9c87a7cb - mathematics_in

Authored by: Jeremy Avigad
Authored at: 2023-06-06 18:35:47 -0700
Committed by: Jeremy Avigad
Committed at: 2023-06-06 18:35:47 -0700
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Update Tue, Jun 6, 2023 6:35:47 PM

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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. 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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 <cite>instructions <https://github.com/leanprover/lean4/blob/master/doc/quickstart.md>_</cite>.</li> - <li>In a terminal, navigate to the folder where you want to put a copy of the - repository, and type <code class="docutils literal notranslate">git clone git@github.com:leanprover-community/mathematics_in_lean.git</code> - to fetch it from github.</li> - <li>Navigate to <code class="docutils literal notranslate">mathematics_in_lean</code>, and execute <code class="docutils literal notranslate">lake exe cache get</code> to fetch a compiled - version of the library, <code class="docutils literal notranslate">mathlib</code>.</li> - <li>Type <code class="docutils literal notranslate">code .</code> to open the folder in <code class="docutils literal notranslate">VS Code</code>. Alternatively, you can run <code class="docutils literal notranslate">VS Code</code> and - choose <code class="docutils literal notranslate">Open Folder</code> from the <code class="docutils literal notranslate">File</code> menu. Be sure to open the folder <code class="docutils literal notranslate">mathematics_in_lean</code>, - not any other folder.</li> - </ol> - Opening any Lean file will simultaneously open this - book in a VS Code window. You can update to a newer version by tying - <code class="docutils literal notranslate">git pull</code> followed by <code class="docutils literal notranslate">lake exe cache get</code> inside - the <code class="docutils literal notranslate">mathematics_in_lean</code> folder. - Alternatively, you can run Lean and VS Code in the cloud, - using <a class="reference external" href="https://gitpod.io/">Gitpod</a>. - You can find instructions as to how to do that on the Mathematics in Lean - <a class="reference external" href="https://github.com/leanprover-community/mathematics_in_lean">project page</a> - on Github. - Each section in this book has an associated Lean file with examples - and exercises. You can find them in the folder <cite>src</cite>, - organized by chapter. We recommend making a copy of that folder - so that you can experiment with the files as you go, - while leaving the originals intact. - The text will often include examples, like this one: - <div class="highlight-lean notranslate"><div class="highlight"><pre>#eval "Hello, World!" - </pre></div> - </div> - You should be able to find the corresponding example in the associated - Lean file. - If you click on the line, VS Code will show you Lean’s feedback in - the <code class="docutils literal notranslate">Lean Goal</code> window, and if you hover - your cursor over the <code class="docutils literal notranslate">#eval</code> command VS Code will show you Lean’s response - to this command in a pop-up window. - You are encouraged to edit the file and try examples of your own. - This book moreover provides lots of challenging exercises for you to try. - Don’t rush past these! - Lean is about doing mathematics interactively, not just reading about it. - Working through the exercises is central to the experience. - You can always compare your solutions to the ones in the <code class="docutils literal notranslate">solutions</code> - folder associated with each section. - </section> - <section id="overview"> - <h2>1.2. Overview<a class="headerlink" href="#overview" title="Permalink to this heading"></a></h2> - Put simply, Lean is a tool for building complex expressions in a formal language - known as dependent type theory. - Every expression has a type, and you can use the <cite>#check</cite> command to - print it. - Some expressions have types like <cite>ℕ</cite> or <cite>ℕ → ℕ</cite>. - These are mathematical objects. - <div class="highlight-lean notranslate"><div class="highlight"><pre>#check 2 + 2 - - def f (x : ℕ) := - x + 3 - - #check f - </pre></div> - </div> - Some expressions have type <cite>Prop</cite>. - These are mathematical statements. - <div class="highlight-lean notranslate"><div class="highlight"><pre>#check 2 + 2 = 4 - - def FermatLastTheorem := - ∀ x y z n : ℕ, n > 2 ∧ x * y * z ≠ 0 → x ^ n + y ^ n ≠ z ^ n - - #check FermatLastTheorem - </pre></div> - </div> - Some expressions have a type, <cite>P</cite>, where <cite>P</cite> itself has type <cite>Prop</cite>. - Such an expression is a proof of the proposition <cite>P</cite>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem easy : 2 + 2 = 4 := - rfl - - #check easy - - theorem hard : FermatLastTheorem := - sorry - - #check hard - </pre></div> - </div> - If you manage to construct an expression of type <cite>fermat_last_theorem</cite> and - Lean accepts it as a term of that type, - you have done something very impressive. - (Using <code class="docutils literal notranslate">sorry</code> is cheating, and Lean knows it.) - So now you know the game. - All that is left to learn are the rules. - This book is complementary to a companion tutorial, - <a class="reference external" href="https://leanprover.github.io/theorem_proving_in_lean/">Theorem Proving in Lean</a>, - which provides a more thorough introduction to the underlying logical framework - and core syntax of Lean. - Theorem Proving in Lean is for