01bfbf50 - mathematics_in

Authored by: Jeremy Avigad
Authored at: 2023-06-07 07:25:39 -0700
Committed by: Jeremy Avigad
Committed at: 2023-06-07 07:25:39 -0700
Object name: 01bfbf50720017954f0748a6e003cab9e55f4046
Parent: 9c87a7cb

Update Wed Jun 7 07:25:39 PDT 2023

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 + <!DOCTYPE html> + <html class="writer-html5" lang="en" > + <head> + <meta charset="utf-8" /><meta name="viewport" content="width=device-width, initial-scale=1" /> + + <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 src="_static/jquery.js"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> + <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> + <script src="_static/doctools.js"></script> + <script src="_static/sphinx_highlight.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" aria-label="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_Structures.html">6. Structures</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> + <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</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" aria-label="Home"></a></li> + <li class="breadcrumb-item active">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 <a class="reference external" href="https://github.com/leanprover/lean4/blob/master/doc/quickstart.md">instructions</a>.</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. 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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" aria-label="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_Structures.html">6. Structures</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> + <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</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" aria-label="Home"></a></li> + <li class="breadcrumb-item active">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 rewriting tactic <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. + 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 you are curious to see all available abreviations, you can hit Ctrl-Shift-p + and then type abbreviations to get access to the <code class="docutils literal notranslate">Lean 4: Show all abbreviations</code> command. + 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. Note that + the left-pointing arrow refers to going from right to left in the identity provided + by <code class="docutils literal notranslate">mul_assoc</code>, it has nothing to do with the left or right side of the 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, using the theorem <cite>sub_self</cite> for the second one: + <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> + Multiple rewrite commands can be carried out with a single command, + by listing the relevant identities separated by commas inside the 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. Lean then 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 all variables. + 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 underscores 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 as long as they follow + purely from the ring axioms, without using any local assumption. + <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">Mathlib.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 Mathlib.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 1, + so in the following example, <code class="docutils literal notranslate">nth_rewrite 2 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> + </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">rfl</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. + 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. 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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_Structures.html">6. Structures</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> + <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</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" aria-label="Home"></a></li> + <li class="breadcrumb-item active">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">FnUb f a</code> and <code class="docutils literal notranslate">FnLb 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">FnUb (fun x ↦ f x + g x) (a + b)</code> + forces Lean to unfold the definition of <code class="docutils literal notranslate">FnUb</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">FnUb</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">FnUb</code> and <code class="docutils literal notranslate">FnLb</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">fnUb_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 fnUb_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 <code class="docutils literal notranslate">fun x => f x + g x</code>. + <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 ubgb + use a + b + apply fnUb_add ubfa ubgb + </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 + 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, ubgb⟩ + exact ⟨a + b, fnUb_add ubfa ubgb⟩ + + example : FnHasUb f → FnHasUb g → FnHasUb fun x => f x + g x := by + rintro ⟨a, ubfa⟩ ⟨b, ubgb⟩ + exact ⟨a + b, fnUb_add ubfa ubgb⟩ + </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, ubgb⟩ => ⟨a + b, fnUb_add ubfa ubgb⟩ + </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="C08_Topology.html#filters">Section 8.1</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. 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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_Structures.html">6. Structures</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. 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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">rfl</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. 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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_Structures.html">6. Structures</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> + <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</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" aria-label="Home"></a></li> + <li class="breadcrumb-item active">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 _root_.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_Structures.html" class="btn btn-neutral float-right" title="6. Structures" 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>
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Hierarchies" href="C07_Hierarchies.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" aria-label="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. Structures</a><ul> + <li class="toctree-l2"><a class="reference internal" href="#defining-structures">6.1. Defining 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_Hierarchies.html">7. Hierarchies</a></li> + <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</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" aria-label="Home"></a></li> + <li class="breadcrumb-item active">6. Structures</li> + <li class="wy-breadcrumbs-aside"> + <a href="_sources/C06_Structures.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="structures"> + <h1>6. Structures<a class="headerlink" href="#structures" 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="defining-structures"> + <h2>6.1. Defining structures<a class="headerlink" href="#defining-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 := + 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 : AddGroup₁ 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 H1 : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ + · ext <;> simp [Int.mod'_eq, mod_def, div_def, norm] <;> ring + have H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y + · calc + norm (x % y) * norm y = norm (x % y * conj y) := by simp only [norm_mul, norm_conj] + _ = abs (Int.mod' (x.re * y.re + x.im * y.im) (norm y)) ^ 2 + + abs (Int.mod' (-(x.re * y.im) + x.im * y.re) (norm y)) ^ 2 := by simp [H1, norm, sq_abs] + _ ≤ (y.norm / 2) ^ 2 + (y.norm / 2) ^ 2 := by gcongr <;> apply Int.abs_mod'_le _ _ norm_y_pos + _ = norm y / 2 * (norm y / 2 * 2) := by ring + _ ≤ norm y / 2 * norm y := by gcongr; apply Int.ediv_mul_le; norm_num + calc norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right H2 norm_y_pos + _ < norm y := by + 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. 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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_Structures.html">6. Structures</a></li> + <li class="toctree-l1 current"><a class="current reference internal" href="#">7. Hierarchies</a><ul> + <li class="toctree-l2"><a class="reference internal" href="#basics">7.1. Basics</a></li> + <li class="toctree-l2"><a class="reference internal" href="#morphisms">7.2. Morphisms</a></li> + <li class="toctree-l2"><a class="reference internal" href="#sub-objects">7.3. Sub-objects</a></li> + </ul> + </li> + <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</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" aria-label="Home"></a></li> + <li class="breadcrumb-item active">7. Hierarchies</li> + <li class="wy-breadcrumbs-aside"> + <a href="_sources/C07_Hierarchies.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="hierarchies"> + <h1>7. Hierarchies<a class="headerlink" href="#hierarchies" title="Permalink to this heading"></a></h1> + We have seen in <a class="reference internal" href="C06_Structures.html#structures">Chapter 6</a> how to define the class + of groups and build instances of this class, and then how to build an instance + of the commutative ring class. But of course there is a hierarchy here: a + commutative ring is in particular an additive group. In this chapter we + will study how to build such hierarchies. They appear in all branches + of mathematics but in this chapter the emphasis will be on algebraic examples. + It may seem premature to discuss how to build hierarchies before more discussions + about using existing hierarchies. But some understanding of the technology underlying + hierarchies is required to use them. So you should probably still read this chapter, + but without trying too hard to remember everything on your first read, then read + the following chapters and come back here for a second reading. + In this chapter, we will redefine (simpler versions of) many things that appear in Mathlib + so we will used indices to distinguish our version. For instance we will have <code class="docutils literal notranslate">Ring₁</code> + as our version of <code class="docutils literal notranslate">Ring</code>. Since we will gradually explain more powerful ways of formalizing + structures, those indices will sometimes grow beyond one. + <section id="basics"> + <h2>7.1. Basics<a class="headerlink" href="#basics" title="Permalink to this heading"></a></h2> + At the very bottom of all hierarchies in Lean, we find data-carrying + classes. The following class records that the given type <code class="docutils literal notranslate">α</code> is endowed with + a distinguished element called <code class="docutils literal notranslate">one</code>. At this stage, it has no property at all. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class One₁ (α : Type) where + /-- The element one -/ + one : α + </pre></div> + </div> + Since we’ll make a much heavier use of classes in this chapter, we need to understand some + more details about what the <code class="docutils literal notranslate">class</code> command is doing. + First, the <code class="docutils literal notranslate">class</code> command above defines a structure <code class="docutils literal notranslate">One₁</code> with parameter <code class="docutils literal notranslate">α : Type</code> and + a single field <code class="docutils literal notranslate">one</code>. It also mark this structure as a class so that arguments of type + <code class="docutils literal notranslate">One₁ α</code> for some type <code class="docutils literal notranslate">α</code> will be inferrable using the instance resolution procedure, + as long as they are marked as instance-implicit, ie appear between square brackets. + Those two effects could also have been achieved using the <code class="docutils literal notranslate">structure</code> command with <code class="docutils literal notranslate">class</code> + attribute, ie writing <code class="docutils literal notranslate">@[class] structure</code> instance of <code class="docutils literal notranslate">class</code>. But the class command also + ensures that <code class="docutils literal notranslate">One₁ α</code> appears as an instance-implicit argument in its own fields. Compare: + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check One₁.one -- One₁.one {α : Type} [self : One₁ α] : α + + @[class] structure One₂ (α : Type) where + /-- The element one -/ + one : α + + #check One₂.one + </pre></div> + </div> + In the second check, we can see that <code class="docutils literal notranslate">self : One₂ α</code> is an explicit argument. + Let us make sure the first version is indeed usable without any explicit argument. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (α : Type) [One₁ α] : α := One₁.one + </pre></div> + </div> + Remark: in the above example, the argument <code class="docutils literal notranslate">One₁ α</code> is marked as instance-implicit, + which is a bit silly since this affects only uses of the declaration and declaration created by + the <code class="docutils literal notranslate">example</code> command cannot be used. However it allows to avoid giving a name to that + argument and, more importantly, it starts installing the good habit of marking <code class="docutils literal notranslate">One₁ α</code> + arguments as instance-implicit. + Another remark is that all this will work only when Lean knows what is <code class="docutils literal notranslate">α</code>. In the above + example, leaving out the type ascription <code class="docutils literal notranslate">: α</code> would generate an error message like: + <code class="docutils literal notranslate">typeclass instance problem is stuck, it is often due to metavariables One₁ (?m.263 α)</code> + where <code class="docutils literal notranslate">?m.263 α</code> means “some type depending on <code class="docutils literal notranslate">α</code>” (and 263 is simply an auto-generated + index that would be useful to distinguish between several unknown things). Another way + to avoid this issue would be to use a type annotation, as in: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (α : Type) [One₁ α] := (One₁.one : α) + </pre></div> + </div> + You may have already encountered that issue when playing with limits of sequences + in <a class="reference internal" href="C03_Logic.html#sequences-and-convergence">Section 3.6</a> if you tried to state for instance that + <code class="docutils literal notranslate">0 < 1</code> without telling Lean whether you meant this inequality to be about natural numbers + or real numbers. + Our next task is to assign a notation to <code class="docutils literal notranslate">One₁.one</code>. This we don’t want collisions + with the builtin notation for <code class="docutils literal notranslate">1</code>, we will use <code class="docutils literal notranslate">𝟙</code>. This is achieved by the following + command where the first line tells Lean to use the documentation + of <code class="docutils literal notranslate">One₁.one</code> as documentation for the symbol <code class="docutils literal notranslate">𝟙</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[inherit_doc] + notation "𝟙" => One₁.one + + example {α : Type} [One₁ α] : α := 𝟙 + + example {α : Type} [One₁ α] : (𝟙 : α) = 𝟙 := rfl + </pre></div> + </div> + We now want a data-carrying class recording a binary operation. We don’t want to choose + between addition and multiplication for now so we’ll use diamond. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class Dia₁ (α : Type) where + dia : α → α → α + + infixl:70 " ⋄ " => Dia₁.dia + </pre></div> + </div> + As in the <code class="docutils literal notranslate">One₁</code> example, the operation has no property at all at this stage. Let us + now define the class of semigroup structures where the operation is denoted by <code class="docutils literal notranslate">⋄</code>. + For now, we define it by hand as a structure with two fields, a <code class="docutils literal notranslate">Dia₁</code> instance and some + <code class="docutils literal notranslate">Prop</code>-valued field <code class="docutils literal notranslate">dia_assoc</code> asserting associativity of <code class="docutils literal notranslate">⋄</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class Semigroup₁ (α : Type) where + toDia₁ : Dia₁ α + /-- Diamond is associative -/ + dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) + </pre></div> + </div> + Note that while stating <cite>dia_assoc</cite>, the previously defined field <cite>toDia₁</cite> is in the local + context hence can be used when Lean searches for an instance of <cite>Dia₁ α</cite> to make sense + of <cite>a ⋄ b</cite>. However this <cite>toDia₁</cite> field does not become part of the type class instances database. + Hence doing <code class="docutils literal notranslate">example {α : Type} [Semigroup₁ α] (a b : α) : α := a ⋄ b</code> would fail with + error message <code class="docutils literal notranslate">failed to synthesize instance Dia₁ α</code>. + We can fix this by adding the <code class="docutils literal notranslate">instance</code> attribute later. + <div class="highlight-lean notranslate"><div class="highlight"><pre>attribute [instance] Semigroup₁.toDia₁ + + example {α : Type} [Semigroup₁ α] (a b : α) : α := a ⋄ b + </pre></div> + </div> + Before building up, we need a more convenient way to extend structures than explicitly + writing fields like <cite>toDia₁</cite> and adding the instance attribute by hand. The <code class="docutils literal notranslate">class</code> + supports this using the <code class="docutils literal notranslate">extends</code> syntax as in: + <div class="highlight-lean notranslate"><div class="highlight"><pre>class Semigroup₂ (α : Type) extends Dia₁ α where + /-- Diamond is associative -/ + dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) + + example {α : Type} [Semigroup₂ α] (a b : α) : α := a ⋄ b + </pre></div> + </div> + Note this syntax is also available in the <code class="docutils literal notranslate">structure</code> command, although it that + case it fixes only the hurdle of writing fields such as <cite>toDia₁</cite> since there + is no instance to define in that case. + Let us now try to combine a diamond operation and a distinguished one with axioms saying + this element is neutral on both sides. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class DiaOneClass₁ (α : Type) extends One₁ α, Dia₁ α where + /-- One is a left neutral element for diamond. -/ + one_dia : ∀ a : α, 𝟙 ⋄ a = a + /-- One is a right neutral element for diamond -/ + dia_one : ∀ a : α, a ⋄ 𝟙 = a + </pre></div> + </div> + In the next example, we tell Lean that <code class="docutils literal notranslate">α</code> has a <code class="docutils literal notranslate">DiaOneClass₁</code> structure and state a + property that uses both a <cite>Dia₁</cite> instance and a <cite>One₁</cite> instance. In order to see how Lean finds + those instances we set a tracing option whose result can be seen in the info view. This result + is rather terse by default but can be expended by clicking one lines ending with black arrows. + It includes failed attempts where Lean tried to find instances before having enough type + information to succceed. The successful attempts do involve the instances generated by the + <code class="docutils literal notranslate">extends</code> syntax. + <div class="highlight-lean notranslate"><div class="highlight"><pre>set_option trace.Meta.synthInstance true in + example {α : Type} [DiaOneClass₁ α] (a b : α) : Prop := a ⋄ b = 𝟙 + </pre></div> + </div> + Note that we don’t need to include extra fields where combining existing classes. Hence we can + define monoids as: + <div class="highlight-lean notranslate"><div class="highlight"><pre>class Monoid₁ (α : Type) extends Semigroup₁ α, DiaOneClass₁ α + </pre></div> + </div> + While the above definition seems straightforward, it hides an important subtlety. Both + <code class="docutils literal notranslate">Semigroup₁ α</code> and <code class="docutils literal notranslate">DiaOneClass₁ α</code> extend <code class="docutils literal notranslate">Dia₁ α</code>, so one could fear that having + a <code class="docutils literal notranslate">Monoid₁ α</code> instance gives two unrelated diamond operations on <code class="docutils literal notranslate">α</code>, one coming from + a field <code class="docutils literal notranslate">Monoid₁.toSemigroup₁</code> and one coming from a field <code class="docutils literal notranslate">Monoid₁.toDiaOneClass₁</code>. + Indeed if we try to build a monoid class by hand using: + <div class="highlight-lean notranslate"><div class="highlight"><pre>class Monoid₂ (α : Type) where + toSemigroup₁ : Semigroup₁ α + toDiaOneClass₁ : DiaOneClass₁ α + </pre></div> + </div> + then we get two completely unrelated diamond operations + <code class="docutils literal notranslate">Monoid₂.toSemigroup₁.toDia₁.dia</code> and <code class="docutils literal notranslate">Monoid₂.toDiaOneClass₁.toDia₁.dia</code>. + The version generated using the <code class="docutils literal notranslate">extends</code> syntax does not have this defect. