ); + } + + } ); + + jQuery.fn.extend( { + + trigger: function( type, data ) { + return this.each( function() { + jQuery.event.trigger( type, data, this ); + } ); + }, + triggerHandler: function( type, data ) { + var elem = this[ 0 ]; + if ( elem ) { + return jQuery.event.trigger( type, data, elem, true ); + } + } + } ); + + + // Support: Firefox <=44 + // Firefox doesn't have focus(in | out) events + // Related ticket - https://bugzilla.mozilla.org/show_bug.cgi?id=687787 + // + // Support: Chrome <=48 - 49, Safari <=9.0 - 9.1 + // focus(in | out) events fire after focus & blur events, + // which is spec violation - http://www.w3.org/TR/DOM-Level-3-Events/#events-focusevent-event-order + // Related ticket - https://bugs.chromium.org/p/chromium/issues/detail?id=449857 + if ( !support.focusin ) { + jQuery.each( { focus: "focusin", blur: "focusout" }, function( orig, fix ) { + + // Attach a single capturing handler on the document while someone wants focusin/focusout + var handler = function( event ) { + jQuery.event.simulate( fix, event.target, jQuery.event.fix( event ) ); + }; + + jQuery.event.special[ fix ] = { + setup: function() { + + // Handle: regular nodes (via `this.ownerDocument`), window + // (via `this.document`) & document (via `this`). + var doc = this.ownerDocument || this.document || this, + attaches = dataPriv.access( doc, fix ); + + if ( !attaches ) { + doc.addEventListener( orig, handler, true ); + } + dataPriv.access( doc, fix, ( attaches || 0 ) + 1 ); + }, + teardown: function() { + var doc = this.ownerDocument || this.document || this, + attaches = dataPriv.access( doc, fix ) - 1; + + if ( !attaches ) { + doc.removeEventListener( orig, handler, true ); + dataPriv.remove( doc, fix ); + + } else { + dataPriv.access( doc, fix, attaches ); + } + } + }; + } ); + } + var location = window.location; + + var nonce = { guid: Date.now() }; + + var rquery = ( /\?/ ); + + + + // Cross-browser xml parsing + jQuery.parseXML = function( data ) { + var xml, parserErrorElem; + if ( !data || typeof data !== "string" ) { + return null; + } + + // Support: IE 9 - 11 only + // IE throws on parseFromString with invalid input. + try { + xml = ( new window.DOMParser() ).parseFromString( data, "text/xml" ); + } catch ( e ) {} + + parserErrorElem = xml && xml.getElementsByTagName( "parsererror" )[ 0 ]; + if ( !xml || parserErrorElem ) { + jQuery.error( "Invalid XML: " + ( + parserErrorElem ? + jQuery.map( parserErrorElem.childNodes, function( el ) { + return el.textContent; + } ).join( "\n" ) : + data + ) ); + } + return xml; + }; + + + var + rbracket = /\[\]$/, + rCRLF = /\r?\n/g, + rsubmitterTypes = /^(?:submit|button|image|reset|file)$/i, + rsubmittable = /^(?:input|select|textarea|keygen)/i; + + function buildParams( prefix, obj, traditional, add ) { + var name; + + if ( Array.isArray( obj ) ) { + + // Serialize array item. + jQuery.each( obj, function( i, v ) { + if ( traditional || rbracket.test( prefix ) ) { + + // Treat each array item as a scalar. + add( prefix, v ); + + } else { + + // Item is non-scalar (array or object), encode its numeric index. + buildParams( + prefix + "[" + ( typeof v === "object" && v != null ? i : "" ) + "]", + v, + traditional, + add + ); + } + } ); + + } else if ( !traditional && toType( obj ) === "object" ) { + + // Serialize object item. + for ( name in obj ) { + buildParams( prefix + "[" + name + "]", obj[ name ], traditional, add ); + } + + } else { + + // Serialize scalar item. + add( prefix, obj ); + } + } + + // Serialize an array of form elements or a set of + // key/values into a query string + jQuery.param = function( a, traditional ) { + var prefix, + s = [], + add = function( key, valueOrFunction ) { + + // If value is a function, invoke it and use its return value + var value = isFunction( valueOrFunction ) ? + valueOrFunction() : + valueOrFunction; + + s[ s.length ] = encodeURIComponent( key ) + "=" + + encodeURIComponent( value == null ? "" : value ); + }; + + if ( a == null ) { + return ""; + } + + // If an array was passed in, assume that it is an array of form elements. + if ( Array.isArray( a ) || ( a.jquery && !jQuery.isPlainObject( a ) ) ) { + + // Serialize the form elements + jQuery.each( a, function() { + add( this.name, this.value ); + } ); + + } else { + + // If traditional, encode the "old" way (the way 1.3.2 or older + // did it), otherwise encode params recursively. + for ( prefix in a ) { + buildParams( prefix, a[ prefix ], traditional, add ); + } + } + + // Return the resulting serialization + return s.join( "&" ); + }; + + jQuery.fn.extend( { + serialize: function() { + return jQuery.param( this.serializeArray() ); + }, + serializeArray: function() { + return this.map( function() { + + // Can add propHook for "elements" to filter or add form elements + var elements = jQuery.prop( this, "elements" ); + return elements ? jQuery.makeArray( elements ) : this; + } ).filter( function() { + var type = this.type; + + // Use .is( ":disabled" ) so that fieldset[disabled] works + return this.name && !jQuery( this ).is( ":disabled" ) && + rsubmittable.test( this.nodeName ) && !rsubmitterTypes.test( type ) && + ( this.checked || !rcheckableType.test( type ) ); + } ).map( function( _i, elem ) { + var val = jQuery( this ).val(); + + if ( val == null ) { + return null; + } + + if ( Array.isArray( val ) ) { + return jQuery.map( val, function( val ) { + return { name: elem.name, value: val.replace( rCRLF, "\r\n" ) }; + } ); + } + + return { name: elem.name, value: val.replace( rCRLF, "\r\n" ) }; + } ).get(); + } + } ); + + + var + r20 = /%20/g, + rhash = /#.*$/, + rantiCache = /([?&])_=[^&]*/, + rheaders = /^(.*?):[ \t]*([^\r\n]*)$/mg, + + // #7653, #8125, #8152: local protocol detection + rlocalProtocol = /^(?:about|app|app-storage|.+-extension|file|res|widget):$/, + rnoContent = /^(?:GET|HEAD)$/, + rprotocol = /^\/\//, + + /* Prefilters + * 1) They are useful to introduce custom dataTypes (see ajax/jsonp.js for an example) + * 2) These are called: + * - BEFORE asking for a transport + * - AFTER param serialization (s.data is a string if s.processData is true) + * 3) key is the dataType + * 4) the catchall symbol "*" can be used + * 5) execution will start with transport dataType and THEN continue down to "*" if needed + */ + prefilters = {}, + + /* Transports bindings + * 1) key is the dataType + * 2) the catchall symbol "*" can be used + * 3) selection will start with transport dataType and THEN go to "*" if needed + */ + transports = {}, + + // Avoid comment-prolog char sequence (#10098); must appease lint and evade compression + allTypes = "*/".concat( "*" ), + + // Anchor tag for parsing the document origin + originAnchor = document.createElement( "a" ); + + originAnchor.href = location.href; + + // Base "constructor" for jQuery.ajaxPrefilter and jQuery.ajaxTransport + function addToPrefiltersOrTransports( structure ) { + + // dataTypeExpression is optional and defaults to "*" + return function( dataTypeExpression, func ) { + + if ( typeof dataTypeExpression !== "string" ) { + func = dataTypeExpression; + dataTypeExpression = "*"; + } + + var dataType, + i = 0, + dataTypes = dataTypeExpression.toLowerCase().match( rnothtmlwhite ) || []; + + if ( isFunction( func ) ) { + + // For each dataType in the dataTypeExpression + while ( ( dataType = dataTypes[ i++ ] ) ) { + + // Prepend if requested + if ( dataType[ 0 ] === "+" ) { + dataType = dataType.slice( 1 ) || "*"; + ( structure[ dataType ] = structure[ dataType ] || [] ).unshift( func ); + + // Otherwise append + } else { + ( structure[ dataType ] = structure[ dataType ] || [] ).push( func ); + } + } + } + }; + } + + // Base inspection function for prefilters and transports + function inspectPrefiltersOrTransports( structure, options, originalOptions, jqXHR ) { + + var inspected = {}, + seekingTransport = ( structure === transports ); + + function inspect( dataType ) { + var selected; + inspected[ dataType ] = true; + jQuery.each( structure[ dataType ] || [], function( _, prefilterOrFactory ) { + var dataTypeOrTransport = prefilterOrFactory( options, originalOptions, jqXHR ); + if ( typeof dataTypeOrTransport === "string" && + !seekingTransport && !inspected[ dataTypeOrTransport ] ) { + + options.dataTypes.unshift( dataTypeOrTransport ); + inspect( dataTypeOrTransport ); + return false; + } else if ( seekingTransport ) { + return !( selected = dataTypeOrTransport ); + } + } ); + return selected; + } + + return inspect( options.dataTypes[ 0 ] ) || !inspected[ "*" ] && inspect( "*" ); + } + + // A special extend for ajax options + // that takes "flat" options (not to be deep extended) + // Fixes #9887 + function ajaxExtend( target, src ) { + var key, deep, + flatOptions = jQuery.ajaxSettings.flatOptions || {}; + + for ( key in src ) { + if ( src[ key ] !== undefined ) { + ( flatOptions[ key ] ? target : ( deep || ( deep = {} ) ) )[ key ] = src[ key ]; + } + } + if ( deep ) { + jQuery.extend( true, target, deep ); + } + + return target; + } + + /* Handles responses to an ajax request: + * - finds the right dataType (mediates between content-type and expected dataType) + * - returns the corresponding response + */ + function ajaxHandleResponses( s, jqXHR, responses ) { + + var ct, type, finalDataType, firstDataType, + contents = s.contents, + dataTypes = s.dataTypes; + + // Remove auto dataType and get content-type in the process + while ( dataTypes[ 0 ] === "*" ) { + dataTypes.shift(); + if ( ct === undefined ) { + ct = s.mimeType || jqXHR.getResponseHeader( "Content-Type" ); + } + } + + // Check if we're dealing with a known content-type + if ( ct ) { + for ( type in contents ) { + if ( contents[ type ] && contents[ type ].test( ct ) ) { + dataTypes.unshift( type ); + break; + } + } + } + + // Check to see if we have a response for the expected dataType + if ( dataTypes[ 0 ] in responses ) { + finalDataType = dataTypes[ 0 ]; + } else { + + // Try convertible dataTypes + for ( type in responses ) { + if ( !dataTypes[ 0 ] || s.converters[ type + " " + dataTypes[ 0 ] ] ) { + finalDataType = type; + break; + } + if ( !firstDataType ) { + firstDataType = type; + } + } + + // Or just use first one + finalDataType = finalDataType || firstDataType; + } + + // If we found a dataType + // We add the dataType to the list if needed + // and return the corresponding response + if ( finalDataType ) { + if ( finalDataType !== dataTypes[ 0 ] ) { + dataTypes.unshift( finalDataType ); + } + return responses[ finalDataType ]; + } + } + + /* Chain conversions given the request and the original response + * Also sets the responseXXX fields on the jqXHR instance + */ + function ajaxConvert( s, response, jqXHR, isSuccess ) { + var conv2, current, conv, tmp, prev, + converters = {}, + + // Work with a copy of dataTypes in case we need to modify it for conversion + dataTypes = s.dataTypes.slice(); + + // Create converters map with lowercased keys + if ( dataTypes[ 1 ] ) { + for ( conv in s.converters ) { + converters[ conv.toLowerCase() ] = s.converters[ conv ]; + } + } + + current = dataTypes.shift(); + + // Convert to each sequential dataType + while ( current ) { + + if ( s.responseFields[ current ] ) { + jqXHR[ s.responseFields[ current ] ] = response; + } + + // Apply the dataFilter if provided + if ( !prev && isSuccess && s.dataFilter ) { + response = s.dataFilter( response, s.dataType ); + } + + prev = current; + current = dataTypes.shift(); + + if ( current ) { + + // There's only work to do if current dataType is non-auto + if ( current === "*" ) { + + current = prev; + + // Convert response if prev dataType is non-auto and differs from current + } else if ( prev !== "*" && prev !== current ) { + + // Seek a direct converter + conv = converters[ prev + " " + current ] || converters[ "* " + current ]; + + // If none found, seek a pair + if ( !conv ) { + for ( conv2 in converters ) { + + // If conv2 outputs current + tmp = conv2.split( " " ); + if ( tmp[ 1 ] === current ) { + + // If prev can be converted to accepted input + conv = converters[ prev + " " + tmp[ 0 ] ] || + converters[ "* " + tmp[ 0 ] ]; + if ( conv ) { + + // Condense equivalence converters + if ( conv === true ) { + conv = converters[ conv2 ]; + + // Otherwise, insert the intermediate dataType + } else if ( converters[ conv2 ] !== true ) { + current = tmp[ 0 ]; + dataTypes.unshift( tmp[ 1 ] ); + } + break; + } + } + } + } + + // Apply converter (if not an equivalence) + if ( conv !== true ) { + + // Unless errors are allowed to bubble, catch and return them + if ( conv && s.throws ) { + response = conv( response ); + } else { + try { + response = conv( response ); + } catch ( e ) { + return { + state: "parsererror", + error: conv ? e : "No conversion from " + prev + " to " + current + }; + } + } + } + } + } + } + + return { state: "success", data: response }; + } + + jQuery.extend( { + + // Counter for holding the number of active queries + active: 0, + + // Last-Modified header cache for next request + lastModified: {}, + etag: {}, + + ajaxSettings: { + url: location.href, + type: "GET", + isLocal: rlocalProtocol.test( location.protocol ), + global: true, + processData: true, + async: true, + contentType: "application/x-www-form-urlencoded; charset=UTF-8", + + /* + timeout: 0, + data: null, + dataType: null, + username: null, + password: null, + cache: null, + throws: false, + traditional: false, + headers: {}, + */ + + accepts: { + "*": allTypes, + text: "text/plain", + html: "text/html", + xml: "application/xml, text/xml", + json: "application/json, text/javascript" + }, + + contents: { + xml: /\bxml\b/, + html: /\bhtml/, + json: /\bjson\b/ + }, + + responseFields: { + xml: "responseXML", + text: "responseText", + json: "responseJSON" + }, + + // Data converters + // Keys separate source (or catchall "*") and destination types with a single space + converters: { + + // Convert anything to text + "* text": String, + + // Text to html (true = no transformation) + "text html": true, + + // Evaluate text as a json expression + "text json": JSON.parse, + + // Parse text as xml + "text xml": jQuery.parseXML + }, + + // For options that shouldn't be deep extended: + // you can add your own custom options here if + // and when you create one that shouldn't be + // deep extended (see ajaxExtend) + flatOptions: { + url: true, + context: true + } + }, + + // Creates a full fledged settings object into target + // with both ajaxSettings and settings fields. + // If target is omitted, writes into ajaxSettings. + ajaxSetup: function( target, settings ) { + return settings ? + + // Building a settings object + ajaxExtend( ajaxExtend( target, jQuery.ajaxSettings ), settings ) : + + // Extending ajaxSettings + ajaxExtend( jQuery.ajaxSettings, target ); + }, + + ajaxPrefilter: addToPrefiltersOrTransports( prefilters ), + ajaxTransport: addToPrefiltersOrTransports( transports ), + + // Main method + ajax: function( url, options ) { + + // If url is an object, simulate pre-1.5 signature + if ( typeof url === "object" ) { + options = url; + url = undefined; + } + + // Force options to be an object + options = options || {}; + + var transport, + + // URL without anti-cache param + cacheURL, + + // Response headers + responseHeadersString, + responseHeaders, + + // timeout handle + timeoutTimer, + + // Url cleanup var + urlAnchor, + + // Request state (becomes false upon send and true upon completion) + completed, + + // To know if global events are to be dispatched + fireGlobals, + + // Loop variable + i, + + // uncached part of the url + uncached, + + // Create the final options object + s = jQuery.ajaxSetup( {}, options ), + + // Callbacks context + callbackContext = s.context || s, + + // Context for global events is callbackContext if it is a DOM node or jQuery collection + globalEventContext = s.context && + ( callbackContext.nodeType || callbackContext.jquery ) ? + jQuery( callbackContext ) : + jQuery.event, + + // Deferreds + deferred = jQuery.Deferred(), + completeDeferred = jQuery.Callbacks( "once memory" ), + + // Status-dependent callbacks + statusCode = s.statusCode || {}, + + // Headers (they are sent all at once) + requestHeaders = {}, + requestHeadersNames = {}, + + // Default abort message + strAbort = "canceled", + + // Fake xhr + jqXHR = { + readyState: 0, + + // Builds headers hashtable if needed + getResponseHeader: function( key ) { + var match; + if ( completed ) { + if ( !responseHeaders ) { + responseHeaders = {}; + while ( ( match = rheaders.exec( responseHeadersString ) ) ) { + responseHeaders[ match[ 1 ].toLowerCase() + " " ] = + ( responseHeaders[ match[ 1 ].toLowerCase() + " " ] || [] ) + .concat( match[ 2 ] ); + } + } + match = responseHeaders[ key.toLowerCase() + " " ]; + } + return match == null ? null : match.join( ", " ); + }, + + // Raw string + getAllResponseHeaders: function() { + return completed ? responseHeadersString : null; + }, + + // Caches the header + setRequestHeader: function( name, value ) { + if ( completed == null ) { + name = requestHeadersNames[ name.toLowerCase() ] = + requestHeadersNames[ name.toLowerCase() ] || name; + requestHeaders[ name ] = value; + } + return this; + }, + + // Overrides response content-type header + overrideMimeType: function( type ) { + if ( completed == null ) { + s.mimeType = type; + } + return this; + }, + + // Status-dependent callbacks + statusCode: function( map ) { + var code; + if ( map ) { + if ( completed ) { + + // Execute the appropriate callbacks + jqXHR.always( map[ jqXHR.status ] ); + } else { + + // Lazy-add the new callbacks in a way that preserves old ones + for ( code in map ) { + statusCode[ code ] = [ statusCode[ code ], map[ code ] ]; + } + } + } + return this; + }, + + // Cancel the request + abort: function( statusText ) { + var finalText = statusText || strAbort; + if ( transport ) { + transport.abort( finalText ); + } + done( 0, finalText ); + return this; + } + }; + + // Attach deferreds + deferred.promise( jqXHR ); + + // Add protocol if not provided (prefilters might expect it) + // Handle falsy url in the settings object (#10093: consistency with old signature) + // We also use the url parameter if available + s.url = ( ( url || s.url || location.href ) + "" ) + .replace( rprotocol, location.protocol + "//" ); + + // Alias method option to type as per ticket #12004 + s.type = options.method || options.type || s.method || s.type; + + // Extract dataTypes list + s.dataTypes = ( s.dataType || "*" ).toLowerCase().match( rnothtmlwhite ) || [ "" ]; + + // A cross-domain request is in order when the origin doesn't match the current origin. + if ( s.crossDomain == null ) { + urlAnchor = document.createElement( "a" ); + + // Support: IE <=8 - 11, Edge 12 - 15 + // IE throws exception on accessing the href property if url is malformed, + // e.g. http://example.com:80x/ + try { + urlAnchor.href = s.url; + + // Support: IE <=8 - 11 only + // Anchor's host property isn't correctly set when s.url is relative + urlAnchor.href = urlAnchor.href; + s.crossDomain = originAnchor.protocol + "//" + originAnchor.host !== + urlAnchor.protocol + "//" + urlAnchor.host; + } catch ( e ) { + + // If there is an error parsing the URL, assume it is crossDomain, + // it can be rejected by the transport if it is invalid + s.crossDomain = true; + } + } + + // Convert data if not already a string + if ( s.data && s.processData && typeof s.data !== "string" ) { + s.data = jQuery.param( s.data, s.traditional ); + } + + // Apply prefilters + inspectPrefiltersOrTransports( prefilters, s, options, jqXHR ); + + // If request was aborted inside a prefilter, stop there + if ( completed ) { + return jqXHR; + } + + // We can fire global events as of now if asked to + // Don't fire events if jQuery.event is undefined in an AMD-usage scenario (#15118) + fireGlobals = jQuery.event && s.global; + + // Watch for a new set of requests + if ( fireGlobals && jQuery.active++ === 0 ) { + jQuery.event.trigger( "ajaxStart" ); + } + + // Uppercase the type + s.type = s.type.toUpperCase(); + + // Determine if request has content + s.hasContent = !rnoContent.test( s.type ); + + // Save the URL in case we're toying with the If-Modified-Since + // and/or If-None-Match header later on + // Remove hash to simplify url manipulation + cacheURL = s.url.replace( rhash, "" ); + + // More options handling for requests with no content + if ( !s.hasContent ) { + + // Remember the hash so we can put it back + uncached = s.url.slice( cacheURL.length ); + + // If data is available and should be processed, append data to url + if ( s.data && ( s.processData || typeof s.data === "string" ) ) { + cacheURL += ( rquery.test( cacheURL ) ? "&" : "?" ) + s.data; + + // #9682: remove data so that it's not used in an eventual retry + delete s.data; + } + + // Add or update anti-cache param if needed + if ( s.cache === false ) { + cacheURL = cacheURL.replace( rantiCache, "$1" ); + uncached = ( rquery.test( cacheURL ) ? "&" : "?" ) + "_=" + ( nonce.guid++ ) + + uncached; + } + + // Put hash and anti-cache on the URL that will be requested (gh-1732) + s.url = cacheURL + uncached; + + // Change '%20' to '+' if this is encoded form body content (gh-2658) + } else if ( s.data && s.processData && + ( s.contentType || "" ).indexOf( "application/x-www-form-urlencoded" ) === 0 ) { + s.data = s.data.replace( r20, "+" ); + } + + // Set the If-Modified-Since and/or If-None-Match header, if in ifModified mode. + if ( s.ifModified ) { + if ( jQuery.lastModified[ cacheURL ] ) { + jqXHR.setRequestHeader( "If-Modified-Since", jQuery.lastModified[ cacheURL ] ); + } + if ( jQuery.etag[ cacheURL ] ) { + jqXHR.setRequestHeader( "If-None-Match", jQuery.etag[ cacheURL ] ); + } + } + + // Set the correct header, if data is being sent + if ( s.data && s.hasContent && s.contentType !== false || options.contentType ) { + jqXHR.setRequestHeader( "Content-Type", s.contentType ); + } + + // Set the Accepts header for the server, depending on the dataType + jqXHR.setRequestHeader( + "Accept", + s.dataTypes[ 0 ] && s.accepts[ s.dataTypes[ 0 ] ] ? + s.accepts[ s.dataTypes[ 0 ] ] + + ( s.dataTypes[ 0 ] !== "*" ? ", " + allTypes + "; q=0.01" : "" ) : + s.accepts[ "*" ] + ); + + // Check for headers option + for ( i in s.headers ) { + jqXHR.setRequestHeader( i, s.headers[ i ] ); + } + + // Allow custom headers/mimetypes and early abort + if ( s.beforeSend && + ( s.beforeSend.call( callbackContext, jqXHR, s ) === false || completed ) ) { + + // Abort if not done already and return + return jqXHR.abort(); + } + + // Aborting is no longer a cancellation + strAbort = "abort"; + + // Install callbacks on deferreds + completeDeferred.add( s.complete ); + jqXHR.done( s.success ); + jqXHR.fail( s.error ); + + // Get transport + transport = inspectPrefiltersOrTransports( transports, s, options, jqXHR ); + + // If no transport, we auto-abort + if ( !transport ) { + done( -1, "No Transport" ); + } else { + jqXHR.readyState = 1; + + // Send global event + if ( fireGlobals ) { + globalEventContext.trigger( "ajaxSend", [ jqXHR, s ] ); + } + + // If request was aborted inside ajaxSend, stop there + if ( completed ) { + return jqXHR; + } + + // Timeout + if ( s.async && s.timeout > 0 ) { + timeoutTimer = window.setTimeout( function() { + jqXHR.abort( "timeout" ); + }, s.timeout ); + } + + try { + completed = false; + transport.send( requestHeaders, done ); + } catch ( e ) { + + // Rethrow post-completion exceptions + if ( completed ) { + throw e; + } + + // Propagate others as results + done( -1, e ); + } + } + + // Callback for when everything is done + function done( status, nativeStatusText, responses, headers ) { + var isSuccess, success, error, response, modified, + statusText = nativeStatusText; + + // Ignore repeat invocations + if ( completed ) { + return; + } + + completed = true; + + // Clear timeout if it exists + if ( timeoutTimer ) { + window.clearTimeout( timeoutTimer ); + } + + // Dereference transport for early garbage collection + // (no matter how long the jqXHR object will be used) + transport = undefined; + + // Cache response headers + responseHeadersString = headers || ""; + + // Set readyState + jqXHR.readyState = status > 0 ? 4 : 0; + + // Determine if successful + isSuccess = status >= 200 && status < 300 || status === 304; + + // Get response data + if ( responses ) { + response = ajaxHandleResponses( s, jqXHR, responses ); + } + + // Use a noop converter for missing script but not if jsonp + if ( !isSuccess && + jQuery.inArray( "script", s.dataTypes ) > -1 && + jQuery.inArray( "json", s.dataTypes ) < 0 ) { + s.converters[ "text script" ] = function() {}; + } + + // Convert no matter what (that way responseXXX fields are always set) + response = ajaxConvert( s, response, jqXHR, isSuccess ); + + // If successful, handle type chaining + if ( isSuccess ) { + + // Set the If-Modified-Since and/or If-None-Match header, if in ifModified mode. + if ( s.ifModified ) { + modified = jqXHR.getResponseHeader( "Last-Modified" ); + if ( modified ) { + jQuery.lastModified[ cacheURL ] = modified; + } + modified = jqXHR.getResponseHeader( "etag" ); + if ( modified ) { + jQuery.etag[ cacheURL ] = modified; + } + } + + // if no content + if ( status === 204 || s.type === "HEAD" ) { + statusText = "nocontent"; + + // if not modified + } else if ( status === 304 ) { + statusText = "notmodified"; + + // If we have data, let's convert it + } else { + statusText = response.state; + success = response.data; + error = response.error; + isSuccess = !error; + } + } else { + + // Extract error from statusText and normalize for non-aborts + error = statusText; + if ( status || !statusText ) { + statusText = "error"; + if ( status < 0 ) { + status = 0; + } + } + } + + // Set data for the fake xhr object + jqXHR.status = status; + jqXHR.statusText = ( nativeStatusText || statusText ) + ""; + + // Success/Error + if ( isSuccess ) { + deferred.resolveWith( callbackContext, [ success, statusText, jqXHR ] ); + } else { + deferred.rejectWith( callbackContext, [ jqXHR, statusText, error ] ); + } + + // Status-dependent callbacks + jqXHR.statusCode( statusCode ); + statusCode = undefined; + + if ( fireGlobals ) { + globalEventContext.trigger( isSuccess ? "ajaxSuccess" : "ajaxError", + [ jqXHR, s, isSuccess ? success : error ] ); + } + + // Complete + completeDeferred.fireWith( callbackContext, [ jqXHR, statusText ] ); + + if ( fireGlobals ) { + globalEventContext.trigger( "ajaxComplete", [ jqXHR, s ] ); + + // Handle the global AJAX counter + if ( !( --jQuery.active ) ) { + jQuery.event.trigger( "ajaxStop" ); + } + } + } + + return jqXHR; + }, + + getJSON: function( url, data, callback ) { + return jQuery.get( url, data, callback, "json" ); + }, + + getScript: function( url, callback ) { + return jQuery.get( url, undefined, callback, "script" ); + } + } ); + + jQuery.each( [ "get", "post" ], function( _i, method ) { + jQuery[ method ] = function( url, data, callback, type ) { + + // Shift arguments if data argument was omitted + if ( isFunction( data ) ) { + type = type || callback; + callback = data; + data = undefined; + } + + // The url can be an options object (which then must have .url) + return jQuery.ajax( jQuery.extend( { + url: url, + type: method, + dataType: type, + data: data, + success: callback + }, jQuery.isPlainObject( url ) && url ) ); + }; + } ); + + jQuery.ajaxPrefilter( function( s ) { + var i; + for ( i in s.headers ) { + if ( i.toLowerCase() === "content-type" ) { + s.contentType = s.headers[ i ] || ""; + } + } + } ); + + + jQuery._evalUrl = function( url, options, doc ) { + return jQuery.ajax( { + url: url, + + // Make this explicit, since user can override this through ajaxSetup (#11264) + type: "GET", + dataType: "script", + cache: true, + async: false, + global: false, + + // Only evaluate the response if it is successful (gh-4126) + // dataFilter is not invoked for failure responses, so using it instead + // of the default converter is kludgy but it works. + converters: { + "text script": function() {} + }, + dataFilter: function( response ) { + jQuery.globalEval( response, options, doc ); + } + } ); + }; + + + jQuery.fn.extend( { + wrapAll: function( html ) { + var wrap; + + if ( this[ 0 ] ) { + if ( isFunction( html ) ) { + html = html.call( this[ 0 ] ); + } + + // The elements to wrap the target around + wrap = jQuery( html, this[ 0 ].ownerDocument ).eq( 0 ).clone( true ); + + if ( this[ 0 ].parentNode ) { + wrap.insertBefore( this[ 0 ] ); + } + + wrap.map( function() { + var elem = this; + + while ( elem.firstElementChild ) { + elem = elem.firstElementChild; + } + + return elem; + } ).append( this ); + } + + return this; + }, + + wrapInner: function( html ) { + if ( isFunction( html ) ) { + return this.each( function( i ) { + jQuery( this ).wrapInner( html.call( this, i ) ); + } ); + } + + return this.each( function() { + var self = jQuery( this ), + contents = self.contents(); + + if ( contents.length ) { + contents.wrapAll( html ); + + } else { + self.append( html ); + } + } ); + }, + + wrap: function( html ) { + var htmlIsFunction = isFunction( html ); + + return this.each( function( i ) { + jQuery( this ).wrapAll( htmlIsFunction ? html.call( this, i ) : html ); + } ); + }, + + unwrap: function( selector ) { + this.parent( selector ).not( "body" ).each( function() { + jQuery( this ).replaceWith( this.childNodes ); + } ); + return this; + } + } ); + + + jQuery.expr.pseudos.hidden = function( elem ) { + return !jQuery.expr.pseudos.visible( elem ); + }; + jQuery.expr.pseudos.visible = function( elem ) { + return !!( elem.offsetWidth || elem.offsetHeight || elem.getClientRects().length ); + }; + + + + + jQuery.ajaxSettings.xhr = function() { + try { + return new window.XMLHttpRequest(); + } catch ( e ) {} + }; + + var xhrSuccessStatus = { + + // File protocol always yields status code 0, assume 200 + 0: 200, + + // Support: IE <=9 only + // #1450: sometimes IE returns 1223 when it should be 204 + 1223: 204 + }, + xhrSupported = jQuery.ajaxSettings.xhr(); + + support.cors = !!xhrSupported && ( "withCredentials" in xhrSupported ); + support.ajax = xhrSupported = !!xhrSupported; + + jQuery.ajaxTransport( function( options ) { + var callback, errorCallback; + + // Cross domain only allowed if supported through XMLHttpRequest + if ( support.cors || xhrSupported && !options.crossDomain ) { + return { + send: function( headers, complete ) { + var i, + xhr = options.xhr(); + + xhr.open( + options.type, + options.url, + options.async, + options.username, + options.password + ); + + // Apply custom fields if provided + if ( options.xhrFields ) { + for ( i in options.xhrFields ) { + xhr[ i ] = options.xhrFields[ i ]; + } + } + + // Override mime type if needed + if ( options.mimeType && xhr.overrideMimeType ) { + xhr.overrideMimeType( options.mimeType ); + } + + // X-Requested-With header + // For cross-domain requests, seeing as conditions for a preflight are + // akin to a jigsaw puzzle, we simply never set it to be sure. + // (it can always be set on a per-request basis or even using ajaxSetup) + // For same-domain requests, won't change header if already provided. + if ( !options.crossDomain && !headers[ "X-Requested-With" ] ) { + headers[ "X-Requested-With" ] = "XMLHttpRequest"; + } + + // Set headers + for ( i in headers ) { + xhr.setRequestHeader( i, headers[ i ] ); + } + + // Callback + callback = function( type ) { + return function() { + if ( callback ) { + callback = errorCallback = xhr.onload = + xhr.onerror = xhr.onabort = xhr.ontimeout = + xhr.onreadystatechange = null; + + if ( type === "abort" ) { + xhr.abort(); + } else if ( type === "error" ) { + + // Support: IE <=9 only + // On a manual native abort, IE9 throws + // errors on any property access that is not readyState + if ( typeof xhr.status !== "number" ) { + complete( 0, "error" ); + } else { + complete( + + // File: protocol always yields status 0; see #8605, #14207 + xhr.status, + xhr.statusText + ); + } + } else { + complete( + xhrSuccessStatus[ xhr.status ] || xhr.status, + xhr.statusText, + + // Support: IE <=9 only + // IE9 has no XHR2 but throws on binary (trac-11426) + // For XHR2 non-text, let the caller handle it (gh-2498) + ( xhr.responseType || "text" ) !== "text" || + typeof xhr.responseText !== "string" ? + { binary: xhr.response } : + { text: xhr.responseText }, + xhr.getAllResponseHeaders() + ); + } + } + }; + }; + + // Listen to events + xhr.onload = callback(); + errorCallback = xhr.onerror = xhr.ontimeout = callback( "error" ); + + // Support: IE 9 only + // Use onreadystatechange to replace onabort + // to handle uncaught aborts + if ( xhr.onabort !== undefined ) { + xhr.onabort = errorCallback; + } else { + xhr.onreadystatechange = function() { + + // Check readyState before timeout as it changes + if ( xhr.readyState === 4 ) { + + // Allow onerror to be called first, + // but that will not handle a native abort + // Also, save errorCallback to a variable + // as xhr.onerror cannot be accessed + window.setTimeout( function() { + if ( callback ) { + errorCallback(); + } + } ); + } + }; + } + + // Create the abort callback + callback = callback( "abort" ); + + try { + + // Do send the request (this may raise an exception) + xhr.send( options.hasContent && options.data || null ); + } catch ( e ) { + + // #14683: Only rethrow if this hasn't been notified as an error yet + if ( callback ) { + throw e; + } + } + }, + + abort: function() { + if ( callback ) { + callback(); + } + } + }; + } + } ); + + + + + // Prevent auto-execution of scripts when no explicit dataType was provided (See gh-2432) + jQuery.ajaxPrefilter( function( s ) { + if ( s.crossDomain ) { + s.contents.script = false; + } + } ); + + // Install script dataType + jQuery.ajaxSetup( { + accepts: { + script: "text/javascript, application/javascript, " + + "application/ecmascript, application/x-ecmascript" + }, + contents: { + script: /\b(?:java|ecma)script\b/ + }, + converters: { + "text script": function( text ) { + jQuery.globalEval( text ); + return text; + } + } + } ); + + // Handle cache's special case and crossDomain + jQuery.ajaxPrefilter( "script", function( s ) { + if ( s.cache === undefined ) { + s.cache = false; + } + if ( s.crossDomain ) { + s.type = "GET"; + } + } ); + + // Bind script tag hack transport + jQuery.ajaxTransport( "script", function( s ) { + + // This transport only deals with cross domain or forced-by-attrs requests + if ( s.crossDomain || s.scriptAttrs ) { + var script, callback; + return { + send: function( _, complete ) { + script = jQuery( "<script>" ) + .attr( s.scriptAttrs || {} ) + .prop( { charset: s.scriptCharset, src: s.url } ) + .on( "load error", callback = function( evt ) { + script.remove(); + callback = null; + if ( evt ) { + complete( evt.type === "error" ? 404 : 200, evt.type ); + } + } ); + + // Use native DOM manipulation to avoid our domManip AJAX trickery + document.head.appendChild( script[ 0 ] ); + }, + abort: function() { + if ( callback ) { + callback(); + } + } + }; + } + } ); + + + + + var oldCallbacks = [], + rjsonp = /(=)\?(?=&|$)|\?\?/; + + // Default jsonp settings + jQuery.ajaxSetup( { + jsonp: "callback", + jsonpCallback: function() { + var callback = oldCallbacks.pop() || ( jQuery.expando + "_" + ( nonce.guid++ ) ); + this[ callback ] = true; + return callback; + } + } ); + + // Detect, normalize options and install callbacks for jsonp requests + jQuery.ajaxPrefilter( "json jsonp", function( s, originalSettings, jqXHR ) { + + var callbackName, overwritten, responseContainer, + jsonProp = s.jsonp !== false && ( rjsonp.test( s.url ) ? + "url" : + typeof s.data === "string" && + ( s.contentType || "" ) + .indexOf( "application/x-www-form-urlencoded" ) === 0 && + rjsonp.test( s.data ) && "data" + ); + + // Handle iff the expected data type is "jsonp" or we have a parameter to set + if ( jsonProp || s.dataTypes[ 0 ] === "jsonp" ) { + + // Get callback name, remembering preexisting value associated with it + callbackName = s.jsonpCallback = isFunction( s.jsonpCallback ) ? + s.jsonpCallback() : + s.jsonpCallback; + + // Insert callback into url or form data + if ( jsonProp ) { + s[ jsonProp ] = s[ jsonProp ].replace( rjsonp, "$1" + callbackName ); + } else if ( s.jsonp !== false ) { + s.url += ( rquery.test( s.url ) ? "&" : "?" ) + s.jsonp + "=" + callbackName; + } + + // Use data converter to retrieve json after script execution + s.converters[ "script json" ] = function() { + if ( !responseContainer ) { + jQuery.error( callbackName + " was not called" ); + } + return responseContainer[ 0 ]; + }; + + // Force json dataType + s.dataTypes[ 0 ] = "json"; + + // Install callback + overwritten = window[ callbackName ]; + window[ callbackName ] = function() { + responseContainer = arguments; + }; + + // Clean-up function (fires after converters) + jqXHR.always( function() { + + // If previous value didn't exist - remove it + if ( overwritten === undefined ) { + jQuery( window ).removeProp( callbackName ); + + // Otherwise restore preexisting value + } else { + window[ callbackName ] = overwritten; + } + + // Save back as free + if ( s[ callbackName ] ) { + + // Make sure that re-using the options doesn't screw things around + s.jsonpCallback = originalSettings.jsonpCallback; + + // Save the callback name for future use + oldCallbacks.push( callbackName ); + } + + // Call if it was a function and we have a response + if ( responseContainer && isFunction( overwritten ) ) { + overwritten( responseContainer[ 0 ] ); + } + + responseContainer = overwritten = undefined; + } ); + + // Delegate to script + return "script"; + } + } ); + + + + + // Support: Safari 8 only + // In Safari 8 documents created via document.implementation.createHTMLDocument + // collapse sibling forms: the second one becomes a child of the first one. + // Because of that, this security measure has to be disabled in Safari 8. + // https://bugs.webkit.org/show_bug.cgi?id=137337 + support.createHTMLDocument = ( function() { + var body = document.implementation.createHTMLDocument( "" ).body; + body.innerHTML = "<form></form><form></form>"; + return body.childNodes.length === 2; + } )(); + + + // Argument "data" should be string of html + // context (optional): If specified, the fragment will be created in this context, + // defaults to document + // keepScripts (optional): If true, will include scripts passed in the html string + jQuery.parseHTML = function( data, context, keepScripts ) { + if ( typeof data !== "string" ) { + return []; + } + if ( typeof context === "boolean" ) { + keepScripts = context; + context = false; + } + + var base, parsed, scripts; + + if ( !context ) { + + // Stop scripts or inline event handlers from being executed immediately + // by using document.implementation + if ( support.createHTMLDocument ) { + context = document.implementation.createHTMLDocument( "" ); + + // Set the base href for the created document + // so any parsed elements with URLs + // are based on the document's URL (gh-2965) + base = context.createElement( "base" ); + base.href = document.location.href; + context.head.appendChild( base ); + } else { + context = document; + } + } + + parsed = rsingleTag.exec( data ); + scripts = !keepScripts && []; + + // Single tag + if ( parsed ) { + return [ context.createElement( parsed[ 1 ] ) ]; + } + + parsed = buildFragment( [ data ], context, scripts ); + + if ( scripts && scripts.length ) { + jQuery( scripts ).remove(); + } + + return jQuery.merge( [], parsed.childNodes ); + }; + + + /** + * Load a url into a page + */ + jQuery.fn.load = function( url, params, callback ) { + var selector, type, response, + self = this, + off = url.indexOf( " " ); + + if ( off > -1 ) { + selector = stripAndCollapse( url.slice( off ) ); + url = url.slice( 0, off ); + } + + // If it's a function + if ( isFunction( params ) ) { + + // We assume that it's the callback + callback = params; + params = undefined; + + // Otherwise, build a param string + } else if ( params && typeof params === "object" ) { + type = "POST"; + } + + // If we have elements to modify, make the request + if ( self.length > 0 ) { + jQuery.ajax( { + url: url, + + // If "type" variable is undefined, then "GET" method will be used. + // Make value of this field explicit since + // user can override it through ajaxSetup method + type: type || "GET", + dataType: "html", + data: params + } ).done( function( responseText ) { + + // Save response for use in complete callback + response = arguments; + + self.html( selector ? + + // If a selector was specified, locate the right elements in a dummy div + // Exclude scripts to avoid IE 'Permission Denied' errors + jQuery( "<div>" ).append( jQuery.parseHTML( responseText ) ).find( selector ) : + + // Otherwise use the full result + responseText ); + + // If the request succeeds, this function gets "data", "status", "jqXHR" + // but they are ignored because response was set above. + // If it fails, this function gets "jqXHR", "status", "error" + } ).always( callback && function( jqXHR, status ) { + self.each( function() { + callback.apply( this, response || [ jqXHR.responseText, status, jqXHR ] ); + } ); + } ); + } + + return this; + }; + + + + + jQuery.expr.pseudos.animated = function( elem ) { + return jQuery.grep( jQuery.timers, function( fn ) { + return elem === fn.elem; + } ).length; + }; + + + + + jQuery.offset = { + setOffset: function( elem, options, i ) { + var curPosition, curLeft, curCSSTop, curTop, curOffset, curCSSLeft, calculatePosition, + position = jQuery.css( elem, "position" ), + curElem = jQuery( elem ), + props = {}; + + // Set position first, in-case top/left are set even on static elem + if ( position === "static" ) { + elem.style.position = "relative"; + } + + curOffset = curElem.offset(); + curCSSTop = jQuery.css( elem, "top" ); + curCSSLeft = jQuery.css( elem, "left" ); + calculatePosition = ( position === "absolute" || position === "fixed" ) && + ( curCSSTop + curCSSLeft ).indexOf( "auto" ) > -1; + + // Need to be able to calculate position if either + // top or left is auto and position is either absolute or fixed + if ( calculatePosition ) { + curPosition = curElem.position(); + curTop = curPosition.top; + curLeft = curPosition.left; + + } else { + curTop = parseFloat( curCSSTop ) || 0; + curLeft = parseFloat( curCSSLeft ) || 0; + } + + if ( isFunction( options ) ) { + + // Use jQuery.extend here to allow modification of coordinates argument (gh-1848) + options = options.call( elem, i, jQuery.extend( {}, curOffset ) ); + } + + if ( options.top != null ) { + props.top = ( options.top - curOffset.top ) + curTop; + } + if ( options.left != null ) { + props.left = ( options.left - curOffset.left ) + curLeft; + } + + if ( "using" in options ) { + options.using.call( elem, props ); + + } else { + curElem.css( props ); + } + } + }; + + jQuery.fn.extend( { + + // offset() relates an element's border box to the document origin + offset: function( options ) { + + // Preserve chaining for setter + if ( arguments.length ) { + return options === undefined ? + this : + this.each( function( i ) { + jQuery.offset.setOffset( this, options, i ); + } ); + } + + var rect, win, + elem = this[ 0 ]; + + if ( !elem ) { + return; + } + + // Return zeros for disconnected and hidden (display: none) elements (gh-2310) + // Support: IE <=11 only + // Running getBoundingClientRect on a + // disconnected node in IE throws an error + if ( !elem.getClientRects().length ) { + return { top: 0, left: 0 }; + } + + // Get document-relative position by adding viewport scroll to viewport-relative gBCR + rect = elem.getBoundingClientRect(); + win = elem.ownerDocument.defaultView; + return { + top: rect.top + win.pageYOffset, + left: rect.left + win.pageXOffset + }; + }, + + // position() relates an element's margin box to its offset parent's padding box + // This corresponds to the behavior of CSS absolute positioning + position: function() { + if ( !this[ 0 ] ) { + return; + } + + var offsetParent, offset, doc, + elem = this[ 0 ], + parentOffset = { top: 0, left: 0 }; + + // position:fixed elements are offset from the viewport, which itself always has zero offset + if ( jQuery.css( elem, "position" ) === "fixed" ) { + + // Assume position:fixed implies availability of getBoundingClientRect + offset = elem.getBoundingClientRect(); + + } else { + offset = this.offset(); + + // Account for the *real* offset parent, which can be the document or its root element + // when a statically positioned element is identified + doc = elem.ownerDocument; + offsetParent = elem.offsetParent || doc.documentElement; + while ( offsetParent && + ( offsetParent === doc.body || offsetParent === doc.documentElement ) && + jQuery.css( offsetParent, "position" ) === "static" ) { + + offsetParent = offsetParent.parentNode; + } + if ( offsetParent && offsetParent !== elem && offsetParent.nodeType === 1 ) { + + // Incorporate borders into its offset, since they are outside its content origin + parentOffset = jQuery( offsetParent ).offset(); + parentOffset.top += jQuery.css( offsetParent, "borderTopWidth", true ); + parentOffset.left += jQuery.css( offsetParent, "borderLeftWidth", true ); + } + } + + // Subtract parent offsets and element margins + return { + top: offset.top - parentOffset.top - jQuery.css( elem, "marginTop", true ), + left: offset.left - parentOffset.left - jQuery.css( elem, "marginLeft", true ) + }; + }, + + // This method will return documentElement in the following cases: + // 1) For the element inside the iframe without offsetParent, this method will return + // documentElement of the parent window + // 2) For the hidden or detached element + // 3) For body or html element, i.e. in case of the html node - it will return itself + // + // but those exceptions were never presented as a real life use-cases + // and might be considered as more preferable results. + // + // This logic, however, is not guaranteed and can change at any point in the future + offsetParent: function() { + return this.map( function() { + var offsetParent = this.offsetParent; + + while ( offsetParent && jQuery.css( offsetParent, "position" ) === "static" ) { + offsetParent = offsetParent.offsetParent; + } + + return offsetParent || documentElement; + } ); + } + } ); + + // Create scrollLeft and scrollTop methods + jQuery.each( { scrollLeft: "pageXOffset", scrollTop: "pageYOffset" }, function( method, prop ) { + var top = "pageYOffset" === prop; + + jQuery.fn[ method ] = function( val ) { + return access( this, function( elem, method, val ) { + + // Coalesce documents and windows + var win; + if ( isWindow( elem ) ) { + win = elem; + } else if ( elem.nodeType === 9 ) { + win = elem.defaultView; + } + + if ( val === undefined ) { + return win ? win[ prop ] : elem[ method ]; + } + + if ( win ) { + win.scrollTo( + !top ? val : win.pageXOffset, + top ? val : win.pageYOffset + ); + + } else { + elem[ method ] = val; + } + }, method, val, arguments.length ); + }; + } ); + + // Support: Safari <=7 - 9.1, Chrome <=37 - 49 + // Add the top/left cssHooks using jQuery.fn.position + // Webkit bug: https://bugs.webkit.org/show_bug.cgi?id=29084 + // Blink bug: https://bugs.chromium.org/p/chromium/issues/detail?id=589347 + // getComputedStyle returns percent when specified for top/left/bottom/right; + // rather than make the css module depend on the offset module, just check for it here + jQuery.each( [ "top", "left" ], function( _i, prop ) { + jQuery.cssHooks[ prop ] = addGetHookIf( support.pixelPosition, + function( elem, computed ) { + if ( computed ) { + computed = curCSS( elem, prop ); + + // If curCSS returns percentage, fallback to offset + return rnumnonpx.test( computed ) ? + jQuery( elem ).position()[ prop ] + "px" : + computed; + } + } + ); + } ); + + + // Create innerHeight, innerWidth, height, width, outerHeight and outerWidth methods + jQuery.each( { Height: "height", Width: "width" }, function( name, type ) { + jQuery.each( { + padding: "inner" + name, + content: type, + "": "outer" + name + }, function( defaultExtra, funcName ) { + + // Margin is only for outerHeight, outerWidth + jQuery.fn[ funcName ] = function( margin, value ) { + var chainable = arguments.length && ( defaultExtra || typeof margin !== "boolean" ), + extra = defaultExtra || ( margin === true || value === true ? "margin" : "border" ); + + return access( this, function( elem, type, value ) { + var doc; + + if ( isWindow( elem ) ) { + + // $( window ).outerWidth/Height return w/h including scrollbars (gh-1729) + return funcName.indexOf( "outer" ) === 0 ? + elem[ "inner" + name ] : + elem.document.documentElement[ "client" + name ]; + } + + // Get document width or height + if ( elem.nodeType === 9 ) { + doc = elem.documentElement; + + // Either scroll[Width/Height] or offset[Width/Height] or client[Width/Height], + // whichever is greatest + return Math.max( + elem.body[ "scroll" + name ], doc[ "scroll" + name ], + elem.body[ "offset" + name ], doc[ "offset" + name ], + doc[ "client" + name ] + ); + } + + return value === undefined ? + + // Get width or height on the element, requesting but not forcing parseFloat + jQuery.css( elem, type, extra ) : + + // Set width or height on the element + jQuery.style( elem, type, value, extra ); + }, type, chainable ? margin : undefined, chainable ); + }; + } ); + } ); + + + jQuery.each( [ + "ajaxStart", + "ajaxStop", + "ajaxComplete", + "ajaxError", + "ajaxSuccess", + "ajaxSend" + ], function( _i, type ) { + jQuery.fn[ type ] = function( fn ) { + return this.on( type, fn ); + }; + } ); + + + + + jQuery.fn.extend( { + + bind: function( types, data, fn ) { + return this.on( types, null, data, fn ); + }, + unbind: function( types, fn ) { + return this.off( types, null, fn ); + }, + + delegate: function( selector, types, data, fn ) { + return this.on( types, selector, data, fn ); + }, + undelegate: function( selector, types, fn ) { + + // ( namespace ) or ( selector, types [, fn] ) + return arguments.length === 1 ? + this.off( selector, "**" ) : + this.off( types, selector || "**", fn ); + }, + + hover: function( fnOver, fnOut ) { + return this.mouseenter( fnOver ).mouseleave( fnOut || fnOver ); + } + } ); + + jQuery.each( + ( "blur focus focusin focusout resize scroll click dblclick " + + "mousedown mouseup mousemove mouseover mouseout mouseenter mouseleave " + + "change select submit keydown keypress keyup contextmenu" ).split( " " ), + function( _i, name ) { + + // Handle event binding + jQuery.fn[ name ] = function( data, fn ) { + return arguments.length > 0 ? + this.on( name, null, data, fn ) : + this.trigger( name ); + }; + } + ); + + + + + // Support: Android <=4.0 only + // Make sure we trim BOM and NBSP + var rtrim = /^[\s\uFEFF\xA0]+|[\s\uFEFF\xA0]+$/g; + + // Bind a function to a context, optionally partially applying any + // arguments. + // jQuery.proxy is deprecated to promote standards (specifically Function#bind) + // However, it is not slated for removal any time soon + jQuery.proxy = function( fn, context ) { + var tmp, args, proxy; + + if ( typeof context === "string" ) { + tmp = fn[ context ]; + context = fn; + fn = tmp; + } + + // Quick check to determine if target is callable, in the spec + // this throws a TypeError, but we will just return undefined. + if ( !isFunction( fn ) ) { + return undefined; + } + + // Simulated bind + args = slice.call( arguments, 2 ); + proxy = function() { + return fn.apply( context || this, args.concat( slice.call( arguments ) ) ); + }; + + // Set the guid of unique handler to the same of original handler, so it can be removed + proxy.guid = fn.guid = fn.guid || jQuery.guid++; + + return proxy; + }; + + jQuery.holdReady = function( hold ) { + if ( hold ) { + jQuery.readyWait++; + } else { + jQuery.ready( true ); + } + }; + jQuery.isArray = Array.isArray; + jQuery.parseJSON = JSON.parse; + jQuery.nodeName = nodeName; + jQuery.isFunction = isFunction; + jQuery.isWindow = isWindow; + jQuery.camelCase = camelCase; + jQuery.type = toType; + + jQuery.now = Date.now; + + jQuery.isNumeric = function( obj ) { + + // As of jQuery 3.0, isNumeric is limited to + // strings and numbers (primitives or objects) + // that can be coerced to finite numbers (gh-2662) + var type = jQuery.type( obj ); + return ( type === "number" || type === "string" ) && + + // parseFloat NaNs numeric-cast false positives ("") + // ...but misinterprets leading-number strings, particularly hex literals ("0x...") + // subtraction forces infinities to NaN + !isNaN( obj - parseFloat( obj ) ); + }; + + jQuery.trim = function( text ) { + return text == null ? + "" : + ( text + "" ).replace( rtrim, "" ); + }; + + + + // Register as a named AMD module, since jQuery can be concatenated with other + // files that may use define, but not via a proper concatenation script that + // understands anonymous AMD modules. A named AMD is safest and most robust + // way to register. Lowercase jquery is used because AMD module names are + // derived from file names, and jQuery is normally delivered in a lowercase + // file name. Do this after creating the global so that if an AMD module wants + // to call noConflict to hide this version of jQuery, it will work. + + // Note that for maximum portability, libraries that are not jQuery should + // declare themselves as anonymous modules, and avoid setting a global if an + // AMD loader is present. jQuery is a special case. For more information, see + // https://github.com/jrburke/requirejs/wiki/Updating-existing-libraries#wiki-anon + + if ( typeof define === "function" && define.amd ) { + define( "jquery", [], function() { + return jQuery; + } ); + } + + + + + var + + // Map over jQuery in case of overwrite + _jQuery = window.jQuery, + + // Map over the $ in case of overwrite + _$ = window.$; + + jQuery.noConflict = function( deep ) { + if ( window.$ === jQuery ) { + window.$ = _$; + } + + if ( deep && window.jQuery === jQuery ) { + window.jQuery = _jQuery; + } + + return jQuery; + }; + + // Expose jQuery and $ identifiers, even in AMD + // (#7102#comment:10, https://github.com/jquery/jquery/pull/557) + // and CommonJS for browser emulators (#13566) + if ( typeof noGlobal === "undefined" ) { + window.jQuery = window.$ = jQuery; + } + + + + + return jQuery; + } );
html/_static/jquery.js (new)
- @@ -0,0 +1,2 @@
 + /*! jQuery v3.6.0 | (c) OpenJS Foundation and other contributors | jquery.org/license */ + !function(e,t){"use strict";"object"==typeof module&&"object"==typeof module.exports?module.exports=e.document?t(e,!0):function(e){if(!e.document)throw new Error("jQuery requires a window with a document");return t(e)}:t(e)}("undefined"!=typeof window?window:this,function(C,e){"use strict";var t=[],r=Object.getPrototypeOf,s=t.slice,g=t.flat?function(e){return t.flat.call(e)}:function(e){return t.concat.apply([],e)},u=t.push,i=t.indexOf,n={},o=n.toString,v=n.hasOwnProperty,a=v.toString,l=a.call(Object),y={},m=function(e){return"function"==typeof e&&"number"!=typeof e.nodeType&&"function"!=typeof e.item},x=function(e){return null!=e&&e===e.window},E=C.document,c={type:!0,src:!0,nonce:!0,noModule:!0};function b(e,t,n){var r,i,o=(n=n||E).createElement("script");if(o.text=e,t)for(r in c)(i=t[r]||t.getAttribute&&t.getAttribute(r))&&o.setAttribute(r,i);n.head.appendChild(o).parentNode.removeChild(o)}function w(e){return null==e?e+"":"object"==typeof e||"function"==typeof e?n[o.call(e)]||"object":typeof e}var f="3.6.0",S=function(e,t){return new S.fn.init(e,t)};function p(e){var t=!!e&&"length"in e&&e.length,n=w(e);return!m(e)&&!x(e)&&("array"===n||0===t||"number"==typeof t&&0<t&&t-1 in e)}S.fn=S.prototype={jquery:f,constructor:S,length:0,toArray:function(){return s.call(this)},get:function(e){return null==e?s.call(this):e<0?this[e+this.length]:this[e]},pushStack:function(e){var t=S.merge(this.constructor(),e);return t.prevObject=this,t},each:function(e){return S.each(this,e)},map:function(n){return this.pushStack(S.map(this,function(e,t){return n.call(e,t,e)}))},slice:function(){return this.pushStack(s.apply(this,arguments))},first:function(){return this.eq(0)},last:function(){return this.eq(-1)},even:function(){return this.pushStack(S.grep(this,function(e,t){return(t+1)%2}))},odd:function(){return this.pushStack(S.grep(this,function(e,t){return t%2}))},eq:function(e){var t=this.length,n=+e+(e<0?t:0);return this.pushStack(0<=n&&n<t?[this[n]]:[])},end:function(){return this.prevObject||this.constructor()},push:u,sort:t.sort,splice:t.splice},S.extend=S.fn.extend=function(){var e,t,n,r,i,o,a=arguments[0]||{},s=1,u=arguments.length,l=!1;for("boolean"==typeof a&&(l=a,a=arguments[s]||{},s++),"object"==typeof a||m(a)||(a={}),s===u&&(a=this,s--);s<u;s++)if(null!=(e=arguments[s]))for(t in e)r=e[t],"__proto__"!==t&&a!==r&&(l&&r&&(S.isPlainObject(r)||(i=Array.isArray(r)))?(n=a[t],o=i&&!Array.isArray(n)?[]:i||S.isPlainObject(n)?n:{},i=!1,a[t]=S.extend(l,o,r)):void 0!==r&&(a[t]=r));return a},S.extend({expando:"jQuery"+(f+Math.random()).replace(/\D/g,""),isReady:!0,error:function(e){throw new Error(e)},noop:function(){},isPlainObject:function(e){var t,n;return!(!e||"[object Object]"!==o.call(e))&&(!(t=r(e))||"function"==typeof(n=v.call(t,"constructor")&&t.constructor)&&a.call(n)===l)},isEmptyObject:function(e){var t;for(t in e)return!1;return!0},globalEval:function(e,t,n){b(e,{nonce:t&&t.nonce},n)},each:function(e,t){var n,r=0;if(p(e)){for(n=e.length;r<n;r++)if(!1===t.call(e[r],r,e[r]))break}else for(r in e)if(!1===t.call(e[r],r,e[r]))break;return e},makeArray:function(e,t){var n=t||[];return null!=e&&(p(Object(e))?S.merge(n,"string"==typeof e?[e]:e):u.call(n,e)),n},inArray:function(e,t,n){return null==t?-1:i.call(t,e,n)},merge:function(e,t){for(var n=+t.length,r=0,i=e.length;r<n;r++)e[i++]=t[r];return e.length=i,e},grep:function(e,t,n){for(var r=[],i=0,o=e.length,a=!n;i<o;i++)!t(e[i],i)!==a&&r.push(e[i]);return r},map:function(e,t,n){var r,i,o=0,a=[];if(p(e))for(r=e.length;o<r;o++)null!=(i=t(e[o],o,n))&&a.push(i);else for(o in e)null!=(i=t(e[o],o,n))&&a.push(i);return g(a)},guid:1,support:y}),"function"==typeof Symbol&&(S.fn[Symbol.iterator]=t[Symbol.iterator]),S.each("Boolean Number String Function Array Date RegExp Object Error Symbol".split(" "),function(e,t){n["[object "+t+"]"]=t.toLowerCase()});var d=function(n){var e,d,b,o,i,h,f,g,w,u,l,T,C,a,E,v,s,c,y,S="sizzle"+1*new Date,p=n.document,k=0,r=0,m=ue(),x=ue(),A=ue(),N=ue(),j=function(e,t){return e===t&&(l=!0),0},D={}.hasOwnProperty,t=[],q=t.pop,L=t.push,H=t.push,O=t.slice,P=function(e,t){for(var n=0,r=e.length;n<r;n++)if(e[n]===t)return n;return-1},R="checked|selected|async|autofocus|autoplay|controls|defer|disabled|hidden|ismap|loop|multiple|open|readonly|required|scoped",M="[\\x20\\t\\r\\n\\f]",I="(?:\\\\[\\da-fA-F]{1,6}"+M+"?|\\\\[^\\r\\n\\f]|[\\w-]|[^\0-\\x7f])+",W="\\["+M+"*("+I+")(?:"+M+"*([*^$|!~]?=)"+M+"*(?:'((?:\\\\.|[^\\\\'])*)'|\"((?:\\\\.|[^\\\\\"])*)\"|("+I+"))|)"+M+"*\\]",F=":("+I+")(?:\$(('((?:\\\\.|[^\\\\'])*)'|\"((?:\\\\.|[^\\\\\"])*)\")|((?:\\\\.|[^\\\\()[\\]]|"+W+")*)|.*)\$|)",B=new RegExp(M+"+","g"),$=new RegExp("^"+M+"+|((?:^|[^\\\\])(?:\\\\.)*)"+M+"+$","g"),_=new RegExp("^"+M+"*,"+M+"*"),z=new RegExp("^"+M+"*([>+~]|"+M+")"+M+"*"),U=new RegExp(M+"|>"),X=new RegExp(F),V=new RegExp("^"+I+"$"),G={ID:new RegExp("^#("+I+")"),CLASS:new RegExp("^\\.("+I+")"),TAG:new RegExp("^("+I+"|[*])"),ATTR:new RegExp("^"+W),PSEUDO:new RegExp("^"+F),CHILD:new RegExp("^:(only|first|last|nth|nth-last)-(child|of-type)(?:\$"+M+"*(even|odd|(([+-]|)(\\d*)n|)"+M+"*(?:([+-]|)"+M+"*(\\d+)|))"+M+"*\$|)","i"),bool:new RegExp("^(?:"+R+")$","i"),needsContext:new RegExp("^"+M+"*[>+~]|:(even|odd|eq|gt|lt|nth|first|last)(?:\$"+M+"*((?:-\\d)?\\d*)"+M+"*\$|)(?=[^-]|$)","i")},Y=/HTML$/i,Q=/^(?:input|select|textarea|button)$/i,J=/^h\d$/i,K=/^[^{]+\{\s*\[native \w/,Z=/^(?:#([\w-]+)|(\w+)|\.([\w-]+))$/,ee=/[+~]/,te=new RegExp("\\\\[\\da-fA-F]{1,6}"+M+"?|\\\\([^\\r\\n\\f])","g"),ne=function(e,t){var n="0x"+e.slice(1)-65536;return t||(n<0?String.fromCharCode(n+65536):String.fromCharCode(n>>10|55296,1023&n|56320))},re=/([\0-\x1f\x7f]|^-?\d)|^-$|[^\0-\x1f\x7f-\uFFFF\w-]/g,ie=function(e,t){return t?"\0"===e?"\ufffd":e.slice(0,-1)+"\\"+e.charCodeAt(e.length-1).toString(16)+" ":"\\"+e},oe=function(){T()},ae=be(function(e){return!0===e.disabled&&"fieldset"===e.nodeName.toLowerCase()},{dir:"parentNode",next:"legend"});try{H.apply(t=O.call(p.childNodes),p.childNodes),t[p.childNodes.length].nodeType}catch(e){H={apply:t.length?function(e,t){L.apply(e,O.call(t))}:function(e,t){var n=e.length,r=0;while(e[n++]=t[r++]);e.length=n-1}}}function se(t,e,n,r){var i,o,a,s,u,l,c,f=e&&e.ownerDocument,p=e?e.nodeType:9;if(n=n||[],"string"!=typeof t||!t||1!==p&&9!==p&&11!==p)return n;if(!r&&(T(e),e=e||C,E)){if(11!==p&&(u=Z.exec(t)))if(i=u[1]){if(9===p){if(!(a=e.getElementById(i)))return n;if(a.id===i)return n.push(a),n}else if(f&&(a=f.getElementById(i))&&y(e,a)&&a.id===i)return n.push(a),n}else{if(u[2])return H.apply(n,e.getElementsByTagName(t)),n;if((i=u[3])&&d.getElementsByClassName&&e.getElementsByClassName)return H.apply(n,e.getElementsByClassName(i)),n}if(d.qsa&&!N[t+" "]&&(!v||!v.test(t))&&(1!==p||"object"!==e.nodeName.toLowerCase())){if(c=t,f=e,1===p&&(U.test(t)||z.test(t))){(f=ee.test(t)&&ye(e.parentNode)||e)===e&&d.scope||((s=e.getAttribute("id"))?s=s.replace(re,ie):e.setAttribute("id",s=S)),o=(l=h(t)).length;while(o--)l[o]=(s?"#"+s:":scope")+" "+xe(l[o]);c=l.join(",")}try{return H.apply(n,f.querySelectorAll(c)),n}catch(e){N(t,!0)}finally{s===S&&e.removeAttribute("id")}}}return g(t.replace($,"$1"),e,n,r)}function ue(){var r=[];return function e(t,n){return r.push(t+" ")>b.cacheLength&&delete e[r.shift()],e[t+" "]=n}}function le(e){return e[S]=!0,e}function ce(e){var t=C.createElement("fieldset");try{return!!e(t)}catch(e){return!1}finally{t.parentNode&&t.parentNode.removeChild(t),t=null}}function fe(e,t){var n=e.split("|"),r=n.length;while(r--)b.attrHandle[n[r]]=t}function pe(e,t){var n=t&&e,r=n&&1===e.nodeType&&1===t.nodeType&&e.sourceIndex-t.sourceIndex;if(r)return r;if(n)while(n=n.nextSibling)if(n===t)return-1;return e?1:-1}function de(t){return function(e){return"input"===e.nodeName.toLowerCase()&&e.type===t}}function he(n){return function(e){var t=e.nodeName.toLowerCase();return("input"===t||"button"===t)&&e.type===n}}function ge(t){return function(e){return"form"in e?e.parentNode&&!1===e.disabled?"label"in e?"label"in e.parentNode?e.parentNode.disabled===t:e.disabled===t:e.isDisabled===t||e.isDisabled!==!t&&ae(e)===t:e.disabled===t:"label"in e&&e.disabled===t}}function ve(a){return le(function(o){return o=+o,le(function(e,t){var n,r=a([],e.length,o),i=r.length;while(i--)e[n=r[i]]&&(e[n]=!(t[n]=e[n]))})})}function ye(e){return e&&"undefined"!=typeof e.getElementsByTagName&&e}for(e in d=se.support={},i=se.isXML=function(e){var t=e&&e.namespaceURI,n=e&&(e.ownerDocument||e).documentElement;return!Y.test(t||n&&n.nodeName||"HTML")},T=se.setDocument=function(e){var t,n,r=e?e.ownerDocument||e:p;return r!=C&&9===r.nodeType&&r.documentElement&&(a=(C=r).documentElement,E=!i(C),p!=C&&(n=C.defaultView)&&n.top!==n&&(n.addEventListener?n.addEventListener("unload",oe,!1):n.attachEvent&&n.attachEvent("onunload",oe)),d.scope=ce(function(e){return a.appendChild(e).appendChild(C.createElement("div")),"undefined"!=typeof e.querySelectorAll&&!e.querySelectorAll(":scope fieldset div").length}),d.attributes=ce(function(e){return e.className="i",!e.getAttribute("className")}),d.getElementsByTagName=ce(function(e){return e.appendChild(C.createComment("")),!e.getElementsByTagName("*").length}),d.getElementsByClassName=K.test(C.getElementsByClassName),d.getById=ce(function(e){return a.appendChild(e).id=S,!C.getElementsByName||!C.getElementsByName(S).length}),d.getById?(b.filter.ID=function(e){var t=e.replace(te,ne);return function(e){return e.getAttribute("id")===t}},b.find.ID=function(e,t){if("undefined"!=typeof t.getElementById&&E){var n=t.getElementById(e);return n?[n]:[]}}):(b.filter.ID=function(e){var n=e.replace(te,ne);return function(e){var t="undefined"!=typeof e.getAttributeNode&&e.getAttributeNode("id");return t&&t.value===n}},b.find.ID=function(e,t){if("undefined"!=typeof t.getElementById&&E){var n,r,i,o=t.getElementById(e);if(o){if((n=o.getAttributeNode("id"))&&n.value===e)return[o];i=t.getElementsByName(e),r=0;while(o=i[r++])if((n=o.getAttributeNode("id"))&&n.value===e)return[o]}return[]}}),b.find.TAG=d.getElementsByTagName?function(e,t){return"undefined"!=typeof t.getElementsByTagName?t.getElementsByTagName(e):d.qsa?t.querySelectorAll(e):void 0}:function(e,t){var n,r=[],i=0,o=t.getElementsByTagName(e);if("*"===e){while(n=o[i++])1===n.nodeType&&r.push(n);return r}return o},b.find.CLASS=d.getElementsByClassName&&function(e,t){if("undefined"!