2e14db9f - mathematics_in

Authored by: Patrick Massot
Authored at: 2023-06-05 07:09:10 +0200
Committed by: Patrick Massot
Committed at: 2023-06-05 07:09:10 +0200
Object name: 2e14db9fcbb75644c986287c0d17380b3ec45c49
Parent: f3109b5d

Update lun. 05 juin 2023 07:09:10 CEST

Changes

196 changed files (+21119/-13467)

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: 88166e69df2da4dbd33869bb72fe3995 + config: 798c3985ff3075064f96be164dceb162 tags: 645f666f9bcd5a90fca523b33c5a78b7
html/C01_Introduction.html
- @@ -1,7 +1,7 @@
 <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + <meta 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>
- @@ -13,11 +13,11 @@
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- @@ -30,11 +30,15 @@
 <nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > - <a href="index.html" class="icon icon-home"> Mathematics in Lean + + + + <a href="index.html" class="icon icon-home"> + Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> - <input type="text" name="q" placeholder="Search docs" /> + <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
- @@ -50,8 +54,10 @@
 <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> - <li class="toctree-l1"><a class="reference internal" href="C06_Abstract_Algebra.html">6. Abstract Algebra</a></li> - <li class="toctree-l1"><a class="reference internal" href="C07_Topology.html">7. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> + <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li>
- @@ -70,8 +76,8 @@
 <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> - <li><a href="index.html" class="icon icon-home"></a> »</li> - <li>1. Introduction</li> + <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> + <li class="breadcrumb-item active">1. Introduction</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C01_Introduction.rst.txt" rel="nofollow"> View page source</a> </li>
html/C02_Basics.html
- @@ -1,7 +1,7 @@
 <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + <meta 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>
- @@ -13,11 +13,11 @@
 <script src="_static/js/html5shiv.min.js"></script> <![endif]--> - <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/jquery.js"></script> - <script src="_static/underscore.js"></script> <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> + <script 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" />
- @@ -31,11 +31,15 @@
 <nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > - <a href="index.html" class="icon icon-home"> Mathematics in Lean + + + + <a href="index.html" class="icon icon-home"> + Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> - <input type="text" name="q" placeholder="Search docs" /> + <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
- @@ -54,8 +58,10 @@
 <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> - <li class="toctree-l1"><a class="reference internal" href="C06_Abstract_Algebra.html">6. Abstract Algebra</a></li> - <li class="toctree-l1"><a class="reference internal" href="C07_Topology.html">7. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> + <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li>
- @@ -74,8 +80,8 @@
 <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> - <li><a href="index.html" class="icon icon-home"></a> »</li> - <li>2. Basics</li> + <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> + <li class="breadcrumb-item active">2. Basics</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C02_Basics.rst.txt" rel="nofollow"> View page source</a> </li>
- @@ -101,7 +107,7 @@ the net result is a proof that the left-hand side of the calculation
 is equal to the right-hand side. In Lean, stating a theorem is tantamount to stating a goal, namely, the goal of proving the theorem. - Lean provides the <code class="docutils literal notranslate">rewrite</code> tactic, abbreviated <code class="docutils literal notranslate">rw</code>, + Lean provides the rewriting tactic <code class="docutils literal notranslate">rw</code>, to replace the left-hand side of an identity by the right-hand side in the goal. If <code class="docutils literal notranslate">a</code>, <code class="docutils literal notranslate">b</code>, and <code class="docutils literal notranslate">c</code> are real numbers, <code class="docutils literal notranslate">mul_assoc a b c</code> is the identity <code class="docutils literal notranslate">a * b * c = a * (b * c)</code>
- @@ -123,15 +129,15 @@ notational conventions and leave out parentheses when Lean does as well.
 imports the theory of the real numbers from <code class="docutils literal notranslate">mathlib</code>. For the sake of brevity, we generally suppress information like this when it - is repeated from example to example. - Clicking the <code class="docutils literal notranslate">try it!</code> button displays the full - example as it is meant to be processed and checked by Lean. + is repeated from example to example. You are welcome to make changes to see what happens. You can type the <code class="docutils literal notranslate">ℝ</code> character as <code class="docutils literal notranslate">\R</code> or <code class="docutils literal notranslate">\real</code> in VS Code. The symbol doesn’t appear until you hit space or the tab key. If you hover over a symbol when reading a Lean file, VS Code will show you the syntax that can be used to enter it. + If you are curious to see all available abreviations, you can hit Ctrl-Shift-p + and then type abbreviations to get access to the <code class="docutils literal notranslate">Lean 4: Show all abbreviations</code> command. If your keyboard does not have an easily accessible backslash, you can change the leading character by changing the <code class="docutils literal notranslate">lean.input.leader</code> setting.
- @@ -170,7 +176,9 @@ in each case replacing <code class="docutils literal notranslate">rw</code> tactic, you can use a left arrow (<code class="docutils literal notranslate">\l</code>) to reverse an identity. For example, <code class="docutils literal notranslate">rw ← mul_assoc a b c</code> - replaces <code class="docutils literal notranslate">a * (b * c)</code> by <code class="docutils literal notranslate">a * b * c</code> in the current goal. + 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
- @@ -208,7 +216,7 @@ and the second with only one argument.
 rw [mul_assoc] </pre></div> </div> - Try these: + Try these, using the theorem <cite>sub_self</cite> for the second one: <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a b c d e f : ℝ) (h : b * c = e * f) : a * b * c * d = a * e * f * d := by sorry
- @@ -216,12 +224,8 @@ and the second with only one argument.
 sorry </pre></div> </div> - For the second one, you can use the theorem <code class="docutils literal notranslate">sub_self</code>, - where <code class="docutils literal notranslate">sub_self a</code> is the identity <code class="docutils literal notranslate">a - a = 0</code>. - We now introduce some useful features of Lean. - First, multiple rewrite commands can be carried out - with a single command, - by listing the relevant identities within square brackets. + Multiple rewrite commands can be carried out with a single command, + by listing the relevant identities separated by commas inside the square brackets. <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by rw [h', ← mul_assoc, h, mul_assoc] </pre></div>
- @@ -229,9 +233,7 @@ by listing the relevant identities within square brackets.
 You still see the incremental progress by placing the cursor after a comma in any list of rewrites. Another trick is that we can declare variables once and for all outside - an example or theorem. - When Lean sees them mentioned in the statement of the theorem, - it includes them automatically. + an example or theorem. Lean then includes them automatically. <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (a b c d e f g : ℝ) example (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by
- @@ -239,8 +241,7 @@ it includes them automatically.
 </pre></div> </div> Inspection of the tactic state at the beginning of the above proof - reveals that Lean indeed included the relevant variables, leaving out - <cite>g</cite> that doesn’t feature in the statement. + reveals that Lean indeed included all variables. We can delimit the scope of the declaration by putting it in a <code class="docutils literal notranslate">section ... end</code> block. Finally, recall from the introduction that Lean provides us with a
- @@ -355,7 +356,8 @@ in the assumption <code class="docutils literal notranslate">h
 because at that point <code class="docutils literal notranslate">hyp</code> matches the goal exactly. We close this section by noting that <code class="docutils literal notranslate">mathlib</code> provides a useful bit of automation with a <code class="docutils literal notranslate">ring</code> tactic, - which is designed to prove identities in any commutative ring. + 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
- @@ -371,11 +373,11 @@ which is designed to prove identities in any commutative ring.
 </pre></div> </div> The <code class="docutils literal notranslate">ring</code> tactic is imported indirectly when we - import <code class="docutils literal notranslate">Data.Real.Basic</code>, + import <code class="docutils literal notranslate">Mathlib.Data.Real.Basic</code>, but we will see in the next section that it can be used for calculations on structures other than the real numbers. It can be imported explicitly with the command - <code class="docutils literal notranslate">import tactic</code>. + <code class="docutils literal notranslate">import Mathlib.Tactic</code>. We will see there are similar tactics for other common kind of algebraic structures. There is a variation of <code class="docutils literal notranslate">rw</code> called <code class="docutils literal notranslate">nth_rewrite</code> that allows you to replace only particular instances of an expression in the goal.
- @@ -387,9 +389,6 @@ occurrence of <code class="docutils literal notranslate">a</sp
 rw [add_mul] </pre></div> </div> - See also <code class="docutils literal notranslate">nth_rewrite_lhs</code> and <code class="docutils literal notranslate">nth_rewrite_rhs</code>. - For a more sophisticated means of rewriting particular subexpressions, - see the <a class="reference external" href="https://leanprover-community.github.io/extras/conv.html">documentation for the conversion tactic</a>. </section> <section id="proving-identities-in-algebraic-structures"> <h2>2.2. Proving Identities in Algebraic Structures<a class="headerlink" href="#proving-identities-in-algebraic-structures" title="Permalink to this heading"></a></h2>
html/C03_Logic.html
- @@ -1,7 +1,7 @@
 <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + <meta 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>3. Logic — Mathematics in Lean 0.1 documentation</title>
- @@ -13,11 +13,11 @@
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- @@ -31,11 +31,15 @@
 <nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > - <a href="index.html" class="icon icon-home"> Mathematics in Lean + + + + <a href="index.html" class="icon icon-home"> + Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> - <input type="text" name="q" placeholder="Search docs" /> + <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
- @@ -55,8 +59,10 @@
 </li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> - <li class="toctree-l1"><a class="reference internal" href="C06_Abstract_Algebra.html">6. Abstract Algebra</a></li> - <li class="toctree-l1"><a class="reference internal" href="C07_Topology.html">7. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> + <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li>
- @@ -75,8 +81,8 @@
 <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> - <li><a href="index.html" class="icon icon-home"></a> »</li> - <li>3. Logic</li> + <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> + <li class="breadcrumb-item active">3. Logic</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C03_Logic.rst.txt" rel="nofollow"> View page source</a> </li>
- @@ -1666,7 +1672,7 @@ everywhere by any linear order <code class="docutils literal notranslate">∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε </pre></div> </div> - In <a class="reference internal" href="C07_Topology.html#filters">Section 7.1</a>, we will see that mathlib has mechanisms + In <a class="reference internal" href="C08_Topology.html#filters">Section 8.1</a>, we will see that mathlib has mechanisms for dealing with convergence in vastly more general terms, not only abstracting away particular features of the domain and codomain,
html/C04_Sets_and_Functions.html
- @@ -1,7 +1,7 @@
 <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + <meta 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>4. Sets and Functions — Mathematics in Lean 0.1 documentation</title>
- @@ -13,11 +13,11 @@
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- @@ -31,11 +31,15 @@
 <nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > - <a href="index.html" class="icon icon-home"> Mathematics in Lean + + + + <a href="index.html" class="icon icon-home"> + Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> - <input type="text" name="q" placeholder="Search docs" /> + <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
- @@ -52,8 +56,10 @@
 </ul> </li> <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> - <li class="toctree-l1"><a class="reference internal" href="C06_Abstract_Algebra.html">6. Abstract Algebra</a></li> - <li class="toctree-l1"><a class="reference internal" href="C07_Topology.html">7. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> + <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li>
- @@ -72,8 +78,8 @@
 <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> - <li><a href="index.html" class="icon icon-home"></a> »</li> - <li>4. Sets and Functions</li> + <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> + <li class="breadcrumb-item active">4. Sets and Functions</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C04_Sets_and_Functions.rst.txt" rel="nofollow"> View page source</a> </li>
html/C05_Number_Theory.html
- @@ -1,7 +1,7 @@
 <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + <meta 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>5. Number Theory — Mathematics in Lean 0.1 documentation</title>
- @@ -13,16 +13,16 @@
 <script src="_static/js/html5shiv.min.js"></script> <![endif]--> - <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/jquery.js"></script> - <script src="_static/underscore.js"></script> <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> + <script 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" /> <link rel="search" title="Search" href="search.html" /> - <link rel="next" title="6. Abstract Algebra" href="C06_Abstract_Algebra.html" /> + <link rel="next" title="6. Structures" href="C06_Structures.html" /> <link rel="prev" title="4. Sets and Functions" href="C04_Sets_and_Functions.html" /> </head>
- @@ -31,11 +31,15 @@
 <nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > - <a href="index.html" class="icon icon-home"> Mathematics in Lean + + + + <a href="index.html" class="icon icon-home"> + Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> - <input type="text" name="q" placeholder="Search docs" /> + <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
- @@ -53,8 +57,10 @@
 <li class="toctree-l2"><a class="reference internal" href="#infinitely-many-primes">5.4. Infinitely Many Primes</a></li> </ul> </li> - <li class="toctree-l1"><a class="reference internal" href="C06_Abstract_Algebra.html">6. Abstract Algebra</a></li> - <li class="toctree-l1"><a class="reference internal" href="C07_Topology.html">7. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> + <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li>
- @@ -73,8 +79,8 @@
 <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> - <li><a href="index.html" class="icon icon-home"></a> »</li> - <li>5. Number Theory</li> + <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> + <li class="breadcrumb-item active">5. Number Theory</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C05_Number_Theory.rst.txt" rel="nofollow"> View page source</a> </li>
- @@ -906,7 +912,7 @@ these methods to prove the two examples below.
 <code class="docutils literal notranslate">n</code> is prime and <code class="docutils literal notranslate">s</code> is a set of primes. To show that, we need the following lemma, which you should be able to prove using the theorem <code class="docutils literal notranslate">Nat.Prime.eq_one_or_self_of_dvd</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem Nat.Prime.eq_of_dvd_of_prime {p q : ℕ} + <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
- @@ -1157,7 +1163,7 @@ feat of formalization.
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- @@ -1,10 +1,10 @@
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- @@ -31,11 +31,15 @@
 <nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > - <a href="index.html" class="icon icon-home"> Mathematics in Lean + + + + <a href="index.html" class="icon icon-home"> + Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> - <input type="text" name="q" placeholder="Search docs" /> + <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
- @@ -47,13 +51,15 @@
 <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> - <li class="toctree-l1 current"><a class="current reference internal" href="#">6. Abstract Algebra</a><ul> - <li class="toctree-l2"><a class="reference internal" href="#structures">6.1. Structures</a></li> + <li class="toctree-l1 current"><a class="current reference internal" href="#">6. Structures</a><ul> + <li class="toctree-l2"><a class="reference internal" href="#defining-structures">6.1. Defining structures</a></li> <li class="toctree-l2"><a class="reference internal" href="#algebraic-structures">6.2. Algebraic Structures</a></li> <li class="toctree-l2"><a class="reference internal" href="#building-the-gaussian-integers">6.3. Building the Gaussian Integers</a></li> </ul> </li> - <li class="toctree-l1"><a class="reference internal" href="C07_Topology.html">7. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> + <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li>
- @@ -72,10 +78,10 @@
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- @@ -83,8 +89,8 @@
 <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> - <section id="abstract-algebra"> - <h1>6. Abstract Algebra<a class="headerlink" href="#abstract-algebra" title="Permalink to this heading"></a></h1> + <section id="structures"> + <h1>6. Structures<a class="headerlink" href="#structures" title="Permalink to this heading"></a></h1> Modern mathematics makes essential use of algebraic structures, which encapsulate patterns that can be instantiated in
- @@ -103,8 +109,8 @@ It will also show you how to define and use
 algebraic structures on your own. For more technical detail, you can consult <a class="reference external" href="https://leanprover.github.io/theorem_proving_in_lean/">Theorem Proving in Lean</a>, and a paper by Anne Baanen, <a class="reference external" href="https://arxiv.org/abs/2202.01629">Use and abuse of instance parameters in the Lean mathematical library</a>. - <section id="structures"> - <h2>6.1. Structures<a class="headerlink" href="#structures" title="Permalink to this heading"></a></h2> + <section id="defining-structures"> + <h2>6.1. Defining structures<a class="headerlink" href="#defining-structures" title="Permalink to this heading"></a></h2> In the broadest sense of the term, a structure is a specification of a collection of data, possibly with constraints that the data is required to satisfy.
- @@ -1545,7 +1551,7 @@ the notions of being prime and being irreducible coincide.
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 + <!DOCTYPE html> + <html class="writer-html5" lang="en" > + <head> + <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" href="_static/pygments.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> + <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="shortcut icon" href="_static/favicon.ico"/> +  + + <script src="_static/jquery.js"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> + <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> + <script src="_static/doctools.js"></script> + <script src="_static/sphinx_highlight.js"></script> + <script src="_static/js/theme.js"></script> + <link rel="index" title="Index" href="genindex.html" /> + <link rel="search" title="Search" href="search.html" /> + <link rel="next" title="8. Topology" href="C08_Topology.html" /> + <link rel="prev" title="6. Structures" href="C06_Structures.html" /> + </head> + + <body class="wy-body-for-nav"> + <div class="wy-grid-for-nav"> + <nav data-toggle="wy-nav-shift" class="wy-nav-side"> + <div class="wy-side-scroll"> + <div class="wy-side-nav-search" > + + + + <a href="index.html" class="icon icon-home"> + Mathematics in Lean + </a> + <div role="search"> + <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> + <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> + <input type="hidden" name="check_keywords" value="yes" /> + <input type="hidden" name="area" value="default" /> + </form> + </div> + </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> + <ul class="current"> + <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a></li> + <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> + <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> + <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> + <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> + <li class="toctree-l1 current"><a class="current reference internal" href="#">7. Hierarchies</a><ul> + <li class="toctree-l2"><a class="reference internal" href="#basics">7.1. Basics</a></li> + <li class="toctree-l2"><a class="reference internal" href="#morphisms">7.2. Morphisms</a></li> + <li class="toctree-l2"><a class="reference internal" href="#sub-objects">7.3. Sub-objects</a></li> + </ul> + </li> + <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> + </ul> + <ul> + <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li> + </ul> + + </div> + </div> + </nav> + + <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > + + <a href="index.html">Mathematics in Lean</a> + </nav> + + <div class="wy-nav-content"> + <div class="rst-content"> + <div role="navigation" aria-label="Page navigation"> + <ul class="wy-breadcrumbs"> + <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> + <li class="breadcrumb-item active">7. Hierarchies</li> + <li class="wy-breadcrumbs-aside"> + <a href="_sources/C07_Hierarchies.rst.txt" rel="nofollow"> View page source</a> + </li> + </ul> + <hr/> + </div> + <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> + <div itemprop="articleBody"> + + <section id="hierarchies"> + <h1>7. Hierarchies<a class="headerlink" href="#hierarchies" title="Permalink to this heading"></a></h1> + We have seen in <a class="reference internal" href="C06_Structures.html#structures">Chapter 6</a> how to define the class + of groups and build instances of this class, and then how to build an instance + of the commutative ring class. But of course there is a hierarchy here: a + commutative ring is in particular an additive group. In this chapter we + will study how to build such hierarchies. They appear in all branches + of mathematics but in this chapter the emphasis will be on algebraic examples. + It may seem premature to discuss how to build hierarchies before more discussions + about using existing hierarchies. But some understanding of the technology underlying + hierarchies is required to use them. So you should probably still read this chapter, + but without trying too hard to remember everything on your first read, then read + the following chapters and come back here for a second reading. + In this chapter, we will redefine (simpler versions of) many things that appear in Mathlib + so we will used indices to distinguish our version. For instance we will have <code class="docutils literal notranslate">Ring₁</code> + as our version of <code class="docutils literal notranslate">Ring</code>. Since we will gradually explain more powerful ways of formalizing + structures, those indices will sometimes grow beyond one. + <section id="basics"> + <h2>7.1. Basics<a class="headerlink" href="#basics" title="Permalink to this heading"></a></h2> + At the very bottom of all hierarchies in Lean, we find data-carrying + classes. The following class records that the given type <code class="docutils literal notranslate">α</code> is endowed with + a distinguished element called <code class="docutils literal notranslate">one</code>. At this stage, it has no property at all. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class One₁ (α : Type) where + /-- The element one -/ + one : α + </pre></div> + </div> + Since we’ll make a much heavier use of classes in this chapter, we need to understand some + more details about what the <code class="docutils literal notranslate">class</code> command is doing. + First, the <code class="docutils literal notranslate">class</code> command above defines a structure <code class="docutils literal notranslate">One₁</code> with parameter <code class="docutils literal notranslate">α : Type</code> and + a single field <code class="docutils literal notranslate">one</code>. It also mark this structure as a class so that arguments of type + <code class="docutils literal notranslate">One₁ α</code> for some type <code class="docutils literal notranslate">α</code> will be inferrable using the instance resolution procedure, + as long as they are marked as instance-implicit, ie appear between square brackets. + Those two effects could also have been achieved using the <code class="docutils literal notranslate">structure</code> command with <code class="docutils literal notranslate">class</code> + attribute, ie writing <code class="docutils literal notranslate">@[class] structure</code> instance of <code class="docutils literal notranslate">class</code>. But the class command also + ensures that <code class="docutils literal notranslate">One₁ α</code> appears as an instance-implicit argument in its own fields. Compare: + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check One₁.one -- One₁.one {α : Type} [self : One₁ α] : α + + @[class] structure One₂ (α : Type) where + /-- The element one -/ + one : α + + #check One₂.one + </pre></div> + </div> + In the second check, we can see that <code class="docutils literal notranslate">self : One₂ α</code> is an explicit argument. + Let us make sure the first version is indeed usable without any explicit argument. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (α : Type) [One₁ α] : α := One₁.one + </pre></div> + </div> + Remark: in the above example, the argument <code class="docutils literal notranslate">One₁ α</code> is marked as instance-implicit, + which is a bit silly since this affects only uses of the declaration and declaration created by + the <code class="docutils literal notranslate">example</code> command cannot be used. However it allows to avoid giving a name to that + argument and, more importantly, it starts installing the good habit of marking <code class="docutils literal notranslate">One₁ α</code> + arguments as instance-implicit. + Another remark is that all this will work only when Lean knows what is <code class="docutils literal notranslate">α</code>. In the above + example, leaving out the type ascription <code class="docutils literal notranslate">: α</code> would generate an error message like: + <code class="docutils literal notranslate">typeclass instance problem is stuck, it is often due to metavariables One₁ (?m.263 α)</code> + where <code class="docutils literal notranslate">?m.263 α</code> means “some type depending on <code class="docutils literal notranslate">α</code>” (and 263 is simply an auto-generated + index that would be useful to distinguish between several unknown things). Another way + to avoid this issue would be to use a type annotation, as in: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (α : Type) [One₁ α] := (One₁.one : α) + </pre></div> + </div> + You may have already encountered that issue when playing with limits of sequences + in <a class="reference internal" href="C03_Logic.html#sequences-and-convergence">Section 3.6</a> if you tried to state for instance that + <code class="docutils literal notranslate">0 < 1</code> without telling Lean whether you meant this inequality to be about natural numbers + or real numbers. + Our next task is to assign a notation to <code class="docutils literal notranslate">One₁.one</code>. This we don’t want collisions + with the builtin notation for <code class="docutils literal notranslate">1</code>, we will use <code class="docutils literal notranslate">𝟙</code>. This is achieved by the following + command where the first line tells Lean to use the documentation + of <code class="docutils literal notranslate">One₁.one</code> as documentation for the symbol <code class="docutils literal notranslate">𝟙</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[inherit_doc] + notation "𝟙" => One₁.one + + example {α : Type} [One₁ α] : α := 𝟙 + + example {α : Type} [One₁ α] : (𝟙 : α) = 𝟙 := rfl + </pre></div> + </div> + We now want a data-carrying class recording a binary operation. We don’t want to choose + between addition and multiplication for now so we’ll use diamond. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class Dia₁ (α : Type) where + dia : α → α → α + + infixl:70 " ⋄ " => Dia₁.dia + </pre></div> + </div> + As in the <code class="docutils literal notranslate">One₁</code> example, the operation has no property at all at this stage. Let us + now define the class of semigroup structures where the operation is denoted by <code class="docutils literal notranslate">⋄</code>. + For now, we define it by hand as a structure with two fields, a <code class="docutils literal notranslate">Dia₁</code> instance and some + <code class="docutils literal notranslate">Prop</code>-valued field <code class="docutils literal notranslate">dia_assoc</code> asserting associativity of <code class="docutils literal notranslate">⋄</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class Semigroup₁ (α : Type) where + toDia₁ : Dia₁ α + /-- Diamond is associative -/ + dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) + </pre></div> + </div> + Note that while stating <cite>dia_assoc</cite>, the previously defined field <cite>toDia₁</cite> is in the local + context hence can be used when Lean searches for an instance of <cite>Dia₁ α</cite> to make sense + of <cite>a ⋄ b</cite>. However this <cite>toDia₁</cite> field does not become part of the type class instances database. + Hence doing <code class="docutils literal notranslate">example {α : Type} [Semigroup₁ α] (a b : α) : α := a ⋄ b</code> would fail with + error message <code class="docutils literal notranslate">failed to synthesize instance Dia₁ α</code>. + We can fix this by adding the <code class="docutils literal notranslate">instance</code> attribute later. + <div class="highlight-lean notranslate"><div class="highlight"><pre>attribute [instance] Semigroup₁.toDia₁ + + example {α : Type} [Semigroup₁ α] (a b : α) : α := a ⋄ b + </pre></div> + </div> + Before building up, we need a more convenient way to extend structures than explicitly + writing fields like <cite>toDia₁</cite> and adding the instance attribute by hand. The <code class="docutils literal notranslate">class</code> + supports this using the <code class="docutils literal notranslate">extends</code> syntax as in: + <div class="highlight-lean notranslate"><div class="highlight"><pre>class Semigroup₂ (α : Type) extends Dia₁ α where + /-- Diamond is associative -/ + dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) + + example {α : Type} [Semigroup₂ α] (a b : α) : α := a ⋄ b + </pre></div> + </div> + Note this syntax is also available in the <code class="docutils literal notranslate">structure</code> command, although it that + case it fixes only the hurdle of writing fields such as <cite>toDia₁</cite> since there + is no instance to define in that case. + Let us now try to combine a diamond operation and a distinguished one with axioms saying + this element is neutral on both sides. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class DiaOneClass₁ (α : Type) extends One₁ α, Dia₁ α where + /-- One is a left neutral element for diamond. -/ + one_dia : ∀ a : α, 𝟙 ⋄ a = a + /-- One is a right neutral element for diamond -/ + dia_one : ∀ a : α, a ⋄ 𝟙 = a + </pre></div> + </div> + In the next example, we tell Lean that <code class="docutils literal notranslate">α</code> has a <code class="docutils literal notranslate">DiaOneClass₁</code> structure and state a + property that uses both a <cite>Dia₁</cite> instance and a <cite>One₁</cite> instance. In order to see how Lean finds + those instances we set a tracing option whose result can be seen in the info view. This result + is rather terse by default but can be expended by clicking one lines ending with black arrows. + It includes failed attempts where Lean tried to find instances before having enough type + information to succceed. The successful attempts do involve the instances generated by the + <code class="docutils literal notranslate">extends</code> syntax. + <div class="highlight-lean notranslate"><div class="highlight"><pre>set_option trace.Meta.synthInstance true in + example {α : Type} [DiaOneClass₁ α] (a b : α) : Prop := a ⋄ b = 𝟙 + </pre></div> + </div> + Note that we don’t need to include extra fields where combining existing classes. Hence we can + define monoids as: + <div class="highlight-lean notranslate"><div class="highlight"><pre>class Monoid₁ (α : Type) extends Semigroup₁ α, DiaOneClass₁ α + </pre></div> + </div> + While the above definition seems straightforward, it hides an important subtlety. Both + <code class="docutils literal notranslate">Semigroup₁ α</code> and <code class="docutils literal notranslate">DiaOneClass₁ α</code> extend <code class="docutils literal notranslate">Dia₁ α</code>, so one could fear that having + a <code class="docutils literal notranslate">Monoid₁ α</code> instance gives two unrelated diamond operations on <code class="docutils literal notranslate">α</code>, one coming from + a field <code class="docutils literal notranslate">Monoid₁.toSemigroup₁</code> and one coming from a field <code class="docutils literal notranslate">Monoid₁.toDiaOneClass₁</code>. + Indeed if we try to build a monoid class by hand using: + <div class="highlight-lean notranslate"><div class="highlight"><pre>class Monoid₂ (α : Type) where + toSemigroup₁ : Semigroup₁ α + toDiaOneClass₁ : DiaOneClass₁ α + </pre></div> + </div> + then we get two completely unrelated diamond operations + <code class="docutils literal notranslate">Monoid₂.toSemigroup₁.toDia₁.dia</code> and <code class="docutils literal notranslate">Monoid₂.toDiaOneClass₁.toDia₁.dia</code>. + The version generated using the <code class="docutils literal notranslate">extends</code> syntax does not have this defect. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {α : Type} [Monoid₁ α] : + (Monoid₁.toSemigroup₁.toDia₁.dia : α → α → α) = Monoid₁.toDiaOneClass₁.toDia₁.dia := rfl + </pre></div> + </div> + So the <code class="docutils literal notranslate">class</code> command did some magic for us. An easy way to see what are the fields of + our classes is to check their constructor. Compare: + <div class="highlight-lean notranslate"><div class="highlight"><pre>/- Monoid₂.mk {α : Type} (toSemigroup₁ : Semigroup₁ α) (toDiaOneClass₁ : DiaOneClass₁ α) : Monoid₂ α -/ + #check Monoid₂.mk + + /- Monoid₁.mk {α : Type} [toSemigroup₁ : Semigroup₁ α] [toOne₁ : One₁ α] (one_dia : ∀ (a : α), 𝟙 ⋄ a = a) (dia_one : ∀ (a : α), a ⋄ 𝟙 = a) : Monoid₁ α -/ + #check Monoid₁.mk + </pre></div> + </div> + So we see that <code class="docutils literal notranslate">Monoid₁</code> takes <code class="docutils literal notranslate">Semigroup₁ α</code> argument as expected but then it won’t + take a would-be overlapping <code class="docutils literal notranslate">DiaOneClass₁ α</code> argument but instead tears it appart and includes + only the non-overlapping parts. And it also auto-generated an instance <code class="docutils literal notranslate">Monoid₁.toDiaOneClass₁</code> + which is not a field but has the expected signature which, from the end-user point of view, + restores the symmetry between the two extended classes <code class="docutils literal notranslate">Semigroup₁</code> and <code class="docutils literal notranslate">DiaOneClass₁</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check Monoid₁.toSemigroup₁ + #check Monoid₁.toDiaOneClass₁ + </pre></div> + </div> + We are now very close to defining groups. We could add to the monoid structure a field asserting + the existence of an inverse for every element. But then we would need to work to access these + inverses. In practice it is more convenient to add it as data. To optimize reusability, + we define a new data-carrying class, and then give it some notation. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class Inv₁ (α : Type) where + /-- The inversion function -/ + inv : α → α + + @[inherit_doc] + postfix:max "⁻¹" => Inv₁.inv + + class Group₁ (G : Type) extends Monoid₁ G, Inv G where + inv_dia : ∀ a : G, a⁻¹ ⋄ a = 𝟙 + </pre></div> + </div> + The above definition may seem too weak, we only ask that <code class="docutils literal notranslate">a⁻¹</code> is a left-inverse of <code class="docutils literal notranslate">a</code>. + But the other side is automatic. In order to prove that, we need a preliminary lemma. + <div class="highlight-lean notranslate"><div class="highlight"><pre>lemma left_inv_eq_right_inv₁ {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by + rw [← DiaOneClass₁.one_dia c, ← hba, Semigroup₁.dia_assoc, hac, DiaOneClass₁.dia_one b] + </pre></div> + </div> + In this lemma, it is pretty annoying to give full names, especially since it requires knowing + which part of the hierarchy provides those facts. One way to fix this is to use the <code class="docutils literal notranslate">export</code> + command to copy those facts as lemmas in the root name space. + <div class="highlight-lean notranslate"><div class="highlight"><pre>export DiaOneClass₁ (one_dia dia_one) + export Semigroup₁ (dia_assoc) + export Group₁ (inv_dia) + </pre></div> + </div> + We can then rewrite the above proof as: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by + rw [← one_dia c, ← hba, dia_assoc, hac, dia_one b] + </pre></div> + </div> + It is now your turn to prove things about our algebraic structures. + <div class="highlight-lean notranslate"><div class="highlight"><pre>lemma inv_eq_of_dia [Group₁ G] {a b : G} (h : a ⋄ b = 𝟙) : a⁻¹ = b := + sorry + + lemma dia_inv [Group₁ G] (a : G) : a ⋄ a⁻¹ = 𝟙 := + sorry + </pre></div> + </div> + At this stage we would like to move on to define rings, but there is a serious issue. + A ring structure on a type contains both an additive group structure and a multiplicative + monoid structure, and some properties about their interaction. But so far we hard-coded + a notation <code class="docutils literal notranslate">⋄</code> for all our operations. More fundamentally, the type class system + assumes every type has only one instance of each type class. There are various + ways to solve this issue. Surprisingly mathlib uses the naive idea to duplicate + everything for additive and multiplicative theories with the help of some code-generating + attribute. Structures and classes are defined in both additive and multiplicative notation + with an attibute <code class="docutils literal notranslate">to_additive</code> linking them. In case of multiple inheritance like for + semi-groups, the auto-generated “symmetry-restoring” instances need also to be marked. + This is a bit technical you don’t need to understand details. The important point is that + lemmas are then only stated in multiplicative notation and marked with the attribute <code class="docutils literal notranslate">to_additive</code> + to generate the additive version as <code class="docutils literal notranslate">left_inv_eq_right_inv'</code> with it’s auto-generated additive + version <code class="docutils literal notranslate">left_neg_eq_right_neg'</code>. In order to check the name of this additive version we + used that <code class="docutils literal notranslate">wathsnew in</code> command on top of <code class="docutils literal notranslate">left_inv_eq_right_inv'</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class AddSemigroup₃ (α : Type) extends Add α where + /-- Multiplication is associative -/ + add_assoc₃ : ∀ a b c : α, a + b + c = a + (b + c) + + @[to_additive AddSemigroup₃] + class Semigroup₃ (α : Type) extends Mul α where + /-- Multiplication is associative -/ + mul_assoc₃ : ∀ a b c : α, a * b * c = a * (b * c) + + class AddMonoid₃ (α : Type) extends AddSemigroup₃ α, AddZeroClass α + + @[to_additive AddMonoid₃] + class Monoid₃ (α : Type) extends Semigroup₃ α, MulOneClass α + + attribute [to_additive existing] Monoid₃.toMulOneClass + + export Semigroup₃ (mul_assoc₃) + export AddSemigroup₃ (add_assoc₃) + + whatsnew in + @[to_additive] + lemma left_inv_eq_right_inv' {M : Type} [Monoid₃ M] {a b c : M} (hba : b * a = 1) (hac : a * c = 1) : b = c := by + rw [← one_mul c, ← hba, mul_assoc₃, hac, mul_one b] + + #check left_neg_eq_right_neg' + </pre></div> + </div> + Equipped with this technology, we can easily define also commutative semigroups, monoids and + groups, and then define rings. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class AddCommSemigroup₃ (α : Type) extends AddSemigroup₃ α where + add_comm : ∀ a b : α, a + b = b + a + + @[to_additive AddCommSemigroup₃] + class CommSemigroup₃ (α : Type) extends Semigroup₃ α where + mul_comm : ∀ a b : α, a * b = b * a + + class AddCommMonoid₃ (α : Type) extends AddMonoid₃ α, AddCommSemigroup₃ α + + @[to_additive AddCommMonoid₃] + class CommMonoid₃ (α : Type) extends Monoid₃ α, CommSemigroup₃ α + + class AddGroup₃ (G : Type) extends AddMonoid₃ G, Neg G where + neg_add : ∀ a : G, -a + a = 0 + + @[to_additive AddGroup₃] + class Group₃ (G : Type) extends Monoid₃ G, Inv G where + inv_mul : ∀ a : G, a⁻¹ * a = 1 + </pre></div> + </div> + We should remember to tagged lemmas with <code class="docutils literal notranslate">simp</code> when approriate. + <div class="highlight-lean notranslate"><div class="highlight"><pre>attribute [simp] Group₃.inv_mul AddGroup₃.neg_add + </pre></div> + </div> + Then we need to repeat ourselves a bit since we switch to standard notations, but at least + <code class="docutils literal notranslate">to_additive</code> does the work of translating from the multiplicative notation to the additive one. + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[to_additive] + lemma inv_eq_of_mul [Group₃ G] {a b : G} (h : a * b = 1) : a⁻¹ = b := + sorry + </pre></div> + </div> + Note that <code class="docutils literal notranslate">to_additive</code> can be ask to tag a lemma with <code class="docutils literal notranslate">simp</code> and propagate that attribute + to the additive version as follows. + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[to_additive (attr := simp)] + lemma Group₃.mul_inv {G : Type} [Group₃ G] {a : G} : a * a⁻¹ = 1 := by + sorry + + @[to_additive] + lemma mul_left_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : a * b = a * c) : b = c := by + sorry + + @[to_additive] + lemma mul_right_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : b*a = c*a) : b = c := by + sorry + + class AddCommGroup₃ (G : Type) extends AddGroup₃ G, AddCommMonoid₃ G + + @[to_additive AddCommGroup₃] + class CommGroup₃ (G : Type) extends Group₃ G, CommMonoid₃ G + </pre></div> + </div> + We are now ready for rings. For demonstration puprposes we won’t assume that addition is + commutative, and then immediately provide an instance of <code class="docutils literal notranslate">AddCommGroup₃</code>. Mathlib does not + play this game, first because in practice this does not make any ring instance easier and + also because Mathlib’s algebraic hierarchy goes through semi-rings which are like rings but without + opposites so that the proof below does not work for them. What we gain here, besides a nice exercise + if you have never seen it, is an example of building an instance using the syntax that allows + to provide a parent structure and some extra fields. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class Ring₃ (R : Type) extends AddGroup₃ R, Monoid₃ R, MulZeroClass R where + /-- Multiplication is left distributive over addition -/ + left_distrib : ∀ a b c : R, a * (b + c) = a * b + a * c + /-- Multiplication is right distributive over addition -/ + right_distrib : ∀ a b c : R, (a + b) * c = a * c + b * c + + instance {R : Type} [Ring₃ R] : AddCommGroup₃ R := + { Ring₃.toAddGroup₃ with + add_comm := by + sorry } + </pre></div> + </div> + Of course we can also build concrete instances, such as a ring structure on integers (of course + the instance below uses that all the work is already done in Mathlib). + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance : Ring₃ ℤ where + add := (· + ·) + add_assoc₃ := add_assoc + zero := 0 + zero_add := by simp + add_zero := by simp + neg := (- ·) + neg_add := by simp + mul := (· * ·) + mul_assoc₃ := mul_assoc + one := 1 + one_mul := by simp + mul_one := by simp + zero_mul := by simp + mul_zero := by simp + left_distrib := Int.mul_add + right_distrib := Int.add_mul + </pre></div> + </div> + As an exercise you can now set up a simple hierarchy for order relations, including a class + for ordered commutative monoids, which have both a partial order and a commutative monoid structure + such that <code class="docutils literal notranslate">∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b</code>. Of course you need to add fields and maybe + <code class="docutils literal notranslate">extends</code> clauses to the following classes. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class LE₁ (α : Type) where + /-- The Less-or-Equal relation. -/ + le : α → α → Prop + + @[inherit_doc] infix:50 " ≤₁ " => LE₁.le + + class Preorder₁ (α : Type) + + class PartialOrder₁ (α : Type) + + class OrderedCommMonoid₁ (α : Type) + + instance : OrderedCommMonoid₁ ℕ where + </pre></div> + </div> + We now want to discuss algebraic structures involving several types. The prime example + is modules over rings. If you don’t know what is a module, you can pretend it means vector space + and think that all our rings are fields. Those structures are commutative additive groups + equipped with a scalar multiplication by elements of some ring. + We first define the data-carrying type class of scalar multiplication by some type <code class="docutils literal notranslate">α</code> on some + type <code class="docutils literal notranslate">β</code>, and give it a right associative notation. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class SMul₃ (α : Type) (β : Type) where + /-- Scalar multiplication -/ + smul : α → β → β + + infixr:73 " • " => SMul₃.smul + </pre></div> + </div> + Then we can define modules (again think about vector spaces if you don’t know what is a module). + <div class="highlight-lean notranslate"><div class="highlight"><pre>class Module₁ (R : Type) [Ring₃ R] (M : Type) [AddCommGroup₃ M] extends SMul₃ R M where + zero_smul : ∀ m : M, (0 : R) • m = 0 + one_smul : ∀ m : M, (1 : R) • m = m + mul_smul : ∀ (a b : R) (m : M), (a * b) • m = a • b • m + add_smul : ∀ (a b : R) (m : M), (a + b) • m = a • m + b • m + smul_add : ∀ (a : R) (m n : M), a • (m + n) = a • m + a • n + </pre></div> + </div> + There is something interesting going on here. While it isn’t too surprising that the + ring structure on <code class="docutils literal notranslate">R</code> is a parameter in this definition, you probably expected <code class="docutils literal notranslate">AddCommGroup3 M</code> + to be part of the <code class="docutils literal notranslate">extends</code> clause just as <code class="docutils literal notranslate">SMul₃ R M</code> is. Trying to do that would lead + to a mysterious sounding error message: + <code class="docutils literal notranslate">cannot find synthesization order for instance Module₁.toAddCommGroup₃ with type (R : Type) → [inst : Ring₃ R] → {M : Type} → [self : Module₁ R M] → AddCommGroup₃ M + all remaining arguments have metavariables: Ring₃ ?R @Module₁ ?R ?inst✝ M</code>. + In order to understand this message, you need to remember that such an <code class="docutils literal notranslate">extends</code> clause would + lead to a field <code class="docutils literal notranslate">Module₃.toAddCommGroup₃</code> marked as an instance. This instance + would have the signature appearing in the error message: + <code class="docutils literal notranslate">(R : Type) → [inst : Ring₃ R] → {M : Type} → [self : Module₁ R M] → AddCommGroup₃ M</code>. + With such an instance in the type class database, each time Lean would look for a + <code class="docutils literal notranslate">AddCommGroup₃ M</code> instance for some <code class="docutils literal notranslate">M</code>, it would need to go hunting for a completely + unspecified type <code class="docutils literal notranslate">R``and a ``Ring₃ R</code> instance before embarking on the main quest of finding a + <code class="docutils literal notranslate">Module₁ R M</code> instance. Those two side-quests are represented by the meta-variables mentionned in + the error message and denoted by <code class="docutils literal notranslate">?R</code> and <code class="docutils literal notranslate">?inst✝</code> there. Such a <code class="docutils literal notranslate">Module₃.toAddCommGroup₃</code> + instance would then be a huge trap for the instance resolution procedure and then <code class="docutils literal notranslate">class</code> command + refuses to set it up. + What about <code class="docutils literal notranslate">extends SMul₃ R M</code> then? That one creates a field + <code class="docutils literal notranslate">Module₁.toSMul₃ : {R : Type} →  [inst : Ring₃ R] → {M : Type} → [inst_1 : AddCommGroup₃ M] → [self : Module₁ R M] → SMul₃ R M</code> + whose end result <code class="docutils literal notranslate">SMul₃ R M</code> mentions both <code class="docutils literal notranslate">R</code> and <code class="docutils literal notranslate">M</code> so this field can + safely be used as an instance. The rule is easy to remember: each class appearing in the + <code class="docutils literal notranslate">extends</code> clause should mention every type appearing in the parameters. + Let us create our first module instance: a ring is a module over itself using its multiplication + as a scalar multiplication. + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance selfModule (R : Type) [Ring₃ R] : Module₁ R R where + smul := fun r s ↦ r*s + zero_smul := zero_mul + one_smul := one_mul + mul_smul := mul_assoc₃ + add_smul := Ring₃.right_distrib + smul_add := Ring₃.left_distrib + </pre></div> + </div> + As a second example, every abelian group is a module over <code class="docutils literal notranslate">ℤ</code> (this is one of the reason to + generalize the theory of vector spaces by allowing non-invertible scalars). First one can define + scalar multiplication by a natural number for any type equipped with a zero and an addition: + <code class="docutils literal notranslate">n • a</code> is defined as <code class="docutils literal notranslate">a + ⋯ + a</code> where <code class="docutils literal notranslate">a</code> appears <code class="docutils literal notranslate">n</code> times. Then this is extended + to scalar multiplication by an integer by ensuring <code class="docutils literal notranslate">(-1) • a = -a</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>def nsmul₁ [Zero M] [Add M] : ℕ → M → M + | 0, _ => 0 + | n + 1, a => a + nsmul₁ n a + + def zsmul₁ {M : Type _} [Zero M] [Add M] [Neg M] : ℤ → M → M + | Int.ofNat n, a => nsmul₁ n a + | Int.negSucc n, a => -nsmul₁ n.succ a + </pre></div> + </div> + Proving this gives rise to a module structure is a bit tedious and not interesting for the + current discussion, so we will sorry all axioms. You are not asked to replace those sorries + with proofs. If you insist on doing it then you will probably want to state and prove several + intermediate lemmas about <code class="docutils literal notranslate">nsmul₁</code> and <code class="docutils literal notranslate">zsmul₁</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance abGrpModule (A : Type) [AddCommGroup₃ A] : Module₁ ℤ A where + smul := zsmul₁ + zero_smul := sorry + one_smul := sorry + mul_smul := sorry + add_smul := sorry + smul_add := sorry + </pre></div> + </div> + A much more important issue is that we now have two module structures over the ring <code class="docutils literal notranslate">ℤ</code> + for <code class="docutils literal notranslate">ℤ</code> itself: <code class="docutils literal notranslate">abGrpModule ℤ</code> since <code class="docutils literal notranslate">ℤ</code> is a abelian group, and <code class="docutils literal notranslate">selfModule ℤ</code> since + <code class="docutils literal notranslate">ℤ</code> is a ring. Those two module structure correspond to the same abelian group structure, + but it is not obvious that they have the same scalar multiplication. They actually do, but + this isn’t true by definition, it requires a proof. This is very bad news for the type class + instance resolution procedure and will lead to very frustating failures for users of this + hierarchy. When directly asked to find an instance, Lean will pick one, and we can see + which one using: + <div class="highlight-lean notranslate"><div class="highlight"><pre>#synth Module₁ ℤ ℤ -- abGrpModule ℤ + </pre></div> + </div> + But in a more indirect context it can happen that Lean infers the one and then gets confused. + This situation is known as a bad diamond. This has nothing to do with the diamond operation + we used above, it refers to the way one can draw the paths from <code class="docutils literal notranslate">ℤ</code> to its <code class="docutils literal notranslate">Module₁ ℤ</code> + going through either <code class="docutils literal notranslate">AddCommGroup₃ ℤ</code> or <code class="docutils literal notranslate">Ring₃ ℤ</code>. + It is important to understand that not all diamonds are bad. In fact there are diamonds everywhere + in mathlib, and also in this chapter. Already at the very beginning we saw one can go + from <code class="docutils literal notranslate">Monoid₁ α</code> to <code class="docutils literal notranslate">Dia₁ α</code> through either <code class="docutils literal notranslate">Semigroup₁ α</code> or <code class="docutils literal notranslate">DiaOneClass₁ α</code> and + thanks to the work done by the <code class="docutils literal notranslate">class</code> command, the resulting two <code class="docutils literal notranslate">Dia₁ α</code> instances + are definitionnaly equal. In particular a diamond having a <code class="docutils literal notranslate">Prop</code>-valued class at the bottom + cannot be bad since any too proofs of the same statement are definitionnaly equal. + But the diamond we created with modules is definitely bad. The offending piece is the <code class="docutils literal notranslate">smul</code> + field which is data, not a proof, and we have two constructions that are not definitionnaly equal. + The robust way of fixing this issue is to make sure that going from a rich structure to a + poor structure is always done by forgetting data, not by defining data. This well-known pattern + as been named “forgetful inheritance” and extensively discussed in + <a class="reference external" href="https://inria.hal.science/hal-02463336">https://inria.hal.science/hal-02463336</a>. + In our concrete case, we can modify the definition of <code class="docutils literal notranslate">AddMonoid₃</code> to include a <code class="docutils literal notranslate">nsmul</code> data + field and some <code class="docutils literal notranslate">Prop</code>-valued fields ensuring this operation is provably the one we constructed + above. Those fields are given default values using <code class="docutils literal notranslate">:=</code> after their type in the definition below. + Thanks to these default values, most instances would be constructed exactly as with our previous + definitions. But in the special case of <code class="docutils literal notranslate">ℤ</code> we will be able to provide specific values. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class AddMonoid₄ (M : Type) extends AddSemigroup₃ M, AddZeroClass M where + /-- Multiplication by a natural number. -/ + nsmul : ℕ → M → M := nsmul₁ + /-- Multiplication by `(0 : ℕ)` gives `0`. -/ + nsmul_zero : ∀ x, nsmul 0 x = 0 := by intros; rfl + /-- Multiplication by `(n + 1 : ℕ)` behaves as expected. -/ + nsmul_succ : ∀ (n : ℕ) (x), nsmul (n + 1) x = x + nsmul n x := by intros; rfl + + instance mySMul {M : Type} [AddMonoid₄ M] : SMul ℕ M := ⟨AddMonoid₄.nsmul⟩ + </pre></div> + </div> + Let us check we can still construct a product monoid instance without providing the <code class="docutils literal notranslate">nsmul</code> + related fields. + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance (M N : Type) [AddMonoid₄ M] [AddMonoid₄ N] : AddMonoid₄ (M × N) where + add := fun p q ↦ (p.1 + q.1, p.2 + q.2) + add_assoc₃ := fun a b c ↦ by ext <;> apply add_assoc₃ + zero := (0, 0) + zero_add := fun a ↦ by ext <;> apply zero_add + add_zero := fun a ↦ by ext <;> apply add_zero + </pre></div> + </div> + And now let us handle the special case of <code class="docutils literal notranslate">ℤ</code> where we want to build <code class="docutils literal notranslate">nsmul</code> using the coercion + of <code class="docutils literal notranslate">ℕ</code> to <code class="docutils literal notranslate">ℤ</code> and the multiplication on <code class="docutils literal notranslate">ℤ</code>. Note in particular how the proof fields + contain more work than in the default value above. + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance : AddMonoid₄ ℤ where + add := (· + ·) + add_assoc₃ := Int.add_assoc + zero := 0 + zero_add := Int.zero_add + add_zero := Int.add_zero + nsmul := fun n m ↦ (n : ℤ) * m + nsmul_zero := Int.zero_mul + nsmul_succ := fun n m ↦ show (n + 1 : ℤ) * m = m + n * m + by rw [Int.add_mul, Int.add_comm, Int.one_mul] + </pre></div> + </div> + Let us check we solved our issue. Because Lean already has a definition of scalar multiplication + of a natural number and an integer, and we want to make sure our instance is used, we won’t use + the <code class="docutils literal notranslate">•</code> notation but call <code class="docutils literal notranslate">SMul.mul</code> and explicitly provide our instance defined above. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (n : ℕ) (m : ℤ) : SMul.smul (self := mySMul) n m = n * m := rfl + </pre></div> + </div> + This story then continues with incorporating a <code class="docutils literal notranslate">zsmul</code> field into the definition of groups + and similar tricks. You are now ready to read the definition of monoids, groups, rings and modules + in mathlib. There are more complicated than what we have seen here, because they are part of a huge + hierarchy, but all principles have been explained above. + As an exercise, you can come back to the order relation hierarchy you built above and try + to incorportate a type class <code class="docutils literal notranslate">LT₁</code> carrying the Less-Than notation <code class="docutils literal notranslate"><₁</code> and make sure + that every preorder comes with a <code class="docutils literal notranslate"><₁</code> which has a default value built from <code class="docutils literal notranslate">≤₁</code> and a + <code class="docutils literal notranslate">Prop</code>-valued field asserting the natural relation between those two comparison operators. + TEXT. + -/ + </section> + <section id="morphisms"> + <h2>7.2. Morphisms<a class="headerlink" href="#morphisms" title="Permalink to this heading"></a></h2> + So far in this chapter, we discussed how to create a hierarchy of mathematical structures. + But defining structures is not really completed until we have morphisms. There are two + main approaches here. The most obvious one is to define a predicate on functions. + <div class="highlight-lean notranslate"><div class="highlight"><pre>def isMonoidHom₁ [Monoid G] [Monoid H] (f : G → H) : Prop := + f 1 = 1 ∧ ∀ g g', f (g * g') = f g * f g' + </pre></div> + </div> + In this definition, it is a bit unpleasant to use a conjunction. In particular users + will need to remember the ordering we chose when they want to access the two conditions. + So we could use a structure instead. + <div class="highlight-lean notranslate"><div class="highlight"><pre>structure isMonoidHom₂ [Monoid G] [Monoid H] (f : G → H) : Prop where + map_one : f 1 = 1 + map_mul : ∀ g g', f (g * g') = f g * f g' + </pre></div> + </div> + Once we are here, it is even tempting to make it a class and use the type class instance resolution + procedure to automatically infer <code class="docutils literal notranslate">isMonoidHom₂</code> for complicated functions out of instances for + simpler functions. For instance a composition of monoid morphisms is a monoid morphism and this + seems like a useful instance. However such an instance would be very tricky for the resolution + procedure since it would need to hunt down <code class="docutils literal notranslate">g ∘ f</code> everywhere. Sometimes it would not + succeed, for instance in <code class="docutils literal notranslate">g (f x)</code>. And sometimes it would succeed too much by seeing + <code class="docutils literal notranslate">id ∘ f</code> or <code class="docutils literal notranslate">g ∘ id</code> everywhere, leading to a diverging instance search. More generally one + must always keep in mind that recognizing which function is applied in a given expression is + a very difficult problem, called the “higher-order unification problem”. + So Mathlib does not use this class approach. + A more fundamental question is whether we use predicates as above (using either a <code class="docutils literal notranslate">def</code> or a + <code class="docutils literal notranslate">structure</code>) or use structures bundling a function and predicates. This is partly a psychological + issue. It is extremely rare to consider a function between monoids that is not a morphism. + It really feels like “monoid morphism” is not an adjective you can assign to a bare function, + it is a noun. On the other hand one can argue that a continuous function between topological spaces + is really a function that happens to be continuous. This is one reason why Mathlib has a + <code class="docutils literal notranslate">Continuous</code> predicate. For instance you can write: + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : Continuous (id : ℝ → ℝ) := continuous_id + </pre></div> + </div> + We still have bundles continuous functions, which are convenient for instance to put a topology + on a space of continuous functions, but they are not the primary tool to work with continuity. + By constrast, morphisms between monoids (or other algebraic structures) are bundled as in: + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] + structure MonoidHom₁ (G H : Type) [Monoid G] [Monoid H] where + toFun : G → H + map_one : toFun 1 = 1 + map_mul : ∀ g g', toFun (g * g') = toFun g * toFun g' + </pre></div> + </div> + Of course we don’t want to type <code class="docutils literal notranslate">toFun</code> everywhere so we register a coercion using + the <code class="docutils literal notranslate">CoeFun</code> type class. Its first argument is the type we want to coerce to a function. + The second argument describes the target function type. In our case it is always <code class="docutils literal notranslate">G → H</code> + for every <code class="docutils literal notranslate">f : MonoidHom₁ G H</code>. We also tag <code class="docutils literal notranslate">MonoidHom₁.toFun</code> with the <code class="docutils literal notranslate">coe</code> attribute to + make sure it is displayed almost invisibly in the tactic state, simply by a <code class="docutils literal notranslate">↑</code> prefix. + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance [Monoid G] [Monoid H] : CoeFun (MonoidHom₁ G H) (fun _ ↦ G → H) where + coe := MonoidHom₁.toFun + + attribute [coe] MonoidHom₁.toFun + </pre></div> + </div> + Let us check we can indeed apply a bundled monoid morphism to an element. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [Monoid G] [Monoid H] (f : MonoidHom₁ G H) : f 1 = 1 := f.map_one + </pre></div> + </div> + We can do the same with other kind of morphisms until we reach ring morphisms. + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] + structure AddMonoidHom₁ (G H : Type) [AddMonoid G] [AddMonoid H] where + toFun : G → H + map_zero : toFun 0 = 0 + map_add : ∀ g g', toFun (g + g') = toFun g + toFun g' + + instance [AddMonoid G] [AddMonoid H] : CoeFun (AddMonoidHom₁ G H) (fun _ ↦ G → H) where + coe := AddMonoidHom₁.toFun + + attribute [coe] AddMonoidHom₁.toFun + + @[ext] + structure RingHom₁ (R S : Type) [Ring R] [Ring S] extends MonoidHom₁ R S, AddMonoidHom₁ R S + </pre></div> + </div> + There are a couple of issues about this approach. A minor one is we don’t quite know where to put + the <code class="docutils literal notranslate">coe</code> attribute since the <code class="docutils literal notranslate">RingHom₁.toFun</code> does not exist, the relevant function is + <code class="docutils literal notranslate">MonoidHom₁.toFun ∘ RingHom₁.toMonoidHom₁</code> which is not a declaration that can be tagged with an + attribute (but we could still define a <code class="docutils literal notranslate">CoeFun  (RingHom₁ R S) (fun _ ↦ R → S)</code> instance). + A much more important one is that lemmas about monoid morphisms won’t directly apply + to ring morphisms. This leaves the alternative of either juggling with <code class="docutils literal notranslate">RingHom₁.toMonoidHom₁</code> + each time we want to apply a monoid morphism lemma or restate every such lemmas for ring morphisms. + Neither option is appealing so Mathlib uses a new hierarchy trick here. The idea is to define + a type class for objects that are at least monoid morphisms, instantiate that class with both monoid + morphisms and ring morphisms and use it to state every lemma. In the definition below, + <code class="docutils literal notranslate">F</code> could be <code class="docutils literal notranslate">MonoidHom₁ M N</code>, or <code class="docutils literal notranslate">RingHom₁ M N</code> if <code class="docutils literal notranslate">M</code> and <code class="docutils literal notranslate">N</code> have a ring structure. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class MonoidHomClass₁ (F : Type) (M N : Type) [Monoid M] [Monoid N] where + toFun : F → M → N + map_one : ∀ f : F, toFun f 1 = 1 + map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' + </pre></div> + </div> + However there is a problem with the above implementation. We haven’t registered a coercion to + function instance yet. Let us try to do it now. + <div class="highlight-lean notranslate"><div class="highlight"><pre>def badInst [Monoid M] [Monoid N] [MonoidHomClass₁ F M N] : CoeFun F (fun _ ↦ M → N) where + coe := MonoidHomClass₁.toFun + </pre></div> + </div> + Making the an instance would be bad. When faced with something like <code class="docutils literal notranslate">f x</code> where the type of <code class="docutils literal notranslate">f</code> + is not a function type, Lean will try to find a <code class="docutils literal notranslate">CoeFun</code> instance to coerce <code class="docutils literal notranslate">f</code> into a function. + The above function has type: + <code class="docutils literal notranslate">{M N F : Type} → [Monoid M] → [Monoid N] → [MonoidHomClass₁ F M N] → CoeFun F (fun x ↦ M → N)</code> + so, when it trying to apply it, it wouldn’t be a priori clear to Lean in which order the unknown + types <code class="docutils literal notranslate">M</code>, <code class="docutils literal notranslate">N</code> and <code class="docutils literal notranslate">F</code> should be inferred. This is a kind of bad instance that is slightly + different from the one we saw already, but it boils down to the same issue: without knowing <code class="docutils literal notranslate">M</code>, + Lean would have to search for a monoid instance on an unknown type, hence hopelessly try every + monoid instance in the database. If you are curious to see the effect of such an instance you + can type <code class="docutils literal notranslate">set_option synthInstance.checkSynthOrder false in</code> on top of the above declaration, + replace <code class="docutils literal notranslate">def badInst</code> with <code class="docutils literal notranslate">instance</code>, and look for random failures in this file. + Here the solution is easy, we need to tell Lean to first search what is <code class="docutils literal notranslate">F</code> and then deduce <code class="docutils literal notranslate">M</code> + and <code class="docutils literal notranslate">N</code>. This is done using the <code class="docutils literal notranslate">outParam</code> function. This function is defined as the identity + function, but is still recognized by the type class machinery and triggers the desired behavior. + Hence we can retry defining our class, paying attention to the <code class="docutils literal notranslate">outParam</code> function: + <div class="highlight-lean notranslate"><div class="highlight"><pre>class MonoidHomClass₂ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] where + toFun : F → M → N + map_one : ∀ f : F, toFun f 1 = 1 + map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' + + instance [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] : CoeFun F (fun _ ↦ M → N) where + coe := MonoidHomClass₂.toFun + + attribute [coe] MonoidHomClass₂.toFun + </pre></div> + </div> + Now we can proceed with our plan to instantiate this class. + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₂ (MonoidHom₁ M N) M N where + toFun := MonoidHom₁.toFun + map_one := fun f ↦ f.map_one + map_mul := fun f ↦ f.map_mul + + instance (R S : Type) [Ring R] [Ring S] : MonoidHomClass₂ (RingHom₁ R S) R S where + toFun := fun f ↦ f.toMonoidHom₁.toFun + map_one := fun f ↦ f.toMonoidHom₁.map_one + map_mul := fun f ↦ f.toMonoidHom₁.map_mul + </pre></div> + </div> + As promised every lemma we prove about <code class="docutils literal notranslate">f : F</code> assuming an instance of <code class="docutils literal notranslate">MonoidHomClass₁ F</code> will + apply both to monoid morphims and ring morphisms. + Let us see an example lemma and check it applies to both situations. + <div class="highlight-lean notranslate"><div class="highlight"><pre>lemma map_inv_of_inv [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] (f : F) {m m' : M} (h : m*m' = 1) : + f m * f m' = 1 := by + rw [← MonoidHomClass₂.map_mul, h, MonoidHomClass₂.map_one] + + example [Monoid M] [Monoid N] (f : MonoidHom₁ M N) {m m' : M} (h : m*m' = 1) : f m * f m' = 1 := + map_inv_of_inv f h + + example [Ring R] [Ring S] (f : RingHom₁ R S) {r r' : R} (h : r*r' = 1) : f r * f r' = 1 := + map_inv_of_inv f h + </pre></div> + </div> + At first sight, it may look like we got back to our old bad idea of making <code class="docutils literal notranslate">MonoidHom₁</code> a class. + But we haven’t. Everything is shifted one level of abstraction up. The type class resolution + procedure won’t be looking for functions, it will be looking for either + <code class="docutils literal notranslate">MonoidHom₁</code> or <code class="docutils literal notranslate">RingHom₁</code>. + One remaining issue with our approach is the presence of repeatitive code around the <code class="docutils literal notranslate">toFun</code> + field and the corresponding <code class="docutils literal notranslate">CoeFun</code> instance and <code class="docutils literal notranslate">coe</code> attribute. It would also be better + to record that this pattern is used only for function with extra properties, meaning that the + coercion to functions should be injective. So Mathlib adds one more layer of abstraction with + the base class <code class="docutils literal notranslate">FunLike</code>. Let us redefine our <code class="docutils literal notranslate">MonoidHomClass</code> on top of this base layer. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class MonoidHomClass₃ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] extends + FunLike F M (fun _ ↦ N) where + map_one : ∀ f : F, f 1 = 1 + map_mul : ∀ (f : F) g g', f (g * g') = f g * f g' + + instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₃ (MonoidHom₁ M N) M N where + coe := MonoidHom₁.toFun + coe_injective' := MonoidHom₁.ext + map_one := MonoidHom₁.map_one + map_mul := MonoidHom₁.map_mul + </pre></div> + </div> + Of course the hierarchy of morphisms does not stop here. We could go on and define a class + <code class="docutils literal notranslate">RingHomClass₃</code> extending <code class="docutils literal notranslate">MonoidHomClass₃</code> and instantiate it on <code class="docutils literal notranslate">RingHom</code> and + then later on <code class="docutils literal notranslate">AlgebraHom</code> (algebras are rings with some extra structure). But we’ve + covered the main formalization ideas used in Mathlib for morphisms and you should be ready + to understand how morphisms are defined in Mathlib. + As an exercise, you should try to define your class of bundled order-preserving function between + ordered types, and then order preserving monoid morphisms. This is for training purposes only. + Like continuous functions, order preserving functions are primarily unbundled in Mathlib where + they are defined by the <code class="docutils literal notranslate">monontone</code> predicate. Of course you need to complete the class + definitions below. + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] + structure OrderPresHom (α β : Type) [LE α] [LE β] where + toFun : α → β + le_of_le : ∀ a a', a ≤ a' → toFun a ≤ toFun a' + + @[ext] + structure OrderPresMonoidHom (M N : Type) [Monoid M] [LE M] [Monoid N] [LE N] extends + MonoidHom₁ M N, OrderPresHom M N + + class OrderPresHomClass (F : Type) (α β : outParam Type) [LE α] [LE β] + + instance (α β : Type) [LE α] [LE β] : OrderPresHomClass (OrderPresHom α β) α β where + + instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : + OrderPresHomClass (OrderPresMonoidHom α β) α β where + + instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : + MonoidHomClass₃ (OrderPresMonoidHom α β) α β + := sorry + </pre></div> + </div> + </section> + <section id="sub-objects"> + <h2>7.3. Sub-objects<a class="headerlink" href="#sub-objects" title="Permalink to this heading"></a></h2> + After defining some algebraic structure and its morphisms, the next step is to consider sets + that inherit this algebraic structure, for instance subgroups or subrings. + This largely overlaps our previous topic. Indeed a set in <code class="docutils literal notranslate">X</code> is implemented as a function from + <code class="docutils literal notranslate">X</code> to <code class="docutils literal notranslate">Prop</code> so sub-objects are function satisfying a certain predicate. + Hence we can reuse of lot of the ideas that led to the <code class="docutils literal notranslate">FunLike</code> class and its descendants. + We won’t reuse <code class="docutils literal notranslate">FunLike</code> itself because this would break the abstraction barrier from <code class="docutils literal notranslate">Set X</code> + to <code class="docutils literal notranslate">X → Prop</code>. Instead there is a <code class="docutils literal notranslate">SetLike</code> class. Instead of wrapping an injection into a + function type, that class wraps an injection into a <code class="docutils literal notranslate">Set</code> type and defines the corresponding + coercion and <code class="docutils literal notranslate">Membership</code> instance. + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] + structure Submonoid₁ (M : Type) [Monoid M] where + /-- The carrier of a submonoid. -/ + carrier : Set M + /-- The product of two elements of a submonoid belongs to the submonoid. -/ + mul_mem {a b} : a ∈ carrier → b ∈ carrier → a * b ∈ carrier + /-- The unit element belongs to the submonoid. -/ + one_mem : 1 ∈ carrier + + /-- Submonoids in `M` can be seen as sets in `M`. -/ + instance [Monoid M] : SetLike (Submonoid₁ M) M where + coe := Submonoid₁.carrier + coe_injective' := Submonoid₁.ext + </pre></div> + </div> + Equipped with the above <code class="docutils literal notranslate">SetLike</code> instance, we can already state naturally that + a submonoid <code class="docutils literal notranslate">N</code> contains <code class="docutils literal notranslate">1</code> without using <code class="docutils literal notranslate">N.carrier</code>. + We can also silently treat <code class="docutils literal notranslate">N</code> as a set in <code class="docutils literal notranslate">M</code> as take its direct image under a map. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [Monoid M] (N : Submonoid₁ M) : 1 ∈ N := N.one_mem + + example [Monoid M] (N : Submonoid₁ M) (α : Type) (f : M → α) := f '' N + </pre></div> + </div> + We also have a coercion to <code class="docutils literal notranslate">Type</code> which uses <code class="docutils literal notranslate">Subtype</code> so, given a submonoid <code class="docutils literal notranslate">N</code> we can write + a parameter <code class="docutils literal notranslate">(x : N)</code> which can be coerced to an element of <code class="docutils literal notranslate">M</code> belonging to <code class="docutils literal notranslate">N</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [Monoid M] (N : Submonoid₁ M) (x : N) : (x : M) ∈ N := x.property + </pre></div> + </div> + Using this coercion to <code class="docutils literal notranslate">Type</code> we can also tackle the task of equipping a submonoid with a + monoid structure. We will use the coercion from the type associated to <code class="docutils literal notranslate">N</code> as above, and the + lemma <code class="docutils literal notranslate">SetCoe.ext</code> asserting this coercion is injective. Both are provided by the <code class="docutils literal notranslate">SetLike</code> + instance. + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance SubMonoid₁Monoid [Monoid M] (N : Submonoid₁ M) : Monoid N where + mul := fun x y ↦ ⟨x*y, N.mul_mem x.property y.property⟩ + mul_assoc := fun x y z ↦ SetCoe.ext (mul_assoc (x : M) y z) + one := ⟨1, N.one_mem⟩ + one_mul := fun x ↦ SetCoe.ext (one_mul (x : M)) + mul_one := fun x ↦ SetCoe.ext (mul_one (x : M)) + </pre></div> + </div> + Note that, in the above instance, instead of using the coercion to <code class="docutils literal notranslate">M</code> and calling the + <code class="docutils literal notranslate">property</code> field, we could have used destructuring binders as follows. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [Monoid M] (N : Submonoid₁ M) : Monoid N where + mul := fun ⟨x, hx⟩ ⟨y, hy⟩ ↦ ⟨x*y, N.mul_mem hx hy⟩ + mul_assoc := fun ⟨x, _⟩ ⟨y, _⟩ ⟨z, _⟩ ↦ SetCoe.ext (mul_assoc x y z) + one := ⟨1, N.one_mem⟩ + one_mul := fun ⟨x, _⟩ ↦ SetCoe.ext (one_mul x) + mul_one := fun ⟨x, _⟩ ↦ SetCoe.ext (mul_one x) + </pre></div> + </div> + In order to apply lemmas about submonoids to subgroups or subrings, we need a class, just + like for morphisms. Note this class take a <code class="docutils literal notranslate">SetLike</code> instance as a parameter so it does not need + a carrier field and can use the membership notation in its fields. + <div class="highlight-lean notranslate"><div class="highlight"><pre>class SubmonoidClass₁ (S : Type) (M : Type) [Monoid M] [SetLike S M] : Prop where + mul_mem : ∀ (s : S) {a b : M}, a ∈ s → b ∈ s → a * b ∈ s + one_mem : ∀ s : S, 1 ∈ s + + instance [Monoid M] : SubmonoidClass₁ (Submonoid₁ M) M where + mul_mem := Submonoid₁.mul_mem + one_mem := Submonoid₁.one_mem + </pre></div> + </div> + As an exercise you should define a <code class="docutils literal notranslate">Subgroup₁</code> structure, endow it with a <code class="docutils literal notranslate">SetLike</code> instance + and a <code class="docutils literal notranslate">SubmonoidClass₁</code> instance, put a <code class="docutils literal notranslate">Group</code> instance on the subtype associated to a + <code class="docutils literal notranslate">Subgroup₁</code> and define a <code class="docutils literal notranslate">SubgroupClass₁</code> class. + Another very important thing to know about subobjects of a given algebraic object in Mathlib + always form a complete lattice, and this structure is used a lot. For instance you may look for + the lemma saying that an intersection of submonoids is a submonoid. But this won’t be a lemma, + this will be an infimum construction. Let us do the case of two submonoids. + <div class="highlight-lean notranslate"><div class="highlight"><pre>instance [Monoid M] : Inf (Submonoid₁ M) := + ⟨fun S₁ S₂ => + { carrier := S₁ ∩ S₂ + one_mem := ⟨S₁.one_mem, S₂.one_mem⟩ + mul_mem := fun ⟨hx, hx'⟩ ⟨hy, hy'⟩ ↦ ⟨S₁.mul_mem hx hy, S₂.mul_mem hx' hy'⟩ }⟩ + </pre></div> + </div> + This allows to get the intersections of two submonoids as a submonoid. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [Monoid M] (N P : Submonoid₁ M) : Submonoid₁ M := N ⊓ P + </pre></div> + </div> + You may think it’s a shame that we had to use the inf symbol <code class="docutils literal notranslate">⊓</code> in the above example instead + of the intersection symbol <code class="docutils literal notranslate">∩</code>. But think about the supremum. The union of two submonoids is not + a submonoid. However submonoids still form a lattice (even a complete one). Actually <code class="docutils literal notranslate">N ⊔ P</code> is + the submonoid generated by the union of <code class="docutils literal notranslate">N</code> and <code class="docutils literal notranslate">P</code> and of course it would be very confusing to + denote it by <code class="docutils literal notranslate">N ∪ P</code>. So you can see the use of <code class="docutils literal notranslate">N ⊓ P</code> as much more consistent. It is also + a lot more consistent across various kind of algebraic structures. It may look a bit weird at first + to see the sum of two vector subspace <code class="docutils literal notranslate">E</code> and <code class="docutils literal notranslate">F</code> denoted by <code class="docutils literal notranslate">E ⊔ F</code> instead of <code class="docutils literal notranslate">E + F</code>. + But you will get used to it. And soon you will consider the <code class="docutils literal notranslate">E + F</code> notation as a distraction + emphasizing the anecdotal fact that elements of <code class="docutils literal notranslate">E ⊔ F</code> can be written as a sum of an element of + <code class="docutils literal notranslate">E</code> and an element of <code class="docutils literal notranslate">F</code> instead of emphasizing the fundamental fact that <code class="docutils literal notranslate">E ⊔ F</code> is the + smallest vector subspace containing both <code class="docutils literal notranslate">E</code> and <code class="docutils literal notranslate">F</code>. + Our last topic for this chapter is that of quotients. Again we want to explain how + convenient notation are built and code duplication is avoided in Mathlib. Here the main device + is the <code class="docutils literal notranslate">HasQuotient</code> class which allows notations like <code class="docutils literal notranslate">M ⧸ N</code>. Beware the quotient symbol + <code class="docutils literal notranslate">⧸</code> is a special unicode character, not a regular ASCII division symbol. + As an example, we will build the quotient of a commutative monoid by a submonoid, leave proofs + to you. + <div class="highlight-lean notranslate"><div class="highlight"><pre>def Submonoid.Setoid [CommMonoid M] (N : Submonoid M) : Setoid M where + r := fun x y ↦ ∃ w ∈ N, ∃ z ∈ N, x*w = y*z + iseqv := { + refl := fun x ↦ ⟨1, N.one_mem, 1, N.one_mem, rfl⟩ + symm := fun ⟨w, hw, z, hz, h⟩ ↦ ⟨z, hz, w, hw, h.symm⟩ + trans := by + sorry + } + + instance [CommMonoid M] : HasQuotient M (Submonoid M) where + quotient' := fun N ↦ Quotient N.Setoid + + def QuotientMonoid.mk [CommMonoid M] (N : Submonoid M) : M → M ⧸ N := Quotient.mk N.Setoid + + instance [CommMonoid M] (N : Submonoid M) : Monoid (M ⧸ N) where + mul := Quotient.map₂' (· * ·) (by + sorry + ) + mul_assoc := by + sorry + one := QuotientMonoid.mk N 1 + one_mul := by + sorry + mul_one := by + sorry + </pre></div> + </div> + -/ + </section> + </section> + + + </div> + </div> + <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> + <a href="C06_Structures.html" class="btn btn-neutral float-left" title="6. 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- @@ -47,25 +51,27 @@
 <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> - <li class="toctree-l1"><a class="reference internal" href="C06_Abstract_Algebra.html">6. Abstract Algebra</a></li> - <li class="toctree-l1 current"><a class="current reference internal" href="#">7. Topology</a><ul> - <li class="toctree-l2"><a class="reference internal" href="#filters">7.1. Filters</a></li> - <li class="toctree-l2"><a class="reference internal" href="#metric-spaces">7.2. Metric spaces</a><ul> - <li class="toctree-l3"><a class="reference internal" href="#convergence-and-continuity">7.2.1. Convergence and continuity</a></li> - <li class="toctree-l3"><a class="reference internal" href="#balls-open-sets-and-closed-sets">7.2.2. Balls, open sets and closed sets</a></li> - <li class="toctree-l3"><a class="reference internal" href="#compactness">7.2.3. Compactness</a></li> - <li class="toctree-l3"><a class="reference internal" href="#uniformly-continuous-functions">7.2.4. Uniformly continuous functions</a></li> - <li class="toctree-l3"><a class="reference internal" href="#completeness">7.2.5. Completeness</a></li> + <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> + <li class="toctree-l1 current"><a class="current reference internal" href="#">8. Topology</a><ul> + <li class="toctree-l2"><a class="reference internal" href="#filters">8.1. Filters</a></li> + <li class="toctree-l2"><a class="reference internal" href="#metric-spaces">8.2. Metric spaces</a><ul> + <li class="toctree-l3"><a class="reference internal" href="#convergence-and-continuity">8.2.1. Convergence and continuity</a></li> + <li class="toctree-l3"><a class="reference internal" href="#balls-open-sets-and-closed-sets">8.2.2. Balls, open sets and closed sets</a></li> + <li class="toctree-l3"><a class="reference internal" href="#compactness">8.2.3. Compactness</a></li> + <li class="toctree-l3"><a class="reference internal" href="#uniformly-continuous-functions">8.2.4. Uniformly continuous functions</a></li> + <li class="toctree-l3"><a class="reference internal" href="#completeness">8.2.5. Completeness</a></li> </ul> </li> - <li class="toctree-l2"><a class="reference internal" href="#topological-spaces">7.3. Topological spaces</a><ul> - <li class="toctree-l3"><a class="reference internal" href="#fundamentals">7.3.1. Fundamentals</a></li> - <li class="toctree-l3"><a class="reference internal" href="#separation-and-countability">7.3.2. Separation and countability</a></li> - <li class="toctree-l3"><a class="reference internal" href="#id5">7.3.3. Compactness</a></li> + <li class="toctree-l2"><a class="reference internal" href="#topological-spaces">8.3. Topological spaces</a><ul> + <li class="toctree-l3"><a class="reference internal" href="#fundamentals">8.3.1. Fundamentals</a></li> + <li class="toctree-l3"><a class="reference internal" href="#separation-and-countability">8.3.2. Separation and countability</a></li> + <li class="toctree-l3"><a class="reference internal" href="#id5">8.3.3. Compactness</a></li> </ul> </li> </ul> </li> + <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li>
- @@ -84,10 +90,10 @@
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- @@ -96,7 +102,7 @@
 <div itemprop="articleBody"> <section id="index-0"> - <h1>7. Topology<a class="headerlink" href="#index-0" title="Permalink to this heading"></a></h1> + <h1>8. Topology<a class="headerlink" href="#index-0" title="Permalink to this heading"></a></h1> Calculus is based on the concept of a function, which is used to model quantities that depend on one another. For example, it is common to study quantities that change over time.
- @@ -110,7 +116,7 @@ as $x$ tends to Section 3.6</a>. Topology is the abstract study of limits and continuity. Having covered the essentials of formalization in Chapters <a class="reference internal" href="C02_Basics.html#basics">2</a> - to <a class="reference internal" href="C06_Abstract_Algebra.html#abstract-algebra">6</a>, + to <a class="reference internal" href="C06_Structures.html#structures">6</a>, in this chapter, we will explain how topological notions are formalized in mathlib. Not only do topological abstractions apply in much greater generality, but that also, somewhat paradoxically, make it easier to reason about limits
- @@ -118,7 +124,7 @@ and continuity in concrete instances.
 Topological notions build on quite a few layers of mathematical structure. The first layer is naive set theory, as described in <a class="reference internal" href="C04_Sets_and_Functions.html#sets-and-functions">Chapter 4</a>. - The next layer is the theory of filters, which we will describe in <a class="reference internal" href="#filters">Section 7.1</a>. + The next layer is the theory of filters, which we will describe in <a class="reference internal" href="#filters">Section 8.1</a>. On top of that, we layer the theories of topological spaces, metric spaces, and a slightly more exotic intermediate notion called a uniform space.
- @@ -164,7 +170,7 @@ 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>7.1. Filters<a class="headerlink" href="#filters" title="Permalink to this heading"></a></h2> + <h2>8.1. Filters<a class="headerlink" href="#filters" title="Permalink to this heading"></a></h2> A filter on a type <code class="docutils literal notranslate">X</code> is a collection of sets of <code class="docutils literal notranslate">X</code> that satisfies three conditions that we will spell out below. The notion supports two related ideas:
- @@ -328,7 +334,7 @@ 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">×ᶠ</code>, which answers this question: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : 𝓝 (x₀, y₀) = 𝓝 x₀ ×ᶠ 𝓝 y₀ := + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : 𝓝 (x₀, y₀) = 𝓝 x₀ ×ˢ 𝓝 y₀ := nhds_prod_eq </pre></div> </div>
- @@ -518,7 +524,7 @@ using the fact that <code class="docutils literal notranslate"> </section> <section id="metric-spaces"> - <h2>7.2. Metric spaces<a class="headerlink" href="#metric-spaces" title="Permalink to this heading"></a></h2> + <h2>8.2. Metric spaces<a class="headerlink" href="#metric-spaces" title="Permalink to this heading"></a></h2> Examples in the previous section focus on sequences of real numbers. In this section we will go up a bit in generality and focus on metric spaces. A metric space is a type <code class="docutils literal notranslate">X</code> equipped with a distance function <code class="docutils literal notranslate">dist : X → X → ℝ</code> which is a generalization of the function <code class="docutils literal notranslate">fun x y ↦ |x - y|</code> from the case where <code class="docutils literal notranslate">X = ℝ</code>.
- @@ -537,7 +543,7 @@ 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>7.2.1. Convergence and continuity<a class="headerlink" href="#convergence-and-continuity" title="Permalink to this heading"></a></h3> + <h3>8.2.1. Convergence and continuity<a class="headerlink" href="#convergence-and-continuity" title="Permalink to this heading"></a></h3> Using distance functions, we can already define convergent sequences and continuous functions between metric spaces. They are actually defined in a more general setting covered in the next section, but we have lemmas recasting the definition is terms of distances.
- @@ -626,7 +632,7 @@ and get our final proof, now bordering obfuscation.
 </div> </section> <section id="balls-open-sets-and-closed-sets"> - <h3>7.2.2. Balls, open sets and closed sets<a class="headerlink" href="#balls-open-sets-and-closed-sets" title="Permalink to this heading"></a></h3> + <h3>8.2.2. Balls, open sets and closed sets<a class="headerlink" href="#balls-open-sets-and-closed-sets" title="Permalink to this heading"></a></h3> Once we have a distance function, the most important geometric definitions are (open) balls and closed balls. <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (r : ℝ)
- @@ -683,7 +689,7 @@ argument so we can invoke <code class="docutils literal notranslate"> </section> <section id="compactness"> - <h3>7.2.3. Compactness<a class="headerlink" href="#compactness" title="Permalink to this heading"></a></h3> + <h3>8.2.3. Compactness<a class="headerlink" href="#compactness" title="Permalink to this heading"></a></h3> Compactness is an important topological notion. It distinguishes subsets of a metric space that enjoy the same kind of properties as segments in reals compared to other intervals: <ul class="simple">
- @@ -726,7 +732,7 @@ are deduced from more general versions, some of which will be discussed in later
 In a compact metric space any closed set is compact, this is <code class="docutils literal notranslate">IsCompact.isClosed</code>. </section> <section id="uniformly-continuous-functions"> - <h3>7.2.4. Uniformly continuous functions<a class="headerlink" href="#uniformly-continuous-functions" title="Permalink to this heading"></a></h3> + <h3>8.2.4. Uniformly continuous functions<a class="headerlink" href="#uniformly-continuous-functions" title="Permalink to this heading"></a></h3> We now turn to uniformity notions on metric spaces : uniformly continuous functions, Cauchy sequences and completeness. Again those are defined in a more general context but we have lemmas in the metric name space to access their elementary definitions. We start with uniform continuity.
- @@ -757,7 +763,7 @@ of the distance function on <code class="docutils literal notranslate"> </section> <section id="completeness"> - <h3>7.2.5. Completeness<a class="headerlink" href="#completeness" title="Permalink to this heading"></a></h3> + <h3>8.2.5. Completeness<a class="headerlink" href="#completeness" title="Permalink to this heading"></a></h3> A Cauchy sequence in a metric space is a sequence whose terms get closer and closer to each other. There are a couple of equivalent ways to state that idea. In particular converging sequences are Cauchy. The converse is true only in so-called complete
- @@ -849,9 +855,9 @@ define something inductively in the middle of a proof using <code class="docutil
 </section> </section> <section id="topological-spaces"> - <h2>7.3. Topological spaces<a class="headerlink" href="#topological-spaces" title="Permalink to this heading"></a></h2> + <h2>8.3. Topological spaces<a class="headerlink" href="#topological-spaces" title="Permalink to this heading"></a></h2> <section id="fundamentals"> - <h3>7.3.1. Fundamentals<a class="headerlink" href="#fundamentals" title="Permalink to this heading"></a></h3> + <h3>8.3.1. Fundamentals<a class="headerlink" href="#fundamentals" title="Permalink to this heading"></a></h3> We now go up in generality and introduce topological spaces. We will review the two main ways to define topological spaces and then explain how the category of topological spaces is much better behaved than the category of metric spaces. Note that we won’t be using mathlib category theory here, only having
- @@ -1040,7 +1046,7 @@ Let us explore that constraint “on papar” using notation </section> <section id="separation-and-countability"> - <h3>7.3.2. Separation and countability<a class="headerlink" href="#separation-and-countability" title="Permalink to this heading"></a></h3> + <h3>8.3.2. Separation and countability<a class="headerlink" href="#separation-and-countability" title="Permalink to this heading"></a></h3> We saw that the category of topological spaces have very nice properties. The price to pay for this is existence of rather pathological topological spaces. There are a number of assumptions you can make on a topological space to ensure its behavior
- @@ -1130,7 +1136,7 @@ of sets can be understood using sequences.
 </div> </section> <section id="id5"> - <h3>7.3.3. Compactness<a class="headerlink" href="#id5" title="Permalink to this heading"></a></h3> + <h3>8.3.3. Compactness<a class="headerlink" href="#id5" title="Permalink to this heading"></a></h3> Let us now discuss how compactness is defined for topological spaces. As usual there are several ways to think about it and mathlib goes for the filter version. We first need to define cluster points of filters. Given a filter <code class="docutils literal notranslate">F</code> on a topological space <code class="docutils literal notranslate">X</code>,
- @@ -1199,8 +1205,8 @@ cover <code class="docutils literal notranslate">s</cod
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Basics</a></li> + <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> + <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> + <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> + <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> + <li class="toctree-l1 current"><a class="current reference internal" href="#">9. Differential Calculus</a><ul> + <li class="toctree-l2"><a class="reference internal" href="#elementary-differential-calculus">9.1. Elementary Differential Calculus</a></li> + <li class="toctree-l2"><a class="reference internal" href="#differential-calculus-in-normed-spaces">9.2. Differential Calculus in Normed Spaces</a><ul> + <li class="toctree-l3"><a class="reference internal" href="#id3">9.2.1. Normed spaces</a></li> + <li class="toctree-l3"><a class="reference internal" href="#continuous-linear-maps">9.2.2. Continuous linear maps</a></li> + <li class="toctree-l3"><a class="reference internal" href="#asymptotic-comparisons">9.2.3. Asymptotic comparisons</a></li> + <li class="toctree-l3"><a class="reference internal" href="#differentiability">9.2.4. Differentiability</a></li> + </ul> + </li> + </ul> + </li> + </ul> + <ul> + <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li> + </ul> + + </div> + </div> + </nav> + + <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" > + + <a href="index.html">Mathematics in Lean</a> + </nav> + + <div class="wy-nav-content"> + <div class="rst-content"> + <div role="navigation" aria-label="Page navigation"> + <ul class="wy-breadcrumbs"> + <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> + <li class="breadcrumb-item active">9. Differential Calculus</li> + <li class="wy-breadcrumbs-aside"> + <a href="_sources/C09_Differential_Calculus.rst.txt" rel="nofollow"> View page source</a> + </li> + </ul> + <hr/> + </div> + <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> + <div itemprop="articleBody"> + + <section id="index-0"> + <h1>9. Differential Calculus<a class="headerlink" href="#index-0" title="Permalink to this heading"></a></h1> + We now consider the formalization of notions from analysis, + starting with differentiation in this chapter + and turning integration and measure theory in the next. + In <a class="reference internal" href="#elementary-differential-calculus">Section 9.1</a>, we stick with the + setting of functions from the real numbers to the real numbers, + which is familiar from any introductory calculus class. + In <a class="reference internal" href="#normed-spaces">Section 9.2</a>, we then consider the notion of a derivative in + a much broader setting. + <section id="elementary-differential-calculus"> + <h2>9.1. Elementary Differential Calculus<a class="headerlink" href="#elementary-differential-calculus" title="Permalink to this heading"></a></h2> + Let <code class="docutils literal notranslate">f</code> be a function from the reals to the reals. There is a difference + between talking about the derivative of <code class="docutils literal notranslate">f</code> at a single point and + talking about the derivative function. + In mathlib, the first notion is represented as follows. + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Real + + /-- The sin function has derivative 1 at 0. -/ + example : HasDerivAt sin 1 0 := by simpa using hasDerivAt_sin 0 + </pre></div> + </div> + We can also express that <code class="docutils literal notranslate">f</code> is differentiable at a point without + specifying its derivative there + by writing <code class="docutils literal notranslate">differentiable_at ℝ</code>. + We specify <code class="docutils literal notranslate">ℝ</code> explicitly because in a slightly more general context, + when talking about functions from <code class="docutils literal notranslate">ℂ</code> to <code class="docutils literal notranslate">ℂ</code>, + we want to be able to distinguish between being differentiable in the real sense + and being differentiable in the sense of the complex derivative. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (x : ℝ) : DifferentiableAt ℝ sin x := + (hasDerivAt_sin x).differentiableAt + </pre></div> + </div> + It would be inconvenient to have to provide a proof of differentiability + every time we want to refer to a derivative. + So mathlib provides a function <code class="docutils literal notranslate">deriv f : ℝ → ℝ</code> that is defined for any + function <code class="docutils literal notranslate">f : ℝ → ℝ</code> + but is defined to take the value <code class="docutils literal notranslate">0</code> at any point where <code class="docutils literal notranslate">f</code> is not differentiable. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {f : ℝ → ℝ} {x a : ℝ} (h : HasDerivAt f a x) : deriv f x = a := + h.deriv + + example {f : ℝ → ℝ} {x : ℝ} (h : ¬DifferentiableAt ℝ f x) : deriv f x = 0 := + deriv_zero_of_not_differentiableAt h + </pre></div> + </div> + Of course there are many lemmas about <code class="docutils literal notranslate">deriv</code> that do require differentiability assumptions. + For instance, you should think about a counterexample to the next lemma without the + differentiability assumptions. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {f g : ℝ → ℝ} {x : ℝ} (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) : + deriv (f + g) x = deriv f x + deriv g x := + deriv_add hf hg + </pre></div> + </div> + Interestingly, however, there are statements that can avoid differentiability + assumptions by taking advantage + of the fact that the value of <code class="docutils literal notranslate">deriv</code> defaults to zero when the function is + not differentiable. + So making sense of the following statement requires knowing the precise + definition of <code class="docutils literal notranslate">deriv</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {f : ℝ → ℝ} {a : ℝ} (h : IsLocalMin f a) : deriv f a = 0 := + h.deriv_eq_zero + </pre></div> + </div> + We can eve state Rolle’s theorem without any differentiability assumptions, which + seems even weirder. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {f : ℝ → ℝ} {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hfI : f a = f b) : + ∃ c ∈ Ioo a b, deriv f c = 0 := + exists_deriv_eq_zero f hab hfc hfI + </pre></div> + </div> + Of course, this trick does not work for the general mean value theorem. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (f : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hf : ContinuousOn f (Icc a b)) + (hf' : DifferentiableOn ℝ f (Ioo a b)) : ∃ c ∈ Ioo a b, deriv f c = (f b - f a) / (b - a) := + exists_deriv_eq_slope f hab hf hf' + </pre></div> + </div> + Lean can automatically compute some simple derivatives using the <code class="docutils literal notranslate">simp</code> tactic. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : deriv (fun x : ℝ ↦ x ^ 5) 6 = 5 * 6 ^ 4 := by simp + + example : deriv sin π = -1 := by simp + </pre></div> + </div> + </section> + <section id="differential-calculus-in-normed-spaces"> + <h2>9.2. Differential Calculus in Normed Spaces<a class="headerlink" href="#differential-calculus-in-normed-spaces" title="Permalink to this heading"></a></h2> + <section id="id3"> + <h3>9.2.1. Normed spaces<a class="headerlink" href="#id3" title="Permalink to this heading"></a></h3> + Differentiation can be generalized beyond <code class="docutils literal notranslate">ℝ</code> using the notion of a + normed vector space, which encapsulates both direction and distance. + We start with the notion of a normed group, which as an additive commutative + group equipped with a real-valued norm function + satisfying the following conditions. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {E : Type _} [NormedAddCommGroup E] + + example (x : E) : 0 ≤ ‖x‖ := + norm_nonneg x + + example {x : E} : ‖x‖ = 0 ↔ x = 0 := + norm_eq_zero + + example (x y : E) : ‖x + y‖ ≤ ‖x‖ + ‖y‖ := + norm_add_le x y + </pre></div> + </div> + Every normed space is a metric space with distance function + $d(x, y) = \| x - y \|$, and hence it is also a topological space. + Lean and mathlib know this. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : MetricSpace E := by infer_instance + + example {X : Type _} [TopologicalSpace X] {f : X → E} (hf : Continuous f) : + Continuous fun x => ‖f x‖ := + hf.norm + </pre></div> + </div> + In order to use the notion of a norm with concepts from linear algebra, + we add the assumption <code class="docutils literal notranslate">normed_space ℝ E</code> on top of <code class="docutils literal notranslate">normed_add_group E</code>. + This stipulates that <code class="docutils literal notranslate">E</code> is a vector space over <code class="docutils literal notranslate">ℝ</code> and that + scalar multiplication satisfies the following condition. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable [NormedSpace ℝ E] + + example (a : ℝ) (x : E) : ‖a • x‖ = |a| * ‖x‖ := + norm_smul a x + </pre></div> + </div> + A complete normed space is known as a Banach space. + Every finite-dimensional vector space is complete. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example [FiniteDimensional ℝ E] : CompleteSpace E := by infer_instance + </pre></div> + </div> + In all the previous examples, we used the real numbers as the base field. + More generally, we can make sense of calculus with a vector space over any + non-discrete normed field. These are fields that are equipped with a + real-valued norm that is multiplicative and has the property that + not every element has norm zero or one + (equivalently, there is an element whose norm is bigger than one). + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (𝕜 : Type _) [NontriviallyNormedField 𝕜] (x y : 𝕜) : ‖x * y‖ = ‖x‖ * ‖y‖ := + norm_mul x y + + example (𝕜 : Type _) [NontriviallyNormedField 𝕜] : ∃ x : 𝕜, 1 < ‖x‖ := + NormedField.exists_one_lt_norm 𝕜 + </pre></div> + </div> + A finite-dimensional vector space over a nondiscrete normed field is + complete as long as the field itself is complete. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (𝕜 : Type _) [NontriviallyNormedField 𝕜] (E : Type _) [NormedAddCommGroup E] + [NormedSpace 𝕜 E] [CompleteSpace 𝕜] [FiniteDimensional 𝕜 E] : CompleteSpace E := + FiniteDimensional.complete 𝕜 E + </pre></div> + </div> + </section> + <section id="continuous-linear-maps"> + <h3>9.2.2. Continuous linear maps<a class="headerlink" href="#continuous-linear-maps" title="Permalink to this heading"></a></h3> + We now turn to the morphisms in the category of normed spaces, namely, + continuous linear maps. + In mathlib, the type of <code class="docutils literal notranslate">𝕜</code>-linear continuous maps between normed spaces + <code class="docutils literal notranslate">E</code> and <code class="docutils literal notranslate">F</code> is written <code class="docutils literal notranslate">E →L[𝕜] F</code>. + They are implemented as bundled maps, which means that an element of this type + a structure that that includes the function itself and the properties + of being linear and continuous. + Lean will insert a coercion so that a continuous linear map can be treated + as a function. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] + [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] + + example : E →L[𝕜] E := + ContinuousLinearMap.id 𝕜 E + + example (f : E →L[𝕜] F) : E → F := + f + + example (f : E →L[𝕜] F) : Continuous f := + f.cont + + example (f : E →L[𝕜] F) (x y : E) : f (x + y) = f x + f y := + f.map_add x y + + example (f : E →L[𝕜] F) (a : 𝕜) (x : E) : f (a • x) = a • f x := + f.map_smul a x + </pre></div> + </div> + Continuous linear maps have an operator norm that is characterized by the + following properties. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (f : E →L[𝕜] F) + + example (x : E) : ‖f x‖ ≤ ‖f‖ * ‖x‖ := + f.le_op_norm x + + example {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ x, ‖f x‖ ≤ M * ‖x‖) : ‖f‖ ≤ M := + f.op_norm_le_bound hMp hM + </pre></div> + </div> + There is also a notion of bundled continuous linear isomorphism. + Their type of such isomorphisms is <code class="docutils literal notranslate">E ≃L[𝕜] F</code>. + As a challenging exercise, you can prove the Banach-Steinhaus theorem, also + known as the Uniform Boundedness Principle. + The principle states that a family of continuous linear maps from a Banach space + into a normed space is pointwise + bounded, then the norms of these linear maps are uniformly bounded. + The main ingredient is Baire’s theorem + <code class="docutils literal notranslate">nonempty_interior_of_Union_of_closed.</code> (You proved a version of this in the topology chapter.) + Minor ingredients include <code class="docutils literal notranslate">continuous_linear_map.op_norm_le_of_shell</code>, + <code class="docutils literal notranslate">interior_subset</code> and <code class="docutils literal notranslate">interior_Inter_subset</code> and <code class="docutils literal notranslate">is_closed_le</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] + [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] + + open Metric + + example {ι : Type _} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) : + ∃ C', ∀ i, ‖g i‖ ≤ C' := by + -- sequence of subsets consisting of those `x : E` with norms `‖g i x‖` bounded by `n` + let e : ℕ → Set E := fun n => ⋂ i : ι, { x : E | ‖g i x‖ ≤ n } + -- each of these sets is closed + have hc : ∀ n : ℕ, IsClosed (e n) + sorry + -- the union is the entire space; this is where we use `h` + have hU : (⋃ n : ℕ, e n) = univ + sorry + /- apply the Baire category theorem to conclude that for some `m : ℕ`, + `e m` contains some `x` -/ + obtain ⟨m, x, hx⟩ : ∃ m, ∃ x, x ∈ interior (e m) := sorry + obtain ⟨ε, ε_pos, hε⟩ : ∃ ε > 0, ball x ε ⊆ interior (e m) := sorry + obtain ⟨k, hk⟩ : ∃ k : 𝕜, 1 < ‖k‖ := sorry + -- show all elements in the ball have norm bounded by `m` after applying any `g i` + have real_norm_le : ∀ z ∈ ball x ε, ∀ (i : ι), ‖g i z‖ ≤ m + sorry + have εk_pos : 0 < ε / ‖k‖ := sorry + refine' ⟨(m + m : ℕ) / (ε / ‖k‖), fun i => ContinuousLinearMap.op_norm_le_of_shell ε_pos _ hk _⟩ + sorry + sorry + </pre></div> + </div> + </section> + <section id="asymptotic-comparisons"> + <h3>9.2.3. Asymptotic comparisons<a class="headerlink" href="#asymptotic-comparisons" title="Permalink to this heading"></a></h3> + Defining differentiability also requires asymptotic comparisons. + Mathlib has an extensive library covering the big O and little o relations, + whose definitions are shown below. + Opening the <code class="docutils literal notranslate">asymptotics</code> locale allows us to use the corresponding + notation. + Here we will only use little o to define differentiability. + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Asymptotics + + open Asymptotics + + example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] (c : ℝ) + (l : Filter α) (f : α → E) (g : α → F) : IsBigOWith c l f g ↔ ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := + isBigOWith_iff + + example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] + (l : Filter α) (f : α → E) (g : α → F) : f =O[l] g ↔ ∃ C, IsBigOWith C l f g := + isBigO_iff_isBigOWith + + example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] + (l : Filter α) (f : α → E) (g : α → F) : f =o[l] g ↔ ∀ C > 0, IsBigOWith C l f g := + isLittleO_iff_forall_isBigOWith + + example {α : Type _} {E : Type _} [NormedAddCommGroup E] (l : Filter α) (f g : α → E) : + f ~[l] g ↔ (f - g) =o[l] g := + Iff.rfl + </pre></div> + </div> + </section> + <section id="differentiability"> + <h3>9.2.4. Differentiability<a class="headerlink" href="#differentiability" title="Permalink to this heading"></a></h3> + We are now ready to discuss differentiable functions between normed spaces. + In analogy the elementary one-dimensional, + mathlib defines a predicate <code class="docutils literal notranslate">has_fderiv_at</code> and a function <code class="docutils literal notranslate">fderiv</code>. + Here the letter + “f” stands for Fréchet. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] + [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] + + example (f : E → F) (f' : E →L[𝕜] F) (x₀ : E) : + HasFDerivAt f f' x₀ ↔ (fun x => f x - f x₀ - f' (x - x₀)) =o[𝓝 x₀] fun x => x - x₀ := + Iff.rfl + + example (f : E → F) (f' : E →L[𝕜] F) (x₀ : E) (hff' : HasFDerivAt f f' x₀) : fderiv 𝕜 f x₀ = f' := + hff'.fderiv + </pre></div> + </div> + We also have iterated derivatives that take values in the type of multilinear maps + <code class="docutils literal notranslate">E [×n]→L[𝕜] F</code>, + and we have continuously differential functions. + The type <code class="docutils literal notranslate">with_top ℕ</code> is <code class="docutils literal notranslate">ℕ</code> with an additional element <code class="docutils literal notranslate">⊤</code> that + is bigger than every natural number. + So $\mathcal{C}^\infty$ functions are functions <code class="docutils literal notranslate">f</code> that satisfy + <code class="docutils literal notranslate">cont_diff 𝕜 ⊤ f</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (n : ℕ) (f : E → F) : E → E[×n]→L[𝕜] F := + iteratedFDeriv 𝕜 n f + + example (n : WithTop ℕ) {f : E → F} : + ContDiff 𝕜 n f ↔ + (∀ m : ℕ, (m : WithTop ℕ) ≤ n → Continuous fun x => iteratedFDeriv 𝕜 m f x) ∧ + ∀ m : ℕ, (m : WithTop ℕ) < n → Differentiable 𝕜 fun x => iteratedFDeriv 𝕜 m f x := + contDiff_iff_continuous_differentiable + </pre></div> + </div> + There is a stricter notion of differentiability called + <code class="docutils literal notranslate">has_strict_fderiv_at</code>, which is used in the statement + of the inverse function theorem and the statement of the implicit function + theorem, both of which are in mathlib. + Over <code class="docutils literal notranslate">ℝ</code> or <code class="docutils literal notranslate">ℂ</code>, continuously differentiable + functions are strictly differentiable. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {𝕂 : Type _} [IsROrC 𝕂] {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕂 E] {F : Type _} + [NormedAddCommGroup F] [NormedSpace 𝕂 F] {f : E → F} {x : E} {n : WithTop ℕ} + (hf : ContDiffAt 𝕂 n f x) (hn : 1 ≤ n) : HasStrictFDerivAt f (fderiv 𝕂 f x) x := + hf.hasStrictFDerivAt hn + </pre></div> + </div> + The local inverse theorem is stated using an operation that produces an + inverse function from a + function and the assumptions that the function is strictly differentiable at a + point <code class="docutils literal notranslate">a</code> and that its derivative is an isomorphism. + The first example below gets this local inverse. + The next one states that it is indeed a local inverse + from the left and from the right, and that it is strictly differentiable. + <div class="highlight-lean notranslate"><div class="highlight"><pre>section LocalInverse + variable [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} + + example (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : F → E := + HasStrictFDerivAt.localInverse f f' a hf + + example (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : + ∀ᶠ x in 𝓝 a, hf.localInverse f f' a (f x) = x := + hf.eventually_left_inverse + + example (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : + ∀ᶠ x in 𝓝 (f a), f (hf.localInverse f f' a x) = x := + hf.eventually_right_inverse + + example [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} + (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : + HasStrictFDerivAt (HasStrictFDerivAt.localInverse f f' a hf) (f'.symm : F →L[𝕜] E) (f a) := + HasStrictFDerivAt.to_localInverse hf + + end LocalInverse + </pre></div> + </div> + This has been only a quick tour of the differential calculus in mathlib. + The library contains many variations that we have not discussed. + For example, you may want to use one-sided derivatives in the + one-dimensional setting. The means to do so are found in mathlib in a more + general context; + see <code class="docutils literal notranslate">HasFDerivWithinAt</code> or the even more general <code class="docutils literal notranslate">HasFDerivAtFilter</code>. + </section> + </section> + </section> + + + </div> + </div> + <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> + <a href="C08_Topology.html" class="btn btn-neutral float-left" title="8. Topology" accesskey="p" rel="prev"> Previous</a> + <a href="genindex.html" class="btn btn-neutral float-right" title="Index" accesskey="n" rel="next">Next </a> + </div> + + <hr/> + + <div role="contentinfo"> + © Copyright 2020, Jeremy Avigad, Kevin Buzzard, Robert Y. 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html/C09_Integration_and_Measure_Theory.html > html/C10_Integration_and_Measure_Theory.html
- @@ -1,7 +1,7 @@
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- @@ -13,11 +13,11 @@
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- @@ -28,11 +28,15 @@
 <nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > - <a href="index.html" class="icon icon-home"> Mathematics in Lean + + + + <a href="index.html" class="icon icon-home"> + Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> - <input type="text" name="q" placeholder="Search docs" /> + <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
- @@ -44,8 +48,10 @@
 <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> - <li class="toctree-l1"><a class="reference internal" href="C06_Abstract_Algebra.html">6. Abstract Algebra</a></li> - <li class="toctree-l1"><a class="reference internal" href="C07_Topology.html">7. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> + <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li>
- @@ -64,10 +70,10 @@
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html/_sources/C06_Abstract_Algebra.rst.txt > html/_sources/C06_Structures.rst.txt
- @@ -1,7 +1,7 @@
 - .. _abstract_algebra: + .. _structures: - Abstract Algebra - ================ + Structures + ========== Modern mathematics makes essential use of algebraic structures,
- @@ -24,6 +24,6 @@ algebraic structures on your own.
 For more technical detail, you can consult `Theorem Proving in Lean <https://leanprover.github.io/theorem_proving_in_lean/>`_, and a paper by Anne Baanen, `Use and abuse of instance parameters in the Lean mathematical library <https://arxiv.org/abs/2202.01629>`_. - .. include:: C06_Abstract_Algebra/S01_Structures.inc - .. include:: C06_Abstract_Algebra/S02_Algebraic_Structures.inc - .. include:: C06_Abstract_Algebra/S03_Building_the_Gaussian_Integers.inc + .. include:: C06_Structures/S01_Structures.inc + .. include:: C06_Structures/S02_Algebraic_Structures.inc + .. include:: C06_Structures/S03_Building_the_Gaussian_Integers.inc
html/_sources/C07_Hierarchies.rst.txt (new)
- @@ -0,0 +1,26 @@
 + .. _hierarchies: + + Hierarchies + =========== + + We have seen in :numref:`Chapter %s <structures>` how to define the class + of groups and build instances of this class, and then how to build an instance + of the commutative ring class. But of course there is a hierarchy here: a + commutative ring is in particular an additive group. In this chapter we + will study how to build such hierarchies. They appear in all branches + of mathematics but in this chapter the emphasis will be on algebraic examples. + + It may seem premature to discuss how to build hierarchies before more discussions + about using existing hierarchies. But some understanding of the technology underlying + hierarchies is required to use them. So you should probably still read this chapter, + but without trying too hard to remember everything on your first read, then read + the following chapters and come back here for a second reading. + + In this chapter, we will redefine (simpler versions of) many things that appear in Mathlib + so we will used indices to distinguish our version. For instance we will have ``Ring₁`` + as our version of ``Ring``. Since we will gradually explain more powerful ways of formalizing + structures, those indices will sometimes grow beyond one. + + .. include:: C07_Hierarchies/S01_Basics.inc + .. include:: C07_Hierarchies/S02_Morphisms.inc + .. include:: C07_Hierarchies/S03_Subobjects.inc
html/_sources/C07_Topology.rst.txt > html/_sources/C08_Topology.rst.txt
- @@ -19,7 +19,7 @@ We have already begun to consider such notions in :numref:`sequences_and_converg
 *Topology* is the abstract study of limits and continuity. Having covered the essentials of formalization in Chapters :numref:`%s <basics>` - to :numref:`%s <abstract_algebra>`, + to :numref:`%s <structures>`, in this chapter, we will explain how topological notions are formalized in mathlib. Not only do topological abstractions apply in much greater generality, but that also, somewhat paradoxically, make it easier to reason about limits
- @@ -77,8 +77,8 @@ Formalizing mathematics requires making the relevant notion of "sameness"
 fully explicit, and that is exactly what Bourbaki's theory of filters manages to do. - .. include:: C07_Topology/S01_Filters.inc + .. include:: C08_Topology/S01_Filters.inc - .. include:: C07_Topology/S02_Metric_Spaces.inc + .. include:: C08_Topology/S02_Metric_Spaces.inc - .. include:: C07_Topology/S03_Topological_Spaces.inc + .. include:: C08_Topology/S03_Topological_Spaces.inc
html/_sources/C08_Differential_Calculus.rst.txt > html/_sources/C09_Differential_Calculus.rst.txt
- @@ -14,6 +14,5 @@ which is familiar from any introductory calculus class.
