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Homework.html
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<link href="coqdoc.css" rel="stylesheet" type="text/css" />
<title></title>
</head>
<body>
<div id="page">
<div id="header">
</div>
<div id="main">
<div class="code">
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="var">HoTT</span> <span class="id" title="var">Join</span> <span class="id" title="var">UnivalenceAxiom</span>.<br/>
<br/>
<br/>
</div>
<div class="doc">
<i>Note</i>: we will use the <span class="inlinecode"><span class="id" title="var">Admitted</span></span> command whenever we do not
give the term for a certain type. We then associate the
following meanings to assertion commands:
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" title="keyword">Fact</span></span> denotes a fact proven in the course or the HoTT book
</li>
<li> <span class="inlinecode"><span class="id" title="var">Proposition</span></span> denotes either a proposition assumed to be true
in the subject, or a partially answered optional question
</li>
<li> <span class="inlinecode"><span class="id" title="keyword">Lemma</span></span>/<span class="inlinecode"><span class="id" title="keyword">Theorem</span></span> denote a partially answered mandatory question
</li>
</ul>
<div class="paragraph"> </div>
We also follow the convention to name <span class="inlinecode"><span class="id" title="var">q_i_j</span></span> the answer to
question j of exercise i.
<div class="paragraph"> </div>
Whenever we use a lemma implemented in the HoTT library,
we will be careful to state which lemma of the course/HoTT book
it corresponds to.
<div class="paragraph"> </div>
<a name="lab1"></a><h1 class="section">Prelude</h1>
<div class="paragraph"> </div>
Notations to match more closely those of the course/subject.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Notation</span> "'ua'" := <span class="id" title="var">equiv_path</span>.<br/>
<span class="id" title="keyword">Notation</span> "'ua^-1'" := <span class="id" title="var">path_universe</span>.<br/>
<br/>
<span class="id" title="keyword">Notation</span> "2" := <span class="id" title="var">Bool</span>.<br/>
<span class="id" title="keyword">Notation</span> "'Σ' A" := (<span class="id" title="var">Susp</span> <span class="id" title="var">A</span>) (<span class="id" title="var">format</span> "'Σ' A", <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 10).<br/>
<span class="id" title="keyword">Notation</span> "'S^' n" := (<span class="id" title="var">Sphere</span> <span class="id" title="var">n</span>) (<span class="id" title="var">format</span> "'S^' n", <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 10).<br/>
<span class="id" title="keyword">Notation</span> "'S*^' n" := (<span class="id" title="var">psphere</span> <span class="id" title="var">n</span>) (<span class="id" title="var">format</span> "'S*^' n", <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 10).<br/>
<span class="id" title="keyword">Infix</span> "×" := <span class="id" title="var">prod</span> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 30).<br/>
<span class="id" title="keyword">Infix</span> "∗" := <span class="id" title="var">Join</span> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 30).<br/>
<span class="id" title="keyword">Infix</span> "≃" := <span class="id" title="var">Equiv</span> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 50).<br/>
<span class="id" title="keyword">Infix</span> "≃*" := <span class="id" title="var">pEquiv</span> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 50).<br/>
<span class="id" title="keyword">Notation</span> "| A |" := (<span class="id" title="var">merely</span> <span class="id" title="var">A</span>) (<span class="id" title="var">format</span> "| A |", <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70).<br/>
<br/>
</div>
<div class="doc">
<a name="lab2"></a><h3 class="section">Spans</h3>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Inductive</span> <span class="id" title="var">Span</span> :=<br/>
| <span class="id" title="var">span</span> {<span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">C</span> : <span class="id" title="keyword">Type</span>} (<span class="id" title="var">f</span> : <span class="id" title="var">B</span> → <span class="id" title="var">A</span>) (<span class="id" title="var">g</span> : <span class="id" title="var">B</span> → <span class="id" title="var">C</span>).<br/>
<br/>
<span class="id" title="keyword">Record</span> <span class="id" title="var">EqSpan'</span><br/>
{<span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">C</span> : <span class="id" title="keyword">Type</span>} (<span class="id" title="var">f</span> : <span class="id" title="var">B</span> → <span class="id" title="var">A</span>) (<span class="id" title="var">g</span> : <span class="id" title="var">B</span> → <span class="id" title="var">C</span>) <br/>
{<span class="id" title="var">A'</span> <span class="id" title="var">B'</span> <span class="id" title="var">C'</span> : <span class="id" title="keyword">Type</span>} (<span class="id" title="var">f'</span> : <span class="id" title="var">B'</span> → <span class="id" title="var">A'</span>) (<span class="id" title="var">g'</span> : <span class="id" title="var">B'</span> → <span class="id" title="var">C'</span>)<br/>
:= {<br/>
<span class="id" title="var">equiv_left</span> : <span class="id" title="var">A</span> ≃ <span class="id" title="var">A'</span>;<br/>
<span class="id" title="var">equiv_apex</span> : <span class="id" title="var">B</span> ≃ <span class="id" title="var">B'</span>;<br/>
<span class="id" title="var">equiv_right</span> : <span class="id" title="var">C</span> ≃ <span class="id" title="var">C'</span>;<br/>
<span class="id" title="var">coherence_left</span> : <span class="id" title="var">equiv_left</span> <span class="id" title="var">o</span> <span class="id" title="var">f</span> == <span class="id" title="var">f'</span> <span class="id" title="var">o</span> <span class="id" title="var">equiv_apex</span>;<br/>
<span class="id" title="var">coherence_right</span> : <span class="id" title="var">equiv_right</span> <span class="id" title="var">o</span> <span class="id" title="var">g</span> == <span class="id" title="var">g'</span> <span class="id" title="var">o</span> <span class="id" title="var">equiv_apex</span>;<br/>
}.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">EqSpan</span> '(<span class="id" title="var">span</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">f</span> <span class="id" title="var">g</span>) '(<span class="id" title="var">span</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">f'</span> <span class="id" title="var">g'</span>) :=<br/>
<span class="id" title="var">EqSpan'</span> <span class="id" title="var">f</span> <span class="id" title="var">g</span> <span class="id" title="var">f'</span> <span class="id" title="var">g'</span>.<br/>
<br/>
<span class="id" title="keyword">Proposition</span> <span class="id" title="var">path_span</span> {<span class="id" title="var">s</span> <span class="id" title="var">s'</span>} :<br/>
<span class="id" title="var">EqSpan</span> <span class="id" title="var">s</span> <span class="id" title="var">s'</span> ≃ (<span class="id" title="var">s</span> = <span class="id" title="var">s'</span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">Admitted</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab3"></a><h1 class="section">Exercise 1</h1>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Exercise_1</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab4"></a><h2 class="section">Question 1</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Question_1</span>.<br/>
<br/>
<span class="id" title="keyword">Context</span> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>).<br/>
<br/>
<span class="id" title="keyword">Let</span> φ : 2 ∗ <span class="id" title="var">A</span> → <span class="id" title="var">Σ</span> <span class="id" title="var">A</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="comment">(* By recursion on joins, that is on pushouts *)</span><br/>
<span class="id" title="var">srapply</span> <span class="id" title="var">Join_rec</span>; <span class="id" title="tactic">simpl</span>.<br/>
<span class="comment">(* φ(left(b)) :≡ … *)</span><br/>
* <span class="id" title="tactic">exact</span> (<br/>