people who prefer to read a user manual cover to cover before - using a new dishwasher. - If you are the kind of person who prefers to hit the start button and - figure out how to activate the potscrubber feature later, - it makes more sense to start here and refer back to - Theorem Proving in Lean as necessary. - Another thing that distinguishes Mathematics in Lean from - Theorem Proving in Lean is that here we place a much greater - emphasis on the use of tactics. - Given that we are trying to build complex expressions, - Lean offers two ways of going about it: - we can write down the expressions themselves - (that is, suitable text descriptions thereof), - or we can provide Lean with instructions as to how to construct them. - For example, the following expression represents a proof of the fact that - if <code class="docutils literal notranslate">n</code> is even then so is <code class="docutils literal notranslate">m * n</code>: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : ∀ m n : Nat, Even n → Even (m * n) := fun m n ⟨k, (hk : n = k + k)⟩ => - have hmn : m * n = m * k + m * k := by rw [hk, mul_add] - show ∃ l, m * n = l + l from ⟨_, hmn⟩ - </pre></div> - </div> - The proof term can be compressed to a single line: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : ∀ m n : Nat, Even n → Even (m * n) := - fun m n ⟨k, hk⟩ => ⟨m * k, by rw [hk, mul_add]⟩ - </pre></div> - </div> - The following is, instead, a tactic-style proof of the same theorem: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : ∀ m n : Nat, Even n → Even (m * n) := by - -- say m and n are natural numbers, and assume n=2*k - rintro m n ⟨k, hk⟩ - -- We need to prove m*n is twice a natural number. Let's show it's twice m*k. - use m * k - -- substitute in for n - rw [hk] - -- and now it's obvious - ring - </pre></div> - </div> - As you enter each line of such a proof in VS Code, - Lean displays the proof state in a separate window, - telling you what facts you have already established and what - tasks remain to prove your theorem. - You can replay the proof by stepping through the lines, - since Lean will continue to show you the state of the proof - at the point where the cursor is. - In this example, you will then see that - the first line of the proof introduces <code class="docutils literal notranslate">m</code> and <code class="docutils literal notranslate">n</code> - (we could have renamed them at that point, if we wanted to), - and also decomposes the hypothesis <code class="docutils literal notranslate">even n</code> to - a <code class="docutils literal notranslate">k</code> and the assumption that <code class="docutils literal notranslate">n = 2 * k</code>. - The second line, <code class="docutils literal notranslate">use m * k</code>, - declares that we are going to show that <code class="docutils literal notranslate">m * n</code> is even by - showing <code class="docutils literal notranslate">m * n = 2 * (m * k)</code>. - The next line uses the <code class="docutils literal notranslate">rewrite</code> tactic - to replace <code class="docutils literal notranslate">n</code> by <code class="docutils literal notranslate">2 * k</code> in the goal, - and the <cite>ring</cite> tactic solves the resulting goal <code class="docutils literal notranslate">m * (2 * k) = 2 * (m * k)</code>. - The ability to build a proof in small steps with incremental feedback - is extremely powerful. For that reason, - tactic proofs are often easier and quicker to write than - proof terms. - There isn’t a sharp distinction between the two: - tactic proofs can be inserted in proof terms, - as we did with the phrase <code class="docutils literal notranslate">by rw [hk, mul_left_comm]</code> in the example above. - We will also see that, conversely, - it is often useful to insert a short proof term in the middle of a tactic proof. - That said, in this book, our emphasis will be on the use of tactics. - In our example, the tactic proof can also be reduced to a one-liner: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : ∀ m n : Nat, Even n → Even (m * n) := by - rintro m n ⟨k, hk⟩; use m * k; rw [hk]; ring - </pre></div> - </div> - Here we have used tactics to carry out small proof steps. - But they can also provide substantial automation, - and justify longer calculations and bigger inferential steps. - For example, we can invoke Lean’s simplifier with - specific rules for simplifying statements about parity to - prove our theorem automatically. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : ∀ m n : Nat, Even n → Even (m * n) := by - intros; simp [*, parity_simps] - </pre></div> - </div> - Another big difference between the two introductions is that - Theorem Proving in Lean depends only on core Lean and its built-in - tactics, whereas Mathematics in Lean is built on top of Lean’s - powerful and ever-growing library, mathlib. - As a result, we can show you how to use some of the mathematical - objects and theorems in the library, - and some of the very useful tactics. - This book is not meant to be used as an overview of the library; - the <a class="reference external" href="https://leanprover-community.github.io/">community</a> - web pages contain extensive documentation. - Rather, our goal is to introduce you to the style of thinking that - underlies that formalization, - so that you are comfortable browsing the library and - finding things on your own. - Interactive theorem proving can be frustrating, - and the learning curve is steep. - But the Lean community is very welcoming to newcomers, - and people are available