  In :numref:`normed_spaces`, we then consider the notion of a derivative in a much broader setting. - .. - .. include:: C08_Differential_Calculus/S01_Elementary_Differential_Calculus.inc - .. include:: C08_Differential_Calculus/S02_Differential_Calculus_in_Normed_Spaces.inc + .. include:: C09_Differential_Calculus/S01_Elementary_Differential_Calculus.inc + .. include:: C09_Differential_Calculus/S02_Differential_Calculus_in_Normed_Spaces.inc
html/_sources/C09_Integration_and_Measure_Theory.rst.txt > html/_sources/C10_Integration_and_Measure_Theory.rst.txt
html/_sources/index.rst.txt
- @@ -12,8 +12,10 @@ Mathematics in Lean
  C03_Logic C04_Sets_and_Functions C05_Number_Theory - C06_Abstract_Algebra - C07_Topology + C06_Structures + C07_Hierarchies + C08_Topology + C09_Differential_Calculus .. toctree:: :hidden:
html/_static/_sphinx_javascript_frameworks_compat.js
- @@ -1,20 +1,9 @@
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html/_static/basic.css
- @@ -4,7 +4,7 @@
  * * Sphinx stylesheet -- basic theme. * - * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS. + * :copyright: Copyright 2007-2023 by the Sphinx team, see AUTHORS. * :license: BSD, see LICENSE for details. * */
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- @@ -335,13 +325,17 @@ p.sidebar-title {
  font-weight: bold; } - div.admonition, div.topic, aside.topic, blockquote { + nav.contents, + aside.topic, + div.admonition, div.topic, blockquote { clear: left; } /* -- topics ---------------------------------------------------------------- */ - div.topic, aside.topic { + nav.contents, + aside.topic, + div.topic { border: 1px solid #ccc; padding: 7px; margin: 10px 0 10px 0;
- @@ -379,16 +373,18 @@ div.body p.centered {
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html/_static/css/badge_only.css
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html/_static/doctools.js
- @@ -4,12 +4,19 @@
 * * Base JavaScript utilities for all Sphinx HTML documentation. * - * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS. + * :copyright: Copyright 2007-2023 by the Sphinx team, see AUTHORS. * :license: BSD, see LICENSE for details. * */ "use strict"; + const BLACKLISTED_KEY_CONTROL_ELEMENTS = new Set([ + "TEXTAREA", + "INPUT", + "SELECT", + "BUTTON", + ]); + const _ready = (callback) => { if (document.readyState !== "loading") { callback();
- @@ -18,73 +25,11 @@ const _ready = (callback) => {
 } }; - /** - * highlight a given string on a node by wrapping it in - * span elements with the given class name. - */ - const _highlight = (node, addItems, text, className) => { - if (node.nodeType === Node.TEXT_NODE) { - const val = node.nodeValue; - const parent = node.parentNode; - const pos = val.toLowerCase().indexOf(text); - if ( - pos >= 0 && - !parent.classList.contains(className) && - !parent.classList.contains("nohighlight") - ) { - let span; - - const closestNode = parent.closest("body, svg, foreignObject"); - const isInSVG = closestNode && closestNode.matches("svg"); - if (isInSVG) { - span = document.createElementNS("http://www.w3.org/2000/svg", "tspan"); - } else { - span = document.createElement("span"); - span.classList.add(className); - } - - span.appendChild(document.createTextNode(val.substr(pos, text.length))); - parent.insertBefore( - span, - parent.insertBefore( - document.createTextNode(val.substr(pos + text.length)), - node.nextSibling - ) - ); - node.nodeValue = val.substr(0, pos); - - if (isInSVG) { - const rect = document.createElementNS( - "http://www.w3.org/2000/svg", - "rect" - ); - const bbox = parent.getBBox(); - rect.x.baseVal.value = bbox.x; - rect.y.baseVal.value = bbox.y; - rect.width.baseVal.value = bbox.width; - rect.height.baseVal.value = bbox.height; - rect.setAttribute("class", className); - addItems.push({ parent: parent, target: rect }); - } - } - } else if (node.matches && !node.matches("button, select, textarea")) { - node.childNodes.forEach((el) => _highlight(el, addItems, text, className)); - } - }; - const _highlightText = (thisNode, text, className) => { - let addItems = []; - _highlight(thisNode, addItems, text, className); - addItems.forEach((obj) => - obj.parent.insertAdjacentElement("beforebegin", obj.target) - ); - }; - /** * Small JavaScript module for the documentation. */ const Documentation = { init: () => { - Documentation.highlightSearchWords(); Documentation.initDomainIndexTable(); Documentation.initOnKeyListeners(); },
- @@ -126,51 +71,6 @@ const Documentation = {
 Documentation.LOCALE = catalog.locale; }, - /** - * highlight the search words provided in the url in the text - */ - highlightSearchWords: () => { - const highlight = - new URLSearchParams(window.location.search).get("highlight") || ""; - const terms = highlight.toLowerCase().split(/\s+/).filter(x => x); - if (terms.length === 0) return; // nothing to do - - // There should never be more than one element matching "div.body" - const divBody = document.querySelectorAll("div.body"); - const body = divBody.length ? divBody[0] : document.querySelector("body"); - window.setTimeout(() => { - terms.forEach((term) => _highlightText(body, term, "highlighted")); - }, 10); - - const searchBox = document.getElementById("searchbox"); - if (searchBox === null) return; - searchBox.appendChild( - document - .createRange() - .createContextualFragment( - '' + - '<a href="javascript:Documentation.hideSearchWords()">' + - Documentation.gettext("Hide Search Matches") + - "</a>" - ) - ); - }, - - /** - * helper function to hide the search marks again - */ - hideSearchWords: () => { - document - .querySelectorAll("#searchbox .highlight-link") - .forEach((el) => el.remove()); - document - .querySelectorAll("span.highlighted") - .forEach((el) => el.classList.remove("highlighted")); - const url = new URL(window.location); - url.searchParams.delete("highlight"); - window.history.replaceState({}, "", url); - }, - /** * helper function to focus on search bar */
- @@ -210,15 +110,11 @@ const Documentation = {
 ) return; - const blacklistedElements = new Set([ - "TEXTAREA", - "INPUT", - "SELECT", - "BUTTON", - ]); document.addEventListener("keydown", (event) => { - if (blacklistedElements.has(document.activeElement.tagName)) return; // bail for input elements - if (event.altKey || event.ctrlKey || event.metaKey) return; // bail with special keys + // bail for input elements + if (BLACKLISTED_KEY_CONTROL_ELEMENTS.has(document.activeElement.tagName)) return; + // bail with special keys + if (event.altKey || event.ctrlKey || event.metaKey) return; if (!event.shiftKey) { switch (event.key) {
- @@ -240,10 +136,6 @@ const Documentation = {
 event.preventDefault(); } break; - case "Escape": - if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS) break; - Documentation.hideSearchWords(); - event.preventDefault(); } }
html/_static/documentation_options.js
- @@ -10,5 +10,5 @@ var DOCUMENTATION_OPTIONS = {
  SOURCELINK_SUFFIX: '.txt', NAVIGATION_WITH_KEYS: false, SHOW_SEARCH_SUMMARY: true, - ENABLE_SEARCH_SHORTCUTS: false, + ENABLE_SEARCH_SHORTCUTS: true, };
html/_static/jquery-3.6.0.js (deleted)
- @@ -1,10881 +0,0 @@
 - /*! - * jQuery JavaScript Library v3.6.0 - * https://jquery.com/ - * - * Includes Sizzle.js - * https://sizzlejs.com/ - * - * Copyright OpenJS Foundation and other contributors - * Released under the MIT license - * https://jquery.org/license - * - * Date: 2021-03-02T17:08Z - */ - ( function( global, factory ) { - - "use strict"; - - if ( typeof module === "object" && typeof module.exports === "object" ) { - - // For CommonJS and CommonJS-like environments where a proper `window` - // is present, execute the factory and get jQuery. - // For environments that do not have a `window` with a `document` - // (such as Node.js), expose a factory as module.exports. - // This accentuates the need for the creation of a real `window`. - // e.g. var jQuery = require("jquery")(window); - // See ticket #14549 for more info. - module.exports = global.document ? - factory( global, true ) : - function( w ) { - if ( !w.document ) { - throw new Error( "jQuery requires a window with a document" ); - } - return factory( w ); - }; - } else { - factory( global ); - } - - // Pass this if window is not defined yet - } )( typeof window !== "undefined" ? window : this, function( window, noGlobal ) { - - // Edge <= 12 - 13+, Firefox <=18 - 45+, IE 10 - 11, Safari 5.1 - 9+, iOS 6 - 9.1 - // throw exceptions when non-strict code (e.g., ASP.NET 4.5) accesses strict mode - // arguments.callee.caller (trac-13335). But as of jQuery 3.0 (2016), strict mode should be common - // enough that all such attempts are guarded in a try block. - "use strict"; - - var arr = []; - - var getProto = Object.getPrototypeOf; - - var slice = arr.slice; - - var flat = arr.flat ? function( array ) { - return arr.flat.call( array ); - } : function( array ) { - return arr.concat.apply( [], array ); - }; - - - var push = arr.push; - - var indexOf = arr.indexOf; - - var class2type = {}; - - var toString = class2type.toString; - - var hasOwn = class2type.hasOwnProperty; - - var fnToString = hasOwn.toString; - - var ObjectFunctionString = fnToString.call( Object ); - - var support = {}; - - var isFunction = function isFunction( obj ) { - - // Support: Chrome <=57, Firefox <=52 - // In some browsers, typeof returns "function" for HTML <object> elements - // (i.e., `typeof document.createElement( "object" ) === "function"`). - // We don't want to classify *any* DOM node as a function. - // Support: QtWeb <=3.8.5, WebKit <=534.34, wkhtmltopdf tool <=0.12.5 - // Plus for old WebKit, typeof returns "function" for HTML collections - // (e.g., `typeof document.getElementsByTagName("div") === "function"`). (gh-4756) - return typeof obj === "function" && typeof obj.nodeType !== "number" && - typeof obj.item !== "function"; - }; - - - var isWindow = function isWindow( obj ) { - return obj != null && obj === obj.window; - }; - - - var document = window.document; - - - - var preservedScriptAttributes = { - type: true, - src: true, - nonce: true, - noModule: true - }; - - function DOMEval( code, node, doc ) { - doc = doc || document; - - var i, val, - script = doc.createElement( "script" ); - - script.text = code; - if ( node ) { - for ( i in preservedScriptAttributes ) { - - // Support: Firefox 64+, Edge 18+ - // Some browsers don't support the "nonce" property on scripts. - // On the other hand, just using `getAttribute` is not enough as - // the `nonce` attribute is reset to an empty string whenever it - // becomes browsing-context connected. - // See https://github.com/whatwg/html/issues/2369 - // See https://html.spec.whatwg.org/#nonce-attributes - // The `node.getAttribute` check was added for the sake of - // `jQuery.globalEval` so that it can fake a nonce-containing node - // via an object. - val = node[ i ] || node.getAttribute && node.getAttribute( i ); - if ( val ) { - script.setAttribute( i, val ); - } - } - } - doc.head.appendChild( script ).parentNode.removeChild( script ); - } - - - function toType( obj ) { - if ( obj == null ) { - return obj + ""; - } - - // Support: Android <=2.3 only (functionish RegExp) - return typeof obj === "object" || typeof obj === "function" ? - class2type[ toString.call( obj ) ] || "object" : - typeof obj; - } - /* global Symbol */ - // Defining this global in .eslintrc.json would create a danger of using the global - // unguarded in another place, it seems safer to define global only for this module - - - - var - version = "3.6.0", - - // Define a local copy of jQuery - jQuery = function( selector, context ) { - - // The jQuery object is actually just the init constructor 'enhanced' - // Need init if jQuery is called (just allow error to be thrown if not included) - return new jQuery.fn.init( selector, context ); - }; - - jQuery.fn = jQuery.prototype = { - - // The current version of jQuery being used - jquery: version, - - constructor: jQuery, - - // The default length of a jQuery object is 0 - length: 0, - - toArray: function() { - return slice.call( this ); - }, - - // Get the Nth element in the matched element set OR - // Get the whole matched element set as a clean array - get: function( num ) { - - // Return all the elements in a clean array - if ( num == null ) { - return slice.call( this ); - } - - // Return just the one element from the set - return num < 0 ? this[ num + this.length ] : this[ num ]; - }, - - // Take an array of elements and push it onto the stack - // (returning the new matched element set) - pushStack: function( elems ) { - - // Build a new jQuery matched element set - var ret = jQuery.merge( this.constructor(), elems ); - - // Add the old object onto the stack (as a reference) - ret.prevObject = this; - - // Return the newly-formed element set - return ret; - }, - - // Execute a callback for every element in the matched set. - each: function( callback ) { - return jQuery.each( this, callback ); - }, - - map: function( callback ) { - return this.pushStack( jQuery.map( this, function( elem, i ) { - return callback.call( elem, i, elem ); - } ) ); - }, - - slice: function() { - return this.pushStack( slice.apply( this, arguments ) ); - }, - - first: function() { - return this.eq( 0 ); - }, - - last: function() { - return this.eq( -1 ); - }, - - even: function() { - return this.pushStack( jQuery.grep( this, function( _elem, i ) { - return ( i + 1 ) % 2; - } ) ); - }, - - odd: function() { - return this.pushStack( jQuery.grep( this, function( _elem, i ) { - return i % 2; - } ) ); - }, - - eq: function( i ) { - var len = this.length, - j = +i + ( i < 0 ? len : 0 ); - return this.pushStack( j >= 0 && j < len ? [ this[ j ] ] : [] ); - }, - - end: function() { - return this.prevObject || this.constructor(); - }, - - // For internal use only. - // Behaves like an Array's method, not like a jQuery method. - push: push, - sort: arr.sort, - splice: arr.splice - }; - - jQuery.extend = jQuery.fn.extend = function() { - var options, name, src, copy, copyIsArray, clone, - target = arguments[ 0 ] || {}, - i = 1, - length = arguments.length, - deep = false; - - // Handle a deep copy situation - if ( typeof target === "boolean" ) { - deep = target; - - // Skip the boolean and the target - target = arguments[ i ] || {}; - i++; - } - - // Handle case when target is a string or something (possible in deep copy) - if ( typeof target !== "object" && !isFunction( target ) ) { - target = {}; - } - - // Extend jQuery itself if only one argument is passed - if ( i === length ) { - target = this; - i--; - } - - for ( ; i < length; i++ ) { - - // Only deal with non-null/undefined values - if ( ( options = arguments[ i ] ) != null ) { - - // Extend the base object - for ( name in options ) { - copy = options[ name ]; - - // Prevent Object.prototype pollution - // Prevent never-ending loop - if ( name === "__proto__" || target === copy ) { - continue; - } - - // Recurse if we're merging plain objects or arrays - if ( deep && copy && ( jQuery.isPlainObject( copy ) || - ( copyIsArray = Array.isArray( copy ) ) ) ) { - src = target[ name ]; - - // Ensure proper type for the source value - if ( copyIsArray && !Array.isArray( src ) ) { - clone = []; - } else if ( !copyIsArray && !jQuery.isPlainObject( src ) ) { - clone = {}; - } else { - clone = src; - } - copyIsArray = false; - - // Never move original objects, clone them - target[ name ] = jQuery.extend( deep, clone, copy ); - - // Don't bring in undefined values - } else if ( copy !== undefined ) { - target[ name ] = copy; - } - } - } - } - - // Return the modified object - return target; - }; - - jQuery.extend( { - - // Unique for each copy of jQuery on the page - expando: "jQuery" + ( version + Math.random() ).replace( /\D/g, "" ), - - // Assume jQuery is ready without the ready module - isReady: true, - - error: function( msg ) { - throw new Error( msg ); - }, - - noop: function() {}, - - isPlainObject: function( obj ) { - var proto, Ctor; - - // Detect obvious negatives - // Use toString instead of jQuery.type to catch host objects - if ( !obj || toString.call( obj ) !== "[object Object]" ) { - return false; - } - - proto = getProto( obj ); - - // Objects with no prototype (e.g., `Object.create( null )`) are plain - if ( !proto ) { - return true; - } - - // Objects with prototype are plain iff they were constructed by a global Object function - Ctor = hasOwn.call( proto, "constructor" ) && proto.constructor; - return typeof Ctor === "function" && fnToString.call( Ctor ) === ObjectFunctionString; - }, - - isEmptyObject: function( obj ) { - var name; - - for ( name in obj ) { - return false; - } - return true; - }, - - // Evaluates a script in a provided context; falls back to the global one - // if not specified. - globalEval: function( code, options, doc ) { - DOMEval( code, { nonce: options && options.nonce }, doc ); - }, - - each: function( obj, callback ) { - var length, i = 0; - - if ( isArrayLike( obj ) ) { - length = obj.length; - for ( ; i < length; i++ ) { - if ( callback.call( obj[ i ], i, obj[ i ] ) === false ) { - break; - } - } - } else { - for ( i in obj ) { - if ( callback.call( obj[ i ], i, obj[ i ] ) === false ) { - break; - } - } - } - - return obj; - }, - - // results is for internal usage only - makeArray: function( arr, results ) { - var ret = results || []; - - if ( arr != null ) { - if ( isArrayLike( Object( arr ) ) ) { - jQuery.merge( ret, - typeof arr === "string" ? - [ arr ] : arr - ); - } else { - push.call( ret, arr ); - } - } - - return ret; - }, - - inArray: function( elem, arr, i ) { - return arr == null ? -1 : indexOf.call( arr, elem, i ); - }, - - // Support: Android <=4.0 only, PhantomJS 1 only - // push.apply(_, arraylike) throws on ancient WebKit - merge: function( first, second ) { - var len = +second.length, - j = 0, - i = first.length; - - for ( ; j < len; j++ ) { - first[ i++ ] = second[ j ]; - } - - first.length = i; - - return first; - }, - - grep: function( elems, callback, invert ) { - var callbackInverse, - matches = [], - i = 0, - length = elems.length, - callbackExpect = !invert; - - // Go through the array, only saving the items - // that pass the validator function - for ( ; i < length; i++ ) { - callbackInverse = !callback( elems[ i ], i ); - if ( callbackInverse !== callbackExpect ) { - matches.push( elems[ i ] ); - } - } - - return matches; - }, - - // arg is for internal usage only - map: function( elems, callback, arg ) { - var length, value, - i = 0, - ret = []; - - // Go through the array, translating each of the items to their new values - if ( isArrayLike( elems ) ) { - length = elems.length; - for ( ; i < length; i++ ) { - value = callback( elems[ i ], i, arg ); - - if ( value != null ) { - ret.push( value ); - } - } - - // Go through every key on the object, - } else { - for ( i in elems ) { - value = callback( elems[ i ], i, arg ); - - if ( value != null ) { - ret.push( value ); - } - } - } - - // Flatten any nested arrays - return flat( ret ); - }, - - // A global GUID counter for objects - guid: 1, - - // jQuery.support is not used in Core but other projects attach their - // properties to it so it needs to exist. - support: support - } ); - - if ( typeof Symbol === "function" ) { - jQuery.fn[ Symbol.iterator ] = arr[ Symbol.iterator ]; - } - - // Populate the class2type map - jQuery.each( "Boolean Number String Function Array Date RegExp Object Error Symbol".split( " " ), - function( _i, name ) { - class2type[ "[object " + name + "]" ] = name.toLowerCase(); - } ); - - function isArrayLike( obj ) { - - // Support: real iOS 8.2 only (not reproducible in simulator) - // `in` check used to prevent JIT error (gh-2145) - // hasOwn isn't used here due to false negatives - // regarding Nodelist length in IE - var length = !!obj && "length" in obj && obj.length, - type = toType( obj ); - - if ( isFunction( obj ) || isWindow( obj ) ) { - return false; - } - - return type === "array" || length === 0 || - typeof length === "number" && length > 0 && ( length - 1 ) in obj; - } - var Sizzle = - /*! - * Sizzle CSS Selector Engine v2.3.6 - * https://sizzlejs.com/ - * - * Copyright JS Foundation and other contributors - * Released under the MIT license - * https://js.foundation/ - * - * Date: 2021-02-16 - */ - ( function( window ) { - var i, - support, - Expr, - getText, - isXML, - tokenize, - compile, - select, - outermostContext, - sortInput, - hasDuplicate, - - // Local document vars - setDocument, - document, - docElem, - documentIsHTML, - rbuggyQSA, - rbuggyMatches, - matches, - contains, - - // Instance-specific data - expando = "sizzle" + 1 * new Date(), - preferredDoc = window.document, - dirruns = 0, - done = 0, - classCache = createCache(), - tokenCache = createCache(), - compilerCache = createCache(), - nonnativeSelectorCache = createCache(), - sortOrder = function( a, b ) { - if ( a === b ) { - hasDuplicate = true; - } - return 0; - }, - - // Instance methods - hasOwn = ( {} ).hasOwnProperty, - arr = [], - pop = arr.pop, - pushNative = arr.push, - push = arr.push, - slice = arr.slice, - - // Use a stripped-down indexOf as it's faster than native - // https://jsperf.com/thor-indexof-vs-for/5 - indexOf = function( list, elem ) { - var i = 0, - len = list.length; - for ( ; i < len; i++ ) { - if ( list[ i ] === elem ) { - return i; - } - } - return -1; - }, - - booleans = "checked|selected|async|autofocus|autoplay|controls|defer|disabled|hidden|" + - "ismap|loop|multiple|open|readonly|required|scoped", - - // Regular expressions - - // http://www.w3.org/TR/css3-selectors/#whitespace - whitespace = "[\\x20\\t\\r\\n\\f]", - - // https://www.w3.org/TR/css-syntax-3/#ident-token-diagram - identifier = "(?:\\\\[\\da-fA-F]{1,6}" + whitespace + - "?|\\\\[^\\r\\n\\f]|[\\w-]|[^\0-\\x7f])+", - - // Attribute selectors: http://www.w3.org/TR/selectors/#attribute-selectors - attributes = "\\[" + whitespace + "*(" + identifier + ")(?:" + whitespace + - - // Operator (capture 2) - "*([*^$|!~]?=)" + whitespace + - - // "Attribute values must be CSS identifiers [capture 5] - // or strings [capture 3 or capture 4]" - "*(?:'((?:\\\\.|[^\\\\'])*)'|\"((?:\\\\.|[^\\\\\"])*)\"|(" + identifier + "))|)" + - whitespace + "*\\]", - - pseudos = ":(" + identifier + ")(?:\$(" + - - // To reduce the number of selectors needing tokenize in the preFilter, prefer arguments: - // 1. quoted (capture 3; capture 4 or capture 5) - "('((?:\\\\.|[^\\\\'])*)'|\"((?:\\\\.|[^\\\\\"])*)\")|" + - - // 2. simple (capture 6) - "((?:\\\\.|[^\\\\()[\\]]|" + attributes + ")*)|" + - - // 3. anything else (capture 2) - ".*" + - ")\$|)", - - // Leading and non-escaped trailing whitespace, capturing some non-whitespace characters preceding the latter - rwhitespace = new RegExp( whitespace + "+", "g" ), - rtrim = new RegExp( "^" + whitespace + "+|((?:^|[^\\\\])(?:\\\\.)*)" + - whitespace + "+$", "g" ), - - rcomma = new RegExp( "^" + whitespace + "*," + whitespace + "*" ), - rcombinators = new RegExp( "^" + whitespace + "*([>+~]|" + whitespace + ")" + whitespace + - "*" ), - rdescend = new RegExp( whitespace + "|>" ), - - rpseudo = new RegExp( pseudos ), - ridentifier = new RegExp( "^" + identifier + "$" ), - - matchExpr = { - "ID": new RegExp( "^#(" + identifier + ")" ), - "CLASS": new RegExp( "^\\.(" + identifier + ")" ), - "TAG": new RegExp( "^(" + identifier + "|[*])" ), - "ATTR": new RegExp( "^" + attributes ), - "PSEUDO": new RegExp( "^" + pseudos ), - "CHILD": new RegExp( "^:(only|first|last|nth|nth-last)-(child|of-type)(?:\$" + - whitespace + "*(even|odd|(([+-]|)(\\d*)n|)" + whitespace + "*(?:([+-]|)" + - whitespace + "*(\\d+)|))" + whitespace + "*\$|)", "i" ), - "bool": new RegExp( "^(?:" + booleans + ")$", "i" ), - - // For use in libraries implementing .is() - // We use this for POS matching in `select` - "needsContext": new RegExp( "^" + whitespace + - "*[>+~]|:(even|odd|eq|gt|lt|nth|first|last)(?:\$" + whitespace + - "*((?:-\\d)?\\d*)" + whitespace + "*\$|)(?=[^-]|$)", "i" ) - }, - - rhtml = /HTML$/i, - rinputs = /^(?:input|select|textarea|button)$/i, - rheader = /^h\d$/i, - - rnative = /^[^{]+\{\s*\[native \w/, - - // Easily-parseable/retrievable ID or TAG or CLASS selectors - rquickExpr = /^(?:#([\w-]+)|(\w+)|\.([\w-]+))$/, - - rsibling = /[+~]/, - - // CSS escapes - // http://www.w3.org/TR/CSS21/syndata.html#escaped-characters - runescape = new RegExp( "\\\\[\\da-fA-F]{1,6}" + whitespace + "?|\\\\([^\\r\\n\\f])", "g" ), - funescape = function( escape, nonHex ) { - var high = "0x" + escape.slice( 1 ) - 0x10000; - - return nonHex ? - - // Strip the backslash prefix from a non-hex escape sequence - nonHex : - - // Replace a hexadecimal escape sequence with the encoded Unicode code point - // Support: IE <=11+ - // For values outside the Basic Multilingual Plane (BMP), manually construct a - // surrogate pair - high < 0 ? - String.fromCharCode( high + 0x10000 ) : - String.fromCharCode( high >> 10 | 0xD800, high & 0x3FF | 0xDC00 ); - }, - - // CSS string/identifier serialization - // https://drafts.csswg.org/cssom/#common-serializing-idioms - rcssescape = /([\0-\x1f\x7f]|^-?\d)|^-$|[^\0-\x1f\x7f-\uFFFF\w-]/g, - fcssescape = function( ch, asCodePoint ) { - if ( asCodePoint ) { - - // U+0000 NULL becomes U+FFFD REPLACEMENT CHARACTER - if ( ch === "\0" ) { - return "\uFFFD"; - } - - // Control characters and (dependent upon position) numbers get escaped as code points - return ch.slice( 0, -1 ) + "\\" + - ch.charCodeAt( ch.length - 1 ).toString( 16 ) + " "; - } - - // Other potentially-special ASCII characters get backslash-escaped - return "\\" + ch; - }, - - // Used for iframes - // See setDocument() - // Removing the function wrapper causes a "Permission Denied" - // error in IE - unloadHandler = function() { - setDocument(); - }, - - inDisabledFieldset = addCombinator( - function( elem ) { - return elem.disabled === true && elem.nodeName.toLowerCase() === "fieldset"; - }, - { dir: "parentNode", next: "legend" } - ); - - // Optimize for push.apply( _, NodeList ) - try { - push.apply( - ( arr = slice.call( preferredDoc.childNodes ) ), - preferredDoc.childNodes - ); - - // Support: Android<4.0 - // Detect silently failing push.apply - // eslint-disable-next-line no-unused-expressions - arr[ preferredDoc.childNodes.length ].nodeType; - } catch ( e ) { - push = { apply: arr.length ? - - // Leverage slice if possible - function( target, els ) { - pushNative.apply( target, slice.call( els ) ); - } : - - // Support: IE<9 - // Otherwise append directly - function( target, els ) { - var j = target.length, - i = 0; - - // Can't trust NodeList.length - while ( ( target[ j++ ] = els[ i++ ] ) ) {} - target.length = j - 1; - } - }; - } - - function Sizzle( selector, context, results, seed ) { - var m, i, elem, nid, match, groups, newSelector, - newContext = context && context.ownerDocument, - - // nodeType defaults to 9, since context defaults to document - nodeType = context ? context.nodeType : 9; - - results = results || []; - - // Return early from calls with invalid selector or context - if ( typeof selector !== "string" || !selector || - nodeType !== 1 && nodeType !== 9 && nodeType !== 11 ) { - - return results; - } - - // Try to shortcut find operations (as opposed to filters) in HTML documents - if ( !seed ) { - setDocument( context ); - context = context || document; - - if ( documentIsHTML ) { - - // If the selector is sufficiently simple, try using a "get*By*" DOM method - // (excepting DocumentFragment context, where the methods don't exist) - if ( nodeType !== 11 && ( match = rquickExpr.exec( selector ) ) ) { - - // ID selector - if ( ( m = match[ 1 ] ) ) { - - // Document context - if ( nodeType === 9 ) { - if ( ( elem = context.getElementById( m ) ) ) { - - // Support: IE, Opera, Webkit - // TODO: identify versions - // getElementById can match elements by name instead of ID - if ( elem.id === m ) { - results.push( elem ); - return results; - } - } else { - return results; - } - - // Element context - } else { - - // Support: IE, Opera, Webkit - // TODO: identify versions - // getElementById can match elements by name instead of ID - if ( newContext && ( elem = newContext.getElementById( m ) ) && - contains( context, elem ) && - elem.id === m ) { - - results.push( elem ); - return results; - } - } - - // Type selector - } else if ( match[ 2 ] ) { - push.apply( results, context.getElementsByTagName( selector ) ); - return results; - - // Class selector - } else if ( ( m = match[ 3 ] ) && support.getElementsByClassName && - context.getElementsByClassName ) { - - push.apply( results, context.getElementsByClassName( m ) ); - return results; - } - } - - // Take advantage of querySelectorAll - if ( support.qsa && - !nonnativeSelectorCache[ selector + " " ] && - ( !rbuggyQSA || !rbuggyQSA.test( selector ) ) && - - // Support: IE 8 only - // Exclude object elements - ( nodeType !== 1 || context.nodeName.toLowerCase() !== "object" ) ) { - - newSelector = selector; - newContext = context; - - // qSA considers elements outside a scoping root when evaluating child or - // descendant combinators, which is not what we want. - // In such cases, we work around the behavior by prefixing every selector in the - // list with an ID selector referencing the scope context. - // The technique has to be used as well when a leading combinator is used - // as such selectors are not recognized by querySelectorAll. - // Thanks to Andrew Dupont for this technique. - if ( nodeType === 1 && - ( rdescend.test( selector ) || rcombinators.test( selector ) ) ) { - - // Expand context for sibling selectors - newContext = rsibling.test( selector ) && testContext( context.parentNode ) || - context; - - // We can use :scope instead of the ID hack if the browser - // supports it & if we're not changing the context. - if ( newContext !== context || !support.scope ) { - - // Capture the context ID, setting it first if necessary - if ( ( nid = context.getAttribute( "id" ) ) ) { - nid = nid.replace( rcssescape, fcssescape ); - } else { - context.setAttribute( "id", ( nid = expando ) ); - } - } - - // Prefix every selector in the list - groups = tokenize( selector ); - i = groups.length; - while ( i-- ) { - groups[ i ] = ( nid ? "#" + nid : ":scope" ) + " " + - toSelector( groups[ i ] ); - } - newSelector = groups.join( "," ); - } - - try { - push.apply( results, - newContext.querySelectorAll( newSelector ) - ); - return results; - } catch ( qsaError ) { - nonnativeSelectorCache( selector, true ); - } finally { - if ( nid === expando ) { - context.removeAttribute( "id" ); - } - } - } - } - } - - // All others - return select( selector.replace( rtrim, "$1" ), context, results, seed ); - } - - /** - * Create key-value caches of limited size - * @returns {function(string, object)} Returns the Object data after storing it on itself with - * property name the (space-suffixed) string and (if the cache is larger than Expr.cacheLength) - * deleting the oldest entry - */ - function createCache() { - var keys = []; - - function cache( key, value ) { - - // Use (key + " ") to avoid collision with native prototype properties (see Issue #157) - if ( keys.push( key + " " ) > Expr.cacheLength ) { - - // Only keep the most recent entries - delete cache[ keys.shift() ]; - } - return ( cache[ key + " " ] = value ); - } - return cache; - } - - /** - * Mark a function for special use by Sizzle - * @param {Function} fn The function to mark - */ - function markFunction( fn ) { - fn[ expando ] = true; - return fn; - } - - /** - * Support testing using an element - * @param {Function} fn Passed the created element and returns a boolean result - */ - function assert( fn ) { - var el = document.createElement( "fieldset" ); - - try { - return !!fn( el ); - } catch ( e ) { - return false; - } finally { - - // Remove from its parent by default - if ( el.parentNode ) { - el.parentNode.removeChild( el ); - } - - // release memory in IE - el = null; - } - } - - /** - * Adds the same handler for all of the specified attrs - * @param {String} attrs Pipe-separated list of attributes - * @param {Function} handler The method that will be applied - */ - function addHandle( attrs, handler ) { - var arr = attrs.split( "|" ), - i = arr.length; - - while ( i-- ) { - Expr.attrHandle[ arr[ i ] ] = handler; - } - } - - /** - * Checks document order of two siblings - * @param {Element} a - * @param {Element} b - * @returns {Number} Returns less than 0 if a precedes b, greater than 0 if a follows b - */ - function siblingCheck( a, b ) { - var cur = b && a, - diff = cur && a.nodeType === 1 && b.nodeType === 1 && - a.sourceIndex - b.sourceIndex; - - // Use IE sourceIndex if available on both nodes - if ( diff ) { - return diff; - } - - // Check if b follows a - if ( cur ) { - while ( ( cur = cur.nextSibling ) ) { - if ( cur === b ) { - return -1; - } - } - } - - return a ? 1 : -1; - } - - /** - * Returns a function to use in pseudos for input types - * @param {String} type - */ - function createInputPseudo( type ) { - return function( elem ) { - var name = elem.nodeName.toLowerCase(); - return name === "input" && elem.type === type; - }; - } - - /** - * Returns a function to use in pseudos for buttons - * @param {String} type - */ - function createButtonPseudo( type ) { - return function( elem ) { - var name = elem.nodeName.toLowerCase(); - return ( name === "input" || name === "button" ) && elem.type === type; - }; - } - - /** - * Returns a function to use in pseudos for :enabled/:disabled - * @param {Boolean} disabled true for :disabled; false for :enabled - */ - function createDisabledPseudo( disabled ) { - - // Known :disabled false positives: fieldset[disabled] > legend:nth-of-type(n+2) :can-disable - return function( elem ) { - - // Only certain elements can match :enabled or :disabled - // https://html.spec.whatwg.org/multipage/scripting.html#selector-enabled - // https://html.spec.whatwg.org/multipage/scripting.html#selector-disabled - if ( "form" in elem ) { - - // Check for inherited disabledness on relevant non-disabled elements: - // * listed form-associated elements in a disabled fieldset - // https://html.spec.whatwg.org/multipage/forms.html#category-listed - // https://html.spec.whatwg.org/multipage/forms.html#concept-fe-disabled - // * option elements in a disabled optgroup - // https://html.spec.whatwg.org/multipage/forms.html#concept-option-disabled - // All such elements have a "form" property. - if ( elem.parentNode && elem.disabled === false ) { - - // Option elements defer to a parent optgroup if present - if ( "label" in elem ) { - if ( "label" in elem.parentNode ) { - return elem.parentNode.disabled === disabled; - } else { - return elem.disabled === disabled; - } - } - - // Support: IE 6 - 11 - // Use the isDisabled shortcut property to check for disabled fieldset ancestors - return elem.isDisabled === disabled || - - // Where there is no isDisabled, check manually - /* jshint -W018 */ - elem.isDisabled !== !disabled && - inDisabledFieldset( elem ) === disabled; - } - - return elem.disabled === disabled; - - // Try to winnow out elements that can't be disabled before trusting the disabled property. - // Some victims get caught in our net (label, legend, menu, track), but it shouldn't - // even exist on them, let alone have a boolean value. - } else if ( "label" in elem ) { - return elem.disabled === disabled; - } - - // Remaining elements are neither :enabled nor :disabled - return false; - }; - } - - /** - * Returns a function to use in pseudos for positionals - * @param {Function} fn - */ - function createPositionalPseudo( fn ) { - return markFunction( function( argument ) { - argument = +argument; - return markFunction( function( seed, matches ) { - var j, - matchIndexes = fn( [], seed.length, argument ), - i = matchIndexes.length; - - // Match elements found at the specified indexes - while ( i-- ) { - if ( seed[ ( j = matchIndexes[ i ] ) ] ) { - seed[ j ] = !( matches[ j ] = seed[ j ] ); - } - } - } ); - } ); - } - - /** - * Checks a node for validity as a Sizzle context - * @param {Element|Object=} context - * @returns {Element|Object|Boolean} The input node if acceptable, otherwise a falsy value - */ - function testContext( context ) { - return context && typeof context.getElementsByTagName !== "undefined" && context; - } - - // Expose support vars for convenience - support = Sizzle.support = {}; - - /** - * Detects XML nodes - * @param {Element|Object} elem An element or a document - * @returns {Boolean} True iff elem is a non-HTML XML node - */ - isXML = Sizzle.isXML = function( elem ) { - var namespace = elem && elem.namespaceURI, - docElem = elem && ( elem.ownerDocument || elem ).documentElement; - - // Support: IE <=8 - // Assume HTML when documentElement doesn't yet exist, such as inside loading iframes - // https://bugs.jquery.com/ticket/4833 - return !rhtml.test( namespace || docElem && docElem.nodeName || "HTML" ); - }; - - /** - * Sets document-related variables once based on the current document - * @param {Element|Object} [doc] An element or document object to use to set the document - * @returns {Object} Returns the current document - */ - setDocument = Sizzle.setDocument = function( node ) { - var hasCompare, subWindow, - doc = node ? node.ownerDocument || node : preferredDoc; - - // Return early if doc is invalid or already selected - // Support: IE 11+, Edge 17 - 18+ - // IE/Edge sometimes throw a "Permission denied" error when strict-comparing - // two documents; shallow comparisons work. - // eslint-disable-next-line eqeqeq - if ( doc == document || doc.nodeType !== 9 || !doc.documentElement ) { - return document; - } - - // Update global variables - document = doc; - docElem = document.documentElement; - documentIsHTML = !isXML( document ); - - // Support: IE 9 - 11+, Edge 12 - 18+ - // Accessing iframe documents after unload throws "permission denied" errors (jQuery #13936) - // Support: IE 11+, Edge 17 - 18+ - // IE/Edge sometimes throw a "Permission denied" error when strict-comparing - // two documents; shallow comparisons work. - // eslint-disable-next-line eqeqeq - if ( preferredDoc != document && - ( subWindow = document.defaultView ) && subWindow.top !== subWindow ) { - - // Support: IE 11, Edge - if ( subWindow.addEventListener ) { - subWindow.addEventListener( "unload", unloadHandler, false ); - - // Support: IE 9 - 10 only - } else if ( subWindow.attachEvent ) { - subWindow.attachEvent( "onunload", unloadHandler ); - } - } - - // Support: IE 8 - 11+, Edge 12 - 18+, Chrome <=16 - 25 only, Firefox <=3.6 - 31 only, - // Safari 4 - 5 only, Opera <=11.6 - 12.x only - // IE/Edge & older browsers don't support the :scope pseudo-class. - // Support: Safari 6.0 only - // Safari 6.0 supports :scope but it's an alias of :root there. - support.scope = assert( function( el ) { - docElem.appendChild( el ).appendChild( document.createElement( "div" ) ); - return typeof el.querySelectorAll !== "undefined" && - !el.querySelectorAll( ":scope fieldset div" ).length; - } ); - - /* Attributes - ---------------------------------------------------------------------- */ - - // Support: IE<8 - // Verify that getAttribute really returns attributes and not properties - // (excepting IE8 booleans) - support.attributes = assert( function( el ) { - el.className = "i"; - return !el.getAttribute( "className" ); - } ); - - /* getElement(s)By* - ---------------------------------------------------------------------- */ - - // Check if getElementsByTagName("*") returns only elements - support.getElementsByTagName = assert( function( el ) { - el.appendChild( document.createComment( "" ) ); - return !el.getElementsByTagName( "*" ).length; - } ); - - // Support: IE<9 - support.getElementsByClassName = rnative.test( document.getElementsByClassName ); - - // Support: IE<10 - // Check if getElementById returns elements by name - // The broken getElementById methods don't pick up programmatically-set names, - // so use a roundabout getElementsByName test - support.getById = assert( function( el ) { - docElem.appendChild( el ).id = expando; - return !document.getElementsByName || !document.getElementsByName( expando ).length; - } ); - - // ID filter and find - if ( support.getById ) { - Expr.filter[ "ID" ] = function( id ) { - var attrId = id.replace( runescape, funescape ); - return function( elem ) { - return elem.getAttribute( "id" ) === attrId; - }; - }; - Expr.find[ "ID" ] = function( id, context ) { - if ( typeof context.getElementById !== "undefined" && documentIsHTML ) { - var elem = context.getElementById( id ); - return elem ? [ elem ] : []; - } - }; - } else { - Expr.filter[ "ID" ] = function( id ) { - var attrId = id.replace( runescape, funescape ); - return function( elem ) { - var node = typeof elem.getAttributeNode !== "undefined" && - elem.getAttributeNode( "id" ); - return node && node.value === attrId; - }; - }; - - // Support: IE 6 - 7 only - // getElementById is not reliable as a find shortcut - Expr.find[ "ID" ] = function( id, context ) { - if ( typeof context.getElementById !== "undefined" && documentIsHTML ) { - var node, i, elems, - elem = context.getElementById( id ); - - if ( elem ) { - - // Verify the id attribute - node = elem.getAttributeNode( "id" ); - if ( node && node.value === id ) { - return [ elem ]; - } - - // Fall back on getElementsByName - elems = context.getElementsByName( id ); - i = 0; - while ( ( elem = elems[ i++ ] ) ) { - node = elem.getAttributeNode( "id" ); - if ( node && node.value === id ) { - return [ elem ]; - } - } - } - - return []; - } - }; - } - - // Tag - Expr.find[ "TAG" ] = support.getElementsByTagName ? - function( tag, context ) { - if ( typeof context.getElementsByTagName !== "undefined" ) { - return context.getElementsByTagName( tag ); - - // DocumentFragment nodes don't have gEBTN - } else if ( support.qsa ) { - return context.querySelectorAll( tag ); - } - } : - - function( tag, context ) { - var elem, - tmp = [], - i = 0, - - // By happy coincidence, a (broken) gEBTN appears on DocumentFragment nodes too - results = context.getElementsByTagName( tag ); - - // Filter out possible comments - if ( tag === "*" ) { - while ( ( elem = results[ i++ ] ) ) { - if ( elem.nodeType === 1 ) { - tmp.push( elem ); - } - } - - return tmp; - } - return results; - }; - - // Class - Expr.find[ "CLASS" ] = support.getElementsByClassName && function( className, context ) { - if ( typeof context.getElementsByClassName !== "undefined" && documentIsHTML ) { - return context.getElementsByClassName( className ); - } - }; - - /* QSA/matchesSelector - ---------------------------------------------------------------------- */ - - // QSA and matchesSelector support - - // matchesSelector(:active) reports false when true (IE9/Opera 11.5) - rbuggyMatches = []; - - // qSa(:focus) reports false when true (Chrome 21) - // We allow this because of a bug in IE8/9 that throws an error - // whenever `document.activeElement` is accessed on an iframe - // So, we allow :focus to pass through QSA all the time to avoid the IE error - // See https://bugs.jquery.com/ticket/13378 - rbuggyQSA = []; - - if ( ( support.qsa = rnative.test( document.querySelectorAll ) ) ) { - - // Build QSA regex - // Regex strategy adopted from Diego Perini - assert( function( el ) { - - var input; - - // Select is set to empty string on purpose - // This is to test IE's treatment of not explicitly - // setting a boolean content attribute, - // since its presence should be enough - // https://bugs.jquery.com/ticket/12359 - docElem.appendChild( el ).innerHTML = "<a id='" + expando + "'></a>" + - "<select id='" + expando + "-\r\\' msallowcapture=''>" + - "<option selected=''></option></select>"; - - // Support: IE8, Opera 11-12.16 - // Nothing should be selected when empty strings follow ^= or $= or *= - // The test attribute must be unknown in Opera but "safe" for WinRT - // https://msdn.microsoft.com/en-us/library/ie/hh465388.aspx#attribute_section - if ( el.querySelectorAll( "[msallowcapture^='']" ).length ) { - rbuggyQSA.push( "[*^$]=" + whitespace + "*(?:''|\"\")" ); - } - - // Support: IE8 - // Boolean attributes and "value" are not treated correctly - if ( !el.querySelectorAll( "[selected]" ).length ) { - rbuggyQSA.push( "\\[" + whitespace + "*(?:value|" + booleans + ")" ); - } - - // Support: Chrome<29, Android<4.4, Safari<7.0+, iOS<7.0+, PhantomJS<1.9.8+ - if ( !el.querySelectorAll( "[id~=" + expando + "-]" ).length ) { - rbuggyQSA.push( "~=" ); - } - - // Support: IE 11+, Edge 15 - 18+ - // IE 11/Edge don't find elements on a `[name='']` query in some cases. - // Adding a temporary attribute to the document before the selection works - // around the issue. - // Interestingly, IE 10 & older don't seem to have the issue. - input = document.createElement( "input" ); - input.setAttribute( "name", "" ); - el.appendChild( input ); - if ( !el.querySelectorAll( "[name='']" ).length ) { - rbuggyQSA.push( "\\[" + whitespace + "*name" + whitespace + "*=" + - whitespace + "*(?:''|\"\")" ); - } - - // Webkit/Opera - :checked should return selected option elements - // http://www.w3.org/TR/2011/REC-css3-selectors-20110929/#checked - // IE8 throws error here and will not see later tests - if ( !el.querySelectorAll( ":checked" ).length ) { - rbuggyQSA.push( ":checked" ); - } - - // Support: Safari 8+, iOS 8+ - // https://bugs.webkit.org/show_bug.cgi?id=136851 - // In-page `selector#id sibling-combinator selector` fails - if ( !el.querySelectorAll( "a#" + expando + "+*" ).length ) { - rbuggyQSA.push( ".#.+[+~]" ); - } - - // Support: Firefox <=3.6 - 5 only - // Old Firefox doesn't throw on a badly-escaped identifier. - el.querySelectorAll( "\\\f" ); - rbuggyQSA.push( "[\\r\\n\\f]" ); - } ); - - assert( function( el ) { - el.innerHTML = "<a href='' disabled='disabled'></a>" + - "<select disabled='disabled'><option/></select>"; - - // Support: Windows 8 Native Apps - // The type and name attributes are restricted during .innerHTML assignment - var input = document.createElement( "input" ); - input.setAttribute( "type", "hidden" ); - el.appendChild( input ).setAttribute( "name", "D" ); - - // Support: IE8 - // Enforce case-sensitivity of name attribute - if ( el.querySelectorAll( "[name=d]" ).length ) { - rbuggyQSA.push( "name" + whitespace + "*[*^$|!~]?=" ); - } - - // FF 3.5 - :enabled/:disabled and hidden elements (hidden elements are still enabled) - // IE8 throws error here and will not see later tests - if ( el.querySelectorAll( ":enabled" ).length !== 2 ) { - rbuggyQSA.push( ":enabled", ":disabled" ); - } - - // Support: IE9-11+ - // IE's :disabled selector does not pick up the children of disabled fieldsets - docElem.appendChild( el ).disabled = true; - if ( el.querySelectorAll( ":disabled" ).length !== 2 ) { - rbuggyQSA.push( ":enabled", ":disabled" ); - } - - // Support: Opera 10 - 11 only - // Opera 10-11 does not throw on post-comma invalid pseudos - el.querySelectorAll( "*,:x" ); - rbuggyQSA.push( ",.*:" ); - } ); - } - - if ( ( support.matchesSelector = rnative.test( ( matches = docElem.matches || - docElem.webkitMatchesSelector || - docElem.mozMatchesSelector || - docElem.oMatchesSelector || - docElem.msMatchesSelector ) ) ) ) { - - assert( function( el ) { - - // Check to see if it's possible to do matchesSelector - // on a disconnected node (IE 9) - support.disconnectedMatch = matches.call( el, "*" ); - - // This should fail with an exception - // Gecko does not error, returns false instead - matches.call( el, "[s!='']:x" ); - rbuggyMatches.push( "!=", pseudos ); - } ); - } - - rbuggyQSA = rbuggyQSA.length && new RegExp( rbuggyQSA.join( "|" ) ); - rbuggyMatches = rbuggyMatches.length && new RegExp( rbuggyMatches.join( "|" ) ); - - /* Contains - ---------------------------------------------------------------------- */ - hasCompare = rnative.test( docElem.compareDocumentPosition ); - - // Element contains another - // Purposefully self-exclusive - // As in, an element does not contain itself - contains = hasCompare || rnative.test( docElem.contains ) ? - function( a, b ) { - var adown = a.nodeType === 9 ? a.documentElement : a, - bup = b && b.parentNode; - return a === bup || !!( bup && bup.nodeType === 1 && ( - adown.contains ? - adown.contains( bup ) : - a.compareDocumentPosition && a.compareDocumentPosition( bup ) & 16 - ) ); - } : - function( a, b ) { - if ( b ) { - while ( ( b = b.parentNode ) ) { - if ( b === a ) { - return true; - } - } - } - return false; - }; - - /* Sorting - ---------------------------------------------------------------------- */ - - // Document order sorting - sortOrder = hasCompare ? - function( a, b ) { - - // Flag for duplicate removal - if ( a === b ) { - hasDuplicate = true; - return 0; - } - - // Sort on method existence if only one input has compareDocumentPosition - var compare = !a.compareDocumentPosition - !b.compareDocumentPosition; - if ( compare ) { - return compare; - } - - // Calculate position if both inputs belong to the same document - // Support: IE 11+, Edge 17 - 18+ - // IE/Edge sometimes throw a "Permission denied" error when strict-comparing - // two documents; shallow comparisons work. - // eslint-disable-next-line eqeqeq - compare = ( a.ownerDocument || a ) == ( b.ownerDocument || b ) ? - a.compareDocumentPosition( b ) : - - // Otherwise we know they are disconnected - 1; - - // Disconnected nodes - if ( compare & 1 || - ( !support.sortDetached && b.compareDocumentPosition( a ) === compare ) ) { - - // Choose the first element that is related to our preferred document - // Support: IE 11+, Edge 17 - 18+ - // IE/Edge sometimes throw a "Permission denied" error when strict-comparing - // two documents; shallow comparisons work. - // eslint-disable-next-line eqeqeq - if ( a == document || a.ownerDocument == preferredDoc && - contains( preferredDoc, a ) ) { - return -1; - } - - // Support: IE 11+, Edge 17 - 18+ - // IE/Edge sometimes throw a "Permission denied" error when strict-comparing - // two documents; shallow comparisons work. - // eslint-disable-next-line eqeqeq - if ( b == document || b.ownerDocument == preferredDoc && - contains( preferredDoc, b ) ) { - return 1; - } - - // Maintain original order - return sortInput ? - ( indexOf( sortInput, a ) - indexOf( sortInput, b ) ) : - 0; - } - - return compare & 4 ? -1 : 1; - } : - function( a, b ) { - - // Exit early if the nodes are identical - if ( a === b ) { - hasDuplicate = true; - return 0; - } - - var cur, - i = 0, - aup = a.parentNode, - bup = b.parentNode, - ap = [ a ], - bp = [ b ]; - - // Parentless nodes are either documents or disconnected - if ( !aup || !bup ) { - - // Support: IE 11+, Edge 17 - 18+ - // IE/Edge sometimes throw a "Permission denied" error when strict-comparing - // two documents; shallow comparisons work. - /* eslint-disable eqeqeq */ - return a == document ? -1 : - b == document ? 1 : - /* eslint-enable eqeqeq */ - aup ? -1 : - bup ? 1 : - sortInput ? - ( indexOf( sortInput, a ) - indexOf( sortInput, b ) ) : - 0; - - // If the nodes are siblings, we can do a quick check - } else if ( aup === bup ) { - return siblingCheck( a, b ); - } - - // Otherwise we need full lists of their ancestors for comparison - cur = a; - while ( ( cur = cur.parentNode ) ) { - ap.unshift( cur ); - } - cur = b; - while ( ( cur = cur.parentNode ) ) { - bp.unshift( cur ); - } - - // Walk down the tree looking for a discrepancy - while ( ap[ i ] === bp[ i ] ) { - i++; - } - - return i ? - - // Do a sibling check if the nodes have a common ancestor - siblingCheck( ap[ i ], bp[ i ] ) : - - // Otherwise nodes in our document sort first - // Support: IE 11+, Edge 17 - 18+ - // IE/Edge sometimes throw a "Permission denied" error when strict-comparing - // two documents; shallow comparisons work. - /* eslint-disable eqeqeq */ - ap[ i ] == preferredDoc ? -1 : - bp[ i ] == preferredDoc ? 1 : - /* eslint-enable eqeqeq */ - 0; - }; - - return document; - }; - - Sizzle.matches = function( expr, elements ) { - return Sizzle( expr, null, null, elements ); - }; - - Sizzle.matchesSelector = function( elem, expr ) { - setDocument( elem ); - - if ( support.matchesSelector && documentIsHTML && - !nonnativeSelectorCache[ expr + " " ] && - ( !rbuggyMatches || !rbuggyMatches.test( expr ) ) && - ( !rbuggyQSA || !rbuggyQSA.test( expr ) ) ) { - - try { - var ret = matches.call( elem, expr ); - - // IE 9's matchesSelector returns false on disconnected nodes - if ( ret || support.disconnectedMatch || - - // As well, disconnected nodes are said to be in a document - // fragment in IE 9 - elem.document && elem.document.nodeType !== 11 ) { - return ret; - } - } catch ( e ) { - nonnativeSelectorCache( expr, true ); - } - } - - return Sizzle( expr, document, null, [ elem ] ).length > 0; - }; - - Sizzle.contains = function( context, elem ) { - - // Set document vars if needed - // Support: IE 11+, Edge 17 - 18+ - // IE/Edge sometimes throw a "Permission denied" error when strict-comparing - // two documents; shallow comparisons work. - // eslint-disable-next-line eqeqeq - if ( ( context.ownerDocument || context ) != document ) { - setDocument( context ); - } - return contains( context, elem ); - }; - - Sizzle.attr = function( elem, name ) { - - // Set document vars if needed - // Support: IE 11+, Edge 17 - 18+ - // IE/Edge sometimes throw a "Permission denied" error when strict-comparing - // two documents; shallow comparisons work. - // eslint-disable-next-line eqeqeq - if ( ( elem.ownerDocument || elem ) != document ) { - setDocument( elem ); - } - - var fn = Expr.attrHandle[ name.toLowerCase() ], - - // Don't get fooled by Object.prototype properties (jQuery #13807) - val = fn && hasOwn.call( Expr.attrHandle, name.toLowerCase() ) ? - fn( elem, name, !documentIsHTML ) : - undefined; - - return val !== undefined ? - val : - support.attributes || !documentIsHTML ? - elem.getAttribute( name ) : - ( val = elem.getAttributeNode( name ) ) && val.specified ? - val.value : - null; - }; - - Sizzle.escape = function( sel ) { - return ( sel + "" ).replace( rcssescape, fcssescape ); - }; - - Sizzle.error = function( msg ) { - throw new Error( "Syntax error, unrecognized expression: " + msg ); - }; - - /** - * Document sorting and removing duplicates - * @param {ArrayLike} results - */ - Sizzle.uniqueSort = function( results ) { - var elem, - duplicates = [], - j = 0, - i = 0; - - // Unless we *know* we can detect duplicates, assume their presence - hasDuplicate = !support.detectDuplicates; - sortInput = !support.sortStable && results.slice( 0 ); - results.sort( sortOrder ); - - if ( hasDuplicate ) { - while ( ( elem = results[ i++ ] ) ) { - if ( elem === results[ i ] ) { - j = duplicates.push( i ); - } - } - while ( j-- ) { - results.splice( duplicates[ j ], 1 ); - } - } - - // Clear input after sorting to release objects - // See https://github.com/jquery/sizzle/pull/225 - sortInput = null; - - return results; - }; - - /** - * Utility function for retrieving the text value of an array of DOM nodes - * @param {Array|Element} elem - */ - getText = Sizzle.getText = function( elem ) { - var node, - ret = "", - i = 0, - nodeType = elem.nodeType; - - if ( !nodeType ) { - - // If no nodeType, this is expected to be an array - while ( ( node = elem[ i++ ] ) ) { - - // Do not traverse comment nodes - ret += getText( node ); - } - } else if ( nodeType === 1 || nodeType === 9 || nodeType === 11 ) { - - // Use textContent for elements - // innerText usage removed for consistency of new lines (jQuery #11153) - if ( typeof elem.textContent === "string" ) { - return elem.textContent; - } else { - - // Traverse its children - for ( elem = elem.firstChild; elem; elem = elem.nextSibling ) { - ret += getText( elem ); - } - } - } else if ( nodeType === 3 || nodeType === 4 ) { - return elem.nodeValue; - } - - // Do not include comment or processing instruction nodes - - return ret; - }; - - Expr = Sizzle.selectors = { - - // Can be adjusted by the user - cacheLength: 50, - - createPseudo: markFunction, - - match: matchExpr, - - attrHandle: {}, - - find: {}, - - relative: { - ">": { dir: "parentNode", first: true }, - " ": { dir: "parentNode" }, - "+": { dir: "previousSibling", first: true }, - "~": { dir: "previousSibling" } - }, - - preFilter: { - "ATTR": function( match ) { - match[ 1 ] = match[ 1 ].replace( runescape, funescape ); - - // Move the given value to match[3] whether quoted or unquoted - match[ 3 ] = ( match[ 3 ] || match[ 4 ] || - match[ 5 ] || "" ).replace( runescape, funescape ); - - if ( match[ 2 ] === "~=" ) { - match[ 3 ] = " " + match[ 3 ] + " "; - } - - return match.slice( 0, 4 ); - }, - - "CHILD": function( match ) { - - /* matches from matchExpr["CHILD"] - 1 type (only|nth|...) - 2 what (child|of-type) - 3 argument (even|odd|\d*|\d*n([+-]\d+)?|...) - 4 xn-component of xn+y argument ([+-]?\d*n|) - 5 sign of xn-component - 6 x of xn-component - 7 sign of y-component - 8 y of y-component - */ - match[ 1 ] = match[ 1 ].toLowerCase(); - - if ( match[ 1 ].slice( 0, 3 ) === "nth" ) { - - // nth-* requires argument - if ( !match[ 3 ] ) { - Sizzle.error( match[ 0 ] ); - } - - // numeric x and y parameters for Expr.filter.CHILD - // remember that false/true cast respectively to 0/1 - match[ 4 ] = +( match[ 4 ] ? - match[ 5 ] + ( match[ 6 ] || 1 ) : - 2 * ( match[ 3 ] === "even" || match[ 3 ] === "odd" ) ); - match[ 5 ] = +( ( match[ 7 ] + match[ 8 ] ) || match[ 3 ] === "odd" ); - - // other types prohibit arguments - } else if ( match[ 3 ] ) { - Sizzle.error( match[ 0 ] ); - } - - return match; - }, - - "PSEUDO": function( match ) { - var excess, - unquoted = !match[ 6 ] && match[ 2 ]; - - if ( matchExpr[ "CHILD" ].test( match[ 0 ] ) ) { - return null; - } - - // Accept quoted arguments as-is - if ( match[ 3 ] ) { - match[ 2 ] = match[ 4 ] || match[ 5 ] || ""; - - // Strip excess characters from unquoted arguments - } else if ( unquoted && rpseudo.test( unquoted ) && - - // Get excess from tokenize (recursively) - ( excess = tokenize( unquoted, true ) ) && - - // advance to the next closing parenthesis - ( excess = unquoted.indexOf( ")", unquoted.length - excess ) - unquoted.length ) ) { - - // excess is a negative index - match[ 0 ] = match[ 0 ].slice( 0, excess ); - match[ 2 ] = unquoted.slice( 0, excess ); - } - - // Return only captures needed by the pseudo filter method (type and argument) - return match.slice( 0, 3 ); - } - }, - - filter: { - - "TAG": function( nodeNameSelector ) { - var nodeName = nodeNameSelector.replace( runescape, funescape ).toLowerCase(); - return nodeNameSelector === "*" ? - function() { - return true; - } : - function( elem ) { - return elem.nodeName && elem.nodeName.toLowerCase() === nodeName; - }; - }, - - "CLASS": function( className ) { - var pattern = classCache[ className + " " ]; - - return pattern || - ( pattern = new RegExp( "(^|" + whitespace + - ")" + className + "(" + whitespace + "|$)" ) ) && classCache( - className, function( elem ) { - return pattern.test( - typeof elem.className === "string" && elem.className || - typeof elem.getAttribute !== "undefined" && - elem.getAttribute( "class" ) || - "" - ); - } ); - }, - - "ATTR": function( name, operator, check ) { - return function( elem ) { - var result = Sizzle.attr( elem, name ); - - if ( result == null ) { - return operator === "!="; - } - if ( !operator ) { - return true; - } - - result += ""; - - /* eslint-disable max-len */ - - return operator === "=" ? result === check : - operator === "!=" ? result !== check : - operator === "^=" ? check && result.indexOf( check ) === 0 : - operator === "*=" ? check && result.indexOf( check ) > -1 : - operator === "$=" ? check && result.slice( -check.length ) === check : - operator === "~=" ? ( " " + result.replace( rwhitespace, " " ) + " " ).indexOf( check ) > -1 : - operator === "|=" ? result === check || result.slice( 0, check.length + 1 ) === check + "-" : - false; - /* eslint-enable max-len */ - - }; - }, - - "CHILD": function( type, what, _argument, first, last ) { - var simple = type.slice( 0, 3 ) !== "nth", - forward = type.slice( -4 ) !== "last", - ofType = what === "of-type"; - - return first === 1 && last === 0 ? - - // Shortcut for :nth-*(n) - function( elem ) { - return !!elem.parentNode; - } : - - function( elem, _context, xml ) { - var cache, uniqueCache, outerCache, node, nodeIndex, start, - dir = simple !== forward ? "nextSibling" : "previousSibling", - parent = elem.parentNode, - name = ofType && elem.nodeName.toLowerCase(), - useCache = !xml && !ofType, - diff = false; - - if ( parent ) { - - // :(first|last|only)-(child|of-type) - if ( simple ) { - while ( dir ) { - node = elem; - while ( ( node = node[ dir ] ) ) { - if ( ofType ? - node.nodeName.toLowerCase() === name : - node.nodeType === 1 ) { - - return false; - } - } - - // Reverse direction for :only-* (if we haven't yet done so) - start = dir = type === "only" && !start && "nextSibling"; - } - return true; - } - - start = [ forward ? parent.firstChild : parent.lastChild ]; - - // non-xml :nth-child(...) stores cache data on `parent` - if ( forward && useCache ) { - - // Seek `elem` from a previously-cached index - - // ...in a gzip-friendly way - node = parent; - outerCache = node[ expando ] || ( node[ expando ] = {} ); - - // Support: IE <9 only - // Defend against cloned attroperties (jQuery gh-1709) - uniqueCache = outerCache[ node.uniqueID ] || - ( outerCache[ node.uniqueID ] = {} ); - - cache = uniqueCache[ type ] || []; - nodeIndex = cache[ 0 ] === dirruns && cache[ 1 ]; - diff = nodeIndex && cache[ 2 ]; - node = nodeIndex && parent.childNodes[ nodeIndex ]; - - while ( ( node = ++nodeIndex && node && node[ dir ] || - - // Fallback to seeking `elem` from the start - ( diff = nodeIndex = 0 ) || start.pop() ) ) { - - // When found, cache indexes on `parent` and break - if ( node.nodeType === 1 && ++diff && node === elem ) { - uniqueCache[ type ] = [ dirruns, nodeIndex, diff ]; - break; - } - } - - } else { - - // Use previously-cached element index if available - if ( useCache ) { - - // ...in a gzip-friendly way - node = elem; - outerCache = node[ expando ] || ( node[ expando ] = {} ); - - // Support: IE <9 only - // Defend against cloned attroperties (jQuery gh-1709) - uniqueCache = outerCache[ node.uniqueID ] || - ( outerCache[ node.uniqueID ] = {} ); - - cache = uniqueCache[ type ] || []; - nodeIndex = cache[ 0 ] === dirruns && cache[ 1 ]; - diff = nodeIndex; - } - - // xml :nth-child(...) - // or :nth-last-child(...) or :nth(-last)?-of-type(...) - if ( diff === false ) { - - // Use the same loop as above to seek `elem` from the start - while ( ( node = ++nodeIndex && node && node[ dir ] || - ( diff = nodeIndex = 0 ) || start.pop() ) ) { - - if ( ( ofType ? - node.nodeName.toLowerCase() === name : - node.nodeType === 1 ) && - ++diff ) { - - // Cache the index of each encountered element - if ( useCache ) { - outerCache = node[ expando ] || - ( node[ expando ] = {} ); - - // Support: IE <9 only - // Defend against cloned attroperties (jQuery gh-1709) - uniqueCache = outerCache[ node.uniqueID ] || - ( outerCache[ node.uniqueID ] = {} ); - - uniqueCache[ type ] = [ dirruns, diff ]; - } - - if ( node === elem ) { - break; - } - } - } - } - } - - // Incorporate the offset, then check against cycle size - diff -= last; - return diff === first || ( diff % first === 0 && diff / first >= 0 ); - } - }; - }, - - "PSEUDO": function( pseudo, argument ) { - - // pseudo-class names are case-insensitive - // http://www.w3.org/TR/selectors/#pseudo-classes - // Prioritize by case sensitivity in case custom pseudos are added with uppercase letters - // Remember that setFilters inherits from pseudos - var args, - fn = Expr.pseudos[ pseudo ] || Expr.setFilters[ pseudo.toLowerCase() ] || - Sizzle.error( "unsupported pseudo: " + pseudo ); - - // The user may use createPseudo to indicate that - // arguments are needed to create the filter function - // just as Sizzle does - if ( fn[ expando ] ) { - return fn( argument ); - } - - // But maintain support for old signatures - if ( fn.length > 1 ) { - args = [ pseudo, pseudo, "", argument ]; - return Expr.setFilters.hasOwnProperty( pseudo.toLowerCase() ) ? - markFunction( function( seed, matches ) { - var idx, - matched = fn( seed, argument ), - i = matched.length; - while ( i-- ) { - idx = indexOf( seed, matched[ i ] ); - seed[ idx ] = !( matches[ idx ] = matched[ i ] ); - } - } ) : - function( elem ) { - return fn( elem, 0, args ); - }; - } - - return fn; - } - }, - - pseudos: { - - // Potentially complex pseudos - "not": markFunction( function( selector ) { - - // Trim the selector passed to compile - // to avoid treating leading and trailing - // spaces as combinators - var input = [], - results = [], - matcher = compile( selector.replace( rtrim, "$1" ) ); - - return matcher[ expando ] ? - markFunction( function( seed, matches, _context, xml ) { - var elem, - unmatched = matcher( seed, null, xml, [] ), - i = seed.length; - - // Match elements unmatched by `matcher` - while ( i-- ) { - if ( ( elem = unmatched[ i ] ) ) { - seed[ i ] = !( matches[ i ] = elem ); - } - } - } ) : - function( elem, _context, xml ) { - input[ 0 ] = elem; - matcher( input, null, xml, results ); - - // Don't keep the element (issue #299) - input[ 0 ] = null; - return !results.pop(); - }; - } ), - - "has": markFunction( function( selector ) { - return function( elem ) { - return Sizzle( selector, elem ).length > 0; - }; - } ), - - "contains": markFunction( function( text ) { - text = text.replace( runescape, funescape ); - return function( elem ) { - return ( elem.textContent || getText( elem ) ).indexOf( text ) > -1; - }; - } ), - - // "Whether an element is represented by a :lang() selector - // is based solely on the element's language value - // being equal to the identifier C, - // or beginning with the identifier C immediately followed by "-". - // The matching of C against the element's language value is performed case-insensitively. - // The identifier C does not have to be a valid language name." - // http://www.w3.org/TR/selectors/#lang-pseudo - "lang": markFunction( function( lang ) { - - // lang value must be a valid identifier - if ( !ridentifier.test( lang || "" ) ) { - Sizzle.error( "unsupported lang: " + lang ); - } - lang = lang.replace( runescape, funescape ).toLowerCase(); - return function( elem ) { - var elemLang; - do { - if ( ( elemLang = documentIsHTML ? - elem.lang : - elem.getAttribute( "xml:lang" ) || elem.getAttribute( "lang" ) ) ) { - - elemLang = elemLang.toLowerCase(); - return elemLang === lang || elemLang.indexOf( lang + "-" ) === 0; - } - } while ( ( elem = elem.parentNode ) && elem.nodeType === 1 ); - return false; - }; - } ), - - // Miscellaneous - "target": function( elem ) { - var hash = window.location && window.location.hash; - return hash && hash.slice( 1 ) === elem.id; - }, - - "root": function( elem ) { - return elem === docElem; - }, - - "focus": function( elem ) { - return elem === document.activeElement && - ( !document.hasFocus || document.hasFocus() ) && - !!( elem.type || elem.href || ~elem.tabIndex ); - }, - - // Boolean properties - "enabled": createDisabledPseudo( false ), - "disabled": createDisabledPseudo( true ), - - "checked": function( elem ) { - - // In CSS3, :checked should return both checked and selected elements - // http://www.w3.org/TR/2011/REC-css3-selectors-20110929/#checked - var nodeName = elem.nodeName.toLowerCase(); - return ( nodeName === "input" && !!elem.checked ) || - ( nodeName === "option" && !!elem.selected ); - }, - - "selected": function( elem ) { - - // Accessing this property makes selected-by-default - // options in Safari work properly - if ( elem.parentNode ) { - // eslint-disable-next-line no-unused-expressions - elem.parentNode.selectedIndex; - } - - return elem.selected === true; - }, - - // Contents - "empty": function( elem ) { - - // http://www.w3.org/TR/selectors/#empty-pseudo - // :empty is negated by element (1) or content nodes (text: 3; cdata: 4; entity ref: 5), - // but not by others (comment: 8; processing instruction: 7; etc.) - // nodeType < 6 works because attributes (2) do not appear as children - for ( elem = elem.firstChild; elem; elem = elem.nextSibling ) { - if ( elem.nodeType < 6 ) { - return false; - } - } - return true; - }, - - "parent": function( elem ) { - return !Expr.pseudos[ "empty" ]( elem ); - }, - - // Element/input types - "header": function( elem ) { - return rheader.test( elem.nodeName ); - }, - - "input": function( elem ) { - return rinputs.test( elem.nodeName ); - }, - - "button": function( elem ) { - var name = elem.nodeName.toLowerCase(); - return name === "input" && elem.type === "button" || name === "button"; - }, - - "text": function( elem ) { - var attr; - return elem.nodeName.toLowerCase() === "input" && - elem.type === "text" && - - // Support: IE<8 - // New HTML5 attribute values (e.g., "search") appear with elem.type === "text" - ( ( attr = elem.getAttribute( "type" ) ) == null || - attr.toLowerCase() === "text" ); - }, - - // Position-in-collection - "first": createPositionalPseudo( function() { - return [ 0 ]; - } ), - - "last": createPositionalPseudo( function( _matchIndexes, length ) { - return [ length - 1 ]; - } ), - - "eq": createPositionalPseudo( function( _matchIndexes, length, argument ) { - return [ argument < 0 ? argument + length : argument ]; - } ), - - "even": createPositionalPseudo( function( matchIndexes, length ) { - var i = 0; - for ( ; i < length; i += 2 ) { - matchIndexes.push( i ); - } - return matchIndexes; - } ), - - "odd": createPositionalPseudo( function( matchIndexes, length ) { - var i = 1; - for ( ; i < length; i += 2 ) { - matchIndexes.push( i ); - } - return matchIndexes; - } ), - - "lt": createPositionalPseudo( function( matchIndexes, length, argument ) { - var i = argument < 0 ? - argument + length : - argument > length ? - length : - argument; - for ( ; --i >= 0; ) { - matchIndexes.push( i ); - } - return matchIndexes; - } ), - - "gt": createPositionalPseudo( function( matchIndexes, length, argument ) { - var i = argument < 0 ? argument + length : argument; - for ( ; ++i < length; ) { - matchIndexes.push( i ); - } - return matchIndexes; - } ) - } - }; - - Expr.pseudos[ "nth" ] = Expr.pseudos[ "eq" ]; - - // Add button/input type pseudos - for ( i in { radio: true, checkbox: true, file: true, password: true, image: true } ) { - Expr.pseudos[ i ] = createInputPseudo( i ); - } - for ( i in { submit: true, reset: true } ) { - Expr.pseudos[ i ] = createButtonPseudo( i ); - } - - // Easy API for creating new setFilters - function setFilters() {} - setFilters.prototype = Expr.filters = Expr.pseudos; - Expr.setFilters = new setFilters(); - - tokenize = Sizzle.tokenize = function( selector, parseOnly ) { - var matched, match, tokens, type, - soFar, groups, preFilters, - cached = tokenCache[ selector + " " ]; - - if ( cached ) { - return parseOnly ? 0 : cached.slice( 0 ); - } - - soFar = selector; - groups = []; - preFilters = Expr.preFilter; - - while ( soFar ) { - - // Comma and first run - if ( !matched || ( match = rcomma.exec( soFar ) ) ) { - if ( match ) { - - // Don't consume trailing commas as valid - soFar = soFar.slice( match[ 0 ].length ) || soFar; - } - groups.push( ( tokens = [] ) ); - } - - matched = false; - - // Combinators - if ( ( match = rcombinators.exec( soFar ) ) ) { - matched = match.shift(); - tokens.push( { - value: matched, - - // Cast descendant combinators to space - type: match[ 0 ].replace( rtrim, " " ) - } ); - soFar = soFar.slice( matched.length ); - } - - // Filters - for ( type in Expr.filter ) { - if ( ( match = matchExpr[ type ].exec( soFar ) ) && ( !preFilters[ type ] || - ( match = preFilters[ type ]( match ) ) ) ) { - matched = match.shift(); - tokens.push( { - value: matched, - type: type, - matches: match - } ); - soFar = soFar.slice( matched.length ); - } - } - - if ( !matched ) { - break; - } - } - - // Return the length of the invalid excess - // if we're just parsing - // Otherwise, throw an error or return tokens - return parseOnly ? - soFar.length : - soFar ? - Sizzle.error( selector ) : - - // Cache the tokens - tokenCache( selector, groups ).slice( 0 ); - }; - - function toSelector( tokens ) { - var i = 0, - len = tokens.length, - selector = ""; - for ( ; i < len; i++ ) { - selector += tokens[ i ].value; - } - return selector; - } - - function addCombinator( matcher, combinator, base ) { - var dir = combinator.dir, - skip = combinator.next, - key = skip || dir, - checkNonElements = base && key === "parentNode", - doneName = done++; - - return combinator.first ? - - // Check against closest ancestor/preceding element - function( elem, context, xml ) { - while ( ( elem = elem[ dir ] ) ) { - if ( elem.nodeType === 1 || checkNonElements ) { - return matcher( elem, context, xml ); - } - } - return false; - } : - - // Check against all ancestor/preceding elements - function( elem, context, xml ) { - var oldCache, uniqueCache, outerCache, - newCache = [ dirruns, doneName ]; - - // We can't set arbitrary data on XML nodes, so they don't benefit from combinator caching - if ( xml ) { - while ( ( elem = elem[ dir ] ) ) { - if ( elem.nodeType === 1 || checkNonElements ) { - if ( matcher( elem, context, xml ) ) { - return true; - } - } - } - } else { - while ( ( elem = elem[ dir ] ) ) { - if ( elem.nodeType === 1 || checkNonElements ) { - outerCache = elem[ expando ] || ( elem[ expando ] = {} ); - - // Support: IE <9 only - // Defend against cloned attroperties (jQuery gh-1709) - uniqueCache = outerCache[ elem.uniqueID ] || - ( outerCache[ elem.uniqueID ] = {} ); - - if ( skip && skip === elem.nodeName.toLowerCase() ) { - elem = elem[ dir ] || elem; - } else if ( ( oldCache = uniqueCache[ key ] ) && - oldCache[ 0 ] === dirruns && oldCache[ 1 ] === doneName ) { - - // Assign to newCache so results back-propagate to previous elements - return ( newCache[ 2 ] = oldCache[ 2 ] ); - } else { - - // Reuse newcache so results back-propagate to previous elements - uniqueCache[ key ] = newCache; - - // A match means we're done; a fail means we have to keep checking - if ( ( newCache[ 2 ] = matcher( elem, context, xml ) ) ) { - return true; - } - } - } - } - } - return false; - }; - } - - function elementMatcher( matchers ) { - return matchers.length > 1 ? - function( elem, context, xml ) { - var i = matchers.length; - while ( i-- ) { - if ( !matchers[ i ]( elem, context, xml ) ) { - return false; - } - } - return true; - } : - matchers[ 0 ]; - } - - function multipleContexts( selector, contexts, results ) { - var i = 0, - len = contexts.length; - for ( ; i < len; i++ ) { - Sizzle( selector, contexts[ i ], results ); - } - return results; - } - - function condense( unmatched, map, filter, context, xml ) { - var elem, - newUnmatched = [], - i = 0, - len = unmatched.length, - mapped = map != null; - - for ( ; i < len; i++ ) { - if ( ( elem = unmatched[ i ] ) ) { - if ( !filter || filter( elem, context, xml ) ) { - newUnmatched.push( elem ); - if ( mapped ) { - map.push( i ); - } - } - } - } - - return newUnmatched; - } - - function setMatcher( preFilter, selector, matcher, postFilter, postFinder, postSelector ) { - if ( postFilter && !postFilter[ expando ] ) { - postFilter = setMatcher( postFilter ); - } - if ( postFinder && !postFinder[ expando ] ) { - postFinder = setMatcher( postFinder, postSelector ); - } - return markFunction( function( seed, results, context, xml ) { - var temp, i, elem, - preMap = [], - postMap = [], - preexisting = results.length, - - // Get initial elements from seed or context - elems = seed || multipleContexts( - selector || "*", - context.nodeType ? [ context ] : context, - [] - ), - - // Prefilter to get matcher input, preserving a map for seed-results synchronization - matcherIn = preFilter && ( seed || !selector ) ? - condense( elems, preMap, preFilter, context, xml ) : - elems, - - matcherOut = matcher ? - - // If we have a postFinder, or filtered seed, or non-seed postFilter or preexisting results, - postFinder || ( seed ? preFilter : preexisting || postFilter ) ? - - // ...intermediate processing is necessary - [] : - - // ...otherwise use results directly - results : - matcherIn; - - // Find primary matches - if ( matcher ) { - matcher( matcherIn, matcherOut, context, xml ); - } - - // Apply postFilter - if ( postFilter ) { - temp = condense( matcherOut, postMap ); - postFilter( temp, [], context, xml ); - - // Un-match failing elements by moving them back to matcherIn - i = temp.length; - while ( i-- ) { - if ( ( elem = temp[ i ] ) ) { - matcherOut[ postMap[ i ] ] = !( matcherIn[ postMap[ i ] ] = elem ); - } - } - } - - if ( seed ) { - if ( postFinder || preFilter ) { - if ( postFinder ) { - - // Get the final matcherOut by condensing this intermediate into postFinder contexts - temp = []; - i = matcherOut.length; - while ( i-- ) { - if ( ( elem = matcherOut[ i ] ) ) { - - // Restore matcherIn since elem is not yet a final match - temp.push( ( matcherIn[ i ] = elem ) ); - } - } - postFinder( null, ( matcherOut = [] ), temp, xml ); - } - - // Move matched elements from seed to results to keep them synchronized - i = matcherOut.length; - while ( i-- ) { - if ( ( elem = matcherOut[ i ] ) && - ( temp = postFinder ? indexOf( seed, elem ) : preMap[ i ] ) > -1 ) { - - seed[ temp ] = !( results[ temp ] = elem ); - } - } - } - - // Add elements to results, through postFinder if defined - } else { - matcherOut = condense( - matcherOut === results ? - matcherOut.splice( preexisting, matcherOut.length ) : - matcherOut - ); - if ( postFinder ) { - postFinder( null, results, matcherOut, xml ); - } else { - push.apply( results, matcherOut ); - } - } - } ); - } - - function matcherFromTokens( tokens ) { - var checkContext, matcher, j, - len = tokens.length, - leadingRelative = Expr.relative[ tokens[ 0 ].type ], - implicitRelative = leadingRelative || Expr.relative[ " " ], - i = leadingRelative ? 1 : 0, - - // The foundational matcher ensures that elements are reachable from top-level context(s) - matchContext = addCombinator( function( elem ) { - return elem === checkContext; - }, implicitRelative, true ), - matchAnyContext = addCombinator( function( elem ) { - return indexOf( checkContext, elem ) > -1; - }, implicitRelative, true ), - matchers = [ function( elem, context, xml ) { - var ret = ( !leadingRelative && ( xml || context !== outermostContext ) ) || ( - ( checkContext = context ).nodeType ? - matchContext( elem, context, xml ) : - matchAnyContext( elem, context, xml ) ); - - // Avoid hanging onto element (issue #299) - checkContext = null; - return ret; - } ]; - - for ( ; i < len; i++ ) { - if ( ( matcher = Expr.relative[ tokens[ i ].type ] ) ) { - matchers = [ addCombinator( elementMatcher( matchers ), matcher ) ]; - } else { - matcher = Expr.filter[ tokens[ i ].type ].apply( null, tokens[ i ].matches ); - - // Return special upon seeing a positional matcher - if ( matcher[ expando ] ) { - - // Find the next relative operator (if any) for proper handling - j = ++i; - for ( ; j < len; j++ ) { - if ( Expr.relative[ tokens[ j ].type ] ) { - break; - } - } - return setMatcher( - i > 1 && elementMatcher( matchers ), - i > 1 && toSelector( - - // If the preceding token was a descendant combinator, insert an implicit any-element `*` - tokens - .slice( 0, i - 1 ) - .concat( { value: tokens[ i - 2 ].type === " " ? "*" : "" } ) - ).replace( rtrim, "$1" ), - matcher, - i < j && matcherFromTokens( tokens.slice( i, j ) ), - j < len && matcherFromTokens( ( tokens = tokens.slice( j ) ) ), - j < len && toSelector( tokens ) - ); - } - matchers.push( matcher ); - } - } - - return elementMatcher( matchers ); - } - - function matcherFromGroupMatchers( elementMatchers, setMatchers ) { - var bySet = setMatchers.length > 0, - byElement = elementMatchers.length > 0, - superMatcher = function( seed, context, xml, results, outermost ) { - var elem, j, matcher, - matchedCount = 0, - i = "0", - unmatched = seed && [], - setMatched = [], - contextBackup = outermostContext, - - // We must always have either seed elements or outermost context - elems = seed || byElement && Expr.find[ "TAG" ]( "*", outermost ), - - // Use integer dirruns iff this is the outermost matcher - dirrunsUnique = ( dirruns += contextBackup == null ? 1 : Math.random() || 0.1 ), - len = elems.length; - - if ( outermost ) { - - // Support: IE 11+, Edge 17 - 18+ - // IE/Edge sometimes throw a "Permission denied" error when strict-comparing - // two documents; shallow comparisons work. - // eslint-disable-next-line eqeqeq - outermostContext = context == document || context || outermost; - } - - // Add elements passing elementMatchers directly to results - // Support: IE<9, Safari - // Tolerate NodeList properties (IE: "length"; Safari: <number>) matching elements by id - for ( ; i !== len && ( elem = elems[ i ] ) != null; i++ ) { - if ( byElement && elem ) { - j = 0; - - // Support: IE 11+, Edge 17 - 18+ - // IE/Edge sometimes throw a "Permission denied" error when strict-comparing - // two documents; shallow comparisons work. - // eslint-disable-next-line eqeqeq - if ( !context && elem.ownerDocument != document ) { - setDocument( elem ); - xml = !documentIsHTML; - } - while ( ( matcher = elementMatchers[ j++ ] ) ) { - if ( matcher( elem, context || document, xml ) ) { - results.push( elem ); - break; - } - } - if ( outermost ) { - dirruns = dirrunsUnique; - } - } - - // Track unmatched elements for set filters - if ( bySet ) { - - // They will have gone through all possible matchers - if ( ( elem = !matcher && elem ) ) { - matchedCount--; - } - - // Lengthen the array for every element, matched or not - if ( seed ) { - unmatched.push( elem ); - } - } - } - - // `i` is now the count of elements visited above, and adding it to `matchedCount` - // makes the latter nonnegative. - matchedCount += i; - - // Apply set filters to unmatched elements - // NOTE: This can be skipped if there are no unmatched elements (i.e., `matchedCount` - // equals `i`), unless we didn't visit _any_ elements in the above loop because we have - // no element matchers and no seed. - // Incrementing an initially-string "0" `i` allows `i` to remain a string only in that - // case, which will result in a "00" `matchedCount` that differs from `i` but is also - // numerically zero. - if ( bySet && i !== matchedCount ) { - j = 0; - while ( ( matcher = setMatchers[ j++ ] ) ) { - matcher( unmatched, setMatched, context, xml ); - } - - if ( seed ) { - - // Reintegrate element matches to eliminate the need for sorting - if ( matchedCount > 0 ) { - while ( i-- ) { - if ( !( unmatched[ i ] || setMatched[ i ] ) ) { - setMatched[ i ] = pop.call( results ); - } - } - } - - // Discard index placeholder values to get only actual matches - setMatched = condense( setMatched ); - } - - // Add matches to results - push.apply( results, setMatched ); - - // Seedless set matches succeeding multiple successful matchers stipulate sorting - if ( outermost && !seed && setMatched.length > 0 && - ( matchedCount + setMatchers.length ) > 1 ) { - - Sizzle.uniqueSort( results ); - } - } - - // Override manipulation of globals by nested matchers - if ( outermost ) { - dirruns = dirrunsUnique; - outermostContext = contextBackup; - } - - return unmatched; - }; - - return bySet ? - markFunction( superMatcher ) : - superMatcher; - } - - compile = Sizzle.compile = function( selector, match /* Internal Use Only */ ) { - var i, - setMatchers = [], - elementMatchers = [], - cached = compilerCache[ selector + " " ]; - - if ( !cached ) { - - // Generate a function of recursive functions that can be used to check each element - if ( !match ) { - match = tokenize( selector ); - } - i = match.length; - while ( i-- ) { - cached = matcherFromTokens( match[ i ] ); - if ( cached[ expando ] ) { - setMatchers.push( cached ); - } else { - elementMatchers.push( cached ); - } - } - - // Cache the compiled function - cached = compilerCache( - selector, - matcherFromGroupMatchers( elementMatchers, setMatchers ) - ); - - // Save selector and tokenization - cached.selector = selector; - } - return cached; - }; - - /** - * A low-level selection function that works with Sizzle's compiled - * selector functions - * @param {String|Function} selector A selector or a pre-compiled - * selector function built with Sizzle.compile - * @param {Element} context - * @param {Array} [results] - * @param {Array} [seed] A set of elements to match against - */ - select = Sizzle.select = function( selector, context, results, seed ) { - var i, tokens, token, type, find, - compiled = typeof selector === "function" && selector, - match = !seed && tokenize( ( selector = compiled.selector || selector ) ); - - results = results || []; - - // Try to minimize operations if there is only one selector in the list and no seed - // (the latter of which guarantees us context) - if ( match.length === 1 ) { - - // Reduce context if the leading compound selector is an ID - tokens = match[ 0 ] = match[ 0 ].slice( 0 ); - if ( tokens.length > 2 && ( token = tokens[ 0 ] ).type === "ID" && - context.nodeType === 9 && documentIsHTML && Expr.relative[ tokens[ 1 ].type ] ) { - - context = ( Expr.find[ "ID" ]( token.matches[ 0 ] - .replace( runescape, funescape ), context ) || [] )[ 0 ]; - if ( !context ) { - return results; - - // Precompiled matchers will still verify ancestry, so step up a level - } else if ( compiled ) { - context = context.parentNode; - } - - selector = selector.slice( tokens.shift().value.length ); - } - - // Fetch a seed set for right-to-left matching - i = matchExpr[ "needsContext" ].test( selector ) ? 0 : tokens.length; - while ( i-- ) { - token = tokens[ i ]; - - // Abort if we hit a combinator - if ( Expr.relative[ ( type = token.type ) ] ) { - break; - } - if ( ( find = Expr.find[ type ] ) ) { - - // Search, expanding context for leading sibling combinators - if ( ( seed = find( - token.matches[ 0 ].replace( runescape, funescape ), - rsibling.test( tokens[ 0 ].type ) && testContext( context.parentNode ) || - context - ) ) ) { - - // If seed is empty or no tokens remain, we can return early - tokens.splice( i, 1 ); - selector = seed.length && toSelector( tokens ); - if ( !selector ) { - push.apply( results, seed ); - return results; - } - - break; - } - } - } - } - - // Compile and execute a filtering function if one is not provided - // Provide `match` to avoid retokenization if we modified the selector above - ( compiled || compile( selector, match ) )( - seed, - context, - !documentIsHTML, - results, - !context || rsibling.test( selector ) && testContext( context.parentNode ) || context - ); - return results; - }; - - // One-time assignments - - // Sort stability - support.sortStable = expando.split( "" ).sort( sortOrder ).join( "" ) === expando; - - // Support: Chrome 14-35+ - // Always assume duplicates if they aren't passed to the comparison function - support.detectDuplicates = !!hasDuplicate; - - // Initialize against the default document - setDocument(); - - // Support: Webkit<537.32 - Safari 6.0.3/Chrome 25 (fixed in Chrome 27) - // Detached nodes confoundingly follow *each other* - support.sortDetached = assert( function( el ) { - - // Should return 1, but returns 4 (following) - return el.compareDocumentPosition( document.createElement( "fieldset" ) ) & 1; - } ); - - // Support: IE<8 - // Prevent attribute/property "interpolation" - // https://msdn.microsoft.com/en-us/library/ms536429%28VS.85%29.aspx - if ( !assert( function( el ) { - el.innerHTML = "<a href='#'></a>"; - return el.firstChild.getAttribute( "href" ) === "#"; - } ) ) { - addHandle( "type|href|height|width", function( elem, name, isXML ) { - if ( !isXML ) { - return elem.getAttribute( name, name.toLowerCase() === "type" ? 1 : 2 ); - } - } ); - } - - // Support: IE<9 - // Use defaultValue in place of getAttribute("value") - if ( !support.attributes || !assert( function( el ) { - el.innerHTML = "<input/>"; - el.firstChild.setAttribute( "value", "" ); - return el.firstChild.getAttribute( "value" ) === ""; - } ) ) { - addHandle( "value", function( elem, _name, isXML ) { - if ( !isXML && elem.nodeName.toLowerCase() === "input" ) { - return elem.defaultValue; - } - } ); - } - - // Support: IE<9 - // Use getAttributeNode to fetch booleans when getAttribute lies - if ( !assert( function( el ) { - return el.getAttribute( "disabled" ) == null; - } ) ) { - addHandle( booleans, function( elem, name, isXML ) { - var val; - if ( !isXML ) { - return elem[ name ] === true ? name.toLowerCase() : - ( val = elem.getAttributeNode( name ) ) && val.specified ? - val.value : - null; - } - } ); - } - - return Sizzle; - - } )( window ); - - - - jQuery.find = Sizzle; - jQuery.expr = Sizzle.selectors; - - // Deprecated - jQuery.expr[ ":" ] = jQuery.expr.pseudos; - jQuery.uniqueSort = jQuery.unique = Sizzle.uniqueSort; - jQuery.text = Sizzle.getText; - jQuery.isXMLDoc = Sizzle.isXML; - jQuery.contains = Sizzle.contains; - jQuery.escapeSelector = Sizzle.escape; - - - - - var dir = function( elem, dir, until ) { - var matched = [], - truncate = until !== undefined; - - while ( ( elem = elem[ dir ] ) && elem.nodeType !== 9 ) { - if ( elem.nodeType === 1 ) { - if ( truncate && jQuery( elem ).is( until ) ) { - break; - } - matched.push( elem ); - } - } - return matched; - }; - - - var siblings = function( n, elem ) { - var matched = []; - - for ( ; n; n = n.nextSibling ) { - if ( n.nodeType === 1 && n !== elem ) { - matched.push( n ); - } - } - - return matched; - }; - - - var rneedsContext = jQuery.expr.match.needsContext; - - - - function nodeName( elem, name ) { - - return elem.nodeName && elem.nodeName.toLowerCase() === name.toLowerCase(); - - } - var rsingleTag = ( /^<([a-z][^\/\0>:\x20\t\r\n\f]*)[\x20\t\r\n\f]*\/?>(?:<\/\1>|)$/i ); - - - - // Implement the identical functionality for filter and not - function winnow( elements, qualifier, not ) { - if ( isFunction( qualifier ) ) { - return jQuery.grep( elements, function( elem, i ) { - return !!qualifier.call( elem, i, elem ) !== not; - } ); - } - - // Single element - if ( qualifier.nodeType ) { - return jQuery.grep( elements, function( elem ) { - return ( elem === qualifier ) !== not; - } ); - } - - // Arraylike of elements (jQuery, arguments, Array) - if ( typeof qualifier !== "string" ) { - return jQuery.grep( elements, function( elem ) { - return ( indexOf.call( qualifier, elem ) > -1 ) !== not; - } ); - } - - // Filtered directly for both simple and complex selectors - return jQuery.filter( qualifier, elements, not ); - } - - jQuery.filter = function( expr, elems, not ) { - var elem = elems[ 0 ]; - - if ( not ) { - expr = ":not(" + expr + ")"; - } - - if ( elems.length === 1 && elem.nodeType === 1 ) { - return jQuery.find.matchesSelector( elem, expr ) ? [ elem ] : []; - } - - return jQuery.find.matches( expr, jQuery.grep( elems, function( elem ) { - return elem.nodeType === 1; - } ) ); - }; - - jQuery.fn.extend( { - find: function( selector ) { - var i, ret, - len = this.length, - self = this; - - if ( typeof selector !== "string" ) { - return this.pushStack( jQuery( selector ).filter( function() { - for ( i = 0; i < len; i++ ) { - if ( jQuery.contains( self[ i ], this ) ) { - return true; - } - } - } ) ); - } - - ret = this.pushStack( [] ); - - for ( i = 0; i < len; i++ ) { - jQuery.find( selector, self[ i ], ret ); - } - - return len > 1 ? jQuery.uniqueSort( ret ) : ret; - }, - filter: function( selector ) { - return this.pushStack( winnow( this, selector || [], false ) ); - }, - not: function( selector ) { - return this.pushStack( winnow( this, selector || [], true ) ); - }, - is: function( selector ) { - return !!winnow( - this, - - // If this is a positional/relative selector, check membership in the returned set - // so $("p:first").is("p:last") won't return true for a doc with two "p". - typeof selector === "string" && rneedsContext.test( selector ) ? - jQuery( selector ) : - selector || [], - false - ).length; - } - } ); - - - // Initialize a jQuery object - - - // A central reference to the root jQuery(document) - var rootjQuery, - - // A simple way to check for HTML strings - // Prioritize #id over <tag> to avoid XSS via location.hash (#9521) - // Strict HTML recognition (#11290: must start with <) - // Shortcut simple #id case for speed - rquickExpr = /^(?:\s*(<[\w\W]+>)[^>]*|#([\w-]+))$/, - - init = jQuery.fn.init = function( selector, context, root ) { - var match, elem; - - // HANDLE: $(""), $(null), $(undefined), $(false) - if ( !selector ) { - return this; - } - - // Method init() accepts an alternate rootjQuery - // so migrate can support jQuery.sub (gh-2101) - root = root || rootjQuery; - - // Handle HTML strings - if ( typeof selector === "string" ) { - if ( selector[ 0 ] === "<" && - selector[ selector.length - 1 ] === ">" && - selector.length >= 3 ) { - - // Assume that strings that start and end with <> are HTML and skip the regex check - match = [ null, selector, null ]; - - } else { - match = rquickExpr.exec( selector ); - } - - // Match html or make sure no context is specified for #id - if ( match && ( match[ 1 ] || !context ) ) { - - // HANDLE: $(html) -> $(array) - if ( match[ 1 ] ) { - context = context instanceof jQuery ? context[ 0 ] : context; - - // Option to run scripts is true for back-compat - // Intentionally let the error be thrown if parseHTML is not present - jQuery.merge( this, jQuery.parseHTML( - match[ 1 ], - context && context.nodeType ? context.ownerDocument || context : document, - true - ) ); - - // HANDLE: $(html, props) - if ( rsingleTag.test( match[ 1 ] ) && jQuery.isPlainObject( context ) ) { - for ( match in context ) { - - // Properties of context are called as methods if possible - if ( isFunction( this[ match ] ) ) { - this[ match ]( context[ match ] ); - - // ...and otherwise set as attributes - } else { - this.attr( match, context[ match ] ); - } - } - } - - return this; - - // HANDLE: $(#id) - } else { - elem = document.getElementById( match[ 2 ] ); - - if ( elem ) { - - // Inject the element directly into the jQuery object - this[ 0 ] = elem; - this.length = 1; - } - return this; - } - - // HANDLE: $(expr, $(...)) - } else if ( !context || context.jquery ) { - return ( context || root ).find( selector ); - - // HANDLE: $(expr, context) - // (which is just equivalent to: $(context).find(expr) - } else { - return this.constructor( context ).find( selector ); - } - - // HANDLE: $(DOMElement) - } else if ( selector.nodeType ) { - this[ 0 ] = selector; - this.length = 1; - return this; - - // HANDLE: $(function) - // Shortcut for document ready - } else if ( isFunction( selector ) ) { - return root.ready !== undefined ? - root.ready( selector ) : - - // Execute immediately if ready is not present - selector( jQuery ); - } - - return jQuery.makeArray( selector, this ); - }; - - // Give the init function the jQuery prototype for later instantiation - init.prototype = jQuery.fn; - - // Initialize central reference - rootjQuery = jQuery( document ); - - - var rparentsprev = /^(?:parents|prev(?:Until|All))/, - - // Methods guaranteed to produce a unique set when starting from a unique set - guaranteedUnique = { - children: true, - contents: true, - next: true, - prev: true - }; - - jQuery.fn.extend( { - has: function( target ) { - var targets = jQuery( target, this ), - l = targets.length; - - return this.filter( function() { - var i = 0; - for ( ; i < l; i++ ) { - if ( jQuery.contains( this, targets[ i ] ) ) { - return true; - } - } - } ); - }, - - closest: function( selectors, context ) { - var cur, - i = 0, - l = this.length, - matched = [], - targets = typeof selectors !== "string" && jQuery( selectors ); - - // Positional selectors never match, since there's no _selection_ context - if ( !rneedsContext.test( selectors ) ) { - for ( ; i < l; i++ ) { - for ( cur = this[ i ]; cur && cur !== context; cur = cur.parentNode ) { - - // Always skip document fragments - if ( cur.nodeType < 11 && ( targets ? - targets.index( cur ) > -1 : - - // Don't pass non-elements to Sizzle - cur.nodeType === 1 && - jQuery.find.matchesSelector( cur, selectors ) ) ) { - - matched.push( cur ); - break; - } - } - } - } - - return this.pushStack( matched.length > 1 ? jQuery.uniqueSort( matched ) : matched ); - }, - - // Determine the position of an element within the set - index: function( elem ) { - - // No argument, return index in parent - if ( !elem ) { - return ( this[ 0 ] && this[ 0 ].parentNode ) ? this.first().prevAll().length : -1; - } - - // Index in selector - if ( typeof elem === "string" ) { - return indexOf.call( jQuery( elem ), this[ 0 ] ); - } - - // Locate the position of the desired element - return indexOf.call( this, - - // If it receives a jQuery object, the first element is used - elem.jquery ? elem[ 0 ] : elem - ); - }, - - add: function( selector, context ) { - return this.pushStack( - jQuery.uniqueSort( - jQuery.merge( this.get(), jQuery( selector, context ) ) - ) - ); - }, - - addBack: function( selector ) { - return this.add( selector == null ? - this.prevObject : this.prevObject.filter( selector ) - ); - } - } ); - - function sibling( cur, dir ) { - while ( ( cur = cur[ dir ] ) && cur.nodeType !== 1 ) {} - return cur; - } - - jQuery.each( { - parent: function( elem ) { - var parent = elem.parentNode; - return parent && parent.nodeType !== 11 ? parent : null; - }, - parents: function( elem ) { - return dir( elem, "parentNode" ); - }, - parentsUntil: function( elem, _i, until ) { - return dir( elem, "parentNode", until ); - }, - next: function( elem ) { - return sibling( elem, "nextSibling" ); - }, - prev: function( elem ) { - return sibling( elem, "previousSibling" ); - }, - nextAll: function( elem ) { - return dir( elem, "nextSibling" ); - }, - prevAll: function( elem ) { - return dir( elem, "previousSibling" ); - }, - nextUntil: function( elem, _i, until ) { - return dir( elem, "nextSibling", until ); - }, - prevUntil: function( elem, _i, until ) { - return dir( elem, "previousSibling", until ); - }, - siblings: function( elem ) { - return siblings( ( elem.parentNode || {} ).firstChild, elem ); - }, - children: function( elem ) { - return siblings( elem.firstChild ); - }, - contents: function( elem ) { - if ( elem.contentDocument != null && - - // Support: IE 11+ - // <object> elements with no `data` attribute has an object - // `contentDocument` with a `null` prototype. - getProto( elem.contentDocument ) ) { - - return elem.contentDocument; - } - - // Support: IE 9 - 11 only, iOS 7 only, Android Browser <=4.3 only - // Treat the template element as a regular one in browsers that - // don't support it. - if ( nodeName( elem, "template" ) ) { - elem = elem.content || elem; - } - - return jQuery.merge( [], elem.childNodes ); - } - }, function( name, fn ) { - jQuery.fn[ name ] = function( until, selector ) { - var matched = jQuery.map( this, fn, until ); - - if ( name.slice( -5 ) !== "Until" ) { - selector = until; - } - - if ( selector && typeof selector === "string" ) { - matched = jQuery.filter( selector, matched ); - } - - if ( this.length > 1 ) { - - // Remove duplicates - if ( !guaranteedUnique[ name ] ) { - jQuery.uniqueSort( matched ); - } - - // Reverse order for parents* and prev-derivatives - if ( rparentsprev.test( name ) ) { - matched.reverse(); - } - } - - return this.pushStack( matched ); - }; - } ); - var rnothtmlwhite = ( /[^\x20\t\r\n\f]+/g ); - - - - // Convert String-formatted options into Object-formatted ones - function createOptions( options ) { - var object = {}; - jQuery.each( options.match( rnothtmlwhite ) || [], function( _, flag ) { - object[ flag ] = true; - } ); - return object; - } - - /* - * Create a callback list using the following parameters: - * - * options: an optional list of space-separated options that will change how - * the callback list behaves or a more traditional option object - * - * By default a callback list will act like an event callback list and can be - * "fired" multiple times. - * - * Possible options: - * - * once: will ensure the callback list can only be fired once (like a Deferred) - * - * memory: will keep track of previous values and will call any callback added - * after the list has been fired right away with the latest "memorized" - * values (like a Deferred) - * - * unique: will ensure a callback can only be added once (no duplicate in the list) - * - * stopOnFalse: interrupt callings when a callback returns false - * - */ - jQuery.Callbacks = function( options ) { - - // Convert options from String-formatted to Object-formatted if needed - // (we check in cache first) - options = typeof options === "string" ? - createOptions( options ) : - jQuery.extend( {}, options ); - - var // Flag to know if list is currently firing - firing, - - // Last fire value for non-forgettable lists - memory, - - // Flag to know if list was already fired - fired, - - // Flag to prevent firing - locked, - - // Actual callback list - list = [], - - // Queue of execution data for repeatable lists - queue = [], - - // Index of currently firing callback (modified by add/remove as needed) - firingIndex = -1, - - // Fire callbacks - fire = function() { - - // Enforce single-firing - locked = locked || options.once; - - // Execute callbacks for all pending executions, - // respecting firingIndex overrides and runtime changes - fired = firing = true; - for ( ; queue.length; firingIndex = -1 ) { - memory = queue.shift(); - while ( ++firingIndex < list.length ) { - - // Run callback and check for early termination - if ( list[ firingIndex ].apply( memory[ 0 ], memory[ 1 ] ) === false && - options.stopOnFalse ) { - - // Jump to end and forget the data so .add doesn't re-fire - firingIndex = list.length; - memory = false; - } - } - } - - // Forget the data if we're done with it - if ( !options.memory ) { - memory = false; - } - - firing = false; - - // Clean up if we're done firing for good - if ( locked ) { - - // Keep an empty list if we have data for future add calls - if ( memory ) { - list = []; - - // Otherwise, this object is spent - } else { - list = ""; - } - } - }, - - // Actual Callbacks object - self = { - - // Add a callback or a collection of callbacks to the list - add: function() { - if ( list ) { - - // If we have memory from a past run, we should fire after adding - if ( memory && !firing ) { - firingIndex = list.length - 1; - queue.push( memory ); - } - - ( function add( args ) { - jQuery.each( args, function( _, arg ) { - if ( isFunction( arg ) ) { - if ( !options.unique || !self.has( arg ) ) { - list.push( arg ); - } - } else if ( arg && arg.length && toType( arg ) !== "string" ) { - - // Inspect recursively - add( arg ); - } - } ); - } )( arguments ); - - if ( memory && !firing ) { - fire(); - } - } - return this; - }, - - // Remove a callback from the list - remove: function() { - jQuery.each( arguments, function( _, arg ) { - var index; - while ( ( index = jQuery.inArray( arg, list, index ) ) > -1 ) { - list.splice( index, 1 ); - - // Handle firing indexes - if ( index <= firingIndex ) { - firingIndex--; - } - } - } ); - return this; - }, - - // Check if a given callback is in the list. - // If no argument is given, return whether or not list has callbacks attached. - has: function( fn ) { - return fn ? - jQuery.inArray( fn, list ) > -1 : - list.length > 0; - }, - - // Remove all callbacks from the list - empty: function() { - if ( list ) { - list = []; - } - return this; - }, - - // Disable .fire and .add - // Abort any current/pending executions - // Clear all callbacks and values - disable: function() { - locked = queue = []; - list = memory = ""; - return this; - }, - disabled: function() { - return !list; - }, - - // Disable .fire - // Also disable .add unless we have memory (since it would have no effect) - // Abort any pending executions - lock: function() { - locked = queue = []; - if ( !memory && !firing ) { - list = memory = ""; - } - return this; - }, - locked: function() { - return !!locked; - }, - - // Call all callbacks with the given context and arguments - fireWith: function( context, args ) { - if ( !locked ) { - args = args || []; - args = [ context, args.slice ? args.slice() : args ]; - queue.push( args ); - if ( !firing ) { - fire(); - } - } - return this; - }, - - // Call all the callbacks with the given arguments - fire: function() { - self.fireWith( this, arguments ); - return this; - }, - - // To know if the callbacks have already been called at least once - fired: function() { - return !!fired; - } - }; - - return self; - }; - - - function Identity( v ) { - return v; - } - function Thrower( ex ) { - throw ex; - } - - function adoptValue( value, resolve, reject, noValue ) { - var method; - - try { - - // Check for promise aspect first to privilege synchronous behavior - if ( value && isFunction( ( method = value.promise ) ) ) { - method.call( value ).done( resolve ).fail( reject ); - - // Other thenables - } else if ( value && isFunction( ( method = value.then ) ) ) { - method.call( value, resolve, reject ); - - // Other non-thenables - } else { - - // Control `resolve` arguments by letting Array#slice cast boolean `noValue` to integer: - // * false: [ value ].slice( 0 ) => resolve( value ) - // * true: [ value ].slice( 1 ) => resolve() - resolve.apply( undefined, [ value ].slice( noValue ) ); - } - - // For Promises/A+, convert exceptions into rejections - // Since jQuery.when doesn't unwrap thenables, we can skip the extra checks appearing in - // Deferred#then to conditionally suppress rejection. - } catch ( value ) { - - // Support: Android 4.0 only - // Strict mode functions invoked without .call/.apply get global-object context - reject.apply( undefined, [ value ] ); - } - } - - jQuery.extend( { - - Deferred: function( func ) { - var tuples = [ - - // action, add listener, callbacks, - // ... .then handlers, argument index, [final state] - [ "notify", "progress", jQuery.Callbacks( "memory" ), - jQuery.Callbacks( "memory" ), 2 ], - [ "resolve", "done", jQuery.Callbacks( "once memory" ), - jQuery.Callbacks( "once memory" ), 0, "resolved" ], - [ "reject", "fail", jQuery.Callbacks( "once memory" ), - jQuery.Callbacks( "once memory" ), 1, "rejected" ] - ], - state = "pending", - promise = { - state: function() { - return state; - }, - always: function() { - deferred.done( arguments ).fail( arguments ); - return this; - }, - "catch": function( fn ) { - return promise.then( null, fn ); - }, - - // Keep pipe for back-compat - pipe: function( /* fnDone, fnFail, fnProgress */ ) { - var fns = arguments; - - return jQuery.Deferred( function( newDefer ) { - jQuery.each( tuples, function( _i, tuple ) { - - // Map tuples (progress, done, fail) to arguments (done, fail, progress) - var fn = isFunction( fns[ tuple[ 4 ] ] ) && fns[ tuple[ 4 ] ]; - - // deferred.progress(function() { bind to newDefer or newDefer.notify }) - // deferred.done(function() { bind to newDefer or newDefer.resolve }) - // deferred.fail(function() { bind to newDefer or newDefer.reject }) - deferred[ tuple[ 1 ] ]( function() { - var returned = fn && fn.apply( this, arguments ); - if ( returned && isFunction( returned.promise ) ) { - returned.promise() - .progress( newDefer.notify ) - .done( newDefer.resolve ) - .fail( newDefer.reject ); - } else { - newDefer[ tuple[ 0 ] + "With" ]( - this, - fn ? [ returned ] : arguments - ); - } - } ); - } ); - fns = null; - } ).promise(); - }, - then: function( onFulfilled, onRejected, onProgress ) { - var maxDepth = 0; - function resolve( depth, deferred, handler, special ) { - return function() { - var that = this, - args = arguments, - mightThrow = function() { - var returned, then; - - // Support: Promises/A+ section 2.3.3.3.3 - // https://promisesaplus.com/#point-59 - // Ignore double-resolution attempts - if ( depth < maxDepth ) { - return; - } - - returned = handler.apply( that, args ); - - // Support: Promises/A+ section 2.3.1 - // https://promisesaplus.com/#point-48 - if ( returned === deferred.promise() ) { - throw new TypeError( "Thenable self-resolution" ); - } - - // Support: Promises/A+ sections 2.3.3.1, 3.5 - // https://promisesaplus.com/#point-54 - // https://promisesaplus.com/#point-75 - // Retrieve `then` only once - then = returned && - - // Support: Promises/A+ section 2.3.4 - // https://promisesaplus.com/#point-64 - // Only check objects and functions for thenability - ( typeof returned === "object" || - typeof returned === "function" ) && - returned.then; - - // Handle a returned thenable - if ( isFunction( then ) ) { - - // Special processors (notify) just wait for resolution - if ( special ) { - then.call( - returned, - resolve( maxDepth, deferred, Identity, special ), - resolve( maxDepth, deferred, Thrower, special ) - ); - - // Normal processors (resolve) also hook into progress - } else { - - // ...and disregard older resolution values - maxDepth++; - - then.call( - returned, - resolve( maxDepth, deferred, Identity, special ), - resolve( maxDepth, deferred, Thrower, special ), - resolve( maxDepth, deferred, Identity, - deferred.notifyWith ) - ); - } - - // Handle all other returned values - } else { - - // Only substitute handlers pass on context - // and multiple values (non-spec behavior) - if ( handler !== Identity ) { - that = undefined; - args = [ returned ]; - } - - // Process the value(s) - // Default process is resolve - ( special || deferred.resolveWith )( that, args ); - } - }, - - // Only normal processors (resolve) catch and reject exceptions - process = special ? - mightThrow : - function() { - try { - mightThrow(); - } catch ( e ) { - - if ( jQuery.Deferred.exceptionHook ) { - jQuery.Deferred.exceptionHook( e, - process.stackTrace ); - } - - // Support: Promises/A+ section 2.3.3.3.4.1 - // https://promisesaplus.com/#point-61 - // Ignore post-resolution exceptions - if ( depth + 1 >= maxDepth ) { - - // Only substitute handlers pass on context - // and multiple values (non-spec behavior) - if ( handler !== Thrower ) { - that = undefined; - args = [ e ]; - } - - deferred.rejectWith( that, args ); - } - } - }; - - // Support: Promises/A+ section 2.3.3.3.1 - // https://promisesaplus.com/#point-57 - // Re-resolve promises immediately to dodge false rejection from - // subsequent errors - if ( depth ) { - process(); - } else { - - // Call an optional hook to record the stack, in case of exception - // since it's otherwise lost when execution goes async - if ( jQuery.Deferred.getStackHook ) { - process.stackTrace = jQuery.Deferred.getStackHook(); - } - window.setTimeout( process ); - } - }; - } - - return jQuery.Deferred( function( newDefer ) { - - // progress_handlers.add( ... ) - tuples[ 0 ][ 3 ].add( - resolve( - 0, - newDefer, - isFunction( onProgress ) ? - onProgress : - Identity, - newDefer.notifyWith - ) - ); - - // fulfilled_handlers.add( ... ) - tuples[ 1 ][ 3 ].add( - resolve( - 0, - newDefer, - isFunction( onFulfilled ) ? - onFulfilled : - Identity - ) - ); - - // rejected_handlers.add( ... ) - tuples[ 2 ][ 3 ].add( - resolve( - 0, - newDefer, - isFunction( onRejected ) ? - onRejected : - Thrower - ) - ); - } ).promise(); - }, - - // Get a promise for this deferred - // If obj is provided, the promise aspect is added to the object - promise: function( obj ) { - return obj != null ? jQuery.extend( obj, promise ) : promise; - } - }, - deferred = {}; - - // Add list-specific methods - jQuery.each( tuples, function( i, tuple ) { - var list = tuple[ 2 ], - stateString = tuple[ 5 ]; - - // promise.progress = list.add - // promise.done = list.add - // promise.fail = list.add - promise[ tuple[ 1 ] ] = list.add; - - // Handle state - if ( stateString ) { - list.add( - function() { - - // state = "resolved" (i.e., fulfilled) - // state = "rejected" - state = stateString; - }, - - // rejected_callbacks.disable - // fulfilled_callbacks.disable - tuples[ 3 - i ][ 2 ].disable, - - // rejected_handlers.disable - // fulfilled_handlers.disable - tuples[ 3 - i ][ 3 ].disable, - - // progress_callbacks.lock - tuples[ 0 ][ 2 ].lock, - - // progress_handlers.lock - tuples[ 0 ][ 3 ].lock - ); - } - - // progress_handlers.fire - // fulfilled_handlers.fire - // rejected_handlers.fire - list.add( tuple[ 3 ].fire ); - - // deferred.notify = function() { deferred.notifyWith(...) } - // deferred.resolve = function() { deferred.resolveWith(...) } - // deferred.reject = function() { deferred.rejectWith(...) } - deferred[ tuple[ 0 ] ] = function() { - deferred[ tuple[ 0 ] + "With" ]( this === deferred ? undefined : this, arguments ); - return this; - }; - - // deferred.notifyWith = list.fireWith - // deferred.resolveWith = list.fireWith - // deferred.rejectWith = list.fireWith - deferred[ tuple[ 0 ] + "With" ] = list.fireWith; - } ); - - // Make the deferred a promise - promise.promise( deferred ); - - // Call given func if any - if ( func ) { - func.call( deferred, deferred ); - } - - // All done! - return deferred; - }, - - // Deferred helper - when: function( singleValue ) { - var - - // count of uncompleted subordinates - remaining = arguments.length, - - // count of unprocessed arguments - i = remaining, - - // subordinate fulfillment data - resolveContexts = Array( i ), - resolveValues = slice.call( arguments ), - - // the primary Deferred - primary = jQuery.Deferred(), - - // subordinate callback factory - updateFunc = function( i ) { - return function( value ) { - resolveContexts[ i ] = this; - resolveValues[ i ] = arguments.length > 1 ? slice.call( arguments ) : value; - if ( !( --remaining ) ) { - primary.resolveWith( resolveContexts, resolveValues ); - } - }; - }; - - // Single- and empty arguments are adopted like Promise.resolve - if ( remaining <= 1 ) { - adoptValue( singleValue, primary.done( updateFunc( i ) ).resolve, primary.reject, - !remaining ); - - // Use .then() to unwrap secondary thenables (cf. gh-3000) - if ( primary.state() === "pending" || - isFunction( resolveValues[ i ] && resolveValues[ i ].then ) ) { - - return primary.then(); - } - } - - // Multiple arguments are aggregated like Promise.all array elements - while ( i-- ) { - adoptValue( resolveValues[ i ], updateFunc( i ), primary.reject ); - } - - return primary.promise(); - } - } ); - - - // These usually indicate a programmer mistake during development, - // warn about them ASAP rather than swallowing them by default. - var rerrorNames = /^(Eval|Internal|Range|Reference|Syntax|Type|URI)Error$/; - - jQuery.Deferred.exceptionHook = function( error, stack ) { - - // Support: IE 8 - 9 only - // Console exists when dev tools are open, which can happen at any time - if ( window.console && window.console.warn && error && rerrorNames.test( error.name ) ) { - window.console.warn( "jQuery.Deferred exception: " + error.message, error.stack, stack ); - } - }; - - - - - jQuery.readyException = function( error ) { - window.setTimeout( function() { - throw error; - } ); - }; - - - - - // The deferred used on DOM ready - var readyList = jQuery.Deferred(); - - jQuery.fn.ready = function( fn ) { - - readyList - .then( fn ) - - // Wrap jQuery.readyException in a function so that the lookup - // happens at the time of error handling instead of callback - // registration. - .catch( function( error ) { - jQuery.readyException( error ); - } ); - - return this; - }; - - jQuery.extend( { - - // Is the DOM ready to be used? Set to true once it occurs. - isReady: false, - - // A counter to track how many items to wait for before - // the ready event fires. See #6781 - readyWait: 1, - - // Handle when the DOM is ready - ready: function( wait ) { - - // Abort if there are pending holds or we're already ready - if ( wait === true ? --jQuery.readyWait : jQuery.isReady ) { - return; - } - - // Remember that the DOM is ready - jQuery.isReady = true; - - // If a normal DOM Ready event fired, decrement, and wait if need be - if ( wait !== true && --jQuery.readyWait > 0 ) { - return; - } - - // If there are functions bound, to execute - readyList.resolveWith( document, [ jQuery ] ); - } - } ); - - jQuery.ready.then = readyList.then; - - // The ready event handler and self cleanup method - function completed() { - document.removeEventListener( "DOMContentLoaded", completed ); - window.removeEventListener( "load", completed ); - jQuery.ready(); - } - - // Catch cases where $(document).ready() is called - // after the browser event has already occurred. - // Support: IE <=9 - 10 only - // Older IE sometimes signals "interactive" too soon - if ( document.readyState === "complete" || - ( document.readyState !== "loading" && !document.documentElement.doScroll ) ) { - - // Handle it asynchronously to allow scripts the opportunity to delay ready - window.setTimeout( jQuery.ready ); - - } else { - - // Use the handy event callback - document.addEventListener( "DOMContentLoaded", completed ); - - // A fallback to window.onload, that will always work - window.addEventListener( "load", completed ); - } - - - - - // Multifunctional method to get and set values of a collection - // The value/s can optionally be executed if it's a function - var access = function( elems, fn, key, value, chainable, emptyGet, raw ) { - var i = 0, - len = elems.length, - bulk = key == null; - - // Sets many values - if ( toType( key ) === "object" ) { - chainable = true; - for ( i in key ) { - access( elems, fn, i, key[ i ], true, emptyGet, raw ); - } - - // Sets one value - } else if ( value !== undefined ) { - chainable = true; - - if ( !isFunction( value ) ) { - raw = true; - } - - if ( bulk ) { - - // Bulk operations run against the entire set - if ( raw ) { - fn.call( elems, value ); - fn = null; - - // ...except when executing function values - } else { - bulk = fn; - fn = function( elem, _key, value ) { - return bulk.call( jQuery( elem ), value ); - }; - } - } - - if ( fn ) { - for ( ; i < len; i++ ) { - fn( - elems[ i ], key, raw ? - value : - value.call( elems[ i ], i, fn( elems[ i ], key ) ) - ); - } - } - } - - if ( chainable ) { - return elems; - } - - // Gets - if ( bulk ) { - return fn.call( elems ); - } - - return len ? fn( elems[ 0 ], key ) : emptyGet; - }; - - - // Matches dashed string for camelizing - var rmsPrefix = /^-ms-/, - rdashAlpha = /-([a-z])/g; - - // Used by camelCase as callback to replace() - function fcamelCase( _all, letter ) { - return letter.toUpperCase(); - } - - // Convert dashed to camelCase; used by the css and data modules - // Support: IE <=9 - 11, Edge 12 - 15 - // Microsoft forgot to hump their vendor prefix (#9572) - function camelCase( string ) { - return string.replace( rmsPrefix, "ms-" ).replace( rdashAlpha, fcamelCase ); - } - var acceptData = function( owner ) { - - // Accepts only: - // - Node - // - Node.ELEMENT_NODE - // - Node.DOCUMENT_NODE - // - Object - // - Any - return owner.nodeType === 1 || owner.nodeType === 9 || !( +owner.nodeType ); - }; - - - - - function Data() { - this.expando = jQuery.expando + Data.uid++; - } - - Data.uid = 1; - - Data.prototype = { - - cache: function( owner ) { - - // Check if the owner object already has a cache - var value = owner[ this.expando ]; - - // If not, create one - if ( !value ) { - value = {}; - - // We can accept data for non-element nodes in modern browsers, - // but we should not, see #8335. - // Always return an empty object. - if ( acceptData( owner ) ) { - - // If it is a node unlikely to be stringify-ed or looped over - // use plain assignment - if ( owner.nodeType ) { - owner[ this.expando ] = value; - - // Otherwise secure it in a non-enumerable property - // configurable must be true to allow the property to be - // deleted when data is removed - } else { - Object.defineProperty( owner, this.expando, { - value: value, - configurable: true - } ); - } - } - } - - return value; - }, - set: function( owner, data, value ) { - var prop, - cache = this.cache( owner ); - - // Handle: [ owner, key, value ] args - // Always use camelCase key (gh-2257) - if ( typeof data === "string" ) { - cache[ camelCase( data ) ] = value; - - // Handle: [ owner, { properties } ] args - } else { - - // Copy the properties one-by-one to the cache object - for ( prop in data ) { - cache[ camelCase( prop ) ] = data[ prop ]; - } - } - return cache; - }, - get: function( owner, key ) { - return key === undefined ? - this.cache( owner ) : - - // Always use camelCase key (gh-2257) - owner[ this.expando ] && owner[ this.expando ][ camelCase( key ) ]; - }, - access: function( owner, key, value ) { - - // In cases where either: - // - // 1. No key was specified - // 2. A string key was specified, but no value provided - // - // Take the "read" path and allow the get method to determine - // which value to return, respectively either: - // - // 1. The entire cache object - // 2. The data stored at the key - // - if ( key === undefined || - ( ( key && typeof key === "string" ) && value === undefined ) ) { - - return this.get( owner, key ); - } - - // When the key is not a string, or both a key and value - // are specified, set or extend (existing objects) with either: - // - // 1. An object of properties - // 2. A key and value - // - this.set( owner, key, value ); - - // Since the "set" path can have two possible entry points - // return the expected data based on which path was taken[*] - return value !== undefined ? value : key; - }, - remove: function( owner, key ) { - var i, - cache = owner[ this.expando ]; - - if ( cache === undefined ) { - return; - } - - if ( key !== undefined ) { - - // Support array or space separated string of keys - if ( Array.isArray( key ) ) { - - // If key is an array of keys... - // We always set camelCase keys, so remove that. - key = key.map( camelCase ); - } else { - key = camelCase( key ); - - // If a key with the spaces exists, use it. - // Otherwise, create an array by matching non-whitespace - key = key in cache ? - [ key ] : - ( key.match( rnothtmlwhite ) || [] ); - } - - i = key.length; - - while ( i-- ) { - delete cache[ key[ i ] ]; - } - } - - // Remove the expando if there's no more data - if ( key === undefined || jQuery.isEmptyObject( cache ) ) { - - // Support: Chrome <=35 - 45 - // Webkit & Blink performance suffers when deleting properties - // from DOM nodes, so set to undefined instead - // https://bugs.chromium.org/p/chromium/issues/detail?id=378607 (bug restricted) - if ( owner.nodeType ) { - owner[ this.expando ] = undefined; - } else { - delete owner[ this.expando ]; - } - } - }, - hasData: function( owner ) { - var cache = owner[ this.expando ]; - return cache !