<span class="id" title="keyword">fun</span> <span class="id" title="var">b</span> ⇒ <span class="id" title="keyword">match</span> <span class="id" title="var">b</span> <span class="id" title="keyword">with</span><br/>
| <span class="id" title="var">false</span> ⇒ <span class="id" title="var">North</span><br/>
| <span class="id" title="var">true</span> ⇒ <span class="id" title="var">South</span><br/>
<span class="id" title="keyword">end</span>).<br/>
<span class="comment">(* φ(right(a)) :≡ … *)</span><br/>
* <span class="id" title="tactic">exact</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <span class="id" title="var">South</span>).<br/>
<span class="comment">(* ap_φ(quo(b,a)) :≡ … *)</span><br/>
* <span class="id" title="tactic">intros</span> <span class="id" title="var">b</span> <span class="id" title="var">a</span>. <span class="id" title="tactic">simpl</span>.<br/>
<span class="comment">(* By recursion/case on b *)</span><br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">b</span>.<br/>
- <span class="id" title="tactic">exact</span> <span class="id" title="var">idpath</span>. <span class="comment">(* <span class="inlinecode"><span class="id" title="var">idpath</span></span> is refl *)</span><br/>
- <span class="id" title="tactic">exact</span> (<span class="id" title="var">merid</span> <span class="id" title="var">a</span>).<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">Let</span> ψ : <span class="id" title="var">Σ</span> <span class="id" title="var">A</span> → 2 ∗ <span class="id" title="var">A</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="comment">(* By recursion on suspensions *)</span><br/>
<span class="id" title="var">srapply</span> <span class="id" title="var">Susp_rec</span>.<br/>
<span class="comment">(* ψ(N) :≡ … *)</span><br/>
* <span class="id" title="tactic">exact</span> (<span class="id" title="var">joinl</span> <span class="id" title="var">false</span>).<br/>
<span class="comment">(* ψ(S) :≡ … *)</span><br/>
* <span class="id" title="tactic">exact</span> (<span class="id" title="var">joinl</span> <span class="id" title="var">true</span>).<br/>
<span class="comment">(* ψ(merid_a) :≡ … *)</span><br/>
* <span class="comment">(* <span class="inlinecode"><span class="id" title="var">jglue</span></span> <span class="inlinecode"><span class="id" title="var">b</span></span> <span class="inlinecode"><span class="id" title="var">a</span></span> is quo(b,a) *)</span><br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">a</span>. <span class="id" title="tactic">exact</span> ((<span class="id" title="var">jglue</span> <span class="id" title="var">false</span> <span class="id" title="var">a</span>) @ (<span class="id" title="var">jglue</span> <span class="id" title="var">true</span> <span class="id" title="var">a</span>)^).<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
</div>
<div class="doc">
For this first proof, we will comment every step in order to
document how "informal" reasoning translates into tactics and lemmas
of the HoTT library. In subsequent proofs, we will only document
parts that use new tactics/lemmas.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Lemma</span> <span class="id" title="var">q_1_1</span> :<br/>
2 ∗ <span class="id" title="var">A</span> ≃ <span class="id" title="var">Σ</span> <span class="id" title="var">A</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">unshelve</span> <span class="id" title="tactic">econstructor</span>.<br/>
<span class="comment">(* We choose φ as the underlying map of the equivalence *)</span><br/>
* <span class="id" title="tactic">exact</span> φ.<br/>
<span class="comment">(* We show that φ is indeed an equivalence, with ψ as inverse *)</span><br/>
* <span class="id" title="tactic">apply</span> <span class="id" title="var">isequiv_adjointify</span> <span class="id" title="keyword">with</span> ψ.<br/>
<span class="comment">(* φ ○ ψ ~ id *)</span><br/>
- <span class="id" title="tactic">unfold</span> <span class="id" title="var">Sect</span>.<br/>
<span class="comment">(* By induction on suspensions *)</span><br/>
<span class="id" title="var">srapply</span> <span class="id" title="var">Susp_ind</span>; <span class="id" title="tactic">simpl</span>.<br/>
<span class="comment">(* φ(ψ(N) = N *)</span><br/>
+ <span class="id" title="tactic">exact</span> <span class="id" title="var">idpath</span>.<br/>
<span class="comment">(* φ(ψ(S) = S *)</span><br/>
+ <span class="id" title="tactic">exact</span> <span class="id" title="var">idpath</span>.<br/>
<span class="comment">(* tr_{merid x}(refl) = refl *)</span><br/>
+ <span class="id" title="tactic">intros</span> <span class="id" title="var">a</span>.<br/>
<span class="comment">(* We use an instance of theorem 2.11.3 of the HoTT book *)</span><br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">transport_paths_FFlr</span>.<br/>
<span class="comment">(* Compute with the definition of ψ *)</span><br/>
<span class="id" title="tactic">unfold</span> ψ. <span class="id" title="tactic">rewrite</span> <span class="id" title="var">Susp_rec_beta_merid</span>.<br/>
<span class="comment">(* Lemma 7 and 8 of the course *)</span><br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">ap_pp</span>, <span class="id" title="var">ap_V</span>.<br/>
<span class="comment">(* Compute with the definition of φ *)</span><br/>
<span class="id" title="tactic">unfold</span> φ. <span class="id" title="tactic">rewrite</span> 2 <span class="id" title="var">Join_rec_beta_jglue</span>.<br/>
<span class="comment">(* The <span class="inlinecode"><span class="id" title="var">hott_simpl</span></span> tactic tries to simplify paths using<br/>
basic homotopies, such as composition with identity and<br/>
inverse paths *)</span><br/>
<span class="id" title="var">hott_simpl</span>.<br/>
<span class="comment">(* ψ ○ φ ~ id *)</span><br/>
- <span class="id" title="tactic">unfold</span> <span class="id" title="var">Sect</span>.<br/>
<span class="comment">(* By induction on joins *)</span><br/>
<span class="id" title="var">srapply</span> <span class="id" title="var">Join_ind</span>; <span class="id" title="tactic">simpl</span>.<br/>
<span class="comment">(* ψ(φ(left(b))) = left(b) *)</span><br/>
+ <span class="id" title="tactic">intros</span> <span class="id" title="var">b</span>.<br/>
<span class="comment">(* By case on b *)</span><br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">b</span>; <span class="id" title="tactic">simpl</span>.<br/>
** <span class="id" title="tactic">exact</span> <span class="id" title="var">idpath</span>.<br/>
** <span class="id" title="tactic">exact</span> <span class="id" title="var">idpath</span>.<br/>
<span class="comment">(* ψ(φ(right(a))) = right(a) *)</span><br/>
+ <span class="id" title="tactic">intros</span> <span class="id" title="var">a</span>. <span class="id" title="tactic">exact</span> (<span class="id" title="var">jglue</span> <span class="id" title="var">true</span> <span class="id" title="var">a</span>).<br/>
<span class="comment">(* tr_{quo(b,a)}(refl) = quo(1,a) *)</span><br/>
+ <span class="id" title="tactic">intros</span> <span class="id" title="var">b</span> <span class="id" title="var">a</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">dp_path_transport</span>.<br/>
<span class="comment">(* Again, same instance of theorem 2.11.3 of the HoTT book *)</span><br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">transport_paths_FFlr</span>.<br/>
<span class="comment">(* Compute with the definition of φ *)</span><br/>
<span class="id" title="tactic">unfold</span> φ. <span class="id" title="tactic">rewrite</span> <span class="id" title="var">Join_rec_beta_jglue</span>.<br/>
<span class="comment">(* By case on b *)</span><br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">b</span>.<br/>
<span class="comment">(* tr_{quo(1,a)}(refl) = quo(1,a) *)</span><br/>
** <span class="id" title="var">hott_simpl</span>.<br/>
<span class="comment">(* tr_{quo(0,a)}(refl) = quo(1,a) *)</span><br/>
** <span class="comment">(* Compute with the definition of ψ *)</span><br/>
<span class="id" title="tactic">unfold</span> ψ. <span class="id" title="tactic">rewrite</span> <span class="id" title="var">Susp_rec_beta_merid</span>.<br/>
<span class="comment">(* Unwhiskering on the right with quo(1,a), that is<br/>
post-composing with quo(1,a) on both sides *)</span><br/>
