Update mer. 30 août 2023 09:46:32 EDT

Changes

4 changed files (+6/-40)

• MIL/C06_Structures/S01_Structures.lean
  • @@ -72,30 +72,14 @@ 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] + -- the same proof still works, but the goal view here is harder to read 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
• html/C06_Structures.html
  • @@ -222,11 +222,9 @@ in a manner similar to the way we defined recursive functions in
    The definitions <code class="docutils literal notranslate"><span class="pre">addAlt</span></code> and <code class="docutils literal notranslate"><span class="pre">addAlt'</span></code> below are essentially the same; the only difference is that we use anonymous constructor notation in the second. - Although it is sometimes convenient to define functions this way, - the definitional properties are not as convenient. - For example, the expressions <code class="docutils literal notranslate"><span class="pre">addAlt</span> <span class="pre">a</span> <span class="pre">b</span></code> and <code class="docutils literal notranslate"><span class="pre">addAlt'</span> <span class="pre">a</span> <span class="pre">b</span></code> - cannot be simplified until we decompose <code class="docutils literal notranslate"><span class="pre">a</span></code> and <code class="docutils literal notranslate"><span class="pre">b</span></code> into - components, which we can do with <code class="docutils literal notranslate"><span class="pre">cases</span></code>, <code class="docutils literal notranslate"><span class="pre">rcases</span></code>, etc.</p> + Although it is sometimes convenient to define functions this way, and structural eta-reduction makes + this alternative definitionally equivalent, it can make things less convenient in later proofs. + In particular, <cite>rw [addAlt]</cite> leaves us with a messier goal view containing a <cite>match</cite> statement.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">addAlt</span> <span class="o">:</span> <span class="n">Point</span> <span class="bp">&#8594;</span> <span class="n">Point</span> <span class="bp">&#8594;</span> <span class="n">Point</span> <span class="bp">|</span> <span class="n">Point.mk</span> <span class="n">x&#8321;</span> <span class="n">y&#8321;</span> <span class="n">z&#8321;</span><span class="o">,</span> <span class="n">Point.mk</span> <span class="n">x&#8322;</span> <span class="n">y&#8322;</span> <span class="n">z&#8322;</span> <span class="bp">=&gt;</span> <span class="o">&#10216;</span><span class="n">x&#8321;</span> <span class="bp">+</span> <span class="n">x&#8322;</span><span class="o">,</span> <span class="n">y&#8321;</span> <span class="bp">+</span> <span class="n">y&#8322;</span><span class="o">,</span> <span class="n">z&#8321;</span> <span class="bp">+</span> <span class="n">z&#8322;</span><span class="o">&#10217;</span>
  • @@ -234,29 +232,13 @@ components, which we can do with <code class="docutils literal notranslate"><spa
    <span class="bp">|</span> <span class="o">&#10216;</span><span class="n">x&#8321;</span><span class="o">,</span> <span class="n">y&#8321;</span><span class="o">,</span> <span class="n">z&#8321;</span><span class="o">&#10217;,</span> <span class="o">&#10216;</span><span class="n">x&#8322;</span><span class="o">,</span> <span class="n">y&#8322;</span><span class="o">,</span> <span class="n">z&#8322;</span><span class="o">&#10217;</span> <span class="bp">=&gt;</span> <span class="o">&#10216;</span><span class="n">x&#8321;</span> <span class="bp">+</span> <span class="n">x&#8322;</span><span class="o">,</span> <span class="n">y&#8321;</span> <span class="bp">+</span> <span class="n">y&#8322;</span><span class="o">,</span> <span class="n">z&#8321;</span> <span class="bp">+</span> <span class="n">z&#8322;</span><span class="o">&#10217;</span> <span class="kd">theorem</span> <span class="n">addAlt_x</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">Point</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="n">a.addAlt</span> <span class="n">b</span><span class="o">)</span><span class="bp">.</span><span class="n">x</span> <span class="bp">=</span> <span class="n">a.x</span> <span class="bp">+</span> <span class="n">b.x</span> <span class="o">:=</span> <span class="kd">by</span> - <span class="n">cases</span> <span class="n">a</span> - <span class="n">cases</span> <span class="n">b</span> <span class="n">rfl</span> <span class="kd">theorem</span> <span class="n">addAlt_comm</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">Point</span><span class="o">)</span> <span class="o">:</span> <span class="n">addAlt</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">addAlt</span> <span class="n">b</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> - <span class="n">rcases</span> <span class="n">a</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">xa</span><span class="o">,</span> <span class="n">ya</span><span class="o">,</span> <span class="n">za</span><span class="o">&#10217;</span> - <span class="n">rcases</span> <span class="n">b</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">xb</span><span class="o">,</span> <span class="n">yb</span><span class="o">,</span> <span class="n">zb</span><span class="o">&#10217;</span> <span class="n">rw</span> <span class="o">[</span><span class="n">addAlt</span><span class="o">,</span> <span class="n">addAlt</span><span