021da55a - mathematics_in

Authored by: Patrick Massot
Authored at: 2024-09-27 15:36:42 +0200
Committed by: Patrick Massot
Committed at: 2024-09-27 15:36:42 +0200
Object name: 021da55aec6a7108fd8d29afc132c665f2979326
Parent: 0be3e8ea

Update ven. 27 sept. 2024 15:36:42 CEST

Changes

29 changed files (+4065/-4184)

MIL/C09_Linear_Algebra/S01_Vector_Spaces.lean
- @@ -153,4 +153,3 @@ example [Fintype ι] : (⨁ i, V i) ≃ₗ[K] (Π i, V i) :=
  linearEquivFunOnFintype K ι V end families -
MIL/C09_Linear_Algebra/S04_Bases.lean
- @@ -261,4 +261,3 @@ example : Cardinal.lift.{v, u} (.mk ι) = Cardinal.lift.{u, v} (.mk ι') :=
  example [FiniteDimensional K V] : (FiniteDimensional.finrank K V : Cardinal) = Module.rank K V := finrank_eq_rank K V -
html/.buildinfo
- @@ -1,4 +1,4 @@
  # Sphinx build info version 1 # This file hashes the configuration used when building these files. When it is not found, a full rebuild will be done. - config: 641199866dfd7d43f34ecf960cb67cbc + config: dfa3c760a9c309561b2de03685612f6b tags: 645f666f9bcd5a90fca523b33c5a78b7
html/C01_Introduction.html
- @@ -1,25 +1,23 @@
 <!DOCTYPE html> - <html class="writer-html5" lang="en" data-content_root="./"> + <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="viewport" content="width=device-width, initial-scale=1" /> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>1. Introduction — Mathematics in Lean 0.1 documentation</title> - <link rel="stylesheet" type="text/css" href="_static/pygments.css?v=80d5e7a1" /> - <link rel="stylesheet" type="text/css" href="_static/css/theme.css?v=19f00094" /> - <link rel="stylesheet" type="text/css" href="_static/css/custom.css?v=0731ccc3" /> - - + <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?v=5d32c60e"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js?v=2cd50e6c"></script> - <script src="_static/documentation_options.js?v=2709fde1"></script> - <script src="_static/doctools.js?v=9a2dae69"></script> - <script src="_static/sphinx_highlight.js?v=dc90522c"></script> + <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" />
- @@ -93,9 +91,9 @@
 <div itemprop="articleBody"> <section id="introduction"> - <h1>1. Introduction<a class="headerlink" href="#introduction" title="Link to this heading"></a></h1> + <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="Link to this heading"></a></h2> + <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.
- @@ -162,7 +160,7 @@ which is why we suggested making a copy.)
 We intend for you to work on the exercises in the <code class="docutils literal notranslate">MIL</code> folder while reading the textbook, which contains explanations, instructions, and hints. The text will often include examples, like this one: - <div class="highlight-lean notranslate"><div class="highlight"><pre>#eval "Hello, World!" + <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
- @@ -182,42 +180,42 @@ You can always compare your solutions to the ones in the <code class="docutils l
 folder associated with each section. </section> <section id="overview"> - <h2>1.2. Overview<a class="headerlink" href="#overview" title="Link to this heading"></a></h2> + <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 + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check 2 + 2 - def f (x : ℕ) := - x + 3 + def f (x : ℕ) := + x + 3 - #check f + #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 + <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 + def FermatLastTheorem := + ∀ x y z n : ℕ, n > 2 ∧ x * y * z ≠ 0 → x ^ n + y ^ n ≠ z ^ n - #check FermatLastTheorem + #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 + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem easy : 2 + 2 = 4 := + rfl - #check easy + #check easy - theorem hard : FermatLastTheorem := - sorry + theorem hard : FermatLastTheorem := + sorry - #check hard + #check hard </pre></div> </div> If you manage to construct an expression of type <code class="docutils literal notranslate">FermatLastTheorem</code> and
- @@ -246,27 +244,27 @@ we can write down the expressions themselves
 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⟩ + <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]⟩ + <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, where lines starting with <code class="docutils literal notranslate">--</code> are comments, hence ignored by Lean: - <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 for `n`, - rw [hk] - -- and now it's obvious. - ring + <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 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,
- @@ -298,8 +296,8 @@ 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 + <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.
- @@ -308,8 +306,8 @@ 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] + <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
html/C02_Basics.html
- @@ -1,25 +1,23 @@
 <!DOCTYPE html> - <html class="writer-html5" lang="en" data-content_root="./"> + <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="viewport" content="width=device-width, initial-scale=1" /> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>2. Basics — Mathematics in Lean 0.1 documentation</title> - <link rel="stylesheet" type="text/css" href="_static/pygments.css?v=80d5e7a1" /> - <link rel="stylesheet" type="text/css" href="_static/css/theme.css?v=19f00094" /> - <link rel="stylesheet" type="text/css" href="_static/css/custom.css?v=0731ccc3" /> - - + <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?v=5d32c60e"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js?v=2cd50e6c"></script> - <script src="_static/documentation_options.js?v=2709fde1"></script> - <script src="_static/doctools.js?v=9a2dae69"></script> - <script src="_static/sphinx_highlight.js?v=dc90522c"></script> + <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 async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" />
- @@ -97,13 +95,13 @@
 <div itemprop="articleBody"> <section id="basics"> - <h1>2. Basics<a class="headerlink" href="#basics" title="Link to this heading"></a></h1> + <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="Link to this heading"></a></h2> + <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,
- @@ -125,9 +123,9 @@ so the left-hand side of <code class="docutils literal notranslate"> 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] + <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> lines at the beginning of the associated examples file
- @@ -151,12 +149,12 @@ Lean reports on the current proof state in the
 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 + <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:
- @@ -183,20 +181,20 @@ For example, <code class="docutils literal notranslate">rw</sp
 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 + <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 + 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] + <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.
- @@ -205,43 +203,43 @@ For example, <code class="docutils literal notranslate">mul_co
 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 + <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 + example (a b c : ℝ) : a * (b * c) = b * (a * c) := by + sorry </pre></div> </div> You can 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] + <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 <code class="docutils literal notranslate">sub_self</code> 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 + <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 + 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] + <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 : ℝ) + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (a b c d e f : ℝ) - example (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by - rw [h', ← mul_assoc, h, mul_assoc] + 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
- @@ -251,16 +249,16 @@ in a <code class="docutils literal notranslate">section
 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) + 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 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 end </pre></div>
- @@ -280,10 +278,10 @@ that <code class="docutils literal notranslate">2 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] + <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
- @@ -291,14 +289,14 @@ 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] + <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>:
- @@ -315,44 +313,44 @@ try changing the indentation in the proof above to see what happens.
 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 + <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 + <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 + <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 + #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 + <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
- @@ -361,18 +359,18 @@ because at that point <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 + <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 * a + 2 * (a * b) + b * b := by + ring - example : (a + b) * (a - b) = a ^ 2 - b ^ 2 := 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 + 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
- @@ -387,14 +385,14 @@ structures.
 Possible matches are enumerated starting with 1, so in the following example, <code class="docutils literal notranslate">nth_rw 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] + <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="Link to this heading"></a></h2> + <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:
- @@ -406,17 +404,17 @@ 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] + <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 (neg_add_cancel : ∀ 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) + #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 (neg_add_cancel : ∀ 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,
- @@ -453,18 +451,18 @@ form a ring in which commutativity usually fails. If we declare <code class="doc
 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) + <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 : 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 * a + 2 * (a * b) + b * b := by ring - example : (a + b) * (a - b) = a ^ 2 - b ^ 2 := 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 + 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.
- @@ -493,17 +491,17 @@ 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] + <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_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, neg_add_cancel] + theorem add_right_neg (a : R) : a + -a = 0 := by rw [add_comm, neg_add_cancel] - #check MyRing.add_zero - #check add_zero + #check MyRing.add_zero + #check add_zero - end MyRing + end MyRing </pre></div> </div> The net effect is that we can temporarily reprove a theorem in the library,
- @@ -518,21 +516,21 @@ This declares <code class="docutils literal notranslate">R</sp
 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, neg_add_cancel, zero_add] + <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, neg_add_cancel, 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 + <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 + <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 + 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.
- @@ -560,10 +558,10 @@ 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] + <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!
- @@ -593,26 +591,26 @@ than <code class="docutils literal notranslate">apply</
 human readers and easier to maintain when the library evolves. 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 + <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 + <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 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_zero : (-0 : R) = 0 := by + apply neg_eq_of_add_eq_zero + rw [add_zero] - theorem neg_neg (a : R) : - -a = a := by - sorry + 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
- @@ -621,16 +619,16 @@ it is impossible for Lean to infer which <code class="docutils literal notransla
 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 + <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 + <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 + 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 “reflexivity”.
- @@ -645,8 +643,8 @@ 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 := by - sorry + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem self_sub (a : R) : a - a = 0 := by + sorry </pre></div> </div> Show that you can prove this using <code class="docutils literal notranslate">rw</code>,
- @@ -657,11 +655,11 @@ using either <code class="docutils literal notranslate">apply<
 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 + <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 := by - sorry + theorem two_mul (a : R) : 2 * a = a + a := by + sorry </pre></div> </div> We close this section by noting that some of the facts about
- @@ -669,11 +667,11 @@ 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] + <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 (neg_add_cancel : ∀ a : A, -a + a = 0) + #check (add_assoc : ∀ a b c : A, a + b + c = a + (b + c)) + #check (zero_add : ∀ a : A, 0 + a = a) + #check (neg_add_cancel : ∀ a : A, -a + a = 0) </pre></div> </div> It is conventional to use additive notation when
- @@ -682,25 +680,25 @@ 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] + <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 (inv_mul_cancel : ∀ a : G, a⁻¹ * a = 1) + #check (mul_assoc : ∀ a b c : G, a * b * c = a * (b * c)) + #check (one_mul : ∀ a : G, 1 * a = a) + #check (inv_mul_cancel : ∀ 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_inv_cancel (a : G) : a * a⁻¹ = 1 := by - sorry + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem mul_inv_cancel (a : G) : a * a⁻¹ = 1 := by + sorry - theorem mul_one (a : G) : a * 1 = a := 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 + 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
- @@ -714,7 +712,7 @@ 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="Link to this heading"></a></h2> + <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,
- @@ -723,8 +721,8 @@ 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) + <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>,
- @@ -737,13 +735,13 @@ 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) + <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) + #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,
- @@ -753,23 +751,23 @@ 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 + <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 := 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 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 := by + apply le_refl - example (x : ℝ) : x ≤ x := - le_refl x + example (x : ℝ) : x ≤ x := + le_refl x </pre></div> </div> In the first example, applying <code class="docutils literal notranslate">le_trans</code>
- @@ -784,26 +782,26 @@ 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) + <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 + <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 + <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 + <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,
- @@ -817,28 +815,28 @@ applying a fact or theorem <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'] + <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 → a ≤ b → log a ≤ log 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 + <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 → a ≤ b → log a ≤ log 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>
- @@ -848,9 +846,9 @@ phrase “if and only if.”
 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 + <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>
- @@ -862,11 +860,11 @@ Here, <code class="docutils literal notranslate">mp</co
 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 + <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>,
- @@ -876,14 +874,14 @@ 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 + <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 : (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 - apply log_le_log h₀ - sorry + example (h : a ≤ b) : log (1 + exp a) ≤ log (1 + exp b) := by + have h₀ : 0 < 1 + exp a := by sorry + apply log_le_log h₀ + sorry </pre></div> </div> From these examples, it should be clear that being able to
- @@ -913,9 +911,9 @@ and you can find similar theorems nearby.</li>
 <li>You can use the <code class="docutils literal notranslate">apply?</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 - -- apply? - exact sq_nonneg a + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : 0 ≤ a ^ 2 := by + -- apply? + exact sq_nonneg a </pre></div> </div> To try out <code class="docutils literal notranslate">apply?</code> in this example,
- @@ -923,23 +921,23 @@ delete the <code class="docutils literal notranslate">exact</s
 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 + <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">apply?</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 + <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 + 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>,
- @@ -959,34 +957,34 @@ we can simply write the proof term <code class="docutils literal notranslate"><s
 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 + <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>. You will also need the <code class="docutils literal notranslate">constructor</code> tactic to split a conjunction to two goals; see <a class="reference internal" href="C03_Logic.html#conjunction-and-biimplication">Section 3.4</a>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : |a * b| ≤ (a ^ 2 + b ^ 2) / 2 := by - sorry + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : |a * b| ≤ (a ^ 2 + b ^ 2) / 2 := by + sorry - #check abs_le'.mpr + #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-examples-using-apply-and-rw"> - <h2>2.4. More examples using apply and rw<a class="headerlink" href="#more-examples-using-apply-and-rw" title="Link to this heading"></a></h2> + <h2>2.4. More examples using apply and rw<a class="headerlink" href="#more-examples-using-apply-and-rw" 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) + <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
- @@ -1011,16 +1009,16 @@ With time, these conventions will become second nature.
 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 + <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
- @@ -1040,15 +1038,15 @@ but using them makes the proof easier to read and maintain.
 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 + <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
- @@ -1063,20 +1061,20 @@ uses <code class="docutils literal notranslate">h 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 + <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> We encourage you to prove the following as exercises. You can use either of the tricks just described to shorten the first. - <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 + <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.
- @@ -1099,10 +1097,10 @@ and in the second case, we have <code class="docutils literal notranslate">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 + <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
- @@ -1113,12 +1111,12 @@ As a hint, you can use the theorem <code class="docutils literal notranslate"><s
 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, |a + b| ≤ |a| + |b|) + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (abs_add : ∀ a b : ℝ, |a + b| ≤ |a| + |b|) </pre></div> </div> Use it to prove the following variant, using also <code class="docutils literal notranslate">add_sub_cancel_right</code>: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : |a| - |b| ≤ |a - b| := - sorry + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : |a| - |b| ≤ |a - b| := + sorry end </pre></div> </div>
- @@ -1133,15 +1131,15 @@ 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 + <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 ∣ y * x * z := by + apply dvd_mul_of_dvd_left + apply dvd_mul_left - example : x ∣ x ^ 2 := by - 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
- @@ -1150,8 +1148,8 @@ forces Lean to expand the definition of <code class="docutils literal notranslat
 <code class="docutils literal notranslate">x^1 * x</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 + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : x ∣ w) : x ∣ y * (x * z) + x ^ 2 + w ^ 2 := by + sorry end </pre></div> </div>
- @@ -1160,18 +1158,18 @@ you need to prove the following:
 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) + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (m n : ℕ) - #check (Nat.gcd_zero_right n : Nat.gcd n 0 = n) - #check (Nat.gcd_zero_left n : Nat.gcd 0 n = n) - #check (Nat.lcm_zero_right n : Nat.lcm n 0 = 0) - #check (Nat.lcm_zero_left n : Nat.lcm 0 n = 0) + #check (Nat.gcd_zero_right n : Nat.gcd n 0 = n) + #check (Nat.gcd_zero_left n : Nat.gcd 0 n = n) + #check (Nat.lcm_zero_right n : Nat.lcm n 0 = 0) + #check (Nat.lcm_zero_left n : Nat.lcm 0 n = 0) </pre></div> </div> 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 : Nat.gcd m n = Nat.gcd n m := by - sorry + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : Nat.gcd m n = Nat.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
- @@ -1182,7 +1180,7 @@ You can use <code class="docutils literal notranslate">_root_.
 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="Link to this heading"></a></h2> + <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,
- @@ -1193,13 +1191,13 @@ For example, a partial order consists of a set with a
 binary relation that is reflexive, transitive, and antisymmetric. 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) + <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) - #check (le_antisymm : x ≤ y → y ≤ x → x = y) + #check x ≤ y + #check (le_refl x : x ≤ x) + #check (le_trans : x ≤ y → y ≤ z → x ≤ z) + #check (le_antisymm : x ≤ y → y ≤ x → x = y) </pre></div> </div> Here we are adopting the Mathlib convention of using
- @@ -1218,14 +1216,14 @@ which acts somewhat like <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) + <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 + 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,”
- @@ -1237,17 +1235,17 @@ 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) + <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) + #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
- @@ -1302,28 +1300,28 @@ which takes <code class="docutils literal notranslate">yexact le_trans inf_le_left inf_le_right</code>, but this requires a lot more planning. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : x ⊓ y = y ⊓ x := by - sorry + <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 ⊓ z = x ⊓ (y ⊓ z) := by + sorry - example : x ⊔ y = y ⊔ x := by - sorry + example : x ⊔ y = y ⊔ x := by + sorry - example : x ⊔ y ⊔ z = x ⊔ (y ⊔ z) := 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 + <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 + 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>.
- @@ -1331,13 +1329,13 @@ using only those axioms:
 <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) + <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 ⊔ z) = x ⊓ y ⊔ x ⊓ z) - #check (inf_sup_right x y z : (x ⊔ y) ⊓ z = x ⊓ z ⊔ y ⊓ z) - #check (sup_inf_left x y z : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) - #check (sup_inf_right x y z : x ⊓ y ⊔ z = (x ⊔ z) ⊓ (y ⊔ z)) + #check (inf_sup_left x y z : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) + #check (inf_sup_right x y z : (x ⊔ y) ⊓ z = x ⊓ z ⊔ y ⊓ z) + #check (sup_inf_left x y z : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) + #check (sup_inf_right x y z : x ⊓ y ⊔ z = (x ⊔ z) ⊓ (y ⊔ z)) </pre></div> </div> The left and right versions are easily shown to be
- @@ -1348,14 +1346,14 @@ 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) + <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 - 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.
- @@ -1363,16 +1361,16 @@ For example, a strict ordered ring consists of a commutative ring toget
 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) + <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) + #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) + <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
- @@ -1381,14 +1379,14 @@ 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 (h : a ≤ b) : 0 ≤ b - a := by - sorry + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : a ≤ b) : 0 ≤ b - a := by + sorry - example (h: 0 ≤ b - a) : a ≤ b := by - sorry + example (h: 0 ≤ b - a) : a ≤ b := by + sorry - example (h : a ≤ b) (h' : 0 ≤ c) : a * c ≤ b * c := by - sorry + example (h : a ≤ b) (h' : 0 ≤ c) : a * c ≤ b * c := by + sorry </pre></div> </div> Finally, here is one last example.
- @@ -1396,19 +1394,19 @@ 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) + <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) + #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 + <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>.
html/C03_Logic.html
- @@ -1,25 +1,23 @@
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- @@ -98,7 +96,7 @@
 <div itemprop="articleBody"> <section id="logic"> - <h1>3. Logic<a class="headerlink" href="#logic" title="Link to this heading"></a></h1> + <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 $y$.”
- @@ -109,15 +107,15 @@ using logical terms like “and,” “or,” “not,”
 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="Link to this heading"></a></h2> + <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 → |x| = x + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check ∀ x : ℝ, 0 ≤ x → |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 → |x| < ε → |y| < ε → |x * y| < ε + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |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>,
- @@ -138,17 +136,17 @@ In particular, if you have proved a theorem of that form,
 you can apply it to objects and hypotheses in the same way. We will use as an example the following statement that we will help you to prove a bit later: - <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem my_lemma : ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |x * y| < ε := - sorry + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem my_lemma : ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |x * y| < ε := + sorry section - variable (a b δ : ℝ) - variable (h₀ : 0 < δ) (h₁ : δ ≤ 1) - variable (ha : |a| < δ) (hb : |b| < δ) + variable (a b δ : ℝ) + variable (h₀ : 0 < δ) (h₁ : δ ≤ 1) + variable (ha : |a| < δ) (hb : |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 + #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>
- @@ -158,15 +156,15 @@ 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 → |x| < ε → |y| < ε → |x * y| < ε := - sorry + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem my_lemma2 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |x * y| < ε := + sorry section - variable (a b δ : ℝ) - variable (h₀ : 0 < δ) (h₁ : δ ≤ 1) - variable (ha : |a| < δ) (hb : |b| < δ) + variable (a b δ : ℝ) + variable (h₀ : 0 < δ) (h₁ : δ ≤ 1) + variable (ha : |a| < δ) (hb : |b| < δ) - #check my_lemma2 h₀ h₁ ha hb + #check my_lemma2 h₀ h₁ ha hb end </pre></div>
- @@ -178,10 +176,10 @@ 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 → |x| < ε → |y| < ε → |x * y| < ε := by - intro x y ε epos ele1 xlt ylt - sorry + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem my_lemma3 : + ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |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;
- @@ -199,14 +197,14 @@ 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 → |x| < ε → |y| < ε → |x * y| < ε := by - intro x y ε epos ele1 xlt ylt - calc - |x * y| = |x| * |y| := sorry - _ ≤ |x| * ε := sorry - _ < 1 * ε := sorry - _ = ε := sorry + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem my_lemma4 : + ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |x * y| < ε := by + intro x y ε epos ele1 xlt ylt + calc + |x * y| = |x| * |y| := sorry + _ ≤ |x| * ε := sorry + _ < 1 * ε := sorry + _ = ε := sorry </pre></div> </div> Finish the proof using the theorems
- @@ -227,22 +225,22 @@ The first says that <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 + <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 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 the function that maps <code class="docutils literal notranslate">x</code> to <code class="docutils literal notranslate">f x + g x</code>. Going from the expression <code class="docutils literal notranslate">f x + g x</code> to this function is called a lambda abstraction in type theory. - <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 + <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>
- @@ -265,15 +263,15 @@ and gives you more control over how the goal is transformed.
 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 + <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 (nnf : FnLb f 0) (nng : FnLb g 0) : FnLb (fun x ↦ f x * g x) 0 := + sorry - example (hfa : FnUb f a) (hgb : FnUb g b) (nng : FnLb g 0) (nna : 0 ≤ a) : - FnUb (fun x ↦ f x * g x) (a * b) := - sorry + example (hfa : FnUb f a) (hgb : 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
- @@ -289,15 +287,15 @@ 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] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type*} {R : Type*} [OrderedCancelAddCommMonoid R] - #check add_le_add + #check add_le_add - def FnUb' (f : α → R) (a : R) : Prop := - ∀ x, f x ≤ a + 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) + 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
- @@ -310,8 +308,8 @@ 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 + <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
- @@ -327,11 +325,11 @@ 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 + <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
- @@ -347,8 +345,8 @@ So the <code class="docutils literal notranslate">intro
 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) + <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
- @@ -362,11 +360,11 @@ 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 + <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 + example (mf : Monotone f) (mg : Monotone g) : Monotone fun x ↦ f (g x) := + sorry </pre></div> </div> Here are some more examples.
- @@ -377,27 +375,27 @@ and odd if $f(-x) = -f(x)$
 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) + <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) + 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 (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 (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) : FnOdd fun x ↦ f x * g x := by + sorry - example (ef : FnEven f) (og : FnOdd g) : FnEven fun x ↦ f (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>
- @@ -437,16 +435,16 @@ we can write <code class="docutils literal notranslate">h</spa
 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) + <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 + example : s ⊆ s := by + intro x xs + exact xs - theorem Subset.refl : s ⊆ s := fun x xs ↦ xs + theorem Subset.refl : s ⊆ s := fun x xs ↦ xs - theorem Subset.trans : r ⊆ s → s ⊆ t → r ⊆ t := by - sorry + 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,
- @@ -457,14 +455,14 @@ 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) + <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 + def SetUb (s : Set α) (a : α) := + ∀ x, x ∈ s → x ≤ a - example (h : SetUb s a) (h' : a ≤ b) : SetUb s b := - sorry + example (h : SetUb s a) (h' : a ≤ b) : SetUb s b := + sorry </pre></div> </div> We close this section with one last important example.
- @@ -479,27 +477,27 @@ We then ask you to show that multiplication by a nonzero
 constant is also injective, using the lemma name in the example as a source of inspiration. Recall you should use Ctrl-space completion after guessing the beginning of a lemma name. - <div class="highlight-lean notranslate"><div class="highlight"><pre>open Function + <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) : 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 + 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} + <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 + 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="Link to this heading"></a></h2> + <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
- @@ -515,29 +513,29 @@ and the <code class="docutils literal notranslate">norm_num</s
 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 + <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> You can give the <code class="docutils literal notranslate">use</code> tactic proofs as well as data: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : ∃ x : ℝ, 2 < x ∧ x < 3 := by - have h1 : 2 < (5 : ℝ) / 2 := by norm_num - have h2 : (5 : ℝ) / 2 < 3 := by norm_num - use 5 / 2, h1, h2 + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : ∃ x : ℝ, 2 < x ∧ x < 3 := by + have h1 : 2 < (5 : ℝ) / 2 := by norm_num + have h2 : (5 : ℝ) / 2 < 3 := by norm_num + use 5 / 2, h1, h2 </pre></div> </div> In fact, the <code class="docutils literal notranslate">use</code> tactic automatically tries to use available assumptions as well. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : ∃ x : ℝ, 2 < x ∧ x < 3 := by - have h : 2 < (5 : ℝ) / 2 ∧ (5 : ℝ) / 2 < 3 := by norm_num - use 5 / 2 + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : ∃ x : ℝ, 2 < x ∧ x < 3 := by + have h : 2 < (5 : ℝ) / 2 ∧ (5 : ℝ) / 2 < 3 := by norm_num + use 5 / 2 </pre></div> </div> Alternatively, we can use Lean’s anonymous constructor notation to construct a proof of an existential quantifier. - <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⟩ + <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> Notice that there is no <code class="docutils literal notranslate">by</code>; here we are giving an explicit proof term.
- @@ -547,8 +545,8 @@ 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⟩ + <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.
- @@ -562,29 +560,29 @@ which say that <code class="docutils literal notranslate">a</s
 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 + <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 FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, a ≤ f x - def FnHasUb (f : ℝ → ℝ) := - ∃ a, FnUb f a + def FnHasUb (f : ℝ → ℝ) := + ∃ a, FnUb f a - def FnHasLb (f : ℝ → ℝ) := - ∃ a, FnLb 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} + <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 - rcases ubf with ⟨a, ubfa⟩ - rcases ubg with ⟨b, ubgb⟩ - use a + b - apply fnUb_add ubfa ubgb + 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⟩ + use a + b + apply fnUb_add ubfa ubgb </pre></div> </div> The <code class="docutils literal notranslate">rcases</code> tactic unpacks the information
- @@ -612,35 +610,35 @@ 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 + <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 + example {c : ℝ} (ubf : FnHasUb f) (h : c ≥ 0) : FnHasUb fun x ↦ c * f x := by + sorry </pre></div> </div> The “r” in <code class="docutils literal notranslate">rcases</code> stands for “recursive,” because it allows us to use arbitrarily complex patterns to unpack nested data. The <code class="docutils literal notranslate">rintro</code> tactic is a combination of <code class="docutils literal notranslate">intro</code> and <code class="docutils literal notranslate">rcases</code>: - <div class="highlight-lean notranslate"><div class="highlight"><pre>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⟩ + <div class="highlight-lean notranslate"><div class="highlight"><pre>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 fun 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⟩ + <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> The task of unpacking information in a hypothesis is so important that Lean and Mathlib provide a number of ways to do it. For example, the <code class="docutils literal notranslate">obtain</code> tactic provides suggestive syntax: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x ↦ f x + g x := by - obtain ⟨a, ubfa⟩ := ubf - obtain ⟨b, ubgb⟩ := ubg - exact ⟨a + b, fnUb_add ubfa ubgb⟩ + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x ↦ f x + g x := by + obtain ⟨a, ubfa⟩ := ubf + obtain ⟨b, ubgb⟩ := ubg + exact ⟨a + b, fnUb_add ubfa ubgb⟩ </pre></div> </div> Think of the first <code class="docutils literal notranslate">obtain</code> instruction as matching the “contents” of <code class="docutils literal notranslate">ubf</code>
- @@ -650,29 +648,29 @@ there is a small difference in that <code class="docutils literal notranslate"><
 when it is done, whereas it is still present after <code class="docutils literal notranslate">obtain</code>. Lean also supports syntax that is similar to that used in other functional programming languages: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x ↦ f x + g x := by - cases ubf - case intro a ubfa => - cases ubg - case intro b ubgb => - exact ⟨a + b, fnUb_add ubfa ubgb⟩ - - example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x ↦ f x + g x := by - cases ubf - next a ubfa => - cases ubg - next b ubgb => - exact ⟨a + b, fnUb_add ubfa ubgb⟩ - - example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x ↦ f x + g x := by - match ubf, ubg with - | ⟨a, ubfa⟩, ⟨b, ubgb⟩ => - exact ⟨a + b, fnUb_add ubfa ubgb⟩ - - example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x ↦ f x + g x := - match ubf, ubg with - | ⟨a, ubfa⟩, ⟨b, ubgb⟩ => - ⟨a + b, fnUb_add ubfa ubgb⟩ + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x ↦ f x + g x := by + cases ubf + case intro a ubfa => + cases ubg + case intro b ubgb => + exact ⟨a + b, fnUb_add ubfa ubgb⟩ + + example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x ↦ f x + g x := by + cases ubf + next a ubfa => + cases ubg + next b ubgb => + exact ⟨a + b, fnUb_add ubfa ubgb⟩ + + example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x ↦ f x + g x := by + match ubf, ubg with + | ⟨a, ubfa⟩, ⟨b, ubgb⟩ => + exact ⟨a + b, fnUb_add ubfa ubgb⟩ + + example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x ↦ f x + g x := + match ubf, ubg with + | ⟨a, ubfa⟩, ⟨b, ubgb⟩ => + ⟨a + b, fnUb_add ubfa ubgb⟩ </pre></div> </div> In the first example, if you put your cursor after <code class="docutils literal notranslate">cases ubf</code>,
- @@ -702,18 +700,18 @@ 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] + <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 + 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, 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,
- @@ -743,23 +741,23 @@ 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 + <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 - rcases divab with ⟨d, beq⟩ - rcases divbc with ⟨e, ceq⟩ - rw [ceq, beq] - use d * e; ring + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (divab : a ∣ b) (divbc : b ∣ c) : a ∣ c := by + rcases divab with ⟨d, beq⟩ + rcases divbc with ⟨e, ceq⟩ + rw [ceq, beq] + use d * e; ring </pre></div> </div> And once again, this provides a nice setting for using
- @@ -767,8 +765,8 @@ For example, divisibility is implicitly an “exists” statement.
 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 + <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 $f : \alpha \to \beta$
- @@ -779,48 +777,48 @@ 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 + <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 using the theorem <code class="docutils literal notranslate">mul_div_cancel₀</code>.: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {c : ℝ} (h : c ≠ 0) : Surjective fun x ↦ c * x := by - sorry + <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, <code class="docutils literal notranslate">field_simp</code>, 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 + <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> The next example uses a surjectivity hypothesis by applying it to a suitable value. Note that you can use <code class="docutils literal notranslate">rcases</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 - rcases h 2 with ⟨x, hx⟩ - use x - rw [hx] - norm_num + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {f : ℝ → ℝ} (h : Surjective f) : ∃ x, f x ^ 2 = 4 := by + rcases 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} + <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 + 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="Link to this heading"></a></h2> + <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
- @@ -841,10 +839,10 @@ which says that we have <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 + <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.
- @@ -861,41 +859,41 @@ by applying <code class="docutils literal notranslate">lt_irre
 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 - rcases fnub with ⟨a, fnuba⟩ - rcases h a with ⟨x, hx⟩ - have : f x ≤ a := fnuba x - linarith + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : ∀ a, ∃ x, f x > a) : ¬FnHasUb f := by + intro fnub + rcases fnub with ⟨a, fnuba⟩ + rcases h a with ⟨x, hx⟩ + have : f x ≤ a := fnuba x + linarith </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 are in the context. 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 + <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 + 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) + <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 + <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 + example (h : a ≤ b) (h' : f b < f a) : ¬Monotone f := by + sorry </pre></div> </div> We can show that the first example in the last snippet
- @@ -903,12 +901,12 @@ cannot be proved if we replace <code class="docutils literal notranslate"> - <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 + <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,
- @@ -920,8 +918,8 @@ Lean will unfold the definition of <code class="docutils literal notranslate"><s
 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 + <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
- @@ -934,19 +932,19 @@ is equivalent to saying that something fails to have property <code class="docut
 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) + <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 - 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
- @@ -958,13 +956,13 @@ 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''⟩ + <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.
- @@ -977,18 +975,18 @@ 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 + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : ¬¬Q) : Q := by + sorry - 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 + <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
- @@ -999,14 +997,14 @@ 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 + <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 - dsimp only [FnHasUb, FnUb] at h - push_neg at h - exact h + example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by + dsimp only [FnHasUb, FnUb] at h + push_neg at h + exact h </pre></div> </div> In the second example, we use dsimp to
- @@ -1021,8 +1019,8 @@ 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 + <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>,
- @@ -1034,14 +1032,14 @@ hypothesis <code class="docutils literal notranslate">h
 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 + <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 + 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
- @@ -1063,16 +1061,16 @@ establishes the goal so we can move on to the next one.
 <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 + <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 := + absurd h (lt_irrefl 0) - example (h : 0 < 0) : a > 37 := by - have h' : ¬0 < 0 := lt_irrefl 0 - contradiction + 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
- @@ -1085,18 +1083,18 @@ such as a pair of the form <code class="docutils literal notranslate">linarith</code> also works. </section> <section id="conjunction-and-iff"> - <h2>3.4. Conjunction and Iff<a class="headerlink" href="#conjunction-and-iff" title="Link to this heading"></a></h2> + <h2>3.4. Conjunction and Iff<a class="headerlink" href="#conjunction-and-iff" 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] + <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
- @@ -1109,14 +1107,14 @@ 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])⟩ + <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⟩ + 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
- @@ -1125,46 +1123,46 @@ You can use the <code class="docutils literal notranslate">rca
 as well as <code class="docutils literal notranslate">rintro</code> or a pattern-matching <code class="docutils literal notranslate">fun</code>, 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 - rcases h with ⟨h₀, h₁⟩ - contrapose! h₁ - exact le_antisymm h₀ h₁ + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by + rcases 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 := 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} : x ≤ y ∧ x ≠ y → ¬y ≤ x := + fun ⟨h₀, h₁⟩ h' ↦ h₁ (le_antisymm h₀ h') </pre></div> </div> In analogy to the <code class="docutils literal notranslate">obtain</code> tactic, there is also a pattern-matching <code class="docutils literal notranslate">have</code>: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by - have ⟨h₀, h₁⟩ := h - contrapose! h₁ - exact le_antisymm h₀ h₁ + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by + have ⟨h₀, h₁⟩ := h + contrapose! h₁ + exact le_antisymm h₀ h₁ </pre></div> </div> In contrast to <code class="docutils literal notranslate">rcases</code>, here the <code class="docutils literal notranslate">have</code> tactic leaves <code class="docutils literal notranslate">h</code> in the context. And even though we won’t use them, once again we have the computer scientists’ pattern-matching syntax: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by - cases h - case intro h₀ h₁ => - contrapose! h₁ - exact le_antisymm h₀ h₁ + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by + cases h + case intro h₀ h₁ => + contrapose! h₁ + exact le_antisymm h₀ h₁ - example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by - cases h - next h₀ h₁ => - contrapose! h₁ - exact le_antisymm h₀ h₁ + example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by + cases h + next h₀ h₁ => + contrapose! h₁ + exact le_antisymm h₀ h₁ - example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by - match h with - | ⟨h₀, h₁⟩ => - contrapose! h₁ - exact le_antisymm h₀ h₁ + example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by + match h with + | ⟨h₀, h₁⟩ => + contrapose! h₁ + exact le_antisymm h₀ h₁ </pre></div> </div> In contrast to using an existential quantifier,
- @@ -1172,47 +1170,47 @@ 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' + <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') + 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 + <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⟩ + <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 := 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 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 + <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 + example : ∃ m n : ℕ, 4 < m ∧ m < n ∧ n < 10 ∧ Nat.Prime m ∧ Nat.Prime n := by + use 5 + use 7 + norm_num - 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} : 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
- @@ -1226,16 +1224,16 @@ You can also use <code class="docutils literal notranslate">ca
 To prove an if-and-only-if statement, you can use <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 + <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)⟩ + 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
- @@ -1243,8 +1241,8 @@ 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 + <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
- @@ -1252,12 +1250,12 @@ two real numbers <code class="docutils literal notranslate">x<
 <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' + <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 + 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.