<span class="id" title="tactic">apply</span> (<span class="id" title="var">cancelR</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> (<span class="id" title="var">jglue</span> <span class="id" title="var">true</span> <span class="id" title="var">a</span>)^).<br/>
<span class="id" title="var">hott_simpl</span>.<br/>
<span class="comment">(* Associativity of composition *)</span><br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">concat_pp_p</span>.<br/>
<span class="id" title="var">hott_simpl</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Question_1</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab5"></a><h2 class="section">Question 2</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Question_2</span>.<br/>
<br/>
<span class="id" title="keyword">Proposition</span> <span class="id" title="var">q_1_2</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">C</span>,<br/>
(<span class="id" title="var">A</span> ∗ <span class="id" title="var">B</span>) ∗ <span class="id" title="var">C</span> ≃ <span class="id" title="var">A</span> ∗ (<span class="id" title="var">B</span> ∗ <span class="id" title="var">C</span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">Admitted</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Question_2</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab6"></a><h2 class="section">Question 3</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Question_3</span>.<br/>
<br/>
<span class="id" title="keyword">Fact</span> <span class="id" title="var">Bool_Sphere_0</span> :<br/>
2 ≃ <span class="id" title="var">S</span>^0.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">Admitted</span>.<br/>
<br/>
</div>
<div class="doc">
We need functoriality of type formers with respect to equivalence,
in order to apply equivalences to their arguments. We could prove
it individually for each type former of interest, but we choose
instead to use univalence to prove it once and forall.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">ae</span> {<span class="id" title="var">X</span> <span class="id" title="var">Y</span>} (<span class="id" title="var">T</span> : <span class="id" title="keyword">Type</span> → <span class="id" title="keyword">Type</span>) :<br/>
<span class="id" title="var">X</span> ≃ <span class="id" title="var">Y</span> → <span class="id" title="var">T</span> <span class="id" title="var">X</span> ≃ <span class="id" title="var">T</span> <span class="id" title="var">Y</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">e</span>.<br/>
<span class="comment">(* <span class="inlinecode"><span class="id" title="var">path_universe_uncurried</span></span> is the map given by univalence *)</span><br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">path_universe_uncurried</span> <span class="id" title="tactic">in</span> <span class="id" title="var">e</span>.<br/>
<span class="comment">(* The <span class="inlinecode"><span class="id" title="var">path_induction</span></span> tactic applies path induction to every<br/>
path found in the context *)</span><br/>
<span class="id" title="var">path_induction</span>.<br/>
<span class="comment">(* <span class="inlinecode"><span class="id" title="var">equiv_idmap</span></span> is the identity equivalence *)</span><br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">equiv_idmap</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
</div>
<div class="doc">
The <span class="inlinecode"><span class="id" title="tactic">rewrite</span></span> tactic of the HoTT library is quite weak since
it does not support equivalences, and does not allow selecting
the occurrences of the term we want to rewrite. Thus we will
compose equivalences manually using the <span class="inlinecode"><span class="id" title="tactic">refine</span></span> tactic and
the <span class="inlinecode"><span class="id" title="var">oE</span></span> composition operator.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">equiv_scope</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <span class="id" title="var">q_1_3</span> :<br/>
<span class="id" title="var">S</span>^1 ∗ <span class="id" title="var">S</span>^1 ≃ <span class="id" title="var">S</span>^3.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="comment">(* Rewrite on the left of ∗ using functoriality *)</span><br/>
<span class="id" title="tactic">refine</span> (<span class="id" title="var">_</span> <span class="id" title="var">oE</span> <span class="id" title="var">ae</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">X</span> ⇒ <span class="id" title="var">X</span> ∗ <span class="id" title="var">_</span>) <span class="id" title="var">_</span>).<br/>
2: { <span class="comment">(* Rewrite under Σ using functoriality (since S^n is the<br/>
definition of the sphere based on iterated suspensions) *)</span><br/>
<span class="id" title="tactic">refine</span> (<span class="id" title="var">ae</span> <span class="id" title="var">Susp</span> <span class="id" title="var">_</span>).<br/>
<span class="comment">(* Rewrite S^0 into 2 *)</span><br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">Bool_Sphere_0</span>^-1. }<br/>
<span class="comment">(* Rewrite on the left of ∗ using functoriality *)</span><br/>
<span class="id" title="tactic">refine</span> (<span class="id" title="var">_</span> <span class="id" title="var">oE</span> <span class="id" title="var">ae</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">X</span> ⇒ <span class="id" title="var">X</span> ∗ <span class="id" title="var">_</span>) <span class="id" title="var">_</span>).<br/>
2: { <span class="comment">(* Rewrite Σ2 into 2 ∗ 2 using <span class="inlinecode"><span class="id" title="var">q_1_1</span></span> *)</span><br/>
<span class="id" title="tactic">exact</span> (<span class="id" title="var">q_1_1</span> <span class="id" title="var">_</span>)^-1. }<br/>
<span class="comment">(* Associativity of ∗ *)</span><br/>
<span class="id" title="tactic">refine</span> (<span class="id" title="var">_</span> <span class="id" title="var">oE</span> <span class="id" title="var">q_1_2</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>).<br/>
<span class="comment">(* Rewrite on the right of ∗ using functoriality *)</span><br/>
<span class="id" title="tactic">refine</span> (<span class="id" title="var">_</span> <span class="id" title="var">oE</span> <span class="id" title="var">ae</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">X</span> ⇒ <span class="id" title="var">_</span> ∗ <span class="id" title="var">X</span>) <span class="id" title="var">_</span>).<br/>
2: { <span class="comment">(* Rewrite 2 ∗ S^1 into ΣS^1 using <span class="inlinecode"><span class="id" title="var">q_1_1</span></span> *)</span><br/>
<span class="id" title="tactic">exact</span> (<span class="id" title="var">q_1_1</span> <span class="id" title="var">_</span>). }<br/>
<span class="comment">(* Rewrite 2 ∗ S^1 into ΣΣS^1 using <span class="inlinecode"><span class="id" title="var">q_1_1</span></span> *)</span><br/>
<span class="id" title="tactic">refine</span> (<span class="id" title="var">_</span> <span class="id" title="var">oE</span> <span class="id" title="var">q_1_1</span> <span class="id" title="var">_</span>).<br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">equiv_idmap</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">Close</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">equiv_scope</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Question_3</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Exercise_1</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab7"></a><h1 class="section">Exercise 2</h1>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Exercise_2</span>.<br/>
<br/>
<span class="id" title="keyword">Context</span> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">P</span> : <span class="id" title="var">Σ</span> <span class="id" title="var">A</span> → <span class="id" title="keyword">Type</span>).<br/>
<br/>
</div>
<div class="doc">
<a name="lab8"></a><h2 class="section">Question 1</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Question_1</span>.<br/>
<br/>
<span class="id" title="keyword">Let</span> ψ '(<span class="id" title="var">q</span>, <span class="id" title="var">a</span>) := <span class="id" title="var">transport</span> <span class="id" title="var">P</span> (<span class="id" title="var">merid</span> <span class="id" title="var">a</span>) <span class="id" title="var">q</span>.<br/>