== undefined && !jQuery.isEmptyObject( cache ); - } - }; - var dataPriv = new Data(); - - var dataUser = new Data(); - - - - // Implementation Summary - // - // 1. Enforce API surface and semantic compatibility with 1.9.x branch - // 2. Improve the module's maintainability by reducing the storage - // paths to a single mechanism. - // 3. Use the same single mechanism to support "private" and "user" data. - // 4. _Never_ expose "private" data to user code (TODO: Drop _data, _removeData) - // 5. Avoid exposing implementation details on user objects (eg. expando properties) - // 6. Provide a clear path for implementation upgrade to WeakMap in 2014 - - var rbrace = /^(?:\{[\w\W]*\}|\[[\w\W]*\])$/, - rmultiDash = /[A-Z]/g; - - function getData( data ) { - if ( data === "true" ) { - return true; - } - - if ( data === "false" ) { - return false; - } - - if ( data === "null" ) { - return null; - } - - // Only convert to a number if it doesn't change the string - if ( data === +data + "" ) { - return +data; - } - - if ( rbrace.test( data ) ) { - return JSON.parse( data ); - } - - return data; - } - - function dataAttr( elem, key, data ) { - var name; - - // If nothing was found internally, try to fetch any - // data from the HTML5 data-* attribute - if ( data === undefined && elem.nodeType === 1 ) { - name = "data-" + key.replace( rmultiDash, "-$&" ).toLowerCase(); - data = elem.getAttribute( name ); - - if ( typeof data === "string" ) { - try { - data = getData( data ); - } catch ( e ) {} - - // Make sure we set the data so it isn't changed later - dataUser.set( elem, key, data ); - } else { - data = undefined; - } - } - return data; - } - - jQuery.extend( { - hasData: function( elem ) { - return dataUser.hasData( elem ) || dataPriv.hasData( elem ); - }, - - data: function( elem, name, data ) { - return dataUser.access( elem, name, data ); - }, - - removeData: function( elem, name ) { - dataUser.remove( elem, name ); - }, - - // TODO: Now that all calls to _data and _removeData have been replaced - // with direct calls to dataPriv methods, these can be deprecated. - _data: function( elem, name, data ) { - return dataPriv.access( elem, name, data ); - }, - - _removeData: function( elem, name ) { - dataPriv.remove( elem, name ); - } - } ); - - jQuery.fn.extend( { - data: function( key, value ) { - var i, name, data, - elem = this[ 0 ], - attrs = elem && elem.attributes; - - // Gets all values - if ( key === undefined ) { - if ( this.length ) { - data = dataUser.get( elem ); - - if ( elem.nodeType === 1 && !dataPriv.get( elem, "hasDataAttrs" ) ) { - i = attrs.length; - while ( i-- ) { - - // Support: IE 11 only - // The attrs elements can be null (#14894) - if ( attrs[ i ] ) { - name = attrs[ i ].name; - if ( name.indexOf( "data-" ) === 0 ) { - name = camelCase( name.slice( 5 ) ); - dataAttr( elem, name, data[ name ] ); - } - } - } - dataPriv.set( elem, "hasDataAttrs", true ); - } - } - - return data; - } - - // Sets multiple values - if ( typeof key === "object" ) { - return this.each( function() { - dataUser.set( this, key ); - } ); - } - - return access( this, function( value ) { - var data; - - // The calling jQuery object (element matches) is not empty - // (and therefore has an element appears at this[ 0 ]) and the - // `value` parameter was not undefined. An empty jQuery object - // will result in `undefined` for elem = this[ 0 ] which will - // throw an exception if an attempt to read a data cache is made. - if ( elem && value === undefined ) { - - // Attempt to get data from the cache - // The key will always be camelCased in Data - data = dataUser.get( elem, key ); - if ( data !== undefined ) { - return data; - } - - // Attempt to "discover" the data in - // HTML5 custom data-* attrs - data = dataAttr( elem, key ); - if ( data !== undefined ) { - return data; - } - - // We tried really hard, but the data doesn't exist. - return; - } - - // Set the data... - this.each( function() { - - // We always store the camelCased key - dataUser.set( this, key, value ); - } ); - }, null, value, arguments.length > 1, null, true ); - }, - - removeData: function( key ) { - return this.each( function() { - dataUser.remove( this, key ); - } ); - } - } ); - - - jQuery.extend( { - queue: function( elem, type, data ) { - var queue; - - if ( elem ) { - type = ( type || "fx" ) + "queue"; - queue = dataPriv.get( elem, type ); - - // Speed up dequeue by getting out quickly if this is just a lookup - if ( data ) { - if ( !queue || Array.isArray( data ) ) { - queue = dataPriv.access( elem, type, jQuery.makeArray( data ) ); - } else { - queue.push( data ); - } - } - return queue || []; - } - }, - - dequeue: function( elem, type ) { - type = type || "fx"; - - var queue = jQuery.queue( elem, type ), - startLength = queue.length, - fn = queue.shift(), - hooks = jQuery._queueHooks( elem, type ), - next = function() { - jQuery.dequeue( elem, type ); - }; - - // If the fx queue is dequeued, always remove the progress sentinel - if ( fn === "inprogress" ) { - fn = queue.shift(); - startLength--; - } - - if ( fn ) { - - // Add a progress sentinel to prevent the fx queue from being - // automatically dequeued - if ( type === "fx" ) { - queue.unshift( "inprogress" ); - } - - // Clear up the last queue stop function - delete hooks.stop; - fn.call( elem, next, hooks ); - } - - if ( !startLength && hooks ) { - hooks.empty.fire(); - } - }, - - // Not public - generate a queueHooks object, or return the current one - _queueHooks: function( elem, type ) { - var key = type + "queueHooks"; - return dataPriv.get( elem, key ) || dataPriv.access( elem, key, { - empty: jQuery.Callbacks( "once memory" ).add( function() { - dataPriv.remove( elem, [ type + "queue", key ] ); - } ) - } ); - } - } ); - - jQuery.fn.extend( { - queue: function( type, data ) { - var setter = 2; - - if ( typeof type !== "string" ) { - data = type; - type = "fx"; - setter--; - } - - if ( arguments.length < setter ) { - return jQuery.queue( this[ 0 ], type ); - } - - return data === undefined ? - this : - this.each( function() { - var queue = jQuery.queue( this, type, data ); - - // Ensure a hooks for this queue - jQuery._queueHooks( this, type ); - - if ( type === "fx" && queue[ 0 ] !== "inprogress" ) { - jQuery.dequeue( this, type ); - } - } ); - }, - dequeue: function( type ) { - return this.each( function() { - jQuery.dequeue( this, type ); - } ); - }, - clearQueue: function( type ) { - return this.queue( type || "fx", [] ); - }, - - // Get a promise resolved when queues of a certain type - // are emptied (fx is the type by default) - promise: function( type, obj ) { - var tmp, - count = 1, - defer = jQuery.Deferred(), - elements = this, - i = this.length, - resolve = function() { - if ( !( --count ) ) { - defer.resolveWith( elements, [ elements ] ); - } - }; - - if ( typeof type !== "string" ) { - obj = type; - type = undefined; - } - type = type || "fx"; - - while ( i-- ) { - tmp = dataPriv.get( elements[ i ], type + "queueHooks" ); - if ( tmp && tmp.empty ) { - count++; - tmp.empty.add( resolve ); - } - } - resolve(); - return defer.promise( obj ); - } - } ); - var pnum = ( /[+-]?(?:\d*\.|)\d+(?:[eE][+-]?\d+|)/ ).source; - - var rcssNum = new RegExp( "^(?:([+-])=|)(" + pnum + ")([a-z%]*)$", "i" ); - - - var cssExpand = [ "Top", "Right", "Bottom", "Left" ]; - - var documentElement = document.documentElement; - - - - var isAttached = function( elem ) { - return jQuery.contains( elem.ownerDocument, elem ); - }, - composed = { composed: true }; - - // Support: IE 9 - 11+, Edge 12 - 18+, iOS 10.0 - 10.2 only - // Check attachment across shadow DOM boundaries when possible (gh-3504) - // Support: iOS 10.0-10.2 only - // Early iOS 10 versions support `attachShadow` but not `getRootNode`, - // leading to errors. We need to check for `getRootNode`. - if ( documentElement.getRootNode ) { - isAttached = function( elem ) { - return jQuery.contains( elem.ownerDocument, elem ) || - elem.getRootNode( composed ) === elem.ownerDocument; - }; - } - var isHiddenWithinTree = function( elem, el ) { - - // isHiddenWithinTree might be called from jQuery#filter function; - // in that case, element will be second argument - elem = el || elem; - - // Inline style trumps all - return elem.style.display === "none" || - elem.style.display === "" && - - // Otherwise, check computed style - // Support: Firefox <=43 - 45 - // Disconnected elements can have computed display: none, so first confirm that elem is - // in the document. - isAttached( elem ) && - - jQuery.css( elem, "display" ) === "none"; - }; - - - - function adjustCSS( elem, prop, valueParts, tween ) { - var adjusted, scale, - maxIterations = 20, - currentValue = tween ? - function() { - return tween.cur(); - } : - function() { - return jQuery.css( elem, prop, "" ); - }, - initial = currentValue(), - unit = valueParts && valueParts[ 3 ] || ( jQuery.cssNumber[ prop ] ? "" : "px" ), - - // Starting value computation is required for potential unit mismatches - initialInUnit = elem.nodeType && - ( jQuery.cssNumber[ prop ] || unit !== "px" && +initial ) && - rcssNum.exec( jQuery.css( elem, prop ) ); - - if ( initialInUnit && initialInUnit[ 3 ] !== unit ) { - - // Support: Firefox <=54 - // Halve the iteration target value to prevent interference from CSS upper bounds (gh-2144) - initial = initial / 2; - - // Trust units reported by jQuery.css - unit = unit || initialInUnit[ 3 ]; - - // Iteratively approximate from a nonzero starting point - initialInUnit = +initial || 1; - - while ( maxIterations-- ) { - - // Evaluate and update our best guess (doubling guesses that zero out). - // Finish if the scale equals or crosses 1 (making the old*new product non-positive). - jQuery.style( elem, prop, initialInUnit + unit ); - if ( ( 1 - scale ) * ( 1 - ( scale = currentValue() / initial || 0.5 ) ) <= 0 ) { - maxIterations = 0; - } - initialInUnit = initialInUnit / scale; - - } - - initialInUnit = initialInUnit * 2; - jQuery.style( elem, prop, initialInUnit + unit ); - - // Make sure we update the tween properties later on - valueParts = valueParts || []; - } - - if ( valueParts ) { - initialInUnit = +initialInUnit || +initial || 0; - - // Apply relative offset (+=/-=) if specified - adjusted = valueParts[ 1 ] ? - initialInUnit + ( valueParts[ 1 ] + 1 ) * valueParts[ 2 ] : - +valueParts[ 2 ]; - if ( tween ) { - tween.unit = unit; - tween.start = initialInUnit; - tween.end = adjusted; - } - } - return adjusted; - } - - - var defaultDisplayMap = {}; - - function getDefaultDisplay( elem ) { - var temp, - doc = elem.ownerDocument, - nodeName = elem.nodeName, - display = defaultDisplayMap[ nodeName ]; - - if ( display ) { - return display; - } - - temp = doc.body.appendChild( doc.createElement( nodeName ) ); - display = jQuery.css( temp, "display" ); - - temp.parentNode.removeChild( temp ); - - if ( display === "none" ) { - display = "block"; - } - defaultDisplayMap[ nodeName ] = display; - - return display; - } - - function showHide( elements, show ) { - var display, elem, - values = [], - index = 0, - length = elements.length; - - // Determine new display value for elements that need to change - for ( ; index < length; index++ ) { - elem = elements[ index ]; - if ( !elem.style ) { - continue; - } - - display = elem.style.display; - if ( show ) { - - // Since we force visibility upon cascade-hidden elements, an immediate (and slow) - // check is required in this first loop unless we have a nonempty display value (either - // inline or about-to-be-restored) - if ( display === "none" ) { - values[ index ] = dataPriv.get( elem, "display" ) || null; - if ( !values[ index ] ) { - elem.style.display = ""; - } - } - if ( elem.style.display === "" && isHiddenWithinTree( elem ) ) { - values[ index ] = getDefaultDisplay( elem ); - } - } else { - if ( display !== "none" ) { - values[ index ] = "none"; - - // Remember what we're overwriting - dataPriv.set( elem, "display", display ); - } - } - } - - // Set the display of the elements in a second loop to avoid constant reflow - for ( index = 0; index < length; index++ ) { - if ( values[ index ] != null ) { - elements[ index ].style.display = values[ index ]; - } - } - - return elements; - } - - jQuery.fn.extend( { - show: function() { - return showHide( this, true ); - }, - hide: function() { - return showHide( this ); - }, - toggle: function( state ) { - if ( typeof state === "boolean" ) { - return state ? this.show() : this.hide(); - } - - return this.each( function() { - if ( isHiddenWithinTree( this ) ) { - jQuery( this ).show(); - } else { - jQuery( this ).hide(); - } - } ); - } - } ); - var rcheckableType = ( /^(?:checkbox|radio)$/i ); - - var rtagName = ( /<([a-z][^\/\0>\x20\t\r\n\f]*)/i ); - - var rscriptType = ( /^$|^module$|\/(?:java|ecma)script/i ); - - - - ( function() { - var fragment = document.createDocumentFragment(), - div = fragment.appendChild( document.createElement( "div" ) ), - input = document.createElement( "input" ); - - // Support: Android 4.0 - 4.3 only - // Check state lost if the name is set (#11217) - // Support: Windows Web Apps (WWA) - // `name` and `type` must use .setAttribute for WWA (#14901) - input.setAttribute( "type", "radio" ); - input.setAttribute( "checked", "checked" ); - input.setAttribute( "name", "t" ); - - div.appendChild( input ); - - // Support: Android <=4.1 only - // Older WebKit doesn't clone checked state correctly in fragments - support.checkClone = div.cloneNode( true ).cloneNode( true ).lastChild.checked; - - // Support: IE <=11 only - // Make sure textarea (and checkbox) defaultValue is properly cloned - div.innerHTML = "<textarea>x</textarea>"; - support.noCloneChecked = !!div.cloneNode( true ).lastChild.defaultValue; - - // Support: IE <=9 only - // IE <=9 replaces <option> tags with their contents when inserted outside of - // the select element. - div.innerHTML = "<option></option>"; - support.option = !!div.lastChild; - } )(); - - - // We have to close these tags to support XHTML (#13200) - var wrapMap = { - - // XHTML parsers do not magically insert elements in the - // same way that tag soup parsers do. So we cannot shorten - // this by omitting <tbody> or other required elements. - thead: [ 1, "<table>", "</table>" ], - col: [ 2, "<table><colgroup>", "</colgroup></table>" ], - tr: [ 2, "<table><tbody>", "</tbody></table>" ], - td: [ 3, "<table><tbody><tr>", "</tr></tbody></table>" ], - - _default: [ 0, "", "" ] - }; - - wrapMap.tbody = wrapMap.tfoot = wrapMap.colgroup = wrapMap.caption = wrapMap.thead; - wrapMap.th = wrapMap.td; - - // Support: IE <=9 only - if ( !support.option ) { - wrapMap.optgroup = wrapMap.option = [ 1, "<select multiple='multiple'>", "</select>" ]; - } - - - function getAll( context, tag ) { - - // Support: IE <=9 - 11 only - // Use typeof to avoid zero-argument method invocation on host objects (#15151) - var ret; - - if ( typeof context.getElementsByTagName !== "undefined" ) { - ret = context.getElementsByTagName( tag || "*" ); - - } else if ( typeof context.querySelectorAll !== "undefined" ) { - ret = context.querySelectorAll( tag || "*" ); - - } else { - ret = []; - } - - if ( tag === undefined || tag && nodeName( context, tag ) ) { - return jQuery.merge( [ context ], ret ); - } - - return ret; - } - - - // Mark scripts as having already been evaluated - function setGlobalEval( elems, refElements ) { - var i = 0, - l = elems.length; - - for ( ; i < l; i++ ) { - dataPriv.set( - elems[ i ], - "globalEval", - !refElements || dataPriv.get( refElements[ i ], "globalEval" ) - ); - } - } - - - var rhtml = /<|&#?\w+;/; - - function buildFragment( elems, context, scripts, selection, ignored ) { - var elem, tmp, tag, wrap, attached, j, - fragment = context.createDocumentFragment(), - nodes = [], - i = 0, - l = elems.length; - - for ( ; i < l; i++ ) { - elem = elems[ i ]; - - if ( elem || elem === 0 ) { - - // Add nodes directly - if ( toType( elem ) === "object" ) { - - // Support: Android <=4.0 only, PhantomJS 1 only - // push.apply(_, arraylike) throws on ancient WebKit - jQuery.merge( nodes, elem.nodeType ? [ elem ] : elem ); - - // Convert non-html into a text node - } else if ( !rhtml.test( elem ) ) { - nodes.push( context.createTextNode( elem ) ); - - // Convert html into DOM nodes - } else { - tmp = tmp || fragment.appendChild( context.createElement( "div" ) ); - - // Deserialize a standard representation - tag = ( rtagName.exec( elem ) || [ "", "" ] )[ 1 ].toLowerCase(); - wrap = wrapMap[ tag ] || wrapMap._default; - tmp.innerHTML = wrap[ 1 ] + jQuery.htmlPrefilter( elem ) + wrap[ 2 ]; - - // Descend through wrappers to the right content - j = wrap[ 0 ]; - while ( j-- ) { - tmp = tmp.lastChild; - } - - // Support: Android <=4.0 only, PhantomJS 1 only - // push.apply(_, arraylike) throws on ancient WebKit - jQuery.merge( nodes, tmp.childNodes ); - - // Remember the top-level container - tmp = fragment.firstChild; - - // Ensure the created nodes are orphaned (#12392) - tmp.textContent = ""; - } - } - } - - // Remove wrapper from fragment - fragment.textContent = ""; - - i = 0; - while ( ( elem = nodes[ i++ ] ) ) { - - // Skip elements already in the context collection (trac-4087) - if ( selection && jQuery.inArray( elem, selection ) > -1 ) { - if ( ignored ) { - ignored.push( elem ); - } - continue; - } - - attached = isAttached( elem ); - - // Append to fragment - tmp = getAll( fragment.appendChild( elem ), "script" ); - - // Preserve script evaluation history - if ( attached ) { - setGlobalEval( tmp ); - } - - // Capture executables - if ( scripts ) { - j = 0; - while ( ( elem = tmp[ j++ ] ) ) { - if ( rscriptType.test( elem.type || "" ) ) { - scripts.push( elem ); - } - } - } - } - - return fragment; - } - - - var rtypenamespace = /^([^.]*)(?:\.(.+)|)/; - - function returnTrue() { - return true; - } - - function returnFalse() { - return false; - } - - // Support: IE <=9 - 11+ - // focus() and blur() are asynchronous, except when they are no-op. - // So expect focus to be synchronous when the element is already active, - // and blur to be synchronous when the element is not already active. - // (focus and blur are always synchronous in other supported browsers, - // this just defines when we can count on it). - function expectSync( elem, type ) { - return ( elem === safeActiveElement() ) === ( type === "focus" ); - } - - // Support: IE <=9 only - // Accessing document.activeElement can throw unexpectedly - // https://bugs.jquery.com/ticket/13393 - function safeActiveElement() { - try { - return document.activeElement; - } catch ( err ) { } - } - - function on( elem, types, selector, data, fn, one ) { - var origFn, type; - - // Types can be a map of types/handlers - if ( typeof types === "object" ) { - - // ( types-Object, selector, data ) - if ( typeof selector !== "string" ) { - - // ( types-Object, data ) - data = data || selector; - selector = undefined; - } - for ( type in types ) { - on( elem, type, selector, data, types[ type ], one ); - } - return elem; - } - - if ( data == null && fn == null ) { - - // ( types, fn ) - fn = selector; - data = selector = undefined; - } else if ( fn == null ) { - if ( typeof selector === "string" ) { - - // ( types, selector, fn ) - fn = data; - data = undefined; - } else { - - // ( types, data, fn ) - fn = data; - data = selector; - selector = undefined; - } - } - if ( fn === false ) { - fn = returnFalse; - } else if ( !fn ) { - return elem; - } - - if ( one === 1 ) { - origFn = fn; - fn = function( event ) { - - // Can use an empty set, since event contains the info - jQuery().off( event ); - return origFn.apply( this, arguments ); - }; - - // Use same guid so caller can remove using origFn - fn.guid = origFn.guid || ( origFn.guid = jQuery.guid++ ); - } - return elem.each( function() { - jQuery.event.add( this, types, fn, data, selector ); - } ); - } - - /* - * Helper functions for managing events -- not part of the public interface. - * Props to Dean Edwards' addEvent library for many of the ideas. - */ - jQuery.event = { - - global: {}, - - add: function( elem, types, handler, data, selector ) { - - var handleObjIn, eventHandle, tmp, - events, t, handleObj, - special, handlers, type, namespaces, origType, - elemData = dataPriv.get( elem ); - - // Only attach events to objects that accept data - if ( !acceptData( elem ) ) { - return; - } - - // Caller can pass in an object of custom data in lieu of the handler - if ( handler.handler ) { - handleObjIn = handler; - handler = handleObjIn.handler; - selector = handleObjIn.selector; - } - - // Ensure that invalid selectors throw exceptions at attach time - // Evaluate against documentElement in case elem is a non-element node (e.g., document) - if ( selector ) { - jQuery.find.matchesSelector( documentElement, selector ); - } - - // Make sure that the handler has a unique ID, used to find/remove it later - if ( !handler.guid ) { - handler.guid = jQuery.guid++; - } - - // Init the element's event structure and main handler, if this is the first - if ( !( events = elemData.events ) ) { - events = elemData.events = Object.create( null ); - } - if ( !( eventHandle = elemData.handle ) ) { - eventHandle = elemData.handle = function( e ) { - - // Discard the second event of a jQuery.event.trigger() and - // when an event is called after a page has unloaded - return typeof jQuery !== "undefined" && jQuery.event.triggered !== e.type ? - jQuery.event.dispatch.apply( elem, arguments ) : undefined; - }; - } - - // Handle multiple events separated by a space - types = ( types || "" ).match( rnothtmlwhite ) || [ "" ]; - t = types.length; - while ( t-- ) { - tmp = rtypenamespace.exec( types[ t ] ) || []; - type = origType = tmp[ 1 ]; - namespaces = ( tmp[ 2 ] || "" ).split( "." ).sort(); - - // There *must* be a type, no attaching namespace-only handlers - if ( !type ) { - continue; - } - - // If event changes its type, use the special event handlers for the changed type - special = jQuery.event.special[ type ] || {}; - - // If selector defined, determine special event api type, otherwise given type - type = ( selector ? special.delegateType : special.bindType ) || type; - - // Update special based on newly reset type - special = jQuery.event.special[ type ] || {}; - - // handleObj is passed to all event handlers - handleObj = jQuery.extend( { - type: type, - origType: origType, - data: data, - handler: handler, - guid: handler.guid, - selector: selector, - needsContext: selector && jQuery.expr.match.needsContext.test( selector ), - namespace: namespaces.join( "." ) - }, handleObjIn ); - - // Init the event handler queue if we're the first - if ( !( handlers = events[ type ] ) ) { - handlers = events[ type ] = []; - handlers.delegateCount = 0; - - // Only use addEventListener if the special events handler returns false - if ( !special.setup || - special.setup.call( elem, data, namespaces, eventHandle ) === false ) { - - if ( elem.addEventListener ) { - elem.addEventListener( type, eventHandle ); - } - } - } - - if ( special.add ) { - special.add.call( elem, handleObj ); - - if ( !handleObj.handler.guid ) { - handleObj.handler.guid = handler.guid; - } - } - - // Add to the element's handler list, delegates in front - if ( selector ) { - handlers.splice( handlers.delegateCount++, 0, handleObj ); - } else { - handlers.push( handleObj ); - } - - // Keep track of which events have ever been used, for event optimization - jQuery.event.global[ type ] = true; - } - - }, - - // Detach an event or set of events from an element - remove: function( elem, types, handler, selector, mappedTypes ) { - - var j, origCount, tmp, - events, t, handleObj, - special, handlers, type, namespaces, origType, - elemData = dataPriv.hasData( elem ) && dataPriv.get( elem ); - - if ( !elemData || !( events = elemData.events ) ) { - return; - } - - // Once for each type.namespace in types; type may be omitted - types = ( types || "" ).match( rnothtmlwhite ) || [ "" ]; - t = types.length; - while ( t-- ) { - tmp = rtypenamespace.exec( types[ t ] ) || []; - type = origType = tmp[ 1 ]; - namespaces = ( tmp[ 2 ] || "" ).split( "." ).sort(); - - // Unbind all events (on this namespace, if provided) for the element - if ( !type ) { - for ( type in events ) { - jQuery.event.remove( elem, type + types[ t ], handler, selector, true ); - } - continue; - } - - special = jQuery.event.special[ type ] || {}; - type = ( selector ? special.delegateType : special.bindType ) || type; - handlers = events[ type ] || []; - tmp = tmp[ 2 ] && - new RegExp( "(^|\\.)" + namespaces.join( "\\.(?:.*\\.|)" ) + "(\\.|$)" ); - - // Remove matching events - origCount = j = handlers.length; - while ( j-- ) { - handleObj = handlers[ j ]; - - if ( ( mappedTypes || origType === handleObj.origType ) && - ( !handler || handler.guid === handleObj.guid ) && - ( !tmp || tmp.test( handleObj.namespace ) ) && - ( !selector || selector === handleObj.selector || - selector === "**" && handleObj.selector ) ) { - handlers.splice( j, 1 ); - - if ( handleObj.selector ) { - handlers.delegateCount--; - } - if ( special.remove ) { - special.remove.call( elem, handleObj ); - } - } - } - - // Remove generic event handler if we removed something and no more handlers exist - // (avoids potential for endless recursion during removal of special event handlers) - if ( origCount && !handlers.length ) { - if ( !special.teardown || - special.teardown.call( elem, namespaces, elemData.handle ) === false ) { - - jQuery.removeEvent( elem, type, elemData.handle ); - } - - delete events[ type ]; - } - } - - // Remove data and the expando if it's no longer used - if ( jQuery.isEmptyObject( events ) ) { - dataPriv.remove( elem, "handle events" ); - } - }, - - dispatch: function( nativeEvent ) { - - var i, j, ret, matched, handleObj, handlerQueue, - args = new Array( arguments.length ), - - // Make a writable jQuery.Event from the native event object - event = jQuery.event.fix( nativeEvent ), - - handlers = ( - dataPriv.get( this, "events" ) || Object.create( null ) - )[ event.type ] || [], - special = jQuery.event.special[ event.type ] || {}; - - // Use the fix-ed jQuery.Event rather than the (read-only) native event - args[ 0 ] = event; - - for ( i = 1; i < arguments.length; i++ ) { - args[ i ] = arguments[ i ]; - } - - event.delegateTarget = this; - - // Call the preDispatch hook for the mapped type, and let it bail if desired - if ( special.preDispatch && special.preDispatch.call( this, event ) === false ) { - return; - } - - // Determine handlers - handlerQueue = jQuery.event.handlers.call( this, event, handlers ); - - // Run delegates first; they may want to stop propagation beneath us - i = 0; - while ( ( matched = handlerQueue[ i++ ] ) && !event.isPropagationStopped() ) { - event.currentTarget = matched.elem; - - j = 0; - while ( ( handleObj = matched.handlers[ j++ ] ) && - !event.isImmediatePropagationStopped() ) { - - // If the event is namespaced, then each handler is only invoked if it is - // specially universal or its namespaces are a superset of the event's. - if ( !event.rnamespace || handleObj.namespace === false || - event.rnamespace.test( handleObj.namespace ) ) { - - event.handleObj = handleObj; - event.data = handleObj.data; - - ret = ( ( jQuery.event.special[ handleObj.origType ] || {} ).handle || - handleObj.handler ).apply( matched.elem, args ); - - if ( ret !== undefined ) { - if ( ( event.result = ret ) === false ) { - event.preventDefault(); - event.stopPropagation(); - } - } - } - } - } - - // Call the postDispatch hook for the mapped type - if ( special.postDispatch ) { - special.postDispatch.call( this, event ); - } - - return event.result; - }, - - handlers: function( event, handlers ) { - var i, handleObj, sel, matchedHandlers, matchedSelectors, - handlerQueue = [], - delegateCount = handlers.delegateCount, - cur = event.target; - - // Find delegate handlers - if ( delegateCount && - - // Support: IE <=9 - // Black-hole SVG <use> instance trees (trac-13180) - cur.nodeType && - - // Support: Firefox <=42 - // Suppress spec-violating clicks indicating a non-primary pointer button (trac-3861) - // https://www.w3.org/TR/DOM-Level-3-Events/#event-type-click - // Support: IE 11 only - // ...but not arrow key "clicks" of radio inputs, which can have `button` -1 (gh-2343) - !( event.type === "click" && event.button >= 1 ) ) { - - for ( ; cur !== this; cur = cur.parentNode || this ) { - - // Don't check non-elements (#13208) - // Don't process clicks on disabled elements (#6911, #8165, #11382, #11764) - if ( cur.nodeType === 1 && !( event.type === "click" && cur.disabled === true ) ) { - matchedHandlers = []; - matchedSelectors = {}; - for ( i = 0; i < delegateCount; i++ ) { - handleObj = handlers[ i ]; - - // Don't conflict with Object.prototype properties (#13203) - sel = handleObj.selector + " "; - - if ( matchedSelectors[ sel ] === undefined ) { - matchedSelectors[ sel ] = handleObj.needsContext ? - jQuery( sel, this ).index( cur ) > -1 : - jQuery.find( sel, this, null, [ cur ] ).length; - } - if ( matchedSelectors[ sel ] ) { - matchedHandlers.push( handleObj ); - } - } - if ( matchedHandlers.length ) { - handlerQueue.push( { elem: cur, handlers: matchedHandlers } ); - } - } - } - } - - // Add the remaining (directly-bound) handlers - cur = this; - if ( delegateCount < handlers.length ) { - handlerQueue.push( { elem: cur, handlers: handlers.slice( delegateCount ) } ); - } - - return handlerQueue; - }, - - addProp: function( name, hook ) { - Object.defineProperty( jQuery.Event.prototype, name, { - enumerable: true, - configurable: true, - - get: isFunction( hook ) ? - function() { - if ( this.originalEvent ) { - return hook( this.originalEvent ); - } - } : - function() { - if ( this.originalEvent ) { - return this.originalEvent[ name ]; - } - }, - - set: function( value ) { - Object.defineProperty( this, name, { - enumerable: true, - configurable: true, - writable: true, - value: value - } ); - } - } ); - }, - - fix: function( originalEvent ) { - return originalEvent[ jQuery.expando ] ? - originalEvent : - new jQuery.Event( originalEvent ); - }, - - special: { - load: { - - // Prevent triggered image.load events from bubbling to window.load - noBubble: true - }, - click: { - - // Utilize native event to ensure correct state for checkable inputs - setup: function( data ) { - - // For mutual compressibility with _default, replace `this` access with a local var. - // `|| data` is dead code meant only to preserve the variable through minification. - var el = this || data; - - // Claim the first handler - if ( rcheckableType.test( el.type ) && - el.click && nodeName( el, "input" ) ) { - - // dataPriv.set( el, "click", ... ) - leverageNative( el, "click", returnTrue ); - } - - // Return false to allow normal processing in the caller - return false; - }, - trigger: function( data ) { - - // For mutual compressibility with _default, replace `this` access with a local var. - // `|| data` is dead code meant only to preserve the variable through minification. - var el = this || data; - - // Force setup before triggering a click - if ( rcheckableType.test( el.type ) && - el.click && nodeName( el, "input" ) ) { - - leverageNative( el, "click" ); - } - - // Return non-false to allow normal event-path propagation - return true; - }, - - // For cross-browser consistency, suppress native .click() on links - // Also prevent it if we're currently inside a leveraged native-event stack - _default: function( event ) { - var target = event.target; - return rcheckableType.test( target.type ) && - target.click && nodeName( target, "input" ) && - dataPriv.get( target, "click" ) || - nodeName( target, "a" ); - } - }, - - beforeunload: { - postDispatch: function( event ) { - - // Support: Firefox 20+ - // Firefox doesn't alert if the returnValue field is not set. - if ( event.result !== undefined && event.originalEvent ) { - event.originalEvent.returnValue = event.result; - } - } - } - } - }; - - // Ensure the presence of an event listener that handles manually-triggered - // synthetic events by interrupting progress until reinvoked in response to - // *native* events that it fires directly, ensuring that state changes have - // already occurred before other listeners are invoked. - function leverageNative( el, type, expectSync ) { - - // Missing expectSync indicates a trigger call, which must force setup through jQuery.event.add - if ( !expectSync ) { - if ( dataPriv.get( el, type ) === undefined ) { - jQuery.event.add( el, type, returnTrue ); - } - return; - } - - // Register the controller as a special universal handler for all event namespaces - dataPriv.set( el, type, false ); - jQuery.event.add( el, type, { - namespace: false, - handler: function( event ) { - var notAsync, result, - saved = dataPriv.get( this, type ); - - if ( ( event.isTrigger & 1 ) && this[ type ] ) { - - // Interrupt processing of the outer synthetic .trigger()ed event - // Saved data should be false in such cases, but might be a leftover capture object - // from an async native handler (gh-4350) - if ( !saved.length ) { - - // Store arguments for use when handling the inner native event - // There will always be at least one argument (an event object), so this array - // will not be confused with a leftover capture object. - saved = slice.call( arguments ); - dataPriv.set( this, type, saved ); - - // Trigger the native event and capture its result - // Support: IE <=9 - 11+ - // focus() and blur() are asynchronous - notAsync = expectSync( this, type ); - this[ type ](); - result = dataPriv.get( this, type ); - if ( saved !== result || notAsync ) { - dataPriv.set( this, type, false ); - } else { - result = {}; - } - if ( saved !== result ) { - - // Cancel the outer synthetic event - event.stopImmediatePropagation(); - event.preventDefault(); - - // Support: Chrome 86+ - // In Chrome, if an element having a focusout handler is blurred by - // clicking outside of it, it invokes the handler synchronously. If - // that handler calls `.remove()` on the element, the data is cleared, - // leaving `result` undefined. We need to guard against this. - return result && result.value; - } - - // If this is an inner synthetic event for an event with a bubbling surrogate - // (focus or blur), assume that the surrogate already propagated from triggering the - // native event and prevent that from happening again here. - // This technically gets the ordering wrong w.r.t. to `.trigger()` (in which the - // bubbling surrogate propagates *after* the non-bubbling base), but that seems - // less bad than duplication. - } else if ( ( jQuery.event.special[ type ] || {} ).delegateType ) { - event.stopPropagation(); - } - - // If this is a native event triggered above, everything is now in order - // Fire an inner synthetic event with the original arguments - } else if ( saved.length ) { - - // ...and capture the result - dataPriv.set( this, type, { - value: jQuery.event.trigger( - - // Support: IE <=9 - 11+ - // Extend with the prototype to reset the above stopImmediatePropagation() - jQuery.extend( saved[ 0 ], jQuery.Event.prototype ), - saved.slice( 1 ), - this - ) - } ); - - // Abort handling of the native event - event.stopImmediatePropagation(); - } - } - } ); - } - - jQuery.removeEvent = function( elem, type, handle ) { - - // This "if" is needed for plain objects - if ( elem.removeEventListener ) { - elem.removeEventListener( type, handle ); - } - }; - - jQuery.Event = function( src, props ) { - - // Allow instantiation without the 'new' keyword - if ( !( this instanceof jQuery.Event ) ) { - return new jQuery.Event( src, props ); - } - - // Event object - if ( src && src.type ) { - this.originalEvent = src; - this.type = src.type; - - // Events bubbling up the document may have been marked as prevented - // by a handler lower down the tree; reflect the correct value. - this.isDefaultPrevented = src.defaultPrevented || - src.defaultPrevented === undefined && - - // Support: Android <=2.3 only - src.returnValue === false ? - returnTrue : - returnFalse; - - // Create target properties - // Support: Safari <=6 - 7 only - // Target should not be a text node (#504, #13143) - this.target = ( src.target && src.target.nodeType === 3 ) ? - src.target.parentNode : - src.target; - - this.currentTarget = src.currentTarget; - this.relatedTarget = src.relatedTarget; - - // Event type - } else { - this.type = src; - } - - // Put explicitly provided properties onto the event object - if ( props ) { - jQuery.extend( this, props ); - } - - // Create a timestamp if incoming event doesn't have one - this.timeStamp = src && src.timeStamp || Date.now(); - - // Mark it as fixed - this[ jQuery.expando ] = true; - }; - - // jQuery.Event is based on DOM3 Events as specified by the ECMAScript Language Binding - // https://www.w3.org/TR/2003/WD-DOM-Level-3-Events-20030331/ecma-script-binding.html - jQuery.Event.prototype = { - constructor: jQuery.Event, - isDefaultPrevented: returnFalse, - isPropagationStopped: returnFalse, - isImmediatePropagationStopped: returnFalse, - isSimulated: false, - - preventDefault: function() { - var e = this.originalEvent; - - this.isDefaultPrevented = returnTrue; - - if ( e && !this.isSimulated ) { - e.preventDefault(); - } - }, - stopPropagation: function() { - var e = this.originalEvent; - - this.isPropagationStopped = returnTrue; - - if ( e && !this.isSimulated ) { - e.stopPropagation(); - } - }, - stopImmediatePropagation: function() { - var e = this.originalEvent; - - this.isImmediatePropagationStopped = returnTrue; - - if ( e && !this.isSimulated ) { - e.stopImmediatePropagation(); - } - - this.stopPropagation(); - } - }; - - // Includes all common event props including KeyEvent and MouseEvent specific props - jQuery.each( { - altKey: true, - bubbles: true, - cancelable: true, - changedTouches: true, - ctrlKey: true, - detail: true, - eventPhase: true, - metaKey: true, - pageX: true, - pageY: true, - shiftKey: true, - view: true, - "char": true, - code: true, - charCode: true, - key: true, - keyCode: true, - button: true, - buttons: true, - clientX: true, - clientY: true, - offsetX: true, - offsetY: true, - pointerId: true, - pointerType: true, - screenX: true, - screenY: true, - targetTouches: true, - toElement: true, - touches: true, - which: true - }, jQuery.event.addProp ); - - jQuery.each( { focus: "focusin", blur: "focusout" }, function( type, delegateType ) { - jQuery.event.special[ type ] = { - - // Utilize native event if possible so blur/focus sequence is correct - setup: function() { - - // Claim the first handler - // dataPriv.set( this, "focus", ... ) - // dataPriv.set( this, "blur", ... ) - leverageNative( this, type, expectSync ); - - // Return false to allow normal processing in the caller - return false; - }, - trigger: function() { - - // Force setup before trigger - leverageNative( this, type ); - - // Return non-false to allow normal event-path propagation - return true; - }, - - // Suppress native focus or blur as it's already being fired - // in leverageNative. - _default: function() { - return true; - }, - - delegateType: delegateType - }; - } ); - - // Create mouseenter/leave events using mouseover/out and event-time checks - // so that event delegation works in jQuery. - // Do the same for pointerenter/pointerleave and pointerover/pointerout - // - // Support: Safari 7 only - // Safari sends mouseenter too often; see: - // https://bugs.chromium.org/p/chromium/issues/detail?id=470258 - // for the description of the bug (it existed in older Chrome versions as well). - jQuery.each( { - mouseenter: "mouseover", - mouseleave: "mouseout", - pointerenter: "pointerover", - pointerleave: "pointerout" - }, function( orig, fix ) { - jQuery.event.special[ orig ] = { - delegateType: fix, - bindType: fix, - - handle: function( event ) { - var ret, - target = this, - related = event.relatedTarget, - handleObj = event.handleObj; - - // For mouseenter/leave call the handler if related is outside the target. - // NB: No relatedTarget if the mouse left/entered the browser window - if ( !related || ( related !== target && !jQuery.contains( target, related ) ) ) { - event.type = handleObj.origType; - ret = handleObj.handler.apply( this, arguments ); - event.type = fix; - } - return ret; - } - }; - } ); - - jQuery.fn.extend( { - - on: function( types, selector, data, fn ) { - return on( this, types, selector, data, fn ); - }, - one: function( types, selector, data, fn ) { - return on( this, types, selector, data, fn, 1 ); - }, - off: function( types, selector, fn ) { - var handleObj, type; - if ( types && types.preventDefault && types.handleObj ) { - - // ( event ) dispatched jQuery.Event - handleObj = types.handleObj; - jQuery( types.delegateTarget ).off( - handleObj.namespace ? - handleObj.origType + "." + handleObj.namespace : - handleObj.origType, - handleObj.selector, - handleObj.handler - ); - return this; - } - if ( typeof types === "object" ) { - - // ( types-object [, selector] ) - for ( type in types ) { - this.off( type, selector, types[ type ] ); - } - return this; - } - if ( selector === false || typeof selector === "function" ) { - - // ( types [, fn] ) - fn = selector; - selector = undefined; - } - if ( fn === false ) { - fn = returnFalse; - } - return this.each( function() { - jQuery.event.remove( this, types, fn, selector ); - } ); - } - } ); - - - var - - // Support: IE <=10 - 11, Edge 12 - 13 only - // In IE/Edge using regex groups here causes severe slowdowns. - // See https://connect.microsoft.com/IE/feedback/details/1736512/ - rnoInnerhtml = /<script|<style|<link/i, - - // checked="checked" or checked - rchecked = /checked\s*(?:[^=]|=\s*.checked.)/i, - rcleanScript = /^\s*<!(?:\[CDATA\[|--)|(?:\]\]|--)>\s*$/g; - - // Prefer a tbody over its parent table for containing new rows - function manipulationTarget( elem, content ) { - if ( nodeName( elem, "table" ) && - nodeName( content.nodeType !== 11 ? content : content.firstChild, "tr" ) ) { - - return jQuery( elem ).children( "tbody" )[ 0 ] || elem; - } - - return elem; - } - - // Replace/restore the type attribute of script elements for safe DOM manipulation - function disableScript( elem ) { - elem.type = ( elem.getAttribute( "type" ) !== null ) + "/" + elem.type; - return elem; - } - function restoreScript( elem ) { - if ( ( elem.type || "" ).slice( 0, 5 ) === "true/" ) { - elem.type = elem.type.slice( 5 ); - } else { - elem.removeAttribute( "type" ); - } - - return elem; - } - - function cloneCopyEvent( src, dest ) { - var i, l, type, pdataOld, udataOld, udataCur, events; - - if ( dest.nodeType !== 1 ) { - return; - } - - // 1. Copy private data: events, handlers, etc. - if ( dataPriv.hasData( src ) ) { - pdataOld = dataPriv.get( src ); - events = pdataOld.events; - - if ( events ) { - dataPriv.remove( dest, "handle events" ); - - for ( type in events ) { - for ( i = 0, l = events[ type ].length; i < l; i++ ) { - jQuery.event.add( dest, type, events[ type ][ i ] ); - } - } - } - } - - // 2. Copy user data - if ( dataUser.hasData( src ) ) { - udataOld = dataUser.access( src ); - udataCur = jQuery.extend( {}, udataOld ); - - dataUser.set( dest, udataCur ); - } - } - - // Fix IE bugs, see support tests - function fixInput( src, dest ) { - var nodeName = dest.nodeName.toLowerCase(); - - // Fails to persist the checked state of a cloned checkbox or radio button. - if ( nodeName === "input" && rcheckableType.test( src.type ) ) { - dest.checked = src.checked; - - // Fails to return the selected option to the default selected state when cloning options - } else if ( nodeName === "input" || nodeName === "textarea" ) { - dest.defaultValue = src.defaultValue; - } - } - - function domManip( collection, args, callback, ignored ) { - - // Flatten any nested arrays - args = flat( args ); - - var fragment, first, scripts, hasScripts, node, doc, - i = 0, - l = collection.length, - iNoClone = l - 1, - value = args[ 0 ], - valueIsFunction = isFunction( value ); - - // We can't cloneNode fragments that contain checked, in WebKit - if ( valueIsFunction || - ( l > 1 && typeof value === "string" && - !support.checkClone && rchecked.test( value ) ) ) { - return collection.each( function( index ) { - var self = collection.eq( index ); - if ( valueIsFunction ) { - args[ 0 ] = value.call( this, index, self.html() ); - } - domManip( self, args, callback, ignored ); - } ); - } - - if ( l ) { - fragment = buildFragment( args, collection[ 0 ].ownerDocument, false, collection, ignored ); - first = fragment.firstChild; - - if ( fragment.childNodes.length === 1 ) { - fragment = first; - } - - // Require either new content or an interest in ignored elements to invoke the callback - if ( first || ignored ) { - scripts = jQuery.map( getAll( fragment, "script" ), disableScript ); - hasScripts = scripts.length; - - // Use the original fragment for the last item - // instead of the first because it can end up - // being emptied incorrectly in certain situations (#8070). - for ( ; i < l; i++ ) { - node = fragment; - - if ( i !== iNoClone ) { - node = jQuery.clone( node, true, true ); - - // Keep references to cloned scripts for later restoration - if ( hasScripts ) { - - // Support: Android <=4.0 only, PhantomJS 1 only - // push.apply(_, arraylike) throws on ancient WebKit - jQuery.merge( scripts, getAll( node, "script" ) ); - } - } - - callback.call( collection[ i ], node, i ); - } - - if ( hasScripts ) { - doc = scripts[ scripts.length - 1 ].ownerDocument; - - // Reenable scripts - jQuery.map( scripts, restoreScript ); - - // Evaluate executable scripts on first document insertion - for ( i = 0; i < hasScripts; i++ ) { - node = scripts[ i ]; - if ( rscriptType.test( node.type || "" ) && - !dataPriv.access( node, "globalEval" ) && - jQuery.contains( doc, node ) ) { - - if ( node.src && ( node.type || "" ).toLowerCase() !== "module" ) { - - // Optional AJAX dependency, but won't run scripts if not present - if ( jQuery._evalUrl && !node.noModule ) { - jQuery._evalUrl( node.src, { - nonce: node.nonce || node.getAttribute( "nonce" ) - }, doc ); - } - } else { - DOMEval( node.textContent.replace( rcleanScript, "" ), node, doc ); - } - } - } - } - } - } - - return collection; - } - - function remove( elem, selector, keepData ) { - var node, - nodes = selector ? jQuery.filter( selector, elem ) : elem, - i = 0; - - for ( ; ( node = nodes[ i ] ) != null; i++ ) { - if ( !keepData && node.nodeType === 1 ) { - jQuery.cleanData( getAll( node ) ); - } - - if ( node.parentNode ) { - if ( keepData && isAttached( node ) ) { - setGlobalEval( getAll( node, "script" ) ); - } - node.parentNode.removeChild( node ); - } - } - - return elem; - } - - jQuery.extend( { - htmlPrefilter: function( html ) { - return html; - }, - - clone: function( elem, dataAndEvents, deepDataAndEvents ) { - var i, l, srcElements, destElements, - clone = elem.cloneNode( true ), - inPage = isAttached( elem ); - - // Fix IE cloning issues - if ( !support.noCloneChecked && ( elem.nodeType === 1 || elem.nodeType === 11 ) && - !jQuery.isXMLDoc( elem ) ) { - - // We eschew Sizzle here for performance reasons: https://jsperf.com/getall-vs-sizzle/2 - destElements = getAll( clone ); - srcElements = getAll( elem ); - - for ( i = 0, l = srcElements.length; i < l; i++ ) { - fixInput( srcElements[ i ], destElements[ i ] ); - } - } - - // Copy the events from the original to the clone - if ( dataAndEvents ) { - if ( deepDataAndEvents ) { - srcElements = srcElements || getAll( elem ); - destElements = destElements || getAll( clone ); - - for ( i = 0, l = srcElements.length; i < l; i++ ) { - cloneCopyEvent( srcElements[ i ], destElements[ i ] ); - } - } else { - cloneCopyEvent( elem, clone ); - } - } - - // Preserve script evaluation history - destElements = getAll( clone, "script" ); - if ( destElements.length > 0 ) { - setGlobalEval( destElements, !inPage && getAll( elem, "script" ) ); - } - - // Return the cloned set - return clone; - }, - - cleanData: function( elems ) { - var data, elem, type, - special = jQuery.event.special, - i = 0; - - for ( ; ( elem = elems[ i ] ) !== undefined; i++ ) { - if ( acceptData( elem ) ) { - if ( ( data = elem[ dataPriv.expando ] ) ) { - if ( data.events ) { - for ( type in data.events ) { - if ( special[ type ] ) { - jQuery.event.remove( elem, type ); - - // This is a shortcut to avoid jQuery.event.remove's overhead - } else { - jQuery.removeEvent( elem, type, data.handle ); - } - } - } - - // Support: Chrome <=35 - 45+ - // Assign undefined instead of using delete, see Data#remove - elem[ dataPriv.expando ] = undefined; - } - if ( elem[ dataUser.expando ] ) { - - // Support: Chrome <=35 - 45+ - // Assign undefined instead of using delete, see Data#remove - elem[ dataUser.expando ] = undefined; - } - } - } - } - } ); - - jQuery.fn.extend( { - detach: function( selector ) { - return remove( this, selector, true ); - }, - - remove: function( selector ) { - return remove( this, selector ); - }, - - text: function( value ) { - return access( this, function( value ) { - return value === undefined ? - jQuery.text( this ) : - this.empty().each( function() { - if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) { - this.textContent = value; - } - } ); - }, null, value, arguments.length ); - }, - - append: function() { - return domManip( this, arguments, function( elem ) { - if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) { - var target = manipulationTarget( this, elem ); - target.appendChild( elem ); - } - } ); - }, - - prepend: function() { - return domManip( this, arguments, function( elem ) { - if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) { - var target = manipulationTarget( this, elem ); - target.insertBefore( elem, target.firstChild ); - } - } ); - }, - - before: function() { - return domManip( this, arguments, function( elem ) { - if ( this.parentNode ) { - this.parentNode.insertBefore( elem, this ); - } - } ); - }, - - after: function() { - return domManip( this, arguments, function( elem ) { - if ( this.parentNode ) { - this.parentNode.insertBefore( elem, this.nextSibling ); - } - } ); - }, - - empty: function() { - var elem, - i = 0; - - for ( ; ( elem = this[ i ] ) != null; i++ ) { - if ( elem.nodeType === 1 ) { - - // Prevent memory leaks - jQuery.cleanData( getAll( elem, false ) ); - - // Remove any remaining nodes - elem.textContent = ""; - } - } - - return this; - }, - - clone: function( dataAndEvents, deepDataAndEvents ) { - dataAndEvents = dataAndEvents == null ? false : dataAndEvents; - deepDataAndEvents = deepDataAndEvents == null ? dataAndEvents : deepDataAndEvents; - - return this.map( function() { - return jQuery.clone( this, dataAndEvents, deepDataAndEvents ); - } ); - }, - - html: function( value ) { - return access( this, function( value ) { - var elem = this[ 0 ] || {}, - i = 0, - l = this.length; - - if ( value === undefined && elem.nodeType === 1 ) { - return elem.innerHTML; - } - - // See if we can take a shortcut and just use innerHTML - if ( typeof value === "string" && !rnoInnerhtml.test( value ) && - !wrapMap[ ( rtagName.exec( value ) || [ "", "" ] )[ 1 ].toLowerCase() ] ) { - - value = jQuery.htmlPrefilter( value ); - - try { - for ( ; i < l; i++ ) { - elem = this[ i ] || {}; - - // Remove element nodes and prevent memory leaks - if ( elem.nodeType === 1 ) { - jQuery.cleanData( getAll( elem, false ) ); - elem.innerHTML = value; - } - } - - elem = 0; - - // If using innerHTML throws an exception, use the fallback method - } catch ( e ) {} - } - - if ( elem ) { - this.empty().append( value ); - } - }, null, value, arguments.length ); - }, - - replaceWith: function() { - var ignored = []; - - // Make the changes, replacing each non-ignored context element with the new content - return domManip( this, arguments, function( elem ) { - var parent = this.parentNode; - - if ( jQuery.inArray( this, ignored ) < 0 ) { - jQuery.cleanData( getAll( this ) ); - if ( parent ) { - parent.replaceChild( elem, this ); - } - } - - // Force callback invocation - }, ignored ); - } - } ); - - jQuery.each( { - appendTo: "append", - prependTo: "prepend", - insertBefore: "before", - insertAfter: "after", - replaceAll: "replaceWith" - }, function( name, original ) { - jQuery.fn[ name ] = function( selector ) { - var elems, - ret = [], - insert = jQuery( selector ), - last = insert.length - 1, - i = 0; - - for ( ; i <= last; i++ ) { - elems = i === last ? this : this.clone( true ); - jQuery( insert[ i ] )[ original ]( elems ); - - // Support: Android <=4.0 only, PhantomJS 1 only - // .get() because push.apply(_, arraylike) throws on ancient WebKit - push.apply( ret, elems.get() ); - } - - return this.pushStack( ret ); - }; - } ); - var rnumnonpx = new RegExp( "^(" + pnum + ")(?!px)[a-z%]+$", "i" ); - - var getStyles = function( elem ) { - - // Support: IE <=11 only, Firefox <=30 (#15098, #14150) - // IE throws on elements created in popups - // FF meanwhile throws on frame elements through "defaultView.getComputedStyle" - var view = elem.ownerDocument.defaultView; - - if ( !view || !view.opener ) { - view = window; - } - - return view.getComputedStyle( elem ); - }; - - var swap = function( elem, options, callback ) { - var ret, name, - old = {}; - - // Remember the old values, and insert the new ones - for ( name in options ) { - old[ name ] = elem.style[ name ]; - elem.style[ name ] = options[ name ]; - } - - ret = callback.call( elem ); - - // Revert the old values - for ( name in options ) { - elem.style[ name ] = old[ name ]; - } - - return ret; - }; - - - var rboxStyle = new RegExp( cssExpand.join( "|" ), "i" ); - - - - ( function() { - - // Executing both pixelPosition & boxSizingReliable tests require only one layout - // so they're executed at the same time to save the second computation. - function computeStyleTests() { - - // This is a singleton, we need to execute it only once - if ( !div ) { - return; - } - - container.style.cssText = "position:absolute;left:-11111px;width:60px;" + - "margin-top:1px;padding:0;border:0"; - div.style.cssText = - "position:relative;display:block;box-sizing:border-box;overflow:scroll;" + - "margin:auto;border:1px;padding:1px;" + - "width:60%;top:1%"; - documentElement.appendChild( container ).appendChild( div ); - - var divStyle = window.getComputedStyle( div ); - pixelPositionVal = divStyle.top !== "1%"; - - // Support: Android 4.0 - 4.3 only, Firefox <=3 - 44 - reliableMarginLeftVal = roundPixelMeasures( divStyle.marginLeft ) === 12; - - // Support: Android 4.0 - 4.3 only, Safari <=9.1 - 10.1, iOS <=7.0 - 9.3 - // Some styles come back with percentage values, even though they shouldn't - div.style.right = "60%"; - pixelBoxStylesVal = roundPixelMeasures( divStyle.right ) === 36; - - // Support: IE 9 - 11 only - // Detect misreporting of content dimensions for box-sizing:border-box elements - boxSizingReliableVal = roundPixelMeasures( divStyle.width ) === 36; - - // Support: IE 9 only - // Detect overflow:scroll screwiness (gh-3699) - // Support: Chrome <=64 - // Don't get tricked when zoom affects offsetWidth (gh-4029) - div.style.position = "absolute"; - scrollboxSizeVal = roundPixelMeasures( div.offsetWidth / 3 ) === 12; - - documentElement.removeChild( container ); - - // Nullify the div so it wouldn't be stored in the memory and - // it will also be a sign that checks already performed - div = null; - } - - function roundPixelMeasures( measure ) { - return Math.round( parseFloat( measure ) ); - } - - var pixelPositionVal, boxSizingReliableVal, scrollboxSizeVal, pixelBoxStylesVal, - reliableTrDimensionsVal, reliableMarginLeftVal, - container = document.createElement( "div" ), - div = document.createElement( "div" ); - - // Finish early in limited (non-browser) environments - if ( !div.style ) { - return; - } - - // Support: IE <=9 - 11 only - // Style of cloned element affects source element cloned (#8908) - div.style.backgroundClip = "content-box"; - div.cloneNode( true ).style.backgroundClip = ""; - support.clearCloneStyle = div.style.backgroundClip === "content-box"; - - jQuery.extend( support, { - boxSizingReliable: function() { - computeStyleTests(); - return boxSizingReliableVal; - }, - pixelBoxStyles: function() { - computeStyleTests(); - return pixelBoxStylesVal; - }, - pixelPosition: function() { - computeStyleTests(); - return pixelPositionVal; - }, - reliableMarginLeft: function() { - computeStyleTests(); - return reliableMarginLeftVal; - }, - scrollboxSize: function() { - computeStyleTests(); - return scrollboxSizeVal; - }, - - // Support: IE 9 - 11+, Edge 15 - 18+ - // IE/Edge misreport `getComputedStyle` of table rows with width/height - // set in CSS while `offset*` properties report correct values. - // Behavior in IE 9 is more subtle than in newer versions & it passes - // some versions of this test; make sure not to make it pass there! - // - // Support: Firefox 70+ - // Only Firefox includes border widths - // in computed dimensions. (gh-4529) - reliableTrDimensions: function() { - var table, tr, trChild, trStyle; - if ( reliableTrDimensionsVal == null ) { - table = document.createElement( "table" ); - tr = document.createElement( "tr" ); - trChild = document.createElement( "div" ); - - table.style.cssText = "position:absolute;left:-11111px;border-collapse:separate"; - tr.style.cssText = "border:1px solid"; - - // Support: Chrome 86+ - // Height set through cssText does not get applied. - // Computed height then comes back as 0. - tr.style.height = "1px"; - trChild.style.height = "9px"; - - // Support: Android 8 Chrome 86+ - // In our bodyBackground.html iframe, - // display for all div elements is set to "inline", - // which causes a problem only in Android 8 Chrome 86. - // Ensuring the div is display: block - // gets around this issue. - trChild.style.display = "block"; - - documentElement - .appendChild( table ) - .appendChild( tr ) - .appendChild( trChild ); - - trStyle = window.getComputedStyle( tr ); - reliableTrDimensionsVal = ( parseInt( trStyle.height, 10 ) + - parseInt( trStyle.borderTopWidth, 10 ) + - parseInt( trStyle.borderBottomWidth, 10 ) ) === tr.offsetHeight; - - documentElement.removeChild( table ); - } - return reliableTrDimensionsVal; - } - } ); - } )(); - - - function curCSS( elem, name, computed ) { - var width, minWidth, maxWidth, ret, - - // Support: Firefox 51+ - // Retrieving style before computed somehow - // fixes an issue with getting wrong values - // on detached elements - style = elem.style; - - computed = computed || getStyles( elem ); - - // getPropertyValue is needed for: - // .css('filter') (IE 9 only, #12537) - // .css('--customProperty) (#3144) - if ( computed ) { - ret = computed.getPropertyValue( name ) || computed[ name ]; - - if ( ret === "" && !isAttached( elem ) ) { - ret = jQuery.style( elem, name ); - } - - // A tribute to the "awesome hack by Dean Edwards" - // Android Browser returns percentage for some values, - // but width seems to be reliably pixels. - // This is against the CSSOM draft spec: - // https://drafts.csswg.org/cssom/#resolved-values - if ( !support.pixelBoxStyles() && rnumnonpx.test( ret ) && rboxStyle.test( name ) ) { - - // Remember the original values - width = style.width; - minWidth = style.minWidth; - maxWidth = style.maxWidth; - - // Put in the new values to get a computed value out - style.minWidth = style.maxWidth = style.width = ret; - ret = computed.width; - - // Revert the changed values - style.width = width; - style.minWidth = minWidth; - style.maxWidth = maxWidth; - } - } - - return ret !== undefined ? - - // Support: IE <=9 - 11 only - // IE returns zIndex value as an integer. - ret + "" : - ret; - } - - - function addGetHookIf( conditionFn, hookFn ) { - - // Define the hook, we'll check on the first run if it's really needed. - return { - get: function() { - if ( conditionFn() ) { - - // Hook not needed (or it's not possible to use it due - // to missing dependency), remove it. - delete this.get; - return; - } - - // Hook needed; redefine it so that the support test is not executed again. - return ( this.get = hookFn ).apply( this, arguments ); - } - }; - } - - - var cssPrefixes = [ "Webkit", "Moz", "ms" ], - emptyStyle = document.createElement( "div" ).style, - vendorProps = {}; - - // Return a vendor-prefixed property or undefined - function vendorPropName( name ) { - - // Check for vendor prefixed names - var capName = name[ 0 ].toUpperCase() + name.slice( 1 ), - i = cssPrefixes.length; - - while ( i-- ) { - name = cssPrefixes[ i ] + capName; - if ( name in emptyStyle ) { - return name; - } - } - } - - // Return a potentially-mapped jQuery.cssProps or vendor prefixed property - function finalPropName( name ) { - var final = jQuery.cssProps[ name ] || vendorProps[ name ]; - - if ( final ) { - return final; - } - if ( name in emptyStyle ) { - return name; - } - return vendorProps[ name ] = vendorPropName( name ) || name; - } - - - var - - // Swappable if display is none or starts with table - // except "table", "table-cell", or "table-caption" - // See here for display values: https://developer.mozilla.org/en-US/docs/CSS/display - rdisplayswap = /^(none|table(?!-c[ea]).+)/, - rcustomProp = /^--/, - cssShow = { position: "absolute", visibility: "hidden", display: "block" }, - cssNormalTransform = { - letterSpacing: "0", - fontWeight: "400" - }; - - function setPositiveNumber( _elem, value, subtract ) { - - // Any relative (+/-) values have already been - // normalized at this point - var matches = rcssNum.exec( value ); - return matches ? - - // Guard against undefined "subtract", e.g., when used as in cssHooks - Math.max( 0, matches[ 2 ] - ( subtract || 0 ) ) + ( matches[ 3 ] || "px" ) : - value; - } - - function boxModelAdjustment( elem, dimension, box, isBorderBox, styles, computedVal ) { - var i = dimension === "width" ? 1 : 0, - extra = 0, - delta = 0; - - // Adjustment may not be necessary - if ( box === ( isBorderBox ? "border" : "content" ) ) { - return 0; - } - - for ( ; i < 4; i += 2 ) { - - // Both box models exclude margin - if ( box === "margin" ) { - delta += jQuery.css( elem, box + cssExpand[ i ], true, styles ); - } - - // If we get here with a content-box, we're seeking "padding" or "border" or "margin" - if ( !isBorderBox ) { - - // Add padding - delta += jQuery.css( elem, "padding" + cssExpand[ i ], true, styles ); - - // For "border" or "margin", add border - if ( box !== "padding" ) { - delta += jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles ); - - // But still keep track of it otherwise - } else { - extra += jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles ); - } - - // If we get here with a border-box (content + padding + border), we're seeking "content" or - // "padding" or "margin" - } else { - - // For "content", subtract padding - if ( box === "content" ) { - delta -= jQuery.css( elem, "padding" + cssExpand[ i ], true, styles ); - } - - // For "content" or "padding", subtract border - if ( box !== "margin" ) { - delta -= jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles ); - } - } - } - - // Account for positive content-box scroll gutter when requested by providing computedVal - if ( !isBorderBox && computedVal >= 0 ) { - - // offsetWidth/offsetHeight is a rounded sum of content, padding, scroll gutter, and border - // Assuming integer scroll gutter, subtract the rest and round down - delta += Math.max( 0, Math.ceil( - elem[ "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 ) ] - - computedVal - - delta - - extra - - 0.5 - - // If offsetWidth/offsetHeight is unknown, then we can't determine content-box scroll gutter - // Use an explicit zero to avoid NaN (gh-3964) - ) ) || 0; - } - - return delta; - } - - function getWidthOrHeight( elem, dimension, extra ) { - - // Start with computed style - var styles = getStyles( elem ), - - // To avoid forcing a reflow, only fetch boxSizing if we need it (gh-4322). - // Fake content-box until we know it's needed to know the true value. - boxSizingNeeded = !support.boxSizingReliable() || extra, - isBorderBox = boxSizingNeeded && - jQuery.css( elem, "boxSizing", false, styles ) === "border-box", - valueIsBorderBox = isBorderBox, - - val = curCSS( elem, dimension, styles ), - offsetProp = "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 ); - - // Support: Firefox <=54 - // Return a confounding non-pixel value or feign ignorance, as appropriate. - if ( rnumnonpx.test( val ) ) { - if ( !extra ) { - return val; - } - val = "auto"; - } - - - // Support: IE 9 - 11 only - // Use offsetWidth/offsetHeight for when box sizing is unreliable. - // In those cases, the computed value can be trusted to be border-box. - if ( ( !support.boxSizingReliable() && isBorderBox || - - // Support: IE 10 - 11+, Edge 15 - 18+ - // IE/Edge misreport `getComputedStyle` of table rows with width/height - // set in CSS while `offset*` properties report correct values. - // Interestingly, in some cases IE 9 doesn't suffer from this issue. - !support.reliableTrDimensions() && nodeName( elem, "tr" ) || - - // Fall back to offsetWidth/offsetHeight when value is "auto" - // This happens for inline elements with no explicit setting (gh-3571) - val === "auto" || - - // Support: Android <=4.1 - 4.3 only - // Also use offsetWidth/offsetHeight for misreported inline dimensions (gh-3602) - !parseFloat( val ) && jQuery.css( elem, "display", false, styles ) === "inline" ) && - - // Make sure the element is visible & connected - elem.getClientRects().length ) { - - isBorderBox = jQuery.css( elem, "boxSizing", false, styles ) === "border-box"; - - // Where available, offsetWidth/offsetHeight approximate border box dimensions. - // Where not available (e.g., SVG), assume unreliable box-sizing and interpret the - // retrieved value as a content box dimension. - valueIsBorderBox = offsetProp in elem; - if ( valueIsBorderBox ) { - val = elem[ offsetProp ]; - } - } - - // Normalize "" and auto - val = parseFloat( val ) || 0; - - // Adjust for the element's box model - return ( val + - boxModelAdjustment( - elem, - dimension, - extra || ( isBorderBox ? "border" : "content" ), - valueIsBorderBox, - styles, - - // Provide the current computed size to request scroll gutter calculation (gh-3589) - val - ) - ) + "px"; - } - - jQuery.extend( { - - // Add in style property hooks for overriding the default - // behavior of getting and setting a style property - cssHooks: { - opacity: { - get: function( elem, computed ) { - if ( computed ) { - - // We should always get a number back from opacity - var ret = curCSS( elem, "opacity" ); - return ret === "" ? "1" : ret; - } - } - } - }, - - // Don't automatically add "px" to these possibly-unitless properties - cssNumber: { - "animationIterationCount": true, - "columnCount": true, - "fillOpacity": true, - "flexGrow": true, - "flexShrink": true, - "fontWeight": true, - "gridArea": true, - "gridColumn": true, - "gridColumnEnd": true, - "gridColumnStart": true, - "gridRow": true, - "gridRowEnd": true, - "gridRowStart": true, - "lineHeight": true, - "opacity": true, - "order": true, - "orphans": true, - "widows": true, - "zIndex": true, - "zoom": true - }, - - // Add in properties whose names you wish to fix before - // setting or getting the value - cssProps: {}, - - // Get and set the style property on a DOM Node - style: function( elem, name, value, extra ) { - - // Don't set styles on text and comment nodes - if ( !elem || elem.nodeType === 3 || elem.nodeType === 8 || !elem.style ) { - return; - } - - // Make sure that we're working with the right name - var ret, type, hooks, - origName = camelCase( name ), - isCustomProp = rcustomProp.test( name ), - style = elem.style; - - // Make sure that we're working with the right name. We don't - // want to query the value if it is a CSS custom property - // since they are user-defined. - if ( !isCustomProp ) { - name = finalPropName( origName ); - } - - // Gets hook for the prefixed version, then unprefixed version - hooks = jQuery.cssHooks[ name ] || jQuery.cssHooks[ origName ]; - - // Check if we're setting a value - if ( value !== undefined ) { - type = typeof value; - - // Convert "+=" or "-=" to relative numbers (#7345) - if ( type === "string" && ( ret = rcssNum.exec( value ) ) && ret[ 1 ] ) { - value = adjustCSS( elem, name, ret ); - - // Fixes bug #9237 - type = "number"; - } - - // Make sure that null and NaN values aren't set (#7116) - if ( value == null || value !== value ) { - return; - } - - // If a number was passed in, add the unit (except for certain CSS properties) - // The isCustomProp check can be removed in jQuery 4.0 when we only auto-append - // "px" to a few hardcoded values. - if ( type === "number" && !isCustomProp ) { - value += ret && ret[ 3 ] || ( jQuery.cssNumber[ origName ] ? "" : "px" ); - } - - // background-* props affect original clone's values - if ( !support.clearCloneStyle && value === "" && name.indexOf( "background" ) === 0 ) { - style[ name ] = "inherit"; - } - - // If a hook was provided, use that value, otherwise just set the specified value - if ( !hooks || !( "set" in hooks ) || - ( value = hooks.set( elem, value, extra ) ) !== undefined ) { - - if ( isCustomProp ) { - style.setProperty( name, value ); - } else { - style[ name ] = value; - } - } - - } else { - - // If a hook was provided get the non-computed value from there - if ( hooks && "get" in hooks && - ( ret = hooks.get( elem, false, extra ) ) !== undefined ) { - - return ret; - } - - // Otherwise just get the value from the style object - return style[ name ]; - } - }, - - css: function( elem, name, extra, styles ) { - var val, num, hooks, - origName = camelCase( name ), - isCustomProp = rcustomProp.test( name ); - - // Make sure that we're working with the right name. We don't - // want to modify the value if it is a CSS custom property - // since they are user-defined. - if ( !isCustomProp ) { - name = finalPropName( origName ); - } - - // Try prefixed name followed by the unprefixed name - hooks = jQuery.cssHooks[ name ] || jQuery.cssHooks[ origName ]; - - // If a hook was provided get the computed value from there - if ( hooks && "get" in hooks ) { - val = hooks.get( elem, true, extra ); - } - - // Otherwise, if a way to get the computed value exists, use that - if ( val === undefined ) { - val = curCSS( elem, name, styles ); - } - - // Convert "normal" to computed value - if ( val === "normal" && name in cssNormalTransform ) { - val = cssNormalTransform[ name ]; - } - - // Make numeric if forced or a qualifier was provided and val looks numeric - if ( extra === "" || extra ) { - num = parseFloat( val ); - return extra === true || isFinite( num ) ? num || 0 : val; - } - - return val; - } - } ); - - jQuery.each( [ "height", "width" ], function( _i, dimension ) { - jQuery.cssHooks[ dimension ] = { - get: function( elem, computed, extra ) { - if ( computed ) { - - // Certain elements can have dimension info if we invisibly show them - // but it must have a current display style that would benefit - return rdisplayswap.test( jQuery.css( elem, "display" ) ) && - - // Support: Safari 8+ - // Table columns in Safari have non-zero offsetWidth & zero - // getBoundingClientRect().width unless display is changed. - // Support: IE <=11 only - // Running getBoundingClientRect on a disconnected node - // in IE throws an error. - ( !elem.getClientRects().length || !elem.getBoundingClientRect().width ) ? - swap( elem, cssShow, function() { - return getWidthOrHeight( elem, dimension, extra ); - } ) : - getWidthOrHeight( elem, dimension, extra ); - } - }, - - set: function( elem, value, extra ) { - var matches, - styles = getStyles( elem ), - - // Only read styles.position if the test has a chance to fail - // to avoid forcing a reflow. - scrollboxSizeBuggy = !support.scrollboxSize() && - styles.position === "absolute", - - // To avoid forcing a reflow, only fetch boxSizing if we need it (gh-3991) - boxSizingNeeded = scrollboxSizeBuggy || extra, - isBorderBox = boxSizingNeeded && - jQuery.css( elem, "boxSizing", false, styles ) === "border-box", - subtract = extra ? - boxModelAdjustment( - elem, - dimension, - extra, - isBorderBox, - styles - ) : - 0; - - // Account for unreliable border-box dimensions by comparing offset* to computed and - // faking a content-box to get border and padding (gh-3699) - if ( isBorderBox && scrollboxSizeBuggy ) { - subtract -= Math.ceil( - elem[ "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 ) ] - - parseFloat( styles[ dimension ] ) - - boxModelAdjustment( elem, dimension, "border", false, styles ) - - 0.5 - ); - } - - // Convert to pixels if value adjustment is needed - if ( subtract && ( matches = rcssNum.exec( value ) ) && - ( matches[ 3 ] || "px" ) !== "px" ) { - - elem.style[ dimension ] = value; - value = jQuery.css( elem, dimension ); - } - - return setPositiveNumber( elem, value, subtract ); - } - }; - } ); - - jQuery.cssHooks.marginLeft = addGetHookIf( support.reliableMarginLeft, - function( elem, computed ) { - if ( computed ) { - return ( parseFloat( curCSS( elem, "marginLeft" ) ) || - elem.getBoundingClientRect().left - - swap( elem, { marginLeft: 0 }, function() { - return elem.getBoundingClientRect().left; - } ) - ) + "px"; - } - } - ); - - // These hooks are used by animate to expand properties - jQuery.each( { - margin: "", - padding: "", - border: "Width" - }, function( prefix, suffix ) { - jQuery.cssHooks[ prefix + suffix ] = { - expand: function( value ) { - var i = 0, - expanded = {}, - - // Assumes a single number if not a string - parts = typeof value === "string" ? value.split( " " ) : [ value ]; - - for ( ; i < 4; i++ ) { - expanded[ prefix + cssExpand[ i ] + suffix ] = - parts[ i ] || parts[ i - 2 ] || parts[ 0 ]; - } - - return expanded; - } - }; - - if ( prefix !== "margin" ) { - jQuery.cssHooks[ prefix + suffix ].set = setPositiveNumber; - } - } ); - - jQuery.fn.extend( { - css: function( name, value ) { - return access( this, function( elem, name, value ) { - var styles, len, - map = {}, - i = 0; - - if ( Array.isArray( name ) ) { - styles = getStyles( elem ); - len = name.length; - - for ( ; i < len; i++ ) { - map[ name[ i ] ] = jQuery.css( elem, name[ i ], false, styles ); - } - - return map; - } - - return value !== undefined ? - jQuery.style( elem, name, value ) : - jQuery.css( elem, name ); - }, name, value, arguments.length > 1 ); - } - } ); - - - function Tween( elem, options, prop, end, easing ) { - return new Tween.prototype.init( elem, options, prop, end, easing ); - } - jQuery.Tween = Tween; - - Tween.prototype = { - constructor: Tween, - init: function( elem, options, prop, end, easing, unit ) { - this.elem = elem; - this.prop = prop; - this.easing = easing || jQuery.easing._default; - this.options = options; - this.start = this.now = this.cur(); - this.end = end; - this.unit = unit || ( jQuery.cssNumber[ prop ] ? "" : "px" ); - }, - cur: function() { - var hooks = Tween.propHooks[ this.prop ]; - - return hooks && hooks.get ? - hooks.get( this ) : - Tween.propHooks._default.get( this ); - }, - run: function( percent ) { - var eased, - hooks = Tween.propHooks[ this.prop ]; - - if ( this.options.duration ) { - this.pos = eased = jQuery.easing[ this.easing ]( - percent, this.options.duration * percent, 0, 1, this.options.duration - ); - } else { - this.pos = eased = percent; - } - this.now = ( this.end - this.start ) * eased + this.start; - - if ( this.options.step ) { - this.options.step.call( this.elem, this.now, this ); - } - - if ( hooks && hooks.set ) { - hooks.set( this ); - } else { - Tween.propHooks._default.set( this ); - } - return this; - } - }; - - Tween.prototype.init.prototype = Tween.prototype; - - Tween.propHooks = { - _default: { - get: function( tween ) { - var result; - - // Use a property on the element directly when it is not a DOM element, - // or when there is no matching style property that exists. - if ( tween.elem.nodeType !== 1 || - tween.elem[ tween.prop ] != null && tween.elem.style[ tween.prop ] == null ) { - return tween.elem[ tween.prop ]; - } - - // Passing an empty string as a 3rd parameter to .css will automatically - // attempt a parseFloat and fallback to a string if the parse fails. - // Simple values such as "10px" are parsed to Float; - // complex values such as "rotate(1rad)" are returned as-is. - result = jQuery.css( tween.elem, tween.prop, "" ); - - // Empty strings, null, undefined and "auto" are converted to 0. - return !result || result === "auto" ? 0 : result; - }, - set: function( tween ) { - - // Use step hook for back compat. - // Use cssHook if its there. - // Use .style if available and use plain properties where available. - if ( jQuery.fx.step[ tween.prop ] ) { - jQuery.fx.step[ tween.prop ]( tween ); - } else if ( tween.elem.nodeType === 1 && ( - jQuery.cssHooks[ tween.prop ] || - tween.elem.style[ finalPropName( tween.prop ) ] != null ) ) { - jQuery.style( tween.elem, tween.prop, tween.now + tween.unit ); - } else { - tween.elem[ tween.prop ] = tween.now; - } - } - } - }; - - // Support: IE <=9 only - // Panic based approach to setting things on disconnected nodes - Tween.propHooks.scrollTop = Tween.propHooks.scrollLeft = { - set: function( tween ) { - if ( tween.elem.nodeType && tween.elem.parentNode ) { - tween.elem[ tween.prop ] = tween.now; - } - } - }; - - jQuery.easing = { - linear: function( p ) { - return p; - }, - swing: function( p ) { - return 0.5 - Math.cos( p * Math.PI ) / 2; - }, - _default: "swing" - }; - - jQuery.fx = Tween.prototype.init; - - // Back compat <1.8 extension point - jQuery.fx.step = {}; - - - - - var - fxNow, inProgress, - rfxtypes = /^(?:toggle|show|hide)$/, - rrun = /queueHooks$/; - - function schedule() { - if ( inProgress ) { - if ( document.hidden === false && window.requestAnimationFrame ) { - window.requestAnimationFrame( schedule ); - } else { - window.setTimeout( schedule, jQuery.fx.interval ); - } - - jQuery.fx.tick(); - } - } - - // Animations created synchronously will run synchronously - function createFxNow() { - window.setTimeout( function() { - fxNow = undefined; - } ); - return ( fxNow = Date.now() ); - } - - // Generate parameters to create a standard animation - function genFx( type, includeWidth ) { - var which, - i = 0, - attrs = { height: type }; - - // If we include width, step value is 1 to do all cssExpand values, - // otherwise step value is 2 to skip over Left and Right - includeWidth = includeWidth ? 1 : 0; - for ( ; i < 4; i += 2 - includeWidth ) { - which = cssExpand[ i ]; - attrs[ "margin" + which ] = attrs[ "padding" + which ] = type; - } - - if ( includeWidth ) { - attrs.opacity = attrs.width = type; - } - - return attrs; - } - - function createTween( value, prop, animation ) { - var tween, - collection = ( Animation.tweeners[ prop ] || [] ).concat( Animation.tweeners[ "*" ] ), - index = 0, - length = collection.length; - for ( ; index < length; index++ ) { - if ( ( tween = collection[ index ].call( animation, prop, value ) ) ) { - - // We're done with this property - return tween; - } - } - } - - function defaultPrefilter( elem, props, opts ) { - var prop, value, toggle, hooks, oldfire, propTween, restoreDisplay, display, - isBox = "width" in props || "height" in props, - anim = this, - orig = {}, - style = elem.style, - hidden = elem.nodeType && isHiddenWithinTree( elem ), - dataShow = dataPriv.get( elem, "fxshow" ); - - // Queue-skipping animations hijack the fx hooks - if ( !opts.queue ) { - hooks = jQuery._queueHooks( elem, "fx" ); - if ( hooks.unqueued == null ) { - hooks.unqueued = 0; - oldfire = hooks.empty.fire; - hooks.empty.fire = function() { - if ( !hooks.unqueued ) { - oldfire(); - } - }; - } - hooks.unqueued++; - - anim.always( function() { - - // Ensure the complete handler is called before this completes - anim.always( function() { - hooks.unqueued--; - if ( !jQuery.queue( elem, "fx" ).length ) { - hooks.empty.fire(); - } - } ); - } ); - } - - // Detect show/hide animations - for ( prop in props ) { - value = props[ prop ]; - if ( rfxtypes.test( value ) ) { - delete props[ prop ]; - toggle = toggle || value === "toggle"; - if ( value === ( hidden ? "hide" : "show" ) ) { - - // Pretend to be hidden if this is a "show" and - // there is still data from a stopped show/hide - if ( value === "show" && dataShow && dataShow[ prop ] !== undefined ) { - hidden = true; - - // Ignore all other no-op show/hide data - } else { - continue; - } - } - orig[ prop ] = dataShow && dataShow[ prop ] || jQuery.style( elem, prop ); - } - } - - // Bail out if this is a no-op like .hide().hide() - propTween = !jQuery.isEmptyObject( props ); - if ( !propTween && jQuery.isEmptyObject( orig ) ) { - return; - } - - // Restrict "overflow" and "display" styles during box animations - if ( isBox && elem.nodeType === 1 ) { - - // Support: IE <=9 - 11, Edge 12 - 15 - // Record all 3 overflow attributes because IE does not infer the shorthand - // from identically-valued overflowX and overflowY and Edge just mirrors - // the overflowX value there. - opts.overflow = [ style.overflow, style.overflowX, style.overflowY ]; - - // Identify a display type, preferring old show/hide data over the CSS cascade - restoreDisplay = dataShow && dataShow.display; - if ( restoreDisplay == null ) { - restoreDisplay = dataPriv.get( elem, "display" ); - } - display = jQuery.css( elem, "display" ); - if ( display === "none" ) { - if ( restoreDisplay ) { - display = restoreDisplay; - } else { - - // Get nonempty value(s) by temporarily forcing visibility - showHide( [ elem ], true ); - restoreDisplay = elem.style.display || restoreDisplay; - display = jQuery.css( elem, "display" ); - showHide( [ elem ] ); - } - } - - // Animate inline elements as inline-block - if ( display === "inline" || display === "inline-block" && restoreDisplay != null ) { - if ( jQuery.css( elem, "float" ) === "none" ) { - - // Restore the original display value at the end of pure show/hide animations - if ( !propTween ) { - anim.done( function() { - style.display = restoreDisplay; - } ); - if ( restoreDisplay == null ) { - display = style.display; - restoreDisplay = display === "none" ? "" : display; - } - } - style.display = "inline-block"; - } - } - } - - if ( opts.overflow ) { - style.overflow = "hidden"; - anim.always( function() { - style.overflow = opts.overflow[ 0 ]; - style.overflowX = opts.overflow[ 1 ]; - style.overflowY = opts.overflow[ 2 ]; - } ); - } - - // Implement show/hide animations - propTween = false; - for ( prop in orig ) { - - // General show/hide setup for this element animation - if ( !propTween ) { - if ( dataShow ) { - if ( "hidden" in dataShow ) { - hidden = dataShow.hidden; - } - } else { - dataShow = dataPriv.access( elem, "fxshow", { display: restoreDisplay } ); - } - - // Store hidden/visible for toggle so `.stop().toggle()` "reverses" - if ( toggle ) { - dataShow.hidden = !hidden; - } - - // Show elements before animating them - if ( hidden ) { - showHide( [ elem ], true ); - } - - /* eslint-disable no-loop-func */ - - anim.done( function() { - - /* eslint-enable no-loop-func */ - - // The final step of a "hide" animation is actually hiding the element - if ( !hidden ) { - showHide( [ elem ] ); - } - dataPriv.remove( elem, "fxshow" ); - for ( prop in orig ) { - jQuery.style( elem, prop, orig[ prop ] ); - } - } ); - } - - // Per-property setup - propTween = createTween( hidden ? dataShow[ prop ] : 0, prop, anim ); - if ( !( prop in dataShow ) ) { - dataShow[ prop ] = propTween.start; - if ( hidden ) { - propTween.end = propTween.start; - propTween.start = 0; - } - } - } - } - - function propFilter( props, specialEasing ) { - var index, name, easing, value, hooks; - - // camelCase, specialEasing and expand cssHook pass - for ( index in props ) { - name = camelCase( index ); - easing = specialEasing[ name ]; - value = props[ index ]; - if ( Array.isArray( value ) ) { - easing = value[ 1 ]; - value = props[ index ] = value[ 0 ]; - } - - if ( index !== name ) { - props[ name ] = value; - delete props[ index ]; - } - - hooks = jQuery.cssHooks[ name ]; - if ( hooks && "expand" in hooks ) { - value = hooks.expand( value ); - delete props[ name ]; - - // Not quite $.extend, this won't overwrite existing keys. - // Reusing 'index' because we have the correct "name" - for ( index in value ) { - if ( !( index in props ) ) { - props[ index ] = value[ index ]; - specialEasing[ index ] = easing; - } - } - } else { - specialEasing[ name ] = easing; - } - } - } - - function Animation( elem, properties, options ) { - var result, - stopped, - index = 0, - length = Animation.prefilters.length, - deferred = jQuery.Deferred().always( function() { - - // Don't match elem in the :animated selector - delete tick.elem; - } ), - tick = function() { - if ( stopped ) { - return false; - } - var currentTime = fxNow || createFxNow(), - remaining = Math.max( 0, animation.startTime + animation.duration - currentTime ), - - // Support: Android 2.3 only - // Archaic crash bug won't allow us to use `1 - ( 0.5 || 0 )` (#12497) - temp = remaining / animation.duration || 0, - percent = 1 - temp, - index = 0, - length = animation.tweens.length; - - for ( ; index < length; index++ ) { - animation.tweens[ index ].run( percent ); - } - - deferred.notifyWith( elem, [ animation, percent, remaining ] ); - - // If there's more to do, yield - if ( percent < 1 && length ) { - return remaining; - } - - // If this was an empty animation, synthesize a final progress notification - if ( !length ) { - deferred.notifyWith( elem, [ animation, 1, 0 ] ); - } - - // Resolve the animation and report its conclusion - deferred.resolveWith( elem, [ animation ] ); - return false; - }, - animation = deferred.promise( { - elem: elem, - props: jQuery.extend( {}, properties ), - opts: jQuery.extend( true, { - specialEasing: {}, - easing: jQuery.easing._default - }, options ), - originalProperties: properties, - originalOptions: options, - startTime: fxNow || createFxNow(), - duration: options.duration, - tweens: [], - createTween: function( prop, end ) { - var tween = jQuery.Tween( elem, animation.opts, prop, end, - animation.opts.specialEasing[ prop ] || animation.opts.easing ); - animation.tweens.push( tween ); - return tween; - }, - stop: function( gotoEnd ) { - var index = 0, - - // If we are going to the end, we want to run all the tweens - // otherwise we skip this part - length = gotoEnd ? animation.tweens.length : 0; - if ( stopped ) { - return this; - } - stopped = true; - for ( ; index < length; index++ ) { - animation.tweens[ index ].run( 1 ); - } - - // Resolve when we played the last frame; otherwise, reject - if ( gotoEnd ) { - deferred.notifyWith( elem, [ animation, 1, 0 ] ); - deferred.resolveWith( elem, [ animation, gotoEnd ] ); - } else { - deferred.rejectWith( elem, [ animation, gotoEnd ] ); - } - return this; - } - } ), - props = animation.props; - - propFilter( props, animation.opts.specialEasing ); - - for ( ; index < length; index++ ) { - result = Animation.prefilters[ index ].call( animation, elem, props, animation.opts ); - if ( result ) { - if ( isFunction( result.stop ) ) { - jQuery._queueHooks( animation.elem, animation.opts.queue ).stop = - result.stop.bind( result ); - } - return result; - } - } - - jQuery.map( props, createTween, animation ); - - if ( isFunction( animation.opts.start ) ) { - animation.opts.start.call( elem, animation ); - } - - // Attach callbacks from options - animation - .progress( animation.opts.progress ) - .done( animation.opts.done, animation.opts.complete ) - .fail( animation.opts.fail ) - .always( animation.opts.always ); - - jQuery.fx.timer( - jQuery.extend( tick, { - elem: elem, - anim: animation, - queue: animation.opts.queue - } ) - ); - - return animation; - } - - jQuery.Animation = jQuery.extend( Animation, { - - tweeners: { - "*": [ function( prop, value ) { - var tween = this.createTween( prop, value ); - adjustCSS( tween.elem, prop, rcssNum.exec( value ), tween ); - return tween; - } ] - }, - - tweener: function( props, callback ) { - if ( isFunction( props ) ) { - callback = props; - props = [ "*" ]; - } else { - props = props.match( rnothtmlwhite ); - } - - var prop, - index = 0, - length = props.length; - - for ( ; index < length; index++ ) { - prop = props[ index ]; - Animation.tweeners[ prop ] = Animation.tweeners[ prop ] || []; - Animation.tweeners[ prop ].unshift( callback ); - } - }, - - prefilters: [ defaultPrefilter ], - - prefilter: function( callback, prepend ) { - if ( prepend ) { - Animation.prefilters.unshift( callback ); - } else { - Animation.prefilters.push( callback ); - } - } - } ); - - jQuery.speed = function( speed, easing, fn ) { - var opt = speed && typeof speed === "object" ? jQuery.extend( {}, speed ) : { - complete: fn || !fn && easing || - isFunction( speed ) && speed, - duration: speed, - easing: fn && easing || easing && !isFunction( easing ) && easing - }; - - // Go to the end state if fx are off - if ( jQuery.fx.off ) { - opt.duration = 0; - - } else { - if ( typeof opt.duration !== "number" ) { - if ( opt.duration in jQuery.fx.speeds ) { - opt.duration = jQuery.fx.speeds[ opt.duration ]; - - } else { - opt.duration = jQuery.fx.speeds._default; - } - } - } - - // Normalize opt.queue - true/undefined/null -> "fx" - if ( opt.queue == null || opt.queue === true ) { - opt.queue = "fx"; - } - - // Queueing - opt.old = opt.complete; - - opt.complete = function() { - if ( isFunction( opt.old ) ) { - opt.old.call( this ); - } - - if ( opt.queue ) { - jQuery.dequeue( this, opt.queue ); - } - }; - - return opt; - }; - - jQuery.fn.extend( { - fadeTo: function( speed, to, easing, callback ) { - - // Show any hidden elements after setting opacity to 0 - return this.filter( isHiddenWithinTree ).css( "opacity", 0 ).show() - - // Animate to the value specified - .end().animate( { opacity: to }, speed, easing, callback ); - }, - animate: function( prop, speed, easing, callback ) { - var empty = jQuery.isEmptyObject( prop ), - optall = jQuery.speed( speed, easing, callback ), - doAnimation = function() { - - // Operate on a copy of prop so per-property easing won't be lost - var anim = Animation( this, jQuery.extend( {}, prop ), optall ); - - // Empty animations, or finishing resolves immediately - if ( empty || dataPriv.get( this, "finish" ) ) { - anim.stop( true ); - } - }; - - doAnimation.finish = doAnimation; - - return empty || optall.queue === false ? - this.each( doAnimation ) : - this.queue( optall.queue, doAnimation ); - }, - stop: function( type, clearQueue, gotoEnd ) { - var stopQueue = function( hooks ) { - var stop = hooks.stop; - delete hooks.stop; - stop( gotoEnd ); - }; - - if ( typeof type !== "string" ) { - gotoEnd = clearQueue; - clearQueue = type; - type = undefined; - } - if ( clearQueue ) { - this.queue( type || "fx", [] ); - } - - return this.each( function() { - var dequeue = true, - index = type != null && type + "queueHooks", - timers = jQuery.timers, - data = dataPriv.get( this ); - - if ( index ) { - if ( data[ index ] && data[ index ].stop ) { - stopQueue( data[ index ] ); - } - } else { - for ( index in data ) { - if ( data[ index ] && data[ index ].stop && rrun.test( index ) ) { - stopQueue( data[ index ] ); - } - } - } - - for ( index = timers.length; index--; ) { - if ( timers[ index ].elem === this && - ( type == null || timers[ index ].queue === type ) ) { - - timers[ index ].anim.stop( gotoEnd ); - dequeue = false; - timers.splice( index, 1 ); - } - } - - // Start the next in the queue if the last step wasn't forced. - // Timers currently will call their complete callbacks, which - // will dequeue but only if they were gotoEnd. - if ( dequeue || !gotoEnd ) { - jQuery.dequeue( this, type ); - } - } ); - }, - finish: function( type ) { - if ( type !== false ) { - type = type || "fx"; - } - return this.each( function() { - var index, - data = dataPriv.get( this ), - queue = data[ type + "queue" ], - hooks = data[ type + "queueHooks" ], - timers = jQuery.timers, - length = queue ? queue.length : 0; - - // Enable finishing flag on private data - data.finish = true; - - // Empty the queue first - jQuery.queue( this, type, [] ); - - if ( hooks && hooks.stop ) { - hooks.stop.call( this, true ); - } - - // Look for any active animations, and finish them - for ( index = timers.length; index--; ) { - if ( timers[ index ].elem === this && timers[ index ].queue === type ) { - timers[ index ].anim.stop( true ); - timers.splice( index, 1 ); - } - } - - // Look for any animations in the old queue and finish them - for ( index = 0; index < length; index++ ) { - if ( queue[ index ] && queue[ index ].finish ) { - queue[ index ].finish.call( this ); - } - } - - // Turn off finishing flag - delete data.finish; - } ); - } - } ); - - jQuery.each( [ "toggle", "show", "hide" ], function( _i, name ) { - var cssFn = jQuery.fn[ name ]; - jQuery.fn[ name ] = function( speed, easing, callback ) { - return speed == null || typeof speed === "boolean" ? - cssFn.apply( this, arguments ) : - this.animate( genFx( name, true ), speed, easing, callback ); - }; - } ); - - // Generate shortcuts for custom animations - jQuery.each( { - slideDown: genFx( "show" ), - slideUp: genFx( "hide" ), - slideToggle: genFx( "toggle" ), - fadeIn: { opacity: "show" }, - fadeOut: { opacity: "hide" }, - fadeToggle: { opacity: "toggle" } - }, function( name, props ) { - jQuery.fn[ name ] = function( speed, easing, callback ) { - return this.animate( props, speed, easing, callback ); - }; - } ); - - jQuery.timers = []; - jQuery.fx.tick = function() { - var timer, - i = 0, - timers = jQuery.timers; - - fxNow = Date.now(); - - for ( ; i < timers.length; i++ ) { - timer = timers[ i ]; - - // Run the timer and safely remove it when done (allowing for external removal) - if ( !timer() && timers[ i ] === timer ) { - timers.splice( i--, 1 ); - } - } - - if ( !timers.length ) { - jQuery.fx.stop(); - } - fxNow = undefined; - }; - - jQuery.fx.timer = function( timer ) { - jQuery.timers.push( timer ); - jQuery.fx.start(); - }; - - jQuery.fx.interval = 13; - jQuery.fx.start = function() { - if ( inProgress ) { - return; - } - - inProgress = true; - schedule(); - }; - - jQuery.fx.stop = function() { - inProgress = null; - }; - - jQuery.fx.speeds = { - slow: 600, - fast: 200, - - // Default speed - _default: 400 - }; - - - // Based off of the plugin by Clint Helfers, with permission. - // https://web.archive.org/web/20100324014747/http://blindsignals.com/index.php/2009/07/jquery-delay/ - jQuery.fn.delay = function( time, type ) { - time = jQuery.fx ? jQuery.fx.speeds[ time ] || time : time; - type = type || "fx"; - - return this.queue( type, function( next, hooks ) { - var timeout = window.setTimeout( next, time ); - hooks.stop = function() { - window.clearTimeout( timeout ); - }; - } ); - }; - - - ( function() { - var input = document.createElement( "input" ), - select = document.createElement( "select" ), - opt = select.appendChild( document.createElement( "option" ) ); - - input.type = "checkbox"; - - // Support: Android <=4.3 only - // Default value for a checkbox should be "on" - support.checkOn = input.value !== ""; - - // Support: IE <=11 only - // Must access selectedIndex to make default options select - support.optSelected = opt.selected; - - // Support: IE <=11 only - // An input loses its value after becoming a radio - input = document.createElement( "input" ); - input.value = "t"; - input.type = "radio"; - support.radioValue = input.value === "t"; - } )(); - - - var boolHook, - attrHandle = jQuery.expr.attrHandle; - - jQuery.fn.extend( { - attr: function( name, value ) { - return access( this, jQuery.attr, name, value, arguments.length > 1 ); - }, - - removeAttr: function( name ) { - return this.each( function() { - jQuery.removeAttr( this, name ); - } ); - } - } ); - - jQuery.extend( { - attr: function( elem, name, value ) { - var ret, hooks, - nType = elem.nodeType; - - // Don't get/set attributes on text, comment and attribute nodes - if ( nType === 3 || nType === 8 || nType === 2 ) { - return; - } - - // Fallback to prop when attributes are not supported - if ( typeof elem.getAttribute === "undefined" ) { - return jQuery.prop( elem, name, value ); - } - - // Attribute hooks are determined by the lowercase version - // Grab necessary hook if one is defined - if ( nType !== 1 || !jQuery.isXMLDoc( elem ) ) { - hooks = jQuery.attrHooks[ name.toLowerCase() ] || - ( jQuery.expr.match.bool.test( name ) ? boolHook : undefined ); - } - - if ( value !== undefined ) { - if ( value === null ) { - jQuery.removeAttr( elem, name ); - return; - } - - if ( hooks && "set" in hooks && - ( ret = hooks.set( elem, value, name ) ) !== undefined ) { - return ret; - } - - elem.setAttribute( name, value + "" ); - return value; - } - - if ( hooks && "get" in hooks && ( ret = hooks.get( elem, name ) ) !== null ) { - return ret; - } - - ret = jQuery.find.attr( elem, name ); - - // Non-existent attributes return null, we normalize to undefined - return ret == null ? undefined : ret; - }, - - attrHooks: { - type: { - set: function( elem, value ) { - if ( !support.radioValue && value === "radio" && - nodeName( elem, "input" ) ) { - var val = elem.value; - elem.setAttribute( "type", value ); - if ( val ) { - elem.value = val; - } - return value; - } - } - } - }, - - removeAttr: function( elem, value ) { - var name, - i = 0, - - // Attribute names can contain non-HTML whitespace characters - // https://html.spec.whatwg.org/multipage/syntax.html#attributes-2 - attrNames = value && value.match( rnothtmlwhite ); - - if ( attrNames && elem.nodeType === 1 ) { - while ( ( name = attrNames[ i++ ] ) ) { - elem.removeAttribute( name ); - } - } - } - } ); - - // Hooks for boolean attributes - boolHook = { - set: function( elem, value, name ) { - if ( value === false ) { - - // Remove boolean attributes when set to false - jQuery.removeAttr( elem, name ); - } else { - elem.setAttribute( name, name ); - } - return name; - } - }; - - jQuery.each( jQuery.expr.match.bool.source.match( /\w+/g ), function( _i, name ) { - var getter = attrHandle[ name ] || jQuery.find.attr; - - attrHandle[ name ] = function( elem, name, isXML ) { - var ret, handle, - lowercaseName = name.toLowerCase(); - - if ( !isXML ) { - - // Avoid an infinite loop by temporarily removing this function from the getter - handle = attrHandle[ lowercaseName ]; - attrHandle[ lowercaseName ] = ret; - ret = getter( elem, name, isXML ) != null ? - lowercaseName : - null; - attrHandle[ lowercaseName ] = handle; - } - return ret; - }; - } ); - - - - - var rfocusable = /^(?:input|select|textarea|button)$/i, - rclickable = /^(?:a|area)$/i; - - jQuery.fn.extend( { - prop: function( name, value ) { - return access( this, jQuery.prop, name, value, arguments.length > 1 ); - }, - - removeProp: function( name ) { - return this.each( function() { - delete this[ jQuery.propFix[ name ] || name ]; - } ); - } - } ); - - jQuery.extend( { - prop: function( elem, name, value ) { - var ret, hooks, - nType = elem.nodeType; - - // Don't get/set properties on text, comment and attribute nodes - if ( nType === 3 || nType === 8 || nType === 2 ) { - return; - } - - if ( nType !== 1 || !jQuery.isXMLDoc( elem ) ) { - - // Fix name and attach hooks - name = jQuery.propFix[ name ] || name; - hooks = jQuery.propHooks[ name ]; - } - - if ( value !== undefined ) { - if ( hooks && "set" in hooks && - ( ret = hooks.set( elem, value, name ) ) !== undefined ) { - return ret; - } - - return ( elem[ name ] = value ); - } - - if ( hooks && "get" in hooks && ( ret = hooks.get( elem, name ) ) !== null ) { - return ret; - } - - return elem[ name ]; - }, - - propHooks: { - tabIndex: { - get: function( elem ) { - - // Support: IE <=9 - 11 only - // elem.tabIndex doesn't always return the - // correct value when it hasn't been explicitly set - // https://web.archive.org/web/20141116233347/http://fluidproject.org/blog/2008/01/09/getting-setting-and-removing-tabindex-values-with-javascript/ - // Use proper attribute retrieval(#12072) - var tabindex = jQuery.find.attr( elem, "tabindex" ); - - if ( tabindex ) { - return parseInt( tabindex, 10 ); - } - - if ( - rfocusable.test( elem.nodeName ) || - rclickable.test( elem.nodeName ) && - elem.href - ) { - return 0; - } - - return -1; - } - } - }, - - propFix: { - "for": "htmlFor", - "class": "className" - } - } ); - - // Support: IE <=11 only - // Accessing the selectedIndex property - // forces the browser to respect setting selected - // on the option - // The getter ensures a default option is selected - // when in an optgroup - // eslint rule "no-unused-expressions" is disabled for this code - // since it considers such accessions noop - if ( !support.optSelected ) { - jQuery.propHooks.selected = { - get: function( elem ) { - - /* eslint no-unused-expressions: "off" */ - - var parent = elem.parentNode; - if ( parent && parent.parentNode ) { - parent.parentNode.selectedIndex; - } - return null; - }, - set: function( elem ) { - - /* eslint no-unused-expressions: "off" */ - - var parent = elem.parentNode; - if ( parent ) { - parent.selectedIndex; - - if ( parent.parentNode ) { - parent.parentNode.selectedIndex; - } - } - } - }; - } - - jQuery.each( [ - "tabIndex", - "readOnly", - "maxLength", - "cellSpacing", - "cellPadding", - "rowSpan", - "colSpan", - "useMap", - "frameBorder", - "contentEditable" - ], function() { - jQuery.propFix[ this.toLowerCase() ] = this; - } ); - - - - - // Strip and collapse whitespace according to HTML spec - // https://infra.spec.whatwg.org/#strip-and-collapse-ascii-whitespace - function stripAndCollapse( value ) { - var tokens = value.match( rnothtmlwhite ) || []; - return tokens.join( " " ); - } - - - function getClass( elem ) { - return elem.getAttribute && elem.getAttribute( "class" ) || ""; - } - - function classesToArray( value ) { - if ( Array.isArray( value ) ) { - return value; - } - if ( typeof value === "string" ) { - return value.match( rnothtmlwhite ) || []; - } - return []; - } - - jQuery.fn.extend( { - addClass: function( value ) { - var classes, elem, cur, curValue, clazz, j, finalValue, - i = 0; - - if ( isFunction( value ) ) { - return this.each( function( j ) { - jQuery( this ).addClass( value.call( this, j, getClass( this ) ) ); - } ); - } - - classes = classesToArray( value ); - - if ( classes.length ) { - while ( ( elem = this[ i++ ] ) ) { - curValue = getClass( elem ); - cur = elem.nodeType === 1 && ( " " + stripAndCollapse( curValue ) + " " ); - - if ( cur ) { - j = 0; - while ( ( clazz = classes[ j++ ] ) ) { - if ( cur.indexOf( " " + clazz + " " ) < 0 ) { - cur += clazz + " "; - } - } - - // Only assign if different to avoid unneeded rendering. - finalValue = stripAndCollapse( cur ); - if ( curValue !== finalValue ) { - elem.setAttribute( "class", finalValue ); - } - } - } - } - - return this; - }, - - removeClass: function( value ) { - var classes, elem, cur, curValue, clazz, j, finalValue, - i = 0; - - if ( isFunction( value ) ) { - return this.each( function( j ) { - jQuery( this ).removeClass( value.call( this, j, getClass( this ) ) ); - } ); - } - - if ( !arguments.length ) { - return this.attr( "class", "" ); - } - - classes = classesToArray( value ); - - if ( classes.length ) { - while ( ( elem = this[ i++ ] ) ) { - curValue = getClass( elem ); - - // This expression is here for better compressibility (see addClass) - cur = elem.nodeType === 1 && ( " " + stripAndCollapse( curValue ) + " " ); - - if ( cur ) { - j = 0; - while ( ( clazz = classes[ j++ ] ) ) { - - // Remove *all* instances - while ( cur.indexOf( " " + clazz + " " ) > -1 ) { - cur = cur.replace( " " + clazz + " ", " " ); - } - } - - // Only assign if different to avoid unneeded rendering. - finalValue = stripAndCollapse( cur ); - if ( curValue !== finalValue ) { - elem.setAttribute( "class", finalValue ); - } - } - } - } - - return this; - }, - - toggleClass: function( value, stateVal ) { - var type = typeof value, - isValidValue = type === "string" || Array.isArray( value ); - - if ( typeof stateVal === "boolean" && isValidValue ) { - return stateVal ? this.addClass( value ) : this.removeClass( value ); - } - - if ( isFunction( value ) ) { - return this.each( function( i ) { - jQuery( this ).toggleClass( - value.call( this, i, getClass( this ), stateVal ), - stateVal - ); - } ); - } - - return this.each( function() { - var className, i, self, classNames; - - if ( isValidValue ) { - - // Toggle individual class names - i = 0; - self = jQuery( this ); - classNames = classesToArray( value ); - - while ( ( className = classNames[ i++ ] ) ) { - - // Check each className given, space separated list - if ( self.hasClass( className ) ) { - self.removeClass( className ); - } else { - self.addClass( className ); - } - } - - // Toggle whole class name - } else if ( value === undefined || type === "boolean" ) { - className = getClass( this ); - if ( className ) { - - // Store className if set - dataPriv.set( this, "__className__", className ); - } - - // If the element has a class name or if we're passed `false`, - // then remove the whole classname (if there was one, the above saved it). - // Otherwise bring back whatever was previously saved (if anything), - // falling back to the empty string if nothing was stored. - if ( this.setAttribute ) { - this.setAttribute( "class", - className || value === false ? - "" : - dataPriv.get( this, "__className__" ) || "" - ); - } - } - } ); - }, - - hasClass: function( selector ) { - var className, elem, - i = 0; - - className = " " + selector + " "; - while ( ( elem = this[ i++ ] ) ) { - if ( elem.nodeType === 1 && - ( " " + stripAndCollapse( getClass( elem ) ) + " " ).indexOf( className ) > -1 ) { - return true; - } - } - - return false; - } - } ); - - - - - var rreturn = /\r/g; - - jQuery.fn.extend( { - val: function( value ) { - var hooks, ret, valueIsFunction, - elem = this[ 0 ]; - - if ( !arguments.length ) { - if ( elem ) { - hooks = jQuery.valHooks[ elem.type ] || - jQuery.valHooks[ elem.nodeName.toLowerCase() ]; - - if ( hooks && - "get" in hooks && - ( ret = hooks.get( elem, "value" ) ) !== undefined - ) { - return ret; - } - - ret = elem.value; - - // Handle most common string cases - if ( typeof ret === "string" ) { - return ret.replace( rreturn, "" ); - } - - // Handle cases where value is null/undef or number - return ret == null ? "" : ret; - } - - return; - } - - valueIsFunction = isFunction( value ); - - return this.each( function( i ) { - var val; - - if ( this.nodeType !== 1 ) { - return; - } - - if ( valueIsFunction ) { - val = value.call( this, i, jQuery( this ).val() ); - } else { - val = value; - } - - // Treat null/undefined as ""; convert numbers to string - if ( val == null ) { - val = ""; - - } else if ( typeof val === "number" ) { - val += ""; - - } else if ( Array.isArray( val ) ) { - val = jQuery.map( val, function( value ) { - return value == null ? "" : value + ""; - } ); - } - - hooks = jQuery.valHooks[ this.type ] || jQuery.valHooks[ this.nodeName.toLowerCase() ]; - - // If set returns undefined, fall back to normal setting - if ( !hooks || !( "set" in hooks ) || hooks.set( this, val, "value" ) === undefined ) { - this.value = val; - } - } ); - } - } ); - - jQuery.extend( { - valHooks: { - option: { - get: function( elem ) { - - var val = jQuery.find.attr( elem, "value" ); - return val != null ? - val : - - // Support: IE <=10 - 11 only - // option.text throws exceptions (#14686, #14858) - // Strip and collapse whitespace - // https://html.spec.whatwg.org/#strip-and-collapse-whitespace - stripAndCollapse( jQuery.text( elem ) ); - } - }, - select: { - get: function( elem ) { - var value, option, i, - options = elem.options, - index = elem.selectedIndex, - one = elem.type === "select-one", - values = one ? null : [], - max = one ? index + 1 : options.length; - - if ( index < 0 ) { - i = max; - - } else { - i = one ? index : 0; - } - - // Loop through all the selected options - for ( ; i < max; i++ ) { - option = options[ i ]; - - // Support: IE <=9 only - // IE8-9 doesn't update selected after form reset (#2551) - if ( ( option.selected || i === index ) && - - // Don't return options that are disabled or in a disabled optgroup - !option.disabled && - ( !option.parentNode.disabled || - !nodeName( option.parentNode, "optgroup" ) ) ) { - - // Get the specific value for the option - value = jQuery( option ).val(); - - // We don't need an array for one selects - if ( one ) { - return value; - } - - // Multi-Selects return an array - values.push( value ); - } - } - - return values; - }, - - set: function( elem, value ) { - var optionSet, option, - options = elem.options, - values = jQuery.makeArray( value ), - i = options.length; - - while ( i-- ) { - option = options[ i ]; - - /* eslint-disable no-cond-assign */ - - if ( option.selected = - jQuery.inArray( jQuery.valHooks.option.get( option ), values ) > -1 - ) { - optionSet = true; - } - - /* eslint-enable no-cond-assign */ - } - - // Force browsers to behave consistently when non-matching value is set - if ( !optionSet ) { - elem.selectedIndex = -1; - } - return values; - } - } - } - } ); - - // Radios and checkboxes getter/setter - jQuery.each( [ "radio", "checkbox" ], function() { - jQuery.valHooks[ this ] = { - set: function( elem, value ) { - if ( Array.isArray( value ) ) { - return ( elem.checked = jQuery.inArray( jQuery( elem ).val(), value ) > -1 ); - } - } - }; - if ( !support.checkOn ) { - jQuery.valHooks[ this ].get = function( elem ) { - return elem.getAttribute( "value" ) === null ? "on" : elem.value; - }; - } - } ); - - - - - // Return jQuery for attributes-only inclusion - - - support.focusin = "onfocusin" in window; - - - var rfocusMorph = /^(?:focusinfocus|focusoutblur)$/, - stopPropagationCallback = function( e ) { - e.stopPropagation(); - }; - - jQuery.extend( jQuery.event, { - - trigger: function( event, data, elem, onlyHandlers ) { - - var i, cur, tmp, bubbleType, ontype, handle, special, lastElement, - eventPath = [ elem || document ], - type = hasOwn.call( event, "type" ) ? event.type : event, - namespaces = hasOwn.call( event, "namespace" ) ? event.namespace.split( "." ) : []; - - cur = lastElement = tmp = elem = elem || document; - - // Don't do events on text and comment nodes - if ( elem.nodeType === 3 || elem.nodeType === 8 ) { - return; - } - - // focus/blur morphs to focusin/out; ensure we're not firing them right now - if ( rfocusMorph.test( type + jQuery.event.triggered ) ) { - return; - } - - if ( type.indexOf( "." ) > -1 ) { - - // Namespaced trigger; create a regexp to match event type in handle() - namespaces = type.split( "." ); - type = namespaces.shift(); - namespaces.sort(); - } - ontype = type.indexOf( ":" ) < 0 && "on" + type; - - // Caller can pass in a jQuery.Event object, Object, or just an event type string - event = event[ jQuery.expando ] ? - event : - new jQuery.Event( type, typeof event === "object" && event ); - - // Trigger bitmask: & 1 for native handlers; & 2 for jQuery (always true) - event.isTrigger = onlyHandlers ? 2 : 3; - event.namespace = namespaces.join( "." ); - event.rnamespace = event.namespace ? - new RegExp( "(^|\\.)" + namespaces.join( "\\.(?:.*\\.|)" ) + "(\\.|$)" ) : - null; - - // Clean up the event in case it is being reused - event.result = undefined; - if ( !event.target ) { - event.target = elem; - } - - // Clone any incoming data and prepend the event, creating the handler arg list - data = data == null ? - [ event ] : - jQuery.makeArray( data, [ event ] ); - - // Allow special events to draw outside the lines - special = jQuery.event.special[ type ] || {}; - if ( !onlyHandlers && special.trigger && special.trigger.apply( elem, data ) === false ) { - return; - } - - // Determine event propagation path in advance, per W3C events spec (#9951) - // Bubble up to document, then to window; watch for a global ownerDocument var (#9724) - if ( !onlyHandlers && !special.noBubble && !isWindow( elem ) ) { - - bubbleType = special.delegateType || type; - if ( !rfocusMorph.test( bubbleType + type ) ) { - cur = cur.parentNode; - } - for ( ; cur; cur = cur.parentNode ) { - eventPath.push( cur ); - tmp = cur; - } - - // Only add window if we got to document (e.g., not plain obj or detached DOM) - if ( tmp === ( elem.ownerDocument || document ) ) { - eventPath.push( tmp.defaultView || tmp.parentWindow || window ); - } - } - - // Fire handlers on the event path - i = 0; - while ( ( cur = eventPath[ i++ ] ) && !event.isPropagationStopped() ) { - lastElement = cur; - event.type = i > 1 ? - bubbleType : - special.bindType || type; - - // jQuery handler - handle = ( dataPriv.get( cur, "events" ) || Object.create( null ) )[ event.type ] && - dataPriv.get( cur, "handle" ); - if ( handle ) { - handle.apply( cur, data ); - } - - // Native handler - handle = ontype && cur[ ontype ]; - if ( handle && handle.apply && acceptData( cur ) ) { - event.result = handle.apply( cur, data ); - if ( event.result === false ) { - event.preventDefault(); - } - } - } - event.type = type; - - // If nobody prevented the default action, do it now - if ( !onlyHandlers && !event.isDefaultPrevented() ) { - - if ( ( !special._default || - special._default.apply( eventPath.pop(), data ) === false ) && - acceptData( elem ) ) { - - // Call a native DOM method on the target with the same name as the event. - // Don't do default actions on window, that's where global variables be (#6170) - if ( ontype && isFunction( elem[ type ] ) && !isWindow( elem ) ) { - - // Don't re-trigger an onFOO event when we call its FOO() method - tmp = elem[ ontype ]; - - if ( tmp ) { - elem[ ontype ] = null; - } - - // Prevent re-triggering of the same event, since we already bubbled it above - jQuery.event.triggered = type; - - if ( event.isPropagationStopped() ) { - lastElement.addEventListener( type, stopPropagationCallback ); - } - - elem[ type ](); - - if ( event.isPropagationStopped() ) { - lastElement.removeEventListener( type, stopPropagationCallback ); - } - - jQuery.event.triggered = undefined; - - if ( tmp ) { - elem[ ontype ] = tmp; - } - } - } - } - - return event.result; - }, - - // Piggyback on a donor event to simulate a different one - // Used only for `focus(in | out)` events - simulate: function( type, elem, event ) { - var e = jQuery.extend( - new jQuery.Event(), - event, - { - type: type, - isSimulated: true - } - ); - - jQuery.event.trigger( e, null, elem ); - } - - } ); - - jQuery.fn.extend( { - - trigger: function( type, data ) { - return this.each( function() { - jQuery.event.trigger( type, data, this ); - } ); - }, - triggerHandler: function( type, data ) { - var elem = this[ 0 ]; - if ( elem ) { - return jQuery.event.trigger( type, data, elem, true ); - } - } - } ); - - - // Support: Firefox <=44 - // Firefox doesn't have focus(in | out) events - // Related ticket - https://bugzilla.mozilla.org/show_bug.cgi?id=687787 - // - // Support: Chrome <=48 - 49, Safari <=9.0 - 9.1 - // focus(in | out) events fire after focus & blur events, - // which is spec violation - http://www.w3.org/TR/DOM-Level-3-Events/#events-focusevent-event-order - // Related ticket - https://bugs.chromium.org/p/chromium/issues/detail?id=449857 - if ( !support.focusin ) { - jQuery.each( { focus: "focusin", blur: "focusout" }, function( orig, fix ) { - - // Attach a single capturing handler on the document while someone wants focusin/focusout - var handler = function( event ) { - jQuery.event.simulate( fix, event.target, jQuery.event.fix( event ) ); - }; - - jQuery.event.special[ fix ] = { - setup: function() { - - // Handle: regular nodes (via `this.ownerDocument`), window - // (via `this.document`) & document (via `this`). - var doc = this.ownerDocument || this.document || this, - attaches = dataPriv.access( doc, fix ); - - if ( !attaches ) { - doc.addEventListener( orig, handler, true ); - } - dataPriv.access( doc, fix, ( attaches || 0 ) + 1 ); - }, - teardown: function() { - var doc = this.ownerDocument || this.document || this, - attaches = dataPriv.access( doc, fix ) - 1; - - if ( !attaches ) { - doc.removeEventListener( orig, handler, true ); - dataPriv.remove( doc, fix ); - - } else { - dataPriv.access( doc, fix, attaches ); - } - } - }; - } ); - } - var location = window.location; - - var nonce = { guid: Date.now() }; - - var rquery = ( /\?/ ); - - - - // Cross-browser xml parsing - jQuery.parseXML = function( data ) { - var xml, parserErrorElem; - if ( !data || typeof data !== "string" ) { - return null; - } - - // Support: IE 9 - 11 only - // IE throws on parseFromString with invalid input. - try { - xml = ( new window.DOMParser() ).parseFromString( data, "text/xml" ); - } catch ( e ) {} - - parserErrorElem = xml && xml.getElementsByTagName( "parsererror" )[ 0 ]; - if ( !xml || parserErrorElem ) { - jQuery.error( "Invalid XML: " + ( - parserErrorElem ? - jQuery.map( parserErrorElem.childNodes, function( el ) { - return el.textContent; - } ).join( "\n" ) : - data - ) ); - } - return xml; - }; - - - var - rbracket = /\[\]$/, - rCRLF = /\r?\n/g, - rsubmitterTypes = /^(?:submit|button|image|reset|file)$/i, - rsubmittable = /^(?:input|select|textarea|keygen)/i; - - function buildParams( prefix, obj, traditional, add ) { - var name; - - if ( Array.isArray( obj ) ) { - - // Serialize array item. - jQuery.each( obj, function( i, v ) { - if ( traditional || rbracket.test( prefix ) ) { - - // Treat each array item as a scalar. - add( prefix, v ); - - } else { - - // Item is non-scalar (array or object), encode its numeric index. - buildParams( - prefix + "[" + ( typeof v === "object" && v != null ? i : "" ) + "]", - v, - traditional, - add - ); - } - } ); - - } else if ( !traditional && toType( obj ) === "object" ) { - - // Serialize object item. - for ( name in obj ) { - buildParams( prefix + "[" + name + "]", obj[ name ], traditional, add ); - } - - } else { - - // Serialize scalar item. - add( prefix, obj ); - } - } - - // Serialize an array of form elements or a set of - // key/values into a query string - jQuery.param = function( a, traditional ) { - var prefix, - s = [], - add = function( key, valueOrFunction ) { - - // If value is a function, invoke it and use its return value - var value = isFunction( valueOrFunction ) ? - valueOrFunction() : - valueOrFunction; - - s[ s.length ] = encodeURIComponent( key ) + "=" + - encodeURIComponent( value == null ? "" : value ); - }; - - if ( a == null ) { - return ""; - } - - // If an array was passed in, assume that it is an array of form elements. - if ( Array.isArray( a ) || ( a.jquery && !jQuery.isPlainObject( a ) ) ) { - - // Serialize the form elements - jQuery.each( a, function() { - add( this.name, this.value ); - } ); - - } else { - - // If traditional, encode the "old" way (the way 1.3.2 or older - // did it), otherwise encode params recursively. - for ( prefix in a ) { - buildParams( prefix, a[ prefix ], traditional, add ); - } - } - - // Return the resulting serialization - return s.join( "&" ); - }; - - jQuery.fn.extend( { - serialize: function() { - return jQuery.param( this.serializeArray() ); - }, - serializeArray: function() { - return this.map( function() { - - // Can add propHook for "elements" to filter or add form elements - var elements = jQuery.prop( this, "elements" ); - return elements ? jQuery.makeArray( elements ) : this; - } ).filter( function() { - var type = this.type; - - // Use .is( ":disabled" ) so that fieldset[disabled] works - return this.name && !jQuery( this ).is( ":disabled" ) && - rsubmittable.test( this.nodeName ) && !rsubmitterTypes.test( type ) && - ( this.checked || !rcheckableType.test( type ) ); - } ).map( function( _i, elem ) { - var val = jQuery( this ).val(); - - if ( val == null ) { - return null; - } - - if ( Array.isArray( val ) ) { - return jQuery.map( val, function( val ) { - return { name: elem.name, value: val.replace( rCRLF, "\r\n" ) }; - } ); - } - - return { name: elem.name, value: val.replace( rCRLF, "\r\n" ) }; - } ).get(); - } - } ); - - - var - r20 = /%20/g, - rhash = /#.*$/, - rantiCache = /([?&])_=[^&]*/, - rheaders = /^(.*?):[ \t]*([^\r\n]*)$/mg, - - // #7653, #8125, #8152: local protocol detection - rlocalProtocol = /^(?:about|app|app-storage|.+-extension|file|res|widget):$/, - rnoContent = /^(?:GET|HEAD)$/, - rprotocol = /^\/\//, - - /* Prefilters - * 1) They are useful to introduce custom dataTypes (see ajax/jsonp.js for an example) - * 2) These are called: - * - BEFORE asking for a transport - * - AFTER param serialization (s.data is a string if s.processData is true) - * 3) key is the dataType - * 4) the catchall symbol "*" can be used - * 5) execution will start with transport dataType and THEN continue down to "*" if needed - */ - prefilters = {}, - - /* Transports bindings - * 1) key is the dataType - * 2) the catchall symbol "*" can be used - * 3) selection will start with transport dataType and THEN go to "*" if needed - */ - transports = {}, - - // Avoid comment-prolog char sequence (#10098); must appease lint and evade compression - allTypes = "*/".concat( "*" ), - - // Anchor tag for parsing the document origin - originAnchor = document.createElement( "a" ); - - originAnchor.href = location.href; - - // Base "constructor" for jQuery.ajaxPrefilter and jQuery.ajaxTransport - function addToPrefiltersOrTransports( structure ) { - - // dataTypeExpression is optional and defaults to "*" - return function( dataTypeExpression, func ) { - - if ( typeof dataTypeExpression !== "string" ) { - func = dataTypeExpression; - dataTypeExpression = "*"; - } - - var dataType, - i = 0, - dataTypes = dataTypeExpression.toLowerCase().match( rnothtmlwhite ) || []; - - if ( isFunction( func ) ) { - - // For each dataType in the dataTypeExpression - while ( ( dataType = dataTypes[ i++ ] ) ) { - - // Prepend if requested - if ( dataType[ 0 ] === "+" ) { - dataType = dataType.slice( 1 ) || "*"; - ( structure[ dataType ] = structure[ dataType ] || [] ).unshift( func ); - - // Otherwise append - } else { - ( structure[ dataType ] = structure[ dataType ] || [] ).push( func ); - } - } - } - }; - } - - // Base inspection function for prefilters and transports - function inspectPrefiltersOrTransports( structure, options, originalOptions, jqXHR ) { - - var inspected = {}, - seekingTransport = ( structure === transports ); - - function inspect( dataType ) { - var selected; - inspected[ dataType ] = true; - jQuery.each( structure[ dataType ] || [], function( _, prefilterOrFactory ) { - var dataTypeOrTransport = prefilterOrFactory( options, originalOptions, jqXHR ); - if ( typeof dataTypeOrTransport === "string" && - !seekingTransport && !inspected[ dataTypeOrTransport ] ) { - - options.dataTypes.unshift( dataTypeOrTransport ); - inspect( dataTypeOrTransport ); - return false; - } else if ( seekingTransport ) { - return !( selected = dataTypeOrTransport ); - } - } ); - return selected; - } - - return inspect( options.dataTypes[ 0 ] ) || !inspected[ "*" ] && inspect( "*" ); - } - - // A special extend for ajax options - // that takes "flat" options (not to be deep extended) - // Fixes #9887 - function ajaxExtend( target, src ) { - var key, deep, - flatOptions = jQuery.ajaxSettings.flatOptions || {}; - - for ( key in src ) { - if ( src[ key ] !