on the - <a class="reference external" href="https://leanprover.zulipchat.com/">Lean Zulip chat group</a> round the clock - to answer questions. - We hope to see you there, and have no doubt that - soon enough you, too, will be able to answer such questions - and contribute to the development of mathlib. - So here is your mission, should you choose to accept it: - dive in, try the exercises, come to Zulip with questions, and have fun. - But be forewarned: - interactive theorem proving will challenge you to think about - mathematics and mathematical reasoning in fundamentally new ways. - Your life may never be the same. - Acknowledgments. We are grateful to Gabriel Ebner for setting up the - infrastructure for running this tutorial in VS Code, - and to Scott Morrison and Mario Carneiro for help porting it from Lean 3. - We are also grateful for help and corrections from - Bryan Gin-ge Chen, Johan Commelin, Mathieu Guay-Paquet, Julian Külshammer, - Giovanni Mascellani, Hunter Monroe, Pietro Monticone, Bartosz Piotrowski, - and Guilherme Silva. - Our work has been partially supported by the Hoskinson Center for - Formal Mathematics. - </section> - </section> - - - </div> - </div> - <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> - <a href="index.html" class="btn btn-neutral float-left" title="Mathematics in Lean" accesskey="p" rel="prev"> Previous</a> - <a href="C02_Basics.html" class="btn btn-neutral float-right" title="2. Basics" accesskey="n" rel="next">Next </a> - </div> - - <hr/> - - <div role="contentinfo"> - © Copyright 2020, Jeremy Avigad, Kevin Buzzard, Robert Y. 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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. Introduction" accesskey="p" rel="prev"> Previous</a> - <a href="C03_Logic.html" class="btn btn-neutral float-right" title="3. Logic" accesskey="n" rel="next">Next </a> - </div> - - <hr/> - - <div role="contentinfo"> - © Copyright 2020, Jeremy Avigad, Kevin Buzzard, Robert Y. Lewis, Patrick Massot. - </div> - - Built with <a href="https://www.sphinx-doc.org/">Sphinx</a> using a - <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a> - provided by <a href="https://readthedocs.org">Read the Docs</a>. - - - </footer> - </div> - </div> - </section> - </div> - <script> - jQuery(function () { - SphinxRtdTheme.Navigation.enable(true); - }); - </script> - - </body> - </html>
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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 (which can also be written <code class="docutils literal notranslate">rintro</code>) - is a combination of <code class="docutils literal notranslate">intro</code> and <code class="docutils literal notranslate">rcases</code>. - These examples illustrate their use: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x => f x + g x := by - rcases ubf with ⟨a, ubfa⟩ - rcases ubg with ⟨b, 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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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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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 : x % y * conj y = - ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by - rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] - ext <;> simp <;> ring - have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by - conv => - lhs - rw [← norm_conj y, ← norm_mul, this, norm] - simp - trans 2 * (y.norm / 2) ^ 2 - · rw [two_mul] - apply add_le_add <;> - · rw [sq_le_sq] - apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) - apply le_abs_self - rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] - apply mul_le_mul_of_nonneg_left - · apply Int.ediv_mul_le - norm_num - apply Int.ediv_nonneg (norm_nonneg y) - norm_num - have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos - apply lt_of_le_of_lt this - apply Int.ediv_lt_of_lt_mul - · norm_num - linarith - </pre></div> - </div> - We are in the home stretch. Our <code class="docutils literal notranslate">norm</code> function maps Gaussian integers to - nonnegative integers. We need a function that maps Gaussian integers to natural - numbers, and we obtain that by composing <code class="docutils literal notranslate">norm</code> with the function - <code class="docutils literal notranslate">Int.natAbs</code>, which maps integers to the natural numbers. - The first of the next two lemmas establishes that mapping the norm to the - natural numbers and back to the integers does not change the value. - The second one re-expresses the fact that the norm is decreasing. - <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := - Int.natAbs_of_nonneg (norm_nonneg _) - - theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : - (x % y).norm.natAbs < y.norm.natAbs := by - apply Int.ofNat_lt.1 - simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] - apply norm_mod_lt x hy - </pre></div> - </div> - We also need to establish the second key property of the norm function - on a Euclidean domain. - <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : - ¬(norm (x * y)).natAbs < (norm x).natAbs := by - apply not_lt_of_ge - rw [norm_mul, Int.natAbs_mul] - apply le_mul_of_one_le_right (Nat.zero_le _) - apply Int.ofNat_le.1 - rw [coe_natAbs_norm] - exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) - </pre></div> - </div> - We can now put it together to show that the Gaussian integers are an - instance of a Euclidean domain. We use the quotient and remainder function we - have defined. - The mathlib definition of a Euclidean domain is more