<br/>
<span class="id" title="keyword">Proposition</span> <span class="id" title="var">q_2_1</span> :<br/>
{<span class="id" title="var">x</span> : <span class="id" title="var">Σ</span> <span class="id" title="var">A</span> & <span class="id" title="var">P</span> <span class="id" title="var">x</span>} ≃ <span class="id" title="var">Pushout</span> <span class="id" title="var">fst</span> ψ.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">Admitted</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Question_1</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Exercise_2</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab9"></a><h1 class="section">Exercise 3</h1>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Exercise_3</span>.<br/>
<br/>
<span class="id" title="keyword">Context</span> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">μ</span> : <span class="id" title="var">A</span> × <span class="id" title="var">A</span> → <span class="id" title="var">A</span>).<br/>
<br/>
<span class="id" title="keyword">Notation</span> "'μ'" := <span class="id" title="var">μ</span>.<br/>
<span class="id" title="keyword">Notation</span> "'μ' ( a , b )" := (<span class="id" title="var">μ</span> (<span class="id" title="var">pair</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span>)) (<span class="id" title="var">format</span> "'μ' ( a , b )").<br/>
<span class="id" title="keyword">Notation</span> "'μ' ( a , '_' )" := (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ <span class="id" title="var">μ</span>(<span class="id" title="var">a</span>,<span class="id" title="var">x</span>)) (<span class="id" title="var">format</span> "'μ' ( a , _ )").<br/>
<span class="id" title="keyword">Notation</span> "'μ' ( '_' , a )" := (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ <span class="id" title="var">μ</span>(<span class="id" title="var">x</span>,<span class="id" title="var">a</span>)) (<span class="id" title="var">format</span> "'μ' ( _ , a )").<br/>
<br/>
<span class="id" title="keyword">Context</span> `(<span class="id" title="keyword">∀</span> <span class="id" title="var">a</span>, <span class="id" title="var">IsEquiv</span> <span class="id" title="var">μ</span>(<span class="id" title="var">a</span>,<span class="id" title="var">_</span>)).<br/>
<span class="id" title="keyword">Context</span> `(<span class="id" title="keyword">∀</span> <span class="id" title="var">a</span>, <span class="id" title="var">IsEquiv</span> <span class="id" title="var">μ</span>(<span class="id" title="var">_</span>,<span class="id" title="var">a</span>)).<br/>
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">P</span> : <span class="id" title="var">Σ</span> <span class="id" title="var">A</span> → <span class="id" title="keyword">Type</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">srapply</span> <span class="id" title="var">Susp_rec</span>; <span class="id" title="tactic">simpl</span>.<br/>
* <span class="id" title="tactic">exact</span> <span class="id" title="var">A</span>.<br/>
* <span class="id" title="tactic">exact</span> <span class="id" title="var">A</span>.<br/>
* <span class="id" title="tactic">intros</span> <span class="id" title="var">a</span>. <span class="id" title="tactic">exact</span> (<span class="id" title="var">ua</span>^-1 <span class="id" title="var">μ</span>(<span class="id" title="var">_</span>,<span class="id" title="var">a</span>)).<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab10"></a><h2 class="section">Question 1</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Question_1</span>.<br/>
<br/>
<span class="id" title="keyword">Fact</span> <span class="id" title="var">hottbook_lemma_2_10_5</span> :<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">B</span> : <span class="id" title="var">A</span> → <span class="id" title="keyword">Type</span>) (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <span class="id" title="var">A</span>) (<span class="id" title="var">p</span> : <span class="id" title="var">x</span> = <span class="id" title="var">y</span>) (<span class="id" title="var">u</span> : <span class="id" title="var">B</span> <span class="id" title="var">x</span>),<br/>
<span class="id" title="var">transport</span> <span class="id" title="var">B</span> <span class="id" title="var">p</span> <span class="id" title="var">u</span> = <span class="id" title="var">ua</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> (<span class="id" title="var">ap</span> <span class="id" title="var">B</span> <span class="id" title="var">p</span>) <span class="id" title="var">u</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">Admitted</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <span class="id" title="var">q_3_1</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> : <span class="id" title="var">A</span>,<br/>
<span class="id" title="var">transport</span> <span class="id" title="var">P</span> (<span class="id" title="var">merid</span> <span class="id" title="var">b</span>) <span class="id" title="var">a</span> = <span class="id" title="var">μ</span>(<span class="id" title="var">a</span>,<span class="id" title="var">b</span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">hottbook_lemma_2_10_5</span>. <span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Susp_rec_beta_merid</span>.<br/>
<span class="comment">(* <span class="inlinecode"><span class="id" title="var">transport_path_universe</span></span> is the propositional computation<br/>
rule of univalence (see section 2.10 of the HoTT book) *)</span><br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">transport_path_universe</span>.<br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">idpath</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Question_1</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab11"></a><h2 class="section">Question 2</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Question_2</span>.<br/>
<br/>
<span class="id" title="keyword">Let</span> ψ '(<span class="id" title="var">q</span>, <span class="id" title="var">a</span>) := <span class="id" title="var">transport</span> <span class="id" title="var">P</span> (<span class="id" title="var">merid</span> <span class="id" title="var">a</span>) <span class="id" title="var">q</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <span class="id" title="var">path_</span>ψ<span class="id" title="var">_μ</span> :<br/>
ψ = <span class="id" title="var">μ</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="comment">(* <span class="inlinecode"><span class="id" title="var">equiv_path_arrow</span></span> is function extensionality *)</span><br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">equiv_path_arrow</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">p</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">p</span> <span class="id" title="keyword">as</span> (<span class="id" title="var">a</span>, <span class="id" title="var">b</span>).<br/>
<span class="id" title="tactic">unfold</span> ψ. <span class="id" title="tactic">apply</span> <span class="id" title="var">q_3_1</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <span class="id" title="var">q_3_2</span> :<br/>
{<span class="id" title="var">x</span> : <span class="id" title="var">Σ</span> <span class="id" title="var">A</span> & <span class="id" title="var">P</span> <span class="id" title="var">x</span>} ≃ <span class="id" title="var">Pushout</span> <span class="id" title="var">fst</span> <span class="id" title="var">μ</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">refine</span> (<span class="id" title="var">_</span> <span class="id" title="var">oE</span> (<span class="id" title="var">q_2_1</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>)).<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">path_</span>ψ<span class="id" title="var">_μ</span>.<br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">equiv_idmap</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Question_2</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab12"></a><h2 class="section">Question 3</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Let</span> <span class="id" title="var">s</span> := @<span class="id" title="var">span</span> <span class="id" title="var">A</span> (<span class="id" title="var">A</span> × <span class="id" title="var">A</span>) <span class="id" title="var">A</span> <span class="id" title="var">fst</span> <span class="id" title="var">μ</span>.<br/>