== undefined ) { - ( flatOptions[ key ] ? target : ( deep || ( deep = {} ) ) )[ key ] = src[ key ]; - } - } - if ( deep ) { - jQuery.extend( true, target, deep ); - } - - return target; - } - - /* Handles responses to an ajax request: - * - finds the right dataType (mediates between content-type and expected dataType) - * - returns the corresponding response - */ - function ajaxHandleResponses( s, jqXHR, responses ) { - - var ct, type, finalDataType, firstDataType, - contents = s.contents, - dataTypes = s.dataTypes; - - // Remove auto dataType and get content-type in the process - while ( dataTypes[ 0 ] === "*" ) { - dataTypes.shift(); - if ( ct === undefined ) { - ct = s.mimeType || jqXHR.getResponseHeader( "Content-Type" ); - } - } - - // Check if we're dealing with a known content-type - if ( ct ) { - for ( type in contents ) { - if ( contents[ type ] && contents[ type ].test( ct ) ) { - dataTypes.unshift( type ); - break; - } - } - } - - // Check to see if we have a response for the expected dataType - if ( dataTypes[ 0 ] in responses ) { - finalDataType = dataTypes[ 0 ]; - } else { - - // Try convertible dataTypes - for ( type in responses ) { - if ( !dataTypes[ 0 ] || s.converters[ type + " " + dataTypes[ 0 ] ] ) { - finalDataType = type; - break; - } - if ( !firstDataType ) { - firstDataType = type; - } - } - - // Or just use first one - finalDataType = finalDataType || firstDataType; - } - - // If we found a dataType - // We add the dataType to the list if needed - // and return the corresponding response - if ( finalDataType ) { - if ( finalDataType !== dataTypes[ 0 ] ) { - dataTypes.unshift( finalDataType ); - } - return responses[ finalDataType ]; - } - } - - /* Chain conversions given the request and the original response - * Also sets the responseXXX fields on the jqXHR instance - */ - function ajaxConvert( s, response, jqXHR, isSuccess ) { - var conv2, current, conv, tmp, prev, - converters = {}, - - // Work with a copy of dataTypes in case we need to modify it for conversion - dataTypes = s.dataTypes.slice(); - - // Create converters map with lowercased keys - if ( dataTypes[ 1 ] ) { - for ( conv in s.converters ) { - converters[ conv.toLowerCase() ] = s.converters[ conv ]; - } - } - - current = dataTypes.shift(); - - // Convert to each sequential dataType - while ( current ) { - - if ( s.responseFields[ current ] ) { - jqXHR[ s.responseFields[ current ] ] = response; - } - - // Apply the dataFilter if provided - if ( !prev && isSuccess && s.dataFilter ) { - response = s.dataFilter( response, s.dataType ); - } - - prev = current; - current = dataTypes.shift(); - - if ( current ) { - - // There's only work to do if current dataType is non-auto - if ( current === "*" ) { - - current = prev; - - // Convert response if prev dataType is non-auto and differs from current - } else if ( prev !== "*" && prev !== current ) { - - // Seek a direct converter - conv = converters[ prev + " " + current ] || converters[ "* " + current ]; - - // If none found, seek a pair - if ( !conv ) { - for ( conv2 in converters ) { - - // If conv2 outputs current - tmp = conv2.split( " " ); - if ( tmp[ 1 ] === current ) { - - // If prev can be converted to accepted input - conv = converters[ prev + " " + tmp[ 0 ] ] || - converters[ "* " + tmp[ 0 ] ]; - if ( conv ) { - - // Condense equivalence converters - if ( conv === true ) { - conv = converters[ conv2 ]; - - // Otherwise, insert the intermediate dataType - } else if ( converters[ conv2 ] !== true ) { - current = tmp[ 0 ]; - dataTypes.unshift( tmp[ 1 ] ); - } - break; - } - } - } - } - - // Apply converter (if not an equivalence) - if ( conv !== true ) { - - // Unless errors are allowed to bubble, catch and return them - if ( conv && s.throws ) { - response = conv( response ); - } else { - try { - response = conv( response ); - } catch ( e ) { - return { - state: "parsererror", - error: conv ? e : "No conversion from " + prev + " to " + current - }; - } - } - } - } - } - } - - return { state: "success", data: response }; - } - - jQuery.extend( { - - // Counter for holding the number of active queries - active: 0, - - // Last-Modified header cache for next request - lastModified: {}, - etag: {}, - - ajaxSettings: { - url: location.href, - type: "GET", - isLocal: rlocalProtocol.test( location.protocol ), - global: true, - processData: true, - async: true, - contentType: "application/x-www-form-urlencoded; charset=UTF-8", - - /* - timeout: 0, - data: null, - dataType: null, - username: null, - password: null, - cache: null, - throws: false, - traditional: false, - headers: {}, - */ - - accepts: { - "*": allTypes, - text: "text/plain", - html: "text/html", - xml: "application/xml, text/xml", - json: "application/json, text/javascript" - }, - - contents: { - xml: /\bxml\b/, - html: /\bhtml/, - json: /\bjson\b/ - }, - - responseFields: { - xml: "responseXML", - text: "responseText", - json: "responseJSON" - }, - - // Data converters - // Keys separate source (or catchall "*") and destination types with a single space - converters: { - - // Convert anything to text - "* text": String, - - // Text to html (true = no transformation) - "text html": true, - - // Evaluate text as a json expression - "text json": JSON.parse, - - // Parse text as xml - "text xml": jQuery.parseXML - }, - - // For options that shouldn't be deep extended: - // you can add your own custom options here if - // and when you create one that shouldn't be - // deep extended (see ajaxExtend) - flatOptions: { - url: true, - context: true - } - }, - - // Creates a full fledged settings object into target - // with both ajaxSettings and settings fields. - // If target is omitted, writes into ajaxSettings. - ajaxSetup: function( target, settings ) { - return settings ? - - // Building a settings object - ajaxExtend( ajaxExtend( target, jQuery.ajaxSettings ), settings ) : - - // Extending ajaxSettings - ajaxExtend( jQuery.ajaxSettings, target ); - }, - - ajaxPrefilter: addToPrefiltersOrTransports( prefilters ), - ajaxTransport: addToPrefiltersOrTransports( transports ), - - // Main method - ajax: function( url, options ) { - - // If url is an object, simulate pre-1.5 signature - if ( typeof url === "object" ) { - options = url; - url = undefined; - } - - // Force options to be an object - options = options || {}; - - var transport, - - // URL without anti-cache param - cacheURL, - - // Response headers - responseHeadersString, - responseHeaders, - - // timeout handle - timeoutTimer, - - // Url cleanup var - urlAnchor, - - // Request state (becomes false upon send and true upon completion) - completed, - - // To know if global events are to be dispatched - fireGlobals, - - // Loop variable - i, - - // uncached part of the url - uncached, - - // Create the final options object - s = jQuery.ajaxSetup( {}, options ), - - // Callbacks context - callbackContext = s.context || s, - - // Context for global events is callbackContext if it is a DOM node or jQuery collection - globalEventContext = s.context && - ( callbackContext.nodeType || callbackContext.jquery ) ? - jQuery( callbackContext ) : - jQuery.event, - - // Deferreds - deferred = jQuery.Deferred(), - completeDeferred = jQuery.Callbacks( "once memory" ), - - // Status-dependent callbacks - statusCode = s.statusCode || {}, - - // Headers (they are sent all at once) - requestHeaders = {}, - requestHeadersNames = {}, - - // Default abort message - strAbort = "canceled", - - // Fake xhr - jqXHR = { - readyState: 0, - - // Builds headers hashtable if needed - getResponseHeader: function( key ) { - var match; - if ( completed ) { - if ( !responseHeaders ) { - responseHeaders = {}; - while ( ( match = rheaders.exec( responseHeadersString ) ) ) { - responseHeaders[ match[ 1 ].toLowerCase() + " " ] = - ( responseHeaders[ match[ 1 ].toLowerCase() + " " ] || [] ) - .concat( match[ 2 ] ); - } - } - match = responseHeaders[ key.toLowerCase() + " " ]; - } - return match == null ? null : match.join( ", " ); - }, - - // Raw string - getAllResponseHeaders: function() { - return completed ? responseHeadersString : null; - }, - - // Caches the header - setRequestHeader: function( name, value ) { - if ( completed == null ) { - name = requestHeadersNames[ name.toLowerCase() ] = - requestHeadersNames[ name.toLowerCase() ] || name; - requestHeaders[ name ] = value; - } - return this; - }, - - // Overrides response content-type header - overrideMimeType: function( type ) { - if ( completed == null ) { - s.mimeType = type; - } - return this; - }, - - // Status-dependent callbacks - statusCode: function( map ) { - var code; - if ( map ) { - if ( completed ) { - - // Execute the appropriate callbacks - jqXHR.always( map[ jqXHR.status ] ); - } else { - - // Lazy-add the new callbacks in a way that preserves old ones - for ( code in map ) { - statusCode[ code ] = [ statusCode[ code ], map[ code ] ]; - } - } - } - return this; - }, - - // Cancel the request - abort: function( statusText ) { - var finalText = statusText || strAbort; - if ( transport ) { - transport.abort( finalText ); - } - done( 0, finalText ); - return this; - } - }; - - // Attach deferreds - deferred.promise( jqXHR ); - - // Add protocol if not provided (prefilters might expect it) - // Handle falsy url in the settings object (#10093: consistency with old signature) - // We also use the url parameter if available - s.url = ( ( url || s.url || location.href ) + "" ) - .replace( rprotocol, location.protocol + "//" ); - - // Alias method option to type as per ticket #12004 - s.type = options.method || options.type || s.method || s.type; - - // Extract dataTypes list - s.dataTypes = ( s.dataType || "*" ).toLowerCase().match( rnothtmlwhite ) || [ "" ]; - - // A cross-domain request is in order when the origin doesn't match the current origin. - if ( s.crossDomain == null ) { - urlAnchor = document.createElement( "a" ); - - // Support: IE <=8 - 11, Edge 12 - 15 - // IE throws exception on accessing the href property if url is malformed, - // e.g. http://example.com:80x/ - try { - urlAnchor.href = s.url; - - // Support: IE <=8 - 11 only - // Anchor's host property isn't correctly set when s.url is relative - urlAnchor.href = urlAnchor.href; - s.crossDomain = originAnchor.protocol + "//" + originAnchor.host !== - urlAnchor.protocol + "//" + urlAnchor.host; - } catch ( e ) { - - // If there is an error parsing the URL, assume it is crossDomain, - // it can be rejected by the transport if it is invalid - s.crossDomain = true; - } - } - - // Convert data if not already a string - if ( s.data && s.processData && typeof s.data !== "string" ) { - s.data = jQuery.param( s.data, s.traditional ); - } - - // Apply prefilters - inspectPrefiltersOrTransports( prefilters, s, options, jqXHR ); - - // If request was aborted inside a prefilter, stop there - if ( completed ) { - return jqXHR; - } - - // We can fire global events as of now if asked to - // Don't fire events if jQuery.event is undefined in an AMD-usage scenario (#15118) - fireGlobals = jQuery.event && s.global; - - // Watch for a new set of requests - if ( fireGlobals && jQuery.active++ === 0 ) { - jQuery.event.trigger( "ajaxStart" ); - } - - // Uppercase the type - s.type = s.type.toUpperCase(); - - // Determine if request has content - s.hasContent = !rnoContent.test( s.type ); - - // Save the URL in case we're toying with the If-Modified-Since - // and/or If-None-Match header later on - // Remove hash to simplify url manipulation - cacheURL = s.url.replace( rhash, "" ); - - // More options handling for requests with no content - if ( !s.hasContent ) { - - // Remember the hash so we can put it back - uncached = s.url.slice( cacheURL.length ); - - // If data is available and should be processed, append data to url - if ( s.data && ( s.processData || typeof s.data === "string" ) ) { - cacheURL += ( rquery.test( cacheURL ) ? "&" : "?" ) + s.data; - - // #9682: remove data so that it's not used in an eventual retry - delete s.data; - } - - // Add or update anti-cache param if needed - if ( s.cache === false ) { - cacheURL = cacheURL.replace( rantiCache, "$1" ); - uncached = ( rquery.test( cacheURL ) ? "&" : "?" ) + "_=" + ( nonce.guid++ ) + - uncached; - } - - // Put hash and anti-cache on the URL that will be requested (gh-1732) - s.url = cacheURL + uncached; - - // Change '%20' to '+' if this is encoded form body content (gh-2658) - } else if ( s.data && s.processData && - ( s.contentType || "" ).indexOf( "application/x-www-form-urlencoded" ) === 0 ) { - s.data = s.data.replace( r20, "+" ); - } - - // Set the If-Modified-Since and/or If-None-Match header, if in ifModified mode. - if ( s.ifModified ) { - if ( jQuery.lastModified[ cacheURL ] ) { - jqXHR.setRequestHeader( "If-Modified-Since", jQuery.lastModified[ cacheURL ] ); - } - if ( jQuery.etag[ cacheURL ] ) { - jqXHR.setRequestHeader( "If-None-Match", jQuery.etag[ cacheURL ] ); - } - } - - // Set the correct header, if data is being sent - if ( s.data && s.hasContent && s.contentType !== false || options.contentType ) { - jqXHR.setRequestHeader( "Content-Type", s.contentType ); - } - - // Set the Accepts header for the server, depending on the dataType - jqXHR.setRequestHeader( - "Accept", - s.dataTypes[ 0 ] && s.accepts[ s.dataTypes[ 0 ] ] ? - s.accepts[ s.dataTypes[ 0 ] ] + - ( s.dataTypes[ 0 ] !== "*" ? ", " + allTypes + "; q=0.01" : "" ) : - s.accepts[ "*" ] - ); - - // Check for headers option - for ( i in s.headers ) { - jqXHR.setRequestHeader( i, s.headers[ i ] ); - } - - // Allow custom headers/mimetypes and early abort - if ( s.beforeSend && - ( s.beforeSend.call( callbackContext, jqXHR, s ) === false || completed ) ) { - - // Abort if not done already and return - return jqXHR.abort(); - } - - // Aborting is no longer a cancellation - strAbort = "abort"; - - // Install callbacks on deferreds - completeDeferred.add( s.complete ); - jqXHR.done( s.success ); - jqXHR.fail( s.error ); - - // Get transport - transport = inspectPrefiltersOrTransports( transports, s, options, jqXHR ); - - // If no transport, we auto-abort - if ( !transport ) { - done( -1, "No Transport" ); - } else { - jqXHR.readyState = 1; - - // Send global event - if ( fireGlobals ) { - globalEventContext.trigger( "ajaxSend", [ jqXHR, s ] ); - } - - // If request was aborted inside ajaxSend, stop there - if ( completed ) { - return jqXHR; - } - - // Timeout - if ( s.async && s.timeout > 0 ) { - timeoutTimer = window.setTimeout( function() { - jqXHR.abort( "timeout" ); - }, s.timeout ); - } - - try { - completed = false; - transport.send( requestHeaders, done ); - } catch ( e ) { - - // Rethrow post-completion exceptions - if ( completed ) { - throw e; - } - - // Propagate others as results - done( -1, e ); - } - } - - // Callback for when everything is done - function done( status, nativeStatusText, responses, headers ) { - var isSuccess, success, error, response, modified, - statusText = nativeStatusText; - - // Ignore repeat invocations - if ( completed ) { - return; - } - - completed = true; - - // Clear timeout if it exists - if ( timeoutTimer ) { - window.clearTimeout( timeoutTimer ); - } - - // Dereference transport for early garbage collection - // (no matter how long the jqXHR object will be used) - transport = undefined; - - // Cache response headers - responseHeadersString = headers || ""; - - // Set readyState - jqXHR.readyState = status > 0 ? 4 : 0; - - // Determine if successful - isSuccess = status >= 200 && status < 300 || status === 304; - - // Get response data - if ( responses ) { - response = ajaxHandleResponses( s, jqXHR, responses ); - } - - // Use a noop converter for missing script but not if jsonp - if ( !isSuccess && - jQuery.inArray( "script", s.dataTypes ) > -1 && - jQuery.inArray( "json", s.dataTypes ) < 0 ) { - s.converters[ "text script" ] = function() {}; - } - - // Convert no matter what (that way responseXXX fields are always set) - response = ajaxConvert( s, response, jqXHR, isSuccess ); - - // If successful, handle type chaining - if ( isSuccess ) { - - // Set the If-Modified-Since and/or If-None-Match header, if in ifModified mode. - if ( s.ifModified ) { - modified = jqXHR.getResponseHeader( "Last-Modified" ); - if ( modified ) { - jQuery.lastModified[ cacheURL ] = modified; - } - modified = jqXHR.getResponseHeader( "etag" ); - if ( modified ) { - jQuery.etag[ cacheURL ] = modified; - } - } - - // if no content - if ( status === 204 || s.type === "HEAD" ) { - statusText = "nocontent"; - - // if not modified - } else if ( status === 304 ) { - statusText = "notmodified"; - - // If we have data, let's convert it - } else { - statusText = response.state; - success = response.data; - error = response.error; - isSuccess = !error; - } - } else { - - // Extract error from statusText and normalize for non-aborts - error = statusText; - if ( status || !statusText ) { - statusText = "error"; - if ( status < 0 ) { - status = 0; - } - } - } - - // Set data for the fake xhr object - jqXHR.status = status; - jqXHR.statusText = ( nativeStatusText || statusText ) + ""; - - // Success/Error - if ( isSuccess ) { - deferred.resolveWith( callbackContext, [ success, statusText, jqXHR ] ); - } else { - deferred.rejectWith( callbackContext, [ jqXHR, statusText, error ] ); - } - - // Status-dependent callbacks - jqXHR.statusCode( statusCode ); - statusCode = undefined; - - if ( fireGlobals ) { - globalEventContext.trigger( isSuccess ? "ajaxSuccess" : "ajaxError", - [ jqXHR, s, isSuccess ? success : error ] ); - } - - // Complete - completeDeferred.fireWith( callbackContext, [ jqXHR, statusText ] ); - - if ( fireGlobals ) { - globalEventContext.trigger( "ajaxComplete", [ jqXHR, s ] ); - - // Handle the global AJAX counter - if ( !( --jQuery.active ) ) { - jQuery.event.trigger( "ajaxStop" ); - } - } - } - - return jqXHR; - }, - - getJSON: function( url, data, callback ) { - return jQuery.get( url, data, callback, "json" ); - }, - - getScript: function( url, callback ) { - return jQuery.get( url, undefined, callback, "script" ); - } - } ); - - jQuery.each( [ "get", "post" ], function( _i, method ) { - jQuery[ method ] = function( url, data, callback, type ) { - - // Shift arguments if data argument was omitted - if ( isFunction( data ) ) { - type = type || callback; - callback = data; - data = undefined; - } - - // The url can be an options object (which then must have .url) - return jQuery.ajax( jQuery.extend( { - url: url, - type: method, - dataType: type, - data: data, - success: callback - }, jQuery.isPlainObject( url ) && url ) ); - }; - } ); - - jQuery.ajaxPrefilter( function( s ) { - var i; - for ( i in s.headers ) { - if ( i.toLowerCase() === "content-type" ) { - s.contentType = s.headers[ i ] || ""; - } - } - } ); - - - jQuery._evalUrl = function( url, options, doc ) { - return jQuery.ajax( { - url: url, - - // Make this explicit, since user can override this through ajaxSetup (#11264) - type: "GET", - dataType: "script", - cache: true, - async: false, - global: false, - - // Only evaluate the response if it is successful (gh-4126) - // dataFilter is not invoked for failure responses, so using it instead - // of the default converter is kludgy but it works. - converters: { - "text script": function() {} - }, - dataFilter: function( response ) { - jQuery.globalEval( response, options, doc ); - } - } ); - }; - - - jQuery.fn.extend( { - wrapAll: function( html ) { - var wrap; - - if ( this[ 0 ] ) { - if ( isFunction( html ) ) { - html = html.call( this[ 0 ] ); - } - - // The elements to wrap the target around - wrap = jQuery( html, this[ 0 ].ownerDocument ).eq( 0 ).clone( true ); - - if ( this[ 0 ].parentNode ) { - wrap.insertBefore( this[ 0 ] ); - } - - wrap.map( function() { - var elem = this; - - while ( elem.firstElementChild ) { - elem = elem.firstElementChild; - } - - return elem; - } ).append( this ); - } - - return this; - }, - - wrapInner: function( html ) { - if ( isFunction( html ) ) { - return this.each( function( i ) { - jQuery( this ).wrapInner( html.call( this, i ) ); - } ); - } - - return this.each( function() { - var self = jQuery( this ), - contents = self.contents(); - - if ( contents.length ) { - contents.wrapAll( html ); - - } else { - self.append( html ); - } - } ); - }, - - wrap: function( html ) { - var htmlIsFunction = isFunction( html ); - - return this.each( function( i ) { - jQuery( this ).wrapAll( htmlIsFunction ? html.call( this, i ) : html ); - } ); - }, - - unwrap: function( selector ) { - this.parent( selector ).not( "body" ).each( function() { - jQuery( this ).replaceWith( this.childNodes ); - } ); - return this; - } - } ); - - - jQuery.expr.pseudos.hidden = function( elem ) { - return !jQuery.expr.pseudos.visible( elem ); - }; - jQuery.expr.pseudos.visible = function( elem ) { - return !!( elem.offsetWidth || elem.offsetHeight || elem.getClientRects().length ); - }; - - - - - jQuery.ajaxSettings.xhr = function() { - try { - return new window.XMLHttpRequest(); - } catch ( e ) {} - }; - - var xhrSuccessStatus = { - - // File protocol always yields status code 0, assume 200 - 0: 200, - - // Support: IE <=9 only - // #1450: sometimes IE returns 1223 when it should be 204 - 1223: 204 - }, - xhrSupported = jQuery.ajaxSettings.xhr(); - - support.cors = !!xhrSupported && ( "withCredentials" in xhrSupported ); - support.ajax = xhrSupported = !!xhrSupported; - - jQuery.ajaxTransport( function( options ) { - var callback, errorCallback; - - // Cross domain only allowed if supported through XMLHttpRequest - if ( support.cors || xhrSupported && !options.crossDomain ) { - return { - send: function( headers, complete ) { - var i, - xhr = options.xhr(); - - xhr.open( - options.type, - options.url, - options.async, - options.username, - options.password - ); - - // Apply custom fields if provided - if ( options.xhrFields ) { - for ( i in options.xhrFields ) { - xhr[ i ] = options.xhrFields[ i ]; - } - } - - // Override mime type if needed - if ( options.mimeType && xhr.overrideMimeType ) { - xhr.overrideMimeType( options.mimeType ); - } - - // X-Requested-With header - // For cross-domain requests, seeing as conditions for a preflight are - // akin to a jigsaw puzzle, we simply never set it to be sure. - // (it can always be set on a per-request basis or even using ajaxSetup) - // For same-domain requests, won't change header if already provided. - if ( !options.crossDomain && !headers[ "X-Requested-With" ] ) { - headers[ "X-Requested-With" ] = "XMLHttpRequest"; - } - - // Set headers - for ( i in headers ) { - xhr.setRequestHeader( i, headers[ i ] ); - } - - // Callback - callback = function( type ) { - return function() { - if ( callback ) { - callback = errorCallback = xhr.onload = - xhr.onerror = xhr.onabort = xhr.ontimeout = - xhr.onreadystatechange = null; - - if ( type === "abort" ) { - xhr.abort(); - } else if ( type === "error" ) { - - // Support: IE <=9 only - // On a manual native abort, IE9 throws - // errors on any property access that is not readyState - if ( typeof xhr.status !== "number" ) { - complete( 0, "error" ); - } else { - complete( - - // File: protocol always yields status 0; see #8605, #14207 - xhr.status, - xhr.statusText - ); - } - } else { - complete( - xhrSuccessStatus[ xhr.status ] || xhr.status, - xhr.statusText, - - // Support: IE <=9 only - // IE9 has no XHR2 but throws on binary (trac-11426) - // For XHR2 non-text, let the caller handle it (gh-2498) - ( xhr.responseType || "text" ) !== "text" || - typeof xhr.responseText !== "string" ? - { binary: xhr.response } : - { text: xhr.responseText }, - xhr.getAllResponseHeaders() - ); - } - } - }; - }; - - // Listen to events - xhr.onload = callback(); - errorCallback = xhr.onerror = xhr.ontimeout = callback( "error" ); - - // Support: IE 9 only - // Use onreadystatechange to replace onabort - // to handle uncaught aborts - if ( xhr.onabort !== undefined ) { - xhr.onabort = errorCallback; - } else { - xhr.onreadystatechange = function() { - - // Check readyState before timeout as it changes - if ( xhr.readyState === 4 ) { - - // Allow onerror to be called first, - // but that will not handle a native abort - // Also, save errorCallback to a variable - // as xhr.onerror cannot be accessed - window.setTimeout( function() { - if ( callback ) { - errorCallback(); - } - } ); - } - }; - } - - // Create the abort callback - callback = callback( "abort" ); - - try { - - // Do send the request (this may raise an exception) - xhr.send( options.hasContent && options.data || null ); - } catch ( e ) { - - // #14683: Only rethrow if this hasn't been notified as an error yet - if ( callback ) { - throw e; - } - } - }, - - abort: function() { - if ( callback ) { - callback(); - } - } - }; - } - } ); - - - - - // Prevent auto-execution of scripts when no explicit dataType was provided (See gh-2432) - jQuery.ajaxPrefilter( function( s ) { - if ( s.crossDomain ) { - s.contents.script = false; - } - } ); - - // Install script dataType - jQuery.ajaxSetup( { - accepts: { - script: "text/javascript, application/javascript, " + - "application/ecmascript, application/x-ecmascript" - }, - contents: { - script: /\b(?:java|ecma)script\b/ - }, - converters: { - "text script": function( text ) { - jQuery.globalEval( text ); - return text; - } - } - } ); - - // Handle cache's special case and crossDomain - jQuery.ajaxPrefilter( "script", function( s ) { - if ( s.cache === undefined ) { - s.cache = false; - } - if ( s.crossDomain ) { - s.type = "GET"; - } - } ); - - // Bind script tag hack transport - jQuery.ajaxTransport( "script", function( s ) { - - // This transport only deals with cross domain or forced-by-attrs requests - if ( s.crossDomain || s.scriptAttrs ) { - var script, callback; - return { - send: function( _, complete ) { - script = jQuery( "<script>" ) - .attr( s.scriptAttrs || {} ) - .prop( { charset: s.scriptCharset, src: s.url } ) - .on( "load error", callback = function( evt ) { - script.remove(); - callback = null; - if ( evt ) { - complete( evt.type === "error" ? 404 : 200, evt.type ); - } - } ); - - // Use native DOM manipulation to avoid our domManip AJAX trickery - document.head.appendChild( script[ 0 ] ); - }, - abort: function() { - if ( callback ) { - callback(); - } - } - }; - } - } ); - - - - - var oldCallbacks = [], - rjsonp = /(=)\?(?=&|$)|\?\?/; - - // Default jsonp settings - jQuery.ajaxSetup( { - jsonp: "callback", - jsonpCallback: function() { - var callback = oldCallbacks.pop() || ( jQuery.expando + "_" + ( nonce.guid++ ) ); - this[ callback ] = true; - return callback; - } - } ); - - // Detect, normalize options and install callbacks for jsonp requests - jQuery.ajaxPrefilter( "json jsonp", function( s, originalSettings, jqXHR ) { - - var callbackName, overwritten, responseContainer, - jsonProp = s.jsonp !== false && ( rjsonp.test( s.url ) ? - "url" : - typeof s.data === "string" && - ( s.contentType || "" ) - .indexOf( "application/x-www-form-urlencoded" ) === 0 && - rjsonp.test( s.data ) && "data" - ); - - // Handle iff the expected data type is "jsonp" or we have a parameter to set - if ( jsonProp || s.dataTypes[ 0 ] === "jsonp" ) { - - // Get callback name, remembering preexisting value associated with it - callbackName = s.jsonpCallback = isFunction( s.jsonpCallback ) ? - s.jsonpCallback() : - s.jsonpCallback; - - // Insert callback into url or form data - if ( jsonProp ) { - s[ jsonProp ] = s[ jsonProp ].replace( rjsonp, "$1" + callbackName ); - } else if ( s.jsonp !== false ) { - s.url += ( rquery.test( s.url ) ? "&" : "?" ) + s.jsonp + "=" + callbackName; - } - - // Use data converter to retrieve json after script execution - s.converters[ "script json" ] = function() { - if ( !responseContainer ) { - jQuery.error( callbackName + " was not called" ); - } - return responseContainer[ 0 ]; - }; - - // Force json dataType - s.dataTypes[ 0 ] = "json"; - - // Install callback - overwritten = window[ callbackName ]; - window[ callbackName ] = function() { - responseContainer = arguments; - }; - - // Clean-up function (fires after converters) - jqXHR.always( function() { - - // If previous value didn't exist - remove it - if ( overwritten === undefined ) { - jQuery( window ).removeProp( callbackName ); - - // Otherwise restore preexisting value - } else { - window[ callbackName ] = overwritten; - } - - // Save back as free - if ( s[ callbackName ] ) { - - // Make sure that re-using the options doesn't screw things around - s.jsonpCallback = originalSettings.jsonpCallback; - - // Save the callback name for future use - oldCallbacks.push( callbackName ); - } - - // Call if it was a function and we have a response - if ( responseContainer && isFunction( overwritten ) ) { - overwritten( responseContainer[ 0 ] ); - } - - responseContainer = overwritten = undefined; - } ); - - // Delegate to script - return "script"; - } - } ); - - - - - // Support: Safari 8 only - // In Safari 8 documents created via document.implementation.createHTMLDocument - // collapse sibling forms: the second one becomes a child of the first one. - // Because of that, this security measure has to be disabled in Safari 8. - // https://bugs.webkit.org/show_bug.cgi?id=137337 - support.createHTMLDocument = ( function() { - var body = document.implementation.createHTMLDocument( "" ).body; - body.innerHTML = "<form></form><form></form>"; - return body.childNodes.length === 2; - } )(); - - - // Argument "data" should be string of html - // context (optional): If specified, the fragment will be created in this context, - // defaults to document - // keepScripts (optional): If true, will include scripts passed in the html string - jQuery.parseHTML = function( data, context, keepScripts ) { - if ( typeof data !== "string" ) { - return []; - } - if ( typeof context === "boolean" ) { - keepScripts = context; - context = false; - } - - var base, parsed, scripts; - - if ( !context ) { - - // Stop scripts or inline event handlers from being executed immediately - // by using document.implementation - if ( support.createHTMLDocument ) { - context = document.implementation.createHTMLDocument( "" ); - - // Set the base href for the created document - // so any parsed elements with URLs - // are based on the document's URL (gh-2965) - base = context.createElement( "base" ); - base.href = document.location.href; - context.head.appendChild( base ); - } else { - context = document; - } - } - - parsed = rsingleTag.exec( data ); - scripts = !keepScripts && []; - - // Single tag - if ( parsed ) { - return [ context.createElement( parsed[ 1 ] ) ]; - } - - parsed = buildFragment( [ data ], context, scripts ); - - if ( scripts && scripts.length ) { - jQuery( scripts ).remove(); - } - - return jQuery.merge( [], parsed.childNodes ); - }; - - - /** - * Load a url into a page - */ - jQuery.fn.load = function( url, params, callback ) { - var selector, type, response, - self = this, - off = url.indexOf( " " ); - - if ( off > -1 ) { - selector = stripAndCollapse( url.slice( off ) ); - url = url.slice( 0, off ); - } - - // If it's a function - if ( isFunction( params ) ) { - - // We assume that it's the callback - callback = params; - params = undefined; - - // Otherwise, build a param string - } else if ( params && typeof params === "object" ) { - type = "POST"; - } - - // If we have elements to modify, make the request - if ( self.length > 0 ) { - jQuery.ajax( { - url: url, - - // If "type" variable is undefined, then "GET" method will be used. - // Make value of this field explicit since - // user can override it through ajaxSetup method - type: type || "GET", - dataType: "html", - data: params - } ).done( function( responseText ) { - - // Save response for use in complete callback - response = arguments; - - self.html( selector ? - - // If a selector was specified, locate the right elements in a dummy div - // Exclude scripts to avoid IE 'Permission Denied' errors - jQuery( "<div>" ).append( jQuery.parseHTML( responseText ) ).find( selector ) : - - // Otherwise use the full result - responseText ); - - // If the request succeeds, this function gets "data", "status", "jqXHR" - // but they are ignored because response was set above. - // If it fails, this function gets "jqXHR", "status", "error" - } ).always( callback && function( jqXHR, status ) { - self.each( function() { - callback.apply( this, response || [ jqXHR.responseText, status, jqXHR ] ); - } ); - } ); - } - - return this; - }; - - - - - jQuery.expr.pseudos.animated = function( elem ) { - return jQuery.grep( jQuery.timers, function( fn ) { - return elem === fn.elem; - } ).length; - }; - - - - - jQuery.offset = { - setOffset: function( elem, options, i ) { - var curPosition, curLeft, curCSSTop, curTop, curOffset, curCSSLeft, calculatePosition, - position = jQuery.css( elem, "position" ), - curElem = jQuery( elem ), - props = {}; - - // Set position first, in-case top/left are set even on static elem - if ( position === "static" ) { - elem.style.position = "relative"; - } - - curOffset = curElem.offset(); - curCSSTop = jQuery.css( elem, "top" ); - curCSSLeft = jQuery.css( elem, "left" ); - calculatePosition = ( position === "absolute" || position === "fixed" ) && - ( curCSSTop + curCSSLeft ).indexOf( "auto" ) > -1; - - // Need to be able to calculate position if either - // top or left is auto and position is either absolute or fixed - if ( calculatePosition ) { - curPosition = curElem.position(); - curTop = curPosition.top; - curLeft = curPosition.left; - - } else { - curTop = parseFloat( curCSSTop ) || 0; - curLeft = parseFloat( curCSSLeft ) || 0; - } - - if ( isFunction( options ) ) { - - // Use jQuery.extend here to allow modification of coordinates argument (gh-1848) - options = options.call( elem, i, jQuery.extend( {}, curOffset ) ); - } - - if ( options.top != null ) { - props.top = ( options.top - curOffset.top ) + curTop; - } - if ( options.left != null ) { - props.left = ( options.left - curOffset.left ) + curLeft; - } - - if ( "using" in options ) { - options.using.call( elem, props ); - - } else { - curElem.css( props ); - } - } - }; - - jQuery.fn.extend( { - - // offset() relates an element's border box to the document origin - offset: function( options ) { - - // Preserve chaining for setter - if ( arguments.length ) { - return options === undefined ? - this : - this.each( function( i ) { - jQuery.offset.setOffset( this, options, i ); - } ); - } - - var rect, win, - elem = this[ 0 ]; - - if ( !elem ) { - return; - } - - // Return zeros for disconnected and hidden (display: none) elements (gh-2310) - // Support: IE <=11 only - // Running getBoundingClientRect on a - // disconnected node in IE throws an error - if ( !elem.getClientRects().length ) { - return { top: 0, left: 0 }; - } - - // Get document-relative position by adding viewport scroll to viewport-relative gBCR - rect = elem.getBoundingClientRect(); - win = elem.ownerDocument.defaultView; - return { - top: rect.top + win.pageYOffset, - left: rect.left + win.pageXOffset - }; - }, - - // position() relates an element's margin box to its offset parent's padding box - // This corresponds to the behavior of CSS absolute positioning - position: function() { - if ( !this[ 0 ] ) { - return; - } - - var offsetParent, offset, doc, - elem = this[ 0 ], - parentOffset = { top: 0, left: 0 }; - - // position:fixed elements are offset from the viewport, which itself always has zero offset - if ( jQuery.css( elem, "position" ) === "fixed" ) { - - // Assume position:fixed implies availability of getBoundingClientRect - offset = elem.getBoundingClientRect(); - - } else { - offset = this.offset(); - - // Account for the *real* offset parent, which can be the document or its root element - // when a statically positioned element is identified - doc = elem.ownerDocument; - offsetParent = elem.offsetParent || doc.documentElement; - while ( offsetParent && - ( offsetParent === doc.body || offsetParent === doc.documentElement ) && - jQuery.css( offsetParent, "position" ) === "static" ) { - - offsetParent = offsetParent.parentNode; - } - if ( offsetParent && offsetParent !== elem && offsetParent.nodeType === 1 ) { - - // Incorporate borders into its offset, since they are outside its content origin - parentOffset = jQuery( offsetParent ).offset(); - parentOffset.top += jQuery.css( offsetParent, "borderTopWidth", true ); - parentOffset.left += jQuery.css( offsetParent, "borderLeftWidth", true ); - } - } - - // Subtract parent offsets and element margins - return { - top: offset.top - parentOffset.top - jQuery.css( elem, "marginTop", true ), - left: offset.left - parentOffset.left - jQuery.css( elem, "marginLeft", true ) - }; - }, - - // This method will return documentElement in the following cases: - // 1) For the element inside the iframe without offsetParent, this method will return - // documentElement of the parent window - // 2) For the hidden or detached element - // 3) For body or html element, i.e. in case of the html node - it will return itself - // - // but those exceptions were never presented as a real life use-cases - // and might be considered as more preferable results. - // - // This logic, however, is not guaranteed and can change at any point in the future - offsetParent: function() { - return this.map( function() { - var offsetParent = this.offsetParent; - - while ( offsetParent && jQuery.css( offsetParent, "position" ) === "static" ) { - offsetParent = offsetParent.offsetParent; - } - - return offsetParent || documentElement; - } ); - } - } ); - - // Create scrollLeft and scrollTop methods - jQuery.each( { scrollLeft: "pageXOffset", scrollTop: "pageYOffset" }, function( method, prop ) { - var top = "pageYOffset" === prop; - - jQuery.fn[ method ] = function( val ) { - return access( this, function( elem, method, val ) { - - // Coalesce documents and windows - var win; - if ( isWindow( elem ) ) { - win = elem; - } else if ( elem.nodeType === 9 ) { - win = elem.defaultView; - } - - if ( val === undefined ) { - return win ? win[ prop ] : elem[ method ]; - } - - if ( win ) { - win.scrollTo( - !top ? val : win.pageXOffset, - top ? val : win.pageYOffset - ); - - } else { - elem[ method ] = val; - } - }, method, val, arguments.length ); - }; - } ); - - // Support: Safari <=7 - 9.1, Chrome <=37 - 49 - // Add the top/left cssHooks using jQuery.fn.position - // Webkit bug: https://bugs.webkit.org/show_bug.cgi?id=29084 - // Blink bug: https://bugs.chromium.org/p/chromium/issues/detail?id=589347 - // getComputedStyle returns percent when specified for top/left/bottom/right; - // rather than make the css module depend on the offset module, just check for it here - jQuery.each( [ "top", "left" ], function( _i, prop ) { - jQuery.cssHooks[ prop ] = addGetHookIf( support.pixelPosition, - function( elem, computed ) { - if ( computed ) { - computed = curCSS( elem, prop ); - - // If curCSS returns percentage, fallback to offset - return rnumnonpx.test( computed ) ? - jQuery( elem ).position()[ prop ] + "px" : - computed; - } - } - ); - } ); - - - // Create innerHeight, innerWidth, height, width, outerHeight and outerWidth methods - jQuery.each( { Height: "height", Width: "width" }, function( name, type ) { - jQuery.each( { - padding: "inner" + name, - content: type, - "": "outer" + name - }, function( defaultExtra, funcName ) { - - // Margin is only for outerHeight, outerWidth - jQuery.fn[ funcName ] = function( margin, value ) { - var chainable = arguments.length && ( defaultExtra || typeof margin !== "boolean" ), - extra = defaultExtra || ( margin === true || value === true ? "margin" : "border" ); - - return access( this, function( elem, type, value ) { - var doc; - - if ( isWindow( elem ) ) { - - // $( window ).outerWidth/Height return w/h including scrollbars (gh-1729) - return funcName.indexOf( "outer" ) === 0 ? - elem[ "inner" + name ] : - elem.document.documentElement[ "client" + name ]; - } - - // Get document width or height - if ( elem.nodeType === 9 ) { - doc = elem.documentElement; - - // Either scroll[Width/Height] or offset[Width/Height] or client[Width/Height], - // whichever is greatest - return Math.max( - elem.body[ "scroll" + name ], doc[ "scroll" + name ], - elem.body[ "offset" + name ], doc[ "offset" + name ], - doc[ "client" + name ] - ); - } - - return value === undefined ? - - // Get width or height on the element, requesting but not forcing parseFloat - jQuery.css( elem, type, extra ) : - - // Set width or height on the element - jQuery.style( elem, type, value, extra ); - }, type, chainable ? margin : undefined, chainable ); - }; - } ); - } ); - - - jQuery.each( [ - "ajaxStart", - "ajaxStop", - "ajaxComplete", - "ajaxError", - "ajaxSuccess", - "ajaxSend" - ], function( _i, type ) { - jQuery.fn[ type ] = function( fn ) { - return this.on( type, fn ); - }; - } ); - - - - - jQuery.fn.extend( { - - bind: function( types, data, fn ) { - return this.on( types, null, data, fn ); - }, - unbind: function( types, fn ) { - return this.off( types, null, fn ); - }, - - delegate: function( selector, types, data, fn ) { - return this.on( types, selector, data, fn ); - }, - undelegate: function( selector, types, fn ) { - - // ( namespace ) or ( selector, types [, fn] ) - return arguments.length === 1 ? - this.off( selector, "**" ) : - this.off( types, selector || "**", fn ); - }, - - hover: function( fnOver, fnOut ) { - return this.mouseenter( fnOver ).mouseleave( fnOut || fnOver ); - } - } ); - - jQuery.each( - ( "blur focus focusin focusout resize scroll click dblclick " + - "mousedown mouseup mousemove mouseover mouseout mouseenter mouseleave " + - "change select submit keydown keypress keyup contextmenu" ).split( " " ), - function( _i, name ) { - - // Handle event binding - jQuery.fn[ name ] = function( data, fn ) { - return arguments.length > 0 ? - this.on( name, null, data, fn ) : - this.trigger( name ); - }; - } - ); - - - - - // Support: Android <=4.0 only - // Make sure we trim BOM and NBSP - var rtrim = /^[\s\uFEFF\xA0]+|[\s\uFEFF\xA0]+$/g; - - // Bind a function to a context, optionally partially applying any - // arguments. - // jQuery.proxy is deprecated to promote standards (specifically Function#bind) - // However, it is not slated for removal any time soon - jQuery.proxy = function( fn, context ) { - var tmp, args, proxy; - - if ( typeof context === "string" ) { - tmp = fn[ context ]; - context = fn; - fn = tmp; - } - - // Quick check to determine if target is callable, in the spec - // this throws a TypeError, but we will just return undefined. - if ( !isFunction( fn ) ) { - return undefined; - } - - // Simulated bind - args = slice.call( arguments, 2 ); - proxy = function() { - return fn.apply( context || this, args.concat( slice.call( arguments ) ) ); - }; - - // Set the guid of unique handler to the same of original handler, so it can be removed - proxy.guid = fn.guid = fn.guid || jQuery.guid++; - - return proxy; - }; - - jQuery.holdReady = function( hold ) { - if ( hold ) { - jQuery.readyWait++; - } else { - jQuery.ready( true ); - } - }; - jQuery.isArray = Array.isArray; - jQuery.parseJSON = JSON.parse; - jQuery.nodeName = nodeName; - jQuery.isFunction = isFunction; - jQuery.isWindow = isWindow; - jQuery.camelCase = camelCase; - jQuery.type = toType; - - jQuery.now = Date.now; - - jQuery.isNumeric = function( obj ) { - - // As of jQuery 3.0, isNumeric is limited to - // strings and numbers (primitives or objects) - // that can be coerced to finite numbers (gh-2662) - var type = jQuery.type( obj ); - return ( type === "number" || type === "string" ) && - - // parseFloat NaNs numeric-cast false positives ("") - // ...but misinterprets leading-number strings, particularly hex literals ("0x...") - // subtraction forces infinities to NaN - !isNaN( obj - parseFloat( obj ) ); - }; - - jQuery.trim = function( text ) { - return text == null ? - "" : - ( text + "" ).replace( rtrim, "" ); - }; - - - - // Register as a named AMD module, since jQuery can be concatenated with other - // files that may use define, but not via a proper concatenation script that - // understands anonymous AMD modules. A named AMD is safest and most robust - // way to register. Lowercase jquery is used because AMD module names are - // derived from file names, and jQuery is normally delivered in a lowercase - // file name. Do this after creating the global so that if an AMD module wants - // to call noConflict to hide this version of jQuery, it will work. - - // Note that for maximum portability, libraries that are not jQuery should - // declare themselves as anonymous modules, and avoid setting a global if an - // AMD loader is present. jQuery is a special case. For more information, see - // https://github.com/jrburke/requirejs/wiki/Updating-existing-libraries#wiki-anon - - if ( typeof define === "function" && define.amd ) { - define( "jquery", [], function() { - return jQuery; - } ); - } - - - - - var - - // Map over jQuery in case of overwrite - _jQuery = window.jQuery, - - // Map over the $ in case of overwrite - _$ = window.$; - - jQuery.noConflict = function( deep ) { - if ( window.$ === jQuery ) { - window.$ = _$; - } - - if ( deep && window.jQuery === jQuery ) { - window.jQuery = _jQuery; - } - - return jQuery; - }; - - // Expose jQuery and $ identifiers, even in AMD - // (#7102#comment:10, https://github.com/jquery/jquery/pull/557) - // and CommonJS for browser emulators (#13566) - if ( typeof noGlobal === "undefined" ) { - window.jQuery = window.$ = jQuery; - } - - - - - return jQuery; - } );
html/_static/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 2007-2022 by the Sphinx team, see AUTHORS. + * :copyright: Copyright 2007-2023 by the Sphinx team, see AUTHORS. * :license: BSD, see LICENSE for details. * */
html/_static/searchtools.js
- @@ -4,7 +4,7 @@
 * * Sphinx JavaScript utilities for the full-text search. * - * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS. + * :copyright: Copyright 2007-2023 by the Sphinx team, see AUTHORS. * :license: BSD, see LICENSE for details. * */
- @@ -57,14 +57,14 @@ const _removeChildren = (element) => {
 const _escapeRegExp = (string) => string.replace(/[.*+\-?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string - const _displayItem = (item, highlightTerms, searchTerms) => { + const _displayItem = (item, searchTerms) => { const docBuilder = DOCUMENTATION_OPTIONS.BUILDER; const docUrlRoot = DOCUMENTATION_OPTIONS.URL_ROOT; const docFileSuffix = DOCUMENTATION_OPTIONS.FILE_SUFFIX; const docLinkSuffix = DOCUMENTATION_OPTIONS.LINK_SUFFIX; const showSearchSummary = DOCUMENTATION_OPTIONS.SHOW_SEARCH_SUMMARY; - const [docName, title, anchor, descr] = item; + const [docName, title, anchor, descr, score, _filename] = item; let listItem = document.createElement("li"); let requestUrl;
- @@ -82,13 +82,12 @@ const _displayItem = (item, highlightTerms, searchTerms) => {
 requestUrl = docUrlRoot + docName + docFileSuffix; linkUrl = docName + docLinkSuffix; } - const params = new URLSearchParams(); - params.set("highlight", [...highlightTerms].join(" ")); let linkEl = listItem.appendChild(document.createElement("a")); - linkEl.href = linkUrl + "?" + params.toString() + anchor; + linkEl.href = linkUrl + anchor; + linkEl.dataset.score = score; linkEl.innerHTML = title; if (descr) - listItem.appendChild(document.createElement("span")).innerText = + listItem.appendChild(document.createElement("span")).innerHTML = " (" + descr + ")"; else if (showSearchSummary) fetch(requestUrl)
- @@ -96,7 +95,7 @@ const _displayItem = (item, highlightTerms, searchTerms) => {
 .then((data) => { if (data) listItem.appendChild( - Search.makeSearchSummary(data, searchTerms, highlightTerms) + Search.makeSearchSummary(data, searchTerms) ); }); Search.output.appendChild(listItem);
- @@ -116,15 +115,14 @@ const _finishSearch = (resultCount) => {
 const _displayNextItem = ( results, resultCount, - highlightTerms, searchTerms ) => { // results left, load the summary and display it // this is intended to be dynamic (don't sub resultsCount) if (results.length) { - _displayItem(results.pop(), highlightTerms, searchTerms); + _displayItem(results.pop(), searchTerms); setTimeout( - () => _displayNextItem(results, resultCount, highlightTerms, searchTerms), + () => _displayNextItem(results, resultCount, searchTerms), 5 ); }
- @@ -155,10 +153,8 @@ const Search = {
 _pulse_status: -1, htmlToText: (htmlString) => { - const htmlElement = document - .createRange() - .createContextualFragment(htmlString); - _removeChildren(htmlElement.querySelectorAll(".headerlink")); + const htmlElement = new DOMParser().parseFromString(htmlString, 'text/html'); + htmlElement.querySelectorAll(".headerlink").forEach((el) => { el.remove() }); const docContent = htmlElement.querySelector('[role="main"]'); if (docContent !== undefined) return docContent.textContent; console.warn(
- @@ -239,6 +235,12 @@ const Search = {
 * 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();
- @@ -266,6 +268,10 @@ const Search = {
 } }); + if (SPHINX_HIGHLIGHT_ENABLED) { // set in sphinx_highlight.js + localStorage.setItem("sphinx_highlight_terms", [...highlightTerms].join(" ")) + } + // console.debug("SEARCH: searching for:"); // console.info("required: ", [...searchTerms]); // console.info("excluded: ", [...excludedTerms]);
- @@ -274,6 +280,40 @@ const Search = {
 let results = []; _removeChildren(document.getElementById("search-progress")); + const queryLower = query.toLowerCase(); + for (const [title, foundTitles] of Object.entries(allTitles)) { + if (title.toLowerCase().includes(queryLower) && (queryLower.length >= title.length/2)) { + for (const [file, id] of foundTitles) { + let score = Math.round(100 * queryLower.length / title.length) + results.push([ + docNames[file], + titles[file] !== title ? `${titles[file]} > ${title}` : title, + id !== null ? "#" + id : "", + null, + score, + filenames[file], + ]); + } + } + } + + // search for explicit entries in index directives + for (const [entry, foundEntries] of Object.entries(indexEntries)) { + if (entry.includes(queryLower) && (queryLower.length >= entry.length/2)) { + for (const [file, id] of foundEntries) { + let score = Math.round(100 * queryLower.length / entry.length) + results.push([ + docNames[file], + titles[file], + id ? "#" + id : "", + null, + score, + filenames[file], + ]); + } + } + } + // lookup as object objectTerms.forEach((term) => results.push(...Search.performObjectSearch(term, objectTerms))
- @@ -320,7 +360,7 @@ const Search = {
 // console.info("search results:", Search.lastresults); // print the results - _displayNextItem(results, results.length, highlightTerms, searchTerms); + _displayNextItem(results, results.length, searchTerms); }, /**
- @@ -401,8 +441,8 @@ const Search = {
 // prepare search const terms = Search._index.terms; const titleTerms = Search._index.titleterms; - const docNames = Search._index.docnames; const filenames = Search._index.filenames; + const docNames = Search._index.docnames; const titles = Search._index.titles; const scoreMap = new Map();
- @@ -499,16 +539,15 @@ const Search = {
 /** * helper function to return a node containing the * search summary for a given text. keywords is a list - * of stemmed words, highlightWords is the list of normal, unstemmed - * words. the first one is used to find the occurrence, the - * latter for highlighting it. + * of stemmed words. */ - makeSearchSummary: (htmlText, keywords, highlightWords) => { - const text = Search.htmlToText(htmlText).toLowerCase(); + makeSearchSummary: (htmlText, keywords) => { + const text = Search.htmlToText(htmlText); if (text === "") return null; + const textLower = text.toLowerCase(); const actualStartPosition = [...keywords] - .map((k) => text.indexOf(k.toLowerCase())) + .map((k) => textLower.indexOf(k.toLowerCase())) .filter((i) => i > -1) .slice(-1)[0]; const startWithContext = Math.max(actualStartPosition - 120, 0);
- @@ -516,13 +555,9 @@ const Search = {
 const top = startWithContext === 0 ? "" : "..."; const tail = startWithContext + 240 < text.length ? "..." : ""; - let summary = document.createElement("div"); + let summary = document.createElement("p"); summary.classList.add("context"); - summary.innerText = top + text.substr(startWithContext, 240).trim() + tail; - - highlightWords.forEach((highlightWord) => - _highlightText(summary, highlightWord, "highlighted") - ); + summary.textContent = top + text.substr(startWithContext, 240).trim() + tail; return summary; },
html/_static/sphinx_highlight.js (new)
- @@ -0,0 +1,144 @@
 + /* Highlighting utilities for Sphinx HTML documentation. */ + "use strict"; + + const SPHINX_HIGHLIGHT_ENABLED = true + + /** + * highlight a given string on a node by wrapping it in + * span elements with the given class name. + */ + const _highlight = (node, addItems, text, className) => { + if (node.nodeType === Node.TEXT_NODE) { + const val = node.nodeValue; + const parent = node.parentNode; + const pos = val.toLowerCase().indexOf(text); + if ( + pos >= 0 && + !parent.classList.contains(className) && + !parent.classList.contains("nohighlight") + ) { + let span; + + const closestNode = parent.closest("body, svg, foreignObject"); + const isInSVG = closestNode && closestNode.matches("svg"); + if (isInSVG) { + span = document.createElementNS("http://www.w3.org/2000/svg", "tspan"); + } else { + span = document.createElement("span"); + span.classList.add(className); + } + + span.appendChild(document.createTextNode(val.substr(pos, text.length))); + parent.insertBefore( + span, + parent.insertBefore( + document.createTextNode(val.substr(pos + text.length)), + node.nextSibling + ) + ); + node.nodeValue = val.substr(0, pos); + + if (isInSVG) { + const rect = document.createElementNS( + "http://www.w3.org/2000/svg", + "rect" + ); + const bbox = parent.getBBox(); + rect.x.baseVal.value = bbox.x; + rect.y.baseVal.value = bbox.y; + rect.width.baseVal.value = bbox.width; + rect.height.baseVal.value = bbox.height; + rect.setAttribute("class", className); + addItems.push({ parent: parent, target: rect }); + } + } + } else if (node.matches && !node.matches("button, select, textarea")) { + node.childNodes.forEach((el) => _highlight(el, addItems, text, className)); + } + }; + const _highlightText = (thisNode, text, className) => { + let addItems = []; + _highlight(thisNode, addItems, text, className); + addItems.forEach((obj) => + obj.parent.insertAdjacentElement("beforebegin", obj.target) + ); + }; + + /** + * Small JavaScript module for the documentation. + */ + const SphinxHighlight = { + + /** + * highlight the search words provided in localstorage in the text + */ + highlightSearchWords: () => { + if (!SPHINX_HIGHLIGHT_ENABLED) return; // bail if no highlight + + // get and clear terms from localstorage + const url = new URL(window.location); + const highlight = + localStorage.getItem("sphinx_highlight_terms") + || url.searchParams.get("highlight") + || ""; + localStorage.removeItem("sphinx_highlight_terms") + url.searchParams.delete("highlight"); + window.history.replaceState({}, "", url); + + // get individual terms from highlight string + const terms = highlight.toLowerCase().split(/\s+/).filter(x => x); + if (terms.length === 0) return; // nothing to do + + // There should never be more than one element matching "div.body" + const divBody = document.querySelectorAll("div.body"); + const body = divBody.length ? divBody[0] : document.querySelector("body"); + window.setTimeout(() => { + terms.forEach((term) => _highlightText(body, term, "highlighted")); + }, 10); + + const searchBox = document.getElementById("searchbox"); + if (searchBox === null) return; + searchBox.appendChild( + document + .createRange() + .createContextualFragment( + '' + + '<a href="javascript:SphinxHighlight.hideSearchWords()">' + + _("Hide Search Matches") + + "</a>" + ) + ); + }, + + /** + * helper function to hide the search marks again + */ + hideSearchWords: () => { + document + .querySelectorAll("#searchbox .highlight-link") + .forEach((el) => el.remove()); + document + .querySelectorAll("span.highlighted") + .forEach((el) => el.classList.remove("highlighted")); + localStorage.removeItem("sphinx_highlight_terms") + }, + + initEscapeListener: () => { + // only install a listener if it is really needed + if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS) return; + + document.addEventListener("keydown", (event) => { + // bail for input elements + if (BLACKLISTED_KEY_CONTROL_ELEMENTS.has(document.activeElement.tagName)) return; + // bail with special keys + if (event.shiftKey || event.altKey || event.ctrlKey || event.metaKey) return; + if (DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS && (event.key === "Escape")) { + SphinxHighlight.hideSearchWords(); + event.preventDefault(); + } + }); + }, + }; + + _ready(SphinxHighlight.highlightSearchWords); + _ready(SphinxHighlight.initEscapeListener);
html/_static/underscore-1.13.1.js (deleted)
- @@ -1,2042 +0,0 @@
 - (function (global, factory) { - typeof exports === 'object' && typeof module !== 'undefined' ? module.exports = factory() : - typeof define === 'function' && define.amd ? define('underscore', factory) : - (global = typeof globalThis !== 'undefined' ? globalThis : global || self, (function () { - var current = global._; - var exports = global._ = factory(); - exports.noConflict = function () { global._ = current; return exports; }; - }())); - }(this, (function () { - // Underscore.js 1.13.1 - // https://underscorejs.org - // (c) 2009-2021 Jeremy Ashkenas, Julian Gonggrijp, and DocumentCloud and Investigative Reporters & Editors - // Underscore may be freely distributed under the MIT license. - - // Current version. - var VERSION = '1.13.1'; - - // Establish the root object, `window` (`self`) in the browser, `global` - // on the server, or `this` in some virtual machines. We use `self` - // instead of `window` for `WebWorker` support. - var root = typeof self == 'object' && self.self === self && self || - typeof global == 'object' && global.global === global && global || - Function('return this')() || - {}; - - // Save bytes in the minified (but not gzipped) version: - var ArrayProto = Array.prototype, ObjProto = Object.prototype; - var SymbolProto = typeof Symbol !== 'undefined' ? Symbol.prototype : null; - - // Create quick reference variables for speed access to core prototypes. - var push = ArrayProto.push, - slice = ArrayProto.slice, - toString = ObjProto.toString, - hasOwnProperty = ObjProto.hasOwnProperty; - - // Modern feature detection. - var supportsArrayBuffer = typeof ArrayBuffer !== 'undefined', - supportsDataView = typeof DataView !== 'undefined'; - - // All **ECMAScript 5+** native function implementations that we hope to use - // are declared here. - var nativeIsArray = Array.isArray, - nativeKeys = Object.keys, - nativeCreate = Object.create, - nativeIsView = supportsArrayBuffer && ArrayBuffer.isView; - - // Create references to these builtin functions because we override them. - var _isNaN = isNaN, - _isFinite = isFinite; - - // Keys in IE < 9 that won't be iterated by `for key in ...` and thus missed. - var hasEnumBug = !{toString: null}.propertyIsEnumerable('toString'); - var nonEnumerableProps = ['valueOf', 'isPrototypeOf', 'toString', - 'propertyIsEnumerable', 'hasOwnProperty', 'toLocaleString']; - - // The largest integer that can be represented exactly. - var MAX_ARRAY_INDEX = Math.pow(2, 53) - 1; - - // Some functions take a variable number of arguments, or a few expected - // arguments at the beginning and then a variable number of values to operate - // on. This helper accumulates all remaining arguments past the function’s - // argument length (or an explicit `startIndex`), into an array that becomes - // the last argument. Similar to ES6’s "rest parameter". - function restArguments(func, startIndex) { - startIndex = startIndex == null ? func.length - 1 : +startIndex; - return function() { - var length = Math.max(arguments.length - startIndex, 0), - rest = Array(length), - index = 0; - for (; index < length; index++) { - rest[index] = arguments[index + startIndex]; - } - switch (startIndex) { - case 0: return func.call(this, rest); - case 1: return func.call(this, arguments[0], rest); - case 2: return func.call(this, arguments[0], arguments[1], rest); - } - var args = Array(startIndex + 1); - for (index = 0; index < startIndex; index++) { - args[index] = arguments[index]; - } - args[startIndex] = rest; - return func.apply(this, args); - }; - } - - // Is a given variable an object? - function isObject(obj) { - var type = typeof obj; - return type === 'function' || type === 'object' && !!obj; - } - - // Is a given value equal to null? - function isNull(obj) { - return obj === null; - } - - // Is a given variable undefined? - function isUndefined(obj) { - return obj === void 0; - } - - // Is a given value a boolean? - function isBoolean(obj) { - return obj === true || obj === false || toString.call(obj) === '[object Boolean]'; - } - - // Is a given value a DOM element? - function isElement(obj) { - return !!(obj && obj.nodeType === 1); - } - - // Internal function for creating a `toString`-based type tester. - function tagTester(name) { - var tag = '[object ' + name + ']'; - return function(obj) { - return toString.call(obj) === tag; - }; - } - - var isString = tagTester('String'); - - var isNumber = tagTester('Number'); - - var isDate = tagTester('Date'); - - var isRegExp = tagTester('RegExp'); - - var isError = tagTester('Error'); - - var isSymbol = tagTester('Symbol'); - - var isArrayBuffer = tagTester('ArrayBuffer'); - - var isFunction = tagTester('Function'); - - // Optimize `isFunction` if appropriate. Work around some `typeof` bugs in old - // v8, IE 11 (#1621), Safari 8 (#1929), and PhantomJS (#2236). - var nodelist = root.document && root.document.childNodes; - if (typeof /./ != 'function' && typeof Int8Array != 'object' && typeof nodelist != 'function') { - isFunction = function(obj) { - return typeof obj == 'function' || false; - }; - } - - var isFunction$1 = isFunction; - - var hasObjectTag = tagTester('Object'); - - // In IE 10 - Edge 13, `DataView` has string tag `'[object Object]'`. - // In IE 11, the most common among them, this problem also applies to - // `Map`, `WeakMap` and `Set`. - var hasStringTagBug = ( - supportsDataView && hasObjectTag(new DataView(new ArrayBuffer(8))) - ), - isIE11 = (typeof Map !== 'undefined' && hasObjectTag(new Map)); - - var isDataView = tagTester('DataView'); - - // In IE 10 - Edge 13, we need a different heuristic - // to determine whether an object is a `DataView`. - function ie10IsDataView(obj) { - return obj != null && isFunction$1(obj.getInt8) && isArrayBuffer(obj.buffer); - } - - var isDataView$1 = (hasStringTagBug ? ie10IsDataView : isDataView); - - // Is a given value an array? - // Delegates to ECMA5's native `Array.isArray`. - var isArray = nativeIsArray || tagTester('Array'); - - // Internal function to check whether `key` is an own property name of `obj`. - function has$1(obj, key) { - return obj != null && hasOwnProperty.call(obj, key); - } - - var isArguments = tagTester('Arguments'); - - // Define a fallback version of the method in browsers (ahem, IE < 9), where - // there isn't any inspectable "Arguments" type. - (function() { - if (!isArguments(arguments)) { - isArguments = function(obj) { - return has$1(obj, 'callee'); - }; - } - }()); - - var isArguments$1 = isArguments; - - // Is a given object a finite number? - function isFinite$1(obj) { - return !isSymbol(obj) && _isFinite(obj) && !isNaN(parseFloat(obj)); - } - - // Is the given value `NaN`? - function isNaN$1(obj) { - return isNumber(obj) && _isNaN(obj); - } - - // Predicate-generating function. Often useful outside of Underscore. - function constant(value) { - return function() { - return value; - }; - } - - // Common internal logic for `isArrayLike` and `isBufferLike`. - function createSizePropertyCheck(getSizeProperty) { - return function(collection) { - var sizeProperty = getSizeProperty(collection); - return typeof sizeProperty == 'number' && sizeProperty >= 0 && sizeProperty <= MAX_ARRAY_INDEX; - } - } - - // Internal helper to generate a function to obtain property `key` from `obj`. - function shallowProperty(key) { - return function(obj) { - return obj == null ? void 0 : obj[key]; - }; - } - - // Internal helper to obtain the `byteLength` property of an object. - var getByteLength = shallowProperty('byteLength'); - - // Internal helper to determine whether we should spend extensive checks against - // `ArrayBuffer` et al. - var isBufferLike = createSizePropertyCheck(getByteLength); - - // Is a given value a typed array? - var typedArrayPattern = /\[object ((I|Ui)nt(8|16|32)|Float(32|64)|Uint8Clamped|Big(I|Ui)nt64)Array\]/; - function isTypedArray(obj) { - // `ArrayBuffer.isView` is the most future-proof, so use it when available. - // Otherwise, fall back on the above regular expression. - return nativeIsView ? (nativeIsView(obj) && !isDataView$1(obj)) : - isBufferLike(obj) && typedArrayPattern.test(toString.call(obj)); - } - - var isTypedArray$1 = supportsArrayBuffer ? isTypedArray : constant(false); - - // Internal helper to obtain the `length` property of an object. - var getLength = shallowProperty('length'); - - // Internal helper to create a simple lookup structure. - // `collectNonEnumProps` used to depend on `_.contains`, but this led to - // circular imports. `emulatedSet` is a one-off solution that only works for - // arrays of strings. - function emulatedSet(keys) { - var hash = {}; - for (var l = keys.length, i = 0; i < l; ++i) hash[keys[i]] = true; - return { - contains: function(key) { return hash[key]; }, - push: function(key) { - hash[key] = true; - return keys.push(key); - } - }; - } - - // Internal helper. Checks `keys` for the presence of keys in IE < 9 that won't - // be iterated by `for key in ...` and thus missed. Extends `keys` in place if - // needed. - function collectNonEnumProps(obj, keys) { - keys = emulatedSet(keys); - var nonEnumIdx = nonEnumerableProps.length; - var constructor = obj.constructor; - var proto = isFunction$1(constructor) && constructor.prototype || ObjProto; - - // Constructor is a special case. - var prop = 'constructor'; - if (has$1(obj, prop) && !keys.contains(prop)) keys.push(prop); - - while (nonEnumIdx--) { - prop = nonEnumerableProps[nonEnumIdx]; - if (prop in obj && obj[prop] !== proto[prop] && !keys.contains(prop)) { - keys.push(prop); - } - } - } - - // Retrieve the names of an object's own properties. - // Delegates to **ECMAScript 5**'s native `Object.keys`. - function keys(obj) { - if (!isObject(obj)) return []; - if (nativeKeys) return nativeKeys(obj); - var keys = []; - for (var key in obj) if (has$1(obj, key)) keys.push(key); - // Ahem, IE < 9. - if (hasEnumBug) collectNonEnumProps(obj, keys); - return keys; - } - - // Is a given array, string, or object empty? - // An "empty" object has no enumerable own-properties. - function isEmpty(obj) { - if (obj == null) return true; - // Skip the more expensive `toString`-based type checks if `obj` has no - // `.length`. - var length = getLength(obj); - if (typeof length == 'number' && ( - isArray(obj) || isString(obj) || isArguments$1(obj) - )) return length === 0; - return getLength(keys(obj)) === 0; - } - - // Returns whether an object has a given set of `key:value` pairs. - function isMatch(object, attrs) { - var _keys = keys(attrs), length = _keys.length; - if (object == null) return !length; - var obj = Object(object); - for (var i = 0; i < length; i++) { - var key = _keys[i]; - if (attrs[key] !== obj[key] || !(key in obj)) return false; - } - return true; - } - - // If Underscore is called as a function, it returns a wrapped object that can - // be used OO-style. This wrapper holds altered versions of all functions added - // through `_.mixin`. Wrapped objects may be chained. - function _$1(obj) { - if (obj instanceof _$1) return obj; - if (!(this instanceof _$1)) return new _$1(obj); - this._wrapped = obj; - } - - _$1.VERSION = VERSION; - - // Extracts the result from a wrapped and chained object. - _$1.prototype.value = function() { - return this._wrapped; - }; - - // Provide unwrapping proxies for some methods used in engine operations - // such as arithmetic and JSON stringification. - _$1.prototype.valueOf = _$1.prototype.toJSON = _$1.prototype.value; - - _$1.prototype.toString = function() { - return String(this._wrapped); - }; - - // Internal function to wrap or shallow-copy an ArrayBuffer, - // typed array or DataView to a new view, reusing the buffer. - function toBufferView(bufferSource) { - return new Uint8Array( - bufferSource.buffer || bufferSource, - bufferSource.byteOffset || 0, - getByteLength(bufferSource) - ); - } - - // We use this string twice, so give it a name for minification. - var tagDataView = '[object DataView]'; - - // Internal recursive comparison function for `_.isEqual`. - function eq(a, b, aStack, bStack) { - // Identical objects are equal. `0 === -0`, but they aren't identical. - // See the [Harmony `egal` proposal](https://wiki.ecmascript.org/doku.php?id=harmony:egal). - if (a === b) return a !== 0 || 1 / a === 1 / b; - // `null` or `undefined` only equal to itself (strict comparison). - if (a == null || b == null) return false; - // `NaN`s are equivalent, but non-reflexive. - if (a !== a) return b !== b; - // Exhaust primitive checks - var type = typeof a; - if (type !== 'function' && type !== 'object' && typeof b != 'object') return false; - return deepEq(a, b, aStack, bStack); - } - - // Internal recursive comparison function for `_.isEqual`. - function deepEq(a, b, aStack, bStack) { - // Unwrap any wrapped objects. - if (a instanceof _$1) a = a._wrapped; - if (b instanceof _$1) b = b._wrapped; - // Compare `[[Class]]` names. - var className = toString.call(a); - if (className !== toString.call(b)) return false; - // Work around a bug in IE 10 - Edge 13. - if (hasStringTagBug && className == '[object Object]' && isDataView$1(a)) { - if (!isDataView$1(b)) return false; - className = tagDataView; - } - switch (className) { - // These types are compared by value. - case '[object RegExp]': - // RegExps are coerced to strings for comparison (Note: '' + /a/i === '/a/i') - case '[object String]': - // Primitives and their corresponding object wrappers are equivalent; thus, `"5"` is - // equivalent to `new String("5")`. - return '' + a === '' + b; - case '[object Number]': - // `NaN`s are equivalent, but non-reflexive. - // Object(NaN) is equivalent to NaN. - if (+a !== +a) return +b !== +b; - // An `egal` comparison is performed for other numeric values. - return +a === 0 ? 1 / +a === 1 / b : +a === +b; - case '[object Date]': - case '[object Boolean]': - // Coerce dates and booleans to numeric primitive values. Dates are compared by their - // millisecond representations. Note that invalid dates with millisecond representations - // of `NaN` are not equivalent. - return +a === +b; - case '[object Symbol]': - return SymbolProto.valueOf.call(a) === SymbolProto.valueOf.call(b); - case '[object ArrayBuffer]': - case tagDataView: - // Coerce to typed array so we can fall through. - return deepEq(toBufferView(a), toBufferView(b), aStack, bStack); - } - - var areArrays = className === '[object Array]'; - if (!areArrays && isTypedArray$1(a)) { - var byteLength = getByteLength(a); - if (byteLength !== getByteLength(b)) return false; - if (a.buffer === b.buffer && a.byteOffset === b.byteOffset) return true; - areArrays = true; - } - if (!areArrays) { - if (typeof a != 'object' || typeof b != 'object') return false; - - // Objects with different constructors are not equivalent, but `Object`s or `Array`s - // from different frames are. - var aCtor = a.constructor, bCtor = b.constructor; - if (aCtor !== bCtor && !(isFunction$1(aCtor) && aCtor instanceof aCtor && - isFunction$1(bCtor) && bCtor instanceof bCtor) - && ('constructor' in a && 'constructor' in b)) { - return false; - } - } - // Assume equality for cyclic structures. The algorithm for detecting cyclic - // structures is adapted from ES 5.1 section 15.12.3, abstract operation `JO`. - - // Initializing stack of traversed objects. - // It's done here since we only need them for objects and arrays comparison. - aStack = aStack || []; - bStack = bStack || []; - var length = aStack.length; - while (length--) { - // Linear search. Performance is inversely proportional to the number of - // unique nested structures. - if (aStack[length] === a) return bStack[length] === b; - } - - // Add the first object to the stack of traversed objects. - aStack.push(a); - bStack.push(b); - - // Recursively compare objects and arrays. - if (areArrays) { - // Compare array lengths to determine if a deep comparison is necessary. - length = a.length; - if (length !== b.length) return false; - // Deep compare the contents, ignoring non-numeric properties. - while (length--) { - if (!eq(a[length], b[length], aStack, bStack)) return false; - } - } else { - // Deep compare objects. - var _keys = keys(a), key; - length = _keys.length; - // Ensure that both objects contain the same number of properties before comparing deep equality. - if (keys(b).length !== length) return false; - while (length--) { - // Deep compare each member - key = _keys[length]; - if (!(has$1(b, key) && eq(a[key], b[key], aStack, bStack))) return false; - } - } - // Remove the first object from the stack of traversed objects. - aStack.pop(); - bStack.pop(); - return true; - } - - // Perform a deep comparison to check if two objects are equal. - function isEqual(a, b) { - return eq(a, b); - } - - // Retrieve all the enumerable property names of an object. - function allKeys(obj) { - if (!isObject(obj)) return []; - var keys = []; - for (var key in obj) keys.push(key); - // Ahem, IE < 9. - if (hasEnumBug) collectNonEnumProps(obj, keys); - return keys; - } - - // Since the regular `Object.prototype.toString` type tests don't work for - // some types in IE 11, we use a fingerprinting heuristic instead, based - // on the methods. It's not great, but it's the best we got. - // The fingerprint method lists are defined below. - function ie11fingerprint(methods) { - var length = getLength(methods); - return function(obj) { - if (obj == null) return false; - // `Map`, `WeakMap` and `Set` have no enumerable keys. - var keys = allKeys(obj); - if (getLength(keys)) return false; - for (var i = 0; i < length; i++) { - if (!isFunction$1(obj[methods[i]])) return false; - } - // If we are testing against `WeakMap`, we need to ensure that - // `obj` doesn't have a `forEach` method in order to distinguish - // it from a regular `Map`. - return methods !== weakMapMethods || !isFunction$1(obj[forEachName]); - }; - } - - // In the interest of compact minification, we write - // each string in the fingerprints only once. - var forEachName = 'forEach', - hasName = 'has', - commonInit = ['clear', 'delete'], - mapTail = ['get', hasName, 'set']; - - // `Map`, `WeakMap` and `Set` each have slightly different - // combinations of the above sublists. - var mapMethods = commonInit.concat(forEachName, mapTail), - weakMapMethods = commonInit.concat(mapTail), - setMethods = ['add'].concat(commonInit, forEachName, hasName); - - var isMap = isIE11 ? ie11fingerprint(mapMethods) : tagTester('Map'); - - var isWeakMap = isIE11 ? ie11fingerprint(weakMapMethods) : tagTester('WeakMap'); - - var isSet = isIE11 ? ie11fingerprint(setMethods) : tagTester('Set'); - - var isWeakSet = tagTester('WeakSet'); - - // Retrieve the values of an object's properties. - function values(obj) { - var _keys = keys(obj); - var length = _keys.length; - var values = Array(length); - for (var i = 0; i < length; i++) { - values[i] = obj[_keys[i]]; - } - return values; - } - - // Convert an object into a list of `[key, value]` pairs. - // The opposite of `_.object` with one argument. - function pairs(obj) { - var _keys = keys(obj); - var length = _keys.length; - var pairs = Array(length); - for (var i = 0; i < length; i++) { - pairs[i] = [_keys[i], obj[_keys[i]]]; - } - return pairs; - } - - // Invert the keys and values of an object. The values must be serializable. - function invert(obj) { - var result = {}; - var _keys = keys(obj); - for (var i = 0, length = _keys.length; i < length; i++) { - result[obj[_keys[i]]] = _keys[i]; - } - return result; - } - - // Return a sorted list of the function names available on the object. - function functions(obj) { - var names = []; - for (var key in obj) { - if (isFunction$1(obj[key])) names.push(key); - } - return names.sort(); - } - - // An internal function for creating assigner functions. - function createAssigner(keysFunc, defaults) { - return function(obj) { - var length = arguments.length; - if (defaults) obj = Object(obj); - if (length < 2 || obj == null) return obj; - for (var index = 1; index < length; index++) { - var source = arguments[index], - keys = keysFunc(source), - l = keys.length; - for (var i = 0; i < l; i++) { - var key = keys[i]; - if (!defaults || obj[key] === void 0) obj[key] = source[key]; - } - } - return obj; - }; - } - - // Extend a given object with all the properties in passed-in object(s). - var extend = createAssigner(allKeys); - - // Assigns a given object with all the own properties in the passed-in - // object(s). - // (https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Object/assign) - var extendOwn = createAssigner(keys); - - // Fill in a given object with default properties. - var defaults = createAssigner(allKeys, true); - - // Create a naked function reference for surrogate-prototype-swapping. - function ctor() { - return function(){}; - } - - // An internal function for creating a new object that inherits from another. - function baseCreate(prototype) { - if (!isObject(prototype)) return {}; - if (nativeCreate) return nativeCreate(prototype); - var Ctor = ctor(); - Ctor.prototype = prototype; - var result = new Ctor; - Ctor.prototype = null; - return result; - } - - // Creates an object that inherits from the given prototype object. - // If additional properties are provided then they will be added to the - // created object. - function create(prototype, props) { - var result = baseCreate(prototype); - if (props) extendOwn(result, props); - return result; - } - - // Create a (shallow-cloned) duplicate of an object. - function clone(obj) { - if (!isObject(obj)) return obj; - return isArray(obj) ? obj.slice() : extend({}, obj); - } - - // Invokes `interceptor` with the `obj` and then returns `obj`. - // The primary purpose of this method is to "tap into" a method chain, in - // order to perform operations on intermediate results within the chain. - function tap(obj, interceptor) { - interceptor(obj); - return obj; - } - - // Normalize a (deep) property `path` to array. - // Like `_.iteratee`, this function can be customized. - function toPath$1(path) { - return isArray(path) ? path : [path]; - } - _$1.toPath = toPath$1; - - // Internal wrapper for `_.toPath` to enable minification. - // Similar to `cb` for `_.iteratee`. - function toPath(path) { - return _$1.toPath(path); - } - - // Internal function to obtain a nested property in `obj` along `path`. - function deepGet(obj, path) { - var length = path.length; - for (var i = 0; i < length; i++) { - if (obj == null) return void 0; - obj = obj[path[i]]; - } - return length ? obj : void 0; - } - - // Get the value of the (deep) property on `path` from `object`. - // If any property in `path` does not exist or if the value is - // `undefined`, return `defaultValue` instead. - // The `path` is normalized through `_.toPath`. - function get(object, path, defaultValue) { - var value = deepGet(object, toPath(path)); - return isUndefined(value) ? defaultValue : value; - } - - // Shortcut function for checking if an object has a given property directly on - // itself (in other words, not on a prototype). Unlike the internal `has` - // function, this public version can also traverse nested properties. - function has(obj, path) { - path = toPath(path); - var length = path.length; - for (var i = 0; i < length; i++) { - var key = path[i]; - if (!has$1(obj, key)) return false; - obj = obj[key]; - } - return !!length; - } - - // Keep the identity function around for default iteratees. - function identity(value) { - return value; - } - - // Returns a predicate for checking whether an object has a given set of - // `key:value` pairs. - function matcher(attrs) { - attrs = extendOwn({}, attrs); - return function(obj) { - return isMatch(obj, attrs); - }; - } - - // Creates a function that, when passed an object, will traverse that object’s - // properties down the given `path`, specified as an array of keys or indices. - function property(path) { - path = toPath(path); - return function(obj) { - return deepGet(obj, path); - }; - } - - // Internal function that returns an efficient (for current engines) version - // of the passed-in callback, to be repeatedly applied in other Underscore - // functions. - function optimizeCb(func, context, argCount) { - if (context === void 0) return func; - switch (argCount == null ? 3 : argCount) { - case 1: return function(value) { - return func.call(context, value); - }; - // The 2-argument case is omitted because we’re not using it. - case 3: return function(value, index, collection) { - return func.call(context, value, index, collection); - }; - case 4: return function(accumulator, value, index, collection) { - return func.call(context, accumulator, value, index, collection); - }; - } - return function() { - return func.apply(context, arguments); - }; - } - - // An internal function to generate callbacks that can be applied to each - // element in a collection, returning the desired result — either `_.identity`, - // an arbitrary callback, a property matcher, or a property accessor. - function baseIteratee(value, context, argCount) { - if (value == null) return identity; - if (isFunction$1(value)) return optimizeCb(value, context, argCount); - if (isObject(value) && !isArray(value)) return matcher(value); - return property(value); - } - - // External wrapper for our callback generator. Users may customize - // `_.iteratee` if they want additional predicate/iteratee shorthand styles. - // This abstraction hides the internal-only `argCount` argument. - function iteratee(value, context) { - return baseIteratee(value, context, Infinity); - } - _$1.iteratee = iteratee; - - // The function we call internally to generate a callback. It invokes - // `_.iteratee` if overridden, otherwise `baseIteratee`. - function cb(value, context, argCount) { - if (_$1.iteratee !== iteratee) return _$1.iteratee(value, context); - return baseIteratee(value, context, argCount); - } - - // Returns the results of applying the `iteratee` to each element of `obj`. - // In contrast to `_.map` it returns an object. - function mapObject(obj, iteratee, context) { - iteratee = cb(iteratee, context); - var _keys = keys(obj), - length = _keys.length, - results = {}; - for (var index = 0; index < length; index++) { - var currentKey = _keys[index]; - results[currentKey] = iteratee(obj[currentKey], currentKey, obj); - } - return results; - } - - // Predicate-generating function. Often useful outside of Underscore. - function noop(){} - - // Generates a function for a given object that returns a given property. - function propertyOf(obj) { - if (obj == null) return noop; - return function(path) { - return get(obj, path); - }; - } - - // Run a function **n** times. - function times(n, iteratee, context) { - var accum = Array(Math.max(0, n)); - iteratee = optimizeCb(iteratee, context, 1); - for (var i = 0; i < n; i++) accum[i] = iteratee(i); - return accum; - } - - // Return a random integer between `min` and `max` (inclusive). - function random(min, max) { - if (max == null) { - max = min; - min = 0; - } - return min + Math.floor(Math.random() * (max - min + 1)); - } - - // A (possibly faster) way to get the current timestamp as an integer. - var now = Date.now || function() { - return new Date().getTime(); - }; - - // Internal helper to generate functions for escaping and unescaping strings - // to/from HTML interpolation. - function createEscaper(map) { - var escaper = function(match) { - return map[match]; - }; - // Regexes for identifying a key that needs to be escaped. - var source = '(?:' + keys(map).join('|') + ')'; - var testRegexp = RegExp(source); - var replaceRegexp = RegExp(source, 'g'); - return function(string) { - string = string == null ? '' : '' + string; - return testRegexp.test(string) ? string.replace(replaceRegexp, escaper) : string; - }; - } - - // Internal list of HTML entities for escaping. - var escapeMap = { - '&': '&', - '<': '<', - '>': '>', - '"': '"', - "'": ''', - '`': '`' - }; - - // Function for escaping strings to HTML interpolation. - var _escape = createEscaper(escapeMap); - - // Internal list of HTML entities for unescaping. - var unescapeMap = invert(escapeMap); - - // Function for unescaping strings from HTML interpolation. - var _unescape = createEscaper(unescapeMap); - - // By default, Underscore uses ERB-style template delimiters. Change the - // following template settings to use alternative delimiters. - var templateSettings = _$1.templateSettings = { - evaluate: /<%([\s\S]+?)%>/g, - interpolate: /<%=([\s\S]+?)%>/g, - escape: /<%-([\s\S]+?)%>/g - }; - - // When customizing `_.templateSettings`, if you don't want to define an - // interpolation, evaluation or escaping regex, we need one that is - // guaranteed not to match. - var noMatch = /(.)^/; - - // Certain characters need to be escaped so that they can be put into a - // string literal. - var escapes = { - "'": "'", - '\\': '\\', - '\r': 'r', - '\n': 'n', - '\u2028': 'u2028', - '\u2029': 'u2029' - }; - - var escapeRegExp = /\\|'|\r|\n|\u2028|\u2029/g; - - function escapeChar(match) { - return '\\' + escapes[match]; - } - - // In order to prevent third-party code injection through - // `_.templateSettings.variable`, we test it against the following regular - // expression. It is intentionally a bit more liberal than just matching valid - // identifiers, but still prevents possible loopholes through defaults or - // destructuring assignment. - var bareIdentifier = /^\s*(\w|\$)+\s*$/; - - // JavaScript micro-templating, similar to John Resig's implementation. - // Underscore templating handles arbitrary delimiters, preserves whitespace, - // and correctly escapes quotes within interpolated code. - // NB: `oldSettings` only exists for backwards compatibility. - function template(text, settings, oldSettings) { - if (!settings && oldSettings) settings = oldSettings; - settings = defaults({}, settings, _$1.templateSettings); - - // Combine delimiters into one regular expression via alternation. - var matcher = RegExp([ - (settings.escape || noMatch).source, - (settings.interpolate || noMatch).source, - (settings.evaluate || noMatch).source - ].join('|') + '|$', 'g'); - - // Compile the template source, escaping string literals appropriately. - var index = 0; - var source = "__p+='"; - text.replace(matcher, function(match, escape, interpolate, evaluate, offset) { - source += text.slice(index, offset).replace(escapeRegExp, escapeChar); - index = offset + match.length; - - if (escape) { - source += "'+\n((__t=(" + escape + "))==null?'':_.escape(__t))+\n'"; - } else if (interpolate) { - source += "'+\n((__t=(" + interpolate + "))==null?'':__t)+\n'"; - } else if (evaluate) { - source += "';\n" + evaluate + "\n__p+='"; - } - - // Adobe VMs need the match returned to produce the correct offset. - return match; - }); - source += "';\n"; - - var argument = settings.variable; - if (argument) { - // Insure against third-party code injection. (CVE-2021-23358) - if (!bareIdentifier.test(argument)) throw new Error( - 'variable is not a bare identifier: ' + argument - ); - } else { - // If a variable is not specified, place data values in local scope. - source = 'with(obj||{}){\n' + source + '}\n'; - argument = 'obj'; - } - - source = "var __t,__p='',__j=Array.prototype.join," + - "print=function(){__p+=__j.call(arguments,'');};\n" + - source + 'return __p;\n'; - - var render; - try { - render = new Function(argument, '_', source); - } catch (e) { - e.source = source; - throw e; - } - - var template = function(data) { - return render.call(this, data, _$1); - }; - - // Provide the compiled source as a convenience for precompilation. - template.source = 'function(' + argument + '){\n' + source + '}'; - - return template; - } - - // Traverses the children of `obj` along `path`. If a child is a function, it - // is invoked with its parent as context. Returns the value of the final - // child, or `fallback` if any child is undefined. - function result(obj, path, fallback) { - path = toPath(path); - var length = path.length; - if (!length) { - return isFunction$1(fallback) ? fallback.call(obj) : fallback; - } - for (var i = 0; i < length; i++) { - var prop = obj == null ? void 0 : obj[path[i]]; - if (prop === void 0) { - prop = fallback; - i = length; // Ensure we don't continue iterating. - } - obj = isFunction$1(prop) ? prop.call(obj) : prop; - } - return obj; - } - - // Generate a unique integer id (unique within the entire client session). - // Useful for temporary DOM ids. - var idCounter = 0; - function uniqueId(prefix) { - var id = ++idCounter + ''; - return prefix ? prefix + id : id; - } - - // Start chaining a wrapped Underscore object. - function chain(obj) { - var instance = _$1(obj); - instance._chain = true; - return instance; - } - - // Internal function to execute `sourceFunc` bound to `context` with optional - // `args`. Determines whether to execute a function as a constructor or as a - // normal function. - function executeBound(sourceFunc, boundFunc, context, callingContext, args) { - if (!(callingContext instanceof boundFunc)) return sourceFunc.apply(context, args); - var self = baseCreate(sourceFunc.prototype); - var result = sourceFunc.apply(self, args); - if (isObject(result)) return result; - return self; - } - - // Partially apply a function by creating a version that has had some of its - // arguments pre-filled, without changing its dynamic `this` context. `_` acts - // as a placeholder by default, allowing any combination of arguments to be - // pre-filled. Set `_.partial.placeholder` for a custom placeholder argument. - var partial = restArguments(function(func, boundArgs) { - var placeholder = partial.placeholder; - var bound = function() { - var position = 0, length = boundArgs.length; - var args = Array(length); - for (var i = 0; i < length; i++) { - args[i] = boundArgs[i] === placeholder ? arguments[position++] : boundArgs[i]; - } - while (position < arguments.length) args.push(arguments[position++]); - return executeBound(func, bound, this, this, args); - }; - return bound; - }); - - partial.placeholder = _$1; - - // Create a function bound to a given object (assigning `this`, and arguments, - // optionally). - var bind = restArguments(function(func, context, args) { - if (!isFunction$1(func)) throw new TypeError('Bind must be called on a function'); - var bound = restArguments(function(callArgs) { - return executeBound(func, bound, context, this, args.concat(callArgs)); - }); - return bound; - }); - - // Internal helper for collection methods to determine whether a collection - // should be iterated as an array or as an object. - // Related: https://people.mozilla.org/~jorendorff/es6-draft.html#sec-tolength - // Avoids a very nasty iOS 8 JIT bug on ARM-64. #2094 - var isArrayLike = createSizePropertyCheck(getLength); - - // Internal implementation of a recursive `flatten` function. - function flatten$1(input, depth, strict, output) { - output = output || []; - if (!depth && depth !== 0) { - depth = Infinity; - } else if (depth <= 0) { - return output.concat(input); - } - var idx = output.length; - for (var i = 0, length = getLength(input); i < length; i++) { - var value = input[i]; - if (isArrayLike(value) && (isArray(value) || isArguments$1(value))) { - // Flatten current level of array or arguments object. - if (depth > 1) { - flatten$1(value, depth - 1, strict, output); - idx = output.length; - } else { - var j = 0, len = value.length; - while (j < len) output[idx++] = value[j++]; - } - } else if (!strict) { - output[idx++] = value; - } - } - return output; - } - - // Bind a number of an object's methods to that object. Remaining arguments - // are the method names to be bound. Useful for ensuring that all callbacks - // defined on an object belong to it. - var bindAll = restArguments(function(obj, keys) { - keys = flatten$1(keys, false, false); - var index = keys.length; - if (index < 1) throw new Error('bindAll must be passed function names'); - while (index--) { - var key = keys[index]; - obj[key] = bind(obj[key], obj); - } - return obj; - }); - - // Memoize an expensive function by storing its results. - function memoize(func, hasher) { - var memoize = function(key) { - var cache = memoize.cache; - var address = '' + (hasher ? hasher.apply(this, arguments) : key); - if (!has$1(cache, address)) cache[address] = func.apply(this, arguments); - return cache[address]; - }; - memoize.cache = {}; - return memoize; - } - - // Delays a function for the given number of milliseconds, and then calls - // it with the arguments supplied. - var delay = restArguments(function(func, wait, args) { - return setTimeout(function() { - return func.apply(null, args); - }, wait); - }); - - // Defers a function, scheduling it to run after the current call stack has - // cleared. - var defer = partial(delay, _$1, 1); - - // Returns a function, that, when invoked, will only be triggered at most once - // during a given window of time. Normally, the throttled function will run - // as much as it can, without ever going more than once per `wait` duration; - // but if you'd like to disable the execution on the leading edge, pass - // `{leading: false}`. To disable execution on the trailing edge, ditto. - function throttle(func, wait, options) { - var timeout, context, args, result; - var previous = 0; - if (!options) options = {}; - - var later = function() { - previous = options.leading === false ? 0 : now(); - timeout = null; - result = func.apply(context, args); - if (!timeout) context = args = null; - }; - - var throttled = function() { - var _now = now(); - if (!previous && options.leading === false) previous = _now; - var remaining = wait - (_now - previous); - context = this; - args = arguments; - if (remaining <= 0 || remaining > wait) { - if (timeout) { - clearTimeout(timeout); - timeout = null; - } - previous = _now; - result = func.apply(context, args); - if (!timeout) context = args = null; - } else if (!timeout && options.trailing !== false) { - timeout = setTimeout(later, remaining); - } - return result; - }; - - throttled.cancel = function() { - clearTimeout(timeout); - previous = 0; - timeout = context = args = null; - }; - - return throttled; - } - - // When a sequence of calls of the returned function ends, the argument - // function is triggered. The end of a sequence is defined by the `wait` - // parameter. If `immediate` is passed, the argument function will be - // triggered at the beginning of the sequence instead of at the end. - function debounce(func, wait, immediate) { - var timeout, previous, args, result, context; - - var later = function() { - var passed = now() - previous; - if (wait > passed) { - timeout = setTimeout(later, wait - passed); - } else { - timeout = null; - if (!immediate) result = func.apply(context, args); - // This check is needed because `func` can recursively invoke `debounced`. - if (!timeout) args = context = null; - } - }; - - var debounced = restArguments(function(_args) { - context = this; - args = _args; - previous = now(); - if (!timeout) { - timeout = setTimeout(later, wait); - if (immediate) result = func.apply(context, args); - } - return result; - }); - - debounced.cancel = function() { - clearTimeout(timeout); - timeout = args = context = null; - }; - - return debounced; - } - - // Returns the first function passed as an argument to the second, - // allowing you to adjust arguments, run code before and after, and - // conditionally execute the original function. - function wrap(func, wrapper) { - return partial(wrapper, func); - } - - // Returns a negated version of the passed-in predicate. - function negate(predicate) { - return function() { - return !predicate.apply(this, arguments); - }; - } - - // Returns a function that is the composition of a list of functions, each - // consuming the return value of the function that follows. - function compose() { - var args = arguments; - var start = args.length - 1; - return function() { - var i = start; - var result = args[start].apply(this, arguments); - while (i--) result = args[i].call(this, result); - return result; - }; - } - - // Returns a function that will only be executed on and after the Nth call. - function after(times, func) { - return function() { - if (--times < 1) { - return func.apply(this, arguments); - } - }; - } - - // Returns a function that will only be executed up to (but not including) the - // Nth call. - function before(times, func) { - var memo; - return function() { - if (--times > 0) { - memo = func.apply(this, arguments); - } - if (times <= 1) func = null; - return memo; - }; - } - - // Returns a function that will be executed at most one time, no matter how - // often you call it. Useful for lazy initialization. - var once = partial(before, 2); - - // Returns the first key on an object that passes a truth test. - function findKey(obj, predicate, context) { - predicate = cb(predicate, context); - var _keys = keys(obj), key; - for (var i = 0, length = _keys.length; i < length; i++) { - key = _keys[i]; - if (predicate(obj[key], key, obj)) return key; - } - } - - // Internal function to generate `_.findIndex` and `_.findLastIndex`. - function createPredicateIndexFinder(dir) { - return function(array, predicate, context) { - predicate = cb(predicate, context); - var length = getLength(array); - var index = dir > 0 ? 0 : length - 1; - for (; index >= 0 && index < length; index += dir) { - if (predicate(array[index], index, array)) return index; - } - return -1; - }; - } - - // Returns the first index on an array-like that passes a truth test. - var findIndex = createPredicateIndexFinder(1); - - // Returns the last index on an array-like that passes a truth test. - var findLastIndex = createPredicateIndexFinder(-1); - - // Use a comparator function to figure out the smallest index at which - // an object should be inserted so as to maintain order. Uses binary search. - function sortedIndex(array, obj, iteratee, context) { - iteratee = cb(iteratee, context, 1); - var value = iteratee(obj); - var low = 0, high = getLength(array); - while (low < high) { - var mid = Math.floor((low + high) / 2); - if (iteratee(array[mid]) < value) low = mid + 1; else high = mid; - } - return low; - } - - // Internal function to generate the `_.indexOf` and `_.lastIndexOf` functions. - function createIndexFinder(dir, predicateFind, sortedIndex) { - return function(array, item, idx) { - var i = 0, length = getLength(array); - if (typeof idx == 'number') { - if (dir > 0) { - i = idx >= 0 ? idx : Math.max(idx + length, i); - } else { - length = idx >= 0 ? Math.min(idx + 1, length) : idx + length + 1; - } - } else if (sortedIndex && idx && length) { - idx = sortedIndex(array, item); - return array[idx] === item ? idx : -1; - } - if (item !== item) { - idx = predicateFind(slice.call(array, i, length), isNaN$1); - return idx >= 0 ? idx + i : -1; - } - for (idx = dir > 0 ? i : length - 1; idx >= 0 && idx < length; idx += dir) { - if (array[idx] === item) return idx; - } - return -1; - }; - } - - // Return the position of the first occurrence of an item in an array, - // or -1 if the item is not included in the array. - // If the array is large and already in sort order, pass `true` - // for **isSorted** to use binary search. - var indexOf = createIndexFinder(1, findIndex, sortedIndex); - - // Return the position of the last occurrence of an item in an array, - // or -1 if the item is not included in the array. - var lastIndexOf = createIndexFinder(-1, findLastIndex); - - // Return the first value which passes a truth test. - function find(obj, predicate, context) { - var keyFinder = isArrayLike(obj) ? findIndex : findKey; - var key = keyFinder(obj, predicate, context); - if (key !== void 0 && key !== -1) return obj[key]; - } - - // Convenience version of a common use case of `_.find`: getting the first - // object containing specific `key:value` pairs. - function findWhere(obj, attrs) { - return find(obj, matcher(attrs)); - } - - // The cornerstone for collection functions, an `each` - // implementation, aka `forEach`. - // Handles raw objects in addition to array-likes. Treats all - // sparse array-likes as if they were dense. - function each(obj, iteratee, context) { - iteratee = optimizeCb(iteratee, context); - var i, length; - if (isArrayLike(obj)) { - for (i = 0, length = obj.length; i < length; i++) { - iteratee(obj[i], i, obj); - } - } else { - var _keys = keys(obj); - for (i = 0, length = _keys.length; i < length; i++) { - iteratee(obj[_keys[i]], _keys[i], obj); - } - } - return obj; - } - - // Return the results of applying the iteratee to each element. - function map(obj, iteratee, context) { - iteratee = cb(iteratee, context); - var _keys = !isArrayLike(obj) && keys(obj), - length = (_keys || obj).length, - results = Array(length); - for (var index = 0; index < length; index++) { - var currentKey = _keys ? _keys[index] : index; - results[index] = iteratee(obj[currentKey], currentKey, obj); - } - return results; - } - - // Internal helper to create a reducing function, iterating left or right. - function createReduce(dir) { - // Wrap code that reassigns argument variables in a separate function than - // the one that accesses `arguments.length` to avoid a perf hit. (#1991) - var reducer = function(obj, iteratee, memo, initial) { - var _keys = !isArrayLike(obj) && keys(obj), - length = (_keys || obj).length, - index = dir > 0 ? 0 : length - 1; - if (!initial) { - memo = obj[_keys ? _keys[index] : index]; - index += dir; - } - for (; index >= 0 && index < length; index += dir) { - var currentKey = _keys ? _keys[index] : index; - memo = iteratee(memo, obj[currentKey], currentKey, obj); - } - return memo; - }; - - return function(obj, iteratee, memo, context) { - var initial = arguments.length >= 3; - return reducer(obj, optimizeCb(iteratee, context, 4), memo, initial); - }; - } - - // **Reduce** builds up a single result from a list of values, aka `inject`, - // or `foldl`. - var reduce = createReduce(1); - - // The right-associative version of reduce, also known as `foldr`. - var reduceRight = createReduce(-1); - - // Return all the elements that pass a truth test. - function filter(obj, predicate, context) { - var results = []; - predicate = cb(predicate, context); - each(obj, function(value, index, list) { - if (predicate(value, index, list)) results.push(value); - }); - return results; - } - - // Return all the elements for which a truth test fails. - function reject(obj, predicate, context) { - return filter(obj, negate(cb(predicate)), context); - } - - // Determine whether all of the elements pass a truth test. - function every(obj, predicate, context) { - predicate = cb(predicate, context); - var _keys = !isArrayLike(obj) && keys(obj), - length = (_keys || obj).length; - for (var index = 0; index < length; index++) { - var currentKey = _keys ? _keys[index] : index; - if (!predicate(obj[currentKey], currentKey, obj)) return false; - } - return true; - } - - // Determine if at least one element in the object passes a truth test. - function some(obj, predicate, context) { - predicate = cb(predicate, context); - var _keys = !isArrayLike(obj) && keys(obj), - length = (_keys || obj).length; - for (var index = 0; index < length; index++) { - var currentKey = _keys ? _keys[index] : index; - if (predicate(obj[currentKey], currentKey, obj)) return true; - } - return false; - } - - // Determine if the array or object contains a given item (using `===`). - function contains(obj, item, fromIndex, guard) { - if (!isArrayLike(obj)) obj = values(obj); - if (typeof fromIndex != 'number' || guard) fromIndex = 0; - return indexOf(obj, item, fromIndex) >= 0; - } - - // Invoke a method (with arguments) on every item in a collection. - var invoke = restArguments(function(obj, path, args) { - var contextPath, func; - if (isFunction$1(path)) { - func = path; - } else { - path = toPath(path); - contextPath = path.slice(0, -1); - path = path[path.length - 1]; - } - return map(obj, function(context) { - var method = func; - if (!method) { - if (contextPath && contextPath.length) { - context = deepGet(context, contextPath); - } - if (context == null) return void 0; - method = context[path]; - } - return method == null ? method : method.apply(context, args); - }); - }); - - // Convenience version of a common use case of `_.map`: fetching a property. - function pluck(obj, key) { - return map(obj, property(key)); - } - - // Convenience version of a common use case of `_.filter`: selecting only - // objects containing specific `key:value` pairs. - function where(obj, attrs) { - return filter(obj, matcher(attrs)); - } - - // Return the maximum element (or element-based computation). - function max(obj, iteratee, context) { - var result = -Infinity, lastComputed = -Infinity, - value, computed; - if (iteratee == null || typeof iteratee == 'number' && typeof obj[0] != 'object' && obj != null) { - obj = isArrayLike(obj) ? obj : values(obj); - for (var i = 0, length = obj.length; i < length; i++) { - value = obj[i]; - if (value != null && value > result) { - result = value; - } - } - } else { - iteratee = cb(iteratee, context); - each(obj, function(v, index, list) { - computed = iteratee(v, index, list); - if (computed > lastComputed || computed === -Infinity && result === -Infinity) { - result = v; - lastComputed = computed; - } - }); - } - return result; - } - - // Return the minimum element (or element-based computation). - function min(obj, iteratee, context) { - var result = Infinity, lastComputed = Infinity, - value, computed; - if (iteratee == null || typeof iteratee == 'number' && typeof obj[0] != 'object' && obj != null) { - obj = isArrayLike(obj) ? obj : values(obj); - for (var i = 0, length = obj.length; i < length; i++) { - value = obj[i]; - if (value != null && value < result) { - result = value; - } - } - } else { - iteratee = cb(iteratee, context); - each(obj, function(v, index, list) { - computed = iteratee(v, index, list); - if (computed < lastComputed || computed === Infinity && result === Infinity) { - result = v; - lastComputed = computed; - } - }); - } - return result; - } - - // Sample **n** random values from a collection using the modern version of the - // [Fisher-Yates shuffle](https://en.wikipedia.org/wiki/Fisher–Yates_shuffle). - // If **n** is not specified, returns a single random element. - // The internal `guard` argument allows it to work with `_.map`. - function sample(obj, n, guard) { - if (n == null || guard) { - if (!isArrayLike(obj)) obj = values(obj); - return obj[random(obj.length - 1)]; - } - var sample = isArrayLike(obj) ? clone(obj) : values(obj); - var length = getLength(sample); - n = Math.max(Math.min(n, length), 0); - var last = length - 1; - for (var index = 0; index < n; index++) { - var rand = random(index, last); - var temp = sample[index]; - sample[index] = sample[rand]; - sample[rand] = temp; - } - return sample.slice(0, n); - } - - // Shuffle a collection. - function shuffle(obj) { - return sample(obj, Infinity); - } - - // Sort the object's values by a criterion produced by an iteratee. - function sortBy(obj, iteratee, context) { - var index = 0; - iteratee = cb(iteratee, context); - return pluck(map(obj, function(value, key, list) { - return { - value: value, - index: index++, - criteria: iteratee(value, key, list) - }; - }).sort(function(left, right) { - var a = left.criteria; - var b = right.criteria; - if (a !== b) { - if (a > b || a === void 0) return 1; - if (a < b || b === void 0) return -1; - } - return left.index - right.index; - }), 'value'); - } - - // An internal function used for aggregate "group by" operations. - function group(behavior, partition) { - return function(obj, iteratee, context) { - var result = partition ? [[], []] : {}; - iteratee = cb(iteratee, context); - each(obj, function(value, index) { - var key = iteratee(value, index, obj); - behavior(result, value, key); - }); - return result; - }; - } - - // Groups the object's values by a criterion. Pass either a string attribute - // to group by, or a function that returns the criterion. - var groupBy = group(function(result, value, key) { - if (has$1(result, key)) result[key].push(value); else result[key] = [value]; - }); - - // Indexes the object's values by a criterion, similar to `_.groupBy`, but for - // when you know that your index values will be unique. - var indexBy = group(function(result, value, key) { - result[key] = value; - }); - - // Counts instances of an object that group by a certain criterion. Pass - // either a string attribute to count by, or a function that returns the - // criterion. - var countBy = group(function(result, value, key) { - if (has$1(result, key)) result[key]++; else result[key] = 1; - }); - - // Split a collection into two arrays: one whose elements all pass the given - // truth test, and one whose elements all do not pass the truth test. - var partition = group(function(result, value, pass) { - result[pass ? 0 : 1].push(value); - }, true); - - // Safely create a real, live array from anything iterable. - var reStrSymbol = /[^\ud800-\udfff]|[\ud800-\udbff][\udc00-\udfff]|[\ud800-\udfff]/g; - function toArray(obj) { - if (!obj) return []; - if (isArray(obj)) return slice.call(obj); - if (isString(obj)) { - // Keep surrogate pair characters together. - return obj.match(reStrSymbol); - } - if (isArrayLike(obj)) return map(obj, identity); - return values(obj); - } - - // Return the number of elements in a collection. - function size(obj) { - if (obj == null) return 0; - return isArrayLike(obj) ? obj.length : keys(obj).length; - } - - // Internal `_.pick` helper function to determine whether `key` is an enumerable - // property name of `obj`. - function keyInObj(value, key, obj) { - return key in obj; - } - - // Return a copy of the object only containing the allowed properties. - var pick = restArguments(function(obj, keys) { - var result = {}, iteratee = keys[0]; - if (obj == null) return result; - if (isFunction$1(iteratee)) { - if (keys.length > 1) iteratee = optimizeCb(iteratee, keys[1]); - keys = allKeys(obj); - } else { - iteratee = keyInObj; - keys = flatten$1(keys, false, false); - obj = Object(obj); - } - for (var i = 0, length = keys.length; i < length; i++) { - var key = keys[i]; - var value = obj[key]; - if (iteratee(value, key, obj)) result[key] = value; - } - return result; - }); - - // Return a copy of the object without the disallowed properties. - var omit = restArguments(function(obj, keys) { - var iteratee = keys[0], context; - if (isFunction$1(iteratee)) { - iteratee = negate(iteratee); - if (keys.length > 1) context = keys[1]; - } else { - keys = map(flatten$1(keys, false, false), String); - iteratee = function(value, key) { - return !contains(keys, key); - }; - } - return pick(obj, iteratee, context); - }); - - // Returns everything but the last entry of the array. Especially useful on - // the arguments object. Passing **n** will return all the values in - // the array, excluding the last N. - function initial(array, n, guard) { - return slice.call(array, 0, Math.max(0, array.length - (n == null || guard ? 1 : n))); - } - - // Get the first element of an array. Passing **n** will return the first N - // values in the array. The **guard** check allows it to work with `_.map`. - function first(array, n, guard) { - if (array == null || array.length < 1) return n == null || guard ? void 0 : []; - if (n == null || guard) return array[0]; - return initial(array, array.length - n); - } - - // Returns everything but the first entry of the `array`. Especially useful on - // the `arguments` object. Passing an **n** will return the rest N values in the - // `array`. - function rest(array, n, guard) { - return slice.call(array, n == null || guard ? 1 : n); - } - - // Get the last element of an array. 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- @@ -12,11 +12,11 @@
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- @@ -27,11 +27,15 @@
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- @@ -43,8 +47,10 @@
 <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> - <li class="toctree-l1"><a class="reference internal" href="C06_Abstract_Algebra.html">6. Abstract Algebra</a></li> - <li class="toctree-l1"><a class="reference internal" href="C07_Topology.html">7. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> + <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> + <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> + <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> </ul> <ul class="current"> <li class="toctree-l1 current"><a class="current reference internal" href="#">Index</a></li>
- @@ -63,8 +69,8 @@
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- @@ -162,7 +168,7 @@
 </li> <li><a href="C03_Logic.html#index-18">constructor</a> </li> - <li><a href="C07_Topology.html#index-3">continuity</a> + <li><a href="C08_Topology.html#index-3">continuity</a> </li> <li><a href="C03_Logic.html#index-17">contradiction</a> </li>
- @@ -178,7 +184,7 @@
 <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C02_Basics.html#index-12">definitional equality</a> </li> - <li><a href="C08_Differential_Calculus.html#index-0">differential calculus</a> + <li><a href="C09_Differential_Calculus.html#index-0">differential calculus</a> </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul>
- @@ -192,6 +198,8 @@
 <h2 id="E">E</h2> <table style="width: 100%" class="indextable genindextable"><tr> <td style="width: 33%; vertical-align: top;"><ul> + <li><a href="C09_Differential_Calculus.html#index-1">elementary calculus</a> + </li> <li><a href="C03_Logic.html#index-4">erw</a> </li> <li><a href="C02_Basics.html#index-5">exact</a>
- @@ -218,7 +226,7 @@
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- @@ -262,7 +270,7 @@
 <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C03_Logic.html#index-5">injective function</a> </li> - <li><a href="C09_Integration_and_Measure_Theory.html#index-0">integration</a> + <li><a href="C10_Integration_and_Measure_Theory.html#index-0">integration</a> </li> <li><a href="C03_Logic.html#index-0">intro</a> </li>
- @@ -298,7 +306,7 @@
 <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C02_Basics.html#index-20">max</a> </li> - <li><a href="C02_Basics.html#index-28">metric space</a>, <a href="C07_Topology.html#index-2">[1]</a> + <li><a href="C02_Basics.html#index-28">metric space</a>, <a href="C08_Topology.html#index-2">[1]</a> </li> </ul></td> <td style="width: 33%; vertical-align: top;"><ul>
- @@ -317,6 +325,8 @@
 </ul></td> <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C02_Basics.html#index-19">norm_num</a> + </li> + <li><a href="C09_Differential_Calculus.html#index-2">normed space</a> </li> </ul></td> </tr></table>
- @@ -431,7 +441,7 @@
 </li> <li><a href="C03_Logic.html#index-18">constructor</a> </li> - <li><a href="C07_Topology.html#index-3">continuity</a> + <li><a href="C08_Topology.html#index-3">continuity</a> </li> <li><a href="C03_Logic.html#index-17">contradiction</a> </li>
- @@ -496,9 +506,9 @@
 <td style="width: 33%; vertical-align: top;"><ul> <li><a href="C03_Logic.html#index-12">this</a> </li> - <li><a href="C07_Topology.html#index-4">topological space</a> + <li><a href="C08_Topology.html#index-4">topological space</a> </li> - <li><a href="C07_Topology.html#index-0">topology</a> + <li><a href="C08_Topology.html#index-0">topology</a> </li> </ul></td> </tr></table>
html/index.html
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  [{"git": {"url": "https://github.com/leanprover-community/mathlib4", "subDir?": null, - "rev": "52d71ad9606cf755e2437cfedfbcc89ce8f2b86a", + "rev": "311c241339518f68f44f2c7ecb3df0e7bdc64853", "name": "mathlib", "inputRev?": "master"}}, {"git":
- @@ -16,12 +16,12 @@
  {"git": {"url": "https://github.com/JLimperg/aesop", "subDir?": null, - "rev": "38bcf8b9e564d23bad55cbfa3c770f135f926d98", + "rev": "ca73109cc40837bc61df8024c9016da4b4f99d4c", "name": "aesop", "inputRev?": "master"}}, {"git": {"url": "https://github.com/leanprover/std4", "subDir?": null, - "rev": "b886042255d5e93b8d0adb780d49b1c8f3482fbb", + "rev": "6932c4ea52914dc6b0488944e367459ddc4d01a6", "name": "std", "inputRev?": "main"}}]}
lake-packages/Qq (new)
- @@ -0,0 +1,1 @@
  + Subproject commit c71f94e34c1cda52eef5c93dc9da409ab2727420
lake-packages/aesop (new)
- @@ -0,0 +1,1 @@
  + Subproject commit 38bcf8b9e564d23bad55cbfa3c770f135f926d98
lake-packages/mathlib (new)
- @@ -0,0 +1,1 @@
  + Subproject commit 52d71ad9606cf755e2437cfedfbcc89ce8f2b86a
lake-packages/std (new)
- @@ -0,0 +1,1 @@
  + Subproject commit b886042255d5e93b8d0adb780d49b1c8f3482fbb
lakefile.lean
- @@ -6,7 +6,7 @@ package mil {
  } @[default_target] - lean_lib «MIL» { + lean_lib MIL { -- add library configuration options here }
lean-toolchain
- @@ -1,1 +1,1 @@
  - leanprover/lean4:nightly-2023-05-16 + leanprover/lean4:nightly-2023-05-31
mathematics_in_lean.pdf
src/01_Introduction/01_Getting_Started.lean (new)
- @@ -0,0 +1,1 @@
  + #eval "Hello, World!"
src/01_Introduction/02_Overview.lean (new)
- @@ -0,0 +1,62 @@
  + import data.nat.basic + import data.nat.parity + import tactic + + open nat + /- These are pieces of data. -/ + + #check 2 + 2 + + def f (x : ℕ) := x + 3 + + #check f + + /- These are propositions, of type `Prop`. -/ + + #check 2 + 2 = 4 + + def fermat_last_theorem := + ∀ x y z n : ℕ, n > 2 ∧ x * y * z ≠ 0 → x^n + y^n ≠ z^n + + #check fermat_last_theorem + + /- These are proofs of propositions. -/ + + theorem easy : 2 + 2 = 4 := rfl + + #check easy + + theorem hard : fermat_last_theorem := sorry + + #check hard + + /- Here are some proofs. -/ + + example : ∀ m n : nat, even n → even (m * n) := + assume m n ⟨k, (hk : n = k + k)⟩, + have hmn : m * n = m * k + m * k, + by rw [hk, mul_add], + show ∃ l, m * n = l + l, + from ⟨_, hmn⟩ + + example : ∀ m n : nat, even n → even (m * n) := + λ m n ⟨k, hk⟩, ⟨m * k, by rw [hk, mul_add]⟩ + + example : ∀ m n : nat, even n → even (m * n) := + begin + -- say m and n are natural numbers, and assume n=2*k + rintros m n ⟨k, hk⟩, + -- We need to prove m*n is twice a natural. Let's show it's twice m*k. + use m * k, + -- substitute in for n + rw hk, + -- and now it's obvious + ring + end + + example : ∀ m n : nat, even n → even (m * n) := + by { rintros m n ⟨k, hk⟩, use m * k, rw hk, ring } + + example : ∀ m n : nat, even n → even (m * n) := + by intros; simp * with parity_simps +
src/01_Introduction/solutions/solutions_01_Getting_Started.lean (new)
src/01_Introduction/solutions/solutions_02_Overview.lean (new)
- @@ -0,0 +1,6 @@
  + import data.nat.basic + import data.nat.parity + import tactic + + open nat + -- There are no exercises in this section.
src/02_Basics/01_Calculating.lean (new)
- @@ -0,0 +1,186 @@
  + import data.real.basic + + /- An example. -/ + + import data.real.basic + example (a b c : ℝ) : (a * b) * c = b * (a * c) := + begin + rw mul_comm a b, + rw mul_assoc b a c + end + + /- Try these.-/ + + example (a b c : ℝ) : (c * b) * a = b * (a * c) := + begin + sorry + end + + example (a b c : ℝ) : a * (b * c) = b * (a * c) := + begin + sorry + end + + /- An example. -/ + + example (a b c : ℝ) : a * b * c = b * c * a := + begin + rw mul_assoc, + rw mul_comm + end + + /- Try doing the first of these without providing any arguments at all, + and the second with only one argument. -/ + + example (a b c : ℝ) : a * (b * c) = b * (c * a) := + begin + sorry + end + + example (a b c : ℝ) : a * (b * c) = b * (a * c) := + begin + sorry + end + + /- Using facts from the local context. -/ + + example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : + a * (b * e) = c * (d * f) := + begin + rw h', + rw ←mul_assoc, + rw h, + rw mul_assoc + end + + /- Try these. For the second one, use the theorem `sub_self`. -/ + + example (a b c d e f : ℝ) (h : b * c = e * f) : + a * b * c * d = a * e * f * d := + begin + sorry + end + + example (a b c d : ℝ) (hyp : c = b * a - d) (hyp' : d = a * b) : c = 0 := + begin + sorry + end + + /- Examples. -/ + + example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : + a * (b * e) = c * (d * f) := + by rw [h', ←mul_assoc, h, mul_assoc] + + section + + variables a b c d e f g : ℝ + + example (h : a * b = c * d) (h' : e = f) : + a * (b * e) = c * (d * f) := + by rw [h', ←mul_assoc, h, mul_assoc] + + end + + section + variables a b c : ℝ + + #check a + #check a + b + #check (a : ℝ) + #check mul_comm a b + #check (mul_comm a b : a * b = b * a) + #check mul_assoc c a b + #check mul_comm a + #check mul_comm + #check @mul_comm + + end + + section + variables a b : ℝ + + example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := + begin + rw [mul_add, add_mul, add_mul], + rw [←add_assoc, add_assoc (a * a)], + rw [mul_comm b a, ←two_mul] + end + + example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := + calc + (a + b) * (a + b) + = a * a + b * a + (a * b + b * b) : + by rw [mul_add, add_mul, add_mul] + ... = a * a + (b * a + a * b) + b * b : + by rw [←add_assoc, add_assoc (a * a)] + ... = a * a + 2 * (a * b) + b * b : + by rw [mul_comm b a, ←two_mul] + + + example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := + calc + (a + b) * (a + b) + = a * a + b * a + (a * b + b * b) : + begin + sorry + end + ... = a * a + (b * a + a * b) + b * b : by sorry + ... = a * a + 2 * (a * b) + b * b : by sorry + end + + /- Try these. For the second, use the theorems listed underneath. -/ + + section + variables a b c d : ℝ + + example : (a + b) * (c + d) = a * c + a * d + b * c + b * d := + sorry + + example (a b : ℝ) : (a + b) * (a - b) = a^2 - b^2 := + begin + sorry + end + + #check pow_two a + #check mul_sub a b c + #check add_mul a b c + #check add_sub a b c + #check sub_sub a b c + #check add_zero a + + end + + /- Examples. -/ + + section + variables a b c d : ℝ + + example (a b c d : ℝ) (hyp : c = d * a + b) (hyp' : b = a * d) : + c = 2 * a * d := + begin + 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 + end + + example : (c * b) * a = b * (a * c) := + by ring + + example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := + by ring + + example : (a + b) * (a - b) = a^2 - b^2 := + by ring + + example (hyp : c = d * a + b) (hyp' : b = a * d) : + c = 2 * a * d := + begin + rw [hyp, hyp'], + ring + end + + end +