general than the one - above in that it allows us to show that remainder decreases with respect - to any well-founded measure. - Comparing the values of a norm function that returns natural numbers is - just one instance of such a measure, - and in that case, the required properties are the theorems - <code class="docutils literal notranslate">natAbs_norm_mod_lt</code> and <code class="docutils literal notranslate">not_norm_mul_left_lt_norm</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>instance : EuclideanDomain gaussInt := - { gaussInt.instCommRing with - quotient := (· / ·) - remainder := (· % ·) - quotient_mul_add_remainder_eq := - fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] - quotient_zero := fun x => by - simp [div_def, norm, Int.div'] - rfl - r := Measure (Int.natAbs ∘ norm) - r_wellFounded := (measure (Int.natAbs ∘ norm)).2 - remainder_lt := natAbs_norm_mod_lt - mul_left_not_lt := not_norm_mul_left_lt_norm } - </pre></div> - </div> - An immediate payoff is that we now know that, in the Gaussian integers, - the notions of being prime and being irreducible coincide. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x : gaussInt) : Irreducible x ↔ Prime x := - PrincipalIdealRing.irreducible_iff_prime - </pre></div> - </div> - </section> - </section> - - - </div> - </div> - <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> - <a href="C05_Number_Theory.html" class="btn btn-neutral float-left" title="5. 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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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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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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 (deleted)
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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 (deleted)
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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 (deleted)
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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 - - - - - - - - - - -
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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
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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
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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
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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
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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 (deleted)
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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 (deleted)
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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
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  - Index - =====
html/_sources/index.rst.txt (deleted)
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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
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html/_static/language_data.js (deleted)
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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 (deleted)
html/_static/plus.png (deleted)
html/_static/pygments.css (deleted)
- @@ -1,74 +0,0 @@
  - pre { line-height: 125%; } - td.linenos .normal { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; } - span.linenos { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; } - td.linenos .special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; } - span.linenos.special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; } - .highlight .hll { background-color: #ffffcc } - .highlight { background: #f8f8f8; } - .highlight .c { color: #3D7B7B; font-style: italic } /* Comment */ - .highlight .err { border: 1px solid #FF0000 } /* Error */ - .highlight .k { color: #008000; font-weight: bold } /* Keyword */ - .highlight .o { color: #666666 } /* Operator */ - .highlight .ch { color: #3D7B7B; font-style: italic } /* Comment.Hashbang */ - .highlight .cm { color: #3D7B7B; font-style: italic } /* Comment.Multiline */ - .highlight .cp { color: #9C6500 } /* Comment.Preproc */ - .highlight .cpf { color: #3D7B7B; font-style: italic } /* Comment.PreprocFile */ - .highlight .c1 { color: #3D7B7B; font-style: italic } /* Comment.Single */ - .highlight .cs { color: #3D7B7B; font-style: italic } /* Comment.Special */ - .highlight .gd { color: #A00000 } /* Generic.Deleted */ - .highlight .ge { font-style: italic } /* Generic.Emph */ - .highlight .gr { color: #E40000 } /* Generic.Error */ - .highlight .gh { color: #000080; font-weight: bold } /* Generic.Heading */ - .highlight .gi { color: #008400 } /* Generic.Inserted */ - .highlight .go { color: #717171 } /* Generic.Output */ - .highlight .gp { color: #000080; font-weight: bold } /* Generic.Prompt */ - .highlight .gs { font-weight: bold } /* Generic.Strong */ - .highlight .gu { color: #800080; font-weight: bold } /* Generic.Subheading */ - .highlight .gt { color: #0044DD } /* Generic.Traceback */ - .highlight .kc { color: #008000; font-weight: bold } /* Keyword.Constant */ - .highlight .kd { color: #008000; font-weight: bold } /* Keyword.Declaration */ - .highlight .kn { color: #008000; font-weight: bold } /* Keyword.Namespace */ - .highlight .kp { color: #008000 } /* Keyword.Pseudo */ - .highlight .kr { color: #008000; font-weight: bold } /* Keyword.Reserved */ - .highlight .kt { color: #B00040 } /* Keyword.Type */ - .highlight .m { color: #666666 } /* Literal.Number */ - .highlight .s { color: #BA2121 } /* Literal.String */ - .highlight .na { color: #687822 } /* Name.Attribute */ - .highlight .nb { color: #008000 } /* Name.Builtin */ - .highlight .nc { color: #0000FF; font-weight: bold } /* Name.Class */ - .highlight .no { color: #880000 } /* Name.Constant */ - .highlight .nd { color: #AA22FF } /* Name.Decorator */ - .highlight .ni { color: #717171; font-weight: bold } /* Name.Entity */ - .highlight .ne { color: #CB3F38; font-weight: bold } /* Name.Exception */ - .highlight .nf { color: #0000FF } /* Name.Function */ - .highlight .nl { color: #767600 } /* Name.Label */ - .highlight .nn { color: #0000FF; font-weight: bold } /* Name.Namespace */ - .highlight .nt { color: #008000; font-weight: bold } /* Name.Tag */ - .highlight .nv { color: #19177C } /* Name.Variable */ - .highlight .ow { color: #AA22FF; font-weight: bold } /* Operator.Word */ - .highlight .w { color: #bbbbbb } /* Text.Whitespace */ - .highlight .mb { color: #666666 } /* Literal.Number.Bin */ - .highlight .mf { color: #666666 } /* Literal.Number.Float */ - .highlight .mh { color: #666666 } /* Literal.Number.Hex */ - .highlight .mi { color: #666666 } /* Literal.Number.Integer */ - .highlight .mo { color: #666666 } /* Literal.Number.Oct */ - .highlight .sa { color: #BA2121 } /* Literal.String.Affix */ - .highlight .sb { color: #BA2121 } /* Literal.String.Backtick */ - .highlight .sc { color: #BA2121 } /* Literal.String.Char */ - .highlight .dl { color: #BA2121 } /* Literal.String.Delimiter */ - .highlight .sd { color: #BA2121; font-style: italic } /* Literal.String.Doc */ - .highlight .s2 { color: #BA2121 } /* Literal.String.Double */ - .highlight .se { color: #AA5D1F; font-weight: bold } /* Literal.String.Escape */ - .highlight .sh { color: #BA2121 } /* Literal.String.Heredoc */ - .highlight .si { color: #A45A77; font-weight: bold } /* Literal.String.Interpol */ - .highlight .sx { color: #008000 } /* Literal.String.Other */ - .highlight .sr { color: #A45A77 } /* Literal.String.Regex */ - .highlight .s1 { color: #BA2121 } /* Literal.String.Single */ - .highlight .ss { color: #19177C } /* Literal.String.Symbol */ - .highlight .bp { color: #008000 } /* Name.Builtin.Pseudo */ - .highlight .fm { color: #0000FF } /* Name.Function.Magic */ - .highlight .vc { color: #19177C } /* Name.Variable.Class */ - .highlight .vg { color: #19177C } /* Name.Variable.Global */ - .highlight .vi { color: #19177C } /* Name.Variable.Instance */ - .highlight .vm { color: #19177C } /* Name.Variable.Magic */ - .highlight .il { color: #666666 } /* Literal.Number.Integer.Long */
html/_static/searchtools.js (deleted)
- @@ -1,566 +0,0 @@
 - /* - * 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 ? 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html/_static/sphinx_highlight.js (deleted)
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html/genindex.html (deleted)
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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=1.0" /> - <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%; 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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. 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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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lake-manifest.json
- @@ -1,27 +1,1 @@
  - {"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"}}]} + ../lake-manifest.json
lakefile.lean
- @@ -1,13 +1,1 @@
  - 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" + ../lakefile.lean
lean-toolchain
- @@ -1,1 +1,1 @@
  - leanprover/lean4:nightly-2023-05-31 + ../lean-toolchain
mathematics_in_lean.pdf
src (new)
- @@ -0,0 +1,1 @@
  + ../src
src/C01_Introduction/S01_Getting_Started.lean (deleted)
- @@ -1,2 +0,0 @@
  - #eval "Hello, World!" -
src/C01_Introduction/S02_Overview.lean (deleted)
- @@ -1,57 +0,0 @@
  - 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 (deleted)
src/C01_Introduction/solutions/Solutions_S02_Overview.lean (deleted)
- @@ -1,7 +0,0 @@
  - 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 (deleted)
- @@ -1,147 +0,0 @@
  - 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 (deleted)
- @@ -1,146 +0,0 @@
  - 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 (deleted)
- @@ -1,130 +0,0 @@
 - 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 (deleted)
- @@ -1,90 +0,0 @@
  - 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 (deleted)
- @@ -1,109 +0,0 @@
 - 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 (deleted)
- @@ -1,33 +0,0 @@
  - 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 (deleted)
- @@ -1,73 +0,0 @@
  - 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 (deleted)
- @@ -1,61 +0,0 @@
 - 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 (deleted)
- @@ -1,116 +0,0 @@
  - 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 (deleted)
- @@ -1,135 +0,0 @@
  - 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 (deleted)
- @@ -1,178 +0,0 @@
 - 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 (deleted)
- @@ -1,131 +0,0 @@
 - 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 (deleted)
- @@ -1,137 +0,0 @@
 - 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 (deleted)
- @@ -1,135 +0,0 @@
 - 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 (deleted)
- @@ -1,101 +0,0 @@
 - 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 (deleted)
- @@ -1,99 +0,0 @@
 - 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 (deleted)
