Update Wed May 22 06:14:40 PM EDT 2024

Changes

11 changed files (+17/-21)

• MIL/C01_Introduction/S02_Overview.lean
  • @@ -1,4 +1,3 @@
    - import Mathlib.Data.Nat.Parity import MIL.Common open Nat
• MIL/C01_Introduction/solutions/Solutions_S02_Overview.lean
  • @@ -1,4 +1,3 @@
    - import Mathlib.Data.Nat.Parity import MIL.Common open Nat
• MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean
  • @@ -12,7 +12,7 @@ variable (x y z : α)
    #check x < y - #check (lt_irrefl x : ¬x < x) + #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)
• MIL/C04_Sets_and_Functions/S01_Sets.lean
  • @@ -1,6 +1,5 @@
    import Mathlib.Data.Set.Lattice import Mathlib.Data.Nat.Prime - import Mathlib.Data.Nat.Parity import MIL.Common section
• MIL/C04_Sets_and_Functions/solutions/Solutions_S01_Sets.lean
  • @@ -1,6 +1,5 @@
    import Mathlib.Data.Set.Lattice import Mathlib.Data.Nat.Prime - import Mathlib.Data.Nat.Parity import MIL.Common section
• html/C02_Basics.html
  • @@ -1204,7 +1204,7 @@ letters like <code class="docutils literal notranslate"><span class="pre">&#945;
    (entered as <code class="docutils literal notranslate"><span class="pre">\a</span></code>, <code class="docutils literal notranslate"><span class="pre">\b</span></code>, and <code class="docutils literal notranslate"><span class="pre">\g</span></code>) for arbitrary types. The library often uses letters like <code class="docutils literal notranslate"><span class="pre">R</span></code> and <code class="docutils literal notranslate"><span class="pre">G</span></code> - for the carries of algebraic structures like rings and groups, + for the carriers of algebraic structures like rings and groups, respectively, but in general Greek letters are used for types, especially when there is little or no structure
  • @@ -1216,7 +1216,7 @@ Saying that <code class="docutils literal notranslate"><span class="pre">x</span
    is equivalent to saying that it is less-than-or-equal to <code class="docutils literal notranslate"><span class="pre">y</span></code> and not equal to <code class="docutils literal notranslate"><span class="pre">y</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">y</span> - <span class="k">#check</span> <span class="o">(</span><span class="n">lt_irrefl</span> <span class="n">x</span> <span class="o">:</span> <span class="bp">&#172;</span><span class="n">x</span> <span class="bp">&lt;</span> <span class="n">x</span><span class="o">)</span> + <span class="k">#check</span> <span class="o">(</span><span class="n">lt_irrefl</span> <span class="n">x</span> <span class="o">:</span> <span class="bp">&#172;</span> <span class="o">(</span><span class="n">x</span> <span class="bp">&lt;</span> <span class="n">x</span><span class="o">))</span> <span class="k">#check</span> <span class="o">(</span><span class="n">lt_trans</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">y</span> <span class="bp">&#8594;</span> <span class="n">y</span> <span class="bp">&lt;</span> <span class="n">z</span> <span class="bp">&#8594;</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">z</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">lt_of_le_of_lt</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">&#8804;</span> <span class="n">y</span> <span class="bp">&#8594;</span> <span class="n">y</span> <span class="bp">&lt;</span> <span class="n">z</span> <span class="bp">&#8594;</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">z</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">lt_of_lt_of_le</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">y</span> <span class="bp">&#8594;</span> <span class="n">y</span> <span class="bp">&#8804;</span> <span class="n">z</span> <span class="bp">&#8594;</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">z</span><span class="o">)</span>
• html/C05_Elementary_Number_Theory.html
  • @@ -184,7 +184,7 @@ giving rise to the theorem <code class="docutils literal notranslate"><span clas