=typeof t.getElementsByClassName&&E)return t.getElementsByClassName(e)},s=[],v=[],(d.qsa=K.test(C.querySelectorAll))&&(ce(function(e){var t;a.appendChild(e).innerHTML="<a id='"+S+"'></a><select id='"+S+"-\r\\' msallowcapture=''><option selected=''></option></select>",e.querySelectorAll("[msallowcapture^='']").length&&v.push("[*^$]="+M+"*(?:''|\"\")"),e.querySelectorAll("[selected]").length||v.push("\\["+M+"*(?:value|"+R+")"),e.querySelectorAll("[id~="+S+"-]").length||v.push("~="),(t=C.createElement("input")).setAttribute("name",""),e.appendChild(t),e.querySelectorAll("[name='']").length||v.push("\\["+M+"*name"+M+"*="+M+"*(?:''|\"\")"),e.querySelectorAll(":checked").length||v.push(":checked"),e.querySelectorAll("a#"+S+"+*").length||v.push(".#.+[+~]"),e.querySelectorAll("\\\f"),v.push("[\\r\\n\\f]")}),ce(function(e){e.innerHTML="<a href='' disabled='disabled'></a><select disabled='disabled'><option/></select>";var t=C.createElement("input");t.setAttribute("type","hidden"),e.appendChild(t).setAttribute("name","D"),e.querySelectorAll("[name=d]").length&&v.push("name"+M+"*[*^$|!~]?="),2!==e.querySelectorAll(":enabled").length&&v.push(":enabled",":disabled"),a.appendChild(e).disabled=!0,2!==e.querySelectorAll(":disabled").length&&v.push(":enabled",":disabled"),e.querySelectorAll("*,:x"),v.push(",.*:")})),(d.matchesSelector=K.test(c=a.matches||a.webkitMatchesSelector||a.mozMatchesSelector||a.oMatchesSelector||a.msMatchesSelector))&&ce(function(e){d.disconnectedMatch=c.call(e,"*"),c.call(e,"[s!='']:x"),s.push("!=",F)}),v=v.length&&new RegExp(v.join("|")),s=s.length&&new RegExp(s.join("|")),t=K.test(a.compareDocumentPosition),y=t||K.test(a.contains)?function(e,t){var n=9===e.nodeType?e.documentElement:e,r=t&&t.parentNode;return e===r||!(!r||1!==r.nodeType||!(n.contains?n.contains(r):e.compareDocumentPosition&&16&e.compareDocumentPosition(r)))}:function(e,t){if(t)while(t=t.parentNode)if(t===e)return!0;return!1},j=t?function(e,t){if(e===t)return l=!0,0;var n=!e.compareDocumentPosition-!t.compareDocumentPosition;return n||(1&(n=(e.ownerDocument||e)==(t.ownerDocument||t)?e.compareDocumentPosition(t):1)||!d.sortDetached&&t.compareDocumentPosition(e)===n?e==C||e.ownerDocument==p&&y(p,e)?-1:t==C||t.ownerDocument==p&&y(p,t)?1:u?P(u,e)-P(u,t):0:4&n?-1:1)}:function(e,t){if(e===t)return l=!0,0;var n,r=0,i=e.parentNode,o=t.parentNode,a=[e],s=[t];if(!i||!o)return e==C?-1:t==C?1:i?-1:o?1:u?P(u,e)-P(u,t):0;if(i===o)return pe(e,t);n=e;while(n=n.parentNode)a.unshift(n);n=t;while(n=n.parentNode)s.unshift(n);while(a[r]===s[r])r++;return r?pe(a[r],s[r]):a[r]==p?-1:s[r]==p?1:0}),C},se.matches=function(e,t){return se(e,null,null,t)},se.matchesSelector=function(e,t){if(T(e),d.matchesSelector&&E&&!N[t+" "]&&(!s||!s.test(t))&&(!v||!v.test(t)))try{var n=c.call(e,t);if(n||d.disconnectedMatch||e.document&&11!==e.document.nodeType)return n}catch(e){N(t,!0)}return 0<se(t,C,null,[e]).length},se.contains=function(e,t){return(e.ownerDocument||e)!=C&&T(e),y(e,t)},se.attr=function(e,t){(e.ownerDocument||e)!=C&&T(e);var n=b.attrHandle[t.toLowerCase()],r=n&&D.call(b.attrHandle,t.toLowerCase())?n(e,t,!E):void 0;return void 0!==r?r:d.attributes||!E?e.getAttribute(t):(r=e.getAttributeNode(t))&&r.specified?r.value:null},se.escape=function(e){return(e+"").replace(re,ie)},se.error=function(e){throw new Error("Syntax error, unrecognized expression: "+e)},se.uniqueSort=function(e){var t,n=[],r=0,i=0;if(l=!d.detectDuplicates,u=!d.sortStable&&e.slice(0),e.sort(j),l){while(t=e[i++])t===e[i]&&(r=n.push(i));while(r--)e.splice(n[r],1)}return u=null,e},o=se.getText=function(e){var t,n="",r=0,i=e.nodeType;if(i){if(1===i||9===i||11===i){if("string"==typeof e.textContent)return e.textContent;for(e=e.firstChild;e;e=e.nextSibling)n+=o(e)}else if(3===i||4===i)return e.nodeValue}else while(t=e[r++])n+=o(t);return n},(b=se.selectors={cacheLength:50,createPseudo:le,match:G,attrHandle:{},find:{},relative:{">":{dir:"parentNode",first:!0}," ":{dir:"parentNode"},"+":{dir:"previousSibling",first:!0},"~":{dir:"previousSibling"}},preFilter:{ATTR:function(e){return e[1]=e[1].replace(te,ne),e[3]=(e[3]||e[4]||e[5]||"").replace(te,ne),"~="===e[2]&&(e[3]=" "+e[3]+" "),e.slice(0,4)},CHILD:function(e){return e[1]=e[1].toLowerCase(),"nth"===e[1].slice(0,3)?(e[3]||se.error(e[0]),e[4]=+(e[4]?e[5]+(e[6]||1):2*("even"===e[3]||"odd"===e[3])),e[5]=+(e[7]+e[8]||"odd"===e[3])):e[3]&&se.error(e[0]),e},PSEUDO:function(e){var t,n=!e[6]&&e[2];return G.CHILD.test(e[0])?null:(e[3]?e[2]=e[4]||e[5]||"":n&&X.test(n)&&(t=h(n,!0))&&(t=n.indexOf(")",n.length-t)-n.length)&&(e[0]=e[0].slice(0,t),e[2]=n.slice(0,t)),e.slice(0,3))}},filter:{TAG:function(e){var t=e.replace(te,ne).toLowerCase();return"*"===e?function(){return!0}:function(e){return e.nodeName&&e.nodeName.toLowerCase()===t}},CLASS:function(e){var t=m[e+" "];return t||(t=new RegExp("(^|"+M+")"+e+"("+M+"|$)"))&&m(e,function(e){return t.test("string"==typeof e.className&&e.className||"undefined"!=typeof e.getAttribute&&e.getAttribute("class")||"")})},ATTR:function(n,r,i){return function(e){var t=se.attr(e,n);return null==t?"!="===r:!r||(t+="","="===r?t===i:"!="===r?t!==i:"^="===r?i&&0===t.indexOf(i):"*="===r?i&&-1<t.indexOf(i):"$="===r?i&&t.slice(-i.length)===i:"~="===r?-1<(" "+t.replace(B," ")+" ").indexOf(i):"|="===r&&(t===i||t.slice(0,i.length+1)===i+"-"))}},CHILD:function(h,e,t,g,v){var y="nth"!==h.slice(0,3),m="last"!==h.slice(-4),x="of-type"===e;return 1===g&&0===v?function(e){return!!e.parentNode}:function(e,t,n){var r,i,o,a,s,u,l=y!==m?"nextSibling":"previousSibling",c=e.parentNode,f=x&&e.nodeName.toLowerCase(),p=!n&&!x,d=!1;if(c){if(y){while(l){a=e;while(a=a[l])if(x?a.nodeName.toLowerCase()===f:1===a.nodeType)return!1;u=l="only"===h&&!u&&"nextSibling"}return!0}if(u=[m?c.firstChild:c.lastChild],m&&p){d=(s=(r=(i=(o=(a=c)[S]||(a[S]={}))[a.uniqueID]||(o[a.uniqueID]={}))[h]||[])[0]===k&&r[1])&&r[2],a=s&&c.childNodes[s];while(a=++s&&a&&a[l]||(d=s=0)||u.pop())if(1===a.nodeType&&++d&&a===e){i[h]=[k,s,d];break}}else if(p&&(d=s=(r=(i=(o=(a=e)[S]||(a[S]={}))[a.uniqueID]||(o[a.uniqueID]={}))[h]||[])[0]===k&&r[1]),!1===d)while(a=++s&&a&&a[l]||(d=s=0)||u.pop())if((x?a.nodeName.toLowerCase()===f:1===a.nodeType)&&++d&&(p&&((i=(o=a[S]||(a[S]={}))[a.uniqueID]||(o[a.uniqueID]={}))[h]=[k,d]),a===e))break;return(d-=v)===g||d%g==0&&0<=d/g}}},PSEUDO:function(e,o){var t,a=b.pseudos[e]||b.setFilters[e.toLowerCase()]||se.error("unsupported pseudo: "+e);return a[S]?a(o):1<a.length?(t=[e,e,"",o],b.setFilters.hasOwnProperty(e.toLowerCase())?le(function(e,t){var n,r=a(e,o),i=r.length;while(i--)e[n=P(e,r[i])]=!(t[n]=r[i])}):function(e){return a(e,0,t)}):a}},pseudos:{not:le(function(e){var r=[],i=[],s=f(e.replace($,"$1"));return s[S]?le(function(e,t,n,r){var i,o=s(e,null,r,[]),a=e.length;while(a--)(i=o[a])&&(e[a]=!(t[a]=i))}):function(e,t,n){return r[0]=e,s(r,null,n,i),r[0]=null,!i.pop()}}),has:le(function(t){return function(e){return 0<se(t,e).length}}),contains:le(function(t){return t=t.replace(te,ne),function(e){return-1<(e.textContent||o(e)).indexOf(t)}}),lang:le(function(n){return V.test(n||"")||se.error("unsupported lang: "+n),n=n.replace(te,ne).toLowerCase(),function(e){var t;do{if(t=E?e.lang:e.getAttribute("xml:lang")||e.getAttribute("lang"))return(t=t.toLowerCase())===n||0===t.indexOf(n+"-")}while((e=e.parentNode)&&1===e.nodeType);return!1}}),target:function(e){var t=n.location&&n.location.hash;return t&&t.slice(1)===e.id},root:function(e){return e===a},focus:function(e){return e===C.activeElement&&(!C.hasFocus||C.hasFocus())&&!!(e.type||e.href||~e.tabIndex)},enabled:ge(!1),disabled:ge(!0),checked:function(e){var t=e.nodeName.toLowerCase();return"input"===t&&!!e.checked||"option"===t&&!!e.selected},selected:function(e){return e.parentNode&&e.parentNode.selectedIndex,!0===e.selected},empty:function(e){for(e=e.firstChild;e;e=e.nextSibling)if(e.nodeType<6)return!1;return!0},parent:function(e){return!b.pseudos.empty(e)},header:function(e){return J.test(e.nodeName)},input:function(e){return Q.test(e.nodeName)},button:function(e){var t=e.nodeName.toLowerCase();return"input"===t&&"button"===e.type||"button"===t},text:function(e){var t;return"input"===e.nodeName.toLowerCase()&&"text"===e.type&&(null==(t=e.getAttribute("type"))||"text"===t.toLowerCase())},first:ve(function(){return[0]}),last:ve(function(e,t){return[t-1]}),eq:ve(function(e,t,n){return[n<0?n+t:n]}),even:ve(function(e,t){for(var n=0;n<t;n+=2)e.push(n);return e}),odd:ve(function(e,t){for(var n=1;n<t;n+=2)e.push(n);return e}),lt:ve(function(e,t,n){for(var r=n<0?n+t:t<n?t:n;0<=--r;)e.push(r);return e}),gt:ve(function(e,t,n){for(var r=n<0?n+t:n;++r<t;)e.push(r);return e})}}).pseudos.nth=b.pseudos.eq,{radio:!0,checkbox:!0,file:!0,password:!0,image:!0})b.pseudos[e]=de(e);for(e in{submit:!0,reset:!0})b.pseudos[e]=he(e);function me(){}function xe(e){for(var t=0,n=e.length,r="";t<n;t++)r+=e[t].value;return r}function be(s,e,t){var u=e.dir,l=e.next,c=l||u,f=t&&"parentNode"===c,p=r++;return e.first?function(e,t,n){while(e=e[u])if(1===e.nodeType||f)return s(e,t,n);return!1}:function(e,t,n){var r,i,o,a=[k,p];if(n){while(e=e[u])if((1===e.nodeType||f)&&s(e,t,n))return!0}else while(e=e[u])if(1===e.nodeType||f)if(i=(o=e[S]||(e[S]={}))[e.uniqueID]||(o[e.uniqueID]={}),l&&l===e.nodeName.toLowerCase())e=e[u]||e;else{if((r=i[c])&&r[0]===k&&r[1]===p)return a[2]=r[2];if((i[c]=a)[2]=s(e,t,n))return!0}return!1}}function we(i){return 1<i.length?function(e,t,n){var r=i.length;while(r--)if(!i[r](e,t,n))return!1;return!0}:i[0]}function Te(e,t,n,r,i){for(var o,a=[],s=0,u=e.length,l=null!=t;s<u;s++)(o=e[s])&&(n&&!n(o,r,i)||(a.push(o),l&&t.push(s)));return a}function Ce(d,h,g,v,y,e){return v&&!v[S]&&(v=Ce(v)),y&&!y[S]&&(y=Ce(y,e)),le(function(e,t,n,r){var i,o,a,s=[],u=[],l=t.length,c=e||function(e,t,n){for(var r=0,i=t.length;r<i;r++)se(e,t[r],n);return n}(h||"*",n.nodeType?[n]:n,[]),f=!d||!e&&h?c:Te(c,s,d,n,r),p=g?y||(e?d:l||v)?[]:t:f;if(g&&g(f,p,n,r),v){i=Te(p,u),v(i,[],n,r),o=i.length;while(o--)(a=i[o])&&(p[u[o]]=!(f[u[o]]=a))}if(e){if(y||d){if(y){i=[],o=p.length;while(o--)(a=p[o])&&i.push(f[o]=a);y(null,p=[],i,r)}o=p.length;while(o--)(a=p[o])&&-1<(i=y?P(e,a):s[o])&&(e[i]=!(t[i]=a))}}else p=Te(p===t?p.splice(l,p.length):p),y?y(null,t,p,r):H.apply(t,p)})}function Ee(e){for(var i,t,n,r=e.length,o=b.relative[e[0].type],a=o||b.relative[" "],s=o?1:0,u=be(function(e){return e===i},a,!0),l=be(function(e){return-1<P(i,e)},a,!0),c=[function(e,t,n){var r=!o&&(n||t!==w)||((i=t).nodeType?u(e,t,n):l(e,t,n));return i=null,r}];s<r;s++)if(t=b.relative[e[s].type])c=[be(we(c),t)];else{if((t=b.filter[e[s].type].apply(null,e[s].matches))[S]){for(n=++s;n<r;n++)if(b.relative[e[n].type])break;return Ce(1<s&&we(c),1<s&&xe(e.slice(0,s-1).concat({value:" "===e[s-2].type?"*":""})).replace($,"$1"),t,s<n&&Ee(e.slice(s,n)),n<r&&Ee(e=e.slice(n)),n<r&&xe(e))}c.push(t)}return we(c)}return me.prototype=b.filters=b.pseudos,b.setFilters=new me,h=se.tokenize=function(e,t){var n,r,i,o,a,s,u,l=x[e+" "];if(l)return t?0:l.slice(0);a=e,s=[],u=b.preFilter;while(a){for(o in n&&!(r=_.exec(a))||(r&&(a=a.slice(r[0].length)||a),s.push(i=[])),n=!1,(r=z.exec(a))&&(n=r.shift(),i.push({value:n,type:r[0].replace($," ")}),a=a.slice(n.length)),b.filter)!(r=G[o].exec(a))||u[o]&&!(r=u[o](r))||(n=r.shift(),i.push({value:n,type:o,matches:r}),a=a.slice(n.length));if(!n)break}return t?a.length:a?se.error(e):x(e,s).slice(0)},f=se.compile=function(e,t){var n,v,y,m,x,r,i=[],o=[],a=A[e+" "];if(!a){t||(t=h(e)),n=t.length;while(n--)(a=Ee(t[n]))[S]?i.push(a):o.push(a);(a=A(e,(v=o,m=0<(y=i).length,x=0<v.length,r=function(e,t,n,r,i){var o,a,s,u=0,l="0",c=e&&[],f=[],p=w,d=e||x&&b.find.TAG("*",i),h=k+=null==p?1:Math.random()||.1,g=d.length;for(i&&(w=t==C||t||i);l!==g&&null!=(o=d[l]);l++){if(x&&o){a=0,t||o.ownerDocument==C||(T(o),n=!E);while(s=v[a++])if(s(o,t||C,n)){r.push(o);break}i&&(k=h)}m&&((o=!s&&o)&&u--,e&&c.push(o))}if(u+=l,m&&l!==u){a=0;while(s=y[a++])s(c,f,t,n);if(e){if(0<u)while(l--)c[l]||f[l]||(f[l]=q.call(r));f=Te(f)}H.apply(r,f),i&&!e&&0<f.length&&1<u+y.length&&se.uniqueSort(r)}return i&&(k=h,w=p),c},m?le(r):r))).selector=e}return a},g=se.select=function(e,t,n,r){var i,o,a,s,u,l="function"==typeof e&&e,c=!r&&h(e=l.selector||e);if(n=n||[],1===c.length){if(2<(o=c[0]=c[0].slice(0)).length&&"ID"===(a=o[0]).type&&9===t.nodeType&&E&&b.relative[o[1].type]){if(!(t=(b.find.ID(a.matches[0].replace(te,ne),t)||[])[0]))return n;l&&(t=t.parentNode),e=e.slice(o.shift().value.length)}i=G.needsContext.test(e)?0:o.length;while(i--){if(a=o[i],b.relative[s=a.type])break;if((u=b.find[s])&&(r=u(a.matches[0].replace(te,ne),ee.test(o[0].type)&&ye(t.parentNode)||t))){if(o.splice(i,1),!(e=r.length&&xe(o)))return H.apply(n,r),n;break}}}return(l||f(e,c))(r,t,!E,n,!t||ee.test(e)&&ye(t.parentNode)||t),n},d.sortStable=S.split("").sort(j).join("")===S,d.detectDuplicates=!!l,T(),d.sortDetached=ce(function(e){return 1&e.compareDocumentPosition(C.createElement("fieldset"))}),ce(function(e){return e.innerHTML="<a href='#'></a>","#"===e.firstChild.getAttribute("href")})||fe("type|href|height|width",function(e,t,n){if(!n)return e.getAttribute(t,"type"===t.toLowerCase()?1:2)}),d.attributes&&ce(function(e){return e.innerHTML="<input/>",e.firstChild.setAttribute("value",""),""===e.firstChild.getAttribute("value")})||fe("value",function(e,t,n){if(!n&&"input"===e.nodeName.toLowerCase())return e.defaultValue}),ce(function(e){return null==e.getAttribute("disabled")})||fe(R,function(e,t,n){var r;if(!n)return!0===e[t]?t.toLowerCase():(r=e.getAttributeNode(t))&&r.specified?r.value:null}),se}(C);S.find=d,S.expr=d.selectors,S.expr[":"]=S.expr.pseudos,S.uniqueSort=S.unique=d.uniqueSort,S.text=d.getText,S.isXMLDoc=d.isXML,S.contains=d.contains,S.escapeSelector=d.escape;var h=function(e,t,n){var r=[],i=void 0!==n;while((e=e[t])&&9!==e.nodeType)if(1===e.nodeType){if(i&&S(e).is(n))break;r.push(e)}return r},T=function(e,t){for(var n=[];e;e=e.nextSibling)1===e.nodeType&&e!==t&&n.push(e);return n},k=S.expr.match.needsContext;function A(e,t){return e.nodeName&&e.nodeName.toLowerCase()===t.toLowerCase()}var N=/^<([a-z][^\/\0>:\x20\t\r\n\f]*)[\x20\t\r\n\f]*\/?>(?:<\/\1>|)$/i;function j(e,n,r){return m(n)?S.grep(e,function(e,t){return!!n.call(e,t,e)!==r}):n.nodeType?S.grep(e,function(e){return e===n!==r}):"string"!=typeof n?S.grep(e,function(e){return-1<i.call(n,e)!==r}):S.filter(n,e,r)}S.filter=function(e,t,n){var r=t[0];return n&&(e=":not("+e+")"),1===t.length&&1===r.nodeType?S.find.matchesSelector(r,e)?[r]:[]:S.find.matches(e,S.grep(t,function(e){return 1===e.nodeType}))},S.fn.extend({find:function(e){var t,n,r=this.length,i=this;if("string"!=typeof e)return this.pushStack(S(e).filter(function(){for(t=0;t<r;t++)if(S.contains(i[t],this))return!0}));for(n=this.pushStack([]),t=0;t<r;t++)S.find(e,i[t],n);return 1<r?S.uniqueSort(n):n},filter:function(e){return this.pushStack(j(this,e||[],!1))},not:function(e){return this.pushStack(j(this,e||[],!0))},is:function(e){return!!j(this,"string"==typeof e&&k.test(e)?S(e):e||[],!1).length}});var D,q=/^(?:\s*(<[\w\W]+>)[^>]*|#([\w-]+))$/;(S.fn.init=function(e,t,n){var r,i;if(!e)return this;if(n=n||D,"string"==typeof e){if(!(r="<"===e[0]&&">"===e[e.length-1]&&3<=e.length?[null,e,null]:q.exec(e))||!r[1]&&t)return!t||t.jquery?(t||n).find(e):this.constructor(t).find(e);if(r[1]){if(t=t instanceof S?t[0]:t,S.merge(this,S.parseHTML(r[1],t&&t.nodeType?t.ownerDocument||t:E,!0)),N.test(r[1])&&S.isPlainObject(t))for(r in t)m(this[r])?this[r](t[r]):this.attr(r,t[r]);return this}return(i=E.getElementById(r[2]))&&(this[0]=i,this.length=1),this}return e.nodeType?(this[0]=e,this.length=1,this):m(e)?void 0!==n.ready?n.ready(e):e(S):S.makeArray(e,this)}).prototype=S.fn,D=S(E);var L=/^(?:parents|prev(?:Until|All))/,H={children:!0,contents:!0,next:!0,prev:!0};function O(e,t){while((e=e[t])&&1!==e.nodeType);return e}S.fn.extend({has:function(e){var t=S(e,this),n=t.length;return this.filter(function(){for(var e=0;e<n;e++)if(S.contains(this,t[e]))return!0})},closest:function(e,t){var n,r=0,i=this.length,o=[],a="string"!=typeof e&&S(e);if(!k.test(e))for(;r<i;r++)for(n=this[r];n&&n!==t;n=n.parentNode)if(n.nodeType<11&&(a?-1<a.index(n):1===n.nodeType&&S.find.matchesSelector(n,e))){o.push(n);break}return this.pushStack(1<o.length?S.uniqueSort(o):o)},index:function(e){return e?"string"==typeof e?i.call(S(e),this[0]):i.call(this,e.jquery?e[0]:e):this[0]&&this[0].parentNode?this.first().prevAll().length:-1},add:function(e,t){return this.pushStack(S.uniqueSort(S.merge(this.get(),S(e,t))))},addBack:function(e){return this.add(null==e?this.prevObject:this.prevObject.filter(e))}}),S.each({parent:function(e){var t=e.parentNode;return t&&11!==t.nodeType?t:null},parents:function(e){return h(e,"parentNode")},parentsUntil:function(e,t,n){return h(e,"parentNode",n)},next:function(e){return O(e,"nextSibling")},prev:function(e){return O(e,"previousSibling")},nextAll:function(e){return h(e,"nextSibling")},prevAll:function(e){return h(e,"previousSibling")},nextUntil:function(e,t,n){return h(e,"nextSibling",n)},prevUntil:function(e,t,n){return h(e,"previousSibling",n)},siblings:function(e){return T((e.parentNode||{}).firstChild,e)},children:function(e){return T(e.firstChild)},contents:function(e){return null!=e.contentDocument&&r(e.contentDocument)?e.contentDocument:(A(e,"template")&&(e=e.content||e),S.merge([],e.childNodes))}},function(r,i){S.fn[r]=function(e,t){var n=S.map(this,i,e);return"Until"!==r.slice(-5)&&(t=e),t&&"string"==typeof t&&(n=S.filter(t,n)),1<this.length&&(H[r]||S.uniqueSort(n),L.test(r)&&n.reverse()),this.pushStack(n)}});var P=/[^\x20\t\r\n\f]+/g;function R(e){return e}function M(e){throw e}function I(e,t,n,r){var i;try{e&&m(i=e.promise)?i.call(e).done(t).fail(n):e&&m(i=e.then)?i.call(e,t,n):t.apply(void 0,[e].slice(r))}catch(e){n.apply(void 0,[e])}}S.Callbacks=function(r){var e,n;r="string"==typeof r?(e=r,n={},S.each(e.match(P)||[],function(e,t){n[t]=!0}),n):S.extend({},r);var i,t,o,a,s=[],u=[],l=-1,c=function(){for(a=a||r.once,o=i=!0;u.length;l=-1){t=u.shift();while(++l<s.length)!1===s[l].apply(t[0],t[1])&&r.stopOnFalse&&(l=s.length,t=!1)}r.memory||(t=!1),i=!1,a&&(s=t?[]:"")},f={add:function(){return s&&(t&&!i&&(l=s.length-1,u.push(t)),function n(e){S.each(e,function(e,t){m(t)?r.unique&&f.has(t)||s.push(t):t&&t.length&&"string"!==w(t)&&n(t)})}(arguments),t&&!i&&c()),this},remove:function(){return S.each(arguments,function(e,t){var n;while(-1<(n=S.inArray(t,s,n)))s.splice(n,1),n<=l&&l--}),this},has:function(e){return e?-1<S.inArray(e,s):0<s.length},empty:function(){return s&&(s=[]),this},disable:function(){return a=u=[],s=t="",this},disabled:function(){return!s},lock:function(){return a=u=[],t||i||(s=t=""),this},locked:function(){return!!a},fireWith:function(e,t){return a||(t=[e,(t=t||[]).slice?t.slice():t],u.push(t),i||c()),this},fire:function(){return f.fireWith(this,arguments),this},fired:function(){return!!o}};return f},S.extend({Deferred:function(e){var o=[["notify","progress",S.Callbacks("memory"),S.Callbacks("memory"),2],["resolve","done",S.Callbacks("once memory"),S.Callbacks("once memory"),0,"resolved"],["reject","fail",S.Callbacks("once memory"),S.Callbacks("once memory"),1,"rejected"]],i="pending",a={state:function(){return i},always:function(){return s.done(arguments).fail(arguments),this},"catch":function(e){return a.then(null,e)},pipe:function(){var i=arguments;return S.Deferred(function(r){S.each(o,function(e,t){var n=m(i[t[4]])&&i[t[4]];s[t[1]](function(){var e=n&&n.apply(this,arguments);e&&m(e.promise)?e.promise().progress(r.notify).done(r.resolve).fail(r.reject):r[t[0]+"With"](this,n?[e]:arguments)})}),i=null}).promise()},then:function(t,n,r){var u=0;function l(i,o,a,s){return function(){var n=this,r=arguments,e=function(){var e,t;if(!(i<u)){if((e=a.apply(n,r))===o.promise())throw new TypeError("Thenable self-resolution");t=e&&("object"==typeof e||"function"==typeof e)&&e.then,m(t)?s?t.call(e,l(u,o,R,s),l(u,o,M,s)):(u++,t.call(e,l(u,o,R,s),l(u,o,M,s),l(u,o,R,o.notifyWith))):(a!==R&&(n=void 0,r=[e]),(s||o.resolveWith)(n,r))}},t=s?e:function(){try{e()}catch(e){S.Deferred.exceptionHook&&S.Deferred.exceptionHook(e,t.stackTrace),u<=i+1&&(a!==M&&(n=void 0,r=[e]),o.rejectWith(n,r))}};i?t():(S.Deferred.getStackHook&&(t.stackTrace=S.Deferred.getStackHook()),C.setTimeout(t))}}return S.Deferred(function(e){o[0][3].add(l(0,e,m(r)?r:R,e.notifyWith)),o[1][3].add(l(0,e,m(t)?t:R)),o[2][3].add(l(0,e,m(n)?n:M))}).promise()},promise:function(e){return null!=e?S.extend(e,a):a}},s={};return S.each(o,function(e,t){var n=t[2],r=t[5];a[t[1]]=n.add,r&&n.add(function(){i=r},o[3-e][2].disable,o[3-e][3].disable,o[0][2].lock,o[0][3].lock),n.add(t[3].fire),s[t[0]]=function(){return s[t[0]+"With"](this===s?void 0:this,arguments),this},s[t[0]+"With"]=n.fireWith}),a.promise(s),e&&e.call(s,s),s},when:function(e){var n=arguments.length,t=n,r=Array(t),i=s.call(arguments),o=S.Deferred(),a=function(t){return function(e){r[t]=this,i[t]=1<arguments.length?s.call(arguments):e,--n||o.resolveWith(r,i)}};if(n<=1&&(I(e,o.done(a(t)).resolve,o.reject,!n),"pending"===o.state()||m(i[t]&&i[t].then)))return o.then();while(t--)I(i[t],a(t),o.reject);return o.promise()}});var W=/^(Eval|Internal|Range|Reference|Syntax|Type|URI)Error$/;S.Deferred.exceptionHook=function(e,t){C.console&&C.console.warn&&e&&W.test(e.name)&&C.console.warn("jQuery.Deferred exception: "+e.message,e.stack,t)},S.readyException=function(e){C.setTimeout(function(){throw e})};var F=S.Deferred();function B(){E.removeEventListener("DOMContentLoaded",B),C.removeEventListener("load",B),S.ready()}S.fn.ready=function(e){return F.then(e)["catch"](function(e){S.readyException(e)}),this},S.extend({isReady:!1,readyWait:1,ready:function(e){(!0===e?--S.readyWait:S.isReady)||(S.isReady=!0)!==e&&0<--S.readyWait||F.resolveWith(E,[S])}}),S.ready.then=F.then,"complete"===E.readyState||"loading"!==E.readyState&&!E.documentElement.doScroll?C.setTimeout(S.ready):(E.addEventListener("DOMContentLoaded",B),C.addEventListener("load",B));var $=function(e,t,n,r,i,o,a){var s=0,u=e.length,l=null==n;if("object"===w(n))for(s in i=!0,n)$(e,t,s,n[s],!0,o,a);else if(void 0!==r&&(i=!0,m(r)||(a=!0),l&&(a?(t.call(e,r),t=null):(l=t,t=function(e,t,n){return l.call(S(e),n)})),t))for(;s<u;s++)t(e[s],n,a?r:r.call(e[s],s,t(e[s],n)));return i?e:l?t.call(e):u?t(e[0],n):o},_=/^-ms-/,z=/-([a-z])/g;function U(e,t){return t.toUpperCase()}function X(e){return e.replace(_,"ms-").replace(z,U)}var V=function(e){return 1===e.nodeType||9===e.nodeType||!+e.nodeType};function G(){this.expando=S.expando+G.uid++}G.uid=1,G.prototype={cache:function(e){var t=e[this.expando];return t||(t={},V(e)&&(e.nodeType?e[this.expando]=t:Object.defineProperty(e,this.expando,{value:t,configurable:!0}))),t},set:function(e,t,n){var r,i=this.cache(e);if("string"==typeof t)i[X(t)]=n;else for(r in t)i[X(r)]=t[r];return i},get:function(e,t){return void 0===t?this.cache(e):e[this.expando]&&e[this.expando][X(t)]},access:function(e,t,n){return void 0===t||t&&"string"==typeof t&&void 0===n?this.get(e,t):(this.set(e,t,n),void 0!==n?n:t)},remove:function(e,t){var n,r=e[this.expando];if(void 0!==r){if(void 0!==t){n=(t=Array.isArray(t)?t.map(X):(t=X(t))in r?[t]:t.match(P)||[]).length;while(n--)delete r[t[n]]}(void 0===t||S.isEmptyObject(r))&&(e.nodeType?e[this.expando]=void 0:delete e[this.expando])}},hasData:function(e){var t=e[this.expando];return void 0!==t&&!S.isEmptyObject(t)}};var Y=new G,Q=new G,J=/^(?:\{[\w\W]*\}|\[[\w\W]*\])$/,K=/[A-Z]/g;function Z(e,t,n){var r,i;if(void 0===n&&1===e.nodeType)if(r="data-"+t.replace(K,"-$&").toLowerCase(),"string"==typeof(n=e.getAttribute(r))){try{n="true"===(i=n)||"false"!==i&&("null"===i?null:i===+i+""?+i:J.test(i)?JSON.parse(i):i)}catch(e){}Q.set(e,t,n)}else n=void 0;return n}S.extend({hasData:function(e){return Q.hasData(e)||Y.hasData(e)},data:function(e,t,n){return Q.access(e,t,n)},removeData:function(e,t){Q.remove(e,t)},_data:function(e,t,n){return Y.access(e,t,n)},_removeData:function(e,t){Y.remove(e,t)}}),S.fn.extend({data:function(n,e){var t,r,i,o=this[0],a=o&&o.attributes;if(void 0===n){if(this.length&&(i=Q.get(o),1===o.nodeType&&!Y.get(o,"hasDataAttrs"))){t=a.length;while(t--)a[t]&&0===(r=a[t].name).indexOf("data-")&&(r=X(r.slice(5)),Z(o,r,i[r]));Y.set(o,"hasDataAttrs",!0)}return i}return"object"==typeof n?this.each(function(){Q.set(this,n)}):$(this,function(e){var t;if(o&&void 0===e)return void 0!==(t=Q.get(o,n))?t:void 0!==(t=Z(o,n))?t:void 0;this.each(function(){Q.set(this,n,e)})},null,e,1<arguments.length,null,!0)},removeData:function(e){return this.each(function(){Q.remove(this,e)})}}),S.extend({queue:function(e,t,n){var r;if(e)return t=(t||"fx")+"queue",r=Y.get(e,t),n&&(!r||Array.isArray(n)?r=Y.access(e,t,S.makeArray(n)):r.push(n)),r||[]},dequeue:function(e,t){t=t||"fx";var n=S.queue(e,t),r=n.length,i=n.shift(),o=S._queueHooks(e,t);"inprogress"===i&&(i=n.shift(),r--),i&&("fx"===t&&n.unshift("inprogress"),delete o.stop,i.call(e,function(){S.dequeue(e,t)},o)),!r&&o&&o.empty.fire()},_queueHooks:function(e,t){var n=t+"queueHooks";return Y.get(e,n)||Y.access(e,n,{empty:S.Callbacks("once memory").add(function(){Y.remove(e,[t+"queue",n])})})}}),S.fn.extend({queue:function(t,n){var e=2;return"string"!=typeof t&&(n=t,t="fx",e--),arguments.length<e?S.queue(this[0],t):void 0===n?this:this.each(function(){var e=S.queue(this,t,n);S._queueHooks(this,t),"fx"===t&&"inprogress"!==e[0]&&S.dequeue(this,t)})},dequeue:function(e){return this.each(function(){S.dequeue(this,e)})},clearQueue:function(e){return this.queue(e||"fx",[])},promise:function(e,t){var n,r=1,i=S.Deferred(),o=this,a=this.length,s=function(){--r||i.resolveWith(o,[o])};"string"!=typeof e&&(t=e,e=void 0),e=e||"fx";while(a--)(n=Y.get(o[a],e+"queueHooks"))&&n.empty&&(r++,n.empty.add(s));return s(),i.promise(t)}});var ee=/[+-]?(?:\d*\.|)\d+(?:[eE][+-]?\d+|)/.source,te=new RegExp("^(?:([+-])=|)("+ee+")([a-z%]*)$","i"),ne=["Top","Right","Bottom","Left"],re=E.documentElement,ie=function(e){return S.contains(e.ownerDocument,e)},oe={composed:!0};re.getRootNode&&(ie=function(e){return S.contains(e.ownerDocument,e)||e.getRootNode(oe)===e.ownerDocument});var ae=function(e,t){return"none"===(e=t||e).style.display||""===e.style.display&&ie(e)&&"none"===S.css(e,"display")};function se(e,t,n,r){var i,o,a=20,s=r?function(){return r.cur()}:function(){return S.css(e,t,"")},u=s(),l=n&&n[3]||(S.cssNumber[t]?"":"px"),c=e.nodeType&&(S.cssNumber[t]||"px"!==l&&+u)&&te.exec(S.css(e,t));if(c&&c[3]!==l){u/=2,l=l||c[3],c=+u||1;while(a--)S.style(e,t,c+l),(1-o)*(1-(o=s()/u||.5))<=0&&(a=0),c/=o;c*=2,S.style(e,t,c+l),n=n||[]}return n&&(c=+c||+u||0,i=n[1]?c+(n[1]+1)*n[2]:+n[2],r&&(r.unit=l,r.start=c,r.end=i)),i}var ue={};function le(e,t){for(var n,r,i,o,a,s,u,l=[],c=0,f=e.length;c<f;c++)(r=e[c]).style&&(n=r.style.display,t?("none"===n&&(l[c]=Y.get(r,"display")||null,l[c]||(r.style.display="")),""===r.style.display&&ae(r)&&(l[c]=(u=a=o=void 0,a=(i=r).ownerDocument,s=i.nodeName,(u=ue[s])||(o=a.body.appendChild(a.createElement(s)),u=S.css(o,"display"),o.parentNode.removeChild(o),"none"===u&&(u="block"),ue[s]=u)))):"none"!==n&&(l[c]="none",Y.set(r,"display",n)));for(c=0;c<f;c++)null!=l[c]&&(e[c].style.display=l[c]);return e}S.fn.extend({show:function(){return le(this,!0)},hide:function(){return le(this)},toggle:function(e){return"boolean"==typeof e?e?this.show():this.hide():this.each(function(){ae(this)?S(this).show():S(this).hide()})}});var ce,fe,pe=/^(?:checkbox|radio)$/i,de=/<([a-z][^\/\0>\x20\t\r\n\f]*)/i,he=/^$|^module$|\/(?:java|ecma)script/i;ce=E.createDocumentFragment().appendChild(E.createElement("div")),(fe=E.createElement("input")).setAttribute("type","radio"),fe.setAttribute("checked","checked"),fe.setAttribute("name","t"),ce.appendChild(fe),y.checkClone=ce.cloneNode(!0).cloneNode(!0).lastChild.checked,ce.innerHTML="<textarea>x</textarea>",y.noCloneChecked=!!ce.cloneNode(!0).lastChild.defaultValue,ce.innerHTML="<option></option>",y.option=!!ce.lastChild;var ge={thead:[1,"<table>","</table>"],col:[2,"<table><colgroup>","</colgroup></table>"],tr:[2,"<table><tbody>","</tbody></table>"],td:[3,"<table><tbody><tr>","</tr></tbody></table>"],_default:[0,"",""]};function ve(e,t){var n;return n="undefined"!=typeof e.getElementsByTagName?e.getElementsByTagName(t||"*"):"undefined"!=typeof e.querySelectorAll?e.querySelectorAll(t||"*"):[],void 0===t||t&&A(e,t)?S.merge([e],n):n}function ye(e,t){for(var n=0,r=e.length;n<r;n++)Y.set(e[n],"globalEval",!t||Y.get(t[n],"globalEval"))}ge.tbody=ge.tfoot=ge.colgroup=ge.caption=ge.thead,ge.th=ge.td,y.option||(ge.optgroup=ge.option=[1,"<select multiple='multiple'>","</select>"]);var me=/<|&#?\w+;/;function xe(e,t,n,r,i){for(var o,a,s,u,l,c,f=t.createDocumentFragment(),p=[],d=0,h=e.length;d<h;d++)if((o=e[d])||0===o)if("object"===w(o))S.merge(p,o.nodeType?[o]:o);else if(me.test(o)){a=a||f.appendChild(t.createElement("div")),s=(de.exec(o)||["",""])[1].toLowerCase(),u=ge[s]||ge._default,a.innerHTML=u[1]+S.htmlPrefilter(o)+u[2],c=u[0];while(c--)a=a.lastChild;S.merge(p,a.childNodes),(a=f.firstChild).textContent=""}else p.push(t.createTextNode(o));f.textContent="",d=0;while(o=p[d++])if(r&&-1<S.inArray(o,r))i&&i.push(o);else if(l=ie(o),a=ve(f.appendChild(o),"script"),l&&ye(a),n){c=0;while(o=a[c++])he.test(o.type||"")&&n.push(o)}return f}var be=/^([^.]*)(?:\.(.+)|)/;function we(){return!0}function Te(){return!1}function Ce(e,t){return e===function(){try{return E.activeElement}catch(e){}}()==("focus"===t)}function Ee(e,t,n,r,i,o){var a,s;if("object"==typeof t){for(s in"string"!=typeof n&&(r=r||n,n=void 0),t)Ee(e,s,n,r,t[s],o);return e}if(null==r&&null==i?(i=n,r=n=void 0):null==i&&("string"==typeof n?(i=r,r=void 0):(i=r,r=n,n=void 0)),!1===i)i=Te;else if(!i)return e;return 1===o&&(a=i,(i=function(e){return S().off(e),a.apply(this,arguments)}).guid=a.guid||(a.guid=S.guid++)),e.each(function(){S.event.add(this,t,i,r,n)})}function Se(e,i,o){o?(Y.set(e,i,!1),S.event.add(e,i,{namespace:!1,handler:function(e){var t,n,r=Y.get(this,i);if(1&e.isTrigger&&this[i]){if(r.length)(S.event.special[i]||{}).delegateType&&e.stopPropagation();else if(r=s.call(arguments),Y.set(this,i,r),t=o(this,i),this[i](),r!==(n=Y.get(this,i))||t?Y.set(this,i,!1):n={},r!==n)return e.stopImmediatePropagation(),e.preventDefault(),n&&n.value}else r.length&&(Y.set(this,i,{value:S.event.trigger(S.extend(r[0],S.Event.prototype),r.slice(1),this)}),e.stopImmediatePropagation())}})):void 0===Y.get(e,i)&&S.event.add(e,i,we)}S.event={global:{},add:function(t,e,n,r,i){var o,a,s,u,l,c,f,p,d,h,g,v=Y.get(t);if(V(t)){n.handler&&(n=(o=n).handler,i=o.selector),i&&S.find.matchesSelector(re,i),n.guid||(n.guid=S.guid++),(u=v.events)||(u=v.events=Object.create(null)),(a=v.handle)||(a=v.handle=function(e){return"undefined"!=typeof S&&S.event.triggered!==e.type?S.event.dispatch.apply(t,arguments):void 0}),l=(e=(e||"").match(P)||[""]).length;while(l--)d=g=(s=be.exec(e[l])||[])[1],h=(s[2]||"").split(".").sort(),d&&(f=S.event.special[d]||{},d=(i?f.delegateType:f.bindType)||d,f=S.event.special[d]||{},c=S.extend({type:d,origType:g,data:r,handler:n,guid:n.guid,selector:i,needsContext:i&&S.expr.match.needsContext.test(i),namespace:h.join(".")},o),(p=u[d])||((p=u[d]=[]).delegateCount=0,f.setup&&!1!==f.setup.call(t,r,h,a)||t.addEventListener&&t.addEventListener(d,a)),f.add&&(f.add.call(t,c),c.handler.guid||(c.handler.guid=n.guid)),i?p.splice(p.delegateCount++,0,c):p.push(c),S.event.global[d]=!0)}},remove:function(e,t,n,r,i){var o,a,s,u,l,c,f,p,d,h,g,v=Y.hasData(e)&&Y.get(e);if(v&&(u=v.events)){l=(t=(t||"").match(P)||[""]).length;while(l--)if(d=g=(s=be.exec(t[l])||[])[1],h=(s[2]||"").split(".").sort(),d){f=S.event.special[d]||{},p=u[d=(r?f.delegateType:f.bindType)||d]||[],s=s[2]&&new RegExp("(^|\\.)"+h.join("\\.(?:.*\\.|)")+"(\\.|$)"),a=o=p.length;while(o--)c=p[o],!i&&g!==c.origType||n&&n.guid!==c.guid||s&&!s.test(c.namespace)||r&&r!==c.selector&&("**"!==r||!c.selector)||(p.splice(o,1),c.selector&&p.delegateCount--,f.remove&&f.remove.call(e,c));a&&!p.length&&(f.teardown&&!1!==f.teardown.call(e,h,v.handle)||S.removeEvent(e,d,v.handle),delete u[d])}else for(d in u)S.event.remove(e,d+t[l],n,r,!0);S.isEmptyObject(u)&&Y.remove(e,"handle events")}},dispatch:function(e){var t,n,r,i,o,a,s=new Array(arguments.length),u=S.event.fix(e),l=(Y.get(this,"events")||Object.create(null))[u.type]||[],c=S.event.special[u.type]||{};for(s[0]=u,t=1;t<arguments.length;t++)s[t]=arguments[t];if(u.delegateTarget=this,!c.preDispatch||!1!==c.preDispatch.call(this,u)){a=S.event.handlers.call(this,u,l),t=0;while((i=a[t++])&&!u.isPropagationStopped()){u.currentTarget=i.elem,n=0;while((o=i.handlers[n++])&&!u.isImmediatePropagationStopped())u.rnamespace&&!1!==o.namespace&&!u.rnamespace.test(o.namespace)||(u.handleObj=o,u.data=o.data,void 0!==(r=((S.event.special[o.origType]||{}).handle||o.handler).apply(i.elem,s))&&!1===(u.result=r)&&(u.preventDefault(),u.stopPropagation()))}return c.postDispatch&&c.postDispatch.call(this,u),u.result}},handlers:function(e,t){var n,r,i,o,a,s=[],u=t.delegateCount,l=e.target;if(u&&l.nodeType&&!("click"===e.type&&1<=e.button))for(;l!==this;l=l.parentNode||this)if(1===l.nodeType&&("click"!==e.type||!0!==l.disabled)){for(o=[],a={},n=0;n<u;n++)void 0===a[i=(r=t[n]).selector+" "]&&(a[i]=r.needsContext?-1<S(i,this).index(l):S.find(i,this,null,[l]).length),a[i]&&o.push(r);o.length&&s.push({elem:l,handlers:o})}return l=this,u<t.length&&s.push({elem:l,handlers:t.slice(u)}),s},addProp:function(t,e){Object.defineProperty(S.Event.prototype,t,{enumerable:!0,configurable:!0,get:m(e)?function(){if(this.originalEvent)return e(this.originalEvent)}:function(){if(this.originalEvent)return this.originalEvent[t]},set:function(e){Object.defineProperty(this,t,{enumerable:!0,configurable:!0,writable:!0,value:e})}})},fix:function(e){return e[S.expando]?e:new S.Event(e)},special:{load:{noBubble:!0},click:{setup:function(e){var t=this||e;return pe.test(t.type)&&t.click&&A(t,"input")&&Se(t,"click",we),!1},trigger:function(e){var t=this||e;return pe.test(t.type)&&t.click&&A(t,"input")&&Se(t,"click"),!0},_default:function(e){var t=e.target;return pe.test(t.type)&&t.click&&A(t,"input")&&Y.get(t,"click")||A(t,"a")}},beforeunload:{postDispatch:function(e){void 0!==e.result&&e.originalEvent&&(e.originalEvent.returnValue=e.result)}}}},S.removeEvent=function(e,t,n){e.removeEventListener&&e.removeEventListener(t,n)},S.Event=function(e,t){if(!(this instanceof S.Event))return new S.Event(e,t);e&&e.type?(this.originalEvent=e,this.type=e.type,this.isDefaultPrevented=e.defaultPrevented||void 0===e.defaultPrevented&&!1===e.returnValue?we:Te,this.target=e.target&&3===e.target.nodeType?e.target.parentNode:e.target,this.currentTarget=e.currentTarget,this.relatedTarget=e.relatedTarget):this.type=e,t&&S.extend(this,t),this.timeStamp=e&&e.timeStamp||Date.now(),this[S.expando]=!0},S.Event.prototype={constructor:S.Event,isDefaultPrevented:Te,isPropagationStopped:Te,isImmediatePropagationStopped:Te,isSimulated:!1,preventDefault:function(){var e=this.originalEvent;this.isDefaultPrevented=we,e&&!this.isSimulated&&e.preventDefault()},stopPropagation:function(){var e=this.originalEvent;this.isPropagationStopped=we,e&&!this.isSimulated&&e.stopPropagation()},stopImmediatePropagation:function(){var e=this.originalEvent;this.isImmediatePropagationStopped=we,e&&!this.isSimulated&&e.stopImmediatePropagation(),this.stopPropagation()}},S.each({altKey:!0,bubbles:!0,cancelable:!0,changedTouches:!0,ctrlKey:!0,detail:!0,eventPhase:!0,metaKey:!0,pageX:!0,pageY:!0,shiftKey:!0,view:!0,"char":!0,code:!0,charCode:!0,key:!0,keyCode:!0,button:!0,buttons:!0,clientX:!0,clientY:!0,offsetX:!0,offsetY:!0,pointerId:!0,pointerType:!0,screenX:!0,screenY:!0,targetTouches:!0,toElement:!0,touches:!0,which:!0},S.event.addProp),S.each({focus:"focusin",blur:"focusout"},function(e,t){S.event.special[e]={setup:function(){return Se(this,e,Ce),!1},trigger:function(){return Se(this,e),!0},_default:function(){return!0},delegateType:t}}),S.each({mouseenter:"mouseover",mouseleave:"mouseout",pointerenter:"pointerover",pointerleave:"pointerout"},function(e,i){S.event.special[e]={delegateType:i,bindType:i,handle:function(e){var t,n=e.relatedTarget,r=e.handleObj;return n&&(n===this||S.contains(this,n))||(e.type=r.origType,t=r.handler.apply(this,arguments),e.type=i),t}}}),S.fn.extend({on:function(e,t,n,r){return Ee(this,e,t,n,r)},one:function(e,t,n,r){return Ee(this,e,t,n,r,1)},off:function(e,t,n){var r,i;if(e&&e.preventDefault&&e.handleObj)return r=e.handleObj,S(e.delegateTarget).off(r.namespace?r.origType+"."+r.namespace:r.origType,r.selector,r.handler),this;if("object"==typeof e){for(i in e)this.off(i,t,e[i]);return this}return!1!==t&&"function"!=typeof t||(n=t,t=void 0),!1===n&&(n=Te),this.each(function(){S.event.remove(this,e,n,t)})}});var ke=/<script|<style|<link/i,Ae=/checked\s*(?:[^=]|=\s*.checked.)/i,Ne=/^\s*<!(?:\[CDATA\[|--)|(?:\]\]|--)>\s*$/g;function je(e,t){return A(e,"table")&&A(11!==t.nodeType?t:t.firstChild,"tr")&&S(e).children("tbody")[0]||e}function De(e){return e.type=(null!==e.getAttribute("type"))+"/"+e.type,e}function qe(e){return"true/"===(e.type||"").slice(0,5)?e.type=e.type.slice(5):e.removeAttribute("type"),e}function Le(e,t){var n,r,i,o,a,s;if(1===t.nodeType){if(Y.hasData(e)&&(s=Y.get(e).events))for(i in Y.remove(t,"handle events"),s)for(n=0,r=s[i].length;n<r;n++)S.event.add(t,i,s[i][n]);Q.hasData(e)&&(o=Q.access(e),a=S.extend({},o),Q.set(t,a))}}function He(n,r,i,o){r=g(r);var e,t,a,s,u,l,c=0,f=n.length,p=f-1,d=r[0],h=m(d);if(h||1<f&&"string"==typeof d&&!y.checkClone&&Ae.test(d))return n.each(function(e){var t=n.eq(e);h&&(r[0]=d.call(this,e,t.html())),He(t,r,i,o)});if(f&&(t=(e=xe(r,n[0].ownerDocument,!1,n,o)).firstChild,1===e.childNodes.length&&(e=t),t||o)){for(s=(a=S.map(ve(e,"script"),De)).length;c<f;c++)u=e,c!==p&&(u=S.clone(u,!0,!0),s&&S.merge(a,ve(u,"script"))),i.call(n[c],u,c);if(s)for(l=a[a.length-1].ownerDocument,S.map(a,qe),c=0;c<s;c++)u=a[c],he.test(u.type||"")&&!Y.access(u,"globalEval")&&S.contains(l,u)&&(u.src&&"module"!==(u.type||"").toLowerCase()?S._evalUrl&&!u.noModule&&S._evalUrl(u.src,{nonce:u.nonce||u.getAttribute("nonce")},l):b(u.textContent.replace(Ne,""),u,l))}return n}function Oe(e,t,n){for(var r,i=t?S.filter(t,e):e,o=0;null!=(r=i[o]);o++)n||1!==r.nodeType||S.cleanData(ve(r)),r.parentNode&&(n&&ie(r)&&ye(ve(r,"script")),r.parentNode.removeChild(r));return e}S.extend({htmlPrefilter:function(e){return e},clone:function(e,t,n){var r,i,o,a,s,u,l,c=e.cloneNode(!0),f=ie(e);if(!(y.noCloneChecked||1!==e.nodeType&&11!==e.nodeType||S.isXMLDoc(e)))for(a=ve(c),r=0,i=(o=ve(e)).length;r<i;r++)s=o[r],u=a[r],void 0,"input"===(l=u.nodeName.toLowerCase())&&pe.test(s.type)?u.checked=s.checked:"input"!==l&&"textarea"!==l||(u.defaultValue=s.defaultValue);if(t)if(n)for(o=o||ve(e),a=a||ve(c),r=0,i=o.length;r<i;r++)Le(o[r],a[r]);else Le(e,c);return 0<(a=ve(c,"script")).length&&ye(a,!f&&ve(e,"script")),c},cleanData:function(e){for(var t,n,r,i=S.event.special,o=0;void 0!==(n=e[o]);o++)if(V(n)){if(t=n[Y.expando]){if(t.events)for(r in t.events)i[r]?S.event.remove(n,r):S.removeEvent(n,r,t.handle);n[Y.expando]=void 0}n[Q.expando]&&(n[Q.expando]=void 0)}}}),S.fn.extend({detach:function(e){return Oe(this,e,!0)},remove:function(e){return Oe(this,e)},text:function(e){return $(this,function(e){return void 0===e?S.text(this):this.empty().each(function(){1!==this.nodeType&&11!==this.nodeType&&9!==this.nodeType||(this.textContent=e)})},null,e,arguments.length)},append:function(){return He(this,arguments,function(e){1!==this.nodeType&&11!==this.nodeType&&9!==this.nodeType||je(this,e).appendChild(e)})},prepend:function(){return He(this,arguments,function(e){if(1===this.nodeType||11===this.nodeType||9===this.nodeType){var t=je(this,e);t.insertBefore(e,t.firstChild)}})},before:function(){return He(this,arguments,function(e){this.parentNode&&this.parentNode.insertBefore(e,this)})},after:function(){return He(this,arguments,function(e){this.parentNode&&this.parentNode.insertBefore(e,this.nextSibling)})},empty:function(){for(var e,t=0;null!=(e=this[t]);t++)1===e.nodeType&&(S.cleanData(ve(e,!1)),e.textContent="");return this},clone:function(e,t){return e=null!=e&&e,t=null==t?e:t,this.map(function(){return S.clone(this,e,t)})},html:function(e){return $(this,function(e){var t=this[0]||{},n=0,r=this.length;if(void 0===e&&1===t.nodeType)return t.innerHTML;if("string"==typeof e&&!ke.test(e)&&!ge[(de.exec(e)||["",""])[1].toLowerCase()]){e=S.htmlPrefilter(e);try{for(;n<r;n++)1===(t=this[n]||{}).nodeType&&(S.cleanData(ve(t,!1)),t.innerHTML=e);t=0}catch(e){}}t&&this.empty().append(e)},null,e,arguments.length)},replaceWith:function(){var n=[];return He(this,arguments,function(e){var t=this.parentNode;S.inArray(this,n)<0&&(S.cleanData(ve(this)),t&&t.replaceChild(e,this))},n)}}),S.each({appendTo:"append",prependTo:"prepend",insertBefore:"before",insertAfter:"after",replaceAll:"replaceWith"},function(e,a){S.fn[e]=function(e){for(var t,n=[],r=S(e),i=r.length-1,o=0;o<=i;o++)t=o===i?this:this.clone(!0),S(r[o])[a](t),u.apply(n,t.get());return this.pushStack(n)}});var Pe=new RegExp("^("+ee+")(?!px)[a-z%]+$","i"),Re=function(e){var t=e.ownerDocument.defaultView;return t&&t.opener||(t=C),t.getComputedStyle(e)},Me=function(e,t,n){var r,i,o={};for(i in t)o[i]=e.style[i],e.style[i]=t[i];for(i in r=n.call(e),t)e.style[i]=o[i];return r},Ie=new RegExp(ne.join("|"),"i");function We(e,t,n){var r,i,o,a,s=e.style;return(n=n||Re(e))&&(""!==(a=n.getPropertyValue(t)||n[t])||ie(e)||(a=S.style(e,t)),!y.pixelBoxStyles()&&Pe.test(a)&&Ie.test(t)&&(r=s.width,i=s.minWidth,o=s.maxWidth,s.minWidth=s.maxWidth=s.width=a,a=n.width,s.width=r,s.minWidth=i,s.maxWidth=o)),void 0!==a?a+"":a}function Fe(e,t){return{get:function(){if(!e())return(this.get=t).apply(this,arguments);delete this.get}}}!function(){function e(){if(l){u.style.cssText="position:absolute;left:-11111px;width:60px;margin-top:1px;padding:0;border:0",l.style.cssText="position:relative;display:block;box-sizing:border-box;overflow:scroll;margin:auto;border:1px;padding:1px;width:60%;top:1%",re.appendChild(u).appendChild(l);var e=C.getComputedStyle(l);n="1%"!==e.top,s=12===t(e.marginLeft),l.style.right="60%",o=36===t(e.right),r=36===t(e.width),l.style.position="absolute",i=12===t(l.offsetWidth/3),re.removeChild(u),l=null}}function t(e){return Math.round(parseFloat(e))}var n,r,i,o,a,s,u=E.createElement("div"),l=E.createElement("div");l.style&&(l.style.backgroundClip="content-box",l.cloneNode(!0).style.backgroundClip="",y.clearCloneStyle="content-box"===l.style.backgroundClip,S.extend(y,{boxSizingReliable:function(){return e(),r},pixelBoxStyles:function(){return e(),o},pixelPosition:function(){return e(),n},reliableMarginLeft:function(){return e(),s},scrollboxSize:function(){return e(),i},reliableTrDimensions:function(){var e,t,n,r;return null==a&&(e=E.createElement("table"),t=E.createElement("tr"),n=E.createElement("div"),e.style.cssText="position:absolute;left:-11111px;border-collapse:separate",t.style.cssText="border:1px solid",t.style.height="1px",n.style.height="9px",n.style.display="block",re.appendChild(e).appendChild(t).appendChild(n),r=C.getComputedStyle(t),a=parseInt(r.height,10)+parseInt(r.borderTopWidth,10)+parseInt(r.borderBottomWidth,10)===t.offsetHeight,re.removeChild(e)),a}}))}();var Be=["Webkit","Moz","ms"],$e=E.createElement("div").style,_e={};function ze(e){var t=S.cssProps[e]||_e[e];return t||(e in $e?e:_e[e]=function(e){var t=e[0].toUpperCase()+e.slice(1),n=Be.length;while(n--)if((e=Be[n]+t)in $e)return e}(e)||e)}var Ue=/^(none|table(?!-c[ea]).+)/,Xe=/^--/,Ve={position:"absolute",visibility:"hidden",display:"block"},Ge={letterSpacing:"0",fontWeight:"400"};function Ye(e,t,n){var r=te.exec(t);return r?Math.max(0,r[2]-(n||0))+(r[3]||"px"):t}function Qe(e,t,n,r,i,o){var a="width"===t?1:0,s=0,u=0;if(n===(r?"border":"content"))return 0;for(;a<4;a+=2)"margin"===n&&(u+=S.css(e,n+ne[a],!0,i)),r?("content"===n&&(u-=S.css(e,"padding"+ne[a],!0,i)),"margin"!==n&&(u-=S.css(e,"border"+ne[a]+"Width",!0,i))):(u+=S.css(e,"padding"+ne[a],!0,i),"padding"!==n?u+=S.css(e,"border"+ne[a]+"Width",!0,i):s+=S.css(e,"border"+ne[a]+"Width",!0,i));return!r&&0<=o&&(u+=Math.max(0,Math.ceil(e["offset"+t[0].toUpperCase()+t.slice(1)]-o-u-s-.5))||0),u}function Je(e,t,n){var r=Re(e),i=(!y.boxSizingReliable()||n)&&"border-box"===S.css(e,"boxSizing",!1,r),o=i,a=We(e,t,r),s="offset"+t[0].toUpperCase()+t.slice(1);if(Pe.test(a)){if(!n)return a;a="auto"}return(!y.boxSizingReliable()&&i||!y.reliableTrDimensions()&&A(e,"tr")||"auto"===a||!parseFloat(a)&&"inline"===S.css(e,"display",!1,r))&&e.getClientRects().length&&(i="border-box"===S.css(e,"boxSizing",!1,r),(o=s in e)&&(a=e[s])),(a=parseFloat(a)||0)+Qe(e,t,n||(i?"border":"content"),o,r,a)+"px"}function Ke(e,t,n,r,i){return new Ke.prototype.init(e,t,n,r,i)}S.extend({cssHooks:{opacity:{get:function(e,t){if(t){var n=We(e,"opacity");return""===n?"1":n}}}},cssNumber:{animationIterationCount:!0,columnCount:!0,fillOpacity:!0,flexGrow:!0,flexShrink:!0,fontWeight:!0,gridArea:!0,gridColumn:!0,gridColumnEnd:!0,gridColumnStart:!0,gridRow:!0,gridRowEnd:!0,gridRowStart:!0,lineHeight:!0,opacity:!0,order:!0,orphans:!0,widows:!0,zIndex:!0,zoom:!0},cssProps:{},style:function(e,t,n,r){if(e&&3!==e.nodeType&&8!==e.nodeType&&e.style){var i,o,a,s=X(t),u=Xe.test(t),l=e.style;if(u||(t=ze(s)),a=S.cssHooks[t]||S.cssHooks[s],void 0===n)return a&&"get"in a&&void 0!==(i=a.get(e,!1,r))?i:l[t];"string"===(o=typeof n)&&(i=te.exec(n))&&i[1]&&(n=se(e,t,i),o="number"),null!=n&&n==n&&("number"!==o||u||(n+=i&&i[3]||(S.cssNumber[s]?"":"px")),y.clearCloneStyle||""!==n||0!==t.indexOf("background")||(l[t]="inherit"),a&&"set"in a&&void 0===(n=a.set(e,n,r))||(u?l.setProperty(t,n):l[t]=n))}},css:function(e,t,n,r){var i,o,a,s=X(t);return Xe.test(t)||(t=ze(s)),(a=S.cssHooks[t]||S.cssHooks[s])&&"get"in a&&(i=a.get(e,!0,n)),void 0===i&&(i=We(e,t,r)),"normal"===i&&t in Ge&&(i=Ge[t]),""===n||n?(o=parseFloat(i),!0===n||isFinite(o)?o||0:i):i}}),S.each(["height","width"],function(e,u){S.cssHooks[u]={get:function(e,t,n){if(t)return!Ue.test(S.css(e,"display"))||e.getClientRects().length&&e.getBoundingClientRect().width?Je(e,u,n):Me(e,Ve,function(){return Je(e,u,n)})},set:function(e,t,n){var r,i=Re(e),o=!y.scrollboxSize()&&"absolute"===i.position,a=(o||n)&&"border-box"===S.css(e,"boxSizing",!1,i),s=n?Qe(e,u,n,a,i):0;return a&&o&&(s-=Math.ceil(e["offset"+u[0].toUpperCase()+u.slice(1)]-parseFloat(i[u])-Qe(e,u,"border",!1,i)-.5)),s&&(r=te.exec(t))&&"px"!==(r[3]||"px")&&(e.style[u]=t,t=S.css(e,u)),Ye(0,t,s)}}}),S.cssHooks.marginLeft=Fe(y.reliableMarginLeft,function(e,t){if(t)return(parseFloat(We(e,"marginLeft"))||e.getBoundingClientRect().left-Me(e,{marginLeft:0},function(){return e.getBoundingClientRect().left}))+"px"}),S.each({margin:"",padding:"",border:"Width"},function(i,o){S.cssHooks[i+o]={expand:function(e){for(var t=0,n={},r="string"==typeof e?e.split(" "):[e];t<4;t++)n[i+ne[t]+o]=r[t]||r[t-2]||r[0];return n}},"margin"!==i&&(S.cssHooks[i+o].set=Ye)}),S.fn.extend({css:function(e,t){return $(this,function(e,t,n){var r,i,o={},a=0;if(Array.isArray(t)){for(r=Re(e),i=t.length;a<i;a++)o[t[a]]=S.css(e,t[a],!1,r);return o}return void 0!==n?S.style(e,t,n):S.css(e,t)},e,t,1<arguments.length)}}),((S.Tween=Ke).prototype={constructor:Ke,init:function(e,t,n,r,i,o){this.elem=e,this.prop=n,this.easing=i||S.easing._default,this.options=t,this.start=this.now=this.cur(),this.end=r,this.unit=o||(S.cssNumber[n]?"":"px")},cur:function(){var e=Ke.propHooks[this.prop];return e&&e.get?e.get(this):Ke.propHooks._default.get(this)},run:function(e){var t,n=Ke.propHooks[this.prop];return this.options.duration?this.pos=t=S.easing[this.easing](e,this.options.duration*e,0,1,this.options.duration):this.pos=t=e,this.now=(this.end-this.start)*t+this.start,this.options.step&&this.options.step.call(this.elem,this.now,this),n&&n.set?n.set(this):Ke.propHooks._default.set(this),this}}).init.prototype=Ke.prototype,(Ke.propHooks={_default:{get:function(e){var t;return 1!==e.elem.nodeType||null!=e.elem[e.prop]&&null==e.elem.style[e.prop]?e.elem[e.prop]:(t=S.css(e.elem,e.prop,""))&&"auto"!==t?t:0},set:function(e){S.fx.step[e.prop]?S.fx.step[e.prop](e):1!==e.elem.nodeType||!S.cssHooks[e.prop]&&null==e.elem.style[ze(e.prop)]?e.elem[e.prop]=e.now:S.style(e.elem,e.prop,e.now+e.unit)}}}).scrollTop=Ke.propHooks.scrollLeft={set:function(e){e.elem.nodeType&&e.elem.parentNode&&(e.elem[e.prop]=e.now)}},S.easing={linear:function(e){return e},swing:function(e){return.5-Math.cos(e*Math.PI)/2},_default:"swing"},S.fx=Ke.prototype.init,S.fx.step={};var Ze,et,tt,nt,rt=/^(?:toggle|show|hide)$/,it=/queueHooks$/;function ot(){et&&(!1===E.hidden&&C.requestAnimationFrame?C.requestAnimationFrame(ot):C.setTimeout(ot,S.fx.interval),S.fx.tick())}function at(){return C.setTimeout(function(){Ze=void 0}),Ze=Date.now()}function st(e,t){var n,r=0,i={height:e};for(t=t?1:0;r<4;r+=2-t)i["margin"+(n=ne[r])]=i["padding"+n]=e;return t&&(i.opacity=i.width=e),i}function ut(e,t,n){for(var r,i=(lt.tweeners[t]||[]).concat(lt.tweeners["*"]),o=0,a=i.length;o<a;o++)if(r=i[o].call(n,t,e))return r}function lt(o,e,t){var n,a,r=0,i=lt.prefilters.length,s=S.Deferred().always(function(){delete u.elem}),u=function(){if(a)return!1;for(var e=Ze||at(),t=Math.max(0,l.startTime+l.duration-e),n=1-(t/l.duration||0),r=0,i=l.tweens.length;r<i;r++)l.tweens[r].run(n);return s.notifyWith(o,[l,n,t]),n<1&&i?t:(i||s.notifyWith(o,[l,1,0]),s.resolveWith(o,[l]),!1)},l=s.promise({elem:o,props:S.extend({},e),opts:S.extend(!0,{specialEasing:{},easing:S.easing._default},t),originalProperties:e,originalOptions:t,startTime:Ze||at(),duration:t.duration,tweens:[],createTween:function(e,t){var n=S.Tween(o,l.opts,e,t,l.opts.specialEasing[e]||l.opts.easing);return l.tweens.push(n),n},stop:function(e){var t=0,n=e?l.tweens.length:0;if(a)return this;for(a=!0;t<n;t++)l.tweens[t].run(1);return e?(s.notifyWith(o,[l,1,0]),s.resolveWith(o,[l,e])):s.rejectWith(o,[l,e]),this}}),c=l.props;for(!function(e,t){var n,r,i,o,a;for(n in e)if(i=t[r=X(n)],o=e[n],Array.isArray(o)&&(i=o[1],o=e[n]=o[0]),n!==r&&(e[r]=o,delete e[n]),(a=S.cssHooks[r])&&"expand"in a)for(n in o=a.expand(o),delete e[r],o)n in e||(e[n]=o[n],t[n]=i);else t[r]=i}(c,l.opts.specialEasing);r<i;r++)if(n=lt.prefilters[r].call(l,o,c,l.opts))return m(n.stop)&&(S._queueHooks(l.elem,l.opts.queue).stop=n.stop.bind(n)),n;return S.map(c,ut,l),m(l.opts.start)&&l.opts.start.call(o,l),l.progress(l.opts.progress).done(l.opts.done,l.opts.complete).fail(l.opts.fail).always(l.opts.always),S.fx.timer(S.extend(u,{elem:o,anim:l,queue:l.opts.queue})),l}S.Animation=S.extend(lt,{tweeners:{"*":[function(e,t){var n=this.createTween(e,t);return se(n.elem,e,te.exec(t),n),n}]},tweener:function(e,t){m(e)?(t=e,e=["*"]):e=e.match(P);for(var n,r=0,i=e.length;r<i;r++)n=e[r],lt.tweeners[n]=lt.tweeners[n]||[],lt.tweeners[n].unshift(t)},prefilters:[function(e,t,n){var r,i,o,a,s,u,l,c,f="width"in t||"height"in t,p=this,d={},h=e.style,g=e.nodeType&&ae(e),v=Y.get(e,"fxshow");for(r in n.queue||(null==(a=S._queueHooks(e,"fx")).unqueued&&(a.unqueued=0,s=a.empty.fire,a.empty.fire=function(){a.unqueued||s()}),a.unqueued++,p.always(function(){p.always(function(){a.unqueued--,S.queue(e,"fx").length||a.empty.fire()})})),t)if(i=t[r],rt.test(i)){if(delete t[r],o=o||"toggle"===i,i===(g?"hide":"show")){if("show"!==i||!v||void 0===v[r])continue;g=!0}d[r]=v&&v[r]||S.style(e,r)}if((u=!S.isEmptyObject(t))||!S.isEmptyObject(d))for(r in f&&1===e.nodeType&&(n.overflow=[h.overflow,h.overflowX,h.overflowY],null==(l=v&&v.display)&&(l=Y.get(e,"display")),"none"===(c=S.css(e,"display"))&&(l?c=l:(le([e],!0),l=e.style.display||l,c=S.css(e,"display"),le([e]))),("inline"===c||"inline-block"===c&&null!=l)&&"none"===S.css(e,"float")&&(u||(p.done(function(){h.display=l}),null==l&&(c=h.display,l="none"===c?"":c)),h.display="inline-block")),n.overflow&&(h.overflow="hidden",p.always(function(){h.overflow=n.overflow[0],h.overflowX=n.overflow[1],h.overflowY=n.overflow[2]})),u=!1,d)u||(v?"hidden"in v&&(g=v.hidden):v=Y.access(e,"fxshow",{display:l}),o&&(v.hidden=!g),g&&le([e],!0),p.done(function(){for(r in g||le([e]),Y.remove(e,"fxshow"),d)S.style(e,r,d[r])})),u=ut(g?v[r]:0,r,p),r in v||(v[r]=u.start,g&&(u.end=u.start,u.start=0))}],prefilter:function(e,t){t?lt.prefilters.unshift(e):lt.prefilters.push(e)}}),S.speed=function(e,t,n){var r=e&&"object"==typeof e?S.extend({},e):{complete:n||!n&&t||m(e)&&e,duration:e,easing:n&&t||t&&!m(t)&&t};return S.fx.off?r.duration=0:"number"!=typeof r.duration&&(r.duration in S.fx.speeds?r.duration=S.fx.speeds[r.duration]:r.duration=S.fx.speeds._default),null!=r.queue&&!0!==r.queue||(r.queue="fx"),r.old=r.complete,r.complete=function(){m(r.old)&&r.old.call(this),r.queue&&S.dequeue(this,r.queue)},r},S.fn.extend({fadeTo:function(e,t,n,r){return this.filter(ae).css("opacity",0).show().end().animate({opacity:t},e,n,r)},animate:function(t,e,n,r){var i=S.isEmptyObject(t),o=S.speed(e,n,r),a=function(){var e=lt(this,S.extend({},t),o);(i||Y.get(this,"finish"))&&e.stop(!0)};return a.finish=a,i||!1===o.queue?this.each(a):this.queue(o.queue,a)},stop:function(i,e,o){var a=function(e){var t=e.stop;delete e.stop,t(o)};return"string"!=typeof i&&(o=e,e=i,i=void 0),e&&this.queue(i||"fx",[]),this.each(function(){var e=!0,t=null!=i&&i+"queueHooks",n=S.timers,r=Y.get(this);if(t)r[t]&&r[t].stop&&a(r[t]);else for(t in r)r[t]&&r[t].stop&&it.test(t)&&a(r[t]);for(t=n.length;t--;)n[t].elem!==this||null!=i&&n[t].queue!==i||(n[t].anim.stop(o),e=!1,n.splice(t,1));!e&&o||S.dequeue(this,i)})},finish:function(a){return!1!==a&&(a=a||"fx"),this.each(function(){var e,t=Y.get(this),n=t[a+"queue"],r=t[a+"queueHooks"],i=S.timers,o=n?n.length:0;for(t.finish=!0,S.queue(this,a,[]),r&&r.stop&&r.stop.call(this,!0),e=i.length;e--;)i[e].elem===this&&i[e].queue===a&&(i[e].anim.stop(!0),i.splice(e,1));for(e=0;e<o;e++)n[e]&&n[e].finish&&n[e].finish.call(this);delete t.finish})}}),S.each(["toggle","show","hide"],function(e,r){var i=S.fn[r];S.fn[r]=function(e,t,n){return null==e||"boolean"==typeof e?i.apply(this,arguments):this.animate(st(r,!0),e,t,n)}}),S.each({slideDown:st("show"),slideUp:st("hide"),slideToggle:st("toggle"),fadeIn:{opacity:"show"},fadeOut:{opacity:"hide"},fadeToggle:{opacity:"toggle"}},function(e,r){S.fn[e]=function(e,t,n){return this.animate(r,e,t,n)}}),S.timers=[],S.fx.tick=function(){var e,t=0,n=S.timers;for(Ze=Date.now();t<n.length;t++)(e=n[t])()||n[t]!==e||n.splice(t--,1);n.length||S.fx.stop(),Ze=void 0},S.fx.timer=function(e){S.timers.push(e),S.fx.start()},S.fx.interval=13,S.fx.start=function(){et||(et=!0,ot())},S.fx.stop=function(){et=null},S.fx.speeds={slow:600,fast:200,_default:400},S.fn.delay=function(r,e){return r=S.fx&&S.fx.speeds[r]||r,e=e||"fx",this.queue(e,function(e,t){var n=C.setTimeout(e,r);t.stop=function(){C.clearTimeout(n)}})},tt=E.createElement("input"),nt=E.createElement("select").appendChild(E.createElement("option")),tt.type="checkbox",y.checkOn=""!==tt.value,y.optSelected=nt.selected,(tt=E.createElement("input")).value="t",tt.type="radio",y.radioValue="t"===tt.value;var ct,ft=S.expr.attrHandle;S.fn.extend({attr:function(e,t){return $(this,S.attr,e,t,1<arguments.length)},removeAttr:function(e){return this.each(function(){S.removeAttr(this,e)})}}),S.extend({attr:function(e,t,n){var r,i,o=e.nodeType;if(3!==o&&8!==o&&2!==o)return"undefined"==typeof e.getAttribute?S.prop(e,t,n):(1===o&&S.isXMLDoc(e)||(i=S.attrHooks[t.toLowerCase()]||(S.expr.match.bool.test(t)?ct:void 0)),void 0!==n?null===n?void S.removeAttr(e,t):i&&"set"in i&&void 0!==(r=i.set(e,n,t))?r:(e.setAttribute(t,n+""),n):i&&"get"in i&&null!==(r=i.get(e,t))?r:null==(r=S.find.attr(e,t))?void 0:r)},attrHooks:{type:{set:function(e,t){if(!y.radioValue&&"radio"===t&&A(e,"input")){var n=e.value;return e.setAttribute("type",t),n&&(e.value=n),t}}}},removeAttr:function(e,t){var n,r=0,i=t&&t.match(P);if(i&&1===e.nodeType)while(n=i[r++])e.removeAttribute(n)}}),ct={set:function(e,t,n){return!1===t?S.removeAttr(e,n):e.setAttribute(n,n),n}},S.each(S.expr.match.bool.source.match(/\w+/g),function(e,t){var a=ft[t]||S.find.attr;ft[t]=function(e,t,n){var r,i,o=t.toLowerCase();return n||(i=ft[o],ft[o]=r,r=null!=a(e,t,n)?o:null,ft[o]=i),r}});var pt=/^(?:input|select|textarea|button)$/i,dt=/^(?:a|area)$/i;function ht(e){return(e.match(P)||[]).join(" ")}function gt(e){return e.getAttribute&&e.getAttribute("class")||""}function vt(e){return Array.isArray(e)?e:"string"==typeof e&&e.match(P)||[]}S.fn.extend({prop:function(e,t){return $(this,S.prop,e,t,1<arguments.length)},removeProp:function(e){return this.each(function(){delete this[S.propFix[e]||e]})}}),S.extend({prop:function(e,t,n){var r,i,o=e.nodeType;if(3!==o&&8!==o&&2!==o)return 1===o&&S.isXMLDoc(e)||(t=S.propFix[t]||t,i=S.propHooks[t]),void 0!==n?i&&"set"in i&&void 0!==(r=i.set(e,n,t))?r:e[t]=n:i&&"get"in i&&null!==(r=i.get(e,t))?r:e[t]},propHooks:{tabIndex:{get:function(e){var t=S.find.attr(e,"tabindex");return t?parseInt(t,10):pt.test(e.nodeName)||dt.test(e.nodeName)&&e.href?0:-1}}},propFix:{"for":"htmlFor","class":"className"}}),y.optSelected||(S.propHooks.selected={get:function(e){var t=e.parentNode;return t&&t.parentNode&&t.parentNode.selectedIndex,null},set:function(e){var t=e.parentNode;t&&(t.selectedIndex,t.parentNode&&t.parentNode.selectedIndex)}}),S.each(["tabIndex","readOnly","maxLength","cellSpacing","cellPadding","rowSpan","colSpan","useMap","frameBorder","contentEditable"],function(){S.propFix[this.toLowerCase()]=this}),S.fn.extend({addClass:function(t){var e,n,r,i,o,a,s,u=0;if(m(t))return this.each(function(e){S(this).addClass(t.call(this,e,gt(this)))});if((e=vt(t)).length)while(n=this[u++])if(i=gt(n),r=1===n.nodeType&&" "+ht(i)+" "){a=0;while(o=e[a++])r.indexOf(" "+o+" ")<0&&(r+=o+" ");i!==(s=ht(r))&&n.setAttribute("class",s)}return this},removeClass:function(t){var e,n,r,i,o,a,s,u=0;if(m(t))return this.each(function(e){S(this).removeClass(t.call(this,e,gt(this)))});if(!arguments.length)return this.attr("class","");if((e=vt(t)).length)while(n=this[u++])if(i=gt(n),r=1===n.nodeType&&" "+ht(i)+" "){a=0;while(o=e[a++])while(-1<r.indexOf(" "+o+" "))r=r.replace(" "+o+" "," ");i!==(s=ht(r))&&n.setAttribute("class",s)}return this},toggleClass:function(i,t){var o=typeof i,a="string"===o||Array.isArray(i);return"boolean"==typeof t&&a?t?this.addClass(i):this.removeClass(i):m(i)?this.each(function(e){S(this).toggleClass(i.call(this,e,gt(this),t),t)}):this.each(function(){var e,t,n,r;if(a){t=0,n=S(this),r=vt(i);while(e=r[t++])n.hasClass(e)?n.removeClass(e):n.addClass(e)}else void 0!==i&&"boolean"!==o||((e=gt(this))&&Y.set(this,"__className__",e),this.setAttribute&&this.setAttribute("class",e||!1===i?"":Y.get(this,"__className__")||""))})},hasClass:function(e){var t,n,r=0;t=" "+e+" ";while(n=this[r++])if(1===n.nodeType&&-1<(" "+ht(gt(n))+" ").indexOf(t))return!0;return!1}});var yt=/\r/g;S.fn.extend({val:function(n){var r,e,i,t=this[0];return arguments.length?(i=m(n),this.each(function(e){var t;1===this.nodeType&&(null==(t=i?n.call(this,e,S(this).val()):n)?t="":"number"==typeof t?t+="":Array.isArray(t)&&(t=S.map(t,function(e){return null==e?"":e+""})),(r=S.valHooks[this.type]||S.valHooks[this.nodeName.toLowerCase()])&&"set"in r&&void 0!==r.set(this,t,"value")||(this.value=t))})):t?(r=S.valHooks[t.type]||S.valHooks[t.nodeName.toLowerCase()])&&"get"in r&&void 0!==(e=r.get(t,"value"))?e:"string"==typeof(e=t.value)?e.replace(yt,""):null==e?"":e:void 0}}),S.extend({valHooks:{option:{get:function(e){var t=S.find.attr(e,"value");return null!=t?t:ht(S.text(e))}},select:{get:function(e){var t,n,r,i=e.options,o=e.selectedIndex,a="select-one"===e.type,s=a?null:[],u=a?o+1:i.length;for(r=o<0?u:a?o:0;r<u;r++)if(((n=i[r]).selected||r===o)&&!n.disabled&&(!n.parentNode.disabled||!A(n.parentNode,"optgroup"))){if(t=S(n).val(),a)return t;s.push(t)}return s},set:function(e,t){var n,r,i=e.options,o=S.makeArray(t),a=i.length;while(a--)((r=i[a]).selected=-1<S.inArray(S.valHooks.option.get(r),o))&&(n=!0);return n||(e.selectedIndex=-1),o}}}}),S.each(["radio","checkbox"],function(){S.valHooks[this]={set:function(e,t){if(Array.isArray(t))return e.checked=-1<S.inArray(S(e).val(),t)}},y.checkOn||(S.valHooks[this].get=function(e){return null===e.getAttribute("value")?"on":e.value})}),y.focusin="onfocusin"in C;var mt=/^(?:focusinfocus|focusoutblur)$/,xt=function(e){e.stopPropagation()};S.extend(S.event,{trigger:function(e,t,n,r){var i,o,a,s,u,l,c,f,p=[n||E],d=v.call(e,"type")?e.type:e,h=v.call(e,"namespace")?e.namespace.split("."):[];if(o=f=a=n=n||E,3!==n.nodeType&&8!==n.nodeType&&!mt.test(d+S.event.triggered)&&(-1<d.indexOf(".")&&(d=(h=d.split(".")).shift(),h.sort()),u=d.indexOf(":")<0&&"on"+d,(e=e[S.expando]?e:new S.Event(d,"object"==typeof e&&e)).isTrigger=r?2:3,e.namespace=h.join("."),e.rnamespace=e.namespace?new RegExp("(^|\\.)"+h.join("\\.(?:.*\\.|)")+"(\\.|$)"):null,e.result=void 0,e.target||(e.target=n),t=null==t?[e]:S.makeArray(t,[e]),c=S.event.special[d]||{},r||!c.trigger||!1!==c.trigger.apply(n,t))){if(!r&&!c.noBubble&&!x(n)){for(s=c.delegateType||d,mt.test(s+d)||(o=o.parentNode);o;o=o.parentNode)p.push(o),a=o;a===(n.ownerDocument||E)&&p.push(a.defaultView||a.parentWindow||C)}i=0;while((o=p[i++])&&!e.isPropagationStopped())f=o,e.type=1<i?s:c.bindType||d,(l=(Y.get(o,"events")||Object.create(null))[e.type]&&Y.get(o,"handle"))&&l.apply(o,t),(l=u&&o[u])&&l.apply&&V(o)&&(e.result=l.apply(o,t),!1===e.result&&e.preventDefault());return e.type=d,r||e.isDefaultPrevented()||c._default&&!1!==c._default.apply(p.pop(),t)||!V(n)||u&&m(n[d])&&!x(n)&&((a=n[u])&&(n[u]=null),S.event.triggered=d,e.isPropagationStopped()&&f.addEventListener(d,xt),n[d](),e.isPropagationStopped()&&f.removeEventListener(d,xt),S.event.triggered=void 0,a&&(n[u]=a)),e.result}},simulate:function(e,t,n){var r=S.extend(new S.Event,n,{type:e,isSimulated:!0});S.event.trigger(r,null,t)}}),S.fn.extend({trigger:function(e,t){return this.each(function(){S.event.trigger(e,t,this)})},triggerHandler:function(e,t){var n=this[0];if(n)return S.event.trigger(e,t,n,!0)}}),y.focusin||S.each({focus:"focusin",blur:"focusout"},function(n,r){var i=function(e){S.event.simulate(r,e.target,S.event.fix(e))};S.event.special[r]={setup:function(){var e=this.ownerDocument||this.document||this,t=Y.access(e,r);t||e.addEventListener(n,i,!0),Y.access(e,r,(t||0)+1)},teardown:function(){var e=this.ownerDocument||this.document||this,t=Y.access(e,r)-1;t?Y.access(e,r,t):(e.removeEventListener(n,i,!0),Y.remove(e,r))}}});var bt=C.location,wt={guid:Date.now()},Tt=/\?/;S.parseXML=function(e){var t,n;if(!e||"string"!=typeof e)return null;try{t=(new C.DOMParser).parseFromString(e,"text/xml")}catch(e){}return n=t&&t.getElementsByTagName("parsererror")[0],t&&!n||S.error("Invalid XML: "+(n?S.map(n.childNodes,function(e){return e.textContent}).join("\n"):e)),t};var Ct=/\[\]$/,Et=/\r?\n/g,St=/^(?:submit|button|image|reset|file)$/i,kt=/^(?:input|select|textarea|keygen)/i;function At(n,e,r,i){var t;if(Array.isArray(e))S.each(e,function(e,t){r||Ct.test(n)?i(n,t):At(n+"["+("object"==typeof t&&null!=t?e:"")+"]",t,r,i)});else if(r||"object"!==w(e))i(n,e);else for(t in e)At(n+"["+t+"]",e[t],r,i)}S.param=function(e,t){var n,r=[],i=function(e,t){var n=m(t)?t():t;r[r.length]=encodeURIComponent(e)+"="+encodeURIComponent(null==n?"":n)};if(null==e)return"";if(Array.isArray(e)||e.jquery&&!S.isPlainObject(e))S.each(e,function(){i(this.name,this.value)});else for(n in e)At(n,e[n],t,i);return r.join("&")},S.fn.extend({serialize:function(){return S.param(this.serializeArray())},serializeArray:function(){return this.map(function(){var e=S.prop(this,"elements");return e?S.makeArray(e):this}).filter(function(){var e=this.type;return this.name&&!S(this).is(":disabled")&&kt.test(this.nodeName)&&!St.test(e)&&(this.checked||!pe.test(e))}).map(function(e,t){var n=S(this).val();return null==n?null:Array.isArray(n)?S.map(n,function(e){return{name:t.name,value:e.replace(Et,"\r\n")}}):{name:t.name,value:n.replace(Et,"\r\n")}}).get()}});var Nt=/%20/g,jt=/#.*$/,Dt=/([?&])_=[^&]*/,qt=/^(.*?):[ \t]*([^\r\n]*)$/gm,Lt=/^(?:GET|HEAD)$/,Ht=/^\/\//,Ot={},Pt={},Rt="*/".concat("*"),Mt=E.createElement("a");function It(o){return function(e,t){"string"!=typeof e&&(t=e,e="*");var n,r=0,i=e.toLowerCase().match(P)||[];if(m(t))while(n=i[r++])"+"===n[0]?(n=n.slice(1)||"*",(o[n]=o[n]||[]).unshift(t)):(o[n]=o[n]||[]).push(t)}}function Wt(t,i,o,a){var s={},u=t===Pt;function l(e){var r;return s[e]=!0,S.each(t[e]||[],function(e,t){var n=t(i,o,a);return"string"!=typeof n||u||s[n]?u?!(r=n):void 0:(i.dataTypes.unshift(n),l(n),!1)}),r}return l(i.dataTypes[0])||!s["*"]&&l("*")}function Ft(e,t){var n,r,i=S.ajaxSettings.flatOptions||{};for(n in t)void 0!==t[n]&&((i[n]?e:r||(r={}))[n]=t[n]);return r&&S.extend(!0,e,r),e}Mt.href=bt.href,S.extend({active:0,lastModified:{},etag:{},ajaxSettings:{url:bt.href,type:"GET",isLocal:/^(?:about|app|app-storage|.