- @@ -1273,14 +1271,14 @@ replace an expression of the form <code class="docutils literal notranslate"><sp
 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) : |x + 3| < 5 → -8 < x ∧ x < 2 := by - rw [abs_lt] - intro h - constructor <;> linarith + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x : ℝ) : |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 + 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
- @@ -1288,13 +1286,13 @@ 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 + <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 + example : ¬Monotone fun x : ℝ ↦ -x := by + sorry </pre></div> </div> The remaining exercises in this section are designed
- @@ -1309,12 +1307,12 @@ 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) + <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 + example : a < b ↔ a ≤ b ∧ a ≠ b := by + rw [lt_iff_le_not_le] + sorry </pre></div> </div> Beyond logical operations, you do not need
- @@ -1330,34 +1328,34 @@ 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) + <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 < 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 + 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="Link to this heading"></a></h2> + <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} + <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) : 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 > 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
- @@ -1368,11 +1366,11 @@ When we write proof terms we can use
 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 + <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 + 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
- @@ -1393,12 +1391,12 @@ the <code class="docutils literal notranslate">rcases</
 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 < |y| → x < y ∨ x < -y := by - rcases 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 + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : x < |y| → x < y ∨ x < -y := by + rcases 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> Notice that the pattern changes from <code class="docutils literal notranslate">⟨h₀, h₁⟩</code> in the case of
- @@ -1419,14 +1417,14 @@ allowing us to split on those two cases.
 syntax for disjunction. Now the <code class="docutils literal notranslate">cases</code> tactic is more attractive, because it allows us to name each <code class="docutils literal notranslate">case</code>, and name the hypothesis that is introduced closer to where it is used. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : x < |y| → x < y ∨ x < -y := by - cases le_or_gt 0 y - case inl h => - rw [abs_of_nonneg h] - intro h; left; exact h - case inr h => - rw [abs_of_neg h] - intro h; right; exact h + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : x < |y| → x < y ∨ x < -y := by + cases le_or_gt 0 y + case inl h => + rw [abs_of_nonneg h] + intro h; left; exact h + case inr h => + rw [abs_of_neg h] + intro h; right; exact h </pre></div> </div> The names <code class="docutils literal notranslate">inl</code> and <code class="docutils literal notranslate">inr</code> are short for “intro left” and “intro right,”
- @@ -1434,23 +1432,23 @@ respectively. Using <code class="docutils literal notranslate">next</code>, or <code class="docutils literal notranslate">match</code>, or even a pattern-matching <code class="docutils literal notranslate">have</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : x < |y| → x < y ∨ x < -y := by - cases le_or_gt 0 y - next h => - rw [abs_of_nonneg h] - intro h; left; exact h - next h => - rw [abs_of_neg h] - intro h; right; exact h - - example : x < |y| → x < y ∨ x < -y := by - match le_or_gt 0 y with - | Or.inl h => - rw [abs_of_nonneg h] - intro h; left; exact h - | Or.inr h => - rw [abs_of_neg h] - intro h; right; exact h + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : x < |y| → x < y ∨ x < -y := by + cases le_or_gt 0 y + next h => + rw [abs_of_nonneg h] + intro h; left; exact h + next h => + rw [abs_of_neg h] + intro h; right; exact h + + example : x < |y| → x < y ∨ x < -y := by + match le_or_gt 0 y with + | Or.inl h => + rw [abs_of_nonneg h] + intro h; left; exact h + | Or.inr h => + rw [abs_of_neg h] + intro h; right; exact h </pre></div> </div> In the case of <code class="docutils literal notranslate">match</code>, we need to use the full names
- @@ -1460,54 +1458,54 @@ cases of a disjunction.
 Try proving the triangle inequality using the 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 + <div class="highlight-lean notranslate"><div class="highlight"><pre>namespace MyAbs - theorem le_abs_self (x : ℝ) : x ≤ |x| := by - sorry + theorem le_abs_self (x : ℝ) : x ≤ |x| := by + sorry - theorem neg_le_abs_self (x : ℝ) : -x ≤ |x| := by - sorry + theorem neg_le_abs_self (x : ℝ) : -x ≤ |x| := by + sorry - theorem abs_add (x y : ℝ) : |x + y| ≤ |x| + |y| := by - sorry + theorem abs_add (x y : ℝ) : |x + y| ≤ |x| + |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 < |y| ↔ x < y ∨ x < -y := by - sorry + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem lt_abs : x < |y| ↔ x < y ∨ x < -y := by + sorry - theorem abs_lt : |x| < y ↔ -y < x ∧ x < y := by - sorry + theorem abs_lt : |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 nested 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 + <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 + <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 + <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>
- @@ -1515,11 +1513,11 @@ tells us that <code class="docutils literal notranslate">x</sp
 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 + <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 + 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
- @@ -1537,14 +1535,14 @@ 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) + <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 = 1) : x = 1 ∨ x = -1 := by + sorry - example (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := 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
- @@ -1556,19 +1554,19 @@ 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 + <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 + <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
- @@ -1580,13 +1578,13 @@ 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 + <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="Link to this heading"></a></h2> + <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>.
- @@ -1596,8 +1594,8 @@ remains within $\varepsilon$
 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, |s n - a| < ε + <div class="highlight-lean notranslate"><div class="highlight"><pre>def ConvergesTo (s : ℕ → ℝ) (a : ℝ) := + ∀ ε > 0, ∃ N, ∀ n ≥ N, |s n - a| < ε </pre></div> </div> The notation <code class="docutils literal notranslate">∀ ε > 0, ...</code> is a convenient abbreviation
- @@ -1617,9 +1615,9 @@ 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 + <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
- @@ -1629,9 +1627,9 @@ 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) : |a| = |a - b + b| := by - congr - ring + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a b : ℝ) : |a| = |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,
- @@ -1647,10 +1645,10 @@ Instead, the <code class="docutils literal notranslate">conver
 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 + <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
- @@ -1659,12 +1657,12 @@ 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 - rw [sub_self, abs_zero] - apply εpos + <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 + 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
- @@ -1688,16 +1686,16 @@ the sequence <code class="docutils literal notranslate">fun</s
 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 -- this line is not needed but cleans up the goal a bit. - have ε2pos : 0 < ε / 2 := by linarith - rcases cs (ε / 2) ε2pos with ⟨Ns, hs⟩ - rcases ct (ε / 2) ε2pos with ⟨Nt, ht⟩ - use max Ns Nt - sorry + <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 -- this line is not needed but cleans up the goal a bit. + have ε2pos : 0 < ε / 2 := by linarith + rcases cs (ε / 2) ε2pos with ⟨Ns, hs⟩ + rcases 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>,
- @@ -1719,27 +1717,27 @@ 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] - ring - rw [h] - ring - have acpos : 0 < |c| := abs_pos.mpr h - sorry + <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] + ring + rw [h] + ring + have acpos : 0 < |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 → |s n| < b := by - rcases cs 1 zero_lt_one with ⟨N, h⟩ - use N, |a| + 1 - sorry + <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 → |s n| < b := by + rcases cs 1 zero_lt_one with ⟨N, h⟩ + use N, |a| + 1 + sorry </pre></div> </div> In fact, the theorem could be strengthened to assert
- @@ -1754,31 +1752,31 @@ To do so, we use the previous theorem to find a <code class="docutils literal no
 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 - rcases ct _ pos₀ with ⟨N₁, h₁⟩ - sorry + <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 + rcases 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 + <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,
- @@ -1786,22 +1784,22 @@ 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 : |a - b| > 0 := by sorry - let ε := |a - b| / 2 - have εpos : ε > 0 := by - change |a - b| / 2 > 0 - linarith - rcases sa ε εpos with ⟨Na, hNa⟩ - rcases sb ε εpos with ⟨Nb, hNb⟩ - let N := max Na Nb - have absa : |s N - a| < ε := by sorry - have absb : |s N - b| < ε := by sorry - have : |a - b| < |a - b| := by sorry - exact lt_irrefl _ this + <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 : |a - b| > 0 := by sorry + let ε := |a - b| / 2 + have εpos : ε > 0 := by + change |a - b| / 2 > 0 + linarith + rcases sa ε εpos with ⟨Na, hNa⟩ + rcases sb ε εpos with ⟨Nb, hNb⟩ + let N := max Na Nb + have absa : |s N - a| < ε := by sorry + have absb : |s N - b| < ε := by sorry + have : |a - b| < |a - b| := by sorry + exact lt_irrefl _ this </pre></div> </div> We close the section with the observation that our proofs can be generalized.
- @@ -1810,10 +1808,10 @@ 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] + <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, |s n - a| < ε + def ConvergesTo' (s : α → ℝ) (a : ℝ) := + ∀ ε > 0, ∃ N, ∀ n ≥ N, |s n - a| < ε </pre></div> </div> In <a class="reference internal" href="C10_Topology.html#filters">Section 10.1</a>, we will see that Mathlib has mechanisms
html/C04_Sets_and_Functions.html
- @@ -1,25 +1,23 @@
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- @@ -95,7 +93,7 @@
 <div itemprop="articleBody"> <section id="sets-and-functions"> - <h1>4. Sets and Functions<a class="headerlink" href="#sets-and-functions" title="Link to this heading"></a></h1> + <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.
- @@ -128,7 +126,7 @@ from real numbers to real numbers.
 The distinction between types and sets 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="Link to this heading"></a></h2> + <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.
- @@ -157,69 +155,69 @@ Unlike <code class="docutils literal notranslate">rw</c
 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 - simp only [mem_setOf] - 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⟩ + <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 + simp only [mem_setOf] + 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⟩ + <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 example also illustrate the phenomenon: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := - fun x ⟨xs, xu⟩ ↦ ⟨h xs, xu⟩ + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := + fun x ⟨xs, xu⟩ ↦ ⟨h xs, xu⟩ </pre></div> </div> 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 - rcases xtu with xt | xu - · left - show x ∈ s ∩ t - exact ⟨xs, xt⟩ - · right - show x ∈ s ∩ u - exact ⟨xs, xu⟩ + <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 + rcases 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⟩ + <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 + <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>,
- @@ -234,28 +232,28 @@ It can be rewritten manually using <code class="docutils literal notranslate"><s
 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 - rcases 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 + <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 + rcases 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 + <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,
- @@ -264,57 +262,57 @@ 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⟩ + <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⟩⟩ + <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> 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] + <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⟩ + <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 + <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 + <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 ∪ s ∩ t = s := by + sorry - example : s \ t ∪ t = s ∪ t := by - sorry + example : s \ t ∪ t = s ∪ t := by + sorry - example : s \ t ∪ t \ s = (s ∪ 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,
- @@ -330,17 +328,17 @@ In other words, sets are really properties, treated as objects.
 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 } + <div class="highlight-lean notranslate"><div class="highlight"><pre>def evens : Set ℕ := + { n | Even n } - def odds : Set ℕ := - { n | ¬Even n } + def odds : Set ℕ := + { n | ¬Even n } - example : evens ∪ odds = univ := by - rw [evens, odds] - ext n - simp [-Nat.not_even_iff_odd] - apply Classical.em + example : evens ∪ odds = univ := by + rw [evens, odds] + ext n + simp [-Nat.not_even_iff_odd] + apply Classical.em </pre></div> </div> You should step through this proof and make sure
- @@ -363,11 +361,11 @@ 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 + <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 + example (x : ℕ) : x ∈ (univ : Set ℕ) := + trivial </pre></div> </div> As an exercise, prove the following inclusion.
- @@ -376,8 +374,8 @@ 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.odd_iff</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : { n | Nat.Prime n } ∩ { n | n > 2 } ⊆ { n | ¬Even n } := by - sorry + <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
- @@ -386,21 +384,21 @@ 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 + <div class="highlight-lean notranslate"><div class="highlight"><pre>#print Prime - #print Nat.Prime + #print Nat.Prime - example (n : ℕ) : Prime n ↔ Nat.Prime n := - Nat.prime_iff.symm + 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 + 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] + <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>,
- @@ -421,28 +419,28 @@ 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) + <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 (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 + example (h : ∃ x ∈ s, ¬Even x ∧ Prime x) : ∃ x ∈ s, Prime x := by + rcases h with ⟨x, xs, _, prime_x⟩ + use x, xs </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) + variable (ssubt : s ⊆ t) - example (h₀ : ∀ x ∈ t, ¬Even x) (h₁ : ∀ x ∈ t, Prime x) : ∀ x ∈ s, ¬Even x ∧ Prime x := by - sorry + 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 + example (h : ∃ x ∈ s, ¬Even x ∧ Prime x) : ∃ x ∈ t, Prime x := by + sorry end </pre></div>
- @@ -458,35 +456,35 @@ 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 + <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
- @@ -497,8 +495,8 @@ the scope of the bound variable extends as far as it can.
 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 + <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,
- @@ -507,23 +505,23 @@ You can unpack their meaning with <code class="docutils literal notranslate"><sp
 and <code class="docutils literal notranslate">mem_iInter₂</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 + <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.
- @@ -531,8 +529,8 @@ If you start typing <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 + <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>,
- @@ -543,41 +541,41 @@ Similarly, their intersection, <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)) + <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_iUnion₂] + simp - example : ⋂₀ s = ⋂ t ∈ s, t := by - ext x - rw [mem_iInter₂] - rfl + 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="Link to this heading"></a></h2> + <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) + <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 + open Function + open Set - example : f ⁻¹' (u ∩ v) = f ⁻¹' u ∩ f ⁻¹' v := by - ext - rfl + 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>,
- @@ -589,25 +587,25 @@ So a hypothesis <code class="docutils literal notranslate">y<
 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 + <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 + <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
- @@ -616,8 +614,8 @@ 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 + <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
- @@ -636,47 +634,47 @@ 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 + <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 : f '' (f ⁻¹' u) ⊆ u := by + sorry - example (h : Surjective f) : u ⊆ f '' (f ⁻¹' 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 : s ⊆ t) : f '' s ⊆ f '' t := by + sorry - example (h : u ⊆ v) : f ⁻¹' u ⊆ f ⁻¹' v := 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 ⁻¹' (u ∪ v) = f ⁻¹' u ∪ f ⁻¹' v := by + sorry - example : f '' (s ∩ t) ⊆ f '' s ∩ f '' t := 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 (h : Injective f) : f '' s ∩ f '' t ⊆ f '' (s ∩ t) := by + sorry - example : 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 ⁻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 ∩ v = f '' (s ∩ f ⁻¹' v) := by + sorry - example : f '' (s ∩ f ⁻¹' u) ⊆ f '' s ∩ u := 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 - 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,
- @@ -687,29 +685,29 @@ 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) + <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 - sorry + example : (f '' ⋃ i, A i) = ⋃ i, f '' A i := by + sorry - example : (f '' ⋂ i, A i) ⊆ ⋂ i, f '' A i := by - sorry + example : (f '' ⋂ i, A i) ⊆ ⋂ i, f '' A i := by + sorry - example (i : I) (injf : Injective f) : (⋂ i, f '' A i) ⊆ f '' ⋂ i, A i := by - sorry + example (i : I) (injf : Injective f) : (⋂ i, f '' A i) ⊆ f '' ⋂ i, A i := by + sorry - example : (f ⁻¹' ⋃ i, B i) = ⋃ i, f ⁻¹' B i := by - sorry + example : (f ⁻¹' ⋃ i, B i) = ⋃ i, f ⁻¹' B i := by + sorry - example : (f ⁻¹' ⋂ i, B i) = ⋂ i, f ⁻¹' B i := by - sorry + example : (f ⁻¹' ⋂ i, B i) = ⋂ i, f ⁻¹' B i := by + sorry </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 _ + <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
- @@ -723,39 +721,39 @@ 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] + <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 + <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 : InjOn (fun x ↦ x ^ 2) { x : ℝ | x ≥ 0 } := by + sorry - example : sqrt '' { x | x ≥ 0 } = { y | y ≥ 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 + 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>,
- @@ -775,16 +773,16 @@ 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">choose</code> operator, illustrated below. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α β : Type*} [Inhabited α] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α β : Type*} [Inhabited α] - #check (default : α) + #check (default : α) - variable (P : α → Prop) (h : ∃ x, P x) + variable (P : α → Prop) (h : ∃ x, P x) - #check Classical.choose h + #check Classical.choose h - example : P (Classical.choose h) := - Classical.choose_spec 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.choose h</code>
- @@ -793,16 +791,16 @@ The theorem <code class="docutils literal notranslate">Classic
 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 + <div class="highlight-lean notranslate"><div class="highlight"><pre>noncomputable section - open Classical + open Classical - def inverse (f : α → β) : β → α := fun y : β ↦ - if h : ∃ x, f x = y then Classical.choose h else default + 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, dif_pos h] - exact Classical.choose_spec h + theorem inverse_spec {f : α → β} (y : β) (h : ∃ x, f x = y) : f (inverse f y) = y := by + rw [inverse, dif_pos h] + exact Classical.choose_spec h </pre></div> </div> The lines <code class="docutils literal notranslate">noncomputable section</code> and <code class="docutils literal notranslate">open Classical</code>
- @@ -835,15 +833,15 @@ 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) + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (f : α → β) - open Function + open Function - example : Injective f ↔ LeftInverse (inverse f) f := - sorry + example : Injective f ↔ LeftInverse (inverse f) f := + sorry - example : Surjective f ↔ RightInverse (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
- @@ -851,24 +849,24 @@ 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 + <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="Link to this heading"></a></h2> + <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.)
- @@ -943,9 +941,9 @@ 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] + <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.
- @@ -961,21 +959,21 @@ in <code class="docutils literal notranslate">β</c
 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) + <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) + <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 sbAux : ℕ → Set α + | 0 => univ \ g '' univ + | n + 1 => g '' (f '' sbAux n) - def sbSet := - ⋃ n, sbAux f g n + def sbSet := + ⋃ n, sbAux f g n </pre></div> </div> The definition <code class="docutils literal notranslate">sbAux</code> is an example of a recursive definition,
- @@ -987,8 +985,8 @@ S_{n+1} &= g(f(S_n)).\end{split}\]</div>
 The definition <code class="docutils literal notranslate">sbSet</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 + <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
- @@ -1007,16 +1005,16 @@ and fill in the remaining parts.
 You will need to use <code class="docutils literal notranslate">invFun_eq</code> at the end. Notice that rewriting with <code class="docutils literal notranslate">sbAux</code> here replaces <code class="docutils literal notranslate">sbAux 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 + <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.
- @@ -1040,32 +1038,32 @@ Applying $g$ to both sides y
 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) : 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 _root_.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 + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem sb_injective (hf : Injective f) : 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 _root_.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.
- @@ -1099,32 +1097,32 @@ The tactic <code class="docutils literal notranslate">rcases</
 and <code class="docutils literal notranslate">g y ∈ sbAux f g (n + 1)</code>. In both cases, calling the simplifier with <code class="docutils literal notranslate">simp [sbAux]</code> applies the corresponding defining equation of <code class="docutils literal notranslate">sbAux</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem sb_surjective (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⟩ - rcases 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 + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem sb_surjective (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⟩ + rcases 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, sb_surjective f g hg⟩ + <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, sb_surjective f g hg⟩ </pre></div> </div> </section>
html/C05_Elementary_Number_Theory.html
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- @@ -95,14 +93,14 @@
 <div itemprop="articleBody"> <section id="elementary-number-theory"> - <h1>5. Elementary Number Theory<a class="headerlink" href="#elementary-number-theory" title="Link to this heading"></a></h1> + <h1>5. Elementary Number Theory<a class="headerlink" href="#elementary-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="Link to this heading"></a></h2> + <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,
- @@ -126,18 +124,18 @@ 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 + <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 := + h - example (m n : Nat) (h : m.Coprime n) : m.gcd n = 1 := by - rw [Nat.Coprime] at h - exact 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.Coprime 12 7 := by norm_num - example : Nat.gcd 12 8 = 4 := 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
- @@ -155,24 +153,24 @@ 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 + <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 + 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 + #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 (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 + example : Nat.Prime 17 := by norm_num -- commonly used - example : Nat.Prime 2 := - Nat.prime_two + example : Nat.Prime 2 := + Nat.prime_two - example : Nat.Prime 3 := - Nat.prime_three + example : Nat.Prime 3 := + Nat.prime_three </pre></div> </div> In the natural numbers, a prime number has the property that it cannot
- @@ -190,16 +188,16 @@ 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">Algebra.Group.Even</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 + <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 + 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 {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
- @@ -216,32 +214,32 @@ You can also use the search engine on the
 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 := - -- apply? suggests the following: - (mul_right_inj' h').mp h + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a b c : Nat) (h : a * b = a * c) (h' : a ≠ 0) : b = c := + -- apply? 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 + <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.
- @@ -251,8 +249,8 @@ At the end of the proof, you’ll need to derive a contradiction from
 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 + <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.
- @@ -274,10 +272,10 @@ are prime, that any <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.primeFactorsList n</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>#check Nat.primeFactorsList - #check Nat.prime_of_mem_primeFactorsList - #check Nat.prod_primeFactorsList - #check Nat.primeFactorsList_unique + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check Nat.primeFactorsList + #check Nat.prime_of_mem_primeFactorsList + #check Nat.prod_primeFactorsList + #check Nat.primeFactorsList_unique </pre></div> </div> You can browse these theorems and others nearby, even though we have not
- @@ -288,20 +286,20 @@ 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 + <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 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 + 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,
- @@ -313,17 +311,17 @@ the three theorems above as a black box.
 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 + <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
- @@ -350,20 +348,20 @@ to finish it off.
 Note that this example does not assume that <code class="docutils literal notranslate">p</code> is prime, but the conclusion is trivial when <code class="docutils literal notranslate">p</code> is not prime since <code class="docutils literal notranslate">r.factorization p</code> is then zero by definition, and the proof works in all cases anyway. - <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} : - k ∣ r.factorization p := by - rcases 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 + 1) * n ^ k).factorization p = - k * n.factorization p + (r + 1).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 + <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} : + k ∣ r.factorization p := by + rcases 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 + 1) * n ^ k).factorization p = + k * n.factorization p + (r + 1).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.
- @@ -405,7 +403,7 @@ 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="Link to this heading"></a></h2> + <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.
- @@ -413,9 +411,9 @@ 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 where - | zero : Nat - | succ (n : Nat) : Nat + <div class="highlight-lean notranslate"><div class="highlight"><pre>inductive Nat where + | zero : Nat + | succ (n : Nat) : Nat </pre></div> </div> You can find this in the library by writing <code class="docutils literal notranslate">#check Nat</code> and
- @@ -429,11 +427,11 @@ 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 + <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 + 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
- @@ -442,9 +440,9 @@ 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 + <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.
- @@ -453,23 +451,23 @@ 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 + <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 + rw [fac] - example : fac 0 = 1 := by - simp [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 := + 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 + rw [fac] - example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := by - simp [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
- @@ -492,26 +490,26 @@ and a required to prove <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 + <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] - rcases Nat.of_le_succ ile with h | h - · apply dvd_mul_of_dvd_right (ih h) - rw [h] - apply dvd_mul_right + <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] + rcases 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
- @@ -521,10 +519,10 @@ 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 using <code class="docutils literal notranslate">pow_succ</code> or <code class="docutils literal notranslate">pow_succ'</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem pow_two_le_fac (n : ℕ) : 2 ^ (n - 1) ≤ fac n := by - rcases n with _ | n - · simp [fac] - sorry + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem pow_two_le_fac (n : ℕ) : 2 ^ (n - 1) ≤ fac n := by + rcases n with _ | n + · simp [fac] + sorry </pre></div> </div> Induction is often used to prove identities involving finite sums and
- @@ -543,50 +541,50 @@ 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) + <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 + #check Finset.sum s f + #check Finset.prod s f - open BigOperators - open Finset + open BigOperators + open Finset - example : s.sum f = ∑ x in s, f x := - rfl + example : s.sum f = ∑ x in s, f x := + rfl - example : s.prod 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).sum f = ∑ x in range n, f x := + rfl - example : (range n).prod 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 of 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 + <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) (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) : ∏ 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 (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 - · simp [fac, prod_range_zero] - simp [fac, ih, prod_range_succ, mul_comm] + <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 + · simp [fac, prod_range_zero] + simp [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
- @@ -600,8 +598,8 @@ 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] + <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
- @@ -615,18 +613,18 @@ 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] - ring + <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] + 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 + <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
- @@ -665,53 +663,53 @@ 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 where - | zero : MyNat - | succ : MyNat → MyNat + <div class="highlight-lean notranslate"><div class="highlight"><pre>inductive MyNat where + | zero : MyNat + | succ : MyNat → MyNat - namespace MyNat + namespace MyNat - def add : MyNat → MyNat → MyNat - | x, zero => x - | x, succ y => succ (add x y) + 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 + 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 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 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_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 + 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>5.3. Infinitely Many Primes<a class="headerlink" href="#infinitely-many-primes" title="Link to this heading"></a></h2> + <h2>5.3. 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
- @@ -731,12 +729,12 @@ 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 + <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>,
- @@ -744,10 +742,10 @@ 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 + <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
- @@ -757,12 +755,12 @@ 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 + <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
- @@ -787,44 +785,44 @@ 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. - <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 - 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 - · rcases ih m mltn mgt2 mp with ⟨p, pp, pdvd⟩ - use p, pp - apply pdvd.trans mdvdn + <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 + 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 + · 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 + <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
- @@ -853,31 +851,31 @@ so they need to be expanded manually using equivalences like
 and <code class="docutils literal notranslate">Finset.mem_sdiff</code>. The <code class="docutils literal notranslate">ext</code> tactic can still be used to 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 + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Finset section - variable {α : Type*} [DecidableEq α] (r s t : Finset α) + 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 + 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 ∪ 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 + simp [subset_iff] + intro x + tauto - example : r ∩ s ∪ r ∩ t = r ∩ (s ∪ t) := by - ext x - simp - tauto + example : r ∩ s ∪ r ∩ t = r ∩ (s ∪ t) := by + ext x + simp + tauto end </pre></div>
- @@ -886,27 +884,27 @@ that every element of one is an element of the other.
 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 + <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 + <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 + <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
- @@ -925,15 +923,15 @@ 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 + <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.
- @@ -943,8 +941,8 @@ we wrote <code class="docutils literal notranslate">{ <
 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 + <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
- @@ -957,25 +955,25 @@ 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 + <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:
- @@ -991,22 +989,22 @@ we can dispense with the assumption.
 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) + <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 + 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
- @@ -1036,7 +1034,7 @@ But in that case, $p$ divide
 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 + <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”
- @@ -1047,66 +1045,66 @@ a small number of cases.
 In the second named theorem, remember that the semicolon means that the subsequent tactic block is applied to all the goals created by the preceding tactic. - <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] + <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] - 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 m % 4 <;> simp [-Nat.mul_mod_mod] - have : n % 4 < 4 := Nat.mod_lt n (by norm_num) - interval_cases n % 4 <;> simp + 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 m % 4 <;> simp [-Nat.mul_mod_mod] + have : n % 4 < 4 := Nat.mod_lt n (by norm_num) + interval_cases n % 4 <;> simp - 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 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 + <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 - induction' n using Nat.strong_induction_on with n 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] - rcases this with h1 | h1 - . sorry - . sorry + <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 + induction' n using Nat.strong_induction_on with n 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] + rcases 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 + <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 + 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
- @@ -1114,31 +1112,31 @@ 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 - rcases 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⟩ - rcases 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 + <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 + rcases 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⟩ + rcases 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
html/C06_Structures.html
- @@ -1,25 +1,23 @@
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- @@ -95,7 +93,7 @@
 <div itemprop="articleBody"> <section id="structures"> - <h1>6. Structures<a class="headerlink" href="#structures" title="Link to this heading"></a></h1> + <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
- @@ -115,42 +113,42 @@ 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="Link to this heading"></a></h2> + <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 + <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 + <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 + 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 + <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 myPoint2 : Point := + ⟨2, -1, 4⟩ - def myPoint3 := - Point.mk 2 (-1) 4 + def myPoint3 := + Point.mk 2 (-1) 4 </pre></div> </div> In the first example, the fields of the structure are named
- @@ -159,12 +157,12 @@ The function <code class="docutils literal notranslate">Point.
 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 + <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 + #check Point'.build 2 (-1) 4 </pre></div> </div> The next two examples show how to define functions on structures.
- @@ -182,23 +180,23 @@ 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 + <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 := + ⟨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 + 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 + #check add myPoint1 myPoint2 + #check myPoint1.add myPoint2 - end Point + end Point - #check Point.add myPoint1 myPoint2 - #check myPoint1.add myPoint2 + #check Point.add myPoint1 myPoint2 + #check myPoint1.add myPoint2 </pre></div> </div> Below we will continue to put definitions in the relevant
- @@ -211,18 +209,18 @@ Below we use the <code class="docutils literal notranslate">pr
 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 + <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] + 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 + <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
- @@ -235,20 +233,20 @@ in the second.
 Although it is sometimes convenient to define functions this way, and structural eta-reduction makes this alternative definitionally equivalent, it can make things less convenient in later proofs. In particular, <cite>rw [addAlt]</cite> leaves us with a messier goal view containing a <cite>match</cite> statement. - <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⟩ + <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⟩ + 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 - rfl + theorem addAlt_x (a b : Point) : (a.addAlt b).x = a.x + b.x := by + rfl - theorem addAlt_comm (a b : Point) : addAlt a b = addAlt b a := by - rw [addAlt, addAlt] - -- the same proof still works, but the goal view here is harder to read - ext <;> dsimp - repeat' apply add_comm + theorem addAlt_comm (a b : Point) : addAlt a b = addAlt b a := by + rw [addAlt, addAlt] + -- the same proof still works, but the goal view here is harder to read + ext <;> dsimp + repeat' apply add_comm </pre></div> </div> Mathematical constructions often involve taking apart bundled information and
- @@ -258,15 +256,15 @@ 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 + <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 + 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 + 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
- @@ -290,44 +288,44 @@ 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 + <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] + <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] + <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>
- @@ -338,9 +336,9 @@ we can take the weighted average $\l
 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 + <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.
- @@ -349,29 +347,29 @@ For example, we can generalize the standard 2-simplex to the standard
 At this stage, you don’t have to know anything about the type <code class="docutils literal notranslate">Fin n</code> 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 + <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
- @@ -384,15 +382,15 @@ 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 + <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) + variable (f : ℝ → ℝ) (linf : IsLinear f) - #check linf.is_additive - #check linf.preserves_mul + #check linf.is_additive + #check linf.preserves_mul end </pre></div>
- @@ -401,11 +399,11 @@ the two components of linearity.
 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 + <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 + 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
- @@ -418,42 +416,42 @@ Any <code class="docutils literal notranslate">x 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 } + <div class="highlight-lean notranslate"><div class="highlight"><pre>def PReal := + { y : ℝ // 0 < y } section - variable (x : PReal) + variable (x : PReal) - #check x.val - #check x.property - #check x.1 - #check x.2 + #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 } + <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 } + 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 + <div class="highlight-lean notranslate"><div class="highlight"><pre>def StdSimplex := Σ n : ℕ, StandardSimplex n section - variable (s : StdSimplex) + variable (s : StdSimplex) - #check s.fst - #check s.snd + #check s.fst + #check s.snd - #check s.1 - #check s.2 + #check s.1 + #check s.2 end </pre></div>
- @@ -476,7 +474,7 @@ 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="Link to this heading"></a></h2> + <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">
- @@ -594,14 +592,14 @@ this is exactly what the <code class="docutils literal notranslate"> 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 - inv_mul_cancel : ∀ x : α, mul (inv x) x = one + <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 + inv_mul_cancel : ∀ 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>.
- @@ -630,9 +628,9 @@ morally the same as the definition of a group that Mathlib uses.
 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 + <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">Mathlib.Algebra.Category.GroupCat.Basic</code>,
- @@ -655,17 +653,17 @@ a function <code class="docutils literal notranslate">f.toFun<
 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) + <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
- @@ -676,20 +674,20 @@ that the property of being in bijective correspondence is an equivalence relatio
 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 + <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 (x : α) : (f.trans g) x = g (f x) := + rfl - example : (f.trans g : α → γ) = g ∘ f := - 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 + <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
- @@ -697,15 +695,15 @@ of equivalences. We orient things so that <code class="docutils literal notransl
 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 - inv_mul_cancel := Equiv.self_trans_symm + <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 + inv_mul_cancel := 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>
- @@ -739,27 +737,27 @@ to the <code class="docutils literal notranslate">Group₁
 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 + <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 + 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 := + ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ - def neg (a : Point) : Point := sorry + def neg (a : Point) : Point := sorry - def zero : Point := sorry + def zero : Point := sorry - def addGroupPoint : AddGroup₁ Point := sorry + def addGroupPoint : AddGroup₁ Point := sorry - end Point + end Point </pre></div> </div> We are making progress.
- @@ -773,21 +771,21 @@ 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) + <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 + #check f * g + #check mul_assoc f g g⁻¹ -- group power, defined for any group - #check g ^ n + #check g ^ n - example : f * g * g⁻¹ = f := by rw [mul_assoc, mul_inv_cancel, mul_one] + example : f * g * g⁻¹ = f := by rw [mul_assoc, mul_inv_cancel, mul_one] - example : f * g * g⁻¹ = f := - mul_inv_cancel_right f g + 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 + 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
- @@ -851,41 +849,41 @@ 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 - inv_mul_cancel : ∀ 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 - inv_mul_cancel := Equiv.self_trans_symm + <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 + inv_mul_cancel : ∀ 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 + inv_mul_cancel := Equiv.self_trans_symm </pre></div> </div> The following illustrates their use. - <div class="highlight-lean notranslate"><div class="highlight"><pre>#check Group₂.mul + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check Group₂.mul - def mySquare {α : Type*} [Group₂ α] (x : α) := - Group₂.mul x x + def mySquare {α : Type*} [Group₂ α] (x : α) := + Group₂.mul x x - #check mySquare + #check mySquare section - variable {β : Type*} (f g : Equiv.Perm β) + variable {β : Type*} (f g : Equiv.Perm β) - example : Group₂.mul f g = g.trans f := - rfl + example : Group₂.mul f g = g.trans f := + rfl - example : mySquare f = f.trans f := - rfl + example : mySquare f = f.trans f := + rfl end </pre></div>
- @@ -918,12 +916,12 @@ 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⟩ + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance : Inhabited Point where default := ⟨0, 0, 0⟩ - #check (default : Point) + #check (default : Point) - example : ([] : List Point).headI = default := - rfl + example : ([] : List Point).headI = default := + rfl </pre></div> </div> The class inference mechanism is also used for generic notation.