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {α : Type} [Monoid₁ α] : + (Monoid₁.toSemigroup₁.toDia₁.dia : α → α → α) = Monoid₁.toDiaOneClass₁.toDia₁.dia := rfl + </pre></div> + </div> + So the <code class="docutils literal notranslate">class</code> command did some magic for us (and the <code class="docutils literal notranslate">structure</code> command would have done it + too). An easy way to see what are the fields of our classes is to check their constructor. Compare: + <div class="highlight-lean notranslate"><div class="highlight"><pre>/- Monoid₂.mk {α : Type} (toSemigroup₁ : Semigroup₁ α) (toDiaOneClass₁ : DiaOneClass₁ α) : Monoid₂ α -/ + #check Monoid₂.mk + + /- Monoid₁.mk {α : Type} [toSemigroup₁ : Semigroup₁ α] [toOne₁ : One₁ α] (one_dia : ∀ (a : α), 𝟙 ⋄ a = a) (dia_one : ∀ (a : α), a ⋄ 𝟙 = a) : Monoid₁ α -/ + #check Monoid₁.mk + </pre></div> + </div> + So we see that <code class="docutils literal notranslate">Monoid₁</code> takes <code class="docutils literal notranslate">Semigroup₁ α</code> argument as expected but then it won’t + take a would-be overlapping <code class="docutils literal notranslate">DiaOneClass₁ α</code> argument but instead tears it appart and includes + only the non-overlapping parts. And it also auto-generated an instance <code class="docutils literal notranslate">Monoid₁.toDiaOneClass₁</code> + which is not a field but has the expected signature which, from the end-user point of view, + restores the symmetry between the two extended classes <code class="docutils literal notranslate">Semigroup₁</code> and <code class="docutils literal notranslate">DiaOneClass₁</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check Monoid₁.toSemigroup₁ + #check Monoid₁.toDiaOneClass₁ + </pre></div> + </div> + We are now very close to defining groups. We could add to the monoid structure a field asserting + the existence of an inverse for every element. But then we would need to work to access these + inverses. In practice it is more convenient to add it as data. To optimize reusability, + we define a new data-carrying class, and then give it some notation. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class Inv₁ (α : Type) where + /-- The inversion function -/ + inv : α → α + + @[inherit_doc] + postfix:max "⁻¹" => Inv₁.inv + + class Group₁ (G : Type) extends Monoid₁ G, Inv G where + inv_dia : ∀ a : G, a⁻¹ ⋄ a = 𝟙 + </pre></div> + </div> + The above definition may seem too weak, we only ask that <code class="docutils literal notranslate">a⁻¹</code> is a left-inverse of <code class="docutils literal notranslate">a</code>. + But the other side is automatic. In order to prove that, we need a preliminary lemma. + <div class="highlight-lean notranslate"><div class="highlight"><pre>lemma left_inv_eq_right_inv₁ {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by + rw [← DiaOneClass₁.one_dia c, ← hba, Semigroup₁.dia_assoc, hac, DiaOneClass₁.dia_one b] + </pre></div> + </div> + In this lemma, it is pretty annoying to give full names, especially since it requires knowing + which part of the hierarchy provides those facts. One way to fix this is to use the <code class="docutils literal notranslate">export</code> + command to copy those facts as lemmas in the root name space. + <div class="highlight-lean notranslate"><div class="highlight"><pre>export DiaOneClass₁ (one_dia dia_one) + export Semigroup₁ (dia_assoc) + export Group₁ (inv_dia) + </pre></div> + </div> + We can then rewrite the above proof as: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by + rw [← one_dia c, ← hba, dia_assoc, hac, dia_one b] + </pre></div> + </div> + It is now your turn to prove things about our algebraic structures. + <div class="highlight-lean notranslate"><div class="highlight"><pre>lemma inv_eq_of_dia [Group₁ G] {a b : G} (h : a ⋄ b = 𝟙) : a⁻¹ = b := + sorry + + lemma dia_inv [Group₁ G] (a : G) : a ⋄ a⁻¹ = 𝟙 := + sorry + </pre></div> + </div> + At this stage we would like to move on to define rings, but there is a serious issue. + A ring structure on a type contains both an additive group structure and a multiplicative + monoid structure, and some properties about their interaction. But so far we hard-coded + a notation <code class="docutils literal notranslate">⋄</code> for all our operations. More fundamentally, the type class system + assumes every type has only one instance of each type class. There are various + ways to solve this issue. Surprisingly mathlib uses the naive idea to duplicate + everything for additive and multiplicative theories with the help of some code-generating + attribute. Structures and classes are defined in both additive and multiplicative notation + with an attibute <code class="docutils literal notranslate">to_additive</code> linking them. In case of multiple inheritance like for + semi-groups, the auto-generated “symmetry-restoring” instances need also to be marked. + This is a bit technical you don’t need to understand details. The important point is that + lemmas are then only stated in multiplicative notation and marked with the attribute <code class="docutils literal notranslate">to_additive</code> + to generate the additive version as <code class="docutils literal notranslate">left_inv_eq_right_inv'</code> with it’s auto-generated additive + version <code class="docutils literal notranslate">left_neg_eq_right_neg'</code>. In order to check the name of this additive version we + used the <code class="docutils literal notranslate">whatsnew in</code> command on top of <code class="docutils literal notranslate">left_inv_eq_right_inv'</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class AddSemigroup₃ (α : Type) extends Add α where + /-- Multiplication is associative -/ + add_assoc₃ : ∀ a b c : α, a + b + c = a + (b + c) + + @[to_additive AddSemigroup₃] + class Semigroup₃ (α : Type) extends Mul α where + /-- Multiplication is associative -/ + mul_assoc₃ : ∀ a b c : α, a * b * c = a * (b * c) + + class AddMonoid₃ (α : Type) extends AddSemigroup₃ α, AddZeroClass α + + @[to_additive AddMonoid₃] + class Monoid₃ (α : Type) extends Semigroup₃ α, MulOneClass α + + attribute [to_additive existing] Monoid₃.toMulOneClass + + export Semigroup₃ (mul_assoc₃) + export AddSemigroup₃ (add_assoc₃) + + whatsnew in + @[to_additive] + lemma left_inv_eq_right_inv' {M : Type} [Monoid₃ M] {a b c : M} (hba : b * a = 1) (hac : a * c = 1) : b = c := by + rw [← one_mul c, ← hba, mul_assoc₃, hac, mul_one b] + + #check left_neg_eq_right_neg' + </pre></div> + </div> + Equipped with this technology, we can easily define also commutative semigroups, monoids and + groups, and then define rings. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class AddCommSemigroup₃ (α : Type) extends AddSemigroup₃ α where + add_comm : ∀ a b : α, a + b = b + a + + @[to_additive AddCommSemigroup₃] + class CommSemigroup₃ (α : Type) extends Semigroup₃ α where + mul_comm : ∀ a b : α, a * b = b * a + + class AddCommMonoid₃ (α : Type) extends AddMonoid₃ α, AddCommSemigroup₃ α + + @[to_additive AddCommMonoid₃] + class CommMonoid₃ (α : Type) extends Monoid₃ α, CommSemigroup₃ α + + class AddGroup₃ (G : Type) extends AddMonoid₃ G, Neg G where + neg_add : ∀ a : G, -a + a = 0 + + @[to_additive AddGroup₃] + class Group₃ (G : Type) extends Monoid₃ G, Inv G where + inv_mul : ∀ a : G, a⁻¹ * a = 1 + </pre></div> + </div> + We should remember to tagged lemmas with <code class="docutils literal notranslate">simp</code> when approriate. + <div class="highlight-lean notranslate"><div class="highlight"><pre>attribute [simp] Group₃.inv_mul AddGroup₃.neg_add + </pre></div> + </div> + Then we need to repeat ourselves a bit since we switch to standard notations, but at least + <code class="docutils literal notranslate">to_additive</code> does the work of translating from the multiplicative notation to the additive one. + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[to_additive] + lemma inv_eq_of_mul [Group₃ G] {a b : G} (h : a * b = 1) : a⁻¹ = b := + sorry + </pre></div> + </div> + Note that <code class="docutils literal notranslate">to_additive</code> can be ask to tag a lemma with <code class="docutils literal notranslate">simp</code> and propagate that attribute + to the additive version as follows. + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[to_additive (attr := simp)] + lemma Group₃.mul_inv {G : Type} [Group₃ G] {a : G} : a * a⁻¹ = 1 := by + sorry + + @[to_additive] + lemma mul_left_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : a * b = a * c) : b = c := by + sorry + + @[to_additive] + lemma mul_right_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : b*a = c*a) : b = c := by + sorry + + class AddCommGroup₃ (G : Type) extends AddGroup₃ G, AddCommMonoid₃ G + + @[to_additive AddCommGroup₃] + class CommGroup₃ (G : Type) extends Group₃ G, CommMonoid₃ G + </pre></div> + </div> + We are now ready for rings. For demonstration puprposes we won’t assume that addition is + commutative, and then immediately provide an instance of <code class="docutils literal notranslate">AddCommGroup₃</code>. Mathlib does not + play this game, first because in practice this does not make any ring instance easier and + also because Mathlib’s algebraic hierarchy goes through semi-rings which are like rings but without + opposites so that the proof below does not work for them. What we gain here, besides a nice exercise + if you have never seen it, is an example of building an instance using the syntax that allows + to provide a parent structure and some extra fields. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class Ring₃ (R : Type) extends AddGroup₃ R, Monoid₃ R, MulZeroClass R where + /-- Multiplication is left distributive over addition -/ + left_distrib : ∀ a b c : R, a * (b + c) = a * b + a * c + /-- Multiplication is right distributive over addition -/ + right_distrib : ∀ a b c : R, (a + b) * c = a * c + b * c + + instance {R : Type} [Ring₃ R] : AddCommGroup₃ R := + { Ring₃.toAddGroup₃ with + add_comm := by + sorry } + </pre></div> + </div> + Of course we can also build concrete instances, such as a ring structure on integers (of course + the instance below uses that all the work is already done in Mathlib). + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance : Ring₃ ℤ where + add := (· + ·) + add_assoc₃ := add_assoc + zero := 0 + zero_add := by simp + add_zero := by simp + neg := (- ·) + neg_add := by simp + mul := (· * ·) + mul_assoc₃ := mul_assoc + one := 1 + one_mul := by simp + mul_one := by simp + zero_mul := by simp + mul_zero := by simp + left_distrib := Int.mul_add + right_distrib := Int.add_mul + </pre></div> + </div> + As an exercise you can now set up a simple hierarchy for order relations, including a class + for ordered commutative monoids, which have both a partial order and a commutative monoid structure + such that <code class="docutils literal notranslate">∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b</code>. Of course you need to add fields and maybe + <code class="docutils literal notranslate">extends</code> clauses to the following classes. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class LE₁ (α : Type) where + /-- The Less-or-Equal relation. -/ + le : α → α → Prop + + @[inherit_doc] infix:50 " ≤₁ " => LE₁.le + + class Preorder₁ (α : Type) + + class PartialOrder₁ (α : Type) + + class OrderedCommMonoid₁ (α : Type) + + instance : OrderedCommMonoid₁ ℕ where + </pre></div> + </div> + We now want to discuss algebraic structures involving several types. The prime example + is modules over rings. If you don’t know what is a module, you can pretend it means vector space + and think that all our rings are fields. Those structures are commutative additive groups + equipped with a scalar multiplication by elements of some ring. + We first define the data-carrying type class of scalar multiplication by some type <code class="docutils literal notranslate">α</code> on some + type <code class="docutils literal notranslate">β</code>, and give it a right associative notation. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class SMul₃ (α : Type) (β : Type) where + /-- Scalar multiplication -/ + smul : α → β → β + + infixr:73 " • " => SMul₃.smul + </pre></div> + </div> + Then we can define modules (again think about vector spaces if you don’t know what is a module). + <div class="highlight-lean notranslate"><div class="highlight"><pre>class Module₁ (R : Type) [Ring₃ R] (M : Type) [AddCommGroup₃ M] extends SMul₃ R M where + zero_smul : ∀ m : M, (0 : R) • m = 0 + one_smul : ∀ m : M, (1 : R) • m = m + mul_smul : ∀ (a b : R) (m : M), (a * b) • m = a • b • m + add_smul : ∀ (a b : R) (m : M), (a + b) • m = a • m + b • m + smul_add : ∀ (a : R) (m n : M), a • (m + n) = a • m + a • n + </pre></div> + </div> + There is something interesting going on here. While it isn’t too surprising that the + ring structure on <code class="docutils literal notranslate">R</code> is a parameter in this definition, you probably expected <code class="docutils literal notranslate">AddCommGroup3 M</code> + to be part of the <code class="docutils literal notranslate">extends</code> clause just as <code class="docutils literal notranslate">SMul₃ R M</code> is. Trying to do that would lead + to a mysterious sounding error message: + <code class="docutils literal notranslate">cannot find synthesization order for instance Module₁.toAddCommGroup₃ with type (R : Type) → [inst : Ring₃ R] → {M : Type} → [self : Module₁ R M] → AddCommGroup₃ M + all remaining arguments have metavariables: Ring₃ ?R @Module₁ ?R ?inst✝ M</code>. + In order to understand this message, you need to remember that such an <code class="docutils literal notranslate">extends</code> clause would + lead to a field <code class="docutils literal notranslate">Module₃.toAddCommGroup₃</code> marked as an instance. This instance + would have the signature appearing in the error message: + <code class="docutils literal notranslate">(R : Type) → [inst : Ring₃ R] → {M : Type} → [self : Module₁ R M] → AddCommGroup₃ M</code>. + With such an instance in the type class database, each time Lean would look for a + <code class="docutils literal notranslate">AddCommGroup₃ M</code> instance for some <code class="docutils literal notranslate">M</code>, it would need to go hunting for a completely + unspecified type <code class="docutils literal notranslate">R``and a ``Ring₃ R</code> instance before embarking on the main quest of finding a + <code class="docutils literal notranslate">Module₁ R M</code> instance. Those two side-quests are represented by the meta-variables mentionned in + the error message and denoted by <code class="docutils literal notranslate">?R</code> and <code class="docutils literal notranslate">?inst✝</code> there. Such a <code class="docutils literal notranslate">Module₃.toAddCommGroup₃</code> + instance would then be a huge trap for the instance resolution procedure and then <code class="docutils literal notranslate">class</code> command + refuses to set it up. + What about <code class="docutils literal notranslate">extends SMul₃ R M</code> then? That one creates a field + <code class="docutils literal notranslate">Module₁.toSMul₃ : {R : Type} →  [inst : Ring₃ R] → {M : Type} → [inst_1 : AddCommGroup₃ M] → [self : Module₁ R M] → SMul₃ R M</code> + whose end result <code class="docutils literal notranslate">SMul₃ R M</code> mentions both <code class="docutils literal notranslate">R</code> and <code class="docutils literal notranslate">M</code> so this field can + safely be used as an instance. The rule is easy to remember: each class appearing in the + <code class="docutils literal notranslate">extends</code> clause should mention every type appearing in the parameters. + Let us create our first module instance: a ring is a module over itself using its multiplication + as a scalar multiplication. + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance selfModule (R : Type) [Ring₃ R] : Module₁ R R where + smul := fun r s ↦ r*s + zero_smul := zero_mul + one_smul := one_mul + mul_smul := mul_assoc₃ + add_smul := Ring₃.right_distrib + smul_add := Ring₃.left_distrib + </pre></div> + </div> + As a second example, every abelian group is a module over <code class="docutils literal notranslate">ℤ</code> (this is one of the reason to + generalize the theory of vector spaces by allowing non-invertible scalars). First one can define + scalar multiplication by a natural number for any type equipped with a zero and an addition: + <code class="docutils literal notranslate">n • a</code> is defined as <code class="docutils literal notranslate">a + ⋯ + a</code> where <code class="docutils literal notranslate">a</code> appears <code class="docutils literal notranslate">n</code> times. Then this is extended + to scalar multiplication by an integer by ensuring <code class="docutils literal notranslate">(-1) • a = -a</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>def nsmul₁ [Zero M] [Add M] : ℕ → M → M + | 0, _ => 0 + | n + 1, a => a + nsmul₁ n a + + def zsmul₁ {M : Type _} [Zero M] [Add M] [Neg M] : ℤ → M → M + | Int.ofNat n, a => nsmul₁ n a + | Int.negSucc n, a => -nsmul₁ n.succ a + </pre></div> + </div> + Proving this gives rise to a module structure is a bit tedious and not interesting for the + current discussion, so we will sorry all axioms. You are not asked to replace those sorries + with proofs. If you insist on doing it then you will probably want to state and prove several + intermediate lemmas about <code class="docutils literal notranslate">nsmul₁</code> and <code class="docutils literal notranslate">zsmul₁</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance abGrpModule (A : Type) [AddCommGroup₃ A] : Module₁ ℤ A where + smul := zsmul₁ + zero_smul := sorry + one_smul := sorry + mul_smul := sorry + add_smul := sorry + smul_add := sorry + </pre></div> + </div> + A much more important issue is that we now have two module structures over the ring <code class="docutils literal notranslate">ℤ</code> + for <code class="docutils literal notranslate">ℤ</code> itself: <code class="docutils literal notranslate">abGrpModule ℤ</code> since <code class="docutils literal notranslate">ℤ</code> is a abelian group, and <code class="docutils literal notranslate">selfModule ℤ</code> since + <code class="docutils literal notranslate">ℤ</code> is a ring. Those two module structure correspond to the same abelian group structure, + but it is not obvious that they have the same scalar multiplication. They actually do, but + this isn’t true by definition, it requires a proof. This is very bad news for the type class + instance resolution procedure and will lead to very frustating failures for users of this + hierarchy. When directly asked to find an instance, Lean will pick one, and we can see + which one using: + <div class="highlight-lean notranslate"><div class="highlight"><pre>#synth Module₁ ℤ ℤ -- abGrpModule ℤ + </pre></div> + </div> + But in a more indirect context it can happen that Lean infers the one and then gets confused. + This situation is known as a bad diamond. This has nothing to do with the diamond operation + we used above, it refers to the way one can draw the paths from <code class="docutils literal notranslate">ℤ</code> to its <code class="docutils literal notranslate">Module₁ ℤ</code> + going through either <code class="docutils literal notranslate">AddCommGroup₃ ℤ</code> or <code class="docutils literal notranslate">Ring₃ ℤ</code>. + It is important to understand that not all diamonds are bad. In fact there are diamonds everywhere + in mathlib, and also in this chapter. Already at the very beginning we saw one can go + from <code class="docutils literal notranslate">Monoid₁ α</code> to <code class="docutils literal notranslate">Dia₁ α</code> through either <code class="docutils literal notranslate">Semigroup₁ α</code> or <code class="docutils literal notranslate">DiaOneClass₁ α</code> and + thanks to the work done by the <code class="docutils literal notranslate">class</code> command, the resulting two <code class="docutils literal notranslate">Dia₁ α</code> instances + are definitionnaly equal. In particular a diamond having a <code class="docutils literal notranslate">Prop</code>-valued class at the bottom + cannot be bad since any too proofs of the same statement are definitionnaly equal. + But the diamond we created with modules is definitely bad. The offending piece is the <code class="docutils literal notranslate">smul</code> + field which is data, not a proof, and we have two constructions that are not definitionnaly equal. + The robust way of fixing this issue is to make sure that going from a rich structure to a + poor structure is always done by forgetting data, not by defining data. This well-known pattern + as been named “forgetful inheritance” and extensively discussed in + <a class="reference external" href="https://inria.hal.science/hal-02463336">https://inria.hal.science/hal-02463336</a>. + In our concrete case, we can modify the definition of <code class="docutils literal notranslate">AddMonoid₃</code> to include a <code class="docutils literal notranslate">nsmul</code> data + field and some <code class="docutils literal notranslate">Prop</code>-valued fields ensuring this operation is provably the one we constructed + above. Those fields are given default values using <code class="docutils literal notranslate">:=</code> after their type in the definition below. + Thanks to these default values, most instances would be constructed exactly as with our previous + definitions. But in the special case of <code class="docutils literal notranslate">ℤ</code> we will be able to provide specific values. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class AddMonoid₄ (M : Type) extends AddSemigroup₃ M, AddZeroClass M where + /-- Multiplication by a natural number. -/ + nsmul : ℕ → M → M := nsmul₁ + /-- Multiplication by `(0 : ℕ)` gives `0`. -/ + nsmul_zero : ∀ x, nsmul 0 x = 0 := by intros; rfl + /-- Multiplication by `(n + 1 : ℕ)` behaves as expected. -/ + nsmul_succ : ∀ (n : ℕ) (x), nsmul (n + 1) x = x + nsmul n x := by intros; rfl + + instance mySMul {M : Type} [AddMonoid₄ M] : SMul ℕ M := ⟨AddMonoid₄.nsmul⟩ + </pre></div> + </div> + Let us check we can still construct a product monoid instance without providing the <code class="docutils literal notranslate">nsmul</code> + related fields. + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance (M N : Type) [AddMonoid₄ M] [AddMonoid₄ N] : AddMonoid₄ (M × N) where + add := fun p q ↦ (p.1 + q.1, p.2 + q.2) + add_assoc₃ := fun a b c ↦ by ext <;> apply add_assoc₃ + zero := (0, 0) + zero_add := fun a ↦ by ext <;> apply zero_add + add_zero := fun a ↦ by ext <;> apply add_zero + </pre></div> + </div> + And now let us handle the special case of <code class="docutils literal notranslate">ℤ</code> where we want to build <code class="docutils literal notranslate">nsmul</code> using the coercion + of <code class="docutils literal notranslate">ℕ</code> to <code class="docutils literal notranslate">ℤ</code> and the multiplication on <code class="docutils literal notranslate">ℤ</code>. Note in particular how the proof fields + contain more work than in the default value above. + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance : AddMonoid₄ ℤ where + add := (· + ·) + add_assoc₃ := Int.add_assoc + zero := 0 + zero_add := Int.zero_add + add_zero := Int.add_zero + nsmul := fun n m ↦ (n : ℤ) * m + nsmul_zero := Int.zero_mul + nsmul_succ := fun n m ↦ show (n + 1 : ℤ) * m = m + n * m + by rw [Int.add_mul, Int.add_comm, Int.one_mul] + </pre></div> + </div> + Let us check we solved our issue. Because Lean already has a definition of scalar multiplication + of a natural number and an integer, and we want to make sure our instance is used, we won’t use + the <code class="docutils literal notranslate">•</code> notation but call <code class="docutils literal notranslate">SMul.mul</code> and explicitly provide our instance defined above. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (n : ℕ) (m : ℤ) : SMul.smul (self := mySMul) n m = n * m := rfl + </pre></div> + </div> + This story then continues with incorporating a <code class="docutils literal notranslate">zsmul</code> field into the definition of groups + and similar tricks. You are now ready to read the definition of monoids, groups, rings and modules + in mathlib. There are more complicated than what we have seen here, because they are part of a huge + hierarchy, but all principles have been explained above. + As an exercise, you can come back to the order relation hierarchy you built above and try + to incorportate a type class <code class="docutils literal notranslate">LT₁</code> carrying the Less-Than notation <code class="docutils literal notranslate"><₁</code> and make sure + that every preorder comes with a <code class="docutils literal notranslate"><₁</code> which has a default value built from <code class="docutils literal notranslate">≤₁</code> and a + <code class="docutils literal notranslate">Prop</code>-valued field asserting the natural relation between those two comparison operators. + TEXT. + -/ + </section> + <section id="morphisms"> + <h2>7.2. Morphisms<a class="headerlink" href="#morphisms" title="Permalink to this heading"></a></h2> + So far in this chapter, we discussed how to create a hierarchy of mathematical structures. + But defining structures is not really completed until we have morphisms. There are two + main approaches here. The most obvious one is to define a predicate on functions. + <div class="highlight-lean notranslate"><div class="highlight"><pre>def isMonoidHom₁ [Monoid G] [Monoid H] (f : G → H) : Prop := + f 1 = 1 ∧ ∀ g g', f (g * g') = f g * f g' + </pre></div> + </div> + In this definition, it is a bit unpleasant to use a conjunction. In particular users + will need to remember the ordering we chose when they want to access the two conditions. + So we could use a structure instead. + <div class="highlight-lean notranslate"><div class="highlight"><pre>structure isMonoidHom₂ [Monoid G] [Monoid H] (f : G → H) : Prop where + map_one : f 1 = 1 + map_mul : ∀ g g', f (g * g') = f g * f g' + </pre></div> + </div> + Once we are here, it is even tempting to make it a class and use the type class instance resolution + procedure to automatically infer <code class="docutils literal notranslate">isMonoidHom₂</code> for complicated functions out of instances for + simpler functions. For instance a composition of monoid morphisms is a monoid morphism and this + seems like a useful instance. However such an instance would be very tricky for the resolution + procedure since it would need to hunt down <code class="docutils literal notranslate">g ∘ f</code> everywhere. Seeing it failing in <code class="docutils literal notranslate">g (f x)</code> + would be very frustrating. More generally one must always keep in mind that recognizing which + function is applied in a given expression is a very difficult problem, called the “higher-order + unification problem”. So Mathlib does not use this class approach. + A more fundamental question is whether we use predicates as above (using either a <code class="docutils literal notranslate">def</code> or a + <code class="docutils literal notranslate">structure</code>) or use structures bundling a function and predicates. This is partly a psychological + issue. It is extremely rare to consider a function between monoids that is not a morphism. + It really feels like “monoid morphism” is not an adjective you can assign to a bare function, + it is a noun. On the other hand one can argue that a continuous function between topological spaces + is really a function that happens to be continuous. This is one reason why Mathlib has a + <code class="docutils literal notranslate">Continuous</code> predicate. For instance you can write: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : Continuous (id : ℝ → ℝ) := continuous_id + </pre></div> + </div> + We still have bundles continuous functions, which are convenient for instance to put a topology + on a space of continuous functions, but they are not the primary tool to work with continuity. + By constrast, morphisms between monoids (or other algebraic structures) are bundled as in: + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] + structure MonoidHom₁ (G H : Type) [Monoid G] [Monoid H] where + toFun : G → H + map_one : toFun 1 = 1 + map_mul : ∀ g g', toFun (g * g') = toFun g * toFun g' + </pre></div> + </div> + Of course we don’t want to type <code class="docutils literal notranslate">toFun</code> everywhere so we register a coercion using + the <code class="docutils literal notranslate">CoeFun</code> type class. Its first argument is the type we want to coerce to a function. + The second argument describes the target function type. In our case it is always <code class="docutils literal notranslate">G → H</code> + for every <code class="docutils literal notranslate">f : MonoidHom₁ G H</code>. We also tag <code class="docutils literal notranslate">MonoidHom₁.toFun</code> with the <code class="docutils literal notranslate">coe</code> attribute to + make sure it is displayed almost invisibly in the tactic state, simply by a <code class="docutils literal notranslate">↑</code> prefix. + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance [Monoid G] [Monoid H] : CoeFun (MonoidHom₁ G H) (fun _ ↦ G → H) where + coe := MonoidHom₁.toFun + + attribute [coe] MonoidHom₁.toFun + </pre></div> + </div> + Let us check we can indeed apply a bundled monoid morphism to an element. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [Monoid G] [Monoid H] (f : MonoidHom₁ G H) : f 1 = 1 := f.map_one + </pre></div> + </div> + We can do the same with other kind of morphisms until we reach ring morphisms. + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] + structure AddMonoidHom₁ (G H : Type) [AddMonoid G] [AddMonoid H] where + toFun : G → H + map_zero : toFun 0 = 0 + map_add : ∀ g g', toFun (g + g') = toFun g + toFun g' + + instance [AddMonoid G] [AddMonoid H] : CoeFun (AddMonoidHom₁ G H) (fun _ ↦ G → H) where + coe := AddMonoidHom₁.toFun + + attribute [coe] AddMonoidHom₁.toFun + + @[ext] + structure RingHom₁ (R S : Type) [Ring R] [Ring S] extends MonoidHom₁ R S, AddMonoidHom₁ R S + </pre></div> + </div> + There are a couple of issues about this approach. A minor one is we don’t quite know where to put + the <code class="docutils literal notranslate">coe</code> attribute since the <code class="docutils literal notranslate">RingHom₁.toFun</code> does not exist, the relevant function is + <code class="docutils literal notranslate">MonoidHom₁.toFun ∘ RingHom₁.toMonoidHom₁</code> which is not a declaration that can be tagged with an + attribute (but we could still define a <code class="docutils literal notranslate">CoeFun  (RingHom₁ R S) (fun _ ↦ R → S)</code> instance). + A much more important one is that lemmas about monoid morphisms won’t directly apply + to ring morphisms. This leaves the alternative of either juggling with <code class="docutils literal notranslate">RingHom₁.toMonoidHom₁</code> + each time we want to apply a monoid morphism lemma or restate every such lemmas for ring morphisms. + Neither option is appealing so Mathlib uses a new hierarchy trick here. The idea is to define + a type class for objects that are at least monoid morphisms, instantiate that class with both monoid + morphisms and ring morphisms and use it to state every lemma. In the definition below, + <code class="docutils literal notranslate">F</code> could be <code class="docutils literal notranslate">MonoidHom₁ M N</code>, or <code class="docutils literal notranslate">RingHom₁ M N</code> if <code class="docutils literal notranslate">M</code> and <code class="docutils literal notranslate">N</code> have a ring structure. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class MonoidHomClass₁ (F : Type) (M N : Type) [Monoid M] [Monoid N] where + toFun : F → M → N + map_one : ∀ f : F, toFun f 1 = 1 + map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' + </pre></div> + </div> + However there is a problem with the above implementation. We haven’t registered a coercion to + function instance yet. Let us try to do it now. + <div class="highlight-lean notranslate"><div class="highlight"><pre>def badInst [Monoid M] [Monoid N] [MonoidHomClass₁ F M N] : CoeFun F (fun _ ↦ M → N) where + coe := MonoidHomClass₁.toFun + </pre></div> + </div> + Making the an instance would be bad. When faced with something like <code class="docutils literal notranslate">f x</code> where the type of <code class="docutils literal notranslate">f</code> + is not a function type, Lean will try to find a <code class="docutils literal notranslate">CoeFun</code> instance to coerce <code class="docutils literal notranslate">f</code> into a function. + The above function has type: + <code class="docutils literal notranslate">{M N F : Type} → [Monoid M] → [Monoid N] → [MonoidHomClass₁ F M N] → CoeFun F (fun x ↦ M → N)</code> + so, when it trying to apply it, it wouldn’t be a priori clear to Lean in which order the unknown + types <code class="docutils literal notranslate">M</code>, <code class="docutils literal notranslate">N</code> and <code class="docutils literal notranslate">F</code> should be inferred. This is a kind of bad instance that is slightly + different from the one we saw already, but it boils down to the same issue: without knowing <code class="docutils literal notranslate">M</code>, + Lean would have to search for a monoid instance on an unknown type, hence hopelessly try every + monoid instance in the database. If you are curious to see the effect of such an instance you + can type <code class="docutils literal notranslate">set_option synthInstance.checkSynthOrder false in</code> on top of the above declaration, + replace <code class="docutils literal notranslate">def badInst</code> with <code class="docutils literal notranslate">instance</code>, and look for random failures in this file. + Here the solution is easy, we need to tell Lean to first search what is <code class="docutils literal notranslate">F</code> and then deduce <code class="docutils literal notranslate">M</code> + and <code class="docutils literal notranslate">N</code>. This is done using the <code class="docutils literal notranslate">outParam</code> function. This function is defined as the identity + function, but is still recognized by the type class machinery and triggers the desired behavior. + Hence we can retry defining our class, paying attention to the <code class="docutils literal notranslate">outParam</code> function: + <div class="highlight-lean notranslate"><div class="highlight"><pre>class MonoidHomClass₂ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] where + toFun : F → M → N + map_one : ∀ f : F, toFun f 1 = 1 + map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' + + instance [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] : CoeFun F (fun _ ↦ M → N) where + coe := MonoidHomClass₂.toFun + + attribute [coe] MonoidHomClass₂.toFun + </pre></div> + </div> + Now we can proceed with our plan to instantiate this class. + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₂ (MonoidHom₁ M N) M N where + toFun := MonoidHom₁.toFun + map_one := fun f ↦ f.map_one + map_mul := fun f ↦ f.map_mul + + instance (R S : Type) [Ring R] [Ring S] : MonoidHomClass₂ (RingHom₁ R S) R S where + toFun := fun f ↦ f.toMonoidHom₁.toFun + map_one := fun f ↦ f.toMonoidHom₁.map_one + map_mul := fun f ↦ f.toMonoidHom₁.map_mul + </pre></div> + </div> + As promised every lemma we prove about <code class="docutils literal notranslate">f : F</code> assuming an instance of <code class="docutils literal notranslate">MonoidHomClass₁ F</code> will + apply both to monoid morphims and ring morphisms. + Let us see an example lemma and check it applies to both situations. + <div class="highlight-lean notranslate"><div class="highlight"><pre>lemma map_inv_of_inv [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] (f : F) {m m' : M} (h : m*m' = 1) : + f m * f m' = 1 := by + rw [← MonoidHomClass₂.map_mul, h, MonoidHomClass₂.map_one] + + example [Monoid M] [Monoid N] (f : MonoidHom₁ M N) {m m' : M} (h : m*m' = 1) : f m * f m' = 1 := + map_inv_of_inv f h + + example [Ring R] [Ring S] (f : RingHom₁ R S) {r r' : R} (h : r*r' = 1) : f r * f r' = 1 := + map_inv_of_inv f h + </pre></div> + </div> + At first sight, it may look like we got back to our old bad idea of making <code class="docutils literal notranslate">MonoidHom₁</code> a class. + But we haven’t. Everything is shifted one level of abstraction up. The type class resolution + procedure won’t be looking for functions, it will be looking for either + <code class="docutils literal notranslate">MonoidHom₁</code> or <code class="docutils literal notranslate">RingHom₁</code>. + One remaining issue with our approach is the presence of repeatitive code around the <code class="docutils literal notranslate">toFun</code> + field and the corresponding <code class="docutils literal notranslate">CoeFun</code> instance and <code class="docutils literal notranslate">coe</code> attribute. It would also be better + to record that this pattern is used only for function with extra properties, meaning that the + coercion to functions should be injective. So Mathlib adds one more layer of abstraction with + the base class <code class="docutils literal notranslate">FunLike</code>. Let us redefine our <code class="docutils literal notranslate">MonoidHomClass</code> on top of this base layer. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class MonoidHomClass₃ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] extends + FunLike F M (fun _ ↦ N) where + map_one : ∀ f : F, f 1 = 1 + map_mul : ∀ (f : F) g g', f (g * g') = f g * f g' + + instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₃ (MonoidHom₁ M N) M N where + coe := MonoidHom₁.toFun + coe_injective' := MonoidHom₁.ext + map_one := MonoidHom₁.map_one + map_mul := MonoidHom₁.map_mul + </pre></div> + </div> + Of course the hierarchy of morphisms does not stop here. We could go on and define a class + <code class="docutils literal notranslate">RingHomClass₃</code> extending <code class="docutils literal notranslate">MonoidHomClass₃</code> and instantiate it on <code class="docutils literal notranslate">RingHom</code> and + then later on <code class="docutils literal notranslate">AlgebraHom</code> (algebras are rings with some extra structure). But we’ve + covered the main formalization ideas used in Mathlib for morphisms and you should be ready + to understand how morphisms are defined in Mathlib. + As an exercise, you should try to define your class of bundled order-preserving function between + ordered types, and then order preserving monoid morphisms. This is for training purposes only. + Like continuous functions, order preserving functions are primarily unbundled in Mathlib where + they are defined by the <code class="docutils literal notranslate">Monotone</code> predicate. Of course you need to complete the class + definitions below. + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] + structure OrderPresHom (α β : Type) [LE α] [LE β] where + toFun : α → β + le_of_le : ∀ a a', a ≤ a' → toFun a ≤ toFun a' + + @[ext] + structure OrderPresMonoidHom (M N : Type) [Monoid M] [LE M] [Monoid N] [LE N] extends + MonoidHom₁ M N, OrderPresHom M N + + class OrderPresHomClass (F : Type) (α β : outParam Type) [LE α] [LE β] + + instance (α β : Type) [LE α] [LE β] : OrderPresHomClass (OrderPresHom α β) α β where + + instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : + OrderPresHomClass (OrderPresMonoidHom α β) α β where + + instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : + MonoidHomClass₃ (OrderPresMonoidHom α β) α β + := sorry + </pre></div> + </div> + </section> + <section id="sub-objects"> + <h2>7.3. Sub-objects<a class="headerlink" href="#sub-objects" title="Permalink to this heading"></a></h2> + After defining some algebraic structure and its morphisms, the next step is to consider sets + that inherit this algebraic structure, for instance subgroups or subrings. + This largely overlaps our previous topic. Indeed a set in <code class="docutils literal notranslate">X</code> is implemented as a function from + <code class="docutils literal notranslate">X</code> to <code class="docutils literal notranslate">Prop</code> so sub-objects are function satisfying a certain predicate. + Hence we can reuse of lot of the ideas that led to the <code class="docutils literal notranslate">FunLike</code> class and its descendants. + We won’t reuse <code class="docutils literal notranslate">FunLike</code> itself because this would break the abstraction barrier from <code class="docutils literal notranslate">Set X</code> + to <code class="docutils literal notranslate">X → Prop</code>. Instead there is a <code class="docutils literal notranslate">SetLike</code> class. Instead of wrapping an injection into a + function type, that class wraps an injection into a <code class="docutils literal notranslate">Set</code> type and defines the corresponding + coercion and <code class="docutils literal notranslate">Membership</code> instance. + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] + structure Submonoid₁ (M : Type) [Monoid M] where + /-- The carrier of a submonoid. -/ + carrier : Set M + /-- The product of two elements of a submonoid belongs to the submonoid. -/ + mul_mem {a b} : a ∈ carrier → b ∈ carrier → a * b ∈ carrier + /-- The unit element belongs to the submonoid. -/ + one_mem : 1 ∈ carrier + + /-- Submonoids in `M` can be seen as sets in `M`. -/ + instance [Monoid M] : SetLike (Submonoid₁ M) M where + coe := Submonoid₁.carrier + coe_injective' := Submonoid₁.ext + </pre></div> + </div> + Equipped with the above <code class="docutils literal notranslate">SetLike</code> instance, we can already state naturally that + a submonoid <code class="docutils literal notranslate">N</code> contains <code class="docutils literal notranslate">1</code> without using <code class="docutils literal notranslate">N.carrier</code>. + We can also silently treat <code class="docutils literal notranslate">N</code> as a set in <code class="docutils literal notranslate">M</code> as take its direct image under a map. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [Monoid M] (N : Submonoid₁ M) : 1 ∈ N := N.one_mem + + example [Monoid M] (N : Submonoid₁ M) (α : Type) (f : M → α) := f '' N + </pre></div> + </div> + We also have a coercion to <code class="docutils literal notranslate">Type</code> which uses <code class="docutils literal notranslate">Subtype</code> so, given a submonoid <code class="docutils literal notranslate">N</code> we can write + a parameter <code class="docutils literal notranslate">(x : N)</code> which can be coerced to an element of <code class="docutils literal notranslate">M</code> belonging to <code class="docutils literal notranslate">N</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [Monoid M] (N : Submonoid₁ M) (x : N) : (x : M) ∈ N := x.property + </pre></div> + </div> + Using this coercion to <code class="docutils literal notranslate">Type</code> we can also tackle the task of equipping a submonoid with a + monoid structure. We will use the coercion from the type associated to <code class="docutils literal notranslate">N</code> as above, and the + lemma <code class="docutils literal notranslate">SetCoe.ext</code> asserting this coercion is injective. Both are provided by the <code class="docutils literal notranslate">SetLike</code> + instance. + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance SubMonoid₁Monoid [Monoid M] (N : Submonoid₁ M) : Monoid N where + mul := fun x y ↦ ⟨x*y, N.mul_mem x.property y.property⟩ + mul_assoc := fun x y z ↦ SetCoe.ext (mul_assoc (x : M) y z) + one := ⟨1, N.one_mem⟩ + one_mul := fun x ↦ SetCoe.ext (one_mul (x : M)) + mul_one := fun x ↦ SetCoe.ext (mul_one (x : M)) + </pre></div> + </div> + Note that, in the above instance, instead of using the coercion to <code class="docutils literal notranslate">M</code> and calling the + <code class="docutils literal notranslate">property</code> field, we could have used destructuring binders as follows. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [Monoid M] (N : Submonoid₁ M) : Monoid N where + mul := fun ⟨x, hx⟩ ⟨y, hy⟩ ↦ ⟨x*y, N.mul_mem hx hy⟩ + mul_assoc := fun ⟨x, _⟩ ⟨y, _⟩ ⟨z, _⟩ ↦ SetCoe.ext (mul_assoc x y z) + one := ⟨1, N.one_mem⟩ + one_mul := fun ⟨x, _⟩ ↦ SetCoe.ext (one_mul x) + mul_one := fun ⟨x, _⟩ ↦ SetCoe.ext (mul_one x) + </pre></div> + </div> + In order to apply lemmas about submonoids to subgroups or subrings, we need a class, just + like for morphisms. Note this class take a <code class="docutils literal notranslate">SetLike</code> instance as a parameter so it does not need + a carrier field and can use the membership notation in its fields. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class SubmonoidClass₁ (S : Type) (M : Type) [Monoid M] [SetLike S M] : Prop where + mul_mem : ∀ (s : S) {a b : M}, a ∈ s → b ∈ s → a * b ∈ s + one_mem : ∀ s : S, 1 ∈ s + + instance [Monoid M] : SubmonoidClass₁ (Submonoid₁ M) M where + mul_mem := Submonoid₁.mul_mem + one_mem := Submonoid₁.one_mem + </pre></div> + </div> + As an exercise you should define a <code class="docutils literal notranslate">Subgroup₁</code> structure, endow it with a <code class="docutils literal notranslate">SetLike</code> instance + and a <code class="docutils literal notranslate">SubmonoidClass₁</code> instance, put a <code class="docutils literal notranslate">Group</code> instance on the subtype associated to a + <code class="docutils literal notranslate">Subgroup₁</code> and define a <code class="docutils literal notranslate">SubgroupClass₁</code> class. + Another very important thing to know about subobjects of a given algebraic object in Mathlib + always form a complete lattice, and this structure is used a lot. For instance you may look for + the lemma saying that an intersection of submonoids is a submonoid. But this won’t be a lemma, + this will be an infimum construction. Let us do the case of two submonoids. + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance [Monoid M] : Inf (Submonoid₁ M) := + ⟨fun S₁ S₂ => + { carrier := S₁ ∩ S₂ + one_mem := ⟨S₁.one_mem, S₂.one_mem⟩ + mul_mem := fun ⟨hx, hx'⟩ ⟨hy, hy'⟩ ↦ ⟨S₁.mul_mem hx hy, S₂.mul_mem hx' hy'⟩ }⟩ + </pre></div> + </div> + This allows to get the intersections of two submonoids as a submonoid. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [Monoid M] (N P : Submonoid₁ M) : Submonoid₁ M := N ⊓ P + </pre></div> + </div> + You may think it’s a shame that we had to use the inf symbol <code class="docutils literal notranslate">⊓</code> in the above example instead + of the intersection symbol <code class="docutils literal notranslate">∩</code>. But think about the supremum. The union of two submonoids is not + a submonoid. However submonoids still form a lattice (even a complete one). Actually <code class="docutils literal notranslate">N ⊔ P</code> is + the submonoid generated by the union of <code class="docutils literal notranslate">N</code> and <code class="docutils literal notranslate">P</code> and of course it would be very confusing to + denote it by <code class="docutils literal notranslate">N ∪ P</code>. So you can see the use of <code class="docutils literal notranslate">N ⊓ P</code> as much more consistent. It is also + a lot more consistent across various kind of algebraic structures. It may look a bit weird at first + to see the sum of two vector subspace <code class="docutils literal notranslate">E</code> and <code class="docutils literal notranslate">F</code> denoted by <code class="docutils literal notranslate">E ⊔ F</code> instead of <code class="docutils literal notranslate">E + F</code>. + But you will get used to it. And soon you will consider the <code class="docutils literal notranslate">E + F</code> notation as a distraction + emphasizing the anecdotal fact that elements of <code class="docutils literal notranslate">E ⊔ F</code> can be written as a sum of an element of + <code class="docutils literal notranslate">E</code> and an element of <code class="docutils literal notranslate">F</code> instead of emphasizing the fundamental fact that <code class="docutils literal notranslate">E ⊔ F</code> is the + smallest vector subspace containing both <code class="docutils literal notranslate">E</code> and <code class="docutils literal notranslate">F</code>. + Our last topic for this chapter is that of quotients. Again we want to explain how + convenient notation are built and code duplication is avoided in Mathlib. Here the main device + is the <code class="docutils literal notranslate">HasQuotient</code> class which allows notations like <code class="docutils literal notranslate">M ⧸ N</code>. Beware the quotient symbol + <code class="docutils literal notranslate">⧸</code> is a special unicode character, not a regular ASCII division symbol. + As an example, we will build the quotient of a commutative monoid by a submonoid, leave proofs + to you. + <div class="highlight-lean notranslate"><div class="highlight"><pre>def Submonoid.Setoid [CommMonoid M] (N : Submonoid M) : Setoid M where + r := fun x y ↦ ∃ w ∈ N, ∃ z ∈ N, x*w = y*z + iseqv := { + refl := fun x ↦ ⟨1, N.one_mem, 1, N.one_mem, rfl⟩ + symm := fun ⟨w, hw, z, hz, h⟩ ↦ ⟨z, hz, w, hw, h.symm⟩ + trans := by + sorry + } + + instance [CommMonoid M] : HasQuotient M (Submonoid M) where + quotient' := fun N ↦ Quotient N.Setoid + + def QuotientMonoid.mk [CommMonoid M] (N : Submonoid M) : M → M ⧸ N := Quotient.mk N.Setoid + + instance [CommMonoid M] (N : Submonoid M) : Monoid (M ⧸ N) where + mul := Quotient.map₂' (· * ·) (by + sorry + ) + mul_assoc := by + sorry + one := QuotientMonoid.mk N 1 + one_mul := by + sorry + mul_one := by + sorry + </pre></div> + </div> + -/ + </section> + </section> + + + </div> + </div> + <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> + <a href="C06_Structures.html" class="btn btn-neutral float-left" title="6. 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Differential Calculus" href="C09_Differential_Calculus.html" /> + <link rel="prev" title="7. Hierarchies" href="C07_Hierarchies.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" aria-label="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_Structures.html">6. Structures</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> + <li class="toctree-l1 current"><a class="current reference internal" href="#">8. Topology</a><ul> + <li class="toctree-l2"><a class="reference internal" href="#filters">8.1. Filters</a></li> + <li class="toctree-l2"><a class="reference internal" href="#metric-spaces">8.2. Metric spaces</a><ul> + <li class="toctree-l3"><a class="reference internal" href="#convergence-and-continuity">8.2.1. Convergence and continuity</a></li> + <li class="toctree-l3"><a class="reference internal" href="#balls-open-sets-and-closed-sets">8.2.2. Balls, open sets and closed sets</a></li> + <li class="toctree-l3"><a class="reference internal" href="#compactness">8.2.3. Compactness</a></li> + <li class="toctree-l3"><a class="reference internal" href="#uniformly-continuous-functions">8.2.4. Uniformly continuous functions</a></li> + <li class="toctree-l3"><a class="reference internal" href="#completeness">8.2.5. Completeness</a></li> + </ul> + </li> + <li class="toctree-l2"><a class="reference internal" href="#topological-spaces">8.3. Topological spaces</a><ul> + <li class="toctree-l3"><a class="reference internal" href="#fundamentals">8.3.1. Fundamentals</a></li> + <li class="toctree-l3"><a class="reference internal" href="#separation-and-countability">8.3.2. Separation and countability</a></li> + <li class="toctree-l3"><a class="reference internal" href="#id5">8.3.3. Compactness</a></li> + </ul> + </li> + </ul> + </li> + <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</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" aria-label="Home"></a></li> + <li class="breadcrumb-item active">8. Topology</li> + <li class="wy-breadcrumbs-aside"> + <a href="_sources/C08_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>8. 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_Structures.html#structures">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 8.1</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>8.1. 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 ×ˢ 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>8.2. 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>8.2.1. 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>8.2.2. 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>8.2.3. 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>8.2.4. 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>8.2.5. 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>8.3. Topological spaces<a class="headerlink" href="#topological-spaces" title="Permalink to this heading"></a></h2> + <section id="fundamentals"> + <h3>8.3.1. 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>8.3.2. 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>8.3.3. 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="C07_Hierarchies.html" class="btn btn-neutral float-left" title="7. 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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_Structures.html">6. Structures</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> + <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> + <li class="toctree-l1 current"><a class="current reference internal" href="#">9. Differential Calculus</a><ul> + <li class="toctree-l2"><a class="reference internal" href="#elementary-differential-calculus">9.1. Elementary Differential Calculus</a></li> + <li class="toctree-l2"><a class="reference internal" href="#differential-calculus-in-normed-spaces">9.2. Differential Calculus in Normed Spaces</a><ul> + <li class="toctree-l3"><a class="reference internal" href="#id3">9.2.1. Normed spaces</a></li> + <li class="toctree-l3"><a class="reference internal" href="#continuous-linear-maps">9.2.2. Continuous linear maps</a></li> + <li class="toctree-l3"><a class="reference internal" href="#asymptotic-comparisons">9.2.3. Asymptotic comparisons</a></li> + <li class="toctree-l3"><a class="reference internal" href="#differentiability">9.2.4. 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Differential Calculus</li> + <li class="wy-breadcrumbs-aside"> + <a href="_sources/C09_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>9. 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 <a class="reference internal" href="#elementary-differential-calculus">Section 9.1</a>, we stick with the + setting of functions from the real numbers to the real numbers, + which is familiar from any introductory calculus class. + In <a class="reference internal" href="#normed-spaces">Section 9.2</a>, we then consider the notion of a derivative in + a much broader setting. + <section id="elementary-differential-calculus"> + <h2>9.1. Elementary Differential Calculus<a class="headerlink" href="#elementary-differential-calculus" title="Permalink to this heading"></a></h2> + Let <code class="docutils literal notranslate">f</code> be a function from the reals to the reals. There is a difference + between talking about the derivative of <code class="docutils literal notranslate">f</code> at a single point and + talking about the derivative function. + In mathlib, the first notion is represented as follows. + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Real + + /-- The sin function has derivative 1 at 0. -/ + example : HasDerivAt sin 1 0 := by simpa using hasDerivAt_sin 0 + </pre></div> + </div> + We can also express that <code class="docutils literal notranslate">f</code> is differentiable at a point without + specifying its derivative there + by writing <code class="docutils literal notranslate">differentiable_at ℝ</code>. + We specify <code class="docutils literal notranslate">ℝ</code> explicitly because in a slightly more general context, + when talking about functions from <code class="docutils literal notranslate">ℂ</code> to <code class="docutils literal notranslate">ℂ</code>, + we want to be able to distinguish between being differentiable in the real sense + and being differentiable in the sense of the complex derivative. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x : ℝ) : DifferentiableAt ℝ sin x := + (hasDerivAt_sin x).differentiableAt + </pre></div> + </div> + It would be inconvenient to have to provide a proof of differentiability + every time we want to refer to a derivative. + So mathlib provides a function <code class="docutils literal notranslate">deriv f : ℝ → ℝ</code> that is defined for any + function <code class="docutils literal notranslate">f : ℝ → ℝ</code> + but is defined to take the value <code class="docutils literal notranslate">0</code> at any point where <code class="docutils literal notranslate">f</code> is not differentiable. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {f : ℝ → ℝ} {x a : ℝ} (h : HasDerivAt f a x) : deriv f x = a := + h.deriv + + example {f : ℝ → ℝ} {x : ℝ} (h : ¬DifferentiableAt ℝ f x) : deriv f x = 0 := + deriv_zero_of_not_differentiableAt h + </pre></div> + </div> + Of course there are many lemmas about <code class="docutils literal notranslate">deriv</code> that do require differentiability assumptions. + For instance, you should think about a counterexample to the next lemma without the + differentiability assumptions. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {f g : ℝ → ℝ} {x : ℝ} (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) : + deriv (f + g) x = deriv f x + deriv g x := + deriv_add hf hg + </pre></div> + </div> + Interestingly, however, there are statements that can avoid differentiability + assumptions by taking advantage + of the fact that the value of <code class="docutils literal notranslate">deriv</code> defaults to zero when the function is + not differentiable. + So making sense of the following statement requires knowing the precise + definition of <code class="docutils literal notranslate">deriv</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {f : ℝ → ℝ} {a : ℝ} (h : IsLocalMin f a) : deriv f a = 0 := + h.deriv_eq_zero + </pre></div> + </div> + We can eve state Rolle’s theorem without any differentiability assumptions, which + seems even weirder. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {f : ℝ → ℝ} {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hfI : f a = f b) : + ∃ c ∈ Ioo a b, deriv f c = 0 := + exists_deriv_eq_zero f hab hfc hfI + </pre></div> + </div> + Of course, this trick does not work for the general mean value theorem. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (f : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hf : ContinuousOn f (Icc a b)) + (hf' : DifferentiableOn ℝ f (Ioo a b)) : ∃ c ∈ Ioo a b, deriv f c = (f b - f a) / (b - a) := + exists_deriv_eq_slope f hab hf hf' + </pre></div> + </div> + Lean can automatically compute some simple derivatives using the <code class="docutils literal notranslate">simp</code> tactic. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : deriv (fun x : ℝ ↦ x ^ 5) 6 = 5 * 6 ^ 4 := by simp + + example : deriv sin π = -1 := by simp + </pre></div> + </div> + </section> + <section id="differential-calculus-in-normed-spaces"> + <h2>9.2. Differential Calculus in Normed Spaces<a class="headerlink" href="#differential-calculus-in-normed-spaces" title="Permalink to this heading"></a></h2> + <section id="id3"> + <h3>9.2.1. Normed spaces<a class="headerlink" href="#id3" title="Permalink to this heading"></a></h3> + Differentiation can be generalized beyond <code class="docutils literal notranslate">ℝ</code> using the notion of a + normed vector space, which encapsulates both direction and distance. + We start with the notion of a normed group, which as an additive commutative + group equipped with a real-valued norm function + satisfying the following conditions. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {E : Type _} [NormedAddCommGroup E] + + example (x : E) : 0 ≤ ‖x‖ := + norm_nonneg x + + example {x : E} : ‖x‖ = 0 ↔ x = 0 := + norm_eq_zero + + example (x y : E) : ‖x + y‖ ≤ ‖x‖ + ‖y‖ := + norm_add_le x y + </pre></div> + </div> + Every normed space is a metric space with distance function + $d(x, y) = \| x - y \|$, and hence it is also a topological space. + Lean and mathlib know this. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : MetricSpace E := by infer_instance + + example {X : Type _} [TopologicalSpace X] {f : X → E} (hf : Continuous f) : + Continuous fun x => ‖f x‖ := + hf.norm + </pre></div> + </div> + In order to use the notion of a norm with concepts from linear algebra, + we add the assumption <code class="docutils literal notranslate">normed_space ℝ E</code> on top of <code class="docutils literal notranslate">normed_add_group E</code>. + This stipulates that <code class="docutils literal notranslate">E</code> is a vector space over <code class="docutils literal notranslate">ℝ</code> and that + scalar multiplication satisfies the following condition. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable [NormedSpace ℝ E] + + example (a : ℝ) (x : E) : ‖a • x‖ = |a| * ‖x‖ := + norm_smul a x + </pre></div> + </div> + A complete normed space is known as a Banach space. + Every finite-dimensional vector space is complete. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [FiniteDimensional ℝ E] : CompleteSpace E := by infer_instance + </pre></div> + </div> + In all the previous examples, we used the real numbers as the base field. + More generally, we can make sense of calculus with a vector space over any + non-discrete normed field. These are fields that are equipped with a + real-valued norm that is multiplicative and has the property that + not every element has norm zero or one + (equivalently, there is an element whose norm is bigger than one). + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (𝕜 : Type _) [NontriviallyNormedField 𝕜] (x y : 𝕜) : ‖x * y‖ = ‖x‖ * ‖y‖ := + norm_mul x y + + example (𝕜 : Type _) [NontriviallyNormedField 𝕜] : ∃ x : 𝕜, 1 < ‖x‖ := + NormedField.exists_one_lt_norm 𝕜 + </pre></div> + </div> + A finite-dimensional vector space over a nondiscrete normed field is + complete as long as the field itself is complete. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (𝕜 : Type _) [NontriviallyNormedField 𝕜] (E : Type _) [NormedAddCommGroup E] + [NormedSpace 𝕜 E] [CompleteSpace 𝕜] [FiniteDimensional 𝕜 E] : CompleteSpace E := + FiniteDimensional.complete 𝕜 E + </pre></div> + </div> + </section> + <section id="continuous-linear-maps"> + <h3>9.2.2. Continuous linear maps<a class="headerlink" href="#continuous-linear-maps" title="Permalink to this heading"></a></h3> + We now turn to the morphisms in the category of normed spaces, namely, + continuous linear maps. + In mathlib, the type of <code class="docutils literal notranslate">𝕜</code>-linear continuous maps between normed spaces + <code class="docutils literal notranslate">E</code> and <code class="docutils literal notranslate">F</code> is written <code class="docutils literal notranslate">E →L[𝕜] F</code>. + They are implemented as bundled maps, which means that an element of this type + a structure that that includes the function itself and the properties + of being linear and continuous. + Lean will insert a coercion so that a continuous linear map can be treated + as a function. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] + [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] + + example : E →L[𝕜] E := + ContinuousLinearMap.id 𝕜 E + + example (f : E →L[𝕜] F) : E → F := + f + + example (f : E →L[𝕜] F) : Continuous f := + f.cont + + example (f : E →L[𝕜] F) (x y : E) : f (x + y) = f x + f y := + f.map_add x y + + example (f : E →L[𝕜] F) (a : 𝕜) (x : E) : f (a • x) = a • f x := + f.map_smul a x + </pre></div> + </div> + Continuous linear maps have an operator norm that is characterized by the + following properties. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (f : E →L[𝕜] F) + + example (x : E) : ‖f x‖ ≤ ‖f‖ * ‖x‖ := + f.le_op_norm x + + example {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ x, ‖f x‖ ≤ M * ‖x‖) : ‖f‖ ≤ M := + f.op_norm_le_bound hMp hM + </pre></div> + </div> + There is also a notion of bundled continuous linear isomorphism. + Their type of such isomorphisms is <code class="docutils literal notranslate">E ≃L[𝕜] F</code>. + As a challenging exercise, you can prove the Banach-Steinhaus theorem, also + known as the Uniform Boundedness Principle. + The principle states that a family of continuous linear maps from a Banach space + into a normed space is pointwise + bounded, then the norms of these linear maps are uniformly bounded. + The main ingredient is Baire’s theorem + <code class="docutils literal notranslate">nonempty_interior_of_Union_of_closed.</code> (You proved a version of this in the topology chapter.) + Minor ingredients include <code class="docutils literal notranslate">continuous_linear_map.op_norm_le_of_shell</code>, + <code class="docutils literal notranslate">interior_subset</code> and <code class="docutils literal notranslate">interior_Inter_subset</code> and <code class="docutils literal notranslate">is_closed_le</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] + [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] + + open Metric + + example {ι : Type _} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) : + ∃ C', ∀ i, ‖g i‖ ≤ C' := by + -- sequence of subsets consisting of those `x : E` with norms `‖g i x‖` bounded by `n` + let e : ℕ → Set E := fun n => ⋂ i : ι, { x : E | ‖g i x‖ ≤ n } + -- each of these sets is closed + have hc : ∀ n : ℕ, IsClosed (e n) + sorry + -- the union is the entire space; this is where we use `h` + have hU : (⋃ n : ℕ, e n) = univ + sorry + /- apply the Baire category theorem to conclude that for some `m : ℕ`, + `e m` contains some `x` -/ + obtain ⟨m, x, hx⟩ : ∃ m, ∃ x, x ∈ interior (e m) := sorry + obtain ⟨ε, ε_pos, hε⟩ : ∃ ε > 0, ball x ε ⊆ interior (e m) := sorry + obtain ⟨k, hk⟩ : ∃ k : 𝕜, 1 < ‖k‖ := sorry + -- show all elements in the ball have norm bounded by `m` after applying any `g i` + have real_norm_le : ∀ z ∈ ball x ε, ∀ (i : ι), ‖g i z‖ ≤ m + sorry + have εk_pos : 0 < ε / ‖k‖ := sorry + refine' ⟨(m + m : ℕ) / (ε / ‖k‖), fun i => ContinuousLinearMap.op_norm_le_of_shell ε_pos _ hk _⟩ + sorry + sorry + </pre></div> + </div> + </section> + <section id="asymptotic-comparisons"> + <h3>9.2.3. Asymptotic comparisons<a class="headerlink" href="#asymptotic-comparisons" title="Permalink to this heading"></a></h3> + Defining differentiability also requires asymptotic comparisons. + Mathlib has an extensive library covering the big O and little o relations, + whose definitions are shown below. + Opening the <code class="docutils literal notranslate">asymptotics</code> locale allows us to use the corresponding + notation. + Here we will only use little o to define differentiability. + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Asymptotics + + open Asymptotics + + example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] (c : ℝ) + (l : Filter α) (f : α → E) (g : α → F) : IsBigOWith c l f g ↔ ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := + isBigOWith_iff + + example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] + (l : Filter α) (f : α → E) (g : α → F) : f =O[l] g ↔ ∃ C, IsBigOWith C l f g := + isBigO_iff_isBigOWith + + example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] + (l : Filter α) (f : α → E) (g : α → F) : f =o[l] g ↔ ∀ C > 0, IsBigOWith C l f g := + isLittleO_iff_forall_isBigOWith + + example {α : Type _} {E : Type _} [NormedAddCommGroup E] (l : Filter α) (f g : α → E) : + f ~[l] g ↔ (f - g) =o[l] g := + Iff.rfl + </pre></div> + </div> + </section> + <section id="differentiability"> + <h3>9.2.4. Differentiability<a class="headerlink" href="#differentiability" title="Permalink to this heading"></a></h3> + We are now ready to discuss differentiable functions between normed spaces. + In analogy the elementary one-dimensional, + mathlib defines a predicate <code class="docutils literal notranslate">has_fderiv_at</code> and a function <code class="docutils literal notranslate">fderiv</code>. + Here the letter + “f” stands for Fréchet. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] + [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] + + example (f : E → F) (f' : E →L[𝕜] F) (x₀ : E) : + HasFDerivAt f f' x₀ ↔ (fun x => f x - f x₀ - f' (x - x₀)) =o[𝓝 x₀] fun x => x - x₀ := + Iff.rfl + + example (f : E → F) (f' : E →L[𝕜] F) (x₀ : E) (hff' : HasFDerivAt f f' x₀) : fderiv 𝕜 f x₀ = f' := + hff'.fderiv + </pre></div> + </div> + We also have iterated derivatives that take values in the type of multilinear maps + <code class="docutils literal notranslate">E [×n]→L[𝕜] F</code>, + and we have continuously differential functions. + The type <code class="docutils literal notranslate">with_top ℕ</code> is <code class="docutils literal notranslate">ℕ</code> with an additional element <code class="docutils literal notranslate">⊤</code> that + is bigger than every natural number. + So $\mathcal{C}^\infty$ functions are functions <code class="docutils literal notranslate">f</code> that satisfy + <code class="docutils literal notranslate">cont_diff 𝕜 ⊤ f</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (n : ℕ) (f : E → F) : E → E[×n]→L[𝕜] F := + iteratedFDeriv 𝕜 n f + + example (n : WithTop ℕ) {f : E → F} : + ContDiff 𝕜 n f ↔ + (∀ m : ℕ, (m : WithTop ℕ) ≤ n → Continuous fun x => iteratedFDeriv 𝕜 m f x) ∧ + ∀ m : ℕ, (m : WithTop ℕ) < n → Differentiable 𝕜 fun x => iteratedFDeriv 𝕜 m f x := + contDiff_iff_continuous_differentiable + </pre></div> + </div> + There is a stricter notion of differentiability called + <code class="docutils literal notranslate">has_strict_fderiv_at</code>, which is used in the statement + of the inverse function theorem and the statement of the implicit function + theorem, both of which are in mathlib. + Over <code class="docutils literal notranslate">ℝ</code> or <code class="docutils literal notranslate">ℂ</code>, continuously differentiable + functions are strictly differentiable. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {𝕂 : Type _} [IsROrC 𝕂] {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕂 E] {F : Type _} + [NormedAddCommGroup F] [NormedSpace 𝕂 F] {f : E → F} {x : E} {n : WithTop ℕ} + (hf : ContDiffAt 𝕂 n f x) (hn : 1 ≤ n) : HasStrictFDerivAt f (fderiv 𝕂 f x) x := + hf.hasStrictFDerivAt hn + </pre></div> + </div> + The local inverse theorem is stated using an operation that produces an + inverse function from a + function and the assumptions that the function is strictly differentiable at a + point <code class="docutils literal notranslate">a</code> and that its derivative is an isomorphism. + The first example below gets this local inverse. + The next one states that it is indeed a local inverse + from the left and from the right, and that it is strictly differentiable. + <div class="highlight-lean notranslate"><div class="highlight"><pre>section LocalInverse + variable [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} + + example (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : F → E := + HasStrictFDerivAt.localInverse f f' a hf + + example (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : + ∀ᶠ x in 𝓝 a, hf.localInverse f f' a (f x) = x := + hf.eventually_left_inverse + + example (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : + ∀ᶠ x in 𝓝 (f a), f (hf.localInverse f f' a x) = x := + hf.eventually_right_inverse + + example [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} + (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : + HasStrictFDerivAt (HasStrictFDerivAt.localInverse f f' a hf) (f'.symm : F →L[𝕜] E) (f a) := + HasStrictFDerivAt.to_localInverse hf + + end LocalInverse + </pre></div> + </div> + This has been only a quick tour of the differential calculus in mathlib. + The library contains many variations that we have not discussed. + For example, you may want to use one-sided derivatives in the + one-dimensional setting. The means to do so are found in mathlib in a more + general context; + see <code class="docutils literal notranslate">HasFDerivWithinAt</code> or the even more general <code class="docutils literal notranslate">HasFDerivAtFilter</code>. + </section> + </section> + </section> + + + </div> + </div> + <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> + <a href="C08_Topology.html" class="btn btn-neutral float-left" title="8. Topology" 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. 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 + <!DOCTYPE html> + <html class="writer-html5" lang="en" > + <head> + <meta charset="utf-8" /><meta name="viewport" content="width=device-width, initial-scale=1" /> + + <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 src="_static/jquery.js"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> + <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> + <script src="_static/doctools.js"></script> + <script src="_static/sphinx_highlight.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" aria-label="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_Structures.html">6. Structures</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> + <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</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" aria-label="Home"></a></li> + <li class="breadcrumb-item active">Integration and Measure Theory</li> + <li class="wy-breadcrumbs-aside"> + <a href="_sources/C10_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>
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html/_sources/C01_Introduction.rst.txt (new)
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  + .. _introduction: + + Introduction + ============ + + .. include:: C01_Introduction/S01_Getting_Started.inc + .. include:: C01_Introduction/S02_Overview.inc
html/_sources/C02_Basics.rst.txt (new)
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  + .. _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)
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  + .. _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)
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  + .. _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)
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  + .. _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_Structures.rst.txt (new)
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 + .. _structures: + + Structures + ========== + + 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_Structures/S01_Structures.inc + .. include:: C06_Structures/S02_Algebraic_Structures.inc + .. include:: C06_Structures/S03_Building_the_Gaussian_Integers.inc
html/_sources/C07_Hierarchies.rst.txt (new)
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 + .. _hierarchies: + + Hierarchies + =========== + + We have seen in :numref:`Chapter %s <structures>` how to define the class + of groups and build instances of this class, and then how to build an instance + of the commutative ring class. But of course there is a hierarchy here: a + commutative ring is in particular an additive group. In this chapter we + will study how to build such hierarchies. They appear in all branches + of mathematics but in this chapter the emphasis will be on algebraic examples. + + It may seem premature to discuss how to build hierarchies before more discussions + about using existing hierarchies. But some understanding of the technology underlying + hierarchies is required to use them. So you should probably still read this chapter, + but without trying too hard to remember everything on your first read, then read + the following chapters and come back here for a second reading. + + In this chapter, we will redefine (simpler versions of) many things that appear in Mathlib + so we will used indices to distinguish our version. For instance we will have ``Ring₁`` + as our version of ``Ring``. Since we will gradually explain more powerful ways of formalizing + structures, those indices will sometimes grow beyond one. + + .. include:: C07_Hierarchies/S01_Basics.inc + .. include:: C07_Hierarchies/S02_Morphisms.inc + .. include:: C07_Hierarchies/S03_Subobjects.inc
html/_sources/C08_Topology.rst.txt (new)
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 + .. _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 <structures>`, + 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:: C08_Topology/S01_Filters.inc + + .. include:: C08_Topology/S02_Metric_Spaces.inc + + .. include:: C08_Topology/S03_Topological_Spaces.inc
html/_sources/C09_Differential_Calculus.rst.txt (new)
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  + .. _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:: C09_Differential_Calculus/S01_Elementary_Differential_Calculus.inc + .. include:: C09_Differential_Calculus/S02_Differential_Calculus_in_Normed_Spaces.inc
html/_sources/C10_Integration_and_Measure_Theory.rst.txt (new)
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  + .. _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)
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  + Index + =====
html/_sources/index.rst.txt (new)
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  + + + Mathematics in Lean + =================== + + .. toctree:: + :numbered: + :maxdepth: 2 + + C01_Introduction + C02_Basics + C03_Logic + C04_Sets_and_Functions + C05_Number_Theory + C06_Structures + C07_Hierarchies + C08_Topology + C09_Differential_Calculus + + .. toctree:: + :hidden: + + genindex
html/_static/_sphinx_javascript_frameworks_compat.js (new)
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+ 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, + nav.contents > :last-child, + aside.topic > :last-child, + div.topic > :last-child, + div.admonition > :last-child { + margin-bottom: 0; + } + + div.sidebar::after, + aside.sidebar::after, + nav.contents::after, + aside.topic::after, + div.topic::after, + div.admonition::after, + blockquote::after { + display: block; + content: ''; + clear: both; + } + + /* -- tables ---------------------------------------------------------------- */ + + table.docutils { + margin-top: 10px; 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html/_static/documentation_options.js (new)
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html/_static/js/theme.js (new)
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html/_static/language_data.js (new)
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 + /* + * 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 2007-2023 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 = /.$/; + w = w.replace(re,""); + } + + // and turn initial Y back to y + if (firstch == "y") + w = firstch.toLowerCase() + w.substr(1); + return w; + } + } +
html/_static/minus.png (new)
html/_static/plus.png (new)
html/_static/pygments.css (new)
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html/_static/searchtools.js (new)
- @@ -0,0 +1,566 @@
 + /* + * searchtools.js + * ~~~~~~~~~~~~~~~~ + * + * Sphinx JavaScript utilities for the full-text search. + * + * :copyright: Copyright 2007-2023 by the Sphinx team, see AUTHORS. + * :license: BSD, see LICENSE for details. + * + */ + "use strict"; + + /** + * Simple result scoring code. + */ + if (typeof Scorer === "undefined") { + var Scorer = { + // Implement the following function to further tweak the score for each result + // The function takes a result array [docname, title, anchor, descr, score, filename] + // and returns the new score. + /* + score: result => { + const [docname, title, anchor, descr, score, filename] = result + return score + }, + */ + + // query matches the full name of an object + objNameMatch: 11, + // or matches in the last dotted part of the object name + objPartialMatch: 6, + // Additive scores depending on the priority of the object + objPrio: { + 0: 15, // used to be importantResults + 1: 5, // used to be objectResults + 2: -5, // used to be unimportantResults + }, + // Used when the priority is not in the mapping. + objPrioDefault: 0, + + // query found in title + title: 15, + partialTitle: 7, + // query found in terms + term: 5, + partialTerm: 2, + }; + } + + const _removeChildren = (element) => { + while (element && element.lastChild) element.removeChild(element.lastChild); + }; + + /** + * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions#escaping + */ + const _escapeRegExp = (string) => + string.replace(/[.*+\-?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string + + const _displayItem = (item, searchTerms) => { + const docBuilder = DOCUMENTATION_OPTIONS.BUILDER; + const docUrlRoot = DOCUMENTATION_OPTIONS.URL_ROOT; + const docFileSuffix = DOCUMENTATION_OPTIONS.FILE_SUFFIX; + const docLinkSuffix = DOCUMENTATION_OPTIONS.LINK_SUFFIX; + const showSearchSummary = DOCUMENTATION_OPTIONS.SHOW_SEARCH_SUMMARY; + + const [docName, title, anchor, descr, score, _filename] = item; + + let listItem = document.createElement("li"); + let requestUrl; + let linkUrl; + if (docBuilder === "dirhtml") { + // dirhtml builder + let dirname = docName + "/"; + if (dirname.match(/\/index\/$/)) + 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; + } + let linkEl = listItem.appendChild(document.createElement("a")); + linkEl.href = linkUrl + anchor; + linkEl.dataset.score = score; + linkEl.innerHTML = title; + if (descr) + listItem.appendChild(document.createElement("span")).innerHTML = + " (" + descr + ")"; + else if (showSearchSummary) + fetch(requestUrl) + .then((responseData) => responseData.text()) + .then((data) => { + if (data) + listItem.appendChild( + Search.makeSearchSummary(data, searchTerms) + ); + }); + 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, + 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(), searchTerms); + setTimeout( + () => _displayNextItem(results, resultCount, 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 = new DOMParser().parseFromString(htmlString, 'text/html'); + htmlElement.querySelectorAll(".headerlink").forEach((el) => { el.remove() }); + 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) => { + const filenames = Search._index.filenames; + const docNames = Search._index.docnames; + const titles = Search._index.titles; + const allTitles = Search._index.alltitles; + const indexEntries = Search._index.indexentries; + + // 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); + } + }); + + if (SPHINX_HIGHLIGHT_ENABLED) { // set in sphinx_highlight.js + localStorage.setItem("sphinx_highlight_terms", [...highlightTerms].join(" ")) + } + + // 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")); + + const queryLower = query.toLowerCase(); + for (const [title, foundTitles] of Object.entries(allTitles)) { + if (title.toLowerCase().includes(queryLower) && (queryLower.length >= title.length/2)) { + for (const [file, id] of foundTitles) { + let score = Math.round(100 * queryLower.length / title.length) + results.push([ + docNames[file], + titles[file] !== title ? `${titles[file]} > ${title}` : title, + id !== null ? "#" + id : "", + null, + score, + filenames[file], + ]); + } + } + } + + // search for explicit entries in index directives + for (const [entry, foundEntries] of Object.entries(indexEntries)) { + if (entry.includes(queryLower) && (queryLower.length >= entry.length/2)) { + for (const [file, id] of foundEntries) { + let score = Math.round(100 * queryLower.length / entry.length) + results.push([ + docNames[file], + titles[file], + id ? "#" + id : "", + null, + score, + filenames[file], + ]); + } + } + } + + // 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, 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 filenames = Search._index.filenames; + const docNames = Search._index.docnames; + 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. + */ + makeSearchSummary: (htmlText, keywords) => { + const text = Search.htmlToText(htmlText); + if (text === "") return null; + + const textLower = text.toLowerCase(); + const actualStartPosition = [...keywords] + .map((k) => textLower.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("p"); + summary.classList.add("context"); + summary.textContent = top + text.substr(startWithContext, 240).trim() + tail; + + return summary; + }, + }; + + _ready(Search.init);
