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@@ -13,7 +13,7 @@
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-pre > code.sourceCode > span { text-indent: -5em; padding-left: 5em; }
+pre > code.sourceCode > span { display: inline-block; text-indent: -5em; padding-left: 5em; }
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 code span.bn { color: #40a070; } /* BaseN */
-code span.bu { } /* BuiltIn */
+code span.bu { color: #008000; } /* BuiltIn */
 code span.cf { color: #007020; font-weight: bold; } /* ControlFlow */
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@@ -63,7 +63,7 @@ code span.er { color: #ff0000; font-weight: bold; } /* Error */
 code span.ex { } /* Extension */
 code span.fl { color: #40a070; } /* Float */
 code span.fu { color: #06287e; } /* Function */
-code span.im { } /* Import */
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@@ -85,103 +85,243 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 <div id="everything">
 <nav id="TOC">
 <ul>
-<li><a href="#lecture-1-and-so-it-begins"><span class="toc-section-number">1</span> Lecture 1: …And so It Begins</a>
+<li><a href="#lecture-1-and-so-it-begins"
+id="toc-lecture-1-and-so-it-begins"><span
+class="toc-section-number">1</span> Lecture 1: …And so It Begins</a>
 <ul>
-<li><a href="#about-the-course"><span class="toc-section-number">1.1</span> About the Course</a></li>
-<li><a href="#read-these"><span class="toc-section-number">1.2</span> Read These</a></li>
-<li><a href="#haskell"><span class="toc-section-number">1.3</span> Haskell</a></li>
-<li><a href="#running-haskell"><span class="toc-section-number">1.4</span> Running Haskell</a></li>
-<li><a href="#lets-start"><span class="toc-section-number">1.5</span> Let’s Start!</a></li>
-<li><a href="#expressions-and-types"><span class="toc-section-number">1.6</span> Expressions and Types</a></li>
-<li><a href="#the-structure-of-a-haskell-program"><span class="toc-section-number">1.7</span> The Structure of a Haskell Program</a></li>
-<li><a href="#working-with-examples"><span class="toc-section-number">1.8</span> Working with Examples</a></li>
-<li><a href="#how-do-i-get-anything-done"><span class="toc-section-number">1.9</span> How Do I Get Anything Done?</a></li>
-<li><a href="#all-together-now"><span class="toc-section-number">1.10</span> All Together Now!</a></li>
-<li><a href="#a-word-about-indentation"><span class="toc-section-number">1.11</span> A Word About Indentation</a></li>
-<li><a href="#quiz"><span class="toc-section-number">1.12</span> Quiz</a></li>
-<li><a href="#working-on-the-exercises"><span class="toc-section-number">1.13</span> Working on the Exercises</a></li>
-<li><a href="#exercises"><span class="toc-section-number">1.14</span> Exercises</a></li>
+<li><a href="#about-the-course" id="toc-about-the-course"><span
+class="toc-section-number">1.1</span> About the Course</a></li>
+<li><a href="#read-these" id="toc-read-these"><span
+class="toc-section-number">1.2</span> Read These</a></li>
+<li><a href="#haskell" id="toc-haskell"><span
+class="toc-section-number">1.3</span> Haskell</a></li>
+<li><a href="#running-haskell" id="toc-running-haskell"><span
+class="toc-section-number">1.4</span> Running Haskell</a></li>
+<li><a href="#lets-start" id="toc-lets-start"><span
+class="toc-section-number">1.5</span> Let’s Start!</a></li>
+<li><a href="#expressions-and-types"
+id="toc-expressions-and-types"><span
+class="toc-section-number">1.6</span> Expressions and Types</a></li>
+<li><a href="#the-structure-of-a-haskell-program"
+id="toc-the-structure-of-a-haskell-program"><span
+class="toc-section-number">1.7</span> The Structure of a Haskell
+Program</a></li>
+<li><a href="#working-with-examples"
+id="toc-working-with-examples"><span
+class="toc-section-number">1.8</span> Working with Examples</a></li>
+<li><a href="#how-do-i-get-anything-done"
+id="toc-how-do-i-get-anything-done"><span
+class="toc-section-number">1.9</span> How Do I Get Anything
+Done?</a></li>
+<li><a href="#all-together-now" id="toc-all-together-now"><span
+class="toc-section-number">1.10</span> All Together Now!</a></li>
+<li><a href="#a-word-about-indentation"
+id="toc-a-word-about-indentation"><span
+class="toc-section-number">1.11</span> A Word About Indentation</a></li>
+<li><a href="#quiz" id="toc-quiz"><span
+class="toc-section-number">1.12</span> Quiz</a></li>
+<li><a href="#working-on-the-exercises"
+id="toc-working-on-the-exercises"><span
+class="toc-section-number">1.13</span> Working on the Exercises</a></li>
+<li><a href="#exercises" id="toc-exercises"><span
+class="toc-section-number">1.14</span> Exercises</a></li>
 </ul></li>
-<li><a href="#lecture-2-either-you-die-a-hero"><span class="toc-section-number">2</span> Lecture 2: Either You Die a Hero…</a>
+<li><a href="#lecture-2-either-you-die-a-hero"
+id="toc-lecture-2-either-you-die-a-hero"><span
+class="toc-section-number">2</span> Lecture 2: Either You Die a
+Hero…</a>
 <ul>
-<li><a href="#recursion-and-helper-functions"><span class="toc-section-number">2.1</span> Recursion and Helper Functions</a></li>
-<li><a href="#guards"><span class="toc-section-number">2.2</span> Guards</a></li>
-<li><a href="#lists"><span class="toc-section-number">2.3</span> Lists</a></li>
-<li><a href="#a-word-about-immutability-1"><span class="toc-section-number">2.4</span> A Word About Immutability</a></li>
-<li><a href="#a-word-about-type-inference-and-polymorphism"><span class="toc-section-number">2.5</span> A Word About Type Inference and Polymorphism</a></li>
-<li><a href="#the-maybe-type"><span class="toc-section-number">2.6</span> The <code>Maybe</code> Type</a></li>
-<li><a href="#sidenote-constructors"><span class="toc-section-number">2.7</span> Sidenote: Constructors</a></li>
-<li><a href="#the-either-type"><span class="toc-section-number">2.8</span> The <code>Either</code> type</a></li>
-<li><a href="#the-case-of-expression"><span class="toc-section-number">2.9</span> The case-of Expression</a></li>
-<li><a href="#recap-pattern-matching"><span class="toc-section-number">2.10</span> Recap: Pattern Matching</a></li>
-<li><a href="#quiz-1"><span class="toc-section-number">2.11</span> Quiz</a></li>
-<li><a href="#exercises-1"><span class="toc-section-number">2.12</span> Exercises</a></li>
+<li><a href="#recursion-and-helper-functions"
+id="toc-recursion-and-helper-functions"><span
+class="toc-section-number">2.1</span> Recursion and Helper
+Functions</a></li>
+<li><a href="#guards" id="toc-guards"><span
+class="toc-section-number">2.2</span> Guards</a></li>
+<li><a href="#lists" id="toc-lists"><span
+class="toc-section-number">2.3</span> Lists</a></li>
+<li><a href="#a-word-about-immutability-1"
+id="toc-a-word-about-immutability-1"><span
+class="toc-section-number">2.4</span> A Word About Immutability</a></li>
+<li><a href="#a-word-about-type-inference-and-polymorphism"
+id="toc-a-word-about-type-inference-and-polymorphism"><span
+class="toc-section-number">2.5</span> A Word About Type Inference and
+Polymorphism</a></li>
+<li><a href="#the-maybe-type" id="toc-the-maybe-type"><span
+class="toc-section-number">2.6</span> The <code>Maybe</code>
+Type</a></li>
+<li><a href="#sidenote-constructors"
+id="toc-sidenote-constructors"><span
+class="toc-section-number">2.7</span> Sidenote: Constructors</a></li>
+<li><a href="#the-either-type" id="toc-the-either-type"><span
+class="toc-section-number">2.8</span> The <code>Either</code>
+type</a></li>
+<li><a href="#the-case-of-expression"
+id="toc-the-case-of-expression"><span
+class="toc-section-number">2.9</span> The case-of Expression</a></li>
+<li><a href="#recap-pattern-matching"
+id="toc-recap-pattern-matching"><span
+class="toc-section-number">2.10</span> Recap: Pattern Matching</a></li>
+<li><a href="#quiz-1" id="toc-quiz-1"><span
+class="toc-section-number">2.11</span> Quiz</a></li>
+<li><a href="#exercises-1" id="toc-exercises-1"><span
+class="toc-section-number">2.12</span> Exercises</a></li>
 </ul></li>
-<li><a href="#lecture-3-catamorphic"><span class="toc-section-number">3</span> Lecture 3: Catamorphic</a>
+<li><a href="#lecture-3-catamorphic"
+id="toc-lecture-3-catamorphic"><span class="toc-section-number">3</span>
+Lecture 3: Catamorphic</a>
 <ul>
-<li><a href="#functional-programming-at-last"><span class="toc-section-number">3.1</span> Functional Programming, at Last</a></li>
-<li><a href="#partial-application"><span class="toc-section-number">3.2</span> Partial Application</a></li>
-<li><a href="#prefix-and-infix-notations"><span class="toc-section-number">3.3</span> Prefix and Infix Notations</a></li>
-<li><a href="#lambdas"><span class="toc-section-number">3.4</span> Lambdas</a></li>
-<li><a href="#sidenote-the-.-and-operators"><span class="toc-section-number">3.5</span> Sidenote: The <code>.</code> and <code>$</code> Operators</a></li>
-<li><a href="#example-rewriting-whatfollows"><span class="toc-section-number">3.6</span> Example: Rewriting <code>whatFollows</code></a></li>
-<li><a href="#more-functional-list-wrangling-examples"><span class="toc-section-number">3.7</span> More Functional List Wrangling Examples</a></li>
-<li><a href="#lists-and-recursion"><span class="toc-section-number">3.8</span> Lists and Recursion</a></li>
-<li><a href="#something-fun-list-comprehensions"><span class="toc-section-number">3.9</span> Something Fun: List Comprehensions</a></li>
-<li><a href="#something-fun-custom-operators"><span class="toc-section-number">3.10</span> Something Fun: Custom Operators</a></li>
-<li><a href="#something-useful-typed-holes"><span class="toc-section-number">3.11</span> Something Useful: Typed Holes</a></li>
-<li><a href="#quiz-2"><span class="toc-section-number">3.12</span> Quiz</a></li>
-<li><a href="#exercises-2"><span class="toc-section-number">3.13</span> Exercises</a></li>
+<li><a href="#functional-programming-at-last"
+id="toc-functional-programming-at-last"><span
+class="toc-section-number">3.1</span> Functional Programming, at
+Last</a></li>
+<li><a href="#partial-application" id="toc-partial-application"><span
+class="toc-section-number">3.2</span> Partial Application</a></li>
+<li><a href="#prefix-and-infix-notations"
+id="toc-prefix-and-infix-notations"><span
+class="toc-section-number">3.3</span> Prefix and Infix
+Notations</a></li>
+<li><a href="#lambdas" id="toc-lambdas"><span
+class="toc-section-number">3.4</span> Lambdas</a></li>
+<li><a href="#sidenote-the-.-and-operators"
+id="toc-sidenote-the-.-and-operators"><span
+class="toc-section-number">3.5</span> Sidenote: The <code>.</code> and
+<code>$</code> Operators</a></li>
+<li><a href="#example-rewriting-whatfollows"
+id="toc-example-rewriting-whatfollows"><span
+class="toc-section-number">3.6</span> Example: Rewriting
+<code>whatFollows</code></a></li>
+<li><a href="#more-functional-list-wrangling-examples"
+id="toc-more-functional-list-wrangling-examples"><span
+class="toc-section-number">3.7</span> More Functional List Wrangling
+Examples</a></li>
+<li><a href="#lists-and-recursion" id="toc-lists-and-recursion"><span
+class="toc-section-number">3.8</span> Lists and Recursion</a></li>
+<li><a href="#something-fun-list-comprehensions"
+id="toc-something-fun-list-comprehensions"><span
+class="toc-section-number">3.9</span> Something Fun: List
+Comprehensions</a></li>
+<li><a href="#something-fun-custom-operators"
+id="toc-something-fun-custom-operators"><span
+class="toc-section-number">3.10</span> Something Fun: Custom
+Operators</a></li>
+<li><a href="#something-useful-typed-holes"
+id="toc-something-useful-typed-holes"><span
+class="toc-section-number">3.11</span> Something Useful: Typed
+Holes</a></li>
+<li><a href="#quiz-2" id="toc-quiz-2"><span
+class="toc-section-number">3.12</span> Quiz</a></li>
+<li><a href="#exercises-2" id="toc-exercises-2"><span
+class="toc-section-number">3.13</span> Exercises</a></li>
 </ul></li>
-<li><a href="#lecture-4-real-classy"><span class="toc-section-number">4</span> Lecture 4: Real Classy</a>
+<li><a href="#lecture-4-real-classy"
+id="toc-lecture-4-real-classy"><span class="toc-section-number">4</span>
+Lecture 4: Real Classy</a>
 <ul>
-<li><a href="#sidenote-tuples"><span class="toc-section-number">4.1</span> Sidenote: Tuples</a></li>
-<li><a href="#interlude-folding"><span class="toc-section-number">4.2</span> Interlude: Folding</a></li>
-<li><a href="#type-classes"><span class="toc-section-number">4.3</span> Type Classes</a></li>
-<li><a href="#type-constraints"><span class="toc-section-number">4.4</span> Type Constraints</a></li>
-<li><a href="#standard-type-classes"><span class="toc-section-number">4.5</span> Standard Type Classes</a></li>
-<li><a href="#more-data-structures"><span class="toc-section-number">4.6</span> More Data Structures</a></li>
-<li><a href="#reading-docs"><span class="toc-section-number">4.7</span> Reading Docs</a></li>
-<li><a href="#quiz-3"><span class="toc-section-number">4.8</span> Quiz</a></li>
-<li><a href="#exercises-3"><span class="toc-section-number">4.9</span> Exercises</a></li>
+<li><a href="#sidenote-tuples" id="toc-sidenote-tuples"><span
+class="toc-section-number">4.1</span> Sidenote: Tuples</a></li>
+<li><a href="#interlude-folding" id="toc-interlude-folding"><span
+class="toc-section-number">4.2</span> Interlude: Folding</a></li>
+<li><a href="#type-classes" id="toc-type-classes"><span
+class="toc-section-number">4.3</span> Type Classes</a></li>
+<li><a href="#type-constraints" id="toc-type-constraints"><span
+class="toc-section-number">4.4</span> Type Constraints</a></li>
+<li><a href="#standard-type-classes"
+id="toc-standard-type-classes"><span
+class="toc-section-number">4.5</span> Standard Type Classes</a></li>
+<li><a href="#more-data-structures" id="toc-more-data-structures"><span
+class="toc-section-number">4.6</span> More Data Structures</a></li>
+<li><a href="#reading-docs" id="toc-reading-docs"><span
+class="toc-section-number">4.7</span> Reading Docs</a></li>
+<li><a href="#quiz-3" id="toc-quiz-3"><span
+class="toc-section-number">4.8</span> Quiz</a></li>
+<li><a href="#exercises-3" id="toc-exercises-3"><span
+class="toc-section-number">4.9</span> Exercises</a></li>
 </ul></li>
-<li><a href="#lecture-5-you-need-string-for-a-knot"><span class="toc-section-number">5</span> Lecture 5: You Need String for a Knot</a>
+<li><a href="#lecture-5-you-need-string-for-a-knot"
+id="toc-lecture-5-you-need-string-for-a-knot"><span
+class="toc-section-number">5</span> Lecture 5: You Need String for a
+Knot</a>
 <ul>
-<li><a href="#algebraic-datatypes"><span class="toc-section-number">5.1</span> Algebraic Datatypes</a></li>
-<li><a href="#type-parameters"><span class="toc-section-number">5.2</span> Type Parameters</a></li>
-<li><a href="#recursive-types"><span class="toc-section-number">5.3</span> Recursive Types</a></li>
-<li><a href="#record-syntax"><span class="toc-section-number">5.4</span> Record Syntax</a></li>
-<li><a href="#algebraic-datatypes-summary"><span class="toc-section-number">5.5</span> Algebraic Datatypes: Summary</a></li>
-<li><a href="#sidenote-other-ways-of-defining-types"><span class="toc-section-number">5.6</span> Sidenote: Other Ways of Defining Types</a></li>
-<li><a href="#how-do-algebraic-datatypes-work"><span class="toc-section-number">5.7</span> How Do Algebraic Datatypes Work?</a></li>
-<li><a href="#quiz-4"><span class="toc-section-number">5.8</span> Quiz</a></li>
-<li><a href="#exercises-4"><span class="toc-section-number">5.9</span> Exercises</a></li>
+<li><a href="#algebraic-datatypes" id="toc-algebraic-datatypes"><span
+class="toc-section-number">5.1</span> Algebraic Datatypes</a></li>
+<li><a href="#type-parameters" id="toc-type-parameters"><span
+class="toc-section-number">5.2</span> Type Parameters</a></li>
+<li><a href="#recursive-types" id="toc-recursive-types"><span
+class="toc-section-number">5.3</span> Recursive Types</a></li>
+<li><a href="#record-syntax" id="toc-record-syntax"><span
+class="toc-section-number">5.4</span> Record Syntax</a></li>
+<li><a href="#algebraic-datatypes-summary"
+id="toc-algebraic-datatypes-summary"><span
+class="toc-section-number">5.5</span> Algebraic Datatypes:
+Summary</a></li>
+<li><a href="#sidenote-other-ways-of-defining-types"
+id="toc-sidenote-other-ways-of-defining-types"><span
+class="toc-section-number">5.6</span> Sidenote: Other Ways of Defining
+Types</a></li>
+<li><a href="#how-do-algebraic-datatypes-work"
+id="toc-how-do-algebraic-datatypes-work"><span
+class="toc-section-number">5.7</span> How Do Algebraic Datatypes
+Work?</a></li>
+<li><a href="#quiz-4" id="toc-quiz-4"><span
+class="toc-section-number">5.8</span> Quiz</a></li>
+<li><a href="#exercises-4" id="toc-exercises-4"><span
+class="toc-section-number">5.9</span> Exercises</a></li>
 </ul></li>
-<li><a href="#lecture-6-working-class-hero"><span class="toc-section-number">6</span> Lecture 6: Working Class Hero</a>
+<li><a href="#lecture-6-working-class-hero"
+id="toc-lecture-6-working-class-hero"><span
+class="toc-section-number">6</span> Lecture 6: Working Class Hero</a>
 <ul>
-<li><a href="#syntax-of-classes-and-instances"><span class="toc-section-number">6.1</span> Syntax of Classes and Instances</a></li>
-<li><a href="#default-implementations"><span class="toc-section-number">6.2</span> Default Implementations</a></li>
-<li><a href="#useful-stuff"><span class="toc-section-number">6.3</span> Useful Stuff</a></li>
-<li><a href="#hierarchies"><span class="toc-section-number">6.4</span> Hierarchies</a></li>
-<li><a href="#quiz-5"><span class="toc-section-number">6.5</span> Quiz</a></li>
-<li><a href="#exercises-5"><span class="toc-section-number">6.6</span> Exercises</a></li>
+<li><a href="#syntax-of-classes-and-instances"
+id="toc-syntax-of-classes-and-instances"><span
+class="toc-section-number">6.1</span> Syntax of Classes and
+Instances</a></li>
+<li><a href="#default-implementations"
+id="toc-default-implementations"><span
+class="toc-section-number">6.2</span> Default Implementations</a></li>
+<li><a href="#useful-stuff" id="toc-useful-stuff"><span
+class="toc-section-number">6.3</span> Useful Stuff</a></li>
+<li><a href="#hierarchies" id="toc-hierarchies"><span
+class="toc-section-number">6.4</span> Hierarchies</a></li>
+<li><a href="#quiz-5" id="toc-quiz-5"><span
+class="toc-section-number">6.5</span> Quiz</a></li>
+<li><a href="#exercises-5" id="toc-exercises-5"><span
+class="toc-section-number">6.6</span> Exercises</a></li>
 </ul></li>
-<li><a href="#lecture-7-new-constellations"><span class="toc-section-number">7</span> Lecture 7: New Constellations</a>
+<li><a href="#lecture-7-new-constellations"
+id="toc-lecture-7-new-constellations"><span
+class="toc-section-number">7</span> Lecture 7: New Constellations</a>
 <ul>
-<li><a href="#modeling-with-boxes"><span class="toc-section-number">7.1</span> Modeling with Boxes</a></li>
-<li><a href="#modeling-with-cases"><span class="toc-section-number">7.2</span> Modeling with Cases</a></li>
-<li><a href="#monoids"><span class="toc-section-number">7.3</span> Monoids</a></li>
-<li><a href="#open-and-closed-abstractions"><span class="toc-section-number">7.4</span> Open and Closed Abstractions</a></li>
-<li><a href="#modeling-with-languages"><span class="toc-section-number">7.5</span> Modeling with Languages</a></li>
-<li><a href="#exercises-6"><span class="toc-section-number">7.6</span> Exercises</a></li>
+<li><a href="#modeling-with-boxes" id="toc-modeling-with-boxes"><span
+class="toc-section-number">7.1</span> Modeling with Boxes</a></li>
+<li><a href="#modeling-with-cases" id="toc-modeling-with-cases"><span
+class="toc-section-number">7.2</span> Modeling with Cases</a></li>
+<li><a href="#monoids" id="toc-monoids"><span
+class="toc-section-number">7.3</span> Monoids</a></li>
+<li><a href="#open-and-closed-abstractions"
+id="toc-open-and-closed-abstractions"><span
+class="toc-section-number">7.4</span> Open and Closed
+Abstractions</a></li>
+<li><a href="#modeling-with-languages"
+id="toc-modeling-with-languages"><span
+class="toc-section-number">7.5</span> Modeling with Languages</a></li>
+<li><a href="#exercises-6" id="toc-exercises-6"><span
+class="toc-section-number">7.6</span> Exercises</a></li>
 </ul></li>
-<li><a href="#lecture-8-the-aftertaste"><span class="toc-section-number">8</span> Lecture 8: The Aftertaste</a>
+<li><a href="#lecture-8-the-aftertaste"
+id="toc-lecture-8-the-aftertaste"><span
+class="toc-section-number">8</span> Lecture 8: The Aftertaste</a>
 <ul>
-<li><a href="#a-taste-of-io"><span class="toc-section-number">8.1</span> A Taste of IO</a></li>
-<li><a href="#summary"><span class="toc-section-number">8.2</span> Summary</a></li>
-<li><a href="#what-next"><span class="toc-section-number">8.3</span> What Next?</a></li>
-<li><a href="#final-project-graphics"><span class="toc-section-number">8.4</span> Final Project: Graphics</a></li>
-<li><a href="#acknowledgements"><span class="toc-section-number">8.5</span> Acknowledgements</a></li>
+<li><a href="#a-taste-of-io" id="toc-a-taste-of-io"><span
+class="toc-section-number">8.1</span> A Taste of IO</a></li>
+<li><a href="#summary" id="toc-summary"><span
+class="toc-section-number">8.2</span> Summary</a></li>
+<li><a href="#what-next" id="toc-what-next"><span
+class="toc-section-number">8.3</span> What Next?</a></li>
+<li><a href="#final-project-graphics"
+id="toc-final-project-graphics"><span
+class="toc-section-number">8.4</span> Final Project: Graphics</a></li>
+<li><a href="#acknowledgements" id="toc-acknowledgements"><span
+class="toc-section-number">8.5</span> Acknowledgements</a></li>
 </ul></li>
 </ul>
 </nav>
@@ -191,104 +331,240 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 <header>
 <h1 class="title">Haskell Mooc, part 1</h1>
 </header>
-<p>by Joel Kaasinen (<a href="https://nitor.com/en">Nitor</a>) and John Lång (University of Helsinki)</p>
-<h1 data-number="1" id="lecture-1-and-so-it-begins"><span class="header-section-number">1</span> Lecture 1: …And so It Begins</h1>
-<h2 data-number="1.1" id="about-the-course"><span class="header-section-number">1.1</span> About the Course</h2>
-<p>This is an online course on Functional Programming that uses the Haskell programming language. You can study at your own pace. All the material and exercises are openly available.</p>
-<p>This course is aimed at beginners who wish to learn functional programming, but also people who have experience with functional programming and want to learn Haskell in particular. The course assumes no previous knowledge, but knowing at least one programming language beforehand will make the course easier.</p>
-<p>Working on the exercises involves knowing how to use the command line, and basic usage of the Git version control system.</p>
-<p>This is part 1 of a two-part course. Part 1 covers the basics of Haskell syntax and features. You will learn about recursion, higher-order functions, algebraic data types and some of Haskell’s advanced features. However, part 1 will stick to pure functional programming, without side-effects. I/O and Monads will be introduced in part 2.</p>
-<p>The course is split into 8 lectures. They are roughly the same size, but some lectures have more material than others. Each lecture set ends with 10-30 small programming exercises on the topics of the lecture.</p>
-<h2 data-number="1.2" id="read-these"><span class="header-section-number">1.2</span> Read These</h2>
-<p>In addition to this course material, the following sources might be useful if you feel like you’re missing examples or explanations.</p>
+<p>by Joel Kaasinen (<a href="https://nitor.com/en">Nitor</a>) and John
+Lång (University of Helsinki)</p>
+<h1 data-number="1" id="lecture-1-and-so-it-begins"><span
+class="header-section-number">1</span> Lecture 1: …And so It Begins</h1>
+<h2 data-number="1.1" id="about-the-course"><span
+class="header-section-number">1.1</span> About the Course</h2>
+<p>This is an online course on Functional Programming that uses the
+Haskell programming language. You can study at your own pace. All the
+material and exercises are openly available.</p>
+<p>This course is aimed at beginners who wish to learn functional
+programming, but also people who have experience with functional
+programming and want to learn Haskell in particular. The course assumes
+no previous knowledge, but knowing at least one programming language
+beforehand will make the course easier.</p>
+<p>Working on the exercises involves knowing how to use the command
+line, and basic usage of the Git version control system.</p>
+<p>This is part 1 of a two-part course. Part 1 covers the basics of
+Haskell syntax and features. You will learn about recursion,
+higher-order functions, algebraic data types and some of Haskell’s
+advanced features. However, part 1 will stick to pure functional
+programming, without side-effects. I/O and Monads will be introduced in
+part 2.</p>
+<p>The course is split into 8 lectures. They are roughly the same size,
+but some lectures have more material than others. Each lecture set ends
+with 10-30 small programming exercises on the topics of the lecture.</p>
+<h2 data-number="1.2" id="read-these"><span
+class="header-section-number">1.2</span> Read These</h2>
+<p>In addition to this course material, the following sources might be
+useful if you feel like you’re missing examples or explanations.</p>
 <ul>
 <li><a href="https://haskell.mooc.fi">The course pages</a></li>
-<li>The course <a href="https://t.me/haskell_mooc_fi">channel on Telegram</a></li>
-<li>The course <a href="https://github.com/moocfi/haskell-mooc">repository on Github</a> contains the exercises and this material</li>
+<li>The course <a href="https://t.me/haskell_mooc_fi">channel on
+Telegram</a></li>
+<li>The course <a
+href="https://github.com/moocfi/haskell-mooc">repository on Github</a>
+contains the exercises and this material</li>
 <li>Additional resources
 <ul>
-<li><a href="https://www.haskell.org/tutorial/">A Gentle Introduction to Haskell</a> - an older and shorter tutorial, but still worth reading</li>
-<li><a href="http://learnyouahaskell.com/chapters">Learn You a Haskell for Great Good!</a> – a nice free introduction to Haskell</li>
-<li><a href="https://www.cs.yale.edu/homes/hudak/SOE/index.htm">The Haskell School of Expression</a> - slightly older but still relevant introduction to functional programming</li>
-<li><a href="https://haskellbook.com/">Haskell Programming from First Principles</a> - $59 e-book on Haskell, slow and long</li>
-<li>The IRC channel <code>#haskell</code> on <a href="https://libera.chat/">libera.chat</a> is a nice place for beginners</li>
+<li><a href="https://www.haskell.org/tutorial/">A Gentle Introduction to
+Haskell</a> - an older and shorter tutorial, but still worth
+reading</li>
+<li><a href="http://learnyouahaskell.com/chapters">Learn You a Haskell
+for Great Good!</a> – a nice free introduction to Haskell</li>
+<li><a href="https://www.cs.yale.edu/homes/hudak/SOE/index.htm">The
+Haskell School of Expression</a> - slightly older but still relevant
+introduction to functional programming</li>
+<li><a href="https://haskellbook.com/">Haskell Programming from First
+Principles</a> - $59 e-book on Haskell, slow and long</li>
+<li>The IRC channel <code>#haskell</code> on <a
+href="https://libera.chat/">libera.chat</a> is a nice place for
+beginners</li>
 </ul></li>
 </ul>
-<h2 data-number="1.3" id="haskell"><span class="header-section-number">1.3</span> Haskell</h2>
+<h2 data-number="1.3" id="haskell"><span
+class="header-section-number">1.3</span> Haskell</h2>
 <p>Haskell is</p>
-<p><strong>Functional</strong> – the basic building blocks of programs are functions. Functions can return functions and take functions as arguments. Also, the only looping construct in Haskell is recursion.</p>
-<p><strong>Pure</strong> - Haskell functions are pure, that is, they don’t have side effects. Side effects mean things like reading a file, printing out text, or changing global variables. All inputs to a function must be in its arguments, and all outputs from a function in its return value. This sounds restricting, but makes reasoning about programs easier, and allows for more optimizations by the compiler.</p>
-<p><strong>Lazy</strong> - values are only evaluated when they are needed. This makes it possible to work with infinite data structures, and also makes pure programs more efficient.</p>
-<p><strong>Strongly typed</strong> - every Haskell value and expression has a type. The compiler checks the types at compile-time and guarantees that no type errors can happen at runtime. This means no AttributeErrors (a la Python), ClassCastExceptions (a la Java) or segmentation faults (a la C). The Haskell type system is very powerful and can help you design better programs.</p>
-<p><strong>Type inferred</strong> - in addition to checking the types, the compiler can deduce the types for most programs. This makes working with a strongly typed language easier. Indeed, most Haskell functions can be written completely without types. However programmers can still give functions and values type annotations to make finding type errors easier. Type annotations also make reading programs easier.</p>
-<p><strong>Garbage-collected</strong> - like most high-level languages these days, Haskell has automatic memory management via garbage collection. This means that the programmer doesn’t need to worry about allocating or freeing memory, the language runtime handles all of it automatically.</p>
-<p><strong>Compiled</strong> - even though we mostly use Haskell via the interactive GHCi environment on this course, Haskell is a compiled language. Haskell programs can be compiled to very efficient binaries, and the GHC compiler is very good at optimising functional code into performant machine code.</p>
-<p>You’ll learn what these terms mean in practice during this course. Don’t worry if some of them sound abstract right now.</p>
-<p>See also: <a href="https://wiki.haskell.org/Functional_programming">The Haskell Wiki page on Functional programming</a>.</p>
-<h3 data-number="1.3.1" id="features"><span class="header-section-number">1.3.1</span> Features</h3>
+<p><strong>Functional</strong> – the basic building blocks of programs
+are functions. Functions can return functions and take functions as
+arguments. Also, the only looping construct in Haskell is recursion.</p>
+<p><strong>Pure</strong> - Haskell functions are pure, that is, they
+don’t have side effects. Side effects mean things like reading a file,
+printing out text, or changing global variables. All inputs to a
+function must be in its arguments, and all outputs from a function in
+its return value. This sounds restricting, but makes reasoning about
+programs easier, and allows for more optimizations by the compiler.</p>
+<p><strong>Lazy</strong> - values are only evaluated when they are
+needed. This makes it possible to work with infinite data structures,
+and also makes pure programs more efficient.</p>
+<p><strong>Strongly typed</strong> - every Haskell value and expression
+has a type. The compiler checks the types at compile-time and guarantees
+that no type errors can happen at runtime. This means no AttributeErrors
+(a la Python), ClassCastExceptions (a la Java) or segmentation faults (a
+la C). The Haskell type system is very powerful and can help you design
+better programs.</p>
+<p><strong>Type inferred</strong> - in addition to checking the types,
+the compiler can deduce the types for most programs. This makes working
+with a strongly typed language easier. Indeed, most Haskell functions
+can be written completely without types. However programmers can still
+give functions and values type annotations to make finding type errors
+easier. Type annotations also make reading programs easier.</p>
+<p><strong>Garbage-collected</strong> - like most high-level languages
+these days, Haskell has automatic memory management via garbage
+collection. This means that the programmer doesn’t need to worry about
+allocating or freeing memory, the language runtime handles all of it
+automatically.</p>
+<p><strong>Compiled</strong> - even though we mostly use Haskell via the
+interactive GHCi environment on this course, Haskell is a compiled
+language. Haskell programs can be compiled to very efficient binaries,
+and the GHC compiler is very good at optimising functional code into
+performant machine code.</p>
+<p>You’ll learn what these terms mean in practice during this course.
+Don’t worry if some of them sound abstract right now.</p>
+<p>See also: <a
+href="https://wiki.haskell.org/Functional_programming">The Haskell Wiki
+page on Functional programming</a>.</p>
+<h3 data-number="1.3.1" id="features"><span
+class="header-section-number">1.3.1</span> Features</h3>
 <p>Here’s a showcase of some of the Haskell’s cool features:</p>
-<p><strong>Higher-order functions</strong> – functions can take functions as arguments:</p>
-<div class="sourceCode" id="cb1"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> <span class="fu">length</span> [<span class="st">&quot;abc&quot;</span>,<span class="st">&quot;abcdef&quot;</span>]</span></code></pre></div>
+<p><strong>Higher-order functions</strong> – functions can take
+functions as arguments:</p>
+<div class="sourceCode" id="cb1"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> <span class="fu">length</span> [<span class="st">&quot;abc&quot;</span>,<span class="st">&quot;abcdef&quot;</span>]</span></code></pre></div>
 <p>This results in <code>[3,6]</code>.</p>
-<p><strong>Anonymous functions aka lambdas</strong> – you can define single-use helper functions without giving them a name</p>
-<div class="sourceCode" id="cb2"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="fu">filter</span> (\x <span class="ot">-&gt;</span> <span class="fu">length</span> x <span class="op">&gt;</span> <span class="dv">1</span>) [<span class="st">&quot;abc&quot;</span>,<span class="st">&quot;d&quot;</span>,<span class="st">&quot;ef&quot;</span>]</span></code></pre></div>
+<p><strong>Anonymous functions aka lambdas</strong> – you can define
+single-use helper functions without giving them a name</p>
+<div class="sourceCode" id="cb2"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="fu">filter</span> (\x <span class="ot">-&gt;</span> <span class="fu">length</span> x <span class="op">&gt;</span> <span class="dv">1</span>) [<span class="st">&quot;abc&quot;</span>,<span class="st">&quot;d&quot;</span>,<span class="st">&quot;ef&quot;</span>]</span></code></pre></div>
 <p>This results in <code>["abc","ef"]</code>.</p>
-<p><strong>Partial application</strong> – you can define new functions by giving another function only some of the arguments it needs. For example this multiplies all elements in a list by 3:</p>
-<div class="sourceCode" id="cb3"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> (<span class="op">*</span><span class="dv">3</span>) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span></code></pre></div>
-<p><strong>Algebraic datatypes</strong> – a syntax for defining datatypes that can contain a number of different cases:</p>
-<div class="sourceCode" id="cb4"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Shape</span> <span class="ot">=</span> <span class="dt">Point</span> <span class="op">|</span> <span class="dt">Rectangle</span> <span class="dt">Double</span> <span class="dt">Double</span> <span class="op">|</span> <span class="dt">Circle</span> <span class="dt">Double</span></span></code></pre></div>
-<p>Now the type <code>Shape</code> can have values like <code>Point</code>, <code>Rectangle 3 6</code> and <code>Circle 5</code></p>
-<p><strong>Pattern matching</strong> – defining functions based on cases that correspond to your data definitions:</p>
-<div class="sourceCode" id="cb5"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a>area <span class="dt">Point</span> <span class="ot">=</span> <span class="dv">0</span></span>
+<p><strong>Partial application</strong> – you can define new functions
+by giving another function only some of the arguments it needs. For
+example this multiplies all elements in a list by 3:</p>
+<div class="sourceCode" id="cb3"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> (<span class="op">*</span><span class="dv">3</span>) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span></code></pre></div>
+<p><strong>Algebraic datatypes</strong> – a syntax for defining
+datatypes that can contain a number of different cases:</p>
+<div class="sourceCode" id="cb4"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Shape</span> <span class="ot">=</span> <span class="dt">Point</span> <span class="op">|</span> <span class="dt">Rectangle</span> <span class="dt">Double</span> <span class="dt">Double</span> <span class="op">|</span> <span class="dt">Circle</span> <span class="dt">Double</span></span></code></pre></div>
+<p>Now the type <code>Shape</code> can have values like
+<code>Point</code>, <code>Rectangle 3 6</code> and
+<code>Circle 5</code></p>
+<p><strong>Pattern matching</strong> – defining functions based on cases
+that correspond to your data definitions:</p>
+<div class="sourceCode" id="cb5"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a>area <span class="dt">Point</span> <span class="ot">=</span> <span class="dv">0</span></span>
 <span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a>area (<span class="dt">Rectangle</span> width height) <span class="ot">=</span> width <span class="op">*</span> height</span>
 <span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a>area (<span class="dt">Circle</span> radius) <span class="ot">=</span> <span class="fu">pi</span> <span class="op">*</span> radius <span class="op">*</span> radius</span></code></pre></div>
-<p><strong>Lists</strong> – Unlike many languages, Haskell has a concise built-in syntax for lists. Lists can be built from other lists using <em>list comprehensions</em>. Here’s a snippet that generates names of even length from a set of options for first and last names:</p>
-<div class="sourceCode" id="cb6"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a>[whole <span class="op">|</span> first <span class="ot">&lt;-</span> [<span class="st">&quot;Eva&quot;</span>, <span class="st">&quot;Mike&quot;</span>],</span>
+<p><strong>Lists</strong> – Unlike many languages, Haskell has a concise
+built-in syntax for lists. Lists can be built from other lists using
+<em>list comprehensions</em>. Here’s a snippet that generates names of
+even length from a set of options for first and last names:</p>
+<div class="sourceCode" id="cb6"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a>[whole <span class="op">|</span> first <span class="ot">&lt;-</span> [<span class="st">&quot;Eva&quot;</span>, <span class="st">&quot;Mike&quot;</span>],</span>
 <span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a>         <span class="fu">last</span> <span class="ot">&lt;-</span> [<span class="st">&quot;Smith&quot;</span>, <span class="st">&quot;Wood&quot;</span>, <span class="st">&quot;Odd&quot;</span>],</span>
 <span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a>         <span class="kw">let</span> whole <span class="ot">=</span> first <span class="op">++</span> <span class="fu">last</span>,</span>
 <span id="cb6-4"><a href="#cb6-4" aria-hidden="true" tabindex="-1"></a>         <span class="fu">even</span> (<span class="fu">length</span> whole)]</span></code></pre></div>
-<p>This results in <code>["EvaSmith","EvaOdd","MikeWood"]</code>. Thanks to the laziness of Haskell, we can even create so-called <em>infinite lists</em>:</p>
-<div class="sourceCode" id="cb7"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a>primes <span class="ot">=</span> [ n <span class="op">|</span> n <span class="ot">&lt;-</span> [<span class="dv">2</span><span class="op">..</span>] , <span class="fu">all</span> (\k <span class="ot">-&gt;</span> n <span class="ot">`mod`</span> k <span class="op">/=</span> <span class="dv">0</span>) [<span class="dv">2</span><span class="op">..</span>n <span class="ot">`div`</span> <span class="dv">2</span>] ]</span></code></pre></div>
+<p>This results in <code>["EvaSmith","EvaOdd","MikeWood"]</code>. Thanks
+to the laziness of Haskell, we can even create so-called <em>infinite
+lists</em>:</p>
+<div class="sourceCode" id="cb7"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a>primes <span class="ot">=</span> [ n <span class="op">|</span> n <span class="ot">&lt;-</span> [<span class="dv">2</span><span class="op">..</span>] , <span class="fu">all</span> (\k <span class="ot">-&gt;</span> n <span class="ot">`mod`</span> k <span class="op">/=</span> <span class="dv">0</span>) [<span class="dv">2</span><span class="op">..</span>n <span class="ot">`div`</span> <span class="dv">2</span>] ]</span></code></pre></div>
 <p>The first ten prime numbers can be then obtained by evaluating</p>
-<div class="sourceCode" id="cb8"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a><span class="fu">take</span> <span class="dv">10</span> primes</span></code></pre></div>
+<div class="sourceCode" id="cb8"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a><span class="fu">take</span> <span class="dv">10</span> primes</span></code></pre></div>
 <p>This evaluates to <code>[2,3,5,7,11,13,17,19,23,29]</code>.</p>
-<p><strong>Parameterized types</strong> – you can define types that are parameterized by other types. For example <code>[Int]</code> is a list of <code>Int</code>s and <code>[Bool]</code> is a list of booleans. You can define typed functions that work on all kinds of lists, for example <code>reverse</code> has the type <code>[a] -&gt; [a]</code> which means it takes a list containing any type <code>a</code>, and returns a list of the same type.</p>
-<p><strong>Type classes</strong> – another form of polymorphism where you can give a function a different implementation depending on the arguments’ types. For example the <code>Show</code> type class defines the function <code>show</code> that can convert values of various types to strings. The <code>Num</code> type class defines arithmetic operators like <code>+</code> that work on all number types (<code>Int</code>, <code>Double</code>, <code>Complex</code>, …).</p>
-<h3 data-number="1.3.2" id="some-history"><span class="header-section-number">1.3.2</span> Some History</h3>
+<p><strong>Parameterized types</strong> – you can define types that are
+parameterized by other types. For example <code>[Int]</code> is a list
+of <code>Int</code>s and <code>[Bool]</code> is a list of booleans. You
+can define typed functions that work on all kinds of lists, for example
+<code>reverse</code> has the type <code>[a] -&gt; [a]</code> which means
+it takes a list containing any type <code>a</code>, and returns a list
+of the same type.</p>
+<p><strong>Type classes</strong> – another form of polymorphism where
+you can give a function a different implementation depending on the
+arguments’ types. For example the <code>Show</code> type class defines
+the function <code>show</code> that can convert values of various types
+to strings. The <code>Num</code> type class defines arithmetic operators
+like <code>+</code> that work on all number types (<code>Int</code>,
+<code>Double</code>, <code>Complex</code>, …).</p>
+<h3 data-number="1.3.2" id="some-history"><span
+class="header-section-number">1.3.2</span> Some History</h3>
 <p>A brief timeline of Haskell:</p>
 <ul>
 <li>1930s: Lambda Calculus</li>
 <li>1950s-1970s: Lisp, Pattern Matching, Scheme, ML</li>
-<li>1978 John Backus: <a href="https://dl.acm.org/doi/pdf/10.1145/359576.359579">Can programming be liberated from the von Neumann style?</a></li>
-<li>1987 A decision was made to unify the field of pure functional languages</li>
+<li>1978 John Backus: <a
+href="https://dl.acm.org/doi/pdf/10.1145/359576.359579">Can programming
+be liberated from the von Neumann style?</a></li>
+<li>1987 A decision was made to unify the field of pure functional
+languages</li>
 <li>1990 Haskell 1.0</li>
 <li>1991 Haskell 1.1 (let-syntax, sections)</li>
 <li>1992 Haskell 1.2, GHC</li>
 <li>1996 Haskell 1.3 (Monads, do-syntax, type system improvements)</li>
 <li>1999 Haskell 98</li>
 <li>2000’s: GHC development, many extensions to the language</li>
-<li>2009 <a href="https://www.haskell.org/onlinereport/haskell2010/">The Haskell 2010 standard</a></li>
+<li>2009 <a href="https://www.haskell.org/onlinereport/haskell2010/">The
+Haskell 2010 standard</a></li>
 <li>2010’s: GHC development, Haskell Platform, Haskell Stack</li>
 </ul>
-<p>The word ‘haskel’ means wisdom in Hebrew, but the name of the Haskell programming language comes from the logician Haskell Curry. The name Haskell comes from the Old Norse words áss (god) and ketill (helmet).</p>
-<h3 data-number="1.3.3" id="uses-of-haskell"><span class="header-section-number">1.3.3</span> Uses of Haskell</h3>
-<p>Here are some examples of software projects that were written in Haskell.</p>
+<p>The word ‘haskel’ means wisdom in Hebrew, but the name of the Haskell
+programming language comes from the logician Haskell Curry. The name
+Haskell comes from the Old Norse words áss (god) and ketill
+(helmet).</p>
+<h3 data-number="1.3.3" id="uses-of-haskell"><span
+class="header-section-number">1.3.3</span> Uses of Haskell</h3>
+<p>Here are some examples of software projects that were written in
+Haskell.</p>
 <ul>
-<li>The <a href="https://en.wikipedia.org/wiki/Darcs">Darcs</a> distributed version control system</li>
-<li>The <a href="https://engineering.fb.com/security/fighting-spam-with-haskell/">Sigma spam-prevention tool at Facebook</a></li>
-<li>The implementations of the <a href="https://www.purescript.org/">PureScript</a> and <a href="https://elm-lang.org/">Elm</a> programming languages are written in Haskell</li>
-<li>The <a href="https://pandoc.org/">Pandoc</a> tool for converting between different document formats – it’s also used to produce this course material</li>
-<li>The <a href="https://postgrest.org/">PostgREST</a> server that exposes a HTTP REST API for a PostgreSQL database</li>
-<li>Functional consulting companies like <a href="https://galois.com/">Galois</a> and <a href="https://well-typed.com/">Well-Typed</a> have a long history of developing critical systems for clients in Haskell</li>
+<li>The <a href="https://en.wikipedia.org/wiki/Darcs">Darcs</a>
+distributed version control system</li>
+<li>The <a
+href="https://engineering.fb.com/security/fighting-spam-with-haskell/">Sigma
+spam-prevention tool at Facebook</a></li>
+<li>The implementations of the <a
+href="https://www.purescript.org/">PureScript</a> and <a
+href="https://elm-lang.org/">Elm</a> programming languages are written
+in Haskell</li>
+<li>The <a href="https://pandoc.org/">Pandoc</a> tool for converting
+between different document formats – it’s also used to produce this
+course material</li>
+<li>The <a href="https://postgrest.org/">PostgREST</a> server that
+exposes a HTTP REST API for a PostgreSQL database</li>
+<li>Functional consulting companies like <a
+href="https://galois.com/">Galois</a> and <a
+href="https://well-typed.com/">Well-Typed</a> have a long history of
+developing critical systems for clients in Haskell</li>
 </ul>
-<p>See <a href="https://wiki.haskell.org/Haskell_in_industry">The Haskell Wiki</a> and <a href="https://serokell.io/blog/top-software-written-in-haskell">this blog post</a> for more!</p>
-<h2 data-number="1.4" id="running-haskell"><span class="header-section-number">1.4</span> Running Haskell</h2>
-<p>The easiest way to get Haskell is to install the <code>stack</code> tool, see <a href="https://haskellstack.org" class="uri">https://haskellstack.org</a>. The exercises on this course are intended to work with Stack, so you should use it for now.</p>
-<p>By the way, if you’re interested in what Stack is, and how it relates to other Haskell tools like Cabal and GHC, <a href="https://www.quora.com/What-is-the-difference-between-Cabal-and-Stack-in-Haskell-projects-Which-one-do-you-recommend-and-why">read more here</a> or <a href="https://docs.haskellstack.org/en/stable/faq/">here</a>. We’ll get back to Haskell packages and using them in detail in part 2 of the course.</p>
-<p>For now, after installing Stack, just run <code>stack ghci</code> to get an interactive Haskell environment.</p>
-<p><strong>Note!</strong> GHC 8.10.7 has a GHCi bug that makes editing lines impossible on ARM-based systems. As a workaround, use <code>TERM=dumb stack ghci</code>. More info <a href="https://gitlab.haskell.org/ghc/ghc/-/issues/20022">here</a>.</p>
-<h2 data-number="1.5" id="lets-start"><span class="header-section-number">1.5</span> Let’s Start!</h2>
-<p>GHCi is the interactive Haskell interpreter. Here’s an example session:</p>
+<p>See <a href="https://wiki.haskell.org/Haskell_in_industry">The
+Haskell Wiki</a> and <a
+href="https://serokell.io/blog/top-software-written-in-haskell">this
+blog post</a> for more!</p>
+<h2 data-number="1.4" id="running-haskell"><span
+class="header-section-number">1.4</span> Running Haskell</h2>
+<p>The easiest way to get Haskell is to install the <code>stack</code>
+tool, see <a href="https://haskellstack.org"
+class="uri">https://haskellstack.org</a>. The exercises on this course
+are intended to work with Stack, so you should use it for now.</p>
+<p>By the way, if you’re interested in what Stack is, and how it relates
+to other Haskell tools like Cabal and GHC, <a
+href="https://www.quora.com/What-is-the-difference-between-Cabal-and-Stack-in-Haskell-projects-Which-one-do-you-recommend-and-why">read
+more here</a> or <a
+href="https://docs.haskellstack.org/en/stable/faq/">here</a>. We’ll get
+back to Haskell packages and using them in detail in part 2 of the
+course.</p>
+<p>For now, after installing Stack, just run <code>stack ghci</code> to
+get an interactive Haskell environment.</p>
+<p><strong>Note!</strong> GHC 8.10.7 has a GHCi bug that makes editing
+lines impossible on ARM-based systems. As a workaround, use
+<code>TERM=dumb stack ghci</code>. More info <a
+href="https://gitlab.haskell.org/ghc/ghc/-/issues/20022">here</a>.</p>
+<h2 data-number="1.5" id="lets-start"><span
+class="header-section-number">1.5</span> Let’s Start!</h2>
+<p>GHCi is the interactive Haskell interpreter. Here’s an example
+session:</p>
 <pre><code>$ stack ghci
 GHCi, version 9.2.8: https://www.haskell.org/ghc/  :? for help
 Prelude&gt; 1+1
@@ -307,34 +583,57 @@ Prelude&gt; :type tail
 tail :: [a] -&gt; [a]
 Prelude&gt; :quit
 Leaving GHCi.</code></pre>
-<p>By the way, the first time you run <code>stack ghci</code> it will download GHC and some libraries, so don’t worry if you see some output and have to wait for a while before getting the <code>Prelude&gt;</code> prompt.</p>
-<p>Let’s walk through this. Don’t worry if you don’t understand things yet, this is just a first brush with expressions and types.</p>
+<p>By the way, the first time you run <code>stack ghci</code> it will
+download GHC and some libraries, so don’t worry if you see some output
+and have to wait for a while before getting the <code>Prelude&gt;</code>
+prompt.</p>
+<p>Let’s walk through this. Don’t worry if you don’t understand things
+yet, this is just a first brush with expressions and types.</p>
 <pre><code>Prelude&gt; 1+1
 2</code></pre>
-<p>The <code>Prelude&gt;</code> is the GHCi prompt. It indicates we can use the functions from the Haskell base library called Prelude. We evaluate 1 plus 1, and the result is 2.</p>
+<p>The <code>Prelude&gt;</code> is the GHCi prompt. It indicates we can
+use the functions from the Haskell base library called Prelude. We
+evaluate 1 plus 1, and the result is 2.</p>
 <pre><code>Prelude&gt; &quot;asdf&quot;
 &quot;asdf&quot;</code></pre>
-<p>Here we evaluate a string literal, and the result is the same string.</p>
+<p>Here we evaluate a string literal, and the result is the same
+string.</p>
 <pre><code>Prelude&gt; reverse &quot;asdf&quot;
 &quot;fdsa&quot;</code></pre>
-<p>Here we compute the reverse of a string by applying the function <code>reverse</code> to the value <code>"asdf"</code>.</p>
+<p>Here we compute the reverse of a string by applying the function
+<code>reverse</code> to the value <code>"asdf"</code>.</p>
 <pre><code>Prelude&gt; :type &quot;asdf&quot;
 &quot;asdf&quot; :: [Char]</code></pre>
-<p>In addition to evaluating expressions we can also ask for their type with the <code>:type</code> (abbreviated <code>:t</code>) GHCi command. The type of <code>"asdf"</code> is a list of characters. Commands that start with <code>:</code> are part of the user interface of GHCi, not part of the Haskell language.</p>
+<p>In addition to evaluating expressions we can also ask for their type
+with the <code>:type</code> (abbreviated <code>:t</code>) GHCi command.
+The type of <code>"asdf"</code> is a list of characters. Commands that
+start with <code>:</code> are part of the user interface of GHCi, not
+part of the Haskell language.</p>
 <pre><code>Prelude&gt; tail &quot;asdf&quot;
 &quot;sdf&quot;
 Prelude&gt; :t tail &quot;asdf&quot;
 tail &quot;asdf&quot; :: [Char]</code></pre>
-<p>The <code>tail</code> function works on lists and returns all except the first element of the list. Here we see <code>tail</code> applied to <code>"asdf"</code>. We also check the type of the expression, and it is a list of characters, as expected.</p>
+<p>The <code>tail</code> function works on lists and returns all except
+the first element of the list. Here we see <code>tail</code> applied to
+<code>"asdf"</code>. We also check the type of the expression, and it is
+a list of characters, as expected.</p>
 <pre><code>Prelude&gt; :t tail
 tail :: [a] -&gt; [a]</code></pre>
-<p>Finally, here’s the type of the <code>tail</code> function. It takes a list of any type as an argument, and returns a list of the same type.</p>
+<p>Finally, here’s the type of the <code>tail</code> function. It takes
+a list of any type as an argument, and returns a list of the same
+type.</p>
 <pre><code>Prelude&gt; :quit
 Leaving GHCi.</code></pre>
 <p>That’s how you quit GHCi.</p>
-<h2 data-number="1.6" id="expressions-and-types"><span class="header-section-number">1.6</span> Expressions and Types</h2>
-<p>Just like we saw in the GHCi example above, <em>expressions</em> and <em>types</em> are the bread and butter of Haskell. In fact, almost everything in a Haskell program is an expression. In particular, there are no <em>statements</em> like in Python, Java or C.</p>
-<p>An expression has a <em>value</em> and a <em>type</em>. We write an expression and its type like this: <code>expression :: type</code>. Here are some examples:</p>
+<h2 data-number="1.6" id="expressions-and-types"><span
+class="header-section-number">1.6</span> Expressions and Types</h2>
+<p>Just like we saw in the GHCi example above, <em>expressions</em> and
+<em>types</em> are the bread and butter of Haskell. In fact, almost
+everything in a Haskell program is an expression. In particular, there
+are no <em>statements</em> like in Python, Java or C.</p>
+<p>An expression has a <em>value</em> and a <em>type</em>. We write an
+expression and its type like this: <code>expression :: type</code>. Here
+are some examples:</p>
 <table>
 <thead>
 <tr class="header">
@@ -361,8 +660,12 @@ Leaving GHCi.</code></pre>
 </tr>
 </tbody>
 </table>
-<h3 data-number="1.6.1" id="syntax-of-expressions"><span class="header-section-number">1.6.1</span> Syntax of Expressions</h3>
-<p>Expressions consist of functions <em>applied</em> to arguments. Functions are <em>applied</em> (i.e. called) by placing the arguments after the name of the function – there is no special syntax for a function call.</p>
+<h3 data-number="1.6.1" id="syntax-of-expressions"><span
+class="header-section-number">1.6.1</span> Syntax of Expressions</h3>
+<p>Expressions consist of functions <em>applied</em> to arguments.
+Functions are <em>applied</em> (i.e. called) by placing the arguments
+after the name of the function – there is no special syntax for a
+function call.</p>
 <table>
 <thead>
 <tr class="header">
@@ -381,7 +684,8 @@ Leaving GHCi.</code></pre>
 </tr>
 </tbody>
 </table>
-<p>Parentheses can be used to <em>group</em> expressions (just like in math and other languages).</p>
+<p>Parentheses can be used to <em>group</em> expressions (just like in
+math and other languages).</p>
 <table>
 <thead>
 <tr class="header">
@@ -408,7 +712,10 @@ Leaving GHCi.</code></pre>
 </tr>
 </tbody>
 </table>
-<p>Some function names are made special characters and they are used as operators: between their arguments instead of before them. Function calls <em>bind tighter</em> than operators, just like multiplication binds tighter than addition.</p>
+<p>Some function names are made special characters and they are used as
+operators: between their arguments instead of before them. Function
+calls <em>bind tighter</em> than operators, just like multiplication
+binds tighter than addition.</p>
 <table>
 <thead>
 <tr class="header">
@@ -431,8 +738,13 @@ Leaving GHCi.</code></pre>
 </tr>
 </tbody>
 </table>
-<p>PS. in Haskell, function application <em>associates left</em>, that is, <code>f g x y</code> is actually the same as <code>(((f g) x) y)</code>. We’ll get back to this topic later. For now you can just think that <code>f g x y</code> is <code>f</code> applied to the arguments <code>g</code>, <code>x</code> and <code>y</code>.</p>
-<h3 data-number="1.6.2" id="syntax-of-types"><span class="header-section-number">1.6.2</span> Syntax of Types</h3>
+<p>PS. in Haskell, function application <em>associates left</em>, that
+is, <code>f g x y</code> is actually the same as
+<code>(((f g) x) y)</code>. We’ll get back to this topic later. For now
+you can just think that <code>f g x y</code> is <code>f</code> applied
+to the arguments <code>g</code>, <code>x</code> and <code>y</code>.</p>
+<h3 data-number="1.6.2" id="syntax-of-types"><span
+class="header-section-number">1.6.2</span> Syntax of Types</h3>
 <p>Here are some basic types of Haskell to get you started.</p>
 <table>
 <thead>
@@ -446,55 +758,87 @@ Leaving GHCi.</code></pre>
 <tbody>
 <tr class="odd">
 <td style="text-align: left;"><code>Int</code></td>
-<td style="text-align: left;"><code>1</code>, <code>2</code>, <code>-3</code></td>
+<td style="text-align: left;"><code>1</code>, <code>2</code>,
+<code>-3</code></td>
 <td style="text-align: left;">Number type (signed, 64bit)</td>
-<td style="text-align: left;"><code>+</code>, <code>-</code>, <code>*</code>, <code>div</code>, <code>mod</code></td>
+<td style="text-align: left;"><code>+</code>, <code>-</code>,
+<code>*</code>, <code>div</code>, <code>mod</code></td>
 </tr>
 <tr class="even">
 <td style="text-align: left;"><code>Integer</code></td>
-<td style="text-align: left;"><code>1</code>, <code>-2</code>, <code>900000000000000000</code></td>
+<td style="text-align: left;"><code>1</code>, <code>-2</code>,
+<code>900000000000000000</code></td>
 <td style="text-align: left;">Unbounded number type</td>
-<td style="text-align: left;"><code>+</code>, <code>-</code>, <code>*</code>, <code>div</code>, <code>mod</code></td>
+<td style="text-align: left;"><code>+</code>, <code>-</code>,
+<code>*</code>, <code>div</code>, <code>mod</code></td>
 </tr>
 <tr class="odd">
 <td style="text-align: left;"><code>Double</code></td>
 <td style="text-align: left;"><code>0.1</code>, <code>1.2e5</code></td>
 <td style="text-align: left;">Floating point numbers</td>
-<td style="text-align: left;"><code>+</code>, <code>-</code>, <code>*</code>, <code>/</code>, <code>sqrt</code></td>
+<td style="text-align: left;"><code>+</code>, <code>-</code>,
+<code>*</code>, <code>/</code>, <code>sqrt</code></td>
 </tr>
 <tr class="even">
 <td style="text-align: left;"><code>Bool</code></td>
 <td style="text-align: left;"><code>True</code>, <code>False</code></td>
 <td style="text-align: left;">Truth values</td>
-<td style="text-align: left;"><code>&amp;&amp;</code>, <code>||</code>, <code>not</code></td>
+<td style="text-align: left;"><code>&amp;&amp;</code>, <code>||</code>,
+<code>not</code></td>
 </tr>
 <tr class="odd">
-<td style="text-align: left;"><code>String</code> aka <code>[Char]</code></td>
+<td style="text-align: left;"><code>String</code> aka
+<code>[Char]</code></td>
 <td style="text-align: left;"><code>"abcd"</code>, <code>""</code></td>
 <td style="text-align: left;">Strings of characters</td>
 <td style="text-align: left;"><code>reverse</code>, <code>++</code></td>
 </tr>
 </tbody>
 </table>
-<p>As you can see, the names of types in Haskell start with a capital letter. Some values like <code>True</code> also start with a capital letter, but variables and functions start with a lower case letter (<code>reverse</code>, <code>not</code>, <code>x</code>). We’ll get back to the meaning of capital letters in Lecture 2.</p>
+<p>As you can see, the names of types in Haskell start with a capital
+letter. Some values like <code>True</code> also start with a capital
+letter, but variables and functions start with a lower case letter
+(<code>reverse</code>, <code>not</code>, <code>x</code>). We’ll get back
+to the meaning of capital letters in Lecture 2.</p>
 <p>Function types are written using the <code>-&gt;</code> syntax:</p>
 <ul>
-<li>A function of one argument: <code>argumentType -&gt; returnType</code></li>
-<li>… of two arguments: <code>argument1Type -&gt; argument2Type -&gt; returnType</code></li>
-<li>… of three arguments: <code>argument1Type -&gt; argument2Type -&gt; argument3Type -&gt; returnType</code></li>
+<li>A function of one argument:
+<code>argumentType -&gt; returnType</code></li>
+<li>… of two arguments:
+<code>argument1Type -&gt; argument2Type -&gt; returnType</code></li>
+<li>… of three arguments:
+<code>argument1Type -&gt; argument2Type -&gt; argument3Type -&gt; returnType</code></li>
 </ul>
 <p>Looks a bit weird, right? We’ll get back to this as well.</p>
-<h3 data-number="1.6.3" id="note-about-misleading-types"><span class="header-section-number">1.6.3</span> Note About Misleading Types</h3>
-<p>Sometimes, the types you see in GHCi are a bit different than what you’d assume. Here are two common cases.</p>
-<div class="sourceCode" id="cb17"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="dv">1</span><span class="op">+</span><span class="dv">1</span></span>
+<h3 data-number="1.6.3" id="note-about-misleading-types"><span
+class="header-section-number">1.6.3</span> Note About Misleading
+Types</h3>
+<p>Sometimes, the types you see in GHCi are a bit different than what
+you’d assume. Here are two common cases.</p>
+<div class="sourceCode" id="cb17"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="dv">1</span><span class="op">+</span><span class="dv">1</span></span>
 <span id="cb17-2"><a href="#cb17-2" aria-hidden="true" tabindex="-1"></a><span class="dv">1</span><span class="op">+</span><span class="dv">1</span><span class="ot"> ::</span> <span class="dt">Num</span> a <span class="ot">=&gt;</span> a</span></code></pre></div>
-<p>For now, you should read the type <code>Num a =&gt; a</code> as “any number type”. In Haskell, number literals are <em>overloaded</em> which means that they can be interpreted as any number type (e.g. <code>Int</code> or <code>Double</code>). We’ll get back to what <code>Num a</code> actually means when we talk about <em>type classes</em> later.</p>
-<div class="sourceCode" id="cb18"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb18-1"><a href="#cb18-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="st">&quot;asdf&quot;</span></span>
+<p>For now, you should read the type <code>Num a =&gt; a</code> as “any
+number type”. In Haskell, number literals are <em>overloaded</em> which
+means that they can be interpreted as any number type
+(e.g. <code>Int</code> or <code>Double</code>). We’ll get back to what
+<code>Num a</code> actually means when we talk about <em>type
+classes</em> later.</p>
+<div class="sourceCode" id="cb18"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb18-1"><a href="#cb18-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="st">&quot;asdf&quot;</span></span>
 <span id="cb18-2"><a href="#cb18-2" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;asdf&quot;</span><span class="ot"> ::</span> [<span class="dt">Char</span>]</span></code></pre></div>
-<p>The type <code>String</code> is just an alias for the type <code>[Char]</code> which means “list of characters”. We’ll get back to lists on the next lecture! In any case, you can use <code>String</code> and <code>[Char]</code> interchangeably, but GHCi will mostly use <code>[Char]</code> when describing types to you.</p>
-<h2 data-number="1.7" id="the-structure-of-a-haskell-program"><span class="header-section-number">1.7</span> The Structure of a Haskell Program</h2>
-<p>Here’s a simple Haskell program that does some arithmetic and prints some values.</p>
-<div class="sourceCode" id="cb19"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb19-1"><a href="#cb19-1" aria-hidden="true" tabindex="-1"></a><span class="kw">module</span> <span class="dt">Gold</span> <span class="kw">where</span></span>
+<p>The type <code>String</code> is just an alias for the type
+<code>[Char]</code> which means “list of characters”. We’ll get back to
+lists on the next lecture! In any case, you can use <code>String</code>
+and <code>[Char]</code> interchangeably, but GHCi will mostly use
+<code>[Char]</code> when describing types to you.</p>
+<h2 data-number="1.7" id="the-structure-of-a-haskell-program"><span
+class="header-section-number">1.7</span> The Structure of a Haskell
+Program</h2>
+<p>Here’s a simple Haskell program that does some arithmetic and prints
+some values.</p>
+<div class="sourceCode" id="cb19"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb19-1"><a href="#cb19-1" aria-hidden="true" tabindex="-1"></a><span class="kw">module</span> <span class="dt">Gold</span> <span class="kw">where</span></span>
 <span id="cb19-2"><a href="#cb19-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb19-3"><a href="#cb19-3" aria-hidden="true" tabindex="-1"></a><span class="co">-- The golden ratio</span></span>
 <span id="cb19-4"><a href="#cb19-4" aria-hidden="true" tabindex="-1"></a><span class="ot">phi ::</span> <span class="dt">Double</span></span>
@@ -508,45 +852,79 @@ Leaving GHCi.</code></pre>
 <span id="cb19-12"><a href="#cb19-12" aria-hidden="true" tabindex="-1"></a>main <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb19-13"><a href="#cb19-13" aria-hidden="true" tabindex="-1"></a>  <span class="fu">print</span> (polynomial phi)</span>
 <span id="cb19-14"><a href="#cb19-14" aria-hidden="true" tabindex="-1"></a>  <span class="fu">print</span> (f phi)</span></code></pre></div>
-<p>If you put this in a file called <code>Gold.hs</code> and run it with (for example) <code>stack runhaskell Gold.hs</code>, you should see this output</p>
+<p>If you put this in a file called <code>Gold.hs</code> and run it with
+(for example) <code>stack runhaskell Gold.hs</code>, you should see this
+output</p>
 <pre><code>0.0
 -1.0</code></pre>
 <p>Let’s walk through the file.</p>
-<div class="sourceCode" id="cb21"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb21-1"><a href="#cb21-1" aria-hidden="true" tabindex="-1"></a><span class="kw">module</span> <span class="dt">Gold</span> <span class="kw">where</span></span></code></pre></div>
-<p>There is one Haskell <em>module</em> per source file. A module consists of <em>definitions</em>.</p>
-<div class="sourceCode" id="cb22"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb22-1"><a href="#cb22-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- The golden ratio</span></span></code></pre></div>
-<p>This is a comment. Comments are not part of the actual program, but text for human readers of the program.</p>
-<div class="sourceCode" id="cb23"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb23-1"><a href="#cb23-1" aria-hidden="true" tabindex="-1"></a><span class="ot">phi ::</span> <span class="dt">Double</span></span>
+<div class="sourceCode" id="cb21"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb21-1"><a href="#cb21-1" aria-hidden="true" tabindex="-1"></a><span class="kw">module</span> <span class="dt">Gold</span> <span class="kw">where</span></span></code></pre></div>
+<p>There is one Haskell <em>module</em> per source file. A module
+consists of <em>definitions</em>.</p>
+<div class="sourceCode" id="cb22"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb22-1"><a href="#cb22-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- The golden ratio</span></span></code></pre></div>
+<p>This is a comment. Comments are not part of the actual program, but
+text for human readers of the program.</p>
+<div class="sourceCode" id="cb23"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb23-1"><a href="#cb23-1" aria-hidden="true" tabindex="-1"></a><span class="ot">phi ::</span> <span class="dt">Double</span></span>
 <span id="cb23-2"><a href="#cb23-2" aria-hidden="true" tabindex="-1"></a>phi <span class="ot">=</span> (<span class="fu">sqrt</span> <span class="dv">5</span> <span class="op">+</span> <span class="dv">1</span>) <span class="op">/</span> <span class="dv">2</span></span></code></pre></div>
-<p>This is a definition of the constant <code>phi</code>, with an accompanying <em>type annotation</em> (also known as a <em>type signature</em>) <code>phi :: Double</code>. The type annotation means that <code>phi</code> has type <code>Double</code>. The line with a equals sign (<code>=</code>) is called an <em>equation</em>. The left hand side of the <code>=</code> is the expression we are defining, and the right hand side of the <code>=</code> is the definition.</p>
-<p>In general a definition (of a function or constant) consists of an optional <em>type annotation</em> and one or more <em>equations</em></p>
-<div class="sourceCode" id="cb24"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb24-1"><a href="#cb24-1" aria-hidden="true" tabindex="-1"></a><span class="ot">polynomial ::</span> <span class="dt">Double</span> <span class="ot">-&gt;</span> <span class="dt">Double</span></span>
+<p>This is a definition of the constant <code>phi</code>, with an
+accompanying <em>type annotation</em> (also known as a <em>type
+signature</em>) <code>phi :: Double</code>. The type annotation means
+that <code>phi</code> has type <code>Double</code>. The line with a
+equals sign (<code>=</code>) is called an <em>equation</em>. The left
+hand side of the <code>=</code> is the expression we are defining, and
+the right hand side of the <code>=</code> is the definition.</p>
+<p>In general a definition (of a function or constant) consists of an
+optional <em>type annotation</em> and one or more <em>equations</em></p>
+<div class="sourceCode" id="cb24"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb24-1"><a href="#cb24-1" aria-hidden="true" tabindex="-1"></a><span class="ot">polynomial ::</span> <span class="dt">Double</span> <span class="ot">-&gt;</span> <span class="dt">Double</span></span>
 <span id="cb24-2"><a href="#cb24-2" aria-hidden="true" tabindex="-1"></a>polynomial x <span class="ot">=</span> x<span class="op">^</span><span class="dv">2</span> <span class="op">-</span> x <span class="op">-</span> <span class="dv">1</span></span></code></pre></div>
-<p>This is the definition of a function called <code>polynomial</code>. It has a type annotation and an equation. Note how an equation for a function differs from the equation of a constant by the presence of a parameter <code>x</code> left of the <code>=</code> sign. Note also that <code>^</code> is the power operator in Haskell, not bitwise xor like in many other languages.</p>
-<div class="sourceCode" id="cb25"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb25-1"><a href="#cb25-1" aria-hidden="true" tabindex="-1"></a>f x <span class="ot">=</span> polynomial (polynomial x)</span></code></pre></div>
-<p>This is the definition of a function called <code>f</code>. Note the lack of type annotation. What is the type of <code>f</code>?</p>
-<div class="sourceCode" id="cb26"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb26-1"><a href="#cb26-1" aria-hidden="true" tabindex="-1"></a>main <span class="ot">=</span> <span class="kw">do</span></span>
+<p>This is the definition of a function called <code>polynomial</code>.
+It has a type annotation and an equation. Note how an equation for a
+function differs from the equation of a constant by the presence of a
+parameter <code>x</code> left of the <code>=</code> sign. Note also that
+<code>^</code> is the power operator in Haskell, not bitwise xor like in
+many other languages.</p>
+<div class="sourceCode" id="cb25"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb25-1"><a href="#cb25-1" aria-hidden="true" tabindex="-1"></a>f x <span class="ot">=</span> polynomial (polynomial x)</span></code></pre></div>
+<p>This is the definition of a function called <code>f</code>. Note the
+lack of type annotation. What is the type of <code>f</code>?</p>
+<div class="sourceCode" id="cb26"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb26-1"><a href="#cb26-1" aria-hidden="true" tabindex="-1"></a>main <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb26-2"><a href="#cb26-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">print</span> (polynomial phi)</span>
 <span id="cb26-3"><a href="#cb26-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">print</span> (f phi)</span></code></pre></div>
-<p>This is a description of what happens when you run the program. It uses do-syntax and the IO Monad. We’ll get back to those in part 2 of the course.</p>
-<h2 data-number="1.8" id="working-with-examples"><span class="header-section-number">1.8</span> Working with Examples</h2>
+<p>This is a description of what happens when you run the program. It
+uses do-syntax and the IO Monad. We’ll get back to those in part 2 of
+the course.</p>
+<h2 data-number="1.8" id="working-with-examples"><span
+class="header-section-number">1.8</span> Working with Examples</h2>
 <p>When you see an example definition like this</p>
-<div class="sourceCode" id="cb27"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb27-1"><a href="#cb27-1" aria-hidden="true" tabindex="-1"></a><span class="ot">polynomial ::</span> <span class="dt">Double</span> <span class="ot">-&gt;</span> <span class="dt">Double</span></span>
+<div class="sourceCode" id="cb27"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb27-1"><a href="#cb27-1" aria-hidden="true" tabindex="-1"></a><span class="ot">polynomial ::</span> <span class="dt">Double</span> <span class="ot">-&gt;</span> <span class="dt">Double</span></span>
 <span id="cb27-2"><a href="#cb27-2" aria-hidden="true" tabindex="-1"></a>polynomial x <span class="ot">=</span> x<span class="op">^</span><span class="dv">2</span> <span class="op">-</span> x <span class="op">-</span> <span class="dv">1</span></span></code></pre></div>
-<p>you should usually play around with it. Start by running it. There are a couple of ways to do this.</p>
+<p>you should usually play around with it. Start by running it. There
+are a couple of ways to do this.</p>
 <p>If a definition fits on one line, you can just define it in GHCi:</p>
 <pre><code>Prelude&gt; polynomial x = x^2 - x - 1
 Prelude&gt; polynomial 3.0
 5.0</code></pre>
-<p>For a multi-line definition, you can either use <code>;</code> to separate lines, or use the special <code>:{ :}</code> syntax to paste a block of code into GHCi:</p>
-<div class="sourceCode" id="cb29"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb29-1"><a href="#cb29-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>{</span>
+<p>For a multi-line definition, you can either use <code>;</code> to
+separate lines, or use the special <code>:{ :}</code> syntax to paste a
+block of code into GHCi:</p>
+<div class="sourceCode" id="cb29"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb29-1"><a href="#cb29-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>{</span>
 <span id="cb29-2"><a href="#cb29-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">|</span><span class="ot"> polynomial ::</span> <span class="dt">Double</span> <span class="ot">-&gt;</span> <span class="dt">Double</span></span>
 <span id="cb29-3"><a href="#cb29-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">|</span> polynomial x <span class="ot">=</span> x<span class="op">^</span><span class="dv">2</span> <span class="op">-</span> x <span class="op">-</span> <span class="dv">1</span></span>
 <span id="cb29-4"><a href="#cb29-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">|</span> <span class="op">:</span>}</span>
 <span id="cb29-5"><a href="#cb29-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> polynomial <span class="fl">3.0</span></span>
 <span id="cb29-6"><a href="#cb29-6" aria-hidden="true" tabindex="-1"></a><span class="fl">5.0</span></span></code></pre></div>
-<p>Finally, you can paste the code into a new or existing <code>.hs</code> file, and then <code>:load</code> it into GHCi. If the file has already been loaded, you can also use <code>:reload</code>.</p>
-<div class="sourceCode" id="cb30"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb30-1"><a href="#cb30-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- first copy and paste the definition into Example.hs, then run GHCi</span></span>
+<p>Finally, you can paste the code into a new or existing
+<code>.hs</code> file, and then <code>:load</code> it into GHCi. If the
+file has already been loaded, you can also use <code>:reload</code>.</p>
+<div class="sourceCode" id="cb30"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb30-1"><a href="#cb30-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- first copy and paste the definition into Example.hs, then run GHCi</span></span>
 <span id="cb30-2"><a href="#cb30-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>load Example.hs</span>
 <span id="cb30-3"><a href="#cb30-3" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span> <span class="kw">of</span> <span class="dv">1</span>] <span class="dt">Compiling</span> <span class="dt">Main</span>             ( Example.hs, interpreted )</span>
 <span id="cb30-4"><a href="#cb30-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Ok</span>, one <span class="kw">module</span> loaded<span class="op">.</span></span>
@@ -558,133 +936,249 @@ Prelude&gt; polynomial 3.0
 <span id="cb30-10"><a href="#cb30-10" aria-hidden="true" tabindex="-1"></a><span class="dt">Ok</span>, one <span class="kw">module</span> loaded<span class="op">.</span></span>
 <span id="cb30-11"><a href="#cb30-11" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Main</span><span class="op">&gt;</span> polynomial <span class="dv">3</span></span>
 <span id="cb30-12"><a href="#cb30-12" aria-hidden="true" tabindex="-1"></a><span class="fl">3.0</span></span></code></pre></div>
-<p>After you’ve run the example, try modifying it, or making another function that is similar but different. You learn programming by programming, not by reading!</p>
-<h3 data-number="1.8.1" id="dealing-with-errors"><span class="header-section-number">1.8.1</span> Dealing with Errors</h3>
-<p>Since Haskell is a typed language, you’ll pretty quickly bump into type errors. Here’s an example of an error during a GHCi session:</p>
-<div class="sourceCode" id="cb31"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb31-1"><a href="#cb31-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="st">&quot;string&quot;</span> <span class="op">++</span> <span class="dt">True</span></span>
+<p>After you’ve run the example, try modifying it, or making another
+function that is similar but different. You learn programming by
+programming, not by reading!</p>
+<h3 data-number="1.8.1" id="dealing-with-errors"><span
+class="header-section-number">1.8.1</span> Dealing with Errors</h3>
+<p>Since Haskell is a typed language, you’ll pretty quickly bump into
+type errors. Here’s an example of an error during a GHCi session:</p>
+<div class="sourceCode" id="cb31"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb31-1"><a href="#cb31-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="st">&quot;string&quot;</span> <span class="op">++</span> <span class="dt">True</span></span>
 <span id="cb31-2"><a href="#cb31-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb31-3"><a href="#cb31-3" aria-hidden="true" tabindex="-1"></a><span class="op">&lt;</span>interactive<span class="op">&gt;:</span><span class="dv">1</span><span class="op">:</span><span class="dv">13</span><span class="op">:</span> <span class="fu">error</span><span class="op">:</span></span>
 <span id="cb31-4"><a href="#cb31-4" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">Couldn&#39;t</span> match expected <span class="kw">type</span> ‘[<span class="dt">Char</span>]’ with actual <span class="kw">type</span> ‘<span class="dt">Bool</span>’</span>
 <span id="cb31-5"><a href="#cb31-5" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">In</span> the second argument <span class="kw">of</span> ‘(<span class="op">++</span>)’, namely ‘<span class="dt">True</span>’</span>
 <span id="cb31-6"><a href="#cb31-6" aria-hidden="true" tabindex="-1"></a>      <span class="dt">In</span> the expression<span class="op">:</span> <span class="st">&quot;string&quot;</span> <span class="op">++</span> <span class="dt">True</span></span>
 <span id="cb31-7"><a href="#cb31-7" aria-hidden="true" tabindex="-1"></a>      <span class="dt">In</span> an equation for ‘it’<span class="op">:</span> it <span class="ot">=</span> <span class="st">&quot;string&quot;</span> <span class="op">++</span> <span class="dt">True</span></span></code></pre></div>
-<p>This is the most common type error, “Couldn’t match expected type”. Even though the error looks long and scary, it’s pretty simple if you just read through it.</p>
+<p>This is the most common type error, “Couldn’t match expected type”.
+Even though the error looks long and scary, it’s pretty simple if you
+just read through it.</p>
 <ul>
-<li><p>The first line of the error message, <code>&lt;interactive&gt;:1:13: error:</code> tells us that the error occurred in GHCi. If we had loaded a file, we might instead get something like <code>Sandbox.hs:3:17: error:</code>, where <code>Sandbox.hs</code> is the name of the file, <code>3</code> is the line number and <code>17</code> is the number of a character in the line.</p></li>
-<li><p>The line <code>• Couldn't match expected type ‘[Char]’ with actual type ‘Bool’</code> tells us that the immediate cause for the error is that there was an expression of type <code>Bool</code>, when GHCi was expecting to find an expression of type <code>[Char]</code>“. The location of this error was indicated in the first line of the error message. Note that the expected type is not always right. Giving type annotations by hand can help debugging typing errors.</p></li>
-<li><p>The line <code>• In the second argument of ‘(++)’, namely ‘True’</code> tells that the expression that had the wrong type was the second argument of the operator <code>(++)</code>. We’ll learn later why it’s surrounded by parentheses.</p></li>
-<li><p>The full expression with the error was <code>"string" ++ True</code>. As mentioned above, <code>String</code> is a type alias for <code>[Char]</code>, the type of character lists. The first argument to <code>++</code> was a list of characters, and since <code>++</code> can only combine two lists of the same type, the second argument should’ve been of type <code>[Char]</code> too.</p></li>
-<li><p>The line <code>In an equation for ‘it’: it = "string" ++ True</code> says that the expression occurred in the definition of the variable <code>it</code>, which is a default variable name that GHCi uses for standalone expressions. If we had a line <code>x = "string" ++ True</code> in a file, or a declaration <code>let x = "string" ++ True</code> in GHCi, GHCi would print <code>In an equation for ‘x’: x = "string" ++ True</code> instead.</p></li>
+<li><p>The first line of the error message,
+<code>&lt;interactive&gt;:1:13: error:</code> tells us that the error
+occurred in GHCi. If we had loaded a file, we might instead get
+something like <code>Sandbox.hs:3:17: error:</code>, where
+<code>Sandbox.hs</code> is the name of the file, <code>3</code> is the
+line number and <code>17</code> is the number of a character in the
+line.</p></li>
+<li><p>The line
+<code>• Couldn't match expected type ‘[Char]’ with actual type ‘Bool’</code>
+tells us that the immediate cause for the error is that there was an
+expression of type <code>Bool</code>, when GHCi was expecting to find an
+expression of type <code>[Char]</code>“. The location of this error was
+indicated in the first line of the error message. Note that the expected
+type is not always right. Giving type annotations by hand can help
+debugging typing errors.</p></li>
+<li><p>The line
+<code>• In the second argument of ‘(++)’, namely ‘True’</code> tells
+that the expression that had the wrong type was the second argument of
+the operator <code>(++)</code>. We’ll learn later why it’s surrounded by
+parentheses.</p></li>
+<li><p>The full expression with the error was
+<code>"string" ++ True</code>. As mentioned above, <code>String</code>
+is a type alias for <code>[Char]</code>, the type of character lists.
+The first argument to <code>++</code> was a list of characters, and
+since <code>++</code> can only combine two lists of the same type, the
+second argument should’ve been of type <code>[Char]</code> too.</p></li>
+<li><p>The line
+<code>In an equation for ‘it’: it = "string" ++ True</code> says that
+the expression occurred in the definition of the variable
+<code>it</code>, which is a default variable name that GHCi uses for
+standalone expressions. If we had a line
+<code>x = "string" ++ True</code> in a file, or a declaration
+<code>let x = "string" ++ True</code> in GHCi, GHCi would print
+<code>In an equation for ‘x’: x = "string" ++ True</code>
+instead.</p></li>
 </ul>
 <p>There are also others types of errors.</p>
-<div class="sourceCode" id="cb32"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb32-1"><a href="#cb32-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">True</span> <span class="op">+</span> <span class="dv">1</span></span>
+<div class="sourceCode" id="cb32"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb32-1"><a href="#cb32-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">True</span> <span class="op">+</span> <span class="dv">1</span></span>
 <span id="cb32-2"><a href="#cb32-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb32-3"><a href="#cb32-3" aria-hidden="true" tabindex="-1"></a><span class="op">&lt;</span>interactive<span class="op">&gt;:</span><span class="dv">6</span><span class="op">:</span><span class="dv">1</span><span class="op">:</span> <span class="fu">error</span><span class="op">:</span></span>
 <span id="cb32-4"><a href="#cb32-4" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">No</span> <span class="kw">instance</span> for (<span class="dt">Num</span> <span class="dt">Bool</span>) arising from a use <span class="kw">of</span> ‘<span class="op">+</span>’</span>
 <span id="cb32-5"><a href="#cb32-5" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">In</span> the expression<span class="op">:</span> <span class="dt">True</span> <span class="op">+</span> <span class="dv">1</span></span>
 <span id="cb32-6"><a href="#cb32-6" aria-hidden="true" tabindex="-1"></a>      <span class="dt">In</span> an equation for ‘it’<span class="op">:</span> it <span class="ot">=</span> <span class="dt">True</span> <span class="op">+</span> <span class="dv">1</span></span></code></pre></div>
-<p>This is the kind of error you get when you try to use a numeric function like <code>+</code> on something that’s not a number.</p>
+<p>This is the kind of error you get when you try to use a numeric
+function like <code>+</code> on something that’s not a number.</p>
 <p>The hardest error to track down is usually this:</p>
-<div class="sourceCode" id="cb33"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb33-1"><a href="#cb33-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">True</span> <span class="op">+</span></span>
+<div class="sourceCode" id="cb33"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb33-1"><a href="#cb33-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">True</span> <span class="op">+</span></span>
 <span id="cb33-2"><a href="#cb33-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb33-3"><a href="#cb33-3" aria-hidden="true" tabindex="-1"></a><span class="op">&lt;</span>interactive<span class="op">&gt;:</span><span class="dv">10</span><span class="op">:</span><span class="dv">7</span><span class="op">:</span> <span class="fu">error</span><span class="op">:</span></span>
 <span id="cb33-4"><a href="#cb33-4" aria-hidden="true" tabindex="-1"></a>    parse <span class="fu">error</span> (possibly incorrect indentation <span class="fu">or</span> mismatched brackets)</span></code></pre></div>
-<p>There are many ways to cause it. Probably you’re missing some characters somewhere. We’ll get back to indentation later in this lecture.</p>
+<p>There are many ways to cause it. Probably you’re missing some
+characters somewhere. We’ll get back to indentation later in this
+lecture.</p>
 <!-- TODO more errors based on feedback -->
-<h3 data-number="1.8.2" id="arithmetic"><span class="header-section-number">1.8.2</span> Arithmetic</h3>
-<p>There’s one thing in Haskell arithmetic that often trips up beginners, and that’s division.</p>
-<p>In Haskell there are two division functions, the <code>/</code> operator and the <code>div</code> function. The <code>div</code> function does integer division:</p>
-<div class="sourceCode" id="cb34"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb34-1"><a href="#cb34-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dv">7</span> <span class="ot">`div`</span> <span class="dv">2</span></span>
+<h3 data-number="1.8.2" id="arithmetic"><span
+class="header-section-number">1.8.2</span> Arithmetic</h3>
+<p>There’s one thing in Haskell arithmetic that often trips up
+beginners, and that’s division.</p>
+<p>In Haskell there are two division functions, the <code>/</code>
+operator and the <code>div</code> function. The <code>div</code>
+function does integer division:</p>
+<div class="sourceCode" id="cb34"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb34-1"><a href="#cb34-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dv">7</span> <span class="ot">`div`</span> <span class="dv">2</span></span>
 <span id="cb34-2"><a href="#cb34-2" aria-hidden="true" tabindex="-1"></a><span class="dv">3</span></span></code></pre></div>
 <p>The <code>/</code> operator performs the usual division:</p>
-<div class="sourceCode" id="cb35"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb35-1"><a href="#cb35-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fl">7.0</span> <span class="op">/</span> <span class="fl">2.0</span></span>
+<div class="sourceCode" id="cb35"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb35-1"><a href="#cb35-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fl">7.0</span> <span class="op">/</span> <span class="fl">2.0</span></span>
 <span id="cb35-2"><a href="#cb35-2" aria-hidden="true" tabindex="-1"></a><span class="fl">3.5</span></span></code></pre></div>
-<p>However, you can only use <code>div</code> on whole number types like <code>Int</code> and <code>Integer</code>, and you can only use <code>/</code> on decimal types like <code>Double</code>. Here’s an example of what happens if you try to mix them up:</p>
-<div class="sourceCode" id="cb36"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb36-1"><a href="#cb36-1" aria-hidden="true" tabindex="-1"></a><span class="ot">halve ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<p>However, you can only use <code>div</code> on whole number types like
+<code>Int</code> and <code>Integer</code>, and you can only use
+<code>/</code> on decimal types like <code>Double</code>. Here’s an
+example of what happens if you try to mix them up:</p>
+<div class="sourceCode" id="cb36"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb36-1"><a href="#cb36-1" aria-hidden="true" tabindex="-1"></a><span class="ot">halve ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb36-2"><a href="#cb36-2" aria-hidden="true" tabindex="-1"></a>halve x <span class="ot">=</span> x <span class="op">/</span> <span class="dv">2</span></span></code></pre></div>
 <pre><code>error:
     • No instance for (Fractional Int) arising from a use of ‘/’
     • In the expression: x / 2
       In an equation for ‘halve’: halve x = x / 2</code></pre>
-<p>Just try to keep this in mind for now. We’ll get back to the difference between <code>/</code> and <code>div</code>, and what <code>Num</code> and <code>Fractional</code> mean when talking about type classes.</p>
-<h2 data-number="1.9" id="how-do-i-get-anything-done"><span class="header-section-number">1.9</span> How Do I Get Anything Done?</h2>
-<p>So far you’ve seen some arithmetic, reversing a string and so on. How does one write actual programs in Haskell? Many of the usual programming constructs like loops, statements and assignment are missing from Haskell. Next, we’ll go through the basic building blocks of Haskell programs:</p>
+<p>Just try to keep this in mind for now. We’ll get back to the
+difference between <code>/</code> and <code>div</code>, and what
+<code>Num</code> and <code>Fractional</code> mean when talking about
+type classes.</p>
+<h2 data-number="1.9" id="how-do-i-get-anything-done"><span
+class="header-section-number">1.9</span> How Do I Get Anything
+Done?</h2>
+<p>So far you’ve seen some arithmetic, reversing a string and so on. How
+does one write actual programs in Haskell? Many of the usual programming
+constructs like loops, statements and assignment are missing from
+Haskell. Next, we’ll go through the basic building blocks of Haskell
+programs:</p>
 <ul>
 <li>Conditionals</li>
 <li>Local definitions</li>
 <li>Pattern matching</li>
 <li>Recursion</li>
 </ul>
-<h3 data-number="1.9.1" id="conditionals"><span class="header-section-number">1.9.1</span> Conditionals</h3>
-<p>In other languages, <code>if</code> is a <em>statement</em>. It doesn’t have a value, it just conditionally executes other statements.</p>
-<p>In Haskell, <code>if</code> is an <em>expression</em>. It has a value. It selects between two other expressions. It corresponds to the <code>?:</code> operator in C or Java.</p>
-<div class="sourceCode" id="cb38"><pre class="sourceCode java"><code class="sourceCode java"><span id="cb38-1"><a href="#cb38-1" aria-hidden="true" tabindex="-1"></a><span class="co">// Java</span></span>
+<h3 data-number="1.9.1" id="conditionals"><span
+class="header-section-number">1.9.1</span> Conditionals</h3>
+<p>In other languages, <code>if</code> is a <em>statement</em>. It
+doesn’t have a value, it just conditionally executes other
+statements.</p>
+<p>In Haskell, <code>if</code> is an <em>expression</em>. It has a
+value. It selects between two other expressions. It corresponds to the
+<code>?:</code> operator in C or Java.</p>
+<div class="sourceCode" id="cb38"><pre
+class="sourceCode java"><code class="sourceCode java"><span id="cb38-1"><a href="#cb38-1" aria-hidden="true" tabindex="-1"></a><span class="co">// Java</span></span>
 <span id="cb38-2"><a href="#cb38-2" aria-hidden="true" tabindex="-1"></a><span class="dt">int</span> price <span class="op">=</span> product<span class="op">.</span><span class="fu">equals</span><span class="op">(</span><span class="st">&quot;milk&quot;</span><span class="op">)</span> <span class="op">?</span> <span class="dv">1</span> <span class="op">:</span> <span class="dv">2</span><span class="op">;</span></span></code></pre></div>
-<p>Python’s conditional expressions are quite close to haskell’s <code>if</code>:</p>
-<div class="sourceCode" id="cb39"><pre class="sourceCode python"><code class="sourceCode python"><span id="cb39-1"><a href="#cb39-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Python</span></span>
+<p>Python’s conditional expressions are quite close to haskell’s
+<code>if</code>:</p>
+<div class="sourceCode" id="cb39"><pre
+class="sourceCode python"><code class="sourceCode python"><span id="cb39-1"><a href="#cb39-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Python</span></span>
 <span id="cb39-2"><a href="#cb39-2" aria-hidden="true" tabindex="-1"></a>price <span class="op">=</span> <span class="dv">1</span> <span class="cf">if</span> product <span class="op">==</span> <span class="st">&quot;milk&quot;</span> <span class="cf">else</span> <span class="dv">2</span></span></code></pre></div>
 <p>This is how the same example looks in Haskell:</p>
-<div class="sourceCode" id="cb40"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb40-1"><a href="#cb40-1" aria-hidden="true" tabindex="-1"></a>price <span class="ot">=</span> <span class="kw">if</span> <span class="fu">product</span> <span class="op">==</span> <span class="st">&quot;milk&quot;</span> <span class="kw">then</span> <span class="dv">1</span> <span class="kw">else</span> <span class="dv">2</span></span></code></pre></div>
-<p>Because Haskell’s <code>if</code> <em>returns</em> a value, you <strong>always</strong> need an <code>else</code>!</p>
-<h4 data-number="1.9.1.1" id="functions-returning-bool"><span class="header-section-number">1.9.1.1</span> Functions Returning <code>Bool</code></h4>
-<p>In order to write if expressions, you need to know how to get values of type <code>Bool</code>. The most common way is comparisons. The usual <code>==</code>, <code>&lt;</code>, <code>&lt;=</code>, <code>&gt;</code> and <code>&gt;=</code> operators work in Haskell. You can do ordered comparisons (<code>&lt;</code>, <code>&gt;</code>) on all sorts of numbers, and equality comparisons (<code>==</code>) on almost anything:</p>
-<div class="sourceCode" id="cb41"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb41-1"><a href="#cb41-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="st">&quot;foo&quot;</span> <span class="op">==</span> <span class="st">&quot;bar&quot;</span></span>
+<div class="sourceCode" id="cb40"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb40-1"><a href="#cb40-1" aria-hidden="true" tabindex="-1"></a>price <span class="ot">=</span> <span class="kw">if</span> <span class="fu">product</span> <span class="op">==</span> <span class="st">&quot;milk&quot;</span> <span class="kw">then</span> <span class="dv">1</span> <span class="kw">else</span> <span class="dv">2</span></span></code></pre></div>
+<p>Because Haskell’s <code>if</code> <em>returns</em> a value, you
+<strong>always</strong> need an <code>else</code>!</p>
+<h4 data-number="1.9.1.1" id="functions-returning-bool"><span
+class="header-section-number">1.9.1.1</span> Functions Returning
+<code>Bool</code></h4>
+<p>In order to write if expressions, you need to know how to get values
+of type <code>Bool</code>. The most common way is comparisons. The usual
+<code>==</code>, <code>&lt;</code>, <code>&lt;=</code>,
+<code>&gt;</code> and <code>&gt;=</code> operators work in Haskell. You
+can do ordered comparisons (<code>&lt;</code>, <code>&gt;</code>) on all
+sorts of numbers, and equality comparisons (<code>==</code>) on almost
+anything:</p>
+<div class="sourceCode" id="cb41"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb41-1"><a href="#cb41-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="st">&quot;foo&quot;</span> <span class="op">==</span> <span class="st">&quot;bar&quot;</span></span>
 <span id="cb41-2"><a href="#cb41-2" aria-hidden="true" tabindex="-1"></a><span class="dt">False</span></span>
 <span id="cb41-3"><a href="#cb41-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fl">5.0</span> <span class="op">&lt;=</span> <span class="fl">7.2</span></span>
 <span id="cb41-4"><a href="#cb41-4" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span>
 <span id="cb41-5"><a href="#cb41-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dv">1</span> <span class="op">==</span> <span class="dv">1</span></span>
 <span id="cb41-6"><a href="#cb41-6" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span></code></pre></div>
-<p>One oddity of Haskell is that the not-equals operator is written <code>/=</code> instead of the usual <code>!=</code>:</p>
-<div class="sourceCode" id="cb42"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb42-1"><a href="#cb42-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dv">2</span> <span class="op">/=</span> <span class="dv">3</span></span>
+<p>One oddity of Haskell is that the not-equals operator is written
+<code>/=</code> instead of the usual <code>!=</code>:</p>
+<div class="sourceCode" id="cb42"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb42-1"><a href="#cb42-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dv">2</span> <span class="op">/=</span> <span class="dv">3</span></span>
 <span id="cb42-2"><a href="#cb42-2" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span>
 <span id="cb42-3"><a href="#cb42-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="st">&quot;bike&quot;</span> <span class="op">/=</span> <span class="st">&quot;bike&quot;</span></span>
 <span id="cb42-4"><a href="#cb42-4" aria-hidden="true" tabindex="-1"></a><span class="dt">False</span></span></code></pre></div>
-<p>Remember that in addition to these comparisons, you can get <code>Bool</code> values out of other <code>Bool</code> values by using the <code>&amp;&amp;</code> (“and”) and <code>||</code> (“or”) operators, and the <code>not</code> function.</p>
-<h4 data-number="1.9.1.2" id="examples"><span class="header-section-number">1.9.1.2</span> Examples</h4>
-<div class="sourceCode" id="cb43"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb43-1"><a href="#cb43-1" aria-hidden="true" tabindex="-1"></a>checkPassword password <span class="ot">=</span> <span class="kw">if</span> password <span class="op">==</span> <span class="st">&quot;swordfish&quot;</span></span>
+<p>Remember that in addition to these comparisons, you can get
+<code>Bool</code> values out of other <code>Bool</code> values by using
+the <code>&amp;&amp;</code> (“and”) and <code>||</code> (“or”)
+operators, and the <code>not</code> function.</p>
+<h4 data-number="1.9.1.2" id="examples"><span
+class="header-section-number">1.9.1.2</span> Examples</h4>
+<div class="sourceCode" id="cb43"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb43-1"><a href="#cb43-1" aria-hidden="true" tabindex="-1"></a>checkPassword password <span class="ot">=</span> <span class="kw">if</span> password <span class="op">==</span> <span class="st">&quot;swordfish&quot;</span></span>
 <span id="cb43-2"><a href="#cb43-2" aria-hidden="true" tabindex="-1"></a>                         <span class="kw">then</span> <span class="st">&quot;You&#39;re in.&quot;</span></span>
 <span id="cb43-3"><a href="#cb43-3" aria-hidden="true" tabindex="-1"></a>                         <span class="kw">else</span> <span class="st">&quot;ACCESS DENIED!&quot;</span></span></code></pre></div>
-<div class="sourceCode" id="cb44"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb44-1"><a href="#cb44-1" aria-hidden="true" tabindex="-1"></a>absoluteValue n <span class="ot">=</span> <span class="kw">if</span> n <span class="op">&lt;</span> <span class="dv">0</span> <span class="kw">then</span> <span class="op">-</span>n <span class="kw">else</span> n</span></code></pre></div>
-<div class="sourceCode" id="cb45"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb45-1"><a href="#cb45-1" aria-hidden="true" tabindex="-1"></a>login user password <span class="ot">=</span> <span class="kw">if</span> user <span class="op">==</span> <span class="st">&quot;unicorn73&quot;</span></span>
+<div class="sourceCode" id="cb44"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb44-1"><a href="#cb44-1" aria-hidden="true" tabindex="-1"></a>absoluteValue n <span class="ot">=</span> <span class="kw">if</span> n <span class="op">&lt;</span> <span class="dv">0</span> <span class="kw">then</span> <span class="op">-</span>n <span class="kw">else</span> n</span></code></pre></div>
+<div class="sourceCode" id="cb45"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb45-1"><a href="#cb45-1" aria-hidden="true" tabindex="-1"></a>login user password <span class="ot">=</span> <span class="kw">if</span> user <span class="op">==</span> <span class="st">&quot;unicorn73&quot;</span></span>
 <span id="cb45-2"><a href="#cb45-2" aria-hidden="true" tabindex="-1"></a>                      <span class="kw">then</span> <span class="kw">if</span> password <span class="op">==</span> <span class="st">&quot;f4bulous!&quot;</span></span>
 <span id="cb45-3"><a href="#cb45-3" aria-hidden="true" tabindex="-1"></a>                           <span class="kw">then</span> <span class="st">&quot;unicorn73 logged in&quot;</span></span>
 <span id="cb45-4"><a href="#cb45-4" aria-hidden="true" tabindex="-1"></a>                           <span class="kw">else</span> <span class="st">&quot;wrong password&quot;</span></span>
 <span id="cb45-5"><a href="#cb45-5" aria-hidden="true" tabindex="-1"></a>                      <span class="kw">else</span> <span class="st">&quot;unknown user&quot;</span></span></code></pre></div>
-<h3 data-number="1.9.2" id="local-definitions"><span class="header-section-number">1.9.2</span> Local Definitions</h3>
-<p>Haskell has two different ways for creating local definitions: <code>let...in</code> and <code>where</code>.</p>
+<h3 data-number="1.9.2" id="local-definitions"><span
+class="header-section-number">1.9.2</span> Local Definitions</h3>
+<p>Haskell has two different ways for creating local definitions:
+<code>let...in</code> and <code>where</code>.</p>
 <p><code>where</code> adds local definitions to a definition:</p>
-<div class="sourceCode" id="cb46"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb46-1"><a href="#cb46-1" aria-hidden="true" tabindex="-1"></a><span class="ot">circleArea ::</span> <span class="dt">Double</span> <span class="ot">-&gt;</span> <span class="dt">Double</span></span>
+<div class="sourceCode" id="cb46"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb46-1"><a href="#cb46-1" aria-hidden="true" tabindex="-1"></a><span class="ot">circleArea ::</span> <span class="dt">Double</span> <span class="ot">-&gt;</span> <span class="dt">Double</span></span>
 <span id="cb46-2"><a href="#cb46-2" aria-hidden="true" tabindex="-1"></a>circleArea r <span class="ot">=</span> <span class="fu">pi</span> <span class="op">*</span> rsquare</span>
 <span id="cb46-3"><a href="#cb46-3" aria-hidden="true" tabindex="-1"></a>    <span class="kw">where</span> <span class="fu">pi</span> <span class="ot">=</span> <span class="fl">3.1415926</span></span>
 <span id="cb46-4"><a href="#cb46-4" aria-hidden="true" tabindex="-1"></a>          rsquare <span class="ot">=</span> r <span class="op">*</span> r</span></code></pre></div>
 <p><code>let...in</code> is an expression:</p>
-<div class="sourceCode" id="cb47"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb47-1"><a href="#cb47-1" aria-hidden="true" tabindex="-1"></a>circleArea r <span class="ot">=</span> <span class="kw">let</span> <span class="fu">pi</span> <span class="ot">=</span> <span class="fl">3.1415926</span></span>
+<div class="sourceCode" id="cb47"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb47-1"><a href="#cb47-1" aria-hidden="true" tabindex="-1"></a>circleArea r <span class="ot">=</span> <span class="kw">let</span> <span class="fu">pi</span> <span class="ot">=</span> <span class="fl">3.1415926</span></span>
 <span id="cb47-2"><a href="#cb47-2" aria-hidden="true" tabindex="-1"></a>                   rsquare <span class="ot">=</span> r <span class="op">*</span> r</span>
 <span id="cb47-3"><a href="#cb47-3" aria-hidden="true" tabindex="-1"></a>               <span class="kw">in</span> <span class="fu">pi</span> <span class="op">*</span> rsquare</span></code></pre></div>
 <p>Local definitions can also be functions:</p>
-<div class="sourceCode" id="cb48"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb48-1"><a href="#cb48-1" aria-hidden="true" tabindex="-1"></a>circleArea r <span class="ot">=</span> <span class="fu">pi</span> <span class="op">*</span> square r</span>
+<div class="sourceCode" id="cb48"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb48-1"><a href="#cb48-1" aria-hidden="true" tabindex="-1"></a>circleArea r <span class="ot">=</span> <span class="fu">pi</span> <span class="op">*</span> square r</span>
 <span id="cb48-2"><a href="#cb48-2" aria-hidden="true" tabindex="-1"></a>    <span class="kw">where</span> <span class="fu">pi</span> <span class="ot">=</span> <span class="fl">3.1415926</span></span>
 <span id="cb48-3"><a href="#cb48-3" aria-hidden="true" tabindex="-1"></a>          square x <span class="ot">=</span> x <span class="op">*</span> x</span></code></pre></div>
-<div class="sourceCode" id="cb49"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb49-1"><a href="#cb49-1" aria-hidden="true" tabindex="-1"></a>circleArea r <span class="ot">=</span> <span class="kw">let</span> <span class="fu">pi</span> <span class="ot">=</span> <span class="fl">3.1415926</span></span>
+<div class="sourceCode" id="cb49"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb49-1"><a href="#cb49-1" aria-hidden="true" tabindex="-1"></a>circleArea r <span class="ot">=</span> <span class="kw">let</span> <span class="fu">pi</span> <span class="ot">=</span> <span class="fl">3.1415926</span></span>
 <span id="cb49-2"><a href="#cb49-2" aria-hidden="true" tabindex="-1"></a>                   square x <span class="ot">=</span> x <span class="op">*</span> x</span>
 <span id="cb49-3"><a href="#cb49-3" aria-hidden="true" tabindex="-1"></a>               <span class="kw">in</span> <span class="fu">pi</span> <span class="op">*</span> square r</span></code></pre></div>
-<p>We’ll get back to the differences between <code>let</code> and <code>where</code>, but mostly you can use which ever you like.</p>
-<h3 data-number="1.9.3" id="a-word-about-immutability"><span class="header-section-number">1.9.3</span> A Word About Immutability</h3>
-<p>Even though things like <code>pi</code> above are often called <em>variables</em>, I’ve chosen to call them <em>definitions</em> here. This is because unlike variables in Python or Java, the values of these definitions can’t be changed. Haskell variables aren’t boxes into which you can put new values, Haskell variables name a value (or rather, an expression) and that’s it.</p>
-<p>We’ll talk about immutability again later on this course, but for now it’s enough to know that things like this don’t work.</p>
-<div class="sourceCode" id="cb50"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb50-1"><a href="#cb50-1" aria-hidden="true" tabindex="-1"></a>increment x <span class="ot">=</span> <span class="kw">let</span> x <span class="ot">=</span> x<span class="op">+</span><span class="dv">1</span></span>
+<p>We’ll get back to the differences between <code>let</code> and
+<code>where</code>, but mostly you can use which ever you like.</p>
+<h3 data-number="1.9.3" id="a-word-about-immutability"><span
+class="header-section-number">1.9.3</span> A Word About
+Immutability</h3>
+<p>Even though things like <code>pi</code> above are often called
+<em>variables</em>, I’ve chosen to call them <em>definitions</em> here.
+This is because unlike variables in Python or Java, the values of these
+definitions can’t be changed. Haskell variables aren’t boxes into which
+you can put new values, Haskell variables name a value (or rather, an
+expression) and that’s it.</p>
+<p>We’ll talk about immutability again later on this course, but for now
+it’s enough to know that things like this don’t work.</p>
+<div class="sourceCode" id="cb50"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb50-1"><a href="#cb50-1" aria-hidden="true" tabindex="-1"></a>increment x <span class="ot">=</span> <span class="kw">let</span> x <span class="ot">=</span> x<span class="op">+</span><span class="dv">1</span></span>
 <span id="cb50-2"><a href="#cb50-2" aria-hidden="true" tabindex="-1"></a>              <span class="kw">in</span> x</span></code></pre></div>
-<p>This is just an infinite loop, because it tries to define a new variable <code>x</code> with the property <code>x = x+1</code>. Thus when evaluating <code>x</code>, Haskell just keeps computing <code>1+1+1+1+...</code> indefinitely.</p>
-<div class="sourceCode" id="cb51"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb51-1"><a href="#cb51-1" aria-hidden="true" tabindex="-1"></a>compute x <span class="ot">=</span> <span class="kw">let</span> a <span class="ot">=</span> x<span class="op">+</span><span class="dv">1</span></span>
+<p>This is just an infinite loop, because it tries to define a new
+variable <code>x</code> with the property <code>x = x+1</code>. Thus
+when evaluating <code>x</code>, Haskell just keeps computing
+<code>1+1+1+1+...</code> indefinitely.</p>
+<div class="sourceCode" id="cb51"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb51-1"><a href="#cb51-1" aria-hidden="true" tabindex="-1"></a>compute x <span class="ot">=</span> <span class="kw">let</span> a <span class="ot">=</span> x<span class="op">+</span><span class="dv">1</span></span>
 <span id="cb51-2"><a href="#cb51-2" aria-hidden="true" tabindex="-1"></a>                a <span class="ot">=</span> a<span class="op">*</span><span class="dv">2</span></span>
 <span id="cb51-3"><a href="#cb51-3" aria-hidden="true" tabindex="-1"></a>            <span class="kw">in</span> a</span></code></pre></div>
 <pre><code>error:
     Conflicting definitions for ‘a’
     Bound at: &lt;interactive&gt;:14:17
               &lt;interactive&gt;:15:17</code></pre>
-<p>Here we get a straightforward error when we’re trying to “update” the value of <code>a</code>.</p>
-<p>As a remark, local definitions can <em>shadow</em> the names of variables defined elsewhere. Shadowing is not a side-effect. Instead, shadowing creates a new variable within a more restricted scope that uses the same name as some variable in the outer scope. For example, all of the functions <code>f</code>, <code>g</code>, and <code>h</code> below are legal:</p>
-<div class="sourceCode" id="cb53"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb53-1"><a href="#cb53-1" aria-hidden="true" tabindex="-1"></a><span class="ot">x ::</span> <span class="dt">Int</span></span>
+<p>Here we get a straightforward error when we’re trying to “update” the
+value of <code>a</code>.</p>
+<p>As a remark, local definitions can <em>shadow</em> the names of
+variables defined elsewhere. Shadowing is not a side-effect. Instead,
+shadowing creates a new variable within a more restricted scope that
+uses the same name as some variable in the outer scope. For example, all
+of the functions <code>f</code>, <code>g</code>, and <code>h</code>
+below are legal:</p>
+<div class="sourceCode" id="cb53"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb53-1"><a href="#cb53-1" aria-hidden="true" tabindex="-1"></a><span class="ot">x ::</span> <span class="dt">Int</span></span>
 <span id="cb53-2"><a href="#cb53-2" aria-hidden="true" tabindex="-1"></a>x <span class="ot">=</span> <span class="dv">5</span></span>
 <span id="cb53-3"><a href="#cb53-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb53-4"><a href="#cb53-4" aria-hidden="true" tabindex="-1"></a><span class="ot">f ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
@@ -695,30 +1189,47 @@ Prelude&gt; polynomial 3.0
 <span id="cb53-9"><a href="#cb53-9" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb53-10"><a href="#cb53-10" aria-hidden="true" tabindex="-1"></a><span class="ot">h ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb53-11"><a href="#cb53-11" aria-hidden="true" tabindex="-1"></a>h x <span class="ot">=</span> x <span class="kw">where</span> x <span class="ot">=</span> <span class="dv">3</span></span></code></pre></div>
-<p>If we apply them to the global constant <code>x</code>, we see the effects of shadowing:</p>
-<div class="sourceCode" id="cb54"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb54-1"><a href="#cb54-1" aria-hidden="true" tabindex="-1"></a>f <span class="dv">1</span> <span class="op">==&gt;</span> <span class="dv">2</span></span>
+<p>If we apply them to the global constant <code>x</code>, we see the
+effects of shadowing:</p>
+<div class="sourceCode" id="cb54"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb54-1"><a href="#cb54-1" aria-hidden="true" tabindex="-1"></a>f <span class="dv">1</span> <span class="op">==&gt;</span> <span class="dv">2</span></span>
 <span id="cb54-2"><a href="#cb54-2" aria-hidden="true" tabindex="-1"></a>g <span class="dv">1</span> <span class="op">==&gt;</span> <span class="dv">6</span></span>
 <span id="cb54-3"><a href="#cb54-3" aria-hidden="true" tabindex="-1"></a>h <span class="dv">1</span> <span class="op">==&gt;</span> <span class="dv">3</span></span>
 <span id="cb54-4"><a href="#cb54-4" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb54-5"><a href="#cb54-5" aria-hidden="true" tabindex="-1"></a>f x <span class="op">==&gt;</span> <span class="dv">10</span></span>
 <span id="cb54-6"><a href="#cb54-6" aria-hidden="true" tabindex="-1"></a>g x <span class="op">==&gt;</span> <span class="dv">6</span></span>
 <span id="cb54-7"><a href="#cb54-7" aria-hidden="true" tabindex="-1"></a>h x <span class="op">==&gt;</span> <span class="dv">3</span></span></code></pre></div>
-<p>It is best to always choose new names for local variables, so that shadowing never happens. That way, the reader of the code will understand where the variables that are used in an expression come from. Note that in the following example, <code>f</code> and <code>g</code> don’t shadow each others’ arguments:</p>
-<div class="sourceCode" id="cb55"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb55-1"><a href="#cb55-1" aria-hidden="true" tabindex="-1"></a><span class="ot">f ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<p>It is best to always choose new names for local variables, so that
+shadowing never happens. That way, the reader of the code will
+understand where the variables that are used in an expression come from.
+Note that in the following example, <code>f</code> and <code>g</code>
+don’t shadow each others’ arguments:</p>
+<div class="sourceCode" id="cb55"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb55-1"><a href="#cb55-1" aria-hidden="true" tabindex="-1"></a><span class="ot">f ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb55-2"><a href="#cb55-2" aria-hidden="true" tabindex="-1"></a>f x <span class="ot">=</span> <span class="dv">2</span> <span class="op">*</span> x <span class="op">+</span> <span class="dv">1</span></span>
 <span id="cb55-3"><a href="#cb55-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb55-4"><a href="#cb55-4" aria-hidden="true" tabindex="-1"></a><span class="ot">g ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb55-5"><a href="#cb55-5" aria-hidden="true" tabindex="-1"></a>g x <span class="ot">=</span> x <span class="op">-</span> <span class="dv">2</span></span></code></pre></div>
-<h3 data-number="1.9.4" id="pattern-matching"><span class="header-section-number">1.9.4</span> Pattern Matching</h3>
-<p>A definition (of a function) can consist of multiple <em>equations</em>. The equations are matched in order against the arguments until a suitable one is found. This is called <em>pattern matching</em>.</p>
-<p>Pattern matching in Haskell is very powerful, and we’ll keep learning new things about it along this course, but here are a couple of first examples:</p>
-<div class="sourceCode" id="cb56"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb56-1"><a href="#cb56-1" aria-hidden="true" tabindex="-1"></a><span class="ot">greet ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<h3 data-number="1.9.4" id="pattern-matching"><span
+class="header-section-number">1.9.4</span> Pattern Matching</h3>
+<p>A definition (of a function) can consist of multiple
+<em>equations</em>. The equations are matched in order against the
+arguments until a suitable one is found. This is called <em>pattern
+matching</em>.</p>
+<p>Pattern matching in Haskell is very powerful, and we’ll keep learning
+new things about it along this course, but here are a couple of first
+examples:</p>
+<div class="sourceCode" id="cb56"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb56-1"><a href="#cb56-1" aria-hidden="true" tabindex="-1"></a><span class="ot">greet ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb56-2"><a href="#cb56-2" aria-hidden="true" tabindex="-1"></a>greet <span class="st">&quot;Finland&quot;</span> name <span class="ot">=</span> <span class="st">&quot;Hei, &quot;</span> <span class="op">++</span> name</span>
 <span id="cb56-3"><a href="#cb56-3" aria-hidden="true" tabindex="-1"></a>greet <span class="st">&quot;Italy&quot;</span>   name <span class="ot">=</span> <span class="st">&quot;Ciao, &quot;</span> <span class="op">++</span> name</span>
 <span id="cb56-4"><a href="#cb56-4" aria-hidden="true" tabindex="-1"></a>greet <span class="st">&quot;England&quot;</span> name <span class="ot">=</span> <span class="st">&quot;How do you do, &quot;</span> <span class="op">++</span> name</span>
 <span id="cb56-5"><a href="#cb56-5" aria-hidden="true" tabindex="-1"></a>greet _         name <span class="ot">=</span> <span class="st">&quot;Hello, &quot;</span> <span class="op">++</span> name</span></code></pre></div>
-<p>The function <code>greet</code> generates a greeting given a country and a name (both <code>String</code>s). It has special cases for three countries, and a default case. This is how it works:</p>
-<div class="sourceCode" id="cb57"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb57-1"><a href="#cb57-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> greet <span class="st">&quot;Finland&quot;</span> <span class="st">&quot;Pekka&quot;</span></span>
+<p>The function <code>greet</code> generates a greeting given a country
+and a name (both <code>String</code>s). It has special cases for three
+countries, and a default case. This is how it works:</p>
+<div class="sourceCode" id="cb57"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb57-1"><a href="#cb57-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> greet <span class="st">&quot;Finland&quot;</span> <span class="st">&quot;Pekka&quot;</span></span>
 <span id="cb57-2"><a href="#cb57-2" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;Hei, Pekka&quot;</span></span>
 <span id="cb57-3"><a href="#cb57-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> greet <span class="st">&quot;England&quot;</span> <span class="st">&quot;Bob&quot;</span></span>
 <span id="cb57-4"><a href="#cb57-4" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;How do you do, Bob&quot;</span></span>
@@ -726,11 +1237,16 @@ Prelude&gt; polynomial 3.0
 <span id="cb57-6"><a href="#cb57-6" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;Ciao, Maria&quot;</span></span>
 <span id="cb57-7"><a href="#cb57-7" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> greet <span class="st">&quot;Greenland&quot;</span> <span class="st">&quot;Jan&quot;</span></span>
 <span id="cb57-8"><a href="#cb57-8" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;Hello, Jan&quot;</span></span></code></pre></div>
-<p>The special pattern <code>_</code> matches anything. It’s usually used for default cases. Because patterns are matched in order, it’s important to (<em>usually</em>) put the <code>_</code> case last. Consider:</p>
-<div class="sourceCode" id="cb58"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb58-1"><a href="#cb58-1" aria-hidden="true" tabindex="-1"></a>brokenGreet _         name <span class="ot">=</span> <span class="st">&quot;Hello, &quot;</span> <span class="op">++</span> name</span>
+<p>The special pattern <code>_</code> matches anything. It’s usually
+used for default cases. Because patterns are matched in order, it’s
+important to (<em>usually</em>) put the <code>_</code> case last.
+Consider:</p>
+<div class="sourceCode" id="cb58"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb58-1"><a href="#cb58-1" aria-hidden="true" tabindex="-1"></a>brokenGreet _         name <span class="ot">=</span> <span class="st">&quot;Hello, &quot;</span> <span class="op">++</span> name</span>
 <span id="cb58-2"><a href="#cb58-2" aria-hidden="true" tabindex="-1"></a>brokenGreet <span class="st">&quot;Finland&quot;</span> name <span class="ot">=</span> <span class="st">&quot;Hei, &quot;</span> <span class="op">++</span> name</span></code></pre></div>
 <p>Now the first case gets selected for all inputs.</p>
-<div class="sourceCode" id="cb59"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb59-1"><a href="#cb59-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> brokenGreet <span class="st">&quot;Finland&quot;</span> <span class="st">&quot;Varpu&quot;</span></span>
+<div class="sourceCode" id="cb59"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb59-1"><a href="#cb59-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> brokenGreet <span class="st">&quot;Finland&quot;</span> <span class="st">&quot;Varpu&quot;</span></span>
 <span id="cb59-2"><a href="#cb59-2" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;Hello, Varpu&quot;</span></span>
 <span id="cb59-3"><a href="#cb59-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> brokenGreet <span class="st">&quot;Sweden&quot;</span> <span class="st">&quot;Ole&quot;</span></span>
 <span id="cb59-4"><a href="#cb59-4" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;Hello, Ole&quot;</span></span></code></pre></div>
@@ -738,42 +1254,64 @@ Prelude&gt; polynomial 3.0
 <pre><code>&lt;interactive&gt;:1:1: warning: [-Woverlapping-patterns]
     Pattern match is redundant
     In an equation for ‘brokenGreet’: brokenGreet &quot;Finland&quot; name = ...</code></pre>
-<p>Some more examples follow. But first let’s introduce the standard library function <code>show</code> that can turn (almost!) anything into a string:</p>
-<div class="sourceCode" id="cb61"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb61-1"><a href="#cb61-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">show</span> <span class="dt">True</span></span>
+<p>Some more examples follow. But first let’s introduce the standard
+library function <code>show</code> that can turn (almost!) anything into
+a string:</p>
+<div class="sourceCode" id="cb61"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb61-1"><a href="#cb61-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">show</span> <span class="dt">True</span></span>
 <span id="cb61-2"><a href="#cb61-2" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;True&quot;</span></span>
 <span id="cb61-3"><a href="#cb61-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">show</span> <span class="dv">3</span></span>
 <span id="cb61-4"><a href="#cb61-4" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;3&quot;</span></span></code></pre></div>
-<p>So, here’s an example of a function with pattern matching and a default case that actually uses the value (instead of just ignoring it with <code>_</code>):</p>
-<div class="sourceCode" id="cb62"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb62-1"><a href="#cb62-1" aria-hidden="true" tabindex="-1"></a><span class="ot">describe ::</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<p>So, here’s an example of a function with pattern matching and a
+default case that actually uses the value (instead of just ignoring it
+with <code>_</code>):</p>
+<div class="sourceCode" id="cb62"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb62-1"><a href="#cb62-1" aria-hidden="true" tabindex="-1"></a><span class="ot">describe ::</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb62-2"><a href="#cb62-2" aria-hidden="true" tabindex="-1"></a>describe <span class="dv">0</span> <span class="ot">=</span> <span class="st">&quot;zero&quot;</span></span>
 <span id="cb62-3"><a href="#cb62-3" aria-hidden="true" tabindex="-1"></a>describe <span class="dv">1</span> <span class="ot">=</span> <span class="st">&quot;one&quot;</span></span>
 <span id="cb62-4"><a href="#cb62-4" aria-hidden="true" tabindex="-1"></a>describe <span class="dv">2</span> <span class="ot">=</span> <span class="st">&quot;an even prime&quot;</span></span>
 <span id="cb62-5"><a href="#cb62-5" aria-hidden="true" tabindex="-1"></a>describe n <span class="ot">=</span> <span class="st">&quot;the number &quot;</span> <span class="op">++</span> <span class="fu">show</span> n</span></code></pre></div>
 <p>This is how it works:</p>
-<div class="sourceCode" id="cb63"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb63-1"><a href="#cb63-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> describe <span class="dv">0</span></span>
+<div class="sourceCode" id="cb63"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb63-1"><a href="#cb63-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> describe <span class="dv">0</span></span>
 <span id="cb63-2"><a href="#cb63-2" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;zero&quot;</span></span>
 <span id="cb63-3"><a href="#cb63-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> describe <span class="dv">2</span></span>
 <span id="cb63-4"><a href="#cb63-4" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;an even prime&quot;</span></span>
 <span id="cb63-5"><a href="#cb63-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> describe <span class="dv">7</span></span>
 <span id="cb63-6"><a href="#cb63-6" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;the number 7&quot;</span></span></code></pre></div>
-<p>You can even pattern match on multiple arguments. Again, the equations are tried in order. Here’s a reimplementation of the <code>login</code> function from earlier:</p>
-<div class="sourceCode" id="cb64"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb64-1"><a href="#cb64-1" aria-hidden="true" tabindex="-1"></a><span class="ot">login ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<p>You can even pattern match on multiple arguments. Again, the
+equations are tried in order. Here’s a reimplementation of the
+<code>login</code> function from earlier:</p>
+<div class="sourceCode" id="cb64"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb64-1"><a href="#cb64-1" aria-hidden="true" tabindex="-1"></a><span class="ot">login ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb64-2"><a href="#cb64-2" aria-hidden="true" tabindex="-1"></a>login <span class="st">&quot;unicorn73&quot;</span> <span class="st">&quot;f4bulous!&quot;</span> <span class="ot">=</span> <span class="st">&quot;unicorn73 logged in&quot;</span></span>
 <span id="cb64-3"><a href="#cb64-3" aria-hidden="true" tabindex="-1"></a>login <span class="st">&quot;unicorn73&quot;</span> _           <span class="ot">=</span> <span class="st">&quot;wrong password&quot;</span></span>
 <span id="cb64-4"><a href="#cb64-4" aria-hidden="true" tabindex="-1"></a>login _           _           <span class="ot">=</span> <span class="st">&quot;unknown user&quot;</span></span></code></pre></div>
-<h3 data-number="1.9.5" id="recursion"><span class="header-section-number">1.9.5</span> Recursion</h3>
-<p>In Haskell, all sorts of loops are implemented with recursion. Function calls are very efficient, so you don’t need to worry about performance. (We’ll talk about performance later).</p>
-<p>Learning how to do simple things with recursion in Haskell will help you use recursion on more complex problems later. Recursion is also often a useful way for thinking about solving harder problems.</p>
-<p>Here’s our first recursive function which computes the factorial. In mathematics, factorial is the product of <em>n</em> first positive integers and is written as <em>n!</em>. The definition of factorial is</p>
+<h3 data-number="1.9.5" id="recursion"><span
+class="header-section-number">1.9.5</span> Recursion</h3>
+<p>In Haskell, all sorts of loops are implemented with recursion.