src/02_Basics/02_Proving_Identities_in_Algebraic_Structures.lean (new)
- @@ -0,0 +1,157 @@
  + import algebra.ring + import data.real.basic + import tactic + + section + variables (R : Type*) [ring R] + + #check (add_assoc : ∀ a b c : R, a + b + c = a + (b + c)) + #check (add_comm : ∀ a b : R, a + b = b + a) + #check (zero_add : ∀ a : R, 0 + a = a) + #check (add_left_neg : ∀ a : R, -a + a = 0) + #check (mul_assoc : ∀ a b c : R, a * b * c = a * (b * c)) + #check (mul_one : ∀ a : R, a * 1 = a) + #check (one_mul : ∀ a : R, 1 * a = a) + #check (mul_add : ∀ a b c : R, a * (b + c) = a * b + a * c) + #check (add_mul : ∀ a b c : R, (a + b) * c = a * c + b * c) + end + + section + variables (R : Type*) [comm_ring R] + variables a b c d : R + + example : (c * b) * a = b * (a * c) := + by ring + + example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := + by ring + + example : (a + b) * (a - b) = a^2 - b^2 := + by ring + + example (hyp : c = d * a + b) (hyp' : b = a * d) : + c = 2 * a * d := + begin + rw [hyp, hyp'], + ring + end + end + + namespace my_ring + variables {R : Type*} [ring R] + + theorem add_zero (a : R) : a + 0 = a := + by rw [add_comm, zero_add] + + theorem add_right_neg (a : R) : a + -a = 0 := + by rw [add_comm, add_left_neg] + + #check @my_ring.add_zero + #check @add_zero + + end my_ring + + namespace my_ring + variables {R : Type*} [ring R] + + theorem neg_add_cancel_left (a b : R) : -a + (a + b) = b := + by rw [←add_assoc, add_left_neg, zero_add] + + /- Prove these: -/ + + theorem add_neg_cancel_right (a b : R) : (a + b) + -b = a := + sorry + + theorem add_left_cancel {a b c : R} (h : a + b = a + c) : b = c := + sorry + + theorem add_right_cancel {a b c : R} (h : a + b = c + b) : a = c := + sorry + + theorem mul_zero (a : R) : a * 0 = 0 := + begin + have h : a * 0 + a * 0 = a * 0 + 0, + { rw [←mul_add, add_zero, add_zero] }, + rw add_left_cancel h + end + + theorem zero_mul (a : R) : 0 * a = 0 := + sorry + + theorem neg_eq_of_add_eq_zero {a b : R} (h : a + b = 0) : -a = b := + sorry + + theorem eq_neg_of_add_eq_zero {a b : R} (h : a + b = 0) : a = -b := + sorry + + theorem neg_zero : (-0 : R) = 0 := + begin + apply neg_eq_of_add_eq_zero, + rw add_zero + end + + theorem neg_neg (a : R) : -(-a) = a := + sorry + + end my_ring + + /- Examples. -/ + + section + variables {R : Type*} [ring R] + + example (a b : R) : a - b = a + -b := + sub_eq_add_neg a b + + end + + example (a b : ℝ) : a - b = a + -b := + rfl + + example (a b : ℝ) : a - b = a + -b := + by reflexivity + + namespace my_ring + + variables {R : Type*} [ring R] + + theorem self_sub (a : R) : a - a = 0 := + sorry + + lemma one_add_one_eq_two : 1 + 1 = (2 : R) := + by refl + + theorem two_mul (a : R) : 2 * a = a + a := + sorry + + end my_ring + + section + variables (A : Type*) [add_group A] + + #check (add_assoc : ∀ a b c : A, a + b + c = a + (b + c)) + #check (zero_add : ∀ a : A, 0 + a = a) + #check (add_left_neg : ∀ a : A, -a + a = 0) + end + + section + variables {G : Type*} [group G] + + #check (mul_assoc : ∀ a b c : G, a * b * c = a * (b * c)) + #check (one_mul : ∀ a : G, 1 * a = a) + #check (mul_left_inv : ∀ a : G, a⁻¹ * a = 1) + + namespace my_group + + theorem mul_right_inv (a : G) : a * a⁻¹ = 1 := + sorry + + theorem mul_one (a : G) : a * 1 = a := + sorry + + theorem mul_inv_rev (a b : G) : (a * b)⁻¹ = b⁻¹ * a ⁻¹ := + sorry + + end my_group + end +
src/02_Basics/03_Using_Theorems_and_Lemmas.lean (new)
- @@ -0,0 +1,159 @@
 + import analysis.special_functions.log.basic + + variables a b c d e : ℝ + open real + + #check (le_refl : ∀ a : ℝ, a ≤ a) + #check (le_trans : a ≤ b → b ≤ c → a ≤ c) + + section + variables (h : a ≤ b) (h' : b ≤ c) + + #check (le_refl : ∀ a : real, a ≤ a) + #check (le_refl a : a ≤ a) + #check (le_trans : a ≤ b → b ≤ c → a ≤ c) + #check (le_trans h : b ≤ c → a ≤ c) + #check (le_trans h h' : a ≤ c) + end + + example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := + begin + apply le_trans, + { apply h₀ }, + apply h₁ + end + + example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := + begin + apply le_trans h₀, + apply h₁ + end + + example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := + by exact 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 exact le_refl x + + example (x : ℝ) : x ≤ x := + le_refl x + + #check (le_refl : ∀ a, a ≤ a) + #check (le_trans : a ≤ b → b ≤ c → a ≤ c) + #check (lt_of_le_of_lt : a ≤ b → b < c → a < c) + #check (lt_of_lt_of_le : a < b → b ≤ c → a < c) + #check (lt_trans : a < b → b < c → a < c) + + /- Try this. -/ + + example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) + (h₃ : d < e) : + a < e := + sorry + + example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) + (h₃ : d < e) : + a < e := + by linarith + + section + example (h : 2 * a ≤ 3 * b) (h' : 1 ≤ a) (h'' : d = 2) : + d + a ≤ 5 * b := + by linarith + end + + example (h : 1 ≤ a) (h' : b ≤ c) : + 2 + a + exp b ≤ 3 * a + exp c := + by linarith [exp_le_exp.mpr h'] + + #check (exp_le_exp : exp a ≤ exp b ↔ a ≤ b) + #check (exp_lt_exp : exp a < exp b ↔ a < b) + #check (log_le_log : 0 < a → 0 < b → (log a ≤ log b ↔ a ≤ b)) + #check (log_lt_log : 0 < a → a < b → log a < log b) + #check (add_le_add : a ≤ b → c ≤ d → a + c ≤ b + d) + #check (add_le_add_left : a ≤ b → ∀ c, c + a ≤ c + b) + #check (add_le_add_right : a ≤ b → ∀ c, a + c ≤ b + c) + #check (add_lt_add_of_le_of_lt : a ≤ b → c < d → a + c < b + d) + #check (add_lt_add_of_lt_of_le : a < b → c ≤ d → a + c < b + d) + #check (add_lt_add_left : a < b → ∀ c, c + a < c + b) + #check (add_lt_add_right : a < b → ∀ c, a + c < b + c) + #check (add_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a + b) + #check (add_pos : 0 < a → 0 < b → 0 < a + b) + #check (add_pos_of_pos_of_nonneg : 0 < a → 0 ≤ b → 0 < a + b) + #check (exp_pos : ∀ a, 0 < exp a) + + #check @add_le_add_left + example (h : a ≤ b) : exp a ≤ exp b := + begin + rw exp_le_exp, + exact h + end + + example (h₀ : a ≤ b) (h₁ : c < d) : a + exp c + e < b + exp d + e := + begin + 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 + end + + example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := + begin + sorry + end + + example : (0 : ℝ) < 1 := + by norm_num + + example (h : a ≤ b) : log (1 + exp a) ≤ log (1 + exp b) := + begin + have h₀ : 0 < 1 + exp a, + { sorry }, + have h₁ : 0 < 1 + exp b, + { sorry }, + apply (log_le_log h₀ h₁).mpr, + sorry + end + + example : 0 ≤ a^2 := + begin + -- library_search, + exact pow_two_nonneg a + end + + example (h : a ≤ b) : c - exp b ≤ c - exp a := + sorry + + example : 2*a*b ≤ a^2 + b^2 := + begin + 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 + end + + example : 2*a*b ≤ a^2 + b^2 := + begin + 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 + end + + example : abs (a*b) ≤ (a^2 + b^2) / 2 := + sorry + + #check abs_le'.mpr +
src/02_Basics/04_More_on_Order_and_Divisibility.lean (new)
- @@ -0,0 +1,93 @@
  + import data.real.basic + + section + variables a b c d : ℝ + + #check (min_le_left a b : min a b ≤ a) + #check (min_le_right a b : min a b ≤ b) + #check (le_min : c ≤ a → c ≤ b → c ≤ min a b) + + example : min a b = min b a := + begin + 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 } + end + + example : min a b = min b a := + begin + have h : ∀ x y, min x y ≤ min y x, + { intros x y, + apply le_min, + apply min_le_right, + apply min_le_left }, + apply le_antisymm, apply h, apply h + end + + example : min a b = min b a := + begin + apply le_antisymm, + repeat { + apply le_min, + apply min_le_right, + apply min_le_left } + end + + example : max a b = max b a := + sorry + + example : min (min a b) c = min a (min b c) := + sorry + + lemma aux : min a b + c ≤ min (a + c) (b + c) := + sorry + + example : min a b + c = min (a + c) (b + c) := + sorry + + #check (abs_add : ∀ a b : ℝ, abs (a + b) ≤ abs a + abs b) + + example : abs a - abs b ≤ abs (a - b) := + sorry + + end + + section + variables w x y z : ℕ + + example (h₀ : x ∣ y) (h₁ : y ∣ z) : x ∣ z := + dvd_trans h₀ h₁ + + example : x ∣ y * x * z := + begin + apply dvd_mul_of_dvd_left, + apply dvd_mul_left + end + + example : x ∣ x^2 := + by apply dvd_mul_right + + example (h : x ∣ w) : x ∣ y * (x * z) + x^2 + w^2 := + sorry + + end + + section + variables m n : ℕ + open nat + + #check (gcd_zero_right n : gcd n 0 = n) + #check (gcd_zero_left n : gcd 0 n = n) + #check (lcm_zero_right n : lcm n 0 = 0) + #check (lcm_zero_left n : lcm 0 n = 0) + + example : gcd m n = gcd n m := + sorry + + end
src/02_Basics/05_Proving_Facts_about_Algebraic_Structures.lean (new)
- @@ -0,0 +1,99 @@
 + import topology.metric_space.basic + + section + variables {α : Type*} [partial_order α] + variables x y z : α + + #check x ≤ y + #check (le_refl x : x ≤ x) + #check (le_trans : x ≤ y → y ≤ z → x ≤ z) + + #check x < y + #check (lt_irrefl x : ¬ x < x) + #check (lt_trans : x < y → y < z → x < z) + #check (lt_of_le_of_lt : x ≤ y → y < z → x < z) + #check (lt_of_lt_of_le : x < y → y ≤ z → x < z) + + example : x < y ↔ x ≤ y ∧ x ≠ y := + lt_iff_le_and_ne + + end + + section + variables {α : Type*} [lattice α] + variables x y z : α + + #check x ⊓ y + #check (inf_le_left : x ⊓ y ≤ x) + #check (inf_le_right : x ⊓ y ≤ y) + #check (le_inf : z ≤ x → z ≤ y → z ≤ x ⊓ y) + + #check x ⊔ y + #check (le_sup_left : x ≤ x ⊔ y) + #check (le_sup_right: y ≤ x ⊔ y) + #check (sup_le : x ≤ z → y ≤ z → x ⊔ y ≤ z) + + example : x ⊓ y = y ⊓ x := sorry + example : x ⊓ y ⊓ z = x ⊓ (y ⊓ z) := sorry + example : x ⊔ y = y ⊔ x := sorry + example : x ⊔ y ⊔ z = x ⊔ (y ⊔ z) := sorry + + theorem absorb1 : x ⊓ (x ⊔ y) = x := sorry + theorem absorb2 : x ⊔ (x ⊓ y) = x := sorry + + end + + section + variables {α : Type*} [distrib_lattice α] + variables x y z : α + + #check (inf_sup_left : x ⊓ (y ⊔ z) = (x ⊓ y) ⊔ (x ⊓ z)) + #check (inf_sup_right : (x ⊔ y) ⊓ z = (x ⊓ z) ⊔ (y ⊓ z)) + #check (sup_inf_left : x ⊔ (y ⊓ z) = (x ⊔ y) ⊓ (x ⊔ z)) + #check (sup_inf_right : (x ⊓ y) ⊔ z = (x ⊔ z) ⊓ (y ⊔ z)) + + end + + section + variables {α : Type*} [lattice α] + variables a b c : α + + example (h : ∀ x y z : α, x ⊓ (y ⊔ z) = (x ⊓ y) ⊔ (x ⊓ z)) : + a ⊔ (b ⊓ c) = (a ⊔ b) ⊓ (a ⊔ c) := + sorry + + example (h : ∀ x y z : α, x ⊔ (y ⊓ z) = (x ⊔ y) ⊓ (x ⊔ z)) : + a ⊓ (b ⊔ c) = (a ⊓ b) ⊔ (a ⊓ c) := + sorry + + end + + section + variables {R : Type*} [ordered_ring R] + variables a b c : R + + #check (add_le_add_left : a ≤ b → ∀ c, c + a ≤ c + b) + #check (mul_pos : 0 < a → 0 < b → 0 < a * b) + + #check (mul_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a * b) + + example : a ≤ b → 0 ≤ b - a := sorry + + example : 0 ≤ b - a → a ≤ b := sorry + + example (h : a ≤ b) (h' : 0 ≤ c) : a * c ≤ b * c := sorry + + end + + section + variables {X : Type*} [metric_space X] + variables x y z : X + + #check (dist_self x : dist x x = 0) + #check (dist_comm x y : dist x y = dist y x) + #check (dist_triangle x y z : dist x z ≤ dist x y + dist y z) + + example (x y : X) : 0 ≤ dist x y := sorry + + end +
src/02_Basics/solutions/solutions_01_Calculating.lean (new)
- @@ -0,0 +1,45 @@
  + import data.real.basic + + example (a b c : ℝ) : (c * b) * a = b * (a * c) := + begin + rw mul_comm c b, + rw mul_assoc b c a, + rw mul_comm c a + end + + example (a b c : ℝ) : a * (b * c) = b * (a * c) := + begin + rw ←mul_assoc a b c, + rw mul_comm a b, + rw mul_assoc b a c + end + + example (a b c : ℝ) : a * (b * c) = b * (c * a) := + begin + rw mul_comm, + rw mul_assoc + end + + example (a b c : ℝ) : a * (b * c) = b * (a * c) := + begin + rw ←mul_assoc, + rw mul_comm a, + rw mul_assoc + end + + example (a b c d e f : ℝ) (h : b * c = e * f) : + a * b * c * d = a * e * f * d := + begin + rw mul_assoc a, + rw h, + rw ←mul_assoc + end + + example (a b c d : ℝ) (hyp : c = b * a - d) (hyp' : d = a * b) : c = 0 := + begin + rw hyp, + rw hyp', + rw mul_comm, + rw sub_self + end +
src/02_Basics/solutions/solutions_02_Proving_Identities_in_Algebraic_Structures.lean (new)
- @@ -0,0 +1,84 @@
  + import algebra.ring + import data.real.basic + import tactic + + namespace my_ring + variables {R : Type*} [ring R] + + theorem add_neg_cancel_right (a b : R) : (a + b) + -b = a := + by rw [add_assoc, add_right_neg, add_zero] + + theorem add_left_cancel {a b c : R} (h : a + b = a + c) : b = c := + by rw [←neg_add_cancel_left a b, h, neg_add_cancel_left] + + theorem add_right_cancel {a b c : R} (h : a + b = c + b) : a = c := + by rw [←add_neg_cancel_right a b, h, add_neg_cancel_right] + + theorem zero_mul (a : R) : 0 * a = 0 := + begin + have h : 0 * a + 0 * a = 0 * a + 0, + { rw [←add_mul, add_zero, add_zero] }, + rw add_left_cancel h + end + + theorem neg_eq_of_add_eq_zero {a b : R} (h : a + b = 0) : -a = b := + by rw [←neg_add_cancel_left a b, h, add_zero] + + theorem eq_neg_of_add_eq_zero {a b : R} (h : a + b = 0) : a = -b := + begin + symmetry, + apply neg_eq_of_add_eq_zero, + rw [add_comm, h] + end + + theorem neg_zero : (-0 : R) = 0 := + begin + apply neg_eq_of_add_eq_zero, + rw add_zero + end + + theorem neg_neg (a : R) : -(-a) = a := + begin + apply neg_eq_of_add_eq_zero, + rw add_left_neg + end + + end my_ring + + namespace my_ring + + variables {R : Type*} [ring R] + + theorem self_sub (a : R) : a - a = 0 := + by rw [sub_eq_add_neg, add_right_neg] + + lemma one_add_one_eq_two : 1 + 1 = (2 : R) := + by refl + + theorem two_mul (a : R) : 2 * a = a + a := + by rw [←one_add_one_eq_two, add_mul, one_mul] + + end my_ring + + section + variables {G : Type*} [group G] + + namespace my_group + + theorem mul_right_inv (a : G) : a * a⁻¹ = 1 := + begin + have h : (a * a⁻¹)⁻¹ * ((a * a⁻¹) * (a * a⁻¹)) = 1, + { rw [mul_assoc, ←mul_assoc a⁻¹ a, mul_left_inv, one_mul, mul_left_inv] }, + rw [←h, ←mul_assoc, mul_left_inv, one_mul] + end + + theorem mul_one (a : G) : a * 1 = a := + by rw [←mul_left_inv a, ←mul_assoc, mul_right_inv, one_mul] + + theorem mul_inv_rev (a b : G) : (a * b)⁻¹ = b⁻¹ * a ⁻¹ := + by rw [←one_mul (b⁻¹ * a⁻¹), ←mul_left_inv (a * b), mul_assoc, mul_assoc, + ←mul_assoc b b⁻¹, mul_right_inv, one_mul, mul_right_inv, mul_one] + + end my_group + end +
src/02_Basics/solutions/solutions_03_Using_Theorems_and_Lemmas.lean (new)
- @@ -0,0 +1,79 @@
 + import analysis.special_functions.log.basic + + variables a b c d e : ℝ + open real + + example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) + (h₃ : d < e) : + a < e := + begin + apply lt_of_le_of_lt h₀, + apply lt_trans h₁, + exact lt_of_le_of_lt h₂ h₃ + end + + example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := + begin + apply add_le_add_left, + rw exp_le_exp, + apply add_le_add_left h₀ + end + + -- an alterantive using `linarith`. + example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := + begin + have : exp (a + d) ≤ exp (a + e), + { rw exp_le_exp, linarith }, + linarith [this] + end + + example (h : a ≤ b) : log (1 + exp a) ≤ log (1 + exp b) := + begin + have h₀ : 0 < 1 + exp a, + { linarith [exp_pos a]}, + have h₁ : 0 < 1 + exp b, + { linarith [exp_pos b] }, + apply (log_le_log h₀ h₁).mpr, + apply add_le_add_left (exp_le_exp.mpr h), + end + + -- SOLUTION. + example (h : a ≤ b) : c - exp b ≤ c - exp a := + begin + apply sub_le_sub_left, + exact exp_le_exp.mpr h + end + + -- alternatively: + example (h : a ≤ b) : c - exp b ≤ c - exp a := + by linarith [exp_le_exp.mpr h] + + theorem fact1 : a*b*2 ≤ a^2 + b^2 := + begin + 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 + end + + theorem fact2 : -(a*b)*2 ≤ a^2 + b^2 := + begin + 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 + end + + example : abs (a*b) ≤ (a^2 + b^2) / 2 := + begin + have h : (0 : ℝ) < 2, + { norm_num }, + apply abs_le'.mpr, + split, + { rw le_div_iff h, + apply fact1 }, + rw le_div_iff h, + apply fact2, + end
src/02_Basics/solutions/solutions_04_More_on_Order_and_Divisibility.lean (new)
- @@ -0,0 +1,134 @@
  + import data.real.basic + + section + variables a b c d : ℝ + + #check (min_le_left a b : min a b ≤ a) + #check (min_le_right a b : min a b ≤ b) + #check (le_min : c ≤ a → c ≤ b → c ≤ min a b) + + example : max a b = max b a := + begin + apply le_antisymm, + repeat { + apply max_le, + apply le_max_right, + apply le_max_left } + end + + example : min (min a b) c = min a (min b c) := + begin + apply le_antisymm, + { apply le_min, + { apply le_trans, + apply min_le_left, + apply min_le_left }, + apply le_min, + { apply le_trans, + apply min_le_left, + apply min_le_right }, + apply min_le_right }, + apply le_min, + { apply le_min, + { apply min_le_left }, + apply le_trans, + apply min_le_right, + apply min_le_left }, + apply le_trans, + apply min_le_right, + apply min_le_right + end + + lemma aux : min a b + c ≤ min (a + c) (b + c) := + begin + apply le_min, + { apply add_le_add_right, + apply min_le_left }, + apply add_le_add_right, + apply min_le_right + end + + example : min a b + c = min (a + c) (b + c) := + begin + apply le_antisymm, + { apply aux }, + have h : min (a + c) (b + c) = min (a + c) (b + c) - c + c, + { rw sub_add_cancel }, + rw h, + apply add_le_add_right, + rw sub_eq_add_neg, + apply le_trans, + apply aux, + rw [add_neg_cancel_right, add_neg_cancel_right] + end + + example : abs a - abs b ≤ abs (a - b) := + calc + abs a - abs b = abs (a - b + b) - abs b : + by rw sub_add_cancel + ... ≤ abs (a - b) + abs b - abs b : + begin + apply sub_le_sub_right, + apply abs_add + end + ... ≤ abs (a - b) : + by rw add_sub_cancel + + -- alternatively + example : abs a - abs b ≤ abs (a - b) := + begin + have h := abs_add (a - b) b, + rw sub_add_cancel at h, + linarith + end + + end + + section + variables w x y z : ℕ + + example (h₀ : x ∣ y) (h₁ : y ∣ z) : x ∣ z := + dvd_trans h₀ h₁ + + example : x ∣ y * x * z := + begin + apply dvd_mul_of_dvd_left, + apply dvd_mul_left + end + + example : x ∣ x^2 := + by apply dvd_mul_right + + example (h : x ∣ w) : x ∣ y * (x * z) + x^2 + w^2 := + begin + apply dvd_add, + { apply dvd_add, + { apply dvd_mul_of_dvd_right, + apply dvd_mul_right }, + apply dvd_mul_right }, + rw pow_two, + apply dvd_mul_of_dvd_right, + exact h + end + + end + + section + variables m n : ℕ + open nat + + #check (gcd_zero_right n : gcd n 0 = n) + #check (gcd_zero_left n : gcd 0 n = n) + #check (lcm_zero_right n : lcm n 0 = 0) + #check (lcm_zero_left n : lcm 0 n = 0) + + example : gcd m n = gcd n m := + begin + apply dvd_antisymm, + repeat { + apply dvd_gcd, + apply gcd_dvd_right, + apply gcd_dvd_left } + end + + end
src/02_Basics/solutions/solutions_05_Proving_Facts_about_Algebraic_Structures.lean (new)
- @@ -0,0 +1,159 @@
  + import topology.metric_space.basic + + section + variables {α : Type*} [lattice α] + variables x y z : α + + example : x ⊓ y = y ⊓ x := + begin + apply le_antisymm, + repeat { + apply le_inf, + { apply inf_le_right }, + apply inf_le_left } + end + + example : x ⊓ y ⊓ z = x ⊓ (y ⊓ z) := + begin + apply le_antisymm, + { apply le_inf, + { apply le_trans, + apply inf_le_left, + apply inf_le_left }, + apply le_inf, + { apply le_trans, + apply inf_le_left, + apply inf_le_right }, + apply inf_le_right }, + apply le_inf, + { apply le_inf, + { apply inf_le_left }, + apply le_trans, + apply inf_le_right, + apply inf_le_left }, + apply le_trans, + apply inf_le_right, + apply inf_le_right + end + + example : x ⊔ y = y ⊔ x := + begin + apply le_antisymm, + repeat { + apply sup_le, + { apply le_sup_right }, + apply le_sup_left } + end + + example : x ⊔ y ⊔ z = x ⊔ (y ⊔ z) := + begin + apply le_antisymm, + { apply sup_le, + { apply sup_le, + apply le_sup_left, + { apply le_trans, + apply @le_sup_left _ _ y z, + apply le_sup_right } }, + apply le_trans, + apply @le_sup_right _ _ y z, + apply le_sup_right }, + apply sup_le, + { apply le_trans, + apply @le_sup_left _ _ x y, + apply le_sup_left }, + apply sup_le, + { apply le_trans, + apply @le_sup_right _ _ x y, + apply le_sup_left }, + apply le_sup_right + end + + theorem absorb1 : x ⊓ (x ⊔ y) = x := + begin + apply le_antisymm, + { apply inf_le_left }, + apply le_inf, + { apply le_refl }, + apply le_sup_left + end + + theorem absorb2 : x ⊔ (x ⊓ y) = x := + begin + apply le_antisymm, + { apply sup_le, + { apply le_refl }, + apply inf_le_left }, + apply le_sup_left + end + + end + + section + variables {α : Type*} [distrib_lattice α] + variables x y z : α + + #check (inf_sup_left : x ⊓ (y ⊔ z) = (x ⊓ y) ⊔ (x ⊓ z)) + #check (inf_sup_right : (x ⊔ y) ⊓ z = (x ⊓ z) ⊔ (y ⊓ z)) + #check (sup_inf_left : x ⊔ (y ⊓ z) = (x ⊔ y) ⊓ (x ⊔ z)) + #check (sup_inf_right : (x ⊓ y) ⊔ z = (x ⊔ z) ⊓ (y ⊔ z)) + + end + + section + variables {α : Type*} [lattice α] + variables a b c : α + + example (h : ∀ x y z : α, x ⊓ (y ⊔ z) = (x ⊓ y) ⊔ (x ⊓ z)) : + a ⊔ (b ⊓ c) = (a ⊔ b) ⊓ (a ⊔ c) := + by rw [h, @inf_comm _ _ (a ⊔ b), absorb1, @inf_comm _ _ (a ⊔ b), h, + ←sup_assoc, @inf_comm _ _ c a, absorb2, inf_comm] + + example (h : ∀ x y z : α, x ⊔ (y ⊓ z) = (x ⊔ y) ⊓ (x ⊔ z)) : + a ⊓ (b ⊔ c) = (a ⊓ b) ⊔ (a ⊓ c) := + by rw [h, @sup_comm _ _ (a ⊓ b), absorb2, @sup_comm _ _ (a ⊓ b), h, + ←inf_assoc, @sup_comm _ _ c a, absorb1, sup_comm] + + end + + section + variables {R : Type*} [ordered_ring R] + variables a b c : R + + theorem aux1 : a ≤ b → 0 ≤ b - a := + begin + intro h, + rw [←sub_self a, sub_eq_add_neg, sub_eq_add_neg, add_comm, add_comm b], + apply add_le_add_left h + end + + theorem aux2 : 0 ≤ b - a → a ≤ b := + begin + intro h, + rw [←add_zero a, ←sub_add_cancel b a, add_comm (b - a)], + apply add_le_add_left h + end + + example (h : a ≤ b) (h' : 0 ≤ c) : a * c ≤ b * c := + begin + have h1 : 0 ≤ (b - a) * c, + { exact mul_nonneg (aux1 _ _ h) h' }, + rw sub_mul at h1, + exact aux2 _ _ h1 + end + + end + + section + variables {X : Type*} [metric_space X] + variables x y z : X + + example (x y : X) : 0 ≤ dist x y := + begin + have : 0 ≤ dist x y + dist y x, + { rw [←dist_self x], + apply dist_triangle }, + linarith [dist_comm x y] + end + + end +
src/03_Logic/01_Implication_and_the_Universal_Quantifier.lean (new)
- @@ -0,0 +1,188 @@
 + import data.real.basic + + #check ∀ x : ℝ, 0 ≤ x → abs x = x + + #check ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε + + lemma my_lemma : ∀ x y ε : ℝ, + 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := + sorry + + section + variables a b δ : ℝ + variables (h₀ : 0 < δ) (h₁ : δ ≤ 1) + variables (ha : abs a < δ) (hb : abs b < δ) + + #check my_lemma a b δ + #check my_lemma a b δ h₀ h₁ + #check my_lemma a b δ h₀ h₁ ha hb + end + + lemma my_lemma2 : ∀ {x y ε : ℝ}, + 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := + sorry + + section + variables a b δ : ℝ + variables (h₀ : 0 < δ) (h₁ : δ ≤ 1) + variables (ha : abs a < δ) (hb : abs b < δ) + + #check my_lemma2 h₀ h₁ ha hb + end + + lemma my_lemma3 : ∀ {x y ε : ℝ}, + 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := + begin + intros x y ε epos ele1 xlt ylt, + sorry + end + + lemma my_lemma4 : ∀ {x y ε : ℝ}, + 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := + begin + intros x y ε epos ele1 xlt ylt, + calc + abs (x * y) = abs x * abs y : sorry + ... ≤ abs x * ε : sorry + ... < 1 * ε : sorry + ... = ε : sorry + end + + def fn_ub (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a + def fn_lb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x + + section + variables (f g : ℝ → ℝ) (a b : ℝ) + + example (hfa : fn_ub f a) (hgb : fn_ub g b) : + fn_ub (λ x, f x + g x) (a + b) := + begin + intro x, + dsimp, + apply add_le_add, + apply hfa, + apply hgb + end + + example (hfa : fn_lb f a) (hgb : fn_lb g b) : + fn_lb (λ x, f x + g x) (a + b) := + sorry + + example (nnf : fn_lb f 0) (nng : fn_lb g 0) : + fn_lb (λ x, f x * g x) 0 := + sorry + + example (hfa : fn_ub f a) (hfb : fn_ub g b) + (nng : fn_lb g 0) (nna : 0 ≤ a) : + fn_ub (λ x, f x * g x) (a * b) := + sorry + + end + + section + variables {α : Type*} {R : Type*} [ordered_cancel_add_comm_monoid R] + + #check @add_le_add + + def fn_ub' (f : α → R) (a : R) : Prop := ∀ x, f x ≤ a + + theorem fn_ub_add {f g : α → R} {a b : R} + (hfa : fn_ub' f a) (hgb : fn_ub' g b) : + fn_ub' (λ x, f x + g x) (a + b) := + λ x, add_le_add (hfa x) (hgb x) + end + + example (f : ℝ → ℝ) (h : monotone f) : + ∀ {a b}, a ≤ b → f a ≤ f b := h + + section + variables (f g : ℝ → ℝ) + + example (mf : monotone f) (mg : monotone g) : + monotone (λ x, f x + g x) := + begin + intros a b aleb, + apply add_le_add, + apply mf aleb, + apply mg aleb + end + + example (mf : monotone f) (mg : monotone g) : + monotone (λ x, f x + g x) := + λ a b aleb, add_le_add (mf aleb) (mg aleb) + + example {c : ℝ} (mf : monotone f) (nnc : 0 ≤ c) : + monotone (λ x, c * f x) := + sorry + + example (mf : monotone f) (mg : monotone g) : + monotone (λ x, f (g x)) := + sorry + + def fn_even (f : ℝ → ℝ) : Prop := ∀ x, f x = f (-x) + def fn_odd (f : ℝ → ℝ) : Prop := ∀ x, f x = - f (-x) + + example (ef : fn_even f) (eg : fn_even g) : fn_even (λ x, f x + g x) := + begin + intro x, + calc + (λ x, f x + g x) x = f x + g x : rfl + ... = f (-x) + g (-x) : by rw [ef, eg] + end + + example (of : fn_odd f) (og : fn_odd g) : fn_even (λ x, f x * g x) := + sorry + + example (ef : fn_even f) (og : fn_odd g) : fn_odd (λ x, f x * g x) := + sorry + + example (ef : fn_even f) (og : fn_odd g) : fn_even (λ x, f (g x)) := + sorry + + end + + section + variables {α : Type*} (r s t : set α) + + example : s ⊆ s := + by { intros x xs, exact xs } + + theorem subset.refl : s ⊆ s := λ x xs, xs + + theorem subset.trans : r ⊆ s → s ⊆ t → r ⊆ t := + sorry + + end + + section + + variables {α : Type*} [partial_order α] + variables (s : set α) (a b : α) + + def set_ub (s : set α) (a : α) := ∀ x, x ∈ s → x ≤ a + + example (h : set_ub s a) (h' : a ≤ b) : set_ub s b := + sorry + + end + + section + open function + + example (c : ℝ) : injective (λ x, x + c) := + begin + intros x₁ x₂ h', + exact (add_left_inj c).mp h', + end + + example {c : ℝ} (h : c ≠ 0) : injective (λ x, c * x) := + sorry + + variables {α : Type*} {β : Type*} {γ : Type*} + variables {g : β → γ} {f : α → β} + + example (injg : injective g) (injf : injective f) : + injective (λ x, g (f x)) := + sorry + + end
src/03_Logic/02_The_Existential_Quantifier.lean (new)
- @@ -0,0 +1,149 @@
 + import data.real.basic + + example : ∃ x : ℝ, 2 < x ∧ x < 3 := + begin + use 5 / 2, + norm_num + end + + example : ∃ x : ℝ, 2 < x ∧ x < 3 := + begin + have h : 2 < (5 : ℝ) / 2 ∧ (5 : ℝ) / 2 < 3, + by norm_num, + exact ⟨5 / 2, h⟩ + end + + example : ∃ x : ℝ, 2 < x ∧ x < 3 := + ⟨5 / 2, by norm_num⟩ + + def fn_ub (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a + def fn_lb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x + + def fn_has_ub (f : ℝ → ℝ) := ∃ a, fn_ub f a + def fn_has_lb (f : ℝ → ℝ) := ∃ a, fn_lb f a + + + theorem fn_ub_add {f g : ℝ → ℝ} {a b : ℝ} + (hfa : fn_ub f a) (hgb : fn_ub g b) : + fn_ub (λ x, f x + g x) (a + b) := + λ x, add_le_add (hfa x) (hgb x) + + section + variables {f g : ℝ → ℝ} + + example (ubf : fn_has_ub f) (ubg : fn_has_ub g) : + fn_has_ub (λ x, f x + g x) := + begin + cases ubf with a ubfa, + cases ubg with b ubfb, + use a + b, + apply fn_ub_add ubfa ubfb + end + + example (lbf : fn_has_lb f) (lbg : fn_has_lb g) : + fn_has_lb (λ x, f x + g x) := + sorry + + example {c : ℝ} (ubf : fn_has_ub f) (h : c ≥ 0): + fn_has_ub (λ x, c * f x) := + sorry + + example (ubf : fn_has_ub f) (ubg : fn_has_ub g) : + fn_has_ub (λ x, f x + g x) := + begin + rcases ubf with ⟨a, ubfa⟩, + rcases ubg with ⟨b, ubfb⟩, + exact ⟨a + b, fn_ub_add ubfa ubfb⟩ + end + + example : fn_has_ub f → fn_has_ub g → + fn_has_ub (λ x, f x + g x) := + begin + rintros ⟨a, ubfa⟩ ⟨b, ubfb⟩, + exact ⟨a + b, fn_ub_add ubfa ubfb⟩ + end + + example : fn_has_ub f → fn_has_ub g → + fn_has_ub (λ x, f x + g x) := + λ ⟨a, ubfa⟩ ⟨b, ubfb⟩, ⟨a + b, fn_ub_add ubfa ubfb⟩ + + end + + section + variables {α : Type*} [comm_ring α] + + def sum_of_squares (x : α) := ∃ a b, x = a^2 + b^2 + + theorem sum_of_squares_mul {x y : α} + (sosx : sum_of_squares x) (sosy : sum_of_squares y) : + sum_of_squares (x * y) := + begin + 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 + end + theorem sum_of_squares_mul' {x y : α} + (sosx : sum_of_squares x) (sosy : sum_of_squares y) : + sum_of_squares (x * y) := + begin + rcases sosx with ⟨a, b, rfl⟩, + rcases sosy with ⟨c, d, rfl⟩, + use [a*c - b*d, a*d + b*c], + ring + end + + end + section + variables {a b c : ℕ} + + example (divab : a ∣ b) (divbc : b ∣ c) : a ∣ c := + begin + cases divab with d beq, + cases divbc with e ceq, + rw [ceq, beq], + use (d * e), ring + end + + example (divab : a ∣ b) (divac : a ∣ c) : a ∣ (b + c) := + sorry + + end + + section + open function + + example {c : ℝ} : surjective (λ x, x + c) := + begin + intro x, + use x - c, + dsimp, ring + end + + example {c : ℝ} (h : c ≠ 0) : surjective (λ x, c * x) := + sorry + + example (x y : ℝ) (h : x - y ≠ 0) : (x^2 - y^2) / (x - y) = x + y := + by { field_simp [h], ring } + + example {f : ℝ → ℝ} (h : surjective f) : ∃ x, (f x)^2 = 4 := + begin + cases h 2 with x hx, + use x, + rw hx, + norm_num + end + + end + + section + open function + variables {α : Type*} {β : Type*} {γ : Type*} + variables {g : β → γ} {f : α → β} + + example (surjg : surjective g) (surjf : surjective f) : + surjective (λ x, g (f x)) := + sorry + + end
src/03_Logic/03_Negation.lean (new)
- @@ -0,0 +1,159 @@
 + import data.real.basic + + section + variables a b : ℝ + + example (h : a < b) : ¬ b < a := + begin + intro h', + have : a < a, + from lt_trans h h', + apply lt_irrefl a this + end + + def fn_ub (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a + def fn_lb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x + + def fn_has_ub (f : ℝ → ℝ) := ∃ a, fn_ub f a + def fn_has_lb (f : ℝ → ℝ) := ∃ a, fn_lb f a + + variable f : ℝ → ℝ + + example (h : ∀ a, ∃ x, f x > a) : ¬ fn_has_ub f := + begin + intros fnub, + cases fnub with a fnuba, + cases h a with x hx, + have : f x ≤ a, + from fnuba x, + linarith + end + + example (h : ∀ a, ∃ x, f x < a) : ¬ fn_has_lb f := + sorry + + example : ¬ fn_has_ub (λ x, x) := + sorry + + #check (not_le_of_gt : a > b → ¬ a ≤ b) + #check (not_lt_of_ge : a ≥ b → ¬ a < b) + #check (lt_of_not_ge : ¬ a ≥ b → a < b) + #check (le_of_not_gt : ¬ a > b → a ≤ b) + + example (h : monotone f) (h' : f a < f b) : a < b := + sorry + + example (h : a ≤ b) (h' : f b < f a) : ¬ monotone f := + sorry + + example : + ¬ ∀ {f : ℝ → ℝ}, monotone f → ∀ {a b}, f a ≤ f b → a ≤ b := + begin + intro h, + let f := λ x : ℝ, (0 : ℝ), + have monof : monotone f, + { sorry }, + have h' : f 1 ≤ f 0, + from le_refl _, + sorry + end + + example (x : ℝ) (h : ∀ ε > 0, x < ε) : x ≤ 0 := + sorry + + end + + section + variables {α : Type*} (P : α → Prop) (Q : Prop) + + example (h : ¬ ∃ x, P x) : ∀ x, ¬ P x := + sorry + + example (h : ∀ x, ¬ P x) : ¬ ∃ x, P x := + sorry + + example (h : ¬ ∀ x, P x) : ∃ x, ¬ P x := + sorry + + example (h : ∃ x, ¬ P x) : ¬ ∀ x, P x := + sorry + + open_locale classical + + example (h : ¬ ∀ x, P x) : ∃ x, ¬ P x := + begin + by_contradiction h', + apply h, + intro x, + show P x, + by_contradiction h'', + exact h' ⟨x, h''⟩ + end + + example (h : ¬ ¬ Q) : Q := + sorry + + example (h : Q) : ¬ ¬ Q := + sorry + + end + + open_locale classical + section + variable (f : ℝ → ℝ) + + example (h : ¬ fn_has_ub f) : ∀ a, ∃ x, f x > a := + sorry + + example (h : ¬ ∀ a, ∃ x, f x > a) : fn_has_ub f := + begin + push_neg at h, + exact h + end + + example (h : ¬ fn_has_ub f) : ∀ a, ∃ x, f x > a := + begin + simp only [fn_has_ub, fn_ub] at h, + push_neg at h, + exact h + end + + example (h : ¬ monotone f) : ∃ x y, x ≤ y ∧ f y < f x := + sorry + + example (h : ¬ fn_has_ub f) : ∀ a, ∃ x, f x > a := + begin + contrapose! h, + exact h + end + + example (x : ℝ) (h : ∀ ε > 0, x ≤ ε) : x ≤ 0 := + begin + contrapose! h, + use x / 2, + split; linarith + end + + end + + section + variable a : ℕ + + example (h : 0 < 0) : a > 37 := + begin + exfalso, + apply lt_irrefl 0 h + end + + example (h : 0 < 0) : a > 37 := + absurd h (lt_irrefl 0) + + example (h : 0 < 0) : a > 37 := + begin + have h' : ¬ 0 < 0, + from lt_irrefl 0, + contradiction + end + + end +
src/03_Logic/04_Conjunction_and_Bi-implication.lean (new)
- @@ -0,0 +1,163 @@
 + import data.real.basic + import data.nat.prime + + example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬ y ≤ x) : x ≤ y ∧ x ≠ y := + begin + split, + { assumption }, + intro h, + apply h₁, + rw h + end + + example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬ y ≤ x) : x ≤ y ∧ x ≠ y := + ⟨h₀, λ h, h₁ (by rw h)⟩ + + example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬ y ≤ x) : x ≤ y ∧ x ≠ y := + begin + have h : x ≠ y, + { contrapose! h₁, + rw h₁ }, + exact ⟨h₀, h⟩ + end + + example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬ y ≤ x := + begin + cases h with h₀ h₁, + contrapose! h₁, + exact le_antisymm h₀ h₁ + end + + example {x y : ℝ} : x ≤ y ∧ x ≠ y → ¬ y ≤ x := + begin + rintros ⟨h₀, h₁⟩ h', + exact h₁ (le_antisymm h₀ h') + end + + example {x y : ℝ} : x ≤ y ∧ x ≠ y → ¬ y ≤ x := + λ ⟨h₀, h₁⟩ h', h₁ (le_antisymm h₀ h') + + example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬ y ≤ x := + begin + intro h', + apply h.right, + exact le_antisymm h.left h' + end + + example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬ y ≤ x := + λ h', h.right (le_antisymm h.left h') + + example {m n : ℕ} (h : m ∣ n ∧ m ≠ n) : + m ∣ n ∧ ¬ n ∣ m := + sorry + + example : ∃ x : ℝ, 2 < x ∧ x < 4 := + ⟨5/2, by norm_num, by norm_num⟩ + + example (x y : ℝ) : (∃ z : ℝ, x < z ∧ z < y) → x < y := + begin + rintros ⟨z, xltz, zlty⟩, + exact lt_trans xltz zlty + end + + example (x y : ℝ) : (∃ z : ℝ, x < z ∧ z < y) → x < y := + λ ⟨z, xltz, zlty⟩, lt_trans xltz zlty + + example : ∃ x : ℝ, 2 < x ∧ x < 4 := + begin + use 5 / 2, + split; norm_num + end + + example : ∃ m n : ℕ, + 4 < m ∧ m < n ∧ n < 10 ∧ nat.prime m ∧ nat.prime n := + begin + use [5, 7], + norm_num + end + + example {x y : ℝ} : x ≤ y ∧ x ≠ y → x ≤ y ∧ ¬ y ≤ x := + begin + rintros ⟨h₀, h₁⟩, + use [h₀, λ h', h₁ (le_antisymm h₀ h')] + end + + example {x y : ℝ} (h : x ≤ y) : ¬ y ≤ x ↔ x ≠ y := + begin + split, + { contrapose!, + rintro rfl, + reflexivity }, + contrapose!, + exact le_antisymm h + end + + example {x y : ℝ} (h : x ≤ y) : ¬ y ≤ x ↔ x ≠ y := + ⟨λ h₀ h₁, h₀ (by rw h₁), λ h₀ h₁, h₀ (le_antisymm h h₁)⟩ + + example {x y : ℝ} : x ≤ y ∧ ¬ y ≤ x ↔ x ≤ y ∧ x ≠ y := + sorry + + theorem aux {x y : ℝ} (h : x^2 + y^2 = 0) : x = 0 := + begin + have h' : x^2 = 0, + { sorry }, + exact pow_eq_zero h' + end + + example (x y : ℝ) : x^2 + y^2 = 0 ↔ x = 0 ∧ y = 0 := + sorry + + section + + example (x y : ℝ) : abs (x + 3) < 5 → -8 < x ∧ x < 2 := + begin + rw abs_lt, + intro h, + split; linarith + end + + example : 3 ∣ nat.gcd 6 15 := + begin + rw nat.dvd_gcd_iff, + split; norm_num + end + + end + + theorem not_monotone_iff {f : ℝ → ℝ}: + ¬ monotone f ↔ ∃ x y, x ≤ y ∧ f x > f y := + by { rw monotone, push_neg } + + example : ¬ monotone (λ x : ℝ, -x) := + sorry + + section + variables {α : Type*} [partial_order α] + variables a b : α + + example : a < b ↔ a ≤ b ∧ a ≠ b := + begin + rw lt_iff_le_not_le, + sorry + end + + end + + section + variables {α : Type*} [preorder α] + variables a b c : α + + example : ¬ a < a := + begin + rw lt_iff_le_not_le, + sorry + end + + example : a < b → b < c → a < c := + begin + simp only [lt_iff_le_not_le], + sorry + end + + end
src/03_Logic/05_Disjunction.lean (new)
- @@ -0,0 +1,108 @@
 + import data.real.basic + + section + variables {x y : ℝ} + + example (h : y > x^2) : y > 0 ∨ y < -1 := + by { left, linarith [pow_two_nonneg x] } + + example (h : -y > x^2 + 1) : y > 0 ∨ y < -1 := + by { right, linarith [pow_two_nonneg x] } + + example (h : y > 0) : y > 0 ∨ y < -1 := + or.inl h + + example (h : y < -1) : y > 0 ∨ y < -1 := + or.inr h + + example : x < abs y → x < y ∨ x < -y := + begin + cases le_or_gt 0 y with h h, + { rw abs_of_nonneg h, + intro h, left, exact h }, + rw abs_of_neg h, + intro h, right, exact h + end + + namespace my_abs + + theorem le_abs_self (x : ℝ) : x ≤ abs x := + sorry + + theorem neg_le_abs_self (x : ℝ) : -x ≤ abs x := + sorry + + theorem abs_add (x y : ℝ) : abs (x + y) ≤ abs x + abs y := + sorry + + theorem lt_abs : x < abs y ↔ x < y ∨ x < -y := + sorry + + theorem abs_lt : abs x < y ↔ - y < x ∧ x < y := + sorry + + end my_abs + end + + example {x : ℝ} (h : x ≠ 0) : x < 0 ∨ x > 0 := + begin + rcases lt_trichotomy x 0 with xlt | xeq | xgt, + { left, exact xlt }, + { contradiction }, + right, exact xgt + end + + example {m n k : ℕ} (h : m ∣ n ∨ m ∣ k) : m ∣ n * k := + begin + rcases h with ⟨a, rfl⟩ | ⟨b, rfl⟩, + { rw [mul_assoc], + apply dvd_mul_right }, + rw [mul_comm, mul_assoc], + apply dvd_mul_right + end + + example {z : ℝ} (h : ∃ x y, z = x^2 + y^2 ∨ z = x^2 + y^2 + 1) : + z ≥ 0 := + sorry + + example {x : ℝ} (h : x^2 = 1) : x = 1 ∨ x = -1 := + sorry + + example {x y : ℝ} (h : x^2 = y^2) : x = y ∨ x = -y := + sorry + + section + variables {R : Type*} [comm_ring R] [is_domain R] + variables (x y : R) + + example (h : x^2 = 1) : x = 1 ∨ x = -1 := + sorry + + example (h : x^2 = y^2) : x = y ∨ x = -y := + sorry + + end + + example (P : Prop) : ¬ ¬ P → P := + begin + intro h, + cases classical.em P, + { assumption }, + contradiction + end + + section + open_locale classical + + example (P : Prop) : ¬ ¬ P → P := + begin + intro h, + by_cases h' : P, + { assumption }, + contradiction + end + + example (P Q : Prop) : (P → Q) ↔ ¬ P ∨ Q := + sorry + + end
src/03_Logic/06_Sequences_and_Convergence.lean (new)
- @@ -0,0 +1,119 @@
 + import data.real.basic + + def converges_to (s : ℕ → ℝ) (a : ℝ) := + ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε + + example : (λ x y : ℝ, (x + y)^2) = (λ x y : ℝ, x^2 + 2*x*y + y^2) := + by { ext, ring } + + example (a b : ℝ) : abs a = abs (a - b + b) := + by { congr, ring } + + example {a : ℝ} (h : 1 < a) : a < a * a := + begin + convert (mul_lt_mul_right _).2 h, + { rw [one_mul] }, + exact lt_trans zero_lt_one h + end + + theorem converges_to_const (a : ℝ) : converges_to (λ x : ℕ, a) a := + begin + intros ε εpos, + use 0, + intros n nge, dsimp, + rw [sub_self, abs_zero], + apply εpos + end + + theorem converges_to_add {s t : ℕ → ℝ} {a b : ℝ} + (cs : converges_to s a) (ct : converges_to t b): + converges_to (λ n, s n + t n) (a + b) := + begin + intros ε εpos, dsimp, + have ε2pos : 0 < ε / 2, + { linarith }, + cases cs (ε / 2) ε2pos with Ns hs, + cases ct (ε / 2) ε2pos with Nt ht, + use max Ns Nt, + sorry + end + + theorem converges_to_mul_const {s : ℕ → ℝ} {a : ℝ} + (c : ℝ) (cs : converges_to s a) : + converges_to (λ n, c * s n) (c * a) := + begin + by_cases h : c = 0, + { convert converges_to_const 0, + { ext, rw [h, zero_mul] }, + rw [h, zero_mul] }, + have acpos : 0 < abs c, + from abs_pos.mpr h, + sorry + end + + theorem exists_abs_le_of_converges_to {s : ℕ → ℝ} {a : ℝ} + (cs : converges_to s a) : + ∃ N b, ∀ n, N ≤ n → abs (s n) < b := + begin + cases cs 1 zero_lt_one with N h, + use [N, abs a + 1], + sorry + end + + lemma aux {s t : ℕ → ℝ} {a : ℝ} + (cs : converges_to s a) (ct : converges_to t 0) : + converges_to (λ n, s n * t n) 0 := + begin + intros ε εpos, dsimp, + rcases exists_abs_le_of_converges_to cs with ⟨N₀, B, h₀⟩, + have Bpos : 0 < B, + from lt_of_le_of_lt (abs_nonneg _) (h₀ N₀ (le_refl _)), + have pos₀ : ε / B > 0, + from div_pos εpos Bpos, + cases ct _ pos₀ with N₁ h₁, + sorry + end + + theorem converges_to_mul {s t : ℕ → ℝ} {a b : ℝ} + (cs : converges_to s a) (ct : converges_to t b): + converges_to (λ n, s n * t n) (a * b) := + begin + have h₁ : converges_to (λ n, s n * (t n - b)) 0, + { apply aux cs, + convert converges_to_add ct (converges_to_const (-b)), + ring }, + convert (converges_to_add h₁ (converges_to_mul_const b cs)), + { ext, ring }, + ring + end + + theorem converges_to_unique {s : ℕ → ℝ} {a b : ℝ} + (sa : converges_to s a) (sb : converges_to s b) : + a = b := + begin + by_contradiction abne, + have : abs (a - b) > 0, + { sorry }, + let ε := abs (a - b) / 2, + have εpos : ε > 0, + { change abs (a - b) / 2 > 0, linarith }, + cases sa ε εpos with Na hNa, + cases sb ε εpos with Nb hNb, + let N := max Na Nb, + have absa : abs (s N - a) < ε, + { sorry }, + have absb : abs (s N - b) < ε, + { sorry }, + have : abs (a - b) < abs (a - b), + { sorry }, + exact lt_irrefl _ this + end + + section + variables {α : Type*} [linear_order α] + + def converges_to' (s : α → ℝ) (a : ℝ) := + ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε + + end +
src/03_Logic/solutions/solutions_01_Implication_and_the_Universal_Quantifier.lean (new)
- @@ -0,0 +1,151 @@
  + import data.real.basic + + def fn_ub (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a + def fn_lb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x + + section + variables (f g : ℝ → ℝ) (a b : ℝ) + + example (hfa : fn_lb f a) (hgb : fn_lb g b) : + fn_lb (λ x, f x + g x) (a + b) := + begin + intro x, + apply add_le_add, + apply hfa, + apply hgb + end + + example (nnf : fn_lb f 0) (nng : fn_lb g 0) : + fn_lb (λ x, f x * g x) 0 := + begin + intro x, + apply mul_nonneg, + apply nnf, + apply nng + end + + example (hfa : fn_ub f a) (hfb : fn_ub g b) + (nng : fn_lb g 0) (nna : 0 ≤ a) : + fn_ub (λ x, f x * g x) (a * b) := + begin + intro x, + apply mul_le_mul, + apply hfa, + apply hfb, + apply nng, + apply nna + end + + end + + section + variables (f g : ℝ → ℝ) + + example {c : ℝ} (mf : monotone f) (nnc : 0 ≤ c) : + monotone (λ x, c * f x) := + begin + intros a b aleb, + apply mul_le_mul_of_nonneg_left _ nnc, + apply mf aleb + end + + example {c : ℝ} (mf : monotone f) (nnc : 0 ≤ c) : + monotone (λ x, c * f x) := + λ a b aleb, mul_le_mul_of_nonneg_left (mf aleb) nnc + + example (mf : monotone f) (mg : monotone g) : + monotone (λ x, f (g x)) := + begin + intros a b aleb, + apply mf, + apply mg, + apply aleb + end + + example (mf : monotone f) (mg : monotone g) : + monotone (λ x, f (g x)) := + λ a b aleb, mf (mg aleb) + + def fn_even (f : ℝ → ℝ) : Prop := ∀ x, f x = f (-x) + def fn_odd (f : ℝ → ℝ) : Prop := ∀ x, f x = - f (-x) + + example (of : fn_odd f) (og : fn_odd g) : fn_even (λ x, f x * g x) := + begin + intro x, + calc + (λ x, f x * g x) x = f x * g x : rfl + ... = f (- x) * g (- x) : by rw [of, og, neg_mul_neg] + end + + example (ef : fn_even f) (og : fn_odd g) : fn_odd (λ x, f x * g x) := + begin + intro x, + dsimp, + rw [ef, og, neg_mul_eq_mul_neg] + end + + example (ef : fn_even f) (og : fn_odd g) : fn_even (λ x, f (g x)) := + begin + intro x, + dsimp, + rw [og, ←ef] + end + + end + + section + variables {α : Type*} (r s t : set α) + + example : r ⊆ s → s ⊆ t → r ⊆ t := + begin + intros rsubs ssubt x xr, + apply ssubt, + apply rsubs, + apply xr + end + + theorem subset.trans : r ⊆ s → s ⊆ t → r ⊆ t := + λ rsubs ssubt x xr, ssubt (rsubs xr) + + end + + section + + variables {α : Type*} [partial_order α] + variables (s : set α) (a b : α) + + def set_ub (s : set α) (a : α) := ∀ x, x ∈ s → x ≤ a + + example (h : set_ub s a) (h' : a ≤ b) : set_ub s b := + begin + intros x xs, + apply le_trans (h x xs) h' + end + + example (h : set_ub s a) (h' : a ≤ b) : set_ub s b := + λ x xs, le_trans (h x xs) h' + + end + + section + open function + + example {c : ℝ} (h : c ≠ 0) : injective (λ x, c * x) := + begin + intros x₁ x₂ h', + apply (mul_right_inj' h).mp h' + end + + variables {α : Type*} {β : Type*} {γ : Type*} + variables {g : β → γ} {f : α → β} + + example (injg : injective g) (injf : injective f) : + injective (λ x, g (f x)) := + begin + intros x₁ x₂ h, + apply injf, + apply injg, + apply h + end + + end
src/03_Logic/solutions/solutions_02_The_Existential_Quantifier.lean (new)
- @@ -0,0 +1,90 @@
  + import data.real.basic + + def fn_ub (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a + def fn_lb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x + + def fn_has_ub (f : ℝ → ℝ) := ∃ a, fn_ub f a + def fn_has_lb (f : ℝ → ℝ) := ∃ a, fn_lb f a + + theorem fn_ub_add {f g : ℝ → ℝ} {a b : ℝ} + (hfa : fn_ub f a) (hgb : fn_ub g b) : + fn_ub (λ x, f x + g x) (a + b) := + λ x, add_le_add (hfa x) (hgb x) + + section + variables {f g : ℝ → ℝ} + + example (lbf : fn_has_lb f) (lbg : fn_has_lb g) : + fn_has_lb (λ x, f x + g x) := + begin + cases lbf with a lbfa, + cases lbg with b lbgb, + use a + b, + intro x, + exact add_le_add (lbfa x) (lbgb x) + end + + example {c : ℝ} (ubf : fn_has_ub f) (h : c ≥ 0): + fn_has_ub (λ x, c * f x) := + begin + cases ubf with a lbfa, + use c * a, + intro x, + exact mul_le_mul_of_nonneg_left (lbfa x) h + end + + end + + section + variables {a b c : ℕ} + + example (divab : a ∣ b) (divbc : b ∣ c) : a ∣ c := + begin + rcases divab with ⟨d, rfl⟩, + rcases divbc with ⟨e, rfl⟩, + use (d * e), ring + end + + example (divab : a ∣ b) (divac : a ∣ c) : a ∣ (b + c) := + begin + rcases divab with ⟨d, rfl⟩, + rcases divac with ⟨e, rfl⟩, + use (d + e), ring + end + + end + + section + open function + + example {c : ℝ} (h : c ≠ 0) : surjective (λ x, c * x) := + begin + intro x, + use x / c, + dsimp, rw [mul_div_cancel' _ h] + end + + example {c : ℝ} (h : c ≠ 0) : surjective (λ x, c * x) := + begin + intro x, + use x / c, + field_simp [h], ring + end + + end + + section + open function + variables {α : Type*} {β : Type*} {γ : Type*} + variables {g : β → γ} {f : α → β} + + example (surjg : surjective g) (surjf : surjective f) : + surjective (λ x, g (f x)) := + begin + intro z, + rcases surjg z with ⟨y, rfl⟩, + rcases surjf y with ⟨x, rfl⟩, + use [x, rfl] + end + + end
src/03_Logic/solutions/solutions_03_Negation.lean (new)
- @@ -0,0 +1,132 @@
 + import data.real.basic + + section + variables a b : ℝ + + def fn_ub (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a + def fn_lb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x + + def fn_has_ub (f : ℝ → ℝ) := ∃ a, fn_ub f a + def fn_has_lb (f : ℝ → ℝ) := ∃ a, fn_lb f a + + variable f : ℝ → ℝ + + example (h : ∀ a, ∃ x, f x < a) : ¬ fn_has_lb f := + begin + rintros ⟨a, ha⟩, + rcases h a with ⟨x, hx⟩, + have := ha x, + linarith + end + + example : ¬ fn_has_ub (λ x, x) := + begin + rintros ⟨a, ha⟩, + have : a + 1 ≤ a := ha (a + 1), + linarith + end + + example (h : monotone f) (h' : f a < f b) : a < b := + begin + apply lt_of_not_ge, + intro h'', + apply absurd h', + apply not_lt_of_ge (h h'') + end + + example (h : a ≤ b) (h' : f b < f a) : ¬ monotone f := + begin + intro h'', + apply absurd h', + apply not_lt_of_ge, + apply h'' h + end + + example : + ¬ ∀ {f : ℝ → ℝ}, monotone f → ∀ {a b}, f a ≤ f b → a ≤ b := + begin + intro h, + let f := λ x : ℝ, (0 : ℝ), + have monof : monotone f, + { intros a b leab, + refl }, + have h' : f 1 ≤ f 0, + from le_refl _, + have : (1 : ℝ) ≤ 0 := h monof h', + linarith + end + + example (x : ℝ) (h : ∀ ε > 0, x < ε) : x ≤ 0 := + begin + apply le_of_not_gt, + intro h', + linarith [h _ h'] + end + + end + + section + variables {α : Type*} (P : α → Prop) (Q : Prop) + + example (h : ¬ ∃ x, P x) : ∀ x, ¬ P x := + begin + intros x Px, + apply h, + use [x, Px] + end + + example (h : ∀ x, ¬ P x) : ¬ ∃ x, P x := + begin + rintros ⟨x, Px⟩, + exact h x Px + end + + example (h : ∃ x, ¬ P x) : ¬ ∀ x, P x := + begin + intro h', + rcases h with ⟨x, nPx⟩, + apply nPx, + apply h' + end + + + example (h : ¬ ¬ Q) : Q := + begin + by_contradiction h', + exact h h' + end + + example (h : Q) : ¬ ¬ Q := + begin + intro h', + exact h' h + end + + end + + open_locale classical + section + variable (f : ℝ → ℝ) + + example (h : ¬ fn_has_ub f) : ∀ a, ∃ x, f x > a := + begin + intro a, + by_contradiction h', + apply h, + use a, + intro x, + apply le_of_not_gt, + intro h'', + apply h', + use [x, h''] + end + + example (h : ¬ monotone f) : ∃ x y, x ≤ y ∧ f y < f x := + begin + rw [monotone] at h, + push_neg at h, + exact h + end + + end +
src/03_Logic/solutions/solutions_04_Conjunction_and_Bi-implication.lean (new)
- @@ -0,0 +1,108 @@
 + import data.real.basic + import data.nat.prime + + example {m n : ℕ} (h : m ∣ n ∧ m ≠ n) : + m ∣ n ∧ ¬ n ∣ m := + begin + cases h with h0 h1, + split, + { exact h0 }, + intro h2, + apply h1, + apply nat.dvd_antisymm h0 h2, + end + + example {x y : ℝ} : x ≤ y ∧ ¬ y ≤ x ↔ x ≤ y ∧ x ≠ y := + begin + split, + { rintros ⟨h0, h1⟩, + split, + { exact h0 }, + intro h2, + apply h1, + rw h2 }, + rintros ⟨h0, h1⟩, + split, + { exact h0 }, + intro h2, + apply h1, + apply le_antisymm h0 h2 + end + + theorem aux {x y : ℝ} (h : x^2 + y^2 = 0) : x = 0 := + begin + have h' : x^2 = 0, + { linarith [pow_two_nonneg x, pow_two_nonneg y] }, + exact pow_eq_zero h' + end + + example (x y : ℝ) : x^2 + y^2 = 0 ↔ x = 0 ∧ y = 0 := + begin + split, + { intro h, + split, + { exact aux h }, + rw add_comm at h, + exact aux h }, + rintros ⟨rfl, rfl⟩, + norm_num + end + + theorem not_monotone_iff {f : ℝ → ℝ}: + ¬ monotone f ↔ ∃ x y, x ≤ y ∧ f x > f y := + by { rw monotone, push_neg } + + example : ¬ monotone (λ x : ℝ, -x) := + begin + rw not_monotone_iff, + use [0, 1], + norm_num + end + + section + variables {α : Type*} [partial_order α] + variables a b : α + + example : a < b ↔ a ≤ b ∧ a ≠ b := + begin + rw lt_iff_le_not_le, + split, + { rintros ⟨h0, h1⟩, + split, + { exact h0 }, + intro h2, + apply h1, + rw h2 }, + rintros ⟨h0, h1⟩, + split, + { exact h0 }, + intro h2, + apply h1, + apply le_antisymm h0 h2 + end + + end + + section + variables {α : Type*} [preorder α] + variables a b c : α + + example : ¬ a < a := + begin + rw lt_iff_le_not_le, + rintros ⟨h0, h1⟩, + exact h1 h0 + end + + example : a < b → b < c → a < c := + begin + simp only [lt_iff_le_not_le], + rintros ⟨h0, h1⟩ ⟨h2, h3⟩, + split, + { apply le_trans h0 h2 }, + intro h4, + apply h1, + apply le_trans h2 h4 + end + + end
src/03_Logic/solutions/solutions_05_Disjunction.lean (new)
- @@ -0,0 +1,162 @@
 + import data.real.basic + + section + variables {x y : ℝ} + + namespace my_abs + + theorem le_abs_self (x : ℝ) : x ≤ abs x := + begin + cases le_or_gt 0 x with h h, + { rw abs_of_nonneg h }, + rw abs_of_neg h, + linarith + end + + theorem neg_le_abs_self (x : ℝ) : -x ≤ abs x := + begin + cases le_or_gt 0 x with h h, + { rw abs_of_nonneg h, + linarith }, + rw abs_of_neg h + end + + theorem abs_add (x y : ℝ) : abs (x + y) ≤ abs x + abs y := + begin + cases le_or_gt 0 (x + y) with h h, + { rw abs_of_nonneg h, + linarith [le_abs_self x, le_abs_self y] }, + rw abs_of_neg h, + linarith [neg_le_abs_self x, neg_le_abs_self y] + end + + theorem lt_abs : x < abs y ↔ x < y ∨ x < -y := + begin + cases le_or_gt 0 y with h h, + { rw abs_of_nonneg h, + split, + { intro h', left, exact h' }, + intro h', + cases h' with h' h', + { exact h' }, + linarith }, + rw abs_of_neg h, + split, + { intro h', right, exact h' }, + intro h', + cases h' with h' h', + { linarith }, + exact h' + end + + theorem abs_lt : abs x < y ↔ - y < x ∧ x < y := + begin + cases le_or_gt 0 x with h h, + { rw abs_of_nonneg h, + split, + { intro h', + split, + { linarith }, + exact h' }, + intro h', + cases h' with h1 h2, + exact h2 }, + rw abs_of_neg h, + split, + { intro h', + split, + { linarith }, + linarith }, + intro h', + linarith + end + + end my_abs + end + + example {z : ℝ} (h : ∃ x y, z = x^2 + y^2 ∨ z = x^2 + y^2 + 1) : + z ≥ 0 := + by { rcases h with ⟨x, y, rfl | rfl⟩; linarith [sq_nonneg x, sq_nonneg y] } + + example {x : ℝ} (h : x^2 = 1) : x = 1 ∨ x = -1 := + begin + have h' : x^2 - 1 = 0, + { rw [h, sub_self] }, + have h'' : (x + 1) * (x - 1) = 0, + { rw ← h', + ring }, + cases eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1, + { right, + exact eq_neg_iff_add_eq_zero.mpr h1 }, + left, + exact eq_of_sub_eq_zero h1 + end + + example {x y : ℝ} (h : x^2 = y^2) : x = y ∨ x = -y := + begin + have h' : x^2 - y^2 = 0, + { rw [h, sub_self] }, + have h'' : (x + y) * (x - y) = 0, + { rw ← h', + ring }, + cases eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1, + { right, + exact eq_neg_iff_add_eq_zero.mpr h1 }, + left, + exact eq_of_sub_eq_zero h1 + end + + section + variables {R : Type*} [comm_ring R] [is_domain R] + variables (x y : R) + + example (h : x^2 = 1) : x = 1 ∨ x = -1 := + begin + have h' : x^2 - 1 = 0, + { rw [h, sub_self] }, + have h'' : (x + 1) * (x - 1) = 0, + { rw ← h', + ring }, + cases eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1, + { right, + exact eq_neg_iff_add_eq_zero.mpr h1 }, + left, + exact eq_of_sub_eq_zero h1 + end + + example (h : x^2 = y^2) : x = y ∨ x = -y := + begin + have h' : x^2 - y^2 = 0, + { rw [h, sub_self] }, + have h'' : (x + y) * (x - y) = 0, + { rw ← h', + ring }, + cases eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1, + { right, + exact eq_neg_iff_add_eq_zero.mpr h1 }, + left, + exact eq_of_sub_eq_zero h1 + end + + end + + section + open_locale classical + + example (P Q : Prop) : (P → Q) ↔ ¬ P ∨ Q := + begin + split, + { intro h, + by_cases h' : P, + { right, + exact h h'}, + left, + exact h' }, + rintros (h | h), + { intro h', + exact absurd h' h }, + intro _, + exact h + end + + end
src/03_Logic/solutions/solutions_06_Sequences_and_Convergence.lean (new)
- @@ -0,0 +1,143 @@
 + import data.real.basic + + def converges_to (s : ℕ → ℝ) (a : ℝ) := + ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε + + theorem converges_to_const (a : ℝ) : converges_to (λ x : ℕ, a) a := + begin + intros ε εpos, + use 0, + intros n nge, dsimp, + rw [sub_self, abs_zero], + apply εpos + end + + theorem converges_to_add {s t : ℕ → ℝ} {a b : ℝ} + (cs : converges_to s a) (ct : converges_to t b): + converges_to (λ n, s n + t n) (a + b) := + begin + intros ε εpos, dsimp, + have ε2pos : 0 < ε / 2, + { linarith }, + cases cs (ε / 2) ε2pos with Ns hs, + cases ct (ε / 2) ε2pos with Nt ht, + use max Ns Nt, + intros n hn, + have ngeNs : n ≥ Ns := le_of_max_le_left hn, + have ngeNt : n ≥ Nt := le_of_max_le_right hn, + calc + |s n + t n - (a + b)| = | s n - a + (t n - b) | : + by { congr, ring } + ... ≤ | s n - a | + | (t n - b) | : + abs_add _ _ + ... < ε / 2 + ε / 2 : add_lt_add (hs n ngeNs) (ht n ngeNt) + ... = ε : by norm_num + end + + theorem converges_to_mul_const {s : ℕ → ℝ} {a : ℝ} + (c : ℝ) (cs : converges_to s a) : + converges_to (λ n, c * s n) (c * a) := + begin + by_cases h : c = 0, + { convert converges_to_const 0, + { ext, rw [h, zero_mul] }, + rw [h, zero_mul] }, + have acpos : 0 < abs c, + from abs_pos.mpr h, + intros ε εpos, dsimp, + have εcpos : 0 < ε / abs c, + { apply div_pos εpos acpos }, + cases cs (ε / abs c) εcpos with Ns hs, + use Ns, + intros n ngt, + calc + |c * s n - c * a| = |c| * |s n - a| : + by { rw [←abs_mul, mul_sub] } + ... < |c| * (ε / |c|) : + mul_lt_mul_of_pos_left (hs n ngt) acpos + ... = ε : mul_div_cancel' _ (ne_of_lt acpos).symm + end + + theorem exists_abs_le_of_converges_to {s : ℕ → ℝ} {a : ℝ} + (cs : converges_to s a) : + ∃ N b, ∀ n, N ≤ n → abs (s n) < b := + begin + cases cs 1 zero_lt_one with N h, + use [N, abs a + 1], + intros n ngt, + calc + |s n| = |s n - a + a| : by { congr, abel } + ... ≤ |s n - a| + |a| : abs_add _ _ + ... < |a| + 1 : by linarith [h n ngt] + end + + lemma aux {s t : ℕ → ℝ} {a : ℝ} + (cs : converges_to s a) (ct : converges_to t 0) : + converges_to (λ n, s n * t n) 0 := + begin + intros ε εpos, dsimp, + rcases exists_abs_le_of_converges_to cs with ⟨N₀, B, h₀⟩, + have Bpos : 0 < B, + from lt_of_le_of_lt (abs_nonneg _) (h₀ N₀ (le_refl _)), + have pos₀ : ε / B > 0, + from div_pos εpos Bpos, + cases ct _ pos₀ with N₁ h₁, + use max N₀ N₁, + intros n ngt, + have ngeN₀ : n ≥ N₀ := le_of_max_le_left ngt, + have ngeN₁ : n ≥ N₁ := le_of_max_le_right ngt, + calc + |s n * t n - 0| = |s n| * |t n - 0| : + by rw [sub_zero, abs_mul, sub_zero] + ... < B * (ε / B) : + mul_lt_mul'' (h₀ n ngeN₀) (h₁ n ngeN₁) (abs_nonneg _) (abs_nonneg _) + ... = ε : mul_div_cancel' _ (ne_of_lt Bpos).symm + end + + theorem converges_to_muL {s t : ℕ → ℝ} {a b : ℝ} + (cs : converges_to s a) (ct : converges_to t b): + converges_to (λ n, s n * t n) (a * b) := + begin + have h₁ : converges_to (λ n, s n * (t n - b)) 0, + { apply aux cs, + convert converges_to_add ct (converges_to_const (-b)), + ring }, + convert (converges_to_add h₁ (converges_to_mul_const b cs)), + { ext, ring }, + ring + end + + theorem converges_to_unique {s : ℕ → ℝ} {a b : ℝ} + (sa : converges_to s a) (sb : converges_to s b) : + a = b := + begin + by_contradiction abne, + have : abs (a - b) > 0, + { apply lt_of_le_of_ne, + { apply abs_nonneg }, + intro h'', + apply abne, + apply eq_of_abs_sub_eq_zero h''.symm, }, + let ε := abs (a - b) / 2, + have εpos : ε > 0, + { change abs (a - b) / 2 > 0, linarith }, + cases sa ε εpos with Na hNa, + cases sb ε εpos with Nb hNb, + let N := max Na Nb, + have absa : abs (s N - a) < ε, + { apply hNa, apply le_max_left }, + have absb : abs (s N - b) < ε, + { apply hNb, apply le_max_right }, + have : abs (a - b) < abs (a - b), + calc + abs (a - b) = abs (- (s N - a) + (s N - b)) : + by { congr, ring } + ... ≤ abs (- (s N - a)) + abs (s N - b) : + abs_add _ _ + ... = abs (s N - a) + abs (s N - b) : + by rw [abs_neg] + ... < ε + ε : add_lt_add absa absb + ... = abs (a - b) : by norm_num, + exact lt_irrefl _ this + end +
src/04_Sets_and_Functions/01_Sets.lean (new)
- @@ -0,0 +1,273 @@
  + import data.set.lattice + import data.nat.parity + import tactic + + section + variable {α : Type*} + variables (s t u : set α) + + open set + + example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := + begin + rw [subset_def, inter_def, inter_def], + rw subset_def at h, + dsimp, + rintros x ⟨xs, xu⟩, + exact ⟨h _ xs, xu⟩, + end + + example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := + begin + simp only [subset_def, mem_inter_eq] at *, + rintros x ⟨xs, xu⟩, + exact ⟨h _ xs, xu⟩, + end + + example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := + begin + intros x xsu, + exact ⟨h xsu.1, xsu.2⟩ + end + + theorem foo (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := + λ x ⟨xs, xu⟩, ⟨h xs, xu⟩ + + example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := + by exact λ x ⟨xs, xu⟩, ⟨h xs, xu⟩ + + example : s ∩ (t ∪ u) ⊆ (s ∩ t) ∪ (s ∩ u) := + begin + intros x hx, + have xs : x ∈ s := hx.1, + have xtu : x ∈ t ∪ u := hx.2, + cases xtu with xt xu, + { left, + show x ∈ s ∩ t, + exact ⟨xs, xt⟩ }, + right, + show x ∈ s ∩ u, + exact ⟨xs, xu⟩ + end + + example : s ∩ (t ∪ u) ⊆ (s ∩ t) ∪ (s ∩ u) := + begin + rintros x ⟨xs, xt | xu⟩, + { left, exact ⟨xs, xt⟩ }, + right, exact ⟨xs, xu⟩ + end + + example : (s ∩ t) ∪ (s ∩ u) ⊆ s ∩ (t ∪ u):= + sorry + + example : s \ t \ u ⊆ s \ (t ∪ u) := + begin + intros x xstu, + have xs : x ∈ s := xstu.1.1, + have xnt : x ∉ t := xstu.1.2, + have xnu : x ∉ u := xstu.2, + split, + { exact xs }, dsimp, + intro xtu, -- x ∈ t ∨ x ∈ u + cases xtu with xt xu, + { show false, from xnt xt }, + show false, from xnu xu + end + + example : s \ t \ u ⊆ s \ (t ∪ u) := + begin + rintros x ⟨⟨xs, xnt⟩, xnu⟩, + use xs, + rintros (xt | xu); contradiction + end + + example : s \ (t ∪ u) ⊆ s \ t \ u := + sorry + + example : s ∩ t = t ∩ s := + begin + ext x, + simp only [mem_inter_eq], + split, + { rintros ⟨xs, xt⟩, exact ⟨xt, xs⟩ }, + rintros ⟨xt, xs⟩, exact ⟨xs, xt⟩ + end + + example : s ∩ t = t ∩ s := + set.ext $ λ x, ⟨λ ⟨xs, xt⟩, ⟨xt, xs⟩, λ ⟨xt, xs⟩, ⟨xs, xt⟩⟩ + + example : s ∩ t = t ∩ s := + by ext x; simp [and.comm] + + example : s ∩ t = t ∩ s := + begin + apply subset.antisymm, + { rintros x ⟨xs, xt⟩, exact ⟨xt, xs⟩ }, + rintros x ⟨xt, xs⟩, exact ⟨xs, xt⟩ + end + + example : s ∩ t = t ∩ s := + subset.antisymm sorry sorry + + example : s ∩ (s ∪ t) = s := + sorry + + example : s ∪ (s ∩ t) = s := + sorry + + example : (s \ t) ∪ t = s ∪ t := + sorry + + example : (s \ t) ∪ (t \ s) = (s ∪ t) \ (s ∩ t) := + sorry + + + def evens : set ℕ := {n | even n} + def odds : set ℕ := {n | ¬ even n} + + example : evens ∪ odds = univ := + begin + rw [evens, odds], + ext n, + simp, + apply classical.em + end + + example (x : ℕ) (h : x ∈ (∅ : set ℕ)) : false := + h + + example (x : ℕ) : x ∈ (univ : set ℕ) := + trivial + + example : { n | nat.prime n } ∩ { n | n > 2} ⊆ { n | ¬ even n } := + sorry + + #print prime + #print nat.prime + + example (n : ℕ) : prime n ↔ nat.prime n := nat.prime_iff.symm + + example (n : ℕ) (h : prime n) : nat.prime n := + by { rw nat.prime_iff, exact h } + + example (n : ℕ) (h : prime n) : nat.prime n := + by rwa nat.prime_iff + + end + section + variables (s t : set ℕ) + + example (h₀ : ∀ x ∈ s, ¬ even x) (h₁ : ∀ x ∈ s, prime x) : + ∀ x ∈ s, ¬ even x ∧ prime x := + begin + intros x xs, + split, + { apply h₀ x xs }, + apply h₁ x xs + end + + example (h : ∃ x ∈ s, ¬ even x ∧ prime x) : + ∃ x ∈ s, prime x := + begin + rcases h with ⟨x, xs, _, prime_x⟩, + use [x, xs, prime_x] + end + + section + variable (ssubt : s ⊆ t) + + include ssubt + + example (h₀ : ∀ x ∈ t, ¬ even x) (h₁ : ∀ x ∈ t, prime x) : + ∀ x ∈ s, ¬ even x ∧ prime x := + sorry + + example (h : ∃ x ∈ s, ¬ even x ∧ prime x) : + ∃ x ∈ t, prime x := + sorry + + end + + end + + section + variables {α I : Type*} + variables A B : I → set α + variable s : set α + open set + + example : s ∩ (⋃ i, A i) = ⋃ i, (A i ∩ s) := + begin + ext x, + simp only [mem_inter_eq, mem_Union], + split, + { rintros ⟨xs, ⟨i, xAi⟩⟩, + exact ⟨i, xAi, xs⟩ }, + rintros ⟨i, xAi, xs⟩, + exact ⟨xs, ⟨i, xAi⟩⟩ + end + + example : (⋂ i, A i ∩ B i) = (⋂ i, A i) ∩ (⋂ i, B i) := + begin + ext x, + simp only [mem_inter_eq, mem_Inter], + split, + { intro h, + split, + { intro i, + exact (h i).1 }, + intro i, + exact (h i).2 }, + rintros ⟨h1, h2⟩ i, + split, + { exact h1 i }, + exact h2 i + end + + open_locale classical + + example : s ∪ (⋂ i, A i) = ⋂ i, (A i ∪ s) := + sorry + + def primes : set ℕ := {x | nat.prime x} + + example : (⋃ p ∈ primes, {x | p^2 ∣ x}) = {x | ∃ p ∈ primes, p^2 ∣ x} := + by { ext, rw mem_Union₂, refl } + + 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} := + begin + intro x, + contrapose!, + simp, + apply nat.exists_prime_and_dvd + end + + example : (⋃ p ∈ primes, {x | x ≤ p}) = univ := + sorry + + end + + section + open set + + variables {α : Type*} (s : set (set α)) + + example : ⋃₀ s = ⋃ t ∈ s, t := + begin + ext x, + rw mem_Union₂, + refl + end + + example : ⋂₀ s = ⋂ t ∈ s, t := + begin + ext x, + rw mem_Inter₂, + refl + end + + end +
src/04_Sets_and_Functions/02_Functions.lean (new)
- @@ -0,0 +1,216 @@
  + import data.set.lattice + import data.set.function + import analysis.special_functions.log.basic + + section + + variables {α β : Type*} + variable f : α → β + variables s t : set α + variables u v : set β + open function + open set + + example : f ⁻¹' (u ∩ v) = f ⁻¹' u ∩ f ⁻¹' v := + by { ext, refl } + + example : f '' (s ∪ t) = f '' s ∪ f '' t := + begin + ext y, split, + { rintros ⟨x, xs | xt, rfl⟩, + { left, use [x, xs] }, + right, use [x, xt] }, + rintros (⟨x, xs, rfl⟩ | ⟨x, xt, rfl⟩), + { use [x, or.inl xs] }, + use [x, or.inr xt] + end + + example : s ⊆ f ⁻¹' (f '' s) := + begin + intros x xs, + show f x ∈ f '' s, + use [x, xs] + end + + example : f '' s ⊆ v ↔ s ⊆ f ⁻¹' v := + sorry + + example (h : injective f) : f ⁻¹' (f '' s) ⊆ s := + sorry + + example : f '' (f⁻¹' u) ⊆ u := + sorry + + example (h : surjective f) : u ⊆ f '' (f⁻¹' u) := + sorry + + example (h : s ⊆ t) : f '' s ⊆ f '' t := + sorry + + example (h : u ⊆ v) : f ⁻¹' u ⊆ f ⁻¹' v := + sorry + + example : f ⁻¹' (u ∪ v) = f ⁻¹' u ∪ f ⁻¹' v := + sorry + + example : f '' (s ∩ t) ⊆ f '' s ∩ f '' t := + sorry + + example (h : injective f) : f '' s ∩ f '' t ⊆ f '' (s ∩ t) := + sorry + + example : f '' s \ f '' t ⊆ f '' (s \ t) := + sorry + + example : f ⁻¹' u \ f ⁻¹' v ⊆ f ⁻¹' (u \ v) := + sorry + + example : f '' s ∩ v = f '' (s ∩ f ⁻¹' v) := + sorry + + example : f '' (s ∩ f ⁻¹' u) ⊆ f '' s ∪ u := + sorry + + example : s ∩ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∩ u) := + sorry + + example : s ∪ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∪ u) := + sorry + + variables {I : Type*} (A : I → set α) (B : I → set β) + + example : f '' (⋃ i, A i) = ⋃ i, f '' A i := + begin + ext y, simp, + split, + { rintros ⟨x, ⟨i, xAi⟩, fxeq⟩, + use [i, x, xAi, fxeq] }, + rintros ⟨i, x, xAi, fxeq⟩, + exact ⟨x, ⟨i, xAi⟩, fxeq⟩ + end + + example : f '' (⋂ i, A i) ⊆ ⋂ i, f '' A i := + begin + intro y, simp, + intros x h fxeq i, + use [x, h i, fxeq], + end + + example (i : I) (injf : injective f) : + (⋂ i, f '' A i) ⊆ f '' (⋂ i, A i) := + begin + intro y, simp, + intro h, + rcases h i with ⟨x, xAi, fxeq⟩, + use x, split, + { intro i', + rcases h i' with ⟨x', x'Ai, fx'eq⟩, + have : f x = f x', by rw [fxeq, fx'eq], + have : x = x', from injf this, + rw this, + exact x'Ai }, + exact fxeq + end + + example : f ⁻¹' (⋃ i, B i) = ⋃ i, f ⁻¹' (B i) := + by { ext x, simp } + + example : f ⁻¹' (⋂ i, B i) = ⋂ i, f ⁻¹' (B i) := + by { ext x, simp } + + end + + section + open set real + + example : inj_on log { x | x > 0 } := + begin + intros 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 + end + + example : range exp = { y | y > 0 } := + begin + ext y, split, + { rintros ⟨x, rfl⟩, + apply exp_pos }, + intro ypos, + use log y, + rw exp_log ypos + end + + example : inj_on sqrt { x | x ≥ 0 } := + sorry + + example : inj_on (λ x, x^2) { x : ℝ | x ≥ 0 } := + sorry + + example : sqrt '' { x | x ≥ 0 } = {y | y ≥ 0} := + sorry + + example : range (λ x, x^2) = {y : ℝ | y ≥ 0} := + sorry + + end + + section + variables {α β : Type*} [inhabited α] + + #check (default : α) + + variables (P : α → Prop) (h : ∃ x, P x) + + #check classical.some h + + example : P (classical.some h) := classical.some_spec h + + noncomputable theory + open_locale classical + + def inverse (f : α → β) : β → α := + λ y : β, if h : ∃ x, f x = y then classical.some h else default + + theorem inverse_spec {f : α → β} (y : β) (h : ∃ x, f x = y) : + f (inverse f y) = y := + begin + rw inverse, dsimp, rw dif_pos h, + exact classical.some_spec h + end + + variable f : α → β + open function + + example : injective f ↔ left_inverse (inverse f) f := + sorry + + example : surjective f ↔ right_inverse (inverse f) f := + sorry + + end + + section + variable {α : Type*} + open function + + theorem Cantor : ∀ f : α → set α, ¬ surjective f := + begin + intros f surjf, + let S := { i | i ∉ f i}, + rcases surjf S with ⟨j, h⟩, + have h₁ : j ∉ f j, + { intro h', + have : j ∉ f j, + { by rwa h at h' }, + contradiction }, + have h₂ : j ∈ S, + sorry, + have h₃ : j ∉ S, + sorry, + contradiction + end + + end
src/04_Sets_and_Functions/03_The_Schroeder_Bernstein_Theorem.lean (new)
- @@ -0,0 +1,108 @@
  + import data.set.lattice + import data.set.function + import tactic + + open set + open function + + noncomputable theory + open_locale classical + + variables {α β : Type*} [nonempty β] + + section + variables (f : α → β) (g : β → α) + + def sb_aux : ℕ → set α + | 0 := univ \ (g '' univ) + | (n + 1) := g '' (f '' sb_aux n) + + def sb_set := ⋃ n, sb_aux f g n + + def sb_fun (x : α) : β := if x ∈ sb_set f g then f x else inv_fun g x + + theorem sb_right_inv {x : α} (hx : x ∉ sb_set f g) : + g (inv_fun g x) = x := + begin + have : x ∈ g '' univ, + { contrapose! hx, + rw [sb_set, mem_Union], + use [0], + rw [sb_aux, mem_diff], + sorry }, + have : ∃ y, g y = x, + { sorry }, + sorry + end + + theorem sb_injective (hf: injective f) (hg : injective g) : + injective (sb_fun f g) := + begin + set A := sb_set f g with A_def, + set h := sb_fun f g with h_def, + intros x₁ x₂, + assume hxeq : h x₁ = h x₂, + show x₁ = x₂, + simp only [h_def, sb_fun, ←A_def] at hxeq, + by_cases xA : x₁ ∈ A ∨ x₂ ∈ A, + { wlog : x₁ ∈ A := xA using [x₁ x₂, x₂ x₁], + have x₂A : x₂ ∈ A, + { apply not_imp_self.mp, + assume x₂nA : x₂ ∉ A, + rw [if_pos xA, if_neg x₂nA] at hxeq, + rw [A_def, sb_set, mem_Union] at xA, + have x₂eq : x₂ = g (f x₁), + { sorry }, + rcases xA with ⟨n, hn⟩, + rw [A_def, sb_set, mem_Union], + use n + 1, + simp [sb_aux], + exact ⟨x₁, hn, x₂eq.symm⟩ }, + sorry }, + push_neg at xA, + sorry + end + + theorem sb_surjective (hf: injective f) (hg : injective g) : + surjective (sb_fun f g) := + begin + set A := sb_set f g with A_def, + set h := sb_fun f g with h_def, + intro y, + by_cases gyA : g y ∈ A, + { rw [A_def, sb_set, mem_Union] at gyA, + rcases gyA with ⟨n, hn⟩, + cases n with n, + { simp [sb_aux] at hn, + contradiction }, + simp [sb_aux] at hn, + rcases hn with ⟨x, xmem, hx⟩, + use x, + have : x ∈ A, + { rw [A_def, sb_set, mem_Union], + exact ⟨n, xmem⟩ }, + simp only [h_def, sb_fun, if_pos this], + exact hg hx }, + sorry + end + + end + + theorem schroeder_bernstein {f : α → β} {g : β → α} + (hf: injective f) (hg : injective g) : + ∃ h : α → β, bijective h := + ⟨sb_fun f g, sb_injective f g hf hg, sb_surjective f g hf hg⟩ + + /- Auxliary information -/ + + section + + variables (g : β → α) (x : α) + + #check (inv_fun g : α → β) + + #check (left_inverse_inv_fun : injective g → left_inverse (inv_fun g) g) + #check (left_inverse_inv_fun : injective g → ∀ y, inv_fun g (g y) = y) + + #check (inv_fun_eq : (∃ y, g y = x) → g (inv_fun g x) = x) + end
src/04_Sets_and_Functions/solutions/solutions_01_Sets.lean (new)
- @@ -0,0 +1,151 @@
  + import data.set.lattice + import data.nat.parity + import tactic + + section + variable {α : Type*} + variables (s t u : set α) + + open set + + example : (s ∩ t) ∪ (s ∩ u) ⊆ s ∩ (t ∪ u):= + begin + rintros x (⟨xs, xt⟩ | ⟨xs, xu⟩), + { use xs, left, exact xt }, + use xs, right, exact xu + end + + example : s \ (t ∪ u) ⊆ s \ t \ u := + begin + rintros x ⟨xs, xntu⟩, + use xs, + { intro xt, exact xntu (or.inl xt) }, + intro xu, + apply xntu (or.inr xu) + end + + example : s ∩ t = t ∩ s := + subset.antisymm (λ x ⟨xs, xt⟩, ⟨xt, xs⟩) (λ x ⟨xt, xs⟩, ⟨xs, xt⟩) + + example : s ∩ (s ∪ t) = s := + begin + ext x, split, + { rintros ⟨xs, _⟩, exact xs }, + intro xs, + use xs, left, exact xs + end + + example : s ∪ (s ∩ t) = s := + begin + ext x, split, + { rintros (xs | ⟨xs, xt⟩); exact xs }, + intro xs, left, exact xs + end + + example : (s \ t) ∪ t = s ∪ t := + begin + ext x, split, + { rintros (⟨xs, nxt⟩ | xt), + { left, exact xs}, + right, exact xt }, + by_cases h : x ∈ t, + { intro _, right, exact h }, + rintros (xs | xt), + { left, use [xs, h] }, + right, use xt + end + + example : (s \ t) ∪ (t \ s) = (s ∪ t) \ (s ∩ t) := + begin + ext x, split, + { rintros (⟨xs, xnt⟩ | ⟨xt, xns⟩), + { split, left, exact xs, + rintros ⟨_, xt⟩, contradiction }, + split , right, exact xt, + rintros ⟨xs, _⟩, contradiction }, + rintros ⟨xs | xt, nxst⟩, + { left, use xs, intro xt, + apply nxst, + split; assumption }, + right, use xt, intro xs, + apply nxst, + split; assumption + end + + example : { n | nat.prime n } ∩ { n | n > 2} ⊆ { n | ¬ even n } := + begin + intro n, + simp, + intro nprime, + cases nat.prime.eq_two_or_odd nprime with h h, + { rw h, intro, linarith }, + rw [nat.even_iff, h], + norm_num + end + + end + section + variables (s t : set ℕ) + + section + variable (ssubt : s ⊆ t) + + include ssubt + + example (h₀ : ∀ x ∈ t, ¬ even x) (h₁ : ∀ x ∈ t, prime x) : + ∀ x ∈ s, ¬ even x ∧ prime x := + begin + intros x xs, + split, + { apply h₀ x (ssubt xs) }, + apply h₁ x (ssubt xs) + end + + example (h : ∃ x ∈ s, ¬ even x ∧ prime x) : + ∃ x ∈ t, prime x := + begin + rcases h with ⟨x, xs, _, px⟩, + use [x, ssubt xs, px] + end + + end + + end + + section + variables {α I : Type*} + variables A B : I → set α + variable s : set α + open set + + example : s ∪ (⋂ i, A i) = ⋂ i, (A i ∪ s) := + begin + ext x, + simp only [mem_union, mem_Inter], + split, + { rintros (xs | xI), + { intro i, right, exact xs }, + intro i, left, exact xI i }, + intro h, + by_cases xs : x ∈ s, + { left, exact xs }, + right, + intro i, + cases h i, + { assumption }, + contradiction + end + + def primes : set ℕ := {x | nat.prime x} + + example : (⋃ p ∈ primes, {x | x ≤ p}) = univ := + begin + apply eq_univ_of_forall, + intro x, + simp, + rcases nat.exists_infinite_primes x with ⟨p, primep, pge⟩, + use [p, pge, primep] + end + + end +
src/04_Sets_and_Functions/solutions/solutions_02_Functions.lean (new)
- @@ -0,0 +1,281 @@
  + import data.set.lattice + import data.set.function + import analysis.special_functions.log.basic + + section + + variables {α β : Type*} + variable f : α → β + variables s t : set α + variables u v : set β + open function + open set + + example : f '' s ⊆ v ↔ s ⊆ f ⁻¹' v := + begin + split, + { intros h x xs, + have : f x ∈ f '' s, + from mem_image_of_mem _ xs, + exact h this }, + intros h y ymem, + rcases ymem with ⟨x, xs, fxeq⟩, + rw ← fxeq, + apply h xs + end + + example (h : injective f) : f ⁻¹' (f '' s) ⊆ s := + begin + rintros x ⟨y, ys, fxeq⟩, + rw ← h fxeq, + exact ys + end + + example : f '' (f⁻¹' u) ⊆ u := + begin + rintros y ⟨x, xmem, rfl⟩, + exact xmem + end + + example (h : surjective f) : u ⊆ f '' (f⁻¹' u) := + begin + intros y yu, + rcases h y with ⟨x, fxeq⟩, + use x, + split, + { show f x ∈ u, + rw fxeq, exact yu }, + exact fxeq + end + + example (h : s ⊆ t) : f '' s ⊆ f '' t := + begin + rintros y ⟨x, xs, fxeq⟩, + use [x, h xs, fxeq] + end + + example (h : u ⊆ v) : f ⁻¹' u ⊆ f ⁻¹' v := + by intro x; apply h + + example : f ⁻¹' (u ∪ v) = f ⁻¹' u ∪ f ⁻¹' v := + by ext x; refl + + example : f '' (s ∩ t) ⊆ f '' s ∩ f '' t := + begin + rintros y ⟨x, ⟨xs, xt⟩, rfl⟩, + use [x, xs, rfl, x, xt, rfl] + end + + example (h : injective f) : f '' s ∩ f '' t ⊆ f '' (s ∩ t) := + begin + rintros y ⟨⟨x₁, x₁s, rfl⟩, ⟨x₂, x₂t, fx₂eq⟩⟩, + use [x₁, x₁s], + rw ← h fx₂eq, + exact x₂t + end + + example : f '' s \ f '' t ⊆ f '' (s \ t) := + begin + rintros y ⟨⟨x₁, x₁s, rfl⟩, h⟩, + use [x₁, x₁s], + intro h', + apply h, + use [x₁, h', rfl] + end + + example : f ⁻¹' u \ f ⁻¹' v ⊆ f ⁻¹' (u \ v) := + λ x, id + + example : f '' s ∩ v = f '' (s ∩ f ⁻¹' v) := + begin + ext y, split, + { rintros ⟨⟨x, xs, rfl⟩, fxv⟩, + use [x, xs, fxv] }, + rintros ⟨x, ⟨⟨xs, fxv⟩, rfl⟩⟩, + use [x, xs, rfl, fxv], + end + + example : f '' (s ∩ f ⁻¹' u) ⊆ f '' s ∩ u := + begin + rintros y ⟨x, ⟨xs, fxu⟩, rfl⟩, + use [x, xs, rfl, fxu], + end + + example : s ∩ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∩ u) := + begin + rintros x ⟨xs, fxu⟩, + use [x, xs, rfl, fxu], + end + + example : s ∪ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∪ u) := + begin + rintros x (xs | fxu), + { left, use [x, xs, rfl] }, + right, use fxu + end + + variables {I : Type*} (A : I → set α) (B : I → set β) + + example : f '' (⋃ i, A i) = ⋃ i, f '' A i := + begin + ext y, simp, + split, + { rintros ⟨x, ⟨i, xAi⟩, fxeq⟩, + use [i, x, xAi, fxeq] }, + rintros ⟨i, x, xAi, fxeq⟩, + exact ⟨x, ⟨i, xAi⟩, fxeq⟩ + end + + example : f '' (⋂ i, A i) ⊆ ⋂ i, f '' A i := + begin + intro y, simp, + intros x h fxeq i, + use [x, h i, fxeq], + end + + example (i : I) (injf : injective f) : (⋂ i, f '' A i) ⊆ f '' (⋂ i, A i) := + begin + intro y, simp, + intro h, + rcases h i with ⟨x, xAi, fxeq⟩, + use x, split, + { intro i', + rcases h i' with ⟨x', x'Ai, fx'eq⟩, + have : f x = f x', by rw [fxeq, fx'eq], + have : x = x', from injf this, + rw this, + exact x'Ai }, + exact fxeq + end + + example : f ⁻¹' (⋃ i, B i) = ⋃ i, f ⁻¹' (B i) := + by { ext x, simp } + + example : f ⁻¹' (⋂ i, B i) = ⋂ i, f ⁻¹' (B i) := + by { ext x, simp } + + end + + section + open set real + + example : inj_on sqrt { x | x ≥ 0 } := + begin + intros x xnonneg y ynonneg, + intro e, + calc + x = (sqrt x)^2 : by rw sq_sqrt xnonneg + ... = (sqrt y)^2 : by rw e + ... = y : by rw sq_sqrt ynonneg + end + + example : inj_on (λ x, x^2) { x : ℝ | x ≥ 0 } := + begin + intros x xnonneg y ynonneg, + intro e, + dsimp at *, + calc + x = sqrt (x^2) : by rw sqrt_sq xnonneg + ... = sqrt (y^2) : by rw e + ... = y : by rw sqrt_sq ynonneg, + end + + example : sqrt '' { x | x ≥ 0 } = {y | y ≥ 0} := + begin + ext y, split, + { rintros ⟨x, ⟨xnonneg, rfl⟩⟩, + apply sqrt_nonneg }, + intro ynonneg, + use y^2, + dsimp at *, + split, + apply pow_nonneg ynonneg, + apply sqrt_sq, + assumption, + end + + example : range (λ x, x^2) = {y : ℝ | y ≥ 0} := + begin + ext y, + split, + { rintros ⟨x, rfl⟩, + dsimp at *, + apply pow_two_nonneg }, + intro ynonneg, + use sqrt y, + exact sq_sqrt ynonneg, + end + + end + + section + variables {α β : Type*} [inhabited α] + + noncomputable theory + open_locale classical + + def inverse (f : α → β) : β → α := + λ y : β, if h : ∃ x, f x = y then classical.some h else default + + theorem inverse_spec {f : α → β} (y : β) (h : ∃ x, f x = y) : + f (inverse f y) = y := + begin + rw inverse, dsimp, rw dif_pos h, + exact classical.some_spec h + end + + variable f : α → β + open function + + example : injective f ↔ left_inverse (inverse f) f := + begin + split, + { intros h y, + apply h, + apply inverse_spec, + use y }, + intros h x1 x2 e, + rw [←h x1, ←h x2, e] + end + + example : injective f ↔ left_inverse (inverse f) f := + ⟨λ h y, h (inverse_spec _ ⟨y, rfl⟩), λ h x1 x2 e, by rw [←h x1, ←h x2, e]⟩ + + example : surjective f ↔ right_inverse (inverse f) f := + begin + split, + { intros h y, + apply inverse_spec, + apply h }, + intros h y, + use (inverse f y), + apply h + end + + example : surjective f ↔ right_inverse (inverse f) f := + ⟨λ h y, inverse_spec _ (h _), λ h y, ⟨inverse f y, h _⟩⟩ + + end + + section + variable {α : Type*} + open function + + theorem Cantor : ∀ f : α → set α, ¬ surjective f := + begin + intros f surjf, + let S := { i | i ∉ f i}, + rcases surjf S with ⟨j, h⟩, + have h₁ : j ∉ f j, + { intro h', + have : j ∉ f j, + by rwa h at h', + contradiction }, + have h₂ : j ∈ S, + from h₁, + have h₃ : j ∉ S, + by rwa h at h₁, + contradiction + end + + end
src/04_Sets_and_Functions/solutions/solutions_03_The_Schroeder_Bernstein_Theorem.lean (new)
- @@ -0,0 +1,94 @@
  + import data.set.lattice + import data.set.function + import tactic + + open set + open function + + noncomputable theory + open_locale classical + + variables {α β : Type*} [nonempty β] + + section + variables (f : α → β) (g : β → α) + + def sb_aux : ℕ → set α + | 0 := univ \ (g '' univ) + | (n + 1) := g '' (f '' sb_aux n) + + def sb_set := ⋃ n, sb_aux f g n + + def sb_fun (x : α) : β := if x ∈ sb_set f g then f x else inv_fun g x + + theorem sb_right_inv {x : α} (hx : x ∉ sb_set f g) : + g (inv_fun g x) = x := + begin + have : x ∈ g '' univ, + { contrapose! hx, + rw [sb_set, mem_Union], + use [0], + rw [sb_aux, mem_diff], + exact ⟨mem_univ _, hx⟩ }, + have : ∃ y, g y = x, + { simp at this, assumption }, + exact inv_fun_eq this + end + + theorem sb_injective (hf: injective f) (hg : injective g) : + injective (sb_fun f g) := + begin + set A := sb_set f g with A_def, + set h := sb_fun f g with h_def, + intros x₁ x₂, + assume hxeq : h x₁ = h x₂, + show x₁ = x₂, + simp only [h_def, sb_fun, ←A_def] at hxeq, + by_cases xA : x₁ ∈ A ∨ x₂ ∈ A, + { wlog : x₁ ∈ A := xA using [x₁ x₂, x₂ x₁], + have x₂A : x₂ ∈ A, + { apply not_imp_self.mp, + assume x₂nA : x₂ ∉ A, + rw [if_pos xA, if_neg x₂nA] at hxeq, + rw [A_def, sb_set, mem_Union] at xA, + have x₂eq : x₂ = g (f x₁), + { rw [hxeq, sb_right_inv f g x₂nA] }, + rcases xA with ⟨n, hn⟩, + rw [A_def, sb_set, mem_Union], + use n + 1, + simp [sb_aux], + exact ⟨x₁, hn, x₂eq.symm⟩ }, + rw [if_pos xA, if_pos x₂A] at hxeq, + exact hf hxeq }, + push_neg at xA, + rw [if_neg xA.1, if_neg xA.2] at hxeq, + rw [←sb_right_inv f g xA.1, hxeq, sb_right_inv f g xA.2] + end + + theorem sb_surjective (hf: injective f) (hg : injective g) : + surjective (sb_fun f g) := + begin + set A := sb_set f g with A_def, + set h := sb_fun f g with h_def, + intro y, + by_cases gyA : g y ∈ A, + { rw [A_def, sb_set, mem_Union] at gyA, + rcases gyA with ⟨n, hn⟩, + cases n with n, + { simp [sb_aux] at hn, + contradiction }, + simp [sb_aux] at hn, + rcases hn with ⟨x, xmem, hx⟩, + use x, + have : x ∈ A, + { rw [A_def, sb_set, mem_Union], + exact ⟨n, xmem⟩ }, + simp only [h_def, sb_fun, if_pos this], + exact hg hx }, + use g y, + simp only [h_def, sb_fun, if_neg gyA], + apply left_inverse_inv_fun hg + end + + end +
src/05_Number_Theory/01_Irrational_Roots.lean (new)
- @@ -0,0 +1,126 @@
 + import data.nat.gcd + import data.real.irrational + + #print nat.coprime + + example (m n : nat) (h : m.coprime n) : m.gcd n = 1 := h + + example (m n : nat) (h : m.coprime n) : m.gcd n = 1 := + by { rw nat.coprime at h, exact h } + + example : nat.coprime 12 7 := by norm_num + example : nat.gcd 12 8 = 4 := by norm_num + + #check @nat.prime_def_lt + + example (p : ℕ) (prime_p : nat.prime p) : 2 ≤ p ∧ ∀ (m : ℕ), m < p → m ∣ p → m = 1 := + by rwa nat.prime_def_lt at prime_p + + #check nat.prime.eq_one_or_self_of_dvd + + example (p : ℕ) (prime_p : nat.prime p) : ∀ (m : ℕ), m ∣ p → m = 1 ∨ m = p := + prime_p.eq_one_or_self_of_dvd + + example : nat.prime 17 := by norm_num + + -- commonly used + example : nat.prime 2 := nat.prime_two + example : nat.prime 3 := nat.prime_three + + #check @nat.prime.dvd_mul + #check nat.prime.dvd_mul nat.prime_two + #check nat.prime_two.dvd_mul + + lemma even_of_even_sqr {m : ℕ} (h : 2 ∣ m^2) : 2 ∣ m := + begin + rw [pow_two, nat.prime_two.dvd_mul] at h, + cases h; assumption + end + + example {m : ℕ} (h : 2 ∣ m^2) : 2 ∣ m := + nat.prime.dvd_of_dvd_pow nat.prime_two h + + example (a b c : nat) (h : a * b = a * c) (h' : a ≠ 0) : + b = c := + begin + -- library_search suggests the following: + exact (mul_right_inj' h').mp h + end + + example {m n : ℕ} (coprime_mn : m.coprime n) : m^2 ≠ 2 * n^2 := + begin + intro sqr_eq, + have : 2 ∣ m, + sorry, + obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this, + have : 2 * (2 * k^2) = 2 * n^2, + { rw [←sqr_eq, meq], ring }, + have : 2 * k^2 = n^2, + sorry, + have : 2 ∣ n, + sorry, + have : 2 ∣ m.gcd n, + sorry, + have : 2 ∣ 1, + sorry, + norm_num at this + end + + example {m n p : ℕ} (coprime_mn : m.coprime n) (prime_p : p.prime) : m^2 ≠ p * n^2 := + sorry + + #check nat.factors + #check nat.prime_of_mem_factors + #check nat.prod_factors + #check nat.factors_unique + + + theorem factorization_mul' {m n : ℕ} (mnez : m ≠ 0) (nnez : n ≠ 0) (p : ℕ) : + (m * n).factorization p = m.factorization p + n.factorization p := + by { rw nat.factorization_mul mnez nnez, refl } + + theorem factorization_pow' (n k p : ℕ) : + (n^k).factorization p = k * n.factorization p := + by { rw nat.factorization_pow, refl } + + theorem nat.prime.factorization' {p : ℕ} (prime_p : p.prime) : + p.factorization p = 1 := + by { rw prime_p.factorization, simp } + + + example {m n p : ℕ} (nnz : n ≠ 0) (prime_p : p.prime) : m^2 ≠ p * n^2 := + begin + intro sqr_eq, + have nsqr_nez : n^2 ≠ 0, + by simpa, + have eq1 : nat.factorization (m^2) p = 2 * m.factorization p, + sorry, + have eq2 : (p * n^2).factorization p = 2 * n.factorization p + 1, + sorry, + have : (2 * m.factorization p) % 2 = (2 * n.factorization p + 1) % 2, + { rw [←eq1, sqr_eq, eq2] }, + rw [add_comm, nat.add_mul_mod_self_left, nat.mul_mod_right] at this, + norm_num at this + end + + example {m n k r : ℕ} (nnz : n ≠ 0) (pow_eq : m^k = r * n^k) + {p : ℕ} (prime_p : p.prime) : k ∣ r.factorization p := + begin + cases r with r, + { simp }, + have npow_nz : n^k ≠ 0 := λ npowz, nnz (pow_eq_zero npowz), + have eq1 : (m^k).factorization p = k * m.factorization p, + sorry, + have eq2 : (r.succ * n^k).factorization p = + k * n.factorization p + r.succ.factorization p, + sorry, + have : r.succ.factorization p = k * m.factorization p - k * n.factorization p, + { rw [←eq1, pow_eq, eq2, add_comm, nat.add_sub_cancel] }, + rw this, + sorry + end + + #check multiplicity + #check @irrational_nrt_of_n_not_dvd_multiplicity + #check irrational_sqrt_two +
src/05_Number_Theory/02_Induction_and_Recursion.lean (new)
- @@ -0,0 +1,151 @@
 + import data.nat.prime + import algebra.big_operators + import tactic + + example (n : nat) : n.succ ≠ nat.zero := nat.succ_ne_zero n + + example (m n : nat) (h : m.succ = n.succ) : m = n := nat.succ.inj h + + def fac : ℕ → ℕ + | 0 := 1 + | (n + 1) := (n + 1) * fac n + + example : fac 0 = 1 := rfl + example : fac 0 = 1 := by rw fac + example : fac 0 = 1 := by simp [fac] + + example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := rfl + example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := by rw fac + example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := by simp [fac] + + theorem fac_pos (n : ℕ) : 0 < fac n := + begin + induction n with n ih, + { rw fac, exact zero_lt_one }, + rw fac, + exact mul_pos n.succ_pos ih, + end + + theorem dvd_fac {i n : ℕ} (ipos : 0 < i) (ile : i ≤ n) : i ∣ fac n := + begin + induction n with n ih, + { exact absurd ipos (not_lt_of_ge ile) }, + rw fac, + cases nat.of_le_succ ile with h h, + { apply dvd_mul_of_dvd_right (ih h) }, + rw h, + apply dvd_mul_right + end + + theorem pow_two_le_fac (n : ℕ) : 2^(n-1) ≤ fac n := + begin + cases n with n, + { simp [fac] }, + sorry + end + + section + + variables {α : Type*} (s : finset ℕ) (f : ℕ → ℕ) (n : ℕ) + + #check finset.sum s f + #check finset.prod s f + + open_locale big_operators + open finset + + example : s.sum f = ∑ x in s, f x := rfl + example : s.prod f = ∏ x in s, f x := rfl + + example : (range n).sum f = ∑ x in range n, f x := rfl + example : (range n).prod f = ∏ x in range n, f x := rfl + + example (f : ℕ → ℕ) : ∑ x in range 0, f x = 0 := + finset.sum_range_zero f + + example (f : ℕ → ℕ) (n : ℕ): ∑ x in range n.succ, f x = (∑ x in range n, f x) + f n := + finset.sum_range_succ f n + + example (f : ℕ → ℕ) : ∏ x in range 0, f x = 1 := + finset.prod_range_zero f + + example (f : ℕ → ℕ) (n : ℕ): ∏ x in range n.succ, f x = (∏ x in range n, f x) * f n := + finset.prod_range_succ f n + + + example (n : ℕ) : fac n = ∏ i in range n, (i + 1) := + begin + induction n with n ih, + { simp [fac] }, + simp [fac, ih, prod_range_succ, mul_comm] + end + + example (a b c d e f : ℕ) : a * ((b * c) * f * (d * e)) = d * (a * f * e) * (c * b) := + by simp [mul_assoc, mul_comm, mul_left_comm] + + theorem sum_id (n : ℕ) : ∑ i in range (n + 1), i = n * (n + 1) / 2 := + begin + symmetry, apply nat.div_eq_of_eq_mul_right (by norm_num : 0 < 2), + induction n with n ih, + { simp }, + rw [finset.sum_range_succ, mul_add 2, ←ih, nat.succ_eq_add_one], + ring + end + + theorem sum_sqr (n : ℕ) : ∑ i in range (n + 1), i^2 = n * (n + 1) * (2 *n + 1) / 6 := + sorry + + end + + inductive my_nat + | zero : my_nat + | succ : my_nat → my_nat + + namespace my_nat + + def add : my_nat → my_nat → my_nat + | x zero := x + | x (succ y) := succ (add x y) + + def mul : my_nat → my_nat → my_nat + | x zero := zero + | x (succ y) := add (mul x y) x + + theorem zero_add (n : my_nat) : add zero n = n := + begin + induction n with n ih, + { refl }, + rw [add, ih] + end + + theorem succ_add (m n : my_nat) : add (succ m) n = succ (add m n) := + begin + induction n with n ih, + { refl }, + rw [add, ih], + refl + end + + theorem add_comm (m n : my_nat) : add m n = add n m := + begin + induction n with n ih, + { rw zero_add, refl }, + rw [add, succ_add, ih] + end + + theorem add_assoc (m n k : my_nat) : add (add m n) k = add m (add n k) := + sorry + + theorem mul_add (m n k : my_nat) : mul m (add n k) = add (mul m n) (mul m k) := + sorry + + theorem zero_mul (n : my_nat) : mul zero n = zero := + sorry + + theorem succ_mul (m n : my_nat) : mul (succ m) n = add (mul m n) n := + sorry + + theorem mul_comm (m n : my_nat) : mul m n = mul n m := + sorry + + end my_nat
src/05_Number_Theory/03_Infinitely_Many_Primes.lean (new)
- @@ -0,0 +1,254 @@
 + import data.nat.prime + import algebra.big_operators + import tactic + + open_locale big_operators + + theorem two_le {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := + begin + cases m, contradiction, + cases m, contradiction, + repeat { apply nat.succ_le_succ }, + apply zero_le + end + + example {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := + begin + by_contradiction h, + push_neg at h, + interval_cases m; contradiction + end + + example {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := + begin + by_contradiction h, + push_neg at h, + revert m h h0 h1, + dec_trivial + end + + example {m : ℕ} (h : m < 2) : m = 0 ∨ m = 1 := + by dec_trivial! + + example {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := + by omega + + theorem exists_prime_factor {n : nat} (h : 2 ≤ n) : + ∃ p : nat, p.prime ∧ p ∣ n := + begin + by_cases np : n.prime, + { use [n, np, dvd_rfl] }, + induction n using nat.strong_induction_on with n ih, + dsimp at ih, + rw nat.prime_def_lt at np, + push_neg at np, + rcases np h with ⟨m, mltn, mdvdn, mne1⟩, + have : m ≠ 0, + { 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, mdvdn] }, + rcases ih m mltn mgt2 mp with ⟨p, pp, pdvd⟩, + use [p, pp, pdvd.trans mdvdn] + end + + theorem primes_infinite : ∀ n, ∃ p > n, nat.prime p := + begin + intro n, + have : 2 ≤ nat.factorial (n + 1) + 1, + sorry, + rcases exists_prime_factor this with ⟨p, pp, pdvd⟩, + refine ⟨p, _, pp⟩, + show p > n, + by_contradiction ple, push_neg at ple, + have : p ∣ nat.factorial (n + 1), + sorry, + have : p ∣ 1, + sorry, + show false, + sorry + end + + open finset + + section + variables {α : Type*} [decidable_eq α] (r s t : finset α) + + example : r ∩ (s ∪ t) ⊆ (r ∩ s) ∪ (r ∩ t) := + begin + rw subset_iff, + intro x, + rw [mem_inter, mem_union, mem_union, mem_inter, mem_inter], + tauto + end + + example : r ∩ (s ∪ t) ⊆ (r ∩ s) ∪ (r ∩ t) := + by { simp [subset_iff], intro x, tauto } + + example : (r ∩ s) ∪ (r ∩ t) ⊆ r ∩ (s ∪ t) := + by { simp [subset_iff], intro x, tauto } + + example : (r ∩ s) ∪ (r ∩ t) = r ∩ (s ∪ t) := + by { ext x, simp, tauto } + + end + + section + variables {α : Type*} [decidable_eq α] (r s t : finset α) + + example : (r ∪ s) ∩ (r ∪ t) = r ∪ (s ∩ t) := + sorry + + example : (r \ s \ t) = r \ (s ∪ t) := + sorry + + end + example (s : finset ℕ) (n : ℕ) (h : n ∈ s) : n ∣ (∏ i in s, i) := + finset.dvd_prod_of_mem _ h + + theorem nat.prime.eq_of_dvd_of_prime {p q : ℕ} + (prime_p : nat.prime p) (prime_q : nat.prime q) (h : p ∣ q) : + p = q := + sorry + + theorem mem_of_dvd_prod_primes {s : finset ℕ} {p : ℕ} (prime_p : p.prime) : + (∀ n ∈ s, nat.prime n) → (p ∣ ∏ n in s, n) → p ∈ s := + begin + intros 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 + end + + example (s : finset ℕ) (x : ℕ) : x ∈ s.filter nat.prime ↔ x ∈ s ∧ x.prime := + mem_filter + + theorem primes_infinite' : ∀ (s : finset nat), ∃ p, nat.prime p ∧ p ∉ s := + begin + intro s, + by_contradiction h, + push_neg at h, + set s' := s.filter nat.prime with s'_def, + have mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ n.prime, + { intro n, + simp [s'_def], + apply h }, + have : 2 ≤ (∏ i in s', i) + 1, + sorry, + rcases exists_prime_factor this with ⟨p, pp, pdvd⟩, + have : p ∣ (∏ i in s', i), + sorry, + have : p ∣ 1, + { convert nat.dvd_sub' pdvd this, simp }, + show false, + sorry + end + + theorem bounded_of_ex_finset (Q : ℕ → Prop): + (∃ s : finset ℕ, ∀ k, Q k → k ∈ s) → ∃ n, ∀ k, Q k → k < n := + begin + rintros ⟨s, hs⟩, + use s.sup id + 1, + intros k Qk, + apply nat.lt_succ_of_le, + show id k ≤ s.sup id, + apply le_sup (hs k Qk) + end + + theorem ex_finset_of_bounded (Q : ℕ → Prop) [decidable_pred Q] : + (∃ n, ∀ k, Q k → k ≤ n) → (∃ s : finset ℕ, ∀ k, Q k ↔ k ∈ s) := + begin + rintros ⟨n, hn⟩, + use (range (n + 1)).filter Q, + intro k, + simp [nat.lt_succ_iff], + exact hn k + end + + example : 27 % 4 = 3 := by norm_num + + example (n : ℕ) : (4 * n + 3) % 4 = 3 := + by { rw [add_comm, nat.add_mul_mod_self_left], norm_num } + + theorem mod_4_eq_3_or_mod_4_eq_3 {m n : ℕ} (h : m * n % 4 = 3) : + m % 4 = 3 ∨ n % 4 = 3 := + begin + revert h, + rw [nat.mul_mod], + have : m % 4 < 4 := nat.mod_lt m (by norm_num), + interval_cases m % 4 with hm; simp [hm], + have : n % 4 < 4 := nat.mod_lt n (by norm_num), + interval_cases n % 4 with hn; simp [hn]; norm_num + end + + theorem two_le_of_mod_4_eq_3 {n : ℕ} (h : n % 4 = 3) : 2 ≤ n := + by apply two_le; { intro neq, rw neq at h, norm_num at h } + + theorem aux {m n : ℕ} (h₀ : m ∣ n) (h₁ : 2 ≤ m) (h₂ : m < n) : + (n / m) ∣ n ∧ n / m < n := + sorry + + theorem exists_prime_factor_mod_4_eq_3 {n : nat} (h : n % 4 = 3) : + ∃ p : nat, p.prime ∧ p ∣ n ∧ p % 4 = 3 := + begin + by_cases np : n.prime, + { use [n, np, dvd_rfl, h] }, + induction n using nat.strong_induction_on with n ih, + dsimp at ih, + rw nat.prime_def_lt at np, + push_neg at np, + rcases np (two_le_of_mod_4_eq_3 h) with ⟨m, mltn, mdvdn, mne1⟩, + have mge2 : 2 ≤ m, + { 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, + { apply mod_4_eq_3_or_mod_4_eq_3, rw [neq, h] }, + cases this with h1 h1, + { sorry }, + sorry + end + + example (m n : ℕ) (s : finset ℕ) (h : m ∈ erase s n) : m ≠ n ∧ m ∈ s := + by rwa mem_erase at h + + example (m n : ℕ) (s : finset ℕ) (h : m ∈ erase s n) : m ≠ n ∧ m ∈ s := + by { simp at h, assumption } + + theorem primes_mod_4_eq_3_infinite : ∀ n, ∃ p > n, nat.prime p ∧ p % 4 = 3 := + begin + by_contradiction h, + push_neg at h, + cases h with n hn, + have : ∃ s : finset nat, ∀ p : ℕ, p.prime ∧ p % 4 = 3 ↔ p ∈ s, + { apply ex_finset_of_bounded, + use n, + contrapose! hn, + rcases hn with ⟨p, ⟨pp, p4⟩, pltn⟩, + exact ⟨p, pltn, pp, p4⟩ }, + cases this with s hs, + have h₀ : 2 ≤ 4 * (∏ i in erase s 3, i) + 3, + sorry, + have h₁ : (4 * (∏ i in erase s 3, i) + 3) % 4 = 3, + sorry, + rcases exists_prime_factor_mod_4_eq_3 h₁ with ⟨p, pp, pdvd, p4eq⟩, + have ps : p ∈ s, + sorry, + have pne3 : p ≠ 3, + sorry, + have : p ∣ 4 * (∏ i in erase s 3, i), + sorry, + have : p ∣ 3, + sorry, + have : p = 3, + sorry, + contradiction + end + +
src/05_Number_Theory/solutions/solutions_01_Irrational_Roots.lean (new)
- @@ -0,0 +1,107 @@
  + import data.nat.gcd + import data.real.irrational + + lemma even_of_even_sqr {m : ℕ} (h : 2 ∣ m^2) : 2 ∣ m := + begin + rw [pow_two, nat.prime_two.dvd_mul] at h, + cases h; assumption + end + + example {m n : ℕ} (coprime_mn : m.coprime n) : m^2 ≠ 2 * n^2 := + begin + intro sqr_eq, + have : 2 ∣ m, + { apply even_of_even_sqr, + rw sqr_eq, + apply dvd_mul_right }, + obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this, + have : 2 * (2 * k^2) = 2 * n^2, + { rw [←sqr_eq, meq], ring }, + have : 2 * k^2 = n^2, + from (mul_right_inj' (by norm_num)).mp this, + have : 2 ∣ n, + { apply even_of_even_sqr, + rw ←this, + apply dvd_mul_right }, + have : 2 ∣ m.gcd n, + by apply nat.dvd_gcd; assumption, + have : 2 ∣ 1, + { convert this, symmetry, exact coprime_mn }, + norm_num at this + end + + example {m n p : ℕ} (coprime_mn : m.coprime n) (prime_p : p.prime) : m^2 ≠ p * n^2 := + begin + intro sqr_eq, + have : p ∣ m, + { apply prime_p.dvd_of_dvd_pow, + rw sqr_eq, + apply dvd_mul_right }, + obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this, + have : p * (p * k^2) = p * n^2, + { rw [←sqr_eq, meq], ring }, + have : p * k^2 = n^2, + { apply (mul_right_inj' _).mp this, + exact prime_p.ne_zero }, + have : p ∣ n, + { apply prime_p.dvd_of_dvd_pow, + rw ←this, + apply dvd_mul_right }, + have : p ∣ nat.gcd m n, + { apply nat.dvd_gcd; assumption }, + have : p ∣ 1, + { convert this, symmetry, exact coprime_mn }, + have : 2 ≤ 1, + { apply prime_p.two_le.trans, + exact nat.le_of_dvd zero_lt_one this }, + norm_num at this + end + + + 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, refl } + + theorem factorization_pow' (n k p : ℕ) : + (n^k).factorization p = k * n.factorization p := + by { rw nat.factorization_pow, refl } + + theorem nat.prime.factorization' {p : ℕ} (prime_p : p.prime) : + p.factorization p = 1 := + by { rw prime_p.factorization, simp } + + + example {m n p : ℕ} (nnz : n ≠ 0) (prime_p : p.prime) : m^2 ≠ p * n^2 := + begin + intro sqr_eq, + have nsqr_nez : n^2 ≠ 0, + by simpa, + have eq1 : nat.factorization (m^2) p = 2 * m.factorization p, + by { rw factorization_pow' }, + have eq2 : (p * n^2).factorization p = 2 * n.factorization p + 1, + { rw [factorization_mul' prime_p.ne_zero nsqr_nez, prime_p.factorization', + factorization_pow', add_comm] }, + have : (2 * m.factorization p) % 2 = (2 * n.factorization p + 1) % 2, + { rw [←eq1, sqr_eq, eq2] }, + rw [add_comm, nat.add_mul_mod_self_left, nat.mul_mod_right] at this, + norm_num at this + end + + example {m n k r : ℕ} (nnz : n ≠ 0) (pow_eq : m^k = r * n^k) + {p : ℕ} (prime_p : p.prime) : k ∣ r.factorization p := + begin + cases r with r, + { simp }, + have npow_nz : n^k ≠ 0 := λ npowz, nnz (pow_eq_zero npowz), + have eq1 : (m^k).factorization p = k * m.factorization p, + by rw factorization_pow', + have eq2 : (r.succ * n^k).factorization p = + k * n.factorization p + r.succ.factorization p, + { rw [factorization_mul' r.succ_ne_zero npow_nz, factorization_pow', + add_comm] }, + have : r.succ.factorization p = k * m.factorization p - k * n.factorization p, + { rw [←eq1, pow_eq, eq2, add_comm, nat.add_sub_cancel] }, + rw this, + apply nat.dvd_sub'; apply nat.dvd_mul_right + end +
src/05_Number_Theory/solutions/solutions_02_Induction_and_Recursion.lean (new)
- @@ -0,0 +1,114 @@
 + import data.nat.prime + import algebra.big_operators + import tactic + + def fac : ℕ → ℕ + | 0 := 1 + | (n + 1) := (n + 1) * fac n + + theorem pow_two_le_fac (n : ℕ) : 2^(n-1) ≤ fac n := + begin + cases n with n, + { simp [fac] }, + induction n with n ih, + { simp [fac] }, + simp at *, + rw [pow_succ, fac], + apply nat.mul_le_mul _ ih, + repeat { apply nat.succ_le_succ }, + apply zero_le + end + + section + + variables {α : Type*} (s : finset ℕ) (f : ℕ → ℕ) (n : ℕ) + + open_locale big_operators + open finset + + theorem sum_sqr (n : ℕ) : ∑ i in range (n + 1), i^2 = n * (n + 1) * (2 *n + 1) / 6 := + begin + symmetry, apply nat.div_eq_of_eq_mul_right (by norm_num : 0 < 6), + induction n with n ih, + { simp }, + rw [finset.sum_range_succ, mul_add 6, ←ih, nat.succ_eq_add_one], + ring + end + + end + + inductive my_nat + | zero : my_nat + | succ : my_nat → my_nat + + namespace my_nat + + def add : my_nat → my_nat → my_nat + | x zero := x + | x (succ y) := succ (add x y) + + def mul : my_nat → my_nat → my_nat + | x zero := zero + | x (succ y) := add (mul x y) x + + theorem zero_add (n : my_nat) : add zero n = n := + begin + induction n with n ih, + { refl }, + rw [add, ih] + end + + theorem succ_add (m n : my_nat) : add (succ m) n = succ (add m n) := + begin + induction n with n ih, + { refl }, + rw [add, ih], + refl + end + + theorem add_comm (m n : my_nat) : add m n = add n m := + begin + induction n with n ih, + { rw zero_add, refl }, + rw [add, succ_add, ih] + end + + theorem add_assoc (m n k : my_nat) : add (add m n) k = add m (add n k) := + begin + induction k with k ih, + { refl }, + rw [add, ih], + refl + end + + theorem mul_add (m n k : my_nat) : mul m (add n k) = add (mul m n) (mul m k) := + begin + induction k with k ih, + { refl }, + rw [add, mul, mul, ih, add_assoc] + end + + theorem zero_mul (n : my_nat) : mul zero n = zero := + begin + induction n with n ih, + { refl }, + rw [mul, ih], + refl + end + + theorem succ_mul (m n : my_nat) : mul (succ m) n = add (mul m n) n := + begin + induction n with n ih, + { refl }, + rw [mul, mul, ih, add_assoc, add_assoc, add_comm n, succ_add], + refl + end + + theorem mul_comm (m n : my_nat) : mul m n = mul n m := + begin + induction n with n ih, + { rw [zero_mul], refl }, + rw [mul, ih, succ_mul] + end + + end my_nat
src/05_Number_Theory/solutions/solutions_03_Infinitely_Many_Primes.lean (new)
- @@ -0,0 +1,246 @@
 + import data.nat.prime + import algebra.big_operators + import tactic + + open_locale big_operators + + theorem two_le {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := + begin + cases m, contradiction, + cases m, contradiction, + repeat { apply nat.succ_le_succ }, + apply zero_le + end + + theorem exists_prime_factor {n : nat} (h : 2 ≤ n) : + ∃ p : nat, p.prime ∧ p ∣ n := + begin + by_cases np : n.prime, + { use [n, np, dvd_rfl] }, + induction n using nat.strong_induction_on with n ih, + dsimp at ih, + rw nat.prime_def_lt at np, + push_neg at np, + rcases np h with ⟨m, mltn, mdvdn, mne1⟩, + have : m ≠ 0, + { 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, mdvdn] }, + rcases ih m mltn mgt2 mp with ⟨p, pp, pdvd⟩, + use [p, pp, pdvd.trans mdvdn] + end + + theorem primes_infinite : ∀ n, ∃ p > n, nat.prime p := + begin + intro n, + have : 2 ≤ nat.factorial (n + 1) + 1, + { apply nat.succ_le_succ, + exact nat.succ_le_of_lt (nat.factorial_pos _) }, + rcases exists_prime_factor this with ⟨p, pp, pdvd⟩, + refine ⟨p, _, pp⟩, + show p > n, + by_contradiction ple, push_neg at ple, + have : p ∣ nat.factorial (n + 1), + { apply nat.dvd_factorial, + apply pp.pos, + linarith }, + have : p ∣ 1, + { convert nat.dvd_sub' pdvd this, simp }, + show false, + have := nat.le_of_dvd zero_lt_one this, + linarith [pp.two_le] + end + + open finset + + section + variables {α : Type*} [decidable_eq α] (r s t : finset α) + + example : (r ∪ s) ∩ (r ∪ t) = r ∪ (s ∩ t) := + begin + ext x, + rw [mem_inter, mem_union, mem_union, mem_union, mem_inter], + tauto + end + + example : (r ∪ s) ∩ (r ∪ t) = r ∪ (s ∩ t) := + by { ext x, simp, tauto } + + example : (r \ s \ t) = r \ (s ∪ t) := + begin + ext x, + rw [mem_sdiff, mem_sdiff, mem_sdiff, mem_union], + tauto + end + + example : (r \ s \ t) = r \ (s ∪ t) := + by { ext x, simp, tauto } + + end + theorem nat.prime.eq_of_dvd_of_prime {p q : ℕ} + (prime_p : nat.prime p) (prime_q : nat.prime q) (h : p ∣ q) : + p = q := + begin + cases prime_q.eq_one_or_self_of_dvd _ h, + { linarith [prime_p.two_le] }, + assumption + end + + 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 := + begin + intros h₀ h₁, + induction s using finset.induction_on with a s ans ih, + { simp at h₁, + linarith [prime_p.two_le] }, + simp [finset.prod_insert ans, prime_p.dvd_mul] at h₀ h₁, + rw mem_insert, + cases h₁ with h₁ h₁, + { left, exact prime_p.eq_of_dvd_of_prime h₀.1 h₁ }, + right, + exact ih h₀.2 h₁ + end + + theorem primes_infinite' : ∀ (s : finset nat), ∃ p, nat.prime p ∧ p ∉ s := + begin + intro s, + by_contradiction h, + push_neg at h, + set s' := s.filter nat.prime with s'_def, + have mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ n.prime, + { intro n, + simp [s'_def], + apply h }, + have : 2 ≤ (∏ i in s', i) + 1, + { apply nat.succ_le_succ, + apply nat.succ_le_of_lt, + apply finset.prod_pos, + intros n ns', + apply (mem_s'.mp ns').pos }, + rcases exists_prime_factor this with ⟨p, pp, pdvd⟩, + have : p ∣ (∏ i in s', i), + { apply dvd_prod_of_mem, + rw mem_s', + apply pp }, + have : p ∣ 1, + { convert nat.dvd_sub' pdvd this, simp }, + show false, + have := nat.le_of_dvd zero_lt_one this, + linarith [pp.two_le] + end + + theorem bounded_of_ex_finset (Q : ℕ → Prop): + (∃ s : finset ℕ, ∀ k, Q k → k ∈ s) → ∃ n, ∀ k, Q k → k < n := + begin + rintros ⟨s, hs⟩, + use s.sup id + 1, + intros k Qk, + apply nat.lt_succ_of_le, + show id k ≤ s.sup id, + apply le_sup (hs k Qk) + end + + theorem ex_finset_of_bounded (Q : ℕ → Prop) [decidable_pred Q] : + (∃ n, ∀ k, Q k → k ≤ n) → (∃ s : finset ℕ, ∀ k, Q k ↔ k ∈ s) := + begin + rintros ⟨n, hn⟩, + use (range (n + 1)).filter Q, + intro k, + simp [nat.lt_succ_iff], + exact hn k + end + + theorem mod_4_eq_3_or_mod_4_eq_3 {m n : ℕ} (h : m * n % 4 = 3) : + m % 4 = 3 ∨ n % 4 = 3 := + begin + revert h, + rw [nat.mul_mod], + have : m % 4 < 4 := nat.mod_lt m (by norm_num), + interval_cases m % 4 with hm; simp [hm], + have : n % 4 < 4 := nat.mod_lt n (by norm_num), + interval_cases n % 4 with hn; simp [hn]; norm_num + end + + theorem two_le_of_mod_4_eq_3 {n : ℕ} (h : n % 4 = 3) : 2 ≤ n := + by apply two_le; { intro neq, rw neq at h, norm_num at h } + + theorem aux {m n : ℕ} (h₀ : m ∣ n) (h₁ : 2 ≤ m) (h₂ : m < n) : + (n / m) ∣ n ∧ n / m < n := + begin + split, + { exact nat.div_dvd_of_dvd h₀ }, + exact nat.div_lt_self (lt_of_le_of_lt (zero_le _) h₂) h₁ + end + + theorem exists_prime_factor_mod_4_eq_3 {n : nat} (h : n % 4 = 3) : + ∃ p : nat, p.prime ∧ p ∣ n ∧ p % 4 = 3 := + begin + by_cases np : n.prime, + { use [n, np, dvd_rfl, h] }, + induction n using nat.strong_induction_on with n ih, + dsimp at ih, + rw nat.prime_def_lt at np, + push_neg at np, + rcases np (two_le_of_mod_4_eq_3 h) with ⟨m, mltn, mdvdn, mne1⟩, + have mge2 : 2 ≤ m, + { 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, + { apply mod_4_eq_3_or_mod_4_eq_3, rw [neq, h] }, + cases this with h1 h1, + { by_cases mp : m.prime, + { use [m, mp, mdvdn, h1] }, + rcases ih m mltn h1 mp with ⟨p, pp, pdvd, p4eq⟩, + use [p, pp, pdvd.trans mdvdn, p4eq] }, + obtain ⟨nmdvdn, nmltn⟩ := aux mdvdn mge2 mltn, + by_cases nmp : (n / m).prime, + { use [n / m, nmp, nmdvdn, h1] }, + rcases ih (n / m) nmltn h1 nmp with ⟨p, pp, pdvd, p4eq⟩, + use [p, pp, pdvd.trans nmdvdn, p4eq] + end + + theorem primes_mod_4_eq_3_infinite : ∀ n, ∃ p > n, nat.prime p ∧ p % 4 = 3 := + begin + by_contradiction h, + push_neg at h, + cases h with n hn, + have : ∃ s : finset nat, ∀ p : ℕ, p.prime ∧ p % 4 = 3 ↔ p ∈ s, + { apply ex_finset_of_bounded, + use n, + contrapose! hn, + rcases hn with ⟨p, ⟨pp, p4⟩, pltn⟩, + exact ⟨p, pltn, pp, p4⟩ }, + cases this with s hs, + have h₀ : 2 ≤ 4 * (∏ i in erase s 3, i) + 3, + { apply le_add_left, norm_num }, + have h₁ : (4 * (∏ i in erase s 3, i) + 3) % 4 = 3, + { rw [add_comm, nat.add_mul_mod_self_left], norm_num }, + rcases exists_prime_factor_mod_4_eq_3 h₁ with ⟨p, pp, pdvd, p4eq⟩, + have ps : p ∈ s, + { rw ←hs p, exact ⟨pp, p4eq⟩ }, + have pne3 : p ≠ 3, + { intro peq, + rw [peq, ←nat.dvd_add_iff_left (dvd_refl 3)] at pdvd, + rw nat.prime_three.dvd_mul at pdvd, + norm_num at pdvd, + have : 3 ∈ s.erase 3, + { apply mem_of_dvd_prod_primes nat.prime_three _ pdvd, + intro n, simp [← hs n], tauto }, + simp at this, + exact this }, + have : p ∣ 4 * (∏ i in erase s 3, i), + { apply dvd_trans _ (dvd_mul_left _ _), + apply dvd_prod_of_mem, + simp, split; assumption }, + have : p ∣ 3, + { convert nat.dvd_sub' pdvd this, simp }, + have : p = 3, + { apply pp.eq_of_dvd_of_prime nat.prime_three this }, + contradiction + end +
src/06_Abstract_Algebra/01_Structures.lean (new)
- @@ -0,0 +1,228 @@
 + import algebra.big_operators.ring + import data.real.basic + + @[ext] structure point := (x : ℝ) (y : ℝ) (z : ℝ) + + #check point.ext + + example (a b : point) (hx : a.x = b.x) (hy : a.y = b.y) (hz : a.z = b.z) : + a = b := + begin + ext, + repeat { assumption } + end + + def my_point1 : point := + { x := 2, + y := -1, + z := 4 } + + def my_point2 := + { point . + x := 2, + y := -1, + z := 4 } + + def my_point3 : point := ⟨2, -1, 4⟩ + + def my_point4 := point.mk 2 (-1) 4 + + structure point' := build :: (x : ℝ) (y : ℝ) (z : ℝ) + + #check point'.build 2 (-1) 4 + + namespace point + + def add (a b : point) : point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ + + def add' (a b : point) : point := + { x := a.x + b.x, + y := a.y + b.y, + z := a.z + b.z } + + #check add my_point1 my_point2 + #check my_point1.add my_point2 + + end point + + #check point.add my_point1 my_point2 + #check my_point1.add my_point2 + + namespace point + + protected theorem add_comm (a b : point) : add a b = add b a := + begin + rw [add, add], + ext; dsimp, + repeat { apply add_comm } + end + + example (a b : point) : add a b = add b a := + by simp [add, add_comm] + + theorem add_x (a b : point) : (a.add b).x = a.x + b.x := rfl + + def add_alt : point → point → point + | (point.mk x₁ y₁ z₁) (point.mk x₂ y₂ z₂) := ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ + + def add_alt' : point → point → point + | ⟨x₁, y₁, z₁⟩ ⟨x₂, y₂, z₂⟩ := ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ + + theorem add_alt_x (a b : point) : (a.add_alt b).x = a.x + b.x := + by { cases a, cases b, refl } + + theorem add_alt_comm (a b : point) : add_alt a b = add_alt b a := + begin + rcases a with ⟨xa, ya, za⟩, + rcases b with ⟨xb, yb, zb⟩, + rw [add_alt, add_alt], + ext; dsimp, + apply add_comm, + repeat { apply add_comm }, + end + + example (a b : point) : add_alt a b = add_alt b a := + begin + rcases a with ⟨xa, ya, za⟩, + rcases b with ⟨xb, yb, zb⟩, + simp [add_alt, add_comm] + end + + example : ∀ a b : point, add_alt a b = add_alt b a := + begin + rintros ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩, + simp [add_alt, add_comm] + end + + example : ∀ a b : point, add a b = add b a := + λ ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩, by simp [add, add_comm] + + protected theorem add_assoc (a b c : point) : + (a.add b).add c = a.add (b.add c) := + 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) := + sorry + + end point + + structure standard_two_simplex := + (x : ℝ) + (y : ℝ) + (z : ℝ) + (x_nonneg : 0 ≤ x) + (y_nonneg : 0 ≤ y) + (z_nonneg : 0 ≤ z) + (sum_eq : x + y + z = 1) + + namespace standard_two_simplex + + def swap_xy (a : standard_two_simplex) : standard_two_simplex := + { x := a.y, + y := a.x, + z := a.z, + x_nonneg := a.y_nonneg, + y_nonneg := a.x_nonneg, + z_nonneg := a.z_nonneg, + sum_eq := by rw [add_comm a.y a.x, a.sum_eq] } + + noncomputable theory + + def midpoint (a b : standard_two_simplex) : standard_two_simplex := + { x := (a.x + b.x) / 2, + y := (a.y + b.y) / 2, + z := (a.z + b.z) / 2, + x_nonneg := div_nonneg (add_nonneg a.x_nonneg b.x_nonneg) (by norm_num), + y_nonneg := div_nonneg (add_nonneg a.y_nonneg b.y_nonneg) (by norm_num), + z_nonneg := div_nonneg (add_nonneg a.z_nonneg b.z_nonneg) (by norm_num), + sum_eq := by { field_simp, linarith [a.sum_eq, b.sum_eq]} } + + def weighted_average (lambda : real) + (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) + (a b : standard_two_simplex) : + standard_two_simplex := + sorry + + end standard_two_simplex + + open_locale big_operators + + structure standard_simplex (n : ℕ) := + (v : fin n → ℝ) + (nonneg : ∀ i : fin n, 0 ≤ v i) + (sum_eq_one : ∑ i, v i = 1) + + namespace standard_simplex + + def midpoint (n : ℕ) (a b : standard_simplex n) : standard_simplex n := + { v := λ i, (a.v i + b.v i) / 2, + nonneg := + begin + intro i, + apply div_nonneg, + { linarith [a.nonneg i, b.nonneg i] }, + norm_num + end, + sum_eq_one := + begin + simp [div_eq_mul_inv, ←finset.sum_mul, finset.sum_add_distrib, + a.sum_eq_one, b.sum_eq_one], + field_simp + end } + + end standard_simplex + + structure is_linear (f : ℝ → ℝ) := + (is_additive : ∀ x y, f (x + y) = f x + f y) + (preserves_mul : ∀ x c, f (c * x) = c * f x) + + section + variables (f : ℝ → ℝ) (linf : is_linear f) + + #check linf.is_additive + #check linf.preserves_mul + + end + + def point'' := ℝ × ℝ × ℝ + + def is_linear' (f : ℝ → ℝ) := + (∀ x y, f (x + y) = f x + f y) ∧ (∀ x c, f (c * x) = c * f x) + + def preal := { y : ℝ // 0 < y } + + section + variable x : preal + + #check x.val + #check x.property + + #check x.1 + #check x.2 + + end + + def standard_two_simplex' := + { 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 standard_simplex' (n : ℕ) := + { v : fin n → ℝ // (∀ i : fin n, 0 ≤ v i) ∧ (∑ i, v i = 1) } + + def std_simplex := Σ n : ℕ, standard_simplex n + + section + variable s : std_simplex + + #check s.fst + #check s.snd + + #check s.1 + #check s.2 + + end + +
src/06_Abstract_Algebra/02_Algebraic_Structures.lean (new)
- @@ -0,0 +1,154 @@
  + import data.real.basic + + structure group₁ (α : Type*) := + (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 x one = x) + (mul_left_inv : ∀ x : α, mul (inv x) x = one) + + structure Group₁ := + (α : Type*) + (str : group₁ α) + + section + variables (α β γ : Type*) + variables (f : α ≃ β) (g : β ≃ γ) + + #check equiv α β + #check (f.to_fun : α → β) + #check (f.inv_fun : β → α) + #check (f.right_inv: ∀ x : β, f (f.inv_fun x) = x) + #check (f.left_inv: ∀ x : α, f.inv_fun (f x) = x) + + #check (equiv.refl α : α ≃ α) + #check (f.symm : β ≃ α) + #check (f.trans g : α ≃ γ) + + example (x : α) : (f.trans g).to_fun x = g.to_fun (f.to_fun x) := rfl + + example (x : α) : (f.trans g) x = g (f x) := rfl + + example : (f.trans g : α → γ) = g ∘ f := rfl + + end + + example (α : Type*) : equiv.perm α = (α ≃ α) := rfl + + def perm_group {α : Type*} : group₁ (equiv.perm α) := + { mul := λ f g, equiv.trans g f, + one := equiv.refl α, + inv := equiv.symm, + mul_assoc := λ f g h, (equiv.trans_assoc _ _ _).symm, + one_mul := equiv.trans_refl, + mul_one := equiv.refl_trans, + mul_left_inv := equiv.self_trans_symm } + + structure add_group₁ (α : Type*) := + (add : α → α → α) + -- fill in the rest + + @[ext] structure point := (x : ℝ) (y : ℝ) (z : ℝ) + + namespace point + + def add (a b : point) : point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ + + def neg (a b : point) : point := sorry + + def zero : point := sorry + + def add_group_point : add_group point := sorry + + end point + + section + variables {α : Type*} (f g : equiv.perm α) (n : ℕ) + + #check f * g + #check mul_assoc f g g⁻¹ + + -- group power, defined for any group + #check g^n + + example : f * g * (g⁻¹) = f := + by { rw [mul_assoc, mul_right_inv, mul_one] } + + example : f * g * (g⁻¹) = f := mul_inv_cancel_right f g + + example {α : Type*} (f g : equiv.perm α) : g.symm.trans (g.trans f) = f := + mul_inv_cancel_right f g + + end + + class group₂ (α : Type*) := + (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 x one = x) + (mul_left_inv : ∀ x : α, mul (inv x) x = one) + + instance {α : Type*} : group₂ (equiv.perm α) := + { mul := λ f g, equiv.trans g f, + one := equiv.refl α, + inv := equiv.symm, + mul_assoc := λ f g h, (equiv.trans_assoc _ _ _).symm, + one_mul := equiv.trans_refl, + mul_one := equiv.refl_trans, + mul_left_inv := equiv.self_trans_symm } + + #check @group₂.mul + + def my_square {α : Type*} [group₂ α] (x : α) := group₂.mul x x + + #check @my_square + + section + variables {β : Type*} (f g : equiv.perm β) + + example : group₂.mul f g = g.trans f := rfl + + example : my_square f = f.trans f := rfl + + end + + instance : inhabited point := { default := ⟨0, 0, 0⟩ } + + #check (default : point) + + example : ([] : list point).head = default := rfl + + instance : has_add point := { add := point.add } + + section + variables x y : point + + #check x + y + + example : x + y = point.add x y := rfl + + end + + instance has_mul_group₂ {α : Type*} [group₂ α] : has_mul α := ⟨group₂.mul⟩ + + instance has_one_group₂ {α : Type*} [group₂ α] : has_one α := ⟨group₂.one⟩ + + instance has_inv_group₂ {α : Type*} [group₂ α] : has_inv α := ⟨group₂.inv⟩ + + section + variables {α : Type*} (f g : equiv.perm α) + + #check f * 1 * g⁻¹ + + def foo: f * 1 * g⁻¹ = g.symm.trans ((equiv.refl α).trans f) := rfl + + end + + class add_group₂ (α : Type*) := + (add : α → α → α) + -- fill in the rest +
src/06_Abstract_Algebra/solutions/solutions_01_Structures.lean (new)
- @@ -0,0 +1,101 @@
  + import algebra.big_operators.ring + import data.real.basic + + @[ext] structure point := (x : ℝ) (y : ℝ) (z : ℝ) + + namespace point + + def add (a b : point) : point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ + + protected theorem add_assoc (a b c : point) : + (a.add b).add c = a.add (b.add c) := + by { simp [add, add_assoc] } + + def smul (r : ℝ) (a : point) : point := + ⟨r * a.x, r * a.y, r * a.z⟩ + + theorem smul_distrib (r : ℝ) (a b : point) : + (smul r a).add (smul r b) = smul r (a.add b) := + by { simp [add, smul, mul_add] } + + end point + + structure standard_two_simplex := + (x : ℝ) + (y : ℝ) + (z : ℝ) + (x_nonneg : 0 ≤ x) + (y_nonneg : 0 ≤ y) + (z_nonneg : 0 ≤ z) + (sum_eq : x + y + z = 1) + + namespace standard_two_simplex + + noncomputable theory + + def weighted_average (lambda : real) + (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) + (a b : standard_two_simplex) : + standard_two_simplex := + { x := lambda * a.x + (1 - lambda) * b.x, + y := lambda * a.y + (1 - lambda) * b.y, + z := lambda * a.z + (1 - lambda) * b.z, + x_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.x_nonneg) + (mul_nonneg (by linarith) b.x_nonneg), + y_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.y_nonneg) + (mul_nonneg (by linarith) b.y_nonneg), + z_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.z_nonneg) + (mul_nonneg (by linarith) b.z_nonneg), + sum_eq := + begin + transitivity (a.x + a.y + a.z) * lambda + (b.x + b.y + b.z) * (1 - lambda), + { ring }, + simp [a.sum_eq, b.sum_eq] + end } + + end standard_two_simplex + + open_locale big_operators + + structure standard_simplex (n : ℕ) := + (v : fin n → ℝ) + (nonneg : ∀ i : fin n, 0 ≤ v i) + (sum_eq_one : ∑ i, v i = 1) + + namespace standard_simplex + + def midpoint (n : ℕ) (a b : standard_simplex n) : standard_simplex n := + { v := λ i, (a.v i + b.v i) / 2, + nonneg := + begin + intro i, + apply div_nonneg, + { linarith [a.nonneg i, b.nonneg i] }, + norm_num + end, + sum_eq_one := + begin + simp [div_eq_mul_inv, ←finset.sum_mul, finset.sum_add_distrib, + a.sum_eq_one, b.sum_eq_one], + field_simp + end } + + end standard_simplex + + namespace standard_simplex + + def weighted_average {n : ℕ} (lambda : real) + (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) + (a b : standard_simplex n) : standard_simplex n := + { v := λ i, lambda * a.v i + (1 - lambda) * b.v i, + nonneg := λ i, add_nonneg (mul_nonneg lambda_nonneg (a.nonneg i)) + (mul_nonneg (by linarith) (b.nonneg i)), + sum_eq_one := + begin + transitivity lambda * (∑ i, a.v i) + (1 - lambda) * (∑ i, b.v i), + { rw [finset.sum_add_distrib, finset.mul_sum, finset.mul_sum] }, + simp [a.sum_eq_one, b.sum_eq_one] + end } + + end standard_simplex +
src/06_Abstract_Algebra/solutions/solutions_02_Algebraic_Structures.lean (new)
- @@ -0,0 +1,65 @@
  + import data.real.basic + + structure add_group₁ (α : Type*) := + (add: α → α → α) + (zero: α) + (neg: α → α) + (add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z)) + (add_zero: ∀ x : α, add x zero = x) + (zero_add: ∀ x : α, add x zero = x) + (add_left_neg : ∀ x : α, add (neg x) x = zero) + + @[ext] structure point := (x : ℝ) (y : ℝ) (z : ℝ) + + namespace point + + def add (a b : point) : point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ + + def neg (a : point) : point := ⟨-a.x, -a.y, -a.z⟩ + + def zero : point := ⟨0, 0, 0⟩ + + def add_group_point : add_group₁ point := + { add := point.add, + zero := point.zero, + neg := point.neg, + add_assoc := by { simp [point.add, add_assoc] }, + add_zero := by { simp [point.add, point.zero], intro, ext; refl }, + zero_add := by { simp [point.add, point.zero], intro, ext; refl }, + add_left_neg := by { simp [point.add, point.neg, point.zero] } } + + end point + + class add_group₂ (α : Type*) := + (add: α → α → α) + (zero: α) + (neg: α → α) + (add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z)) + (add_zero: ∀ x : α, add x zero = x) + (zero_add: ∀ x : α, add x zero = x) + (add_left_neg : ∀ x : α, add (neg x) x = zero) + + instance has_add_add_group₂ {α : Type*} [add_group₂ α] : + has_add α := ⟨add_group₂.add⟩ + + instance has_zero_add_group₂ {α : Type*} [add_group₂ α] : + has_zero α := ⟨add_group₂.zero⟩ + + instance has_neg_add_group₂ {α : Type*} [add_group₂ α] : + has_neg α := ⟨add_group₂.neg⟩ + + instance : add_group₂ point := + { add := point.add, + zero := point.zero, + neg := point.neg, + add_assoc := by { simp [point.add, add_assoc] }, + add_zero := by { simp [point.add, point.zero], intro, ext; refl }, + zero_add := by { simp [point.add, point.zero], intro, ext; refl }, + add_left_neg := by { simp [point.add, point.neg, point.zero] } } + + section + variables (x y : point) + + #check x + -y + 0 + end +
src/07_Topology/01_Filters.lean (new)
- @@ -0,0 +1,106 @@
 + import topology.instances.real + + open set filter + open_locale topological_space filter + + def principal {α : Type*} (s : set α) : filter α := + { sets := {t | s ⊆ t}, + univ_sets := sorry, + sets_of_superset := sorry, + inter_sets := sorry} + + example : filter ℕ := + { sets := {s | ∃ a, ∀ b, a ≤ b → b ∈ s}, + univ_sets := sorry, + sets_of_superset := sorry, + inter_sets := sorry } + + def tendsto₁ {X Y : Type*} (f : X → Y) (F : filter X) (G : filter Y) := + ∀ V ∈ G, f ⁻¹' V ∈ F + + def tendsto₂ {X Y : Type*} (f : X → Y) (F : filter X) (G : filter Y) := + map f F ≤ G + + example {X Y : Type*} (f : X → Y) (F : filter X) (G : filter Y) : + tendsto₂ f F G ↔ tendsto₁ f F G := iff.rfl + + #check (@filter.map_mono : ∀ {α β} {m : α → β}, monotone (map m)) + #check (@filter.map_map : ∀ {α β γ} {f : filter α} {m : α → β} {m' : β → γ}, + map m' (map m f) = map (m' ∘ m) f) + + example {X Y Z : Type*} {F : filter X} {G : filter Y} {H : filter Z} {f : X → Y} {g : Y → Z} + (hf : tendsto₁ f F G) (hg : tendsto₁ g G H) : tendsto₁ (g ∘ f) F H := + sorry + + variables (f : ℝ → ℝ) (x₀ y₀ : ℝ) + + #check comap (coe : ℚ → ℝ) (𝓝 x₀) + #check tendsto (f ∘ coe) (comap (coe : ℚ → ℝ) (𝓝 x₀)) (𝓝 y₀) + + section + variables {α β γ : Type*} (F : filter α) {m : γ → β} {n : β → α} + + #check (comap_comap : comap m (comap n F) = comap (n ∘ m) F) + end + + example : 𝓝 (x₀, y₀) = 𝓝 x₀ ×ᶠ 𝓝 y₀ := nhds_prod_eq + + #check le_inf_iff + + example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : + tendsto f at_top (𝓝 (x₀, y₀)) ↔ + tendsto (prod.fst ∘ f) at_top (𝓝 x₀) ∧ tendsto (prod.snd ∘ f) at_top (𝓝 y₀) := + sorry + + example (x₀ : ℝ) : has_basis (𝓝 x₀) (λ ε : ℝ, 0 < ε) (λ ε, Ioo (x₀ - ε) (x₀ + ε)) := + nhds_basis_Ioo_pos x₀ + + example (u : ℕ → ℝ) (x₀ : ℝ) : + tendsto u at_top (𝓝 x₀) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, u n ∈ Ioo (x₀ - ε) (x₀ + ε) := + begin + have : at_top.has_basis (λ n : ℕ, true) Ici := at_top_basis, + rw this.tendsto_iff (nhds_basis_Ioo_pos x₀), + simp + end + + example (P Q : ℕ → Prop) (hP : ∀ᶠ n in at_top, P n) (hQ : ∀ᶠ n in at_top, Q n) : + ∀ᶠ n in at_top, P n ∧ Q n := hP.and hQ + + example (u v : ℕ → ℝ) (h : ∀ᶠ n in at_top, u n = v n) (x₀ : ℝ) : + tendsto u at_top (𝓝 x₀) ↔ tendsto v at_top (𝓝 x₀) := + tendsto_congr' h + + example (u v : ℕ → ℝ) (h : u =ᶠ[at_top] v) (x₀ : ℝ) : + tendsto u at_top (𝓝 x₀) ↔ tendsto v at_top (𝓝 x₀) := + tendsto_congr' h + + #check @eventually_of_forall + #check @eventually.mono + #check @eventually.and + + example (P Q R : ℕ → Prop) (hP : ∀ᶠ n in at_top, P n) (hQ : ∀ᶠ n in at_top, Q n) + (hR : ∀ᶠ n in at_top, P n ∧ Q n → R n) : + ∀ᶠ n in at_top, R n := + begin + apply (hP.and (hQ.and hR)).mono, + rintros n ⟨h, h', h''⟩, + exact h'' ⟨h, h'⟩ + end + + example (P Q R : ℕ → Prop) (hP : ∀ᶠ n in at_top, P n) (hQ : ∀ᶠ n in at_top, Q n) + (hR : ∀ᶠ n in at_top, P n ∧ Q n → R n) : + ∀ᶠ n in at_top, R n := + begin + filter_upwards [hP, hQ, hR], + intros n h h' h'', + exact h'' ⟨h, h'⟩ + end + + #check mem_closure_iff_cluster_pt + #check le_principal_iff + #check ne_bot_of_le + + example (u : ℕ → ℝ) (M : set ℝ) (x : ℝ) + (hux : tendsto u at_top (𝓝 x)) (huM : ∀ᶠ n in at_top, u n ∈ M) : x ∈ closure M := + sorry +
src/07_Topology/02_Metric_Spaces.lean (new)
- @@ -0,0 +1,282 @@