html/_static/sphinx_highlight.js (new)
- @@ -0,0 +1,144 @@
 + /* Highlighting utilities for Sphinx HTML documentation. */ + "use strict"; + + const SPHINX_HIGHLIGHT_ENABLED = true + + /** + * 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 SphinxHighlight = { + + /** + * highlight the search words provided in localstorage in the text + */ + highlightSearchWords: () => { + if (!SPHINX_HIGHLIGHT_ENABLED) return; // bail if no highlight + + // get and clear terms from localstorage + const url = new URL(window.location); + const highlight = + localStorage.getItem("sphinx_highlight_terms") + || url.searchParams.get("highlight") + || ""; + localStorage.removeItem("sphinx_highlight_terms") + url.searchParams.delete("highlight"); + window.history.replaceState({}, "", url); + + // get individual terms from highlight string + 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:SphinxHighlight.hideSearchWords()">' + + _("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")); + localStorage.removeItem("sphinx_highlight_terms") + }, + + initEscapeListener: () => { + // only install a listener if it is really needed + if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS) return; + + document.addEventListener("keydown", (event) => { + // bail for input elements + if (BLACKLISTED_KEY_CONTROL_ELEMENTS.has(document.activeElement.tagName)) return; + // bail with special keys + if (event.shiftKey || event.altKey || event.ctrlKey || event.metaKey) return; + if (DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS && (event.key === "Escape")) { + SphinxHighlight.hideSearchWords(); + event.preventDefault(); + } + }); + }, + }; + + _ready(SphinxHighlight.highlightSearchWords); + _ready(SphinxHighlight.initEscapeListener);
html/genindex.html (new)
- @@ -0,0 +1,553 @@
 + <!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 src="_static/jquery.js"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> + <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> + <script src="_static/doctools.js"></script> + <script src="_static/sphinx_highlight.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" aria-label="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_Structures.html">6. Structures</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> + <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</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" aria-label="Home"></a></li> + <li class="breadcrumb-item active">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="C08_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="C09_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="C09_Differential_Calculus.html#index-1">elementary calculus</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-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="C08_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="C10_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="C08_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> + <li><a href="C09_Differential_Calculus.html#index-2">normed space</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="C08_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="C08_Topology.html#index-4">topological space</a> + </li> + <li><a href="C08_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. 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 + <!DOCTYPE html> + <html class="writer-html5" lang="en" > + <head> + <meta charset="utf-8" /><meta name="viewport" content="width=device-width, initial-scale=1" /> + + <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 src="_static/jquery.js"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> + <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> + <script src="_static/doctools.js"></script> + <script src="_static/sphinx_highlight.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" aria-label="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_Structures.html">6. Structures</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> + <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</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" aria-label="Home"></a></li> + <li class="breadcrumb-item active">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_Structures.html">6. Structures</a><ul> + <li class="toctree-l2"><a class="reference internal" href="C06_Structures.html#defining-structures">6.1. Defining structures</a></li> + <li class="toctree-l2"><a class="reference internal" href="C06_Structures.html#algebraic-structures">6.2. Algebraic Structures</a></li> + <li class="toctree-l2"><a class="reference internal" href="C06_Structures.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_Hierarchies.html">7. Hierarchies</a><ul> + <li class="toctree-l2"><a class="reference internal" href="C07_Hierarchies.html#basics">7.1. Basics</a></li> + <li class="toctree-l2"><a class="reference internal" href="C07_Hierarchies.html#morphisms">7.2. Morphisms</a></li> + <li class="toctree-l2"><a class="reference internal" href="C07_Hierarchies.html#sub-objects">7.3. Sub-objects</a></li> + </ul> + </li> + <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a><ul> + <li class="toctree-l2"><a class="reference internal" href="C08_Topology.html#filters">8.1. Filters</a></li> + <li class="toctree-l2"><a class="reference internal" href="C08_Topology.html#metric-spaces">8.2. Metric spaces</a></li> + <li class="toctree-l2"><a class="reference internal" href="C08_Topology.html#topological-spaces">8.3. Topological spaces</a></li> + </ul> + </li> + <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a><ul> + <li class="toctree-l2"><a class="reference internal" href="C09_Differential_Calculus.html#elementary-differential-calculus">9.1. Elementary Differential Calculus</a></li> + <li class="toctree-l2"><a class="reference internal" href="C09_Differential_Calculus.html#differential-calculus-in-normed-spaces">9.2. Differential Calculus in Normed 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. 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[[2, "index-24"]], "field_simp": [[2, "index-11"]], "from": [[2, "index-12"]], "injective function": [[2, "index-5"]], "intro": [[2, "index-0"]], "lambda abstraction": [[2, "index-1"]], "left": [[2, "index-21"]], "let": [[2, "index-13"]], "monotone function": [[2, "index-3"]], "push_neg": [[2, "index-15"]], "rcases": [[2, "index-9"]], "right": [[2, "index-21"]], "rintro": [[2, "index-9"]], "simp": [[2, "index-20"], [3, "index-1"]], "surjective function": [[2, "index-10"]], "tactic": [[2, "index-11"]], "this": [[2, "index-12"]], "use": [[2, "index-6"]], "bounded quantifiers": [[3, "index-3"]], "rwa": [[3, "index-2"]], "set operations": [[3, "index-0"]], "filter": [[7, "index-1"]], "continuity": [[7, "index-3"]], "topological space": [[7, "index-4"]], "topology": [[7, "index-0"]], "differential calculus": [[8, "index-0"]], "elementary calculus": [[8, "index-1"]], "normed space": [[8, "index-2"]], "integration": [[9, "index-0"]]}})
lake-manifest.json
- @@ -1,1 +1,27 @@
  - ../lake-manifest.json + {"version": 4, + "packagesDir": "lake-packages", + "packages": + [{"git": + {"url": "https://github.com/leanprover-community/mathlib4", + "subDir?": null, + "rev": "2db7650f98b1ea3543e48b6b02aa9112e33bb68b", + "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": "ca73109cc40837bc61df8024c9016da4b4f99d4c", + "name": "aesop", + "inputRev?": "master"}}, + {"git": + {"url": "https://github.com/leanprover/std4", + "subDir?": null, + "rev": "6932c4ea52914dc6b0488944e367459ddc4d01a6", + "name": "std", + "inputRev?": "main"}}]}
lakefile.lean
- @@ -1,1 +1,13 @@
  - ../lakefile.lean + 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
- @@ -1,1 +1,1 @@
  - ../lean-toolchain + leanprover/lean4:nightly-2023-05-31
mathematics_in_lean.pdf
src (deleted)
- @@ -1,1 +0,0 @@
  - ../src
src/C01_Introduction/S01_Getting_Started.lean (new)
- @@ -0,0 +1,2 @@
  + #eval "Hello, World!" +
src/C01_Introduction/S02_Overview.lean (new)
- @@ -0,0 +1,57 @@
  + 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 +1,147 @@
  + 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] + + + 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 +1,90 @@
  + import Mathlib.Data.Real.Basic + + namespace C02S04 + 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 +1,116 @@
  + import Mathlib.Data.Real.Basic + + namespace C02S04 + 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 +1,178 @@
 + import Mathlib.Data.Real.Basic + + namespace C03S01 + #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 fnUb_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 + + namespace C03S02 + 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 ubgb + use a + b + apply fnUb_add ubfa ubgb + + 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, ubgb⟩ + exact ⟨a + b, fnUb_add ubfa ubgb⟩ + + example : FnHasUb f → FnHasUb g → FnHasUb fun x => f x + g x := by + rintro ⟨a, ubfa⟩ ⟨b, ubgb⟩ + exact ⟨a + b, fnUb_add ubfa ubgb⟩ + + example : FnHasUb f → FnHasUb g → FnHasUb fun x => f x + g x := + fun ⟨a, ubfa⟩ ⟨b, ubgb⟩ => ⟨a + b, fnUb_add ubfa ubgb⟩ + + 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 +1,137 @@
 + import Mathlib.Data.Real.Basic + + namespace C03S03 + 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 +1,135 @@
 + import Mathlib.Data.Real.Basic + import Mathlib.Data.Nat.Prime + import Mathlib.Tactic.NormNum + + namespace C03S04 + 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 +1,101 @@
 + import Mathlib.Data.Real.Basic + + namespace C03S05 + 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 +1,99 @@
 + import Mathlib.Data.Real.Basic + + namespace C03S06 + 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 +1,129 @@
  + import Mathlib.Data.Real.Basic + + namespace C03S01 + 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 + + namespace C03S02 + 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 +1,111 @@
 + import Mathlib.Data.Real.Basic + + namespace C03S03 + 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 +1,94 @@
 + import Mathlib.Data.Real.Basic + import Mathlib.Data.Nat.Prime + import Mathlib.Tactic.NormNum + + namespace C03S04 + 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 +1,141 @@
 + import Mathlib.Data.Real.Basic + + namespace C03S05 + 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 +1,127 @@
 + import Mathlib.Data.Real.Basic + + namespace C03S06 + 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 +1,213 @@
  + 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 +1,98 @@
  + 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 +1,252 @@
  + 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 +1,121 @@
 + 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 +1,233 @@
 + import Mathlib.Data.Nat.Prime + import Mathlib.Algebra.BigOperators.Order + import Mathlib.Tactic + import Mathlib.Tactic.IntervalCases + + open BigOperators + + namespace C05S03 + 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 _root_.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 +1,105 @@
 + 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 +1,242 @@
 + import Mathlib.Data.Nat.Prime + import Mathlib.Algebra.BigOperators.Order + import Mathlib.Tactic + import Mathlib.Tactic.IntervalCases + + open BigOperators + + namespace C05S03 + 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 _root_.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_Structures/S01_Structures.lean (new)
- @@ -0,0 +1,224 @@
 + import Mathlib.Algebra.BigOperators.Ring + import Mathlib.Data.Real.Basic + + namespace C06S01 + 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_Structures/S02_Algebraic_Structures.lean (new)
- @@ -0,0 +1,171 @@
  + import Mathlib.Data.Real.Basic + + namespace C06S02 + 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_Structures/S03_Building_the_Gaussian_Integers.lean (new)
- @@ -0,0 +1,273 @@
 + 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 + + @[simp] + theorem sub_re (x y : gaussInt) : (x - y).re = x.re - y.re := + rfl + + @[simp] + theorem sub_im (x y : gaussInt) : (x - y).im = x.im - y.im := + rfl + + 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 H1 : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ + · ext <;> simp [Int.mod'_eq, mod_def, div_def, norm] <;> ring + have H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y + · calc + norm (x % y) * norm y = norm (x % y * conj y) := by simp only [norm_mul, norm_conj] + _ = abs (Int.mod' (x.re * y.re + x.im * y.im) (norm y)) ^ 2 + + abs (Int.mod' (-(x.re * y.im) + x.im * y.re) (norm y)) ^ 2 := by simp [H1, norm, sq_abs] + _ ≤ (y.norm / 2) ^ 2 + (y.norm / 2) ^ 2 := by gcongr <;> apply Int.abs_mod'_le _ _ norm_y_pos + _ = norm y / 2 * (norm y / 2 * 2) := by ring + _ ≤ norm y / 2 * norm y := by gcongr; apply Int.ediv_mul_le; norm_num + calc norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right H2 norm_y_pos + _ < norm y := by + 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_Structures/solutions/Solutions_S01_Structures.lean (new)
- @@ -0,0 +1,99 @@
  + import Mathlib.Algebra.BigOperators.Ring + import Mathlib.Data.Real.Basic + + namespace C06S01 + 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_Structures/solutions/Solutions_S02_Algebraic_Structures.lean (new)
- @@ -0,0 +1,74 @@
  + import Mathlib.Data.Real.Basic + + namespace C06S02 + 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_Structures/solutions/Solutions_S03_Building_the_Gaussian_Integers.lean (new)
- @@ -0,0 +1,287 @@
 + 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 + + @[simp] + theorem sub_re (x y : gaussInt) : (x - y).re = x.re - y.re := + rfl + + @[simp] + theorem sub_im (x y : gaussInt) : (x - y).im = x.im - y.im := + rfl + + 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 H1 : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ + · ext <;> simp [Int.mod'_eq, mod_def, div_def, norm] <;> ring + have H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y + · calc + norm (x % y) * norm y = norm (x % y * conj y) := by simp only [norm_mul, norm_conj] + _ = abs (Int.mod' (x.re * y.re + x.im * y.im) (norm y)) ^ 2 + + abs (Int.mod' (-(x.re * y.im) + x.im * y.re) (norm y)) ^ 2 := by simp [H1, norm, sq_abs] + _ ≤ (y.norm / 2) ^ 2 + (y.norm / 2) ^ 2 := by gcongr <;> apply Int.abs_mod'_le _ _ norm_y_pos + _ = norm y / 2 * (norm y / 2 * 2) := by ring + _ ≤ norm y / 2 * norm y := by gcongr; apply Int.ediv_mul_le; norm_num + calc norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right H2 norm_y_pos + _ < norm y := by + 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_Hierarchies/S01_Basics.lean (new)
- @@ -0,0 +1,308 @@
 + import Mathlib.Algebra.BigOperators.Ring + import Mathlib.Data.Real.Basic + + + class One₁ (α : Type) where + /-- The element one -/ + one : α + + + #check One₁.one -- One₁.one {α : Type} [self : One₁ α] : α + + @[class] structure One₂ (α : Type) where + /-- The element one -/ + one : α + + #check One₂.one + + + example (α : Type) [One₁ α] : α := One₁.one + + example (α : Type) [One₁ α] := (One₁.one : α) + + @[inherit_doc] + notation "𝟙" => One₁.one + + example {α : Type} [One₁ α] : α := 𝟙 + + example {α : Type} [One₁ α] : (𝟙 : α) = 𝟙 := rfl + + + class Dia₁ (α : Type) where + dia : α → α → α + + infixl:70 " ⋄ " => Dia₁.dia + + + class Semigroup₁ (α : Type) where + toDia₁ : Dia₁ α + /-- Diamond is associative -/ + dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) + + + attribute [instance] Semigroup₁.toDia₁ + + example {α : Type} [Semigroup₁ α] (a b : α) : α := a ⋄ b + + + class Semigroup₂ (α : Type) extends Dia₁ α where + /-- Diamond is associative -/ + dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) + + example {α : Type} [Semigroup₂ α] (a b : α) : α := a ⋄ b + + class DiaOneClass₁ (α : Type) extends One₁ α, Dia₁ α where + /-- One is a left neutral element for diamond. -/ + one_dia : ∀ a : α, 𝟙 ⋄ a = a + /-- One is a right neutral element for diamond -/ + dia_one : ∀ a : α, a ⋄ 𝟙 = a + + + + set_option trace.Meta.synthInstance true in + example {α : Type} [DiaOneClass₁ α] (a b : α) : Prop := a ⋄ b = 𝟙 + + + class Monoid₁ (α : Type) extends Semigroup₁ α, DiaOneClass₁ α + + + class Monoid₂ (α : Type) where + toSemigroup₁ : Semigroup₁ α + toDiaOneClass₁ : DiaOneClass₁ α + + + example {α : Type} [Monoid₁ α] : + (Monoid₁.toSemigroup₁.toDia₁.dia : α → α → α) = Monoid₁.toDiaOneClass₁.toDia₁.dia := rfl + + + /- Monoid₂.mk {α : Type} (toSemigroup₁ : Semigroup₁ α) (toDiaOneClass₁ : DiaOneClass₁ α) : Monoid₂ α -/ + #check Monoid₂.mk + + /- Monoid₁.mk {α : Type} [toSemigroup₁ : Semigroup₁ α] [toOne₁ : One₁ α] (one_dia : ∀ (a : α), 𝟙 ⋄ a = a) (dia_one : ∀ (a : α), a ⋄ 𝟙 = a) : Monoid₁ α -/ + #check Monoid₁.mk + + + #check Monoid₁.toSemigroup₁ + #check Monoid₁.toDiaOneClass₁ + + + class Inv₁ (α : Type) where + /-- The inversion function -/ + inv : α → α + + @[inherit_doc] + postfix:max "⁻¹" => Inv₁.inv + + class Group₁ (G : Type) extends Monoid₁ G, Inv G where + inv_dia : ∀ a : G, a⁻¹ ⋄ a = 𝟙 + + + lemma left_inv_eq_right_inv₁ {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by + rw [← DiaOneClass₁.one_dia c, ← hba, Semigroup₁.dia_assoc, hac, DiaOneClass₁.dia_one b] + + + export DiaOneClass₁ (one_dia dia_one) + export Semigroup₁ (dia_assoc) + export Group₁ (inv_dia) + + + example {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by + rw [← one_dia c, ← hba, dia_assoc, hac, dia_one b] + + + lemma inv_eq_of_dia [Group₁ G] {a b : G} (h : a ⋄ b = 𝟙) : a⁻¹ = b := + sorry + + lemma dia_inv [Group₁ G] (a : G) : a ⋄ a⁻¹ = 𝟙 := + sorry + + + + + class AddSemigroup₃ (α : Type) extends Add α where + /-- Multiplication is associative -/ + add_assoc₃ : ∀ a b c : α, a + b + c = a + (b + c) + + @[to_additive AddSemigroup₃] + class Semigroup₃ (α : Type) extends Mul α where + /-- Multiplication is associative -/ + mul_assoc₃ : ∀ a b c : α, a * b * c = a * (b * c) + + class AddMonoid₃ (α : Type) extends AddSemigroup₃ α, AddZeroClass α + + @[to_additive AddMonoid₃] + class Monoid₃ (α : Type) extends Semigroup₃ α, MulOneClass α + + attribute [to_additive existing] Monoid₃.toMulOneClass + + export Semigroup₃ (mul_assoc₃) + export AddSemigroup₃ (add_assoc₃) + + whatsnew in + @[to_additive] + lemma left_inv_eq_right_inv' {M : Type} [Monoid₃ M] {a b c : M} (hba : b * a = 1) (hac : a * c = 1) : b = c := by + rw [← one_mul c, ← hba, mul_assoc₃, hac, mul_one b] + + #check left_neg_eq_right_neg' + + class AddCommSemigroup₃ (α : Type) extends AddSemigroup₃ α where + add_comm : ∀ a b : α, a + b = b + a + + @[to_additive AddCommSemigroup₃] + class CommSemigroup₃ (α : Type) extends Semigroup₃ α where + mul_comm : ∀ a b : α, a * b = b * a + + class AddCommMonoid₃ (α : Type) extends AddMonoid₃ α, AddCommSemigroup₃ α + + @[to_additive AddCommMonoid₃] + class CommMonoid₃ (α : Type) extends Monoid₃ α, CommSemigroup₃ α + + class AddGroup₃ (G : Type) extends