- @@ -933,15 +931,15 @@ a binary function on <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 + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance : Add Point where add := Point.add section - variable (x y : Point) + variable (x y : Point) - #check x + y + #check x + y - example : x + y = Point.add x y := - rfl + example : x + y = Point.add x y := + rfl end </pre></div>
- @@ -955,22 +953,22 @@ 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⟩ + <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 hasOneGroup₂ {α : Type*} [Group₂ α] : One α := + ⟨Group₂.one⟩ - instance hasInvGroup₂ {α : Type*} [Group₂ α] : Inv α := - ⟨Group₂.inv⟩ + instance hasInvGroup₂ {α : Type*} [Group₂ α] : Inv α := + ⟨Group₂.inv⟩ section - variable {α : Type*} (f g : Equiv.Perm α) + variable {α : Type*} (f g : Equiv.Perm α) - #check f * 1 * g⁻¹ + #check f * 1 * g⁻¹ - def foo : f * 1 * g⁻¹ = g.symm.trans ((Equiv.refl α).trans f) := - rfl + def foo : f * 1 * g⁻¹ = g.symm.trans ((Equiv.refl α).trans f) := + rfl end </pre></div>
- @@ -1009,9 +1007,9 @@ using the classes <code class="docutils literal notranslate">A
 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 + <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
- @@ -1025,7 +1023,7 @@ 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="Link to this heading"></a></h2> + <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
- @@ -1037,10 +1035,10 @@ 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 + <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,
- @@ -1052,20 +1050,20 @@ be a square root of $-1$. Th
 \[\begin{split}(a + bi) (c + di) & = ac + bci + adi + bd i^2 \\ & = (ac - bd) + (bc + ad)i.\end{split}\]</div> This explains the definition of <code class="docutils literal notranslate">Mul</code> below. - <div class="highlight-lean notranslate"><div class="highlight"><pre>instance : Zero GaussInt := - ⟨⟨0, 0⟩⟩ + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance : Zero GaussInt := + ⟨⟨0, 0⟩⟩ - instance : One GaussInt := - ⟨⟨1, 0⟩⟩ + instance : One GaussInt := + ⟨⟨1, 0⟩⟩ - instance : Add GaussInt := - ⟨fun x y ↦ ⟨x.re + y.re, x.im + y.im⟩⟩ + instance : Add GaussInt := + ⟨fun x y ↦ ⟨x.re + y.re, x.im + y.im⟩⟩ - instance : Neg GaussInt := - ⟨fun x ↦ ⟨-x.re, -x.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⟩⟩ + 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
- @@ -1077,64 +1075,64 @@ files associated with this chapter, these definitions are made in the
 <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 + <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 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 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 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 + 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 + <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 zero_im : (0 : GaussInt).im = 0 := + rfl - @[simp] - theorem one_re : (1 : GaussInt).re = 1 := - rfl + @[simp] + theorem one_re : (1 : GaussInt).re = 1 := + rfl - @[simp] - theorem one_im : (1 : GaussInt).im = 0 := - 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_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 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_re (x : GaussInt) : (-x).re = -x.re := + rfl - @[simp] - theorem neg_im (x : GaussInt) : (-x).im = -x.im := - 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_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 + @[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
- @@ -1158,63 +1156,63 @@ to reduce the identities to their real and imaginary components,
 simplifying, and, if necessary, carrying out the relevant ring calculation in the integers. Note that we could easily avoid repeating all this code, but this is not the topic of the current discussion. - <div class="highlight-lean notranslate"><div class="highlight"><pre>instance instCommRing : CommRing GaussInt where - zero := 0 - one := 1 - add := (· + ·) - neg x := -x - mul := (· * ·) - nsmul := nsmulRec - zsmul := zsmulRec - add_assoc := by - intros - ext <;> simp <;> ring - zero_add := by - intro - ext <;> simp - add_zero := by - intro - ext <;> simp - neg_add_cancel := 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 := by - intros - ext <;> simp - mul_zero := by - intros - ext <;> simp + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance instCommRing : CommRing GaussInt where + zero := 0 + one := 1 + add := (· + ·) + neg x := -x + mul := (· * ·) + nsmul := nsmulRec + zsmul := zsmulRec + add_assoc := by + intros + ext <;> simp <;> ring + zero_add := by + intro + ext <;> simp + add_zero := by + intro + ext <;> simp + neg_add_cancel := 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 := by + intros + ext <;> simp + mul_zero := by + intros + ext <;> simp </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 that 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 + <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
- @@ -1232,14 +1230,14 @@ In that case, we can take $q$$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) + <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 → 0 ≤ a % b := + Int.emod_nonneg a - example (a b : ℤ) : b ≠ 0 → a % b < |b| := - Int.emod_lt a + example (a b : ℤ) : b ≠ 0 → a % b < |b| := + Int.emod_lt a </pre></div> </div> In an arbitrary ring, an element $a$ is said to be a unit if it divides
- @@ -1327,38 +1325,38 @@ 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) : |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 + <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) : |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] + <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 + <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
- @@ -1366,33 +1364,33 @@ in the <code class="docutils literal notranslate">GaussInt</sp
 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 + <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⟩ + <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_re (x : GaussInt) : (conj x).re = x.re := + rfl - @[simp] - theorem conj_im (x : GaussInt) : (conj x).im = -x.im := - 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] + theorem norm_conj (x : GaussInt) : norm (conj x) = norm x := by simp [norm] </pre></div> </div> Finally, we define division for the Gaussian integers
- @@ -1405,23 +1403,23 @@ then the real and imaginary parts of <code class="docutils literal notranslate">
 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)⟩⟩ + <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)⟩ + <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 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 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
- @@ -1449,24 +1447,24 @@ So we have
 as required. This messy calculation is carried out in the next proof. We encourage you to step through the details and see if you can find a nicer argument. - <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem norm_mod_lt (x : GaussInt) {y : GaussInt} (hy : y ≠ 0) : - (x % y).norm < y.norm := by - have norm_y_pos : 0 < norm y := by rwa [norm_pos] - have H1 : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ - · ext <;> simp [Int.mod'_eq, mod_def, div_def, norm] <;> ring - have H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y - · calc - norm (x % y) * norm y = norm (x % y * conj y) := by simp only [norm_mul, norm_conj] - _ = |Int.mod' (x.re * y.re + x.im * y.im) (norm y)| ^ 2 - + |Int.mod' (-(x.re * y.im) + x.im * y.re) (norm y)| ^ 2 := by simp [H1, norm, sq_abs] - _ ≤ (y.norm / 2) ^ 2 + (y.norm / 2) ^ 2 := by gcongr <;> apply Int.abs_mod'_le _ _ norm_y_pos - _ = norm y / 2 * (norm y / 2 * 2) := by ring - _ ≤ norm y / 2 * norm y := by gcongr; apply Int.ediv_mul_le; norm_num - calc norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right H2 norm_y_pos - _ < norm y := by - apply Int.ediv_lt_of_lt_mul - · norm_num - · linarith + <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem norm_mod_lt (x : GaussInt) {y : GaussInt} (hy : y ≠ 0) : + (x % y).norm < y.norm := by + have norm_y_pos : 0 < norm y := by rwa [norm_pos] + have H1 : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ + · ext <;> simp [Int.mod'_eq, mod_def, div_def, norm] <;> ring + have H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y + · calc + norm (x % y) * norm y = norm (x % y * conj y) := by simp only [norm_mul, norm_conj] + _ = |Int.mod' (x.re * y.re + x.im * y.im) (norm y)| ^ 2 + + |Int.mod' (-(x.re * y.im) + x.im * y.re) (norm y)| ^ 2 := by simp [H1, norm, sq_abs] + _ ≤ (y.norm / 2) ^ 2 + (y.norm / 2) ^ 2 := by gcongr <;> apply Int.abs_mod'_le _ _ norm_y_pos + _ = norm y / 2 * (norm y / 2 * 2) := by ring + _ ≤ norm y / 2 * norm y := by gcongr; apply Int.ediv_mul_le; norm_num + calc norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right H2 norm_y_pos + _ < norm y := by + apply Int.ediv_lt_of_lt_mul + · norm_num + · linarith </pre></div> </div> We are in the home stretch. Our <code class="docutils literal notranslate">norm</code> function maps Gaussian integers to
- @@ -1476,26 +1474,26 @@ numbers, and we obtain that by composing <code class="docutils literal notransla
 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.natCast_natAbs, abs_of_nonneg, norm_nonneg] - apply norm_mod_lt x hy + <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.natCast_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) + <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
- @@ -1508,25 +1506,25 @@ 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] ; ring - quotient_zero := fun x ↦ by - simp [div_def, norm, Int.div'] - rfl - r := (measure (Int.natAbs ∘ norm)).1 - r_wellFounded := (measure (Int.natAbs ∘ norm)).2 - remainder_lt := natAbs_norm_mod_lt - mul_left_not_lt := not_norm_mul_left_lt_norm } + <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] ; ring + quotient_zero := fun x ↦ by + simp [div_def, norm, Int.div'] + rfl + r := (measure (Int.natAbs ∘ norm)).1 + 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 := - irreducible_iff_prime + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x : GaussInt) : Irreducible x ↔ Prime x := + irreducible_iff_prime </pre></div> </div> </section>
html/C07_Hierarchies.html
- @@ -1,25 +1,23 @@
 <!DOCTYPE html> - <html class="writer-html5" lang="en" data-content_root="./"> + <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="viewport" content="width=device-width, initial-scale=1" /> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>7. Hierarchies — Mathematics in Lean 0.1 documentation</title> - <link rel="stylesheet" type="text/css" href="_static/pygments.css?v=80d5e7a1" /> - <link rel="stylesheet" type="text/css" href="_static/css/theme.css?v=19f00094" /> - <link rel="stylesheet" type="text/css" href="_static/css/custom.css?v=0731ccc3" /> - - + <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?v=5d32c60e"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js?v=2cd50e6c"></script> - <script src="_static/documentation_options.js?v=2709fde1"></script> - <script src="_static/doctools.js?v=9a2dae69"></script> - <script src="_static/sphinx_highlight.js?v=dc90522c"></script> + <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" />
- @@ -94,7 +92,7 @@
 <div itemprop="articleBody"> <section id="hierarchies"> - <h1>7. Hierarchies<a class="headerlink" href="#hierarchies" title="Link to this heading"></a></h1> + <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
- @@ -111,13 +109,13 @@ so we will used indices to distinguish our version. For instance we will have <c
 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="Link to this heading"></a></h2> + <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 + <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
- @@ -129,18 +127,18 @@ as long as they are marked as instance-implicit, ie appear between square bracke
 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 + <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 + @[class] structure One₂ (α : Type) where + /-- The element one -/ + one : α - #check One₂.one + #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 + <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,
- @@ -154,7 +152,7 @@ example, leaving out the type ascription <code class="docutils literal notransla
 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) + <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
- @@ -165,30 +163,30 @@ or real numbers.
 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 + <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] : α := 𝟙 - example {α : Type} [One₁ α] : (𝟙 : α) = 𝟙 := rfl + 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 + <div class="highlight-lean notranslate"><div class="highlight"><pre>class Dia₁ (α : Type) where + dia : α → α → α - infixl:70 " ⋄ " => Dia₁.dia + 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) + <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
- @@ -197,19 +195,19 @@ of <cite>a ⋄ b</cite>. However this <cite>toDia₁</cite> field does n
 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 + <div class="highlight-lean notranslate"><div class="highlight"><pre>attribute [instance] Semigroup₁.toDia₁ - example {α : Type} [Semigroup₁ α] (a b : α) : α := a ⋄ b + 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) + <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 + 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
- @@ -217,11 +215,11 @@ case it fixes only the hurdle of writing fields such as <cite>toDia₁</cite
 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 + <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
- @@ -231,13 +229,13 @@ is rather terse by default but it can be expanded by clicking on lines ending wi
 It includes failed attempts where Lean tried to find instances before having enough type information to succeed. 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 + <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 + <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
- @@ -245,25 +243,25 @@ define monoids as:
 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 + <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 + <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 + #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.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
- @@ -271,50 +269,50 @@ take a would-be overlapping <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 + <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 + <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 + @[inherit_doc] + postfix:max "⁻¹" => Inv₁.inv - class Group₁ (G : Type) extends Monoid₁ G, Inv₁ G where - inv_dia : ∀ a : G, a⁻¹ ⋄ a = 𝟙 + class Group₁ (G : Type) extends Monoid₁ G, Inv₁ 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] + <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) + <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] + <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 + <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 + 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.
- @@ -332,84 +330,84 @@ lemmas are then only stated in multiplicative notation and marked with the attri
 to generate the additive version as <code class="docutils literal notranslate">left_inv_eq_right_inv'</code> with its 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 + <div class="highlight-lean notranslate"><div class="highlight"><pre>class AddSemigroup₃ (α : Type) extends Add α where /-- Addition is associative -/ - add_assoc₃ : ∀ a b c : α, a + b + c = a + (b + c) + add_assoc₃ : ∀ a b c : α, a + b + c = a + (b + c) - @[to_additive AddSemigroup₃] - class Semigroup₃ (α : Type) extends Mul α where + @[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) + mul_assoc₃ : ∀ a b c : α, a * b * c = a * (b * c) - class AddMonoid₃ (α : Type) extends AddSemigroup₃ α, AddZeroClass α + class AddMonoid₃ (α : Type) extends AddSemigroup₃ α, AddZeroClass α - @[to_additive AddMonoid₃] - class Monoid₃ (α : Type) extends Semigroup₃ α, MulOneClass α + @[to_additive AddMonoid₃] + class Monoid₃ (α : Type) extends Semigroup₃ α, MulOneClass α - attribute [to_additive existing] Monoid₃.toMulOneClass + attribute [to_additive existing] Monoid₃.toMulOneClass - export Semigroup₃ (mul_assoc₃) - export AddSemigroup₃ (add_assoc₃) + 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] + 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' + #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 + <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 + @[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 + class AddCommMonoid₃ (α : Type) extends AddMonoid₃ α, AddCommSemigroup₃ α - @[to_additive AddCommMonoid₃] - class CommMonoid₃ (α : Type) extends Monoid₃ α, CommSemigroup₃ 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 + 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 + @[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 tag lemmas with <code class="docutils literal notranslate">simp</code> when appropriate. - <div class="highlight-lean notranslate"><div class="highlight"><pre>attribute [simp] Group₃.inv_mul AddGroup₃.neg_add + <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 + <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 asked 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 + <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_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 + @[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 + class AddCommGroup₃ (G : Type) extends AddGroup₃ G, AddCommMonoid₃ G - @[to_additive AddCommGroup₃] - class CommGroup₃ (G : Type) extends Group₃ G, CommMonoid₃ G + @[to_additive AddCommGroup₃] + class CommGroup₃ (G : Type) extends Group₃ G, CommMonoid₃ G </pre></div> </div> We are now ready for rings. For demonstration purposes we won’t assume that addition is
- @@ -419,56 +417,56 @@ also because Mathlib’s algebraic hierarchy goes through semirings which ar
 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 + <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 } + 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 + <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 + <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 + @[inherit_doc] infix:50 " ≤₁ " => LE₁.le - class Preorder₁ (α : Type) + class Preorder₁ (α : Type) - class PartialOrder₁ (α : Type) + class PartialOrder₁ (α : Type) - class OrderedCommMonoid₁ (α : Type) + class OrderedCommMonoid₁ (α : Type) - instance : OrderedCommMonoid₁ ℕ where + instance : OrderedCommMonoid₁ ℕ where </pre></div> </div> We now want to discuss algebraic structures involving several types. The prime example
- @@ -477,20 +475,20 @@ and think that all our rings are fields. Those structures are commutative additi
 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 + <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 + 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 + <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
- @@ -517,13 +515,13 @@ safely be used as an instance. The rule is easy to remember: each class appearin
 <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 + <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
- @@ -531,26 +529,26 @@ generalize the theory of vector spaces by allowing non-invertible scalars). Firs
 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 + <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 + 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 + <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>
- @@ -561,7 +559,7 @@ this isn’t true by definition, it requires a proof. This is very bad news
 instance resolution procedure and will lead to very frustrating 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 + <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.
- @@ -585,46 +583,46 @@ field and some <code class="docutils literal notranslate">Prop
 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 + <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⟩ + 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 + <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] + <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 + <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
- @@ -638,20 +636,20 @@ that every preorder comes with a <code class="docutils literal notranslate"><spa
 -/ </section> <section id="morphisms"> - <h2>7.2. Morphisms<a class="headerlink" href="#morphisms" title="Link to this heading"></a></h2> + <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' + <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' + <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
- @@ -669,17 +667,17 @@ It really feels like “monoid morphism” is not an adjective you can a
 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 + <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 contrast, 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' + <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
- @@ -687,30 +685,30 @@ the <code class="docutils literal notranslate">CoeFun</
 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 + <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 + 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 + <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' + <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 + instance [AddMonoid G] [AddMonoid H] : CoeFun (AddMonoidHom₁ G H) (fun _ ↦ G → H) where + coe := AddMonoidHom₁.toFun - attribute [coe] AddMonoidHom₁.toFun + attribute [coe] AddMonoidHom₁.toFun - @[ext] - structure RingHom₁ (R S : Type) [Ring R] [Ring S] extends MonoidHom₁ R S, AddMonoidHom₁ R S + @[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
- @@ -724,16 +722,16 @@ Neither option is appealing so Mathlib uses a new hierarchy trick here. The idea
 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' + <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 + <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 this 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>
- @@ -751,41 +749,41 @@ replace <code class="docutils literal notranslate">def
 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' + <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 + instance [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] : CoeFun F (fun _ ↦ M → N) where + coe := MonoidHomClass₂.toFun - attribute [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 + <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 + 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 morphisms 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] + <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 [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 + 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.
- @@ -798,16 +796,16 @@ to record that this pattern is used only for functions with extra properties, me
 coercion to functions should be injective. So Mathlib adds one more layer of abstraction with the base class <code class="docutils literal notranslate">DFunLike</code> (where “DFun” stands for dependent function). 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 - DFunLike 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' + <div class="highlight-lean notranslate"><div class="highlight"><pre>class MonoidHomClass₃ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] extends + DFunLike 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 + 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
- @@ -820,30 +818,30 @@ ordered types, and then order preserving monoid morphisms. This is for training
 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' + <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 + @[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] + 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] [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] : + 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 + 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="Link to this heading"></a></h2> + <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 with our previous topic. Indeed a set in <code class="docutils literal notranslate">X</code> is implemented as a function from
- @@ -853,66 +851,66 @@ We won’t reuse <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 + <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 + 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 + <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 + 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 + <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)) + <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) + <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 + <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 + 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
- @@ -922,15 +920,15 @@ and a <code class="docutils literal notranslate">SubmonoidClas
 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'⟩ }⟩ + <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 + <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
- @@ -952,31 +950,31 @@ is the <code class="docutils literal notranslate">HasQuotient<
 to you. In the last example, you can use <code class="docutils literal notranslate">Setoid.refl</code> but it won’t automatically pick up the relevant <code class="docutils literal notranslate">Setoid</code> structure. You can fix this issue by providing all arguments using the <code class="docutils literal notranslate">@</code> syntax, as in <code class="docutils literal notranslate">@Setoid.refl M N.Setoid</code>. - <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 + <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>
html/C08_Groups_and_Rings.html
- @@ -1,25 +1,23 @@
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- @@ -107,7 +105,7 @@
 <div itemprop="articleBody"> <section id="groups-and-rings"> - <h1>8. Groups and Rings<a class="headerlink" href="#groups-and-rings" title="Link to this heading"></a></h1> + <h1>8. Groups and Rings<a class="headerlink" href="#groups-and-rings" title="Permalink to this heading"></a></h1> We saw in <a class="reference internal" href="C02_Basics.html#proving-identities-in-algebraic-structures">Section 2.2</a> how to reason about operations in groups and rings. Later, in <a class="reference internal" href="C06_Structures.html#section-algebraic-structures">Section 6.2</a>, we saw how to define abstract algebraic structures, such as group structures, as well as concrete instances
- @@ -122,9 +120,9 @@ decisions behind the way the topics are treated.
 So making sense of some of the examples may require reviewing the background from <a class="reference internal" href="C07_Hierarchies.html#hierarchies">Chapter 7</a>. <section id="monoids-and-groups"> - <h2>8.1. Monoids and Groups<a class="headerlink" href="#monoids-and-groups" title="Link to this heading"></a></h2> - <section id="monoids-and-their-morphisms"> - <h3>8.1.1. Monoids and their morphisms<a class="headerlink" href="#monoids-and-their-morphisms" title="Link to this heading"></a></h3> + <h2>8.1. Monoids and Groups<a class="headerlink" href="#monoids-and-groups" title="Permalink to this heading"></a></h2> + <section id="monoids-and-their-morphisms"> + <h3>8.1.1. Monoids and their morphisms<a class="headerlink" href="#monoids-and-their-morphisms" title="Permalink to this heading"></a></h3> Courses in abstract algebra often start with groups and then progress to rings, fields, and vector spaces. This involves some contortions when discussing multiplication on rings since the multiplication operation does not come from a group structure
- @@ -146,9 +144,9 @@ argument (in other words, in square brackets).
 By default, <code class="docutils literal notranslate">Monoid</code> uses multiplicative notation for the operation; for additive notation use <code class="docutils literal notranslate">AddMonoid</code> instead. The commutative versions of these structures add the prefix <code class="docutils literal notranslate">Comm</code> before <code class="docutils literal notranslate">Monoid</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {M : Type*} [Monoid M] (x : M) : x * 1 = x := mul_one x + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {M : Type*} [Monoid M] (x : M) : x * 1 = x := mul_one x - example {M : Type*} [AddCommMonoid M] (x y : M) : x + y = y + x := add_comm x y + example {M : Type*} [AddCommMonoid M] (x y : M) : x + y = y + x := add_comm x y </pre></div> </div> Note that although <code class="docutils literal notranslate">AddMonoid</code> is found in the library,
- @@ -157,51 +155,51 @@ it is generally confusing to use additive notation with a non-commutative operat
 <code class="docutils literal notranslate">M →* N</code>. Lean will automatically see such a morphism as a function from <code class="docutils literal notranslate">M</code> to <code class="docutils literal notranslate">N</code> when we apply it to elements of <code class="docutils literal notranslate">M</code>. The additive version is called <code class="docutils literal notranslate">AddMonoidHom</code> and written <code class="docutils literal notranslate">M →+ N</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {M N : Type*} [Monoid M] [Monoid N] (x y : M) (f : M →* N) : f (x * y) = f x * f y := - f.map_mul x y + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {M N : Type*} [Monoid M] [Monoid N] (x y : M) (f : M →* N) : f (x * y) = f x * f y := + f.map_mul x y - example {M N : Type*} [AddMonoid M] [AddMonoid N] (f : M →+ N) : f 0 = 0 := - f.map_zero + example {M N : Type*} [AddMonoid M] [AddMonoid N] (f : M →+ N) : f 0 = 0 := + f.map_zero </pre></div> </div> These morphisms are bundled maps, i.e. they package together a map and some of its properties. Remember that <a class="reference internal" href="C07_Hierarchies.html#section-hierarchies-morphisms">Section 7.2</a> explains bundled maps; here we simply note the slightly unfortunate consequence that we cannot use ordinary function composition to compose maps. Instead, we need to use <code class="docutils literal notranslate">MonoidHom.comp</code> and <code class="docutils literal notranslate">AddMonoidHom.comp</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {M N P : Type*} [AddMonoid M] [AddMonoid N] [AddMonoid P] - (f : M →+ N) (g : N →+ P) : M →+ P := g.comp f + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {M N P : Type*} [AddMonoid M] [AddMonoid N] [AddMonoid P] + (f : M →+ N) (g : N →+ P) : M →+ P := g.comp f </pre></div> </div> </section> <section id="groups-and-their-morphisms"> - <h3>8.1.2. Groups and their morphisms<a class="headerlink" href="#groups-and-their-morphisms" title="Link to this heading"></a></h3> + <h3>8.1.2. Groups and their morphisms<a class="headerlink" href="#groups-and-their-morphisms" title="Permalink to this heading"></a></h3> We will have much more to say about groups, which are monoids with the extra property that every element has an inverse. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (x : G) : x * x⁻¹ = 1 := mul_inv_cancel x + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (x : G) : x * x⁻¹ = 1 := mul_inv_cancel x </pre></div> </div> Similar to the <code class="docutils literal notranslate">ring</code> tactic that we saw earlier, there is a <code class="docutils literal notranslate">group</code> tactic that proves any identity that holds in any group. (Equivalently, it proves the identities that hold in free groups.) - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (x y z : G) : x * (y * z) * (x * z)⁻¹ * (x * y * x⁻¹)⁻¹ = 1 := by - group + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (x y z : G) : x * (y * z) * (x * z)⁻¹ * (x * y * x⁻¹)⁻¹ = 1 := by + group </pre></div> </div> There is also a tactic for identities in commutative additive groups called <code class="docutils literal notranslate">abel</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [AddCommGroup G] (x y z : G) : z + x + (y - z - x) = y := by - abel + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [AddCommGroup G] (x y z : G) : z + x + (y - z - x) = y := by + abel </pre></div> </div> Interestingly, a group morphism is nothing more than a monoid morphism between groups. So we can copy and paste one of our earlier examples, replacing <code class="docutils literal notranslate">Monoid</code> with <code class="docutils literal notranslate">Group</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G H : Type*} [Group G] [Group H] (x y : G) (f : G →* H) : f (x * y) = f x * f y := - f.map_mul x y + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G H : Type*} [Group G] [Group H] (x y : G) (f : G →* H) : f (x * y) = f x * f y := + f.map_mul x y </pre></div> </div> Of course we do get some new properties, such as this one: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G H : Type*} [Group G] [Group H] (x : G) (f : G →* H) : f (x⁻¹) = (f x)⁻¹ := - f.map_inv x + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G H : Type*} [Group G] [Group H] (x : G) (f : G →* H) : f (x⁻¹) = (f x)⁻¹ := + f.map_inv x </pre></div> </div> You may be worried that constructing group morphisms will require us to do unnecessary work since
- @@ -209,9 +207,9 @@ the definition of monoid morphism enforces that neutral elements are sent to neu
 while this is automatic in the case of group morphisms. In practice the extra work is not hard, but, to avoid it, there is a function building a group morphism from a function between groups that is compatible with the composition laws. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G H : Type*} [Group G] [Group H] (f : G → H) (h : ∀ x y, f (x * y) = f x * f y) : - G →* H := - MonoidHom.mk' f h + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G H : Type*} [Group G] [Group H] (f : G → H) (h : ∀ x y, f (x * y) = f x * f y) : + G →* H := + MonoidHom.mk' f h </pre></div> </div> There is also a type <code class="docutils literal notranslate">MulEquiv</code> of group (or monoid) isomorphisms denoted by <code class="docutils literal notranslate">≃*</code> (and
- @@ -222,31 +220,31 @@ the identity isomorphism of <code class="docutils literal notranslate">f.symm</code> and <code class="docutils literal notranslate">f.trans g</code> respectively. Elements of this type are automatically coerced to morphisms and functions when necessary. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G H : Type*} [Group G] [Group H] (f : G ≃* H) : - f.trans f.symm = MulEquiv.refl G := - f.self_trans_symm + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G H : Type*} [Group G] [Group H] (f : G ≃* H) : + f.trans f.symm = MulEquiv.refl G := + f.self_trans_symm </pre></div> </div> One can use <code class="docutils literal notranslate">MulEquiv.ofBijective</code> to build an isomorphism from a bijective morphism. Doing so makes the inverse function noncomputable. - <div class="highlight-lean notranslate"><div class="highlight"><pre>noncomputable example {G H : Type*} [Group G] [Group H] - (f : G →* H) (h : Function.Bijective f) : - G ≃* H := - MulEquiv.ofBijective f h + <div class="highlight-lean notranslate"><div class="highlight"><pre>noncomputable example {G H : Type*} [Group G] [Group H] + (f : G →* H) (h : Function.Bijective f) : + G ≃* H := + MulEquiv.ofBijective f h </pre></div> </div> </section> <section id="subgroups"> - <h3>8.1.3. Subgroups<a class="headerlink" href="#subgroups" title="Link to this heading"></a></h3> + <h3>8.1.3. Subgroups<a class="headerlink" href="#subgroups" title="Permalink to this heading"></a></h3> Just as group morphisms are bundled, a subgroup of <code class="docutils literal notranslate">G</code> is also a bundled structure consisting of a set in <code class="docutils literal notranslate">G</code> with the relevant closure properties. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (H : Subgroup G) {x y : G} (hx : x ∈ H) (hy : y ∈ H) : - x * y ∈ H := - H.mul_mem hx hy + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (H : Subgroup G) {x y : G} (hx : x ∈ H) (hy : y ∈ H) : + x * y ∈ H := + H.mul_mem hx hy - example {G : Type*} [Group G] (H : Subgroup G) {x : G} (hx : x ∈ H) : - x⁻¹ ∈ H := - H.inv_mem hx + example {G : Type*} [Group G] (H : Subgroup G) {x : G} (hx : x ∈ H) : + x⁻¹ ∈ H := + H.inv_mem hx </pre></div> </div> In the example above, it is important to understand that <code class="docutils literal notranslate">Subgroup G</code> is the type of subgroups
- @@ -259,29 +257,29 @@ equal in the same way it is used to prove that two sets are equal.
 To state and prove, for example, that <code class="docutils literal notranslate">ℤ</code> is an additive subgroup of <code class="docutils literal notranslate">ℚ</code>, what we really want is to construct a term of type <code class="docutils literal notranslate">AddSubgroup ℚ</code> whose projection to <code class="docutils literal notranslate">Set ℚ</code> is <code class="docutils literal notranslate">ℤ</code>, or, more precisely, the image of <code class="docutils literal notranslate">ℤ</code> in <code class="docutils literal notranslate">ℚ</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : AddSubgroup ℚ where - carrier := Set.range ((↑) : ℤ → ℚ) - add_mem' := by - rintro _ _ ⟨n, rfl⟩ ⟨m, rfl⟩ - use n + m - simp - zero_mem' := by - use 0 - simp - neg_mem' := by - rintro _ ⟨n, rfl⟩ - use -n - simp + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : AddSubgroup ℚ where + carrier := Set.range ((↑) : ℤ → ℚ) + add_mem' := by + rintro _ _ ⟨n, rfl⟩ ⟨m, rfl⟩ + use n + m + simp + zero_mem' := by + use 0 + simp + neg_mem' := by + rintro _ ⟨n, rfl⟩ + use -n + simp </pre></div> </div> Using type classes, Mathlib knows that a subgroup of a group inherits a group structure. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (H : Subgroup G) : Group H := inferInstance + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (H : Subgroup G) : Group H := inferInstance </pre></div> </div> This example is subtle. The object <code class="docutils literal notranslate">H</code> is not a type, but Lean automatically coerces it to a type by interpreting it as a subtype of <code class="docutils literal notranslate">G</code>. So the above example can be restated more explicitly as: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (H : Subgroup G) : Group {x : G // x ∈ H} := inferInstance + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (H : Subgroup G) : Group {x : G // x ∈ H} := inferInstance </pre></div> </div> An important benefit of having a type <code class="docutils literal notranslate">Subgroup G</code> instead of a predicate
- @@ -293,8 +291,8 @@ have used the lattice operation <code class="docutils literal notranslate"> Let us check that the set underlying the infimum of two subgroups is indeed, by definition, their intersection. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (H H' : Subgroup G) : - ((H ⊓ H' : Subgroup G) : Set G) = (H : Set G) ∩ (H' : Set G) := rfl + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (H H' : Subgroup G) : + ((H ⊓ H' : Subgroup G) : Set G) = (H : Set G) ∩ (H' : Set G) := rfl </pre></div> </div> It may look strange to have a different notation for what amounts to the intersection of the
- @@ -302,36 +300,36 @@ underlying sets, but the correspondence does not carry over to the supremum oper
 union, since a union of subgroups is not, in general, a subgroup. Instead one needs to use the subgroup generated by the union, which is done using <code class="docutils literal notranslate">Subgroup.closure</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (H H' : Subgroup G) : - ((H ⊔ H' : Subgroup G) : Set G) = Subgroup.closure ((H : Set G) ∪ (H' : Set G)) := by - rw [Subgroup.sup_eq_closure] + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (H H' : Subgroup G) : + ((H ⊔ H' : Subgroup G) : Set G) = Subgroup.closure ((H : Set G) ∪ (H' : Set G)) := by + rw [Subgroup.sup_eq_closure] </pre></div> </div> Another subtlety is that <code class="docutils literal notranslate">G</code> itself does not have type <code class="docutils literal notranslate">Subgroup G</code>, so we need a way to talk about <code class="docutils literal notranslate">G</code> seen as a subgroup of <code class="docutils literal notranslate">G</code>. This is also provided by the lattice structure: the full subgroup is the top element of this lattice. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (x : G) : x ∈ (⊤ : Subgroup G) := trivial + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (x : G) : x ∈ (⊤ : Subgroup G) := trivial </pre></div> </div> Similarly the bottom element of this lattice is the subgroup whose only element is the neutral element. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (x : G) : x ∈ (⊥ : Subgroup G) ↔ x = 1 := Subgroup.mem_bot + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (x : G) : x ∈ (⊥ : Subgroup G) ↔ x = 1 := Subgroup.mem_bot </pre></div> </div> As an exercise in manipulating groups and subgroups, you can define the conjugate of a subgroup by an element of the ambient group. - <div class="highlight-lean notranslate"><div class="highlight"><pre>def conjugate {G : Type*} [Group G] (x : G) (H : Subgroup G) : Subgroup G where - carrier := {a : G | ∃ h, h ∈ H ∧ a = x * h * x⁻¹} - one_mem' := by - dsimp - sorry - inv_mem' := by - dsimp - sorry - mul_mem' := by - dsimp - sorry + <div class="highlight-lean notranslate"><div class="highlight"><pre>def conjugate {G : Type*} [Group G] (x : G) (H : Subgroup G) : Subgroup G where + carrier := {a : G | ∃ h, h ∈ H ∧ a = x * h * x⁻¹} + one_mem' := by + dsimp + sorry + inv_mem' := by + dsimp + sorry + mul_mem' := by + dsimp + sorry </pre></div> </div> Tying the previous two topics together, one can push forward and pull back subgroups using
- @@ -339,55 +337,55 @@ group morphisms. The naming convention in Mathlib is to call those operations <c
 and <code class="docutils literal notranslate">comap</code>. These are not the common mathematical terms, but they have the advantage of being shorter than “pushforward” and “direct image.” - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G H : Type*} [Group G] [Group H] (G' : Subgroup G) (f : G →* H) : Subgroup H := - Subgroup.map f G' + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G H : Type*} [Group G] [Group H] (G' : Subgroup G) (f : G →* H) : Subgroup H := + Subgroup.map f G' - example {G H : Type*} [Group G] [Group H] (H' : Subgroup H) (f : G →* H) : Subgroup G := - Subgroup.comap f H' + example {G H : Type*} [Group G] [Group H] (H' : Subgroup H) (f : G →* H) : Subgroup G := + Subgroup.comap f H' - #check Subgroup.mem_map - #check Subgroup.mem_comap + #check Subgroup.mem_map + #check Subgroup.mem_comap </pre></div> </div> In particular, the preimage of the bottom subgroup under a morphism <code class="docutils literal notranslate">f</code> is a subgroup called the kernel of <code class="docutils literal notranslate">f</code>, and the range of <code class="docutils literal notranslate">f</code> is also a subgroup. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G H : Type*} [Group G] [Group H] (f : G →* H) (g : G) : - g ∈ MonoidHom.ker f ↔ f g = 1 := - f.mem_ker + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G H : Type*} [Group G] [Group H] (f : G →* H) (g : G) : + g ∈ MonoidHom.ker f ↔ f g = 1 := + f.mem_ker - example {G H : Type*} [Group G] [Group H] (f : G →* H) (h : H) : - h ∈ MonoidHom.range f ↔ ∃ g : G, f g = h := - f.mem_range + example {G H : Type*} [Group G] [Group H] (f : G →* H) (h : H) : + h ∈ MonoidHom.range f ↔ ∃ g : G, f g = h := + f.mem_range </pre></div> </div> As exercises in manipulating group morphisms and subgroups, let us prove some elementary properties. They are already proved in Mathlib, so do not use <code class="docutils literal notranslate">exact?</code> too quickly if you want to benefit from these exercises. - <div class="highlight-lean notranslate"><div class="highlight"><pre>section exercises - variable {G H : Type*} [Group G] [Group H] + <div class="highlight-lean notranslate"><div class="highlight"><pre>section exercises + variable {G H : Type*} [Group G] [Group H] - open Subgroup + open Subgroup - example (φ : G →* H) (S T : Subgroup H) (hST : S ≤ T) : comap φ S ≤ comap φ T := by - sorry + example (φ : G →* H) (S T : Subgroup H) (hST : S ≤ T) : comap φ S ≤ comap φ T := by + sorry - example (φ : G →* H) (S T : Subgroup G) (hST : S ≤ T) : map φ S ≤ map φ T := by - sorry + example (φ : G →* H) (S T : Subgroup G) (hST : S ≤ T) : map φ S ≤ map φ T := by + sorry - variable {K : Type*} [Group K] + variable {K : Type*} [Group K] -- Remember you can use the `ext` tactic to prove an equality of subgroups. - example (φ : G →* H) (ψ : H →* K) (U : Subgroup K) : - comap (ψ.comp φ) U = comap φ (comap ψ U) := by - sorry + example (φ : G →* H) (ψ : H →* K) (U : Subgroup K) : + comap (ψ.comp φ) U = comap φ (comap ψ U) := by + sorry -- Pushing a subgroup along one homomorphism and then another is equal to -- pushing it forward along the composite of the homomorphisms. - example (φ : G →* H) (ψ : H →* K) (S : Subgroup G) : - map (ψ.comp φ) S = map ψ (S.map φ) := by - sorry + example (φ : G →* H) (ψ : H →* K) (S : Subgroup G) : + map (ψ.comp φ) S = map ψ (S.map φ) := by + sorry - end exercises + end exercises </pre></div> </div> Let us finish this introduction to subgroups in Mathlib with two very classical results.