+Function calls are very efficient, so you don’t need to worry about
+performance. (We’ll talk about performance later).</p>
+<p>Learning how to do simple things with recursion in Haskell will help
+you use recursion on more complex problems later. Recursion is also
+often a useful way for thinking about solving harder problems.</p>
+<p>Here’s our first recursive function which computes the factorial. In
+mathematics, factorial is the product of <em>n</em> first positive
+integers and is written as <em>n!</em>. The definition of factorial
+is</p>
 <blockquote>
 <p><em>n! = n * (n-1) * … * 1</em></p>
 </blockquote>
-<p>For example, <em>4! = 4*3*2*1 = 24</em>. Well anyway, here’s the Haskell implementation of factorial:</p>
-<div class="sourceCode" id="cb65"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb65-1"><a href="#cb65-1" aria-hidden="true" tabindex="-1"></a><span class="ot">factorial ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<p>For example, <em>4! = 4*3*2*1 = 24</em>. Well anyway, here’s the
+Haskell implementation of factorial:</p>
+<div class="sourceCode" id="cb65"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb65-1"><a href="#cb65-1" aria-hidden="true" tabindex="-1"></a><span class="ot">factorial ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb65-2"><a href="#cb65-2" aria-hidden="true" tabindex="-1"></a>factorial <span class="dv">1</span> <span class="ot">=</span> <span class="dv">1</span></span>
 <span id="cb65-3"><a href="#cb65-3" aria-hidden="true" tabindex="-1"></a>factorial n <span class="ot">=</span> n <span class="op">*</span> factorial (n<span class="op">-</span><span class="dv">1</span>)</span></code></pre></div>
-<p>This is how it works. We use <code>==&gt;</code> to mean “evaluates to”.</p>
-<div class="sourceCode" id="cb66"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb66-1"><a href="#cb66-1" aria-hidden="true" tabindex="-1"></a>factorial <span class="dv">3</span></span>
+<p>This is how it works. We use <code>==&gt;</code> to mean “evaluates
+to”.</p>
+<div class="sourceCode" id="cb66"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb66-1"><a href="#cb66-1" aria-hidden="true" tabindex="-1"></a>factorial <span class="dv">3</span></span>
 <span id="cb66-2"><a href="#cb66-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">3</span> <span class="op">*</span> factorial (<span class="dv">3</span><span class="op">-</span><span class="dv">1</span>)</span>
 <span id="cb66-3"><a href="#cb66-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">3</span> <span class="op">*</span> factorial <span class="dv">2</span></span>
 <span id="cb66-4"><a href="#cb66-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">3</span> <span class="op">*</span> <span class="dv">2</span> <span class="op">*</span> factorial <span class="dv">1</span></span>
@@ -781,20 +1319,30 @@ Prelude&gt; polynomial 3.0
 <span id="cb66-6"><a href="#cb66-6" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">6</span></span></code></pre></div>
 <p>What happens when you evaluate <code>factorial (-1)</code>?</p>
 <p>Here’s another example:</p>
-<div class="sourceCode" id="cb67"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb67-1"><a href="#cb67-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- compute the sum 1^2+2^2+3^2+...+n^2</span></span>
+<div class="sourceCode" id="cb67"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb67-1"><a href="#cb67-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- compute the sum 1^2+2^2+3^2+...+n^2</span></span>
 <span id="cb67-2"><a href="#cb67-2" aria-hidden="true" tabindex="-1"></a>squareSum <span class="dv">0</span> <span class="ot">=</span> <span class="dv">0</span></span>
 <span id="cb67-3"><a href="#cb67-3" aria-hidden="true" tabindex="-1"></a>squareSum n <span class="ot">=</span> n<span class="op">^</span><span class="dv">2</span> <span class="op">+</span> squareSum (n<span class="op">-</span><span class="dv">1</span>)</span></code></pre></div>
-<p>A function can call itself recursively multiple times. As an example let’s consider the <em>Fibonacci sequence</em> from mathematics. The Fibonacci sequence is a sequence of integers with the following definition.</p>
+<p>A function can call itself recursively multiple times. As an example
+let’s consider the <em>Fibonacci sequence</em> from mathematics. The
+Fibonacci sequence is a sequence of integers with the following
+definition.</p>
 <blockquote>
-<p>The sequence starts with 1, 1. To get the next element of the sequence, sum the previous two elements of the sequence.</p>
+<p>The sequence starts with 1, 1. To get the next element of the
+sequence, sum the previous two elements of the sequence.</p>
 </blockquote>
-<p>The first elements of the Fibonacci sequence are 1, 1, 2, 3, 5, 8, 13 and so on. Here’s a function <code>fibonacci</code> which computes the <code>n</code>th element in the Fibonacci sequence. Note how it mirrors the mathematical definition.</p>
-<div class="sourceCode" id="cb68"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb68-1"><a href="#cb68-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Fibonacci numbers, slow version</span></span>
+<p>The first elements of the Fibonacci sequence are 1, 1, 2, 3, 5, 8, 13
+and so on. Here’s a function <code>fibonacci</code> which computes the
+<code>n</code>th element in the Fibonacci sequence. Note how it mirrors
+the mathematical definition.</p>
+<div class="sourceCode" id="cb68"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb68-1"><a href="#cb68-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Fibonacci numbers, slow version</span></span>
 <span id="cb68-2"><a href="#cb68-2" aria-hidden="true" tabindex="-1"></a>fibonacci <span class="dv">1</span> <span class="ot">=</span> <span class="dv">1</span></span>
 <span id="cb68-3"><a href="#cb68-3" aria-hidden="true" tabindex="-1"></a>fibonacci <span class="dv">2</span> <span class="ot">=</span> <span class="dv">1</span></span>
 <span id="cb68-4"><a href="#cb68-4" aria-hidden="true" tabindex="-1"></a>fibonacci n <span class="ot">=</span> fibonacci (n<span class="op">-</span><span class="dv">2</span>) <span class="op">+</span> fibonacci (n<span class="op">-</span><span class="dv">1</span>)</span></code></pre></div>
 <p>Here’s how <code>fibonacci 5</code> evaluates:</p>
-<div class="sourceCode" id="cb69"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb69-1"><a href="#cb69-1" aria-hidden="true" tabindex="-1"></a>fibonacci <span class="dv">5</span></span>
+<div class="sourceCode" id="cb69"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb69-1"><a href="#cb69-1" aria-hidden="true" tabindex="-1"></a>fibonacci <span class="dv">5</span></span>
 <span id="cb69-2"><a href="#cb69-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> fibonacci <span class="dv">3</span>                 <span class="op">+</span> fibonacci <span class="dv">4</span></span>
 <span id="cb69-3"><a href="#cb69-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> (fibonacci <span class="dv">1</span> <span class="op">+</span> fibonacci <span class="dv">2</span>) <span class="op">+</span> fibonacci <span class="dv">4</span></span>
 <span id="cb69-4"><a href="#cb69-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> (    <span class="dv">1</span>       <span class="op">+</span>       <span class="dv">1</span>    ) <span class="op">+</span> fibonacci <span class="dv">4</span></span>
@@ -802,7 +1350,13 @@ Prelude&gt; polynomial 3.0
 <span id="cb69-6"><a href="#cb69-6" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> (    <span class="dv">1</span>       <span class="op">+</span>       <span class="dv">1</span>    ) <span class="op">+</span> (fibonacci <span class="dv">2</span> <span class="op">+</span> (fibonacci <span class="dv">1</span> <span class="op">+</span> fibonacci <span class="dv">2</span>))</span>
 <span id="cb69-7"><a href="#cb69-7" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> (    <span class="dv">1</span>       <span class="op">+</span>       <span class="dv">1</span>    ) <span class="op">+</span> (    <span class="dv">1</span>       <span class="op">+</span> (    <span class="dv">1</span>       <span class="op">+</span>     <span class="dv">1</span>      ))</span>
 <span id="cb69-8"><a href="#cb69-8" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">5</span></span></code></pre></div>
-<p>Note how <code>fibonacci 3</code> gets evaluated twice and <code>fibonacci 2</code> three times. This is not the most efficient implementation of the <code>fibonacci</code> function. We’ll get back to this in the next lecture. Another way to think about the evaluation of the fibonacci function is to visualize it as a tree (we abbreviate <code>fibonacci</code> as <code>fib</code>): <!-- There could be some animation or a clickable interactive element for
+<p>Note how <code>fibonacci 3</code> gets evaluated twice and
+<code>fibonacci 2</code> three times. This is not the most efficient
+implementation of the <code>fibonacci</code> function. We’ll get back to
+this in the next lecture. Another way to think about the evaluation of
+the fibonacci function is to visualize it as a tree (we abbreviate
+<code>fibonacci</code> as <code>fib</code>):
+<!-- There could be some animation or a clickable interactive element for
      cycling through these images: --></p>
 <p><img src="img/Fibonacci-step1.svg" /></p>
 <p><img src="img/Fibonacci-step2.svg" /></p>
@@ -844,20 +1398,34 @@ fib 5        ==>          |           |
                                                   1       1
 ```
 -->
-<p>This tree then exaclty corresponds with the expression <code>(1 + 1) + (1 + (1 + 1))</code>. Recursion can often produce chain-like, tree-like, nested, or loopy structures and computations. Recursion is one of the main techniques in functional programming, so it’s worth spending some effort in learning it.</p>
+<p>This tree then exaclty corresponds with the expression
+<code>(1 + 1) + (1 + (1 + 1))</code>. Recursion can often produce
+chain-like, tree-like, nested, or loopy structures and computations.
+Recursion is one of the main techniques in functional programming, so
+it’s worth spending some effort in learning it.</p>
 <!-- ![A fancy diagram on Fibonacci Numbers](img/Fibonacci.svg) -->
-<h2 data-number="1.10" id="all-together-now"><span class="header-section-number">1.10</span> All Together Now!</h2>
-<p>Finally, here’s a complete Haskell module that uses ifs, pattern matching, local defintions and recursion. The module is interested in the <a href="https://en.wikipedia.org/wiki/Collatz_conjecture"><em>Collatz conjecture</em></a>, a famous open problem in mathematics. It asks:</p>
+<h2 data-number="1.10" id="all-together-now"><span
+class="header-section-number">1.10</span> All Together Now!</h2>
+<p>Finally, here’s a complete Haskell module that uses ifs, pattern
+matching, local defintions and recursion. The module is interested in
+the <a
+href="https://en.wikipedia.org/wiki/Collatz_conjecture"><em>Collatz
+conjecture</em></a>, a famous open problem in mathematics. It asks:</p>
 <blockquote>
-<p>Does the Collatz sequence eventually reach 1 for all positive integer initial values?</p>
+<p>Does the Collatz sequence eventually reach 1 for all positive integer
+initial values?</p>
 </blockquote>
-<p>The Collatz sequence is defined by taking any number as a starting value, and then repeatedly performing the following operation:</p>
+<p>The Collatz sequence is defined by taking any number as a starting
+value, and then repeatedly performing the following operation:</p>
 <ul>
 <li>if the number is even, divide it by two</li>
 <li>if the number is odd, triple it and add one</li>
 </ul>
-<p>As an example, the Collatz sequence for 3 is: 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1 … As you can see, once the number reaches 1, it gets caught in a loop.</p>
-<div class="sourceCode" id="cb70"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb70-1"><a href="#cb70-1" aria-hidden="true" tabindex="-1"></a><span class="kw">module</span> <span class="dt">Collatz</span> <span class="kw">where</span></span>
+<p>As an example, the Collatz sequence for 3 is: 3, 10, 5, 16, 8, 4, 2,
+1, 4, 2, 1, 4, 2, 1 … As you can see, once the number reaches 1, it gets
+caught in a loop.</p>
+<div class="sourceCode" id="cb70"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb70-1"><a href="#cb70-1" aria-hidden="true" tabindex="-1"></a><span class="kw">module</span> <span class="dt">Collatz</span> <span class="kw">where</span></span>
 <span id="cb70-2"><a href="#cb70-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb70-3"><a href="#cb70-3" aria-hidden="true" tabindex="-1"></a><span class="co">-- one step of the Collatz sequence</span></span>
 <span id="cb70-4"><a href="#cb70-4" aria-hidden="true" tabindex="-1"></a><span class="ot">step ::</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> <span class="dt">Integer</span></span>
@@ -893,7 +1461,8 @@ Prelude&gt; :load Collatz.hs
 [1 of 1] Compiling Collatz          ( Collatz.hs, interpreted )
 Ok, one module loaded.
 *Collatz&gt;</code></pre>
-<p>Let’s verify that our program computes the start of the Collatz sequence for 3 correctly.</p>
+<p>Let’s verify that our program computes the start of the Collatz
+sequence for 3 correctly.</p>
 <pre><code>*Collatz&gt; step 3
 10
 *Collatz&gt; step 10
@@ -903,7 +1472,8 @@ Ok, one module loaded.
 <p>How many steps does it take for 3 to reach 1?</p>
 <pre><code>*Collatz&gt; collatz 3
 7</code></pre>
-<p>What’s the longest Collatz sequence for a starting value under 10? What about 100?</p>
+<p>What’s the longest Collatz sequence for a starting value under 10?
+What about 100?</p>
 <pre><code>*Collatz&gt; longest 10
 9
 *Collatz&gt; longest 100
@@ -913,16 +1483,23 @@ Ok, one module loaded.
 19
 *Collatz&gt; collatz 97
 118</code></pre>
-<h2 data-number="1.11" id="a-word-about-indentation"><span class="header-section-number">1.11</span> A Word About Indentation</h2>
-<p>The previous examples have been fancily indented. In Haskell indentation matters, a bit like in Python. The complete set of rules for indentation is hard to describe, but you should get along fine with these rules of thumb:</p>
+<h2 data-number="1.11" id="a-word-about-indentation"><span
+class="header-section-number">1.11</span> A Word About Indentation</h2>
+<p>The previous examples have been fancily indented. In Haskell
+indentation matters, a bit like in Python. The complete set of rules for
+indentation is hard to describe, but you should get along fine with
+these rules of thumb:</p>
 <ol type="1">
 <li>Things that are grouped together start from the same column</li>
-<li>If an expression (or equation) has to be split on to many lines, increase indentation</li>
+<li>If an expression (or equation) has to be split on to many lines,
+increase indentation</li>
 </ol>
-<p>While you can get away with using tabs, it is highly recommended to use spaces for all indenting.</p>
+<p>While you can get away with using tabs, it is highly recommended to
+use spaces for all indenting.</p>
 <p>Some examples are in order.</p>
 <p>These all are ok:</p>
-<div class="sourceCode" id="cb76"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb76-1"><a href="#cb76-1" aria-hidden="true" tabindex="-1"></a>i x <span class="ot">=</span> <span class="kw">let</span> y <span class="ot">=</span> x<span class="op">+</span>x<span class="op">+</span>x<span class="op">+</span>x<span class="op">+</span>x<span class="op">+</span>x <span class="kw">in</span> <span class="fu">div</span> y <span class="dv">5</span></span>
+<div class="sourceCode" id="cb76"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb76-1"><a href="#cb76-1" aria-hidden="true" tabindex="-1"></a>i x <span class="ot">=</span> <span class="kw">let</span> y <span class="ot">=</span> x<span class="op">+</span>x<span class="op">+</span>x<span class="op">+</span>x<span class="op">+</span>x<span class="op">+</span>x <span class="kw">in</span> <span class="fu">div</span> y <span class="dv">5</span></span>
 <span id="cb76-2"><a href="#cb76-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb76-3"><a href="#cb76-3" aria-hidden="true" tabindex="-1"></a><span class="co">-- let and in are grouped together, an expression is split</span></span>
 <span id="cb76-4"><a href="#cb76-4" aria-hidden="true" tabindex="-1"></a>j x <span class="ot">=</span> <span class="kw">let</span> y <span class="ot">=</span> x<span class="op">+</span>x<span class="op">+</span>x</span>
@@ -939,7 +1516,8 @@ Ok, one module loaded.
 <span id="cb76-15"><a href="#cb76-15" aria-hidden="true" tabindex="-1"></a>    a <span class="ot">=</span> <span class="dv">1</span></span>
 <span id="cb76-16"><a href="#cb76-16" aria-hidden="true" tabindex="-1"></a>    b <span class="ot">=</span> <span class="dv">1</span></span></code></pre></div>
 <p>These are not ok:</p>
-<div class="sourceCode" id="cb77"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb77-1"><a href="#cb77-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- indentation not increased even though expression split on many lines</span></span>
+<div class="sourceCode" id="cb77"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb77-1"><a href="#cb77-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- indentation not increased even though expression split on many lines</span></span>
 <span id="cb77-2"><a href="#cb77-2" aria-hidden="true" tabindex="-1"></a>i x <span class="ot">=</span> <span class="kw">let</span> y <span class="ot">=</span> x<span class="op">+</span>x<span class="op">+</span>x<span class="op">+</span>x<span class="op">+</span>x<span class="op">+</span>x</span>
 <span id="cb77-3"><a href="#cb77-3" aria-hidden="true" tabindex="-1"></a><span class="kw">in</span> <span class="fu">div</span> y <span class="dv">5</span></span>
 <span id="cb77-4"><a href="#cb77-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -964,12 +1542,22 @@ Ok, one module loaded.
 <span id="cb77-23"><a href="#cb77-23" aria-hidden="true" tabindex="-1"></a><span class="kw">where</span></span>
 <span id="cb77-24"><a href="#cb77-24" aria-hidden="true" tabindex="-1"></a>  a <span class="ot">=</span> <span class="dv">1</span></span>
 <span id="cb77-25"><a href="#cb77-25" aria-hidden="true" tabindex="-1"></a>  b <span class="ot">=</span> <span class="dv">1</span></span></code></pre></div>
-<p>If you make a mistake with the indentation, you’ll typically get a parse error like this:</p>
+<p>If you make a mistake with the indentation, you’ll typically get a
+parse error like this:</p>
 <pre><code>Indent.hs:2:1: error: parse error on input ‘where’</code></pre>
-<p>The error includes the line number, so just go over that line again. If you can’t seem to get indentation to work, try putting everything on just one long line at first.</p>
-<h2 data-number="1.12" id="quiz"><span class="header-section-number">1.12</span> Quiz</h2>
-<p>At the end of each lecture you’ll find a quiz like this. The quizes aren’t graded, they’re just here to help you check you’ve understood the chapter. You can check your answer by clicking on an option. You’ll see a green background if you were right, a red one if you were wrong. Feel free to guess as many times as you want, just make sure you understand why the right option is right in the end.</p>
-<p>What is the Haskell equivalent of the C/Java/Python expression <code>combine(prettify(lawn),construct(house,concrete))</code>?</p>
+<p>The error includes the line number, so just go over that line again.
+If you can’t seem to get indentation to work, try putting everything on
+just one long line at first.</p>
+<h2 data-number="1.12" id="quiz"><span
+class="header-section-number">1.12</span> Quiz</h2>
+<p>At the end of each lecture you’ll find a quiz like this. The quizes
+aren’t graded, they’re just here to help you check you’ve understood the
+chapter. You can check your answer by clicking on an option. You’ll see
+a green background if you were right, a red one if you were wrong. Feel
+free to guess as many times as you want, just make sure you understand
+why the right option is right in the end.</p>
+<p>What is the Haskell equivalent of the C/Java/Python expression
+<code>combine(prettify(lawn),construct(house,concrete))</code>?</p>
 <ol class="quiz">
 <li>
 <code>combine prettify (lawn) construct (house concerete)</code>
@@ -981,7 +1569,8 @@ Ok, one module loaded.
 <code>combine (prettify lawn) (construct house concrete)</code>
 </li>
 </ol>
-What is the C/Java/Python equivalent of the Haskell expression <code>send metric (double population + increase)</code>?
+What is the C/Java/Python equivalent of the Haskell expression
+<code>send metric (double population + increase)</code>?
 <ol class="quiz">
 <li>
 <code>send(metric(double(population+increase)))</code>
@@ -1017,16 +1606,20 @@ It’s impossible to reuse the name of a variable
 It’s possible to reassign a value to a variable
 </li>
 <li class="correct">
-An <code>if</code> always requires both <code>then</code> and <code>else</code>
+An <code>if</code> always requires both <code>then</code> and
+<code>else</code>
 </li>
 </ol>
-<p>What does the function <code>f x = if even (x + 1) then x + 1 else f (x - 1)</code> do?</p>
+<p>What does the function
+<code>f x = if even (x + 1) then x + 1 else f (x - 1)</code> do?</p>
 <ol class="quiz">
 <li class="correct">
-Maps every value <code>x</code> to the least even number greater than or equal to <code>x</code>
+Maps every value <code>x</code> to the least even number greater than or
+equal to <code>x</code>
 </li>
 <li>
-Maps every value <code>x</code> to the greatest even number less than or equal to <code>x</code>
+Maps every value <code>x</code> to the greatest even number less than or
+equal to <code>x</code>
 </li>
 <li>
 Maps every value to itself
@@ -1056,19 +1649,45 @@ Because <code>div</code> is not defined for the type <code>Int</code>
 Because <code>`...`</code> is used for delimiting strings.
 </li>
 </ol>
-<h2 data-number="1.13" id="working-on-the-exercises"><span class="header-section-number">1.13</span> Working on the Exercises</h2>
-<p>The course materials, including exercises, are available in a Git repository on GitHub at <a href="https://github.com/moocfi/haskell-mooc" class="uri">https://github.com/moocfi/haskell-mooc</a>. If you’re not familiar with Git, see <a href="https://help.github.com/en/github/creating-cloning-and-archiving-repositories/cloning-a-repository">GitHub’s instructions on cloning a repository</a>.</p>
-<p>Once you’ve cloned the <code>haskell-mooc</code> repository, go into the <code>exercises</code> directory. To download and build dependencies needed for running the exercise tests (such as the correct version of GHC and various libraries), run following command in your terminal:</p>
-<div class="sourceCode" id="cb79"><pre class="sourceCode sh"><code class="sourceCode bash"><span id="cb79-1"><a href="#cb79-1" aria-hidden="true" tabindex="-1"></a><span class="ex">$</span> stack build</span></code></pre></div>
-<p>Do note that the dependencies are multiple gigabytes and it will take a while for the command to finish.</p>
-<p><strong>Note!</strong> Here are some fixes for common problems with <code>stack build</code>:</p>
+<h2 data-number="1.13" id="working-on-the-exercises"><span
+class="header-section-number">1.13</span> Working on the Exercises</h2>
+<p>The course materials, including exercises, are available in a Git
+repository on GitHub at <a href="https://github.com/moocfi/haskell-mooc"
+class="uri">https://github.com/moocfi/haskell-mooc</a>. If you’re not
+familiar with Git, see <a
+href="https://help.github.com/en/github/creating-cloning-and-archiving-repositories/cloning-a-repository">GitHub’s
+instructions on cloning a repository</a>.</p>
+<p>Once you’ve cloned the <code>haskell-mooc</code> repository, go into
+the <code>exercises</code> directory. To download and build dependencies
+needed for running the exercise tests (such as the correct version of
+GHC and various libraries), run following command in your terminal:</p>
+<div class="sourceCode" id="cb79"><pre
+class="sourceCode sh"><code class="sourceCode bash"><span id="cb79-1"><a href="#cb79-1" aria-hidden="true" tabindex="-1"></a><span class="ex">$</span> stack build</span></code></pre></div>
+<p>Do note that the dependencies are multiple gigabytes and it will take
+a while for the command to finish.</p>
+<p><strong>Note!</strong> Here are some fixes for common problems with
+<code>stack build</code>:</p>
 <ul>
-<li>If you get an error like <code>While building package zlib-0.6.2.3</code>, you need to install the zlib library headers. The right command for Ubuntu is <code>sudo apt install zlib1g-dev</code>.</li>
-<li>If you get an error like <code>Downloading lts-18.18 build plan ... RedownloadInvalidResponse</code>, your version of stack is too old. Run <code>stack upgrade</code> to get a newer one.</li>
+<li>If you get an error like
+<code>While building package zlib-0.6.2.3</code>, you need to install
+the zlib library headers. The right command for Ubuntu is
+<code>sudo apt install zlib1g-dev</code>.</li>
+<li>If you get an error like
+<code>Downloading lts-18.18 build plan ... RedownloadInvalidResponse</code>,
+your version of stack is too old. Run <code>stack upgrade</code> to get
+a newer one.</li>
 </ul>
-<p>There are primarily two types of files in the <code>exercises</code> directory: exercise sets named <code>SetNX.hs</code> and accompanying test program for the exercises named <code>SetNXTest.hs</code>. Both are Haskell source files, but only the exercise file should to be edited when solving the exercises. Instructions to all individual exercises are embedded in the exercise file as comments.</p>
-<p>Use the tests file to check your answers. For example when you have solved some of the exercises in <code>Set1.hs</code>, run the following command:</p>
-<div class="sourceCode" id="cb80"><pre class="sourceCode sh"><code class="sourceCode bash"><span id="cb80-1"><a href="#cb80-1" aria-hidden="true" tabindex="-1"></a><span class="ex">$</span> stack runhaskell Set1Test.hs</span></code></pre></div>
+<p>There are primarily two types of files in the <code>exercises</code>
+directory: exercise sets named <code>SetNX.hs</code> and accompanying
+test program for the exercises named <code>SetNXTest.hs</code>. Both are
+Haskell source files, but only the exercise file should to be edited
+when solving the exercises. Instructions to all individual exercises are
+embedded in the exercise file as comments.</p>
+<p>Use the tests file to check your answers. For example when you have
+solved some of the exercises in <code>Set1.hs</code>, run the following
+command:</p>
+<div class="sourceCode" id="cb80"><pre
+class="sourceCode sh"><code class="sourceCode bash"><span id="cb80-1"><a href="#cb80-1" aria-hidden="true" tabindex="-1"></a><span class="ex">$</span> stack runhaskell Set1Test.hs</span></code></pre></div>
 <p>The output of the tests looks something like this:</p>
 <pre><code>===== EXERCISE 1
 +++++ Pass
@@ -1117,7 +1736,9 @@ quadruple 1
 1101111111111111111
 18 / 19</code></pre>
 <p>In the example above, I’ve made a mistake in exercise 3.</p>
-<p>To make debugging faster and more straightforward, I can load my exercise file in GHCi, which allows me to evaluate any top-level function manually. For instance I can verify the above mistake by:</p>
+<p>To make debugging faster and more straightforward, I can load my
+exercise file in GHCi, which allows me to evaluate any top-level
+function manually. For instance I can verify the above mistake by:</p>
 <pre><code>$ stack ghci Set1.hs
 GHCi, version 9.2.8: https://www.haskell.org/ghc/  :? for help
 [1 of 2] Compiling Mooc.Todo        ( Mooc/Todo.hs, interpreted )
@@ -1125,29 +1746,55 @@ GHCi, version 9.2.8: https://www.haskell.org/ghc/  :? for help
 Ok, two modules loaded.
 *Set1&gt; quadruple 1
 2</code></pre>
-<p>Once you’re done with an exercise set, you can turn it in on the <a href="https://haskell.mooc.fi/submit.php">Submit page</a> on the course pages. After that you can see the results of your submission on the <a href="https://haskell.mooc.fi/results.php">Results page</a> and your total score on the <a href="https://haskell.mooc.fi/status.php">My status page</a>.</p>
-<p><strong>Note!</strong> You may turn in an exercise set as many times as you want.</p>
-<p><strong>Note!</strong> If you don’t want to use Stack or can’t get it working, you should also be able to run the tests with <a href="https://www.haskell.org/cabal/download.html">Cabal</a> like this:</p>
-<div class="sourceCode" id="cb83"><pre class="sourceCode sh"><code class="sourceCode bash"><span id="cb83-1"><a href="#cb83-1" aria-hidden="true" tabindex="-1"></a><span class="ex">$</span> cabal v2-build</span>
+<p>Once you’re done with an exercise set, you can turn it in on the <a
+href="https://haskell.mooc.fi/submit.php">Submit page</a> on the course
+pages. After that you can see the results of your submission on the <a
+href="https://haskell.mooc.fi/results.php">Results page</a> and your
+total score on the <a href="https://haskell.mooc.fi/status.php">My
+status page</a>.</p>
+<p><strong>Note!</strong> You may turn in an exercise set as many times
+as you want.</p>
+<p><strong>Note!</strong> If you don’t want to use Stack or can’t get it
+working, you should also be able to run the tests with <a
+href="https://www.haskell.org/cabal/download.html">Cabal</a> like
+this:</p>
+<div class="sourceCode" id="cb83"><pre
+class="sourceCode sh"><code class="sourceCode bash"><span id="cb83-1"><a href="#cb83-1" aria-hidden="true" tabindex="-1"></a><span class="ex">$</span> cabal v2-build</span>
 <span id="cb83-2"><a href="#cb83-2" aria-hidden="true" tabindex="-1"></a><span class="ex">$</span> cabal v2-exec runhaskell Set1Test.hs</span></code></pre></div>
-<h3 data-number="1.13.1" id="model-solutions"><span class="header-section-number">1.13.1</span> Model Solutions</h3>
-<p>Once you’ve successfully completed all the exercises in a set, you can view the model solutions on the <a href="https://haskell.mooc.fi/status">My status page</a>. It’s useful to glance at the model solutions, they might show you a technique you’ve missed!</p>
-<h2 data-number="1.14" id="exercises"><span class="header-section-number">1.14</span> Exercises</h2>
+<h3 data-number="1.13.1" id="model-solutions"><span
+class="header-section-number">1.13.1</span> Model Solutions</h3>
+<p>Once you’ve successfully completed all the exercises in a set, you
+can view the model solutions on the <a
+href="https://haskell.mooc.fi/status">My status page</a>. It’s useful to
+glance at the model solutions, they might show you a technique you’ve
+missed!</p>
+<h2 data-number="1.14" id="exercises"><span
+class="header-section-number">1.14</span> Exercises</h2>
 <ul>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set1.hs">Set1</a></li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set1.hs">Set1</a></li>
 </ul>
-<h1 data-number="2" id="lecture-2-either-you-die-a-hero"><span class="header-section-number">2</span> Lecture 2: Either You Die a Hero…</h1>
+<h1 data-number="2" id="lecture-2-either-you-die-a-hero"><span
+class="header-section-number">2</span> Lecture 2: Either You Die a
+Hero…</h1>
 <ul>
 <li>More about recursion</li>
 <li>Guards</li>
 <li>More types: lists, <code>Maybe</code>, <code>Either</code></li>
 <li>Polymorphism</li>
 </ul>
-<h2 data-number="2.1" id="recursion-and-helper-functions"><span class="header-section-number">2.1</span> Recursion and Helper Functions</h2>
-<p>Often you’ll find you need helper variables in recursion to keep track of things. You can get them by defining a helper function with more arguments. Analogy: arguments of the helper function are variables you update in your loop.</p>
-<p>Here’s an example of how you would convert a loop (in Java or Python) into a recursive helper function in Haskell.</p>
+<h2 data-number="2.1" id="recursion-and-helper-functions"><span
+class="header-section-number">2.1</span> Recursion and Helper
+Functions</h2>
+<p>Often you’ll find you need helper variables in recursion to keep
+track of things. You can get them by defining a helper function with
+more arguments. Analogy: arguments of the helper function are variables
+you update in your loop.</p>
+<p>Here’s an example of how you would convert a loop (in Java or Python)
+into a recursive helper function in Haskell.</p>
 <p>Java:</p>
-<div class="sourceCode" id="cb84"><pre class="sourceCode java"><code class="sourceCode java"><span id="cb84-1"><a href="#cb84-1" aria-hidden="true" tabindex="-1"></a><span class="kw">public</span> <span class="bu">String</span> <span class="fu">repeatString</span><span class="op">(</span><span class="dt">int</span> n<span class="op">,</span> <span class="bu">String</span> str<span class="op">)</span> <span class="op">{</span></span>
+<div class="sourceCode" id="cb84"><pre
+class="sourceCode java"><code class="sourceCode java"><span id="cb84-1"><a href="#cb84-1" aria-hidden="true" tabindex="-1"></a><span class="kw">public</span> <span class="bu">String</span> <span class="fu">repeatString</span><span class="op">(</span><span class="dt">int</span> n<span class="op">,</span> <span class="bu">String</span> str<span class="op">)</span> <span class="op">{</span></span>
 <span id="cb84-2"><a href="#cb84-2" aria-hidden="true" tabindex="-1"></a>    <span class="bu">String</span> result <span class="op">=</span> <span class="st">&quot;&quot;</span><span class="op">;</span></span>
 <span id="cb84-3"><a href="#cb84-3" aria-hidden="true" tabindex="-1"></a>    <span class="cf">while</span> <span class="op">(</span>n<span class="op">&gt;</span><span class="dv">0</span><span class="op">)</span> <span class="op">{</span></span>
 <span id="cb84-4"><a href="#cb84-4" aria-hidden="true" tabindex="-1"></a>        result <span class="op">=</span> result<span class="op">+</span>str<span class="op">;</span></span>
@@ -1156,29 +1803,38 @@ Ok, two modules loaded.
 <span id="cb84-7"><a href="#cb84-7" aria-hidden="true" tabindex="-1"></a>    <span class="cf">return</span> result<span class="op">;</span></span>
 <span id="cb84-8"><a href="#cb84-8" aria-hidden="true" tabindex="-1"></a><span class="op">}</span></span></code></pre></div>
 <p>Python:</p>
-<div class="sourceCode" id="cb85"><pre class="sourceCode python"><code class="sourceCode python"><span id="cb85-1"><a href="#cb85-1" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> repeatString(n, <span class="bu">str</span>):</span>
+<div class="sourceCode" id="cb85"><pre
+class="sourceCode python"><code class="sourceCode python"><span id="cb85-1"><a href="#cb85-1" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> repeatString(n, <span class="bu">str</span>):</span>
 <span id="cb85-2"><a href="#cb85-2" aria-hidden="true" tabindex="-1"></a>    result <span class="op">=</span> <span class="st">&quot;&quot;</span></span>
 <span id="cb85-3"><a href="#cb85-3" aria-hidden="true" tabindex="-1"></a>    <span class="cf">while</span> n<span class="op">&gt;</span><span class="dv">0</span>:</span>
 <span id="cb85-4"><a href="#cb85-4" aria-hidden="true" tabindex="-1"></a>        result <span class="op">=</span> result<span class="op">+</span><span class="bu">str</span></span>
 <span id="cb85-5"><a href="#cb85-5" aria-hidden="true" tabindex="-1"></a>        n <span class="op">=</span> n<span class="op">-</span><span class="dv">1</span></span>
 <span id="cb85-6"><a href="#cb85-6" aria-hidden="true" tabindex="-1"></a>    <span class="cf">return</span> result</span></code></pre></div>
 <p>Haskell:</p>
-<div class="sourceCode" id="cb86"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb86-1"><a href="#cb86-1" aria-hidden="true" tabindex="-1"></a>repeatString n str <span class="ot">=</span> repeatHelper n str <span class="st">&quot;&quot;</span></span>
+<div class="sourceCode" id="cb86"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb86-1"><a href="#cb86-1" aria-hidden="true" tabindex="-1"></a>repeatString n str <span class="ot">=</span> repeatHelper n str <span class="st">&quot;&quot;</span></span>
 <span id="cb86-2"><a href="#cb86-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb86-3"><a href="#cb86-3" aria-hidden="true" tabindex="-1"></a>repeatHelper n str result <span class="ot">=</span> <span class="kw">if</span> (n<span class="op">==</span><span class="dv">0</span>)</span>
 <span id="cb86-4"><a href="#cb86-4" aria-hidden="true" tabindex="-1"></a>                            <span class="kw">then</span> result</span>
 <span id="cb86-5"><a href="#cb86-5" aria-hidden="true" tabindex="-1"></a>                            <span class="kw">else</span> repeatHelper (n<span class="op">-</span><span class="dv">1</span>) str (result<span class="op">++</span>str)</span></code></pre></div>
-<div class="sourceCode" id="cb87"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb87-1"><a href="#cb87-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> repeatString <span class="dv">3</span> <span class="st">&quot;ABC&quot;</span></span>
+<div class="sourceCode" id="cb87"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb87-1"><a href="#cb87-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> repeatString <span class="dv">3</span> <span class="st">&quot;ABC&quot;</span></span>
 <span id="cb87-2"><a href="#cb87-2" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;ABCABCABC&quot;</span></span></code></pre></div>
-<p>You might have noticed that the Java and Python implementations look a bit weird since they use while loops instead of for loops. This is because this way the conversion to Haskell is more straightforward.</p>
-<p>This can be made a bit tidier by using pattern matching instead of an <code>if</code>:</p>
-<div class="sourceCode" id="cb88"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb88-1"><a href="#cb88-1" aria-hidden="true" tabindex="-1"></a>repeatString n str <span class="ot">=</span> repeatHelper n str <span class="st">&quot;&quot;</span></span>
+<p>You might have noticed that the Java and Python implementations look
+a bit weird since they use while loops instead of for loops. This is
+because this way the conversion to Haskell is more straightforward.</p>
+<p>This can be made a bit tidier by using pattern matching instead of an
+<code>if</code>:</p>
+<div class="sourceCode" id="cb88"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb88-1"><a href="#cb88-1" aria-hidden="true" tabindex="-1"></a>repeatString n str <span class="ot">=</span> repeatHelper n str <span class="st">&quot;&quot;</span></span>
 <span id="cb88-2"><a href="#cb88-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb88-3"><a href="#cb88-3" aria-hidden="true" tabindex="-1"></a>repeatHelper <span class="dv">0</span> _   result <span class="ot">=</span> result</span>
 <span id="cb88-4"><a href="#cb88-4" aria-hidden="true" tabindex="-1"></a>repeatHelper n str result <span class="ot">=</span> repeatHelper (n<span class="op">-</span><span class="dv">1</span>) str (result<span class="op">++</span>str)</span></code></pre></div>
-<p>Here’s another example with more variables: computing fibonacci numbers efficiently.</p>
+<p>Here’s another example with more variables: computing fibonacci
+numbers efficiently.</p>
 <p>Java:</p>
-<div class="sourceCode" id="cb89"><pre class="sourceCode java"><code class="sourceCode java"><span id="cb89-1"><a href="#cb89-1" aria-hidden="true" tabindex="-1"></a><span class="kw">public</span> <span class="dt">int</span> <span class="fu">fibonacci</span><span class="op">(</span><span class="dt">int</span> n<span class="op">)</span> <span class="op">{</span></span>
+<div class="sourceCode" id="cb89"><pre
+class="sourceCode java"><code class="sourceCode java"><span id="cb89-1"><a href="#cb89-1" aria-hidden="true" tabindex="-1"></a><span class="kw">public</span> <span class="dt">int</span> <span class="fu">fibonacci</span><span class="op">(</span><span class="dt">int</span> n<span class="op">)</span> <span class="op">{</span></span>
 <span id="cb89-2"><a href="#cb89-2" aria-hidden="true" tabindex="-1"></a>    <span class="dt">int</span> a <span class="op">=</span> <span class="dv">0</span><span class="op">;</span></span>
 <span id="cb89-3"><a href="#cb89-3" aria-hidden="true" tabindex="-1"></a>    <span class="dt">int</span> b <span class="op">=</span> <span class="dv">1</span><span class="op">;</span></span>
 <span id="cb89-4"><a href="#cb89-4" aria-hidden="true" tabindex="-1"></a>    <span class="cf">while</span> <span class="op">(</span>n<span class="op">&gt;</span><span class="dv">1</span><span class="op">)</span> <span class="op">{</span></span>
@@ -1190,7 +1846,8 @@ Ok, two modules loaded.
 <span id="cb89-10"><a href="#cb89-10" aria-hidden="true" tabindex="-1"></a>    <span class="cf">return</span> b<span class="op">;</span></span>
 <span id="cb89-11"><a href="#cb89-11" aria-hidden="true" tabindex="-1"></a><span class="op">}</span></span></code></pre></div>
 <p>Python:</p>
-<div class="sourceCode" id="cb90"><pre class="sourceCode python"><code class="sourceCode python"><span id="cb90-1"><a href="#cb90-1" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> fibonacci(n):</span>
+<div class="sourceCode" id="cb90"><pre
+class="sourceCode python"><code class="sourceCode python"><span id="cb90-1"><a href="#cb90-1" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> fibonacci(n):</span>
 <span id="cb90-2"><a href="#cb90-2" aria-hidden="true" tabindex="-1"></a>    a <span class="op">=</span> <span class="dv">0</span></span>
 <span id="cb90-3"><a href="#cb90-3" aria-hidden="true" tabindex="-1"></a>    b <span class="op">=</span> <span class="dv">1</span></span>
 <span id="cb90-4"><a href="#cb90-4" aria-hidden="true" tabindex="-1"></a>    <span class="cf">while</span> n<span class="op">&gt;</span><span class="dv">1</span>:</span>
@@ -1200,41 +1857,75 @@ Ok, two modules loaded.
 <span id="cb90-8"><a href="#cb90-8" aria-hidden="true" tabindex="-1"></a>        n <span class="op">=</span> n<span class="op">-</span><span class="dv">1</span></span>
 <span id="cb90-9"><a href="#cb90-9" aria-hidden="true" tabindex="-1"></a>    <span class="cf">return</span> b</span></code></pre></div>
 <p>Haskell:</p>
-<div class="sourceCode" id="cb91"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb91-1"><a href="#cb91-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- fibonacci numbers, fast version</span></span>
+<div class="sourceCode" id="cb91"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb91-1"><a href="#cb91-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- fibonacci numbers, fast version</span></span>
 <span id="cb91-2"><a href="#cb91-2" aria-hidden="true" tabindex="-1"></a><span class="ot">fibonacci ::</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> <span class="dt">Integer</span></span>
 <span id="cb91-3"><a href="#cb91-3" aria-hidden="true" tabindex="-1"></a>fibonacci n <span class="ot">=</span> fibonacci&#39; <span class="dv">0</span> <span class="dv">1</span> n</span>
 <span id="cb91-4"><a href="#cb91-4" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb91-5"><a href="#cb91-5" aria-hidden="true" tabindex="-1"></a><span class="ot">fibonacci&#39; ::</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> <span class="dt">Integer</span></span>
 <span id="cb91-6"><a href="#cb91-6" aria-hidden="true" tabindex="-1"></a>fibonacci&#39; a b <span class="dv">1</span> <span class="ot">=</span> b</span>
 <span id="cb91-7"><a href="#cb91-7" aria-hidden="true" tabindex="-1"></a>fibonacci&#39; a b n <span class="ot">=</span> fibonacci&#39; b (a<span class="op">+</span>b) (n<span class="op">-</span><span class="dv">1</span>)</span></code></pre></div>
-<p>Take a while to study these and note how the Haskell recursion has the same format as the loop.</p>
-<p>Sidenote: Haskell programs often use the apostrophe to name helper functions and alternative versions of functions. Thus the name <code>fibonacci'</code> for the helper function above. Names like <code>foo'</code> are usually read <em>foo prime</em> (like in mathematics).</p>
-<p>I said earlier that this version of fibonacci is more efficient. Can you see why? The answer is that there are less recursive calls. The expression <code>fibonacci' _ _ n</code> calls <code>fibonacci' _ _ (n-1)</code> once, and this means that we can compute <code>fibonacci' _ _ n</code> in <code>n</code> steps.</p>
-<p>This type of recursion where a function just directly calls itself with different arguments is called <em>tail recursion</em>. As you’ve seen above, tail recursion corresponds to loops. This is why tail recursion is often fast: the compiler can generate a loop in machine code when it sees tail recursion.</p>
-<h2 data-number="2.2" id="guards"><span class="header-section-number">2.2</span> Guards</h2>
-<p>Before we move on to new types, let’s go over one more piece of Haskell syntax.</p>
-<p>The <code>if then else</code> is often a bit cumbersome, especially when you have multiple cases. An easier alternative is Haskell’s <em>conditional definition</em> or <em>guarded definition</em>. This is a bit like pattern matching in that you have multiple equations, but you can have arbitrary code deciding which equation to use. Guarded definitions look like this:</p>
-<div class="sourceCode" id="cb92"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb92-1"><a href="#cb92-1" aria-hidden="true" tabindex="-1"></a>f x y z</span>
+<p>Take a while to study these and note how the Haskell recursion has
+the same format as the loop.</p>
+<p>Sidenote: Haskell programs often use the apostrophe to name helper
+functions and alternative versions of functions. Thus the name
+<code>fibonacci'</code> for the helper function above. Names like
+<code>foo'</code> are usually read <em>foo prime</em> (like in
+mathematics).</p>
+<p>I said earlier that this version of fibonacci is more efficient. Can
+you see why? The answer is that there are less recursive calls. The
+expression <code>fibonacci' _ _ n</code> calls
+<code>fibonacci' _ _ (n-1)</code> once, and this means that we can
+compute <code>fibonacci' _ _ n</code> in <code>n</code> steps.</p>
+<p>This type of recursion where a function just directly calls itself
+with different arguments is called <em>tail recursion</em>. As you’ve
+seen above, tail recursion corresponds to loops. This is why tail
+recursion is often fast: the compiler can generate a loop in machine
+code when it sees tail recursion.</p>
+<h2 data-number="2.2" id="guards"><span
+class="header-section-number">2.2</span> Guards</h2>
+<p>Before we move on to new types, let’s go over one more piece of
+Haskell syntax.</p>
+<p>The <code>if then else</code> is often a bit cumbersome, especially
+when you have multiple cases. An easier alternative is Haskell’s
+<em>conditional definition</em> or <em>guarded definition</em>. This is
+a bit like pattern matching in that you have multiple equations, but you
+can have arbitrary code deciding which equation to use. Guarded
+definitions look like this:</p>
+<div class="sourceCode" id="cb92"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb92-1"><a href="#cb92-1" aria-hidden="true" tabindex="-1"></a>f x y z</span>
 <span id="cb92-2"><a href="#cb92-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> condition1 <span class="ot">=</span> something</span>
 <span id="cb92-3"><a href="#cb92-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> condition2 <span class="ot">=</span> other</span>
 <span id="cb92-4"><a href="#cb92-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> <span class="fu">otherwise</span>  <span class="ot">=</span> somethingother</span></code></pre></div>
-<p>A condition can be any expression of type <code>Bool</code>. The first condition that evaluates to <code>True</code> is chosen. The word <code>otherwise</code> is just an alias for <code>True</code>. It is used to mark the default case.</p>
-<h3 data-number="2.2.1" id="examples-1"><span class="header-section-number">2.2.1</span> Examples</h3>
-<p>Here are some examples of using guards. First off, we have a function that describes the given number. Note how it is important to have the <code>"Two"</code> case before the <code>"Even"</code> case.</p>
-<div class="sourceCode" id="cb93"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb93-1"><a href="#cb93-1" aria-hidden="true" tabindex="-1"></a><span class="ot">describe ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<p>A condition can be any expression of type <code>Bool</code>. The
+first condition that evaluates to <code>True</code> is chosen. The word
+<code>otherwise</code> is just an alias for <code>True</code>. It is
+used to mark the default case.</p>
+<h3 data-number="2.2.1" id="examples-1"><span
+class="header-section-number">2.2.1</span> Examples</h3>
+<p>Here are some examples of using guards. First off, we have a function
+that describes the given number. Note how it is important to have the
+<code>"Two"</code> case before the <code>"Even"</code> case.</p>
+<div class="sourceCode" id="cb93"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb93-1"><a href="#cb93-1" aria-hidden="true" tabindex="-1"></a><span class="ot">describe ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb93-2"><a href="#cb93-2" aria-hidden="true" tabindex="-1"></a>describe n</span>
 <span id="cb93-3"><a href="#cb93-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> n<span class="op">==</span><span class="dv">2</span>      <span class="ot">=</span> <span class="st">&quot;Two&quot;</span></span>
 <span id="cb93-4"><a href="#cb93-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> <span class="fu">even</span> n    <span class="ot">=</span> <span class="st">&quot;Even&quot;</span></span>
 <span id="cb93-5"><a href="#cb93-5" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> n<span class="op">==</span><span class="dv">3</span>      <span class="ot">=</span> <span class="st">&quot;Three&quot;</span></span>
 <span id="cb93-6"><a href="#cb93-6" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> n<span class="op">&gt;</span><span class="dv">100</span>     <span class="ot">=</span> <span class="st">&quot;Big!!&quot;</span></span>
 <span id="cb93-7"><a href="#cb93-7" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> <span class="fu">otherwise</span> <span class="ot">=</span> <span class="st">&quot;The number &quot;</span><span class="op">++</span><span class="fu">show</span> n</span></code></pre></div>
-<p>Here is factorial, implemented with guards instead of pattern matching. Unlike the pattern-matching version, this one doesn’t loop forever with negative inputs.</p>
-<div class="sourceCode" id="cb94"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb94-1"><a href="#cb94-1" aria-hidden="true" tabindex="-1"></a>factorial n</span>
+<p>Here is factorial, implemented with guards instead of pattern
+matching. Unlike the pattern-matching version, this one doesn’t loop
+forever with negative inputs.</p>
+<div class="sourceCode" id="cb94"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb94-1"><a href="#cb94-1" aria-hidden="true" tabindex="-1"></a>factorial n</span>
 <span id="cb94-2"><a href="#cb94-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> n<span class="op">&lt;</span><span class="dv">0</span>       <span class="ot">=</span> <span class="op">-</span><span class="dv">1</span></span>
 <span id="cb94-3"><a href="#cb94-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> n<span class="op">==</span><span class="dv">0</span>      <span class="ot">=</span> <span class="dv">1</span></span>
 <span id="cb94-4"><a href="#cb94-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> <span class="fu">otherwise</span> <span class="ot">=</span> n <span class="op">*</span> factorial (n<span class="op">-</span><span class="dv">1</span>)</span></code></pre></div>
-<p>You can even combine guards with pattern matching. Here’s the implementation of a simple age guessing game:</p>
-<div class="sourceCode" id="cb95"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb95-1"><a href="#cb95-1" aria-hidden="true" tabindex="-1"></a><span class="ot">guessAge ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<p>You can even combine guards with pattern matching. Here’s the
+implementation of a simple age guessing game:</p>
+<div class="sourceCode" id="cb95"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb95-1"><a href="#cb95-1" aria-hidden="true" tabindex="-1"></a><span class="ot">guessAge ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb95-2"><a href="#cb95-2" aria-hidden="true" tabindex="-1"></a>guessAge <span class="st">&quot;Griselda&quot;</span> age</span>
 <span id="cb95-3"><a href="#cb95-3" aria-hidden="true" tabindex="-1"></a>    <span class="op">|</span> age <span class="op">&lt;</span> <span class="dv">47</span> <span class="ot">=</span> <span class="st">&quot;Too low!&quot;</span></span>
 <span id="cb95-4"><a href="#cb95-4" aria-hidden="true" tabindex="-1"></a>    <span class="op">|</span> age <span class="op">&gt;</span> <span class="dv">47</span> <span class="ot">=</span> <span class="st">&quot;Too high!&quot;</span></span>
@@ -1244,7 +1935,8 @@ Ok, two modules loaded.
 <span id="cb95-8"><a href="#cb95-8" aria-hidden="true" tabindex="-1"></a>    <span class="op">|</span> age <span class="op">&gt;</span> <span class="dv">12</span> <span class="ot">=</span> <span class="st">&quot;Too high!&quot;</span></span>
 <span id="cb95-9"><a href="#cb95-9" aria-hidden="true" tabindex="-1"></a>    <span class="op">|</span> <span class="fu">otherwise</span> <span class="ot">=</span> <span class="st">&quot;Correct!&quot;</span></span>
 <span id="cb95-10"><a href="#cb95-10" aria-hidden="true" tabindex="-1"></a>guessAge name age <span class="ot">=</span> <span class="st">&quot;Wrong name!&quot;</span></span></code></pre></div>
-<div class="sourceCode" id="cb96"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb96-1"><a href="#cb96-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> guessAge <span class="st">&quot;Griselda&quot;</span> <span class="dv">30</span></span>
+<div class="sourceCode" id="cb96"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb96-1"><a href="#cb96-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> guessAge <span class="st">&quot;Griselda&quot;</span> <span class="dv">30</span></span>
 <span id="cb96-2"><a href="#cb96-2" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;Too low!&quot;</span></span>
 <span id="cb96-3"><a href="#cb96-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> guessAge <span class="st">&quot;Griselda&quot;</span> <span class="dv">60</span></span>
 <span id="cb96-4"><a href="#cb96-4" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;Too high!&quot;</span></span>
@@ -1254,20 +1946,37 @@ Ok, two modules loaded.
 <span id="cb96-8"><a href="#cb96-8" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;Wrong name!&quot;</span></span>
 <span id="cb96-9"><a href="#cb96-9" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> guessAge <span class="st">&quot;Hansel&quot;</span> <span class="dv">10</span></span>
 <span id="cb96-10"><a href="#cb96-10" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;Too low!&quot;</span></span></code></pre></div>
-<h2 data-number="2.3" id="lists"><span class="header-section-number">2.3</span> Lists</h2>
-<p>So far we’ve always worked with single values like number or booleans. Strings contain multiple characters, but still in some sense a string is just one piece of information. In order to be able to do actual programming, we need to handle variable numbers of items. For this we need <em>data structures</em>.</p>
-<p>The basic datastructure in Haskell is the list. Lists are used to store multiple values of the same type (in other words, Haskell lists are homogeneous). This is what a list literal looks like:</p>
-<div class="sourceCode" id="cb97"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb97-1"><a href="#cb97-1" aria-hidden="true" tabindex="-1"></a>[<span class="dv">0</span>,<span class="dv">3</span>,<span class="dv">4</span>,<span class="dv">1</span><span class="op">+</span><span class="dv">1</span>]</span></code></pre></div>
-<p>A list type is written as <code>[Element]</code>, where <code>Element</code> is the type of the lists elements. Here are some more list expressions and their types:</p>
-<div class="sourceCode" id="cb98"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb98-1"><a href="#cb98-1" aria-hidden="true" tabindex="-1"></a>[<span class="dt">True</span>,<span class="dt">True</span>,<span class="dt">False</span>]<span class="ot"> ::</span> [<span class="dt">Bool</span>]</span>
+<h2 data-number="2.3" id="lists"><span
+class="header-section-number">2.3</span> Lists</h2>
+<p>So far we’ve always worked with single values like number or
+booleans. Strings contain multiple characters, but still in some sense a
+string is just one piece of information. In order to be able to do
+actual programming, we need to handle variable numbers of items. For
+this we need <em>data structures</em>.</p>
+<p>The basic datastructure in Haskell is the list. Lists are used to
+store multiple values of the same type (in other words, Haskell lists
+are homogeneous). This is what a list literal looks like:</p>
+<div class="sourceCode" id="cb97"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb97-1"><a href="#cb97-1" aria-hidden="true" tabindex="-1"></a>[<span class="dv">0</span>,<span class="dv">3</span>,<span class="dv">4</span>,<span class="dv">1</span><span class="op">+</span><span class="dv">1</span>]</span></code></pre></div>
+<p>A list type is written as <code>[Element]</code>, where
+<code>Element</code> is the type of the lists elements. Here are some
+more list expressions and their types:</p>
+<div class="sourceCode" id="cb98"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb98-1"><a href="#cb98-1" aria-hidden="true" tabindex="-1"></a>[<span class="dt">True</span>,<span class="dt">True</span>,<span class="dt">False</span>]<span class="ot"> ::</span> [<span class="dt">Bool</span>]</span>
 <span id="cb98-2"><a href="#cb98-2" aria-hidden="true" tabindex="-1"></a>[<span class="st">&quot;Moi&quot;</span>,<span class="st">&quot;Hei&quot;</span>]<span class="ot"> ::</span> [<span class="dt">String</span>]</span>
 <span id="cb98-3"><a href="#cb98-3" aria-hidden="true" tabindex="-1"></a>[]<span class="ot"> ::</span> [a]                   <span class="co">-- more about this later</span></span>
 <span id="cb98-4"><a href="#cb98-4" aria-hidden="true" tabindex="-1"></a>[[<span class="dv">1</span>,<span class="dv">2</span>],[<span class="dv">3</span>,<span class="dv">4</span>]]<span class="ot"> ::</span> [[<span class="dt">Int</span>]]    <span class="co">-- a list of lists</span></span>
 <span id="cb98-5"><a href="#cb98-5" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span><span class="op">..</span><span class="dv">7</span>]<span class="ot"> ::</span> [<span class="dt">Int</span>]             <span class="co">-- range syntax, value [1,2,3,4,5,6,7]</span></span></code></pre></div>
-<p>Haskell lists are implemented as singly-linked lists. We’ll return to this later.</p>
-<h3 data-number="2.3.1" id="list-operations"><span class="header-section-number">2.3.1</span> List Operations</h3>
-<p>The Haskell standard library comes with lots of functions that operate on lists. Here are some of the most important ones, together with their types. We’ll get back to what <code>[a]</code> actually means in a second, but for now you can imagine it means “any list”.</p>
-<div class="sourceCode" id="cb99"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb99-1"><a href="#cb99-1" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span><span class="ot"> ::</span> [a] <span class="ot">-&gt;</span> a            <span class="co">-- returns the first element</span></span>
+<p>Haskell lists are implemented as singly-linked lists. We’ll return to
+this later.</p>
+<h3 data-number="2.3.1" id="list-operations"><span
+class="header-section-number">2.3.1</span> List Operations</h3>
+<p>The Haskell standard library comes with lots of functions that
+operate on lists. Here are some of the most important ones, together
+with their types. We’ll get back to what <code>[a]</code> actually means
+in a second, but for now you can imagine it means “any list”.</p>
+<div class="sourceCode" id="cb99"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb99-1"><a href="#cb99-1" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span><span class="ot"> ::</span> [a] <span class="ot">-&gt;</span> a            <span class="co">-- returns the first element</span></span>
 <span id="cb99-2"><a href="#cb99-2" aria-hidden="true" tabindex="-1"></a><span class="fu">last</span><span class="ot"> ::</span> [a] <span class="ot">-&gt;</span> a            <span class="co">-- returns the last element</span></span>
 <span id="cb99-3"><a href="#cb99-3" aria-hidden="true" tabindex="-1"></a><span class="fu">tail</span><span class="ot"> ::</span> [a] <span class="ot">-&gt;</span> [a]          <span class="co">-- returns everything except the first element</span></span>
 <span id="cb99-4"><a href="#cb99-4" aria-hidden="true" tabindex="-1"></a><span class="fu">init</span><span class="ot"> ::</span> [a] <span class="ot">-&gt;</span> [a]          <span class="co">-- returns everything except the last element</span></span>
@@ -1278,35 +1987,61 @@ Ok, two modules loaded.
 <span id="cb99-9"><a href="#cb99-9" aria-hidden="true" tabindex="-1"></a><span class="fu">reverse</span><span class="ot"> ::</span> [a] <span class="ot">-&gt;</span> [a]       <span class="co">-- reverse a list</span></span>
 <span id="cb99-10"><a href="#cb99-10" aria-hidden="true" tabindex="-1"></a><span class="fu">null</span><span class="ot"> ::</span> [a] <span class="ot">-&gt;</span> <span class="dt">Bool</span>         <span class="co">-- is this list empty?</span></span>
 <span id="cb99-11"><a href="#cb99-11" aria-hidden="true" tabindex="-1"></a><span class="fu">length</span><span class="ot"> ::</span> [a] <span class="ot">-&gt;</span> <span class="dt">Int</span>        <span class="co">-- the length of a list</span></span></code></pre></div>
-<p>Sidenote: the last two operations (<code>null</code> and <code>length</code>) actually have more generic types, but here I’m pretending that you can only use them on lists.</p>
+<p>Sidenote: the last two operations (<code>null</code> and
+<code>length</code>) actually have more generic types, but here I’m
+pretending that you can only use them on lists.</p>
 <p>Lists can be compared with the familiar <code>==</code> operator.</p>
 <p>Do you remember this from our first GHCi session?</p>
-<div class="sourceCode" id="cb100"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb100-1"><a href="#cb100-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="st">&quot;asdf&quot;</span></span>
+<div class="sourceCode" id="cb100"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb100-1"><a href="#cb100-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="st">&quot;asdf&quot;</span></span>
 <span id="cb100-2"><a href="#cb100-2" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;asdf&quot;</span><span class="ot"> ::</span> [<span class="dt">Char</span>]</span></code></pre></div>
-<p>This means that <code>String</code> is just an alias for <code>[Char]</code>, which means string is a list of characters. This means you can use all list operations on strings!</p>
-<p>Some list operations come from the module <code>Data.List</code>. You can import a module in code or in GHCi with the <code>import Data.List</code> syntax. One example is the <code>sort</code> function which sorts a list:</p>
-<div class="sourceCode" id="cb101"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb101-1"><a href="#cb101-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="dt">Data.List</span></span>
+<p>This means that <code>String</code> is just an alias for
+<code>[Char]</code>, which means string is a list of characters. This
+means you can use all list operations on strings!</p>
+<p>Some list operations come from the module <code>Data.List</code>. You
+can import a module in code or in GHCi with the
+<code>import Data.List</code> syntax. One example is the
+<code>sort</code> function which sorts a list:</p>
+<div class="sourceCode" id="cb101"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb101-1"><a href="#cb101-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="dt">Data.List</span></span>
 <span id="cb101-2"><a href="#cb101-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.List</span><span class="op">&gt;</span> <span class="fu">sort</span> [<span class="dv">1</span>,<span class="dv">0</span>,<span class="dv">5</span>,<span class="dv">3</span>]</span>
 <span id="cb101-3"><a href="#cb101-3" aria-hidden="true" tabindex="-1"></a>[<span class="dv">0</span>,<span class="dv">1</span>,<span class="dv">3</span>,<span class="dv">5</span>]</span></code></pre></div>
 <p>Note how the set of imported modules shows up in the GHCi prompt.</p>
-<h3 data-number="2.3.2" id="examples-2"><span class="header-section-number">2.3.2</span> Examples</h3>
-<p>Here are some examples of working with lists. In this case, instead of showing you output from GHCi, I merely use <code>==&gt;</code> to show what an expression evaluates to.</p>
+<h3 data-number="2.3.2" id="examples-2"><span
+class="header-section-number">2.3.2</span> Examples</h3>
+<p>Here are some examples of working with lists. In this case, instead
+of showing you output from GHCi, I merely use <code>==&gt;</code> to
+show what an expression evaluates to.</p>
 <p>Indexing a list:</p>
-<div class="sourceCode" id="cb102"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb102-1"><a href="#cb102-1" aria-hidden="true" tabindex="-1"></a>[<span class="dv">7</span>,<span class="dv">10</span>,<span class="dv">4</span>,<span class="dv">5</span>] <span class="op">!!</span> <span class="dv">2</span></span>
+<div class="sourceCode" id="cb102"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb102-1"><a href="#cb102-1" aria-hidden="true" tabindex="-1"></a>[<span class="dv">7</span>,<span class="dv">10</span>,<span class="dv">4</span>,<span class="dv">5</span>] <span class="op">!!</span> <span class="dv">2</span></span>
 <span id="cb102-2"><a href="#cb102-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">4</span></span></code></pre></div>
-<p>Defining a function that discards the 3rd and 4th elements of a list using <code>take</code> and <code>drop</code>:</p>
-<div class="sourceCode" id="cb103"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb103-1"><a href="#cb103-1" aria-hidden="true" tabindex="-1"></a>f xs <span class="ot">=</span> <span class="fu">take</span> <span class="dv">2</span> xs <span class="op">++</span> <span class="fu">drop</span> <span class="dv">4</span> xs</span></code></pre></div>
-<div class="sourceCode" id="cb104"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb104-1"><a href="#cb104-1" aria-hidden="true" tabindex="-1"></a>f [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>,<span class="dv">5</span>,<span class="dv">6</span>]  <span class="op">==&gt;</span>  [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">5</span>,<span class="dv">6</span>]</span>
+<p>Defining a function that discards the 3rd and 4th elements of a list
+using <code>take</code> and <code>drop</code>:</p>
+<div class="sourceCode" id="cb103"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb103-1"><a href="#cb103-1" aria-hidden="true" tabindex="-1"></a>f xs <span class="ot">=</span> <span class="fu">take</span> <span class="dv">2</span> xs <span class="op">++</span> <span class="fu">drop</span> <span class="dv">4</span> xs</span></code></pre></div>
+<div class="sourceCode" id="cb104"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb104-1"><a href="#cb104-1" aria-hidden="true" tabindex="-1"></a>f [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>,<span class="dv">5</span>,<span class="dv">6</span>]  <span class="op">==&gt;</span>  [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">5</span>,<span class="dv">6</span>]</span>
 <span id="cb104-2"><a href="#cb104-2" aria-hidden="true" tabindex="-1"></a>f [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]        <span class="op">==&gt;</span>  [<span class="dv">1</span>,<span class="dv">2</span>]</span></code></pre></div>
-<p>Rotating a list by taking the first element and moving it to the end:</p>
-<div class="sourceCode" id="cb105"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb105-1"><a href="#cb105-1" aria-hidden="true" tabindex="-1"></a>g xs <span class="ot">=</span> <span class="fu">tail</span> xs <span class="op">++</span> [<span class="fu">head</span> xs]</span></code></pre></div>
-<div class="sourceCode" id="cb106"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb106-1"><a href="#cb106-1" aria-hidden="true" tabindex="-1"></a>g [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]      <span class="op">==&gt;</span>  [<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">1</span>]</span>
+<p>Rotating a list by taking the first element and moving it to the
+end:</p>
+<div class="sourceCode" id="cb105"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb105-1"><a href="#cb105-1" aria-hidden="true" tabindex="-1"></a>g xs <span class="ot">=</span> <span class="fu">tail</span> xs <span class="op">++</span> [<span class="fu">head</span> xs]</span></code></pre></div>
+<div class="sourceCode" id="cb106"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb106-1"><a href="#cb106-1" aria-hidden="true" tabindex="-1"></a>g [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]      <span class="op">==&gt;</span>  [<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">1</span>]</span>
 <span id="cb106-2"><a href="#cb106-2" aria-hidden="true" tabindex="-1"></a>g (g [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>])  <span class="op">==&gt;</span>  [<span class="dv">3</span>,<span class="dv">1</span>,<span class="dv">2</span>]</span></code></pre></div>
 <p>Here’s an example of the range syntax:</p>
-<div class="sourceCode" id="cb107"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb107-1"><a href="#cb107-1" aria-hidden="true" tabindex="-1"></a><span class="fu">reverse</span> [<span class="dv">1</span><span class="op">..</span><span class="dv">4</span>] <span class="op">==&gt;</span> [<span class="dv">4</span>,<span class="dv">3</span>,<span class="dv">2</span>,<span class="dv">1</span>]</span></code></pre></div>
-<h2 data-number="2.4" id="a-word-about-immutability-1"><span class="header-section-number">2.4</span> A Word About Immutability</h2>
-<p>Because Haskell is pure, it also means that functions can’t <em>mutate</em> (change) their inputs. Mutation is a side effect, and Haskell functions are only allowed output via their return value. This means that Haskell list functions always return a new list. In practice:</p>
-<div class="sourceCode" id="cb108"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb108-1"><a href="#cb108-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> list <span class="ot">=</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>]</span>
+<div class="sourceCode" id="cb107"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb107-1"><a href="#cb107-1" aria-hidden="true" tabindex="-1"></a><span class="fu">reverse</span> [<span class="dv">1</span><span class="op">..</span><span class="dv">4</span>] <span class="op">==&gt;</span> [<span class="dv">4</span>,<span class="dv">3</span>,<span class="dv">2</span>,<span class="dv">1</span>]</span></code></pre></div>
+<h2 data-number="2.4" id="a-word-about-immutability-1"><span
+class="header-section-number">2.4</span> A Word About Immutability</h2>
+<p>Because Haskell is pure, it also means that functions can’t
+<em>mutate</em> (change) their inputs. Mutation is a side effect, and
+Haskell functions are only allowed output via their return value. This
+means that Haskell list functions always return a new list. In
+practice:</p>
+<div class="sourceCode" id="cb108"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb108-1"><a href="#cb108-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> list <span class="ot">=</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>]</span>
 <span id="cb108-2"><a href="#cb108-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">reverse</span> list</span>
 <span id="cb108-3"><a href="#cb108-3" aria-hidden="true" tabindex="-1"></a>[<span class="dv">4</span>,<span class="dv">3</span>,<span class="dv">2</span>,<span class="dv">1</span>]</span>
 <span id="cb108-4"><a href="#cb108-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> list</span>
@@ -1315,17 +2050,43 @@ Ok, two modules loaded.
 <span id="cb108-7"><a href="#cb108-7" aria-hidden="true" tabindex="-1"></a>[<span class="dv">3</span>,<span class="dv">4</span>]</span>
 <span id="cb108-8"><a href="#cb108-8" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> list</span>
 <span id="cb108-9"><a href="#cb108-9" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>]</span></code></pre></div>
-<p>This might seem very inefficient but it turns out it can be both performant and quite useful. We’ll get back to how Haskell datastructures work in a later lecture.</p>
-<h2 data-number="2.5" id="a-word-about-type-inference-and-polymorphism"><span class="header-section-number">2.5</span> A Word About Type Inference and Polymorphism</h2>
-<p>So what does a type like <code>head :: [a] -&gt; a</code> mean? It means given a list that contains elements of any type <code>a</code>, the return value will be of the same type <code>a</code>.</p>
-<p>In this type, <code>a</code> is a <em>type variable</em>. Type variables are types that start with a small letter, e.g. <code>a</code>, <code>b</code>, <code>thisIsATypeVariable</code>. A type variable means a type that is not yet known, or in other words a type that could be anything. Type variables can turn into <em>concrete types</em> (e.g. <code>Bool</code>) by the process of <em>type inference</em> (also called <em>unification</em>).</p>
-<p>Let’s have a look at some examples. If we apply <code>head</code> to a list of booleans, type inference will compare the type of head’s argument, <code>[a]</code>, with the type of the actual argument, <code>[Bool]</code> and deduce that <code>a</code> must be <code>Bool</code>. This means that the return type of <code>head</code> will in this case also be <code>Bool</code>.</p>
-<div class="sourceCode" id="cb109"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb109-1"><a href="#cb109-1" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span><span class="ot"> ::</span> [a] <span class="ot">-&gt;</span> a</span>
+<p>This might seem very inefficient but it turns out it can be both
+performant and quite useful. We’ll get back to how Haskell
+datastructures work in a later lecture.</p>
+<h2 data-number="2.5"
+id="a-word-about-type-inference-and-polymorphism"><span
+class="header-section-number">2.5</span> A Word About Type Inference and
+Polymorphism</h2>
+<p>So what does a type like <code>head :: [a] -&gt; a</code> mean? It
+means given a list that contains elements of any type <code>a</code>,
+the return value will be of the same type <code>a</code>.</p>
+<p>In this type, <code>a</code> is a <em>type variable</em>. Type
+variables are types that start with a small letter, e.g. <code>a</code>,
+<code>b</code>, <code>thisIsATypeVariable</code>. A type variable means
+a type that is not yet known, or in other words a type that could be
+anything. Type variables can turn into <em>concrete types</em>
+(e.g. <code>Bool</code>) by the process of <em>type inference</em> (also
+called <em>unification</em>).</p>
+<p>Let’s have a look at some examples. If we apply <code>head</code> to
+a list of booleans, type inference will compare the type of head’s
+argument, <code>[a]</code>, with the type of the actual argument,
+<code>[Bool]</code> and deduce that <code>a</code> must be
+<code>Bool</code>. This means that the return type of <code>head</code>
+will in this case also be <code>Bool</code>.</p>
+<div class="sourceCode" id="cb109"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb109-1"><a href="#cb109-1" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span><span class="ot"> ::</span> [a] <span class="ot">-&gt;</span> a</span>
 <span id="cb109-2"><a href="#cb109-2" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span> [<span class="dt">True</span>,<span class="dt">False</span>]<span class="ot"> ::</span> <span class="dt">Bool</span></span></code></pre></div>
-<p>The function <code>tail</code> takes a list, and returns a list of the same type. If we apply <code>tail</code> to a list of booleans, the return value will also be a list of booleans.</p>
-<div class="sourceCode" id="cb110"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb110-1"><a href="#cb110-1" aria-hidden="true" tabindex="-1"></a><span class="fu">tail</span><span class="ot"> ::</span> [a] <span class="ot">-&gt;</span> [a]</span>
+<p>The function <code>tail</code> takes a list, and returns a list of
+the same type. If we apply <code>tail</code> to a list of booleans, the
+return value will also be a list of booleans.</p>
+<div class="sourceCode" id="cb110"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb110-1"><a href="#cb110-1" aria-hidden="true" tabindex="-1"></a><span class="fu">tail</span><span class="ot"> ::</span> [a] <span class="ot">-&gt;</span> [a]</span>
 <span id="cb110-2"><a href="#cb110-2" aria-hidden="true" tabindex="-1"></a><span class="fu">tail</span> [<span class="dt">True</span>,<span class="dt">False</span>]<span class="ot"> ::</span> [<span class="dt">Bool</span>]</span></code></pre></div>
-<p>If types don’t match, we get a type error. Consider the operator <code>++</code> which takes two lists of the same type, as we can see from its type <code>[a] -&gt; [a] -&gt; [a]</code>. If we try to apply <code>++</code> to a list of booleans and a list of characters we get an error. This is what happens in GHCi:</p>
+<p>If types don’t match, we get a type error. Consider the operator
+<code>++</code> which takes two lists of the same type, as we can see
+from its type <code>[a] -&gt; [a] -&gt; [a]</code>. If we try to apply
+<code>++</code> to a list of booleans and a list of characters we get an
+error. This is what happens in GHCi:</p>
 <pre><code>Prelude&gt; [True,False] ++ &quot;Moi&quot;
 
 &lt;interactive&gt;:1:16:
@@ -1334,30 +2095,70 @@ Ok, two modules loaded.
       Inferred type: [Char]
     In the second argument of `(++)&#39;, namely `&quot;Moi&quot;&#39;
     In the expression: [True, False] ++ &quot;Moi&quot;</code></pre>
-<p>Type inference is really powerful. It uses the simple process of unification to get us types for practically any Haskell expression. Consider these two functions:</p>
-<div class="sourceCode" id="cb112"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb112-1"><a href="#cb112-1" aria-hidden="true" tabindex="-1"></a>f xs ys <span class="ot">=</span> [<span class="fu">head</span> xs, <span class="fu">head</span> ys]</span>
+<p>Type inference is really powerful. It uses the simple process of
+unification to get us types for practically any Haskell expression.
+Consider these two functions:</p>
+<div class="sourceCode" id="cb112"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb112-1"><a href="#cb112-1" aria-hidden="true" tabindex="-1"></a>f xs ys <span class="ot">=</span> [<span class="fu">head</span> xs, <span class="fu">head</span> ys]</span>
 <span id="cb112-2"><a href="#cb112-2" aria-hidden="true" tabindex="-1"></a>g zs <span class="ot">=</span> f <span class="st">&quot;Moi&quot;</span> zs</span></code></pre></div>
-<p>We can ask GHCi for their types, and we will see that type inference has figured out that the two arguments to <code>f</code> must have the same type, since their heads get put into the same list.</p>
-<div class="sourceCode" id="cb113"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb113-1"><a href="#cb113-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t f</span>
+<p>We can ask GHCi for their types, and we will see that type inference
+has figured out that the two arguments to <code>f</code> must have the
+same type, since their heads get put into the same list.</p>
+<div class="sourceCode" id="cb113"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb113-1"><a href="#cb113-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t f</span>
 <span id="cb113-2"><a href="#cb113-2" aria-hidden="true" tabindex="-1"></a><span class="ot">f ::</span> [a] <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [a]</span></code></pre></div>
-<p>The function <code>g</code>, which fixed one of the arguments of <code>f</code> to a string (i.e. <code>[Char]</code>) gets a narrower type. Type inference has decided that the argument <code>zs</code> to <code>g</code> must also have type <code>[Char]</code>, since otherwise the type of <code>f</code> would not match the call to <code>f</code>.</p>
-<div class="sourceCode" id="cb114"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb114-1"><a href="#cb114-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t g</span>
+<p>The function <code>g</code>, which fixed one of the arguments of
+<code>f</code> to a string (i.e. <code>[Char]</code>) gets a narrower
+type. Type inference has decided that the argument <code>zs</code> to
+<code>g</code> must also have type <code>[Char]</code>, since otherwise
+the type of <code>f</code> would not match the call to
+<code>f</code>.</p>
+<div class="sourceCode" id="cb114"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb114-1"><a href="#cb114-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t g</span>
 <span id="cb114-2"><a href="#cb114-2" aria-hidden="true" tabindex="-1"></a><span class="ot">g ::</span> [<span class="dt">Char</span>] <span class="ot">-&gt;</span> [<span class="dt">Char</span>]</span></code></pre></div>
-<h3 data-number="2.5.1" id="sidenote-some-terminology"><span class="header-section-number">2.5.1</span> Sidenote: Some Terminology</h3>
-<p>In a type like <code>[Char]</code> we call <code>Char</code> a <em>type parameter</em>. A type like the list type that needs a type parameter is called a <em>parameterized type</em>.</p>
-<p>The fact that a function like <code>head</code> can be used with many different types of arguments is called <em>polymorphism</em>. The <code>head</code> function is said to be <em>polymorphic</em>. There are many forms of polymorphism, and this Haskell form that uses type variables is called <em>parametric polymorphism</em>.</p>
-<h3 data-number="2.5.2" id="sidenote-type-annotations"><span class="header-section-number">2.5.2</span> Sidenote: Type Annotations</h3>
-<p>Since Haskell has type inference, you don’t need to give any type annotations. However even though type annotations aren’t required, there are multiple reasons to add them:</p>
+<h3 data-number="2.5.1" id="sidenote-some-terminology"><span
+class="header-section-number">2.5.1</span> Sidenote: Some
+Terminology</h3>
+<p>In a type like <code>[Char]</code> we call <code>Char</code> a
+<em>type parameter</em>. A type like the list type that needs a type
+parameter is called a <em>parameterized type</em>.</p>
+<p>The fact that a function like <code>head</code> can be used with many
+different types of arguments is called <em>polymorphism</em>. The
+<code>head</code> function is said to be <em>polymorphic</em>. There are
+many forms of polymorphism, and this Haskell form that uses type
+variables is called <em>parametric polymorphism</em>.</p>
+<h3 data-number="2.5.2" id="sidenote-type-annotations"><span
+class="header-section-number">2.5.2</span> Sidenote: Type
+Annotations</h3>
+<p>Since Haskell has type inference, you don’t need to give any type
+annotations. However even though type annotations aren’t required, there
+are multiple reasons to add them:</p>
 <ol type="1">
 <li>They act as documentation</li>
-<li>They act as assertions that the compiler checks: help you spot mistakes</li>
-<li>You can use type annotations to give a function a narrower type than Haskell infers</li>
+<li>They act as assertions that the compiler checks: help you spot
+mistakes</li>
+<li>You can use type annotations to give a function a narrower type than
+Haskell infers</li>
 </ol>
-<p>A good rule of thumb is to give top-level definitions type annotations.</p>
-<h2 data-number="2.6" id="the-maybe-type"><span class="header-section-number">2.6</span> The <code>Maybe</code> Type</h2>
-<p>In addition to the list type, Haskell has other parameterized types too. Let’s look at a very common and useful one: the <code>Maybe</code> type.</p>
-<p>Sometimes an operation doesn’t have a valid return value (E.g. division by zero.). We have a couple of options in this situation. We can use an error value, like <code>-1</code>. This is a bit ugly, not always possible. We can throw an exception. This is impure. In some other languages we would return a special null value that exists in (almost) all types. However Haskell does not have a null.</p>
-<p>The solution Haskell offers us instead is to change our return type to a <code>Maybe</code> type. This is pure, safe and neat. The type <code>Maybe a</code> has two <em>constructors</em>: <code>Nothing</code> and <code>Just</code>. <code>Nothing</code> is just a constant, but <code>Just</code> takes a parameter. More concretely:</p>
+<p>A good rule of thumb is to give top-level definitions type
+annotations.</p>
+<h2 data-number="2.6" id="the-maybe-type"><span
+class="header-section-number">2.6</span> The <code>Maybe</code>
+Type</h2>
+<p>In addition to the list type, Haskell has other parameterized types
+too. Let’s look at a very common and useful one: the <code>Maybe</code>
+type.</p>
+<p>Sometimes an operation doesn’t have a valid return value (E.g.
+division by zero.). We have a couple of options in this situation. We
+can use an error value, like <code>-1</code>. This is a bit ugly, not
+always possible. We can throw an exception. This is impure. In some
+other languages we would return a special null value that exists in
+(almost) all types. However Haskell does not have a null.</p>
+<p>The solution Haskell offers us instead is to change our return type
+to a <code>Maybe</code> type. This is pure, safe and neat. The type
+<code>Maybe a</code> has two <em>constructors</em>: <code>Nothing</code>
+and <code>Just</code>. <code>Nothing</code> is just a constant, but
+<code>Just</code> takes a parameter. More concretely:</p>
 <table>
 <thead>
 <tr class="header">
@@ -1368,21 +2169,31 @@ Ok, two modules loaded.
 <tbody>
 <tr class="odd">
 <td style="text-align: left;"><code>Maybe Bool</code></td>
-<td style="text-align: left;"><code>Nothing</code>, <code>Just False</code>, <code>Just True</code></td>
+<td style="text-align: left;"><code>Nothing</code>,
+<code>Just False</code>, <code>Just True</code></td>
 </tr>
 <tr class="even">
 <td style="text-align: left;"><code>Maybe Int</code></td>
-<td style="text-align: left;"><code>Nothing</code>, <code>Just 0</code>, <code>Just 1</code>, …</td>
+<td style="text-align: left;"><code>Nothing</code>, <code>Just 0</code>,
+<code>Just 1</code>, …</td>
 </tr>
 <tr class="odd">
 <td style="text-align: left;"><code>Maybe [Int]</code></td>
-<td style="text-align: left;"><code>Nothing</code>, <code>Just []</code>, <code>Just [1,1337]</code>, …</td>
+<td style="text-align: left;"><code>Nothing</code>,
+<code>Just []</code>, <code>Just [1,1337]</code>, …</td>
 </tr>
 </tbody>
 </table>
-<p>You can think of <code>Maybe a</code> as being a bit like <code>[a]</code> except there can only be 0 or 1 elements, not more. Alternatively, you can think of <code>Maybe a</code> introducing a null value to the type <code>a</code>. If you’re familiar with Java, <code>Maybe Integer</code> is the Haskell equivalent of Java’s <code>Optional&lt;Integer&gt;</code>.</p>
-<p>You can create <code>Maybe</code> values by either specifying <code>Nothing</code> or <code>Just someOtherValue</code>:</p>
-<div class="sourceCode" id="cb115"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb115-1"><a href="#cb115-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="dt">Nothing</span></span>
+<p>You can think of <code>Maybe a</code> as being a bit like
+<code>[a]</code> except there can only be 0 or 1 elements, not more.
+Alternatively, you can think of <code>Maybe a</code> introducing a null
+value to the type <code>a</code>. If you’re familiar with Java,
+<code>Maybe Integer</code> is the Haskell equivalent of Java’s
+<code>Optional&lt;Integer&gt;</code>.</p>
+<p>You can create <code>Maybe</code> values by either specifying
+<code>Nothing</code> or <code>Just someOtherValue</code>:</p>
+<div class="sourceCode" id="cb115"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb115-1"><a href="#cb115-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="dt">Nothing</span></span>
 <span id="cb115-2"><a href="#cb115-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Nothing</span><span class="ot"> ::</span> <span class="dt">Maybe</span> a</span>
 <span id="cb115-3"><a href="#cb115-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">Just</span> <span class="st">&quot;a camel&quot;</span></span>
 <span id="cb115-4"><a href="#cb115-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Just</span> <span class="st">&quot;a camel&quot;</span></span>
@@ -1392,21 +2203,27 @@ Ok, two modules loaded.
 <span id="cb115-8"><a href="#cb115-8" aria-hidden="true" tabindex="-1"></a><span class="dt">Just</span> <span class="dt">True</span></span>
 <span id="cb115-9"><a href="#cb115-9" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="dt">Just</span> <span class="dt">True</span></span>
 <span id="cb115-10"><a href="#cb115-10" aria-hidden="true" tabindex="-1"></a><span class="dt">Just</span> <span class="dt">True</span><span class="ot"> ::</span> <span class="dt">Maybe</span> <span class="dt">Bool</span></span></code></pre></div>
-<div class="sourceCode" id="cb116"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb116-1"><a href="#cb116-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- given a password, return (Just username) if login succeeds, Nothing otherwise</span></span>
+<div class="sourceCode" id="cb116"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb116-1"><a href="#cb116-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- given a password, return (Just username) if login succeeds, Nothing otherwise</span></span>
 <span id="cb116-2"><a href="#cb116-2" aria-hidden="true" tabindex="-1"></a><span class="ot">login ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">String</span></span>
 <span id="cb116-3"><a href="#cb116-3" aria-hidden="true" tabindex="-1"></a>login <span class="st">&quot;f4bulous!&quot;</span> <span class="ot">=</span> <span class="dt">Just</span> <span class="st">&quot;unicorn73&quot;</span></span>
 <span id="cb116-4"><a href="#cb116-4" aria-hidden="true" tabindex="-1"></a>login <span class="st">&quot;swordfish&quot;</span> <span class="ot">=</span> <span class="dt">Just</span> <span class="st">&quot;megahacker&quot;</span></span>
 <span id="cb116-5"><a href="#cb116-5" aria-hidden="true" tabindex="-1"></a>login _           <span class="ot">=</span> <span class="dt">Nothing</span></span></code></pre></div>
-<p>You use a <code>Maybe</code> value by pattern matching on it. Usually you define patterns for the <code>Nothing</code> and <code>Just something</code> cases. Some examples:</p>
-<div class="sourceCode" id="cb117"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb117-1"><a href="#cb117-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Multiply an Int with a Maybe Int. Nothing is treated as no multiplication at all.</span></span>
+<p>You use a <code>Maybe</code> value by pattern matching on it. Usually
+you define patterns for the <code>Nothing</code> and
+<code>Just something</code> cases. Some examples:</p>
+<div class="sourceCode" id="cb117"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb117-1"><a href="#cb117-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Multiply an Int with a Maybe Int. Nothing is treated as no multiplication at all.</span></span>
 <span id="cb117-2"><a href="#cb117-2" aria-hidden="true" tabindex="-1"></a><span class="ot">perhapsMultiply ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb117-3"><a href="#cb117-3" aria-hidden="true" tabindex="-1"></a>perhapsMultiply i <span class="dt">Nothing</span> <span class="ot">=</span> i</span>
 <span id="cb117-4"><a href="#cb117-4" aria-hidden="true" tabindex="-1"></a>perhapsMultiply i (<span class="dt">Just</span> j) <span class="ot">=</span> i<span class="op">*</span>j   <span class="co">-- Note how j denotes the value inside the Just</span></span></code></pre></div>
-<div class="sourceCode" id="cb118"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb118-1"><a href="#cb118-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> perhapsMultiply <span class="dv">3</span> <span class="dt">Nothing</span></span>
+<div class="sourceCode" id="cb118"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb118-1"><a href="#cb118-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> perhapsMultiply <span class="dv">3</span> <span class="dt">Nothing</span></span>
 <span id="cb118-2"><a href="#cb118-2" aria-hidden="true" tabindex="-1"></a><span class="dv">3</span></span>
 <span id="cb118-3"><a href="#cb118-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> perhapsMultiply <span class="dv">3</span> (<span class="dt">Just</span> <span class="dv">2</span>)</span>
 <span id="cb118-4"><a href="#cb118-4" aria-hidden="true" tabindex="-1"></a><span class="dv">6</span></span></code></pre></div>
-<div class="sourceCode" id="cb119"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb119-1"><a href="#cb119-1" aria-hidden="true" tabindex="-1"></a><span class="ot">intOrZero ::</span> <span class="dt">Maybe</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<div class="sourceCode" id="cb119"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb119-1"><a href="#cb119-1" aria-hidden="true" tabindex="-1"></a><span class="ot">intOrZero ::</span> <span class="dt">Maybe</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb119-2"><a href="#cb119-2" aria-hidden="true" tabindex="-1"></a>intOrZero <span class="dt">Nothing</span> <span class="ot">=</span> <span class="dv">0</span></span>
 <span id="cb119-3"><a href="#cb119-3" aria-hidden="true" tabindex="-1"></a>intOrZero (<span class="dt">Just</span> i) <span class="ot">=</span> i</span>
 <span id="cb119-4"><a href="#cb119-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -1415,16 +2232,35 @@ Ok, two modules loaded.
 <span id="cb119-7"><a href="#cb119-7" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb119-8"><a href="#cb119-8" aria-hidden="true" tabindex="-1"></a><span class="ot">headOrZero ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb119-9"><a href="#cb119-9" aria-hidden="true" tabindex="-1"></a>headOrZero xs <span class="ot">=</span> intOrZero (safeHead xs)</span></code></pre></div>
-<div class="sourceCode" id="cb120"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb120-1"><a href="#cb120-1" aria-hidden="true" tabindex="-1"></a>headOrZero []  <span class="op">==&gt;</span> intOrZero (safeHead [])  <span class="op">==&gt;</span> intOrZero <span class="dt">Nothing</span>  <span class="op">==&gt;</span> <span class="dv">0</span></span>
+<div class="sourceCode" id="cb120"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb120-1"><a href="#cb120-1" aria-hidden="true" tabindex="-1"></a>headOrZero []  <span class="op">==&gt;</span> intOrZero (safeHead [])  <span class="op">==&gt;</span> intOrZero <span class="dt">Nothing</span>  <span class="op">==&gt;</span> <span class="dv">0</span></span>
 <span id="cb120-2"><a href="#cb120-2" aria-hidden="true" tabindex="-1"></a>headOrZero [<span class="dv">1</span>] <span class="op">==&gt;</span> intOrZero (safeHead [<span class="dv">1</span>]) <span class="op">==&gt;</span> intOrZero (<span class="dt">Just</span> <span class="dv">1</span>) <span class="op">==&gt;</span> <span class="dv">1</span></span></code></pre></div>
-<h2 data-number="2.7" id="sidenote-constructors"><span class="header-section-number">2.7</span> Sidenote: Constructors</h2>
-<p>As you can see above, we can pattern match on the constructors of <code>Maybe</code>: <code>Just</code> and <code>Nothing</code>. We’ll get back to what constructors mean later. For now it’s enough to note that constructors are special values that start with a capital letter that you can pattern match on.</p>
-<p>Other constructors that we’ve already seen include the constructors of <code>Bool</code> – <code>True</code> and <code>False</code>. We’ll introduce the constructors of the list type on the next lecture.</p>
-<p>Constructors can be used just like Haskell values. Constructors that take no arguments like <code>Nothing</code>, and <code>False</code> are just constants. Constructors like <code>Just</code> that take an argument behave like functions. They even have function types!</p>
+<h2 data-number="2.7" id="sidenote-constructors"><span
+class="header-section-number">2.7</span> Sidenote: Constructors</h2>
+<p>As you can see above, we can pattern match on the constructors of
+<code>Maybe</code>: <code>Just</code> and <code>Nothing</code>. We’ll
+get back to what constructors mean later. For now it’s enough to note
+that constructors are special values that start with a capital letter
+that you can pattern match on.</p>
+<p>Other constructors that we’ve already seen include the constructors
+of <code>Bool</code> – <code>True</code> and <code>False</code>. We’ll
+introduce the constructors of the list type on the next lecture.</p>
+<p>Constructors can be used just like Haskell values. Constructors that
+take no arguments like <code>Nothing</code>, and <code>False</code> are
+just constants. Constructors like <code>Just</code> that take an
+argument behave like functions. They even have function types!</p>
 <pre><code>Prelude&gt; :t Just
 Just :: a -&gt; Maybe a</code></pre>
-<h2 data-number="2.8" id="the-either-type"><span class="header-section-number">2.8</span> The <code>Either</code> type</h2>
-<p>Sometimes it would be nice if you could add an error message or something to <code>Nothing</code>. That’s why we have the <code>Either</code> type. The <code>Either</code> type takes two type arguments. The type <code>Either a b</code> has two constructors: <code>Left</code> and <code>Right</code>. Both take an argument, <code>Left</code> an argument of type <code>a</code> and <code>Right</code> an argument of type <code>b</code>.</p>
+<h2 data-number="2.8" id="the-either-type"><span
+class="header-section-number">2.8</span> The <code>Either</code>
+type</h2>
+<p>Sometimes it would be nice if you could add an error message or
+something to <code>Nothing</code>. That’s why we have the
+<code>Either</code> type. The <code>Either</code> type takes two type
+arguments. The type <code>Either a b</code> has two constructors:
+<code>Left</code> and <code>Right</code>. Both take an argument,
+<code>Left</code> an argument of type <code>a</code> and
+<code>Right</code> an argument of type <code>b</code>.</p>
 <table>
 <thead>
 <tr class="header">
@@ -1435,48 +2271,79 @@ Just :: a -&gt; Maybe a</code></pre>
 <tbody>
 <tr class="odd">
 <td style="text-align: left;"><code>Either Int Bool</code></td>
-<td style="text-align: left;"><code>Left 0</code>, <code>Left 1</code>, <code>Right False</code>, <code>Right True</code>, …</td>
+<td style="text-align: left;"><code>Left 0</code>, <code>Left 1</code>,
+<code>Right False</code>, <code>Right True</code>, …</td>
 </tr>
 <tr class="even">
 <td style="text-align: left;"><code>Either String [Int]</code></td>
-<td style="text-align: left;"><code>Left "asdf"</code>, <code>Right [0,1,2]</code>, …</td>
+<td style="text-align: left;"><code>Left "asdf"</code>,
+<code>Right [0,1,2]</code>, …</td>
 </tr>
 <tr class="odd">
 <td style="text-align: left;"><code>Either Integer Integer</code></td>
-<td style="text-align: left;"><code>Left 0</code>, <code>Right 0</code>, <code>Left 1</code>, <code>Right 1</code>, …</td>
+<td style="text-align: left;"><code>Left 0</code>, <code>Right 0</code>,
+<code>Left 1</code>, <code>Right 1</code>, …</td>
 </tr>
 </tbody>
 </table>
-<p>Here’s a simple example: a <code>readInt</code> function that only knows a couple of numbers and returns a descriptive error for the rest. Note the Haskell convention of using <code>Left</code> for errors and <code>Right</code> for success.</p>
-<div class="sourceCode" id="cb122"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb122-1"><a href="#cb122-1" aria-hidden="true" tabindex="-1"></a><span class="fu">readInt</span><span class="ot"> ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Either</span> <span class="dt">String</span> <span class="dt">Int</span></span>
+<p>Here’s a simple example: a <code>readInt</code> function that only
+knows a couple of numbers and returns a descriptive error for the rest.
+Note the Haskell convention of using <code>Left</code> for errors and
+<code>Right</code> for success.</p>
+<div class="sourceCode" id="cb122"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb122-1"><a href="#cb122-1" aria-hidden="true" tabindex="-1"></a><span class="fu">readInt</span><span class="ot"> ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Either</span> <span class="dt">String</span> <span class="dt">Int</span></span>
 <span id="cb122-2"><a href="#cb122-2" aria-hidden="true" tabindex="-1"></a><span class="fu">readInt</span> <span class="st">&quot;0&quot;</span> <span class="ot">=</span> <span class="dt">Right</span> <span class="dv">0</span></span>
 <span id="cb122-3"><a href="#cb122-3" aria-hidden="true" tabindex="-1"></a><span class="fu">readInt</span> <span class="st">&quot;1&quot;</span> <span class="ot">=</span> <span class="dt">Right</span> <span class="dv">1</span></span>
 <span id="cb122-4"><a href="#cb122-4" aria-hidden="true" tabindex="-1"></a><span class="fu">readInt</span> s <span class="ot">=</span> <span class="dt">Left</span> (<span class="st">&quot;Unsupported string: &quot;</span> <span class="op">++</span> s)</span></code></pre></div>
-<p>Sidenote: the constructors of <code>Either</code> are called <code>Left</code> and <code>Right</code> because they refer to the left and right type arguments of <code>Either</code>. Note how in <code>Either a b</code>, <code>a</code> is the left argument and <code>b</code> is the right argument. Thus <code>Left</code> contains a value of type <code>a</code> and likewise <code>Right</code> of type <code>b</code>. The convention of using <code>Right</code> for success is probably simply because right also means correct. No offense is intended to left-handed people.</p>
-<p>Here’s another example: pattern matching an <code>Either</code>. Just like with <code>Maybe</code>, there are two patterns for an <code>Either</code>, one for each constructor.</p>
-<div class="sourceCode" id="cb123"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb123-1"><a href="#cb123-1" aria-hidden="true" tabindex="-1"></a><span class="ot">iWantAString ::</span> <span class="dt">Either</span> <span class="dt">Int</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<p>Sidenote: the constructors of <code>Either</code> are called
+<code>Left</code> and <code>Right</code> because they refer to the left
+and right type arguments of <code>Either</code>. Note how in
+<code>Either a b</code>, <code>a</code> is the left argument and
+<code>b</code> is the right argument. Thus <code>Left</code> contains a
+value of type <code>a</code> and likewise <code>Right</code> of type
+<code>b</code>. The convention of using <code>Right</code> for success
+is probably simply because right also means correct. No offense is
+intended to left-handed people.</p>
+<p>Here’s another example: pattern matching an <code>Either</code>. Just
+like with <code>Maybe</code>, there are two patterns for an
+<code>Either</code>, one for each constructor.</p>
+<div class="sourceCode" id="cb123"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb123-1"><a href="#cb123-1" aria-hidden="true" tabindex="-1"></a><span class="ot">iWantAString ::</span> <span class="dt">Either</span> <span class="dt">Int</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb123-2"><a href="#cb123-2" aria-hidden="true" tabindex="-1"></a>iWantAString (<span class="dt">Right</span> str)   <span class="ot">=</span> str</span>
 <span id="cb123-3"><a href="#cb123-3" aria-hidden="true" tabindex="-1"></a>iWantAString (<span class="dt">Left</span> number) <span class="ot">=</span> <span class="fu">show</span> number</span></code></pre></div>
-<p>As you recall, Haskell lists can only contain elements of the same type. You can’t have a value like <code>[1,"foo",2]</code>. However, you can use a type like <code>Either</code> to represent lists that can contain two different types of values. For example we could track the number of people on a lecture, with a possibility of adding an explanation if a value is missing:</p>
+<p>As you recall, Haskell lists can only contain elements of the same
+type. You can’t have a value like <code>[1,"foo",2]</code>. However, you
+can use a type like <code>Either</code> to represent lists that can
+contain two different types of values. For example we could track the
+number of people on a lecture, with a possibility of adding an
+explanation if a value is missing:</p>
 <pre><code>lectureParticipants :: [Either String Int]
 lectureParticipants = [Right 10, Right 13, Left &quot;easter vacation&quot;, Right 17, Left &quot;lecturer was sick&quot;, Right 3]</code></pre>
-<h2 data-number="2.9" id="the-case-of-expression"><span class="header-section-number">2.9</span> The case-of Expression</h2>
-<p>We’ve seen pattern matching in function arguments, but there’s also a way to pattern match in an expression. It looks like this:</p>
-<div class="sourceCode" id="cb125"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb125-1"><a href="#cb125-1" aria-hidden="true" tabindex="-1"></a><span class="kw">case</span> <span class="op">&lt;</span>value<span class="op">&gt;</span> <span class="kw">of</span> <span class="op">&lt;</span><span class="kw">pattern</span><span class="op">&gt;</span> <span class="ot">-&gt;</span> <span class="op">&lt;</span>expression<span class="op">&gt;</span></span>
+<h2 data-number="2.9" id="the-case-of-expression"><span
+class="header-section-number">2.9</span> The case-of Expression</h2>
+<p>We’ve seen pattern matching in function arguments, but there’s also a
+way to pattern match in an expression. It looks like this:</p>
+<div class="sourceCode" id="cb125"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb125-1"><a href="#cb125-1" aria-hidden="true" tabindex="-1"></a><span class="kw">case</span> <span class="op">&lt;</span>value<span class="op">&gt;</span> <span class="kw">of</span> <span class="op">&lt;</span><span class="kw">pattern</span><span class="op">&gt;</span> <span class="ot">-&gt;</span> <span class="op">&lt;</span>expression<span class="op">&gt;</span></span>
 <span id="cb125-2"><a href="#cb125-2" aria-hidden="true" tabindex="-1"></a>                <span class="op">&lt;</span><span class="kw">pattern</span><span class="op">&gt;</span> <span class="ot">-&gt;</span> <span class="op">&lt;</span>expression<span class="op">&gt;</span></span></code></pre></div>
-<p>As an example let’s rewrite the <code>describe</code> example from the first lecture using <code>case</code>:</p>
-<div class="sourceCode" id="cb126"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb126-1"><a href="#cb126-1" aria-hidden="true" tabindex="-1"></a><span class="ot">describe ::</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<p>As an example let’s rewrite the <code>describe</code> example from
+the first lecture using <code>case</code>:</p>
+<div class="sourceCode" id="cb126"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb126-1"><a href="#cb126-1" aria-hidden="true" tabindex="-1"></a><span class="ot">describe ::</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb126-2"><a href="#cb126-2" aria-hidden="true" tabindex="-1"></a>describe <span class="dv">0</span> <span class="ot">=</span> <span class="st">&quot;zero&quot;</span></span>
 <span id="cb126-3"><a href="#cb126-3" aria-hidden="true" tabindex="-1"></a>describe <span class="dv">1</span> <span class="ot">=</span> <span class="st">&quot;one&quot;</span></span>
 <span id="cb126-4"><a href="#cb126-4" aria-hidden="true" tabindex="-1"></a>describe <span class="dv">2</span> <span class="ot">=</span> <span class="st">&quot;an even prime&quot;</span></span>
 <span id="cb126-5"><a href="#cb126-5" aria-hidden="true" tabindex="-1"></a>describe n <span class="ot">=</span> <span class="st">&quot;the number &quot;</span> <span class="op">++</span> <span class="fu">show</span> n</span></code></pre></div>
-<div class="sourceCode" id="cb127"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb127-1"><a href="#cb127-1" aria-hidden="true" tabindex="-1"></a><span class="ot">describe ::</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<div class="sourceCode" id="cb127"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb127-1"><a href="#cb127-1" aria-hidden="true" tabindex="-1"></a><span class="ot">describe ::</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb127-2"><a href="#cb127-2" aria-hidden="true" tabindex="-1"></a>describe n <span class="ot">=</span> <span class="kw">case</span> n <span class="kw">of</span> <span class="dv">0</span> <span class="ot">-&gt;</span> <span class="st">&quot;zero&quot;</span></span>
 <span id="cb127-3"><a href="#cb127-3" aria-hidden="true" tabindex="-1"></a>                       <span class="dv">1</span> <span class="ot">-&gt;</span> <span class="st">&quot;one&quot;</span></span>
 <span id="cb127-4"><a href="#cb127-4" aria-hidden="true" tabindex="-1"></a>                       <span class="dv">2</span> <span class="ot">-&gt;</span> <span class="st">&quot;an even prime&quot;</span></span>
 <span id="cb127-5"><a href="#cb127-5" aria-hidden="true" tabindex="-1"></a>                       n <span class="ot">-&gt;</span> <span class="st">&quot;the number &quot;</span> <span class="op">++</span> <span class="fu">show</span> n</span></code></pre></div>
-<p>A more interesting example is when the value we’re pattern matching on is not a function argument. For example:</p>
-<div class="sourceCode" id="cb128"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb128-1"><a href="#cb128-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- parse country code into country name, returns Nothing if code not recognized</span></span>
+<p>A more interesting example is when the value we’re pattern matching
+on is not a function argument. For example:</p>
+<div class="sourceCode" id="cb128"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb128-1"><a href="#cb128-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- parse country code into country name, returns Nothing if code not recognized</span></span>
 <span id="cb128-2"><a href="#cb128-2" aria-hidden="true" tabindex="-1"></a><span class="ot">parseCountry ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">String</span></span>
 <span id="cb128-3"><a href="#cb128-3" aria-hidden="true" tabindex="-1"></a>parseCountry <span class="st">&quot;FI&quot;</span> <span class="ot">=</span> <span class="dt">Just</span> <span class="st">&quot;Finland&quot;</span></span>
 <span id="cb128-4"><a href="#cb128-4" aria-hidden="true" tabindex="-1"></a>parseCountry <span class="st">&quot;SE&quot;</span> <span class="ot">=</span> <span class="dt">Just</span> <span class="st">&quot;Sweden&quot;</span></span>
@@ -1485,44 +2352,62 @@ lectureParticipants = [Right 10, Right 13, Left &quot;easter vacation&quot;, Rig
 <span id="cb128-7"><a href="#cb128-7" aria-hidden="true" tabindex="-1"></a><span class="ot">flyTo ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb128-8"><a href="#cb128-8" aria-hidden="true" tabindex="-1"></a>flyTo countryCode <span class="ot">=</span> <span class="kw">case</span> parseCountry countryCode <span class="kw">of</span> <span class="dt">Just</span> country <span class="ot">-&gt;</span> <span class="st">&quot;You&#39;re flying to &quot;</span> <span class="op">++</span> country</span>
 <span id="cb128-9"><a href="#cb128-9" aria-hidden="true" tabindex="-1"></a>                                                     <span class="dt">Nothing</span> <span class="ot">-&gt;</span> <span class="st">&quot;You&#39;re not flying anywhere&quot;</span></span></code></pre></div>
-<div class="sourceCode" id="cb129"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb129-1"><a href="#cb129-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> flyTo <span class="st">&quot;FI&quot;</span></span>
+<div class="sourceCode" id="cb129"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb129-1"><a href="#cb129-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> flyTo <span class="st">&quot;FI&quot;</span></span>
 <span id="cb129-2"><a href="#cb129-2" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;You&#39;re flying to Finland&quot;</span></span>
 <span id="cb129-3"><a href="#cb129-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> flyTo <span class="st">&quot;DE&quot;</span></span>
 <span id="cb129-4"><a href="#cb129-4" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;You&#39;re not flying anywhere&quot;</span></span></code></pre></div>
-<p>We could write the <code>flyTo</code> function using a helper function for pattern matching instead of using the case-of expression:</p>
-<div class="sourceCode" id="cb130"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb130-1"><a href="#cb130-1" aria-hidden="true" tabindex="-1"></a><span class="ot">flyTo ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<p>We could write the <code>flyTo</code> function using a helper
+function for pattern matching instead of using the case-of
+expression:</p>
+<div class="sourceCode" id="cb130"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb130-1"><a href="#cb130-1" aria-hidden="true" tabindex="-1"></a><span class="ot">flyTo ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb130-2"><a href="#cb130-2" aria-hidden="true" tabindex="-1"></a>flyTo countryCode <span class="ot">=</span> handleResult (parseCountry countryCode)</span>
 <span id="cb130-3"><a href="#cb130-3" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> handleResult (<span class="dt">Just</span> country) <span class="ot">=</span> <span class="st">&quot;You&#39;re flying to &quot;</span> <span class="op">++</span> country</span>
 <span id="cb130-4"><a href="#cb130-4" aria-hidden="true" tabindex="-1"></a>        handleResult <span class="dt">Nothing</span>        <span class="ot">=</span> <span class="st">&quot;You&#39;re not flying anywhere&quot;</span></span></code></pre></div>
-<p>In fact, a case-of expression can always be replaced with a helper function. Here’s one more example, written in both ways:</p>
-<div class="sourceCode" id="cb131"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb131-1"><a href="#cb131-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- given a sentence, decide whether it is a statement, question or exclamation</span></span>
+<p>In fact, a case-of expression can always be replaced with a helper
+function. Here’s one more example, written in both ways:</p>
+<div class="sourceCode" id="cb131"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb131-1"><a href="#cb131-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- given a sentence, decide whether it is a statement, question or exclamation</span></span>
 <span id="cb131-2"><a href="#cb131-2" aria-hidden="true" tabindex="-1"></a><span class="ot">sentenceType ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb131-3"><a href="#cb131-3" aria-hidden="true" tabindex="-1"></a>sentenceType sentence <span class="ot">=</span> <span class="kw">case</span> <span class="fu">last</span> sentence <span class="kw">of</span> <span class="ch">&#39;.&#39;</span> <span class="ot">-&gt;</span> <span class="st">&quot;statement&quot;</span></span>
 <span id="cb131-4"><a href="#cb131-4" aria-hidden="true" tabindex="-1"></a>                                              <span class="ch">&#39;?&#39;</span> <span class="ot">-&gt;</span> <span class="st">&quot;question&quot;</span></span>
 <span id="cb131-5"><a href="#cb131-5" aria-hidden="true" tabindex="-1"></a>                                              <span class="ch">&#39;!&#39;</span> <span class="ot">-&gt;</span> <span class="st">&quot;exclamation&quot;</span></span>
 <span id="cb131-6"><a href="#cb131-6" aria-hidden="true" tabindex="-1"></a>                                              _   <span class="ot">-&gt;</span> <span class="st">&quot;not a sentence&quot;</span></span></code></pre></div>
-<div class="sourceCode" id="cb132"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb132-1"><a href="#cb132-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- same function, helper function instead of case-of</span></span>
+<div class="sourceCode" id="cb132"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb132-1"><a href="#cb132-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- same function, helper function instead of case-of</span></span>
 <span id="cb132-2"><a href="#cb132-2" aria-hidden="true" tabindex="-1"></a>sentenceType sentence <span class="ot">=</span> classify (<span class="fu">last</span> sentence)</span>
 <span id="cb132-3"><a href="#cb132-3" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> classify <span class="ch">&#39;.&#39;</span> <span class="ot">=</span> <span class="st">&quot;statement&quot;</span></span>
 <span id="cb132-4"><a href="#cb132-4" aria-hidden="true" tabindex="-1"></a>        classify <span class="ch">&#39;?&#39;</span> <span class="ot">=</span> <span class="st">&quot;question&quot;</span></span>
 <span id="cb132-5"><a href="#cb132-5" aria-hidden="true" tabindex="-1"></a>        classify <span class="ch">&#39;!&#39;</span> <span class="ot">=</span> <span class="st">&quot;exclamation&quot;</span></span>
 <span id="cb132-6"><a href="#cb132-6" aria-hidden="true" tabindex="-1"></a>        classify _   <span class="ot">=</span> <span class="st">&quot;not a sentence&quot;</span></span></code></pre></div>
-<div class="sourceCode" id="cb133"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb133-1"><a href="#cb133-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> sentenceType <span class="st">&quot;This is Haskell.&quot;</span></span>
+<div class="sourceCode" id="cb133"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb133-1"><a href="#cb133-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> sentenceType <span class="st">&quot;This is Haskell.&quot;</span></span>
 <span id="cb133-2"><a href="#cb133-2" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;statement&quot;</span></span>
 <span id="cb133-3"><a href="#cb133-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> sentenceType <span class="st">&quot;This is Haskell!&quot;</span></span>
 <span id="cb133-4"><a href="#cb133-4" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;exclamation&quot;</span></span></code></pre></div>
-<h3 data-number="2.9.1" id="when-to-use-case-expressions"><span class="header-section-number">2.9.1</span> When to Use Case Expressions</h3>
-<p>You might be asking, what is the point of having another pattern matching syntax. Well, <code>case</code> expressions have some advantages over equations which we’ll discuss next.</p>
-<p>Firstly, and perhaps most importantly, <code>case</code> expressions enable us to pattern match against function outputs. We might want to write early morning motivational messages to working (lazy) Haskellers:</p>
-<div class="sourceCode" id="cb134"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb134-1"><a href="#cb134-1" aria-hidden="true" tabindex="-1"></a><span class="ot">motivate ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<h3 data-number="2.9.1" id="when-to-use-case-expressions"><span
+class="header-section-number">2.9.1</span> When to Use Case
+Expressions</h3>
+<p>You might be asking, what is the point of having another pattern
+matching syntax. Well, <code>case</code> expressions have some
+advantages over equations which we’ll discuss next.</p>
+<p>Firstly, and perhaps most importantly, <code>case</code> expressions
+enable us to pattern match against function outputs. We might want to
+write early morning motivational messages to working (lazy)
+Haskellers:</p>
+<div class="sourceCode" id="cb134"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb134-1"><a href="#cb134-1" aria-hidden="true" tabindex="-1"></a><span class="ot">motivate ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb134-2"><a href="#cb134-2" aria-hidden="true" tabindex="-1"></a>motivate <span class="st">&quot;Monday&quot;</span>    <span class="ot">=</span> <span class="st">&quot;Have a nice week at work!&quot;</span></span>
 <span id="cb134-3"><a href="#cb134-3" aria-hidden="true" tabindex="-1"></a>motivate <span class="st">&quot;Tuesday&quot;</span>   <span class="ot">=</span> <span class="st">&quot;You&#39;re one day closer to weekend!&quot;</span></span>
 <span id="cb134-4"><a href="#cb134-4" aria-hidden="true" tabindex="-1"></a>motivate <span class="st">&quot;Wednesday&quot;</span> <span class="ot">=</span> <span class="st">&quot;3 more day(s) until the weekend!&quot;</span></span>
 <span id="cb134-5"><a href="#cb134-5" aria-hidden="true" tabindex="-1"></a>motivate <span class="st">&quot;Thursday&quot;</span>  <span class="ot">=</span> <span class="st">&quot;2 more day(s) until the weekend!&quot;</span></span>
 <span id="cb134-6"><a href="#cb134-6" aria-hidden="true" tabindex="-1"></a>motivate <span class="st">&quot;Friday&quot;</span>    <span class="ot">=</span> <span class="st">&quot;1 more day(s) until the weekend!&quot;</span></span>
 <span id="cb134-7"><a href="#cb134-7" aria-hidden="true" tabindex="-1"></a>motivate _           <span class="ot">=</span> <span class="st">&quot;Relax! You don&#39;t need to work today!&quot;</span></span></code></pre></div>
-<p>Using a <code>case</code> expression we can run a helper function against the argument and pattern match on the result:</p>
-<div class="sourceCode" id="cb135"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb135-1"><a href="#cb135-1" aria-hidden="true" tabindex="-1"></a><span class="ot">motivate ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<p>Using a <code>case</code> expression we can run a helper function
+against the argument and pattern match on the result:</p>
+<div class="sourceCode" id="cb135"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb135-1"><a href="#cb135-1" aria-hidden="true" tabindex="-1"></a><span class="ot">motivate ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb135-2"><a href="#cb135-2" aria-hidden="true" tabindex="-1"></a>motivate day <span class="ot">=</span> <span class="kw">case</span> distanceToSunday day <span class="kw">of</span></span>
 <span id="cb135-3"><a href="#cb135-3" aria-hidden="true" tabindex="-1"></a>  <span class="dv">6</span> <span class="ot">-&gt;</span> <span class="st">&quot;Have a nice week at work!&quot;</span></span>
 <span id="cb135-4"><a href="#cb135-4" aria-hidden="true" tabindex="-1"></a>  <span class="dv">5</span> <span class="ot">-&gt;</span> <span class="st">&quot;You&#39;re one day closer to weekend!&quot;</span></span>
@@ -1530,27 +2415,40 @@ lectureParticipants = [Right 10, Right 13, Left &quot;easter vacation&quot;, Rig
 <span id="cb135-6"><a href="#cb135-6" aria-hidden="true" tabindex="-1"></a>       <span class="kw">then</span> <span class="fu">show</span> (n <span class="op">-</span> <span class="dv">1</span>) <span class="op">++</span> <span class="st">&quot; more day(s) until the weekend!&quot;</span></span>
 <span id="cb135-7"><a href="#cb135-7" aria-hidden="true" tabindex="-1"></a>       <span class="kw">else</span> <span class="st">&quot;Relax! You don&#39;t need to work today!&quot;</span></span></code></pre></div>
 <p>By the way, there’s also a third way, guards:</p>
-<div class="sourceCode" id="cb136"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb136-1"><a href="#cb136-1" aria-hidden="true" tabindex="-1"></a><span class="ot">motivate ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<div class="sourceCode" id="cb136"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb136-1"><a href="#cb136-1" aria-hidden="true" tabindex="-1"></a><span class="ot">motivate ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb136-2"><a href="#cb136-2" aria-hidden="true" tabindex="-1"></a>motivate day</span>
 <span id="cb136-3"><a href="#cb136-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> n <span class="op">==</span> <span class="dv">6</span> <span class="ot">=</span> <span class="st">&quot;Have a nice week at work!&quot;</span></span>
 <span id="cb136-4"><a href="#cb136-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> n <span class="op">==</span> <span class="dv">5</span> <span class="ot">=</span> <span class="st">&quot;You&#39;re one day closer to weekend!&quot;</span></span>
 <span id="cb136-5"><a href="#cb136-5" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> n <span class="op">&gt;</span> <span class="dv">1</span> <span class="ot">=</span> <span class="fu">show</span> (n <span class="op">-</span> <span class="dv">1</span>) <span class="op">++</span> <span class="st">&quot; more day(s) until the weekend!&quot;</span></span>
 <span id="cb136-6"><a href="#cb136-6" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> <span class="fu">otherwise</span> <span class="ot">=</span> <span class="st">&quot;Relax! You don&#39;t need to work today!&quot;</span></span>
 <span id="cb136-7"><a href="#cb136-7" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> n <span class="ot">=</span> distanceToSunday day</span></code></pre></div>
-<p>We’ll see in a moment how we can define <code>distanceToSunday</code> using equations and <code>case</code> expressions.</p>
-<p>Secondly, if a helper function needs to be shared among many patterns, then equations don’t work. For example:</p>
-<div class="sourceCode" id="cb137"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb137-1"><a href="#cb137-1" aria-hidden="true" tabindex="-1"></a><span class="ot">area ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Double</span> <span class="ot">-&gt;</span> <span class="dt">Double</span></span>
+<p>We’ll see in a moment how we can define <code>distanceToSunday</code>
+using equations and <code>case</code> expressions.</p>
+<p>Secondly, if a helper function needs to be shared among many
+patterns, then equations don’t work. For example:</p>
+<div class="sourceCode" id="cb137"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb137-1"><a href="#cb137-1" aria-hidden="true" tabindex="-1"></a><span class="ot">area ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Double</span> <span class="ot">-&gt;</span> <span class="dt">Double</span></span>
 <span id="cb137-2"><a href="#cb137-2" aria-hidden="true" tabindex="-1"></a>area <span class="st">&quot;square&quot;</span> x <span class="ot">=</span> square x</span>
 <span id="cb137-3"><a href="#cb137-3" aria-hidden="true" tabindex="-1"></a>area <span class="st">&quot;circle&quot;</span> x <span class="ot">=</span> <span class="fu">pi</span> <span class="op">*</span> square x</span>
 <span id="cb137-4"><a href="#cb137-4" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> square x <span class="ot">=</span> x <span class="op">*</span> x</span></code></pre></div>
-<p>This won’t compile because a the <code>where</code> clause only appends to the <code>"circle"</code> case, so the <code>square</code> helper function is not available in the <code>"square"</code> case. On the other hand, we can write</p>
-<div class="sourceCode" id="cb138"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb138-1"><a href="#cb138-1" aria-hidden="true" tabindex="-1"></a><span class="ot">area ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Double</span> <span class="ot">-&gt;</span> <span class="dt">Double</span></span>
+<p>This won’t compile because a the <code>where</code> clause only
+appends to the <code>"circle"</code> case, so the <code>square</code>
+helper function is not available in the <code>"square"</code> case. On
+the other hand, we can write</p>
+<div class="sourceCode" id="cb138"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb138-1"><a href="#cb138-1" aria-hidden="true" tabindex="-1"></a><span class="ot">area ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Double</span> <span class="ot">-&gt;</span> <span class="dt">Double</span></span>
 <span id="cb138-2"><a href="#cb138-2" aria-hidden="true" tabindex="-1"></a>area shape x <span class="ot">=</span> <span class="kw">case</span> shape <span class="kw">of</span></span>
 <span id="cb138-3"><a href="#cb138-3" aria-hidden="true" tabindex="-1"></a>  <span class="st">&quot;square&quot;</span> <span class="ot">-&gt;</span> square x</span>
 <span id="cb138-4"><a href="#cb138-4" aria-hidden="true" tabindex="-1"></a>  <span class="st">&quot;circle&quot;</span> <span class="ot">-&gt;</span> <span class="fu">pi</span> <span class="op">*</span> square x</span>
 <span id="cb138-5"><a href="#cb138-5" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> square x <span class="ot">=</span> x<span class="op">*</span>x</span></code></pre></div>
-<p>Thirdly, <code>case</code> expressions may help to write more concise code in a situation where a (long) function name would have to be repeated multiple times using equations. As we saw above, we might need a function which measures the distance between a given day and Sunday:</p>
-<div class="sourceCode" id="cb139"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb139-1"><a href="#cb139-1" aria-hidden="true" tabindex="-1"></a><span class="ot">distanceToSunday ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<p>Thirdly, <code>case</code> expressions may help to write more concise
+code in a situation where a (long) function name would have to be
+repeated multiple times using equations. As we saw above, we might need
+a function which measures the distance between a given day and
+Sunday:</p>
+<div class="sourceCode" id="cb139"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb139-1"><a href="#cb139-1" aria-hidden="true" tabindex="-1"></a><span class="ot">distanceToSunday ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb139-2"><a href="#cb139-2" aria-hidden="true" tabindex="-1"></a>distanceToSunday <span class="st">&quot;Monday&quot;</span>    <span class="ot">=</span> <span class="dv">6</span></span>
 <span id="cb139-3"><a href="#cb139-3" aria-hidden="true" tabindex="-1"></a>distanceToSunday <span class="st">&quot;Tuesday&quot;</span>   <span class="ot">=</span> <span class="dv">5</span></span>
 <span id="cb139-4"><a href="#cb139-4" aria-hidden="true" tabindex="-1"></a>distanceToSunday <span class="st">&quot;Wednesday&quot;</span> <span class="ot">=</span> <span class="dv">4</span></span>
@@ -1558,8 +2456,10 @@ lectureParticipants = [Right 10, Right 13, Left &quot;easter vacation&quot;, Rig
 <span id="cb139-6"><a href="#cb139-6" aria-hidden="true" tabindex="-1"></a>distanceToSunday <span class="st">&quot;Friday&quot;</span>    <span class="ot">=</span> <span class="dv">2</span></span>
 <span id="cb139-7"><a href="#cb139-7" aria-hidden="true" tabindex="-1"></a>distanceToSunday <span class="st">&quot;Saturday&quot;</span>  <span class="ot">=</span> <span class="dv">1</span></span>
 <span id="cb139-8"><a href="#cb139-8" aria-hidden="true" tabindex="-1"></a>distanceToSunday <span class="st">&quot;Sunday&quot;</span>    <span class="ot">=</span> <span class="dv">0</span></span></code></pre></div>
-<p>Using a <code>case</code> expression leads into much more concise implementation:</p>
-<div class="sourceCode" id="cb140"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb140-1"><a href="#cb140-1" aria-hidden="true" tabindex="-1"></a><span class="ot">distanceToSunday ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<p>Using a <code>case</code> expression leads into much more concise
+implementation:</p>
+<div class="sourceCode" id="cb140"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb140-1"><a href="#cb140-1" aria-hidden="true" tabindex="-1"></a><span class="ot">distanceToSunday ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb140-2"><a href="#cb140-2" aria-hidden="true" tabindex="-1"></a>distanceToSunday d <span class="ot">=</span> <span class="kw">case</span> d <span class="kw">of</span></span>
 <span id="cb140-3"><a href="#cb140-3" aria-hidden="true" tabindex="-1"></a>  <span class="st">&quot;Monday&quot;</span>    <span class="ot">-&gt;</span> <span class="dv">6</span></span>
 <span id="cb140-4"><a href="#cb140-4" aria-hidden="true" tabindex="-1"></a>  <span class="st">&quot;Tuesday&quot;</span>   <span class="ot">-&gt;</span> <span class="dv">5</span></span>
@@ -1568,26 +2468,43 @@ lectureParticipants = [Right 10, Right 13, Left &quot;easter vacation&quot;, Rig
 <span id="cb140-7"><a href="#cb140-7" aria-hidden="true" tabindex="-1"></a>  <span class="st">&quot;Friday&quot;</span>    <span class="ot">-&gt;</span> <span class="dv">2</span></span>
 <span id="cb140-8"><a href="#cb140-8" aria-hidden="true" tabindex="-1"></a>  <span class="st">&quot;Saturday&quot;</span>  <span class="ot">-&gt;</span> <span class="dv">1</span></span>
 <span id="cb140-9"><a href="#cb140-9" aria-hidden="true" tabindex="-1"></a>  <span class="st">&quot;Sunday&quot;</span>    <span class="ot">-&gt;</span> <span class="dv">0</span></span></code></pre></div>
-<p>These three benefits make the <code>case</code> expression a versatile tool in a Haskeller’s toolbox. It’s worth remembering how <code>case</code> works.</p>
-<p>(Representing weekdays as strings may get the job done, but it’s not the perfect solution. What happens if we apply <code>motivate</code> to <code>"monday"</code> (with all letters in lower case) or <code>"keskiviikko"</code>? In Lecture 5, we will learn a better way to represent things like weekdays.)</p>
-<h2 data-number="2.10" id="recap-pattern-matching"><span class="header-section-number">2.10</span> Recap: Pattern Matching</h2>
+<p>These three benefits make the <code>case</code> expression a
+versatile tool in a Haskeller’s toolbox. It’s worth remembering how
+<code>case</code> works.</p>
+<p>(Representing weekdays as strings may get the job done, but it’s not
+the perfect solution. What happens if we apply <code>motivate</code> to
+<code>"monday"</code> (with all letters in lower case) or
+<code>"keskiviikko"</code>? In Lecture 5, we will learn a better way to
+represent things like weekdays.)</p>
+<h2 data-number="2.10" id="recap-pattern-matching"><span
+class="header-section-number">2.10</span> Recap: Pattern Matching</h2>
 <p>Things you can use as patterns:</p>
 <ul>
-<li><code>Int</code> and <code>Integer</code> constants like <code>(-1)</code>, <code>0</code>, <code>1</code>, <code>2</code>, …</li>
-<li><code>Bool</code> values <code>True</code> and <code>False</code></li>
+<li><code>Int</code> and <code>Integer</code> constants like
+<code>(-1)</code>, <code>0</code>, <code>1</code>, <code>2</code>,
+…</li>
+<li><code>Bool</code> values <code>True</code> and
+<code>False</code></li>
 <li><code>Char</code> constants: <code>'a'</code>, <code>'b'</code></li>
-<li><code>String</code> constants: <code>"abc"</code>, <code>""</code></li>
-<li><code>Maybe</code> constructors: <code>Nothing</code>, <code>(Just x)</code></li>
-<li><code>Either</code> constructors: <code>(Left x)</code>, <code>(Right y)</code></li>
-<li>The special <code>_</code> pattern which means “anything, I don’t care”</li>
-<li>Combinations of these patterns, like for example <code>(Just 1)</code></li>
-<li>We’ll learn about other patterns, for example lists, in the next lectures.</li>
+<li><code>String</code> constants: <code>"abc"</code>,
+<code>""</code></li>
+<li><code>Maybe</code> constructors: <code>Nothing</code>,
+<code>(Just x)</code></li>
+<li><code>Either</code> constructors: <code>(Left x)</code>,
+<code>(Right y)</code></li>
+<li>The special <code>_</code> pattern which means “anything, I don’t
+care”</li>
+<li>Combinations of these patterns, like for example
+<code>(Just 1)</code></li>
+<li>We’ll learn about other patterns, for example lists, in the next
+lectures.</li>
 </ul>
 <p>Places where you can use patterns:</p>
 <ul>
 <li>Defining a function with equations:</li>
 </ul>
-<div class="sourceCode" id="cb141"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb141-1"><a href="#cb141-1" aria-hidden="true" tabindex="-1"></a><span class="ot">f ::</span> <span class="dt">Bool</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<div class="sourceCode" id="cb141"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb141-1"><a href="#cb141-1" aria-hidden="true" tabindex="-1"></a><span class="ot">f ::</span> <span class="dt">Bool</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb141-2"><a href="#cb141-2" aria-hidden="true" tabindex="-1"></a>f <span class="dt">False</span> <span class="dt">Nothing</span>  <span class="ot">=</span> <span class="dv">1</span></span>
 <span id="cb141-3"><a href="#cb141-3" aria-hidden="true" tabindex="-1"></a>f <span class="dt">False</span> _        <span class="ot">=</span> <span class="dv">2</span></span>
 <span id="cb141-4"><a href="#cb141-4" aria-hidden="true" tabindex="-1"></a>f <span class="dt">True</span>  (<span class="dt">Just</span> i) <span class="ot">=</span> i</span>
@@ -1595,27 +2512,39 @@ lectureParticipants = [Right 10, Right 13, Left &quot;easter vacation&quot;, Rig
 <ul>
 <li>In the <code>case of</code> expression:</li>
 </ul>
-<div class="sourceCode" id="cb142"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb142-1"><a href="#cb142-1" aria-hidden="true" tabindex="-1"></a><span class="kw">case</span> number <span class="kw">of</span> <span class="dv">0</span> <span class="ot">-&gt;</span> <span class="st">&quot;zero&quot;</span></span>
+<div class="sourceCode" id="cb142"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb142-1"><a href="#cb142-1" aria-hidden="true" tabindex="-1"></a><span class="kw">case</span> number <span class="kw">of</span> <span class="dv">0</span> <span class="ot">-&gt;</span> <span class="st">&quot;zero&quot;</span></span>
 <span id="cb142-2"><a href="#cb142-2" aria-hidden="true" tabindex="-1"></a>               <span class="dv">1</span> <span class="ot">-&gt;</span> <span class="st">&quot;one&quot;</span></span>
 <span id="cb142-3"><a href="#cb142-3" aria-hidden="true" tabindex="-1"></a>               _ <span class="ot">-&gt;</span> <span class="st">&quot;not zero or one&quot;</span></span></code></pre></div>
-<p>The only thing you really <em>need</em> pattern matching for is <em>getting the value</em> inside a <code>Just</code>, <code>Left</code> or <code>Right</code> constructor. Here are two more examples of this:</p>
-<div class="sourceCode" id="cb143"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb143-1"><a href="#cb143-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- getElement (Just i) gets the ith element (counting from zero) of a list, getElement Nothing gets the last element</span></span>
+<p>The only thing you really <em>need</em> pattern matching for is
+<em>getting the value</em> inside a <code>Just</code>, <code>Left</code>
+or <code>Right</code> constructor. Here are two more examples of
+this:</p>
+<div class="sourceCode" id="cb143"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb143-1"><a href="#cb143-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- getElement (Just i) gets the ith element (counting from zero) of a list, getElement Nothing gets the last element</span></span>
 <span id="cb143-2"><a href="#cb143-2" aria-hidden="true" tabindex="-1"></a><span class="ot">getElement ::</span> <span class="dt">Maybe</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> a</span>
 <span id="cb143-3"><a href="#cb143-3" aria-hidden="true" tabindex="-1"></a>getElement (<span class="dt">Just</span> i) xs <span class="ot">=</span> xs <span class="op">!!</span> i</span>
 <span id="cb143-4"><a href="#cb143-4" aria-hidden="true" tabindex="-1"></a>getElement <span class="dt">Nothing</span> xs <span class="ot">=</span> <span class="fu">last</span> xs</span></code></pre></div>
-<div class="sourceCode" id="cb144"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb144-1"><a href="#cb144-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> getElement <span class="dt">Nothing</span> <span class="st">&quot;hurray!&quot;</span></span>
+<div class="sourceCode" id="cb144"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb144-1"><a href="#cb144-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> getElement <span class="dt">Nothing</span> <span class="st">&quot;hurray!&quot;</span></span>
 <span id="cb144-2"><a href="#cb144-2" aria-hidden="true" tabindex="-1"></a><span class="ch">&#39;!&#39;</span></span>
 <span id="cb144-3"><a href="#cb144-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> getElement (<span class="dt">Just</span> <span class="dv">3</span>) [<span class="dv">5</span>,<span class="dv">6</span>,<span class="dv">7</span>,<span class="dv">8</span>,<span class="dv">9</span>]</span>
 <span id="cb144-4"><a href="#cb144-4" aria-hidden="true" tabindex="-1"></a><span class="dv">8</span></span></code></pre></div>
-<div class="sourceCode" id="cb145"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb145-1"><a href="#cb145-1" aria-hidden="true" tabindex="-1"></a><span class="ot">direction ::</span> <span class="dt">Either</span> <span class="dt">Int</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<div class="sourceCode" id="cb145"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb145-1"><a href="#cb145-1" aria-hidden="true" tabindex="-1"></a><span class="ot">direction ::</span> <span class="dt">Either</span> <span class="dt">Int</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb145-2"><a href="#cb145-2" aria-hidden="true" tabindex="-1"></a>direction (<span class="dt">Left</span> i) <span class="ot">=</span> <span class="st">&quot;you should go left &quot;</span> <span class="op">++</span> <span class="fu">show</span> i <span class="op">++</span> <span class="st">&quot; meters!&quot;</span></span>
 <span id="cb145-3"><a href="#cb145-3" aria-hidden="true" tabindex="-1"></a>direction (<span class="dt">Right</span> i) <span class="ot">=</span> <span class="st">&quot;you should go right &quot;</span> <span class="op">++</span> <span class="fu">show</span> i <span class="op">++</span> <span class="st">&quot; meters!&quot;</span></span></code></pre></div>
-<div class="sourceCode" id="cb146"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb146-1"><a href="#cb146-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> direction (<span class="dt">Left</span> <span class="dv">3</span>)</span>
+<div class="sourceCode" id="cb146"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb146-1"><a href="#cb146-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> direction (<span class="dt">Left</span> <span class="dv">3</span>)</span>
 <span id="cb146-2"><a href="#cb146-2" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;you should go left 3 meters!&quot;</span></span>
 <span id="cb146-3"><a href="#cb146-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> direction (<span class="dt">Right</span> <span class="dv">5</span>)</span>
 <span id="cb146-4"><a href="#cb146-4" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;you should go right 5 meters!&quot;</span></span></code></pre></div>
-<p>Other uses (that we’ve seen so far!) of pattern matching can also be accomplished with the <code>==</code> operator. However, things like <code>x==Nothing</code> won’t work in all cases. We’ll find out why when we talk about type classes in lecture 4.</p>
-<h2 data-number="2.11" id="quiz-1"><span class="header-section-number">2.11</span> Quiz</h2>
+<p>Other uses (that we’ve seen so far!) of pattern matching can also be
+accomplished with the <code>==</code> operator. However, things like
+<code>x==Nothing</code> won’t work in all cases. We’ll find out why when
+we talk about type classes in lecture 4.</p>
+<h2 data-number="2.11" id="quiz-1"><span
+class="header-section-number">2.11</span> Quiz</h2>
 <p>How many values does <code>f x = [x,x]</code> return?</p>
 <ol class="quiz">
 <li>
@@ -1626,7 +2555,8 @@ One
 <li>
 Two
 </ol>
-<p>Why does the expression <code>Nothing 1</code> cause a type error?</p>
+<p>Why does the expression <code>Nothing 1</code> cause a type
+error?</p>
 <ol class="quiz">
 <li class="correct">
 Because <code>Nothing</code> takes no arguments
@@ -1638,7 +2568,8 @@ Because <code>Nothing</code> returns nothing
 Because <code>Nothing</code> is a constructor
 </li>
 </ol>
-<p>What is the type of the function <code>f x y = if x &amp;&amp; y then Right x else Left "foo"</code>?</p>
+<p>What is the type of the function
+<code>f x y = if x &amp;&amp; y then Right x else Left "foo"</code>?</p>
 <ol class="quiz">
 <li>
 <code>Bool -&gt; Bool -&gt; Either Bool String</code>
@@ -1650,7 +2581,8 @@ Because <code>Nothing</code> is a constructor
 <code>Bool -&gt; Bool -&gt; Either String Bool</code>
 </li>
 </ol>
-<p>Which of the following functions could have the type <code>Bool -&gt; Int -&gt; [Bool]</code></p>
+<p>Which of the following functions could have the type
+<code>Bool -&gt; Int -&gt; [Bool]</code></p>
 <ol class="quiz">
 <li>
 <code>f x y = [0, y]</code>
@@ -1662,7 +2594,8 @@ Because <code>Nothing</code> is a constructor
 <code>f x y = [y, True]</code>
 </li>
 </ol>
-<p>What is the type of this function? <code>justBoth a b = [Just a, Just b]</code></p>
+<p>What is the type of this function?