<span class="id" title="keyword">Let</span> <span class="id" title="var">s'</span> := @<span class="id" title="var">span</span> <span class="id" title="var">A</span> (<span class="id" title="var">A</span> × <span class="id" title="var">A</span>) <span class="id" title="var">A</span> <span class="id" title="var">fst</span> <span class="id" title="var">snd</span>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Question_3</span>.<br/>
<br/>
<span class="id" title="keyword">Let</span> <span class="id" title="var">ε</span> : <span class="id" title="var">A</span> × <span class="id" title="var">A</span> → <span class="id" title="var">A</span> × <span class="id" title="var">A</span> := <span class="id" title="keyword">fun</span> '(<span class="id" title="var">a</span>, <span class="id" title="var">b</span>) ⇒ (<span class="id" title="var">a</span>, <span class="id" title="var">μ</span>(<span class="id" title="var">a</span>,<span class="id" title="var">_</span>) <span class="id" title="var">b</span>).<br/>
<span class="id" title="keyword">Let</span> <span class="id" title="var">ε_inv</span> : <span class="id" title="var">A</span> × <span class="id" title="var">A</span> → <span class="id" title="var">A</span> × <span class="id" title="var">A</span> := <span class="id" title="keyword">fun</span> '(<span class="id" title="var">a</span>, <span class="id" title="var">b</span>) ⇒ (<span class="id" title="var">a</span>, <span class="id" title="var">μ</span>(<span class="id" title="var">a</span>,<span class="id" title="var">_</span>)^-1 <span class="id" title="var">b</span>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <span class="id" title="var">q_3_3</span> :<br/>
<span class="id" title="var">s</span> = <span class="id" title="var">s'</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="comment">(* We use the analysis of equality between spans *)</span><br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">path_span</span>. <span class="id" title="var">unshelve</span> <span class="id" title="tactic">econstructor</span>.<br/>
<span class="comment">(* Left equivalence *)</span><br/>
* <span class="id" title="tactic">exact</span> <span class="id" title="var">equiv_idmap</span>.<br/>
<span class="comment">(* Apex equivalence *)</span><br/>
* <span class="id" title="var">unshelve</span> <span class="id" title="tactic">econstructor</span>.<br/>
- <span class="id" title="tactic">exact</span> <span class="id" title="var">ε</span>.<br/>
- <span class="id" title="tactic">apply</span> <span class="id" title="var">isequiv_adjointify</span> <span class="id" title="keyword">with</span> <span class="id" title="var">ε_inv</span>.<br/>
+ <span class="id" title="tactic">unfold</span> <span class="id" title="var">Sect</span>. <span class="id" title="tactic">intros</span> [<span class="id" title="var">a</span> <span class="id" title="var">b</span>]. <span class="id" title="tactic">unfold</span> <span class="id" title="var">ε</span>, <span class="id" title="var">ε_inv</span>.<br/>
<span class="comment">(* μ(a,_) is an equivalence *)</span><br/>
<span class="id" title="tactic">rewrite</span> (<span class="id" title="var">eisretr</span> <span class="id" title="var">μ</span>(<span class="id" title="var">a</span>,<span class="id" title="var">_</span>)).<br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">idpath</span>.<br/>
+ <span class="id" title="tactic">unfold</span> <span class="id" title="var">Sect</span>. <span class="id" title="tactic">intros</span> [<span class="id" title="var">a</span> <span class="id" title="var">b</span>]. <span class="id" title="tactic">unfold</span> <span class="id" title="var">ε</span>, <span class="id" title="var">ε_inv</span>.<br/>
<span class="comment">(* μ(a,_) is an equivalence *)</span><br/>
<span class="id" title="tactic">rewrite</span> (<span class="id" title="var">eissect</span> <span class="id" title="var">μ</span>(<span class="id" title="var">a</span>,<span class="id" title="var">_</span>)).<br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">idpath</span>.<br/>
<span class="comment">(* Right equivalence *)</span><br/>
* <span class="id" title="tactic">exact</span> <span class="id" title="var">equiv_idmap</span>.<br/>
<span class="comment">(* Left coherence *)</span><br/>
* <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">equiv_path_arrow</span>. <span class="id" title="tactic">exact</span> <span class="id" title="var">idpath</span>.<br/>
<span class="comment">(* Right coherence *)</span><br/>
* <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">equiv_path_arrow</span>. <span class="id" title="tactic">exact</span> <span class="id" title="var">idpath</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Question_3</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab13"></a><h2 class="section">Question 4</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Question_4</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <span class="id" title="var">q_3_4</span> :<br/>
{<span class="id" title="var">x</span> : <span class="id" title="var">Σ</span> <span class="id" title="var">A</span> & <span class="id" title="var">P</span> <span class="id" title="var">x</span>} ≃ <span class="id" title="var">A</span> ∗ <span class="id" title="var">A</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">refine</span> (<span class="id" title="var">_</span> <span class="id" title="var">oE</span> <span class="id" title="var">q_3_2</span>).<br/>
<span class="comment">(* <span class="inlinecode"><span class="id" title="var">equiv_pushout</span></span> gives an analysis of equivalence between pushouts<br/>
as defined in the HoTT library, which will allow us to reduce it<br/>
to the analysis of equality between spans of <span class="inlinecode"><span class="id" title="var">q_3_3</span></span> *)</span><br/>
<span class="id" title="tactic">refine</span> (<span class="id" title="var">_</span> <span class="id" title="var">oE</span> <span class="id" title="var">equiv_pushout</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>).<br/>
* <span class="id" title="tactic">exact</span> <span class="id" title="var">equiv_idmap</span>.<br/>
<span class="comment">(* Left coherence... *)</span><br/>
* <span class="id" title="tactic">eapply</span> <span class="id" title="var">coherence_left</span>.<br/>
<span class="comment">(* ...and right coherence... *)</span><br/>
* <span class="id" title="tactic">eapply</span> <span class="id" title="var">coherence_right</span>.<br/>
<span class="id" title="var">Unshelve</span>.<br/>
<span class="comment">(* ...that we get from <span class="inlinecode"><span class="id" title="var">q_3_3</span></span> *)</span><br/>
<span class="id" title="tactic">apply</span> (<span class="id" title="var">path_span</span>^-1 <span class="id" title="var">q_3_3</span>).<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Question_4</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Exercise_3</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab14"></a><h1 class="section">Exercise 4</h1>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Exercise_4</span>.<br/>
<br/>
<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">pointed_scope</span>.<br/>
<br/>
<span class="id" title="keyword">Class</span> <span class="id" title="var">IsHType</span> (<span class="id" title="var">carrier</span> : <span class="id" title="var">pType</span>) := {<br/>
<span class="id" title="var">mult</span> : <span class="id" title="var">carrier</span> × <span class="id" title="var">carrier</span> → <span class="id" title="var">carrier</span>;<br/>
<span class="id" title="var">unit_left</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <span class="id" title="var">mult</span> (<span class="id" title="var">point</span> <span class="id" title="var">carrier</span>, <span class="id" title="var">x</span>) = <span class="id" title="var">x</span>;<br/>
<span class="id" title="var">unit_right</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <span class="id" title="var">mult</span> (<span class="id" title="var">x</span>, <span class="id" title="var">point</span> <span class="id" title="var">carrier</span>) = <span class="id" title="var">x</span>;<br/>
}.<br/>
<br/>