- @@ -1,129 +0,0 @@
  - 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 (deleted)
- @@ -1,81 +0,0 @@
  - 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 (deleted)
- @@ -1,111 +0,0 @@
 - 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 (deleted)
- @@ -1,94 +0,0 @@
 - 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 (deleted)
- @@ -1,141 +0,0 @@
 - 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 (deleted)
- @@ -1,127 +0,0 @@
 - 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 (deleted)
- @@ -1,247 +0,0 @@
 - 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 (deleted)
- @@ -1,213 +0,0 @@
  - 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 (deleted)
- @@ -1,98 +0,0 @@
  - 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 (deleted)
- @@ -1,158 +0,0 @@
 - 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 (deleted)
- @@ -1,252 +0,0 @@
  - 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 (deleted)
- @@ -1,88 +0,0 @@
  - 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 (deleted)
- @@ -1,121 +0,0 @@
 - 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 (deleted)
- @@ -1,148 +0,0 @@
 - 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 (deleted)
- @@ -1,233 +0,0 @@
 - 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 (deleted)
- @@ -1,105 +0,0 @@
 - 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 (deleted)
- @@ -1,98 +0,0 @@
 - 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 (deleted)
- @@ -1,242 +0,0 @@
 - 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 (deleted)
- @@ -1,224 +0,0 @@
 - 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 (deleted)
- @@ -1,171 +0,0 @@
  - 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 (deleted)
- @@ -1,276 +0,0 @@
 - import Mathlib.Data.Int.Basic - import Mathlib.Algebra.EuclideanDomain.Basic - import Mathlib.RingTheory.PrincipalIdealDomain - import Mathlib.Tactic - - @[ext] - structure gaussInt where - re : ℤ - im : ℤ - - namespace gaussInt - - instance : Zero gaussInt := - ⟨⟨0, 0⟩⟩ - - instance : One gaussInt := - ⟨⟨1, 0⟩⟩ - - instance : Add gaussInt := - ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ - - instance : Neg gaussInt := - ⟨fun x => ⟨-x.re, -x.im⟩⟩ - - instance : Mul gaussInt := - ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ - - theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := - rfl - - theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := - rfl - - theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := - rfl - - theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := - rfl - - theorem mul_def (x y : gaussInt) : - x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := - rfl - - @[simp] - theorem zero_re : (0 : gaussInt).re = 0 := - rfl - - @[simp] - theorem zero_im : (0 : gaussInt).im = 0 := - rfl - - @[simp] - theorem one_re : (1 : gaussInt).re = 1 := - rfl - - @[simp] - theorem one_im : (1 : gaussInt).im = 0 := - rfl - - @[simp] - theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := - rfl - - @[simp] - theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := - rfl - - @[simp] - theorem neg_re (x : gaussInt) : (-x).re = -x.re := - rfl - - @[simp] - theorem neg_im (x : gaussInt) : (-x).im = -x.im := - rfl - - @[simp] - theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := - rfl - - @[simp] - theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := - rfl - - instance instCommRing : CommRing gaussInt where - zero := 0 - one := 1 - add := (· + ·) - neg x := -x - mul := (· * ·) - add_assoc := by - intros - ext <;> simp <;> ring - zero_add := by - intro - ext <;> simp - add_zero := by - intro - ext <;> simp - add_left_neg := by - intro - ext <;> simp - add_comm := by - intros - ext <;> simp <;> ring - mul_assoc := by - intros - ext <;> simp <;> ring - one_mul := by - intro - ext <;> simp - mul_one := by - intro - ext <;> simp - left_distrib := by - intros - ext <;> simp <;> ring - right_distrib := by - intros - ext <;> simp <;> ring - mul_comm := by - intros - ext <;> simp <;> ring - zero_mul := sorry - mul_zero := sorry - - instance : Nontrivial gaussInt := by - use 0, 1 - rw [Ne, gaussInt.ext_iff] - simp - - end gaussInt - - example (a b : ℤ) : a = b * (a / b) + a % b := - Eq.symm <| Int.ediv_add_emod a b - - example (a b : ℤ) : b ≠ 0 → 0 ≤ a % b := - Int.emod_nonneg a - - example (a b : ℤ) : b ≠ 0 → a % b < abs b := - Int.emod_lt a - - namespace Int - - def div' (a b : ℤ) := - (a + b / 2) / b - - def mod' (a b : ℤ) := - (a + b / 2) % b - b / 2 - - theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by - rw [div', mod'] - linarith [Int.ediv_add_emod (a + b / 2) b] - - theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by - rw [mod', abs_le] - constructor - · linarith [Int.emod_nonneg (a + b / 2) h.ne'] - have := Int.emod_lt_of_pos (a + b / 2) h - have := Int.ediv_add_emod b 2 - have := Int.emod_lt_of_pos b zero_lt_two - revert this; intro this -- FIXME, this should not be needed - linarith - - theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] - - end Int - - theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : - x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by - sorry - namespace gaussInt - - def norm (x : gaussInt) := - x.re ^ 2 + x.im ^ 2 - - @[simp] - theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by - sorry - theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by - sorry - theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by - sorry - theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by - sorry - def conj (x : gaussInt) : gaussInt := - ⟨x.re, -x.im⟩ - - @[simp] - theorem conj_re (x : gaussInt) : (conj x).re = x.re := - rfl - - @[simp] - theorem conj_im (x : gaussInt) : (conj x).im = -x.im := - rfl - - theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] - - instance : Div gaussInt := - ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ - - instance : Mod gaussInt := - ⟨fun x y => x - y * (x / y)⟩ - - theorem div_def (x y : gaussInt) : - x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := - rfl - - theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := - rfl - - theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : - (x % y).norm < y.norm := by - have norm_y_pos : 0 < norm y := by rwa [norm_pos] - have : x % y * conj y = - ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by - rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] - ext <;> simp <;> ring - have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by - conv => - lhs - rw [← norm_conj y, ← norm_mul, this, norm] - simp - trans 2 * (y.norm / 2) ^ 2 - · rw [two_mul] - apply add_le_add <;> - · rw [sq_le_sq] - apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) - apply le_abs_self - rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] - apply mul_le_mul_of_nonneg_left - · apply Int.ediv_mul_le - norm_num - apply Int.ediv_nonneg (norm_nonneg y) - norm_num - have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos - apply lt_of_le_of_lt this - apply Int.ediv_lt_of_lt_mul - · norm_num - linarith - - theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := - Int.natAbs_of_nonneg (norm_nonneg _) - - theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : - (x % y).norm.natAbs < y.norm.natAbs := by - apply Int.ofNat_lt.1 - simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] - apply norm_mod_lt x hy - - theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : - ¬(norm (x * y)).natAbs < (norm x).natAbs := by - apply not_lt_of_ge - rw [norm_mul, Int.natAbs_mul] - apply le_mul_of_one_le_right (Nat.zero_le _) - apply Int.ofNat_le.1 - rw [coe_natAbs_norm] - exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) - - instance : EuclideanDomain gaussInt := - { gaussInt.instCommRing with - quotient := (· / ·) - remainder := (· % ·) - quotient_mul_add_remainder_eq := - fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] - quotient_zero := fun x => by - simp [div_def, norm, Int.div'] - rfl - r := Measure (Int.natAbs ∘ norm) - r_wellFounded := (measure (Int.natAbs ∘ norm)).2 - remainder_lt := natAbs_norm_mod_lt - mul_left_not_lt := not_norm_mul_left_lt_norm } - - example (x : gaussInt) : Irreducible x ↔ Prime x := - PrincipalIdealRing.irreducible_iff_prime - - end gaussInt
src/C06_Structures/solutions/Solutions_S01_Structures.lean (deleted)
- @@ -1,99 +0,0 @@
  - 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 (deleted)
- @@ -1,74 +0,0 @@
  - 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 (deleted)
- @@ -1,290 +0,0 @@
 - import Mathlib.Data.Int.Basic - import Mathlib.Algebra.EuclideanDomain.Basic - import Mathlib.RingTheory.PrincipalIdealDomain - import Mathlib.Tactic - - @[ext] - structure gaussInt where - re : ℤ - im : ℤ - - namespace gaussInt - - instance : Zero gaussInt := - ⟨⟨0, 0⟩⟩ - - instance : One gaussInt := - ⟨⟨1, 0⟩⟩ - - instance : Add gaussInt := - ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ - - instance : Neg gaussInt := - ⟨fun x => ⟨-x.re, -x.im⟩⟩ - - instance : Mul gaussInt := - ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ - - theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := - rfl - - theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := - rfl - - theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := - rfl - - theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := - rfl - - theorem mul_def (x y : gaussInt) : - x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := - rfl - - @[simp] - theorem zero_re : (0 : gaussInt).re = 0 := - rfl - - @[simp] - theorem zero_im : (0 : gaussInt).im = 0 := - rfl - - @[simp] - theorem one_re : (1 : gaussInt).re = 1 := - rfl - - @[simp] - theorem one_im : (1 : gaussInt).im = 0 := - rfl - - @[simp] - theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := - rfl - - @[simp] - theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := - rfl - - @[simp] - theorem neg_re (x : gaussInt) : (-x).re = -x.re := - rfl - - @[simp] - theorem neg_im (x : gaussInt) : (-x).im = -x.im := - rfl - - @[simp] - theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := - rfl - - @[simp] - theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := - rfl - - instance instCommRing : CommRing gaussInt where - zero := 0 - one := 1 - add := (· + ·) - neg x := -x - mul := (· * ·) - add_assoc := by - intros - ext <;> simp <;> ring - zero_add := by - intro - ext <;> simp - add_zero := by - intro - ext <;> simp - add_left_neg := by - intro - ext <;> simp - add_comm := by - intros - ext <;> simp <;> ring - mul_assoc := by - intros - ext <;> simp <;> ring - one_mul := by - intro - ext <;> simp - mul_one := by - intro - ext <;> simp - left_distrib := by - intros - ext <;> simp <;> ring - right_distrib := by - intros - ext <;> simp <;> ring - mul_comm := by - intros - ext <;> simp <;> ring - zero_mul := sorry - mul_zero := sorry - - instance : Nontrivial gaussInt := by - use 0, 1 - rw [Ne, gaussInt.ext_iff] - simp - - end gaussInt - - namespace Int - - def div' (a b : ℤ) := - (a + b / 2) / b - - def mod' (a b : ℤ) := - (a + b / 2) % b - b / 2 - - theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by - rw [div', mod'] - linarith [Int.ediv_add_emod (a + b / 