+-extension|file|res|widget):$/.test(bt.protocol),global:!0,processData:!0,async:!0,contentType:"application/x-www-form-urlencoded; charset=UTF-8",accepts:{"*":Rt,text:"text/plain",html:"text/html",xml:"application/xml, text/xml",json:"application/json, text/javascript"},contents:{xml:/\bxml\b/,html:/\bhtml/,json:/\bjson\b/},responseFields:{xml:"responseXML",text:"responseText",json:"responseJSON"},converters:{"* text":String,"text html":!0,"text json":JSON.parse,"text xml":S.parseXML},flatOptions:{url:!0,context:!0}},ajaxSetup:function(e,t){return t?Ft(Ft(e,S.ajaxSettings),t):Ft(S.ajaxSettings,e)},ajaxPrefilter:It(Ot),ajaxTransport:It(Pt),ajax:function(e,t){"object"==typeof e&&(t=e,e=void 0),t=t||{};var c,f,p,n,d,r,h,g,i,o,v=S.ajaxSetup({},t),y=v.context||v,m=v.context&&(y.nodeType||y.jquery)?S(y):S.event,x=S.Deferred(),b=S.Callbacks("once memory"),w=v.statusCode||{},a={},s={},u="canceled",T={readyState:0,getResponseHeader:function(e){var t;if(h){if(!n){n={};while(t=qt.exec(p))n[t[1].toLowerCase()+" "]=(n[t[1].toLowerCase()+" "]||[]).concat(t[2])}t=n[e.toLowerCase()+" "]}return null==t?null:t.join(", ")},getAllResponseHeaders:function(){return h?p:null},setRequestHeader:function(e,t){return null==h&&(e=s[e.toLowerCase()]=s[e.toLowerCase()]||e,a[e]=t),this},overrideMimeType:function(e){return null==h&&(v.mimeType=e),this},statusCode:function(e){var t;if(e)if(h)T.always(e[T.status]);else for(t in e)w[t]=[w[t],e[t]];return this},abort:function(e){var t=e||u;return c&&c.abort(t),l(0,t),this}};if(x.promise(T),v.url=((e||v.url||bt.href)+"").replace(Ht,bt.protocol+"//"),v.type=t.method||t.type||v.method||v.type,v.dataTypes=(v.dataType||"*").toLowerCase().match(P)||[""],null==v.crossDomain){r=E.createElement("a");try{r.href=v.url,r.href=r.href,v.crossDomain=Mt.protocol+"//"+Mt.host!=r.protocol+"//"+r.host}catch(e){v.crossDomain=!0}}if(v.data&&v.processData&&"string"!=typeof v.data&&(v.data=S.param(v.data,v.traditional)),Wt(Ot,v,t,T),h)return T;for(i in(g=S.event&&v.global)&&0==S.active++&&S.event.trigger("ajaxStart"),v.type=v.type.toUpperCase(),v.hasContent=!Lt.test(v.type),f=v.url.replace(jt,""),v.hasContent?v.data&&v.processData&&0===(v.contentType||"").indexOf("application/x-www-form-urlencoded")&&(v.data=v.data.replace(Nt,"+")):(o=v.url.slice(f.length),v.data&&(v.processData||"string"==typeof v.data)&&(f+=(Tt.test(f)?"&":"?")+v.data,delete v.data),!1===v.cache&&(f=f.replace(Dt,"$1"),o=(Tt.test(f)?"&":"?")+"_="+wt.guid+++o),v.url=f+o),v.ifModified&&(S.lastModified[f]&&T.setRequestHeader("If-Modified-Since",S.lastModified[f]),S.etag[f]&&T.setRequestHeader("If-None-Match",S.etag[f])),(v.data&&v.hasContent&&!1!==v.contentType||t.contentType)&&T.setRequestHeader("Content-Type",v.contentType),T.setRequestHeader("Accept",v.dataTypes[0]&&v.accepts[v.dataTypes[0]]?v.accepts[v.dataTypes[0]]+("*"!==v.dataTypes[0]?", "+Rt+"; q=0.01":""):v.accepts["*"]),v.headers)T.setRequestHeader(i,v.headers[i]);if(v.beforeSend&&(!1===v.beforeSend.call(y,T,v)||h))return T.abort();if(u="abort",b.add(v.complete),T.done(v.success),T.fail(v.error),c=Wt(Pt,v,t,T)){if(T.readyState=1,g&&m.trigger("ajaxSend",[T,v]),h)return T;v.async&&0<v.timeout&&(d=C.setTimeout(function(){T.abort("timeout")},v.timeout));try{h=!1,c.send(a,l)}catch(e){if(h)throw e;l(-1,e)}}else l(-1,"No Transport");function l(e,t,n,r){var i,o,a,s,u,l=t;h||(h=!0,d&&C.clearTimeout(d),c=void 0,p=r||"",T.readyState=0<e?4:0,i=200<=e&&e<300||304===e,n&&(s=function(e,t,n){var r,i,o,a,s=e.contents,u=e.dataTypes;while("*"===u[0])u.shift(),void 0===r&&(r=e.mimeType||t.getResponseHeader("Content-Type"));if(r)for(i in s)if(s[i]&&s[i].test(r)){u.unshift(i);break}if(u[0]in n)o=u[0];else{for(i in n){if(!u[0]||e.converters[i+" "+u[0]]){o=i;break}a||(a=i)}o=o||a}if(o)return o!==u[0]&&u.unshift(o),n[o]}(v,T,n)),!i&&-1<S.inArray("script",v.dataTypes)&&S.inArray("json",v.dataTypes)<0&&(v.converters["text script"]=function(){}),s=function(e,t,n,r){var i,o,a,s,u,l={},c=e.dataTypes.slice();if(c[1])for(a in e.converters)l[a.toLowerCase()]=e.converters[a];o=c.shift();while(o)if(e.responseFields[o]&&(n[e.responseFields[o]]=t),!u&&r&&e.dataFilter&&(t=e.dataFilter(t,e.dataType)),u=o,o=c.shift())if("*"===o)o=u;else if("*"!==u&&u!==o){if(!(a=l[u+" "+o]||l["* "+o]))for(i in l)if((s=i.split(" "))[1]===o&&(a=l[u+" "+s[0]]||l["* "+s[0]])){!0===a?a=l[i]:!0!==l[i]&&(o=s[0],c.unshift(s[1]));break}if(!0!==a)if(a&&e["throws"])t=a(t);else try{t=a(t)}catch(e){return{state:"parsererror",error:a?e:"No conversion from "+u+" to "+o}}}return{state:"success",data:t}}(v,s,T,i),i?(v.ifModified&&((u=T.getResponseHeader("Last-Modified"))&&(S.lastModified[f]=u),(u=T.getResponseHeader("etag"))&&(S.etag[f]=u)),204===e||"HEAD"===v.type?l="nocontent":304===e?l="notmodified":(l=s.state,o=s.data,i=!(a=s.error))):(a=l,!e&&l||(l="error",e<0&&(e=0))),T.status=e,T.statusText=(t||l)+"",i?x.resolveWith(y,[o,l,T]):x.rejectWith(y,[T,l,a]),T.statusCode(w),w=void 0,g&&m.trigger(i?"ajaxSuccess":"ajaxError",[T,v,i?o:a]),b.fireWith(y,[T,l]),g&&(m.trigger("ajaxComplete",[T,v]),--S.active||S.event.trigger("ajaxStop")))}return T},getJSON:function(e,t,n){return S.get(e,t,n,"json")},getScript:function(e,t){return S.get(e,void 0,t,"script")}}),S.each(["get","post"],function(e,i){S[i]=function(e,t,n,r){return m(t)&&(r=r||n,n=t,t=void 0),S.ajax(S.extend({url:e,type:i,dataType:r,data:t,success:n},S.isPlainObject(e)&&e))}}),S.ajaxPrefilter(function(e){var t;for(t in e.headers)"content-type"===t.toLowerCase()&&(e.contentType=e.headers[t]||"")}),S._evalUrl=function(e,t,n){return S.ajax({url:e,type:"GET",dataType:"script",cache:!0,async:!1,global:!1,converters:{"text script":function(){}},dataFilter:function(e){S.globalEval(e,t,n)}})},S.fn.extend({wrapAll:function(e){var t;return this[0]&&(m(e)&&(e=e.call(this[0])),t=S(e,this[0].ownerDocument).eq(0).clone(!0),this[0].parentNode&&t.insertBefore(this[0]),t.map(function(){var e=this;while(e.firstElementChild)e=e.firstElementChild;return e}).append(this)),this},wrapInner:function(n){return m(n)?this.each(function(e){S(this).wrapInner(n.call(this,e))}):this.each(function(){var e=S(this),t=e.contents();t.length?t.wrapAll(n):e.append(n)})},wrap:function(t){var n=m(t);return this.each(function(e){S(this).wrapAll(n?t.call(this,e):t)})},unwrap:function(e){return this.parent(e).not("body").each(function(){S(this).replaceWith(this.childNodes)}),this}}),S.expr.pseudos.hidden=function(e){return!S.expr.pseudos.visible(e)},S.expr.pseudos.visible=function(e){return!!(e.offsetWidth||e.offsetHeight||e.getClientRects().length)},S.ajaxSettings.xhr=function(){try{return new C.XMLHttpRequest}catch(e){}};var Bt={0:200,1223:204},$t=S.ajaxSettings.xhr();y.cors=!!$t&&"withCredentials"in $t,y.ajax=$t=!!$t,S.ajaxTransport(function(i){var o,a;if(y.cors||$t&&!i.crossDomain)return{send:function(e,t){var n,r=i.xhr();if(r.open(i.type,i.url,i.async,i.username,i.password),i.xhrFields)for(n in i.xhrFields)r[n]=i.xhrFields[n];for(n in i.mimeType&&r.overrideMimeType&&r.overrideMimeType(i.mimeType),i.crossDomain||e["X-Requested-With"]||(e["X-Requested-With"]="XMLHttpRequest"),e)r.setRequestHeader(n,e[n]);o=function(e){return function(){o&&(o=a=r.onload=r.onerror=r.onabort=r.ontimeout=r.onreadystatechange=null,"abort"===e?r.abort():"error"===e?"number"!=typeof r.status?t(0,"error"):t(r.status,r.statusText):t(Bt[r.status]||r.status,r.statusText,"text"!==(r.responseType||"text")||"string"!=typeof r.responseText?{binary:r.response}:{text:r.responseText},r.getAllResponseHeaders()))}},r.onload=o(),a=r.onerror=r.ontimeout=o("error"),void 0!==r.onabort?r.onabort=a:r.onreadystatechange=function(){4===r.readyState&&C.setTimeout(function(){o&&a()})},o=o("abort");try{r.send(i.hasContent&&i.data||null)}catch(e){if(o)throw e}},abort:function(){o&&o()}}}),S.ajaxPrefilter(function(e){e.crossDomain&&(e.contents.script=!1)}),S.ajaxSetup({accepts:{script:"text/javascript, application/javascript, application/ecmascript, application/x-ecmascript"},contents:{script:/\b(?:java|ecma)script\b/},converters:{"text script":function(e){return S.globalEval(e),e}}}),S.ajaxPrefilter("script",function(e){void 0===e.cache&&(e.cache=!1),e.crossDomain&&(e.type="GET")}),S.ajaxTransport("script",function(n){var r,i;if(n.crossDomain||n.scriptAttrs)return{send:function(e,t){r=S("<script>").attr(n.scriptAttrs||{}).prop({charset:n.scriptCharset,src:n.url}).on("load error",i=function(e){r.remove(),i=null,e&&t("error"===e.type?404:200,e.type)}),E.head.appendChild(r[0])},abort:function(){i&&i()}}});var _t,zt=[],Ut=/(=)\?(?=&|$)|\?\?/;S.ajaxSetup({jsonp:"callback",jsonpCallback:function(){var e=zt.pop()||S.expando+"_"+wt.guid++;return this[e]=!0,e}}),S.ajaxPrefilter("json jsonp",function(e,t,n){var r,i,o,a=!1!==e.jsonp&&(Ut.test(e.url)?"url":"string"==typeof e.data&&0===(e.contentType||"").indexOf("application/x-www-form-urlencoded")&&Ut.test(e.data)&&"data");if(a||"jsonp"===e.dataTypes[0])return r=e.jsonpCallback=m(e.jsonpCallback)?e.jsonpCallback():e.jsonpCallback,a?e[a]=e[a].replace(Ut,"$1"+r):!1!==e.jsonp&&(e.url+=(Tt.test(e.url)?"&":"?")+e.jsonp+"="+r),e.converters["script json"]=function(){return o||S.error(r+" was not called"),o[0]},e.dataTypes[0]="json",i=C[r],C[r]=function(){o=arguments},n.always(function(){void 0===i?S(C).removeProp(r):C[r]=i,e[r]&&(e.jsonpCallback=t.jsonpCallback,zt.push(r)),o&&m(i)&&i(o[0]),o=i=void 0}),"script"}),y.createHTMLDocument=((_t=E.implementation.createHTMLDocument("").body).innerHTML="<form></form><form></form>",2===_t.childNodes.length),S.parseHTML=function(e,t,n){return"string"!=typeof e?[]:("boolean"==typeof t&&(n=t,t=!1),t||(y.createHTMLDocument?((r=(t=E.implementation.createHTMLDocument("")).createElement("base")).href=E.location.href,t.head.appendChild(r)):t=E),o=!n&&[],(i=N.exec(e))?[t.createElement(i[1])]:(i=xe([e],t,o),o&&o.length&&S(o).remove(),S.merge([],i.childNodes)));var r,i,o},S.fn.load=function(e,t,n){var r,i,o,a=this,s=e.indexOf(" ");return-1<s&&(r=ht(e.slice(s)),e=e.slice(0,s)),m(t)?(n=t,t=void 0):t&&"object"==typeof t&&(i="POST"),0<a.length&&S.ajax({url:e,type:i||"GET",dataType:"html",data:t}).done(function(e){o=arguments,a.html(r?S("<div>").append(S.parseHTML(e)).find(r):e)}).always(n&&function(e,t){a.each(function(){n.apply(this,o||[e.responseText,t,e])})}),this},S.expr.pseudos.animated=function(t){return S.grep(S.timers,function(e){return t===e.elem}).length},S.offset={setOffset:function(e,t,n){var r,i,o,a,s,u,l=S.css(e,"position"),c=S(e),f={};"static"===l&&(e.style.position="relative"),s=c.offset(),o=S.css(e,"top"),u=S.css(e,"left"),("absolute"===l||"fixed"===l)&&-1<(o+u).indexOf("auto")?(a=(r=c.position()).top,i=r.left):(a=parseFloat(o)||0,i=parseFloat(u)||0),m(t)&&(t=t.call(e,n,S.extend({},s))),null!=t.top&&(f.top=t.top-s.top+a),null!=t.left&&(f.left=t.left-s.left+i),"using"in t?t.using.call(e,f):c.css(f)}},S.fn.extend({offset:function(t){if(arguments.length)return void 0===t?this:this.each(function(e){S.offset.setOffset(this,t,e)});var e,n,r=this[0];return r?r.getClientRects().length?(e=r.getBoundingClientRect(),n=r.ownerDocument.defaultView,{top:e.top+n.pageYOffset,left:e.left+n.pageXOffset}):{top:0,left:0}:void 0},position:function(){if(this[0]){var e,t,n,r=this[0],i={top:0,left:0};if("fixed"===S.css(r,"position"))t=r.getBoundingClientRect();else{t=this.offset(),n=r.ownerDocument,e=r.offsetParent||n.documentElement;while(e&&(e===n.body||e===n.documentElement)&&"static"===S.css(e,"position"))e=e.parentNode;e&&e!==r&&1===e.nodeType&&((i=S(e).offset()).top+=S.css(e,"borderTopWidth",!0),i.left+=S.css(e,"borderLeftWidth",!0))}return{top:t.top-i.top-S.css(r,"marginTop",!0),left:t.left-i.left-S.css(r,"marginLeft",!0)}}},offsetParent:function(){return this.map(function(){var e=this.offsetParent;while(e&&"static"===S.css(e,"position"))e=e.offsetParent;return e||re})}}),S.each({scrollLeft:"pageXOffset",scrollTop:"pageYOffset"},function(t,i){var o="pageYOffset"===i;S.fn[t]=function(e){return $(this,function(e,t,n){var r;if(x(e)?r=e:9===e.nodeType&&(r=e.defaultView),void 0===n)return r?r[i]:e[t];r?r.scrollTo(o?r.pageXOffset:n,o?n:r.pageYOffset):e[t]=n},t,e,arguments.length)}}),S.each(["top","left"],function(e,n){S.cssHooks[n]=Fe(y.pixelPosition,function(e,t){if(t)return t=We(e,n),Pe.test(t)?S(e).position()[n]+"px":t})}),S.each({Height:"height",Width:"width"},function(a,s){S.each({padding:"inner"+a,content:s,"":"outer"+a},function(r,o){S.fn[o]=function(e,t){var n=arguments.length&&(r||"boolean"!=typeof e),i=r||(!0===e||!0===t?"margin":"border");return $(this,function(e,t,n){var r;return x(e)?0===o.indexOf("outer")?e["inner"+a]:e.document.documentElement["client"+a]:9===e.nodeType?(r=e.documentElement,Math.max(e.body["scroll"+a],r["scroll"+a],e.body["offset"+a],r["offset"+a],r["client"+a])):void 0===n?S.css(e,t,i):S.style(e,t,n,i)},s,n?e:void 0,n)}})}),S.each(["ajaxStart","ajaxStop","ajaxComplete","ajaxError","ajaxSuccess","ajaxSend"],function(e,t){S.fn[t]=function(e){return this.on(t,e)}}),S.fn.extend({bind:function(e,t,n){return this.on(e,null,t,n)},unbind:function(e,t){return this.off(e,null,t)},delegate:function(e,t,n,r){return this.on(t,e,n,r)},undelegate:function(e,t,n){return 1===arguments.length?this.off(e,"**"):this.off(t,e||"**",n)},hover:function(e,t){return this.mouseenter(e).mouseleave(t||e)}}),S.each("blur focus focusin focusout resize scroll click dblclick mousedown mouseup mousemove mouseover mouseout mouseenter mouseleave change select submit keydown keypress keyup contextmenu".split(" "),function(e,n){S.fn[n]=function(e,t){return 0<arguments.length?this.on(n,null,e,t):this.trigger(n)}});var Xt=/^[\s\uFEFF\xA0]+|[\s\uFEFF\xA0]+$/g;S.proxy=function(e,t){var n,r,i;if("string"==typeof t&&(n=e[t],t=e,e=n),m(e))return r=s.call(arguments,2),(i=function(){return e.apply(t||this,r.concat(s.call(arguments)))}).guid=e.guid=e.guid||S.guid++,i},S.holdReady=function(e){e?S.readyWait++:S.ready(!0)},S.isArray=Array.isArray,S.parseJSON=JSON.parse,S.nodeName=A,S.isFunction=m,S.isWindow=x,S.camelCase=X,S.type=w,S.now=Date.now,S.isNumeric=function(e){var t=S.type(e);return("number"===t||"string"===t)&&!isNaN(e-parseFloat(e))},S.trim=function(e){return null==e?"":(e+"").replace(Xt,"")},"function"==typeof define&&define.amd&&define("jquery",[],function(){return S});var Vt=C.jQuery,Gt=C.$;return S.noConflict=function(e){return C.$===S&&(C.$=Gt),e&&C.jQuery===S&&(C.jQuery=Vt),S},"undefined"==typeof e&&(C.jQuery=C.$=S),S});
html/_static/js/badge_only.js (new)
- @@ -0,0 +1,1 @@
  + !function(e){var t={};function r(n){if(t[n])return t[n].exports;var o=t[n]={i:n,l:!1,exports:{}};return e[n].call(o.exports,o,o.exports,r),o.l=!0,o.exports}r.m=e,r.c=t,r.d=function(e,t,n){r.o(e,t)||Object.defineProperty(e,t,{enumerable:!0,get:n})},r.r=function(e){"undefined"!=typeof Symbol&&Symbol.toStringTag&&Object.defineProperty(e,Symbol.toStringTag,{value:"Module"}),Object.defineProperty(e,"__esModule",{value:!0})},r.t=function(e,t){if(1&t&&(e=r(e)),8&t)return e;if(4&t&&"object"==typeof e&&e&&e.__esModule)return e;var n=Object.create(null);if(r.r(n),Object.defineProperty(n,"default",{enumerable:!0,value:e}),2&t&&"string"!=typeof e)for(var o in e)r.d(n,o,function(t){return e[t]}.bind(null,o));return n},r.n=function(e){var t=e&&e.__esModule?function(){return e.default}:function(){return e};return r.d(t,"a",t),t},r.o=function(e,t){return Object.prototype.hasOwnProperty.call(e,t)},r.p="",r(r.s=4)}({4:function(e,t,r){}});
html/_static/js/html5shiv-printshiv.min.js (new)
- @@ -0,0 +1,4 @@
 + /** + * @preserve HTML5 Shiv 3.7.3-pre | @afarkas @jdalton @jon_neal @rem | MIT/GPL2 Licensed + */ + !function(a,b){function c(a,b){var c=a.createElement("p"),d=a.getElementsByTagName("head")[0]||a.documentElement;return c.innerHTML="x<style>"+b+"</style>",d.insertBefore(c.lastChild,d.firstChild)}function d(){var a=y.elements;return"string"==typeof a?a.split(" "):a}function e(a,b){var c=y.elements;"string"!=typeof c&&(c=c.join(" ")),"string"!=typeof a&&(a=a.join(" ")),y.elements=c+" "+a,j(b)}function f(a){var b=x[a[v]];return b||(b={},w++,a[v]=w,x[w]=b),b}function g(a,c,d){if(c||(c=b),q)return c.createElement(a);d||(d=f(c));var e;return e=d.cache[a]?d.cache[a].cloneNode():u.test(a)?(d.cache[a]=d.createElem(a)).cloneNode():d.createElem(a),!e.canHaveChildren||t.test(a)||e.tagUrn?e:d.frag.appendChild(e)}function h(a,c){if(a||(a=b),q)return a.createDocumentFragment();c=c||f(a);for(var e=c.frag.cloneNode(),g=0,h=d(),i=h.length;i>g;g++)e.createElement(h[g]);return e}function i(a,b){b.cache||(b.cache={},b.createElem=a.createElement,b.createFrag=a.createDocumentFragment,b.frag=b.createFrag()),a.createElement=function(c){return y.shivMethods?g(c,a,b):b.createElem(c)},a.createDocumentFragment=Function("h,f","return function(){var n=f.cloneNode(),c=n.createElement;h.shivMethods&&("+d().join().replace(/[\w\-:]+/g,function(a){return b.createElem(a),b.frag.createElement(a),'c("'+a+'")'})+");return n}")(y,b.frag)}function j(a){a||(a=b);var d=f(a);return!y.shivCSS||p||d.hasCSS||(d.hasCSS=!!c(a,"article,aside,dialog,figcaption,figure,footer,header,hgroup,main,nav,section{display:block}mark{background:#FF0;color:#000}template{display:none}")),q||i(a,d),a}function k(a){for(var b,c=a.getElementsByTagName("*"),e=c.length,f=RegExp("^(?:"+d().join("|")+")$","i"),g=[];e--;)b=c[e],f.test(b.nodeName)&&g.push(b.applyElement(l(b)));return g}function l(a){for(var b,c=a.attributes,d=c.length,e=a.ownerDocument.createElement(A+":"+a.nodeName);d--;)b=c[d],b.specified&&e.setAttribute(b.nodeName,b.nodeValue);return e.style.cssText=a.style.cssText,e}function m(a){for(var b,c=a.split("{"),e=c.length,f=RegExp("(^|[\\s,>+~])("+d().join("|")+")(?=[[\\s,>+~#.:]|$)","gi"),g="$1"+A+"\\:$2";e--;)b=c[e]=c[e].split("}"),b[b.length-1]=b[b.length-1].replace(f,g),c[e]=b.join("}");return c.join("{")}function n(a){for(var b=a.length;b--;)a[b].removeNode()}function o(a){function b(){clearTimeout(g._removeSheetTimer),d&&d.removeNode(!0),d=null}var d,e,g=f(a),h=a.namespaces,i=a.parentWindow;return!B||a.printShived?a:("undefined"==typeof h[A]&&h.add(A),i.attachEvent("onbeforeprint",function(){b();for(var f,g,h,i=a.styleSheets,j=[],l=i.length,n=Array(l);l--;)n[l]=i[l];for(;h=n.pop();)if(!h.disabled&&z.test(h.media)){try{f=h.imports,g=f.length}catch(o){g=0}for(l=0;g>l;l++)n.push(f[l]);try{j.push(h.cssText)}catch(o){}}j=m(j.reverse().join("")),e=k(a),d=c(a,j)}),i.attachEvent("onafterprint",function(){n(e),clearTimeout(g._removeSheetTimer),g._removeSheetTimer=setTimeout(b,500)}),a.printShived=!0,a)}var p,q,r="3.7.3",s=a.html5||{},t=/^<|^(?:button|map|select|textarea|object|iframe|option|optgroup)$/i,u=/^(?:a|b|code|div|fieldset|h1|h2|h3|h4|h5|h6|i|label|li|ol|p|q|span|strong|style|table|tbody|td|th|tr|ul)$/i,v="_html5shiv",w=0,x={};!function(){try{var a=b.createElement("a");a.innerHTML="<xyz></xyz>",p="hidden"in a,q=1==a.childNodes.length||function(){b.createElement("a");var a=b.createDocumentFragment();return"undefined"==typeof a.cloneNode||"undefined"==typeof a.createDocumentFragment||"undefined"==typeof a.createElement}()}catch(c){p=!0,q=!0}}();var y={elements:s.elements||"abbr article aside audio bdi canvas data datalist details dialog figcaption figure footer header hgroup main mark meter nav output picture progress section summary template time video",version:r,shivCSS:s.shivCSS!==!1,supportsUnknownElements:q,shivMethods:s.shivMethods!==!1,type:"default",shivDocument:j,createElement:g,createDocumentFragment:h,addElements:e};a.html5=y,j(b);var z=/^$|\b(?:all|print)\b/,A="html5shiv",B=!q&&function(){var c=b.documentElement;return!("undefined"==typeof b.namespaces||"undefined"==typeof b.parentWindow||"undefined"==typeof c.applyElement||"undefined"==typeof c.removeNode||"undefined"==typeof a.attachEvent)}();y.type+=" print",y.shivPrint=o,o(b),"object"==typeof module&&module.exports&&(module.exports=y)}("undefined"!=typeof window?window:this,document);
html/_static/js/html5shiv.min.js (new)
- @@ -0,0 +1,4 @@
 + /** + * @preserve HTML5 Shiv 3.7.3 | @afarkas @jdalton @jon_neal @rem | MIT/GPL2 Licensed + */ + !function(a,b){function c(a,b){var c=a.createElement("p"),d=a.getElementsByTagName("head")[0]||a.documentElement;return c.innerHTML="x<style>"+b+"</style>",d.insertBefore(c.lastChild,d.firstChild)}function d(){var a=t.elements;return"string"==typeof a?a.split(" "):a}function e(a,b){var c=t.elements;"string"!=typeof c&&(c=c.join(" ")),"string"!=typeof a&&(a=a.join(" ")),t.elements=c+" "+a,j(b)}function f(a){var b=s[a[q]];return b||(b={},r++,a[q]=r,s[r]=b),b}function g(a,c,d){if(c||(c=b),l)return c.createElement(a);d||(d=f(c));var e;return e=d.cache[a]?d.cache[a].cloneNode():p.test(a)?(d.cache[a]=d.createElem(a)).cloneNode():d.createElem(a),!e.canHaveChildren||o.test(a)||e.tagUrn?e:d.frag.appendChild(e)}function h(a,c){if(a||(a=b),l)return a.createDocumentFragment();c=c||f(a);for(var e=c.frag.cloneNode(),g=0,h=d(),i=h.length;i>g;g++)e.createElement(h[g]);return e}function i(a,b){b.cache||(b.cache={},b.createElem=a.createElement,b.createFrag=a.createDocumentFragment,b.frag=b.createFrag()),a.createElement=function(c){return t.shivMethods?g(c,a,b):b.createElem(c)},a.createDocumentFragment=Function("h,f","return function(){var n=f.cloneNode(),c=n.createElement;h.shivMethods&&("+d().join().replace(/[\w\-:]+/g,function(a){return b.createElem(a),b.frag.createElement(a),'c("'+a+'")'})+");return n}")(t,b.frag)}function j(a){a||(a=b);var d=f(a);return!t.shivCSS||k||d.hasCSS||(d.hasCSS=!!c(a,"article,aside,dialog,figcaption,figure,footer,header,hgroup,main,nav,section{display:block}mark{background:#FF0;color:#000}template{display:none}")),l||i(a,d),a}var k,l,m="3.7.3-pre",n=a.html5||{},o=/^<|^(?:button|map|select|textarea|object|iframe|option|optgroup)$/i,p=/^(?:a|b|code|div|fieldset|h1|h2|h3|h4|h5|h6|i|label|li|ol|p|q|span|strong|style|table|tbody|td|th|tr|ul)$/i,q="_html5shiv",r=0,s={};!function(){try{var a=b.createElement("a");a.innerHTML="<xyz></xyz>",k="hidden"in a,l=1==a.childNodes.length||function(){b.createElement("a");var a=b.createDocumentFragment();return"undefined"==typeof a.cloneNode||"undefined"==typeof a.createDocumentFragment||"undefined"==typeof a.createElement}()}catch(c){k=!0,l=!0}}();var t={elements:n.elements||"abbr article aside audio bdi canvas data datalist details dialog figcaption figure footer header hgroup main mark meter nav output picture progress section summary template time video",version:m,shivCSS:n.shivCSS!==!1,supportsUnknownElements:l,shivMethods:n.shivMethods!==!1,type:"default",shivDocument:j,createElement:g,createDocumentFragment:h,addElements:e};a.html5=t,j(b),"object"==typeof module&&module.exports&&(module.exports=t)}("undefined"!=typeof window?window:this,document);
html/_static/js/theme.js (new)
- @@ -0,0 +1,1 @@
 + !function(n){var e={};function t(i){if(e[i])return e[i].exports;var o=e[i]={i:i,l:!1,exports:{}};return n[i].call(o.exports,o,o.exports,t),o.l=!0,o.exports}t.m=n,t.c=e,t.d=function(n,e,i){t.o(n,e)||Object.defineProperty(n,e,{enumerable:!0,get:i})},t.r=function(n){"undefined"!=typeof Symbol&&Symbol.toStringTag&&Object.defineProperty(n,Symbol.toStringTag,{value:"Module"}),Object.defineProperty(n,"__esModule",{value:!0})},t.t=function(n,e){if(1&e&&(n=t(n)),8&e)return n;if(4&e&&"object"==typeof n&&n&&n.__esModule)return n;var i=Object.create(null);if(t.r(i),Object.defineProperty(i,"default",{enumerable:!0,value:n}),2&e&&"string"!=typeof n)for(var o in n)t.d(i,o,function(e){return n[e]}.bind(null,o));return i},t.n=function(n){var e=n&&n.__esModule?function(){return n.default}:function(){return n};return t.d(e,"a",e),e},t.o=function(n,e){return Object.prototype.hasOwnProperty.call(n,e)},t.p="",t(t.s=0)}([function(n,e,t){t(1),n.exports=t(3)},function(n,e,t){(function(){var e="undefined"!=typeof window?window.jQuery:t(2);n.exports.ThemeNav={navBar:null,win:null,winScroll:!1,winResize:!1,linkScroll:!1,winPosition:0,winHeight:null,docHeight:null,isRunning:!1,enable:function(n){var t=this;void 0===n&&(n=!0),t.isRunning||(t.isRunning=!0,e((function(e){t.init(e),t.reset(),t.win.on("hashchange",t.reset),n&&t.win.on("scroll",(function(){t.linkScroll||t.winScroll||(t.winScroll=!0,requestAnimationFrame((function(){t.onScroll()})))})),t.win.on("resize",(function(){t.winResize||(t.winResize=!0,requestAnimationFrame((function(){t.onResize()})))})),t.onResize()})))},enableSticky:function(){this.enable(!0)},init:function(n){n(document);var e=this;this.navBar=n("div.wy-side-scroll:first"),this.win=n(window),n(document).on("click","[data-toggle='wy-nav-top']",(function(){n("[data-toggle='wy-nav-shift']").toggleClass("shift"),n("[data-toggle='rst-versions']").toggleClass("shift")})).on("click",".wy-menu-vertical .current ul li a",(function(){var t=n(this);n("[data-toggle='wy-nav-shift']").removeClass("shift"),n("[data-toggle='rst-versions']").toggleClass("shift"),e.toggleCurrent(t),e.hashChange()})).on("click","[data-toggle='rst-current-version']",(function(){n("[data-toggle='rst-versions']").toggleClass("shift-up")})),n("table.docutils:not(.field-list,.footnote,.citation)").wrap("<div class='wy-table-responsive'></div>"),n("table.docutils.footnote").wrap("<div class='wy-table-responsive footnote'></div>"),n("table.docutils.citation").wrap("<div class='wy-table-responsive citation'></div>"),n(".wy-menu-vertical ul").not(".simple").siblings("a").each((function(){var t=n(this);expand=n('<button class="toctree-expand" title="Open/close menu"></button>'),expand.on("click",(function(n){return e.toggleCurrent(t),n.stopPropagation(),!1})),t.prepend(expand)}))},reset:function(){var n=encodeURI(window.location.hash)||"#";try{var e=$(".wy-menu-vertical"),t=e.find('[href="'+n+'"]');if(0===t.length){var i=$('.document [id="'+n.substring(1)+'"]').closest("div.section");0===(t=e.find('[href="#'+i.attr("id")+'"]')).length&&(t=e.find('[href="#"]'))}if(t.length>0){$(".wy-menu-vertical .current").removeClass("current").attr("aria-expanded","false"),t.addClass("current").attr("aria-expanded","true"),t.closest("li.toctree-l1").parent().addClass("current").attr("aria-expanded","true");for(let n=1;n<=10;n++)t.closest("li.toctree-l"+n).addClass("current").attr("aria-expanded","true");t[0].scrollIntoView()}}catch(n){console.log("Error expanding nav for anchor",n)}},onScroll:function(){this.winScroll=!1;var n=this.win.scrollTop(),e=n+this.winHeight,t=this.navBar.scrollTop()+(n-this.winPosition);n<0||e>this.docHeight||(this.navBar.scrollTop(t),this.winPosition=n)},onResize:function(){this.winResize=!1,this.winHeight=this.win.height(),this.docHeight=$(document).height()},hashChange:function(){this.linkScroll=!0,this.win.one("hashchange",(function(){this.linkScroll=!1}))},toggleCurrent:function(n){var e=n.closest("li");e.siblings("li.current").removeClass("current").attr("aria-expanded","false"),e.siblings().find("li.current").removeClass("current").attr("aria-expanded","false");var t=e.find("> ul li");t.length&&(t.removeClass("current").attr("aria-expanded","false"),e.toggleClass("current").attr("aria-expanded",(function(n,e){return"true"==e?"false":"true"})))}},"undefined"!=typeof window&&(window.SphinxRtdTheme={Navigation:n.exports.ThemeNav,StickyNav:n.exports.ThemeNav}),function(){for(var n=0,e=["ms","moz","webkit","o"],t=0;t<e.length&&!window.requestAnimationFrame;++t)window.requestAnimationFrame=window[e[t]+"RequestAnimationFrame"],window.cancelAnimationFrame=window[e[t]+"CancelAnimationFrame"]||window[e[t]+"CancelRequestAnimationFrame"];window.requestAnimationFrame||(window.requestAnimationFrame=function(e,t){var i=(new Date).getTime(),o=Math.max(0,16-(i-n)),r=window.setTimeout((function(){e(i+o)}),o);return n=i+o,r}),window.cancelAnimationFrame||(window.cancelAnimationFrame=function(n){clearTimeout(n)})}()}).call(window)},function(n,e){n.exports=jQuery},function(n,e,t){}]);
html/_static/language_data.js (new)
- @@ -0,0 +1,199 @@
 + /* + * language_data.js + * ~~~~~~~~~~~~~~~~ + * + * This script contains the language-specific data used by searchtools.js, + * namely the list of stopwords, stemmer, scorer and splitter. + * + * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS. + * :license: BSD, see LICENSE for details. + * + */ + + var stopwords = ["a", "and", "are", "as", "at", "be", "but", "by", "for", "if", "in", "into", "is", "it", "near", "no", "not", "of", "on", "or", "such", "that", "the", "their", "then", "there", "these", "they", "this", "to", "was", "will", "with"]; + + + /* Non-minified version is copied as a separate JS file, is available */ + + /** + * Porter Stemmer + */ + var Stemmer = function() { + + var step2list = { + ational: 'ate', + tional: 'tion', + enci: 'ence', + anci: 'ance', + izer: 'ize', + bli: 'ble', + alli: 'al', + entli: 'ent', + eli: 'e', + ousli: 'ous', + ization: 'ize', + ation: 'ate', + ator: 'ate', + alism: 'al', + iveness: 'ive', + fulness: 'ful', + ousness: 'ous', + aliti: 'al', + iviti: 'ive', + biliti: 'ble', + logi: 'log' + }; + + var step3list = { + icate: 'ic', + ative: '', + alize: 'al', + iciti: 'ic', + ical: 'ic', + ful: '', + ness: '' + }; + + var c = "[^aeiou]"; // consonant + var v = "[aeiouy]"; // vowel + var C = c + "[^aeiouy]*"; // consonant sequence + var V = v + "[aeiou]*"; // vowel sequence + + var mgr0 = "^(" + C + ")?" + V + C; // [C]VC... is m>0 + var meq1 = "^(" + C + ")?" + V + C + "(" + V + ")?$"; // [C]VC[V] is m=1 + var mgr1 = "^(" + C + ")?" + V + C + V + C; // [C]VCVC... is m>1 + var s_v = "^(" + C + ")?" + v; // vowel in stem + + this.stemWord = function (w) { + var stem; + var suffix; + var firstch; + var origword = w; + + if (w.length < 3) + return w; + + var re; + var re2; + var re3; + var re4; + + firstch = w.substr(0,1); + if (firstch == "y") + w = firstch.toUpperCase() + w.substr(1); + + // Step 1a + re = /^(.+?)(ss|i)es$/; + re2 = /^(.+?)([^s])s$/; + + if (re.test(w)) + w = w.replace(re,"$1$2"); + else if (re2.test(w)) + w = w.replace(re2,"$1$2"); + + // Step 1b + re = /^(.+?)eed$/; + re2 = /^(.+?)(ed|ing)$/; + if (re.test(w)) { + var fp = re.exec(w); + re = new RegExp(mgr0); + if (re.test(fp[1])) { + re = /.$/; + w = w.replace(re,""); + } + } + else if (re2.test(w)) { + var fp = re2.exec(w); + stem = fp[1]; + re2 = new RegExp(s_v); + if (re2.test(stem)) { + w = stem; + re2 = /(at|bl|iz)$/; + re3 = new RegExp("([^aeiouylsz])\\1$"); + re4 = new RegExp("^" + C + v + "[^aeiouwxy]$"); + if (re2.test(w)) + w = w + "e"; + else if (re3.test(w)) { + re = /.$/; + w = w.replace(re,""); + } + else if (re4.test(w)) + w = w + "e"; + } + } + + // Step 1c + re = /^(.+?)y$/; + if (re.test(w)) { + var fp = re.exec(w); + stem = fp[1]; + re = new RegExp(s_v); + if (re.test(stem)) + w = stem + "i"; + } + + // Step 2 + re = /^(.+?)(ational|tional|enci|anci|izer|bli|alli|entli|eli|ousli|ization|ation|ator|alism|iveness|fulness|ousness|aliti|iviti|biliti|logi)$/; + if (re.test(w)) { + var fp = re.exec(w); + stem = fp[1]; + suffix = fp[2]; + re = new RegExp(mgr0); + if (re.test(stem)) + w = stem + step2list[suffix]; + } + + // Step 3 + re = /^(.+?)(icate|ative|alize|iciti|ical|ful|ness)$/; + if (re.test(w)) { + var fp = re.exec(w); + stem = fp[1]; + suffix = fp[2]; + re = new RegExp(mgr0); + if (re.test(stem)) + w = stem + step3list[suffix]; + } + + // Step 4 + re = /^(.+?)(al|ance|ence|er|ic|able|ible|ant|ement|ment|ent|ou|ism|ate|iti|ous|ive|ize)$/; + re2 = /^(.+?)(s|t)(ion)$/; + if (re.test(w)) { + var fp = re.exec(w); + stem = fp[1]; + re = new RegExp(mgr1); + if (re.test(stem)) + w = stem; + } + else if (re2.test(w)) { + var fp = re2.exec(w); + stem = fp[1] + fp[2]; + re2 = new RegExp(mgr1); + if (re2.test(stem)) + w = stem; + } + + // Step 5 + re = /^(.+?)e$/; + if (re.test(w)) { + var fp = re.exec(w); + stem = fp[1]; + re = new RegExp(mgr1); + re2 = new RegExp(meq1); + re3 = new RegExp("^" + C + v + "[^aeiouwxy]$"); + if (re.test(stem) || (re2.test(stem) && !(re3.test(stem)))) + w = stem; + } + re = /ll$/; + re2 = new RegExp(mgr1); + if (re.test(w) && re2.test(w)) { + re = /.$/; + w = w.replace(re,""); + } + + // and turn initial Y back to y + if (firstch == "y") + w = firstch.toLowerCase() + w.substr(1); + return w; + } + } +
html/_static/minus.png (new)
html/_static/plus.png (new)
html/_static/pygments.css (new)
- @@ -0,0 +1,74 @@
  + pre { line-height: 125%; } + td.linenos .normal { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; } + span.linenos { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; } + td.linenos .special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; } + span.linenos.special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; } + .highlight .hll { background-color: #ffffcc } + .highlight { background: #f8f8f8; } + .highlight .c { color: #3D7B7B; font-style: italic } /* Comment */ + .highlight .err { border: 1px solid #FF0000 } /* Error */ + .highlight .k { color: #008000; font-weight: bold } /* Keyword */ + .highlight .o { color: #666666 } /* Operator */ + .highlight .ch { color: #3D7B7B; font-style: italic } /* Comment.Hashbang */ + .highlight .cm { color: #3D7B7B; font-style: italic } /* Comment.Multiline */ + .highlight .cp { color: #9C6500 } /* Comment.Preproc */ + .highlight .cpf { color: #3D7B7B; font-style: italic } /* Comment.PreprocFile */ + .highlight .c1 { color: #3D7B7B; font-style: italic } /* Comment.Single */ + .highlight .cs { color: #3D7B7B; font-style: italic } /* Comment.Special */ + .highlight .gd { color: #A00000 } /* Generic.Deleted */ + .highlight .ge { font-style: italic } /* Generic.Emph */ + .highlight .gr { color: #E40000 } /* Generic.Error */ + .highlight .gh { color: #000080; font-weight: bold } /* Generic.Heading */ + .highlight .gi { color: #008400 } /* Generic.Inserted */ + .highlight .go { color: #717171 } /* Generic.Output */ + .highlight .gp { color: #000080; font-weight: bold } /* Generic.Prompt */ + .highlight .gs { font-weight: bold } /* Generic.Strong */ + .highlight .gu { color: #800080; font-weight: bold } /* Generic.Subheading */ + .highlight .gt { color: #0044DD } /* Generic.Traceback */ + .highlight .kc { color: #008000; font-weight: bold } /* Keyword.Constant */ + .highlight .kd { color: #008000; font-weight: bold } /* Keyword.Declaration */ + .highlight .kn { color: #008000; font-weight: bold } /* Keyword.Namespace */ + .highlight .kp { color: #008000 } /* Keyword.Pseudo */ + .highlight .kr { color: #008000; font-weight: bold } /* Keyword.Reserved */ + .highlight .kt { color: #B00040 } /* Keyword.Type */ + .highlight .m { color: #666666 } /* Literal.Number */ + .highlight .s { color: #BA2121 } /* Literal.String */ + .highlight .na { color: #687822 } /* Name.Attribute */ + .highlight .nb { color: #008000 } /* Name.Builtin */ + .highlight .nc { color: #0000FF; font-weight: bold } /* Name.Class */ + .highlight .no { color: #880000 } /* Name.Constant */ + .highlight .nd { color: #AA22FF } /* Name.Decorator */ + .highlight .ni { color: #717171; font-weight: bold } /* Name.Entity */ + .highlight .ne { color: #CB3F38; font-weight: bold } /* Name.Exception */ + .highlight .nf { color: #0000FF } /* Name.Function */ + .highlight .nl { color: #767600 } /* Name.Label */ + .highlight .nn { color: #0000FF; font-weight: bold } /* Name.Namespace */ + .highlight .nt { color: #008000; font-weight: bold } /* Name.Tag */ + .highlight .nv { color: #19177C } /* Name.Variable */ + .highlight .ow { color: #AA22FF; font-weight: bold } /* Operator.Word */ + .highlight .w { color: #bbbbbb } /* Text.Whitespace */ + .highlight .mb { color: #666666 } /* Literal.Number.Bin */ + .highlight .mf { color: #666666 } /* Literal.Number.Float */ + .highlight .mh { color: #666666 } /* Literal.Number.Hex */ + .highlight .mi { color: #666666 } /* Literal.Number.Integer */ + .highlight .mo { color: #666666 } /* Literal.Number.Oct */ + .highlight .sa { color: #BA2121 } /* Literal.String.Affix */ + .highlight .sb { color: #BA2121 } /* Literal.String.Backtick */ + .highlight .sc { color: #BA2121 } /* Literal.String.Char */ + .highlight .dl { color: #BA2121 } /* Literal.String.Delimiter */ + .highlight .sd { color: #BA2121; font-style: italic } /* Literal.String.Doc */ + .highlight .s2 { color: #BA2121 } /* Literal.String.Double */ + .highlight .se { color: #AA5D1F; font-weight: bold } /* Literal.String.Escape */ + .highlight .sh { color: #BA2121 } /* Literal.String.Heredoc */ + .highlight .si { color: #A45A77; font-weight: bold } /* Literal.String.Interpol */ + .highlight .sx { color: #008000 } /* Literal.String.Other */ + .highlight .sr { color: #A45A77 } /* Literal.String.Regex */ + .highlight .s1 { color: #BA2121 } /* Literal.String.Single */ + .highlight .ss { color: #19177C } /* Literal.String.Symbol */ + .highlight .bp { color: #008000 } /* Name.Builtin.Pseudo */ + .highlight .fm { color: #0000FF } /* Name.Function.Magic */ + .highlight .vc { color: #19177C } /* Name.Variable.Class */ + .highlight .vg { color: #19177C } /* Name.Variable.Global */ + .highlight .vi { color: #19177C } /* Name.Variable.Instance */ + .highlight .vm { color: #19177C } /* Name.Variable.Magic */ + .highlight .il { color: #666666 } /* Literal.Number.Integer.Long */
html/_static/searchtools.js (new)
- @@ -0,0 +1,531 @@
 + /* + * searchtools.js + * ~~~~~~~~~~~~~~~~ + * + * Sphinx JavaScript utilities for the full-text search. + * + * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS. + * :license: BSD, see LICENSE for details. + * + */ + "use strict"; + + /** + * Simple result scoring code. + */ + if (typeof Scorer === "undefined") { + var Scorer = { + // Implement the following function to further tweak the score for each result + // The function takes a result array [docname, title, anchor, descr, score, filename] + // and returns the new score. + /* + score: result => { + const [docname, title, anchor, descr, score, filename] = result + return score + }, + */ + + // query matches the full name of an object + objNameMatch: 11, + // or matches in the last dotted part of the object name + objPartialMatch: 6, + // Additive scores depending on the priority of the object + objPrio: { + 0: 15, // used to be importantResults + 1: 5, // used to be objectResults + 2: -5, // used to be unimportantResults + }, + // Used when the priority is not in the mapping. + objPrioDefault: 0, + + // query found in title + title: 15, + partialTitle: 7, + // query found in terms + term: 5, + partialTerm: 2, + }; + } + + const _removeChildren = (element) => { + while (element && element.lastChild) element.removeChild(element.lastChild); + }; + + /** + * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions#escaping + */ + const _escapeRegExp = (string) => + string.replace(/[.*+\-?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string + + const _displayItem = (item, highlightTerms, searchTerms) => { + const docBuilder = DOCUMENTATION_OPTIONS.BUILDER; + const docUrlRoot = DOCUMENTATION_OPTIONS.URL_ROOT; + const docFileSuffix = DOCUMENTATION_OPTIONS.FILE_SUFFIX; + const docLinkSuffix = DOCUMENTATION_OPTIONS.LINK_SUFFIX; + const showSearchSummary = DOCUMENTATION_OPTIONS.SHOW_SEARCH_SUMMARY; + + const [docName, title, anchor, descr] = item; + + let listItem = document.createElement("li"); + let requestUrl; + let linkUrl; + if (docBuilder === "dirhtml") { + // dirhtml builder + let dirname = docName + "/"; + if (dirname.match(/\/index\/$/)) + dirname = dirname.substring(0, dirname.length - 6); + else if (dirname === "index/") dirname = ""; + requestUrl = docUrlRoot + dirname; + linkUrl = requestUrl; + } else { + // normal html builders + requestUrl = docUrlRoot + docName + docFileSuffix; + linkUrl = docName + docLinkSuffix; + } + const params = new URLSearchParams(); + params.set("highlight", [...highlightTerms].join(" ")); + let linkEl = listItem.appendChild(document.createElement("a")); + linkEl.href = linkUrl + "?" + params.toString() + anchor; + linkEl.innerHTML = title; + if (descr) + listItem.appendChild(document.createElement("span")).innerText = + " (" + descr + ")"; + else if (showSearchSummary) + fetch(requestUrl) + .then((responseData) => responseData.text()) + .then((data) => { + if (data) + listItem.appendChild( + Search.makeSearchSummary(data, searchTerms, highlightTerms) + ); + }); + Search.output.appendChild(listItem); + }; + const _finishSearch = (resultCount) => { + Search.stopPulse(); + Search.title.innerText = _("Search Results"); + if (!resultCount) + Search.status.innerText = Documentation.gettext( + "Your search did not match any documents. Please make sure that all words are spelled correctly and that you've selected enough categories." + ); + else + Search.status.innerText = _( + `Search finished, found ${resultCount} page(s) matching the search query.` + ); + }; + const _displayNextItem = ( + results, + resultCount, + highlightTerms, + searchTerms + ) => { + // results left, load the summary and display it + // this is intended to be dynamic (don't sub resultsCount) + if (results.length) { + _displayItem(results.pop(), highlightTerms, searchTerms); + setTimeout( + () => _displayNextItem(results, resultCount, highlightTerms, searchTerms), + 5 + ); + } + // search finished, update title and status message + else _finishSearch(resultCount); + }; + + /** + * Default splitQuery function. Can be overridden in ``sphinx.search`` with a + * custom function per language. + * + * The regular expression works by splitting the string on consecutive characters + * that are not Unicode letters, numbers, underscores, or emoji characters. + * This is the same as ``\W+`` in Python, preserving the surrogate pair area. + */ + if (typeof splitQuery === "undefined") { + var splitQuery = (query) => query + .split(/[^\p{Letter}\p{Number}_\p{Emoji_Presentation}]+/gu) + .filter(term => term) // remove remaining empty strings + } + + /** + * Search Module + */ + const Search = { + _index: null, + _queued_query: null, + _pulse_status: -1, + + htmlToText: (htmlString) => { + const htmlElement = document + .createRange() + .createContextualFragment(htmlString); + _removeChildren(htmlElement.querySelectorAll(".headerlink")); + const docContent = htmlElement.querySelector('[role="main"]'); + if (docContent !== undefined) return docContent.textContent; + console.warn( + "Content block not found. Sphinx search tries to obtain it via '[role=main]'. Could you check your theme or template." + ); + return ""; + }, + + init: () => { + const query = new URLSearchParams(window.location.search).get("q"); + document + .querySelectorAll('input[name="q"]') + .forEach((el) => (el.value = query)); + if (query) Search.performSearch(query); + }, + + loadIndex: (url) => + (document.body.appendChild(document.createElement("script")).src = url), + + setIndex: (index) => { + Search._index = index; + if (Search._queued_query !== null) { + const query = Search._queued_query; + Search._queued_query = null; + Search.query(query); + } + }, + + hasIndex: () => Search._index !== null, + + deferQuery: (query) => (Search._queued_query = query), + + stopPulse: () => (Search._pulse_status = -1), + + startPulse: () => { + if (Search._pulse_status >= 0) return; + + const pulse = () => { + Search._pulse_status = (Search._pulse_status + 1) % 4; + Search.dots.innerText = ".".repeat(Search._pulse_status); + if (Search._pulse_status >= 0) window.setTimeout(pulse, 500); + }; + pulse(); + }, + + /** + * perform a search for something (or wait until index is loaded) + */ + performSearch: (query) => { + // create the required interface elements + const searchText = document.createElement("h2"); + searchText.textContent = _("Searching"); + const searchSummary = document.createElement("p"); + searchSummary.classList.add("search-summary"); + searchSummary.innerText = ""; + const searchList = document.createElement("ul"); + searchList.classList.add("search"); + + const out = document.getElementById("search-results"); + Search.title = out.appendChild(searchText); + Search.dots = Search.title.appendChild(document.createElement("span")); + Search.status = out.appendChild(searchSummary); + Search.output = out.appendChild(searchList); + + const searchProgress = document.getElementById("search-progress"); + // Some themes don't use the search progress node + if (searchProgress) { + searchProgress.innerText = _("Preparing search..."); + } + Search.startPulse(); + + // index already loaded, the browser was quick! + if (Search.hasIndex()) Search.query(query); + else Search.deferQuery(query); + }, + + /** + * execute search (requires search index to be loaded) + */ + query: (query) => { + // stem the search terms and add them to the correct list + const stemmer = new Stemmer(); + const searchTerms = new Set(); + const excludedTerms = new Set(); + const highlightTerms = new Set(); + const objectTerms = new Set(splitQuery(query.toLowerCase().trim())); + splitQuery(query.trim()).forEach((queryTerm) => { + const queryTermLower = queryTerm.toLowerCase(); + + // maybe skip this "word" + // stopwords array is from language_data.js + if ( + stopwords.indexOf(queryTermLower) !== -1 || + queryTerm.match(/^\d+$/) + ) + return; + + // stem the word + let word = stemmer.stemWord(queryTermLower); + // select the correct list + if (word[0] === "-") excludedTerms.add(word.substr(1)); + else { + searchTerms.add(word); + highlightTerms.add(queryTermLower); + } + }); + + // console.debug("SEARCH: searching for:"); + // console.info("required: ", [...searchTerms]); + // console.info("excluded: ", [...excludedTerms]); + + // array of [docname, title, anchor, descr, score, filename] + let results = []; + _removeChildren(document.getElementById("search-progress")); + + // lookup as object + objectTerms.forEach((term) => + results.push(...Search.performObjectSearch(term, objectTerms)) + ); + + // lookup as search terms in fulltext + results.push(...Search.performTermsSearch(searchTerms, excludedTerms)); + + // let the scorer override scores with a custom scoring function + if (Scorer.score) results.forEach((item) => (item[4] = Scorer.score(item))); + + // now sort the results by score (in opposite order of appearance, since the + // display function below uses pop() to retrieve items) and then + // alphabetically + results.sort((a, b) => { + const leftScore = a[4]; + const rightScore = b[4]; + if (leftScore === rightScore) { + // same score: sort alphabetically + const leftTitle = a[1].toLowerCase(); + const rightTitle = b[1].toLowerCase(); + if (leftTitle === rightTitle) return 0; + return leftTitle > rightTitle ? -1 : 1; // inverted is intentional + } + return leftScore > rightScore ? 1 : -1; + }); + + // remove duplicate search results + // note the reversing of results, so that in the case of duplicates, the highest-scoring entry is kept + let seen = new Set(); + results = results.reverse().reduce((acc, result) => { + let resultStr = result.slice(0, 4).concat([result[5]]).map(v => String(v)).join(','); + if (!seen.has(resultStr)) { + acc.push(result); + seen.add(resultStr); + } + return acc; + }, []); + + results = results.reverse(); + + // for debugging + //Search.lastresults = results.slice(); // a copy + // console.info("search results:", Search.lastresults); + + // print the results + _displayNextItem(results, results.length, highlightTerms, searchTerms); + }, + + /** + * search for object names + */ + performObjectSearch: (object, objectTerms) => { + const filenames = Search._index.filenames; + const docNames = Search._index.docnames; + const objects = Search._index.objects; + const objNames = Search._index.objnames; + const titles = Search._index.titles; + + const results = []; + + const objectSearchCallback = (prefix, match) => { + const name = match[4] + const fullname = (prefix ? prefix + "." : "") + name; + const fullnameLower = fullname.toLowerCase(); + if (fullnameLower.indexOf(object) < 0) return; + + let score = 0; + const parts = fullnameLower.split("."); + + // check for different match types: exact matches of full name or + // "last name" (i.e. last dotted part) + if (fullnameLower === object || parts.slice(-1)[0] === object) + score += Scorer.objNameMatch; + else if (parts.slice(-1)[0].indexOf(object) > -1) + score += Scorer.objPartialMatch; // matches in last name + + const objName = objNames[match[1]][2]; + const title = titles[match[0]]; + + // If more than one term searched for, we require other words to be + // found in the name/title/description + const otherTerms = new Set(objectTerms); + otherTerms.delete(object); + if (otherTerms.size > 0) { + const haystack = `${prefix} ${name} ${objName} ${title}`.toLowerCase(); + if ( + [...otherTerms].some((otherTerm) => haystack.indexOf(otherTerm) < 0) + ) + return; + } + + let anchor = match[3]; + if (anchor === "") anchor = fullname; + else if (anchor === "-") anchor = objNames[match[1]][1] + "-" + fullname; + + const descr = objName + _(", in ") + title; + + // add custom score for some objects according to scorer + if (Scorer.objPrio.hasOwnProperty(match[2])) + score += Scorer.objPrio[match[2]]; + else score += Scorer.objPrioDefault; + + results.push([ + docNames[match[0]], + fullname, + "#" + anchor, + descr, + score, + filenames[match[0]], + ]); + }; + Object.keys(objects).forEach((prefix) => + objects[prefix].forEach((array) => + objectSearchCallback(prefix, array) + ) + ); + return results; + }, + + /** + * search for full-text terms in the index + */ + performTermsSearch: (searchTerms, excludedTerms) => { + // prepare search + const terms = Search._index.terms; + const titleTerms = Search._index.titleterms; + const docNames = Search._index.docnames; + const filenames = Search._index.filenames; + const titles = Search._index.titles; + + const scoreMap = new Map(); + const fileMap = new Map(); + + // perform the search on the required terms + searchTerms.forEach((word) => { + const files = []; + const arr = [ + { files: terms[word], score: Scorer.term }, + { files: titleTerms[word], score: Scorer.title }, + ]; + // add support for partial matches + if (word.length > 2) { + const escapedWord = _escapeRegExp(word); + Object.keys(terms).forEach((term) => { + if (term.match(escapedWord) && !terms[word]) + arr.push({ files: terms[term], score: Scorer.partialTerm }); + }); + Object.keys(titleTerms).forEach((term) => { + if (term.match(escapedWord) && !titleTerms[word]) + arr.push({ files: titleTerms[word], score: Scorer.partialTitle }); + }); + } + + // no match but word was a required one + if (arr.every((record) => record.files === undefined)) return; + + // found search word in contents + arr.forEach((record) => { + if (record.files === undefined) return; + + let recordFiles = record.files; + if (recordFiles.length === undefined) recordFiles = [recordFiles]; + files.push(...recordFiles); + + // set score for the word in each file + recordFiles.forEach((file) => { + if (!scoreMap.has(file)) scoreMap.set(file, {}); + scoreMap.get(file)[word] = record.score; + }); + }); + + // create the mapping + files.forEach((file) => { + if (fileMap.has(file) && fileMap.get(file).indexOf(word) === -1) + fileMap.get(file).push(word); + else fileMap.set(file, [word]); + }); + }); + + // now check if the files don't contain excluded terms + const results = []; + for (const [file, wordList] of fileMap) { + // check if all requirements are matched + + // as search terms with length < 3 are discarded + const filteredTermCount = [...searchTerms].filter( + (term) => term.length > 2 + ).length; + if ( + wordList.length !== searchTerms.size && + wordList.length !== filteredTermCount + ) + continue; + + // ensure that none of the excluded terms is in the search result + if ( + [...excludedTerms].some( + (term) => + terms[term] === file || + titleTerms[term] === file || + (terms[term] || []).includes(file) || + (titleTerms[term] || []).includes(file) + ) + ) + break; + + // select one (max) score for the file. + const score = Math.max(...wordList.map((w) => scoreMap.get(file)[w])); + // add result to the result list + results.push([ + docNames[file], + titles[file], + "", + null, + score, + filenames[file], + ]); + } + return results; + }, + + /** + * helper function to return a node containing the + * search summary for a given text. keywords is a list + * of stemmed words, highlightWords is the list of normal, unstemmed + * words. the first one is used to find the occurrence, the + * latter for highlighting it. + */ + makeSearchSummary: (htmlText, keywords, highlightWords) => { + const text = Search.htmlToText(htmlText).toLowerCase(); + if (text === "") return null; + + const actualStartPosition = [...keywords] + .map((k) => text.indexOf(k.toLowerCase())) + .filter((i) => i > -1) + .slice(-1)[0]; + const startWithContext = Math.max(actualStartPosition - 120, 0); + + const top = startWithContext === 0 ? "" : "..."; + const tail = startWithContext + 240 < text.length ? "..." : ""; + + let summary = document.createElement("div"); + summary.classList.add("context"); + summary.innerText = top + text.substr(startWithContext, 240).trim() + tail; + + highlightWords.forEach((highlightWord) => + _highlightText(summary, highlightWord, "highlighted") + ); + + return summary; + }, + }; + + _ready(Search.init);
html/_static/underscore-1.13.1.js (new)
- @@ -0,0 +1,2042 @@
 + (function (global, factory) { + typeof exports === 'object' && typeof module !== 'undefined' ? module.exports = factory() : + typeof define === 'function' && define.amd ? define('underscore', factory) : + (global = typeof globalThis !== 'undefined' ? globalThis : global || self, (function () { + var current = global._; + var exports = global._ = factory(); + exports.noConflict = function () { global._ = current; return exports; }; + }())); + }(this, (function () { + // Underscore.js 1.13.1 + // https://underscorejs.org + // (c) 2009-2021 Jeremy Ashkenas, Julian Gonggrijp, and DocumentCloud and Investigative Reporters & Editors + // Underscore may be freely distributed under the MIT license. + + // Current version. + var VERSION = '1.13.1'; + + // Establish the root object, `window` (`self`) in the browser, `global` + // on the server, or `this` in some virtual machines. We use `self` + // instead of `window` for `WebWorker` support. + var root = typeof self == 'object' && self.self === self && self || + typeof global == 'object' && global.global === global && global || + Function('return this')() || + {}; + + // Save bytes in the minified (but not gzipped) version: + var ArrayProto = Array.prototype, ObjProto = Object.prototype; + var SymbolProto = typeof Symbol !== 'undefined' ? Symbol.prototype : null; + + // Create quick reference variables for speed access to core prototypes. + var push = ArrayProto.push, + slice = ArrayProto.slice, + toString = ObjProto.toString, + hasOwnProperty = ObjProto.hasOwnProperty; + + // Modern feature detection. + var supportsArrayBuffer = typeof ArrayBuffer !== 'undefined', + supportsDataView = typeof DataView !== 'undefined'; + + // All **ECMAScript 5+** native function implementations that we hope to use + // are declared here. + var nativeIsArray = Array.isArray, + nativeKeys = Object.keys, + nativeCreate = Object.create, + nativeIsView = supportsArrayBuffer && ArrayBuffer.isView; + + // Create references to these builtin functions because we override them. + var _isNaN = isNaN, + _isFinite = isFinite; + + // Keys in IE < 9 that won't be iterated by `for key in ...` and thus missed. + var hasEnumBug = !{toString: null}.propertyIsEnumerable('toString'); + var nonEnumerableProps = ['valueOf', 'isPrototypeOf', 'toString', + 'propertyIsEnumerable', 'hasOwnProperty', 'toLocaleString']; + + // The largest integer that can be represented exactly. + var MAX_ARRAY_INDEX = Math.pow(2, 53) - 1; + + // Some functions take a variable number of arguments, or a few expected + // arguments at the beginning and then a variable number of values to operate + // on. This helper accumulates all remaining arguments past the function’s + // argument length (or an explicit `startIndex`), into an array that becomes + // the last argument. Similar to ES6’s "rest parameter". + function restArguments(func, startIndex) { + startIndex = startIndex == null ? func.length - 1 : +startIndex; + return function() { + var length = Math.max(arguments.length - startIndex, 0), + rest = Array(length), + index = 0; + for (; index < length; index++) { + rest[index] = arguments[index + startIndex]; + } + switch (startIndex) { + case 0: return func.call(this, rest); + case 1: return func.call(this, arguments[0], rest); + case 2: return func.call(this, arguments[0], arguments[1], rest); + } + var args = Array(startIndex + 1); + for (index = 0; index < startIndex; index++) { + args[index] = arguments[index]; + } + args[startIndex] = rest; + return func.apply(this, args); + }; + } + + // Is a given variable an object? + function isObject(obj) { + var type = typeof obj; + return type === 'function' || type === 'object' && !!obj; + } + + // Is a given value equal to null? + function isNull(obj) { + return obj === null; + } + + // Is a given variable undefined? + function isUndefined(obj) { + return obj === void 0; + } + + // Is a given value a boolean? + function isBoolean(obj) { + return obj === true || obj === false || toString.call(obj) === '[object Boolean]'; + } + + // Is a given value a DOM element? + function isElement(obj) { + return !!(obj && obj.nodeType === 1); + } + + // Internal function for creating a `toString`-based type tester. + function tagTester(name) { + var tag = '[object ' + name + ']'; + return function(obj) { + return toString.call(obj) === tag; + }; + } + + var isString = tagTester('String'); + + var isNumber = tagTester('Number'); + + var isDate = tagTester('Date'); + + var isRegExp = tagTester('RegExp'); + + var isError = tagTester('Error'); + + var isSymbol = tagTester('Symbol'); + + var isArrayBuffer = tagTester('ArrayBuffer'); + + var isFunction = tagTester('Function'); + + // Optimize `isFunction` if appropriate. Work around some `typeof` bugs in old + // v8, IE 11 (#1621), Safari 8 (#1929), and PhantomJS (#2236). + var nodelist = root.document && root.document.childNodes; + if (typeof /./ != 'function' && typeof Int8Array != 'object' && typeof nodelist != 'function') { + isFunction = function(obj) { + return typeof obj == 'function' || false; + }; + } + + var isFunction$1 = isFunction; + + var hasObjectTag = tagTester('Object'); + + // In IE 10 - Edge 13, `DataView` has string tag `'[object Object]'`. + // In IE 11, the most common among them, this problem also applies to + // `Map`, `WeakMap` and `Set`. + var hasStringTagBug = ( + supportsDataView && hasObjectTag(new DataView(new ArrayBuffer(8))) + ), + isIE11 = (typeof Map !== 'undefined' && hasObjectTag(new Map)); + + var isDataView = tagTester('DataView'); + + // In IE 10 - Edge 13, we need a different heuristic + // to determine whether an object is a `DataView`. + function ie10IsDataView(obj) { + return obj != null && isFunction$1(obj.getInt8) && isArrayBuffer(obj.buffer); + } + + var isDataView$1 = (hasStringTagBug ? ie10IsDataView : isDataView); + + // Is a given value an array? + // Delegates to ECMA5's native `Array.isArray`. + var isArray = nativeIsArray || tagTester('Array'); + + // Internal function to check whether `key` is an own property name of `obj`. + function has$1(obj, key) { + return obj != null && hasOwnProperty.call(obj, key); + } + + var isArguments = tagTester('Arguments'); + + // Define a fallback version of the method in browsers (ahem, IE < 9), where + // there isn't any inspectable "Arguments" type. + (function() { + if (!isArguments(arguments)) { + isArguments = function(obj) { + return has$1(obj, 'callee'); + }; + } + }()); + + var isArguments$1 = isArguments; + + // Is a given object a finite number? + function isFinite$1(obj) { + return !isSymbol(obj) && _isFinite(obj) && !isNaN(parseFloat(obj)); + } + + // Is the given value `NaN`? + function isNaN$1(obj) { + return isNumber(obj) && _isNaN(obj); + } + + // Predicate-generating function. Often useful outside of Underscore. + function constant(value) { + return function() { + return value; + }; + } + + // Common internal logic for `isArrayLike` and `isBufferLike`. + function createSizePropertyCheck(getSizeProperty) { + return function(collection) { + var sizeProperty = getSizeProperty(collection); + return typeof sizeProperty == 'number' && sizeProperty >= 0 && sizeProperty <= MAX_ARRAY_INDEX; + } + } + + // Internal helper to generate a function to obtain property `key` from `obj`. + function shallowProperty(key) { + return function(obj) { + return obj == null ? void 0 : obj[key]; + }; + } + + // Internal helper to obtain the `byteLength` property of an object. + var getByteLength = shallowProperty('byteLength'); + + // Internal helper to determine whether we should spend extensive checks against + // `ArrayBuffer` et al. + var isBufferLike = createSizePropertyCheck(getByteLength); + + // Is a given value a typed array? + var typedArrayPattern = /\[object ((I|Ui)nt(8|16|32)|Float(32|64)|Uint8Clamped|Big(I|Ui)nt64)Array\]/; + function isTypedArray(obj) { + // `ArrayBuffer.isView` is the most future-proof, so use it when available. + // Otherwise, fall back on the above regular expression. + return nativeIsView ? (nativeIsView(obj) && !isDataView$1(obj)) : + isBufferLike(obj) && typedArrayPattern.test(toString.call(obj)); + } + + var isTypedArray$1 = supportsArrayBuffer ? isTypedArray : constant(false); + + // Internal helper to obtain the `length` property of an object. + var getLength = shallowProperty('length'); + + // Internal helper to create a simple lookup structure. + // `collectNonEnumProps` used to depend on `_.contains`, but this led to + // circular imports. `emulatedSet` is a one-off solution that only works for + // arrays of strings. + function emulatedSet(keys) { + var hash = {}; + for (var l = keys.length, i = 0; i < l; ++i) hash[keys[i]] = true; + return { + contains: function(key) { return hash[key]; }, + push: function(key) { + hash[key] = true; + return keys.push(key); + } + }; + } + + // Internal helper. Checks `keys` for the presence of keys in IE < 9 that won't + // be iterated by `for key in ...` and thus missed. Extends `keys` in place if + // needed. + function collectNonEnumProps(obj, keys) { + keys = emulatedSet(keys); + var nonEnumIdx = nonEnumerableProps.length; + var constructor = obj.constructor; + var proto = isFunction$1(constructor) && constructor.prototype || ObjProto; + + // Constructor is a special case. + var prop = 'constructor'; + if (has$1(obj, prop) && !keys.contains(prop)) keys.push(prop); + + while (nonEnumIdx--) { + prop = nonEnumerableProps[nonEnumIdx]; + if (prop in obj && obj[prop] !== proto[prop] && !keys.contains(prop)) { + keys.push(prop); + } + } + } + + // Retrieve the names of an object's own properties. + // Delegates to **ECMAScript 5**'s native `Object.keys`. + function keys(obj) { + if (!isObject(obj)) return []; + if (nativeKeys) return nativeKeys(obj); + var keys = []; + for (var key in obj) if (has$1(obj, key)) keys.push(key); + // Ahem, IE < 9. + if (hasEnumBug) collectNonEnumProps(obj, keys); + return keys; + } + + // Is a given array, string, or object empty? + // An "empty" object has no enumerable own-properties. + function isEmpty(obj) { + if (obj == null) return true; + // Skip the more expensive `toString`-based type checks if `obj` has no + // `.length`. + var length = getLength(obj); + if (typeof length == 'number' && ( + isArray(obj) || isString(obj) || isArguments$1(obj) + )) return length === 0; + return getLength(keys(obj)) === 0; + } + + // Returns whether an object has a given set of `key:value` pairs. + function isMatch(object, attrs) { + var _keys = keys(attrs), length = _keys.length; + if (object == null) return !length; + var obj = Object(object); + for (var i = 0; i < length; i++) { + var key = _keys[i]; + if (attrs[key] !== obj[key] || !(key in obj)) return false; + } + return true; + } + + // If Underscore is called as a function, it returns a wrapped object that can + // be used OO-style. This wrapper holds altered versions of all functions added + // through `_.mixin`. Wrapped objects may be chained. + function _$1(obj) { + if (obj instanceof _$1) return obj; + if (!(this instanceof _$1)) return new _$1(obj); + this._wrapped = obj; + } + + _$1.VERSION = VERSION; + + // Extracts the result from a wrapped and chained object. + _$1.prototype.value = function() { + return this._wrapped; + }; + + // Provide unwrapping proxies for some methods used in engine operations + // such as arithmetic and JSON stringification. + _$1.prototype.valueOf = _$1.prototype.toJSON = _$1.prototype.value; + + _$1.prototype.toString = function() { + return String(this._wrapped); + }; + + // Internal function to wrap or shallow-copy an ArrayBuffer, + // typed array or DataView to a new view, reusing the buffer. + function toBufferView(bufferSource) { + return new Uint8Array( + bufferSource.buffer || bufferSource, + bufferSource.byteOffset || 0, + getByteLength(bufferSource) + ); + } + + // We use this string twice, so give it a name for minification. + var tagDataView = '[object DataView]'; + + // Internal recursive comparison function for `_.isEqual`. + function eq(a, b, aStack, bStack) { + // Identical objects are equal. `0 === -0`, but they aren't identical. + // See the [Harmony `egal` proposal](https://wiki.ecmascript.org/doku.php?id=harmony:egal). + if (a === b) return a !== 0 || 1 / a === 1 / b; + // `null` or `undefined` only equal to itself (strict comparison). + if (a == null || b == null) return false; + // `NaN`s are equivalent, but non-reflexive. + if (a !== a) return b !== b; + // Exhaust primitive checks + var type = typeof a; + if (type !== 'function' && type !