<span class="id" title="keyword">Class</span> <span class="id" title="var">IsConnected</span> <span class="id" title="var">A</span> := <span class="id" title="var">connected</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <span class="id" title="var">A</span>, |<span class="id" title="var">x</span> = <span class="id" title="var">y</span>|.<br/>
<br/>
<span class="id" title="keyword">Context</span> (<span class="id" title="var">A</span> : <span class="id" title="var">pType</span>) `{<span class="id" title="var">IsHType</span> <span class="id" title="var">A</span>} `{<span class="id" title="var">IsConnected</span> <span class="id" title="var">A</span>}.<br/>
<br/>
<span class="id" title="keyword">Notation</span> <span class="id" title="var">e</span> := (<span class="id" title="var">point</span> <span class="id" title="var">A</span>).<br/>
<span class="id" title="keyword">Notation</span> "'μ'" := <span class="id" title="var">mult</span>.<br/>
<span class="id" title="keyword">Notation</span> "'μ' ( a , b )" := (<span class="id" title="var">μ</span> (<span class="id" title="var">pair</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span>)) (<span class="id" title="var">format</span> "'μ' ( a , b )").<br/>
<span class="id" title="keyword">Notation</span> "'μ' ( a , '_' )" := (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ <span class="id" title="var">μ</span>(<span class="id" title="var">a</span>,<span class="id" title="var">x</span>)) (<span class="id" title="var">format</span> "'μ' ( a , _ )").<br/>
<span class="id" title="keyword">Notation</span> "'μ' ( '_' , a )" := (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ <span class="id" title="var">μ</span>(<span class="id" title="var">x</span>,<span class="id" title="var">a</span>)) (<span class="id" title="var">format</span> "'μ' ( _ , a )").<br/>
<br/>
</div>
<div class="doc">
<a name="lab15"></a><h2 class="section">Question 1</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Question_1</span>.<br/>
<br/>
</div>
<div class="doc">
I could not find a proper induction principle for the
propositional truncation in the HoTT library, thus we
assume the one seen in course.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Fact</span> <span class="id" title="var">merely_ind</span> :<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">P</span> : |<span class="id" title="var">A</span>| → <span class="id" title="keyword">Type</span>),<br/>
(<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <span class="id" title="var">A</span>, <span class="id" title="var">P</span> (<span class="id" title="var">tr</span> <span class="id" title="var">x</span>)) →<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : |<span class="id" title="var">A</span>|, <span class="id" title="var">P</span> <span class="id" title="var">x</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">Admitted</span>.<br/>
<br/>
</div>
<div class="doc">
A consequence of the induction principle is that
for any type B, assuming |B| amounts to assuming B.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Lemma</span> <span class="id" title="var">destruct_merely</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">B</span>,<br/>
|<span class="id" title="var">B</span>| → <span class="id" title="var">B</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">B</span>. <span class="id" title="tactic">apply</span> (<span class="id" title="var">merely_ind</span> <span class="id" title="var">B</span>). <span class="id" title="tactic">exact</span> <span class="id" title="var">idmap</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <span class="id" title="var">q_4_1</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <span class="id" title="var">A</span>,<br/>
|<span class="id" title="var">μ</span>(<span class="id" title="var">x</span>,<span class="id" title="var">_</span>) = <span class="id" title="var">idmap</span>| × |<span class="id" title="var">μ</span>(<span class="id" title="var">_</span>,<span class="id" title="var">x</span>) = <span class="id" title="var">idmap</span>|.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span>. <span class="id" title="tactic">split</span>.<br/>
* <span class="id" title="tactic">apply</span> <span class="id" title="var">tr</span>.<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">equiv_path_arrow</span>.<br/>
<span class="id" title="tactic">unfold</span> "==". <span class="id" title="tactic">intros</span> <span class="id" title="var">a</span>.<br/>
<span class="id" title="tactic">pose</span> <span class="id" title="var">proof</span> (<span class="id" title="var">connected</span> <span class="id" title="var">μ</span>(<span class="id" title="var">x</span>,<span class="id" title="var">a</span>) <span class="id" title="var">a</span>).<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">destruct_merely</span>.<br/>
* <span class="id" title="tactic">apply</span> <span class="id" title="var">tr</span>.<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">equiv_path_arrow</span>.<br/>
<span class="id" title="tactic">unfold</span> "==". <span class="id" title="tactic">intros</span> <span class="id" title="var">a</span>.<br/>
<span class="id" title="tactic">pose</span> <span class="id" title="var">proof</span> (<span class="id" title="var">connected</span> <span class="id" title="var">μ</span>(<span class="id" title="var">a</span>,<span class="id" title="var">x</span>) <span class="id" title="var">a</span>).<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">destruct_merely</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Question_1</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab16"></a><h2 class="section">Question 2</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Instance</span> <span class="id" title="var">isequiv_μ_left</span> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <span class="id" title="var">IsEquiv</span> <span class="id" title="var">μ</span>(<span class="id" title="var">x</span>,<span class="id" title="var">_</span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span>. <span class="id" title="tactic">destruct</span> (<span class="id" title="var">q_4_1</span> <span class="id" title="var">x</span>) <span class="id" title="keyword">as</span> [<span class="id" title="var">Hl</span> <span class="id" title="var">Hr</span>].<br/>
<span class="id" title="tactic">apply</span> (<span class="id" title="var">destruct_merely</span> <span class="id" title="var">_</span>) <span class="id" title="tactic">in</span> <span class="id" title="var">Hl</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hl</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">isequiv_idmap</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">Instance</span> <span class="id" title="var">isequiv_μ_right</span> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <span class="id" title="var">IsEquiv</span> <span class="id" title="var">μ</span>(<span class="id" title="var">_</span>,<span class="id" title="var">x</span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span>. <span class="id" title="tactic">destruct</span> (<span class="id" title="var">q_4_1</span> <span class="id" title="var">x</span>) <span class="id" title="keyword">as</span> [<span class="id" title="var">Hl</span> <span class="id" title="var">Hr</span>].<br/>
<span class="id" title="tactic">apply</span> (<span class="id" title="var">destruct_merely</span> <span class="id" title="var">_</span>) <span class="id" title="tactic">in</span> <span class="id" title="var">Hr</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hr</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">isequiv_idmap</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab17"></a><h2 class="section">Question 3</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Question_3</span>.<br/>
<br/>
</div>
<div class="doc">
Short name for the equivalence obtained in <span class="inlinecode"><span class="id" title="var">q_3_4</span></span>,
instanciated to our H-type <span class="inlinecode"><span class="id" title="var">A</span></span>.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Let</span> <span class="id" title="var">ε</span> := (<span class="id" title="var">q_3_4</span> <span class="id" title="var">A</span> <span class="id" title="var">μ</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>).<br/>
<br/>
</div>
<div class="doc">
First, we need a few definitions to make our operations
live in the pointed universe.