AddMonoid₃ G, Neg G where + neg_add : ∀ a : G, -a + a = 0 + + @[to_additive AddGroup₃] + class Group₃ (G : Type) extends Monoid₃ G, Inv G where + inv_mul : ∀ a : G, a⁻¹ * a = 1 + + + attribute [simp] Group₃.inv_mul AddGroup₃.neg_add + + + + @[to_additive] + lemma inv_eq_of_mul [Group₃ G] {a b : G} (h : a * b = 1) : a⁻¹ = b := + sorry + + + @[to_additive (attr := simp)] + lemma Group₃.mul_inv {G : Type} [Group₃ G] {a : G} : a * a⁻¹ = 1 := by + sorry + + @[to_additive] + lemma mul_left_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : a * b = a * c) : b = c := by + sorry + + @[to_additive] + lemma mul_right_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : b*a = c*a) : b = c := by + sorry + + class AddCommGroup₃ (G : Type) extends AddGroup₃ G, AddCommMonoid₃ G + + @[to_additive AddCommGroup₃] + class CommGroup₃ (G : Type) extends Group₃ G, CommMonoid₃ G + + + + class Ring₃ (R : Type) extends AddGroup₃ R, Monoid₃ R, MulZeroClass R where + /-- Multiplication is left distributive over addition -/ + left_distrib : ∀ a b c : R, a * (b + c) = a * b + a * c + /-- Multiplication is right distributive over addition -/ + right_distrib : ∀ a b c : R, (a + b) * c = a * c + b * c + + instance {R : Type} [Ring₃ R] : AddCommGroup₃ R := + { Ring₃.toAddGroup₃ with + add_comm := by + sorry } + + instance : Ring₃ ℤ where + add := (· + ·) + add_assoc₃ := add_assoc + zero := 0 + zero_add := by simp + add_zero := by simp + neg := (- ·) + neg_add := by simp + mul := (· * ·) + mul_assoc₃ := mul_assoc + one := 1 + one_mul := by simp + mul_one := by simp + zero_mul := by simp + mul_zero := by simp + left_distrib := Int.mul_add + right_distrib := Int.add_mul + + class LE₁ (α : Type) where + /-- The Less-or-Equal relation. -/ + le : α → α → Prop + + @[inherit_doc] infix:50 " ≤₁ " => LE₁.le + + class Preorder₁ (α : Type) + + class PartialOrder₁ (α : Type) + + class OrderedCommMonoid₁ (α : Type) + + instance : OrderedCommMonoid₁ ℕ where + + class SMul₃ (α : Type) (β : Type) where + /-- Scalar multiplication -/ + smul : α → β → β + + infixr:73 " • " => SMul₃.smul + + + class Module₁ (R : Type) [Ring₃ R] (M : Type) [AddCommGroup₃ M] extends SMul₃ R M where + zero_smul : ∀ m : M, (0 : R) • m = 0 + one_smul : ∀ m : M, (1 : R) • m = m + mul_smul : ∀ (a b : R) (m : M), (a * b) • m = a • b • m + add_smul : ∀ (a b : R) (m : M), (a + b) • m = a • m + b • m + smul_add : ∀ (a : R) (m n : M), a • (m + n) = a • m + a • n + + instance selfModule (R : Type) [Ring₃ R] : Module₁ R R where + smul := fun r s ↦ r*s + zero_smul := zero_mul + one_smul := one_mul + mul_smul := mul_assoc₃ + add_smul := Ring₃.right_distrib + smul_add := Ring₃.left_distrib + + def nsmul₁ [Zero M] [Add M] : ℕ → M → M + | 0, _ => 0 + | n + 1, a => a + nsmul₁ n a + + def zsmul₁ {M : Type _} [Zero M] [Add M] [Neg M] : ℤ → M → M + | Int.ofNat n, a => nsmul₁ n a + | Int.negSucc n, a => -nsmul₁ n.succ a + + instance abGrpModule (A : Type) [AddCommGroup₃ A] : Module₁ ℤ A where + smul := zsmul₁ + zero_smul := sorry + one_smul := sorry + mul_smul := sorry + add_smul := sorry + smul_add := sorry + + #synth Module₁ ℤ ℤ -- abGrpModule ℤ + + + class AddMonoid₄ (M : Type) extends AddSemigroup₃ M, AddZeroClass M where + /-- Multiplication by a natural number. -/ + nsmul : ℕ → M → M := nsmul₁ + /-- Multiplication by `(0 : ℕ)` gives `0`. -/ + nsmul_zero : ∀ x, nsmul 0 x = 0 := by intros; rfl + /-- Multiplication by `(n + 1 : ℕ)` behaves as expected. -/ + nsmul_succ : ∀ (n : ℕ) (x), nsmul (n + 1) x = x + nsmul n x := by intros; rfl + + instance mySMul {M : Type} [AddMonoid₄ M] : SMul ℕ M := ⟨AddMonoid₄.nsmul⟩ + + instance (M N : Type) [AddMonoid₄ M] [AddMonoid₄ N] : AddMonoid₄ (M × N) where + add := fun p q ↦ (p.1 + q.1, p.2 + q.2) + add_assoc₃ := fun a b c ↦ by ext <;> apply add_assoc₃ + zero := (0, 0) + zero_add := fun a ↦ by ext <;> apply zero_add + add_zero := fun a ↦ by ext <;> apply add_zero + + instance : AddMonoid₄ ℤ where + add := (· + ·) + add_assoc₃ := Int.add_assoc + zero := 0 + zero_add := Int.zero_add + add_zero := Int.add_zero + nsmul := fun n m ↦ (n : ℤ) * m + nsmul_zero := Int.zero_mul + nsmul_succ := fun n m ↦ show (n + 1 : ℤ) * m = m + n * m + by rw [Int.add_mul, Int.add_comm, Int.one_mul] + + example (n : ℕ) (m : ℤ) : SMul.smul (self := mySMul) n m = n * m := rfl
src/C07_Hierarchies/S02_Morphisms.lean (new)
- @@ -0,0 +1,114 @@
  + import Mathlib.Topology.Instances.Real + + + def isMonoidHom₁ [Monoid G] [Monoid H] (f : G → H) : Prop := + f 1 = 1 ∧ ∀ g g', f (g * g') = f g * f g' + structure isMonoidHom₂ [Monoid G] [Monoid H] (f : G → H) : Prop where + map_one : f 1 = 1 + map_mul : ∀ g g', f (g * g') = f g * f g' + example : Continuous (id : ℝ → ℝ) := continuous_id + @[ext] + structure MonoidHom₁ (G H : Type) [Monoid G] [Monoid H] where + toFun : G → H + map_one : toFun 1 = 1 + map_mul : ∀ g g', toFun (g * g') = toFun g * toFun g' + + instance [Monoid G] [Monoid H] : CoeFun (MonoidHom₁ G H) (fun _ ↦ G → H) where + coe := MonoidHom₁.toFun + + attribute [coe] MonoidHom₁.toFun + + + example [Monoid G] [Monoid H] (f : MonoidHom₁ G H) : f 1 = 1 := f.map_one + + @[ext] + structure AddMonoidHom₁ (G H : Type) [AddMonoid G] [AddMonoid H] where + toFun : G → H + map_zero : toFun 0 = 0 + map_add : ∀ g g', toFun (g + g') = toFun g + toFun g' + + instance [AddMonoid G] [AddMonoid H] : CoeFun (AddMonoidHom₁ G H) (fun _ ↦ G → H) where + coe := AddMonoidHom₁.toFun + + attribute [coe] AddMonoidHom₁.toFun + + @[ext] + structure RingHom₁ (R S : Type) [Ring R] [Ring S] extends MonoidHom₁ R S, AddMonoidHom₁ R S + + + + class MonoidHomClass₁ (F : Type) (M N : Type) [Monoid M] [Monoid N] where + toFun : F → M → N + map_one : ∀ f : F, toFun f 1 = 1 + map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' + + + def badInst [Monoid M] [Monoid N] [MonoidHomClass₁ F M N] : CoeFun F (fun _ ↦ M → N) where + coe := MonoidHomClass₁.toFun + + + class MonoidHomClass₂ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] where + toFun : F → M → N + map_one : ∀ f : F, toFun f 1 = 1 + map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' + + instance [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] : CoeFun F (fun _ ↦ M → N) where + coe := MonoidHomClass₂.toFun + + attribute [coe] MonoidHomClass₂.toFun + + + instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₂ (MonoidHom₁ M N) M N where + toFun := MonoidHom₁.toFun + map_one := fun f ↦ f.map_one + map_mul := fun f ↦ f.map_mul + + instance (R S : Type) [Ring R] [Ring S] : MonoidHomClass₂ (RingHom₁ R S) R S where + toFun := fun f ↦ f.toMonoidHom₁.toFun + map_one := fun f ↦ f.toMonoidHom₁.map_one + map_mul := fun f ↦ f.toMonoidHom₁.map_mul + + + lemma map_inv_of_inv [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] (f : F) {m m' : M} (h : m*m' = 1) : + f m * f m' = 1 := by + rw [← MonoidHomClass₂.map_mul, h, MonoidHomClass₂.map_one] + + example [Monoid M] [Monoid N] (f : MonoidHom₁ M N) {m m' : M} (h : m*m' = 1) : f m * f m' = 1 := + map_inv_of_inv f h + + example [Ring R] [Ring S] (f : RingHom₁ R S) {r r' : R} (h : r*r' = 1) : f r * f r' = 1 := + map_inv_of_inv f h + + + + class MonoidHomClass₃ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] extends + FunLike F M (fun _ ↦ N) where + map_one : ∀ f : F, f 1 = 1 + map_mul : ∀ (f : F) g g', f (g * g') = f g * f g' + + instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₃ (MonoidHom₁ M N) M N where + coe := MonoidHom₁.toFun + coe_injective' := MonoidHom₁.ext + map_one := MonoidHom₁.map_one + map_mul := MonoidHom₁.map_mul + + + @[ext] + structure OrderPresHom (α β : Type) [LE α] [LE β] where + toFun : α → β + le_of_le : ∀ a a', a ≤ a' → toFun a ≤ toFun a' + + @[ext] + structure OrderPresMonoidHom (M N : Type) [Monoid M] [LE M] [Monoid N] [LE N] extends + MonoidHom₁ M N, OrderPresHom M N + + class OrderPresHomClass (F : Type) (α β : outParam Type) [LE α] [LE β] + + instance (α β : Type) [LE α] [LE β] : OrderPresHomClass (OrderPresHom α β) α β where + + instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : + OrderPresHomClass (OrderPresMonoidHom α β) α β where + + instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : + MonoidHomClass₃ (OrderPresMonoidHom α β) α β + := sorry
src/C07_Hierarchies/S03_Subobjects.lean (new)
- @@ -0,0 +1,88 @@
  + import Mathlib.GroupTheory.QuotientGroup + + + + @[ext] + structure Submonoid₁ (M : Type) [Monoid M] where + /-- The carrier of a submonoid. -/ + carrier : Set M + /-- The product of two elements of a submonoid belongs to the submonoid. -/ + mul_mem {a b} : a ∈ carrier → b ∈ carrier → a * b ∈ carrier + /-- The unit element belongs to the submonoid. -/ + one_mem : 1 ∈ carrier + + /-- Submonoids in `M` can be seen as sets in `M`. -/ + instance [Monoid M] : SetLike (Submonoid₁ M) M where + coe := Submonoid₁.carrier + coe_injective' := Submonoid₁.ext + + + + example [Monoid M] (N : Submonoid₁ M) : 1 ∈ N := N.one_mem + + example [Monoid M] (N : Submonoid₁ M) (α : Type) (f : M → α) := f '' N + + + example [Monoid M] (N : Submonoid₁ M) (x : N) : (x : M) ∈ N := x.property + + + instance SubMonoid₁Monoid [Monoid M] (N : Submonoid₁ M) : Monoid N where + mul := fun x y ↦ ⟨x*y, N.mul_mem x.property y.property⟩ + mul_assoc := fun x y z ↦ SetCoe.ext (mul_assoc (x : M) y z) + one := ⟨1, N.one_mem⟩ + one_mul := fun x ↦ SetCoe.ext (one_mul (x : M)) + mul_one := fun x ↦ SetCoe.ext (mul_one (x : M)) + + + example [Monoid M] (N : Submonoid₁ M) : Monoid N where + mul := fun ⟨x, hx⟩ ⟨y, hy⟩ ↦ ⟨x*y, N.mul_mem hx hy⟩ + mul_assoc := fun ⟨x, _⟩ ⟨y, _⟩ ⟨z, _⟩ ↦ SetCoe.ext (mul_assoc x y z) + one := ⟨1, N.one_mem⟩ + one_mul := fun ⟨x, _⟩ ↦ SetCoe.ext (one_mul x) + mul_one := fun ⟨x, _⟩ ↦ SetCoe.ext (mul_one x) + + + class SubmonoidClass₁ (S : Type) (M : Type) [Monoid M] [SetLike S M] : Prop where + mul_mem : ∀ (s : S) {a b : M}, a ∈ s → b ∈ s → a * b ∈ s + one_mem : ∀ s : S, 1 ∈ s + + instance [Monoid M] : SubmonoidClass₁ (Submonoid₁ M) M where + mul_mem := Submonoid₁.mul_mem + one_mem := Submonoid₁.one_mem + + + instance [Monoid M] : Inf (Submonoid₁ M) := + ⟨fun S₁ S₂ => + { carrier := S₁ ∩ S₂ + one_mem := ⟨S₁.one_mem, S₂.one_mem⟩ + mul_mem := fun ⟨hx, hx'⟩ ⟨hy, hy'⟩ ↦ ⟨S₁.mul_mem hx hy, S₂.mul_mem hx' hy'⟩ }⟩ + + + example [Monoid M] (N P : Submonoid₁ M) : Submonoid₁ M := N ⊓ P + + + def Submonoid.Setoid [CommMonoid M] (N : Submonoid M) : Setoid M where + r := fun x y ↦ ∃ w ∈ N, ∃ z ∈ N, x*w = y*z + iseqv := { + refl := fun x ↦ ⟨1, N.one_mem, 1, N.one_mem, rfl⟩ + symm := fun ⟨w, hw, z, hz, h⟩ ↦ ⟨z, hz, w, hw, h.symm⟩ + trans := by + sorry + } + + instance [CommMonoid M] : HasQuotient M (Submonoid M) where + quotient' := fun N ↦ Quotient N.Setoid + + def QuotientMonoid.mk [CommMonoid M] (N : Submonoid M) : M → M ⧸ N := Quotient.mk N.Setoid + + instance [CommMonoid M] (N : Submonoid M) : Monoid (M ⧸ N) where + mul := Quotient.map₂' (· * ·) (by + sorry + ) + mul_assoc := by + sorry + one := QuotientMonoid.mk N 1 + one_mul := by + sorry + mul_one := by + sorry
src/C07_Hierarchies/solutions/Solutions_S01_Basics.lean (new)
- @@ -0,0 +1,348 @@
 + import Mathlib.Algebra.BigOperators.Ring + import Mathlib.Data.Real.Basic + + + class One₁ (α : Type) where + /-- The element one -/ + one : α + + + #check One₁.one -- One₁.one {α : Type} [self : One₁ α] : α + + @[class] structure One₂ (α : Type) where + /-- The element one -/ + one : α + + #check One₂.one + + + example (α : Type) [One₁ α] : α := One₁.one + + example (α : Type) [One₁ α] := (One₁.one : α) + + @[inherit_doc] + notation "𝟙" => One₁.one + + example {α : Type} [One₁ α] : α := 𝟙 + + example {α : Type} [One₁ α] : (𝟙 : α) = 𝟙 := rfl + + + class Dia₁ (α : Type) where + dia : α → α → α + + infixl:70 " ⋄ " => Dia₁.dia + + + class Semigroup₁ (α : Type) where + toDia₁ : Dia₁ α + /-- Diamond is associative -/ + dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) + + + attribute [instance] Semigroup₁.toDia₁ + + example {α : Type} [Semigroup₁ α] (a b : α) : α := a ⋄ b + + + class Semigroup₂ (α : Type) extends Dia₁ α where + /-- Diamond is associative -/ + dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) + + example {α : Type} [Semigroup₂ α] (a b : α) : α := a ⋄ b + + class DiaOneClass₁ (α : Type) extends One₁ α, Dia₁ α where + /-- One is a left neutral element for diamond. -/ + one_dia : ∀ a : α, 𝟙 ⋄ a = a + /-- One is a right neutral element for diamond -/ + dia_one : ∀ a : α, a ⋄ 𝟙 = a + + + + set_option trace.Meta.synthInstance true in + example {α : Type} [DiaOneClass₁ α] (a b : α) : Prop := a ⋄ b = 𝟙 + + + class Monoid₁ (α : Type) extends Semigroup₁ α, DiaOneClass₁ α + + + class Monoid₂ (α : Type) where + toSemigroup₁ : Semigroup₁ α + toDiaOneClass₁ : DiaOneClass₁ α + + + example {α : Type} [Monoid₁ α] : + (Monoid₁.toSemigroup₁.toDia₁.dia : α → α → α) = Monoid₁.toDiaOneClass₁.toDia₁.dia := rfl + + + /- Monoid₂.mk {α : Type} (toSemigroup₁ : Semigroup₁ α) (toDiaOneClass₁ : DiaOneClass₁ α) : Monoid₂ α -/ + #check Monoid₂.mk + + /- Monoid₁.mk {α : Type} [toSemigroup₁ : Semigroup₁ α] [toOne₁ : One₁ α] (one_dia : ∀ (a : α), 𝟙 ⋄ a = a) (dia_one : ∀ (a : α), a ⋄ 𝟙 = a) : Monoid₁ α -/ + #check Monoid₁.mk + + + #check Monoid₁.toSemigroup₁ + #check Monoid₁.toDiaOneClass₁ + + + class Inv₁ (α : Type) where + /-- The inversion function -/ + inv : α → α + + @[inherit_doc] + postfix:max "⁻¹" => Inv₁.inv + + class Group₁ (G : Type) extends Monoid₁ G, Inv G where + inv_dia : ∀ a : G, a⁻¹ ⋄ a = 𝟙 + + + lemma left_inv_eq_right_inv₁ {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by + rw [← DiaOneClass₁.one_dia c, ← hba, Semigroup₁.dia_assoc, hac, DiaOneClass₁.dia_one b] + + + export DiaOneClass₁ (one_dia dia_one) + export Semigroup₁ (dia_assoc) + export Group₁ (inv_dia) + + + example {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by + rw [← one_dia c, ← hba, dia_assoc, hac, dia_one b] + + + lemma inv_eq_of_dia [Group₁ G] {a b : G} (h : a ⋄ b = 𝟙) : a⁻¹ = b := + left_inv_eq_right_inv₁ (inv_dia a) h + + lemma dia_inv [Group₁ G] (a : G) : a ⋄ a⁻¹ = 𝟙 := + by rw [← inv_dia a⁻¹, inv_eq_of_dia (inv_dia a)] + + + + + + class AddSemigroup₃ (α : Type) extends Add α where + /-- Multiplication is associative -/ + add_assoc₃ : ∀ a b c : α, a + b + c = a + (b + c) + + @[to_additive AddSemigroup₃] + class Semigroup₃ (α : Type) extends Mul α where + /-- Multiplication is associative -/ + mul_assoc₃ : ∀ a b c : α, a * b * c = a * (b * c) + + class AddMonoid₃ (α : Type) extends AddSemigroup₃ α, AddZeroClass α + + @[to_additive AddMonoid₃] + class Monoid₃ (α : Type) extends Semigroup₃ α, MulOneClass α + + attribute [to_additive existing] Monoid₃.toMulOneClass + + export Semigroup₃ (mul_assoc₃) + export AddSemigroup₃ (add_assoc₃) + + whatsnew in + @[to_additive] + lemma left_inv_eq_right_inv' {M : Type} [Monoid₃ M] {a b c : M} (hba : b * a = 1) (hac : a * c = 1) : b = c := by + rw [← one_mul c, ← hba, mul_assoc₃, hac, mul_one b] + + #check left_neg_eq_right_neg' + + class AddCommSemigroup₃ (α : Type) extends AddSemigroup₃ α where + add_comm : ∀ a b : α, a + b = b + a + + @[to_additive AddCommSemigroup₃] + class CommSemigroup₃ (α : Type) extends Semigroup₃ α where + mul_comm : ∀ a b : α, a * b = b * a + + class AddCommMonoid₃ (α : Type) extends AddMonoid₃ α, AddCommSemigroup₃ α + + @[to_additive AddCommMonoid₃] + class CommMonoid₃ (α : Type) extends Monoid₃ α, CommSemigroup₃ α + + class AddGroup₃ (G : Type) extends AddMonoid₃ G, Neg G where + neg_add : ∀ a : G, -a + a = 0 + + @[to_additive AddGroup₃] + class Group₃ (G : Type) extends Monoid₃ G, Inv G where + inv_mul : ∀ a : G, a⁻¹ * a = 1 + + + attribute [simp] Group₃.inv_mul AddGroup₃.neg_add + + + + @[to_additive] + lemma inv_eq_of_mul [Group₃ G] {a b : G} (h : a * b = 1) : a⁻¹ = b := + left_inv_eq_right_inv' (Group₃.inv_mul a) h + + + @[to_additive (attr := simp)] + lemma Group₃.mul_inv {G : Type} [Group₃ G] {a : G} : a * a⁻¹ = 1 := by + rw [← inv_mul a⁻¹, inv_eq_of_mul (inv_mul a)] + + @[to_additive] + lemma mul_left_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : a * b = a * c) : b = c := by + simpa [← mul_assoc₃] using congr_arg (a⁻¹ * ·) h + + @[to_additive] + lemma mul_right_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : b*a = c*a) : b = c := by + simpa [mul_assoc₃] using congr_arg (· * a⁻¹) h + + class AddCommGroup₃ (G : Type) extends AddGroup₃ G, AddCommMonoid₃ G + + @[to_additive AddCommGroup₃] + class CommGroup₃ (G : Type) extends Group₃ G, CommMonoid₃ G + + + + class Ring₃ (R : Type) extends AddGroup₃ R, Monoid₃ R, MulZeroClass R where + /-- Multiplication is left distributive over addition -/ + left_distrib : ∀ a b c : R, a * (b + c) = a * b + a * c + /-- Multiplication is right distributive over addition -/ + right_distrib : ∀ a b c : R, (a + b) * c = a * c + b * c + + instance {R : Type} [Ring₃ R] : AddCommGroup₃ R := + { Ring₃.toAddGroup₃ with + add_comm := by + intro a b + have : a + (a + b + b) = a + (b + a + b) := calc + a + (a + b + b) = (a + a) + (b + b) := by simp [add_assoc₃, add_assoc₃] + _ = (1 * a + 1 * a) + (1 * b + 1 * b) := by simp + _ = (1 + 1) * a + (1 + 1) * b := by simp [Ring₃.right_distrib] + _ = (1 + 1) * (a + b) := by simp [Ring₃.left_distrib] + _ = 1 * (a + b) + 1 * (a + b) := by simp [Ring₃.right_distrib] + _ = (a + b) + (a + b) := by simp + _ = a + (b + a + b) := by simp [add_assoc₃] + exact add_right_cancel₃ (add_left_cancel₃ this) } + + instance : Ring₃ ℤ where + add := (· + ·) + add_assoc₃ := add_assoc + zero := 0 + zero_add := by simp + add_zero := by simp + neg := (- ·) + neg_add := by simp + mul := (· * ·) + mul_assoc₃ := mul_assoc + one := 1 + one_mul := by simp + mul_one := by simp + zero_mul := by simp + mul_zero := by simp + left_distrib := Int.mul_add + right_distrib := Int.add_mul + + class LE₁ (α : Type) where + /-- The Less-or-Equal relation. -/ + le : α → α → Prop + + @[inherit_doc] infix:50 " ≤₁ " => LE₁.le + + class Preorder₁ (α : Type) + extends LE₁ α where + le_refl : ∀ a : α, a ≤₁ a + le_trans : ∀ a b c : α, a ≤₁ b → b ≤₁ c → a ≤₁ c + + class PartialOrder₁ (α : Type) + extends Preorder₁ α where + le_antisymm : ∀ a b : α, a ≤₁ b → b ≤₁ a → a = b + + class OrderedCommMonoid₁ (α : Type) + extends PartialOrder₁ α, CommMonoid₃ α where + mul_of_le : ∀ a b : α, a ≤₁ b → ∀ c : α, c * a ≤₁ c * b + + instance : OrderedCommMonoid₁ ℕ where + le := (· ≤ ·) + le_refl := fun _ ↦ le_rfl + le_trans := fun _ _ _ ↦ le_trans + le_antisymm := fun _ _ ↦ le_antisymm + mul := (· * ·) + mul_assoc₃ := mul_assoc + one := 1 + one_mul := one_mul + mul_one := mul_one + mul_comm := mul_comm + mul_of_le := fun _ _ h c ↦ Nat.mul_le_mul_left c h + + class SMul₃ (α : Type) (β : Type) where + /-- Scalar multiplication -/ + smul : α → β → β + + infixr:73 " • " => SMul₃.smul + + + class Module₁ (R : Type) [Ring₃ R] (M : Type) [AddCommGroup₃ M] extends SMul₃ R M where + zero_smul : ∀ m : M, (0 : R) • m = 0 + one_smul : ∀ m : M, (1 : R) • m = m + mul_smul : ∀ (a b : R) (m : M), (a * b) • m = a • b • m + add_smul : ∀ (a b : R) (m : M), (a + b) • m = a • m + b • m + smul_add : ∀ (a : R) (m n : M), a • (m + n) = a • m + a • n + + instance selfModule (R : Type) [Ring₃ R] : Module₁ R R where + smul := fun r s ↦ r*s + zero_smul := zero_mul + one_smul := one_mul + mul_smul := mul_assoc₃ + add_smul := Ring₃.right_distrib + smul_add := Ring₃.left_distrib + + def nsmul₁ [Zero M] [Add M] : ℕ → M → M + | 0, _ => 0 + | n + 1, a => a + nsmul₁ n a + + def zsmul₁ {M : Type _} [Zero M] [Add M] [Neg M] : ℤ → M → M + | Int.ofNat n, a => nsmul₁ n a + | Int.negSucc n, a => -nsmul₁ n.succ a + + instance abGrpModule (A : Type) [AddCommGroup₃ A] : Module₁ ℤ A where + smul := zsmul₁ + zero_smul := sorry + one_smul := sorry + mul_smul := sorry + add_smul := sorry + smul_add := sorry + + #synth Module₁ ℤ ℤ -- abGrpModule ℤ + + + class AddMonoid₄ (M : Type) extends AddSemigroup₃ M, AddZeroClass M where + /-- Multiplication by a natural number. -/ + nsmul : ℕ → M → M := nsmul₁ + /-- Multiplication by `(0 : ℕ)` gives `0`. -/ + nsmul_zero : ∀ x, nsmul 0 x = 0 := by intros; rfl + /-- Multiplication by `(n + 1 : ℕ)` behaves as expected. -/ + nsmul_succ : ∀ (n : ℕ) (x), nsmul (n + 1) x = x + nsmul n x := by intros; rfl + + instance mySMul {M : Type} [AddMonoid₄ M] : SMul ℕ M := ⟨AddMonoid₄.nsmul⟩ + + instance (M N : Type) [AddMonoid₄ M] [AddMonoid₄ N] : AddMonoid₄ (M × N) where + add := fun p q ↦ (p.1 + q.1, p.2 + q.2) + add_assoc₃ := fun a b c ↦ by ext <;> apply add_assoc₃ + zero := (0, 0) + zero_add := fun a ↦ by ext <;> apply zero_add + add_zero := fun a ↦ by ext <;> apply add_zero + + instance : AddMonoid₄ ℤ where + add := (· + ·) + add_assoc₃ := Int.add_assoc + zero := 0 + zero_add := Int.zero_add + add_zero := Int.add_zero + nsmul := fun n m ↦ (n : ℤ) * m + nsmul_zero := Int.zero_mul + nsmul_succ := fun n m ↦ show (n + 1 : ℤ) * m = m + n * m + by rw [Int.add_mul, Int.add_comm, Int.one_mul] + + example (n : ℕ) (m : ℤ) : SMul.smul (self := mySMul) n m = n * m := rfl + + class LT₁ (α : Type) where + /-- The Less-Than relation -/ + lt : α → α → Prop + + @[inherit_doc] infix:50 " <₁ " => LT₁.lt + + class PreOrder₂ (α : Type) extends LE₁ α, LT₁ α where + le_refl : ∀ a : α, a ≤₁ a + le_trans : ∀ a b c : α, a ≤₁ b → b ≤₁ c → a ≤₁ c + lt := fun a b => a ≤₁ b ∧ ¬b ≤₁ a + lt_iff_le_not_le : ∀ a b : α, a <₁ b ↔ a ≤₁ b ∧ ¬b ≤₁ a := by intros; rfl