== 'object' && typeof b != 'object') return false; + return deepEq(a, b, aStack, bStack); + } + + // Internal recursive comparison function for `_.isEqual`. + function deepEq(a, b, aStack, bStack) { + // Unwrap any wrapped objects. + if (a instanceof _$1) a = a._wrapped; + if (b instanceof _$1) b = b._wrapped; + // Compare `[[Class]]` names. + var className = toString.call(a); + if (className !== toString.call(b)) return false; + // Work around a bug in IE 10 - Edge 13. + if (hasStringTagBug && className == '[object Object]' && isDataView$1(a)) { + if (!isDataView$1(b)) return false; + className = tagDataView; + } + switch (className) { + // These types are compared by value. + case '[object RegExp]': + // RegExps are coerced to strings for comparison (Note: '' + /a/i === '/a/i') + case '[object String]': + // Primitives and their corresponding object wrappers are equivalent; thus, `"5"` is + // equivalent to `new String("5")`. + return '' + a === '' + b; + case '[object Number]': + // `NaN`s are equivalent, but non-reflexive. + // Object(NaN) is equivalent to NaN. + if (+a !== +a) return +b !== +b; + // An `egal` comparison is performed for other numeric values. + return +a === 0 ? 1 / +a === 1 / b : +a === +b; + case '[object Date]': + case '[object Boolean]': + // Coerce dates and booleans to numeric primitive values. Dates are compared by their + // millisecond representations. Note that invalid dates with millisecond representations + // of `NaN` are not equivalent. + return +a === +b; + case '[object Symbol]': + return SymbolProto.valueOf.call(a) === SymbolProto.valueOf.call(b); + case '[object ArrayBuffer]': + case tagDataView: + // Coerce to typed array so we can fall through. + return deepEq(toBufferView(a), toBufferView(b), aStack, bStack); + } + + var areArrays = className === '[object Array]'; + if (!areArrays && isTypedArray$1(a)) { + var byteLength = getByteLength(a); + if (byteLength !== getByteLength(b)) return false; + if (a.buffer === b.buffer && a.byteOffset === b.byteOffset) return true; + areArrays = true; + } + if (!areArrays) { + if (typeof a != 'object' || typeof b != 'object') return false; + + // Objects with different constructors are not equivalent, but `Object`s or `Array`s + // from different frames are. + var aCtor = a.constructor, bCtor = b.constructor; + if (aCtor !== bCtor && !(isFunction$1(aCtor) && aCtor instanceof aCtor && + isFunction$1(bCtor) && bCtor instanceof bCtor) + && ('constructor' in a && 'constructor' in b)) { + return false; + } + } + // Assume equality for cyclic structures. The algorithm for detecting cyclic + // structures is adapted from ES 5.1 section 15.12.3, abstract operation `JO`. + + // Initializing stack of traversed objects. + // It's done here since we only need them for objects and arrays comparison. + aStack = aStack || []; + bStack = bStack || []; + var length = aStack.length; + while (length--) { + // Linear search. Performance is inversely proportional to the number of + // unique nested structures. + if (aStack[length] === a) return bStack[length] === b; + } + + // Add the first object to the stack of traversed objects. + aStack.push(a); + bStack.push(b); + + // Recursively compare objects and arrays. + if (areArrays) { + // Compare array lengths to determine if a deep comparison is necessary. + length = a.length; + if (length !== b.length) return false; + // Deep compare the contents, ignoring non-numeric properties. + while (length--) { + if (!eq(a[length], b[length], aStack, bStack)) return false; + } + } else { + // Deep compare objects. + var _keys = keys(a), key; + length = _keys.length; + // Ensure that both objects contain the same number of properties before comparing deep equality. + if (keys(b).length !== length) return false; + while (length--) { + // Deep compare each member + key = _keys[length]; + if (!(has$1(b, key) && eq(a[key], b[key], aStack, bStack))) return false; + } + } + // Remove the first object from the stack of traversed objects. + aStack.pop(); + bStack.pop(); + return true; + } + + // Perform a deep comparison to check if two objects are equal. + function isEqual(a, b) { + return eq(a, b); + } + + // Retrieve all the enumerable property names of an object. + function allKeys(obj) { + if (!isObject(obj)) return []; + var keys = []; + for (var key in obj) keys.push(key); + // Ahem, IE < 9. + if (hasEnumBug) collectNonEnumProps(obj, keys); + return keys; + } + + // Since the regular `Object.prototype.toString` type tests don't work for + // some types in IE 11, we use a fingerprinting heuristic instead, based + // on the methods. It's not great, but it's the best we got. + // The fingerprint method lists are defined below. + function ie11fingerprint(methods) { + var length = getLength(methods); + return function(obj) { + if (obj == null) return false; + // `Map`, `WeakMap` and `Set` have no enumerable keys. + var keys = allKeys(obj); + if (getLength(keys)) return false; + for (var i = 0; i < length; i++) { + if (!isFunction$1(obj[methods[i]])) return false; + } + // If we are testing against `WeakMap`, we need to ensure that + // `obj` doesn't have a `forEach` method in order to distinguish + // it from a regular `Map`. + return methods !== weakMapMethods || !isFunction$1(obj[forEachName]); + }; + } + + // In the interest of compact minification, we write + // each string in the fingerprints only once. + var forEachName = 'forEach', + hasName = 'has', + commonInit = ['clear', 'delete'], + mapTail = ['get', hasName, 'set']; + + // `Map`, `WeakMap` and `Set` each have slightly different + // combinations of the above sublists. + var mapMethods = commonInit.concat(forEachName, mapTail), + weakMapMethods = commonInit.concat(mapTail), + setMethods = ['add'].concat(commonInit, forEachName, hasName); + + var isMap = isIE11 ? ie11fingerprint(mapMethods) : tagTester('Map'); + + var isWeakMap = isIE11 ? ie11fingerprint(weakMapMethods) : tagTester('WeakMap'); + + var isSet = isIE11 ? ie11fingerprint(setMethods) : tagTester('Set'); + + var isWeakSet = tagTester('WeakSet'); + + // Retrieve the values of an object's properties. + function values(obj) { + var _keys = keys(obj); + var length = _keys.length; + var values = Array(length); + for (var i = 0; i < length; i++) { + values[i] = obj[_keys[i]]; + } + return values; + } + + // Convert an object into a list of `[key, value]` pairs. + // The opposite of `_.object` with one argument. + function pairs(obj) { + var _keys = keys(obj); + var length = _keys.length; + var pairs = Array(length); + for (var i = 0; i < length; i++) { + pairs[i] = [_keys[i], obj[_keys[i]]]; + } + return pairs; + } + + // Invert the keys and values of an object. The values must be serializable. + function invert(obj) { + var result = {}; + var _keys = keys(obj); + for (var i = 0, length = _keys.length; i < length; i++) { + result[obj[_keys[i]]] = _keys[i]; + } + return result; + } + + // Return a sorted list of the function names available on the object. + function functions(obj) { + var names = []; + for (var key in obj) { + if (isFunction$1(obj[key])) names.push(key); + } + return names.sort(); + } + + // An internal function for creating assigner functions. + function createAssigner(keysFunc, defaults) { + return function(obj) { + var length = arguments.length; + if (defaults) obj = Object(obj); + if (length < 2 || obj == null) return obj; + for (var index = 1; index < length; index++) { + var source = arguments[index], + keys = keysFunc(source), + l = keys.length; + for (var i = 0; i < l; i++) { + var key = keys[i]; + if (!defaults || obj[key] === void 0) obj[key] = source[key]; + } + } + return obj; + }; + } + + // Extend a given object with all the properties in passed-in object(s). + var extend = createAssigner(allKeys); + + // Assigns a given object with all the own properties in the passed-in + // object(s). + // (https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Object/assign) + var extendOwn = createAssigner(keys); + + // Fill in a given object with default properties. + var defaults = createAssigner(allKeys, true); + + // Create a naked function reference for surrogate-prototype-swapping. + function ctor() { + return function(){}; + } + + // An internal function for creating a new object that inherits from another. + function baseCreate(prototype) { + if (!isObject(prototype)) return {}; + if (nativeCreate) return nativeCreate(prototype); + var Ctor = ctor(); + Ctor.prototype = prototype; + var result = new Ctor; + Ctor.prototype = null; + return result; + } + + // Creates an object that inherits from the given prototype object. + // If additional properties are provided then they will be added to the + // created object. + function create(prototype, props) { + var result = baseCreate(prototype); + if (props) extendOwn(result, props); + return result; + } + + // Create a (shallow-cloned) duplicate of an object. + function clone(obj) { + if (!isObject(obj)) return obj; + return isArray(obj) ? obj.slice() : extend({}, obj); + } + + // Invokes `interceptor` with the `obj` and then returns `obj`. + // The primary purpose of this method is to "tap into" a method chain, in + // order to perform operations on intermediate results within the chain. + function tap(obj, interceptor) { + interceptor(obj); + return obj; + } + + // Normalize a (deep) property `path` to array. + // Like `_.iteratee`, this function can be customized. + function toPath$1(path) { + return isArray(path) ? path : [path]; + } + _$1.toPath = toPath$1; + + // Internal wrapper for `_.toPath` to enable minification. + // Similar to `cb` for `_.iteratee`. + function toPath(path) { + return _$1.toPath(path); + } + + // Internal function to obtain a nested property in `obj` along `path`. + function deepGet(obj, path) { + var length = path.length; + for (var i = 0; i < length; i++) { + if (obj == null) return void 0; + obj = obj[path[i]]; + } + return length ? obj : void 0; + } + + // Get the value of the (deep) property on `path` from `object`. + // If any property in `path` does not exist or if the value is + // `undefined`, return `defaultValue` instead. + // The `path` is normalized through `_.toPath`. + function get(object, path, defaultValue) { + var value = deepGet(object, toPath(path)); + return isUndefined(value) ? defaultValue : value; + } + + // Shortcut function for checking if an object has a given property directly on + // itself (in other words, not on a prototype). Unlike the internal `has` + // function, this public version can also traverse nested properties. + function has(obj, path) { + path = toPath(path); + var length = path.length; + for (var i = 0; i < length; i++) { + var key = path[i]; + if (!has$1(obj, key)) return false; + obj = obj[key]; + } + return !!length; + } + + // Keep the identity function around for default iteratees. + function identity(value) { + return value; + } + + // Returns a predicate for checking whether an object has a given set of + // `key:value` pairs. + function matcher(attrs) { + attrs = extendOwn({}, attrs); + return function(obj) { + return isMatch(obj, attrs); + }; + } + + // Creates a function that, when passed an object, will traverse that object’s + // properties down the given `path`, specified as an array of keys or indices. + function property(path) { + path = toPath(path); + return function(obj) { + return deepGet(obj, path); + }; + } + + // Internal function that returns an efficient (for current engines) version + // of the passed-in callback, to be repeatedly applied in other Underscore + // functions. + function optimizeCb(func, context, argCount) { + if (context === void 0) return func; + switch (argCount == null ? 3 : argCount) { + case 1: return function(value) { + return func.call(context, value); + }; + // The 2-argument case is omitted because we’re not using it. + case 3: return function(value, index, collection) { + return func.call(context, value, index, collection); + }; + case 4: return function(accumulator, value, index, collection) { + return func.call(context, accumulator, value, index, collection); + }; + } + return function() { + return func.apply(context, arguments); + }; + } + + // An internal function to generate callbacks that can be applied to each + // element in a collection, returning the desired result — either `_.identity`, + // an arbitrary callback, a property matcher, or a property accessor. + function baseIteratee(value, context, argCount) { + if (value == null) return identity; + if (isFunction$1(value)) return optimizeCb(value, context, argCount); + if (isObject(value) && !isArray(value)) return matcher(value); + return property(value); + } + + // External wrapper for our callback generator. Users may customize + // `_.iteratee` if they want additional predicate/iteratee shorthand styles. + // This abstraction hides the internal-only `argCount` argument. + function iteratee(value, context) { + return baseIteratee(value, context, Infinity); + } + _$1.iteratee = iteratee; + + // The function we call internally to generate a callback. It invokes + // `_.iteratee` if overridden, otherwise `baseIteratee`. + function cb(value, context, argCount) { + if (_$1.iteratee !== iteratee) return _$1.iteratee(value, context); + return baseIteratee(value, context, argCount); + } + + // Returns the results of applying the `iteratee` to each element of `obj`. + // In contrast to `_.map` it returns an object. + function mapObject(obj, iteratee, context) { + iteratee = cb(iteratee, context); + var _keys = keys(obj), + length = _keys.length, + results = {}; + for (var index = 0; index < length; index++) { + var currentKey = _keys[index]; + results[currentKey] = iteratee(obj[currentKey], currentKey, obj); + } + return results; + } + + // Predicate-generating function. Often useful outside of Underscore. + function noop(){} + + // Generates a function for a given object that returns a given property. + function propertyOf(obj) { + if (obj == null) return noop; + return function(path) { + return get(obj, path); + }; + } + + // Run a function **n** times. + function times(n, iteratee, context) { + var accum = Array(Math.max(0, n)); + iteratee = optimizeCb(iteratee, context, 1); + for (var i = 0; i < n; i++) accum[i] = iteratee(i); + return accum; + } + + // Return a random integer between `min` and `max` (inclusive). + function random(min, max) { + if (max == null) { + max = min; + min = 0; + } + return min + Math.floor(Math.random() * (max - min + 1)); + } + + // A (possibly faster) way to get the current timestamp as an integer. + var now = Date.now || function() { + return new Date().getTime(); + }; + + // Internal helper to generate functions for escaping and unescaping strings + // to/from HTML interpolation. + function createEscaper(map) { + var escaper = function(match) { + return map[match]; + }; + // Regexes for identifying a key that needs to be escaped. + var source = '(?:' + keys(map).join('|') + ')'; + var testRegexp = RegExp(source); + var replaceRegexp = RegExp(source, 'g'); + return function(string) { + string = string == null ? '' : '' + string; + return testRegexp.test(string) ? string.replace(replaceRegexp, escaper) : string; + }; + } + + // Internal list of HTML entities for escaping. + var escapeMap = { + '&': '&', + '<': '<', + '>': '>', + '"': '"', + "'": ''', + '`': '`' + }; + + // Function for escaping strings to HTML interpolation. + var _escape = createEscaper(escapeMap); + + // Internal list of HTML entities for unescaping. + var unescapeMap = invert(escapeMap); + + // Function for unescaping strings from HTML interpolation. + var _unescape = createEscaper(unescapeMap); + + // By default, Underscore uses ERB-style template delimiters. Change the + // following template settings to use alternative delimiters. + var templateSettings = _$1.templateSettings = { + evaluate: /<%([\s\S]+?)%>/g, + interpolate: /<%=([\s\S]+?)%>/g, + escape: /<%-([\s\S]+?)%>/g + }; + + // When customizing `_.templateSettings`, if you don't want to define an + // interpolation, evaluation or escaping regex, we need one that is + // guaranteed not to match. + var noMatch = /(.)^/; + + // Certain characters need to be escaped so that they can be put into a + // string literal. + var escapes = { + "'": "'", + '\\': '\\', + '\r': 'r', + '\n': 'n', + '\u2028': 'u2028', + '\u2029': 'u2029' + }; + + var escapeRegExp = /\\|'|\r|\n|\u2028|\u2029/g; + + function escapeChar(match) { + return '\\' + escapes[match]; + } + + // In order to prevent third-party code injection through + // `_.templateSettings.variable`, we test it against the following regular + // expression. It is intentionally a bit more liberal than just matching valid + // identifiers, but still prevents possible loopholes through defaults or + // destructuring assignment. + var bareIdentifier = /^\s*(\w|\$)+\s*$/; + + // JavaScript micro-templating, similar to John Resig's implementation. + // Underscore templating handles arbitrary delimiters, preserves whitespace, + // and correctly escapes quotes within interpolated code. + // NB: `oldSettings` only exists for backwards compatibility. + function template(text, settings, oldSettings) { + if (!settings && oldSettings) settings = oldSettings; + settings = defaults({}, settings, _$1.templateSettings); + + // Combine delimiters into one regular expression via alternation. + var matcher = RegExp([ + (settings.escape || noMatch).source, + (settings.interpolate || noMatch).source, + (settings.evaluate || noMatch).source + ].join('|') + '|$', 'g'); + + // Compile the template source, escaping string literals appropriately. + var index = 0; + var source = "__p+='"; + text.replace(matcher, function(match, escape, interpolate, evaluate, offset) { + source += text.slice(index, offset).replace(escapeRegExp, escapeChar); + index = offset + match.length; + + if (escape) { + source += "'+\n((__t=(" + escape + "))==null?'':_.escape(__t))+\n'"; + } else if (interpolate) { + source += "'+\n((__t=(" + interpolate + "))==null?'':__t)+\n'"; + } else if (evaluate) { + source += "';\n" + evaluate + "\n__p+='"; + } + + // Adobe VMs need the match returned to produce the correct offset. + return match; + }); + source += "';\n"; + + var argument = settings.variable; + if (argument) { + // Insure against third-party code injection. (CVE-2021-23358) + if (!bareIdentifier.test(argument)) throw new Error( + 'variable is not a bare identifier: ' + argument + ); + } else { + // If a variable is not specified, place data values in local scope. + source = 'with(obj||{}){\n' + source + '}\n'; + argument = 'obj'; + } + + source = "var __t,__p='',__j=Array.prototype.join," + + "print=function(){__p+=__j.call(arguments,'');};\n" + + source + 'return __p;\n'; + + var render; + try { + render = new Function(argument, '_', source); + } catch (e) { + e.source = source; + throw e; + } + + var template = function(data) { + return render.call(this, data, _$1); + }; + + // Provide the compiled source as a convenience for precompilation. + template.source = 'function(' + argument + '){\n' + source + '}'; + + return template; + } + + // Traverses the children of `obj` along `path`. If a child is a function, it + // is invoked with its parent as context. Returns the value of the final + // child, or `fallback` if any child is undefined. + function result(obj, path, fallback) { + path = toPath(path); + var length = path.length; + if (!length) { + return isFunction$1(fallback) ? fallback.call(obj) : fallback; + } + for (var i = 0; i < length; i++) { + var prop = obj == null ? void 0 : obj[path[i]]; + if (prop === void 0) { + prop = fallback; + i = length; // Ensure we don't continue iterating. + } + obj = isFunction$1(prop) ? prop.call(obj) : prop; + } + return obj; + } + + // Generate a unique integer id (unique within the entire client session). + // Useful for temporary DOM ids. + var idCounter = 0; + function uniqueId(prefix) { + var id = ++idCounter + ''; + return prefix ? prefix + id : id; + } + + // Start chaining a wrapped Underscore object. + function chain(obj) { + var instance = _$1(obj); + instance._chain = true; + return instance; + } + + // Internal function to execute `sourceFunc` bound to `context` with optional + // `args`. Determines whether to execute a function as a constructor or as a + // normal function. + function executeBound(sourceFunc, boundFunc, context, callingContext, args) { + if (!(callingContext instanceof boundFunc)) return sourceFunc.apply(context, args); + var self = baseCreate(sourceFunc.prototype); + var result = sourceFunc.apply(self, args); + if (isObject(result)) return result; + return self; + } + + // Partially apply a function by creating a version that has had some of its + // arguments pre-filled, without changing its dynamic `this` context. `_` acts + // as a placeholder by default, allowing any combination of arguments to be + // pre-filled. Set `_.partial.placeholder` for a custom placeholder argument. + var partial = restArguments(function(func, boundArgs) { + var placeholder = partial.placeholder; + var bound = function() { + var position = 0, length = boundArgs.length; + var args = Array(length); + for (var i = 0; i < length; i++) { + args[i] = boundArgs[i] === placeholder ? arguments[position++] : boundArgs[i]; + } + while (position < arguments.length) args.push(arguments[position++]); + return executeBound(func, bound, this, this, args); + }; + return bound; + }); + + partial.placeholder = _$1; + + // Create a function bound to a given object (assigning `this`, and arguments, + // optionally). + var bind = restArguments(function(func, context, args) { + if (!isFunction$1(func)) throw new TypeError('Bind must be called on a function'); + var bound = restArguments(function(callArgs) { + return executeBound(func, bound, context, this, args.concat(callArgs)); + }); + return bound; + }); + + // Internal helper for collection methods to determine whether a collection + // should be iterated as an array or as an object. + // Related: https://people.mozilla.org/~jorendorff/es6-draft.html#sec-tolength + // Avoids a very nasty iOS 8 JIT bug on ARM-64. #2094 + var isArrayLike = createSizePropertyCheck(getLength); + + // Internal implementation of a recursive `flatten` function. + function flatten$1(input, depth, strict, output) { + output = output || []; + if (!depth && depth !== 0) { + depth = Infinity; + } else if (depth <= 0) { + return output.concat(input); + } + var idx = output.length; + for (var i = 0, length = getLength(input); i < length; i++) { + var value = input[i]; + if (isArrayLike(value) && (isArray(value) || isArguments$1(value))) { + // Flatten current level of array or arguments object. + if (depth > 1) { + flatten$1(value, depth - 1, strict, output); + idx = output.length; + } else { + var j = 0, len = value.length; + while (j < len) output[idx++] = value[j++]; + } + } else if (!strict) { + output[idx++] = value; + } + } + return output; + } + + // Bind a number of an object's methods to that object. Remaining arguments + // are the method names to be bound. Useful for ensuring that all callbacks + // defined on an object belong to it. + var bindAll = restArguments(function(obj, keys) { + keys = flatten$1(keys, false, false); + var index = keys.length; + if (index < 1) throw new Error('bindAll must be passed function names'); + while (index--) { + var key = keys[index]; + obj[key] = bind(obj[key], obj); + } + return obj; + }); + + // Memoize an expensive function by storing its results. + function memoize(func, hasher) { + var memoize = function(key) { + var cache = memoize.cache; + var address = '' + (hasher ? hasher.apply(this, arguments) : key); + if (!has$1(cache, address)) cache[address] = func.apply(this, arguments); + return cache[address]; + }; + memoize.cache = {}; + return memoize; + } + + // Delays a function for the given number of milliseconds, and then calls + // it with the arguments supplied. + var delay = restArguments(function(func, wait, args) { + return setTimeout(function() { + return func.apply(null, args); + }, wait); + }); + + // Defers a function, scheduling it to run after the current call stack has + // cleared. + var defer = partial(delay, _$1, 1); + + // Returns a function, that, when invoked, will only be triggered at most once + // during a given window of time. Normally, the throttled function will run + // as much as it can, without ever going more than once per `wait` duration; + // but if you'd like to disable the execution on the leading edge, pass + // `{leading: false}`. To disable execution on the trailing edge, ditto. + function throttle(func, wait, options) { + var timeout, context, args, result; + var previous = 0; + if (!options) options = {}; + + var later = function() { + previous = options.leading === false ? 0 : now(); + timeout = null; + result = func.apply(context, args); + if (!timeout) context = args = null; + }; + + var throttled = function() { + var _now = now(); + if (!previous && options.leading === false) previous = _now; + var remaining = wait - (_now - previous); + context = this; + args = arguments; + if (remaining <= 0 || remaining > wait) { + if (timeout) { + clearTimeout(timeout); + timeout = null; + } + previous = _now; + result = func.apply(context, args); + if (!timeout) context = args = null; + } else if (!timeout && options.trailing !== false) { + timeout = setTimeout(later, remaining); + } + return result; + }; + + throttled.cancel = function() { + clearTimeout(timeout); + previous = 0; + timeout = context = args = null; + }; + + return throttled; + } + + // When a sequence of calls of the returned function ends, the argument + // function is triggered. The end of a sequence is defined by the `wait` + // parameter. If `immediate` is passed, the argument function will be + // triggered at the beginning of the sequence instead of at the end. + function debounce(func, wait, immediate) { + var timeout, previous, args, result, context; + + var later = function() { + var passed = now() - previous; + if (wait > passed) { + timeout = setTimeout(later, wait - passed); + } else { + timeout = null; + if (!immediate) result = func.apply(context, args); + // This check is needed because `func` can recursively invoke `debounced`. + if (!timeout) args = context = null; + } + }; + + var debounced = restArguments(function(_args) { + context = this; + args = _args; + previous = now(); + if (!timeout) { + timeout = setTimeout(later, wait); + if (immediate) result = func.apply(context, args); + } + return result; + }); + + debounced.cancel = function() { + clearTimeout(timeout); + timeout = args = context = null; + }; + + return debounced; + } + + // Returns the first function passed as an argument to the second, + // allowing you to adjust arguments, run code before and after, and + // conditionally execute the original function. + function wrap(func, wrapper) { + return partial(wrapper, func); + } + + // Returns a negated version of the passed-in predicate. + function negate(predicate) { + return function() { + return !predicate.apply(this, arguments); + }; + } + + // Returns a function that is the composition of a list of functions, each + // consuming the return value of the function that follows. + function compose() { + var args = arguments; + var start = args.length - 1; + return function() { + var i = start; + var result = args[start].apply(this, arguments); + while (i--) result = args[i].call(this, result); + return result; + }; + } + + // Returns a function that will only be executed on and after the Nth call. + function after(times, func) { + return function() { + if (--times < 1) { + return func.apply(this, arguments); + } + }; + } + + // Returns a function that will only be executed up to (but not including) the + // Nth call. + function before(times, func) { + var memo; + return function() { + if (--times > 0) { + memo = func.apply(this, arguments); + } + if (times <= 1) func = null; + return memo; + }; + } + + // Returns a function that will be executed at most one time, no matter how + // often you call it. Useful for lazy initialization. + var once = partial(before, 2); + + // Returns the first key on an object that passes a truth test. + function findKey(obj, predicate, context) { + predicate = cb(predicate, context); + var _keys = keys(obj), key; + for (var i = 0, length = _keys.length; i < length; i++) { + key = _keys[i]; + if (predicate(obj[key], key, obj)) return key; + } + } + + // Internal function to generate `_.findIndex` and `_.findLastIndex`. + function createPredicateIndexFinder(dir) { + return function(array, predicate, context) { + predicate = cb(predicate, context); + var length = getLength(array); + var index = dir > 0 ? 0 : length - 1; + for (; index >= 0 && index < length; index += dir) { + if (predicate(array[index], index, array)) return index; + } + return -1; + }; + } + + // Returns the first index on an array-like that passes a truth test. + var findIndex = createPredicateIndexFinder(1); + + // Returns the last index on an array-like that passes a truth test. + var findLastIndex = createPredicateIndexFinder(-1); + + // Use a comparator function to figure out the smallest index at which + // an object should be inserted so as to maintain order. Uses binary search. + function sortedIndex(array, obj, iteratee, context) { + iteratee = cb(iteratee, context, 1); + var value = iteratee(obj); + var low = 0, high = getLength(array); + while (low < high) { + var mid = Math.floor((low + high) / 2); + if (iteratee(array[mid]) < value) low = mid + 1; else high = mid; + } + return low; + } + + // Internal function to generate the `_.indexOf` and `_.lastIndexOf` functions. + function createIndexFinder(dir, predicateFind, sortedIndex) { + return function(array, item, idx) { + var i = 0, length = getLength(array); + if (typeof idx == 'number') { + if (dir > 0) { + i = idx >= 0 ? idx : Math.max(idx + length, i); + } else { + length = idx >= 0 ? Math.min(idx + 1, length) : idx + length + 1; + } + } else if (sortedIndex && idx && length) { + idx = sortedIndex(array, item); + return array[idx] === item ? idx : -1; + } + if (item !== item) { + idx = predicateFind(slice.call(array, i, length), isNaN$1); + return idx >= 0 ? idx + i : -1; + } + for (idx = dir > 0 ? i : length - 1; idx >= 0 && idx < length; idx += dir) { + if (array[idx] === item) return idx; + } + return -1; + }; + } + + // Return the position of the first occurrence of an item in an array, + // or -1 if the item is not included in the array. + // If the array is large and already in sort order, pass `true` + // for **isSorted** to use binary search. + var indexOf = createIndexFinder(1, findIndex, sortedIndex); + + // Return the position of the last occurrence of an item in an array, + // or -1 if the item is not included in the array. + var lastIndexOf = createIndexFinder(-1, findLastIndex); + + // Return the first value which passes a truth test. + function find(obj, predicate, context) { + var keyFinder = isArrayLike(obj) ? findIndex : findKey; + var key = keyFinder(obj, predicate, context); + if (key !== void 0 && key !== -1) return obj[key]; + } + + // Convenience version of a common use case of `_.find`: getting the first + // object containing specific `key:value` pairs. + function findWhere(obj, attrs) { + return find(obj, matcher(attrs)); + } + + // The cornerstone for collection functions, an `each` + // implementation, aka `forEach`. + // Handles raw objects in addition to array-likes. Treats all + // sparse array-likes as if they were dense. + function each(obj, iteratee, context) { + iteratee = optimizeCb(iteratee, context); + var i, length; + if (isArrayLike(obj)) { + for (i = 0, length = obj.length; i < length; i++) { + iteratee(obj[i], i, obj); + } + } else { + var _keys = keys(obj); + for (i = 0, length = _keys.length; i < length; i++) { + iteratee(obj[_keys[i]], _keys[i], obj); + } + } + return obj; + } + + // Return the results of applying the iteratee to each element. + function map(obj, iteratee, context) { + iteratee = cb(iteratee, context); + var _keys = !isArrayLike(obj) && keys(obj), + length = (_keys || obj).length, + results = Array(length); + for (var index = 0; index < length; index++) { + var currentKey = _keys ? _keys[index] : index; + results[index] = iteratee(obj[currentKey], currentKey, obj); + } + return results; + } + + // Internal helper to create a reducing function, iterating left or right. + function createReduce(dir) { + // Wrap code that reassigns argument variables in a separate function than + // the one that accesses `arguments.length` to avoid a perf hit. (#1991) + var reducer = function(obj, iteratee, memo, initial) { + var _keys = !isArrayLike(obj) && keys(obj), + length = (_keys || obj).length, + index = dir > 0 ? 0 : length - 1; + if (!initial) { + memo = obj[_keys ? _keys[index] : index]; + index += dir; + } + for (; index >= 0 && index < length; index += dir) { + var currentKey = _keys ? _keys[index] : index; + memo = iteratee(memo, obj[currentKey], currentKey, obj); + } + return memo; + }; + + return function(obj, iteratee, memo, context) { + var initial = arguments.length >= 3; + return reducer(obj, optimizeCb(iteratee, context, 4), memo, initial); + }; + } + + // **Reduce** builds up a single result from a list of values, aka `inject`, + // or `foldl`. + var reduce = createReduce(1); + + // The right-associative version of reduce, also known as `foldr`. + var reduceRight = createReduce(-1); + + // Return all the elements that pass a truth test. + function filter(obj, predicate, context) { + var results = []; + predicate = cb(predicate, context); + each(obj, function(value, index, list) { + if (predicate(value, index, list)) results.push(value); + }); + return results; + } + + // Return all the elements for which a truth test fails. + function reject(obj, predicate, context) { + return filter(obj, negate(cb(predicate)), context); + } + + // Determine whether all of the elements pass a truth test. + function every(obj, predicate, context) { + predicate = cb(predicate, context); + var _keys = !isArrayLike(obj) && keys(obj), + length = (_keys || obj).length; + for (var index = 0; index < length; index++) { + var currentKey = _keys ? _keys[index] : index; + if (!predicate(obj[currentKey], currentKey, obj)) return false; + } + return true; + } + + // Determine if at least one element in the object passes a truth test. + function some(obj, predicate, context) { + predicate = cb(predicate, context); + var _keys = !isArrayLike(obj) && keys(obj), + length = (_keys || obj).length; + for (var index = 0; index < length; index++) { + var currentKey = _keys ? _keys[index] : index; + if (predicate(obj[currentKey], currentKey, obj)) return true; + } + return false; + } + + // Determine if the array or object contains a given item (using `===`). + function contains(obj, item, fromIndex, guard) { + if (!isArrayLike(obj)) obj = values(obj); + if (typeof fromIndex != 'number' || guard) fromIndex = 0; + return indexOf(obj, item, fromIndex) >= 0; + } + + // Invoke a method (with arguments) on every item in a collection. + var invoke = restArguments(function(obj, path, args) { + var contextPath, func; + if (isFunction$1(path)) { + func = path; + } else { + path = toPath(path); + contextPath = path.slice(0, -1); + path = path[path.length - 1]; + } + return map(obj, function(context) { + var method = func; + if (!method) { + if (contextPath && contextPath.length) { + context = deepGet(context, contextPath); + } + if (context == null) return void 0; + method = context[path]; + } + return method == null ? method : method.apply(context, args); + }); + }); + + // Convenience version of a common use case of `_.map`: fetching a property. + function pluck(obj, key) { + return map(obj, property(key)); + } + + // Convenience version of a common use case of `_.filter`: selecting only + // objects containing specific `key:value` pairs. + function where(obj, attrs) { + return filter(obj, matcher(attrs)); + } + + // Return the maximum element (or element-based computation). + function max(obj, iteratee, context) { + var result = -Infinity, lastComputed = -Infinity, + value, computed; + if (iteratee == null || typeof iteratee == 'number' && typeof obj[0] != 'object' && obj != null) { + obj = isArrayLike(obj) ? obj : values(obj); + for (var i = 0, length = obj.length; i < length; i++) { + value = obj[i]; + if (value != null && value > result) { + result = value; + } + } + } else { + iteratee = cb(iteratee, context); + each(obj, function(v, index, list) { + computed = iteratee(v, index, list); + if (computed > lastComputed || computed === -Infinity && result === -Infinity) { + result = v; + lastComputed = computed; + } + }); + } + return result; + } + + // Return the minimum element (or element-based computation). + function min(obj, iteratee, context) { + var result = Infinity, lastComputed = Infinity, + value, computed; + if (iteratee == null || typeof iteratee == 'number' && typeof obj[0] != 'object' && obj != null) { + obj = isArrayLike(obj) ? obj : values(obj); + for (var i = 0, length = obj.length; i < length; i++) { + value = obj[i]; + if (value != null && value < result) { + result = value; + } + } + } else { + iteratee = cb(iteratee, context); + each(obj, function(v, index, list) { + computed = iteratee(v, index, list); + if (computed < lastComputed || computed === Infinity && result === Infinity) { + result = v; + lastComputed = computed; + } + }); + } + return result; + } + + // Sample **n** random values from a collection using the modern version of the + // [Fisher-Yates shuffle](https://en.wikipedia.org/wiki/Fisher–Yates_shuffle). + // If **n** is not specified, returns a single random element. + // The internal `guard` argument allows it to work with `_.map`. + function sample(obj, n, guard) { + if (n == null || guard) { + if (!isArrayLike(obj)) obj = values(obj); + return obj[random(obj.length - 1)]; + } + var sample = isArrayLike(obj) ? clone(obj) : values(obj); + var length = getLength(sample); + n = Math.max(Math.min(n, length), 0); + var last = length - 1; + for (var index = 0; index < n; index++) { + var rand = random(index, last); + var temp = sample[index]; + sample[index] = sample[rand]; + sample[rand] = temp; + } + return sample.slice(0, n); + } + + // Shuffle a collection. + function shuffle(obj) { + return sample(obj, Infinity); + } + + // Sort the object's values by a criterion produced by an iteratee. + function sortBy(obj, iteratee, context) { + var index = 0; + iteratee = cb(iteratee, context); + return pluck(map(obj, function(value, key, list) { + return { + value: value, + index: index++, + criteria: iteratee(value, key, list) + }; + }).sort(function(left, right) { + var a = left.criteria; + var b = right.criteria; + if (a !== b) { + if (a > b || a === void 0) return 1; + if (a < b || b === void 0) return -1; + } + return left.index - right.index; + }), 'value'); + } + + // An internal function used for aggregate "group by" operations. + function group(behavior, partition) { + return function(obj, iteratee, context) { + var result = partition ? [[], []] : {}; + iteratee = cb(iteratee, context); + each(obj, function(value, index) { + var key = iteratee(value, index, obj); + behavior(result, value, key); + }); + return result; + }; + } + + // Groups the object's values by a criterion. Pass either a string attribute + // to group by, or a function that returns the criterion. + var groupBy = group(function(result, value, key) { + if (has$1(result, key)) result[key].push(value); else result[key] = [value]; + }); + + // Indexes the object's values by a criterion, similar to `_.groupBy`, but for + // when you know that your index values will be unique. + var indexBy = group(function(result, value, key) { + result[key] = value; + }); + + // Counts instances of an object that group by a certain criterion. Pass + // either a string attribute to count by, or a function that returns the + // criterion. + var countBy = group(function(result, value, key) { + if (has$1(result, key)) result[key]++; else result[key] = 1; + }); + + // Split a collection into two arrays: one whose elements all pass the given + // truth test, and one whose elements all do not pass the truth test. + var partition = group(function(result, value, pass) { + result[pass ? 0 : 1].push(value); + }, true); + + // Safely create a real, live array from anything iterable. + var reStrSymbol = /[^\ud800-\udfff]|[\ud800-\udbff][\udc00-\udfff]|[\ud800-\udfff]/g; + function toArray(obj) { + if (!obj) return []; + if (isArray(obj)) return slice.call(obj); + if (isString(obj)) { + // Keep surrogate pair characters together. + return obj.match(reStrSymbol); + } + if (isArrayLike(obj)) return map(obj, identity); + return values(obj); + } + + // Return the number of elements in a collection. + function size(obj) { + if (obj == null) return 0; + return isArrayLike(obj) ? obj.length : keys(obj).length; + } + + // Internal `_.pick` helper function to determine whether `key` is an enumerable + // property name of `obj`. + function keyInObj(value, key, obj) { + return key in obj; + } + + // Return a copy of the object only containing the allowed properties. + var pick = restArguments(function(obj, keys) { + var result = {}, iteratee = keys[0]; + if (obj == null) return result; + if (isFunction$1(iteratee)) { + if (keys.length > 1) iteratee = optimizeCb(iteratee, keys[1]); + keys = allKeys(obj); + } else { + iteratee = keyInObj; + keys = flatten$1(keys, false, false); + obj = Object(obj); + } + for (var i = 0, length = keys.length; i < length; i++) { + var key = keys[i]; + var value = obj[key]; + if (iteratee(value, key, obj)) result[key] = value; + } + return result; + }); + + // Return a copy of the object without the disallowed properties. + var omit = restArguments(function(obj, keys) { + var iteratee = keys[0], context; + if (isFunction$1(iteratee)) { + iteratee = negate(iteratee); + if (keys.length > 1) context = keys[1]; + } else { + keys = map(flatten$1(keys, false, false), String); + iteratee = function(value, key) { + return !contains(keys, key); + }; + } + return pick(obj, iteratee, context); + }); + + // Returns everything but the last entry of the array. Especially useful on + // the arguments object. Passing **n** will return all the values in + // the array, excluding the last N. + function initial(array, n, guard) { + return slice.call(array, 0, Math.max(0, array.length - (n == null || guard ? 1 : n))); + } + + // Get the first element of an array. Passing **n** will return the first N + // values in the array. The **guard** check allows it to work with `_.map`. + function first(array, n, guard) { + if (array == null || array.length < 1) return n == null || guard ? void 0 : []; + if (n == null || guard) return array[0]; + return initial(array, array.length - n); + } + + // Returns everything but the first entry of the `array`. Especially useful on + // the `arguments` object. Passing an **n** will return the rest N values in the + // `array`. + function rest(array, n, guard) { + return slice.call(array, n == null || guard ? 1 : n); + } + + // Get the last element of an array. Passing **n** will return the last N + // values in the array. + function last(array, n, guard) { + if (array == null || array.length < 1) return n == null || guard ? void 0 : []; + if (n == null || guard) return array[array.length - 1]; + return rest(array, Math.max(0, array.length - n)); + } + + // Trim out all falsy values from an array. + function compact(array) { + return filter(array, Boolean); + } + + // Flatten out an array, either recursively (by default), or up to `depth`. + // Passing `true` or `false` as `depth` means `1` or `Infinity`, respectively. + function flatten(array, depth) { + return flatten$1(array, depth, false); + } + + // Take the difference between one array and a number of other arrays. + // Only the elements present in just the first array will remain. + var difference = restArguments(function(array, rest) { + rest = flatten$1(rest, true, true); + return filter(array, function(value){ + return !contains(rest, value); + }); + }); + + // Return a version of the array that does not contain the specified value(s). + var without = restArguments(function(array, otherArrays) { + return difference(array, otherArrays); + }); + + // Produce a duplicate-free version of the array. If the array has already + // been sorted, you have the option of using a faster algorithm. + // The faster algorithm will not work with an iteratee if the iteratee + // is not a one-to-one function, so providing an iteratee will disable + // the faster algorithm. + function uniq(array, isSorted, iteratee, context) { + if (!isBoolean(isSorted)) { + context = iteratee; + iteratee = isSorted; + isSorted = false; + } + if (iteratee != null) iteratee = cb(iteratee, context); + var result = []; + var seen = []; + for (var i = 0, length = getLength(array); i < length; i++) { + var value = array[i], + computed = iteratee ? iteratee(value, i, array) : value; + if (isSorted && !iteratee) { + if (!i || seen !== computed) result.push(value); + seen = computed; + } else if (iteratee) { + if (!contains(seen, computed)) { + seen.push(computed); + result.push(value); + } + } else if (!contains(result, value)) { + result.push(value); + } + } + return result; + } + + // Produce an array that contains the union: each distinct element from all of + // the passed-in arrays. + var union = restArguments(function(arrays) { + return uniq(flatten$1(arrays, true, true)); + }); + + // Produce an array that contains every item shared between all the + // passed-in arrays. + function intersection(array) { + var result = []; + var argsLength = arguments.length; + for (var i = 0, length = getLength(array); i < length; i++) { + var item = array[i]; + if (contains(result, item)) continue; + var j; + for (j = 1; j < argsLength; j++) { + if (!contains(arguments[j], item)) break; + } + if (j === argsLength) result.push(item); + } + return result; + } + + // Complement of zip. Unzip accepts an array of arrays and groups + // each array's elements on shared indices. + function unzip(array) { + var length = array && max(array, getLength).length || 0; + var result = Array(length); + + for (var index = 0; index < length; index++) { + result[index] = pluck(array, index); + } + return result; + } + + // Zip together multiple lists into a single array -- elements that share + // an index go together. + var zip = restArguments(unzip); + + // Converts lists into objects. Pass either a single array of `[key, value]` + // pairs, or two parallel arrays of the same length -- one of keys, and one of + // the corresponding values. Passing by pairs is the reverse of `_.pairs`. + function object(list, values) { + var result = {}; + for (var i = 0, length = getLength(list); i < length; i++) { + if (values) { + result[list[i]] = values[i]; + } else { + result[list[i][0]] = list[i][1]; + } + } + return result; + } + + // Generate an integer Array containing an arithmetic progression. A port of + // the native Python `range()` function. See + // [the Python documentation](https://docs.python.org/library/functions.html#range). + function range(start, stop, step) { + if (stop == null) { + stop = start || 0; + start = 0; + } + if (!step) { + step = stop < start ? -1 : 1; + } + + var length = Math.max(Math.ceil((stop - start) / step), 0); + var range = Array(length); + + for (var idx = 0; idx < length; idx++, start += step) { + range[idx] = start; + } + + return range; + } + + // Chunk a single array into multiple arrays, each containing `count` or fewer + // items. + function chunk(array, count) { + if (count == null || count < 1) return []; + var result = []; + var i = 0, length = array.length; + while (i < length) { + result.push(slice.call(array, i, i += count)); + } + return result; + } + + // Helper function to continue chaining intermediate results. + function chainResult(instance, obj) { + return instance._chain ? _$1(obj).chain() : obj; + } + + // Add your own custom functions to the Underscore object. + function mixin(obj) { + each(functions(obj), function(name) { + var func = _$1[name] = obj[name]; + _$1.prototype[name] = function() { + var args = [this._wrapped]; + push.apply(args, arguments); + return chainResult(this, func.apply(_$1, args)); + }; + }); + return _$1; + } + + // Add all mutator `Array` functions to the wrapper. + each(['pop', 'push', 'reverse', 'shift', 'sort', 'splice', 'unshift'], function(name) { + var method = ArrayProto[name]; + _$1.prototype[name] = function() { + var obj = this._wrapped; + if (obj != null) { + method.apply(obj, arguments); + if ((name === 'shift' || name === 'splice') && obj.length === 0) { + delete obj[0]; + } + } + return chainResult(this, obj); + }; + }); + + // Add all accessor `Array` functions to the wrapper. + each(['concat', 'join', 'slice'], function(name) { + var method = ArrayProto[name]; + _$1.prototype[name] = function() { + var obj = this._wrapped; + if (obj != null) obj = method.apply(obj, arguments); + return chainResult(this, obj); + }; + }); + + // Named Exports + + var allExports = { + __proto__: null, + VERSION: VERSION, + restArguments: restArguments, + isObject: isObject, + isNull: isNull, + isUndefined: isUndefined, + isBoolean: isBoolean, + isElement: isElement, + isString: isString, + isNumber: isNumber, + isDate: isDate, + isRegExp: isRegExp, + isError: isError, + isSymbol: isSymbol, + isArrayBuffer: isArrayBuffer, + isDataView: isDataView$1, + isArray: isArray, + isFunction: isFunction$1, + isArguments: isArguments$1, + isFinite: isFinite$1, + isNaN: isNaN$1, + isTypedArray: isTypedArray$1, + isEmpty: isEmpty, + isMatch: isMatch, + isEqual: isEqual, + isMap: isMap, + isWeakMap: isWeakMap, + isSet: isSet, + isWeakSet: isWeakSet, + keys: keys, + allKeys: allKeys, + values: values, + pairs: pairs, + invert: invert, + functions: functions, + methods: functions, + extend: extend, + extendOwn: extendOwn, + assign: extendOwn, + defaults: defaults, + create: create, + clone: clone, + tap: tap, + get: get, + has: has, + mapObject: mapObject, + identity: identity, + constant: constant, + noop: noop, + toPath: toPath$1, + property: property, + propertyOf: propertyOf, + matcher: matcher, + matches: matcher, + times: times, + random: random, + now: now, + escape: _escape, + unescape: _unescape, + templateSettings: templateSettings, + template: template, + result: result, + uniqueId: uniqueId, + chain: chain, + iteratee: iteratee, + partial: partial, + bind: bind, + bindAll: bindAll, + memoize: memoize, + delay: delay, + defer: defer, + throttle: throttle, + debounce: debounce, + wrap: wrap, + negate: negate, + compose: compose, + after: after, + before: before, + once: once, + findKey: findKey, + findIndex: findIndex, + findLastIndex: findLastIndex, + sortedIndex: sortedIndex, + indexOf: indexOf, + lastIndexOf: lastIndexOf, + find: find, + detect: find, + findWhere: findWhere, + each: each, + forEach: each, + map: map, + collect: map, + reduce: reduce, + foldl: reduce, + inject: reduce, + reduceRight: reduceRight, + foldr: reduceRight, + filter: filter, + select: filter, + reject: reject, + every: every, + all: every, + some: some, + any: some, + contains: contains, + includes: contains, + include: contains, + invoke: invoke, + pluck: pluck, + where: where, + max: max, + min: min, + shuffle: shuffle, + sample: sample, + sortBy: sortBy, + groupBy: groupBy, + indexBy: indexBy, + countBy: countBy, + partition: partition, + toArray: toArray, + size: size, + pick: pick, + omit: omit, + first: first, + head: first, + take: first, + initial: initial, + last: last, + rest: rest, + tail: rest, + drop: rest, + compact: compact, + flatten: flatten, + without: without, + uniq: uniq, + unique: uniq, + union: union, + intersection: intersection, + difference: difference, + unzip: unzip, + transpose: unzip, + zip: zip, + object: object, + range: range, + chunk: chunk, + mixin: mixin, + 'default': _$1 + }; + + // Default Export + + // Add all of the Underscore functions to the wrapper object. + var _ = mixin(allExports); + // Legacy Node.js API. + _._ = _; + + return _; + + }))); + //# sourceMappingURL=underscore-umd.js.map
html/_static/underscore.js (new)
- @@ -0,0 +1,6 @@
 + !function(n,r){"object"==typeof exports&&"undefined"!=typeof module?module.exports=r():"function"==typeof define&&define.amd?define("underscore",r):(n="undefined"!=typeof globalThis?globalThis:n||self,function(){var t=n._,e=n._=r();e.noConflict=function(){return n._=t,e}}())}(this,(function(){ + // Underscore.js 1.13.1 + // https://underscorejs.org + // (c) 2009-2021 Jeremy Ashkenas, Julian Gonggrijp, and DocumentCloud and Investigative Reporters & Editors + // Underscore may be freely distributed under the MIT license. + var n="1.13.1",r="object"==typeof self&&self.self===self&&self||"object"==typeof global&&global.global===global&&global||Function("return this")()||{},t=Array.prototype,e=Object.prototype,u="undefined"!=typeof Symbol?Symbol.prototype:null,o=t.push,i=t.slice,a=e.toString,f=e.hasOwnProperty,c="undefined"!=typeof ArrayBuffer,l="undefined"!=typeof DataView,s=Array.isArray,p=Object.keys,v=Object.create,h=c&&ArrayBuffer.isView,y=isNaN,d=isFinite,g=!{toString:null}.propertyIsEnumerable("toString"),b=["valueOf","isPrototypeOf","toString","propertyIsEnumerable","hasOwnProperty","toLocaleString"],m=Math.pow(2,53)-1;function j(n,r){return r=null==r?n.length-1:+r,function(){for(var t=Math.max(arguments.length-r,0),e=Array(t),u=0;u<t;u++)e[u]=arguments[u+r];switch(r){case 0:return n.call(this,e);case 1:return n.call(this,arguments[0],e);case 2:return n.call(this,arguments[0],arguments[1],e)}var o=Array(r+1);for(u=0;u<r;u++)o[u]=arguments[u];return o[r]=e,n.apply(this,o)}}function _(n){var r=typeof n;return"function"===r||"object"===r&&!!n}function w(n){return void 0===n}function A(n){return!0===n||!1===n||"[object Boolean]"===a.call(n)}function x(n){var r="[object "+n+"]";return function(n){return a.call(n)===r}}var S=x("String"),O=x("Number"),M=x("Date"),E=x("RegExp"),B=x("Error"),N=x("Symbol"),I=x("ArrayBuffer"),T=x("Function"),k=r.document&&r.document.childNodes;"function"!=typeof/./&&"object"!=typeof Int8Array&&"function"!=typeof k&&(T=function(n){return"function"==typeof n||!1});var D=T,R=x("Object"),F=l&&R(new DataView(new ArrayBuffer(8))),V="undefined"!=typeof Map&&R(new Map),P=x("DataView");var q=F?function(n){return null!=n&&D(n.getInt8)&&I(n.buffer)}:P,U=s||x("Array");function W(n,r){return null!=n&&f.call(n,r)}var z=x("Arguments");!function(){z(arguments)||(z=function(n){return W(n,"callee")})}();var L=z;function $(n){return O(n)&&y(n)}function C(n){return function(){return n}}function K(n){return function(r){var t=n(r);return"number"==typeof t&&t>=0&&t<=m}}function J(n){return function(r){return null==r?void 0:r[n]}}var G=J("byteLength"),H=K(G),Q=/\[object ((I|Ui)nt(8|16|32)|Float(32|64)|Uint8Clamped|Big(I|Ui)nt64)Array\]/;var X=c?function(n){return h?h(n)&&!q(n):H(n)&&Q.test(a.call(n))}:C(!1),Y=J("length");function Z(n,r){r=function(n){for(var r={},t=n.length,e=0;e<t;++e)r[n[e]]=!0;return{contains:function(n){return r[n]},push:function(t){return r[t]=!0,n.push(t)}}}(r);var t=b.length,u=n.constructor,o=D(u)&&u.prototype||e,i="constructor";for(W(n,i)&&!r.contains(i)&&r.push(i);t--;)(i=b[t])in n&&n[i]!==o[i]&&!r.contains(i)&&r.push(i)}function nn(n){if(!_(n))return[];if(p)return p(n);var r=[];for(var t in n)W(n,t)&&r.push(t);return g&&Z(n,r),r}function rn(n,r){var t=nn(r),e=t.length;if(null==n)return!e;for(var u=Object(n),o=0;o<e;o++){var i=t[o];if(r[i]!==u[i]||!(i in u))return!1}return!0}function tn(n){return n instanceof tn?n:this instanceof tn?void(this._wrapped=n):new tn(n)}function en(n){return new Uint8Array(n.buffer||n,n.byteOffset||0,G(n))}tn.VERSION=n,tn.prototype.value=function(){return this._wrapped},tn.prototype.valueOf=tn.prototype.toJSON=tn.prototype.value,tn.prototype.toString=function(){return String(this._wrapped)};var un="[object DataView]";function on(n,r,t,e){if(n===r)return 0!==n||1/n==1/r;if(null==n||null==r)return!1;if(n!=n)return r!=r;var o=typeof n;return("function"===o||"object"===o||"object"==typeof r)&&function n(r,t,e,o){r instanceof tn&&(r=r._wrapped);t instanceof tn&&(t=t._wrapped);var i=a.call(r);if(i!==a.call(t))return!1;if(F&&"[object Object]"==i&&q(r)){if(!q(t))return!1;i=un}switch(i){case"[object RegExp]":case"[object String]":return""+r==""+t;case"[object Number]":return+r!=+r?+t!=+t:0==+r?1/+r==1/t:+r==+t;case"[object Date]":case"[object Boolean]":return+r==+t;case"[object Symbol]":return u.valueOf.call(r)===u.valueOf.call(t);case"[object ArrayBuffer]":case un:return n(en(r),en(t),e,o)}var f="[object Array]"===i;if(!f&&X(r)){if(G(r)!==G(t))return!1;if(r.buffer===t.buffer&&r.byteOffset===t.byteOffset)return!0;f=!0}if(!f){if("object"!=typeof r||"object"!=typeof t)return!1;var c=r.constructor,l=t.constructor;if(c!==l&&!(D(c)&&c instanceof c&&D(l)&&l instanceof l)&&"constructor"in r&&"constructor"in t)return!1}o=o||[];var s=(e=e||[]).length;for(;s--;)if(e[s]===r)return o[s]===t;if(e.push(r),o.push(t),f){if((s=r.length)!==t.length)return!1;for(;s--;)if(!on(r[s],t[s],e,o))return!1}else{var p,v=nn(r);if(s=v.length,nn(t).length!==s)return!1;for(;s--;)if(p=v[s],!W(t,p)||!on(r[p],t[p],e,o))return!1}return e.pop(),o.pop(),!0}(n,r,t,e)}function an(n){if(!_(n))return[];var r=[];for(var t in n)r.push(t);return g&&Z(n,r),r}function fn(n){var r=Y(n);return function(t){if(null==t)return!1;var e=an(t);if(Y(e))return!1;for(var u=0;u<r;u++)if(!D(t[n[u]]))return!1;return n!==hn||!D(t[cn])}}var cn="forEach",ln="has",sn=["clear","delete"],pn=["get",ln,"set"],vn=sn.concat(cn,pn),hn=sn.concat(pn),yn=["add"].concat(sn,cn,ln),dn=V?fn(vn):x("Map"),gn=V?fn(hn):x("WeakMap"),bn=V?fn(yn):x("Set"),mn=x("WeakSet");function jn(n){for(var r=nn(n),t=r.length,e=Array(t),u=0;u<t;u++)e[u]=n[r[u]];return e}function _n(n){for(var r={},t=nn(n),e=0,u=t.length;e<u;e++)r[n[t[e]]]=t[e];return r}function wn(n){var r=[];for(var t in n)D(n[t])&&r.push(t);return r.sort()}function An(n,r){return function(t){var e=arguments.length;if(r&&(t=Object(t)),e<2||null==t)return t;for(var u=1;u<e;u++)for(var o=arguments[u],i=n(o),a=i.length,f=0;f<a;f++){var c=i[f];r&&void 0!==t[c]||(t[c]=o[c])}return t}}var xn=An(an),Sn=An(nn),On=An(an,!0);function Mn(n){if(!_(n))return{};if(v)return v(n);var r=function(){};r.prototype=n;var t=new r;return r.prototype=null,t}function En(n){return _(n)?U(n)?n.slice():xn({},n):n}function Bn(n){return U(n)?n:[n]}function Nn(n){return tn.toPath(n)}function In(n,r){for(var t=r.length,e=0;e<t;e++){if(null==n)return;n=n[r[e]]}return t?n:void 0}function Tn(n,r,t){var e=In(n,Nn(r));return w(e)?t:e}function kn(n){return n}function Dn(n){return n=Sn({},n),function(r){return rn(r,n)}}function Rn(n){return n=Nn(n),function(r){return In(r,n)}}function Fn(n,r,t){if(void 0===r)return n;switch(null==t?3:t){case 1:return function(t){return n.call(r,t)};case 3:return function(t,e,u){return n.call(r,t,e,u)};case 4:return function(t,e,u,o){return n.call(r,t,e,u,o)}}return function(){return n.apply(r,arguments)}}function Vn(n,r,t){return null==n?kn:D(n)?Fn(n,r,t):_(n)&&!U(n)?Dn(n):Rn(n)}function Pn(n,r){return Vn(n,r,1/0)}function qn(n,r,t){return tn.iteratee!==Pn?tn.iteratee(n,r):Vn(n,r,t)}function Un(){}function Wn(n,r){return null==r&&(r=n,n=0),n+Math.floor(Math.random()*(r-n+1))}tn.toPath=Bn,tn.iteratee=Pn;var zn=Date.now||function(){return(new Date).getTime()};function Ln(n){var r=function(r){return n[r]},t="(?:"+nn(n).join("|")+")",e=RegExp(t),u=RegExp(t,"g");return function(n){return n=null==n?"":""+n,e.test(n)?n.replace(u,r):n}}var $n={"&":"&","<":"<",">":">",'"':""","'":"'","`":"`"},Cn=Ln($n),Kn=Ln(_n($n)),Jn=tn.templateSettings={evaluate:/<%([\s\S]+?)%>/g,interpolate:/<%=([\s\S]+?)%>/g,escape:/<%-([\s\S]+?)%>/g},Gn=/(.)^/,Hn={"'":"'","\\":"\\","\r":"r","\n":"n","\u2028":"u2028","\u2029":"u2029"},Qn=/\\|'|\r|\n|\u2028|\u2029/g;function Xn(n){return"\\"+Hn[n]}var Yn=/^\s*(\w|\$)+\s*$/;var Zn=0;function nr(n,r,t,e,u){if(!(e instanceof r))return n.apply(t,u);var o=Mn(n.prototype),i=n.apply(o,u);return _(i)?i:o}var rr=j((function(n,r){var t=rr.placeholder,e=function(){for(var u=0,o=r.length,i=Array(o),a=0;a<o;a++)i[a]=r[a]===t?arguments[u++]:r[a];for(;u<arguments.length;)i.push(arguments[u++]);return nr(n,e,this,this,i)};return e}));rr.placeholder=tn;var tr=j((function(n,r,t){if(!D(n))throw new TypeError("Bind must be called on a function");var e=j((function(u){return nr(n,e,r,this,t.concat(u))}));return e})),er=K(Y);function ur(n,r,t,e){if(e=e||[],r||0===r){if(r<=0)return e.concat(n)}else r=1/0;for(var u=e.length,o=0,i=Y(n);o<i;o++){var a=n[o];if(er(a)&&(U(a)||L(a)))if(r>1)ur(a,r-1,t,e),u=e.length;else for(var f=0,c=a.length;f<c;)e[u++]=a[f++];else t||(e[u++]=a)}return e}var or=j((function(n,r){var t=(r=ur(r,!1,!1)).length;if(t<1)throw new Error("bindAll must be passed function names");for(;t--;){var e=r[t];n[e]=tr(n[e],n)}return n}));var ir=j((function(n,r,t){return setTimeout((function(){return n.apply(null,t)}),r)})),ar=rr(ir,tn,1);function fr(n){return function(){return!n.apply(this,arguments)}}function cr(n,r){var t;return function(){return--n>0&&(t=r.apply(this,arguments)),n<=1&&(r=null),t}}var lr=rr(cr,2);function sr(n,r,t){r=qn(r,t);for(var e,u=nn(n),o=0,i=u.length;o<i;o++)if(r(n[e=u[o]],e,n))return e}function pr(n){return function(r,t,e){t=qn(t,e);for(var u=Y(r),o=n>0?0:u-1;o>=0&&o<u;o+=n)if(t(r[o],o,r))return o;return-1}}var vr=pr(1),hr=pr(-1);function yr(n,r,t,e){for(var u=(t=qn(t,e,1))(r),o=0,i=Y(n);o<i;){var a=Math.floor((o+i)/2);t(n[a])<u?o=a+1:i=a}return o}function dr(n,r,t){return function(e,u,o){var a=0,f=Y(e);if("number"==typeof o)n>0?a=o>=0?o:Math.max(o+f,a):f=o>=0?Math.min(o+1,f):o+f+1;else if(t&&o&&f)return e[o=t(e,u)]===u?o:-1;if(u!=u)return(o=r(i.call(e,a,f),$))>=0?o+a:-1;for(o=n>0?a:f-1;o>=0&&o<f;o+=n)if(e[o]===u)return o;return-1}}var gr=dr(1,vr,yr),br=dr(-1,hr);function mr(n,r,t){var e=(er(n)?vr:sr)(n,r,t);if(void 0!==e&&-1!==e)return n[e]}function jr(n,r,t){var e,u;if(r=Fn(r,t),er(n))for(e=0,u=n.length;e<u;e++)r(n[e],e,n);else{var o=nn(n);for(e=0,u=o.length;e<u;e++)r(n[o[e]],o[e],n)}return n}function _r(n,r,t){r=qn(r,t);for(var e=!er(n)&&nn(n),u=(e||n).length,o=Array(u),i=0;i<u;i++){var a=e?e[i]:i;o[i]=r(n[a],a,n)}return o}function wr(n){var r=function(r,t,e,u){var o=!er(r)&&nn(r),i=(o||r).length,a=n>0?0:i-1;for(u||(e=r[o?o[a]:a],a+=n);a>=0&&a<i;a+=n){var f=o?o[a]:a;e=t(e,r[f],f,r)}return e};return function(n,t,e,u){var o=arguments.length>=3;return r(n,Fn(t,u,4),e,o)}}var Ar=wr(1),xr=wr(-1);function Sr(n,r,t){var e=[];return r=qn(r,t),jr(n,(function(n,t,u){r(n,t,u)&&e.push(n)})),e}function Or(n,r,t){r=qn(r,t);for(var e=!er(n)&&nn(n),u=(e||n).length,o=0;o<u;o++){var i=e?e[o]:o;if(!r(n[i],i,n))return!1}return!0}function Mr(n,r,t){r=qn(r,t);for(var e=!er(n)&&nn(n),u=(e||n).length,o=0;o<u;o++){var i=e?e[o]:o;if(r(n[i],i,n))return!0}return!1}function Er(n,r,t,e){return er(n)||(n=jn(n)),("number"!=typeof t||e)&&(t=0),gr(n,r,t)>=0}var Br=j((function(n,r,t){var e,u;return D(r)?u=r:(r=Nn(r),e=r.slice(0,-1),r=r[r.length-1]),_r(n,(function(n){var o=u;if(!o){if(e&&e.length&&(n=In(n,e)),null==n)return;o=n[r]}return null==o?o:o.apply(n,t)}))}));function Nr(n,r){return _r(n,Rn(r))}function Ir(n,r,t){var e,u,o=-1/0,i=-1/0;if(null==r||"number"==typeof r&&"object"!=typeof n[0]&&null!=n)for(var a=0,f=(n=er(n)?n:jn(n)).length;a<f;a++)null!=(e=n[a])&&e>o&&(o=e);else r=qn(r,t),jr(n,(function(n,t,e){((u=r(n,t,e))>i||u===-1/0&&o===-1/0)&&(o=n,i=u)}));return o}function Tr(n,r,t){if(null==r||t)return er(n)||(n=jn(n)),n[Wn(n.length-1)];var e=er(n)?En(n):jn(n),u=Y(e);r=Math.max(Math.min(r,u),0);for(var o=u-1,i=0;i<r;i++){var a=Wn(i,o),f=e[i];e[i]=e[a],e[a]=f}return e.slice(0,r)}function kr(n,r){return function(t,e,u){var o=r?[[],[]]:{};return e=qn(e,u),jr(t,(function(r,u){var i=e(r,u,t);n(o,r,i)})),o}}var Dr=kr((function(n,r,t){W(n,t)?n[t].push(r):n[t]=[r]})),Rr=kr((function(n,r,t){n[t]=r})),Fr=kr((function(n,r,t){W(n,t)?n[t]++:n[t]=1})),Vr=kr((function(n,r,t){n[t?0:1].push(r)}),!0),Pr=/[^\ud800-\udfff]|[\ud800-\udbff][\udc00-\udfff]|[\ud800-\udfff]/g;function qr(n,r,t){return r in t}var Ur=j((function(n,r){var t={},e=r[0];if(null==n)return t;D(e)?(r.length>1&&(e=Fn(e,r[1])),r=an(n)):(e=qr,r=ur(r,!1,!1),n=Object(n));for(var u=0,o=r.length;u<o;u++){var i=r[u],a=n[i];e(a,i,n)&&(t[i]=a)}return t})),Wr=j((function(n,r){var t,e=r[0];return D(e)?(e=fr(e),r.length>1&&(t=r[1])):(r=_r(ur(r,!1,!1),String),e=function(n,t){return!Er(r,t)}),Ur(n,e,t)}));function zr(n,r,t){return i.call(n,0,Math.max(0,n.length-(null==r||t?1:r)))}function Lr(n,r,t){return null==n||n.length<1?null==r||t?void 0:[]:null==r||t?n[0]:zr(n,n.length-r)}function $r(n,r,t){return i.call(n,null==r||t?1:r)}var Cr=j((function(n,r){return r=ur(r,!0,!0),Sr(n,(function(n){return!Er(r,n)}))})),Kr=j((function(n,r){return Cr(n,r)}));function Jr(n,r,t,e){A(r)||(e=t,t=r,r=!1),null!=t&&(t=qn(t,e));for(var u=[],o=[],i=0,a=Y(n);i<a;i++){var f=n[i],c=t?t(f,i,n):f;r&&!t?(i&&o===c||u.push(f),o=c):t?Er(o,c)||(o.push(c),u.push(f)):Er(u,f)||u.push(f)}return u}var Gr=j((function(n){return Jr(ur(n,!0,!0))}));function Hr(n){for(var r=n&&Ir(n,Y).length||0,t=Array(r),e=0;e<r;e++)t[e]=Nr(n,e);return t}var Qr=j(Hr);function Xr(n,r){return n._chain?tn(r).chain():r}function Yr(n){return jr(wn(n),(function(r){var t=tn[r]=n[r];tn.prototype[r]=function(){var n=[this._wrapped];return o.apply(n,arguments),Xr(this,t.apply(tn,n))}})),tn}jr(["pop","push","reverse","shift","sort","splice","unshift"],(function(n){var r=t[n];tn.prototype[n]=function(){var t=this._wrapped;return null!=t&&(r.apply(t,arguments),"shift"!==n&&"splice"!==n||0!==t.length||delete t[0]),Xr(this,t)}})),jr(["concat","join","slice"],(function(n){var r=t[n];tn.prototype[n]=function(){var n=this._wrapped;return null!=n&&(n=r.apply(n,arguments)),Xr(this,n)}}));var Zr=Yr({__proto__:null,VERSION:n,restArguments:j,isObject:_,isNull:function(n){return null===n},isUndefined:w,isBoolean:A,isElement:function(n){return!(!n||1!==n.nodeType)},isString:S,isNumber:O,isDate:M,isRegExp:E,isError:B,isSymbol:N,isArrayBuffer:I,isDataView:q,isArray:U,isFunction:D,isArguments:L,isFinite:function(n){return!N(n)&&d(n)&&!isNaN(parseFloat(n))},isNaN:$,isTypedArray:X,isEmpty:function(n){if(null==n)return!0;var r=Y(n);return"number"==typeof r&&(U(n)||S(n)||L(n))?0===r:0===Y(nn(n))},isMatch:rn,isEqual:function(n,r){return on(n,r)},isMap:dn,isWeakMap:gn,isSet:bn,isWeakSet:mn,keys:nn,allKeys:an,values:jn,pairs:function(n){for(var r=nn(n),t=r.length,e=Array(t),u=0;u<t;u++)e[u]=[r[u],n[r[u]]];return e},invert:_n,functions:wn,methods:wn,extend:xn,extendOwn:Sn,assign:Sn,defaults:On,create:function(n,r){var t=Mn(n);return r&&Sn(t,r),t},clone:En,tap:function(n,r){return r(n),n},get:Tn,has:function(n,r){for(var t=(r=Nn(r)).length,e=0;e<t;e++){var u=r[e];if(!W(n,u))return!1;n=n[u]}return!!t},mapObject:function(n,r,t){r=qn(r,t);for(var e=nn(n),u=e.length,o={},i=0;i<u;i++){var a=e[i];o[a]=r(n[a],a,n)}return o},identity:kn,constant:C,noop:Un,toPath:Bn,property:Rn,propertyOf:function(n){return null==n?Un:function(r){return Tn(n,r)}},matcher:Dn,matches:Dn,times:function(n,r,t){var e=Array(Math.max(0,n));r=Fn(r,t,1);for(var u=0;u<n;u++)e[u]=r(u);return e},random:Wn,now:zn,escape:Cn,unescape:Kn,templateSettings:Jn,template:function(n,r,t){!r&&t&&(r=t),r=On({},r,tn.templateSettings);var e=RegExp([(r.escape||Gn).source,(r.interpolate||Gn).source,(r.evaluate||Gn).source].join("|")+"|$","g"),u=0,o="__p+='";n.replace(e,(function(r,t,e,i,a){return o+=n.slice(u,a).replace(Qn,Xn),u=a+r.length,t?o+="'+\n((__t=("+t+"))==null?'':_.escape(__t))+\n'":e?o+="'+\n((__t=("+e+"))==null?'':__t)+\n'":i&&(o+="';\n"+i+"\n__p+='"),r})),o+="';\n";var i,a=r.variable;if(a){if(!Yn.test(a))throw new Error("variable is not a bare identifier: "+a)}else o="with(obj||{}){\n"+o+"}\n",a="obj";o="var __t,__p='',__j=Array.prototype.join,"+"print=function(){__p+=__j.call(arguments,'');};\n"+o+"return __p;\n";try{i=new Function(a,"_",o)}catch(n){throw n.source=o,n}var f=function(n){return i.call(this,n,tn)};return f.source="function("+a+"){\n"+o+"}",f},result:function(n,r,t){var e=(r=Nn(r)).length;if(!e)return D(t)?t.call(n):t;for(var u=0;u<e;u++){var o=null==n?void 0:n[r[u]];void 0===o&&(o=t,u=e),n=D(o)?o.call(n):o}return n},uniqueId:function(n){var r=++Zn+"";return n?n+r:r},chain:function(n){var r=tn(n);return r._chain=!0,r},iteratee:Pn,partial:rr,bind:tr,bindAll:or,memoize:function(n,r){var t=function(e){var u=t.cache,o=""+(r?r.apply(this,arguments):e);return W(u,o)||(u[o]=n.apply(this,arguments)),u[o]};return t.cache={},t},delay:ir,defer:ar,throttle:function(n,r,t){var e,u,o,i,a=0;t||(t={});var f=function(){a=!1===t.leading?0:zn(),e=null,i=n.apply(u,o),e||(u=o=null)},c=function(){var c=zn();a||!1!==t.leading||(a=c);var l=r-(c-a);return u=this,o=arguments,l<=0||l>r?(e&&(clearTimeout(e),e=null),a=c,i=n.apply(u,o),e||(u=o=null)):e||!1===t.trailing||(e=setTimeout(f,l)),i};return c.cancel=function(){clearTimeout(e),a=0,e=u=o=null},c},debounce:function(n,r,t){var e,u,o,i,a,f=function(){var c=zn()-u;r>c?e=setTimeout(f,r-c):(e=null,t||(i=n.apply(a,o)),e||(o=a=null))},c=j((function(c){return a=this,o=c,u=zn(),e||(e=setTimeout(f,r),t&&(i=n.apply(a,o))),i}));return c.cancel=function(){clearTimeout(e),e=o=a=null},c},wrap:function(n,r){return rr(r,n)},negate:fr,compose:function(){var n=arguments,r=n.length-1;return function(){for(var t=r,e=n[r].apply(this,arguments);t--;)e=n[t].call(this,e);return e}},after:function(n,r){return function(){if(--n<1)return r.apply(this,arguments)}},before:cr,once:lr,findKey:sr,findIndex:vr,findLastIndex:hr,sortedIndex:yr,indexOf:gr,lastIndexOf:br,find:mr,detect:mr,findWhere:function(n,r){return mr(n,Dn(r))},each:jr,forEach:jr,map:_r,collect:_r,reduce:Ar,foldl:Ar,inject:Ar,reduceRight:xr,foldr:xr,filter:Sr,select:Sr,reject:function(n,r,t){return Sr(n,fr(qn(r)),t)},every:Or,all:Or,some:Mr,any:Mr,contains:Er,includes:Er,include:Er,invoke:Br,pluck:Nr,where:function(n,r){return Sr(n,Dn(r))},max:Ir,min:function(n,r,t){var e,u,o=1/0,i=1/0;if(null==r||"number"==typeof r&&"object"!=typeof n[0]&&null!=n)for(var a=0,f=(n=er(n)?n:jn(n)).length;a<f;a++)null!=(e=n[a])&&e<o&&(o=e);else r=qn(r,t),jr(n,(function(n,t,e){((u=r(n,t,e))<i||u===1/0&&o===1/0)&&(o=n,i=u)}));return o},shuffle:function(n){return Tr(n,1/0)},sample:Tr,sortBy:function(n,r,t){var e=0;return r=qn(r,t),Nr(_r(n,(function(n,t,u){return{value:n,index:e++,criteria:r(n,t,u)}})).sort((function(n,r){var t=n.criteria,e=r.criteria;if(t!==e){if(t>e||void 0===t)return 1;if(t<e||void 0===e)return-1}return n.index-r.index})),"value")},groupBy:Dr,indexBy:Rr,countBy:Fr,partition:Vr,toArray:function(n){return n?U(n)?i.call(n):S(n)?n.match(Pr):er(n)?_r(n,kn):jn(n):[]},size:function(n){return null==n?0:er(n)?n.length:nn(n).length},pick:Ur,omit:Wr,first:Lr,head:Lr,take:Lr,initial:zr,last:function(n,r,t){return null==n||n.length<1?null==r||t?void 0:[]:null==r||t?n[n.length-1]:$r(n,Math.max(0,n.length-r))},rest:$r,tail:$r,drop:$r,compact:function(n){return Sr(n,Boolean)},flatten:function(n,r){return ur(n,r,!1)},without:Kr,uniq:Jr,unique:Jr,union:Gr,intersection:function(n){for(var r=[],t=arguments.length,e=0,u=Y(n);e<u;e++){var o=n[e];if(!Er(r,o)){var i;for(i=1;i<t&&Er(arguments[i],o);i++);i===t&&r.push(o)}}return r},difference:Cr,unzip:Hr,transpose:Hr,zip:Qr,object:function(n,r){for(var t={},e=0,u=Y(n);e<u;e++)r?t[n[e]]=r[e]:t[n[e][0]]=n[e][1];return t},range:function(n,r,t){null==r&&(r=n||0,n=0),t||(t=r<n?-1:1);for(var e=Math.max(Math.ceil((r-n)/t),0),u=Array(e),o=0;o<e;o++,n+=t)u[o]=n;return u},chunk:function(n,r){if(null==r||r<1)return[];for(var t=[],e=0,u=n.length;e<u;)t.push(i.call(n,e,e+=r));return t},mixin:Yr,default:tn});return Zr._=Zr,Zr}));