- @@ -395,47 +393,47 @@ Lagrange theorem states the cardinality of a subgroup of a finite group divides
 the group. Sylow’s first theorem is a famous partial converse to Lagrange’s theorem. While this corner of Mathlib is partly set up to allow computation, we can tell Lean to use nonconstructive logic anyway using the following <code class="docutils literal notranslate">open scoped</code> command. - <div class="highlight-lean notranslate"><div class="highlight"><pre>open scoped Classical + <div class="highlight-lean notranslate"><div class="highlight"><pre>open scoped Classical - example {G : Type*} [Group G] (G' : Subgroup G) : Nat.card G' ∣ Nat.card G := - ⟨G'.index, mul_comm G'.index _ ▸ G'.index_mul_card.symm⟩ + example {G : Type*} [Group G] (G' : Subgroup G) : Nat.card G' ∣ Nat.card G := + ⟨G'.index, mul_comm G'.index _ ▸ G'.index_mul_card.symm⟩ - open Subgroup + open Subgroup - example {G : Type*} [Group G] [Finite G] (p : ℕ) {n : ℕ} [Fact p.Prime] - (hdvd : p ^ n ∣ Nat.card G) : ∃ K : Subgroup G, Nat.card K = p ^ n := - Sylow.exists_subgroup_card_pow_prime p hdvd + example {G : Type*} [Group G] [Finite G] (p : ℕ) {n : ℕ} [Fact p.Prime] + (hdvd : p ^ n ∣ Nat.card G) : ∃ K : Subgroup G, Nat.card K = p ^ n := + Sylow.exists_subgroup_card_pow_prime p hdvd </pre></div> </div> The next two exercises derive a corollary of Lagrange’s lemma. (This is also already in Mathlib, so do not use <code class="docutils literal notranslate">exact?</code> too quickly.) - <div class="highlight-lean notranslate"><div class="highlight"><pre>lemma eq_bot_iff_card {G : Type*} [Group G] {H : Subgroup G} : - H = ⊥ ↔ Nat.card H = 1 := by - suffices (∀ x ∈ H, x = 1) ↔ ∃ x ∈ H, ∀ a ∈ H, a = x by - simpa [eq_bot_iff_forall, Nat.card_eq_one_iff_exists] - sorry + <div class="highlight-lean notranslate"><div class="highlight"><pre>lemma eq_bot_iff_card {G : Type*} [Group G] {H : Subgroup G} : + H = ⊥ ↔ Nat.card H = 1 := by + suffices (∀ x ∈ H, x = 1) ↔ ∃ x ∈ H, ∀ a ∈ H, a = x by + simpa [eq_bot_iff_forall, Nat.card_eq_one_iff_exists] + sorry - #check card_dvd_of_le + #check card_dvd_of_le - lemma inf_bot_of_coprime {G : Type*} [Group G] (H K : Subgroup G) - (h : (Nat.card H).Coprime (Nat.card K)) : H ⊓ K = ⊥ := by - sorry + lemma inf_bot_of_coprime {G : Type*} [Group G] (H K : Subgroup G) + (h : (Nat.card H).Coprime (Nat.card K)) : H ⊓ K = ⊥ := by + sorry </pre></div> </div> </section> <section id="concrete-groups"> - <h3>8.1.4. Concrete groups<a class="headerlink" href="#concrete-groups" title="Link to this heading"></a></h3> + <h3>8.1.4. Concrete groups<a class="headerlink" href="#concrete-groups" title="Permalink to this heading"></a></h3> One can also manipulate concrete groups in Mathlib, although this is typically more complicated than working with the abstract theory. For instance, given any type <code class="docutils literal notranslate">X</code>, the group of permutations of <code class="docutils literal notranslate">X</code> is <code class="docutils literal notranslate">Equiv.Perm X</code>. In particular the symmetric group $\mathfrak{S}_n$ is <code class="docutils literal notranslate">Equiv.Perm (Fin n)</code>. One can state abstract results about this group, for instance saying that <code class="docutils literal notranslate">Equiv.Perm X</code> is generated by cycles if <code class="docutils literal notranslate">X</code> is finite. - <div class="highlight-lean notranslate"><div class="highlight"><pre>open Equiv + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Equiv - example {X : Type*} [Finite X] : Subgroup.closure {σ : Perm X | Perm.IsCycle σ} = ⊤ := - Perm.closure_isCycle + example {X : Type*} [Finite X] : Subgroup.closure {σ : Perm X | Perm.IsCycle σ} = ⊤ := + Perm.closure_isCycle </pre></div> </div> One can be fully concrete and compute actual products of cycles. Below we use the <code class="docutils literal notranslate">#simp</code> command,
- @@ -443,20 +441,20 @@ which calls the <code class="docutils literal notranslate">sim
 cyclic permutation. In the example, the result is a permutation of <code class="docutils literal notranslate">ℕ</code>. One could use a type ascription such as <code class="docutils literal notranslate">(1 : Fin 5)</code> on the first number appearing to make it a computation in <code class="docutils literal notranslate">Perm (Fin 5)</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>#simp [mul_assoc] c[1, 2, 3] * c[2, 3, 4] + <div class="highlight-lean notranslate"><div class="highlight"><pre>#simp [mul_assoc] c[1, 2, 3] * c[2, 3, 4] </pre></div> </div> Another way to work with concrete groups is to use free groups and group presentations. The free group on a type <code class="docutils literal notranslate">α</code> is <code class="docutils literal notranslate">FreeGroup α</code> and the inclusion map is <code class="docutils literal notranslate">FreeGroup.of : α → FreeGroup α</code>. For instance let us define a type <code class="docutils literal notranslate">S</code> with three elements denoted by <code class="docutils literal notranslate">a</code>, <code class="docutils literal notranslate">b</code> and <code class="docutils literal notranslate">c</code>, and the element <code class="docutils literal notranslate">ab⁻¹</code> of the corresponding free group. - <div class="highlight-lean notranslate"><div class="highlight"><pre>section FreeGroup + <div class="highlight-lean notranslate"><div class="highlight"><pre>section FreeGroup - inductive S | a | b | c + inductive S | a | b | c - open S + open S - def myElement : FreeGroup S := (.of a) * (.of b)⁻¹ + def myElement : FreeGroup S := (.of a) * (.of b)⁻¹ </pre></div> </div> Note that we gave the expected type of the definition so that Lean knows that <code class="docutils literal notranslate">.of</code> means
- @@ -464,10 +462,10 @@ by <code class="docutils literal notranslate">a</code>,
 The universal property of free groups is embodied as the equivalence <code class="docutils literal notranslate">FreeGroup.lift</code>. For example, let us define the group morphism from <code class="docutils literal notranslate">FreeGroup S</code> to <code class="docutils literal notranslate">Perm (Fin 5)</code> that sends <code class="docutils literal notranslate">a</code> to <code class="docutils literal notranslate">c[1, 2, 3]</code>, <code class="docutils literal notranslate">b</code> to <code class="docutils literal notranslate">c[2, 3, 1]</code>, and <code class="docutils literal notranslate">c</code> to <code class="docutils literal notranslate">c[2, 3]</code>, - <div class="highlight-lean notranslate"><div class="highlight"><pre>def myMorphism : FreeGroup S →* Perm (Fin 5) := - FreeGroup.lift fun | .a => c[1, 2, 3] - | .b => c[2, 3, 1] - | .c => c[2, 3] + <div class="highlight-lean notranslate"><div class="highlight"><pre>def myMorphism : FreeGroup S →* Perm (Fin 5) := + FreeGroup.lift fun | .a => c[1, 2, 3] + | .b => c[2, 3, 1] + | .c => c[2, 3] </pre></div> </div> As a last concrete example, let us see how to define a group generated by a single element whose
- @@ -479,30 +477,30 @@ i.e. a set of elements of some free group, and returns a group that is this free
 by a normal subgroup generated by relations. (We will see how to handle more general quotients in <a class="reference internal" href="#quotient-groups">Section 8.1.6</a>.) Since we somehow hide this behind a definition, we use <code class="docutils literal notranslate">deriving Group</code> to force creation of a group instance on <code class="docutils literal notranslate">myGroup</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>def myGroup := PresentedGroup {.of () ^ 3} deriving Group + <div class="highlight-lean notranslate"><div class="highlight"><pre>def myGroup := PresentedGroup {.of () ^ 3} deriving Group </pre></div> </div> The universal property of presented groups ensures that morphisms out of this group can be built from functions that send the relations to the neutral element of the target group. So we need such a function and a proof that the condition holds. Then we can feed this proof to <code class="docutils literal notranslate">PresentedGroup.toGroup</code> to get the desired group morphism. - <div class="highlight-lean notranslate"><div class="highlight"><pre>def myMap : Unit → Perm (Fin 5) - | () => c[1, 2, 3] + <div class="highlight-lean notranslate"><div class="highlight"><pre>def myMap : Unit → Perm (Fin 5) + | () => c[1, 2, 3] - lemma compat_myMap : - ∀ r ∈ ({.of () ^ 3} : Set (FreeGroup Unit)), FreeGroup.lift myMap r = 1 := by - rintro _ rfl - simp - decide + lemma compat_myMap : + ∀ r ∈ ({.of () ^ 3} : Set (FreeGroup Unit)), FreeGroup.lift myMap r = 1 := by + rintro _ rfl + simp + decide - def myNewMorphism : myGroup →* Perm (Fin 5) := PresentedGroup.toGroup compat_myMap + def myNewMorphism : myGroup →* Perm (Fin 5) := PresentedGroup.toGroup compat_myMap - end FreeGroup + end FreeGroup </pre></div> </div> </section> <section id="group-actions"> - <h3>8.1.5. Group actions<a class="headerlink" href="#group-actions" title="Link to this heading"></a></h3> + <h3>8.1.5. Group actions<a class="headerlink" href="#group-actions" title="Permalink to this heading"></a></h3> One important way that group theory interacts with the rest of mathematics is through the use of group actions. An action of a group <code class="docutils literal notranslate">G</code> on some type <code class="docutils literal notranslate">X</code> is nothing more than a morphism from <code class="docutils literal notranslate">G</code> to
- @@ -514,31 +512,31 @@ requires some contortions, such as defining type synonyms, each of which carries
 type class instances. This allows us in particular to use <code class="docutils literal notranslate">g • x</code> to denote the action of a group element <code class="docutils literal notranslate">g</code> on a point <code class="docutils literal notranslate">x</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>noncomputable section GroupActions + <div class="highlight-lean notranslate"><div class="highlight"><pre>noncomputable section GroupActions - example {G X : Type*} [Group G] [MulAction G X] (g g': G) (x : X) : - g • (g' • x) = (g * g') • x := - (mul_smul g g' x).symm + example {G X : Type*} [Group G] [MulAction G X] (g g': G) (x : X) : + g • (g' • x) = (g * g') • x := + (mul_smul g g' x).symm </pre></div> </div> There is also a version for additive group called <code class="docutils literal notranslate">AddAction</code>, where the action is denoted by <code class="docutils literal notranslate">+ᵥ</code>. This is used for instance in the definition of affine spaces. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G X : Type*} [AddGroup G] [AddAction G X] (g g' : G) (x : X) : - g +ᵥ (g' +ᵥ x) = (g + g') +ᵥ x := - (add_vadd g g' x).symm + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G X : Type*} [AddGroup G] [AddAction G X] (g g' : G) (x : X) : + g +ᵥ (g' +ᵥ x) = (g + g') +ᵥ x := + (add_vadd g g' x).symm </pre></div> </div> The underlying group morphism is called <code class="docutils literal notranslate">MulAction.toPermHom</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>open MulAction + <div class="highlight-lean notranslate"><div class="highlight"><pre>open MulAction - example {G X : Type*} [Group G] [MulAction G X] : G →* Equiv.Perm X := - toPermHom G X + example {G X : Type*} [Group G] [MulAction G X] : G →* Equiv.Perm X := + toPermHom G X </pre></div> </div> As an illustration let us see how to define the Cayley isomorphism embedding of any group <code class="docutils literal notranslate">G</code> into a permutation group, namely <code class="docutils literal notranslate">Perm G</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>def CayleyIsoMorphism (G : Type*) [Group G] : G ≃* (toPermHom G G).range := - Equiv.Perm.subgroupOfMulAction G G + <div class="highlight-lean notranslate"><div class="highlight"><pre>def CayleyIsoMorphism (G : Type*) [Group G] : G ≃* (toPermHom G G).range := + Equiv.Perm.subgroupOfMulAction G G </pre></div> </div> Note that nothing before the above definition required having a group rather than a monoid (or any
- @@ -546,7 +544,7 @@ type endowed with a multiplication operation really).
 The group condition really enters the picture when we will want to partition <code class="docutils literal notranslate">X</code> into orbits. The corresponding equivalence relation on <code class="docutils literal notranslate">X</code> is called <code class="docutils literal notranslate">MulAction.orbitRel</code>. It is not declared as a global instance. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G X : Type*} [Group G] [MulAction G X] : Setoid X := orbitRel G X + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G X : Type*} [Group G] [MulAction G X] : Setoid X := orbitRel G X </pre></div> </div> Using this we can state that <code class="docutils literal notranslate">X</code> is partitioned into orbits under the action of <code class="docutils literal notranslate">G</code>.
- @@ -555,9 +553,9 @@ More precisely, we get a bijection between <code class="docutils literal notrans
 where <code class="docutils literal notranslate">Quotient.out' ω</code> simply chooses an element that projects to <code class="docutils literal notranslate">ω</code>. Recall that elements of this dependent product are pairs <code class="docutils literal notranslate">⟨ω, x⟩</code> where the type <code class="docutils literal notranslate">orbit G (Quotient.out' ω)</code> of <code class="docutils literal notranslate">x</code> depends on <code class="docutils literal notranslate">ω</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G X : Type*} [Group G] [MulAction G X] : - X ≃ (ω : orbitRel.Quotient G X) × (orbit G (Quotient.out' ω)) := - MulAction.selfEquivSigmaOrbits G X + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G X : Type*} [Group G] [MulAction G X] : + X ≃ (ω : orbitRel.Quotient G X) × (orbit G (Quotient.out' ω)) := + MulAction.selfEquivSigmaOrbits G X </pre></div> </div> In particular, when X is finite, this can be combined with <code class="docutils literal notranslate">Fintype.card_congr</code> and
- @@ -567,59 +565,59 @@ Furthermore, the orbits are in bijection with the quotient of <code class="docut
 stabilizers by left translation. This action of a subgroup by left-translation is used to define quotients of a group by a subgroup with notation <cite>/</cite> so we can use the following concise statement. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G X : Type*} [Group G] [MulAction G X] (x : X) : - orbit G x ≃ G ⧸ stabilizer G x := - MulAction.orbitEquivQuotientStabilizer G x + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G X : Type*} [Group G] [MulAction G X] (x : X) : + orbit G x ≃ G ⧸ stabilizer G x := + MulAction.orbitEquivQuotientStabilizer G x </pre></div> </div> An important special case of combining the above two results is when <code class="docutils literal notranslate">X</code> is a group <code class="docutils literal notranslate">G</code> equipped with the action of a subgroup <code class="docutils literal notranslate">H</code> by translation. In this case all stabilizers are trivial so every orbit is in bijection with <code class="docutils literal notranslate">H</code> and we get: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (H : Subgroup G) : G ≃ (G ⧸ H) × H := - groupEquivQuotientProdSubgroup + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (H : Subgroup G) : G ≃ (G ⧸ H) × H := + groupEquivQuotientProdSubgroup </pre></div> </div> This is the conceptual variant of the version of Lagrange theorem that we saw above. Note this version makes no finiteness assumption. As an exercise for this section, let us build the action of a group on its subgroup by conjugation, using our definition of <code class="docutils literal notranslate">conjugate</code> from a previous exercise. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {G : Type*} [Group G] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {G : Type*} [Group G] - lemma conjugate_one (H : Subgroup G) : conjugate 1 H = H := by - sorry + lemma conjugate_one (H : Subgroup G) : conjugate 1 H = H := by + sorry - instance : MulAction G (Subgroup G) where - smul := conjugate - one_smul := by - sorry - mul_smul := by - sorry + instance : MulAction G (Subgroup G) where + smul := conjugate + one_smul := by + sorry + mul_smul := by + sorry - end GroupActions + end GroupActions </pre></div> </div> </section> <section id="quotient-groups"> - <h3>8.1.6. Quotient groups<a class="headerlink" href="#quotient-groups" title="Link to this heading"></a></h3> + <h3>8.1.6. Quotient groups<a class="headerlink" href="#quotient-groups" title="Permalink to this heading"></a></h3> In the above discussion of subgroups acting on groups, we saw the quotient <code class="docutils literal notranslate">G ⧸ H</code> appear. In general this is only a type. It can be endowed with a group structure such that the quotient map is a group morphism if and only if <code class="docutils literal notranslate">H</code> is a normal subgroup (and this group structure is then unique). The normality assumption is a type class <code class="docutils literal notranslate">Subgroup.Normal</code> so that type class inference can use it to derive the group structure on the quotient. - <div class="highlight-lean notranslate"><div class="highlight"><pre>noncomputable section QuotientGroup + <div class="highlight-lean notranslate"><div class="highlight"><pre>noncomputable section QuotientGroup - example {G : Type*} [Group G] (H : Subgroup G) [H.Normal] : Group (G ⧸ H) := inferInstance + example {G : Type*} [Group G] (H : Subgroup G) [H.Normal] : Group (G ⧸ H) := inferInstance - example {G : Type*} [Group G] (H : Subgroup G) [H.Normal] : G →* G ⧸ H := - QuotientGroup.mk' H + example {G : Type*} [Group G] (H : Subgroup G) [H.Normal] : G →* G ⧸ H := + QuotientGroup.mk' H </pre></div> </div> The universal property of quotient groups is accessed through <code class="docutils literal notranslate">QuotientGroup.lift</code>: a group morphism <code class="docutils literal notranslate">φ</code> descends to <code class="docutils literal notranslate">G ⧸ N</code> as soon as its kernel contains <code class="docutils literal notranslate">N</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (N : Subgroup G) [N.Normal] {M : Type*} - [Group M] (φ : G →* M) (h : N ≤ MonoidHom.ker φ) : G ⧸ N →* M := - QuotientGroup.lift N φ h + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (N : Subgroup G) [N.Normal] {M : Type*} + [Group M] (φ : G →* M) (h : N ≤ MonoidHom.ker φ) : G ⧸ N →* M := + QuotientGroup.lift N φ h </pre></div> </div> The fact that the target group is called <code class="docutils literal notranslate">M</code> is the above snippet is a clue that having a
- @@ -627,9 +625,9 @@ monoid structure on <code class="docutils literal notranslate">An important special case is when <code class="docutils literal notranslate">N = ker φ</code>. In that case the descended morphism is injective and we get a group isomorphism onto its image. This result is often called the first isomorphism theorem. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] {M : Type*} [Group M] (φ : G →* M) : - G ⧸ MonoidHom.ker φ →* MonoidHom.range φ := - QuotientGroup.quotientKerEquivRange φ + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] {M : Type*} [Group M] (φ : G →* M) : + G ⧸ MonoidHom.ker φ →* MonoidHom.range φ := + QuotientGroup.quotientKerEquivRange φ </pre></div> </div> Applying the universal property to a composition of a morphism <code class="docutils literal notranslate">φ : G →* G'</code>
- @@ -639,18 +637,18 @@ The condition required on <code class="docutils literal notranslate">N'</code>.” But this is equivalent to asking that <code class="docutils literal notranslate">φ</code> should pull <code class="docutils literal notranslate">N'</code> back over <code class="docutils literal notranslate">N</code>, and the latter condition is nicer to work with since the definition of pullback does not involve an existential quantifier. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G G': Type*} [Group G] [Group G'] - {N : Subgroup G} [N.Normal] {N' : Subgroup G'} [N'.Normal] - {φ : G →* G'} (h : N ≤ Subgroup.comap φ N') : G ⧸ N →* G' ⧸ N':= - QuotientGroup.map N N' φ h + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G G': Type*} [Group G] [Group G'] + {N : Subgroup G} [N.Normal] {N' : Subgroup G'} [N'.Normal] + {φ : G →* G'} (h : N ≤ Subgroup.comap φ N') : G ⧸ N →* G' ⧸ N':= + QuotientGroup.map N N' φ h </pre></div> </div> One subtle point to keep in mind is that the type <code class="docutils literal notranslate">G ⧸ N</code> really depends on <code class="docutils literal notranslate">N</code> (up to definitional equality), so having a proof that two normal subgroups <code class="docutils literal notranslate">N</code> and <code class="docutils literal notranslate">M</code> are equal is not enough to make the corresponding quotients equal. However the universal properties does give an isomorphism in this case. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] {M N : Subgroup G} [M.Normal] - [N.Normal] (h : M = N) : G ⧸ M ≃* G ⧸ N := QuotientGroup.quotientMulEquivOfEq h + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] {M N : Subgroup G} [M.Normal] + [N.Normal] (h : M = N) : G ⧸ M ≃* G ⧸ N := QuotientGroup.quotientMulEquivOfEq h </pre></div> </div> As a final series of exercises for this section, we will prove that if <code class="docutils literal notranslate">H</code> and <code class="docutils literal notranslate">K</code> are disjoint
- @@ -660,58 +658,58 @@ then <code class="docutils literal notranslate">G</code
 We start with playing a bit with Lagrange’s lemma, without assuming the subgroups are normal or disjoint. <div class="highlight-lean notranslate"><div class="highlight"><pre>section - variable {G : Type*} [Group G] {H K : Subgroup G} + variable {G : Type*} [Group G] {H K : Subgroup G} - open MonoidHom + open MonoidHom - #check Nat.card_pos -- The nonempty argument will be automatically inferred for subgroups - #check Subgroup.index_eq_card - #check Subgroup.index_mul_card - #check Nat.eq_of_mul_eq_mul_right + #check Nat.card_pos -- The nonempty argument will be automatically inferred for subgroups + #check Subgroup.index_eq_card + #check Subgroup.index_mul_card + #check Nat.eq_of_mul_eq_mul_right - lemma aux_card_eq [Finite G] (h' : Nat.card G = Nat.card H * Nat.card K) : - Nat.card (G ⧸ H) = Nat.card K := by - sorry + lemma aux_card_eq [Finite G] (h' : Nat.card G = Nat.card H * Nat.card K) : + Nat.card (G ⧸ H) = Nat.card K := by + sorry </pre></div> </div> From now on, we assume that our subgroups are normal and disjoint, and we assume the cardinality condition. Now we construct the first building block of the desired isomorphism. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable [H.Normal] [K.Normal] [Fintype G] (h : Disjoint H K) - (h' : Nat.card G = Nat.card H * Nat.card K) + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable [H.Normal] [K.Normal] [Fintype G] (h : Disjoint H K) + (h' : Nat.card G = Nat.card H * Nat.card K) - #check Nat.bijective_iff_injective_and_card - #check ker_eq_bot_iff - #check restrict - #check ker_restrict + #check Nat.bijective_iff_injective_and_card + #check ker_eq_bot_iff + #check restrict + #check ker_restrict - def iso₁ [Fintype G] (h : Disjoint H K) (h' : Nat.card G = Nat.card H * Nat.card K) : K ≃* G ⧸ H := by - sorry + def iso₁ [Fintype G] (h : Disjoint H K) (h' : Nat.card G = Nat.card H * Nat.card K) : K ≃* G ⧸ H := by + sorry </pre></div> </div> Now we can define our second building block. We will need <code class="docutils literal notranslate">MonoidHom.prod</code>, which builds a morphism from <code class="docutils literal notranslate">G₀</code> to <code class="docutils literal notranslate">G₁ × G₂</code> out of morphisms from <code class="docutils literal notranslate">G₀</code> to <code class="docutils literal notranslate">G₁</code> and <code class="docutils literal notranslate">G₂</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>def iso₂ : G ≃* (G ⧸ K) × (G ⧸ H) := by - sorry + <div class="highlight-lean notranslate"><div class="highlight"><pre>def iso₂ : G ≃* (G ⧸ K) × (G ⧸ H) := by + sorry </pre></div> </div> We are ready to put all pieces together. - <div class="highlight-lean notranslate"><div class="highlight"><pre>#check MulEquiv.prodCongr + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check MulEquiv.prodCongr - def finalIso : G ≃* H × K := - sorry + def finalIso : G ≃* H × K := + sorry </pre></div> </div> </section> </section> <section id="rings"> - <h2>8.2. Rings<a class="headerlink" href="#rings" title="Link to this heading"></a></h2> + <h2>8.2. Rings<a class="headerlink" href="#rings" title="Permalink to this heading"></a></h2> <section id="rings-their-units-morphisms-and-subrings"> - <h3>8.2.1. Rings, their units, morphisms and subrings<a class="headerlink" href="#rings-their-units-morphisms-and-subrings" title="Link to this heading"></a></h3> + <h3>8.2.1. Rings, their units, morphisms and subrings<a class="headerlink" href="#rings-their-units-morphisms-and-subrings" title="Permalink to this heading"></a></h3> The type of ring structures on a type <code class="docutils literal notranslate">R</code> is <code class="docutils literal notranslate">Ring R</code>. The variant where multiplication is assumed to be commutative is <code class="docutils literal notranslate">CommRing R</code>. We have already seen that the <code class="docutils literal notranslate">ring</code> tactic will prove any equality that follows from the axioms of a commutative ring. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] (x y : R) : (x + y) ^ 2 = x ^ 2 + y ^ 2 + 2 * x * y := by ring + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] (x y : R) : (x + y) ^ 2 = x ^ 2 + y ^ 2 + 2 * x * y := by ring </pre></div> </div> More exotic variants do not require that the addition on <code class="docutils literal notranslate">R</code> forms a group but only an additive
- @@ -721,7 +719,7 @@ of functions taking values in the natural numbers.
 Another important example is the type of ideals in a ring, which will be discussed below. The name of the <code class="docutils literal notranslate">ring</code> tactic is doubly misleading, since it assumes commutativity but works in semirings as well. In other words, it applies to any <code class="docutils literal notranslate">CommSemiring</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x y : ℕ) : (x + y) ^ 2 = x ^ 2 + y ^ 2 + 2 * x * y := by ring + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x y : ℕ) : (x + y) ^ 2 = x ^ 2 + y ^ 2 + 2 * x * y := by ring </pre></div> </div> There are also versions of the ring and semiring classes that do not assume the existence of a
- @@ -738,33 +736,33 @@ This implementation detail is relevant mainly when defining computable functions
 situations one can use <code class="docutils literal notranslate">IsUnit.unit {x : M} : IsUnit x → Mˣ</code> to build a unit. In the commutative case, one also has <code class="docutils literal notranslate">Units.mkOfMulEqOne (x y : M) : x * y = 1 → Mˣ</code> which builds <code class="docutils literal notranslate">x</code> seen as unit. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x : ℤˣ) : x = 1 ∨ x = -1 := Int.units_eq_one_or x + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x : ℤˣ) : x = 1 ∨ x = -1 := Int.units_eq_one_or x - example {M : Type*} [Monoid M] (x : Mˣ) : (x : M) * x⁻¹ = 1 := Units.mul_inv x + example {M : Type*} [Monoid M] (x : Mˣ) : (x : M) * x⁻¹ = 1 := Units.mul_inv x - example {M : Type*} [Monoid M] : Group Mˣ := inferInstance + example {M : Type*} [Monoid M] : Group Mˣ := inferInstance </pre></div> </div> The type of ring morphisms between two (semi)-rings <code class="docutils literal notranslate">R</code> and <code class="docutils literal notranslate">S</code> is <code class="docutils literal notranslate">RingHom R S</code>, with notation <code class="docutils literal notranslate">R →+* S</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R S : Type*} [Ring R] [Ring S] (f : R →+* S) (x y : R) : - f (x + y) = f x + f y := f.map_add x y + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R S : Type*} [Ring R] [Ring S] (f : R →+* S) (x y : R) : + f (x + y) = f x + f y := f.map_add x y - example {R S : Type*} [Ring R] [Ring S] (f : R →+* S) : Rˣ →* Sˣ := - Units.map f + example {R S : Type*} [Ring R] [Ring S] (f : R →+* S) : Rˣ →* Sˣ := + Units.map f </pre></div> </div> The isomorphism variant is <code class="docutils literal notranslate">RingEquiv</code>, with notation <code class="docutils literal notranslate">≃+*</code>. As with submonoids and subgroups, there is a <code class="docutils literal notranslate">Subring R</code> type for subrings of a ring <code class="docutils literal notranslate">R</code>, but this type is a lot less useful than the type of subgroups since one cannot quotient a ring by a subring. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [Ring R] (S : Subring R) : Ring S := inferInstance + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [Ring R] (S : Subring R) : Ring S := inferInstance </pre></div> </div> Also notice that <code class="docutils literal notranslate">RingHom.range</code> produces a subring. </section> <section id="ideals-and-quotients"> - <h3>8.2.2. Ideals and quotients<a class="headerlink" href="#ideals-and-quotients" title="Link to this heading"></a></h3> + <h3>8.2.2. Ideals and quotients<a class="headerlink" href="#ideals-and-quotients" title="Permalink to this heading"></a></h3> For historical reasons, Mathlib only has a theory of ideals for commutative rings. (The ring library was originally developed to make quick progress toward the foundations of modern algebraic geometry.) So in this section we will work with commutative (semi)rings.
- @@ -775,40 +773,40 @@ ideals. But anonymous projection notation won’t always work as expected. F
 one cannot replace <code class="docutils literal notranslate">Ideal.Quotient.mk I</code> by <code class="docutils literal notranslate">I.Quotient.mk</code> in the snippet below because there are two <code class="docutils literal notranslate">.``s and so it will parse as ``(Ideal.Quotient I).mk</code>; but <code class="docutils literal notranslate">Ideal.Quotient</code> by itself doesn’t exist. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] (I : Ideal R) : R →+* R ⧸ I := - Ideal.Quotient.mk I + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] (I : Ideal R) : R →+* R ⧸ I := + Ideal.Quotient.mk I - example {R : Type*} [CommRing R] {a : R} {I : Ideal R} : - Ideal.Quotient.mk I a = 0 ↔ a ∈ I := - Ideal.Quotient.eq_zero_iff_mem + example {R : Type*} [CommRing R] {a : R} {I : Ideal R} : + Ideal.Quotient.mk I a = 0 ↔ a ∈ I := + Ideal.Quotient.eq_zero_iff_mem </pre></div> </div> The universal property of quotient rings is <code class="docutils literal notranslate">Ideal.Quotient.lift</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R S : Type*} [CommRing R] [CommRing S] (I : Ideal R) (f : R →+* S) - (H : I ≤ RingHom.ker f) : R ⧸ I →+* S := - Ideal.Quotient.lift I f H + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R S : Type*} [CommRing R] [CommRing S] (I : Ideal R) (f : R →+* S) + (H : I ≤ RingHom.ker f) : R ⧸ I →+* S := + Ideal.Quotient.lift I f H </pre></div> </div> In particular it leads to the first isomorphism theorem for rings. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R S : Type*} [CommRing R] [CommRing S](f : R →+* S) : - R ⧸ RingHom.ker f ≃+* f.range := - RingHom.quotientKerEquivRange f + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R S : Type*} [CommRing R] [CommRing S](f : R →+* S) : + R ⧸ RingHom.ker f ≃+* f.range := + RingHom.quotientKerEquivRange f </pre></div> </div> Ideals form a complete lattice structure with the inclusion relation, as well as a semiring structure. These two structures interact nicely. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {R : Type*} [CommRing R] {I J : Ideal R} + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {R : Type*} [CommRing R] {I J : Ideal R} - example : I + J = I ⊔ J := rfl + example : I + J = I ⊔ J := rfl - example {x : R} : x ∈ I + J ↔ ∃ a ∈ I, ∃ b ∈ J, a + b = x := by - simp [Submodule.mem_sup] + example {x : R} : x ∈ I + J ↔ ∃ a ∈ I, ∃ b ∈ J, a + b = x := by + simp [Submodule.mem_sup] - example : I * J ≤ J := Ideal.mul_le_left + example : I * J ≤ J := Ideal.mul_le_left - example : I * J ≤ I := Ideal.mul_le_right + example : I * J ≤ I := Ideal.mul_le_right - example : I * J ≤ I ⊓ J := Ideal.mul_le_inf + example : I * J ≤ I ⊓ J := Ideal.mul_le_inf </pre></div> </div> One can use ring morphisms to push ideals forward and pull them back using <code class="docutils literal notranslate">Ideal.map</code> and
- @@ -816,128 +814,128 @@ structure. These two structures interact nicely.