    <p>We can use this fact to establish a key property in the argument above: if the square of a number is even, then that number is even as well. - Mathlib defines the predicate <code class="docutils literal notranslate"><span class="pre">Even</span></code> in <code class="docutils literal notranslate"><span class="pre">Data.Nat.Parity</span></code>, + Mathlib defines the predicate <code class="docutils literal notranslate"><span class="pre">Even</span></code> in <code class="docutils literal notranslate"><span class="pre">Algebra.Group.Even</span></code>, but for reasons that will become clear below, we will simply use <code class="docutils literal notranslate"><span class="pre">2</span> <span class="pre">&#8739;</span> <span class="pre">m</span></code> to express that <code class="docutils literal notranslate"><span class="pre">m</span></code> is even.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="n">Nat.Prime.dvd_mul</span>
• html/searchindex.js
  • @@ -1,1 +1,1 @@
    - Search.setIndex({"docnames": ["C01_Introduction", "C02_Basics", "C03_Logic", "C04_Sets_and_Functions", "C05_Elementary_Number_Theory", "C06_Structures", "C07_Hierarchies", "C08_Groups_and_Rings", "C09_Topology", "C10_Differential_Calculus", "C11_Integration_and_Measure_Theory", "genindex", "index"], "filenames": ["C01_Introduction.rst", "C02_Basics.rst", "C03_Logic.rst", "C04_Sets_and_Functions.rst", "C05_Elementary_Number_Theory.rst", "C06_Structures.rst", "C07_Hierarchies.rst", "C08_Groups_and_Rings.rst", "C09_Topology.rst", "C10_Differential_Calculus.rst", "C11_Integration_and_Measure_Theory.rst", "genindex.rst", "index.rst"], "titles": ["<span class=\"section-number\">1. </span>Introduction", "<span class=\"section-number\">2. </span>Basics", "<span class=\"section-number\">3. </span>Logic", "<span class=\"section-number\">4. </span>Sets and Functions", "<span class=\"section-number\">5. </span>Elementary Number Theory", "<span class=\"section-number\">6. </span>Structures", "<span class=\"section-number\">7. </span>Hierarchies", "<span class=\"section-number\">8. </span>Groups and Rings", "<span class=\"section-number\">9. </span>Topology", "<span class=\"section-number\">10. </span>Differential Calculus", "<span class=\"section-number\">11. </span>Integration and Measure Theory", "Index", "Mathematics in Lean"], "terms": {"The": [0, 1, 4, 5, 6, 7, 8, 9, 10, 12], "goal": [0, 1, 2, 3, 4, 5, 8], "thi": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], "book": [0, 2, 8], "i": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], "teach": 0, "you": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], "formal": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], "mathemat": [0, 1, 2, 3, 4, 5, 6, 7, 8], "us": [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12], "lean": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], "4": [0, 1, 2, 4, 5, 7, 8, 9], "interact": [0, 5, 6, 7, 10], "proof": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], "assist": [0, 2, 5], "It": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], "assum": [0, 1, 2, 3, 4, 6, 7, 8], "know": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], "some": [0, 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"rewrite": [[1, "index-0"]], "rfl": [[1, "index-13"]], "ring": [[1, "index-6"]], "ring (algebraic structure)": [[1, "index-7"], [7, "index-4"]], "ring (tactic)": [[1, "index-6"]], "rw": [[1, "index-0"], [1, "index-4"]], "rw and rewrite": [[1, "index-0"], [1, "index-4"]], "show": [[1, "index-22"]], "tactics": [[1, "index-0"], [1, "index-11"], [1, "index-12"], [1, "index-13"], [1, "index-15"], [1, "index-17"], [1, "index-18"], [1, "index-20"], [1, "index-22"], [1, "index-23"], [1, "index-3"], [1, "index-4"], [1, "index-5"], [1, "index-6"], [2, "index-0"], [2, "index-12"], [2, "index-13"], [2, "index-14"], [2, "index-15"], [2, "index-16"], [2, "index-17"], [2, "index-18"], [2, "index-19"], [2, "index-2"], [2, "index-20"], [2, "index-21"], [2, "index-23"], [2, "index-24"], [2, "index-25"], [2, "index-26"], [2, "index-4"], [2, "index-6"], [2, "index-8"], [2, "index-9"], [3, "index-1"], [3, "index-2"], [4, "index-0"], [7, "index-2"], [7, "index-3"], [8, "index-3"]], "absurd": [[2, "index-17"]], "anonymous