+<code>justBoth a b = [Just a, Just b]</code></p>
 <ol class="quiz">
 <li>
 <code>a -&gt; b -&gt; [Maybe a, Maybe b]</code>
@@ -1677,141 +2610,237 @@ Because <code>Nothing</code> is a constructor
 <code>a -&gt; a -&gt; [Maybe a]</code>
 </li>
 </ol>
-<h2 data-number="2.12" id="exercises-1"><span class="header-section-number">2.12</span> Exercises</h2>
+<h2 data-number="2.12" id="exercises-1"><span
+class="header-section-number">2.12</span> Exercises</h2>
 <ul>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set2a.hs">Set2a</a></li>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set2b.hs">Set2b</a></li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set2a.hs">Set2a</a></li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set2b.hs">Set2b</a></li>
 </ul>
-<h1 data-number="3" id="lecture-3-catamorphic"><span class="header-section-number">3</span> Lecture 3: Catamorphic</h1>
+<h1 data-number="3" id="lecture-3-catamorphic"><span
+class="header-section-number">3</span> Lecture 3: Catamorphic</h1>
 <!-- TODO name of lecture is too obscure, and also folds are only introduced in Lecture 4 -->
 <ul>
 <li>Lists, lists, lists</li>
 <li>Functionality</li>
 <li>A bit about types</li>
 </ul>
-<h2 data-number="3.1" id="functional-programming-at-last"><span class="header-section-number">3.1</span> Functional Programming, at Last</h2>
-<p>Now with lists and polymorphism in our toolbox, we can finally start to look at functional programming.</p>
-<p>In Haskell a function is a value, just like a number or a list is. Functions can be passed as parameters to other functions. Here’s a toy example. The function <code>applyTo1</code> takes a function of type <code>Int-&gt;Int</code>, applies it to the number <code>1</code>, and returns the result.</p>
-<div class="sourceCode" id="cb147"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb147-1"><a href="#cb147-1" aria-hidden="true" tabindex="-1"></a><span class="ot">applyTo1 ::</span> (<span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span>) <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<h2 data-number="3.1" id="functional-programming-at-last"><span
+class="header-section-number">3.1</span> Functional Programming, at
+Last</h2>
+<p>Now with lists and polymorphism in our toolbox, we can finally start
+to look at functional programming.</p>
+<p>In Haskell a function is a value, just like a number or a list is.
+Functions can be passed as parameters to other functions. Here’s a toy
+example. The function <code>applyTo1</code> takes a function of type
+<code>Int-&gt;Int</code>, applies it to the number <code>1</code>, and
+returns the result.</p>
+<div class="sourceCode" id="cb147"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb147-1"><a href="#cb147-1" aria-hidden="true" tabindex="-1"></a><span class="ot">applyTo1 ::</span> (<span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span>) <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb147-2"><a href="#cb147-2" aria-hidden="true" tabindex="-1"></a>applyTo1 f <span class="ot">=</span> f <span class="dv">1</span></span></code></pre></div>
-<p>Let’s define a simple function of type <code>Int-&gt;Int</code> and see <code>applyTo1</code> in action.</p>
-<div class="sourceCode" id="cb148"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb148-1"><a href="#cb148-1" aria-hidden="true" tabindex="-1"></a><span class="ot">addThree ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<p>Let’s define a simple function of type <code>Int-&gt;Int</code> and
+see <code>applyTo1</code> in action.</p>
+<div class="sourceCode" id="cb148"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb148-1"><a href="#cb148-1" aria-hidden="true" tabindex="-1"></a><span class="ot">addThree ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb148-2"><a href="#cb148-2" aria-hidden="true" tabindex="-1"></a>addThree x <span class="ot">=</span> x <span class="op">+</span> <span class="dv">3</span></span></code></pre></div>
-<div class="sourceCode" id="cb149"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb149-1"><a href="#cb149-1" aria-hidden="true" tabindex="-1"></a>applyTo1 addThree</span>
+<div class="sourceCode" id="cb149"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb149-1"><a href="#cb149-1" aria-hidden="true" tabindex="-1"></a>applyTo1 addThree</span>
 <span id="cb149-2"><a href="#cb149-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> addThree <span class="dv">1</span></span>
 <span id="cb149-3"><a href="#cb149-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">1</span> <span class="op">+</span> <span class="dv">3</span></span>
 <span id="cb149-4"><a href="#cb149-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">4</span></span></code></pre></div>
 <p>Let’s go back to the type annotation for <code>applyTo1</code>.</p>
-<div class="sourceCode" id="cb150"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb150-1"><a href="#cb150-1" aria-hidden="true" tabindex="-1"></a><span class="ot">applyTo1 ::</span> (<span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span>) <span class="ot">-&gt;</span> <span class="dt">Int</span></span></code></pre></div>
-<p>The parentheses are needed because the type <code>Int -&gt; Int -&gt; Int</code> would be the type of a function taking two <code>Int</code> arguments. More on this later.</p>
-<p>Let’s look at a slightly more interesting example. This time we’ll implement a polymorphic function <code>doTwice</code>. Note how we can use it with various types of values and functions.</p>
-<div class="sourceCode" id="cb151"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb151-1"><a href="#cb151-1" aria-hidden="true" tabindex="-1"></a><span class="ot">doTwice ::</span> (a <span class="ot">-&gt;</span> a) <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
+<div class="sourceCode" id="cb150"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb150-1"><a href="#cb150-1" aria-hidden="true" tabindex="-1"></a><span class="ot">applyTo1 ::</span> (<span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span>) <span class="ot">-&gt;</span> <span class="dt">Int</span></span></code></pre></div>
+<p>The parentheses are needed because the type
+<code>Int -&gt; Int -&gt; Int</code> would be the type of a function
+taking two <code>Int</code> arguments. More on this later.</p>
+<p>Let’s look at a slightly more interesting example. This time we’ll
+implement a polymorphic function <code>doTwice</code>. Note how we can
+use it with various types of values and functions.</p>
+<div class="sourceCode" id="cb151"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb151-1"><a href="#cb151-1" aria-hidden="true" tabindex="-1"></a><span class="ot">doTwice ::</span> (a <span class="ot">-&gt;</span> a) <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb151-2"><a href="#cb151-2" aria-hidden="true" tabindex="-1"></a>doTwice f x <span class="ot">=</span> f (f x)</span></code></pre></div>
-<div class="sourceCode" id="cb152"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb152-1"><a href="#cb152-1" aria-hidden="true" tabindex="-1"></a>doTwice addThree <span class="dv">1</span></span>
+<div class="sourceCode" id="cb152"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb152-1"><a href="#cb152-1" aria-hidden="true" tabindex="-1"></a>doTwice addThree <span class="dv">1</span></span>
 <span id="cb152-2"><a href="#cb152-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> addThree (addThree <span class="dv">1</span>)</span>
 <span id="cb152-3"><a href="#cb152-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">7</span></span>
 <span id="cb152-4"><a href="#cb152-4" aria-hidden="true" tabindex="-1"></a>doTwice <span class="fu">tail</span> <span class="st">&quot;abcd&quot;</span></span>
 <span id="cb152-5"><a href="#cb152-5" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="fu">tail</span> (<span class="fu">tail</span> <span class="st">&quot;abcd&quot;</span>)</span>
 <span id="cb152-6"><a href="#cb152-6" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="st">&quot;cd&quot;</span></span></code></pre></div>
-<div class="sourceCode" id="cb153"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb153-1"><a href="#cb153-1" aria-hidden="true" tabindex="-1"></a><span class="ot">makeCool ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<div class="sourceCode" id="cb153"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb153-1"><a href="#cb153-1" aria-hidden="true" tabindex="-1"></a><span class="ot">makeCool ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb153-2"><a href="#cb153-2" aria-hidden="true" tabindex="-1"></a>makeCool str <span class="ot">=</span> <span class="st">&quot;WOW &quot;</span> <span class="op">++</span> str <span class="op">++</span> <span class="st">&quot;!&quot;</span></span></code></pre></div>
-<div class="sourceCode" id="cb154"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb154-1"><a href="#cb154-1" aria-hidden="true" tabindex="-1"></a>doTwice makeCool <span class="st">&quot;Haskell&quot;</span></span>
+<div class="sourceCode" id="cb154"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb154-1"><a href="#cb154-1" aria-hidden="true" tabindex="-1"></a>doTwice makeCool <span class="st">&quot;Haskell&quot;</span></span>
 <span id="cb154-2"><a href="#cb154-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="st">&quot;WOW WOW Haskell!!&quot;</span></span></code></pre></div>
-<h3 data-number="3.1.1" id="functional-programming-on-lists"><span class="header-section-number">3.1.1</span> Functional Programming on Lists</h3>
-<p>That was a bit boring. Luckily there are many useful list functions that take functions as arguments. By the way, functions that take functions as arguments (or return functions) are often called <em>higher-order functions</em>.</p>
-<p>The most famous of these list-processing higher-order functions is <code>map</code>. It gives you a new list by applying the given function to all elements of a list.</p>
-<div class="sourceCode" id="cb155"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb155-1"><a href="#cb155-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [b]</span></code></pre></div>
-<div class="sourceCode" id="cb156"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb156-1"><a href="#cb156-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> addThree [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
+<h3 data-number="3.1.1" id="functional-programming-on-lists"><span
+class="header-section-number">3.1.1</span> Functional Programming on
+Lists</h3>
+<p>That was a bit boring. Luckily there are many useful list functions
+that take functions as arguments. By the way, functions that take
+functions as arguments (or return functions) are often called
+<em>higher-order functions</em>.</p>
+<p>The most famous of these list-processing higher-order functions is
+<code>map</code>. It gives you a new list by applying the given function
+to all elements of a list.</p>
+<div class="sourceCode" id="cb155"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb155-1"><a href="#cb155-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [b]</span></code></pre></div>
+<div class="sourceCode" id="cb156"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb156-1"><a href="#cb156-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> addThree [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb156-2"><a href="#cb156-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="dv">4</span>,<span class="dv">5</span>,<span class="dv">6</span>]</span></code></pre></div>
-<p>The partner in crime for <code>map</code> is <code>filter</code>. Instead of transforming all elements of a list, <code>filter</code> drops some elements of a list and keeps others. In other words, <code>filter</code> selects the elements from a list that fulfill a condition.</p>
-<div class="sourceCode" id="cb157"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb157-1"><a href="#cb157-1" aria-hidden="true" tabindex="-1"></a><span class="fu">filter</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> <span class="dt">Bool</span>) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [a]</span></code></pre></div>
+<p>The partner in crime for <code>map</code> is <code>filter</code>.
+Instead of transforming all elements of a list, <code>filter</code>
+drops some elements of a list and keeps others. In other words,
+<code>filter</code> selects the elements from a list that fulfill a
+condition.</p>
+<div class="sourceCode" id="cb157"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb157-1"><a href="#cb157-1" aria-hidden="true" tabindex="-1"></a><span class="fu">filter</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> <span class="dt">Bool</span>) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [a]</span></code></pre></div>
 <p>Here’s an example: selecting the positive elements from a list</p>
-<div class="sourceCode" id="cb158"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb158-1"><a href="#cb158-1" aria-hidden="true" tabindex="-1"></a><span class="ot">positive ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
+<div class="sourceCode" id="cb158"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb158-1"><a href="#cb158-1" aria-hidden="true" tabindex="-1"></a><span class="ot">positive ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb158-2"><a href="#cb158-2" aria-hidden="true" tabindex="-1"></a>positive x <span class="ot">=</span> x<span class="op">&gt;</span><span class="dv">0</span></span></code></pre></div>
-<div class="sourceCode" id="cb159"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb159-1"><a href="#cb159-1" aria-hidden="true" tabindex="-1"></a><span class="fu">filter</span> positive [<span class="dv">0</span>,<span class="dv">1</span>,<span class="op">-</span><span class="dv">1</span>,<span class="dv">3</span>,<span class="op">-</span><span class="dv">3</span>]</span>
+<div class="sourceCode" id="cb159"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb159-1"><a href="#cb159-1" aria-hidden="true" tabindex="-1"></a><span class="fu">filter</span> positive [<span class="dv">0</span>,<span class="dv">1</span>,<span class="op">-</span><span class="dv">1</span>,<span class="dv">3</span>,<span class="op">-</span><span class="dv">3</span>]</span>
 <span id="cb159-2"><a href="#cb159-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="dv">1</span>,<span class="dv">3</span>]</span></code></pre></div>
-<p>Note how both the type signatures of <code>map</code> and <code>filter</code> use polymorphism. They work on all kinds of lists. The type of <code>map</code> even uses two type parameters! Here are some examples of type inference using <code>map</code> and <code>filter</code>.</p>
-<div class="sourceCode" id="cb160"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb160-1"><a href="#cb160-1" aria-hidden="true" tabindex="-1"></a>onlyPositive xs <span class="ot">=</span> <span class="fu">filter</span> positive xs</span>
+<p>Note how both the type signatures of <code>map</code> and
+<code>filter</code> use polymorphism. They work on all kinds of lists.
+The type of <code>map</code> even uses two type parameters! Here are
+some examples of type inference using <code>map</code> and
+<code>filter</code>.</p>
+<div class="sourceCode" id="cb160"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb160-1"><a href="#cb160-1" aria-hidden="true" tabindex="-1"></a>onlyPositive xs <span class="ot">=</span> <span class="fu">filter</span> positive xs</span>
 <span id="cb160-2"><a href="#cb160-2" aria-hidden="true" tabindex="-1"></a>mapBooleans f <span class="ot">=</span> <span class="fu">map</span> f [<span class="dt">False</span>,<span class="dt">True</span>]</span></code></pre></div>
-<div class="sourceCode" id="cb161"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb161-1"><a href="#cb161-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t onlyPositive</span>
+<div class="sourceCode" id="cb161"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb161-1"><a href="#cb161-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t onlyPositive</span>
 <span id="cb161-2"><a href="#cb161-2" aria-hidden="true" tabindex="-1"></a><span class="ot">onlyPositive ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> [<span class="dt">Int</span>]</span>
 <span id="cb161-3"><a href="#cb161-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t mapBooleans</span>
 <span id="cb161-4"><a href="#cb161-4" aria-hidden="true" tabindex="-1"></a><span class="ot">mapBooleans ::</span> (<span class="dt">Bool</span> <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> [b]</span>
 <span id="cb161-5"><a href="#cb161-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t mapBooleans <span class="fu">not</span></span>
 <span id="cb161-6"><a href="#cb161-6" aria-hidden="true" tabindex="-1"></a>mapBooleans<span class="ot"> not ::</span> [<span class="dt">Bool</span>]</span></code></pre></div>
-<p>One more thing: remember how constructors were just functions? That means you can pass them as arguments to other functions!</p>
-<div class="sourceCode" id="cb162"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb162-1"><a href="#cb162-1" aria-hidden="true" tabindex="-1"></a>wrapJust xs <span class="ot">=</span> <span class="fu">map</span> <span class="dt">Just</span> xs</span></code></pre></div>
-<div class="sourceCode" id="cb163"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb163-1"><a href="#cb163-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t wrapJust</span>
+<p>One more thing: remember how constructors were just functions? That
+means you can pass them as arguments to other functions!</p>
+<div class="sourceCode" id="cb162"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb162-1"><a href="#cb162-1" aria-hidden="true" tabindex="-1"></a>wrapJust xs <span class="ot">=</span> <span class="fu">map</span> <span class="dt">Just</span> xs</span></code></pre></div>
+<div class="sourceCode" id="cb163"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb163-1"><a href="#cb163-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t wrapJust</span>
 <span id="cb163-2"><a href="#cb163-2" aria-hidden="true" tabindex="-1"></a><span class="ot">wrapJust ::</span> [a] <span class="ot">-&gt;</span> [<span class="dt">Maybe</span> a]</span>
 <span id="cb163-3"><a href="#cb163-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> wrapJust [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb163-4"><a href="#cb163-4" aria-hidden="true" tabindex="-1"></a>[<span class="dt">Just</span> <span class="dv">1</span>,<span class="dt">Just</span> <span class="dv">2</span>,<span class="dt">Just</span> <span class="dv">3</span>]</span></code></pre></div>
-<h3 data-number="3.1.2" id="examples-of-functional-programming-on-lists"><span class="header-section-number">3.1.2</span> Examples of Functional Programming on Lists</h3>
-<p>How many “palindrome numbers” are between <code>1</code> and <code>n</code>?</p>
-<div class="sourceCode" id="cb164"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb164-1"><a href="#cb164-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- a predicate that checks if a string is a palindrome</span></span>
+<h3 data-number="3.1.2"
+id="examples-of-functional-programming-on-lists"><span
+class="header-section-number">3.1.2</span> Examples of Functional
+Programming on Lists</h3>
+<p>How many “palindrome numbers” are between <code>1</code> and
+<code>n</code>?</p>
+<div class="sourceCode" id="cb164"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb164-1"><a href="#cb164-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- a predicate that checks if a string is a palindrome</span></span>
 <span id="cb164-2"><a href="#cb164-2" aria-hidden="true" tabindex="-1"></a><span class="ot">palindrome ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb164-3"><a href="#cb164-3" aria-hidden="true" tabindex="-1"></a>palindrome str <span class="ot">=</span> str <span class="op">==</span> <span class="fu">reverse</span> str</span>
 <span id="cb164-4"><a href="#cb164-4" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb164-5"><a href="#cb164-5" aria-hidden="true" tabindex="-1"></a><span class="co">-- palindromes n takes all numbers from 1 to n, converts them to strings using show, and keeps only palindromes</span></span>
 <span id="cb164-6"><a href="#cb164-6" aria-hidden="true" tabindex="-1"></a><span class="ot">palindromes ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> [<span class="dt">String</span>]</span>
 <span id="cb164-7"><a href="#cb164-7" aria-hidden="true" tabindex="-1"></a>palindromes n <span class="ot">=</span> <span class="fu">filter</span> palindrome (<span class="fu">map</span> <span class="fu">show</span> [<span class="dv">1</span><span class="op">..</span>n])</span></code></pre></div>
-<div class="sourceCode" id="cb165"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb165-1"><a href="#cb165-1" aria-hidden="true" tabindex="-1"></a>palindrome <span class="st">&quot;1331&quot;</span> <span class="op">==&gt;</span> <span class="dt">True</span></span>
+<div class="sourceCode" id="cb165"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb165-1"><a href="#cb165-1" aria-hidden="true" tabindex="-1"></a>palindrome <span class="st">&quot;1331&quot;</span> <span class="op">==&gt;</span> <span class="dt">True</span></span>
 <span id="cb165-2"><a href="#cb165-2" aria-hidden="true" tabindex="-1"></a>palindromes <span class="dv">150</span> <span class="op">==&gt;</span></span>
 <span id="cb165-3"><a href="#cb165-3" aria-hidden="true" tabindex="-1"></a>  [<span class="st">&quot;1&quot;</span>,<span class="st">&quot;2&quot;</span>,<span class="st">&quot;3&quot;</span>,<span class="st">&quot;4&quot;</span>,<span class="st">&quot;5&quot;</span>,<span class="st">&quot;6&quot;</span>,<span class="st">&quot;7&quot;</span>,<span class="st">&quot;8&quot;</span>,<span class="st">&quot;9&quot;</span>,</span>
 <span id="cb165-4"><a href="#cb165-4" aria-hidden="true" tabindex="-1"></a>   <span class="st">&quot;11&quot;</span>,<span class="st">&quot;22&quot;</span>,<span class="st">&quot;33&quot;</span>,<span class="st">&quot;44&quot;</span>,<span class="st">&quot;55&quot;</span>,<span class="st">&quot;66&quot;</span>,<span class="st">&quot;77&quot;</span>,<span class="st">&quot;88&quot;</span>,<span class="st">&quot;99&quot;</span>,</span>
 <span id="cb165-5"><a href="#cb165-5" aria-hidden="true" tabindex="-1"></a>   <span class="st">&quot;101&quot;</span>,<span class="st">&quot;111&quot;</span>,<span class="st">&quot;121&quot;</span>,<span class="st">&quot;131&quot;</span>,<span class="st">&quot;141&quot;</span>]</span>
 <span id="cb165-6"><a href="#cb165-6" aria-hidden="true" tabindex="-1"></a><span class="fu">length</span> (palindromes <span class="dv">9999</span>) <span class="op">==&gt;</span> <span class="dv">198</span></span></code></pre></div>
-<p>How many words in a string start with “a”? This uses the function <code>words</code> from the module <code>Data.List</code> that splits a string into words.</p>
-<div class="sourceCode" id="cb166"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb166-1"><a href="#cb166-1" aria-hidden="true" tabindex="-1"></a><span class="ot">countAWords ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<p>How many words in a string start with “a”? This uses the function
+<code>words</code> from the module <code>Data.List</code> that splits a
+string into words.</p>
+<div class="sourceCode" id="cb166"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb166-1"><a href="#cb166-1" aria-hidden="true" tabindex="-1"></a><span class="ot">countAWords ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb166-2"><a href="#cb166-2" aria-hidden="true" tabindex="-1"></a>countAWords string <span class="ot">=</span> <span class="fu">length</span> (<span class="fu">filter</span> startsWithA (<span class="fu">words</span> string))</span>
 <span id="cb166-3"><a href="#cb166-3" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> startsWithA s <span class="ot">=</span> <span class="fu">head</span> s <span class="op">==</span> <span class="ch">&#39;a&#39;</span></span></code></pre></div>
-<div class="sourceCode" id="cb167"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb167-1"><a href="#cb167-1" aria-hidden="true" tabindex="-1"></a>countAWords <span class="st">&quot;does anyone want an apple?&quot;</span></span>
+<div class="sourceCode" id="cb167"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb167-1"><a href="#cb167-1" aria-hidden="true" tabindex="-1"></a>countAWords <span class="st">&quot;does anyone want an apple?&quot;</span></span>
 <span id="cb167-2"><a href="#cb167-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">3</span></span></code></pre></div>
-<p>The function <code>tails</code> from <code>Data.List</code> returns the list of all suffixes (“tails”) of a list. We can use <code>tails</code> for many string processing tasks. Here’s how <code>tails</code> works:</p>
-<div class="sourceCode" id="cb168"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb168-1"><a href="#cb168-1" aria-hidden="true" tabindex="-1"></a>tails <span class="st">&quot;echo&quot;</span></span>
+<p>The function <code>tails</code> from <code>Data.List</code> returns
+the list of all suffixes (“tails”) of a list. We can use
+<code>tails</code> for many string processing tasks. Here’s how
+<code>tails</code> works:</p>
+<div class="sourceCode" id="cb168"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb168-1"><a href="#cb168-1" aria-hidden="true" tabindex="-1"></a>tails <span class="st">&quot;echo&quot;</span></span>
 <span id="cb168-2"><a href="#cb168-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="st">&quot;echo&quot;</span>,<span class="st">&quot;cho&quot;</span>,<span class="st">&quot;ho&quot;</span>,<span class="st">&quot;o&quot;</span>,<span class="st">&quot;&quot;</span>]</span></code></pre></div>
-<p>Here’s an example where we find what characters come after a given character in a string. First of all, we use <code>tails</code>, <code>map</code> and <code>take</code> to get all substrings of a certain length:</p>
-<div class="sourceCode" id="cb169"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb169-1"><a href="#cb169-1" aria-hidden="true" tabindex="-1"></a><span class="ot">substringsOfLength ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> [<span class="dt">String</span>]</span>
+<p>Here’s an example where we find what characters come after a given
+character in a string. First of all, we use <code>tails</code>,
+<code>map</code> and <code>take</code> to get all substrings of a
+certain length:</p>
+<div class="sourceCode" id="cb169"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb169-1"><a href="#cb169-1" aria-hidden="true" tabindex="-1"></a><span class="ot">substringsOfLength ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> [<span class="dt">String</span>]</span>
 <span id="cb169-2"><a href="#cb169-2" aria-hidden="true" tabindex="-1"></a>substringsOfLength n string <span class="ot">=</span> <span class="fu">map</span> shorten (tails string)</span>
 <span id="cb169-3"><a href="#cb169-3" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> shorten s <span class="ot">=</span> <span class="fu">take</span> n s</span></code></pre></div>
-<div class="sourceCode" id="cb170"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb170-1"><a href="#cb170-1" aria-hidden="true" tabindex="-1"></a>substringsOfLength <span class="dv">3</span> <span class="st">&quot;hello&quot;</span></span>
+<div class="sourceCode" id="cb170"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb170-1"><a href="#cb170-1" aria-hidden="true" tabindex="-1"></a>substringsOfLength <span class="dv">3</span> <span class="st">&quot;hello&quot;</span></span>
 <span id="cb170-2"><a href="#cb170-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="st">&quot;hel&quot;</span>,<span class="st">&quot;ell&quot;</span>,<span class="st">&quot;llo&quot;</span>,<span class="st">&quot;lo&quot;</span>,<span class="st">&quot;o&quot;</span>,<span class="st">&quot;&quot;</span>]</span></code></pre></div>
-<p>There’s some shorter substrings left at the end (can you see why?), but they’re fine for our purposes right now. Now that we have <code>substringsOfLength</code>, we can implement the function <code>whatFollows c k s</code> that finds all the occurrences of the character <code>c</code> in the string <code>s</code>, and outputs the <code>k</code> letters that come after these occurrences.</p>
-<div class="sourceCode" id="cb171"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb171-1"><a href="#cb171-1" aria-hidden="true" tabindex="-1"></a><span class="ot">whatFollows ::</span> <span class="dt">Char</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> [<span class="dt">String</span>]</span>
+<p>There’s some shorter substrings left at the end (can you see why?),
+but they’re fine for our purposes right now. Now that we have
+<code>substringsOfLength</code>, we can implement the function
+<code>whatFollows c k s</code> that finds all the occurrences of the
+character <code>c</code> in the string <code>s</code>, and outputs the
+<code>k</code> letters that come after these occurrences.</p>
+<div class="sourceCode" id="cb171"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb171-1"><a href="#cb171-1" aria-hidden="true" tabindex="-1"></a><span class="ot">whatFollows ::</span> <span class="dt">Char</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> [<span class="dt">String</span>]</span>
 <span id="cb171-2"><a href="#cb171-2" aria-hidden="true" tabindex="-1"></a>whatFollows c k string <span class="ot">=</span> <span class="fu">map</span> <span class="fu">tail</span> (<span class="fu">filter</span> match (substringsOfLength (k<span class="op">+</span><span class="dv">1</span>) string))</span>
 <span id="cb171-3"><a href="#cb171-3" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> match sub <span class="ot">=</span> <span class="fu">take</span> <span class="dv">1</span> sub <span class="op">==</span> [c]</span></code></pre></div>
-<div class="sourceCode" id="cb172"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb172-1"><a href="#cb172-1" aria-hidden="true" tabindex="-1"></a>whatFollows <span class="ch">&#39;a&#39;</span> <span class="dv">2</span> <span class="st">&quot;abracadabra&quot;</span></span>
+<div class="sourceCode" id="cb172"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb172-1"><a href="#cb172-1" aria-hidden="true" tabindex="-1"></a>whatFollows <span class="ch">&#39;a&#39;</span> <span class="dv">2</span> <span class="st">&quot;abracadabra&quot;</span></span>
 <span id="cb172-2"><a href="#cb172-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="st">&quot;br&quot;</span>,<span class="st">&quot;ca&quot;</span>,<span class="st">&quot;da&quot;</span>,<span class="st">&quot;br&quot;</span>,<span class="st">&quot;&quot;</span>]</span></code></pre></div>
-<h2 data-number="3.2" id="partial-application"><span class="header-section-number">3.2</span> Partial Application</h2>
-<p>When using higher-order functions you can find yourself defining lots of small helper functions, like <code>addThree</code> or <code>shorten</code> in the previous examples. This is a bit of a chore in the long run, but luckily Haskell’s functions behave a bit weirdly…</p>
+<h2 data-number="3.2" id="partial-application"><span
+class="header-section-number">3.2</span> Partial Application</h2>
+<p>When using higher-order functions you can find yourself defining lots
+of small helper functions, like <code>addThree</code> or
+<code>shorten</code> in the previous examples. This is a bit of a chore
+in the long run, but luckily Haskell’s functions behave a bit
+weirdly…</p>
 <p>Let’s start in GHCi:</p>
-<div class="sourceCode" id="cb173"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb173-1"><a href="#cb173-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> add a b <span class="ot">=</span> a<span class="op">+</span>b</span>
+<div class="sourceCode" id="cb173"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb173-1"><a href="#cb173-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> add a b <span class="ot">=</span> a<span class="op">+</span>b</span>
 <span id="cb173-2"><a href="#cb173-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> add <span class="dv">1</span> <span class="dv">5</span></span>
 <span id="cb173-3"><a href="#cb173-3" aria-hidden="true" tabindex="-1"></a><span class="dv">6</span></span>
 <span id="cb173-4"><a href="#cb173-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> addThree <span class="ot">=</span> add <span class="dv">3</span></span>
 <span id="cb173-5"><a href="#cb173-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> addThree <span class="dv">2</span></span>
 <span id="cb173-6"><a href="#cb173-6" aria-hidden="true" tabindex="-1"></a><span class="dv">5</span></span></code></pre></div>
-<p>So, we’ve defined <code>add</code>, a function of two arguments, and only given it one argument. The result is not a type error but a new function. The new function just stores (or remembers) the given argument, waits for another argument, and then gives both to <code>add</code>.</p>
+<p>So, we’ve defined <code>add</code>, a function of two arguments, and
+only given it one argument. The result is not a type error but a new
+function. The new function just stores (or remembers) the given
+argument, waits for another argument, and then gives both to
+<code>add</code>.</p>
 <pre><code>Prelude&gt; map addThree [1,2,3]
 [4,5,6]
 Prelude&gt; map (add 3) [1,2,3]
 [4,5,6]</code></pre>
-<p>Here we can see that we don’t even need to give a name to the function returned by <code>add 3</code>. We can just use it anywhere where a function of one argument is expected.</p>
-<p>This is called <em>partial application</em>. All functions in Haskell behave like this. Let’s have a closer look. Here’s a function that takes many arguments.</p>
-<div class="sourceCode" id="cb175"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb175-1"><a href="#cb175-1" aria-hidden="true" tabindex="-1"></a><span class="ot">between ::</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
+<p>Here we can see that we don’t even need to give a name to the
+function returned by <code>add 3</code>. We can just use it anywhere
+where a function of one argument is expected.</p>
+<p>This is called <em>partial application</em>. All functions in Haskell
+behave like this. Let’s have a closer look. Here’s a function that takes
+many arguments.</p>
+<div class="sourceCode" id="cb175"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb175-1"><a href="#cb175-1" aria-hidden="true" tabindex="-1"></a><span class="ot">between ::</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb175-2"><a href="#cb175-2" aria-hidden="true" tabindex="-1"></a>between lo high x <span class="ot">=</span> x <span class="op">&lt;</span> high <span class="op">&amp;&amp;</span> x <span class="op">&gt;</span> lo</span></code></pre></div>
-<div class="sourceCode" id="cb176"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb176-1"><a href="#cb176-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> between <span class="dv">3</span> <span class="dv">7</span> <span class="dv">5</span></span>
+<div class="sourceCode" id="cb176"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb176-1"><a href="#cb176-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> between <span class="dv">3</span> <span class="dv">7</span> <span class="dv">5</span></span>
 <span id="cb176-2"><a href="#cb176-2" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span>
 <span id="cb176-3"><a href="#cb176-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> between <span class="dv">3</span> <span class="dv">6</span> <span class="dv">8</span></span>
 <span id="cb176-4"><a href="#cb176-4" aria-hidden="true" tabindex="-1"></a><span class="dt">False</span></span></code></pre></div>
-<p>We can give <code>between</code> less arguments and get back new functions, just like we saw with <code>add</code>:</p>
-<div class="sourceCode" id="cb177"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb177-1"><a href="#cb177-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> (between <span class="dv">1</span> <span class="dv">5</span>) <span class="dv">2</span></span>
+<p>We can give <code>between</code> less arguments and get back new
+functions, just like we saw with <code>add</code>:</p>
+<div class="sourceCode" id="cb177"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb177-1"><a href="#cb177-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> (between <span class="dv">1</span> <span class="dv">5</span>) <span class="dv">2</span></span>
 <span id="cb177-2"><a href="#cb177-2" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span>
 <span id="cb177-3"><a href="#cb177-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">let</span> f <span class="ot">=</span> between <span class="dv">1</span> <span class="dv">5</span> <span class="kw">in</span> f <span class="dv">2</span></span>
 <span id="cb177-4"><a href="#cb177-4" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span>
 <span id="cb177-5"><a href="#cb177-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">map</span> (between <span class="dv">1</span> <span class="dv">3</span>) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb177-6"><a href="#cb177-6" aria-hidden="true" tabindex="-1"></a>[<span class="dt">False</span>,<span class="dt">True</span>,<span class="dt">False</span>]</span></code></pre></div>
-<p>Look at the types of partially applying <code>between</code>. They behave neatly, with arguments disappearing one by one from the type as values are added to the expression.</p>
-<div class="sourceCode" id="cb178"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb178-1"><a href="#cb178-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t between</span>
+<p>Look at the types of partially applying <code>between</code>. They
+behave neatly, with arguments disappearing one by one from the type as
+values are added to the expression.</p>
+<div class="sourceCode" id="cb178"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb178-1"><a href="#cb178-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t between</span>
 <span id="cb178-2"><a href="#cb178-2" aria-hidden="true" tabindex="-1"></a><span class="ot">between ::</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb178-3"><a href="#cb178-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t between <span class="dv">1</span></span>
 <span id="cb178-4"><a href="#cb178-4" aria-hidden="true" tabindex="-1"></a>between <span class="dv">1</span><span class="ot"> ::</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
@@ -1819,30 +2848,58 @@ Prelude&gt; map (add 3) [1,2,3]
 <span id="cb178-6"><a href="#cb178-6" aria-hidden="true" tabindex="-1"></a>between <span class="dv">1</span> <span class="dv">2</span><span class="ot"> ::</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb178-7"><a href="#cb178-7" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t between <span class="dv">1</span> <span class="dv">2</span> <span class="dv">3</span></span>
 <span id="cb178-8"><a href="#cb178-8" aria-hidden="true" tabindex="-1"></a>between <span class="dv">1</span> <span class="dv">2</span> <span class="dv">3</span><span class="ot"> ::</span> <span class="dt">Bool</span></span></code></pre></div>
-<p>Actually, when we write a type like <code>Integer -&gt; Integer -&gt; Integer -&gt; Bool</code>, it means <code>Integer -&gt; (Integer -&gt; (Integer -&gt; Bool))</code>. That is, a multi-argument function is just a function that returns a function. Similarly, an expression like <code>between 1 2 3</code> is the same as <code>((between 1) 2) 3</code>, so passing multiple arguments to a function happens via multiple single-argument calls. Representing multi-argument functions like this is called <em>currying</em> (after the logician Haskell Curry). Currying is what makes partial application possible.</p>
-<p>Here’s another example of using partial application with <code>map</code>:</p>
-<div class="sourceCode" id="cb179"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb179-1"><a href="#cb179-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> (<span class="fu">drop</span> <span class="dv">1</span>) [<span class="st">&quot;Hello&quot;</span>,<span class="st">&quot;World!&quot;</span>]</span>
+<p>Actually, when we write a type like
+<code>Integer -&gt; Integer -&gt; Integer -&gt; Bool</code>, it means
+<code>Integer -&gt; (Integer -&gt; (Integer -&gt; Bool))</code>. That
+is, a multi-argument function is just a function that returns a
+function. Similarly, an expression like <code>between 1 2 3</code> is
+the same as <code>((between 1) 2) 3</code>, so passing multiple
+arguments to a function happens via multiple single-argument calls.
+Representing multi-argument functions like this is called
+<em>currying</em> (after the logician Haskell Curry). Currying is what
+makes partial application possible.</p>
+<p>Here’s another example of using partial application with
+<code>map</code>:</p>
+<div class="sourceCode" id="cb179"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb179-1"><a href="#cb179-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> (<span class="fu">drop</span> <span class="dv">1</span>) [<span class="st">&quot;Hello&quot;</span>,<span class="st">&quot;World!&quot;</span>]</span>
 <span id="cb179-2"><a href="#cb179-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="st">&quot;ello&quot;</span>,<span class="st">&quot;orld!&quot;</span>]</span></code></pre></div>
-<p>In addition to normal functions, partial application also works with operators. With operators you can choose whether you apply the left or the right argument. (Partially applied operators are also called <em>sections</em> or <em>operator sections</em>). Some examples:</p>
-<div class="sourceCode" id="cb180"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb180-1"><a href="#cb180-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">map</span> (<span class="op">*</span><span class="dv">2</span>) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
+<p>In addition to normal functions, partial application also works with
+operators. With operators you can choose whether you apply the left or
+the right argument. (Partially applied operators are also called
+<em>sections</em> or <em>operator sections</em>). Some examples:</p>
+<div class="sourceCode" id="cb180"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb180-1"><a href="#cb180-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">map</span> (<span class="op">*</span><span class="dv">2</span>) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb180-2"><a href="#cb180-2" aria-hidden="true" tabindex="-1"></a>[<span class="dv">2</span>,<span class="dv">4</span>,<span class="dv">6</span>]</span>
 <span id="cb180-3"><a href="#cb180-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">map</span> (<span class="dv">2</span><span class="op">*</span>) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb180-4"><a href="#cb180-4" aria-hidden="true" tabindex="-1"></a>[<span class="dv">2</span>,<span class="dv">4</span>,<span class="dv">6</span>]</span>
 <span id="cb180-5"><a href="#cb180-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">map</span> (<span class="dv">1</span><span class="op">/</span>) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>,<span class="dv">5</span>]</span>
 <span id="cb180-6"><a href="#cb180-6" aria-hidden="true" tabindex="-1"></a>[<span class="fl">1.0</span>,<span class="fl">0.5</span>,<span class="fl">0.3333333333333333</span>,<span class="fl">0.25</span>,<span class="fl">0.2</span>]</span></code></pre></div>
-<h2 data-number="3.3" id="prefix-and-infix-notations"><span class="header-section-number">3.3</span> Prefix and Infix Notations</h2>
-<p>Normal Haskell operators are applied with <em>prefix notation</em>, which is just a fancy way to say that the function name comes before the arguments. In contrast, operators are applied with <em>infix notation</em> – the name of the function comes between the arguments.</p>
-<p>An infix operator can be converted into a prefix function by adding parentheses around it. For instance,</p>
-<div class="sourceCode" id="cb181"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb181-1"><a href="#cb181-1" aria-hidden="true" tabindex="-1"></a>(<span class="op">+</span>) <span class="dv">1</span> <span class="dv">2</span> <span class="op">==&gt;</span> <span class="dv">1</span> <span class="op">+</span> <span class="dv">2</span> <span class="op">==&gt;</span> <span class="dv">3</span></span></code></pre></div>
-<p>This is useful especially when an operator needs to be passed as an argument to another function.</p>
-<p>As an example, the function <code>zipWith</code> takes two lists, a binary function, and joins the lists using the function. We can use <code>zipWith (+)</code> to sum two lists, element-by-element:</p>
-<div class="sourceCode" id="cb182"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb182-1"><a href="#cb182-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="fu">zipWith</span></span>
+<h2 data-number="3.3" id="prefix-and-infix-notations"><span
+class="header-section-number">3.3</span> Prefix and Infix Notations</h2>
+<p>Normal Haskell operators are applied with <em>prefix notation</em>,
+which is just a fancy way to say that the function name comes before the
+arguments. In contrast, operators are applied with <em>infix
+notation</em> – the name of the function comes between the
+arguments.</p>
+<p>An infix operator can be converted into a prefix function by adding
+parentheses around it. For instance,</p>
+<div class="sourceCode" id="cb181"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb181-1"><a href="#cb181-1" aria-hidden="true" tabindex="-1"></a>(<span class="op">+</span>) <span class="dv">1</span> <span class="dv">2</span> <span class="op">==&gt;</span> <span class="dv">1</span> <span class="op">+</span> <span class="dv">2</span> <span class="op">==&gt;</span> <span class="dv">3</span></span></code></pre></div>
+<p>This is useful especially when an operator needs to be passed as an
+argument to another function.</p>
+<p>As an example, the function <code>zipWith</code> takes two lists, a
+binary function, and joins the lists using the function. We can use
+<code>zipWith (+)</code> to sum two lists, element-by-element:</p>
+<div class="sourceCode" id="cb182"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb182-1"><a href="#cb182-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="fu">zipWith</span></span>
 <span id="cb182-2"><a href="#cb182-2" aria-hidden="true" tabindex="-1"></a><span class="fu">zipWith</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> c) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [b] <span class="ot">-&gt;</span> [c]</span>
 <span id="cb182-3"><a href="#cb182-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">zipWith</span> (<span class="op">+</span>) [<span class="dv">0</span>,<span class="dv">2</span>,<span class="dv">5</span>] [<span class="dv">1</span>,<span class="dv">3</span>,<span class="dv">3</span>]</span>
 <span id="cb182-4"><a href="#cb182-4" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span>,<span class="dv">5</span>,<span class="dv">8</span>]</span></code></pre></div>
-<p>Without the ability to turn an operator into a function, we’d have to use a helper function – such as <code>add</code> above.</p>
+<p>Without the ability to turn an operator into a function, we’d have to
+use a helper function – such as <code>add</code> above.</p>
 <p>Note that omitting the parentheses leads into a type error:</p>
-<div class="sourceCode" id="cb183"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb183-1"><a href="#cb183-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">zipWith</span> <span class="op">+</span> [<span class="dv">0</span>,<span class="dv">2</span>,<span class="dv">5</span>,<span class="dv">3</span>] [<span class="dv">1</span>,<span class="dv">3</span>,<span class="dv">3</span>]</span>
+<div class="sourceCode" id="cb183"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb183-1"><a href="#cb183-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">zipWith</span> <span class="op">+</span> [<span class="dv">0</span>,<span class="dv">2</span>,<span class="dv">5</span>,<span class="dv">3</span>] [<span class="dv">1</span>,<span class="dv">3</span>,<span class="dv">3</span>]</span>
 <span id="cb183-2"><a href="#cb183-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb183-3"><a href="#cb183-3" aria-hidden="true" tabindex="-1"></a><span class="op">&lt;</span>interactive<span class="op">&gt;:</span><span class="dv">1</span><span class="op">:</span><span class="dv">11</span><span class="op">:</span> <span class="fu">error</span><span class="op">:</span></span>
 <span id="cb183-4"><a href="#cb183-4" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">Couldn&#39;t</span> match expected <span class="kw">type</span> ‘[<span class="dt">Integer</span>]</span>
@@ -1855,18 +2912,38 @@ Prelude&gt; map (add 3) [1,2,3]
 <span id="cb183-11"><a href="#cb183-11" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">Relevant</span> bindings include</span>
 <span id="cb183-12"><a href="#cb183-12" aria-hidden="true" tabindex="-1"></a><span class="ot">        it ::</span> (a <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> c) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [b] <span class="ot">-&gt;</span> [c]</span>
 <span id="cb183-13"><a href="#cb183-13" aria-hidden="true" tabindex="-1"></a>          (bound at <span class="op">&lt;</span>interactive<span class="op">&gt;:</span><span class="dv">1</span><span class="op">:</span><span class="dv">1</span>)</span></code></pre></div>
-<p>The reason for this weird-looking error is that GHCi got confused and thought that we were somehow trying to add <code>zipWith</code> and <code>[0,2,5,3] [1,3,3]</code> together. Logically, it deduced that <code>[0,2,5,3]</code> must be a function since it’s being applied to <code>[1,3,3]</code> (remember that functions bind tighter than operators).</p>
-<p>Unfortunately, error messages can sometimes be obscure, since the compiler cannot always know the “real” cause of the error (which is in this case was omitting the parentheses). Weird error messages are frustrating, but only the programmer knows what was the original intent behind the code.</p>
-<p>Another nice feature of Haskell is the syntax for applying a binary function as if it was an infix operator, by surrounding it with backticks (`). For example:</p>
-<div class="sourceCode" id="cb184"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb184-1"><a href="#cb184-1" aria-hidden="true" tabindex="-1"></a><span class="dv">6</span> <span class="ot">`div`</span> <span class="dv">2</span> <span class="op">==&gt;</span> <span class="fu">div</span> <span class="dv">6</span> <span class="dv">2</span> <span class="op">==&gt;</span> <span class="dv">3</span></span>
+<p>The reason for this weird-looking error is that GHCi got confused and
+thought that we were somehow trying to add <code>zipWith</code> and
+<code>[0,2,5,3] [1,3,3]</code> together. Logically, it deduced that
+<code>[0,2,5,3]</code> must be a function since it’s being applied to
+<code>[1,3,3]</code> (remember that functions bind tighter than
+operators).</p>
+<p>Unfortunately, error messages can sometimes be obscure, since the
+compiler cannot always know the “real” cause of the error (which is in
+this case was omitting the parentheses). Weird error messages are
+frustrating, but only the programmer knows what was the original intent
+behind the code.</p>
+<p>Another nice feature of Haskell is the syntax for applying a binary
+function as if it was an infix operator, by surrounding it with
+backticks (`). For example:</p>
+<div class="sourceCode" id="cb184"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb184-1"><a href="#cb184-1" aria-hidden="true" tabindex="-1"></a><span class="dv">6</span> <span class="ot">`div`</span> <span class="dv">2</span> <span class="op">==&gt;</span> <span class="fu">div</span> <span class="dv">6</span> <span class="dv">2</span> <span class="op">==&gt;</span> <span class="dv">3</span></span>
 <span id="cb184-2"><a href="#cb184-2" aria-hidden="true" tabindex="-1"></a>(<span class="op">+</span><span class="dv">1</span>) <span class="ot">`map`</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] <span class="op">==&gt;</span> <span class="fu">map</span> (<span class="op">+</span><span class="dv">1</span>) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] <span class="op">==&gt;</span> [<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>]</span></code></pre></div>
-<h2 data-number="3.4" id="lambdas"><span class="header-section-number">3.4</span> Lambdas</h2>
-<p>The last spanner we need in our functional programming toolbox is λ (lambda). Lambda expressions are <em>anonymous functions</em>. Consider a situation where you need a function only once, for example in an expression like</p>
-<div class="sourceCode" id="cb185"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb185-1"><a href="#cb185-1" aria-hidden="true" tabindex="-1"></a><span class="kw">let</span> big x <span class="ot">=</span> x<span class="op">&gt;</span><span class="dv">7</span> <span class="kw">in</span> <span class="fu">filter</span> big [<span class="dv">1</span>,<span class="dv">10</span>,<span class="dv">100</span>]</span></code></pre></div>
-<p>A lambda expression allows us to write this directly, without defining a name (<code>big</code>) for the helper function:</p>
-<div class="sourceCode" id="cb186"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb186-1"><a href="#cb186-1" aria-hidden="true" tabindex="-1"></a><span class="fu">filter</span> (\x <span class="ot">-&gt;</span> x<span class="op">&gt;</span><span class="dv">7</span>) [<span class="dv">1</span>,<span class="dv">10</span>,<span class="dv">100</span>]</span></code></pre></div>
+<h2 data-number="3.4" id="lambdas"><span
+class="header-section-number">3.4</span> Lambdas</h2>
+<p>The last spanner we need in our functional programming toolbox is λ
+(lambda). Lambda expressions are <em>anonymous functions</em>. Consider
+a situation where you need a function only once, for example in an
+expression like</p>
+<div class="sourceCode" id="cb185"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb185-1"><a href="#cb185-1" aria-hidden="true" tabindex="-1"></a><span class="kw">let</span> big x <span class="ot">=</span> x<span class="op">&gt;</span><span class="dv">7</span> <span class="kw">in</span> <span class="fu">filter</span> big [<span class="dv">1</span>,<span class="dv">10</span>,<span class="dv">100</span>]</span></code></pre></div>
+<p>A lambda expression allows us to write this directly, without
+defining a name (<code>big</code>) for the helper function:</p>
+<div class="sourceCode" id="cb186"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb186-1"><a href="#cb186-1" aria-hidden="true" tabindex="-1"></a><span class="fu">filter</span> (\x <span class="ot">-&gt;</span> x<span class="op">&gt;</span><span class="dv">7</span>) [<span class="dv">1</span>,<span class="dv">10</span>,<span class="dv">100</span>]</span></code></pre></div>
 <p>Here are some more examples in GHCi:</p>
-<div class="sourceCode" id="cb187"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb187-1"><a href="#cb187-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> (\x <span class="ot">-&gt;</span> x<span class="op">*</span>x) <span class="dv">3</span></span>
+<div class="sourceCode" id="cb187"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb187-1"><a href="#cb187-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> (\x <span class="ot">-&gt;</span> x<span class="op">*</span>x) <span class="dv">3</span></span>
 <span id="cb187-2"><a href="#cb187-2" aria-hidden="true" tabindex="-1"></a><span class="dv">9</span></span>
 <span id="cb187-3"><a href="#cb187-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> (\x <span class="ot">-&gt;</span> <span class="fu">reverse</span> x <span class="op">==</span> x) <span class="st">&quot;ABBA&quot;</span></span>
 <span id="cb187-4"><a href="#cb187-4" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span>
@@ -1874,124 +2951,221 @@ Prelude&gt; map (add 3) [1,2,3]
 <span id="cb187-6"><a href="#cb187-6" aria-hidden="true" tabindex="-1"></a>[<span class="st">&quot;ABBA&quot;</span>,<span class="st">&quot;otto&quot;</span>,<span class="st">&quot;anna&quot;</span>]</span>
 <span id="cb187-7"><a href="#cb187-7" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> (\x y <span class="ot">-&gt;</span> x<span class="op">^</span><span class="dv">2</span><span class="op">+</span>y<span class="op">^</span><span class="dv">2</span>) <span class="dv">2</span> <span class="dv">3</span>           <span class="co">-- multiple arguments</span></span>
 <span id="cb187-8"><a href="#cb187-8" aria-hidden="true" tabindex="-1"></a><span class="dv">13</span></span></code></pre></div>
-<p>The Haskell syntax for lambdas is a bit surprising. The backslash character (<code>\</code>) stands for the greek letter lambda (λ). The Haskell expression <code>\x -&gt; x+1</code> is trying to mimic the mathematical notation <em>λx. x+1</em>. Other languages use syntax like <code>x =&gt; x+1</code> (JavaScript) or <code>lambda x: x+1</code> (Python).</p>
-<p><strong>Note!</strong> You never <em>need</em> to use a lambda expression. You can always instead define the function normally using <code>let</code> or <code>where</code>.</p>
-<p>By the way, lambda expressions are quite powerful constructs which have a deep theory of their own, known as <a href="https://en.wikipedia.org/wiki/Lambda_calculus">Lambda calculus</a>. Some even consider purely functional programming languages such as Haskell to be typed extensions of Lambda calculus with extra syntax.</p>
-<h2 data-number="3.5" id="sidenote-the-.-and-operators"><span class="header-section-number">3.5</span> Sidenote: The <code>.</code> and <code>$</code> Operators</h2>
-<p>The two most common operators in Haskell codebases are probably <code>.</code> and <code>$</code>. They are useful when writing code that uses higher-order functions. The first of these, the <code>.</code> operator, is the <em>function composition</em> operator. Here’s its type</p>
-<div class="sourceCode" id="cb188"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb188-1"><a href="#cb188-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(.) ::</span> (b <span class="ot">-&gt;</span> c) <span class="ot">-&gt;</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> c</span></code></pre></div>
+<p>The Haskell syntax for lambdas is a bit surprising. The backslash
+character (<code>\</code>) stands for the greek letter lambda (λ). The
+Haskell expression <code>\x -&gt; x+1</code> is trying to mimic the
+mathematical notation <em>λx. x+1</em>. Other languages use syntax like
+<code>x =&gt; x+1</code> (JavaScript) or <code>lambda x: x+1</code>
+(Python).</p>
+<p><strong>Note!</strong> You never <em>need</em> to use a lambda
+expression. You can always instead define the function normally using
+<code>let</code> or <code>where</code>.</p>
+<p>By the way, lambda expressions are quite powerful constructs which
+have a deep theory of their own, known as <a
+href="https://en.wikipedia.org/wiki/Lambda_calculus">Lambda
+calculus</a>. Some even consider purely functional programming languages
+such as Haskell to be typed extensions of Lambda calculus with extra
+syntax.</p>
+<h2 data-number="3.5" id="sidenote-the-.-and-operators"><span
+class="header-section-number">3.5</span> Sidenote: The <code>.</code>
+and <code>$</code> Operators</h2>
+<p>The two most common operators in Haskell codebases are probably
+<code>.</code> and <code>$</code>. They are useful when writing code
+that uses higher-order functions. The first of these, the <code>.</code>
+operator, is the <em>function composition</em> operator. Here’s its
+type</p>
+<div class="sourceCode" id="cb188"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb188-1"><a href="#cb188-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(.) ::</span> (b <span class="ot">-&gt;</span> c) <span class="ot">-&gt;</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> c</span></code></pre></div>
 <p>And here’s what it does</p>
 <pre><code>(f.g) x ==&gt; f (g x)</code></pre>
-<p>You can use function composition to build functions out of other functions, without mentioning any arguments. For example:</p>
-<div class="sourceCode" id="cb190"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb190-1"><a href="#cb190-1" aria-hidden="true" tabindex="-1"></a>double x <span class="ot">=</span> <span class="dv">2</span><span class="op">*</span>x</span>
+<p>You can use function composition to build functions out of other
+functions, without mentioning any arguments. For example:</p>
+<div class="sourceCode" id="cb190"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb190-1"><a href="#cb190-1" aria-hidden="true" tabindex="-1"></a>double x <span class="ot">=</span> <span class="dv">2</span><span class="op">*</span>x</span>
 <span id="cb190-2"><a href="#cb190-2" aria-hidden="true" tabindex="-1"></a>quadruple <span class="ot">=</span> double <span class="op">.</span> double  <span class="co">-- computes 2*(2*x) == 4*x</span></span>
 <span id="cb190-3"><a href="#cb190-3" aria-hidden="true" tabindex="-1"></a>f <span class="ot">=</span> quadruple <span class="op">.</span> (<span class="op">+</span><span class="dv">1</span>)         <span class="co">-- computes 4*(x+1)</span></span>
 <span id="cb190-4"><a href="#cb190-4" aria-hidden="true" tabindex="-1"></a>g <span class="ot">=</span> (<span class="op">+</span><span class="dv">1</span>) <span class="op">.</span> quadruple         <span class="co">-- computes 4*x+1</span></span>
 <span id="cb190-5"><a href="#cb190-5" aria-hidden="true" tabindex="-1"></a>third <span class="ot">=</span> <span class="fu">head</span> <span class="op">.</span> <span class="fu">tail</span> <span class="op">.</span> <span class="fu">tail</span>   <span class="co">-- fetches the third element of a list</span></span></code></pre></div>
-<p>We can also reimplement <code>doTwice</code> using <code>(.)</code>. Note how we can use <code>doTwice</code> both as applied only to a function, or as applied to a function and a value.</p>
-<div class="sourceCode" id="cb191"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb191-1"><a href="#cb191-1" aria-hidden="true" tabindex="-1"></a><span class="ot">doTwice ::</span> (a <span class="ot">-&gt;</span> a) <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
+<p>We can also reimplement <code>doTwice</code> using <code>(.)</code>.
+Note how we can use <code>doTwice</code> both as applied only to a
+function, or as applied to a function and a value.</p>
+<div class="sourceCode" id="cb191"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb191-1"><a href="#cb191-1" aria-hidden="true" tabindex="-1"></a><span class="ot">doTwice ::</span> (a <span class="ot">-&gt;</span> a) <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb191-2"><a href="#cb191-2" aria-hidden="true" tabindex="-1"></a>doTwice f <span class="ot">=</span> f <span class="op">.</span> f</span></code></pre></div>
-<div class="sourceCode" id="cb192"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb192-1"><a href="#cb192-1" aria-hidden="true" tabindex="-1"></a><span class="kw">let</span> ttail <span class="ot">=</span> doTwice <span class="fu">tail</span></span>
+<div class="sourceCode" id="cb192"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb192-1"><a href="#cb192-1" aria-hidden="true" tabindex="-1"></a><span class="kw">let</span> ttail <span class="ot">=</span> doTwice <span class="fu">tail</span></span>
 <span id="cb192-2"><a href="#cb192-2" aria-hidden="true" tabindex="-1"></a><span class="kw">in</span> ttail [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>]</span>
 <span id="cb192-3"><a href="#cb192-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="dv">3</span>,<span class="dv">4</span>]</span>
 <span id="cb192-4"><a href="#cb192-4" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb192-5"><a href="#cb192-5" aria-hidden="true" tabindex="-1"></a>(doTwice <span class="fu">tail</span>) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>] <span class="op">==&gt;</span> [<span class="dv">3</span>,<span class="dv">4</span>]</span>
 <span id="cb192-6"><a href="#cb192-6" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb192-7"><a href="#cb192-7" aria-hidden="true" tabindex="-1"></a>doTwice <span class="fu">tail</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>] <span class="op">==&gt;</span> [<span class="dv">3</span>,<span class="dv">4</span>]</span></code></pre></div>
-<p>Often function composition is not used when defining a new function, but instead to avoid defining a helper function. For instance, consider the difference between these two expressions:</p>
-<div class="sourceCode" id="cb193"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb193-1"><a href="#cb193-1" aria-hidden="true" tabindex="-1"></a><span class="kw">let</span> notEmpty x <span class="ot">=</span> <span class="fu">not</span> (<span class="fu">null</span> x)</span>
+<p>Often function composition is not used when defining a new function,
+but instead to avoid defining a helper function. For instance, consider
+the difference between these two expressions:</p>
+<div class="sourceCode" id="cb193"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb193-1"><a href="#cb193-1" aria-hidden="true" tabindex="-1"></a><span class="kw">let</span> notEmpty x <span class="ot">=</span> <span class="fu">not</span> (<span class="fu">null</span> x)</span>
 <span id="cb193-2"><a href="#cb193-2" aria-hidden="true" tabindex="-1"></a><span class="kw">in</span> <span class="fu">filter</span> notEmpty [[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>],[],[<span class="dv">4</span>]]</span>
 <span id="cb193-3"><a href="#cb193-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>],[<span class="dv">4</span>]]</span></code></pre></div>
-<div class="sourceCode" id="cb194"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb194-1"><a href="#cb194-1" aria-hidden="true" tabindex="-1"></a><span class="fu">filter</span> (<span class="fu">not</span> <span class="op">.</span> <span class="fu">null</span>) [[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>],[],[<span class="dv">4</span>]]</span>
+<div class="sourceCode" id="cb194"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb194-1"><a href="#cb194-1" aria-hidden="true" tabindex="-1"></a><span class="fu">filter</span> (<span class="fu">not</span> <span class="op">.</span> <span class="fu">null</span>) [[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>],[],[<span class="dv">4</span>]]</span>
 <span id="cb194-2"><a href="#cb194-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>],[<span class="dv">4</span>]]</span></code></pre></div>
-<p>The other operator, <code>$</code> is more subtle. Let’s look at its type.</p>
-<div class="sourceCode" id="cb195"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb195-1"><a href="#cb195-1" aria-hidden="true" tabindex="-1"></a><span class="ot">($) ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> b</span></code></pre></div>
-<p>It takes a function of type <code>a -&gt; b</code> and a value of type <code>a</code>, and returns a value of type <code>b</code>. In other words, it’s a function application operator. The expression <code>f $ x</code> is the same as <code>f x</code>. This seems pretty useless, but it means that the <code>$</code> operator can be used to eliminate parentheses! These expressions are the same:</p>
-<div class="sourceCode" id="cb196"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb196-1"><a href="#cb196-1" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span> (<span class="fu">reverse</span> <span class="st">&quot;abcd&quot;</span>)</span>
+<p>The other operator, <code>$</code> is more subtle. Let’s look at its
+type.</p>
+<div class="sourceCode" id="cb195"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb195-1"><a href="#cb195-1" aria-hidden="true" tabindex="-1"></a><span class="ot">($) ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> b</span></code></pre></div>
+<p>It takes a function of type <code>a -&gt; b</code> and a value of
+type <code>a</code>, and returns a value of type <code>b</code>. In
+other words, it’s a function application operator. The expression
+<code>f $ x</code> is the same as <code>f x</code>. This seems pretty
+useless, but it means that the <code>$</code> operator can be used to
+eliminate parentheses! These expressions are the same:</p>
+<div class="sourceCode" id="cb196"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb196-1"><a href="#cb196-1" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span> (<span class="fu">reverse</span> <span class="st">&quot;abcd&quot;</span>)</span>
 <span id="cb196-2"><a href="#cb196-2" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span> <span class="op">$</span> <span class="fu">reverse</span> <span class="st">&quot;abcd&quot;</span></span></code></pre></div>
-<p>This isn’t that impressive when it’s used to eliminate one pair of parentheses, but together <code>.</code> and <code>$</code> can eliminate lots of them! For example, we can rewrite</p>
-<div class="sourceCode" id="cb197"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb197-1"><a href="#cb197-1" aria-hidden="true" tabindex="-1"></a><span class="fu">reverse</span> (<span class="fu">map</span> <span class="fu">head</span> (<span class="fu">map</span> <span class="fu">reverse</span> ([<span class="st">&quot;Haskell&quot;</span>,<span class="st">&quot;pro&quot;</span>] <span class="op">++</span> [<span class="st">&quot;dodo&quot;</span>,<span class="st">&quot;lyric&quot;</span>])))</span></code></pre></div>
+<p>This isn’t that impressive when it’s used to eliminate one pair of
+parentheses, but together <code>.</code> and <code>$</code> can
+eliminate lots of them! For example, we can rewrite</p>
+<div class="sourceCode" id="cb197"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb197-1"><a href="#cb197-1" aria-hidden="true" tabindex="-1"></a><span class="fu">reverse</span> (<span class="fu">map</span> <span class="fu">head</span> (<span class="fu">map</span> <span class="fu">reverse</span> ([<span class="st">&quot;Haskell&quot;</span>,<span class="st">&quot;pro&quot;</span>] <span class="op">++</span> [<span class="st">&quot;dodo&quot;</span>,<span class="st">&quot;lyric&quot;</span>])))</span></code></pre></div>
 <p>as</p>
-<div class="sourceCode" id="cb198"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb198-1"><a href="#cb198-1" aria-hidden="true" tabindex="-1"></a>(<span class="fu">reverse</span> <span class="op">.</span> <span class="fu">map</span> <span class="fu">head</span> <span class="op">.</span> <span class="fu">map</span> <span class="fu">reverse</span>) ([<span class="st">&quot;Haskell&quot;</span>,<span class="st">&quot;pro&quot;</span>] <span class="op">++</span> [<span class="st">&quot;dodo&quot;</span>,<span class="st">&quot;lyric&quot;</span>])</span></code></pre></div>
+<div class="sourceCode" id="cb198"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb198-1"><a href="#cb198-1" aria-hidden="true" tabindex="-1"></a>(<span class="fu">reverse</span> <span class="op">.</span> <span class="fu">map</span> <span class="fu">head</span> <span class="op">.</span> <span class="fu">map</span> <span class="fu">reverse</span>) ([<span class="st">&quot;Haskell&quot;</span>,<span class="st">&quot;pro&quot;</span>] <span class="op">++</span> [<span class="st">&quot;dodo&quot;</span>,<span class="st">&quot;lyric&quot;</span>])</span></code></pre></div>
 <p>and then</p>
-<div class="sourceCode" id="cb199"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb199-1"><a href="#cb199-1" aria-hidden="true" tabindex="-1"></a><span class="fu">reverse</span> <span class="op">.</span> <span class="fu">map</span> <span class="fu">head</span> <span class="op">.</span> <span class="fu">map</span> <span class="fu">reverse</span> <span class="op">$</span> [<span class="st">&quot;Haskell&quot;</span>,<span class="st">&quot;pro&quot;</span>] <span class="op">++</span> [<span class="st">&quot;dodo&quot;</span>,<span class="st">&quot;lyric&quot;</span>]</span></code></pre></div>
-<p>Sometimes the operators <code>.</code> and <code>$</code> are useful as functions in their own right. For example, a list of functions can be applied to an argument using map and a section of <code>$</code>:</p>
-<div class="sourceCode" id="cb200"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb200-1"><a href="#cb200-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> (<span class="op">$</span><span class="st">&quot;string&quot;</span>) [<span class="fu">reverse</span>, <span class="fu">take</span> <span class="dv">2</span>, <span class="fu">drop</span> <span class="dv">2</span>]</span>
+<div class="sourceCode" id="cb199"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb199-1"><a href="#cb199-1" aria-hidden="true" tabindex="-1"></a><span class="fu">reverse</span> <span class="op">.</span> <span class="fu">map</span> <span class="fu">head</span> <span class="op">.</span> <span class="fu">map</span> <span class="fu">reverse</span> <span class="op">$</span> [<span class="st">&quot;Haskell&quot;</span>,<span class="st">&quot;pro&quot;</span>] <span class="op">++</span> [<span class="st">&quot;dodo&quot;</span>,<span class="st">&quot;lyric&quot;</span>]</span></code></pre></div>
+<p>Sometimes the operators <code>.</code> and <code>$</code> are useful
+as functions in their own right. For example, a list of functions can be
+applied to an argument using map and a section of <code>$</code>:</p>
+<div class="sourceCode" id="cb200"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb200-1"><a href="#cb200-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> (<span class="op">$</span><span class="st">&quot;string&quot;</span>) [<span class="fu">reverse</span>, <span class="fu">take</span> <span class="dv">2</span>, <span class="fu">drop</span> <span class="dv">2</span>]</span>
 <span id="cb200-2"><a href="#cb200-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="fu">reverse</span> <span class="op">$</span> <span class="st">&quot;string&quot;</span>, <span class="fu">take</span> <span class="dv">2</span> <span class="op">$</span> <span class="st">&quot;string&quot;</span>, <span class="fu">drop</span> <span class="dv">2</span> <span class="op">$</span> <span class="st">&quot;string&quot;</span>]</span>
 <span id="cb200-3"><a href="#cb200-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="fu">reverse</span> <span class="st">&quot;string&quot;</span>, <span class="fu">take</span> <span class="dv">2</span> <span class="st">&quot;string&quot;</span>, <span class="fu">drop</span> <span class="dv">2</span> <span class="st">&quot;string&quot;</span>]</span>
 <span id="cb200-4"><a href="#cb200-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="st">&quot;gnirts&quot;</span>, <span class="st">&quot;st&quot;</span>, <span class="st">&quot;ring&quot;</span>]</span></code></pre></div>
 <!-- `zipWith (.)` would be an example on using `.` as a function. -->
-<p>If this seems complicated, don’t worry. You don’t need to use <code>.</code> and <code>$</code> in your own code until you’re comfortable with them. However, you’ll bump into <code>.</code> and <code>$</code> when reading Haskell examples and code on the internet, so it’s good to know about them. <a href="https://typeclasses.com/featured/dollar">This article</a> might also help.</p>
-<h2 data-number="3.6" id="example-rewriting-whatfollows"><span class="header-section-number">3.6</span> Example: Rewriting <code>whatFollows</code></h2>
-<p>Now, let’s rewrite the <code>whatFollows</code> example from earlier using the tools we just saw. Here’s the original version:</p>
-<div class="sourceCode" id="cb201"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb201-1"><a href="#cb201-1" aria-hidden="true" tabindex="-1"></a><span class="ot">substringsOfLength ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> [<span class="dt">String</span>]</span>
+<p>If this seems complicated, don’t worry. You don’t need to use
+<code>.</code> and <code>$</code> in your own code until you’re
+comfortable with them. However, you’ll bump into <code>.</code> and
+<code>$</code> when reading Haskell examples and code on the internet,
+so it’s good to know about them. <a
+href="https://typeclasses.com/featured/dollar">This article</a> might
+also help.</p>
+<h2 data-number="3.6" id="example-rewriting-whatfollows"><span
+class="header-section-number">3.6</span> Example: Rewriting
+<code>whatFollows</code></h2>
+<p>Now, let’s rewrite the <code>whatFollows</code> example from earlier
+using the tools we just saw. Here’s the original version:</p>
+<div class="sourceCode" id="cb201"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb201-1"><a href="#cb201-1" aria-hidden="true" tabindex="-1"></a><span class="ot">substringsOfLength ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> [<span class="dt">String</span>]</span>
 <span id="cb201-2"><a href="#cb201-2" aria-hidden="true" tabindex="-1"></a>substringsOfLength n string <span class="ot">=</span> <span class="fu">map</span> shorten (tails string)</span>
 <span id="cb201-3"><a href="#cb201-3" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> shorten s <span class="ot">=</span> <span class="fu">take</span> n s</span>
 <span id="cb201-4"><a href="#cb201-4" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb201-5"><a href="#cb201-5" aria-hidden="true" tabindex="-1"></a><span class="ot">whatFollows ::</span> <span class="dt">Char</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> [<span class="dt">String</span>]</span>
 <span id="cb201-6"><a href="#cb201-6" aria-hidden="true" tabindex="-1"></a>whatFollows c k string <span class="ot">=</span> <span class="fu">map</span> <span class="fu">tail</span> (<span class="fu">filter</span> match (substringsOfLength (k<span class="op">+</span><span class="dv">1</span>) string))</span>
 <span id="cb201-7"><a href="#cb201-7" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> match sub <span class="ot">=</span> <span class="fu">take</span> <span class="dv">1</span> sub <span class="op">==</span> [c]</span></code></pre></div>
-<p>To get started, let’s get rid of the helper function <code>substringsOfLength</code> and move all the code to <code>whatFollows</code>:</p>
-<div class="sourceCode" id="cb202"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb202-1"><a href="#cb202-1" aria-hidden="true" tabindex="-1"></a>whatFollows c k string <span class="ot">=</span> <span class="fu">map</span> <span class="fu">tail</span> (<span class="fu">filter</span> match (<span class="fu">map</span> shorten (tails string)))</span>
+<p>To get started, let’s get rid of the helper function
+<code>substringsOfLength</code> and move all the code to
+<code>whatFollows</code>:</p>
+<div class="sourceCode" id="cb202"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb202-1"><a href="#cb202-1" aria-hidden="true" tabindex="-1"></a>whatFollows c k string <span class="ot">=</span> <span class="fu">map</span> <span class="fu">tail</span> (<span class="fu">filter</span> match (<span class="fu">map</span> shorten (tails string)))</span>
 <span id="cb202-2"><a href="#cb202-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> shorten s <span class="ot">=</span> <span class="fu">take</span> (k<span class="op">+</span><span class="dv">1</span>) s</span>
 <span id="cb202-3"><a href="#cb202-3" aria-hidden="true" tabindex="-1"></a>        match sub <span class="ot">=</span> <span class="fu">take</span> <span class="dv">1</span> sub <span class="op">==</span> [c]</span></code></pre></div>
-<p>Now let’s use partial application instead of defining <code>shorten</code>:</p>
-<div class="sourceCode" id="cb203"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb203-1"><a href="#cb203-1" aria-hidden="true" tabindex="-1"></a>whatFollows c k string <span class="ot">=</span> <span class="fu">map</span> <span class="fu">tail</span> (<span class="fu">filter</span> match (<span class="fu">map</span> (<span class="fu">take</span> (k<span class="op">+</span><span class="dv">1</span>)) (tails string)))</span>
+<p>Now let’s use partial application instead of defining
+<code>shorten</code>:</p>
+<div class="sourceCode" id="cb203"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb203-1"><a href="#cb203-1" aria-hidden="true" tabindex="-1"></a>whatFollows c k string <span class="ot">=</span> <span class="fu">map</span> <span class="fu">tail</span> (<span class="fu">filter</span> match (<span class="fu">map</span> (<span class="fu">take</span> (k<span class="op">+</span><span class="dv">1</span>)) (tails string)))</span>
 <span id="cb203-2"><a href="#cb203-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> match sub <span class="ot">=</span> <span class="fu">take</span> <span class="dv">1</span> sub <span class="op">==</span> [c]</span></code></pre></div>
-<p>Let’s use <code>.</code> and <code>$</code> to eliminate some of those parentheses:</p>
-<div class="sourceCode" id="cb204"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb204-1"><a href="#cb204-1" aria-hidden="true" tabindex="-1"></a>whatFollows c k string <span class="ot">=</span> <span class="fu">map</span> <span class="fu">tail</span> <span class="op">.</span> <span class="fu">filter</span> match <span class="op">.</span> <span class="fu">map</span> (<span class="fu">take</span> (k<span class="op">+</span><span class="dv">1</span>)) <span class="op">$</span> tails string</span>
+<p>Let’s use <code>.</code> and <code>$</code> to eliminate some of
+those parentheses:</p>
+<div class="sourceCode" id="cb204"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb204-1"><a href="#cb204-1" aria-hidden="true" tabindex="-1"></a>whatFollows c k string <span class="ot">=</span> <span class="fu">map</span> <span class="fu">tail</span> <span class="op">.</span> <span class="fu">filter</span> match <span class="op">.</span> <span class="fu">map</span> (<span class="fu">take</span> (k<span class="op">+</span><span class="dv">1</span>)) <span class="op">$</span> tails string</span>
 <span id="cb204-2"><a href="#cb204-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> match sub <span class="ot">=</span> <span class="fu">take</span> <span class="dv">1</span> sub <span class="op">==</span> [c]</span></code></pre></div>
 <p>We can also replace <code>match</code> with a lambda:</p>
-<div class="sourceCode" id="cb205"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb205-1"><a href="#cb205-1" aria-hidden="true" tabindex="-1"></a>whatFollows c k string <span class="ot">=</span> <span class="fu">map</span> <span class="fu">tail</span> <span class="op">.</span> <span class="fu">filter</span> (\sub <span class="ot">-&gt;</span> <span class="fu">take</span> <span class="dv">1</span> sub <span class="op">==</span> [c]) <span class="op">.</span> <span class="fu">map</span> (<span class="fu">take</span> (k<span class="op">+</span><span class="dv">1</span>)) <span class="op">$</span> tails string</span></code></pre></div>
-<p>Finally, we don’t need to mention the <code>string</code> parameter at all, since we can just express <code>whatFollows</code> as a composition of <code>map</code>, <code>filter</code>, <code>map</code> and <code>tails</code>:</p>
-<div class="sourceCode" id="cb206"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb206-1"><a href="#cb206-1" aria-hidden="true" tabindex="-1"></a>whatFollows c k <span class="ot">=</span> <span class="fu">map</span> <span class="fu">tail</span> <span class="op">.</span> <span class="fu">filter</span> (\sub <span class="ot">-&gt;</span> <span class="fu">take</span> <span class="dv">1</span> sub <span class="op">==</span> [c]) <span class="op">.</span> <span class="fu">map</span> (<span class="fu">take</span> (k<span class="op">+</span><span class="dv">1</span>)) <span class="op">.</span> tails</span></code></pre></div>
-<p>We can even go a bit further by rewriting the lambda using an operator section</p>
-<div class="sourceCode" id="cb207"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb207-1"><a href="#cb207-1" aria-hidden="true" tabindex="-1"></a>    \sub <span class="ot">-&gt;</span> <span class="fu">take</span> <span class="dv">1</span> sub <span class="op">==</span> [c]</span>
+<div class="sourceCode" id="cb205"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb205-1"><a href="#cb205-1" aria-hidden="true" tabindex="-1"></a>whatFollows c k string <span class="ot">=</span> <span class="fu">map</span> <span class="fu">tail</span> <span class="op">.</span> <span class="fu">filter</span> (\sub <span class="ot">-&gt;</span> <span class="fu">take</span> <span class="dv">1</span> sub <span class="op">==</span> [c]) <span class="op">.</span> <span class="fu">map</span> (<span class="fu">take</span> (k<span class="op">+</span><span class="dv">1</span>)) <span class="op">$</span> tails string</span></code></pre></div>
+<p>Finally, we don’t need to mention the <code>string</code> parameter
+at all, since we can just express <code>whatFollows</code> as a
+composition of <code>map</code>, <code>filter</code>, <code>map</code>
+and <code>tails</code>:</p>
+<div class="sourceCode" id="cb206"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb206-1"><a href="#cb206-1" aria-hidden="true" tabindex="-1"></a>whatFollows c k <span class="ot">=</span> <span class="fu">map</span> <span class="fu">tail</span> <span class="op">.</span> <span class="fu">filter</span> (\sub <span class="ot">-&gt;</span> <span class="fu">take</span> <span class="dv">1</span> sub <span class="op">==</span> [c]) <span class="op">.</span> <span class="fu">map</span> (<span class="fu">take</span> (k<span class="op">+</span><span class="dv">1</span>)) <span class="op">.</span> tails</span></code></pre></div>
+<p>We can even go a bit further by rewriting the lambda using an
+operator section</p>
+<div class="sourceCode" id="cb207"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb207-1"><a href="#cb207-1" aria-hidden="true" tabindex="-1"></a>    \sub <span class="ot">-&gt;</span> <span class="fu">take</span> <span class="dv">1</span> sub <span class="op">==</span> [c]</span>
 <span id="cb207-2"><a href="#cb207-2" aria-hidden="true" tabindex="-1"></a><span class="op">===</span> \sub <span class="ot">-&gt;</span> (<span class="op">==</span>[c]) (<span class="fu">take</span> <span class="dv">1</span> sub)</span>
 <span id="cb207-3"><a href="#cb207-3" aria-hidden="true" tabindex="-1"></a><span class="op">===</span> \sub <span class="ot">-&gt;</span> (<span class="op">==</span>[c]) ((<span class="fu">take</span> <span class="dv">1</span>) sub)</span>
 <span id="cb207-4"><a href="#cb207-4" aria-hidden="true" tabindex="-1"></a><span class="op">===</span> \sub <span class="ot">-&gt;</span> ((<span class="op">==</span>[c]) <span class="op">.</span> (<span class="fu">take</span> <span class="dv">1</span>)) sub</span>
 <span id="cb207-5"><a href="#cb207-5" aria-hidden="true" tabindex="-1"></a><span class="op">===</span> ((<span class="op">==</span>[c]) <span class="op">.</span> (<span class="fu">take</span> <span class="dv">1</span>))</span>
 <span id="cb207-6"><a href="#cb207-6" aria-hidden="true" tabindex="-1"></a><span class="op">===</span> ((<span class="op">==</span>[c]) <span class="op">.</span> <span class="fu">take</span> <span class="dv">1</span>)</span></code></pre></div>
 <p>Now what we have left is:</p>
-<div class="sourceCode" id="cb208"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb208-1"><a href="#cb208-1" aria-hidden="true" tabindex="-1"></a>whatFollows c k <span class="ot">=</span> <span class="fu">map</span> <span class="fu">tail</span> <span class="op">.</span> <span class="fu">filter</span> ((<span class="op">==</span>[c]) <span class="op">.</span> <span class="fu">take</span> <span class="dv">1</span>) <span class="op">.</span> <span class="fu">map</span> (<span class="fu">take</span> (k<span class="op">+</span><span class="dv">1</span>)) <span class="op">.</span> tails</span></code></pre></div>
-<p>This is a somewhat extreme version of the function, but when used in moderation the techniques shown here can make code easier to read.</p>
-<h2 data-number="3.7" id="more-functional-list-wrangling-examples"><span class="header-section-number">3.7</span> More Functional List Wrangling Examples</h2>
-<p>Here are some more examples of functional programming with lists. Let’s start by introducing a couple of new list functions:</p>
-<div class="sourceCode" id="cb209"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb209-1"><a href="#cb209-1" aria-hidden="true" tabindex="-1"></a><span class="fu">takeWhile</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> <span class="dt">Bool</span>) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [a]   <span class="co">-- take elements from a list as long as they satisfy a predicate</span></span>
+<div class="sourceCode" id="cb208"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb208-1"><a href="#cb208-1" aria-hidden="true" tabindex="-1"></a>whatFollows c k <span class="ot">=</span> <span class="fu">map</span> <span class="fu">tail</span> <span class="op">.</span> <span class="fu">filter</span> ((<span class="op">==</span>[c]) <span class="op">.</span> <span class="fu">take</span> <span class="dv">1</span>) <span class="op">.</span> <span class="fu">map</span> (<span class="fu">take</span> (k<span class="op">+</span><span class="dv">1</span>)) <span class="op">.</span> tails</span></code></pre></div>
+<p>This is a somewhat extreme version of the function, but when used in
+moderation the techniques shown here can make code easier to read.</p>
+<h2 data-number="3.7" id="more-functional-list-wrangling-examples"><span
+class="header-section-number">3.7</span> More Functional List Wrangling
+Examples</h2>
+<p>Here are some more examples of functional programming with lists.
+Let’s start by introducing a couple of new list functions:</p>
+<div class="sourceCode" id="cb209"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb209-1"><a href="#cb209-1" aria-hidden="true" tabindex="-1"></a><span class="fu">takeWhile</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> <span class="dt">Bool</span>) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [a]   <span class="co">-- take elements from a list as long as they satisfy a predicate</span></span>
 <span id="cb209-2"><a href="#cb209-2" aria-hidden="true" tabindex="-1"></a><span class="fu">dropWhile</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> <span class="dt">Bool</span>) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [a]   <span class="co">-- drop elements from a list as long as they satisfy a predicate</span></span></code></pre></div>
-<div class="sourceCode" id="cb210"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb210-1"><a href="#cb210-1" aria-hidden="true" tabindex="-1"></a><span class="fu">takeWhile</span> <span class="fu">even</span> [<span class="dv">2</span>,<span class="dv">4</span>,<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]   <span class="op">==&gt;</span> [<span class="dv">2</span>,<span class="dv">4</span>]</span>
+<div class="sourceCode" id="cb210"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb210-1"><a href="#cb210-1" aria-hidden="true" tabindex="-1"></a><span class="fu">takeWhile</span> <span class="fu">even</span> [<span class="dv">2</span>,<span class="dv">4</span>,<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]   <span class="op">==&gt;</span> [<span class="dv">2</span>,<span class="dv">4</span>]</span>
 <span id="cb210-2"><a href="#cb210-2" aria-hidden="true" tabindex="-1"></a><span class="fu">dropWhile</span> <span class="fu">even</span> [<span class="dv">2</span>,<span class="dv">4</span>,<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]   <span class="op">==&gt;</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span></code></pre></div>
-<p>There’s also the function <code>elem</code>, which can be used to check if a list contains an element:</p>
-<div class="sourceCode" id="cb211"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb211-1"><a href="#cb211-1" aria-hidden="true" tabindex="-1"></a><span class="fu">elem</span> <span class="dv">3</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]   <span class="op">==&gt;</span> <span class="dt">True</span></span>
+<p>There’s also the function <code>elem</code>, which can be used to
+check if a list contains an element:</p>
+<div class="sourceCode" id="cb211"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb211-1"><a href="#cb211-1" aria-hidden="true" tabindex="-1"></a><span class="fu">elem</span> <span class="dv">3</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]   <span class="op">==&gt;</span> <span class="dt">True</span></span>
 <span id="cb211-2"><a href="#cb211-2" aria-hidden="true" tabindex="-1"></a><span class="fu">elem</span> <span class="dv">4</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]   <span class="op">==&gt;</span> <span class="dt">False</span></span></code></pre></div>
-<p>Using these, we can implement a function <code>findSubstring</code> that finds the earliest and longest substring in a string that consist only of the given characters.</p>
-<div class="sourceCode" id="cb212"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb212-1"><a href="#cb212-1" aria-hidden="true" tabindex="-1"></a><span class="ot">findSubstring ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<p>Using these, we can implement a function <code>findSubstring</code>
+that finds the earliest and longest substring in a string that consist
+only of the given characters.</p>
+<div class="sourceCode" id="cb212"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb212-1"><a href="#cb212-1" aria-hidden="true" tabindex="-1"></a><span class="ot">findSubstring ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb212-2"><a href="#cb212-2" aria-hidden="true" tabindex="-1"></a>findSubstring chars <span class="ot">=</span> <span class="fu">takeWhile</span> (\x <span class="ot">-&gt;</span> <span class="fu">elem</span> x chars)</span>
 <span id="cb212-3"><a href="#cb212-3" aria-hidden="true" tabindex="-1"></a>                      <span class="op">.</span> <span class="fu">dropWhile</span> (\x <span class="ot">-&gt;</span> <span class="fu">not</span> <span class="op">$</span> <span class="fu">elem</span> x chars)</span></code></pre></div>
-<div class="sourceCode" id="cb213"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb213-1"><a href="#cb213-1" aria-hidden="true" tabindex="-1"></a>findSubstring <span class="st">&quot;a&quot;</span> <span class="st">&quot;bbaabaaaab&quot;</span>              <span class="op">==&gt;</span> <span class="st">&quot;aa&quot;</span></span>
+<div class="sourceCode" id="cb213"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb213-1"><a href="#cb213-1" aria-hidden="true" tabindex="-1"></a>findSubstring <span class="st">&quot;a&quot;</span> <span class="st">&quot;bbaabaaaab&quot;</span>              <span class="op">==&gt;</span> <span class="st">&quot;aa&quot;</span></span>
 <span id="cb213-2"><a href="#cb213-2" aria-hidden="true" tabindex="-1"></a>findSubstring <span class="st">&quot;abcd&quot;</span> <span class="st">&quot;xxxyyyzabaaxxabcd&quot;</span>    <span class="op">==&gt;</span> <span class="st">&quot;abaa&quot;</span></span></code></pre></div>
 <!-- `zipWith` has already been introduced in section 3.3 above. -->
-<p>The function <code>zipWith</code> lets you combine two lists element-by-element:</p>
-<div class="sourceCode" id="cb214"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb214-1"><a href="#cb214-1" aria-hidden="true" tabindex="-1"></a><span class="fu">zipWith</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> c) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [b] <span class="ot">-&gt;</span> [c]</span></code></pre></div>
-<div class="sourceCode" id="cb215"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb215-1"><a href="#cb215-1" aria-hidden="true" tabindex="-1"></a><span class="fu">zipWith</span> (<span class="op">++</span>) [<span class="st">&quot;John&quot;</span>,<span class="st">&quot;Mary&quot;</span>] [<span class="st">&quot;Smith&quot;</span>,<span class="st">&quot;Cooper&quot;</span>]</span>
+<p>The function <code>zipWith</code> lets you combine two lists
+element-by-element:</p>
+<div class="sourceCode" id="cb214"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb214-1"><a href="#cb214-1" aria-hidden="true" tabindex="-1"></a><span class="fu">zipWith</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> c) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [b] <span class="ot">-&gt;</span> [c]</span></code></pre></div>
+<div class="sourceCode" id="cb215"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb215-1"><a href="#cb215-1" aria-hidden="true" tabindex="-1"></a><span class="fu">zipWith</span> (<span class="op">++</span>) [<span class="st">&quot;John&quot;</span>,<span class="st">&quot;Mary&quot;</span>] [<span class="st">&quot;Smith&quot;</span>,<span class="st">&quot;Cooper&quot;</span>]</span>
 <span id="cb215-2"><a href="#cb215-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="st">&quot;JohnSmith&quot;</span>,<span class="st">&quot;MaryCooper&quot;</span>]</span>
 <span id="cb215-3"><a href="#cb215-3" aria-hidden="true" tabindex="-1"></a><span class="fu">zipWith</span> <span class="fu">take</span> [<span class="dv">4</span>,<span class="dv">3</span>] [<span class="st">&quot;Hello&quot;</span>,<span class="st">&quot;Warden&quot;</span>]</span>
 <span id="cb215-4"><a href="#cb215-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="st">&quot;Hell&quot;</span>,<span class="st">&quot;War&quot;</span>]</span></code></pre></div>
-<p>Sometimes with higher-order functions it’s useful to have a function that does nothing. The function <code>id :: a -&gt; a</code> is the identity function and just returns its argument.</p>
-<div class="sourceCode" id="cb216"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb216-1"><a href="#cb216-1" aria-hidden="true" tabindex="-1"></a><span class="fu">id</span> <span class="dv">3</span> <span class="op">==&gt;</span> <span class="dv">3</span></span>
+<p>Sometimes with higher-order functions it’s useful to have a function
+that does nothing. The function <code>id :: a -&gt; a</code> is the
+identity function and just returns its argument.</p>
+<div class="sourceCode" id="cb216"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb216-1"><a href="#cb216-1" aria-hidden="true" tabindex="-1"></a><span class="fu">id</span> <span class="dv">3</span> <span class="op">==&gt;</span> <span class="dv">3</span></span>
 <span id="cb216-2"><a href="#cb216-2" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> <span class="fu">id</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] <span class="op">==&gt;</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span></code></pre></div>
-<p>This seems a bit useless, but you can use it for example with <code>filter</code> or <code>dropWhile</code>:</p>
-<div class="sourceCode" id="cb217"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb217-1"><a href="#cb217-1" aria-hidden="true" tabindex="-1"></a><span class="fu">filter</span> <span class="fu">id</span> [<span class="dt">True</span>,<span class="dt">False</span>,<span class="dt">True</span>,<span class="dt">True</span>]  <span class="op">==&gt;</span>  [<span class="dt">True</span>,<span class="dt">True</span>,<span class="dt">True</span>]</span>
+<p>This seems a bit useless, but you can use it for example with
+<code>filter</code> or <code>dropWhile</code>:</p>
+<div class="sourceCode" id="cb217"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb217-1"><a href="#cb217-1" aria-hidden="true" tabindex="-1"></a><span class="fu">filter</span> <span class="fu">id</span> [<span class="dt">True</span>,<span class="dt">False</span>,<span class="dt">True</span>,<span class="dt">True</span>]  <span class="op">==&gt;</span>  [<span class="dt">True</span>,<span class="dt">True</span>,<span class="dt">True</span>]</span>
 <span id="cb217-2"><a href="#cb217-2" aria-hidden="true" tabindex="-1"></a><span class="fu">dropWhile</span> <span class="fu">id</span> [<span class="dt">True</span>,<span class="dt">True</span>,<span class="dt">False</span>,<span class="dt">True</span>,<span class="dt">False</span>]  <span class="op">==&gt;</span>  [<span class="dt">False</span>,<span class="dt">True</span>,<span class="dt">False</span>]</span></code></pre></div>
-<p>Another very simple but sometimes crucial function is the constant function, <code>const :: a -&gt; b -&gt; a</code>. It always returns its first argument:</p>
-<div class="sourceCode" id="cb218"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb218-1"><a href="#cb218-1" aria-hidden="true" tabindex="-1"></a><span class="fu">const</span> <span class="dv">3</span> <span class="dt">True</span> <span class="op">==&gt;</span> <span class="dv">3</span></span>
+<p>Another very simple but sometimes crucial function is the constant
+function, <code>const :: a -&gt; b -&gt; a</code>. It always returns its
+first argument:</p>
+<div class="sourceCode" id="cb218"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb218-1"><a href="#cb218-1" aria-hidden="true" tabindex="-1"></a><span class="fu">const</span> <span class="dv">3</span> <span class="dt">True</span> <span class="op">==&gt;</span> <span class="dv">3</span></span>
 <span id="cb218-2"><a href="#cb218-2" aria-hidden="true" tabindex="-1"></a><span class="fu">const</span> <span class="dv">3</span> <span class="dv">0</span>    <span class="op">==&gt;</span> <span class="dv">3</span></span></code></pre></div>
-<p>When partially applied it can be used when you need a function that always returns the same value:</p>
-<div class="sourceCode" id="cb219"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb219-1"><a href="#cb219-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> (<span class="fu">const</span> <span class="dv">5</span>) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>] <span class="op">==&gt;</span> [<span class="dv">5</span>,<span class="dv">5</span>,<span class="dv">5</span>,<span class="dv">5</span>]</span>
+<p>When partially applied it can be used when you need a function that
+always returns the same value:</p>
+<div class="sourceCode" id="cb219"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb219-1"><a href="#cb219-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> (<span class="fu">const</span> <span class="dv">5</span>) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>] <span class="op">==&gt;</span> [<span class="dv">5</span>,<span class="dv">5</span>,<span class="dv">5</span>,<span class="dv">5</span>]</span>
 <span id="cb219-2"><a href="#cb219-2" aria-hidden="true" tabindex="-1"></a><span class="fu">filter</span> (<span class="fu">const</span> <span class="dt">True</span>) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>] <span class="op">==&gt;</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>]</span></code></pre></div>
-<h2 data-number="3.8" id="lists-and-recursion"><span class="header-section-number">3.8</span> Lists and Recursion</h2>
+<h2 data-number="3.8" id="lists-and-recursion"><span
+class="header-section-number">3.8</span> Lists and Recursion</h2>
 <p>Here’s a new operator, <code>:</code></p>
-<div class="sourceCode" id="cb220"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb220-1"><a href="#cb220-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dv">1</span><span class="op">:</span>[]</span>
+<div class="sourceCode" id="cb220"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb220-1"><a href="#cb220-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dv">1</span><span class="op">:</span>[]</span>
 <span id="cb220-2"><a href="#cb220-2" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span>]</span>
 <span id="cb220-3"><a href="#cb220-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dv">1</span><span class="op">:</span>[<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb220-4"><a href="#cb220-4" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
@@ -2001,9 +3175,19 @@ Prelude&gt; map (add 3) [1,2,3]
 <span id="cb220-8"><a href="#cb220-8" aria-hidden="true" tabindex="-1"></a><span class="dv">1</span></span>
 <span id="cb220-9"><a href="#cb220-9" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t (<span class="op">:</span>)</span>
 <span id="cb220-10"><a href="#cb220-10" aria-hidden="true" tabindex="-1"></a><span class="ot">(:) ::</span> a <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [a]</span></code></pre></div>
-<p>The <code>:</code> operator builds a list out of a head and a tail. In other words, <code>x : xs</code> is the same as <code>[x] ++ xs</code>. Why do we need an operator for this?</p>
-<p>Actually, <code>:</code> is the <em>constructor</em> for lists: it returns a new linked list node. The other list constructor is <code>[]</code>, the empty list. All lists are built using <code>:</code> and <code>[]</code>. The familiar <code>[x,y,z]</code> syntax is actually just a nicer way to write <code>x:y:z:[]</code>, or even more explicitly, <code>x:(y:(z:[]))</code>. In fact <code>(++)</code> is defined in terms of <code>:</code> and recursion in the standard library.</p>
-<p>Here’s a picture of how <code>[1,2,3]</code> is structured in memory:</p>
+<p>The <code>:</code> operator builds a list out of a head and a tail.
+In other words, <code>x : xs</code> is the same as
+<code>[x] ++ xs</code>. Why do we need an operator for this?</p>
+<p>Actually, <code>:</code> is the <em>constructor</em> for lists: it
+returns a new linked list node. The other list constructor is
+<code>[]</code>, the empty list. All lists are built using
+<code>:</code> and <code>[]</code>. The familiar <code>[x,y,z]</code>
+syntax is actually just a nicer way to write <code>x:y:z:[]</code>, or
+even more explicitly, <code>x:(y:(z:[]))</code>. In fact
+<code>(++)</code> is defined in terms of <code>:</code> and recursion in
+the standard library.</p>
+<p>Here’s a picture of how <code>[1,2,3]</code> is structured in
+memory:</p>
 <p><img src="img/list123.svg" /></p>
 <!--
 ```
@@ -2016,89 +3200,138 @@ Prelude&gt; map (add 3) [1,2,3]
        3    []
 ```
 -->
-<h3 data-number="3.8.1" id="building-a-list"><span class="header-section-number">3.8.1</span> Building a List</h3>
-<p>Using <code>:</code> we can define recursive functions that build lists. For example here’s a function that builds lists like <code>[3,2,1]</code>:</p>
-<div class="sourceCode" id="cb221"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb221-1"><a href="#cb221-1" aria-hidden="true" tabindex="-1"></a>descend <span class="dv">0</span> <span class="ot">=</span> []</span>
+<h3 data-number="3.8.1" id="building-a-list"><span
+class="header-section-number">3.8.1</span> Building a List</h3>
+<p>Using <code>:</code> we can define recursive functions that build
+lists. For example here’s a function that builds lists like
+<code>[3,2,1]</code>:</p>
+<div class="sourceCode" id="cb221"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb221-1"><a href="#cb221-1" aria-hidden="true" tabindex="-1"></a>descend <span class="dv">0</span> <span class="ot">=</span> []</span>
 <span id="cb221-2"><a href="#cb221-2" aria-hidden="true" tabindex="-1"></a>descend n <span class="ot">=</span> n <span class="op">:</span> descend (n<span class="op">-</span><span class="dv">1</span>)</span></code></pre></div>
-<div class="sourceCode" id="cb222"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb222-1"><a href="#cb222-1" aria-hidden="true" tabindex="-1"></a>descend <span class="dv">4</span> <span class="op">==&gt;</span> [<span class="dv">4</span>,<span class="dv">3</span>,<span class="dv">2</span>,<span class="dv">1</span>]</span></code></pre></div>
-<p>Here’s a function that builds a list by iterating a function <code>n</code> times:</p>
-<div class="sourceCode" id="cb223"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb223-1"><a href="#cb223-1" aria-hidden="true" tabindex="-1"></a><span class="fu">iterate</span> f <span class="dv">0</span> x <span class="ot">=</span> [x]</span>
+<div class="sourceCode" id="cb222"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb222-1"><a href="#cb222-1" aria-hidden="true" tabindex="-1"></a>descend <span class="dv">4</span> <span class="op">==&gt;</span> [<span class="dv">4</span>,<span class="dv">3</span>,<span class="dv">2</span>,<span class="dv">1</span>]</span></code></pre></div>
+<p>Here’s a function that builds a list by iterating a function
+<code>n</code> times:</p>
+<div class="sourceCode" id="cb223"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb223-1"><a href="#cb223-1" aria-hidden="true" tabindex="-1"></a><span class="fu">iterate</span> f <span class="dv">0</span> x <span class="ot">=</span> [x]</span>
 <span id="cb223-2"><a href="#cb223-2" aria-hidden="true" tabindex="-1"></a><span class="fu">iterate</span> f n x <span class="ot">=</span> x <span class="op">:</span> <span class="fu">iterate</span> f (n<span class="op">-</span><span class="dv">1</span>) (f x)</span></code></pre></div>
-<div class="sourceCode" id="cb224"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb224-1"><a href="#cb224-1" aria-hidden="true" tabindex="-1"></a><span class="fu">iterate</span> (<span class="op">*</span><span class="dv">2</span>) <span class="dv">4</span> <span class="dv">3</span> <span class="op">==&gt;</span> [<span class="dv">3</span>,<span class="dv">6</span>,<span class="dv">12</span>,<span class="dv">24</span>,<span class="dv">48</span>]</span>
+<div class="sourceCode" id="cb224"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb224-1"><a href="#cb224-1" aria-hidden="true" tabindex="-1"></a><span class="fu">iterate</span> (<span class="op">*</span><span class="dv">2</span>) <span class="dv">4</span> <span class="dv">3</span> <span class="op">==&gt;</span> [<span class="dv">3</span>,<span class="dv">6</span>,<span class="dv">12</span>,<span class="dv">24</span>,<span class="dv">48</span>]</span>
 <span id="cb224-2"><a href="#cb224-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb224-3"><a href="#cb224-3" aria-hidden="true" tabindex="-1"></a><span class="kw">let</span> xs <span class="ot">=</span> <span class="st">&quot;terve&quot;</span></span>
 <span id="cb224-4"><a href="#cb224-4" aria-hidden="true" tabindex="-1"></a><span class="kw">in</span> <span class="fu">iterate</span> <span class="fu">tail</span> (<span class="fu">length</span> xs) xs</span>
 <span id="cb224-5"><a href="#cb224-5" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="st">&quot;terve&quot;</span>,<span class="st">&quot;erve&quot;</span>,<span class="st">&quot;rve&quot;</span>,<span class="st">&quot;ve&quot;</span>,<span class="st">&quot;e&quot;</span>,<span class="st">&quot;&quot;</span>]</span></code></pre></div>
-<p>Here’s a more complicated example: splitting a string into pieces at a given character:</p>
-<div class="sourceCode" id="cb225"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb225-1"><a href="#cb225-1" aria-hidden="true" tabindex="-1"></a><span class="ot">split ::</span> <span class="dt">Char</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> [<span class="dt">String</span>]</span>
+<p>Here’s a more complicated example: splitting a string into pieces at
+a given character:</p>
+<div class="sourceCode" id="cb225"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb225-1"><a href="#cb225-1" aria-hidden="true" tabindex="-1"></a><span class="ot">split ::</span> <span class="dt">Char</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> [<span class="dt">String</span>]</span>
 <span id="cb225-2"><a href="#cb225-2" aria-hidden="true" tabindex="-1"></a>split c [] <span class="ot">=</span> []</span>
 <span id="cb225-3"><a href="#cb225-3" aria-hidden="true" tabindex="-1"></a>split c xs <span class="ot">=</span> start <span class="op">:</span> split c (<span class="fu">drop</span> <span class="dv">1</span> rest)</span>
 <span id="cb225-4"><a href="#cb225-4" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> start <span class="ot">=</span> <span class="fu">takeWhile</span> (<span class="op">/=</span>c) xs</span>
 <span id="cb225-5"><a href="#cb225-5" aria-hidden="true" tabindex="-1"></a>        rest <span class="ot">=</span> <span class="fu">dropWhile</span> (<span class="op">/=</span>c) xs</span></code></pre></div>
-<div class="sourceCode" id="cb226"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb226-1"><a href="#cb226-1" aria-hidden="true" tabindex="-1"></a>split <span class="ch">&#39;x&#39;</span> <span class="st">&quot;fooxxbarxquux&quot;</span>   <span class="op">==&gt;</span>   [<span class="st">&quot;foo&quot;</span>,<span class="st">&quot;&quot;</span>,<span class="st">&quot;bar&quot;</span>,<span class="st">&quot;quu&quot;</span>]</span></code></pre></div>
-<h3 data-number="3.8.2" id="pattern-matching-for-lists"><span class="header-section-number">3.8.2</span> Pattern Matching for Lists</h3>
-<p>Last lecture, it was said that constructors are things that can be pattern matched on. Above, it was divulged that the constructors for the list type are <code>:</code> and <code>[]</code>. We can put one and one together and guess that we can pattern match on <code>:</code> and <code>[]</code>. This is true! Here’s how you can define your own versions of <code>head</code> and <code>tail</code> using pattern matching:</p>
-<div class="sourceCode" id="cb227"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb227-1"><a href="#cb227-1" aria-hidden="true" tabindex="-1"></a><span class="ot">myhead ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<div class="sourceCode" id="cb226"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb226-1"><a href="#cb226-1" aria-hidden="true" tabindex="-1"></a>split <span class="ch">&#39;x&#39;</span> <span class="st">&quot;fooxxbarxquux&quot;</span>   <span class="op">==&gt;</span>   [<span class="st">&quot;foo&quot;</span>,<span class="st">&quot;&quot;</span>,<span class="st">&quot;bar&quot;</span>,<span class="st">&quot;quu&quot;</span>]</span></code></pre></div>
+<h3 data-number="3.8.2" id="pattern-matching-for-lists"><span
+class="header-section-number">3.8.2</span> Pattern Matching for
+Lists</h3>
+<p>Last lecture, it was said that constructors are things that can be
+pattern matched on. Above, it was divulged that the constructors for the
+list type are <code>:</code> and <code>[]</code>. We can put one and one
+together and guess that we can pattern match on <code>:</code> and
+<code>[]</code>. This is true! Here’s how you can define your own
+versions of <code>head</code> and <code>tail</code> using pattern
+matching:</p>
+<div class="sourceCode" id="cb227"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb227-1"><a href="#cb227-1" aria-hidden="true" tabindex="-1"></a><span class="ot">myhead ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb227-2"><a href="#cb227-2" aria-hidden="true" tabindex="-1"></a>myhead [] <span class="ot">=</span> <span class="op">-</span><span class="dv">1</span></span>
 <span id="cb227-3"><a href="#cb227-3" aria-hidden="true" tabindex="-1"></a>myhead (first<span class="op">:</span>rest) <span class="ot">=</span> first</span>
 <span id="cb227-4"><a href="#cb227-4" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb227-5"><a href="#cb227-5" aria-hidden="true" tabindex="-1"></a><span class="ot">mytail ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> [<span class="dt">Int</span>]</span>
 <span id="cb227-6"><a href="#cb227-6" aria-hidden="true" tabindex="-1"></a>mytail [] <span class="ot">=</span> []</span>
 <span id="cb227-7"><a href="#cb227-7" aria-hidden="true" tabindex="-1"></a>mytail (first<span class="op">:</span>rest) <span class="ot">=</span> rest</span></code></pre></div>
-<p>You can <em>nest</em> patterns. That is, you can pattern match more than one element from the start of a list. In this example, we use the pattern <code>(a:b:_)</code> which is the same as <code>(a:(b:_))</code>:</p>
-<div class="sourceCode" id="cb228"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb228-1"><a href="#cb228-1" aria-hidden="true" tabindex="-1"></a><span class="ot">sumFirstTwo ::</span> [<span class="dt">Integer</span>] <span class="ot">-&gt;</span> <span class="dt">Integer</span></span>
+<p>You can <em>nest</em> patterns. That is, you can pattern match more
+than one element from the start of a list. In this example, we use the
+pattern <code>(a:b:_)</code> which is the same as
+<code>(a:(b:_))</code>:</p>
+<div class="sourceCode" id="cb228"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb228-1"><a href="#cb228-1" aria-hidden="true" tabindex="-1"></a><span class="ot">sumFirstTwo ::</span> [<span class="dt">Integer</span>] <span class="ot">-&gt;</span> <span class="dt">Integer</span></span>
 <span id="cb228-2"><a href="#cb228-2" aria-hidden="true" tabindex="-1"></a><span class="co">-- this equation gets used for lists of length at least two</span></span>
 <span id="cb228-3"><a href="#cb228-3" aria-hidden="true" tabindex="-1"></a>sumFirstTwo (a<span class="op">:</span>b<span class="op">:</span>_) <span class="ot">=</span> a<span class="op">+</span>b</span>
 <span id="cb228-4"><a href="#cb228-4" aria-hidden="true" tabindex="-1"></a><span class="co">-- this equation gets used for all other lists (i.e. lists of length 0 or 1)</span></span>
 <span id="cb228-5"><a href="#cb228-5" aria-hidden="true" tabindex="-1"></a>sumFirstTwo _       <span class="ot">=</span> <span class="dv">0</span></span></code></pre></div>
-<div class="sourceCode" id="cb229"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb229-1"><a href="#cb229-1" aria-hidden="true" tabindex="-1"></a>sumFirstTwo [<span class="dv">1</span>]      <span class="op">==&gt;</span> <span class="dv">0</span></span>
+<div class="sourceCode" id="cb229"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb229-1"><a href="#cb229-1" aria-hidden="true" tabindex="-1"></a>sumFirstTwo [<span class="dv">1</span>]      <span class="op">==&gt;</span> <span class="dv">0</span></span>
 <span id="cb229-2"><a href="#cb229-2" aria-hidden="true" tabindex="-1"></a>sumFirstTwo [<span class="dv">1</span>,<span class="dv">2</span>]    <span class="op">==&gt;</span> <span class="dv">3</span></span>
 <span id="cb229-3"><a href="#cb229-3" aria-hidden="true" tabindex="-1"></a>sumFirstTwo [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">4</span>]  <span class="op">==&gt;</span> <span class="dv">3</span></span></code></pre></div>
 <p>Here’s an example that uses many different list patterns:</p>
-<div class="sourceCode" id="cb230"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb230-1"><a href="#cb230-1" aria-hidden="true" tabindex="-1"></a><span class="ot">describeList ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<div class="sourceCode" id="cb230"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb230-1"><a href="#cb230-1" aria-hidden="true" tabindex="-1"></a><span class="ot">describeList ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb230-2"><a href="#cb230-2" aria-hidden="true" tabindex="-1"></a>describeList []         <span class="ot">=</span> <span class="st">&quot;an empty list&quot;</span></span>
 <span id="cb230-3"><a href="#cb230-3" aria-hidden="true" tabindex="-1"></a>describeList (x<span class="op">:</span>[])     <span class="ot">=</span> <span class="st">&quot;a list with one element&quot;</span></span>
 <span id="cb230-4"><a href="#cb230-4" aria-hidden="true" tabindex="-1"></a>describeList (x<span class="op">:</span>y<span class="op">:</span>[])   <span class="ot">=</span> <span class="st">&quot;a list with two elements&quot;</span></span>
 <span id="cb230-5"><a href="#cb230-5" aria-hidden="true" tabindex="-1"></a>describeList (x<span class="op">:</span>y<span class="op">:</span>z<span class="op">:</span>xs) <span class="ot">=</span> <span class="st">&quot;a list with at least three elements&quot;</span></span></code></pre></div>
-<div class="sourceCode" id="cb231"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb231-1"><a href="#cb231-1" aria-hidden="true" tabindex="-1"></a>describeList [<span class="dv">1</span>,<span class="dv">3</span>]        <span class="op">==&gt;</span> <span class="st">&quot;a list with two elements&quot;</span></span>
+<div class="sourceCode" id="cb231"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb231-1"><a href="#cb231-1" aria-hidden="true" tabindex="-1"></a>describeList [<span class="dv">1</span>,<span class="dv">3</span>]        <span class="op">==&gt;</span> <span class="st">&quot;a list with two elements&quot;</span></span>
 <span id="cb231-2"><a href="#cb231-2" aria-hidden="true" tabindex="-1"></a>describeList [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>,<span class="dv">5</span>]  <span class="op">==&gt;</span> <span class="st">&quot;a list with at least three elements&quot;</span></span></code></pre></div>
-<p>List patterns that end with <code>:[]</code> can be typed out as list literals. That is, just like <code>[1,2,3]</code> is the same value as <code>1:2:3:[]</code>, the pattern <code>[x,y]</code> is the same as the pattern <code>x:y:[]</code>. Let’s rewrite that previous example.</p>
-<div class="sourceCode" id="cb232"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb232-1"><a href="#cb232-1" aria-hidden="true" tabindex="-1"></a><span class="ot">describeList ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<p>List patterns that end with <code>:[]</code> can be typed out as list
+literals. That is, just like <code>[1,2,3]</code> is the same value as
+<code>1:2:3:[]</code>, the pattern <code>[x,y]</code> is the same as the
+pattern <code>x:y:[]</code>. Let’s rewrite that previous example.</p>
+<div class="sourceCode" id="cb232"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb232-1"><a href="#cb232-1" aria-hidden="true" tabindex="-1"></a><span class="ot">describeList ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb232-2"><a href="#cb232-2" aria-hidden="true" tabindex="-1"></a>describeList []         <span class="ot">=</span> <span class="st">&quot;an empty list&quot;</span></span>
 <span id="cb232-3"><a href="#cb232-3" aria-hidden="true" tabindex="-1"></a>describeList [x]        <span class="ot">=</span> <span class="st">&quot;a list with exactly one element&quot;</span></span>
 <span id="cb232-4"><a href="#cb232-4" aria-hidden="true" tabindex="-1"></a>describeList [x,y]      <span class="ot">=</span> <span class="st">&quot;a list with exactly two elements&quot;</span></span>
 <span id="cb232-5"><a href="#cb232-5" aria-hidden="true" tabindex="-1"></a>describeList (x<span class="op">:</span>y<span class="op">:</span>z<span class="op">:</span>xs) <span class="ot">=</span> <span class="st">&quot;a list with at least three elements&quot;</span></span></code></pre></div>
-<p>Another way we can nest patterns is pattern matching on the head while pattern matching on a list. For example this function checks if a list starts with <code>0</code>:</p>
-<div class="sourceCode" id="cb233"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb233-1"><a href="#cb233-1" aria-hidden="true" tabindex="-1"></a><span class="ot">startsWithZero ::</span> [<span class="dt">Integer</span>] <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
+<p>Another way we can nest patterns is pattern matching on the head
+while pattern matching on a list. For example this function checks if a
+list starts with <code>0</code>:</p>
+<div class="sourceCode" id="cb233"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb233-1"><a href="#cb233-1" aria-hidden="true" tabindex="-1"></a><span class="ot">startsWithZero ::</span> [<span class="dt">Integer</span>] <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb233-2"><a href="#cb233-2" aria-hidden="true" tabindex="-1"></a>startsWithZero (<span class="dv">0</span><span class="op">:</span>xs) <span class="ot">=</span> <span class="dt">True</span></span>
 <span id="cb233-3"><a href="#cb233-3" aria-hidden="true" tabindex="-1"></a>startsWithZero (x<span class="op">:</span>xs) <span class="ot">=</span> <span class="dt">False</span></span>
 <span id="cb233-4"><a href="#cb233-4" aria-hidden="true" tabindex="-1"></a>startsWithZero []     <span class="ot">=</span> <span class="dt">False</span></span></code></pre></div>
 <!-- The last two cases could be merged as `startsWithZero _ = False`. -->
-<h3 data-number="3.8.3" id="consuming-a-list"><span class="header-section-number">3.8.3</span> Consuming a List</h3>
-<p>Using pattern matching and recursion, we can recursively process a whole list. Here’s how you sum all the numbers in a list:</p>
-<div class="sourceCode" id="cb234"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb234-1"><a href="#cb234-1" aria-hidden="true" tabindex="-1"></a><span class="ot">sumNumbers ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<h3 data-number="3.8.3" id="consuming-a-list"><span
+class="header-section-number">3.8.3</span> Consuming a List</h3>
+<p>Using pattern matching and recursion, we can recursively process a
+whole list. Here’s how you sum all the numbers in a list:</p>
+<div class="sourceCode" id="cb234"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb234-1"><a href="#cb234-1" aria-hidden="true" tabindex="-1"></a><span class="ot">sumNumbers ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb234-2"><a href="#cb234-2" aria-hidden="true" tabindex="-1"></a>sumNumbers [] <span class="ot">=</span> <span class="dv">0</span></span>
 <span id="cb234-3"><a href="#cb234-3" aria-hidden="true" tabindex="-1"></a>sumNumbers (x<span class="op">:</span>xs) <span class="ot">=</span> x <span class="op">+</span> sumNumbers xs</span></code></pre></div>
-<p>Here’s how you compute the largest number in a list, this time using a helper function.</p>
-<div class="sourceCode" id="cb235"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb235-1"><a href="#cb235-1" aria-hidden="true" tabindex="-1"></a><span class="ot">myMaximum ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<p>Here’s how you compute the largest number in a list, this time using
+a helper function.</p>
+<div class="sourceCode" id="cb235"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb235-1"><a href="#cb235-1" aria-hidden="true" tabindex="-1"></a><span class="ot">myMaximum ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb235-2"><a href="#cb235-2" aria-hidden="true" tabindex="-1"></a>myMaximum [] <span class="ot">=</span> <span class="dv">0</span>       <span class="co">-- actually this should be some sort of error...</span></span>
 <span id="cb235-3"><a href="#cb235-3" aria-hidden="true" tabindex="-1"></a>myMaximum (x<span class="op">:</span>xs) <span class="ot">=</span> go x xs</span>
 <span id="cb235-4"><a href="#cb235-4" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> go biggest [] <span class="ot">=</span> biggest</span>
 <span id="cb235-5"><a href="#cb235-5" aria-hidden="true" tabindex="-1"></a>        go biggest (x<span class="op">:</span>xs) <span class="ot">=</span> go (<span class="fu">max</span> biggest x) xs</span></code></pre></div>
-<p><strong>Note!</strong>, “<code>go</code>” is just a cute name for the helper function here. It’s not special syntax.</p>
-<p>It’s often convenient to use nested patterns while consuming a list. Here’s an example that counts how many <code>Nothing</code> values occur in a list of <code>Maybe</code>s:</p>
-<div class="sourceCode" id="cb236"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb236-1"><a href="#cb236-1" aria-hidden="true" tabindex="-1"></a><span class="ot">countNothings ::</span> [<span class="dt">Maybe</span> a] <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<p><strong>Note!</strong>, “<code>go</code>” is just a cute name for the
+helper function here. It’s not special syntax.</p>
+<p>It’s often convenient to use nested patterns while consuming a list.