 the latter is more convenient to use since it does not involve an existential quantifier. This explains why it is used to state the condition that allows us to build morphisms between quotient rings. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R S : Type*} [CommRing R] [CommRing S] (I : Ideal R) (J : Ideal S) (f : R →+* S) - (H : I ≤ Ideal.comap f J) : R ⧸ I →+* S ⧸ J := - Ideal.quotientMap J f H + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R S : Type*} [CommRing R] [CommRing S] (I : Ideal R) (J : Ideal S) (f : R →+* S) + (H : I ≤ Ideal.comap f J) : R ⧸ I →+* S ⧸ J := + Ideal.quotientMap J f H </pre></div> </div> One subtle point is that the type <code class="docutils literal notranslate">R ⧸ I</code> really depends on <code class="docutils literal notranslate">I</code> (up to definitional equality), so having a proof that two ideals <code class="docutils literal notranslate">I</code> and <code class="docutils literal notranslate">J</code> are equal is not enough to make the corresponding quotients equal. However, the universal properties do provide an isomorphism in this case. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J := - Ideal.quotEquivOfEq h + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J := + Ideal.quotEquivOfEq h </pre></div> </div> We can now present the Chinese remainder isomorphism as an example. Pay attention to the difference between the indexed infimum symbol <code class="docutils literal notranslate">⨅</code> and the big product of types symbol <code class="docutils literal notranslate">Π</code>. Depending on your font, those can be pretty hard to distinguish. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] (f : ι → Ideal R) - (hf : ∀ i j, i ≠ j → IsCoprime (f i) (f j)) : (R ⧸ ⨅ i, f i) ≃+* Π i, R ⧸ f i := - Ideal.quotientInfRingEquivPiQuotient f hf + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] (f : ι → Ideal R) + (hf : ∀ i j, i ≠ j → IsCoprime (f i) (f j)) : (R ⧸ ⨅ i, f i) ≃+* Π i, R ⧸ f i := + Ideal.quotientInfRingEquivPiQuotient f hf </pre></div> </div> The elementary version of the Chinese remainder theorem, a statement about <code class="docutils literal notranslate">ZMod</code>, can be easily deduced from the previous one: - <div class="highlight-lean notranslate"><div class="highlight"><pre>open BigOperators PiNotation + <div class="highlight-lean notranslate"><div class="highlight"><pre>open BigOperators PiNotation - example {ι : Type*} [Fintype ι] (a : ι → ℕ) (coprime : ∀ i j, i ≠ j → (a i).Coprime (a j)) : - ZMod (∏ i, a i) ≃+* Π i, ZMod (a i) := - ZMod.prodEquivPi a coprime + example {ι : Type*} [Fintype ι] (a : ι → ℕ) (coprime : ∀ i j, i ≠ j → (a i).Coprime (a j)) : + ZMod (∏ i, a i) ≃+* Π i, ZMod (a i) := + ZMod.prodEquivPi a coprime </pre></div> </div> As a series of exercises, we will reprove the Chinese remainder theorem in the general case. We first need to define the map appearing in the theorem, as a ring morphism, using the universal property of quotient rings. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {ι R : Type*} [CommRing R] - open Ideal Quotient Function + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {ι R : Type*} [CommRing R] + open Ideal Quotient Function - #check Pi.ringHom - #check ker_Pi_Quotient_mk + #check Pi.ringHom + #check ker_Pi_Quotient_mk /-- The homomorphism from ``R ⧸ ⨅ i, I i`` to ``Π i, R ⧸ I i`` featured in the Chinese Remainder Theorem. -/ - def chineseMap (I : ι → Ideal R) : (R ⧸ ⨅ i, I i) →+* Π i, R ⧸ I i := - sorry + def chineseMap (I : ι → Ideal R) : (R ⧸ ⨅ i, I i) →+* Π i, R ⧸ I i := + sorry </pre></div> </div> Make sure the following next two lemmas can be proven by <code class="docutils literal notranslate">rfl</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>lemma chineseMap_mk (I : ι → Ideal R) (x : R) : - chineseMap I (Quotient.mk _ x) = fun i : ι ↦ Ideal.Quotient.mk (I i) x := - sorry + <div class="highlight-lean notranslate"><div class="highlight"><pre>lemma chineseMap_mk (I : ι → Ideal R) (x : R) : + chineseMap I (Quotient.mk _ x) = fun i : ι ↦ Ideal.Quotient.mk (I i) x := + sorry - lemma chineseMap_mk' (I : ι → Ideal R) (x : R) (i : ι) : - chineseMap I (mk _ x) i = mk (I i) x := - sorry + lemma chineseMap_mk' (I : ι → Ideal R) (x : R) (i : ι) : + chineseMap I (mk _ x) i = mk (I i) x := + sorry </pre></div> </div> The next lemma proves the easy half of the Chinese remainder theorem, without any assumption on the family of ideals. The proof is less than one line long. - <div class="highlight-lean notranslate"><div class="highlight"><pre>#check injective_lift_iff + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check injective_lift_iff - lemma chineseMap_inj (I : ι → Ideal R) : Injective (chineseMap I) := by - sorry + lemma chineseMap_inj (I : ι → Ideal R) : Injective (chineseMap I) := by + sorry </pre></div> </div> We are now ready for the heart of the theorem, which will show the surjectivity of our <code class="docutils literal notranslate">chineseMap</code>. First we need to know the different ways one can express the coprimality (also called co-maximality assumption). Only the first two will be needed below. - <div class="highlight-lean notranslate"><div class="highlight"><pre>#check IsCoprime - #check isCoprime_iff_add - #check isCoprime_iff_exists - #check isCoprime_iff_sup_eq - #check isCoprime_iff_codisjoint + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check IsCoprime + #check isCoprime_iff_add + #check isCoprime_iff_exists + #check isCoprime_iff_sup_eq + #check isCoprime_iff_codisjoint </pre></div> </div> We take the opportunity to use induction on <code class="docutils literal notranslate">Finset</code>. Relevant lemmas on <code class="docutils literal notranslate">Finset</code> are given below. Remember that the <code class="docutils literal notranslate">ring</code> tactic works for semirings and that the ideals of a ring form a semiring. - <div class="highlight-lean notranslate"><div class="highlight"><pre>#check Finset.mem_insert_of_mem - #check Finset.mem_insert_self - - theorem isCoprime_Inf {I : Ideal R} {J : ι → Ideal R} {s : Finset ι} - (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by - classical - simp_rw [isCoprime_iff_add] at * - induction s using Finset.induction with - | empty => - simp - | @insert i s _ hs => - rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] - set K := ⨅ j ∈ s, J j - calc - 1 = I + K := sorry - _ = I + K * (I + J i) := sorry - _ = (1 + K) * I + K * J i := sorry - _ ≤ I + K ⊓ J i := sorry + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check Finset.mem_insert_of_mem + #check Finset.mem_insert_self + + theorem isCoprime_Inf {I : Ideal R} {J : ι → Ideal R} {s : Finset ι} + (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by + classical + simp_rw [isCoprime_iff_add] at * + induction s using Finset.induction with + | empty => + simp + | @insert i s _ hs => + rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] + set K := ⨅ j ∈ s, J j + calc + 1 = I + K := sorry + _ = I + K * (I + J i) := sorry + _ = (1 + K) * I + K * J i := sorry + _ ≤ I + K ⊓ J i := sorry </pre></div> </div> We can now prove surjectivity of the map appearing in the Chinese remainder theorem. - <div class="highlight-lean notranslate"><div class="highlight"><pre>lemma chineseMap_surj [Fintype ι] {I : ι → Ideal R} - (hI : ∀ i j, i ≠ j → IsCoprime (I i) (I j)) : Surjective (chineseMap I) := by - classical - intro g - choose f hf using fun i ↦ Ideal.Quotient.mk_surjective (g i) - have key : ∀ i, ∃ e : R, mk (I i) e = 1 ∧ ∀ j, j ≠ i → mk (I j) e = 0 := by - intro i - have hI' : ∀ j ∈ ({i} : Finset ι)ᶜ, IsCoprime (I i) (I j) := by - sorry - sorry - choose e he using key - use mk _ (∑ i, f i * e i) - sorry + <div class="highlight-lean notranslate"><div class="highlight"><pre>lemma chineseMap_surj [Fintype ι] {I : ι → Ideal R} + (hI : ∀ i j, i ≠ j → IsCoprime (I i) (I j)) : Surjective (chineseMap I) := by + classical + intro g + choose f hf using fun i ↦ Ideal.Quotient.mk_surjective (g i) + have key : ∀ i, ∃ e : R, mk (I i) e = 1 ∧ ∀ j, j ≠ i → mk (I j) e = 0 := by + intro i + have hI' : ∀ j ∈ ({i} : Finset ι)ᶜ, IsCoprime (I i) (I j) := by + sorry + sorry + choose e he using key + use mk _ (∑ i, f i * e i) + sorry </pre></div> </div> Now all the pieces come together in the following: - <div class="highlight-lean notranslate"><div class="highlight"><pre>noncomputable def chineseIso [Fintype ι] (f : ι → Ideal R) - (hf : ∀ i j, i ≠ j → IsCoprime (f i) (f j)) : (R ⧸ ⨅ i, f i) ≃+* Π i, R ⧸ f i := - { Equiv.ofBijective _ ⟨chineseMap_inj f, chineseMap_surj hf⟩, - chineseMap f with } + <div class="highlight-lean notranslate"><div class="highlight"><pre>noncomputable def chineseIso [Fintype ι] (f : ι → Ideal R) + (hf : ∀ i j, i ≠ j → IsCoprime (f i) (f j)) : (R ⧸ ⨅ i, f i) ≃+* Π i, R ⧸ f i := + { Equiv.ofBijective _ ⟨chineseMap_inj f, chineseMap_surj hf⟩, + chineseMap f with } </pre></div> </div> </section> <section id="algebras-and-polynomials"> - <h3>8.2.3. Algebras and polynomials<a class="headerlink" href="#algebras-and-polynomials" title="Link to this heading"></a></h3> + <h3>8.2.3. Algebras and polynomials<a class="headerlink" href="#algebras-and-polynomials" title="Permalink to this heading"></a></h3> Given a commutative (semi)ring <code class="docutils literal notranslate">R</code>, an algebra over <code class="docutils literal notranslate">R</code> is a semiring <code class="docutils literal notranslate">A</code> equipped with a ring morphism whose image commutes with every element of <code class="docutils literal notranslate">A</code>. This is encoded as a type class <code class="docutils literal notranslate">Algebra R A</code>.
- @@ -949,13 +947,13 @@ Note that this notion of algebra is sometimes called an associative unital a
 existence of more general notions of algebra. The fact that <code class="docutils literal notranslate">algebraMap R A</code> is ring morphism packages together a lot of properties of scalar multiplication, such as the following: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R A : Type*} [CommRing R] [Ring A] [Algebra R A] (r r' : R) (a : A) : - (r + r') • a = r • a + r' • a := - add_smul r r' a + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R A : Type*} [CommRing R] [Ring A] [Algebra R A] (r r' : R) (a : A) : + (r + r') • a = r • a + r' • a := + add_smul r r' a - example {R A : Type*} [CommRing R] [Ring A] [Algebra R A] (r r' : R) (a : A) : - (r * r') • a = r • r' • a := - mul_smul r r' a + example {R A : Type*} [CommRing R] [Ring A] [Algebra R A] (r r' : R) (a : A) : + (r * r') • a = r • r' • a := + mul_smul r r' a </pre></div> </div> The morphisms between two <code class="docutils literal notranslate">R</code>-algebras <code class="docutils literal notranslate">A</code> and <code class="docutils literal notranslate">B</code> are ring morphisms
- @@ -970,11 +968,11 @@ which can be written as <code class="docutils literal notranslate">R</code> to <code class="docutils literal notranslate">R[X]</code> is denoted by <code class="docutils literal notranslate">C</code>, which stands for “constant” since the corresponding polynomial functions are always constant. The indeterminate is denoted by <code class="docutils literal notranslate">X</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>open Polynomial + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Polynomial - example {R : Type*} [CommRing R] : R[X] := X + example {R : Type*} [CommRing R] : R[X] := X - example {R : Type*} [CommRing R] (r : R) := X - C r + example {R : Type*} [CommRing R] (r : R) := X - C r </pre></div> </div> In the first example above, it is crucial that we give Lean the expected type since it cannot be
- @@ -983,15 +981,15 @@ algebra can be inferred from our use of <code class="docutils literal notranslat
 Because <code class="docutils literal notranslate">C</code> is a ring morphism from <code class="docutils literal notranslate">R</code> to <code class="docutils literal notranslate">R[X]</code>, we can use all ring morphisms lemmas such as <code class="docutils literal notranslate">map_zero</code>, <code class="docutils literal notranslate">map_one</code>, <code class="docutils literal notranslate">map_mul</code>, and <code class="docutils literal notranslate">map_pow</code> before computing in the ring <code class="docutils literal notranslate">R[X]</code>. For example: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] (r : R) : (X + C r) * (X - C r) = X ^ 2 - C (r ^ 2) := by - rw [C.map_pow] - ring + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] (r : R) : (X + C r) * (X - C r) = X ^ 2 - C (r ^ 2) := by + rw [C.map_pow] + ring </pre></div> </div> You can access coefficients using <code class="docutils literal notranslate">Polynomial.coeff</code> - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] (r:R) : (C r).coeff 0 = r := by simp + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] (r:R) : (C r).coeff 0 = r := by simp - example {R : Type*} [CommRing R] : (X ^ 2 + 2 * X + C 3 : R[X]).coeff 1 = 2 := by simp + example {R : Type*} [CommRing R] : (X ^ 2 + 2 * X + C 3 : R[X]).coeff 1 = 2 := by simp </pre></div> </div> Defining the degree of a polynomial is always tricky because of the special case of the zero
- @@ -1003,36 +1001,36 @@ degree of the zero polynomial, and it is absorbent for addition. (It is almost a
 multiplication, except that <code class="docutils literal notranslate">⊥ * 0 = 0</code>.) Morally speaking, the <code class="docutils literal notranslate">degree</code> version is the correct one. For instance, it allows us to state the expected formula for the degree of a product (assuming the base ring has no zero divisor). - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [Semiring R] [NoZeroDivisors R] {p q : R[X]} : - degree (p * q) = degree p + degree q := - Polynomial.degree_mul + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [Semiring R] [NoZeroDivisors R] {p q : R[X]} : + degree (p * q) = degree p + degree q := + Polynomial.degree_mul </pre></div> </div> Whereas the version for <code class="docutils literal notranslate">natDegree</code> needs to assume non-zero polynomials. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [Semiring R] [NoZeroDivisors R] {p q : R[X]} (hp : p ≠ 0) (hq : q ≠ 0) : - natDegree (p * q) = natDegree p + natDegree q := - Polynomial.natDegree_mul hp hq + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [Semiring R] [NoZeroDivisors R] {p q : R[X]} (hp : p ≠ 0) (hq : q ≠ 0) : + natDegree (p * q) = natDegree p + natDegree q := + Polynomial.natDegree_mul hp hq </pre></div> </div> However, <code class="docutils literal notranslate">ℕ</code> is much nicer to use than <code class="docutils literal notranslate">WithBot ℕ</code>, so Mathlib makes both versions available and provides lemmas to convert between them. Also, <code class="docutils literal notranslate">natDegree</code> is the more convenient definition to use when computing the degree of a composition. Composition of polynomial is <code class="docutils literal notranslate">Polynomial.comp</code> and we have: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [Semiring R] [NoZeroDivisors R] {p q : R[X]} : - natDegree (comp p q) = natDegree p * natDegree q := - Polynomial.natDegree_comp + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [Semiring R] [NoZeroDivisors R] {p q : R[X]} : + natDegree (comp p q) = natDegree p * natDegree q := + Polynomial.natDegree_comp </pre></div> </div> Polynomials give rise to polynomial functions: any polynomial can be evaluated on <code class="docutils literal notranslate">R</code> using <code class="docutils literal notranslate">Polynomial.eval</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] (P: R[X]) (x : R) := P.eval x + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] (P: R[X]) (x : R) := P.eval x - example {R : Type*} [CommRing R] (r : R) : (X - C r).eval r = 0 := by simp + example {R : Type*} [CommRing R] (r : R) : (X - C r).eval r = 0 := by simp </pre></div> </div> In particular, there is a predicate, <code class="docutils literal notranslate">IsRoot</code>, that holds for elements <code class="docutils literal notranslate">r</code> in <code class="docutils literal notranslate">R</code> where a polynomial vanishes. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] (P : R[X]) (r : R) : IsRoot P r ↔ P.eval r = 0 := Iff.rfl + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] (P : R[X]) (r : R) : IsRoot P r ↔ P.eval r = 0 := Iff.rfl </pre></div> </div> We would like to say that, assuming <code class="docutils literal notranslate">R</code> has no zero divisor, a polynomial has at most as many
- @@ -1042,12 +1040,12 @@ So Mathlib defines <code class="docutils literal notranslate">
 i.e. the finite set that is defined to be empty if <code class="docutils literal notranslate">P</code> is zero and the roots of <code class="docutils literal notranslate">P</code>, with multiplicities, otherwise. This is defined only when the underlying ring is a domain since otherwise the definition does not have good properties. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] [IsDomain R] (r : R) : (X - C r).roots = {r} := - roots_X_sub_C r + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R : Type*} [CommRing R] [IsDomain R] (r : R) : (X - C r).roots = {r} := + roots_X_sub_C r - example {R : Type*} [CommRing R] [IsDomain R] (r : R) (n : ℕ): - ((X - C r) ^ n).roots = n • {r} := - by simp + example {R : Type*} [CommRing R] [IsDomain R] (r : R) (n : ℕ): + ((X - C r) ^ n).roots = n • {r} := + by simp </pre></div> </div> Both <code class="docutils literal notranslate">Polynomial.eval</code> and <code class="docutils literal notranslate">Polynomial.roots</code> consider only the coefficients ring. They do not
- @@ -1058,38 +1056,38 @@ every element of <code class="docutils literal notranslate">a<
 has a coercion to functions, one can apply it to a polynomial. But <code class="docutils literal notranslate">aeval</code> does not have a polynomial as an argument, so one cannot use dot notation like in <code class="docutils literal notranslate">P.eval</code> above. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : aeval Complex.I (X ^ 2 + 1 : ℝ[X]) = 0 := by simp + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : aeval Complex.I (X ^ 2 + 1 : ℝ[X]) = 0 := by simp </pre></div> </div> The function corresponding to <code class="docutils literal notranslate">roots</code> in this context is <code class="docutils literal notranslate">aroots</code> which takes a polynomial and then an algebra and outputs a multiset (with the same caveat about the zero polynomial as for <code class="docutils literal notranslate">roots</code>). - <div class="highlight-lean notranslate"><div class="highlight"><pre>open Complex Polynomial - - example : aroots (X ^ 2 + 1 : ℝ[X]) ℂ = {Complex.I, -I} := by - suffices roots (X ^ 2 + 1 : ℂ[X]) = {I, -I} by simpa [aroots_def] - have factored : (X ^ 2 + 1 : ℂ[X]) = (X - C I) * (X - C (-I)) := by - have key : (C I * C I : ℂ[X]) = -1 := by simp [← C_mul] - rw [C_neg] - linear_combination key - have p_ne_zero : (X - C I) * (X - C (-I)) ≠ 0 := by - intro H - apply_fun eval 0 at H - simp [eval] at H - simp only [factored, roots_mul p_ne_zero, roots_X_sub_C] - rfl + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Complex Polynomial + + example : aroots (X ^ 2 + 1 : ℝ[X]) ℂ = {Complex.I, -I} := by + suffices roots (X ^ 2 + 1 : ℂ[X]) = {I, -I} by simpa [aroots_def] + have factored : (X ^ 2 + 1 : ℂ[X]) = (X - C I) * (X - C (-I)) := by + have key : (C I * C I : ℂ[X]) = -1 := by simp [← C_mul] + rw [C_neg] + linear_combination key + have p_ne_zero : (X - C I) * (X - C (-I)) ≠ 0 := by + intro H + apply_fun eval 0 at H + simp [eval] at H + simp only [factored, roots_mul p_ne_zero, roots_X_sub_C] + rfl -- Mathlib knows about D'Alembert-Gauss theorem: ``ℂ`` is algebraically closed. - example : IsAlgClosed ℂ := inferInstance + example : IsAlgClosed ℂ := inferInstance </pre></div> </div> More generally, given an ring morphism <code class="docutils literal notranslate">f : R →+* S</code> one can evaluate <code class="docutils literal notranslate">P : R[X]</code> at a point in <code class="docutils literal notranslate">S</code> using <code class="docutils literal notranslate">Polynomial.eval₂</code>. This one produces an actual function from <code class="docutils literal notranslate">R[X]</code> to <code class="docutils literal notranslate">S</code> since it does not assume the existence of a <code class="docutils literal notranslate">Algebra R S</code> instance, so dot notation works as you would expect. - <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (Complex.ofReal : ℝ →+* ℂ) + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (Complex.ofReal : ℝ →+* ℂ) - example : (X ^ 2 + 1 : ℝ[X]).eval₂ Complex.ofReal Complex.I = 0 := by simp + example : (X ^ 2 + 1 : ℝ[X]).eval₂ Complex.ofReal Complex.I = 0 := by simp </pre></div> </div> Let us end by mentioning multivariate polynomials briefly. Given a commutative semiring <code class="docutils literal notranslate">R</code>,
- @@ -1099,9 +1097,9 @@ a type <code class="docutils literal notranslate">σX i</code>.) For instance, if we want two indeterminates we can use <code class="docutils literal notranslate">Fin 2</code> as <code class="docutils literal notranslate">σ</code> and write the polynomial defining the unit circle in $\mathbb{R}^2`$ as: - <div class="highlight-lean notranslate"><div class="highlight"><pre>open MvPolynomial + <div class="highlight-lean notranslate"><div class="highlight"><pre>open MvPolynomial - def circleEquation : MvPolynomial (Fin 2) ℝ := X 0 ^ 2 + X 1 ^ 2 - 1 + def circleEquation : MvPolynomial (Fin 2) ℝ := X 0 ^ 2 + X 1 ^ 2 - 1 </pre></div> </div> Recall that function application has a very high precedence so the expression above is read as
- @@ -1109,7 +1107,7 @@ For instance, if we want two indeterminates we can use
 We can evaluate it to make sure the point with coordinates $(1, 0)$ is on the circle. Recall the <code class="docutils literal notranslate">![...]</code> notation denotes elements of <code class="docutils literal notranslate">Fin n → X</code> for some natural number <code class="docutils literal notranslate">n</code> determined by the number of arguments and some type <code class="docutils literal notranslate">X</code> determined by the type of arguments. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : MvPolynomial.eval ![0, 1] circleEquation = 0 := by simp [circleEquation] + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : MvPolynomial.eval ![0, 1] circleEquation = 0 := by simp [circleEquation] </pre></div> </div> </section>
html/C09_Linear_Algebra.html
- @@ -1,25 +1,23 @@
 <!DOCTYPE html> - <html class="writer-html5" lang="en" data-content_root="./"> + <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="viewport" content="width=device-width, initial-scale=1" /> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>9. Linear algebra — Mathematics in Lean 0.1 documentation</title> - <link rel="stylesheet" type="text/css" href="_static/pygments.css?v=80d5e7a1" /> - <link rel="stylesheet" type="text/css" href="_static/css/theme.css?v=19f00094" /> - <link rel="stylesheet" type="text/css" href="_static/css/custom.css?v=0731ccc3" /> - - + <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?v=5d32c60e"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js?v=2cd50e6c"></script> - <script src="_static/documentation_options.js?v=2709fde1"></script> - <script src="_static/doctools.js?v=9a2dae69"></script> - <script src="_static/sphinx_highlight.js?v=dc90522c"></script> + <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 async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" />
- @@ -113,19 +111,19 @@
 <div itemprop="articleBody"> <section id="linear-algebra"> - <h1>9. Linear algebra<a class="headerlink" href="#linear-algebra" title="Link to this heading"></a></h1> + <h1>9. Linear algebra<a class="headerlink" href="#linear-algebra" title="Permalink to this heading"></a></h1> <section id="vector-spaces-and-linear-maps"> - <h2>9.1. Vector spaces and linear maps<a class="headerlink" href="#vector-spaces-and-linear-maps" title="Link to this heading"></a></h2> + <h2>9.1. Vector spaces and linear maps<a class="headerlink" href="#vector-spaces-and-linear-maps" title="Permalink to this heading"></a></h2> <section id="vector-spaces"> - <h3>9.1.1. Vector spaces<a class="headerlink" href="#vector-spaces" title="Link to this heading"></a></h3> + <h3>9.1.1. Vector spaces<a class="headerlink" href="#vector-spaces" title="Permalink to this heading"></a></h3> We will start directly abstract linear algebra, taking place in a vector space over any field. However you can find information about matrices in <a class="reference internal" href="#matrices">Section 9.4.1</a> which does not logically depend on this abstract theory. Mathlib actually deals with a more general version of linear algebra involving the word module, - but for now we will pretend this is only an excentric spelling habit. + but for now we will pretend this is only an eccentric spelling habit. The way to say “let $K$ be a field and let $V$ be a vector space over $K$” (and make them implicit arguments to later results) is: - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {K : Type*} [Field K] {V : Type*} [AddCommGroup V] [Module K V] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {K : Type*} [Field K] {V : Type*} [AddCommGroup V] [Module K V] </pre></div> </div> We explained in <a class="reference internal" href="C07_Hierarchies.html#hierarchies">Chapter 7</a> why we need two separate
- @@ -135,7 +133,7 @@ Mathematically we want to say that having a <span class="math notranslate nohigh
 implies having an additive commutative group structure. We could tell this to Lean. But then whenever Lean would need to find such a group structure on a type $V$, it would go hunting for vector space - structures using a completely unspecified field $K$ that cannot be infered + structures using a completely unspecified field $K$ that cannot be inferred from $V$. This would be very bad for the type class synthesis system. The multiplication of a vector <cite>v</cite> by a scalar <cite>a</cite> is denoted by
- @@ -146,14 +144,14 @@ following from the axioms of vector spaces and fields, in the same way the
 <cite>ring</cite> tactic is used in commutative rings or the <cite>group</cite> tactic is used in groups. But it is still useful to remember than scalar multiplication is abbreviated <cite>smul</cite> in lemma names. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a : K) (u v : V) : a • (u + v) = a • u + a • v := - smul_add a u v + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a : K) (u v : V) : a • (u + v) = a • u + a • v := + smul_add a u v - example (a b : K) (u : V) : (a + b) • u = a • u + b • u := - add_smul a b u + example (a b : K) (u : V) : (a + b) • u = a • u + b • u := + add_smul a b u - example (a b : K) (u : V) : a • b • u = b • a • u := - smul_comm a b u + example (a b : K) (u : V) : a • b • u = b • a • u := + smul_comm a b u </pre></div> </div> As a quick note for more advanced readers, let us point out that, as suggested by
- @@ -162,14 +160,14 @@ rings.
 In fact it even covers semi-modules over semi-rings. If you think you do not need this level of generality, you can meditate the following example that nicely captures a lot of algebraic rules about ideals acting on submodules: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R M : Type*} [CommSemiring R] [AddCommMonoid M] [Module R M] : - Module (Ideal R) (Submodule R M) := - inferInstance + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {R M : Type*} [CommSemiring R] [AddCommMonoid M] [Module R M] : + Module (Ideal R) (Submodule R M) := + inferInstance </pre></div> </div> </section> <section id="linear-maps"> - <h3>9.1.2. Linear maps<a class="headerlink" href="#linear-maps" title="Link to this heading"></a></h3> + <h3>9.1.2. Linear maps<a class="headerlink" href="#linear-maps" title="Permalink to this heading"></a></h3> Next we need linear maps. Like group morphisms, linear maps in Mathlib are bundled maps, ie packages made of a map and proofs of its linearity properties. Those bundled maps are converted to ordinary functions when applied.
- @@ -181,31 +179,31 @@ But this is crucial when several fields come into play.
 For instance real-linear maps from $ℂ$ to $ℂ$ are every map $z ↦ az + b\bar{z}$ while only the maps $z ↦ az$ are complex linear, and this difference is crucial in complex analysis. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {W : Type*} [AddCommGroup W] [Module K W] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {W : Type*} [AddCommGroup W] [Module K W] - variable (φ : V →ₗ[K] W) + variable (φ : V →ₗ[K] W) - example (a : K) (v : V) : φ (a • v) = a • φ v := - map_smul φ a v + example (a : K) (v : V) : φ (a • v) = a • φ v := + map_smul φ a v - example (v w : V) : φ (v + w) = φ v + φ w := - map_add φ v w + example (v w : V) : φ (v + w) = φ v + φ w := + map_add φ v w </pre></div> </div> Note that <code class="docutils literal notranslate">V →ₗ[K] W</code> itself carries interesting algebraic structures (this is part of the motivation for bundling those maps). It is a <code class="docutils literal notranslate">K</code>-vector space so we can add linear maps and multiply them by scalars. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (ψ : V →ₗ[K] W) + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (ψ : V →ₗ[K] W) - #check (2 • φ + ψ : V →ₗ[K] W) + #check (2 • φ + ψ : V →ₗ[K] W) </pre></div> </div> One down-side of using bundled maps is that we cannot use ordinary function composition. We need to use <code class="docutils literal notranslate">LinearMap.comp</code> or the notation <code class="docutils literal notranslate">∘ₗ</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (θ : W →ₗ[K] V) + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (θ : W →ₗ[K] V) - #check (φ.comp θ : W →ₗ[K] W) - #check (φ ∘ₗ θ : W →ₗ[K] W) + #check (φ.comp θ : W →ₗ[K] W) + #check (φ ∘ₗ θ : W →ₗ[K] W) </pre></div> </div> There are two main ways to construct linear maps.
- @@ -213,10 +211,10 @@ First we can build the structure by providing the function and the linearity pro
 As usual, this is facilitated by the structure code action: you can type <code class="docutils literal notranslate">example : V →ₗ[K] V := _</code> and use the code action “Generate a skeleton” attached to the underscore. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : V →ₗ[K] V where - toFun v := 3 • v - map_add' _ _ := smul_add .. - map_smul' _ _ := smul_comm .. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : V →ₗ[K] V where + toFun v := 3 • v + map_add' _ _ := smul_add .. + map_smul' _ _ := smul_comm .. </pre></div> </div> You may wonder why the proof fields of <code class="docutils literal notranslate">LinearMap</code> have names ending with a prime.
- @@ -226,12 +224,12 @@ and <code class="docutils literal notranslate">LinearMap.map_s
 This is not yet the end of the story. One also want a version of <code class="docutils literal notranslate">map_add</code> that applies to any (bundled) map preserving addition, such as additive group morphisms, linear maps, continuous linear maps, <code class="docutils literal notranslate">K</code>-algebra maps etc… This one is <code class="docutils literal notranslate">map_add</code> (in the root namespace). - The intermediate version, <code class="docutils literal notranslate">LinearMap.map_add</code> is a bit redudant but allows to use dot notation, which + The intermediate version, <code class="docutils literal notranslate">LinearMap.map_add</code> is a bit redundant but allows to use dot notation, which can be nice sometimes. A similar story exists for <code class="docutils literal notranslate">map_smul</code>, and the general framework is explained in <a class="reference internal" href="C07_Hierarchies.html#hierarchies">Chapter 7</a>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (φ.map_add' : ∀ x y : V, φ.toFun (x + y) = φ.toFun x + φ.toFun y) - #check (φ.map_add : ∀ x y : V, φ (x + y) = φ x + φ y) - #check (map_add φ : ∀ x y : V, φ (x + y) = φ x + φ y) + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (φ.map_add' : ∀ x y : V, φ.toFun (x + y) = φ.toFun x + φ.toFun y) + #check (φ.map_add : ∀ x y : V, φ (x + y) = φ x + φ y) + #check (map_add φ : ∀ x y : V, φ (x + y) = φ x + φ y) </pre></div> </div> One can also build linear maps from the ones that are already defined in Mathlib
- @@ -243,8 +241,8 @@ for Lean to infer <code class="docutils literal notranslate">V
 But also <code class="docutils literal notranslate">LinearMap.lsmul K V</code> is an interesting object by itself: it has type <code class="docutils literal notranslate">K →ₗ[K] V →ₗ[K] V</code>, meaning it is a <code class="docutils literal notranslate">K</code>-linear map from <code class="docutils literal notranslate">K</code> —seen as a vector space over itself— to the space of <code class="docutils literal notranslate">K</code>-linear maps from <code class="docutils literal notranslate">V</code> to <code class="docutils literal notranslate">V</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (LinearMap.lsmul K V 3 : V →ₗ[K] V) - #check (LinearMap.lsmul K V : K →ₗ[K] V →ₗ[K] V) + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (LinearMap.lsmul K V 3 : V →ₗ[K] V) + #check (LinearMap.lsmul K V : K →ₗ[K] V →ₗ[K] V) </pre></div> </div> There is also a type <code class="docutils literal notranslate">LinearEquiv</code> of linear isomorphisms denoted by <code class="docutils literal notranslate">V ≃ₗ[K] W</code>.
- @@ -252,21 +250,21 @@ The inverse of <code class="docutils literal notranslate">f</s
 composition of <code class="docutils literal notranslate">f</code> and <code class="docutils literal notranslate">g</code> is <code class="docutils literal notranslate">f.trans g</code> also denoted by <code class="docutils literal notranslate">f ≪≫ₗ g</code>, and the identity isomorphism of <code class="docutils literal notranslate">V</code> is <code class="docutils literal notranslate">LinearEquiv.refl K V</code>. Elements of this type are automatically coerced to morphisms and functions when necessary. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (f : V ≃ₗ[K] W) : f ≪≫ₗ f.symm = LinearEquiv.refl K V := - f.self_trans_symm + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (f : V ≃ₗ[K] W) : f ≪≫ₗ f.symm = LinearEquiv.refl K V := + f.self_trans_symm </pre></div> </div> One can use <code class="docutils literal notranslate">LinearEquiv.ofBijective</code> to build an isomorphism from a bijective morphism. Doing so makes the inverse function noncomputable. - <div class="highlight-lean notranslate"><div class="highlight"><pre>noncomputable example (f : V →ₗ[K] W) (h : Function.Bijective f) : V ≃ₗ[K] W := - .ofBijective f h + <div class="highlight-lean notranslate"><div class="highlight"><pre>noncomputable example (f : V →ₗ[K] W) (h : Function.Bijective f) : V ≃ₗ[K] W := + .ofBijective f h </pre></div> </div> Note that in the above example, Lean uses the announced type to understand that <code class="docutils literal notranslate">.ofBijective</code> refers to <code class="docutils literal notranslate">LinearEquiv.ofBijective</code> (without needing to open any namespace). </section> <section id="sums-and-products-of-vector-spaces"> - <h3>9.1.3. Sums and products of vector spaces<a class="headerlink" href="#sums-and-products-of-vector-spaces" title="Link to this heading"></a></h3> + <h3>9.1.3. Sums and products of vector spaces<a class="headerlink" href="#sums-and-products-of-vector-spaces" title="Permalink to this heading"></a></h3> We can build new vector spaces out of old ones using direct sums and direct products. Let us start with two vectors spaces. In this case there is no difference between sum and product,
- @@ -276,57 +274,57 @@ and projections) as linear maps, as well as the universal properties constructin
 into products and out of sums (if you are not familiar with the category-theoretic distinction between sums and products, you can simply ignore the universal property vocabulary and focus on the types of the following examples). - <div class="highlight-lean notranslate"><div class="highlight"><pre>section binary_product + <div class="highlight-lean notranslate"><div class="highlight"><pre>section binary_product - variable {W : Type*} [AddCommGroup W] [Module K W] - variable {U : Type*} [AddCommGroup U] [Module K U] - variable {T : Type*} [AddCommGroup T] [Module K T] + variable {W : Type*} [AddCommGroup W] [Module K W] + variable {U : Type*} [AddCommGroup U] [Module K U] + variable {T : Type*} [AddCommGroup T] [Module K T] -- First projection map - example : V × W →ₗ[K] V := LinearMap.fst K V W + example : V × W →ₗ[K] V := LinearMap.fst K V W -- Second projection map - example : V × W →ₗ[K] W := LinearMap.snd K V W + example : V × W →ₗ[K] W := LinearMap.snd K V W -- Universal property of the product - example (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K] V × W := LinearMap.prod φ ψ + example (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K] V × W := LinearMap.prod φ ψ -- The product map does the expected thing, first component - example (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl + example (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl -- The product map does the expected thing, second component - example (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl + example (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl -- We can also combine maps in parallel - example (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ + example (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ -- This is simply done by combining the projections with the universal property - example (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : - φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl + example (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : + φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl -- First inclusion map - example : V →ₗ[K] V × W := LinearMap.inl K V W + example : V →ₗ[K] V × W := LinearMap.inl K V W -- Second inclusion map - example : W →ₗ[K] V × W := LinearMap.inr K V W + example : W →ₗ[K] V × W := LinearMap.inr K V W -- Universal property of the sum (aka coproduct) - example (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ + example (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ -- The coproduct map does the expected thing, first component - example (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ := - LinearMap.coprod_inl φ ψ + example (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ := + LinearMap.coprod_inl φ ψ -- The coproduct map does the expected thing, second component - example (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ := - LinearMap.coprod_inr φ ψ + example (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ := + LinearMap.coprod_inr φ ψ -- The coproduct map is defined in the expected way - example (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) : - φ.coprod ψ (v, w) = φ v + ψ w := - rfl + example (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) : + φ.coprod ψ (v, w) = φ v + ψ w := + rfl - end binary_product + end binary_product </pre></div> </div> Let us now turn to sums and products of arbitrary families of vector spaces.
- @@ -335,57 +333,57 @@ properties of sums and products.