html/genindex.html (new)
- @@ -0,0 +1,543 @@
 + <!DOCTYPE html> + <html class="writer-html5" lang="en" > + <head> + <meta charset="utf-8" /> + <meta name="viewport" content="width=device-width, initial-scale=1.0" /> + <title>Index — Mathematics in Lean 0.1 documentation</title> + <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="shortcut icon" href="_static/favicon.ico"/> +  + + <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> + <script src="_static/jquery.js"></script> + <script src="_static/underscore.js"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> + <script src="_static/doctools.js"></script> + <script src="_static/js/theme.js"></script> + <link rel="index" title="Index" href="#" /> + <link rel="search" title="Search" href="search.html" /> + </head> + + <body class="wy-body-for-nav"> + <div class="wy-grid-for-nav"> + <nav data-toggle="wy-nav-shift" class="wy-nav-side"> + <div class="wy-side-scroll"> + <div class="wy-side-nav-search" > + <a href="index.html" class="icon icon-home"> Mathematics in Lean + </a> + <div role="search"> + <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> + <input type="text" name="q" placeholder="Search docs" /> + <input type="hidden" name="check_keywords" value="yes" /> + <input type="hidden" name="area" value="default" /> + </form> + </div> + </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> + <ul> + <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a></li> + <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> + <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> + <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> + <li class="toctree-l1"><a class="reference internal" href="C06_Abstract_Algebra.html">6. Abstract Algebra</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Topology.html">7. Topology</a></li> + </ul> + <ul class="current"> + <li class="toctree-l1 current"><a class="current reference internal" href="#">Index</a></li> + </ul> + + </div> + </div> + </nav> + + <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > + + <a href="index.html">Mathematics in Lean</a> + </nav> + + <div class="wy-nav-content"> + <div class="rst-content"> + <div role="navigation" aria-label="Page navigation"> + <ul class="wy-breadcrumbs"> + <li><a href="index.html" class="icon icon-home"></a> »</li> + <li>Index</li> + <li class="wy-breadcrumbs-aside"> + </li> + </ul> + <hr/> + </div> + <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> + <div itemprop="articleBody"> + + + <h1 id="index">Index</h1> + + <div class="genindex-jumpbox"> + <a href="#A">A</a> + | <a href="#B">B</a> + | <a href="#C">C</a> + | <a href="#D">D</a> + | <a href="#E">E</a> + | <a href="#F">F</a> + | <a href="#G">G</a> + | <a href="#H">H</a> + | <a href="#I">I</a> + | <a href="#L">L</a> + | <a href="#M">M</a> + | <a href="#N">N</a> + | <a href="#O">O</a> + | <a href="#P">P</a> + | <a href="#R">R</a> + | <a href="#S">S</a> + | <a href="#T">T</a> + | <a href="#U">U</a> + + </div> + <h2 id="A">A</h2> + <table style="width: 100%" class="indextable genindextable"><tr> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C02_Basics.html#index-23">absolute value</a> + </li> + <li><a href="C03_Logic.html#index-17">absurd</a> + </li> + </ul></td> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C03_Logic.html#index-7">anonymous constructor</a> + </li> + <li><a href="C02_Basics.html#index-16">apply</a> + </li> + <li><a href="C03_Logic.html#index-19">assumption</a> + </li> + </ul></td> + </tr></table> + + <h2 id="B">B</h2> + <table style="width: 100%" class="indextable genindextable"><tr> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C04_Sets_and_Functions.html#index-3">bounded quantifiers</a> + </li> + </ul></td> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C03_Logic.html#index-23">by_cases</a> + </li> + <li><a href="C03_Logic.html#index-14">by_contra</a> + </li> + </ul></td> + </tr></table> + + <h2 id="C">C</h2> + <table style="width: 100%" class="indextable genindextable"><tr> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C02_Basics.html#index-3">calc</a> + </li> + <li><a href="C03_Logic.html#index-8">cases</a> + </li> + <li><a href="C03_Logic.html#index-2">change</a> + </li> + <li><a href="C01_Introduction.html#index-0">check</a> + </li> + <li> + command + + <ul> + <li><a href="C02_Basics.html#index-9">open</a> + </li> + </ul></li> + <li> + commands + + <ul> + <li><a href="C01_Introduction.html#index-0">check</a> + </li> + </ul></li> + </ul></td> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C02_Basics.html#index-8">commutative ring</a> + </li> + <li><a href="C03_Logic.html#index-25">congr</a> + </li> + <li><a href="C03_Logic.html#index-18">constructor</a> + </li> + <li><a href="C07_Topology.html#index-3">continuity</a> + </li> + <li><a href="C03_Logic.html#index-17">contradiction</a> + </li> + <li><a href="C03_Logic.html#index-16">contrapose</a> + </li> + <li><a href="C03_Logic.html#index-26">convert</a> + </li> + </ul></td> + </tr></table> + + <h2 id="D">D</h2> + <table style="width: 100%" class="indextable genindextable"><tr> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C02_Basics.html#index-12">definitional equality</a> + </li> + <li><a href="C08_Differential_Calculus.html#index-0">differential calculus</a> + </li> + </ul></td> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C02_Basics.html#index-24">divisibility</a> + </li> + <li><a href="C03_Logic.html#index-2">dsimp</a> + </li> + </ul></td> + </tr></table> + + <h2 id="E">E</h2> + <table style="width: 100%" class="indextable genindextable"><tr> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C03_Logic.html#index-4">erw</a> + </li> + <li><a href="C02_Basics.html#index-5">exact</a> + </li> + <li><a href="C03_Logic.html#index-22">excluded middle</a> + </li> + </ul></td> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C03_Logic.html#index-17">exfalso</a> + </li> + <li><a href="C02_Basics.html#index-18">exponential</a> + </li> + <li><a href="C03_Logic.html#index-24">ext</a> + </li> + <li><a href="C03_Logic.html#index-24">extensionality</a> + </li> + </ul></td> + </tr></table> + + <h2 id="F">F</h2> + <table style="width: 100%" class="indextable genindextable"><tr> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C03_Logic.html#index-11">field_simp</a> + </li> + </ul></td> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C07_Topology.html#index-1">Filter</a> + </li> + <li><a href="C03_Logic.html#index-12">from</a> + </li> + </ul></td> + </tr></table> + + <h2 id="G">G</h2> + <table style="width: 100%" class="indextable genindextable"><tr> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C02_Basics.html#index-25">gcd</a> + </li> + <li><a href="C02_Basics.html#index-2">goal</a> + </li> + </ul></td> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C02_Basics.html#index-13">group (algebraic structure)</a> + + <ul> + <li><a href="C02_Basics.html#index-14">(tactic)</a> + </li> + </ul></li> + </ul></td> + </tr></table> + + <h2 id="H">H</h2> + <table style="width: 100%" class="indextable genindextable"><tr> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C02_Basics.html#index-11">have</a>, <a href="C03_Logic.html#index-12">[1]</a> + </li> + </ul></td> + </tr></table> + + <h2 id="I">I</h2> + <table style="width: 100%" class="indextable genindextable"><tr> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C02_Basics.html#index-10">implicit argument</a> + </li> + <li><a href="C02_Basics.html#index-15">inequalities</a> + </li> + </ul></td> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C03_Logic.html#index-5">injective function</a> + </li> + <li><a href="C09_Integration_and_Measure_Theory.html#index-0">integration</a> + </li> + <li><a href="C03_Logic.html#index-0">intro</a> + </li> + </ul></td> + </tr></table> + + <h2 id="L">L</h2> + <table style="width: 100%" class="indextable genindextable"><tr> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C03_Logic.html#index-1">lambda abstraction</a> + </li> + <li><a href="C02_Basics.html#index-27">lattice</a> + </li> + <li><a href="C02_Basics.html#index-25">lcm</a> + </li> + <li><a href="C03_Logic.html#index-21">left</a> + </li> + </ul></td> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C03_Logic.html#index-13">let</a> + </li> + <li><a href="C02_Basics.html#index-17">linarith</a> + </li> + <li><a href="C02_Basics.html#index-2">local context</a> + </li> + <li><a href="C02_Basics.html#index-18">logarithm</a> + </li> + </ul></td> + </tr></table> + + <h2 id="M">M</h2> + <table style="width: 100%" class="indextable genindextable"><tr> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C02_Basics.html#index-20">max</a> + </li> + <li><a href="C02_Basics.html#index-28">metric space</a>, <a href="C07_Topology.html#index-2">[1]</a> + </li> + </ul></td> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C02_Basics.html#index-20">min</a> + </li> + <li><a href="C03_Logic.html#index-3">monotone function</a> + </li> + </ul></td> + </tr></table> + + <h2 id="N">N</h2> + <table style="width: 100%" class="indextable genindextable"><tr> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C02_Basics.html#index-9">namespace</a> + </li> + </ul></td> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C02_Basics.html#index-19">norm_num</a> + </li> + </ul></td> + </tr></table> + + <h2 id="O">O</h2> + <table style="width: 100%" class="indextable genindextable"><tr> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C02_Basics.html#index-9">open</a> + </li> + </ul></td> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C02_Basics.html#index-26">order relation</a> + </li> + </ul></td> + </tr></table> + + <h2 id="P">P</h2> + <table style="width: 100%" class="indextable genindextable"><tr> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C02_Basics.html#index-26">partial order</a> + </li> + </ul></td> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C02_Basics.html#index-2">proof state</a> + </li> + <li><a href="C03_Logic.html#index-15">push_neg</a> + </li> + </ul></td> + </tr></table> + + <h2 id="R">R</h2> + <table style="width: 100%" class="indextable genindextable"><tr> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C03_Logic.html#index-9">rcases</a> + </li> + <li><a href="C02_Basics.html#index-1">real numbers</a> + </li> + <li><a href="C02_Basics.html#index-12">reflexivity</a> + </li> + <li><a href="C02_Basics.html#index-22">repeat</a> + </li> + <li><a href="C02_Basics.html#index-0">rewrite</a> + </li> + <li><a href="C02_Basics.html#index-12">rfl</a> + </li> + </ul></td> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C03_Logic.html#index-21">right</a> + </li> + <li><a href="C02_Basics.html#index-7">ring (algebraic structure)</a> + + <ul> + <li><a href="C02_Basics.html#index-6">(tactic)</a> + </li> + </ul></li> + <li><a href="C03_Logic.html#index-9">rintro</a> + </li> + <li><a href="C02_Basics.html#index-0">rw</a>, <a href="C02_Basics.html#index-4">[1]</a> + </li> + <li><a href="C04_Sets_and_Functions.html#index-2">rwa</a> + </li> + </ul></td> + </tr></table> + + <h2 id="S">S</h2> + <table style="width: 100%" class="indextable genindextable"><tr> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C04_Sets_and_Functions.html#index-0">set operations</a> + </li> + <li><a href="C02_Basics.html#index-21">show</a> + </li> + </ul></td> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C03_Logic.html#index-20">simp</a>, <a href="C04_Sets_and_Functions.html#index-1">[1]</a> + </li> + <li><a href="C03_Logic.html#index-10">surjective function</a> + </li> + </ul></td> + </tr></table> + + <h2 id="T">T</h2> + <table style="width: 100%" class="indextable genindextable"><tr> + <td style="width: 33%; vertical-align: top;"><ul> + <li> + tactic + + <ul> + <li><a href="C03_Logic.html#index-11">field_simp</a> + </li> + </ul></li> + <li> + tactics + + <ul> + <li><a href="C02_Basics.html#index-14">abel</a> + </li> + <li><a href="C02_Basics.html#index-16">apply</a> + </li> + <li><a href="C03_Logic.html#index-19">assumption</a> + </li> + <li><a href="C03_Logic.html#index-23">by_cases</a> + </li> + <li><a href="C03_Logic.html#index-14">by_contra and by_contradiction</a> + </li> + <li><a href="C02_Basics.html#index-3">calc</a> + </li> + <li><a href="C03_Logic.html#index-8">cases</a> + </li> + <li><a href="C03_Logic.html#index-2">change</a> + </li> + <li><a href="C03_Logic.html#index-25">congr</a> + </li> + <li><a href="C03_Logic.html#index-18">constructor</a> + </li> + <li><a href="C07_Topology.html#index-3">continuity</a> + </li> + <li><a href="C03_Logic.html#index-17">contradiction</a> + </li> + <li><a href="C03_Logic.html#index-16">contrapose</a> + </li> + <li><a href="C03_Logic.html#index-26">convert</a> + </li> + <li><a href="C03_Logic.html#index-2">dsimp</a> + </li> + <li><a href="C03_Logic.html#index-4">erw</a> + </li> + <li><a href="C02_Basics.html#index-5">exact</a> + </li> + <li><a href="C03_Logic.html#index-17">exfalso</a> + </li> + <li><a href="C03_Logic.html#index-24">ext</a> + </li> + <li><a href="C03_Logic.html#index-12">from</a> + </li> + <li><a href="C02_Basics.html#index-14">group</a> + </li> + <li><a href="C02_Basics.html#index-11">have</a>, <a href="C03_Logic.html#index-12">[1]</a> + </li> + <li><a href="C03_Logic.html#index-0">intro</a> + </li> + <li><a href="C03_Logic.html#index-21">left</a> + </li> + <li><a href="C03_Logic.html#index-13">let</a> + </li> + <li><a href="C02_Basics.html#index-17">linarith</a> + </li> + <li><a href="C02_Basics.html#index-14">noncomm_ring</a> + </li> + <li><a href="C02_Basics.html#index-19">norm_num</a> + </li> + <li><a href="C03_Logic.html#index-15">push_neg</a> + </li> + <li><a href="C03_Logic.html#index-9">rcases</a> + </li> + <li><a href="C02_Basics.html#index-12">refl and reflexivity</a> + </li> + <li><a href="C02_Basics.html#index-22">repeat</a> + </li> + <li><a href="C03_Logic.html#index-21">right</a> + </li> + <li><a href="C02_Basics.html#index-6">ring</a> + </li> + <li><a href="C03_Logic.html#index-9">rintro</a> + </li> + <li><a href="C02_Basics.html#index-0">rw and rewrite</a>, <a href="C02_Basics.html#index-4">[1]</a> + </li> + <li><a href="C04_Sets_and_Functions.html#index-2">rwa</a> + </li> + <li><a href="C02_Basics.html#index-21">show</a> + </li> + <li><a href="C03_Logic.html#index-20">simp</a>, <a href="C04_Sets_and_Functions.html#index-1">[1]</a> + </li> + <li><a href="C03_Logic.html#index-6">use</a> + </li> + </ul></li> + </ul></td> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C03_Logic.html#index-12">this</a> + </li> + <li><a href="C07_Topology.html#index-4">topological space</a> + </li> + <li><a href="C07_Topology.html#index-0">topology</a> + </li> + </ul></td> + </tr></table> + + <h2 id="U">U</h2> + <table style="width: 100%" class="indextable genindextable"><tr> + <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C03_Logic.html#index-6">use</a> + </li> + </ul></td> + </tr></table> + + + + </div> + </div> + <footer> + + <hr/> + + <div role="contentinfo"> + © Copyright 2020, Jeremy Avigad, Kevin Buzzard, Robert Y. Lewis, Patrick Massot. + </div> + + Built with <a href="https://www.sphinx-doc.org/">Sphinx</a> using a + <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a> + provided by <a href="https://readthedocs.org">Read the Docs</a>. + + + </footer> + </div> + </div> + </section> + </div> + <script> + jQuery(function () { + SphinxRtdTheme.Navigation.enable(true); + }); + </script> + + </body> + </html>
html/index.html (new)
- @@ -0,0 +1,166 @@
 + <!DOCTYPE html> + <html class="writer-html5" lang="en" > + <head> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + + <meta name="viewport" content="width=device-width, initial-scale=1.0" /> + <title>Mathematics in Lean — Mathematics in Lean 0.1 documentation</title> + <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="shortcut icon" href="_static/favicon.ico"/> +  + + <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> + <script src="_static/jquery.js"></script> + <script src="_static/underscore.js"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> + <script src="_static/doctools.js"></script> + <script src="_static/js/theme.js"></script> + <link rel="index" title="Index" href="genindex.html" /> + <link rel="search" title="Search" href="search.html" /> + <link rel="next" title="1. Introduction" href="C01_Introduction.html" /> + </head> + + <body class="wy-body-for-nav"> + <div class="wy-grid-for-nav"> + <nav data-toggle="wy-nav-shift" class="wy-nav-side"> + <div class="wy-side-scroll"> + <div class="wy-side-nav-search" > + <a href="#" class="icon icon-home"> Mathematics in Lean + </a> + <div role="search"> + <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> + <input type="text" name="q" placeholder="Search docs" /> + <input type="hidden" name="check_keywords" value="yes" /> + <input type="hidden" name="area" value="default" /> + </form> + </div> + </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> + <ul> + <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a></li> + <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> + <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> + <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> + <li class="toctree-l1"><a class="reference internal" href="C06_Abstract_Algebra.html">6. Abstract Algebra</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Topology.html">7. Topology</a></li> + </ul> + <ul> + <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li> + </ul> + + </div> + </div> + </nav> + + <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > + + <a href="#">Mathematics in Lean</a> + </nav> + + <div class="wy-nav-content"> + <div class="rst-content"> + <div role="navigation" aria-label="Page navigation"> + <ul class="wy-breadcrumbs"> + <li><a href="#" class="icon icon-home"></a> »</li> + <li>Mathematics in Lean</li> + <li class="wy-breadcrumbs-aside"> + <a href="_sources/index.rst.txt" rel="nofollow"> View page source</a> + </li> + </ul> + <hr/> + </div> + <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> + <div itemprop="articleBody"> + + <section id="mathematics-in-lean"> + <h1>Mathematics in Lean<a class="headerlink" href="#mathematics-in-lean" title="Permalink to this heading"></a></h1> + <div class="toctree-wrapper compound"> + <ul> + <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a><ul> + <li class="toctree-l2"><a class="reference internal" href="C01_Introduction.html#getting-started">1.1. Getting Started</a></li> + <li class="toctree-l2"><a class="reference internal" href="C01_Introduction.html#overview">1.2. Overview</a></li> + </ul> + </li> + <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a><ul> + <li class="toctree-l2"><a class="reference internal" href="C02_Basics.html#calculating">2.1. Calculating</a></li> + <li class="toctree-l2"><a class="reference internal" href="C02_Basics.html#proving-identities-in-algebraic-structures">2.2. Proving Identities in Algebraic Structures</a></li> + <li class="toctree-l2"><a class="reference internal" href="C02_Basics.html#using-theorems-and-lemmas">2.3. Using Theorems and Lemmas</a></li> + <li class="toctree-l2"><a class="reference internal" href="C02_Basics.html#more-on-order-and-divisibility">2.4. More on Order and Divisibility</a></li> + <li class="toctree-l2"><a class="reference internal" href="C02_Basics.html#proving-facts-about-algebraic-structures">2.5. Proving Facts about Algebraic Structures</a></li> + </ul> + </li> + <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a><ul> + <li class="toctree-l2"><a class="reference internal" href="C03_Logic.html#implication-and-the-universal-quantifier">3.1. Implication and the Universal Quantifier</a></li> + <li class="toctree-l2"><a class="reference internal" href="C03_Logic.html#the-existential-quantifier">3.2. The Existential Quantifier</a></li> + <li class="toctree-l2"><a class="reference internal" href="C03_Logic.html#negation">3.3. Negation</a></li> + <li class="toctree-l2"><a class="reference internal" href="C03_Logic.html#conjunction-and-bi-implication">3.4. Conjunction and Bi-implication</a></li> + <li class="toctree-l2"><a class="reference internal" href="C03_Logic.html#disjunction">3.5. Disjunction</a></li> + <li class="toctree-l2"><a class="reference internal" href="C03_Logic.html#sequences-and-convergence">3.6. Sequences and Convergence</a></li> + </ul> + </li> + <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a><ul> + <li class="toctree-l2"><a class="reference internal" href="C04_Sets_and_Functions.html#sets">4.1. Sets</a></li> + <li class="toctree-l2"><a class="reference internal" href="C04_Sets_and_Functions.html#functions">4.2. Functions</a></li> + <li class="toctree-l2"><a class="reference internal" href="C04_Sets_and_Functions.html#the-schroder-bernstein-theorem">4.3. The Schröder-Bernstein Theorem</a></li> + </ul> + </li> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a><ul> + <li class="toctree-l2"><a class="reference internal" href="C05_Number_Theory.html#irrational-roots">5.1. Irrational Roots</a></li> + <li class="toctree-l2"><a class="reference internal" href="C05_Number_Theory.html#induction-and-recursion">5.2. Induction and Recursion</a></li> + <li class="toctree-l2"><a class="reference internal" href="C05_Number_Theory.html#id2">5.3. Induction and Recursion</a></li> + <li class="toctree-l2"><a class="reference internal" href="C05_Number_Theory.html#infinitely-many-primes">5.4. Infinitely Many Primes</a></li> + </ul> + </li> + <li class="toctree-l1"><a class="reference internal" href="C06_Abstract_Algebra.html">6. Abstract Algebra</a><ul> + <li class="toctree-l2"><a class="reference internal" href="C06_Abstract_Algebra.html#structures">6.1. Structures</a></li> + <li class="toctree-l2"><a class="reference internal" href="C06_Abstract_Algebra.html#algebraic-structures">6.2. Algebraic Structures</a></li> + <li class="toctree-l2"><a class="reference internal" href="C06_Abstract_Algebra.html#building-the-gaussian-integers">6.3. Building the Gaussian Integers</a></li> + </ul> + </li> + <li class="toctree-l1"><a class="reference internal" href="C07_Topology.html">7. Topology</a><ul> + <li class="toctree-l2"><a class="reference internal" href="C07_Topology.html#filters">7.1. Filters</a></li> + <li class="toctree-l2"><a class="reference internal" href="C07_Topology.html#metric-spaces">7.2. Metric spaces</a></li> + <li class="toctree-l2"><a class="reference internal" href="C07_Topology.html#topological-spaces">7.3. Topological spaces</a></li> + </ul> + </li> + </ul> + </div> + <div class="toctree-wrapper compound"> + </div> + </section> + + + </div> + </div> + <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> + <a href="C01_Introduction.html" class="btn btn-neutral float-right" title="1. Introduction" accesskey="n" rel="next">Next </a> + </div> + + <hr/> + + <div role="contentinfo"> + © Copyright 2020, Jeremy Avigad, Kevin Buzzard, Robert Y. Lewis, Patrick Massot. + </div> + + Built with <a href="https://www.sphinx-doc.org/">Sphinx</a> using a + <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a> + provided by <a href="https://readthedocs.org">Read the Docs</a>. + + + </footer> + </div> + </div> + </section> + </div> + <script> + jQuery(function () { + SphinxRtdTheme.Navigation.enable(true); + }); + </script> + + </body> + </html>
html/objects.inv (new)
html/search.html (new)
- @@ -0,0 +1,126 @@
 + <!DOCTYPE html> + <html class="writer-html5" lang="en" > + <head> + <meta charset="utf-8" /> + <meta name="viewport" content="width=device-width, initial-scale=1.0" /> + <title>Search — Mathematics in Lean 0.1 documentation</title> + <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="shortcut icon" href="_static/favicon.ico"/> + +  + + <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> + <script src="_static/jquery.js"></script> + <script src="_static/underscore.js"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> + <script src="_static/doctools.js"></script> + <script src="_static/js/theme.js"></script> + <script src="_static/searchtools.js"></script> + <script src="_static/language_data.js"></script> + <link rel="index" title="Index" href="genindex.html" /> + <link rel="search" title="Search" href="#" /> + </head> + + <body class="wy-body-for-nav"> + <div class="wy-grid-for-nav"> + <nav data-toggle="wy-nav-shift" class="wy-nav-side"> + <div class="wy-side-scroll"> + <div class="wy-side-nav-search" > + <a href="index.html" class="icon icon-home"> Mathematics in Lean + </a> + <div role="search"> + <form id="rtd-search-form" class="wy-form" action="#" method="get"> + <input type="text" name="q" placeholder="Search docs" /> + <input type="hidden" name="check_keywords" value="yes" /> + <input type="hidden" name="area" value="default" /> + </form> + </div> + </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> + <ul> + <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a></li> + <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> + <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> + <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> + <li class="toctree-l1"><a class="reference internal" href="C06_Abstract_Algebra.html">6. Abstract Algebra</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Topology.html">7. Topology</a></li> + </ul> + <ul> + <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li> + </ul> + + </div> + </div> + </nav> + + <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > + + <a href="index.html">Mathematics in Lean</a> + </nav> + + <div class="wy-nav-content"> + <div class="rst-content"> + <div role="navigation" aria-label="Page navigation"> + <ul class="wy-breadcrumbs"> + <li><a href="index.html" class="icon icon-home"></a> »</li> + <li>Search</li> + <li class="wy-breadcrumbs-aside"> + </li> + </ul> + <hr/> + </div> + <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> + <div itemprop="articleBody"> + + <noscript> + <div id="fallback" class="admonition warning"> + + Please activate JavaScript to enable the search functionality. + + </div> + </noscript> + + + <div id="search-results"> + + </div> + + </div> + </div> + <footer> + + <hr/> + + <div role="contentinfo"> + © Copyright 2020, Jeremy Avigad, Kevin Buzzard, Robert Y. Lewis, Patrick Massot. + </div> + + Built with <a href="https://www.sphinx-doc.org/">Sphinx</a> using a + <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a> + provided by <a href="https://readthedocs.org">Read the Docs</a>. + + + </footer> + </div> + </div> + </section> + </div> + <script> + jQuery(function () { + SphinxRtdTheme.Navigation.enable(true); + }); + </script> + <script> + jQuery(function() { Search.loadIndex("searchindex.js"); }); + </script> + + <script id="searchindexloader"></script> + + + + </body> + </html>
html/searchindex.js (new)
- @@ -0,0 +1,1 @@
 + Search.setIndex({"docnames": ["C01_Introduction", "C02_Basics", "C03_Logic", "C04_Sets_and_Functions", "C05_Number_Theory", "C06_Abstract_Algebra", "C07_Topology", "C08_Differential_Calculus", "C09_Integration_and_Measure_Theory", "genindex", "index"], "filenames": ["C01_Introduction.rst", "C02_Basics.rst", "C03_Logic.rst", "C04_Sets_and_Functions.rst", "C05_Number_Theory.rst", "C06_Abstract_Algebra.rst", "C07_Topology.rst", "C08_Differential_Calculus.rst", "C09_Integration_and_Measure_Theory.rst", "genindex.rst", "index.rst"], "titles": ["1. Introduction", "2. Basics", "3. Logic", "4. Sets and Functions", "5. Number Theory", "6. Abstract Algebra", "7. Topology", "Differential Calculus", "Integration and Measure Theory", "Index", "Mathematics in Lean"], "terms": {"The": [0, 1, 4, 5, 6, 10], "goal": [0, 1, 2, 3, 4, 5, 6], "thi": [0, 1, 2, 3, 4, 5, 6, 7], "book": [0, 6], "teach": 0, "you": [0, 1, 2, 3, 4, 5, 6], "formal": [0, 1, 2, 3, 4, 5, 6, 7], "mathemat": [0, 1, 2, 3, 4, 5, 6], "us": [0, 2, 3, 4, 5, 6, 10], "lean": [0, 1, 2, 3, 4, 5, 6], "4": [0, 1, 2, 4, 5, 6], "interact": [0, 5], "proof": [0, 1, 2, 3, 4, 5, 6], "assist": [0, 5], "It": [0, 1, 2, 3, 4, 5, 6], "assum": [0, 1, 2, 3, 4, 6], "know": [0, 1, 2, 3, 4, 5, 6], "some": [0, 1, 2, 3, 4, 5, 6], "doe": [0, 1, 2, 3, 4, 5, 6], "requir": [0, 1, 2, 3, 4, 5, 6], "much": [0, 2, 4, 6, 7], "although": [0, 3, 4, 5, 6], "we": [0, 1, 2, 3, 4, 5, 6, 7], "cover": [0, 1, 3, 6], "exampl": [0, 1, 2, 3, 4, 5, 6], "rang": [0, 2, 3, 4, 6], "from": [0, 1, 2, 3, 4, 5, 6, 7], "number": [0, 1, 2, 3, 5, 6, 7, 10], "theori": [0, 1, 2, 3, 6, 7, 10], "measur": [0, 5, 6, 7], "analysi": [0, 5, 7], "focu": [0, 1, 6], "elementari": [0, 3, 4, 5, 6], "aspect": [0, 5], "those": [0, 1, 2, 3, 5, 6], "field": [0, 1, 5], "hope": 0, "thei": [0, 1, 2, 3, 4, 5, 6], "ar": [0, 1, 2, 3, 4, 5, 6], "familiar": [0, 1, 4, 5, 6, 7], "can": [0, 1, 2, 3, 4, 5, 6], "pick": 0, "them": [0, 1, 2, 3, 4, 5, 6], "up": [0, 2, 3, 4, 5, 6], "go": [0, 1, 2, 3, 4, 6], "also": [0, 1, 2, 3, 4, 5, 6], "don": [0, 1, 2, 3, 4, 5, 6], "t": [0, 1, 2, 3, 4, 5, 6], "presuppos": 0, "ani": [0, 1, 2, 3, 4, 5, 6, 7], "background": 0, "seen": [0, 1, 2, 4, 5, 6], "kind": [0, 1, 6], "comput": [0, 3, 4, 5], "program": [0, 1], "write": [0, 1, 2, 3, 4, 5, 6], "definit": [0, 1, 2, 3, 4, 5, 6], "theorem": [0, 2, 4, 5, 6, 10], "regiment": 0, "languag": [0, 1, 3], "like": [0, 1, 2, 3, 4, 5, 6], "understand": [0, 2, 3, 4, 5, 6], "In": [0, 1, 2, 3, 4, 5, 6, 7], "return": [0, 2, 3, 4, 5, 6], "provid": [0, 1, 2, 3, 4, 5, 6], "feedback": 0, "inform": [0, 1, 2, 3, 5, 6], "interpret": [0, 1, 4, 5, 6], "express": [0, 1, 2, 3, 4, 5, 6], "guarante": [0, 3, 5], "well": [0, 1, 2, 3, 4, 5, 6], "form": [0, 1, 2, 3, 4, 5, 6], "ultim": 0, "certifi": 0, "correct": [0, 1], "our": [0, 1, 2, 3, 4, 5, 6], "learn": [0, 1, 2, 4, 5], "more": [0, 2, 3, 4, 5, 6, 10], "about": [0, 2, 3, 4, 5, 6, 10], "project": [0, 4, 5, 6], "page": [0, 1, 4], "commun": [0, 4], "web": [0, 1, 4], "tutori": 0, "base": [0, 3, 4, 6], "s": [0, 1, 2, 3, 4, 5, 6], "larg": [0, 6], "ever": [0, 6], "grow": 0, "librari": [0, 1, 2, 3, 4, 5, 6], "mathlib": [0, 1, 2, 3, 4, 5, 6], "strongli": 0, "recommend": [0, 1, 3], "take": [0, 1, 2, 3, 4, 5, 6], "look": [0, 1, 2, 3, 4, 6], "zulip": [0, 4], "onlin": 0, "chat": 0, "group": [0, 1, 2, 3, 5, 6], "haven": [0, 2, 6], "alreadi": [0, 1, 2, 3, 4, 5, 6], "ll": [0, 1, 2, 4, 6], "find": [0, 1, 2, 4, 5, 6], "live": [0, 5], "welcom": [0, 1], "enthusiast": 0, "happi": 0, "answer": [0, 5, 6], "question": [0, 4, 6], "offer": [0, 2, 5, 6], "moral": [0, 5], "support": [0, 1, 2, 3, 4, 5, 6], "read": [0, 1, 4, 6], "pdf": 0, "html": 0, "version": [0, 1, 2, 3, 4, 5, 6], "design": [0, 1, 2, 3, 5, 6], "run": 0, "insid": [0, 1, 2, 3], "vs": [0, 1, 2, 3, 5], "code": [0, 1, 2, 3, 5], "editor": [0, 1], "To": [0, 1, 2, 3, 4, 5, 6], "instal": 0, "follow": [0, 1, 2, 3, 4, 5, 6], "instruct": [0, 1, 6], "http": 0, "github": [0, 1], "com": 0, "leanprov": 0, "lean4": 0, "blob": 0, "master": [0, 1, 4], "doc": 0, "quickstart": 0, "md": 0, "_": [0, 1, 2, 3, 4, 5, 6], "termin": 0, "navig": 0, "folder": 0, "where": [0, 1, 2, 3, 4, 5, 6], "want": [0, 1, 2, 3, 4, 5, 6], "put": [0, 1, 2, 3, 4, 5, 6], "copi": [0, 3, 4, 6], "repositori": [0, 1], "type": [0, 1, 2, 3, 4, 5, 6], "git": 0, "clone": 0, "mathematics_in_lean": 0, "fetch": 0, "execut": 0, "lake": 0, "ex": [0, 2], "cach": 0, "compil": 0, "open": [0, 1, 2, 3, 4, 5], "altern": [0, 2, 3, 4, 5, 6], "choos": [0, 2, 3, 4, 5, 6], "file": [0, 1, 4, 5, 6], "menu": [0, 1], "Be": [0, 1, 3], "sure": [0, 1, 2, 3, 5], "other": [0, 1, 2, 3, 4, 5, 6], "simultan": 0, "window": [0, 1, 2, 5], "updat": 0, "newer": 0, "ty": 0, "pull": [0, 6], "cloud": 0, "gitpod": 0, "how": [0, 1, 2, 3, 4, 5, 6], "do": [0, 1, 2, 3, 4, 5, 6], "each": [0, 1, 2, 3, 4, 5, 6], "section": [0, 1, 2, 3, 4, 5, 6], "ha": [0, 1, 2, 3, 4, 5, 6], "an": [0, 1, 2, 3, 4, 5, 6], "associ": [0, 1, 2, 4, 5, 6], "exercis": [0, 1, 2, 3, 5, 6], "src": 0, "organ": 0, "chapter": [0, 1, 2, 3, 4, 5, 6, 7], "make": [0, 1, 2, 3, 4, 5, 6], "so": [0, 1, 2, 3, 4, 5, 6], "experi": [0, 1, 6], "while": [0, 1, 2, 6], "leav": [0, 1, 2, 3, 5], "origin": [0, 1, 5], "intact": 0, "text": [0, 3, 5, 6], "often": [0, 1, 2, 3, 4, 5, 6], "includ": [0, 1, 2, 3, 4, 6], "one": [0, 1, 2, 3, 4, 5, 6], "eval": 0, "hello": 0, "world": 0, "should": [0, 1, 2, 3, 4, 5, 6], "abl": [0, 1, 2, 3, 4, 5], "correspond": [0, 1, 2, 3, 4, 5, 6], "If": [0, 1, 2, 3, 4, 5, 6], "click": [0, 1, 3, 4, 5], "line": [0, 1, 2, 3, 4], "show": [0, 1, 2, 3, 4, 5, 6], "hover": [0, 1, 2, 3, 4, 5], "your": [0, 1, 2, 3, 5, 6], "cursor": [0, 1, 2], "over": [0, 1, 2, 3, 4, 5, 6], "command": [0, 1, 2, 3, 4, 5], "respons": [0, 1], "pop": 0, "encourag": [0, 1, 2, 3, 4, 5], "edit": 0, "try": [0, 1, 2, 3, 4, 5, 6], "own": [0, 1, 4, 5], "moreov": [0, 1, 4, 5, 6], "lot": [0, 3, 5, 6], "challeng": [0, 1, 2, 3, 5], "rush": 0, "past": [0, 1], "just": [0, 1, 2, 3, 4, 5, 6], "work": [0, 1, 2, 3, 4, 5, 6], "through": [0, 1, 3, 4, 5, 6], "central": [0, 4], "alwai": [0, 1, 2, 3, 5], "compar": [0, 5, 6], "solut": [0, 1, 4], "ones": [0, 1, 2, 3, 5], "simpli": [0, 1, 2, 3, 4, 5, 6], "tool": [0, 1, 2, 5], "build": [0, 2, 4, 6, 10], "complex": [0, 1, 2, 4, 5], "known": [0, 1, 2, 3, 4, 5, 6], "depend": [0, 2, 3, 4, 5, 6], "everi": [0, 1, 2, 3, 4, 5, 6], "check": [0, 1, 2, 3, 4, 5, 6], "print": [0, 3, 4, 5], "have": [0, 1, 2, 3, 4, 5, 6], "\u2115": [0, 1, 2, 3, 4, 5, 6], "These": [0, 1, 2, 3, 4, 5], "object": [0, 1, 2, 3, 4, 5], "2": [0, 1, 2, 3, 4, 5, 6], "def": [0, 2, 3, 4, 5, 6], "f": [0, 1, 2, 3, 4, 5, 6], "x": [0, 1, 2, 3, 4, 5, 6], "3": [0, 1, 2, 3, 4, 5, 6], "prop": [0, 2, 3, 4, 5, 6], "statement": [0, 1, 2, 3, 4, 5, 6], "fermatlasttheorem": 0, "y": [0, 1, 2, 3, 4, 5, 6], "z": [0, 1, 2, 5, 6], "n": [0, 1, 2, 3, 4, 5, 6], "0": [0, 1, 2, 3, 4, 5, 6], "p": [0, 2, 3, 4, 5, 6], "itself": [0, 3, 5, 6], "Such": [0, 1, 2], "proposit": [0, 2, 3, 4, 5], "easi": [0, 4, 5, 6], "rfl": [0, 1, 2, 3, 4, 5, 6], "hard": [0, 1, 3, 5], "sorri": [0, 1, 2, 3, 4, 5, 6], "manag": [0, 1, 4, 5, 6], "construct": [0, 2, 3, 4, 5, 6], "fermat_last_theorem": 0, "accept": [0, 1, 6], "term": [0, 1, 2, 3, 4, 5, 6], "done": [0, 2, 3, 4, 6], "someth": [0, 1, 2, 4, 6], "veri": [0, 2, 4, 6], "impress": 0, "cheat": [0, 1], "now": [0, 1, 2, 3, 4, 5, 6, 7], "game": 0, "all": [0, 1, 2, 3, 4, 5, 6], "left": [0, 1, 2, 3, 4, 5, 6], "rule": [0, 2, 4, 5, 6], "complementari": 0, "companion": [0, 1], "prove": [0, 2, 3, 4, 5, 6, 10], "which": [0, 1, 2, 3, 4, 5, 6, 7], "thorough": 0, "underli": [0, 1, 5], "logic": [0, 1, 3, 4, 5, 10], "framework": 0, "core": [0, 4, 5], "syntax": [0, 1, 3, 4], "peopl": [0, 1], "who": [0, 6], "prefer": [0, 1, 3], "user": [0, 2], "manual": [0, 3, 4], "befor": [0, 1, 2, 3, 4, 5], "new": [0, 1, 2, 3, 4, 5, 6], "dishwash": 0, "person": 0, "hit": [0, 1], "button": [0, 1], "figur": [0, 1, 2, 5], "out": [0, 1, 2, 3, 4, 5, 6], "activ": 0, "potscrubb": 0, "featur": [0, 1, 2, 4], "later": [0, 1, 2, 4, 5, 6], "sens": [0, 2, 3, 4, 5, 6], "here": [0, 1, 2, 3, 4, 5, 6], "refer": [0, 1, 2, 4, 5, 6], "back": [0, 1, 2, 3, 5, 6], "necessari": [0, 2, 4, 5], "anoth": [0, 1, 2, 3, 4, 5, 6], "thing": [0, 1, 2, 3, 4, 5, 6], "distinguish": [0, 5, 6], "place": [0, 1, 2, 5, 6], "greater": [0, 1, 2, 3, 4, 6], "emphasi": 0, "tactic": [0, 1, 2, 3, 4, 5, 6], "given": [0, 1, 2, 3, 4, 5, 6], "two": [0, 1, 2, 3, 4, 5, 6], "wai": [0, 1, 2, 3, 4, 5, 6], "down": [0, 3, 4, 6], "themselv": [0, 4, 5], "suitabl": [0, 1, 2, 5, 6], "descript": [0, 1, 2, 4, 6], "thereof": 0, "For": [0, 1, 2, 3, 4, 5, 6], "repres": [0, 1, 2, 3, 4, 5], "fact": [0, 2, 3, 4, 5, 6, 10], "even": [0, 1, 2, 3, 4, 5, 6], "m": [0, 1, 2, 4, 6], "nat": [0, 1, 2, 3, 4, 5, 6], "fun": [0, 2, 3, 4, 5, 6], "k": [0, 2, 4, 5, 6], "hk": 0, "hmn": 0, "rw": [0, 1, 2, 3, 4, 5, 6], "mul_add": [0, 1, 4], "l": [0, 1], "compress": [0, 1, 6], "singl": [0, 1, 2, 3, 4, 5], "instead": [0, 1, 2, 3, 4, 5, 6], "style": [0, 1], "same": [0, 1, 2, 3, 4, 5, 6], "sai": [0, 1, 2, 3, 4, 5, 6], "natur": [0, 1, 2, 3, 4, 5, 6], "rintro": [0, 2, 3, 4, 5, 6], "need": [0, 1, 2, 3, 4, 5, 6], "twice": [0, 6], "let": [0, 1, 2, 3, 4, 5, 6], "substitut": [0, 2], "obviou": [0, 4, 5], "ring": [0, 1, 2, 3, 4, 5], "As": [0, 1, 2, 3, 4, 5, 6], "enter": [0, 1, 2, 3, 6], "displai": [0, 1, 2], "state": [0, 1, 2, 3, 4, 6], "separ": [0, 1, 2, 4, 5], "tell": [0, 2, 3, 4, 5], "what": [0, 1, 2, 3, 4, 5, 6], "establish": [0, 1, 2, 3, 4, 5], "task": [0, 2, 4, 5, 6], "remain": [0, 1, 2, 3, 5, 6], "replai": 0, "step": [0, 1, 2, 3, 4, 5], "sinc": [0, 1, 2, 3, 4, 5, 6], "continu": [0, 1, 4, 5], "point": [0, 1, 2, 3, 5, 6], "see": [0, 1, 2, 3, 4, 5, 6], "first": [0, 1, 2, 3, 4, 5, 6], "introduc": [0, 1, 2, 3, 4, 6], "could": [0, 1, 2, 4, 5, 6], "renam": 0, "decompos": [0, 3, 5], "hypothesi": [0, 1, 2, 3, 4], "assumpt": [0, 1, 2, 3, 4, 5, 6], "second": [0, 1, 2, 3, 4, 5, 6], "declar": [0, 1, 4, 5], "next": [0, 1, 2, 3, 4, 5, 6, 7], "rewrit": [0, 1, 2, 3, 4, 5, 6], "replac": [0, 1, 2, 3, 4], "solv": [0, 1, 2, 3, 4, 5, 6], "result": [0, 1, 2, 3, 4, 5, 6], "abil": 0, "small": [0, 4, 5, 6], "increment": [0, 1], "extrem": [0, 6], "power": [0, 2, 3, 4, 5], "reason": [0, 1, 2, 4, 5, 6], "easier": [0, 1, 3, 4, 6], "quicker": 0, "than": [0, 1, 2, 3, 4, 5, 6], "There": [0, 1, 2, 3, 4, 6], "isn": [0, 2, 3, 4], "sharp": 0, "distinct": [0, 2, 3, 4, 5, 6], "between": [0, 1, 2, 3, 4, 5, 6], "insert": [0, 2, 4, 5, 6], "did": [0, 2, 6], "phrase": [0, 1, 2, 4, 5], "mul_left_comm": [0, 4], "abov": [0, 1, 2, 3, 4, 5, 6], "convers": [0, 1, 2, 4, 6], "short": [0, 1, 2, 3, 4, 5], "middl": [0, 1, 2, 6], "That": [0, 4, 5], "said": [0, 2, 4, 5], "reduc": [0, 2, 3, 4, 5], "liner": 0, "carri": [0, 1, 2, 3, 4, 5, 6], "But": [0, 1, 2, 3, 4, 5, 6], "substanti": 0, "autom": [0, 1, 5], "justifi": [0, 1, 2, 6], "longer": [0, 4, 5], "calcul": [0, 2, 4, 5, 10], "bigger": [0, 3], "inferenti": 0, "invok": [0, 1, 6], "simplifi": [0, 2, 3, 4, 5, 6], "specif": [0, 1, 3, 4, 5], "pariti": [0, 4], "automat": [0, 1, 2, 3, 4, 5, 6], "intro": [0, 1, 2, 3, 4, 5, 6], "simp": [0, 2, 3, 4, 5, 6], "parity_simp": 0, "big": [0, 5, 6], "differ": [0, 1, 2, 3, 4, 5, 6], "onli": [0, 1, 2, 3, 4, 5, 6], "its": [0, 1, 2, 3, 4, 5, 6], "built": [0, 2], "wherea": [0, 1, 3, 5, 6], "top": [0, 6], "meant": [0, 1, 2], "contain": [0, 2, 3, 4, 5, 6], "extens": [0, 1, 6], "document": [0, 1, 3, 4], "rather": [0, 1, 2, 4, 5, 6], "think": [0, 1, 2, 3, 4, 5, 6], "comfort": [0, 2], "brows": [0, 1, 4], "frustrat": 0, "curv": 0, "steep": 0, "newcom": 0, "avail": [0, 4, 5], "round": [0, 1, 5], "clock": 0, "doubt": 0, "soon": [0, 2, 6], "enough": [0, 1, 2, 3, 4, 6], "too": [0, 1, 3, 5, 6], "contribut": [0, 3], "develop": [0, 1, 4], "mission": 0, "dive": 0, "come": [0, 1, 2, 3, 4, 5, 6], "forewarn": 0, "fundament": [0, 4, 5], "life": [0, 2], "mai": [0, 1, 2, 3, 5, 6], "never": 0, "acknowledg": 0, "grate": [0, 5], "gabriel": 0, "ebner": 0, "set": [0, 1, 2, 4, 5, 7, 10], "infrastructur": 0, "scott": 0, "morrison": 0, "mario": 0, "carneiro": 0, "help": [0, 1, 2, 3, 4, 5, 6], "port": 0, "bryan": 0, "gin": 0, "ge": [0, 2, 5], "chen": 0, "johan": 0, "commelin": 0, "mathieu": 0, "guai": 0, "paquet": 0, "julian": 0, "k\u00fclshammer": 0, "giovanni": 0, "mascellani": 0, "hunter": 0, "monro": 0, "pietro": 0, "monticon": 0, "bartosz": 0, "piotrowski": 0, "guilherm": 0, "silva": 0, "been": [0, 1, 2, 4, 5], "partial": [0, 1, 2, 3, 5], "hoskinson": 0, "center": [0, 6], "nut": 1, "bolt": 1, "appli": [1, 2, 3, 4, 5, 6], "gener": [1, 2, 3, 4, 5, 6], "without": [1, 2, 4, 5, 6], "when": [1, 2, 3, 4, 5, 6], "net": 1, "hand": [1, 2, 3, 4, 5, 6], "side": [1, 2, 3, 4, 5, 6], "equal": [1, 2, 3, 4, 5, 6], "right": [1, 2, 3, 4, 5, 6], "tantamount": [1, 3], "name": [1, 2, 3, 4, 5, 6], "abbrevi": [1, 2, 3, 5], "b": [1, 2, 3, 4, 5, 6], "c": [1, 2, 4, 5, 6], "real": [1, 2, 3, 4, 5, 6, 7], "mul_assoc": [1, 2, 4, 5], "mul_comm": [1, 2, 4, 5], "elimin": 1, "explicitli": [1, 2, 3, 5, 6], "purpos": [1, 2, 3, 4, 5, 6], "illustr": [1, 2, 3, 4, 5], "multipl": [1, 2, 3, 4, 5], "written": [1, 2, 3, 4, 5, 6], "howev": [1, 2, 5, 6], "good": [1, 2, 3, 4, 5, 6], "mind": [1, 2, 5], "notat": [1, 2, 3, 4, 5, 6], "convent": [1, 5], "parenthes": [1, 2, 3, 4], "\u211d": [1, 2, 3, 5, 6], "import": [1, 2, 3, 4, 5, 6], "begin": [1, 2, 3, 4, 5], "sake": [1, 4], "breviti": [1, 5], "suppress": 1, "repeat": [1, 4, 5], "full": [1, 3, 4, 5, 6], "process": [1, 3], "chang": [1, 2, 4, 5, 6], "happen": [1, 6], "charact": [1, 3], "r": [1, 2, 4, 5, 6], "symbol": [1, 2, 5], "doesn": [1, 2, 6], "appear": [1, 2, 5, 6], "until": [1, 2, 4, 5, 6], "space": [1, 3, 5, 10], "tab": [1, 2, 3, 4], "kei": [1, 4, 5, 6], "keyboard": 1, "easili": [1, 4, 5], "access": [1, 3, 5, 6], "backslash": [1, 3], "lead": [1, 2, 4, 5, 6], "input": [1, 3, 6], "leader": 1, "report": 1, "current": [1, 2], "infoview": 1, "move": [1, 2, 6], "A": [1, 2, 3, 4, 5, 6], "typic": 1, "might": [1, 2, 3, 4, 5], "1": [1, 2, 3, 4, 5, 6], "h\u2081": [1, 2, 3, 4], "prime": [1, 2, 3, 5, 10], "h\u2082": [1, 3, 4], "h\u2083": [1, 3], "denot": [1, 3, 4, 6], "context": [1, 2, 4, 5, 6], "plai": [1, 3, 4, 6], "three": [1, 2, 4, 5, 6], "label": [1, 2], "everyth": [1, 2, 6], "identifi": [1, 2, 4], "subscript": 1, "h": [1, 2, 3, 4, 5, 6], "legal": 1, "would": [1, 2, 3, 4, 5, 6], "h1": [1, 3, 4], "h2": [1, 3], "h3": 1, "foo": [1, 3, 5], "bar": [1, 2], "baz": 1, "last": [1, 2, 3, 4, 5, 6], "sometim": [1, 2, 3, 4, 5, 6], "target": [1, 6], "combin": [1, 2, 5, 6], "practic": [1, 2, 3, 6], "intend": [1, 2, 5], "mean": [1, 2, 3, 4, 5, 6], "usual": [1, 2, 3, 4, 5, 6], "clear": [1, 2, 4, 5], "case": [1, 2, 3, 4, 5, 6], "With": [1, 3, 4, 6], "arrow": [1, 2, 6], "revers": [1, 2, 3, 5, 6], "argument": [1, 2, 3, 4, 5, 6], "tri": [1, 2, 4], "match": [1, 2, 5], "pattern": [1, 2, 3, 5], "local": [1, 2, 6], "d": [1, 2, 4, 5], "e": [1, 2, 3, 4, 5, 6], "hyp": 1, "sub_self": [1, 2], "list": [1, 2, 3, 4, 5], "relev": [1, 2, 4, 5, 6], "within": [1, 2, 6], "squar": [1, 2, 4, 5], "bracket": [1, 2, 5], "still": [1, 2, 3, 4, 5, 6], "progress": [1, 5], "after": [1, 2, 4, 6], "comma": 1, "trick": [1, 2, 4], "variabl": [1, 2, 3, 4, 5, 6], "onc": [1, 2, 3, 4, 5, 6], "outsid": [1, 4], "mention": [1, 2, 3, 5, 6], "g": [1, 2, 3, 5, 6], "inspect": 1, "reveal": 1, "inde": [1, 5, 6], "delimit": 1, "scope": [1, 2, 3, 5], "end": [1, 2, 3, 4, 5, 6], "block": [1, 4], "final": [1, 2, 3, 5, 6], "recal": [1, 2, 4, 5, 6], "introduct": [1, 2, 6, 10], "determin": [1, 5, 6], "both": [1, 2, 3, 4, 5, 6], "expect": [1, 2, 3, 4, 5, 6], "rais": [1, 4], "error": [1, 2, 3], "explain": [1, 2, 3, 4, 5, 6], "output": 1, "meanwhil": [1, 4], "two_mul": [1, 5], "add_mul": 1, "distribut": [1, 4, 5], "addit": [1, 2, 4, 5, 6], "add_assoc": [1, 4, 5], "precis": [1, 3, 6], "possibl": [1, 2, 3, 4, 5, 6], "calc": [1, 2, 3, 6], "keyword": [1, 2, 4, 5], "notic": [1, 2, 3, 4, 5, 6], "finicki": 1, "dot": [1, 6], "underscir": 1, "justif": [1, 2], "format": 1, "indic": [1, 2, 3, 5], "indent": 1, "One": [1, 2, 3, 4, 5, 6], "outlin": [1, 2, 4, 6], "modulo": [1, 4, 5], "individu": 1, "pure": [1, 6], "littl": [1, 2, 6], "underneath": [1, 3], "pow_two": [1, 4, 5], "mul_sub": 1, "add_sub": 1, "sub_sub": 1, "add_zero": [1, 5], "perform": [1, 2, 3], "exact": [1, 2, 3, 4, 5, 6], "becaus": [1, 2, 3, 4, 5, 6], "exactli": [1, 2, 3, 5, 6], "close": [1, 2, 3], "note": [1, 2, 4, 5, 6], "bit": [1, 4, 6], "commut": [1, 2, 3, 4, 5], "indirectli": 1, "data": [1, 2, 4, 5, 6], "similar": [1, 2, 3, 5, 6], "common": [1, 2, 3, 4, 5, 6], "variat": [1, 3, 4, 5, 6], "call": [1, 2, 3, 4, 5, 6], "nth_rewrit": 1, "allow": [1, 2, 3, 4, 5, 6], "particular": [1, 2, 5, 6], "instanc": [1, 2, 3, 4, 5, 6], "enumer": [1, 2], "start": [1, 2, 3, 4, 6, 7, 10], "zero": [1, 2, 3, 4, 5, 6], "occurr": 1, "nth_rw": 1, "nth_rewrite_lh": 1, "nth_rewrite_rh": 1, "sophist": [1, 6], "subexpress": 1, "consist": [1, 3, 5, 6], "collect": [1, 3, 5, 6], "oper": [1, 2, 3, 4, 5, 6], "time": [1, 4, 5, 6], "constant": [1, 2], "mapsto": [1, 5], "abelian": [1, 5], "negat": [1, 5, 10], "invers": [1, 3, 5], "axiom": [1, 3, 5, 6], "add_comm": [1, 4, 5], "zero_add": [1, 4, 5], "add_left_neg": [1, 5], "mul_on": [1, 5], "one_mul": [1, 2, 5], "being": [1, 3, 5, 6], "suffic": [1, 2, 3, 6], "give": [1, 2, 3, 4, 6], "element": [1, 2, 3, 4, 5, 6], "concret": [1, 2, 4, 5, 6], "integ": [1, 2, 4, 10], "abstract": [1, 2, 6, 10], "character": [1, 3, 4, 6], "axiomat": [1, 2, 3, 5], "train": 1, "recogn": [1, 2, 4, 5, 6], "appropri": [1, 2, 3, 5], "\u2124": [1, 4, 5], "ration": [1, 2, 4, 6], "\u211a": [1, 4, 6], "\u2102": [1, 5], "extend": [1, 3, 4, 5, 6], "Not": [1, 5, 6], "properti": [1, 2, 3, 4, 5, 6], "hold": [1, 2, 3, 4, 5, 6], "arbitrari": [1, 2, 3, 4, 5], "taken": [1, 2], "cours": [1, 2, 4, 6], "linear": [1, 2, 5, 6], "matric": [1, 5], "fail": [1, 2, 3, 4, 5, 6], "commr": [1, 2, 5], "unchang": [1, 2], "linarith": [1, 2, 4, 5], "permiss": 1, "strike": [1, 2], "balanc": 1, "concis": [1, 5], "readabl": [1, 2, 3, 4, 6], "strengthen": [1, 2, 4], "skill": [1, 2, 3, 4], "deriv": [1, 2, 4, 7], "most": [1, 2, 3, 4, 5, 6], "content": [1, 4, 6], "organiz": 1, "mechan": [1, 2, 5], "namespac": [1, 2, 3, 4, 5, 6], "shorter": [1, 3, 4, 6], "avoid": [1, 3, 4, 5], "due": [1, 3], "clash": 1, "myre": 1, "add_right_neg": 1, "effect": [1, 3, 6], "temporarili": [1, 2], "reprov": 1, "care": [1, 2, 3, 5], "earlier": 1, "pai": [1, 2, 6], "attent": [1, 2, 4, 6], "curli": [1, 2], "implicit": [1, 2, 5, 6], "moment": [1, 2, 5], "worri": [1, 3, 4, 5], "neg_add_cancel_left": 1, "add_neg_cancel_right": 1, "add_left_cancel": 1, "add_right_cancel": 1, "plan": [1, 6], "brace": 1, "imagin": 1, "situat": [1, 3, 6], "draw": [1, 5], "conclus": [1, 2, 4], "hypothes": [1, 2, 3, 4, 5], "redund": [1, 5, 6], "few": [1, 2, 3, 4, 6], "extra": [1, 2, 3, 5, 6], "oner": 1, "complic": [1, 2], "tediou": [1, 2], "mark": [1, 2, 6], "suppos": [1, 2, 3, 4, 5, 6], "infer": [1, 2, 3, 5], "mul_zero": [1, 5], "serv": [1, 3, 4, 5], "therefor": [1, 4, 5, 6], "promot": 1, "modular": 1, "subproof": 1, "wa": [1, 3], "except": [1, 5], "ad": [1, 2, 3, 4, 5, 6], "free": [1, 6], "At": [1, 2, 4, 5], "rememb": [1, 2, 3, 4, 5, 6], "zero_mul": [1, 2, 4, 5], "By": [1, 3, 5], "neg_eq_of_add_eq_zero": 1, "eq_neg_of_add_eq_zero": 1, "neg_zero": 1, "neg_neg": 1, "had": 1, "annot": [1, 3, 4, 5], "third": [1, 2, 3, 5, 6], "specifi": [1, 2, 3, 4, 5, 6], "imposs": 1, "default": [1, 2, 3, 4, 5], "subtract": [1, 4, 5], "provabl": [1, 3, 6], "sub_eq_add_neg": [1, 5], "On": [1, 2, 3, 5, 6], "defin": [1, 2, 3, 4, 5, 6], "reflex": [1, 2], "present": [1, 2, 4, 6], "forc": [1, 2, 3], "unfold": [1, 2, 3, 4, 5, 6], "refl": [1, 2, 3, 5], "deal": [1, 2, 3, 4, 5, 6], "equat": [1, 2, 3, 4, 5], "interchang": 1, "self_sub": 1, "either": [1, 2, 3, 5, 6], "effort": [1, 4], "one_add_one_eq_two": 1, "norm_num": [1, 2, 4, 5], "strength": 1, "weaker": [1, 2], "notion": [1, 2, 3, 4, 5, 6, 7], "addgroup": 1, "otherwis": [1, 2, 3, 4, 5], "variant": [1, 2, 6], "addcommgroup": 1, "commgroup": 1, "mul_left_inv": [1, 5], "\u00b9": [1, 3, 5, 6], "feel": [1, 2, 6], "cocki": 1, "helper": 1, "along": [1, 4, 6], "hint": [1, 2], "mul_right_inv": [1, 5], "mul_inv_rev": 1, "non": [1, 3, 6], "abel": 1, "noncomm_r": 1, "seem": [1, 2, 4, 6], "odd": [1, 2, 3, 4], "partli": 1, "histor": 1, "conveni": [1, 2, 3, 5, 6], "great": [1, 5], "sort": [1, 3, 4], "inequ": [1, 2, 6], "le": [1, 5], "whenev": [1, 2, 4, 5], "consid": [1, 2, 3, 4, 5, 6, 7], "le_refl": [1, 2], "le_tran": [1, 2, 5], "detail": [1, 2, 3, 4, 5], "unless": [1, 2, 5], "realli": [1, 2, 3, 4, 6], "insist": 1, "discuss": [1, 2, 4, 5, 6], "implic": [1, 6, 10], "h\u2080": [1, 2, 3, 4, 6], "creat": 1, "option": [1, 2, 4, 5, 6], "visibl": 1, "must": [1, 5, 6], "complet": [1, 2, 3, 4], "decreas": [1, 5], "fourth": [1, 2], "mode": [1, 2, 5], "entir": [1, 3, 5, 6], "lt_of_le_of_lt": [1, 2, 5], "lt_of_lt_of_l": 1, "lt_tran": [1, 2], "togeth": [1, 2, 3, 4, 5, 6], "handl": [1, 3, 4], "arithmet": 1, "5": [1, 2, 5, 6], "pass": [1, 5], "exp_le_exp": 1, "mpr": [1, 2, 5, 6], "exp": [1, 3], "applic": [1, 2, 4, 5], "function": [1, 2, 4, 5, 7, 10], "compound": [1, 2, 6], "pars": [1, 3], "exp_lt_exp": 1, "log_le_log": 1, "log": [1, 3], "log_lt_log": 1, "add_le_add": [1, 2, 5], "add_le_add_left": 1, "add_le_add_right": 1, "add_lt_add_of_le_of_lt": 1, "add_lt_add_of_lt_of_l": 1, "add_lt_add_left": 1, "add_lt_add_right": 1, "add_nonneg": [1, 5], "add_po": 1, "add_pos_of_pos_of_nonneg": 1, "exp_po": [1, 3], "bi": [1, 5, 10], "lr": 1, "iff": [1, 3, 6], "connect": [1, 3, 5, 6], "equival": [1, 2, 3, 4, 5, 6], "mp": [1, 2, 3, 4, 6], "forward": [1, 2, 5, 6], "direct": [1, 2, 3, 6], "stand": [1, 2, 5, 6], "modu": 1, "ponen": 1, "respect": [1, 2, 3, 4, 5, 6], "thu": [1, 3, 4, 5, 6], "again": [1, 2, 3, 4, 5, 6], "numer": [1, 4, 5], "constitut": 1, "part": [1, 2, 3, 4, 5, 6], "strategi": [1, 2, 4], "api": 1, "reli": [1, 2, 3, 5, 6], "guess": [1, 2, 3, 4, 5], "a_of_b_of_c": 1, "approxim": 1, "loud": 1, "probabl": [1, 5, 6], "add_l": 1, "choic": [1, 2, 3, 5, 6], "exist": [1, 2, 3, 6], "jump": [1, 4, 5, 6], "nearbi": [1, 4], "library_search": [1, 4], "sq_nonneg": 1, "delet": [1, 2, 3, 4], "uncom": 1, "previou": [1, 2, 3, 5, 6], "suggest": [1, 2, 4, 5, 6], "long": [1, 2, 3, 5], "better": [1, 3, 4, 5, 6], "confirm": [1, 2, 3, 4], "finish": [1, 2, 3, 4], "job": 1, "pow_two_nonneg": [1, 2], "tend": [1, 6], "around": [1, 3, 6], "binari": [1, 2, 4, 5], "increas": 1, "worth": [1, 2, 5], "definition": [1, 4, 5, 6], "principl": [1, 2, 3, 4], "favor": [1, 2, 5], "timesav": 1, "clever": 1, "involv": [1, 2, 4, 5, 6], "nice": [1, 2, 4, 6], "idea": [1, 2, 3, 5, 6], "abs_l": [1, 5], "ab": [1, 2, 5], "congratul": [1, 2, 4], "becom": [1, 4, 5, 6], "min": [1, 2, 6], "uniqu": [1, 2, 3, 4, 5, 6], "min_le_left": 1, "min_le_right": 1, "le_min": 1, "max": [1, 2, 6], "pair": [1, 2, 4, 5, 6], "act": 1, "curri": 1, "logician": 1, "haskel": 1, "get": [1, 2, 3, 4, 6, 10], "bind": [1, 3], "tighter": [1, 3], "infix": 1, "le_antisymm": [1, 2], "less": [1, 2, 3, 4, 5, 6], "usag": 1, "inconsist": 1, "outer": 1, "level": [1, 2], "nest": [1, 2], "maintain": 1, "bother": [1, 3], "repetit": 1, "foreshadow": 1, "univers": [1, 3, 5, 6, 10], "quantifi": [1, 3, 4, 5, 6, 10], "desir": [1, 3, 6], "implicitli": [1, 2], "mani": [1, 2, 3, 5, 6, 10], "whether": [1, 2, 3, 4, 6], "Of": [1, 2, 4, 6], "interest": [1, 2, 3, 5, 6], "vice": [1, 3], "versa": [1, 3], "word": [1, 2, 3, 4, 5, 6], "switch": 1, "transit": [1, 2, 5, 6], "total": 1, "satisfi": [1, 3, 4, 5, 6], "disjunct": [1, 3, 10], "stick": [1, 2, 6, 7], "split": [1, 2, 3, 4], "aux": [1, 2, 4, 6], "valu": [1, 2, 3, 4, 5, 6], "yield": [1, 2, 3, 4, 5, 6], "made": [1, 2, 3, 5, 6], "manifest": [1, 6], "triangl": [1, 2, 5], "abs_add": [1, 2], "sub_add_cancel": [1, 5], "relat": [1, 2, 3, 5, 6], "ordinari": [1, 2, 4, 5, 6], "unicod": [1, 3], "obtain": [1, 2, 4, 5, 6], "dvd": 1, "dvd_tran": 1, "dvd_mul_of_dvd_left": 1, "dvd_mul_left": 1, "expon": 1, "expand": [1, 2, 3, 4, 5], "w": 1, "greatest": [1, 5, 6], "divisor": [1, 2, 4, 5], "gcd": [1, 2, 4], "least": [1, 5, 6], "lcm": 1, "analog": [1, 2, 3, 4, 5, 6], "divid": [1, 2, 4, 5], "gcd_zero_right": 1, "gcd_zero_left": 1, "lcm_zero_right": 1, "lcm_zero_left": 1, "similarli": [1, 2, 3, 4, 5], "prefix": [1, 4], "dvd_antisymm": 1, "complain": 1, "ambigu": [1, 5], "_root_": 1, "saw": [1, 5, 6], "govern": [1, 5], "class": [1, 4, 5, 6, 7], "describ": [1, 2, 3, 4, 5, 6], "\u03b1": [1, 2, 3, 4, 5, 6], "partialord": [1, 2], "adopt": 1, "letter": [1, 5], "\u03b2": [1, 2, 3, 5, 6], "\u03b3": [1, 2, 5, 6], "greek": [1, 4], "especi": [1, 5, 6], "strict": [1, 2], "somewhat": [1, 3, 6], "lt_irrefl": [1, 2], "lt_iff_le_and_n": 1, "lattic": [1, 5, 6], "inf_le_left": [1, 6], "inf_le_right": 1, "le_inf": 1, "le_sup_left": 1, "le_sup_right": 1, "sup_l": 1, "lower": [1, 2, 4], "bound": [1, 2, 3, 4, 5, 6], "upper": [1, 2], "glb": 1, "lub": 1, "infimum": [1, 6], "supremum": [1, 4], "inf": [1, 6], "sup": [1, 4, 6], "further": [1, 6], "matter": [1, 2], "meet": [1, 2, 3, 5], "join": [1, 5], "keep": [1, 5], "dictionari": 1, "subset": [1, 2, 3, 5, 6], "domain": [1, 2, 3, 4, 5, 6], "boolean": 1, "truth": [1, 4], "fals": [1, 2, 3, 4], "true": [1, 2, 3, 6], "posit": [1, 3, 4, 5, 6], "subspac": 1, "vector": [1, 3], "intersect": [1, 3, 5, 6], "sum": [1, 2, 4, 5, 6], "inclus": [1, 3, 6], "topolog": [1, 5, 10], "union": [1, 3, 4, 5], "inf_comm": 1, "inf_assoc": 1, "sup_comm": 1, "sup_assoc": 1, "absorpt": 1, "law": 1, "absorb1": 1, "absorb2": 1, "found": [1, 5], "inf_sup_self": 1, "sup_inf_self": 1, "distriblattic": 1, "inf_sup_left": 1, "inf_sup_right": 1, "sup_inf_left": 1, "sup_inf_right": 1, "shown": [1, 5], "explicit": [1, 2, 5, 6], "nondistribut": 1, "finit": [1, 3, 4, 5, 6], "impli": [1, 2, 3, 4, 5, 6], "larger": [1, 6], "carrier": [1, 5], "compat": [1, 6], "strictorderedr": 1, "mul_po": [1, 4], "mul_nonneg": 1, "coupl": [1, 2, 6], "metric": [1, 5, 10], "equip": [1, 3, 5, 6], "distanc": [1, 2, 6], "dist": [1, 6], "map": [1, 2, 3, 5, 6], "metricspac": [1, 6], "dist_self": 1, "dist_comm": [1, 6], "dist_triangl": [1, 6], "nonneg": [1, 5], "nonneg_of_mul_nonneg_left": 1, "dist_nonneg": [1, 6], "dealt": 2, "basic": [2, 4, 5, 6, 10], "simpl": [2, 4, 5], "absolut": 2, "\u03b5": [2, 6], "though": [2, 3, 4, 5], "treat": [2, 3, 5, 6], "my_lemma": 2, "\u03b4": [2, 6], "hb": [2, 6], "subsequ": [2, 4, 6], "lemma": [2, 4, 5, 6, 10], "my_lemma2": 2, "stage": [2, 5], "my_lemma3": 2, "epo": 2, "ele1": 2, "xlt": 2, "ylt": 2, "essenti": [2, 3, 5, 6], "colon": 2, "why": [2, 4, 6], "off": [2, 3, 4], "my_lemma4": 2, "abs_mul": 2, "mul_le_mul": 2, "abs_nonneg": 2, "mul_lt_mul_right": 2, "extract": [2, 6], "hidden": 2, "expos": [2, 6], "predic": [2, 3, 4, 6], "fn_ub": 2, "fn_lb": 2, "fnub": 2, "fnlb": 2, "hfa": 2, "hgb": 2, "dsimp": [2, 3, 4, 5], "simplif": [2, 3, 4], "contract": 2, "anyhow": 2, "control": 2, "transform": [2, 3], "rest": [2, 3, 5, 6], "routin": 2, "nnf": 2, "nng": 2, "hfb": 2, "nna": 2, "order": [2, 3, 4, 5, 6, 10], "codomain": [2, 4, 6], "structur": [2, 4, 6, 10], "monoid": [2, 3, 4], "fn_ub_add": 2, "orderedcanceladdcommmonoid": 2, "high": 2, "monoton": [2, 6], "nondecreas": [2, 4], "placehold": 2, "Or": [2, 3], "backward": [2, 6], "subgoal": 2, "mf": 2, "mg": 2, "aleb": 2, "lambda": [2, 5], "underscor": [2, 3], "flag": [2, 3], "squiggli": 2, "marker": 2, "nnc": 2, "bbb": [2, 5], "fneven": 2, "fnodd": 2, "ef": 2, "eg": 2, "og": 2, "shorten": 2, "rid": 2, "won": [2, 4, 6], "cannot": [2, 3, 4, 5, 6], "contrari": 2, "syntact": 2, "reduct": [2, 3], "erw": 2, "harder": 2, "spot": 2, "rudimentari": 2, "foundat": [2, 3, 4, 5], "impos": 2, "restrict": [2, 3, 4, 6], "talk": [2, 4, 6], "assert": [2, 3], "ask": [2, 4, 5, 6], "xs": [2, 3], "tran": [2, 4, 5], "setub": 2, "inject": [2, 3, 4, 6], "x_1": [2, 3], "x_2": [2, 3], "x\u2081": [2, 3, 5, 6], "x\u2082": [2, 3, 5], "add": [2, 4, 5, 6], "nonzero": [2, 4, 5], "add_left_inj": 2, "composit": [2, 3, 5, 6], "injg": 2, "injf": [2, 3], "canon": [2, 3, 4, 5], "exhibit": [2, 6], "anonym": [2, 3, 4, 5], "constructor": [2, 3, 4, 5], "angl": 2, "whatev": [2, 4], "certain": [2, 6], "fnhasub": 2, "fnhaslb": 2, "fnub_add": 2, "gun": 2, "ubf": 2, "ubg": 2, "ubfa": 2, "ubfb": 2, "unpack": [2, 3], "claus": 2, "whose": [2, 4, 5, 6], "els": [2, 3, 4], "turn": [2, 3, 4, 5, 6, 7], "directli": [2, 4, 5, 6], "lbf": 2, "lbg": 2, "cousin": 2, "rcase": [2, 3, 4, 5, 6], "flexibl": [2, 6], "recurs": [2, 3, 5, 10], "harm": [2, 3], "swiss": 2, "armi": 2, "knive": 2, "wide": 2, "old": [2, 5], "chestnut": 2, "product": [2, 4, 5, 6], "magic": [2, 5], "verifi": 2, "sumofsquar": 2, "sumofsquares_mul": 2, "sosx": 2, "sosi": 2, "xeq": [2, 3], "yeq": 2, "insight": 2, "motiv": 2, "gaussian": [2, 10], "i": [2, 3, 4, 5, 6], "sqrt": [2, 3, 4, 5], "norm": [2, 5, 6], "reflect": 2, "di": [2, 5], "xy": [2, 5], "cryptic": 2, "easiest": [2, 4], "perspicu": 2, "divis": [2, 4, 5, 10], "divab": 2, "divbc": 2, "beq": 2, "ceq": 2, "And": [2, 4, 6], "pretti": [2, 6], "Then": [2, 3, 5, 6], "divac": 2, "alpha": [2, 3], "beta": [2, 3], "surject": [2, 3, 6], "yourself": [2, 5], "field_simp": [2, 5], "denomin": [2, 4, 5], "div_mul_cancel": 2, "hx": [2, 3, 5], "method": [2, 3, 4, 5], "surjg": 2, "surjf": [2, 3], "strictli": 2, "contradict": [2, 3, 4], "speak": [2, 5, 6], "irreflex": 2, "asymmetri": 2, "lt_asymm": 2, "sugar": 2, "eventu": [2, 3, 6], "fnuba": 2, "not_le_of_gt": 2, "not_lt_of_g": [2, 4, 5], "lt_of_not_g": 2, "le_of_not_gt": 2, "snippet": [2, 4, 5], "counterexampl": 2, "monof": 2, "noth": [2, 3, 4, 5], "four": [2, 5], "valid": 2, "far": [2, 3, 6], "q": [2, 4, 5, 6], "straightforward": 2, "difficult": 2, "conclud": 2, "nonexist": 2, "contradictori": 2, "classic": [2, 3, 4], "by_contra": [2, 4], "not_not": 2, "front": 2, "push": [2, 4, 6], "inward": 2, "facilit": [2, 6], "push_neg": [2, 3, 4], "restat": 2, "contrapos": [2, 3, 4], "yet": [2, 3, 4, 5, 6], "semicolon": [2, 4], "falso": 2, "anyth": [2, 5, 6], "elim": 2, "strang": [2, 4], "fairli": 2, "reach": 2, "37": 2, "exfalso": 2, "absurd": [2, 4], "slick": 2, "drop": 2, "manner": [2, 5], "contrast": [2, 3, 5, 6], "compon": [2, 3, 5, 6], "techniqu": [2, 6], "variou": [2, 3, 5], "xltz": 2, "zlty": 2, "10": 2, "7": [2, 4, 6], "behav": [2, 3, 6], "roughli": [2, 3, 4], "friend": [2, 5], "were": [2, 4, 5, 6], "inscrut": [2, 3], "gadget": [2, 6], "auxiliari": [2, 6], "pow_eq_zero": [2, 4], "doubl": [2, 3], "symmetr": [2, 3], "abs_lt": 2, "dvd_gcd_iff": 2, "8": [2, 4, 6], "6": [2, 4, 5, 6], "15": 2, "below": [2, 3, 4, 5, 6], "not_monotone_iff": 2, "antisymmetr": 2, "aris": 2, "preorder": [2, 6], "pre": 2, "lt_iff_le_not_l": 2, "beyond": 2, "repeatedli": [2, 6], "instanti": [2, 5, 6], "inl": [2, 3], "inr": [2, 3], "produc": 2, "branch": [2, 3], "le_or_gt": 2, "abs_of_nonneg": [2, 5], "abs_of_neg": 2, "immedi": [2, 3, 5], "myab": 2, "le_abs_self": [2, 5], "neg_le_abs_self": 2, "enjoi": [2, 3, 4, 6], "pun": 2, "lt_ab": 2, "genuin": 2, "vertic": [2, 5], "lt_trichotomi": 2, "xgt": 2, "dvd_mul_right": [2, 4], "eq_zero_or_eq_zero_of_mul_eq_zero": 2, "nontrivi": [2, 3, 4, 5, 6], "integr": [2, 7], "isdomain": 2, "em": [2, 3], "exclud": [2, 4], "by_cas": [2, 3, 4], "dispos": 2, "s_0": [2, 3], "s_1": 2, "s_2": 2, "ldot": [2, 3, 4], "varepsilon": [2, 6], "s_n": [2, 3], "render": [2, 4], "convergesto": 2, "ext": [2, 3, 4, 5], "enabl": [2, 3, 4, 5], "actual": [2, 3, 4, 6], "u": [2, 3, 5, 6], "v": [2, 3, 5, 6], "congr": 2, "reconcil": 