+Here’s an example that counts how many <code>Nothing</code> values occur
+in a list of <code>Maybe</code>s:</p>
+<div class="sourceCode" id="cb236"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb236-1"><a href="#cb236-1" aria-hidden="true" tabindex="-1"></a><span class="ot">countNothings ::</span> [<span class="dt">Maybe</span> a] <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb236-2"><a href="#cb236-2" aria-hidden="true" tabindex="-1"></a>countNothings [] <span class="ot">=</span> <span class="dv">0</span></span>
 <span id="cb236-3"><a href="#cb236-3" aria-hidden="true" tabindex="-1"></a>countNothings (<span class="dt">Nothing</span> <span class="op">:</span> xs) <span class="ot">=</span> <span class="dv">1</span> <span class="op">+</span> countNothings xs</span>
 <span id="cb236-4"><a href="#cb236-4" aria-hidden="true" tabindex="-1"></a>countNothings (<span class="dt">Just</span> _  <span class="op">:</span> xs) <span class="ot">=</span> countNothings xs</span></code></pre></div>
-<div class="sourceCode" id="cb237"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb237-1"><a href="#cb237-1" aria-hidden="true" tabindex="-1"></a>countNothings [<span class="dt">Nothing</span>,<span class="dt">Just</span> <span class="dv">1</span>,<span class="dt">Nothing</span>]  <span class="op">==&gt;</span>  <span class="dv">2</span></span></code></pre></div>
-<h3 data-number="3.8.4" id="building-and-consuming-a-list"><span class="header-section-number">3.8.4</span> Building and Consuming a List</h3>
-<p>Now that we can build and consume lists, let’s do both of them at the same time. This function doubles all elements in a list.</p>
-<div class="sourceCode" id="cb238"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb238-1"><a href="#cb238-1" aria-hidden="true" tabindex="-1"></a><span class="ot">doubleList ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> [<span class="dt">Int</span>]</span>
+<div class="sourceCode" id="cb237"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb237-1"><a href="#cb237-1" aria-hidden="true" tabindex="-1"></a>countNothings [<span class="dt">Nothing</span>,<span class="dt">Just</span> <span class="dv">1</span>,<span class="dt">Nothing</span>]  <span class="op">==&gt;</span>  <span class="dv">2</span></span></code></pre></div>
+<h3 data-number="3.8.4" id="building-and-consuming-a-list"><span
+class="header-section-number">3.8.4</span> Building and Consuming a
+List</h3>
+<p>Now that we can build and consume lists, let’s do both of them at the
+same time. This function doubles all elements in a list.</p>
+<div class="sourceCode" id="cb238"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb238-1"><a href="#cb238-1" aria-hidden="true" tabindex="-1"></a><span class="ot">doubleList ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> [<span class="dt">Int</span>]</span>
 <span id="cb238-2"><a href="#cb238-2" aria-hidden="true" tabindex="-1"></a>doubleList [] <span class="ot">=</span> []</span>
 <span id="cb238-3"><a href="#cb238-3" aria-hidden="true" tabindex="-1"></a>doubleList (x<span class="op">:</span>xs) <span class="ot">=</span> <span class="dv">2</span><span class="op">*</span>x <span class="op">:</span> doubleList xs</span></code></pre></div>
 <p>It evaluates like this:</p>
-<div class="sourceCode" id="cb239"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb239-1"><a href="#cb239-1" aria-hidden="true" tabindex="-1"></a>doubleList [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
+<div class="sourceCode" id="cb239"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb239-1"><a href="#cb239-1" aria-hidden="true" tabindex="-1"></a>doubleList [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb239-2"><a href="#cb239-2" aria-hidden="true" tabindex="-1"></a><span class="op">===</span> doubleList (<span class="dv">1</span><span class="op">:</span>(<span class="dv">2</span><span class="op">:</span>(<span class="dv">3</span><span class="op">:</span>[])))</span>
 <span id="cb239-3"><a href="#cb239-3" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="dv">2</span><span class="op">*</span><span class="dv">1</span> <span class="op">:</span> doubleList (<span class="dv">2</span><span class="op">:</span>(<span class="dv">3</span><span class="op">:</span>[]))</span>
 <span id="cb239-4"><a href="#cb239-4" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="dv">2</span><span class="op">*</span><span class="dv">1</span> <span class="op">:</span> (<span class="dv">2</span><span class="op">*</span><span class="dv">2</span> <span class="op">:</span> doubleList (<span class="dv">3</span><span class="op">:</span>[]))</span>
@@ -2106,76 +3339,158 @@ Prelude&gt; map (add 3) [1,2,3]
 <span id="cb239-6"><a href="#cb239-6" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="dv">2</span><span class="op">*</span><span class="dv">1</span> <span class="op">:</span> (<span class="dv">2</span><span class="op">*</span><span class="dv">2</span> <span class="op">:</span> (<span class="dv">2</span><span class="op">*</span><span class="dv">3</span> <span class="op">:</span> []))</span>
 <span id="cb239-7"><a href="#cb239-7" aria-hidden="true" tabindex="-1"></a><span class="op">===</span> [<span class="dv">2</span><span class="op">*</span><span class="dv">1</span>, <span class="dv">2</span><span class="op">*</span><span class="dv">2</span>, <span class="dv">2</span><span class="op">*</span><span class="dv">3</span>]</span>
 <span id="cb239-8"><a href="#cb239-8" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> [<span class="dv">2</span>,<span class="dv">4</span>,<span class="dv">6</span>]</span></code></pre></div>
-<p>Once you know pattern matching for lists, it’s straightforward to define <code>map</code> and <code>filter</code>. Actually, let’s just look at the GHC standard library implementations. <a href="https://hackage.haskell.org/package/base-4.16.4.0/docs/src/GHC.Base.html#map">Here’s map</a>:</p>
-<div class="sourceCode" id="cb240"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb240-1"><a href="#cb240-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [b]</span>
+<p>Once you know pattern matching for lists, it’s straightforward to
+define <code>map</code> and <code>filter</code>. Actually, let’s just
+look at the GHC standard library implementations. <a
+href="https://hackage.haskell.org/package/base-4.16.4.0/docs/src/GHC.Base.html#map">Here’s
+map</a>:</p>
+<div class="sourceCode" id="cb240"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb240-1"><a href="#cb240-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [b]</span>
 <span id="cb240-2"><a href="#cb240-2" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> _ []     <span class="ot">=</span> []</span>
 <span id="cb240-3"><a href="#cb240-3" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> f (x<span class="op">:</span>xs) <span class="ot">=</span> f x <span class="op">:</span> <span class="fu">map</span> f xs</span></code></pre></div>
-<p>and <a href="https://hackage.haskell.org/package/base-4.16.4.0/docs/src/GHC.List.html#filter">here’s filter</a>:</p>
-<div class="sourceCode" id="cb241"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb241-1"><a href="#cb241-1" aria-hidden="true" tabindex="-1"></a><span class="fu">filter</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> <span class="dt">Bool</span>) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [a]</span>
+<p>and <a
+href="https://hackage.haskell.org/package/base-4.16.4.0/docs/src/GHC.List.html#filter">here’s
+filter</a>:</p>
+<div class="sourceCode" id="cb241"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb241-1"><a href="#cb241-1" aria-hidden="true" tabindex="-1"></a><span class="fu">filter</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> <span class="dt">Bool</span>) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [a]</span>
 <span id="cb241-2"><a href="#cb241-2" aria-hidden="true" tabindex="-1"></a><span class="fu">filter</span> _pred []    <span class="ot">=</span> []</span>
 <span id="cb241-3"><a href="#cb241-3" aria-hidden="true" tabindex="-1"></a><span class="fu">filter</span> <span class="fu">pred</span> (x<span class="op">:</span>xs)</span>
 <span id="cb241-4"><a href="#cb241-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> <span class="fu">pred</span> x         <span class="ot">=</span> x <span class="op">:</span> <span class="fu">filter</span> <span class="fu">pred</span> xs</span>
 <span id="cb241-5"><a href="#cb241-5" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> <span class="fu">otherwise</span>      <span class="ot">=</span> <span class="fu">filter</span> <span class="fu">pred</span> xs</span></code></pre></div>
-<p>(<strong>Note!</strong> Naming the argument <code>_pred</code> is a way to tell the reader of the code that this argument is unused. It could have been just <code>_</code> as well.)</p>
-<h3 data-number="3.8.5" id="tail-recursion-and-lists"><span class="header-section-number">3.8.5</span> Tail Recursion and Lists</h3>
-<p>When a recursive function evaluates to a new call to that same function with different arguments, it is called <em>tail-recursive</em>. (The recursive call is said to be in <em>tail position</em>.) This is the type of recursion that corresponds to an imperative loop. We’ve already seen many examples of tail-recursive functions, but we haven’t really contrasted the two ways for writing the same function. This is <code>sumNumbers</code> from earlier in this lecture:</p>
-<div class="sourceCode" id="cb242"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb242-1"><a href="#cb242-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Not tail recursive!</span></span>
+<p>(<strong>Note!</strong> Naming the argument <code>_pred</code> is a
+way to tell the reader of the code that this argument is unused. It
+could have been just <code>_</code> as well.)</p>
+<h3 data-number="3.8.5" id="tail-recursion-and-lists"><span
+class="header-section-number">3.8.5</span> Tail Recursion and Lists</h3>
+<p>When a recursive function evaluates to a new call to that same
+function with different arguments, it is called <em>tail-recursive</em>.
+(The recursive call is said to be in <em>tail position</em>.) This is
+the type of recursion that corresponds to an imperative loop. We’ve
+already seen many examples of tail-recursive functions, but we haven’t
+really contrasted the two ways for writing the same function. This is
+<code>sumNumbers</code> from earlier in this lecture:</p>
+<div class="sourceCode" id="cb242"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb242-1"><a href="#cb242-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Not tail recursive!</span></span>
 <span id="cb242-2"><a href="#cb242-2" aria-hidden="true" tabindex="-1"></a><span class="ot">sumNumbers ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb242-3"><a href="#cb242-3" aria-hidden="true" tabindex="-1"></a>sumNumbers [] <span class="ot">=</span> <span class="dv">0</span></span>
 <span id="cb242-4"><a href="#cb242-4" aria-hidden="true" tabindex="-1"></a>sumNumbers (x<span class="op">:</span>xs) <span class="ot">=</span> x <span class="op">+</span> sumNumbers xs</span></code></pre></div>
-<p>In the second equation the function <code>+</code> is at the top level, i.e. in tail position. The recursive call to <code>sumNumbers</code> is an argument of <code>+</code>. This is <code>sumNumbers</code> written using a tail recursive helper function:</p>
-<div class="sourceCode" id="cb243"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb243-1"><a href="#cb243-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Tail recursive version</span></span>
+<p>In the second equation the function <code>+</code> is at the top
+level, i.e. in tail position. The recursive call to
+<code>sumNumbers</code> is an argument of <code>+</code>. This is
+<code>sumNumbers</code> written using a tail recursive helper
+function:</p>
+<div class="sourceCode" id="cb243"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb243-1"><a href="#cb243-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Tail recursive version</span></span>
 <span id="cb243-2"><a href="#cb243-2" aria-hidden="true" tabindex="-1"></a><span class="ot">sumNumbers ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb243-3"><a href="#cb243-3" aria-hidden="true" tabindex="-1"></a>sumNumbers xs <span class="ot">=</span> go <span class="dv">0</span> xs</span>
 <span id="cb243-4"><a href="#cb243-4" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> go <span class="fu">sum</span> [] <span class="ot">=</span> <span class="fu">sum</span></span>
 <span id="cb243-5"><a href="#cb243-5" aria-hidden="true" tabindex="-1"></a>        go <span class="fu">sum</span> (x<span class="op">:</span>xs) <span class="ot">=</span> go (<span class="fu">sum</span><span class="op">+</span>x) xs</span></code></pre></div>
-<p>Note the second equation of <code>go</code>: it has the recursive call to <code>go</code> at the top level, i.e. in tail position. The <code>+</code> is now in an argument to <code>go</code>.</p>
-<p>For a function like <code>sumNumbers</code> that produces a single value (a number), it doesn’t really matter which form of recursion you choose. The non-tail-recursive function is easier to read, while the tail-recursive one can be easier to come up with. You can try writing a function both ways. The tail-recursive form might be more efficient, but that depends on many details. We’ll talk more about Haskell performance in part 2 of this course.</p>
-<p>However, when you’re returning a list there is a big difference between these two forms. Consider the function <code>doubleList</code> from earlier. Here it is again, implemented first directly, and then via a tail-recursive helper function.</p>
-<div class="sourceCode" id="cb244"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb244-1"><a href="#cb244-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Not tail recursive!</span></span>
+<p>Note the second equation of <code>go</code>: it has the recursive
+call to <code>go</code> at the top level, i.e. in tail position. The
+<code>+</code> is now in an argument to <code>go</code>.</p>
+<p>For a function like <code>sumNumbers</code> that produces a single
+value (a number), it doesn’t really matter which form of recursion you
+choose. The non-tail-recursive function is easier to read, while the
+tail-recursive one can be easier to come up with. You can try writing a
+function both ways. The tail-recursive form might be more efficient, but
+that depends on many details. We’ll talk more about Haskell performance
+in part 2 of this course.</p>
+<p>However, when you’re returning a list there is a big difference
+between these two forms. Consider the function <code>doubleList</code>
+from earlier. Here it is again, implemented first directly, and then via
+a tail-recursive helper function.</p>
+<div class="sourceCode" id="cb244"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb244-1"><a href="#cb244-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Not tail recursive!</span></span>
 <span id="cb244-2"><a href="#cb244-2" aria-hidden="true" tabindex="-1"></a><span class="ot">doubleList ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> [<span class="dt">Int</span>]</span>
 <span id="cb244-3"><a href="#cb244-3" aria-hidden="true" tabindex="-1"></a>doubleList [] <span class="ot">=</span> []</span>
 <span id="cb244-4"><a href="#cb244-4" aria-hidden="true" tabindex="-1"></a>doubleList (x<span class="op">:</span>xs) <span class="ot">=</span> <span class="dv">2</span><span class="op">*</span>x <span class="op">:</span> doubleList xs</span></code></pre></div>
-<div class="sourceCode" id="cb245"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb245-1"><a href="#cb245-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Tail recursive version</span></span>
+<div class="sourceCode" id="cb245"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb245-1"><a href="#cb245-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Tail recursive version</span></span>
 <span id="cb245-2"><a href="#cb245-2" aria-hidden="true" tabindex="-1"></a><span class="ot">doubleList ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> [<span class="dt">Int</span>]</span>
 <span id="cb245-3"><a href="#cb245-3" aria-hidden="true" tabindex="-1"></a>doubleList xs <span class="ot">=</span> go [] xs</span>
 <span id="cb245-4"><a href="#cb245-4" aria-hidden="true" tabindex="-1"></a>    <span class="kw">where</span> go result [] <span class="ot">=</span> result</span>
 <span id="cb245-5"><a href="#cb245-5" aria-hidden="true" tabindex="-1"></a>          go result (x<span class="op">:</span>xs) <span class="ot">=</span> go (result<span class="op">++</span>[<span class="dv">2</span><span class="op">*</span>x]) xs</span></code></pre></div>
-<p>Here the direct version is much more efficient. The <code>(:)</code> operator works in constant time, whereas the <code>(++)</code> operator needs to walk the whole list, needing linear time. Thus the direct version uses linear time (<em>O(n)</em>) with respect to the length of the list, while the tail-recursive version is quadratic (<em>O(n²)</em>)!</p>
-<p>One might be tempted to fix this by using <code>(:)</code> in the tail-recursive version, but then the list would get generated in the reverse order. This could be fixed with an application of <code>reverse</code>, but that would make the resulting function quite complicated.</p>
-<p>There is another reason to prefer the direct version: laziness. We’ll get back to laziness in part 2 of the course, but for now it’s enough for you to know that <strong>the direct way of generating a list is simpler, more efficient and more idiomatic</strong>. You should try to practice it in the exercises. Check out the standard library implementations of <code>map</code> and <code>filter</code> above, even they produce the list directly without tail recursion!</p>
-<h2 data-number="3.9" id="something-fun-list-comprehensions"><span class="header-section-number">3.9</span> Something Fun: List Comprehensions</h2>
-<p>Haskell has <em>list comprehensions</em>, a nice syntax for defining lists that combines the power of <code>map</code> and <code>filter</code>. You might be familiar with Python’s list comprehensions already. Haskell’s work pretty much the same way, but their syntax is a bit different.</p>
+<p>Here the direct version is much more efficient. The <code>(:)</code>
+operator works in constant time, whereas the <code>(++)</code> operator
+needs to walk the whole list, needing linear time. Thus the direct
+version uses linear time (<em>O(n)</em>) with respect to the length of
+the list, while the tail-recursive version is quadratic
+(<em>O(n²)</em>)!</p>
+<p>One might be tempted to fix this by using <code>(:)</code> in the
+tail-recursive version, but then the list would get generated in the
+reverse order. This could be fixed with an application of
+<code>reverse</code>, but that would make the resulting function quite
+complicated.</p>
+<p>There is another reason to prefer the direct version: laziness. We’ll
+get back to laziness in part 2 of the course, but for now it’s enough
+for you to know that <strong>the direct way of generating a list is
+simpler, more efficient and more idiomatic</strong>. You should try to
+practice it in the exercises. Check out the standard library
+implementations of <code>map</code> and <code>filter</code> above, even
+they produce the list directly without tail recursion!</p>
+<h2 data-number="3.9" id="something-fun-list-comprehensions"><span
+class="header-section-number">3.9</span> Something Fun: List
+Comprehensions</h2>
+<p>Haskell has <em>list comprehensions</em>, a nice syntax for defining
+lists that combines the power of <code>map</code> and
+<code>filter</code>. You might be familiar with Python’s list
+comprehensions already. Haskell’s work pretty much the same way, but
+their syntax is a bit different.</p>
 <p>Mapping:</p>
-<div class="sourceCode" id="cb246"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb246-1"><a href="#cb246-1" aria-hidden="true" tabindex="-1"></a>[<span class="dv">2</span><span class="op">*</span>i <span class="op">|</span> i<span class="ot">&lt;-</span>[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]]</span>
+<div class="sourceCode" id="cb246"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb246-1"><a href="#cb246-1" aria-hidden="true" tabindex="-1"></a>[<span class="dv">2</span><span class="op">*</span>i <span class="op">|</span> i<span class="ot">&lt;-</span>[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]]</span>
 <span id="cb246-2"><a href="#cb246-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="dv">2</span>,<span class="dv">4</span>,<span class="dv">6</span>]</span></code></pre></div>
 <p>Filtering:</p>
-<div class="sourceCode" id="cb247"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb247-1"><a href="#cb247-1" aria-hidden="true" tabindex="-1"></a>[i <span class="op">|</span> i <span class="ot">&lt;-</span> [<span class="dv">1</span><span class="op">..</span><span class="dv">7</span>], <span class="fu">even</span> i]</span>
+<div class="sourceCode" id="cb247"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb247-1"><a href="#cb247-1" aria-hidden="true" tabindex="-1"></a>[i <span class="op">|</span> i <span class="ot">&lt;-</span> [<span class="dv">1</span><span class="op">..</span><span class="dv">7</span>], <span class="fu">even</span> i]</span>
 <span id="cb247-2"><a href="#cb247-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="dv">2</span>,<span class="dv">4</span>,<span class="dv">6</span>]</span></code></pre></div>
 <p>In general, these two forms are equivalent:</p>
-<div class="sourceCode" id="cb248"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb248-1"><a href="#cb248-1" aria-hidden="true" tabindex="-1"></a>[f x <span class="op">|</span> x <span class="ot">&lt;-</span> lis, p x]</span>
+<div class="sourceCode" id="cb248"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb248-1"><a href="#cb248-1" aria-hidden="true" tabindex="-1"></a>[f x <span class="op">|</span> x <span class="ot">&lt;-</span> lis, p x]</span>
 <span id="cb248-2"><a href="#cb248-2" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> f (<span class="fu">filter</span> p lis)</span></code></pre></div>
-<p>List comprehensions can do even more. You can iterate over multiple lists:</p>
-<div class="sourceCode" id="cb249"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb249-1"><a href="#cb249-1" aria-hidden="true" tabindex="-1"></a>[ first <span class="op">++</span> <span class="st">&quot; &quot;</span> <span class="op">++</span> <span class="fu">last</span> <span class="op">|</span> first <span class="ot">&lt;-</span> [<span class="st">&quot;John&quot;</span>, <span class="st">&quot;Mary&quot;</span>], <span class="fu">last</span> <span class="ot">&lt;-</span> [<span class="st">&quot;Smith&quot;</span>,<span class="st">&quot;Cooper&quot;</span>] ]</span>
+<p>List comprehensions can do even more. You can iterate over multiple
+lists:</p>
+<div class="sourceCode" id="cb249"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb249-1"><a href="#cb249-1" aria-hidden="true" tabindex="-1"></a>[ first <span class="op">++</span> <span class="st">&quot; &quot;</span> <span class="op">++</span> <span class="fu">last</span> <span class="op">|</span> first <span class="ot">&lt;-</span> [<span class="st">&quot;John&quot;</span>, <span class="st">&quot;Mary&quot;</span>], <span class="fu">last</span> <span class="ot">&lt;-</span> [<span class="st">&quot;Smith&quot;</span>,<span class="st">&quot;Cooper&quot;</span>] ]</span>
 <span id="cb249-2"><a href="#cb249-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="st">&quot;John Smith&quot;</span>,<span class="st">&quot;John Cooper&quot;</span>,<span class="st">&quot;Mary Smith&quot;</span>,<span class="st">&quot;Mary Cooper&quot;</span>]</span></code></pre></div>
 <p>You can make local definitions:</p>
-<div class="sourceCode" id="cb250"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb250-1"><a href="#cb250-1" aria-hidden="true" tabindex="-1"></a>[ reversed <span class="op">|</span> word <span class="ot">&lt;-</span> [<span class="st">&quot;this&quot;</span>,<span class="st">&quot;is&quot;</span>,<span class="st">&quot;a&quot;</span>,<span class="st">&quot;string&quot;</span>], <span class="kw">let</span> reversed <span class="ot">=</span> <span class="fu">reverse</span> word ]</span>
+<div class="sourceCode" id="cb250"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb250-1"><a href="#cb250-1" aria-hidden="true" tabindex="-1"></a>[ reversed <span class="op">|</span> word <span class="ot">&lt;-</span> [<span class="st">&quot;this&quot;</span>,<span class="st">&quot;is&quot;</span>,<span class="st">&quot;a&quot;</span>,<span class="st">&quot;string&quot;</span>], <span class="kw">let</span> reversed <span class="ot">=</span> <span class="fu">reverse</span> word ]</span>
 <span id="cb250-2"><a href="#cb250-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="st">&quot;siht&quot;</span>,<span class="st">&quot;si&quot;</span>,<span class="st">&quot;a&quot;</span>,<span class="st">&quot;gnirts&quot;</span>]</span></code></pre></div>
 <p>You can even do pattern matching in list comprehensions!</p>
-<div class="sourceCode" id="cb251"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb251-1"><a href="#cb251-1" aria-hidden="true" tabindex="-1"></a>firstLetters string <span class="ot">=</span> [ char <span class="op">|</span> (char<span class="op">:</span>_) <span class="ot">&lt;-</span> <span class="fu">words</span> string ]</span></code></pre></div>
-<div class="sourceCode" id="cb252"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb252-1"><a href="#cb252-1" aria-hidden="true" tabindex="-1"></a>firstLetters <span class="st">&quot;Hello World!&quot;</span></span>
+<div class="sourceCode" id="cb251"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb251-1"><a href="#cb251-1" aria-hidden="true" tabindex="-1"></a>firstLetters string <span class="ot">=</span> [ char <span class="op">|</span> (char<span class="op">:</span>_) <span class="ot">&lt;-</span> <span class="fu">words</span> string ]</span></code></pre></div>
+<div class="sourceCode" id="cb252"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb252-1"><a href="#cb252-1" aria-hidden="true" tabindex="-1"></a>firstLetters <span class="st">&quot;Hello World!&quot;</span></span>
 <span id="cb252-2"><a href="#cb252-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="st">&quot;HW&quot;</span></span></code></pre></div>
-<h2 data-number="3.10" id="something-fun-custom-operators"><span class="header-section-number">3.10</span> Something Fun: Custom Operators</h2>
-<p>In Haskell an <em>operator</em> is anything built from the characters <code>!#$%&amp;*+./&lt;=&gt;?@\^|-~</code>. Operators can be defined just like functions (note the slightly different type annotation):</p>
-<div class="sourceCode" id="cb253"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb253-1"><a href="#cb253-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(&lt;+&gt;) ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> [<span class="dt">Int</span>]</span>
+<h2 data-number="3.10" id="something-fun-custom-operators"><span
+class="header-section-number">3.10</span> Something Fun: Custom
+Operators</h2>
+<p>In Haskell an <em>operator</em> is anything built from the characters
+<code>!#$%&amp;*+./&lt;=&gt;?@\^|-~</code>. Operators can be defined
+just like functions (note the slightly different type annotation):</p>
+<div class="sourceCode" id="cb253"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb253-1"><a href="#cb253-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(&lt;+&gt;) ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> [<span class="dt">Int</span>]</span>
 <span id="cb253-2"><a href="#cb253-2" aria-hidden="true" tabindex="-1"></a>xs <span class="op">&lt;+&gt;</span> ys <span class="ot">=</span> <span class="fu">zipWith</span> (<span class="op">+</span>) xs ys</span></code></pre></div>
-<div class="sourceCode" id="cb254"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb254-1"><a href="#cb254-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(+++) ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<div class="sourceCode" id="cb254"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb254-1"><a href="#cb254-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(+++) ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb254-2"><a href="#cb254-2" aria-hidden="true" tabindex="-1"></a>a <span class="op">+++</span> b <span class="ot">=</span> a <span class="op">++</span> <span class="st">&quot; &quot;</span> <span class="op">++</span> b</span></code></pre></div>
-<h2 data-number="3.11" id="something-useful-typed-holes"><span class="header-section-number">3.11</span> Something Useful: Typed Holes</h2>
-<p>Sometimes when writing Haskell it can be tricky to find expressions that have the right type. Luckily, the compiler can help you here! A feature called <em>Typed Holes</em> lets you leave blanks in your code, and the compiler will tell you what type the expression in the blank should have.</p>
-<p>Blanks can look like <code>_</code> or <code>_name</code>. They can be confused with the “anything goes” pattern <code>_</code>, but the difference is that a hole occurs on the <em>right side</em> of a <code>=</code>, while an anything goes pattern occurs on the <em>left side</em> of a <code>=</code>.</p>
+<h2 data-number="3.11" id="something-useful-typed-holes"><span
+class="header-section-number">3.11</span> Something Useful: Typed
+Holes</h2>
+<p>Sometimes when writing Haskell it can be tricky to find expressions
+that have the right type. Luckily, the compiler can help you here! A
+feature called <em>Typed Holes</em> lets you leave blanks in your code,
+and the compiler will tell you what type the expression in the blank
+should have.</p>
+<p>Blanks can look like <code>_</code> or <code>_name</code>. They can
+be confused with the “anything goes” pattern <code>_</code>, but the
+difference is that a hole occurs on the <em>right side</em> of a
+<code>=</code>, while an anything goes pattern occurs on the <em>left
+side</em> of a <code>=</code>.</p>
 <p>Let’s start with a simple example in GHCi:</p>
-<div class="sourceCode" id="cb255"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb255-1"><a href="#cb255-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">filter</span> _hole [<span class="dt">True</span>,<span class="dt">False</span>]</span>
+<div class="sourceCode" id="cb255"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb255-1"><a href="#cb255-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">filter</span> _hole [<span class="dt">True</span>,<span class="dt">False</span>]</span>
 <span id="cb255-2"><a href="#cb255-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb255-3"><a href="#cb255-3" aria-hidden="true" tabindex="-1"></a><span class="op">&lt;</span>interactive<span class="op">&gt;:</span> <span class="fu">error</span><span class="op">:</span></span>
 <span id="cb255-4"><a href="#cb255-4" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">Found</span> hole<span class="op">:</span><span class="ot"> _hole ::</span> <span class="dt">Bool</span> <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
@@ -2192,49 +3507,77 @@ Prelude&gt; map (add 3) [1,2,3]
 <span id="cb255-15"><a href="#cb255-15" aria-hidden="true" tabindex="-1"></a><span class="ot">        id ::</span> <span class="kw">forall</span> a<span class="op">.</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb255-16"><a href="#cb255-16" aria-hidden="true" tabindex="-1"></a>          with <span class="fu">id</span> <span class="op">@</span><span class="dt">Bool</span></span>
 <span id="cb255-17"><a href="#cb255-17" aria-hidden="true" tabindex="-1"></a>          (imported from ‘<span class="dt">Prelude</span>’ (<span class="fu">and</span> originally defined <span class="kw">in</span> ‘<span class="dt">GHC.Base</span>’))</span></code></pre></div>
-<p>The important part of this message is the very first line. This tells you what type Haskell is expecting for the hole.</p>
+<p>The important part of this message is the very first line. This tells
+you what type Haskell is expecting for the hole.</p>
 <pre><code>&lt;interactive&gt;: error:
     • Found hole: _hole :: Bool -&gt; Bool</code></pre>
-<p>The rest of the error message offers some suggestions for the value of <code>_hole</code>, for example <code>id</code> and <code>not</code>.</p>
-<p>Let’s look at a longer example, where we try to implement a function that filters a list using a list of booleans:</p>
-<div class="sourceCode" id="cb257"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb257-1"><a href="#cb257-1" aria-hidden="true" tabindex="-1"></a>keepElements [<span class="dv">5</span>,<span class="dv">6</span>,<span class="dv">7</span>,<span class="dv">8</span>] [<span class="dt">True</span>,<span class="dt">False</span>,<span class="dt">True</span>,<span class="dt">False</span>] <span class="op">==&gt;</span> [<span class="dv">5</span>,<span class="dv">7</span>]</span></code></pre></div>
-<p>We’ll start with <code>zip</code> since we know that pairs up the elements of the two lists nicely. We add a typed hole <code>_doIt</code> and call it with the result of <code>zip</code> to see what we need to do next.</p>
-<div class="sourceCode" id="cb258"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb258-1"><a href="#cb258-1" aria-hidden="true" tabindex="-1"></a><span class="ot">keepElements ::</span> [a] <span class="ot">-&gt;</span> [<span class="dt">Bool</span>] <span class="ot">-&gt;</span> [a]</span>
+<p>The rest of the error message offers some suggestions for the value
+of <code>_hole</code>, for example <code>id</code> and
+<code>not</code>.</p>
+<p>Let’s look at a longer example, where we try to implement a function
+that filters a list using a list of booleans:</p>
+<div class="sourceCode" id="cb257"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb257-1"><a href="#cb257-1" aria-hidden="true" tabindex="-1"></a>keepElements [<span class="dv">5</span>,<span class="dv">6</span>,<span class="dv">7</span>,<span class="dv">8</span>] [<span class="dt">True</span>,<span class="dt">False</span>,<span class="dt">True</span>,<span class="dt">False</span>] <span class="op">==&gt;</span> [<span class="dv">5</span>,<span class="dv">7</span>]</span></code></pre></div>
+<p>We’ll start with <code>zip</code> since we know that pairs up the
+elements of the two lists nicely. We add a typed hole <code>_doIt</code>
+and call it with the result of <code>zip</code> to see what we need to
+do next.</p>
+<div class="sourceCode" id="cb258"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb258-1"><a href="#cb258-1" aria-hidden="true" tabindex="-1"></a><span class="ot">keepElements ::</span> [a] <span class="ot">-&gt;</span> [<span class="dt">Bool</span>] <span class="ot">-&gt;</span> [a]</span>
 <span id="cb258-2"><a href="#cb258-2" aria-hidden="true" tabindex="-1"></a>keepElements xs bs <span class="ot">=</span> _doIt (<span class="fu">zip</span> xs bs)</span></code></pre></div>
-<div class="sourceCode" id="cb259"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb259-1"><a href="#cb259-1" aria-hidden="true" tabindex="-1"></a><span class="op">&lt;</span>interactive<span class="op">&gt;:</span> <span class="fu">error</span><span class="op">:</span></span>
+<div class="sourceCode" id="cb259"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb259-1"><a href="#cb259-1" aria-hidden="true" tabindex="-1"></a><span class="op">&lt;</span>interactive<span class="op">&gt;:</span> <span class="fu">error</span><span class="op">:</span></span>
 <span id="cb259-2"><a href="#cb259-2" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">Found</span> hole<span class="op">:</span><span class="ot"> _doIt ::</span> [(a, <span class="dt">Bool</span>)] <span class="ot">-&gt;</span> [a]</span>
 <span id="cb259-3"><a href="#cb259-3" aria-hidden="true" tabindex="-1"></a>    <span class="op">...</span></span></code></pre></div>
-<p>That looks like something that could be done with <code>map</code>. Let’s see what happens:</p>
-<div class="sourceCode" id="cb260"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb260-1"><a href="#cb260-1" aria-hidden="true" tabindex="-1"></a><span class="ot">keepElements ::</span> [a] <span class="ot">-&gt;</span> [<span class="dt">Bool</span>] <span class="ot">-&gt;</span> [a]</span>
+<p>That looks like something that could be done with <code>map</code>.
+Let’s see what happens:</p>
+<div class="sourceCode" id="cb260"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb260-1"><a href="#cb260-1" aria-hidden="true" tabindex="-1"></a><span class="ot">keepElements ::</span> [a] <span class="ot">-&gt;</span> [<span class="dt">Bool</span>] <span class="ot">-&gt;</span> [a]</span>
 <span id="cb260-2"><a href="#cb260-2" aria-hidden="true" tabindex="-1"></a>keepElements xs bs <span class="ot">=</span> <span class="fu">map</span> _f (<span class="fu">zip</span> xs bs)</span></code></pre></div>
-<div class="sourceCode" id="cb261"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb261-1"><a href="#cb261-1" aria-hidden="true" tabindex="-1"></a><span class="op">&lt;</span>interactive<span class="op">&gt;:</span> <span class="fu">error</span><span class="op">:</span></span>
+<div class="sourceCode" id="cb261"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb261-1"><a href="#cb261-1" aria-hidden="true" tabindex="-1"></a><span class="op">&lt;</span>interactive<span class="op">&gt;:</span> <span class="fu">error</span><span class="op">:</span></span>
 <span id="cb261-2"><a href="#cb261-2" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">Found</span> hole<span class="op">:</span><span class="ot"> _f ::</span> (a, <span class="dt">Bool</span>) <span class="ot">-&gt;</span> a</span>
 <span id="cb261-3"><a href="#cb261-3" aria-hidden="true" tabindex="-1"></a>    <span class="op">...</span></span>
 <span id="cb261-4"><a href="#cb261-4" aria-hidden="true" tabindex="-1"></a>      <span class="dt">Valid</span> hole fits include</span>
 <span id="cb261-5"><a href="#cb261-5" aria-hidden="true" tabindex="-1"></a><span class="ot">        fst ::</span> <span class="kw">forall</span> a b<span class="op">.</span> (a, b) <span class="ot">-&gt;</span> a</span></code></pre></div>
-<p>Great! GHC reminded us of the function <code>fst</code> that grabs the first out of a pair. Are we done now?</p>
-<div class="sourceCode" id="cb262"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb262-1"><a href="#cb262-1" aria-hidden="true" tabindex="-1"></a><span class="ot">keepElements ::</span> [a] <span class="ot">-&gt;</span> [<span class="dt">Bool</span>] <span class="ot">-&gt;</span> [a]</span>
+<p>Great! GHC reminded us of the function <code>fst</code> that grabs
+the first out of a pair. Are we done now?</p>
+<div class="sourceCode" id="cb262"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb262-1"><a href="#cb262-1" aria-hidden="true" tabindex="-1"></a><span class="ot">keepElements ::</span> [a] <span class="ot">-&gt;</span> [<span class="dt">Bool</span>] <span class="ot">-&gt;</span> [a]</span>
 <span id="cb262-2"><a href="#cb262-2" aria-hidden="true" tabindex="-1"></a>keepElements xs bs <span class="ot">=</span> <span class="fu">map</span> <span class="fu">fst</span> (<span class="fu">zip</span> xs bs)</span></code></pre></div>
-<div class="sourceCode" id="cb263"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb263-1"><a href="#cb263-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> keepElements [<span class="dv">5</span>,<span class="dv">6</span>,<span class="dv">7</span>,<span class="dv">8</span>] [<span class="dt">True</span>,<span class="dt">False</span>,<span class="dt">True</span>,<span class="dt">False</span>]</span>
+<div class="sourceCode" id="cb263"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb263-1"><a href="#cb263-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> keepElements [<span class="dv">5</span>,<span class="dv">6</span>,<span class="dv">7</span>,<span class="dv">8</span>] [<span class="dt">True</span>,<span class="dt">False</span>,<span class="dt">True</span>,<span class="dt">False</span>]</span>
 <span id="cb263-2"><a href="#cb263-2" aria-hidden="true" tabindex="-1"></a>[<span class="dv">5</span>,<span class="dv">6</span>,<span class="dv">7</span>,<span class="dv">8</span>]</span></code></pre></div>
-<p>Oh right, we’ve forgotten to do the filtering part. Let’s try a typed hole again:</p>
-<div class="sourceCode" id="cb264"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb264-1"><a href="#cb264-1" aria-hidden="true" tabindex="-1"></a><span class="ot">keepElements ::</span> [a] <span class="ot">-&gt;</span> [<span class="dt">Bool</span>] <span class="ot">-&gt;</span> [a]</span>
+<p>Oh right, we’ve forgotten to do the filtering part. Let’s try a typed
+hole again:</p>
+<div class="sourceCode" id="cb264"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb264-1"><a href="#cb264-1" aria-hidden="true" tabindex="-1"></a><span class="ot">keepElements ::</span> [a] <span class="ot">-&gt;</span> [<span class="dt">Bool</span>] <span class="ot">-&gt;</span> [a]</span>
 <span id="cb264-2"><a href="#cb264-2" aria-hidden="true" tabindex="-1"></a>keepElements xs bs <span class="ot">=</span> <span class="fu">map</span> <span class="fu">fst</span> (<span class="fu">filter</span> _predicate (<span class="fu">zip</span> xs bs))</span></code></pre></div>
-<div class="sourceCode" id="cb265"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb265-1"><a href="#cb265-1" aria-hidden="true" tabindex="-1"></a><span class="op">&lt;</span>interactive<span class="op">&gt;:</span> <span class="fu">error</span><span class="op">:</span></span>
+<div class="sourceCode" id="cb265"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb265-1"><a href="#cb265-1" aria-hidden="true" tabindex="-1"></a><span class="op">&lt;</span>interactive<span class="op">&gt;:</span> <span class="fu">error</span><span class="op">:</span></span>
 <span id="cb265-2"><a href="#cb265-2" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">Found</span> hole<span class="op">:</span><span class="ot"> _predicate ::</span> (a, <span class="dt">Bool</span>) <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb265-3"><a href="#cb265-3" aria-hidden="true" tabindex="-1"></a>    <span class="op">...</span></span>
 <span id="cb265-4"><a href="#cb265-4" aria-hidden="true" tabindex="-1"></a>      <span class="dt">Valid</span> hole fits include</span>
 <span id="cb265-5"><a href="#cb265-5" aria-hidden="true" tabindex="-1"></a><span class="ot">        snd ::</span> <span class="kw">forall</span> a b<span class="op">.</span> (a, b) <span class="ot">-&gt;</span> b</span>
 <span id="cb265-6"><a href="#cb265-6" aria-hidden="true" tabindex="-1"></a>        <span class="op">...</span></span>
 <span id="cb265-7"><a href="#cb265-7" aria-hidden="true" tabindex="-1"></a>        <span class="op">...</span> lots <span class="kw">of</span> other suggestions</span></code></pre></div>
-<p>Again GHC has reminded us of a function that seems to do the right thing: just grab the second element out of the tuple. Now our function is finished and works as expected.</p>
-<div class="sourceCode" id="cb266"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb266-1"><a href="#cb266-1" aria-hidden="true" tabindex="-1"></a><span class="ot">keepElements ::</span> [a] <span class="ot">-&gt;</span> [<span class="dt">Bool</span>] <span class="ot">-&gt;</span> [a]</span>
+<p>Again GHC has reminded us of a function that seems to do the right
+thing: just grab the second element out of the tuple. Now our function
+is finished and works as expected.</p>
+<div class="sourceCode" id="cb266"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb266-1"><a href="#cb266-1" aria-hidden="true" tabindex="-1"></a><span class="ot">keepElements ::</span> [a] <span class="ot">-&gt;</span> [<span class="dt">Bool</span>] <span class="ot">-&gt;</span> [a]</span>
 <span id="cb266-2"><a href="#cb266-2" aria-hidden="true" tabindex="-1"></a>keepElements xs bs <span class="ot">=</span> <span class="fu">map</span> <span class="fu">fst</span> (<span class="fu">filter</span> <span class="fu">snd</span> (<span class="fu">zip</span> xs bs))</span></code></pre></div>
-<div class="sourceCode" id="cb267"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb267-1"><a href="#cb267-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> keepElements [<span class="dv">5</span>,<span class="dv">6</span>,<span class="dv">7</span>,<span class="dv">8</span>] [<span class="dt">True</span>,<span class="dt">False</span>,<span class="dt">True</span>,<span class="dt">False</span>]</span>
+<div class="sourceCode" id="cb267"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb267-1"><a href="#cb267-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> keepElements [<span class="dv">5</span>,<span class="dv">6</span>,<span class="dv">7</span>,<span class="dv">8</span>] [<span class="dt">True</span>,<span class="dt">False</span>,<span class="dt">True</span>,<span class="dt">False</span>]</span>
 <span id="cb267-2"><a href="#cb267-2" aria-hidden="true" tabindex="-1"></a>[<span class="dv">5</span>,<span class="dv">7</span>]</span></code></pre></div>
-<p><strong>Remember typed holes</strong> when you get stuck with type errors when working on the exercises! Try replacing a function or variable with a typed hole. It might help you figure out what you need.</p>
-<h2 data-number="3.12" id="quiz-2"><span class="header-section-number">3.12</span> Quiz</h2>
-<p>What’s the type of this function? <code>both p q x = p x &amp;&amp; q x</code></p>
+<p><strong>Remember typed holes</strong> when you get stuck with type
+errors when working on the exercises! Try replacing a function or
+variable with a typed hole. It might help you figure out what you
+need.</p>
+<h2 data-number="3.12" id="quiz-2"><span
+class="header-section-number">3.12</span> Quiz</h2>
+<p>What’s the type of this function?
+<code>both p q x = p x &amp;&amp; q x</code></p>
 <ol class="quiz">
 <li>
 <code>a -&gt; Bool -&gt; a -&gt; Bool -&gt; a -&gt; Bool</code>
@@ -2246,7 +3589,8 @@ Prelude&gt; map (add 3) [1,2,3]
 <code>(a -&gt; Bool) -&gt; (b -&gt; Bool) -&gt; c -&gt; Bool</code>
 </li>
 </ol>
-<p>What’s the (most general) type of this function? <code>applyInOut f g x = f (g (f x))</code></p>
+<p>What’s the (most general) type of this function?
+<code>applyInOut f g x = f (g (f x))</code></p>
 <ol class="quiz">
 <li class="correct">
 <code>(a -&gt; b) -&gt; (b -&gt; a) -&gt; a -&gt; b</code>
@@ -2258,7 +3602,8 @@ Prelude&gt; map (add 3) [1,2,3]
 <code>(a -&gt; a) -&gt; (a -&gt; a) -&gt; a -&gt; a</code>
 </li>
 </ol>
-<p>Which one of the following functions adds its first argument to the second?</p>
+<p>Which one of the following functions adds its first argument to the
+second?</p>
 <ol class="quiz">
 <li>
 <code>f x x = x + x</code>
@@ -2270,7 +3615,8 @@ Prelude&gt; map (add 3) [1,2,3]
 <code>f = \x y -&gt; x + x</code>
 </li>
 </ol>
-<p>Which one of the following functions does not satisfy <code>f 1 ==&gt; 1</code>?</p>
+<p>Which one of the following functions does not satisfy
+<code>f 1 ==&gt; 1</code>?</p>
 <ol class="quiz">
 <li>
 <code>f x = (\y -&gt; y) x</code>
@@ -2321,7 +3667,8 @@ Returns all except the first and last elements of a list
 Returns the second element of a list
 </li>
 </ol>
-<p>What is the result of <code>reverse $ take 5 . tail $ "This is a test"</code>?</p>
+<p>What is the result of
+<code>reverse $ take 5 . tail $ "This is a test"</code>?</p>
 <ol class="quiz">
 <li class="correct">
 <code>"i sih"</code>
@@ -2333,7 +3680,8 @@ Returns the second element of a list
 A type error
 </li>
 </ol>
-<p>If <code>f :: a -&gt; b</code>, then what is the type of <code>map (.f)</code>?</p>
+<p>If <code>f :: a -&gt; b</code>, then what is the type of
+<code>map (.f)</code>?</p>
 <ol class="quiz">
 <li class="correct">
 <code>[b -&gt; c] -&gt; [a -&gt; c]</code>
@@ -2348,7 +3696,8 @@ A type error
 <code>[a] -&gt; [b]</code>
 </li>
 </ol>
-<p>What is the type of the leftmost <code>id</code> in <code>id id</code>?</p>
+<p>What is the type of the leftmost <code>id</code> in
+<code>id id</code>?</p>
 <ol class="quiz">
 <li>
 unspecified
@@ -2377,22 +3726,43 @@ unspecified
 <code>a -&gt; b -&gt; c -&gt; a</code>
 </li>
 </ol>
-<h2 data-number="3.13" id="exercises-2"><span class="header-section-number">3.13</span> Exercises</h2>
+<h2 data-number="3.13" id="exercises-2"><span
+class="header-section-number">3.13</span> Exercises</h2>
 <ul>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set3a.hs">Set3a</a>: normal list exercises</li>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set3b.hs">Set3b</a>: list recursion exercises</li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set3a.hs">Set3a</a>:
+normal list exercises</li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set3b.hs">Set3b</a>:
+list recursion exercises</li>
 </ul>
-<h3 data-number="3.13.1" id="common-errors"><span class="header-section-number">3.13.1</span> Common Errors</h3>
-<div class="sourceCode" id="cb268"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb268-1"><a href="#cb268-1" aria-hidden="true" tabindex="-1"></a><span class="dt">No</span> <span class="kw">instance</span> for (<span class="dt">Eq</span> a) arising from a use <span class="kw">of</span> ‘<span class="op">==</span>’</span></code></pre></div>
-<p>You’ve probably tried to use <code>x==Nothing</code> to check if a value is <code>Nothing</code>. Use pattern matching instead. The reason for this error is that values of type <code>Maybe a</code> can’t be compared because Haskell doesn’t know how to compare values of the polymorphic type <code>a</code>. You’ll find more about this in the next lecture. Use pattern matching instead of <code>==</code> for now.</p>
-<h1 data-number="4" id="lecture-4-real-classy"><span class="header-section-number">4</span> Lecture 4: Real Classy</h1>
+<h3 data-number="3.13.1" id="common-errors"><span
+class="header-section-number">3.13.1</span> Common Errors</h3>
+<div class="sourceCode" id="cb268"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb268-1"><a href="#cb268-1" aria-hidden="true" tabindex="-1"></a><span class="dt">No</span> <span class="kw">instance</span> for (<span class="dt">Eq</span> a) arising from a use <span class="kw">of</span> ‘<span class="op">==</span>’</span></code></pre></div>
+<p>You’ve probably tried to use <code>x==Nothing</code> to check if a
+value is <code>Nothing</code>. Use pattern matching instead. The reason
+for this error is that values of type <code>Maybe a</code> can’t be
+compared because Haskell doesn’t know how to compare values of the
+polymorphic type <code>a</code>. You’ll find more about this in the next
+lecture. Use pattern matching instead of <code>==</code> for now.</p>
+<h1 data-number="4" id="lecture-4-real-classy"><span
+class="header-section-number">4</span> Lecture 4: Real Classy</h1>
 <ul>
 <li>Tuples</li>
 <li>Type classes</li>
 <li>Data structures: Map, Array</li>
 </ul>
-<h2 data-number="4.1" id="sidenote-tuples"><span class="header-section-number">4.1</span> Sidenote: Tuples</h2>
-<p>Before we dive into type classes, let’s introduce the last remaining built-in datatype in Haskell: the tuple. <em>Tuples</em> or <em>pairs</em> (or triples, quadruples, etc) are a way of bundling a couple of values of different types together. You can think of tuples as fixed-length lists (just like Python’s tuples). Unlike lists, each element in the tuple can have a different type. The types of the elements are reflected in the type of the tuple. Here are some examples of tuple types and values:</p>
+<h2 data-number="4.1" id="sidenote-tuples"><span
+class="header-section-number">4.1</span> Sidenote: Tuples</h2>
+<p>Before we dive into type classes, let’s introduce the last remaining
+built-in datatype in Haskell: the tuple. <em>Tuples</em> or
+<em>pairs</em> (or triples, quadruples, etc) are a way of bundling a
+couple of values of different types together. You can think of tuples as
+fixed-length lists (just like Python’s tuples). Unlike lists, each
+element in the tuple can have a different type. The types of the
+elements are reflected in the type of the tuple. Here are some examples
+of tuple types and values:</p>
 <table>
 <thead>
 <tr class="header">
@@ -2415,36 +3785,53 @@ unspecified
 </tr>
 </tbody>
 </table>
-<p>To get values out of tuples, you can use the functions <code>fst</code> and <code>snd</code>:</p>
-<div class="sourceCode" id="cb269"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb269-1"><a href="#cb269-1" aria-hidden="true" tabindex="-1"></a><span class="fu">fst</span><span class="ot"> ::</span> (a, b) <span class="ot">-&gt;</span> a</span>
+<p>To get values out of tuples, you can use the functions
+<code>fst</code> and <code>snd</code>:</p>
+<div class="sourceCode" id="cb269"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb269-1"><a href="#cb269-1" aria-hidden="true" tabindex="-1"></a><span class="fu">fst</span><span class="ot"> ::</span> (a, b) <span class="ot">-&gt;</span> a</span>
 <span id="cb269-2"><a href="#cb269-2" aria-hidden="true" tabindex="-1"></a><span class="fu">snd</span><span class="ot"> ::</span> (a, b) <span class="ot">-&gt;</span> b</span></code></pre></div>
-<p>You can also pattern match on tuples. This is often the most convenient way, and also works for tuples of larger sizes. The <code>fst</code> and <code>snd</code> functions work only on pairs.</p>
-<p>Tuples are very useful in combination with lists. Here are some examples using the <code>zip</code>, <code>unzip</code> and <code>partition</code> functions from the <code>Data.List</code> module.</p>
-<div class="sourceCode" id="cb270"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb270-1"><a href="#cb270-1" aria-hidden="true" tabindex="-1"></a><span class="fu">zip</span><span class="ot"> ::</span> [a] <span class="ot">-&gt;</span> [b] <span class="ot">-&gt;</span> [(a, b)]    <span class="co">-- two lists to list of pairs</span></span>
+<p>You can also pattern match on tuples. This is often the most
+convenient way, and also works for tuples of larger sizes. The
+<code>fst</code> and <code>snd</code> functions work only on pairs.</p>
+<p>Tuples are very useful in combination with lists. Here are some
+examples using the <code>zip</code>, <code>unzip</code> and
+<code>partition</code> functions from the <code>Data.List</code>
+module.</p>
+<div class="sourceCode" id="cb270"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb270-1"><a href="#cb270-1" aria-hidden="true" tabindex="-1"></a><span class="fu">zip</span><span class="ot"> ::</span> [a] <span class="ot">-&gt;</span> [b] <span class="ot">-&gt;</span> [(a, b)]    <span class="co">-- two lists to list of pairs</span></span>
 <span id="cb270-2"><a href="#cb270-2" aria-hidden="true" tabindex="-1"></a><span class="fu">unzip</span><span class="ot"> ::</span> [(a, b)] <span class="ot">-&gt;</span> ([a], [b])  <span class="co">-- list of pairs to pair of lists</span></span>
 <span id="cb270-3"><a href="#cb270-3" aria-hidden="true" tabindex="-1"></a><span class="ot">partition ::</span> (a <span class="ot">-&gt;</span> <span class="dt">Bool</span>) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> ([a], [a])    <span class="co">-- elements that satisfy and don&#39;t satisfy a predicate</span></span></code></pre></div>
 <!-- How about `zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]`? -->
-<div class="sourceCode" id="cb271"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb271-1"><a href="#cb271-1" aria-hidden="true" tabindex="-1"></a><span class="fu">zip</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] [<span class="dt">True</span>,<span class="dt">False</span>,<span class="dt">True</span>]</span>
+<div class="sourceCode" id="cb271"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb271-1"><a href="#cb271-1" aria-hidden="true" tabindex="-1"></a><span class="fu">zip</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] [<span class="dt">True</span>,<span class="dt">False</span>,<span class="dt">True</span>]</span>
 <span id="cb271-2"><a href="#cb271-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [(<span class="dv">1</span>,<span class="dt">True</span>),(<span class="dv">2</span>,<span class="dt">False</span>),(<span class="dv">3</span>,<span class="dt">True</span>)]</span>
 <span id="cb271-3"><a href="#cb271-3" aria-hidden="true" tabindex="-1"></a><span class="fu">unzip</span> [(<span class="st">&quot;Fred&quot;</span>,<span class="dv">1</span>), (<span class="st">&quot;Jack&quot;</span>,<span class="dv">10</span>), (<span class="st">&quot;Helen&quot;</span>,<span class="dv">13</span>)]</span>
 <span id="cb271-4"><a href="#cb271-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> ([<span class="st">&quot;Fred&quot;</span>,<span class="st">&quot;Jack&quot;</span>,<span class="st">&quot;Helen&quot;</span>],[<span class="dv">1</span>,<span class="dv">10</span>,<span class="dv">13</span>])</span>
 <span id="cb271-5"><a href="#cb271-5" aria-hidden="true" tabindex="-1"></a>partition (<span class="op">&gt;</span><span class="dv">0</span>) [<span class="op">-</span><span class="dv">1</span>,<span class="dv">1</span>,<span class="op">-</span><span class="dv">4</span>,<span class="dv">3</span>,<span class="dv">2</span>,<span class="dv">0</span>]</span>
 <span id="cb271-6"><a href="#cb271-6" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> ([<span class="dv">1</span>,<span class="dv">3</span>,<span class="dv">2</span>],[<span class="op">-</span><span class="dv">1</span>,<span class="op">-</span><span class="dv">4</span>,<span class="dv">0</span>])</span></code></pre></div>
 <p>Here’s an example of pattern matching on tuples:</p>
-<div class="sourceCode" id="cb272"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb272-1"><a href="#cb272-1" aria-hidden="true" tabindex="-1"></a><span class="ot">swap ::</span> (a,b) <span class="ot">-&gt;</span> (b,a)</span>
+<div class="sourceCode" id="cb272"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb272-1"><a href="#cb272-1" aria-hidden="true" tabindex="-1"></a><span class="ot">swap ::</span> (a,b) <span class="ot">-&gt;</span> (b,a)</span>
 <span id="cb272-2"><a href="#cb272-2" aria-hidden="true" tabindex="-1"></a>swap (x,y) <span class="ot">=</span> (y,x)</span></code></pre></div>
-<p>Here’s an example of pattern matching on tuples and lists at the same time:</p>
-<div class="sourceCode" id="cb273"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb273-1"><a href="#cb273-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- sum all numbers that are paired with True</span></span>
+<p>Here’s an example of pattern matching on tuples and lists at the same
+time:</p>
+<div class="sourceCode" id="cb273"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb273-1"><a href="#cb273-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- sum all numbers that are paired with True</span></span>
 <span id="cb273-2"><a href="#cb273-2" aria-hidden="true" tabindex="-1"></a><span class="ot">sumIf ::</span> [(<span class="dt">Bool</span>,<span class="dt">Int</span>)] <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb273-3"><a href="#cb273-3" aria-hidden="true" tabindex="-1"></a>sumIf [] <span class="ot">=</span> <span class="dv">0</span></span>
 <span id="cb273-4"><a href="#cb273-4" aria-hidden="true" tabindex="-1"></a>sumIf ((<span class="dt">True</span>,x)<span class="op">:</span>xs) <span class="ot">=</span> x <span class="op">+</span> sumIf xs</span>
 <span id="cb273-5"><a href="#cb273-5" aria-hidden="true" tabindex="-1"></a>sumIf ((<span class="dt">False</span>,_)<span class="op">:</span>xs) <span class="ot">=</span> sumIf xs</span></code></pre></div>
-<div class="sourceCode" id="cb274"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb274-1"><a href="#cb274-1" aria-hidden="true" tabindex="-1"></a>sumIf [(<span class="dt">True</span>,<span class="dv">1</span>),(<span class="dt">False</span>,<span class="dv">10</span>),(<span class="dt">True</span>,<span class="dv">100</span>)]</span>
+<div class="sourceCode" id="cb274"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb274-1"><a href="#cb274-1" aria-hidden="true" tabindex="-1"></a>sumIf [(<span class="dt">True</span>,<span class="dv">1</span>),(<span class="dt">False</span>,<span class="dv">10</span>),(<span class="dt">True</span>,<span class="dv">100</span>)]</span>
 <span id="cb274-2"><a href="#cb274-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">101</span></span></code></pre></div>
-<h2 data-number="4.2" id="interlude-folding"><span class="header-section-number">4.2</span> Interlude: Folding</h2>
+<h2 data-number="4.2" id="interlude-folding"><span
+class="header-section-number">4.2</span> Interlude: Folding</h2>
 <!-- TODO: should this be in the same section as foldable? -->
-<p>Consider the functions <code>sumNumbers :: [Int] -&gt; Int</code>, <code>myMaximum :: [Int] -&gt; Int</code>, and <code>countNothings :: [Maybe a] -&gt; Int</code> again.</p>
-<div class="sourceCode" id="cb275"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb275-1"><a href="#cb275-1" aria-hidden="true" tabindex="-1"></a><span class="ot">sumNumbers ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<p>Consider the functions <code>sumNumbers :: [Int] -&gt; Int</code>,
+<code>myMaximum :: [Int] -&gt; Int</code>, and
+<code>countNothings :: [Maybe a] -&gt; Int</code> again.</p>
+<div class="sourceCode" id="cb275"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb275-1"><a href="#cb275-1" aria-hidden="true" tabindex="-1"></a><span class="ot">sumNumbers ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb275-2"><a href="#cb275-2" aria-hidden="true" tabindex="-1"></a>sumNumbers [] <span class="ot">=</span> <span class="dv">0</span></span>
 <span id="cb275-3"><a href="#cb275-3" aria-hidden="true" tabindex="-1"></a>sumNumbers (x<span class="op">:</span>xs) <span class="ot">=</span> x <span class="op">+</span> sumNumbers xs</span>
 <span id="cb275-4"><a href="#cb275-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -2458,16 +3845,29 @@ unspecified
 <span id="cb275-12"><a href="#cb275-12" aria-hidden="true" tabindex="-1"></a>countNothings [] <span class="ot">=</span> <span class="dv">0</span></span>
 <span id="cb275-13"><a href="#cb275-13" aria-hidden="true" tabindex="-1"></a>countNothings (<span class="dt">Nothing</span> <span class="op">:</span> xs) <span class="ot">=</span> <span class="dv">1</span> <span class="op">+</span> countNothings xs</span>
 <span id="cb275-14"><a href="#cb275-14" aria-hidden="true" tabindex="-1"></a>countNothings (<span class="dt">Just</span> _  <span class="op">:</span> xs) <span class="ot">=</span> countNothings xs</span></code></pre></div>
-<p>They have one common characteristic. They take a list and produce a value that depends on the values of the elements in the given list. They “crunch” or <em>fold</em> a list of many values into a single value.</p>
-<p>Prelude has a function called <code>foldr</code>, which performs a <em>right associative fold</em> over a <code>Foldable</code> data type. We’ll learn more about <code>Foldable</code> soon. At this point, it suffices to think of lists, so we define</p>
-<div class="sourceCode" id="cb276"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb276-1"><a href="#cb276-1" aria-hidden="true" tabindex="-1"></a><span class="fu">foldr</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> b</span>
+<p>They have one common characteristic. They take a list and produce a
+value that depends on the values of the elements in the given list. They
+“crunch” or <em>fold</em> a list of many values into a single value.</p>
+<p>Prelude has a function called <code>foldr</code>, which performs a
+<em>right associative fold</em> over a <code>Foldable</code> data type.
+We’ll learn more about <code>Foldable</code> soon. At this point, it
+suffices to think of lists, so we define</p>
+<div class="sourceCode" id="cb276"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb276-1"><a href="#cb276-1" aria-hidden="true" tabindex="-1"></a><span class="fu">foldr</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> b</span>
 <span id="cb276-2"><a href="#cb276-2" aria-hidden="true" tabindex="-1"></a><span class="fu">foldr</span> f y []     <span class="ot">=</span> y</span>
 <span id="cb276-3"><a href="#cb276-3" aria-hidden="true" tabindex="-1"></a><span class="fu">foldr</span> f y (x<span class="op">:</span>xs) <span class="ot">=</span> f x (<span class="fu">foldr</span> f y xs)</span></code></pre></div>
-<p>What this definition says, is that for an empty list <code>[] :: [a]</code>, <code>foldr</code> returns the default value <code>y :: b</code>. For any other list <code>x : xs</code>, <code>foldr</code> applies <code>f</code> to <code>x</code> and the result of <code>foldr f y xs</code> (i.e. folding over the rest of the list). It’s a simple definition by recursion.</p>
-<p>In other words, <code>foldr</code> calls its argument function <code>f</code> repeatedly with two arguments.</p>
+<p>What this definition says, is that for an empty list
+<code>[] :: [a]</code>, <code>foldr</code> returns the default value
+<code>y :: b</code>. For any other list <code>x : xs</code>,
+<code>foldr</code> applies <code>f</code> to <code>x</code> and the
+result of <code>foldr f y xs</code> (i.e. folding over the rest of the
+list). It’s a simple definition by recursion.</p>
+<p>In other words, <code>foldr</code> calls its argument function
+<code>f</code> repeatedly with two arguments.</p>
 <ul>
 <li>The first argument is the current element from the list.</li>
-<li>The second argument is what <code>f</code> returned for the rest of the list.</li>
+<li>The second argument is what <code>f</code> returned for the rest of
+the list.</li>
 </ul>
 <p>Consider the list <code>[1,2,3]</code>:</p>
 <p><img src="img/list123.svg" /></p>
@@ -2482,8 +3882,10 @@ unspecified
        3    []
 ```
 -->
-<p>The expression <code>foldr (+) 0 [1,2,3]</code> evaluates as follows:</p>
-<div class="sourceCode" id="cb277"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb277-1"><a href="#cb277-1" aria-hidden="true" tabindex="-1"></a><span class="fu">foldr</span> (<span class="op">+</span>) <span class="dv">0</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] <span class="op">==&gt;</span> <span class="fu">foldr</span> (<span class="op">+</span>) <span class="dv">0</span> (<span class="dv">1</span><span class="op">:</span><span class="dv">2</span><span class="op">:</span><span class="dv">3</span><span class="op">:</span>[])</span>
+<p>The expression <code>foldr (+) 0 [1,2,3]</code> evaluates as
+follows:</p>
+<div class="sourceCode" id="cb277"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb277-1"><a href="#cb277-1" aria-hidden="true" tabindex="-1"></a><span class="fu">foldr</span> (<span class="op">+</span>) <span class="dv">0</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] <span class="op">==&gt;</span> <span class="fu">foldr</span> (<span class="op">+</span>) <span class="dv">0</span> (<span class="dv">1</span><span class="op">:</span><span class="dv">2</span><span class="op">:</span><span class="dv">3</span><span class="op">:</span>[])</span>
 <span id="cb277-2"><a href="#cb277-2" aria-hidden="true" tabindex="-1"></a>                    <span class="op">==&gt;</span> <span class="dv">1</span> <span class="op">+</span> (<span class="fu">foldr</span> (<span class="op">+</span>) <span class="dv">0</span> (<span class="dv">2</span><span class="op">:</span><span class="dv">3</span><span class="op">:</span>[]))</span>
 <span id="cb277-3"><a href="#cb277-3" aria-hidden="true" tabindex="-1"></a>                    <span class="op">==&gt;</span> <span class="dv">1</span> <span class="op">+</span> (<span class="dv">2</span> <span class="op">+</span> (<span class="fu">foldr</span> (<span class="op">+</span>) <span class="dv">0</span> (<span class="dv">3</span><span class="op">:</span>[])))</span>
 <span id="cb277-4"><a href="#cb277-4" aria-hidden="true" tabindex="-1"></a>                    <span class="op">==&gt;</span> <span class="dv">1</span> <span class="op">+</span> (<span class="dv">2</span> <span class="op">+</span> (<span class="dv">3</span> <span class="op">+</span> (<span class="fu">foldr</span> (<span class="op">+</span>) <span class="dv">0</span> [])))</span>
@@ -2501,14 +3903,38 @@ unspecified
        3     0
 ```
 -->
-<p>One way to think about <code>foldr f y xs</code> is that it replaces the <code>(:)</code> operation with <code>f</code> and <code>[]</code> with <code>y</code>. In this case, <code>f</code> was <code>(+)</code> and <code>y</code> was <code>0</code>. If you write out how <code>sumNumbers [1,2,3]</code> behaves, you’ll notice that it performs the same computation as <code>foldr (+) 0 [1,2,3]</code> does! More generally:</p>
-<div class="sourceCode" id="cb278"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb278-1"><a href="#cb278-1" aria-hidden="true" tabindex="-1"></a>sumNumbers xs <span class="op">==</span> <span class="fu">foldr</span> (<span class="op">+</span>) <span class="dv">0</span> xs</span></code></pre></div>
-<p>Those more experienced with math may notice that we can prove this claim by <em>induction</em>: Firstly, <code>sumNumbers [] ==&gt; 0</code> and <code>foldr (+) 0 [] ==&gt; 0</code>, so in the base case <code>sumNumbers [] == foldr (+) 0 []</code>. Next, we may assume as our induction hypothesis that <code>sumNumbers xs == foldr (+) 0 xs</code> for any list <code>xs</code>. Then, for the list <code>x:xs</code>, we have <code>sumNumbers (x:xs) ==&gt; x + sumNumbers xs</code>. Hence, <code>foldr (+) 0 (x:xs) ==&gt; x + foldr (+) 0 xs ==&gt; x + sumNumbers xs</code> by induction hypothesis. Therefore, by induction, the equation holds.</p>
-<p>You don’t need to read, write, or understand induction proofs in this course, but perhaps it is reassuring to know that properties and equalities of functions in Haskell can be (in principle) analysed mathematically, because Haskell is such a nice language. (Equalities and properties can be analysed in any programming language, but for Haskell, this analysis is especially convenient because Haskell is pure.)</p>
+<p>One way to think about <code>foldr f y xs</code> is that it replaces
+the <code>(:)</code> operation with <code>f</code> and <code>[]</code>
+with <code>y</code>. In this case, <code>f</code> was <code>(+)</code>
+and <code>y</code> was <code>0</code>. If you write out how
+<code>sumNumbers [1,2,3]</code> behaves, you’ll notice that it performs
+the same computation as <code>foldr (+) 0 [1,2,3]</code> does! More
+generally:</p>
+<div class="sourceCode" id="cb278"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb278-1"><a href="#cb278-1" aria-hidden="true" tabindex="-1"></a>sumNumbers xs <span class="op">==</span> <span class="fu">foldr</span> (<span class="op">+</span>) <span class="dv">0</span> xs</span></code></pre></div>
+<p>Those more experienced with math may notice that we can prove this
+claim by <em>induction</em>: Firstly,
+<code>sumNumbers [] ==&gt; 0</code> and
+<code>foldr (+) 0 [] ==&gt; 0</code>, so in the base case
+<code>sumNumbers [] == foldr (+) 0 []</code>. Next, we may assume as our
+induction hypothesis that <code>sumNumbers xs == foldr (+) 0 xs</code>
+for any list <code>xs</code>. Then, for the list <code>x:xs</code>, we
+have <code>sumNumbers (x:xs) ==&gt; x + sumNumbers xs</code>. Hence,
+<code>foldr (+) 0 (x:xs) ==&gt; x + foldr (+) 0 xs ==&gt; x + sumNumbers xs</code>
+by induction hypothesis. Therefore, by induction, the equation
+holds.</p>
+<p>You don’t need to read, write, or understand induction proofs in this
+course, but perhaps it is reassuring to know that properties and
+equalities of functions in Haskell can be (in principle) analysed
+mathematically, because Haskell is such a nice language. (Equalities and
+properties can be analysed in any programming language, but for Haskell,
+this analysis is especially convenient because Haskell is pure.)</p>
 <p>Another folding example is the <code>map</code> function:</p>
-<div class="sourceCode" id="cb279"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb279-1"><a href="#cb279-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> g xs <span class="ot">=</span> <span class="fu">foldr</span> helper [] xs</span>
+<div class="sourceCode" id="cb279"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb279-1"><a href="#cb279-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> g xs <span class="ot">=</span> <span class="fu">foldr</span> helper [] xs</span>
 <span id="cb279-2"><a href="#cb279-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> helper y ys <span class="ot">=</span> g y <span class="op">:</span> ys</span></code></pre></div>
-<p>To see why this works, consider what <code>foldr helper [] [x1,x2,..,xn]</code> does:</p>
+<p>To see why this works, consider what
+<code>foldr helper [] [x1,x2,..,xn]</code> does:</p>
 <p><img src="img/foldr1.svg" /></p>
 <!-- Do these lists actually need to be finite? -->
 <!--
@@ -2526,7 +3952,8 @@ unspecified
              xn    []                                xn    y
 ```
 -->
-<p>Now, since <code>helper x xs ==&gt; g x : xs</code> for every <code>x</code> and <code>xs</code>, we get that:</p>
+<p>Now, since <code>helper x xs ==&gt; g x : xs</code> for every
+<code>x</code> and <code>xs</code>, we get that:</p>
 <p><img src="img/foldr2.svg" /></p>
 <!--
 ```
@@ -2543,29 +3970,72 @@ unspecified
              xn    y                                g xn   y
 ```
 -->
-<p>The resulting list, <code>[ g x1, g x2, g x3, ..., g xn ]</code>, is then exactly what we would have gotten with <code>map g xs</code>. (This could have been also proved by induction as we did for <code>sumNumbers</code>.) The lesson to take away is that folding is a particular, yet quite general, way to apply some transformation recursively into some structure (e.g. a list).</p>
-<h2 data-number="4.3" id="type-classes"><span class="header-section-number">4.3</span> Type Classes</h2>
-<p>How can Haskell’s <code>+</code> work on both <code>Int</code>s and <code>Double</code>s? Why can I compare all sorts of things with <code>==</code>? We’ve briefly mentioned constrained types earlier. Let’s see what they really mean. Let’s look at the types of <code>==</code> and <code>+</code>.</p>
-<div class="sourceCode" id="cb280"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb280-1"><a href="#cb280-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(==) ::</span> (<span class="dt">Eq</span> a) <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Bool</span></span></code></pre></div>
-<p>The type <code>(Eq a) =&gt; a -&gt; a -&gt; Bool</code> means: <em>for all types <code>a</code> that belong to the class <code>Eq</code>, this is a function of type <code>a -&gt; a -&gt; Bool</code></em>. That is, if the type <code>a</code> is a member of the class <code>Eq</code>, you can give two values of type <code>a</code> to <code>==</code> and get a <code>Bool</code> result.</p>
-<div class="sourceCode" id="cb281"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb281-1"><a href="#cb281-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(+) ::</span> (<span class="dt">Num</span> a) <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span></code></pre></div>
-<p>Similarly, the type <code>(Num a) =&gt; a -&gt; a -&gt; a</code> means: <em>for all types <code>a</code> that belong to the class <code>Num</code>, this is a function of type <code>a -&gt; a -&gt; a</code></em>. That is, you can give two values of the same type <code>a</code> to <code>+</code> and get out a third value of type <code>a</code>, as long as <code>a</code> is a member of <code>Num</code>.</p>
-<p><code>Num</code> and <code>Eq</code> are type classes. A <em>type class</em> is a way to group together types that support similar operations.</p>
-<p><strong>Note!</strong> A type class is a collection of types. It doesn’t have much to do with the classes of object oriented programming! In some situations, type classes can act like <em>interfaces</em> in object oriented programming. Unfortunately the functions in a type class are often called <em>methods</em>, adding to the confusion.</p>
-<p>PS. remember how using type variables for polymorphism was called <em>parametric polymorphism</em>? The fancy word for what type classes achieve is <em>ad-hoc polymorphism</em>. The difference is that with parametric polymorphism the function (e.g. <code>head</code>) has the same implementation for all types, whereas with ad-hoc polymorphisms there are multiple implementations (consider <code>==</code> on numbers and strings).</p>
-<h2 data-number="4.4" id="type-constraints"><span class="header-section-number">4.4</span> Type Constraints</h2>
-<p>When you’re working with a concrete type (not a type variable), you can just use type class functions (in this case, <code>(==)</code>):</p>
-<div class="sourceCode" id="cb282"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb282-1"><a href="#cb282-1" aria-hidden="true" tabindex="-1"></a><span class="ot">f ::</span> (<span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span>) <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
+<p>The resulting list, <code>[ g x1, g x2, g x3, ..., g xn ]</code>, is
+then exactly what we would have gotten with <code>map g xs</code>. (This
+could have been also proved by induction as we did for
+<code>sumNumbers</code>.) The lesson to take away is that folding is a
+particular, yet quite general, way to apply some transformation
+recursively into some structure (e.g. a list).</p>
+<h2 data-number="4.3" id="type-classes"><span
+class="header-section-number">4.3</span> Type Classes</h2>
+<p>How can Haskell’s <code>+</code> work on both <code>Int</code>s and
+<code>Double</code>s? Why can I compare all sorts of things with
+<code>==</code>? We’ve briefly mentioned constrained types earlier.
+Let’s see what they really mean. Let’s look at the types of
+<code>==</code> and <code>+</code>.</p>
+<div class="sourceCode" id="cb280"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb280-1"><a href="#cb280-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(==) ::</span> (<span class="dt">Eq</span> a) <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Bool</span></span></code></pre></div>
+<p>The type <code>(Eq a) =&gt; a -&gt; a -&gt; Bool</code> means:
+<em>for all types <code>a</code> that belong to the class
+<code>Eq</code>, this is a function of type
+<code>a -&gt; a -&gt; Bool</code></em>. That is, if the type
+<code>a</code> is a member of the class <code>Eq</code>, you can give
+two values of type <code>a</code> to <code>==</code> and get a
+<code>Bool</code> result.</p>
+<div class="sourceCode" id="cb281"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb281-1"><a href="#cb281-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(+) ::</span> (<span class="dt">Num</span> a) <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span></code></pre></div>
+<p>Similarly, the type <code>(Num a) =&gt; a -&gt; a -&gt; a</code>
+means: <em>for all types <code>a</code> that belong to the class
+<code>Num</code>, this is a function of type
+<code>a -&gt; a -&gt; a</code></em>. That is, you can give two values of
+the same type <code>a</code> to <code>+</code> and get out a third value
+of type <code>a</code>, as long as <code>a</code> is a member of
+<code>Num</code>.</p>
+<p><code>Num</code> and <code>Eq</code> are type classes. A <em>type
+class</em> is a way to group together types that support similar
+operations.</p>
+<p><strong>Note!</strong> A type class is a collection of types. It
+doesn’t have much to do with the classes of object oriented programming!
+In some situations, type classes can act like <em>interfaces</em> in
+object oriented programming. Unfortunately the functions in a type class
+are often called <em>methods</em>, adding to the confusion.</p>
+<p>PS. remember how using type variables for polymorphism was called
+<em>parametric polymorphism</em>? The fancy word for what type classes
+achieve is <em>ad-hoc polymorphism</em>. The difference is that with
+parametric polymorphism the function (e.g. <code>head</code>) has the
+same implementation for all types, whereas with ad-hoc polymorphisms
+there are multiple implementations (consider <code>==</code> on numbers
+and strings).</p>
+<h2 data-number="4.4" id="type-constraints"><span
+class="header-section-number">4.4</span> Type Constraints</h2>
+<p>When you’re working with a concrete type (not a type variable), you
+can just use type class functions (in this case, <code>(==)</code>):</p>
+<div class="sourceCode" id="cb282"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb282-1"><a href="#cb282-1" aria-hidden="true" tabindex="-1"></a><span class="ot">f ::</span> (<span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span>) <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb282-2"><a href="#cb282-2" aria-hidden="true" tabindex="-1"></a>f g x <span class="ot">=</span> x <span class="op">==</span> g x</span></code></pre></div>
-<p>Of course, if the type in question isn’t a member of the right class, you get an error. For example:</p>
-<div class="sourceCode" id="cb283"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb283-1"><a href="#cb283-1" aria-hidden="true" tabindex="-1"></a><span class="ot">addTrue ::</span> <span class="dt">Bool</span> <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
+<p>Of course, if the type in question isn’t a member of the right class,
+you get an error. For example:</p>
+<div class="sourceCode" id="cb283"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb283-1"><a href="#cb283-1" aria-hidden="true" tabindex="-1"></a><span class="ot">addTrue ::</span> <span class="dt">Bool</span> <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb283-2"><a href="#cb283-2" aria-hidden="true" tabindex="-1"></a>addTrue b <span class="ot">=</span> b <span class="op">+</span> <span class="dt">True</span></span></code></pre></div>
 <pre><code>error:
     • No instance for (Num Bool) arising from a use of ‘+’
     • In the expression: b + True
       In an equation for ‘addTrue’: addTrue b = b + True</code></pre>
-<p>However in a <em>polymorphic</em> function, you need to add <em>type constraints</em>. This doesn’t work:</p>
-<div class="sourceCode" id="cb285"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb285-1"><a href="#cb285-1" aria-hidden="true" tabindex="-1"></a><span class="ot">f ::</span> (a <span class="ot">-&gt;</span> a) <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
+<p>However in a <em>polymorphic</em> function, you need to add <em>type
+constraints</em>. This doesn’t work:</p>
+<div class="sourceCode" id="cb285"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb285-1"><a href="#cb285-1" aria-hidden="true" tabindex="-1"></a><span class="ot">f ::</span> (a <span class="ot">-&gt;</span> a) <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb285-2"><a href="#cb285-2" aria-hidden="true" tabindex="-1"></a>f g x <span class="ot">=</span> x <span class="op">==</span> g x</span></code></pre></div>
 <p>Luckily the error is nice:</p>
 <pre><code>error:
@@ -2576,23 +4046,37 @@ unspecified
             f :: (a -&gt; a) -&gt; a -&gt; Bool
     • In the expression: x == g x
       In an equation for ‘f’: f g x = x == g x</code></pre>
-<p>To signal that <code>f</code> only works on types that are members of the <code>Eq</code> class, we add a type constraint <code>(Eq a) =&gt;</code> to the type annotation.</p>
-<div class="sourceCode" id="cb287"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb287-1"><a href="#cb287-1" aria-hidden="true" tabindex="-1"></a><span class="ot">f ::</span> (<span class="dt">Eq</span> a) <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> a) <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
+<p>To signal that <code>f</code> only works on types that are members of
+the <code>Eq</code> class, we add a type constraint
+<code>(Eq a) =&gt;</code> to the type annotation.</p>
+<div class="sourceCode" id="cb287"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb287-1"><a href="#cb287-1" aria-hidden="true" tabindex="-1"></a><span class="ot">f ::</span> (<span class="dt">Eq</span> a) <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> a) <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb287-2"><a href="#cb287-2" aria-hidden="true" tabindex="-1"></a>f g x <span class="ot">=</span> x <span class="op">==</span> g x</span></code></pre></div>
-<p>If you don’t have a type annotation, <em>type inference</em> can provide the constraints!</p>
-<div class="sourceCode" id="cb288"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb288-1"><a href="#cb288-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> f g x <span class="ot">=</span> x <span class="op">==</span> g x</span>
+<p>If you don’t have a type annotation, <em>type inference</em> can
+provide the constraints!</p>
+<div class="sourceCode" id="cb288"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb288-1"><a href="#cb288-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> f g x <span class="ot">=</span> x <span class="op">==</span> g x</span>
 <span id="cb288-2"><a href="#cb288-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span><span class="kw">type</span> f</span>
 <span id="cb288-3"><a href="#cb288-3" aria-hidden="true" tabindex="-1"></a><span class="ot">f ::</span> (<span class="dt">Eq</span> a) <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> a) <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Bool</span></span></code></pre></div>
 <p>You can also have multiple constraints:</p>
-<div class="sourceCode" id="cb289"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb289-1"><a href="#cb289-1" aria-hidden="true" tabindex="-1"></a><span class="ot">bothPairsEqual ::</span> (<span class="dt">Eq</span> a, <span class="dt">Eq</span> b) <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
+<div class="sourceCode" id="cb289"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb289-1"><a href="#cb289-1" aria-hidden="true" tabindex="-1"></a><span class="ot">bothPairsEqual ::</span> (<span class="dt">Eq</span> a, <span class="dt">Eq</span> b) <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb289-2"><a href="#cb289-2" aria-hidden="true" tabindex="-1"></a>bothPairsEqual left1 left2 right1 right2 <span class="ot">=</span> left1 <span class="op">==</span> left2 <span class="op">&amp;&amp;</span> right1 <span class="op">==</span> right2</span></code></pre></div>
-<h2 data-number="4.5" id="standard-type-classes"><span class="header-section-number">4.5</span> Standard Type Classes</h2>
-<p>Here are some standard Haskell type classes you should know about.</p>
-<h3 data-number="4.5.1" id="eq"><span class="header-section-number">4.5.1</span> <code>Eq</code></h3>
-<p>We already saw the <code>Eq</code> class for equality comparisons. Here are the basic operations of the <code>Eq</code> class and some examples of their use. As you can see pretty much all the types we’ve seen so far, except for functions, are members of <code>Eq</code>.</p>
-<div class="sourceCode" id="cb290"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb290-1"><a href="#cb290-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(==) ::</span> <span class="dt">Eq</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
+<h2 data-number="4.5" id="standard-type-classes"><span
+class="header-section-number">4.5</span> Standard Type Classes</h2>
+<p>Here are some standard Haskell type classes you should know
+about.</p>
+<h3 data-number="4.5.1" id="eq"><span
+class="header-section-number">4.5.1</span> <code>Eq</code></h3>
+<p>We already saw the <code>Eq</code> class for equality comparisons.
+Here are the basic operations of the <code>Eq</code> class and some
+examples of their use. As you can see pretty much all the types we’ve
+seen so far, except for functions, are members of <code>Eq</code>.</p>
+<div class="sourceCode" id="cb290"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb290-1"><a href="#cb290-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(==) ::</span> <span class="dt">Eq</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb290-2"><a href="#cb290-2" aria-hidden="true" tabindex="-1"></a><span class="ot">(/=) ::</span> <span class="dt">Eq</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Bool</span></span></code></pre></div>
-<div class="sourceCode" id="cb291"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb291-1"><a href="#cb291-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dv">1</span> <span class="op">==</span> <span class="dv">2</span></span>
+<div class="sourceCode" id="cb291"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb291-1"><a href="#cb291-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dv">1</span> <span class="op">==</span> <span class="dv">2</span></span>
 <span id="cb291-2"><a href="#cb291-2" aria-hidden="true" tabindex="-1"></a><span class="dt">False</span></span>
 <span id="cb291-3"><a href="#cb291-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dv">1</span> <span class="op">/=</span> <span class="dv">2</span></span>
 <span id="cb291-4"><a href="#cb291-4" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span>
@@ -2608,22 +4092,31 @@ unspecified
 <span id="cb291-14"><a href="#cb291-14" aria-hidden="true" tabindex="-1"></a>        (<span class="fu">maybe</span> you haven&#39;t applied a function to enough arguments<span class="op">?</span>)</span>
 <span id="cb291-15"><a href="#cb291-15" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">In</span> the expression<span class="op">:</span> (\ x <span class="ot">-&gt;</span> x <span class="op">+</span> <span class="dv">1</span>) <span class="op">==</span> (\ x <span class="ot">-&gt;</span> x <span class="op">+</span> <span class="dv">2</span>)</span>
 <span id="cb291-16"><a href="#cb291-16" aria-hidden="true" tabindex="-1"></a>      <span class="dt">In</span> an equation for ‘it’<span class="op">:</span> it <span class="ot">=</span> (\ x <span class="ot">-&gt;</span> x <span class="op">+</span> <span class="dv">1</span>) <span class="op">==</span> (\ x <span class="ot">-&gt;</span> x <span class="op">+</span> <span class="dv">2</span>)</span></code></pre></div>
-<p>There are some other useful functions that use the <code>Eq</code> class, like <code>nub</code> from the module <code>Data.List</code>.</p>
-<div class="sourceCode" id="cb292"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb292-1"><a href="#cb292-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="dt">Data.List</span></span>
+<p>There are some other useful functions that use the <code>Eq</code>
+class, like <code>nub</code> from the module <code>Data.List</code>.</p>
+<div class="sourceCode" id="cb292"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb292-1"><a href="#cb292-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="dt">Data.List</span></span>
 <span id="cb292-2"><a href="#cb292-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.List</span><span class="op">&gt;</span> <span class="op">:</span>t nub</span>
 <span id="cb292-3"><a href="#cb292-3" aria-hidden="true" tabindex="-1"></a><span class="ot">nub ::</span> <span class="dt">Eq</span> a <span class="ot">=&gt;</span> [a] <span class="ot">-&gt;</span> [a]</span>
 <span id="cb292-4"><a href="#cb292-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.List</span><span class="op">&gt;</span> nub [<span class="dv">3</span>,<span class="dv">5</span>,<span class="dv">3</span>,<span class="dv">1</span>,<span class="dv">1</span>]      <span class="co">-- eliminates duplicates</span></span>
 <span id="cb292-5"><a href="#cb292-5" aria-hidden="true" tabindex="-1"></a>[<span class="dv">3</span>,<span class="dv">5</span>,<span class="dv">1</span>]</span></code></pre></div>
-<h3 data-number="4.5.2" id="ord"><span class="header-section-number">4.5.2</span> <code>Ord</code></h3>
-<p>The <code>Ord</code> class is for ordering (less than, greater than). Again, here are the basic operations and some examples of their use. Note the new <code>Ordering</code> type. It has values <code>LT</code> for “less than”, <code>EQ</code> for “equal” and <code>GT</code> for “greater than”.</p>
-<div class="sourceCode" id="cb293"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb293-1"><a href="#cb293-1" aria-hidden="true" tabindex="-1"></a><span class="fu">compare</span><span class="ot"> ::</span> <span class="dt">Ord</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Ordering</span></span>
+<h3 data-number="4.5.2" id="ord"><span
+class="header-section-number">4.5.2</span> <code>Ord</code></h3>
+<p>The <code>Ord</code> class is for ordering (less than, greater than).
+Again, here are the basic operations and some examples of their use.
+Note the new <code>Ordering</code> type. It has values <code>LT</code>
+for “less than”, <code>EQ</code> for “equal” and <code>GT</code> for
+“greater than”.</p>
+<div class="sourceCode" id="cb293"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb293-1"><a href="#cb293-1" aria-hidden="true" tabindex="-1"></a><span class="fu">compare</span><span class="ot"> ::</span> <span class="dt">Ord</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Ordering</span></span>
 <span id="cb293-2"><a href="#cb293-2" aria-hidden="true" tabindex="-1"></a><span class="ot">(&lt;) ::</span> <span class="dt">Ord</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb293-3"><a href="#cb293-3" aria-hidden="true" tabindex="-1"></a><span class="ot">(&gt;) ::</span> <span class="dt">Ord</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb293-4"><a href="#cb293-4" aria-hidden="true" tabindex="-1"></a><span class="ot">(&gt;=) ::</span> <span class="dt">Ord</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb293-5"><a href="#cb293-5" aria-hidden="true" tabindex="-1"></a><span class="ot">(&lt;=) ::</span> <span class="dt">Ord</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb293-6"><a href="#cb293-6" aria-hidden="true" tabindex="-1"></a><span class="fu">max</span><span class="ot"> ::</span> <span class="dt">Ord</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb293-7"><a href="#cb293-7" aria-hidden="true" tabindex="-1"></a><span class="fu">min</span><span class="ot"> ::</span> <span class="dt">Ord</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span></code></pre></div>
-<div class="sourceCode" id="cb294"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb294-1"><a href="#cb294-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">compare</span> <span class="dv">1</span> <span class="dv">1</span>                <span class="co">-- 1 is EQual to 1</span></span>
+<div class="sourceCode" id="cb294"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb294-1"><a href="#cb294-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">compare</span> <span class="dv">1</span> <span class="dv">1</span>                <span class="co">-- 1 is EQual to 1</span></span>
 <span id="cb294-2"><a href="#cb294-2" aria-hidden="true" tabindex="-1"></a><span class="dt">EQ</span></span>
 <span id="cb294-3"><a href="#cb294-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">compare</span> <span class="dv">1</span> <span class="dv">3</span>                <span class="co">-- 1 is Less Than 3</span></span>
 <span id="cb294-4"><a href="#cb294-4" aria-hidden="true" tabindex="-1"></a><span class="dt">LT</span></span>
@@ -2639,16 +4132,21 @@ unspecified
 <span id="cb294-14"><a href="#cb294-14" aria-hidden="true" tabindex="-1"></a><span class="dt">False</span></span>
 <span id="cb294-15"><a href="#cb294-15" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] <span class="op">&gt;</span> [<span class="dv">1</span>,<span class="dv">1</span>]</span>
 <span id="cb294-16"><a href="#cb294-16" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span></code></pre></div>
-<p>When we can compare values, we can also sort lists of them. The function <code>sort</code> from <code>Data.List</code> works on all types that belong to the <code>Ord</code> class.</p>
-<div class="sourceCode" id="cb295"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb295-1"><a href="#cb295-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="dt">Data.List</span></span>
+<p>When we can compare values, we can also sort lists of them. The
+function <code>sort</code> from <code>Data.List</code> works on all
+types that belong to the <code>Ord</code> class.</p>
+<div class="sourceCode" id="cb295"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb295-1"><a href="#cb295-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="dt">Data.List</span></span>
 <span id="cb295-2"><a href="#cb295-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.List</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="fu">sort</span></span>
 <span id="cb295-3"><a href="#cb295-3" aria-hidden="true" tabindex="-1"></a><span class="fu">sort</span><span class="ot"> ::</span> <span class="dt">Ord</span> a <span class="ot">=&gt;</span> [a] <span class="ot">-&gt;</span> [a]</span>
 <span id="cb295-4"><a href="#cb295-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.List</span><span class="op">&gt;</span> <span class="fu">sort</span> [<span class="dv">6</span>,<span class="dv">1</span>,<span class="dv">4</span>,<span class="dv">8</span>,<span class="dv">2</span>]</span>
 <span id="cb295-5"><a href="#cb295-5" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">4</span>,<span class="dv">6</span>,<span class="dv">8</span>]</span>
 <span id="cb295-6"><a href="#cb295-6" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.List</span><span class="op">&gt;</span> <span class="fu">sort</span> <span class="st">&quot;black sphinx of quartz, judge my vow!&quot;</span>     <span class="co">-- remember, strings are lists!</span></span>
 <span id="cb295-7"><a href="#cb295-7" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;      !,aabcdefghijklmnoopqrstuuvwxyz&quot;</span></span></code></pre></div>
-<p>As a last example, let’s sort a list of lists according to length. We’ll need two helper functions:</p>
-<div class="sourceCode" id="cb296"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb296-1"><a href="#cb296-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- from the module Data.Ord</span></span>
+<p>As a last example, let’s sort a list of lists according to length.
+We’ll need two helper functions:</p>
+<div class="sourceCode" id="cb296"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb296-1"><a href="#cb296-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- from the module Data.Ord</span></span>
 <span id="cb296-2"><a href="#cb296-2" aria-hidden="true" tabindex="-1"></a><span class="co">-- compares two values &quot;through&quot; the function f</span></span>
 <span id="cb296-3"><a href="#cb296-3" aria-hidden="true" tabindex="-1"></a><span class="ot">comparing ::</span> (<span class="dt">Ord</span> a) <span class="ot">=&gt;</span> (b <span class="ot">-&gt;</span> a) <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> <span class="dt">Ordering</span></span>
 <span id="cb296-4"><a href="#cb296-4" aria-hidden="true" tabindex="-1"></a>comparing f x y <span class="ot">=</span> <span class="fu">compare</span> (f x) (f y)</span>
@@ -2656,14 +4154,21 @@ unspecified
 <span id="cb296-6"><a href="#cb296-6" aria-hidden="true" tabindex="-1"></a><span class="co">-- from the module Data.List</span></span>
 <span id="cb296-7"><a href="#cb296-7" aria-hidden="true" tabindex="-1"></a><span class="co">-- sorts a list using the given comparison function</span></span>
 <span id="cb296-8"><a href="#cb296-8" aria-hidden="true" tabindex="-1"></a><span class="ot">sortBy ::</span> (a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Ordering</span>) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [a]</span></code></pre></div>
-<p>Now the implementation of <code>sortByLength</code> is straightforward:</p>
-<div class="sourceCode" id="cb297"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb297-1"><a href="#cb297-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- sorts lists by their length</span></span>
+<p>Now the implementation of <code>sortByLength</code> is
+straightforward:</p>
+<div class="sourceCode" id="cb297"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb297-1"><a href="#cb297-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- sorts lists by their length</span></span>
 <span id="cb297-2"><a href="#cb297-2" aria-hidden="true" tabindex="-1"></a><span class="ot">sortByLength ::</span> [[a]] <span class="ot">-&gt;</span> [[a]]</span>
 <span id="cb297-3"><a href="#cb297-3" aria-hidden="true" tabindex="-1"></a>sortByLength <span class="ot">=</span> sortBy (comparing <span class="fu">length</span>)</span></code></pre></div>
-<div class="sourceCode" id="cb298"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb298-1"><a href="#cb298-1" aria-hidden="true" tabindex="-1"></a>sortByLength [[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>],[<span class="dv">4</span>,<span class="dv">5</span>],[<span class="dv">4</span>,<span class="dv">5</span>,<span class="dv">6</span>,<span class="dv">7</span>]]   <span class="op">==&gt;</span>  [[<span class="dv">4</span>,<span class="dv">5</span>],[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>],[<span class="dv">4</span>,<span class="dv">5</span>,<span class="dv">6</span>,<span class="dv">7</span>]]</span></code></pre></div>
-<h3 data-number="4.5.3" id="num-integral-fractional-floating"><span class="header-section-number">4.5.3</span> <code>Num</code>, <code>Integral</code>, <code>Fractional</code>, <code>Floating</code></h3>
+<div class="sourceCode" id="cb298"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb298-1"><a href="#cb298-1" aria-hidden="true" tabindex="-1"></a>sortByLength [[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>],[<span class="dv">4</span>,<span class="dv">5</span>],[<span class="dv">4</span>,<span class="dv">5</span>,<span class="dv">6</span>,<span class="dv">7</span>]]   <span class="op">==&gt;</span>  [[<span class="dv">4</span>,<span class="dv">5</span>],[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>],[<span class="dv">4</span>,<span class="dv">5</span>,<span class="dv">6</span>,<span class="dv">7</span>]]</span></code></pre></div>
+<h3 data-number="4.5.3" id="num-integral-fractional-floating"><span
+class="header-section-number">4.5.3</span> <code>Num</code>,
+<code>Integral</code>, <code>Fractional</code>,
+<code>Floating</code></h3>
 <p>The <code>Num</code> class contains integer arithmetic:</p>
-<div class="sourceCode" id="cb299"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb299-1"><a href="#cb299-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(+) ::</span> <span class="dt">Num</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
+<div class="sourceCode" id="cb299"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb299-1"><a href="#cb299-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(+) ::</span> <span class="dt">Num</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb299-2"><a href="#cb299-2" aria-hidden="true" tabindex="-1"></a><span class="ot">(-) ::</span> <span class="dt">Num</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb299-3"><a href="#cb299-3" aria-hidden="true" tabindex="-1"></a><span class="ot">(*) ::</span> <span class="dt">Num</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb299-4"><a href="#cb299-4" aria-hidden="true" tabindex="-1"></a><span class="fu">negate</span><span class="ot"> ::</span> <span class="dt">Num</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a    <span class="co">-- 0-x</span></span>
@@ -2671,55 +4176,109 @@ unspecified
 <span id="cb299-6"><a href="#cb299-6" aria-hidden="true" tabindex="-1"></a><span class="fu">signum</span><span class="ot"> ::</span> <span class="dt">Num</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a    <span class="co">-- -1 for negative values, 0 for 0, +1 for positive values</span></span>
 <span id="cb299-7"><a href="#cb299-7" aria-hidden="true" tabindex="-1"></a><span class="fu">fromInteger</span><span class="ot"> ::</span> <span class="dt">Num</span> a <span class="ot">=&gt;</span> <span class="dt">Integer</span> <span class="ot">-&gt;</span> a</span></code></pre></div>
 <p><code>Num</code> also shows up in the types of integer literals:</p>
-<div class="sourceCode" id="cb300"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb300-1"><a href="#cb300-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="dv">12</span></span>
+<div class="sourceCode" id="cb300"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb300-1"><a href="#cb300-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="dv">12</span></span>
 <span id="cb300-2"><a href="#cb300-2" aria-hidden="true" tabindex="-1"></a><span class="dv">12</span><span class="ot"> ::</span> <span class="dt">Num</span> p <span class="ot">=&gt;</span> p</span></code></pre></div>
-<p>This means that a literal like <code>12</code> can be interpreted as a member of any type implementing <code>Num</code>. When GHC reads a number literal like, <code>12</code> it produces code that corresponds to <code>fromIntegral 12</code>.</p>
-<div class="sourceCode" id="cb301"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb301-1"><a href="#cb301-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dv">1</span><span class="ot"> ::</span> <span class="dt">Int</span></span>
+<p>This means that a literal like <code>12</code> can be interpreted as
+a member of any type implementing <code>Num</code>. When GHC reads a
+number literal like, <code>12</code> it produces code that corresponds
+to <code>fromIntegral 12</code>.</p>
+<div class="sourceCode" id="cb301"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb301-1"><a href="#cb301-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dv">1</span><span class="ot"> ::</span> <span class="dt">Int</span></span>
 <span id="cb301-2"><a href="#cb301-2" aria-hidden="true" tabindex="-1"></a><span class="dv">1</span></span>
 <span id="cb301-3"><a href="#cb301-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dv">1</span><span class="ot"> ::</span> <span class="dt">Double</span></span>
 <span id="cb301-4"><a href="#cb301-4" aria-hidden="true" tabindex="-1"></a><span class="fl">1.0</span></span>
 <span id="cb301-5"><a href="#cb301-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">fromIntegral</span> <span class="dv">1</span><span class="ot"> ::</span> <span class="dt">Double</span></span>
 <span id="cb301-6"><a href="#cb301-6" aria-hidden="true" tabindex="-1"></a><span class="fl">1.0</span></span></code></pre></div>
-<p><code>Integral</code> is the class of types that represent whole numbers, like <code>Int</code> and <code>Integer</code>. The most interesting functions are <code>div</code> and <code>mod</code> for integer division and remainder. All types that belong to <code>Integral</code> also belong to <code>Num</code>.</p>
-<div class="sourceCode" id="cb302"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb302-1"><a href="#cb302-1" aria-hidden="true" tabindex="-1"></a><span class="fu">div</span><span class="ot"> ::</span> <span class="dt">Integral</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
+<p><code>Integral</code> is the class of types that represent whole
+numbers, like <code>Int</code> and <code>Integer</code>. The most
+interesting functions are <code>div</code> and <code>mod</code> for
+integer division and remainder. All types that belong to
+<code>Integral</code> also belong to <code>Num</code>.</p>
+<div class="sourceCode" id="cb302"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb302-1"><a href="#cb302-1" aria-hidden="true" tabindex="-1"></a><span class="fu">div</span><span class="ot"> ::</span> <span class="dt">Integral</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb302-2"><a href="#cb302-2" aria-hidden="true" tabindex="-1"></a><span class="fu">mod</span><span class="ot"> ::</span> <span class="dt">Integral</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span></code></pre></div>
-<p><code>Fractional</code> is the class for types that have division. All types that belong to <code>Fractional</code> also belong to <code>Num</code>.</p>
-<div class="sourceCode" id="cb303"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb303-1"><a href="#cb303-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(/) ::</span> <span class="dt">Fractional</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span></code></pre></div>
-<p><code>Floating</code> contains some additional operations that only make sense for floating point numbers. All types that belong to <code>Floating</code> also belong to <code>Fractional</code> (and to <code>Num</code>).</p>
-<div class="sourceCode" id="cb304"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb304-1"><a href="#cb304-1" aria-hidden="true" tabindex="-1"></a><span class="fu">sqrt</span><span class="ot"> ::</span> <span class="dt">Floating</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a</span>
+<p><code>Fractional</code> is the class for types that have division.
+All types that belong to <code>Fractional</code> also belong to
+<code>Num</code>.</p>
+<div class="sourceCode" id="cb303"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb303-1"><a href="#cb303-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(/) ::</span> <span class="dt">Fractional</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span></code></pre></div>
+<p><code>Floating</code> contains some additional operations that only
+make sense for floating point numbers. All types that belong to
+<code>Floating</code> also belong to <code>Fractional</code> (and to
+<code>Num</code>).</p>
+<div class="sourceCode" id="cb304"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb304-1"><a href="#cb304-1" aria-hidden="true" tabindex="-1"></a><span class="fu">sqrt</span><span class="ot"> ::</span> <span class="dt">Floating</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb304-2"><a href="#cb304-2" aria-hidden="true" tabindex="-1"></a><span class="fu">sin</span><span class="ot"> ::</span> <span class="dt">Floating</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a</span></code></pre></div>
-<h3 data-number="4.5.4" id="read-and-show"><span class="header-section-number">4.5.4</span> <code>Read</code> and <code>Show</code></h3>
-<p>The <code>Show</code> and <code>Read</code> classes are for the functions <code>show</code> and <code>read</code>, that convert values to and from Strings.</p>
-<div class="sourceCode" id="cb305"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb305-1"><a href="#cb305-1" aria-hidden="true" tabindex="-1"></a><span class="fu">show</span><span class="ot"> ::</span> <span class="dt">Show</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<h3 data-number="4.5.4" id="read-and-show"><span
+class="header-section-number">4.5.4</span> <code>Read</code> and
+<code>Show</code></h3>
+<p>The <code>Show</code> and <code>Read</code> classes are for the
+functions <code>show</code> and <code>read</code>, that convert values
+to and from Strings.</p>
+<div class="sourceCode" id="cb305"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb305-1"><a href="#cb305-1" aria-hidden="true" tabindex="-1"></a><span class="fu">show</span><span class="ot"> ::</span> <span class="dt">Show</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb305-2"><a href="#cb305-2" aria-hidden="true" tabindex="-1"></a><span class="fu">read</span><span class="ot"> ::</span> <span class="dt">Read</span> a <span class="ot">=&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> a</span></code></pre></div>
-<div class="sourceCode" id="cb306"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb306-1"><a href="#cb306-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">show</span> <span class="dv">3</span></span>
+<div class="sourceCode" id="cb306"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb306-1"><a href="#cb306-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">show</span> <span class="dv">3</span></span>
 <span id="cb306-2"><a href="#cb306-2" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;3&quot;</span></span>
 <span id="cb306-3"><a href="#cb306-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">read</span> <span class="st">&quot;3&quot;</span><span class="ot"> ::</span> <span class="dt">Int</span></span>
 <span id="cb306-4"><a href="#cb306-4" aria-hidden="true" tabindex="-1"></a><span class="dv">3</span></span>
 <span id="cb306-5"><a href="#cb306-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">read</span> <span class="st">&quot;3&quot;</span><span class="ot"> ::</span> <span class="dt">Double</span></span>
 <span id="cb306-6"><a href="#cb306-6" aria-hidden="true" tabindex="-1"></a><span class="fl">3.0</span></span></code></pre></div>
-<p>As you can see above, you often need to use a type annotation with <code>read</code> so that the compiler can choose the right implementation.</p>
-<h3 data-number="4.5.5" id="sidenote-foldable"><span class="header-section-number">4.5.5</span> Sidenote: <code>Foldable</code></h3>
-<p>One more thing! You might remember that it was mentioned earlier that the type of <code>length</code> isn’t <code>[a] -&gt; Int</code> but something more general. Let’s have a look:</p>
-<div class="sourceCode" id="cb307"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb307-1"><a href="#cb307-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="fu">length</span></span>
+<p>As you can see above, you often need to use a type annotation with
+<code>read</code> so that the compiler can choose the right
+implementation.</p>
+<h3 data-number="4.5.5" id="sidenote-foldable"><span
+class="header-section-number">4.5.5</span> Sidenote:
+<code>Foldable</code></h3>
+<p>One more thing! You might remember that it was mentioned earlier that
+the type of <code>length</code> isn’t <code>[a] -&gt; Int</code> but
+something more general. Let’s have a look:</p>
+<div class="sourceCode" id="cb307"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb307-1"><a href="#cb307-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="fu">length</span></span>
 <span id="cb307-2"><a href="#cb307-2" aria-hidden="true" tabindex="-1"></a><span class="fu">length</span><span class="ot"> ::</span> <span class="dt">Foldable</span> t <span class="ot">=&gt;</span> t a <span class="ot">-&gt;</span> <span class="dt">Int</span></span></code></pre></div>
-<p>This type looks a bit different than the ones we’ve seen before. The type variable <code>t</code> has an argument <code>a</code>. We’ll look at type classes like this in more detail in part 2, but here’s a crash course.</p>
-<p>What <code>Foldable</code> represents is types that you can fold over. The true type of <code>foldr</code> is:</p>
-<div class="sourceCode" id="cb308"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb308-1"><a href="#cb308-1" aria-hidden="true" tabindex="-1"></a><span class="fu">foldr</span><span class="ot"> ::</span> <span class="dt">Foldable</span> t <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> t a <span class="ot">-&gt;</span> b</span></code></pre></div>
-<p>We’ve succesfully used the fact that lists are <code>Foldable</code> since we’ve managed to use <code>length</code> and <code>foldr</code> on lists. However, <code>Maybe</code> is also <code>Foldable</code>! The <code>Foldable</code> instance for <code>Maybe</code> just pretends that values of <code>Maybe a</code> are like lists of length 0 or 1:</p>
-<div class="sourceCode" id="cb309"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb309-1"><a href="#cb309-1" aria-hidden="true" tabindex="-1"></a><span class="fu">foldr</span> (<span class="op">+</span>) <span class="dv">1</span> <span class="dt">Nothing</span>   <span class="op">==&gt;</span> <span class="dv">1</span></span>
+<p>This type looks a bit different than the ones we’ve seen before. The
+type variable <code>t</code> has an argument <code>a</code>. We’ll look
+at type classes like this in more detail in part 2, but here’s a crash
+course.</p>
+<p>What <code>Foldable</code> represents is types that you can fold
+over. The true type of <code>foldr</code> is:</p>
+<div class="sourceCode" id="cb308"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb308-1"><a href="#cb308-1" aria-hidden="true" tabindex="-1"></a><span class="fu">foldr</span><span class="ot"> ::</span> <span class="dt">Foldable</span> t <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> t a <span class="ot">-&gt;</span> b</span></code></pre></div>
+<p>We’ve succesfully used the fact that lists are <code>Foldable</code>
+since we’ve managed to use <code>length</code> and <code>foldr</code> on
+lists. However, <code>Maybe</code> is also <code>Foldable</code>! The
+<code>Foldable</code> instance for <code>Maybe</code> just pretends that
+values of <code>Maybe a</code> are like lists of length 0 or 1:</p>
+<div class="sourceCode" id="cb309"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb309-1"><a href="#cb309-1" aria-hidden="true" tabindex="-1"></a><span class="fu">foldr</span> (<span class="op">+</span>) <span class="dv">1</span> <span class="dt">Nothing</span>   <span class="op">==&gt;</span> <span class="dv">1</span></span>
 <span id="cb309-2"><a href="#cb309-2" aria-hidden="true" tabindex="-1"></a><span class="fu">foldr</span> (<span class="op">+</span>) <span class="dv">1</span> (<span class="dt">Just</span> <span class="dv">3</span>)  <span class="op">==&gt;</span> <span class="dv">4</span></span>
 <span id="cb309-3"><a href="#cb309-3" aria-hidden="true" tabindex="-1"></a><span class="fu">length</span> <span class="dt">Nothing</span>        <span class="op">==&gt;</span> <span class="dv">0</span></span>
 <span id="cb309-4"><a href="#cb309-4" aria-hidden="true" tabindex="-1"></a><span class="fu">length</span> (<span class="dt">Just</span> <span class="ch">&#39;a&#39;</span>)     <span class="op">==&gt;</span> <span class="dv">1</span></span></code></pre></div>
 <p>We’ll meet some more foldable types next.</p>
-<h2 data-number="4.6" id="more-data-structures"><span class="header-section-number">4.6</span> More Data Structures</h2>
-<p>Now that we are familiar with the standard type classes, we can look at one of their applications: the <code>Map</code> and <code>Array</code> data structures.</p>
-<h3 data-number="4.6.1" id="data.map"><span class="header-section-number">4.6.1</span> <code>Data.Map</code></h3>
-<p>The <code>Data.Map</code> module defines the <code>Map</code> type. Maps are search trees for key-value pairs. One way to look at this is that a value of type <code>Map k v</code> is roughly the same as a value of type <code>[(k,v)]</code>, a list of pairs. However, the operations on a map are more efficient than operations on a list.</p>
-<p>Since <code>Data.Map</code> contains some function with the same names as <code>Prelude</code> functions, the namespace needs to be imported <em>qualified</em>:</p>
-<div class="sourceCode" id="cb310"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb310-1"><a href="#cb310-1" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="kw">qualified</span> <span class="dt">Data.Map</span> <span class="kw">as</span> <span class="dt">Map</span></span></code></pre></div>
-<p>Now we can refer to the map type as <code>Map.Map</code>, and to various map functions like <code>Map.insert</code>. Here are the most important functions for maps:</p>
-<div class="sourceCode" id="cb311"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb311-1"><a href="#cb311-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Create a Map from a list of key-value pairs</span></span>
+<h2 data-number="4.6" id="more-data-structures"><span
+class="header-section-number">4.6</span> More Data Structures</h2>
+<p>Now that we are familiar with the standard type classes, we can look
+at one of their applications: the <code>Map</code> and
+<code>Array</code> data structures.</p>
+<h3 data-number="4.6.1" id="data.map"><span
+class="header-section-number">4.6.1</span> <code>Data.Map</code></h3>
+<p>The <code>Data.Map</code> module defines the <code>Map</code> type.
+Maps are search trees for key-value pairs. One way to look at this is
+that a value of type <code>Map k v</code> is roughly the same as a value
+of type <code>[(k,v)]</code>, a list of pairs. However, the operations
+on a map are more efficient than operations on a list.</p>
+<p>Since <code>Data.Map</code> contains some function with the same
+names as <code>Prelude</code> functions, the namespace needs to be
+imported <em>qualified</em>:</p>
+<div class="sourceCode" id="cb310"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb310-1"><a href="#cb310-1" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="kw">qualified</span> <span class="dt">Data.Map</span> <span class="kw">as</span> <span class="dt">Map</span></span></code></pre></div>
+<p>Now we can refer to the map type as <code>Map.Map</code>, and to
+various map functions like <code>Map.insert</code>. Here are the most
+important functions for maps:</p>
+<div class="sourceCode" id="cb311"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb311-1"><a href="#cb311-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Create a Map from a list of key-value pairs</span></span>
 <span id="cb311-2"><a href="#cb311-2" aria-hidden="true" tabindex="-1"></a>Map.fromList<span class="ot"> ::</span> <span class="dt">Ord</span> k <span class="ot">=&gt;</span> [(k, a)] <span class="ot">-&gt;</span> <span class="dt">Map.Map</span> k a</span>
 <span id="cb311-3"><a href="#cb311-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb311-4"><a href="#cb311-4" aria-hidden="true" tabindex="-1"></a><span class="co">-- Insert a value into a map. Overrides any previous value with the same key.</span></span>
@@ -2731,9 +4290,16 @@ unspecified
 <span id="cb311-10"><a href="#cb311-10" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb311-11"><a href="#cb311-11" aria-hidden="true" tabindex="-1"></a><span class="co">-- An empty map</span></span>
 <span id="cb311-12"><a href="#cb311-12" aria-hidden="true" tabindex="-1"></a>Map.empty<span class="ot"> ::</span> <span class="dt">Map.Map</span> k a</span></code></pre></div>
-<p>The <code>Ord</code> constraint for the key type of the map is needed because maps are implemented as <em>ordered binary search trees</em>.</p>
-<p>Note that like all Haskell values, maps are <em>immutable</em> meaning you can’t change a map once you define it. However, map operations like <code>insert</code> produce a <em>new</em> map. To perform multiple map operations you need to reuse the return value. Here’s a GHCi session operating on a map.</p>
-<div class="sourceCode" id="cb312"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb312-1"><a href="#cb312-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="kw">qualified</span> <span class="dt">Data.Map</span> <span class="kw">as</span> <span class="dt">Map</span></span>
+<p>The <code>Ord</code> constraint for the key type of the map is needed
+because maps are implemented as <em>ordered binary search
+trees</em>.</p>
+<p>Note that like all Haskell values, maps are <em>immutable</em>
+meaning you can’t change a map once you define it. However, map
+operations like <code>insert</code> produce a <em>new</em> map. To
+perform multiple map operations you need to reuse the return value.
+Here’s a GHCi session operating on a map.</p>
+<div class="sourceCode" id="cb312"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb312-1"><a href="#cb312-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="kw">qualified</span> <span class="dt">Data.Map</span> <span class="kw">as</span> <span class="dt">Map</span></span>
 <span id="cb312-2"><a href="#cb312-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Map</span><span class="op">&gt;</span> values <span class="ot">=</span> Map.fromList [(<span class="st">&quot;z&quot;</span>,<span class="dv">3</span>),(<span class="st">&quot;w&quot;</span>,<span class="dv">4</span>)]</span>
 <span id="cb312-3"><a href="#cb312-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Map</span><span class="op">&gt;</span> Map.lookup <span class="st">&quot;z&quot;</span> values</span>
 <span id="cb312-4"><a href="#cb312-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Just</span> <span class="dv">3</span></span>
@@ -2746,46 +4312,93 @@ unspecified
 <span id="cb312-11"><a href="#cb312-11" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Map</span><span class="op">&gt;</span> Map.insert <span class="st">&quot;x&quot;</span> <span class="dv">1</span> (Map.insert <span class="st">&quot;y&quot;</span> <span class="dv">2</span> values)   <span class="co">-- two insertions</span></span>
 <span id="cb312-12"><a href="#cb312-12" aria-hidden="true" tabindex="-1"></a>fromList [(<span class="st">&quot;w&quot;</span>,<span class="dv">4</span>),(<span class="st">&quot;x&quot;</span>,<span class="dv">1</span>),(<span class="st">&quot;y&quot;</span>,<span class="dv">2</span>),(<span class="st">&quot;z&quot;</span>,<span class="dv">3</span>)]</span>
 <span id="cb312-13"><a href="#cb312-13" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Map</span><span class="op">&gt;</span></span></code></pre></div>
-<p>Here’s an example of representing a bank as a <code>Map String Int</code> (map from account name to account balance), and withdrawing some money from an account:</p>
-<div class="sourceCode" id="cb313"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb313-1"><a href="#cb313-1" aria-hidden="true" tabindex="-1"></a><span class="ot">withdraw ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Map.Map</span> <span class="dt">String</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Map.Map</span> <span class="dt">String</span> <span class="dt">Int</span></span>
+<p>Here’s an example of representing a bank as a
+<code>Map String Int</code> (map from account name to account balance),
+and withdrawing some money from an account:</p>
+<div class="sourceCode" id="cb313"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb313-1"><a href="#cb313-1" aria-hidden="true" tabindex="-1"></a><span class="ot">withdraw ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Map.Map</span> <span class="dt">String</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Map.Map</span> <span class="dt">String</span> <span class="dt">Int</span></span>
 <span id="cb313-2"><a href="#cb313-2" aria-hidden="true" tabindex="-1"></a>withdraw account amount bank <span class="ot">=</span></span>
 <span id="cb313-3"><a href="#cb313-3" aria-hidden="true" tabindex="-1"></a>  <span class="kw">case</span> Map.lookup account bank <span class="kw">of</span></span>
 <span id="cb313-4"><a href="#cb313-4" aria-hidden="true" tabindex="-1"></a>    <span class="dt">Nothing</span>  <span class="ot">-&gt;</span> bank                                   <span class="co">-- account not found, no change</span></span>
 <span id="cb313-5"><a href="#cb313-5" aria-hidden="true" tabindex="-1"></a>    <span class="dt">Just</span> <span class="fu">sum</span> <span class="ot">-&gt;</span> Map.insert account (<span class="fu">sum</span><span class="op">-</span>amount) bank   <span class="co">-- set new balance</span></span></code></pre></div>
-<p>Here’s how you might use the <code>withdraw</code> function in <code>GHCi</code>. Note how the maps get printed as <code>fromList</code> invocations. Also note how calling <code>withdraw ... bank</code> returns a <em>new</em> bank and doesn’t change the existing bank.</p>
-<div class="sourceCode" id="cb314"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb314-1"><a href="#cb314-1" aria-hidden="true" tabindex="-1"></a><span class="dt">GHCi</span><span class="op">&gt;</span> bank <span class="ot">=</span> Map.fromList [(<span class="st">&quot;Bob&quot;</span>,<span class="dv">100</span>),(<span class="st">&quot;Mike&quot;</span>,<span class="dv">50</span>)]</span>
+<p>Here’s how you might use the <code>withdraw</code> function in
+<code>GHCi</code>. Note how the maps get printed as
+<code>fromList</code> invocations. Also note how calling
+<code>withdraw ... bank</code> returns a <em>new</em> bank and doesn’t
+change the existing bank.</p>
+<div class="sourceCode" id="cb314"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb314-1"><a href="#cb314-1" aria-hidden="true" tabindex="-1"></a><span class="dt">GHCi</span><span class="op">&gt;</span> bank <span class="ot">=</span> Map.fromList [(<span class="st">&quot;Bob&quot;</span>,<span class="dv">100</span>),(<span class="st">&quot;Mike&quot;</span>,<span class="dv">50</span>)]</span>
 <span id="cb314-2"><a href="#cb314-2" aria-hidden="true" tabindex="-1"></a><span class="dt">GHCi</span><span class="op">&gt;</span> withdraw <span class="st">&quot;Bob&quot;</span> <span class="dv">80</span> bank</span>
 <span id="cb314-3"><a href="#cb314-3" aria-hidden="true" tabindex="-1"></a>fromList [(<span class="st">&quot;Bob&quot;</span>,<span class="dv">20</span>),(<span class="st">&quot;Mike&quot;</span>,<span class="dv">50</span>)]</span>
 <span id="cb314-4"><a href="#cb314-4" aria-hidden="true" tabindex="-1"></a><span class="dt">GHCi</span><span class="op">&gt;</span> bank                         <span class="co">-- note immutability</span></span>
 <span id="cb314-5"><a href="#cb314-5" aria-hidden="true" tabindex="-1"></a>fromList [(<span class="st">&quot;Bob&quot;</span>,<span class="dv">100</span>),(<span class="st">&quot;Mike&quot;</span>,<span class="dv">50</span>)]</span>
 <span id="cb314-6"><a href="#cb314-6" aria-hidden="true" tabindex="-1"></a><span class="dt">GHCi</span><span class="op">&gt;</span> withdraw <span class="st">&quot;Bozo&quot;</span> <span class="dv">1000</span> bank</span>
 <span id="cb314-7"><a href="#cb314-7" aria-hidden="true" tabindex="-1"></a>fromList [(<span class="st">&quot;Bob&quot;</span>,<span class="dv">100</span>),(<span class="st">&quot;Mike&quot;</span>,<span class="dv">50</span>)]</span></code></pre></div>
-<p><code>Data.Map</code> defines all sorts of useful higher-order functions for updating maps. We can rewrite the <code>withdraw</code> function using <code>Data.Map.adjust</code>:</p>
-<div class="sourceCode" id="cb315"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb315-1"><a href="#cb315-1" aria-hidden="true" tabindex="-1"></a><span class="ot">withdraw ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Map.Map</span> <span class="dt">String</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Map.Map</span> <span class="dt">String</span> <span class="dt">Int</span></span>
+<p><code>Data.Map</code> defines all sorts of useful higher-order
+functions for updating maps. We can rewrite the <code>withdraw</code>
+function using <code>Data.Map.adjust</code>:</p>
+<div class="sourceCode" id="cb315"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb315-1"><a href="#cb315-1" aria-hidden="true" tabindex="-1"></a><span class="ot">withdraw ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Map.Map</span> <span class="dt">String</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Map.Map</span> <span class="dt">String</span> <span class="dt">Int</span></span>
 <span id="cb315-2"><a href="#cb315-2" aria-hidden="true" tabindex="-1"></a>withdraw account amount bank <span class="ot">=</span> Map.adjust (\x <span class="ot">-&gt;</span> x<span class="op">-</span>amount) account bank</span></code></pre></div>
-<p><strong>Note!</strong> There are separate <code>Data.Map.Strict</code> and <code>Data.Map.Lazy</code> implementations. When you import <code>Data.Map</code> you get <code>Data.Map.Lazy</code>. You can find the documentation for all the <code>Data.Map</code> functions in <a href="https://hackage.haskell.org/package/containers-0.6.5.1/docs/Data-Map-Lazy.html">the docs for <code>Data.Map.Lazy</code></a>. We won’t go into their differences here, but mostly you should use <code>Data.Map.Strict</code> in real code.</p>
-<h3 data-number="4.6.2" id="data.array"><span class="header-section-number">4.6.2</span> <code>Data.Array</code></h3>
-<p>Another type that works kind of like a list but is more efficient for some operations is the array. Arrays are familiar from many other programming languages, but Haskell arrays are a bit different.</p>
-<p>Unlike the <code>Data.Map</code> module, the <code>Data.Array</code> can just be imported normally:</p>
-<div class="sourceCode" id="cb316"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb316-1"><a href="#cb316-1" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="dt">Data.Array</span></span></code></pre></div>
-<p>Now we can look at the type of the <code>array</code> function that constructs an array.</p>
-<div class="sourceCode" id="cb317"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb317-1"><a href="#cb317-1" aria-hidden="true" tabindex="-1"></a><span class="ot">array ::</span> <span class="dt">Ix</span> i <span class="ot">=&gt;</span> (i, i) <span class="ot">-&gt;</span> [(i, e)] <span class="ot">-&gt;</span> <span class="dt">Array</span> i e</span></code></pre></div>
-<p>There are a couple of things to notice here. First of all, the <code>Array</code> type is parameterized by <em>two</em> types: the index type and the element type. Most other programming languages only parameterize arrays with the element type, but the index type is always <code>int</code>. In Haskell, we can have, for example, an <code>Array Char Int</code>: an array indexed by characters, or <code>Array Bool String</code>, an array indexed by booleans, or even <code>Array (Int,Int) Int</code>, a two-dimensional array of ints.</p>
-<p>Not all types can be index types. Only types that are similar to integers are suitable. That is the reason for the <code>Ix i</code> class constraint. The <code>Ix</code> class collects all the types that can be used as array indices.</p>
-<p>Secondly, the <code>array</code> function takes an extra <code>(i,i)</code> parameter. These are the minimum and maximum indices of the array. Unlike some other languages, where arrays always start at index 0 or 1, in Haskell you can define an array that starts from 7 and goes to 11. So here’s that array:</p>
-<div class="sourceCode" id="cb318"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb318-1"><a href="#cb318-1" aria-hidden="true" tabindex="-1"></a><span class="ot">myArray ::</span> <span class="dt">Array</span> <span class="dt">Int</span> <span class="dt">String</span></span>
+<p><strong>Note!</strong> There are separate
+<code>Data.Map.Strict</code> and <code>Data.Map.Lazy</code>
+implementations. When you import <code>Data.Map</code> you get
+<code>Data.Map.Lazy</code>. You can find the documentation for all the
+<code>Data.Map</code> functions in <a
+href="https://hackage.haskell.org/package/containers-0.6.5.1/docs/Data-Map-Lazy.html">the
+docs for <code>Data.Map.Lazy</code></a>. We won’t go into their
+differences here, but mostly you should use <code>Data.Map.Strict</code>
+in real code.</p>
+<h3 data-number="4.6.2" id="data.array"><span
+class="header-section-number">4.6.2</span> <code>Data.Array</code></h3>
+<p>Another type that works kind of like a list but is more efficient for
+some operations is the array. Arrays are familiar from many other
+programming languages, but Haskell arrays are a bit different.</p>
+<p>Unlike the <code>Data.Map</code> module, the <code>Data.Array</code>
+can just be imported normally:</p>
+<div class="sourceCode" id="cb316"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb316-1"><a href="#cb316-1" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="dt">Data.Array</span></span></code></pre></div>
+<p>Now we can look at the type of the <code>array</code> function that
+constructs an array.</p>
+<div class="sourceCode" id="cb317"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb317-1"><a href="#cb317-1" aria-hidden="true" tabindex="-1"></a><span class="ot">array ::</span> <span class="dt">Ix</span> i <span class="ot">=&gt;</span> (i, i) <span class="ot">-&gt;</span> [(i, e)] <span class="ot">-&gt;</span> <span class="dt">Array</span> i e</span></code></pre></div>
+<p>There are a couple of things to notice here. First of all, the
+<code>Array</code> type is parameterized by <em>two</em> types: the
+index type and the element type. Most other programming languages only
+parameterize arrays with the element type, but the index type is always
+<code>int</code>. In Haskell, we can have, for example, an
+<code>Array Char Int</code>: an array indexed by characters, or
+<code>Array Bool String</code>, an array indexed by booleans, or even
+<code>Array (Int,Int) Int</code>, a two-dimensional array of ints.</p>
+<p>Not all types can be index types. Only types that are similar to
+integers are suitable. That is the reason for the <code>Ix i</code>
+class constraint. The <code>Ix</code> class collects all the types that
+can be used as array indices.</p>
+<p>Secondly, the <code>array</code> function takes an extra
+<code>(i,i)</code> parameter. These are the minimum and maximum indices
+of the array. Unlike some other languages, where arrays always start at
+index 0 or 1, in Haskell you can define an array that starts from 7 and
+goes to 11. So here’s that array:</p>
+<div class="sourceCode" id="cb318"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb318-1"><a href="#cb318-1" aria-hidden="true" tabindex="-1"></a><span class="ot">myArray ::</span> <span class="dt">Array</span> <span class="dt">Int</span> <span class="dt">String</span></span>
 <span id="cb318-2"><a href="#cb318-2" aria-hidden="true" tabindex="-1"></a>myArray <span class="ot">=</span> array (<span class="dv">7</span>,<span class="dv">11</span>) [(<span class="dv">7</span>,<span class="st">&quot;seven&quot;</span>), (<span class="dv">8</span>,<span class="st">&quot;eight&quot;</span>), (<span class="dv">9</span>,<span class="st">&quot;nine&quot;</span>), (<span class="dv">10</span>,<span class="st">&quot;ten&quot;</span>), (<span class="dv">11</span>,<span class="st">&quot;ELEVEN&quot;</span>)]</span></code></pre></div>
-<p>Listing all the indices and elements in order can be a bit cumbersome, so there’s also the <code>listArray</code> constructor that just takes a list of elements in order:</p>
-<div class="sourceCode" id="cb319"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb319-1"><a href="#cb319-1" aria-hidden="true" tabindex="-1"></a><span class="ot">listArray ::</span> <span class="dt">Ix</span> i <span class="ot">=&gt;</span> (i, i) <span class="ot">-&gt;</span> [e] <span class="ot">-&gt;</span> <span class="dt">Array</span> i e</span></code></pre></div>
-<div class="sourceCode" id="cb320"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb320-1"><a href="#cb320-1" aria-hidden="true" tabindex="-1"></a><span class="ot">myArray ::</span> <span class="dt">Array</span> <span class="dt">Int</span> <span class="dt">String</span></span>
+<p>Listing all the indices and elements in order can be a bit
+cumbersome, so there’s also the <code>listArray</code> constructor that
+just takes a list of elements in order:</p>
+<div class="sourceCode" id="cb319"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb319-1"><a href="#cb319-1" aria-hidden="true" tabindex="-1"></a><span class="ot">listArray ::</span> <span class="dt">Ix</span> i <span class="ot">=&gt;</span> (i, i) <span class="ot">-&gt;</span> [e] <span class="ot">-&gt;</span> <span class="dt">Array</span> i e</span></code></pre></div>
+<div class="sourceCode" id="cb320"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb320-1"><a href="#cb320-1" aria-hidden="true" tabindex="-1"></a><span class="ot">myArray ::</span> <span class="dt">Array</span> <span class="dt">Int</span> <span class="dt">String</span></span>
 <span id="cb320-2"><a href="#cb320-2" aria-hidden="true" tabindex="-1"></a>myArray <span class="ot">=</span> listArray (<span class="dv">7</span>,<span class="dv">11</span>) [<span class="st">&quot;seven&quot;</span>, <span class="st">&quot;eight&quot;</span>, <span class="st">&quot;nine&quot;</span>, <span class="st">&quot;ten&quot;</span>, <span class="st">&quot;ELEVEN&quot;</span>]</span></code></pre></div>
 <p>Arrays are used with two new operators:</p>
-<div class="sourceCode" id="cb321"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb321-1"><a href="#cb321-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Array lookup</span></span>
+<div class="sourceCode" id="cb321"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb321-1"><a href="#cb321-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Array lookup</span></span>
 <span id="cb321-2"><a href="#cb321-2" aria-hidden="true" tabindex="-1"></a><span class="ot">(!) ::</span> <span class="dt">Ix</span> i <span class="ot">=&gt;</span> <span class="dt">Array</span> i e <span class="ot">-&gt;</span> i <span class="ot">-&gt;</span> e</span>
 <span id="cb321-3"><a href="#cb321-3" aria-hidden="true" tabindex="-1"></a><span class="co">-- Array update</span></span>
 <span id="cb321-4"><a href="#cb321-4" aria-hidden="true" tabindex="-1"></a><span class="ot">(//) ::</span> <span class="dt">Ix</span> i <span class="ot">=&gt;</span> <span class="dt">Array</span> i e <span class="ot">-&gt;</span> [(i, e)] <span class="ot">-&gt;</span> <span class="dt">Array</span> i e</span></code></pre></div>
 <p>Here’s an example <code>GHCi</code> session:</p>
-<div class="sourceCode" id="cb322"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb322-1"><a href="#cb322-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="dt">Data.Array</span></span>
+<div class="sourceCode" id="cb322"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb322-1"><a href="#cb322-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="dt">Data.Array</span></span>
 <span id="cb322-2"><a href="#cb322-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.Array</span><span class="op">&gt;</span> myArray <span class="ot">=</span> listArray (<span class="dv">7</span>,<span class="dv">11</span>) [<span class="st">&quot;seven&quot;</span>, <span class="st">&quot;eight&quot;</span>, <span class="st">&quot;nine&quot;</span>, <span class="st">&quot;ten&quot;</span>, <span class="st">&quot;ELEVEN&quot;</span>]</span>
 <span id="cb322-3"><a href="#cb322-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.Array</span><span class="op">&gt;</span> myArray</span>
 <span id="cb322-4"><a href="#cb322-4" aria-hidden="true" tabindex="-1"></a>array (<span class="dv">7</span>,<span class="dv">11</span>) [(<span class="dv">7</span>,<span class="st">&quot;seven&quot;</span>),(<span class="dv">8</span>,<span class="st">&quot;eight&quot;</span>),(<span class="dv">9</span>,<span class="st">&quot;nine&quot;</span>),(<span class="dv">10</span>,<span class="st">&quot;ten&quot;</span>),(<span class="dv">11</span>,<span class="st">&quot;ELEVEN&quot;</span>)]</span>
@@ -2793,29 +4406,79 @@ unspecified
 <span id="cb322-6"><a href="#cb322-6" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;eight&quot;</span></span>
 <span id="cb322-7"><a href="#cb322-7" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.Array</span><span class="op">&gt;</span> myArray <span class="op">//</span> [(<span class="dv">8</span>,<span class="st">&quot;ocho&quot;</span>),(<span class="dv">9</span>,<span class="st">&quot;nueve&quot;</span>)]</span>
 <span id="cb322-8"><a href="#cb322-8" aria-hidden="true" tabindex="-1"></a>array (<span class="dv">7</span>,<span class="dv">11</span>) [(<span class="dv">7</span>,<span class="st">&quot;seven&quot;</span>),(<span class="dv">8</span>,<span class="st">&quot;ocho&quot;</span>),(<span class="dv">9</span>,<span class="st">&quot;nueve&quot;</span>),(<span class="dv">10</span>,<span class="st">&quot;ten&quot;</span>),(<span class="dv">11</span>,<span class="st">&quot;ELEVEN&quot;</span>)]</span></code></pre></div>
-<p>You might be wondering why the <code>(//)</code> operator does multiple updates at once. The reason is the main weakness of Haskell arrays: immutability. Since arrays can’t be changed in place, <code>(//)</code> must copy the whole array. This is why in Haskell it’s often preferable to use lists or maps to store data that needs to be updated. However, arrays may still be useful when constructed once and then used for a large number of lookups. We’ll get back to how Haskell data structures work in the next lecture.</p>
-<p><strong>Note!</strong> In this course we’ll use only <code>Array</code>, a simple array type that’s specified in the Haskell standard. There are many other array types like the mutable <code>IOArray</code> and the somewhat obscure <code>DiffArray</code>. There are also type classes for arrays like <code>IArray</code> and <code>MArray</code>. In addition to arrays there is a <a href="https://hackage.haskell.org/package/vector">wide family of <code>Vector</code> types</a> that can be more practical than <code>Array</code> for real programs.</p>
-<h3 data-number="4.6.3" id="sidenote-folding-over-maps-arrays"><span class="header-section-number">4.6.3</span> Sidenote: Folding over Maps &amp; Arrays</h3>
-<p>The <code>Map</code> and <code>Array</code> type are instances of <code>Foldable</code> just like lists are! This means you can use functions like <code>length</code> and <code>foldr</code> on them:</p>
-<div class="sourceCode" id="cb323"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb323-1"><a href="#cb323-1" aria-hidden="true" tabindex="-1"></a><span class="fu">length</span> (array (<span class="dv">7</span>,<span class="dv">11</span>) [(<span class="dv">7</span>,<span class="st">&quot;seven&quot;</span>),(<span class="dv">8</span>,<span class="st">&quot;eight&quot;</span>),(<span class="dv">9</span>,<span class="st">&quot;nine&quot;</span>),(<span class="dv">10</span>,<span class="st">&quot;ten&quot;</span>),(<span class="dv">11</span>,<span class="st">&quot;ELEVEN&quot;</span>)])</span>
+<p>You might be wondering why the <code>(//)</code> operator does
+multiple updates at once. The reason is the main weakness of Haskell
+arrays: immutability. Since arrays can’t be changed in place,
+<code>(//)</code> must copy the whole array. This is why in Haskell it’s
+often preferable to use lists or maps to store data that needs to be
+updated. However, arrays may still be useful when constructed once and
+then used for a large number of lookups. We’ll get back to how Haskell
+data structures work in the next lecture.</p>
+<p><strong>Note!</strong> In this course we’ll use only
+<code>Array</code>, a simple array type that’s specified in the Haskell
+standard. There are many other array types like the mutable
+<code>IOArray</code> and the somewhat obscure <code>DiffArray</code>.