<div class="paragraph"> </div>
<a name="lab18"></a><h3 class="section">Pointed Σ-types</h3>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">psig</span> {<span class="id" title="var">X</span> : <span class="id" title="var">pType</span>}<br/>
(<span class="id" title="var">P</span> : <span class="id" title="var">X</span> → <span class="id" title="keyword">Type</span>) `{<span class="id" title="var">pointP</span> : <span class="id" title="var">IsPointed</span> (<span class="id" title="var">P</span> (<span class="id" title="var">point</span> <span class="id" title="var">X</span>))} : <span class="id" title="var">pType</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">unshelve</span> <span class="id" title="tactic">econstructor</span>.<br/>
* <span class="id" title="tactic">exact</span> (<span class="id" title="var">sig</span> <span class="id" title="var">P</span>).<br/>
* <span class="id" title="tactic">apply</span> <span class="id" title="var">ispointed_sigma</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab19"></a><h3 class="section">Pointed joins</h3>
<div class="paragraph"> </div>
Note that we do not use a general definition for pointed joins:
instead we only define the pointed <span class="inlinecode"><span class="id" title="var">A</span></span> <span class="inlinecode">∗</span> <span class="inlinecode"><span class="id" title="var">A</span></span>, so that we can choose
a specific basepoint (this will allow us to prove ε to be a
pointed equivalence, and not just a "mere" equivalence.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">pjoin</span> : <span class="id" title="var">pType</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">unshelve</span> <span class="id" title="tactic">econstructor</span>.<br/>
* <span class="id" title="tactic">exact</span> (<span class="id" title="var">A</span> ∗ <span class="id" title="var">A</span>).<br/>
* <span class="id" title="tactic">exact</span> (<span class="id" title="var">ε</span> (<span class="id" title="var">North</span>; <span class="id" title="var">e</span>)).<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab20"></a><h3 class="section">Fiber sequences</h3>
<div class="paragraph"> </div>
We also need to define fiber sequences, since they are
not implemented in the HoTT library.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">fib</span> {<span class="id" title="var">X</span> <span class="id" title="var">Y</span> : <span class="id" title="var">pType</span>} (<span class="id" title="var">f</span> : <span class="id" title="var">X</span> ->* <span class="id" title="var">Y</span>) : <span class="id" title="var">pType</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">unshelve</span> <span class="id" title="tactic">econstructor</span>.<br/>
* <span class="id" title="tactic">exact</span> (<span class="id" title="var">hfiber</span> <span class="id" title="var">f</span> (<span class="id" title="var">point</span> <span class="id" title="var">Y</span>)).<br/>
* <span class="id" title="tactic">exact</span> (<span class="id" title="var">point</span> <span class="id" title="var">X</span>; <span class="id" title="var">point_eq</span> <span class="id" title="var">f</span>).<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">fib_proj</span> {<span class="id" title="var">X</span> <span class="id" title="var">Y</span> : <span class="id" title="var">pType</span>} (<span class="id" title="var">f</span> : <span class="id" title="var">X</span> ->* <span class="id" title="var">Y</span>) : <span class="id" title="var">fib</span> <span class="id" title="var">f</span> ->* <span class="id" title="var">X</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">unshelve</span> <span class="id" title="tactic">econstructor</span>.<br/>
* <span class="id" title="tactic">intros</span> <span class="id" title="var">x</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">x</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">x</span> <span class="id" title="var">_</span>]. <span class="id" title="tactic">exact</span> <span class="id" title="var">x</span>.<br/>
* <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">exact</span> <span class="id" title="var">idpath</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">IsFiberSeq</span> {<span class="id" title="var">X</span> <span class="id" title="var">Y</span> <span class="id" title="var">Z</span> : <span class="id" title="var">pType</span>} (<span class="id" title="var">f</span> : <span class="id" title="var">X</span> ->* <span class="id" title="var">Y</span>) (<span class="id" title="var">g</span> : <span class="id" title="var">Y</span> ->* <span class="id" title="var">Z</span>) :=<br/>
{<span class="id" title="var">p</span> : <span class="id" title="var">X</span> = <span class="id" title="var">fib</span> <span class="id" title="var">g</span> & <span class="id" title="var">transport</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">B</span> ⇒ <span class="id" title="var">B</span> ->* <span class="id" title="var">Y</span>) <span class="id" title="var">p</span> <span class="id" title="var">f</span> = <span class="id" title="var">fib_proj</span> <span class="id" title="var">g</span>}.<br/>
<br/>
<span class="id" title="keyword">Record</span> <span class="id" title="var">FiberSeq</span> (<span class="id" title="var">X</span> <span class="id" title="var">Y</span> <span class="id" title="var">Z</span> : <span class="id" title="var">pType</span>) := {<br/>
<span class="id" title="var">fiberseq_proj</span> : <span class="id" title="var">X</span> ->* <span class="id" title="var">Y</span>;<br/>
<span class="id" title="var">fiberseq_map</span> : <span class="id" title="var">Y</span> ->* <span class="id" title="var">Z</span>;<br/>
<span class="id" title="var">isfiberseq</span> : <span class="id" title="var">IsFiberSeq</span> <span class="id" title="var">fiberseq_proj</span> <span class="id" title="var">fiberseq_map</span>;<br/>
}.<br/>
<br/>
</div>
<div class="doc">
One can make a fiber sequence from a fibration, i.e.