2, "peel": 2, "convert": [2, 4], "quit": [2, 6], "zero_lt_on": [2, 4], "fill": [2, 3, 4, 5], "convergesto_const": 2, "\u03b5po": [2, 6], "nge": 2, "abs_zero": 2, "save": [2, 3], "troubl": [2, 3, 5], "pen": 2, "paper": [2, 3, 5, 6], "ns": 2, "nt": 2, "maximum": [2, 4, 6], "implement": [2, 5], "convergesto_add": 2, "cs": 2, "ct": 2, "\u03b52po": 2, "hs": [2, 4, 6], "ht": 2, "le_of_max_le_left": 2, "le_of_max_le_right": 2, "tricki": [2, 3, 4], "convergesto_mul_const": 2, "mulzeroclass": 2, "acpo": 2, "abs_po": 2, "independ": [2, 3, 5], "exists_abs_le_of_convergesto": 2, "strong": [2, 4], "n\u2080": 2, "bpo": [2, 6], "pos\u2080": 2, "div_po": 2, "n\u2081": 2, "convergesto_mul": 2, "sketch": [2, 3, 4, 6], "limit": [2, 6], "bold": 2, "scratch": 2, "convergesto_uniqu": 2, "sa": 2, "sb": 2, "abn": 2, "na": 2, "hna": 2, "nb": 2, "hnb": 2, "absa": 2, "absb": 2, "observ": [2, 4, 6], "everywher": [2, 3, 6], "linearord": 2, "vastli": 2, "awai": [2, 5], "vocabulari": 3, "uniform": [3, 6], "primit": 3, "conceptu": 3, "advantag": [3, 4, 5], "overload": 3, "verbos": 3, "system": [3, 4, 5], "wrong": 3, "theoret": [3, 6], "ss": 3, "sub": [3, 6], "cap": 3, "un": 3, "cup": 3, "univ": [3, 6], "empti": [3, 4, 5, 6], "member": [3, 6], "membership": [3, 4], "mem": 3, "notin": 3, "ident": [3, 4, 5, 6, 10], "databas": [3, 4, 6], "unlik": [3, 4], "existenti": [3, 10], "subset_def": 3, "inter_def": 3, "xu": 3, "mem_inter_iff": 3, "xsu": 3, "phenomenon": 3, "quirk": 3, "pitfal": 3, "heavili": [3, 6], "fall": 3, "union_def": 3, "mem_union": [3, 4], "xtu": 3, "xt": 3, "unnecessari": 3, "clearer": [3, 5], "correctli": 3, "special": [3, 4, 5, 6], "rewritten": 3, "diff_eq": 3, "mem_diff": 3, "xstu": 3, "xnt": 3, "xnu": 3, "extension": [3, 5], "unsurprisingli": 3, "dollar": 3, "sign": [3, 6], "and_comm": 3, "antisymm": 3, "hood": [3, 5], "builder": 3, "trivial": [3, 4, 6], "eq_two_or_odd": 3, "even_iff": 3, "confus": [3, 4, 6], "fortun": 3, "agre": 3, "prime_iff": 3, "symm": [3, 4, 5, 6], "rwa": [3, 4, 5], "signific": 3, "ball": 3, "bex": 3, "bex_def": 3, "prime_x": 3, "slight": 3, "ssubt": 3, "index": [3, 6], "model": [3, 6], "sequenc": [3, 6, 10], "a_0": 3, "a_1": 3, "a_2": 3, "mem_iunion": 3, "xai": 3, "mem_iint": 3, "mem_union\u2082": 3, "mem_inter\u2082": 3, "mem_iunion\u2082": 3, "exists_prime_and_dvd": 3, "eq_univ": 3, "eq_univ_of_foral": 3, "exists_infinite_prim": 3, "\u2080": 3, "sunion": 3, "sinter": 3, "relationship": [3, 4], "mem_iinter\u2082": 3, "sunion_eq_biunion": 3, "sinter_eq_biint": 3, "preimag": [3, 6], "imag": [3, 4, 6], "tripl": 3, "tag": 3, "mem_image_of_mem": 3, "galoi": [3, 5, 6], "image_subset_iff": 3, "represent": [3, 4, 5], "asid": 3, "raini": 3, "dai": 3, "behavior": [3, 6], "nonempti": [3, 6], "condit": [3, 4, 6], "fxeq": 3, "ai": 3, "fx": 3, "eq": [3, 5], "injon": 3, "theme": 3, "rel": [3, 5], "relativ": 3, "xpo": 3, "ypo": 3, "exp_log": 3, "ingredi": [3, 5, 6], "assign": [3, 5], "inhabit": [3, 5, 6], "appeal": 3, "choose_spec": 3, "some_spec": 3, "noncomput": [3, 5], "inverse_spec": 3, "dif_po": 3, "dif_neg": 3, "fulli": [3, 6], "alon": 3, "leftinvers": 3, "rightinvers": 3, "hack": 3, "half": 3, "dozen": 3, "condens": 3, "cantor": 3, "famou": 3, "miss": [3, 4], "j": [3, 5], "intuit": [3, 6], "cardin": 3, "biject": [3, 5], "nineteenth": 3, "centuri": 3, "infinit": [3, 6, 10], "dedekind": 3, "quickli": 3, "behind": 3, "problem": [3, 4, 5, 6], "shade": 3, "region": 3, "diagram": 3, "circ": [3, 5], "scale": 3, "inner": 3, "smaller": [3, 4, 6], "concentr": 3, "unshad": 3, "compos": [3, 5, 6], "disjoint": 3, "sound": [3, 5], "plausibl": 3, "delic": 3, "improv": [3, 4], "confid": 3, "invfun": [3, 5], "leftinverse_invfun": 3, "invfun_eq": 3, "sbaux": 3, "sbset": 3, "sb_aux": 3, "s_": 3, "sb_set": 3, "bigcup_": 3, "mathbb": [3, 4, 5], "sbfun": 3, "complement": [3, 6], "outermost": 3, "setminu": 3, "inv_fun": 3, "inv_fun_eq": 3, "sb_right_inv": 3, "goe": [3, 4, 5, 6], "henc": [3, 4, 5, 6], "neither": [3, 5], "nor": [3, 5], "sb_inject": 3, "hf": [3, 6], "hg": [3, 6], "a_def": 3, "h_def": 3, "hxeq": 3, "xa": [3, 5], "wlog": 3, "x\u2081a": 3, "resolve_left": 3, "x\u2082a": 3, "not_imp_self": 3, "x\u2082na": 3, "if_po": 3, "if_neg": 3, "x\u2082eq": 3, "hn": [3, 4, 6], "sb_fun": 3, "bring": [3, 4, 6], "tradeoff": 3, "encapsul": [3, 5], "symmetri": 3, "dwell": 3, "succ": [3, 4], "sb_surject": 3, "gya": 3, "xmem": 3, "sweet": 3, "schroeder_bernstein": 3, "substant": 4, "ancient": 4, "fraction": 4, "lowest": 4, "2c": 4, "4c": 4, "factor": [4, 5], "coprim": 4, "smart": 4, "12": 4, "encount": 4, "algebra": [4, 6, 10], "prime_def_lt": 4, "eq_one_or_self_of_dvd": 4, "prime_p": 4, "17": 4, "commonli": [4, 5], "prime_two": 4, "prime_thre": 4, "broader": [4, 7], "irreduc": [4, 5], "coincid": [4, 5, 6], "rise": 4, "dvd_mul": 4, "even_of_even_sqr": 4, "dvd_of_dvd_pow": 4, "proce": 4, "profici": 4, "ctrl": [4, 5], "search": [4, 5], "engin": 4, "hesit": 4, "mul_right_inj": 4, "heart": 4, "irration": 4, "dvd_gcd": 4, "coprime_mn": 4, "sqr_eq": 4, "meq": 4, "dvd_iff_exists_eq_mul_left": 4, "two_l": 4, "le_of_dvd": 4, "approach": [4, 5, 6], "quick": 4, "ne": [4, 5], "occur": 4, "suffici": [4, 6], "permut": 4, "prime_of_mem_factor": 4, "prod_factor": 4, "factors_uniqu": 4, "factorization_mul": 4, "mnez": 4, "nnez": 4, "factorization_pow": 4, "black": 4, "box": 4, "simpa": 4, "nnz": 4, "nsqr_nez": 4, "eq1": 4, "eq2": 4, "add_mul_mod_self_left": 4, "mul_mod_right": 4, "count_factors_mul_of_po": 4, "successor": 4, "succ_ne_zero": 4, "npow_nz": 4, "dvd_sub": 4, "pow_eq": 4, "npowz": 4, "add_sub_cancel": 4, "understood": [4, 6], "quotient": [4, 5, 6], "pictur": [4, 5], "mediat": 4, "headach": 4, "contend": 4, "issu": [4, 5, 6], "th": 4, "topic": 4, "enat": 4, "infin": [4, 6], "appreci": 4, "role": [4, 5, 6], "_section_induction_and_recurs": 4, "writ": 4, "datatyp": 4, "freeli": 4, "translat": [4, 6], "mathematician": [4, 6], "inj": 4, "factori": 4, "fac": 4, "ih": 4, "fac_po": 4, "succ_po": 4, "dvd_fac": 4, "ipo": 4, "il": 4, "of_le_succ": 4, "dvd_mul_of_dvd_right": 4, "crude": 4, "remaind": [4, 5], "pow_two_le_fac": 4, "finset": [4, 5, 6], "bigoper": [4, 5], "prod": [4, 6], "sum_range_zero": 4, "sum_range_succ": 4, "summat": 4, "prod_range_zero": 4, "prod_range_succ": 4, "deserv": 4, "comment": 4, "danger": [4, 5], "ordinarili": [4, 5], "loop": 4, "indefinit": 4, "fix": [4, 6], "placement": 4, "re": [4, 5], "handi": 4, "sum_id": 4, "div_eq_of_eq_mul_right": 4, "succ_eq_add_on": 4, "sum_sqr": 4, "mynat": 4, "thumb": 4, "decid": [4, 6], "preced": 4, "truncat": 4, "exponenti": 4, "cut": 4, "predecessor": 4, "pred": 4, "mul": [4, 5, 6], "succ_add": 4, "succ_mul": 4, "explor": [4, 6], "standard": [4, 5, 6], "formul": [4, 6], "quirki": 4, "among": 4, "annoi": 4, "h0": 4, "succ_le_succ": 4, "zero_l": [4, 5], "interval_cas": 4, "interv": [4, 6], "decis": 4, "procedur": 4, "revert": [4, 5], "minfac": 4, "smallest": [4, 6], "strong_induction_on": 4, "subsum": 4, "exists_prime_factor": 4, "np": 4, "mltn": 4, "mdvdn": 4, "mne1": 4, "mz": 4, "zero_dvd_iff": 4, "mgt2": 4, "pp": 4, "pdvd": 4, "factorial_po": 4, "dvd_factori": 4, "primes_infinit": 4, "refin": 4, "ple": 4, "p_1": 4, "p_n": 4, "prod_": 4, "p_i": [4, 6], "computation": 4, "test": 4, "decidableeq": 4, "abandon": 4, "ourselv": 4, "subset_iff": 4, "mem_int": 4, "mem_sdiff": 4, "tauto": 4, "dispens": 4, "tautolog": 4, "dvd_prod_of_mem": 4, "eq_of_dvd_of_prim": 4, "prime_q": 4, "preserv": [4, 6], "induction_on": 4, "singleton": 4, "prod_empti": 4, "prod_insert": 4, "mem_of_dvd_prod_prim": 4, "mem_insert": 4, "wrote": 4, "filter": [4, 10], "mem_filt": 4, "aim": 4, "prod_po": 4, "_def": 4, "mem_": 4, "id": [4, 6], "bounded_of_ex_finset": 4, "qk": 4, "lt_succ_of_l": 4, "le_sup": 4, "ex_finset_of_bound": 4, "decidablepr": 4, "lt_succ_iff": 4, "congruent": 4, "p_k": 4, "loss": 4, "27": 4, "mod_4_eq_3_or_mod_4_eq_3": 4, "mul_mod": 4, "mod_lt": 4, "hm": 4, "two_le_of_mod_4_eq_3": 4, "neq": 4, "div_dvd_of_dvd": 4, "div_lt_self": 4, "piec": [4, 5, 6], "exists_prime_factor_mod_4_eq_3": 4, "dvd_rfl": 4, "mge2": 4, "mul_div_cancel": 4, "home": [4, 5], "stretch": [4, 5], "remov": [4, 6], "eras": 4, "mem_eras": 4, "readi": [4, 6], "dvd_add_iff_left": 4, "primes_mod_4_eq_3_infinit": 4, "p4": 4, "pltn": 4, "p4eq": 4, "ps": 4, "pne3": 4, "seriou": [4, 6], "feat": 4, "modern": 5, "subject": 5, "mysteri": 5, "technic": 5, "consult": 5, "ann": 5, "baanen": 5, "abus": 5, "paramet": 5, "broadest": 5, "constraint": [5, 6], "bundl": [5, 6], "tupl": 5, "hy": 5, "hz": 5, "mypoint1": 5, "mypoint2": 5, "mypoint3": 5, "mk": [5, 6], "former": 5, "latter": [5, 6], "quot": [5, 6], "protect": 5, "intern": 5, "add_x": 5, "addalt": 5, "etc": [5, 6], "y\u2081": 5, "z\u2081": 5, "y\u2082": 5, "z\u2082": 5, "addalt_x": 5, "addalt_comm": 5, "ya": 5, "za": 5, "xb": 5, "yb": 5, "zb": 5, "apart": 5, "effici": [5, 6], "scalar": 5, "smul": 5, "smul_distrib": 5, "road": 5, "link": 5, "belong": [5, 6], "simplex": 5, "convinc": 5, "equilater": 5, "interior": 5, "standardtwosimplex": 5, "x_nonneg": 5, "y_nonneg": 5, "z_nonneg": 5, "sum_eq": 5, "swap": 5, "swapxi": 5, "interestingli": 5, "midpoint": 5, "div_nonneg": 5, "weight": 5, "averag": 5, "weightedaverag": 5, "lambda_nonneg": 5, "lambda_l": 5, "fin": 5, "standardsimplex": 5, "sum_eq_on": 5, "div_eq_mul_inv": 5, "sum_mul": 5, "sum_add_distrib": 5, "mul_sum": 5, "manipul": [5, 6], "islinear": 5, "is_addit": 5, "preserves_mul": 5, "linf": 5, "subtyp": [5, 6], "preal": 5, "val": 5, "sigma": 5, "wherebi": [5, 6], "stdsimplex": 5, "\u03c3": 5, "fst": [5, 6], "snd": [5, 6], "custom": 5, "robust": 5, "interfac": 5, "redefin": 5, "accessor": 5, "weav": 5, "rich": 5, "interconnect": 5, "hierarchi": 5, "clarifi": 5, "antireflex": 5, "cdot": 5, "mathcal": 5, "proxi": 5, "bipartit": 5, "graph": 5, "categori": [5, 6], "morphism": 5, "basi": [5, 6], "discret": 5, "inherit": 5, "polynomi": 5, "coeffici": 5, "dual": [5, 6], "accommod": 5, "almost": [5, 6], "marriag": 5, "heaven": 5, "group\u2081": 5, "inv": 5, "struc": 5, "counterpart": 5, "chosen": 5, "assur": 5, "groupcat": 5, "group\u2081cat": 5, "str": 5, "capit": 5, "roman": 5, "equiv": 5, "tofun": 5, "right_inv": 5, "left_inv": 5, "creativ": 5, "evid": 5, "coercion": [5, 6], "omit": 5, "perm": 5, "under": [5, 6], "orient": 5, "permgroup": 5, "trans_assoc": 5, "trans_refl": 5, "refl_tran": 5, "self_trans_symm": 5, "grouptheori": 5, "g_1": 5, "g_2": 5, "g_3": 5, "tightli": 5, "isomorph": 5, "additivegroup": 5, "Its": 5, "neg": [5, 6], "reproduc": 5, "accompani": 5, "addgroup\u2081": 5, "scheme": 5, "add_group_point": 5, "add_group\u2081": 5, "arrang": 5, "mul_inv_cancel_right": 5, "achiev": [5, 6], "silent": 5, "regist": 5, "grp": 5, "contextu": 5, "cue": 5, "synthes": 5, "whole": 5, "_inst_1": 5, "candid": 5, "group\u2082": 5, "mysquar": 5, "my_squar": 5, "remark": 5, "head": 5, "store": 5, "headi": 5, "hasmulgroup\u2082": 5, "hasonegroup\u2082": 5, "hasinvgroup\u2082": 5, "suppli": 5, "accord": 5, "capabl": 5, "chain": 5, "recent": 5, "prioriti": 5, "bad": [5, 6], "artifici": 5, "addgroup\u2082": 5, "subtl": [5, 6], "configur": 5, "invis": 5, "wise": 5, "euclidean": 5, "terminolog": 5, "mid": 5, "imaginari": 5, "gaussint": 5, "im": 5, "pointwis": [5, 6], "root": [5, 10], "ac": 5, "bci": 5, "adi": 5, "bd": 5, "bc": 5, "hasmul": 5, "zero_def": 5, "one_def": 5, "add_def": 5, "neg_def": 5, "mul_def": 5, "zero_r": 5, "zero_im": 5, "one_r": 5, "one_im": 5, "add_r": 5, "add_im": 5, "neg_r": 5, "neg_im": 5, "mul_r": 5, "mul_im": 5, "surprisingli": 5, "concept": [5, 6], "light": 5, "bulb": 5, "skeleton": [5, 6], "scari": 5, "entri": 5, "instcommr": 5, "left_distrib": 5, "right_distrib": 5, "ext_iff": 5, "bq": 5, "archetyp": 5, "int": 5, "ediv_add_emod": 5, "emod_nonneg": 5, "emod_lt": 5, "unit": [5, 6], "algorithm": 5, "conjug": 5, "frac": 5, "nearest": 5, "size": 5, "vi": 5, "multipli": 5, "view": [5, 6], "emb": 5, "forth": 5, "quadrat": 5, "gaussian_int": 5, "stai": 5, "face": [5, 6], "machineri": [5, 6], "adapt": 5, "invest": 5, "pragmat": 5, "seek": 5, "heather": 5, "macbeth": 5, "eleg": 5, "div": 5, "mod": 5, "_add_mod": 5, "abs_mod": 5, "_le": 5, "emod_lt_of_po": 5, "zero_lt_two": 5, "fixm": 5, "_eq": 5, "sq_add_sq_eq_zero": 5, "linearorderedr": 5, "norm_nonneg": 5, "norm_eq_zero": 5, "norm_po": 5, "norm_mul": 5, "conj": 5, "conj_r": 5, "conj_im": 5, "norm_conj": 5, "bespok": 5, "quad": 5, "record": 5, "div_def": 5, "mod_def": 5, "messi": 5, "nicer": [5, 6], "norm_mod_lt": 5, "norm_y_po": 5, "sub_mul": 5, "conv": 5, "lh": 5, "sq_le_sq": 5, "mul_le_mul_of_nonneg_left": 5, "ediv_mul_l": 5, "ediv_nonneg": 5, "le_of_mul_le_mul_right": 5, "ediv_lt_of_lt_mul": 5, "natab": 5, "coe_natabs_norm": 5, "natabs_of_nonneg": 5, "natabs_norm_mod_lt": 5, "ofnat_lt": 5, "coe_natab": 5, "not_norm_mul_left_lt_norm": 5, "natabs_mul": 5, "le_mul_of_one_le_right": 5, "ofnat_l": 5, "add_one_le_of_lt": 5, "euclideandomain": 5, "quotient_mul_add_remainder_eq": 5, "quotient_zero": 5, "r_wellfound": 5, "remainder_lt": 5, "mul_left_not_lt": 5, "payoff": 5, "principalidealr": 5, "irreducible_iff_prim": 5, "calculu": 6, "quantiti": 6, "studi": 6, "begun": 6, "paradox": 6, "layer": 6, "naiv": 6, "slightli": 6, "exot": 6, "intermedi": 6, "x\u2080": 6, "convention": 6, "eight": 6, "varieti": 6, "wish": 6, "64": 6, "y\u2080": 6, "z\u2080": 6, "paragraph": 6, "512": 6, "bourbaki": 6, "spell": 6, "dualli": 6, "arbitrarili": 6, "neighborhood": 6, "at_top": 6, "\ud835\udcdd": 6, "\ud835\udce4": 6, "entourag": 6, "\u03bc": 6, "a_": 6, "univ_set": 6, "sets_of_superset": 6, "inter_set": 6, "blur": 6, "princip": 6, "\ud835\udcdf": 6, "demonstr": 6, "opportun": 6, "x_0": 6, "ioo": 6, "tendsto\u2081": 6, "tendsto": 6, "lim_": 6, "sourc": 6, "abstractli": 6, "hide": 6, "salient": 6, "pushforward": 6, "f_": 6, "ve": 6, "tendsto\u2082": 6, "via": 6, "promis": 6, "leverag": 6, "map_mono": 6, "map_map": 6, "shot": 6, "256": 6, "pullback": 6, "comap": 6, "map_le_iff_le_comap": 6, "contravari": 6, "comap_comap": 6, "shift": 6, "plane": 6, "\u1da0": 6, "nhds_prod_eq": 6, "aforement": 6, "le_inf_iff": 6, "attop": 6, "bottom": 6, "shouldn": 6, "prohibit": 6, "global": 6, "precondit": 6, "closur": 6, "ne_bot": 6, "tour": 6, "claim": 6, "recaptur": 6, "superfici": 6, "stronger": 6, "famili": 6, "\u03b9": 6, "select": 6, "hasbasi": 6, "nhds_basis_ioo_po": 6, "has_basi": 6, "tendsto_iff": 6, "reformul": 6, "ici": 6, "attop_basi": 6, "knew": 6, "gave": 6, "n_p": 6, "n_q": 6, "tiresom": 6, "unpleas": 6, "superscript": 6, "hp": 6, "hq": 6, "eventually_eq": 6, "tendsto_congr": 6, "review": 6, "ensur": 6, "eventually_of_foral": 6, "mono": 6, "item": 6, "filter_upward": 6, "hr": 6, "reader": 6, "ae": 6, "aka": 6, "occasion": 6, "frequent": 6, "mem_closure_of_tendsto": 6, "cluster_pt": 6, "mem_closure_iff_clusterpt": 6, "le_principal_iff": 6, "nebot_of_l": 6, "hux": 6, "hum": 6, "dist_eq_zero": 6, "emetricspac": 6, "pseudometricspac": 6, "pseudoemetricspac": 6, "journei": 6, "recast": 6, "tendsto_attop": 6, "continuous_iff": 6, "devot": 6, "uncurri": 6, "slow": 6, "continuous_fst": 6, "comp": 6, "assembl": 6, "prod_mk": 6, "continuous_snd": 6, "continuous_dist": 6, "clunki": 6, "crucial": 6, "gradual": 6, "elabor": 6, "refus": 6, "prod_map": 6, "sad": 6, "wrap": 6, "border": 6, "obfusc": 6, "continuous_pow": 6, "continuous_id": 6, "continuousat": 6, "continuousat_iff": 6, "geometr": 6, "closedbal": 6, "radiu": 6, "mem_ball_self": 6, "mem_closedball_self": 6, "isopen": 6, "isopen_iff": 6, "Their": 6, "isclos": 6, "s\u1d9c": 6, "isopen_compl_iff": 6, "hu": 6, "mem_of_tendsto": 6, "mem_closure_iff": 6, "mem_closure_iff_seq_limit": 6, "main": 6, "nhds_basis_bal": 6, "nhds_basis_closedbal": 6, "mem_iff": 6, "segment": 6, "somewher": 6, "continuouson": 6, "minimum": 6, "deduc": 6, "iscompact": 6, "icc": 6, "iscompact_icc": 6, "\u03c6": 6, "strictmono": 6, "tendsto_subseq": 6, "exists_forall_l": 6, "exists_forall_g": 6, "compactspac": 6, "iscompact_univ": 6, "cauchi": 6, "uniformcontinu": 6, "uniformcontinuous_iff": 6, "clearli": 6, "isclosed_l": 6, "eq_empty_or_nonempti": 6, "attain": 6, "closer": 6, "cauchyseq": 6, "cauchyseq_iff": 6, "completespac": 6, "cauchyseq_tendsto_of_complet": 6, "criterion": 6, "explan": 6, "tendsto_pow_attop_nhds_0_of_lt_1": 6, "dist_le_range_sum_dist": 6, "cauchyseq_of_le_geometric_two": 6, "\u03b5_po": 6, "le_iff_exists_add": 6, "boss": 6, "bair": 6, "exclam": 6, "induct": [6, 10], "rec_on": 6, "ho": 6, "hd": 6, "dens": 6, "densiti": 6, "\u03b4po": 6, "hpo": 6, "hball": 6, "mem_closure_iff_nhds_basi": 6, "recon": 6, "rpo": 6, "rb": 6, "incl": 6, "cdist": 6, "ylim": 6, "yball": 6, "categor": 6, "ignor": 6, "topologicalspac": 6, "isopen_univ": 6, "isopen_empti": 6, "isopen_iunion": 6, "fintyp": 6, "isopen_iint": 6, "continuous_def": 6, "attach": 6, "filteri": 6, "sent": 6, "mem_nhds_iff": 6, "weird": 6, "digress": 6, "pure_le_nhd": 6, "eventually_eventually_nhd": 6, "topological_spac": 6, "mk_of_nhd": 6, "nhds_mk_of_nhd": 6, "clean": 6, "fonctori": 6, "induc": 6, "sensibl": 6, "uncount": 6, "relatedli": 6, "coinduc": 6, "t_x": 6, "t_y": 6, "coinduced_le_iff_le_induc": 6, "covari": 6, "coinduced_compos": 6, "induced_compos": 6, "topological_structur": 6, "primarili": 6, "ie": 6, "focus": 6, "nhd": 6, "opposit": 6, "recov": 6, "foward": 6, "continuous_iff_coinduced_l": 6, "g_": 6, "t_z": 6, "wasn": 6, "\u03c0": 6, "papar": 6, "t_": 6, "x_i": 6, "pi": 6, "defect": 6, "functori": 6, "price": 6, "patholog": 6, "t2_space": 6, "hausdorff": 6, "regular": 6, "t2space": 6, "tendsto_nhds_uniqu": 6, "regularspac": 6, "closed_nhds_basi": 6, "nhds_basis_open": 6, "denseinduc": 6, "continuousat_extend": 6, "funni": 6, "_in": 6, "thank": 6, "nhds_induc": 6, "is_open": 6, "fortiori": 6, "f_cont": 6, "tendsto_right_iff": 6, "firstcountabletopolog": 6, "sever": 6, "cluster": 6, "clusterpt": 6, "nebot": 6, "hfx": 6, "push_pul": 6, "of_map": 6, "f_ne": 6, "f_le": 6, "map_eq": 6, "hne": 6, "hle": 6, "huo": 6, "hsu": 6, "elim_finite_subcov": 6, "elementary_differential_calculu": 7, "introductori": 7, "normed_spac": 7, "overview": 10, "conjunct": 10, "converg": 10, "schr\u00f6der": 10, "bernstein": 10, "irrat": 10}, "objects": {}, "objtypes": {}, "objnames": {}, "titleterms": {"introduct": 0, "get": 0, "start": 0, "overview": 0, "basic": 1, "calcul": 1, "prove": 1, "ident": 1, "algebra": [1, 5], "structur": [1, 5], "us": 1, "theorem": [1, 3], "lemma": 1, "more": 1, "order": 1, "divis": 1, "fact": 1, "about": 1, "logic": 2, "implic": 2, "univers": 2, "quantifi": 2, "The": [2, 3], "existenti": 2, "negat": 2, "conjunct": 2, "bi": 2, "disjunct": 2, "sequenc": 2, "converg": [2, 6], "set": [3, 6], "function": [3, 6], "schr\u00f6der": 3, "bernstein": 3, "number": 4, "theori": [4, 8], "irrat": 4, "root": 4, "induct": 4, "recurs": 4, "infinit": 4, "mani": 4, "prime": 4, "abstract": 5, "build": 5, "gaussian": 5, "integ": 5, "topolog": 6, "filter": 6, "metric": 6, "space": 6, "continu": 6, "ball": 6, "open": 6, "close": 6, "compact": 6, "uniformli": 6, "complet": 6, "fundament": 6, "separ": 6, "countabl": 6, "differenti": 7, "calculu": 7, "integr": 8, "measur": 8, "index": 9, "mathemat": 10, "lean": 10}, "envversion": {"sphinx.domains.c": 2, "sphinx.domains.changeset": 1, "sphinx.domains.citation": 1, "sphinx.domains.cpp": 6, "sphinx.domains.index": 1, "sphinx.domains.javascript": 2, "sphinx.domains.math": 2, "sphinx.domains.python": 3, "sphinx.domains.rst": 2, "sphinx.domains.std": 2, "sphinx": 56}})
lake-manifest.json (new)
- @@ -0,0 +1,27 @@
  + {"version": 4, + "packagesDir": "lake-packages", + "packages": + [{"git": + {"url": "https://github.com/leanprover-community/mathlib4", + "subDir?": null, + "rev": "52d71ad9606cf755e2437cfedfbcc89ce8f2b86a", + "name": "mathlib", + "inputRev?": "master"}}, + {"git": + {"url": "https://github.com/gebner/quote4", + "subDir?": null, + "rev": "c71f94e34c1cda52eef5c93dc9da409ab2727420", + "name": "Qq", + "inputRev?": "master"}}, + {"git": + {"url": "https://github.com/JLimperg/aesop", + "subDir?": null, + "rev": "38bcf8b9e564d23bad55cbfa3c770f135f926d98", + "name": "aesop", + "inputRev?": "master"}}, + {"git": + {"url": "https://github.com/leanprover/std4", + "subDir?": null, + "rev": "b886042255d5e93b8d0adb780d49b1c8f3482fbb", + "name": "std", + "inputRev?": "main"}}]}
lakefile.lean (new)
- @@ -0,0 +1,13 @@
  + import Lake + open Lake DSL + + package mil { + -- add package configuration options here + } + + @[default_target] + lean_lib «MIL» { + -- add library configuration options here + } + + require mathlib from git "https://github.com/leanprover-community/mathlib4"@"master"
lean-toolchain (new)
- @@ -0,0 +1,1 @@
  + leanprover/lean4:nightly-2023-05-16
mathematics_in_lean.pdf (new)
src/C01_Introduction/S01_Getting_Started.lean (new)
- @@ -0,0 +1,2 @@
  + #eval "Hello, World!" +
src/C01_Introduction/S02_Overview.lean (new)
- @@ -0,0 +1,56 @@
  + import Mathlib.Data.Nat.Basic + import Mathlib.Data.Nat.Parity + import Mathlib.Tactic + + open Nat + + -- These are pieces of data. + #check 2 + 2 + + def f (x : ℕ) := + x + 3 + + #check f + + -- These are propositions, of type `Prop`. + #check 2 + 2 = 4 + + def FermatLastTheorem := + ∀ x y z n : ℕ, n > 2 ∧ x * y * z ≠ 0 → x ^ n + y ^ n ≠ z ^ n + + #check FermatLastTheorem + + -- These are proofs of propositions. + theorem easy : 2 + 2 = 4 := + rfl + + #check easy + + theorem hard : FermatLastTheorem := + sorry + + #check hard + + -- Here are some proofs. + example : ∀ m n : Nat, Even n → Even (m * n) := fun m n ⟨k, (hk : n = k + k)⟩ => + have hmn : m * n = m * k + m * k := by rw [hk, mul_add] + show ∃ l, m * n = l + l from ⟨_, hmn⟩ + + example : ∀ m n : Nat, Even n → Even (m * n) := fun m n ⟨k, hk⟩ => ⟨m * k, by rw [hk, mul_add]⟩ + + example : ∀ m n : Nat, Even n → Even (m * n) := by + -- say m and n are natural numbers, and assume n=2*k + rintro m n ⟨k, hk⟩ + -- We need to prove m*n is twice a natural number. Let's show it's twice m*k. + use m * k + -- substitute in for n + rw [hk] + -- and now it's obvious + ring + + example : ∀ m n : Nat, Even n → Even (m * n) := by + rintro m n ⟨k, hk⟩; use m * k; rw [hk]; ring + + example : ∀ m n : Nat, Even n → Even (m * n) := by + intros; simp [*, parity_simps] +
src/C01_Introduction/solutions/Solutions_S01_Getting_Started.lean (new)
src/C01_Introduction/solutions/Solutions_S02_Overview.lean (new)
- @@ -0,0 +1,7 @@
  + import Mathlib.Data.Nat.Basic + import Mathlib.Data.Nat.Parity + import Mathlib.Tactic + + open Nat + + -- There are no exercises in this section.
src/C02_Basics/S01_Calculating.lean (new)
- @@ -0,0 +1,148 @@
  + import Mathlib.Data.Real.Basic + import Mathlib.Data.Real.Basic + + -- An example. + example (a b c : ℝ) : a * b * c = b * (a * c) := by + rw [mul_comm a b] + rw [mul_assoc b a c] + + -- Try these. + example (a b c : ℝ) : c * b * a = b * (a * c) := by + sorry + + example (a b c : ℝ) : a * (b * c) = b * (a * c) := by + sorry + + -- An example. + example (a b c : ℝ) : a * b * c = b * c * a := by + rw [mul_assoc] + rw [mul_comm] + + /- Try doing the first of these without providing any arguments at all, + and the second with only one argument. -/ + example (a b c : ℝ) : a * (b * c) = b * (c * a) := by + sorry + + example (a b c : ℝ) : a * (b * c) = b * (a * c) := by + sorry + + -- Using facts from the local context. + example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by + rw [h'] + rw [← mul_assoc] + rw [h] + rw [mul_assoc] + + -- Try these. For the second one, use the theorem `sub_self`. + example (a b c d e f : ℝ) (h : b * c = e * f) : a * b * c * d = a * e * f * d := by + sorry + + example (a b c d : ℝ) (hyp : c = b * a - d) (hyp' : d = a * b) : c = 0 := by + sorry + + -- Examples. + example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by + rw [h', ← mul_assoc, h, mul_assoc] + + section + + variable (a b c d e f g : ℝ) + + example (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by + rw [h', ← mul_assoc, h, mul_assoc] + + end + + section + variable (a b c : ℝ) + + #check a + #check a + b + #check (a : ℝ) + #check mul_comm a b + #check (mul_comm a b : a * b = b * a) + #check mul_assoc c a b + #check mul_comm a + #check mul_comm + #check @mul_comm + + end + + section + variable (a b : ℝ) + + example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by + rw [mul_add, add_mul, add_mul] + rw [← add_assoc, add_assoc (a * a)] + rw [mul_comm b a, ← two_mul] + + example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := + calc + (a + b) * (a + b) = a * a + b * a + (a * b + b * b) := by + rw [mul_add, add_mul, add_mul] + _ = a * a + (b * a + a * b) + b * b := by + rw [← add_assoc, add_assoc (a * a)] + _ = a * a + 2 * (a * b) + b * b := by + rw [mul_comm b a, ← two_mul] + + example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := + calc + (a + b) * (a + b) = a * a + b * a + (a * b + b * b) := by + sorry + _ = a * a + (b * a + a * b) + b * b := by + sorry + _ = a * a + 2 * (a * b) + b * b := by + sorry + + end + + -- Try these. For the second, use the theorems listed underneath. + section + variable (a b c d : ℝ) + + example : (a + b) * (c + d) = a * c + a * d + b * c + b * d := by + sorry + + example (a b : ℝ) : (a + b) * (a - b) = a ^ 2 - b ^ 2 := by + sorry + + #check pow_two a + #check mul_sub a b c + #check add_mul a b c + #check add_sub a b c + #check sub_sub a b c + #check add_zero a + + end + + -- Examples. + + section + variable (a b c d : ℝ) + + example (a b c d : ℝ) (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by + rw [hyp'] at hyp + rw [mul_comm d a] at hyp + rw [← two_mul (a * d)] at hyp + rw [← mul_assoc 2 a d] at hyp + exact hyp + + example : c * b * a = b * (a * c) := by + ring + + example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by + ring + + example : (a + b) * (a - b) = a ^ 2 - b ^ 2 := by + ring + + example (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by + rw [hyp, hyp'] + ring + + end + + example (a b c : ℕ) (h : a + b = c) : (a + b) * (a + b) = a * c + b * c := by + nth_rw 2 [h] + rw [add_mul] +
src/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean (new)
- @@ -0,0 +1,146 @@
  + import Mathlib.Algebra.Ring.Defs + import Mathlib.Data.Real.Basic + import Mathlib.Tactic + + section + variable (R : Type _) [Ring R] + + #check (add_assoc : ∀ a b c : R, a + b + c = a + (b + c)) + #check (add_comm : ∀ a b : R, a + b = b + a) + #check (zero_add : ∀ a : R, 0 + a = a) + #check (add_left_neg : ∀ a : R, -a + a = 0) + #check (mul_assoc : ∀ a b c : R, a * b * c = a * (b * c)) + #check (mul_one : ∀ a : R, a * 1 = a) + #check (one_mul : ∀ a : R, 1 * a = a) + #check (mul_add : ∀ a b c : R, a * (b + c) = a * b + a * c) + #check (add_mul : ∀ a b c : R, (a + b) * c = a * c + b * c) + + end + + section + variable (R : Type _) [CommRing R] + variable (a b c d : R) + + example : c * b * a = b * (a * c) := by ring + + example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by ring + + example : (a + b) * (a - b) = a ^ 2 - b ^ 2 := by ring + + example (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by + rw [hyp, hyp'] + ring + + end + + namespace MyRing + variable {R : Type _} [Ring R] + + theorem add_zero (a : R) : a + 0 = a := by rw [add_comm, zero_add] + + theorem add_right_neg (a : R) : a + -a = 0 := by rw [add_comm, add_left_neg] + + #check @MyRing.add_zero + #check @add_zero + + end MyRing + + namespace MyRing + variable {R : Type _} [Ring R] + + theorem neg_add_cancel_left (a b : R) : -a + (a + b) = b := by + rw [← add_assoc, add_left_neg, zero_add] + + -- Prove these: + theorem add_neg_cancel_right (a b : R) : a + b + -b = a := by + sorry + + theorem add_left_cancel {a b c : R} (h : a + b = a + c) : b = c := by + sorry + + theorem add_right_cancel {a b c : R} (h : a + b = c + b) : a = c := by + sorry + + theorem mul_zero (a : R) : a * 0 = 0 := by + have h : a * 0 + a * 0 = a * 0 + 0 := by + rw [← mul_add, add_zero, add_zero] + rw [add_left_cancel h] + + theorem zero_mul (a : R) : 0 * a = 0 := by + sorry + + theorem neg_eq_of_add_eq_zero {a b : R} (h : a + b = 0) : -a = b := by + sorry + + theorem eq_neg_of_add_eq_zero {a b : R} (h : a + b = 0) : a = -b := by + sorry + + theorem neg_zero : (-0 : R) = 0 := by + apply neg_eq_of_add_eq_zero + rw [add_zero] + + theorem neg_neg (a : R) : - -a = a := by + sorry + + end MyRing + + -- Examples. + section + variable {R : Type _} [Ring R] + + example (a b : R) : a - b = a + -b := + sub_eq_add_neg a b + + end + + example (a b : ℝ) : a - b = a + -b := + rfl + + example (a b : ℝ) : a - b = a + -b := by + rfl + + namespace MyRing + variable {R : Type _} [Ring R] + + theorem self_sub (a : R) : a - a = 0 := + sorry + + theorem one_add_one_eq_two : 1 + 1 = (2 : R) := by + norm_num + + theorem two_mul (a : R) : 2 * a = a + a := + sorry + + end MyRing + + section + variable (A : Type _) [AddGroup A] + + #check (add_assoc : ∀ a b c : A, a + b + c = a + (b + c)) + #check (zero_add : ∀ a : A, 0 + a = a) + #check (add_left_neg : ∀ a : A, -a + a = 0) + + end + + section + variable {G : Type _} [Group G] + + #check (mul_assoc : ∀ a b c : G, a * b * c = a * (b * c)) + #check (one_mul : ∀ a : G, 1 * a = a) + #check (mul_left_inv : ∀ a : G, a⁻¹ * a = 1) + + namespace MyGroup + + theorem mul_right_inv (a : G) : a * a⁻¹ = 1 := by + sorry + + theorem mul_one (a : G) : a * 1 = a := by + sorry + + theorem mul_inv_rev (a b : G) : (a * b)⁻¹ = b⁻¹ * a⁻¹ := by + sorry + + end MyGroup + + end +
src/C02_Basics/S03_Using_Theorems_and_Lemmas.lean (new)
- @@ -0,0 +1,130 @@
 + import Mathlib.Analysis.SpecialFunctions.Log.Basic + import Mathlib.Tactic + + variable (a b c d e : ℝ) + open Real + + #check (le_refl : ∀ a : ℝ, a ≤ a) + #check (le_trans : a ≤ b → b ≤ c → a ≤ c) + + section + variable (h : a ≤ b) (h' : b ≤ c) + + #check (le_refl : ∀ a : Real, a ≤ a) + #check (le_refl a : a ≤ a) + #check (le_trans : a ≤ b → b ≤ c → a ≤ c) + #check (le_trans h : b ≤ c → a ≤ c) + #check (le_trans h h' : a ≤ c) + + end + + example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := by + apply le_trans + · apply h₀ + . apply h₁ + + example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := by + apply le_trans h₀ + apply h₁ + + example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := + le_trans h₀ h₁ + + example (x : ℝ) : x ≤ x := by + apply le_refl + + example (x : ℝ) : x ≤ x := + le_refl x + + #check (le_refl : ∀ a, a ≤ a) + #check (le_trans : a ≤ b → b ≤ c → a ≤ c) + #check (lt_of_le_of_lt : a ≤ b → b < c → a < c) + #check (lt_of_lt_of_le : a < b → b ≤ c → a < c) + #check (lt_trans : a < b → b < c → a < c) + + -- Try this. + example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) (h₃ : d < e) : a < e := by + sorry + + example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) (h₃ : d < e) : a < e := by + linarith + + section + + example (h : 2 * a ≤ 3 * b) (h' : 1 ≤ a) (h'' : d = 2) : d + a ≤ 5 * b := by + linarith + + end + + example (h : 1 ≤ a) (h' : b ≤ c) : 2 + a + exp b ≤ 3 * a + exp c := by + linarith [exp_le_exp.mpr h'] + + #check (exp_le_exp : exp a ≤ exp b ↔ a ≤ b) + #check (exp_lt_exp : exp a < exp b ↔ a < b) + #check (log_le_log : 0 < a → 0 < b → (log a ≤ log b ↔ a ≤ b)) + #check (log_lt_log : 0 < a → a < b → log a < log b) + #check (add_le_add : a ≤ b → c ≤ d → a + c ≤ b + d) + #check (add_le_add_left : a ≤ b → ∀ c, c + a ≤ c + b) + #check (add_le_add_right : a ≤ b → ∀ c, a + c ≤ b + c) + #check (add_lt_add_of_le_of_lt : a ≤ b → c < d → a + c < b + d) + #check (add_lt_add_of_lt_of_le : a < b → c ≤ d → a + c < b + d) + #check (add_lt_add_left : a < b → ∀ c, c + a < c + b) + #check (add_lt_add_right : a < b → ∀ c, a + c < b + c) + #check (add_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a + b) + #check (add_pos : 0 < a → 0 < b → 0 < a + b) + #check (add_pos_of_pos_of_nonneg : 0 < a → 0 ≤ b → 0 < a + b) + #check (exp_pos : ∀ a, 0 < exp a) + #check @add_le_add_left + + example (h : a ≤ b) : exp a ≤ exp b := by + rw [exp_le_exp] + exact h + + example (h₀ : a ≤ b) (h₁ : c < d) : a + exp c + e < b + exp d + e := by + apply add_lt_add_of_lt_of_le + · apply add_lt_add_of_le_of_lt h₀ + apply exp_lt_exp.mpr h₁ + apply le_refl + + example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := by sorry + + example : (0 : ℝ) < 1 := by norm_num + + example (h : a ≤ b) : log (1 + exp a) ≤ log (1 + exp b) := by + have h₀ : 0 < 1 + exp a := by sorry + have h₁ : 0 < 1 + exp b := by sorry + apply (log_le_log h₀ h₁).mpr + sorry + + example : 0 ≤ a ^ 2 := by + -- library_search + exact sq_nonneg a + + example (h : a ≤ b) : c - exp b ≤ c - exp a := by + sorry + + example : 2 * a * b ≤ a ^ 2 + b ^ 2 := by + have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 + calc + a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring + _ ≥ 0 := by apply pow_two_nonneg + + calc + 2 * a * b = 2 * a * b + 0 := by ring + _ ≤ 2 * a * b + (a ^ 2 - 2 * a * b + b ^ 2) := + add_le_add (le_refl _) h + _ = a ^ 2 + b ^ 2 := by ring + + + example : 2 * a * b ≤ a ^ 2 + b ^ 2 := by + have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 + calc + a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring + _ ≥ 0 := by apply pow_two_nonneg + linarith + + example : abs (a * b) ≤ (a ^ 2 + b ^ 2) / 2 := by + sorry + + #check abs_le'.mpr +
src/C02_Basics/S04_More_on_Order_and_Divisibility.lean (new)
- @@ -0,0 +1,94 @@
  + import Mathlib.Data.Real.Basic + + section + variable (a b c d : ℝ) + + #check (min_le_left a b : min a b ≤ a) + #check (min_le_right a b : min a b ≤ b) + #check (le_min : c ≤ a → c ≤ b → c ≤ min a b) + + example : min a b = min b a := by + apply le_antisymm + · show min a b ≤ min b a + apply le_min + · apply min_le_right + apply min_le_left + · show min b a ≤ min a b + apply le_min + · apply min_le_right + apply min_le_left + + example : min a b = min b a := by + have h : ∀ x y : ℝ, min x y ≤ min y x := by + intro x y + apply le_min + apply min_le_right + apply min_le_left + apply le_antisymm + apply h + apply h + + example : min a b = min b a := by + apply le_antisymm + repeat + apply le_min + apply min_le_right + apply min_le_left + + example : max a b = max b a := by + sorry + + example : min (min a b) c = min a (min b c) := by + sorry + + theorem aux : min a b + c ≤ min (a + c) (b + c) := by + sorry + + example : min a b + c = min (a + c) (b + c) := by + sorry + + #check (abs_add : ∀ a b : ℝ, abs (a + b) ≤ abs a + abs b) + + example : abs a - abs b ≤ abs (a - b) := + sorry + + end + + section + variable (w x y z : ℕ) + + example (h₀ : x ∣ y) (h₁ : y ∣ z) : x ∣ z := + dvd_trans h₀ h₁ + + example : x ∣ y * x * z := by + apply dvd_mul_of_dvd_left + apply dvd_mul_left + + example : x ∣ x ^ 2 := by + apply dvd_mul_left + + example (h : x ∣ w) : x ∣ y * (x * z) + x ^ 2 + w ^ 2 := by + sorry + + end + + section + + variable (m n : ℕ) + + open Nat + + #check (gcd_zero_right n : gcd n 0 = n) + + #check (gcd_zero_left n : gcd 0 n = n) + + #check (lcm_zero_right n : lcm n 0 = 0) + + #check (lcm_zero_left n : lcm 0 n = 0) + + example : gcd m n = gcd n m := by + sorry + + end + +
src/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean (new)
- @@ -0,0 +1,109 @@
 + import Mathlib.Topology.MetricSpace.Basic + + section + variable {α : Type _} [PartialOrder α] + variable (x y z : α) + + #check x ≤ y + #check (le_refl x : x ≤ x) + #check (le_trans : x ≤ y → y ≤ z → x ≤ z) + + #check x < y + #check (lt_irrefl x : ¬x < x) + #check (lt_trans : x < y → y < z → x < z) + #check (lt_of_le_of_lt : x ≤ y → y < z → x < z) + #check (lt_of_lt_of_le : x < y → y ≤ z → x < z) + + example : x < y ↔ x ≤ y ∧ x ≠ y := + lt_iff_le_and_ne + + end + + section + variable {α : Type _} [Lattice α] + variable (x y z : α) + + #check x ⊓ y + #check (inf_le_left : x ⊓ y ≤ x) + #check (inf_le_right : x ⊓ y ≤ y) + #check (le_inf : z ≤ x → z ≤ y → z ≤ x ⊓ y) + #check x ⊔ y + #check (le_sup_left : x ≤ x ⊔ y) + #check (le_sup_right : y ≤ x ⊔ y) + #check (sup_le : x ≤ z → y ≤ z → x ⊔ y ≤ z) + + example : x ⊓ y = y ⊓ x := by + sorry + + example : x ⊓ y ⊓ z = x ⊓ (y ⊓ z) := by + sorry + + example : x ⊔ y = y ⊔ x := by + sorry + + example : x ⊔ y ⊔ z = x ⊔ (y ⊔ z) := by + sorry + + theorem absorb1 : x ⊓ (x ⊔ y) = x := by + sorry + + theorem absorb2 : x ⊔ x ⊓ y = x := by + sorry + + end + + section + variable {α : Type _} [DistribLattice α] + variable (x y z : α) + + #check (inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) + #check (inf_sup_right : (x ⊔ y) ⊓ z = x ⊓ z ⊔ y ⊓ z) + #check (sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) + #check (sup_inf_right : x ⊓ y ⊔ z = (x ⊔ z) ⊓ (y ⊔ z)) + end + + section + variable {α : Type _} [Lattice α] + variable (a b c : α) + + example (h : ∀ x y z : α, x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) : a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c) := by + sorry + + example (h : ∀ x y z : α, x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) : a ⊓ (b ⊔ c) = a ⊓ b ⊔ a ⊓ c := by + sorry + + end + + section + variable {R : Type _} [StrictOrderedRing R] + variable (a b c : R) + + #check (add_le_add_left : a ≤ b → ∀ c, c + a ≤ c + b) + #check (mul_pos : 0 < a → 0 < b → 0 < a * b) + + #check (mul_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a * b) + + example : a ≤ b → 0 ≤ b - a := by + sorry + + example : 0 ≤ b - a → a ≤ b := by + sorry + + example (h : a ≤ b) (h' : 0 ≤ c) : a * c ≤ b * c := by + sorry + + end + + section + variable {X : Type _} [MetricSpace X] + variable (x y z : X) + + #check (dist_self x : dist x x = 0) + #check (dist_comm x y : dist x y = dist y x) + #check (dist_triangle x y z : dist x z ≤ dist x y + dist y z) + + example (x y : X) : 0 ≤ dist x y := by + sorry + + end +
src/C02_Basics/solutions/Solutions_S01_Calculating.lean (new)
- @@ -0,0 +1,33 @@
  + import Mathlib.Data.Real.Basic + import Mathlib.Data.Real.Basic + + example (a b c : ℝ) : c * b * a = b * (a * c) := by + rw [mul_comm c b] + rw [mul_assoc b c a] + rw [mul_comm c a] + + example (a b c : ℝ) : a * (b * c) = b * (a * c) := by + rw [← mul_assoc a b c] + rw [mul_comm a b] + rw [mul_assoc b a c] + + example (a b c : ℝ) : a * (b * c) = b * (c * a) := by + rw [mul_comm] + rw [mul_assoc] + + example (a b c : ℝ) : a * (b * c) = b * (a * c) := by + rw [← mul_assoc] + rw [mul_comm a] + rw [mul_assoc] + + example (a b c d e f : ℝ) (h : b * c = e * f) : a * b * c * d = a * e * f * d := by + rw [mul_assoc a] + rw [h] + rw [← mul_assoc] + + example (a b c d : ℝ) (hyp : c = b * a - d) (hyp' : d = a * b) : c = 0 := by + rw [hyp] + rw [hyp'] + rw [mul_comm] + rw [sub_self] +
src/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean (new)
- @@ -0,0 +1,73 @@
  + import Mathlib.Algebra.Ring.Defs + import Mathlib.Data.Real.Basic + import Mathlib.Tactic + + namespace MyRing + variable {R : Type _} [Ring R] + + theorem add_neg_cancel_right (a b : R) : a + b + -b = a := by + rw [add_assoc, add_right_neg, add_zero] + + theorem add_left_cancel {a b c : R} (h : a + b = a + c) : b = c := by + rw [← neg_add_cancel_left a b, h, neg_add_cancel_left] + + theorem add_right_cancel {a b c : R} (h : a + b = c + b) : a = c := by + rw [← add_neg_cancel_right a b, h, add_neg_cancel_right] + + theorem zero_mul (a : R) : 0 * a = 0 := by + have h : 0 * a + 0 * a = 0 * a + 0 := by rw [← add_mul, add_zero, add_zero] + rw [add_left_cancel h] + + theorem neg_eq_of_add_eq_zero {a b : R} (h : a + b = 0) : -a = b := by + rw [← neg_add_cancel_left a b, h, add_zero] + + theorem eq_neg_of_add_eq_zero {a b : R} (h : a + b = 0) : a = -b := by + symm + apply neg_eq_of_add_eq_zero + rw [add_comm, h] + + theorem neg_zero : (-0 : R) = 0 := by + apply neg_eq_of_add_eq_zero + rw [add_zero] + + theorem neg_neg (a : R) : - -a = a := by + apply neg_eq_of_add_eq_zero + rw [add_left_neg] + + end MyRing + + namespace MyRing + variable {R : Type _} [Ring R] + + theorem self_sub (a : R) : a - a = 0 := by + rw [sub_eq_add_neg, add_right_neg] + + theorem one_add_one_eq_two : 1 + 1 = (2 : R) := by + norm_num + + theorem two_mul (a : R) : 2 * a = a + a := by + rw [← one_add_one_eq_two, add_mul, one_mul] + + end MyRing + + section + variable {G : Type _} [Group G] + + namespace MyGroup + + theorem mul_right_inv (a : G) : a * a⁻¹ = 1 := by + have h : (a * a⁻¹)⁻¹ * (a * a⁻¹ * (a * a⁻¹)) = 1 := by + rw [mul_assoc, ← mul_assoc a⁻¹ a, mul_left_inv, one_mul, mul_left_inv] + rw [← h, ← mul_assoc, mul_left_inv, one_mul] + + theorem mul_one (a : G) : a * 1 = a := by + rw [← mul_left_inv a, ← mul_assoc, mul_right_inv, one_mul] + + theorem mul_inv_rev (a b : G) : (a * b)⁻¹ = b⁻¹ * a⁻¹ := by + rw [← one_mul (b⁻¹ * a⁻¹), ← mul_left_inv (a * b), mul_assoc, mul_assoc, ← mul_assoc b b⁻¹, + mul_right_inv, one_mul, mul_right_inv, mul_one] + + end MyGroup + + end +
src/C02_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean (new)
- @@ -0,0 +1,61 @@
 + import Mathlib.Analysis.SpecialFunctions.Log.Basic + import Mathlib.Tactic + + variable (a b c d e : ℝ) + open Real + + example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) (h₃ : d < e) : a < e := by + apply lt_of_le_of_lt h₀ + apply lt_trans h₁ + exact lt_of_le_of_lt h₂ h₃ + + example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := by + apply add_le_add_left + rw [exp_le_exp] + apply add_le_add_left h₀ + + -- an alternative using `linarith`. + example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := by + have : exp (a + d) ≤ exp (a + e) := by + rw [exp_le_exp] + linarith + linarith [this] + + example (h : a ≤ b) : log (1 + exp a) ≤ log (1 + exp b) := by + have h₀ : 0 < 1 + exp a := by linarith [exp_pos a] + have h₁ : 0 < 1 + exp b := by linarith [exp_pos b] + apply (log_le_log h₀ h₁).mpr + apply add_le_add_left (exp_le_exp.mpr h) + + -- SOLUTION. + example (h : a ≤ b) : c - exp b ≤ c - exp a := by + apply sub_le_sub_left + exact exp_le_exp.mpr h + + -- alternatively: + example (h : a ≤ b) : c - exp b ≤ c - exp a := by + linarith [exp_le_exp.mpr h] + + theorem fact1 : a * b * 2 ≤ a ^ 2 + b ^ 2 := by + have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 + calc + a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring + _ ≥ 0 := by apply pow_two_nonneg + linarith + + theorem fact2 : -(a * b) * 2 ≤ a ^ 2 + b ^ 2 := by + have h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2 + calc + a ^ 2 + 2 * a * b + b ^ 2 = (a + b) ^ 2 := by ring + _ ≥ 0 := by apply pow_two_nonneg + linarith + + example : abs (a * b) ≤ (a ^ 2 + b ^ 2) / 2 := by + have h : (0 : ℝ) < 2 := by norm_num + apply abs_le'.mpr + constructor + · rw [le_div_iff h] + apply fact1 + rw [le_div_iff h] + apply fact2 +
src/C02_Basics/solutions/Solutions_S04_More_on_Order_and_Divisibility.lean (new)
- @@ -0,0 +1,120 @@
  + import Mathlib.Data.Real.Basic + + section + variable (a b c d : ℝ) + + #check (min_le_left a b : min a b ≤ a) + #check (min_le_right a b : min a b ≤ b) + #check (le_min : c ≤ a → c ≤ b → c ≤ min a b) + + example : max a b = max b a := by + apply le_antisymm + repeat' + apply max_le + apply le_max_right + apply le_max_left + + example : min (min a b) c = min a (min b c) := by + apply le_antisymm + · apply le_min + · apply le_trans + apply min_le_left + apply min_le_left + apply le_min + · apply le_trans + apply min_le_left + apply min_le_right + apply min_le_right + apply le_min + · apply le_min + · apply min_le_left + apply le_trans + apply min_le_right + apply min_le_left + apply le_trans + apply min_le_right + apply min_le_right + + theorem aux : min a b + c ≤ min (a + c) (b + c) := by + apply le_min + · apply add_le_add_right + apply min_le_left + apply add_le_add_right + apply min_le_right + + example : min a b + c = min (a + c) (b + c) := by + apply le_antisymm + · apply aux + have h : min (a + c) (b + c) = min (a + c) (b + c) - c + c := by rw [sub_add_cancel] + rw [h] + apply add_le_add_right + rw [sub_eq_add_neg] + apply le_trans + apply aux + rw [add_neg_cancel_right, add_neg_cancel_right] + + example : abs a - abs b ≤ abs (a - b) := + calc + abs a - abs b = abs (a - b + b) - abs b := by rw [sub_add_cancel] + _ ≤ abs (a - b) + abs b - abs b := by + apply sub_le_sub_right + apply abs_add + _ ≤ abs (a - b) := by rw [add_sub_cancel] + + + -- alternatively + example : abs a - abs b ≤ abs (a - b) := by + have h := abs_add (a - b) b + rw [sub_add_cancel] at h + linarith + + end + + section + variable (w x y z : ℕ) + + example (h₀ : x ∣ y) (h₁ : y ∣ z) : x ∣ z := + dvd_trans h₀ h₁ + + example : x ∣ y * x * z := by + apply dvd_mul_of_dvd_left + apply dvd_mul_left + + example : x ∣ x ^ 2 := by + apply dvd_mul_left + + example (h : x ∣ w) : x ∣ y * (x * z) + x ^ 2 + w ^ 2 := by + apply dvd_add + · apply dvd_add + · apply dvd_mul_of_dvd_right + apply dvd_mul_right + apply dvd_mul_left + rw [pow_two] + apply dvd_mul_of_dvd_right + exact h + + end + + section + + variable (m n : ℕ) + + open Nat + + #check (gcd_zero_right n : gcd n 0 = n) + + #check (gcd_zero_left n : gcd 0 n = n) + + #check (lcm_zero_right n : lcm n 0 = 0) + + #check (lcm_zero_left n : lcm 0 n = 0) + + example : gcd m n = gcd n m := by + apply _root_.dvd_antisymm + repeat' + apply dvd_gcd + apply gcd_dvd_right + apply gcd_dvd_left + + end +
src/C02_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean (new)
- @@ -0,0 +1,135 @@
  + import Mathlib.Topology.MetricSpace.Basic + + section + variable {α : Type _} [Lattice α] + variable (x y z : α) + + example : x ⊓ y = y ⊓ x := by + apply le_antisymm + repeat' + apply le_inf + · apply inf_le_right + apply inf_le_left + + example : x ⊓ y ⊓ z = x ⊓ (y ⊓ z) := by + apply le_antisymm + · apply le_inf + · apply le_trans + apply inf_le_left + apply inf_le_left + apply le_inf + · apply le_trans + apply inf_le_left + apply inf_le_right + apply inf_le_right + apply le_inf + · apply le_inf + · apply inf_le_left + apply le_trans + apply inf_le_right + apply inf_le_left + apply le_trans + apply inf_le_right + apply inf_le_right + + example : x ⊔ y = y ⊔ x := by + apply le_antisymm + repeat' + apply sup_le + · apply le_sup_right + apply le_sup_left + + example : x ⊔ y ⊔ z = x ⊔ (y ⊔ z) := by + apply le_antisymm + · apply sup_le + · apply sup_le + apply le_sup_left + · apply le_trans + apply @le_sup_left _ _ y z + apply le_sup_right + apply le_trans + apply @le_sup_right _ _ y z + apply le_sup_right + apply sup_le + · apply le_trans + apply @le_sup_left _ _ x y + apply le_sup_left + apply sup_le + · apply le_trans + apply @le_sup_right _ _ x y + apply le_sup_left + apply le_sup_right + + theorem absorb1 : x ⊓ (x ⊔ y) = x := by + apply le_antisymm + · apply inf_le_left + apply le_inf + · apply le_refl + apply le_sup_left + + theorem absorb2 : x ⊔ x ⊓ y = x := by + apply le_antisymm + · apply sup_le + · apply le_refl + apply inf_le_left + apply le_sup_left + + end + + section + variable {α : Type _} [DistribLattice α] + variable (x y z : α) + + #check (inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) + #check (inf_sup_right : (x ⊔ y) ⊓ z = x ⊓ z ⊔ y ⊓ z) + #check (sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) + #check (sup_inf_right : x ⊓ y ⊔ z = (x ⊔ z) ⊓ (y ⊔ z)) + end + + section + variable {α : Type _} [Lattice α] + variable (a b c : α) + + example (h : ∀ x y z : α, x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) : a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c) := by + rw [h, @inf_comm _ _ (a ⊔ b), absorb1, @inf_comm _ _ (a ⊔ b), h, ← sup_assoc, @inf_comm _ _ c a, + absorb2, inf_comm] + + example (h : ∀ x y z : α, x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) : a ⊓ (b ⊔ c) = a ⊓ b ⊔ a ⊓ c := by + rw [h, @sup_comm _ _ (a ⊓ b), absorb2, @sup_comm _ _ (a ⊓ b), h, ← inf_assoc, @sup_comm _ _ c a, + absorb1, sup_comm] + + end + + section + variable {R : Type _} [StrictOrderedRing R] + variable (a b c : R) + + theorem aux1 : a ≤ b → 0 ≤ b - a := by + intro h + rw [← sub_self a, sub_eq_add_neg, sub_eq_add_neg, add_comm, add_comm b] + apply add_le_add_left h + + theorem aux2 : 0 ≤ b - a → a ≤ b := by + intro h + rw [← add_zero a, ← sub_add_cancel b a, add_comm (b - a)] + apply add_le_add_left h + + example (h : a ≤ b) (h' : 0 ≤ c) : a * c ≤ b * c := by + have h1 : 0 ≤ (b - a) * c := mul_nonneg (aux1 _ _ h) h' + rw [sub_mul] at h1 + exact aux2 _ _ h1 + + end + + section + variable {X : Type _} [MetricSpace X] + variable (x y z : X) + + example (x y : X) : 0 ≤ dist x y :=by + have : 0 ≤ dist x y + dist y x := by + rw [← dist_self x] + apply dist_triangle + linarith [dist_comm x y] + + end +
src/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean (new)
- @@ -0,0 +1,175 @@
 + import Mathlib.Data.Real.Basic + + #check ∀ x : ℝ, 0 ≤ x → abs x = x + + #check ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε + + theorem my_lemma : ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := + sorry + + section + variable (a b δ : ℝ) + variable (h₀ : 0 < δ) (h₁ : δ ≤ 1) + variable (ha : abs a < δ) (hb : abs b < δ) + + #check my_lemma a b δ + #check my_lemma a b δ h₀ h₁ + #check my_lemma a b δ h₀ h₁ ha hb + + end + + theorem my_lemma2 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := + sorry + + section + variable (a b δ : ℝ) + variable (h₀ : 0 < δ) (h₁ : δ ≤ 1) + variable (ha : abs a < δ) (hb : abs b < δ) + + #check my_lemma2 h₀ h₁ ha hb + + end + + theorem my_lemma3 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := by + intro x y ε epos ele1 xlt ylt + sorry + + theorem my_lemma4 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := by + intro x y ε epos ele1 xlt ylt + calc + abs (x * y) = abs x * abs y := sorry + _ ≤ abs x * ε := sorry + _ < 1 * ε := sorry + _ = ε := sorry + + def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, f x ≤ a + + def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, a ≤ f x + + section + variable (f g : ℝ → ℝ) (a b : ℝ) + + example (hfa : FnUb f a) (hgb : FnUb g b) : FnUb (fun x ↦ f x + g x) (a + b) := by + intro x + dsimp + apply add_le_add + apply hfa + apply hgb + + example (hfa : FnLb f a) (hgb : FnLb g b) : FnLb (fun x => f x + g x) (a + b) := + sorry + + example (nnf : FnLb f 0) (nng : FnLb g 0) : FnLb (fun x => f x * g x) 0 := + sorry + + example (hfa : FnUb f a) (hfb : FnUb g b) (nng : FnLb g 0) (nna : 0 ≤ a) : + FnUb (fun x => f x * g x) (a * b) := + sorry + + end + + section + variable {α : Type _} {R : Type _} [OrderedCancelAddCommMonoid R] + + #check @add_le_add + + def FnUb' (f : α → R) (a : R) : Prop := + ∀ x, f x ≤ a + + theorem fn_ub_add {f g : α → R} {a b : R} (hfa : FnUb' f a) (hgb : FnUb' g b) : + FnUb' (fun x => f x + g x) (a + b) := fun x => add_le_add (hfa x) (hgb x) + + end + + example (f : ℝ → ℝ) (h : Monotone f) : ∀ {a b}, a ≤ b → f a ≤ f b := + @h + + section + variable (f g : ℝ → ℝ) + + example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f x + g x := by + intro a b aleb + apply add_le_add + apply mf aleb + apply mg aleb + + example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f x + g x := + fun a b aleb => add_le_add (mf aleb) (mg aleb) + + example {c : ℝ} (mf : Monotone f) (nnc : 0 ≤ c) : Monotone fun x => c * f x := + sorry + + example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f (g x) := + sorry + + def FnEven (f : ℝ → ℝ) : Prop := + ∀ x, f x = f (-x) + + def FnOdd (f : ℝ → ℝ) : Prop := + ∀ x, f x = -f (-x) + + example (ef : FnEven f) (eg : FnEven g) : FnEven fun x => f x + g x := by + intro x + calc + (fun x => f x + g x) x = f x + g x := rfl + _ = f (-x) + g (-x) := by rw [ef, eg] + + + example (of : FnOdd f) (og : FnOdd g) : FnEven fun x => f x * g x := by + sorry + + example (ef : FnEven f) (og : FnOdd g) : FnOdd fun x => f x * g x := by + sorry + + example (ef : FnEven f) (og : FnOdd g) : FnEven fun x => f (g x) := by + sorry + + end + + section + + variable {α : Type _} (r s t : Set α) + + example : s ⊆ s := by + intro x xs + exact xs + + theorem Subset.refl : s ⊆ s := fun x xs => xs + + theorem Subset.trans : r ⊆ s → s ⊆ t → r ⊆ t := by + sorry + + end + + section + variable {α : Type _} [PartialOrder α] + variable (s : Set α) (a b : α) + + def SetUb (s : Set α) (a : α) := + ∀ x, x ∈ s → x ≤ a + + example (h : SetUb s a) (h' : a ≤ b) : SetUb s b := + sorry + + end + + section + + open Function + + example (c : ℝ) : Injective fun x => x + c := by + intro x₁ x₂ h' + exact (add_left_inj c).mp h' + + example {c : ℝ} (h : c ≠ 0) : Injective fun x => c * x := by + sorry + + variable {α : Type _} {β : Type _} {γ : Type _} + variable {g : β → γ} {f : α → β} + + example (injg : Injective g) (injf : Injective f) : Injective fun x => g (f x) := by + sorry + + end
src/C03_Logic/S02_The_Existential_Quantifier.lean (new)
- @@ -0,0 +1,131 @@
 + import Mathlib.Data.Real.Basic + + example : ∃ x : ℝ, 2 < x ∧ x < 3 := by + use 5 / 2 + norm_num + + example : ∃ x : ℝ, 2 < x ∧ x < 3 := + have h : 2 < (5 : ℝ) / 2 ∧ (5 : ℝ) / 2 < 3 := by norm_num + ⟨5 / 2, h⟩ + + example : ∃ x : ℝ, 2 < x ∧ x < 3 := + ⟨5 / 2, by norm_num⟩ + + def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, f x ≤ a + + def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, a ≤ f x + + def FnHasUb (f : ℝ → ℝ) := + ∃ a, FnUb f a + + def FnHasLb (f : ℝ → ℝ) := + ∃ a, FnLb f a + + theorem fnUb_add {f g : ℝ → ℝ} {a b : ℝ} (hfa : FnUb f a) (hgb : FnUb g b) : + FnUb (fun x => f x + g x) (a + b) := + fun x => add_le_add (hfa x) (hgb x) + + section + + variable {f g : ℝ → ℝ} + + example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x => f x + g x := by + cases' ubf with a ubfa + cases' ubg with b ubfb + use a + b + apply fnUb_add ubfa ubfb + + example (lbf : FnHasLb f) (lbg : FnHasLb g) : FnHasLb fun x => f x + g x := by + sorry + + example {c : ℝ} (ubf : FnHasUb f) (h : c ≥ 0) : FnHasUb fun x => c * f x := by + sorry + + example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x => f x + g x := by + rcases ubf with ⟨a, ubfa⟩ + rcases ubg with ⟨b, ubfb⟩ + exact ⟨a + b, fnUb_add ubfa ubfb⟩ + + example : FnHasUb f → FnHasUb g → FnHasUb fun x => f x + g x := by + rintro ⟨a, ubfa⟩ ⟨b, ubfb⟩ + exact ⟨a + b, fnUb_add ubfa ubfb⟩ + + example : FnHasUb f → FnHasUb g → FnHasUb fun x => f x + g x := + fun ⟨a, ubfa⟩ ⟨b, ubfb⟩ => ⟨a + b, fnUb_add ubfa ubfb⟩ + + end + + section + + variable {α : Type _} [CommRing α] + + def SumOfSquares (x : α) := + ∃ a b, x = a ^ 2 + b ^ 2 + + theorem sumOfSquares_mul {x y : α} (sosx : SumOfSquares x) (sosy : SumOfSquares y) : + SumOfSquares (x * y) := by + rcases sosx with ⟨a, b, xeq⟩ + rcases sosy with ⟨c, d, yeq⟩ + rw [xeq, yeq] + use a * c - b * d, a * d + b * c + ring + + theorem sumOfSquares_mul' {x y : α} (sosx : SumOfSquares x) (sosy : SumOfSquares y) : + SumOfSquares (x * y) := by + rcases sosx with ⟨a, b, rfl⟩ + rcases sosy with ⟨c, d, rfl⟩ + use a * c - b * d, a * d + b * c + ring + + end + + section + variable {a b c : ℕ} + + example (divab : a ∣ b) (divbc : b ∣ c) : a ∣ c := by + cases' divab with d beq + cases' divbc with e ceq + rw [ceq, beq] + use d * e; ring + + example (divab : a ∣ b) (divac : a ∣ c) : a ∣ b + c := by + sorry + + end + + section + + open Function + + example {c : ℝ} : Surjective fun x => x + c := by + intro x + use x - c + dsimp; ring + + example {c : ℝ} (h : c ≠ 0) : Surjective fun x => c * x := by + sorry + + example (x y : ℝ) (h : x - y ≠ 0) : (x ^ 2 - y ^ 2) / (x - y) = x + y := by + field_simp [h] + ring + + example {f : ℝ → ℝ} (h : Surjective f) : ∃ x, f x ^ 2 = 4 := by + cases' h 2 with x hx + use x + rw [hx] + norm_num + + end + + section + open Function + variable {α : Type _} {β : Type _} {γ : Type _} + variable {g : β → γ} {f : α → β} + + example (surjg : Surjective g) (surjf : Surjective f) : Surjective fun x => g (f x) := by + sorry + + end +
src/C03_Logic/S03_Negation.lean (new)
- @@ -0,0 +1,136 @@
 + import Mathlib.Data.Real.Basic + + section + variable (a b : ℝ) + + example (h : a < b) : ¬b < a := by + intro h' + have : a < a := lt_trans h h' + apply lt_irrefl a this + + def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, f x ≤ a + + def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, a ≤ f x + + def FnHasUb (f : ℝ → ℝ) := + ∃ a, FnUb f a + + def FnHasLb (f : ℝ → ℝ) := + ∃ a, FnLb f a + + variable (f : ℝ → ℝ) + + example (h : ∀ a, ∃ x, f x > a) : ¬FnHasUb f := by + intro fnub + cases' fnub with a fnuba + cases' h a with x hx + have : f x ≤ a := fnuba x + linarith + + example (h : ∀ a, ∃ x, f x < a) : ¬FnHasLb f := + sorry + + example : ¬FnHasUb fun x => x := + sorry + + #check (not_le_of_gt : a > b → ¬a ≤ b) + #check (not_lt_of_ge : a ≥ b → ¬a < b) + #check (lt_of_not_ge : ¬a ≥ b → a < b) + #check (le_of_not_gt : ¬a > b → a ≤ b) + + example (h : Monotone f) (h' : f a < f b) : a (0 : ℝ) + have monof : Monotone f := by sorry + have h' : f 1 ≤ f 0 := le_refl _ + sorry + + example (x : ℝ) (h : ∀ ε > 0, x < ε) : x ≤ 0 := by + sorry + + end + + section + variable {α : Type _} (P : α → Prop) (Q : Prop) + + example (h : ¬∃ x, P x) : ∀ x, ¬P x := by + sorry + + example (h : ∀ x, ¬P x) : ¬∃ x, P x := by + sorry + + example (h : ¬∀ x, P x) : ∃ x, ¬P x := by + sorry + + example (h : ∃ x, ¬P x) : ¬∀ x, P x := by + sorry + + example (h : ¬∀ x, P x) : ∃ x, ¬P x := by + by_contra h' + apply h + intro x + show P x + by_contra h'' + exact h' ⟨x, h''⟩ + + example (h : ¬¬Q) : Q := by + sorry + + example (h : Q) : ¬¬Q := by + sorry + + end + + section + variable (f : ℝ → ℝ) + + example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by + sorry + + example (h : ¬∀ a, ∃ x, f x > a) : FnHasUb f := by + push_neg at h + exact h + + example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by + simp only [FnHasUb, FnUb] at h + push_neg at h + exact h + + example (h : ¬Monotone f) : ∃ x y, x ≤ y ∧ f y < f x := by + sorry + + example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by + contrapose! h + exact h + + example (x : ℝ) (h : ∀ ε > 0, x ≤ ε) : x ≤ 0 := by + contrapose! h + use x / 2 + constructor <;> linarith + + end + + section + variable (a : ℕ) + + example (h : 0 < 0) : a > 37 := by + exfalso + apply lt_irrefl 0 h + + example (h : 0 < 0) : a > 37 := + absurd h (lt_irrefl 0) + + example (h : 0 < 0) : a > 37 := by + have h' : ¬0 < 0 := lt_irrefl 0 + contradiction + + end +
src/C03_Logic/S04_Conjunction_and_Bi-implication.lean (new)
- @@ -0,0 +1,134 @@
 + import Mathlib.Data.Real.Basic + import Mathlib.Data.Nat.Prime + import Mathlib.Tactic.NormNum + + example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y := by + constructor + · assumption + intro h + apply h₁ + rw [h] + + example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y := + ⟨h₀, fun h => h₁ (by rw [h])⟩ + + example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y := + have h : x ≠ y := by + contrapose! h₁ + rw [h₁] + ⟨h₀, h⟩ + + example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by + cases' h with h₀ h₁ + contrapose! h₁ + exact le_antisymm h₀ h₁ + + example {x y : ℝ} : x ≤ y ∧ x ≠ y → ¬y ≤ x := by + rintro ⟨h₀, h₁⟩ h' + exact h₁ (le_antisymm h₀ h') + + example {x y : ℝ} : x ≤ y ∧ x ≠ y → ¬y ≤ x := + fun ⟨h₀, h₁⟩ h' => h₁ (le_antisymm h₀ h') + + example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by + intro h' + apply h.right + exact le_antisymm h.left h' + + example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := + fun h' => h.right (le_antisymm h.left h') + + example {m n : ℕ} (h : m ∣ n ∧ m ≠ n) : m ∣ n ∧ ¬n ∣ m := + sorry + + example : ∃ x : ℝ, 2 < x ∧ x < 4 := + ⟨5 / 2, by norm_num, by norm_num⟩ + + example (x y : ℝ) : (∃ z : ℝ, x < z ∧ z < y) → x < y := by + rintro ⟨z, xltz, zlty⟩ + exact lt_trans xltz zlty + + example (x y : ℝ) : (∃ z : ℝ, x < z ∧ z < y) → x < y := + fun ⟨z, xltz, zlty⟩ => lt_trans xltz zlty + + example : ∃ x : ℝ, 2 < x ∧ x < 4 := by + use 5 / 2 + constructor <;> norm_num + + example : ∃ m n : ℕ, 4 < m ∧ m < n ∧ n < 10 ∧ Nat.Prime m ∧ Nat.Prime n := by + use 5 + use 7 + norm_num + sorry + + example {x y : ℝ} : x ≤ y ∧ x ≠ y → x ≤ y ∧ ¬y ≤ x := by + rintro ⟨h₀, h₁⟩ + use h₀ + exact fun h' => h₁ (le_antisymm h₀ h') + + example {x y : ℝ} (h : x ≤ y) : ¬y ≤ x ↔ x ≠ y := by + constructor + · contrapose! + rintro rfl + rfl + contrapose! + exact le_antisymm h + + example {x y : ℝ} (h : x ≤ y) : ¬y ≤ x ↔ x ≠ y := + ⟨fun h₀ h₁ => h₀ (by rw [h₁]), fun h₀ h₁ => h₀ (le_antisymm h h₁)⟩ + + example {x y : ℝ} : x ≤ y ∧ ¬y ≤ x ↔ x ≤ y ∧ x ≠ y := + sorry + + theorem aux {x y : ℝ} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := + have h' : x ^ 2 = 0 := by sorry + pow_eq_zero h' + + example (x y : ℝ) : x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := + sorry + + section + + example (x : ℝ) : abs (x + 3) < 5 → -8 < x ∧ x < 2 := by + rw [abs_lt] + intro h + constructor <;> linarith + + example : 3 ∣ Nat.gcd 6 15 := by + rw [Nat.dvd_gcd_iff] + constructor <;> norm_num + + end + + theorem not_monotone_iff {f : ℝ → ℝ} : ¬Monotone f ↔ ∃ x y, x ≤ y ∧ f x > f y := by + rw [Monotone] + push_neg + rfl + + example : ¬Monotone fun x : ℝ => -x := by + sorry + + section + variable {α : Type _} [PartialOrder α] + variable (a b : α) + + example : a < b ↔ a ≤ b ∧ a ≠ b := by + rw [lt_iff_le_not_le] + sorry + + end + + section + variable {α : Type _} [Preorder α] + variable (a b c : α) + + example : ¬a < a := by + rw [lt_iff_le_not_le] + sorry + + example : a < b → b < c → a < c := by + simp only [lt_iff_le_not_le] + sorry + + end +
src/C03_Logic/S05_Disjunction.lean (new)
- @@ -0,0 +1,100 @@
 + import Mathlib.Data.Real.Basic + + section + + variable {x y : ℝ} + + example (h : y > x ^ 2) : y > 0 ∨ y < -1 := by + left + linarith [pow_two_nonneg x] + + example (h : -y > x ^ 2 + 1) : y > 0 ∨ y < -1 := by + right + linarith [pow_two_nonneg x] + + example (h : y > 0) : y > 0 ∨ y < -1 := + Or.inl h + + example (h : y < -1) : y > 0 ∨ y < -1 := + Or.inr h + + example : x < abs y → x < y ∨ x < -y := by + cases' le_or_gt 0 y with h h + · rw [abs_of_nonneg h] + intro h + left + exact h + rw [abs_of_neg h] + intro h; right; exact h + + namespace MyAbs + + theorem le_abs_self (x : ℝ) : x ≤ abs x := by + sorry + + theorem neg_le_abs_self (x : ℝ) : -x ≤ abs x := by + sorry + + theorem abs_add (x y : ℝ) : abs (x + y) ≤ abs x + abs y := by + sorry + + theorem lt_abs : x < abs y ↔ x < y ∨ x < -y := by + sorry + + theorem abs_lt : abs x < y ↔ -y < x ∧ x < y := by + sorry + + end MyAbs + + end + + example {x : ℝ} (h : x ≠ 0) : x < 0 ∨ x > 0 := by + rcases lt_trichotomy x 0 with (xlt | xeq | xgt) + · left + exact xlt + · contradiction + right; exact xgt + + example {m n k : ℕ} (h : m ∣ n ∨ m ∣ k) : m ∣ n * k := by + rcases h with (⟨a, rfl⟩ | ⟨b, rfl⟩) + · rw [mul_assoc] + apply dvd_mul_right + rw [mul_comm, mul_assoc] + apply dvd_mul_right + + example {z : ℝ} (h : ∃ x y, z = x ^ 2 + y ^ 2 ∨ z = x ^ 2 + y ^ 2 + 1) : z ≥ 0 := by + sorry + + example {x : ℝ} (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by + sorry + + example {x y : ℝ} (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by + sorry + + section + variable {R : Type _} [CommRing R] [IsDomain R] + variable (x y : R) + + example (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by + sorry + + example (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by + sorry + + end + + example (P : Prop) : ¬¬P → P := by + intro h + cases em P + · assumption + contradiction + + example (P : Prop) : ¬¬P → P := by + intro h + by_cases h' : P + · assumption + contradiction + + example (P Q : Prop) : P → Q ↔ ¬P ∨ Q := by + sorry +
src/C03_Logic/S06_Sequences_and_Convergence.lean (new)
- @@ -0,0 +1,95 @@
 + import Mathlib.Data.Real.Basic + + def ConvergesTo (s : ℕ → ℝ) (a : ℝ) := + ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε + + example : (fun x y : ℝ => (x + y) ^ 2) = fun x y : ℝ => x ^ 2 + 2 * x * y + y ^ 2 := by + ext + ring + + example (a b : ℝ) : abs a = abs (a - b + b) := by + congr + ring + + example {a : ℝ} (h : 1 < a) : a < a * a := by + convert(mul_lt_mul_right _).2 h + · rw [one_mul] + exact lt_trans zero_lt_one h + + theorem convergesTo_const (a : ℝ) : ConvergesTo (fun x : ℕ => a) a := by + intro ε εpos + use 0 + intro n nge; dsimp + rw [sub_self, abs_zero] + apply εpos + + theorem convergesTo_add {s t : ℕ → ℝ} {a b : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t b) : + ConvergesTo (fun n => s n + t n) (a + b) := by + intro ε εpos + dsimp + have ε2pos : 0 < ε / 2 := by linarith + cases' cs (ε / 2) ε2pos with Ns hs + cases' ct (ε / 2) ε2pos with Nt ht + use max Ns Nt + sorry + + theorem convergesTo_mul_const {s : ℕ → ℝ} {a : ℝ} (c : ℝ) (cs : ConvergesTo s a) : + ConvergesTo (fun n => c * s n) (c * a) := by + by_cases h : c = 0 + · convert convergesTo_const 0 + · rw [h, MulZeroClass.zero_mul] + rw [h, MulZeroClass.zero_mul] + have acpos : 0 < abs c := abs_pos.mpr h + sorry + + theorem exists_abs_le_of_convergesTo {s : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) : + ∃ N b, ∀ n, N ≤ n → abs (s n) s n * t n) 0 := by + intro ε εpos + dsimp + rcases exists_abs_le_of_convergesTo cs with ⟨N₀, B, h₀⟩ + have Bpos : 0 0 := div_pos εpos Bpos + cases' ct _ pos₀ with N₁ h₁ + sorry + + theorem convergesTo_mul {s t : ℕ → ℝ} {a b : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t b) : + ConvergesTo (fun n => s n * t n) (a * b) := by + have h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 := by + apply aux cs + convert convergesTo_add ct (convergesTo_const (-b)) + ring + have := convergesTo_add h₁ (convergesTo_mul_const b cs) + convert convergesTo_add h₁ (convergesTo_mul_const b cs) using 1 + · ext; ring + ring + + theorem convergesTo_unique {s : ℕ → ℝ} {a b : ℝ} (sa : ConvergesTo s a) (sb : ConvergesTo s b) : + a = b := by + by_contra abne + have : abs (a - b) > 0 := by sorry + let ε := abs (a - b) / 2 + have εpos : ε > 0 := by + change abs (a - b) / 2 > 0 + linarith + cases' sa ε εpos with Na hNa + cases' sb ε εpos with Nb hNb + let N := max Na Nb + have absa : abs (s N - a) < ε := by sorry + have absb : abs (s N - b) < ε := by sorry + have : abs (a - b) < abs (a - b) := by sorry + exact lt_irrefl _ this + + section + variable {α : Type _} [LinearOrder α] + + def ConvergesTo' (s : α → ℝ) (a : ℝ) := + ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε + + end +