+There are also type classes for arrays like <code>IArray</code> and
+<code>MArray</code>. In addition to arrays there is a <a
+href="https://hackage.haskell.org/package/vector">wide family of
+<code>Vector</code> types</a> that can be more practical than
+<code>Array</code> for real programs.</p>
+<h3 data-number="4.6.3" id="sidenote-folding-over-maps-arrays"><span
+class="header-section-number">4.6.3</span> Sidenote: Folding over Maps
+&amp; Arrays</h3>
+<p>The <code>Map</code> and <code>Array</code> type are instances of
+<code>Foldable</code> just like lists are! This means you can use
+functions like <code>length</code> and <code>foldr</code> on them:</p>
+<div class="sourceCode" id="cb323"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb323-1"><a href="#cb323-1" aria-hidden="true" tabindex="-1"></a><span class="fu">length</span> (array (<span class="dv">7</span>,<span class="dv">11</span>) [(<span class="dv">7</span>,<span class="st">&quot;seven&quot;</span>),(<span class="dv">8</span>,<span class="st">&quot;eight&quot;</span>),(<span class="dv">9</span>,<span class="st">&quot;nine&quot;</span>),(<span class="dv">10</span>,<span class="st">&quot;ten&quot;</span>),(<span class="dv">11</span>,<span class="st">&quot;ELEVEN&quot;</span>)])</span>
 <span id="cb323-2"><a href="#cb323-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">5</span></span>
 <span id="cb323-3"><a href="#cb323-3" aria-hidden="true" tabindex="-1"></a><span class="fu">foldr</span> (<span class="op">+</span>) <span class="dv">0</span> (Map.fromList [(<span class="st">&quot;banana&quot;</span>,<span class="dv">3</span>),(<span class="st">&quot;egg&quot;</span>,<span class="dv">7</span>)])</span>
 <span id="cb323-4"><a href="#cb323-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">10</span></span></code></pre></div>
-<h2 data-number="4.7" id="reading-docs"><span class="header-section-number">4.7</span> Reading Docs</h2>
-<p>Haskell libraries tend to have pretty good docs. We’ve linked to docs via Hackage (<a href="https://hackage.haskell.org" class="uri">https://hackage.haskell.org</a>) previously, but it’s important to know how to find the docs by yourself too. The tool for generating Haskell documentation is called <em>Haddock</em> so sometimes Haskell docs are referred to as <em>haddocks</em>.</p>
-<p>Hackage is the Haskell package repository (just like <a href="https://pypi.org/">PyPI</a> for Python, Maven Central for Java or <a href="https://npmjs.com">NPM</a> for Javascript). In addition to the actual packages, it hosts documentation for them. Most of the modules that we use on this course are in the package called <code>base</code>. You can browse the docs for the base package at <a href="https://hackage.haskell.org/package/base-4.16.4.0/" class="uri">https://hackage.haskell.org/package/base-4.16.4.0/</a>.</p>
-<p>When you’re not quite sure where the function you’re looking for is, Hoogle (<a href="https://hoogle.haskell.org/" class="uri">https://hoogle.haskell.org/</a>) can help. Hoogle is a search engine for Haskell documentation. It is a great resource when you need to check what was the type of <code>foldr</code> or which packages contain a function named <code>reverse</code>.</p>
-<p>Finally, since this course is using the <code>stack</code> tool, you can also browse the documentation for the libraries stack has installed for you with the commands</p>
+<h2 data-number="4.7" id="reading-docs"><span
+class="header-section-number">4.7</span> Reading Docs</h2>
+<p>Haskell libraries tend to have pretty good docs. We’ve linked to docs
+via Hackage (<a href="https://hackage.haskell.org"
+class="uri">https://hackage.haskell.org</a>) previously, but it’s
+important to know how to find the docs by yourself too. The tool for
+generating Haskell documentation is called <em>Haddock</em> so sometimes
+Haskell docs are referred to as <em>haddocks</em>.</p>
+<p>Hackage is the Haskell package repository (just like <a
+href="https://pypi.org/">PyPI</a> for Python, Maven Central for Java or
+<a href="https://npmjs.com">NPM</a> for Javascript). In addition to the
+actual packages, it hosts documentation for them. Most of the modules
+that we use on this course are in the package called <code>base</code>.
+You can browse the docs for the base package at <a
+href="https://hackage.haskell.org/package/base-4.16.4.0/"
+class="uri">https://hackage.haskell.org/package/base-4.16.4.0/</a>.</p>
+<p>When you’re not quite sure where the function you’re looking for is,
+Hoogle (<a href="https://hoogle.haskell.org/"
+class="uri">https://hoogle.haskell.org/</a>) can help. Hoogle is a
+search engine for Haskell documentation. It is a great resource when you
+need to check what was the type of <code>foldr</code> or which packages
+contain a function named <code>reverse</code>.</p>
+<p>Finally, since this course is using the <code>stack</code> tool, you
+can also browse the documentation for the libraries stack has installed
+for you with the commands</p>
 <pre><code>stack haddock --open
 stack haddock --open &lt;package&gt;</code></pre>
-<p>This has the added benefit of getting exactly the right version of the documentation.</p>
-<p>In summary, here are the main ways of reading Haskell library documentation:</p>
+<p>This has the added benefit of getting exactly the right version of
+the documentation.</p>
+<p>In summary, here are the main ways of reading Haskell library
+documentation:</p>
 <ul>
-<li>If you know the name of the package you browse the docs via <a href="https://hackage.haskell.org/" class="uri">https://hackage.haskell.org/</a>.</li>
-<li>If you know the name of the function you can find it using <a href="https://hoogle.haskell.org/" class="uri">https://hoogle.haskell.org/</a>.</li>
-<li>If you’re using <code>stack</code> you can use <code>stack haddock --open</code> or <code>stack haddock --open &lt;package&gt;</code> to open docs in your browser.</li>
+<li>If you know the name of the package you browse the docs via <a
+href="https://hackage.haskell.org/"
+class="uri">https://hackage.haskell.org/</a>.</li>
+<li>If you know the name of the function you can find it using <a
+href="https://hoogle.haskell.org/"
+class="uri">https://hoogle.haskell.org/</a>.</li>
+<li>If you’re using <code>stack</code> you can use
+<code>stack haddock --open</code> or
+<code>stack haddock --open &lt;package&gt;</code> to open docs in your
+browser.</li>
 </ul>
-<h2 data-number="4.8" id="quiz-3"><span class="header-section-number">4.8</span> Quiz</h2>
+<h2 data-number="4.8" id="quiz-3"><span
+class="header-section-number">4.8</span> Quiz</h2>
 <p>What is the type of <code>swap . swap</code>?</p>
 <ol class="quiz">
 <li class="correct">
@@ -2840,7 +4503,8 @@ stack haddock --open &lt;package&gt;</code></pre>
 <code>(a -&gt; b) -&gt; (b -&gt; a) -&gt; a -&gt; (b, a)</code>
 </li>
 </ol>
-<p>What is the type of <code>\t -&gt; (fst . fst $ t, (snd . fst $ t, snd t))</code>?</p>
+<p>What is the type of
+<code>\t -&gt; (fst . fst $ t, (snd . fst $ t, snd t))</code>?</p>
 <ol class="quiz">
 <li>
 <code>(a, (b, c)) -&gt; (a, (b, c))</code>
@@ -2852,7 +4516,8 @@ stack haddock --open &lt;package&gt;</code></pre>
 <code>((a, b), c) -&gt; (a, (b, c))</code>
 </li>
 </ol>
-<p>What does the function <code>foldr (\x xs -&gt; xs ++ [x]) []</code> do?</p>
+<p>What does the function <code>foldr (\x xs -&gt; xs ++ [x]) []</code>
+do?</p>
 <ol class="quiz">
 <li>
 It doesn’t change its input list at all
@@ -2864,7 +4529,8 @@ It changes the associativity of a list from left to right
 It reverses its input list
 </li>
 </ol>
-<p>What does the function <code>foldr (\(x, y) zs -&gt; x : y : zs) []</code> do?</p>
+<p>What does the function
+<code>foldr (\(x, y) zs -&gt; x : y : zs) []</code> do?</p>
 <ol class="quiz">
 <li>
 It turns a list of pairs into a pair of lists
@@ -2876,7 +4542,8 @@ It turns a pair of lists into a list of pairs
 It turns a list of pairs into a list of elements
 </li>
 </ol>
-<p>What is the type of <code>foldr (\n b -&gt; n == 3 &amp;&amp; b)</code>?</p>
+<p>What is the type of
+<code>foldr (\n b -&gt; n == 3 &amp;&amp; b)</code>?</p>
 <ol class="quiz">
 <li class="correct">
 <code>(Foldable t, Eq a, Num a) =&gt; Bool -&gt; t a -&gt; Bool</code>
@@ -2888,7 +4555,8 @@ It turns a list of pairs into a list of elements
 <code>(Foldable t, Eq a, Num a) =&gt; Bool -&gt; [ a ] -&gt; Bool</code>
 </li>
 </ol>
-<p>What is the type of <code>\x -&gt; case x of (True, "Foo") -&gt; show True ++ "Foo"</code>?</p>
+<p>What is the type of
+<code>\x -&gt; case x of (True, "Foo") -&gt; show True ++ "Foo"</code>?</p>
 <ol class="quiz">
 <li>
 <code>Either Bool String -&gt; String</code>
@@ -2901,22 +4569,34 @@ It turns a list of pairs into a list of elements
 </li>
 </ol>
 <!-- TODO more, maybe some Array and Map questions -->
-<h2 data-number="4.9" id="exercises-3"><span class="header-section-number">4.9</span> Exercises</h2>
+<h2 data-number="4.9" id="exercises-3"><span
+class="header-section-number">4.9</span> Exercises</h2>
 <ul>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set4a.hs">Set4a</a>: type classes</li>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set4b.hs">Set4b</a>: folds</li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set4a.hs">Set4a</a>:
+type classes</li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set4b.hs">Set4b</a>:
+folds</li>
 </ul>
-<h1 data-number="5" id="lecture-5-you-need-string-for-a-knot"><span class="header-section-number">5</span> Lecture 5: You Need String for a Knot</h1>
+<h1 data-number="5" id="lecture-5-you-need-string-for-a-knot"><span
+class="header-section-number">5</span> Lecture 5: You Need String for a
+Knot</h1>
 <ul>
 <li>Type system</li>
 <li>Defining custom types</li>
 </ul>
-<h2 data-number="5.1" id="algebraic-datatypes"><span class="header-section-number">5.1</span> Algebraic Datatypes</h2>
-<p>Haskell has a system called <em>algebraic datatypes</em> for defining new types. This sounds fancy, but is rather simple. Let’s dive in by looking at the standard library definitions of some familiar types:</p>
-<div class="sourceCode" id="cb325"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb325-1"><a href="#cb325-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Bool</span> <span class="ot">=</span> <span class="dt">True</span> <span class="op">|</span> <span class="dt">False</span></span>
+<h2 data-number="5.1" id="algebraic-datatypes"><span
+class="header-section-number">5.1</span> Algebraic Datatypes</h2>
+<p>Haskell has a system called <em>algebraic datatypes</em> for defining
+new types. This sounds fancy, but is rather simple. Let’s dive in by
+looking at the standard library definitions of some familiar types:</p>
+<div class="sourceCode" id="cb325"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb325-1"><a href="#cb325-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Bool</span> <span class="ot">=</span> <span class="dt">True</span> <span class="op">|</span> <span class="dt">False</span></span>
 <span id="cb325-2"><a href="#cb325-2" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Ordering</span> <span class="ot">=</span> <span class="dt">LT</span> <span class="op">|</span> <span class="dt">EQ</span> <span class="op">|</span> <span class="dt">GT</span></span></code></pre></div>
 <p>With this syntax you too can define types:</p>
-<div class="sourceCode" id="cb326"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb326-1"><a href="#cb326-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- definition of a type with three values</span></span>
+<div class="sourceCode" id="cb326"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb326-1"><a href="#cb326-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- definition of a type with three values</span></span>
 <span id="cb326-2"><a href="#cb326-2" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Color</span> <span class="ot">=</span> <span class="dt">Red</span> <span class="op">|</span> <span class="dt">Green</span> <span class="op">|</span> <span class="dt">Blue</span></span>
 <span id="cb326-3"><a href="#cb326-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb326-4"><a href="#cb326-4" aria-hidden="true" tabindex="-1"></a><span class="co">-- a function that uses pattern matching on our new type</span></span>
@@ -2924,29 +4604,48 @@ It turns a list of pairs into a list of elements
 <span id="cb326-6"><a href="#cb326-6" aria-hidden="true" tabindex="-1"></a>rgb <span class="dt">Red</span> <span class="ot">=</span> [<span class="dv">1</span>,<span class="dv">0</span>,<span class="dv">0</span>]</span>
 <span id="cb326-7"><a href="#cb326-7" aria-hidden="true" tabindex="-1"></a>rgb <span class="dt">Green</span> <span class="ot">=</span> [<span class="dv">0</span>,<span class="dv">1</span>,<span class="dv">0</span>]</span>
 <span id="cb326-8"><a href="#cb326-8" aria-hidden="true" tabindex="-1"></a>rgb <span class="dt">Blue</span> <span class="ot">=</span> [<span class="dv">0</span>,<span class="dv">0</span>,<span class="dv">1</span>]</span></code></pre></div>
-<div class="sourceCode" id="cb327"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb327-1"><a href="#cb327-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="dt">Red</span></span>
+<div class="sourceCode" id="cb327"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb327-1"><a href="#cb327-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="dt">Red</span></span>
 <span id="cb327-2"><a href="#cb327-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Red</span><span class="ot"> ::</span> <span class="dt">Color</span></span>
 <span id="cb327-3"><a href="#cb327-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t [<span class="dt">Red</span>,<span class="dt">Blue</span>,<span class="dt">Green</span>]</span>
 <span id="cb327-4"><a href="#cb327-4" aria-hidden="true" tabindex="-1"></a>[<span class="dt">Red</span>,<span class="dt">Blue</span>,<span class="dt">Green</span>]<span class="ot"> ::</span> [<span class="dt">Color</span>]</span>
 <span id="cb327-5"><a href="#cb327-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> rgb <span class="dt">Red</span></span>
 <span id="cb327-6"><a href="#cb327-6" aria-hidden="true" tabindex="-1"></a>[<span class="fl">1.0</span>,<span class="fl">0.0</span>,<span class="fl">0.0</span>]</span></code></pre></div>
-<h3 data-number="5.1.1" id="fields"><span class="header-section-number">5.1.1</span> Fields</h3>
-<p>Types like <code>Bool</code>, <code>Ordering</code> and <code>Color</code> that just list a bunch of constants are called <em>enumerations</em> or <em>enums</em> in Haskell and other languages. Enums are useful, but you need other types as well. Here we define a type for reports containing an id number, a title, and a body:</p>
-<div class="sourceCode" id="cb328"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb328-1"><a href="#cb328-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Report</span> <span class="ot">=</span> <span class="dt">ConstructReport</span> <span class="dt">Int</span> <span class="dt">String</span> <span class="dt">String</span></span></code></pre></div>
+<h3 data-number="5.1.1" id="fields"><span
+class="header-section-number">5.1.1</span> Fields</h3>
+<p>Types like <code>Bool</code>, <code>Ordering</code> and
+<code>Color</code> that just list a bunch of constants are called
+<em>enumerations</em> or <em>enums</em> in Haskell and other languages.
+Enums are useful, but you need other types as well. Here we define a
+type for reports containing an id number, a title, and a body:</p>
+<div class="sourceCode" id="cb328"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb328-1"><a href="#cb328-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Report</span> <span class="ot">=</span> <span class="dt">ConstructReport</span> <span class="dt">Int</span> <span class="dt">String</span> <span class="dt">String</span></span></code></pre></div>
 <p>This is how you create a report:</p>
-<div class="sourceCode" id="cb329"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb329-1"><a href="#cb329-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="dt">ConstructReport</span> <span class="dv">1</span> <span class="st">&quot;Title&quot;</span> <span class="st">&quot;This is the body.&quot;</span></span>
+<div class="sourceCode" id="cb329"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb329-1"><a href="#cb329-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="dt">ConstructReport</span> <span class="dv">1</span> <span class="st">&quot;Title&quot;</span> <span class="st">&quot;This is the body.&quot;</span></span>
 <span id="cb329-2"><a href="#cb329-2" aria-hidden="true" tabindex="-1"></a><span class="dt">ConstructReport</span> <span class="dv">1</span> <span class="st">&quot;Title&quot;</span> <span class="st">&quot;This is the body.&quot;</span><span class="ot"> ::</span> <span class="dt">Report</span></span></code></pre></div>
 <p>You can access the fields with pattern matching:</p>
-<div class="sourceCode" id="cb330"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb330-1"><a href="#cb330-1" aria-hidden="true" tabindex="-1"></a><span class="ot">reportContents ::</span> <span class="dt">Report</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<div class="sourceCode" id="cb330"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb330-1"><a href="#cb330-1" aria-hidden="true" tabindex="-1"></a><span class="ot">reportContents ::</span> <span class="dt">Report</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb330-2"><a href="#cb330-2" aria-hidden="true" tabindex="-1"></a>reportContents (<span class="dt">ConstructReport</span> <span class="fu">id</span> title contents) <span class="ot">=</span> contents</span>
 <span id="cb330-3"><a href="#cb330-3" aria-hidden="true" tabindex="-1"></a><span class="ot">setReportContents ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Report</span> <span class="ot">-&gt;</span> <span class="dt">Report</span></span>
 <span id="cb330-4"><a href="#cb330-4" aria-hidden="true" tabindex="-1"></a>setReportContents contents (<span class="dt">ConstructReport</span> <span class="fu">id</span> title _contents) <span class="ot">=</span> <span class="dt">ConstructReport</span> <span class="fu">id</span> title contents</span></code></pre></div>
-<h3 data-number="5.1.2" id="constructors"><span class="header-section-number">5.1.2</span> Constructors</h3>
-<p>The things on the right hand side of a <code>data</code> declaration are called <em>constructors</em>. <code>True</code>, <code>False</code>, <code>Red</code> and <code>ConstructReport</code> are all examples of constructors. A type can have multiple constructors, and a constructor can have zero or more fields.</p>
-<p>Here is a datatype for a standard playing card. It has five constructors, of which <code>Joker</code> has zero fields and the others have one field.</p>
-<div class="sourceCode" id="cb331"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb331-1"><a href="#cb331-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Card</span> <span class="ot">=</span> <span class="dt">Joker</span> <span class="op">|</span> <span class="dt">Heart</span> <span class="dt">Int</span> <span class="op">|</span> <span class="dt">Club</span> <span class="dt">Int</span> <span class="op">|</span> <span class="dt">Spade</span> <span class="dt">Int</span> <span class="op">|</span> <span class="dt">Diamond</span> <span class="dt">Int</span></span></code></pre></div>
-<p>Constructors with fields have function type and can be used wherever functions can:</p>
-<div class="sourceCode" id="cb332"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb332-1"><a href="#cb332-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="dt">Heart</span></span>
+<h3 data-number="5.1.2" id="constructors"><span
+class="header-section-number">5.1.2</span> Constructors</h3>
+<p>The things on the right hand side of a <code>data</code> declaration
+are called <em>constructors</em>. <code>True</code>, <code>False</code>,
+<code>Red</code> and <code>ConstructReport</code> are all examples of
+constructors. A type can have multiple constructors, and a constructor
+can have zero or more fields.</p>
+<p>Here is a datatype for a standard playing card. It has five
+constructors, of which <code>Joker</code> has zero fields and the others
+have one field.</p>
+<div class="sourceCode" id="cb331"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb331-1"><a href="#cb331-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Card</span> <span class="ot">=</span> <span class="dt">Joker</span> <span class="op">|</span> <span class="dt">Heart</span> <span class="dt">Int</span> <span class="op">|</span> <span class="dt">Club</span> <span class="dt">Int</span> <span class="op">|</span> <span class="dt">Spade</span> <span class="dt">Int</span> <span class="op">|</span> <span class="dt">Diamond</span> <span class="dt">Int</span></span></code></pre></div>
+<p>Constructors with fields have function type and can be used wherever
+functions can:</p>
+<div class="sourceCode" id="cb332"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb332-1"><a href="#cb332-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="dt">Heart</span></span>
 <span id="cb332-2"><a href="#cb332-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Heart</span><span class="ot"> ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Card</span></span>
 <span id="cb332-3"><a href="#cb332-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="dt">Club</span></span>
 <span id="cb332-4"><a href="#cb332-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Club</span><span class="ot"> ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Card</span></span>
@@ -2954,9 +4653,13 @@ It turns a list of pairs into a list of elements
 <span id="cb332-6"><a href="#cb332-6" aria-hidden="true" tabindex="-1"></a>[<span class="dt">Heart</span> <span class="dv">1</span>,<span class="dt">Heart</span> <span class="dv">2</span>,<span class="dt">Heart</span> <span class="dv">3</span>]</span>
 <span id="cb332-7"><a href="#cb332-7" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> (<span class="dt">Heart</span> <span class="op">.</span> (\x <span class="ot">-&gt;</span> x<span class="op">+</span><span class="dv">1</span>)) <span class="dv">3</span></span>
 <span id="cb332-8"><a href="#cb332-8" aria-hidden="true" tabindex="-1"></a><span class="dt">Heart</span> <span class="dv">4</span></span></code></pre></div>
-<h3 data-number="5.1.3" id="sidenote-deriving"><span class="header-section-number">5.1.3</span> Sidenote: Deriving</h3>
-<p>By the way, there’s something missing from our <code>Card</code> type. Look at how it behaves compared to <code>Ordering</code> and <code>Bool</code>:</p>
-<div class="sourceCode" id="cb333"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb333-1"><a href="#cb333-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">EQ</span></span>
+<h3 data-number="5.1.3" id="sidenote-deriving"><span
+class="header-section-number">5.1.3</span> Sidenote: Deriving</h3>
+<p>By the way, there’s something missing from our <code>Card</code>
+type. Look at how it behaves compared to <code>Ordering</code> and
+<code>Bool</code>:</p>
+<div class="sourceCode" id="cb333"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb333-1"><a href="#cb333-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">EQ</span></span>
 <span id="cb333-2"><a href="#cb333-2" aria-hidden="true" tabindex="-1"></a><span class="dt">EQ</span></span>
 <span id="cb333-3"><a href="#cb333-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">True</span></span>
 <span id="cb333-4"><a href="#cb333-4" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span>
@@ -2966,40 +4669,78 @@ It turns a list of pairs into a list of elements
 <span id="cb333-8"><a href="#cb333-8" aria-hidden="true" tabindex="-1"></a>      arising from a use <span class="kw">of</span> <span class="ot">`print&#39; at &lt;interactive&gt;:1:0-4</span></span>
 <span id="cb333-9"><a href="#cb333-9" aria-hidden="true" tabindex="-1"></a><span class="ot">    Possible fix: add an instance declaration for (Show Card)</span></span>
 <span id="cb333-10"><a href="#cb333-10" aria-hidden="true" tabindex="-1"></a><span class="ot">    In a stmt of a &#39;do&#39; expression: print it</span></span></code></pre></div>
-<p>The problem is that Haskell does not know how to print the types we defined. As the error says, they are not part of the <code>Show</code> class. The easy solution is to just add a <code>deriving Show</code> after the type definition:</p>
-<div class="sourceCode" id="cb334"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb334-1"><a href="#cb334-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Card</span> <span class="ot">=</span> <span class="dt">Joker</span> <span class="op">|</span> <span class="dt">Heart</span> <span class="dt">Int</span> <span class="op">|</span> <span class="dt">Club</span> <span class="dt">Int</span> <span class="op">|</span> <span class="dt">Spade</span> <span class="dt">Int</span> <span class="op">|</span> <span class="dt">Diamond</span> <span class="dt">Int</span></span>
+<p>The problem is that Haskell does not know how to print the types we
+defined. As the error says, they are not part of the <code>Show</code>
+class. The easy solution is to just add a <code>deriving Show</code>
+after the type definition:</p>
+<div class="sourceCode" id="cb334"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb334-1"><a href="#cb334-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Card</span> <span class="ot">=</span> <span class="dt">Joker</span> <span class="op">|</span> <span class="dt">Heart</span> <span class="dt">Int</span> <span class="op">|</span> <span class="dt">Club</span> <span class="dt">Int</span> <span class="op">|</span> <span class="dt">Spade</span> <span class="dt">Int</span> <span class="op">|</span> <span class="dt">Diamond</span> <span class="dt">Int</span></span>
 <span id="cb334-2"><a href="#cb334-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">deriving</span> <span class="dt">Show</span></span></code></pre></div>
-<div class="sourceCode" id="cb335"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb335-1"><a href="#cb335-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">Joker</span></span>
+<div class="sourceCode" id="cb335"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb335-1"><a href="#cb335-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">Joker</span></span>
 <span id="cb335-2"><a href="#cb335-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Joker</span></span></code></pre></div>
-<p>The <code>deriving</code> syntax is a way to automatically make your class a member of certain basic type classes, most notably <code>Read</code>, <code>Show</code> and <code>Eq</code>. We’ll talk more about what this means later.</p>
-<h3 data-number="5.1.4" id="algebraic"><span class="header-section-number">5.1.4</span> Algebraic?</h3>
-<p>So why are these datatypes called algebraic? This is because, theoretically speaking, each datatype can be a <em>sum</em> of constructors, and each constructor is a <em>product</em> of fields. It makes sense to think of these as sums and products for many reasons, one being that we can count the possible values of each type this way:</p>
-<div class="sourceCode" id="cb336"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb336-1"><a href="#cb336-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Bool</span> <span class="ot">=</span> <span class="dt">True</span> <span class="op">|</span> <span class="dt">False</span>            <span class="co">-- corresponds to 1+1. Has 2 possible values.</span></span>
+<p>The <code>deriving</code> syntax is a way to automatically make your
+class a member of certain basic type classes, most notably
+<code>Read</code>, <code>Show</code> and <code>Eq</code>. We’ll talk
+more about what this means later.</p>
+<h3 data-number="5.1.4" id="algebraic"><span
+class="header-section-number">5.1.4</span> Algebraic?</h3>
+<p>So why are these datatypes called algebraic? This is because,
+theoretically speaking, each datatype can be a <em>sum</em> of
+constructors, and each constructor is a <em>product</em> of fields. It
+makes sense to think of these as sums and products for many reasons, one
+being that we can count the possible values of each type this way:</p>
+<div class="sourceCode" id="cb336"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb336-1"><a href="#cb336-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Bool</span> <span class="ot">=</span> <span class="dt">True</span> <span class="op">|</span> <span class="dt">False</span>            <span class="co">-- corresponds to 1+1. Has 2 possible values.</span></span>
 <span id="cb336-2"><a href="#cb336-2" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">TwoBools</span> <span class="ot">=</span> <span class="dt">TwoBools</span> <span class="dt">Bool</span> <span class="dt">Bool</span>  <span class="co">-- corresponds to Bool*Bool, i.e. 2*2. Has 4 possible values.</span></span>
 <span id="cb336-3"><a href="#cb336-3" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Complex</span> <span class="ot">=</span> <span class="dt">Two</span> <span class="dt">Bool</span> <span class="dt">Bool</span> <span class="op">|</span> <span class="dt">One</span> <span class="dt">Bool</span> <span class="op">|</span> <span class="dt">None</span></span>
 <span id="cb336-4"><a href="#cb336-4" aria-hidden="true" tabindex="-1"></a>                                    <span class="co">-- corresponds to Bool*Bool+Bool+1 = 2*2+2+1 = 7. Has 7 possible values.</span></span></code></pre></div>
-<p>There is a rich theory of algebraic datatypes. If you’re interested, you might find more info <a href="https://codewords.recurse.com/issues/three/algebra-and-calculus-of-algebraic-data-types">here</a> or <a href="https://www.cis.upenn.edu/~sweirich/papers/yorgey-thesis.pdf">here</a>.</p>
-<h2 data-number="5.2" id="type-parameters"><span class="header-section-number">5.2</span> Type Parameters</h2>
-<p>We introduced type parameters and parametric polymorphism when introducing lists in Lecture 2. Since then, we’ve seen other parameterized types like <code>Maybe</code> and <code>Either</code>. Now we’ll learn how we can define our own parameterized types.</p>
-<h3 data-number="5.2.1" id="defining-parameterized-types"><span class="header-section-number">5.2.1</span> Defining Parameterized Types</h3>
+<p>There is a rich theory of algebraic datatypes. If you’re interested,
+you might find more info <a
+href="https://codewords.recurse.com/issues/three/algebra-and-calculus-of-algebraic-data-types">here</a>
+or <a
+href="https://www.cis.upenn.edu/~sweirich/papers/yorgey-thesis.pdf">here</a>.</p>
+<h2 data-number="5.2" id="type-parameters"><span
+class="header-section-number">5.2</span> Type Parameters</h2>
+<p>We introduced type parameters and parametric polymorphism when
+introducing lists in Lecture 2. Since then, we’ve seen other
+parameterized types like <code>Maybe</code> and <code>Either</code>. Now
+we’ll learn how we can define our own parameterized types.</p>
+<h3 data-number="5.2.1" id="defining-parameterized-types"><span
+class="header-section-number">5.2.1</span> Defining Parameterized
+Types</h3>
 <p>The definition for <code>Maybe</code> is:</p>
-<div class="sourceCode" id="cb337"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb337-1"><a href="#cb337-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Maybe</span> a <span class="ot">=</span> <span class="dt">Nothing</span> <span class="op">|</span> <span class="dt">Just</span> a</span></code></pre></div>
-<p>What’s <code>a</code>? We define a parameterized type by mentioning a <em>type variable</em> (<code>a</code> in this case) on the left side of the <code>=</code> sign. We can then use the same type variable in fields for our constructors. This is analogous to polymorphic functions. Instead of defining separate functions</p>
-<div class="sourceCode" id="cb338"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb338-1"><a href="#cb338-1" aria-hidden="true" tabindex="-1"></a><span class="ot">headInt ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<div class="sourceCode" id="cb337"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb337-1"><a href="#cb337-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Maybe</span> a <span class="ot">=</span> <span class="dt">Nothing</span> <span class="op">|</span> <span class="dt">Just</span> a</span></code></pre></div>
+<p>What’s <code>a</code>? We define a parameterized type by mentioning a
+<em>type variable</em> (<code>a</code> in this case) on the left side of
+the <code>=</code> sign. We can then use the same type variable in
+fields for our constructors. This is analogous to polymorphic functions.
+Instead of defining separate functions</p>
+<div class="sourceCode" id="cb338"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb338-1"><a href="#cb338-1" aria-hidden="true" tabindex="-1"></a><span class="ot">headInt ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb338-2"><a href="#cb338-2" aria-hidden="true" tabindex="-1"></a><span class="ot">headBool ::</span> [<span class="dt">Bool</span>] <span class="ot">-&gt;</span> <span class="dt">Bool</span></span></code></pre></div>
-<p>and so on, we define one function <code>head :: [a] -&gt; a</code> that works for all types <code>a</code>. Similarly, instead of defining multiple types</p>
-<div class="sourceCode" id="cb339"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb339-1"><a href="#cb339-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">MaybeInt</span> <span class="ot">=</span> <span class="dt">NothingInt</span> <span class="op">|</span> <span class="dt">JustInt</span> <span class="dt">Int</span></span>
+<p>and so on, we define one function <code>head :: [a] -&gt; a</code>
+that works for all types <code>a</code>. Similarly, instead of defining
+multiple types</p>
+<div class="sourceCode" id="cb339"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb339-1"><a href="#cb339-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">MaybeInt</span> <span class="ot">=</span> <span class="dt">NothingInt</span> <span class="op">|</span> <span class="dt">JustInt</span> <span class="dt">Int</span></span>
 <span id="cb339-2"><a href="#cb339-2" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">MaybeBool</span> <span class="ot">=</span> <span class="dt">NothingBool</span> <span class="op">|</span> <span class="dt">JustBool</span> <span class="dt">Bool</span></span></code></pre></div>
-<p>we define one type <code>Maybe a</code> that works for all types <code>a</code>.</p>
-<p>Here’s our first own parameterized type <code>Described</code>. The values of type <code>Described a</code> contain a value of type <code>a</code> and a <code>String</code> description.</p>
-<div class="sourceCode" id="cb340"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb340-1"><a href="#cb340-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Described</span> a <span class="ot">=</span> <span class="dt">Describe</span> a <span class="dt">String</span></span>
+<p>we define one type <code>Maybe a</code> that works for all types
+<code>a</code>.</p>
+<p>Here’s our first own parameterized type <code>Described</code>. The
+values of type <code>Described a</code> contain a value of type
+<code>a</code> and a <code>String</code> description.</p>
+<div class="sourceCode" id="cb340"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb340-1"><a href="#cb340-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Described</span> a <span class="ot">=</span> <span class="dt">Describe</span> a <span class="dt">String</span></span>
 <span id="cb340-2"><a href="#cb340-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb340-3"><a href="#cb340-3" aria-hidden="true" tabindex="-1"></a><span class="ot">getValue ::</span> <span class="dt">Described</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb340-4"><a href="#cb340-4" aria-hidden="true" tabindex="-1"></a>getValue (<span class="dt">Describe</span> x _) <span class="ot">=</span> x</span>
 <span id="cb340-5"><a href="#cb340-5" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb340-6"><a href="#cb340-6" aria-hidden="true" tabindex="-1"></a><span class="ot">getDescription ::</span> <span class="dt">Described</span> a <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb340-7"><a href="#cb340-7" aria-hidden="true" tabindex="-1"></a>getDescription (<span class="dt">Describe</span> _ desc) <span class="ot">=</span> desc</span></code></pre></div>
-<div class="sourceCode" id="cb341"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb341-1"><a href="#cb341-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="dt">Describe</span></span>
+<div class="sourceCode" id="cb341"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb341-1"><a href="#cb341-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="dt">Describe</span></span>
 <span id="cb341-2"><a href="#cb341-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Describe</span><span class="ot"> ::</span> a <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Described</span> a</span>
 <span id="cb341-3"><a href="#cb341-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="dt">Describe</span> <span class="dt">True</span> <span class="st">&quot;This is true&quot;</span></span>
 <span id="cb341-4"><a href="#cb341-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Describe</span> <span class="dt">True</span> <span class="st">&quot;This is true&quot;</span><span class="ot"> ::</span> <span class="dt">Described</span> <span class="dt">Bool</span></span>
@@ -3007,34 +4748,62 @@ It turns a list of pairs into a list of elements
 <span id="cb341-6"><a href="#cb341-6" aria-hidden="true" tabindex="-1"></a><span class="dv">3</span></span>
 <span id="cb341-7"><a href="#cb341-7" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> getDescription (<span class="dt">Describe</span> <span class="dv">3</span> <span class="st">&quot;a number&quot;</span>)</span>
 <span id="cb341-8"><a href="#cb341-8" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;a number&quot;</span></span></code></pre></div>
-<h3 data-number="5.2.2" id="syntactic-note"><span class="header-section-number">5.2.2</span> Syntactic Note</h3>
-<p>In the above definitions, we’ve used <code>a</code> as a type variable. However any word that starts with a lower case letter is fine. We could have defined <code>Maybe</code> like this:</p>
-<div class="sourceCode" id="cb342"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb342-1"><a href="#cb342-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Maybe</span> theType <span class="ot">=</span> <span class="dt">Nothing</span> <span class="op">|</span> <span class="dt">Just</span> theType</span></code></pre></div>
+<h3 data-number="5.2.2" id="syntactic-note"><span
+class="header-section-number">5.2.2</span> Syntactic Note</h3>
+<p>In the above definitions, we’ve used <code>a</code> as a type
+variable. However any word that starts with a lower case letter is fine.
+We could have defined <code>Maybe</code> like this:</p>
+<div class="sourceCode" id="cb342"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb342-1"><a href="#cb342-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Maybe</span> theType <span class="ot">=</span> <span class="dt">Nothing</span> <span class="op">|</span> <span class="dt">Just</span> theType</span></code></pre></div>
 <p>The rules for Haskell identifiers are:</p>
 <ul>
-<li>Type variables and names for functions and values start lower case (e.g. <code>a</code>, <code>map</code>, <code>xs</code>)</li>
-<li>Type names and constructor names start with upper case (e.g. <code>Maybe</code>, <code>Just</code>, <code>Card</code>, <code>Heart</code>)</li>
+<li>Type variables and names for functions and values start lower case
+(e.g. <code>a</code>, <code>map</code>, <code>xs</code>)</li>
+<li>Type names and constructor names start with upper case
+(e.g. <code>Maybe</code>, <code>Just</code>, <code>Card</code>,
+<code>Heart</code>)</li>
 </ul>
-<p>Note that a type and its constructor can have the same name. This is very common in Haskell code for types that only have one constructor. In this material we try to avoid it to avoid confusion. Here are some examples:</p>
-<div class="sourceCode" id="cb343"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb343-1"><a href="#cb343-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Pair</span> a <span class="ot">=</span> <span class="dt">Pair</span> a a</span>
+<p>Note that a type and its constructor can have the same name. This is
+very common in Haskell code for types that only have one constructor. In
+this material we try to avoid it to avoid confusion. Here are some
+examples:</p>
+<div class="sourceCode" id="cb343"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb343-1"><a href="#cb343-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Pair</span> a <span class="ot">=</span> <span class="dt">Pair</span> a a</span>
 <span id="cb343-2"><a href="#cb343-2" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Report</span> <span class="ot">=</span> <span class="dt">Report</span> <span class="dt">Int</span> <span class="dt">String</span> <span class="dt">String</span></span></code></pre></div>
-<div class="sourceCode" id="cb344"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb344-1"><a href="#cb344-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="dt">Pair</span></span>
+<div class="sourceCode" id="cb344"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb344-1"><a href="#cb344-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="dt">Pair</span></span>
 <span id="cb344-2"><a href="#cb344-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Pair</span><span class="ot"> ::</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Pair</span> a</span></code></pre></div>
-<p>Beware of mixing up types and constructors. Luckily types and constructors can never occur in the same context, so you get a nice error:</p>
-<div class="sourceCode" id="cb345"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb345-1"><a href="#cb345-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">Maybe</span>                              <span class="co">-- trying to use a type name as a value</span></span>
+<p>Beware of mixing up types and constructors. Luckily types and
+constructors can never occur in the same context, so you get a nice
+error:</p>
+<div class="sourceCode" id="cb345"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb345-1"><a href="#cb345-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">Maybe</span>                              <span class="co">-- trying to use a type name as a value</span></span>
 <span id="cb345-2"><a href="#cb345-2" aria-hidden="true" tabindex="-1"></a><span class="op">&lt;</span>interactive<span class="op">&gt;:</span><span class="dv">1</span><span class="op">:</span><span class="dv">1</span><span class="op">:</span> <span class="fu">error</span><span class="op">:</span></span>
 <span id="cb345-3"><a href="#cb345-3" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">Data</span> constructor <span class="fu">not</span> <span class="kw">in</span> scope<span class="op">:</span> <span class="dt">Maybe</span></span>
 <span id="cb345-4"><a href="#cb345-4" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb345-5"><a href="#cb345-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span><span class="ot"> undefined ::</span> <span class="dt">Nothing</span>               <span class="co">-- trying to use a constructor as a type</span></span>
 <span id="cb345-6"><a href="#cb345-6" aria-hidden="true" tabindex="-1"></a><span class="op">&lt;</span>interactive<span class="op">&gt;:</span><span class="dv">2</span><span class="op">:</span><span class="dv">14</span><span class="op">:</span> <span class="fu">error</span><span class="op">:</span></span>
 <span id="cb345-7"><a href="#cb345-7" aria-hidden="true" tabindex="-1"></a>    <span class="dt">Not</span> <span class="kw">in</span> scope<span class="op">:</span> <span class="kw">type</span> constructor <span class="fu">or</span> <span class="kw">class</span> ‘<span class="dt">Nothing</span>’</span></code></pre></div>
-<h3 data-number="5.2.3" id="sidenote-multiple-type-parameters"><span class="header-section-number">5.2.3</span> Sidenote: Multiple Type Parameters</h3>
-<p>Types can have multiple type parameters. The syntax is similar to defining functions with many arguments. Here’s the definition of the standard <code>Either</code> type:</p>
-<div class="sourceCode" id="cb346"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb346-1"><a href="#cb346-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Either</span> a b <span class="ot">=</span> <span class="dt">Left</span> a <span class="op">|</span> <span class="dt">Right</span> b</span></code></pre></div>
-<h2 data-number="5.3" id="recursive-types"><span class="header-section-number">5.3</span> Recursive Types</h2>
-<p>So far, all of the types we’ve defined have been of constant size. We can represent one report or one colour, but how could we represent a collection of things? We could use lists of course, but could we define a list type ourselves?</p>
-<p>Just like Haskell functions, Haskell data types can be <em>recursive</em>. This is no weirder than having an object in Java or Python that refers to another object of the same class. This is how you define a list of integers:</p>
-<div class="sourceCode" id="cb347"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb347-1"><a href="#cb347-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">IntList</span> <span class="ot">=</span> <span class="dt">Empty</span> <span class="op">|</span> <span class="dt">Node</span> <span class="dt">Int</span> <span class="dt">IntList</span></span>
+<h3 data-number="5.2.3" id="sidenote-multiple-type-parameters"><span
+class="header-section-number">5.2.3</span> Sidenote: Multiple Type
+Parameters</h3>
+<p>Types can have multiple type parameters. The syntax is similar to
+defining functions with many arguments. Here’s the definition of the
+standard <code>Either</code> type:</p>
+<div class="sourceCode" id="cb346"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb346-1"><a href="#cb346-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Either</span> a b <span class="ot">=</span> <span class="dt">Left</span> a <span class="op">|</span> <span class="dt">Right</span> b</span></code></pre></div>
+<h2 data-number="5.3" id="recursive-types"><span
+class="header-section-number">5.3</span> Recursive Types</h2>
+<p>So far, all of the types we’ve defined have been of constant size. We
+can represent one report or one colour, but how could we represent a
+collection of things? We could use lists of course, but could we define
+a list type ourselves?</p>
+<p>Just like Haskell functions, Haskell data types can be
+<em>recursive</em>. This is no weirder than having an object in Java or
+Python that refers to another object of the same class. This is how you
+define a list of integers:</p>
+<div class="sourceCode" id="cb347"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb347-1"><a href="#cb347-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">IntList</span> <span class="ot">=</span> <span class="dt">Empty</span> <span class="op">|</span> <span class="dt">Node</span> <span class="dt">Int</span> <span class="dt">IntList</span></span>
 <span id="cb347-2"><a href="#cb347-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">deriving</span> <span class="dt">Show</span></span>
 <span id="cb347-3"><a href="#cb347-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb347-4"><a href="#cb347-4" aria-hidden="true" tabindex="-1"></a><span class="ot">ihead ::</span> <span class="dt">IntList</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
@@ -3046,26 +4815,39 @@ It turns a list of pairs into a list of elements
 <span id="cb347-10"><a href="#cb347-10" aria-hidden="true" tabindex="-1"></a><span class="ot">ilength ::</span> <span class="dt">IntList</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb347-11"><a href="#cb347-11" aria-hidden="true" tabindex="-1"></a>ilength <span class="dt">Empty</span> <span class="ot">=</span> <span class="dv">0</span></span>
 <span id="cb347-12"><a href="#cb347-12" aria-hidden="true" tabindex="-1"></a>ilength (<span class="dt">Node</span> _ t) <span class="ot">=</span> <span class="dv">1</span> <span class="op">+</span> ilength t</span></code></pre></div>
-<p>We can use the functions defined above to work with lists of integers:</p>
-<div class="sourceCode" id="cb348"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb348-1"><a href="#cb348-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> ihead (<span class="dt">Node</span> <span class="dv">3</span> (<span class="dt">Node</span> <span class="dv">5</span> (<span class="dt">Node</span> <span class="dv">4</span> <span class="dt">Empty</span>)))</span>
+<p>We can use the functions defined above to work with lists of
+integers:</p>
+<div class="sourceCode" id="cb348"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb348-1"><a href="#cb348-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> ihead (<span class="dt">Node</span> <span class="dv">3</span> (<span class="dt">Node</span> <span class="dv">5</span> (<span class="dt">Node</span> <span class="dv">4</span> <span class="dt">Empty</span>)))</span>
 <span id="cb348-2"><a href="#cb348-2" aria-hidden="true" tabindex="-1"></a><span class="dv">3</span></span>
 <span id="cb348-3"><a href="#cb348-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> itail (<span class="dt">Node</span> <span class="dv">3</span> (<span class="dt">Node</span> <span class="dv">5</span> (<span class="dt">Node</span> <span class="dv">4</span> <span class="dt">Empty</span>)))</span>
 <span id="cb348-4"><a href="#cb348-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Node</span> <span class="dv">5</span> (<span class="dt">Node</span> <span class="dv">4</span> <span class="dt">Empty</span>)</span>
 <span id="cb348-5"><a href="#cb348-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> ilength (<span class="dt">Node</span> <span class="dv">3</span> (<span class="dt">Node</span> <span class="dv">5</span> (<span class="dt">Node</span> <span class="dv">4</span> <span class="dt">Empty</span>)))</span>
 <span id="cb348-6"><a href="#cb348-6" aria-hidden="true" tabindex="-1"></a><span class="dv">3</span></span></code></pre></div>
-<p>Note that we can’t put values other than <code>Int</code>s inside our <code>IntList</code>:</p>
-<div class="sourceCode" id="cb349"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb349-1"><a href="#cb349-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">Node</span> <span class="dt">False</span> <span class="dt">Empty</span></span>
+<p>Note that we can’t put values other than <code>Int</code>s inside our
+<code>IntList</code>:</p>
+<div class="sourceCode" id="cb349"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb349-1"><a href="#cb349-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">Node</span> <span class="dt">False</span> <span class="dt">Empty</span></span>
 <span id="cb349-2"><a href="#cb349-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb349-3"><a href="#cb349-3" aria-hidden="true" tabindex="-1"></a><span class="op">&lt;</span>interactive<span class="op">&gt;:</span><span class="dv">3</span><span class="op">:</span><span class="dv">6</span><span class="op">:</span> <span class="fu">error</span><span class="op">:</span></span>
 <span id="cb349-4"><a href="#cb349-4" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">Couldn&#39;t</span> match expected <span class="kw">type</span> ‘<span class="dt">Int</span>’ with actual <span class="kw">type</span> ‘<span class="dt">Bool</span>’</span>
 <span id="cb349-5"><a href="#cb349-5" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">In</span> the first argument <span class="kw">of</span> ‘<span class="dt">Node</span>’, namely ‘<span class="dt">False</span>’</span>
 <span id="cb349-6"><a href="#cb349-6" aria-hidden="true" tabindex="-1"></a>      <span class="dt">In</span> the expression<span class="op">:</span> <span class="dt">Node</span> <span class="dt">False</span> <span class="dt">Empty</span></span>
 <span id="cb349-7"><a href="#cb349-7" aria-hidden="true" tabindex="-1"></a>      <span class="dt">In</span> an equation for ‘it’<span class="op">:</span> it <span class="ot">=</span> <span class="dt">Node</span> <span class="dt">False</span> <span class="dt">Empty</span></span></code></pre></div>
-<p>To be able to put any type of element in our list, let’s do the same thing with a type parameter. This is the same as the built in type <code>[a]</code>, but with slightly clunkier syntax:</p>
-<div class="sourceCode" id="cb350"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb350-1"><a href="#cb350-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">List</span> a <span class="ot">=</span> <span class="dt">Empty</span> <span class="op">|</span> <span class="dt">Node</span> a (<span class="dt">List</span> a)</span>
+<p>To be able to put any type of element in our list, let’s do the same
+thing with a type parameter. This is the same as the built in type
+<code>[a]</code>, but with slightly clunkier syntax:</p>
+<div class="sourceCode" id="cb350"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb350-1"><a href="#cb350-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">List</span> a <span class="ot">=</span> <span class="dt">Empty</span> <span class="op">|</span> <span class="dt">Node</span> a (<span class="dt">List</span> a)</span>
 <span id="cb350-2"><a href="#cb350-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">deriving</span> <span class="dt">Show</span></span></code></pre></div>
-<p>Note how we need to pass the the type parameter <code>a</code> onwards in the recursion. We need to write <code>Node a (List a)</code> instead of <code>Node a List</code>. The <code>Node</code> constructor has two arguments. The first has type <code>a</code>, and the second has type <code>List a</code>. Here are the reimplementations of some standard list functions for our <code>List</code> type:</p>
-<div class="sourceCode" id="cb351"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb351-1"><a href="#cb351-1" aria-hidden="true" tabindex="-1"></a><span class="ot">lhead ::</span> <span class="dt">List</span> a <span class="ot">-&gt;</span> a</span>
+<p>Note how we need to pass the the type parameter <code>a</code>
+onwards in the recursion. We need to write <code>Node a (List a)</code>
+instead of <code>Node a List</code>. The <code>Node</code> constructor
+has two arguments. The first has type <code>a</code>, and the second has
+type <code>List a</code>. Here are the reimplementations of some
+standard list functions for our <code>List</code> type:</p>
+<div class="sourceCode" id="cb351"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb351-1"><a href="#cb351-1" aria-hidden="true" tabindex="-1"></a><span class="ot">lhead ::</span> <span class="dt">List</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb351-2"><a href="#cb351-2" aria-hidden="true" tabindex="-1"></a>lhead (<span class="dt">Node</span> h _) <span class="ot">=</span> h</span>
 <span id="cb351-3"><a href="#cb351-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb351-4"><a href="#cb351-4" aria-hidden="true" tabindex="-1"></a><span class="ot">ltail ::</span> <span class="dt">List</span> a <span class="ot">-&gt;</span> <span class="dt">List</span> a</span>
@@ -3078,14 +4860,17 @@ It turns a list of pairs into a list of elements
 <span id="cb351-11"><a href="#cb351-11" aria-hidden="true" tabindex="-1"></a><span class="ot">llength ::</span> <span class="dt">List</span> a <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb351-12"><a href="#cb351-12" aria-hidden="true" tabindex="-1"></a>llength <span class="dt">Empty</span> <span class="ot">=</span> <span class="dv">0</span></span>
 <span id="cb351-13"><a href="#cb351-13" aria-hidden="true" tabindex="-1"></a>llength (<span class="dt">Node</span> _ t) <span class="ot">=</span> <span class="dv">1</span> <span class="op">+</span> llength t</span></code></pre></div>
-<div class="sourceCode" id="cb352"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb352-1"><a href="#cb352-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> lhead (<span class="dt">Node</span> <span class="dt">True</span> <span class="dt">Empty</span>)</span>
+<div class="sourceCode" id="cb352"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb352-1"><a href="#cb352-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> lhead (<span class="dt">Node</span> <span class="dt">True</span> <span class="dt">Empty</span>)</span>
 <span id="cb352-2"><a href="#cb352-2" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span>
 <span id="cb352-3"><a href="#cb352-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> ltail (<span class="dt">Node</span> <span class="dt">True</span> (<span class="dt">Node</span> <span class="dt">False</span> <span class="dt">Empty</span>))</span>
 <span id="cb352-4"><a href="#cb352-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Node</span> <span class="dt">False</span> <span class="dt">Empty</span></span>
 <span id="cb352-5"><a href="#cb352-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> lnull <span class="dt">Empty</span></span>
 <span id="cb352-6"><a href="#cb352-6" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span></code></pre></div>
-<p>Note that just like with normal Haskell lists, we can’t have elements of different types in the same list:</p>
-<div class="sourceCode" id="cb353"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb353-1"><a href="#cb353-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">Node</span> <span class="dt">True</span> (<span class="dt">Node</span> <span class="st">&quot;foo&quot;</span> <span class="dt">Empty</span>)</span>
+<p>Note that just like with normal Haskell lists, we can’t have elements
+of different types in the same list:</p>
+<div class="sourceCode" id="cb353"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb353-1"><a href="#cb353-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">Node</span> <span class="dt">True</span> (<span class="dt">Node</span> <span class="st">&quot;foo&quot;</span> <span class="dt">Empty</span>)</span>
 <span id="cb353-2"><a href="#cb353-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb353-3"><a href="#cb353-3" aria-hidden="true" tabindex="-1"></a><span class="op">&lt;</span>interactive<span class="op">&gt;:</span><span class="dv">5</span><span class="op">:</span><span class="dv">12</span><span class="op">:</span> <span class="fu">error</span><span class="op">:</span></span>
 <span id="cb353-4"><a href="#cb353-4" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">Couldn&#39;t</span> match <span class="kw">type</span> ‘[<span class="dt">Char</span>]’ with ‘<span class="dt">Bool</span>’</span>
@@ -3094,11 +4879,17 @@ It turns a list of pairs into a list of elements
 <span id="cb353-7"><a href="#cb353-7" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">In</span> the second argument <span class="kw">of</span> ‘<span class="dt">Node</span>’, namely ‘(<span class="dt">Node</span> <span class="st">&quot;foo&quot;</span> <span class="dt">Empty</span>)’</span>
 <span id="cb353-8"><a href="#cb353-8" aria-hidden="true" tabindex="-1"></a>      <span class="dt">In</span> the expression<span class="op">:</span> <span class="dt">Node</span> <span class="dt">True</span> (<span class="dt">Node</span> <span class="st">&quot;foo&quot;</span> <span class="dt">Empty</span>)</span>
 <span id="cb353-9"><a href="#cb353-9" aria-hidden="true" tabindex="-1"></a>      <span class="dt">In</span> an equation for ‘it’<span class="op">:</span> it <span class="ot">=</span> <span class="dt">Node</span> <span class="dt">True</span> (<span class="dt">Node</span> <span class="st">&quot;foo&quot;</span> <span class="dt">Empty</span>)</span></code></pre></div>
-<h3 data-number="5.3.1" id="example-growing-a-tree"><span class="header-section-number">5.3.1</span> Example: Growing a Tree</h3>
+<h3 data-number="5.3.1" id="example-growing-a-tree"><span
+class="header-section-number">5.3.1</span> Example: Growing a Tree</h3>
 <p>Just like a list, we can also represent a binary tree:</p>
-<div class="sourceCode" id="cb354"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb354-1"><a href="#cb354-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Tree</span> a <span class="ot">=</span> <span class="dt">Node</span> a (<span class="dt">Tree</span> a) (<span class="dt">Tree</span> a) <span class="op">|</span> <span class="dt">Empty</span></span></code></pre></div>
-<p>Our tree contains nodes, which contain a value of type <code>a</code> and two child trees, and empty trees.</p>
-<p>In case you’re not familiar with binary trees, they’re a data structure that’s often used as the basis for other data structures (<code>Data.Map</code> is based on trees!). Binary trees are often drawn as (upside-down) pictures, like this:</p>
+<div class="sourceCode" id="cb354"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb354-1"><a href="#cb354-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Tree</span> a <span class="ot">=</span> <span class="dt">Node</span> a (<span class="dt">Tree</span> a) (<span class="dt">Tree</span> a) <span class="op">|</span> <span class="dt">Empty</span></span></code></pre></div>
+<p>Our tree contains nodes, which contain a value of type <code>a</code>
+and two child trees, and empty trees.</p>
+<p>In case you’re not familiar with binary trees, they’re a data
+structure that’s often used as the basis for other data structures
+(<code>Data.Map</code> is based on trees!). Binary trees are often drawn
+as (upside-down) pictures, like this:</p>
 <p><img src="img/binaryTree.svg" /></p>
 <!--
 ```haskell
@@ -3109,16 +4900,27 @@ It turns a list of pairs into a list of elements
  2   3
 ```
 -->
-<p>The highest node in the tree is called the <em>root</em> (<code>0</code> in this case), and the nodes with no children are called <code>leaves</code> (<code>2</code>, <code>3</code> and <code>4</code> in this case). We can define this tree using our <code>Tree</code> type like this:</p>
-<div class="sourceCode" id="cb355"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb355-1"><a href="#cb355-1" aria-hidden="true" tabindex="-1"></a><span class="ot">example ::</span> <span class="dt">Tree</span> <span class="dt">Int</span></span>
+<p>The highest node in the tree is called the <em>root</em>
+(<code>0</code> in this case), and the nodes with no children are called
+<code>leaves</code> (<code>2</code>, <code>3</code> and <code>4</code>
+in this case). We can define this tree using our <code>Tree</code> type
+like this:</p>
+<div class="sourceCode" id="cb355"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb355-1"><a href="#cb355-1" aria-hidden="true" tabindex="-1"></a><span class="ot">example ::</span> <span class="dt">Tree</span> <span class="dt">Int</span></span>
 <span id="cb355-2"><a href="#cb355-2" aria-hidden="true" tabindex="-1"></a>example <span class="ot">=</span> (<span class="dt">Node</span> <span class="dv">0</span> (<span class="dt">Node</span> <span class="dv">1</span> (<span class="dt">Node</span> <span class="dv">2</span> <span class="dt">Empty</span> <span class="dt">Empty</span>)</span>
 <span id="cb355-3"><a href="#cb355-3" aria-hidden="true" tabindex="-1"></a>                          (<span class="dt">Node</span> <span class="dv">3</span> <span class="dt">Empty</span> <span class="dt">Empty</span>))</span>
 <span id="cb355-4"><a href="#cb355-4" aria-hidden="true" tabindex="-1"></a>                  (<span class="dt">Node</span> <span class="dv">4</span> <span class="dt">Empty</span> <span class="dt">Empty</span>))</span></code></pre></div>
-<p>The height of a binary tree is length of the longest path from the root to a leaf. In Haskell terms, it’s how many nested levels of <code>Node</code> constructors you need to build the tree. The height of our example tree is 3. Here’s a function that computes the height of a tree:</p>
-<div class="sourceCode" id="cb356"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb356-1"><a href="#cb356-1" aria-hidden="true" tabindex="-1"></a><span class="ot">treeHeight ::</span> <span class="dt">Tree</span> a <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<p>The height of a binary tree is length of the longest path from the
+root to a leaf. In Haskell terms, it’s how many nested levels of
+<code>Node</code> constructors you need to build the tree. The height of
+our example tree is 3. Here’s a function that computes the height of a
+tree:</p>
+<div class="sourceCode" id="cb356"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb356-1"><a href="#cb356-1" aria-hidden="true" tabindex="-1"></a><span class="ot">treeHeight ::</span> <span class="dt">Tree</span> a <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb356-2"><a href="#cb356-2" aria-hidden="true" tabindex="-1"></a>treeHeight <span class="dt">Empty</span> <span class="ot">=</span> <span class="dv">0</span></span>
 <span id="cb356-3"><a href="#cb356-3" aria-hidden="true" tabindex="-1"></a>treeHeight (<span class="dt">Node</span> _ l r) <span class="ot">=</span> <span class="dv">1</span> <span class="op">+</span> <span class="fu">max</span> (treeHeight l) (treeHeight r)</span></code></pre></div>
-<div class="sourceCode" id="cb357"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb357-1"><a href="#cb357-1" aria-hidden="true" tabindex="-1"></a>treeHeight <span class="dt">Empty</span> <span class="op">==&gt;</span> <span class="dv">0</span></span>
+<div class="sourceCode" id="cb357"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb357-1"><a href="#cb357-1" aria-hidden="true" tabindex="-1"></a>treeHeight <span class="dt">Empty</span> <span class="op">==&gt;</span> <span class="dv">0</span></span>
 <span id="cb357-2"><a href="#cb357-2" aria-hidden="true" tabindex="-1"></a>treeHeight (<span class="dt">Node</span> <span class="dv">2</span> <span class="dt">Empty</span> <span class="dt">Empty</span>)</span>
 <span id="cb357-3"><a href="#cb357-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">1</span> <span class="op">+</span> <span class="fu">max</span> (treeHeight <span class="dt">Empty</span>) (treeHeight <span class="dt">Empty</span>)</span>
 <span id="cb357-4"><a href="#cb357-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">1</span> <span class="op">+</span> <span class="fu">max</span> <span class="dv">0</span> <span class="dv">0</span></span>
@@ -3131,8 +4933,12 @@ It turns a list of pairs into a list of elements
 <span id="cb357-11"><a href="#cb357-11" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">1</span> <span class="op">+</span> <span class="fu">max</span> (treeHeight (<span class="dt">Node</span> <span class="dv">1</span> <span class="dt">Empty</span> (<span class="dt">Node</span> <span class="dv">2</span> <span class="dt">Empty</span> <span class="dt">Empty</span>))) (treeHeight <span class="dt">Empty</span>)</span>
 <span id="cb357-12"><a href="#cb357-12" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">1</span> <span class="op">+</span> <span class="fu">max</span> <span class="dv">2</span> <span class="dv">0</span></span>
 <span id="cb357-13"><a href="#cb357-13" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">3</span></span></code></pre></div>
-<p>In case you’re familiar with <em>binary search trees</em>, here are the definitions of the lookup and insert opertions for a binary search tree. If you don’t know what I’m talking about, you don’t need to understand this.</p>
-<div class="sourceCode" id="cb358"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb358-1"><a href="#cb358-1" aria-hidden="true" tabindex="-1"></a><span class="fu">lookup</span><span class="ot"> ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Tree</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
+<p>In case you’re familiar with <em>binary search trees</em>, here are
+the definitions of the lookup and insert opertions for a binary search
+tree. If you don’t know what I’m talking about, you don’t need to
+understand this.</p>
+<div class="sourceCode" id="cb358"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb358-1"><a href="#cb358-1" aria-hidden="true" tabindex="-1"></a><span class="fu">lookup</span><span class="ot"> ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Tree</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb358-2"><a href="#cb358-2" aria-hidden="true" tabindex="-1"></a><span class="fu">lookup</span> x <span class="dt">Empty</span> <span class="ot">=</span> <span class="dt">False</span></span>
 <span id="cb358-3"><a href="#cb358-3" aria-hidden="true" tabindex="-1"></a><span class="fu">lookup</span> x (<span class="dt">Node</span> y l r)</span>
 <span id="cb358-4"><a href="#cb358-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> x <span class="op">&lt;</span> y <span class="ot">=</span> <span class="fu">lookup</span> x l</span>
@@ -3145,85 +4951,137 @@ It turns a list of pairs into a list of elements
 <span id="cb358-11"><a href="#cb358-11" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> x <span class="op">&lt;</span> y <span class="ot">=</span> <span class="dt">Node</span> y (insert x l) r</span>
 <span id="cb358-12"><a href="#cb358-12" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> x <span class="op">&gt;</span> y <span class="ot">=</span> <span class="dt">Node</span> y l (insert x r)</span>
 <span id="cb358-13"><a href="#cb358-13" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> <span class="fu">otherwise</span> <span class="ot">=</span> <span class="dt">Node</span> y l r</span></code></pre></div>
-<h2 data-number="5.4" id="record-syntax"><span class="header-section-number">5.4</span> Record Syntax</h2>
-<p>If some fields need to be accessed often, it can be convenient to have helper functions for reading those fields. For instance, the type <code>Person</code> might have multiple fields:</p>
-<div class="sourceCode" id="cb359"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb359-1"><a href="#cb359-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Person</span> <span class="ot">=</span> <span class="dt">MkPerson</span> <span class="dt">String</span> <span class="dt">Int</span> <span class="dt">String</span> <span class="dt">String</span> <span class="dt">String</span> <span class="kw">deriving</span> <span class="dt">Show</span></span></code></pre></div>
+<h2 data-number="5.4" id="record-syntax"><span
+class="header-section-number">5.4</span> Record Syntax</h2>
+<p>If some fields need to be accessed often, it can be convenient to
+have helper functions for reading those fields. For instance, the type
+<code>Person</code> might have multiple fields:</p>
+<div class="sourceCode" id="cb359"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb359-1"><a href="#cb359-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Person</span> <span class="ot">=</span> <span class="dt">MkPerson</span> <span class="dt">String</span> <span class="dt">Int</span> <span class="dt">String</span> <span class="dt">String</span> <span class="dt">String</span> <span class="kw">deriving</span> <span class="dt">Show</span></span></code></pre></div>
 <p>A list of persons might look like the following:</p>
-<div class="sourceCode" id="cb360"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb360-1"><a href="#cb360-1" aria-hidden="true" tabindex="-1"></a><span class="ot">people ::</span> [<span class="dt">Person</span>]</span>
+<div class="sourceCode" id="cb360"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb360-1"><a href="#cb360-1" aria-hidden="true" tabindex="-1"></a><span class="ot">people ::</span> [<span class="dt">Person</span>]</span>
 <span id="cb360-2"><a href="#cb360-2" aria-hidden="true" tabindex="-1"></a>people <span class="ot">=</span> [ <span class="dt">MkPerson</span> <span class="st">&quot;Jane Doe&quot;</span> <span class="dv">21</span> <span class="st">&quot;Houston&quot;</span> <span class="st">&quot;Texas&quot;</span> <span class="st">&quot;Engineer&quot;</span></span>
 <span id="cb360-3"><a href="#cb360-3" aria-hidden="true" tabindex="-1"></a>         , <span class="dt">MkPerson</span> <span class="st">&quot;Maija Meikäläinen&quot;</span> <span class="dv">35</span> <span class="st">&quot;Rovaniemi&quot;</span> <span class="st">&quot;Finland&quot;</span> <span class="st">&quot;Engineer&quot;</span></span>
 <span id="cb360-4"><a href="#cb360-4" aria-hidden="true" tabindex="-1"></a>         , <span class="dt">MkPerson</span> <span class="st">&quot;Mauno Mutikainen&quot;</span> <span class="dv">27</span> <span class="st">&quot;Turku&quot;</span> <span class="st">&quot;Finland&quot;</span> <span class="st">&quot;Mathematician&quot;</span></span>
 <span id="cb360-5"><a href="#cb360-5" aria-hidden="true" tabindex="-1"></a>         ]</span></code></pre></div>
 <p>Suppose that we need to find all engineers from Finland:</p>
-<div class="sourceCode" id="cb361"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb361-1"><a href="#cb361-1" aria-hidden="true" tabindex="-1"></a><span class="ot">query ::</span> [<span class="dt">Person</span>] <span class="ot">-&gt;</span> [<span class="dt">Person</span>]</span>
+<div class="sourceCode" id="cb361"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb361-1"><a href="#cb361-1" aria-hidden="true" tabindex="-1"></a><span class="ot">query ::</span> [<span class="dt">Person</span>] <span class="ot">-&gt;</span> [<span class="dt">Person</span>]</span>
 <span id="cb361-2"><a href="#cb361-2" aria-hidden="true" tabindex="-1"></a>query [] <span class="ot">=</span> []</span>
 <span id="cb361-3"><a href="#cb361-3" aria-hidden="true" tabindex="-1"></a>query ((<span class="dt">MkPerson</span> name age town state profession)<span class="op">:</span>xs)</span>
 <span id="cb361-4"><a href="#cb361-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> state <span class="op">==</span> <span class="st">&quot;Finland&quot;</span> <span class="op">&amp;&amp;</span> profession <span class="op">==</span> <span class="st">&quot;Engineer&quot;</span> <span class="ot">=</span></span>
 <span id="cb361-5"><a href="#cb361-5" aria-hidden="true" tabindex="-1"></a>      (<span class="dt">MkPerson</span> name age town state profession) <span class="op">:</span> query xs</span>
 <span id="cb361-6"><a href="#cb361-6" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> <span class="fu">otherwise</span> <span class="ot">=</span> query xs</span></code></pre></div>
 <p>Thus,</p>
-<div class="sourceCode" id="cb362"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb362-1"><a href="#cb362-1" aria-hidden="true" tabindex="-1"></a>query people <span class="op">==&gt;</span> [<span class="dt">MkPerson</span> <span class="st">&quot;Maija Meikäläinen&quot;</span> <span class="dv">35</span> <span class="st">&quot;Rovaniemi&quot;</span> <span class="st">&quot;Finland&quot;</span> <span class="st">&quot;Engineer&quot;</span>]</span></code></pre></div>
-<p>Note that the types of the fields give little information on what is the intended content in those fields. We need to remember in all places in the code that <code>town</code> goes before <code>state</code> and not vice versa.</p>
-<p>Haskell has a feature called <em>record syntax</em> that is helpful in these kinds of cases. The datatype <code>Person</code> can be defined as a record:</p>
-<div class="sourceCode" id="cb363"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb363-1"><a href="#cb363-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Person</span> <span class="ot">=</span> <span class="dt">MkPerson</span> {<span class="ot"> name ::</span> <span class="dt">String</span>,<span class="ot"> age ::</span> <span class="dt">Int</span>,<span class="ot"> town ::</span> <span class="dt">String</span>,<span class="ot"> state ::</span> <span class="dt">String</span>,<span class="ot"> profession ::</span> <span class="dt">String</span>}</span>
+<div class="sourceCode" id="cb362"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb362-1"><a href="#cb362-1" aria-hidden="true" tabindex="-1"></a>query people <span class="op">==&gt;</span> [<span class="dt">MkPerson</span> <span class="st">&quot;Maija Meikäläinen&quot;</span> <span class="dv">35</span> <span class="st">&quot;Rovaniemi&quot;</span> <span class="st">&quot;Finland&quot;</span> <span class="st">&quot;Engineer&quot;</span>]</span></code></pre></div>
+<p>Note that the types of the fields give little information on what is
+the intended content in those fields. We need to remember in all places
+in the code that <code>town</code> goes before <code>state</code> and
+not vice versa.</p>
+<p>Haskell has a feature called <em>record syntax</em> that is helpful
+in these kinds of cases. The datatype <code>Person</code> can be defined
+as a record:</p>
+<div class="sourceCode" id="cb363"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb363-1"><a href="#cb363-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Person</span> <span class="ot">=</span> <span class="dt">MkPerson</span> {<span class="ot"> name ::</span> <span class="dt">String</span>,<span class="ot"> age ::</span> <span class="dt">Int</span>,<span class="ot"> town ::</span> <span class="dt">String</span>,<span class="ot"> state ::</span> <span class="dt">String</span>,<span class="ot"> profession ::</span> <span class="dt">String</span>}</span>
 <span id="cb363-2"><a href="#cb363-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">deriving</span> <span class="dt">Show</span></span></code></pre></div>
-<p>We can still define values of <code>Person</code> normally, but the <code>Show</code> instance prints the field names for us:</p>
-<div class="sourceCode" id="cb364"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb364-1"><a href="#cb364-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">MkPerson</span> <span class="st">&quot;Jane Doe&quot;</span> <span class="dv">21</span> <span class="st">&quot;Houston&quot;</span> <span class="st">&quot;Texas&quot;</span> <span class="st">&quot;Engineer&quot;</span></span>
+<p>We can still define values of <code>Person</code> normally, but the
+<code>Show</code> instance prints the field names for us:</p>
+<div class="sourceCode" id="cb364"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb364-1"><a href="#cb364-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">MkPerson</span> <span class="st">&quot;Jane Doe&quot;</span> <span class="dv">21</span> <span class="st">&quot;Houston&quot;</span> <span class="st">&quot;Texas&quot;</span> <span class="st">&quot;Engineer&quot;</span></span>
 <span id="cb364-2"><a href="#cb364-2" aria-hidden="true" tabindex="-1"></a><span class="dt">MkPerson</span> {name <span class="ot">=</span> <span class="st">&quot;Jane Doe&quot;</span>, age <span class="ot">=</span> <span class="dv">21</span>, town <span class="ot">=</span> <span class="st">&quot;Houston&quot;</span>, state <span class="ot">=</span> <span class="st">&quot;Texas&quot;</span>, profession <span class="ot">=</span> <span class="st">&quot;Engineer&quot;</span>}</span></code></pre></div>
-<p>However, we can also define values using record syntax. Note how the fields don’t need to be in any specific order now that they have names.</p>
-<div class="sourceCode" id="cb365"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb365-1"><a href="#cb365-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">MkPerson</span> {name <span class="ot">=</span> <span class="st">&quot;Jane Doe&quot;</span>, town <span class="ot">=</span> <span class="st">&quot;Houston&quot;</span>, profession <span class="ot">=</span> <span class="st">&quot;Engineer&quot;</span>, state <span class="ot">=</span> <span class="st">&quot;Texas&quot;</span>, age <span class="ot">=</span> <span class="dv">21</span>}</span>
+<p>However, we can also define values using record syntax. Note how the
+fields don’t need to be in any specific order now that they have
+names.</p>
+<div class="sourceCode" id="cb365"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb365-1"><a href="#cb365-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">MkPerson</span> {name <span class="ot">=</span> <span class="st">&quot;Jane Doe&quot;</span>, town <span class="ot">=</span> <span class="st">&quot;Houston&quot;</span>, profession <span class="ot">=</span> <span class="st">&quot;Engineer&quot;</span>, state <span class="ot">=</span> <span class="st">&quot;Texas&quot;</span>, age <span class="ot">=</span> <span class="dv">21</span>}</span>
 <span id="cb365-2"><a href="#cb365-2" aria-hidden="true" tabindex="-1"></a><span class="dt">MkPerson</span> {name <span class="ot">=</span> <span class="st">&quot;Jane Doe&quot;</span>, age <span class="ot">=</span> <span class="dv">21</span>, town <span class="ot">=</span> <span class="st">&quot;Houston&quot;</span>, state <span class="ot">=</span> <span class="st">&quot;Texas&quot;</span>, profession <span class="ot">=</span> <span class="st">&quot;Engineer&quot;</span>}</span></code></pre></div>
-<p>Most importantly, We get <em>accessor functions</em> for the fields for free:</p>
-<div class="sourceCode" id="cb366"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb366-1"><a href="#cb366-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t profession</span>
+<p>Most importantly, We get <em>accessor functions</em> for the fields
+for free:</p>
+<div class="sourceCode" id="cb366"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb366-1"><a href="#cb366-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t profession</span>
 <span id="cb366-2"><a href="#cb366-2" aria-hidden="true" tabindex="-1"></a><span class="ot">profession ::</span> <span class="dt">Person</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb366-3"><a href="#cb366-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> profession (<span class="dt">MkPerson</span> <span class="st">&quot;Jane Doe&quot;</span> <span class="dv">21</span> <span class="st">&quot;Houston&quot;</span> <span class="st">&quot;Texas&quot;</span> <span class="st">&quot;Engineer&quot;</span>)</span>
 <span id="cb366-4"><a href="#cb366-4" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;Engineer&quot;</span></span></code></pre></div>
-<p>We can now rewrite the query function using these accessor functions:</p>
-<div class="sourceCode" id="cb367"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb367-1"><a href="#cb367-1" aria-hidden="true" tabindex="-1"></a><span class="ot">query ::</span> [<span class="dt">Person</span>] <span class="ot">-&gt;</span> [<span class="dt">Person</span>]</span>
+<p>We can now rewrite the query function using these accessor
+functions:</p>
+<div class="sourceCode" id="cb367"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb367-1"><a href="#cb367-1" aria-hidden="true" tabindex="-1"></a><span class="ot">query ::</span> [<span class="dt">Person</span>] <span class="ot">-&gt;</span> [<span class="dt">Person</span>]</span>
 <span id="cb367-2"><a href="#cb367-2" aria-hidden="true" tabindex="-1"></a>query []     <span class="ot">=</span> []</span>
 <span id="cb367-3"><a href="#cb367-3" aria-hidden="true" tabindex="-1"></a>query (x<span class="op">:</span>xs)</span>
 <span id="cb367-4"><a href="#cb367-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> state x <span class="op">==</span> <span class="st">&quot;Finland&quot;</span> <span class="op">&amp;&amp;</span> profession x <span class="op">==</span> <span class="st">&quot;Engineer&quot;</span> <span class="ot">=</span></span>
 <span id="cb367-5"><a href="#cb367-5" aria-hidden="true" tabindex="-1"></a>      x <span class="op">:</span> query xs</span>
 <span id="cb367-6"><a href="#cb367-6" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> <span class="fu">otherwise</span> <span class="ot">=</span> query xs</span></code></pre></div>
 <p>You’ll probably agree that the code looks more pleasant now.</p>
-<h2 data-number="5.5" id="algebraic-datatypes-summary"><span class="header-section-number">5.5</span> Algebraic Datatypes: Summary</h2>
+<h2 data-number="5.5" id="algebraic-datatypes-summary"><span
+class="header-section-number">5.5</span> Algebraic Datatypes:
+Summary</h2>
 <ul>
 <li>Types are defined like this</li>
 </ul>
-<div class="sourceCode" id="cb368"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb368-1"><a href="#cb368-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">TypeName</span> <span class="ot">=</span> <span class="dt">ConstructorName</span> <span class="dt">FieldType</span> <span class="dt">FieldType2</span> <span class="op">|</span> <span class="dt">AnotherConstructor</span> <span class="dt">FieldType3</span> <span class="op">|</span> <span class="dt">OneMoreCons</span></span></code></pre></div>
+<div class="sourceCode" id="cb368"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb368-1"><a href="#cb368-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">TypeName</span> <span class="ot">=</span> <span class="dt">ConstructorName</span> <span class="dt">FieldType</span> <span class="dt">FieldType2</span> <span class="op">|</span> <span class="dt">AnotherConstructor</span> <span class="dt">FieldType3</span> <span class="op">|</span> <span class="dt">OneMoreCons</span></span></code></pre></div>
 <ul>
 <li>… or like this if we’re using type variables</li>
 </ul>
-<div class="sourceCode" id="cb369"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb369-1"><a href="#cb369-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">TypeName</span> variable <span class="ot">=</span> <span class="dt">Cons1</span> variable <span class="dt">Type1</span> <span class="op">|</span> <span class="dt">Cons2</span> <span class="dt">Type2</span> variable</span></code></pre></div>
+<div class="sourceCode" id="cb369"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb369-1"><a href="#cb369-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">TypeName</span> variable <span class="ot">=</span> <span class="dt">Cons1</span> variable <span class="dt">Type1</span> <span class="op">|</span> <span class="dt">Cons2</span> <span class="dt">Type2</span> variable</span></code></pre></div>
 <ul>
 <li>You can have one or more constructors</li>
 <li>Each constructor can have zero or more fields</li>
-<li>Constructors start with upper case, type variables with lower case</li>
+<li>Constructors start with upper case, type variables with lower
+case</li>
 <li>Values are handled with pattern matching:</li>
 </ul>
-<div class="sourceCode" id="cb370"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb370-1"><a href="#cb370-1" aria-hidden="true" tabindex="-1"></a>foo (<span class="dt">ConstructorName</span> a b) <span class="ot">=</span> a<span class="op">+</span>b</span>
+<div class="sourceCode" id="cb370"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb370-1"><a href="#cb370-1" aria-hidden="true" tabindex="-1"></a>foo (<span class="dt">ConstructorName</span> a b) <span class="ot">=</span> a<span class="op">+</span>b</span>
 <span id="cb370-2"><a href="#cb370-2" aria-hidden="true" tabindex="-1"></a>foo (<span class="dt">AnotherConstructor</span> _) <span class="ot">=</span> <span class="dv">0</span></span>
 <span id="cb370-3"><a href="#cb370-3" aria-hidden="true" tabindex="-1"></a>foo <span class="dt">OneMoreCons</span> <span class="ot">=</span> <span class="dv">7</span></span></code></pre></div>
 <ul>
 <li>Constructors are just functions:</li>
 </ul>
-<div class="sourceCode" id="cb371"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb371-1"><a href="#cb371-1" aria-hidden="true" tabindex="-1"></a><span class="dt">ConstructorName</span><span class="ot"> ::</span> <span class="dt">FieldType</span> <span class="ot">-&gt;</span> <span class="dt">FieldType2</span> <span class="ot">-&gt;</span> <span class="dt">TypeName</span></span>
+<div class="sourceCode" id="cb371"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb371-1"><a href="#cb371-1" aria-hidden="true" tabindex="-1"></a><span class="dt">ConstructorName</span><span class="ot"> ::</span> <span class="dt">FieldType</span> <span class="ot">-&gt;</span> <span class="dt">FieldType2</span> <span class="ot">-&gt;</span> <span class="dt">TypeName</span></span>
 <span id="cb371-2"><a href="#cb371-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Cons1</span><span class="ot"> ::</span> a <span class="ot">-&gt;</span> <span class="dt">Type1</span> <span class="ot">-&gt;</span> <span class="dt">TypeName</span> a</span></code></pre></div>
 <ul>
 <li>You can also define datatypes using record syntax:</li>
 </ul>
-<div class="sourceCode" id="cb372"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb372-1"><a href="#cb372-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">TypeName</span> <span class="ot">=</span> <span class="dt">Constructor</span> {<span class="ot"> field1 ::</span> <span class="dt">Field1Type</span>,<span class="ot"> field2 ::</span> <span class="dt">Field2Type</span> }</span></code></pre></div>
-<p>This gives you accessor functions like <code>field1 :: TypeName -&gt; Field1Type</code> for free.</p>
-<h2 data-number="5.6" id="sidenote-other-ways-of-defining-types"><span class="header-section-number">5.6</span> Sidenote: Other Ways of Defining Types</h2>
-<p>In addition to the <code>data</code> keyword, there are two additional ways of defining types in Haskell.</p>
-<p>The <code>newtype</code> keyword works like <code>data</code>, but you can only have a single constructor with a single field. It’s sometimes wise to use <code>newtype</code> for performance reasons, but we’ll get back to those in part 2.</p>
-<p>The <code>type</code> keyword introduces a <em>type alias</em>. Type aliases don’t affect type checking, they just offer a shorthand for writing types. For example the familiar <code>String</code> type is an alias for <code>[Char]</code>:</p>
-<div class="sourceCode" id="cb373"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb373-1"><a href="#cb373-1" aria-hidden="true" tabindex="-1"></a><span class="kw">type</span> <span class="dt">String</span> <span class="ot">=</span> [<span class="dt">Char</span>]</span></code></pre></div>
-<p>This means that whenever the compiler reads <code>String</code>, it just immediately replaces it with <code>[Char]</code>. Type aliases seem useful, but they can easily make reading type errors harder.</p>
-<h2 data-number="5.7" id="how-do-algebraic-datatypes-work"><span class="header-section-number">5.7</span> How Do Algebraic Datatypes Work?</h2>
-<p>Remember how lists were represented in memory as linked lists? Let’s look in more detail at what algebraic datatypes look like in memory.</p>
-<p>Haskell data forms <em>directed graphs</em> in memory. Every constructor is a node, every field is an edge. Names (of variables) are pointers into this graph. Different names can <em>share</em> parts of the structure. Here’s an example with lists. Note how the last two elements of <code>x</code> are shared with <code>y</code> and <code>z</code>.</p>
-<div class="sourceCode" id="cb374"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb374-1"><a href="#cb374-1" aria-hidden="true" tabindex="-1"></a><span class="kw">let</span> x <span class="ot">=</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>]</span>
+<div class="sourceCode" id="cb372"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb372-1"><a href="#cb372-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">TypeName</span> <span class="ot">=</span> <span class="dt">Constructor</span> {<span class="ot"> field1 ::</span> <span class="dt">Field1Type</span>,<span class="ot"> field2 ::</span> <span class="dt">Field2Type</span> }</span></code></pre></div>
+<p>This gives you accessor functions like
+<code>field1 :: TypeName -&gt; Field1Type</code> for free.</p>
+<h2 data-number="5.6" id="sidenote-other-ways-of-defining-types"><span
+class="header-section-number">5.6</span> Sidenote: Other Ways of
+Defining Types</h2>
+<p>In addition to the <code>data</code> keyword, there are two
+additional ways of defining types in Haskell.</p>
+<p>The <code>newtype</code> keyword works like <code>data</code>, but
+you can only have a single constructor with a single field. It’s
+sometimes wise to use <code>newtype</code> for performance reasons, but
+we’ll get back to those in part 2.</p>
+<p>The <code>type</code> keyword introduces a <em>type alias</em>. Type
+aliases don’t affect type checking, they just offer a shorthand for
+writing types. For example the familiar <code>String</code> type is an
+alias for <code>[Char]</code>:</p>
+<div class="sourceCode" id="cb373"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb373-1"><a href="#cb373-1" aria-hidden="true" tabindex="-1"></a><span class="kw">type</span> <span class="dt">String</span> <span class="ot">=</span> [<span class="dt">Char</span>]</span></code></pre></div>
+<p>This means that whenever the compiler reads <code>String</code>, it
+just immediately replaces it with <code>[Char]</code>. Type aliases seem
+useful, but they can easily make reading type errors harder.</p>
+<h2 data-number="5.7" id="how-do-algebraic-datatypes-work"><span
+class="header-section-number">5.7</span> How Do Algebraic Datatypes
+Work?</h2>
+<p>Remember how lists were represented in memory as linked lists? Let’s
+look in more detail at what algebraic datatypes look like in memory.</p>
+<p>Haskell data forms <em>directed graphs</em> in memory. Every
+constructor is a node, every field is an edge. Names (of variables) are
+pointers into this graph. Different names can <em>share</em> parts of
+the structure. Here’s an example with lists. Note how the last two
+elements of <code>x</code> are shared with <code>y</code> and
+<code>z</code>.</p>
+<div class="sourceCode" id="cb374"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb374-1"><a href="#cb374-1" aria-hidden="true" tabindex="-1"></a><span class="kw">let</span> x <span class="ot">=</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>]</span>
 <span id="cb374-2"><a href="#cb374-2" aria-hidden="true" tabindex="-1"></a>    y <span class="ot">=</span> <span class="fu">drop</span> <span class="dv">2</span> x</span>
 <span id="cb374-3"><a href="#cb374-3" aria-hidden="true" tabindex="-1"></a>    z <span class="ot">=</span> <span class="dv">5</span><span class="op">:</span>y</span></code></pre></div>
 <p><img src="img/DAG1.svg" /></p>
@@ -3239,11 +5097,19 @@ let x = [1,2,3,4]        (1:) -> (2:) -> (3:) -> (4:) -> []
                               z -> (5:) --+
 ```
 -->
-<p>What happens when you make a new version of a datastructure is called <em>path copying</em>. Since Haskell data is immutable, the changed parts of the datastructure get copied, while the unchanged parts can be shared between the old and new versions.</p>
+<p>What happens when you make a new version of a datastructure is called
+<em>path copying</em>. Since Haskell data is immutable, the changed
+parts of the datastructure get copied, while the unchanged parts can be
+shared between the old and new versions.</p>
 <p>Consider the definition of <code>++</code>:</p>
-<div class="sourceCode" id="cb375"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb375-1"><a href="#cb375-1" aria-hidden="true" tabindex="-1"></a>[]     <span class="op">++</span> ys <span class="ot">=</span> ys</span>
+<div class="sourceCode" id="cb375"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb375-1"><a href="#cb375-1" aria-hidden="true" tabindex="-1"></a>[]     <span class="op">++</span> ys <span class="ot">=</span> ys</span>
 <span id="cb375-2"><a href="#cb375-2" aria-hidden="true" tabindex="-1"></a>(x<span class="op">:</span>xs) <span class="op">++</span> ys <span class="ot">=</span> x<span class="op">:</span>(xs <span class="op">++</span> ys)</span></code></pre></div>
-<p>We are making a copy of the first argument while we walk it. For every <code>:</code> constructor in the first input list, we are creating a new <code>:</code> constructor in the output list. The second argument can be shared. It is not used at all in the recursion. Visually:</p>
+<p>We are making a copy of the first argument while we walk it. For
+every <code>:</code> constructor in the first input list, we are
+creating a new <code>:</code> constructor in the output list. The second
+argument can be shared. It is not used at all in the recursion.
+Visually:</p>
 <p><img src="img/DAG2.svg" /></p>
 <!--
 ```haskell
@@ -3255,9 +5121,16 @@ let x = [1,2,3,4]        (1:) -> (2:) -> (3:) -> (4:) -> []
 xs ++ ys -> (1:) -> (2:) -> (3:) --+
 ```
 -->
-<p>One more way to think about it is this: we want to change the <code>tail</code> pointer of the list element <code>(3:)</code>. That means we need to make a new <code>(3:)</code>. However the <code>(2:)</code> points to the <code>(3:)</code> so we need a new copy of the <code>(2:)</code> as well. Likewise for <code>(1:)</code>.</p>
-<p>The graphs that we get when working with lists are fairly simple. As a more involved example, here is what happens in memory when we run the binary tree insertion example from earlier in this lecture.</p>
-<div class="sourceCode" id="cb376"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb376-1"><a href="#cb376-1" aria-hidden="true" tabindex="-1"></a><span class="ot">insert ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Tree</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Tree</span> <span class="dt">Int</span></span>
+<p>One more way to think about it is this: we want to change the
+<code>tail</code> pointer of the list element <code>(3:)</code>. That
+means we need to make a new <code>(3:)</code>. However the
+<code>(2:)</code> points to the <code>(3:)</code> so we need a new copy
+of the <code>(2:)</code> as well. Likewise for <code>(1:)</code>.</p>
+<p>The graphs that we get when working with lists are fairly simple. As
+a more involved example, here is what happens in memory when we run the
+binary tree insertion example from earlier in this lecture.</p>
+<div class="sourceCode" id="cb376"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb376-1"><a href="#cb376-1" aria-hidden="true" tabindex="-1"></a><span class="ot">insert ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Tree</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Tree</span> <span class="dt">Int</span></span>
 <span id="cb376-2"><a href="#cb376-2" aria-hidden="true" tabindex="-1"></a>insert x <span class="dt">Empty</span> <span class="ot">=</span> <span class="dt">Node</span> x <span class="dt">Empty</span> <span class="dt">Empty</span></span>
 <span id="cb376-3"><a href="#cb376-3" aria-hidden="true" tabindex="-1"></a>insert x (<span class="dt">Node</span> y l r)</span>
 <span id="cb376-4"><a href="#cb376-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> x <span class="op">&lt;</span> y <span class="ot">=</span> <span class="dt">Node</span> y (insert x l) r</span>
@@ -3277,8 +5150,11 @@ Empty  Node 4  Empty  Empty    Node 6    Empty
    Empty   Empty           Empty   Empty
 ```
 --></p>
-<p>Note how the old and the new tree share the subtree with 3 and 4 since it wasn’t changed, but the node 7 that was “changed” and all nodes above it get copied.</p>
-<h2 data-number="5.8" id="quiz-4"><span class="header-section-number">5.8</span> Quiz</h2>
+<p>Note how the old and the new tree share the subtree with 3 and 4
+since it wasn’t changed, but the node 7 that was “changed” and all nodes
+above it get copied.</p>
+<h2 data-number="5.8" id="quiz-4"><span
+class="header-section-number">5.8</span> Quiz</h2>
 <p>Why can’t we <code>map Nothing</code>?</p>
 <ol class="quiz">
 <li class="correct">
@@ -3291,7 +5167,9 @@ Because <code>Nothing</code> returns nothing
 Because <code>Nothing</code> is a constructor.
 </li>
 </ol>
-<p>If we define <code>data Boing = Frick String Boing (Int -&gt; Bool)</code>, what is the type of <code>Frick</code>?</p>
+<p>If we define
+<code>data Boing = Frick String Boing (Int -&gt; Bool)</code>, what is
+the type of <code>Frick</code>?</p>
 <ol class="quiz">
 <li>
 <code>Boing</code>
@@ -3303,7 +5181,9 @@ Because <code>Nothing</code> is a constructor.
 <code>String -&gt; Boing -&gt; (Int -&gt; Bool) -&gt; Boing</code>
 </li>
 </ol>
-<p>If we define <code>data ThreeLists a b c = ThreeLists [a] [b] [c]</code>, what is the type of the constructor <code>ThreeLists</code>?</p>
+<p>If we define
+<code>data ThreeLists a b c = ThreeLists [a] [b] [c]</code>, what is the
+type of the constructor <code>ThreeLists</code>?</p>
 <ol class="quiz">
 <li>
 <code>[a] -&gt; [b] -&gt; [c] -&gt; ThreeLists</code>
@@ -3318,7 +5198,9 @@ Because <code>Nothing</code> is a constructor.
 <code>[a] -&gt; [b] -&gt; [c] -&gt; ThreeLists [a] [b] [c]</code>
 </li>
 </ol>
-<p>If we define <code>data TwoLists a b = TwoList {aList :: [a], bList :: [b]}</code>, what is the type of the function <code>aList</code>?</p>
+<p>If we define
+<code>data TwoLists a b = TwoList {aList :: [a], bList :: [b]}</code>,
+what is the type of the function <code>aList</code>?</p>
 <ol class="quiz">
 <li>
 <code>aList</code> is not a function, it is a field
@@ -3333,41 +5215,69 @@ Because <code>Nothing</code> is a constructor.
 <code>[a]</code>
 </li>
 </ol>
-<h2 data-number="5.9" id="exercises-4"><span class="header-section-number">5.9</span> Exercises</h2>
+<h2 data-number="5.9" id="exercises-4"><span
+class="header-section-number">5.9</span> Exercises</h2>
 <ul>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set5a.hs">Set5a</a>: using and defining algebraic datatypes</li>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set5b.hs">Set5b</a>: playing with binary trees</li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set5a.hs">Set5a</a>:
+using and defining algebraic datatypes</li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set5b.hs">Set5b</a>:
+playing with binary trees</li>
 </ul>
-<h1 data-number="6" id="lecture-6-working-class-hero"><span class="header-section-number">6</span> Lecture 6: Working Class Hero</h1>
-<p>We’ve seen class constraints like <code>Eq a =&gt;</code> in types. We know how to use existing classes with existing types. But how do we use existing classes with our own types? How can we define our own classes?</p>
-<p>Here’s how to make your own type a member of the <code>Eq</code> class:</p>
-<div class="sourceCode" id="cb377"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb377-1"><a href="#cb377-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Color</span> <span class="ot">=</span> <span class="dt">Black</span> <span class="op">|</span> <span class="dt">White</span></span>
+<h1 data-number="6" id="lecture-6-working-class-hero"><span
+class="header-section-number">6</span> Lecture 6: Working Class
+Hero</h1>
+<p>We’ve seen class constraints like <code>Eq a =&gt;</code> in types.
+We know how to use existing classes with existing types. But how do we
+use existing classes with our own types? How can we define our own
+classes?</p>
+<p>Here’s how to make your own type a member of the <code>Eq</code>
+class:</p>
+<div class="sourceCode" id="cb377"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb377-1"><a href="#cb377-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Color</span> <span class="ot">=</span> <span class="dt">Black</span> <span class="op">|</span> <span class="dt">White</span></span>
 <span id="cb377-2"><a href="#cb377-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb377-3"><a href="#cb377-3" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Eq</span> <span class="dt">Color</span> <span class="kw">where</span></span>
 <span id="cb377-4"><a href="#cb377-4" aria-hidden="true" tabindex="-1"></a>  <span class="dt">Black</span> <span class="op">==</span> <span class="dt">Black</span>  <span class="ot">=</span> <span class="dt">True</span></span>
 <span id="cb377-5"><a href="#cb377-5" aria-hidden="true" tabindex="-1"></a>  <span class="dt">White</span> <span class="op">==</span> <span class="dt">White</span>  <span class="ot">=</span> <span class="dt">True</span></span>
 <span id="cb377-6"><a href="#cb377-6" aria-hidden="true" tabindex="-1"></a>  _     <span class="op">==</span> _      <span class="ot">=</span> <span class="dt">False</span></span></code></pre></div>
-<p>A class instance is an <code>instance</code> block that contains definitions for the functions in that class. Here we define how <code>==</code> works on <code>Color</code>.</p>
-<h2 data-number="6.1" id="syntax-of-classes-and-instances"><span class="header-section-number">6.1</span> Syntax of Classes and Instances</h2>
-<p>A type class is defined using <code>class</code> syntax. The functions in the class are given types. Here’s a class <code>Size</code> that contains one function, <code>size</code>:</p>
-<div class="sourceCode" id="cb378"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb378-1"><a href="#cb378-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Size</span> a <span class="kw">where</span></span>
+<p>A class instance is an <code>instance</code> block that contains
+definitions for the functions in that class. Here we define how
+<code>==</code> works on <code>Color</code>.</p>
+<h2 data-number="6.1" id="syntax-of-classes-and-instances"><span
+class="header-section-number">6.1</span> Syntax of Classes and
+Instances</h2>
+<p>A type class is defined using <code>class</code> syntax. The
+functions in the class are given types. Here’s a class <code>Size</code>
+that contains one function, <code>size</code>:</p>
+<div class="sourceCode" id="cb378"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb378-1"><a href="#cb378-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Size</span> a <span class="kw">where</span></span>
 <span id="cb378-2"><a href="#cb378-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  size ::</span> a <span class="ot">-&gt;</span> <span class="dt">Int</span></span></code></pre></div>
-<p>Instances of a class are defined with <code>instance</code> syntax we’ve just seen. Here is how we make <code>Int</code> and <code>[a]</code> members of the <code>Size</code> class:</p>
-<div class="sourceCode" id="cb379"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb379-1"><a href="#cb379-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Size</span> <span class="dt">Int</span> <span class="kw">where</span></span>
+<p>Instances of a class are defined with <code>instance</code> syntax
+we’ve just seen. Here is how we make <code>Int</code> and
+<code>[a]</code> members of the <code>Size</code> class:</p>
+<div class="sourceCode" id="cb379"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb379-1"><a href="#cb379-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Size</span> <span class="dt">Int</span> <span class="kw">where</span></span>
 <span id="cb379-2"><a href="#cb379-2" aria-hidden="true" tabindex="-1"></a>  size x <span class="ot">=</span> <span class="fu">abs</span> x</span>
 <span id="cb379-3"><a href="#cb379-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb379-4"><a href="#cb379-4" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Size</span> [a] <span class="kw">where</span></span>
 <span id="cb379-5"><a href="#cb379-5" aria-hidden="true" tabindex="-1"></a>  size xs <span class="ot">=</span> <span class="fu">length</span> xs</span></code></pre></div>
-<p>Our class <code>Size</code> behaves just like existing type classes. We can use <code>size</code> anywhere where a function can be used, and Haskell can infer types with <code>Size</code> constraints for us:</p>
-<div class="sourceCode" id="cb380"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb380-1"><a href="#cb380-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t size</span>
+<p>Our class <code>Size</code> behaves just like existing type classes.
+We can use <code>size</code> anywhere where a function can be used, and
+Haskell can infer types with <code>Size</code> constraints for us:</p>
+<div class="sourceCode" id="cb380"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb380-1"><a href="#cb380-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t size</span>
 <span id="cb380-2"><a href="#cb380-2" aria-hidden="true" tabindex="-1"></a><span class="ot">size ::</span> <span class="dt">Size</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb380-3"><a href="#cb380-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> size [<span class="dt">True</span>,<span class="dt">False</span>]</span>
 <span id="cb380-4"><a href="#cb380-4" aria-hidden="true" tabindex="-1"></a><span class="dv">2</span></span>
 <span id="cb380-5"><a href="#cb380-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> sizeBoth a b <span class="ot">=</span> [size a, size b]</span>
 <span id="cb380-6"><a href="#cb380-6" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t sizeBoth</span>
 <span id="cb380-7"><a href="#cb380-7" aria-hidden="true" tabindex="-1"></a><span class="ot">sizeBoth ::</span> (<span class="dt">Size</span> a1, <span class="dt">Size</span> a2) <span class="ot">=&gt;</span> a1 <span class="ot">-&gt;</span> a2 <span class="ot">-&gt;</span> [<span class="dt">Int</span>]</span></code></pre></div>
-<p>A class can contain multiple functions, and even constants. Here we define a new version of the <code>Size</code> class with more content.</p>
-<div class="sourceCode" id="cb381"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb381-1"><a href="#cb381-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Size</span> a <span class="kw">where</span></span>
+<p>A class can contain multiple functions, and even constants. Here we
+define a new version of the <code>Size</code> class with more
+content.</p>
+<div class="sourceCode" id="cb381"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb381-1"><a href="#cb381-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Size</span> a <span class="kw">where</span></span>
 <span id="cb381-2"><a href="#cb381-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  empty ::</span> a</span>
 <span id="cb381-3"><a href="#cb381-3" aria-hidden="true" tabindex="-1"></a><span class="ot">  size ::</span> a <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb381-4"><a href="#cb381-4" aria-hidden="true" tabindex="-1"></a><span class="ot">  sameSize ::</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
@@ -3384,9 +5294,15 @@ Because <code>Nothing</code> is a constructor.
 <span id="cb381-15"><a href="#cb381-15" aria-hidden="true" tabindex="-1"></a>  empty <span class="ot">=</span> []</span>
 <span id="cb381-16"><a href="#cb381-16" aria-hidden="true" tabindex="-1"></a>  size xs <span class="ot">=</span> <span class="fu">length</span> xs</span>
 <span id="cb381-17"><a href="#cb381-17" aria-hidden="true" tabindex="-1"></a>  sameSize x y <span class="ot">=</span> size x <span class="op">==</span> size y</span></code></pre></div>
-<h2 data-number="6.2" id="default-implementations"><span class="header-section-number">6.2</span> Default Implementations</h2>
-<p>Did you notice how in the previous example we gave <code>sameSize</code> the same definition in both instances? This is a very common occurrence, and it’s why Haskell classes can have <em>default implementations</em>. As a first example, here’s an <code>Example</code> type class for giving example values of types.</p>
-<div class="sourceCode" id="cb382"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb382-1"><a href="#cb382-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Example</span> a <span class="kw">where</span></span>
+<h2 data-number="6.2" id="default-implementations"><span
+class="header-section-number">6.2</span> Default Implementations</h2>
+<p>Did you notice how in the previous example we gave
+<code>sameSize</code> the same definition in both instances? This is a
+very common occurrence, and it’s why Haskell classes can have
+<em>default implementations</em>. As a first example, here’s an
+<code>Example</code> type class for giving example values of types.</p>
+<div class="sourceCode" id="cb382"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb382-1"><a href="#cb382-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Example</span> a <span class="kw">where</span></span>
 <span id="cb382-2"><a href="#cb382-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  example ::</span> a           <span class="co">-- the main example for the type `a`</span></span>
 <span id="cb382-3"><a href="#cb382-3" aria-hidden="true" tabindex="-1"></a><span class="ot">  examples ::</span> [a]        <span class="co">-- a short list of examples</span></span>
 <span id="cb382-4"><a href="#cb382-4" aria-hidden="true" tabindex="-1"></a>  examples <span class="ot">=</span> [example]   <span class="co">-- ...defaulting to just the main example</span></span>
@@ -3397,8 +5313,14 @@ Because <code>Nothing</code> is a constructor.
 <span id="cb382-9"><a href="#cb382-9" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb382-10"><a href="#cb382-10" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Example</span> <span class="dt">Bool</span> <span class="kw">where</span></span>
 <span id="cb382-11"><a href="#cb382-11" aria-hidden="true" tabindex="-1"></a>  example <span class="ot">=</span> <span class="dt">True</span></span></code></pre></div>
-<p>Here’s how <code>Example</code> works. Note how the default implementation of <code>examples</code> got used in the <code>Bool</code> case but not in the <code>Int</code> case. Also note the need for explicit type signatures to tell GHCi which instance we’re interested in. Without them, we would get an “Ambiguous type variable” error.</p>
-<div class="sourceCode" id="cb383"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb383-1"><a href="#cb383-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span><span class="ot"> example ::</span> <span class="dt">Bool</span></span>
+<p>Here’s how <code>Example</code> works. Note how the default
+implementation of <code>examples</code> got used in the
+<code>Bool</code> case but not in the <code>Int</code> case. Also note
+the need for explicit type signatures to tell GHCi which instance we’re
+interested in. Without them, we would get an “Ambiguous type variable”
+error.</p>
+<div class="sourceCode" id="cb383"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb383-1"><a href="#cb383-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span><span class="ot"> example ::</span> <span class="dt">Bool</span></span>
 <span id="cb383-2"><a href="#cb383-2" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span>
 <span id="cb383-3"><a href="#cb383-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span><span class="ot"> example ::</span> <span class="dt">Int</span></span>
 <span id="cb383-4"><a href="#cb383-4" aria-hidden="true" tabindex="-1"></a><span class="dv">1</span></span>
@@ -3406,17 +5328,37 @@ Because <code>Nothing</code> is a constructor.
 <span id="cb383-6"><a href="#cb383-6" aria-hidden="true" tabindex="-1"></a>[<span class="dt">True</span>]</span>
 <span id="cb383-7"><a href="#cb383-7" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span><span class="ot"> examples ::</span> [<span class="dt">Int</span>]</span>
 <span id="cb383-8"><a href="#cb383-8" aria-hidden="true" tabindex="-1"></a>[<span class="dv">0</span>,<span class="dv">1</span>,<span class="dv">2</span>]</span></code></pre></div>
-<p>The standard type classes use lots of default implementations to make implementing the classes easy. Here is the standard definitions for <code>Eq</code> (formatted for readability).</p>
-<div class="sourceCode" id="cb384"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb384-1"><a href="#cb384-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Eq</span> a <span class="kw">where</span></span>
+<p>The standard type classes use lots of default implementations to make
+implementing the classes easy. Here is the standard definitions for
+<code>Eq</code> (formatted for readability).</p>
+<div class="sourceCode" id="cb384"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb384-1"><a href="#cb384-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Eq</span> a <span class="kw">where</span></span>
 <span id="cb384-2"><a href="#cb384-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  (==) ::</span>  a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb384-3"><a href="#cb384-3" aria-hidden="true" tabindex="-1"></a>  x <span class="op">==</span> y  <span class="ot">=</span> <span class="fu">not</span> (x <span class="op">/=</span> y)</span>
 <span id="cb384-4"><a href="#cb384-4" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb384-5"><a href="#cb384-5" aria-hidden="true" tabindex="-1"></a><span class="ot">  (/=) ::</span>  a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb384-6"><a href="#cb384-6" aria-hidden="true" tabindex="-1"></a>  x <span class="op">/=</span> y  <span class="ot">=</span> <span class="fu">not</span> (x <span class="op">==</span> y)</span></code></pre></div>
-<p>Note how both operations have a default implementation in terms of the other. This means we could define an <code>Eq</code> instance with no content at all, but the resulting functions would just recurse forever. In practice, we want to define at least one of <code>==</code> and <code>/=</code>.</p>
-<p>When there are lots of default implementations, it can be hard to know which functions you need to implement yourself. For this reason class documentation usually mentions the <em>minimal complete definition</em>. For <code>Eq</code>, <a href="https://hackage.haskell.org/package/base-4.16.4.0/docs/Prelude.html#t:Eq">the docs say</a> “Minimal complete definition: either == or /=.”</p>
-<p>Let’s look at <code>Ord</code> next. <code>Ord</code> has 7 operations, all with default implementations in terms of each other. By the way, note the quirky way of defining multiple type signatures at once. It’s okay, it’s a feature of Haskell, this is how <code>Ord</code> is defined in <a href="https://www.haskell.org/onlinereport/haskell2010/haskellch6.html#x13-1270006.3">the standard</a>. (We’ll get back to what the <code>(Eq a) =&gt;</code> part means soon.)</p>
-<div class="sourceCode" id="cb385"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb385-1"><a href="#cb385-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span>  (<span class="dt">Eq</span> a) <span class="ot">=&gt;</span> <span class="dt">Ord</span> a  <span class="kw">where</span></span>
+<p>Note how both operations have a default implementation in terms of
+the other. This means we could define an <code>Eq</code> instance with
+no content at all, but the resulting functions would just recurse
+forever. In practice, we want to define at least one of <code>==</code>
+and <code>/=</code>.</p>
+<p>When there are lots of default implementations, it can be hard to
+know which functions you need to implement yourself. For this reason
+class documentation usually mentions the <em>minimal complete
+definition</em>. For <code>Eq</code>, <a
+href="https://hackage.haskell.org/package/base-4.16.4.0/docs/Prelude.html#t:Eq">the
+docs say</a> “Minimal complete definition: either == or /=.”</p>
+<p>Let’s look at <code>Ord</code> next. <code>Ord</code> has 7
+operations, all with default implementations in terms of each other. By
+the way, note the quirky way of defining multiple type signatures at
+once. It’s okay, it’s a feature of Haskell, this is how <code>Ord</code>
+is defined in <a
+href="https://www.haskell.org/onlinereport/haskell2010/haskellch6.html#x13-1270006.3">the
+standard</a>. (We’ll get back to what the <code>(Eq a) =&gt;</code> part
+means soon.)</p>
+<div class="sourceCode" id="cb385"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb385-1"><a href="#cb385-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span>  (<span class="dt">Eq</span> a) <span class="ot">=&gt;</span> <span class="dt">Ord</span> a  <span class="kw">where</span></span>
 <span id="cb385-2"><a href="#cb385-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  compare              ::</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Ordering</span></span>
 <span id="cb385-3"><a href="#cb385-3" aria-hidden="true" tabindex="-1"></a>  (<span class="op">&lt;</span>), (<span class="op">&lt;=</span>), (<span class="op">&gt;=</span>),<span class="ot"> (&gt;) ::</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb385-4"><a href="#cb385-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">max</span>,<span class="ot"> min             ::</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
@@ -3434,20 +5376,33 @@ Because <code>Nothing</code> is a constructor.
 <span id="cb385-16"><a href="#cb385-16" aria-hidden="true" tabindex="-1"></a>          <span class="op">|</span> <span class="fu">otherwise</span> <span class="ot">=</span>  x</span>
 <span id="cb385-17"><a href="#cb385-17" aria-hidden="true" tabindex="-1"></a>  <span class="fu">min</span> x y <span class="op">|</span> x <span class="op">&lt;=</span> y    <span class="ot">=</span>  x</span>
 <span id="cb385-18"><a href="#cb385-18" aria-hidden="true" tabindex="-1"></a>          <span class="op">|</span> <span class="fu">otherwise</span> <span class="ot">=</span>  y</span></code></pre></div>
-<p>With this definition it’s really hard to know what the minimal complete definition is. Luckily the <a href="https://hackage.haskell.org/package/base-4.16.4.0/docs/Prelude.html#t:Ord">docs tell us</a> “Minimal complete definition: either compare or &lt;=.”</p>
-<p>As a final word on default implementations, if there is never a need to override the default definition, the function can be moved out of the class for simplicity. Consider a class like <code>Combine</code> below:</p>
-<div class="sourceCode" id="cb386"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb386-1"><a href="#cb386-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Combine</span> a <span class="kw">where</span></span>
+<p>With this definition it’s really hard to know what the minimal
+complete definition is. Luckily the <a
+href="https://hackage.haskell.org/package/base-4.16.4.0/docs/Prelude.html#t:Ord">docs
+tell us</a> “Minimal complete definition: either compare or &lt;=.”</p>
+<p>As a final word on default implementations, if there is never a need
+to override the default definition, the function can be moved out of the
+class for simplicity. Consider a class like <code>Combine</code>
+below:</p>
+<div class="sourceCode" id="cb386"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb386-1"><a href="#cb386-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Combine</span> a <span class="kw">where</span></span>
 <span id="cb386-2"><a href="#cb386-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  combine ::</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb386-3"><a href="#cb386-3" aria-hidden="true" tabindex="-1"></a><span class="ot">  combine3 ::</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb386-4"><a href="#cb386-4" aria-hidden="true" tabindex="-1"></a>  combine3 x y z <span class="ot">=</span> combine x (combine y z)</span></code></pre></div>
-<p>It’s hard to think of a case where <code>combine3</code> would be given any other definition, so why not move it out of the class:</p>
-<div class="sourceCode" id="cb387"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb387-1"><a href="#cb387-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Combine</span> a <span class="kw">where</span></span>
+<p>It’s hard to think of a case where <code>combine3</code> would be
+given any other definition, so why not move it out of the class:</p>
+<div class="sourceCode" id="cb387"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb387-1"><a href="#cb387-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Combine</span> a <span class="kw">where</span></span>
 <span id="cb387-2"><a href="#cb387-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  combine ::</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb387-3"><a href="#cb387-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb387-4"><a href="#cb387-4" aria-hidden="true" tabindex="-1"></a><span class="ot">combine3 ::</span> <span class="dt">Combine</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb387-5"><a href="#cb387-5" aria-hidden="true" tabindex="-1"></a>combine3 x y z <span class="ot">=</span> combine x (combine y z)</span></code></pre></div>
-<p>As an example, here are the <code>Eq</code> and <code>Ord</code> instances for a simple pair type. Note how the definition uses the minimal complete definition rules by only defining <code>==</code> and <code>&lt;=</code>.</p>
-<div class="sourceCode" id="cb388"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb388-1"><a href="#cb388-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">IntPair</span> <span class="ot">=</span> <span class="dt">IntPair</span> <span class="dt">Int</span> <span class="dt">Int</span></span>
+<p>As an example, here are the <code>Eq</code> and <code>Ord</code>
+instances for a simple pair type. Note how the definition uses the
+minimal complete definition rules by only defining <code>==</code> and
+<code>&lt;=</code>.</p>
+<div class="sourceCode" id="cb388"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb388-1"><a href="#cb388-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">IntPair</span> <span class="ot">=</span> <span class="dt">IntPair</span> <span class="dt">Int</span> <span class="dt">Int</span></span>
 <span id="cb388-2"><a href="#cb388-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">deriving</span> <span class="dt">Show</span></span>
 <span id="cb388-3"><a href="#cb388-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb388-4"><a href="#cb388-4" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Eq</span> <span class="dt">IntPair</span> <span class="kw">where</span></span>
@@ -3458,7 +5413,8 @@ Because <code>Nothing</code> is a constructor.
 <span id="cb388-9"><a href="#cb388-9" aria-hidden="true" tabindex="-1"></a>     <span class="op">|</span> a1<span class="op">&lt;</span>b1     <span class="ot">=</span> <span class="dt">True</span></span>
 <span id="cb388-10"><a href="#cb388-10" aria-hidden="true" tabindex="-1"></a>     <span class="op">|</span> a1<span class="op">&gt;</span>b1     <span class="ot">=</span> <span class="dt">False</span></span>
 <span id="cb388-11"><a href="#cb388-11" aria-hidden="true" tabindex="-1"></a>     <span class="op">|</span> <span class="fu">otherwise</span> <span class="ot">=</span> a2<span class="op">&lt;=</span>b2</span></code></pre></div>
-<div class="sourceCode" id="cb389"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb389-1"><a href="#cb389-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Main</span><span class="op">&gt;</span> (<span class="dt">IntPair</span> <span class="dv">1</span> <span class="dv">2</span>) <span class="op">&lt;</span> (<span class="dt">IntPair</span> <span class="dv">2</span> <span class="dv">3</span>)</span>
+<div class="sourceCode" id="cb389"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb389-1"><a href="#cb389-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Main</span><span class="op">&gt;</span> (<span class="dt">IntPair</span> <span class="dv">1</span> <span class="dv">2</span>) <span class="op">&lt;</span> (<span class="dt">IntPair</span> <span class="dv">2</span> <span class="dv">3</span>)</span>
 <span id="cb389-2"><a href="#cb389-2" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span>
 <span id="cb389-3"><a href="#cb389-3" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Main</span><span class="op">&gt;</span> (<span class="dt">IntPair</span> <span class="dv">1</span> <span class="dv">2</span>) <span class="op">&gt;</span> (<span class="dt">IntPair</span> <span class="dv">2</span> <span class="dv">3</span>)</span>
 <span id="cb389-4"><a href="#cb389-4" aria-hidden="true" tabindex="-1"></a><span class="dt">False</span></span>
@@ -3466,21 +5422,37 @@ Because <code>Nothing</code> is a constructor.
 <span id="cb389-6"><a href="#cb389-6" aria-hidden="true" tabindex="-1"></a><span class="dt">LT</span></span>
 <span id="cb389-7"><a href="#cb389-7" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Main</span> <span class="dt">Data.List</span><span class="op">&gt;</span> <span class="fu">sort</span> [<span class="dt">IntPair</span> <span class="dv">1</span> <span class="dv">1</span>,<span class="dt">IntPair</span> <span class="dv">1</span> <span class="dv">4</span>,<span class="dt">IntPair</span> <span class="dv">2</span> <span class="dv">1</span>,<span class="dt">IntPair</span> <span class="dv">2</span> <span class="dv">2</span>]</span>
 <span id="cb389-8"><a href="#cb389-8" aria-hidden="true" tabindex="-1"></a>[<span class="dt">IntPair</span> <span class="dv">1</span> <span class="dv">1</span>,<span class="dt">IntPair</span> <span class="dv">1</span> <span class="dv">4</span>,<span class="dt">IntPair</span> <span class="dv">2</span> <span class="dv">1</span>,<span class="dt">IntPair</span> <span class="dv">2</span> <span class="dv">2</span>]</span></code></pre></div>
-<h2 data-number="6.3" id="useful-stuff"><span class="header-section-number">6.3</span> Useful Stuff</h2>
-<h3 data-number="6.3.1" id="deriving"><span class="header-section-number">6.3.1</span> Deriving</h3>
-<p>As we’ve seen many times already, <code>deriving</code> is a way to get automatically generated class instances. The <code>Read</code> and <code>Show</code> classes should pretty much always be derived to get the standard behaviour. The derived instance for <code>Eq</code> is typically what you want. It requires constructors and fields to match.</p>
-<p>The derived <code>Ord</code> instance might not be what you want. It orders constructors left-to-right, and then compares fields inside constructors left-to-right. An example:</p>
-<div class="sourceCode" id="cb390"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb390-1"><a href="#cb390-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Person</span> <span class="ot">=</span> <span class="dt">Dead</span> <span class="op">|</span> <span class="dt">Alive</span> <span class="dt">String</span> <span class="dt">Int</span></span>
+<h2 data-number="6.3" id="useful-stuff"><span
+class="header-section-number">6.3</span> Useful Stuff</h2>
+<h3 data-number="6.3.1" id="deriving"><span
+class="header-section-number">6.3.1</span> Deriving</h3>
+<p>As we’ve seen many times already, <code>deriving</code> is a way to
+get automatically generated class instances. The <code>Read</code> and
+<code>Show</code> classes should pretty much always be derived to get
+the standard behaviour. The derived instance for <code>Eq</code> is
+typically what you want. It requires constructors and fields to
+match.</p>
+<p>The derived <code>Ord</code> instance might not be what you want. It
+orders constructors left-to-right, and then compares fields inside
+constructors left-to-right. An example:</p>
+<div class="sourceCode" id="cb390"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb390-1"><a href="#cb390-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Person</span> <span class="ot">=</span> <span class="dt">Dead</span> <span class="op">|</span> <span class="dt">Alive</span> <span class="dt">String</span> <span class="dt">Int</span></span>
 <span id="cb390-2"><a href="#cb390-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">deriving</span> (<span class="dt">Show</span>, <span class="dt">Eq</span>, <span class="dt">Ord</span>)</span></code></pre></div>
-<div class="sourceCode" id="cb391"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb391-1"><a href="#cb391-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">Dead</span> <span class="op">&lt;</span> <span class="dt">Alive</span> <span class="st">&quot;Bob&quot;</span> <span class="dv">35</span>                   <span class="co">-- constructors are ordered left-to-right</span></span>
+<div class="sourceCode" id="cb391"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb391-1"><a href="#cb391-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">Dead</span> <span class="op">&lt;</span> <span class="dt">Alive</span> <span class="st">&quot;Bob&quot;</span> <span class="dv">35</span>                   <span class="co">-- constructors are ordered left-to-right</span></span>
 <span id="cb391-2"><a href="#cb391-2" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span>
 <span id="cb391-3"><a href="#cb391-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">Alive</span> <span class="st">&quot;Barbara&quot;</span> <span class="dv">35</span> <span class="op">&lt;</span> <span class="dt">Alive</span> <span class="st">&quot;Clive&quot;</span> <span class="dv">17</span>   <span class="co">-- names are compared before ages</span></span>
 <span id="cb391-4"><a href="#cb391-4" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span>
 <span id="cb391-5"><a href="#cb391-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">Alive</span> <span class="st">&quot;Clive&quot;</span> <span class="dv">17</span> <span class="op">&lt;</span> <span class="dt">Alive</span> <span class="st">&quot;Clive&quot;</span> <span class="dv">30</span>     <span class="co">-- finally, ages are compared if names match</span></span>
 <span id="cb391-6"><a href="#cb391-6" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span></code></pre></div>
-<h3 data-number="6.3.2" id="asking-ghci-about-classes"><span class="header-section-number">6.3.2</span> Asking GHCi About Classes</h3>
-<p>You can use the <code>:info</code> command in GHCi to get the contents and instances of a class. These days the info even includes the minimal complete definition (see the MINIMAL pragma). For example:</p>
-<div class="sourceCode" id="cb392"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb392-1"><a href="#cb392-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>info <span class="dt">Num</span></span>
+<h3 data-number="6.3.2" id="asking-ghci-about-classes"><span
+class="header-section-number">6.3.2</span> Asking GHCi About
+Classes</h3>
+<p>You can use the <code>:info</code> command in GHCi to get the
+contents and instances of a class. These days the info even includes the
+minimal complete definition (see the MINIMAL pragma). For example:</p>
+<div class="sourceCode" id="cb392"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb392-1"><a href="#cb392-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>info <span class="dt">Num</span></span>
 <span id="cb392-2"><a href="#cb392-2" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Num</span> a <span class="kw">where</span></span>
 <span id="cb392-3"><a href="#cb392-3" aria-hidden="true" tabindex="-1"></a><span class="ot">  (+) ::</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb392-4"><a href="#cb392-4" aria-hidden="true" tabindex="-1"></a><span class="ot">  (-) ::</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
@@ -3496,27 +5468,42 @@ Because <code>Nothing</code> is a constructor.