 + import topology.instances.real + import analysis.normed_space.banach_steinhaus + + open set filter + open_locale topological_space filter + + + variables {X : Type*} [metric_space X] (a b c : X) + + #check (dist a b : ℝ) + + #check (dist_nonneg : 0 ≤ dist a b) + + #check (dist_eq_zero : dist a b = 0 ↔ a = b) + + #check (dist_comm a b : dist a b = dist b a) + + #check (dist_triangle a b c : dist a c ≤ dist a b + dist b c) + + + -- Note the next three lines are not quoted, their purpose is to make sure those things don't get renamed while we're looking elsewhere. + #check emetric_space + #check pseudo_metric_space + #check pseudo_emetric_space + + + + example {u : ℕ → X} {a : X} : + tendsto u at_top (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := + metric.tendsto_at_top + + example {X Y : Type*} [metric_space X] [metric_space 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*} [metric_space X] [metric_space Y] {f : X → Y} (hf : continuous f) : + continuous (λ p : X × X, dist (f p.1) (f p.2)) := + by continuity + + + example {X Y : Type*} [metric_space X] [metric_space Y] {f : X → Y} (hf : continuous f) : + continuous (λ p : X × X, dist (f p.1) (f p.2)) := + continuous_dist.comp ((hf.comp continuous_fst).prod_mk (hf.comp continuous_snd)) + + + + + example {X Y : Type*} [metric_space X] [metric_space Y] {f : X → Y} (hf : continuous f) : + continuous (λ p : X × X, dist (f p.1) (f p.2)) := + begin + apply continuous.dist, + exact hf.comp continuous_fst, + exact hf.comp continuous_snd + end + + example {X Y : Type*} [metric_space X] [metric_space Y] {f : X → Y} (hf : continuous f) : + continuous (λ p : X × X, dist (f p.1) (f p.2)) := + (hf.comp continuous_fst).dist (hf.comp continuous_snd) + + + + example {X Y : Type*} [metric_space X] [metric_space Y] {f : X → Y} (hf : continuous f) : + continuous (λ p : X × X, dist (f p.1) (f p.2)) := + hf.fst'.dist hf.snd' + + + + example {f : ℝ → X} (hf : continuous f) : continuous (λ x : ℝ, f (x^2 + x)) := + sorry + + + + + example {X Y : Type*} [metric_space X] [metric_space Y] (f : X → Y) (a : X) : + continuous_at f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x}, dist x a < δ → dist (f x) (f a) < ε := + metric.continuous_at_iff + + + + variables r : ℝ + + example : metric.ball a r = {b | dist b a < r} := rfl + + example : metric.closed_ball a r = {b | dist b a ≤ r} := rfl + + + + example (hr : 0 < r) : a ∈ metric.ball a r := metric.mem_ball_self hr + + example (hr : 0 ≤ r) : a ∈ metric.closed_ball a r := metric.mem_closed_ball_self hr + + + example (s : set X) : is_open s ↔ ∀ x ∈ s, ∃ ε > 0, metric.ball x ε ⊆ s := + metric.is_open_iff + + + + example {s : set X} : is_closed s ↔ is_open sᶜ := + is_open_compl_iff.symm + + example {s : set X} (hs : is_closed s) {u : ℕ → X} (hu : tendsto u at_top (𝓝 a)) + (hus : ∀ n, u n ∈ s) : a ∈ s := + hs.mem_of_tendsto hu (eventually_of_forall hus) + + example {s : set X} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, a ∈ metric.ball b ε := + metric.mem_closure_iff + + + example {u : ℕ → X} (hu : tendsto u at_top (𝓝 a)) {s : set X} (hs : ∀ n, u n ∈ s) : + a ∈ closure s := + sorry + + + + example {x : X} {s : set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, metric.ball x ε ⊆ s := + metric.nhds_basis_ball.mem_iff + + example {x : X} {s : set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, metric.closed_ball x ε ⊆ s := + metric.nhds_basis_closed_ball.mem_iff + + + + + example : is_compact (set.Icc 0 1 : set ℝ) := + is_compact_Icc + + example {s : set X} (hs : is_compact s) {u : ℕ → X} (hu : ∀ n, u n ∈ s) : + ∃ a ∈ s, ∃ φ : ℕ → ℕ, strict_mono φ ∧ tendsto (u ∘ φ) at_top (𝓝 a) := + hs.tendsto_subseq hu + + example {s : set X} (hs : is_compact s) (hs' : s.nonempty) + {f : X → ℝ} (hfs : continuous_on f s) : + ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y := + hs.exists_forall_le hs' hfs + + example {s : set X} (hs : is_compact s) (hs' : s.nonempty) + {f : X → ℝ} (hfs : continuous_on f s) : + ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x := + hs.exists_forall_ge hs' hfs + + example {s : set X} (hs : is_compact s) : is_closed s := + hs.is_closed + + + + example {X : Type*} [metric_space X] [compact_space X] : is_compact (univ : set X) := + compact_univ + + + #check is_compact.is_closed + + + example {X : Type*} [metric_space X] {Y : Type*} [metric_space Y] {f : X → Y} : + uniform_continuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : X}, dist a b < δ → dist (f a) (f b) < ε := + metric.uniform_continuous_iff + + + + example {X : Type*} [metric_space X] [compact_space X] {Y : Type*} [metric_space Y] + {f : X → Y} (hf : continuous f) : uniform_continuous f := + sorry + + + -- SOLUTIONs: + example {X : Type*} [metric_space X] [compact_space X] {Y : Type*} [metric_space Y] + {f : X → Y} (hf : continuous f) : uniform_continuous f := + begin + rw metric.uniform_continuous_iff, + intros ε ε_pos, + let φ : X × X → ℝ := λ p, dist (f p.1) (f p.2), + have φ_cont : continuous φ := hf.fst'.dist hf.snd', + let K := { p : X × X | ε ≤ φ p }, + have K_closed : is_closed K := is_closed_le continuous_const φ_cont, + have K_cpct : is_compact K := K_closed.is_compact, + cases eq_empty_or_nonempty K with hK hK, + { use [1, by norm_num], + intros x y hxy, + have : (x, y) ∉ K, by simp [hK], + simpa [K] }, + { rcases K_cpct.exists_forall_le hK continuous_dist.continuous_on with ⟨⟨x₀, x₁⟩, xx_in, H⟩, + use dist x₀ x₁, + split, + { change _ < _, + rw dist_pos, + intro h, + have : ε ≤ 0, by simpa [*] using xx_in, + linarith }, + { intros x x', + contrapose!, + intros hxx', + exact H (x, x') hxx' } }, + end + + + + example (u : ℕ → X) : cauchy_seq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε := + metric.cauchy_seq_iff + + example (u : ℕ → X) : cauchy_seq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, dist (u n) (u N) < ε := + metric.cauchy_seq_iff' + + + example [complete_space X] (u : ℕ → X) (hu : cauchy_seq u) : ∃ x, tendsto u at_top (𝓝 x) := + cauchy_seq_tendsto_of_complete hu + + + + open_locale big_operators + open finset + + + lemma cauchy_seq_of_le_geometric_two' {u : ℕ → X} (hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : + cauchy_seq u := + begin + rw metric.cauchy_seq_iff', + intros ε ε_pos, + obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε, + { sorry }, + use N, + intros n hn, + obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn, + calc dist (u (N + k)) (u N) = dist (u (N+0)) (u (N + k)) : sorry + ... ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) : sorry + ... ≤ ∑ i in range k, (1/2 : ℝ)^(N+i) : sorry + ... = 1/2^N*∑ i in range k, (1 / 2) ^ i : sorry + ... ≤ 1/2^N*2 : sorry + ... < ε : sorry + end + + + open metric + + example [complete_space X] (f : ℕ → set X) (ho : ∀ n, is_open (f n)) (hd : ∀ n, dense (f n)) : dense (⋂n, f n) := + begin + let B : ℕ → ℝ := λ 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 + `closed_ball center radius` is included both in `f n` and in `closed_ball 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) ∧ closed_ball y r ⊆ (closed_ball x δ) ∩ f n, + { sorry }, + choose! center radius Hpos HB Hball using this, + intros x, + rw mem_closure_iff_nhds_basis nhds_basis_closed_ball, + intros 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 + `closed_ball (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) := λn, nat.rec_on n (prod.mk x (min ε (B 0))) + (λn p, prod.mk (center n p.1 p.2) (radius n p.1 p.2)), + let c : ℕ → X := λ n, (F n).1, + let r : ℕ → ℝ := λ n, (F n).2, + have rpos : ∀ n, 0 < r n, + { sorry }, + + have rB : ∀n, r n ≤ B n, + { sorry }, + have incl : ∀n, closed_ball (c (n+1)) (r (n+1)) ⊆ (closed_ball (c n) (r n)) ∩ (f n), + { sorry }, + have cdist : ∀ n, dist (c n) (c (n+1)) ≤ B n, + { sorry }, + have : cauchy_seq c, from cauchy_seq_of_le_geometric_two' cdist, + -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. + rcases cauchy_seq_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, closed_ball (c m) (r m) ⊆ closed_ball (c n) (r n), + { sorry }, + have yball : ∀n, y ∈ closed_ball (c n) (r n), + { sorry }, + sorry + end + + +
src/07_Topology/solutions/solutions_01_Filters.lean (new)
- @@ -0,0 +1,75 @@
  + import topology.instances.real + + open set filter + open_locale topological_space filter + + -- In the next example we could use `tauto` in each proof instead of knowing the lemmas + example {α : Type*} (s : set α) : filter α := + { sets := {t | s ⊆ t}, + univ_sets := subset_univ s, + sets_of_superset := λ U V hU hUV, subset.trans hU hUV, + inter_sets := λ U V hU hV, subset_inter hU hV } + + example : filter ℕ := + { sets := {s | ∃ a, ∀ b, a ≤ b → b ∈ s}, + univ_sets := begin + use 42, + finish, + end, + sets_of_superset := begin + rintros U V ⟨N, hN⟩ hUV, + use N, + tauto, + end, + inter_sets := begin + rintros U V ⟨N, hN⟩ ⟨N', hN'⟩, + use max N N', + intros b hb, + rw max_le_iff at hb, + split ; tauto, + end } + + def tendsto₁ {X Y : Type*} (f : X → Y) (F : filter X) (G : filter Y) := + ∀ V ∈ G, f ⁻¹' V ∈ F + + example {X Y Z : Type*} {F : filter X} {G : filter Y} {H : filter Z} {f : X → Y} {g : Y → Z} + (hf : tendsto₁ f F G) (hg : tendsto₁ g G H) : tendsto₁ (g ∘ f) F H := + calc map (g ∘ f) F = map g (map f F) : by rw map_map + ... ≤ map g G : map_mono hf + ... ≤ H : hg + + example {X Y Z : Type*} {F : filter X} {G : filter Y} {H : filter Z} {f : X → Y} {g : Y → Z} + (hf : tendsto₁ f F G) (hg : tendsto₁ g G H) : tendsto₁ (g ∘ f) F H := + begin + intros V hV, + rw preimage_comp, + apply hf, + apply hg, + exact hV + end + + example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : + tendsto f at_top (𝓝 (x₀, y₀)) ↔ + tendsto (prod.fst ∘ f) at_top (𝓝 x₀) ∧ tendsto (prod.snd ∘ f) at_top (𝓝 y₀) := + calc + tendsto f at_top (𝓝 (x₀, y₀)) ↔ map f at_top ≤ 𝓝 (x₀, y₀) : iff.rfl + ... ↔ map f at_top ≤ 𝓝 x₀ ×ᶠ 𝓝 y₀ : by rw nhds_prod_eq + ... ↔ map f at_top ≤ (comap prod.fst (𝓝 x₀) ⊓ comap prod.snd (𝓝 y₀)) : iff.rfl + ... ↔ map f at_top ≤ comap prod.fst (𝓝 x₀) ∧ map f at_top ≤ (comap prod.snd (𝓝 y₀)) : le_inf_iff + ... ↔ map prod.fst (map f at_top) ≤ 𝓝 x₀ ∧ map prod.snd (map f at_top) ≤ 𝓝 y₀ : + by rw [← map_le_iff_le_comap, ← map_le_iff_le_comap] + ... ↔ map (prod.fst ∘ f) at_top ≤ 𝓝 x₀ ∧ map (prod.snd ∘ f) at_top ≤ 𝓝 y₀ : by rw [map_map, map_map] + + -- an alternative solution + example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : + tendsto f at_top (𝓝 (x₀, y₀)) ↔ + tendsto (prod.fst ∘ f) at_top (𝓝 x₀) ∧ tendsto (prod.snd ∘ f) at_top (𝓝 y₀) := + begin + rw nhds_prod_eq, + unfold tendsto filter.prod, + rw [le_inf_iff, ← map_le_iff_le_comap, map_map, ← map_le_iff_le_comap, map_map] + end + + example (u : ℕ → ℝ) (M : set ℝ) (x : ℝ) + (hux : tendsto u at_top (𝓝 x)) (huM : ∀ᶠ n in at_top, u n ∈ M) : x ∈ closure M := + mem_closure_iff_cluster_pt.mpr (ne_bot_of_le $ le_inf hux $ le_principal_iff.mpr huM)
src/07_Topology/solutions/solutions_02_Metric_Spaces.lean (new)
- @@ -0,0 +1,135 @@
 + import topology.instances.real + import analysis.normed_space.banach_steinhaus + + open set filter + open_locale topological_space filter + + example {f : ℝ → X} (hf : continuous f) : continuous (λ x : ℝ, f (x^2 + x)) := + hf.comp $ (continuous_pow 2).add continuous_id + + + example {u : ℕ → X} (hu : tendsto u at_top (𝓝 a)) {s : set X} (hs : ∀ n, u n ∈ s) : + a ∈ closure s := + begin + rw metric.tendsto_at_top at hu, + rw metric.mem_closure_iff, + intros ε ε_pos, + rcases hu ε ε_pos with ⟨N, hN⟩, + refine ⟨u N, hs _, _⟩, + rw dist_comm, + exact hN N le_rfl + end + + + example {u : ℕ → X} (hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : + cauchy_seq u := + begin + rw metric.cauchy_seq_iff', + intros ε ε_pos, + obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε, + { have : tendsto (λ N : ℕ, (1 / 2 ^ N * 2 : ℝ)) at_top (𝓝 0), + { rw ← zero_mul (2 : ℝ), + apply tendsto.mul, + simp_rw ← one_div_pow (2 : ℝ), + apply tendsto_pow_at_top_nhds_0_of_lt_1 ; linarith, + exact tendsto_const_nhds }, + rcases (at_top_basis.tendsto_iff (nhds_basis_Ioo_pos (0 : ℝ))).mp this ε ε_pos with ⟨N, H, hN⟩, + exact ⟨N, by simpa using (hN N le_rfl).2⟩ }, + use N, + intros n hn, + obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn, + calc dist (u (N + k)) (u N) = dist (u (N+0)) (u (N + k)) : by rw [dist_comm, add_zero] + ... ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) : dist_le_range_sum_dist (λ i, u (N+i)) k + ... ≤ ∑ i in range k, (1/2 : ℝ) ^ (N+i) : sum_le_sum (λ i hi, hu $ N+i) + ... = 1/2^N*∑ i in range k, (1 / 2)^i : by simp_rw [← one_div_pow, pow_add, ← mul_sum] + ... ≤ 1/2^N*2 : mul_le_mul_of_nonneg_left (sum_geometric_two_le _) (one_div_nonneg.mpr (pow_nonneg (zero_le_two : (0 : ℝ) ≤ 2)_)) + ... < ε : hN + end + + example [complete_space X] (f : ℕ → set X) (ho : ∀ n, is_open (f n)) (hd : ∀ n, dense (f n)) : dense (⋂n, f n) := + begin + let B : ℕ → ℝ := λ n, (1/2)^n, + have Bpos : ∀ n, 0 < B n, from λ n, (pow_pos sorry n), + /- Translate the density assumption into two functions `center` and `radius` associating + to any n, x, δ, δpos a center and a positive radius such that + `closed_ball center radius` is included both in `f n` and in `closed_ball 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) ∧ closed_ball y r ⊆ (closed_ball x δ) ∩ f n, + { intros n x δ δpos, + have : x ∈ closure (f n) := hd n x, + rcases metric.mem_closure_iff.1 this (δ/2) (half_pos δpos) with ⟨y, ys, xy⟩, + rw dist_comm at xy, + obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closed_ball y r ⊆ f n := + nhds_basis_closed_ball.mem_iff.1 (is_open_iff_mem_nhds.1 (ho n) y ys), + refine ⟨y, min (min (δ/2) r) (B (n+1)), _, _, λz hz, ⟨_, _⟩⟩, + show 0 < min (min (δ / 2) r) (B (n+1)), + from lt_min (lt_min (half_pos δpos) rpos) (Bpos (n+1)), + show min (min (δ / 2) r) (B (n+1)) ≤ B (n+1), from min_le_right _ _, + show z ∈ closed_ball x δ, from calc + dist z x ≤ dist z y + dist y x : dist_triangle _ _ _ + ... ≤ (min (min (δ / 2) r) (B (n+1))) + (δ/2) : add_le_add hz xy.le + ... ≤ δ/2 + δ/2 : add_le_add_right ((min_le_left _ _).trans (min_le_left _ _)) _ + ... = δ : add_halves δ, + show z ∈ f n, from hr (calc + dist z y ≤ min (min (δ / 2) r) (B (n+1)) : hz + ... ≤ r : (min_le_left _ _).trans (min_le_right _ _)) }, + choose! center radius Hpos HB Hball using this, + refine λ x, (mem_closure_iff_nhds_basis nhds_basis_closed_ball).2 (λ 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 + `closed_ball (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) := λn, nat.rec_on n (prod.mk x (min ε (B 0))) + (λn p, prod.mk (center n p.1 p.2) (radius n p.1 p.2)), + let c : ℕ → X := λ n, (F n).1, + let r : ℕ → ℝ := λ n, (F n).2, + have rpos : ∀ n, 0 < r n, + { assume n, + induction n with n hn, + exact lt_min εpos (Bpos 0), + exact Hpos n (c n) (r n) hn }, + + have rB : ∀n, r n ≤ B n, + { assume n, + induction n with n hn, + exact min_le_right _ _, + exact HB n (c n) (r n) (rpos n) }, + have incl : ∀n, closed_ball (c (n+1)) (r (n+1)) ⊆ (closed_ball (c n) (r n)) ∩ (f n) := + λ n, Hball n (c n) (r n) (rpos n), + have cdist : ∀ n, dist (c n) (c (n+1)) ≤ B n, + { assume n, + rw dist_comm, + have A : c (n+1) ∈ closed_ball (c (n+1)) (r (n+1)) := mem_closed_ball_self (rpos $ n +1).le, + have I := calc + closed_ball (c (n+1)) (r (n+1)) ⊆ closed_ball (c n) (r n) : + (incl n).trans (inter_subset_left _ _) + ... ⊆ closed_ball (c n) (B n) : closed_ball_subset_closed_ball (rB n), + exact I A }, + have : cauchy_seq c, + from cauchy_seq_of_le_geometric_two' cdist, + -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. + rcases cauchy_seq_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, closed_ball (c m) (r m) ⊆ closed_ball (c n) (r n), + { assume n, + refine nat.le_induction _ (λm hnm h, _), + { exact subset.rfl }, + { exact (incl m).trans ((set.inter_subset_left _ _).trans h) }}, + have yball : ∀n, y ∈ closed_ball (c n) (r n), + { assume n, + refine is_closed_ball.mem_of_tendsto ylim _, + refine (filter.eventually_ge_at_top n).mono (λ m hm, _), + exact I n m hm (mem_closed_ball_self (rpos _).le) }, + split, + { suffices : ∀ n, y ∈ f n, by rwa set.mem_Inter, + intro n, + have : closed_ball (c (n+1)) (r (n+1)) ⊆ f n := subset.trans (incl n) (inter_subset_right _ _), + exact this (yball (n+1)) }, + calc dist y x ≤ r 0 : yball 0 + ... ≤ ε : min_le_left _ _, + end +
src/C02_Basics/S01_Calculating.lean
- @@ -33,7 +33,7 @@ example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : a * (b * e) = c *
  rw [h] rw [mul_assoc] - -- Try these. For the second one, use the theorem `sub_self`. + example (a b c d e f : ℝ) (h : b * c = e * f) : a * b * c * d = a * e * f * d := by sorry
- @@ -145,4 +145,3 @@ end
  example (a b c : ℕ) (h : a + b = c) : (a + b) * (a + b) = a * c + b * c := by nth_rw 2 [h] rw [add_mul] -
src/C02_Basics/S04_More_on_Order_and_Divisibility.lean
- @@ -1,5 +1,6 @@
  import Mathlib.Data.Real.Basic + namespace C02S04 section variable (a b c d : ℝ)
src/C02_Basics/solutions/Solutions_S04_More_on_Order_and_Divisibility.lean
- @@ -1,5 +1,6 @@
  import Mathlib.Data.Real.Basic + namespace C02S04 section variable (a b c d : ℝ)
src/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean
- @@ -1,5 +1,6 @@
 import Mathlib.Data.Real.Basic + namespace C03S01 #check ∀ x : ℝ, 0 ≤ x → abs x = x #check ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε
src/C03_Logic/S02_The_Existential_Quantifier.lean
- @@ -1,5 +1,6 @@
 import Mathlib.Data.Real.Basic + namespace C03S02 example : ∃ x : ℝ, 2 < x ∧ x < 3 := by use 5 / 2 norm_num
src/C03_Logic/S03_Negation.lean
- @@ -1,5 +1,6 @@
  import Mathlib.Data.Real.Basic + namespace C03S03 section variable (a b : ℝ)
src/C03_Logic/S04_Conjunction_and_Bi-implication.lean
- @@ -2,6 +2,7 @@ import Mathlib.Data.Real.Basic
  import Mathlib.Data.Nat.Prime import Mathlib.Tactic.NormNum + namespace C03S04 example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y := by constructor · assumption
src/C03_Logic/S05_Disjunction.lean
- @@ -1,5 +1,6 @@
  import Mathlib.Data.Real.Basic + namespace C03S05 section variable {x y : ℝ}
src/C03_Logic/S06_Sequences_and_Convergence.lean
- @@ -1,5 +1,6 @@
 import Mathlib.Data.Real.Basic + namespace C03S06 def ConvergesTo (s : ℕ → ℝ) (a : ℝ) := ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε
src/C03_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean
- @@ -1,5 +1,6 @@
  import Mathlib.Data.Real.Basic + namespace C03S01 def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a
src/C03_Logic/solutions/Solutions_S02_The_Existential_Quantifier.lean
- @@ -1,5 +1,6 @@
  import Mathlib.Data.Real.Basic + namespace C03S02 def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a
src/C03_Logic/solutions/Solutions_S03_Negation.lean
- @@ -1,5 +1,6 @@
  import Mathlib.Data.Real.Basic + namespace C03S03 section variable (a b : ℝ)
src/C03_Logic/solutions/Solutions_S04_Conjunction_and_Bi-implication.lean
- @@ -2,6 +2,7 @@ import Mathlib.Data.Real.Basic
  import Mathlib.Data.Nat.Prime import Mathlib.Tactic.NormNum + namespace C03S04 example {m n : ℕ} (h : m ∣ n ∧ m ≠ n) : m ∣ n ∧ ¬n ∣ m := by cases' h with h0 h1 constructor
src/C03_Logic/solutions/Solutions_S05_Disjunction.lean
- @@ -1,5 +1,6 @@
  import Mathlib.Data.Real.Basic + namespace C03S05 section variable {x y : ℝ}
src/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean
- @@ -1,5 +1,6 @@
 import Mathlib.Data.Real.Basic + namespace C03S06 def ConvergesTo (s : ℕ → ℝ) (a : ℝ) := ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε
src/C05_Number_Theory/S03_Infinitely_Many_Primes.lean
- @@ -5,6 +5,7 @@ import Mathlib.Tactic.IntervalCases
  open BigOperators + namespace C05S03 theorem two_le {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by cases m; contradiction case succ m =>
- @@ -100,7 +101,7 @@ end
  example (s : Finset ℕ) (n : ℕ) (h : n ∈ s) : n ∣ ∏ i in s, i := Finset.dvd_prod_of_mem _ h - theorem Nat.Prime.eq_of_dvd_of_prime {p q : ℕ} + 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
src/C05_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean
- @@ -5,6 +5,7 @@ import Mathlib.Tactic.IntervalCases
  open BigOperators + namespace C05S03 theorem two_le {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by cases m; contradiction case succ m =>
- @@ -80,7 +81,7 @@ example : (r \ s) \ t = r \ (s ∪ t) := by
  end - theorem Nat.Prime.eq_of_dvd_of_prime {p q : ℕ} + theorem _root_.Nat.Prime.eq_of_dvd_of_prime {p q : ℕ} (prime_p : Nat.Prime p) (prime_q : Nat.Prime q) (h : p ∣ q) : p = q := by cases prime_q.eq_one_or_self_of_dvd _ h
src/C06_Abstract_Algebra/S01_Structures.lean
- @@ -1,6 +1,7 @@
  import Mathlib.Algebra.BigOperators.Ring import Mathlib.Data.Real.Basic + namespace C06S01 noncomputable section @[ext]
src/C06_Abstract_Algebra/S02_Algebraic_Structures.lean
- @@ -1,5 +1,6 @@
  import Mathlib.Data.Real.Basic + namespace C06S02 structure Group₁ (α : Type _) where mul : α → α → α one : α
src/C06_Abstract_Algebra/solutions/Solutions_S01_Structures.lean
- @@ -1,6 +1,7 @@
  import Mathlib.Algebra.BigOperators.Ring import Mathlib.Data.Real.Basic + namespace C06S01 noncomputable section @[ext]
src/C06_Abstract_Algebra/solutions/Solutions_S02_Algebraic_Structures.lean
- @@ -1,5 +1,6 @@
  import Mathlib.Data.Real.Basic + namespace C06S02 structure AddGroup₁ (α : Type _) where add : α → α → α zero : α
src/C06_Structures/S01_Structures.lean (new)
- @@ -0,0 +1,224 @@
 + import Mathlib.Algebra.BigOperators.Ring + import Mathlib.Data.Real.Basic + + namespace C06S01 + noncomputable section + + @[ext] + structure Point where + x : ℝ + y : ℝ + z : ℝ + + #check Point.ext + + example (a b : Point) (hx : a.x = b.x) (hy : a.y = b.y) (hz : a.z = b.z) : a = b := by + ext + repeat' assumption + + def myPoint1 : Point where + x := 2 + y := -1 + z := 4 + + def myPoint2 : Point := + ⟨2, -1, 4⟩ + + def myPoint3 := + Point.mk 2 (-1) 4 + + structure Point' where build :: + x : ℝ + y : ℝ + z : ℝ + + #check Point'.build 2 (-1) 4 + + namespace Point + + def add (a b : Point) : Point := + ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ + + def add' (a b : Point) : Point where + x := a.x + b.x + y := a.y + b.y + z := a.z + b.z + + #check add myPoint1 myPoint2 + #check myPoint1.add myPoint2 + + end Point + + #check Point.add myPoint1 myPoint2 + #check myPoint1.add myPoint2 + + namespace Point + + protected theorem add_comm (a b : Point) : add a b = add b a := by + rw [add, add] + ext <;> dsimp + repeat' apply add_comm + + example (a b : Point) : add a b = add b a := by simp [add, add_comm] + + theorem add_x (a b : Point) : (a.add b).x = a.x + b.x := + rfl + + def addAlt : Point → Point → Point + | Point.mk x₁ y₁ z₁, Point.mk x₂ y₂ z₂ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ + + def addAlt' : Point → Point → Point + | ⟨x₁, y₁, z₁⟩, ⟨x₂, y₂, z₂⟩ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ + + theorem addAlt_x (a b : Point) : (a.addAlt b).x = a.x + b.x := by + cases a + cases b + rfl + + theorem addAlt_comm (a b : Point) : addAlt a b = addAlt b a := by + rcases a with ⟨xa, ya, za⟩ + rcases b with ⟨xb, yb, zb⟩ + rw [addAlt, addAlt] + ext <;> dsimp + apply add_comm + repeat' apply add_comm + + example (a b : Point) : addAlt a b = addAlt b a := by + rcases a with ⟨xa, ya, za⟩ + rcases b with ⟨xb, yb, zb⟩ + simp [addAlt, add_comm] + + example : ∀ a b : Point, addAlt a b = addAlt b a := by + rintro ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ + simp [addAlt, add_comm] + + example : ∀ a b : Point, add a b = add b a := fun ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ => by + simp [add, add_comm] + + protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by + sorry + + def smul (r : ℝ) (a : Point) : Point := + sorry + + theorem smul_distrib (r : ℝ) (a b : Point) : + (smul r a).add (smul r b) = smul r (a.add b) := by + sorry + + end Point + + structure StandardTwoSimplex where + x : ℝ + y : ℝ + z : ℝ + x_nonneg : 0 ≤ x + y_nonneg : 0 ≤ y + z_nonneg : 0 ≤ z + sum_eq : x + y + z = 1 + + namespace StandardTwoSimplex + + def swapXy (a : StandardTwoSimplex) : StandardTwoSimplex + where + x := a.y + y := a.x + z := a.z + x_nonneg := a.y_nonneg + y_nonneg := a.x_nonneg + z_nonneg := a.z_nonneg + sum_eq := by rw [add_comm a.y a.x, a.sum_eq] + + noncomputable section + + def midpoint (a b : StandardTwoSimplex) : StandardTwoSimplex + where + x := (a.x + b.x) / 2 + y := (a.y + b.y) / 2 + z := (a.z + b.z) / 2 + x_nonneg := div_nonneg (add_nonneg a.x_nonneg b.x_nonneg) (by norm_num) + y_nonneg := div_nonneg (add_nonneg a.y_nonneg b.y_nonneg) (by norm_num) + z_nonneg := div_nonneg (add_nonneg a.z_nonneg b.z_nonneg) (by norm_num) + sum_eq := by field_simp; linarith [a.sum_eq, b.sum_eq] + + def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) + (a b : StandardTwoSimplex) : StandardTwoSimplex := + sorry + + end + + end StandardTwoSimplex + + open BigOperators + + structure StandardSimplex (n : ℕ) where + V : Fin n → ℝ + NonNeg : ∀ i : Fin n, 0 ≤ V i + sum_eq_one : (∑ i, V i) = 1 + + namespace StandardSimplex + + def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n + where + V i := (a.V i + b.V i) / 2 + NonNeg := by + intro i + apply div_nonneg + · linarith [a.NonNeg i, b.NonNeg i] + norm_num + sum_eq_one := by + simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, + a.sum_eq_one, b.sum_eq_one] + field_simp + + end StandardSimplex + + structure IsLinear (f : ℝ → ℝ) where + is_additive : ∀ x y, f (x + y) = f x + f y + preserves_mul : ∀ x c, f (c * x) = c * f x + + section + variable (f : ℝ → ℝ) (linf : IsLinear f) + + #check linf.is_additive + #check linf.preserves_mul + + end + + def Point'' := + ℝ × ℝ × ℝ + + def IsLinear' (f : ℝ → ℝ) := + (∀ x y, f (x + y) = f x + f y) ∧ ∀ x c, f (c * x) = c * f x + + def PReal := + { y : ℝ // 0 < y } + + section + variable (x : PReal) + + #check x.val + #check x.property + #check x.1 + #check x.2 + + end + + def StandardTwoSimplex' := + { p : ℝ × ℝ × ℝ // 0 ≤ p.1 ∧ 0 ≤ p.2.1 ∧ 0 ≤ p.2.2 ∧ p.1 + p.2.1 + p.2.2 = 1 } + + def StandardSimplex' (n : ℕ) := + { v : Fin n → ℝ // (∀ i : Fin n, 0 ≤ v i) ∧ (∑ i, v i) = 1 } + + def StdSimplex := Σ n : ℕ, StandardSimplex n + + section + variable (s : StdSimplex) + + #check s.fst + #check s.snd + + #check s.1 + #check s.2 + + end +
src/C06_Structures/S02_Algebraic_Structures.lean (new)
- @@ -0,0 +1,171 @@
  + import Mathlib.Data.Real.Basic + + namespace C06S02 + structure Group₁ (α : Type _) where + mul : α → α → α + one : α + inv : α → α + mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) + mul_one : ∀ x : α, mul x one = x + one_mul : ∀ x : α, mul one x = x + mul_left_inv : ∀ x : α, mul (inv x) x = one + + structure Group₁Cat where + α : Type _ + str : Group₁ α + + section + variable (α β γ : Type _) + variable (f : α ≃ β) (g : β ≃ γ) + + #check Equiv α β + #check (f.toFun : α → β) + #check (f.invFun : β → α) + #check (f.right_inv : ∀ x : β, f (f.invFun x) = x) + #check (f.left_inv : ∀ x : α, f.invFun (f x) = x) + #check (Equiv.refl α : α ≃ α) + #check (f.symm : β ≃ α) + #check (f.trans g : α ≃ γ) + + example (x : α) : (f.trans g).toFun x = g.toFun (f.toFun x) := + rfl + + example (x : α) : (f.trans g) x = g (f x) := + rfl + + example : (f.trans g : α → γ) = g ∘ f := + rfl + + end + + example (α : Type _) : Equiv.Perm α = (α ≃ α) := + rfl + + def permGroup {α : Type _} : Group₁ (Equiv.Perm α) + where + mul f g := Equiv.trans g f + one := Equiv.refl α + inv := Equiv.symm + mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm + one_mul := Equiv.trans_refl + mul_one := Equiv.refl_trans + mul_left_inv := Equiv.self_trans_symm + + structure AddGroup₁ (α : Type _) where + (add : α → α → α) + -- fill in the rest + @[ext] + structure Point where + x : ℝ + y : ℝ + z : ℝ + + namespace Point + + def add (a b : Point) : Point := + ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ + + def neg (a : Point) : Point := sorry + + def zero : Point := sorry + + def add_group_point : AddGroup₁ point := sorry + + end Point + + section + variable {α : Type _} (f g : Equiv.Perm α) (n : ℕ) + + #check f * g + #check mul_assoc f g g⁻¹ + + -- group power, defined for any group + #check g ^ n + + example : f * g * g⁻¹ = f := by rw [mul_assoc, mul_right_inv, mul_one] + + example : f * g * g⁻¹ = f := + mul_inv_cancel_right f g + + example {α : Type _} (f g : Equiv.Perm α) : g.symm.trans (g.trans f) = f := + mul_inv_cancel_right f g + + end + + class Group₂ (α : Type _) where + mul : α → α → α + one : α + inv : α → α + mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) + mul_one : ∀ x : α, mul x one = x + one_mul : ∀ x : α, mul one x = x + mul_left_inv : ∀ x : α, mul (inv x) x = one + + instance {α : Type _} : Group₂ (Equiv.Perm α) where + mul f g := Equiv.trans g f + one := Equiv.refl α + inv := Equiv.symm + mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm + one_mul := Equiv.trans_refl + mul_one := Equiv.refl_trans + mul_left_inv := Equiv.self_trans_symm + + #check @Group₂.mul + + def mySquare {α : Type _} [Group₂ α] (x : α) := + Group₂.mul x x + + #check @mySquare + + section + variable {β : Type _} (f g : Equiv.Perm β) + + example : Group₂.mul f g = g.trans f := + rfl + + example : mySquare f = f.trans f := + rfl + + end + + instance : Inhabited Point where default := ⟨0, 0, 0⟩ + + #check (default : Point) + + example : ([] : List Point).headI = default := + rfl + + instance : Add Point where add := Point.add + + section + variable (x y : Point) + + #check x + y + + example : x + y = Point.add x y := + rfl + + end + + instance hasMulGroup₂ {α : Type _} [Group₂ α] : Mul α := + ⟨Group₂.mul⟩ + + instance hasOneGroup₂ {α : Type _} [Group₂ α] : One α := + ⟨Group₂.one⟩ + + instance hasInvGroup₂ {α : Type _} [Group₂ α] : Inv α := + ⟨Group₂.inv⟩ + + section + variable {α : Type _} (f g : Equiv.Perm α) + + #check f * 1 * g⁻¹ + + def foo : f * 1 * g⁻¹ = g.symm.trans ((Equiv.refl α).trans f) := + rfl + + end + + class AddGroup₂ (α : Type _) where + add : α → α → α + -- fill in the rest
src/C06_Structures/S03_Building_the_Gaussian_Integers.lean (new)
- @@ -0,0 +1,276 @@
 + import Mathlib.Data.Int.Basic + import Mathlib.Algebra.EuclideanDomain.Basic + import Mathlib.RingTheory.PrincipalIdealDomain + import Mathlib.Tactic + + @[ext] + structure gaussInt where + re : ℤ + im : ℤ + + namespace gaussInt + + instance : Zero gaussInt := + ⟨⟨0, 0⟩⟩ + + instance : One gaussInt := + ⟨⟨1, 0⟩⟩ + + instance : Add gaussInt := + ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ + + instance : Neg gaussInt := + ⟨fun x => ⟨-x.re, -x.im⟩⟩ + + instance : Mul gaussInt := + ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ + + theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := + rfl + + theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := + rfl + + theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := + rfl + + theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := + rfl + + theorem mul_def (x y : gaussInt) : + x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := + rfl + + @[simp] + theorem zero_re : (0 : gaussInt).re = 0 := + rfl + + @[simp] + theorem zero_im : (0 : gaussInt).im = 0 := + rfl + + @[simp] + theorem one_re : (1 : gaussInt).re = 1 := + rfl + + @[simp] + theorem one_im : (1 : gaussInt).im = 0 := + rfl + + @[simp] + theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := + rfl + + @[simp] + theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := + rfl + + @[simp] + theorem neg_re (x : gaussInt) : (-x).re = -x.re := + rfl + + @[simp] + theorem neg_im (x : gaussInt) : (-x).im = -x.im := + rfl + + @[simp] + theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := + rfl + + @[simp] + theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := + rfl + + instance instCommRing : CommRing gaussInt where + zero := 0 + one := 1 + add := (· + ·) + neg x := -x + mul := (· * ·) + add_assoc := by + intros + ext <;> simp <;> ring + zero_add := by + intro + ext <;> simp + add_zero := by + intro + ext <;> simp + add_left_neg := by + intro + ext <;> simp + add_comm := by + intros + ext <;> simp <;> ring + mul_assoc := by + intros + ext <;> simp <;> ring + one_mul := by + intro + ext <;> simp + mul_one := by + intro + ext <;> simp + left_distrib := by + intros + ext <;> simp <;> ring + right_distrib := by + intros + ext <;> simp <;> ring + mul_comm := by + intros + ext <;> simp <;> ring + zero_mul := sorry + mul_zero := sorry + + instance : Nontrivial gaussInt := by + use 0, 1 + rw [Ne, gaussInt.ext_iff] + simp + + end gaussInt + + example (a b : ℤ) : a = b * (a / b) + a % b := + Eq.symm <| Int.ediv_add_emod a b + + example (a b : ℤ) : b ≠ 0 → 0 ≤ a % b := + Int.emod_nonneg a + + example (a b : ℤ) : b ≠ 0 → a % b < abs b := + Int.emod_lt a + + namespace Int + + def div' (a b : ℤ) := + (a + b / 2) / b + + def mod' (a b : ℤ) := + (a + b / 2) % b - b / 2 + + theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by + rw [div', mod'] + linarith [Int.ediv_add_emod (a + b / 2) b] + + theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by + rw [mod', abs_le] + constructor + · linarith [Int.emod_nonneg (a + b / 2) h.ne'] + have := Int.emod_lt_of_pos (a + b / 2) h + have := Int.ediv_add_emod b 2 + have := Int.emod_lt_of_pos b zero_lt_two + revert this; intro this -- FIXME, this should not be needed + linarith + + theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] + + end Int + + theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : + x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by + sorry + namespace gaussInt + + def norm (x : gaussInt) := + x.re ^ 2 + x.im ^ 2 + + @[simp] + theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by + sorry + theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by + sorry + theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by + sorry + theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by + sorry + def conj (x : gaussInt) : gaussInt := + ⟨x.re, -x.im⟩ + + @[simp] + theorem conj_re (x : gaussInt) : (conj x).re = x.re := + rfl + + @[simp] + theorem conj_im (x : gaussInt) : (conj x).im = -x.im := + rfl + + theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] + + instance : Div gaussInt := + ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ + + instance : Mod gaussInt := + ⟨fun x y => x - y * (x / y)⟩ + + theorem div_def (x y : gaussInt) : + x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := + rfl + + theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := + rfl + + theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : + (x % y).norm < y.norm := by + have norm_y_pos : 0 < norm y := by rwa [norm_pos] + have : x % y * conj y = + ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by + rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] + ext <;> simp <;> ring + have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by + conv => + lhs + rw [← norm_conj y, ← norm_mul, this, norm] + simp + trans 2 * (y.norm / 2) ^ 2 + · rw [two_mul] + apply add_le_add <;> + · rw [sq_le_sq] + apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) + apply le_abs_self + rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] + apply mul_le_mul_of_nonneg_left + · apply Int.ediv_mul_le + norm_num + apply Int.ediv_nonneg (norm_nonneg y) + norm_num + have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos + apply lt_of_le_of_lt this + apply Int.ediv_lt_of_lt_mul + · norm_num + linarith + + theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := + Int.natAbs_of_nonneg (norm_nonneg _) + + theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : + (x % y).norm.natAbs < y.norm.natAbs := by + apply Int.ofNat_lt.1 + simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] + apply norm_mod_lt x hy + + theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : + ¬(norm (x * y)).natAbs < (norm x).natAbs := by + apply not_lt_of_ge + rw [norm_mul, Int.natAbs_mul] + apply le_mul_of_one_le_right (Nat.zero_le _) + apply Int.ofNat_le.1 + rw [coe_natAbs_norm] + exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) + + instance : EuclideanDomain gaussInt := + { gaussInt.instCommRing with + quotient := (· / ·) + remainder := (· % ·) + quotient_mul_add_remainder_eq := + fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] + quotient_zero := fun x => by + simp [div_def, norm, Int.div'] + rfl + r := Measure (Int.natAbs ∘ norm) + r_wellFounded := (measure (Int.natAbs ∘ norm)).2 + remainder_lt := natAbs_norm_mod_lt + mul_left_not_lt := not_norm_mul_left_lt_norm } + + example (x : gaussInt) : Irreducible x ↔ Prime x := + PrincipalIdealRing.irreducible_iff_prime + + end gaussInt
src/C06_Structures/solutions/Solutions_S01_Structures.lean (new)
- @@ -0,0 +1,99 @@
  + import Mathlib.Algebra.BigOperators.Ring + import Mathlib.Data.Real.Basic + + namespace C06S01 + noncomputable section + + @[ext] + structure Point where + x : ℝ + y : ℝ + z : ℝ + + namespace Point + + def add (a b : Point) : Point := + ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ + + protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by + simp [add, add_assoc] + + def smul (r : ℝ) (a : Point) : Point := + ⟨r * a.x, r * a.y, r * a.z⟩ + + theorem smul_distrib (r : ℝ) (a b : Point) : + (smul r a).add (smul r b) = smul r (a.add b) := by + simp [add, smul, mul_add] + + end Point + + structure StandardTwoSimplex where + x : ℝ + y : ℝ + z : ℝ + x_nonneg : 0 ≤ x + y_nonneg : 0 ≤ y + z_nonneg : 0 ≤ z + sum_eq : x + y + z = 1 + + namespace StandardTwoSimplex + + noncomputable section + + def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) + (a b : StandardTwoSimplex) : StandardTwoSimplex + where + x := lambda * a.x + (1 - lambda) * b.x + y := lambda * a.y + (1 - lambda) * b.y + z := lambda * a.z + (1 - lambda) * b.z + x_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.x_nonneg) (mul_nonneg (by linarith) b.x_nonneg) + y_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.y_nonneg) (mul_nonneg (by linarith) b.y_nonneg) + z_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.z_nonneg) (mul_nonneg (by linarith) b.z_nonneg) + sum_eq := by + trans (a.x + a.y + a.z) * lambda + (b.x + b.y + b.z) * (1 - lambda) + · ring + simp [a.sum_eq, b.sum_eq] + + end + + end StandardTwoSimplex + + open BigOperators + + structure StandardSimplex (n : ℕ) where + V : Fin n → ℝ + NonNeg : ∀ i : Fin n, 0 ≤ V i + sum_eq_one : (∑ i, V i) = 1 + + namespace StandardSimplex + + def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n + where + V i := (a.V i + b.V i) / 2 + NonNeg := by + intro i + apply div_nonneg + · linarith [a.NonNeg i, b.NonNeg i] + norm_num + sum_eq_one := by + simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, + a.sum_eq_one, b.sum_eq_one] + field_simp + + end StandardSimplex + + namespace StandardSimplex + + def weightedAverage {n : ℕ} (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) + (a b : StandardSimplex n) : StandardSimplex n + where + V i := lambda * a.V i + (1 - lambda) * b.V i + NonNeg i := + add_nonneg (mul_nonneg lambda_nonneg (a.NonNeg i)) (mul_nonneg (by linarith) (b.NonNeg i)) + sum_eq_one := by + trans (lambda * ∑ i, a.V i) + (1 - lambda) * ∑ i, b.V i + · rw [Finset.sum_add_distrib, Finset.mul_sum, Finset.mul_sum] + simp [a.sum_eq_one, b.sum_eq_one] + + end StandardSimplex +
src/C06_Structures/solutions/Solutions_S02_Algebraic_Structures.lean (new)
- @@ -0,0 +1,74 @@
  + import Mathlib.Data.Real.Basic + + namespace C06S02 + structure AddGroup₁ (α : Type _) where + add : α → α → α + zero : α + neg : α → α + add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z) + add_zero : ∀ x : α, add x zero = x + zero_add : ∀ x : α, add x zero = x + add_left_neg : ∀ x : α, add (neg x) x = zero + + @[ext] + structure Point where + x : ℝ + y : ℝ + z : ℝ + + namespace Point + + def add (a b : Point) : Point := + ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ + + def neg (a : Point) : Point := + ⟨-a.x, -a.y, -a.z⟩ + + def zero : Point := + ⟨0, 0, 0⟩ + + def addGroupPoint : AddGroup₁ Point where + add := Point.add + zero := Point.zero + neg := Point.neg + add_assoc := by simp [Point.add, add_assoc] + add_zero := by simp [Point.add, Point.zero] + zero_add := by simp [Point.add, Point.zero] + add_left_neg := by simp [Point.add, Point.neg, Point.zero] + + end Point + + class AddGroup₂ (α : Type _) where + add : α → α → α + zero : α + neg : α → α + add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z) + add_zero : ∀ x : α, add x zero = x + zero_add : ∀ x : α, add x zero = x + add_left_neg : ∀ x : α, add (neg x) x = zero + + instance hasAddAddGroup₂ {α : Type _} [AddGroup₂ α] : Add α := + ⟨AddGroup₂.add⟩ + + instance hasZeroAddGroup₂ {α : Type _} [AddGroup₂ α] : Zero α := + ⟨AddGroup₂.zero⟩ + + instance hasNegAddGroup₂ {α : Type _} [AddGroup₂ α] : Neg α := + ⟨AddGroup₂.neg⟩ + + instance : AddGroup₂ Point where + add := Point.add + zero := Point.zero + neg := Point.neg + add_assoc := by simp [Point.add, add_assoc] + add_zero := by simp [Point.add, Point.zero] + zero_add := by simp [Point.add, Point.zero] + add_left_neg := by simp [Point.add, Point.neg, Point.zero] + + section + variable (x y : Point) + + #check x + -y + 0 + + end +
src/C06_Structures/solutions/Solutions_S03_Building_the_Gaussian_Integers.lean (new)
- @@ -0,0 +1,290 @@
 + import Mathlib.Data.Int.Basic + import Mathlib.Algebra.EuclideanDomain.Basic + import Mathlib.RingTheory.PrincipalIdealDomain + import Mathlib.Tactic + + @[ext] + structure gaussInt where + re : ℤ + im : ℤ + + namespace gaussInt + + instance : Zero gaussInt := + ⟨⟨0, 0⟩⟩ + + instance : One gaussInt := + ⟨⟨1, 0⟩⟩ + + instance : Add gaussInt := + ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ + + instance : Neg gaussInt := + ⟨fun x => ⟨-x.re, -x.im⟩⟩ + + instance : Mul gaussInt := + ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ + + theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := + rfl + + theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := + rfl + + theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := + rfl + + theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := + rfl + + theorem mul_def (x y : gaussInt) : + x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := + rfl + + @[simp] + theorem zero_re : (0 : gaussInt).re = 0 := + rfl + + @[simp] + theorem zero_im : (0 : gaussInt).im = 0 := + rfl + + @[simp] + theorem one_re : (1 : gaussInt).re = 1 := + rfl + + @[simp] + theorem one_im : (1 : gaussInt).im = 0 := + rfl + + @[simp] + theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := + rfl + + @[simp] + theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := + rfl + + @[simp] + theorem neg_re (x : gaussInt) : (-x).re = -x.re := + rfl + + @[simp] + theorem neg_im (x : gaussInt) : (-x).im = -x.im := + rfl + + @[simp] + theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := + rfl + + @[simp] + theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := + rfl + + instance instCommRing : CommRing gaussInt where + zero := 0 + one := 1 + add := (· + ·) + neg x := -x + mul := (· * ·) + add_assoc := by + intros + ext <;> simp <;> ring + zero_add := by + intro + ext <;> simp + add_zero := by + intro + ext <;> simp + add_left_neg := by + intro + ext <;> simp + add_comm := by + intros + ext <;> simp <;> ring + mul_assoc := by + intros + ext <;> simp <;> ring + one_mul := by + intro + ext <;> simp + mul_one := by + intro + ext <;> simp + left_distrib := by + intros + ext <;> simp <;> ring + right_distrib := by + intros + ext <;> simp <;> ring + mul_comm := by + intros + ext <;> simp <;> ring + zero_mul := sorry + mul_zero := sorry + + instance : Nontrivial gaussInt := by + use 0, 1 + rw [Ne, gaussInt.ext_iff] + simp + + end gaussInt + + namespace Int + + def div' (a b : ℤ) := + (a + b / 2) / b + + def mod' (a b : ℤ) := + (a + b / 2) % b - b / 2 + + theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by + rw [div', mod'] + linarith [Int.ediv_add_emod (a + b / 2) b] + + theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by + rw [mod', abs_le] + constructor + · linarith [Int.emod_nonneg (a + b / 2) h.ne'] + have := Int.emod_lt_of_pos (a + b / 2) h + have := Int.ediv_add_emod b 2 + have := Int.emod_lt_of_pos b zero_lt_two + revert this; intro this -- FIXME, this should not be needed + linarith + + theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] + + end Int + + private theorem aux {α : Type _} [LinearOrderedRing α] {x y : α} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := + haveI h' : x ^ 2 = 0 := by + apply le_antisymm _ (sq_nonneg x) + rw [← h] + apply le_add_of_nonneg_right (sq_nonneg y) + pow_eq_zero h' + + theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : + x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by + constructor + · intro h + constructor + · exact aux h + rw [add_comm] at h + exact aux h + rintro ⟨rfl, rfl⟩ + norm_num + + namespace gaussInt + + def norm (x : gaussInt) := + x.re ^ 2 + x.im ^ 2 + + @[simp] + theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by + apply add_nonneg <;> + apply sq_nonneg + + theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by + rw [norm, sq_add_sq_eq_zero, gaussInt.ext_iff] + rfl + + theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by + rw [lt_iff_le_and_ne, ne_comm, Ne, norm_eq_zero] + simp [norm_nonneg] + + theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by + simp [norm] + ring + + def conj (x : gaussInt) : gaussInt := + ⟨x.re, -x.im⟩ + + @[simp] + theorem conj_re (x : gaussInt) : (conj x).re = x.re := + rfl + + @[simp] + theorem conj_im (x : gaussInt) : (conj x).im = -x.im := + rfl + + theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] + + instance : Div gaussInt := + ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ + + instance : Mod gaussInt := + ⟨fun x y => x - y * (x / y)⟩ + + theorem div_def (x y : gaussInt) : + x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := + rfl + + theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := + rfl + + theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : + (x % y).norm < y.norm := by + have norm_y_pos : 0 < norm y := by rwa [norm_pos] + have : x % y * conj y = + ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by + rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] + ext <;> simp <;> ring + have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by + conv => + lhs + rw [← norm_conj y, ← norm_mul, this, norm] + simp + trans 2 * (y.norm / 2) ^ 2 + · rw [two_mul] + apply add_le_add <;> + · rw [sq_le_sq] + apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) + apply le_abs_self + rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] + apply mul_le_mul_of_nonneg_left + · apply Int.ediv_mul_le + norm_num + apply Int.ediv_nonneg (norm_nonneg y) + norm_num + have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos + apply lt_of_le_of_lt this + apply Int.ediv_lt_of_lt_mul + · norm_num + linarith + + theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := + Int.natAbs_of_nonneg (norm_nonneg _) + + theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : + (x % y).norm.natAbs < y.norm.natAbs := by + apply Int.ofNat_lt.1 + simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] + apply norm_mod_lt x hy + + theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : + ¬(norm (x * y)).natAbs < (norm x).natAbs := by + apply not_lt_of_ge + rw [norm_mul, Int.natAbs_mul] + apply le_mul_of_one_le_right (Nat.zero_le _) + apply Int.ofNat_le.1 + rw [coe_natAbs_norm] + exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) + + instance : EuclideanDomain gaussInt := + { gaussInt.instCommRing with + quotient := (· / ·) + remainder := (· % ·) + quotient_mul_add_remainder_eq := + fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] + quotient_zero := fun x => by + simp [div_def, norm, Int.div'] + rfl + r := Measure (Int.natAbs ∘ norm) + r_wellFounded := (measure (Int.natAbs ∘ norm)).2 + remainder_lt := natAbs_norm_mod_lt + mul_left_not_lt := not_norm_mul_left_lt_norm } + + example (x : gaussInt) : Irreducible x ↔ Prime x := + PrincipalIdealRing.irreducible_iff_prime + + end gaussInt
src/C07_Hierarchies/S01_Basics.lean (new)
- @@ -0,0 +1,308 @@
 + import Mathlib.Algebra.BigOperators.Ring + import Mathlib.Data.Real.Basic + + + class One₁ (α : Type) where + /-- The element one -/ + one : α + + + #check One₁.one -- One₁.one {α : Type} [self : One₁ α] : α + + @[class] structure One₂ (α : Type) where + /-- The element one -/ + one : α + + #check One₂.one + + + example (α : Type) [One₁ α] : α := One₁.one + + example (α : Type) [One₁ α] := (One₁.one : α) + + @[inherit_doc] + notation "𝟙" => One₁.one + + example {α : Type} [One₁ α] : α := 𝟙 + + example {α : Type} [One₁ α] : (𝟙 : α) = 𝟙 := rfl + + + class Dia₁ (α : Type) where + dia : α → α → α + + infixl:70 " ⋄ " => Dia₁.dia + + + class Semigroup₁ (α : Type) where + toDia₁ : Dia₁ α + /-- Diamond is associative -/ + dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) + + + attribute [instance] Semigroup₁.toDia₁ + + example {α : Type} [Semigroup₁ α] (a b : α) : α := a ⋄ b + + + class Semigroup₂ (α : Type) extends Dia₁ α where + /-- Diamond is associative -/ + dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) + + example {α : Type} [Semigroup₂ α] (a b : α) : α := a ⋄ b + + class DiaOneClass₁ (α : Type) extends One₁ α, Dia₁ α where + /-- One is a left neutral element for diamond. -/ + one_dia : ∀ a : α, 𝟙 ⋄ a = a + /-- One is a right neutral element for diamond -/ + dia_one : ∀ a : α, a ⋄ 𝟙 = a + + + + set_option trace.Meta.synthInstance true in + example {α : Type} [DiaOneClass₁ α] (a b : α) : Prop := a ⋄ b = 𝟙 + + + class Monoid₁ (α : Type) extends Semigroup₁ α, DiaOneClass₁ α + + + class Monoid₂ (α : Type) where + toSemigroup₁ : Semigroup₁ α + toDiaOneClass₁ : DiaOneClass₁ α + + + example {α : Type} [Monoid₁ α] : + (Monoid₁.toSemigroup₁.toDia₁.dia : α → α → α) = Monoid₁.toDiaOneClass₁.toDia₁.dia := rfl + + + /- Monoid₂.mk {α : Type} (toSemigroup₁ : Semigroup₁ α) (toDiaOneClass₁ : DiaOneClass₁ α) : Monoid₂ α -/ + #check Monoid₂.mk + + /- Monoid₁.mk {α : Type} [toSemigroup₁ : Semigroup₁ α] [toOne₁ : One₁ α] (one_dia : ∀ (a : α), 𝟙 ⋄ a = a) (dia_one : ∀ (a : α), a ⋄ 𝟙 = a) : Monoid₁ α -/ + #check Monoid₁.mk + + + #check Monoid₁.toSemigroup₁ + #check Monoid₁.toDiaOneClass₁ + + + class Inv₁ (α : Type) where + /-- The inversion function -/ + inv : α → α + + @[inherit_doc] + postfix:max "⁻¹" => Inv₁.inv + + class Group₁ (G : Type) extends Monoid₁ G, Inv G where + inv_dia : ∀ a : G, a⁻¹ ⋄ a = 𝟙 + + + lemma left_inv_eq_right_inv₁ {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by + rw [← DiaOneClass₁.one_dia c, ← hba, Semigroup₁.dia_assoc, hac, DiaOneClass₁.dia_one b] + + + export DiaOneClass₁ (one_dia dia_one) + export Semigroup₁ (dia_assoc) + export Group₁ (inv_dia) + + + example {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by + rw [← one_dia c, ← hba, dia_assoc, hac, dia_one b] + + + lemma inv_eq_of_dia [Group₁ G] {a b : G} (h : a ⋄ b = 𝟙) : a⁻¹ = b := + sorry + + lemma dia_inv [Group₁ G] (a : G) : a ⋄ a⁻¹ = 𝟙 := + sorry + + + + + class AddSemigroup₃ (α : Type) extends Add α where + /-- Multiplication is associative -/ + add_assoc₃ : ∀ a b c : α, a + b + c = a + (b + c) + + @[to_additive AddSemigroup₃] + class Semigroup₃ (α : Type) extends Mul α where + /-- Multiplication is associative -/ + mul_assoc₃ : ∀ a b c : α, a * b * c = a * (b * c) + + class AddMonoid₃ (α : Type) extends AddSemigroup₃ α, AddZeroClass α + + @[to_additive AddMonoid₃] + class Monoid₃ (α : Type) extends Semigroup₃ α, MulOneClass α + + attribute [to_additive existing] Monoid₃.toMulOneClass + + export Semigroup₃ (mul_assoc₃) + export AddSemigroup₃ (add_assoc₃) + + whatsnew in + @[to_additive] + lemma left_inv_eq_right_inv' {M : Type} [Monoid₃ M] {a b c : M} (hba : b * a = 1) (hac : a * c = 1) : b = c := by + rw [← one_mul c, ← hba, mul_assoc₃, hac, mul_one b] + + #check left_neg_eq_right_neg' + + class AddCommSemigroup₃ (α : Type) extends AddSemigroup₃ α where + add_comm : ∀ a b : α, a + b = b + a + + @[to_additive AddCommSemigroup₃] + class CommSemigroup₃ (α : Type) extends Semigroup₃ α where + mul_comm : ∀ a b : α, a * b = b * a + + class AddCommMonoid₃ (α : Type) extends AddMonoid₃ α, AddCommSemigroup₃ α + + @[to_additive AddCommMonoid₃] + class CommMonoid₃ (α : Type) extends Monoid₃ α, CommSemigroup₃ α + + class AddGroup₃ (G : Type) extends AddMonoid₃ G, Neg G where + neg_add : ∀ a : G, -a + a = 0 + + @[to_additive AddGroup₃] + class Group₃ (G : Type) extends Monoid₃ G, Inv G where + inv_mul : ∀ a : G, a⁻¹ * a = 1 + + + attribute [simp] Group₃.inv_mul AddGroup₃.neg_add + + + + @[to_additive] + lemma inv_eq_of_mul [Group₃ G] {a b : G} (h : a * b = 1) : a⁻¹ = b := + sorry + + + @[to_additive (attr := simp)] + lemma Group₃.mul_inv {G : Type} [Group₃ G] {a : G} : a * a⁻¹ = 1 := by + sorry + + @[to_additive] + lemma mul_left_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : a * b = a * c) : b = c := by + sorry + + @[to_additive] + lemma mul_right_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : b*a = c*a) : b = c := by + sorry + + class AddCommGroup₃ (G : Type) extends AddGroup₃ G, AddCommMonoid₃ G + + @[to_additive AddCommGroup₃] + class CommGroup₃ (G : Type) extends Group₃ G, CommMonoid₃ G + + + + class Ring₃ (R : Type) extends AddGroup₃ R, Monoid₃ R, MulZeroClass R where + /-- Multiplication is left distributive over addition -/ + left_distrib : ∀ a b c : R, a * (b + c) = a * b + a * c + /-- Multiplication is right distributive over addition -/ + right_distrib : ∀ a b c : R, (a + b) * c = a * c + b * c + + instance {R : Type} [Ring₃ R] : AddCommGroup₃ R := + { Ring₃.toAddGroup₃ with + add_comm := by + sorry } + + instance : Ring₃ ℤ where + add := (· + ·) + add_assoc₃ := add_assoc + zero := 0 + zero_add := by simp + add_zero := by simp + neg := (- ·) + neg_add := by simp + mul := (· * ·) + mul_assoc₃ := mul_assoc + one := 1 + one_mul := by simp + mul_one := by simp + zero_mul := by simp + mul_zero := by simp + left_distrib := Int.mul_add + right_distrib := Int.add_mul + + class LE₁ (α : Type) where + /-- The Less-or-Equal relation. -/ + le : α → α → Prop + + @[inherit_doc] infix:50 " ≤₁ " => LE₁.le + + class Preorder₁ (α : Type) + + class PartialOrder₁ (α : Type) + + class OrderedCommMonoid₁ (α : Type) + + instance : OrderedCommMonoid₁ ℕ where + + class SMul₃ (α : Type) (β : Type) where + /-- Scalar multiplication -/ + smul : α → β → β + + infixr:73 " • " => SMul₃.smul + + + class Module₁ (R : Type) [Ring₃ R] (M : Type) [AddCommGroup₃ M] extends SMul₃ R M where + zero_smul : ∀ m : M, (0 : R) • m = 0 + one_smul : ∀ m : M, (1 : R) • m = m + mul_smul : ∀ (a b : R) (m : M), (a * b) • m = a • b • m + add_smul : ∀ (a b : R) (m : M), (a + b) • m = a • m + b • m + smul_add : ∀ (a : R) (m n : M), a • (m + n) = a • m + a • n + + instance selfModule (R : Type) [Ring₃ R] : Module₁ R R where + smul := fun r s ↦ r*s + zero_smul := zero_mul + one_smul := one_mul + mul_smul := mul_assoc₃ + add_smul := Ring₃.right_distrib + smul_add := Ring₃.left_distrib + + def nsmul₁ [Zero M] [Add M] : ℕ → M → M + | 0, _ => 0 + | n + 1, a => a + nsmul₁ n a + + def zsmul₁ {M : Type _} [Zero M] [Add M] [Neg M] : ℤ → M → M + | Int.ofNat n, a => nsmul₁ n a + | Int.negSucc n, a => -nsmul₁ n.succ a + + instance abGrpModule (A : Type) [AddCommGroup₃ A] : Module₁ ℤ A where + smul := zsmul₁ + zero_smul := sorry + one_smul := sorry + mul_smul := sorry + add_smul := sorry + smul_add := sorry + + #synth Module₁ ℤ ℤ -- abGrpModule ℤ + + + class AddMonoid₄ (M : Type) extends AddSemigroup₃ M, AddZeroClass M where + /-- Multiplication by a natural number. -/ + nsmul : ℕ → M → M := nsmul₁ + /-- Multiplication by `(0 : ℕ)` gives `0`. -/ + nsmul_zero : ∀ x, nsmul 0 x = 0 := by intros; rfl + /-- Multiplication by `(n + 1 : ℕ)` behaves as expected. -/ + nsmul_succ : ∀ (n : ℕ) (x), nsmul (n + 1) x = x + nsmul n x := by intros; rfl + + instance mySMul {M : Type} [AddMonoid₄ M] : SMul ℕ M := ⟨AddMonoid₄.nsmul⟩ + + instance (M N : Type) [AddMonoid₄ M] [AddMonoid₄ N] : AddMonoid₄ (M × N) where + add := fun p q ↦ (p.1 + q.1, p.2 + q.2) + add_assoc₃ := fun a b c ↦ by ext <;> apply add_assoc₃ + zero := (0, 0) + zero_add := fun a ↦ by ext <;> apply zero_add + add_zero := fun a ↦ by ext <;> apply add_zero + + instance : AddMonoid₄ ℤ where + add := (· + ·) + add_assoc₃ := Int.add_assoc + zero := 0 + zero_add := Int.zero_add + add_zero := Int.add_zero + nsmul := fun n m ↦ (n : ℤ) * m + nsmul_zero := Int.zero_mul + nsmul_succ := fun n m ↦ show (n + 1 : ℤ) * m = m + n * m + by rw [Int.add_mul, Int.add_comm, Int.one_mul] + + example (n : ℕ) (m : ℤ) : SMul.smul (self := mySMul) n m = n * m := rfl
src/C07_Hierarchies/S02_Morphisms.lean (new)
- @@ -0,0 +1,114 @@
  + import Mathlib.Topology.Instances.Real + + + def isMonoidHom₁ [Monoid G] [Monoid H] (f : G → H) : Prop := + f 1 = 1 ∧ ∀ g g', f (g * g') = f g * f g' + structure isMonoidHom₂ [Monoid G] [Monoid H] (f : G → H) : Prop where + map_one : f 1 = 1 + map_mul : ∀ g g', f (g * g') = f g * f g' + example : Continuous (id : ℝ → ℝ) := continuous_id + @[ext] + structure MonoidHom₁ (G H : Type) [Monoid G] [Monoid H] where + toFun : G → H + map_one : toFun 1 = 1 + map_mul : ∀ g g', toFun (g * g') = toFun g * toFun g' + + instance [Monoid G] [Monoid H] : CoeFun (MonoidHom₁ G H) (fun _ ↦ G → H) where + coe := MonoidHom₁.toFun + + attribute [coe] MonoidHom₁.toFun + + + example [Monoid G] [Monoid H] (f : MonoidHom₁ G H) : f 1 = 1 := f.map_one + + @[ext] + structure AddMonoidHom₁ (G H : Type) [AddMonoid G] [AddMonoid H] where + toFun : G → H + map_zero : toFun 0 = 0 + map_add : ∀ g g', toFun (g + g') = toFun g + toFun g' + + instance [AddMonoid G] [AddMonoid H] : CoeFun (AddMonoidHom₁ G H) (fun _ ↦ G → H) where + coe := AddMonoidHom₁.toFun + + attribute [coe] AddMonoidHom₁.toFun + + @[ext] + structure RingHom₁ (R S : Type) [Ring R] [Ring S] extends MonoidHom₁ R S, AddMonoidHom₁ R S + + + + class MonoidHomClass₁ (F : Type) (M N : Type) [Monoid M] [Monoid N] where + toFun : F → M → N + map_one : ∀ f : F, toFun f 1 = 1 + map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' + + + def badInst [Monoid M] [Monoid N] [MonoidHomClass₁ F M N] : CoeFun F (fun _ ↦ M → N) where + coe := MonoidHomClass₁.toFun + + + class MonoidHomClass₂ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] where + toFun : F → M → N + map_one : ∀ f : F, toFun f 1 = 1 + map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' + + instance [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] : CoeFun F (fun _ ↦ M → N) where + coe := MonoidHomClass₂.toFun + + attribute [coe] MonoidHomClass₂.toFun + + + instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₂ (MonoidHom₁ M N) M N where + toFun := MonoidHom₁.toFun + map_one := fun f ↦ f.map_one + map_mul := fun f ↦ f.map_mul + + instance (R S : Type) [Ring R] [Ring S] : MonoidHomClass₂ (RingHom₁ R S) R S where + toFun := fun f ↦ f.toMonoidHom₁.toFun + map_one := fun f ↦ f.toMonoidHom₁.map_one + map_mul := fun f ↦ f.toMonoidHom₁.map_mul + + + lemma map_inv_of_inv [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] (f : F) {m m' : M} (h : m*m' = 1) : + f m * f m' = 1 := by + rw [← MonoidHomClass₂.map_mul, h, MonoidHomClass₂.map_one] + + example [Monoid M] [Monoid N] (f : MonoidHom₁ M N) {m m' : M} (h : m*m' = 1) : f m * f m' = 1 := + map_inv_of_inv f h + + example [Ring R] [Ring S] (f : RingHom₁ R S) {r r' : R} (h : r*r' = 1) : f r * f r' = 1 := + map_inv_of_inv f h + + + + class MonoidHomClass₃ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] extends + FunLike F M (fun _ ↦ N) where + map_one : ∀ f : F, f 1 = 1 + map_mul : ∀ (f : F) g g', f (g * g') = f g * f g' + + instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₃ (MonoidHom₁ M N) M N where + coe := MonoidHom₁.toFun + coe_injective' := MonoidHom₁.ext + map_one := MonoidHom₁.map_one + map_mul := MonoidHom₁.map_mul + + + @[ext] + structure OrderPresHom (α β : Type) [LE α] [LE β] where + toFun : α → β + le_of_le : ∀ a a', a ≤ a' → toFun a ≤ toFun a' + + @[ext] + structure OrderPresMonoidHom (M N : Type) [Monoid M] [LE M] [Monoid N] [LE N] extends + MonoidHom₁ M N, OrderPresHom M N + + class OrderPresHomClass (F : Type) (α β : outParam Type) [LE α] [LE β] + + instance (α β : Type) [LE α] [LE β] : OrderPresHomClass (OrderPresHom α β) α β where + + instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : + OrderPresHomClass (OrderPresMonoidHom α β) α β where + + instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : + MonoidHomClass₃ (OrderPresMonoidHom α β) α β + := sorry
src/C07_Hierarchies/S03_Subobjects.lean (new)
- @@ -0,0 +1,88 @@
  + import Mathlib.GroupTheory.QuotientGroup + + + + @[ext] + structure Submonoid₁ (M : Type) [Monoid M] where + /-- The carrier of a submonoid. -/ + carrier : Set M + /-- The product of two elements of a submonoid belongs to the submonoid. -/ + mul_mem {a b} : a ∈ carrier → b ∈ carrier → a * b ∈ carrier + /-- The unit element belongs to the submonoid. -/ + one_mem : 1 ∈ carrier + + /-- Submonoids in `M` can be seen as sets in `M`. -/ + instance [Monoid M] : SetLike (Submonoid₁ M) M where + coe := Submonoid₁.carrier + coe_injective' := Submonoid₁.ext + + + + example [Monoid M] (N : Submonoid₁ M) : 1 ∈ N := N.one_mem + + example [Monoid M] (N : Submonoid₁ M) (α : Type) (f : M → α) := f '' N + + + example [Monoid M] (N : Submonoid₁ M) (x : N) : (x : M) ∈ N := x.property + + + instance SubMonoid₁Monoid [Monoid M] (N : Submonoid₁ M) : Monoid N where + mul := fun x y ↦ ⟨x*y, N.mul_mem x.property y.property⟩ + mul_assoc := fun x y z ↦ SetCoe.ext (mul_assoc (x : M) y z) + one := ⟨1, N.one_mem⟩ + one_mul := fun x ↦ SetCoe.ext (one_mul (x : M)) + mul_one := fun x ↦ SetCoe.ext (mul_one (x : M)) + + + example [Monoid M] (N : Submonoid₁ M) : Monoid N where + mul := fun ⟨x, hx⟩ ⟨y, hy⟩ ↦ ⟨x*y, N.mul_mem hx hy⟩ + mul_assoc := fun ⟨x, _⟩ ⟨y, _⟩ ⟨z, _⟩ ↦ SetCoe.ext (mul_assoc x y z) + one := ⟨1, N.one_mem⟩ + one_mul := fun ⟨x, _⟩ ↦ SetCoe.ext (one_mul x) + mul_one := fun ⟨x, _⟩ ↦ SetCoe.ext (mul_one x) + + + class SubmonoidClass₁ (S : Type) (M : Type) [Monoid M] [SetLike S M] : Prop where + mul_mem : ∀ (s : S) {a b : M}, a ∈ s → b ∈ s → a * b ∈ s + one_mem : ∀ s : S, 1 ∈ s + + instance [Monoid M] : SubmonoidClass₁ (Submonoid₁ M) M where + mul_mem := Submonoid₁.mul_mem + one_mem := Submonoid₁.one_mem + + + instance [Monoid M] : Inf (Submonoid₁ M) := + ⟨fun S₁ S₂ => + { carrier := S₁ ∩ S₂ + one_mem := ⟨S₁.one_mem, S₂.one_mem⟩ + mul_mem := fun ⟨hx, hx'⟩ ⟨hy, hy'⟩ ↦ ⟨S₁.mul_mem hx hy, S₂.mul_mem hx' hy'⟩ }⟩ + + + example [Monoid M] (N P : Submonoid₁ M) : Submonoid₁ M := N ⊓ P + + + def Submonoid.Setoid [CommMonoid M] (N : Submonoid M) : Setoid M where + r := fun x y ↦ ∃ w ∈ N, ∃ z ∈ N, x*w = y*z + iseqv := { + refl := fun x ↦ ⟨1, N.one_mem, 1, N.one_mem, rfl⟩ + symm := fun ⟨w, hw, z, hz, h⟩ ↦ ⟨z, hz, w, hw, h.symm⟩ + trans := by + sorry + } + + instance [CommMonoid M] : HasQuotient M (Submonoid M) where + quotient' := fun N ↦ Quotient N.Setoid + + def QuotientMonoid.mk [CommMonoid M] (N : Submonoid M) : M → M ⧸ N := Quotient.mk N.Setoid + + instance [CommMonoid M] (N : Submonoid M) : Monoid (M ⧸ N) where + mul := Quotient.map₂' (· * ·) (by + sorry + ) + mul_assoc := by + sorry + one := QuotientMonoid.mk N 1 + one_mul := by + sorry + mul_one := by + sorry
src/C07_Hierarchies/solutions/Solutions_S01_Basics.lean (new)
- @@ -0,0 +1,348 @@
 + import Mathlib.Algebra.BigOperators.Ring + import Mathlib.Data.Real.Basic + + + class One₁ (α : Type) where + /-- The element one -/ + one : α + + + #check One₁.one -- One₁.one {α : Type} [self : One₁ α] : α + + @[class] structure One₂ (α : Type) where + /-- The element one -/ + one : α + + #check One₂.one + + + example (α : Type) [One₁ α] : α := One₁.one + + example (α : Type) [One₁ α] := (One₁.one : α) + + @[inherit_doc] + notation "𝟙" => One₁.one + + example {α : Type} [One₁ α] : α := 𝟙 + + example {α : Type} [One₁ α] : (𝟙 : α) = 𝟙 := rfl + + + class Dia₁ (α : Type) where + dia : α → α → α + + infixl:70 " ⋄ " => Dia₁.dia + + + class Semigroup₁ (α : Type) where + toDia₁ : Dia₁ α + /-- Diamond is associative -/ + dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) + + + attribute [instance] Semigroup₁.toDia₁ + + example {α : Type} [Semigroup₁ α] (a b : α) : α := a ⋄ b + + + class Semigroup₂ (α : Type) extends Dia₁ α where + /-- Diamond is associative -/ + dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) + + example {α : Type} [Semigroup₂ α] (a b : α) : α := a ⋄ b + + class DiaOneClass₁ (α : Type) extends One₁ α, Dia₁ α where + /-- One is a left neutral element for diamond. -/ + one_dia : ∀ a : α, 𝟙 ⋄ a = a + /-- One is a right neutral element for diamond -/ + dia_one : ∀ a : α, a ⋄ 𝟙 = a + + + + set_option trace.Meta.synthInstance true in + example {α : Type} [DiaOneClass₁ α] (a b : α) : Prop := a ⋄ b = 𝟙 + + + class Monoid₁ (α : Type) extends Semigroup₁ α, DiaOneClass₁ α + + + class Monoid₂ (α : Type) where + toSemigroup₁ : Semigroup₁ α + toDiaOneClass₁ : DiaOneClass₁ α + + + example {α : Type} [Monoid₁ α] : + (Monoid₁.toSemigroup₁.toDia₁.dia : α → α → α) = Monoid₁.toDiaOneClass₁.toDia₁.dia := rfl + + + /- Monoid₂.mk {α : Type} (toSemigroup₁ : Semigroup₁ α) (toDiaOneClass₁ : DiaOneClass₁ α) : Monoid₂ α -/ + #check Monoid₂.mk + + /- Monoid₁.mk {α : Type} [toSemigroup₁ : Semigroup₁ α] [toOne₁ : One₁ α] (one_dia : ∀ (a : α), 𝟙 ⋄ a = a) (dia_one : ∀ (a : α), a ⋄ 𝟙 = a) : Monoid₁ α -/ + #check Monoid₁.mk + + + #check Monoid₁.toSemigroup₁ + #check Monoid₁.toDiaOneClass₁ + + + class Inv₁ (α : Type) where + /-- The inversion function -/ + inv : α → α + + @[inherit_doc] + postfix:max "⁻¹" => Inv₁.inv + + class Group₁ (G : Type) extends Monoid₁ G, Inv G where + inv_dia : ∀ a : G, a⁻¹ ⋄ a = 𝟙 + + + lemma left_inv_eq_right_inv₁ {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by + rw [← DiaOneClass₁.one_dia c, ← hba, Semigroup₁.dia_assoc, hac, DiaOneClass₁.dia_one b] + + + export DiaOneClass₁ (one_dia dia_one) + export Semigroup₁ (dia_assoc) + export Group₁ (inv_dia) + + + example {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by + rw [← one_dia c, ← hba, dia_assoc, hac, dia_one b] + + + lemma inv_eq_of_dia [Group₁ G] {a b : G} (h : a ⋄ b = 𝟙) : a⁻¹ = b := + left_inv_eq_right_inv₁ (inv_dia a) h + + lemma dia_inv [Group₁ G] (a : G) : a ⋄ a⁻¹ = 𝟙 := + by rw [← inv_dia a⁻¹, inv_eq_of_dia (inv_dia a)] + + + + + + class AddSemigroup₃ (α : Type) extends Add α where + /-- Multiplication is associative -/ + add_assoc₃ : ∀ a b c : α, a + b + c = a + (b + c) + + @[to_additive AddSemigroup₃] + class Semigroup₃ (α : Type) extends Mul α where + /-- Multiplication is associative -/ + mul_assoc₃ : ∀ a b c : α, a * b * c = a * (b * c) + + class AddMonoid₃ (α : Type) extends AddSemigroup₃ α, AddZeroClass α + + @[to_additive AddMonoid₃] + class Monoid₃ (α : Type) extends Semigroup₃ α, MulOneClass α + + attribute [to_additive existing] Monoid₃.toMulOneClass + + export Semigroup₃ (mul_assoc₃) + export AddSemigroup₃ (add_assoc₃) + + whatsnew in + @[to_additive] + lemma left_inv_eq_right_inv' {M : Type} [Monoid₃ M] {a b c : M} (hba : b * a = 1) (hac : a * c = 1) : b = c := by + rw [← one_mul c, ← hba, mul_assoc₃, hac, mul_one b] + + #check left_neg_eq_right_neg' + + class AddCommSemigroup₃ (α : Type) extends AddSemigroup₃ α where + add_comm : ∀ a b : α, a + b = b + a + + @[to_additive AddCommSemigroup₃] + class CommSemigroup₃ (α : Type) extends Semigroup₃ α where + mul_comm : ∀ a b : α, a * b = b * a + + class AddCommMonoid₃ (α : Type) extends AddMonoid₃ α, AddCommSemigroup₃ α + + @[to_additive AddCommMonoid₃] + class CommMonoid₃ (α : Type) extends Monoid₃ α, CommSemigroup₃ α + + class AddGroup₃ (G : Type) extends AddMonoid₃ G, Neg G where + neg_add : ∀ a : G, -a + a = 0 + + @[to_additive AddGroup₃] + class Group₃ (G : Type) extends Monoid₃ G, Inv G where + inv_mul : ∀ a : G, a⁻¹ * a = 1 + + + attribute [simp] Group₃.inv_mul AddGroup₃.neg_add + + + + @[to_additive] + lemma inv_eq_of_mul [Group₃ G] {a b : G} (h : a * b = 1) : a⁻¹ = b := + left_inv_eq_right_inv' (Group₃.inv_mul a) h + + + @[to_additive (attr := simp)] + lemma Group₃.mul_inv {G : Type} [Group₃ G] {a : G} : a * a⁻¹ = 1 := by + rw [← inv_mul a⁻¹, inv_eq_of_mul (inv_mul a)] + + @[to_additive] + lemma mul_left_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : a * b = a * c) : b = c := by + simpa [← mul_assoc₃] using congr_arg (a⁻¹ * ·) h + + @[to_additive] + lemma mul_right_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : b*a = c*a) : b = c := by + simpa [mul_assoc₃] using congr_arg (· * a⁻¹) h + + class AddCommGroup₃ (G : Type) extends AddGroup₃ G, AddCommMonoid₃ G + + @[to_additive AddCommGroup₃] + class CommGroup₃ (G : Type) extends Group₃ G, CommMonoid₃ G + + + + class Ring₃ (R : Type) extends AddGroup₃ R, Monoid₃ R, MulZeroClass R where + /-- Multiplication is left distributive over addition -/ + left_distrib : ∀ a b c : R, a * (b + c) = a * b + a * c + /-- Multiplication is right distributive over addition -/ + right_distrib : ∀ a b c : R, (a + b) * c = a * c + b * c + + instance {R : Type} [Ring₃ R] : AddCommGroup₃ R := + { Ring₃.toAddGroup₃ with + add_comm := by + intro a b + have : a + (a + b + b) = a + (b + a + b) := calc + a + (a + b + b) = (a + a) + (b + b) := by simp [add_assoc₃, add_assoc₃] + _ = (1 * a + 1 * a) + (1 * b + 1 * b) := by simp + _ = (1 + 1) * a + (1 + 1) * b := by simp [Ring₃.right_distrib] + _ = (1 + 1) * (a + b) := by simp [Ring₃.left_distrib] + _ = 1 * (a + b) + 1 * (a + b) := by simp [Ring₃.right_distrib] + _ = (a + b) + (a + b) := by simp + _ = a + (b + a + b) := by simp [add_assoc₃] + exact add_right_cancel₃ (add_left_cancel₃ this) } + + instance : Ring₃ ℤ where + add := (· + ·) + add_assoc₃ := add_assoc + zero := 0 + zero_add := by simp + add_zero := by simp + neg := (- ·) + neg_add := by simp + mul := (· * ·) + mul_assoc₃ := mul_assoc + one := 1 + one_mul := by simp + mul_one := by simp + zero_mul := by simp + mul_zero := by simp + left_distrib := Int.mul_add + right_distrib := Int.add_mul + + class LE₁ (α : Type) where + /-- The Less-or-Equal relation. -/ + le : α → α → Prop + + @[inherit_doc] infix:50 " ≤₁ " => LE₁.le + + class Preorder₁ (α : Type) + extends LE₁ α where + le_refl : ∀ a : α, a ≤₁ a + le_trans : ∀ a b c : α, a ≤₁ b → b ≤₁ c → a ≤₁ c + + class PartialOrder₁ (α : Type) + extends Preorder₁ α where + le_antisymm : ∀ a b : α, a ≤₁ b → b ≤₁ a → a = b + + class OrderedCommMonoid₁ (α : Type) + extends PartialOrder₁ α, CommMonoid₃ α where + mul_of_le : ∀ a b : α, a ≤₁ b → ∀ c : α, c * a ≤₁ c * b + + instance : OrderedCommMonoid₁ ℕ where + le := (· ≤ ·) + le_refl := fun _ ↦ le_rfl + le_trans := fun _ _ _ ↦ le_trans + le_antisymm := fun _ _ ↦ le_antisymm + mul := (· * ·) + mul_assoc₃ := mul_assoc + one := 1 + one_mul := one_mul + mul_one := mul_one + mul_comm := mul_comm + mul_of_le := fun _ _ h c ↦ Nat.mul_le_mul_left c h + + class SMul₃ (α : Type) (β : Type) where + /-- Scalar multiplication -/ + smul : α → β → β + + infixr:73 " • " => SMul₃.smul + + + class Module₁ (R : Type) [Ring₃ R] (M : Type) [AddCommGroup₃ M] extends SMul₃ R M where + zero_smul : ∀ m : M, (0 : R) • m = 0 + one_smul : ∀ m : M, (1 : R) • m = m + mul_smul : ∀ (a b : R) (m : M), (a * b) • m = a • b • m + add_smul : ∀ (a b : R) (m : M), (a + b) • m = a • m + b • m + smul_add : ∀ (a : R) (m n : M), a • (m + n) = a • m + a • n + + instance selfModule (R : Type) [Ring₃ R] : Module₁ R R where + smul := fun r s ↦ r*s + zero_smul := zero_mul + one_smul := one_mul + mul_smul := mul_assoc₃ + add_smul := Ring₃.right_distrib + smul_add := Ring₃.left_distrib + + def nsmul₁ [Zero M] [Add M] : ℕ → M → M + | 0, _ => 0 + | n + 1, a => a + nsmul₁ n a + + def zsmul₁ {M : Type _} [Zero M] [Add M] [Neg M] : ℤ → M → M + | Int.ofNat n, a => nsmul₁ n a + | Int.negSucc n, a => -nsmul₁ n.succ a + + instance abGrpModule (A : Type) [AddCommGroup₃ A] : Module₁ ℤ A where + smul := zsmul₁ + zero_smul := sorry + one_smul := sorry + mul_smul := sorry + add_smul := sorry + smul_add := sorry + + #synth Module₁ ℤ ℤ -- abGrpModule ℤ + + + class AddMonoid₄ (M : Type) extends AddSemigroup₃ M, AddZeroClass M where + /-- Multiplication by a natural number. -/ + nsmul : ℕ → M → M := nsmul₁ + /-- Multiplication by `(0 : ℕ)` gives `0`. -/ + nsmul_zero : ∀ x, nsmul 0 x = 0 := by intros; rfl + /-- Multiplication by `(n + 1 : ℕ)` behaves as expected. -/ + nsmul_succ : ∀ (n : ℕ) (x), nsmul (n + 1) x = x + nsmul n x := by intros; rfl + + instance mySMul {M : Type} [AddMonoid₄ M] : SMul ℕ M := ⟨AddMonoid₄.nsmul⟩ + + instance (M N : Type) [AddMonoid₄ M] [AddMonoid₄ N] : AddMonoid₄ (M × N) where + add := fun p q ↦ (p.1 + q.1, p.2 + q.2) + add_assoc₃ := fun a b c ↦ by ext <;> apply add_assoc₃ + zero := (0, 0) + zero_add := fun a ↦ by ext <;> apply zero_add + add_zero := fun a ↦ by ext <;> apply add_zero + + instance : AddMonoid₄ ℤ where + add := (· + ·) + add_assoc₃ := Int.add_assoc + zero := 0 + zero_add := Int.zero_add + add_zero := Int.add_zero + nsmul := fun n m ↦ (n : ℤ) * m + nsmul_zero := Int.zero_mul + nsmul_succ := fun n m ↦ show (n + 1 : ℤ) * m = m + n * m + by rw [Int.add_mul, Int.add_comm, Int.one_mul] + + example (n : ℕ) (m : ℤ) : SMul.smul (self := mySMul) n m = n * m := rfl + + class LT₁ (α : Type) where + /-- The Less-Than relation -/ + lt : α → α → Prop + + @[inherit_doc] infix:50 " <₁ " => LT₁.lt + + class PreOrder₂ (α : Type) extends LE₁ α, LT₁ α where + le_refl : ∀ a : α, a ≤₁ a + le_trans : ∀ a b c : α, a ≤₁ b → b ≤₁ c → a ≤₁ c + lt := fun a b => a ≤₁ b ∧ ¬b ≤₁ a + lt_iff_le_not_le : ∀ a b : α, a <₁ b ↔ a ≤₁ b ∧ ¬b ≤₁ a := by intros; rfl
src/C07_Hierarchies/solutions/Solutions_S02_Morphisms.lean (new)
- @@ -0,0 +1,126 @@
  + import Mathlib.Topology.Instances.Real + + + def isMonoidHom₁ [Monoid G] [Monoid H] (f : G → H) : Prop := + f 1 = 1 ∧ ∀ g g', f (g * g') = f g * f g' + structure isMonoidHom₂ [Monoid G] [Monoid H] (f : G → H) : Prop where + map_one : f 1 = 1 + map_mul : ∀ g g', f (g * g') = f g * f g' + example : Continuous (id : ℝ → ℝ) := continuous_id + @[ext] + structure MonoidHom₁ (G H : Type) [Monoid G] [Monoid H] where + toFun : G → H + map_one : toFun 1 = 1 + map_mul : ∀ g g', toFun (g * g') = toFun g * toFun g' + + instance [Monoid G] [Monoid H] : CoeFun (MonoidHom₁ G H) (fun _ ↦ G → H) where + coe := MonoidHom₁.toFun + + attribute [coe] MonoidHom₁.toFun + + + example [Monoid G] [Monoid H] (f : MonoidHom₁ G H) : f 1 = 1 := f.map_one + + @[ext] + structure AddMonoidHom₁ (G H : Type) [AddMonoid G] [AddMonoid H] where + toFun : G → H + map_zero : toFun 0 = 0 + map_add : ∀ g g', toFun (g + g') = toFun g + toFun g' + + instance [AddMonoid G] [AddMonoid H] : CoeFun (AddMonoidHom₁ G H) (fun _ ↦ G → H) where + coe := AddMonoidHom₁.toFun + + attribute [coe] AddMonoidHom₁.toFun + + @[ext] + structure RingHom₁ (R S : Type) [Ring R] [Ring S] extends MonoidHom₁ R S, AddMonoidHom₁ R S + + + + class MonoidHomClass₁ (F : Type) (M N : Type) [Monoid M] [Monoid N] where + toFun : F → M → N + map_one : ∀ f : F, toFun f 1 = 1 + map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' + + + def badInst [Monoid M] [Monoid N] [MonoidHomClass₁ F M N] : CoeFun F (fun _ ↦ M → N) where + coe := MonoidHomClass₁.toFun + + + class MonoidHomClass₂ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] where + toFun : F → M → N + map_one : ∀ f : F, toFun f 1 = 1 + map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' + + instance [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] : CoeFun F (fun _ ↦ M → N) where + coe := MonoidHomClass₂.toFun + + attribute [coe] MonoidHomClass₂.toFun + + + instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₂ (MonoidHom₁ M N) M N where + toFun := MonoidHom₁.toFun + map_one := fun f ↦ f.map_one + map_mul := fun f ↦ f.map_mul + + instance (R S : Type) [Ring R] [Ring S] : MonoidHomClass₂ (RingHom₁ R S) R S where + toFun := fun f ↦ f.toMonoidHom₁.toFun + map_one := fun f ↦ f.toMonoidHom₁.map_one + map_mul := fun f ↦ f.toMonoidHom₁.map_mul + + + lemma map_inv_of_inv [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] (f : F) {m m' : M} (h : m*m' = 1) : + f m * f m' = 1 := by + rw [← MonoidHomClass₂.map_mul, h, MonoidHomClass₂.map_one] + + example [Monoid M] [Monoid N] (f : MonoidHom₁ M N) {m m' : M} (h : m*m' = 1) : f m * f m' = 1 := + map_inv_of_inv f h + + example [Ring R] [Ring S] (f : RingHom₁ R S) {r r' : R} (h : r*r' = 1) : f r * f r' = 1 := + map_inv_of_inv f h + + + + class MonoidHomClass₃ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] extends + FunLike F M (fun _ ↦ N) where + map_one : ∀ f : F, f 1 = 1 + map_mul : ∀ (f : F) g g', f (g * g') = f g * f g' + + instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₃ (MonoidHom₁ M N) M N where + coe := MonoidHom₁.toFun + coe_injective' := MonoidHom₁.ext + map_one := MonoidHom₁.map_one + map_mul := MonoidHom₁.map_mul + + + @[ext] + structure OrderPresHom (α β : Type) [LE α] [LE β] where + toFun : α → β + le_of_le : ∀ a a', a ≤ a' → toFun a ≤ toFun a' + + @[ext] + structure OrderPresMonoidHom (M N : Type) [Monoid M] [LE M] [Monoid N] [LE N] extends + MonoidHom₁ M N, OrderPresHom M N + + class OrderPresHomClass (F : Type) (α β : outParam Type) [LE α] [LE β] + extends FunLike F α (fun _ ↦ β) where + le_of_le : ∀ (f : F) a a', a ≤ a' → f a ≤ f a' + + instance (α β : Type) [LE α] [LE β] : OrderPresHomClass (OrderPresHom α β) α β where + coe := OrderPresHom.toFun + coe_injective' := OrderPresHom.ext + le_of_le := OrderPresHom.le_of_le + + instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : + OrderPresHomClass (OrderPresMonoidHom α β) α β where + coe := fun f ↦ f.toOrderPresHom.toFun + coe_injective' := OrderPresMonoidHom.ext + le_of_le := fun f ↦ f.toOrderPresHom.le_of_le + + instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : + MonoidHomClass₃ (OrderPresMonoidHom α β) α β + where + coe := fun f ↦ f.toOrderPresHom.toFun + coe_injective' := OrderPresMonoidHom.ext + map_one := fun f ↦ f.toMonoidHom₁.map_one + map_mul := fun f ↦ f.toMonoidHom₁.map_mul
src/C07_Hierarchies/solutions/Solutions_S03_Subobjects.lean (new)
- @@ -0,0 +1,126 @@
  + import Mathlib.GroupTheory.QuotientGroup + + + + @[ext] + structure Submonoid₁ (M : Type) [Monoid M] where + /-- The carrier of a submonoid. -/ + carrier : Set M + /-- The product of two elements of a submonoid belongs to the submonoid. -/ + mul_mem {a b} : a ∈ carrier → b ∈ carrier → a * b ∈ carrier + /-- The unit element belongs to the submonoid. -/ + one_mem : 1 ∈ carrier + + /-- Submonoids in `M` can be seen as sets in `M`. -/ + instance [Monoid M] : SetLike (Submonoid₁ M) M where + coe := Submonoid₁.carrier + coe_injective' := Submonoid₁.ext + + + + example [Monoid M] (N : Submonoid₁ M) : 1 ∈ N := N.one_mem + + example [Monoid M] (N : Submonoid₁ M) (α : Type) (f : M → α) := f '' N + + + example [Monoid M] (N : Submonoid₁ M) (x : N) : (x : M) ∈ N := x.property + + + instance SubMonoid₁Monoid [Monoid M] (N : Submonoid₁ M) : Monoid N where + mul := fun x y ↦ ⟨x*y, N.mul_mem x.property y.property⟩ + mul_assoc := fun x y z ↦ SetCoe.ext (mul_assoc (x : M) y z) + one := ⟨1, N.one_mem⟩ + one_mul := fun x ↦ SetCoe.ext (one_mul (x : M)) + mul_one := fun x ↦ SetCoe.ext (mul_one (x : M)) + + + example [Monoid M] (N : Submonoid₁ M) : Monoid N where + mul := fun ⟨x, hx⟩ ⟨y, hy⟩ ↦ ⟨x*y, N.mul_mem hx hy⟩ + mul_assoc := fun ⟨x, _⟩ ⟨y, _⟩ ⟨z, _⟩ ↦ SetCoe.ext (mul_assoc x y z) + one := ⟨1, N.one_mem⟩ + one_mul := fun ⟨x, _⟩ ↦ SetCoe.ext (one_mul x) + mul_one := fun ⟨x, _⟩ ↦ SetCoe.ext (mul_one x) + + + class SubmonoidClass₁ (S : Type) (M : Type) [Monoid M] [SetLike S M] : Prop where + mul_mem : ∀ (s : S) {a b : M}, a ∈ s → b ∈ s → a * b ∈ s + one_mem : ∀ s : S, 1 ∈ s + + instance [Monoid M] : SubmonoidClass₁ (Submonoid₁ M) M where + mul_mem := Submonoid₁.mul_mem + one_mem := Submonoid₁.one_mem + + @[ext] + structure Subgroup₁ (G : Type) [Group G] extends Submonoid₁ G where + /-- The inverse of an element of a subgroup belongs to the subgroup. -/ + inv_mem {a} : a ∈ carrier → a⁻¹ ∈ carrier + + + /-- Subgroups in `M` can be seen as sets in `M`. -/ + instance [Group G] : SetLike (Subgroup₁ G) G where + coe := fun H ↦ H.toSubmonoid₁.carrier + coe_injective' := Subgroup₁.ext + + instance [Group G] (H : Subgroup₁ G) : Group H := + { SubMonoid₁Monoid H.toSubmonoid₁ with + inv := fun x ↦ ⟨x⁻¹, H.inv_mem x.property⟩ + mul_left_inv := fun x ↦ SetCoe.ext (mul_left_inv (x : G)) } + + class SubgroupClass₁ (S : Type _) (G : Type) [Group G] [SetLike S G] + extends SubmonoidClass₁ S G : Prop where + inv_mem : ∀ (s : S) {a : G}, a ∈ s → a⁻¹ ∈ s + + instance [Group G] : SubmonoidClass₁ (Subgroup₁ G) G where + mul_mem := fun H ↦ H.toSubmonoid₁.mul_mem + one_mem := fun H ↦ H.toSubmonoid₁.one_mem + + instance [Group G] : SubgroupClass₁ (Subgroup₁ G) G := + { (inferInstance : SubmonoidClass₁ (Subgroup₁ G) G) with + inv_mem := Subgroup₁.inv_mem } + + + instance [Monoid M] : Inf (Submonoid₁ M) := + ⟨fun S₁ S₂ => + { carrier := S₁ ∩ S₂ + one_mem := ⟨S₁.one_mem, S₂.one_mem⟩ + mul_mem := fun ⟨hx, hx'⟩ ⟨hy, hy'⟩ ↦ ⟨S₁.mul_mem hx hy, S₂.mul_mem hx' hy'⟩ }⟩ + + + example [Monoid M] (N P : Submonoid₁ M) : Submonoid₁ M := N ⊓ P + + + def Submonoid.Setoid [CommMonoid M] (N : Submonoid M) : Setoid M where + r := fun x y ↦ ∃ w ∈ N, ∃ z ∈ N, x*w = y*z + iseqv := { + refl := fun x ↦ ⟨1, N.one_mem, 1, N.one_mem, rfl⟩ + symm := fun ⟨w, hw, z, hz, h⟩ ↦ ⟨z, hz, w, hw, h.symm⟩ + trans := by + rintro a b c ⟨w, hw, z, hz, h⟩ ⟨w', hw', z', hz', h'⟩ + refine ⟨w*w', N.mul_mem hw hw', z*z', N.mul_mem hz hz', ?_⟩ + rw [← mul_assoc, h, mul_comm b, mul_assoc, h', ← mul_assoc, mul_comm z, mul_assoc] + } + + instance [CommMonoid M] : HasQuotient M (Submonoid M) where + quotient' := fun N ↦ Quotient N.Setoid + + def QuotientMonoid.mk [CommMonoid M] (N : Submonoid M) : M → M ⧸ N := Quotient.mk N.Setoid + + instance [CommMonoid M] (N : Submonoid M) : Monoid (M ⧸ N) where + mul := Quotient.map₂' (· * ·) (by + rintro a₁ b₁ ⟨w, hw, z, hz, ha⟩ a₂ b₂ ⟨w', hw', z', hz', hb⟩ + refine ⟨w*w', N.mul_mem hw hw', z*z', N.mul_mem hz hz', ?_⟩ + rw [mul_comm w, ← mul_assoc, mul_assoc a₁, hb, mul_comm, ← mul_assoc, mul_comm w, ha, + mul_assoc, mul_comm z, mul_assoc b₂, mul_comm z', mul_assoc] + ) + mul_assoc := by + rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ + apply Quotient.sound + dsimp only + rw [mul_assoc] + apply @Setoid.refl M N.Setoid + one := QuotientMonoid.mk N 1 + one_mul := by + rintro ⟨a⟩ ; apply Quotient.sound ; dsimp only ; rw [one_mul] ; apply @Setoid.refl M N.Setoid + mul_one := by + rintro ⟨a⟩ ; apply Quotient.sound ; dsimp only ; rw [mul_one] ; apply @Setoid.refl M N.Setoid +
src/C07_Topology/S01_Filters.lean
- @@ -48,7 +48,7 @@ variable {α β γ : Type _} (F : Filter α) {m : γ → β} {n : β → α}
  end - example : 𝓝 (x₀, y₀) = 𝓝 x₀ ×ᶠ 𝓝 y₀ := + example : 𝓝 (x₀, y₀) = 𝓝 x₀ ×ˢ 𝓝 y₀ := nhds_prod_eq #check le_inf_iff
src/C07_Topology/solutions/Solutions_S01_Filters.lean
- @@ -49,7 +49,7 @@ example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) :
  Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := calc Tendsto f atTop (𝓝 (x₀, y₀)) ↔ map f atTop ≤ 𝓝 (x₀, y₀) := Iff.rfl - _ ↔ map f atTop ≤ 𝓝 x₀ ×ᶠ 𝓝 y₀ := by rw [nhds_prod_eq] + _ ↔ map f atTop ≤ 𝓝 x₀ ×ˢ 𝓝 y₀ := by rw [nhds_prod_eq] _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ⊓ comap Prod.snd (𝓝 y₀) := Iff.rfl _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ∧ map f atTop ≤ comap Prod.snd (𝓝 y₀) := le_inf_iff _ ↔ map Prod.fst (map f atTop) ≤ 𝓝 x₀ ∧ map Prod.snd (map f atTop) ≤ 𝓝 y₀ := by
- @@ -63,8 +63,8 @@ example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) :
  Tendsto f atTop (𝓝 (x₀, y₀)) ↔ Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := by rw [nhds_prod_eq] - unfold Tendsto Filter.prod - rw [le_inf_iff, ← map_le_iff_le_comap, map_map, ← map_le_iff_le_comap, map_map] + unfold Tendsto SProd.sprod Filter.instSProd Filter.prod + erw [le_inf_iff, ← map_le_iff_le_comap, map_map, ← map_le_iff_le_comap, map_map] example (u : ℕ → ℝ) (M : Set ℝ) (x : ℝ) (hux : Tendsto u atTop (𝓝 x)) (huM : ∀ᶠ n in atTop, u n ∈ M) : x ∈ closure M :=
src/C08_Topology/S01_Filters.lean (new)
- @@ -0,0 +1,105 @@
 + import Mathlib.Topology.Instances.Real + + open Set Filter Topology + + def principal {α : Type _} (s : Set α) : Filter α + where + sets := { t | s ⊆ t } + univ_sets := sorry + sets_of_superset := sorry + inter_sets := sorry + + example : Filter ℕ := + { sets := { s | ∃ a, ∀ b, a ≤ b → b ∈ s } + univ_sets := sorry + sets_of_superset := sorry + inter_sets := sorry } + + def Tendsto₁ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := + ∀ V ∈ G, f ⁻¹' V ∈ F + + def Tendsto₂ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := + map f F ≤ G + + example {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) : + Tendsto₂ f F G ↔ Tendsto₁ f F G := + Iff.rfl + + #check (@Filter.map_mono : ∀ {α β} {m : α → β}, Monotone (map m)) + + #check + (@Filter.map_map : + ∀ {α β γ} {f : Filter α} {m : α → β} {m' : β → γ}, map m' (map m f) = map (m' ∘ m) f) + + example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} + (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := + sorry + + variable (f : ℝ → ℝ) (x₀ y₀ : ℝ) + + #check comap ((↑) : ℚ → ℝ) (𝓝 x₀) + + #check Tendsto (f ∘ (↑)) (comap ((↑) : ℚ → ℝ) (𝓝 x₀)) (𝓝 y₀) + + section + variable {α β γ : Type _} (F : Filter α) {m : γ → β} {n : β → α} + + #check (comap_comap : comap m (comap n F) = comap (n ∘ m) F) + + end + + example : 𝓝 (x₀, y₀) = 𝓝 x₀ ×ˢ 𝓝 y₀ := + nhds_prod_eq + + #check le_inf_iff + + example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : + Tendsto f atTop (𝓝 (x₀, y₀)) ↔ + Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := + sorry + + example (x₀ : ℝ) : HasBasis (𝓝 x₀) (fun ε : ℝ => 0 < ε) fun ε => Ioo (x₀ - ε) (x₀ + ε) := + nhds_basis_Ioo_pos x₀ + + example (u : ℕ → ℝ) (x₀ : ℝ) : + Tendsto u atTop (𝓝 x₀) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, u n ∈ Ioo (x₀ - ε) (x₀ + ε) := by + have : atTop.HasBasis (fun _ : ℕ => True) Ici := atTop_basis + rw [this.tendsto_iff (nhds_basis_Ioo_pos x₀)] + simp + + example (P Q : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in atTop, Q n) : + ∀ᶠ n in atTop, P n ∧ Q n := + hP.and hQ + + example (u v : ℕ → ℝ) (h : ∀ᶠ n in atTop, u n = v n) (x₀ : ℝ) : + Tendsto u atTop (𝓝 x₀) ↔ Tendsto v atTop (𝓝 x₀) := + tendsto_congr' h + + example (u v : ℕ → ℝ) (h : u =ᶠ[atTop] v) (x₀ : ℝ) : + Tendsto u atTop (𝓝 x₀) ↔ Tendsto v atTop (𝓝 x₀) := + tendsto_congr' h + + #check @eventually_of_forall + #check @Eventually.mono + #check @Eventually.and + + example (P Q R : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in atTop, Q n) + (hR : ∀ᶠ n in atTop, P n ∧ Q n → R n) : ∀ᶠ n in atTop, R n := by + apply (hP.and (hQ.and hR)).mono + rintro n ⟨h, h', h''⟩ + exact h'' ⟨h, h'⟩ + + example (P Q R : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in atTop, Q n) + (hR : ∀ᶠ n in atTop, P n ∧ Q n → R n) : ∀ᶠ n in atTop, R n := by + filter_upwards [hP, hQ, hR] + intro n h h' h'' + exact h'' ⟨h, h'⟩ + + #check mem_closure_iff_clusterPt + #check le_principal_iff + #check neBot_of_le + + example (u : ℕ → ℝ) (M : Set ℝ) (x : ℝ) (hux : Tendsto u atTop (𝓝 x)) + (huM : ∀ᶠ n in atTop, u n ∈ M) : x ∈ closure M := + sorry +
src/C08_Topology/S02_Metric_Spaces.lean (new)
- @@ -0,0 +1,206 @@
 + import Mathlib.Topology.Instances.Real + import Mathlib.Analysis.NormedSpace.BanachSteinhaus + + open Set Filter + open Topology Filter + + variable {X : Type _} [MetricSpace X] (a b c : X) + + #check (dist a b : ℝ) + #check (dist_nonneg : 0 ≤ dist a b) + #check (dist_eq_zero : dist a b = 0 ↔ a = b) + #check (dist_comm a b : dist a b = dist b a) + #check (dist_triangle a b c : dist a c ≤ dist a b + dist b c) + + -- Note the next three lines are not quoted, their purpose is to make sure those things don't get renamed while we're looking elsewhere. + #check EMetricSpace + #check PseudoMetricSpace + #check PseudoEMetricSpace + + example {u : ℕ → X} {a : X} : + Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := + Metric.tendsto_atTop + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} : + Continuous f ↔ + ∀ x : X, ∀ ε > 0, ∃ δ > 0, ∀ x', dist x' x < δ → dist (f x') (f x) < ε := + Metric.continuous_iff + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := by continuity + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := + continuous_dist.comp ((hf.comp continuous_fst).prod_mk (hf.comp continuous_snd)) + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := by + apply Continuous.dist + exact hf.comp continuous_fst + exact hf.comp continuous_snd + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := + (hf.comp continuous_fst).dist (hf.comp continuous_snd) + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := + hf.fst'.dist hf.snd' + + example {f : ℝ → X} (hf : Continuous f) : Continuous fun x : ℝ => f (x ^ 2 + x) := + sorry + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] (f : X → Y) (a : X) : + ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x}, dist x a < δ → dist (f x) (f a) < ε := + Metric.continuousAt_iff + + variable (r : ℝ) + + example : Metric.ball a r = { b | dist b a < r } := + rfl + + example : Metric.closedBall a r = { b | dist b a ≤ r } := + rfl + + example (hr : 0 < r) : a ∈ Metric.ball a r := + Metric.mem_ball_self hr + + example (hr : 0 ≤ r) : a ∈ Metric.closedBall a r := + Metric.mem_closedBall_self hr + + example (s : Set X) : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s := + Metric.isOpen_iff + + example {s : Set X} : IsClosed s ↔ IsOpen (sᶜ) := + isOpen_compl_iff.symm + + example {s : Set X} (hs : IsClosed s) {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) + (hus : ∀ n, u n ∈ s) : a ∈ s := + hs.mem_of_tendsto hu (eventually_of_forall hus) + + example {s : Set X} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, a ∈ Metric.ball b ε := + Metric.mem_closure_iff + + example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : + a ∈ closure s := + sorry + + example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.ball x ε ⊆ s := + Metric.nhds_basis_ball.mem_iff + + example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.closedBall x ε ⊆ s := + Metric.nhds_basis_closedBall.mem_iff + + example : IsCompact (Set.Icc 0 1 : Set ℝ) := + isCompact_Icc + + example {s : Set X} (hs : IsCompact s) {u : ℕ → X} (hu : ∀ n, u n ∈ s) : + ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := + hs.tendsto_subseq hu + + example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} + (hfs : ContinuousOn f s) : + ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y := + hs.exists_forall_le hs' hfs + + example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} + (hfs : ContinuousOn f s) : + ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x := + hs.exists_forall_ge hs' hfs + + example {s : Set X} (hs : IsCompact s) : IsClosed s := + hs.isClosed + + example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := + isCompact_univ + + #check IsCompact.isClosed + + example {X : Type _} [MetricSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} : + UniformContinuous f ↔ + ∀ ε > 0, ∃ δ > 0, ∀ {a b : X}, dist a b < δ → dist (f a) (f b) < ε := + Metric.uniformContinuous_iff + + example {X : Type _} [MetricSpace X] [CompactSpace X] + {Y : Type _} [MetricSpace Y] {f : X → Y} + (hf : Continuous f) : UniformContinuous f := + sorry + + example (u : ℕ → X) : + CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε := + Metric.cauchySeq_iff + + example (u : ℕ → X) : + CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, dist (u n) (u N) < ε := + Metric.cauchySeq_iff' + + example [CompleteSpace X] (u : ℕ → X) (hu : CauchySeq u) : + ∃ x, Tendsto u atTop (𝓝 x) := + cauchySeq_tendsto_of_complete hu + + open BigOperators + + open Finset + + theorem cauchySeq_of_le_geometric_two' {u : ℕ → X} + (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by + rw [Metric.cauchySeq_iff'] + intro ε ε_pos + obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by sorry + use N + intro n hn + obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn + calc + dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := sorry + _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := sorry + _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := sorry + _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2) ^ i := sorry + _ ≤ 1 / 2 ^ N * 2 := sorry + _ < ε := sorry + + + open Metric + + example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) : + Dense (⋂ n, f n) := by + let B : ℕ → ℝ := fun n => (1 / 2) ^ n + have Bpos : ∀ n, 0 < B n + sorry + /- Translate the density assumption into two functions `center` and `radius` associating + to any n, x, δ, δpos a center and a positive radius such that + `closedBall center radius` is included both in `f n` and in `closedBall x δ`. + We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ + have : + ∀ (n : ℕ) (x : X), + ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := + by sorry + choose! center radius Hpos HB Hball using this + intro x + rw [mem_closure_iff_nhds_basis nhds_basis_closedBall] + intro ε εpos + /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` + belonging to all `f n`. For this, we construct inductively a sequence + `F n = (c n, r n)` such that the closed ball `closedBall (c n) (r n)` is included + in the previous ball and in `f n`, and such that `r n` is small enough to ensure + that `c n` is a Cauchy sequence. Then `c n` converges to a limit which belongs + to all the `f n`. -/ + let F : ℕ → X × ℝ := fun n => + Nat.recOn n (Prod.mk x (min ε (B 0))) + fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2) + let c : ℕ → X := fun n => (F n).1 + let r : ℕ → ℝ := fun n => (F n).2 + have rpos : ∀ n, 0 < r n := by sorry + have rB : ∀ n, r n ≤ B n := by sorry + have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := by + sorry + have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by sorry + have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist + -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. + rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩ + -- this point `y` will be the desired point. We will check that it belongs to all + -- `f n` and to `ball x ε`. + use y + have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry + have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry + sorry +
src/C08_Topology/S03_Topological_Spaces.lean (new)
- @@ -0,0 +1,155 @@
 + import Mathlib.Topology.Instances.Real + import Mathlib.Analysis.NormedSpace.BanachSteinhaus + + open Set Filter Topology + + section + variable {X : Type _} [TopologicalSpace X] + + example : IsOpen (univ : Set X) := + isOpen_univ + + example : IsOpen (∅ : Set X) := + isOpen_empty + + example {ι : Type _} {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋃ i, s i) := + isOpen_iUnion hs + + example {ι : Type _} [Fintype ι] {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : + IsOpen (⋂ i, s i) := + isOpen_iInter hs + + variable {Y : Type _} [TopologicalSpace Y] + + example {f : X → Y} : Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) := + continuous_def + + example {f : X → Y} {x : X} : ContinuousAt f x ↔ map f (𝓝 x) ≤ 𝓝 (f x) := + Iff.rfl + + example {f : X → Y} {x : X} : ContinuousAt f x ↔ ∀ U ∈ 𝓝 (f x), ∀ᶠ x in 𝓝 x, f x ∈ U := + Iff.rfl + + example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := + mem_nhds_iff + + example (x : X) : pure x ≤ 𝓝 x := + pure_le_nhds x + + example (x : X) (P : X → Prop) (h : ∀ᶠ y in 𝓝 x, P y) : P x := + pure_le_nhds x h + + example {P : X → Prop} {x : X} (h : ∀ᶠ y in 𝓝 x, P y) : ∀ᶠ y in 𝓝 x, ∀ᶠ z in 𝓝 y, P z := + eventually_eventually_nhds.mpr h + + #check TopologicalSpace.mkOfNhds + + #check TopologicalSpace.nhds_mkOfNhds + + example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) + (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → ∀ᶠ y in n a, ∀ᶠ x in n y, p x) : + ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := + sorry + + end + + -- BOTH. + variable {X Y : Type _} + + example (f : X → Y) : TopologicalSpace X → TopologicalSpace Y := + TopologicalSpace.coinduced f + + example (f : X → Y) : TopologicalSpace Y → TopologicalSpace X := + TopologicalSpace.induced f + + example (f : X → Y) (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) : + TopologicalSpace.coinduced f T_X ≤ T_Y ↔ T_X ≤ TopologicalSpace.induced f T_Y := + coinduced_le_iff_le_induced + + #check coinduced_compose + + #check induced_compose + + example {T T' : TopologicalSpace X} : T ≤ T' ↔ ∀ s, T'.IsOpen s → T.IsOpen s := + Iff.rfl + + example (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) (f : X → Y) : + Continuous f ↔ TopologicalSpace.coinduced f T_X ≤ T_Y := + continuous_iff_coinduced_le + + example {Z : Type _} (f : X → Y) (T_X : TopologicalSpace X) (T_Z : TopologicalSpace Z) + (g : Y → Z) : + @Continuous Y Z (TopologicalSpace.coinduced f T_X) T_Z g ↔ + @Continuous X Z T_X T_Z (g ∘ f) := by + rw [continuous_iff_coinduced_le, coinduced_compose, continuous_iff_coinduced_le] + + example (ι : Type _) (X : ι → Type _) (T_X : ∀ i, TopologicalSpace <| X i) : + (Pi.topologicalSpace : TopologicalSpace (∀ i, X i)) = + ⨅ i, TopologicalSpace.induced (fun x => x i) (T_X i) := + rfl + + example [TopologicalSpace X] [T2Space X] {u : ℕ → X} {a b : X} (ha : Tendsto u atTop (𝓝 a)) + (hb : Tendsto u atTop (𝓝 b)) : a = b := + tendsto_nhds_unique ha hb + + example [TopologicalSpace X] [RegularSpace X] (a : X) : + (𝓝 a).HasBasis (fun s : Set X => s ∈ 𝓝 a ∧ IsClosed s) id := + closed_nhds_basis a + + example [TopologicalSpace X] {x : X} : + (𝓝 x).HasBasis (fun t : Set X => t ∈ 𝓝 x ∧ IsOpen t) id := + nhds_basis_opens' x + + theorem aux {X Y A : Type _} [TopologicalSpace X] {c : A → X} + {f : A → Y} {x : X} {F : Filter Y} + (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : + ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := + sorry + + example [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace Y] {A : Set X} + (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f) + (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) <| 𝓝 x) <| 𝓝 c) : + ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := + sorry + + #check @HasBasis.tendsto_right_iff + + example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] + {s : Set X} {a : X} : + a ∈ closure s ↔ ∃ u : ℕ → X, (∀ n, u n ∈ s) ∧ Tendsto u atTop (𝓝 a) := + mem_closure_iff_seq_limit + + variable [TopologicalSpace X] + + example {F : Filter X} {x : X} : ClusterPt x F ↔ NeBot (𝓝 x ⊓ F) := + Iff.rfl + + example {s : Set X} : + IsCompact s ↔ ∀ (F : Filter X) [NeBot F], F ≤ 𝓟 s → ∃ a ∈ s, ClusterPt a F := + Iff.rfl + + example [TopologicalSpace.FirstCountableTopology X] {s : Set X} {u : ℕ → X} (hs : IsCompact s) + (hu : ∀ n, u n ∈ s) : ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := + hs.tendsto_subseq hu + + variable [TopologicalSpace Y] + + example {x : X} {F : Filter X} {G : Filter Y} (H : ClusterPt x F) {f : X → Y} + (hfx : ContinuousAt f x) (hf : Tendsto f F G) : ClusterPt (f x) G := + ClusterPt.map H hfx hf + + example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) : + IsCompact (f '' s) := by + intro F F_ne F_le + have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by sorry + have Hne : (𝓟 s ⊓ comap f F).NeBot := by sorry + have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left + sorry + + example {ι : Type _} {s : Set X} (hs : IsCompact s) (U : ι → Set X) (hUo : ∀ i, IsOpen (U i)) + (hsU : s ⊆ ⋃ i, U i) : ∃ t : Finset ι, s ⊆ ⋃ i ∈ t, U i := + hs.elim_finite_subcover U hUo hsU + + example [CompactSpace X] : IsCompact (univ : Set X) := + isCompact_univ +
src/C08_Topology/solutions/Solutions_S01_Filters.lean (new)
- @@ -0,0 +1,71 @@
 + import Mathlib.Topology.Instances.Real + + open Set Filter Topology + + -- In the next example we could use `tauto` in each proof instead of knowing the lemmas + example {α : Type _} (s : Set α) : Filter α := + { sets := { t | s ⊆ t } + univ_sets := subset_univ s + sets_of_superset := fun hU hUV => Subset.trans hU hUV + inter_sets := fun hU hV => subset_inter hU hV } + + example : Filter ℕ := + { sets := { s | ∃ a, ∀ b, a ≤ b → b ∈ s } + univ_sets := by + use 42 + simp + sets_of_superset := by + rintro U V ⟨N, hN⟩ hUV + use N + tauto + inter_sets := by + rintro U V ⟨N, hN⟩ ⟨N', hN'⟩ + use max N N' + intro b hb + rw [max_le_iff] at hb + constructor <;> tauto } + + def Tendsto₁ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := + ∀ V ∈ G, f ⁻¹' V ∈ F + + example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} + (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := + calc + map (g ∘ f) F = map g (map f F) := by rw [map_map] + _ ≤ map g G := (map_mono hf) + _ ≤ H := hg + + + example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} + (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := by + intro V hV + rw [preimage_comp] + apply hf + apply hg + exact hV + + example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : + Tendsto f atTop (𝓝 (x₀, y₀)) ↔ + Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := + calc + Tendsto f atTop (𝓝 (x₀, y₀)) ↔ map f atTop ≤ 𝓝 (x₀, y₀) := Iff.rfl + _ ↔ map f atTop ≤ 𝓝 x₀ ×ˢ 𝓝 y₀ := by rw [nhds_prod_eq] + _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ⊓ comap Prod.snd (𝓝 y₀) := Iff.rfl + _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ∧ map f atTop ≤ comap Prod.snd (𝓝 y₀) := le_inf_iff + _ ↔ map Prod.fst (map f atTop) ≤ 𝓝 x₀ ∧ map Prod.snd (map f atTop) ≤ 𝓝 y₀ := by + rw [← map_le_iff_le_comap, ← map_le_iff_le_comap] + _ ↔ map (Prod.fst ∘ f) atTop ≤ 𝓝 x₀ ∧ map (Prod.snd ∘ f) atTop ≤ 𝓝 y₀ := by + rw [map_map, map_map] + + + -- an alternative solution + example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : + Tendsto f atTop (𝓝 (x₀, y₀)) ↔ + Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := by + rw [nhds_prod_eq] + unfold Tendsto SProd.sprod Filter.instSProd Filter.prod + erw [le_inf_iff, ← map_le_iff_le_comap, map_map, ← map_le_iff_le_comap, map_map] + + example (u : ℕ → ℝ) (M : Set ℝ) (x : ℝ) (hux : Tendsto u atTop (𝓝 x)) + (huM : ∀ᶠ n in atTop, u n ∈ M) : x ∈ closure M := + mem_closure_iff_clusterPt.mpr (neBot_of_le <| le_inf hux <| le_principal_iff.mpr huM)
src/C08_Topology/solutions/Solutions_S02_Metric_Spaces.lean (new)
- @@ -0,0 +1,371 @@
 + import Mathlib.Topology.Instances.Real + import Mathlib.Analysis.NormedSpace.BanachSteinhaus + + open Set Filter + open Topology Filter + + variable {X : Type _} [MetricSpace X] (a b c : X) + + #check (dist a b : ℝ) + #check (dist_nonneg : 0 ≤ dist a b) + #check (dist_eq_zero : dist a b = 0 ↔ a = b) + #check (dist_comm a b : dist a b = dist b a) + #check (dist_triangle a b c : dist a c ≤ dist a b + dist b c) + + -- Note the next three lines are not quoted, their purpose is to make sure those things don't get renamed while we're looking elsewhere. + #check EMetricSpace + #check PseudoMetricSpace + #check PseudoEMetricSpace + + example {u : ℕ → X} {a : X} : + Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := + Metric.tendsto_atTop + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} : + Continuous f ↔ + ∀ x : X, ∀ ε > 0, ∃ δ > 0, ∀ x', dist x' x < δ → dist (f x') (f x) < ε := + Metric.continuous_iff + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := by continuity + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := + continuous_dist.comp ((hf.comp continuous_fst).prod_mk (hf.comp continuous_snd)) + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := by + apply Continuous.dist + exact hf.comp continuous_fst + exact hf.comp continuous_snd + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := + (hf.comp continuous_fst).dist (hf.comp continuous_snd) + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := + hf.fst'.dist hf.snd' + + example {f : ℝ → X} (hf : Continuous f) : Continuous fun x : ℝ => f (x ^ 2 + x) := + sorry + + example {f : ℝ → X} (hf : Continuous f) : Continuous fun x : ℝ => f (x ^ 2 + x) := + hf.comp <| (continuous_pow 2).add continuous_id + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] (f : X → Y) (a : X) : + ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x}, dist x a < δ → dist (f x) (f a) < ε := + Metric.continuousAt_iff + + variable (r : ℝ) + + example : Metric.ball a r = { b | dist b a < r } := + rfl + + example : Metric.closedBall a r = { b | dist b a ≤ r } := + rfl + + example (hr : 0 < r) : a ∈ Metric.ball a r := + Metric.mem_ball_self hr + + example (hr : 0 ≤ r) : a ∈ Metric.closedBall a r := + Metric.mem_closedBall_self hr + + example (s : Set X) : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s := + Metric.isOpen_iff + + example {s : Set X} : IsClosed s ↔ IsOpen (sᶜ) := + isOpen_compl_iff.symm + + example {s : Set X} (hs : IsClosed s) {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) + (hus : ∀ n, u n ∈ s) : a ∈ s := + hs.mem_of_tendsto hu (eventually_of_forall hus) + + example {s : Set X} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, a ∈ Metric.ball b ε := + Metric.mem_closure_iff + + example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : + a ∈ closure s := + sorry + + example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := by + rw [Metric.tendsto_atTop] at hu + rw [Metric.mem_closure_iff] + intro ε ε_pos + rcases hu ε ε_pos with ⟨N, hN⟩ + refine' ⟨u N, hs _, _⟩ + rw [dist_comm] + exact hN N le_rfl + + example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.ball x ε ⊆ s := + Metric.nhds_basis_ball.mem_iff + + example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.closedBall x ε ⊆ s := + Metric.nhds_basis_closedBall.mem_iff + + example : IsCompact (Set.Icc 0 1 : Set ℝ) := + isCompact_Icc + + example {s : Set X} (hs : IsCompact s) {u : ℕ → X} (hu : ∀ n, u n ∈ s) : + ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := + hs.tendsto_subseq hu + + example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} + (hfs : ContinuousOn f s) : + ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y := + hs.exists_forall_le hs' hfs + + example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} + (hfs : ContinuousOn f s) : + ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x := + hs.exists_forall_ge hs' hfs + + example {s : Set X} (hs : IsCompact s) : IsClosed s := + hs.isClosed + + example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := + isCompact_univ + + #check IsCompact.isClosed + + example {X : Type _} [MetricSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} : + UniformContinuous f ↔ + ∀ ε > 0, ∃ δ > 0, ∀ {a b : X}, dist a b < δ → dist (f a) (f b) < ε := + Metric.uniformContinuous_iff + + example {X : Type _} [MetricSpace X] [CompactSpace X] + {Y : Type _} [MetricSpace Y] {f : X → Y} + (hf : Continuous f) : UniformContinuous f := + sorry + + example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} + (hf : Continuous f) : UniformContinuous f := by + rw [Metric.uniformContinuous_iff] + intro ε ε_pos + let φ : X × X → ℝ := fun p => dist (f p.1) (f p.2) + have φ_cont : Continuous φ := hf.fst'.dist hf.snd' + let K := { p : X × X | ε ≤ φ p } + have K_closed : IsClosed K := isClosed_le continuous_const φ_cont + have K_cpct : IsCompact K := K_closed.isCompact + cases' eq_empty_or_nonempty K with hK hK + · use 1, by norm_num + intro x y _ + have : (x, y) ∉ K := by simp [hK] + simpa using this + · rcases K_cpct.exists_forall_le hK continuous_dist.continuousOn with ⟨⟨x₀, x₁⟩, xx_in, H⟩ + use dist x₀ x₁ + constructor + · change _ < _ + rw [dist_pos] + intro h + have : ε ≤ 0 := by simpa [*] using xx_in + linarith + · intro x x' + contrapose! + intro hxx' + exact H (x, x') hxx' + + example (u : ℕ → X) : + CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε := + Metric.cauchySeq_iff + + example (u : ℕ → X) : + CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, dist (u n) (u N) < ε := + Metric.cauchySeq_iff' + + example [CompleteSpace X] (u : ℕ → X) (hu : CauchySeq u) : + ∃ x, Tendsto u atTop (𝓝 x) := + cauchySeq_tendsto_of_complete hu + + open BigOperators + + open Finset + + theorem cauchySeq_of_le_geometric_two' {u : ℕ → X} + (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by + rw [Metric.cauchySeq_iff'] + intro ε ε_pos + obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by sorry + use N + intro n hn + obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn + calc + dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := sorry + _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := sorry + _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := sorry + _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2) ^ i := sorry + _ ≤ 1 / 2 ^ N * 2 := sorry + _ < ε := sorry + + + example {u : ℕ → X} (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by + rw [Metric.cauchySeq_iff'] + intro ε ε_pos + obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by + have : Tendsto (fun N : ℕ => (1 / 2 ^ N * 2 : ℝ)) atTop (𝓝 0) := by + rw [← MulZeroClass.zero_mul (2 : ℝ)] + apply Tendsto.mul + simp_rw [← one_div_pow (2 : ℝ)] + apply tendsto_pow_atTop_nhds_0_of_lt_1 <;> linarith + exact tendsto_const_nhds + rcases(atTop_basis.tendsto_iff (nhds_basis_Ioo_pos (0 : ℝ))).mp this ε ε_pos with ⟨N, _, hN⟩ + exact ⟨N, by simpa using (hN N left_mem_Ici).2⟩ + use N + intro n hn + obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn + calc + dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := by rw [dist_comm, add_zero] + _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := + (dist_le_range_sum_dist (fun i => u (N + i)) k) + _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := (sum_le_sum fun i _ => hu <| N + i) + _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2 : ℝ) ^ i := by simp_rw [← one_div_pow, pow_add, ← mul_sum] + _ ≤ 1 / 2 ^ N * 2 := + (mul_le_mul_of_nonneg_left (sum_geometric_two_le _) + (one_div_nonneg.mpr (pow_nonneg (zero_le_two : (0 : ℝ) ≤ 2) _))) + _ < ε := hN + + + open Metric + + example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) : + Dense (⋂ n, f n) := by + let B : ℕ → ℝ := fun n => (1 / 2) ^ n + have Bpos : ∀ n, 0 < B n + sorry + /- Translate the density assumption into two functions `center` and `radius` associating + to any n, x, δ, δpos a center and a positive radius such that + `closedBall center radius` is included both in `f n` and in `closedBall x δ`. + We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ + have : + ∀ (n : ℕ) (x : X), + ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := + by sorry + choose! center radius Hpos HB Hball using this + intro x + rw [mem_closure_iff_nhds_basis nhds_basis_closedBall] + intro ε εpos + /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` + belonging to all `f n`. For this, we construct inductively a sequence + `F n = (c n, r n)` such that the closed ball `closedBall (c n) (r n)` is included + in the previous ball and in `f n`, and such that `r n` is small enough to ensure + that `c n` is a Cauchy sequence. Then `c n` converges to a limit which belongs + to all the `f n`. -/ + let F : ℕ → X × ℝ := fun n => + Nat.recOn n (Prod.mk x (min ε (B 0))) + fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2) + let c : ℕ → X := fun n => (F n).1 + let r : ℕ → ℝ := fun n => (F n).2 + have rpos : ∀ n, 0 < r n := by sorry + have rB : ∀ n, r n ≤ B n := by sorry + have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := by + sorry + have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by sorry + have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist + -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. + rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩ + -- this point `y` will be the desired point. We will check that it belongs to all + -- `f n` and to `ball x ε`. + use y + have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry + have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry + sorry + + example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) : + Dense (⋂ n, f n) := by + let B : ℕ → ℝ := fun n => (1 / 2) ^ n + have Bpos : ∀ n, 0 pow_pos sorry n + /- Translate the density assumption into two functions `center` and `radius` associating + to any n, x, δ, δpos a center and a positive radius such that + `closedBall center radius` is included both in `f n` and in `closedBall x δ`. + We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ + have : + ∀ (n : ℕ) (x : X), + ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := by + intro n x δ δpos + have : x ∈ closure (f n) := hd n x + rcases Metric.mem_closure_iff.1 this (δ / 2) (half_pos δpos) with ⟨y, ys, xy⟩ + rw [dist_comm] at xy + obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closedBall y r ⊆ f n := + nhds_basis_closedBall.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys) + refine' ⟨y, min (min (δ / 2) r) (B (n + 1)), _, _, fun z hz => ⟨_, _⟩⟩ + show 0 < min (min (δ / 2) r) (B (n + 1)) + exact lt_min (lt_min (half_pos δpos) rpos) (Bpos (n + 1)) + show min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1) + exact min_le_right _ _ + show z ∈ closedBall x δ + exact + calc + dist z x ≤ dist z y + dist y x := dist_triangle _ _ _ + _ ≤ min (min (δ / 2) r) (B (n + 1)) + δ / 2 := (add_le_add hz xy.le) + _ ≤ δ / 2 + δ / 2 := (add_le_add_right ((min_le_left _ _).trans (min_le_left _ _)) _) + _ = δ := add_halves δ + + show z ∈ f n + exact + hr + (calc + dist z y ≤ min (min (δ / 2) r) (B (n + 1)) := hz + _ ≤ r := (min_le_left _ _).trans (min_le_right _ _) + ) + choose! center radius Hpos HB Hball using this + refine' fun x => (mem_closure_iff_nhds_basis nhds_basis_closedBall).2 fun ε εpos => _ + /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all + `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball + `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that + `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a + limit which belongs to all the `f n`. -/ + let F : ℕ → X × ℝ := fun n => + Nat.recOn n (Prod.mk x (min ε (B 0))) fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2) + let c : ℕ → X := fun n => (F n).1 + let r : ℕ → ℝ := fun n => (F n).2 + have rpos : ∀ n, 0 < r n := by + intro n + induction' n with n hn + exact lt_min εpos (Bpos 0) + exact Hpos n (c n) (r n) hn + have rB : ∀ n, r n ≤ B n := by + intro n + induction' n with n hn + exact min_le_right _ _ + exact HB n (c n) (r n) (rpos n) + have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := fun n => + Hball n (c n) (r n) (rpos n) + have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by + intro n + rw [dist_comm] + have A : c (n + 1) ∈ closedBall (c (n + 1)) (r (n + 1)) := + mem_closedBall_self (rpos <| n + 1).le + have I := + calc + closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) := + (incl n).trans (inter_subset_left _ _) + _ ⊆ closedBall (c n) (B n) := closedBall_subset_closedBall (rB n) + + exact I A + have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist + -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. + rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩ + -- this point `y` will be the desired point. We will check that it belongs to all + -- `f n` and to `ball x ε`. + use y + have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by + intro n + refine' Nat.le_induction _ fun m hnm h => _ + · exact Subset.rfl + · exact (incl m).trans ((Set.inter_subset_left _ _).trans h) + have yball : ∀ n, y ∈ closedBall (c n) (r n) := by + intro n + refine' isClosed_ball.mem_of_tendsto ylim _ + refine' (Filter.eventually_ge_atTop n).mono fun m hm => _ + exact I n m hm (mem_closedBall_self (rpos _).le) + constructor + · suffices ∀ n, y ∈ f n by rwa [Set.mem_iInter] + intro n + have : closedBall (c (n + 1)) (r (n + 1)) ⊆ f n := + Subset.trans (incl n) (inter_subset_right _ _) + exact this (yball (n + 1)) + calc + dist y x ≤ r 0 := yball 0 + _ ≤ ε := min_le_left _ _ + +
src/C08_Topology/solutions/Solutions_S03_Topological_Spaces.lean (new)
- @@ -0,0 +1,207 @@
 + import Mathlib.Topology.Instances.Real + import Mathlib.Analysis.NormedSpace.BanachSteinhaus + + open Set Filter Topology + + section + variable {X : Type _} [TopologicalSpace X] + + example : IsOpen (univ : Set X) := + isOpen_univ + + example : IsOpen (∅ : Set X) := + isOpen_empty + + example {ι : Type _} {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋃ i, s i) := + isOpen_iUnion hs + + example {ι : Type _} [Fintype ι] {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : + IsOpen (⋂ i, s i) := + isOpen_iInter hs + + variable {Y : Type _} [TopologicalSpace Y] + + example {f : X → Y} : Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) := + continuous_def + + example {f : X → Y} {x : X} : ContinuousAt f x ↔ map f (𝓝 x) ≤ 𝓝 (f x) := + Iff.rfl + + example {f : X → Y} {x : X} : ContinuousAt f x ↔ ∀ U ∈ 𝓝 (f x), ∀ᶠ x in 𝓝 x, f x ∈ U := + Iff.rfl + + example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := + mem_nhds_iff + + example (x : X) : pure x ≤ 𝓝 x := + pure_le_nhds x + + example (x : X) (P : X → Prop) (h : ∀ᶠ y in 𝓝 x, P y) : P x := + pure_le_nhds x h + + example {P : X → Prop} {x : X} (h : ∀ᶠ y in 𝓝 x, P y) : ∀ᶠ y in 𝓝 x, ∀ᶠ z in 𝓝 y, P z := + eventually_eventually_nhds.mpr h + + #check TopologicalSpace.mkOfNhds + + #check TopologicalSpace.nhds_mkOfNhds + + example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) + (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → ∀ᶠ y in n a, ∀ᶠ x in n y, p x) : + ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := + sorry + + example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) + (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → ∀ᶠ y in n a, ∀ᶠ x in n y, p x) : + ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := by + intro a s s_in + refine' ⟨{ y | s ∈ n y }, H a (fun x => x ∈ s) s_in, _, by tauto⟩ + rintro y (hy : s ∈ n y) + exact H₀ y hy + + end + + -- BOTH. + variable {X Y : Type _} + + example (f : X → Y) : TopologicalSpace X → TopologicalSpace Y := + TopologicalSpace.coinduced f + + example (f : X → Y) : TopologicalSpace Y → TopologicalSpace X := + TopologicalSpace.induced f + + example (f : X → Y) (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) : + TopologicalSpace.coinduced f T_X ≤ T_Y ↔ T_X ≤ TopologicalSpace.induced f T_Y := + coinduced_le_iff_le_induced + + #check coinduced_compose + + #check induced_compose + + example {T T' : TopologicalSpace X} : T ≤ T' ↔ ∀ s, T'.IsOpen s → T.IsOpen s := + Iff.rfl + + example (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) (f : X → Y) : + Continuous f ↔ TopologicalSpace.coinduced f T_X ≤ T_Y := + continuous_iff_coinduced_le + + example {Z : Type _} (f : X → Y) (T_X : TopologicalSpace X) (T_Z : TopologicalSpace Z) + (g : Y → Z) : + @Continuous Y Z (TopologicalSpace.coinduced f T_X) T_Z g ↔ + @Continuous X Z T_X T_Z (g ∘ f) := by + rw [continuous_iff_coinduced_le, coinduced_compose, continuous_iff_coinduced_le] + + example (ι : Type _) (X : ι → Type _) (T_X : ∀ i, TopologicalSpace <| X i) : + (Pi.topologicalSpace : TopologicalSpace (∀ i, X i)) = + ⨅ i, TopologicalSpace.induced (fun x => x i) (T_X i) := + rfl + + example [TopologicalSpace X] [T2Space X] {u : ℕ → X} {a b : X} (ha : Tendsto u atTop (𝓝 a)) + (hb : Tendsto u atTop (𝓝 b)) : a = b := + tendsto_nhds_unique ha hb + + example [TopologicalSpace X] [RegularSpace X] (a : X) : + (𝓝 a).HasBasis (fun s : Set X => s ∈ 𝓝 a ∧ IsClosed s) id := + closed_nhds_basis a + + example [TopologicalSpace X] {x : X} : + (𝓝 x).HasBasis (fun t : Set X => t ∈ 𝓝 x ∧ IsOpen t) id := + nhds_basis_opens' x + + theorem aux {X Y A : Type _} [TopologicalSpace X] {c : A → X} + {f : A → Y} {x : X} {F : Filter Y} + (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : + ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := + sorry + + example {X Y A : Type _} [TopologicalSpace X] {c : A → X} + {f : A → Y} {x : X} {F : Filter Y} + (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : + ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := by + simpa [and_assoc] using ((nhds_basis_opens' x).comap c).tendsto_left_iff.mp h V' V'_in + + example [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace Y] {A : Set X} + (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f) + (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) <| 𝓝 x) <| 𝓝 c) : + ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := + sorry + + #check @HasBasis.tendsto_right_iff + + example [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {A : Set X} (hA : ∀ x, x ∈ closure A) + {f : A → Y} (f_cont : Continuous f) + (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) <| 𝓝 x) <| 𝓝 c) : + ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := by + choose φ hφ using hf + use φ + constructor + · rw [continuous_iff_continuousAt] + intro x + suffices ∀ V' ∈ 𝓝 (φ x), IsClosed V' → φ ⁻¹' V' ∈ 𝓝 x by + simpa [ContinuousAt, (closed_nhds_basis (φ x)).tendsto_right_iff] + intro V' V'_in V'_closed + obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, IsOpen V ∧ (↑) ⁻¹' V ⊆ f ⁻¹' V' := aux (hφ x) V'_in + suffices : ∀ y ∈ V, φ y ∈ V' + exact mem_of_superset V_in this + intro y y_in + have hVx : V ∈ 𝓝 y := V_op.mem_nhds y_in + haveI : (comap ((↑) : A → X) (𝓝 y)).NeBot := by simpa [mem_closure_iff_comap_neBot] using hA y + apply V'_closed.mem_of_tendsto (hφ y) + exact mem_of_superset (preimage_mem_comap hVx) hV + · intro a + have lim : Tendsto f (𝓝 a) (𝓝 <| φ a) := by simpa [nhds_induced] using hφ a + exact tendsto_nhds_unique lim f_cont.continuousAt + + example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] + {s : Set X} {a : X} : + a ∈ closure s ↔ ∃ u : ℕ → X, (∀ n, u n ∈ s) ∧ Tendsto u atTop (𝓝 a) := + mem_closure_iff_seq_limit + + variable [TopologicalSpace X] + + example {F : Filter X} {x : X} : ClusterPt x F ↔ NeBot (𝓝 x ⊓ F) := + Iff.rfl + + example {s : Set X} : + IsCompact s ↔ ∀ (F : Filter X) [NeBot F], F ≤ 𝓟 s → ∃ a ∈ s, ClusterPt a F := + Iff.rfl + + example [TopologicalSpace.FirstCountableTopology X] {s : Set X} {u : ℕ → X} (hs : IsCompact s) + (hu : ∀ n, u n ∈ s) : ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := + hs.tendsto_subseq hu + + variable [TopologicalSpace Y] + + example {x : X} {F : Filter X} {G : Filter Y} (H : ClusterPt x F) {f : X → Y} + (hfx : ContinuousAt f x) (hf : Tendsto f F G) : ClusterPt (f x) G := + ClusterPt.map H hfx hf + + example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) : + IsCompact (f '' s) := by + intro F F_ne F_le + have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by sorry + have Hne : (𝓟 s ⊓ comap f F).NeBot := by sorry + have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left + sorry + + example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) : + IsCompact (f '' s) := by + intro F F_ne F_le + have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by rw [Filter.push_pull, map_principal] + have Hne : (𝓟 s ⊓ comap f F).NeBot := by + apply NeBot.of_map + rwa [map_eq, inf_of_le_right F_le] + have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left + rcases hs Hle with ⟨x, x_in, hx⟩ + refine' ⟨f x, mem_image_of_mem f x_in, _⟩ + apply hx.map hf.continuousAt + rw [Tendsto, map_eq] + exact inf_le_right + + example {ι : Type _} {s : Set X} (hs : IsCompact s) (U : ι → Set X) (hUo : ∀ i, IsOpen (U i)) + (hsU : s ⊆ ⋃ i, U i) : ∃ t : Finset ι, s ⊆ ⋃ i ∈ t, U i := + hs.elim_finite_subcover U hUo hsU + + example [CompactSpace X] : IsCompact (univ : Set X) := + isCompact_univ +
src/C09_Differential_Calculus/S01_Elementary_Differential_Calculus.lean (new)
- @@ -0,0 +1,40 @@