src/C03_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean (new)
- @@ -0,0 +1,128 @@
  + import Mathlib.Data.Real.Basic + + def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, f x ≤ a + + def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, a ≤ f x + + section + variable (f g : ℝ → ℝ) (a b : ℝ) + + example (hfa : FnLb f a) (hgb : FnLb g b) : FnLb (fun x => f x + g x) (a + b) := by + intro x + apply add_le_add + apply hfa + apply hgb + + example (nnf : FnLb f 0) (nng : FnLb g 0) : FnLb (fun x => f x * g x) 0 := by + intro x + apply mul_nonneg + apply nnf + apply nng + + example (hfa : FnUb f a) (hfb : FnUb g b) (nng : FnLb g 0) (nna : 0 ≤ a) : + FnUb (fun x => f x * g x) (a * b) := by + intro x + apply mul_le_mul + apply hfa + apply hfb + apply nng + apply nna + + end + + section + variable (f g : ℝ → ℝ) + + example {c : ℝ} (mf : Monotone f) (nnc : 0 ≤ c) : Monotone fun x => c * f x := by + intro a b aleb + apply mul_le_mul_of_nonneg_left _ nnc + apply mf aleb + + example {c : ℝ} (mf : Monotone f) (nnc : 0 ≤ c) : Monotone fun x => c * f x := + fun a b aleb => mul_le_mul_of_nonneg_left (mf aleb) nnc + + example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f (g x) := by + intro a b aleb + apply mf + apply mg + apply aleb + + example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f (g x) := + fun a b aleb => mf (mg aleb) + + def FnEven (f : ℝ → ℝ) : Prop := + ∀ x, f x = f (-x) + + def FnOdd (f : ℝ → ℝ) : Prop := + ∀ x, f x = -f (-x) + + example (of : FnOdd f) (og : FnOdd g) : FnEven fun x => f x * g x := by + intro x + calc + (fun x => f x * g x) x = f x * g x := rfl + _ = f (-x) * g (-x) := by rw [of, og, neg_mul_neg] + + + example (ef : FnEven f) (og : FnOdd g) : FnOdd fun x => f x * g x := by + intro x + dsimp + rw [ef, og, neg_mul_eq_mul_neg] + + example (ef : FnEven f) (og : FnOdd g) : FnEven fun x => f (g x) := by + intro x + dsimp + rw [og, ← ef] + + end + + section + + variable {α : Type _} (r s t : Set α) + + example : r ⊆ s → s ⊆ t → r ⊆ t := by + intro rsubs ssubt x xr + apply ssubt + apply rsubs + apply xr + + theorem Subset.trans : r ⊆ s → s ⊆ t → r ⊆ t := + fun rsubs ssubt x xr => ssubt (rsubs xr) + + end + + section + variable {α : Type _} [PartialOrder α] + variable (s : Set α) (a b : α) + + def SetUb (s : Set α) (a : α) := + ∀ x, x ∈ s → x ≤ a + + example (h : SetUb s a) (h' : a ≤ b) : SetUb s b := by + intro x xs + apply le_trans (h x xs) h' + + example (h : SetUb s a) (h' : a ≤ b) : SetUb s b := + fun x xs => le_trans (h x xs) h' + + end + + section + + open Function + + example {c : ℝ} (h : c ≠ 0) : Injective fun x => c * x := by + intro x₁ x₂ h' + apply (mul_right_inj' h).mp h' + + variable {α : Type _} {β : Type _} {γ : Type _} + variable {g : β → γ} {f : α → β} + + example (injg : Injective g) (injf : Injective f) : Injective fun x => g (f x) := by + intro x₁ x₂ h + apply injf + apply injg + apply h + + end
src/C03_Logic/solutions/Solutions_S02_The_Existential_Quantifier.lean (new)
- @@ -0,0 +1,81 @@
  + import Mathlib.Data.Real.Basic + + def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, f x ≤ a + + def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, a ≤ f x + + def FnHasUb (f : ℝ → ℝ) := + ∃ a, FnUb f a + + def FnHasLb (f : ℝ → ℝ) := + ∃ a, FnLb f a + + theorem fnUb_add {f g : ℝ → ℝ} {a b : ℝ} (hfa : FnUb f a) (hgb : FnUb g b) : + FnUb (fun x => f x + g x) (a + b) := + fun x => add_le_add (hfa x) (hgb x) + + section + + variable {f g : ℝ → ℝ} + + example (lbf : FnHasLb f) (lbg : FnHasLb g) : FnHasLb fun x => f x + g x := by + cases' lbf with a lbfa + cases' lbg with b lbgb + use a + b + intro x + exact add_le_add (lbfa x) (lbgb x) + + example {c : ℝ} (ubf : FnHasUb f) (h : c ≥ 0) : FnHasUb fun x => c * f x := by + cases' ubf with a lbfa + use c * a + intro x + exact mul_le_mul_of_nonneg_left (lbfa x) h + + end + + section + variable {a b c : ℕ} + + example (divab : a ∣ b) (divbc : b ∣ c) : a ∣ c := by + rcases divab with ⟨d, rfl⟩ + rcases divbc with ⟨e, rfl⟩ + use d * e; ring + + example (divab : a ∣ b) (divac : a ∣ c) : a ∣ b + c := by + rcases divab with ⟨d, rfl⟩ + rcases divac with ⟨e, rfl⟩ + use d + e; ring + + end + + section + + open Function + + example {c : ℝ} (h : c ≠ 0) : Surjective fun x => c * x := by + intro x + use x / c + dsimp; rw [mul_div_cancel' _ h] + + example {c : ℝ} (h : c ≠ 0) : Surjective fun x => c * x := by + intro x + use x / c + field_simp [h] ; ring + + end + + section + open Function + variable {α : Type _} {β : Type _} {γ : Type _} + variable {g : β → γ} {f : α → β} + + example (surjg : Surjective g) (surjf : Surjective f) : Surjective fun x => g (f x) := by + intro z + rcases surjg z with ⟨y, rfl⟩ + rcases surjf y with ⟨x, rfl⟩ + use x + + end +
src/C03_Logic/solutions/Solutions_S03_Negation.lean (new)
- @@ -0,0 +1,110 @@
 + import Mathlib.Data.Real.Basic + + section + variable (a b : ℝ) + + def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, f x ≤ a + + def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, a ≤ f x + + def FnHasUb (f : ℝ → ℝ) := + ∃ a, FnUb f a + + def FnHasLb (f : ℝ → ℝ) := + ∃ a, FnLb f a + + variable (f : ℝ → ℝ) + + example (h : ∀ a, ∃ x, f x < a) : ¬FnHasLb f := by + rintro ⟨a, ha⟩ + rcases h a with ⟨x, hx⟩ + have := ha x + linarith + + example : ¬FnHasUb fun x => x := by + rintro ⟨a, ha⟩ + have : a + 1 ≤ a := ha (a + 1) + linarith + + example (h : Monotone f) (h' : f a < f b) : a < b := by + apply lt_of_not_ge + intro h'' + apply absurd h' + apply not_lt_of_ge (h h'') + + example (h : a ≤ b) (h' : f b < f a) : ¬Monotone f := by + intro h'' + apply absurd h' + apply not_lt_of_ge + apply h'' h + + example : ¬∀ {f : ℝ → ℝ}, Monotone f → ∀ {a b}, f a ≤ f b → a ≤ b := by + intro h + let f := fun x : ℝ => (0 : ℝ) + have monof : Monotone f := by + intro a b leab + rfl + have h' : f 1 ≤ f 0 := le_refl _ + have : (1 : ℝ) ≤ 0 := h monof h' + linarith + + example (x : ℝ) (h : ∀ ε > 0, x < ε) : x ≤ 0 := by + apply le_of_not_gt + intro h' + linarith [h _ h'] + + end + + section + variable {α : Type _} (P : α → Prop) (Q : Prop) + + example (h : ¬∃ x, P x) : ∀ x, ¬P x := by + intro x Px + apply h + use x + exact Px + + example (h : ∀ x, ¬P x) : ¬∃ x, P x := by + rintro ⟨x, Px⟩ + exact h x Px + + example (h : ∃ x, ¬P x) : ¬∀ x, P x := by + intro h' + rcases h with ⟨x, nPx⟩ + apply nPx + apply h' + + example (h : ¬¬Q) : Q := by + by_contra h' + exact h h' + + example (h : Q) : ¬¬Q := by + intro h' + exact h' h + + end + + section + variable (f : ℝ → ℝ) + + example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by + intro a + by_contra h' + apply h + use a + intro x + apply le_of_not_gt + intro h'' + apply h' + use x + exact h'' + + example (h : ¬Monotone f) : ∃ x y, x ≤ y ∧ f y < f x := by + rw [Monotone] at h + push_neg at h + exact h + + end +
src/C03_Logic/solutions/Solutions_S04_Conjunction_and_Bi-implication.lean (new)
- @@ -0,0 +1,93 @@
 + import Mathlib.Data.Real.Basic + import Mathlib.Data.Nat.Prime + import Mathlib.Tactic.NormNum + + example {m n : ℕ} (h : m ∣ n ∧ m ≠ n) : m ∣ n ∧ ¬n ∣ m := by + cases' h with h0 h1 + constructor + · exact h0 + intro h2 + apply h1 + apply Nat.dvd_antisymm h0 h2 + + example {x y : ℝ} : x ≤ y ∧ ¬y ≤ x ↔ x ≤ y ∧ x ≠ y := by + constructor + · rintro ⟨h0, h1⟩ + constructor + · exact h0 + intro h2 + apply h1 + rw [h2] + rintro ⟨h0, h1⟩ + constructor + · exact h0 + intro h2 + apply h1 + apply le_antisymm h0 h2 + + theorem aux {x y : ℝ} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := + have h' : x ^ 2 = 0 := by linarith [pow_two_nonneg x, pow_two_nonneg y] + pow_eq_zero h' + + example (x y : ℝ) : x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by + constructor + · intro h + constructor + · exact aux h + rw [add_comm] at h + exact aux h + rintro ⟨rfl, rfl⟩ + norm_num + + theorem not_monotone_iff {f : ℝ → ℝ} : ¬Monotone f ↔ ∃ x y, x ≤ y ∧ f x > f y := by + rw [Monotone] + push_neg + rfl + + example : ¬Monotone fun x : ℝ => -x := by + rw [not_monotone_iff] + use 0, 1 + norm_num + + section + variable {α : Type _} [PartialOrder α] + variable (a b : α) + + example : a < b ↔ a ≤ b ∧ a ≠ b := by + rw [lt_iff_le_not_le] + constructor + · rintro ⟨h0, h1⟩ + constructor + · exact h0 + intro h2 + apply h1 + rw [h2] + rintro ⟨h0, h1⟩ + constructor + · exact h0 + intro h2 + apply h1 + apply le_antisymm h0 h2 + + end + + section + variable {α : Type _} [Preorder α] + variable (a b c : α) + + example : ¬a < a := by + rw [lt_iff_le_not_le] + rintro ⟨h0, h1⟩ + exact h1 h0 + + example : a < b → b < c → a < c := by + simp only [lt_iff_le_not_le] + rintro ⟨h0, h1⟩ ⟨h2, h3⟩ + constructor + · apply le_trans h0 h2 + intro h4 + apply h1 + apply le_trans h2 h4 + + end +
src/C03_Logic/solutions/Solutions_S05_Disjunction.lean (new)
- @@ -0,0 +1,140 @@
 + import Mathlib.Data.Real.Basic + + section + + variable {x y : ℝ} + + namespace MyAbs + + theorem le_abs_self (x : ℝ) : x ≤ abs x := by + cases' le_or_gt 0 x with h h + · rw [abs_of_nonneg h] + rw [abs_of_neg h] + linarith + + theorem neg_le_abs_self (x : ℝ) : -x ≤ abs x := by + cases' le_or_gt 0 x with h h + · rw [abs_of_nonneg h] + linarith + rw [abs_of_neg h] + + theorem abs_add (x y : ℝ) : abs (x + y) ≤ abs x + abs y := by + cases' le_or_gt 0 (x + y) with h h + · rw [abs_of_nonneg h] + linarith [le_abs_self x, le_abs_self y] + rw [abs_of_neg h] + linarith [neg_le_abs_self x, neg_le_abs_self y] + + theorem lt_abs : x < abs y ↔ x < y ∨ x < -y := by + cases' le_or_gt 0 y with h h + · rw [abs_of_nonneg h] + constructor + · intro h' + left + exact h' + intro h' + cases' h' with h' h' + · exact h' + linarith + rw [abs_of_neg h] + constructor + · intro h' + right + exact h' + intro h' + cases' h' with h' h' + · linarith + exact h' + + theorem abs_lt : abs x < y ↔ -y < x ∧ x < y := by + cases' le_or_gt 0 x with h h + · rw [abs_of_nonneg h] + constructor + · intro h' + constructor + · linarith + exact h' + intro h' + cases' h' with h1 h2 + exact h2 + rw [abs_of_neg h] + constructor + · intro h' + constructor + · linarith + linarith + intro h' + linarith + + end MyAbs + + end + + example {z : ℝ} (h : ∃ x y, z = x ^ 2 + y ^ 2 ∨ z = x ^ 2 + y ^ 2 + 1) : z ≥ 0 := by + rcases h with ⟨x, y, rfl | rfl⟩ <;> linarith [sq_nonneg x, sq_nonneg y] + + example {x : ℝ} (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by + have h' : x ^ 2 - 1 = 0 := by rw [h, sub_self] + have h'' : (x + 1) * (x - 1) = 0 := by + rw [← h'] + ring + cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1 + · right + exact eq_neg_iff_add_eq_zero.mpr h1 + left + exact eq_of_sub_eq_zero h1 + + example {x y : ℝ} (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by + have h' : x ^ 2 - y ^ 2 = 0 := by rw [h, sub_self] + have h'' : (x + y) * (x - y) = 0 := by + rw [← h'] + ring + cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1 + · right + exact eq_neg_iff_add_eq_zero.mpr h1 + left + exact eq_of_sub_eq_zero h1 + + section + variable {R : Type _} [CommRing R] [IsDomain R] + variable (x y : R) + + example (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by + have h' : x ^ 2 - 1 = 0 := by rw [h, sub_self] + have h'' : (x + 1) * (x - 1) = 0 := by + rw [← h'] + ring + cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1 + · right + exact eq_neg_iff_add_eq_zero.mpr h1 + left + exact eq_of_sub_eq_zero h1 + + example (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by + have h' : x ^ 2 - y ^ 2 = 0 := by rw [h, sub_self] + have h'' : (x + y) * (x - y) = 0 := by + rw [← h'] + ring + cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1 + · right + exact eq_neg_iff_add_eq_zero.mpr h1 + left + exact eq_of_sub_eq_zero h1 + + end + + example (P Q : Prop) : P → Q ↔ ¬P ∨ Q := by + constructor + · intro h + by_cases h' : P + · right + exact h h' + left + exact h' + rintro (h | h) + · intro h' + exact absurd h' h + intro + exact h + +
src/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean (new)
- @@ -0,0 +1,123 @@
 + import Mathlib.Data.Real.Basic + + def ConvergesTo (s : ℕ → ℝ) (a : ℝ) := + ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε + + theorem convergesTo_const (a : ℝ) : ConvergesTo (fun x : ℕ => a) a := by + intro ε εpos + use 0 + intro n nge; dsimp + rw [sub_self, abs_zero] + apply εpos + + theorem convergesTo_add {s t : ℕ → ℝ} {a b : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t b) : + ConvergesTo (fun n => s n + t n) (a + b) := by + intro ε εpos + dsimp + have ε2pos : 0 < ε / 2 := by linarith + cases' cs (ε / 2) ε2pos with Ns hs + cases' ct (ε / 2) ε2pos with Nt ht + use max Ns Nt + intro n hn + have ngeNs : n ≥ Ns := le_of_max_le_left hn + have ngeNt : n ≥ Nt := le_of_max_le_right hn + calc + |s n + t n - (a + b)| = |s n - a + (t n - b)| := by + congr + ring + _ ≤ |s n - a| + |t n - b| := (abs_add _ _) + _ < ε / 2 + ε / 2 := (add_lt_add (hs n ngeNs) (ht n ngeNt)) + _ = ε := by norm_num + + theorem convergesTo_mul_const {s : ℕ → ℝ} {a : ℝ} (c : ℝ) (cs : ConvergesTo s a) : + ConvergesTo (fun n => c * s n) (c * a) := by + by_cases h : c = 0 + · convert convergesTo_const 0 + · rw [h, MulZeroClass.zero_mul] + rw [h, MulZeroClass.zero_mul] + have acpos : 0 < abs c := abs_pos.mpr h + intro ε εpos + dsimp + have εcpos : 0 < ε / abs c := by apply div_pos εpos acpos + cases' cs (ε / abs c) εcpos with Ns hs + use Ns + intro n ngt + calc + |c * s n - c * a| = |c| * |s n - a| := by rw [← abs_mul, mul_sub] + _ < |c| * (ε / |c|) := (mul_lt_mul_of_pos_left (hs n ngt) acpos) + _ = ε := mul_div_cancel' _ (ne_of_lt acpos).symm + + theorem exists_abs_le_of_convergesTo {s : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) : + ∃ N b, ∀ n, N ≤ n → abs (s n) < b := by + cases' cs 1 zero_lt_one with N h + use N, abs a + 1 + intro n ngt + calc + |s n| = |s n - a + a| := by + congr + abel + _ ≤ |s n - a| + |a| := (abs_add _ _) + _ < |a| + 1 := by linarith [h n ngt] + + theorem aux {s t : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t 0) : + ConvergesTo (fun n => s n * t n) 0 := by + intro ε εpos + dsimp + rcases exists_abs_le_of_convergesTo cs with ⟨N₀, B, h₀⟩ + have Bpos : 0 0 := div_pos εpos Bpos + cases' ct _ pos₀ with N₁ h₁ + use max N₀ N₁ + intro n ngt + have ngeN₀ : n ≥ N₀ := le_of_max_le_left ngt + have ngeN₁ : n ≥ N₁ := le_of_max_le_right ngt + calc + |s n * t n - 0| = |s n| * |t n - 0| := by rw [sub_zero, abs_mul, sub_zero] + _ < B * (ε / B) := (mul_lt_mul'' (h₀ n ngeN₀) (h₁ n ngeN₁) (abs_nonneg _) (abs_nonneg _)) + _ = ε := mul_div_cancel' _ (ne_of_lt Bpos).symm + + theorem convergesTo_mul {s t : ℕ → ℝ} {a b : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t b) : + ConvergesTo (fun n => s n * t n) (a * b) := by + have h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 := by + apply aux cs + convert convergesTo_add ct (convergesTo_const (-b)) + ring + have := convergesTo_add h₁ (convergesTo_mul_const b cs) + convert convergesTo_add h₁ (convergesTo_mul_const b cs) using 1 + · ext; ring + ring + + theorem convergesTo_unique {s : ℕ → ℝ} {a b : ℝ} (sa : ConvergesTo s a) (sb : ConvergesTo s b) : + a = b := by + by_contra abne + have : abs (a - b) > 0 := by + apply lt_of_le_of_ne + · apply abs_nonneg + intro h'' + apply abne + apply eq_of_abs_sub_eq_zero h''.symm + let ε := abs (a - b) / 2 + have εpos : ε > 0 := by + change abs (a - b) / 2 > 0 + linarith + cases' sa ε εpos with Na hNa + cases' sb ε εpos with Nb hNb + let N := max Na Nb + have absa : abs (s N - a) < ε := by + apply hNa + apply le_max_left + have absb : abs (s N - b) < ε := by + apply hNb + apply le_max_right + have : abs (a - b) < abs (a - b) + calc + abs (a - b) = abs (-(s N - a) + (s N - b)) := by + congr + ring + _ ≤ abs (-(s N - a)) + abs (s N - b) := (abs_add _ _) + _ = abs (s N - a) + abs (s N - b) := by rw [abs_neg] + _ < ε + ε := (add_lt_add absa absb) + _ = abs (a - b) := by norm_num + + exact lt_irrefl _ this +
src/C04_Sets_and_Functions/S01_Sets.lean (new)
- @@ -0,0 +1,247 @@
 + import Mathlib.Data.Set.Lattice + import Mathlib.Data.Nat.Prime + import Mathlib.Data.Nat.Parity + import Mathlib.Tactic + + section + variable {α : Type _} + variable (s t u : Set α) + open Set + + example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := by + rw [subset_def, inter_def, inter_def] + rw [subset_def] at h + dsimp + rintro x ⟨xs, xu⟩ + exact ⟨h _ xs, xu⟩ + + example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := by + simp only [subset_def, mem_inter_iff] at * + rintro x ⟨xs, xu⟩ + exact ⟨h _ xs, xu⟩ + + example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := by + intro x xsu + exact ⟨h xsu.1, xsu.2⟩ + + theorem foo (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := + fun x ⟨xs, xu⟩ => ⟨h xs, xu⟩ + + example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := + fun x ⟨xs, xu⟩ => ⟨h xs, xu⟩ + + example : s ∩ (t ∪ u) ⊆ s ∩ t ∪ s ∩ u := by + intro x hx + have xs : x ∈ s := hx.1 + have xtu : x ∈ t ∪ u := hx.2 + cases' xtu with xt xu + · left + show x ∈ s ∩ t + exact ⟨xs, xt⟩ + right + show x ∈ s ∩ u + exact ⟨xs, xu⟩ + + example : s ∩ (t ∪ u) ⊆ s ∩ t ∪ s ∩ u := by + rintro x ⟨xs, xt | xu⟩ + · left + exact ⟨xs, xt⟩ + right; exact ⟨xs, xu⟩ + + example : s ∩ t ∪ s ∩ u ⊆ s ∩ (t ∪ u) := by + sorry + example : (s \ t) \ u ⊆ s \ (t ∪ u) := by + intro x xstu + have xs : x ∈ s := xstu.1.1 + have xnt : x ∉ t := xstu.1.2 + have xnu : x ∉ u := xstu.2 + constructor + · exact xs + intro xtu + -- x ∈ t ∨ x ∈ u + cases' xtu with xt xu + · show False + exact xnt xt + show False; exact xnu xu + + example : (s \ t) \ u ⊆ s \ (t ∪ u) := by + rintro x ⟨⟨xs, xnt⟩, xnu⟩ + use xs + rintro (xt | xu) <;> contradiction + + example : s \ (t ∪ u) ⊆ (s \ t) \ u := by + sorry + example : s ∩ t = t ∩ s := by + ext x + simp only [mem_inter_iff] + constructor + · rintro ⟨xs, xt⟩ + exact ⟨xt, xs⟩ + rintro ⟨xt, xs⟩; exact ⟨xs, xt⟩ + + example : s ∩ t = t ∩ s := + Set.ext fun x => ⟨fun ⟨xs, xt⟩ => ⟨xt, xs⟩, fun ⟨xt, xs⟩ => ⟨xs, xt⟩⟩ + + example : s ∩ t = t ∩ s := by ext x; simp [and_comm] + + example : s ∩ t = t ∩ s := by + apply Subset.antisymm + · rintro x ⟨xs, xt⟩ + exact ⟨xt, xs⟩ + rintro x ⟨xt, xs⟩; exact ⟨xs, xt⟩ + + example : s ∩ t = t ∩ s := + Subset.antisymm sorry sorry + example : s ∩ (s ∪ t) = s := by + sorry + + example : s ∪ s ∩ t = s := by + sorry + + example : s \ t ∪ t = s ∪ t := by + sorry + + example : s \ t ∪ t \ s = (s ∪ t) \ (s ∩ t) := by + sorry + + def evens : Set ℕ := + { n | Even n } + + def odds : Set ℕ := + { n | ¬Even n } + + example : evens ∪ odds = univ := by + rw [evens, odds] + ext n + simp + apply Classical.em + + example (x : ℕ) (h : x ∈ (∅ : Set ℕ)) : False := + h + + example (x : ℕ) : x ∈ (univ : Set ℕ) := + trivial + + example : { n | Nat.Prime n } ∩ { n | n > 2 } ⊆ { n | ¬Even n } := by + sorry + + #print Prime + + #print Nat.Prime + + example (n : ℕ) : Prime n ↔ Nat.Prime n := + Nat.prime_iff.symm + + example (n : ℕ) (h : Prime n) : Nat.Prime n := by + rw [Nat.prime_iff] + exact h + + example (n : ℕ) (h : Prime n) : Nat.Prime n := by + rwa [Nat.prime_iff] + + end + + section + + variable (s t : Set ℕ) + + example (h₀ : ∀ x ∈ s, ¬Even x) (h₁ : ∀ x ∈ s, Prime x) : ∀ x ∈ s, ¬Even x ∧ Prime x := by + intro x xs + constructor + · apply h₀ x xs + apply h₁ x xs + + example (h : ∃ x ∈ s, ¬Even x ∧ Prime x) : ∃ x ∈ s, Prime x := by + rcases h with ⟨x, xs, _, prime_x⟩ + use x, xs + exact prime_x + + section + variable (ssubt : s ⊆ t) + + example (h₀ : ∀ x ∈ t, ¬Even x) (h₁ : ∀ x ∈ t, Prime x) : ∀ x ∈ s, ¬Even x ∧ Prime x := by + sorry + + example (h : ∃ x ∈ s, ¬Even x ∧ Prime x) : ∃ x ∈ t, Prime x := by + sorry + + end + + end + + section + variable {α I : Type _} + variable (A B : I → Set α) + variable (s : Set α) + + open Set + + example : (s ∩ ⋃ i, A i) = ⋃ i, A i ∩ s := by + ext x + simp only [mem_inter_iff, mem_iUnion] + constructor + · rintro ⟨xs, ⟨i, xAi⟩⟩ + exact ⟨i, xAi, xs⟩ + rintro ⟨i, xAi, xs⟩ + exact ⟨xs, ⟨i, xAi⟩⟩ + + example : (⋂ i, A i ∩ B i) = (⋂ i, A i) ∩ ⋂ i, B i := by + ext x + simp only [mem_inter_iff, mem_iInter] + constructor + · intro h + constructor + · intro i + exact (h i).1 + intro i + exact (h i).2 + rintro ⟨h1, h2⟩ i + constructor + · exact h1 i + exact h2 i + + + example : (s ∪ ⋂ i, A i) = ⋂ i, A i ∪ s := by + sorry + + def primes : Set ℕ := + { x | Nat.Prime x } + + example : (⋃ p ∈ primes, { x | p ^ 2 ∣ x }) = { x | ∃ p ∈ primes, p ^ 2 ∣ x } :=by + ext + rw [mem_iUnion₂] + simp + + example : (⋃ p ∈ primes, { x | p ^ 2 ∣ x }) = { x | ∃ p ∈ primes, p ^ 2 ∣ x } := by + ext + simp + + example : (⋂ p ∈ primes, { x | ¬p ∣ x }) ⊆ { x | x = 1 } := by + intro x + contrapose! + simp + apply Nat.exists_prime_and_dvd + + example : (⋃ p ∈ primes, { x | x ≤ p }) = univ := by + sorry + + end + + section + + open Set + + variable {α : Type _} (s : Set (Set α)) + + example : ⋃₀ s = ⋃ t ∈ s, t := by + ext x + rw [mem_iUnion₂] + simp + + example : ⋂₀ s = ⋂ t ∈ s, t := by + ext x + rw [mem_iInter₂] + rfl + + end +
src/C04_Sets_and_Functions/S02_Functions.lean (new)
- @@ -0,0 +1,214 @@
  + import Mathlib.Data.Set.Lattice + import Mathlib.Data.Set.Function + import Mathlib.Analysis.SpecialFunctions.Log.Basic + + section + + variable {α β : Type _} + variable (f : α → β) + variable (s t : Set α) + variable (u v : Set β) + + open Function + open Set + + example : f ⁻¹' (u ∩ v) = f ⁻¹' u ∩ f ⁻¹' v := by + ext + rfl + + example : f '' (s ∪ t) = f '' s ∪ f '' t := by + ext y; constructor + · rintro ⟨x, xs | xt, rfl⟩ + · left + use x, xs + right + use x, xt + rintro (⟨x, xs, rfl⟩ | ⟨x, xt, rfl⟩) + · use x, Or.inl xs + use x, Or.inr xt + + example : s ⊆ f ⁻¹' (f '' s) := by + intro x xs + show f x ∈ f '' s + use x, xs + + example : f '' s ⊆ v ↔ s ⊆ f ⁻¹' v := by + sorry + + example (h : Injective f) : f ⁻¹' (f '' s) ⊆ s := by + sorry + + example : f '' (f ⁻¹' u) ⊆ u := by + sorry + + example (h : Surjective f) : u ⊆ f '' (f ⁻¹' u) := by + sorry + + example (h : s ⊆ t) : f '' s ⊆ f '' t := by + sorry + + example (h : u ⊆ v) : f ⁻¹' u ⊆ f ⁻¹' v := by + sorry + + example : f ⁻¹' (u ∪ v) = f ⁻¹' u ∪ f ⁻¹' v := by + sorry + + example : f '' (s ∩ t) ⊆ f '' s ∩ f '' t := by + sorry + + example (h : Injective f) : f '' s ∩ f '' t ⊆ f '' (s ∩ t) := by + sorry + + example : f '' s \ f '' t ⊆ f '' (s \ t) := by + sorry + + example : f ⁻¹' u \ f ⁻¹' v ⊆ f ⁻¹' (u \ v) := by + sorry + + example : f '' s ∩ v = f '' (s ∩ f ⁻¹' v) := by + sorry + + example : f '' (s ∩ f ⁻¹' u) ⊆ f '' s ∪ u := by + sorry + + example : s ∩ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∩ u) := by + sorry + + example : s ∪ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∪ u) := by + sorry + + variable {I : Type _} (A : I → Set α) (B : I → Set β) + + example : (f '' ⋃ i, A i) = ⋃ i, f '' A i := by + ext y; simp + constructor + · rintro ⟨x, ⟨i, xAi⟩, fxeq⟩ + use i, x + exact ⟨xAi, fxeq⟩ + rintro ⟨i, x, xAi, fxeq⟩ + exact ⟨x, ⟨i, xAi⟩, fxeq⟩ + + example : (f '' ⋂ i, A i) ⊆ ⋂ i, f '' A i := by + intro y; simp + intro x h fxeq i + use x + exact ⟨h i, fxeq⟩ + + example (i : I) (injf : Injective f) : (⋂ i, f '' A i) ⊆ f '' ⋂ i, A i := by + intro y; simp + intro h + rcases h i with ⟨x, xAi, fxeq⟩ + use x; constructor + · intro i' + rcases h i' with ⟨x', x'Ai, fx'eq⟩ + have : f x = f x' := by rw [fxeq, fx'eq] + have : x = x' := injf this + rw [this] + exact x'Ai + exact fxeq + + example : (f ⁻¹' ⋃ i, B i) = ⋃ i, f ⁻¹' B i := by + ext x + simp + + example : (f ⁻¹' ⋂ i, B i) = ⋂ i, f ⁻¹' B i := by + ext x + simp + + example : InjOn f s ↔ ∀ x₁ ∈ s, ∀ x₂ ∈ s, f x₁ = f x₂ → x₁ = x₂ := + Iff.refl _ + + end + + section + + open Set Real + + example : InjOn log { x | x > 0 } := by + intro x xpos y ypos + intro e + -- log x = log y + calc + x = exp (log x) := by rw [exp_log xpos] + _ = exp (log y) := by rw [e] + _ = y := by rw [exp_log ypos] + + + example : range exp = { y | y > 0 } := by + ext y; constructor + · rintro ⟨x, rfl⟩ + apply exp_pos + intro ypos + use log y + rw [exp_log ypos] + + example : InjOn sqrt { x | x ≥ 0 } := by + sorry + + example : InjOn (fun x => x ^ 2) { x : ℝ | x ≥ 0 } := by + sorry + + example : sqrt '' { x | x ≥ 0 } = { y | y ≥ 0 } := by + sorry + + example : (range fun x => x ^ 2) = { y : ℝ | y ≥ 0 } := by + sorry + + end + + section + variable {α β : Type _} [Inhabited α] + + #check (default : α) + + variable (P : α → Prop) (h : ∃ x, P x) + + #check Classical.choose h + + example : P (Classical.choose h) := + Classical.choose_spec h + + noncomputable section + + open Classical + + def inverse (f : α → β) : β → α := fun y : β => + if h : ∃ x, f x = y then Classical.choose h else default + + theorem inverse_spec {f : α → β} (y : β) (h : ∃ x, f x = y) : f (inverse f y) = y := by + rw [inverse]; dsimp; rw [dif_pos h] + exact Classical.choose_spec h + + variable (f : α → β) + + open Function + + example : Injective f ↔ LeftInverse (inverse f) f := + sorry + + example : Surjective f ↔ RightInverse (inverse f) f := + sorry + + end + + section + variable {α : Type _} + open Function + + theorem Cantor : ∀ f : α → Set α, ¬Surjective f := by + intro f surjf + let S := { i | i ∉ f i } + rcases surjf S with ⟨j, h⟩ + have h₁ : j ∉ f j := by + intro h' + have : j ∉ f j := by rwa [h] at h' + contradiction + have h₂ : j ∈ S + sorry + have h₃ : j ∉ S + sorry + contradiction + + -- COMMENTS: TODO: improve this + end +
src/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean (new)
- @@ -0,0 +1,99 @@
  + import Mathlib.Data.Set.Lattice + import Mathlib.Data.Set.Function + import Mathlib.Tactic + + open Set + + open Function + + noncomputable section + open Classical + variable {α β : Type _} [Nonempty β] + + section + variable (f : α → β) (g : β → α) + + def sbAux : ℕ → Set α + | 0 => univ \ g '' univ + | n + 1 => g '' (f '' sbAux n) + + def sbSet := + ⋃ n, sbAux f g n + + def sbFun (x : α) : β := + if x ∈ sbSet f g then f x else invFun g x + + theorem sb_right_inv {x : α} (hx : x ∉ sbSet f g) : g (invFun g x) = x := by + have : x ∈ g '' univ := by + contrapose! hx + rw [sbSet, mem_iUnion] + use 0 + rw [sbAux, mem_diff] + sorry + have : ∃ y, g y = x := by + sorry + sorry + + theorem sb_injective (hf : Injective f) (hg : Injective g) : Injective (sbFun f g) := by + set A := sbSet f g with A_def + set h := sbFun f g with h_def + intro x₁ x₂ + intro (hxeq : h x₁ = h x₂) + show x₁ = x₂ + simp only [h_def, sbFun, ← A_def] at hxeq + by_cases xA : x₁ ∈ A ∨ x₂ ∈ A + · wlog x₁A : x₁ ∈ A generalizing x₁ x₂ hxeq xA + · symm + apply this hxeq.symm xA.symm (xA.resolve_left x₁A) + have x₂A : x₂ ∈ A := by + apply not_imp_self.mp + intro (x₂nA : x₂ ∉ A) + rw [if_pos x₁A, if_neg x₂nA] at hxeq + rw [A_def, sbSet, mem_iUnion] at x₁A + have x₂eq : x₂ = g (f x₁) := by + sorry + rcases x₁A with ⟨n, hn⟩ + rw [A_def, sbSet, mem_iUnion] + use n + 1 + simp [sbAux] + exact ⟨x₁, hn, x₂eq.symm⟩ + sorry + push_neg at xA + sorry + + theorem sb_surjective (hf : Injective f) (hg : Injective g) : Surjective (sbFun f g) := by + set A := sbSet f g with A_def + set h := sbFun f g with h_def + intro y + by_cases gyA : g y ∈ A + · rw [A_def, sbSet, mem_iUnion] at gyA + rcases gyA with ⟨n, hn⟩ + cases' n with n + · simp [sbAux] at hn + simp [sbAux] at hn + rcases hn with ⟨x, xmem, hx⟩ + use x + have : x ∈ A := by + rw [A_def, sbSet, mem_iUnion] + exact ⟨n, xmem⟩ + simp only [h_def, sbFun, if_pos this] + exact hg hx + sorry + + end + + theorem schroeder_bernstein {f : α → β} {g : β → α} (hf : Injective f) (hg : Injective g) : + ∃ h : α → β, Bijective h := + ⟨sbFun f g, sb_injective f g hf hg, sb_surjective f g hf hg⟩ + + -- Auxiliary information + section + variable (g : β → α) (x : α) + + #check (invFun g : α → β) + #check (leftInverse_invFun : Injective g → LeftInverse (invFun g) g) + #check (leftInverse_invFun : Injective g → ∀ y, invFun g (g y) = y) + #check (invFun_eq : (∃ y, g y = x) → g (invFun g x) = x) + + end +
src/C04_Sets_and_Functions/solutions/Solutions_S01_Sets.lean (new)
- @@ -0,0 +1,158 @@
 + import Mathlib.Data.Set.Lattice + import Mathlib.Data.Nat.Prime + import Mathlib.Data.Nat.Parity + import Mathlib.Tactic + + section + variable {α : Type _} + variable (s t u : Set α) + open Set + + example : s ∩ t ∪ s ∩ u ⊆ s ∩ (t ∪ u) := by + rintro x (⟨xs, xt⟩ | ⟨xs, xu⟩) + · use xs + left + exact xt + use xs; right; exact xu + + example : s \ (t ∪ u) ⊆ (s \ t) \ u := by + rintro x ⟨xs, xntu⟩ + constructor + use xs + · intro xt + exact xntu (Or.inl xt) + intro xu + apply xntu (Or.inr xu) + + example : s ∩ t = t ∩ s := + Subset.antisymm + (fun x ⟨xs, xt⟩ => ⟨xt, xs⟩) fun x ⟨xt, xs⟩ => ⟨xs, xt⟩ + + example : s ∩ (s ∪ t) = s := by + ext x; constructor + · rintro ⟨xs, _⟩ + exact xs + intro xs + use xs; left; exact xs + + example : s ∪ s ∩ t = s := by + ext x; constructor + · rintro (xs | ⟨xs, xt⟩) <;> exact xs + intro xs; left; exact xs + + example : s \ t ∪ t = s ∪ t := by + ext x; constructor + · rintro (⟨xs, nxt⟩ | xt) + · left + exact xs + right + exact xt + by_cases h : x ∈ t + · intro + right + exact h + rintro (xs | xt) + · left + use xs + exact h + right; exact xt + + example : s \ t ∪ t \ s = (s ∪ t) \ (s ∩ t) := by + ext x; constructor + · rintro (⟨xs, xnt⟩ | ⟨xt, xns⟩) + · constructor + left + exact xs + rintro ⟨_, xt⟩ + contradiction + constructor + right + exact xt + rintro ⟨xs, _⟩ + contradiction + rintro ⟨xs | xt, nxst⟩ + · left + use xs + intro xt + apply nxst + constructor <;> assumption + right; use xt; intro xs + apply nxst + constructor <;> assumption + + example : { n | Nat.Prime n } ∩ { n | n > 2 } ⊆ { n | ¬Even n } := by + intro n + simp + intro nprime + cases' Nat.Prime.eq_two_or_odd nprime with h h + · rw [h] + intro + linarith + rw [Nat.even_iff, h] + norm_num + + end + + section + + variable (s t : Set ℕ) + + section + variable (ssubt : s ⊆ t) + + example (h₀ : ∀ x ∈ t, ¬Even x) (h₁ : ∀ x ∈ t, Prime x) : ∀ x ∈ s, ¬Even x ∧ Prime x := by + intro x xs + constructor + · apply h₀ x (ssubt xs) + apply h₁ x (ssubt xs) + + example (h : ∃ x ∈ s, ¬Even x ∧ Prime x) : ∃ x ∈ t, Prime x := by + rcases h with ⟨x, xs, _, px⟩ + use x, ssubt xs + exact px + + end + + end + + section + variable {α I : Type _} + variable (A B : I → Set α) + variable (s : Set α) + + open Set + + example : (s ∪ ⋂ i, A i) = ⋂ i, A i ∪ s := by + ext x + simp only [mem_union, mem_iInter] + constructor + · rintro (xs | xI) + · intro i + right + exact xs + intro i + left + exact xI i + intro h + by_cases xs : x ∈ s + · left + exact xs + right + intro i + cases h i + · assumption + contradiction + + def primes : Set ℕ := + { x | Nat.Prime x } + + example : (⋃ p ∈ primes, { x | x ≤ p }) = univ := by + apply eq_univ_of_forall + intro x + simp + rcases Nat.exists_infinite_primes x with ⟨p, primep, pge⟩ + use p, pge + exact primep + + end +
src/C04_Sets_and_Functions/solutions/Solutions_S02_Functions.lean (new)
- @@ -0,0 +1,253 @@
  + import Mathlib.Data.Set.Lattice + import Mathlib.Data.Set.Function + import Mathlib.Analysis.SpecialFunctions.Log.Basic + + section + + variable {α β : Type _} + variable (f : α → β) + variable (s t : Set α) + variable (u v : Set β) + + open Function + open Set + + example : f '' s ⊆ v ↔ s ⊆ f ⁻¹' v := by + constructor + · intro h x xs + have : f x ∈ f '' s := mem_image_of_mem _ xs + exact h this + intro h y ymem + rcases ymem with ⟨x, xs, fxeq⟩ + rw [← fxeq] + apply h xs + + example (h : Injective f) : f ⁻¹' (f '' s) ⊆ s := by + rintro x ⟨y, ys, fxeq⟩ + rw [← h fxeq] + exact ys + + example : f '' (f ⁻¹' u) ⊆ u := by + rintro y ⟨x, xmem, rfl⟩ + exact xmem + + example (h : Surjective f) : u ⊆ f '' (f ⁻¹' u) := by + intro y yu + rcases h y with ⟨x, fxeq⟩ + use x + constructor + · show f x ∈ u + rw [fxeq] + exact yu + exact fxeq + + example (h : s ⊆ t) : f '' s ⊆ f '' t := by + rintro y ⟨x, xs, fxeq⟩ + use x, h xs + exact fxeq + + example (h : u ⊆ v) : f ⁻¹' u ⊆ f ⁻¹' v := by + intro x; apply h + + example : f ⁻¹' (u ∪ v) = f ⁻¹' u ∪ f ⁻¹' v := by + ext x; rfl + + example : f '' (s ∩ t) ⊆ f '' s ∩ f '' t := by + rintro y ⟨x, ⟨xs, xt⟩, rfl⟩ + constructor + . use x, xs + . use x, xt + + example (h : Injective f) : f '' s ∩ f '' t ⊆ f '' (s ∩ t) := by + rintro y ⟨⟨x₁, x₁s, rfl⟩, ⟨x₂, x₂t, fx₂eq⟩⟩ + use x₁ + constructor + . use x₁s + rw [← h fx₂eq] + exact x₂t + . rfl + + example : f '' s \ f '' t ⊆ f '' (s \ t) := by + rintro y ⟨⟨x₁, x₁s, rfl⟩, h⟩ + use x₁ + constructor + . constructor + . exact x₁s + . intro h' + apply h + use x₁, h' + . rfl + + example : f ⁻¹' u \ f ⁻¹' v ⊆ f ⁻¹' (u \ v) := + fun x => id + + example : f '' s ∩ v = f '' (s ∩ f ⁻¹' v) := by + ext y; constructor + · rintro ⟨⟨x, xs, rfl⟩, fxv⟩ + use x, ⟨xs, fxv⟩ + rintro ⟨x, ⟨⟨xs, fxv⟩, rfl⟩⟩ + exact ⟨⟨x, xs, rfl⟩, fxv⟩ + + example : f '' (s ∩ f ⁻¹' u) ⊆ f '' s ∩ u := by + rintro y ⟨x, ⟨xs, fxu⟩, rfl⟩ + exact ⟨⟨x, xs, rfl⟩, fxu⟩ + + example : s ∩ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∩ u) := by + rintro x ⟨xs, fxu⟩ + exact ⟨⟨x, xs, rfl⟩, fxu⟩ + + example : s ∪ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∪ u) := by + rintro x (xs | fxu) + · left + exact ⟨x, xs, rfl⟩ + right; exact fxu + + variable {I : Type _} (A : I → Set α) (B : I → Set β) + + example : (f '' ⋃ i, A i) = ⋃ i, f '' A i := by + ext y; simp + constructor + · rintro ⟨x, ⟨i, xAi⟩, fxeq⟩ + use i, x + exact ⟨xAi, fxeq⟩ + rintro ⟨i, x, xAi, fxeq⟩ + exact ⟨x, ⟨i, xAi⟩, fxeq⟩ + + example : (f '' ⋂ i, A i) ⊆ ⋂ i, f '' A i := by + intro y; simp + intro x h fxeq i + use x + exact ⟨h i, fxeq⟩ + + example (i : I) (injf : Injective f) : (⋂ i, f '' A i) ⊆ f '' ⋂ i, A i := by + intro y; simp + intro h + rcases h i with ⟨x, xAi, fxeq⟩ + use x; constructor + · intro i' + rcases h i' with ⟨x', x'Ai, fx'eq⟩ + have : f x = f x' := by rw [fxeq, fx'eq] + have : x = x' := injf this + rw [this] + exact x'Ai + exact fxeq + + example : (f ⁻¹' ⋃ i, B i) = ⋃ i, f ⁻¹' B i := by + ext x + simp + + example : (f ⁻¹' ⋂ i, B i) = ⋂ i, f ⁻¹' B i := by + ext x + simp + + end + + section + + open Set Real + + example : InjOn sqrt { x | x ≥ 0 } := by + intro x xnonneg y ynonneg + intro e + calc + x = sqrt x ^ 2 := by rw [sq_sqrt xnonneg] + _ = sqrt y ^ 2 := by rw [e] + _ = y := by rw [sq_sqrt ynonneg] + + + example : InjOn (fun x => x ^ 2) { x : ℝ | x ≥ 0 } := by + intro x xnonneg y ynonneg + intro e + dsimp at * + calc + x = sqrt (x ^ 2) := by rw [sqrt_sq xnonneg] + _ = sqrt (y ^ 2) := by rw [e] + _ = y := by rw [sqrt_sq ynonneg] + + + example : sqrt '' { x | x ≥ 0 } = { y | y ≥ 0 } := by + ext y; constructor + · rintro ⟨x, ⟨xnonneg, rfl⟩⟩ + apply sqrt_nonneg + intro ynonneg + use y ^ 2 + dsimp at * + constructor + apply pow_nonneg ynonneg + apply sqrt_sq + assumption + + example : (range fun x => x ^ 2) = { y : ℝ | y ≥ 0 } := by + ext y + constructor + · rintro ⟨x, rfl⟩ + dsimp at * + apply pow_two_nonneg + intro ynonneg + use sqrt y + exact sq_sqrt ynonneg + + end + + section + variable {α β : Type _} [Inhabited α] + + noncomputable section + + open Classical + + def inverse (f : α → β) : β → α := fun y : β => + if h : ∃ x, f x = y then Classical.choose h else default + + theorem inverse_spec {f : α → β} (y : β) (h : ∃ x, f x = y) : f (inverse f y) = y := by + rw [inverse]; dsimp; rw [dif_pos h] + exact Classical.choose_spec h + + variable (f : α → β) + + open Function + + example : Injective f ↔ LeftInverse (inverse f) f := by + constructor + · intro h y + apply h + apply inverse_spec + use y + intro h x1 x2 e + rw [← h x1, ← h x2, e] + + example : Injective f ↔ LeftInverse (inverse f) f := + ⟨fun h y => h (inverse_spec _ ⟨y, rfl⟩), fun h x1 x2 e => by rw [← h x1, ← h x2, e]⟩ + + example : Surjective f ↔ RightInverse (inverse f) f := by + constructor + · intro h y + apply inverse_spec + apply h + intro h y + use inverse f y + apply h + + example : Surjective f ↔ RightInverse (inverse f) f := + ⟨fun h y => inverse_spec _ (h _), fun h y => ⟨inverse f y, h _⟩⟩ + + end + + section + variable {α : Type _} + open Function + + theorem Cantor : ∀ f : α → Set α, ¬Surjective f := by + intro f surjf + let S := { i | i ∉ f i } + rcases surjf S with ⟨j, h⟩ + have h₁ : j ∉ f j := by + intro h' + have : j ∉ f j := by rwa [h] at h' + contradiction + have h₂ : j ∈ S := h₁ + have h₃ : j ∉ S := by rwa [h] at h₁ + contradiction + + end +
src/C04_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean (new)
- @@ -0,0 +1,88 @@
  + import Mathlib.Data.Set.Lattice + import Mathlib.Data.Set.Function + import Mathlib.Tactic + + open Set + open Function + + noncomputable section + open Classical + variable {α β : Type _} [Nonempty β] + + section + variable (f : α → β) (g : β → α) + + def sbAux : ℕ → Set α + | 0 => univ \ g '' univ + | n + 1 => g '' (f '' sbAux n) + + def sbSet := + ⋃ n, sbAux f g n + + def sbFun (x : α) : β := + if x ∈ sbSet f g then f x else invFun g x + + theorem sb_right_inv {x : α} (hx : x ∉ sbSet f g) : g (invFun g x) = x := by + have : x ∈ g '' univ := by + contrapose! hx + rw [sbSet, mem_iUnion] + use 0 + rw [sbAux, mem_diff] + exact ⟨mem_univ _, hx⟩ + have : ∃ y, g y = x := by + simp at this + assumption + exact invFun_eq this + + theorem sb_injective (hf : Injective f) (hg : Injective g) : Injective (sbFun f g) := by + set A := sbSet f g with A_def + set h := sbFun f g with h_def + intro x₁ x₂ + intro (hxeq : h x₁ = h x₂) + show x₁ = x₂ + simp only [h_def, sbFun, ← A_def] at hxeq + by_cases xA : x₁ ∈ A ∨ x₂ ∈ A + · wlog x₁A : x₁ ∈ A generalizing x₁ x₂ hxeq xA + · symm + apply this hxeq.symm xA.symm (xA.resolve_left x₁A) + have x₂A : x₂ ∈ A := by + apply not_imp_self.mp + intro (x₂nA : x₂ ∉ A) + rw [if_pos x₁A, if_neg x₂nA] at hxeq + rw [A_def, sbSet, mem_iUnion] at x₁A + have x₂eq : x₂ = g (f x₁) := by + rw [hxeq, sb_right_inv f g x₂nA] + rcases x₁A with ⟨n, hn⟩ + rw [A_def, sbSet, mem_iUnion] + use n + 1 + simp [sbAux] + exact ⟨x₁, hn, x₂eq.symm⟩ + rw [if_pos x₁A, if_pos x₂A] at hxeq + exact hf hxeq + push_neg at xA + rw [if_neg xA.1, if_neg xA.2] at hxeq + rw [← sb_right_inv f g xA.1, hxeq, sb_right_inv f g xA.2] + + theorem sb_surjective (hf : Injective f) (hg : Injective g) : Surjective (sbFun f g) := by + set A := sbSet f g with A_def + set h := sbFun f g with h_def + intro y + by_cases gyA : g y ∈ A + · rw [A_def, sbSet, mem_iUnion] at gyA + rcases gyA with ⟨n, hn⟩ + cases' n with n + · simp [sbAux] at hn + simp [sbAux] at hn + rcases hn with ⟨x, xmem, hx⟩ + use x + have : x ∈ A := by + rw [A_def, sbSet, mem_iUnion] + exact ⟨n, xmem⟩ + simp only [h_def, sbFun, if_pos this] + exact hg hx + use g y + simp only [h_def, sbFun, if_neg gyA] + apply leftInverse_invFun hg + + end +
src/C05_Number_Theory/S01_Irrational_Roots.lean (new)
- @@ -0,0 +1,117 @@
 + import Mathlib.Data.Nat.Factorization.Basic + import Mathlib.Data.Nat.Prime + import Mathlib.Tactic.NormNum.GCD + import Mathlib.Tactic.NormNum.Prime + + #print Nat.coprime + + example (m n : Nat) (h : m.coprime n) : m.gcd n = 1 := + h + + example (m n : Nat) (h : m.coprime n) : m.gcd n = 1 := by + rw [Nat.coprime] at h + exact h + + example : Nat.coprime 12 7 := by norm_num + + example : Nat.gcd 12 8 = 4 := by norm_num + + #check @Nat.prime_def_lt + + example (p : ℕ) (prime_p : Nat.Prime p) : 2 ≤ p ∧ ∀ m : ℕ, m < p → m ∣ p → m = 1 := by + rwa [Nat.prime_def_lt] at prime_p + + #check Nat.Prime.eq_one_or_self_of_dvd + + example (p : ℕ) (prime_p : Nat.Prime p) : ∀ m : ℕ, m ∣ p → m = 1 ∨ m = p := + prime_p.eq_one_or_self_of_dvd + + example : Nat.Prime 17 := by norm_num + + -- commonly used + example : Nat.Prime 2 := + Nat.prime_two + + example : Nat.Prime 3 := + Nat.prime_three + + #check @Nat.Prime.dvd_mul + #check Nat.Prime.dvd_mul Nat.prime_two + #check Nat.prime_two.dvd_mul + + theorem even_of_even_sqr {m : ℕ} (h : 2 ∣ m ^ 2) : 2 ∣ m := by + rw [pow_two, Nat.prime_two.dvd_mul] at h + cases h <;> assumption + + example {m : ℕ} (h : 2 ∣ m ^ 2) : 2 ∣ m := + Nat.Prime.dvd_of_dvd_pow Nat.prime_two h + + example (a b c : Nat) (h : a * b = a * c) (h' : a ≠ 0) : b = c := + -- library_search suggests the following: + (mul_right_inj' h').mp h + + example {m n : ℕ} (coprime_mn : m.coprime n) : m ^ 2 ≠ 2 * n ^ 2 := by + intro sqr_eq + have : 2 ∣ m := by + sorry + obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this + have : 2 * (2 * k ^ 2) = 2 * n ^ 2 := by + rw [← sqr_eq, meq] + ring + have : 2 * k ^ 2 = n ^ 2 := + sorry + have : 2 ∣ n := by + sorry + have : 2 ∣ m.gcd n := by + sorry + have : 2 ∣ 1 := by + sorry + norm_num at this + + example {m n p : ℕ} (coprime_mn : m.coprime n) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by + sorry + #check Nat.factors + #check Nat.prime_of_mem_factors + #check Nat.prod_factors + #check Nat.factors_unique + + theorem factorization_mul' {m n : ℕ} (mnez : m ≠ 0) (nnez : n ≠ 0) (p : ℕ) : + (m * n).factorization p = m.factorization p + n.factorization p := by + rw [Nat.factorization_mul mnez nnez] + rfl + + theorem factorization_pow' (n k p : ℕ) : (n ^ k).factorization p = k * n.factorization p := by + rw [Nat.factorization_pow] + rfl + + theorem Nat.Prime.factorization' {p : ℕ} (prime_p : p.Prime) : p.factorization p = 1 := by + rw [prime_p.factorization] + simp + + example {m n p : ℕ} (nnz : n ≠ 0) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by + intro sqr_eq + have nsqr_nez : n ^ 2 ≠ 0 := by simpa + have eq1 : Nat.factorization (m ^ 2) p = 2 * m.factorization p := by + sorry + have eq2 : (p * n ^ 2).factorization p = 2 * n.factorization p + 1 := by + sorry + have : 2 * m.factorization p % 2 = (2 * n.factorization p + 1) % 2 := by rw [← eq1, sqr_eq, eq2] + rw [add_comm, Nat.add_mul_mod_self_left, Nat.mul_mod_right] at this + norm_num at this + + example {m n k r : ℕ} (nnz : n ≠ 0) (pow_eq : m ^ k = r * n ^ k) {p : ℕ} (prime_p : p.Prime) : + k ∣ r.factorization p := by + cases' r with r + · simp + have npow_nz : n ^ k ≠ 0 := fun npowz => nnz (pow_eq_zero npowz) + have eq1 : (m ^ k).factorization p = k * m.factorization p := by + sorry + have eq2 : (r.succ * n ^ k).factorization p = k * n.factorization p + r.succ.factorization p := by + sorry + have : r.succ.factorization p = k * m.factorization p - k * n.factorization p := by + rw [← eq1, pow_eq, eq2, add_comm, Nat.add_sub_cancel] + rw [this] + sorry + + #check multiplicity +
src/C05_Number_Theory/S02_Induction_and_Recursion.lean (new)
- @@ -0,0 +1,148 @@
 + import Mathlib.Data.Nat.GCD.Basic + import Mathlib.Algebra.BigOperators.Basic + import Mathlib.Tactic + + example (n : Nat) : n.succ ≠ Nat.zero := + Nat.succ_ne_zero n + + example (m n : Nat) (h : m.succ = n.succ) : m = n := + Nat.succ.inj h + + def fac : ℕ → ℕ + | 0 => 1 + | n + 1 => (n + 1) * fac n + + example : fac 0 = 1 := + rfl + + example : fac 0 = 1 := by + rw [fac] + + example : fac 0 = 1 := by + simp [fac] + + example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := + rfl + + example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := by + rw [fac] + + example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := by + simp [fac] + + theorem fac_pos (n : ℕ) : 0 < fac n := by + induction' n with n ih + · rw [fac] + exact zero_lt_one + rw [fac] + exact mul_pos n.succ_pos ih + + theorem dvd_fac {i n : ℕ} (ipos : 0 < i) (ile : i ≤ n) : i ∣ fac n := by + induction' n with n ih + · exact absurd ipos (not_lt_of_ge ile) + rw [fac] + cases' Nat.of_le_succ ile with h h + · apply dvd_mul_of_dvd_right (ih h) + rw [h] + apply dvd_mul_right + + theorem pow_two_le_fac (n : ℕ) : 2 ^ (n - 1) ≤ fac n := by + cases' n with n + · simp [fac] + sorry + section + + variable {α : Type _} (s : Finset ℕ) (f : ℕ → ℕ) (n : ℕ) + + #check Finset.sum s f + #check Finset.prod s f + + open BigOperators + open Finset + + example : s.sum f = ∑ x in s, f x := + rfl + + example : s.prod f = ∏ x in s, f x := + rfl + + example : (range n).sum f = ∑ x in range n, f x := + rfl + + example : (range n).prod f = ∏ x in range n, f x := + rfl + + example (f : ℕ → ℕ) : (∑ x in range 0, f x) = 0 := + Finset.sum_range_zero f + + example (f : ℕ → ℕ) (n : ℕ) : (∑ x in range n.succ, f x) = (∑ x in range n, f x) + f n := + Finset.sum_range_succ f n + + example (f : ℕ → ℕ) : (∏ x in range 0, f x) = 1 := + Finset.prod_range_zero f + + example (f : ℕ → ℕ) (n : ℕ) : (∏ x in range n.succ, f x) = (∏ x in range n, f x) * f n := + Finset.prod_range_succ f n + + example (n : ℕ) : fac n = ∏ i in range n, (i + 1) := by + induction' n with n ih + · rw [fac, prod_range_zero] + rw [fac, ih, prod_range_succ, mul_comm] + + example (a b c d e f : ℕ) : a * (b * c * f * (d * e)) = d * (a * f * e) * (c * b) := by + simp [mul_assoc, mul_comm, mul_left_comm] + + theorem sum_id (n : ℕ) : (∑ i in range (n + 1), i) = n * (n + 1) / 2 := by + symm; apply Nat.div_eq_of_eq_mul_right (by norm_num : 0 < 2) + induction' n with n ih + · simp + rw [Finset.sum_range_succ, mul_add 2, ← ih, Nat.succ_eq_add_one] + ring + + theorem sum_sqr (n : ℕ) : (∑ i in range (n + 1), i ^ 2) = n * (n + 1) * (2 * n + 1) / 6 := by + sorry + end + + inductive MyNat + | zero : MyNat + | succ : MyNat → MyNat + + namespace MyNat + + def add : MyNat → MyNat → MyNat + | x, zero => x + | x, succ y => succ (add x y) + + def mul : MyNat → MyNat → MyNat + | x, zero => zero + | x, succ y => add (mul x y) x + + theorem zero_add (n : MyNat) : add zero n = n := by + induction' n with n ih + · rfl + rw [add, ih] + + theorem succ_add (m n : MyNat) : add (succ m) n = succ (add m n) := by + induction' n with n ih + · rfl + rw [add, ih] + rfl + + theorem add_comm (m n : MyNat) : add m n = add n m := by + induction' n with n ih + · rw [zero_add] + rfl + rw [add, succ_add, ih] + + theorem add_assoc (m n k : MyNat) : add (add m n) k = add m (add n k) := by + sorry + theorem mul_add (m n k : MyNat) : mul m (add n k) = add (mul m n) (mul m k) := by + sorry + theorem zero_mul (n : MyNat) : mul zero n = zero := by + sorry + theorem succ_mul (m n : MyNat) : mul (succ m) n = add (mul m n) n := by + sorry + theorem mul_comm (m n : MyNat) : mul m n = mul n m := by + sorry + end MyNat +
src/C05_Number_Theory/S03_Infinitely_Many_Primes.lean (new)
- @@ -0,0 +1,229 @@
 + import Mathlib.Data.Nat.Prime + import Mathlib.Algebra.BigOperators.Order + import Mathlib.Tactic + import Mathlib.Tactic.IntervalCases + + open BigOperators + + theorem two_le {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by + cases m; contradiction + case succ m => + cases m; contradiction + repeat' apply Nat.succ_le_succ + apply zero_le + + example {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by + by_contra h + push_neg at h + interval_cases m <;> contradiction + + example {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by + by_contra h + push_neg at h + revert h0 h1 + revert h m + decide + + theorem exists_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ p : Nat, p.Prime ∧ p ∣ n := by + by_cases np : n.Prime + · use n, np + induction' n using Nat.strong_induction_on with n ih + dsimp at ih + rw [Nat.prime_def_lt] at np + push_neg at np + rcases np h with ⟨m, mltn, mdvdn, mne1⟩ + have : m ≠ 0 := by + intro mz + rw [mz, zero_dvd_iff] at mdvdn + linarith + have mgt2 : 2 ≤ m := two_le this mne1 + by_cases mp : m.Prime + · use m, mp + exact mdvdn + . rcases ih m mltn mgt2 mp with ⟨p, pp, pdvd⟩ + use p, pp + apply pdvd.trans mdvdn + + theorem primes_infinite : ∀ n, ∃ p > n, Nat.Prime p := by + intro n + have : 2 ≤ Nat.factorial (n + 1) + 1 := by + sorry + rcases exists_prime_factor this with ⟨p, pp, pdvd⟩ + refine' ⟨p, _, pp⟩ + show p > n + by_contra ple + push_neg at ple + have : p ∣ Nat.factorial (n + 1) := by + sorry + have : p ∣ 1 := by + sorry + show False + sorry + open Finset + + section + variable {α : Type _} [DecidableEq α] (r s t : Finset α) + + example : r ∩ (s ∪ t) ⊆ r ∩ s ∪ r ∩ t := by + rw [subset_iff] + intro x + rw [mem_inter, mem_union, mem_union, mem_inter, mem_inter] + tauto + + example : r ∩ (s ∪ t) ⊆ r ∩ s ∪ r ∩ t := by + simp [subset_iff] + intro x + tauto + + example : r ∩ s ∪ r ∩ t ⊆ r ∩ (s ∪ t) := by + simp [subset_iff] + intro x + tauto + + example : r ∩ s ∪ r ∩ t = r ∩ (s ∪ t) := by + ext x + simp + tauto + + end + + section + variable {α : Type _} [DecidableEq α] (r s t : Finset α) + + example : (r ∪ s) ∩ (r ∪ t) = r ∪ s ∩ t := by + sorry + example : (r \ s) \ t = r \ (s ∪ t) := by + sorry + end + + example (s : Finset ℕ) (n : ℕ) (h : n ∈ s) : n ∣ ∏ i in s, i := + Finset.dvd_prod_of_mem _ h + + theorem Nat.Prime.eq_of_dvd_of_prime {p q : ℕ} (prime_p : Nat.Prime p) (prime_q : Nat.Prime q) + (h : p ∣ q) : p = q := by + sorry + theorem mem_of_dvd_prod_primes {s : Finset ℕ} {p : ℕ} (prime_p : p.Prime) : + (∀ n ∈ s, Nat.Prime n) → (p ∣ ∏ n in s, n) → p ∈ s := by + intro h₀ h₁ + induction' s using Finset.induction_on with a s ans ih + · simp at h₁ + linarith [prime_p.two_le] + simp [Finset.prod_insert ans, prime_p.dvd_mul] at h₀ h₁ + rw [mem_insert] + sorry + example (s : Finset ℕ) (x : ℕ) : x ∈ s.filter Nat.Prime ↔ x ∈ s ∧ x.Prime := + mem_filter + + theorem primes_infinite' : ∀ s : Finset Nat, ∃ p, Nat.Prime p ∧ p ∉ s := by + intro s + by_contra h + push_neg at h + set s' := s.filter Nat.Prime with s'_def + have mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ n.Prime := by + intro n + simp [s'_def] + apply h + have : 2 ≤ (∏ i in s', i) + 1 := by + sorry + rcases exists_prime_factor this with ⟨p, pp, pdvd⟩ + have : p ∣ ∏ i in s', i := by + sorry + have : p ∣ 1 := by + convert Nat.dvd_sub' pdvd this + simp + show False + sorry + theorem bounded_of_ex_finset (Q : ℕ → Prop) : + (∃ s : Finset ℕ, ∀ k, Q k → k ∈ s) → ∃ n, ∀ k, Q k → k < n := by + rintro ⟨s, hs⟩ + use s.sup id + 1 + intro k Qk + apply Nat.lt_succ_of_le + show id k ≤ s.sup id + apply le_sup (hs k Qk) + + theorem ex_finset_of_bounded (Q : ℕ → Prop) [DecidablePred Q] : + (∃ n, ∀ k, Q k → k ≤ n) → ∃ s : Finset ℕ, ∀ k, Q k ↔ k ∈ s := by + rintro ⟨n, hn⟩ + use (range (n + 1)).filter Q + intro k + simp [Nat.lt_succ_iff] + exact hn k + + example : 27 % 4 = 3 := by norm_num + + example (n : ℕ) : (4 * n + 3) % 4 = 3 := by + rw [add_comm, Nat.add_mul_mod_self_left] + norm_num + + theorem mod_4_eq_3_or_mod_4_eq_3 {m n : ℕ} (h : m * n % 4 = 3) : m % 4 = 3 ∨ n % 4 = 3 := by + revert h + rw [Nat.mul_mod] + have : m % 4 < 4 := Nat.mod_lt m (by norm_num) + interval_cases hm : m % 4 <;> simp [hm] + have : n % 4 < 4 := Nat.mod_lt n (by norm_num) + interval_cases hn : n % 4 <;> simp [hn] + + theorem two_le_of_mod_4_eq_3 {n : ℕ} (h : n % 4 = 3) : 2 ≤ n := by + apply two_le <;> + · intro neq + rw [neq] at h + norm_num at h + + theorem aux {m n : ℕ} (h₀ : m ∣ n) (h₁ : 2 ≤ m) (h₂ : m < n) : n / m ∣ n ∧ n / m < n := by + sorry + theorem exists_prime_factor_mod_4_eq_3 {n : Nat} (h : n % 4 = 3) : + ∃ p : Nat, p.Prime ∧ p ∣ n ∧ p % 4 = 3 := by + by_cases np : n.Prime + · use n + exact ⟨np, dvd_rfl, h⟩ + induction' n using Nat.strong_induction_on with n ih + dsimp at ih + rw [Nat.prime_def_lt] at np + push_neg at np + rcases np (two_le_of_mod_4_eq_3 h) with ⟨m, mltn, mdvdn, mne1⟩ + have mge2 : 2 ≤ m := by + apply two_le _ mne1 + intro mz + rw [mz, zero_dvd_iff] at mdvdn + linarith + have neq : m * (n / m) = n := Nat.mul_div_cancel' mdvdn + have : m % 4 = 3 ∨ n / m % 4 = 3 := by + apply mod_4_eq_3_or_mod_4_eq_3 + rw [neq, h] + cases' this with h1 h1 + . sorry + sorry + example (m n : ℕ) (s : Finset ℕ) (h : m ∈ erase s n) : m ≠ n ∧ m ∈ s := by + rwa [mem_erase] at h + + example (m n : ℕ) (s : Finset ℕ) (h : m ∈ erase s n) : m ≠ n ∧ m ∈ s := by + simp at h + assumption + + theorem primes_mod_4_eq_3_infinite : ∀ n, ∃ p > n, Nat.Prime p ∧ p % 4 = 3 := by + by_contra h + push_neg at h + cases' h with n hn + have : ∃ s : Finset Nat, ∀ p : ℕ, p.Prime ∧ p % 4 = 3 ↔ p ∈ s := by + apply ex_finset_of_bounded + use n + contrapose! hn + rcases hn with ⟨p, ⟨pp, p4⟩, pltn⟩ + exact ⟨p, pltn, pp, p4⟩ + cases' this with s hs + have h₁ : ((4 * ∏ i in erase s 3, i) + 3) % 4 = 3 := by + sorry + rcases exists_prime_factor_mod_4_eq_3 h₁ with ⟨p, pp, pdvd, p4eq⟩ + have ps : p ∈ s := by + sorry + have pne3 : p ≠ 3 := by + sorry + have : p ∣ 4 * ∏ i in erase s 3, i := by + sorry + have : p ∣ 3 := by + sorry + have : p = 3 := by + sorry + contradiction +
src/C05_Number_Theory/solutions/Solutions_S01_Irrational_Roots.lean (new)
- @@ -0,0 +1,101 @@
 + import Mathlib.Data.Nat.Factorization.Basic + import Mathlib.Data.Nat.Prime + import Mathlib.Tactic.NormNum.GCD + import Mathlib.Tactic.NormNum.Prime + + theorem even_of_even_sqr {m : ℕ} (h : 2 ∣ m ^ 2) : 2 ∣ m := by + rw [pow_two, Nat.prime_two.dvd_mul] at h + cases h <;> assumption + + example {m n : ℕ} (coprime_mn : m.coprime n) : m ^ 2 ≠ 2 * n ^ 2 := by + intro sqr_eq + have : 2 ∣ m := by + apply even_of_even_sqr + rw [sqr_eq] + apply dvd_mul_right + obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this + have : 2 * (2 * k ^ 2) = 2 * n ^ 2 := by + rw [← sqr_eq, meq] + ring + have : 2 * k ^ 2 = n ^ 2 := + (mul_right_inj' (by norm_num)).mp this + have : 2 ∣ n := by + apply even_of_even_sqr + rw [← this] + apply dvd_mul_right + have : 2 ∣ m.gcd n := by + apply Nat.dvd_gcd <;> + assumption + have : 2 ∣ 1 := by + convert this + symm + exact coprime_mn + norm_num at this + + example {m n p : ℕ} (coprime_mn : m.coprime n) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by + intro sqr_eq + have : p ∣ m := by + apply prime_p.dvd_of_dvd_pow + rw [sqr_eq] + apply dvd_mul_right + obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this + have : p * (p * k ^ 2) = p * n ^ 2 := by + rw [← sqr_eq, meq] + ring + have : p * k ^ 2 = n ^ 2 := by + apply (mul_right_inj' _).mp this + exact prime_p.ne_zero + have : p ∣ n := by + apply prime_p.dvd_of_dvd_pow + rw [← this] + apply dvd_mul_right + have : p ∣ Nat.gcd m n := by apply Nat.dvd_gcd <;> assumption + have : p ∣ 1 := by + convert this + symm + exact coprime_mn + have : 2 ≤ 1 := by + apply prime_p.two_le.trans + exact Nat.le_of_dvd zero_lt_one this + norm_num at this + + theorem factorization_mul' {m n : ℕ} (mnez : m ≠ 0) (nnez : n ≠ 0) (p : ℕ) : + (m * n).factorization p = m.factorization p + n.factorization p := by + rw [Nat.factorization_mul mnez nnez] + rfl + + theorem factorization_pow' (n k p : ℕ) : (n ^ k).factorization p = k * n.factorization p := by + rw [Nat.factorization_pow] + rfl + + theorem Nat.Prime.factorization' {p : ℕ} (prime_p : p.Prime) : p.factorization p = 1 := by + rw [prime_p.factorization] + simp + + example {m n p : ℕ} (nnz : n ≠ 0) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by + intro sqr_eq + have nsqr_nez : n ^ 2 ≠ 0 := by simpa + have eq1 : Nat.factorization (m ^ 2) p = 2 * m.factorization p := by + rw [factorization_pow'] + have eq2 : (p * n ^ 2).factorization p = 2 * n.factorization p + 1 := by + rw [factorization_mul' prime_p.ne_zero nsqr_nez, prime_p.factorization', factorization_pow', + add_comm] + have : 2 * m.factorization p % 2 = (2 * n.factorization p + 1) % 2 := by rw [← eq1, sqr_eq, eq2] + rw [add_comm, Nat.add_mul_mod_self_left, Nat.mul_mod_right] at this + norm_num at this + + example {m n k r : ℕ} (nnz : n ≠ 0) (pow_eq : m ^ k = r * n ^ k) {p : ℕ} (prime_p : p.Prime) : + k ∣ r.factorization p := by + cases' r with r + · simp + have npow_nz : n ^ k ≠ 0 := fun npowz => nnz (pow_eq_zero npowz) + have eq1 : (m ^ k).factorization p = k * m.factorization p := by + rw [factorization_pow'] + have eq2 : (r.succ * n ^ k).factorization p = k * n.factorization p + r.succ.factorization p := by + rw [factorization_mul' r.succ_ne_zero npow_nz, factorization_pow', add_comm] + have : r.succ.factorization p = k * m.factorization p - k * n.factorization p := by + rw [← eq1, pow_eq, eq2, add_comm, Nat.add_sub_cancel] + rw [this] + apply Nat.dvd_sub' <;> + apply Nat.dvd_mul_right +
src/C05_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean (new)
- @@ -0,0 +1,98 @@
 + import Mathlib.Data.Nat.GCD.Basic + import Mathlib.Algebra.BigOperators.Basic + import Mathlib.Tactic + + def fac : ℕ → ℕ + | 0 => 1 + | n + 1 => (n + 1) * fac n + + theorem pow_two_le_fac (n : ℕ) : 2 ^ (n - 1) ≤ fac n := by + cases' n with n + · simp [fac] + induction' n with n ih + · simp [fac] + simp at * + rw [pow_succ, fac] + apply Nat.mul_le_mul _ ih + repeat' apply Nat.succ_le_succ + apply zero_le + + section + + variable {α : Type _} (s : Finset ℕ) (f : ℕ → ℕ) (n : ℕ) + + open BigOperators + open Finset + + theorem sum_sqr (n : ℕ) : (∑ i in range (n + 1), i ^ 2) = n * (n + 1) * (2 * n + 1) / 6 := by + symm; + apply Nat.div_eq_of_eq_mul_right (by norm_num : 0 < 6) + induction' n with n ih + · simp + rw [Finset.sum_range_succ, mul_add 6, ← ih, Nat.succ_eq_add_one] + ring + + end + + inductive MyNat + | zero : MyNat + | succ : MyNat → MyNat + + namespace MyNat + + def add : MyNat → MyNat → MyNat + | x, zero => x + | x, succ y => succ (add x y) + + def mul : MyNat → MyNat → MyNat + | x, zero => zero + | x, succ y => add (mul x y) x + + theorem zero_add (n : MyNat) : add zero n = n := by + induction' n with n ih + · rfl + rw [add, ih] + + theorem succ_add (m n : MyNat) : add (succ m) n = succ (add m n) := by + induction' n with n ih + · rfl + rw [add, ih] + rfl + + theorem add_comm (m n : MyNat) : add m n = add n m := by + induction' n with n ih + · rw [zero_add] + rfl + rw [add, succ_add, ih] + + theorem add_assoc (m n k : MyNat) : add (add m n) k = add m (add n k) := by + induction' k with k ih + · rfl + rw [add, ih] + rfl + + theorem mul_add (m n k : MyNat) : mul m (add n k) = add (mul m n) (mul m k) := by + induction' k with k ih + · rfl + rw [add, mul, mul, ih, add_assoc] + + theorem zero_mul (n : MyNat) : mul zero n = zero := by + induction' n with n ih + · rfl + rw [mul, ih] + rfl + + theorem succ_mul (m n : MyNat) : mul (succ m) n = add (mul m n) n := by + induction' n with n ih + · rfl + rw [mul, mul, ih, add_assoc, add_assoc, add_comm n, succ_add] + rfl + + theorem mul_comm (m n : MyNat) : mul m n = mul n m := by + induction' n with n ih + · rw [zero_mul] + rfl + rw [mul, ih, succ_mul] + + end MyNat +
src/C05_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean (new)
- @@ -0,0 +1,240 @@
 + import Mathlib.Data.Nat.Prime + import Mathlib.Algebra.BigOperators.Order + import Mathlib.Tactic + import Mathlib.Tactic.IntervalCases + + open BigOperators + + theorem two_le {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by + cases m; contradiction + case succ m => + cases m; contradiction + repeat' apply Nat.succ_le_succ + apply zero_le + + theorem exists_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ p : Nat, p.Prime ∧ p ∣ n := by + by_cases np : n.Prime + · use n, np + induction' n using Nat.strong_induction_on with n ih + dsimp at ih + rw [Nat.prime_def_lt] at np + push_neg at np + rcases np h with ⟨m, mltn, mdvdn, mne1⟩ + have : m ≠ 0 := by + intro mz + rw [mz, zero_dvd_iff] at mdvdn + linarith + have mgt2 : 2 ≤ m := two_le this mne1 + by_cases mp : m.Prime + · use m, mp + exact mdvdn + . rcases ih m mltn mgt2 mp with ⟨p, pp, pdvd⟩ + use p, pp + apply pdvd.trans mdvdn + + theorem primes_infinite : ∀ n, ∃ p > n, Nat.Prime p := by + intro n + have : 2 ≤ Nat.factorial (n + 1) + 1 := by + apply Nat.succ_le_succ + exact Nat.succ_le_of_lt (Nat.factorial_pos _) + rcases exists_prime_factor this with ⟨p, pp, pdvd⟩ + refine' ⟨p, _, pp⟩ + show p > n + by_contra ple + push_neg at ple + have : p ∣ Nat.factorial (n + 1) := by + apply Nat.dvd_factorial + apply pp.pos + linarith + have : p ∣ 1 := by + convert Nat.dvd_sub' pdvd this + simp + show False + have := Nat.le_of_dvd zero_lt_one this + linarith [pp.two_le] + + open Finset + + section + variable {α : Type _} [DecidableEq α] (r s t : Finset α) + + example : (r ∪ s) ∩ (r ∪ t) = r ∪ s ∩ t := by + ext x + rw [mem_inter, mem_union, mem_union, mem_union, mem_inter] + tauto + + example : (r ∪ s) ∩ (r ∪ t) = r ∪ s ∩ t := by + ext x + simp + tauto + + example : (r \ s) \ t = r \ (s ∪ t) := by + ext x + rw [mem_sdiff, mem_sdiff, mem_sdiff, mem_union] + tauto + + example : (r \ s) \ t = r \ (s ∪ t) := by + ext x + simp + tauto + + end + + theorem Nat.Prime.eq_of_dvd_of_prime {p q : ℕ} (prime_p : Nat.Prime p) (prime_q : Nat.Prime q) + (h : p ∣ q) : p = q := by + cases prime_q.eq_one_or_self_of_dvd _ h + · linarith [prime_p.two_le] + assumption + + theorem mem_of_dvd_prod_primes {s : Finset ℕ} {p : ℕ} (prime_p : p.Prime) : + (∀ n ∈ s, Nat.Prime n) → (p ∣ ∏ n in s, n) → p ∈ s := by + intro h₀ h₁ + induction' s using Finset.induction_on with a s ans ih + · simp at h₁ + linarith [prime_p.two_le] + simp [Finset.prod_insert ans, prime_p.dvd_mul] at h₀ h₁ + rw [mem_insert] + cases' h₁ with h₁ h₁ + · left + exact prime_p.eq_of_dvd_of_prime h₀.1 h₁ + right + exact ih h₀.2 h₁ + + theorem primes_infinite' : ∀ s : Finset Nat, ∃ p, Nat.Prime p ∧ p ∉ s := by + intro s + by_contra h + push_neg at h + set s' := s.filter Nat.Prime with s'_def + have mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ n.Prime := by + intro n + simp [s'_def] + apply h + have : 2 ≤ (∏ i in s', i) + 1 := by + apply Nat.succ_le_succ + apply Nat.succ_le_of_lt + apply Finset.prod_pos + intro n ns' + apply (mem_s'.mp ns').pos + rcases exists_prime_factor this with ⟨p, pp, pdvd⟩ + have : p ∣ ∏ i in s', i := by + apply dvd_prod_of_mem + rw [mem_s'] + apply pp + have : p ∣ 1 := by + convert Nat.dvd_sub' pdvd this + simp + show False + have := Nat.le_of_dvd zero_lt_one this + linarith [pp.two_le] + + theorem bounded_of_ex_finset (Q : ℕ → Prop) : + (∃ s : Finset ℕ, ∀ k, Q k → k ∈ s) → ∃ n, ∀ k, Q k → k < n := by + rintro ⟨s, hs⟩ + use s.sup id + 1 + intro k Qk + apply Nat.lt_succ_of_le + show id k ≤ s.sup id + apply le_sup (hs k Qk) + + theorem ex_finset_of_bounded (Q : ℕ → Prop) [DecidablePred Q] : + (∃ n, ∀ k, Q k → k ≤ n) → ∃ s : Finset ℕ, ∀ k, Q k ↔ k ∈ s := by + rintro ⟨n, hn⟩ + use (range (n + 1)).filter Q + intro k + simp [Nat.lt_succ_iff] + exact hn k + + theorem mod_4_eq_3_or_mod_4_eq_3 {m n : ℕ} (h : m * n % 4 = 3) : m % 4 = 3 ∨ n % 4 = 3 := by + revert h + rw [Nat.mul_mod] + have : m % 4 < 4 := Nat.mod_lt m (by norm_num) + interval_cases hm : m % 4 <;> simp [hm] + have : n % 4 < 4 := Nat.mod_lt n (by norm_num) + interval_cases hn : n % 4 <;> simp [hn] + + theorem two_le_of_mod_4_eq_3 {n : ℕ} (h : n % 4 = 3) : 2 ≤ n := by + apply two_le <;> + · intro neq + rw [neq] at h + norm_num at h + + theorem aux {m n : ℕ} (h₀ : m ∣ n) (h₁ : 2 ≤ m) (h₂ : m < n) : n / m ∣ n ∧ n / m < n := by + constructor + · exact Nat.div_dvd_of_dvd h₀ + exact Nat.div_lt_self (lt_of_le_of_lt (zero_le _) h₂) h₁ + + theorem exists_prime_factor_mod_4_eq_3 {n : Nat} (h : n % 4 = 3) : + ∃ p : Nat, p.Prime ∧ p ∣ n ∧ p % 4 = 3 := by + by_cases np : n.Prime + · use n + exact ⟨np, dvd_rfl, h⟩ + induction' n using Nat.strong_induction_on with n ih + dsimp at ih + rw [Nat.prime_def_lt] at np + push_neg at np + rcases np (two_le_of_mod_4_eq_3 h) with ⟨m, mltn, mdvdn, mne1⟩ + have mge2 : 2 ≤ m := by + apply two_le _ mne1 + intro mz + rw [mz, zero_dvd_iff] at mdvdn + linarith + have neq : m * (n / m) = n := Nat.mul_div_cancel' mdvdn + have : m % 4 = 3 ∨ n / m % 4 = 3 := by + apply mod_4_eq_3_or_mod_4_eq_3 + rw [neq, h] + cases' this with h1 h1 + · by_cases mp : m.Prime + · use m + exact ⟨mp, mdvdn, h1⟩ + rcases ih m mltn h1 mp with ⟨p, pp, pdvd, p4eq⟩ + use p + exact ⟨pp, pdvd.trans mdvdn, p4eq⟩ + obtain ⟨nmdvdn, nmltn⟩ := aux mdvdn mge2 mltn + by_cases nmp : (n / m).Prime + · use n / m + exact ⟨nmp, nmdvdn, h1⟩ + rcases ih (n / m) nmltn h1 nmp with ⟨p, pp, pdvd, p4eq⟩ + use p + exact ⟨pp, pdvd.trans nmdvdn, p4eq⟩ + + theorem primes_mod_4_eq_3_infinite : ∀ n, ∃ p > n, Nat.Prime p ∧ p % 4 = 3 := by + by_contra h + push_neg at h + cases' h with n hn + have : ∃ s : Finset Nat, ∀ p : ℕ, p.Prime ∧ p % 4 = 3 ↔ p ∈ s := by + apply ex_finset_of_bounded + use n + contrapose! hn + rcases hn with ⟨p, ⟨pp, p4⟩, pltn⟩ + exact ⟨p, pltn, pp, p4⟩ + cases' this with s hs + have h₁ : ((4 * ∏ i in erase s 3, i) + 3) % 4 = 3 := by + rw [add_comm, Nat.add_mul_mod_self_left] + norm_num + rcases exists_prime_factor_mod_4_eq_3 h₁ with ⟨p, pp, pdvd, p4eq⟩ + have ps : p ∈ s := by + rw [← hs p] + exact ⟨pp, p4eq⟩ + have pne3 : p ≠ 3 := by + intro peq + rw [peq, ← Nat.dvd_add_iff_left (dvd_refl 3)] at pdvd + rw [Nat.prime_three.dvd_mul] at pdvd + norm_num at pdvd + have : 3 ∈ s.erase 3 := by + apply mem_of_dvd_prod_primes Nat.prime_three _ pdvd + intro n + simp [← hs n] + tauto + simp at this + have : p ∣ 4 * ∏ i in erase s 3, i := by + apply dvd_trans _ (dvd_mul_left _ _) + apply dvd_prod_of_mem + simp + constructor <;> assumption + have : p ∣ 3 := by + convert Nat.dvd_sub' pdvd this + simp + have : p = 3 := by + apply pp.eq_of_dvd_of_prime Nat.prime_three this + contradiction +
src/C06_Abstract_Algebra/S01_Structures.lean (new)
- @@ -0,0 +1,219 @@