 <span id="cb392-14"><a href="#cb392-14" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Num</span> <span class="dt">Int</span> <span class="co">-- Defined in ‘GHC.Num’</span></span>
 <span id="cb392-15"><a href="#cb392-15" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Num</span> <span class="dt">Float</span> <span class="co">-- Defined in ‘GHC.Float’</span></span>
 <span id="cb392-16"><a href="#cb392-16" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Num</span> <span class="dt">Double</span> <span class="co">-- Defined in ‘GHC.Float’</span></span></code></pre></div>
-<h2 data-number="6.4" id="hierarchies"><span class="header-section-number">6.4</span> Hierarchies</h2>
-<p>Both classes and instances can form <em>hierarchies</em>. This means that a class or instance depends on another class or instance.</p>
-<h3 data-number="6.4.1" id="instance-hierarchies"><span class="header-section-number">6.4.1</span> Instance Hierarchies</h3>
-<p>Let’s try to define an <code>Eq</code> instance for a simple pair type:</p>
-<div class="sourceCode" id="cb393"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb393-1"><a href="#cb393-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Pair</span> a <span class="ot">=</span> <span class="dt">MakePair</span> a a</span>
+<h2 data-number="6.4" id="hierarchies"><span
+class="header-section-number">6.4</span> Hierarchies</h2>
+<p>Both classes and instances can form <em>hierarchies</em>. This means
+that a class or instance depends on another class or instance.</p>
+<h3 data-number="6.4.1" id="instance-hierarchies"><span
+class="header-section-number">6.4.1</span> Instance Hierarchies</h3>
+<p>Let’s try to define an <code>Eq</code> instance for a simple pair
+type:</p>
+<div class="sourceCode" id="cb393"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb393-1"><a href="#cb393-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Pair</span> a <span class="ot">=</span> <span class="dt">MakePair</span> a a</span>
 <span id="cb393-2"><a href="#cb393-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">deriving</span> <span class="dt">Show</span></span>
 <span id="cb393-3"><a href="#cb393-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb393-4"><a href="#cb393-4" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Eq</span> (<span class="dt">Pair</span> a) <span class="kw">where</span></span>
 <span id="cb393-5"><a href="#cb393-5" aria-hidden="true" tabindex="-1"></a>  (<span class="dt">MakePair</span> x y) <span class="op">==</span> (<span class="dt">MakePair</span> a b)   <span class="ot">=</span>   x<span class="op">==</span>a <span class="op">&amp;&amp;</span> y<span class="op">==</span>b</span></code></pre></div>
-<div class="sourceCode" id="cb394"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb394-1"><a href="#cb394-1" aria-hidden="true" tabindex="-1"></a><span class="fu">error</span><span class="op">:</span></span>
+<div class="sourceCode" id="cb394"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb394-1"><a href="#cb394-1" aria-hidden="true" tabindex="-1"></a><span class="fu">error</span><span class="op">:</span></span>
 <span id="cb394-2"><a href="#cb394-2" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">No</span> <span class="kw">instance</span> for (<span class="dt">Eq</span> a) arising from a use <span class="kw">of</span> ‘<span class="op">==</span>’</span>
 <span id="cb394-3"><a href="#cb394-3" aria-hidden="true" tabindex="-1"></a>      <span class="dt">Possible</span> fix<span class="op">:</span> add (<span class="dt">Eq</span> a) to the context <span class="kw">of</span> the <span class="kw">instance</span> declaration</span>
 <span id="cb394-4"><a href="#cb394-4" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">In</span> the first argument <span class="kw">of</span> ‘(<span class="op">&amp;&amp;</span>)’, namely ‘x <span class="op">==</span> a’</span>
 <span id="cb394-5"><a href="#cb394-5" aria-hidden="true" tabindex="-1"></a>      <span class="dt">In</span> the expression<span class="op">:</span> x <span class="op">==</span> a <span class="op">&amp;&amp;</span> y <span class="op">==</span> b</span>
 <span id="cb394-6"><a href="#cb394-6" aria-hidden="true" tabindex="-1"></a>      <span class="dt">In</span> an equation for ‘<span class="op">==</span>’<span class="op">:</span></span>
 <span id="cb394-7"><a href="#cb394-7" aria-hidden="true" tabindex="-1"></a>          (<span class="dt">MakePair</span> x y) <span class="op">==</span> (<span class="dt">MakePair</span> a b) <span class="ot">=</span> x <span class="op">==</span> a <span class="op">&amp;&amp;</span> y <span class="op">==</span> b</span></code></pre></div>
-<p>The compiler is trying to tell us that our <code>Eq (Pair a)</code> instance needs an <code>Eq a</code> instance to work. How can we compare pairs of values of type <code>a</code> if we can’t compare values of type <code>a</code>? To solve this we need to add a type constraint to the instance declaration, just like we’ve added type constraints to function definitions.</p>
-<div class="sourceCode" id="cb395"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb395-1"><a href="#cb395-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Eq</span> a <span class="ot">=&gt;</span> <span class="dt">Eq</span> (<span class="dt">Pair</span> a) <span class="kw">where</span></span>
+<p>The compiler is trying to tell us that our <code>Eq (Pair a)</code>
+instance needs an <code>Eq a</code> instance to work. How can we compare
+pairs of values of type <code>a</code> if we can’t compare values of
+type <code>a</code>? To solve this we need to add a type constraint to
+the instance declaration, just like we’ve added type constraints to
+function definitions.</p>
+<div class="sourceCode" id="cb395"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb395-1"><a href="#cb395-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Eq</span> a <span class="ot">=&gt;</span> <span class="dt">Eq</span> (<span class="dt">Pair</span> a) <span class="kw">where</span></span>
 <span id="cb395-2"><a href="#cb395-2" aria-hidden="true" tabindex="-1"></a>  (<span class="dt">MakePair</span> x y) <span class="op">==</span> (<span class="dt">MakePair</span> a b)   <span class="ot">=</span>   x<span class="op">==</span>a <span class="op">&amp;&amp;</span> y<span class="op">==</span>b</span></code></pre></div>
-<p>Now we can compare pairs, as long as the element type is comparable. However, we can’t compare, say, pairs of functions, since functions don’t have an Eq instance.</p>
-<div class="sourceCode" id="cb396"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb396-1"><a href="#cb396-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">MakePair</span> <span class="dv">1</span> <span class="dv">1</span> <span class="op">==</span> <span class="dt">MakePair</span> <span class="dv">1</span> <span class="dv">1</span></span>
+<p>Now we can compare pairs, as long as the element type is comparable.
+However, we can’t compare, say, pairs of functions, since functions
+don’t have an Eq instance.</p>
+<div class="sourceCode" id="cb396"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb396-1"><a href="#cb396-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">MakePair</span> <span class="dv">1</span> <span class="dv">1</span> <span class="op">==</span> <span class="dt">MakePair</span> <span class="dv">1</span> <span class="dv">1</span></span>
 <span id="cb396-2"><a href="#cb396-2" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span>
 <span id="cb396-3"><a href="#cb396-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="dt">MakePair</span> <span class="fu">reverse</span> <span class="fu">reverse</span> <span class="op">==</span> <span class="dt">MakePair</span> <span class="fu">reverse</span> <span class="fu">reverse</span></span>
 <span id="cb396-4"><a href="#cb396-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -3527,49 +5514,78 @@ Because <code>Nothing</code> is a constructor.
 <span id="cb396-9"><a href="#cb396-9" aria-hidden="true" tabindex="-1"></a>        <span class="dt">MakePair</span> <span class="fu">reverse</span> <span class="fu">reverse</span> <span class="op">==</span> <span class="dt">MakePair</span> <span class="fu">reverse</span> <span class="fu">reverse</span></span>
 <span id="cb396-10"><a href="#cb396-10" aria-hidden="true" tabindex="-1"></a>      <span class="dt">In</span> an equation for ‘it’<span class="op">:</span></span>
 <span id="cb396-11"><a href="#cb396-11" aria-hidden="true" tabindex="-1"></a>          it <span class="ot">=</span> <span class="dt">MakePair</span> <span class="fu">reverse</span> <span class="fu">reverse</span> <span class="op">==</span> <span class="dt">MakePair</span> <span class="fu">reverse</span> <span class="fu">reverse</span></span></code></pre></div>
-<p>Let’s continue with another example. Here’s a simple type class and an instance</p>
-<div class="sourceCode" id="cb397"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb397-1"><a href="#cb397-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Check</span> a <span class="kw">where</span></span>
+<p>Let’s continue with another example. Here’s a simple type class and
+an instance</p>
+<div class="sourceCode" id="cb397"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb397-1"><a href="#cb397-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Check</span> a <span class="kw">where</span></span>
 <span id="cb397-2"><a href="#cb397-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  check ::</span> a <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb397-3"><a href="#cb397-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb397-4"><a href="#cb397-4" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Check</span> <span class="dt">Int</span> <span class="kw">where</span></span>
 <span id="cb397-5"><a href="#cb397-5" aria-hidden="true" tabindex="-1"></a>  check x <span class="ot">=</span> x <span class="op">&gt;</span> <span class="dv">0</span></span></code></pre></div>
-<p>Now we can write a function that checks a list. We use the standard library function <code>and :: [Bool] -&gt; Bool</code> that checks if a list is all <code>True</code>s.</p>
-<div class="sourceCode" id="cb398"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb398-1"><a href="#cb398-1" aria-hidden="true" tabindex="-1"></a><span class="ot">checkAll ::</span> <span class="dt">Check</span> a <span class="ot">=&gt;</span> [a] <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
+<p>Now we can write a function that checks a list. We use the standard
+library function <code>and :: [Bool] -&gt; Bool</code> that checks if a
+list is all <code>True</code>s.</p>
+<div class="sourceCode" id="cb398"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb398-1"><a href="#cb398-1" aria-hidden="true" tabindex="-1"></a><span class="ot">checkAll ::</span> <span class="dt">Check</span> a <span class="ot">=&gt;</span> [a] <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb398-2"><a href="#cb398-2" aria-hidden="true" tabindex="-1"></a>checkAll xs <span class="ot">=</span> <span class="fu">and</span> (<span class="fu">map</span> check xs)</span></code></pre></div>
-<p>In order to turn this into a <code>Check [a]</code> instance, we need to add a constraint to the instance declaration. Our <code>Check [a]</code> is based on the <code>Check a</code> instance.</p>
-<div class="sourceCode" id="cb399"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb399-1"><a href="#cb399-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Check</span> a <span class="ot">=&gt;</span> <span class="dt">Check</span> [a] <span class="kw">where</span></span>
+<p>In order to turn this into a <code>Check [a]</code> instance, we need
+to add a constraint to the instance declaration. Our
+<code>Check [a]</code> is based on the <code>Check a</code>
+instance.</p>
+<div class="sourceCode" id="cb399"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb399-1"><a href="#cb399-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Check</span> a <span class="ot">=&gt;</span> <span class="dt">Check</span> [a] <span class="kw">where</span></span>
 <span id="cb399-2"><a href="#cb399-2" aria-hidden="true" tabindex="-1"></a>  check xs <span class="ot">=</span> <span class="fu">and</span> (<span class="fu">map</span> check xs)</span></code></pre></div>
-<p>This means that our <code>Check [a]</code> instance is only valid when there is a corresponding <code>Check a</code> instance. For example, if we try to invoke a <code>Check [Bool]</code> instance, we get an error about the missing <code>Check Bool</code> instance:</p>
-<div class="sourceCode" id="cb400"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb400-1"><a href="#cb400-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> check [<span class="dt">True</span>,<span class="dt">False</span>]</span>
+<p>This means that our <code>Check [a]</code> instance is only valid
+when there is a corresponding <code>Check a</code> instance. For
+example, if we try to invoke a <code>Check [Bool]</code> instance, we
+get an error about the missing <code>Check Bool</code> instance:</p>
+<div class="sourceCode" id="cb400"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb400-1"><a href="#cb400-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> check [<span class="dt">True</span>,<span class="dt">False</span>]</span>
 <span id="cb400-2"><a href="#cb400-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb400-3"><a href="#cb400-3" aria-hidden="true" tabindex="-1"></a><span class="op">&lt;</span>interactive<span class="op">&gt;:</span><span class="dv">1</span><span class="op">:</span><span class="dv">1</span><span class="op">:</span> <span class="fu">error</span><span class="op">:</span></span>
 <span id="cb400-4"><a href="#cb400-4" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">No</span> <span class="kw">instance</span> for (<span class="dt">Check</span> <span class="dt">Bool</span>) arising from a use <span class="kw">of</span> ‘check’</span>
 <span id="cb400-5"><a href="#cb400-5" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">In</span> the expression<span class="op">:</span> check [<span class="dt">True</span>, <span class="dt">False</span>]</span>
 <span id="cb400-6"><a href="#cb400-6" aria-hidden="true" tabindex="-1"></a>      <span class="dt">In</span> an equation for ‘it’<span class="op">:</span> it <span class="ot">=</span> check [<span class="dt">True</span>, <span class="dt">False</span>]</span></code></pre></div>
-<p>Also, if we try to define <code>Check [a]</code> instance without the constraint, we get an error (with a pretty good suggestion!)</p>
-<div class="sourceCode" id="cb401"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb401-1"><a href="#cb401-1" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">No</span> <span class="kw">instance</span> for (<span class="dt">Check</span> a) arising from a use <span class="kw">of</span> ‘check’</span>
+<p>Also, if we try to define <code>Check [a]</code> instance without the
+constraint, we get an error (with a pretty good suggestion!)</p>
+<div class="sourceCode" id="cb401"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb401-1"><a href="#cb401-1" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">No</span> <span class="kw">instance</span> for (<span class="dt">Check</span> a) arising from a use <span class="kw">of</span> ‘check’</span>
 <span id="cb401-2"><a href="#cb401-2" aria-hidden="true" tabindex="-1"></a>      <span class="dt">Possible</span> fix<span class="op">:</span></span>
 <span id="cb401-3"><a href="#cb401-3" aria-hidden="true" tabindex="-1"></a>        add (<span class="dt">Check</span> a) to the context <span class="kw">of</span> the <span class="kw">instance</span> declaration</span></code></pre></div>
-<p>Finally, sometimes multiple constraints are needed. Consider for example the <code>Eq</code> instance for <code>Either</code>:</p>
-<div class="sourceCode" id="cb402"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb402-1"><a href="#cb402-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> (<span class="dt">Eq</span> a, <span class="dt">Eq</span> b) <span class="ot">=&gt;</span> <span class="dt">Eq</span> (<span class="dt">Either</span> a b) <span class="kw">where</span></span>
+<p>Finally, sometimes multiple constraints are needed. Consider for
+example the <code>Eq</code> instance for <code>Either</code>:</p>
+<div class="sourceCode" id="cb402"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb402-1"><a href="#cb402-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> (<span class="dt">Eq</span> a, <span class="dt">Eq</span> b) <span class="ot">=&gt;</span> <span class="dt">Eq</span> (<span class="dt">Either</span> a b) <span class="kw">where</span></span>
 <span id="cb402-2"><a href="#cb402-2" aria-hidden="true" tabindex="-1"></a>  <span class="dt">Left</span> x  <span class="op">==</span> <span class="dt">Left</span> y   <span class="ot">=</span>  x<span class="op">==</span>y</span>
 <span id="cb402-3"><a href="#cb402-3" aria-hidden="true" tabindex="-1"></a>  <span class="dt">Right</span> x <span class="op">==</span> <span class="dt">Right</span> y  <span class="ot">=</span>  x<span class="op">==</span>y</span>
 <span id="cb402-4"><a href="#cb402-4" aria-hidden="true" tabindex="-1"></a>  _       <span class="op">==</span> _        <span class="ot">=</span>  <span class="dt">False</span></span></code></pre></div>
-<h3 data-number="6.4.2" id="class-hierarchy"><span class="header-section-number">6.4.2</span> Class Hierarchy</h3>
-<p>Respectively, a class can depend on another class. This is useful for instance when you want to use functions from another class in your default implementations:</p>
-<div class="sourceCode" id="cb403"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb403-1"><a href="#cb403-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Size</span> a <span class="kw">where</span></span>
+<h3 data-number="6.4.2" id="class-hierarchy"><span
+class="header-section-number">6.4.2</span> Class Hierarchy</h3>
+<p>Respectively, a class can depend on another class. This is useful for
+instance when you want to use functions from another class in your
+default implementations:</p>
+<div class="sourceCode" id="cb403"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb403-1"><a href="#cb403-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Size</span> a <span class="kw">where</span></span>
 <span id="cb403-2"><a href="#cb403-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  size ::</span> a <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb403-3"><a href="#cb403-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb403-4"><a href="#cb403-4" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Size</span> a <span class="ot">=&gt;</span> <span class="dt">SizeBoth</span> a <span class="kw">where</span></span>
 <span id="cb403-5"><a href="#cb403-5" aria-hidden="true" tabindex="-1"></a><span class="ot">  sizeBoth ::</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb403-6"><a href="#cb403-6" aria-hidden="true" tabindex="-1"></a>  sizeBoth x y <span class="ot">=</span> size x <span class="op">+</span> size y</span></code></pre></div>
-<p>In cases like this we say <code>SizeBoth</code> is a <em>subclass</em> of <code>Size</code>. Note again the confusion with object oriented programming. Examples of subclasses in the standard library include:</p>
-<div class="sourceCode" id="cb404"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb404-1"><a href="#cb404-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Eq</span> a <span class="ot">=&gt;</span> <span class="dt">Ord</span> a <span class="kw">where</span></span>
+<p>In cases like this we say <code>SizeBoth</code> is a
+<em>subclass</em> of <code>Size</code>. Note again the confusion with
+object oriented programming. Examples of subclasses in the standard
+library include:</p>
+<div class="sourceCode" id="cb404"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb404-1"><a href="#cb404-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Eq</span> a <span class="ot">=&gt;</span> <span class="dt">Ord</span> a <span class="kw">where</span></span>
 <span id="cb404-2"><a href="#cb404-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">...</span></span>
 <span id="cb404-3"><a href="#cb404-3" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Num</span> a <span class="ot">=&gt;</span> <span class="dt">Fractional</span> a <span class="kw">where</span></span>
 <span id="cb404-4"><a href="#cb404-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">...</span></span></code></pre></div>
-<p>Another way to look at subclasses is that if you have a <code>class Main a =&gt; Sub a</code>, you must provide an <code>instance Main MyType</code> in order to be able to declare <code>instance Sub MyType</code>.</p>
-<h2 data-number="6.5" id="quiz-5"><span class="header-section-number">6.5</span> Quiz</h2>
+<p>Another way to look at subclasses is that if you have a
+<code>class Main a =&gt; Sub a</code>, you must provide an
+<code>instance Main MyType</code> in order to be able to declare
+<code>instance Sub MyType</code>.</p>
+<h2 data-number="6.5" id="quiz-5"><span
+class="header-section-number">6.5</span> Quiz</h2>
 <p>What are the functions in the <code>Eq</code> class?</p>
 <ol class="quiz">
 <li class="correct">
@@ -3582,7 +5598,8 @@ Because <code>Nothing</code> is a constructor.
 <code>(==)</code>, <code>(&lt;)</code>, <code>(&gt;)</code>
 </li>
 </ol>
-<p>For which of the following classes can we get automatic instances with <code>deriving</code>?</p>
+<p>For which of the following classes can we get automatic instances
+with <code>deriving</code>?</p>
 <ol class="quiz">
 <li>
 <code>Num</code>
@@ -3594,8 +5611,10 @@ Because <code>Nothing</code> is a constructor.
 <code>Size</code>
 </li>
 </ol>
-<p>Given the following definition of the class <code>BitOperations</code></p>
-<div class="sourceCode" id="cb405"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb405-1"><a href="#cb405-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">BitOperations</span> a <span class="kw">where</span></span>
+<p>Given the following definition of the class
+<code>BitOperations</code></p>
+<div class="sourceCode" id="cb405"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb405-1"><a href="#cb405-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">BitOperations</span> a <span class="kw">where</span></span>
 <span id="cb405-2"><a href="#cb405-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  bitNot ::</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb405-3"><a href="#cb405-3" aria-hidden="true" tabindex="-1"></a>  bitNot x <span class="ot">=</span> bitNand x x</span>
 <span id="cb405-4"><a href="#cb405-4" aria-hidden="true" tabindex="-1"></a><span class="ot">  bitAnd ::</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
@@ -3604,7 +5623,8 @@ Because <code>Nothing</code> is a constructor.
 <span id="cb405-7"><a href="#cb405-7" aria-hidden="true" tabindex="-1"></a>  bitOr x y <span class="ot">=</span> bitNot (bitAnd (bitNot x) (bitNot y))</span>
 <span id="cb405-8"><a href="#cb405-8" aria-hidden="true" tabindex="-1"></a><span class="ot">  bitNand ::</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb405-9"><a href="#cb405-9" aria-hidden="true" tabindex="-1"></a>  bitNand x y <span class="ot">=</span> bitNot (bitAnd x y)</span></code></pre></div>
-<p>which set of operations is <em>not</em> a minimal complete definition of <code>BitOperations</code>?</p>
+<p>which set of operations is <em>not</em> a minimal complete definition
+of <code>BitOperations</code>?</p>
 <ol class="quiz">
 <li>
 <code>bitNand, bitAnd</code>
@@ -3619,51 +5639,90 @@ Because <code>Nothing</code> is a constructor.
 <code>bitNot, bitOr</code>
 </li>
 </ol>
-<p>The declaration <code>instance Num a =&gt; Eq (Pair a)</code> tells me that</p>
+<p>The declaration <code>instance Num a =&gt; Eq (Pair a)</code> tells
+me that</p>
 <ol class="quiz">
 <li>
 All instances of <code>Num</code> are instances of <code>Eq</code>
 </li>
 <li class="correct">
-<code>Pair a</code> is an instance of <code>Eq</code> if <code>a</code> is an instance of <code>Num</code>
+<code>Pair a</code> is an instance of <code>Eq</code> if <code>a</code>
+is an instance of <code>Num</code>
 </li>
 <li>
-The instance <code>Eq (Pair a)</code> inherits the instance <code>Num a</code>
+The instance <code>Eq (Pair a)</code> inherits the instance
+<code>Num a</code>
 </li>
 </ol>
-<p>The declaration <code>class Num a =&gt; Fractional a</code> tells me that</p>
+<p>The declaration <code>class Num a =&gt; Fractional a</code> tells me
+that</p>
 <ol class="quiz">
 <li class="correct">
-All instances of <code>Fractional</code> must be instances of <code>Num</code>
+All instances of <code>Fractional</code> must be instances of
+<code>Num</code>
 </li>
 <li>
-All instances of <code>Num</code> must be instances of <code>Fractional</code>
+All instances of <code>Num</code> must be instances of
+<code>Fractional</code>
 </li>
 <li>
-If I define an instance for <code>Fractional</code>, I also get an instance for <code>Num</code>
+If I define an instance for <code>Fractional</code>, I also get an
+instance for <code>Num</code>
 </li>
 <li>
-If I define an instance for <code>Num</code>, I also get an instance for <code>Fractional</code>
+If I define an instance for <code>Num</code>, I also get an instance for
+<code>Fractional</code>
 </li>
 </ol>
-<h2 data-number="6.6" id="exercises-5"><span class="header-section-number">6.6</span> Exercises</h2>
+<h2 data-number="6.6" id="exercises-5"><span
+class="header-section-number">6.6</span> Exercises</h2>
 <ul>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set6.hs">Set6</a>: defining classes and instances</li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set6.hs">Set6</a>:
+defining classes and instances</li>
 </ul>
-<h1 data-number="7" id="lecture-7-new-constellations"><span class="header-section-number">7</span> Lecture 7: New Constellations</h1>
-<p>This lecture offers an introduction to <em>design patterns</em> for <em>typed functional programming</em>. These patterns are both useful when writing Haskell programs, and offer a nice arena for practicing skills from the previous lectures.</p>
-<h2 data-number="7.1" id="modeling-with-boxes"><span class="header-section-number">7.1</span> Modeling with Boxes</h2>
-<p>Sometimes you don’t need a new type, but instead can just reuse a standard type. For example, repesenting car register plate numbers with <code>String</code>. However, if your code is full of <code>String</code>s, it can be easy to accidentally mix up e.g. a car’s model and registration in a function like <code>registerCar :: String -&gt; String -&gt; CarRegistry -&gt; CarRegistry</code>.</p>
-<p>For situations like this it’s common to create a new type that just contains a <code>String</code> (a “boxed” string):</p>
-<div class="sourceCode" id="cb406"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb406-1"><a href="#cb406-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Plate</span> <span class="ot">=</span> <span class="dt">Plate</span> <span class="dt">String</span></span>
+<h1 data-number="7" id="lecture-7-new-constellations"><span
+class="header-section-number">7</span> Lecture 7: New
+Constellations</h1>
+<p>This lecture offers an introduction to <em>design patterns</em> for
+<em>typed functional programming</em>. These patterns are both useful
+when writing Haskell programs, and offer a nice arena for practicing
+skills from the previous lectures.</p>
+<h2 data-number="7.1" id="modeling-with-boxes"><span
+class="header-section-number">7.1</span> Modeling with Boxes</h2>
+<p>Sometimes you don’t need a new type, but instead can just reuse a
+standard type. For example, repesenting car register plate numbers with
+<code>String</code>. However, if your code is full of
+<code>String</code>s, it can be easy to accidentally mix up e.g. a car’s
+model and registration in a function like
+<code>registerCar :: String -&gt; String -&gt; CarRegistry -&gt; CarRegistry</code>.</p>
+<p>For situations like this it’s common to create a new type that just
+contains a <code>String</code> (a “boxed” string):</p>
+<div class="sourceCode" id="cb406"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb406-1"><a href="#cb406-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Plate</span> <span class="ot">=</span> <span class="dt">Plate</span> <span class="dt">String</span></span>
 <span id="cb406-2"><a href="#cb406-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">deriving</span> (<span class="dt">Show</span>, <span class="dt">Eq</span>)</span></code></pre></div>
-<p>We can now give <code>registerCar</code> a slightly nicer type, <code>String -&gt; Plate -&gt; CarRegistry -&gt; CarRegistry</code>. Additionally, we can restrict the operations that are possible on <code>Plate</code>s to a subset of those that are possible on strings. For example, there is no need to combine the register plate numbers of two cars. Thus we don’t need to offer a function <code>concatPlates :: Plate -&gt; Plate -&gt; Plate</code>. We can also define a <em>smart constructor</em> for <code>Plate</code> that checks that the register number is in the correct format:</p>
-<div class="sourceCode" id="cb407"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb407-1"><a href="#cb407-1" aria-hidden="true" tabindex="-1"></a><span class="ot">parsePlate ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">Plate</span></span>
+<p>We can now give <code>registerCar</code> a slightly nicer type,
+<code>String -&gt; Plate -&gt; CarRegistry -&gt; CarRegistry</code>.
+Additionally, we can restrict the operations that are possible on
+<code>Plate</code>s to a subset of those that are possible on strings.
+For example, there is no need to combine the register plate numbers of
+two cars. Thus we don’t need to offer a function
+<code>concatPlates :: Plate -&gt; Plate -&gt; Plate</code>. We can also
+define a <em>smart constructor</em> for <code>Plate</code> that checks
+that the register number is in the correct format:</p>
+<div class="sourceCode" id="cb407"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb407-1"><a href="#cb407-1" aria-hidden="true" tabindex="-1"></a><span class="ot">parsePlate ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">Plate</span></span>
 <span id="cb407-2"><a href="#cb407-2" aria-hidden="true" tabindex="-1"></a>parsePlate string</span>
 <span id="cb407-3"><a href="#cb407-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> correctPlateNumber string  <span class="ot">=</span>  <span class="dt">Just</span> (<span class="dt">Plate</span> string)</span>
 <span id="cb407-4"><a href="#cb407-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> <span class="fu">otherwise</span>                  <span class="ot">=</span>  <span class="dt">Nothing</span></span></code></pre></div>
-<p>Here’s another example: representing money. If we just store money as <code>Int</code>s, the compiler won’t protect us from mistakes like multiplying money with money. If instead we implement our own <code>Money</code> type that wraps <code>Int</code>, we get type safety. Additionally, we can encapsulate the fact that money is represented as an integer amount of cents.</p>
-<div class="sourceCode" id="cb408"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb408-1"><a href="#cb408-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Money</span> <span class="ot">=</span> <span class="dt">Money</span> <span class="dt">Int</span></span>
+<p>Here’s another example: representing money. If we just store money as
+<code>Int</code>s, the compiler won’t protect us from mistakes like
+multiplying money with money. If instead we implement our own
+<code>Money</code> type that wraps <code>Int</code>, we get type safety.
+Additionally, we can encapsulate the fact that money is represented as
+an integer amount of cents.</p>
+<div class="sourceCode" id="cb408"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb408-1"><a href="#cb408-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Money</span> <span class="ot">=</span> <span class="dt">Money</span> <span class="dt">Int</span></span>
 <span id="cb408-2"><a href="#cb408-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">deriving</span> <span class="dt">Show</span></span>
 <span id="cb408-3"><a href="#cb408-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb408-4"><a href="#cb408-4" aria-hidden="true" tabindex="-1"></a><span class="ot">renderMoney ::</span> <span class="dt">Money</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
@@ -3677,7 +5736,8 @@ If I define an instance for <code>Num</code>, I also get an instance for <code>F
 <span id="cb408-12"><a href="#cb408-12" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb408-13"><a href="#cb408-13" aria-hidden="true" tabindex="-1"></a><span class="ot">addVat ::</span> <span class="dt">Money</span> <span class="ot">-&gt;</span> <span class="dt">Money</span></span>
 <span id="cb408-14"><a href="#cb408-14" aria-hidden="true" tabindex="-1"></a>addVat m <span class="ot">=</span> m <span class="op">+!</span> scale m <span class="fl">0.24</span></span></code></pre></div>
-<div class="sourceCode" id="cb409"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb409-1"><a href="#cb409-1" aria-hidden="true" tabindex="-1"></a>renderMoney (<span class="dt">Money</span> <span class="dv">100</span> <span class="op">+!</span> <span class="dt">Money</span> <span class="dv">150</span>)</span>
+<div class="sourceCode" id="cb409"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb409-1"><a href="#cb409-1" aria-hidden="true" tabindex="-1"></a>renderMoney (<span class="dt">Money</span> <span class="dv">100</span> <span class="op">+!</span> <span class="dt">Money</span> <span class="dv">150</span>)</span>
 <span id="cb409-2"><a href="#cb409-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="st">&quot;2.5&quot;</span></span>
 <span id="cb409-3"><a href="#cb409-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb409-4"><a href="#cb409-4" aria-hidden="true" tabindex="-1"></a>scale (<span class="dt">Money</span> <span class="dv">299</span>) <span class="fl">0.24</span></span>
@@ -3685,11 +5745,22 @@ If I define an instance for <code>Num</code>, I also get an instance for <code>F
 <span id="cb409-6"><a href="#cb409-6" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb409-7"><a href="#cb409-7" aria-hidden="true" tabindex="-1"></a>addVat (<span class="dt">Money</span> <span class="dv">299</span>)</span>
 <span id="cb409-8"><a href="#cb409-8" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Money</span> <span class="dv">371</span></span></code></pre></div>
-<p><strong>Note!</strong> If you’re familiar with Object-Oriented Programming, this is a bit like encapsulation.</p>
-<h2 data-number="7.2" id="modeling-with-cases"><span class="header-section-number">7.2</span> Modeling with Cases</h2>
-<p>Haskell’s algebraic datatypes are really powerful at modeling things based on <em>cases</em>. It’s often useful to think of types as defining the set of possible cases, and functions <em>handling</em> those cases (often via pattern matching). Let’s look at two examples.</p>
-<p>Since it’s so easy to define custom types in Haskell, it’s quite convenient to use more descriptive types instead of booleans or strings. Consider a list of persons. In some other language if you wanted to sort the persons into ascending order by name you might use a call like <code>sortPersons(persons, "name", true)</code>. In Haskell you can do this instead:</p>
-<div class="sourceCode" id="cb410"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb410-1"><a href="#cb410-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Person</span> <span class="ot">=</span> <span class="dt">Person</span> {<span class="ot">name ::</span> <span class="dt">String</span>,<span class="ot"> age ::</span> <span class="dt">Int</span>}</span>
+<p><strong>Note!</strong> If you’re familiar with Object-Oriented
+Programming, this is a bit like encapsulation.</p>
+<h2 data-number="7.2" id="modeling-with-cases"><span
+class="header-section-number">7.2</span> Modeling with Cases</h2>
+<p>Haskell’s algebraic datatypes are really powerful at modeling things
+based on <em>cases</em>. It’s often useful to think of types as defining
+the set of possible cases, and functions <em>handling</em> those cases
+(often via pattern matching). Let’s look at two examples.</p>
+<p>Since it’s so easy to define custom types in Haskell, it’s quite
+convenient to use more descriptive types instead of booleans or strings.
+Consider a list of persons. In some other language if you wanted to sort
+the persons into ascending order by name you might use a call like
+<code>sortPersons(persons, "name", true)</code>. In Haskell you can do
+this instead:</p>
+<div class="sourceCode" id="cb410"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb410-1"><a href="#cb410-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Person</span> <span class="ot">=</span> <span class="dt">Person</span> {<span class="ot">name ::</span> <span class="dt">String</span>,<span class="ot"> age ::</span> <span class="dt">Int</span>}</span>
 <span id="cb410-2"><a href="#cb410-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">deriving</span> <span class="dt">Show</span></span>
 <span id="cb410-3"><a href="#cb410-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb410-4"><a href="#cb410-4" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">SortOrder</span> <span class="ot">=</span> <span class="dt">Ascending</span> <span class="op">|</span> <span class="dt">Descending</span></span>
@@ -3704,78 +5775,161 @@ If I define an instance for <code>Num</code>, I also get an instance for <code>F
 <span id="cb410-13"><a href="#cb410-13" aria-hidden="true" tabindex="-1"></a>sortPersons field <span class="dt">Descending</span> ps <span class="ot">=</span> <span class="fu">reverse</span> (sortByField field ps)</span>
 <span id="cb410-14"><a href="#cb410-14" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb410-15"><a href="#cb410-15" aria-hidden="true" tabindex="-1"></a>persons <span class="ot">=</span> [<span class="dt">Person</span> <span class="st">&quot;Fridolf&quot;</span> <span class="dv">73</span>, <span class="dt">Person</span> <span class="st">&quot;Greta&quot;</span> <span class="dv">60</span>, <span class="dt">Person</span> <span class="st">&quot;Hans&quot;</span> <span class="dv">65</span>]</span></code></pre></div>
-<div class="sourceCode" id="cb411"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb411-1"><a href="#cb411-1" aria-hidden="true" tabindex="-1"></a>sortPersons <span class="dt">Name</span> <span class="dt">Ascending</span> persons</span>
+<div class="sourceCode" id="cb411"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb411-1"><a href="#cb411-1" aria-hidden="true" tabindex="-1"></a>sortPersons <span class="dt">Name</span> <span class="dt">Ascending</span> persons</span>
 <span id="cb411-2"><a href="#cb411-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="dt">Person</span> {name <span class="ot">=</span> <span class="st">&quot;Fridolf&quot;</span>, age <span class="ot">=</span> <span class="dv">73</span>},<span class="dt">Person</span> {name <span class="ot">=</span> <span class="st">&quot;Greta&quot;</span>, age <span class="ot">=</span> <span class="dv">60</span>},<span class="dt">Person</span> {name <span class="ot">=</span> <span class="st">&quot;Hans&quot;</span>, age <span class="ot">=</span> <span class="dv">65</span>}]</span>
 <span id="cb411-3"><a href="#cb411-3" aria-hidden="true" tabindex="-1"></a>sortPersons <span class="dt">Age</span> <span class="dt">Descending</span> persons</span>
 <span id="cb411-4"><a href="#cb411-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="dt">Person</span> {name <span class="ot">=</span> <span class="st">&quot;Fridolf&quot;</span>, age <span class="ot">=</span> <span class="dv">73</span>},<span class="dt">Person</span> {name <span class="ot">=</span> <span class="st">&quot;Hans&quot;</span>, age <span class="ot">=</span> <span class="dv">65</span>},<span class="dt">Person</span> {name <span class="ot">=</span> <span class="st">&quot;Greta&quot;</span>, age <span class="ot">=</span> <span class="dv">60</span>}]</span></code></pre></div>
-<p>Note how you can’t accidentally typo the field name (unlike with strings), and how you don’t need to remember whether <code>true</code> refers to ascending or descending order.</p>
-<p>Let’s move on to the next example. Many Haskell functions don’t work with empty lists (consider <code>head []</code>). If you’re writing code that needs to track whether lists are possibly empty or guaranteed to not be empty, you can use the <code>NonEmpty</code> type from the <a href="https://hackage.haskell.org/package/base-4.16.4.0/docs/Data-List-NonEmpty.html">Data.List.NonEmpty</a> module.</p>
+<p>Note how you can’t accidentally typo the field name (unlike with
+strings), and how you don’t need to remember whether <code>true</code>
+refers to ascending or descending order.</p>
+<p>Let’s move on to the next example. Many Haskell functions don’t work
+with empty lists (consider <code>head []</code>). If you’re writing code
+that needs to track whether lists are possibly empty or guaranteed to
+not be empty, you can use the <code>NonEmpty</code> type from the <a
+href="https://hackage.haskell.org/package/base-4.16.4.0/docs/Data-List-NonEmpty.html">Data.List.NonEmpty</a>
+module.</p>
 <p>Consider the definition of <code>NonEmpty</code>:</p>
-<div class="sourceCode" id="cb412"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb412-1"><a href="#cb412-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">NonEmpty</span> a <span class="ot">=</span> a <span class="op">:|</span> [a]</span></code></pre></div>
-<p>Here the type represents a <em>lack of</em> cases. The type <code>NonEmpty a</code> will always consist of a value of type <code>a</code>, and some further <code>a</code>s, collected in a list. Here are some example values of <code>NonEmpty Int</code>:</p>
-<div class="sourceCode" id="cb413"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb413-1"><a href="#cb413-1" aria-hidden="true" tabindex="-1"></a><span class="dv">1</span> <span class="op">:|</span> [<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>]</span>
+<div class="sourceCode" id="cb412"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb412-1"><a href="#cb412-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">NonEmpty</span> a <span class="ot">=</span> a <span class="op">:|</span> [a]</span></code></pre></div>
+<p>Here the type represents a <em>lack of</em> cases. The type
+<code>NonEmpty a</code> will always consist of a value of type
+<code>a</code>, and some further <code>a</code>s, collected in a list.
+Here are some example values of <code>NonEmpty Int</code>:</p>
+<div class="sourceCode" id="cb413"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb413-1"><a href="#cb413-1" aria-hidden="true" tabindex="-1"></a><span class="dv">1</span> <span class="op">:|</span> [<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>]</span>
 <span id="cb413-2"><a href="#cb413-2" aria-hidden="true" tabindex="-1"></a><span class="dv">1</span> <span class="op">:|</span> []</span></code></pre></div>
-<p>By the way, this is also an example of an <em>infix constructor</em>. We’ve already met another infix constructor earlier, the list constructor <code>(:)</code>. Any operator that begins with a colon (the <code>:</code> character) can be used as an infix constructor. We can pattern match on <code>(:|)</code> just like on <code>(:)</code>, as you’ll see in the examples below.</p>
-<p>Here are the functions that convert between normal lists and nonempty lists. Note how we can’t have a function <code>[a] -&gt; NonEmpty a</code>, but must instead use <code>Maybe</code> to represent the possibility that the list was, indeed, empty. Note also how <code>toList</code> has only one equation, we can’t have a <code>toList []</code> situation due to the type <code>NonEmpty</code>.</p>
-<div class="sourceCode" id="cb414"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb414-1"><a href="#cb414-1" aria-hidden="true" tabindex="-1"></a><span class="ot">nonEmpty ::</span> [a] <span class="ot">-&gt;</span> <span class="dt">Maybe</span> (<span class="dt">NonEmpty</span> a)</span>
+<p>By the way, this is also an example of an <em>infix constructor</em>.
+We’ve already met another infix constructor earlier, the list
+constructor <code>(:)</code>. Any operator that begins with a colon (the
+<code>:</code> character) can be used as an infix constructor. We can
+pattern match on <code>(:|)</code> just like on <code>(:)</code>, as
+you’ll see in the examples below.</p>
+<p>Here are the functions that convert between normal lists and nonempty
+lists. Note how we can’t have a function
+<code>[a] -&gt; NonEmpty a</code>, but must instead use
+<code>Maybe</code> to represent the possibility that the list was,
+indeed, empty. Note also how <code>toList</code> has only one equation,
+we can’t have a <code>toList []</code> situation due to the type
+<code>NonEmpty</code>.</p>
+<div class="sourceCode" id="cb414"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb414-1"><a href="#cb414-1" aria-hidden="true" tabindex="-1"></a><span class="ot">nonEmpty ::</span> [a] <span class="ot">-&gt;</span> <span class="dt">Maybe</span> (<span class="dt">NonEmpty</span> a)</span>
 <span id="cb414-2"><a href="#cb414-2" aria-hidden="true" tabindex="-1"></a>nonEmpty [] <span class="ot">=</span> <span class="dt">Nothing</span></span>
 <span id="cb414-3"><a href="#cb414-3" aria-hidden="true" tabindex="-1"></a>nonEmpty (x<span class="op">:</span>xs) <span class="ot">=</span> <span class="dt">Just</span> (x <span class="op">:|</span> xs)</span>
 <span id="cb414-4"><a href="#cb414-4" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb414-5"><a href="#cb414-5" aria-hidden="true" tabindex="-1"></a><span class="ot">toList ::</span> <span class="dt">NonEmpty</span> a <span class="ot">-&gt;</span> [a]</span>
 <span id="cb414-6"><a href="#cb414-6" aria-hidden="true" tabindex="-1"></a>toList (x <span class="op">:|</span> xs) <span class="ot">=</span> x <span class="op">:</span> xs</span></code></pre></div>
-<div class="sourceCode" id="cb415"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb415-1"><a href="#cb415-1" aria-hidden="true" tabindex="-1"></a>nonEmpty [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]     <span class="op">==&gt;</span>  <span class="dt">Just</span> (<span class="dv">1</span> <span class="op">:|</span> [<span class="dv">2</span>,<span class="dv">3</span>])</span>
+<div class="sourceCode" id="cb415"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb415-1"><a href="#cb415-1" aria-hidden="true" tabindex="-1"></a>nonEmpty [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]     <span class="op">==&gt;</span>  <span class="dt">Just</span> (<span class="dv">1</span> <span class="op">:|</span> [<span class="dv">2</span>,<span class="dv">3</span>])</span>
 <span id="cb415-2"><a href="#cb415-2" aria-hidden="true" tabindex="-1"></a>nonEmpty [<span class="dv">1</span>]         <span class="op">==&gt;</span>  <span class="dt">Just</span> (<span class="dv">1</span> <span class="op">:|</span> [])</span>
 <span id="cb415-3"><a href="#cb415-3" aria-hidden="true" tabindex="-1"></a>nonEmpty []          <span class="op">==&gt;</span>  <span class="dt">Nothing</span></span>
 <span id="cb415-4"><a href="#cb415-4" aria-hidden="true" tabindex="-1"></a>toList (<span class="dv">1</span> <span class="op">:|</span> [<span class="dv">2</span>,<span class="dv">3</span>])  <span class="op">==&gt;</span>  [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span></code></pre></div>
-<p>Here are <code>head</code> and <code>last</code> implemented for <code>NonEmpty</code>:</p>
-<div class="sourceCode" id="cb416"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb416-1"><a href="#cb416-1" aria-hidden="true" tabindex="-1"></a>neHead (x <span class="op">:|</span> _) <span class="ot">=</span> x</span>
+<p>Here are <code>head</code> and <code>last</code> implemented for
+<code>NonEmpty</code>:</p>
+<div class="sourceCode" id="cb416"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb416-1"><a href="#cb416-1" aria-hidden="true" tabindex="-1"></a>neHead (x <span class="op">:|</span> _) <span class="ot">=</span> x</span>
 <span id="cb416-2"><a href="#cb416-2" aria-hidden="true" tabindex="-1"></a>neLast (x <span class="op">:|</span> []) <span class="ot">=</span> x</span>
 <span id="cb416-3"><a href="#cb416-3" aria-hidden="true" tabindex="-1"></a>neLast (_ <span class="op">:|</span> xs) <span class="ot">=</span> <span class="fu">last</span> xs</span></code></pre></div>
-<div class="sourceCode" id="cb417"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb417-1"><a href="#cb417-1" aria-hidden="true" tabindex="-1"></a>neHead (<span class="dv">1</span><span class="op">:|</span>[<span class="dv">2</span>,<span class="dv">3</span>])  <span class="op">==&gt;</span>  <span class="dv">1</span></span>
+<div class="sourceCode" id="cb417"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb417-1"><a href="#cb417-1" aria-hidden="true" tabindex="-1"></a>neHead (<span class="dv">1</span><span class="op">:|</span>[<span class="dv">2</span>,<span class="dv">3</span>])  <span class="op">==&gt;</span>  <span class="dv">1</span></span>
 <span id="cb417-2"><a href="#cb417-2" aria-hidden="true" tabindex="-1"></a>neLast (<span class="dv">1</span><span class="op">:|</span>[<span class="dv">2</span>,<span class="dv">3</span>])  <span class="op">==&gt;</span>  <span class="dv">3</span></span></code></pre></div>
-<p>By the way, these functions are available as <code>Data.List.NonEmpty.head</code> and <code>Data.List.NonEmpty.last</code> along with many other useful functions.</p>
-<p>In summary, if you write types that represent all possible cases for your values, and then write functions that handle those cases, your code will be simple and correct.</p>
-<h2 data-number="7.3" id="monoids"><span class="header-section-number">7.3</span> Monoids</h2>
+<p>By the way, these functions are available as
+<code>Data.List.NonEmpty.head</code> and
+<code>Data.List.NonEmpty.last</code> along with many other useful
+functions.</p>
+<p>In summary, if you write types that represent all possible cases for
+your values, and then write functions that handle those cases, your code
+will be simple and correct.</p>
+<h2 data-number="7.3" id="monoids"><span
+class="header-section-number">7.3</span> Monoids</h2>
 <!-- TODO: Give an example of a semigroup that is not a monoid. -->
 <!-- TODO: move to the end of the lecture? -->
-<p>A pattern that comes up surprisingly often in functional programming is the <em>monoid</em> (not to be confused with a <em>monad</em>!). Explanations of monoids are often very mathematical, but the idea is simple: combining things.</p>
-<h3 data-number="7.3.1" id="associative-operations"><span class="header-section-number">7.3.1</span> Associative Operations</h3>
-<p>Many functions and operators we use are <em>associative</em>. This is just a fancy way of saying they don’t need parentheses. For example, all of these expressions have the value 16 because addition is associative:</p>
-<div class="sourceCode" id="cb418"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb418-1"><a href="#cb418-1" aria-hidden="true" tabindex="-1"></a>(<span class="dv">1</span> <span class="op">+</span>  <span class="dv">3</span>) <span class="op">+</span> (<span class="dv">5</span> <span class="op">+</span> <span class="dv">7</span>)</span>
+<p>A pattern that comes up surprisingly often in functional programming
+is the <em>monoid</em> (not to be confused with a <em>monad</em>!).
+Explanations of monoids are often very mathematical, but the idea is
+simple: combining things.</p>
+<h3 data-number="7.3.1" id="associative-operations"><span
+class="header-section-number">7.3.1</span> Associative Operations</h3>
+<p>Many functions and operators we use are <em>associative</em>. This is
+just a fancy way of saying they don’t need parentheses. For example, all
+of these expressions have the value 16 because addition is
+associative:</p>
+<div class="sourceCode" id="cb418"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb418-1"><a href="#cb418-1" aria-hidden="true" tabindex="-1"></a>(<span class="dv">1</span> <span class="op">+</span>  <span class="dv">3</span>) <span class="op">+</span> (<span class="dv">5</span> <span class="op">+</span> <span class="dv">7</span>)</span>
 <span id="cb418-2"><a href="#cb418-2" aria-hidden="true" tabindex="-1"></a> <span class="dv">1</span> <span class="op">+</span> (<span class="dv">3</span>  <span class="op">+</span> (<span class="dv">5</span> <span class="op">+</span> <span class="dv">7</span>))</span>
 <span id="cb418-3"><a href="#cb418-3" aria-hidden="true" tabindex="-1"></a> <span class="dv">1</span>  <span class="op">+</span> <span class="dv">3</span>  <span class="op">+</span>  <span class="dv">5</span> <span class="op">+</span> <span class="dv">7</span></span></code></pre></div>
-<p>Examples of associative operations are easy to come by in Haskell. For example the <code>++</code> operator for catenating lists is associative: it doesn’t matter whether you do <code>([1] ++ [2,3]) ++ [4]</code> or <code>[1] ++ ([2,3] ++ [4])</code> – the result is <code>[1,2,3,4]</code>.</p>
-<p>Another great example is the function composition operator. Both <code>(head . tail) . tail</code> and <code>head . (tail . tail)</code> compute the third element of a list.</p>
-<p>However not all operators are associative. The most familiar examples are subtraction and exponentiation. <code>(1-2)-3</code> is <code>-4</code> but <code>1-(2-3)</code> is <code>2</code>. Similarly, <code>(2^3)^2</code> is <code>64</code> while <code>2^(3^2)</code> is 512. One needs to be careful with parentheses when using operators that are not associative.</p>
-<p>Another operator that’s not associative is the list constructor, <code>(:)</code>. This time the reason is even more fundamental: while <code>True:(False:[])</code> is ok, <code>(True:False):[]</code> does not even type! In order for an operation to be associative, it needs to take two arguments of the same type.</p>
-<p>In addition to operators, functions can also be associative. The syntax looks a bit different, but a function <code>f</code> is associative if these are the same:</p>
-<div class="sourceCode" id="cb419"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb419-1"><a href="#cb419-1" aria-hidden="true" tabindex="-1"></a>f x (f y z)</span>
+<p>Examples of associative operations are easy to come by in Haskell.
+For example the <code>++</code> operator for catenating lists is
+associative: it doesn’t matter whether you do
+<code>([1] ++ [2,3]) ++ [4]</code> or <code>[1] ++ ([2,3] ++ [4])</code>
+– the result is <code>[1,2,3,4]</code>.</p>
+<p>Another great example is the function composition operator. Both
+<code>(head . tail) . tail</code> and <code>head . (tail . tail)</code>
+compute the third element of a list.</p>
+<p>However not all operators are associative. The most familiar examples
+are subtraction and exponentiation. <code>(1-2)-3</code> is
+<code>-4</code> but <code>1-(2-3)</code> is <code>2</code>. Similarly,
+<code>(2^3)^2</code> is <code>64</code> while <code>2^(3^2)</code> is
+512. One needs to be careful with parentheses when using operators that
+are not associative.</p>
+<p>Another operator that’s not associative is the list constructor,
+<code>(:)</code>. This time the reason is even more fundamental: while
+<code>True:(False:[])</code> is ok, <code>(True:False):[]</code> does
+not even type! In order for an operation to be associative, it needs to
+take two arguments of the same type.</p>
+<p>In addition to operators, functions can also be associative. The
+syntax looks a bit different, but a function <code>f</code> is
+associative if these are the same:</p>
+<div class="sourceCode" id="cb419"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb419-1"><a href="#cb419-1" aria-hidden="true" tabindex="-1"></a>f x (f y z)</span>
 <span id="cb419-2"><a href="#cb419-2" aria-hidden="true" tabindex="-1"></a>f (f x y) z</span></code></pre></div>
-<p>Two widely-used associative functions are the <code>min</code> and <code>max</code> functions:</p>
-<div class="sourceCode" id="cb420"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb420-1"><a href="#cb420-1" aria-hidden="true" tabindex="-1"></a><span class="fu">min</span> <span class="dv">2</span> (<span class="fu">min</span> <span class="dv">1</span> <span class="dv">3</span>) <span class="op">==&gt;</span> <span class="dv">1</span></span>
+<p>Two widely-used associative functions are the <code>min</code> and
+<code>max</code> functions:</p>
+<div class="sourceCode" id="cb420"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb420-1"><a href="#cb420-1" aria-hidden="true" tabindex="-1"></a><span class="fu">min</span> <span class="dv">2</span> (<span class="fu">min</span> <span class="dv">1</span> <span class="dv">3</span>) <span class="op">==&gt;</span> <span class="dv">1</span></span>
 <span id="cb420-2"><a href="#cb420-2" aria-hidden="true" tabindex="-1"></a><span class="fu">min</span> (<span class="fu">min</span> <span class="dv">2</span> <span class="dv">1</span>) <span class="dv">3</span> <span class="op">==&gt;</span> <span class="dv">1</span></span>
 <span id="cb420-3"><a href="#cb420-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb420-4"><a href="#cb420-4" aria-hidden="true" tabindex="-1"></a><span class="fu">max</span> <span class="dv">2</span> (<span class="fu">max</span> <span class="dv">1</span> <span class="dv">3</span>) <span class="op">==&gt;</span> <span class="dv">3</span></span>
 <span id="cb420-5"><a href="#cb420-5" aria-hidden="true" tabindex="-1"></a><span class="fu">max</span> (<span class="fu">max</span> <span class="dv">2</span> <span class="dv">1</span>) <span class="dv">3</span> <span class="op">==&gt;</span> <span class="dv">3</span></span></code></pre></div>
-<h3 data-number="7.3.2" id="semigroups"><span class="header-section-number">7.3.2</span> Semigroups</h3>
-<p>Mathematically speaking, an associative function (or operator) forms a <em>semigroup</em>. Haskell has a type class <code>Semigroup</code> (defined in the module <code>Data.Semigroup</code>) that can be used when a type has one clear associative operation.</p>
-<div class="sourceCode" id="cb421"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb421-1"><a href="#cb421-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Semigroup</span> a <span class="kw">where</span></span>
+<h3 data-number="7.3.2" id="semigroups"><span
+class="header-section-number">7.3.2</span> Semigroups</h3>
+<p>Mathematically speaking, an associative function (or operator) forms
+a <em>semigroup</em>. Haskell has a type class <code>Semigroup</code>
+(defined in the module <code>Data.Semigroup</code>) that can be used
+when a type has one clear associative operation.</p>
+<div class="sourceCode" id="cb421"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb421-1"><a href="#cb421-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Semigroup</span> a <span class="kw">where</span></span>
 <span id="cb421-2"><a href="#cb421-2" aria-hidden="true" tabindex="-1"></a>  <span class="co">-- An associative operation.</span></span>
 <span id="cb421-3"><a href="#cb421-3" aria-hidden="true" tabindex="-1"></a><span class="ot">  (&lt;&gt;) ::</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span></code></pre></div>
-<p>Lists are an instance of <code>Semigroup</code> with <code>(++)</code> as <code>(&lt;&gt;)</code>:</p>
-<div class="sourceCode" id="cb422"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb422-1"><a href="#cb422-1" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span>] <span class="op">&lt;&gt;</span> [<span class="dv">2</span>,<span class="dv">3</span>] <span class="op">&lt;&gt;</span> [<span class="dv">4</span>]  <span class="op">==&gt;</span>  [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>]</span></code></pre></div>
-<p>Types that have multiple different associative operators usually aren’t made an instance of Semigroup. An example is <code>Int</code>, which has many associative functions like <code>+</code>, <code>*</code> and <code>max</code>. Instead, the Haskell standard library uses boxing (see earlier in this lecture). Here are the definitions for <code>Sum</code> and <code>Product</code>:</p>
-<div class="sourceCode" id="cb423"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb423-1"><a href="#cb423-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Sum</span> a <span class="ot">=</span> <span class="dt">Sum</span> a</span>
+<p>Lists are an instance of <code>Semigroup</code> with
+<code>(++)</code> as <code>(&lt;&gt;)</code>:</p>
+<div class="sourceCode" id="cb422"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb422-1"><a href="#cb422-1" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span>] <span class="op">&lt;&gt;</span> [<span class="dv">2</span>,<span class="dv">3</span>] <span class="op">&lt;&gt;</span> [<span class="dv">4</span>]  <span class="op">==&gt;</span>  [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>]</span></code></pre></div>
+<p>Types that have multiple different associative operators usually
+aren’t made an instance of Semigroup. An example is <code>Int</code>,
+which has many associative functions like <code>+</code>, <code>*</code>
+and <code>max</code>. Instead, the Haskell standard library uses boxing
+(see earlier in this lecture). Here are the definitions for
+<code>Sum</code> and <code>Product</code>:</p>
+<div class="sourceCode" id="cb423"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb423-1"><a href="#cb423-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Sum</span> a <span class="ot">=</span> <span class="dt">Sum</span> a</span>
 <span id="cb423-2"><a href="#cb423-2" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Num</span> a <span class="ot">=&gt;</span> <span class="dt">Semigroup</span> (<span class="dt">Sum</span> a) <span class="kw">where</span></span>
 <span id="cb423-3"><a href="#cb423-3" aria-hidden="true" tabindex="-1"></a>  <span class="dt">Sum</span> a <span class="op">&lt;&gt;</span> <span class="dt">Sum</span> b  <span class="ot">=</span>  <span class="dt">Sum</span> (a<span class="op">+</span>b)</span>
 <span id="cb423-4"><a href="#cb423-4" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb423-5"><a href="#cb423-5" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Product</span> a <span class="ot">=</span> <span class="dt">Product</span> a</span>
 <span id="cb423-6"><a href="#cb423-6" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Num</span> a <span class="ot">=&gt;</span> <span class="dt">Semigroup</span> (<span class="dt">Product</span> a) <span class="kw">where</span></span>
 <span id="cb423-7"><a href="#cb423-7" aria-hidden="true" tabindex="-1"></a>  <span class="dt">Product</span> a <span class="op">&lt;&gt;</span> <span class="dt">Product</span> b   <span class="ot">=</span>  <span class="dt">Product</span> (a<span class="op">*</span>b)</span></code></pre></div>
-<p>By the way, this is another benefit of boxing things: being able to declare different type class instances!</p>
-<p>Note how the <code>Num a</code> constraint lets us use <code>Num</code> operations like <code>+</code> and <code>*</code> on the contained type <code>a</code>. We can have values like <code>Sum "abc" :: Sum String</code>, but they won’t have a <code>Semigroup</code> instance!</p>
-<p>Similarly, we have box types <code>Min</code> and <code>Max</code>. Let’s play around in GHCi a bit:</p>
-<div class="sourceCode" id="cb424"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb424-1"><a href="#cb424-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="dt">Data.Semigroup</span></span>
+<p>By the way, this is another benefit of boxing things: being able to
+declare different type class instances!</p>
+<p>Note how the <code>Num a</code> constraint lets us use
+<code>Num</code> operations like <code>+</code> and <code>*</code> on
+the contained type <code>a</code>. We can have values like
+<code>Sum "abc" :: Sum String</code>, but they won’t have a
+<code>Semigroup</code> instance!</p>
+<p>Similarly, we have box types <code>Min</code> and <code>Max</code>.
+Let’s play around in GHCi a bit:</p>
+<div class="sourceCode" id="cb424"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb424-1"><a href="#cb424-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="dt">Data.Semigroup</span></span>
 <span id="cb424-2"><a href="#cb424-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.Semigroup</span><span class="op">&gt;</span> <span class="dt">Product</span> (<span class="dv">2</span><span class="ot">::</span><span class="dt">Int</span>) <span class="op">&lt;&gt;</span> <span class="dt">Product</span> <span class="dv">3</span> <span class="op">&lt;&gt;</span> <span class="dt">Product</span> <span class="dv">1</span></span>
 <span id="cb424-3"><a href="#cb424-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Product</span> {getProduct <span class="ot">=</span> <span class="dv">6</span>}</span>
 <span id="cb424-4"><a href="#cb424-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.Semigroup</span><span class="op">&gt;</span> <span class="dt">Sum</span> <span class="dv">3</span> <span class="op">&lt;&gt;</span> <span class="dt">Sum</span> <span class="dv">5</span> <span class="op">&lt;&gt;</span> <span class="dt">Sum</span> <span class="dv">7</span></span>
@@ -3786,9 +5940,14 @@ If I define an instance for <code>Num</code>, I also get an instance for <code>F
 <span id="cb424-9"><a href="#cb424-9" aria-hidden="true" tabindex="-1"></a><span class="dt">Min</span> {getMin <span class="ot">=</span> <span class="dv">3</span>}</span>
 <span id="cb424-10"><a href="#cb424-10" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.Semigroup</span><span class="op">&gt;</span> <span class="dt">Max</span> <span class="dv">4</span> <span class="op">&lt;&gt;</span> <span class="dt">Max</span> <span class="dv">3</span> <span class="op">&lt;&gt;</span> <span class="dt">Max</span> <span class="dv">5</span></span>
 <span id="cb424-11"><a href="#cb424-11" aria-hidden="true" tabindex="-1"></a><span class="dt">Max</span> {getMax <span class="ot">=</span> <span class="dv">5</span>}</span></code></pre></div>
-<h3 data-number="7.3.3" id="finally-monoids"><span class="header-section-number">7.3.3</span> Finally, Monoids</h3>
-<p>If we listen to the mathematicians for a moment again, a <em>monoid</em> is a semigroup with a <em>neutral element</em>. A neutral element is a zero: an element that does nothing when combined with other elements. Here are some examples:</p>
-<div class="sourceCode" id="cb425"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb425-1"><a href="#cb425-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- 0 is the neutral element of (+)</span></span>
+<h3 data-number="7.3.3" id="finally-monoids"><span
+class="header-section-number">7.3.3</span> Finally, Monoids</h3>
+<p>If we listen to the mathematicians for a moment again, a
+<em>monoid</em> is a semigroup with a <em>neutral element</em>. A
+neutral element is a zero: an element that does nothing when combined
+with other elements. Here are some examples:</p>
+<div class="sourceCode" id="cb425"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb425-1"><a href="#cb425-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- 0 is the neutral element of (+)</span></span>
 <span id="cb425-2"><a href="#cb425-2" aria-hidden="true" tabindex="-1"></a><span class="dv">3</span> <span class="op">+</span> <span class="dv">0</span>        <span class="op">==&gt;</span>  <span class="dv">3</span></span>
 <span id="cb425-3"><a href="#cb425-3" aria-hidden="true" tabindex="-1"></a><span class="dv">0</span> <span class="op">+</span> <span class="dv">3</span>        <span class="op">==&gt;</span>  <span class="dv">3</span></span>
 <span id="cb425-4"><a href="#cb425-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -3799,12 +5958,16 @@ If I define an instance for <code>Num</code>, I also get an instance for <code>F
 <span id="cb425-9"><a href="#cb425-9" aria-hidden="true" tabindex="-1"></a><span class="co">-- [] is the neutral element of (++)</span></span>
 <span id="cb425-10"><a href="#cb425-10" aria-hidden="true" tabindex="-1"></a>[] <span class="op">++</span> [<span class="dv">1</span>,<span class="dv">2</span>]  <span class="op">==&gt;</span>  [<span class="dv">1</span>,<span class="dv">2</span>]</span>
 <span id="cb425-11"><a href="#cb425-11" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span>,<span class="dv">2</span>] <span class="op">++</span> []  <span class="op">==&gt;</span>  [<span class="dv">1</span>,<span class="dv">2</span>]</span></code></pre></div>
-<p>The Haskell type class <code>Monoid</code> (from the module <code>Data.Monoid</code>) represents monoids.</p>
-<div class="sourceCode" id="cb426"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb426-1"><a href="#cb426-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Semigroup</span> a <span class="ot">=&gt;</span> <span class="dt">Monoid</span> a <span class="kw">where</span></span>
+<p>The Haskell type class <code>Monoid</code> (from the module
+<code>Data.Monoid</code>) represents monoids.</p>
+<div class="sourceCode" id="cb426"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb426-1"><a href="#cb426-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Semigroup</span> a <span class="ot">=&gt;</span> <span class="dt">Monoid</span> a <span class="kw">where</span></span>
 <span id="cb426-2"><a href="#cb426-2" aria-hidden="true" tabindex="-1"></a>  <span class="co">-- The neutral element</span></span>
 <span id="cb426-3"><a href="#cb426-3" aria-hidden="true" tabindex="-1"></a><span class="ot">  mempty ::</span> a</span></code></pre></div>
-<p>Here are the <code>Monoid</code> instances corresponding to our three examples of neutral elements:</p>
-<div class="sourceCode" id="cb427"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb427-1"><a href="#cb427-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Num</span> a <span class="ot">=&gt;</span> <span class="dt">Monoid</span> (<span class="dt">Sum</span> a) <span class="kw">where</span></span>
+<p>Here are the <code>Monoid</code> instances corresponding to our three
+examples of neutral elements:</p>
+<div class="sourceCode" id="cb427"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb427-1"><a href="#cb427-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Num</span> a <span class="ot">=&gt;</span> <span class="dt">Monoid</span> (<span class="dt">Sum</span> a) <span class="kw">where</span></span>
 <span id="cb427-2"><a href="#cb427-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mempty</span> <span class="ot">=</span> <span class="dt">Sum</span> <span class="dv">0</span></span>
 <span id="cb427-3"><a href="#cb427-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb427-4"><a href="#cb427-4" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Num</span> a <span class="ot">=&gt;</span> <span class="dt">Monoid</span> (<span class="dt">Product</span> a) <span class="kw">where</span></span>
@@ -3812,36 +5975,75 @@ If I define an instance for <code>Num</code>, I also get an instance for <code>F
 <span id="cb427-6"><a href="#cb427-6" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb427-7"><a href="#cb427-7" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Monoid</span> [] <span class="kw">where</span></span>
 <span id="cb427-8"><a href="#cb427-8" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mempty</span> <span class="ot">=</span> []</span></code></pre></div>
-<p>So, what is a monoid for a programmer? A type forms a monoid if there’s a way of combining two elements of the type together so that parenthesis don’t matter, and there’s a also an “empty element” that can be combined with things without changing them. When thought of like this, monoids come up in programming quite often!</p>
-<h3 data-number="7.3.4" id="why"><span class="header-section-number">7.3.4</span> Why?</h3>
-<p>What use is this <code>Monoid</code> class? Can’t we just write <code>1 + 2</code> instead of <code>Sum 1 &lt;&gt; Sum 2</code>? We can, yes, but some library functions work on all <code>Monoid</code> types.</p>
-<p>The reason we want both a neutral element and an associative binary operator is that those are the exact two things we need in order to <em>reduce</em> or <em>fold</em> multiple elements into one value. This is the job of:</p>
-<div class="sourceCode" id="cb428"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb428-1"><a href="#cb428-1" aria-hidden="true" tabindex="-1"></a><span class="fu">mconcat</span><span class="ot"> ::</span> <span class="dt">Monoid</span> a <span class="ot">=&gt;</span> [a] <span class="ot">-&gt;</span> a</span></code></pre></div>
-<p>Sidenote: one way to define <code>mconcat</code> is <code>foldr (&lt;&gt;) mempty</code>. Do you remember <code>foldr</code>?</p>
-<p>Let’s look at why we need the properties of <code>Monoid</code> to implement <code>mconcat</code>. Firstly, we need <code>mempty</code> to handle empty lists:</p>
-<div class="sourceCode" id="cb429"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb429-1"><a href="#cb429-1" aria-hidden="true" tabindex="-1"></a><span class="fu">mconcat</span> []<span class="ot"> ::</span> <span class="dt">Sum</span> <span class="dt">Int</span>          <span class="op">==&gt;</span>  <span class="dt">Sum</span> <span class="dv">0</span></span></code></pre></div>
-<p>Secondly, we need associativity to be able to reduce a list <code>[x,y,z]</code> to a unique value. If <code>&lt;&gt;</code> were not associative, we would have two possible values for <code>mconcat [x,y,z]</code>, namely <code>(x&lt;&gt;y)&lt;&gt;z</code> and <code>x&lt;&gt;(y&lt;&gt;z)</code>.</p>
-<p>The most useful <code>Monoid</code> function is <code>foldMap</code>:</p>
-<div class="sourceCode" id="cb430"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb430-1"><a href="#cb430-1" aria-hidden="true" tabindex="-1"></a><span class="fu">foldMap</span><span class="ot"> ::</span> (<span class="dt">Foldable</span> t, <span class="dt">Monoid</span> m) <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> m) <span class="ot">-&gt;</span> t a <span class="ot">-&gt;</span> m</span></code></pre></div>
+<p>So, what is a monoid for a programmer? A type forms a monoid if
+there’s a way of combining two elements of the type together so that
+parenthesis don’t matter, and there’s a also an “empty element” that can
+be combined with things without changing them. When thought of like
+this, monoids come up in programming quite often!</p>
+<h3 data-number="7.3.4" id="why"><span
+class="header-section-number">7.3.4</span> Why?</h3>
+<p>What use is this <code>Monoid</code> class? Can’t we just write
+<code>1 + 2</code> instead of <code>Sum 1 &lt;&gt; Sum 2</code>? We can,
+yes, but some library functions work on all <code>Monoid</code>
+types.</p>
+<p>The reason we want both a neutral element and an associative binary
+operator is that those are the exact two things we need in order to
+<em>reduce</em> or <em>fold</em> multiple elements into one value. This
+is the job of:</p>
+<div class="sourceCode" id="cb428"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb428-1"><a href="#cb428-1" aria-hidden="true" tabindex="-1"></a><span class="fu">mconcat</span><span class="ot"> ::</span> <span class="dt">Monoid</span> a <span class="ot">=&gt;</span> [a] <span class="ot">-&gt;</span> a</span></code></pre></div>
+<p>Sidenote: one way to define <code>mconcat</code> is
+<code>foldr (&lt;&gt;) mempty</code>. Do you remember
+<code>foldr</code>?</p>
+<p>Let’s look at why we need the properties of <code>Monoid</code> to
+implement <code>mconcat</code>. Firstly, we need <code>mempty</code> to
+handle empty lists:</p>
+<div class="sourceCode" id="cb429"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb429-1"><a href="#cb429-1" aria-hidden="true" tabindex="-1"></a><span class="fu">mconcat</span> []<span class="ot"> ::</span> <span class="dt">Sum</span> <span class="dt">Int</span>          <span class="op">==&gt;</span>  <span class="dt">Sum</span> <span class="dv">0</span></span></code></pre></div>
+<p>Secondly, we need associativity to be able to reduce a list
+<code>[x,y,z]</code> to a unique value. If <code>&lt;&gt;</code> were
+not associative, we would have two possible values for
+<code>mconcat [x,y,z]</code>, namely <code>(x&lt;&gt;y)&lt;&gt;z</code>
+and <code>x&lt;&gt;(y&lt;&gt;z)</code>.</p>
+<p>The most useful <code>Monoid</code> function is
+<code>foldMap</code>:</p>
+<div class="sourceCode" id="cb430"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb430-1"><a href="#cb430-1" aria-hidden="true" tabindex="-1"></a><span class="fu">foldMap</span><span class="ot"> ::</span> (<span class="dt">Foldable</span> t, <span class="dt">Monoid</span> m) <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> m) <span class="ot">-&gt;</span> t a <span class="ot">-&gt;</span> m</span></code></pre></div>
 <p>That type signature looks scary, but concrete cases are simpler:</p>
-<div class="sourceCode" id="cb431"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb431-1"><a href="#cb431-1" aria-hidden="true" tabindex="-1"></a><span class="fu">foldMap</span> <span class="dt">Max</span> [<span class="dv">1</span><span class="ot">::</span><span class="dt">Int</span>,<span class="dv">4</span>,<span class="dv">2</span>]  <span class="op">==&gt;</span>  <span class="dt">Max</span> <span class="dv">4</span></span>
+<div class="sourceCode" id="cb431"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb431-1"><a href="#cb431-1" aria-hidden="true" tabindex="-1"></a><span class="fu">foldMap</span> <span class="dt">Max</span> [<span class="dv">1</span><span class="ot">::</span><span class="dt">Int</span>,<span class="dv">4</span>,<span class="dv">2</span>]  <span class="op">==&gt;</span>  <span class="dt">Max</span> <span class="dv">4</span></span>
 <span id="cb431-2"><a href="#cb431-2" aria-hidden="true" tabindex="-1"></a><span class="fu">foldMap</span> <span class="dt">Product</span> [<span class="dv">1</span><span class="ot">::</span><span class="dt">Int</span>,<span class="dv">4</span>,<span class="dv">2</span>]  <span class="op">==&gt;</span>  <span class="dt">Product</span> <span class="dv">8</span></span>
 <span id="cb431-3"><a href="#cb431-3" aria-hidden="true" tabindex="-1"></a><span class="co">-- We need the ::Int to avoid an &quot;Ambiguous type variable&quot; error when printing the result</span></span></code></pre></div>
-<p>Let’s break down that type. We know that an example of a <code>Foldable t =&gt; t a</code> type is <code>[a]</code>, so we can rewrite the type as</p>
-<div class="sourceCode" id="cb432"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb432-1"><a href="#cb432-1" aria-hidden="true" tabindex="-1"></a><span class="ot">foldMap&#39; ::</span> <span class="dt">Monoid</span> m <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> m) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> m</span></code></pre></div>
+<p>Let’s break down that type. We know that an example of a
+<code>Foldable t =&gt; t a</code> type is <code>[a]</code>, so we can
+rewrite the type as</p>
+<div class="sourceCode" id="cb432"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb432-1"><a href="#cb432-1" aria-hidden="true" tabindex="-1"></a><span class="ot">foldMap&#39; ::</span> <span class="dt">Monoid</span> m <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> m) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> m</span></code></pre></div>
 <p>We can build this function out of functions we already know:</p>
-<div class="sourceCode" id="cb433"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb433-1"><a href="#cb433-1" aria-hidden="true" tabindex="-1"></a>foldMap&#39; f xs <span class="ot">=</span> <span class="fu">mconcat</span> (<span class="fu">map</span> f xs)</span></code></pre></div>
-<p>Oh, by the way, thanks to the <code>(Monoid a, Monoid b) =&gt; Monoid (a,b)</code> instance we can even compute the maximum and product in one pass:</p>
-<div class="sourceCode" id="cb434"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb434-1"><a href="#cb434-1" aria-hidden="true" tabindex="-1"></a><span class="fu">foldMap</span> (\x <span class="ot">-&gt;</span> (<span class="dt">Max</span> x, <span class="dt">Product</span> x)) [<span class="dv">1</span><span class="ot">::</span><span class="dt">Int</span>,<span class="dv">4</span>,<span class="dv">2</span>]  <span class="op">==&gt;</span>  (<span class="dt">Max</span> <span class="dv">4</span>, <span class="dt">Product</span> <span class="dv">8</span>)</span></code></pre></div>
-<p>Note, you don’t need to use monoids in your own code, but you’ll eventually bump into them when using Haskell libraries so it’s good to know what they are.</p>
-<h3 data-number="7.3.5" id="how"><span class="header-section-number">7.3.5</span> How?</h3>
-<p>Due to various historical and performance reasons, the definition of the <code>Monoid</code> and <code>Semigroup</code> classes aren’t just</p>
-<div class="sourceCode" id="cb435"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb435-1"><a href="#cb435-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Semigroup</span> a <span class="kw">where</span></span>
+<div class="sourceCode" id="cb433"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb433-1"><a href="#cb433-1" aria-hidden="true" tabindex="-1"></a>foldMap&#39; f xs <span class="ot">=</span> <span class="fu">mconcat</span> (<span class="fu">map</span> f xs)</span></code></pre></div>
+<p>Oh, by the way, thanks to the
+<code>(Monoid a, Monoid b) =&gt; Monoid (a,b)</code> instance we can
+even compute the maximum and product in one pass:</p>
+<div class="sourceCode" id="cb434"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb434-1"><a href="#cb434-1" aria-hidden="true" tabindex="-1"></a><span class="fu">foldMap</span> (\x <span class="ot">-&gt;</span> (<span class="dt">Max</span> x, <span class="dt">Product</span> x)) [<span class="dv">1</span><span class="ot">::</span><span class="dt">Int</span>,<span class="dv">4</span>,<span class="dv">2</span>]  <span class="op">==&gt;</span>  (<span class="dt">Max</span> <span class="dv">4</span>, <span class="dt">Product</span> <span class="dv">8</span>)</span></code></pre></div>
+<p>Note, you don’t need to use monoids in your own code, but you’ll
+eventually bump into them when using Haskell libraries so it’s good to
+know what they are.</p>
+<h3 data-number="7.3.5" id="how"><span
+class="header-section-number">7.3.5</span> How?</h3>
+<p>Due to various historical and performance reasons, the definition of
+the <code>Monoid</code> and <code>Semigroup</code> classes aren’t
+just</p>
+<div class="sourceCode" id="cb435"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb435-1"><a href="#cb435-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Semigroup</span> a <span class="kw">where</span></span>
 <span id="cb435-2"><a href="#cb435-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  (&lt;&gt;) ::</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb435-3"><a href="#cb435-3" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Semigroup</span> a <span class="ot">=&gt;</span> <span class="dt">Monoid</span> a <span class="kw">where</span></span>
 <span id="cb435-4"><a href="#cb435-4" aria-hidden="true" tabindex="-1"></a><span class="ot">  mempty ::</span> a</span></code></pre></div>
-<p>Although you can mostly pretend they are. The actual definitions are:</p>
-<div class="sourceCode" id="cb436"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb436-1"><a href="#cb436-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Semigroup</span> a <span class="kw">where</span></span>
+<p>Although you can mostly pretend they are. The actual definitions
+are:</p>
+<div class="sourceCode" id="cb436"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb436-1"><a href="#cb436-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Semigroup</span> a <span class="kw">where</span></span>
 <span id="cb436-2"><a href="#cb436-2" aria-hidden="true" tabindex="-1"></a>  <span class="co">-- | An associative operation.</span></span>
 <span id="cb436-3"><a href="#cb436-3" aria-hidden="true" tabindex="-1"></a><span class="ot">  (&lt;&gt;) ::</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb436-4"><a href="#cb436-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -3852,7 +6054,8 @@ If I define an instance for <code>Num</code>, I also get an instance for <code>F
 <span id="cb436-9"><a href="#cb436-9" aria-hidden="true" tabindex="-1"></a>  <span class="co">-- Combine a value with itself using &lt;&gt;, n times</span></span>
 <span id="cb436-10"><a href="#cb436-10" aria-hidden="true" tabindex="-1"></a><span class="ot">  stimes ::</span> <span class="dt">Integral</span> b <span class="ot">=&gt;</span> b <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb436-11"><a href="#cb436-11" aria-hidden="true" tabindex="-1"></a>  stimes n x <span class="ot">=</span> <span class="op">...</span> <span class="co">-- default implementation omitted</span></span></code></pre></div>
-<div class="sourceCode" id="cb437"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb437-1"><a href="#cb437-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Semigroup</span> a <span class="ot">=&gt;</span> <span class="dt">Monoid</span> a <span class="kw">where</span></span>
+<div class="sourceCode" id="cb437"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb437-1"><a href="#cb437-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Semigroup</span> a <span class="ot">=&gt;</span> <span class="dt">Monoid</span> a <span class="kw">where</span></span>
 <span id="cb437-2"><a href="#cb437-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  mempty  ::</span> a</span>
 <span id="cb437-3"><a href="#cb437-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb437-4"><a href="#cb437-4" aria-hidden="true" tabindex="-1"></a><span class="ot">  mappend ::</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> a</span>
@@ -3861,22 +6064,35 @@ If I define an instance for <code>Num</code>, I also get an instance for <code>F
 <span id="cb437-7"><a href="#cb437-7" aria-hidden="true" tabindex="-1"></a>  <span class="co">-- Combine elements of a list with &lt;&gt;</span></span>
 <span id="cb437-8"><a href="#cb437-8" aria-hidden="true" tabindex="-1"></a><span class="ot">  mconcat ::</span> [a] <span class="ot">-&gt;</span> a</span>
 <span id="cb437-9"><a href="#cb437-9" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mconcat</span> <span class="ot">=</span> <span class="op">...</span> <span class="co">-- default implementation omitted</span></span></code></pre></div>
-<p>As you can see, all the operations except <code>&lt;&gt;</code> and <code>mempty</code> have default definitions, so a normal <code>Monoid</code> instance declaration looks just like this:</p>
-<div class="sourceCode" id="cb438"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb438-1"><a href="#cb438-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Semigroup</span> <span class="dt">MyType</span> <span class="kw">where</span></span>
+<p>As you can see, all the operations except <code>&lt;&gt;</code> and
+<code>mempty</code> have default definitions, so a normal
+<code>Monoid</code> instance declaration looks just like this:</p>
+<div class="sourceCode" id="cb438"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb438-1"><a href="#cb438-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Semigroup</span> <span class="dt">MyType</span> <span class="kw">where</span></span>
 <span id="cb438-2"><a href="#cb438-2" aria-hidden="true" tabindex="-1"></a>  x <span class="op">&lt;&gt;</span> y <span class="ot">=</span> <span class="op">...</span></span>
 <span id="cb438-3"><a href="#cb438-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb438-4"><a href="#cb438-4" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Monoid</span> <span class="dt">MyType</span> <span class="kw">where</span></span>
 <span id="cb438-5"><a href="#cb438-5" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mempty</span> <span class="ot">=</span> <span class="op">...</span></span></code></pre></div>
-<h2 data-number="7.4" id="open-and-closed-abstractions"><span class="header-section-number">7.4</span> Open and Closed Abstractions</h2>
-<p>A question novice Haskell programmers often ask (or at least should ask!) is: when should I use type classes? This section offers one answer.</p>
-<p>Let’s look at a concrete example. A vehicle can be either a car or an airplane. We can model this with algebraic datatypes (as we’ve seen earlier in this chapter), but also with type classes. Here’s the datatype version:</p>
-<div class="sourceCode" id="cb439"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb439-1"><a href="#cb439-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Vehicle</span> <span class="ot">=</span> <span class="dt">Car</span> <span class="dt">String</span> <span class="op">|</span> <span class="dt">Airplane</span> <span class="dt">String</span></span>
+<h2 data-number="7.4" id="open-and-closed-abstractions"><span
+class="header-section-number">7.4</span> Open and Closed
+Abstractions</h2>
+<p>A question novice Haskell programmers often ask (or at least should
+ask!) is: when should I use type classes? This section offers one
+answer.</p>
+<p>Let’s look at a concrete example. A vehicle can be either a car or an
+airplane. We can model this with algebraic datatypes (as we’ve seen
+earlier in this chapter), but also with type classes. Here’s the
+datatype version:</p>
+<div class="sourceCode" id="cb439"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb439-1"><a href="#cb439-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Vehicle</span> <span class="ot">=</span> <span class="dt">Car</span> <span class="dt">String</span> <span class="op">|</span> <span class="dt">Airplane</span> <span class="dt">String</span></span>
 <span id="cb439-2"><a href="#cb439-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb439-3"><a href="#cb439-3" aria-hidden="true" tabindex="-1"></a><span class="ot">sound ::</span> <span class="dt">Vehicle</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb439-4"><a href="#cb439-4" aria-hidden="true" tabindex="-1"></a>sound (<span class="dt">Car</span> _) <span class="ot">=</span> <span class="st">&quot;brum brum&quot;</span></span>
 <span id="cb439-5"><a href="#cb439-5" aria-hidden="true" tabindex="-1"></a>sound (<span class="dt">Airplane</span> _) <span class="ot">=</span> <span class="st">&quot;zooooom&quot;</span></span></code></pre></div>
-<p>Here’s the class version. Note how each case gets its own datatype, which are collected together in a type class.</p>
-<div class="sourceCode" id="cb440"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb440-1"><a href="#cb440-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Car</span> <span class="ot">=</span> <span class="dt">Car</span> <span class="dt">String</span></span>
+<p>Here’s the class version. Note how each case gets its own datatype,
+which are collected together in a type class.</p>
+<div class="sourceCode" id="cb440"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb440-1"><a href="#cb440-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Car</span> <span class="ot">=</span> <span class="dt">Car</span> <span class="dt">String</span></span>
 <span id="cb440-2"><a href="#cb440-2" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Airplane</span> <span class="ot">=</span> <span class="dt">Airplane</span> <span class="dt">String</span></span>
 <span id="cb440-3"><a href="#cb440-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb440-4"><a href="#cb440-4" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">VehicleClass</span> a <span class="kw">where</span></span>
@@ -3887,28 +6103,50 @@ If I define an instance for <code>Num</code>, I also get an instance for <code>F
 <span id="cb440-9"><a href="#cb440-9" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb440-10"><a href="#cb440-10" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">VehicleClass</span> <span class="dt">Airplane</span> <span class="kw">where</span></span>
 <span id="cb440-11"><a href="#cb440-11" aria-hidden="true" tabindex="-1"></a>  sound (<span class="dt">Airplane</span> _) <span class="ot">=</span> <span class="st">&quot;zooooom&quot;</span></span></code></pre></div>
-<p>What is the difference between these solutions? The data-based solution is <em>closed</em>, meaning the set of cases is fixed and we can handle all of them in one place. The class-based solution is <em>open</em>, meaning we can add new cases, even in other modules.</p>
-<p>An open abstraction is nice when we want extensibility. In the class-based solution, another module could define a bike:</p>
-<div class="sourceCode" id="cb441"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb441-1"><a href="#cb441-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Bike</span> <span class="ot">=</span> <span class="dt">Bike</span> <span class="dt">String</span></span>
+<p>What is the difference between these solutions? The data-based
+solution is <em>closed</em>, meaning the set of cases is fixed and we
+can handle all of them in one place. The class-based solution is
+<em>open</em>, meaning we can add new cases, even in other modules.</p>
+<p>An open abstraction is nice when we want extensibility. In the
+class-based solution, another module could define a bike:</p>
+<div class="sourceCode" id="cb441"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb441-1"><a href="#cb441-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Bike</span> <span class="ot">=</span> <span class="dt">Bike</span> <span class="dt">String</span></span>
 <span id="cb441-2"><a href="#cb441-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb441-3"><a href="#cb441-3" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">VehicleClass</span> <span class="dt">Bike</span> <span class="kw">where</span></span>
 <span id="cb441-4"><a href="#cb441-4" aria-hidden="true" tabindex="-1"></a>  sound (<span class="dt">Bike</span> _) <span class="ot">=</span> <span class="st">&quot;whirrr&quot;</span></span></code></pre></div>
-<p>A closed abstraction is good when we want to know that we’ve handled all cases, consider for example the function <code>canCollide</code> which checks whether two vehicles can collide:</p>
-<div class="sourceCode" id="cb442"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb442-1"><a href="#cb442-1" aria-hidden="true" tabindex="-1"></a><span class="ot">canCollide ::</span> <span class="dt">Vehicle</span> <span class="ot">-&gt;</span> <span class="dt">Vehicle</span> <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
+<p>A closed abstraction is good when we want to know that we’ve handled
+all cases, consider for example the function <code>canCollide</code>
+which checks whether two vehicles can collide:</p>
+<div class="sourceCode" id="cb442"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb442-1"><a href="#cb442-1" aria-hidden="true" tabindex="-1"></a><span class="ot">canCollide ::</span> <span class="dt">Vehicle</span> <span class="ot">-&gt;</span> <span class="dt">Vehicle</span> <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb442-2"><a href="#cb442-2" aria-hidden="true" tabindex="-1"></a>canCollide (<span class="dt">Car</span> _)      (<span class="dt">Car</span> _)      <span class="ot">=</span> <span class="dt">True</span></span>
 <span id="cb442-3"><a href="#cb442-3" aria-hidden="true" tabindex="-1"></a>canCollide (<span class="dt">Airplane</span> _) (<span class="dt">Airplane</span> _) <span class="ot">=</span> <span class="dt">True</span></span>
 <span id="cb442-4"><a href="#cb442-4" aria-hidden="true" tabindex="-1"></a>canCollide _            _            <span class="ot">=</span> <span class="dt">False</span></span></code></pre></div>
-<p>This would be very hard to implement reliably in the class-based solution. Consider for example how collision checks between <code>Bike</code>s and <code>Car</code>s would get handled.</p>
-<h2 data-number="7.5" id="modeling-with-languages"><span class="header-section-number">7.5</span> Modeling with Languages</h2>
-<p>Sometimes it’s useful to implement a mini programming language for describing parts of your software. The fancy term for these is an <em>Embedded Domain-Specific Language (EDSL)</em>. Haskell is well suited to modeling and interpreting languages. The expressions of the language are represented using (often recursive) algebraic data types. The language can be <em>interpreted</em> (that is, evaluated or run) by a recursive function.</p>
-<p>Here’s an example of a language for describing price computations for products in a web shop.</p>
-<div class="sourceCode" id="cb443"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb443-1"><a href="#cb443-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Discount</span> <span class="ot">=</span> <span class="dt">DiscountPercent</span> <span class="dt">Int</span>         <span class="co">-- A percentage discount</span></span>
+<p>This would be very hard to implement reliably in the class-based
+solution. Consider for example how collision checks between
+<code>Bike</code>s and <code>Car</code>s would get handled.</p>
+<h2 data-number="7.5" id="modeling-with-languages"><span
+class="header-section-number">7.5</span> Modeling with Languages</h2>
+<p>Sometimes it’s useful to implement a mini programming language for
+describing parts of your software. The fancy term for these is an
+<em>Embedded Domain-Specific Language (EDSL)</em>. Haskell is well
+suited to modeling and interpreting languages. The expressions of the
+language are represented using (often recursive) algebraic data types.
+The language can be <em>interpreted</em> (that is, evaluated or run) by
+a recursive function.</p>
+<p>Here’s an example of a language for describing price computations for
+products in a web shop.</p>
+<div class="sourceCode" id="cb443"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb443-1"><a href="#cb443-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Discount</span> <span class="ot">=</span> <span class="dt">DiscountPercent</span> <span class="dt">Int</span>         <span class="co">-- A percentage discount</span></span>
 <span id="cb443-2"><a href="#cb443-2" aria-hidden="true" tabindex="-1"></a>              <span class="op">|</span> <span class="dt">DiscountConstant</span> <span class="dt">Int</span>        <span class="co">-- A constant discount</span></span>
 <span id="cb443-3"><a href="#cb443-3" aria-hidden="true" tabindex="-1"></a>              <span class="op">|</span> <span class="dt">MinimumPrice</span> <span class="dt">Int</span>            <span class="co">-- Set a minimum price</span></span>
 <span id="cb443-4"><a href="#cb443-4" aria-hidden="true" tabindex="-1"></a>              <span class="op">|</span> <span class="dt">ForCustomer</span> <span class="dt">String</span> <span class="dt">Discount</span> <span class="co">-- Discounts can be conditional</span></span>
 <span id="cb443-5"><a href="#cb443-5" aria-hidden="true" tabindex="-1"></a>              <span class="op">|</span> <span class="dt">Many</span> [<span class="dt">Discount</span>]             <span class="co">-- Apply a number of discounts in row</span></span></code></pre></div>
-<p>The language is interpreted by the function <code>applyDiscount</code> that takes a customer name, a price, a discount, and returns a price.</p>
-<div class="sourceCode" id="cb444"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb444-1"><a href="#cb444-1" aria-hidden="true" tabindex="-1"></a><span class="ot">applyDiscount ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Discount</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<p>The language is interpreted by the function
+<code>applyDiscount</code> that takes a customer name, a price, a
+discount, and returns a price.</p>
+<div class="sourceCode" id="cb444"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb444-1"><a href="#cb444-1" aria-hidden="true" tabindex="-1"></a><span class="ot">applyDiscount ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Discount</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb444-2"><a href="#cb444-2" aria-hidden="true" tabindex="-1"></a>applyDiscount _        price (<span class="dt">DiscountPercent</span> percent) <span class="ot">=</span> price <span class="op">-</span> (price <span class="op">*</span> percent) <span class="ot">`div`</span> <span class="dv">100</span></span>
 <span id="cb444-3"><a href="#cb444-3" aria-hidden="true" tabindex="-1"></a>applyDiscount _        price (<span class="dt">DiscountConstant</span> discount) <span class="ot">=</span> price <span class="op">-</span> discount</span>
 <span id="cb444-4"><a href="#cb444-4" aria-hidden="true" tabindex="-1"></a>applyDiscount _        price (<span class="dt">MinimumPrice</span> minPrice) <span class="ot">=</span> <span class="fu">max</span> price minPrice</span>
@@ -3918,8 +6156,10 @@ If I define an instance for <code>Num</code>, I also get an instance for <code>F
 <span id="cb444-8"><a href="#cb444-8" aria-hidden="true" tabindex="-1"></a>applyDiscount customer price (<span class="dt">Many</span> discounts) <span class="ot">=</span> go price discounts</span>
 <span id="cb444-9"><a href="#cb444-9" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> go p [] <span class="ot">=</span> p</span>
 <span id="cb444-10"><a href="#cb444-10" aria-hidden="true" tabindex="-1"></a>        go p (d<span class="op">:</span>ds) <span class="ot">=</span> go (applyDiscount customer p d) ds</span></code></pre></div>
-<p>Here we apply a discount chain of -50%, -$30 with a minimum price of $35:</p>
-<div class="sourceCode" id="cb445"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb445-1"><a href="#cb445-1" aria-hidden="true" tabindex="-1"></a>applyDiscount <span class="st">&quot;Bob&quot;</span> <span class="dv">120</span> (<span class="dt">DiscountPercent</span> <span class="dv">50</span>)</span>
+<p>Here we apply a discount chain of -50%, -$30 with a minimum price of
+$35:</p>
+<div class="sourceCode" id="cb445"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb445-1"><a href="#cb445-1" aria-hidden="true" tabindex="-1"></a>applyDiscount <span class="st">&quot;Bob&quot;</span> <span class="dv">120</span> (<span class="dt">DiscountPercent</span> <span class="dv">50</span>)</span>
 <span id="cb445-2"><a href="#cb445-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">60</span></span>
 <span id="cb445-3"><a href="#cb445-3" aria-hidden="true" tabindex="-1"></a>applyDiscount <span class="st">&quot;Bob&quot;</span> <span class="dv">60</span> (<span class="dt">DiscountConstant</span> <span class="dv">30</span>)</span>
 <span id="cb445-4"><a href="#cb445-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">30</span></span>
@@ -3928,42 +6168,82 @@ If I define an instance for <code>Num</code>, I also get an instance for <code>F
 <span id="cb445-7"><a href="#cb445-7" aria-hidden="true" tabindex="-1"></a>applyDiscount <span class="st">&quot;Bob&quot;</span> <span class="dv">120</span> (<span class="dt">Many</span> [<span class="dt">DiscountPercent</span> <span class="dv">50</span>, <span class="dt">DiscountConstant</span> <span class="dv">30</span>, <span class="dt">MinimumPrice</span> <span class="dv">35</span>])</span>
 <span id="cb445-8"><a href="#cb445-8" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">35</span></span></code></pre></div>
 <p>Here we have different discounts for Ssarah and Yvonne:</p>
-<div class="sourceCode" id="cb446"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb446-1"><a href="#cb446-1" aria-hidden="true" tabindex="-1"></a>applyDiscount <span class="st">&quot;Yvonne&quot;</span> <span class="dv">100</span> (<span class="dt">Many</span> [<span class="dt">ForCustomer</span> <span class="st">&quot;Yvonne&quot;</span> (<span class="dt">DiscountConstant</span> <span class="dv">10</span>), <span class="dt">ForCustomer</span> <span class="st">&quot;Ssarah&quot;</span> (<span class="dt">DiscountConstant</span> <span class="dv">20</span>)])</span>
+<div class="sourceCode" id="cb446"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb446-1"><a href="#cb446-1" aria-hidden="true" tabindex="-1"></a>applyDiscount <span class="st">&quot;Yvonne&quot;</span> <span class="dv">100</span> (<span class="dt">Many</span> [<span class="dt">ForCustomer</span> <span class="st">&quot;Yvonne&quot;</span> (<span class="dt">DiscountConstant</span> <span class="dv">10</span>), <span class="dt">ForCustomer</span> <span class="st">&quot;Ssarah&quot;</span> (<span class="dt">DiscountConstant</span> <span class="dv">20</span>)])</span>
 <span id="cb446-2"><a href="#cb446-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">90</span></span>
 <span id="cb446-3"><a href="#cb446-3" aria-hidden="true" tabindex="-1"></a>applyDiscount <span class="st">&quot;Ssarah&quot;</span> <span class="dv">100</span> (<span class="dt">Many</span> [<span class="dt">ForCustomer</span> <span class="st">&quot;Yvonne&quot;</span> (<span class="dt">DiscountConstant</span> <span class="dv">10</span>), <span class="dt">ForCustomer</span> <span class="st">&quot;Ssarah&quot;</span> (<span class="dt">DiscountConstant</span> <span class="dv">20</span>)])</span>
 <span id="cb446-4"><a href="#cb446-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">80</span></span></code></pre></div>
-<p>As you can see, even a simple <code>Discount</code> type can generate complex behaviours because it is self-referential (recursive). Using <code>Discount</code> we are able to represent the discount logic of our webshop as <em>data</em> instead of writing code.</p>
-<p>There are multiple reasons for representing logic as data instead of code. Unlike code, data can easily be stored in a file or database, or even transmitted over the network. We can also use the same data for multiple purposes, for example we could visualize the discount rules in an administration user interface.</p>
+<p>As you can see, even a simple <code>Discount</code> type can generate
+complex behaviours because it is self-referential (recursive). Using
+<code>Discount</code> we are able to represent the discount logic of our
+webshop as <em>data</em> instead of writing code.</p>
+<p>There are multiple reasons for representing logic as data instead of
+code. Unlike code, data can easily be stored in a file or database, or
+even transmitted over the network. We can also use the same data for
+multiple purposes, for example we could visualize the discount rules in
+an administration user interface.</p>
 <!-- ## TODO: Quiz -->
-<h2 data-number="7.6" id="exercises-6"><span class="header-section-number">7.6</span> Exercises</h2>
+<h2 data-number="7.6" id="exercises-6"><span
+class="header-section-number">7.6</span> Exercises</h2>
 <ul>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set7.hs">Set7</a></li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set7.hs">Set7</a></li>
 </ul>
-<h1 data-number="8" id="lecture-8-the-aftertaste"><span class="header-section-number">8</span> Lecture 8: The Aftertaste</h1>
-<h2 data-number="8.1" id="a-taste-of-io"><span class="header-section-number">8.1</span> A Taste of IO</h2>
-<p>This course has been centered around pure functional programming. We’ve done lots of arithmetic, reversed lists, worked with binary trees, but so far we haven’t been able to affect the world outside our GHCi.</p>
-<p>Things like reading input, writing to a file, or talking over the network are <em>side effects</em>. Side effects can’t be represented with pure functional code. A function like</p>
-<div class="sourceCode" id="cb447"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb447-1"><a href="#cb447-1" aria-hidden="true" tabindex="-1"></a><span class="ot">readInputFromTheUser ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span></code></pre></div>
-<p>can’t be pure, because if it were, <code>readInputFromUser "What is your name?"</code> would always have to return the same result. However, representing side-effects and impurity in a pure language <em>is</em> possible. There are many ways of doing it, and the Haskell way is to use <em>Monads</em>.</p>
-<p>Monads are reputedly difficult to understand. That is probably because they are so abstract. I think it’s best to focus on practical and concrete cases first. Here’s a taste of the <code>IO</code> Monad, which you can use for all sorts of side effects in Haskell.</p>
+<h1 data-number="8" id="lecture-8-the-aftertaste"><span
+class="header-section-number">8</span> Lecture 8: The Aftertaste</h1>
+<h2 data-number="8.1" id="a-taste-of-io"><span
+class="header-section-number">8.1</span> A Taste of IO</h2>
+<p>This course has been centered around pure functional programming.
+We’ve done lots of arithmetic, reversed lists, worked with binary trees,
+but so far we haven’t been able to affect the world outside our
+GHCi.</p>
+<p>Things like reading input, writing to a file, or talking over the
+network are <em>side effects</em>. Side effects can’t be represented
+with pure functional code. A function like</p>
+<div class="sourceCode" id="cb447"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb447-1"><a href="#cb447-1" aria-hidden="true" tabindex="-1"></a><span class="ot">readInputFromTheUser ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span></code></pre></div>
+<p>can’t be pure, because if it were,
+<code>readInputFromUser "What is your name?"</code> would always have to
+return the same result. However, representing side-effects and impurity
+in a pure language <em>is</em> possible. There are many ways of doing
+it, and the Haskell way is to use <em>Monads</em>.</p>
+<p>Monads are reputedly difficult to understand. That is probably
+because they are so abstract. I think it’s best to focus on practical
+and concrete cases first. Here’s a taste of the <code>IO</code> Monad,
+which you can use for all sorts of side effects in Haskell.</p>
 <p>Let’s start!</p>
-<div class="sourceCode" id="cb448"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb448-1"><a href="#cb448-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="fu">getLine</span></span>
+<div class="sourceCode" id="cb448"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb448-1"><a href="#cb448-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="fu">getLine</span></span>
 <span id="cb448-2"><a href="#cb448-2" aria-hidden="true" tabindex="-1"></a><span class="fu">getLine</span><span class="ot"> ::</span> <span class="dt">IO</span> <span class="dt">String</span></span>
 <span id="cb448-3"><a href="#cb448-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> line <span class="ot">&lt;-</span> <span class="fu">getLine</span></span></code></pre></div>
 <pre><code>another line</code></pre>
-<div class="sourceCode" id="cb450"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb450-1"><a href="#cb450-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t line</span>
+<div class="sourceCode" id="cb450"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb450-1"><a href="#cb450-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t line</span>
 <span id="cb450-2"><a href="#cb450-2" aria-hidden="true" tabindex="-1"></a><span class="ot">line ::</span> <span class="dt">String</span></span>
 <span id="cb450-3"><a href="#cb450-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> line</span>
 <span id="cb450-4"><a href="#cb450-4" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;another line&quot;</span></span>
 <span id="cb450-5"><a href="#cb450-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">reverse</span> line</span>
 <span id="cb450-6"><a href="#cb450-6" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;enil rehtona&quot;</span></span></code></pre></div>
-<p>What we’ve seen here is the <em>IO action</em> <code>getLine</code>. It has type <code>IO String</code>. This means that GHCi can <em>execute</em> the action to produce a value of type <code>String</code>. When we enter <code>line &lt;- getLine</code> into GHCi, we mean:</p>
+<p>What we’ve seen here is the <em>IO action</em> <code>getLine</code>.