 Note that the direct sum notation is scoped to the <code class="docutils literal notranslate">DirectSum</code> namespace, and that the universal property of direct sums requires decidable equality on the indexing type (this is somehow an implementation accident). - <div class="highlight-lean notranslate"><div class="highlight"><pre>section families - open DirectSum + <div class="highlight-lean notranslate"><div class="highlight"><pre>section families + open DirectSum - variable {ι : Type*} [DecidableEq ι] - (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module K (V i)] + variable {ι : Type*} [DecidableEq ι] + (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module K (V i)] -- The universal property of the direct sum assembles maps from the summands to build -- a map from the direct sum - example (φ : Π i, (V i →ₗ[K] W)) : (⨁ i, V i) →ₗ[K] W := - DirectSum.toModule K ι W φ + example (φ : Π i, (V i →ₗ[K] W)) : (⨁ i, V i) →ₗ[K] W := + DirectSum.toModule K ι W φ -- The universal property of the direct product assembles maps into the factors -- to build a map into the direct product - example (φ : Π i, (W →ₗ[K] V i)) : W →ₗ[K] (Π i, V i) := - LinearMap.pi φ + example (φ : Π i, (W →ₗ[K] V i)) : W →ₗ[K] (Π i, V i) := + LinearMap.pi φ -- The projection maps from the product - example (i : ι) : (Π j, V j) →ₗ[K] V i := LinearMap.proj i + example (i : ι) : (Π j, V j) →ₗ[K] V i := LinearMap.proj i -- The inclusion maps into the sum - example (i : ι) : V i →ₗ[K] (⨁ i, V i) := DirectSum.lof K ι V i + example (i : ι) : V i →ₗ[K] (⨁ i, V i) := DirectSum.lof K ι V i -- The inclusion maps into the product - example (i : ι) : V i →ₗ[K] (Π i, V i) := LinearMap.single K V i + example (i : ι) : V i →ₗ[K] (Π i, V i) := LinearMap.single K V i -- In case `ι` is a finite type, there is an isomorphism between the sum and product. - example [Fintype ι] : (⨁ i, V i) ≃ₗ[K] (Π i, V i) := - linearEquivFunOnFintype K ι V + example [Fintype ι] : (⨁ i, V i) ≃ₗ[K] (Π i, V i) := + linearEquivFunOnFintype K ι V - end families + end families </pre></div> </div> </section> </section> <section id="subspaces-and-quotients"> - <h2>9.2. Subspaces and quotients<a class="headerlink" href="#subspaces-and-quotients" title="Link to this heading"></a></h2> + <h2>9.2. Subspaces and quotients<a class="headerlink" href="#subspaces-and-quotients" title="Permalink to this heading"></a></h2> <section id="subspaces"> - <h3>9.2.1. Subspaces<a class="headerlink" href="#subspaces" title="Link to this heading"></a></h3> + <h3>9.2.1. Subspaces<a class="headerlink" href="#subspaces" title="Permalink to this heading"></a></h3> Just as linear maps are bundled, a linear subspace of <code class="docutils literal notranslate">V</code> is also a bundled structure consisting of a set in <code class="docutils literal notranslate">V</code>, called the carrier of the subspace, with the relevant closure properties. Again the word module appears instead of vector space because of the more general context that Mathlib actually uses for linear algebra. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {K : Type*} [Field K] {V : Type*} [AddCommGroup V] [Module K V] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {K : Type*} [Field K] {V : Type*} [AddCommGroup V] [Module K V] - example (U : Submodule K V) {x y : V} (hx : x ∈ U) (hy : y ∈ U) : - x + y ∈ U := - U.add_mem hx hy + example (U : Submodule K V) {x y : V} (hx : x ∈ U) (hy : y ∈ U) : + x + y ∈ U := + U.add_mem hx hy - example (U : Submodule K V) {x : V} (hx : x ∈ U) (a : K) : - a • x ∈ U := - U.smul_mem a hx + example (U : Submodule K V) {x : V} (hx : x ∈ U) (a : K) : + a • x ∈ U := + U.smul_mem a hx </pre></div> </div> In the example above, it is important to understand that <code class="docutils literal notranslate">Submodule K V</code> is the type of <code class="docutils literal notranslate">K</code>-linear
- @@ -398,19 +396,19 @@ equal in the same way it is used to prove that two sets are equal.
 To state and prove, for example, that <code class="docutils literal notranslate">ℝ</code> is a <code class="docutils literal notranslate">ℝ</code>-linear subspace of <code class="docutils literal notranslate">ℂ</code>, what we really want is to construct a term of type <code class="docutils literal notranslate">Submodule ℝ ℂ</code> whose projection to <code class="docutils literal notranslate">Set ℂ</code> is <code class="docutils literal notranslate">ℝ</code>, or, more precisely, the image of <code class="docutils literal notranslate">ℝ</code> in <code class="docutils literal notranslate">ℂ</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>noncomputable example : Submodule ℝ ℂ where - carrier := Set.range ((↑) : ℝ → ℂ) - add_mem' := by - rintro _ _ ⟨n, rfl⟩ ⟨m, rfl⟩ - use n + m - simp - zero_mem' := by - use 0 - simp - smul_mem' := by - rintro c - ⟨a, rfl⟩ - use c*a - simp + <div class="highlight-lean notranslate"><div class="highlight"><pre>noncomputable example : Submodule ℝ ℂ where + carrier := Set.range ((↑) : ℝ → ℂ) + add_mem' := by + rintro _ _ ⟨n, rfl⟩ ⟨m, rfl⟩ + use n + m + simp + zero_mem' := by + use 0 + simp + smul_mem' := by + rintro c - ⟨a, rfl⟩ + use c*a + simp </pre></div> </div> The prime at the end of proof fields in <code class="docutils literal notranslate">Submodule</code> are analogous to the one in <code class="docutils literal notranslate">LinearMap</code>.
- @@ -422,32 +420,32 @@ a subspace by a linear map (of course we will see below that Mathlib already kno
 Remember that <code class="docutils literal notranslate">Set.mem_preimage</code> can be used to rewrite a statement involving membership and preimage. This is the only lemma you will need in addition to the lemmas discussed above about <code class="docutils literal notranslate">LinearMap</code> and <code class="docutils literal notranslate">Submodule</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>def preimage {W : Type*} [AddCommGroup W] [Module K W] (φ : V →ₗ[K] W) (H : Submodule K W) : - Submodule K V where - carrier := φ ⁻¹' H - zero_mem' := by - dsimp - sorry - add_mem' := by - sorry - smul_mem' := by - dsimp - sorry + <div class="highlight-lean notranslate"><div class="highlight"><pre>def preimage {W : Type*} [AddCommGroup W] [Module K W] (φ : V →ₗ[K] W) (H : Submodule K W) : + Submodule K V where + carrier := φ ⁻¹' H + zero_mem' := by + dsimp + sorry + add_mem' := by + sorry + smul_mem' := by + dsimp + sorry </pre></div> </div> Using type classes, Mathlib knows that a subspace of a vector space inherits a vector space structure. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (U : Submodule K V) : Module K U := inferInstance + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (U : Submodule K V) : Module K U := inferInstance </pre></div> </div> This example is subtle. The object <code class="docutils literal notranslate">U</code> is not a type, but Lean automatically coerces it to a type by interpreting it as a subtype of <code class="docutils literal notranslate">V</code>. So the above example can be restated more explicitly as: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (U : Submodule K V) : Module K {x : V // x ∈ U} := inferInstance + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (U : Submodule K V) : Module K {x : V // x ∈ U} := inferInstance </pre></div> </div> </section> <section id="complete-lattice-structure-and-internal-direct-sums"> - <h3>9.2.2. Complete lattice structure and internal direct sums<a class="headerlink" href="#complete-lattice-structure-and-internal-direct-sums" title="Link to this heading"></a></h3> + <h3>9.2.2. Complete lattice structure and internal direct sums<a class="headerlink" href="#complete-lattice-structure-and-internal-direct-sums" title="Permalink to this heading"></a></h3> An important benefit of having a type <code class="docutils literal notranslate">Submodule K V</code> instead of a predicate <code class="docutils literal notranslate">IsSubmodule : Set V → Prop</code> is that one can easily endow <code class="docutils literal notranslate">Submodule K V</code> with additional structure. Importantly, it has the structure of a complete lattice structure with respect to
- @@ -457,8 +455,8 @@ use the lattice operation <code class="docutils literal notranslate"> Let us check that the set underlying the infimum of two subspaces is indeed, by definition, their intersection. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (H H' : Submodule K V) : - ((H ⊓ H' : Submodule K V) : Set V) = (H : Set V) ∩ (H' : Set V) := rfl + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (H H' : Submodule K V) : + ((H ⊓ H' : Submodule K V) : Set V) = (H : Set V) ∩ (H' : Set V) := rfl </pre></div> </div> It may look strange to have a different notation for what amounts to the intersection of the
- @@ -466,21 +464,21 @@ underlying sets, but the correspondence does not carry over to the supremum oper
 union, since a union of subspaces is not, in general, a subspace. Instead one needs to use the subspace generated by the union, which is done using <code class="docutils literal notranslate">Submodule.span</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (H H' : Submodule K V) : - ((H ⊔ H' : Submodule K V) : Set V) = Submodule.span K ((H : Set V) ∪ (H' : Set V)) := by - simp [Submodule.span_union] + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (H H' : Submodule K V) : + ((H ⊔ H' : Submodule K V) : Set V) = Submodule.span K ((H : Set V) ∪ (H' : Set V)) := by + simp [Submodule.span_union] </pre></div> </div> Another subtlety is that <code class="docutils literal notranslate">V</code> itself does not have type <code class="docutils literal notranslate">Submodule K V</code>, so we need a way to talk about <code class="docutils literal notranslate">V</code> seen as a subspace of <code class="docutils literal notranslate">V</code>. This is also provided by the lattice structure: the full subspace is the top element of this lattice. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x : V) : x ∈ (⊤ : Submodule K V) := trivial + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x : V) : x ∈ (⊤ : Submodule K V) := trivial </pre></div> </div> Similarly the bottom element of this lattice is the subspace whose only element is the zero element. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x : V) : x ∈ (⊥ : Submodule K V) ↔ x = 0 := Submodule.mem_bot K + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x : V) : x ∈ (⊥ : Submodule K V) ↔ x = 0 := Submodule.mem_bot K </pre></div> </div> In particular we can discuss the case of subspaces that are in (internal) direct sum.
- @@ -488,41 +486,41 @@ In the case of two subspaces, we use the general purpose predicate <code class="
 which makes sense for any bounded partially ordered type. In the case of general families of subspaces we use <code class="docutils literal notranslate">DirectSum.IsInternal</code>. <div class="highlight-lean notranslate"><div class="highlight"><pre>-- If two subspaces are in direct sum then they span the whole space. - example (U V : Submodule K V) (h : IsCompl U V) : - U ⊔ V = ⊤ := h.sup_eq_top + example (U V : Submodule K V) (h : IsCompl U V) : + U ⊔ V = ⊤ := h.sup_eq_top -- If two subspaces are in direct sum then they intersect only at zero. - example (U V : Submodule K V) (h : IsCompl U V) : - U ⊓ V = ⊥ := h.inf_eq_bot + example (U V : Submodule K V) (h : IsCompl U V) : + U ⊓ V = ⊥ := h.inf_eq_bot section - open DirectSum - variable {ι : Type*} [DecidableEq ι] + open DirectSum + variable {ι : Type*} [DecidableEq ι] -- If subspaces are in direct sum then they span the whole space. - example (U : ι → Submodule K V) (h : DirectSum.IsInternal U) : - ⨆ i, U i = ⊤ := h.submodule_iSup_eq_top + example (U : ι → Submodule K V) (h : DirectSum.IsInternal U) : + ⨆ i, U i = ⊤ := h.submodule_iSup_eq_top -- If subspaces are in direct sum then they pairwise intersect only at zero. - example {ι : Type*} [DecidableEq ι] (U : ι → Submodule K V) (h : DirectSum.IsInternal U) - {i j : ι} (hij : i ≠ j) : U i ⊓ U j = ⊥ := - (h.submodule_independent.pairwiseDisjoint hij).eq_bot + example {ι : Type*} [DecidableEq ι] (U : ι → Submodule K V) (h : DirectSum.IsInternal U) + {i j : ι} (hij : i ≠ j) : U i ⊓ U j = ⊥ := + (h.submodule_independent.pairwiseDisjoint hij).eq_bot -- Those conditions characterize direct sums. - #check DirectSum.isInternal_submodule_iff_independent_and_iSup_eq_top + #check DirectSum.isInternal_submodule_iff_independent_and_iSup_eq_top -- The relation with external direct sums: if a family of subspaces is -- in internal direct sum then the map from their external direct sum into `V` -- is a linear isomorphism. - noncomputable example {ι : Type*} [DecidableEq ι] (U : ι → Submodule K V) - (h : DirectSum.IsInternal U) : (⨁ i, U i) ≃ₗ[K] V := - LinearEquiv.ofBijective (coeLinearMap U) h + noncomputable example {ι : Type*} [DecidableEq ι] (U : ι → Submodule K V) + (h : DirectSum.IsInternal U) : (⨁ i, U i) ≃ₗ[K] V := + LinearEquiv.ofBijective (coeLinearMap U) h end </pre></div> </div> </section> <section id="subspace-spanned-by-a-set"> - <h3>9.2.3. Subspace spanned by a set<a class="headerlink" href="#subspace-spanned-by-a-set" title="Link to this heading"></a></h3> + <h3>9.2.3. Subspace spanned by a set<a class="headerlink" href="#subspace-spanned-by-a-set" title="Permalink to this heading"></a></h3> In addition to building subspaces out of existing subspaces, we can build them out of any set <code class="docutils literal notranslate">s</code> using <code class="docutils literal notranslate">Submodule.span K s</code> which builds the smallest subspace containing <code class="docutils literal notranslate">s</code>.
- @@ -530,11 +528,11 @@ On paper it is common to use that this space is made of all linear combinations
 <code class="docutils literal notranslate">s</code>. But it is often more efficient to use its universal property expressed by <code class="docutils literal notranslate">Submodule.span_le</code>, and the whole theory of Galois connections. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {s : Set V} (E : Submodule K V) : Submodule.span K s ≤ E ↔ s ⊆ E := - Submodule.span_le + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {s : Set V} (E : Submodule K V) : Submodule.span K s ≤ E ↔ s ⊆ E := + Submodule.span_le - example : GaloisInsertion (Submodule.span K) ((↑) : Submodule K V → Set V) := - Submodule.gi K V + example : GaloisInsertion (Submodule.span K) ((↑) : Submodule K V → Set V) := + Submodule.gi K V </pre></div> </div> When those are not enough, one can use the relevant induction principle
- @@ -545,28 +543,28 @@ to figure out the predicate we are interested in.
 As an exercise, let us reprove one implication of <code class="docutils literal notranslate">Submodule.mem_sup</code>. Remember that you can use the <cite>module</cite> tactic to close goals that follow from the axioms relating the various algebraic operations on <code class="docutils literal notranslate">V</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {S T : Submodule K V} {x : V} (h : x ∈ S ⊔ T) : - ∃ s ∈ S, ∃ t ∈ T, x = s + t := by - rw [← S.span_eq, ← T.span_eq, ← Submodule.span_union] at h - apply Submodule.span_induction h (p := fun x ↦ ∃ s ∈ S, ∃ t ∈ T, x = s + t) - sorry + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {S T : Submodule K V} {x : V} (h : x ∈ S ⊔ T) : + ∃ s ∈ S, ∃ t ∈ T, x = s + t := by + rw [← S.span_eq, ← T.span_eq, ← Submodule.span_union] at h + apply Submodule.span_induction h (p := fun x ↦ ∃ s ∈ S, ∃ t ∈ T, x = s + t) + sorry </pre></div> </div> </section> <section id="pushing-and-pulling-subspaces"> - <h3>9.2.4. Pushing and pulling subspaces<a class="headerlink" href="#pushing-and-pulling-subspaces" title="Link to this heading"></a></h3> + <h3>9.2.4. Pushing and pulling subspaces<a class="headerlink" href="#pushing-and-pulling-subspaces" title="Permalink to this heading"></a></h3> As promised earlier, we now describe how to push and pull subspaces by linear maps. As usual in Mathlib, the first operation is called <code class="docutils literal notranslate">map</code> and the second one is called <code class="docutils literal notranslate">comap</code>. <div class="highlight-lean notranslate"><div class="highlight"><pre>section - variable {W : Type*} [AddCommGroup W] [Module K W] (φ : V →ₗ[K] W) + variable {W : Type*} [AddCommGroup W] [Module K W] (φ : V →ₗ[K] W) - variable (E : Submodule K V) in - #check (Submodule.map φ E : Submodule K W) + variable (E : Submodule K V) in + #check (Submodule.map φ E : Submodule K W) - variable (F : Submodule K W) in - #check (Submodule.comap φ F : Submodule K V) + variable (F : Submodule K W) in + #check (Submodule.comap φ F : Submodule K V) </pre></div> </div> Note those live in the <code class="docutils literal notranslate">Submodule</code> namespace so one can use dot notation and write
- @@ -574,9 +572,9 @@ As usual in Mathlib, the first operation is called <code class="docutils literal
 Mathlib contributors use this spelling). In particular the range and kernel of a linear map are subspaces. Those special cases are important enough to get declarations. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : LinearMap.range φ = .map φ ⊤ := LinearMap.range_eq_map φ + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : LinearMap.range φ = .map φ ⊤ := LinearMap.range_eq_map φ - example : LinearMap.ker φ = .comap φ ⊥ := Submodule.comap_bot φ -- or `rfl` + example : LinearMap.ker φ = .comap φ ⊥ := Submodule.comap_bot φ -- or `rfl` </pre></div> </div> Note that we cannot write <code class="docutils literal notranslate">φ.ker</code> instead of <code class="docutils literal notranslate">LinearMap.ker φ</code> because <code class="docutils literal notranslate">LinearMap.ker</code> also
- @@ -586,109 +584,109 @@ However we were able to use the other flavor of dot notation in the right-hand s
 Lean expects a term with type <code class="docutils literal notranslate">Submodule K V</code> after elaborating the left-hand side, it interprets <code class="docutils literal notranslate">.comap</code> as <code class="docutils literal notranslate">Submodule.comap</code>. The following lemmas give the key relations between those submodule and the properties of <code class="docutils literal notranslate">φ</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>open Function LinearMap + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Function LinearMap - example : Injective φ ↔ ker φ = ⊥ := ker_eq_bot.symm + example : Injective φ ↔ ker φ = ⊥ := ker_eq_bot.symm - example : Surjective φ ↔ range φ = ⊤ := range_eq_top.symm + example : Surjective φ ↔ range φ = ⊤ := range_eq_top.symm </pre></div> </div> As an exercise, let us prove the Galois connection property for <code class="docutils literal notranslate">map</code> and <code class="docutils literal notranslate">comap</code>. One can use the following lemmas but this is not required since they are true by definition. - <div class="highlight-lean notranslate"><div class="highlight"><pre>#check Submodule.mem_map_of_mem - #check Submodule.mem_map - #check Submodule.mem_comap + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check Submodule.mem_map_of_mem + #check Submodule.mem_map + #check Submodule.mem_comap - example (E : Submodule K V) (F : Submodule K W) : - Submodule.map φ E ≤ F ↔ E ≤ Submodule.comap φ F := by - sorry + example (E : Submodule K V) (F : Submodule K W) : + Submodule.map φ E ≤ F ↔ E ≤ Submodule.comap φ F := by + sorry </pre></div> </div> </section> <section id="quotient-spaces"> - <h3>9.2.5. Quotient spaces<a class="headerlink" href="#quotient-spaces" title="Link to this heading"></a></h3> + <h3>9.2.5. Quotient spaces<a class="headerlink" href="#quotient-spaces" title="Permalink to this heading"></a></h3> Quotient vector spaces use the general quotient notation (typed with <code class="docutils literal notranslate">\quot</code>, not the ordinary <code class="docutils literal notranslate">/</code>). The projection onto a quotient space is <code class="docutils literal notranslate">Submodule.mkQ</code> and the universal property is <code class="docutils literal notranslate">Submodule.liftQ</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (E : Submodule K V) + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (E : Submodule K V) - example : Module K (V ⧸ E) := inferInstance + example : Module K (V ⧸ E) := inferInstance - example : V →ₗ[K] V ⧸ E := E.mkQ + example : V →ₗ[K] V ⧸ E := E.mkQ - example : ker E.mkQ = E := E.ker_mkQ + example : ker E.mkQ = E := E.ker_mkQ - example : range E.mkQ = ⊤ := E.range_mkQ + example : range E.mkQ = ⊤ := E.range_mkQ - example (hφ : E ≤ ker φ) : V ⧸ E →ₗ[K] W := E.liftQ φ hφ + example (hφ : E ≤ ker φ) : V ⧸ E →ₗ[K] W := E.liftQ φ hφ - example (F : Submodule K W) (hφ : E ≤ .comap φ F) : V ⧸ E →ₗ[K] W ⧸ F := E.mapQ F φ hφ + example (F : Submodule K W) (hφ : E ≤ .comap φ F) : V ⧸ E →ₗ[K] W ⧸ F := E.mapQ F φ hφ - noncomputable example : (V ⧸ LinearMap.ker φ) ≃ₗ[K] range φ := φ.quotKerEquivRange + noncomputable example : (V ⧸ LinearMap.ker φ) ≃ₗ[K] range φ := φ.quotKerEquivRange </pre></div> </div> - As an exercise, let us prove the correspondance theorem for subspaces of quotient spaces. + As an exercise, let us prove the correspondence theorem for subspaces of quotient spaces. Mathlib knows a slightly more precise version as <code class="docutils literal notranslate">Submodule.comapMkQRelIso</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>open Submodule + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Submodule - #check Submodule.map_comap_eq - #check Submodule.comap_map_eq + #check Submodule.map_comap_eq + #check Submodule.comap_map_eq - example : Submodule K (V ⧸ E) ≃ { F : Submodule K V // E ≤ F } where - toFun := sorry - invFun := sorry - left_inv := sorry - right_inv := sorry + example : Submodule K (V ⧸ E) ≃ { F : Submodule K V // E ≤ F } where + toFun := sorry + invFun := sorry + left_inv := sorry + right_inv := sorry </pre></div> </div> </section> </section> <section id="endomorphisms"> - <h2>9.3. Endomorphisms<a class="headerlink" href="#endomorphisms" title="Link to this heading"></a></h2> + <h2>9.3. Endomorphisms<a class="headerlink" href="#endomorphisms" title="Permalink to this heading"></a></h2> An important special case of linear maps are endomorphisms: linear maps from a vector space to itself. They are interesting because they form a <code class="docutils literal notranslate">K</code>-algebra. In particular we can evaluate polynomials with coefficients in <code class="docutils literal notranslate">K</code> on them, and they can have eigenvalues and eigenvectors. - Mathlib uses the abreviation <code class="docutils literal notranslate">Module.End K V := V →ₗ[K] V</code> which is convenient when + Mathlib uses the abbreviation <code class="docutils literal notranslate">Module.End K V := V →ₗ[K] V</code> which is convenient when using a lot of these (especially after opening the <code class="docutils literal notranslate">Module</code> namespace). - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {K : Type*} [Field K] {V : Type*} [AddCommGroup V] [Module K V] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {K : Type*} [Field K] {V : Type*} [AddCommGroup V] [Module K V] - variable {W : Type*} [AddCommGroup W] [Module K W] + variable {W : Type*} [AddCommGroup W] [Module K W] - open Polynomial Module LinearMap + open Polynomial Module LinearMap - example (φ ψ : End K V) : φ * ψ = φ ∘ₗ ψ := - LinearMap.mul_eq_comp φ ψ -- `rfl` would also work + example (φ ψ : End K V) : φ * ψ = φ ∘ₗ ψ := + LinearMap.mul_eq_comp φ ψ -- `rfl` would also work -- evaluating `P` on `φ` - example (P : K[X]) (φ : End K V) : V →ₗ[K] V := - aeval φ P + example (P : K[X]) (φ : End K V) : V →ₗ[K] V := + aeval φ P -- evaluating `X` on `φ` gives back `φ` - example (φ : End K V) : aeval φ (X : K[X]) = φ := - aeval_X φ + example (φ : End K V) : aeval φ (X : K[X]) = φ := + aeval_X φ </pre></div> </div> As an exercise manipulating endomorphisms, subspaces and polynomials, let us prove the (binary) kernels lemma: for any endomorphism $φ$ and any two relatively prime polynomials $P$ and $Q$, we have $\ker P(φ) ⊕ \ker Q(φ) = \ker \big(PQ(φ)\big)$. Note that <code class="docutils literal notranslate">IsCoprime x y</code> is defined as <code class="docutils literal notranslate">∃ a b, a * x + b * y = 1</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>#check Submodule.eq_bot_iff - #check Submodule.mem_inf - #check LinearMap.mem_ker + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check Submodule.eq_bot_iff + #check Submodule.mem_inf + #check LinearMap.mem_ker - example (P Q : K[X]) (h : IsCoprime P Q) (φ : End K V) : ker (aeval φ P) ⊓ ker (aeval φ Q) = ⊥ := by - sorry + example (P Q : K[X]) (h : IsCoprime P Q) (φ : End K V) : ker (aeval φ P) ⊓ ker (aeval φ Q) = ⊥ := by + sorry - #check Submodule.add_mem_sup - #check map_mul - #check LinearMap.mul_apply - #check LinearMap.ker_le_ker_comp + #check Submodule.add_mem_sup + #check map_mul + #check LinearMap.mul_apply + #check LinearMap.ker_le_ker_comp - example (P Q : K[X]) (h : IsCoprime P Q) (φ : End K V) : - ker (aeval φ P) ⊔ ker (aeval φ Q) = ker (aeval φ (P*Q)) := by - sorry + example (P Q : K[X]) (h : IsCoprime P Q) (φ : End K V) : + ker (aeval φ P) ⊔ ker (aeval φ Q) = ker (aeval φ (P*Q)) := by + sorry </pre></div> </div> We now move to the discussions of eigenspaces and eigenvalues. By the definition, the eigenspace
- @@ -696,47 +694,47 @@ associated to an endomorphism $&#966
 The first thing to understand is how to spell \(aId$. We could use <code class="docutils literal notranslate">a • LinearMap.id</code>, but Mathlib uses <cite>algebraMap K (End K V) a</cite> which directly plays nicely with the <code class="docutils literal notranslate">K</code>-algebra structure. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a : K) : algebraMap K (End K V) a = a • LinearMap.id := rfl + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a : K) : algebraMap K (End K V) a = a • LinearMap.id := rfl </pre></div> </div> Then the next thing to note is that eigenspaces are defined for all values of <code class="docutils literal notranslate">a</code>, although they are interesting only when they are non-zero. However an eigenvector is, by definition, a non-zero element of an eigenspace. The corresponding predicate is <code class="docutils literal notranslate">End.HasEigenvector</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (φ : End K V) (a : K) : - φ.eigenspace a = LinearMap.ker (φ - algebraMap K (End K V) a) := - rfl + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (φ : End K V) (a : K) : + φ.eigenspace a = LinearMap.ker (φ - algebraMap K (End K V) a) := + rfl </pre></div> </div> Then there is a predicate <code class="docutils literal notranslate">End.HasEigenvalue</code> and the corresponding subtype <code class="docutils literal notranslate">End.Eigenvalues</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (φ : End K V) (a : K) : φ.HasEigenvalue a ↔ φ.eigenspace a ≠ ⊥ := - Iff.rfl + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (φ : End K V) (a : K) : φ.HasEigenvalue a ↔ φ.eigenspace a ≠ ⊥ := + Iff.rfl - example (φ : End K V) (a : K) : φ.HasEigenvalue a ↔ ∃ v, φ.HasEigenvector a v := - ⟨End.HasEigenvalue.exists_hasEigenvector, fun ⟨_, hv⟩ ↦ φ.hasEigenvalue_of_hasEigenvector hv⟩ + example (φ : End K V) (a : K) : φ.HasEigenvalue a ↔ ∃ v, φ.HasEigenvector a v := + ⟨End.HasEigenvalue.exists_hasEigenvector, fun ⟨_, hv⟩ ↦ φ.hasEigenvalue_of_hasEigenvector hv⟩ - example (φ : End K V) : φ.Eigenvalues = {a // φ.HasEigenvalue a} := - rfl + example (φ : End K V) : φ.Eigenvalues = {a // φ.HasEigenvalue a} := + rfl -- Eigenvalue are roots of the minimal polynomial - example (φ : End K V) (a : K) : φ.HasEigenvalue a → (minpoly K φ).IsRoot a := - φ.isRoot_of_hasEigenvalue + example (φ : End K V) (a : K) : φ.HasEigenvalue a → (minpoly K φ).IsRoot a := + φ.isRoot_of_hasEigenvalue -- In finite dimension, the converse is also true (we will discuss dimension below) - example [FiniteDimensional K V] (φ : End K V) (a : K) : - φ.HasEigenvalue a ↔ (minpoly K φ).IsRoot a := - φ.hasEigenvalue_iff_isRoot + example [FiniteDimensional K V] (φ : End K V) (a : K) : + φ.HasEigenvalue a ↔ (minpoly K φ).IsRoot a := + φ.hasEigenvalue_iff_isRoot -- Cayley-Hamilton - example [FiniteDimensional K V] (φ : End K V) : aeval φ φ.charpoly = 0 := - φ.aeval_self_charpoly + example [FiniteDimensional K V] (φ : End K V) : aeval φ φ.charpoly = 0 := + φ.aeval_self_charpoly </pre></div> </div> </section> <section id="matrices-bases-and-dimension"> - <h2>9.4. Matrices, bases and dimension<a class="headerlink" href="#matrices-bases-and-dimension" title="Link to this heading"></a></h2> + <h2>9.4. Matrices, bases and dimension<a class="headerlink" href="#matrices-bases-and-dimension" title="Permalink to this heading"></a></h2> <section id="matrices"> - <h3>9.4.1. Matrices<a class="headerlink" href="#matrices" title="Link to this heading"></a></h3> + <h3>9.4.1. Matrices<a class="headerlink" href="#matrices" title="Permalink to this heading"></a></h3> Before introducing bases for abstract vector spaces, we go back to the much more elementary setup of linear algebra in $K^n$ for some field $K$. Here the main objects are vectors and matrices.
- @@ -745,21 +743,21 @@ For concrete matrices we can use the <code class="docutils literal notranslate">
 and components of lines are separated by colons. When entries have a computable type such as <code class="docutils literal notranslate">ℕ</code> or <code class="docutils literal notranslate">ℚ</code>, we can use the <code class="docutils literal notranslate">eval</code> command to play with basic operations. - <div class="highlight-lean notranslate"><div class="highlight"><pre>section matrices + <div class="highlight-lean notranslate"><div class="highlight"><pre>section matrices -- Adding vectors - #eval !![1, 2] + !![3, 4] -- !![4, 6] + #eval !![1, 2] + !![3, 4] -- !![4, 6] -- Adding matrices - #eval !![1, 2; 3, 4] + !![3, 4; 5, 6] -- !![4, 6; 8, 10] + #eval !![1, 2; 3, 4] + !![3, 4; 5, 6] -- !![4, 6; 8, 10] -- Multiplying matrices - #eval !![1, 2; 3, 4] * !![3, 4; 5, 6] -- !![4, 6; 8, 10] + #eval !![1, 2; 3, 4] * !![3, 4; 5, 6] -- !![4, 6; 8, 10] </pre></div> </div> It is important to understand that this use of <code class="docutils literal notranslate">#eval</code> is interesting only for exploration, it is not meant to replace a computer algebra system such as Sage. - The data representation used here for matrices is not computationaly + The data representation used here for matrices is not computationally efficient in any way. It uses functions instead of arrays and is optimized for proving, not computing. The virtual machine used by <code class="docutils literal notranslate">#eval</code> is also not optimized for this use.
- @@ -769,16 +767,16 @@ from the left (resp. right) interprets the vector as a row (resp. column) vector
 This corresponds to operations <code class="docutils literal notranslate">Matrix.vecMul</code>, with notation <code class="docutils literal notranslate">ᵥ*</code> and <code class="docutils literal notranslate">Matrix.mulVec</code>, with notation ` <cite>*ᵥ`</cite>. Those notations are scoped in the <code class="docutils literal notranslate">Matrix</code> namespace that we therefore need to open. - <div class="highlight-lean notranslate"><div class="highlight"><pre>open Matrix + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Matrix -- matrices acting on vectors on the left - #eval !![1, 2; 3, 4] *ᵥ ![1, 1] -- ![3, 7] + #eval !![1, 2; 3, 4] *ᵥ ![1, 1] -- ![3, 7] -- matrices acting on vectors on the left, resulting in a size one matrix - #eval !![1, 2] *ᵥ ![1, 1] -- ![3] + #eval !![1, 2] *ᵥ ![1, 1] -- ![3] -- matrices acting on vectors on the right - #eval ![1, 1, 1] ᵥ* !![1, 2; 3, 4; 5, 6] -- ![9, 12] + #eval ![1, 1, 1] ᵥ* !![1, 2; 3, 4; 5, 6] -- ![9, 12] </pre></div> </div> In order to generate matrices with identical rows or columns specified by a vector, we
- @@ -786,24 +784,24 @@ use <code class="docutils literal notranslate">Matrix.row</spa
 rows or columns and the vector. For instance one can get single row or single column matrixes (more precisely matrices whose rows or columns are indexed by <code class="docutils literal notranslate">Fin 1</code>). - <div class="highlight-lean notranslate"><div class="highlight"><pre>#eval row (Fin 1) ![1, 2] -- !![1, 2] + <div class="highlight-lean notranslate"><div class="highlight"><pre>#eval row (Fin 1) ![1, 2] -- !![1, 2] - #eval col (Fin 1) ![1, 2] -- !![1; 2] + #eval col (Fin 1) ![1, 2] -- !![1; 2] </pre></div> </div> Other familiar operations include the vector dot product, matrix transpose, and, for square matrices, determinant and trace. <div class="highlight-lean notranslate"><div class="highlight"><pre>-- vector dot product - #eval ![1, 2] ⬝ᵥ ![3, 4] -- `11` + #eval ![1, 2] ⬝ᵥ ![3, 4] -- `11` -- matrix transpose - #eval !![1, 2; 3, 4]ᵀ -- `!![1, 3; 2, 4]` + #eval !![1, 2; 3, 4]ᵀ -- `!![1, 3; 2, 4]` -- determinant - #eval !![(1 : ℤ), 2; 3, 4].det -- `-2` + #eval !![(1 : ℤ), 2; 3, 4].det -- `-2` -- trace - #eval !![(1 : ℤ), 2; 3, 4].trace -- `5` + #eval !![(1 : ℤ), 2; 3, 4].trace -- `5` </pre></div> </div> When entries do not have a computable type, for instance if they are real numbers, we cannot
- @@ -812,14 +810,14 @@ considerably expanding the trusted code base (ie the part of Lean that you need
 checking proofs). So it is good to also use the <code class="docutils literal notranslate">simp</code> and <code class="docutils literal notranslate">norm_num</code> tactics in proofs, or their command counter-part for quick exploration. - <div class="highlight-lean notranslate"><div class="highlight"><pre>#simp !![(1 : ℝ), 2; 3, 4].det -- `4 - 2*3` + <div class="highlight-lean notranslate"><div class="highlight"><pre>#simp !![(1 : ℝ), 2; 3, 4].det -- `4 - 2*3` - #norm_num !![(1 : ℝ), 2; 3, 4].det -- `-2` + #norm_num !![(1 : ℝ), 2; 3, 4].det -- `-2` - #norm_num !![(1 : ℝ), 2; 3, 4].trace -- `5` + #norm_num !![(1 : ℝ), 2; 3, 4].trace -- `5` - variable (a b c d : ℝ) in - #simp !![a, b; c, d].det -- `a * d – b * c` + variable (a b c d : ℝ) in + #simp !![a, b; c, d].det -- `a * d – b * c` </pre></div> </div> The next important operation on square matrices is inversion.
- @@ -830,7 +828,7 @@ the zero matrix for non-invertible matrices.
 and, for any matrix <code class="docutils literal notranslate">A</code>, <code class="docutils literal notranslate">A⁻¹</code> is defined as <code class="docutils literal notranslate">Ring.inverse A.det • A.adjugate</code>. According to Cramer’s rule, this is indeed the inverse of <code class="docutils literal notranslate">A</code> when the determinant of <code class="docutils literal notranslate">A</code> is not zero. - <div class="highlight-lean notranslate"><div class="highlight"><pre>#norm_num [Matrix.inv_def] !![(1 : ℝ), 2; 3, 4]⁻¹ -- !![-2, 1; 3 / 2, -(1 / 2)] + <div class="highlight-lean notranslate"><div class="highlight"><pre>#norm_num [Matrix.inv_def] !![(1 : ℝ), 2; 3, 4]⁻¹ -- !![-2, 1; 3 / 2, -(1 / 2)] </pre></div> </div> Of course this definition is really useful only for invertible matrices.