a type family <span class="inlinecode"><span class="id" title="var">C</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" title="var">X</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="keyword">Type</span></span>, whenever <span class="inlinecode"><span class="id" title="var">X</span></span> is pointed
by <span class="inlinecode">∗</span> and <span class="inlinecode"><span class="id" title="var">C</span>(∗)</span> is pointed as well.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Proposition</span> <span class="id" title="var">fibration_fiberseq</span> {<span class="id" title="var">X</span> : <span class="id" title="var">pType</span>}<br/>
(<span class="id" title="var">C</span> : <span class="id" title="var">X</span> → <span class="id" title="keyword">Type</span>) `{<span class="id" title="var">ispointed_C</span> : <span class="id" title="var">IsPointed</span> (<span class="id" title="var">C</span> (<span class="id" title="var">point</span> <span class="id" title="var">X</span>))} :<br/>
@<span class="id" title="var">FiberSeq</span> {| <span class="id" title="var">pointed_type</span> := <span class="id" title="var">C</span> (<span class="id" title="var">point</span> <span class="id" title="var">X</span>);<br/>
<span class="id" title="var">ispointed_type</span> := <span class="id" title="var">ispointed_C</span> |}<br/>
(<span class="id" title="var">psig</span> <span class="id" title="var">C</span>) <span class="id" title="var">X</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">Admitted</span>.<br/>
<br/>
</div>
<div class="doc">
The trick then is to define the fibration <span class="inlinecode"><span class="id" title="var">C</span></span> so that
it coincides with the <span class="inlinecode"><span class="id" title="var">P</span></span> from exercise 3.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">C</span> : <span class="id" title="var">psusp</span> <span class="id" title="var">A</span> → <span class="id" title="keyword">Type</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">srapply</span> <span class="id" title="var">Susp_rec</span>; <span class="id" title="tactic">simpl</span>.<br/>
* <span class="id" title="tactic">exact</span> <span class="id" title="var">A</span>.<br/>
* <span class="id" title="tactic">exact</span> <span class="id" title="var">A</span>.<br/>
* <span class="id" title="tactic">intros</span> <span class="id" title="var">a</span>. <span class="id" title="tactic">exact</span> (<span class="id" title="var">ua</span>^-1 <span class="id" title="var">μ</span>(<span class="id" title="var">_</span>,<span class="id" title="var">a</span>)).<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">Instance</span> <span class="id" title="var">ispointed_C</span> : <span class="id" title="var">IsPointed</span> (<span class="id" title="var">C</span> (<span class="id" title="var">point</span> (<span class="id" title="var">psusp</span> <span class="id" title="var">A</span>))) := <span class="id" title="var">e</span>.<br/>
<br/>
</div>
<div class="doc">
We use C to get a fiber sequence
<div class="paragraph"> </div>
<span class="inlinecode"><span class="id" title="var">A</span></span> <span class="inlinecode">→∗</span> <span class="inlinecode">(<span class="id" title="var">x</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" title="var">ΣA</span>)</span> <span class="inlinecode">×</span> <span class="inlinecode"><span class="id" title="var">C</span>(<span class="id" title="var">x</span>)</span> <span class="inlinecode">→∗</span> <span class="inlinecode"><span class="id" title="var">ΣA</span></span>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Let</span> σ := <span class="id" title="var">fibration_fiberseq</span> <span class="id" title="var">C</span>.<br/>
<br/>
</div>
<div class="doc">
Intuitively, it only remains to "rewrite" <span class="inlinecode">(<span class="id" title="var">x</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" title="var">ΣA</span>)</span> <span class="inlinecode">×</span> <span class="inlinecode"><span class="id" title="var">C</span>(<span class="id" title="var">x</span>)</span>
into <span class="inlinecode"><span class="id" title="var">A</span></span> <span class="inlinecode">∗</span> <span class="inlinecode"><span class="id" title="var">A</span></span> using ε to conclude. But we cannot do this
directly because we only have an equivalence of non-pointed
types, but sequences are defined on pointed types.
<div class="paragraph"> </div>
So we start by building a pointed equivalence from ε.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">ε_pointed</span> :<br/>
<span class="id" title="var">psig</span> <span class="id" title="var">C</span> ≃* <span class="id" title="var">pjoin</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">unshelve</span> <span class="id" title="tactic">econstructor</span>.<br/>
* <span class="id" title="tactic">exact</span> (<span class="id" title="var">pmap_from_point</span> <span class="id" title="var">ε</span> (<span class="id" title="var">North</span>; <span class="id" title="var">e</span>)).<br/>
* <span class="id" title="tactic">apply</span> <span class="id" title="var">isequiv_adjointify</span> <span class="id" title="keyword">with</span> <span class="id" title="var">ε</span>^-1.<br/>
- <span class="id" title="tactic">apply</span> <span class="id" title="var">eisretr</span>.<br/>
- <span class="id" title="tactic">apply</span> <span class="id" title="var">eissect</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
</div>
<div class="doc">
Then we rewrite, and that's it!
<div class="paragraph"> </div>
In fact we have to cheat a little bit: we use the lemma <span class="inlinecode"><span class="id" title="var">path_ptype</span></span>
to turn the pointed equivalence into a path in the universe of
pointed types, which follows from univalence with a bit of work.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Lemma</span> <span class="id" title="var">q_4_3</span> :<br/>
@<span class="id" title="var">FiberSeq</span> <span class="id" title="var">A</span> (<span class="id" title="var">pjoin</span>) (<span class="id" title="var">psusp</span> <span class="id" title="var">A</span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- (<span class="id" title="var">path_ptype</span> <span class="id" title="var">ε_pointed</span>).<br/>
<span class="id" title="tactic">exact</span> σ.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Question_3</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Exercise_4</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab21"></a><h1 class="section">Exercise 5</h1>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Exercise_5</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab22"></a><h2 class="section">Question 1</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Question_1</span>.<br/>
<br/>
</div>
<div class="doc">
<span class="inlinecode"><span class="id" title="var">S1</span></span> is the inductive definition of the circle, not to be confused
with <span class="inlinecode"><span class="id" title="var">S</span>^1</span> which is the definition based on iterated suspension
(though the two definitions are equivalent).
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Definition</span> ψ' : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <span class="id" title="var">S1</span>, <span class="id" title="var">x</span> = <span class="id" title="var">x</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="comment">(* By induction on circles *)</span><br/>
<span class="id" title="var">srapply</span> <span class="id" title="var">S1_ind</span>; <span class="id" title="tactic">simpl</span>.<br/>
* <span class="id" title="tactic">exact</span> <span class="id" title="var">loop</span>.<br/>
* <span class="comment">(* Lemma 2.11.2 of the HoTT book *)</span><br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">transport_paths_lr</span>.<br/>
<span class="id" title="var">hott_simpl</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Question_1</span>.<br/>