+It has type <code>IO String</code>. This means that GHCi can
+<em>execute</em> the action to produce a value of type
+<code>String</code>. When we enter <code>line &lt;- getLine</code> into
+GHCi, we mean:</p>
 <blockquote>
-<p>Execute the IO action <code>getLine</code>, and give the result the name <code>line</code>.</p>
+<p>Execute the IO action <code>getLine</code>, and give the result the
+name <code>line</code>.</p>
 </blockquote>
-<p>After we’ve received <code>line</code>, it’s a pure <code>String</code> value and we can work with it normally.</p>
-<p>Some IO actions take parameters. For example <code>putStrLn :: String -&gt; IO ()</code> takes a <code>String</code> and returns an <code>IO</code> action that prints that string. The <code>()</code> type is a special type that only has one value, <code>()</code>. In this case <code>IO ()</code> means that this IO always produces the same empty value <code>()</code>. You can run IO actions by just</p>
-<div class="sourceCode" id="cb451"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb451-1"><a href="#cb451-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="fu">putStrLn</span></span>
+<p>After we’ve received <code>line</code>, it’s a pure
+<code>String</code> value and we can work with it normally.</p>
+<p>Some IO actions take parameters. For example
+<code>putStrLn :: String -&gt; IO ()</code> takes a <code>String</code>
+and returns an <code>IO</code> action that prints that string. The
+<code>()</code> type is a special type that only has one value,
+<code>()</code>. In this case <code>IO ()</code> means that this IO
+always produces the same empty value <code>()</code>. You can run IO
+actions by just</p>
+<div class="sourceCode" id="cb451"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb451-1"><a href="#cb451-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="fu">putStrLn</span></span>
 <span id="cb451-2"><a href="#cb451-2" aria-hidden="true" tabindex="-1"></a><span class="fu">putStrLn</span><span class="ot"> ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span>
 <span id="cb451-3"><a href="#cb451-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="fu">putStrLn</span> <span class="st">&quot;hello&quot;</span></span>
 <span id="cb451-4"><a href="#cb451-4" aria-hidden="true" tabindex="-1"></a><span class="fu">putStrLn</span> <span class="st">&quot;hello&quot;</span><span class="ot"> ::</span> <span class="dt">IO</span> ()</span>
@@ -3971,11 +6251,16 @@ If I define an instance for <code>Num</code>, I also get an instance for <code>F
 <span id="cb451-6"><a href="#cb451-6" aria-hidden="true" tabindex="-1"></a>hello</span>
 <span id="cb451-7"><a href="#cb451-7" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> val</span>
 <span id="cb451-8"><a href="#cb451-8" aria-hidden="true" tabindex="-1"></a>()</span></code></pre></div>
-<p>If you don’t need the return value of an IO action, you can run it <em>in GHCi</em> without the <code>&lt;-</code>:</p>
-<div class="sourceCode" id="cb452"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb452-1"><a href="#cb452-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">putStrLn</span> <span class="st">&quot;hello&quot;</span></span>
+<p>If you don’t need the return value of an IO action, you can run it
+<em>in GHCi</em> without the <code>&lt;-</code>:</p>
+<div class="sourceCode" id="cb452"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb452-1"><a href="#cb452-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">putStrLn</span> <span class="st">&quot;hello&quot;</span></span>
 <span id="cb452-2"><a href="#cb452-2" aria-hidden="true" tabindex="-1"></a>hello</span></code></pre></div>
-<p>You can build your own IO actions by combining other actions with <em>do-notation</em>. A <code>do</code> block lists IO actions that are executed in order.</p>
-<div class="sourceCode" id="cb453"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb453-1"><a href="#cb453-1" aria-hidden="true" tabindex="-1"></a><span class="ot">printTwoThings ::</span> <span class="dt">IO</span> ()</span>
+<p>You can build your own IO actions by combining other actions with
+<em>do-notation</em>. A <code>do</code> block lists IO actions that are
+executed in order.</p>
+<div class="sourceCode" id="cb453"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb453-1"><a href="#cb453-1" aria-hidden="true" tabindex="-1"></a><span class="ot">printTwoThings ::</span> <span class="dt">IO</span> ()</span>
 <span id="cb453-2"><a href="#cb453-2" aria-hidden="true" tabindex="-1"></a>printTwoThings <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb453-3"><a href="#cb453-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">putStrLn</span> <span class="st">&quot;Hello!&quot;</span></span>
 <span id="cb453-4"><a href="#cb453-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">putStrLn</span> <span class="st">&quot;How are you?&quot;</span></span>
@@ -3985,40 +6270,66 @@ If I define an instance for <code>Num</code>, I also get an instance for <code>F
 <span id="cb453-8"><a href="#cb453-8" aria-hidden="true" tabindex="-1"></a>  <span class="fu">putStrLn</span> <span class="st">&quot;What&#39;s your name?&quot;</span></span>
 <span id="cb453-9"><a href="#cb453-9" aria-hidden="true" tabindex="-1"></a>  name <span class="ot">&lt;-</span> <span class="fu">getLine</span></span>
 <span id="cb453-10"><a href="#cb453-10" aria-hidden="true" tabindex="-1"></a>  <span class="fu">putStrLn</span> (<span class="st">&quot;Hello, &quot;</span> <span class="op">++</span> name)</span></code></pre></div>
-<div class="sourceCode" id="cb454"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb454-1"><a href="#cb454-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> printTwoThings</span>
+<div class="sourceCode" id="cb454"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb454-1"><a href="#cb454-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> printTwoThings</span>
 <span id="cb454-2"><a href="#cb454-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Hello</span><span class="op">!</span></span>
 <span id="cb454-3"><a href="#cb454-3" aria-hidden="true" tabindex="-1"></a><span class="dt">How</span> are you<span class="op">?</span></span>
 <span id="cb454-4"><a href="#cb454-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> greet</span>
 <span id="cb454-5"><a href="#cb454-5" aria-hidden="true" tabindex="-1"></a><span class="dt">What&#39;s</span> your name<span class="op">?</span></span>
 <span id="cb454-6"><a href="#cb454-6" aria-hidden="true" tabindex="-1"></a><span class="dt">Seraphim</span></span>
 <span id="cb454-7"><a href="#cb454-7" aria-hidden="true" tabindex="-1"></a><span class="dt">Hello</span>, <span class="dt">Seraphim</span></span></code></pre></div>
-<h3 data-number="8.1.1" id="what-about-purity"><span class="header-section-number">8.1.1</span> What About Purity?</h3>
-<p>It feels as if we can just do side effects where ever we want with these IO actions. However, it’s important to remember the distinction between <em>defining</em> an IO action and <em>executing</em> it.</p>
+<h3 data-number="8.1.1" id="what-about-purity"><span
+class="header-section-number">8.1.1</span> What About Purity?</h3>
+<p>It feels as if we can just do side effects where ever we want with
+these IO actions. However, it’s important to remember the distinction
+between <em>defining</em> an IO action and <em>executing</em> it.</p>
 <p>Let’s try to print while mapping over a list</p>
-<div class="sourceCode" id="cb455"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb455-1"><a href="#cb455-1" aria-hidden="true" tabindex="-1"></a><span class="ot">printAndIncrement ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<div class="sourceCode" id="cb455"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb455-1"><a href="#cb455-1" aria-hidden="true" tabindex="-1"></a><span class="ot">printAndIncrement ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb455-2"><a href="#cb455-2" aria-hidden="true" tabindex="-1"></a>printAndIncrement x <span class="ot">=</span> x<span class="op">+</span><span class="dv">1</span></span>
 <span id="cb455-3"><a href="#cb455-3" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> action <span class="ot">=</span> <span class="fu">putStrLn</span> <span class="st">&quot;got a number!&quot;</span></span></code></pre></div>
-<div class="sourceCode" id="cb456"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb456-1"><a href="#cb456-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">map</span> printAndIncrement [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
+<div class="sourceCode" id="cb456"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb456-1"><a href="#cb456-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">map</span> printAndIncrement [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb456-2"><a href="#cb456-2" aria-hidden="true" tabindex="-1"></a>[<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>]</span></code></pre></div>
-<p>This didn’t print anything, because even though we defined our <code>action</code>, it wasn’t given to GHCi for execution. Because <code>printAndIncrement</code> returns an <code>Int</code>, it can’t return an action. Ok, let’s try another approach:</p>
-<div class="sourceCode" id="cb457"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb457-1"><a href="#cb457-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">length</span> (<span class="fu">map</span> <span class="fu">putStrLn</span> [<span class="st">&quot;string1&quot;</span>,<span class="st">&quot;string2&quot;</span>])</span>
+<p>This didn’t print anything, because even though we defined our
+<code>action</code>, it wasn’t given to GHCi for execution. Because
+<code>printAndIncrement</code> returns an <code>Int</code>, it can’t
+return an action. Ok, let’s try another approach:</p>
+<div class="sourceCode" id="cb457"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb457-1"><a href="#cb457-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">length</span> (<span class="fu">map</span> <span class="fu">putStrLn</span> [<span class="st">&quot;string1&quot;</span>,<span class="st">&quot;string2&quot;</span>])</span>
 <span id="cb457-2"><a href="#cb457-2" aria-hidden="true" tabindex="-1"></a><span class="dv">2</span></span></code></pre></div>
 <p>That didn’t print anything either! Let’s see why:</p>
-<div class="sourceCode" id="cb458"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb458-1"><a href="#cb458-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="fu">map</span> <span class="fu">putStrLn</span> [<span class="st">&quot;string1&quot;</span>,<span class="st">&quot;string2&quot;</span>]</span>
+<div class="sourceCode" id="cb458"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb458-1"><a href="#cb458-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="fu">map</span> <span class="fu">putStrLn</span> [<span class="st">&quot;string1&quot;</span>,<span class="st">&quot;string2&quot;</span>]</span>
 <span id="cb458-2"><a href="#cb458-2" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> <span class="fu">putStrLn</span> [<span class="st">&quot;string1&quot;</span>,<span class="st">&quot;string2&quot;</span>]<span class="ot"> ::</span> [<span class="dt">IO</span> ()]</span>
 <span id="cb458-3"><a href="#cb458-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="fu">length</span> (<span class="fu">map</span> <span class="fu">putStrLn</span> [<span class="st">&quot;string1&quot;</span>,<span class="st">&quot;string2&quot;</span>])</span>
 <span id="cb458-4"><a href="#cb458-4" aria-hidden="true" tabindex="-1"></a><span class="fu">length</span> (<span class="fu">map</span> <span class="fu">putStrLn</span> [<span class="st">&quot;string1&quot;</span>,<span class="st">&quot;string2&quot;</span>])<span class="ot"> ::</span> <span class="dt">Int</span></span></code></pre></div>
-<p>We generated a list of IO actions and computed the length of the list. Defining IO actions is pure, it’s <em>running</em> them that causes side effects. Since the type of our expression was <code>Int</code>, no <code>IO</code> actions could land in GHCi and be executed.</p>
+<p>We generated a list of IO actions and computed the length of the
+list. Defining IO actions is pure, it’s <em>running</em> them that
+causes side effects. Since the type of our expression was
+<code>Int</code>, no <code>IO</code> actions could land in GHCi and be
+executed.</p>
 <p>If we instead return an IO action, it does get run:</p>
-<div class="sourceCode" id="cb459"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb459-1"><a href="#cb459-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="fu">head</span> (<span class="fu">map</span> <span class="fu">putStrLn</span> [<span class="st">&quot;string1&quot;</span>,<span class="st">&quot;string2&quot;</span>])</span>
+<div class="sourceCode" id="cb459"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb459-1"><a href="#cb459-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="fu">head</span> (<span class="fu">map</span> <span class="fu">putStrLn</span> [<span class="st">&quot;string1&quot;</span>,<span class="st">&quot;string2&quot;</span>])</span>
 <span id="cb459-2"><a href="#cb459-2" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span> (<span class="fu">map</span> <span class="fu">putStrLn</span> [<span class="st">&quot;string1&quot;</span>,<span class="st">&quot;string2&quot;</span>])<span class="ot"> ::</span> <span class="dt">IO</span> ()</span>
 <span id="cb459-3"><a href="#cb459-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">head</span> (<span class="fu">map</span> <span class="fu">putStrLn</span> [<span class="st">&quot;string1&quot;</span>,<span class="st">&quot;string2&quot;</span>])</span>
 <span id="cb459-4"><a href="#cb459-4" aria-hidden="true" tabindex="-1"></a>string1</span></code></pre></div>
-<p>Here too, the code that produces the action <code>putStrLn "string1"</code> is pure, it’s only after the IO action is executed by GHCi that we see the printed string. And as you can see, the other IO action, <code>putStrLn "string2"</code>, never got run.</p>
-<p>If this feels complicated, don’t worry. We’ll get back to this on part 2 of the course.</p>
-<h3 data-number="8.1.2" id="what-about-haskell-programs"><span class="header-section-number">8.1.2</span> What About Haskell Programs?</h3>
-<p>We know that GHCi can run IO actions. What about actual Haskell programs? The way Haskell programs work is that the IO action called <code>main</code> gets executed when the program is run. Recall our example program from Lecture 1.</p>
-<div class="sourceCode" id="cb460"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb460-1"><a href="#cb460-1" aria-hidden="true" tabindex="-1"></a><span class="kw">module</span> <span class="dt">Gold</span> <span class="kw">where</span></span>
+<p>Here too, the code that produces the action
+<code>putStrLn "string1"</code> is pure, it’s only after the IO action
+is executed by GHCi that we see the printed string. And as you can see,
+the other IO action, <code>putStrLn "string2"</code>, never got run.</p>
+<p>If this feels complicated, don’t worry. We’ll get back to this on
+part 2 of the course.</p>
+<h3 data-number="8.1.2" id="what-about-haskell-programs"><span
+class="header-section-number">8.1.2</span> What About Haskell
+Programs?</h3>
+<p>We know that GHCi can run IO actions. What about actual Haskell
+programs? The way Haskell programs work is that the IO action called
+<code>main</code> gets executed when the program is run. Recall our
+example program from Lecture 1.</p>
+<div class="sourceCode" id="cb460"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb460-1"><a href="#cb460-1" aria-hidden="true" tabindex="-1"></a><span class="kw">module</span> <span class="dt">Gold</span> <span class="kw">where</span></span>
 <span id="cb460-2"><a href="#cb460-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb460-3"><a href="#cb460-3" aria-hidden="true" tabindex="-1"></a><span class="co">-- The golden ratio</span></span>
 <span id="cb460-4"><a href="#cb460-4" aria-hidden="true" tabindex="-1"></a><span class="ot">phi ::</span> <span class="dt">Double</span></span>
@@ -4032,44 +6343,80 @@ If I define an instance for <code>Num</code>, I also get an instance for <code>F
 <span id="cb460-12"><a href="#cb460-12" aria-hidden="true" tabindex="-1"></a>main <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb460-13"><a href="#cb460-13" aria-hidden="true" tabindex="-1"></a>  <span class="fu">print</span> (polynomial phi)</span>
 <span id="cb460-14"><a href="#cb460-14" aria-hidden="true" tabindex="-1"></a>  <span class="fu">print</span> (f phi)</span></code></pre></div>
-<p>Here we see some pure code and a <code>main</code> IO action that prints two things (<code>print</code> is just <code>putStrLn</code> combined with <code>show</code>).</p>
-<p>We can place this code in a file called <code>Gold.hs</code>, compile it into an executable, and run it:</p>
-<div class="sourceCode" id="cb461"><pre class="sourceCode sh"><code class="sourceCode bash"><span id="cb461-1"><a href="#cb461-1" aria-hidden="true" tabindex="-1"></a><span class="ex">$</span> ghc <span class="at">-main-is</span> Gold Gold.hs</span>
+<p>Here we see some pure code and a <code>main</code> IO action that
+prints two things (<code>print</code> is just <code>putStrLn</code>
+combined with <code>show</code>).</p>
+<p>We can place this code in a file called <code>Gold.hs</code>, compile
+it into an executable, and run it:</p>
+<div class="sourceCode" id="cb461"><pre
+class="sourceCode sh"><code class="sourceCode bash"><span id="cb461-1"><a href="#cb461-1" aria-hidden="true" tabindex="-1"></a><span class="ex">$</span> ghc <span class="at">-main-is</span> Gold Gold.hs</span>
 <span id="cb461-2"><a href="#cb461-2" aria-hidden="true" tabindex="-1"></a><span class="ex">[1</span> of 1] Compiling Gold             <span class="er">(</span> <span class="ex">Gold.hs,</span> Gold.o <span class="kw">)</span></span>
 <span id="cb461-3"><a href="#cb461-3" aria-hidden="true" tabindex="-1"></a><span class="ex">Linking</span> Gold ...</span>
 <span id="cb461-4"><a href="#cb461-4" aria-hidden="true" tabindex="-1"></a><span class="ex">$</span> ./Gold</span>
 <span id="cb461-5"><a href="#cb461-5" aria-hidden="true" tabindex="-1"></a><span class="ex">0.0</span></span>
 <span id="cb461-6"><a href="#cb461-6" aria-hidden="true" tabindex="-1"></a><span class="ex">-1.0</span></span></code></pre></div>
-<h2 data-number="8.2" id="summary"><span class="header-section-number">8.2</span> Summary</h2>
-<p>So far, we’ve learned about Haskell’s syntax and types, quite a bit of functional programming and about some language features like type classes.</p>
-<p>We’ve also seen some type-oriented programming, and even gotten a taste of I/O in Haskell.</p>
-<p>Now you know how to write a real computer program in Haskell, but there’s still much to learn.</p>
-<h2 data-number="8.3" id="what-next"><span class="header-section-number">8.3</span> What Next?</h2>
-<p><a href="https://haskell.mooc.fi/part2">Part 2 of the course</a> is now out! Part 2 will cover topics like Monads, IO and how Haskell works under the hood. We’ll also get to do some real world programming with networks and databases. Oh and testing in Haskell is also covered.</p>
-<p>If you don’t feel like jumping into part 2 right now, here are some other Haskell resources you should be able to follow now:</p>
+<h2 data-number="8.2" id="summary"><span
+class="header-section-number">8.2</span> Summary</h2>
+<p>So far, we’ve learned about Haskell’s syntax and types, quite a bit
+of functional programming and about some language features like type
+classes.</p>
+<p>We’ve also seen some type-oriented programming, and even gotten a
+taste of I/O in Haskell.</p>
+<p>Now you know how to write a real computer program in Haskell, but
+there’s still much to learn.</p>
+<h2 data-number="8.3" id="what-next"><span
+class="header-section-number">8.3</span> What Next?</h2>
+<p><a href="https://haskell.mooc.fi/part2">Part 2 of the course</a> is
+now out! Part 2 will cover topics like Monads, IO and how Haskell works
+under the hood. We’ll also get to do some real world programming with
+networks and databases. Oh and testing in Haskell is also covered.</p>
+<p>If you don’t feel like jumping into part 2 right now, here are some
+other Haskell resources you should be able to follow now:</p>
 <ul>
-<li><a href="https://book.realworldhaskell.org/read/">Real World Haskell</a> - free e-book on advanced topics</li>
-<li><a href="https://web.archive.org/web/20220513191346/http://dev.stephendiehl.com/hask/">What I Wish I Knew When Learning Haskell</a></li>
-<li><a href="https://en.wikibooks.org/wiki/Haskell">The Haskell Wikibook</a></li>
-<li><a href="https://www.haskell.org/">The Haskell Website</a> has links to talks and presentations</li>
-<li><a href="https://kowainik.github.io/posts/haskell-mini-patterns">Haskell mini-patterns handbook</a> - more stuff like Lecture 7</li>
+<li><a href="https://book.realworldhaskell.org/read/">Real World
+Haskell</a> - free e-book on advanced topics</li>
+<li><a
+href="https://web.archive.org/web/20220513191346/http://dev.stephendiehl.com/hask/">What
+I Wish I Knew When Learning Haskell</a></li>
+<li><a href="https://en.wikibooks.org/wiki/Haskell">The Haskell
+Wikibook</a></li>
+<li><a href="https://www.haskell.org/">The Haskell Website</a> has links
+to talks and presentations</li>
+<li><a
+href="https://kowainik.github.io/posts/haskell-mini-patterns">Haskell
+mini-patterns handbook</a> - more stuff like Lecture 7</li>
 </ul>
-<p>I also recommend working on some programming problems in Haskell, like the ones from:</p>
+<p>I also recommend working on some programming problems in Haskell,
+like the ones from:</p>
 <ul>
-<li><a href="https://adventofcode.com/2019/events">Advent of Code</a> – nice varying puzzles, start off easy and get harder</li>
-<li><a href="https://www.spoj.com/">Sphere Online Judge</a> – algorithm problems, good variety of difficulties</li>
-<li><a href="https://projecteuler.net/">Project Euler</a> – mathematical programming puzzles</li>
+<li><a href="https://adventofcode.com/2019/events">Advent of Code</a> –
+nice varying puzzles, start off easy and get harder</li>
+<li><a href="https://www.spoj.com/">Sphere Online Judge</a> – algorithm
+problems, good variety of difficulties</li>
+<li><a href="https://projecteuler.net/">Project Euler</a> – mathematical
+programming puzzles</li>
 </ul>
-<p>You can also keep extending your final project and maybe generate some cool art in Haskell.</p>
-<p>In any case – thank you so much for tagging along, and we hope you have a great rest of the year!</p>
-<h2 data-number="8.4" id="final-project-graphics"><span class="header-section-number">8.4</span> Final Project: Graphics</h2>
-<p>Open up the exercise file <a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set8.hs"><code>Set8.hs</code></a> and follow the instructions there. Have fun!</p>
-<h2 data-number="8.5" id="acknowledgements"><span class="header-section-number">8.5</span> Acknowledgements</h2>
-<p>This course was made possible by <a href="https://nitor.com/en">Nitor</a> who donated hours and hours of Joel’s working time for this project. Thank you! Check out our <a href="https://www.nitor.com/en/jobs">open positions</a> if you’re interested in working somewhere that values continuous learning.</p>
+<p>You can also keep extending your final project and maybe generate
+some cool art in Haskell.</p>
+<p>In any case – thank you so much for tagging along, and we hope you
+have a great rest of the year!</p>
+<h2 data-number="8.4" id="final-project-graphics"><span
+class="header-section-number">8.4</span> Final Project: Graphics</h2>
+<p>Open up the exercise file <a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set8.hs"><code>Set8.hs</code></a>
+and follow the instructions there. Have fun!</p>
+<h2 data-number="8.5" id="acknowledgements"><span
+class="header-section-number">8.5</span> Acknowledgements</h2>
+<p>This course was made possible by <a
+href="https://nitor.com/en">Nitor</a> who donated hours and hours of
+Joel’s working time for this project. Thank you! Check out our <a
+href="https://www.nitor.com/en/jobs">open positions</a> if you’re
+interested in working somewhere that values continuous learning.</p>
 <p>Thanks to the whole Haskell Mooc team, especially</p>
 <ul>
 <li>John Lång for help on the material</li>
-<li>Antti Laaksonen for setting up the course and helping with arrangements</li>
+<li>Antti Laaksonen for setting up the course and helping with
+arrangements</li>
 </ul>
 </div>
 </div>
diff --git a/part2.html b/part2.html
index 88b625a..193690a 100644
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+++ b/part2.html
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@@ -85,106 +85,247 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 <div id="everything">
 <nav id="TOC">
 <ul>
-<li><a href="#lecture-9-recap-of-part-1"><span class="toc-section-number">9</span> Lecture 9: Recap of Part 1</a>
+<li><a href="#lecture-9-recap-of-part-1"
+id="toc-lecture-9-recap-of-part-1"><span
+class="toc-section-number">9</span> Lecture 9: Recap of Part 1</a>
 <ul>
-<li><a href="#types"><span class="toc-section-number">9.1</span> Types</a></li>
-<li><a href="#functions"><span class="toc-section-number">9.2</span> Functions</a></li>
-<li><a href="#functional-programming"><span class="toc-section-number">9.3</span> Functional Programming</a></li>
-<li><a href="#recursion"><span class="toc-section-number">9.4</span> Recursion</a></li>
-<li><a href="#type-classes"><span class="toc-section-number">9.5</span> Type Classes</a></li>
-<li><a href="#quiz"><span class="toc-section-number">9.6</span> Quiz</a></li>
-<li><a href="#working-on-the-exercises"><span class="toc-section-number">9.7</span> Working on the Exercises</a></li>
-<li><a href="#exercises"><span class="toc-section-number">9.8</span> Exercises</a></li>
+<li><a href="#types" id="toc-types"><span
+class="toc-section-number">9.1</span> Types</a></li>
+<li><a href="#functions" id="toc-functions"><span
+class="toc-section-number">9.2</span> Functions</a></li>
+<li><a href="#functional-programming"
+id="toc-functional-programming"><span
+class="toc-section-number">9.3</span> Functional Programming</a></li>
+<li><a href="#recursion" id="toc-recursion"><span
+class="toc-section-number">9.4</span> Recursion</a></li>
+<li><a href="#type-classes" id="toc-type-classes"><span
+class="toc-section-number">9.5</span> Type Classes</a></li>
+<li><a href="#quiz" id="toc-quiz"><span
+class="toc-section-number">9.6</span> Quiz</a></li>
+<li><a href="#working-on-the-exercises"
+id="toc-working-on-the-exercises"><span
+class="toc-section-number">9.7</span> Working on the Exercises</a></li>
+<li><a href="#exercises" id="toc-exercises"><span
+class="toc-section-number">9.8</span> Exercises</a></li>
 </ul></li>
-<li><a href="#lecture-10-reductionism"><span class="toc-section-number">10</span> Lecture 10: Reductionism</a>
+<li><a href="#lecture-10-reductionism"
+id="toc-lecture-10-reductionism"><span
+class="toc-section-number">10</span> Lecture 10: Reductionism</a>
 <ul>
-<li><a href="#laziness-purity"><span class="toc-section-number">10.1</span> Laziness &amp; Purity</a></li>
-<li><a href="#equational-reasoning"><span class="toc-section-number">10.2</span> Equational Reasoning</a></li>
-<li><a href="#infinite-lists"><span class="toc-section-number">10.3</span> Infinite Lists</a></li>
-<li><a href="#how-does-haskell-work"><span class="toc-section-number">10.4</span> How does Haskell Work?</a></li>
-<li><a href="#working-with-infinite-lists"><span class="toc-section-number">10.5</span> Working with Infinite Lists</a></li>
-<li><a href="#interlude-adding-strictness"><span class="toc-section-number">10.6</span> Interlude: Adding Strictness</a></li>
-<li><a href="#newtype-declarations"><span class="toc-section-number">10.7</span> Newtype Declarations</a></li>
-<li><a href="#something-fun-tying-the-knot"><span class="toc-section-number">10.8</span> Something Fun: Tying the Knot</a></li>
-<li><a href="#something-fun-debug.trace"><span class="toc-section-number">10.9</span> Something Fun: Debug.Trace</a></li>
-<li><a href="#quiz-1"><span class="toc-section-number">10.10</span> Quiz</a></li>
-<li><a href="#exercises-1"><span class="toc-section-number">10.11</span> Exercises</a></li>
+<li><a href="#laziness-purity" id="toc-laziness-purity"><span
+class="toc-section-number">10.1</span> Laziness &amp; Purity</a></li>
+<li><a href="#equational-reasoning" id="toc-equational-reasoning"><span
+class="toc-section-number">10.2</span> Equational Reasoning</a></li>
+<li><a href="#infinite-lists" id="toc-infinite-lists"><span
+class="toc-section-number">10.3</span> Infinite Lists</a></li>
+<li><a href="#how-does-haskell-work"
+id="toc-how-does-haskell-work"><span
+class="toc-section-number">10.4</span> How does Haskell Work?</a></li>
+<li><a href="#working-with-infinite-lists"
+id="toc-working-with-infinite-lists"><span
+class="toc-section-number">10.5</span> Working with Infinite
+Lists</a></li>
+<li><a href="#interlude-adding-strictness"
+id="toc-interlude-adding-strictness"><span
+class="toc-section-number">10.6</span> Interlude: Adding
+Strictness</a></li>
+<li><a href="#newtype-declarations" id="toc-newtype-declarations"><span
+class="toc-section-number">10.7</span> Newtype Declarations</a></li>
+<li><a href="#something-fun-tying-the-knot"
+id="toc-something-fun-tying-the-knot"><span
+class="toc-section-number">10.8</span> Something Fun: Tying the
+Knot</a></li>
+<li><a href="#something-fun-debug.trace"
+id="toc-something-fun-debug.trace"><span
+class="toc-section-number">10.9</span> Something Fun:
+Debug.Trace</a></li>
+<li><a href="#quiz-1" id="toc-quiz-1"><span
+class="toc-section-number">10.10</span> Quiz</a></li>
+<li><a href="#exercises-1" id="toc-exercises-1"><span
+class="toc-section-number">10.11</span> Exercises</a></li>
 </ul></li>
-<li><a href="#lecture-11-realworld---arealworld"><span class="toc-section-number">11</span> Lecture 11: <code>RealWorld -&gt; (a,RealWorld)</code></a>
+<li><a href="#lecture-11-realworld---arealworld"
+id="toc-lecture-11-realworld---arealworld"><span
+class="toc-section-number">11</span> Lecture 11:
+<code>RealWorld -&gt; (a,RealWorld)</code></a>
 <ul>
-<li><a href="#contents"><span class="toc-section-number">11.1</span> Contents</a></li>
-<li><a href="#youve-been-fooled"><span class="toc-section-number">11.2</span> You’ve Been Fooled!</a></li>
-<li><a href="#the-subtle-return"><span class="toc-section-number">11.3</span> The Subtle <code>return</code></a></li>
-<li><a href="#do-and-types"><span class="toc-section-number">11.4</span> <code>do</code> and Types</a></li>
-<li><a href="#control-structures"><span class="toc-section-number">11.5</span> Control Structures</a></li>
-<li><a href="#a-word-about-do-and-indentation"><span class="toc-section-number">11.6</span> A Word About <code>do</code> and Indentation</a></li>
-<li><a href="#lets-write-a-program"><span class="toc-section-number">11.7</span> Let’s Write a Program</a></li>
-<li><a href="#what-does-it-all-mean"><span class="toc-section-number">11.8</span> What Does It All Mean?</a></li>
-<li><a href="#one-more-thing-ioref"><span class="toc-section-number">11.9</span> One More Thing: IORef</a></li>
-<li><a href="#summary-of-io"><span class="toc-section-number">11.10</span> Summary of IO</a></li>
-<li><a href="#quiz-2"><span class="toc-section-number">11.11</span> Quiz</a></li>
-<li><a href="#exercises-2"><span class="toc-section-number">11.12</span> Exercises</a></li>
+<li><a href="#contents" id="toc-contents"><span
+class="toc-section-number">11.1</span> Contents</a></li>
+<li><a href="#youve-been-fooled" id="toc-youve-been-fooled"><span
+class="toc-section-number">11.2</span> You’ve Been Fooled!</a></li>
+<li><a href="#the-subtle-return" id="toc-the-subtle-return"><span
+class="toc-section-number">11.3</span> The Subtle
+<code>return</code></a></li>
+<li><a href="#do-and-types" id="toc-do-and-types"><span
+class="toc-section-number">11.4</span> <code>do</code> and
+Types</a></li>
+<li><a href="#control-structures" id="toc-control-structures"><span
+class="toc-section-number">11.5</span> Control Structures</a></li>
+<li><a href="#a-word-about-do-and-indentation"
+id="toc-a-word-about-do-and-indentation"><span
+class="toc-section-number">11.6</span> A Word About <code>do</code> and
+Indentation</a></li>
+<li><a href="#lets-write-a-program" id="toc-lets-write-a-program"><span
+class="toc-section-number">11.7</span> Let’s Write a Program</a></li>
+<li><a href="#what-does-it-all-mean"
+id="toc-what-does-it-all-mean"><span
+class="toc-section-number">11.8</span> What Does It All Mean?</a></li>
+<li><a href="#one-more-thing-ioref" id="toc-one-more-thing-ioref"><span
+class="toc-section-number">11.9</span> One More Thing: IORef</a></li>
+<li><a href="#summary-of-io" id="toc-summary-of-io"><span
+class="toc-section-number">11.10</span> Summary of IO</a></li>
+<li><a href="#quiz-2" id="toc-quiz-2"><span
+class="toc-section-number">11.11</span> Quiz</a></li>
+<li><a href="#exercises-2" id="toc-exercises-2"><span
+class="toc-section-number">11.12</span> Exercises</a></li>
 </ul></li>
-<li><a href="#lecture-12-fmap-fmap-fmap"><span class="toc-section-number">12</span> Lecture 12: fmap fmap fmap</a>
+<li><a href="#lecture-12-fmap-fmap-fmap"
+id="toc-lecture-12-fmap-fmap-fmap"><span
+class="toc-section-number">12</span> Lecture 12: fmap fmap fmap</a>
 <ul>
-<li><a href="#contents-1"><span class="toc-section-number">12.1</span> Contents</a></li>
-<li><a href="#functors"><span class="toc-section-number">12.2</span> Functors</a></li>
-<li><a href="#lawful-instances"><span class="toc-section-number">12.3</span> Lawful Instances</a></li>
-<li><a href="#sidenote-kinds"><span class="toc-section-number">12.4</span> Sidenote: Kinds</a></li>
-<li><a href="#foldable-again"><span class="toc-section-number">12.5</span> <code>Foldable</code>, Again</a></li>
-<li><a href="#recap"><span class="toc-section-number">12.6</span> Recap</a></li>
-<li><a href="#quiz-3"><span class="toc-section-number">12.7</span> Quiz</a></li>
-<li><a href="#exercises-3"><span class="toc-section-number">12.8</span> Exercises</a></li>
+<li><a href="#contents-1" id="toc-contents-1"><span
+class="toc-section-number">12.1</span> Contents</a></li>
+<li><a href="#functors" id="toc-functors"><span
+class="toc-section-number">12.2</span> Functors</a></li>
+<li><a href="#lawful-instances" id="toc-lawful-instances"><span
+class="toc-section-number">12.3</span> Lawful Instances</a></li>
+<li><a href="#sidenote-kinds" id="toc-sidenote-kinds"><span
+class="toc-section-number">12.4</span> Sidenote: Kinds</a></li>
+<li><a href="#foldable-again" id="toc-foldable-again"><span
+class="toc-section-number">12.5</span> <code>Foldable</code>,
+Again</a></li>
+<li><a href="#recap" id="toc-recap"><span
+class="toc-section-number">12.6</span> Recap</a></li>
+<li><a href="#quiz-3" id="toc-quiz-3"><span
+class="toc-section-number">12.7</span> Quiz</a></li>
+<li><a href="#exercises-3" id="toc-exercises-3"><span
+class="toc-section-number">12.8</span> Exercises</a></li>
 </ul></li>
-<li><a href="#lecture-13-a-monoid-in-the-category-of-problems"><span class="toc-section-number">13</span> Lecture 13: A Monoid in the Category of Problems</a>
+<li><a href="#lecture-13-a-monoid-in-the-category-of-problems"
+id="toc-lecture-13-a-monoid-in-the-category-of-problems"><span
+class="toc-section-number">13</span> Lecture 13: A Monoid in the
+Category of Problems</a>
 <ul>
-<li><a href="#example-1-maybes"><span class="toc-section-number">13.1</span> Example 1: Maybes</a></li>
-<li><a href="#example-2-logging"><span class="toc-section-number">13.2</span> Example 2: Logging</a></li>
-<li><a href="#example-3-keeping-state"><span class="toc-section-number">13.3</span> Example 3: Keeping State</a></li>
-<li><a href="#finally-the-monad-type-class"><span class="toc-section-number">13.4</span> Finally: The Monad Type Class</a></li>
-<li><a href="#maybe-is-a-monad"><span class="toc-section-number">13.5</span> Maybe is a Monad!</a></li>
-<li><a href="#the-return-of-do"><span class="toc-section-number">13.6</span> The Return of <code>do</code></a></li>
-<li><a href="#logger-is-a-monad"><span class="toc-section-number">13.7</span> Logger is a Monad!</a></li>
-<li><a href="#the-state-monad"><span class="toc-section-number">13.8</span> The State Monad</a></li>
-<li><a href="#the-return-of-mapm"><span class="toc-section-number">13.9</span> The Return of <code>mapM</code></a></li>
-<li><a href="#monads-are-functors"><span class="toc-section-number">13.10</span> Monads are Functors</a></li>
-<li><a href="#one-more-monad"><span class="toc-section-number">13.11</span> One More Monad</a></li>
-<li><a href="#oh-right-io"><span class="toc-section-number">13.12</span> Oh Right, IO</a></li>
-<li><a href="#monads-in-other-languages"><span class="toc-section-number">13.13</span> Monads in Other Languages</a></li>
-<li><a href="#monads-wrap-up"><span class="toc-section-number">13.14</span> Monads: Wrap-up</a></li>
-<li><a href="#sidenote-standard-haskell"><span class="toc-section-number">13.15</span> Sidenote: Standard Haskell</a></li>
-<li><a href="#quiz-4"><span class="toc-section-number">13.16</span> Quiz</a></li>
-<li><a href="#exercises-4"><span class="toc-section-number">13.17</span> Exercises</a></li>
+<li><a href="#example-1-maybes" id="toc-example-1-maybes"><span
+class="toc-section-number">13.1</span> Example 1: Maybes</a></li>
+<li><a href="#example-2-logging" id="toc-example-2-logging"><span
+class="toc-section-number">13.2</span> Example 2: Logging</a></li>
+<li><a href="#example-3-keeping-state"
+id="toc-example-3-keeping-state"><span
+class="toc-section-number">13.3</span> Example 3: Keeping State</a></li>
+<li><a href="#finally-the-monad-type-class"
+id="toc-finally-the-monad-type-class"><span
+class="toc-section-number">13.4</span> Finally: The Monad Type
+Class</a></li>
+<li><a href="#maybe-is-a-monad" id="toc-maybe-is-a-monad"><span
+class="toc-section-number">13.5</span> Maybe is a Monad!</a></li>
+<li><a href="#the-return-of-do" id="toc-the-return-of-do"><span
+class="toc-section-number">13.6</span> The Return of
+<code>do</code></a></li>
+<li><a href="#logger-is-a-monad" id="toc-logger-is-a-monad"><span
+class="toc-section-number">13.7</span> Logger is a Monad!</a></li>
+<li><a href="#the-state-monad" id="toc-the-state-monad"><span
+class="toc-section-number">13.8</span> The State Monad</a></li>
+<li><a href="#the-return-of-mapm" id="toc-the-return-of-mapm"><span
+class="toc-section-number">13.9</span> The Return of
+<code>mapM</code></a></li>
+<li><a href="#monads-are-functors" id="toc-monads-are-functors"><span
+class="toc-section-number">13.10</span> Monads are Functors</a></li>
+<li><a href="#one-more-monad" id="toc-one-more-monad"><span
+class="toc-section-number">13.11</span> One More Monad</a></li>
+<li><a href="#oh-right-io" id="toc-oh-right-io"><span
+class="toc-section-number">13.12</span> Oh Right, IO</a></li>
+<li><a href="#monads-in-other-languages"
+id="toc-monads-in-other-languages"><span
+class="toc-section-number">13.13</span> Monads in Other
+Languages</a></li>
+<li><a href="#monads-wrap-up" id="toc-monads-wrap-up"><span
+class="toc-section-number">13.14</span> Monads: Wrap-up</a></li>
+<li><a href="#sidenote-standard-haskell"
+id="toc-sidenote-standard-haskell"><span
+class="toc-section-number">13.15</span> Sidenote: Standard
+Haskell</a></li>
+<li><a href="#quiz-4" id="toc-quiz-4"><span
+class="toc-section-number">13.16</span> Quiz</a></li>
+<li><a href="#exercises-4" id="toc-exercises-4"><span
+class="toc-section-number">13.17</span> Exercises</a></li>
 </ul></li>
-<li><a href="#lecture-14-lets-use-some-libraries"><span class="toc-section-number">14</span> Lecture 14: Let’s Use Some Libraries!</a>
+<li><a href="#lecture-14-lets-use-some-libraries"
+id="toc-lecture-14-lets-use-some-libraries"><span
+class="toc-section-number">14</span> Lecture 14: Let’s Use Some
+Libraries!</a>
 <ul>
-<li><a href="#text-and-bytestring"><span class="toc-section-number">14.1</span> <code>Text</code> and <code>ByteString</code></a></li>
-<li><a href="#monads-recap"><span class="toc-section-number">14.2</span> Monads: Recap</a></li>
-<li><a href="#writing-a-http-server-wai-and-warp"><span class="toc-section-number">14.3</span> Writing a HTTP Server: WAI and Warp</a></li>
-<li><a href="#working-with-a-database-sqlite-simple"><span class="toc-section-number">14.4</span> Working With a Database: sqlite-simple</a></li>
-<li><a href="#exercises-5"><span class="toc-section-number">14.5</span> Exercises</a></li>
+<li><a href="#text-and-bytestring" id="toc-text-and-bytestring"><span
+class="toc-section-number">14.1</span> <code>Text</code> and
+<code>ByteString</code></a></li>
+<li><a href="#monads-recap" id="toc-monads-recap"><span
+class="toc-section-number">14.2</span> Monads: Recap</a></li>
+<li><a href="#writing-a-http-server-wai-and-warp"
+id="toc-writing-a-http-server-wai-and-warp"><span
+class="toc-section-number">14.3</span> Writing a HTTP Server: WAI and
+Warp</a></li>
+<li><a href="#working-with-a-database-sqlite-simple"
+id="toc-working-with-a-database-sqlite-simple"><span
+class="toc-section-number">14.4</span> Working With a Database:
+sqlite-simple</a></li>
+<li><a href="#exercises-5" id="toc-exercises-5"><span
+class="toc-section-number">14.5</span> Exercises</a></li>
 </ul></li>
-<li><a href="#lecture-15-youre-valid-even-without-monads"><span class="toc-section-number">15</span> Lecture 15: You’re Valid Even Without Monads</a>
+<li><a href="#lecture-15-youre-valid-even-without-monads"
+id="toc-lecture-15-youre-valid-even-without-monads"><span
+class="toc-section-number">15</span> Lecture 15: You’re Valid Even
+Without Monads</a>
 <ul>
-<li><a href="#introduction-to-applicatives"><span class="toc-section-number">15.1</span> Introduction to Applicatives</a></li>
-<li><a href="#the-list-applicative"><span class="toc-section-number">15.2</span> The List Applicative</a></li>
-<li><a href="#new-operators"><span class="toc-section-number">15.3</span> New Operators</a></li>
-<li><a href="#the-validation-applicative"><span class="toc-section-number">15.4</span> The Validation Applicative</a></li>
-<li><a href="#validating-lists-traverse"><span class="toc-section-number">15.5</span> Validating Lists: <code>traverse</code></a></li>
-<li><a href="#sidenote-traversable"><span class="toc-section-number">15.6</span> Sidenote: <code>Traversable</code></a></li>
-<li><a href="#dealing-with-failure-alternative"><span class="toc-section-number">15.7</span> Dealing with Failure: <code>Alternative</code></a></li>
-<li><a href="#sidenote-applicatives-in-context"><span class="toc-section-number">15.8</span> Sidenote: Applicatives in Context</a></li>
-<li><a href="#quiz-5"><span class="toc-section-number">15.9</span> Quiz</a></li>
-<li><a href="#exercises-6"><span class="toc-section-number">15.10</span> Exercises</a></li>
+<li><a href="#introduction-to-applicatives"
+id="toc-introduction-to-applicatives"><span
+class="toc-section-number">15.1</span> Introduction to
+Applicatives</a></li>
+<li><a href="#the-list-applicative" id="toc-the-list-applicative"><span
+class="toc-section-number">15.2</span> The List Applicative</a></li>
+<li><a href="#new-operators" id="toc-new-operators"><span
+class="toc-section-number">15.3</span> New Operators</a></li>
+<li><a href="#the-validation-applicative"
+id="toc-the-validation-applicative"><span
+class="toc-section-number">15.4</span> The Validation
+Applicative</a></li>
+<li><a href="#validating-lists-traverse"
+id="toc-validating-lists-traverse"><span
+class="toc-section-number">15.5</span> Validating Lists:
+<code>traverse</code></a></li>
+<li><a href="#sidenote-traversable" id="toc-sidenote-traversable"><span
+class="toc-section-number">15.6</span> Sidenote:
+<code>Traversable</code></a></li>
+<li><a href="#dealing-with-failure-alternative"
+id="toc-dealing-with-failure-alternative"><span
+class="toc-section-number">15.7</span> Dealing with Failure:
+<code>Alternative</code></a></li>
+<li><a href="#sidenote-applicatives-in-context"
+id="toc-sidenote-applicatives-in-context"><span
+class="toc-section-number">15.8</span> Sidenote: Applicatives in
+Context</a></li>
+<li><a href="#quiz-5" id="toc-quiz-5"><span
+class="toc-section-number">15.9</span> Quiz</a></li>
+<li><a href="#exercises-6" id="toc-exercises-6"><span
+class="toc-section-number">15.10</span> Exercises</a></li>
 </ul></li>
-<li><a href="#lecture-16-odds-and-ends"><span class="toc-section-number">16</span> Lecture 16: Odds and Ends</a>
+<li><a href="#lecture-16-odds-and-ends"
+id="toc-lecture-16-odds-and-ends"><span
+class="toc-section-number">16</span> Lecture 16: Odds and Ends</a>
 <ul>
-<li><a href="#testing-with-quickcheck"><span class="toc-section-number">16.1</span> Testing with QuickCheck</a></li>
-<li><a href="#phantom-types"><span class="toc-section-number">16.2</span> Phantom Types</a></li>
-<li><a href="#simultaneity"><span class="toc-section-number">16.3</span> Simultaneity</a></li>
-<li><a href="#exercises-7"><span class="toc-section-number">16.4</span> Exercises</a></li>
-<li><a href="#where-to-go-from-here"><span class="toc-section-number">16.5</span> Where to Go From Here?</a></li>
-<li><a href="#acknowledgements"><span class="toc-section-number">16.6</span> Acknowledgements</a></li>
+<li><a href="#testing-with-quickcheck"
+id="toc-testing-with-quickcheck"><span
+class="toc-section-number">16.1</span> Testing with QuickCheck</a></li>
+<li><a href="#phantom-types" id="toc-phantom-types"><span
+class="toc-section-number">16.2</span> Phantom Types</a></li>
+<li><a href="#simultaneity" id="toc-simultaneity"><span
+class="toc-section-number">16.3</span> Simultaneity</a></li>
+<li><a href="#exercises-7" id="toc-exercises-7"><span
+class="toc-section-number">16.4</span> Exercises</a></li>
+<li><a href="#where-to-go-from-here"
+id="toc-where-to-go-from-here"><span
+class="toc-section-number">16.5</span> Where to Go From Here?</a></li>
+<li><a href="#acknowledgements" id="toc-acknowledgements"><span
+class="toc-section-number">16.6</span> Acknowledgements</a></li>
 </ul></li>
 </ul>
 </nav>
@@ -194,12 +335,15 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 <header>
 <h1 class="title">Haskell Mooc, part 2</h1>
 </header>
-<p>by Joel Kaasinen (<a href="https://nitor.com/en">Nitor</a>) and John Lång (University of Helsinki)</p>
-<section id="lecture-9-recap-of-part-1" data-number="1">
-<h1 data-number="9"><span class="header-section-number">9</span> Lecture 9: Recap of Part 1</h1>
-<p>This lecture goes over the basic parts of Haskell introduced in part 1 of the course: types, values, pattern matching, functions and recursion.</p>
-<section id="types" data-number="1.1">
-<h2 data-number="9.1"><span class="header-section-number">9.1</span> Types</h2>
+<p>by Joel Kaasinen (<a href="https://nitor.com/en">Nitor</a>) and John
+Lång (University of Helsinki)</p>
+<h1 data-number="9" id="lecture-9-recap-of-part-1"><span
+class="header-section-number">9</span> Lecture 9: Recap of Part 1</h1>
+<p>This lecture goes over the basic parts of Haskell introduced in part
+1 of the course: types, values, pattern matching, functions and
+recursion.</p>
+<h2 data-number="9.1" id="types"><span
+class="header-section-number">9.1</span> Types</h2>
 <p>Remember the primitive types of Haskell? Here they are:</p>
 <table>
 <colgroup>
@@ -221,33 +365,41 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 <td style="text-align: left;">Truth values</td>
 </tr>
 <tr class="even">
-<td style="text-align: left;"><code>0</code>, <code>1</code>, <code>20</code>, <code>-37</code>, …</td>
+<td style="text-align: left;"><code>0</code>, <code>1</code>,
+<code>20</code>, <code>-37</code>, …</td>
 <td style="text-align: left;"><code>Int</code></td>
 <td style="text-align: left;">Whole numbers</td>
 </tr>
 <tr class="odd">
-<td style="text-align: left;"><code>'A'</code>, <code>'a'</code>, <code>'!'</code>, …</td>
+<td style="text-align: left;"><code>'A'</code>, <code>'a'</code>,
+<code>'!'</code>, …</td>
 <td style="text-align: left;"><code>Char</code></td>
 <td style="text-align: left;">Characters</td>
 </tr>
 <tr class="even">
-<td style="text-align: left;"><code>""</code>, <code>"abcd"</code>, …</td>
+<td style="text-align: left;"><code>""</code>, <code>"abcd"</code>,
+…</td>
 <td style="text-align: left;"><code>String</code></td>
-<td style="text-align: left;">Strings, which are actually just lists of characters, <code>[Char]</code></td>
+<td style="text-align: left;">Strings, which are actually just lists of
+characters, <code>[Char]</code></td>
 </tr>
 <tr class="odd">
-<td style="text-align: left;"><code>0.0</code>, <code>-3.2</code>, <code>12.3</code>, …</td>
+<td style="text-align: left;"><code>0.0</code>, <code>-3.2</code>,
+<code>12.3</code>, …</td>
 <td style="text-align: left;"><code>Double</code></td>
 <td style="text-align: left;">Floating-point numbers</td>
 </tr>
 <tr class="even">
 <td style="text-align: left;"><code>()</code></td>
 <td style="text-align: left;"><code>()</code></td>
-<td style="text-align: left;">The so-called unit type with only one value</td>
+<td style="text-align: left;">The so-called unit type with only one
+value</td>
 </tr>
 </tbody>
 </table>
-<p>It’s possible to combine these primitive types in various ways to form more complex types. Function types, tuple types and list types are examples of types that combine other types.</p>
+<p>It’s possible to combine these primitive types in various ways to
+form more complex types. Function types, tuple types and list types are
+examples of types that combine other types.</p>
 <table>
 <colgroup>
 <col style="width: 41%" />
@@ -263,29 +415,38 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 </thead>
 <tbody>
 <tr class="odd">
-<td style="text-align: left;"><code>(1,2)</code>, <code>(True,'c')</code>, …</td>
+<td style="text-align: left;"><code>(1,2)</code>,
+<code>(True,'c')</code>, …</td>
 <td style="text-align: left;"><code>(a, b)</code></td>
-<td style="text-align: left;">A pair of a value of type <code>a</code> and a value of type <code>b</code></td>
+<td style="text-align: left;">A pair of a value of type <code>a</code>
+and a value of type <code>b</code></td>
 </tr>
 <tr class="even">
-<td style="text-align: left;"><code>(1,2,3)</code>, <code>(1,2,'c')</code>, …</td>
+<td style="text-align: left;"><code>(1,2,3)</code>,
+<code>(1,2,'c')</code>, …</td>
 <td style="text-align: left;"><code>(a, b, c)</code></td>
-<td style="text-align: left;">A triple of values (of types <code>a</code>, <code>b</code> and <code>c</code>)</td>
+<td style="text-align: left;">A triple of values (of types
+<code>a</code>, <code>b</code> and <code>c</code>)</td>
 </tr>
 <tr class="odd">
-<td style="text-align: left;"><code>[]</code>, <code>[1,2,3,4]</code>, …</td>
+<td style="text-align: left;"><code>[]</code>, <code>[1,2,3,4]</code>,
+…</td>
 <td style="text-align: left;"><code>[a]</code></td>
 <td style="text-align: left;">List of values of type <code>a</code></td>
 </tr>
 <tr class="even">
-<td style="text-align: left;"><code>not</code>, <code>reverse</code>, <code>\x -&gt; 1</code>, <code>\x -&gt; x</code>, …</td>
+<td style="text-align: left;"><code>not</code>, <code>reverse</code>,
+<code>\x -&gt; 1</code>, <code>\x -&gt; x</code>, …</td>
 <td style="text-align: left;"><code>a -&gt; b</code></td>
-<td style="text-align: left;">Function from type <code>a</code> to type <code>b</code></td>
+<td style="text-align: left;">Function from type <code>a</code> to type
+<code>b</code></td>
 </tr>
 </tbody>
 </table>
-<p>There’s one more powerful mechanism for creating more types: <em>Algebraic datatypes</em> (ADTs). Some examples include:</p>
-<div class="sourceCode" id="cb1"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Enumeration types</span></span>
+<p>There’s one more powerful mechanism for creating more types:
+<em>Algebraic datatypes</em> (ADTs). Some examples include:</p>
+<div class="sourceCode" id="cb1"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Enumeration types</span></span>
 <span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Bool</span> <span class="ot">=</span> <span class="dt">True</span> <span class="op">|</span> <span class="dt">False</span></span>
 <span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Color</span> <span class="ot">=</span> <span class="dt">Red</span> <span class="op">|</span> <span class="dt">Green</span> <span class="op">|</span> <span class="dt">Blue</span></span>
 <span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -307,6 +468,10 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 <span id="cb1-20"><a href="#cb1-20" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">MultiTree</span> a <span class="ot">=</span> <span class="dt">MultiTree</span> a [<span class="dt">MultiTree</span> a]     <span class="co">-- Note the list</span></span></code></pre></div>
 <p>Values of these types include:</p>
 <table>
+<colgroup>
+<col style="width: 82%" />
+<col style="width: 17%" />
+</colgroup>
 <thead>
 <tr class="header">
 <th style="text-align: left;">Values</th>
@@ -319,7 +484,8 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 <td style="text-align: left;"><code>Bool</code></td>
 </tr>
 <tr class="even">
-<td style="text-align: left;"><code>Red</code>, <code>Green</code>, <code>Blue</code></td>
+<td style="text-align: left;"><code>Red</code>, <code>Green</code>,
+<code>Blue</code></td>
 <td style="text-align: left;"><code>Color</code></td>
 </tr>
 <tr class="odd">
@@ -335,44 +501,69 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 <td style="text-align: left;"><code>PairOf Int</code></td>
 </tr>
 <tr class="even">
-<td style="text-align: left;"><code>Empty</code>, <code>Node 3 (Node 4 Empty)</code></td>
+<td style="text-align: left;"><code>Empty</code>,
+<code>Node 3 (Node 4 Empty)</code></td>
 <td style="text-align: left;"><code>IntList</code></td>
 </tr>
 <tr class="odd">
-<td style="text-align: left;"><code>Nothing</code>, <code>Just 3</code>, <code>Just 4</code>, …</td>
+<td style="text-align: left;"><code>Nothing</code>, <code>Just 3</code>,
+<code>Just 4</code>, …</td>
 <td style="text-align: left;"><code>Maybe Int</code></td>
 </tr>
 <tr class="even">
-<td style="text-align: left;"><code>Nothing</code>, <code>Just 'c'</code>, <code>Just 'd'</code>, …</td>
+<td style="text-align: left;"><code>Nothing</code>,
+<code>Just 'c'</code>, <code>Just 'd'</code>, …</td>
 <td style="text-align: left;"><code>Maybe Char</code></td>
 </tr>
 <tr class="odd">
-<td style="text-align: left;"><code>Left "foo"</code>, <code>Right 13</code>, …</td>
+<td style="text-align: left;"><code>Left "foo"</code>,
+<code>Right 13</code>, …</td>
 <td style="text-align: left;"><code>Either String Int</code></td>
 </tr>
 <tr class="even">
-<td style="text-align: left;"><code>Nil</code>, <code>Cons True Nil</code>, <code>Cons True (Cons False Nil)</code> …</td>
+<td style="text-align: left;"><code>Nil</code>,
+<code>Cons True Nil</code>, <code>Cons True (Cons False Nil)</code>
+…</td>
 <td style="text-align: left;"><code>List Bool</code></td>
 </tr>
 <tr class="odd">
-<td style="text-align: left;"><code>Leaf 7</code>, <code>Node 1 (Leaf 0) (Leaf 2)</code>, …</td>
+<td style="text-align: left;"><code>Leaf 7</code>,
+<code>Node 1 (Leaf 0) (Leaf 2)</code>, …</td>
 <td style="text-align: left;"><code>Tree Int</code></td>
 </tr>
 <tr class="even">
-<td style="text-align: left;"><code>MultiTree 'a' [MultiTree 'b' [], MultiTree 'c' []]]</code>, …</td>
+<td
+style="text-align: left;"><code>MultiTree 'a' [MultiTree 'b' [], MultiTree 'c' []]]</code>,
+…</td>
 <td style="text-align: left;"><code>MultiTree Char</code></td>
 </tr>
 </tbody>
 </table>
-<p>You can combine parameterized types in complex ways, for example with something like <code>Either [String-&gt;String] (Maybe String, Int)</code>.</p>
-<p>The names of concrete types start with capital letters. Lowercase letters are used for <em>type variables</em> which indicate <em>parametric polymorphism</em>: functions and values that can have multiple types. Here are some examples of polymorphic function types:</p>
-<div class="sourceCode" id="cb2"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a>[a] <span class="ot">-&gt;</span> [a]    <span class="co">-- function from list of any type, to list of the same type</span></span>
+<p>You can combine parameterized types in complex ways, for example with
+something like
+<code>Either [String-&gt;String] (Maybe String, Int)</code>.</p>
+<p>The names of concrete types start with capital letters. Lowercase
+letters are used for <em>type variables</em> which indicate
+<em>parametric polymorphism</em>: functions and values that can have
+multiple types. Here are some examples of polymorphic function
+types:</p>
+<div class="sourceCode" id="cb2"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a>[a] <span class="ot">-&gt;</span> [a]    <span class="co">-- function from list of any type, to list of the same type</span></span>
 <span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a>[a] <span class="ot">-&gt;</span> a      <span class="co">-- function from list of any type, to the element type</span></span>
 <span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a>(a,b) <span class="ot">-&gt;</span> [a]  <span class="co">-- function from tuple to list</span></span></code></pre></div>
-<section id="more-about-lists" data-number="1.1.1">
-<h3 data-number="9.1.1"><span class="header-section-number">9.1.1</span> More About Lists</h3>
-<p>List literals can be written in using the familiar <code>[x,y,z]</code> syntax. However, that notation is just a shorthand as lists are actually built up of the list constructors <code>[]</code> and <code>(:)</code>. These constructors are also used when pattern matching lists. Here are some examples of lists:</p>
+<h3 data-number="9.1.1" id="more-about-lists"><span
+class="header-section-number">9.1.1</span> More About Lists</h3>
+<p>List literals can be written in using the familiar
+<code>[x,y,z]</code> syntax. However, that notation is just a shorthand
+as lists are actually built up of the list constructors <code>[]</code>
+and <code>(:)</code>. These constructors are also used when pattern
+matching lists. Here are some examples of lists:</p>
 <table>
+<colgroup>
+<col style="width: 34%" />
+<col style="width: 56%" />
+<col style="width: 9%" />
+</colgroup>
 <thead>
 <tr class="header">
 <th style="text-align: left;">Abbreviation</th>
@@ -394,12 +585,17 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 <tr class="odd">
 <td style="text-align: left;"><code>"foo"</code></td>
 <td style="text-align: left;"><code>'f':'o':'o':[]</code></td>
-<td style="text-align: left;"><code>[Char]</code>, also known as <code>String</code></td>
+<td style="text-align: left;"><code>[Char]</code>, also known as
+<code>String</code></td>
 </tr>
 </tbody>
 </table>
 <p>There’s also a range syntax for lists:</p>
 <table>
+<colgroup>
+<col style="width: 23%" />
+<col style="width: 76%" />
+</colgroup>
 <thead>
 <tr class="header">
 <th style="text-align: left;">Range</th>
@@ -409,7 +605,8 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 <tbody>
 <tr class="odd">
 <td style="text-align: left;"><code>['a' .. 'z']</code></td>
-<td style="text-align: left;"><code>"abcdefghijklmnopqrstuvwxyz"</code></td>
+<td
+style="text-align: left;"><code>"abcdefghijklmnopqrstuvwxyz"</code></td>
 </tr>
 <tr class="even">
 <td style="text-align: left;"><code>[0 .. 9]</code></td>
@@ -421,7 +618,8 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 </tr>
 <tr class="even">
 <td style="text-align: left;"><code>[x .. y]</code></td>
-<td style="text-align: left;">everything from <code>x</code> to <code>y</code></td>
+<td style="text-align: left;">everything from <code>x</code> to
+<code>y</code></td>
 </tr>
 <tr class="odd">
 <td style="text-align: left;"><code>[9 .. 3]</code></td>
@@ -429,7 +627,8 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 </tr>
 <tr class="even">
 <td style="text-align: left;"><code>[y, y-1 .. x]</code></td>
-<td style="text-align: left;">everything from <code>y</code> to <code>x</code> in decreasing order</td>
+<td style="text-align: left;">everything from <code>y</code> to
+<code>x</code> in decreasing order</td>
 </tr>
 <tr class="odd">
 <td style="text-align: left;"><code>[9,8 .. 3]</code></td>
@@ -437,7 +636,8 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 </tr>
 </tbody>
 </table>
-<p><em>List comprehensions</em> are another powerful way to create lists:</p>
+<p><em>List comprehensions</em> are another powerful way to create
+lists:</p>
 <table>
 <thead>
 <tr class="header">
@@ -451,36 +651,52 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 <td style="text-align: left;"><code>[1,8,27]</code></td>
 </tr>
 <tr class="even">
-<td style="text-align: left;"><code>[x^2 + y^2 | x &lt;- [1..3], y &lt;- [1..2]]</code></td>
+<td
+style="text-align: left;"><code>[x^2 + y^2 | x &lt;- [1..3], y &lt;- [1..2]]</code></td>
 <td style="text-align: left;"><code>[2,5,5,8,10,13]</code></td>
 </tr>
 <tr class="odd">
-<td style="text-align: left;"><code>[y | x &lt;- [1..10], let y = x^2, even x, y&lt;50]</code></td>
+<td
+style="text-align: left;"><code>[y | x &lt;- [1..10], let y = x^2, even x, y&lt;50]</code></td>
 <td style="text-align: left;"><code>[4,16,36]</code></td>
 </tr>
 <tr class="even">
-<td style="text-align: left;"><code>[c | c &lt;- "Hello, World!", elem c ['a'..'z']]</code></td>
+<td
+style="text-align: left;"><code>[c | c &lt;- "Hello, World!", elem c ['a'..'z']]</code></td>
 <td style="text-align: left;"><code>"elloorld"</code></td>
 </tr>
 </tbody>
 </table>
-<p>In general, <code>[f x | x &lt;- xs, p x]</code> is the same as <code>map f (filter p xs)</code>. Also, <code>[y | x &lt;- xs, let y = f x]</code> is the same as <code>[f x | x &lt;- xs]</code>. Any combination of <code>&lt;-</code>, <code>let</code> and <code>[f x | ...]</code> is possible.</p>
-<p>Just one more note on the syntax. Recall that <code>(:)</code> associates to the right, e.g. <code>True:False:[]</code> is the same as <code>True:(False:[])</code>. (In fact, <code>(True:False):[]</code> is not even a list, because <code>True:False</code> attempts to add <code>True</code> in front of <code>False</code> which is not a list.)</p>
-</section>
-</section>
-<section id="functions" data-number="1.2">
-<h2 data-number="9.2"><span class="header-section-number">9.2</span> Functions</h2>
+<p>In general, <code>[f x | x &lt;- xs, p x]</code> is the same as
+<code>map f (filter p xs)</code>. Also,
+<code>[y | x &lt;- xs, let y = f x]</code> is the same as
+<code>[f x | x &lt;- xs]</code>. Any combination of <code>&lt;-</code>,
+<code>let</code> and <code>[f x | ...]</code> is possible.</p>
+<p>Just one more note on the syntax. Recall that <code>(:)</code>
+associates to the right, e.g. <code>True:False:[]</code> is the same as
+<code>True:(False:[])</code>. (In fact, <code>(True:False):[]</code> is
+not even a list, because <code>True:False</code> attempts to add
+<code>True</code> in front of <code>False</code> which is not a
+list.)</p>
+<h2 data-number="9.2" id="functions"><span
+class="header-section-number">9.2</span> Functions</h2>
 <p>The basic form of function definition is:</p>
-<div class="sourceCode" id="cb3"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="ot">functionName ::</span> argumentType <span class="ot">-&gt;</span> returnType</span>
+<div class="sourceCode" id="cb3"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="ot">functionName ::</span> argumentType <span class="ot">-&gt;</span> returnType</span>
 <span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a>functionName argument <span class="ot">=</span> returnValue</span></code></pre></div>
 <p>For example:</p>
-<div class="sourceCode" id="cb4"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="ot">repeatString ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<div class="sourceCode" id="cb4"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="ot">repeatString ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a>repeatString s <span class="ot">=</span> s <span class="op">++</span> s</span></code></pre></div>
-<p>Functions taking multiple arguments are defined in a similar manner. Note how the type of a multi-argument function looks like.</p>
-<div class="sourceCode" id="cb5"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="ot">surroundString ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<p>Functions taking multiple arguments are defined in a similar manner.
+Note how the type of a multi-argument function looks like.</p>
+<div class="sourceCode" id="cb5"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="ot">surroundString ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a>surroundString around s <span class="ot">=</span> around <span class="op">++</span> s <span class="op">++</span> around</span></code></pre></div>
-<p>Functions can be polymorphic, can take multiple arguments, and can even take functions as arguments. Here are more examples:</p>
-<div class="sourceCode" id="cb6"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="fu">id</span><span class="ot"> ::</span> a <span class="ot">-&gt;</span> a</span>
+<p>Functions can be polymorphic, can take multiple arguments, and can
+even take functions as arguments. Here are more examples:</p>
+<div class="sourceCode" id="cb6"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="fu">id</span><span class="ot"> ::</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a><span class="fu">id</span> x <span class="ot">=</span> x</span>
 <span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb6-4"><a href="#cb6-4" aria-hidden="true" tabindex="-1"></a><span class="fu">const</span><span class="ot"> ::</span> a <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> a</span>
@@ -488,8 +704,12 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 <span id="cb6-6"><a href="#cb6-6" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb6-7"><a href="#cb6-7" aria-hidden="true" tabindex="-1"></a><span class="fu">flip</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> c) <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> c</span>
 <span id="cb6-8"><a href="#cb6-8" aria-hidden="true" tabindex="-1"></a><span class="fu">flip</span> f x y <span class="ot">=</span> f y x</span></code></pre></div>
-<p>More sophisticated functions can be defined using <em>pattern matching</em>. We can pattern match on the <em>constructors</em> of algebraic datatypes like <code>Maybe</code>, and also lists with the constructors <code>[]</code> and <code>(:)</code>.</p>
-<div class="sourceCode" id="cb7"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="ot">swap ::</span> (a,b) <span class="ot">-&gt;</span> (b,a)</span>
+<p>More sophisticated functions can be defined using <em>pattern
+matching</em>. We can pattern match on the <em>constructors</em> of
+algebraic datatypes like <code>Maybe</code>, and also lists with the
+constructors <code>[]</code> and <code>(:)</code>.</p>
+<div class="sourceCode" id="cb7"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="ot">swap ::</span> (a,b) <span class="ot">-&gt;</span> (b,a)</span>
 <span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a>swap (x,y) <span class="ot">=</span> (y,x)</span>
 <span id="cb7-3"><a href="#cb7-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb7-4"><a href="#cb7-4" aria-hidden="true" tabindex="-1"></a><span class="fu">maybe</span><span class="ot"> ::</span> b <span class="ot">-&gt;</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> <span class="dt">Maybe</span> a <span class="ot">-&gt;</span> b</span>
@@ -499,8 +719,13 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 <span id="cb7-8"><a href="#cb7-8" aria-hidden="true" tabindex="-1"></a><span class="ot">safeHead ::</span> [a] <span class="ot">-&gt;</span> <span class="dt">Maybe</span> a</span>
 <span id="cb7-9"><a href="#cb7-9" aria-hidden="true" tabindex="-1"></a>safeHead []    <span class="ot">=</span> <span class="dt">Nothing</span></span>
 <span id="cb7-10"><a href="#cb7-10" aria-hidden="true" tabindex="-1"></a>safeHead (x<span class="op">:</span>_) <span class="ot">=</span> <span class="dt">Just</span> x</span></code></pre></div>
-<p>Yet more sophistication can be achieved with <em>guards</em>. Guards let you define a function case-by-case based on tests of type <code>Bool</code>. Guards are useful in situations where pattern matching can’t be used. Of course, guards can also be combined with pattern matching:</p>
-<div class="sourceCode" id="cb8"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a><span class="ot">myAbs ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<p>Yet more sophistication can be achieved with <em>guards</em>. Guards
+let you define a function case-by-case based on tests of type
+<code>Bool</code>. Guards are useful in situations where pattern
+matching can’t be used. Of course, guards can also be combined with
+pattern matching:</p>
+<div class="sourceCode" id="cb8"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a><span class="ot">myAbs ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb8-2"><a href="#cb8-2" aria-hidden="true" tabindex="-1"></a>myAbs x</span>
 <span id="cb8-3"><a href="#cb8-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> x <span class="op">&lt;</span> <span class="dv">0</span>     <span class="ot">=</span> <span class="op">-</span>x</span>
 <span id="cb8-4"><a href="#cb8-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> <span class="fu">otherwise</span> <span class="ot">=</span> x</span>
@@ -515,13 +740,19 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 <span id="cb8-13"><a href="#cb8-13" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> money <span class="op">&lt;</span> <span class="fl">3.2</span>   <span class="ot">=</span> <span class="st">&quot;You don&#39;t have enough money for a banana&quot;</span></span>
 <span id="cb8-14"><a href="#cb8-14" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> <span class="fu">otherwise</span>     <span class="ot">=</span> <span class="st">&quot;You bought a banana&quot;</span></span>
 <span id="cb8-15"><a href="#cb8-15" aria-hidden="true" tabindex="-1"></a>buy <span class="fu">product</span>  _    <span class="ot">=</span> <span class="st">&quot;No such product: &quot;</span> <span class="op">++</span> <span class="fu">product</span></span></code></pre></div>
-<p><em>Case expressions</em> let us pattern match inside functions. They are useful in situations where the result of one function depends on the result of another and we want to match the pattern on the output of the other function:</p>
-<div class="sourceCode" id="cb9"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a><span class="ot">divDefault ::</span> <span class="dt">Double</span> <span class="ot">-&gt;</span> <span class="dt">Double</span> <span class="ot">-&gt;</span> <span class="dt">Double</span> <span class="ot">-&gt;</span> <span class="dt">Double</span></span>
+<p><em>Case expressions</em> let us pattern match inside functions. They
+are useful in situations where the result of one function depends on the
+result of another and we want to match the pattern on the output of the
+other function:</p>
+<div class="sourceCode" id="cb9"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a><span class="ot">divDefault ::</span> <span class="dt">Double</span> <span class="ot">-&gt;</span> <span class="dt">Double</span> <span class="ot">-&gt;</span> <span class="dt">Double</span> <span class="ot">-&gt;</span> <span class="dt">Double</span></span>
 <span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a>divDefault x y def <span class="ot">=</span> <span class="kw">case</span> safeDiv x y <span class="kw">of</span></span>
 <span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a>  <span class="dt">Nothing</span> <span class="ot">-&gt;</span> def</span>
 <span id="cb9-4"><a href="#cb9-4" aria-hidden="true" tabindex="-1"></a>  <span class="dt">Just</span> w  <span class="ot">-&gt;</span> w</span></code></pre></div>
-<p>Let-expressions enable <em>local definitions</em>. Where-clauses work similarly to <code>let</code>s. For example:</p>
-<div class="sourceCode" id="cb10"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a><span class="ot">circleArea ::</span> <span class="dt">Double</span> <span class="ot">-&gt;</span> <span class="dt">Double</span></span>
+<p>Let-expressions enable <em>local definitions</em>. Where-clauses work
+similarly to <code>let</code>s. For example:</p>
+<div class="sourceCode" id="cb10"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a><span class="ot">circleArea ::</span> <span class="dt">Double</span> <span class="ot">-&gt;</span> <span class="dt">Double</span></span>
 <span id="cb10-2"><a href="#cb10-2" aria-hidden="true" tabindex="-1"></a>circleArea r <span class="ot">=</span> <span class="kw">let</span> <span class="fu">pi</span> <span class="ot">=</span> <span class="fl">3.1415926</span></span>
 <span id="cb10-3"><a href="#cb10-3" aria-hidden="true" tabindex="-1"></a>                   square x <span class="ot">=</span> x <span class="op">*</span> x</span>
 <span id="cb10-4"><a href="#cb10-4" aria-hidden="true" tabindex="-1"></a>               <span class="kw">in</span> <span class="fu">pi</span> <span class="op">*</span> square r</span>
@@ -530,19 +761,37 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 <span id="cb10-7"><a href="#cb10-7" aria-hidden="true" tabindex="-1"></a>circleArea&#39; r <span class="ot">=</span> <span class="fu">pi</span> <span class="op">*</span> square r</span>
 <span id="cb10-8"><a href="#cb10-8" aria-hidden="true" tabindex="-1"></a>    <span class="kw">where</span> <span class="fu">pi</span> <span class="ot">=</span> <span class="fl">3.1415926</span></span>
 <span id="cb10-9"><a href="#cb10-9" aria-hidden="true" tabindex="-1"></a>          square x <span class="ot">=</span> x <span class="op">*</span> x</span></code></pre></div>
-<p><em>Lambda expressions</em> are another occasionally useful syntax for defining functions. Lambda expressions represent anonymous (unnamed) functions. They can be used for defining local functions that are typically used only once.</p>
-<div class="sourceCode" id="cb11"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a><span class="ot">incrementAll ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> [<span class="dt">Int</span>]</span>
+<p><em>Lambda expressions</em> are another occasionally useful syntax
+for defining functions. Lambda expressions represent anonymous (unnamed)
+functions. They can be used for defining local functions that are
+typically used only once.</p>
+<div class="sourceCode" id="cb11"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a><span class="ot">incrementAll ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> [<span class="dt">Int</span>]</span>
 <span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a>incrementAll xs <span class="ot">=</span> <span class="fu">map</span> (\x <span class="ot">-&gt;</span> x <span class="op">+</span> <span class="dv">1</span>) xs</span></code></pre></div>
-<p>Note that <code>f x = y</code> is the same thing as <code>f = \x -&gt; y</code>.</p>
-<p>Finally, binary operators have <em>sections</em>. Sections are partially applied operators. The section of an operator is obtained by writing the operator and one of its arguments in parentheses. For example, <code>(*2)</code> multiplies its argument by <code>2</code> from the right, e.g. <code>(*2) 5 ==&gt; 5 * 2</code>. A fractional number (e.g. a <code>Double</code>) can be inverted with the section <code>(1/)</code>, e.g. <code>(1/) 2 ==&gt; 0.5</code>.</p>
-<div class="sourceCode" id="cb12"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a><span class="ot">incrementAll&#39; ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> [<span class="dt">Int</span>]</span>
+<p>Note that <code>f x = y</code> is the same thing as
+<code>f = \x -&gt; y</code>.</p>
+<p>Finally, binary operators have <em>sections</em>. Sections are
+partially applied operators. The section of an operator is obtained by
+writing the operator and one of its arguments in parentheses. For
+example, <code>(*2)</code> multiplies its argument by <code>2</code>
+from the right, e.g. <code>(*2) 5 ==&gt; 5 * 2</code>. A fractional
+number (e.g. a <code>Double</code>) can be inverted with the section
+<code>(1/)</code>, e.g. <code>(1/) 2 ==&gt; 0.5</code>.</p>
+<div class="sourceCode" id="cb12"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a><span class="ot">incrementAll&#39; ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> [<span class="dt">Int</span>]</span>
 <span id="cb12-2"><a href="#cb12-2" aria-hidden="true" tabindex="-1"></a>incrementAll&#39; xs <span class="ot">=</span> <span class="fu">map</span> (<span class="op">+</span><span class="dv">1</span>) xs</span></code></pre></div>
-</section>
-<section id="functional-programming" data-number="1.3">
-<h2 data-number="9.3"><span class="header-section-number">9.3</span> Functional Programming</h2>
-<p>Haskell is a functional programming language, which means that functions can be passed in as arguments and returned from functions. As a programming paradigm, functional programming aims to build programs by combining simple functions together to form larger and larger ones.</p>
-<p>The most often presented example of functional programming is functional list manipulation using <em>higher-order functions</em> (functions that take functions as arguments) like <code>map</code> and <code>filter</code>. Here’s one example from part 1:</p>
-<div class="sourceCode" id="cb13"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- a predicate that checks if a string is a palindrome</span></span>
+<h2 data-number="9.3" id="functional-programming"><span
+class="header-section-number">9.3</span> Functional Programming</h2>
+<p>Haskell is a functional programming language, which means that
+functions can be passed in as arguments and returned from functions. As
+a programming paradigm, functional programming aims to build programs by
+combining simple functions together to form larger and larger ones.</p>
+<p>The most often presented example of functional programming is
+functional list manipulation using <em>higher-order functions</em>
+(functions that take functions as arguments) like <code>map</code> and
+<code>filter</code>. Here’s one example from part 1:</p>
+<div class="sourceCode" id="cb13"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- a predicate that checks if a string is a palindrome</span></span>
 <span id="cb13-2"><a href="#cb13-2" aria-hidden="true" tabindex="-1"></a><span class="ot">palindrome ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb13-3"><a href="#cb13-3" aria-hidden="true" tabindex="-1"></a>palindrome str <span class="ot">=</span> str <span class="op">==</span> <span class="fu">reverse</span> str</span>
 <span id="cb13-4"><a href="#cb13-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -550,27 +799,34 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 <span id="cb13-6"><a href="#cb13-6" aria-hidden="true" tabindex="-1"></a><span class="co">-- strings using show, and keeps only palindromes</span></span>
 <span id="cb13-7"><a href="#cb13-7" aria-hidden="true" tabindex="-1"></a><span class="ot">palindromes ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> [<span class="dt">String</span>]</span>
 <span id="cb13-8"><a href="#cb13-8" aria-hidden="true" tabindex="-1"></a>palindromes n <span class="ot">=</span> <span class="fu">filter</span> palindrome (<span class="fu">map</span> <span class="fu">show</span> [<span class="dv">1</span><span class="op">..</span>n])</span></code></pre></div>
-<div class="sourceCode" id="cb14"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a>palindromes <span class="dv">150</span></span>
+<div class="sourceCode" id="cb14"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a>palindromes <span class="dv">150</span></span>
 <span id="cb14-2"><a href="#cb14-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="st">&quot;1&quot;</span>,<span class="st">&quot;2&quot;</span>,<span class="st">&quot;3&quot;</span>,<span class="st">&quot;4&quot;</span>,<span class="st">&quot;5&quot;</span>,<span class="st">&quot;6&quot;</span>,<span class="st">&quot;7&quot;</span>,<span class="st">&quot;8&quot;</span>,<span class="st">&quot;9&quot;</span>,</span>
 <span id="cb14-3"><a href="#cb14-3" aria-hidden="true" tabindex="-1"></a>       <span class="st">&quot;11&quot;</span>,<span class="st">&quot;22&quot;</span>,<span class="st">&quot;33&quot;</span>,<span class="st">&quot;44&quot;</span>,<span class="st">&quot;55&quot;</span>,<span class="st">&quot;66&quot;</span>,<span class="st">&quot;77&quot;</span>,<span class="st">&quot;88&quot;</span>,<span class="st">&quot;99&quot;</span>,</span>
 <span id="cb14-4"><a href="#cb14-4" aria-hidden="true" tabindex="-1"></a>       <span class="st">&quot;101&quot;</span>,<span class="st">&quot;111&quot;</span>,<span class="st">&quot;121&quot;</span>,<span class="st">&quot;131&quot;</span>,<span class="st">&quot;141&quot;</span>]</span></code></pre></div>
-<p>We also encountered other functional programming patterns in part 1, like <em>partial application</em>:</p>
-<div class="sourceCode" id="cb15"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> (<span class="fu">take</span> <span class="dv">3</span>) [[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>,<span class="dv">5</span>],[<span class="dv">6</span>,<span class="dv">7</span>,<span class="dv">8</span>,<span class="dv">9</span>,<span class="dv">0</span>]]</span>
+<p>We also encountered other functional programming patterns in part 1,
+like <em>partial application</em>:</p>
+<div class="sourceCode" id="cb15"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> (<span class="fu">take</span> <span class="dv">3</span>) [[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>,<span class="dv">5</span>],[<span class="dv">6</span>,<span class="dv">7</span>,<span class="dv">8</span>,<span class="dv">9</span>,<span class="dv">0</span>]]</span>
 <span id="cb15-2"><a href="#cb15-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>],[<span class="dv">6</span>,<span class="dv">7</span>,<span class="dv">8</span>]]</span></code></pre></div>
 <p>Also, <em>function composition</em>:</p>
-<div class="sourceCode" id="cb16"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb16-1"><a href="#cb16-1" aria-hidden="true" tabindex="-1"></a>(<span class="fu">map</span> <span class="fu">reverse</span> <span class="op">.</span> <span class="fu">filter</span> (<span class="op">/=</span><span class="st">&quot;Smith&quot;</span>)) [<span class="st">&quot;Jones&quot;</span>,<span class="st">&quot;Smith&quot;</span>,<span class="st">&quot;White&quot;</span>]</span>
+<div class="sourceCode" id="cb16"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb16-1"><a href="#cb16-1" aria-hidden="true" tabindex="-1"></a>(<span class="fu">map</span> <span class="fu">reverse</span> <span class="op">.</span> <span class="fu">filter</span> (<span class="op">/=</span><span class="st">&quot;Smith&quot;</span>)) [<span class="st">&quot;Jones&quot;</span>,<span class="st">&quot;Smith&quot;</span>,<span class="st">&quot;White&quot;</span>]</span>
 <span id="cb16-2"><a href="#cb16-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="st">&quot;senoJ&quot;</span>,<span class="st">&quot;etihW&quot;</span>]</span>
 <span id="cb16-3"><a href="#cb16-3" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> (<span class="fu">negate</span> <span class="op">.</span> <span class="fu">sum</span>) [[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>],[<span class="dv">5</span>]]</span>
 <span id="cb16-4"><a href="#cb16-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="op">-</span><span class="dv">6</span>,<span class="op">-</span><span class="dv">5</span>]</span></code></pre></div>
 <p>And finally, <em>folds</em>:</p>
-<div class="sourceCode" id="cb17"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a><span class="fu">foldr</span> (<span class="op">*</span>) <span class="dv">1</span> [<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>]  <span class="op">==&gt;</span> <span class="dv">24</span></span>
+<div class="sourceCode" id="cb17"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a><span class="fu">foldr</span> (<span class="op">*</span>) <span class="dv">1</span> [<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>]  <span class="op">==&gt;</span> <span class="dv">24</span></span>
 <span id="cb17-2"><a href="#cb17-2" aria-hidden="true" tabindex="-1"></a><span class="fu">foldr</span> <span class="fu">max</span> <span class="dv">0</span> [<span class="dv">1</span>,<span class="dv">3</span>,<span class="dv">7</span>]  <span class="op">==&gt;</span> <span class="dv">7</span></span>
 <span id="cb17-3"><a href="#cb17-3" aria-hidden="true" tabindex="-1"></a><span class="fu">foldr</span> (<span class="op">++</span>) <span class="st">&quot;&quot;</span> [<span class="st">&quot;abc&quot;</span>,<span class="st">&quot;de&quot;</span>,<span class="st">&quot;f&quot;</span>] <span class="op">==&gt;</span> <span class="st">&quot;abcdef&quot;</span></span></code></pre></div>
-</section>
-<section id="recursion" data-number="1.4">
-<h2 data-number="9.4"><span class="header-section-number">9.4</span> Recursion</h2>
-<p>To implement a function that uses repetition in Haskell, you need recursion. Haskell has no loops like other programming languages. Here are some simple recursive functions in Haskell:</p>
-<div class="sourceCode" id="cb18"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb18-1"><a href="#cb18-1" aria-hidden="true" tabindex="-1"></a><span class="ot">repeatString ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<h2 data-number="9.4" id="recursion"><span
+class="header-section-number">9.4</span> Recursion</h2>
+<p>To implement a function that uses repetition in Haskell, you need
+recursion. Haskell has no loops like other programming languages. Here
+are some simple recursive functions in Haskell:</p>
+<div class="sourceCode" id="cb18"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb18-1"><a href="#cb18-1" aria-hidden="true" tabindex="-1"></a><span class="ot">repeatString ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb18-2"><a href="#cb18-2" aria-hidden="true" tabindex="-1"></a>repeatString <span class="dv">0</span> s <span class="ot">=</span> <span class="st">&quot;&quot;</span></span>
 <span id="cb18-3"><a href="#cb18-3" aria-hidden="true" tabindex="-1"></a>repeatString n s <span class="ot">=</span> s <span class="op">++</span> repeatString (n<span class="op">-</span><span class="dv">1</span>) s</span>
 <span id="cb18-4"><a href="#cb18-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -583,8 +839,11 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 <span id="cb18-11"><a href="#cb18-11" aria-hidden="true" tabindex="-1"></a>safeLast []     <span class="ot">=</span> <span class="dt">Nothing</span></span>
 <span id="cb18-12"><a href="#cb18-12" aria-hidden="true" tabindex="-1"></a>safeLast [x]    <span class="ot">=</span> <span class="dt">Just</span> x</span>
 <span id="cb18-13"><a href="#cb18-13" aria-hidden="true" tabindex="-1"></a>safeLast (x<span class="op">:</span>xs) <span class="ot">=</span> safeLast xs</span></code></pre></div>
-<p>To consume or produce a list you often need recursion. Here are the implementations of <code>map</code> and <code>filter</code> as examples of recursive list processing:</p>
-<div class="sourceCode" id="cb19"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb19-1"><a href="#cb19-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [b]</span>
+<p>To consume or produce a list you often need recursion. Here are the
+implementations of <code>map</code> and <code>filter</code> as examples
+of recursive list processing:</p>
+<div class="sourceCode" id="cb19"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb19-1"><a href="#cb19-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [b]</span>
 <span id="cb19-2"><a href="#cb19-2" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> _ []     <span class="ot">=</span> []</span>
 <span id="cb19-3"><a href="#cb19-3" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> f (x<span class="op">:</span>xs) <span class="ot">=</span> f x <span class="op">:</span> <span class="fu">map</span> f xs</span>
 <span id="cb19-4"><a href="#cb19-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -593,25 +852,30 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 <span id="cb19-7"><a href="#cb19-7" aria-hidden="true" tabindex="-1"></a><span class="fu">filter</span> <span class="fu">pred</span> (x<span class="op">:</span>xs)</span>
 <span id="cb19-8"><a href="#cb19-8" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> <span class="fu">pred</span> x         <span class="ot">=</span> x <span class="op">:</span> <span class="fu">filter</span> <span class="fu">pred</span> xs</span>
 <span id="cb19-9"><a href="#cb19-9" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> <span class="fu">otherwise</span>      <span class="ot">=</span> <span class="fu">filter</span> <span class="fu">pred</span> xs</span></code></pre></div>
-<p>Sometimes, a recursive helper function is needed in case you need to keep track of multiple pieces of data.</p>
-<div class="sourceCode" id="cb20"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb20-1"><a href="#cb20-1" aria-hidden="true" tabindex="-1"></a><span class="ot">sumNumbers ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<p>Sometimes, a recursive helper function is needed in case you need to
+keep track of multiple pieces of data.</p>
+<div class="sourceCode" id="cb20"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb20-1"><a href="#cb20-1" aria-hidden="true" tabindex="-1"></a><span class="ot">sumNumbers ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb20-2"><a href="#cb20-2" aria-hidden="true" tabindex="-1"></a>sumNumbers xs <span class="ot">=</span> go <span class="dv">0</span> xs</span>
 <span id="cb20-3"><a href="#cb20-3" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> go <span class="fu">sum</span> [] <span class="ot">=</span> <span class="fu">sum</span></span>
 <span id="cb20-4"><a href="#cb20-4" aria-hidden="true" tabindex="-1"></a>        go <span class="fu">sum</span> (x<span class="op">:</span>xs) <span class="ot">=</span> go (<span class="fu">sum</span><span class="op">+</span>x) xs</span></code></pre></div>
-<p>Here’s a final example utilizing guards, pattern matching, a helper function and recursion:</p>
-<div class="sourceCode" id="cb21"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb21-1"><a href="#cb21-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- split a string into pieces at the given character</span></span>
+<p>Here’s a final example utilizing guards, pattern matching, a helper
+function and recursion:</p>
+<div class="sourceCode" id="cb21"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb21-1"><a href="#cb21-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- split a string into pieces at the given character</span></span>
 <span id="cb21-2"><a href="#cb21-2" aria-hidden="true" tabindex="-1"></a><span class="ot">mySplit ::</span> <span class="dt">Char</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> [<span class="dt">String</span>]</span>
 <span id="cb21-3"><a href="#cb21-3" aria-hidden="true" tabindex="-1"></a>mySplit c xs <span class="ot">=</span> helper [] xs</span>
 <span id="cb21-4"><a href="#cb21-4" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> helper piece [] <span class="ot">=</span> [piece]</span>
 <span id="cb21-5"><a href="#cb21-5" aria-hidden="true" tabindex="-1"></a>        helper piece (y<span class="op">:</span>ys)</span>
 <span id="cb21-6"><a href="#cb21-6" aria-hidden="true" tabindex="-1"></a>          <span class="op">|</span> c <span class="op">==</span> y    <span class="ot">=</span> piece <span class="op">:</span> helper [] ys</span>
 <span id="cb21-7"><a href="#cb21-7" aria-hidden="true" tabindex="-1"></a>          <span class="op">|</span> <span class="fu">otherwise</span> <span class="ot">=</span> helper (piece<span class="op">++</span>[y]) ys</span></code></pre></div>
-<div class="sourceCode" id="cb22"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb22-1"><a href="#cb22-1" aria-hidden="true" tabindex="-1"></a>mySplit <span class="ch">&#39;-&#39;</span> <span class="st">&quot;a-bcd-ef&quot;</span>  <span class="op">==&gt;</span>  [<span class="st">&quot;a&quot;</span>,<span class="st">&quot;bcd&quot;</span>,<span class="st">&quot;ef&quot;</span>]</span></code></pre></div>
-</section>
-<section id="type-classes" data-number="1.5">
-<h2 data-number="9.5"><span class="header-section-number">9.5</span> Type Classes</h2>
+<div class="sourceCode" id="cb22"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb22-1"><a href="#cb22-1" aria-hidden="true" tabindex="-1"></a>mySplit <span class="ch">&#39;-&#39;</span> <span class="st">&quot;a-bcd-ef&quot;</span>  <span class="op">==&gt;</span>  [<span class="st">&quot;a&quot;</span>,<span class="st">&quot;bcd&quot;</span>,<span class="st">&quot;ef&quot;</span>]</span></code></pre></div>
+<h2 data-number="9.5" id="type-classes"><span
+class="header-section-number">9.5</span> Type Classes</h2>
 <p>The following functions are <em>parametrically polymorphic</em>:</p>
-<div class="sourceCode" id="cb23"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb23-1"><a href="#cb23-1" aria-hidden="true" tabindex="-1"></a><span class="fu">id</span><span class="ot"> ::</span> a <span class="ot">-&gt;</span> a</span>
+<div class="sourceCode" id="cb23"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb23-1"><a href="#cb23-1" aria-hidden="true" tabindex="-1"></a><span class="fu">id</span><span class="ot"> ::</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb23-2"><a href="#cb23-2" aria-hidden="true" tabindex="-1"></a><span class="fu">id</span> x <span class="ot">=</span> x</span>
 <span id="cb23-3"><a href="#cb23-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb23-4"><a href="#cb23-4" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span><span class="ot"> ::</span> [a] <span class="ot">-&gt;</span> a</span>
@@ -619,56 +883,90 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 <span id="cb23-6"><a href="#cb23-6" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb23-7"><a href="#cb23-7" aria-hidden="true" tabindex="-1"></a><span class="fu">fst</span><span class="ot"> ::</span> (a,b) <span class="ot">-&gt;</span> a</span>
 <span id="cb23-8"><a href="#cb23-8" aria-hidden="true" tabindex="-1"></a><span class="fu">fst</span> (x,y) <span class="ot">=</span> x</span></code></pre></div>
-<p>Parametrically polymorphic functions always work the same way, no matter what types we’re working with. This means that we can’t define a special implementation of <code>id</code> just for the <code>Int</code> type, or <code>fst</code> for the type <code>(Bool, String)</code>.</p>
-<p>By contrast, <em>ad hoc polymorphism</em> allows different types to have different implementations of the same function. Ad hoc polymorphism in Haskell can be achieved by defining a <em>type class</em> and then declaring <em>instances</em> of that type class for various types. Ad hoc polymorphism is a handy way for expressing a common set of operations, even when the implementation of the operations depends on the type they’re acting on.</p>
-<p>Functions that use ad hoc polymorphism have a <em>class constraint</em> in their types. Here are some examples:</p>
-<div class="sourceCode" id="cb24"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb24-1"><a href="#cb24-1" aria-hidden="true" tabindex="-1"></a><span class="fu">negate</span><span class="ot"> ::</span> <span class="dt">Num</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a</span>
+<p>Parametrically polymorphic functions always work the same way, no
+matter what types we’re working with. This means that we can’t define a
+special implementation of <code>id</code> just for the <code>Int</code>
+type, or <code>fst</code> for the type <code>(Bool, String)</code>.</p>
+<p>By contrast, <em>ad hoc polymorphism</em> allows different types to
+have different implementations of the same function. Ad hoc polymorphism
+in Haskell can be achieved by defining a <em>type class</em> and then
+declaring <em>instances</em> of that type class for various types. Ad
+hoc polymorphism is a handy way for expressing a common set of
+operations, even when the implementation of the operations depends on
+the type they’re acting on.</p>
+<p>Functions that use ad hoc polymorphism have a <em>class
+constraint</em> in their types. Here are some examples:</p>
+<div class="sourceCode" id="cb24"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb24-1"><a href="#cb24-1" aria-hidden="true" tabindex="-1"></a><span class="fu">negate</span><span class="ot"> ::</span> <span class="dt">Num</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a</span>
 <span id="cb24-2"><a href="#cb24-2" aria-hidden="true" tabindex="-1"></a><span class="ot">(==) ::</span> <span class="dt">Eq</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb24-3"><a href="#cb24-3" aria-hidden="true" tabindex="-1"></a><span class="fu">sort</span><span class="ot"> ::</span> <span class="dt">Ord</span> a <span class="ot">=&gt;</span> [a] <span class="ot">-&gt;</span> [a]</span></code></pre></div>
-<p>A type like <code>Num a =&gt; a -&gt; a</code> means: for any type <code>X</code> that’s a member of the <code>Num</code> class, this function has type <code>X -&gt; X</code>. In other words, we can invoke <code>negate</code> on any number type, but not on other types:</p>
-<div class="sourceCode" id="cb25"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb25-1"><a href="#cb25-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">negate</span> <span class="dv">1</span></span>
+<p>A type like <code>Num a =&gt; a -&gt; a</code> means: for any type
+<code>X</code> that’s a member of the <code>Num</code> class, this
+function has type <code>X -&gt; X</code>. In other words, we can invoke
+<code>negate</code> on any number type, but not on other types:</p>
+<div class="sourceCode" id="cb25"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb25-1"><a href="#cb25-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">negate</span> <span class="dv">1</span></span>
 <span id="cb25-2"><a href="#cb25-2" aria-hidden="true" tabindex="-1"></a><span class="op">-</span><span class="dv">1</span></span>
 <span id="cb25-3"><a href="#cb25-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">negate</span> <span class="fl">1.0</span></span>
 <span id="cb25-4"><a href="#cb25-4" aria-hidden="true" tabindex="-1"></a><span class="op">-</span><span class="fl">1.0</span></span>
 <span id="cb25-5"><a href="#cb25-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">negate</span> <span class="dt">True</span></span>
 <span id="cb25-6"><a href="#cb25-6" aria-hidden="true" tabindex="-1"></a><span class="op">&lt;</span>interactive<span class="op">&gt;:</span><span class="dv">3</span><span class="op">:</span><span class="dv">1</span><span class="op">:</span> <span class="fu">error</span><span class="op">:</span></span>
 <span id="cb25-7"><a href="#cb25-7" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">No</span> <span class="kw">instance</span> for (<span class="dt">Num</span> <span class="dt">Bool</span>) arising from a use <span class="kw">of</span> ‘negate’</span></code></pre></div>
-<p>Here’s a summary of some useful type classes from the standard library.</p>
+<p>Here’s a summary of some useful type classes from the standard
+library.</p>
 <ul>
 <li>Comparison
 <ul>
-<li><code>Eq</code> is for equality comparison. It contains the <code>==</code> operator</li>
-<li><code>Ord</code> is for order comparison. It contains the ordered comparison operators like <code>&lt;</code> and <code>=&gt;</code>, and functions like <code>max</code> and <code>min</code>.</li>
+<li><code>Eq</code> is for equality comparison. It contains the
+<code>==</code> operator</li>
+<li><code>Ord</code> is for order comparison. It contains the ordered
+comparison operators like <code>&lt;</code> and <code>=&gt;</code>, and
+functions like <code>max</code> and <code>min</code>.</li>
 </ul></li>
 <li>Numbers
 <ul>
-<li><code>Num</code> is for all number types. It contains <code>+</code>, <code>-</code>, <code>*</code> and <code>negate</code>.</li>
-<li><code>Integral</code> is for whole number types. Most notably, it contains integer division <code>div</code>.</li>
-<li><code>Fractional</code> is for number types that support division, <code>/</code></li>
+<li><code>Num</code> is for all number types. It contains
+<code>+</code>, <code>-</code>, <code>*</code> and
+<code>negate</code>.</li>
+<li><code>Integral</code> is for whole number types. Most notably, it
+contains integer division <code>div</code>.</li>
+<li><code>Fractional</code> is for number types that support division,
+<code>/</code></li>
 </ul></li>
 <li>Converting values to strings
 <ul>
-<li><code>Show</code> contains the function <code>show :: Show a =&gt; a -&gt; String</code> that converts values to strings</li>
-<li><code>Read</code> contains the function <code>read :: Read a =&gt; String -&gt; a</code> that is the inverse of <code>show</code></li>
+<li><code>Show</code> contains the function
+<code>show :: Show a =&gt; a -&gt; String</code> that converts values to
+strings</li>
+<li><code>Read</code> contains the function
+<code>read :: Read a =&gt; String -&gt; a</code> that is the inverse of
+<code>show</code></li>
 </ul></li>
 </ul>
-<p>Sometimes, multiple class constraints are needed. For instance here:</p>
-<div class="sourceCode" id="cb26"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb26-1"><a href="#cb26-1" aria-hidden="true" tabindex="-1"></a><span class="ot">sumTwoSmallest ::</span> (<span class="dt">Num</span> a, <span class="dt">Ord</span> a) <span class="ot">=&gt;</span> [a] <span class="ot">-&gt;</span> a</span>
+<p>Sometimes, multiple class constraints are needed. For instance
+here:</p>
+<div class="sourceCode" id="cb26"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb26-1"><a href="#cb26-1" aria-hidden="true" tabindex="-1"></a><span class="ot">sumTwoSmallest ::</span> (<span class="dt">Num</span> a, <span class="dt">Ord</span> a) <span class="ot">=&gt;</span> [a] <span class="ot">-&gt;</span> a</span>
 <span id="cb26-2"><a href="#cb26-2" aria-hidden="true" tabindex="-1"></a>sumTwoSmallest xs <span class="ot">=</span> <span class="kw">let</span> (a<span class="op">:</span>b<span class="op">:</span>_) <span class="ot">=</span> <span class="fu">sort</span> xs</span>
 <span id="cb26-3"><a href="#cb26-3" aria-hidden="true" tabindex="-1"></a>                    <span class="kw">in</span> a<span class="op">+</span>b</span></code></pre></div>
-<p>Now that we’ve seen some classes and types, let’s look at the syntax of declaring classes and instances. Here are two class definitions:</p>
-<div class="sourceCode" id="cb27"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb27-1"><a href="#cb27-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Sized</span> a <span class="kw">where</span></span>
+<p>Now that we’ve seen some classes and types, let’s look at the syntax
+of declaring classes and instances. Here are two class definitions:</p>
+<div class="sourceCode" id="cb27"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb27-1"><a href="#cb27-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Sized</span> a <span class="kw">where</span></span>
 <span id="cb27-2"><a href="#cb27-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  empty ::</span> a        <span class="co">-- a thing with size 0</span></span>
 <span id="cb27-3"><a href="#cb27-3" aria-hidden="true" tabindex="-1"></a><span class="ot">  size ::</span> a <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb27-4"><a href="#cb27-4" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb27-5"><a href="#cb27-5" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Eq</span> a <span class="kw">where</span></span>
 <span id="cb27-6"><a href="#cb27-6" aria-hidden="true" tabindex="-1"></a><span class="ot">  (==) ::</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Bool</span></span></code></pre></div>
 <p>Consider the following data structures:</p>
-<div class="sourceCode" id="cb28"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb28-1"><a href="#cb28-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Numbers</span> <span class="ot">=</span> <span class="dt">None</span> <span class="op">|</span> <span class="dt">One</span> <span class="dt">Int</span> <span class="op">|</span> <span class="dt">Two</span> <span class="dt">Int</span> <span class="dt">Int</span></span>
+<div class="sourceCode" id="cb28"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb28-1"><a href="#cb28-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Numbers</span> <span class="ot">=</span> <span class="dt">None</span> <span class="op">|</span> <span class="dt">One</span> <span class="dt">Int</span> <span class="op">|</span> <span class="dt">Two</span> <span class="dt">Int</span> <span class="dt">Int</span></span>
 <span id="cb28-2"><a href="#cb28-2" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">IntList</span> <span class="ot">=</span> <span class="dt">Nil</span> <span class="op">|</span> <span class="dt">ListNode</span> <span class="dt">Int</span> <span class="dt">IntList</span></span>
 <span id="cb28-3"><a href="#cb28-3" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Tree</span> a <span class="ot">=</span> <span class="dt">Leaf</span> <span class="op">|</span> <span class="dt">Node</span> a (<span class="dt">Tree</span> a) (<span class="dt">Tree</span> a)</span></code></pre></div>
-<p>All of these have sizes that we can count, but we need to perform the operation differently:</p>
-<div class="sourceCode" id="cb29"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb29-1"><a href="#cb29-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Sized</span> <span class="dt">Numbers</span> <span class="kw">where</span></span>
+<p>All of these have sizes that we can count, but we need to perform the
+operation differently:</p>
+<div class="sourceCode" id="cb29"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb29-1"><a href="#cb29-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Sized</span> <span class="dt">Numbers</span> <span class="kw">where</span></span>
 <span id="cb29-2"><a href="#cb29-2" aria-hidden="true" tabindex="-1"></a>  empty <span class="ot">=</span> <span class="dt">None</span></span>
 <span id="cb29-3"><a href="#cb29-3" aria-hidden="true" tabindex="-1"></a>  size <span class="dt">None</span>      <span class="ot">=</span> <span class="dv">0</span></span>
 <span id="cb29-4"><a href="#cb29-4" aria-hidden="true" tabindex="-1"></a>  size (<span class="dt">One</span> _)   <span class="ot">=</span> <span class="dv">1</span></span>
@@ -683,8 +981,10 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 <span id="cb29-13"><a href="#cb29-13" aria-hidden="true" tabindex="-1"></a>  empty <span class="ot">=</span> <span class="dt">Leaf</span></span>
 <span id="cb29-14"><a href="#cb29-14" aria-hidden="true" tabindex="-1"></a>  size <span class="dt">Leaf</span> <span class="ot">=</span> <span class="dv">0</span></span>
 <span id="cb29-15"><a href="#cb29-15" aria-hidden="true" tabindex="-1"></a>  size (<span class="dt">Node</span> _ left right) <span class="ot">=</span> <span class="dv">1</span> <span class="op">+</span> size left <span class="op">+</span> size right</span></code></pre></div>
-<p>We can also easily declare <code>Eq</code> instances for <code>Numbers</code> and <code>IntList</code>:</p>
-<div class="sourceCode" id="cb30"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb30-1"><a href="#cb30-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Eq</span> <span class="dt">Numbers</span> <span class="kw">where</span></span>
+<p>We can also easily declare <code>Eq</code> instances for
+<code>Numbers</code> and <code>IntList</code>:</p>
+<div class="sourceCode" id="cb30"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb30-1"><a href="#cb30-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Eq</span> <span class="dt">Numbers</span> <span class="kw">where</span></span>
 <span id="cb30-2"><a href="#cb30-2" aria-hidden="true" tabindex="-1"></a>  <span class="dt">None</span>      <span class="op">==</span> <span class="dt">None</span>       <span class="ot">=</span> <span class="dt">True</span></span>
 <span id="cb30-3"><a href="#cb30-3" aria-hidden="true" tabindex="-1"></a>  (<span class="dt">One</span> x)   <span class="op">==</span> (<span class="dt">One</span> y)    <span class="ot">=</span> x<span class="op">==</span>y</span>
 <span id="cb30-4"><a href="#cb30-4" aria-hidden="true" tabindex="-1"></a>  (<span class="dt">Two</span> x y) <span class="op">==</span> (<span class="dt">Two</span> z w)  <span class="ot">=</span> x<span class="op">==</span>z <span class="op">&amp;&amp;</span> y<span class="op">==</span>w</span>
@@ -694,25 +994,34 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
 <span id="cb30-8"><a href="#cb30-8" aria-hidden="true" tabindex="-1"></a>  <span class="dt">Nil</span>             <span class="op">==</span> <span class="dt">Nil</span>               <span class="ot">=</span> <span class="dt">True</span></span>
 <span id="cb30-9"><a href="#cb30-9" aria-hidden="true" tabindex="-1"></a>  (<span class="dt">ListNode</span> x xs) <span class="op">==</span> (<span class="dt">ListNode</span> y ys)   <span class="ot">=</span> x <span class="op">==</span> y <span class="op">&amp;&amp;</span> xs <span class="op">==</span> ys</span>
 <span id="cb30-10"><a href="#cb30-10" aria-hidden="true" tabindex="-1"></a>  _               <span class="op">==</span> _                 <span class="ot">=</span> <span class="dt">False</span></span></code></pre></div>
-<p>However, since the <code>Tree</code> datatype is parameterized over the element type <code>a</code>, we need an <code>Eq a</code> instance in order to have an <code>Eq (Tree a)</code> instance. This is achieved by adding a class constraint to the instance declaration. This is called an <em>instance hierarchy</em>.</p>
-<div class="sourceCode" id="cb31"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb31-1"><a href="#cb31-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Eq</span> a <span class="ot">=&gt;</span> <span class="dt">Eq</span> (<span class="dt">Tree</span> a) <span class="kw">where</span></span>
+<p>However, since the <code>Tree</code> datatype is parameterized over
+the element type <code>a</code>, we need an <code>Eq a</code> instance
+in order to have an <code>Eq (Tree a)</code> instance. This is achieved
+by adding a class constraint to the instance declaration. This is called
+an <em>instance hierarchy</em>.</p>
+<div class="sourceCode" id="cb31"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb31-1"><a href="#cb31-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Eq</span> a <span class="ot">=&gt;</span> <span class="dt">Eq</span> (<span class="dt">Tree</span> a) <span class="kw">where</span></span>
 <span id="cb31-2"><a href="#cb31-2" aria-hidden="true" tabindex="-1"></a>  <span class="dt">Leaf</span>         <span class="op">==</span> <span class="dt">Leaf</span>              <span class="ot">=</span> <span class="dt">True</span></span>
 <span id="cb31-3"><a href="#cb31-3" aria-hidden="true" tabindex="-1"></a>  (<span class="dt">Node</span> x l r) <span class="op">==</span> (<span class="dt">Node</span> x&#39; l&#39; r&#39;)   <span class="ot">=</span> x <span class="op">==</span> x&#39; <span class="op">&amp;&amp;</span> l <span class="op">==</span> l&#39; <span class="op">&amp;&amp;</span> r <span class="op">==</span> r&#39;</span>
 <span id="cb31-4"><a href="#cb31-4" aria-hidden="true" tabindex="-1"></a>  _            <span class="op">==</span> _                 <span class="ot">=</span> <span class="dt">False</span></span></code></pre></div>
-<section id="deriving" data-number="1.5.1">
-<h3 data-number="9.5.1"><span class="header-section-number">9.5.1</span> Deriving</h3>
-<p>Some standard type classes, most notably <code>Show</code>, <code>Read</code>, <code>Eq</code> and <code>Ord</code> can be <em>derived</em>, that is, you can ask the compiler to generate automatic instances for you. For example we could have derived all of these classes for our earlier <code>Numbers</code> example.</p>
-<div class="sourceCode" id="cb32"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb32-1"><a href="#cb32-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Numbers</span> <span class="ot">=</span> <span class="dt">None</span> <span class="op">|</span> <span class="dt">One</span> <span class="dt">Int</span> <span class="op">|</span> <span class="dt">Two</span> <span class="dt">Int</span> <span class="dt">Int</span></span>
+<h3 data-number="9.5.1" id="deriving"><span
+class="header-section-number">9.5.1</span> Deriving</h3>
+<p>Some standard type classes, most notably <code>Show</code>,
+<code>Read</code>, <code>Eq</code> and <code>Ord</code> can be
+<em>derived</em>, that is, you can ask the compiler to generate
+automatic instances for you. For example we could have derived all of
+these classes for our earlier <code>Numbers</code> example.</p>
+<div class="sourceCode" id="cb32"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb32-1"><a href="#cb32-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Numbers</span> <span class="ot">=</span> <span class="dt">None</span> <span class="op">|</span> <span class="dt">One</span> <span class="dt">Int</span> <span class="op">|</span> <span class="dt">Two</span> <span class="dt">Int</span> <span class="dt">Int</span></span>
 <span id="cb32-2"><a href="#cb32-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">deriving</span> (<span class="dt">Show</span>, <span class="dt">Read</span>, <span class="dt">Eq</span>, <span class="dt">Ord</span>)</span></code></pre></div>
-<div class="sourceCode" id="cb33"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb33-1"><a href="#cb33-1" aria-hidden="true" tabindex="-1"></a><span class="dt">None</span> <span class="op">==</span> <span class="dt">One</span> <span class="dv">1</span>       <span class="op">==&gt;</span> <span class="dt">False</span></span>
+<div class="sourceCode" id="cb33"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb33-1"><a href="#cb33-1" aria-hidden="true" tabindex="-1"></a><span class="dt">None</span> <span class="op">==</span> <span class="dt">One</span> <span class="dv">1</span>       <span class="op">==&gt;</span> <span class="dt">False</span></span>
 <span id="cb33-2"><a href="#cb33-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Two</span> <span class="dv">1</span> <span class="dv">2</span> <span class="op">==</span> <span class="dt">Two</span> <span class="dv">1</span> <span class="dv">2</span>  <span class="op">==&gt;</span> <span class="dt">True</span></span>
 <span id="cb33-3"><a href="#cb33-3" aria-hidden="true" tabindex="-1"></a><span class="dt">None</span> <span class="op">&lt;</span> <span class="dt">Two</span> <span class="dv">1</span> <span class="dv">2</span>      <span class="op">==&gt;</span> <span class="dt">True</span></span>
 <span id="cb33-4"><a href="#cb33-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Two</span> <span class="dv">1</span> <span class="dv">3</span> <span class="op">&lt;</span> <span class="dt">Two</span> <span class="dv">1</span> <span class="dv">2</span>   <span class="op">==&gt;</span> <span class="dt">False</span></span>
 <span id="cb33-5"><a href="#cb33-5" aria-hidden="true" tabindex="-1"></a><span class="fu">show</span> (<span class="dt">Two</span> <span class="dv">1</span> <span class="dv">3</span>)      <span class="op">==&gt;</span> <span class="st">&quot;Two 1 3&quot;</span></span></code></pre></div>
-</section>
-</section>
-<section id="quiz" data-number="1.6">
-<h2 data-number="9.6"><span class="header-section-number">9.6</span> Quiz</h2>
+<h2 data-number="9.6" id="quiz"><span
+class="header-section-number">9.6</span> Quiz</h2>
 <p>What is the type of <code>('c',not)</code></p>
 <ol class="quiz">
 <li>
@@ -750,7 +1059,8 @@ It is a type error.
 </li>
 </ol>
 <p>Which of these is a value of the following type?</p>
-<div class="sourceCode" id="cb34"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb34-1"><a href="#cb34-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">T</span> <span class="ot">=</span> <span class="dt">X</span> <span class="dt">Int</span> <span class="op">|</span> <span class="dt">Y</span> <span class="dt">String</span> <span class="dt">String</span> <span class="op">|</span> <span class="dt">Z</span> <span class="dt">T</span></span></code></pre></div>
+<div class="sourceCode" id="cb34"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb34-1"><a href="#cb34-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">T</span> <span class="ot">=</span> <span class="dt">X</span> <span class="dt">Int</span> <span class="op">|</span> <span class="dt">Y</span> <span class="dt">String</span> <span class="dt">String</span> <span class="op">|</span> <span class="dt">Z</span> <span class="dt">T</span></span></code></pre></div>
 <ol class="quiz">
 <li>
 <code>X "foo"</code>
@@ -766,7 +1076,8 @@ It is a type error.
 </li>
 </ol>
 <p>What is the type of this function?</p>
-<div class="sourceCode" id="cb35"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb35-1"><a href="#cb35-1" aria-hidden="true" tabindex="-1"></a>f (_<span class="op">:</span><span class="dt">Just</span> x<span class="op">:</span>_) <span class="ot">=</span> x</span>
+<div class="sourceCode" id="cb35"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb35-1"><a href="#cb35-1" aria-hidden="true" tabindex="-1"></a>f (_<span class="op">:</span><span class="dt">Just</span> x<span class="op">:</span>_) <span class="ot">=</span> x</span>
 <span id="cb35-2"><a href="#cb35-2" aria-hidden="true" tabindex="-1"></a>f _            <span class="ot">=</span> <span class="dt">False</span></span></code></pre></div>
 <ol class="quiz">
 <li>
@@ -783,7 +1094,8 @@ It is a type error.
 </li>
 </ol>
 <p>What is the type of this function?</p>
-<div class="sourceCode" id="cb36"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb36-1"><a href="#cb36-1" aria-hidden="true" tabindex="-1"></a>f x y <span class="ot">=</span> x<span class="op">-</span>y <span class="op">==</span> <span class="dv">0</span></span></code></pre></div>
+<div class="sourceCode" id="cb36"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb36-1"><a href="#cb36-1" aria-hidden="true" tabindex="-1"></a>f x y <span class="ot">=</span> x<span class="op">-</span>y <span class="op">==</span> <span class="dv">0</span></span></code></pre></div>
 <ol class="quiz">
 <li class="correct">
 <code>(Num a, Eq a) =&gt; a -&gt; a -&gt; Bool</code>
@@ -798,7 +1110,8 @@ It is a type error.
 <code>a -&gt; a -&gt; Bool</code>
 </li>
 </ol>
-<p>Which of the following types can <code>x</code> have in order for <code>x (&amp;&amp;) y</code> to not be a type error?</p>
+<p>Which of the following types can <code>x</code> have in order for
+<code>x (&amp;&amp;) y</code> to not be a type error?</p>
 <ol class="quiz">
 <li>
 <code>Bool</code>
@@ -810,62 +1123,109 @@ It is a type error.
 <code>(Bool -&gt; Bool -&gt; Bool) -&gt; Bool -&gt; Bool</code>
 </li>
 </ol>
-</section>
-<section id="working-on-the-exercises" data-number="1.7">
-<h2 data-number="9.7"><span class="header-section-number">9.7</span> Working on the Exercises</h2>
-<p>Here’s a short recap of how to work on the exercises. The system is the same as for part 1 of the course.</p>
+<h2 data-number="9.7" id="working-on-the-exercises"><span
+class="header-section-number">9.7</span> Working on the Exercises</h2>
+<p>Here’s a short recap of how to work on the exercises. The system is
+the same as for part 1 of the course.</p>
 <ol type="1">
-<li>Clone the GitHub repository <a href="https://github.com/moocfi/haskell-mooc" class="uri">https://github.com/moocfi/haskell-mooc</a></li>
+<li>Clone the GitHub repository <a
+href="https://github.com/moocfi/haskell-mooc"
+class="uri">https://github.com/moocfi/haskell-mooc</a></li>
 <li>Go to the <code>exercises/</code> directory</li>
 <li>Run <code>stack build</code> to download dependencies</li>
 <li>Edit the file <code>Set9a.hs</code></li>
-<li>Check your answers by running <code>stack runhaskell Set9aTest.hs</code></li>
-<li>Return your answers on the <a href="https://haskell.mooc.fi/submit.php">Submit page</a></li>
-<li>Repeat as necessary. You can work on the exercise sets in any order, and you can return the sets as many times as you want!</li>
-<li>You can see total score on the <a href="https://haskell.mooc.fi/status.php">My status page</a></li>
+<li>Check your answers by running
+<code>stack runhaskell Set9aTest.hs</code></li>
+<li>Return your answers on the <a
+href="https://haskell.mooc.fi/submit.php">Submit page</a></li>
+<li>Repeat as necessary. You can work on the exercise sets in any order,
+and you can return the sets as many times as you want!</li>
+<li>You can see total score on the <a
+href="https://haskell.mooc.fi/status.php">My status page</a></li>
 </ol>
-</section>
-<section id="exercises" data-number="1.8">
-<h2 data-number="9.8"><span class="header-section-number">9.8</span> Exercises</h2>
+<h2 data-number="9.8" id="exercises"><span
+class="header-section-number">9.8</span> Exercises</h2>
 <ul>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set9a.hs">Set9a</a> - small exercises that recap part 1 of the course</li>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set9b.hs">Set9b</a> - let’s solve the N Queens puzzle!</li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set9a.hs">Set9a</a>
+- small exercises that recap part 1 of the course</li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set9b.hs">Set9b</a>
+- let’s solve the N Queens puzzle!</li>
 </ul>
-</section>
-</section>
-<section id="lecture-10-reductionism" data-number="2">
-<h1 data-number="10"><span class="header-section-number">10</span> Lecture 10: Reductionism</h1>
+<h1 data-number="10" id="lecture-10-reductionism"><span
+class="header-section-number">10</span> Lecture 10: Reductionism</h1>
 <ul>
 <li>Purity</li>
 <li>Laziness</li>
 <li>Haskell evaluation</li>
 </ul>
-<section id="laziness-purity" data-number="2.1">
-<h2 data-number="10.1"><span class="header-section-number">10.1</span> Laziness &amp; Purity</h2>
-<p>Purity and laziness were mentioned as key features of Haskell in the beginning of part 1. Let’s take a closer look at them.</p>
-<p>Haskell is a <em>pure</em> functional language. This means that the value <code>f x y</code> is always the same for given <code>x</code> and <code>y</code>. In other words, the values of <code>x</code> and <code>y</code> uniquely determine the value of <code>f x y</code>. This property is also called <em>referential transparency</em>.</p>
-<p>Purity also means that there are no side effects: you can’t have the evaluation of <code>f x y</code> read a line from the user - the line would be different on different invocations of <code>f</code> and would affect the return value, breaking referential transparency! Obviously you need side effects to actually get something done. We’ll get back to how Haskell handles side effects later.</p>
-<p>Haskell is a <em>lazy</em> language. This means that a value is not evaluated if it is not needed. An example illustrates this best. Consider these two functions:</p>
-<div class="sourceCode" id="cb37"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb37-1"><a href="#cb37-1" aria-hidden="true" tabindex="-1"></a>f x <span class="ot">=</span> f x   <span class="co">-- infinite recursion</span></span>
+<h2 data-number="10.1" id="laziness-purity"><span
+class="header-section-number">10.1</span> Laziness &amp; Purity</h2>
+<p>Purity and laziness were mentioned as key features of Haskell in the
+beginning of part 1. Let’s take a closer look at them.</p>
+<p>Haskell is a <em>pure</em> functional language. This means that the
+value <code>f x y</code> is always the same for given <code>x</code> and
+<code>y</code>. In other words, the values of <code>x</code> and
+<code>y</code> uniquely determine the value of <code>f x y</code>. This
+property is also called <em>referential transparency</em>.</p>
+<p>Purity also means that there are no side effects: you can’t have the
+evaluation of <code>f x y</code> read a line from the user - the line
+would be different on different invocations of <code>f</code> and would
+affect the return value, breaking referential transparency! Obviously
+you need side effects to actually get something done. We’ll get back to
+how Haskell handles side effects later.</p>
+<p>Haskell is a <em>lazy</em> language. This means that a value is not
+evaluated if it is not needed. An example illustrates this best.
+Consider these two functions:</p>
+<div class="sourceCode" id="cb37"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb37-1"><a href="#cb37-1" aria-hidden="true" tabindex="-1"></a>f x <span class="ot">=</span> f x   <span class="co">-- infinite recursion</span></span>
 <span id="cb37-2"><a href="#cb37-2" aria-hidden="true" tabindex="-1"></a>g x y <span class="ot">=</span> x</span></code></pre></div>
-<p>Evaluating <code>f 1</code> does not halt due to the infinite recursion. However, this works:</p>
-<div class="sourceCode" id="cb38"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb38-1"><a href="#cb38-1" aria-hidden="true" tabindex="-1"></a>g <span class="dv">2</span> (f <span class="dv">1</span>)  <span class="op">==&gt;</span>  <span class="dv">2</span></span></code></pre></div>
-<p>Laziness is not a problem because Haskell is pure. Only the result of the function matters, not the side effects. So if the result of a function is not used, we can simply not evaluate it without changing the meaning (semantics) of a program. Well okay, sometimes we get a terminating program instead of one that goes on for ever, but adding laziness never makes a functioning Haskell program break.</p>
-<p>If you’re interested in the theory behind this, check out the <a href="https://en.wikipedia.org/wiki/Church%E2%80%93Rosser_theorem">Church-Rosser theorem</a> or the Haskell Wiki article <a href="https://wiki.haskell.org/Lazy_vs._non-strict">Lazy vs. non-strict</a>.</p>
-</section>
-<section id="equational-reasoning" data-number="2.2">
-<h2 data-number="10.2"><span class="header-section-number">10.2</span> Equational Reasoning</h2>
-<p>Referential transparency, the feature that an expression always returns the same value for the same inputs, is a very powerful property that we can leverage to <em>reason about programs</em>.</p>
-<p>In a C-style language, we might write a procedure that may not always return the same value for the same arguments:</p>
+<p>Evaluating <code>f 1</code> does not halt due to the infinite
+recursion. However, this works:</p>
+<div class="sourceCode" id="cb38"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb38-1"><a href="#cb38-1" aria-hidden="true" tabindex="-1"></a>g <span class="dv">2</span> (f <span class="dv">1</span>)  <span class="op">==&gt;</span>  <span class="dv">2</span></span></code></pre></div>
+<p>Laziness is not a problem because Haskell is pure. Only the result of
+the function matters, not the side effects. So if the result of a
+function is not used, we can simply not evaluate it without changing the
+meaning (semantics) of a program. Well okay, sometimes we get a
+terminating program instead of one that goes on for ever, but adding
+laziness never makes a functioning Haskell program break.</p>
+<p>If you’re interested in the theory behind this, check out the <a
+href="https://en.wikipedia.org/wiki/Church%E2%80%93Rosser_theorem">Church-Rosser
+theorem</a> or the Haskell Wiki article <a
+href="https://wiki.haskell.org/Lazy_vs._non-strict">Lazy
+vs. non-strict</a>.</p>
+<h2 data-number="10.2" id="equational-reasoning"><span
+class="header-section-number">10.2</span> Equational Reasoning</h2>
+<p>Referential transparency, the feature that an expression always
+returns the same value for the same inputs, is a very powerful property
+that we can leverage to <em>reason about programs</em>.</p>
+<p>In a C-style language, we might write a procedure that may not always
+return the same value for the same arguments:</p>
 <pre><code>int c = 0;
 int funny(int x) {
   return x + c++;
 }</code></pre>
-<p>The expression <code>c++</code> increments the value of <code>c</code> and returns the old value of <code>c</code>. The next time it is evaluated, the value of <code>c</code> has increased by one. This means that depending on the current value of <code>c</code>, <code>funny(0)</code> might return <code>0</code>, <code>1</code>, <code>2</code>, or any other integer value. (It might even return negative values if <code>c</code> overflows!)</p>
-<p>In some situations this kind of behaviour with side-effects may be useful, but there are also times when it is more important to be able to reason about the code easily. The advantage of pure functions is that they can be analyzed using basic mathematical techniques. Sometimes applying math to our functions can even reveal simplifications or optimisations we otherwise wouldn’t have thought about.</p>
+<p>The expression <code>c++</code> increments the value of
+<code>c</code> and returns the old value of <code>c</code>. The next
+time it is evaluated, the value of <code>c</code> has increased by one.
+This means that depending on the current value of <code>c</code>,
+<code>funny(0)</code> might return <code>0</code>, <code>1</code>,
+<code>2</code>, or any other integer value. (It might even return
+negative values if <code>c</code> overflows!)</p>
+<p>In some situations this kind of behaviour with side-effects may be
+useful, but there are also times when it is more important to be able to
+reason about the code easily. The advantage of pure functions is that
+they can be analyzed using basic mathematical techniques. Sometimes
+applying math to our functions can even reveal simplifications or
+optimisations we otherwise wouldn’t have thought about.</p>
 <p>Consider the following expression:</p>
-<div class="sourceCode" id="cb40"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb40-1"><a href="#cb40-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> (<span class="op">+</span><span class="dv">1</span>) <span class="op">.</span> <span class="fu">reverse</span> <span class="op">.</span> <span class="fu">map</span> (<span class="op">-</span><span class="dv">1</span>)</span></code></pre></div>
-<p>This expression can be simplified to just <code>reverse</code>. We begin by establishing some helpful facts (or <em>lemmas</em>). First, suppose that we know</p>
+<div class="sourceCode" id="cb40"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb40-1"><a href="#cb40-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> (<span class="op">+</span><span class="dv">1</span>) <span class="op">.</span> <span class="fu">reverse</span> <span class="op">.</span> <span class="fu">map</span> (<span class="op">-</span><span class="dv">1</span>)</span></code></pre></div>
+<p>This expression can be simplified to just <code>reverse</code>. We
+begin by establishing some helpful facts (or <em>lemmas</em>). First,
+suppose that we know</p>
 <ol type="1">
 <li><code>map id          === id</code></li>
 <li><code>map f . map g   === map (f.g)</code></li>
@@ -875,17 +1235,26 @@ int funny(int x) {
 <ol start="4" type="1">
 <li><code>(+1) . (-1) === id</code></li>
 </ol>
-<p>We can prove fact 4 by reasoning about how <code>(+1) . (-1)</code> behaves for an arbitrary input <code>x</code>:</p>
-<div class="sourceCode" id="cb41"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb41-1"><a href="#cb41-1" aria-hidden="true" tabindex="-1"></a>((<span class="op">+</span><span class="dv">1</span>) <span class="op">.</span> (<span class="op">-</span><span class="dv">1</span>)) x <span class="op">===</span> ((<span class="op">+</span><span class="dv">1</span>) ((<span class="op">-</span><span class="dv">1</span>) x))</span>
+<p>We can prove fact 4 by reasoning about how <code>(+1) . (-1)</code>
+behaves for an arbitrary input <code>x</code>:</p>
+<div class="sourceCode" id="cb41"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb41-1"><a href="#cb41-1" aria-hidden="true" tabindex="-1"></a>((<span class="op">+</span><span class="dv">1</span>) <span class="op">.</span> (<span class="op">-</span><span class="dv">1</span>)) x <span class="op">===</span> ((<span class="op">+</span><span class="dv">1</span>) ((<span class="op">-</span><span class="dv">1</span>) x))</span>
 <span id="cb41-2"><a href="#cb41-2" aria-hidden="true" tabindex="-1"></a>                <span class="op">===</span> ((<span class="op">+</span><span class="dv">1</span>) (x <span class="op">-</span> <span class="dv">1</span>))</span>
 <span id="cb41-3"><a href="#cb41-3" aria-hidden="true" tabindex="-1"></a>                <span class="op">===</span> (x <span class="op">-</span> <span class="dv">1</span>) <span class="op">+</span> <span class="dv">1</span></span>
 <span id="cb41-4"><a href="#cb41-4" aria-hidden="true" tabindex="-1"></a>                <span class="op">===</span> x</span>
 <span id="cb41-5"><a href="#cb41-5" aria-hidden="true" tabindex="-1"></a>                <span class="op">===</span> <span class="fu">id</span> x</span></code></pre></div>
-<p>Because we didn’t assume anything about <code>x</code>, we may conclude that the above chain of equations holds for <em>every</em> <code>x</code>. Thus,</p>
-<div class="sourceCode" id="cb42"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb42-1"><a href="#cb42-1" aria-hidden="true" tabindex="-1"></a>(<span class="op">+</span><span class="dv">1</span>) <span class="op">.</span> (<span class="op">-</span><span class="dv">1</span>) <span class="op">===</span> <span class="fu">id</span></span></code></pre></div>
-<p>For those who are familiar with the technique of <em>proof by induction</em>, it is a fun exercise to prove the first three facts also. This course doesn’t discuss induction proofs, though, so don’t sweat if you don’t know induction.</p>
+<p>Because we didn’t assume anything about <code>x</code>, we may
+conclude that the above chain of equations holds for <em>every</em>
+<code>x</code>. Thus,</p>
+<div class="sourceCode" id="cb42"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb42-1"><a href="#cb42-1" aria-hidden="true" tabindex="-1"></a>(<span class="op">+</span><span class="dv">1</span>) <span class="op">.</span> (<span class="op">-</span><span class="dv">1</span>) <span class="op">===</span> <span class="fu">id</span></span></code></pre></div>
+<p>For those who are familiar with the technique of <em>proof by
+induction</em>, it is a fun exercise to prove the first three facts
+also. This course doesn’t discuss induction proofs, though, so don’t
+sweat if you don’t know induction.</p>
 <p>Now, from facts 1-4 it follows that</p>
-<div class="sourceCode" id="cb43"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb43-1"><a href="#cb43-1" aria-hidden="true" tabindex="-1"></a>    <span class="fu">map</span> (<span class="op">+</span><span class="dv">1</span>) <span class="op">.</span> <span class="fu">reverse</span> <span class="op">.</span> <span class="fu">map</span> (<span class="op">-</span><span class="dv">1</span>)</span>
+<div class="sourceCode" id="cb43"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb43-1"><a href="#cb43-1" aria-hidden="true" tabindex="-1"></a>    <span class="fu">map</span> (<span class="op">+</span><span class="dv">1</span>) <span class="op">.</span> <span class="fu">reverse</span> <span class="op">.</span> <span class="fu">map</span> (<span class="op">-</span><span class="dv">1</span>)</span>
 <span id="cb43-2"><a href="#cb43-2" aria-hidden="true" tabindex="-1"></a><span class="op">===</span> <span class="fu">map</span> (<span class="op">+</span><span class="dv">1</span>) <span class="op">.</span> (<span class="fu">reverse</span> <span class="op">.</span> <span class="fu">map</span> (<span class="op">-</span><span class="dv">1</span>))    <span class="co">-- By associativity of (.)</span></span>
 <span id="cb43-3"><a href="#cb43-3" aria-hidden="true" tabindex="-1"></a><span class="op">===</span> <span class="fu">map</span> (<span class="op">+</span><span class="dv">1</span>) <span class="op">.</span> (<span class="fu">map</span> (<span class="op">-</span><span class="dv">1</span>) <span class="op">.</span> <span class="fu">reverse</span>)    <span class="co">-- By fact 3</span></span>
 <span id="cb43-4"><a href="#cb43-4" aria-hidden="true" tabindex="-1"></a><span class="op">===</span> (<span class="fu">map</span> (<span class="op">+</span><span class="dv">1</span>) <span class="op">.</span> <span class="fu">map</span> (<span class="op">-</span><span class="dv">1</span>)) <span class="op">.</span> <span class="fu">reverse</span>    <span class="co">-- By associativity of (.)</span></span>
@@ -893,56 +1262,93 @@ int funny(int x) {
 <span id="cb43-6"><a href="#cb43-6" aria-hidden="true" tabindex="-1"></a><span class="op">===</span> <span class="fu">map</span> <span class="fu">id</span> <span class="op">.</span> <span class="fu">reverse</span>                   <span class="co">-- By fact 4</span></span>
 <span id="cb43-7"><a href="#cb43-7" aria-hidden="true" tabindex="-1"></a><span class="op">===</span> <span class="fu">id</span> <span class="op">.</span> <span class="fu">reverse</span>                       <span class="co">-- By fact 1</span></span>
 <span id="cb43-8"><a href="#cb43-8" aria-hidden="true" tabindex="-1"></a><span class="op">===</span> <span class="fu">reverse</span>                            <span class="co">-- By the definition of id</span></span></code></pre></div>
-<p>This course won’t go into details about proving things about programs, but it’s good to know that pure functional programming is very compatible with analysis like this.</p>
-</section>
-<section id="infinite-lists" data-number="2.3">
-<h2 data-number="10.3"><span class="header-section-number">10.3</span> Infinite Lists</h2>
-<p>The benefits of laziness are best demonstrated with some examples involving <em>infinite lists</em>. Let’s start with <code>repeat 1</code>, which generates an infinite list of <code>1</code>s. If we try to tell GHCi to print the value <code>repeat 1</code>, it will just keep printing <code>1</code>s for ever until we interrupt it using Control-C:</p>
-<div class="sourceCode" id="cb44"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb44-1"><a href="#cb44-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">repeat</span> <span class="dv">1</span></span>
+<p>This course won’t go into details about proving things about
+programs, but it’s good to know that pure functional programming is very
+compatible with analysis like this.</p>
+<h2 data-number="10.3" id="infinite-lists"><span
+class="header-section-number">10.3</span> Infinite Lists</h2>
+<p>The benefits of laziness are best demonstrated with some examples
+involving <em>infinite lists</em>. Let’s start with
+<code>repeat 1</code>, which generates an infinite list of
+<code>1</code>s. If we try to tell GHCi to print the value
+<code>repeat 1</code>, it will just keep printing <code>1</code>s for
+ever until we interrupt it using Control-C:</p>
+<div class="sourceCode" id="cb44"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb44-1"><a href="#cb44-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">repeat</span> <span class="dv">1</span></span>
 <span id="cb44-2"><a href="#cb44-2" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span></span>
 <span id="cb44-3"><a href="#cb44-3" aria-hidden="true" tabindex="-1"></a><span class="op">^</span><span class="dt">C</span></span></code></pre></div>
-<p>However, due to laziness, we can work with infinite lists and write computations that end. We just need to use a finite number of elements from the infinite list. Here are some examples:</p>
-<div class="sourceCode" id="cb45"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb45-1"><a href="#cb45-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">take</span> <span class="dv">10</span> <span class="op">$</span> <span class="fu">repeat</span> <span class="dv">1</span></span>
+<p>However, due to laziness, we can work with infinite lists and write
+computations that end. We just need to use a finite number of elements
+from the infinite list. Here are some examples:</p>
+<div class="sourceCode" id="cb45"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb45-1"><a href="#cb45-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">take</span> <span class="dv">10</span> <span class="op">$</span> <span class="fu">repeat</span> <span class="dv">1</span></span>
 <span id="cb45-2"><a href="#cb45-2" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>]</span>
 <span id="cb45-3"><a href="#cb45-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">take</span> <span class="dv">20</span> <span class="op">$</span> <span class="fu">repeat</span> <span class="dv">1</span></span>
 <span id="cb45-4"><a href="#cb45-4" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">1</span>]</span>
 <span id="cb45-5"><a href="#cb45-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">repeat</span> <span class="dv">1</span> <span class="op">!!</span> <span class="dv">13337</span></span>
 <span id="cb45-6"><a href="#cb45-6" aria-hidden="true" tabindex="-1"></a><span class="dv">1</span></span></code></pre></div>
-<p>An infinite list that just repeats one element can sometimes be necessary, but it’s kind of pointless. Let’s look at some more useful infinite lists next. You can use the <code>[n..]</code> syntax to generate an infinite list of numbers, starting from <code>n</code>:</p>
-<div class="sourceCode" id="cb46"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb46-1"><a href="#cb46-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">take</span> <span class="dv">20</span> [<span class="dv">0</span><span class="op">..</span>]</span>
+<p>An infinite list that just repeats one element can sometimes be
+necessary, but it’s kind of pointless. Let’s look at some more useful
+infinite lists next. You can use the <code>[n..]</code> syntax to
+generate an infinite list of numbers, starting from <code>n</code>:</p>
+<div class="sourceCode" id="cb46"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb46-1"><a href="#cb46-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">take</span> <span class="dv">20</span> [<span class="dv">0</span><span class="op">..</span>]</span>
 <span id="cb46-2"><a href="#cb46-2" aria-hidden="true" tabindex="-1"></a>[<span class="dv">0</span>,<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>,<span class="dv">5</span>,<span class="dv">6</span>,<span class="dv">7</span>,<span class="dv">8</span>,<span class="dv">9</span>,<span class="dv">10</span>,<span class="dv">11</span>,<span class="dv">12</span>,<span class="dv">13</span>,<span class="dv">14</span>,<span class="dv">15</span>,<span class="dv">16</span>,<span class="dv">17</span>,<span class="dv">18</span>,<span class="dv">19</span>]</span>
 <span id="cb46-3"><a href="#cb46-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">take</span> <span class="dv">10</span> <span class="op">.</span> <span class="fu">map</span> (<span class="dv">2</span><span class="op">^</span>) <span class="op">$</span> [<span class="dv">0</span><span class="op">..</span>]</span>
 <span id="cb46-4"><a href="#cb46-4" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">4</span>,<span class="dv">8</span>,<span class="dv">16</span>,<span class="dv">32</span>,<span class="dv">64</span>,<span class="dv">128</span>,<span class="dv">256</span>,<span class="dv">512</span>]</span></code></pre></div>
-<p>The function <code>cycle</code> repeats elements from the given list over and over again. It can be useful when dealing with rotations or cycles.</p>
-<div class="sourceCode" id="cb47"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb47-1"><a href="#cb47-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">take</span> <span class="dv">21</span> <span class="op">$</span> <span class="fu">cycle</span> <span class="st">&quot;asdf&quot;</span></span>
+<p>The function <code>cycle</code> repeats elements from the given list
+over and over again. It can be useful when dealing with rotations or
+cycles.</p>
+<div class="sourceCode" id="cb47"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb47-1"><a href="#cb47-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">take</span> <span class="dv">21</span> <span class="op">$</span> <span class="fu">cycle</span> <span class="st">&quot;asdf&quot;</span></span>
 <span id="cb47-2"><a href="#cb47-2" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;asdfasdfasdfasdfasdfa&quot;</span></span>
 <span id="cb47-3"><a href="#cb47-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">take</span> <span class="dv">4</span> <span class="op">.</span> <span class="fu">map</span> (<span class="fu">take</span> <span class="dv">4</span>) <span class="op">.</span> tails <span class="op">$</span> <span class="fu">cycle</span> <span class="st">&quot;asdf&quot;</span></span>
 <span id="cb47-4"><a href="#cb47-4" aria-hidden="true" tabindex="-1"></a>[<span class="st">&quot;asdf&quot;</span>,<span class="st">&quot;sdfa&quot;</span>,<span class="st">&quot;dfas&quot;</span>,<span class="st">&quot;fasd&quot;</span>]</span></code></pre></div>
-<section id="example-transaction-numbers" data-number="2.3.1">
-<h3 data-number="10.3.1"><span class="header-section-number">10.3.1</span> Example: Transaction Numbers</h3>
-<p>As a more concrete example of how <code>cycle</code> is useful, let’s look at computing the check digit of Finnish bank transfer transaction numbers (<a href="https://fi.wikipedia.org/wiki/Tilisiirto#Viitenumero">viitenumero</a>). A transaction number consists of any number of digits, followed by a single check digit. The check digit is checked by multiplying the digits (from right to left) with the numbers 7, 3, 1, 7, 3, 1 and so on, and summing the results. If the result of the sum <em>plus the check digit</em> is divisible by 10, the number is valid.</p>
-<p>Here’s a concrete example. <code>116127</code> is a valid transaction number. The computation goes like this:</p>
+<h3 data-number="10.3.1" id="example-transaction-numbers"><span
+class="header-section-number">10.3.1</span> Example: Transaction
+Numbers</h3>
+<p>As a more concrete example of how <code>cycle</code> is useful, let’s
+look at computing the check digit of Finnish bank transfer transaction
+numbers (<a
+href="https://fi.wikipedia.org/wiki/Tilisiirto#Viitenumero">viitenumero</a>).
+A transaction number consists of any number of digits, followed by a
+single check digit. The check digit is checked by multiplying the digits
+(from right to left) with the numbers 7, 3, 1, 7, 3, 1 and so on, and
+summing the results. If the result of the sum <em>plus the check
+digit</em> is divisible by 10, the number is valid.</p>
+<p>Here’s a concrete example. <code>116127</code> is a valid transaction
+number. The computation goes like this:</p>
 <pre><code>digits:       1  1  6  1  2
               *  *  *  *  *
 multipliers:  3  7  1  3  7
               3+ 7+ 6+ 3+14 = 33
 check digit is 7, 33+7=40 is divisible by 10, valid</code></pre>
-<p>Here’s the Haskell code for a transaction number checker. Note how we use use the infinite list <code>cycle [7,3,1]</code> for the multipliers.</p>
-<div class="sourceCode" id="cb49"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb49-1"><a href="#cb49-1" aria-hidden="true" tabindex="-1"></a><span class="ot">viitenumeroCheck ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
+<p>Here’s the Haskell code for a transaction number checker. Note how we
+use use the infinite list <code>cycle [7,3,1]</code> for the
+multipliers.</p>
+<div class="sourceCode" id="cb49"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb49-1"><a href="#cb49-1" aria-hidden="true" tabindex="-1"></a><span class="ot">viitenumeroCheck ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb49-2"><a href="#cb49-2" aria-hidden="true" tabindex="-1"></a>viitenumeroCheck allDigits <span class="ot">=</span> <span class="fu">mod</span> (checksum<span class="op">+</span>checkDigit) <span class="dv">10</span> <span class="op">==</span> <span class="dv">0</span></span>
 <span id="cb49-3"><a href="#cb49-3" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> (checkDigit<span class="op">:</span>digits) <span class="ot">=</span> <span class="fu">reverse</span> allDigits</span>
 <span id="cb49-4"><a href="#cb49-4" aria-hidden="true" tabindex="-1"></a>        multipliers <span class="ot">=</span> <span class="fu">cycle</span> [<span class="dv">7</span>,<span class="dv">3</span>,<span class="dv">1</span>]</span>
 <span id="cb49-5"><a href="#cb49-5" aria-hidden="true" tabindex="-1"></a>        checksum <span class="ot">=</span> <span class="fu">sum</span> <span class="op">$</span> <span class="fu">zipWith</span> (<span class="op">*</span>) multipliers digits</span></code></pre></div>
-<div class="sourceCode" id="cb50"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb50-1"><a href="#cb50-1" aria-hidden="true" tabindex="-1"></a>viitenumeroCheck [<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">6</span>,<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">7</span>]  <span class="op">==&gt;</span> <span class="dt">True</span></span>
+<div class="sourceCode" id="cb50"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb50-1"><a href="#cb50-1" aria-hidden="true" tabindex="-1"></a>viitenumeroCheck [<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">6</span>,<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">7</span>]  <span class="op">==&gt;</span> <span class="dt">True</span></span>
 <span id="cb50-2"><a href="#cb50-2" aria-hidden="true" tabindex="-1"></a>viitenumeroCheck [<span class="dv">1</span>,<span class="dv">1</span>,<span class="dv">6</span>,<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">8</span>]  <span class="op">==&gt;</span> <span class="dt">False</span></span></code></pre></div>
-</section>
-<section id="example-finding-a-power" data-number="2.3.2">
-<h3 data-number="10.3.2"><span class="header-section-number">10.3.2</span> Example: Finding a Power</h3>
-<p>Finally, here’s how you would find the first power of 3 that’s larger than 100.</p>
-<div class="sourceCode" id="cb51"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb51-1"><a href="#cb51-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">head</span> <span class="op">.</span> <span class="fu">filter</span> (<span class="op">&gt;</span><span class="dv">100</span>) <span class="op">$</span> <span class="fu">map</span> (<span class="dv">3</span><span class="op">^</span>) [<span class="dv">0</span><span class="op">..</span>]</span>
+<h3 data-number="10.3.2" id="example-finding-a-power"><span
+class="header-section-number">10.3.2</span> Example: Finding a
+Power</h3>
+<p>Finally, here’s how you would find the first power of 3 that’s larger
+than 100.</p>
+<div class="sourceCode" id="cb51"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb51-1"><a href="#cb51-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">head</span> <span class="op">.</span> <span class="fu">filter</span> (<span class="op">&gt;</span><span class="dv">100</span>) <span class="op">$</span> <span class="fu">map</span> (<span class="dv">3</span><span class="op">^</span>) [<span class="dv">0</span><span class="op">..</span>]</span>
 <span id="cb51-2"><a href="#cb51-2" aria-hidden="true" tabindex="-1"></a><span class="dv">243</span></span></code></pre></div>
-<p>Let’s go through how this works step by step. Note how map and filter are processing the list lazily, one element at a time, as needed. This is similar to how <em>generators</em> or <em>iterators</em> work in languages like Python or Java.</p>
-<div class="sourceCode" id="cb52"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb52-1"><a href="#cb52-1" aria-hidden="true" tabindex="-1"></a>    <span class="fu">head</span> (<span class="fu">filter</span> (<span class="op">&gt;</span><span class="dv">100</span>) (<span class="fu">map</span> (<span class="dv">3</span><span class="op">^</span>) [<span class="dv">0</span><span class="op">..</span>]))</span>
+<p>Let’s go through how this works step by step. Note how map and filter
+are processing the list lazily, one element at a time, as needed. This
+is similar to how <em>generators</em> or <em>iterators</em> work in
+languages like Python or Java.</p>
+<div class="sourceCode" id="cb52"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb52-1"><a href="#cb52-1" aria-hidden="true" tabindex="-1"></a>    <span class="fu">head</span> (<span class="fu">filter</span> (<span class="op">&gt;</span><span class="dv">100</span>) (<span class="fu">map</span> (<span class="dv">3</span><span class="op">^</span>) [<span class="dv">0</span><span class="op">..</span>]))</span>
 <span id="cb52-2"><a href="#cb52-2" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">head</span> (<span class="fu">filter</span> (<span class="op">&gt;</span><span class="dv">100</span>) (<span class="fu">map</span> (<span class="dv">3</span><span class="op">^</span>) (<span class="dv">0</span><span class="op">:</span>[<span class="dv">1</span><span class="op">..</span>])))   <span class="co">-- evaluate first element of the lazy list</span></span>
 <span id="cb52-3"><a href="#cb52-3" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">head</span> (<span class="fu">filter</span> (<span class="op">&gt;</span><span class="dv">100</span>) (<span class="dv">1</span> <span class="op">:</span> <span class="fu">map</span> (<span class="dv">3</span><span class="op">^</span>) [<span class="dv">1</span><span class="op">..</span>]))   <span class="co">-- map processes the element</span></span>
 <span id="cb52-4"><a href="#cb52-4" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">head</span> (<span class="fu">filter</span> (<span class="op">&gt;</span><span class="dv">100</span>) (<span class="fu">map</span> (<span class="dv">3</span><span class="op">^</span>) [<span class="dv">1</span><span class="op">..</span>]))       <span class="co">-- filter drops the element</span></span>
@@ -956,19 +1362,27 @@ check digit is 7, 33+7=40 is divisible by 10, valid</code></pre>
 <span id="cb52-12"><a href="#cb52-12" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">head</span> (<span class="fu">filter</span> (<span class="op">&gt;</span><span class="dv">100</span>) (<span class="dv">243</span> <span class="op">:</span> <span class="fu">map</span> (<span class="dv">3</span><span class="op">^</span>) [<span class="dv">6</span><span class="op">..</span>])) <span class="co">-- map processes</span></span>
 <span id="cb52-13"><a href="#cb52-13" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">head</span> (<span class="dv">243</span> <span class="op">:</span> <span class="fu">filter</span> (<span class="op">&gt;</span><span class="dv">100</span>) (<span class="fu">map</span> (<span class="dv">3</span><span class="op">^</span>) [<span class="dv">6</span><span class="op">..</span>])) <span class="co">-- filter lets the value through</span></span>
 <span id="cb52-14"><a href="#cb52-14" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="dv">243</span>                                         <span class="co">-- head returns the result</span></span></code></pre></div>
-</section>
-</section>
-<section id="how-does-haskell-work" data-number="2.4">
-<h2 data-number="10.4"><span class="header-section-number">10.4</span> How does Haskell Work?</h2>
-<p>Laziness will probably feel a bit magical to you right now. You might wonder how it can be implemented. Haskell evaluation is remarkably simple, it’s just different than what you might be used to. Let’s dig in.</p>
-<p>In most other programming languages (like Java, C or Python), evaluation proceeds inside-out. Arguments to functions are evaluated before the function.</p>
-<p>Haskell evaluation proceeds outside-in instead of inside-out. The definition of the outermost function in an expression is applied without evaluating any arguments. Here’s a concrete example with toy functions <code>f</code> and <code>g</code>:</p>
-<div class="sourceCode" id="cb53"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb53-1"><a href="#cb53-1" aria-hidden="true" tabindex="-1"></a><span class="ot">g ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<h2 data-number="10.4" id="how-does-haskell-work"><span
+class="header-section-number">10.4</span> How does Haskell Work?</h2>
+<p>Laziness will probably feel a bit magical to you right now. You might
+wonder how it can be implemented. Haskell evaluation is remarkably
+simple, it’s just different than what you might be used to. Let’s dig
+in.</p>
+<p>In most other programming languages (like Java, C or Python),
+evaluation proceeds inside-out. Arguments to functions are evaluated
+before the function.</p>
+<p>Haskell evaluation proceeds outside-in instead of inside-out. The
+definition of the outermost function in an expression is applied without
+evaluating any arguments. Here’s a concrete example with toy functions
+<code>f</code> and <code>g</code>:</p>
+<div class="sourceCode" id="cb53"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb53-1"><a href="#cb53-1" aria-hidden="true" tabindex="-1"></a><span class="ot">g ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb53-2"><a href="#cb53-2" aria-hidden="true" tabindex="-1"></a>g x y <span class="ot">=</span> y<span class="op">+</span><span class="dv">1</span></span>
 <span id="cb53-3"><a href="#cb53-3" aria-hidden="true" tabindex="-1"></a><span class="ot">f ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb53-4"><a href="#cb53-4" aria-hidden="true" tabindex="-1"></a>f a b c <span class="ot">=</span> g (a<span class="op">*</span><span class="dv">1000</span>) c</span></code></pre></div>
 <p>Inside-out (normal) evaluation:</p>
-<div class="sourceCode" id="cb54"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb54-1"><a href="#cb54-1" aria-hidden="true" tabindex="-1"></a>f <span class="dv">1</span> (<span class="dv">1234</span><span class="op">*</span><span class="dv">1234</span>) <span class="dv">2</span></span>
+<div class="sourceCode" id="cb54"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb54-1"><a href="#cb54-1" aria-hidden="true" tabindex="-1"></a>f <span class="dv">1</span> (<span class="dv">1234</span><span class="op">*</span><span class="dv">1234</span>) <span class="dv">2</span></span>
 <span id="cb54-2"><a href="#cb54-2" aria-hidden="true" tabindex="-1"></a>  <span class="co">-- evaluate arguments to f</span></span>
 <span id="cb54-3"><a href="#cb54-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> f <span class="dv">1</span> <span class="dv">1522756</span> <span class="dv">2</span></span>
 <span id="cb54-4"><a href="#cb54-4" aria-hidden="true" tabindex="-1"></a>  <span class="co">-- evaluate f</span></span>
@@ -979,29 +1393,42 @@ check digit is 7, 33+7=40 is divisible by 10, valid</code></pre>
 <span id="cb54-9"><a href="#cb54-9" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">2</span><span class="op">+</span><span class="dv">1</span></span>
 <span id="cb54-10"><a href="#cb54-10" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">3</span></span></code></pre></div>
 <p>Haskell outside-in evaluation:</p>
-<div class="sourceCode" id="cb55"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb55-1"><a href="#cb55-1" aria-hidden="true" tabindex="-1"></a>f <span class="dv">1</span> (<span class="dv">1234</span><span class="op">*</span><span class="dv">1234</span>) <span class="dv">2</span></span>
+<div class="sourceCode" id="cb55"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb55-1"><a href="#cb55-1" aria-hidden="true" tabindex="-1"></a>f <span class="dv">1</span> (<span class="dv">1234</span><span class="op">*</span><span class="dv">1234</span>) <span class="dv">2</span></span>
 <span id="cb55-2"><a href="#cb55-2" aria-hidden="true" tabindex="-1"></a>  <span class="co">-- evaluate f without evaluating arguments</span></span>
 <span id="cb55-3"><a href="#cb55-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> g (<span class="dv">1</span><span class="op">*</span><span class="dv">1000</span>) <span class="dv">2</span></span>
 <span id="cb55-4"><a href="#cb55-4" aria-hidden="true" tabindex="-1"></a>  <span class="co">-- evaluate g without evaluating arguments</span></span>
 <span id="cb55-5"><a href="#cb55-5" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">2</span><span class="op">+</span><span class="dv">1</span></span>
 <span id="cb55-6"><a href="#cb55-6" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">3</span></span></code></pre></div>
-<p>Note how the unused calculations <code>1234*1234</code> and <code>1*1000</code> didn’t get evaluated. This is why laziness is often helpful.</p>
-<section id="pattern-matching-drives-evaluation" data-number="2.4.1">
-<h3 data-number="10.4.1"><span class="header-section-number">10.4.1</span> Pattern Matching Drives Evaluation</h3>
-<p>Let’s look at a more involved example, with pattern matching and more complex data (lists). Pattern matching drives Haskell evaluation in a very concrete way, as we’ll see. Here are some functions that we’ll use. They’re familiar from the Prelude, but I’ll give them simple definitions.</p>
-<div class="sourceCode" id="cb56"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb56-1"><a href="#cb56-1" aria-hidden="true" tabindex="-1"></a><span class="fu">not</span> <span class="dt">True</span> <span class="ot">=</span> <span class="dt">False</span></span>
+<p>Note how the unused calculations <code>1234*1234</code> and
+<code>1*1000</code> didn’t get evaluated. This is why laziness is often
+helpful.</p>
+<h3 data-number="10.4.1" id="pattern-matching-drives-evaluation"><span
+class="header-section-number">10.4.1</span> Pattern Matching Drives
+Evaluation</h3>
+<p>Let’s look at a more involved example, with pattern matching and more
+complex data (lists). Pattern matching drives Haskell evaluation in a
+very concrete way, as we’ll see. Here are some functions that we’ll use.