2) b] - - theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by - rw [mod', abs_le] - constructor - · linarith [Int.emod_nonneg (a + b / 2) h.ne'] - have := Int.emod_lt_of_pos (a + b / 2) h - have := Int.ediv_add_emod b 2 - have := Int.emod_lt_of_pos b zero_lt_two - revert this; intro this -- FIXME, this should not be needed - linarith - - theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] - - end Int - - private theorem aux {α : Type _} [LinearOrderedRing α] {x y : α} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := - haveI h' : x ^ 2 = 0 := by - apply le_antisymm _ (sq_nonneg x) - rw [← h] - apply le_add_of_nonneg_right (sq_nonneg y) - pow_eq_zero h' - - theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : - x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by - constructor - · intro h - constructor - · exact aux h - rw [add_comm] at h - exact aux h - rintro ⟨rfl, rfl⟩ - norm_num - - namespace gaussInt - - def norm (x : gaussInt) := - x.re ^ 2 + x.im ^ 2 - - @[simp] - theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by - apply add_nonneg <;> - apply sq_nonneg - - theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by - rw [norm, sq_add_sq_eq_zero, gaussInt.ext_iff] - rfl - - theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by - rw [lt_iff_le_and_ne, ne_comm, Ne, norm_eq_zero] - simp [norm_nonneg] - - theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by - simp [norm] - ring - - def conj (x : gaussInt) : gaussInt := - ⟨x.re, -x.im⟩ - - @[simp] - theorem conj_re (x : gaussInt) : (conj x).re = x.re := - rfl - - @[simp] - theorem conj_im (x : gaussInt) : (conj x).im = -x.im := - rfl - - theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] - - instance : Div gaussInt := - ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ - - instance : Mod gaussInt := - ⟨fun x y => x - y * (x / y)⟩ - - theorem div_def (x y : gaussInt) : - x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := - rfl - - theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := - rfl - - theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : - (x % y).norm < y.norm := by - have norm_y_pos : 0 < norm y := by rwa [norm_pos] - have : x % y * conj y = - ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by - rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] - ext <;> simp <;> ring - have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by - conv => - lhs - rw [← norm_conj y, ← norm_mul, this, norm] - simp - trans 2 * (y.norm / 2) ^ 2 - · rw [two_mul] - apply add_le_add <;> - · rw [sq_le_sq] - apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) - apply le_abs_self - rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] - apply mul_le_mul_of_nonneg_left - · apply Int.ediv_mul_le - norm_num - apply Int.ediv_nonneg (norm_nonneg y) - norm_num - have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos - apply lt_of_le_of_lt this - apply Int.ediv_lt_of_lt_mul - · norm_num - linarith - - theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := - Int.natAbs_of_nonneg (norm_nonneg _) - - theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : - (x % y).norm.natAbs < y.norm.natAbs := by - apply Int.ofNat_lt.1 - simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] - apply norm_mod_lt x hy - - theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : - ¬(norm (x * y)).natAbs < (norm x).natAbs := by - apply not_lt_of_ge - rw [norm_mul, Int.natAbs_mul] - apply le_mul_of_one_le_right (Nat.zero_le _) - apply Int.ofNat_le.1 - rw [coe_natAbs_norm] - exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) - - instance : EuclideanDomain gaussInt := - { gaussInt.instCommRing with - quotient := (· / ·) - remainder := (· % ·) - quotient_mul_add_remainder_eq := - fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] - quotient_zero := fun x => by - simp [div_def, norm, Int.div'] - rfl - r := Measure (Int.natAbs ∘ norm) - r_wellFounded := (measure (Int.natAbs ∘ norm)).2 - remainder_lt := natAbs_norm_mod_lt - mul_left_not_lt := not_norm_mul_left_lt_norm } - - example (x : gaussInt) : Irreducible x ↔ Prime x := - PrincipalIdealRing.irreducible_iff_prime - - end gaussInt
src/C07_Hierarchies/S01_Basics.lean (deleted)
- @@ -1,308 +0,0 @@
 - 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 (deleted)
- @@ -1,114 +0,0 @@
  - 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 (deleted)
- @@ -1,88 +0,0 @@
  - 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 (deleted)
- @@ -1,348 +0,0 @@
 - 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 (deleted)
- @@ -1,126 +0,0 @@
  - 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 (deleted)
- @@ -1,126 +0,0 @@
  - 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 (deleted)
- @@ -1,105 +0,0 @@
 - 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 (deleted)
- @@ -1,206 +0,0 @@
 - 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 (deleted)
- @@ -1,155 +0,0 @@
 - 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 (deleted)
- @@ -1,71 +0,0 @@
 - 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 (deleted)
- @@ -1,371 +0,0 @@
 - 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 (deleted)
- @@ -1,207 +0,0 @@
 - 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 (deleted)
- @@ -1,40 +0,0 @@
 - 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 (deleted)
- @@ -1,185 +0,0 @@
 - 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 (deleted)
- @@ -1,8 +0,0 @@
  - 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 (deleted)
- @@ -1,62 +0,0 @@
 - 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 -