- @@ -839,18 +837,18 @@ For instance, the <code class="docutils literal notranslate">s
 lemma which has an <code class="docutils literal notranslate">Invertible</code> type-class assumption, so it will trigger only if this can be found by the type-class synthesis system. Here we make this fact available using a <code class="docutils literal notranslate">have</code> statement. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : !![(1 : ℝ), 2; 3, 4]⁻¹ * !![(1 : ℝ), 2; 3, 4] = 1 := by - have : Invertible !![(1 : ℝ), 2; 3, 4] := by - apply Matrix.invertibleOfIsUnitDet - norm_num - simp + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : !![(1 : ℝ), 2; 3, 4]⁻¹ * !![(1 : ℝ), 2; 3, 4] = 1 := by + have : Invertible !![(1 : ℝ), 2; 3, 4] := by + apply Matrix.invertibleOfIsUnitDet + norm_num + simp </pre></div> </div> In this fully concrete case, we could also use the <code class="docutils literal notranslate">norm_num</code> machinery, and <code class="docutils literal notranslate">apply?</code> to find the final line: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : !![(1 : ℝ), 2; 3, 4]⁻¹ * !![(1 : ℝ), 2; 3, 4] = 1 := by - norm_num [Matrix.inv_def] - exact one_fin_two.symm + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : !![(1 : ℝ), 2; 3, 4]⁻¹ * !![(1 : ℝ), 2; 3, 4] = 1 := by + norm_num [Matrix.inv_def] + exact one_fin_two.symm </pre></div> </div> All the concrete matrices above have their rows and columns indexed by <code class="docutils literal notranslate">Fin n</code> for
- @@ -874,32 +872,32 @@ and accept the statement as meaningful, and then prove it by checking all entrie
 on <code class="docutils literal notranslate">Fin 2 → Fin 2 → ℤ</code> but the matrix multiplication on <code class="docutils literal notranslate">Matrix (Fin 2) (Fin 2) ℤ</code>. <div class="highlight-lean notranslate"><div class="highlight"><pre>section - example : (fun _ ↦ 1 : Fin 2 → Fin 2 → ℤ) = !![1, 1; 1, 1] := by - ext i j - fin_cases i <;> fin_cases j <;> rfl + example : (fun _ ↦ 1 : Fin 2 → Fin 2 → ℤ) = !![1, 1; 1, 1] := by + ext i j + fin_cases i <;> fin_cases j <;> rfl - example : (fun _ ↦ 1 : Fin 2 → Fin 2 → ℤ) * (fun _ ↦ 1 : Fin 2 → Fin 2 → ℤ) = !![1, 1; 1, 1] := by - ext i j - fin_cases i <;> fin_cases j <;> rfl + example : (fun _ ↦ 1 : Fin 2 → Fin 2 → ℤ) * (fun _ ↦ 1 : Fin 2 → Fin 2 → ℤ) = !![1, 1; 1, 1] := by + ext i j + fin_cases i <;> fin_cases j <;> rfl - example : !![1, 1; 1, 1] * !![1, 1; 1, 1] = !![2, 2; 2, 2] := by - norm_num + example : !![1, 1; 1, 1] * !![1, 1; 1, 1] = !![2, 2; 2, 2] := by + norm_num </pre></div> </div> In order to define matrices as functions without loosing the benefits of <code class="docutils literal notranslate">Matrix</code> for type class synthesis, we can use the equivalence <code class="docutils literal notranslate">Matrix.of</code> between functions - and matrices. This equivalence is secretely defined using <code class="docutils literal notranslate">Equiv.refl</code>. + and matrices. This equivalence is secretly defined using <code class="docutils literal notranslate">Equiv.refl</code>. For instance we can define Vandermonde matrices corresponding to a vector <code class="docutils literal notranslate">v</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {n : ℕ} (v : Fin n → ℝ) : - Matrix.vandermonde v = Matrix.of (fun i j : Fin n ↦ v i ^ (j : ℕ)) := - rfl + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {n : ℕ} (v : Fin n → ℝ) : + Matrix.vandermonde v = Matrix.of (fun i j : Fin n ↦ v i ^ (j : ℕ)) := + rfl end - end matrices + end matrices </pre></div> </div> </section> <section id="bases"> - <h3>9.4.2. Bases<a class="headerlink" href="#bases" title="Link to this heading"></a></h3> + <h3>9.4.2. Bases<a class="headerlink" href="#bases" title="Permalink to this heading"></a></h3> We now want to discuss bases of vector spaces. Informally there are many ways to define this notion. One can use a universal property. One can say a basis is a family of vectors that is linearly independent and spanning.
- @@ -920,23 +918,23 @@ Evaluating such a function coming from a basis <code class="docutils literal not
 <code class="docutils literal notranslate">i : ι</code> returns the component (or coordinate) of <code class="docutils literal notranslate">v</code> on the <code class="docutils literal notranslate">i</code>-th basis vector. The type of bases indexed by a type <code class="docutils literal notranslate">ι</code> of <code class="docutils literal notranslate">V</code> as a <code class="docutils literal notranslate">K</code> vector space is <code class="docutils literal notranslate">Basis ι K V</code>. The isomorphism is called <code class="docutils literal notranslate">Basis.repr</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {K : Type*} [Field K] {V : Type*} [AddCommGroup V] [Module K V] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {K : Type*} [Field K] {V : Type*} [AddCommGroup V] [Module K V] section - variable {ι : Type*} (B : Basis ι K V) (v : V) (i : ι) + variable {ι : Type*} (B : Basis ι K V) (v : V) (i : ι) -- The basis vector with index ``i`` - #check (B i : V) + #check (B i : V) -- the linear isomorphism with the model space given by ``B`` - #check (B.repr : V ≃ₗ[K] ι →₀ K) + #check (B.repr : V ≃ₗ[K] ι →₀ K) -- the component function of ``v`` - #check (B.repr v : ι →₀ K) + #check (B.repr v : ι →₀ K) -- the component of ``v`` with index ``i`` - #check (B.repr v i : K) + #check (B.repr v i : K) </pre></div> </div> Instead of starting with such an isomorphism, one can start with a family <code class="docutils literal notranslate">b</code> of vectors that is
- @@ -946,15 +944,15 @@ Here <code class="docutils literal notranslate">⊤
 This spelling looks a bit tortuous, but we will see below that it is almost equivalent by definition to the more readable <code class="docutils literal notranslate">∀ v, v ∈ Submodule.span K (Set.range b)</code> (the underscores in the snippet below refers to the useless information <code class="docutils literal notranslate">v ∈ ⊤</code>). - <div class="highlight-lean notranslate"><div class="highlight"><pre>noncomputable example (b : ι → V) (b_indep : LinearIndependent K b) - (b_spans : ∀ v, v ∈ Submodule.span K (Set.range b)) : Basis ι K V := - Basis.mk b_indep (fun v _ ↦ b_spans v) + <div class="highlight-lean notranslate"><div class="highlight"><pre>noncomputable example (b : ι → V) (b_indep : LinearIndependent K b) + (b_spans : ∀ v, v ∈ Submodule.span K (Set.range b)) : Basis ι K V := + Basis.mk b_indep (fun v _ ↦ b_spans v) -- The family of vectors underlying the above basis is indeed ``b``. - example (b : ι → V) (b_indep : LinearIndependent K b) - (b_spans : ∀ v, v ∈ Submodule.span K (Set.range b)) (i : ι) : - Basis.mk b_indep (fun v _ ↦ b_spans v) i = b i := - Basis.mk_apply b_indep (fun v _ ↦ b_spans v) i + example (b : ι → V) (b_indep : LinearIndependent K b) + (b_spans : ∀ v, v ∈ Submodule.span K (Set.range b)) (i : ι) : + Basis.mk b_indep (fun v _ ↦ b_spans v) i = b i := + Basis.mk_apply b_indep (fun v _ ↦ b_spans v) i </pre></div> </div> In particular the model vector space <code class="docutils literal notranslate">ι →₀ K</code> has a so-called canonical basis whose <code class="docutils literal notranslate">repr</code>
- @@ -964,27 +962,27 @@ function evaluated on any vector is the identity isomorphism. It is called
 vanish expect for a single input value. More precisely the basis vector indexed by <code class="docutils literal notranslate">i : ι</code> is <code class="docutils literal notranslate">Finsupp.single i 1</code> which is the finitely supported function taking value <code class="docutils literal notranslate">1</code> at <code class="docutils literal notranslate">i</code> and <code class="docutils literal notranslate">0</code> everywhere else. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable [DecidableEq ι] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable [DecidableEq ι] - example : Finsupp.basisSingleOne.repr = LinearEquiv.refl K (ι →₀ K) := - rfl + example : Finsupp.basisSingleOne.repr = LinearEquiv.refl K (ι →₀ K) := + rfl - example (i : ι) : Finsupp.basisSingleOne i = Finsupp.single i 1 := - rfl + example (i : ι) : Finsupp.basisSingleOne i = Finsupp.single i 1 := + rfl </pre></div> </div> - The story of finitely supported functions is uneeded when the indexing type is finite. + The story of finitely supported functions is unneeded when the indexing type is finite. In this case we can use the simpler <code class="docutils literal notranslate">Pi.basisFun</code> which gives a basis of the whole <code class="docutils literal notranslate">ι → K</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example [Finite ι] (x : ι → K) (i : ι) : (Pi.basisFun K ι).repr x i = x i := by - simp + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [Finite ι] (x : ι → K) (i : ι) : (Pi.basisFun K ι).repr x i = x i := by + simp </pre></div> </div> Going back to the general case of bases of abstract vector spaces, we can express any vector as a linear combination of basis vectors. Let us first see the easy case of finite bases. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example [Fintype ι] : ∑ i : ι, B.repr v i • (B i) = v := - B.sum_repr v + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [Fintype ι] : ∑ i : ι, B.repr v i • (B i) = v := + B.sum_repr v </pre></div> </div> When <code class="docutils literal notranslate">ι</code> is not finite, the above statement makes no sense a priori: we cannot take a sum over <code class="docutils literal notranslate">ι</code>.
- @@ -997,24 +995,24 @@ function <code class="docutils literal notranslate">f</
 sum over the support of <code class="docutils literal notranslate">c</code> of the scalar multiplication <code class="docutils literal notranslate">c • f</code>. In particular, we can replace it by a sum over any finite set containing the support of <code class="docutils literal notranslate">c</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (c : ι →₀ K) (f : ι → V) (s : Finset ι) (h : c.support ⊆ s) : - Finsupp.linearCombination K f c = ∑ i ∈ s, c i • f i := - Finsupp.linearCombination_apply_of_mem_supported K h + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (c : ι →₀ K) (f : ι → V) (s : Finset ι) (h : c.support ⊆ s) : + Finsupp.linearCombination K f c = ∑ i ∈ s, c i • f i := + Finsupp.linearCombination_apply_of_mem_supported K h </pre></div> </div> One could also assume that <code class="docutils literal notranslate">f</code> is finitely supported and still get a well defined sum. But the choice made by <code class="docutils literal notranslate">Finsupp.linearCombination</code> is the one relevant to our basis discussion since it allows to state the generalization of <code class="docutils literal notranslate">Basis.sum_repr</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : Finsupp.linearCombination K B (B.repr v) = v := - B.linearCombination_repr v + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : Finsupp.linearCombination K B (B.repr v) = v := + B.linearCombination_repr v </pre></div> </div> One could wonder why <code class="docutils literal notranslate">K</code> is an explicit argument here, despite the fact it can be inferred from the type of <code class="docutils literal notranslate">c</code>. The point is that the partially applied <code class="docutils literal notranslate">Finsupp.linearCombination K f</code> is interesting in itself. It is not a bare function from <code class="docutils literal notranslate">ι →₀ K</code> to <code class="docutils literal notranslate">V</code> but a <code class="docutils literal notranslate">K</code>-linear map. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (f : ι → V) in - #check (Finsupp.linearCombination K f : (ι →₀ K) →ₗ[K] V) + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (f : ι → V) in + #check (Finsupp.linearCombination K f : (ι →₀ K) →ₗ[K] V) </pre></div> </div> The above subtlety also explain why dot notation cannot be used to write
- @@ -1034,41 +1032,41 @@ This isomorphism is characterized by the fact that it sends any function <code c
 to a linear map sending the basis vector <code class="docutils literal notranslate">B i</code> to <code class="docutils literal notranslate">u i</code>, for every <code class="docutils literal notranslate">i : ι</code>. <div class="highlight-lean notranslate"><div class="highlight"><pre>section - variable {W : Type*} [AddCommGroup W] [Module K W] - (φ : V →ₗ[K] W) (u : ι → W) + variable {W : Type*} [AddCommGroup W] [Module K W] + (φ : V →ₗ[K] W) (u : ι → W) - #check (B.constr K : (ι → W) ≃ₗ[K] (V →ₗ[K] W)) + #check (B.constr K : (ι → W) ≃ₗ[K] (V →ₗ[K] W)) - #check (B.constr K u : V →ₗ[K] W) + #check (B.constr K u : V →ₗ[K] W) - example (i : ι) : B.constr K u (B i) = u i := - B.constr_basis K u i + example (i : ι) : B.constr K u (B i) = u i := + B.constr_basis K u i </pre></div> </div> This property is indeed characteristic because linear maps are determined by their values on bases: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (φ ψ : V →ₗ[K] W) (h : ∀ i, φ (B i) = ψ (B i)) : φ = ψ := - B.ext h + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (φ ψ : V →ₗ[K] W) (h : ∀ i, φ (B i) = ψ (B i)) : φ = ψ := + B.ext h </pre></div> </div> If we also have a basis <code class="docutils literal notranslate">B'</code> on the target space then we can identify linear maps with matrices. This identification is a <code class="docutils literal notranslate">K</code>-linear isomorphism. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {ι' : Type*} (B' : Basis ι' K W) [Fintype ι] [DecidableEq ι] [Fintype ι'] [DecidableEq ι'] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {ι' : Type*} (B' : Basis ι' K W) [Fintype ι] [DecidableEq ι] [Fintype ι'] [DecidableEq ι'] - open LinearMap + open LinearMap - #check (toMatrix B B' : (V →ₗ[K] W) ≃ₗ[K] Matrix ι' ι K) + #check (toMatrix B B' : (V →ₗ[K] W) ≃ₗ[K] Matrix ι' ι K) - open Matrix -- get access to the ``*ᵥ`` notation for multiplication between matrices and vectors. + open Matrix -- get access to the ``*ᵥ`` notation for multiplication between matrices and vectors. - example (φ : V →ₗ[K] W) (v : V) : (toMatrix B B' φ) *ᵥ (B.repr v) = B'.repr (φ v) := - toMatrix_mulVec_repr B B' φ v + example (φ : V →ₗ[K] W) (v : V) : (toMatrix B B' φ) *ᵥ (B.repr v) = B'.repr (φ v) := + toMatrix_mulVec_repr B B' φ v - variable {ι'' : Type*} (B'' : Basis ι'' K W) [Fintype ι''] [DecidableEq ι''] + variable {ι'' : Type*} (B'' : Basis ι'' K W) [Fintype ι''] [DecidableEq ι''] - example (φ : V →ₗ[K] W) : (toMatrix B B'' φ) = (toMatrix B' B'' .id) * (toMatrix B B' φ) := by - simp + example (φ : V →ₗ[K] W) : (toMatrix B B'' φ) = (toMatrix B' B'' .id) * (toMatrix B B' φ) := by + simp end </pre></div>
- @@ -1081,31 +1079,31 @@ This would then need to be complemented using that bases all have isomorphic ind
 get the full result. Of course Mathlib already knows this, and <code class="docutils literal notranslate">simp</code> can close the goal immediately, so you shouldn’t use it too soon, but rather use the provided lemmas. - <div class="highlight-lean notranslate"><div class="highlight"><pre>open Module LinearMap Matrix + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Module LinearMap Matrix -- Some lemmas coming from the fact that `LinearMap.toMatrix` is an algebra morphism. - #check toMatrix_comp - #check id_comp - #check comp_id - #check toMatrix_id + #check toMatrix_comp + #check id_comp + #check comp_id + #check toMatrix_id -- Some lemmas coming from the fact that ``Matrix.det`` is a multiplicative monoid morphism. - #check Matrix.det_mul - #check Matrix.det_one - - example [Fintype ι] (B' : Basis ι K V) (φ : End K V) : - (toMatrix B B φ).det = (toMatrix B' B' φ).det := by - set M := toMatrix B B φ - set M' := toMatrix B' B' φ - set P := (toMatrix B B') LinearMap.id - set P' := (toMatrix B' B) LinearMap.id - sorry + #check Matrix.det_mul + #check Matrix.det_one + + example [Fintype ι] (B' : Basis ι K V) (φ : End K V) : + (toMatrix B B φ).det = (toMatrix B' B' φ).det := by + set M := toMatrix B B φ + set M' := toMatrix B' B' φ + set P := (toMatrix B B') LinearMap.id + set P' := (toMatrix B' B) LinearMap.id + sorry end </pre></div> </div> </section> <section id="dimension"> - <h3>9.4.3. Dimension<a class="headerlink" href="#dimension" title="Link to this heading"></a></h3> + <h3>9.4.3. Dimension<a class="headerlink" href="#dimension" title="Permalink to this heading"></a></h3> Returning to the case of a single vector space, bases are also useful to define the concept of dimension. Here again, there is the elementary case of finite-dimensional vector spaces.
- @@ -1113,19 +1111,19 @@ For such spaces we expect a dimension which is a natural number.
 This is <code class="docutils literal notranslate">FiniteDimensional.finrank</code>. It takes the base field as an explicit argument since a given abelian group can be a vector space over different fields. <div class="highlight-lean notranslate"><div class="highlight"><pre>section - #check (FiniteDimensional.finrank K V : ℕ) + #check (FiniteDimensional.finrank K V : ℕ) -- `Fin n → K` is the archetypical space with dimension `n` over `K`. - example (n : ℕ) : FiniteDimensional.finrank K (Fin n → K) = n := - FiniteDimensional.finrank_fin_fun K + example (n : ℕ) : FiniteDimensional.finrank K (Fin n → K) = n := + FiniteDimensional.finrank_fin_fun K -- Seen as a vector space over itself, `ℂ` has dimension one. - example : FiniteDimensional.finrank ℂ ℂ = 1 := - FiniteDimensional.finrank_self ℂ + example : FiniteDimensional.finrank ℂ ℂ = 1 := + FiniteDimensional.finrank_self ℂ -- But as a real vector space it has dimension two. - example : FiniteDimensional.finrank ℝ ℂ = 2 := - Complex.finrank_real_complex + example : FiniteDimensional.finrank ℝ ℂ = 2 := + Complex.finrank_real_complex </pre></div> </div> Note that <code class="docutils literal notranslate">FiniteDimensional.finrank</code> is defined for any vector space. It returns
- @@ -1133,8 +1131,8 @@ zero for infinite dimensional vector spaces, just as division by zero returns ze
 Of course many lemmas require a finite dimension assumption. This is the role of the <code class="docutils literal notranslate">FiniteDimensional</code> typeclass. For instance, think about how the next example fails without this assumption. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example [FiniteDimensional K V] : 0 < FiniteDimensional.finrank K V ↔ Nontrivial V := - FiniteDimensional.finrank_pos_iff + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [FiniteDimensional K V] : 0 < FiniteDimensional.finrank K V ↔ Nontrivial V := + FiniteDimensional.finrank_pos_iff </pre></div> </div> In the above statement, <code class="docutils literal notranslate">Nontrivial V</code> means <code class="docutils literal notranslate">V</code> has at least two different elements.
- @@ -1143,44 +1141,44 @@ This is fine when using it from left to right, but not when using it from right
 because Lean has no way to guess <code class="docutils literal notranslate">K</code> from the statement <code class="docutils literal notranslate">Nontrivial V</code>. In that case it is useful to use the name argument syntax, after checking that the lemma is stated over a ring named <code class="docutils literal notranslate">R</code>. So we can write: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example [FiniteDimensional K V] (h : 0 < FiniteDimensional.finrank K V) : Nontrivial V := by - apply (FiniteDimensional.finrank_pos_iff (R := K)).1 - exact h + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [FiniteDimensional K V] (h : 0 < FiniteDimensional.finrank K V) : Nontrivial V := by + apply (FiniteDimensional.finrank_pos_iff (R := K)).1 + exact h </pre></div> </div> The above spelling is strange because we already have <code class="docutils literal notranslate">h</code> as an assumption, so we could just as well give the full proof <code class="docutils literal notranslate">FiniteDimensional.finrank_pos_iff.1 h</code> but it is good to know for more complicated cases. By definition, <code class="docutils literal notranslate">FiniteDimensional K V</code> can be read from any basis. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {ι : Type*} (B : Basis ι K V) + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {ι : Type*} (B : Basis ι K V) - example [Finite ι] : FiniteDimensional K V := FiniteDimensional.of_fintype_basis B + example [Finite ι] : FiniteDimensional K V := FiniteDimensional.of_fintype_basis B - example [FiniteDimensional K V] : Finite ι := - (FiniteDimensional.fintypeBasisIndex B).finite + example [FiniteDimensional K V] : Finite ι := + (FiniteDimensional.fintypeBasisIndex B).finite end </pre></div> </div> Using that the subtype corresponding to a linear subspace has a vector space structure, we can talk about the dimension of a subspace. <div class="highlight-lean notranslate"><div class="highlight"><pre>section - variable (E F : Submodule K V) [FiniteDimensional K V] + variable (E F : Submodule K V) [FiniteDimensional K V] - open FiniteDimensional + open FiniteDimensional - example : finrank K (E ⊔ F : Submodule K V) + finrank K (E ⊓ F : Submodule K V) = - finrank K E + finrank K F := - Submodule.finrank_sup_add_finrank_inf_eq E F + example : finrank K (E ⊔ F : Submodule K V) + finrank K (E ⊓ F : Submodule K V) = + finrank K E + finrank K F := + Submodule.finrank_sup_add_finrank_inf_eq E F - example : finrank K E ≤ finrank K V := Submodule.finrank_le E + example : finrank K E ≤ finrank K V := Submodule.finrank_le E </pre></div> </div> In the first statement above, the purpose of the type ascriptions is to make sure that coercion to <code class="docutils literal notranslate">Type*</code> does not trigger too early. We are now ready for an exercise about <code class="docutils literal notranslate">finrank</code> and subspaces. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : finrank K V < finrank K E + finrank K F) : - Nontrivial (E ⊓ F : Submodule K V) := by - sorry + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : finrank K V < finrank K E + finrank K F) : + Nontrivial (E ⊓ F : Submodule K V) := by + sorry end </pre></div> </div>
- @@ -1219,28 +1217,28 @@ a basis of <code class="docutils literal notranslate">V
 So instead it is defined as the supremum <code class="docutils literal notranslate">Module.rank K V</code> of cardinals of all linearly independent sets in <code class="docutils literal notranslate">V</code>. If <code class="docutils literal notranslate">V</code> has universe level <code class="docutils literal notranslate">u</code> then its rank has type <code class="docutils literal notranslate">Cardinal.{u}</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>#check V -- Type u_2 - #check Module.rank K V -- Cardinal.{u_2} + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check V -- Type u_2 + #check Module.rank K V -- Cardinal.{u_2} </pre></div> </div> One can still relate this definition to bases. Indeed there is also a commutative <code class="docutils literal notranslate">max</code> operation on universe levels, and given two universe levels <code class="docutils literal notranslate">u</code> and <code class="docutils literal notranslate">v</code> there is an operation <code class="docutils literal notranslate">Cardinal.lift.{u, v} : Cardinal.{v} → Cardinal.{max v u}</code> that allows to put cardinals in a common universe and state the dimension theorem. - <div class="highlight-lean notranslate"><div class="highlight"><pre>universe u v -- `u` and `v` will denote universe levels + <div class="highlight-lean notranslate"><div class="highlight"><pre>universe u v -- `u` and `v` will denote universe levels - variable {ι : Type u} (B : Basis ι K V) - {ι' : Type v} (B' : Basis ι' K V) + variable {ι : Type u} (B : Basis ι K V) + {ι' : Type v} (B' : Basis ι' K V) - example : Cardinal.lift.{v, u} (.mk ι) = Cardinal.lift.{u, v} (.mk ι') := - mk_eq_mk_of_basis B B' + example : Cardinal.lift.{v, u} (.mk ι) = Cardinal.lift.{u, v} (.mk ι') := + mk_eq_mk_of_basis B B' </pre></div> </div> We can relate the finite dimensional case to this discussion using the coercion - from natural numbers to finite cardinals (or more precisly the finite cardinals which live in <code class="docutils literal notranslate">Cardinal.{v}</code> where <code class="docutils literal notranslate">v</code> is the universe level of <code class="docutils literal notranslate">V</code>). - <div class="highlight-lean notranslate"><div class="highlight"><pre>example [FiniteDimensional K V] : - (FiniteDimensional.finrank K V : Cardinal) = Module.rank K V := - finrank_eq_rank K V + from natural numbers to finite cardinals (or more precisely the finite cardinals which live in <code class="docutils literal notranslate">Cardinal.{v}</code> where <code class="docutils literal notranslate">v</code> is the universe level of <code class="docutils literal notranslate">V</code>). + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [FiniteDimensional K V] : + (FiniteDimensional.finrank K V : Cardinal) = Module.rank K V := + finrank_eq_rank K V </pre></div> </div> </section>
html/C10_Topology.html
- @@ -1,25 +1,23 @@
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- @@ -106,8 +104,8 @@
 <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> - <section id="index-0"> - <h1>10. Topology<a class="headerlink" href="#index-0" title="Link to this heading"></a></h1> + <section id="index-0"> + <h1>10. 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.
- @@ -175,7 +173,7 @@ 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>10.1. Filters<a class="headerlink" href="#filters" title="Link to this heading"></a></h2> + <h2>10.1. 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:
- @@ -218,21 +216,21 @@ One manifestation of this view is that we can associate to any <code class="docu
 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 + <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">atTop : 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 } + <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>.
- @@ -243,8 +241,8 @@ defined in Mathlib as <code class="docutils literal notranslate">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 + <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 atTop (𝓝 x)</code> is equivalent to saying that the sequence <code class="docutils literal notranslate">u : ℕ → ℝ</code>
- @@ -264,12 +262,12 @@ In this examples file we’ve opened the <code class="docutils literal notra
 <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 + <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 + 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
- @@ -285,11 +283,11 @@ The inequality means the “direct image under <code class="docutils literal
 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)) + <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) + (@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
- @@ -298,9 +296,9 @@ 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 + <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.
- @@ -319,19 +317,19 @@ This operation could be used to provided another formulation of <code class="doc
 <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) + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (f : ℝ → ℝ) (x₀ y₀ : ℝ) - #check comap ((↑) : ℚ → ℝ) (𝓝 x₀) + #check comap ((↑) : ℚ → ℝ) (𝓝 x₀) - #check Tendsto (f ∘ (↑)) (comap ((↑) : ℚ → ℝ) (𝓝 x₀)) (𝓝 y₀) + #check Tendsto (f ∘ (↑)) (comap ((↑) : ℚ → ℝ) (𝓝 x₀)) (𝓝 y₀) </pre></div> </div> The pullback operation is also compatible with composition, but it is 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} + 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) + #check (comap_comap : comap m (comap n F) = comap (n ∘ m) F) end </pre></div>
- @@ -339,8 +337,8 @@ which is to say, it reverses the order of the arguments.
 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">×ˢ</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 + <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:
- @@ -354,12 +352,12 @@ Thus the <code class="docutils literal notranslate">inf
 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 + <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 + 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,
- @@ -412,8 +410,8 @@ a predicate on <code class="docutils literal notranslate">&#95
 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 + <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">atTop</code>. The lemma
- @@ -422,11 +420,11 @@ us to reformulate a statement of the form <code class="docutils literal notransl
 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 + <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
- @@ -446,9 +444,9 @@ but we can use the more suggestive notation <code class="docutils literal notran
 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 + <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
- @@ -459,13 +457,13 @@ two sequences of real numbers, and let us show that if
 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">EventuallyEq</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 + <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 + 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>.
- @@ -475,25 +473,25 @@ Given <code class="docutils literal notranslate">F <spa
 <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 + <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'⟩ + <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] with 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] with 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
- @@ -518,29 +516,29 @@ topology library.
 See if you can prove it using the quoted lemmas, using the fact that <code class="docutils literal notranslate">ClusterPt x F</code> means <code class="docutils literal notranslate">(𝓝 x ⊓ F).NeBot</code> and that, by definition, the assumption <code class="docutils literal notranslate">∀ᶠ n in atTop, u n ∈ M</code> means <code class="docutils literal notranslate">M ∈ map u atTop</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>#check mem_closure_iff_clusterPt - #check le_principal_iff - #check neBot_of_le + <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 + 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>10.2. Metric spaces<a class="headerlink" href="#metric-spaces" title="Link to this heading"></a></h2> + <h2>10.2. 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) + <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) + #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.
- @@ -548,18 +546,18 @@ They are called <code class="docutils literal notranslate">EMe
 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>10.2.1. Convergence and continuity<a class="headerlink" href="#convergence-and-continuity" title="Link to this heading"></a></h3> + <h3>10.2.1. 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 in 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 + <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 + 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, so we end up proving a lot of continuity results and there
- @@ -567,8 +565,8 @@ is a <code class="docutils literal notranslate">continuity</sp
 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 + <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
- @@ -582,9 +580,9 @@ We can do the same for the second component to get continuity of <code class="do
 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)) + <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,
- @@ -600,15 +598,15 @@ composition and refuse to apply this lemma. It is especially bad at this when pr
 <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 + <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) + 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
- @@ -618,66 +616,66 @@ as an alternate proof term <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' + <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 + <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 + <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>10.2.2. Balls, open sets and closed sets<a class="headerlink" href="#balls-open-sets-and-closed-sets" title="Link to this heading"></a></h3> + <h3>10.2.2. 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) + <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.ball a r = { b | dist b a < r } := + rfl - example : Metric.closedBall 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 + <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 + 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 in 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 + <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 closed set containing it. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {s : Set X} : IsClosed s ↔ IsOpen (sᶜ) := - isOpen_compl_iff.symm + <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} (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 {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 + <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.
- @@ -685,16 +683,16 @@ In the metric space context, the crucial point is that balls provide bases for t
 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 + <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 + 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>10.2.3. Compactness<a class="headerlink" href="#compactness" title="Link to this heading"></a></h3> + <h3>10.2.3. 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">
- @@ -708,43 +706,43 @@ claims for compact sets in general metric spaces. In the second statement we onl
 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 + <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) {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_isMinOn hs' hfs + 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_isMinOn 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_isMaxOn 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_isMaxOn hs' hfs - example {s : Set X} (hs : IsCompact s) : IsClosed s := - hs.isClosed + example {s : Set X} (hs : IsCompact s) : IsClosed s := + hs.isClosed </pre></div> </div> We can also specify that a metric spaces is 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 + <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">IsClosed.isCompact</code>. </section> <section id="uniformly-continuous-functions"> - <h3>10.2.4. Uniformly continuous functions<a class="headerlink" href="#uniformly-continuous-functions" title="Link to this heading"></a></h3> + <h3>10.2.4. 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 + <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
- @@ -760,109 +758,109 @@ And <code class="docutils literal notranslate">K</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 + <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>10.2.5. Completeness<a class="headerlink" href="#completeness" title="Link to this heading"></a></h3> + <h3>10.2.5. 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 + <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 (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 + 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_zero_of_lt_one</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 : ℝ) ^ i := sorry - _ ≤ 1 / 2 ^ N * 2 := sorry - _ < ε := sorry + <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 : ℝ) ^ 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 + <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 + 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` + 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 + 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>10.3. Topological spaces<a class="headerlink" href="#topological-spaces" title="Link to this heading"></a></h2> + <h2>10.3. Topological spaces<a class="headerlink" href="#topological-spaces" title="Permalink to this heading"></a></h2> <section id="fundamentals"> - <h3>10.3.1. Fundamentals<a class="headerlink" href="#fundamentals" title="Link to this heading"></a></h3> + <h3>10.3.1. 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
- @@ -872,28 +870,28 @@ remember the notion of open sets (or equivalently the notion of closed sets). Fr
 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] + variable {X : Type*} [TopologicalSpace X] - example : IsOpen (univ : Set X) := - isOpen_univ + example : IsOpen (univ : Set X) := + isOpen_univ - example : IsOpen (∅ : Set X) := - isOpen_empty + 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*} {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_of_finite hs + example {ι : Type*} [Fintype ι] {s : ι → Set X} (hs : ∀ i, IsOpen (s i)) : + IsOpen (⋂ i, s i) := + isOpen_iInter_of_finite 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] + <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 + 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
- @@ -911,22 +909,22 @@ that <code class="docutils literal notranslate">f 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 is 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 + <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 + <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 + <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
- @@ -934,17 +932,17 @@ 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 + <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 := - h.self_of_nhds + example (x : X) (P : X → Prop) (h : ∀ᶠ y in 𝓝 x, P y) : P x := + h.self_of_nhds </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 + <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
- @@ -952,10 +950,10 @@ structure on <code class="docutils literal notranslate">X</spa
 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">TopologicalSpace.nhds_mkOfNhds</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 + <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">TopologicalSpace.mkOfNhds</code> is not so frequently used, but it still good to know in what
- @@ -977,17 +975,17 @@ sequences of functions is a respectable notion of convergence. But there is no d
 a map <code class="docutils literal notranslate">f : X → (ℝ → ℝ)</code> is continuous if and only if <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*} + <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 X → TopologicalSpace Y := + TopologicalSpace.coinduced f - example (f : X → Y) : TopologicalSpace Y → TopologicalSpace X := - TopologicalSpace.induced 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 + 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.
- @@ -1003,14 +1001,14 @@ on neighborhoods more than open sets so, for any <code class="docutils literal n
 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">TopologicalSpace 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 + <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-forward (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 + <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
- @@ -1021,11 +1019,11 @@ a function $g : Y → Z$
 \[\begin{split}g \text{ continuous } &⇔ g_*(f_*T_X) ≤ T_Z \\ &⇔ (g ∘ f)_* T_X ≤ T_Z \\ &⇔ g ∘ f \text{ continuous}\end{split}\]</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] + <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
- @@ -1042,17 +1040,17 @@ Let us explore that constraint “on paper” using notation 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 + <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>10.3.2. Separation and countability<a class="headerlink" href="#separation-and-countability" title="Link to this heading"></a></h3> + <h3>10.3.2. 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
- @@ -1060,19 +1058,19 @@ is closer to what metric spaces do. The most important is <code class="docutils
 that will ensure that limits are unique. A stronger separation property is <code class="docutils literal notranslate">T3Space</code> that ensures in addition the <cite>RegularSpace</cite> property: each point has a basis of closed neighborhoods. - <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 + <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 + 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 + <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.
- @@ -1090,11 +1088,11 @@ The assumption “tends to $x$<
 <code class="docutils literal notranslate">comap (↑) (𝓝 x)</code>. Let’s first prove 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' := by - sorry + <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' := by + sorry </pre></div> </div> Let’s now turn to the main proof of the extension by continuity theorem.
- @@ -1120,28 +1118,28 @@ Because we know <code class="docutils literal notranslate">Ten
 <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 where the 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] [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 - sorry + <div class="highlight-lean notranslate"><div class="highlight"><pre>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 + sorry - #check HasBasis.tendsto_right_iff + #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 basis. 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] [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 + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [TopologicalSpace X] [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>10.3.3. Compactness<a class="headerlink" href="#id5" title="Link to this heading"></a></h3> + <h3>10.3.3. 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>,
- @@ -1149,14 +1147,14 @@ a point <code class="docutils literal notranslate">x <s
 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] + <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 {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 {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 atTop</code>, the image under <code class="docutils literal notranslate">u : ℕ → X</code> of <code class="docutils literal notranslate">atTop</code>, the generalized set
- @@ -1165,36 +1163,36 @@ large enough. Saying that <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 [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 + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [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] + <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 + 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">NeBot.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 + <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 + <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> </section>
html/C11_Differential_Calculus.html
- @@ -1,25 +1,23 @@
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- @@ -99,8 +97,8 @@
 <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> - <section id="index-0"> - <h1>11. Differential Calculus<a class="headerlink" href="#index-0" title="Link to this heading"></a></h1> + <section id="index-0"> + <h1>11. 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.
- @@ -110,15 +108,15 @@ which is familiar from any introductory calculus class.