 + import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv + import Mathlib.Analysis.Calculus.MeanValue + + open Set Filter + open Topology Filter Classical Real + + noncomputable section + + open Real + + /-- The sin function has derivative 1 at 0. -/ + example : HasDerivAt sin 1 0 := by simpa using hasDerivAt_sin 0 + + example (x : ℝ) : DifferentiableAt ℝ sin x := + (hasDerivAt_sin x).differentiableAt + + example {f : ℝ → ℝ} {x a : ℝ} (h : HasDerivAt f a x) : deriv f x = a := + h.deriv + + example {f : ℝ → ℝ} {x : ℝ} (h : ¬DifferentiableAt ℝ f x) : deriv f x = 0 := + deriv_zero_of_not_differentiableAt h + + example {f g : ℝ → ℝ} {x : ℝ} (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) : + deriv (f + g) x = deriv f x + deriv g x := + deriv_add hf hg + + example {f : ℝ → ℝ} {a : ℝ} (h : IsLocalMin f a) : deriv f a = 0 := + h.deriv_eq_zero + + example {f : ℝ → ℝ} {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hfI : f a = f b) : + ∃ c ∈ Ioo a b, deriv f c = 0 := + exists_deriv_eq_zero f hab hfc hfI + + example (f : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hf : ContinuousOn f (Icc a b)) + (hf' : DifferentiableOn ℝ f (Ioo a b)) : ∃ c ∈ Ioo a b, deriv f c = (f b - f a) / (b - a) := + exists_deriv_eq_slope f hab hf hf' + + example : deriv (fun x : ℝ ↦ x ^ 5) 6 = 5 * 6 ^ 4 := by simp + + example : deriv sin π = -1 := by simp
src/C09_Differential_Calculus/S02_Differential_Calculus_in_Normed_Spaces.lean (new)
- @@ -0,0 +1,185 @@
 + import Mathlib.Analysis.NormedSpace.BanachSteinhaus + import Mathlib.Analysis.NormedSpace.FiniteDimension + import Mathlib.Analysis.Calculus.Inverse + import Mathlib.Analysis.Calculus.ContDiff + import Mathlib.Analysis.Calculus.FDeriv.Prod + + + open Set Filter + + open Topology Filter + + noncomputable section + + section + + variable {E : Type _} [NormedAddCommGroup E] + + example (x : E) : 0 ≤ ‖x‖ := + norm_nonneg x + + example {x : E} : ‖x‖ = 0 ↔ x = 0 := + norm_eq_zero + + example (x y : E) : ‖x + y‖ ≤ ‖x‖ + ‖y‖ := + norm_add_le x y + + example : MetricSpace E := by infer_instance + + example {X : Type _} [TopologicalSpace X] {f : X → E} (hf : Continuous f) : + Continuous fun x => ‖f x‖ := + hf.norm + + variable [NormedSpace ℝ E] + + example (a : ℝ) (x : E) : ‖a • x‖ = |a| * ‖x‖ := + norm_smul a x + + example [FiniteDimensional ℝ E] : CompleteSpace E := by infer_instance + + example (𝕜 : Type _) [NontriviallyNormedField 𝕜] (x y : 𝕜) : ‖x * y‖ = ‖x‖ * ‖y‖ := + norm_mul x y + + example (𝕜 : Type _) [NontriviallyNormedField 𝕜] : ∃ x : 𝕜, 1 < ‖x‖ := + NormedField.exists_one_lt_norm 𝕜 + + example (𝕜 : Type _) [NontriviallyNormedField 𝕜] (E : Type _) [NormedAddCommGroup E] + [NormedSpace 𝕜 E] [CompleteSpace 𝕜] [FiniteDimensional 𝕜 E] : CompleteSpace E := + FiniteDimensional.complete 𝕜 E + + end + + section + + variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] + [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] + + example : E →L[𝕜] E := + ContinuousLinearMap.id 𝕜 E + + example (f : E →L[𝕜] F) : E → F := + f + + example (f : E →L[𝕜] F) : Continuous f := + f.cont + + example (f : E →L[𝕜] F) (x y : E) : f (x + y) = f x + f y := + f.map_add x y + + example (f : E →L[𝕜] F) (a : 𝕜) (x : E) : f (a • x) = a • f x := + f.map_smul a x + + variable (f : E →L[𝕜] F) + + example (x : E) : ‖f x‖ ≤ ‖f‖ * ‖x‖ := + f.le_op_norm x + + example {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ x, ‖f x‖ ≤ M * ‖x‖) : ‖f‖ ≤ M := + f.op_norm_le_bound hMp hM + + end + + section + + variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] + [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] + + open Metric + + example {ι : Type _} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) : + ∃ C', ∀ i, ‖g i‖ ≤ C' := by + -- sequence of subsets consisting of those `x : E` with norms `‖g i x‖` bounded by `n` + let e : ℕ → Set E := fun n => ⋂ i : ι, { x : E | ‖g i x‖ ≤ n } + -- each of these sets is closed + have hc : ∀ n : ℕ, IsClosed (e n) + sorry + -- the union is the entire space; this is where we use `h` + have hU : (⋃ n : ℕ, e n) = univ + sorry + /- apply the Baire category theorem to conclude that for some `m : ℕ`, + `e m` contains some `x` -/ + obtain ⟨m, x, hx⟩ : ∃ m, ∃ x, x ∈ interior (e m) := sorry + obtain ⟨ε, ε_pos, hε⟩ : ∃ ε > 0, ball x ε ⊆ interior (e m) := sorry + obtain ⟨k, hk⟩ : ∃ k : 𝕜, 1 < ‖k‖ := sorry + -- show all elements in the ball have norm bounded by `m` after applying any `g i` + have real_norm_le : ∀ z ∈ ball x ε, ∀ (i : ι), ‖g i z‖ ≤ m + sorry + have εk_pos : 0 < ε / ‖k‖ := sorry + refine' ⟨(m + m : ℕ) / (ε / ‖k‖), fun i => ContinuousLinearMap.op_norm_le_of_shell ε_pos _ hk _⟩ + sorry + sorry + + end + + open Asymptotics + + open Asymptotics + + example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] (c : ℝ) + (l : Filter α) (f : α → E) (g : α → F) : IsBigOWith c l f g ↔ ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := + isBigOWith_iff + + example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] + (l : Filter α) (f : α → E) (g : α → F) : f =O[l] g ↔ ∃ C, IsBigOWith C l f g := + isBigO_iff_isBigOWith + + example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] + (l : Filter α) (f : α → E) (g : α → F) : f =o[l] g ↔ ∀ C > 0, IsBigOWith C l f g := + isLittleO_iff_forall_isBigOWith + + example {α : Type _} {E : Type _} [NormedAddCommGroup E] (l : Filter α) (f g : α → E) : + f ~[l] g ↔ (f - g) =o[l] g := + Iff.rfl + + section + + variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] + [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] + + example (f : E → F) (f' : E →L[𝕜] F) (x₀ : E) : + HasFDerivAt f f' x₀ ↔ (fun x => f x - f x₀ - f' (x - x₀)) =o[𝓝 x₀] fun x => x - x₀ := + Iff.rfl + + example (f : E → F) (f' : E →L[𝕜] F) (x₀ : E) (hff' : HasFDerivAt f f' x₀) : fderiv 𝕜 f x₀ = f' := + hff'.fderiv + + example (n : ℕ) (f : E → F) : E → E[×n]→L[𝕜] F := + iteratedFDeriv 𝕜 n f + + example (n : WithTop ℕ) {f : E → F} : + ContDiff 𝕜 n f ↔ + (∀ m : ℕ, (m : WithTop ℕ) ≤ n → Continuous fun x => iteratedFDeriv 𝕜 m f x) ∧ + ∀ m : ℕ, (m : WithTop ℕ) < n → Differentiable 𝕜 fun x => iteratedFDeriv 𝕜 m f x := + contDiff_iff_continuous_differentiable + + example {𝕂 : Type _} [IsROrC 𝕂] {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕂 E] {F : Type _} + [NormedAddCommGroup F] [NormedSpace 𝕂 F] {f : E → F} {x : E} {n : WithTop ℕ} + (hf : ContDiffAt 𝕂 n f x) (hn : 1 ≤ n) : HasStrictFDerivAt f (fderiv 𝕂 f x) x := + hf.hasStrictFDerivAt hn + + section LocalInverse + variable [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} + + example (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : F → E := + HasStrictFDerivAt.localInverse f f' a hf + + example (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : + ∀ᶠ x in 𝓝 a, hf.localInverse f f' a (f x) = x := + hf.eventually_left_inverse + + example (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : + ∀ᶠ x in 𝓝 (f a), f (hf.localInverse f f' a x) = x := + hf.eventually_right_inverse + + example [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} + (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : + HasStrictFDerivAt (HasStrictFDerivAt.localInverse f f' a hf) (f'.symm : F →L[𝕜] E) (f a) := + HasStrictFDerivAt.to_localInverse hf + + end LocalInverse + + #check HasFDerivWithinAt + + #check HasFDerivAtFilter + + end
src/C09_Differential_Calculus/solutions/Solutions_S01_Elementary_Differential_Calculus.lean (new)
- @@ -0,0 +1,8 @@
  + import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv + import Mathlib.Analysis.Calculus.MeanValue + + open Set Filter + open Topology Filter Classical Real + + noncomputable section +
src/C09_Differential_Calculus/solutions/Solutions_S02_Differential_Calculus_in_Normed_Spaces.lean (new)
- @@ -0,0 +1,62 @@
 + import Mathlib.Analysis.NormedSpace.BanachSteinhaus + import Mathlib.Analysis.NormedSpace.FiniteDimension + import Mathlib.Analysis.Calculus.Inverse + import Mathlib.Analysis.Calculus.ContDiff + import Mathlib.Analysis.Calculus.FDeriv.Prod + + + open Set Filter + + open Topology Filter + + noncomputable section + + section + + variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] + [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] + + open Metric + + example {ι : Type _} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) : + ∃ C', ∀ i, ‖g i‖ ≤ C' := by + -- sequence of subsets consisting of those `x : E` with norms `‖g i x‖` bounded by `n` + let e : ℕ → Set E := fun n => ⋂ i : ι, { x : E | ‖g i x‖ ≤ n } + -- each of these sets is closed + have hc : ∀ n : ℕ, IsClosed (e n) := fun i => + isClosed_iInter fun i => isClosed_le (g i).cont.norm continuous_const + -- the union is the entire space; this is where we use `h` + have hU : (⋃ n : ℕ, e n) = univ := by + refine' eq_univ_of_forall fun x => _ + cases' h x with C hC + obtain ⟨m, hm⟩ := exists_nat_ge C + exact ⟨e m, mem_range_self m, mem_iInter.mpr fun i => le_trans (hC i) hm⟩ + /- apply the Baire category theorem to conclude that for some `m : ℕ`, + `e m` contains some `x` -/ + obtain ⟨m : ℕ, x : E, hx : x ∈ interior (e m)⟩ := nonempty_interior_of_iUnion_of_closed hc hU + obtain ⟨ε, ε_pos, hε : ball x ε ⊆ interior (e m)⟩ := isOpen_iff.mp isOpen_interior x hx + obtain ⟨k : 𝕜, hk : 1 < ‖k‖⟩ := NormedField.exists_one_lt_norm 𝕜 + -- show all elements in the ball have norm bounded by `m` after applying any `g i` + have real_norm_le : ∀ z ∈ ball x ε, ∀ (i : ι), ‖g i z‖ ≤ m := by + intro z hz i + replace hz := mem_iInter.mp (interior_iInter_subset _ (hε hz)) i + apply interior_subset hz + have εk_pos : 0 < ε / ‖k‖ := div_pos ε_pos (zero_lt_one.trans hk) + refine' ⟨(m + m : ℕ) / (ε / ‖k‖), fun i => ContinuousLinearMap.op_norm_le_of_shell ε_pos _ hk _⟩ + · exact div_nonneg (Nat.cast_nonneg _) εk_pos.le + intro y le_y y_lt + calc + ‖g i y‖ = ‖g i (y + x) - g i x‖ := by rw [(g i).map_add, add_sub_cancel] + _ ≤ ‖g i (y + x)‖ + ‖g i x‖ := (norm_sub_le _ _) + _ ≤ m + m := + (add_le_add (real_norm_le (y + x) (by rwa [add_comm, add_mem_ball_iff_norm]) i) + (real_norm_le x (mem_ball_self ε_pos) i)) + _ = (m + m : ℕ) := by norm_cast + _ ≤ (m + m : ℕ) * (‖y‖ / (ε / ‖k‖)) := + (le_mul_of_one_le_right (Nat.cast_nonneg _) + ((one_le_div <| div_pos ε_pos (zero_lt_one.trans hk)).2 le_y)) + _ = (m + m : ℕ) / (ε / ‖k‖) * ‖y‖ := (mul_comm_div _ _ _).symm + + + end +
src/C_Abstract_Algebra/S01_Structures.lean (new)
- @@ -0,0 +1,227 @@
 + import Mathlib.Algebra.BigOperators.Ring + import Mathlib.Data.Real.Basic + + noncomputable section + + @[ext] + structure Point where + x : ℝ + y : ℝ + z : ℝ + + #check Point.ext + + example (a b : Point) (hx : a.x = b.x) (hy : a.y = b.y) (hz : a.z = b.z) : a = b := by + ext + repeat' assumption + + def myPoint1 : Point where + x := 2 + y := -1 + z := 4 + + def myPoint2 : Point := + ⟨2, -1, 4⟩ + + def myPoint3 := + Point.mk 2 (-1) 4 + + structure Point' where build :: + x : ℝ + y : ℝ + z : ℝ + + #check Point'.build 2 (-1) 4 + + namespace Point + + def add (a b : Point) : Point := + ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ + + def add' (a b : Point) : Point where + x := a.x + b.x + y := a.y + b.y + z := a.z + b.z + + #check add myPoint1 myPoint2 + + #check myPoint1.add myPoint2 + + end Point + + #check Point.add myPoint1 myPoint2 + + #check myPoint1.add myPoint2 + + namespace Point + + protected theorem add_comm (a b : Point) : add a b = add b a := by + rw [add, add] + ext <;> dsimp + repeat' apply add_comm + + example (a b : Point) : add a b = add b a := by simp [add, add_comm] + + theorem add_x (a b : Point) : (a.add b).x = a.x + b.x := + rfl + + def addAlt : Point → Point → Point + | Point.mk x₁ y₁ z₁, Point.mk x₂ y₂ z₂ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ + + def addAlt' : Point → Point → Point + | ⟨x₁, y₁, z₁⟩, ⟨x₂, y₂, z₂⟩ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ + + theorem addAlt_x (a b : Point) : (a.addAlt b).x = a.x + b.x := by + cases a + cases b + rfl + + theorem addAlt_comm (a b : Point) : addAlt a b = addAlt b a := by + rcases a with ⟨xa, ya, za⟩ + rcases b with ⟨xb, yb, zb⟩ + rw [addAlt, addAlt] + ext <;> dsimp + apply add_comm + repeat' apply add_comm + + example (a b : Point) : addAlt a b = addAlt b a := by + rcases a with ⟨xa, ya, za⟩ + rcases b with ⟨xb, yb, zb⟩ + simp [addAlt, add_comm] + + example : ∀ a b : Point, addAlt a b = addAlt b a := by + rintro ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ + simp [addAlt, add_comm] + + example : ∀ a b : Point, add a b = add b a := fun ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ => by + simp [add, add_comm] + + protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by + sorry + def smul (r : ℝ) (a : Point) : Point := + sorry + theorem smul_distrib (r : ℝ) (a b : Point) : (smul r a).add (smul r b) = smul r (a.add b) := by + sorry + end Point + + structure StandardTwoSimplex where + x : ℝ + y : ℝ + z : ℝ + x_nonneg : 0 ≤ x + y_nonneg : 0 ≤ y + z_nonneg : 0 ≤ z + sum_eq : x + y + z = 1 + + namespace StandardTwoSimplex + + def swapXy (a : StandardTwoSimplex) : StandardTwoSimplex + where + x := a.y + y := a.x + z := a.z + x_nonneg := a.y_nonneg + y_nonneg := a.x_nonneg + z_nonneg := a.z_nonneg + sum_eq := by rw [add_comm a.y a.x, a.sum_eq] + + noncomputable section + + def midpoint (a b : StandardTwoSimplex) : StandardTwoSimplex + where + x := (a.x + b.x) / 2 + y := (a.y + b.y) / 2 + z := (a.z + b.z) / 2 + x_nonneg := div_nonneg (add_nonneg a.x_nonneg b.x_nonneg) (by norm_num) + y_nonneg := div_nonneg (add_nonneg a.y_nonneg b.y_nonneg) (by norm_num) + z_nonneg := div_nonneg (add_nonneg a.z_nonneg b.z_nonneg) (by norm_num) + sum_eq := by field_simp; linarith [a.sum_eq, b.sum_eq] + + def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) + (a b : StandardTwoSimplex) : StandardTwoSimplex + where + sorry + end + + end StandardTwoSimplex + + open BigOperators + + structure StandardSimplex (n : ℕ) where + V : Fin n → ℝ + NonNeg : ∀ i : Fin n, 0 ≤ V i + sum_eq_one : (∑ i, V i) = 1 + + namespace StandardSimplex + + def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n + where + V i := (a.V i + b.V i) / 2 + NonNeg := by + intro i + apply div_nonneg + · linarith [a.NonNeg i, b.NonNeg i] + norm_num + sum_eq_one := by + simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one] + field_simp + + end StandardSimplex + + structure IsLinear (f : ℝ → ℝ) where + is_additive : ∀ x y, f (x + y) = f x + f y + preserves_mul : ∀ x c, f (c * x) = c * f x + + section + + variable (f : ℝ → ℝ) (linf : IsLinear f) + + #check linf.is_additive + + #check linf.preserves_mul + + end + + def Point'' := + ℝ × ℝ × ℝ + + def IsLinear' (f : ℝ → ℝ) := + (∀ x y, f (x + y) = f x + f y) ∧ ∀ x c, f (c * x) = c * f x + + def PReal := + { y : ℝ // 0 < y } + + section + + variable (x : PReal) + + #check x.val + + #check x.property + + #check x.1 + + #check x.2 + + end + + def StandardTwoSimplex' := + { p : ℝ × ℝ × ℝ // 0 ≤ p.1 ∧ 0 ≤ p.2.1 ∧ 0 ≤ p.2.2 ∧ p.1 + p.2.1 + p.2.2 = 1 } + + def StandardSimplex' (n : ℕ) := + { v : Fin n → ℝ // (∀ i : Fin n, 0 ≤ v i) ∧ (∑ i, v i) = 1 } + + def StdSimplex := Σ n : ℕ, StandardSimplex n + + section + + variable (s : StdSimplex) + + #check s.fst + #check s.snd + + #check s.1 + #check s.2 + + end +
src/C_Abstract_Algebra/S02_Algebraic_Structures.lean (new)
- @@ -0,0 +1,172 @@
  + import Mathlib.Data.Real.Basic + + structure Group₁ (α : Type _) where + mul : α → α → α + one : α + inv : α → α + mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) + mul_one : ∀ x : α, mul x one = x + one_mul : ∀ x : α, mul one x = x + mul_left_inv : ∀ x : α, mul (inv x) x = one + + structure Group₁Cat where + α : Type _ + str : Group₁ α + + section + + variable (α β γ : Type _) + variable (f : α ≃ β) (g : β ≃ γ) + + #check Equiv α β + #check (f.toFun : α → β) + #check (f.invFun : β → α) + #check (f.right_inv : ∀ x : β, f (f.invFun x) = x) + #check (f.left_inv : ∀ x : α, f.invFun (f x) = x) + #check (Equiv.refl α : α ≃ α) + #check (f.symm : β ≃ α) + #check (f.trans g : α ≃ γ) + + example (x : α) : (f.trans g).toFun x = g.toFun (f.toFun x) := + rfl + + example (x : α) : (f.trans g) x = g (f x) := + rfl + + example : (f.trans g : α → γ) = g ∘ f := + rfl + + end + + example (α : Type _) : Equiv.Perm α = (α ≃ α) := + rfl + + def permGroup {α : Type _} : Group₁ (Equiv.Perm α) + where + mul f g := Equiv.trans g f + one := Equiv.refl α + inv := Equiv.symm + mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm + one_mul := Equiv.trans_refl + mul_one := Equiv.refl_trans + mul_left_inv := Equiv.self_trans_symm + + structure AddGroup₁ (α : Type _) where + (add : α → α → α) + -- fill in the rest + @[ext] + structure Point where + x : ℝ + y : ℝ + z : ℝ + + namespace Point + + def add (a b : Point) : Point := + ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ + + def neg (a : point) : point := sorry + + def zero : point := sorry + + def add_group_point : add_group₁ point := sorry + end Point + + section + + variable {α : Type _} (f g : Equiv.Perm α) (n : ℕ) + + #check f * g + #check mul_assoc f g g⁻¹ + + -- group power, defined for any group + #check g ^ n + + example : f * g * g⁻¹ = f := by rw [mul_assoc, mul_right_inv, mul_one] + + example : f * g * g⁻¹ = f := + mul_inv_cancel_right f g + + example {α : Type _} (f g : Equiv.Perm α) : g.symm.trans (g.trans f) = f := + mul_inv_cancel_right f g + + end + + class Group₂ (α : Type _) where + mul : α → α → α + one : α + inv : α → α + mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) + mul_one : ∀ x : α, mul x one = x + one_mul : ∀ x : α, mul one x = x + mul_left_inv : ∀ x : α, mul (inv x) x = one + + instance {α : Type _} : Group₂ (Equiv.Perm α) where + mul f g := Equiv.trans g f + one := Equiv.refl α + inv := Equiv.symm + mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm + one_mul := Equiv.trans_refl + mul_one := Equiv.refl_trans + mul_left_inv := Equiv.self_trans_symm + + #check @Group₂.mul + + def mySquare {α : Type _} [Group₂ α] (x : α) := + Group₂.mul x x + + #check @mySquare + + section + variable {β : Type _} (f g : Equiv.Perm β) + + example : Group₂.mul f g = g.trans f := + rfl + + example : mySquare f = f.trans f := + rfl + + end + + instance : Inhabited Point where default := ⟨0, 0, 0⟩ + + #check (default : Point) + + example : ([] : List Point).headI = default := + rfl + + instance : Add Point where add := Point.add + + section + variable (x y : Point) + + #check x + y + + example : x + y = Point.add x y := + rfl + + end + + instance hasMulGroup₂ {α : Type _} [Group₂ α] : Mul α := + ⟨Group₂.mul⟩ + + instance hasOneGroup₂ {α : Type _} [Group₂ α] : One α := + ⟨Group₂.one⟩ + + instance hasInvGroup₂ {α : Type _} [Group₂ α] : Inv α := + ⟨Group₂.inv⟩ + + section + + variable {α : Type _} (f g : Equiv.Perm α) + + #check f * 1 * g⁻¹ + + def foo : f * 1 * g⁻¹ = g.symm.trans ((Equiv.refl α).trans f) := + rfl + + end + + class AddGroup₂ (α : Type _) where + add : α → α → α + -- fill in the rest
src/C_Abstract_Algebra/S03_Building_the_Gaussian_Integers.lean (new)
- @@ -0,0 +1,271 @@
 + import Mathlib.Data.Int.Basic + import Mathlib.Algebra.EuclideanDomain.Basic + import Mathlib.RingTheory.PrincipalIdealDomain + import Mathlib.Tactic + + @[ext] + structure gaussInt where + re : ℤ + im : ℤ + + namespace gaussInt + + instance : Zero gaussInt := + ⟨⟨0, 0⟩⟩ + + instance : One gaussInt := + ⟨⟨1, 0⟩⟩ + + instance : Add gaussInt := + ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ + + instance : Neg gaussInt := + ⟨fun x => ⟨-x.re, -x.im⟩⟩ + + instance : Mul gaussInt := + ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ + + theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := + rfl + + theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := + rfl + + theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := + rfl + + theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := + rfl + + theorem mul_def (x y : gaussInt) : x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := + rfl + + @[simp] + theorem zero_re : (0 : gaussInt).re = 0 := + rfl + + @[simp] + theorem zero_im : (0 : gaussInt).im = 0 := + rfl + + @[simp] + theorem one_re : (1 : gaussInt).re = 1 := + rfl + + @[simp] + theorem one_im : (1 : gaussInt).im = 0 := + rfl + + @[simp] + theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := + rfl + + @[simp] + theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := + rfl + + @[simp] + theorem neg_re (x : gaussInt) : (-x).re = -x.re := + rfl + + @[simp] + theorem neg_im (x : gaussInt) : (-x).im = -x.im := + rfl + + @[simp] + theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := + rfl + + @[simp] + theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := + rfl + + instance instCommRing : CommRing gaussInt where + zero := 0 + one := 1 + add := (· + ·) + neg x := -x + mul := (· * ·) + add_assoc := by + intros + ext <;> simp <;> ring + zero_add := by + intros + ext <;> simp + add_zero := by + intros + ext <;> simp + add_left_neg := by + intros + ext <;> simp + add_comm := by + intros + ext <;> simp <;> ring + mul_assoc := by + intros + ext <;> simp <;> ring + one_mul := by + intros + ext <;> simp + mul_one := by + intros + ext <;> simp + left_distrib := by + intros + ext <;> simp <;> ring + right_distrib := by + intros + ext <;> simp <;> ring + mul_comm := by + intros + ext <;> simp <;> ring + zero_mul := sorry + mul_zero := sorry + + instance : Nontrivial gaussInt := by + use 0, 1 + rw [Ne, gaussInt.ext_iff] + simp + + end gaussInt + + example (a b : ℤ) : a = b * (a / b) + a % b := + Eq.symm <| Int.ediv_add_emod a b + + example (a b : ℤ) : b ≠ 0 → 0 ≤ a % b := + Int.emod_nonneg a + + example (a b : ℤ) : b ≠ 0 → a % b < abs b := + Int.emod_lt a + + namespace Int + + def div' (a b : ℤ) := + (a + b / 2) / b + + def mod' (a b : ℤ) := + (a + b / 2) % b - b / 2 + + theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by + rw [div', mod'] + linarith [Int.ediv_add_emod (a + b / 2) b] + + theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by + rw [mod', abs_le] + constructor + · linarith [Int.emod_nonneg (a + b / 2) h.ne'] + have := Int.emod_lt_of_pos (a + b / 2) h + have := Int.ediv_add_emod b 2 + have := Int.emod_lt_of_pos b zero_lt_two + revert this; intro this -- FIXME, this should not be needed + linarith + + theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] + + end Int + + theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : + x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by + sorry + namespace gaussInt + + def norm (x : gaussInt) := + x.re ^ 2 + x.im ^ 2 + + @[simp] + theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by + sorry + theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by + sorry + theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by + sorry + theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by + sorry + def conj (x : gaussInt) : gaussInt := + ⟨x.re, -x.im⟩ + + @[simp] + theorem conj_re (x : gaussInt) : (conj x).re = x.re := + rfl + + @[simp] + theorem conj_im (x : gaussInt) : (conj x).im = -x.im := + rfl + + theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] + + instance : Div gaussInt := + ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ + + instance : Mod gaussInt := + ⟨fun x y => x - y * (x / y)⟩ + + theorem div_def (x y : gaussInt) : + x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := + rfl + + theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := + rfl + + theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : (x % y).norm < y.norm := by + have norm_y_pos : 0 < norm y := by rwa [norm_pos] + have : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by + rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] + ext <;> simp <;> ring + have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by + conv => + lhs + rw [← norm_conj y, ← norm_mul, this, norm] + simp + trans 2 * (y.norm / 2) ^ 2 + · rw [two_mul] + apply add_le_add <;> + · rw [sq_le_sq] + apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) + apply le_abs_self + rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] + apply mul_le_mul_of_nonneg_left + · apply Int.ediv_mul_le + norm_num + apply Int.ediv_nonneg (norm_nonneg y) + norm_num + have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos + apply lt_of_le_of_lt this + apply Int.ediv_lt_of_lt_mul + · norm_num + linarith + + theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := + Int.natAbs_of_nonneg (norm_nonneg _) + + theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : (x % y).norm.natAbs < y.norm.natAbs := by + apply Int.ofNat_lt.1 + simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] + apply norm_mod_lt x hy + + theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : + ¬(norm (x * y)).natAbs < (norm x).natAbs := by + apply not_lt_of_ge + rw [norm_mul, Int.natAbs_mul] + apply le_mul_of_one_le_right (Nat.zero_le _) + apply Int.ofNat_le.1 + rw [coe_natAbs_norm] + exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) + + instance : EuclideanDomain gaussInt := + { gaussInt.instCommRing with + quotient := (· / ·) + remainder := (· % ·) + quotient_mul_add_remainder_eq := fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] + quotient_zero := fun x => by + simp [div_def, norm, Int.div'] + rfl + r := Measure (Int.natAbs ∘ norm) + r_wellFounded := (measure (Int.natAbs ∘ norm)).2 + remainder_lt := natAbs_norm_mod_lt + mul_left_not_lt := not_norm_mul_left_lt_norm } + + example (x : gaussInt) : Irreducible x ↔ Prime x := + PrincipalIdealRing.irreducible_iff_prime + + end gaussInt
src/C_Abstract_Algebra/solutions/Solutions_S01_Structures.lean (new)
- @@ -0,0 +1,96 @@
  + import Mathlib.Algebra.BigOperators.Ring + import Mathlib.Data.Real.Basic + + noncomputable section + + @[ext] + structure Point where + x : ℝ + y : ℝ + z : ℝ + + namespace Point + + def add (a b : Point) : Point := + ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ + + protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by + simp [add, add_assoc] + + def smul (r : ℝ) (a : Point) : Point := + ⟨r * a.x, r * a.y, r * a.z⟩ + + theorem smul_distrib (r : ℝ) (a b : Point) : (smul r a).add (smul r b) = smul r (a.add b) := by + simp [add, smul, mul_add] + + end Point + + structure StandardTwoSimplex where + x : ℝ + y : ℝ + z : ℝ + x_nonneg : 0 ≤ x + y_nonneg : 0 ≤ y + z_nonneg : 0 ≤ z + sum_eq : x + y + z = 1 + + namespace StandardTwoSimplex + + noncomputable section + + def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) + (a b : StandardTwoSimplex) : StandardTwoSimplex + where + x := lambda * a.x + (1 - lambda) * b.x + y := lambda * a.y + (1 - lambda) * b.y + z := lambda * a.z + (1 - lambda) * b.z + x_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.x_nonneg) (mul_nonneg (by linarith) b.x_nonneg) + y_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.y_nonneg) (mul_nonneg (by linarith) b.y_nonneg) + z_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.z_nonneg) (mul_nonneg (by linarith) b.z_nonneg) + sum_eq := by + trans (a.x + a.y + a.z) * lambda + (b.x + b.y + b.z) * (1 - lambda) + · ring + simp [a.sum_eq, b.sum_eq] + + end + + end StandardTwoSimplex + + open BigOperators + + structure StandardSimplex (n : ℕ) where + V : Fin n → ℝ + NonNeg : ∀ i : Fin n, 0 ≤ V i + sum_eq_one : (∑ i, V i) = 1 + + namespace StandardSimplex + + def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n + where + V i := (a.V i + b.V i) / 2 + NonNeg := by + intro i + apply div_nonneg + · linarith [a.NonNeg i, b.NonNeg i] + norm_num + sum_eq_one := by + simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one] + field_simp + + end StandardSimplex + + namespace StandardSimplex + + def weightedAverage {n : ℕ} (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) + (a b : StandardSimplex n) : StandardSimplex n + where + V i := lambda * a.V i + (1 - lambda) * b.V i + NonNeg i := + add_nonneg (mul_nonneg lambda_nonneg (a.NonNeg i)) (mul_nonneg (by linarith) (b.NonNeg i)) + sum_eq_one := by + trans (lambda * ∑ i, a.V i) + (1 - lambda) * ∑ i, b.V i + · rw [Finset.sum_add_distrib, Finset.mul_sum, Finset.mul_sum] + simp [a.sum_eq_one, b.sum_eq_one] + + end StandardSimplex +
src/C_Abstract_Algebra/solutions/Solutions_S02_Algebraic_Structures.lean (new)
- @@ -0,0 +1,73 @@
  + import Mathlib.Data.Real.Basic + + structure AddGroup₁ (α : Type _) where + add : α → α → α + zero : α + neg : α → α + add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z) + add_zero : ∀ x : α, add x zero = x + zero_add : ∀ x : α, add x zero = x + add_left_neg : ∀ x : α, add (neg x) x = zero + + @[ext] + structure Point where + x : ℝ + y : ℝ + z : ℝ + + namespace Point + + def add (a b : Point) : Point := + ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ + + def neg (a : Point) : Point := + ⟨-a.x, -a.y, -a.z⟩ + + def zero : Point := + ⟨0, 0, 0⟩ + + def addGroupPoint : AddGroup₁ Point where + add := Point.add + zero := Point.zero + neg := Point.neg + add_assoc := by simp [Point.add, add_assoc] + add_zero := by simp [Point.add, Point.zero] + zero_add := by simp [Point.add, Point.zero] + add_left_neg := by simp [Point.add, Point.neg, Point.zero] + + end Point + + class AddGroup₂ (α : Type _) where + add : α → α → α + zero : α + neg : α → α + add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z) + add_zero : ∀ x : α, add x zero = x + zero_add : ∀ x : α, add x zero = x + add_left_neg : ∀ x : α, add (neg x) x = zero + + instance hasAddAddGroup₂ {α : Type _} [AddGroup₂ α] : Add α := + ⟨AddGroup₂.add⟩ + + instance hasZeroAddGroup₂ {α : Type _} [AddGroup₂ α] : Zero α := + ⟨AddGroup₂.zero⟩ + + instance hasNegAddGroup₂ {α : Type _} [AddGroup₂ α] : Neg α := + ⟨AddGroup₂.neg⟩ + + instance : AddGroup₂ Point where + add := Point.add + zero := Point.zero + neg := Point.neg + add_assoc := by simp [Point.add, add_assoc] + add_zero := by simp [Point.add, Point.zero] + zero_add := by simp [Point.add, Point.zero] + add_left_neg := by simp [Point.add, Point.neg, Point.zero] + + section + variable (x y : Point) + + #check x + -y + 0 + + end +
src/C_Abstract_Algebra/solutions/Solutions_S03_Building_the_Gaussian_Integers.lean (new)
- @@ -0,0 +1,285 @@
 + import Mathlib.Data.Int.Basic + import Mathlib.Algebra.EuclideanDomain.Basic + import Mathlib.RingTheory.PrincipalIdealDomain + import Mathlib.Tactic + + @[ext] + structure gaussInt where + re : ℤ + im : ℤ + + namespace gaussInt + + instance : Zero gaussInt := + ⟨⟨0, 0⟩⟩ + + instance : One gaussInt := + ⟨⟨1, 0⟩⟩ + + instance : Add gaussInt := + ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ + + instance : Neg gaussInt := + ⟨fun x => ⟨-x.re, -x.im⟩⟩ + + instance : Mul gaussInt := + ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ + + theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := + rfl + + theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := + rfl + + theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := + rfl + + theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := + rfl + + theorem mul_def (x y : gaussInt) : x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := + rfl + + @[simp] + theorem zero_re : (0 : gaussInt).re = 0 := + rfl + + @[simp] + theorem zero_im : (0 : gaussInt).im = 0 := + rfl + + @[simp] + theorem one_re : (1 : gaussInt).re = 1 := + rfl + + @[simp] + theorem one_im : (1 : gaussInt).im = 0 := + rfl + + @[simp] + theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := + rfl + + @[simp] + theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := + rfl + + @[simp] + theorem neg_re (x : gaussInt) : (-x).re = -x.re := + rfl + + @[simp] + theorem neg_im (x : gaussInt) : (-x).im = -x.im := + rfl + + @[simp] + theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := + rfl + + @[simp] + theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := + rfl + + instance instCommRing : CommRing gaussInt where + zero := 0 + one := 1 + add := (· + ·) + neg x := -x + mul := (· * ·) + add_assoc := by + intros + ext <;> simp <;> ring + zero_add := by + intros + ext <;> simp + add_zero := by + intros + ext <;> simp + add_left_neg := by + intros + ext <;> simp + add_comm := by + intros + ext <;> simp <;> ring + mul_assoc := by + intros + ext <;> simp <;> ring + one_mul := by + intros + ext <;> simp + mul_one := by + intros + ext <;> simp + left_distrib := by + intros + ext <;> simp <;> ring + right_distrib := by + intros + ext <;> simp <;> ring + mul_comm := by + intros + ext <;> simp <;> ring + zero_mul := sorry + mul_zero := sorry + + instance : Nontrivial gaussInt := by + use 0, 1 + rw [Ne, gaussInt.ext_iff] + simp + + end gaussInt + + namespace Int + + def div' (a b : ℤ) := + (a + b / 2) / b + + def mod' (a b : ℤ) := + (a + b / 2) % b - b / 2 + + theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by + rw [div', mod'] + linarith [Int.ediv_add_emod (a + b / 2) b] + + theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by + rw [mod', abs_le] + constructor + · linarith [Int.emod_nonneg (a + b / 2) h.ne'] + have := Int.emod_lt_of_pos (a + b / 2) h + have := Int.ediv_add_emod b 2 + have := Int.emod_lt_of_pos b zero_lt_two + revert this; intro this -- FIXME, this should not be needed + linarith + + theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] + + end Int + + private theorem aux {α : Type _} [LinearOrderedRing α] {x y : α} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := + haveI h' : x ^ 2 = 0 := by + apply le_antisymm _ (sq_nonneg x) + rw [← h] + apply le_add_of_nonneg_right (sq_nonneg y) + pow_eq_zero h' + + theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : + x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by + constructor + · intro h + constructor + · exact aux h + rw [add_comm] at h + exact aux h + rintro ⟨rfl, rfl⟩ + norm_num + + namespace gaussInt + + def norm (x : gaussInt) := + x.re ^ 2 + x.im ^ 2 + + @[simp] + theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by + apply add_nonneg <;> + apply sq_nonneg + + theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by + rw [norm, sq_add_sq_eq_zero, gaussInt.ext_iff] + rfl + + theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by + rw [lt_iff_le_and_ne, ne_comm, Ne, norm_eq_zero] + simp [norm_nonneg] + + theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by + simp [norm] + ring + + def conj (x : gaussInt) : gaussInt := + ⟨x.re, -x.im⟩ + + @[simp] + theorem conj_re (x : gaussInt) : (conj x).re = x.re := + rfl + + @[simp] + theorem conj_im (x : gaussInt) : (conj x).im = -x.im := + rfl + + theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] + + instance : Div gaussInt := + ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ + + instance : Mod gaussInt := + ⟨fun x y => x - y * (x / y)⟩ + + theorem div_def (x y : gaussInt) : + x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := + rfl + + theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := + rfl + + theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : (x % y).norm < y.norm := by + have norm_y_pos : 0 < norm y := by rwa [norm_pos] + have : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by + rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] + ext <;> simp <;> ring + have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by + conv => + lhs + rw [← norm_conj y, ← norm_mul, this, norm] + simp + trans 2 * (y.norm / 2) ^ 2 + · rw [two_mul] + apply add_le_add <;> + · rw [sq_le_sq] + apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) + apply le_abs_self + rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] + apply mul_le_mul_of_nonneg_left + · apply Int.ediv_mul_le + norm_num + apply Int.ediv_nonneg (norm_nonneg y) + norm_num + have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos + apply lt_of_le_of_lt this + apply Int.ediv_lt_of_lt_mul + · norm_num + linarith + + theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := + Int.natAbs_of_nonneg (norm_nonneg _) + + theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : (x % y).norm.natAbs < y.norm.natAbs := by + apply Int.ofNat_lt.1 + simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] + apply norm_mod_lt x hy + + theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : + ¬(norm (x * y)).natAbs < (norm x).natAbs := by + apply not_lt_of_ge + rw [norm_mul, Int.natAbs_mul] + apply le_mul_of_one_le_right (Nat.zero_le _) + apply Int.ofNat_le.1 + rw [coe_natAbs_norm] + exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) + + instance : EuclideanDomain gaussInt := + { gaussInt.instCommRing with + quotient := (· / ·) + remainder := (· % ·) + quotient_mul_add_remainder_eq := fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] + quotient_zero := fun x => by + simp [div_def, norm, Int.div'] + rfl + r := Measure (Int.natAbs ∘ norm) + r_wellFounded := (measure (Int.natAbs ∘ norm)).2 + remainder_lt := natAbs_norm_mod_lt + mul_left_not_lt := not_norm_mul_left_lt_norm } + + example (x : gaussInt) : Irreducible x ↔ Prime x := + PrincipalIdealRing.irreducible_iff_prime + + end gaussInt
src/C_Basics/S01_Calculating.lean (new)
- @@ -0,0 +1,164 @@
  + import Mathlib.Data.Real.Basic + import Mathlib.Data.Real.Basic + + -- An example. + example (a b c : ℝ) : a * b * c = b * (a * c) := by + rw [mul_comm a b] + rw [mul_assoc b a c] + + -- Try these. + example (a b c : ℝ) : c * b * a = b * (a * c) := by + sorry + + example (a b c : ℝ) : a * (b * c) = b * (a * c) := by + sorry + + -- An example. + example (a b c : ℝ) : a * b * c = b * c * a := by + rw [mul_assoc] + rw [mul_comm] + + /- Try doing the first of these without providing any arguments at all, + and the second with only one argument. -/ + example (a b c : ℝ) : a * (b * c) = b * (c * a) := by + sorry + + example (a b c : ℝ) : a * (b * c) = b * (a * c) := by + sorry + + -- Using facts from the local context. + example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by + rw [h'] + rw [← mul_assoc] + rw [h] + rw [mul_assoc] + + -- Try these. For the second one, use the theorem `sub_self`. + example (a b c d e f : ℝ) (h : b * c = e * f) : a * b * c * d = a * e * f * d := by + sorry + + example (a b c d : ℝ) (hyp : c = b * a - d) (hyp' : d = a * b) : c = 0 := by + sorry + + -- Examples. + example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by + rw [h', ← mul_assoc, h, mul_assoc] + + section + + variable (a b c d e f g : ℝ) + + example (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by + rw [h', ← mul_assoc, h, mul_assoc] + + end + + section + + variable (a b c : ℝ) + + #check a + + #check a + b + + #check (a : ℝ) + + #check mul_comm a b + + #check (mul_comm a b : a * b = b * a) + + #check mul_assoc c a b + + #check mul_comm a + + #check mul_comm + + #check @mul_comm + + end + + section + + variable (a b : ℝ) + + example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by + rw [mul_add, add_mul, add_mul] + rw [← add_assoc, add_assoc (a * a)] + rw [mul_comm b a, ← two_mul] + + example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := + calc + (a + b) * (a + b) = a * a + b * a + (a * b + b * b) := by + rw [mul_add, add_mul, add_mul] + _ = a * a + (b * a + a * b) + b * b := by + rw [← add_assoc, add_assoc (a * a)] + _ = a * a + 2 * (a * b) + b * b := by + rw [mul_comm b a, ← two_mul] + + example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := + calc + (a + b) * (a + b) = a * a + b * a + (a * b + b * b) := by + sorry + _ = a * a + (b * a + a * b) + b * b := by + sorry + _ = a * a + 2 * (a * b) + b * b := by + sorry + + end + + -- Try these. For the second, use the theorems listed underneath. + section + + variable (a b c d : ℝ) + + example : (a + b) * (c + d) = a * c + a * d + b * c + b * d := by + sorry + + example (a b : ℝ) : (a + b) * (a - b) = a ^ 2 - b ^ 2 := by + sorry + + #check pow_two a + + #check mul_sub a b c + + #check add_mul a b c + + #check add_sub a b c + + #check sub_sub a b c + + #check add_zero a + + end + + -- Examples. + section + + variable (a b c d : ℝ) + + example (a b c d : ℝ) (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by + rw [hyp'] at hyp + rw [mul_comm d a] at hyp + rw [← two_mul (a * d)] at hyp + rw [← mul_assoc 2 a d] at hyp + exact hyp + + example : c * b * a = b * (a * c) := by + ring + + example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by + ring + + example : (a + b) * (a - b) = a ^ 2 - b ^ 2 := by + ring + + example (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by + rw [hyp, hyp'] + ring + + end + + example (a b c : ℕ) (h : a + b = c) : (a + b) * (a + b) = a * c + b * c := by + nth_rw 2 [h] + rw [add_mul] +
src/C_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean (new)
- @@ -0,0 +1,168 @@
  + import Mathlib.Algebra.Ring.Defs + import Mathlib.Data.Real.Basic + import Mathlib.Tactic + + section + + variable (R : Type _) [Ring R] + + #check (add_assoc : ∀ a b c : R, a + b + c = a + (b + c)) + + #check (add_comm : ∀ a b : R, a + b = b + a) + + #check (zero_add : ∀ a : R, 0 + a = a) + + #check (add_left_neg : ∀ a : R, -a + a = 0) + + #check (mul_assoc : ∀ a b c : R, a * b * c = a * (b * c)) + + #check (mul_one : ∀ a : R, a * 1 = a) + + #check (one_mul : ∀ a : R, 1 * a = a) + + #check (mul_add : ∀ a b c : R, a * (b + c) = a * b + a * c) + + #check (add_mul : ∀ a b c : R, (a + b) * c = a * c + b * c) + + end + + section + + variable (R : Type _) [CommRing R] + + variable (a b c d : R) + + example : c * b * a = b * (a * c) := by ring + + example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by ring + + example : (a + b) * (a - b) = a ^ 2 - b ^ 2 := by ring + + example (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by + rw [hyp, hyp'] + ring + + end + + namespace MyRing + + variable {R : Type _} [Ring R] + + theorem add_zero (a : R) : a + 0 = a := by rw [add_comm, zero_add] + + theorem add_right_neg (a : R) : a + -a = 0 := by rw [add_comm, add_left_neg] + + #check @MyRing.add_zero + + #check @add_zero + + end MyRing + + namespace MyRing + + variable {R : Type _} [Ring R] + + theorem neg_add_cancel_left (a b : R) : -a + (a + b) = b := by + rw [← add_assoc, add_left_neg, zero_add] + + -- Prove these: + theorem add_neg_cancel_right (a b : R) : a + b + -b = a := by + sorry + + theorem add_left_cancel {a b c : R} (h : a + b = a + c) : b = c := by + sorry + + theorem add_right_cancel {a b c : R} (h : a + b = c + b) : a = c := by + sorry + + theorem mul_zero (a : R) : a * 0 = 0 := by + have h : a * 0 + a * 0 = a * 0 + 0 := by + rw [← mul_add, add_zero, add_zero] + rw [add_left_cancel h] + + theorem zero_mul (a : R) : 0 * a = 0 := by + sorry + + theorem neg_eq_of_add_eq_zero {a b : R} (h : a + b = 0) : -a = b := by + sorry + + theorem eq_neg_of_add_eq_zero {a b : R} (h : a + b = 0) : a = -b := by + sorry + + theorem neg_zero : (-0 : R) = 0 := by + apply neg_eq_of_add_eq_zero + rw [add_zero] + + theorem neg_neg (a : R) : - -a = a := by + sorry + + end MyRing + + -- Examples. + section + + variable {R : Type _} [Ring R] + + example (a b : R) : a - b = a + -b := + sub_eq_add_neg a b + + end + + example (a b : ℝ) : a - b = a + -b := + rfl + + example (a b : ℝ) : a - b = a + -b := by + rfl + + namespace MyRing + + variable {R : Type _} [Ring R] + + theorem self_sub (a : R) : a - a = 0 := + sorry + + theorem one_add_one_eq_two : 1 + 1 = (2 : R) := by + norm_num + + theorem two_mul (a : R) : 2 * a = a + a := + sorry + + end MyRing + + section + + variable (A : Type _) [AddGroup A] + + #check (add_assoc : ∀ a b c : A, a + b + c = a + (b + c)) + + #check (zero_add : ∀ a : A, 0 + a = a) + + #check (add_left_neg : ∀ a : A, -a + a = 0) + + end + + section + + variable {G : Type _} [Group G] + + #check (mul_assoc : ∀ a b c : G, a * b * c = a * (b * c)) + + #check (one_mul : ∀ a : G, 1 * a = a) + + #check (mul_left_inv : ∀ a : G, a⁻¹ * a = 1) + + namespace MyGroup + + theorem mul_right_inv (a : G) : a * a⁻¹ = 1 := by + sorry + + theorem mul_one (a : G) : a * 1 = a := by + sorry + + theorem mul_inv_rev (a b : G) : (a * b)⁻¹ = b⁻¹ * a⁻¹ := by + sorry + + end MyGroup + + end +
src/C_Basics/S03_Using_Theorems_and_Lemmas.lean (new)
- @@ -0,0 +1,156 @@
 + import Mathlib.Analysis.SpecialFunctions.Log.Basic + import Mathlib.Tactic + + variable (a b c d e : ℝ) + + open Real + + #check (le_refl : ∀ a : ℝ, a ≤ a) + + #check (le_trans : a ≤ b → b ≤ c → a ≤ c) + + section + + variable (h : a ≤ b) (h' : b ≤ c) + + #check (le_refl : ∀ a : Real, a ≤ a) + + #check (le_refl a : a ≤ a) + + #check (le_trans : a ≤ b → b ≤ c → a ≤ c) + + #check (le_trans h : b ≤ c → a ≤ c) + + #check (le_trans h h' : a ≤ c) + + end + + example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := by + apply le_trans + · apply h₀ + . apply h₁ + + example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := by + apply le_trans h₀ + apply h₁ + + example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := + le_trans h₀ h₁ + + example (x : ℝ) : x ≤ x := by + apply le_refl + + example (x : ℝ) : x ≤ x := + le_refl x + + #check (le_refl : ∀ a, a ≤ a) + + #check (le_trans : a ≤ b → b ≤ c → a ≤ c) + + #check (lt_of_le_of_lt : a ≤ b → b < c → a < c) + + #check (lt_of_lt_of_le : a < b → b ≤ c → a < c) + + #check (lt_trans : a < b → b < c → a < c) + + -- Try this. + example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) (h₃ : d < e) : a < e := by + sorry + + example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) (h₃ : d < e) : a < e := by + linarith + + section + + example (h : 2 * a ≤ 3 * b) (h' : 1 ≤ a) (h'' : d = 2) : d + a ≤ 5 * b := by + linarith + + end + + example (h : 1 ≤ a) (h' : b ≤ c) : 2 + a + exp b ≤ 3 * a + exp c := by + linarith [exp_le_exp.mpr h'] + + #check (exp_le_exp : exp a ≤ exp b ↔ a ≤ b) + + #check (exp_lt_exp : exp a < exp b ↔ a < b) + + #check (log_le_log : 0 < a → 0 < b → (log a ≤ log b ↔ a ≤ b)) + + #check (log_lt_log : 0 < a → a < b → log a < log b) + + #check (add_le_add : a ≤ b → c ≤ d → a + c ≤ b + d) + + #check (add_le_add_left : a ≤ b → ∀ c, c + a ≤ c + b) + + #check (add_le_add_right : a ≤ b → ∀ c, a + c ≤ b + c) + + #check (add_lt_add_of_le_of_lt : a ≤ b → c < d → a + c < b + d) + + #check (add_lt_add_of_lt_of_le : a < b → c ≤ d → a + c < b + d) + + #check (add_lt_add_left : a < b → ∀ c, c + a < c + b) + + #check (add_lt_add_right : a < b → ∀ c, a + c < b + c) + + #check (add_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a + b) + + #check (add_pos : 0 < a → 0 < b → 0 < a + b) + + #check (add_pos_of_pos_of_nonneg : 0 < a → 0 ≤ b → 0 < a + b) + + #check (exp_pos : ∀ a, 0 < exp a) + + #check @add_le_add_left + + example (h : a ≤ b) : exp a ≤ exp b := by + rw [exp_le_exp] + exact h + + example (h₀ : a ≤ b) (h₁ : c < d) : a + exp c + e < b + exp d + e := by + apply add_lt_add_of_lt_of_le + · apply add_lt_add_of_le_of_lt h₀ + apply exp_lt_exp.mpr h₁ + apply le_refl + + example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := by sorry + + example : (0 : ℝ) < 1 := by norm_num + + example (h : a ≤ b) : log (1 + exp a) ≤ log (1 + exp b) := by + have h₀ : 0 < 1 + exp a := by sorry + have h₁ : 0 < 1 + exp b := by sorry + apply (log_le_log h₀ h₁).mpr + sorry + + example : 0 ≤ a ^ 2 := by + -- library_search + exact sq_nonneg a + + example (h : a ≤ b) : c - exp b ≤ c - exp a := by + sorry + + example : 2 * a * b ≤ a ^ 2 + b ^ 2 := by + have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 + calc + a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring + _ ≥ 0 := by apply pow_two_nonneg + + calc + 2 * a * b = 2 * a * b + 0 := by ring + _ ≤ 2 * a * b + (a ^ 2 - 2 * a * b + b ^ 2) := + add_le_add (le_refl _) h + _ = a ^ 2 + b ^ 2 := by ring + + + example : 2 * a * b ≤ a ^ 2 + b ^ 2 := by + have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 + calc + a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring + _ ≥ 0 := by apply pow_two_nonneg + linarith + + example : abs (a * b) ≤ (a ^ 2 + b ^ 2) / 2 := by + sorry + + #check abs_le'.mpr +
src/C_Basics/S04_More_on_Order_and_Divisibility.lean (new)
- @@ -0,0 +1,98 @@
  + import Mathlib.Data.Real.Basic + + section + + variable (a b c d : ℝ) + + #check (min_le_left a b : min a b ≤ a) + + #check (min_le_right a b : min a b ≤ b) + + #check (le_min : c ≤ a → c ≤ b → c ≤ min a b) + + example : min a b = min b a := by + apply le_antisymm + · show min a b ≤ min b a + apply le_min + · apply min_le_right + apply min_le_left + · show min b a ≤ min a b + apply le_min + · apply min_le_right + apply min_le_left + + example : min a b = min b a := by + have h : ∀ x y : ℝ, min x y ≤ min y x := by + intro x y + apply le_min + apply min_le_right + apply min_le_left + apply le_antisymm + apply h + apply h + + example : min a b = min b a := by + apply le_antisymm + repeat + apply le_min + apply min_le_right + apply min_le_left + + example : max a b = max b a := by + sorry + + example : min (min a b) c = min a (min b c) := by + sorry + + theorem aux : min a b + c ≤ min (a + c) (b + c) := by + sorry + + example : min a b + c = min (a + c) (b + c) := by + sorry + + #check (abs_add : ∀ a b : ℝ, abs (a + b) ≤ abs a + abs b) + + example : abs a - abs b ≤ abs (a - b) := + sorry + + end + + section + + variable (w x y z : ℕ) + + example (h₀ : x ∣ y) (h₁ : y ∣ z) : x ∣ z := + dvd_trans h₀ h₁ + + example : x ∣ y * x * z := by + apply dvd_mul_of_dvd_left + apply dvd_mul_left + + example : x ∣ x ^ 2 := by + apply dvd_mul_left + + example (h : x ∣ w) : x ∣ y * (x * z) + x ^ 2 + w ^ 2 := by + sorry + + end + + section + + variable (m n : ℕ) + + open Nat + + #check (gcd_zero_right n : gcd n 0 = n) + + #check (gcd_zero_left n : gcd 0 n = n) + + #check (lcm_zero_right n : lcm n 0 = 0) + + #check (lcm_zero_left n : lcm 0 n = 0) + + example : gcd m n = gcd n m := by + sorry + + end + +
src/C_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean (new)
- @@ -0,0 +1,141 @@
 + import Mathlib.Topology.MetricSpace.Basic + + section + + variable {α : Type _} [PartialOrder α] + + variable (x y z : α) + + #check x ≤ y + + #check (le_refl x : x ≤ x) + + #check (le_trans : x ≤ y → y ≤ z → x ≤ z) + + #check x < y + + #check (lt_irrefl x : ¬x < x) + + #check (lt_trans : x < y → y < z → x < z) + + #check (lt_of_le_of_lt : x ≤ y → y < z → x < z) + + #check (lt_of_lt_of_le : x < y → y ≤ z → x < z) + + example : x < y ↔ x ≤ y ∧ x ≠ y := + lt_iff_le_and_ne + + end + + section + + variable {α : Type _} [Lattice α] + + variable (x y z : α) + + #check x ⊓ y + + #check (inf_le_left : x ⊓ y ≤ x) + + #check (inf_le_right : x ⊓ y ≤ y) + + #check (le_inf : z ≤ x → z ≤ y → z ≤ x ⊓ y) + + #check x ⊔ y + + #check (le_sup_left : x ≤ x ⊔ y) + + #check (le_sup_right : y ≤ x ⊔ y) + + #check (sup_le : x ≤ z → y ≤ z → x ⊔ y ≤ z) + + example : x ⊓ y = y ⊓ x := by + sorry + + example : x ⊓ y ⊓ z = x ⊓ (y ⊓ z) := by + sorry + + example : x ⊔ y = y ⊔ x := by + sorry + + example : x ⊔ y ⊔ z = x ⊔ (y ⊔ z) := by + sorry + + theorem absorb1 : x ⊓ (x ⊔ y) = x := by + sorry + + theorem absorb2 : x ⊔ x ⊓ y = x := by + sorry + + end + + section + + variable {α : Type _} [DistribLattice α] + + variable (x y z : α) + + #check (inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) + + #check (inf_sup_right : (x ⊔ y) ⊓ z = x ⊓ z ⊔ y ⊓ z) + + #check (sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) + + #check (sup_inf_right : x ⊓ y ⊔ z = (x ⊔ z) ⊓ (y ⊔ z)) + + end + + section + + variable {α : Type _} [Lattice α] + + variable (a b c : α) + + example (h : ∀ x y z : α, x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) : a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c) := by + sorry + + example (h : ∀ x y z : α, x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) : a ⊓ (b ⊔ c) = a ⊓ b ⊔ a ⊓ c := by + sorry + + end + + section + + variable {R : Type _} [StrictOrderedRing R] + + variable (a b c : R) + + #check (add_le_add_left : a ≤ b → ∀ c, c + a ≤ c + b) + + #check (mul_pos : 0 < a → 0 < b → 0 < a * b) + + #check (mul_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a * b) + + example : a ≤ b → 0 ≤ b - a := by + sorry + + example : 0 ≤ b - a → a ≤ b := by + sorry + + example (h : a ≤ b) (h' : 0 ≤ c) : a * c ≤ b * c := by + sorry + + end + + section + + variable {X : Type _} [MetricSpace X] + + variable (x y z : X) + + #check (dist_self x : dist x x = 0) + + #check (dist_comm x y : dist x y = dist y x) + + #check (dist_triangle x y z : dist x z ≤ dist x y + dist y z) + + example (x y : X) : 0 ≤ dist x y := by + sorry + + end +
src/C_Basics/solutions/Solutions_S01_Calculating.lean (new)
- @@ -0,0 +1,33 @@
  + import Mathlib.Data.Real.Basic + import Mathlib.Data.Real.Basic + + example (a b c : ℝ) : c * b * a = b * (a * c) := by + rw [mul_comm c b] + rw [mul_assoc b c a] + rw [mul_comm c a] + + example (a b c : ℝ) : a * (b * c) = b * (a * c) := by + rw [← mul_assoc a b c] + rw [mul_comm a b] + rw [mul_assoc b a c] + + example (a b c : ℝ) : a * (b * c) = b * (c * a) := by + rw [mul_comm] + rw [mul_assoc] + + example (a b c : ℝ) : a * (b * c) = b * (a * c) := by + rw [← mul_assoc] + rw [mul_comm a] + rw [mul_assoc] + + example (a b c d e f : ℝ) (h : b * c = e * f) : a * b * c * d = a * e * f * d := by + rw [mul_assoc a] + rw [h] + rw [← mul_assoc] + + example (a b c d : ℝ) (hyp : c = b * a - d) (hyp' : d = a * b) : c = 0 := by + rw [hyp] + rw [hyp'] + rw [mul_comm] + rw [sub_self] +
src/C_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean (new)
- @@ -0,0 +1,76 @@
  + import Mathlib.Algebra.Ring.Defs + import Mathlib.Data.Real.Basic + import Mathlib.Tactic + + namespace MyRing + + variable {R : Type _} [Ring R] + + theorem add_neg_cancel_right (a b : R) : a + b + -b = a := by + rw [add_assoc, add_right_neg, add_zero] + + theorem add_left_cancel {a b c : R} (h : a + b = a + c) : b = c := by + rw [← neg_add_cancel_left a b, h, neg_add_cancel_left] + + theorem add_right_cancel {a b c : R} (h : a + b = c + b) : a = c := by + rw [← add_neg_cancel_right a b, h, add_neg_cancel_right] + + theorem zero_mul (a : R) : 0 * a = 0 := by + have h : 0 * a + 0 * a = 0 * a + 0 := by rw [← add_mul, add_zero, add_zero] + rw [add_left_cancel h] + + theorem neg_eq_of_add_eq_zero {a b : R} (h : a + b = 0) : -a = b := by + rw [← neg_add_cancel_left a b, h, add_zero] + + theorem eq_neg_of_add_eq_zero {a b : R} (h : a + b = 0) : a = -b := by + symm + apply neg_eq_of_add_eq_zero + rw [add_comm, h] + + theorem neg_zero : (-0 : R) = 0 := by + apply neg_eq_of_add_eq_zero + rw [add_zero] + + theorem neg_neg (a : R) : - -a = a := by + apply neg_eq_of_add_eq_zero + rw [add_left_neg] + + end MyRing + + namespace MyRing + + variable {R : Type _} [Ring R] + + theorem self_sub (a : R) : a - a = 0 := by + rw [sub_eq_add_neg, add_right_neg] + + theorem one_add_one_eq_two : 1 + 1 = (2 : R) := by + norm_num + + theorem two_mul (a : R) : 2 * a = a + a := by + rw [← one_add_one_eq_two, add_mul, one_mul] + + end MyRing + + section + + variable {G : Type _} [Group G] + + namespace MyGroup + + theorem mul_right_inv (a : G) : a * a⁻¹ = 1 := by + have h : (a * a⁻¹)⁻¹ * (a * a⁻¹ * (a * a⁻¹)) = 1 := by + rw [mul_assoc, ← mul_assoc a⁻¹ a, mul_left_inv, one_mul, mul_left_inv] + rw [← h, ← mul_assoc, mul_left_inv, one_mul] + + theorem mul_one (a : G) : a * 1 = a := by + rw [← mul_left_inv a, ← mul_assoc, mul_right_inv, one_mul] + + theorem mul_inv_rev (a b : G) : (a * b)⁻¹ = b⁻¹ * a⁻¹ := by + rw [← one_mul (b⁻¹ * a⁻¹), ← mul_left_inv (a * b), mul_assoc, mul_assoc, ← mul_assoc b b⁻¹, + mul_right_inv, one_mul, mul_right_inv, mul_one] + + end MyGroup + + end +
src/C_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean (new)
- @@ -0,0 +1,62 @@
 + import Mathlib.Analysis.SpecialFunctions.Log.Basic + import Mathlib.Tactic + + variable (a b c d e : ℝ) + + open Real + + example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) (h₃ : d < e) : a < e := by + apply lt_of_le_of_lt h₀ + apply lt_trans h₁ + exact lt_of_le_of_lt h₂ h₃ + + example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := by + apply add_le_add_left + rw [exp_le_exp] + apply add_le_add_left h₀ + + -- an alternative using `linarith`. + example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := by + have : exp (a + d) ≤ exp (a + e) := by + rw [exp_le_exp] + linarith + linarith [this] + + example (h : a ≤ b) : log (1 + exp a) ≤ log (1 + exp b) := by + have h₀ : 0 < 1 + exp a := by linarith [exp_pos a] + have h₁ : 0 < 1 + exp b := by linarith [exp_pos b] + apply (log_le_log h₀ h₁).mpr + apply add_le_add_left (exp_le_exp.mpr h) + + -- SOLUTION. + example (h : a ≤ b) : c - exp b ≤ c - exp a := by + apply sub_le_sub_left + exact exp_le_exp.mpr h + + -- alternatively: + example (h : a ≤ b) : c - exp b ≤ c - exp a := by + linarith [exp_le_exp.mpr h] + + theorem fact1 : a * b * 2 ≤ a ^ 2 + b ^ 2 := by + have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 + calc + a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring + _ ≥ 0 := by apply pow_two_nonneg + linarith + + theorem fact2 : -(a * b) * 2 ≤ a ^ 2 + b ^ 2 := by + have h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2 + calc + a ^ 2 + 2 * a * b + b ^ 2 = (a + b) ^ 2 := by ring + _ ≥ 0 := by apply pow_two_nonneg + linarith + + example : abs (a * b) ≤ (a ^ 2 + b ^ 2) / 2 := by + have h : (0 : ℝ) < 2 := by norm_num + apply abs_le'.mpr + constructor + · rw [le_div_iff h] + apply fact1 + rw [le_div_iff h] + apply fact2 +
src/C_Basics/solutions/Solutions_S04_More_on_Order_and_Divisibility.lean (new)
- @@ -0,0 +1,124 @@
  + import Mathlib.Data.Real.Basic + + section + + variable (a b c d : ℝ) + + #check (min_le_left a b : min a b ≤ a) + + #check (min_le_right a b : min a b ≤ b) + + #check (le_min : c ≤ a → c ≤ b → c ≤ min a b) + + example : max a b = max b a := by + apply le_antisymm + repeat' + apply max_le + apply le_max_right + apply le_max_left + + example : min (min a b) c = min a (min b c) := by + apply le_antisymm + · apply le_min + · apply le_trans + apply min_le_left + apply min_le_left + apply le_min + · apply le_trans + apply min_le_left + apply min_le_right + apply min_le_right + apply le_min + · apply le_min + · apply min_le_left + apply le_trans + apply min_le_right + apply min_le_left + apply le_trans + apply min_le_right + apply min_le_right + + theorem aux : min a b + c ≤ min (a + c) (b + c) := by + apply le_min + · apply add_le_add_right + apply min_le_left + apply add_le_add_right + apply min_le_right + + example : min a b + c = min (a + c) (b + c) := by + apply le_antisymm + · apply aux + have h : min (a + c) (b + c) = min (a + c) (b + c) - c + c := by rw [sub_add_cancel] + rw [h] + apply add_le_add_right + rw [sub_eq_add_neg] + apply le_trans + apply aux + rw [add_neg_cancel_right, add_neg_cancel_right] + + example : abs a - abs b ≤ abs (a - b) := + calc + abs a - abs b = abs (a - b + b) - abs b := by rw [sub_add_cancel] + _ ≤ abs (a - b) + abs b - abs b := by + apply sub_le_sub_right + apply abs_add + _ ≤ abs (a - b) := by rw [add_sub_cancel] + + + -- alternatively + example : abs a - abs b ≤ abs (a - b) := by + have h := abs_add (a - b) b + rw [sub_add_cancel] at h + linarith + + end + + section + + variable (w x y z : ℕ) + + example (h₀ : x ∣ y) (h₁ : y ∣ z) : x ∣ z := + dvd_trans h₀ h₁ + + example : x ∣ y * x * z := by + apply dvd_mul_of_dvd_left + apply dvd_mul_left + + example : x ∣ x ^ 2 := by + apply dvd_mul_left + + example (h : x ∣ w) : x ∣ y * (x * z) + x ^ 2 + w ^ 2 := by + apply dvd_add + · apply dvd_add + · apply dvd_mul_of_dvd_right + apply dvd_mul_right + apply dvd_mul_left + rw [pow_two] + apply dvd_mul_of_dvd_right + exact h + + end + + section + + variable (m n : ℕ) + + open Nat + + #check (gcd_zero_right n : gcd n 0 = n) + + #check (gcd_zero_left n : gcd 0 n = n) + + #check (lcm_zero_right n : lcm n 0 = 0) + + #check (lcm_zero_left n : lcm 0 n = 0) + + example : gcd m n = gcd n m := by + apply _root_.dvd_antisymm + repeat' + apply dvd_gcd + apply gcd_dvd_right + apply gcd_dvd_left + + end +
src/C_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean (new)
- @@ -0,0 +1,149 @@
  + import Mathlib.Topology.MetricSpace.Basic + + section + + variable {α : Type _} [Lattice α] + + variable (x y z : α) + + example : x ⊓ y = y ⊓ x := by + apply le_antisymm + repeat' + apply le_inf + · apply inf_le_right + apply inf_le_left + + example : x ⊓ y ⊓ z = x ⊓ (y ⊓ z) := by + apply le_antisymm + · apply le_inf + · apply le_trans + apply inf_le_left + apply inf_le_left + apply le_inf + · apply le_trans + apply inf_le_left + apply inf_le_right + apply inf_le_right + apply le_inf + · apply le_inf + · apply inf_le_left + apply le_trans + apply inf_le_right + apply inf_le_left + apply le_trans + apply inf_le_right + apply inf_le_right + + example : x ⊔ y = y ⊔ x := by + apply le_antisymm + repeat' + apply sup_le + · apply le_sup_right + apply le_sup_left + + example : x ⊔ y ⊔ z = x ⊔ (y ⊔ z) := by + apply le_antisymm + · apply sup_le + · apply sup_le + apply le_sup_left + · apply le_trans + apply @le_sup_left _ _ y z + apply le_sup_right + apply le_trans + apply @le_sup_right _ _ y z + apply le_sup_right + apply sup_le + · apply le_trans + apply @le_sup_left _ _ x y + apply le_sup_left + apply sup_le + · apply le_trans + apply @le_sup_right _ _ x y + apply le_sup_left + apply le_sup_right + + theorem absorb1 : x ⊓ (x ⊔ y) = x := by + apply le_antisymm + · apply inf_le_left + apply le_inf + · apply le_refl + apply le_sup_left + + theorem absorb2 : x ⊔ x ⊓ y = x := by + apply le_antisymm + · apply sup_le + · apply le_refl + apply inf_le_left + apply le_sup_left + + end + + section + + variable {α : Type _} [DistribLattice α] + + variable (x y z : α) + + #check (inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) + + #check (inf_sup_right : (x ⊔ y) ⊓ z = x ⊓ z ⊔ y ⊓ z) + + #check (sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) + + #check (sup_inf_right : x ⊓ y ⊔ z = (x ⊔ z) ⊓ (y ⊔ z)) + + end + + section + + variable {α : Type _} [Lattice α] + + variable (a b c : α) + + example (h : ∀ x y z : α, x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) : a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c) := by + rw [h, @inf_comm _ _ (a ⊔ b), absorb1, @inf_comm _ _ (a ⊔ b), h, ← sup_assoc, @inf_comm _ _ c a, + absorb2, inf_comm] + + example (h : ∀ x y z : α, x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) : a ⊓ (b ⊔ c) = a ⊓ b ⊔ a ⊓ c := by + rw [h, @sup_comm _ _ (a ⊓ b), absorb2, @sup_comm _ _ (a ⊓ b), h, ← inf_assoc, @sup_comm _ _ c a, + absorb1, sup_comm] + + end + + section + + variable {R : Type _} [StrictOrderedRing R] + + variable (a b c : R) + + theorem aux1 : a ≤ b → 0 ≤ b - a := by + intro h + rw [← sub_self a, sub_eq_add_neg, sub_eq_add_neg, add_comm, add_comm b] + apply add_le_add_left h + + theorem aux2 : 0 ≤ b - a → a ≤ b := by + intro h + rw [← add_zero a, ← sub_add_cancel b a, add_comm (b - a)] + apply add_le_add_left h + + example (h : a ≤ b) (h' : 0 ≤ c) : a * c ≤ b * c := by + have h1 : 0 ≤ (b - a) * c := mul_nonneg (aux1 _ _ h) h' + rw [sub_mul] at h1 + exact aux2 _ _ h1 + + end + + section + + variable {X : Type _} [MetricSpace X] + + variable (x y z : X) + + example (x y : X) : 0 ≤ dist x y :=by + have : 0 ≤ dist x y + dist y x := by + rw [← dist_self x] + apply dist_triangle + linarith [dist_comm x y] + + end +
src/C_Hierarchies/S01_Basics.lean (new)
- @@ -0,0 +1,308 @@
 + import Mathlib.Algebra.BigOperators.Ring + import Mathlib.Data.Real.Basic + + + class One₁ (α : Type) where + /-- The element one -/ + one : α + + + #check One₁.one -- One₁.one {α : Type} [self : One₁ α] : α + + @[class] structure One₂ (α : Type) where + /-- The element one -/ + one : α + + #check One₂.one + + + example (α : Type) [One₁ α] : α := One₁.one + + example (α : Type) [One₁ α] := (One₁.one : α) + + @[inherit_doc] + notation "𝟙" => One₁.one + + example {α : Type} [One₁ α] : α := 𝟙 + + example {α : Type} [One₁ α] : (𝟙 : α) = 𝟙 := rfl + + + class Dia₁ (α : Type) where + dia : α → α → α + + infixl:70 " ⋄ " => Dia₁.dia + + + class Semigroup₁ (α : Type) where + toDia₁ : Dia₁ α + /-- Diamond is associative -/ + dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) + + + attribute [instance] Semigroup₁.toDia₁ + + example {α : Type} [Semigroup₁ α] (a b : α) : α := a ⋄ b + + + class Semigroup₂ (α : Type) extends Dia₁ α where + /-- Diamond is associative -/ + dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) + + example {α : Type} [Semigroup₂ α] (a b : α) : α := a ⋄ b + + class DiaOneClass₁ (α : Type) extends One₁ α, Dia₁ α where + /-- One is a left neutral element for diamond. -/ + one_dia : ∀ a : α, 𝟙 ⋄ a = a + /-- One is a right neutral element for diamond -/ + dia_one : ∀ a : α, a ⋄ 𝟙 = a + + + + set_option trace.Meta.synthInstance true in + example {α : Type} [DiaOneClass₁ α] (a b : α) : Prop := a ⋄ b = 𝟙 + + + class Monoid₁ (α : Type) extends Semigroup₁ α, DiaOneClass₁ α + + + class Monoid₂ (α : Type) where + toSemigroup₁ : Semigroup₁ α + toDiaOneClass₁ : DiaOneClass₁ α + + + example {α : Type} [Monoid₁ α] : + (Monoid₁.toSemigroup₁.toDia₁.dia : α → α → α) = Monoid₁.toDiaOneClass₁.toDia₁.dia := rfl + + + /- Monoid₂.mk {α : Type} (toSemigroup₁ : Semigroup₁ α) (toDiaOneClass₁ : DiaOneClass₁ α) : Monoid₂ α -/ + #check Monoid₂.mk + + /- Monoid₁.mk {α : Type} [toSemigroup₁ : Semigroup₁ α] [toOne₁ : One₁ α] (one_dia : ∀ (a : α), 𝟙 ⋄ a = a) (dia_one : ∀ (a : α), a ⋄ 𝟙 = a) : Monoid₁ α -/ + #check Monoid₁.mk + + + #check Monoid₁.toSemigroup₁ + #check Monoid₁.toDiaOneClass₁ + + + class Inv₁ (α : Type) where + /-- The inversion function -/ + inv : α → α + + @[inherit_doc] + postfix:max "⁻¹" => Inv₁.inv + + class Group₁ (G : Type) extends Monoid₁ G, Inv G where + inv_dia : ∀ a : G, a⁻¹ ⋄ a = 𝟙 + + + lemma left_inv_eq_right_inv₁ {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by + rw [← DiaOneClass₁.one_dia c, ← hba, Semigroup₁.dia_assoc, hac, DiaOneClass₁.dia_one b] + + + export DiaOneClass₁ (one_dia dia_one) + export Semigroup₁ (dia_assoc) + export Group₁ (inv_dia) + + + example {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by + rw [← one_dia c, ← hba, dia_assoc, hac, dia_one b] + + + lemma inv_eq_of_dia [Group₁ G] {a b : G} (h : a ⋄ b = 𝟙) : a⁻¹ = b := + sorry + + lemma dia_inv [Group₁ G] (a : G) : a ⋄ a⁻¹ = 𝟙 := + sorry + + + + + class AddSemigroup₃ (α : Type) extends Add α where + /-- Multiplication is associative -/ + add_assoc₃ : ∀ a b c : α, a + b + c = a + (b + c) + + @[to_additive AddSemigroup₃] + class Semigroup₃ (α : Type) extends Mul α where + /-- Multiplication is associative -/ + mul_assoc₃ : ∀ a b c : α, a * b * c = a * (b * c) + + class AddMonoid₃ (α : Type) extends AddSemigroup₃ α, AddZeroClass α + + @[to_additive AddMonoid₃] + class Monoid₃ (α : Type) extends Semigroup₃ α, MulOneClass α + + attribute [to_additive existing] Monoid₃.toMulOneClass + + export Semigroup₃ (mul_assoc₃) + export AddSemigroup₃ (add_assoc₃) + + whatsnew in + @[to_additive] + lemma left_inv_eq_right_inv' {M : Type} [Monoid₃ M] {a b c : M} (hba : b * a = 1) (hac : a * c = 1) : b = c := by + rw [← one_mul c, ← hba, mul_assoc₃, hac, mul_one b] + + #check left_neg_eq_right_neg' + + class AddCommSemigroup₃ (α : Type) extends AddSemigroup₃ α where + add_comm : ∀ a b : α, a + b = b + a + + @[to_additive AddCommSemigroup₃] + class CommSemigroup₃ (α : Type) extends Semigroup₃ α where + mul_comm : ∀ a b : α, a * b = b * a + + class AddCommMonoid₃ (α : Type) extends AddMonoid₃ α, AddCommSemigroup₃ α + + @[to_additive AddCommMonoid₃] + class CommMonoid₃ (α : Type) extends Monoid₃ α, CommSemigroup₃ α + + class AddGroup₃ (G : Type) extends AddMonoid₃ G, Neg G where + neg_add : ∀ a : G, -a + a = 0 + + @[to_additive AddGroup₃] + class Group₃ (G : Type) extends Monoid₃ G, Inv G where + inv_mul : ∀ a : G, a⁻¹ * a = 1 + + + attribute [simp] Group₃.inv_mul AddGroup₃.neg_add + + + + @[to_additive] + lemma inv_eq_of_mul [Group₃ G] {a b : G} (h : a * b = 1) : a⁻¹ = b := + sorry + + + @[to_additive (attr := simp)] + lemma Group₃.mul_inv {G : Type} [Group₃ G] {a : G} : a * a⁻¹ = 1 := by + sorry + + @[to_additive] + lemma mul_left_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : a * b = a * c) : b = c := by + sorry + + @[to_additive] + lemma mul_right_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : b*a = c*a) : b = c := by + sorry + + class AddCommGroup₃ (G : Type) extends AddGroup₃ G, AddCommMonoid₃ G + + @[to_additive AddCommGroup₃] + class CommGroup₃ (G : Type) extends Group₃ G, CommMonoid₃ G + + + + class Ring₃ (R : Type) extends AddGroup₃ R, Monoid₃ R, MulZeroClass R where + /-- Multiplication is left distributive over addition -/ + left_distrib : ∀ a b c : R, a * (b + c) = a * b + a * c + /-- Multiplication is right distributive over addition -/ + right_distrib : ∀ a b c : R, (a + b) * c = a * c + b * c + + instance {R : Type} [Ring₃ R] : AddCommGroup₃ R := + { Ring₃.toAddGroup₃ with + add_comm := by + sorry } + + instance : Ring₃ ℤ where + add := (· + ·) + add_assoc₃ := add_assoc + zero := 0 + zero_add := by simp + add_zero := by simp + neg := (- ·) + neg_add := by simp + mul := (· * ·) + mul_assoc₃ := mul_assoc + one := 1 + one_mul := by simp + mul_one := by simp + zero_mul := by simp + mul_zero := by simp + left_distrib := Int.mul_add + right_distrib := Int.add_mul + + class LE₁ (α : Type) where + /-- The Less-or-Equal relation. -/ + le : α → α → Prop + + @[inherit_doc] infix:50 " ≤₁ " => LE₁.le + + class Preorder₁ (α : Type) + + class PartialOrder₁ (α : Type) + + class OrderedCommMonoid₁ (α : Type) + + instance : OrderedCommMonoid₁ ℕ where + + class SMul₃ (α : Type) (β : Type) where + /-- Scalar multiplication -/ + smul : α → β → β + + infixr:73 " • " => SMul₃.smul + + + class Module₁ (R : Type) [Ring₃ R] (M : Type) [AddCommGroup₃ M] extends SMul₃ R M where + zero_smul : ∀ m : M, (0 : R) • m = 0 + one_smul : ∀ m : M, (1 : R) • m = m + mul_smul : ∀ (a b : R) (m : M), (a * b) • m = a • b • m + add_smul : ∀ (a b : R) (m : M), (a + b) • m = a • m + b • m + smul_add : ∀ (a : R) (m n : M), a • (m + n) = a • m + a • n + + instance selfModule (R : Type) [Ring₃ R] : Module₁ R R where + smul := fun r s ↦ r*s + zero_smul := zero_mul + one_smul := one_mul + mul_smul := mul_assoc₃ + add_smul := Ring₃.right_distrib + smul_add := Ring₃.left_distrib + + def nsmul₁ [Zero M] [Add M] : ℕ → M → M + | 0, _ => 0 + | n + 1, a => a + nsmul₁ n a + + def zsmul₁ {M : Type _} [Zero M] [Add M] [Neg M] : ℤ → M → M + | Int.ofNat n, a => nsmul₁ n a + | Int.negSucc n, a => -nsmul₁ n.succ a + + instance abGrpModule (A : Type) [AddCommGroup₃ A] : Module₁ ℤ A where + smul := zsmul₁ + zero_smul := sorry + one_smul := sorry + mul_smul := sorry + add_smul := sorry + smul_add := sorry + + #synth Module₁ ℤ ℤ -- abGrpModule ℤ + + + class AddMonoid₄ (M : Type) extends AddSemigroup₃ M, AddZeroClass M where + /-- Multiplication by a natural number. -/ + nsmul : ℕ → M → M := nsmul₁ + /-- Multiplication by `(0 : ℕ)` gives `0`. -/ + nsmul_zero : ∀ x, nsmul 0 x = 0 := by intros; rfl + /-- Multiplication by `(n + 1 : ℕ)` behaves as expected. -/ + nsmul_succ : ∀ (n : ℕ) (x), nsmul (n + 1) x = x + nsmul n x := by intros; rfl + + instance mySMul {M : Type} [AddMonoid₄ M] : SMul ℕ M := ⟨AddMonoid₄.nsmul⟩ + + instance (M N : Type) [AddMonoid₄ M] [AddMonoid₄ N] : AddMonoid₄ (M × N) where + add := fun p q ↦ (p.1 + q.1, p.2 + q.2) + add_assoc₃ := fun a b c ↦ by ext <;> apply add_assoc₃ + zero := (0, 0) + zero_add := fun a ↦ by ext <;> apply zero_add + add_zero := fun a ↦ by ext <;> apply add_zero + + instance : AddMonoid₄ ℤ where + add := (· + ·) + add_assoc₃ := Int.add_assoc + zero := 0 + zero_add := Int.zero_add + add_zero := Int.add_zero + nsmul := fun n m ↦ (n : ℤ) * m + nsmul_zero := Int.zero_mul + nsmul_succ := fun n m ↦ show (n + 1 : ℤ) * m = m + n * m + by rw [Int.add_mul, Int.add_comm, Int.one_mul] + + example (n : ℕ) (m : ℤ) : SMul.smul (self := mySMul) n m = n * m := rfl
src/C_Hierarchies/S02_Morphisms.lean (new)
- @@ -0,0 +1,114 @@
  + import Mathlib.Topology.Instances.Real + + + def isMonoidHom₁ [Monoid G] [Monoid H] (f : G → H) : Prop := + f 1 = 1 ∧ ∀ g g', f (g * g') = f g * f g' + structure isMonoidHom₂ [Monoid G] [Monoid H] (f : G → H) : Prop where + map_one : f 1 = 1 + map_mul : ∀ g g', f (g * g') = f g * f g' + example : Continuous (id : ℝ → ℝ) := continuous_id + @[ext] + structure MonoidHom₁ (G H : Type) [Monoid G] [Monoid H] where + toFun : G → H + map_one : toFun 1 = 1 + map_mul : ∀ g g', toFun (g * g') = toFun g * toFun g' + + instance [Monoid G] [Monoid H] : CoeFun (MonoidHom₁ G H) (fun _ ↦ G → H) where + coe := MonoidHom₁.toFun + + attribute [coe] MonoidHom₁.toFun + + + example [Monoid G] [Monoid H] (f : MonoidHom₁ G H) : f 1 = 1 := f.map_one + + @[ext] + structure AddMonoidHom₁ (G H : Type) [AddMonoid G] [AddMonoid H] where + toFun : G → H + map_zero : toFun 0 = 0 + map_add : ∀ g g', toFun (g + g') = toFun g + toFun g' + + instance [AddMonoid G] [AddMonoid H] : CoeFun (AddMonoidHom₁ G H) (fun _ ↦ G → H) where + coe := AddMonoidHom₁.toFun + + attribute [coe] AddMonoidHom₁.toFun + + @[ext] + structure RingHom₁ (R S : Type) [Ring R] [Ring S] extends MonoidHom₁ R S, AddMonoidHom₁ R S + + + + class MonoidHomClass₁ (F : Type) (M N : Type) [Monoid M] [Monoid N] where + toFun : F → M → N + map_one : ∀ f : F, toFun f 1 = 1 + map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' + + + def badInst [Monoid M] [Monoid N] [MonoidHomClass₁ F M N] : CoeFun F (fun _ ↦ M → N) where + coe := MonoidHomClass₁.toFun + + + class MonoidHomClass₂ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] where + toFun : F → M → N + map_one : ∀ f : F, toFun f 1 = 1 + map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' + + instance [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] : CoeFun F (fun _ ↦ M → N) where + coe := MonoidHomClass₂.toFun + + attribute [coe] MonoidHomClass₂.toFun + + + instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₂ (MonoidHom₁ M N) M N where + toFun := MonoidHom₁.toFun + map_one := fun f ↦ f.map_one + map_mul := fun f ↦ f.map_mul + + instance (R S : Type) [Ring R] [Ring S] : MonoidHomClass₂ (RingHom₁ R S) R S where + toFun := fun f ↦ f.toMonoidHom₁.toFun + map_one := fun f ↦ f.toMonoidHom₁.map_one + map_mul := fun f ↦ f.toMonoidHom₁.map_mul + + + lemma map_inv_of_inv [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] (f : F) {m m' : M} (h : m*m' = 1) : + f m * f m' = 1 := by + rw [← MonoidHomClass₂.map_mul, h, MonoidHomClass₂.map_one] + + example [Monoid M] [Monoid N] (f : MonoidHom₁ M N) {m m' : M} (h : m*m' = 1) : f m * f m' = 1 := + map_inv_of_inv f h + + example [Ring R] [Ring S] (f : RingHom₁ R S) {r r' : R} (h : r*r' = 1) : f r * f r' = 1 := + map_inv_of_inv f h + + + + class MonoidHomClass₃ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] extends + FunLike F M (fun _ ↦ N) where + map_one : ∀ f : F, f 1 = 1 + map_mul : ∀ (f : F) g g', f (g * g') = f g * f g' + + instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₃ (MonoidHom₁ M N) M N where + coe := MonoidHom₁.toFun + coe_injective' := MonoidHom₁.ext + map_one := MonoidHom₁.map_one + map_mul := MonoidHom₁.map_mul + + + @[ext] + structure OrderPresHom (α β : Type) [LE α] [LE β] where + toFun : α → β + le_of_le : ∀ a a', a ≤ a' → toFun a ≤ toFun a' + + @[ext] + structure OrderPresMonoidHom (M N : Type) [Monoid M] [LE M] [Monoid N] [LE N] extends + MonoidHom₁ M N, OrderPresHom M N + + class OrderPresHomClass (F : Type) (α β : outParam Type) [LE α] [LE β] + + instance (α β : Type) [LE α] [LE β] : OrderPresHomClass (OrderPresHom α β) α β where + + instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : + OrderPresHomClass (OrderPresMonoidHom α β) α β where + + instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : + MonoidHomClass₃ (OrderPresMonoidHom α β) α β + := sorry
src/C_Hierarchies/S03_Subobjects.lean (new)
- @@ -0,0 +1,88 @@
  + import Mathlib.GroupTheory.QuotientGroup + + + + @[ext] + structure Submonoid₁ (M : Type) [Monoid M] where + /-- The carrier of a submonoid. -/ + carrier : Set M + /-- The product of two elements of a submonoid belongs to the submonoid. -/ + mul_mem {a b} : a ∈ carrier → b ∈ carrier → a * b ∈ carrier + /-- The unit element belongs to the submonoid. -/ + one_mem : 1 ∈ carrier + + /-- Submonoids in `M` can be seen as sets in `M`. -/ + instance [Monoid M] : SetLike (Submonoid₁ M) M where + coe := Submonoid₁.carrier + coe_injective' := Submonoid₁.ext + + + + example [Monoid M] (N : Submonoid₁ M) : 1 ∈ N := N.one_mem + + example [Monoid M] (N : Submonoid₁ M) (α : Type) (f : M → α) := f '' N + + + example [Monoid M] (N : Submonoid₁ M) (x : N) : (x : M) ∈ N := x.property + + + instance SubMonoid₁Monoid [Monoid M] (N : Submonoid₁ M) : Monoid N where + mul := fun x y ↦ ⟨x*y, N.mul_mem x.property y.property⟩ + mul_assoc := fun x y z ↦ SetCoe.ext (mul_assoc (x : M) y z) + one := ⟨1, N.one_mem⟩ + one_mul := fun x ↦ SetCoe.ext (one_mul (x : M)) + mul_one := fun x ↦ SetCoe.ext (mul_one (x : M)) + + + example [Monoid M] (N : Submonoid₁ M) : Monoid N where + mul := fun ⟨x, hx⟩ ⟨y, hy⟩ ↦ ⟨x*y, N.mul_mem hx hy⟩ + mul_assoc := fun ⟨x, _⟩ ⟨y, _⟩ ⟨z, _⟩ ↦ SetCoe.ext (mul_assoc x y z) + one := ⟨1, N.one_mem⟩ + one_mul := fun ⟨x, _⟩ ↦ SetCoe.ext (one_mul x) + mul_one := fun ⟨x, _⟩ ↦ SetCoe.ext (mul_one x) + + + class SubmonoidClass₁ (S : Type) (M : Type) [Monoid M] [SetLike S M] : Prop where + mul_mem : ∀ (s : S) {a b : M}, a ∈ s → b ∈ s → a * b ∈ s + one_mem : ∀ s : S, 1 ∈ s + + instance [Monoid M] : SubmonoidClass₁ (Submonoid₁ M) M where + mul_mem := Submonoid₁.mul_mem + one_mem := Submonoid₁.one_mem + + + instance [Monoid M] : Inf (Submonoid₁ M) := + ⟨fun S₁ S₂ => + { carrier := S₁ ∩ S₂ + one_mem := ⟨S₁.one_mem, S₂.one_mem⟩ + mul_mem := fun ⟨hx, hx'⟩ ⟨hy, hy'⟩ ↦ ⟨S₁.mul_mem hx hy, S₂.mul_mem hx' hy'⟩ }⟩ + + + example [Monoid M] (N P : Submonoid₁ M) : Submonoid₁ M := N ⊓ P + + + def Submonoid.Setoid [CommMonoid M] (N : Submonoid M) : Setoid M where + r := fun x y ↦ ∃ w ∈ N, ∃ z ∈ N, x*w = y*z + iseqv := { + refl := fun x ↦ ⟨1, N.one_mem, 1, N.one_mem, rfl⟩ + symm := fun ⟨w, hw, z, hz, h⟩ ↦ ⟨z, hz, w, hw, h.symm⟩ + trans := by + sorry + } + + instance [CommMonoid M] : HasQuotient M (Submonoid M) where + quotient' := fun N ↦ Quotient N.Setoid + + def QuotientMonoid.mk [CommMonoid M] (N : Submonoid M) : M → M ⧸ N := Quotient.mk N.Setoid + + instance [CommMonoid M] (N : Submonoid M) : Monoid (M ⧸ N) where + mul := Quotient.map₂' (· * ·) (by + sorry + ) + mul_assoc := by + sorry + one := QuotientMonoid.mk N 1 + one_mul := by + sorry + mul_one := by + sorry
src/C_Hierarchies/solutions/Solutions_S01_Basics.lean (new)
- @@ -0,0 +1,348 @@
 + import Mathlib.Algebra.BigOperators.Ring + import Mathlib.Data.Real.Basic + + + class One₁ (α : Type) where + /-- The element one -/ + one : α + + + #check One₁.one -- One₁.one {α : Type} [self : One₁ α] : α + + @[class] structure One₂ (α : Type) where + /-- The element one -/ + one : α + + #check One₂.one + + + example (α : Type) [One₁ α] : α := One₁.one + + example (α : Type) [One₁ α] := (One₁.one : α) + + @[inherit_doc] + notation "𝟙" => One₁.one + + example {α : Type} [One₁ α] : α := 𝟙 + + example {α : Type} [One₁ α] : (𝟙 : α) = 𝟙 := rfl + + + class Dia₁ (α : Type) where + dia : α → α → α + + infixl:70 " ⋄ " => Dia₁.dia + + + class Semigroup₁ (α : Type) where + toDia₁ : Dia₁ α + /-- Diamond is associative -/ + dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) + + + attribute [instance] Semigroup₁.toDia₁ + + example {α : Type} [Semigroup₁ α] (a b : α) : α := a ⋄ b + + + class Semigroup₂ (α : Type) extends Dia₁ α where + /-- Diamond is associative -/ + dia_assoc : ∀ a b c : α, a ⋄ b ⋄ c = a ⋄ (b ⋄ c) + + example {α : Type} [Semigroup₂ α] (a b : α) : α := a ⋄ b + + class DiaOneClass₁ (α : Type) extends One₁ α, Dia₁ α where + /-- One is a left neutral element for diamond. -/ + one_dia : ∀ a : α, 𝟙 ⋄ a = a + /-- One is a right neutral element for diamond -/ + dia_one : ∀ a : α, a ⋄ 𝟙 = a + + + + set_option trace.Meta.synthInstance true in + example {α : Type} [DiaOneClass₁ α] (a b : α) : Prop := a ⋄ b = 𝟙 + + + class Monoid₁ (α : Type) extends Semigroup₁ α, DiaOneClass₁ α + + + class Monoid₂ (α : Type) where + toSemigroup₁ : Semigroup₁ α + toDiaOneClass₁ : DiaOneClass₁ α + + + example {α : Type} [Monoid₁ α] : + (Monoid₁.toSemigroup₁.toDia₁.dia : α → α → α) = Monoid₁.toDiaOneClass₁.toDia₁.dia := rfl + + + /- Monoid₂.mk {α : Type} (toSemigroup₁ : Semigroup₁ α) (toDiaOneClass₁ : DiaOneClass₁ α) : Monoid₂ α -/ + #check Monoid₂.mk + + /- Monoid₁.mk {α : Type} [toSemigroup₁ : Semigroup₁ α] [toOne₁ : One₁ α] (one_dia : ∀ (a : α), 𝟙 ⋄ a = a) (dia_one : ∀ (a : α), a ⋄ 𝟙 = a) : Monoid₁ α -/ + #check Monoid₁.mk + + + #check Monoid₁.toSemigroup₁ + #check Monoid₁.toDiaOneClass₁ + + + class Inv₁ (α : Type) where + /-- The inversion function -/ + inv : α → α + + @[inherit_doc] + postfix:max "⁻¹" => Inv₁.inv + + class Group₁ (G : Type) extends Monoid₁ G, Inv G where + inv_dia : ∀ a : G, a⁻¹ ⋄ a = 𝟙 + + + lemma left_inv_eq_right_inv₁ {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by + rw [← DiaOneClass₁.one_dia c, ← hba, Semigroup₁.dia_assoc, hac, DiaOneClass₁.dia_one b] + + + export DiaOneClass₁ (one_dia dia_one) + export Semigroup₁ (dia_assoc) + export Group₁ (inv_dia) + + + example {M : Type} [Monoid₁ M] {a b c : M} (hba : b ⋄ a = 𝟙) (hac : a ⋄ c = 𝟙) : b = c := by + rw [← one_dia c, ← hba, dia_assoc, hac, dia_one b] + + + lemma inv_eq_of_dia [Group₁ G] {a b : G} (h : a ⋄ b = 𝟙) : a⁻¹ = b := + left_inv_eq_right_inv₁ (inv_dia a) h + + lemma dia_inv [Group₁ G] (a : G) : a ⋄ a⁻¹ = 𝟙 := + by rw [← inv_dia a⁻¹, inv_eq_of_dia (inv_dia a)] + + + + + + class AddSemigroup₃ (α : Type) extends Add α where + /-- Multiplication is associative -/ + add_assoc₃ : ∀ a b c : α, a + b + c = a + (b + c) + + @[to_additive AddSemigroup₃] + class Semigroup₃ (α : Type) extends Mul α where + /-- Multiplication is associative -/ + mul_assoc₃ : ∀ a b c : α, a * b * c = a * (b * c) + + class AddMonoid₃ (α : Type) extends AddSemigroup₃ α, AddZeroClass α + + @[to_additive AddMonoid₃] + class Monoid₃ (α : Type) extends Semigroup₃ α, MulOneClass α + + attribute [to_additive existing] Monoid₃.toMulOneClass + + export Semigroup₃ (mul_assoc₃) + export AddSemigroup₃ (add_assoc₃) + + whatsnew in + @[to_additive] + lemma left_inv_eq_right_inv' {M : Type} [Monoid₃ M] {a b c : M} (hba : b * a = 1) (hac : a * c = 1) : b = c := by + rw [← one_mul c, ← hba, mul_assoc₃, hac, mul_one b] + + #check left_neg_eq_right_neg' + + class AddCommSemigroup₃ (α : Type) extends AddSemigroup₃ α where + add_comm : ∀ a b : α, a + b = b + a + + @[to_additive AddCommSemigroup₃] + class CommSemigroup₃ (α : Type) extends Semigroup₃ α where + mul_comm : ∀ a b : α, a * b = b * a + + class AddCommMonoid₃ (α : Type) extends AddMonoid₃ α, AddCommSemigroup₃ α + + @[to_additive AddCommMonoid₃] + class CommMonoid₃ (α : Type) extends Monoid₃ α, CommSemigroup₃ α + + class AddGroup₃ (G : Type) extends AddMonoid₃ G, Neg G where + neg_add : ∀ a : G, -a + a = 0 + + @[to_additive AddGroup₃] + class Group₃ (G : Type) extends Monoid₃ G, Inv G where + inv_mul : ∀ a : G, a⁻¹ * a = 1 + + + attribute [simp] Group₃.inv_mul AddGroup₃.neg_add + + + + @[to_additive] + lemma inv_eq_of_mul [Group₃ G] {a b : G} (h : a * b = 1) : a⁻¹ = b := + left_inv_eq_right_inv' (Group₃.inv_mul a) h + + + @[to_additive (attr := simp)] + lemma Group₃.mul_inv {G : Type} [Group₃ G] {a : G} : a * a⁻¹ = 1 := by + rw [← inv_mul a⁻¹, inv_eq_of_mul (inv_mul a)] + + @[to_additive] + lemma mul_left_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : a * b = a * c) : b = c := by + simpa [← mul_assoc₃] using congr_arg (a⁻¹ * ·) h + + @[to_additive] + lemma mul_right_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : b*a = c*a) : b = c := by + simpa [mul_assoc₃] using congr_arg (· * a⁻¹) h + + class AddCommGroup₃ (G : Type) extends AddGroup₃ G, AddCommMonoid₃ G + + @[to_additive AddCommGroup₃] + class CommGroup₃ (G : Type) extends Group₃ G, CommMonoid₃ G + + + + class Ring₃ (R : Type) extends AddGroup₃ R, Monoid₃ R, MulZeroClass R where + /-- Multiplication is left distributive over addition -/ + left_distrib : ∀ a b c : R, a * (b + c) = a * b + a * c + /-- Multiplication is right distributive over addition -/ + right_distrib : ∀ a b c : R, (a + b) * c = a * c + b * c + + instance {R : Type} [Ring₃ R] : AddCommGroup₃ R := + { Ring₃.toAddGroup₃ with + add_comm := by + intro a b + have : a + (a + b + b) = a + (b + a + b) := calc + a + (a + b + b) = (a + a) + (b + b) := by simp [add_assoc₃, add_assoc₃] + _ = (1 * a + 1 * a) + (1 * b + 1 * b) := by simp + _ = (1 + 1) * a + (1 + 1) * b := by simp [Ring₃.right_distrib] + _ = (1 + 1) * (a + b) := by simp [Ring₃.left_distrib] + _ = 1 * (a + b) + 1 * (a + b) := by simp [Ring₃.right_distrib] + _ = (a + b) + (a + b) := by simp + _ = a + (b + a + b) := by simp [add_assoc₃] + exact add_right_cancel₃ (add_left_cancel₃ this) } + + instance : Ring₃ ℤ where + add := (· + ·) + add_assoc₃ := add_assoc + zero := 0 + zero_add := by simp + add_zero := by simp + neg := (- ·) + neg_add := by simp + mul := (· * ·) + mul_assoc₃ := mul_assoc + one := 1 + one_mul := by simp + mul_one := by simp + zero_mul := by simp + mul_zero := by simp + left_distrib := Int.mul_add + right_distrib := Int.add_mul + + class LE₁ (α : Type) where + /-- The Less-or-Equal relation. -/ + le : α → α → Prop + + @[inherit_doc] infix:50 " ≤₁ " => LE₁.le + + class Preorder₁ (α : Type) + extends LE₁ α where + le_refl : ∀ a : α, a ≤₁ a + le_trans : ∀ a b c : α, a ≤₁ b → b ≤₁ c → a ≤₁ c + + class PartialOrder₁ (α : Type) + extends Preorder₁ α where + le_antisymm : ∀ a b : α, a ≤₁ b → b ≤₁ a → a = b + + class OrderedCommMonoid₁ (α : Type) + extends PartialOrder₁ α, CommMonoid₃ α where + mul_of_le : ∀ a b : α, a ≤₁ b → ∀ c : α, c * a ≤₁ c * b + + instance : OrderedCommMonoid₁ ℕ where + le := (· ≤ ·) + le_refl := fun _ ↦ le_rfl + le_trans := fun _ _ _ ↦ le_trans + le_antisymm := fun _ _ ↦ le_antisymm + mul := (· * ·) + mul_assoc₃ := mul_assoc + one := 1 + one_mul := one_mul + mul_one := mul_one + mul_comm := mul_comm + mul_of_le := fun _ _ h c ↦ Nat.mul_le_mul_left c h + + class SMul₃ (α : Type) (β : Type) where + /-- Scalar multiplication -/ + smul : α → β → β + + infixr:73 " • " => SMul₃.smul + + + class Module₁ (R : Type) [Ring₃ R] (M : Type) [AddCommGroup₃ M] extends SMul₃ R M where + zero_smul : ∀ m : M, (0 : R) • m = 0 + one_smul : ∀ m : M, (1 : R) • m = m + mul_smul : ∀ (a b : R) (m : M), (a * b) • m = a • b • m + add_smul : ∀ (a b : R) (m : M), (a + b) • m = a • m + b • m + smul_add : ∀ (a : R) (m n : M), a • (m + n) = a • m + a • n + + instance selfModule (R : Type) [Ring₃ R] : Module₁ R R where + smul := fun r s ↦ r*s + zero_smul := zero_mul + one_smul := one_mul + mul_smul := mul_assoc₃ + add_smul := Ring₃.right_distrib + smul_add := Ring₃.left_distrib + + def nsmul₁ [Zero M] [Add M] : ℕ → M → M + | 0, _ => 0 + | n + 1, a => a + nsmul₁ n a + + def zsmul₁ {M : Type _} [Zero M] [Add M] [Neg M] : ℤ → M → M + | Int.ofNat n, a => nsmul₁ n a + | Int.negSucc n, a => -nsmul₁ n.succ a + + instance abGrpModule (A : Type) [AddCommGroup₃ A] : Module₁ ℤ A where + smul := zsmul₁ + zero_smul := sorry + one_smul := sorry + mul_smul := sorry + add_smul := sorry + smul_add := sorry + + #synth Module₁ ℤ ℤ -- abGrpModule ℤ + + + class AddMonoid₄ (M : Type) extends AddSemigroup₃ M, AddZeroClass M where + /-- Multiplication by a natural number. -/ + nsmul : ℕ → M → M := nsmul₁ + /-- Multiplication by `(0 : ℕ)` gives `0`. -/ + nsmul_zero : ∀ x, nsmul 0 x = 0 := by intros; rfl + /-- Multiplication by `(n + 1 : ℕ)` behaves as expected. -/ + nsmul_succ : ∀ (n : ℕ) (x), nsmul (n + 1) x = x + nsmul n x := by intros; rfl + + instance mySMul {M : Type} [AddMonoid₄ M] : SMul ℕ M := ⟨AddMonoid₄.nsmul⟩ + + instance (M N : Type) [AddMonoid₄ M] [AddMonoid₄ N] : AddMonoid₄ (M × N) where + add := fun p q ↦ (p.1 + q.1, p.2 + q.2) + add_assoc₃ := fun a b c ↦ by ext <;> apply add_assoc₃ + zero := (0, 0) + zero_add := fun a ↦ by ext <;> apply zero_add + add_zero := fun a ↦ by ext <;> apply add_zero + + instance : AddMonoid₄ ℤ where + add := (· + ·) + add_assoc₃ := Int.add_assoc + zero := 0 + zero_add := Int.zero_add + add_zero := Int.add_zero + nsmul := fun n m ↦ (n : ℤ) * m + nsmul_zero := Int.zero_mul + nsmul_succ := fun n m ↦ show (n + 1 : ℤ) * m = m + n * m + by rw [Int.add_mul, Int.add_comm, Int.one_mul] + + example (n : ℕ) (m : ℤ) : SMul.smul (self := mySMul) n m = n * m := rfl + + class LT₁ (α : Type) where + /-- The Less-Than relation -/ + lt : α → α → Prop + + @[inherit_doc] infix:50 " <₁ " => LT₁.lt + + class PreOrder₂ (α : Type) extends LE₁ α, LT₁ α where + le_refl : ∀ a : α, a ≤₁ a + le_trans : ∀ a b c : α, a ≤₁ b → b ≤₁ c → a ≤₁ c + lt := fun a b => a ≤₁ b ∧ ¬b ≤₁ a + lt_iff_le_not_le : ∀ a b : α, a <₁ b ↔ a ≤₁ b ∧ ¬b ≤₁ a := by intros; rfl
src/C_Hierarchies/solutions/Solutions_S02_Morphisms.lean (new)
- @@ -0,0 +1,126 @@
  + import Mathlib.Topology.Instances.Real + + + def isMonoidHom₁ [Monoid G] [Monoid H] (f : G → H) : Prop := + f 1 = 1 ∧ ∀ g g', f (g * g') = f g * f g' + structure isMonoidHom₂ [Monoid G] [Monoid H] (f : G → H) : Prop where + map_one : f 1 = 1 + map_mul : ∀ g g', f (g * g') = f g * f g' + example : Continuous (id : ℝ → ℝ) := continuous_id + @[ext] + structure MonoidHom₁ (G H : Type) [Monoid G] [Monoid H] where + toFun : G → H + map_one : toFun 1 = 1 + map_mul : ∀ g g', toFun (g * g') = toFun g * toFun g' + + instance [Monoid G] [Monoid H] : CoeFun (MonoidHom₁ G H) (fun _ ↦ G → H) where + coe := MonoidHom₁.toFun + + attribute [coe] MonoidHom₁.toFun + + + example [Monoid G] [Monoid H] (f : MonoidHom₁ G H) : f 1 = 1 := f.map_one + + @[ext] + structure AddMonoidHom₁ (G H : Type) [AddMonoid G] [AddMonoid H] where + toFun : G → H + map_zero : toFun 0 = 0 + map_add : ∀ g g', toFun (g + g') = toFun g + toFun g' + + instance [AddMonoid G] [AddMonoid H] : CoeFun (AddMonoidHom₁ G H) (fun _ ↦ G → H) where + coe := AddMonoidHom₁.toFun + + attribute [coe] AddMonoidHom₁.toFun + + @[ext] + structure RingHom₁ (R S : Type) [Ring R] [Ring S] extends MonoidHom₁ R S, AddMonoidHom₁ R S + + + + class MonoidHomClass₁ (F : Type) (M N : Type) [Monoid M] [Monoid N] where + toFun : F → M → N + map_one : ∀ f : F, toFun f 1 = 1 + map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' + + + def badInst [Monoid M] [Monoid N] [MonoidHomClass₁ F M N] : CoeFun F (fun _ ↦ M → N) where + coe := MonoidHomClass₁.toFun + + + class MonoidHomClass₂ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] where + toFun : F → M → N + map_one : ∀ f : F, toFun f 1 = 1 + map_mul : ∀ f g g', toFun f (g * g') = toFun f g * toFun f g' + + instance [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] : CoeFun F (fun _ ↦ M → N) where + coe := MonoidHomClass₂.toFun + + attribute [coe] MonoidHomClass₂.toFun + + + instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₂ (MonoidHom₁ M N) M N where + toFun := MonoidHom₁.toFun + map_one := fun f ↦ f.map_one + map_mul := fun f ↦ f.map_mul + + instance (R S : Type) [Ring R] [Ring S] : MonoidHomClass₂ (RingHom₁ R S) R S where + toFun := fun f ↦ f.toMonoidHom₁.toFun + map_one := fun f ↦ f.toMonoidHom₁.map_one + map_mul := fun f ↦ f.toMonoidHom₁.map_mul + + + lemma map_inv_of_inv [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] (f : F) {m m' : M} (h : m*m' = 1) : + f m * f m' = 1 := by + rw [← MonoidHomClass₂.map_mul, h, MonoidHomClass₂.map_one] + + example [Monoid M] [Monoid N] (f : MonoidHom₁ M N) {m m' : M} (h : m*m' = 1) : f m * f m' = 1 := + map_inv_of_inv f h + + example [Ring R] [Ring S] (f : RingHom₁ R S) {r r' : R} (h : r*r' = 1) : f r * f r' = 1 := + map_inv_of_inv f h + + + + class MonoidHomClass₃ (F : Type) (M N : outParam Type) [Monoid M] [Monoid N] extends + FunLike F M (fun _ ↦ N) where + map_one : ∀ f : F, f 1 = 1 + map_mul : ∀ (f : F) g g', f (g * g') = f g * f g' + + instance (M N : Type) [Monoid M] [Monoid N] : MonoidHomClass₃ (MonoidHom₁ M N) M N where + coe := MonoidHom₁.toFun + coe_injective' := MonoidHom₁.ext + map_one := MonoidHom₁.map_one + map_mul := MonoidHom₁.map_mul + + + @[ext] + structure OrderPresHom (α β : Type) [LE α] [LE β] where + toFun : α → β + le_of_le : ∀ a a', a ≤ a' → toFun a ≤ toFun a' + + @[ext] + structure OrderPresMonoidHom (M N : Type) [Monoid M] [LE M] [Monoid N] [LE N] extends + MonoidHom₁ M N, OrderPresHom M N + + class OrderPresHomClass (F : Type) (α β : outParam Type) [LE α] [LE β] + extends FunLike F α (fun _ ↦ β) where + le_of_le : ∀ (f : F) a a', a ≤ a' → f a ≤ f a' + + instance (α β : Type) [LE α] [LE β] : OrderPresHomClass (OrderPresHom α β) α β where + coe := OrderPresHom.toFun + coe_injective' := OrderPresHom.ext + le_of_le := OrderPresHom.le_of_le + + instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : + OrderPresHomClass (OrderPresMonoidHom α β) α β where + coe := fun f ↦ f.toOrderPresHom.toFun + coe_injective' := OrderPresMonoidHom.ext + le_of_le := fun f ↦ f.toOrderPresHom.le_of_le + + instance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] : + MonoidHomClass₃ (OrderPresMonoidHom α β) α β + where + coe := fun f ↦ f.toOrderPresHom.toFun + coe_injective' := OrderPresMonoidHom.ext + map_one := fun f ↦ f.toMonoidHom₁.map_one + map_mul := fun f ↦ f.toMonoidHom₁.map_mul
src/C_Hierarchies/solutions/Solutions_S03_Subobjects.lean (new)
- @@ -0,0 +1,126 @@
  + import Mathlib.GroupTheory.QuotientGroup + + + + @[ext] + structure Submonoid₁ (M : Type) [Monoid M] where + /-- The carrier of a submonoid. -/ + carrier : Set M + /-- The product of two elements of a submonoid belongs to the submonoid. -/ + mul_mem {a b} : a ∈ carrier → b ∈ carrier → a * b ∈ carrier + /-- The unit element belongs to the submonoid. -/ + one_mem : 1 ∈ carrier + + /-- Submonoids in `M` can be seen as sets in `M`. -/ + instance [Monoid M] : SetLike (Submonoid₁ M) M where + coe := Submonoid₁.carrier + coe_injective' := Submonoid₁.ext + + + + example [Monoid M] (N : Submonoid₁ M) : 1 ∈ N := N.one_mem + + example [Monoid M] (N : Submonoid₁ M) (α : Type) (f : M → α) := f '' N + + + example [Monoid M] (N : Submonoid₁ M) (x : N) : (x : M) ∈ N := x.property + + + instance SubMonoid₁Monoid [Monoid M] (N : Submonoid₁ M) : Monoid N where + mul := fun x y ↦ ⟨x*y, N.mul_mem x.property y.property⟩ + mul_assoc := fun x y z ↦ SetCoe.ext (mul_assoc (x : M) y z) + one := ⟨1, N.one_mem⟩ + one_mul := fun x ↦ SetCoe.ext (one_mul (x : M)) + mul_one := fun x ↦ SetCoe.ext (mul_one (x : M)) + + + example [Monoid M] (N : Submonoid₁ M) : Monoid N where + mul := fun ⟨x, hx⟩ ⟨y, hy⟩ ↦ ⟨x*y, N.mul_mem hx hy⟩ + mul_assoc := fun ⟨x, _⟩ ⟨y, _⟩ ⟨z, _⟩ ↦ SetCoe.ext (mul_assoc x y z) + one := ⟨1, N.one_mem⟩ + one_mul := fun ⟨x, _⟩ ↦ SetCoe.ext (one_mul x) + mul_one := fun ⟨x, _⟩ ↦ SetCoe.ext (mul_one x) + + + class SubmonoidClass₁ (S : Type) (M : Type) [Monoid M] [SetLike S M] : Prop where + mul_mem : ∀ (s : S) {a b : M}, a ∈ s → b ∈ s → a * b ∈ s + one_mem : ∀ s : S, 1 ∈ s + + instance [Monoid M] : SubmonoidClass₁ (Submonoid₁ M) M where + mul_mem := Submonoid₁.mul_mem + one_mem := Submonoid₁.one_mem + + @[ext] + structure Subgroup₁ (G : Type) [Group G] extends Submonoid₁ G where + /-- The inverse of an element of a subgroup belongs to the subgroup. -/ + inv_mem {a} : a ∈ carrier → a⁻¹ ∈ carrier + + + /-- Subgroups in `M` can be seen as sets in `M`. -/ + instance [Group G] : SetLike (Subgroup₁ G) G where + coe := fun H ↦ H.toSubmonoid₁.carrier + coe_injective' := Subgroup₁.ext + + instance [Group G] (H : Subgroup₁ G) : Group H := + { SubMonoid₁Monoid H.toSubmonoid₁ with + inv := fun x ↦ ⟨x⁻¹, H.inv_mem x.property⟩ + mul_left_inv := fun x ↦ SetCoe.ext (mul_left_inv (x : G)) } + + class SubgroupClass₁ (S : Type _) (G : Type) [Group G] [SetLike S G] + extends SubmonoidClass₁ S G : Prop where + inv_mem : ∀ (s : S) {a : G}, a ∈ s → a⁻¹ ∈ s + + instance [Group G] : SubmonoidClass₁ (Subgroup₁ G) G where + mul_mem := fun H ↦ H.toSubmonoid₁.mul_mem + one_mem := fun H ↦ H.toSubmonoid₁.one_mem + + instance [Group G] : SubgroupClass₁ (Subgroup₁ G) G := + { (inferInstance : SubmonoidClass₁ (Subgroup₁ G) G) with + inv_mem := Subgroup₁.inv_mem } + + + instance [Monoid M] : Inf (Submonoid₁ M) := + ⟨fun S₁ S₂ => + { carrier := S₁ ∩ S₂ + one_mem := ⟨S₁.one_mem, S₂.one_mem⟩ + mul_mem := fun ⟨hx, hx'⟩ ⟨hy, hy'⟩ ↦ ⟨S₁.mul_mem hx hy, S₂.mul_mem hx' hy'⟩ }⟩ + + + example [Monoid M] (N P : Submonoid₁ M) : Submonoid₁ M := N ⊓ P + + + def Submonoid.Setoid [CommMonoid M] (N : Submonoid M) : Setoid M where + r := fun x y ↦ ∃ w ∈ N, ∃ z ∈ N, x*w = y*z + iseqv := { + refl := fun x ↦ ⟨1, N.one_mem, 1, N.one_mem, rfl⟩ + symm := fun ⟨w, hw, z, hz, h⟩ ↦ ⟨z, hz, w, hw, h.symm⟩ + trans := by + rintro a b c ⟨w, hw, z, hz, h⟩ ⟨w', hw', z', hz', h'⟩ + refine ⟨w*w', N.mul_mem hw hw', z*z', N.mul_mem hz hz', ?_⟩ + rw [← mul_assoc, h, mul_comm b, mul_assoc, h', ← mul_assoc, mul_comm z, mul_assoc] + } + + instance [CommMonoid M] : HasQuotient M (Submonoid M) where + quotient' := fun N ↦ Quotient N.Setoid + + def QuotientMonoid.mk [CommMonoid M] (N : Submonoid M) : M → M ⧸ N := Quotient.mk N.Setoid + + instance [CommMonoid M] (N : Submonoid M) : Monoid (M ⧸ N) where + mul := Quotient.map₂' (· * ·) (by + rintro a₁ b₁ ⟨w, hw, z, hz, ha⟩ a₂ b₂ ⟨w', hw', z', hz', hb⟩ + refine ⟨w*w', N.mul_mem hw hw', z*z', N.mul_mem hz hz', ?_⟩ + rw [mul_comm w, ← mul_assoc, mul_assoc a₁, hb, mul_comm, ← mul_assoc, mul_comm w, ha, + mul_assoc, mul_comm z, mul_assoc b₂, mul_comm z', mul_assoc] + ) + mul_assoc := by + rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ + apply Quotient.sound + dsimp only + rw [mul_assoc] + apply @Setoid.refl M N.Setoid + one := QuotientMonoid.mk N 1 + one_mul := by + rintro ⟨a⟩ ; apply Quotient.sound ; dsimp only ; rw [one_mul] ; apply @Setoid.refl M N.Setoid + mul_one := by + rintro ⟨a⟩ ; apply Quotient.sound ; dsimp only ; rw [mul_one] ; apply @Setoid.refl M N.Setoid +
src/C_Introduction/S01_Getting_Started.lean (new)
- @@ -0,0 +1,2 @@
  + #eval "Hello, World!" +
src/C_Introduction/S02_Overview.lean (new)
- @@ -0,0 +1,56 @@
  + import Mathlib.Data.Nat.Basic + import Mathlib.Data.Nat.Parity + import Mathlib.Tactic + + open Nat + + -- These are pieces of data. + #check 2 + 2 + + def f (x : ℕ) := + x + 3 + + #check f + + -- These are propositions, of type `Prop`. + #check 2 + 2 = 4 + + def FermatLastTheorem := + ∀ x y z n : ℕ, n > 2 ∧ x * y * z ≠ 0 → x ^ n + y ^ n ≠ z ^ n + + #check FermatLastTheorem + + -- These are proofs of propositions. + theorem easy : 2 + 2 = 4 := + rfl + + #check easy + + theorem hard : FermatLastTheorem := + sorry + + #check hard + + -- Here are some proofs. + example : ∀ m n : Nat, Even n → Even (m * n) := fun m n ⟨k, (hk : n = k + k)⟩ => + have hmn : m * n = m * k + m * k := by rw [hk, mul_add] + show ∃ l, m * n = l + l from ⟨_, hmn⟩ + + example : ∀ m n : Nat, Even n → Even (m * n) := fun m n ⟨k, hk⟩ => ⟨m * k, by rw [hk, mul_add]⟩ + + example : ∀ m n : Nat, Even n → Even (m * n) := by + -- say m and n are natural numbers, and assume n=2*k + rintro m n ⟨k, hk⟩ + -- We need to prove m*n is twice a natural number. Let's show it's twice m*k. + use m * k + -- substitute in for n + rw [hk] + -- and now it's obvious + ring + + example : ∀ m n : Nat, Even n → Even (m * n) := by + rintro m n ⟨k, hk⟩; use m * k; rw [hk]; ring + + example : ∀ m n : Nat, Even n → Even (m * n) := by + intros; simp [*, parity_simps] +
src/C_Introduction/solutions/Solutions_S01_Getting_Started.lean (new)
src/C_Introduction/solutions/Solutions_S02_Overview.lean (new)
- @@ -0,0 +1,7 @@
  + import Mathlib.Data.Nat.Basic + import Mathlib.Data.Nat.Parity + import Mathlib.Tactic + + open Nat + + -- There are no exercises in this section.
src/C_Logic/S01_Implication_and_the_Universal_Quantifier.lean (new)
- @@ -0,0 +1,178 @@
 + import Mathlib.Data.Real.Basic + + #check ∀ x : ℝ, 0 ≤ x → abs x = x + + #check ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε + + theorem my_lemma : ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := + sorry + + section + variable (a b δ : ℝ) + variable (h₀ : 0 < δ) (h₁ : δ ≤ 1) + variable (ha : abs a < δ) (hb : abs b < δ) + + #check my_lemma a b δ + #check my_lemma a b δ h₀ h₁ + #check my_lemma a b δ h₀ h₁ ha hb + + end + + theorem my_lemma2 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := + sorry + + section + variable (a b δ : ℝ) + variable (h₀ : 0 < δ) (h₁ : δ ≤ 1) + variable (ha : abs a < δ) (hb : abs b < δ) + + #check my_lemma2 h₀ h₁ ha hb + + end + + theorem my_lemma3 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := by + intro x y ε epos ele1 xlt ylt + sorry + + theorem my_lemma4 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := by + intro x y ε epos ele1 xlt ylt + calc + abs (x * y) = abs x * abs y := sorry + _ ≤ abs x * ε := sorry + _ < 1 * ε := sorry + _ = ε := sorry + + def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, f x ≤ a + + def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, a ≤ f x + + section + variable (f g : ℝ → ℝ) (a b : ℝ) + + example (hfa : FnUb f a) (hgb : FnUb g b) : FnUb (fun x ↦ f x + g x) (a + b) := by + intro x + dsimp + apply add_le_add + apply hfa + apply hgb + + example (hfa : FnLb f a) (hgb : FnLb g b) : FnLb (fun x => f x + g x) (a + b) := + sorry + + example (nnf : FnLb f 0) (nng : FnLb g 0) : FnLb (fun x => f x * g x) 0 := + sorry + + example (hfa : FnUb f a) (hfb : FnUb g b) (nng : FnLb g 0) (nna : 0 ≤ a) : + FnUb (fun x => f x * g x) (a * b) := + sorry + + end + + section + variable {α : Type _} {R : Type _} [OrderedCancelAddCommMonoid R] + + #check @add_le_add + + def FnUb' (f : α → R) (a : R) : Prop := + ∀ x, f x ≤ a + + theorem fn_ub_add {f g : α → R} {a b : R} (hfa : FnUb' f a) (hgb : FnUb' g b) : + FnUb' (fun x => f x + g x) (a + b) := fun x => add_le_add (hfa x) (hgb x) + + end + + example (f : ℝ → ℝ) (h : Monotone f) : ∀ {a b}, a ≤ b → f a ≤ f b := + @h + + section + variable (f g : ℝ → ℝ) + + example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f x + g x := by + intro a b aleb + apply add_le_add + apply mf aleb + apply mg aleb + + example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f x + g x := + fun a b aleb => add_le_add (mf aleb) (mg aleb) + + example {c : ℝ} (mf : Monotone f) (nnc : 0 ≤ c) : Monotone fun x => c * f x := + sorry + + example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f (g x) := + sorry + + def FnEven (f : ℝ → ℝ) : Prop := + ∀ x, f x = f (-x) + + def FnOdd (f : ℝ → ℝ) : Prop := + ∀ x, f x = -f (-x) + + example (ef : FnEven f) (eg : FnEven g) : FnEven fun x => f x + g x := by + intro x + calc + (fun x => f x + g x) x = f x + g x := rfl + _ = f (-x) + g (-x) := by rw [ef, eg] + + + example (of : FnOdd f) (og : FnOdd g) : FnEven fun x => f x * g x := by + sorry + + example (ef : FnEven f) (og : FnOdd g) : FnOdd fun x => f x * g x := by + sorry + + example (ef : FnEven f) (og : FnOdd g) : FnEven fun x => f (g x) := by + sorry + + end + + section + + variable {α : Type _} (r s t : Set α) + + example : s ⊆ s := by + intro x xs + exact xs + + theorem Subset.refl : s ⊆ s := fun x xs => xs + + theorem Subset.trans : r ⊆ s → s ⊆ t → r ⊆ t := by + sorry + + end + + section + + variable {α : Type _} [PartialOrder α] + + variable (s : Set α) (a b : α) + + def SetUb (s : Set α) (a : α) := + ∀ x, x ∈ s → x ≤ a + + example (h : SetUb s a) (h' : a ≤ b) : SetUb s b := + sorry + + end + + section + + open Function + + example (c : ℝ) : Injective fun x => x + c := by + intro x₁ x₂ h' + exact (add_left_inj c).mp h' + + example {c : ℝ} (h : c ≠ 0) : Injective fun x => c * x := by + sorry + + variable {α : Type _} {β : Type _} {γ : Type _} + + variable {g : β → γ} {f : α → β} + + example (injg : Injective g) (injf : Injective f) : Injective fun x => g (f x) := by + sorry + + end
src/C_Logic/S02_The_Existential_Quantifier.lean (new)
- @@ -0,0 +1,135 @@
 + import Mathlib.Data.Real.Basic + + example : ∃ x : ℝ, 2 < x ∧ x < 3 := by + use 5 / 2 + norm_num + + example : ∃ x : ℝ, 2 < x ∧ x < 3 := + have h : 2 < (5 : ℝ) / 2 ∧ (5 : ℝ) / 2 < 3 := by norm_num + ⟨5 / 2, h⟩ + + example : ∃ x : ℝ, 2 < x ∧ x < 3 := + ⟨5 / 2, by norm_num⟩ + + def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, f x ≤ a + + def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, a ≤ f x + + def FnHasUb (f : ℝ → ℝ) := + ∃ a, FnUb f a + + def FnHasLb (f : ℝ → ℝ) := + ∃ a, FnLb f a + + theorem fnUb_add {f g : ℝ → ℝ} {a b : ℝ} (hfa : FnUb f a) (hgb : FnUb g b) : + FnUb (fun x => f x + g x) (a + b) := + fun x => add_le_add (hfa x) (hgb x) + + section + + variable {f g : ℝ → ℝ} + + example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x => f x + g x := by + cases' ubf with a ubfa + cases' ubg with b ubfb + use a + b + apply fnUb_add ubfa ubfb + + example (lbf : FnHasLb f) (lbg : FnHasLb g) : FnHasLb fun x => f x + g x := by + sorry + + example {c : ℝ} (ubf : FnHasUb f) (h : c ≥ 0) : FnHasUb fun x => c * f x := by + sorry + + example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x => f x + g x := by + rcases ubf with ⟨a, ubfa⟩ + rcases ubg with ⟨b, ubfb⟩ + exact ⟨a + b, fnUb_add ubfa ubfb⟩ + + example : FnHasUb f → FnHasUb g → FnHasUb fun x => f x + g x := by + rintro ⟨a, ubfa⟩ ⟨b, ubfb⟩ + exact ⟨a + b, fnUb_add ubfa ubfb⟩ + + example : FnHasUb f → FnHasUb g → FnHasUb fun x => f x + g x := + fun ⟨a, ubfa⟩ ⟨b, ubfb⟩ => ⟨a + b, fnUb_add ubfa ubfb⟩ + + end + + section + + variable {α : Type _} [CommRing α] + + def SumOfSquares (x : α) := + ∃ a b, x = a ^ 2 + b ^ 2 + + theorem sumOfSquares_mul {x y : α} (sosx : SumOfSquares x) (sosy : SumOfSquares y) : + SumOfSquares (x * y) := by + rcases sosx with ⟨a, b, xeq⟩ + rcases sosy with ⟨c, d, yeq⟩ + rw [xeq, yeq] + use a * c - b * d, a * d + b * c + ring + + theorem sumOfSquares_mul' {x y : α} (sosx : SumOfSquares x) (sosy : SumOfSquares y) : + SumOfSquares (x * y) := by + rcases sosx with ⟨a, b, rfl⟩ + rcases sosy with ⟨c, d, rfl⟩ + use a * c - b * d, a * d + b * c + ring + + end + + section + + variable {a b c : ℕ} + + example (divab : a ∣ b) (divbc : b ∣ c) : a ∣ c := by + cases' divab with d beq + cases' divbc with e ceq + rw [ceq, beq] + use d * e; ring + + example (divab : a ∣ b) (divac : a ∣ c) : a ∣ b + c := by + sorry + + end + + section + + open Function + + example {c : ℝ} : Surjective fun x => x + c := by + intro x + use x - c + dsimp; ring + + example {c : ℝ} (h : c ≠ 0) : Surjective fun x => c * x := by + sorry + + example (x y : ℝ) (h : x - y ≠ 0) : (x ^ 2 - y ^ 2) / (x - y) = x + y := by + field_simp [h] + ring + + example {f : ℝ → ℝ} (h : Surjective f) : ∃ x, f x ^ 2 = 4 := by + cases' h 2 with x hx + use x + rw [hx] + norm_num + + end + + section + + open Function + + variable {α : Type _} {β : Type _} {γ : Type _} + + variable {g : β → γ} {f : α → β} + + example (surjg : Surjective g) (surjf : Surjective f) : Surjective fun x => g (f x) := by + sorry + + end +
src/C_Logic/S03_Negation.lean (new)
- @@ -0,0 +1,143 @@
 + import Mathlib.Data.Real.Basic + + section + + variable (a b : ℝ) + + example (h : a < b) : ¬b < a := by + intro h' + have : a < a := lt_trans h h' + apply lt_irrefl a this + + def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, f x ≤ a + + def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, a ≤ f x + + def FnHasUb (f : ℝ → ℝ) := + ∃ a, FnUb f a + + def FnHasLb (f : ℝ → ℝ) := + ∃ a, FnLb f a + + variable (f : ℝ → ℝ) + + example (h : ∀ a, ∃ x, f x > a) : ¬FnHasUb f := by + intro fnub + cases' fnub with a fnuba + cases' h a with x hx + have : f x ≤ a := fnuba x + linarith + + example (h : ∀ a, ∃ x, f x < a) : ¬FnHasLb f := + sorry + + example : ¬FnHasUb fun x => x := + sorry + + #check (not_le_of_gt : a > b → ¬a ≤ b) + + #check (not_lt_of_ge : a ≥ b → ¬a < b) + + #check (lt_of_not_ge : ¬a ≥ b → a < b) + + #check (le_of_not_gt : ¬a > b → a ≤ b) + + example (h : Monotone f) (h' : f a < f b) : a (0 : ℝ) + have monof : Monotone f := by sorry + have h' : f 1 ≤ f 0 := le_refl _ + sorry + + example (x : ℝ) (h : ∀ ε > 0, x < ε) : x ≤ 0 := by + sorry + + end + + section + + variable {α : Type _} (P : α → Prop) (Q : Prop) + + example (h : ¬∃ x, P x) : ∀ x, ¬P x := by + sorry + + example (h : ∀ x, ¬P x) : ¬∃ x, P x := by + sorry + + example (h : ¬∀ x, P x) : ∃ x, ¬P x := by + sorry + + example (h : ∃ x, ¬P x) : ¬∀ x, P x := by + sorry + + example (h : ¬∀ x, P x) : ∃ x, ¬P x := by + by_contra h' + apply h + intro x + show P x + by_contra h'' + exact h' ⟨x, h''⟩ + + example (h : ¬¬Q) : Q := by + sorry + + example (h : Q) : ¬¬Q := by + sorry + + end + + section + + variable (f : ℝ → ℝ) + + example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by + sorry + + example (h : ¬∀ a, ∃ x, f x > a) : FnHasUb f := by + push_neg at h + exact h + + example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by + simp only [FnHasUb, FnUb] at h + push_neg at h + exact h + + example (h : ¬Monotone f) : ∃ x y, x ≤ y ∧ f y < f x := by + sorry + + example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by + contrapose! h + exact h + + example (x : ℝ) (h : ∀ ε > 0, x ≤ ε) : x ≤ 0 := by + contrapose! h + use x / 2 + constructor <;> linarith + + end + + section + + variable (a : ℕ) + + example (h : 0 < 0) : a > 37 := by + exfalso + apply lt_irrefl 0 h + + example (h : 0 < 0) : a > 37 := + absurd h (lt_irrefl 0) + + example (h : 0 < 0) : a > 37 := by + have h' : ¬0 < 0 := lt_irrefl 0 + contradiction + + end +
src/C_Logic/S04_Conjunction_and_Bi-implication.lean (new)
- @@ -0,0 +1,138 @@
 + import Mathlib.Data.Real.Basic + import Mathlib.Data.Nat.Prime + import Mathlib.Tactic.NormNum + + example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y := by + constructor + · assumption + intro h + apply h₁ + rw [h] + + example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y := + ⟨h₀, fun h => h₁ (by rw [h])⟩ + + example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y := + have h : x ≠ y := by + contrapose! h₁ + rw [h₁] + ⟨h₀, h⟩ + + example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by + cases' h with h₀ h₁ + contrapose! h₁ + exact le_antisymm h₀ h₁ + + example {x y : ℝ} : x ≤ y ∧ x ≠ y → ¬y ≤ x := by + rintro ⟨h₀, h₁⟩ h' + exact h₁ (le_antisymm h₀ h') + + example {x y : ℝ} : x ≤ y ∧ x ≠ y → ¬y ≤ x := + fun ⟨h₀, h₁⟩ h' => h₁ (le_antisymm h₀ h') + + example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by + intro h' + apply h.right + exact le_antisymm h.left h' + + example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := + fun h' => h.right (le_antisymm h.left h') + + example {m n : ℕ} (h : m ∣ n ∧ m ≠ n) : m ∣ n ∧ ¬n ∣ m := + sorry + + example : ∃ x : ℝ, 2 < x ∧ x < 4 := + ⟨5 / 2, by norm_num, by norm_num⟩ + + example (x y : ℝ) : (∃ z : ℝ, x < z ∧ z < y) → x < y := by + rintro ⟨z, xltz, zlty⟩ + exact lt_trans xltz zlty + + example (x y : ℝ) : (∃ z : ℝ, x < z ∧ z < y) → x < y := + fun ⟨z, xltz, zlty⟩ => lt_trans xltz zlty + + example : ∃ x : ℝ, 2 < x ∧ x < 4 := by + use 5 / 2 + constructor <;> norm_num + + example : ∃ m n : ℕ, 4 < m ∧ m < n ∧ n < 10 ∧ Nat.Prime m ∧ Nat.Prime n := by + use 5 + use 7 + norm_num + sorry + + example {x y : ℝ} : x ≤ y ∧ x ≠ y → x ≤ y ∧ ¬y ≤ x := by + rintro ⟨h₀, h₁⟩ + use h₀ + exact fun h' => h₁ (le_antisymm h₀ h') + + example {x y : ℝ} (h : x ≤ y) : ¬y ≤ x ↔ x ≠ y := by + constructor + · contrapose! + rintro rfl + rfl + contrapose! + exact le_antisymm h + + example {x y : ℝ} (h : x ≤ y) : ¬y ≤ x ↔ x ≠ y := + ⟨fun h₀ h₁ => h₀ (by rw [h₁]), fun h₀ h₁ => h₀ (le_antisymm h h₁)⟩ + + example {x y : ℝ} : x ≤ y ∧ ¬y ≤ x ↔ x ≤ y ∧ x ≠ y := + sorry + + theorem aux {x y : ℝ} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := + have h' : x ^ 2 = 0 := by sorry + pow_eq_zero h' + + example (x y : ℝ) : x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := + sorry + + section + + example (x : ℝ) : abs (x + 3) < 5 → -8 < x ∧ x < 2 := by + rw [abs_lt] + intro h + constructor <;> linarith + + example : 3 ∣ Nat.gcd 6 15 := by + rw [Nat.dvd_gcd_iff] + constructor <;> norm_num + + end + + theorem not_monotone_iff {f : ℝ → ℝ} : ¬Monotone f ↔ ∃ x y, x ≤ y ∧ f x > f y := by + rw [Monotone] + push_neg + rfl + + example : ¬Monotone fun x : ℝ => -x := by + sorry + + section + + variable {α : Type _} [PartialOrder α] + + variable (a b : α) + + example : a < b ↔ a ≤ b ∧ a ≠ b := by + rw [lt_iff_le_not_le] + sorry + + end + + section + + variable {α : Type _} [Preorder α] + + variable (a b c : α) + + example : ¬a < a := by + rw [lt_iff_le_not_le] + sorry + + example : a < b → b < c → a < c := by + simp only [lt_iff_le_not_le] + sorry + + end +
src/C_Logic/S05_Disjunction.lean (new)
- @@ -0,0 +1,102 @@
 + import Mathlib.Data.Real.Basic + + section + + variable {x y : ℝ} + + example (h : y > x ^ 2) : y > 0 ∨ y < -1 := by + left + linarith [pow_two_nonneg x] + + example (h : -y > x ^ 2 + 1) : y > 0 ∨ y < -1 := by + right + linarith [pow_two_nonneg x] + + example (h : y > 0) : y > 0 ∨ y < -1 := + Or.inl h + + example (h : y < -1) : y > 0 ∨ y < -1 := + Or.inr h + + example : x < abs y → x < y ∨ x < -y := by + cases' le_or_gt 0 y with h h + · rw [abs_of_nonneg h] + intro h + left + exact h + rw [abs_of_neg h] + intro h; right; exact h + + namespace MyAbs + + theorem le_abs_self (x : ℝ) : x ≤ abs x := by + sorry + + theorem neg_le_abs_self (x : ℝ) : -x ≤ abs x := by + sorry + + theorem abs_add (x y : ℝ) : abs (x + y) ≤ abs x + abs y := by + sorry + + theorem lt_abs : x < abs y ↔ x < y ∨ x < -y := by + sorry + + theorem abs_lt : abs x < y ↔ -y < x ∧ x < y := by + sorry + + end MyAbs + + end + + example {x : ℝ} (h : x ≠ 0) : x < 0 ∨ x > 0 := by + rcases lt_trichotomy x 0 with (xlt | xeq | xgt) + · left + exact xlt + · contradiction + right; exact xgt + + example {m n k : ℕ} (h : m ∣ n ∨ m ∣ k) : m ∣ n * k := by + rcases h with (⟨a, rfl⟩ | ⟨b, rfl⟩) + · rw [mul_assoc] + apply dvd_mul_right + rw [mul_comm, mul_assoc] + apply dvd_mul_right + + example {z : ℝ} (h : ∃ x y, z = x ^ 2 + y ^ 2 ∨ z = x ^ 2 + y ^ 2 + 1) : z ≥ 0 := by + sorry + + example {x : ℝ} (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by + sorry + + example {x y : ℝ} (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by + sorry + + section + + variable {R : Type _} [CommRing R] [IsDomain R] + + variable (x y : R) + + example (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by + sorry + + example (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by + sorry + + end + + example (P : Prop) : ¬¬P → P := by + intro h + cases em P + · assumption + contradiction + + example (P : Prop) : ¬¬P → P := by + intro h + by_cases h' : P + · assumption + contradiction + + example (P Q : Prop) : P → Q ↔ ¬P ∨ Q := by + sorry +
src/C_Logic/S06_Sequences_and_Convergence.lean (new)
- @@ -0,0 +1,96 @@
 + import Mathlib.Data.Real.Basic + + def ConvergesTo (s : ℕ → ℝ) (a : ℝ) := + ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε + + example : (fun x y : ℝ => (x + y) ^ 2) = fun x y : ℝ => x ^ 2 + 2 * x * y + y ^ 2 := by + ext + ring + + example (a b : ℝ) : abs a = abs (a - b + b) := by + congr + ring + + example {a : ℝ} (h : 1 < a) : a < a * a := by + convert(mul_lt_mul_right _).2 h + · rw [one_mul] + exact lt_trans zero_lt_one h + + theorem convergesTo_const (a : ℝ) : ConvergesTo (fun x : ℕ => a) a := by + intro ε εpos + use 0 + intro n nge; dsimp + rw [sub_self, abs_zero] + apply εpos + + theorem convergesTo_add {s t : ℕ → ℝ} {a b : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t b) : + ConvergesTo (fun n => s n + t n) (a + b) := by + intro ε εpos + dsimp + have ε2pos : 0 < ε / 2 := by linarith + cases' cs (ε / 2) ε2pos with Ns hs + cases' ct (ε / 2) ε2pos with Nt ht + use max Ns Nt + sorry + + theorem convergesTo_mul_const {s : ℕ → ℝ} {a : ℝ} (c : ℝ) (cs : ConvergesTo s a) : + ConvergesTo (fun n => c * s n) (c * a) := by + by_cases h : c = 0 + · convert convergesTo_const 0 + · rw [h, MulZeroClass.zero_mul] + rw [h, MulZeroClass.zero_mul] + have acpos : 0 < abs c := abs_pos.mpr h + sorry + + theorem exists_abs_le_of_convergesTo {s : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) : + ∃ N b, ∀ n, N ≤ n → abs (s n) s n * t n) 0 := by + intro ε εpos + dsimp + rcases exists_abs_le_of_convergesTo cs with ⟨N₀, B, h₀⟩ + have Bpos : 0 0 := div_pos εpos Bpos + cases' ct _ pos₀ with N₁ h₁ + sorry + + theorem convergesTo_mul {s t : ℕ → ℝ} {a b : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t b) : + ConvergesTo (fun n => s n * t n) (a * b) := by + have h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 := by + apply aux cs + convert convergesTo_add ct (convergesTo_const (-b)) + ring + have := convergesTo_add h₁ (convergesTo_mul_const b cs) + convert convergesTo_add h₁ (convergesTo_mul_const b cs) using 1 + · ext; ring + ring + + theorem convergesTo_unique {s : ℕ → ℝ} {a b : ℝ} (sa : ConvergesTo s a) (sb : ConvergesTo s b) : + a = b := by + by_contra abne + have : abs (a - b) > 0 := by sorry + let ε := abs (a - b) / 2 + have εpos : ε > 0 := by + change abs (a - b) / 2 > 0 + linarith + cases' sa ε εpos with Na hNa + cases' sb ε εpos with Nb hNb + let N := max Na Nb + have absa : abs (s N - a) < ε := by sorry + have absb : abs (s N - b) < ε := by sorry + have : abs (a - b) < abs (a - b) := by sorry + exact lt_irrefl _ this + + section + + variable {α : Type _} [LinearOrder α] + + def ConvergesTo' (s : α → ℝ) (a : ℝ) := + ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε + + end +
src/C_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean (new)
- @@ -0,0 +1,131 @@
  + import Mathlib.Data.Real.Basic + + def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, f x ≤ a + + def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, a ≤ f x + + section + variable (f g : ℝ → ℝ) (a b : ℝ) + + example (hfa : FnLb f a) (hgb : FnLb g b) : FnLb (fun x => f x + g x) (a + b) := by + intro x + apply add_le_add + apply hfa + apply hgb + + example (nnf : FnLb f 0) (nng : FnLb g 0) : FnLb (fun x => f x * g x) 0 := by + intro x + apply mul_nonneg + apply nnf + apply nng + + example (hfa : FnUb f a) (hfb : FnUb g b) (nng : FnLb g 0) (nna : 0 ≤ a) : + FnUb (fun x => f x * g x) (a * b) := by + intro x + apply mul_le_mul + apply hfa + apply hfb + apply nng + apply nna + + end + + section + variable (f g : ℝ → ℝ) + + example {c : ℝ} (mf : Monotone f) (nnc : 0 ≤ c) : Monotone fun x => c * f x := by + intro a b aleb + apply mul_le_mul_of_nonneg_left _ nnc + apply mf aleb + + example {c : ℝ} (mf : Monotone f) (nnc : 0 ≤ c) : Monotone fun x => c * f x := + fun a b aleb => mul_le_mul_of_nonneg_left (mf aleb) nnc + + example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f (g x) := by + intro a b aleb + apply mf + apply mg + apply aleb + + example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f (g x) := + fun a b aleb => mf (mg aleb) + + def FnEven (f : ℝ → ℝ) : Prop := + ∀ x, f x = f (-x) + + def FnOdd (f : ℝ → ℝ) : Prop := + ∀ x, f x = -f (-x) + + example (of : FnOdd f) (og : FnOdd g) : FnEven fun x => f x * g x := by + intro x + calc + (fun x => f x * g x) x = f x * g x := rfl + _ = f (-x) * g (-x) := by rw [of, og, neg_mul_neg] + + + example (ef : FnEven f) (og : FnOdd g) : FnOdd fun x => f x * g x := by + intro x + dsimp + rw [ef, og, neg_mul_eq_mul_neg] + + example (ef : FnEven f) (og : FnOdd g) : FnEven fun x => f (g x) := by + intro x + dsimp + rw [og, ← ef] + + end + + section + + variable {α : Type _} (r s t : Set α) + + example : r ⊆ s → s ⊆ t → r ⊆ t := by + intro rsubs ssubt x xr + apply ssubt + apply rsubs + apply xr + + theorem Subset.trans : r ⊆ s → s ⊆ t → r ⊆ t := + fun rsubs ssubt x xr => ssubt (rsubs xr) + + end + + section + + variable {α : Type _} [PartialOrder α] + + variable (s : Set α) (a b : α) + + def SetUb (s : Set α) (a : α) := + ∀ x, x ∈ s → x ≤ a + + example (h : SetUb s a) (h' : a ≤ b) : SetUb s b := by + intro x xs + apply le_trans (h x xs) h' + + example (h : SetUb s a) (h' : a ≤ b) : SetUb s b := + fun x xs => le_trans (h x xs) h' + + end + + section + + open Function + + example {c : ℝ} (h : c ≠ 0) : Injective fun x => c * x := by + intro x₁ x₂ h' + apply (mul_right_inj' h).mp h' + + variable {α : Type _} {β : Type _} {γ : Type _} + + variable {g : β → γ} {f : α → β} + + example (injg : Injective g) (injf : Injective f) : Injective fun x => g (f x) := by + intro x₁ x₂ h + apply injf + apply injg + apply h + + end
src/C_Logic/solutions/Solutions_S02_The_Existential_Quantifier.lean (new)
- @@ -0,0 +1,85 @@
  + import Mathlib.Data.Real.Basic + + def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, f x ≤ a + + def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, a ≤ f x + + def FnHasUb (f : ℝ → ℝ) := + ∃ a, FnUb f a + + def FnHasLb (f : ℝ → ℝ) := + ∃ a, FnLb f a + + theorem fnUb_add {f g : ℝ → ℝ} {a b : ℝ} (hfa : FnUb f a) (hgb : FnUb g b) : + FnUb (fun x => f x + g x) (a + b) := + fun x => add_le_add (hfa x) (hgb x) + + section + + variable {f g : ℝ → ℝ} + + example (lbf : FnHasLb f) (lbg : FnHasLb g) : FnHasLb fun x => f x + g x := by + cases' lbf with a lbfa + cases' lbg with b lbgb + use a + b + intro x + exact add_le_add (lbfa x) (lbgb x) + + example {c : ℝ} (ubf : FnHasUb f) (h : c ≥ 0) : FnHasUb fun x => c * f x := by + cases' ubf with a lbfa + use c * a + intro x + exact mul_le_mul_of_nonneg_left (lbfa x) h + + end + + section + + variable {a b c : ℕ} + + example (divab : a ∣ b) (divbc : b ∣ c) : a ∣ c := by + rcases divab with ⟨d, rfl⟩ + rcases divbc with ⟨e, rfl⟩ + use d * e; ring + + example (divab : a ∣ b) (divac : a ∣ c) : a ∣ b + c := by + rcases divab with ⟨d, rfl⟩ + rcases divac with ⟨e, rfl⟩ + use d + e; ring + + end + + section + + open Function + + example {c : ℝ} (h : c ≠ 0) : Surjective fun x => c * x := by + intro x + use x / c + dsimp; rw [mul_div_cancel' _ h] + + example {c : ℝ} (h : c ≠ 0) : Surjective fun x => c * x := by + intro x + use x / c + field_simp [h] ; ring + + end + + section + + open Function + + variable {α : Type _} {β : Type _} {γ : Type _} + + variable {g : β → γ} {f : α → β} + + example (surjg : Surjective g) (surjf : Surjective f) : Surjective fun x => g (f x) := by + intro z + rcases surjg z with ⟨y, rfl⟩ + rcases surjf y with ⟨x, rfl⟩ + use x + + end +
src/C_Logic/solutions/Solutions_S03_Negation.lean (new)
- @@ -0,0 +1,113 @@
 + import Mathlib.Data.Real.Basic + + section + + variable (a b : ℝ) + + def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, f x ≤ a + + def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := + ∀ x, a ≤ f x + + def FnHasUb (f : ℝ → ℝ) := + ∃ a, FnUb f a + + def FnHasLb (f : ℝ → ℝ) := + ∃ a, FnLb f a + + variable (f : ℝ → ℝ) + + example (h : ∀ a, ∃ x, f x < a) : ¬FnHasLb f := by + rintro ⟨a, ha⟩ + rcases h a with ⟨x, hx⟩ + have := ha x + linarith + + example : ¬FnHasUb fun x => x := by + rintro ⟨a, ha⟩ + have : a + 1 ≤ a := ha (a + 1) + linarith + + example (h : Monotone f) (h' : f a < f b) : a < b := by + apply lt_of_not_ge + intro h'' + apply absurd h' + apply not_lt_of_ge (h h'') + + example (h : a ≤ b) (h' : f b < f a) : ¬Monotone f := by + intro h'' + apply absurd h' + apply not_lt_of_ge + apply h'' h + + example : ¬∀ {f : ℝ → ℝ}, Monotone f → ∀ {a b}, f a ≤ f b → a ≤ b := by + intro h + let f := fun x : ℝ => (0 : ℝ) + have monof : Monotone f := by + intro a b leab + rfl + have h' : f 1 ≤ f 0 := le_refl _ + have : (1 : ℝ) ≤ 0 := h monof h' + linarith + + example (x : ℝ) (h : ∀ ε > 0, x < ε) : x ≤ 0 := by + apply le_of_not_gt + intro h' + linarith [h _ h'] + + end + + section + + variable {α : Type _} (P : α → Prop) (Q : Prop) + + example (h : ¬∃ x, P x) : ∀ x, ¬P x := by + intro x Px + apply h + use x + exact Px + + example (h : ∀ x, ¬P x) : ¬∃ x, P x := by + rintro ⟨x, Px⟩ + exact h x Px + + example (h : ∃ x, ¬P x) : ¬∀ x, P x := by + intro h' + rcases h with ⟨x, nPx⟩ + apply nPx + apply h' + + example (h : ¬¬Q) : Q := by + by_contra h' + exact h h' + + example (h : Q) : ¬¬Q := by + intro h' + exact h' h + + end + + section + + variable (f : ℝ → ℝ) + + example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by + intro a + by_contra h' + apply h + use a + intro x + apply le_of_not_gt + intro h'' + apply h' + use x + exact h'' + + example (h : ¬Monotone f) : ∃ x y, x ≤ y ∧ f y < f x := by + rw [Monotone] at h + push_neg at h + exact h + + end +