 + import Mathlib.Algebra.BigOperators.Ring + import Mathlib.Data.Real.Basic + + noncomputable section + + @[ext] + structure Point where + x : ℝ + y : ℝ + z : ℝ + + #check Point.ext + + example (a b : Point) (hx : a.x = b.x) (hy : a.y = b.y) (hz : a.z = b.z) : a = b := by + ext + repeat' assumption + + def myPoint1 : Point where + x := 2 + y := -1 + z := 4 + + def myPoint2 : Point := + ⟨2, -1, 4⟩ + + def myPoint3 := + Point.mk 2 (-1) 4 + + structure Point' where build :: + x : ℝ + y : ℝ + z : ℝ + + #check Point'.build 2 (-1) 4 + + namespace Point + + def add (a b : Point) : Point := + ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ + + def add' (a b : Point) : Point where + x := a.x + b.x + y := a.y + b.y + z := a.z + b.z + + #check add myPoint1 myPoint2 + #check myPoint1.add myPoint2 + + end Point + + #check Point.add myPoint1 myPoint2 + #check myPoint1.add myPoint2 + + namespace Point + + protected theorem add_comm (a b : Point) : add a b = add b a := by + rw [add, add] + ext <;> dsimp + repeat' apply add_comm + + example (a b : Point) : add a b = add b a := by simp [add, add_comm] + + theorem add_x (a b : Point) : (a.add b).x = a.x + b.x := + rfl + + def addAlt : Point → Point → Point + | Point.mk x₁ y₁ z₁, Point.mk x₂ y₂ z₂ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ + + def addAlt' : Point → Point → Point + | ⟨x₁, y₁, z₁⟩, ⟨x₂, y₂, z₂⟩ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ + + theorem addAlt_x (a b : Point) : (a.addAlt b).x = a.x + b.x := by + cases a + cases b + rfl + + theorem addAlt_comm (a b : Point) : addAlt a b = addAlt b a := by + rcases a with ⟨xa, ya, za⟩ + rcases b with ⟨xb, yb, zb⟩ + rw [addAlt, addAlt] + ext <;> dsimp + apply add_comm + repeat' apply add_comm + + example (a b : Point) : addAlt a b = addAlt b a := by + rcases a with ⟨xa, ya, za⟩ + rcases b with ⟨xb, yb, zb⟩ + simp [addAlt, add_comm] + + example : ∀ a b : Point, addAlt a b = addAlt b a := by + rintro ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ + simp [addAlt, add_comm] + + example : ∀ a b : Point, add a b = add b a := fun ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ => by + simp [add, add_comm] + + protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by + sorry + def smul (r : ℝ) (a : Point) : Point := + sorry + theorem smul_distrib (r : ℝ) (a b : Point) : (smul r a).add (smul r b) = smul r (a.add b) := by + sorry + + end Point + + structure StandardTwoSimplex where + x : ℝ + y : ℝ + z : ℝ + x_nonneg : 0 ≤ x + y_nonneg : 0 ≤ y + z_nonneg : 0 ≤ z + sum_eq : x + y + z = 1 + + namespace StandardTwoSimplex + + def swapXy (a : StandardTwoSimplex) : StandardTwoSimplex + where + x := a.y + y := a.x + z := a.z + x_nonneg := a.y_nonneg + y_nonneg := a.x_nonneg + z_nonneg := a.z_nonneg + sum_eq := by rw [add_comm a.y a.x, a.sum_eq] + + noncomputable section + + def midpoint (a b : StandardTwoSimplex) : StandardTwoSimplex + where + x := (a.x + b.x) / 2 + y := (a.y + b.y) / 2 + z := (a.z + b.z) / 2 + x_nonneg := div_nonneg (add_nonneg a.x_nonneg b.x_nonneg) (by norm_num) + y_nonneg := div_nonneg (add_nonneg a.y_nonneg b.y_nonneg) (by norm_num) + z_nonneg := div_nonneg (add_nonneg a.z_nonneg b.z_nonneg) (by norm_num) + sum_eq := by field_simp; linarith [a.sum_eq, b.sum_eq] + + def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) + (a b : StandardTwoSimplex) : StandardTwoSimplex := + sorry + + end + + end StandardTwoSimplex + + open BigOperators + + structure StandardSimplex (n : ℕ) where + V : Fin n → ℝ + NonNeg : ∀ i : Fin n, 0 ≤ V i + sum_eq_one : (∑ i, V i) = 1 + + namespace StandardSimplex + + def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n + where + V i := (a.V i + b.V i) / 2 + NonNeg := by + intro i + apply div_nonneg + · linarith [a.NonNeg i, b.NonNeg i] + norm_num + sum_eq_one := by + simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one] + field_simp + + end StandardSimplex + + structure IsLinear (f : ℝ → ℝ) where + is_additive : ∀ x y, f (x + y) = f x + f y + preserves_mul : ∀ x c, f (c * x) = c * f x + + section + variable (f : ℝ → ℝ) (linf : IsLinear f) + + #check linf.is_additive + #check linf.preserves_mul + + end + + def Point'' := + ℝ × ℝ × ℝ + + def IsLinear' (f : ℝ → ℝ) := + (∀ x y, f (x + y) = f x + f y) ∧ ∀ x c, f (c * x) = c * f x + + def PReal := + { y : ℝ // 0 < y } + + section + variable (x : PReal) + + #check x.val + #check x.property + #check x.1 + #check x.2 + + end + + def StandardTwoSimplex' := + { p : ℝ × ℝ × ℝ // 0 ≤ p.1 ∧ 0 ≤ p.2.1 ∧ 0 ≤ p.2.2 ∧ p.1 + p.2.1 + p.2.2 = 1 } + + def StandardSimplex' (n : ℕ) := + { v : Fin n → ℝ // (∀ i : Fin n, 0 ≤ v i) ∧ (∑ i, v i) = 1 } + + def StdSimplex := Σ n : ℕ, StandardSimplex n + + section + variable (s : StdSimplex) + + #check s.fst + #check s.snd + + #check s.1 + #check s.2 + + end +
src/C06_Abstract_Algebra/S02_Algebraic_Structures.lean (new)
- @@ -0,0 +1,170 @@
  + import Mathlib.Data.Real.Basic + + structure Group₁ (α : Type _) where + mul : α → α → α + one : α + inv : α → α + mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) + mul_one : ∀ x : α, mul x one = x + one_mul : ∀ x : α, mul one x = x + mul_left_inv : ∀ x : α, mul (inv x) x = one + + structure Group₁Cat where + α : Type _ + str : Group₁ α + + section + variable (α β γ : Type _) + variable (f : α ≃ β) (g : β ≃ γ) + + #check Equiv α β + #check (f.toFun : α → β) + #check (f.invFun : β → α) + #check (f.right_inv : ∀ x : β, f (f.invFun x) = x) + #check (f.left_inv : ∀ x : α, f.invFun (f x) = x) + #check (Equiv.refl α : α ≃ α) + #check (f.symm : β ≃ α) + #check (f.trans g : α ≃ γ) + + example (x : α) : (f.trans g).toFun x = g.toFun (f.toFun x) := + rfl + + example (x : α) : (f.trans g) x = g (f x) := + rfl + + example : (f.trans g : α → γ) = g ∘ f := + rfl + + end + + example (α : Type _) : Equiv.Perm α = (α ≃ α) := + rfl + + def permGroup {α : Type _} : Group₁ (Equiv.Perm α) + where + mul f g := Equiv.trans g f + one := Equiv.refl α + inv := Equiv.symm + mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm + one_mul := Equiv.trans_refl + mul_one := Equiv.refl_trans + mul_left_inv := Equiv.self_trans_symm + + structure AddGroup₁ (α : Type _) where + (add : α → α → α) + -- fill in the rest + @[ext] + structure Point where + x : ℝ + y : ℝ + z : ℝ + + namespace Point + + def add (a b : Point) : Point := + ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ + + def neg (a : Point) : Point := sorry + + def zero : Point := sorry + + def add_group_point : AddGroup₁ Point := sorry + end Point + + section + + variable {α : Type _} (f g : Equiv.Perm α) (n : ℕ) + + #check f * g + #check mul_assoc f g g⁻¹ + + -- group power, defined for any group + #check g ^ n + + example : f * g * g⁻¹ = f := by rw [mul_assoc, mul_right_inv, mul_one] + + example : f * g * g⁻¹ = f := + mul_inv_cancel_right f g + + example {α : Type _} (f g : Equiv.Perm α) : g.symm.trans (g.trans f) = f := + mul_inv_cancel_right f g + + end + + class Group₂ (α : Type _) where + mul : α → α → α + one : α + inv : α → α + mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) + mul_one : ∀ x : α, mul x one = x + one_mul : ∀ x : α, mul one x = x + mul_left_inv : ∀ x : α, mul (inv x) x = one + + instance {α : Type _} : Group₂ (Equiv.Perm α) where + mul f g := Equiv.trans g f + one := Equiv.refl α + inv := Equiv.symm + mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm + one_mul := Equiv.trans_refl + mul_one := Equiv.refl_trans + mul_left_inv := Equiv.self_trans_symm + + #check @Group₂.mul + + def mySquare {α : Type _} [Group₂ α] (x : α) := + Group₂.mul x x + + #check @mySquare + + section + variable {β : Type _} (f g : Equiv.Perm β) + + example : Group₂.mul f g = g.trans f := + rfl + + example : mySquare f = f.trans f := + rfl + + end + + instance : Inhabited Point where default := ⟨0, 0, 0⟩ + + #check (default : Point) + + example : ([] : List Point).headI = default := + rfl + + instance : Add Point where add := Point.add + + section + variable (x y : Point) + + #check x + y + + example : x + y = Point.add x y := + rfl + + end + + instance hasMulGroup₂ {α : Type _} [Group₂ α] : Mul α := + ⟨Group₂.mul⟩ + + instance hasOneGroup₂ {α : Type _} [Group₂ α] : One α := + ⟨Group₂.one⟩ + + instance hasInvGroup₂ {α : Type _} [Group₂ α] : Inv α := + ⟨Group₂.inv⟩ + + section + variable {α : Type _} (f g : Equiv.Perm α) + + #check f * 1 * g⁻¹ + + def foo : f * 1 * g⁻¹ = g.symm.trans ((Equiv.refl α).trans f) := + rfl + + end + + class AddGroup₂ (α : Type _) where + add : α → α → α + -- fill in the rest
src/C06_Abstract_Algebra/S03_Building_the_Gaussian_Integers.lean (new)
- @@ -0,0 +1,271 @@
 + import Mathlib.Data.Int.Basic + import Mathlib.Algebra.EuclideanDomain.Basic + import Mathlib.RingTheory.PrincipalIdealDomain + import Mathlib.Tactic + + @[ext] + structure gaussInt where + re : ℤ + im : ℤ + + namespace gaussInt + + instance : Zero gaussInt := + ⟨⟨0, 0⟩⟩ + + instance : One gaussInt := + ⟨⟨1, 0⟩⟩ + + instance : Add gaussInt := + ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ + + instance : Neg gaussInt := + ⟨fun x => ⟨-x.re, -x.im⟩⟩ + + instance : Mul gaussInt := + ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ + + theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := + rfl + + theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := + rfl + + theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := + rfl + + theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := + rfl + + theorem mul_def (x y : gaussInt) : x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := + rfl + + @[simp] + theorem zero_re : (0 : gaussInt).re = 0 := + rfl + + @[simp] + theorem zero_im : (0 : gaussInt).im = 0 := + rfl + + @[simp] + theorem one_re : (1 : gaussInt).re = 1 := + rfl + + @[simp] + theorem one_im : (1 : gaussInt).im = 0 := + rfl + + @[simp] + theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := + rfl + + @[simp] + theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := + rfl + + @[simp] + theorem neg_re (x : gaussInt) : (-x).re = -x.re := + rfl + + @[simp] + theorem neg_im (x : gaussInt) : (-x).im = -x.im := + rfl + + @[simp] + theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := + rfl + + @[simp] + theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := + rfl + + instance instCommRing : CommRing gaussInt where + zero := 0 + one := 1 + add := (· + ·) + neg x := -x + mul := (· * ·) + add_assoc := by + intros + ext <;> simp <;> ring + zero_add := by + intro + ext <;> simp + add_zero := by + intro + ext <;> simp + add_left_neg := by + intro + ext <;> simp + add_comm := by + intros + ext <;> simp <;> ring + mul_assoc := by + intros + ext <;> simp <;> ring + one_mul := by + intro + ext <;> simp + mul_one := by + intro + ext <;> simp + left_distrib := by + intros + ext <;> simp <;> ring + right_distrib := by + intros + ext <;> simp <;> ring + mul_comm := by + intros + ext <;> simp <;> ring + zero_mul := sorry + mul_zero := sorry + + instance : Nontrivial gaussInt := by + use 0, 1 + rw [Ne, gaussInt.ext_iff] + simp + + end gaussInt + + example (a b : ℤ) : a = b * (a / b) + a % b := + Eq.symm <| Int.ediv_add_emod a b + + example (a b : ℤ) : b ≠ 0 → 0 ≤ a % b := + Int.emod_nonneg a + + example (a b : ℤ) : b ≠ 0 → a % b < abs b := + Int.emod_lt a + + namespace Int + + def div' (a b : ℤ) := + (a + b / 2) / b + + def mod' (a b : ℤ) := + (a + b / 2) % b - b / 2 + + theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by + rw [div', mod'] + linarith [Int.ediv_add_emod (a + b / 2) b] + + theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by + rw [mod', abs_le] + constructor + · linarith [Int.emod_nonneg (a + b / 2) h.ne'] + have := Int.emod_lt_of_pos (a + b / 2) h + have := Int.ediv_add_emod b 2 + have := Int.emod_lt_of_pos b zero_lt_two + revert this; intro this -- FIXME, this should not be needed + linarith + + theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] + + end Int + + theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : + x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by + sorry + namespace gaussInt + + def norm (x : gaussInt) := + x.re ^ 2 + x.im ^ 2 + + @[simp] + theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by + sorry + theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by + sorry + theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by + sorry + theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by + sorry + def conj (x : gaussInt) : gaussInt := + ⟨x.re, -x.im⟩ + + @[simp] + theorem conj_re (x : gaussInt) : (conj x).re = x.re := + rfl + + @[simp] + theorem conj_im (x : gaussInt) : (conj x).im = -x.im := + rfl + + theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] + + instance : Div gaussInt := + ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ + + instance : Mod gaussInt := + ⟨fun x y => x - y * (x / y)⟩ + + theorem div_def (x y : gaussInt) : + x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := + rfl + + theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := + rfl + + theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : (x % y).norm < y.norm := by + have norm_y_pos : 0 < norm y := by rwa [norm_pos] + have : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by + rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] + ext <;> simp <;> ring + have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by + conv => + lhs + rw [← norm_conj y, ← norm_mul, this, norm] + simp + trans 2 * (y.norm / 2) ^ 2 + · rw [two_mul] + apply add_le_add <;> + · rw [sq_le_sq] + apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) + apply le_abs_self + rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] + apply mul_le_mul_of_nonneg_left + · apply Int.ediv_mul_le + norm_num + apply Int.ediv_nonneg (norm_nonneg y) + norm_num + have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos + apply lt_of_le_of_lt this + apply Int.ediv_lt_of_lt_mul + · norm_num + linarith + + theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := + Int.natAbs_of_nonneg (norm_nonneg _) + + theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : (x % y).norm.natAbs < y.norm.natAbs := by + apply Int.ofNat_lt.1 + simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] + apply norm_mod_lt x hy + + theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : + ¬(norm (x * y)).natAbs < (norm x).natAbs := by + apply not_lt_of_ge + rw [norm_mul, Int.natAbs_mul] + apply le_mul_of_one_le_right (Nat.zero_le _) + apply Int.ofNat_le.1 + rw [coe_natAbs_norm] + exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) + + instance : EuclideanDomain gaussInt := + { gaussInt.instCommRing with + quotient := (· / ·) + remainder := (· % ·) + quotient_mul_add_remainder_eq := fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] + quotient_zero := fun x => by + simp [div_def, norm, Int.div'] + rfl + r := Measure (Int.natAbs ∘ norm) + r_wellFounded := (measure (Int.natAbs ∘ norm)).2 + remainder_lt := natAbs_norm_mod_lt + mul_left_not_lt := not_norm_mul_left_lt_norm } + + example (x : gaussInt) : Irreducible x ↔ Prime x := + PrincipalIdealRing.irreducible_iff_prime + + end gaussInt
src/C06_Abstract_Algebra/solutions/Solutions_S01_Structures.lean (new)
- @@ -0,0 +1,96 @@
  + import Mathlib.Algebra.BigOperators.Ring + import Mathlib.Data.Real.Basic + + noncomputable section + + @[ext] + structure Point where + x : ℝ + y : ℝ + z : ℝ + + namespace Point + + def add (a b : Point) : Point := + ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ + + protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by + simp [add, add_assoc] + + def smul (r : ℝ) (a : Point) : Point := + ⟨r * a.x, r * a.y, r * a.z⟩ + + theorem smul_distrib (r : ℝ) (a b : Point) : (smul r a).add (smul r b) = smul r (a.add b) := by + simp [add, smul, mul_add] + + end Point + + structure StandardTwoSimplex where + x : ℝ + y : ℝ + z : ℝ + x_nonneg : 0 ≤ x + y_nonneg : 0 ≤ y + z_nonneg : 0 ≤ z + sum_eq : x + y + z = 1 + + namespace StandardTwoSimplex + + noncomputable section + + def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) + (a b : StandardTwoSimplex) : StandardTwoSimplex + where + x := lambda * a.x + (1 - lambda) * b.x + y := lambda * a.y + (1 - lambda) * b.y + z := lambda * a.z + (1 - lambda) * b.z + x_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.x_nonneg) (mul_nonneg (by linarith) b.x_nonneg) + y_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.y_nonneg) (mul_nonneg (by linarith) b.y_nonneg) + z_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.z_nonneg) (mul_nonneg (by linarith) b.z_nonneg) + sum_eq := by + trans (a.x + a.y + a.z) * lambda + (b.x + b.y + b.z) * (1 - lambda) + · ring + simp [a.sum_eq, b.sum_eq] + + end + + end StandardTwoSimplex + + open BigOperators + + structure StandardSimplex (n : ℕ) where + V : Fin n → ℝ + NonNeg : ∀ i : Fin n, 0 ≤ V i + sum_eq_one : (∑ i, V i) = 1 + + namespace StandardSimplex + + def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n + where + V i := (a.V i + b.V i) / 2 + NonNeg := by + intro i + apply div_nonneg + · linarith [a.NonNeg i, b.NonNeg i] + norm_num + sum_eq_one := by + simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one] + field_simp + + end StandardSimplex + + namespace StandardSimplex + + def weightedAverage {n : ℕ} (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) + (a b : StandardSimplex n) : StandardSimplex n + where + V i := lambda * a.V i + (1 - lambda) * b.V i + NonNeg i := + add_nonneg (mul_nonneg lambda_nonneg (a.NonNeg i)) (mul_nonneg (by linarith) (b.NonNeg i)) + sum_eq_one := by + trans (lambda * ∑ i, a.V i) + (1 - lambda) * ∑ i, b.V i + · rw [Finset.sum_add_distrib, Finset.mul_sum, Finset.mul_sum] + simp [a.sum_eq_one, b.sum_eq_one] + + end StandardSimplex +
src/C06_Abstract_Algebra/solutions/Solutions_S02_Algebraic_Structures.lean (new)
- @@ -0,0 +1,73 @@
  + import Mathlib.Data.Real.Basic + + structure AddGroup₁ (α : Type _) where + add : α → α → α + zero : α + neg : α → α + add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z) + add_zero : ∀ x : α, add x zero = x + zero_add : ∀ x : α, add x zero = x + add_left_neg : ∀ x : α, add (neg x) x = zero + + @[ext] + structure Point where + x : ℝ + y : ℝ + z : ℝ + + namespace Point + + def add (a b : Point) : Point := + ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ + + def neg (a : Point) : Point := + ⟨-a.x, -a.y, -a.z⟩ + + def zero : Point := + ⟨0, 0, 0⟩ + + def addGroupPoint : AddGroup₁ Point where + add := Point.add + zero := Point.zero + neg := Point.neg + add_assoc := by simp [Point.add, add_assoc] + add_zero := by simp [Point.add, Point.zero] + zero_add := by simp [Point.add, Point.zero] + add_left_neg := by simp [Point.add, Point.neg, Point.zero] + + end Point + + class AddGroup₂ (α : Type _) where + add : α → α → α + zero : α + neg : α → α + add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z) + add_zero : ∀ x : α, add x zero = x + zero_add : ∀ x : α, add x zero = x + add_left_neg : ∀ x : α, add (neg x) x = zero + + instance hasAddAddGroup₂ {α : Type _} [AddGroup₂ α] : Add α := + ⟨AddGroup₂.add⟩ + + instance hasZeroAddGroup₂ {α : Type _} [AddGroup₂ α] : Zero α := + ⟨AddGroup₂.zero⟩ + + instance hasNegAddGroup₂ {α : Type _} [AddGroup₂ α] : Neg α := + ⟨AddGroup₂.neg⟩ + + instance : AddGroup₂ Point where + add := Point.add + zero := Point.zero + neg := Point.neg + add_assoc := by simp [Point.add, add_assoc] + add_zero := by simp [Point.add, Point.zero] + zero_add := by simp [Point.add, Point.zero] + add_left_neg := by simp [Point.add, Point.neg, Point.zero] + + section + variable (x y : Point) + + #check x + -y + 0 + + end +
src/C06_Abstract_Algebra/solutions/Solutions_S03_Building_the_Gaussian_Integers.lean (new)
- @@ -0,0 +1,285 @@
 + import Mathlib.Data.Int.Basic + import Mathlib.Algebra.EuclideanDomain.Basic + import Mathlib.RingTheory.PrincipalIdealDomain + import Mathlib.Tactic + + @[ext] + structure gaussInt where + re : ℤ + im : ℤ + + namespace gaussInt + + instance : Zero gaussInt := + ⟨⟨0, 0⟩⟩ + + instance : One gaussInt := + ⟨⟨1, 0⟩⟩ + + instance : Add gaussInt := + ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ + + instance : Neg gaussInt := + ⟨fun x => ⟨-x.re, -x.im⟩⟩ + + instance : Mul gaussInt := + ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ + + theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := + rfl + + theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := + rfl + + theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := + rfl + + theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := + rfl + + theorem mul_def (x y : gaussInt) : x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := + rfl + + @[simp] + theorem zero_re : (0 : gaussInt).re = 0 := + rfl + + @[simp] + theorem zero_im : (0 : gaussInt).im = 0 := + rfl + + @[simp] + theorem one_re : (1 : gaussInt).re = 1 := + rfl + + @[simp] + theorem one_im : (1 : gaussInt).im = 0 := + rfl + + @[simp] + theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := + rfl + + @[simp] + theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := + rfl + + @[simp] + theorem neg_re (x : gaussInt) : (-x).re = -x.re := + rfl + + @[simp] + theorem neg_im (x : gaussInt) : (-x).im = -x.im := + rfl + + @[simp] + theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := + rfl + + @[simp] + theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := + rfl + + instance instCommRing : CommRing gaussInt where + zero := 0 + one := 1 + add := (· + ·) + neg x := -x + mul := (· * ·) + add_assoc := by + intros + ext <;> simp <;> ring + zero_add := by + intro + ext <;> simp + add_zero := by + intro + ext <;> simp + add_left_neg := by + intro + ext <;> simp + add_comm := by + intros + ext <;> simp <;> ring + mul_assoc := by + intros + ext <;> simp <;> ring + one_mul := by + intro + ext <;> simp + mul_one := by + intro + ext <;> simp + left_distrib := by + intros + ext <;> simp <;> ring + right_distrib := by + intros + ext <;> simp <;> ring + mul_comm := by + intros + ext <;> simp <;> ring + zero_mul := sorry + mul_zero := sorry + + instance : Nontrivial gaussInt := by + use 0, 1 + rw [Ne, gaussInt.ext_iff] + simp + + end gaussInt + + namespace Int + + def div' (a b : ℤ) := + (a + b / 2) / b + + def mod' (a b : ℤ) := + (a + b / 2) % b - b / 2 + + theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by + rw [div', mod'] + linarith [Int.ediv_add_emod (a + b / 2) b] + + theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by + rw [mod', abs_le] + constructor + · linarith [Int.emod_nonneg (a + b / 2) h.ne'] + have := Int.emod_lt_of_pos (a + b / 2) h + have := Int.ediv_add_emod b 2 + have := Int.emod_lt_of_pos b zero_lt_two + revert this; intro this -- FIXME, this should not be needed + linarith + + theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] + + end Int + + private theorem aux {α : Type _} [LinearOrderedRing α] {x y : α} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := + haveI h' : x ^ 2 = 0 := by + apply le_antisymm _ (sq_nonneg x) + rw [← h] + apply le_add_of_nonneg_right (sq_nonneg y) + pow_eq_zero h' + + theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : + x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by + constructor + · intro h + constructor + · exact aux h + rw [add_comm] at h + exact aux h + rintro ⟨rfl, rfl⟩ + norm_num + + namespace gaussInt + + def norm (x : gaussInt) := + x.re ^ 2 + x.im ^ 2 + + @[simp] + theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by + apply add_nonneg <;> + apply sq_nonneg + + theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by + rw [norm, sq_add_sq_eq_zero, gaussInt.ext_iff] + rfl + + theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by + rw [lt_iff_le_and_ne, ne_comm, Ne, norm_eq_zero] + simp [norm_nonneg] + + theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by + simp [norm] + ring + + def conj (x : gaussInt) : gaussInt := + ⟨x.re, -x.im⟩ + + @[simp] + theorem conj_re (x : gaussInt) : (conj x).re = x.re := + rfl + + @[simp] + theorem conj_im (x : gaussInt) : (conj x).im = -x.im := + rfl + + theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] + + instance : Div gaussInt := + ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ + + instance : Mod gaussInt := + ⟨fun x y => x - y * (x / y)⟩ + + theorem div_def (x y : gaussInt) : + x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := + rfl + + theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := + rfl + + theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : (x % y).norm < y.norm := by + have norm_y_pos : 0 < norm y := by rwa [norm_pos] + have : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by + rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] + ext <;> simp <;> ring + have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by + conv => + lhs + rw [← norm_conj y, ← norm_mul, this, norm] + simp + trans 2 * (y.norm / 2) ^ 2 + · rw [two_mul] + apply add_le_add <;> + · rw [sq_le_sq] + apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) + apply le_abs_self + rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] + apply mul_le_mul_of_nonneg_left + · apply Int.ediv_mul_le + norm_num + apply Int.ediv_nonneg (norm_nonneg y) + norm_num + have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos + apply lt_of_le_of_lt this + apply Int.ediv_lt_of_lt_mul + · norm_num + linarith + + theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := + Int.natAbs_of_nonneg (norm_nonneg _) + + theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : (x % y).norm.natAbs < y.norm.natAbs := by + apply Int.ofNat_lt.1 + simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] + apply norm_mod_lt x hy + + theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : + ¬(norm (x * y)).natAbs < (norm x).natAbs := by + apply not_lt_of_ge + rw [norm_mul, Int.natAbs_mul] + apply le_mul_of_one_le_right (Nat.zero_le _) + apply Int.ofNat_le.1 + rw [coe_natAbs_norm] + exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) + + instance : EuclideanDomain gaussInt := + { gaussInt.instCommRing with + quotient := (· / ·) + remainder := (· % ·) + quotient_mul_add_remainder_eq := fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] + quotient_zero := fun x => by + simp [div_def, norm, Int.div'] + rfl + r := Measure (Int.natAbs ∘ norm) + r_wellFounded := (measure (Int.natAbs ∘ norm)).2 + remainder_lt := natAbs_norm_mod_lt + mul_left_not_lt := not_norm_mul_left_lt_norm } + + example (x : gaussInt) : Irreducible x ↔ Prime x := + PrincipalIdealRing.irreducible_iff_prime + + end gaussInt
src/C07_Topology/S01_Filters.lean (new)
- @@ -0,0 +1,109 @@
 + import Mathlib.Topology.Instances.Real + + open Set Filter Topology + + def principal {α : Type _} (s : Set α) : Filter α + where + sets := { t | s ⊆ t } + univ_sets := sorry + sets_of_superset := sorry + inter_sets := sorry + + example : Filter ℕ := + { sets := { s | ∃ a, ∀ b, a ≤ b → b ∈ s } + univ_sets := sorry + sets_of_superset := sorry + inter_sets := sorry } + + def Tendsto₁ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := + ∀ V ∈ G, f ⁻¹' V ∈ F + + def Tendsto₂ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := + map f F ≤ G + + example {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) : + Tendsto₂ f F G ↔ Tendsto₁ f F G := + Iff.rfl + + #check (@Filter.map_mono : ∀ {α β} {m : α → β}, Monotone (map m)) + + #check + (@Filter.map_map : + ∀ {α β γ} {f : Filter α} {m : α → β} {m' : β → γ}, map m' (map m f) = map (m' ∘ m) f) + + example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} + (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := + sorry + + variable (f : ℝ → ℝ) (x₀ y₀ : ℝ) + + #check comap ((↑) : ℚ → ℝ) (𝓝 x₀) + + #check Tendsto (f ∘ (↑)) (comap ((↑) : ℚ → ℝ) (𝓝 x₀)) (𝓝 y₀) + + section + variable {α β γ : Type _} (F : Filter α) {m : γ → β} {n : β → α} + + #check (comap_comap : comap m (comap n F) = comap (n ∘ m) F) + + end + + example : 𝓝 (x₀, y₀) = 𝓝 x₀ ×ᶠ 𝓝 y₀ := + nhds_prod_eq + + #check le_inf_iff + + example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : + Tendsto f atTop (𝓝 (x₀, y₀)) ↔ + Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := + sorry + + example (x₀ : ℝ) : HasBasis (𝓝 x₀) (fun ε : ℝ => 0 < ε) fun ε => Ioo (x₀ - ε) (x₀ + ε) := + nhds_basis_Ioo_pos x₀ + + example (u : ℕ → ℝ) (x₀ : ℝ) : + Tendsto u atTop (𝓝 x₀) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, u n ∈ Ioo (x₀ - ε) (x₀ + ε) := by + have : atTop.HasBasis (fun _ : ℕ => True) Ici := atTop_basis + rw [this.tendsto_iff (nhds_basis_Ioo_pos x₀)] + simp + + example (P Q : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in atTop, Q n) : + ∀ᶠ n in atTop, P n ∧ Q n := + hP.and hQ + + example (u v : ℕ → ℝ) (h : ∀ᶠ n in atTop, u n = v n) (x₀ : ℝ) : + Tendsto u atTop (𝓝 x₀) ↔ Tendsto v atTop (𝓝 x₀) := + tendsto_congr' h + + example (u v : ℕ → ℝ) (h : u =ᶠ[atTop] v) (x₀ : ℝ) : + Tendsto u atTop (𝓝 x₀) ↔ Tendsto v atTop (𝓝 x₀) := + tendsto_congr' h + + #check @eventually_of_forall + + #check @Eventually.mono + + #check @Eventually.and + + example (P Q R : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in atTop, Q n) + (hR : ∀ᶠ n in atTop, P n ∧ Q n → R n) : ∀ᶠ n in atTop, R n := by + apply (hP.and (hQ.and hR)).mono + rintro n ⟨h, h', h''⟩ + exact h'' ⟨h, h'⟩ + + example (P Q R : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in atTop, Q n) + (hR : ∀ᶠ n in atTop, P n ∧ Q n → R n) : ∀ᶠ n in atTop, R n := by + filter_upwards [hP, hQ, hR] + intro n h h' h'' + exact h'' ⟨h, h'⟩ + + #check mem_closure_iff_clusterPt + + #check le_principal_iff + + #check neBot_of_le + + example (u : ℕ → ℝ) (M : Set ℝ) (x : ℝ) (hux : Tendsto u atTop (𝓝 x)) + (huM : ∀ᶠ n in atTop, u n ∈ M) : x ∈ closure M := + sorry +
src/C07_Topology/S02_Metric_Spaces.lean (new)
- @@ -0,0 +1,199 @@
 + import Mathlib.Topology.Instances.Real + import Mathlib.Analysis.NormedSpace.BanachSteinhaus + + open Set Filter + open Topology Filter + + variable {X : Type _} [MetricSpace X] (a b c : X) + + #check (dist a b : ℝ) + + #check (dist_nonneg : 0 ≤ dist a b) + + #check (dist_eq_zero : dist a b = 0 ↔ a = b) + + #check (dist_comm a b : dist a b = dist b a) + + #check (dist_triangle a b c : dist a c ≤ dist a b + dist b c) + + -- Note the next three lines are not quoted, their purpose is to make sure those things don't get renamed while we're looking elsewhere. + #check EMetricSpace + + #check PseudoMetricSpace + + #check PseudoEMetricSpace + + example {u : ℕ → X} {a : X} : Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := + Metric.tendsto_atTop + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} : + Continuous f ↔ ∀ x : X, ∀ ε > 0, ∃ δ > 0, ∀ x', dist x' x < δ → dist (f x') (f x) < ε := + Metric.continuous_iff + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := by continuity + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := + continuous_dist.comp ((hf.comp continuous_fst).prod_mk (hf.comp continuous_snd)) + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := by + apply Continuous.dist + exact hf.comp continuous_fst + exact hf.comp continuous_snd + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := + (hf.comp continuous_fst).dist (hf.comp continuous_snd) + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := + hf.fst'.dist hf.snd' + + example {f : ℝ → X} (hf : Continuous f) : Continuous fun x : ℝ => f (x ^ 2 + x) := + sorry + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] (f : X → Y) (a : X) : + ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x}, dist x a < δ → dist (f x) (f a) < ε := + Metric.continuousAt_iff + + variable (r : ℝ) + + example : Metric.ball a r = { b | dist b a < r } := + rfl + + example : Metric.closedBall a r = { b | dist b a ≤ r } := + rfl + + example (hr : 0 < r) : a ∈ Metric.ball a r := + Metric.mem_ball_self hr + + example (hr : 0 ≤ r) : a ∈ Metric.closedBall a r := + Metric.mem_closedBall_self hr + + example (s : Set X) : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s := + Metric.isOpen_iff + + example {s : Set X} : IsClosed s ↔ IsOpen (sᶜ) := + isOpen_compl_iff.symm + + example {s : Set X} (hs : IsClosed s) {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) + (hus : ∀ n, u n ∈ s) : a ∈ s := + hs.mem_of_tendsto hu (eventually_of_forall hus) + + example {s : Set X} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, a ∈ Metric.ball b ε := + Metric.mem_closure_iff + + example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := + sorry + + example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.ball x ε ⊆ s := + Metric.nhds_basis_ball.mem_iff + + example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.closedBall x ε ⊆ s := + Metric.nhds_basis_closedBall.mem_iff + + example : IsCompact (Set.Icc 0 1 : Set ℝ) := + isCompact_Icc + + example {s : Set X} (hs : IsCompact s) {u : ℕ → X} (hu : ∀ n, u n ∈ s) : + ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := + hs.tendsto_subseq hu + + example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : + ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y := + hs.exists_forall_le hs' hfs + + example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : + ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x := + hs.exists_forall_ge hs' hfs + + example {s : Set X} (hs : IsCompact s) : IsClosed s := + hs.isClosed + + example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := + isCompact_univ + + #check IsCompact.isClosed + + example {X : Type _} [MetricSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} : + UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : X}, dist a b < δ → dist (f a) (f b) < ε := + Metric.uniformContinuous_iff + + example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} + (hf : Continuous f) : UniformContinuous f := + sorry + + example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε := + Metric.cauchySeq_iff + + example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, dist (u n) (u N) < ε := + Metric.cauchySeq_iff' + + example [CompleteSpace X] (u : ℕ → X) (hu : CauchySeq u) : ∃ x, Tendsto u atTop (𝓝 x) := + cauchySeq_tendsto_of_complete hu + + open BigOperators + + open Finset + + theorem cauchySeq_of_le_geometric_two' {u : ℕ → X} + (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by + rw [Metric.cauchySeq_iff'] + intro ε ε_pos + obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by sorry + use N + intro n hn + obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn + calc + dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := sorry + _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := sorry + _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := sorry + _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2) ^ i := sorry + _ ≤ 1 / 2 ^ N * 2 := sorry + _ < ε := sorry + + + open Metric + + example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) : + Dense (⋂ n, f n) := by + let B : ℕ → ℝ := fun n => (1 / 2) ^ n + have Bpos : ∀ n, 0 < B n + sorry + /- Translate the density assumption into two functions `center` and `radius` associating + to any n, x, δ, δpos a center and a positive radius such that + `closedBall center radius` is included both in `f n` and in `closedBall x δ`. + We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ + have : + ∀ (n : ℕ) (x : X), + ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := + by sorry + choose! center radius Hpos HB Hball using this + intro x + rw [mem_closure_iff_nhds_basis nhds_basis_closedBall] + intro ε εpos + /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all + `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball + `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that + `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a + limit which belongs to all the `f n`. -/ + let F : ℕ → X × ℝ := fun n => + Nat.recOn n (Prod.mk x (min ε (B 0))) fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2) + let c : ℕ → X := fun n => (F n).1 + let r : ℕ → ℝ := fun n => (F n).2 + have rpos : ∀ n, 0 < r n := by sorry + have rB : ∀ n, r n ≤ B n := by sorry + have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := by sorry + have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by sorry + have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist + -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. + rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩ + -- this point `y` will be the desired point. We will check that it belongs to all + -- `f n` and to `ball x ε`. + use y + have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry + have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry + sorry +
src/C07_Topology/S03_Topological_Spaces.lean (new)
- @@ -0,0 +1,150 @@
 + import Mathlib.Topology.Instances.Real + import Mathlib.Analysis.NormedSpace.BanachSteinhaus + + open Set Filter + + open Topology Filter + + section + variable {X : Type _} [TopologicalSpace X] + + example : IsOpen (univ : Set X) := + isOpen_univ + + example : IsOpen (∅ : Set X) := + isOpen_empty + + example {ι : Type _} {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋃ i, s i) := + isOpen_iUnion hs + + example {ι : Type _} [Fintype ι] {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋂ i, s i) := + isOpen_iInter hs + + variable {Y : Type _} [TopologicalSpace Y] + + example {f : X → Y} : Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) := + continuous_def + + example {f : X → Y} {x : X} : ContinuousAt f x ↔ map f (𝓝 x) ≤ 𝓝 (f x) := + Iff.rfl + + example {f : X → Y} {x : X} : ContinuousAt f x ↔ ∀ U ∈ 𝓝 (f x), ∀ᶠ x in 𝓝 x, f x ∈ U := + Iff.rfl + + example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := + mem_nhds_iff + + example (x : X) : pure x ≤ 𝓝 x := + pure_le_nhds x + + example (x : X) (P : X → Prop) (h : ∀ᶠ y in 𝓝 x, P y) : P x := + pure_le_nhds x h + + example {P : X → Prop} {x : X} (h : ∀ᶠ y in 𝓝 x, P y) : ∀ᶠ y in 𝓝 x, ∀ᶠ z in 𝓝 y, P z := + eventually_eventually_nhds.mpr h + + #check TopologicalSpace.mkOfNhds + + #check TopologicalSpace.nhds_mkOfNhds + + example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) + (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → ∀ᶠ y in n a, ∀ᶠ x in n y, p x) : + ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := + sorry + + end + + -- BOTH. + variable {X Y : Type _} + + example (f : X → Y) : TopologicalSpace X → TopologicalSpace Y := + TopologicalSpace.coinduced f + + example (f : X → Y) : TopologicalSpace Y → TopologicalSpace X := + TopologicalSpace.induced f + + example (f : X → Y) (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) : + TopologicalSpace.coinduced f T_X ≤ T_Y ↔ T_X ≤ TopologicalSpace.induced f T_Y := + coinduced_le_iff_le_induced + + #check coinduced_compose + + #check induced_compose + + example {T T' : TopologicalSpace X} : T ≤ T' ↔ ∀ s, T'.IsOpen s → T.IsOpen s := + Iff.rfl + + example (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) (f : X → Y) : + Continuous f ↔ TopologicalSpace.coinduced f T_X ≤ T_Y := + continuous_iff_coinduced_le + + example {Z : Type _} (f : X → Y) (T_X : TopologicalSpace X) (T_Z : TopologicalSpace Z) (g : Y → Z) : + @Continuous Y Z (TopologicalSpace.coinduced f T_X) T_Z g ↔ @Continuous X Z T_X T_Z (g ∘ f) := by + rw [continuous_iff_coinduced_le, coinduced_compose, continuous_iff_coinduced_le] + + example (ι : Type _) (X : ι → Type _) (T_X : ∀ i, TopologicalSpace <| X i) : + (Pi.topologicalSpace : TopologicalSpace (∀ i, X i)) = + ⨅ i, TopologicalSpace.induced (fun x => x i) (T_X i) := + rfl + + example [TopologicalSpace X] [T2Space X] {u : ℕ → X} {a b : X} (ha : Tendsto u atTop (𝓝 a)) + (hb : Tendsto u atTop (𝓝 b)) : a = b := + tendsto_nhds_unique ha hb + + example [TopologicalSpace X] [RegularSpace X] (a : X) : + (𝓝 a).HasBasis (fun s : Set X => s ∈ 𝓝 a ∧ IsClosed s) id := + closed_nhds_basis a + + example [TopologicalSpace X] {x : X} : (𝓝 x).HasBasis (fun t : Set X => t ∈ 𝓝 x ∧ IsOpen t) id := + nhds_basis_opens' x + + theorem aux {X Y A : Type _} [TopologicalSpace X] {c : A → X} {f : A → Y} {x : X} {F : Filter Y} + (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : + ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := + sorry + + example [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace Y] {A : Set X} + (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f) + (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) <| 𝓝 x) <| 𝓝 c) : + ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := + sorry + + #check @HasBasis.tendsto_right_iff + + example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] {s : Set X} {a : X} : + a ∈ closure s ↔ ∃ u : ℕ → X, (∀ n, u n ∈ s) ∧ Tendsto u atTop (𝓝 a) := + mem_closure_iff_seq_limit + + variable [TopologicalSpace X] + + example {F : Filter X} {x : X} : ClusterPt x F ↔ NeBot (𝓝 x ⊓ F) := + Iff.rfl + + example {s : Set X} : IsCompact s ↔ ∀ (F : Filter X) [NeBot F], F ≤ 𝓟 s → ∃ a ∈ s, ClusterPt a F := + Iff.rfl + + example [TopologicalSpace.FirstCountableTopology X] {s : Set X} {u : ℕ → X} (hs : IsCompact s) + (hu : ∀ n, u n ∈ s) : ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := + hs.tendsto_subseq hu + + variable [TopologicalSpace Y] + + example {x : X} {F : Filter X} {G : Filter Y} (H : ClusterPt x F) {f : X → Y} + (hfx : ContinuousAt f x) (hf : Tendsto f F G) : ClusterPt (f x) G := + ClusterPt.map H hfx hf + + example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) : + IsCompact (f '' s) := by + intro F F_ne F_le + have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by sorry + have Hne : (𝓟 s ⊓ comap f F).NeBot := by sorry + have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left + sorry + + example {ι : Type _} {s : Set X} (hs : IsCompact s) (U : ι → Set X) (hUo : ∀ i, IsOpen (U i)) + (hsU : s ⊆ ⋃ i, U i) : ∃ t : Finset ι, s ⊆ ⋃ i ∈ t, U i := + hs.elim_finite_subcover U hUo hsU + + example [CompactSpace X] : IsCompact (univ : Set X) := + isCompact_univ +
src/C07_Topology/solutions/Solutions_S01_Filters.lean (new)
- @@ -0,0 +1,71 @@
 + import Mathlib.Topology.Instances.Real + + open Set Filter Topology + + -- In the next example we could use `tauto` in each proof instead of knowing the lemmas + example {α : Type _} (s : Set α) : Filter α := + { sets := { t | s ⊆ t } + univ_sets := subset_univ s + sets_of_superset := fun hU hUV => Subset.trans hU hUV + inter_sets := fun hU hV => subset_inter hU hV } + + example : Filter ℕ := + { sets := { s | ∃ a, ∀ b, a ≤ b → b ∈ s } + univ_sets := by + use 42 + simp + sets_of_superset := by + rintro U V ⟨N, hN⟩ hUV + use N + tauto + inter_sets := by + rintro U V ⟨N, hN⟩ ⟨N', hN'⟩ + use max N N' + intro b hb + rw [max_le_iff] at hb + constructor <;> tauto } + + def Tendsto₁ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := + ∀ V ∈ G, f ⁻¹' V ∈ F + + example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} + (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := + calc + map (g ∘ f) F = map g (map f F) := by rw [map_map] + _ ≤ map g G := (map_mono hf) + _ ≤ H := hg + + + example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} + (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := by + intro V hV + rw [preimage_comp] + apply hf + apply hg + exact hV + + example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : + Tendsto f atTop (𝓝 (x₀, y₀)) ↔ + Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := + calc + Tendsto f atTop (𝓝 (x₀, y₀)) ↔ map f atTop ≤ 𝓝 (x₀, y₀) := Iff.rfl + _ ↔ map f atTop ≤ 𝓝 x₀ ×ᶠ 𝓝 y₀ := by rw [nhds_prod_eq] + _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ⊓ comap Prod.snd (𝓝 y₀) := Iff.rfl + _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ∧ map f atTop ≤ comap Prod.snd (𝓝 y₀) := le_inf_iff + _ ↔ map Prod.fst (map f atTop) ≤ 𝓝 x₀ ∧ map Prod.snd (map f atTop) ≤ 𝓝 y₀ := by + rw [← map_le_iff_le_comap, ← map_le_iff_le_comap] + _ ↔ map (Prod.fst ∘ f) atTop ≤ 𝓝 x₀ ∧ map (Prod.snd ∘ f) atTop ≤ 𝓝 y₀ := by + rw [map_map, map_map] + + + -- an alternative solution + example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : + Tendsto f atTop (𝓝 (x₀, y₀)) ↔ + Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := by + rw [nhds_prod_eq] + unfold Tendsto Filter.prod + rw [le_inf_iff, ← map_le_iff_le_comap, map_map, ← map_le_iff_le_comap, map_map] + + example (u : ℕ → ℝ) (M : Set ℝ) (x : ℝ) (hux : Tendsto u atTop (𝓝 x)) + (huM : ∀ᶠ n in atTop, u n ∈ M) : x ∈ closure M := + mem_closure_iff_clusterPt.mpr (neBot_of_le <| le_inf hux <| le_principal_iff.mpr huM)
src/C07_Topology/solutions/Solutions_S02_Metric_Spaces.lean (new)
- @@ -0,0 +1,364 @@
 + import Mathlib.Topology.Instances.Real + import Mathlib.Analysis.NormedSpace.BanachSteinhaus + + open Set Filter + open Topology Filter + + variable {X : Type _} [MetricSpace X] (a b c : X) + + #check (dist a b : ℝ) + + #check (dist_nonneg : 0 ≤ dist a b) + + #check (dist_eq_zero : dist a b = 0 ↔ a = b) + + #check (dist_comm a b : dist a b = dist b a) + + #check (dist_triangle a b c : dist a c ≤ dist a b + dist b c) + + -- Note the next three lines are not quoted, their purpose is to make sure those things don't get renamed while we're looking elsewhere. + #check EMetricSpace + + #check PseudoMetricSpace + + #check PseudoEMetricSpace + + example {u : ℕ → X} {a : X} : Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := + Metric.tendsto_atTop + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} : + Continuous f ↔ ∀ x : X, ∀ ε > 0, ∃ δ > 0, ∀ x', dist x' x < δ → dist (f x') (f x) < ε := + Metric.continuous_iff + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := by continuity + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := + continuous_dist.comp ((hf.comp continuous_fst).prod_mk (hf.comp continuous_snd)) + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := by + apply Continuous.dist + exact hf.comp continuous_fst + exact hf.comp continuous_snd + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := + (hf.comp continuous_fst).dist (hf.comp continuous_snd) + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := + hf.fst'.dist hf.snd' + + example {f : ℝ → X} (hf : Continuous f) : Continuous fun x : ℝ => f (x ^ 2 + x) := + sorry + + example {f : ℝ → X} (hf : Continuous f) : Continuous fun x : ℝ => f (x ^ 2 + x) := + hf.comp <| (continuous_pow 2).add continuous_id + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] (f : X → Y) (a : X) : + ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x}, dist x a < δ → dist (f x) (f a) < ε := + Metric.continuousAt_iff + + variable (r : ℝ) + + example : Metric.ball a r = { b | dist b a < r } := + rfl + + example : Metric.closedBall a r = { b | dist b a ≤ r } := + rfl + + example (hr : 0 < r) : a ∈ Metric.ball a r := + Metric.mem_ball_self hr + + example (hr : 0 ≤ r) : a ∈ Metric.closedBall a r := + Metric.mem_closedBall_self hr + + example (s : Set X) : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s := + Metric.isOpen_iff + + example {s : Set X} : IsClosed s ↔ IsOpen (sᶜ) := + isOpen_compl_iff.symm + + example {s : Set X} (hs : IsClosed s) {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) + (hus : ∀ n, u n ∈ s) : a ∈ s := + hs.mem_of_tendsto hu (eventually_of_forall hus) + + example {s : Set X} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, a ∈ Metric.ball b ε := + Metric.mem_closure_iff + + example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := + sorry + + example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := by + rw [Metric.tendsto_atTop] at hu + rw [Metric.mem_closure_iff] + intro ε ε_pos + rcases hu ε ε_pos with ⟨N, hN⟩ + refine' ⟨u N, hs _, _⟩ + rw [dist_comm] + exact hN N le_rfl + + example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.ball x ε ⊆ s := + Metric.nhds_basis_ball.mem_iff + + example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.closedBall x ε ⊆ s := + Metric.nhds_basis_closedBall.mem_iff + + example : IsCompact (Set.Icc 0 1 : Set ℝ) := + isCompact_Icc + + example {s : Set X} (hs : IsCompact s) {u : ℕ → X} (hu : ∀ n, u n ∈ s) : + ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := + hs.tendsto_subseq hu + + example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : + ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y := + hs.exists_forall_le hs' hfs + + example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : + ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x := + hs.exists_forall_ge hs' hfs + + example {s : Set X} (hs : IsCompact s) : IsClosed s := + hs.isClosed + + example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := + isCompact_univ + + #check IsCompact.isClosed + + example {X : Type _} [MetricSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} : + UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : X}, dist a b < δ → dist (f a) (f b) < ε := + Metric.uniformContinuous_iff + + example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} + (hf : Continuous f) : UniformContinuous f := + sorry + + example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} + (hf : Continuous f) : UniformContinuous f := by + rw [Metric.uniformContinuous_iff] + intro ε ε_pos + let φ : X × X → ℝ := fun p => dist (f p.1) (f p.2) + have φ_cont : Continuous φ := hf.fst'.dist hf.snd' + let K := { p : X × X | ε ≤ φ p } + have K_closed : IsClosed K := isClosed_le continuous_const φ_cont + have K_cpct : IsCompact K := K_closed.isCompact + cases' eq_empty_or_nonempty K with hK hK + · use 1, by norm_num + intro x y _ + have : (x, y) ∉ K := by simp [hK] + simpa using this + · rcases K_cpct.exists_forall_le hK continuous_dist.continuousOn with ⟨⟨x₀, x₁⟩, xx_in, H⟩ + use dist x₀ x₁ + constructor + · change _ < _ + rw [dist_pos] + intro h + have : ε ≤ 0 := by simpa [*] using xx_in + linarith + · intro x x' + contrapose! + intro hxx' + exact H (x, x') hxx' + + example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε := + Metric.cauchySeq_iff + + example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, dist (u n) (u N) < ε := + Metric.cauchySeq_iff' + + example [CompleteSpace X] (u : ℕ → X) (hu : CauchySeq u) : ∃ x, Tendsto u atTop (𝓝 x) := + cauchySeq_tendsto_of_complete hu + + open BigOperators + + open Finset + + theorem cauchySeq_of_le_geometric_two' {u : ℕ → X} + (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by + rw [Metric.cauchySeq_iff'] + intro ε ε_pos + obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by sorry + use N + intro n hn + obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn + calc + dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := sorry + _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := sorry + _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := sorry + _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2) ^ i := sorry + _ ≤ 1 / 2 ^ N * 2 := sorry + _ < ε := sorry + + + example {u : ℕ → X} (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by + rw [Metric.cauchySeq_iff'] + intro ε ε_pos + obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by + have : Tendsto (fun N : ℕ => (1 / 2 ^ N * 2 : ℝ)) atTop (𝓝 0) := by + rw [← MulZeroClass.zero_mul (2 : ℝ)] + apply Tendsto.mul + simp_rw [← one_div_pow (2 : ℝ)] + apply tendsto_pow_atTop_nhds_0_of_lt_1 <;> linarith + exact tendsto_const_nhds + rcases(atTop_basis.tendsto_iff (nhds_basis_Ioo_pos (0 : ℝ))).mp this ε ε_pos with ⟨N, _, hN⟩ + exact ⟨N, by simpa using (hN N left_mem_Ici).2⟩ + use N + intro n hn + obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn + calc + dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := by rw [dist_comm, add_zero] + _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := + (dist_le_range_sum_dist (fun i => u (N + i)) k) + _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := (sum_le_sum fun i _ => hu <| N + i) + _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2 : ℝ) ^ i := by simp_rw [← one_div_pow, pow_add, ← mul_sum] + _ ≤ 1 / 2 ^ N * 2 := + (mul_le_mul_of_nonneg_left (sum_geometric_two_le _) + (one_div_nonneg.mpr (pow_nonneg (zero_le_two : (0 : ℝ) ≤ 2) _))) + _ < ε := hN + + + open Metric + + example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) : + Dense (⋂ n, f n) := by + let B : ℕ → ℝ := fun n => (1 / 2) ^ n + have Bpos : ∀ n, 0 < B n + sorry + /- Translate the density assumption into two functions `center` and `radius` associating + to any n, x, δ, δpos a center and a positive radius such that + `closedBall center radius` is included both in `f n` and in `closedBall x δ`. + We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ + have : + ∀ (n : ℕ) (x : X), + ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := + by sorry + choose! center radius Hpos HB Hball using this + intro x + rw [mem_closure_iff_nhds_basis nhds_basis_closedBall] + intro ε εpos + /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all + `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball + `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that + `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a + limit which belongs to all the `f n`. -/ + let F : ℕ → X × ℝ := fun n => + Nat.recOn n (Prod.mk x (min ε (B 0))) fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2) + let c : ℕ → X := fun n => (F n).1 + let r : ℕ → ℝ := fun n => (F n).2 + have rpos : ∀ n, 0 < r n := by sorry + have rB : ∀ n, r n ≤ B n := by sorry + have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := by sorry + have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by sorry + have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist + -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. + rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩ + -- this point `y` will be the desired point. We will check that it belongs to all + -- `f n` and to `ball x ε`. + use y + have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry + have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry + sorry + + example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) : + Dense (⋂ n, f n) := by + let B : ℕ → ℝ := fun n => (1 / 2) ^ n + have Bpos : ∀ n, 0 pow_pos sorry n + /- Translate the density assumption into two functions `center` and `radius` associating + to any n, x, δ, δpos a center and a positive radius such that + `closedBall center radius` is included both in `f n` and in `closedBall x δ`. + We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ + have : + ∀ (n : ℕ) (x : X), + ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := by + intro n x δ δpos + have : x ∈ closure (f n) := hd n x + rcases Metric.mem_closure_iff.1 this (δ / 2) (half_pos δpos) with ⟨y, ys, xy⟩ + rw [dist_comm] at xy + obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closedBall y r ⊆ f n := + nhds_basis_closedBall.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys) + refine' ⟨y, min (min (δ / 2) r) (B (n + 1)), _, _, fun z hz => ⟨_, _⟩⟩ + show 0 < min (min (δ / 2) r) (B (n + 1)) + exact lt_min (lt_min (half_pos δpos) rpos) (Bpos (n + 1)) + show min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1) + exact min_le_right _ _ + show z ∈ closedBall x δ + exact + calc + dist z x ≤ dist z y + dist y x := dist_triangle _ _ _ + _ ≤ min (min (δ / 2) r) (B (n + 1)) + δ / 2 := (add_le_add hz xy.le) + _ ≤ δ / 2 + δ / 2 := (add_le_add_right ((min_le_left _ _).trans (min_le_left _ _)) _) + _ = δ := add_halves δ + + show z ∈ f n + exact + hr + (calc + dist z y ≤ min (min (δ / 2) r) (B (n + 1)) := hz + _ ≤ r := (min_le_left _ _).trans (min_le_right _ _) + ) + choose! center radius Hpos HB Hball using this + refine' fun x => (mem_closure_iff_nhds_basis nhds_basis_closedBall).2 fun ε εpos => _ + /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all + `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball + `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that + `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a + limit which belongs to all the `f n`. -/ + let F : ℕ → X × ℝ := fun n => + Nat.recOn n (Prod.mk x (min ε (B 0))) fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2) + let c : ℕ → X := fun n => (F n).1 + let r : ℕ → ℝ := fun n => (F n).2 + have rpos : ∀ n, 0 < r n := by + intro n + induction' n with n hn + exact lt_min εpos (Bpos 0) + exact Hpos n (c n) (r n) hn + have rB : ∀ n, r n ≤ B n := by + intro n + induction' n with n hn + exact min_le_right _ _ + exact HB n (c n) (r n) (rpos n) + have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := fun n => + Hball n (c n) (r n) (rpos n) + have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by + intro n + rw [dist_comm] + have A : c (n + 1) ∈ closedBall (c (n + 1)) (r (n + 1)) := + mem_closedBall_self (rpos <| n + 1).le + have I := + calc + closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) := + (incl n).trans (inter_subset_left _ _) + _ ⊆ closedBall (c n) (B n) := closedBall_subset_closedBall (rB n) + + exact I A + have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist + -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. + rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩ + -- this point `y` will be the desired point. We will check that it belongs to all + -- `f n` and to `ball x ε`. + use y + have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by + intro n + refine' Nat.le_induction _ fun m hnm h => _ + · exact Subset.rfl + · exact (incl m).trans ((Set.inter_subset_left _ _).trans h) + have yball : ∀ n, y ∈ closedBall (c n) (r n) := by + intro n + refine' isClosed_ball.mem_of_tendsto ylim _ + refine' (Filter.eventually_ge_atTop n).mono fun m hm => _ + exact I n m hm (mem_closedBall_self (rpos _).le) + constructor + · suffices ∀ n, y ∈ f n by rwa [Set.mem_iInter] + intro n + have : closedBall (c (n + 1)) (r (n + 1)) ⊆ f n := + Subset.trans (incl n) (inter_subset_right _ _) + exact this (yball (n + 1)) + calc + dist y x ≤ r 0 := yball 0 + _ ≤ ε := min_le_left _ _ + +
src/C07_Topology/solutions/Solutions_S03_Topological_Spaces.lean (new)
- @@ -0,0 +1,201 @@
 + import Mathlib.Topology.Instances.Real + import Mathlib.Analysis.NormedSpace.BanachSteinhaus + + open Set Filter + + open Topology Filter + + section + variable {X : Type _} [TopologicalSpace X] + + example : IsOpen (univ : Set X) := + isOpen_univ + + example : IsOpen (∅ : Set X) := + isOpen_empty + + example {ι : Type _} {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋃ i, s i) := + isOpen_iUnion hs + + example {ι : Type _} [Fintype ι] {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋂ i, s i) := + isOpen_iInter hs + + variable {Y : Type _} [TopologicalSpace Y] + + example {f : X → Y} : Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) := + continuous_def + + example {f : X → Y} {x : X} : ContinuousAt f x ↔ map f (𝓝 x) ≤ 𝓝 (f x) := + Iff.rfl + + example {f : X → Y} {x : X} : ContinuousAt f x ↔ ∀ U ∈ 𝓝 (f x), ∀ᶠ x in 𝓝 x, f x ∈ U := + Iff.rfl + + example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := + mem_nhds_iff + + example (x : X) : pure x ≤ 𝓝 x := + pure_le_nhds x + + example (x : X) (P : X → Prop) (h : ∀ᶠ y in 𝓝 x, P y) : P x := + pure_le_nhds x h + + example {P : X → Prop} {x : X} (h : ∀ᶠ y in 𝓝 x, P y) : ∀ᶠ y in 𝓝 x, ∀ᶠ z in 𝓝 y, P z := + eventually_eventually_nhds.mpr h + + #check TopologicalSpace.mkOfNhds + + #check TopologicalSpace.nhds_mkOfNhds + + example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) + (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → ∀ᶠ y in n a, ∀ᶠ x in n y, p x) : + ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := + sorry + + example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) + (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → ∀ᶠ y in n a, ∀ᶠ x in n y, p x) : + ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := by + intro a s s_in + refine' ⟨{ y | s ∈ n y }, H a (fun x => x ∈ s) s_in, _, by tauto⟩ + rintro y (hy : s ∈ n y) + exact H₀ y hy + + end + + -- BOTH. + variable {X Y : Type _} + + example (f : X → Y) : TopologicalSpace X → TopologicalSpace Y := + TopologicalSpace.coinduced f + + example (f : X → Y) : TopologicalSpace Y → TopologicalSpace X := + TopologicalSpace.induced f + + example (f : X → Y) (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) : + TopologicalSpace.coinduced f T_X ≤ T_Y ↔ T_X ≤ TopologicalSpace.induced f T_Y := + coinduced_le_iff_le_induced + + #check coinduced_compose + + #check induced_compose + + example {T T' : TopologicalSpace X} : T ≤ T' ↔ ∀ s, T'.IsOpen s → T.IsOpen s := + Iff.rfl + + example (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) (f : X → Y) : + Continuous f ↔ TopologicalSpace.coinduced f T_X ≤ T_Y := + continuous_iff_coinduced_le + + example {Z : Type _} (f : X → Y) (T_X : TopologicalSpace X) (T_Z : TopologicalSpace Z) (g : Y → Z) : + @Continuous Y Z (TopologicalSpace.coinduced f T_X) T_Z g ↔ @Continuous X Z T_X T_Z (g ∘ f) := by + rw [continuous_iff_coinduced_le, coinduced_compose, continuous_iff_coinduced_le] + + example (ι : Type _) (X : ι → Type _) (T_X : ∀ i, TopologicalSpace <| X i) : + (Pi.topologicalSpace : TopologicalSpace (∀ i, X i)) = + ⨅ i, TopologicalSpace.induced (fun x => x i) (T_X i) := + rfl + + example [TopologicalSpace X] [T2Space X] {u : ℕ → X} {a b : X} (ha : Tendsto u atTop (𝓝 a)) + (hb : Tendsto u atTop (𝓝 b)) : a = b := + tendsto_nhds_unique ha hb + + example [TopologicalSpace X] [RegularSpace X] (a : X) : + (𝓝 a).HasBasis (fun s : Set X => s ∈ 𝓝 a ∧ IsClosed s) id := + closed_nhds_basis a + + example [TopologicalSpace X] {x : X} : (𝓝 x).HasBasis (fun t : Set X => t ∈ 𝓝 x ∧ IsOpen t) id := + nhds_basis_opens' x + + theorem aux {X Y A : Type _} [TopologicalSpace X] {c : A → X} {f : A → Y} {x : X} {F : Filter Y} + (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : + ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := + sorry + + example {X Y A : Type _} [TopologicalSpace X] {c : A → X} {f : A → Y} {x : X} {F : Filter Y} + (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : + ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := by + simpa [and_assoc] using ((nhds_basis_opens' x).comap c).tendsto_left_iff.mp h V' V'_in + + example [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace Y] {A : Set X} + (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f) + (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) <| 𝓝 x) <| 𝓝 c) : + ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := + sorry + + #check @HasBasis.tendsto_right_iff + + example [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {A : Set X} (hA : ∀ x, x ∈ closure A) + {f : A → Y} (f_cont : Continuous f) + (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) <| 𝓝 x) <| 𝓝 c) : + ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := by + choose φ hφ using hf + use φ + constructor + · rw [continuous_iff_continuousAt] + intro x + suffices ∀ V' ∈ 𝓝 (φ x), IsClosed V' → φ ⁻¹' V' ∈ 𝓝 x by + simpa [ContinuousAt, (closed_nhds_basis (φ x)).tendsto_right_iff] + intro V' V'_in V'_closed + obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, IsOpen V ∧ (↑) ⁻¹' V ⊆ f ⁻¹' V' := aux (hφ x) V'_in + suffices : ∀ y ∈ V, φ y ∈ V' + exact mem_of_superset V_in this + intro y y_in + have hVx : V ∈ 𝓝 y := V_op.mem_nhds y_in + haveI : (comap ((↑) : A → X) (𝓝 y)).NeBot := by simpa [mem_closure_iff_comap_neBot] using hA y + apply V'_closed.mem_of_tendsto (hφ y) + exact mem_of_superset (preimage_mem_comap hVx) hV + · intro a + have lim : Tendsto f (𝓝 a) (𝓝 <| φ a) := by simpa [nhds_induced] using hφ a + exact tendsto_nhds_unique lim f_cont.continuousAt + + example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] {s : Set X} {a : X} : + a ∈ closure s ↔ ∃ u : ℕ → X, (∀ n, u n ∈ s) ∧ Tendsto u atTop (𝓝 a) := + mem_closure_iff_seq_limit + + variable [TopologicalSpace X] + + example {F : Filter X} {x : X} : ClusterPt x F ↔ NeBot (𝓝 x ⊓ F) := + Iff.rfl + + example {s : Set X} : IsCompact s ↔ ∀ (F : Filter X) [NeBot F], F ≤ 𝓟 s → ∃ a ∈ s, ClusterPt a F := + Iff.rfl + + example [TopologicalSpace.FirstCountableTopology X] {s : Set X} {u : ℕ → X} (hs : IsCompact s) + (hu : ∀ n, u n ∈ s) : ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := + hs.tendsto_subseq hu + + variable [TopologicalSpace Y] + + example {x : X} {F : Filter X} {G : Filter Y} (H : ClusterPt x F) {f : X → Y} + (hfx : ContinuousAt f x) (hf : Tendsto f F G) : ClusterPt (f x) G := + ClusterPt.map H hfx hf + + example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) : + IsCompact (f '' s) := by + intro F F_ne F_le + have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by sorry + have Hne : (𝓟 s ⊓ comap f F).NeBot := by sorry + have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left + sorry + + example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) : + IsCompact (f '' s) := by + intro F F_ne F_le + have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by rw [Filter.push_pull, map_principal] + have Hne : (𝓟 s ⊓ comap f F).NeBot := by + apply NeBot.of_map + rwa [map_eq, inf_of_le_right F_le] + have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left + rcases hs Hle with ⟨x, x_in, hx⟩ + refine' ⟨f x, mem_image_of_mem f x_in, _⟩ + apply hx.map hf.continuousAt + rw [Tendsto, map_eq] + exact inf_le_right + + example {ι : Type _} {s : Set X} (hs : IsCompact s) (U : ι → Set X) (hUo : ∀ i, IsOpen (U i)) + (hsU : s ⊆ ⋃ i, U i) : ∃ t : Finset ι, s ⊆ ⋃ i ∈ t, U i := + hs.elim_finite_subcover U hUo hsU + + example [CompactSpace X] : IsCompact (univ : Set X) := + isCompact_univ +