 In <a class="reference internal" href="#normed-spaces">Section 11.2</a>, we then consider the notion of a derivative in a much broader setting. <section id="elementary-differential-calculus"> - <h2>11.1. Elementary Differential Calculus<a class="headerlink" href="#elementary-differential-calculus" title="Link to this heading"></a></h2> + <h2>11.1. 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 + <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 + 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
- @@ -128,8 +126,8 @@ We specify <code class="docutils literal notranslate">ℝ<
 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 + <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
- @@ -137,19 +135,19 @@ 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 + <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 + 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 + <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
- @@ -158,76 +156,76 @@ of the fact that the value 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 + <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 even state Rolle’s theorem without any differentiability assumptions, which seems even weirder. - <div class="highlight-lean notranslate"><div class="highlight"><pre>open Set + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Set - 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 hab hfc hfI + 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 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' + <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 + <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 + example : deriv sin π = -1 := by simp </pre></div> </div> </section> <section id="differential-calculus-in-normed-spaces"> - <h2>11.2. Differential Calculus in Normed Spaces<a class="headerlink" href="#differential-calculus-in-normed-spaces" title="Link to this heading"></a></h2> + <h2>11.2. 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>11.2.1. Normed spaces<a class="headerlink" href="#id3" title="Link to this heading"></a></h3> + <h3>11.2.1. 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 is 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] + <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) : 0 ≤ ‖x‖ := + norm_nonneg x - example {x : E} : ‖x‖ = 0 ↔ x = 0 := - norm_eq_zero + example {x : E} : ‖x‖ = 0 ↔ x = 0 := + norm_eq_zero - example (x y : E) : ‖x + y‖ ≤ ‖x‖ + ‖y‖ := - norm_add_le x y + 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 + <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 + 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">NormedSpace ℝ E</code> on top of <code class="docutils literal notranslate">NormedAddGroup 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] + <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 + 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 + <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.
- @@ -236,23 +234,23 @@ More generally, we can make sense of calculus with a vector space over any
 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 + <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 + 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 nontrivially 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 + <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>11.2.2. Continuous linear maps<a class="headerlink" href="#continuous-linear-maps" title="Link to this heading"></a></h3> + <h3>11.2.2. 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
- @@ -262,34 +260,34 @@ 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] + <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 : E →L[𝕜] E := + ContinuousLinearMap.id 𝕜 E - example (f : E →L[𝕜] F) : E → F := - f + 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) : 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) (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 + 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) + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (f : E →L[𝕜] F) - example (x : E) : ‖f x‖ ≤ ‖f‖ * ‖x‖ := - f.le_opNorm x + example (x : E) : ‖f x‖ ≤ ‖f‖ * ‖x‖ := + f.le_opNorm x - example {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ x, ‖f x‖ ≤ M * ‖x‖) : ‖f‖ ≤ M := - f.opNorm_le_bound hMp hM + example {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ x, ‖f x‖ ≤ M * ‖x‖) : ‖f‖ ≤ M := + f.opNorm_le_bound hMp hM </pre></div> </div> There is also a notion of bundled continuous linear isomorphism.
- @@ -303,82 +301,82 @@ The main ingredient is Baire’s theorem
 <code class="docutils literal notranslate">nonempty_interior_of_iUnion_of_closed</code>. (You proved a version of this in the topology chapter.) Minor ingredients include <code class="docutils literal notranslate">continuous_linear_map.opNorm_le_of_shell</code>, <code class="docutils literal notranslate">interior_subset</code> and <code class="docutils literal notranslate">interior_iInter_subset</code> and <code class="docutils literal notranslate">isClosed_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`, + <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.opNorm_le_of_shell ε_pos ?_ hk ?_⟩ - sorry - sorry + 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.opNorm_le_of_shell ε_pos ?_ hk ?_⟩ + sorry + sorry </pre></div> </div> </section> <section id="asymptotic-comparisons"> - <h3>11.2.3. Asymptotic comparisons<a class="headerlink" href="#asymptotic-comparisons" title="Link to this heading"></a></h3> + <h3>11.2.3. 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 + <div class="highlight-lean notranslate"><div class="highlight"><pre>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] (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, 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*} [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 + 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>11.2.4. Differentiability<a class="headerlink" href="#differentiability" title="Link to this heading"></a></h3> + <h3>11.2.4. 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">HasFDerivAt</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>open Topology + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Topology - variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E] - [NormedSpace 𝕜 E] {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F] + 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 := - hasFDerivAtFilter_iff_isLittleO .. + 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 := + hasFDerivAtFilter_iff_isLittleO .. - example (f : E → F) (f' : E →L[𝕜] F) (x₀ : E) (hff' : HasFDerivAt f f' x₀) : fderiv 𝕜 f x₀ = f' := - hff'.fderiv + 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
- @@ -388,14 +386,14 @@ The type <code class="docutils literal notranslate">WithTop</s
 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">ContDiff 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 + <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
- @@ -404,10 +402,10 @@ 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*} [RCLike 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 + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {𝕂 : Type*} [RCLike 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
- @@ -417,26 +415,26 @@ point <code class="docutils literal notranslate">a</cod
 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} + <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 → 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 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 (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 {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 + example {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 + end LocalInverse </pre></div> </div> This has been only a quick tour of the differential calculus in Mathlib.
html/C12_Integration_and_Measure_Theory.html
- @@ -1,25 +1,23 @@
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- @@ -94,22 +92,22 @@
 <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> - <section id="index-0"> - <h1>12. Integration and Measure Theory<a class="headerlink" href="#index-0" title="Link to this heading"></a></h1> + <section id="index-0"> + <h1>12. Integration and Measure Theory<a class="headerlink" href="#index-0" title="Permalink to this heading"></a></h1> <section id="elementary-integration"> - <h2>12.1. Elementary Integration<a class="headerlink" href="#elementary-integration" title="Link to this heading"></a></h2> + <h2>12.1. Elementary Integration<a class="headerlink" href="#elementary-integration" title="Permalink to this heading"></a></h2> We first focus on integration of functions on finite intervals in <code class="docutils literal notranslate">ℝ</code>. We can integrate elementary functions. - <div class="highlight-lean notranslate"><div class="highlight"><pre>open MeasureTheory intervalIntegral + <div class="highlight-lean notranslate"><div class="highlight"><pre>open MeasureTheory intervalIntegral - open Interval + open Interval -- this introduces the notation `[[a, b]]` for the segment from `min a b` to `max a b` - example (a b : ℝ) : (∫ x in a..b, x) = (b ^ 2 - a ^ 2) / 2 := - integral_id + example (a b : ℝ) : (∫ x in a..b, x) = (b ^ 2 - a ^ 2) / 2 := + integral_id - example {a b : ℝ} (h : (0 : ℝ) ∉ [[a, b]]) : (∫ x in a..b, 1 / x) = Real.log (b / a) := - integral_one_div h + example {a b : ℝ} (h : (0 : ℝ) ∉ [[a, b]]) : (∫ x in a..b, 1 / x) = Real.log (b / a) := + integral_one_div h </pre></div> </div> The fundamental theorem of calculus relates integration and differentiation.
- @@ -118,25 +116,25 @@ says that integration provides an inverse to differentiation and the second one
 specifies how to compute integrals of derivatives. (These two parts are very closely related, but their optimal versions, which are not shown here, are not equivalent.) - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (f : ℝ → ℝ) (hf : Continuous f) (a b : ℝ) : deriv (fun u ↦ ∫ x : ℝ in a..u, f x) b = f b := - (integral_hasStrictDerivAt_right (hf.intervalIntegrable _ _) (hf.stronglyMeasurableAtFilter _ _) - hf.continuousAt).hasDerivAt.deriv + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (f : ℝ → ℝ) (hf : Continuous f) (a b : ℝ) : deriv (fun u ↦ ∫ x : ℝ in a..u, f x) b = f b := + (integral_hasStrictDerivAt_right (hf.intervalIntegrable _ _) (hf.stronglyMeasurableAtFilter _ _) + hf.continuousAt).hasDerivAt.deriv - example {f : ℝ → ℝ} {a b : ℝ} {f' : ℝ → ℝ} (h : ∀ x ∈ [[a, b]], HasDerivAt f (f' x) x) - (h' : IntervalIntegrable f' volume a b) : (∫ y in a..b, f' y) = f b - f a := - integral_eq_sub_of_hasDerivAt h h' + example {f : ℝ → ℝ} {a b : ℝ} {f' : ℝ → ℝ} (h : ∀ x ∈ [[a, b]], HasDerivAt f (f' x) x) + (h' : IntervalIntegrable f' volume a b) : (∫ y in a..b, f' y) = f b - f a := + integral_eq_sub_of_hasDerivAt h h' </pre></div> </div> Convolution is also defined in Mathlib and its basic properties are proved. - <div class="highlight-lean notranslate"><div class="highlight"><pre>open Convolution + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Convolution - example (f : ℝ → ℝ) (g : ℝ → ℝ) : f ⋆ g = fun x ↦ ∫ t, f t * g (x - t) := - rfl + example (f : ℝ → ℝ) (g : ℝ → ℝ) : f ⋆ g = fun x ↦ ∫ t, f t * g (x - t) := + rfl </pre></div> </div> </section> <section id="measure-theory"> - <h2>12.2. Measure Theory<a class="headerlink" href="#measure-theory" title="Link to this heading"></a></h2> + <h2>12.2. Measure Theory<a class="headerlink" href="#measure-theory" title="Permalink to this heading"></a></h2> The general context for integration in Mathlib is measure theory. Even the elementary integrals of the previous section are in fact Bochner integrals. Bochner integration is a generalization of Lebesgue integration where the target space can be any Banach space,
- @@ -152,28 +150,28 @@ Note that these axioms are redundant; if you <code class="docutils literal notra
 you will see the ones that Mathlib uses. As the examples below show, countability assumptions can be expressed using the <code class="docutils literal notranslate">Encodable</code> type class. - <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type*} [MeasurableSpace α] + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type*} [MeasurableSpace α] - example : MeasurableSet (∅ : Set α) := - MeasurableSet.empty + example : MeasurableSet (∅ : Set α) := + MeasurableSet.empty - example : MeasurableSet (univ : Set α) := - MeasurableSet.univ + example : MeasurableSet (univ : Set α) := + MeasurableSet.univ - example {s : Set α} (hs : MeasurableSet s) : MeasurableSet (sᶜ) := - hs.compl + example {s : Set α} (hs : MeasurableSet s) : MeasurableSet (sᶜ) := + hs.compl - example : Encodable ℕ := by infer_instance + example : Encodable ℕ := by infer_instance - example (n : ℕ) : Encodable (Fin n) := by infer_instance + example (n : ℕ) : Encodable (Fin n) := by infer_instance - variable {ι : Type*} [Encodable ι] + variable {ι : Type*} [Encodable ι] - example {f : ι → Set α} (h : ∀ b, MeasurableSet (f b)) : MeasurableSet (⋃ b, f b) := - MeasurableSet.iUnion h + example {f : ι → Set α} (h : ∀ b, MeasurableSet (f b)) : MeasurableSet (⋃ b, f b) := + MeasurableSet.iUnion h - example {f : ι → Set α} (h : ∀ b, MeasurableSet (f b)) : MeasurableSet (⋂ b, f b) := - MeasurableSet.iInter h + example {f : ι → Set α} (h : ∀ b, MeasurableSet (f b)) : MeasurableSet (⋂ b, f b) := + MeasurableSet.iInter h </pre></div> </div> Once a type is measurable, we can measure it. On paper, a measure on a set
- @@ -187,18 +185,18 @@ So we extend the measure to any set <code class="docutils literal notranslate"><
 as the infimum of measures of measurable sets containing <code class="docutils literal notranslate">s</code>. Of course, many lemmas still require measurability assumptions, but not all. - <div class="highlight-lean notranslate"><div class="highlight"><pre>open MeasureTheory - variable {μ : Measure α} + <div class="highlight-lean notranslate"><div class="highlight"><pre>open MeasureTheory + variable {μ : Measure α} - example (s : Set α) : μ s = ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), μ t := - measure_eq_iInf s + example (s : Set α) : μ s = ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), μ t := + measure_eq_iInf s - example (s : ι → Set α) : μ (⋃ i, s i) ≤ ∑' i, μ (s i) := - measure_iUnion_le s + example (s : ι → Set α) : μ (⋃ i, s i) ≤ ∑' i, μ (s i) := + measure_iUnion_le s - example {f : ℕ → Set α} (hmeas : ∀ i, MeasurableSet (f i)) (hdis : Pairwise (Disjoint on f)) : - μ (⋃ i, f i) = ∑' i, μ (f i) := - μ.m_iUnion hmeas hdis + example {f : ℕ → Set α} (hmeas : ∀ i, MeasurableSet (f i)) (hdis : Pairwise (Disjoint on f)) : + μ (⋃ i, f i) = ∑' i, μ (f i) := + μ.m_iUnion hmeas hdis </pre></div> </div> Once a type has a measure associated with it, we say that a property <code class="docutils literal notranslate">P</code>
- @@ -207,13 +205,13 @@ has measure 0.
 The collection of properties that hold almost everywhere form a filter, but Mathlib introduces special notation for saying that a property holds almost everywhere. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {P : α → Prop} : (∀ᵐ x ∂μ, P x) ↔ ∀ᶠ x in ae μ, P x := - Iff.rfl + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {P : α → Prop} : (∀ᵐ x ∂μ, P x) ↔ ∀ᶠ x in ae μ, P x := + Iff.rfl </pre></div> </div> </section> <section id="integration"> - <h2>12.3. Integration<a class="headerlink" href="#integration" title="Link to this heading"></a></h2> + <h2>12.3. Integration<a class="headerlink" href="#integration" title="Permalink to this heading"></a></h2> Now that we have measurable spaces and measures we can consider integrals. As explained above, Mathlib uses a very general notion of integration that allows any Banach space as the target.
- @@ -223,11 +221,11 @@ that an integral is equal to zero if the function in question is
 not integrable. Most lemmas having to do with integrals have integrability assumptions. <div class="highlight-lean notranslate"><div class="highlight"><pre>section - variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : α → E} + variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : α → E} - example {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) : - ∫ a, f a + g a ∂μ = ∫ a, f a ∂μ + ∫ a, g a ∂μ := - integral_add hf hg + example {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) : + ∫ a, f a + g a ∂μ = ∫ a, f a ∂μ + ∫ a, g a ∂μ := + integral_add hf hg </pre></div> </div> As an example of the complex interactions between our various conventions, let us see how to integrate constant functions.
- @@ -237,41 +235,41 @@ the point at infinity, to zero.
 For any <code class="docutils literal notranslate">s : Set α</code>, if <code class="docutils literal notranslate">μ s = ⊤</code>, then nonzero constant functions are not integrable on <code class="docutils literal notranslate">s</code>. In that case, their integrals are equal to zero by definition, as is <code class="docutils literal notranslate">(μ s).toReal</code>. So in all cases we have the following lemma. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {s : Set α} (c : E) : ∫ x in s, c ∂μ = (μ s).toReal • c := - setIntegral_const c + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {s : Set α} (c : E) : ∫ x in s, c ∂μ = (μ s).toReal • c := + setIntegral_const c </pre></div> </div> We now quickly explain how to access the most important theorems in integration theory, starting with the dominated convergence theorem. There are several versions in Mathlib, and here we only show the most basic one. - <div class="highlight-lean notranslate"><div class="highlight"><pre>open Filter + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Filter - example {F : ℕ → α → E} {f : α → E} (bound : α → ℝ) (hmeas : ∀ n, AEStronglyMeasurable (F n) μ) - (hint : Integrable bound μ) (hbound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) - (hlim : ∀ᵐ a ∂μ, Tendsto (fun n : ℕ ↦ F n a) atTop (𝓝 (f a))) : - Tendsto (fun n ↦ ∫ a, F n a ∂μ) atTop (𝓝 (∫ a, f a ∂μ)) := - tendsto_integral_of_dominated_convergence bound hmeas hint hbound hlim + example {F : ℕ → α → E} {f : α → E} (bound : α → ℝ) (hmeas : ∀ n, AEStronglyMeasurable (F n) μ) + (hint : Integrable bound μ) (hbound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) + (hlim : ∀ᵐ a ∂μ, Tendsto (fun n : ℕ ↦ F n a) atTop (𝓝 (f a))) : + Tendsto (fun n ↦ ∫ a, F n a ∂μ) atTop (𝓝 (∫ a, f a ∂μ)) := + tendsto_integral_of_dominated_convergence bound hmeas hint hbound hlim </pre></div> </div> Then we have Fubini’s theorem for integrals on product type. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {α : Type*} [MeasurableSpace α] {μ : Measure α} [SigmaFinite μ] {β : Type*} - [MeasurableSpace β] {ν : Measure β} [SigmaFinite ν] (f : α × β → E) - (hf : Integrable f (μ.prod ν)) : ∫ z, f z ∂ μ.prod ν = ∫ x, ∫ y, f (x, y) ∂ν ∂μ := - integral_prod f hf + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {α : Type*} [MeasurableSpace α] {μ : Measure α} [SigmaFinite μ] {β : Type*} + [MeasurableSpace β] {ν : Measure β} [SigmaFinite ν] (f : α × β → E) + (hf : Integrable f (μ.prod ν)) : ∫ z, f z ∂ μ.prod ν = ∫ x, ∫ y, f (x, y) ∂ν ∂μ := + integral_prod f hf </pre></div> </div> There is a very general version of convolution that applies to any continuous bilinear form. - <div class="highlight-lean notranslate"><div class="highlight"><pre>open Convolution + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Convolution - variable {𝕜 : Type*} {G : Type*} {E : Type*} {E' : Type*} {F : Type*} [NormedAddCommGroup E] - [NormedAddCommGroup E'] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] - [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] [MeasurableSpace G] [NormedSpace ℝ F] [CompleteSpace F] - [Sub G] + variable {𝕜 : Type*} {G : Type*} {E : Type*} {E' : Type*} {F : Type*} [NormedAddCommGroup E] + [NormedAddCommGroup E'] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] + [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] [MeasurableSpace G] [NormedSpace ℝ F] [CompleteSpace F] + [Sub G] - example (f : G → E) (g : G → E') (L : E →L[𝕜] E' →L[𝕜] F) (μ : Measure G) : - f ⋆[L, μ] g = fun x ↦ ∫ t, L (f t) (g (x - t)) ∂μ := - rfl + example (f : G → E) (g : G → E') (L : E →L[𝕜] E' →L[𝕜] F) (μ : Measure G) : + f ⋆[L, μ] g = fun x ↦ ∫ t, L (f t) (g (x - t)) ∂μ := + rfl </pre></div> </div> Finally, Mathlib has a very general version of the change-of-variables formula.
- @@ -279,13 +277,13 @@ In the statement below, <code class="docutils literal notranslate">$\sigma$-algebra on <code class="docutils literal notranslate">E</code> is generated by the open sets of <code class="docutils literal notranslate">E</code>, and <code class="docutils literal notranslate">IsAddHaarMeasure μ</code> means that the measure <code class="docutils literal notranslate">μ</code> is left-invariant, gives finite mass to compact sets, and give positive mass to open sets. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] - [MeasurableSpace E] [BorelSpace E] (μ : Measure E) [μ.IsAddHaarMeasure] {F : Type*} - [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {s : Set E} {f : E → E} - {f' : E → E →L[ℝ] E} (hs : MeasurableSet s) - (hf : ∀ x : E, x ∈ s → HasFDerivWithinAt f (f' x) s x) (h_inj : InjOn f s) (g : E → F) : - ∫ x in f '' s, g x ∂μ = ∫ x in s, |(f' x).det| • g (f x) ∂μ := - integral_image_eq_integral_abs_det_fderiv_smul μ hs hf h_inj g + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] + [MeasurableSpace E] [BorelSpace E] (μ : Measure E) [μ.IsAddHaarMeasure] {F : Type*} + [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {s : Set E} {f : E → E} + {f' : E → E →L[ℝ] E} (hs : MeasurableSet s) + (hf : ∀ x : E, x ∈ s → HasFDerivWithinAt f (f' x) s x) (h_inj : InjOn f s) (g : E → F) : + ∫ x in f '' s, g x ∂μ = ∫ x in s, |(f' x).det| • g (f x) ∂μ := + integral_image_eq_integral_abs_det_fderiv_smul μ hs hf h_inj g </pre></div> </div> </section>
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.versionmodified{font-style:italic}@media screen and (max-width:480px){.rst-content .sidebar{width:100%}}span[id*=MathJax-Span]{color:#404040}.math{text-align:center}@font-face{font-family:Lato;src:url(fonts/lato-normal.woff2?bd03a2cc277bbbc338d464e679fe9942) format("woff2"),url(fonts/lato-normal.woff?27bd77b9162d388cb8d4c4217c7c5e2a) format("woff");font-weight:400;font-style:normal;font-display:block}@font-face{font-family:Lato;src:url(fonts/lato-bold.woff2?cccb897485813c7c256901dbca54ecf2) format("woff2"),url(fonts/lato-bold.woff?d878b6c29b10beca227e9eef4246111b) format("woff");font-weight:700;font-style:normal;font-display:block}@font-face{font-family:Lato;src:url(fonts/lato-bold-italic.woff2?0b6bb6725576b072c5d0b02ecdd1900d) format("woff2"),url(fonts/lato-bold-italic.woff?9c7e4e9eb485b4a121c760e61bc3707c) format("woff");font-weight:700;font-style:italic;font-display:block}@font-face{font-family:Lato;src:url(fonts/lato-normal-italic.woff2?4eb103b4d12be57cb1d040ed5e162e9d) 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html/_static/doctools.js
- @@ -4,7 +4,7 @@
  * * Base JavaScript utilities for all Sphinx HTML documentation. * - * :copyright: Copyright 2007-2024 by the Sphinx team, see AUTHORS. + * :copyright: Copyright 2007-2023 by the Sphinx team, see AUTHORS. * :license: BSD, see LICENSE for details. * */
html/_static/documentation_options.js
- @@ -1,4 +1,5 @@
  - const DOCUMENTATION_OPTIONS = { + var DOCUMENTATION_OPTIONS = { + URL_ROOT: document.getElementById("documentation_options").getAttribute('data-url_root'), VERSION: '0.1', LANGUAGE: 'en', COLLAPSE_INDEX: false,
html/_static/language_data.js
- @@ -5,7 +5,7 @@
  * This script contains the language-specific data used by searchtools.js, * namely the list of stopwords, stemmer, scorer and splitter. * - * :copyright: Copyright 2007-2024 by the Sphinx team, see AUTHORS. + * :copyright: Copyright 2007-2023 by the Sphinx team, see AUTHORS. * :license: BSD, see LICENSE for details. * */
- @@ -13,7 +13,7 @@
  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, if available */ + /* Non-minified version is copied as a separate JS file, is available */ /** * Porter Stemmer
html/_static/pygments.css
- @@ -17,7 +17,6 @@ span.linenos.special { color: #000000; background-color: #ffffc0; padding-left:
  .highlight .cs { color: #3D7B7B; font-style: italic } /* Comment.Special */ .highlight .gd { color: #A00000 } /* Generic.Deleted */ .highlight .ge { font-style: italic } /* Generic.Emph */ - .highlight .ges { font-weight: bold; font-style: italic } /* Generic.EmphStrong */ .highlight .gr { color: #E40000 } /* Generic.Error */ .highlight .gh { color: #000080; font-weight: bold } /* Generic.Heading */ .highlight .gi { color: #008400 } /* Generic.Inserted */
html/_static/searchtools.js
- @@ -4,7 +4,7 @@
  * * Sphinx JavaScript utilities for the full-text search. * - * :copyright: Copyright 2007-2024 by the Sphinx team, see AUTHORS. + * :copyright: Copyright 2007-2023 by the Sphinx team, see AUTHORS. * :license: BSD, see LICENSE for details. * */
- @@ -57,12 +57,12 @@ const _removeChildren = (element) => {
  const _escapeRegExp = (string) => string.replace(/[.*+\-?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string - const _displayItem = (item, searchTerms, highlightTerms) => { + const _displayItem = (item, searchTerms) => { const docBuilder = DOCUMENTATION_OPTIONS.BUILDER; + const docUrlRoot = DOCUMENTATION_OPTIONS.URL_ROOT; const docFileSuffix = DOCUMENTATION_OPTIONS.FILE_SUFFIX; const docLinkSuffix = DOCUMENTATION_OPTIONS.LINK_SUFFIX; const showSearchSummary = DOCUMENTATION_OPTIONS.SHOW_SEARCH_SUMMARY; - const contentRoot = document.documentElement.dataset.content_root; const [docName, title, anchor, descr, score, _filename] = item;
- @@ -75,35 +75,28 @@ const _displayItem = (item, searchTerms, highlightTerms) => {
  if (dirname.match(/\/index\/$/)) dirname = dirname.substring(0, dirname.length - 6); else if (dirname === "index/") dirname = ""; - requestUrl = contentRoot + dirname; + requestUrl = docUrlRoot + dirname; linkUrl = requestUrl; } else { // normal html builders - requestUrl = contentRoot + docName + docFileSuffix; + 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) { + if (descr) listItem.appendChild(document.createElement("span")).innerHTML = " (" + descr + ")"; - // highlight search terms in the description - if (SPHINX_HIGHLIGHT_ENABLED) // set in sphinx_highlight.js - highlightTerms.forEach((term) => _highlightText(listItem, term, "highlighted")); - } else if (showSearchSummary) fetch(requestUrl) .then((responseData) => responseData.text()) .then((data) => { if (data) listItem.appendChild( - Search.makeSearchSummary(data, searchTerms, anchor) + Search.makeSearchSummary(data, searchTerms) ); - // highlight search terms in the summary - if (SPHINX_HIGHLIGHT_ENABLED) // set in sphinx_highlight.js - highlightTerms.forEach((term) => _highlightText(listItem, term, "highlighted")); }); Search.output.appendChild(listItem); };
- @@ -116,43 +109,26 @@ const _finishSearch = (resultCount) => {
  ); else Search.status.innerText = _( - "Search finished, found ${resultCount} page(s) matching the search query." - ).replace('${resultCount}', resultCount); + `Search finished, found ${resultCount} page(s) matching the search query.` + ); }; const _displayNextItem = ( results, resultCount, - searchTerms, - highlightTerms, + searchTerms ) => { // results left, load the summary and display it // this is intended to be dynamic (don't sub resultsCount) if (results.length) { - _displayItem(results.pop(), searchTerms, highlightTerms); + _displayItem(results.pop(), searchTerms); setTimeout( - () => _displayNextItem(results, resultCount, searchTerms, highlightTerms), + () => _displayNextItem(results, resultCount, searchTerms), 5 ); } // search finished, update title and status message else _finishSearch(resultCount); }; - // Helper function used by query() to order search results. - // Each input is an array of [docname, title, anchor, descr, score, filename]. - // Order the results by score (in opposite order of appearance, since the - // `_displayNextItem` function uses pop() to retrieve items) and then alphabetically. - const _orderResultsByScoreThenName = (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; - }; /** * Default splitQuery function. Can be overridden in ``sphinx.search`` with a
- @@ -176,26 +152,13 @@ const Search = {
  _queued_query: null, _pulse_status: -1, - htmlToText: (htmlString, anchor) => { + htmlToText: (htmlString) => { const htmlElement = new DOMParser().parseFromString(htmlString, 'text/html'); - for (const removalQuery of [".headerlink", "script", "style"]) { - htmlElement.querySelectorAll(removalQuery).forEach((el) => { el.remove() }); - } - if (anchor) { - const anchorContent = htmlElement.querySelector(`[role="main"] ${anchor}`); - if (anchorContent) return anchorContent.textContent; - - console.warn( - `Anchored content block not found. Sphinx search tries to obtain it via DOM query '[role=main] ${anchor}'. Check your theme or template.` - ); - } - - // if anchor not specified or not found, fall back to main content + htmlElement.querySelectorAll(".headerlink").forEach((el) => { el.remove() }); const docContent = htmlElement.querySelector('[role="main"]'); - if (docContent) return docContent.textContent; - + if (docContent !== undefined) return docContent.textContent; console.warn( - "Content block not found. Sphinx search tries to obtain it via DOM query '[role=main]'. Check your theme or template." + "Content block not found. Sphinx search tries to obtain it via '[role=main]'. Could you check your theme or template." ); return ""; },
- @@ -268,7 +231,16 @@ const Search = {
  else Search.deferQuery(query); }, - _parseQuery: (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();
- @@ -304,38 +276,21 @@ const Search = {
  // console.info("required: ", [...searchTerms]); // console.info("excluded: ", [...excludedTerms]); - return [query, searchTerms, excludedTerms, highlightTerms, objectTerms]; - }, - - /** - * execute search (requires search index to be loaded) - */ - _performSearch: (query, searchTerms, excludedTerms, highlightTerms, objectTerms) => { - 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; - - // Collect multiple result groups to be sorted separately and then ordered. - // Each is an array of [docname, title, anchor, descr, score, filename]. - const normalResults = []; - const nonMainIndexResults = []; - + // array of [docname, title, anchor, descr, score, filename] + let results = []; _removeChildren(document.getElementById("search-progress")); - const queryLower = query.toLowerCase().trim(); + const queryLower = query.toLowerCase(); for (const [title, foundTitles] of Object.entries(allTitles)) { - if (title.toLowerCase().trim().includes(queryLower) && (queryLower.length >= title.length/2)) { + if (title.toLowerCase().includes(queryLower) && (queryLower.length >= title.length/2)) { for (const [file, id] of foundTitles) { - const score = Math.round(Scorer.title * queryLower.length / title.length); - const boost = titles[file] === title ? 1 : 0; // add a boost for document titles - normalResults.push([ + 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 + boost, + score, filenames[file], ]); }
- @@ -345,47 +300,46 @@ const Search = {
  // 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, isMain] of foundEntries) { - const score = Math.round(100 * queryLower.length / entry.length); - const result = [ + 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], - ]; - if (isMain) { - normalResults.push(result); - } else { - nonMainIndexResults.push(result); - } + ]); } } } // lookup as object objectTerms.forEach((term) => - normalResults.push(...Search.performObjectSearch(term, objectTerms)) + results.push(...Search.performObjectSearch(term, objectTerms)) ); // lookup as search terms in fulltext - normalResults.push(...Search.performTermsSearch(searchTerms, excludedTerms)); + results.push(...Search.performTermsSearch(searchTerms, excludedTerms)); // let the scorer override scores with a custom scoring function - if (Scorer.score) { - normalResults.forEach((item) => (item[4] = Scorer.score(item))); - nonMainIndexResults.forEach((item) => (item[4] = Scorer.score(item))); - } - - // Sort each group of results by score and then alphabetically by name. - normalResults.sort(_orderResultsByScoreThenName); - nonMainIndexResults.sort(_orderResultsByScoreThenName); - - // Combine the result groups in (reverse) order. - // Non-main index entries are typically arbitrary cross-references, - // so display them after other results. - let results = [...nonMainIndexResults, ...normalResults]; + 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
- @@ -399,19 +353,14 @@ const Search = {
  return acc; }, []); - return results.reverse(); - }, - - query: (query) => { - const [searchQuery, searchTerms, excludedTerms, highlightTerms, objectTerms] = Search._parseQuery(query); - const results = Search._performSearch(searchQuery, searchTerms, excludedTerms, highlightTerms, objectTerms); + 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, highlightTerms); + _displayNextItem(results, results.length, searchTerms); }, /**
- @@ -509,18 +458,14 @@ const Search = {
  // add support for partial matches if (word.length > 2) { const escapedWord = _escapeRegExp(word); - if (!terms.hasOwnProperty(word)) { - Object.keys(terms).forEach((term) => { - if (term.match(escapedWord)) - arr.push({ files: terms[term], score: Scorer.partialTerm }); - }); - } - if (!titleTerms.hasOwnProperty(word)) { - Object.keys(titleTerms).forEach((term) => { - if (term.match(escapedWord)) - arr.push({ files: titleTerms[term], score: Scorer.partialTitle }); - }); - } + Object.keys(terms).forEach((term) => { + if (term.match(escapedWord) && !terms[word]) + arr.push({ files: terms[term], score: Scorer.partialTerm }); + }); + Object.keys(titleTerms).forEach((term) => { + if (term.match(escapedWord) && !titleTerms[word]) + arr.push({ files: titleTerms[word], score: Scorer.partialTitle }); + }); } // no match but word was a required one
- @@ -543,8 +488,9 @@ const Search = {
  // create the mapping files.forEach((file) => { - if (!fileMap.has(file)) fileMap.set(file, [word]); - else if (fileMap.get(file).indexOf(word) === -1) fileMap.get(file).push(word); + if (fileMap.has(file) && fileMap.get(file).indexOf(word) === -1) + fileMap.get(file).push(word); + else fileMap.set(file, [word]); }); });
- @@ -595,8 +541,8 @@ const Search = {
  * search summary for a given text. keywords is a list * of stemmed words. */ - makeSearchSummary: (htmlText, keywords, anchor) => { - const text = Search.htmlToText(htmlText, anchor); + makeSearchSummary: (htmlText, keywords) => { + const text = Search.htmlToText(htmlText); if (text === "") return null; const textLower = text.toLowerCase();
html/_static/sphinx_highlight.js
- @@ -29,19 +29,14 @@ const _highlight = (node, addItems, text, className) => {
  } span.appendChild(document.createTextNode(val.substr(pos, text.length))); - const rest = document.createTextNode(val.substr(pos + text.length)); parent.insertBefore( span, parent.insertBefore( - rest, + document.createTextNode(val.substr(pos + text.length)), node.nextSibling ) ); node.nodeValue = val.substr(0, pos); - /* There may be more occurrences of search term in this node. So call this - * function recursively on the remaining fragment. - */ - _highlight(rest, addItems, text, className); if (isInSVG) { const rect = document.createElementNS(
- @@ -145,10 +140,5 @@ const SphinxHighlight = {
  }, }; - _ready(() => { - /* Do not call highlightSearchWords() when we are on the search page. - * It will highlight words from the *previous* search query. - */ - if (typeof Search === "undefined") SphinxHighlight.highlightSearchWords(); - SphinxHighlight.initEscapeListener(); - }); + _ready(SphinxHighlight.highlightSearchWords); + _ready(SphinxHighlight.initEscapeListener);
html/genindex.html
- @@ -1,24 +1,22 @@
 <!DOCTYPE html> - <html class="writer-html5" lang="en" data-content_root="./"> + <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" type="text/css" href="_static/pygments.css?v=80d5e7a1" /> - <link rel="stylesheet" type="text/css" href="_static/css/theme.css?v=19f00094" /> - <link rel="stylesheet" type="text/css" href="_static/css/custom.css?v=0731ccc3" /> - - + <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?v=5d32c60e"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js?v=2cd50e6c"></script> - <script src="_static/documentation_options.js?v=2709fde1"></script> - <script src="_static/doctools.js?v=9a2dae69"></script> - <script src="_static/sphinx_highlight.js?v=dc90522c"></script> + <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" />
html/index.html
- @@ -1,26 +1,23 @@
 <!DOCTYPE html> - <html class="writer-html5" lang="en" data-content_root="./"> + <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="viewport" content="width=device-width, initial-scale=1" /> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>Mathematics in Lean — Mathematics in Lean 0.1 documentation</title> - <link rel="stylesheet" type="text/css" href="_static/pygments.css?v=80d5e7a1" /> - <link rel="stylesheet" type="text/css" href="_static/css/theme.css?v=19f00094" /> - <link rel="stylesheet" type="text/css" href="_static/css/custom.css?v=0731ccc3" /> - - + <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?v=5d32c60e"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js?v=2cd50e6c"></script> - <script src="_static/documentation_options.js?v=2709fde1"></script> - <script src="_static/doctools.js?v=9a2dae69"></script> - <script src="_static/sphinx_highlight.js?v=dc90522c"></script> - <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> + <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" />
- @@ -89,7 +86,7 @@
 <div itemprop="articleBody"> <section id="mathematics-in-lean"> - <h1>Mathematics in Lean<a class="headerlink" href="#mathematics-in-lean" title="Link to this heading"></a></h1> + <h1>Mathematics in Lean<a class="headerlink" href="#mathematics-in-lean" title="Permalink to this heading"></a></h1> <div class="toctree-wrapper compound"> <ul> <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a><ul>
html/objects.inv
html/search.html
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Introduction", "2. Basics", "3. Logic", "4. Sets and Functions", "5. Elementary Number Theory", "6. Structures", "7. Hierarchies", "8. Groups and Rings", "9. Linear algebra", "10. Topology", "11. Differential Calculus", "12. 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