src/C_Logic/solutions/Solutions_S04_Conjunction_and_Bi-implication.lean (new)
- @@ -0,0 +1,97 @@
 + import Mathlib.Data.Real.Basic + import Mathlib.Data.Nat.Prime + import Mathlib.Tactic.NormNum + + example {m n : ℕ} (h : m ∣ n ∧ m ≠ n) : m ∣ n ∧ ¬n ∣ m := by + cases' h with h0 h1 + constructor + · exact h0 + intro h2 + apply h1 + apply Nat.dvd_antisymm h0 h2 + + example {x y : ℝ} : x ≤ y ∧ ¬y ≤ x ↔ x ≤ y ∧ x ≠ y := by + constructor + · rintro ⟨h0, h1⟩ + constructor + · exact h0 + intro h2 + apply h1 + rw [h2] + rintro ⟨h0, h1⟩ + constructor + · exact h0 + intro h2 + apply h1 + apply le_antisymm h0 h2 + + theorem aux {x y : ℝ} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := + have h' : x ^ 2 = 0 := by linarith [pow_two_nonneg x, pow_two_nonneg y] + pow_eq_zero h' + + example (x y : ℝ) : x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by + constructor + · intro h + constructor + · exact aux h + rw [add_comm] at h + exact aux h + rintro ⟨rfl, rfl⟩ + norm_num + + theorem not_monotone_iff {f : ℝ → ℝ} : ¬Monotone f ↔ ∃ x y, x ≤ y ∧ f x > f y := by + rw [Monotone] + push_neg + rfl + + example : ¬Monotone fun x : ℝ => -x := by + rw [not_monotone_iff] + use 0, 1 + norm_num + + section + + variable {α : Type _} [PartialOrder α] + + variable (a b : α) + + example : a < b ↔ a ≤ b ∧ a ≠ b := by + rw [lt_iff_le_not_le] + constructor + · rintro ⟨h0, h1⟩ + constructor + · exact h0 + intro h2 + apply h1 + rw [h2] + rintro ⟨h0, h1⟩ + constructor + · exact h0 + intro h2 + apply h1 + apply le_antisymm h0 h2 + + end + + section + + variable {α : Type _} [Preorder α] + + variable (a b c : α) + + example : ¬a < a := by + rw [lt_iff_le_not_le] + rintro ⟨h0, h1⟩ + exact h1 h0 + + example : a < b → b < c → a < c := by + simp only [lt_iff_le_not_le] + rintro ⟨h0, h1⟩ ⟨h2, h3⟩ + constructor + · apply le_trans h0 h2 + intro h4 + apply h1 + apply le_trans h2 h4 + + end +
src/C_Logic/solutions/Solutions_S05_Disjunction.lean (new)
- @@ -0,0 +1,142 @@
 + import Mathlib.Data.Real.Basic + + section + + variable {x y : ℝ} + + namespace MyAbs + + theorem le_abs_self (x : ℝ) : x ≤ abs x := by + cases' le_or_gt 0 x with h h + · rw [abs_of_nonneg h] + rw [abs_of_neg h] + linarith + + theorem neg_le_abs_self (x : ℝ) : -x ≤ abs x := by + cases' le_or_gt 0 x with h h + · rw [abs_of_nonneg h] + linarith + rw [abs_of_neg h] + + theorem abs_add (x y : ℝ) : abs (x + y) ≤ abs x + abs y := by + cases' le_or_gt 0 (x + y) with h h + · rw [abs_of_nonneg h] + linarith [le_abs_self x, le_abs_self y] + rw [abs_of_neg h] + linarith [neg_le_abs_self x, neg_le_abs_self y] + + theorem lt_abs : x < abs y ↔ x < y ∨ x < -y := by + cases' le_or_gt 0 y with h h + · rw [abs_of_nonneg h] + constructor + · intro h' + left + exact h' + intro h' + cases' h' with h' h' + · exact h' + linarith + rw [abs_of_neg h] + constructor + · intro h' + right + exact h' + intro h' + cases' h' with h' h' + · linarith + exact h' + + theorem abs_lt : abs x < y ↔ -y < x ∧ x < y := by + cases' le_or_gt 0 x with h h + · rw [abs_of_nonneg h] + constructor + · intro h' + constructor + · linarith + exact h' + intro h' + cases' h' with h1 h2 + exact h2 + rw [abs_of_neg h] + constructor + · intro h' + constructor + · linarith + linarith + intro h' + linarith + + end MyAbs + + end + + example {z : ℝ} (h : ∃ x y, z = x ^ 2 + y ^ 2 ∨ z = x ^ 2 + y ^ 2 + 1) : z ≥ 0 := by + rcases h with ⟨x, y, rfl | rfl⟩ <;> linarith [sq_nonneg x, sq_nonneg y] + + example {x : ℝ} (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by + have h' : x ^ 2 - 1 = 0 := by rw [h, sub_self] + have h'' : (x + 1) * (x - 1) = 0 := by + rw [← h'] + ring + cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1 + · right + exact eq_neg_iff_add_eq_zero.mpr h1 + left + exact eq_of_sub_eq_zero h1 + + example {x y : ℝ} (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by + have h' : x ^ 2 - y ^ 2 = 0 := by rw [h, sub_self] + have h'' : (x + y) * (x - y) = 0 := by + rw [← h'] + ring + cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1 + · right + exact eq_neg_iff_add_eq_zero.mpr h1 + left + exact eq_of_sub_eq_zero h1 + + section + + variable {R : Type _} [CommRing R] [IsDomain R] + + variable (x y : R) + + example (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by + have h' : x ^ 2 - 1 = 0 := by rw [h, sub_self] + have h'' : (x + 1) * (x - 1) = 0 := by + rw [← h'] + ring + cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1 + · right + exact eq_neg_iff_add_eq_zero.mpr h1 + left + exact eq_of_sub_eq_zero h1 + + example (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by + have h' : x ^ 2 - y ^ 2 = 0 := by rw [h, sub_self] + have h'' : (x + y) * (x - y) = 0 := by + rw [← h'] + ring + cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1 + · right + exact eq_neg_iff_add_eq_zero.mpr h1 + left + exact eq_of_sub_eq_zero h1 + + end + + example (P Q : Prop) : P → Q ↔ ¬P ∨ Q := by + constructor + · intro h + by_cases h' : P + · right + exact h h' + left + exact h' + rintro (h | h) + · intro h' + exact absurd h' h + intro + exact h + +
src/C_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean (new)
- @@ -0,0 +1,126 @@
 + import Mathlib.Data.Real.Basic + + def ConvergesTo (s : ℕ → ℝ) (a : ℝ) := + ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε + + theorem convergesTo_const (a : ℝ) : ConvergesTo (fun x : ℕ => a) a := by + intro ε εpos + use 0 + intro n nge; dsimp + rw [sub_self, abs_zero] + apply εpos + + theorem convergesTo_add {s t : ℕ → ℝ} {a b : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t b) : + ConvergesTo (fun n => s n + t n) (a + b) := by + intro ε εpos + dsimp + have ε2pos : 0 < ε / 2 := by linarith + cases' cs (ε / 2) ε2pos with Ns hs + cases' ct (ε / 2) ε2pos with Nt ht + use max Ns Nt + intro n hn + have ngeNs : n ≥ Ns := le_of_max_le_left hn + have ngeNt : n ≥ Nt := le_of_max_le_right hn + calc + |s n + t n - (a + b)| = |s n - a + (t n - b)| := by + congr + ring + _ ≤ |s n - a| + |t n - b| := (abs_add _ _) + _ < ε / 2 + ε / 2 := (add_lt_add (hs n ngeNs) (ht n ngeNt)) + _ = ε := by norm_num + + + theorem convergesTo_mul_const {s : ℕ → ℝ} {a : ℝ} (c : ℝ) (cs : ConvergesTo s a) : + ConvergesTo (fun n => c * s n) (c * a) := by + by_cases h : c = 0 + · convert convergesTo_const 0 + · rw [h, MulZeroClass.zero_mul] + rw [h, MulZeroClass.zero_mul] + have acpos : 0 < abs c := abs_pos.mpr h + intro ε εpos + dsimp + have εcpos : 0 < ε / abs c := by apply div_pos εpos acpos + cases' cs (ε / abs c) εcpos with Ns hs + use Ns + intro n ngt + calc + |c * s n - c * a| = |c| * |s n - a| := by rw [← abs_mul, mul_sub] + _ < |c| * (ε / |c|) := (mul_lt_mul_of_pos_left (hs n ngt) acpos) + _ = ε := mul_div_cancel' _ (ne_of_lt acpos).symm + + + theorem exists_abs_le_of_converges_to {s : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) : + ∃ N b, ∀ n, N ≤ n → abs (s n) < b := by + cases' cs 1 zero_lt_one with N h + use N, abs a + 1 + intro n ngt + calc + |s n| = |s n - a + a| := by + congr + abel + _ ≤ |s n - a| + |a| := (abs_add _ _) + _ < |a| + 1 := by linarith [h n ngt] + + + theorem aux {s t : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t 0) : + ConvergesTo (fun n => s n * t n) 0 := by + intro ε εpos + dsimp + rcases exists_abs_le_of_convergesTo cs with ⟨N₀, B, h₀⟩ + have Bpos : 0 0 := div_pos εpos Bpos + cases' ct _ pos₀ with N₁ h₁ + use max N₀ N₁ + intro n ngt + have ngeN₀ : n ≥ N₀ := le_of_max_le_left ngt + have ngeN₁ : n ≥ N₁ := le_of_max_le_right ngt + calc + |s n * t n - 0| = |s n| * |t n - 0| := by rw [sub_zero, abs_mul, sub_zero] + _ < B * (ε / B) := (mul_lt_mul'' (h₀ n ngeN₀) (h₁ n ngeN₁) (abs_nonneg _) (abs_nonneg _)) + _ = ε := mul_div_cancel' _ (ne_of_lt Bpos).symm + + theorem convergesTo_mul {s t : ℕ → ℝ} {a b : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t b) : + ConvergesTo (fun n => s n * t n) (a * b) := by + have h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 := by + apply aux cs + convert convergesTo_add ct (convergesTo_const (-b)) + ring + have := convergesTo_add h₁ (convergesTo_mul_const b cs) + convert convergesTo_add h₁ (convergesTo_mul_const b cs) using 1 + · ext; ring + ring + + theorem convergesTo_unique {s : ℕ → ℝ} {a b : ℝ} (sa : ConvergesTo s a) (sb : ConvergesTo s b) : + a = b := by + by_contra abne + have : abs (a - b) > 0 := by + apply lt_of_le_of_ne + · apply abs_nonneg + intro h'' + apply abne + apply eq_of_abs_sub_eq_zero h''.symm + let ε := abs (a - b) / 2 + have εpos : ε > 0 := by + change abs (a - b) / 2 > 0 + linarith + cases' sa ε εpos with Na hNa + cases' sb ε εpos with Nb hNb + let N := max Na Nb + have absa : abs (s N - a) < ε := by + apply hNa + apply le_max_left + have absb : abs (s N - b) < ε := by + apply hNb + apply le_max_right + have : abs (a - b) < abs (a - b) + calc + abs (a - b) = abs (-(s N - a) + (s N - b)) := by + congr + ring + _ ≤ abs (-(s N - a)) + abs (s N - b) := (abs_add _ _) + _ = abs (s N - a) + abs (s N - b) := by rw [abs_neg] + _ < ε + ε := (add_lt_add absa absb) + _ = abs (a - b) := by norm_num + + exact lt_irrefl _ this +
src/C_Number_Theory/S01_Irrational_Roots.lean (new)
- @@ -0,0 +1,118 @@
 + import Mathlib.Data.Nat.Factorization.Basic + import Mathlib.Data.Nat.Prime + import Mathlib.Tactic.NormNum.GCD + import Mathlib.Tactic.NormNum.Prime + + #print Nat.coprime + + example (m n : Nat) (h : m.coprime n) : m.gcd n = 1 := + h + + example (m n : Nat) (h : m.coprime n) : m.gcd n = 1 := by + rw [Nat.coprime] at h + exact h + + example : Nat.coprime 12 7 := by norm_num + + example : Nat.gcd 12 8 = 4 := by norm_num + + #check @Nat.prime_def_lt + + example (p : ℕ) (prime_p : Nat.Prime p) : 2 ≤ p ∧ ∀ m : ℕ, m < p → m ∣ p → m = 1 := by + rwa [Nat.prime_def_lt] at prime_p + + #check Nat.Prime.eq_one_or_self_of_dvd + + example (p : ℕ) (prime_p : Nat.Prime p) : ∀ m : ℕ, m ∣ p → m = 1 ∨ m = p := + prime_p.eq_one_or_self_of_dvd + + example : Nat.Prime 17 := by norm_num + + -- commonly used + example : Nat.Prime 2 := + Nat.prime_two + + example : Nat.Prime 3 := + Nat.prime_three + + #check @Nat.Prime.dvd_mul + + #check Nat.Prime.dvd_mul Nat.prime_two + + #check Nat.prime_two.dvd_mul + + theorem even_of_even_sqr {m : ℕ} (h : 2 ∣ m ^ 2) : 2 ∣ m := by + rw [pow_two, Nat.prime_two.dvd_mul] at h + cases h <;> assumption + + example {m : ℕ} (h : 2 ∣ m ^ 2) : 2 ∣ m := + Nat.Prime.dvd_of_dvd_pow Nat.prime_two h + + example (a b c : Nat) (h : a * b = a * c) (h' : a ≠ 0) : b = c := + -- library_search suggests the following: + (mul_right_inj' h').mp h + + example {m n : ℕ} (coprime_mn : m.coprime n) : m ^ 2 ≠ 2 * n ^ 2 := by + intro sqr_eq + have : 2 ∣ m := by + sorry, + obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this + have : 2 * (2 * k ^ 2) = 2 * n ^ 2 := by + rw [← sqr_eq, meq] + ring + have : 2 * k ^ 2 = n ^ 2 := + sorry, + have : 2 ∣ n := by + sorry, + have : 2 ∣ m.gcd n := by + sorry, + have : 2 ∣ 1 := by + sorry, + norm_num at this + + example {m n p : ℕ} (coprime_mn : m.coprime n) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by + sorry + #check Nat.factors + #check Nat.prime_of_mem_factors + #check Nat.prod_factors + #check Nat.factors_unique + + theorem factorization_mul' {m n : ℕ} (mnez : m ≠ 0) (nnez : n ≠ 0) (p : ℕ) : + (m * n).factorization p = m.factorization p + n.factorization p := by + rw [Nat.factorization_mul mnez nnez] + rfl + + theorem factorization_pow' (n k p : ℕ) : (n ^ k).factorization p = k * n.factorization p := by + rw [Nat.factorization_pow] + rfl + + theorem Nat.Prime.factorization' {p : ℕ} (prime_p : p.Prime) : p.factorization p = 1 := by + rw [prime_p.factorization] + simp + + example {m n p : ℕ} (nnz : n ≠ 0) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by + intro sqr_eq + have nsqr_nez : n ^ 2 ≠ 0 := by simpa + have eq1 : Nat.factorization (m ^ 2) p = 2 * m.factorization p := by + sorry, + have eq2 : (p * n ^ 2).factorization p = 2 * n.factorization p + 1 := by + sorry, + have : 2 * m.factorization p % 2 = (2 * n.factorization p + 1) % 2 := by rw [← eq1, sqr_eq, eq2] + rw [add_comm, Nat.add_mul_mod_self_left, Nat.mul_mod_right] at this + norm_num at this + + example {m n k r : ℕ} (nnz : n ≠ 0) (pow_eq : m ^ k = r * n ^ k) {p : ℕ} (prime_p : p.Prime) : + k ∣ r.factorization p := by + cases' r with r + · simp + have npow_nz : n ^ k ≠ 0 := fun npowz => nnz (pow_eq_zero npowz) + have eq1 : (m ^ k).factorization p = k * m.factorization p := by + sorry, + have eq2 : (r.succ * n ^ k).factorization p = k * n.factorization p + r.succ.factorization p := by + sorry, + have : r.succ.factorization p = k * m.factorization p - k * n.factorization p := by + rw [← eq1, pow_eq, eq2, add_comm, Nat.add_sub_cancel] + rw [this] + sorry + #check multiplicity +
src/C_Number_Theory/S02_Induction_and_Recursion.lean (new)
- @@ -0,0 +1,148 @@
 + import Mathlib.Data.Nat.GCD.Basic + import Mathlib.Algebra.BigOperators.Basic + import Mathlib.Tactic + + example (n : Nat) : n.succ ≠ Nat.zero := + Nat.succ_ne_zero n + + example (m n : Nat) (h : m.succ = n.succ) : m = n := + Nat.succ.inj h + + def fac : ℕ → ℕ + | 0 => 1 + | n + 1 => (n + 1) * fac n + + example : fac 0 = 1 := + rfl + + example : fac 0 = 1 := by + rw [fac] + + example : fac 0 = 1 := by + simp [fac] + + example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := + rfl + + example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := by + rw [fac] + + example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := by + simp [fac] + + theorem fac_pos (n : ℕ) : 0 < fac n := by + induction' n with n ih + · rw [fac] + exact zero_lt_one + rw [fac] + exact mul_pos n.succ_pos ih + + theorem dvd_fac {i n : ℕ} (ipos : 0 < i) (ile : i ≤ n) : i ∣ fac n := by + induction' n with n ih + · exact absurd ipos (not_lt_of_ge ile) + rw [fac] + cases' Nat.of_le_succ ile with h h + · apply dvd_mul_of_dvd_right (ih h) + rw [h] + apply dvd_mul_right + + theorem pow_two_le_fac (n : ℕ) : 2 ^ (n - 1) ≤ fac n := by + cases' n with n + · simp [fac] + sorry + section + + variable {α : Type _} (s : Finset ℕ) (f : ℕ → ℕ) (n : ℕ) + + #check Finset.sum s f + #check Finset.prod s f + + open BigOperators + open Finset + + example : s.sum f = ∑ x in s, f x := + rfl + + example : s.prod f = ∏ x in s, f x := + rfl + + example : (range n).sum f = ∑ x in range n, f x := + rfl + + example : (range n).prod f = ∏ x in range n, f x := + rfl + + example (f : ℕ → ℕ) : (∑ x in range 0, f x) = 0 := + Finset.sum_range_zero f + + example (f : ℕ → ℕ) (n : ℕ) : (∑ x in range n.succ, f x) = (∑ x in range n, f x) + f n := + Finset.sum_range_succ f n + + example (f : ℕ → ℕ) : (∏ x in range 0, f x) = 1 := + Finset.prod_range_zero f + + example (f : ℕ → ℕ) (n : ℕ) : (∏ x in range n.succ, f x) = (∏ x in range n, f x) * f n := + Finset.prod_range_succ f n + + example (n : ℕ) : fac n = ∏ i in range n, (i + 1) := by + induction' n with n ih + · rw [fac, prod_range_zero] + rw [fac, ih, prod_range_succ, mul_comm] + + example (a b c d e f : ℕ) : a * (b * c * f * (d * e)) = d * (a * f * e) * (c * b) := by + simp [mul_assoc, mul_comm, mul_left_comm] + + theorem sum_id (n : ℕ) : (∑ i in range (n + 1), i) = n * (n + 1) / 2 := by + symm; apply Nat.div_eq_of_eq_mul_right (by norm_num : 0 < 2) + induction' n with n ih + · simp + rw [Finset.sum_range_succ, mul_add 2, ← ih, Nat.succ_eq_add_one] + ring + + theorem sum_sqr (n : ℕ) : (∑ i in range (n + 1), i ^ 2) = n * (n + 1) * (2 * n + 1) / 6 := by + sorry + end + + inductive MyNat + | zero : MyNat + | succ : MyNat → MyNat + + namespace MyNat + + def add : MyNat → MyNat → MyNat + | x, zero => x + | x, succ y => succ (add x y) + + def mul : MyNat → MyNat → MyNat + | x, zero => zero + | x, succ y => add (mul x y) x + + theorem zero_add (n : MyNat) : add zero n = n := by + induction' n with n ih + · rfl + rw [add, ih] + + theorem succ_add (m n : MyNat) : add (succ m) n = succ (add m n) := by + induction' n with n ih + · rfl + rw [add, ih] + rfl + + theorem add_comm (m n : MyNat) : add m n = add n m := by + induction' n with n ih + · rw [zero_add] + rfl + rw [add, succ_add, ih] + + theorem add_assoc (m n k : MyNat) : add (add m n) k = add m (add n k) := by + sorry + theorem mul_add (m n k : MyNat) : mul m (add n k) = add (mul m n) (mul m k) := by + sorry + theorem zero_mul (n : MyNat) : mul zero n = zero := by + sorry + theorem succ_mul (m n : MyNat) : mul (succ m) n = add (mul m n) n := by + sorry + theorem mul_comm (m n : MyNat) : mul m n = mul n m := by + sorry + end MyNat +
src/C_Number_Theory/S03_Infinitely_Many_Primes.lean (new)
- @@ -0,0 +1,231 @@
 + import Mathlib.Data.Nat.Prime + import Mathlib.Algebra.BigOperators.Order + import Mathlib.Tactic + import Mathlib.Tactic.IntervalCases + + open BigOperators + + theorem two_le {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by + cases m; contradiction + case succ m => + cases m; contradiction + repeat' apply Nat.succ_le_succ + apply zero_le + + example {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by + by_contra h + push_neg at h + interval_cases m <;> contradiction + + example {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by + by_contra h + push_neg at h + revert h0 h1 + revert h m + decide + + theorem exists_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ p : Nat, p.Prime ∧ p ∣ n := by + by_cases np : n.Prime + · use n, np + induction' n using Nat.strong_induction_on with n ih + dsimp at ih + rw [Nat.prime_def_lt] at np + push_neg at np + rcases np h with ⟨m, mltn, mdvdn, mne1⟩ + have : m ≠ 0 := by + intro mz + rw [mz, zero_dvd_iff] at mdvdn + linarith + have mgt2 : 2 ≤ m := two_le this mne1 + by_cases mp : m.Prime + · use m, mp + exact mdvdn + . rcases ih m mltn mgt2 mp with ⟨p, pp, pdvd⟩ + use p, pp + apply pdvd.trans mdvdn + + theorem primes_infinite : ∀ n, ∃ p > n, Nat.Prime p := by + intro n + have : 2 ≤ Nat.factorial (n + 1) + 1 := by + sorry, + rcases exists_prime_factor this with ⟨p, pp, pdvd⟩ + refine' ⟨p, _, pp⟩ + show p > n + by_contra ple + push_neg at ple + have : p ∣ Nat.factorial (n + 1) := by + sorry, + have : p ∣ 1 := by + sorry, + show False + sorry + open Finset + + section + + variable {α : Type _} [DecidableEq α] (r s t : Finset α) + + example : r ∩ (s ∪ t) ⊆ r ∩ s ∪ r ∩ t := by + rw [subset_iff] + intro x + rw [mem_inter, mem_union, mem_union, mem_inter, mem_inter] + tauto + + example : r ∩ (s ∪ t) ⊆ r ∩ s ∪ r ∩ t := by + simp [subset_iff] + intro x + tauto + + example : r ∩ s ∪ r ∩ t ⊆ r ∩ (s ∪ t) := by + simp [subset_iff] + intro x + tauto + + example : r ∩ s ∪ r ∩ t = r ∩ (s ∪ t) := by + ext x + simp + tauto + + end + + section + + variable {α : Type _} [DecidableEq α] (r s t : Finset α) + + example : (r ∪ s) ∩ (r ∪ t) = r ∪ s ∩ t := by + sorry + example : (r \ s) \ t = r \ (s ∪ t) := by + sorry + end + + example (s : Finset ℕ) (n : ℕ) (h : n ∈ s) : n ∣ ∏ i in s, i := + Finset.dvd_prod_of_mem _ h + + theorem Nat.Prime.eq_of_dvd_of_prime {p q : ℕ} (prime_p : Nat.Prime p) (prime_q : Nat.Prime q) + (h : p ∣ q) : p = q := by + sorry + theorem mem_of_dvd_prod_primes {s : Finset ℕ} {p : ℕ} (prime_p : p.Prime) : + (∀ n ∈ s, Nat.Prime n) → (p ∣ ∏ n in s, n) → p ∈ s := by + intro h₀ h₁ + induction' s using Finset.induction_on with a s ans ih + · simp at h₁ + linarith [prime_p.two_le] + simp [Finset.prod_insert ans, prime_p.dvd_mul] at h₀ h₁ + rw [mem_insert] + sorry + example (s : Finset ℕ) (x : ℕ) : x ∈ s.filter Nat.Prime ↔ x ∈ s ∧ x.Prime := + mem_filter + + theorem primes_infinite' : ∀ s : Finset Nat, ∃ p, Nat.Prime p ∧ p ∉ s := by + intro s + by_contra h + push_neg at h + set s' := s.filter Nat.Prime with s'_def + have mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ n.Prime := by + intro n + simp [s'_def] + apply h + have : 2 ≤ (∏ i in s', i) + 1 := by + sorry, + rcases exists_prime_factor this with ⟨p, pp, pdvd⟩ + have : p ∣ ∏ i in s', i := by + sorry, + have : p ∣ 1 := by + convert Nat.dvd_sub' pdvd this + simp + show False + sorry + theorem bounded_of_ex_finset (Q : ℕ → Prop) : + (∃ s : Finset ℕ, ∀ k, Q k → k ∈ s) → ∃ n, ∀ k, Q k → k < n := by + rintro ⟨s, hs⟩ + use s.sup id + 1 + intro k Qk + apply Nat.lt_succ_of_le + show id k ≤ s.sup id + apply le_sup (hs k Qk) + + theorem ex_finset_of_bounded (Q : ℕ → Prop) [DecidablePred Q] : + (∃ n, ∀ k, Q k → k ≤ n) → ∃ s : Finset ℕ, ∀ k, Q k ↔ k ∈ s := by + rintro ⟨n, hn⟩ + use (range (n + 1)).filter Q + intro k + simp [Nat.lt_succ_iff] + exact hn k + + example : 27 % 4 = 3 := by norm_num + + example (n : ℕ) : (4 * n + 3) % 4 = 3 := by + rw [add_comm, Nat.add_mul_mod_self_left] + norm_num + + theorem mod_4_eq_3_or_mod_4_eq_3 {m n : ℕ} (h : m * n % 4 = 3) : m % 4 = 3 ∨ n % 4 = 3 := by + revert h + rw [Nat.mul_mod] + have : m % 4 < 4 := Nat.mod_lt m (by norm_num) + interval_cases hm : m % 4 <;> simp [hm] + have : n % 4 < 4 := Nat.mod_lt n (by norm_num) + interval_cases hn : n % 4 <;> simp [hn] + + theorem two_le_of_mod_4_eq_3 {n : ℕ} (h : n % 4 = 3) : 2 ≤ n := by + apply two_le <;> + · intro neq + rw [neq] at h + norm_num at h + + theorem aux {m n : ℕ} (h₀ : m ∣ n) (h₁ : 2 ≤ m) (h₂ : m < n) : n / m ∣ n ∧ n / m < n := by + sorry + theorem exists_prime_factor_mod_4_eq_3 {n : Nat} (h : n % 4 = 3) : + ∃ p : Nat, p.Prime ∧ p ∣ n ∧ p % 4 = 3 := by + by_cases np : n.Prime + · use n + exact ⟨np, dvd_rfl, h⟩ + induction' n using Nat.strong_induction_on with n ih + dsimp at ih + rw [Nat.prime_def_lt] at np + push_neg at np + rcases np (two_le_of_mod_4_eq_3 h) with ⟨m, mltn, mdvdn, mne1⟩ + have mge2 : 2 ≤ m := by + apply two_le _ mne1 + intro mz + rw [mz, zero_dvd_iff] at mdvdn + linarith + have neq : m * (n / m) = n := Nat.mul_div_cancel' mdvdn + have : m % 4 = 3 ∨ n / m % 4 = 3 := by + apply mod_4_eq_3_or_mod_4_eq_3 + rw [neq, h] + cases' this with h1 h1 + . sorry + sorry + example (m n : ℕ) (s : Finset ℕ) (h : m ∈ erase s n) : m ≠ n ∧ m ∈ s := by + rwa [mem_erase] at h + + example (m n : ℕ) (s : Finset ℕ) (h : m ∈ erase s n) : m ≠ n ∧ m ∈ s := by + simp at h + assumption + + theorem primes_mod_4_eq_3_infinite : ∀ n, ∃ p > n, Nat.Prime p ∧ p % 4 = 3 := by + by_contra h + push_neg at h + cases' h with n hn + have : ∃ s : Finset Nat, ∀ p : ℕ, p.Prime ∧ p % 4 = 3 ↔ p ∈ s := by + apply ex_finset_of_bounded + use n + contrapose! hn + rcases hn with ⟨p, ⟨pp, p4⟩, pltn⟩ + exact ⟨p, pltn, pp, p4⟩ + cases' this with s hs + have h₁ : ((4 * ∏ i in erase s 3, i) + 3) % 4 = 3 := by + sorry, + rcases exists_prime_factor_mod_4_eq_3 h₁ with ⟨p, pp, pdvd, p4eq⟩ + have ps : p ∈ s := by + sorry, + have pne3 : p ≠ 3 := by + sorry, + have : p ∣ 4 * ∏ i in erase s 3, i := by + sorry, + have : p ∣ 3 := by + sorry, + have : p = 3 := by + sorry, + contradiction +
src/C_Number_Theory/solutions/Solutions_S01_Irrational_Roots.lean (new)
- @@ -0,0 +1,101 @@
 + import Mathlib.Data.Nat.Factorization.Basic + import Mathlib.Data.Nat.Prime + import Mathlib.Tactic.NormNum.GCD + import Mathlib.Tactic.NormNum.Prime + + theorem even_of_even_sqr {m : ℕ} (h : 2 ∣ m ^ 2) : 2 ∣ m := by + rw [pow_two, Nat.prime_two.dvd_mul] at h + cases h <;> assumption + + example {m n : ℕ} (coprime_mn : m.coprime n) : m ^ 2 ≠ 2 * n ^ 2 := by + intro sqr_eq + have : 2 ∣ m := by + apply even_of_even_sqr + rw [sqr_eq] + apply dvd_mul_right + obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this + have : 2 * (2 * k ^ 2) = 2 * n ^ 2 := by + rw [← sqr_eq, meq] + ring + have : 2 * k ^ 2 = n ^ 2 := + (mul_right_inj' (by norm_num)).mp this + have : 2 ∣ n := by + apply even_of_even_sqr + rw [← this] + apply dvd_mul_right + have : 2 ∣ m.gcd n := by + apply Nat.dvd_gcd <;> + assumption + have : 2 ∣ 1 := by + convert this + symm + exact coprime_mn + norm_num at this + + example {m n p : ℕ} (coprime_mn : m.coprime n) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by + intro sqr_eq + have : p ∣ m := by + apply prime_p.dvd_of_dvd_pow + rw [sqr_eq] + apply dvd_mul_right + obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this + have : p * (p * k ^ 2) = p * n ^ 2 := by + rw [← sqr_eq, meq] + ring + have : p * k ^ 2 = n ^ 2 := by + apply (mul_right_inj' _).mp this + exact prime_p.ne_zero + have : p ∣ n := by + apply prime_p.dvd_of_dvd_pow + rw [← this] + apply dvd_mul_right + have : p ∣ Nat.gcd m n := by apply Nat.dvd_gcd <;> assumption + have : p ∣ 1 := by + convert this + symm + exact coprime_mn + have : 2 ≤ 1 := by + apply prime_p.two_le.trans + exact Nat.le_of_dvd zero_lt_one this + norm_num at this + + theorem factorization_mul' {m n : ℕ} (mnez : m ≠ 0) (nnez : n ≠ 0) (p : ℕ) : + (m * n).factorization p = m.factorization p + n.factorization p := by + rw [Nat.factorization_mul mnez nnez] + rfl + + theorem factorization_pow' (n k p : ℕ) : (n ^ k).factorization p = k * n.factorization p := by + rw [Nat.factorization_pow] + rfl + + theorem Nat.Prime.factorization' {p : ℕ} (prime_p : p.Prime) : p.factorization p = 1 := by + rw [prime_p.factorization] + simp + + example {m n p : ℕ} (nnz : n ≠ 0) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by + intro sqr_eq + have nsqr_nez : n ^ 2 ≠ 0 := by simpa + have eq1 : Nat.factorization (m ^ 2) p = 2 * m.factorization p := by + rw [factorization_pow'] + have eq2 : (p * n ^ 2).factorization p = 2 * n.factorization p + 1 := by + rw [factorization_mul' prime_p.ne_zero nsqr_nez, prime_p.factorization', factorization_pow', + add_comm] + have : 2 * m.factorization p % 2 = (2 * n.factorization p + 1) % 2 := by rw [← eq1, sqr_eq, eq2] + rw [add_comm, Nat.add_mul_mod_self_left, Nat.mul_mod_right] at this + norm_num at this + + example {m n k r : ℕ} (nnz : n ≠ 0) (pow_eq : m ^ k = r * n ^ k) {p : ℕ} (prime_p : p.Prime) : + k ∣ r.factorization p := by + cases' r with r + · simp + have npow_nz : n ^ k ≠ 0 := fun npowz => nnz (pow_eq_zero npowz) + have eq1 : (m ^ k).factorization p = k * m.factorization p := by + rw [factorization_pow'] + have eq2 : (r.succ * n ^ k).factorization p = k * n.factorization p + r.succ.factorization p := by + rw [factorization_mul' r.succ_ne_zero npow_nz, factorization_pow', add_comm] + have : r.succ.factorization p = k * m.factorization p - k * n.factorization p := by + rw [← eq1, pow_eq, eq2, add_comm, Nat.add_sub_cancel] + rw [this] + apply Nat.dvd_sub' <;> + apply Nat.dvd_mul_right +
src/C_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean (new)
- @@ -0,0 +1,98 @@
 + import Mathlib.Data.Nat.GCD.Basic + import Mathlib.Algebra.BigOperators.Basic + import Mathlib.Tactic + + def fac : ℕ → ℕ + | 0 => 1 + | n + 1 => (n + 1) * fac n + + theorem pow_two_le_fac (n : ℕ) : 2 ^ (n - 1) ≤ fac n := by + cases' n with n + · simp [fac] + induction' n with n ih + · simp [fac] + simp at * + rw [pow_succ, fac] + apply Nat.mul_le_mul _ ih + repeat' apply Nat.succ_le_succ + apply zero_le + + section + + variable {α : Type _} (s : Finset ℕ) (f : ℕ → ℕ) (n : ℕ) + + open BigOperators + open Finset + + theorem sum_sqr (n : ℕ) : (∑ i in range (n + 1), i ^ 2) = n * (n + 1) * (2 * n + 1) / 6 := by + symm; + apply Nat.div_eq_of_eq_mul_right (by norm_num : 0 < 6) + induction' n with n ih + · simp + rw [Finset.sum_range_succ, mul_add 6, ← ih, Nat.succ_eq_add_one] + ring + + end + + inductive MyNat + | zero : MyNat + | succ : MyNat → MyNat + + namespace MyNat + + def add : MyNat → MyNat → MyNat + | x, zero => x + | x, succ y => succ (add x y) + + def mul : MyNat → MyNat → MyNat + | x, zero => zero + | x, succ y => add (mul x y) x + + theorem zero_add (n : MyNat) : add zero n = n := by + induction' n with n ih + · rfl + rw [add, ih] + + theorem succ_add (m n : MyNat) : add (succ m) n = succ (add m n) := by + induction' n with n ih + · rfl + rw [add, ih] + rfl + + theorem add_comm (m n : MyNat) : add m n = add n m := by + induction' n with n ih + · rw [zero_add] + rfl + rw [add, succ_add, ih] + + theorem add_assoc (m n k : MyNat) : add (add m n) k = add m (add n k) := by + induction' k with k ih + · rfl + rw [add, ih] + rfl + + theorem mul_add (m n k : MyNat) : mul m (add n k) = add (mul m n) (mul m k) := by + induction' k with k ih + · rfl + rw [add, mul, mul, ih, add_assoc] + + theorem zero_mul (n : MyNat) : mul zero n = zero := by + induction' n with n ih + · rfl + rw [mul, ih] + rfl + + theorem succ_mul (m n : MyNat) : mul (succ m) n = add (mul m n) n := by + induction' n with n ih + · rfl + rw [mul, mul, ih, add_assoc, add_assoc, add_comm n, succ_add] + rfl + + theorem mul_comm (m n : MyNat) : mul m n = mul n m := by + induction' n with n ih + · rw [zero_mul] + rfl + rw [mul, ih, succ_mul] + + end MyNat +
src/C_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean (new)
- @@ -0,0 +1,241 @@
 + import Mathlib.Data.Nat.Prime + import Mathlib.Algebra.BigOperators.Order + import Mathlib.Tactic + import Mathlib.Tactic.IntervalCases + + open BigOperators + + theorem two_le {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by + cases m; contradiction + case succ m => + cases m; contradiction + repeat' apply Nat.succ_le_succ + apply zero_le + + theorem exists_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ p : Nat, p.Prime ∧ p ∣ n := by + by_cases np : n.Prime + · use n, np + induction' n using Nat.strong_induction_on with n ih + dsimp at ih + rw [Nat.prime_def_lt] at np + push_neg at np + rcases np h with ⟨m, mltn, mdvdn, mne1⟩ + have : m ≠ 0 := by + intro mz + rw [mz, zero_dvd_iff] at mdvdn + linarith + have mgt2 : 2 ≤ m := two_le this mne1 + by_cases mp : m.Prime + · use m, mp + exact mdvdn + . rcases ih m mltn mgt2 mp with ⟨p, pp, pdvd⟩ + use p, pp + apply pdvd.trans mdvdn + + theorem primes_infinite : ∀ n, ∃ p > n, Nat.Prime p := by + intro n + have : 2 ≤ Nat.factorial (n + 1) + 1 := by + apply Nat.succ_le_succ + exact Nat.succ_le_of_lt (Nat.factorial_pos _) + rcases exists_prime_factor this with ⟨p, pp, pdvd⟩ + refine' ⟨p, _, pp⟩ + show p > n + by_contra ple + push_neg at ple + have : p ∣ Nat.factorial (n + 1) := by + apply Nat.dvd_factorial + apply pp.pos + linarith + have : p ∣ 1 := by + convert Nat.dvd_sub' pdvd this + simp + show False + have := Nat.le_of_dvd zero_lt_one this + linarith [pp.two_le] + + open Finset + + section + + variable {α : Type _} [DecidableEq α] (r s t : Finset α) + + example : (r ∪ s) ∩ (r ∪ t) = r ∪ s ∩ t := by + ext x + rw [mem_inter, mem_union, mem_union, mem_union, mem_inter] + tauto + + example : (r ∪ s) ∩ (r ∪ t) = r ∪ s ∩ t := by + ext x + simp + tauto + + example : (r \ s) \ t = r \ (s ∪ t) := by + ext x + rw [mem_sdiff, mem_sdiff, mem_sdiff, mem_union] + tauto + + example : (r \ s) \ t = r \ (s ∪ t) := by + ext x + simp + tauto + + end + + theorem Nat.Prime.eq_of_dvd_of_prime {p q : ℕ} (prime_p : Nat.Prime p) (prime_q : Nat.Prime q) + (h : p ∣ q) : p = q := by + cases prime_q.eq_one_or_self_of_dvd _ h + · linarith [prime_p.two_le] + assumption + + theorem mem_of_dvd_prod_primes {s : Finset ℕ} {p : ℕ} (prime_p : p.Prime) : + (∀ n ∈ s, Nat.Prime n) → (p ∣ ∏ n in s, n) → p ∈ s := by + intro h₀ h₁ + induction' s using Finset.induction_on with a s ans ih + · simp at h₁ + linarith [prime_p.two_le] + simp [Finset.prod_insert ans, prime_p.dvd_mul] at h₀ h₁ + rw [mem_insert] + cases' h₁ with h₁ h₁ + · left + exact prime_p.eq_of_dvd_of_prime h₀.1 h₁ + right + exact ih h₀.2 h₁ + + theorem primes_infinite' : ∀ s : Finset Nat, ∃ p, Nat.Prime p ∧ p ∉ s := by + intro s + by_contra h + push_neg at h + set s' := s.filter Nat.Prime with s'_def + have mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ n.Prime := by + intro n + simp [s'_def] + apply h + have : 2 ≤ (∏ i in s', i) + 1 := by + apply Nat.succ_le_succ + apply Nat.succ_le_of_lt + apply Finset.prod_pos + intro n ns' + apply (mem_s'.mp ns').pos + rcases exists_prime_factor this with ⟨p, pp, pdvd⟩ + have : p ∣ ∏ i in s', i := by + apply dvd_prod_of_mem + rw [mem_s'] + apply pp + have : p ∣ 1 := by + convert Nat.dvd_sub' pdvd this + simp + show False + have := Nat.le_of_dvd zero_lt_one this + linarith [pp.two_le] + + theorem bounded_of_ex_finset (Q : ℕ → Prop) : + (∃ s : Finset ℕ, ∀ k, Q k → k ∈ s) → ∃ n, ∀ k, Q k → k < n := by + rintro ⟨s, hs⟩ + use s.sup id + 1 + intro k Qk + apply Nat.lt_succ_of_le + show id k ≤ s.sup id + apply le_sup (hs k Qk) + + theorem ex_finset_of_bounded (Q : ℕ → Prop) [DecidablePred Q] : + (∃ n, ∀ k, Q k → k ≤ n) → ∃ s : Finset ℕ, ∀ k, Q k ↔ k ∈ s := by + rintro ⟨n, hn⟩ + use (range (n + 1)).filter Q + intro k + simp [Nat.lt_succ_iff] + exact hn k + + theorem mod_4_eq_3_or_mod_4_eq_3 {m n : ℕ} (h : m * n % 4 = 3) : m % 4 = 3 ∨ n % 4 = 3 := by + revert h + rw [Nat.mul_mod] + have : m % 4 < 4 := Nat.mod_lt m (by norm_num) + interval_cases hm : m % 4 <;> simp [hm] + have : n % 4 < 4 := Nat.mod_lt n (by norm_num) + interval_cases hn : n % 4 <;> simp [hn] + + theorem two_le_of_mod_4_eq_3 {n : ℕ} (h : n % 4 = 3) : 2 ≤ n := by + apply two_le <;> + · intro neq + rw [neq] at h + norm_num at h + + theorem aux {m n : ℕ} (h₀ : m ∣ n) (h₁ : 2 ≤ m) (h₂ : m < n) : n / m ∣ n ∧ n / m < n := by + constructor + · exact Nat.div_dvd_of_dvd h₀ + exact Nat.div_lt_self (lt_of_le_of_lt (zero_le _) h₂) h₁ + + theorem exists_prime_factor_mod_4_eq_3 {n : Nat} (h : n % 4 = 3) : + ∃ p : Nat, p.Prime ∧ p ∣ n ∧ p % 4 = 3 := by + by_cases np : n.Prime + · use n + exact ⟨np, dvd_rfl, h⟩ + induction' n using Nat.strong_induction_on with n ih + dsimp at ih + rw [Nat.prime_def_lt] at np + push_neg at np + rcases np (two_le_of_mod_4_eq_3 h) with ⟨m, mltn, mdvdn, mne1⟩ + have mge2 : 2 ≤ m := by + apply two_le _ mne1 + intro mz + rw [mz, zero_dvd_iff] at mdvdn + linarith + have neq : m * (n / m) = n := Nat.mul_div_cancel' mdvdn + have : m % 4 = 3 ∨ n / m % 4 = 3 := by + apply mod_4_eq_3_or_mod_4_eq_3 + rw [neq, h] + cases' this with h1 h1 + · by_cases mp : m.Prime + · use m + exact ⟨mp, mdvdn, h1⟩ + rcases ih m mltn h1 mp with ⟨p, pp, pdvd, p4eq⟩ + use p + exact ⟨pp, pdvd.trans mdvdn, p4eq⟩ + obtain ⟨nmdvdn, nmltn⟩ := aux mdvdn mge2 mltn + by_cases nmp : (n / m).Prime + · use n / m + exact ⟨nmp, nmdvdn, h1⟩ + rcases ih (n / m) nmltn h1 nmp with ⟨p, pp, pdvd, p4eq⟩ + use p + exact ⟨pp, pdvd.trans nmdvdn, p4eq⟩ + + theorem primes_mod_4_eq_3_infinite : ∀ n, ∃ p > n, Nat.Prime p ∧ p % 4 = 3 := by + by_contra h + push_neg at h + cases' h with n hn + have : ∃ s : Finset Nat, ∀ p : ℕ, p.Prime ∧ p % 4 = 3 ↔ p ∈ s := by + apply ex_finset_of_bounded + use n + contrapose! hn + rcases hn with ⟨p, ⟨pp, p4⟩, pltn⟩ + exact ⟨p, pltn, pp, p4⟩ + cases' this with s hs + have h₁ : ((4 * ∏ i in erase s 3, i) + 3) % 4 = 3 := by + rw [add_comm, Nat.add_mul_mod_self_left] + norm_num + rcases exists_prime_factor_mod_4_eq_3 h₁ with ⟨p, pp, pdvd, p4eq⟩ + have ps : p ∈ s := by + rw [← hs p] + exact ⟨pp, p4eq⟩ + have pne3 : p ≠ 3 := by + intro peq + rw [peq, ← Nat.dvd_add_iff_left (dvd_refl 3)] at pdvd + rw [Nat.prime_three.dvd_mul] at pdvd + norm_num at pdvd + have : 3 ∈ s.erase 3 := by + apply mem_of_dvd_prod_primes Nat.prime_three _ pdvd + intro n + simp [← hs n] + tauto + simp at this + have : p ∣ 4 * ∏ i in erase s 3, i := by + apply dvd_trans _ (dvd_mul_left _ _) + apply dvd_prod_of_mem + simp + constructor <;> assumption + have : p ∣ 3 := by + convert Nat.dvd_sub' pdvd this + simp + have : p = 3 := by + apply pp.eq_of_dvd_of_prime Nat.prime_three this + contradiction +
src/C_Sets_and_Functions/S01_Sets.lean (new)
- @@ -0,0 +1,254 @@
 + import Mathlib.Data.Set.Lattice + import Mathlib.Data.Nat.Prime + import Mathlib.Data.Nat.Parity + import Mathlib.Tactic + + section + + variable {α : Type _} + + variable (s t u : Set α) + + open Set + + example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := by + rw [subset_def, inter_def, inter_def] + rw [subset_def] at h + dsimp + rintro x ⟨xs, xu⟩ + exact ⟨h _ xs, xu⟩ + + example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := by + simp only [subset_def, mem_inter_iff] at * + rintro x ⟨xs, xu⟩ + exact ⟨h _ xs, xu⟩ + + example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := by + intro x xsu + exact ⟨h xsu.1, xsu.2⟩ + + theorem foo (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := + fun x ⟨xs, xu⟩ => ⟨h xs, xu⟩ + + example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := + fun x ⟨xs, xu⟩ => ⟨h xs, xu⟩ + + example : s ∩ (t ∪ u) ⊆ s ∩ t ∪ s ∩ u := by + intro x hx + have xs : x ∈ s := hx.1 + have xtu : x ∈ t ∪ u := hx.2 + cases' xtu with xt xu + · left + show x ∈ s ∩ t + exact ⟨xs, xt⟩ + right + show x ∈ s ∩ u + exact ⟨xs, xu⟩ + + example : s ∩ (t ∪ u) ⊆ s ∩ t ∪ s ∩ u := by + rintro x ⟨xs, xt | xu⟩ + · left + exact ⟨xs, xt⟩ + right; exact ⟨xs, xu⟩ + + example : s ∩ t ∪ s ∩ u ⊆ s ∩ (t ∪ u) := by + sorry + example : (s \ t) \ u ⊆ s \ (t ∪ u) := by + intro x xstu + have xs : x ∈ s := xstu.1.1 + have xnt : x ∉ t := xstu.1.2 + have xnu : x ∉ u := xstu.2 + constructor + · exact xs + intro xtu + -- x ∈ t ∨ x ∈ u + cases' xtu with xt xu + · show False + exact xnt xt + show False; exact xnu xu + + example : (s \ t) \ u ⊆ s \ (t ∪ u) := by + rintro x ⟨⟨xs, xnt⟩, xnu⟩ + use xs + rintro (xt | xu) <;> contradiction + + example : s \ (t ∪ u) ⊆ (s \ t) \ u := by + sorry + example : s ∩ t = t ∩ s := by + ext x + simp only [mem_inter_iff] + constructor + · rintro ⟨xs, xt⟩ + exact ⟨xt, xs⟩ + rintro ⟨xt, xs⟩; exact ⟨xs, xt⟩ + + example : s ∩ t = t ∩ s := + Set.ext fun x => ⟨fun ⟨xs, xt⟩ => ⟨xt, xs⟩, fun ⟨xt, xs⟩ => ⟨xs, xt⟩⟩ + + example : s ∩ t = t ∩ s := by ext x; simp [and_comm] + + example : s ∩ t = t ∩ s := by + apply Subset.antisymm + · rintro x ⟨xs, xt⟩ + exact ⟨xt, xs⟩ + rintro x ⟨xt, xs⟩; exact ⟨xs, xt⟩ + + example : s ∩ t = t ∩ s := + Subset.antisymm sorry sorry + example : s ∩ (s ∪ t) = s := by + sorry + + example : s ∪ s ∩ t = s := by + sorry + + example : s \ t ∪ t = s ∪ t := by + sorry + + example : s \ t ∪ t \ s = (s ∪ t) \ (s ∩ t) := by + sorry + + def evens : Set ℕ := + { n | Even n } + + def odds : Set ℕ := + { n | ¬Even n } + + example : evens ∪ odds = univ := by + rw [evens, odds] + ext n + simp + apply Classical.em + + example (x : ℕ) (h : x ∈ (∅ : Set ℕ)) : False := + h + + example (x : ℕ) : x ∈ (univ : Set ℕ) := + trivial + + example : { n | Nat.Prime n } ∩ { n | n > 2 } ⊆ { n | ¬Even n } := by + sorry + + #print Prime + + #print Nat.Prime + + example (n : ℕ) : Prime n ↔ Nat.Prime n := + Nat.prime_iff.symm + + example (n : ℕ) (h : Prime n) : Nat.Prime n := by + rw [Nat.prime_iff] + exact h + + example (n : ℕ) (h : Prime n) : Nat.Prime n := by + rwa [Nat.prime_iff] + + end + + section + + variable (s t : Set ℕ) + + example (h₀ : ∀ x ∈ s, ¬Even x) (h₁ : ∀ x ∈ s, Prime x) : ∀ x ∈ s, ¬Even x ∧ Prime x := by + intro x xs + constructor + · apply h₀ x xs + apply h₁ x xs + + example (h : ∃ x ∈ s, ¬Even x ∧ Prime x) : ∃ x ∈ s, Prime x := by + rcases h with ⟨x, xs, _, prime_x⟩ + use x, xs + exact prime_x + + section + + variable (ssubt : s ⊆ t) + + example (h₀ : ∀ x ∈ t, ¬Even x) (h₁ : ∀ x ∈ t, Prime x) : ∀ x ∈ s, ¬Even x ∧ Prime x := by + sorry + + example (h : ∃ x ∈ s, ¬Even x ∧ Prime x) : ∃ x ∈ t, Prime x := by + sorry + + end + + end + + section + + variable {α I : Type _} + + variable (A B : I → Set α) + + variable (s : Set α) + + open Set + + example : (s ∩ ⋃ i, A i) = ⋃ i, A i ∩ s := by + ext x + simp only [mem_inter_iff, mem_iUnion] + constructor + · rintro ⟨xs, ⟨i, xAi⟩⟩ + exact ⟨i, xAi, xs⟩ + rintro ⟨i, xAi, xs⟩ + exact ⟨xs, ⟨i, xAi⟩⟩ + + example : (⋂ i, A i ∩ B i) = (⋂ i, A i) ∩ ⋂ i, B i := by + ext x + simp only [mem_inter_iff, mem_iInter] + constructor + · intro h + constructor + · intro i + exact (h i).1 + intro i + exact (h i).2 + rintro ⟨h1, h2⟩ i + constructor + · exact h1 i + exact h2 i + + + example : (s ∪ ⋂ i, A i) = ⋂ i, A i ∪ s := by + sorry + + def primes : Set ℕ := + { x | Nat.Prime x } + + example : (⋃ p ∈ primes, { x | p ^ 2 ∣ x }) = { x | ∃ p ∈ primes, p ^ 2 ∣ x } :=by + ext + rw [mem_iUnion₂] + simp + + example : (⋃ p ∈ primes, { x | p ^ 2 ∣ x }) = { x | ∃ p ∈ primes, p ^ 2 ∣ x } := by + ext + simp + + example : (⋂ p ∈ primes, { x | ¬p ∣ x }) ⊆ { x | x = 1 } := by + intro x + contrapose! + simp + apply Nat.exists_prime_and_dvd + + example : (⋃ p ∈ primes, { x | x ≤ p }) = univ := by + sorry + + end + + section + + open Set + + variable {α : Type _} (s : Set (Set α)) + + example : ⋃₀ s = ⋃ t ∈ s, t := by + ext x + rw [mem_iUnion₂] + simp + + example : ⋂₀ s = ⋂ t ∈ s, t := by + ext x + rw [mem_iInter₂] + rfl + + end +
src/C_Sets_and_Functions/S02_Functions.lean (new)
- @@ -0,0 +1,217 @@
  + import Mathlib.Data.Set.Lattice + import Mathlib.Data.Set.Function + import Mathlib.Analysis.SpecialFunctions.Log.Basic + + section + + variable {α β : Type _} + variable (f : α → β) + variable (s t : Set α) + variable (u v : Set β) + + open Function + open Set + + example : f ⁻¹' (u ∩ v) = f ⁻¹' u ∩ f ⁻¹' v := by + ext + rfl + + example : f '' (s ∪ t) = f '' s ∪ f '' t := by + ext y; constructor + · rintro ⟨x, xs | xt, rfl⟩ + · left + use x, xs + right + use x, xt + rintro (⟨x, xs, rfl⟩ | ⟨x, xt, rfl⟩) + · use x, Or.inl xs + use x, Or.inr xt + + example : s ⊆ f ⁻¹' (f '' s) := by + intro x xs + show f x ∈ f '' s + use x, xs + + example : f '' s ⊆ v ↔ s ⊆ f ⁻¹' v := by + sorry + + example (h : Injective f) : f ⁻¹' (f '' s) ⊆ s := by + sorry + + example : f '' (f ⁻¹' u) ⊆ u := by + sorry + + example (h : Surjective f) : u ⊆ f '' (f ⁻¹' u) := by + sorry + + example (h : s ⊆ t) : f '' s ⊆ f '' t := by + sorry + + example (h : u ⊆ v) : f ⁻¹' u ⊆ f ⁻¹' v := by + sorry + + example : f ⁻¹' (u ∪ v) = f ⁻¹' u ∪ f ⁻¹' v := by + sorry + + example : f '' (s ∩ t) ⊆ f '' s ∩ f '' t := by + sorry + + example (h : Injective f) : f '' s ∩ f '' t ⊆ f '' (s ∩ t) := by + sorry + + example : f '' s \ f '' t ⊆ f '' (s \ t) := by + sorry + + example : f ⁻¹' u \ f ⁻¹' v ⊆ f ⁻¹' (u \ v) := by + sorry + + example : f '' s ∩ v = f '' (s ∩ f ⁻¹' v) := by + sorry + + example : f '' (s ∩ f ⁻¹' u) ⊆ f '' s ∪ u := by + sorry + + example : s ∩ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∩ u) := by + sorry + + example : s ∪ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∪ u) := by + sorry + + variable {I : Type _} (A : I → Set α) (B : I → Set β) + + example : (f '' ⋃ i, A i) = ⋃ i, f '' A i := by + ext y; simp + constructor + · rintro ⟨x, ⟨i, xAi⟩, fxeq⟩ + use i, x + exact ⟨xAi, fxeq⟩ + rintro ⟨i, x, xAi, fxeq⟩ + exact ⟨x, ⟨i, xAi⟩, fxeq⟩ + + example : (f '' ⋂ i, A i) ⊆ ⋂ i, f '' A i := by + intro y; simp + intro x h fxeq i + use x + exact ⟨h i, fxeq⟩ + + example (i : I) (injf : Injective f) : (⋂ i, f '' A i) ⊆ f '' ⋂ i, A i := by + intro y; simp + intro h + rcases h i with ⟨x, xAi, fxeq⟩ + use x; constructor + · intro i' + rcases h i' with ⟨x', x'Ai, fx'eq⟩ + have : f x = f x' := by rw [fxeq, fx'eq] + have : x = x' := injf this + rw [this] + exact x'Ai + exact fxeq + + example : (f ⁻¹' ⋃ i, B i) = ⋃ i, f ⁻¹' B i := by + ext x + simp + + example : (f ⁻¹' ⋂ i, B i) = ⋂ i, f ⁻¹' B i := by + ext x + simp + + example : InjOn f s ↔ ∀ x₁ ∈ s, ∀ x₂ ∈ s, f x₁ = f x₂ → x₁ = x₂ := + Iff.refl _ + + end + + section + + open Set Real + + example : InjOn log { x | x > 0 } := by + intro x xpos y ypos + intro e + -- log x = log y + calc + x = exp (log x) := by rw [exp_log xpos] + _ = exp (log y) := by rw [e] + _ = y := by rw [exp_log ypos] + + + example : range exp = { y | y > 0 } := by + ext y; constructor + · rintro ⟨x, rfl⟩ + apply exp_pos + intro ypos + use log y + rw [exp_log ypos] + + example : InjOn sqrt { x | x ≥ 0 } := by + sorry + + example : InjOn (fun x => x ^ 2) { x : ℝ | x ≥ 0 } := by + sorry + + example : sqrt '' { x | x ≥ 0 } = { y | y ≥ 0 } := by + sorry + + example : (range fun x => x ^ 2) = { y : ℝ | y ≥ 0 } := by + sorry + + end + + section + + variable {α β : Type _} [Inhabited α] + + #check (default : α) + + variable (P : α → Prop) (h : ∃ x, P x) + + #check Classical.choose h + + example : P (Classical.choose h) := + Classical.choose_spec h + + noncomputable section + + open Classical + + def inverse (f : α → β) : β → α := fun y : β => + if h : ∃ x, f x = y then Classical.choose h else default + + theorem inverse_spec {f : α → β} (y : β) (h : ∃ x, f x = y) : f (inverse f y) = y := by + rw [inverse]; dsimp; rw [dif_pos h] + exact Classical.choose_spec h + + variable (f : α → β) + + open Function + + example : Injective f ↔ LeftInverse (inverse f) f := + sorry + + example : Surjective f ↔ RightInverse (inverse f) f := + sorry + + end + + section + + variable {α : Type _} + + open Function + + theorem Cantor : ∀ f : α → Set α, ¬Surjective f := by + intro f surjf + let S := { i | i ∉ f i } + rcases surjf S with ⟨j, h⟩ + have h₁ : j ∉ f j := by + intro h' + have : j ∉ f j := by rwa [h] at h' + contradiction + have h₂ : j ∈ S + sorry + have h₃ : j ∉ S + sorry + contradiction + + -- COMMENTS: TODO: improve this + end +
src/C_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean (new)
- @@ -0,0 +1,99 @@
  + import Mathlib.Data.Set.Lattice + import Mathlib.Data.Set.Function + import Mathlib.Tactic + + open Set + + open Function + + noncomputable section + + open Classical + + variable {α β : Type _} [Nonempty β] + + section + + variable (f : α → β) (g : β → α) + + def sbAux : ℕ → Set α + | 0 => univ \ g '' univ + | n + 1 => g '' (f '' sbAux n) + + def sbSet := + ⋃ n, sbAux f g n + + def sbFun (x : α) : β := + if x ∈ sbSet f g then f x else invFun g x + + theorem sb_right_inv {x : α} (hx : x ∉ sbSet f g) : g (invFun g x) = x := by + have : x ∈ g '' univ := by + contrapose! hx + rw [sbSet, mem_iUnion] + use 0 + rw [sbAux, mem_diff] + sorry }, + have : ∃ y, g y = x := by + { sorry }, + sorry + theorem sb_injective (hf : Injective f) (hg : Injective g) : Injective (sbFun f g) := by + set A := sbSet f g with A_def + set h := sbFun f g with h_def + intro x₁ x₂ + intro (hxeq : h x₁ = h x₂) + show x₁ = x₂ + simp only [h_def, sbFun, ← A_def] at hxeq + by_cases xA : x₁ ∈ A ∨ x₂ ∈ A + · wlog x₁A : x₁ ∈ A generalizing x₁ x₂ hxeq xA + · symm + apply this hxeq.symm xA.symm (xA.resolve_left x₁A) + have x₂A : x₂ ∈ A := by + apply not_imp_self.mp + intro (x₂nA : x₂ ∉ A) + rw [if_pos x₁A, if_neg x₂nA] at hxeq + rw [A_def, sbSet, mem_iUnion] at x₁A + have x₂eq : x₂ = g (f x₁) := by + . sorry + rcases x₁A with ⟨n, hn⟩ + rw [A_def, sbSet, mem_iUnion] + use n + 1 + simp [sbAux] + exact ⟨x₁, hn, x₂eq.symm⟩ + . sorry, + push_neg at xA + sorry + theorem sb_surjective (hf : Injective f) (hg : Injective g) : Surjective (sbFun f g) := by + set A := sbSet f g with A_def + set h := sbFun f g with h_def + intro y + by_cases gyA : g y ∈ A + · rw [A_def, sbSet, mem_iUnion] at gyA + rcases gyA with ⟨n, hn⟩ + cases' n with n + · simp [sbAux] at hn + simp [sbAux] at hn + rcases hn with ⟨x, xmem, hx⟩ + use x + have : x ∈ A := by + rw [A_def, sbSet, mem_iUnion] + exact ⟨n, xmem⟩ + simp only [h_def, sbFun, if_pos this] + exact hg hx + sorry + end + + theorem schroeder_bernstein {f : α → β} {g : β → α} (hf : Injective f) (hg : Injective g) : + ∃ h : α → β, Bijective h := + ⟨sbFun f g, sb_injective f g hf hg, sb_surjective f g hf hg⟩ + + -- Auxiliary information + section + variable (g : β → α) (x : α) + + #check (invFun g : α → β) + #check (leftInverse_invFun : Injective g → LeftInverse (invFun g) g) + #check (leftInverse_invFun : Injective g → ∀ y, invFun g (g y) = y) + #check (invFun_eq : (∃ y, g y = x) → g (invFun g x) = x) + + end +
src/C_Sets_and_Functions/solutions/Solutions_S01_Sets.lean (new)
- @@ -0,0 +1,165 @@
 + import Mathlib.Data.Set.Lattice + import Mathlib.Data.Nat.Prime + import Mathlib.Data.Nat.Parity + import Mathlib.Tactic + + section + + variable {α : Type _} + + variable (s t u : Set α) + + open Set + + example : s ∩ t ∪ s ∩ u ⊆ s ∩ (t ∪ u) := by + rintro x (⟨xs, xt⟩ | ⟨xs, xu⟩) + · use xs + left + exact xt + use xs; right; exact xu + + example : s \ (t ∪ u) ⊆ (s \ t) \ u := by + rintro x ⟨xs, xntu⟩ + constructor + use xs + · intro xt + exact xntu (Or.inl xt) + intro xu + apply xntu (Or.inr xu) + + example : s ∩ t = t ∩ s := + Subset.antisymm + (fun x ⟨xs, xt⟩ => ⟨xt, xs⟩) fun x ⟨xt, xs⟩ => ⟨xs, xt⟩ + + example : s ∩ (s ∪ t) = s := by + ext x; constructor + · rintro ⟨xs, _⟩ + exact xs + intro xs + use xs; left; exact xs + + example : s ∪ s ∩ t = s := by + ext x; constructor + · rintro (xs | ⟨xs, xt⟩) <;> exact xs + intro xs; left; exact xs + + example : s \ t ∪ t = s ∪ t := by + ext x; constructor + · rintro (⟨xs, nxt⟩ | xt) + · left + exact xs + right + exact xt + by_cases h : x ∈ t + · intro + right + exact h + rintro (xs | xt) + · left + use xs + exact h + right; exact xt + + example : s \ t ∪ t \ s = (s ∪ t) \ (s ∩ t) := by + ext x; constructor + · rintro (⟨xs, xnt⟩ | ⟨xt, xns⟩) + · constructor + left + exact xs + rintro ⟨_, xt⟩ + contradiction + constructor + right + exact xt + rintro ⟨xs, _⟩ + contradiction + rintro ⟨xs | xt, nxst⟩ + · left + use xs + intro xt + apply nxst + constructor <;> assumption + right; use xt; intro xs + apply nxst + constructor <;> assumption + + example : { n | Nat.Prime n } ∩ { n | n > 2 } ⊆ { n | ¬Even n } := by + intro n + simp + intro nprime + cases' Nat.Prime.eq_two_or_odd nprime with h h + · rw [h] + intro + linarith + rw [Nat.even_iff, h] + norm_num + + end + + section + + variable (s t : Set ℕ) + + section + + variable (ssubt : s ⊆ t) + + example (h₀ : ∀ x ∈ t, ¬Even x) (h₁ : ∀ x ∈ t, Prime x) : ∀ x ∈ s, ¬Even x ∧ Prime x := by + intro x xs + constructor + · apply h₀ x (ssubt xs) + apply h₁ x (ssubt xs) + + example (h : ∃ x ∈ s, ¬Even x ∧ Prime x) : ∃ x ∈ t, Prime x := by + rcases h with ⟨x, xs, _, px⟩ + use x, ssubt xs + exact px + + end + + end + + section + + variable {α I : Type _} + + variable (A B : I → Set α) + + variable (s : Set α) + + open Set + + example : (s ∪ ⋂ i, A i) = ⋂ i, A i ∪ s := by + ext x + simp only [mem_union, mem_iInter] + constructor + · rintro (xs | xI) + · intro i + right + exact xs + intro i + left + exact xI i + intro h + by_cases xs : x ∈ s + · left + exact xs + right + intro i + cases h i + · assumption + contradiction + + def primes : Set ℕ := + { x | Nat.Prime x } + + example : (⋃ p ∈ primes, { x | x ≤ p }) = univ := by + apply eq_univ_of_forall + intro x + simp + rcases Nat.exists_infinite_primes x with ⟨p, primep, pge⟩ + use p, pge + exact primep + + end +
src/C_Sets_and_Functions/solutions/Solutions_S02_Functions.lean (new)
- @@ -0,0 +1,256 @@
  + import Mathlib.Data.Set.Lattice + import Mathlib.Data.Set.Function + import Mathlib.Analysis.SpecialFunctions.Log.Basic + + section + + variable {α β : Type _} + variable (f : α → β) + variable (s t : Set α) + variable (u v : Set β) + + open Function + open Set + + example : f '' s ⊆ v ↔ s ⊆ f ⁻¹' v := by + constructor + · intro h x xs + have : f x ∈ f '' s := mem_image_of_mem _ xs + exact h this + intro h y ymem + rcases ymem with ⟨x, xs, fxeq⟩ + rw [← fxeq] + apply h xs + + example (h : Injective f) : f ⁻¹' (f '' s) ⊆ s := by + rintro x ⟨y, ys, fxeq⟩ + rw [← h fxeq] + exact ys + + example : f '' (f ⁻¹' u) ⊆ u := by + rintro y ⟨x, xmem, rfl⟩ + exact xmem + + example (h : Surjective f) : u ⊆ f '' (f ⁻¹' u) := by + intro y yu + rcases h y with ⟨x, fxeq⟩ + use x + constructor + · show f x ∈ u + rw [fxeq] + exact yu + exact fxeq + + example (h : s ⊆ t) : f '' s ⊆ f '' t := by + rintro y ⟨x, xs, fxeq⟩ + use x, h xs + exact fxeq + + example (h : u ⊆ v) : f ⁻¹' u ⊆ f ⁻¹' v := by + intro x; apply h + + example : f ⁻¹' (u ∪ v) = f ⁻¹' u ∪ f ⁻¹' v := by + ext x; rfl + + example : f '' (s ∩ t) ⊆ f '' s ∩ f '' t := by + rintro y ⟨x, ⟨xs, xt⟩, rfl⟩ + constructor + . use x, xs + . use x, xt + + example (h : Injective f) : f '' s ∩ f '' t ⊆ f '' (s ∩ t) := by + rintro y ⟨⟨x₁, x₁s, rfl⟩, ⟨x₂, x₂t, fx₂eq⟩⟩ + use x₁ + constructor + . use x₁s + rw [← h fx₂eq] + exact x₂t + . rfl + + example : f '' s \ f '' t ⊆ f '' (s \ t) := by + rintro y ⟨⟨x₁, x₁s, rfl⟩, h⟩ + use x₁ + constructor + . constructor + . exact x₁s + . intro h' + apply h + use x₁, h' + . rfl + + example : f ⁻¹' u \ f ⁻¹' v ⊆ f ⁻¹' (u \ v) := + fun x => id + + example : f '' s ∩ v = f '' (s ∩ f ⁻¹' v) := by + ext y; constructor + · rintro ⟨⟨x, xs, rfl⟩, fxv⟩ + use x, ⟨xs, fxv⟩ + rintro ⟨x, ⟨⟨xs, fxv⟩, rfl⟩⟩ + exact ⟨⟨x, xs, rfl⟩, fxv⟩ + + example : f '' (s ∩ f ⁻¹' u) ⊆ f '' s ∩ u := by + rintro y ⟨x, ⟨xs, fxu⟩, rfl⟩ + exact ⟨⟨x, xs, rfl⟩, fxu⟩ + + example : s ∩ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∩ u) := by + rintro x ⟨xs, fxu⟩ + exact ⟨⟨x, xs, rfl⟩, fxu⟩ + + example : s ∪ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∪ u) := by + rintro x (xs | fxu) + · left + exact ⟨x, xs, rfl⟩ + right; exact fxu + + variable {I : Type _} (A : I → Set α) (B : I → Set β) + + example : (f '' ⋃ i, A i) = ⋃ i, f '' A i := by + ext y; simp + constructor + · rintro ⟨x, ⟨i, xAi⟩, fxeq⟩ + use i, x + exact ⟨xAi, fxeq⟩ + rintro ⟨i, x, xAi, fxeq⟩ + exact ⟨x, ⟨i, xAi⟩, fxeq⟩ + + example : (f '' ⋂ i, A i) ⊆ ⋂ i, f '' A i := by + intro y; simp + intro x h fxeq i + use x + exact ⟨h i, fxeq⟩ + + example (i : I) (injf : Injective f) : (⋂ i, f '' A i) ⊆ f '' ⋂ i, A i := by + intro y; simp + intro h + rcases h i with ⟨x, xAi, fxeq⟩ + use x; constructor + · intro i' + rcases h i' with ⟨x', x'Ai, fx'eq⟩ + have : f x = f x' := by rw [fxeq, fx'eq] + have : x = x' := injf this + rw [this] + exact x'Ai + exact fxeq + + example : (f ⁻¹' ⋃ i, B i) = ⋃ i, f ⁻¹' B i := by + ext x + simp + + example : (f ⁻¹' ⋂ i, B i) = ⋂ i, f ⁻¹' B i := by + ext x + simp + + end + + section + + open Set Real + + example : InjOn sqrt { x | x ≥ 0 } := by + intro x xnonneg y ynonneg + intro e + calc + x = sqrt x ^ 2 := by rw [sq_sqrt xnonneg] + _ = sqrt y ^ 2 := by rw [e] + _ = y := by rw [sq_sqrt ynonneg] + + + example : InjOn (fun x => x ^ 2) { x : ℝ | x ≥ 0 } := by + intro x xnonneg y ynonneg + intro e + dsimp at * + calc + x = sqrt (x ^ 2) := by rw [sqrt_sq xnonneg] + _ = sqrt (y ^ 2) := by rw [e] + _ = y := by rw [sqrt_sq ynonneg] + + + example : sqrt '' { x | x ≥ 0 } = { y | y ≥ 0 } := by + ext y; constructor + · rintro ⟨x, ⟨xnonneg, rfl⟩⟩ + apply sqrt_nonneg + intro ynonneg + use y ^ 2 + dsimp at * + constructor + apply pow_nonneg ynonneg + apply sqrt_sq + assumption + + example : (range fun x => x ^ 2) = { y : ℝ | y ≥ 0 } := by + ext y + constructor + · rintro ⟨x, rfl⟩ + dsimp at * + apply pow_two_nonneg + intro ynonneg + use sqrt y + exact sq_sqrt ynonneg + + end + + section + + variable {α β : Type _} [Inhabited α] + + noncomputable section + + open Classical + + def inverse (f : α → β) : β → α := fun y : β => + if h : ∃ x, f x = y then Classical.choose h else default + + theorem inverse_spec {f : α → β} (y : β) (h : ∃ x, f x = y) : f (inverse f y) = y := by + rw [inverse]; dsimp; rw [dif_pos h] + exact Classical.choose_spec h + + variable (f : α → β) + + open Function + + example : Injective f ↔ LeftInverse (inverse f) f := by + constructor + · intro h y + apply h + apply inverse_spec + use y + intro h x1 x2 e + rw [← h x1, ← h x2, e] + + example : Injective f ↔ LeftInverse (inverse f) f := + ⟨fun h y => h (inverse_spec _ ⟨y, rfl⟩), fun h x1 x2 e => by rw [← h x1, ← h x2, e]⟩ + + example : Surjective f ↔ RightInverse (inverse f) f := by + constructor + · intro h y + apply inverse_spec + apply h + intro h y + use inverse f y + apply h + + example : Surjective f ↔ RightInverse (inverse f) f := + ⟨fun h y => inverse_spec _ (h _), fun h y => ⟨inverse f y, h _⟩⟩ + + end + + section + + variable {α : Type _} + + open Function + + theorem Cantor : ∀ f : α → Set α, ¬Surjective f := by + intro f surjf + let S := { i | i ∉ f i } + rcases surjf S with ⟨j, h⟩ + have h₁ : j ∉ f j := by + intro h' + have : j ∉ f j := by rwa [h] at h' + contradiction + have h₂ : j ∈ S := h₁ + have h₃ : j ∉ S := by rwa [h] at h₁ + contradiction + + end +
src/C_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean (new)
- @@ -0,0 +1,92 @@
  + import Mathlib.Data.Set.Lattice + import Mathlib.Data.Set.Function + import Mathlib.Tactic + + open Set + + open Function + + noncomputable section + + open Classical + + variable {α β : Type _} [Nonempty β] + + section + + variable (f : α → β) (g : β → α) + + def sbAux : ℕ → Set α + | 0 => univ \ g '' univ + | n + 1 => g '' (f '' sbAux n) + + def sbSet := + ⋃ n, sbAux f g n + + def sbFun (x : α) : β := + if x ∈ sbSet f g then f x else invFun g x + + theorem sb_right_inv {x : α} (hx : x ∉ sbSet f g) : g (invFun g x) = x := by + have : x ∈ g '' univ := by + contrapose! hx + rw [sbSet, mem_iUnion] + use 0 + rw [sbAux, mem_diff] + exact ⟨mem_univ _, hx⟩ + have : ∃ y, g y = x := by + simp at this + assumption + exact invFun_eq this + + theorem sb_injective (hf : Injective f) (hg : Injective g) : Injective (sbFun f g) := by + set A := sbSet f g with A_def + set h := sbFun f g with h_def + intro x₁ x₂ + intro (hxeq : h x₁ = h x₂) + show x₁ = x₂ + simp only [h_def, sbFun, ← A_def] at hxeq + by_cases xA : x₁ ∈ A ∨ x₂ ∈ A + · wlog x₁A : x₁ ∈ A generalizing x₁ x₂ hxeq xA + · symm + apply this hxeq.symm xA.symm (xA.resolve_left x₁A) + have x₂A : x₂ ∈ A := by + apply not_imp_self.mp + intro (x₂nA : x₂ ∉ A) + rw [if_pos x₁A, if_neg x₂nA] at hxeq + rw [A_def, sbSet, mem_iUnion] at x₁A + have x₂eq : x₂ = g (f x₁) := by + rw [hxeq, sb_right_inv f g x₂nA] + rcases x₁A with ⟨n, hn⟩ + rw [A_def, sbSet, mem_iUnion] + use n + 1 + simp [sbAux] + exact ⟨x₁, hn, x₂eq.symm⟩ + rw [if_pos x₁A, if_pos x₂A] at hxeq + exact hf hxeq + push_neg at xA + rw [if_neg xA.1, if_neg xA.2] at hxeq + rw [← sb_right_inv f g xA.1, hxeq, sb_right_inv f g xA.2] + + theorem sb_surjective (hf : Injective f) (hg : Injective g) : Surjective (sbFun f g) := by + set A := sbSet f g with A_def + set h := sbFun f g with h_def + intro y + by_cases gyA : g y ∈ A + · rw [A_def, sbSet, mem_iUnion] at gyA + rcases gyA with ⟨n, hn⟩ + cases' n with n + · simp [sbAux] at hn + simp [sbAux] at hn + rcases hn with ⟨x, xmem, hx⟩ + use x + have : x ∈ A := by + rw [A_def, sbSet, mem_iUnion] + exact ⟨n, xmem⟩ + simp only [h_def, sbFun, if_pos this] + exact hg hx + use g y + simp only [h_def, sbFun, if_neg gyA] + apply leftInverse_invFun hg + + end +
src/C_Structures/S01_Structures.lean (new)
- @@ -0,0 +1,227 @@
 + import Mathlib.Algebra.BigOperators.Ring + import Mathlib.Data.Real.Basic + + noncomputable section + + @[ext] + structure Point where + x : ℝ + y : ℝ + z : ℝ + + #check Point.ext + + example (a b : Point) (hx : a.x = b.x) (hy : a.y = b.y) (hz : a.z = b.z) : a = b := by + ext + repeat' assumption + + def myPoint1 : Point where + x := 2 + y := -1 + z := 4 + + def myPoint2 : Point := + ⟨2, -1, 4⟩ + + def myPoint3 := + Point.mk 2 (-1) 4 + + structure Point' where build :: + x : ℝ + y : ℝ + z : ℝ + + #check Point'.build 2 (-1) 4 + + namespace Point + + def add (a b : Point) : Point := + ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ + + def add' (a b : Point) : Point where + x := a.x + b.x + y := a.y + b.y + z := a.z + b.z + + #check add myPoint1 myPoint2 + + #check myPoint1.add myPoint2 + + end Point + + #check Point.add myPoint1 myPoint2 + + #check myPoint1.add myPoint2 + + namespace Point + + protected theorem add_comm (a b : Point) : add a b = add b a := by + rw [add, add] + ext <;> dsimp + repeat' apply add_comm + + example (a b : Point) : add a b = add b a := by simp [add, add_comm] + + theorem add_x (a b : Point) : (a.add b).x = a.x + b.x := + rfl + + def addAlt : Point → Point → Point + | Point.mk x₁ y₁ z₁, Point.mk x₂ y₂ z₂ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ + + def addAlt' : Point → Point → Point + | ⟨x₁, y₁, z₁⟩, ⟨x₂, y₂, z₂⟩ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ + + theorem addAlt_x (a b : Point) : (a.addAlt b).x = a.x + b.x := by + cases a + cases b + rfl + + theorem addAlt_comm (a b : Point) : addAlt a b = addAlt b a := by + rcases a with ⟨xa, ya, za⟩ + rcases b with ⟨xb, yb, zb⟩ + rw [addAlt, addAlt] + ext <;> dsimp + apply add_comm + repeat' apply add_comm + + example (a b : Point) : addAlt a b = addAlt b a := by + rcases a with ⟨xa, ya, za⟩ + rcases b with ⟨xb, yb, zb⟩ + simp [addAlt, add_comm] + + example : ∀ a b : Point, addAlt a b = addAlt b a := by + rintro ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ + simp [addAlt, add_comm] + + example : ∀ a b : Point, add a b = add b a := fun ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ => by + simp [add, add_comm] + + protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by + sorry + def smul (r : ℝ) (a : Point) : Point := + sorry + theorem smul_distrib (r : ℝ) (a b : Point) : (smul r a).add (smul r b) = smul r (a.add b) := by + sorry + end Point + + structure StandardTwoSimplex where + x : ℝ + y : ℝ + z : ℝ + x_nonneg : 0 ≤ x + y_nonneg : 0 ≤ y + z_nonneg : 0 ≤ z + sum_eq : x + y + z = 1 + + namespace StandardTwoSimplex + + def swapXy (a : StandardTwoSimplex) : StandardTwoSimplex + where + x := a.y + y := a.x + z := a.z + x_nonneg := a.y_nonneg + y_nonneg := a.x_nonneg + z_nonneg := a.z_nonneg + sum_eq := by rw [add_comm a.y a.x, a.sum_eq] + + noncomputable section + + def midpoint (a b : StandardTwoSimplex) : StandardTwoSimplex + where + x := (a.x + b.x) / 2 + y := (a.y + b.y) / 2 + z := (a.z + b.z) / 2 + x_nonneg := div_nonneg (add_nonneg a.x_nonneg b.x_nonneg) (by norm_num) + y_nonneg := div_nonneg (add_nonneg a.y_nonneg b.y_nonneg) (by norm_num) + z_nonneg := div_nonneg (add_nonneg a.z_nonneg b.z_nonneg) (by norm_num) + sum_eq := by field_simp; linarith [a.sum_eq, b.sum_eq] + + def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) + (a b : StandardTwoSimplex) : StandardTwoSimplex + where + sorry + end + + end StandardTwoSimplex + + open BigOperators + + structure StandardSimplex (n : ℕ) where + V : Fin n → ℝ + NonNeg : ∀ i : Fin n, 0 ≤ V i + sum_eq_one : (∑ i, V i) = 1 + + namespace StandardSimplex + + def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n + where + V i := (a.V i + b.V i) / 2 + NonNeg := by + intro i + apply div_nonneg + · linarith [a.NonNeg i, b.NonNeg i] + norm_num + sum_eq_one := by + simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one] + field_simp + + end StandardSimplex + + structure IsLinear (f : ℝ → ℝ) where + is_additive : ∀ x y, f (x + y) = f x + f y + preserves_mul : ∀ x c, f (c * x) = c * f x + + section + + variable (f : ℝ → ℝ) (linf : IsLinear f) + + #check linf.is_additive + + #check linf.preserves_mul + + end + + def Point'' := + ℝ × ℝ × ℝ + + def IsLinear' (f : ℝ → ℝ) := + (∀ x y, f (x + y) = f x + f y) ∧ ∀ x c, f (c * x) = c * f x + + def PReal := + { y : ℝ // 0 < y } + + section + + variable (x : PReal) + + #check x.val + + #check x.property + + #check x.1 + + #check x.2 + + end + + def StandardTwoSimplex' := + { p : ℝ × ℝ × ℝ // 0 ≤ p.1 ∧ 0 ≤ p.2.1 ∧ 0 ≤ p.2.2 ∧ p.1 + p.2.1 + p.2.2 = 1 } + + def StandardSimplex' (n : ℕ) := + { v : Fin n → ℝ // (∀ i : Fin n, 0 ≤ v i) ∧ (∑ i, v i) = 1 } + + def StdSimplex := Σ n : ℕ, StandardSimplex n + + section + + variable (s : StdSimplex) + + #check s.fst + #check s.snd + + #check s.1 + #check s.2 + + end +
src/C_Structures/S02_Algebraic_Structures.lean (new)
- @@ -0,0 +1,172 @@
  + import Mathlib.Data.Real.Basic + + structure Group₁ (α : Type _) where + mul : α → α → α + one : α + inv : α → α + mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) + mul_one : ∀ x : α, mul x one = x + one_mul : ∀ x : α, mul one x = x + mul_left_inv : ∀ x : α, mul (inv x) x = one + + structure Group₁Cat where + α : Type _ + str : Group₁ α + + section + + variable (α β γ : Type _) + variable (f : α ≃ β) (g : β ≃ γ) + + #check Equiv α β + #check (f.toFun : α → β) + #check (f.invFun : β → α) + #check (f.right_inv : ∀ x : β, f (f.invFun x) = x) + #check (f.left_inv : ∀ x : α, f.invFun (f x) = x) + #check (Equiv.refl α : α ≃ α) + #check (f.symm : β ≃ α) + #check (f.trans g : α ≃ γ) + + example (x : α) : (f.trans g).toFun x = g.toFun (f.toFun x) := + rfl + + example (x : α) : (f.trans g) x = g (f x) := + rfl + + example : (f.trans g : α → γ) = g ∘ f := + rfl + + end + + example (α : Type _) : Equiv.Perm α = (α ≃ α) := + rfl + + def permGroup {α : Type _} : Group₁ (Equiv.Perm α) + where + mul f g := Equiv.trans g f + one := Equiv.refl α + inv := Equiv.symm + mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm + one_mul := Equiv.trans_refl + mul_one := Equiv.refl_trans + mul_left_inv := Equiv.self_trans_symm + + structure AddGroup₁ (α : Type _) where + (add : α → α → α) + -- fill in the rest + @[ext] + structure Point where + x : ℝ + y : ℝ + z : ℝ + + namespace Point + + def add (a b : Point) : Point := + ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ + + def neg (a : point) : point := sorry + + def zero : point := sorry + + def add_group_point : add_group₁ point := sorry + end Point + + section + + variable {α : Type _} (f g : Equiv.Perm α) (n : ℕ) + + #check f * g + #check mul_assoc f g g⁻¹ + + -- group power, defined for any group + #check g ^ n + + example : f * g * g⁻¹ = f := by rw [mul_assoc, mul_right_inv, mul_one] + + example : f * g * g⁻¹ = f := + mul_inv_cancel_right f g + + example {α : Type _} (f g : Equiv.Perm α) : g.symm.trans (g.trans f) = f := + mul_inv_cancel_right f g + + end + + class Group₂ (α : Type _) where + mul : α → α → α + one : α + inv : α → α + mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) + mul_one : ∀ x : α, mul x one = x + one_mul : ∀ x : α, mul one x = x + mul_left_inv : ∀ x : α, mul (inv x) x = one + + instance {α : Type _} : Group₂ (Equiv.Perm α) where + mul f g := Equiv.trans g f + one := Equiv.refl α + inv := Equiv.symm + mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm + one_mul := Equiv.trans_refl + mul_one := Equiv.refl_trans + mul_left_inv := Equiv.self_trans_symm + + #check @Group₂.mul + + def mySquare {α : Type _} [Group₂ α] (x : α) := + Group₂.mul x x + + #check @mySquare + + section + variable {β : Type _} (f g : Equiv.Perm β) + + example : Group₂.mul f g = g.trans f := + rfl + + example : mySquare f = f.trans f := + rfl + + end + + instance : Inhabited Point where default := ⟨0, 0, 0⟩ + + #check (default : Point) + + example : ([] : List Point).headI = default := + rfl + + instance : Add Point where add := Point.add + + section + variable (x y : Point) + + #check x + y + + example : x + y = Point.add x y := + rfl + + end + + instance hasMulGroup₂ {α : Type _} [Group₂ α] : Mul α := + ⟨Group₂.mul⟩ + + instance hasOneGroup₂ {α : Type _} [Group₂ α] : One α := + ⟨Group₂.one⟩ + + instance hasInvGroup₂ {α : Type _} [Group₂ α] : Inv α := + ⟨Group₂.inv⟩ + + section + + variable {α : Type _} (f g : Equiv.Perm α) + + #check f * 1 * g⁻¹ + + def foo : f * 1 * g⁻¹ = g.symm.trans ((Equiv.refl α).trans f) := + rfl + + end + + class AddGroup₂ (α : Type _) where + add : α → α → α + -- fill in the rest
src/C_Structures/S03_Building_the_Gaussian_Integers.lean (new)
- @@ -0,0 +1,271 @@
 + import Mathlib.Data.Int.Basic + import Mathlib.Algebra.EuclideanDomain.Basic + import Mathlib.RingTheory.PrincipalIdealDomain + import Mathlib.Tactic + + @[ext] + structure gaussInt where + re : ℤ + im : ℤ + + namespace gaussInt + + instance : Zero gaussInt := + ⟨⟨0, 0⟩⟩ + + instance : One gaussInt := + ⟨⟨1, 0⟩⟩ + + instance : Add gaussInt := + ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ + + instance : Neg gaussInt := + ⟨fun x => ⟨-x.re, -x.im⟩⟩ + + instance : Mul gaussInt := + ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ + + theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := + rfl + + theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := + rfl + + theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := + rfl + + theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := + rfl + + theorem mul_def (x y : gaussInt) : x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := + rfl + + @[simp] + theorem zero_re : (0 : gaussInt).re = 0 := + rfl + + @[simp] + theorem zero_im : (0 : gaussInt).im = 0 := + rfl + + @[simp] + theorem one_re : (1 : gaussInt).re = 1 := + rfl + + @[simp] + theorem one_im : (1 : gaussInt).im = 0 := + rfl + + @[simp] + theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := + rfl + + @[simp] + theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := + rfl + + @[simp] + theorem neg_re (x : gaussInt) : (-x).re = -x.re := + rfl + + @[simp] + theorem neg_im (x : gaussInt) : (-x).im = -x.im := + rfl + + @[simp] + theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := + rfl + + @[simp] + theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := + rfl + + instance instCommRing : CommRing gaussInt where + zero := 0 + one := 1 + add := (· + ·) + neg x := -x + mul := (· * ·) + add_assoc := by + intros + ext <;> simp <;> ring + zero_add := by + intros + ext <;> simp + add_zero := by + intros + ext <;> simp + add_left_neg := by + intros + ext <;> simp + add_comm := by + intros + ext <;> simp <;> ring + mul_assoc := by + intros + ext <;> simp <;> ring + one_mul := by + intros + ext <;> simp + mul_one := by + intros + ext <;> simp + left_distrib := by + intros + ext <;> simp <;> ring + right_distrib := by + intros + ext <;> simp <;> ring + mul_comm := by + intros + ext <;> simp <;> ring + zero_mul := sorry + mul_zero := sorry + + instance : Nontrivial gaussInt := by + use 0, 1 + rw [Ne, gaussInt.ext_iff] + simp + + end gaussInt + + example (a b : ℤ) : a = b * (a / b) + a % b := + Eq.symm <| Int.ediv_add_emod a b + + example (a b : ℤ) : b ≠ 0 → 0 ≤ a % b := + Int.emod_nonneg a + + example (a b : ℤ) : b ≠ 0 → a % b < abs b := + Int.emod_lt a + + namespace Int + + def div' (a b : ℤ) := + (a + b / 2) / b + + def mod' (a b : ℤ) := + (a + b / 2) % b - b / 2 + + theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by + rw [div', mod'] + linarith [Int.ediv_add_emod (a + b / 2) b] + + theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by + rw [mod', abs_le] + constructor + · linarith [Int.emod_nonneg (a + b / 2) h.ne'] + have := Int.emod_lt_of_pos (a + b / 2) h + have := Int.ediv_add_emod b 2 + have := Int.emod_lt_of_pos b zero_lt_two + revert this; intro this -- FIXME, this should not be needed + linarith + + theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] + + end Int + + theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : + x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by + sorry + namespace gaussInt + + def norm (x : gaussInt) := + x.re ^ 2 + x.im ^ 2 + + @[simp] + theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by + sorry + theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by + sorry + theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by + sorry + theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by + sorry + def conj (x : gaussInt) : gaussInt := + ⟨x.re, -x.im⟩ + + @[simp] + theorem conj_re (x : gaussInt) : (conj x).re = x.re := + rfl + + @[simp] + theorem conj_im (x : gaussInt) : (conj x).im = -x.im := + rfl + + theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] + + instance : Div gaussInt := + ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ + + instance : Mod gaussInt := + ⟨fun x y => x - y * (x / y)⟩ + + theorem div_def (x y : gaussInt) : + x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := + rfl + + theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := + rfl + + theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : (x % y).norm < y.norm := by + have norm_y_pos : 0 < norm y := by rwa [norm_pos] + have : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by + rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] + ext <;> simp <;> ring + have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by + conv => + lhs + rw [← norm_conj y, ← norm_mul, this, norm] + simp + trans 2 * (y.norm / 2) ^ 2 + · rw [two_mul] + apply add_le_add <;> + · rw [sq_le_sq] + apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) + apply le_abs_self + rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] + apply mul_le_mul_of_nonneg_left + · apply Int.ediv_mul_le + norm_num + apply Int.ediv_nonneg (norm_nonneg y) + norm_num + have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos + apply lt_of_le_of_lt this + apply Int.ediv_lt_of_lt_mul + · norm_num + linarith + + theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := + Int.natAbs_of_nonneg (norm_nonneg _) + + theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : (x % y).norm.natAbs < y.norm.natAbs := by + apply Int.ofNat_lt.1 + simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] + apply norm_mod_lt x hy + + theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : + ¬(norm (x * y)).natAbs < (norm x).natAbs := by + apply not_lt_of_ge + rw [norm_mul, Int.natAbs_mul] + apply le_mul_of_one_le_right (Nat.zero_le _) + apply Int.ofNat_le.1 + rw [coe_natAbs_norm] + exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) + + instance : EuclideanDomain gaussInt := + { gaussInt.instCommRing with + quotient := (· / ·) + remainder := (· % ·) + quotient_mul_add_remainder_eq := fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] + quotient_zero := fun x => by + simp [div_def, norm, Int.div'] + rfl + r := Measure (Int.natAbs ∘ norm) + r_wellFounded := (measure (Int.natAbs ∘ norm)).2 + remainder_lt := natAbs_norm_mod_lt + mul_left_not_lt := not_norm_mul_left_lt_norm } + + example (x : gaussInt) : Irreducible x ↔ Prime x := + PrincipalIdealRing.irreducible_iff_prime + + end gaussInt
src/C_Structures/solutions/Solutions_S01_Structures.lean (new)
- @@ -0,0 +1,96 @@
  + import Mathlib.Algebra.BigOperators.Ring + import Mathlib.Data.Real.Basic + + noncomputable section + + @[ext] + structure Point where + x : ℝ + y : ℝ + z : ℝ + + namespace Point + + def add (a b : Point) : Point := + ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ + + protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by + simp [add, add_assoc] + + def smul (r : ℝ) (a : Point) : Point := + ⟨r * a.x, r * a.y, r * a.z⟩ + + theorem smul_distrib (r : ℝ) (a b : Point) : (smul r a).add (smul r b) = smul r (a.add b) := by + simp [add, smul, mul_add] + + end Point + + structure StandardTwoSimplex where + x : ℝ + y : ℝ + z : ℝ + x_nonneg : 0 ≤ x + y_nonneg : 0 ≤ y + z_nonneg : 0 ≤ z + sum_eq : x + y + z = 1 + + namespace StandardTwoSimplex + + noncomputable section + + def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) + (a b : StandardTwoSimplex) : StandardTwoSimplex + where + x := lambda * a.x + (1 - lambda) * b.x + y := lambda * a.y + (1 - lambda) * b.y + z := lambda * a.z + (1 - lambda) * b.z + x_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.x_nonneg) (mul_nonneg (by linarith) b.x_nonneg) + y_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.y_nonneg) (mul_nonneg (by linarith) b.y_nonneg) + z_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.z_nonneg) (mul_nonneg (by linarith) b.z_nonneg) + sum_eq := by + trans (a.x + a.y + a.z) * lambda + (b.x + b.y + b.z) * (1 - lambda) + · ring + simp [a.sum_eq, b.sum_eq] + + end + + end StandardTwoSimplex + + open BigOperators + + structure StandardSimplex (n : ℕ) where + V : Fin n → ℝ + NonNeg : ∀ i : Fin n, 0 ≤ V i + sum_eq_one : (∑ i, V i) = 1 + + namespace StandardSimplex + + def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n + where + V i := (a.V i + b.V i) / 2 + NonNeg := by + intro i + apply div_nonneg + · linarith [a.NonNeg i, b.NonNeg i] + norm_num + sum_eq_one := by + simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one] + field_simp + + end StandardSimplex + + namespace StandardSimplex + + def weightedAverage {n : ℕ} (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) + (a b : StandardSimplex n) : StandardSimplex n + where + V i := lambda * a.V i + (1 - lambda) * b.V i + NonNeg i := + add_nonneg (mul_nonneg lambda_nonneg (a.NonNeg i)) (mul_nonneg (by linarith) (b.NonNeg i)) + sum_eq_one := by + trans (lambda * ∑ i, a.V i) + (1 - lambda) * ∑ i, b.V i + · rw [Finset.sum_add_distrib, Finset.mul_sum, Finset.mul_sum] + simp [a.sum_eq_one, b.sum_eq_one] + + end StandardSimplex +
src/C_Structures/solutions/Solutions_S02_Algebraic_Structures.lean (new)
- @@ -0,0 +1,73 @@
  + import Mathlib.Data.Real.Basic + + structure AddGroup₁ (α : Type _) where + add : α → α → α + zero : α + neg : α → α + add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z) + add_zero : ∀ x : α, add x zero = x + zero_add : ∀ x : α, add x zero = x + add_left_neg : ∀ x : α, add (neg x) x = zero + + @[ext] + structure Point where + x : ℝ + y : ℝ + z : ℝ + + namespace Point + + def add (a b : Point) : Point := + ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ + + def neg (a : Point) : Point := + ⟨-a.x, -a.y, -a.z⟩ + + def zero : Point := + ⟨0, 0, 0⟩ + + def addGroupPoint : AddGroup₁ Point where + add := Point.add + zero := Point.zero + neg := Point.neg + add_assoc := by simp [Point.add, add_assoc] + add_zero := by simp [Point.add, Point.zero] + zero_add := by simp [Point.add, Point.zero] + add_left_neg := by simp [Point.add, Point.neg, Point.zero] + + end Point + + class AddGroup₂ (α : Type _) where + add : α → α → α + zero : α + neg : α → α + add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z) + add_zero : ∀ x : α, add x zero = x + zero_add : ∀ x : α, add x zero = x + add_left_neg : ∀ x : α, add (neg x) x = zero + + instance hasAddAddGroup₂ {α : Type _} [AddGroup₂ α] : Add α := + ⟨AddGroup₂.add⟩ + + instance hasZeroAddGroup₂ {α : Type _} [AddGroup₂ α] : Zero α := + ⟨AddGroup₂.zero⟩ + + instance hasNegAddGroup₂ {α : Type _} [AddGroup₂ α] : Neg α := + ⟨AddGroup₂.neg⟩ + + instance : AddGroup₂ Point where + add := Point.add + zero := Point.zero + neg := Point.neg + add_assoc := by simp [Point.add, add_assoc] + add_zero := by simp [Point.add, Point.zero] + zero_add := by simp [Point.add, Point.zero] + add_left_neg := by simp [Point.add, Point.neg, Point.zero] + + section + variable (x y : Point) + + #check x + -y + 0 + + end +
src/C_Structures/solutions/Solutions_S03_Building_the_Gaussian_Integers.lean (new)
- @@ -0,0 +1,285 @@
 + import Mathlib.Data.Int.Basic + import Mathlib.Algebra.EuclideanDomain.Basic + import Mathlib.RingTheory.PrincipalIdealDomain + import Mathlib.Tactic + + @[ext] + structure gaussInt where + re : ℤ + im : ℤ + + namespace gaussInt + + instance : Zero gaussInt := + ⟨⟨0, 0⟩⟩ + + instance : One gaussInt := + ⟨⟨1, 0⟩⟩ + + instance : Add gaussInt := + ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ + + instance : Neg gaussInt := + ⟨fun x => ⟨-x.re, -x.im⟩⟩ + + instance : Mul gaussInt := + ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ + + theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := + rfl + + theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := + rfl + + theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := + rfl + + theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := + rfl + + theorem mul_def (x y : gaussInt) : x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := + rfl + + @[simp] + theorem zero_re : (0 : gaussInt).re = 0 := + rfl + + @[simp] + theorem zero_im : (0 : gaussInt).im = 0 := + rfl + + @[simp] + theorem one_re : (1 : gaussInt).re = 1 := + rfl + + @[simp] + theorem one_im : (1 : gaussInt).im = 0 := + rfl + + @[simp] + theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := + rfl + + @[simp] + theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := + rfl + + @[simp] + theorem neg_re (x : gaussInt) : (-x).re = -x.re := + rfl + + @[simp] + theorem neg_im (x : gaussInt) : (-x).im = -x.im := + rfl + + @[simp] + theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := + rfl + + @[simp] + theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := + rfl + + instance instCommRing : CommRing gaussInt where + zero := 0 + one := 1 + add := (· + ·) + neg x := -x + mul := (· * ·) + add_assoc := by + intros + ext <;> simp <;> ring + zero_add := by + intros + ext <;> simp + add_zero := by + intros + ext <;> simp + add_left_neg := by + intros + ext <;> simp + add_comm := by + intros + ext <;> simp <;> ring + mul_assoc := by + intros + ext <;> simp <;> ring + one_mul := by + intros + ext <;> simp + mul_one := by + intros + ext <;> simp + left_distrib := by + intros + ext <;> simp <;> ring + right_distrib := by + intros + ext <;> simp <;> ring + mul_comm := by + intros + ext <;> simp <;> ring + zero_mul := sorry + mul_zero := sorry + + instance : Nontrivial gaussInt := by + use 0, 1 + rw [Ne, gaussInt.ext_iff] + simp + + end gaussInt + + namespace Int + + def div' (a b : ℤ) := + (a + b / 2) / b + + def mod' (a b : ℤ) := + (a + b / 2) % b - b / 2 + + theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by + rw [div', mod'] + linarith [Int.ediv_add_emod (a + b / 2) b] + + theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by + rw [mod', abs_le] + constructor + · linarith [Int.emod_nonneg (a + b / 2) h.ne'] + have := Int.emod_lt_of_pos (a + b / 2) h + have := Int.ediv_add_emod b 2 + have := Int.emod_lt_of_pos b zero_lt_two + revert this; intro this -- FIXME, this should not be needed + linarith + + theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] + + end Int + + private theorem aux {α : Type _} [LinearOrderedRing α] {x y : α} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := + haveI h' : x ^ 2 = 0 := by + apply le_antisymm _ (sq_nonneg x) + rw [← h] + apply le_add_of_nonneg_right (sq_nonneg y) + pow_eq_zero h' + + theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : + x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by + constructor + · intro h + constructor + · exact aux h + rw [add_comm] at h + exact aux h + rintro ⟨rfl, rfl⟩ + norm_num + + namespace gaussInt + + def norm (x : gaussInt) := + x.re ^ 2 + x.im ^ 2 + + @[simp] + theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by + apply add_nonneg <;> + apply sq_nonneg + + theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by + rw [norm, sq_add_sq_eq_zero, gaussInt.ext_iff] + rfl + + theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by + rw [lt_iff_le_and_ne, ne_comm, Ne, norm_eq_zero] + simp [norm_nonneg] + + theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by + simp [norm] + ring + + def conj (x : gaussInt) : gaussInt := + ⟨x.re, -x.im⟩ + + @[simp] + theorem conj_re (x : gaussInt) : (conj x).re = x.re := + rfl + + @[simp] + theorem conj_im (x : gaussInt) : (conj x).im = -x.im := + rfl + + theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] + + instance : Div gaussInt := + ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ + + instance : Mod gaussInt := + ⟨fun x y => x - y * (x / y)⟩ + + theorem div_def (x y : gaussInt) : + x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := + rfl + + theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := + rfl + + theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : (x % y).norm < y.norm := by + have norm_y_pos : 0 < norm y := by rwa [norm_pos] + have : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by + rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] + ext <;> simp <;> ring + have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by + conv => + lhs + rw [← norm_conj y, ← norm_mul, this, norm] + simp + trans 2 * (y.norm / 2) ^ 2 + · rw [two_mul] + apply add_le_add <;> + · rw [sq_le_sq] + apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) + apply le_abs_self + rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] + apply mul_le_mul_of_nonneg_left + · apply Int.ediv_mul_le + norm_num + apply Int.ediv_nonneg (norm_nonneg y) + norm_num + have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos + apply lt_of_le_of_lt this + apply Int.ediv_lt_of_lt_mul + · norm_num + linarith + + theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := + Int.natAbs_of_nonneg (norm_nonneg _) + + theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : (x % y).norm.natAbs < y.norm.natAbs := by + apply Int.ofNat_lt.1 + simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] + apply norm_mod_lt x hy + + theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : + ¬(norm (x * y)).natAbs < (norm x).natAbs := by + apply not_lt_of_ge + rw [norm_mul, Int.natAbs_mul] + apply le_mul_of_one_le_right (Nat.zero_le _) + apply Int.ofNat_le.1 + rw [coe_natAbs_norm] + exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) + + instance : EuclideanDomain gaussInt := + { gaussInt.instCommRing with + quotient := (· / ·) + remainder := (· % ·) + quotient_mul_add_remainder_eq := fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] + quotient_zero := fun x => by + simp [div_def, norm, Int.div'] + rfl + r := Measure (Int.natAbs ∘ norm) + r_wellFounded := (measure (Int.natAbs ∘ norm)).2 + remainder_lt := natAbs_norm_mod_lt + mul_left_not_lt := not_norm_mul_left_lt_norm } + + example (x : gaussInt) : Irreducible x ↔ Prime x := + PrincipalIdealRing.irreducible_iff_prime + + end gaussInt
src/C_Topology/S01_Filters.lean (new)
- @@ -0,0 +1,110 @@
 + import Mathlib.Topology.Instances.Real + + open Set Filter Topology + + def principal {α : Type _} (s : Set α) : Filter α + where + sets := { t | s ⊆ t } + univ_sets := sorry + sets_of_superset := sorry + inter_sets := sorry + + example : Filter ℕ := + { sets := { s | ∃ a, ∀ b, a ≤ b → b ∈ s } + univ_sets := sorry + sets_of_superset := sorry + inter_sets := sorry } + + def Tendsto₁ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := + ∀ V ∈ G, f ⁻¹' V ∈ F + + def Tendsto₂ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := + map f F ≤ G + + example {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) : + Tendsto₂ f F G ↔ Tendsto₁ f F G := + Iff.rfl + + #check (@Filter.map_mono : ∀ {α β} {m : α → β}, Monotone (map m)) + + #check + (@Filter.map_map : + ∀ {α β γ} {f : Filter α} {m : α → β} {m' : β → γ}, map m' (map m f) = map (m' ∘ m) f) + + example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} + (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := + sorry + + variable (f : ℝ → ℝ) (x₀ y₀ : ℝ) + + #check comap ((↑) : ℚ → ℝ) (𝓝 x₀) + + #check Tendsto (f ∘ (↑)) (comap ((↑) : ℚ → ℝ) (𝓝 x₀)) (𝓝 y₀) + + section + + variable {α β γ : Type _} (F : Filter α) {m : γ → β} {n : β → α} + + #check (comap_comap : comap m (comap n F) = comap (n ∘ m) F) + + end + + example : 𝓝 (x₀, y₀) = 𝓝 x₀ ×ᶠ 𝓝 y₀ := + nhds_prod_eq + + #check le_inf_iff + + example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : + Tendsto f atTop (𝓝 (x₀, y₀)) ↔ + Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := + sorry + + example (x₀ : ℝ) : HasBasis (𝓝 x₀) (fun ε : ℝ => 0 < ε) fun ε => Ioo (x₀ - ε) (x₀ + ε) := + nhds_basis_Ioo_pos x₀ + + example (u : ℕ → ℝ) (x₀ : ℝ) : + Tendsto u atTop (𝓝 x₀) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, u n ∈ Ioo (x₀ - ε) (x₀ + ε) := by + have : atTop.HasBasis (fun _ : ℕ => True) Ici := atTop_basis + rw [this.tendsto_iff (nhds_basis_Ioo_pos x₀)] + simp + + example (P Q : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in atTop, Q n) : + ∀ᶠ n in atTop, P n ∧ Q n := + hP.and hQ + + example (u v : ℕ → ℝ) (h : ∀ᶠ n in atTop, u n = v n) (x₀ : ℝ) : + Tendsto u atTop (𝓝 x₀) ↔ Tendsto v atTop (𝓝 x₀) := + tendsto_congr' h + + example (u v : ℕ → ℝ) (h : u =ᶠ[atTop] v) (x₀ : ℝ) : + Tendsto u atTop (𝓝 x₀) ↔ Tendsto v atTop (𝓝 x₀) := + tendsto_congr' h + + #check @eventually_of_forall + + #check @Eventually.mono + + #check @Eventually.and + + example (P Q R : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in atTop, Q n) + (hR : ∀ᶠ n in atTop, P n ∧ Q n → R n) : ∀ᶠ n in atTop, R n := by + apply (hP.and (hQ.and hR)).mono + rintro n ⟨h, h', h''⟩ + exact h'' ⟨h, h'⟩ + + example (P Q R : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in atTop, Q n) + (hR : ∀ᶠ n in atTop, P n ∧ Q n → R n) : ∀ᶠ n in atTop, R n := by + filter_upwards [hP, hQ, hR] + intro n h h' h'' + exact h'' ⟨h, h'⟩ + + #check mem_closure_iff_clusterPt + + #check le_principal_iff + + #check neBot_of_le + + example (u : ℕ → ℝ) (M : Set ℝ) (x : ℝ) (hux : Tendsto u atTop (𝓝 x)) + (huM : ∀ᶠ n in atTop, u n ∈ M) : x ∈ closure M := + sorry +
src/C_Topology/S02_Metric_Spaces.lean (new)
- @@ -0,0 +1,200 @@
 + import Mathlib.Topology.Instances.Real + import Mathlib.Analysis.NormedSpace.BanachSteinhaus + + open Set Filter + + open Topology Filter + + variable {X : Type _} [MetricSpace X] (a b c : X) + + #check (dist a b : ℝ) + + #check (dist_nonneg : 0 ≤ dist a b) + + #check (dist_eq_zero : dist a b = 0 ↔ a = b) + + #check (dist_comm a b : dist a b = dist b a) + + #check (dist_triangle a b c : dist a c ≤ dist a b + dist b c) + + -- Note the next three lines are not quoted, their purpose is to make sure those things don't get renamed while we're looking elsewhere. + #check EMetricSpace + + #check PseudoMetricSpace + + #check PseudoEMetricSpace + + example {u : ℕ → X} {a : X} : Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := + Metric.tendsto_atTop + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} : + Continuous f ↔ ∀ x : X, ∀ ε > 0, ∃ δ > 0, ∀ x', dist x' x < δ → dist (f x') (f x) < ε := + Metric.continuous_iff + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := by continuity + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := + continuous_dist.comp ((hf.comp continuous_fst).prod_mk (hf.comp continuous_snd)) + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := by + apply Continuous.dist + exact hf.comp continuous_fst + exact hf.comp continuous_snd + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := + (hf.comp continuous_fst).dist (hf.comp continuous_snd) + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := + hf.fst'.dist hf.snd' + + example {f : ℝ → X} (hf : Continuous f) : Continuous fun x : ℝ => f (x ^ 2 + x) := + sorry + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] (f : X → Y) (a : X) : + ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x}, dist x a < δ → dist (f x) (f a) < ε := + Metric.continuousAt_iff + + variable (r : ℝ) + + example : Metric.ball a r = { b | dist b a < r } := + rfl + + example : Metric.closedBall a r = { b | dist b a ≤ r } := + rfl + + example (hr : 0 < r) : a ∈ Metric.ball a r := + Metric.mem_ball_self hr + + example (hr : 0 ≤ r) : a ∈ Metric.closedBall a r := + Metric.mem_closedBall_self hr + + example (s : Set X) : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s := + Metric.isOpen_iff + + example {s : Set X} : IsClosed s ↔ IsOpen (sᶜ) := + isOpen_compl_iff.symm + + example {s : Set X} (hs : IsClosed s) {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) + (hus : ∀ n, u n ∈ s) : a ∈ s := + hs.mem_of_tendsto hu (eventually_of_forall hus) + + example {s : Set X} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, a ∈ Metric.ball b ε := + Metric.mem_closure_iff + + example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := + sorry + + example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.ball x ε ⊆ s := + Metric.nhds_basis_ball.mem_iff + + example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.closedBall x ε ⊆ s := + Metric.nhds_basis_closedBall.mem_iff + + example : IsCompact (Set.Icc 0 1 : Set ℝ) := + isCompact_Icc + + example {s : Set X} (hs : IsCompact s) {u : ℕ → X} (hu : ∀ n, u n ∈ s) : + ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := + hs.tendsto_subseq hu + + example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : + ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y := + hs.exists_forall_le hs' hfs + + example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : + ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x := + hs.exists_forall_ge hs' hfs + + example {s : Set X} (hs : IsCompact s) : IsClosed s := + hs.isClosed + + example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := + isCompact_univ + + #check IsCompact.isClosed + + example {X : Type _} [MetricSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} : + UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : X}, dist a b < δ → dist (f a) (f b) < ε := + Metric.uniformContinuous_iff + + example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} + (hf : Continuous f) : UniformContinuous f := + sorry + + example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε := + Metric.cauchySeq_iff + + example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, dist (u n) (u N) < ε := + Metric.cauchySeq_iff' + + example [CompleteSpace X] (u : ℕ → X) (hu : CauchySeq u) : ∃ x, Tendsto u atTop (𝓝 x) := + cauchySeq_tendsto_of_complete hu + + open BigOperators + + open Finset + + theorem cauchySeq_of_le_geometric_two' {u : ℕ → X} + (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by + rw [Metric.cauchySeq_iff'] + intro ε ε_pos + obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by sorry + use N + intro n hn + obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn + calc + dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := sorry + _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := sorry + _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := sorry + _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2) ^ i := sorry + _ ≤ 1 / 2 ^ N * 2 := sorry + _ < ε := sorry + + + open Metric + + example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) : + Dense (⋂ n, f n) := by + let B : ℕ → ℝ := fun n => (1 / 2) ^ n + have Bpos : ∀ n, 0 < B n + sorry + /- Translate the density assumption into two functions `center` and `radius` associating + to any n, x, δ, δpos a center and a positive radius such that + `closedBall center radius` is included both in `f n` and in `closedBall x δ`. + We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ + have : + ∀ (n : ℕ) (x : X), + ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := + by sorry + choose! center radius Hpos HB Hball using this + intro x + rw [mem_closure_iff_nhds_basis nhds_basis_closedBall] + intro ε εpos + /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all + `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball + `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that + `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a + limit which belongs to all the `f n`. -/ + let F : ℕ → X × ℝ := fun n => + Nat.recOn n (Prod.mk x (min ε (B 0))) fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2) + let c : ℕ → X := fun n => (F n).1 + let r : ℕ → ℝ := fun n => (F n).2 + have rpos : ∀ n, 0 < r n := by sorry + have rB : ∀ n, r n ≤ B n := by sorry + have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := by sorry + have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by sorry + have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist + -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. + rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩ + -- this point `y` will be the desired point. We will check that it belongs to all + -- `f n` and to `ball x ε`. + use y + have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry + have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry + sorry +
src/C_Topology/S03_Topological_Spaces.lean (new)
- @@ -0,0 +1,151 @@
 + import Mathlib.Topology.Instances.Real + import Mathlib.Analysis.NormedSpace.BanachSteinhaus + + open Set Filter + + open Topology Filter + + section + + variable {X : Type _} [TopologicalSpace X] + + example : IsOpen (univ : Set X) := + isOpen_univ + + example : IsOpen (∅ : Set X) := + isOpen_empty + + example {ι : Type _} {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋃ i, s i) := + isOpen_iUnion hs + + example {ι : Type _} [Fintype ι] {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋂ i, s i) := + isOpen_iInter hs + + variable {Y : Type _} [TopologicalSpace Y] + + example {f : X → Y} : Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) := + continuous_def + + example {f : X → Y} {x : X} : ContinuousAt f x ↔ map f (𝓝 x) ≤ 𝓝 (f x) := + Iff.rfl + + example {f : X → Y} {x : X} : ContinuousAt f x ↔ ∀ U ∈ 𝓝 (f x), ∀ᶠ x in 𝓝 x, f x ∈ U := + Iff.rfl + + example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := + mem_nhds_iff + + example (x : X) : pure x ≤ 𝓝 x := + pure_le_nhds x + + example (x : X) (P : X → Prop) (h : ∀ᶠ y in 𝓝 x, P y) : P x := + pure_le_nhds x h + + example {P : X → Prop} {x : X} (h : ∀ᶠ y in 𝓝 x, P y) : ∀ᶠ y in 𝓝 x, ∀ᶠ z in 𝓝 y, P z := + eventually_eventually_nhds.mpr h + + #check TopologicalSpace.mkOfNhds + + #check TopologicalSpace.nhds_mkOfNhds + + example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) + (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → ∀ᶠ y in n a, ∀ᶠ x in n y, p x) : + ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := + sorry + + end + + -- BOTH. + variable {X Y : Type _} + + example (f : X → Y) : TopologicalSpace X → TopologicalSpace Y := + TopologicalSpace.coinduced f + + example (f : X → Y) : TopologicalSpace Y → TopologicalSpace X := + TopologicalSpace.induced f + + example (f : X → Y) (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) : + TopologicalSpace.coinduced f T_X ≤ T_Y ↔ T_X ≤ TopologicalSpace.induced f T_Y := + coinduced_le_iff_le_induced + + #check coinduced_compose + + #check induced_compose + + example {T T' : TopologicalSpace X} : T ≤ T' ↔ ∀ s, T'.IsOpen s → T.IsOpen s := + Iff.rfl + + example (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) (f : X → Y) : + Continuous f ↔ TopologicalSpace.coinduced f T_X ≤ T_Y := + continuous_iff_coinduced_le + + example {Z : Type _} (f : X → Y) (T_X : TopologicalSpace X) (T_Z : TopologicalSpace Z) (g : Y → Z) : + @Continuous Y Z (TopologicalSpace.coinduced f T_X) T_Z g ↔ @Continuous X Z T_X T_Z (g ∘ f) := by + rw [continuous_iff_coinduced_le, coinduced_compose, continuous_iff_coinduced_le] + + example (ι : Type _) (X : ι → Type _) (T_X : ∀ i, TopologicalSpace <| X i) : + (Pi.topologicalSpace : TopologicalSpace (∀ i, X i)) = + ⨅ i, TopologicalSpace.induced (fun x => x i) (T_X i) := + rfl + + example [TopologicalSpace X] [T2Space X] {u : ℕ → X} {a b : X} (ha : Tendsto u atTop (𝓝 a)) + (hb : Tendsto u atTop (𝓝 b)) : a = b := + tendsto_nhds_unique ha hb + + example [TopologicalSpace X] [RegularSpace X] (a : X) : + (𝓝 a).HasBasis (fun s : Set X => s ∈ 𝓝 a ∧ IsClosed s) id := + closed_nhds_basis a + + example [TopologicalSpace X] {x : X} : (𝓝 x).HasBasis (fun t : Set X => t ∈ 𝓝 x ∧ IsOpen t) id := + nhds_basis_opens' x + + theorem aux {X Y A : Type _} [TopologicalSpace X] {c : A → X} {f : A → Y} {x : X} {F : Filter Y} + (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : + ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := + sorry + + example [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace Y] {A : Set X} + (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f) + (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) <| 𝓝 x) <| 𝓝 c) : + ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := + sorry + + #check @HasBasis.tendsto_right_iff + + example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] {s : Set X} {a : X} : + a ∈ closure s ↔ ∃ u : ℕ → X, (∀ n, u n ∈ s) ∧ Tendsto u atTop (𝓝 a) := + mem_closure_iff_seq_limit + + variable [TopologicalSpace X] + + example {F : Filter X} {x : X} : ClusterPt x F ↔ NeBot (𝓝 x ⊓ F) := + Iff.rfl + + example {s : Set X} : IsCompact s ↔ ∀ (F : Filter X) [NeBot F], F ≤ 𝓟 s → ∃ a ∈ s, ClusterPt a F := + Iff.rfl + + example [TopologicalSpace.FirstCountableTopology X] {s : Set X} {u : ℕ → X} (hs : IsCompact s) + (hu : ∀ n, u n ∈ s) : ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := + hs.tendsto_subseq hu + + variable [TopologicalSpace Y] + + example {x : X} {F : Filter X} {G : Filter Y} (H : ClusterPt x F) {f : X → Y} + (hfx : ContinuousAt f x) (hf : Tendsto f F G) : ClusterPt (f x) G := + ClusterPt.map H hfx hf + + example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) : + IsCompact (f '' s) := by + intro F F_ne F_le + have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by sorry + have Hne : (𝓟 s ⊓ comap f F).NeBot := by sorry + have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left + sorry + + example {ι : Type _} {s : Set X} (hs : IsCompact s) (U : ι → Set X) (hUo : ∀ i, IsOpen (U i)) + (hsU : s ⊆ ⋃ i, U i) : ∃ t : Finset ι, s ⊆ ⋃ i ∈ t, U i := + hs.elim_finite_subcover U hUo hsU + + example [CompactSpace X] : IsCompact (univ : Set X) := + isCompact_univ +
src/C_Topology/solutions/Solutions_S01_Filters.lean (new)
- @@ -0,0 +1,71 @@
 + import Mathlib.Topology.Instances.Real + + open Set Filter Topology + + -- In the next example we could use `tauto` in each proof instead of knowing the lemmas + example {α : Type _} (s : Set α) : Filter α := + { sets := { t | s ⊆ t } + univ_sets := subset_univ s + sets_of_superset := fun hU hUV => Subset.trans hU hUV + inter_sets := fun hU hV => subset_inter hU hV } + + example : Filter ℕ := + { sets := { s | ∃ a, ∀ b, a ≤ b → b ∈ s } + univ_sets := by + use 42 + simp + sets_of_superset := by + rintro U V ⟨N, hN⟩ hUV + use N + tauto + inter_sets := by + rintro U V ⟨N, hN⟩ ⟨N', hN'⟩ + use max N N' + intro b hb + rw [max_le_iff] at hb + constructor <;> tauto } + + def Tendsto₁ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := + ∀ V ∈ G, f ⁻¹' V ∈ F + + example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} + (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := + calc + map (g ∘ f) F = map g (map f F) := by rw [map_map] + _ ≤ map g G := (map_mono hf) + _ ≤ H := hg + + + example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} + (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := by + intro V hV + rw [preimage_comp] + apply hf + apply hg + exact hV + + example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : + Tendsto f atTop (𝓝 (x₀, y₀)) ↔ + Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := + calc + Tendsto f atTop (𝓝 (x₀, y₀)) ↔ map f atTop ≤ 𝓝 (x₀, y₀) := Iff.rfl + _ ↔ map f atTop ≤ 𝓝 x₀ ×ᶠ 𝓝 y₀ := by rw [nhds_prod_eq] + _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ⊓ comap Prod.snd (𝓝 y₀) := Iff.rfl + _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ∧ map f atTop ≤ comap Prod.snd (𝓝 y₀) := le_inf_iff + _ ↔ map Prod.fst (map f atTop) ≤ 𝓝 x₀ ∧ map Prod.snd (map f atTop) ≤ 𝓝 y₀ := by + rw [← map_le_iff_le_comap, ← map_le_iff_le_comap] + _ ↔ map (Prod.fst ∘ f) atTop ≤ 𝓝 x₀ ∧ map (Prod.snd ∘ f) atTop ≤ 𝓝 y₀ := by + rw [map_map, map_map] + + + -- an alternative solution + example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : + Tendsto f atTop (𝓝 (x₀, y₀)) ↔ + Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := by + rw [nhds_prod_eq] + unfold Tendsto Filter.prod + rw [le_inf_iff, ← map_le_iff_le_comap, map_map, ← map_le_iff_le_comap, map_map] + + example (u : ℕ → ℝ) (M : Set ℝ) (x : ℝ) (hux : Tendsto u atTop (𝓝 x)) + (huM : ∀ᶠ n in atTop, u n ∈ M) : x ∈ closure M := + mem_closure_iff_clusterPt.mpr (neBot_of_le <| le_inf hux <| le_principal_iff.mpr huM)
src/C_Topology/solutions/Solutions_S02_Metric_Spaces.lean (new)
- @@ -0,0 +1,365 @@
 + import Mathlib.Topology.Instances.Real + import Mathlib.Analysis.NormedSpace.BanachSteinhaus + + open Set Filter + + open Topology Filter + + variable {X : Type _} [MetricSpace X] (a b c : X) + + #check (dist a b : ℝ) + + #check (dist_nonneg : 0 ≤ dist a b) + + #check (dist_eq_zero : dist a b = 0 ↔ a = b) + + #check (dist_comm a b : dist a b = dist b a) + + #check (dist_triangle a b c : dist a c ≤ dist a b + dist b c) + + -- Note the next three lines are not quoted, their purpose is to make sure those things don't get renamed while we're looking elsewhere. + #check EMetricSpace + + #check PseudoMetricSpace + + #check PseudoEMetricSpace + + example {u : ℕ → X} {a : X} : Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := + Metric.tendsto_atTop + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} : + Continuous f ↔ ∀ x : X, ∀ ε > 0, ∃ δ > 0, ∀ x', dist x' x < δ → dist (f x') (f x) < ε := + Metric.continuous_iff + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := by continuity + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := + continuous_dist.comp ((hf.comp continuous_fst).prod_mk (hf.comp continuous_snd)) + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := by + apply Continuous.dist + exact hf.comp continuous_fst + exact hf.comp continuous_snd + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := + (hf.comp continuous_fst).dist (hf.comp continuous_snd) + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : + Continuous fun p : X × X => dist (f p.1) (f p.2) := + hf.fst'.dist hf.snd' + + example {f : ℝ → X} (hf : Continuous f) : Continuous fun x : ℝ => f (x ^ 2 + x) := + sorry + + example {f : ℝ → X} (hf : Continuous f) : Continuous fun x : ℝ => f (x ^ 2 + x) := + hf.comp <| (continuous_pow 2).add continuous_id + + example {X Y : Type _} [MetricSpace X] [MetricSpace Y] (f : X → Y) (a : X) : + ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x}, dist x a < δ → dist (f x) (f a) < ε := + Metric.continuousAt_iff + + variable (r : ℝ) + + example : Metric.ball a r = { b | dist b a < r } := + rfl + + example : Metric.closedBall a r = { b | dist b a ≤ r } := + rfl + + example (hr : 0 < r) : a ∈ Metric.ball a r := + Metric.mem_ball_self hr + + example (hr : 0 ≤ r) : a ∈ Metric.closedBall a r := + Metric.mem_closedBall_self hr + + example (s : Set X) : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s := + Metric.isOpen_iff + + example {s : Set X} : IsClosed s ↔ IsOpen (sᶜ) := + isOpen_compl_iff.symm + + example {s : Set X} (hs : IsClosed s) {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) + (hus : ∀ n, u n ∈ s) : a ∈ s := + hs.mem_of_tendsto hu (eventually_of_forall hus) + + example {s : Set X} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, a ∈ Metric.ball b ε := + Metric.mem_closure_iff + + example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := + sorry + + example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := by + rw [Metric.tendsto_atTop] at hu + rw [Metric.mem_closure_iff] + intro ε ε_pos + rcases hu ε ε_pos with ⟨N, hN⟩ + refine' ⟨u N, hs _, _⟩ + rw [dist_comm] + exact hN N le_rfl + + example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.ball x ε ⊆ s := + Metric.nhds_basis_ball.mem_iff + + example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.closedBall x ε ⊆ s := + Metric.nhds_basis_closedBall.mem_iff + + example : IsCompact (Set.Icc 0 1 : Set ℝ) := + isCompact_Icc + + example {s : Set X} (hs : IsCompact s) {u : ℕ → X} (hu : ∀ n, u n ∈ s) : + ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := + hs.tendsto_subseq hu + + example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : + ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y := + hs.exists_forall_le hs' hfs + + example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : + ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x := + hs.exists_forall_ge hs' hfs + + example {s : Set X} (hs : IsCompact s) : IsClosed s := + hs.isClosed + + example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := + isCompact_univ + + #check IsCompact.isClosed + + example {X : Type _} [MetricSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} : + UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : X}, dist a b < δ → dist (f a) (f b) < ε := + Metric.uniformContinuous_iff + + example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} + (hf : Continuous f) : UniformContinuous f := + sorry + + example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} + (hf : Continuous f) : UniformContinuous f := by + rw [Metric.uniformContinuous_iff] + intro ε ε_pos + let φ : X × X → ℝ := fun p => dist (f p.1) (f p.2) + have φ_cont : Continuous φ := hf.fst'.dist hf.snd' + let K := { p : X × X | ε ≤ φ p } + have K_closed : IsClosed K := isClosed_le continuous_const φ_cont + have K_cpct : IsCompact K := K_closed.isCompact + cases' eq_empty_or_nonempty K with hK hK + · use 1, by norm_num + intro x y _ + have : (x, y) ∉ K := by simp [hK] + simpa using this + · rcases K_cpct.exists_forall_le hK continuous_dist.continuousOn with ⟨⟨x₀, x₁⟩, xx_in, H⟩ + use dist x₀ x₁ + constructor + · change _ < _ + rw [dist_pos] + intro h + have : ε ≤ 0 := by simpa [*] using xx_in + linarith + · intro x x' + contrapose! + intro hxx' + exact H (x, x') hxx' + + example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε := + Metric.cauchySeq_iff + + example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, dist (u n) (u N) < ε := + Metric.cauchySeq_iff' + + example [CompleteSpace X] (u : ℕ → X) (hu : CauchySeq u) : ∃ x, Tendsto u atTop (𝓝 x) := + cauchySeq_tendsto_of_complete hu + + open BigOperators + + open Finset + + theorem cauchySeq_of_le_geometric_two' {u : ℕ → X} + (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by + rw [Metric.cauchySeq_iff'] + intro ε ε_pos + obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by sorry + use N + intro n hn + obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn + calc + dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := sorry + _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := sorry + _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := sorry + _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2) ^ i := sorry + _ ≤ 1 / 2 ^ N * 2 := sorry + _ < ε := sorry + + + example {u : ℕ → X} (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by + rw [Metric.cauchySeq_iff'] + intro ε ε_pos + obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by + have : Tendsto (fun N : ℕ => (1 / 2 ^ N * 2 : ℝ)) atTop (𝓝 0) := by + rw [← MulZeroClass.zero_mul (2 : ℝ)] + apply Tendsto.mul + simp_rw [← one_div_pow (2 : ℝ)] + apply tendsto_pow_atTop_nhds_0_of_lt_1 <;> linarith + exact tendsto_const_nhds + rcases(atTop_basis.tendsto_iff (nhds_basis_Ioo_pos (0 : ℝ))).mp this ε ε_pos with ⟨N, _, hN⟩ + exact ⟨N, by simpa using (hN N left_mem_Ici).2⟩ + use N + intro n hn + obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn + calc + dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := by rw [dist_comm, add_zero] + _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := + (dist_le_range_sum_dist (fun i => u (N + i)) k) + _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := (sum_le_sum fun i _ => hu <| N + i) + _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2 : ℝ) ^ i := by simp_rw [← one_div_pow, pow_add, ← mul_sum] + _ ≤ 1 / 2 ^ N * 2 := + (mul_le_mul_of_nonneg_left (sum_geometric_two_le _) + (one_div_nonneg.mpr (pow_nonneg (zero_le_two : (0 : ℝ) ≤ 2) _))) + _ < ε := hN + + + open Metric + + example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) : + Dense (⋂ n, f n) := by + let B : ℕ → ℝ := fun n => (1 / 2) ^ n + have Bpos : ∀ n, 0 < B n + sorry + /- Translate the density assumption into two functions `center` and `radius` associating + to any n, x, δ, δpos a center and a positive radius such that + `closedBall center radius` is included both in `f n` and in `closedBall x δ`. + We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ + have : + ∀ (n : ℕ) (x : X), + ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := + by sorry + choose! center radius Hpos HB Hball using this + intro x + rw [mem_closure_iff_nhds_basis nhds_basis_closedBall] + intro ε εpos + /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all + `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball + `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that + `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a + limit which belongs to all the `f n`. -/ + let F : ℕ → X × ℝ := fun n => + Nat.recOn n (Prod.mk x (min ε (B 0))) fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2) + let c : ℕ → X := fun n => (F n).1 + let r : ℕ → ℝ := fun n => (F n).2 + have rpos : ∀ n, 0 < r n := by sorry + have rB : ∀ n, r n ≤ B n := by sorry + have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := by sorry + have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by sorry + have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist + -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. + rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩ + -- this point `y` will be the desired point. We will check that it belongs to all + -- `f n` and to `ball x ε`. + use y + have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry + have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry + sorry + + example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) : + Dense (⋂ n, f n) := by + let B : ℕ → ℝ := fun n => (1 / 2) ^ n + have Bpos : ∀ n, 0 pow_pos sorry n + /- Translate the density assumption into two functions `center` and `radius` associating + to any n, x, δ, δpos a center and a positive radius such that + `closedBall center radius` is included both in `f n` and in `closedBall x δ`. + We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ + have : + ∀ (n : ℕ) (x : X), + ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := by + intro n x δ δpos + have : x ∈ closure (f n) := hd n x + rcases Metric.mem_closure_iff.1 this (δ / 2) (half_pos δpos) with ⟨y, ys, xy⟩ + rw [dist_comm] at xy + obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closedBall y r ⊆ f n := + nhds_basis_closedBall.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys) + refine' ⟨y, min (min (δ / 2) r) (B (n + 1)), _, _, fun z hz => ⟨_, _⟩⟩ + show 0 < min (min (δ / 2) r) (B (n + 1)) + exact lt_min (lt_min (half_pos δpos) rpos) (Bpos (n + 1)) + show min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1) + exact min_le_right _ _ + show z ∈ closedBall x δ + exact + calc + dist z x ≤ dist z y + dist y x := dist_triangle _ _ _ + _ ≤ min (min (δ / 2) r) (B (n + 1)) + δ / 2 := (add_le_add hz xy.le) + _ ≤ δ / 2 + δ / 2 := (add_le_add_right ((min_le_left _ _).trans (min_le_left _ _)) _) + _ = δ := add_halves δ + + show z ∈ f n + exact + hr + (calc + dist z y ≤ min (min (δ / 2) r) (B (n + 1)) := hz + _ ≤ r := (min_le_left _ _).trans (min_le_right _ _) + ) + choose! center radius Hpos HB Hball using this + refine' fun x => (mem_closure_iff_nhds_basis nhds_basis_closedBall).2 fun ε εpos => _ + /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all + `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball + `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that + `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a + limit which belongs to all the `f n`. -/ + let F : ℕ → X × ℝ := fun n => + Nat.recOn n (Prod.mk x (min ε (B 0))) fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2) + let c : ℕ → X := fun n => (F n).1 + let r : ℕ → ℝ := fun n => (F n).2 + have rpos : ∀ n, 0 < r n := by + intro n + induction' n with n hn + exact lt_min εpos (Bpos 0) + exact Hpos n (c n) (r n) hn + have rB : ∀ n, r n ≤ B n := by + intro n + induction' n with n hn + exact min_le_right _ _ + exact HB n (c n) (r n) (rpos n) + have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := fun n => + Hball n (c n) (r n) (rpos n) + have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by + intro n + rw [dist_comm] + have A : c (n + 1) ∈ closedBall (c (n + 1)) (r (n + 1)) := + mem_closedBall_self (rpos <| n + 1).le + have I := + calc + closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) := + (incl n).trans (inter_subset_left _ _) + _ ⊆ closedBall (c n) (B n) := closedBall_subset_closedBall (rB n) + + exact I A + have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist + -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. + rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩ + -- this point `y` will be the desired point. We will check that it belongs to all + -- `f n` and to `ball x ε`. + use y + have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by + intro n + refine' Nat.le_induction _ fun m hnm h => _ + · exact Subset.rfl + · exact (incl m).trans ((Set.inter_subset_left _ _).trans h) + have yball : ∀ n, y ∈ closedBall (c n) (r n) := by + intro n + refine' isClosed_ball.mem_of_tendsto ylim _ + refine' (Filter.eventually_ge_atTop n).mono fun m hm => _ + exact I n m hm (mem_closedBall_self (rpos _).le) + constructor + · suffices ∀ n, y ∈ f n by rwa [Set.mem_iInter] + intro n + have : closedBall (c (n + 1)) (r (n + 1)) ⊆ f n := + Subset.trans (incl n) (inter_subset_right _ _) + exact this (yball (n + 1)) + calc + dist y x ≤ r 0 := yball 0 + _ ≤ ε := min_le_left _ _ + +
src/C_Topology/solutions/Solutions_S03_Topological_Spaces.lean (new)
- @@ -0,0 +1,202 @@
 + import Mathlib.Topology.Instances.Real + import Mathlib.Analysis.NormedSpace.BanachSteinhaus + + open Set Filter + + open Topology Filter + + section + + variable {X : Type _} [TopologicalSpace X] + + example : IsOpen (univ : Set X) := + isOpen_univ + + example : IsOpen (∅ : Set X) := + isOpen_empty + + example {ι : Type _} {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋃ i, s i) := + isOpen_iUnion hs + + example {ι : Type _} [Fintype ι] {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋂ i, s i) := + isOpen_iInter hs + + variable {Y : Type _} [TopologicalSpace Y] + + example {f : X → Y} : Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) := + continuous_def + + example {f : X → Y} {x : X} : ContinuousAt f x ↔ map f (𝓝 x) ≤ 𝓝 (f x) := + Iff.rfl + + example {f : X → Y} {x : X} : ContinuousAt f x ↔ ∀ U ∈ 𝓝 (f x), ∀ᶠ x in 𝓝 x, f x ∈ U := + Iff.rfl + + example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := + mem_nhds_iff + + example (x : X) : pure x ≤ 𝓝 x := + pure_le_nhds x + + example (x : X) (P : X → Prop) (h : ∀ᶠ y in 𝓝 x, P y) : P x := + pure_le_nhds x h + + example {P : X → Prop} {x : X} (h : ∀ᶠ y in 𝓝 x, P y) : ∀ᶠ y in 𝓝 x, ∀ᶠ z in 𝓝 y, P z := + eventually_eventually_nhds.mpr h + + #check TopologicalSpace.mkOfNhds + + #check TopologicalSpace.nhds_mkOfNhds + + example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) + (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → ∀ᶠ y in n a, ∀ᶠ x in n y, p x) : + ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := + sorry + + example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) + (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → ∀ᶠ y in n a, ∀ᶠ x in n y, p x) : + ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := by + intro a s s_in + refine' ⟨{ y | s ∈ n y }, H a (fun x => x ∈ s) s_in, _, by tauto⟩ + rintro y (hy : s ∈ n y) + exact H₀ y hy + + end + + -- BOTH. + variable {X Y : Type _} + + example (f : X → Y) : TopologicalSpace X → TopologicalSpace Y := + TopologicalSpace.coinduced f + + example (f : X → Y) : TopologicalSpace Y → TopologicalSpace X := + TopologicalSpace.induced f + + example (f : X → Y) (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) : + TopologicalSpace.coinduced f T_X ≤ T_Y ↔ T_X ≤ TopologicalSpace.induced f T_Y := + coinduced_le_iff_le_induced + + #check coinduced_compose + + #check induced_compose + + example {T T' : TopologicalSpace X} : T ≤ T' ↔ ∀ s, T'.IsOpen s → T.IsOpen s := + Iff.rfl + + example (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) (f : X → Y) : + Continuous f ↔ TopologicalSpace.coinduced f T_X ≤ T_Y := + continuous_iff_coinduced_le + + example {Z : Type _} (f : X → Y) (T_X : TopologicalSpace X) (T_Z : TopologicalSpace Z) (g : Y → Z) : + @Continuous Y Z (TopologicalSpace.coinduced f T_X) T_Z g ↔ @Continuous X Z T_X T_Z (g ∘ f) := by + rw [continuous_iff_coinduced_le, coinduced_compose, continuous_iff_coinduced_le] + + example (ι : Type _) (X : ι → Type _) (T_X : ∀ i, TopologicalSpace <| X i) : + (Pi.topologicalSpace : TopologicalSpace (∀ i, X i)) = + ⨅ i, TopologicalSpace.induced (fun x => x i) (T_X i) := + rfl + + example [TopologicalSpace X] [T2Space X] {u : ℕ → X} {a b : X} (ha : Tendsto u atTop (𝓝 a)) + (hb : Tendsto u atTop (𝓝 b)) : a = b := + tendsto_nhds_unique ha hb + + example [TopologicalSpace X] [RegularSpace X] (a : X) : + (𝓝 a).HasBasis (fun s : Set X => s ∈ 𝓝 a ∧ IsClosed s) id := + closed_nhds_basis a + + example [TopologicalSpace X] {x : X} : (𝓝 x).HasBasis (fun t : Set X => t ∈ 𝓝 x ∧ IsOpen t) id := + nhds_basis_opens' x + + theorem aux {X Y A : Type _} [TopologicalSpace X] {c : A → X} {f : A → Y} {x : X} {F : Filter Y} + (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : + ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := + sorry + + example {X Y A : Type _} [TopologicalSpace X] {c : A → X} {f : A → Y} {x : X} {F : Filter Y} + (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : + ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := by + simpa [and_assoc] using ((nhds_basis_opens' x).comap c).tendsto_left_iff.mp h V' V'_in + + example [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace Y] {A : Set X} + (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f) + (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) <| 𝓝 x) <| 𝓝 c) : + ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := + sorry + + #check @HasBasis.tendsto_right_iff + + example [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {A : Set X} (hA : ∀ x, x ∈ closure A) + {f : A → Y} (f_cont : Continuous f) + (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) <| 𝓝 x) <| 𝓝 c) : + ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := by + choose φ hφ using hf + use φ + constructor + · rw [continuous_iff_continuousAt] + intro x + suffices ∀ V' ∈ 𝓝 (φ x), IsClosed V' → φ ⁻¹' V' ∈ 𝓝 x by + simpa [ContinuousAt, (closed_nhds_basis (φ x)).tendsto_right_iff] + intro V' V'_in V'_closed + obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, IsOpen V ∧ (↑) ⁻¹' V ⊆ f ⁻¹' V' := aux (hφ x) V'_in + suffices : ∀ y ∈ V, φ y ∈ V' + exact mem_of_superset V_in this + intro y y_in + have hVx : V ∈ 𝓝 y := V_op.mem_nhds y_in + haveI : (comap ((↑) : A → X) (𝓝 y)).NeBot := by simpa [mem_closure_iff_comap_neBot] using hA y + apply V'_closed.mem_of_tendsto (hφ y) + exact mem_of_superset (preimage_mem_comap hVx) hV + · intro a + have lim : Tendsto f (𝓝 a) (𝓝 <| φ a) := by simpa [nhds_induced] using hφ a + exact tendsto_nhds_unique lim f_cont.continuousAt + + example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] {s : Set X} {a : X} : + a ∈ closure s ↔ ∃ u : ℕ → X, (∀ n, u n ∈ s) ∧ Tendsto u atTop (𝓝 a) := + mem_closure_iff_seq_limit + + variable [TopologicalSpace X] + + example {F : Filter X} {x : X} : ClusterPt x F ↔ NeBot (𝓝 x ⊓ F) := + Iff.rfl + + example {s : Set X} : IsCompact s ↔ ∀ (F : Filter X) [NeBot F], F ≤ 𝓟 s → ∃ a ∈ s, ClusterPt a F := + Iff.rfl + + example [TopologicalSpace.FirstCountableTopology X] {s : Set X} {u : ℕ → X} (hs : IsCompact s) + (hu : ∀ n, u n ∈ s) : ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := + hs.tendsto_subseq hu + + variable [TopologicalSpace Y] + + example {x : X} {F : Filter X} {G : Filter Y} (H : ClusterPt x F) {f : X → Y} + (hfx : ContinuousAt f x) (hf : Tendsto f F G) : ClusterPt (f x) G := + ClusterPt.map H hfx hf + + example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) : + IsCompact (f '' s) := by + intro F F_ne F_le + have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by sorry + have Hne : (𝓟 s ⊓ comap f F).NeBot := by sorry + have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left + sorry + + example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) : + IsCompact (f '' s) := by + intro F F_ne F_le + have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by rw [Filter.push_pull, map_principal] + have Hne : (𝓟 s ⊓ comap f F).NeBot := by + apply NeBot.of_map + rwa [map_eq, inf_of_le_right F_le] + have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left + rcases hs Hle with ⟨x, x_in, hx⟩ + refine' ⟨f x, mem_image_of_mem f x_in, _⟩ + apply hx.map hf.continuousAt + rw [Tendsto, map_eq] + exact inf_le_right + + example {ι : Type _} {s : Set X} (hs : IsCompact s) (U : ι → Set X) (hUo : ∀ i, IsOpen (U i)) + (hsU : s ⊆ ⋃ i, U i) : ∃ t : Finset ι, s ⊆ ⋃ i ∈ t, U i := + hs.elim_finite_subcover U hUo hsU + + example [CompactSpace X] : IsCompact (univ : Set X) := + isCompact_univ +