constructor": [[2, "index-7"]], "assumption": [[2, "index-19"]], "by_cases": [[2, "index-23"]], "by_contra": [[2, "index-14"]], "by_contra and by_contradiction": [[2, "index-14"]], "cases": [[2, "index-8"]], "change": [[2, "index-2"]], "congr": [[2, "index-25"]], "constructor": [[2, "index-18"]], "contradiction": [[2, "index-17"]], "contrapose": [[2, "index-16"]], "convert": [[2, "index-26"]], "dsimp": [[2, "index-2"]], "erw": [[2, "index-4"]], "excluded middle": [[2, "index-22"]], "exfalso": [[2, "index-17"]], "ext": [[2, "index-24"]], "extensionality": [[2, "index-24"]], "field_simp": [[2, "index-11"]], "from": [[2, "index-12"]], "injective function": [[2, "index-5"]], "intro": [[2, "index-0"]], "lambda abstraction": [[2, "index-1"]], "left": [[2, "index-21"]], "let": [[2, "index-13"]], "monotone function": [[2, "index-3"]], "push_neg": [[2, "index-15"]], "rcases": [[2, "index-9"]], "right": [[2, "index-21"]], "rintro": [[2, "index-9"]], "simp": [[2, "index-20"], [3, "index-1"]], "surjective function": [[2, "index-10"]], "tactic": [[2, "index-11"]], "this": [[2, "index-12"]], "use": [[2, "index-6"]], "bounded quantifiers": [[3, "index-3"]], "rwa": [[3, "index-2"]], "set operations": [[3, "index-0"]], "decide": [[4, "index-0"]], "monoid": [[7, "index-0"]], "filter": [[8, "index-1"]], "continuity": [[8, "index-3"]], "topological space": [[8, "index-4"]], "topology": [[8, "index-0"]], "differential calculus": [[9, "index-0"]], "elementary calculus": [[9, "index-1"]], "normed space": [[9, "index-2"]], "integration": [[10, "index-0"], [10, "index-1"]], "measure theory": [[10, "index-2"]]}})
• lake-manifest.json
  • @@ -1,11 +1,11 @@
    {"version": 7, "packagesDir": ".lake/packages", "packages": - [{"url": "https://github.com/leanprover/std4", + [{"url": "https://github.com/leanprover-community/batteries", "type": "git", "subDir": null, - "rev": "3025cb124492b423070f20cf0a70636f757d117f", - "name": "std", + "rev": "7b3c48b58fa0ae1c8f27889bdb086ea5e4b27b06", + "name": "batteries", "manifestFile": "lake-manifest.json", "inputRev": "main", "inherited": true,
  • @@ -13,7 +13,7 @@
    {"url": "https://github.com/leanprover-community/quote4", "type": "git", "subDir": null, - "rev": "64365c656d5e1bffa127d2a1795f471529ee0178", + "rev": "53156671405fbbd5402ed17a79bd129b961bd8d6", "name": "Qq", "manifestFile": "lake-manifest.json", "inputRev": "master",
  • @@ -22,19 +22,19 @@
    {"url": "https://github.com/leanprover-community/aesop", "type": "git", "subDir": null, - "rev": "0a21a48c286c4a4703c0be6ad2045f601f31b1d0", + "rev": "e8c8a42642ceb5af33708b79ae8a3148b681c236", "name": "aesop", "manifestFile": "lake-manifest.json", "inputRev": "master", "inherited": true, - "configFile": "lakefile.lean"}, + "configFile": "lakefile.toml"}, {"url": "https://github.com/leanprover-community/ProofWidgets4", "type": "git", "subDir": null, - "rev": "fe1eff53bd0838c657aa6126fe4dd75ad9939d9a", + "rev": "e6b6247c61280c77ade6bbf0bc3c66a44fe2e0c5", "name": "proofwidgets", "manifestFile": "lake-manifest.json", - "inputRev": "v0.0.35", + "inputRev": "v0.0.36", "inherited": true, "configFile": "lakefile.lean"}, {"url": "https://github.com/leanprover/lean4-cli",
  • @@ -49,16 +49,16 @@
    {"url": "https://github.com/leanprover-community/import-graph.git", "type": "git", "subDir": null, - "rev": "188eb34fcf1125e89d651ad462d02598219718ca", + "rev": "35e38eb320982cfd2fcc864e0e0467ca223c8cdb", "name": "importGraph", "manifestFile": "lake-manifest.json", "inputRev": "main", "inherited": true, - "configFile": "lakefile.lean"}, + "configFile": "lakefile.toml"}, {"url": "https://github.com/leanprover-community/mathlib4", "type": "git", "subDir": null, - "rev": "db651742f2c631e5b8525e9aabcf3d61ed094a4a", + "rev": "dfedb0f929c2a8336ca48766c2902651313864ba", "name": "mathlib", "manifestFile": "lake-manifest.json", "inputRev": "master",
• lean-toolchain
  • @@ -1,1 +1,1 @@
    - leanprover/lean4:v4.8.0-rc1 + leanprover/lean4:v4.8.0-rc2
• mathematics_in_lean.pdf