class="o">]</span> + <span class="c1">-- the same proof still works, but the goal view here is harder to read</span> <span class="n">ext</span> <span class="bp">&lt;;&gt;</span> <span class="n">dsimp</span> - <span class="n">apply</span> <span class="n">add_comm</span> <span class="n">repeat&#39;</span> <span class="n">apply</span> <span class="n">add_comm</span> - - <span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">Point</span><span class="o">)</span> <span class="o">:</span> <span class="n">addAlt</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">addAlt</span> <span class="n">b</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> - <span class="n">rcases</span> <span class="n">a</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">xa</span><span class="o">,</span> <span class="n">ya</span><span class="o">,</span> <span class="n">za</span><span class="o">&#10217;</span> - <span class="n">rcases</span> <span class="n">b</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">xb</span><span class="o">,</span> <span class="n">yb</span><span class="o">,</span> <span class="n">zb</span><span class="o">&#10217;</span> - <span class="n">simp</span> <span class="o">[</span><span class="n">addAlt</span><span class="o">,</span> <span class="n">add_comm</span><span class="o">]</span> - - <span class="kd">example</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">Point</span><span class="o">,</span> <span class="n">addAlt</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">addAlt</span> <span class="n">b</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> - <span class="n">rintro</span> <span class="o">&#10216;</span><span class="n">xa</span><span class="o">,</span> <span class="n">ya</span><span class="o">,</span> <span class="n">za</span><span class="o">&#10217;</span> <span class="o">&#10216;</span><span class="n">xb</span><span class="o">,</span> <span class="n">yb</span><span class="o">,</span> <span class="n">zb</span><span class="o">&#10217;</span> - <span class="n">simp</span> <span class="o">[</span><span class="n">addAlt</span><span class="o">,</span> <span class="n">add_comm</span><span class="o">]</span> - - <span class="kd">example</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">Point</span><span class="o">,</span> <span class="n">add</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">add</span> <span class="n">b</span> <span class="n">a</span> <span class="o">:=</span> <span class="k">fun</span> <span class="o">&#10216;</span><span class="n">xa</span><span class="o">,</span> <span class="n">ya</span><span class="o">,</span> <span class="n">za</span><span class="o">&#10217;</span> <span class="o">&#10216;</span><span class="n">xb</span><span class="o">,</span> <span class="n">yb</span><span class="o">,</span> <span class="n">zb</span><span class="o">&#10217;</span> <span class="bp">&#8614;</span> <span class="kd">by</span> - <span class="n">simp</span> <span class="o">[</span><span class="n">add</span><span class="o">,</span> <span class="n">add_comm</span><span class="o">]</span> </pre></div> </div> <p>Mathematical constructions often involve taking apart bundled information and
• html/searchindex.js
  • @@ -1,1 +1,1 @@
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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>Topology","<span class=\"section-number\">9. </span>Differential Calculus","<span class=\"section-number\">10. </span>Integration and Measure Theory","Index","Mathematics in Lean"],titleterms:{"function":[3,7],"schr\u00f6der":3,The:[2,3],about:1,algebra:[1,5],appli:1,asymptot:8,ball:7,basic:[1,6],bernstein:3,build:5,calcul:1,calculu:8,close:7,compact:7,comparison:8,complet:7,conjunct:2,continu:[7,8],converg:[2,7],countabl:7,defin:5,differenti:8,disjunct:2,elementari:[4,8,9],exampl:1,existenti:2,fact:1,filter:7,fundament:7,gaussian:5,get:0,hierarchi:6,ident:1,iff:2,implic:2,index:10,induct:4,infinit:4,integ:5,integr:9,introduct:0,irrat:4,lean:11,lemma:1,linear:8,logic:2,mani:4,map:8,mathemat:11,measur:9,metric:7,more:1,morphism:6,negat:2,norm:8,number:4,object:6,open:7,overview:0,prime:4,prove:1,quantifi:2,recurs:4,root:4,rw:1,separ:7,sequenc:2,set:[3,7],space:[7,8],start:0,structur:[1,5],sub:6,theorem:[1,3],theori:[4,9],topolog:7,uniformli:7,univers:2,us:1}}) + 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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>Topology","<span class=\"section-number\">9. </span>Differential Calculus","<span class=\"section-number\">10. </span>Integration and Measure Theory","Index","Mathematics in Lean"],titleterms:{"function":[3,7],"schr\u00f6der":3,The:[2,3],about:1,algebra:[1,5],appli:1,asymptot:8,ball:7,basic:[1,6],bernstein:3,build:5,calcul:1,calculu:8,close:7,compact:7,comparison:8,complet:7,conjunct:2,continu:[7,8],converg:[2,7],countabl:7,defin:5,differenti:8,disjunct:2,elementari:[4,8,9],exampl:1,existenti:2,fact:1,filter:7,fundament:7,gaussian:5,get:0,hierarchi:6,ident:1,iff:2,implic:2,index:10,induct:4,infinit:4,integ:5,integr:9,introduct:0,irrat:4,lean:11,lemma:1,linear:8,logic:2,mani:4,map:8,mathemat:11,measur:9,metric:7,more:1,morphism:6,negat:2,norm:8,number:4,object:6,open:7,overview:0,prime:4,prove:1,quantifi:2,recurs:4,root:4,rw:1,separ:7,sequenc:2,set:[3,7],space:[7,8],start:0,structur:[1,5],sub:6,theorem:[1,3],theori:[4,9],topolog:7,uniformli:7,univers:2,us:1}})
• mathematics_in_lean.pdf