src/C07_Hierarchies/solutions/Solutions_S02_Morphisms.lean (new)
- @@ -0,0 +1,126 @@
  + import Mathlib.Topology.Instances.Real + + + def isMonoidHom₁ [Monoid G] [Monoid H] (f : G → H) : Prop := + f 1 = 1 ∧ ∀ g g', f (g * g') = f g * f g' + structure isMonoidHom₂ [Monoid G] [Monoid H] (f : G → H) : Prop where + map_one : f 1 = 1 + map_mul : ∀ g g', f (g * g') = f g * f g' + example : Continuous (id : ℝ → ℝ) := continuous_id + @[ext] + structure MonoidHom₁ (G H : Type) [Monoid G] [Monoid H] where + toFun : G → H + map_one : toFun 1 = 1 + map_mul : ∀ g g', toFun (g * g') = toFun g * toFun g' + + instance [Monoid G] [Monoid H] : CoeFun (MonoidHom₁ G H) (fun _ ↦ G → H) where + coe := MonoidHom₁.toFun + + attribute [coe] MonoidHom₁.toFun + + + example [Monoid G] [Monoid H] (f : MonoidHom₁ G H) : f 1 = 1 := f.map_one + + @[ext] + structure AddMonoidHom₁ (G H : Type) [AddMonoid G] [AddMonoid H] where + toFun : G → H + map_zero : toFun 0 = 0 + map_add : ∀ g g', toFun (g + g') = toFun g + toFun g' + + instance [AddMonoid G] [AddMonoid H] : CoeFun (AddMonoidHom₁ G H) (fun _ ↦ G → H) where + coe := AddMonoidHom₁.toFun + + attribute [coe] AddMonoidHom₁.toFun + + @[ext] + structure RingHom₁ (R S : Type) [Ring R] [Ring S] extends MonoidHom₁ R S, AddMonoidHom₁ R S + + + + class MonoidHomClass₁ (F : Type) (M N : Type) [Monoid M] [Monoid N] where + toFun : F → M → N + map_one : ∀ f : F, toFun f 1 = 1 + map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' + + + def badInst [Monoid M] [Monoid N] [MonoidHomClass₁ F M N] : CoeFun F (fun _ ↦ M → N) where + coe := MonoidHomClass₁.toFun + + + class MonoidHomClass₂ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] where + toFun : F → M → N + map_one : ∀ f : F, toFun f 1 = 1 + map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' + + instance [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] : CoeFun F (fun _ ↦ M → N) where + coe := MonoidHomClass₂.toFun + + attribute [coe] MonoidHomClass₂.toFun + + + instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₂ (MonoidHom₁ M N) M N where + toFun := MonoidHom₁.toFun + map_one := fun f ↦ f.map_one + map_mul := fun f ↦ f.map_mul + + instance (R S : Type) [Ring R] [Ring S] : MonoidHomClass₂ (RingHom₁ R S) R S where + toFun := fun f ↦ f.toMonoidHom₁.toFun + map_one := fun f ↦ f.toMonoidHom₁.map_one + map_mul := fun f ↦ f.toMonoidHom₁.map_mul + + + lemma map_inv_of_inv [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] (f : F) {m m' : M} (h : m*m' = 1) : + f m * f m' = 1 := by + rw [← MonoidHomClass₂.map_mul, h, MonoidHomClass₂.map_one] + + example [Monoid M] [Monoid N] (f : MonoidHom₁ M N) {m m' : M} (h : m*m' = 1) : f m * f m' = 1 := + map_inv_of_inv f h + + example [Ring R] [Ring S] (f : RingHom₁ R S) {r r' : R} (h : r*r' = 1) : f r * f r' = 1 := + map_inv_of_inv f h + + + + class MonoidHomClass₃ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] extends + FunLike F M (fun _ ↦ N) where + map_one : ∀ f : F, f 1 = 1 + map_mul : ∀ (f : F) g g', f (g * g') = f g * f g' + + instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₃ (MonoidHom₁ M N) M N where + coe := MonoidHom₁.toFun + coe_injective' := MonoidHom₁.ext + map_one := MonoidHom₁.map_one + map_mul := MonoidHom₁.map_mul + + + @[ext] + structure OrderPresHom (α β : Type) [LE α] [LE β] where + toFun : α → β + le_of_le : ∀ a a', a ≤ a' → toFun a ≤ toFun a' + + @[ext] + structure OrderPresMonoidHom (M N : Type) [Monoid M] [LE M] [Monoid N] [LE N] extends + MonoidHom₁ M N, OrderPresHom M N + + class OrderPresHomClass (F : Type) (α β : outParam Type) [LE α] [LE β] + extends FunLike F α (fun _ ↦ β) where + le_of_le : ∀ (f : F) a a', a ≤ a' → f a ≤ f a' + + instance (α β : Type) [LE α] [LE β] : OrderPresHomClass (OrderPresHom α β) α β where + coe := OrderPresHom.toFun + coe_injective' := OrderPresHom.ext + le_of_le := OrderPresHom.le_of_le + + instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : + OrderPresHomClass (OrderPresMonoidHom α β) α β where + coe := fun f ↦ f.toOrderPresHom.toFun + coe_injective' := OrderPresMonoidHom.ext + le_of_le := fun f ↦ f.toOrderPresHom.le_of_le + + instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : + MonoidHomClass₃ (OrderPresMonoidHom α β) α β + where + coe := fun f ↦ f.toOrderPresHom.toFun + coe_injective' := OrderPresMonoidHom.ext + map_one := fun f ↦ f.toMonoidHom₁.map_one + map_mul := fun f ↦ f.toMonoidHom₁.map_mul
src/C07_Hierarchies/solutions/Solutions_S03_Subobjects.lean (new)
- @@ -0,0 +1,126 @@
  + import Mathlib.GroupTheory.QuotientGroup + + + + @[ext] + structure Submonoid₁ (M : Type) [Monoid M] where + /-- The carrier of a submonoid. -/ + carrier : Set M + /-- The product of two elements of a submonoid belongs to the submonoid. -/ + mul_mem {a b} : a ∈ carrier → b ∈ carrier → a * b ∈ carrier + /-- The unit element belongs to the submonoid. -/ + one_mem : 1 ∈ carrier + + /-- Submonoids in `M` can be seen as sets in `M`. -/ + instance [Monoid M] : SetLike (Submonoid₁ M) M where + coe := Submonoid₁.carrier + coe_injective' := Submonoid₁.ext + + + + example [Monoid M] (N : Submonoid₁ M) : 1 ∈ N := N.one_mem + + example [Monoid M] (N : Submonoid₁ M) (α : Type) (f : M → α) := f '' N + + + example [Monoid M] (N : Submonoid₁ M) (x : N) : (x : M) ∈ N := x.property + + + instance SubMonoid₁Monoid [Monoid M] (N : Submonoid₁ M) : Monoid N where + mul := fun x y ↦ ⟨x*y, N.mul_mem x.property y.property⟩ + mul_assoc := fun x y z ↦ SetCoe.ext (mul_assoc (x : M) y z) + one := ⟨1, N.one_mem⟩ + one_mul := fun x ↦ SetCoe.ext (one_mul (x : M)) + mul_one := fun x ↦ SetCoe.ext (mul_one (x : M)) + + + example [Monoid M] (N : Submonoid₁ M) : Monoid N where + mul := fun ⟨x, hx⟩ ⟨y, hy⟩ ↦ ⟨x*y, N.mul_mem hx hy⟩ + mul_assoc := fun ⟨x, _⟩ ⟨y, _⟩ ⟨z, _⟩ ↦ SetCoe.ext (mul_assoc x y z) + one := ⟨1, N.one_mem⟩ + one_mul := fun ⟨x, _⟩ ↦ SetCoe.ext (one_mul x) + mul_one := fun ⟨x, _⟩ ↦ SetCoe.ext (mul_one x) + + + class SubmonoidClass₁ (S : Type) (M : Type) [Monoid M] [SetLike S M] : Prop where + mul_mem : ∀ (s : S) {a b : M}, a ∈ s → b ∈ s → a * b ∈ s + one_mem : ∀ s : S, 1 ∈ s + + instance [Monoid M] : SubmonoidClass₁ (Submonoid₁ M) M where + mul_mem := Submonoid₁.mul_mem + one_mem := Submonoid₁.one_mem + + @[ext] + structure Subgroup₁ (G : Type) [Group G] extends Submonoid₁ G where + /-- The inverse of an element of a subgroup belongs to the subgroup. -/ + inv_mem {a} : a ∈ carrier → a⁻¹ ∈ carrier + + + /-- Subgroups in `M` can be seen as sets in `M`. -/ + instance [Group G] : SetLike (Subgroup₁ G) G where + coe := fun H ↦ H.toSubmonoid₁.carrier + coe_injective' := Subgroup₁.ext + + instance [Group G] (H : Subgroup₁ G) : Group H := + { SubMonoid₁Monoid H.toSubmonoid₁ with + inv := fun x ↦ ⟨x⁻¹, H.inv_mem x.property⟩ + mul_left_inv := fun x ↦ SetCoe.ext (mul_left_inv (x : G)) } + + class SubgroupClass₁ (S : Type _) (G : Type) [Group G] [SetLike S G] + extends SubmonoidClass₁ S G : Prop where + inv_mem : ∀ (s : S) {a : G}, a ∈ s → a⁻¹ ∈ s + + instance [Group G] : SubmonoidClass₁ (Subgroup₁ G) G where + mul_mem := fun H ↦ H.toSubmonoid₁.mul_mem + one_mem := fun H ↦ H.toSubmonoid₁.one_mem + + instance [Group G] : SubgroupClass₁ (Subgroup₁ G) G := + { (inferInstance : SubmonoidClass₁ (Subgroup₁ G) G) with + inv_mem := Subgroup₁.inv_mem } + + + instance [Monoid M] : Inf (Submonoid₁ M) := + ⟨fun S₁ S₂ => + { carrier := S₁ ∩ S₂ + one_mem := ⟨S₁.one_mem, S₂.one_mem⟩ + mul_mem := fun ⟨hx, hx'⟩ ⟨hy, hy'⟩ ↦ ⟨S₁.mul_mem hx hy, S₂.mul_mem hx' hy'⟩ }⟩ + + + example [Monoid M] (N P : Submonoid₁ M) : Submonoid₁ M := N ⊓ P + + + def Submonoid.Setoid [CommMonoid M] (N : Submonoid M) : Setoid M where + r := fun x y ↦ ∃ w ∈ N, ∃ z ∈ N, x*w = y*z + iseqv := { + refl := fun x ↦ ⟨1, N.one_mem, 1, N.one_mem, rfl⟩ + symm := fun ⟨w, hw, z, hz, h⟩ ↦ ⟨z, hz, w, hw, h.symm⟩ + trans := by + rintro a b c ⟨w, hw, z, hz, h⟩ ⟨w', hw', z', hz', h'⟩ + refine ⟨w*w', N.mul_mem hw hw', z*z', N.mul_mem hz hz', ?_⟩ + rw [← mul_assoc, h, mul_comm b, mul_assoc, h', ← mul_assoc, mul_comm z, mul_assoc] + } + + instance [CommMonoid M] : HasQuotient M (Submonoid M) where + quotient' := fun N ↦ Quotient N.Setoid + + def QuotientMonoid.mk [CommMonoid M] (N : Submonoid M) : M → M ⧸ N := Quotient.mk N.Setoid + + instance [CommMonoid M] (N : Submonoid M) : Monoid (M ⧸ N) where + mul := Quotient.map₂' (· * ·) (by + rintro a₁ b₁ ⟨w, hw, z, hz, ha⟩ a₂ b₂ ⟨w', hw', z', hz', hb⟩ + refine ⟨w*w', N.mul_mem hw hw', z*z', N.mul_mem hz hz', ?_⟩ + rw [mul_comm w, ← mul_assoc, mul_assoc a₁, hb, mul_comm, ← mul_assoc, mul_comm w, ha, + mul_assoc, mul_comm z, mul_assoc b₂, mul_comm z', mul_assoc] + ) + mul_assoc := by + rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ + apply Quotient.sound + dsimp only + rw [mul_assoc] + apply @Setoid.refl M N.Setoid + one := QuotientMonoid.mk N 1 + one_mul := by + rintro ⟨a⟩ ; apply Quotient.sound ; dsimp only ; rw [one_mul] ; apply @Setoid.refl M N.Setoid + mul_one := by + rintro ⟨a⟩ ; apply Quotient.sound ; dsimp only ; rw [mul_one] ; apply @Setoid.refl M N.Setoid +
src/C08_Topology/S01_Filters.lean (new)
- @@ -0,0 +1,105 @@
 + 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 ×ˢ 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/C08_Topology/S02_Metric_Spaces.lean (new)
- @@ -0,0 +1,206 @@
 + 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/C08_Topology/S03_Topological_Spaces.lean (new)
- @@ -0,0 +1,155 @@
 + import Mathlib.Topology.Instances.Real + import Mathlib.Analysis.NormedSpace.BanachSteinhaus + + open Set Filter Topology + + 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/C08_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 ×ˢ 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 SProd.sprod Filter.instSProd Filter.prod + erw [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/C08_Topology/solutions/Solutions_S02_Metric_Spaces.lean (new)
- @@ -0,0 +1,371 @@
 + 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/C08_Topology/solutions/Solutions_S03_Topological_Spaces.lean (new)
- @@ -0,0 +1,207 @@
 + import Mathlib.Topology.Instances.Real + import Mathlib.Analysis.NormedSpace.BanachSteinhaus + + open Set Filter Topology + + 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 +
src/C09_Differential_Calculus/S01_Elementary_Differential_Calculus.lean (new)
- @@ -0,0 +1,40 @@
 + import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv + import Mathlib.Analysis.Calculus.MeanValue + + open Set Filter + open Topology Filter Classical Real + + noncomputable section + + open Real + + /-- The sin function has derivative 1 at 0. -/ + example : HasDerivAt sin 1 0 := by simpa using hasDerivAt_sin 0 + + example (x : ℝ) : DifferentiableAt ℝ sin x := + (hasDerivAt_sin x).differentiableAt + + example {f : ℝ → ℝ} {x a : ℝ} (h : HasDerivAt f a x) : deriv f x = a := + h.deriv + + example {f : ℝ → ℝ} {x : ℝ} (h : ¬DifferentiableAt ℝ f x) : deriv f x = 0 := + deriv_zero_of_not_differentiableAt h + + example {f g : ℝ → ℝ} {x : ℝ} (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) : + deriv (f + g) x = deriv f x + deriv g x := + deriv_add hf hg + + example {f : ℝ → ℝ} {a : ℝ} (h : IsLocalMin f a) : deriv f a = 0 := + h.deriv_eq_zero + + example {f : ℝ → ℝ} {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hfI : f a = f b) : + ∃ c ∈ Ioo a b, deriv f c = 0 := + exists_deriv_eq_zero f hab hfc hfI + + example (f : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hf : ContinuousOn f (Icc a b)) + (hf' : DifferentiableOn ℝ f (Ioo a b)) : ∃ c ∈ Ioo a b, deriv f c = (f b - f a) / (b - a) := + exists_deriv_eq_slope f hab hf hf' + + example : deriv (fun x : ℝ ↦ x ^ 5) 6 = 5 * 6 ^ 4 := by simp + + example : deriv sin π = -1 := by simp
src/C09_Differential_Calculus/S02_Differential_Calculus_in_Normed_Spaces.lean (new)
- @@ -0,0 +1,185 @@
 + import Mathlib.Analysis.NormedSpace.BanachSteinhaus + import Mathlib.Analysis.NormedSpace.FiniteDimension + import Mathlib.Analysis.Calculus.Inverse + import Mathlib.Analysis.Calculus.ContDiff + import Mathlib.Analysis.Calculus.FDeriv.Prod + + + open Set Filter + + open Topology Filter + + noncomputable section + + section + + variable {E : Type _} [NormedAddCommGroup E] + + example (x : E) : 0 ≤ ‖x‖ := + norm_nonneg x + + example {x : E} : ‖x‖ = 0 ↔ x = 0 := + norm_eq_zero + + example (x y : E) : ‖x + y‖ ≤ ‖x‖ + ‖y‖ := + norm_add_le x y + + example : MetricSpace E := by infer_instance + + example {X : Type _} [TopologicalSpace X] {f : X → E} (hf : Continuous f) : + Continuous fun x => ‖f x‖ := + hf.norm + + variable [NormedSpace ℝ E] + + example (a : ℝ) (x : E) : ‖a • x‖ = |a| * ‖x‖ := + norm_smul a x + + example [FiniteDimensional ℝ E] : CompleteSpace E := by infer_instance + + example (𝕜 : Type _) [NontriviallyNormedField 𝕜] (x y : 𝕜) : ‖x * y‖ = ‖x‖ * ‖y‖ := + norm_mul x y + + example (𝕜 : Type _) [NontriviallyNormedField 𝕜] : ∃ x : 𝕜, 1 < ‖x‖ := + NormedField.exists_one_lt_norm 𝕜 + + example (𝕜 : Type _) [NontriviallyNormedField 𝕜] (E : Type _) [NormedAddCommGroup E] + [NormedSpace 𝕜 E] [CompleteSpace 𝕜] [FiniteDimensional 𝕜 E] : CompleteSpace E := + FiniteDimensional.complete 𝕜 E + + end + + section + + variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] + [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] + + example : E →L[𝕜] E := + ContinuousLinearMap.id 𝕜 E + + example (f : E →L[𝕜] F) : E → F := + f + + example (f : E →L[𝕜] F) : Continuous f := + f.cont + + example (f : E →L[𝕜] F) (x y : E) : f (x + y) = f x + f y := + f.map_add x y + + example (f : E →L[𝕜] F) (a : 𝕜) (x : E) : f (a • x) = a • f x := + f.map_smul a x + + variable (f : E →L[𝕜] F) + + example (x : E) : ‖f x‖ ≤ ‖f‖ * ‖x‖ := + f.le_op_norm x + + example {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ x, ‖f x‖ ≤ M * ‖x‖) : ‖f‖ ≤ M := + f.op_norm_le_bound hMp hM + + end + + section + + variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] + [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] + + open Metric + + example {ι : Type _} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) : + ∃ C', ∀ i, ‖g i‖ ≤ C' := by + -- sequence of subsets consisting of those `x : E` with norms `‖g i x‖` bounded by `n` + let e : ℕ → Set E := fun n => ⋂ i : ι, { x : E | ‖g i x‖ ≤ n } + -- each of these sets is closed + have hc : ∀ n : ℕ, IsClosed (e n) + sorry + -- the union is the entire space; this is where we use `h` + have hU : (⋃ n : ℕ, e n) = univ + sorry + /- apply the Baire category theorem to conclude that for some `m : ℕ`, + `e m` contains some `x` -/ + obtain ⟨m, x, hx⟩ : ∃ m, ∃ x, x ∈ interior (e m) := sorry + obtain ⟨ε, ε_pos, hε⟩ : ∃ ε > 0, ball x ε ⊆ interior (e m) := sorry + obtain ⟨k, hk⟩ : ∃ k : 𝕜, 1 < ‖k‖ := sorry + -- show all elements in the ball have norm bounded by `m` after applying any `g i` + have real_norm_le : ∀ z ∈ ball x ε, ∀ (i : ι), ‖g i z‖ ≤ m + sorry + have εk_pos : 0 < ε / ‖k‖ := sorry + refine' ⟨(m + m : ℕ) / (ε / ‖k‖), fun i => ContinuousLinearMap.op_norm_le_of_shell ε_pos _ hk _⟩ + sorry + sorry + + end + + open Asymptotics + + open Asymptotics + + example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] (c : ℝ) + (l : Filter α) (f : α → E) (g : α → F) : IsBigOWith c l f g ↔ ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := + isBigOWith_iff + + example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] + (l : Filter α) (f : α → E) (g : α → F) : f =O[l] g ↔ ∃ C, IsBigOWith C l f g := + isBigO_iff_isBigOWith + + example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] + (l : Filter α) (f : α → E) (g : α → F) : f =o[l] g ↔ ∀ C > 0, IsBigOWith C l f g := + isLittleO_iff_forall_isBigOWith + + example {α : Type _} {E : Type _} [NormedAddCommGroup E] (l : Filter α) (f g : α → E) : + f ~[l] g ↔ (f - g) =o[l] g := + Iff.rfl + + section + + variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] + [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] + + example (f : E → F) (f' : E →L[𝕜] F) (x₀ : E) : + HasFDerivAt f f' x₀ ↔ (fun x => f x - f x₀ - f' (x - x₀)) =o[𝓝 x₀] fun x => x - x₀ := + Iff.rfl + + example (f : E → F) (f' : E →L[𝕜] F) (x₀ : E) (hff' : HasFDerivAt f f' x₀) : fderiv 𝕜 f x₀ = f' := + hff'.fderiv + + example (n : ℕ) (f : E → F) : E → E[×n]→L[𝕜] F := + iteratedFDeriv 𝕜 n f + + example (n : WithTop ℕ) {f : E → F} : + ContDiff 𝕜 n f ↔ + (∀ m : ℕ, (m : WithTop ℕ) ≤ n → Continuous fun x => iteratedFDeriv 𝕜 m f x) ∧ + ∀ m : ℕ, (m : WithTop ℕ) < n → Differentiable 𝕜 fun x => iteratedFDeriv 𝕜 m f x := + contDiff_iff_continuous_differentiable + + example {𝕂 : Type _} [IsROrC 𝕂] {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕂 E] {F : Type _} + [NormedAddCommGroup F] [NormedSpace 𝕂 F] {f : E → F} {x : E} {n : WithTop ℕ} + (hf : ContDiffAt 𝕂 n f x) (hn : 1 ≤ n) : HasStrictFDerivAt f (fderiv 𝕂 f x) x := + hf.hasStrictFDerivAt hn + + section LocalInverse + variable [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} + + example (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : F → E := + HasStrictFDerivAt.localInverse f f' a hf + + example (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : + ∀ᶠ x in 𝓝 a, hf.localInverse f f' a (f x) = x := + hf.eventually_left_inverse + + example (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : + ∀ᶠ x in 𝓝 (f a), f (hf.localInverse f f' a x) = x := + hf.eventually_right_inverse + + example [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} + (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : + HasStrictFDerivAt (HasStrictFDerivAt.localInverse f f' a hf) (f'.symm : F →L[𝕜] E) (f a) := + HasStrictFDerivAt.to_localInverse hf + + end LocalInverse + + #check HasFDerivWithinAt + + #check HasFDerivAtFilter + + end
src/C09_Differential_Calculus/solutions/Solutions_S01_Elementary_Differential_Calculus.lean (new)
- @@ -0,0 +1,8 @@
  + import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv + import Mathlib.Analysis.Calculus.MeanValue + + open Set Filter + open Topology Filter Classical Real + + noncomputable section +
src/C09_Differential_Calculus/solutions/Solutions_S02_Differential_Calculus_in_Normed_Spaces.lean (new)
- @@ -0,0 +1,62 @@
 + import Mathlib.Analysis.NormedSpace.BanachSteinhaus + import Mathlib.Analysis.NormedSpace.FiniteDimension + import Mathlib.Analysis.Calculus.Inverse + import Mathlib.Analysis.Calculus.ContDiff + import Mathlib.Analysis.Calculus.FDeriv.Prod + + + open Set Filter + + open Topology Filter + + noncomputable section + + section + + variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] + [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] + + open Metric + + example {ι : Type _} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) : + ∃ C', ∀ i, ‖g i‖ ≤ C' := by + -- sequence of subsets consisting of those `x : E` with norms `‖g i x‖` bounded by `n` + let e : ℕ → Set E := fun n => ⋂ i : ι, { x : E | ‖g i x‖ ≤ n } + -- each of these sets is closed + have hc : ∀ n : ℕ, IsClosed (e n) := fun i => + isClosed_iInter fun i => isClosed_le (g i).cont.norm continuous_const + -- the union is the entire space; this is where we use `h` + have hU : (⋃ n : ℕ, e n) = univ := by + refine' eq_univ_of_forall fun x => _ + cases' h x with C hC + obtain ⟨m, hm⟩ := exists_nat_ge C + exact ⟨e m, mem_range_self m, mem_iInter.mpr fun i => le_trans (hC i) hm⟩ + /- apply the Baire category theorem to conclude that for some `m : ℕ`, + `e m` contains some `x` -/ + obtain ⟨m : ℕ, x : E, hx : x ∈ interior (e m)⟩ := nonempty_interior_of_iUnion_of_closed hc hU + obtain ⟨ε, ε_pos, hε : ball x ε ⊆ interior (e m)⟩ := isOpen_iff.mp isOpen_interior x hx + obtain ⟨k : 𝕜, hk : 1 < ‖k‖⟩ := NormedField.exists_one_lt_norm 𝕜 + -- show all elements in the ball have norm bounded by `m` after applying any `g i` + have real_norm_le : ∀ z ∈ ball x ε, ∀ (i : ι), ‖g i z‖ ≤ m := by + intro z hz i + replace hz := mem_iInter.mp (interior_iInter_subset _ (hε hz)) i + apply interior_subset hz + have εk_pos : 0 < ε / ‖k‖ := div_pos ε_pos (zero_lt_one.trans hk) + refine' ⟨(m + m : ℕ) / (ε / ‖k‖), fun i => ContinuousLinearMap.op_norm_le_of_shell ε_pos _ hk _⟩ + · exact div_nonneg (Nat.cast_nonneg _) εk_pos.le + intro y le_y y_lt + calc + ‖g i y‖ = ‖g i (y + x) - g i x‖ := by rw [(g i).map_add, add_sub_cancel] + _ ≤ ‖g i (y + x)‖ + ‖g i x‖ := (norm_sub_le _ _) + _ ≤ m + m := + (add_le_add (real_norm_le (y + x) (by rwa [add_comm, add_mem_ball_iff_norm]) i) + (real_norm_le x (mem_ball_self ε_pos) i)) + _ = (m + m : ℕ) := by norm_cast + _ ≤ (m + m : ℕ) * (‖y‖ / (ε / ‖k‖)) := + (le_mul_of_one_le_right (Nat.cast_nonneg _) + ((one_le_div <| div_pos ε_pos (zero_lt_one.trans hk)).2 le_y)) + _ = (m + m : ℕ) / (ε / ‖k‖) * ‖y‖ := (mul_comm_div _ _ _).symm + + + end +