e3465109 - mathematics_in

Authored by: Patrick Massot
Authored at: 2023-06-05 15:39:20 +0200
Committed by: Patrick Massot
Committed at: 2023-06-05 15:39:20 +0200
Object name: e3465109fa6e750afc850e9429d7a5b3b86e7721
Parent: 2e14db9f

Update lun. 05 juin 2023 15:39:20 CEST

Changes

3 changed files (+9/-11)

html/C07_Hierarchies.html
- @@ -252,8 +252,8 @@ a field <code class="docutils literal notranslate">Monoid&#832
 (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: + So the <code class="docutils literal notranslate">class</code> command did some magic for us (and the <code class="docutils literal notranslate">structure</code> command would have done it + too). An easy way to see what are the fields of our classes is to check their constructor. Compare: <div class="highlight-lean notranslate"><div class="highlight"><pre>/- Monoid₂.mk {α : Type} (toSemigroup₁ : Semigroup₁ α) (toDiaOneClass₁ : DiaOneClass₁ α) : Monoid₂ α -/ #check Monoid₂.mk
- @@ -326,7 +326,7 @@ This is a bit technical you don’t need to understand details. The importan
 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>. + used the <code class="docutils literal notranslate">whatsnew in</code> command on top of <code class="docutils literal notranslate">left_inv_eq_right_inv'</code>. <div class="highlight-lean notranslate"><div class="highlight"><pre>class AddSemigroup₃ (α : Type) extends Add α where /-- Multiplication is associative -/ add_assoc₃ : ∀ a b c : α, a + b + c = a + (b + c)
- @@ -654,12 +654,10 @@ So we could use a structure instead.
 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. + procedure since it would need to hunt down <code class="docutils literal notranslate">g ∘ f</code> everywhere. Seeing it failing in <code class="docutils literal notranslate">g (f x)</code> + would be very frustrating. More generally one must always keep in mind that recognizing which + function is applied in a given expression is a very difficult problem, called the “higher-order + unification problem”. So Mathlib does not use this class approach. A more fundamental question is whether we use predicates as above (using either a <code class="docutils literal notranslate">def</code> or a <code class="docutils literal notranslate">structure</code>) or use structures bundling a function and predicates. This is partly a psychological issue. It is extremely rare to consider a function between monoids that is not a morphism.
- @@ -815,7 +813,7 @@ 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 + they are defined by the <code class="docutils literal notranslate">Monotone</code> predicate. Of course you need to complete the class definitions below. <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] structure OrderPresHom (α β : Type) [LE α] [LE β] where
html/searchindex.js
- @@ -1,1 +1,1 @@
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mathematics_in_lean.pdf