+They’re familiar from the Prelude, but I’ll give them simple
+definitions.</p>
+<div class="sourceCode" id="cb56"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb56-1"><a href="#cb56-1" aria-hidden="true" tabindex="-1"></a><span class="fu">not</span> <span class="dt">True</span> <span class="ot">=</span> <span class="dt">False</span></span>
 <span id="cb56-2"><a href="#cb56-2" aria-hidden="true" tabindex="-1"></a><span class="fu">not</span> <span class="dt">False</span> <span class="ot">=</span> <span class="dt">True</span></span>
 <span id="cb56-3"><a href="#cb56-3" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> f [] <span class="ot">=</span> []</span>
 <span id="cb56-4"><a href="#cb56-4" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> f (x<span class="op">:</span>xs) <span class="ot">=</span> f x <span class="op">:</span> <span class="fu">map</span> f xs</span>
 <span id="cb56-5"><a href="#cb56-5" aria-hidden="true" tabindex="-1"></a><span class="fu">length</span> [] <span class="ot">=</span> <span class="dv">0</span></span>
 <span id="cb56-6"><a href="#cb56-6" aria-hidden="true" tabindex="-1"></a><span class="fu">length</span> (x<span class="op">:</span>xs) <span class="ot">=</span> <span class="dv">1</span><span class="op">+</span><span class="fu">length</span> xs</span></code></pre></div>
 <p>Here’s the inside-out evaluation of an expression:</p>
-<div class="sourceCode" id="cb57"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb57-1"><a href="#cb57-1" aria-hidden="true" tabindex="-1"></a><span class="fu">length</span> (<span class="fu">map</span> <span class="fu">not</span> (<span class="dt">True</span><span class="op">:</span><span class="dt">False</span><span class="op">:</span>[]))</span>
+<div class="sourceCode" id="cb57"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb57-1"><a href="#cb57-1" aria-hidden="true" tabindex="-1"></a><span class="fu">length</span> (<span class="fu">map</span> <span class="fu">not</span> (<span class="dt">True</span><span class="op">:</span><span class="dt">False</span><span class="op">:</span>[]))</span>
 <span id="cb57-2"><a href="#cb57-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="fu">length</span> (<span class="fu">not</span> <span class="dt">True</span> <span class="op">:</span> <span class="fu">not</span> <span class="dt">False</span> <span class="op">:</span> [])  <span class="co">-- evaluate call to map</span></span>
 <span id="cb57-3"><a href="#cb57-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="fu">length</span> (<span class="dt">False</span><span class="op">:</span><span class="dt">True</span><span class="op">:</span>[])              <span class="co">-- evaluate calls to not</span></span>
 <span id="cb57-4"><a href="#cb57-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">2</span></span></code></pre></div>
-<p>Here’s how the evaluation proceeds in Haskell. Note how it’s not strictly outside-in, since we sometimes need to evaluate inside arguments to be able to know which pattern is matched.</p>
-<div class="sourceCode" id="cb58"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb58-1"><a href="#cb58-1" aria-hidden="true" tabindex="-1"></a><span class="fu">length</span> (<span class="fu">map</span> <span class="fu">not</span> (<span class="dt">True</span><span class="op">:</span><span class="dt">False</span><span class="op">:</span>[]))</span>
+<p>Here’s how the evaluation proceeds in Haskell. Note how it’s not
+strictly outside-in, since we sometimes need to evaluate inside
+arguments to be able to know which pattern is matched.</p>
+<div class="sourceCode" id="cb58"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb58-1"><a href="#cb58-1" aria-hidden="true" tabindex="-1"></a><span class="fu">length</span> (<span class="fu">map</span> <span class="fu">not</span> (<span class="dt">True</span><span class="op">:</span><span class="dt">False</span><span class="op">:</span>[]))</span>
 <span id="cb58-2"><a href="#cb58-2" aria-hidden="true" tabindex="-1"></a>  <span class="co">-- We can&#39;t evaluate length since we don&#39;t know which equation of length applies,</span></span>
 <span id="cb58-3"><a href="#cb58-3" aria-hidden="true" tabindex="-1"></a>  <span class="co">-- so we look at length&#39;s argument. We can apply the second equation of map, so we do.</span></span>
 <span id="cb58-4"><a href="#cb58-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="fu">length</span> (<span class="fu">not</span> <span class="dt">True</span> <span class="op">:</span> <span class="fu">map</span> <span class="fu">not</span> (<span class="dt">False</span><span class="op">:</span>[]))</span>
@@ -1025,25 +1452,55 @@ check digit is 7, 33+7=40 is divisible by 10, valid</code></pre>
 <span id="cb58-22"><a href="#cb58-22" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">1</span> <span class="op">+</span> <span class="dv">1</span></span>
 <span id="cb58-23"><a href="#cb58-23" aria-hidden="true" tabindex="-1"></a>  <span class="co">-- Finally we evaluate the outer +</span></span>
 <span id="cb58-24"><a href="#cb58-24" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dv">2</span></span></code></pre></div>
-<p>Note that we didn’t need to evaluate any of the <code>not</code> applications.</p>
-<p>Let’s introduce some terminology. We say that pattern matching <em>forces</em> evaluation. When Haskell evaluates something, it evaluates it to something called <em>weak head normal form (WHNF)</em>. WHNF basically means <em>a value that can be pattern matched on</em>. An expression is in WHNF if it can’t be evaluated on <em>its top level</em>. This means it either:</p>
+<p>Note that we didn’t need to evaluate any of the <code>not</code>
+applications.</p>
+<p>Let’s introduce some terminology. We say that pattern matching
+<em>forces</em> evaluation. When Haskell evaluates something, it
+evaluates it to something called <em>weak head normal form (WHNF)</em>.
+WHNF basically means <em>a value that can be pattern matched on</em>. An
+expression is in WHNF if it can’t be evaluated on <em>its top
+level</em>. This means it either:</p>
 <ul>
 <li>is a constant, for example: <code>1</code></li>
-<li>has a constructor at the top level, for example: <code>False</code>, <code>Just (1+1)</code>, <code>0:filter f xs</code></li>
+<li>has a constructor at the top level, for example: <code>False</code>,
+<code>Just (1+1)</code>, <code>0:filter f xs</code></li>
 <li>is a function, for example: <code>(\x -&gt; 1+x)</code></li>
 </ul>
-<p>The most notable class of expressions that is <em>not</em> in WHNF is function applications. If an expression consists of a function (that is not a constructor), applied to some arguments, it is not in WHNF. We must evaluate it in order to get something pattern matchable.</p>
-<p>In the previous example we couldn’t pick an equation for <code>length</code> in <code>length (map not (False:[]))</code>. The argument <code>(map not ...)</code> is not in WHNF, so it can’t be pattern matched on. Thus we need to evaluate it. When we apply the second equation of <code>map</code>, we get <code>length (not False : map not [])</code>, and now the argument to length is in WHNF since there is a constructor, <code>(:)</code>, at the top level. This is a bit more evident if we switch from infix to prefix notation and write the argument to <code>length</code> as <code>(:) (not False) (map not [])</code>.</p>
-<p>In practice, pattern matching is not the only thing that forces evaluation. Primitives like <code>(+)</code> also force their arguments.</p>
-<p>Instead of forcing, some sources talk about <em>strictness</em>, we can say for instance that <code>(+)</code> is <em>strict in both arguments</em>.</p>
-</section>
-<section id="a-word-about-sharing" data-number="2.4.2">
-<h3 data-number="10.4.2"><span class="header-section-number">10.4.2</span> A Word About Sharing</h3>
-<p>There’s one more thing about Haskell evaluation. Any time you give a value a <em>name</em>, it gets <em>shared</em>. This means that every occurrence of the name points at the same (potentially unevaluated) expression. When the expression gets evaluated, all occurrences of the name see the result.</p>
+<p>The most notable class of expressions that is <em>not</em> in WHNF is
+function applications. If an expression consists of a function (that is
+not a constructor), applied to some arguments, it is not in WHNF. We
+must evaluate it in order to get something pattern matchable.</p>
+<p>In the previous example we couldn’t pick an equation for
+<code>length</code> in <code>length (map not (False:[]))</code>. The
+argument <code>(map not ...)</code> is not in WHNF, so it can’t be
+pattern matched on. Thus we need to evaluate it. When we apply the
+second equation of <code>map</code>, we get
+<code>length (not False : map not [])</code>, and now the argument to
+length is in WHNF since there is a constructor, <code>(:)</code>, at the
+top level. This is a bit more evident if we switch from infix to prefix
+notation and write the argument to <code>length</code> as
+<code>(:) (not False) (map not [])</code>.</p>
+<p>In practice, pattern matching is not the only thing that forces
+evaluation. Primitives like <code>(+)</code> also force their
+arguments.</p>
+<p>Instead of forcing, some sources talk about <em>strictness</em>, we
+can say for instance that <code>(+)</code> is <em>strict in both
+arguments</em>.</p>
+<h3 data-number="10.4.2" id="a-word-about-sharing"><span
+class="header-section-number">10.4.2</span> A Word About Sharing</h3>
+<p>There’s one more thing about Haskell evaluation. Any time you give a
+value a <em>name</em>, it gets <em>shared</em>. This means that every
+occurrence of the name points at the same (potentially unevaluated)
+expression. When the expression gets evaluated, all occurrences of the
+name see the result.</p>
 <p>Let’s look at a very simple example.</p>
-<div class="sourceCode" id="cb59"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb59-1"><a href="#cb59-1" aria-hidden="true" tabindex="-1"></a>square x <span class="ot">=</span> x<span class="op">*</span>x</span></code></pre></div>
-<p>Based on the previous sections, you might imagine evaluation works like the following. The evaluation is first represented textually, and then visually, as an expression tree.</p>
-<div class="sourceCode" id="cb60"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb60-1"><a href="#cb60-1" aria-hidden="true" tabindex="-1"></a>square (<span class="dv">2</span><span class="op">+</span><span class="dv">2</span>)</span>
+<div class="sourceCode" id="cb59"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb59-1"><a href="#cb59-1" aria-hidden="true" tabindex="-1"></a>square x <span class="ot">=</span> x<span class="op">*</span>x</span></code></pre></div>
+<p>Based on the previous sections, you might imagine evaluation works
+like the following. The evaluation is first represented textually, and
+then visually, as an expression tree.</p>
+<div class="sourceCode" id="cb60"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb60-1"><a href="#cb60-1" aria-hidden="true" tabindex="-1"></a>square (<span class="dv">2</span><span class="op">+</span><span class="dv">2</span>)</span>
 <span id="cb60-2"><a href="#cb60-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> (<span class="dv">2</span><span class="op">+</span><span class="dv">2</span>) <span class="op">*</span> (<span class="dv">2</span><span class="op">+</span><span class="dv">2</span>)   <span class="co">-- definition of square</span></span>
 <span id="cb60-3"><a href="#cb60-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span>   <span class="dv">4</span>   <span class="op">*</span> (<span class="dv">2</span><span class="op">+</span><span class="dv">2</span>)   <span class="co">-- (*) forces left argument</span></span>
 <span id="cb60-4"><a href="#cb60-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span>   <span class="dv">4</span>   <span class="op">*</span>   <span class="dv">4</span>     <span class="co">-- (*) forces right argument</span></span>
@@ -1058,8 +1515,17 @@ square   ==>   *     ==>     *    ==>    *    ==>   16
 ```
 -->
 <p><img src="img/square1.svg" /></p>
-<p>However, what really happens is that the expression <code>2+2</code> named by the variable <code>x</code> is only computed once. The result of the evaluation is then shared between the two occurrences of <code>x</code> inside <code>square</code>. So here’s the correct evaluation, first textually, and then visually. Note how now instead of an expression tree, we have an <em>expression graph</em>. This is why Haskell evaluation is sometimes called <a href="https://en.wikipedia.org/wiki/Graph_reduction"><em>graph reduction</em></a>.</p>
-<div class="sourceCode" id="cb61"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb61-1"><a href="#cb61-1" aria-hidden="true" tabindex="-1"></a>square (<span class="dv">2</span><span class="op">+</span><span class="dv">2</span>)</span>
+<p>However, what really happens is that the expression <code>2+2</code>
+named by the variable <code>x</code> is only computed once. The result
+of the evaluation is then shared between the two occurrences of
+<code>x</code> inside <code>square</code>. So here’s the correct
+evaluation, first textually, and then visually. Note how now instead of
+an expression tree, we have an <em>expression graph</em>. This is why
+Haskell evaluation is sometimes called <a
+href="https://en.wikipedia.org/wiki/Graph_reduction"><em>graph
+reduction</em></a>.</p>
+<div class="sourceCode" id="cb61"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb61-1"><a href="#cb61-1" aria-hidden="true" tabindex="-1"></a>square (<span class="dv">2</span><span class="op">+</span><span class="dv">2</span>)</span>
 <span id="cb61-2"><a href="#cb61-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> (<span class="dv">2</span><span class="op">+</span><span class="dv">2</span>) <span class="op">*</span> (<span class="dv">2</span><span class="op">+</span><span class="dv">2</span>)</span>
 <span id="cb61-3"><a href="#cb61-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span>   <span class="dv">4</span>   <span class="op">*</span>   <span class="dv">4</span></span>
 <span id="cb61-4"><a href="#cb61-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span>      <span class="dv">16</span></span></code></pre></div>
@@ -1074,38 +1540,53 @@ square   ==>   *   ==>   *   ==>  16
 ```
 -->
 <p><img src="img/square2.svg" /></p>
-<p>As another example, consider the function <code>f</code> below and its evaluation.</p>
-<div class="sourceCode" id="cb62"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb62-1"><a href="#cb62-1" aria-hidden="true" tabindex="-1"></a><span class="ot">f ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<p>As another example, consider the function <code>f</code> below and
+its evaluation.</p>
+<div class="sourceCode" id="cb62"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb62-1"><a href="#cb62-1" aria-hidden="true" tabindex="-1"></a><span class="ot">f ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb62-2"><a href="#cb62-2" aria-hidden="true" tabindex="-1"></a>f i <span class="ot">=</span> <span class="kw">if</span> i<span class="op">&gt;</span><span class="dv">10</span> <span class="kw">then</span> <span class="dv">10</span> <span class="kw">else</span> i</span></code></pre></div>
-<div class="sourceCode" id="cb63"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb63-1"><a href="#cb63-1" aria-hidden="true" tabindex="-1"></a>                  _______shared________</span>
+<div class="sourceCode" id="cb63"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb63-1"><a href="#cb63-1" aria-hidden="true" tabindex="-1"></a>                  _______shared________</span>
 <span id="cb63-2"><a href="#cb63-2" aria-hidden="true" tabindex="-1"></a>                 <span class="op">|</span>                     <span class="op">|</span></span>
 <span id="cb63-3"><a href="#cb63-3" aria-hidden="true" tabindex="-1"></a>f (<span class="dv">1</span><span class="op">+</span><span class="dv">1</span>) <span class="op">==&gt;</span> <span class="kw">if</span> (<span class="dv">1</span><span class="op">+</span><span class="dv">1</span>)<span class="op">&gt;</span><span class="dv">10</span> <span class="kw">then</span> <span class="dv">10</span> <span class="kw">else</span> (<span class="dv">1</span><span class="op">+</span><span class="dv">1</span>)</span>
 <span id="cb63-4"><a href="#cb63-4" aria-hidden="true" tabindex="-1"></a>        <span class="op">==&gt;</span> <span class="kw">if</span> <span class="dv">2</span><span class="op">&gt;</span><span class="dv">10</span> <span class="kw">then</span> <span class="dv">10</span> <span class="kw">else</span> <span class="dv">2</span></span>
 <span id="cb63-5"><a href="#cb63-5" aria-hidden="true" tabindex="-1"></a>        <span class="op">==&gt;</span> <span class="kw">if</span> <span class="dt">False</span> <span class="kw">then</span> <span class="dv">10</span> <span class="kw">else</span> <span class="dv">2</span></span>
 <span id="cb63-6"><a href="#cb63-6" aria-hidden="true" tabindex="-1"></a>        <span class="op">==&gt;</span> <span class="dv">2</span></span></code></pre></div>
-<p>Haskell does not compute <code>1+1</code> twice because it was named, and the name was used twice. We can contrast this with another function that takes two arguments:</p>
-<div class="sourceCode" id="cb64"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb64-1"><a href="#cb64-1" aria-hidden="true" tabindex="-1"></a><span class="ot">g ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<p>Haskell does not compute <code>1+1</code> twice because it was named,
+and the name was used twice. We can contrast this with another function
+that takes two arguments:</p>
+<div class="sourceCode" id="cb64"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb64-1"><a href="#cb64-1" aria-hidden="true" tabindex="-1"></a><span class="ot">g ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb64-2"><a href="#cb64-2" aria-hidden="true" tabindex="-1"></a>g i j <span class="ot">=</span> <span class="kw">if</span> i<span class="op">&gt;</span><span class="dv">10</span> <span class="kw">then</span> <span class="dv">10</span> <span class="kw">else</span> j</span></code></pre></div>
-<div class="sourceCode" id="cb65"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb65-1"><a href="#cb65-1" aria-hidden="true" tabindex="-1"></a>                        ______no sharing_____</span>
+<div class="sourceCode" id="cb65"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb65-1"><a href="#cb65-1" aria-hidden="true" tabindex="-1"></a>                        ______no sharing_____</span>
 <span id="cb65-2"><a href="#cb65-2" aria-hidden="true" tabindex="-1"></a>                       <span class="op">|</span>                     <span class="op">|</span></span>
 <span id="cb65-3"><a href="#cb65-3" aria-hidden="true" tabindex="-1"></a>g (<span class="dv">1</span><span class="op">+</span><span class="dv">1</span>) (<span class="dv">1</span><span class="op">+</span><span class="dv">1</span>) <span class="op">==&gt;</span> <span class="kw">if</span> (<span class="dv">1</span><span class="op">+</span><span class="dv">1</span>)<span class="op">&gt;</span><span class="dv">10</span> <span class="kw">then</span> <span class="dv">10</span> <span class="kw">else</span> (<span class="dv">1</span><span class="op">+</span><span class="dv">1</span>)</span>
 <span id="cb65-4"><a href="#cb65-4" aria-hidden="true" tabindex="-1"></a>              <span class="op">==&gt;</span> <span class="kw">if</span> <span class="dv">2</span><span class="op">&gt;</span><span class="dv">10</span> <span class="kw">then</span> <span class="dv">10</span> <span class="kw">else</span> (<span class="dv">1</span><span class="op">+</span><span class="dv">1</span>)</span>
 <span id="cb65-5"><a href="#cb65-5" aria-hidden="true" tabindex="-1"></a>              <span class="op">==&gt;</span> <span class="kw">if</span> <span class="dt">False</span> <span class="kw">then</span> <span class="dv">10</span> <span class="kw">else</span> (<span class="dv">1</span><span class="op">+</span><span class="dv">1</span>)</span>
 <span id="cb65-6"><a href="#cb65-6" aria-hidden="true" tabindex="-1"></a>              <span class="op">==&gt;</span> (<span class="dv">1</span><span class="op">+</span><span class="dv">1</span>)</span>
 <span id="cb65-7"><a href="#cb65-7" aria-hidden="true" tabindex="-1"></a>              <span class="op">==&gt;</span> <span class="dv">2</span></span></code></pre></div>
-<p>Here we have two different names for equivalent expressions, and Haskell doesn’t magically share them. Automatically sharing equivalent expressions is an optimization called <em>Common Subexpression Elimination (CSE)</em>. You can learn a bit more <a href="https://wiki.haskell.org/GHC_optimisations#Common_subexpression_elimination">CSE and Haskell here</a>.</p>
+<p>Here we have two different names for equivalent expressions, and
+Haskell doesn’t magically share them. Automatically sharing equivalent
+expressions is an optimization called <em>Common Subexpression
+Elimination (CSE)</em>. You can learn a bit more <a
+href="https://wiki.haskell.org/GHC_optimisations#Common_subexpression_elimination">CSE
+and Haskell here</a>.</p>
 <p>You can name things via</p>
 <ul>
 <li>Function arguments</li>
 <li><code>let ... in ...</code></li>
 <li><code>where</code></li>
 </ul>
-<p>Combined with laziness, sharing means that <em>a name gets evaluated at most once</em>.</p>
-</section>
-<section id="further-examples" data-number="2.4.3">
-<h3 data-number="10.4.3"><span class="header-section-number">10.4.3</span> Further Examples</h3>
-<p>You’ll find below a slightly contrived recursive definition of the function <code>even</code>. It will illustrate the concepts of forcing and sharing.</p>
-<div class="sourceCode" id="cb66"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb66-1"><a href="#cb66-1" aria-hidden="true" tabindex="-1"></a><span class="fu">not</span><span class="ot"> ::</span> <span class="dt">Bool</span> <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
+<p>Combined with laziness, sharing means that <em>a name gets evaluated
+at most once</em>.</p>
+<h3 data-number="10.4.3" id="further-examples"><span
+class="header-section-number">10.4.3</span> Further Examples</h3>
+<p>You’ll find below a slightly contrived recursive definition of the
+function <code>even</code>. It will illustrate the concepts of forcing
+and sharing.</p>
+<div class="sourceCode" id="cb66"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb66-1"><a href="#cb66-1" aria-hidden="true" tabindex="-1"></a><span class="fu">not</span><span class="ot"> ::</span> <span class="dt">Bool</span> <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb66-2"><a href="#cb66-2" aria-hidden="true" tabindex="-1"></a><span class="fu">not</span> <span class="dt">True</span> <span class="ot">=</span> <span class="dt">False</span></span>
 <span id="cb66-3"><a href="#cb66-3" aria-hidden="true" tabindex="-1"></a><span class="fu">not</span> <span class="dt">False</span> <span class="ot">=</span> <span class="dt">True</span></span>
 <span id="cb66-4"><a href="#cb66-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -1115,13 +1596,19 @@ square   ==>   *   ==>   *   ==>  16
 <span id="cb66-8"><a href="#cb66-8" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb66-9"><a href="#cb66-9" aria-hidden="true" tabindex="-1"></a><span class="fu">even</span><span class="ot"> ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb66-10"><a href="#cb66-10" aria-hidden="true" tabindex="-1"></a><span class="fu">even</span> x  <span class="ot">=</span>  (x <span class="op">==</span> <span class="dv">0</span>)  <span class="op">||</span>  <span class="fu">not</span> (<span class="fu">even</span> (x<span class="op">-</span><span class="dv">1</span>))</span></code></pre></div>
-<p>Firstly, note that <code>||</code> forces its left argument, but not its right argument. (In other words, <code>||</code> is <em>strict in its left argument</em>.) This is because we only need to evaluate the left argument of <code>||</code> in order to know which equation applies. This means by extension that <code>even</code> forces its first argument:</p>
+<p>Firstly, note that <code>||</code> forces its left argument, but not
+its right argument. (In other words, <code>||</code> is <em>strict in
+its left argument</em>.) This is because we only need to evaluate the
+left argument of <code>||</code> in order to know which equation
+applies. This means by extension that <code>even</code> forces its first
+argument:</p>
 <ul>
 <li><code>||</code> forces <code>x==0</code></li>
 <li><code>x==0</code> forces <code>x</code></li>
 </ul>
 <p>Now let’s evaluate the expression <code>even 2</code> to WHNF.</p>
-<div class="sourceCode" id="cb67"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb67-1"><a href="#cb67-1" aria-hidden="true" tabindex="-1"></a><span class="fu">even</span> <span class="dv">2</span></span>
+<div class="sourceCode" id="cb67"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb67-1"><a href="#cb67-1" aria-hidden="true" tabindex="-1"></a><span class="fu">even</span> <span class="dv">2</span></span>
 <span id="cb67-2"><a href="#cb67-2" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="dv">2</span> <span class="op">==</span> <span class="dv">0</span>  <span class="op">||</span>  <span class="fu">not</span> (<span class="fu">even</span> (<span class="dv">2</span><span class="op">-</span><span class="dv">1</span>))                    <span class="co">-- apply definition of even</span></span>
 <span id="cb67-3"><a href="#cb67-3" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="dt">False</span>   <span class="op">||</span>  <span class="fu">not</span> (<span class="fu">even</span> (<span class="dv">2</span><span class="op">-</span><span class="dv">1</span>))                    <span class="co">-- || forces its first argument</span></span>
 <span id="cb67-4"><a href="#cb67-4" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">not</span> (<span class="fu">even</span> (<span class="dv">2</span><span class="op">-</span><span class="dv">1</span>))                                <span class="co">-- second equation of ||</span></span>
@@ -1135,10 +1622,14 @@ square   ==>   *   ==>   *   ==>  16
 <span id="cb67-12"><a href="#cb67-12" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">not</span> (<span class="fu">not</span> <span class="dt">True</span>)</span>
 <span id="cb67-13"><a href="#cb67-13" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">not</span> <span class="dt">False</span></span>
 <span id="cb67-14"><a href="#cb67-14" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="dt">True</span></span></code></pre></div>
-<p>Note that with this alternate definition <code>even</code> would not have worked. Can you tell why?</p>
-<div class="sourceCode" id="cb68"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb68-1"><a href="#cb68-1" aria-hidden="true" tabindex="-1"></a>even&#39; x <span class="ot">=</span>  <span class="fu">not</span> (even&#39; (x<span class="op">-</span><span class="dv">1</span>))  <span class="op">||</span>  x <span class="op">==</span> <span class="dv">0</span></span></code></pre></div>
-<p>Now we can really understand what’s going on in the infinite list example from earlier. Let’s use these definitions:</p>
-<div class="sourceCode" id="cb69"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb69-1"><a href="#cb69-1" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span> (x<span class="op">:</span>_) <span class="ot">=</span> x</span>
+<p>Note that with this alternate definition <code>even</code> would not
+have worked. Can you tell why?</p>
+<div class="sourceCode" id="cb68"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb68-1"><a href="#cb68-1" aria-hidden="true" tabindex="-1"></a>even&#39; x <span class="ot">=</span>  <span class="fu">not</span> (even&#39; (x<span class="op">-</span><span class="dv">1</span>))  <span class="op">||</span>  x <span class="op">==</span> <span class="dv">0</span></span></code></pre></div>
+<p>Now we can really understand what’s going on in the infinite list
+example from earlier. Let’s use these definitions:</p>
+<div class="sourceCode" id="cb69"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb69-1"><a href="#cb69-1" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span> (x<span class="op">:</span>_) <span class="ot">=</span> x</span>
 <span id="cb69-2"><a href="#cb69-2" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span> [] <span class="ot">=</span> <span class="op">-</span><span class="dv">1</span></span>
 <span id="cb69-3"><a href="#cb69-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb69-4"><a href="#cb69-4" aria-hidden="true" tabindex="-1"></a><span class="fu">filter</span> p [] <span class="ot">=</span> []</span>
@@ -1152,7 +1643,8 @@ square   ==>   *   ==>   *   ==>  16
 <span id="cb69-12"><a href="#cb69-12" aria-hidden="true" tabindex="-1"></a><span class="co">-- [0..] is syntax sugar for enumFrom 0</span></span>
 <span id="cb69-13"><a href="#cb69-13" aria-hidden="true" tabindex="-1"></a><span class="fu">enumFrom</span> n <span class="ot">=</span> n <span class="op">:</span> <span class="fu">enumFrom</span> (n<span class="op">+</span><span class="dv">1</span>)</span></code></pre></div>
 <p>And here we go:</p>
-<div class="sourceCode" id="cb70"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb70-1"><a href="#cb70-1" aria-hidden="true" tabindex="-1"></a>    <span class="fu">head</span> (<span class="fu">filter</span> (<span class="op">&gt;</span><span class="dv">100</span>) (<span class="fu">map</span> (<span class="dv">3</span><span class="op">^</span>) [<span class="dv">0</span><span class="op">..</span>]))</span>
+<div class="sourceCode" id="cb70"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb70-1"><a href="#cb70-1" aria-hidden="true" tabindex="-1"></a>    <span class="fu">head</span> (<span class="fu">filter</span> (<span class="op">&gt;</span><span class="dv">100</span>) (<span class="fu">map</span> (<span class="dv">3</span><span class="op">^</span>) [<span class="dv">0</span><span class="op">..</span>]))</span>
 <span id="cb70-2"><a href="#cb70-2" aria-hidden="true" tabindex="-1"></a><span class="op">===</span> <span class="fu">head</span> (<span class="fu">filter</span> (<span class="op">&gt;</span><span class="dv">100</span>) (<span class="fu">map</span> (<span class="dv">3</span><span class="op">^</span>) (<span class="fu">enumFrom</span> <span class="dv">0</span>)))</span>
 <span id="cb70-3"><a href="#cb70-3" aria-hidden="true" tabindex="-1"></a><span class="co">-- head forces filter, which forces map, which forces enumFrom. We apply the definition of enumFrom.</span></span>
 <span id="cb70-4"><a href="#cb70-4" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">head</span> (<span class="fu">filter</span> (<span class="op">&gt;</span><span class="dv">100</span>) (<span class="fu">map</span> (<span class="dv">3</span><span class="op">^</span>) (<span class="dv">0</span><span class="op">:</span>[<span class="dv">1</span><span class="op">..</span>])))</span>
@@ -1185,18 +1677,33 @@ square   ==>   *   ==>   *   ==>  16
 <span id="cb70-31"><a href="#cb70-31" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">head</span> (<span class="dv">243</span> <span class="op">:</span> <span class="fu">filter</span> (<span class="op">&gt;</span><span class="dv">100</span>) (<span class="fu">map</span> (<span class="dv">3</span><span class="op">^</span>) [<span class="dv">6</span><span class="op">..</span>]))</span>
 <span id="cb70-32"><a href="#cb70-32" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="dv">243</span></span></code></pre></div>
 <p>Whew.</p>
-</section>
-</section>
-<section id="working-with-infinite-lists" data-number="2.5">
-<h2 data-number="10.5"><span class="header-section-number">10.5</span> Working with Infinite Lists</h2>
-<p>Functions that work with lists often have the best performance when they’re written in such a way that they utilize laziness. One way to try to accomplish this is to write list-handling functions that work well with infinite lists.</p>
-<p>To write a function that transforms an infinite list, you need to write a function that only looks at a limited prefix of the input list, then outputs a <code>(:)</code> constructor, and then recurses. Here’s a first example.</p>
-<div class="sourceCode" id="cb71"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb71-1"><a href="#cb71-1" aria-hidden="true" tabindex="-1"></a><span class="ot">everySecond ::</span> [a] <span class="ot">-&gt;</span> [a]</span>
+<h2 data-number="10.5" id="working-with-infinite-lists"><span
+class="header-section-number">10.5</span> Working with Infinite
+Lists</h2>
+<p>Functions that work with lists often have the best performance when
+they’re written in such a way that they utilize laziness. One way to try
+to accomplish this is to write list-handling functions that work well
+with infinite lists.</p>
+<p>To write a function that transforms an infinite list, you need to
+write a function that only looks at a limited prefix of the input list,
+then outputs a <code>(:)</code> constructor, and then recurses. Here’s a
+first example.</p>
+<div class="sourceCode" id="cb71"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb71-1"><a href="#cb71-1" aria-hidden="true" tabindex="-1"></a><span class="ot">everySecond ::</span> [a] <span class="ot">-&gt;</span> [a]</span>
 <span id="cb71-2"><a href="#cb71-2" aria-hidden="true" tabindex="-1"></a>everySecond [] <span class="ot">=</span> []</span>
 <span id="cb71-3"><a href="#cb71-3" aria-hidden="true" tabindex="-1"></a>everySecond (x<span class="op">:</span>y<span class="op">:</span>xs) <span class="ot">=</span> x <span class="op">:</span> everySecond xs</span></code></pre></div>
-<div class="sourceCode" id="cb72"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb72-1"><a href="#cb72-1" aria-hidden="true" tabindex="-1"></a><span class="fu">take</span> <span class="dv">10</span> (everySecond [<span class="dv">0</span><span class="op">..</span>])  <span class="op">==&gt;</span>  [<span class="dv">0</span>,<span class="dv">2</span>,<span class="dv">4</span>,<span class="dv">6</span>,<span class="dv">8</span>,<span class="dv">10</span>,<span class="dv">12</span>,<span class="dv">14</span>,<span class="dv">16</span>,<span class="dv">18</span>]</span></code></pre></div>
-<p>A good heuristic for writing functions that work well with infinite lists is: can the <code>head</code> of the result be evaluated cheaply? Here are two examples of functions that don’t work with infinite inputs. In the case of <code>mapTailRecursive</code>, the problem is that it needs to process the whole input before being in WHNF. In the case of <code>myDrop</code>, the problem is that it uses the function <code>length</code>, doesn’t work for infinite lists since it tries to iterate until the end of the list.</p>
-<div class="sourceCode" id="cb73"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb73-1"><a href="#cb73-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [b]</span>
+<div class="sourceCode" id="cb72"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb72-1"><a href="#cb72-1" aria-hidden="true" tabindex="-1"></a><span class="fu">take</span> <span class="dv">10</span> (everySecond [<span class="dv">0</span><span class="op">..</span>])  <span class="op">==&gt;</span>  [<span class="dv">0</span>,<span class="dv">2</span>,<span class="dv">4</span>,<span class="dv">6</span>,<span class="dv">8</span>,<span class="dv">10</span>,<span class="dv">12</span>,<span class="dv">14</span>,<span class="dv">16</span>,<span class="dv">18</span>]</span></code></pre></div>
+<p>A good heuristic for writing functions that work well with infinite
+lists is: can the <code>head</code> of the result be evaluated cheaply?
+Here are two examples of functions that don’t work with infinite inputs.
+In the case of <code>mapTailRecursive</code>, the problem is that it
+needs to process the whole input before being in WHNF. In the case of
+<code>myDrop</code>, the problem is that it uses the function
+<code>length</code>, doesn’t work for infinite lists since it tries to
+iterate until the end of the list.</p>
+<div class="sourceCode" id="cb73"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb73-1"><a href="#cb73-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [b]</span>
 <span id="cb73-2"><a href="#cb73-2" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> _ []     <span class="ot">=</span> []</span>
 <span id="cb73-3"><a href="#cb73-3" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> f (x<span class="op">:</span>xs) <span class="ot">=</span> f x <span class="op">:</span> <span class="fu">map</span> f xs</span>
 <span id="cb73-4"><a href="#cb73-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -1204,14 +1711,16 @@ square   ==>   *   ==>   *   ==>  16
 <span id="cb73-6"><a href="#cb73-6" aria-hidden="true" tabindex="-1"></a>mapTailRecursive f xs <span class="ot">=</span> go xs []</span>
 <span id="cb73-7"><a href="#cb73-7" aria-hidden="true" tabindex="-1"></a>    <span class="kw">where</span> go (x<span class="op">:</span>xs) res <span class="ot">=</span> go xs (res<span class="op">++</span>[f x])</span>
 <span id="cb73-8"><a href="#cb73-8" aria-hidden="true" tabindex="-1"></a>          go []     res <span class="ot">=</span> res</span></code></pre></div>
-<div class="sourceCode" id="cb74"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb74-1"><a href="#cb74-1" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span> (<span class="fu">map</span> <span class="fu">succ</span> [<span class="dv">0</span><span class="op">..</span>]) <span class="op">==&gt;</span> <span class="fu">head</span> (<span class="fu">succ</span> <span class="dv">0</span> <span class="op">:</span> <span class="fu">map</span> <span class="fu">succ</span> [<span class="dv">1</span><span class="op">..</span>]) <span class="op">==&gt;</span> <span class="fu">succ</span> <span class="dv">0</span> <span class="op">==&gt;</span> <span class="dv">1</span></span>
+<div class="sourceCode" id="cb74"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb74-1"><a href="#cb74-1" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span> (<span class="fu">map</span> <span class="fu">succ</span> [<span class="dv">0</span><span class="op">..</span>]) <span class="op">==&gt;</span> <span class="fu">head</span> (<span class="fu">succ</span> <span class="dv">0</span> <span class="op">:</span> <span class="fu">map</span> <span class="fu">succ</span> [<span class="dv">1</span><span class="op">..</span>]) <span class="op">==&gt;</span> <span class="fu">succ</span> <span class="dv">0</span> <span class="op">==&gt;</span> <span class="dv">1</span></span>
 <span id="cb74-2"><a href="#cb74-2" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span> (mapTailRecursive <span class="fu">succ</span> [<span class="dv">0</span><span class="op">..</span>])</span>
 <span id="cb74-3"><a href="#cb74-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="fu">head</span> (go [<span class="dv">0</span><span class="op">..</span>] [])</span>
 <span id="cb74-4"><a href="#cb74-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="fu">head</span> (go [<span class="dv">1</span><span class="op">..</span>] ([]<span class="op">++</span>[<span class="fu">succ</span> <span class="dv">0</span>]))</span>
 <span id="cb74-5"><a href="#cb74-5" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="fu">head</span> (go [<span class="dv">2</span><span class="op">..</span>] ([]<span class="op">++</span>[<span class="fu">succ</span> <span class="dv">0</span>]<span class="op">++</span>[<span class="fu">succ</span> <span class="dv">1</span>]))</span>
 <span id="cb74-6"><a href="#cb74-6" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="fu">head</span> (go [<span class="dv">3</span><span class="op">..</span>] ([]<span class="op">++</span>[<span class="fu">succ</span> <span class="dv">0</span>]<span class="op">++</span>[<span class="fu">succ</span> <span class="dv">1</span>]<span class="op">++</span>[<span class="fu">succ</span> <span class="dv">2</span>]))</span>
 <span id="cb74-7"><a href="#cb74-7" aria-hidden="true" tabindex="-1"></a>  <span class="co">--  never terminates</span></span></code></pre></div>
-<div class="sourceCode" id="cb75"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb75-1"><a href="#cb75-1" aria-hidden="true" tabindex="-1"></a><span class="fu">drop</span><span class="ot"> ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [a]</span>
+<div class="sourceCode" id="cb75"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb75-1"><a href="#cb75-1" aria-hidden="true" tabindex="-1"></a><span class="fu">drop</span><span class="ot"> ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [a]</span>
 <span id="cb75-2"><a href="#cb75-2" aria-hidden="true" tabindex="-1"></a><span class="fu">drop</span> <span class="dv">0</span> xs <span class="ot">=</span> xs</span>
 <span id="cb75-3"><a href="#cb75-3" aria-hidden="true" tabindex="-1"></a><span class="fu">drop</span> _ [] <span class="ot">=</span> []</span>
 <span id="cb75-4"><a href="#cb75-4" aria-hidden="true" tabindex="-1"></a><span class="fu">drop</span> n (x<span class="op">:</span>xs) <span class="ot">=</span> <span class="fu">drop</span> (n<span class="op">-</span><span class="dv">1</span>) xs</span>
@@ -1219,41 +1728,66 @@ square   ==>   *   ==>   *   ==>  16
 <span id="cb75-6"><a href="#cb75-6" aria-hidden="true" tabindex="-1"></a><span class="ot">myDrop ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [a]</span>
 <span id="cb75-7"><a href="#cb75-7" aria-hidden="true" tabindex="-1"></a>myDrop <span class="dv">0</span> xs <span class="ot">=</span> xs</span>
 <span id="cb75-8"><a href="#cb75-8" aria-hidden="true" tabindex="-1"></a>myDrop n xs <span class="ot">=</span> <span class="kw">if</span> n <span class="op">&gt;</span> <span class="fu">length</span> xs <span class="kw">then</span> [] <span class="kw">else</span> myDrop (n<span class="op">-</span><span class="dv">1</span>) (<span class="fu">tail</span> xs)</span></code></pre></div>
-<div class="sourceCode" id="cb76"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb76-1"><a href="#cb76-1" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span> (<span class="fu">drop</span> <span class="dv">2</span> [<span class="dv">0</span><span class="op">..</span>]) <span class="op">==&gt;</span> <span class="fu">head</span> (<span class="fu">drop</span> <span class="dv">1</span> [<span class="dv">1</span><span class="op">..</span>]) <span class="op">==&gt;</span> <span class="fu">head</span> (<span class="fu">drop</span> <span class="dv">0</span> [<span class="dv">2</span><span class="op">..</span>]) <span class="op">==&gt;</span> <span class="fu">head</span> [<span class="dv">2</span><span class="op">..</span>] <span class="op">==&gt;</span> <span class="dv">2</span></span>
+<div class="sourceCode" id="cb76"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb76-1"><a href="#cb76-1" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span> (<span class="fu">drop</span> <span class="dv">2</span> [<span class="dv">0</span><span class="op">..</span>]) <span class="op">==&gt;</span> <span class="fu">head</span> (<span class="fu">drop</span> <span class="dv">1</span> [<span class="dv">1</span><span class="op">..</span>]) <span class="op">==&gt;</span> <span class="fu">head</span> (<span class="fu">drop</span> <span class="dv">0</span> [<span class="dv">2</span><span class="op">..</span>]) <span class="op">==&gt;</span> <span class="fu">head</span> [<span class="dv">2</span><span class="op">..</span>] <span class="op">==&gt;</span> <span class="dv">2</span></span>
 <span id="cb76-2"><a href="#cb76-2" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span> (myDrop <span class="dv">2</span> [<span class="dv">0</span><span class="op">..</span>])</span>
 <span id="cb76-3"><a href="#cb76-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="fu">head</span> (<span class="kw">if</span> n <span class="op">&gt;</span> <span class="fu">length</span> [<span class="dv">0</span><span class="op">..</span>] <span class="kw">then</span> [] <span class="kw">else</span> myDrop (n<span class="op">-</span><span class="dv">1</span>) (<span class="fu">tail</span> [<span class="dv">0</span><span class="op">..</span>]))</span>
 <span id="cb76-4"><a href="#cb76-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="fu">head</span> (<span class="kw">if</span> n <span class="op">&gt;</span> <span class="dv">1</span><span class="op">+</span><span class="fu">length</span> [<span class="dv">1</span><span class="op">..</span>] <span class="kw">then</span> [] <span class="kw">else</span> myDrop (n<span class="op">-</span><span class="dv">1</span>) (<span class="fu">tail</span> [<span class="dv">0</span><span class="op">..</span>]))</span>
 <span id="cb76-5"><a href="#cb76-5" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="fu">head</span> (<span class="kw">if</span> n <span class="op">&gt;</span> <span class="dv">1</span><span class="op">+</span><span class="dv">1</span><span class="op">+</span><span class="fu">length</span> [<span class="dv">2</span><span class="op">..</span>] <span class="kw">then</span> [] <span class="kw">else</span> myDrop (n<span class="op">-</span><span class="dv">1</span>) (<span class="fu">tail</span> [<span class="dv">0</span><span class="op">..</span>]))</span>
 <span id="cb76-6"><a href="#cb76-6" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="fu">head</span> (<span class="kw">if</span> n <span class="op">&gt;</span> <span class="dv">1</span><span class="op">+</span><span class="dv">1</span><span class="op">+</span><span class="dv">1</span><span class="op">+</span><span class="fu">length</span> [<span class="dv">3</span><span class="op">..</span>] <span class="kw">then</span> [] <span class="kw">else</span> myDrop (n<span class="op">-</span><span class="dv">1</span>) (<span class="fu">tail</span> [<span class="dv">0</span><span class="op">..</span>]))</span>
 <span id="cb76-7"><a href="#cb76-7" aria-hidden="true" tabindex="-1"></a>  <span class="co">--  never terminates</span></span></code></pre></div>
-<p>Pretty much all the list functions in the standard library are written in this form, for example:</p>
-<div class="sourceCode" id="cb77"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb77-1"><a href="#cb77-1" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span> (<span class="fu">takeWhile</span> (<span class="op">&gt;=</span><span class="dv">0</span>) [<span class="dv">0</span><span class="op">..</span>]) <span class="op">==&gt;</span> <span class="dv">0</span></span>
+<p>Pretty much all the list functions in the standard library are
+written in this form, for example:</p>
+<div class="sourceCode" id="cb77"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb77-1"><a href="#cb77-1" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span> (<span class="fu">takeWhile</span> (<span class="op">&gt;=</span><span class="dv">0</span>) [<span class="dv">0</span><span class="op">..</span>]) <span class="op">==&gt;</span> <span class="dv">0</span></span>
 <span id="cb77-2"><a href="#cb77-2" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span> (<span class="fu">concat</span> (<span class="fu">repeat</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>])) <span class="op">==&gt;</span> <span class="dv">1</span></span>
 <span id="cb77-3"><a href="#cb77-3" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span> (<span class="fu">zip</span> [<span class="dv">0</span><span class="op">..</span>] [<span class="dv">2</span><span class="op">..</span>]) <span class="op">==&gt;</span> (<span class="dv">0</span>,<span class="dv">2</span>)</span>
 <span id="cb77-4"><a href="#cb77-4" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span> (<span class="fu">filter</span> <span class="fu">even</span> [<span class="dv">3</span><span class="op">..</span>]) <span class="op">==&gt;</span> <span class="dv">4</span></span></code></pre></div>
-</section>
-<section id="interlude-adding-strictness" data-number="2.6">
-<h2 data-number="10.6"><span class="header-section-number">10.6</span> Interlude: Adding Strictness</h2>
-<p>Remember <code>foldr</code> from part 1? Let’s have a look at its cousin <code>foldl</code>. Here’s the definition of <code>foldl</code> for lists (it’s actually part of the <code>Foldable</code> typeclass and so works for various other types too). While <code>foldr</code> processes a list right-to-left, <code>foldl</code> processes a list left-to-right. To be a bit more exact, <code>foldr</code> <em>associates to the right</em> while <code>foldl</code> <em>associates to the left</em>. Note the difference in the next example:</p>
-<div class="sourceCode" id="cb78"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb78-1"><a href="#cb78-1" aria-hidden="true" tabindex="-1"></a><span class="fu">foldr</span> (<span class="op">+</span>) <span class="dv">0</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]  <span class="op">==&gt;</span>  <span class="dv">1</span><span class="op">+</span>(<span class="dv">2</span><span class="op">+</span>(<span class="dv">3</span><span class="op">+</span><span class="dv">0</span>))</span>
+<h2 data-number="10.6" id="interlude-adding-strictness"><span
+class="header-section-number">10.6</span> Interlude: Adding
+Strictness</h2>
+<p>Remember <code>foldr</code> from part 1? Let’s have a look at its
+cousin <code>foldl</code>. Here’s the definition of <code>foldl</code>
+for lists (it’s actually part of the <code>Foldable</code> typeclass and
+so works for various other types too). While <code>foldr</code>
+processes a list right-to-left, <code>foldl</code> processes a list
+left-to-right. To be a bit more exact, <code>foldr</code> <em>associates
+to the right</em> while <code>foldl</code> <em>associates to the
+left</em>. Note the difference in the next example:</p>
+<div class="sourceCode" id="cb78"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb78-1"><a href="#cb78-1" aria-hidden="true" tabindex="-1"></a><span class="fu">foldr</span> (<span class="op">+</span>) <span class="dv">0</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]  <span class="op">==&gt;</span>  <span class="dv">1</span><span class="op">+</span>(<span class="dv">2</span><span class="op">+</span>(<span class="dv">3</span><span class="op">+</span><span class="dv">0</span>))</span>
 <span id="cb78-2"><a href="#cb78-2" aria-hidden="true" tabindex="-1"></a><span class="fu">foldl</span> (<span class="op">+</span>) <span class="dv">0</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]  <span class="op">==&gt;</span>  ((<span class="dv">0</span><span class="op">+</span><span class="dv">1</span>)<span class="op">+</span><span class="dv">2</span>)<span class="op">+</span><span class="dv">3</span></span></code></pre></div>
-<p>Here are the definitions of <code>foldl</code> and <code>foldr</code>:</p>
-<div class="sourceCode" id="cb79"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb79-1"><a href="#cb79-1" aria-hidden="true" tabindex="-1"></a><span class="fu">foldl</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> a) <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> [b] <span class="ot">-&gt;</span> a</span>
+<p>Here are the definitions of <code>foldl</code> and
+<code>foldr</code>:</p>
+<div class="sourceCode" id="cb79"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb79-1"><a href="#cb79-1" aria-hidden="true" tabindex="-1"></a><span class="fu">foldl</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> a) <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> [b] <span class="ot">-&gt;</span> a</span>
 <span id="cb79-2"><a href="#cb79-2" aria-hidden="true" tabindex="-1"></a><span class="fu">foldl</span> f z [] <span class="ot">=</span> z</span>
 <span id="cb79-3"><a href="#cb79-3" aria-hidden="true" tabindex="-1"></a><span class="fu">foldl</span> f z (x<span class="op">:</span>xs) <span class="ot">=</span> <span class="fu">foldl</span> f (f z x) xs</span></code></pre></div>
-<div class="sourceCode" id="cb80"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb80-1"><a href="#cb80-1" aria-hidden="true" tabindex="-1"></a><span class="fu">foldr</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> b</span>
+<div class="sourceCode" id="cb80"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb80-1"><a href="#cb80-1" aria-hidden="true" tabindex="-1"></a><span class="fu">foldr</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> b</span>
 <span id="cb80-2"><a href="#cb80-2" aria-hidden="true" tabindex="-1"></a><span class="fu">foldr</span> f y []     <span class="ot">=</span> y</span>
 <span id="cb80-3"><a href="#cb80-3" aria-hidden="true" tabindex="-1"></a><span class="fu">foldr</span> f y (x<span class="op">:</span>xs) <span class="ot">=</span> f x (<span class="fu">foldr</span> f y xs)</span></code></pre></div>
-<p>As <code>foldr f y (x:xs) ==&gt; f x (foldr f y xs)</code>, it enables lazy evaluation to focus on <code>f</code> on the second step. Hence, <code>foldr</code> works nicely with lazy or short-circuiting operations:</p>
-<div class="sourceCode" id="cb81"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb81-1"><a href="#cb81-1" aria-hidden="true" tabindex="-1"></a>    <span class="fu">foldr</span> (<span class="op">&amp;&amp;</span>) <span class="dt">True</span> [<span class="dt">False</span>,<span class="dt">False</span>,<span class="dt">False</span>]</span>
+<p>As <code>foldr f y (x:xs) ==&gt; f x (foldr f y xs)</code>, it
+enables lazy evaluation to focus on <code>f</code> on the second step.
+Hence, <code>foldr</code> works nicely with lazy or short-circuiting
+operations:</p>
+<div class="sourceCode" id="cb81"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb81-1"><a href="#cb81-1" aria-hidden="true" tabindex="-1"></a>    <span class="fu">foldr</span> (<span class="op">&amp;&amp;</span>) <span class="dt">True</span> [<span class="dt">False</span>,<span class="dt">False</span>,<span class="dt">False</span>]</span>
 <span id="cb81-2"><a href="#cb81-2" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="dt">False</span> <span class="op">&amp;&amp;</span> (<span class="fu">foldr</span> (<span class="op">&amp;&amp;</span>) <span class="dt">True</span> [<span class="dt">False</span>,<span class="dt">False</span>])</span>
 <span id="cb81-3"><a href="#cb81-3" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="dt">False</span></span></code></pre></div>
-<div class="sourceCode" id="cb82"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb82-1"><a href="#cb82-1" aria-hidden="true" tabindex="-1"></a>    <span class="fu">head</span> (<span class="fu">foldr</span> (<span class="op">++</span>) [] [<span class="st">&quot;Hello&quot;</span>,<span class="st">&quot;World&quot;</span>,<span class="st">&quot;lorem&quot;</span>,<span class="st">&quot;ipsum&quot;</span>])</span>
+<div class="sourceCode" id="cb82"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb82-1"><a href="#cb82-1" aria-hidden="true" tabindex="-1"></a>    <span class="fu">head</span> (<span class="fu">foldr</span> (<span class="op">++</span>) [] [<span class="st">&quot;Hello&quot;</span>,<span class="st">&quot;World&quot;</span>,<span class="st">&quot;lorem&quot;</span>,<span class="st">&quot;ipsum&quot;</span>])</span>
 <span id="cb82-2"><a href="#cb82-2" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">head</span> (<span class="st">&quot;Hello&quot;</span> <span class="op">++</span> (<span class="fu">foldr</span> (<span class="op">++</span>) [] [<span class="st">&quot;World&quot;</span>,<span class="st">&quot;lorem&quot;</span>,<span class="st">&quot;ipsum&quot;</span>]))</span>
 <span id="cb82-3"><a href="#cb82-3" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">head</span> (<span class="ch">&#39;H&#39;</span><span class="op">:</span>(<span class="st">&quot;ello&quot;</span> <span class="op">++</span> (<span class="fu">foldr</span> (<span class="op">++</span>) [] [<span class="st">&quot;World&quot;</span>,<span class="st">&quot;lorem&quot;</span>,<span class="st">&quot;ipsum&quot;</span>])))</span>
 <span id="cb82-4"><a href="#cb82-4" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="ch">&#39;H&#39;</span></span></code></pre></div>
-<p>However <code>foldl</code> needs to process the whole list in order to produce a (WHNF) value. The reason is that <code>foldl</code> remains in the leftmost-outermost position for as long as its list argument remains non-empty. This makes <code>foldl</code> the priority for lazy evaluation. Only after the list becomes empty does the evaluation proceed into simplifying the folded values.</p>
-<div class="sourceCode" id="cb83"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb83-1"><a href="#cb83-1" aria-hidden="true" tabindex="-1"></a>    <span class="fu">foldl</span> (<span class="op">&amp;&amp;</span>) <span class="dt">True</span> [<span class="dt">False</span>,<span class="dt">False</span>,<span class="dt">False</span>]</span>
+<p>However <code>foldl</code> needs to process the whole list in order
+to produce a (WHNF) value. The reason is that <code>foldl</code> remains
+in the leftmost-outermost position for as long as its list argument
+remains non-empty. This makes <code>foldl</code> the priority for lazy
+evaluation. Only after the list becomes empty does the evaluation
+proceed into simplifying the folded values.</p>
+<div class="sourceCode" id="cb83"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb83-1"><a href="#cb83-1" aria-hidden="true" tabindex="-1"></a>    <span class="fu">foldl</span> (<span class="op">&amp;&amp;</span>) <span class="dt">True</span> [<span class="dt">False</span>,<span class="dt">False</span>,<span class="dt">False</span>]</span>
 <span id="cb83-2"><a href="#cb83-2" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">foldl</span> (<span class="op">&amp;&amp;</span>) (<span class="dt">True</span><span class="op">&amp;&amp;</span><span class="dt">False</span>) [<span class="dt">False</span>,<span class="dt">False</span>]</span>
 <span id="cb83-3"><a href="#cb83-3" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">foldl</span> (<span class="op">&amp;&amp;</span>) ((<span class="dt">True</span><span class="op">&amp;&amp;</span><span class="dt">False</span>)<span class="op">&amp;&amp;</span><span class="dt">False</span>) [<span class="dt">False</span>]</span>
 <span id="cb83-4"><a href="#cb83-4" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">foldl</span> (<span class="op">&amp;&amp;</span>) (((<span class="dt">True</span><span class="op">&amp;&amp;</span><span class="dt">False</span>)<span class="op">&amp;&amp;</span><span class="dt">False</span>)<span class="op">&amp;&amp;</span><span class="dt">False</span>) []</span>
@@ -1261,7 +1795,8 @@ square   ==>   *   ==>   *   ==>  16
 <span id="cb83-6"><a href="#cb83-6" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> (    <span class="dt">False</span>    <span class="op">&amp;&amp;</span><span class="dt">False</span>)<span class="op">&amp;&amp;</span><span class="dt">False</span></span>
 <span id="cb83-7"><a href="#cb83-7" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span>              <span class="dt">False</span>    <span class="op">&amp;&amp;</span><span class="dt">False</span></span>
 <span id="cb83-8"><a href="#cb83-8" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span>                      <span class="dt">False</span></span></code></pre></div>
-<div class="sourceCode" id="cb84"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb84-1"><a href="#cb84-1" aria-hidden="true" tabindex="-1"></a>    <span class="fu">head</span> (<span class="fu">foldl</span> (<span class="op">++</span>) [] [<span class="st">&quot;Hello&quot;</span>,<span class="st">&quot;World&quot;</span>,<span class="st">&quot;lorem&quot;</span>,<span class="st">&quot;ipsum&quot;</span>])</span>
+<div class="sourceCode" id="cb84"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb84-1"><a href="#cb84-1" aria-hidden="true" tabindex="-1"></a>    <span class="fu">head</span> (<span class="fu">foldl</span> (<span class="op">++</span>) [] [<span class="st">&quot;Hello&quot;</span>,<span class="st">&quot;World&quot;</span>,<span class="st">&quot;lorem&quot;</span>,<span class="st">&quot;ipsum&quot;</span>])</span>
 <span id="cb84-2"><a href="#cb84-2" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">head</span> (<span class="fu">foldl</span> (<span class="op">++</span>) ([]<span class="op">++</span><span class="st">&quot;Hello&quot;</span>) [<span class="st">&quot;World&quot;</span>,<span class="st">&quot;lorem&quot;</span>,<span class="st">&quot;ipsum&quot;</span>])</span>
 <span id="cb84-3"><a href="#cb84-3" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">head</span> (<span class="fu">foldl</span> (<span class="op">++</span>) (([]<span class="op">++</span><span class="st">&quot;Hello&quot;</span>)<span class="op">++</span><span class="st">&quot;World&quot;</span>) [<span class="st">&quot;lorem&quot;</span>,<span class="st">&quot;ipsum&quot;</span>])</span>
 <span id="cb84-4"><a href="#cb84-4" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">head</span> (<span class="fu">foldl</span> (<span class="op">++</span>) ((([]<span class="op">++</span><span class="st">&quot;Hello&quot;</span>)<span class="op">++</span><span class="st">&quot;World&quot;</span>)<span class="op">++</span><span class="st">&quot;lorem&quot;</span>) [<span class="st">&quot;ipsum&quot;</span>])</span>
@@ -1280,8 +1815,14 @@ square   ==>   *   ==>   *   ==>  16
 <span id="cb84-17"><a href="#cb84-17" aria-hidden="true" tabindex="-1"></a><span class="co">-- now the last ++ can operate</span></span>
 <span id="cb84-18"><a href="#cb84-18" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">head</span> (<span class="ch">&#39;H&#39;</span><span class="op">:</span>(__<span class="op">++</span><span class="st">&quot;ipsum&quot;</span>))</span>
 <span id="cb84-19"><a href="#cb84-19" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="ch">&#39;H&#39;</span></span></code></pre></div>
-<p>So why use <code>foldl</code> at all? Let’s return to our first fold example again. Now, since <code>+</code> is a strict operation, both types of fold need to build up an expression with lots of <code>+</code>s. The Haskell implementation needs to track this expression in memory, which is why a problem like this is called a <em>space leak</em>.</p>
-<div class="sourceCode" id="cb85"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb85-1"><a href="#cb85-1" aria-hidden="true" tabindex="-1"></a>    <span class="fu">foldr</span> (<span class="op">+</span>) <span class="dv">0</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
+<p>So why use <code>foldl</code> at all? Let’s return to our first fold
+example again. Now, since <code>+</code> is a strict operation, both
+types of fold need to build up an expression with lots of
+<code>+</code>s. The Haskell implementation needs to track this
+expression in memory, which is why a problem like this is called a
+<em>space leak</em>.</p>
+<div class="sourceCode" id="cb85"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb85-1"><a href="#cb85-1" aria-hidden="true" tabindex="-1"></a>    <span class="fu">foldr</span> (<span class="op">+</span>) <span class="dv">0</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb85-2"><a href="#cb85-2" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="dv">1</span> <span class="op">+</span> <span class="fu">foldr</span> (<span class="op">+</span>) <span class="dv">0</span> [<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb85-3"><a href="#cb85-3" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="dv">1</span> <span class="op">+</span> (<span class="dv">2</span> <span class="op">+</span> <span class="fu">foldr</span> (<span class="op">+</span>) <span class="dv">0</span> [<span class="dv">3</span>])</span>
 <span id="cb85-4"><a href="#cb85-4" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="dv">1</span> <span class="op">+</span> (<span class="dv">2</span> <span class="op">+</span> (<span class="dv">3</span> <span class="op">+</span> <span class="fu">foldr</span> (<span class="op">+</span>) <span class="dv">0</span> []))</span>
@@ -1289,7 +1830,8 @@ square   ==>   *   ==>   *   ==>  16
 <span id="cb85-6"><a href="#cb85-6" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="dv">1</span> <span class="op">+</span> (<span class="dv">2</span> <span class="op">+</span> <span class="dv">3</span>)</span>
 <span id="cb85-7"><a href="#cb85-7" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="dv">1</span> <span class="op">+</span> <span class="dv">5</span></span>
 <span id="cb85-8"><a href="#cb85-8" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="dv">6</span></span></code></pre></div>
-<div class="sourceCode" id="cb86"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb86-1"><a href="#cb86-1" aria-hidden="true" tabindex="-1"></a>    <span class="fu">foldl</span> (<span class="op">+</span>) <span class="dv">0</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
+<div class="sourceCode" id="cb86"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb86-1"><a href="#cb86-1" aria-hidden="true" tabindex="-1"></a>    <span class="fu">foldl</span> (<span class="op">+</span>) <span class="dv">0</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb86-2"><a href="#cb86-2" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">foldl</span> (<span class="op">+</span>) (<span class="dv">0</span><span class="op">+</span><span class="dv">1</span>) [<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb86-3"><a href="#cb86-3" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">foldl</span> (<span class="op">+</span>) ((<span class="dv">0</span><span class="op">+</span><span class="dv">1</span>)<span class="op">+</span><span class="dv">2</span>) [<span class="dv">3</span>]</span>
 <span id="cb86-4"><a href="#cb86-4" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">foldl</span> (<span class="op">+</span>) (((<span class="dv">0</span><span class="op">+</span><span class="dv">1</span>)<span class="op">+</span><span class="dv">2</span>)<span class="op">+</span><span class="dv">3</span>) []</span>
@@ -1297,8 +1839,11 @@ square   ==>   *   ==>   *   ==>  16
 <span id="cb86-6"><a href="#cb86-6" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> (  <span class="dv">1</span>  <span class="op">+</span><span class="dv">2</span>)<span class="op">+</span><span class="dv">3</span></span>
 <span id="cb86-7"><a href="#cb86-7" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span>       <span class="dv">3</span>  <span class="op">+</span><span class="dv">3</span></span>
 <span id="cb86-8"><a href="#cb86-8" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span>          <span class="dv">6</span></span></code></pre></div>
-<p>Now let’s instead look at what happens when we use <code>foldl'</code>, a version of <code>foldl</code> that forces its second argument!</p>
-<div class="sourceCode" id="cb87"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb87-1"><a href="#cb87-1" aria-hidden="true" tabindex="-1"></a>    foldl&#39; (<span class="op">+</span>) <span class="dv">0</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
+<p>Now let’s instead look at what happens when we use
+<code>foldl'</code>, a version of <code>foldl</code> that forces its
+second argument!</p>
+<div class="sourceCode" id="cb87"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb87-1"><a href="#cb87-1" aria-hidden="true" tabindex="-1"></a>    foldl&#39; (<span class="op">+</span>) <span class="dv">0</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb87-2"><a href="#cb87-2" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> foldl&#39; (<span class="op">+</span>) (<span class="dv">0</span><span class="op">+</span><span class="dv">1</span>) [<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb87-3"><a href="#cb87-3" aria-hidden="true" tabindex="-1"></a><span class="co">-- force second argument</span></span>
 <span id="cb87-4"><a href="#cb87-4" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> foldl&#39; (<span class="op">+</span>) <span class="dv">1</span> [<span class="dv">2</span>,<span class="dv">3</span>]</span>
@@ -1309,13 +1854,21 @@ square   ==>   *   ==>   *   ==>  16
 <span id="cb87-9"><a href="#cb87-9" aria-hidden="true" tabindex="-1"></a><span class="co">-- force second argument</span></span>
 <span id="cb87-10"><a href="#cb87-10" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> foldl&#39; (<span class="op">+</span>) <span class="dv">6</span> []</span>
 <span id="cb87-11"><a href="#cb87-11" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="dv">6</span></span></code></pre></div>
-<p>Now the work is performed incrementally while scanning the list. No space leak! Sometimes too much laziness can cause space leaks, and a bit of strictness can fix them.</p>
-<p>You can find <code>foldl'</code> in the <code>Data.List</code> module, and it works just like this. But how could one implement <code>foldl'</code>? We certainly know by now how to do it for a specific type, say <code>Int</code>. We just add a pattern match on the second argument that doesn’t change the semantics of the function.</p>
-<div class="sourceCode" id="cb88"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb88-1"><a href="#cb88-1" aria-hidden="true" tabindex="-1"></a><span class="ot">foldl&#39;Int ::</span> (<span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span>) <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<p>Now the work is performed incrementally while scanning the list. No
+space leak! Sometimes too much laziness can cause space leaks, and a bit
+of strictness can fix them.</p>
+<p>You can find <code>foldl'</code> in the <code>Data.List</code>
+module, and it works just like this. But how could one implement
+<code>foldl'</code>? We certainly know by now how to do it for a
+specific type, say <code>Int</code>. We just add a pattern match on the
+second argument that doesn’t change the semantics of the function.</p>
+<div class="sourceCode" id="cb88"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb88-1"><a href="#cb88-1" aria-hidden="true" tabindex="-1"></a><span class="ot">foldl&#39;Int ::</span> (<span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span>) <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb88-2"><a href="#cb88-2" aria-hidden="true" tabindex="-1"></a>foldl&#39;Int f z [] <span class="ot">=</span> z</span>
 <span id="cb88-3"><a href="#cb88-3" aria-hidden="true" tabindex="-1"></a>foldl&#39;Int f <span class="dv">0</span> (x<span class="op">:</span>xs) <span class="ot">=</span> foldl&#39;Int f (f <span class="dv">0</span> x) xs</span>
 <span id="cb88-4"><a href="#cb88-4" aria-hidden="true" tabindex="-1"></a>foldl&#39;Int f z (x<span class="op">:</span>xs) <span class="ot">=</span> foldl&#39;Int f (f z x) xs</span></code></pre></div>
-<div class="sourceCode" id="cb89"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb89-1"><a href="#cb89-1" aria-hidden="true" tabindex="-1"></a>    foldl&#39;Int (<span class="op">+</span>) <span class="dv">0</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
+<div class="sourceCode" id="cb89"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb89-1"><a href="#cb89-1" aria-hidden="true" tabindex="-1"></a>    foldl&#39;Int (<span class="op">+</span>) <span class="dv">0</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb89-2"><a href="#cb89-2" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> foldl&#39;Int (<span class="op">+</span>) (<span class="dv">0</span><span class="op">+</span><span class="dv">1</span>) [<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb89-3"><a href="#cb89-3" aria-hidden="true" tabindex="-1"></a><span class="co">-- to be able to pick between the second and third equations, (0+1) is forced</span></span>
 <span id="cb89-4"><a href="#cb89-4" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> foldl&#39;Int (<span class="op">+</span>) <span class="dv">1</span> [<span class="dv">2</span>,<span class="dv">3</span>]</span>
@@ -1326,8 +1879,15 @@ square   ==>   *   ==>   *   ==>  16
 <span id="cb89-9"><a href="#cb89-9" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> foldl&#39;Int (<span class="op">+</span>) (<span class="dv">3</span><span class="op">+</span><span class="dv">3</span>) []</span>
 <span id="cb89-10"><a href="#cb89-10" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="dv">3</span><span class="op">+</span><span class="dv">3</span></span>
 <span id="cb89-11"><a href="#cb89-11" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="dv">6</span></span></code></pre></div>
-<p>To write a generic implementation of <code>foldl'</code> we need to introduce a new built-in function, <code>seq</code>. The call <code>seq a b</code> evaluates to <code>b</code> but forces <code>a</code> into WHNF. Here are some examples of using <code>seq</code> in GHCi. To demonstrate what gets evaluated, we use the special value <code>undefined</code>, which causes an error if something tries to evaluate it into WHNF.</p>
-<div class="sourceCode" id="cb90"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb90-1"><a href="#cb90-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">seq</span> (<span class="fu">not</span> <span class="dt">True</span>) <span class="dv">3</span></span>
+<p>To write a generic implementation of <code>foldl'</code> we need to
+introduce a new built-in function, <code>seq</code>. The call
+<code>seq a b</code> evaluates to <code>b</code> but forces
+<code>a</code> into WHNF. Here are some examples of using
+<code>seq</code> in GHCi. To demonstrate what gets evaluated, we use the
+special value <code>undefined</code>, which causes an error if something
+tries to evaluate it into WHNF.</p>
+<div class="sourceCode" id="cb90"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb90-1"><a href="#cb90-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">seq</span> (<span class="fu">not</span> <span class="dt">True</span>) <span class="dv">3</span></span>
 <span id="cb90-2"><a href="#cb90-2" aria-hidden="true" tabindex="-1"></a><span class="dv">3</span></span>
 <span id="cb90-3"><a href="#cb90-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">seq</span> <span class="fu">undefined</span> <span class="dv">3</span></span>
 <span id="cb90-4"><a href="#cb90-4" aria-hidden="true" tabindex="-1"></a><span class="op">***</span> <span class="dt">Exception</span><span class="op">:</span> Prelude.undefined</span>
@@ -1337,11 +1897,15 @@ square   ==>   *   ==>   *   ==>  16
 <span id="cb90-8"><a href="#cb90-8" aria-hidden="true" tabindex="-1"></a><span class="op">***</span> <span class="dt">Exception</span><span class="op">:</span> Prelude.undefined</span>
 <span id="cb90-9"><a href="#cb90-9" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">let</span> f x <span class="ot">=</span> f x <span class="kw">in</span> <span class="fu">seq</span> (f <span class="dv">3</span>) <span class="dv">3</span></span>
 <span id="cb90-10"><a href="#cb90-10" aria-hidden="true" tabindex="-1"></a><span class="co">-- ...infinite recursion</span></span></code></pre></div>
-<p>As an example of using <code>seq</code> in a function, here’s a version of <code>head</code> that doesn’t work for infinite lists (since it evaluates the last element of the list):</p>
-<div class="sourceCode" id="cb91"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb91-1"><a href="#cb91-1" aria-hidden="true" tabindex="-1"></a><span class="ot">strictHead ::</span> [a] <span class="ot">-&gt;</span> a</span>
+<p>As an example of using <code>seq</code> in a function, here’s a
+version of <code>head</code> that doesn’t work for infinite lists (since
+it evaluates the last element of the list):</p>
+<div class="sourceCode" id="cb91"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb91-1"><a href="#cb91-1" aria-hidden="true" tabindex="-1"></a><span class="ot">strictHead ::</span> [a] <span class="ot">-&gt;</span> a</span>
 <span id="cb91-2"><a href="#cb91-2" aria-hidden="true" tabindex="-1"></a>strictHead xs <span class="ot">=</span> <span class="fu">seq</span> (<span class="fu">last</span> xs) (<span class="fu">head</span> xs)</span></code></pre></div>
 <p>Let’s play around with it in GHCi:</p>
-<div class="sourceCode" id="cb92"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb92-1"><a href="#cb92-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">head</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
+<div class="sourceCode" id="cb92"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb92-1"><a href="#cb92-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">head</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb92-2"><a href="#cb92-2" aria-hidden="true" tabindex="-1"></a><span class="dv">1</span></span>
 <span id="cb92-3"><a href="#cb92-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> strictHead [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb92-4"><a href="#cb92-4" aria-hidden="true" tabindex="-1"></a><span class="dv">1</span></span>
@@ -1353,12 +1917,18 @@ square   ==>   *   ==>   *   ==>  16
 <span id="cb92-10"><a href="#cb92-10" aria-hidden="true" tabindex="-1"></a><span class="dv">1</span></span>
 <span id="cb92-11"><a href="#cb92-11" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> strictHead [<span class="dv">1</span><span class="op">..</span>]</span>
 <span id="cb92-12"><a href="#cb92-12" aria-hidden="true" tabindex="-1"></a><span class="co">-- ...infinite recursion</span></span></code></pre></div>
-<p>Finally, here’s a definition for <code>foldl'</code>. Note how we need to introduce sharing of a new variable, <code>z'</code>, to be able to make <code>seq</code> evaluate the new value and then use it in a recursive call. The new definition is also used in a more detailed evaluation of <code>foldl' (+) 0 [1,2,3]</code> below.</p>
-<div class="sourceCode" id="cb93"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb93-1"><a href="#cb93-1" aria-hidden="true" tabindex="-1"></a><span class="ot">foldl&#39; ::</span> (a <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> a) <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> [b] <span class="ot">-&gt;</span> a</span>
+<p>Finally, here’s a definition for <code>foldl'</code>. Note how we
+need to introduce sharing of a new variable, <code>z'</code>, to be able
+to make <code>seq</code> evaluate the new value and then use it in a
+recursive call. The new definition is also used in a more detailed
+evaluation of <code>foldl' (+) 0 [1,2,3]</code> below.</p>
+<div class="sourceCode" id="cb93"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb93-1"><a href="#cb93-1" aria-hidden="true" tabindex="-1"></a><span class="ot">foldl&#39; ::</span> (a <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> a) <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> [b] <span class="ot">-&gt;</span> a</span>
 <span id="cb93-2"><a href="#cb93-2" aria-hidden="true" tabindex="-1"></a>foldl&#39; f z [] <span class="ot">=</span> z</span>
 <span id="cb93-3"><a href="#cb93-3" aria-hidden="true" tabindex="-1"></a>foldl&#39; f z (x<span class="op">:</span>xs) <span class="ot">=</span> <span class="kw">let</span> z&#39; <span class="ot">=</span> f z x</span>
 <span id="cb93-4"><a href="#cb93-4" aria-hidden="true" tabindex="-1"></a>                    <span class="kw">in</span> <span class="fu">seq</span> z&#39; (foldl&#39; f z&#39; xs)</span></code></pre></div>
-<div class="sourceCode" id="cb94"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb94-1"><a href="#cb94-1" aria-hidden="true" tabindex="-1"></a>    foldl&#39; (<span class="op">+</span>) <span class="dv">0</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
+<div class="sourceCode" id="cb94"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb94-1"><a href="#cb94-1" aria-hidden="true" tabindex="-1"></a>    foldl&#39; (<span class="op">+</span>) <span class="dv">0</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb94-2"><a href="#cb94-2" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">seq</span> (<span class="dv">0</span><span class="op">+</span><span class="dv">1</span>) (foldl&#39; (<span class="op">+</span>) (<span class="dv">0</span><span class="op">+</span><span class="dv">1</span>) [<span class="dv">2</span>,<span class="dv">3</span>])  <span class="co">-- seq forces first argument</span></span>
 <span id="cb94-3"><a href="#cb94-3" aria-hidden="true" tabindex="-1"></a>          <span class="op">|</span>                 <span class="op">|</span></span>
 <span id="cb94-4"><a href="#cb94-4" aria-hidden="true" tabindex="-1"></a>          <span class="op">+-----</span>sharing<span class="co">-----&#39;</span></span>
@@ -1372,18 +1942,41 @@ square   ==>   *   ==>   *   ==>  16
 <span id="cb94-12"><a href="#cb94-12" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="fu">seq</span>   <span class="dv">6</span>   (foldl&#39; (<span class="op">+</span>)   <span class="dv">6</span>   [])</span>
 <span id="cb94-13"><a href="#cb94-13" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> foldl&#39; (<span class="op">+</span>)   <span class="dv">6</span>   []</span>
 <span id="cb94-14"><a href="#cb94-14" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="dv">6</span></span></code></pre></div>
-<p>We won’t dive deeper into this subject on this course, but it’s important that you’re aware that <code>seq</code> exists. You can find more about <code>seq</code> on <a href="https://wiki.haskell.org/Seq">the Haskell Wiki</a> and learn more about when it is necessary to add strictness in <a href="https://book.realworldhaskell.org/read/profiling-and-optimization.html">Real World Haskell</a>. Often it’s nicer to use <em>bang patterns</em> instead of <code>seq</code>, as discussed by <a href="https://www.fpcomplete.com/blog/2017/09/all-about-strictness/">FPComplete</a> and <a href="https://book.realworldhaskell.org/read/profiling-and-optimization.html">Real World Haskell</a>.</p>
-</section>
-<section id="newtype-declarations" data-number="2.7">
-<h2 data-number="10.7"><span class="header-section-number">10.7</span> Newtype Declarations</h2>
-<p>Recall lecture 7. Sometimes we need boxed types. There’s a special keyword <code>newtype</code> that can be used instead of <code>data</code> when a boxed type is needed. <code>newtype</code> expects exactly one constructor, with exactly one field. For instance,</p>
-<div class="sourceCode" id="cb95"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb95-1"><a href="#cb95-1" aria-hidden="true" tabindex="-1"></a><span class="kw">newtype</span> <span class="dt">Money</span> <span class="ot">=</span> <span class="dt">Cents</span> <span class="dt">Int</span></span></code></pre></div>
+<p>We won’t dive deeper into this subject on this course, but it’s
+important that you’re aware that <code>seq</code> exists. You can find
+more about <code>seq</code> on <a
+href="https://wiki.haskell.org/Seq">the Haskell Wiki</a> and learn more
+about when it is necessary to add strictness in <a
+href="https://book.realworldhaskell.org/read/profiling-and-optimization.html">Real
+World Haskell</a>. Often it’s nicer to use <em>bang patterns</em>
+instead of <code>seq</code>, as discussed by <a
+href="https://www.fpcomplete.com/blog/2017/09/all-about-strictness/">FPComplete</a>
+and <a
+href="https://book.realworldhaskell.org/read/profiling-and-optimization.html">Real
+World Haskell</a>.</p>
+<h2 data-number="10.7" id="newtype-declarations"><span
+class="header-section-number">10.7</span> Newtype Declarations</h2>
+<p>Recall lecture 7. Sometimes we need boxed types. There’s a special
+keyword <code>newtype</code> that can be used instead of
+<code>data</code> when a boxed type is needed. <code>newtype</code>
+expects exactly one constructor, with exactly one field. For
+instance,</p>
+<div class="sourceCode" id="cb95"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb95-1"><a href="#cb95-1" aria-hidden="true" tabindex="-1"></a><span class="kw">newtype</span> <span class="dt">Money</span> <span class="ot">=</span> <span class="dt">Cents</span> <span class="dt">Int</span></span></code></pre></div>
 <p>However, the following won’t work, you need <code>data</code>:</p>
-<div class="sourceCode" id="cb96"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb96-1"><a href="#cb96-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- the compiler won&#39;t accept these!</span></span>
+<div class="sourceCode" id="cb96"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb96-1"><a href="#cb96-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- the compiler won&#39;t accept these!</span></span>
 <span id="cb96-2"><a href="#cb96-2" aria-hidden="true" tabindex="-1"></a><span class="kw">newtype</span> <span class="dt">Currency</span> <span class="ot">=</span> <span class="dt">Dollars</span> <span class="dt">Int</span> <span class="op">|</span> <span class="dt">Euros</span> <span class="dt">Int</span></span>
 <span id="cb96-3"><a href="#cb96-3" aria-hidden="true" tabindex="-1"></a><span class="kw">newtype</span> <span class="dt">Money</span> <span class="ot">=</span> <span class="dt">Money</span> <span class="dt">Int</span> <span class="dt">Int</span></span></code></pre></div>
-<p>So what’s the difference? In terms of writing code, nothing. You work with a <code>newtype</code> exactly as you would with a <code>data</code>. However, the memory layout is different. Using <code>data</code> introduces an indirection layer (the constructor), but using <code>newtype</code> doesn’t. The indirection for <code>data</code> is necessary to support multiple constructors and multiple fields. An illustration:</p>
-<div class="sourceCode" id="cb97"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb97-1"><a href="#cb97-1" aria-hidden="true" tabindex="-1"></a>code<span class="op">:</span>                                 memory<span class="op">:</span></span>
+<p>So what’s the difference? In terms of writing code, nothing. You work
+with a <code>newtype</code> exactly as you would with a
+<code>data</code>. However, the memory layout is different. Using
+<code>data</code> introduces an indirection layer (the constructor), but
+using <code>newtype</code> doesn’t. The indirection for
+<code>data</code> is necessary to support multiple constructors and
+multiple fields. An illustration:</p>
+<div class="sourceCode" id="cb97"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb97-1"><a href="#cb97-1" aria-hidden="true" tabindex="-1"></a>code<span class="op">:</span>                                 memory<span class="op">:</span></span>
 <span id="cb97-2"><a href="#cb97-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb97-3"><a href="#cb97-3" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Money</span> <span class="ot">=</span> <span class="dt">Cents</span> <span class="dt">Int</span>                x <span class="op">--&gt;</span> <span class="dt">Cents</span> <span class="op">--&gt;</span> <span class="dv">100</span></span>
 <span id="cb97-4"><a href="#cb97-4" aria-hidden="true" tabindex="-1"></a>x <span class="ot">=</span> <span class="dt">Cents</span> <span class="dv">100</span></span>
@@ -1391,8 +1984,15 @@ square   ==>   *   ==>   *   ==>  16
 <span id="cb97-6"><a href="#cb97-6" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb97-7"><a href="#cb97-7" aria-hidden="true" tabindex="-1"></a><span class="kw">newtype</span> <span class="dt">Money</span> <span class="ot">=</span> <span class="dt">Cents</span> <span class="dt">Int</span>             x <span class="op">--&gt;</span> <span class="dv">100</span></span>
 <span id="cb97-8"><a href="#cb97-8" aria-hidden="true" tabindex="-1"></a>x <span class="ot">=</span> <span class="dt">Cents</span> <span class="dv">100</span></span></code></pre></div>
-<p>This difference has many repercussions. First of all, <code>newtype</code> is more efficient: the type can be said to “disappear” when compiling. The type is still checked though, so you get type safety without any performance impact. Secondly, newtypes are <em>strict</em>. Concretely, this means that <code>Money x</code> is in weak head normal form only if <code>x</code> is in WHNF. This can be witnessed in GHCi:</p>
-<div class="sourceCode" id="cb98"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb98-1"><a href="#cb98-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- if we use data, Cents undefined is in WHNF</span></span>
+<p>This difference has many repercussions. First of all,
+<code>newtype</code> is more efficient: the type can be said to
+“disappear” when compiling. The type is still checked though, so you get
+type safety without any performance impact. Secondly, newtypes are
+<em>strict</em>. Concretely, this means that <code>Money x</code> is in
+weak head normal form only if <code>x</code> is in WHNF. This can be
+witnessed in GHCi:</p>
+<div class="sourceCode" id="cb98"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb98-1"><a href="#cb98-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- if we use data, Cents undefined is in WHNF</span></span>
 <span id="cb98-2"><a href="#cb98-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">data</span> <span class="dt">Money</span> <span class="ot">=</span> <span class="dt">Cents</span> <span class="dt">Int</span></span>
 <span id="cb98-3"><a href="#cb98-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">seq</span> (<span class="dt">Cents</span> <span class="fu">undefined</span>) <span class="dt">True</span></span>
 <span id="cb98-4"><a href="#cb98-4" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span>
@@ -1401,36 +2001,57 @@ square   ==>   *   ==>   *   ==>  16
 <span id="cb98-7"><a href="#cb98-7" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">newtype</span> <span class="dt">Money</span> <span class="ot">=</span> <span class="dt">Cents</span> <span class="dt">Int</span></span>
 <span id="cb98-8"><a href="#cb98-8" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">seq</span> (<span class="dt">Cents</span> <span class="fu">undefined</span>) <span class="dt">True</span></span>
 <span id="cb98-9"><a href="#cb98-9" aria-hidden="true" tabindex="-1"></a><span class="op">***</span> <span class="dt">Exception</span><span class="op">:</span> Prelude.undefined</span></code></pre></div>
-<p>So when should you use <code>newtype</code>? In general it’s best to use <code>newtype</code> whenever you have a single-field single-constructor datatype. However nothing will go catastrophically wrong if you always use <code>data</code>. The <code>newtype</code> pattern is also often used when you need to define a different type class instance for a type. Here’s an example that defines a number type with an inverted ordering</p>
-<div class="sourceCode" id="cb99"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb99-1"><a href="#cb99-1" aria-hidden="true" tabindex="-1"></a><span class="kw">newtype</span> <span class="dt">Inverted</span> <span class="ot">=</span> <span class="dt">Inverted</span> <span class="dt">Int</span></span>
+<p>So when should you use <code>newtype</code>? In general it’s best to
+use <code>newtype</code> whenever you have a single-field
+single-constructor datatype. However nothing will go catastrophically
+wrong if you always use <code>data</code>. The <code>newtype</code>
+pattern is also often used when you need to define a different type
+class instance for a type. Here’s an example that defines a number type
+with an inverted ordering</p>
+<div class="sourceCode" id="cb99"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb99-1"><a href="#cb99-1" aria-hidden="true" tabindex="-1"></a><span class="kw">newtype</span> <span class="dt">Inverted</span> <span class="ot">=</span> <span class="dt">Inverted</span> <span class="dt">Int</span></span>
 <span id="cb99-2"><a href="#cb99-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">deriving</span> (<span class="dt">Show</span>, <span class="dt">Eq</span>)</span>
 <span id="cb99-3"><a href="#cb99-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb99-4"><a href="#cb99-4" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Ord</span> <span class="dt">Inverted</span> <span class="kw">where</span></span>
 <span id="cb99-5"><a href="#cb99-5" aria-hidden="true" tabindex="-1"></a>  <span class="fu">compare</span> (<span class="dt">Inverted</span> i) (<span class="dt">Inverted</span> j) <span class="ot">=</span> <span class="fu">compare</span> j i</span></code></pre></div>
-<div class="sourceCode" id="cb100"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb100-1"><a href="#cb100-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.List</span><span class="op">&gt;</span> <span class="fu">sort</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
+<div class="sourceCode" id="cb100"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb100-1"><a href="#cb100-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.List</span><span class="op">&gt;</span> <span class="fu">sort</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb100-2"><a href="#cb100-2" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb100-3"><a href="#cb100-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.List</span><span class="op">&gt;</span> <span class="fu">sort</span> [<span class="dt">Inverted</span> <span class="dv">1</span>,<span class="dt">Inverted</span> <span class="dv">2</span>,<span class="dt">Inverted</span> <span class="dv">3</span>]</span>
 <span id="cb100-4"><a href="#cb100-4" aria-hidden="true" tabindex="-1"></a>[<span class="dt">Inverted</span> <span class="dv">3</span>,<span class="dt">Inverted</span> <span class="dv">2</span>,<span class="dt">Inverted</span> <span class="dv">1</span>]</span></code></pre></div>
-</section>
-<section id="something-fun-tying-the-knot" data-number="2.8">
-<h2 data-number="10.8"><span class="header-section-number">10.8</span> Something Fun: Tying the Knot</h2>
+<h2 data-number="10.8" id="something-fun-tying-the-knot"><span
+class="header-section-number">10.8</span> Something Fun: Tying the
+Knot</h2>
 <!-- TODO: Should we discuss recursion and corecursion? Like
 generating and consuming lists or streams. -->
-<p>Now that we know about sharing and path copying, we can make our own <em>cyclic datastructures</em>. Remember the <code>cycle</code> examples from the list lecture?</p>
-<div class="sourceCode" id="cb101"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb101-1"><a href="#cb101-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">take</span> <span class="dv">21</span> <span class="op">$</span> <span class="fu">cycle</span> <span class="st">&quot;asdf&quot;</span></span>
+<p>Now that we know about sharing and path copying, we can make our own
+<em>cyclic datastructures</em>. Remember the <code>cycle</code> examples
+from the list lecture?</p>
+<div class="sourceCode" id="cb101"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb101-1"><a href="#cb101-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">take</span> <span class="dv">21</span> <span class="op">$</span> <span class="fu">cycle</span> <span class="st">&quot;asdf&quot;</span></span>
 <span id="cb101-2"><a href="#cb101-2" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;asdfasdfasdfasdfasdfa&quot;</span></span></code></pre></div>
 <p>This is what it looks like in memory:</p>
 <p><img src="img/DAG4.svg" /></p>
-<p>Earlier it was said that Haskell data forms directed graphs in memory. This is an example of a directed graph with a cycle.</p>
-<p>How can we define structures like this? We just give a value a name, and refer to that name in the value itself. That is, the value is <em>recursive</em> or <em>self-referential</em>. This trick is known as <em>tying the knot</em>. A simple example:</p>
-<div class="sourceCode" id="cb102"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb102-1"><a href="#cb102-1" aria-hidden="true" tabindex="-1"></a>  code                     memory</span>
+<p>Earlier it was said that Haskell data forms directed graphs in
+memory. This is an example of a directed graph with a cycle.</p>
+<p>How can we define structures like this? We just give a value a name,
+and refer to that name in the value itself. That is, the value is
+<em>recursive</em> or <em>self-referential</em>. This trick is known as
+<em>tying the knot</em>. A simple example:</p>
+<div class="sourceCode" id="cb102"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb102-1"><a href="#cb102-1" aria-hidden="true" tabindex="-1"></a>  code                     memory</span>
 <span id="cb102-2"><a href="#cb102-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb102-3"><a href="#cb102-3" aria-hidden="true" tabindex="-1"></a><span class="kw">let</span> xs <span class="ot">=</span> <span class="dv">1</span><span class="op">:</span><span class="dv">2</span><span class="op">:</span>xs      xs <span class="ot">-&gt;</span> (<span class="dv">1</span><span class="op">:</span>) <span class="ot">-&gt;</span> (<span class="dv">2</span><span class="op">:</span>) <span class="op">-+</span></span>
 <span id="cb102-4"><a href="#cb102-4" aria-hidden="true" tabindex="-1"></a> <span class="kw">in</span> xs                      <span class="op">^</span>            <span class="op">|</span></span>
 <span id="cb102-5"><a href="#cb102-5" aria-hidden="true" tabindex="-1"></a>                            <span class="op">+------------+</span></span></code></pre></div>
-<p>Note how we use the name <code>xs</code> inside the definition of <code>xs</code>. When we make a recursive definition like this, sharing causes it to turn into a cyclic structure in memory.</p>
-<p>A more fun example: a simple adventure game where the world is a self-referential structure. Note how the cyclic structure is built with local definitions that refer to each other.</p>
-<div class="sourceCode" id="cb103"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb103-1"><a href="#cb103-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Room</span> <span class="ot">=</span> <span class="dt">Room</span> <span class="dt">String</span> [(<span class="dt">String</span>,<span class="dt">Room</span>)]</span>
+<p>Note how we use the name <code>xs</code> inside the definition of
+<code>xs</code>. When we make a recursive definition like this, sharing
+causes it to turn into a cyclic structure in memory.</p>
+<p>A more fun example: a simple adventure game where the world is a
+self-referential structure. Note how the cyclic structure is built with
+local definitions that refer to each other.</p>
+<div class="sourceCode" id="cb103"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb103-1"><a href="#cb103-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Room</span> <span class="ot">=</span> <span class="dt">Room</span> <span class="dt">String</span> [(<span class="dt">String</span>,<span class="dt">Room</span>)]</span>
 <span id="cb103-2"><a href="#cb103-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb103-3"><a href="#cb103-3" aria-hidden="true" tabindex="-1"></a><span class="ot">describe ::</span> <span class="dt">Room</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb103-4"><a href="#cb103-4" aria-hidden="true" tabindex="-1"></a>describe (<span class="dt">Room</span> s _) <span class="ot">=</span> s</span>
@@ -1453,7 +2074,8 @@ generating and consuming lists or streams. -->
 <span id="cb103-21"><a href="#cb103-21" aria-hidden="true" tabindex="-1"></a>play room [] <span class="ot">=</span> [describe room]</span>
 <span id="cb103-22"><a href="#cb103-22" aria-hidden="true" tabindex="-1"></a>play room (d<span class="op">:</span>ds) <span class="ot">=</span> <span class="kw">case</span> move room d <span class="kw">of</span> <span class="dt">Nothing</span> <span class="ot">-&gt;</span> [describe room]</span>
 <span id="cb103-23"><a href="#cb103-23" aria-hidden="true" tabindex="-1"></a>                                       <span class="dt">Just</span> r <span class="ot">-&gt;</span> describe room <span class="op">:</span> play r ds</span></code></pre></div>
-<div class="sourceCode" id="cb104"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb104-1"><a href="#cb104-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> play world [<span class="st">&quot;Stay&quot;</span>,<span class="st">&quot;Enter cave&quot;</span>,<span class="st">&quot;Go deeper&quot;</span>,<span class="st">&quot;Go back&quot;</span>,<span class="st">&quot;Go deeper&quot;</span>,<span class="st">&quot;Go right&quot;</span>]</span>
+<div class="sourceCode" id="cb104"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb104-1"><a href="#cb104-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> play world [<span class="st">&quot;Stay&quot;</span>,<span class="st">&quot;Enter cave&quot;</span>,<span class="st">&quot;Go deeper&quot;</span>,<span class="st">&quot;Go back&quot;</span>,<span class="st">&quot;Go deeper&quot;</span>,<span class="st">&quot;Go right&quot;</span>]</span>
 <span id="cb104-2"><a href="#cb104-2" aria-hidden="true" tabindex="-1"></a>[<span class="st">&quot;It&#39;s a flowery meadow next to a cliff.&quot;</span>,</span>
 <span id="cb104-3"><a href="#cb104-3" aria-hidden="true" tabindex="-1"></a> <span class="st">&quot;It&#39;s a flowery meadow next to a cliff.&quot;</span>,</span>
 <span id="cb104-4"><a href="#cb104-4" aria-hidden="true" tabindex="-1"></a> <span class="st">&quot;You are in a cave&quot;</span>,</span>
@@ -1461,7 +2083,8 @@ generating and consuming lists or streams. -->
 <span id="cb104-6"><a href="#cb104-6" aria-hidden="true" tabindex="-1"></a> <span class="st">&quot;You are in a cave&quot;</span>,</span>
 <span id="cb104-7"><a href="#cb104-7" aria-hidden="true" tabindex="-1"></a> <span class="st">&quot;This is a very dark tunnel. It seems you can either go left or right.&quot;</span>,</span>
 <span id="cb104-8"><a href="#cb104-8" aria-hidden="true" tabindex="-1"></a> <span class="st">&quot;A green light from a computer terminal floods the room. The terminal says &lt;&lt;loop&gt;&gt;.&quot;</span>]</span></code></pre></div>
-<p>Here’s what the <code>world</code> of the game looks like in memory:</p>
+<p>Here’s what the <code>world</code> of the game looks like in
+memory:</p>
 <pre><code>               ,-----------------,
                v                 |
          +-----------------------|-----------------+
@@ -1481,13 +2104,23 @@ tunnel--&gt;|Room &quot;This is...&quot; [&quot;Go back&quot; o, &quot;Go left&q
          +---------------------+                +-----------------------------|-+
 pit-----&gt;|Room &quot;You fall...&quot; []|    treasure---&gt;|Room &quot;A green...&quot; [&quot;Go back&quot; o]|
          +---------------------+                +-------------------------------+</code></pre>
-<p>We’ve now seen three types of recursion. Recursive functions call themselves. Recursive types allow us to express arbitarily large structures. Recursive values are one way to implement infinite structures.</p>
-</section>
-<section id="something-fun-debug.trace" data-number="2.9">
-<h2 data-number="10.9"><span class="header-section-number">10.9</span> Something Fun: Debug.Trace</h2>
-<p>Even though Haskell is a pure programming language, we can sometimes gain insights by sprinkling in a bit of impurity.</p>
-<p>We can use the function <code>trace :: String -&gt; a -&gt; a</code> from the module <code>Debug.Trace</code> to peek into Haskell evaluation. The expression <code>trace "message" x</code> is the same as <code>x</code>, but prints <code>message</code> when it is evaluated (forced). We can use <code>trace</code> to witness the laziness of the <code>||</code> operator:</p>
-<div class="sourceCode" id="cb106"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb106-1"><a href="#cb106-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="dt">Debug.Trace</span></span>
+<p>We’ve now seen three types of recursion. Recursive functions call
+themselves. Recursive types allow us to express arbitarily large
+structures. Recursive values are one way to implement infinite
+structures.</p>
+<h2 data-number="10.9" id="something-fun-debug.trace"><span
+class="header-section-number">10.9</span> Something Fun:
+Debug.Trace</h2>
+<p>Even though Haskell is a pure programming language, we can sometimes
+gain insights by sprinkling in a bit of impurity.</p>
+<p>We can use the function <code>trace :: String -&gt; a -&gt; a</code>
+from the module <code>Debug.Trace</code> to peek into Haskell
+evaluation. The expression <code>trace "message" x</code> is the same as
+<code>x</code>, but prints <code>message</code> when it is evaluated
+(forced). We can use <code>trace</code> to witness the laziness of the
+<code>||</code> operator:</p>
+<div class="sourceCode" id="cb106"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb106-1"><a href="#cb106-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="dt">Debug.Trace</span></span>
 <span id="cb106-2"><a href="#cb106-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Debug.Trace</span><span class="op">&gt;</span> trace <span class="st">&quot;a&quot;</span> <span class="dt">True</span></span>
 <span id="cb106-3"><a href="#cb106-3" aria-hidden="true" tabindex="-1"></a>a</span>
 <span id="cb106-4"><a href="#cb106-4" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span>
@@ -1498,8 +2131,14 @@ pit-----&gt;|Room &quot;You fall...&quot; []|    treasure---&gt;|Room &quot;A gr
 <span id="cb106-9"><a href="#cb106-9" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Debug.Trace</span><span class="op">&gt;</span> trace <span class="st">&quot;a&quot;</span> <span class="dt">True</span> <span class="op">||</span> trace <span class="st">&quot;b&quot;</span> <span class="dt">True</span></span>
 <span id="cb106-10"><a href="#cb106-10" aria-hidden="true" tabindex="-1"></a>a</span>
 <span id="cb106-11"><a href="#cb106-11" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span></code></pre></div>
-<p>We can also have a look at when list elements are evaluated. Note how <code>length</code> doesn’t need to evaluate the elements of the list, and <code>sum</code> needs to evaluate all of them. (To be precise, <code>head xs</code> doesn’t actually evaluate the first element of <code>xs</code>, but returns it to GHCi, which evaluates it in order to show it.)</p>
-<div class="sourceCode" id="cb107"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb107-1"><a href="#cb107-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Debug.Trace</span><span class="op">&gt;</span> <span class="fu">head</span> [trace <span class="st">&quot;first&quot;</span> <span class="dv">1</span>, trace <span class="st">&quot;second&quot;</span> <span class="dv">2</span>, trace <span class="st">&quot;third&quot;</span> <span class="dv">3</span>]</span>
+<p>We can also have a look at when list elements are evaluated. Note how
+<code>length</code> doesn’t need to evaluate the elements of the list,
+and <code>sum</code> needs to evaluate all of them. (To be precise,
+<code>head xs</code> doesn’t actually evaluate the first element of
+<code>xs</code>, but returns it to GHCi, which evaluates it in order to
+show it.)</p>
+<div class="sourceCode" id="cb107"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb107-1"><a href="#cb107-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Debug.Trace</span><span class="op">&gt;</span> <span class="fu">head</span> [trace <span class="st">&quot;first&quot;</span> <span class="dv">1</span>, trace <span class="st">&quot;second&quot;</span> <span class="dv">2</span>, trace <span class="st">&quot;third&quot;</span> <span class="dv">3</span>]</span>
 <span id="cb107-2"><a href="#cb107-2" aria-hidden="true" tabindex="-1"></a>first</span>
 <span id="cb107-3"><a href="#cb107-3" aria-hidden="true" tabindex="-1"></a><span class="dv">1</span></span>
 <span id="cb107-4"><a href="#cb107-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Debug.Trace</span><span class="op">&gt;</span> <span class="fu">last</span> [trace <span class="st">&quot;first&quot;</span> <span class="dv">1</span>, trace <span class="st">&quot;second&quot;</span> <span class="dv">2</span>, trace <span class="st">&quot;third&quot;</span> <span class="dv">3</span>]</span>
@@ -1512,8 +2151,16 @@ pit-----&gt;|Room &quot;You fall...&quot; []|    treasure---&gt;|Room &quot;A gr
 <span id="cb107-11"><a href="#cb107-11" aria-hidden="true" tabindex="-1"></a>second</span>
 <span id="cb107-12"><a href="#cb107-12" aria-hidden="true" tabindex="-1"></a>first</span>
 <span id="cb107-13"><a href="#cb107-13" aria-hidden="true" tabindex="-1"></a><span class="dv">6</span></span></code></pre></div>
-<p><code>Debug.Trace</code> also offers useful variants of <code>trace</code>. A notable one is <code>traceShowId x</code> which prints <code>show x</code> and evaluates to <code>x</code>. Let’s verify the evaluation of our previous head-filter-map example using <code>traceShowId</code>. Note how even though we map <code>traceShowId</code> over the infinite list <code>[0..]</code>, only 6 values are actually evaluated. The last 243 is the returned value, not a trace print.</p>
-<div class="sourceCode" id="cb108"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb108-1"><a href="#cb108-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Debug.Trace</span><span class="op">&gt;</span> <span class="fu">head</span> (<span class="fu">filter</span> (<span class="op">&gt;</span><span class="dv">100</span>) (<span class="fu">map</span> (\x <span class="ot">-&gt;</span> traceShowId (<span class="dv">3</span><span class="op">^</span>x)) [<span class="dv">0</span><span class="op">..</span>]))</span>
+<p><code>Debug.Trace</code> also offers useful variants of
+<code>trace</code>. A notable one is <code>traceShowId x</code> which
+prints <code>show x</code> and evaluates to <code>x</code>. Let’s verify
+the evaluation of our previous head-filter-map example using
+<code>traceShowId</code>. Note how even though we map
+<code>traceShowId</code> over the infinite list <code>[0..]</code>, only
+6 values are actually evaluated. The last 243 is the returned value, not
+a trace print.</p>
+<div class="sourceCode" id="cb108"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb108-1"><a href="#cb108-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Debug.Trace</span><span class="op">&gt;</span> <span class="fu">head</span> (<span class="fu">filter</span> (<span class="op">&gt;</span><span class="dv">100</span>) (<span class="fu">map</span> (\x <span class="ot">-&gt;</span> traceShowId (<span class="dv">3</span><span class="op">^</span>x)) [<span class="dv">0</span><span class="op">..</span>]))</span>
 <span id="cb108-2"><a href="#cb108-2" aria-hidden="true" tabindex="-1"></a><span class="dv">1</span></span>
 <span id="cb108-3"><a href="#cb108-3" aria-hidden="true" tabindex="-1"></a><span class="dv">3</span></span>
 <span id="cb108-4"><a href="#cb108-4" aria-hidden="true" tabindex="-1"></a><span class="dv">9</span></span>
@@ -1521,18 +2168,24 @@ pit-----&gt;|Room &quot;You fall...&quot; []|    treasure---&gt;|Room &quot;A gr
 <span id="cb108-6"><a href="#cb108-6" aria-hidden="true" tabindex="-1"></a><span class="dv">81</span></span>
 <span id="cb108-7"><a href="#cb108-7" aria-hidden="true" tabindex="-1"></a><span class="dv">243</span></span>
 <span id="cb108-8"><a href="#cb108-8" aria-hidden="true" tabindex="-1"></a><span class="dv">243</span></span></code></pre></div>
-<p><code>Debug.Trace</code> is especially useful when you have an infinite recursion bug. Here’s an example:</p>
-<div class="sourceCode" id="cb109"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb109-1"><a href="#cb109-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- computes sums like 7+5+3+1</span></span>
+<p><code>Debug.Trace</code> is especially useful when you have an
+infinite recursion bug. Here’s an example:</p>
+<div class="sourceCode" id="cb109"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb109-1"><a href="#cb109-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- computes sums like 7+5+3+1</span></span>
 <span id="cb109-2"><a href="#cb109-2" aria-hidden="true" tabindex="-1"></a><span class="ot">sumEverySecond ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb109-3"><a href="#cb109-3" aria-hidden="true" tabindex="-1"></a>sumEverySecond <span class="dv">0</span> <span class="ot">=</span> <span class="dv">0</span></span>
 <span id="cb109-4"><a href="#cb109-4" aria-hidden="true" tabindex="-1"></a>sumEverySecond n <span class="ot">=</span> n <span class="op">+</span> sumEverySecond (n<span class="op">-</span><span class="dv">2</span>)</span></code></pre></div>
-<div class="sourceCode" id="cb110"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb110-1"><a href="#cb110-1" aria-hidden="true" tabindex="-1"></a>sumEverySecond <span class="dv">6</span> <span class="op">==&gt;</span> <span class="dv">12</span></span>
+<div class="sourceCode" id="cb110"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb110-1"><a href="#cb110-1" aria-hidden="true" tabindex="-1"></a>sumEverySecond <span class="dv">6</span> <span class="op">==&gt;</span> <span class="dv">12</span></span>
 <span id="cb110-2"><a href="#cb110-2" aria-hidden="true" tabindex="-1"></a>sumEverySecond <span class="dv">7</span> <span class="op">==&gt;</span> doesn&#39;t terminate</span></code></pre></div>
-<p>We can debug this by adding a <code>trace</code> to wrap the whole recursive case.</p>
-<div class="sourceCode" id="cb111"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb111-1"><a href="#cb111-1" aria-hidden="true" tabindex="-1"></a><span class="ot">sumEverySecond ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<p>We can debug this by adding a <code>trace</code> to wrap the whole
+recursive case.</p>
+<div class="sourceCode" id="cb111"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb111-1"><a href="#cb111-1" aria-hidden="true" tabindex="-1"></a><span class="ot">sumEverySecond ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb111-2"><a href="#cb111-2" aria-hidden="true" tabindex="-1"></a>sumEverySecond <span class="dv">0</span> <span class="ot">=</span> <span class="dv">0</span></span>
 <span id="cb111-3"><a href="#cb111-3" aria-hidden="true" tabindex="-1"></a>sumEverySecond n <span class="ot">=</span> trace (<span class="st">&quot;sumEverySecond &quot;</span><span class="op">++</span><span class="fu">show</span> n) (n <span class="op">+</span> sumEverySecond (n<span class="op">-</span><span class="dv">2</span>))</span></code></pre></div>
-<div class="sourceCode" id="cb112"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb112-1"><a href="#cb112-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Debug.Trace</span><span class="op">&gt;</span> sumEverySecond <span class="dv">6</span></span>
+<div class="sourceCode" id="cb112"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb112-1"><a href="#cb112-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Debug.Trace</span><span class="op">&gt;</span> sumEverySecond <span class="dv">6</span></span>
 <span id="cb112-2"><a href="#cb112-2" aria-hidden="true" tabindex="-1"></a>sumEverySecond <span class="dv">6</span></span>
 <span id="cb112-3"><a href="#cb112-3" aria-hidden="true" tabindex="-1"></a>sumEverySecond <span class="dv">4</span></span>
 <span id="cb112-4"><a href="#cb112-4" aria-hidden="true" tabindex="-1"></a>sumEverySecond <span class="dv">2</span></span>
@@ -1546,16 +2199,23 @@ pit-----&gt;|Room &quot;You fall...&quot; []|    treasure---&gt;|Room &quot;A gr
 <span id="cb112-12"><a href="#cb112-12" aria-hidden="true" tabindex="-1"></a>sumEverySecond <span class="op">-</span><span class="dv">3</span></span>
 <span id="cb112-13"><a href="#cb112-13" aria-hidden="true" tabindex="-1"></a>sumEverySecond <span class="op">-</span><span class="dv">5</span></span>
 <span id="cb112-14"><a href="#cb112-14" aria-hidden="true" tabindex="-1"></a><span class="co">-- and so on</span></span></code></pre></div>
-<p>A ha! The problem is that our recursion base case of <code>sumEverySecond 0</code> is not enough to stop the recursion.</p>
-<p>Finally, a word of caution. Using <code>trace</code>, and especially <code>traceShowId</code>, can cause things that would not otherwise get evaluated to get evaluated. For example:</p>
-<div class="sourceCode" id="cb113"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb113-1"><a href="#cb113-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Debug.Trace</span><span class="op">&gt;</span> traceHead xs <span class="ot">=</span> <span class="fu">head</span> (traceShowId xs)</span>
+<p>A ha! The problem is that our recursion base case of
+<code>sumEverySecond 0</code> is not enough to stop the recursion.</p>
+<p>Finally, a word of caution. Using <code>trace</code>, and especially
+<code>traceShowId</code>, can cause things that would not otherwise get
+evaluated to get evaluated. For example:</p>
+<div class="sourceCode" id="cb113"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb113-1"><a href="#cb113-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Debug.Trace</span><span class="op">&gt;</span> traceHead xs <span class="ot">=</span> <span class="fu">head</span> (traceShowId xs)</span>
 <span id="cb113-2"><a href="#cb113-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Debug.Trace</span><span class="op">&gt;</span> traceHead [<span class="dv">0</span><span class="op">..</span>]</span>
 <span id="cb113-3"><a href="#cb113-3" aria-hidden="true" tabindex="-1"></a><span class="co">-- never terminates since it&#39;s trying to show an infinite list</span></span></code></pre></div>
-<p>So feel free to use <code>Debug.Trace</code> when working on the exercises, but try to leave <code>trace</code> calls out of your final answers. Some exercise sets check your imports and disallow <code>Debug.Trace</code>.</p>
-<p>We’ll see a more principled way of dealing with side effects in the next lecture!</p>
-</section>
-<section id="quiz-1" data-number="2.10">
-<h2 data-number="10.10"><span class="header-section-number">10.10</span> Quiz</h2>
+<p>So feel free to use <code>Debug.Trace</code> when working on the
+exercises, but try to leave <code>trace</code> calls out of your final
+answers. Some exercise sets check your imports and disallow
+<code>Debug.Trace</code>.</p>
+<p>We’ll see a more principled way of dealing with side effects in the
+next lecture!</p>
+<h2 data-number="10.10" id="quiz-1"><span
+class="header-section-number">10.10</span> Quiz</h2>
 <p>Which of these statements is true?</p>
 <ol class="quiz">
 <li class="correct">
@@ -1568,7 +2228,8 @@ pit-----&gt;|Room &quot;You fall...&quot; []|    treasure---&gt;|Room &quot;A gr
 <code>reverse . id === id</code>
 </li>
 </ol>
-<p>Which of these is an infinite list that starts with <code>[0,1,2,1,2,1,2...]</code>?</p>
+<p>Which of these is an infinite list that starts with
+<code>[0,1,2,1,2,1,2...]</code>?</p>
 <ol class="quiz">
 <li>
 <code>cycle [0,1,2]</code>
@@ -1584,7 +2245,8 @@ pit-----&gt;|Room &quot;You fall...&quot; []|    treasure---&gt;|Room &quot;A gr
 </li>
 </ol>
 <p>What’s the next step when evaluating this expression?</p>
-<div class="sourceCode" id="cb114"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb114-1"><a href="#cb114-1" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span> (<span class="fu">map</span> <span class="fu">not</span> (<span class="dt">True</span><span class="op">:</span><span class="dt">False</span><span class="op">:</span>[]))</span></code></pre></div>
+<div class="sourceCode" id="cb114"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb114-1"><a href="#cb114-1" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span> (<span class="fu">map</span> <span class="fu">not</span> (<span class="dt">True</span><span class="op">:</span><span class="dt">False</span><span class="op">:</span>[]))</span></code></pre></div>
 <ol class="quiz">
 <li>
 <code>head (False : True : [])</code>
@@ -1615,7 +2277,8 @@ pit-----&gt;|Room &quot;You fall...&quot; []|    treasure---&gt;|Room &quot;A gr
 </li>
 </ol>
 <p>Which of these statements about the following function is true?</p>
-<div class="sourceCode" id="cb115"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb115-1"><a href="#cb115-1" aria-hidden="true" tabindex="-1"></a>f <span class="dv">0</span> x <span class="ot">=</span> <span class="dv">1</span><span class="op">+</span>x</span>
+<div class="sourceCode" id="cb115"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb115-1"><a href="#cb115-1" aria-hidden="true" tabindex="-1"></a>f <span class="dv">0</span> x <span class="ot">=</span> <span class="dv">1</span><span class="op">+</span>x</span>
 <span id="cb115-2"><a href="#cb115-2" aria-hidden="true" tabindex="-1"></a>f _ x <span class="ot">=</span> <span class="dv">2</span><span class="op">+</span>x</span></code></pre></div>
 <ol class="quiz">
 <li>
@@ -1632,7 +2295,8 @@ None of the above
 </li>
 </ol>
 <p>Does this function work with infinite lists as input? Why?</p>
-<div class="sourceCode" id="cb116"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb116-1"><a href="#cb116-1" aria-hidden="true" tabindex="-1"></a>f [] <span class="ot">=</span> []</span>
+<div class="sourceCode" id="cb116"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb116-1"><a href="#cb116-1" aria-hidden="true" tabindex="-1"></a>f [] <span class="ot">=</span> []</span>
 <span id="cb116-2"><a href="#cb116-2" aria-hidden="true" tabindex="-1"></a>f (x<span class="op">:</span>xs) <span class="ot">=</span> x <span class="op">:</span> <span class="fu">map</span> <span class="fu">not</span> xs</span></code></pre></div>
 <ol class="quiz">
 <li>
@@ -1642,20 +2306,23 @@ No, because it includes a <code>[]</code> case, which is never reached.
 No, because it uses <code>map</code>, which evaluates the whole list.
 </li>
 <li class="correct">
-Yes, because it only looks at the first element of the list before producing a WHNF value.
+Yes, because it only looks at the first element of the list before
+producing a WHNF value.
 </li>
 <li>
 Yes, because it calls <code>map</code>, which works with infinite lists.
 </li>
 </ol>
 <p>What about this one?</p>
-<div class="sourceCode" id="cb117"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb117-1"><a href="#cb117-1" aria-hidden="true" tabindex="-1"></a>f xs <span class="ot">=</span> <span class="fu">map</span> (<span class="op">+</span>(<span class="fu">sum</span> xs)) xs</span></code></pre></div>
+<div class="sourceCode" id="cb117"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb117-1"><a href="#cb117-1" aria-hidden="true" tabindex="-1"></a>f xs <span class="ot">=</span> <span class="fu">map</span> (<span class="op">+</span>(<span class="fu">sum</span> xs)) xs</span></code></pre></div>
 <ol class="quiz">
 <li>
 No, because it uses <code>map</code>, which evaluates the whole list.
 </li>
 <li class="correct">
-No, because computing the <code>head</code> of the result needs the whole input list.
+No, because computing the <code>head</code> of the result needs the
+whole input list.
 </li>
 <li>
 Yes, because it doesn’t includes a <code>[]</code> case
@@ -1664,36 +2331,42 @@ Yes, because it doesn’t includes a <code>[]</code> case
 Yes, because it calls <code>map</code>, which works with infinite lists.
 </li>
 </ol>
-</section>
-<section id="exercises-1" data-number="2.11">
-<h2 data-number="10.11"><span class="header-section-number">10.11</span> Exercises</h2>
+<h2 data-number="10.11" id="exercises-1"><span
+class="header-section-number">10.11</span> Exercises</h2>
 <ul>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set10a.hs">Set10a</a></li>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set10b.hs">Set10b</a></li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set10a.hs">Set10a</a></li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set10b.hs">Set10b</a></li>
 </ul>
-</section>
-</section>
-<section id="lecture-11-realworld---arealworld" data-number="3">
-<h1 data-number="11"><span class="header-section-number">11</span> Lecture 11: <code>RealWorld -&gt; (a,RealWorld)</code></h1>
-<section id="contents" data-number="3.1">
-<h2 data-number="11.1"><span class="header-section-number">11.1</span> Contents</h2>
+<h1 data-number="11" id="lecture-11-realworld---arealworld"><span
+class="header-section-number">11</span> Lecture 11:
+<code>RealWorld -&gt; (a,RealWorld)</code></h1>
+<h2 data-number="11.1" id="contents"><span
+class="header-section-number">11.1</span> Contents</h2>
 <ul>
 <li>IO</li>
 </ul>
-</section>
-<section id="youve-been-fooled" data-number="3.2">
-<h2 data-number="11.2"><span class="header-section-number">11.2</span> You’ve Been Fooled!</h2>
-<p>Forget what we talked about functional programming and purity. Actually, Haskell is the <em>world’s best imperative programming language</em>! Let’s start:</p>
-<div class="sourceCode" id="cb118"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb118-1"><a href="#cb118-1" aria-hidden="true" tabindex="-1"></a>questionnaire <span class="ot">=</span> <span class="kw">do</span></span>
+<h2 data-number="11.2" id="youve-been-fooled"><span
+class="header-section-number">11.2</span> You’ve Been Fooled!</h2>
+<p>Forget what we talked about functional programming and purity.
+Actually, Haskell is the <em>world’s best imperative programming
+language</em>! Let’s start:</p>
+<div class="sourceCode" id="cb118"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb118-1"><a href="#cb118-1" aria-hidden="true" tabindex="-1"></a>questionnaire <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb118-2"><a href="#cb118-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">putStrLn</span> <span class="st">&quot;Write something!&quot;</span></span>
 <span id="cb118-3"><a href="#cb118-3" aria-hidden="true" tabindex="-1"></a>  s <span class="ot">&lt;-</span> <span class="fu">getLine</span></span>
 <span id="cb118-4"><a href="#cb118-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">putStrLn</span> (<span class="st">&quot;You wrote: &quot;</span><span class="op">++</span>s)</span></code></pre></div>
-<div class="sourceCode" id="cb119"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb119-1"><a href="#cb119-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> questionnaire</span>
+<div class="sourceCode" id="cb119"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb119-1"><a href="#cb119-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> questionnaire</span>
 <span id="cb119-2"><a href="#cb119-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Write</span> something<span class="op">!</span></span>
 <span id="cb119-3"><a href="#cb119-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Haskell</span><span class="op">!</span></span>
 <span id="cb119-4"><a href="#cb119-4" aria-hidden="true" tabindex="-1"></a><span class="dt">You</span> wrote<span class="op">:</span> <span class="dt">Haskell</span><span class="op">!</span></span></code></pre></div>
-<p>Reading input and writing output was easy enough. We can also read stuff over the network. Here’s a complete Haskell program that fetches a some words from a URL using HTTP and prints them.</p>
-<div class="sourceCode" id="cb120"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb120-1"><a href="#cb120-1" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="dt">Network.HTTP</span></span>
+<p>Reading input and writing output was easy enough. We can also read
+stuff over the network. Here’s a complete Haskell program that fetches a
+some words from a URL using HTTP and prints them.</p>
+<div class="sourceCode" id="cb120"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb120-1"><a href="#cb120-1" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="dt">Network.HTTP</span></span>
 <span id="cb120-2"><a href="#cb120-2" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="dt">Control.Monad</span></span>
 <span id="cb120-3"><a href="#cb120-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb120-4"><a href="#cb120-4" aria-hidden="true" tabindex="-1"></a>main <span class="ot">=</span> <span class="kw">do</span></span>
@@ -1702,18 +2375,26 @@ Yes, because it calls <code>map</code>, which works with infinite lists.
 <span id="cb120-7"><a href="#cb120-7" aria-hidden="true" tabindex="-1"></a>  forM_ (<span class="fu">words</span> body) <span class="op">$</span> \w <span class="ot">-&gt;</span> <span class="kw">do</span></span>
 <span id="cb120-8"><a href="#cb120-8" aria-hidden="true" tabindex="-1"></a>     <span class="fu">putStr</span> <span class="st">&quot;word: &quot;</span></span>
 <span id="cb120-9"><a href="#cb120-9" aria-hidden="true" tabindex="-1"></a>     <span class="fu">putStrLn</span> w</span></code></pre></div>
-<p>You can find this program in the course repository as <a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Examples/FetchWords.hs"><code>exercises/Examples/FetchWords.hs</code></a>, and you can run it like this:</p>
+<p>You can find this program in the course repository as <a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Examples/FetchWords.hs"><code>exercises/Examples/FetchWords.hs</code></a>,
+and you can run it like this:</p>
 <pre><code>$ cd exercises/Examples
 $ stack runhaskell FetchWords.hs
 word: haskell
 word: for
 word: ever</code></pre>
 <p>What’s going on here? Let’s look at the types:</p>
-<div class="sourceCode" id="cb122"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb122-1"><a href="#cb122-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="fu">putStrLn</span></span>
+<div class="sourceCode" id="cb122"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb122-1"><a href="#cb122-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="fu">putStrLn</span></span>
 <span id="cb122-2"><a href="#cb122-2" aria-hidden="true" tabindex="-1"></a><span class="fu">putStrLn</span><span class="ot"> ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span>
 <span id="cb122-3"><a href="#cb122-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="fu">getLine</span></span>
 <span id="cb122-4"><a href="#cb122-4" aria-hidden="true" tabindex="-1"></a><span class="fu">getLine</span><span class="ot"> ::</span> <span class="dt">IO</span> <span class="dt">String</span></span></code></pre></div>
-<p>A value of type <code>IO a</code> is an <em>operation</em> that <em>produces</em> a value of type <code>a</code>. So <code>getLine</code> is an IO operation that produces a string. The <code>()</code> type is the so called <em>unit type</em>, its only value is <code>()</code>. It’s mostly used when an IO operation doesn’t return anything (but rather just has side effects).</p>
+<p>A value of type <code>IO a</code> is an <em>operation</em> that
+<em>produces</em> a value of type <code>a</code>. So
+<code>getLine</code> is an IO operation that produces a string. The
+<code>()</code> type is the so called <em>unit type</em>, its only value
+is <code>()</code>. It’s mostly used when an IO operation doesn’t return
+anything (but rather just has side effects).</p>
 <p>A comparison with Java (method) types might help:</p>
 <table>
 <thead>
@@ -1732,8 +2413,10 @@ word: ever</code></pre>
 <td style="text-align: left;"><code>int getSomething()</code></td>
 </tr>
 <tr class="odd">
-<td style="text-align: left;"><code>force :: a -&gt; b -&gt; IO ()</code></td>
-<td style="text-align: left;"><code>void force(a arg0, b arg1)</code></td>
+<td
+style="text-align: left;"><code>force :: a -&gt; b -&gt; IO ()</code></td>
+<td
+style="text-align: left;"><code>void force(a arg0, b arg1)</code></td>
 </tr>
 <tr class="even">
 <td style="text-align: left;"><code>mogrify :: c -&gt; IO d</code></td>
@@ -1741,33 +2424,48 @@ word: ever</code></pre>
 </tr>
 </tbody>
 </table>
-<p>IO operations can be combined into bigger operations using <em>do-notation</em>.</p>
-<div class="sourceCode" id="cb123"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb123-1"><a href="#cb123-1" aria-hidden="true" tabindex="-1"></a><span class="kw">do</span> operation</span>
+<p>IO operations can be combined into bigger operations using
+<em>do-notation</em>.</p>
+<div class="sourceCode" id="cb123"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb123-1"><a href="#cb123-1" aria-hidden="true" tabindex="-1"></a><span class="kw">do</span> operation</span>
 <span id="cb123-2"><a href="#cb123-2" aria-hidden="true" tabindex="-1"></a>   operation arg</span>
 <span id="cb123-3"><a href="#cb123-3" aria-hidden="true" tabindex="-1"></a>   variable <span class="ot">&lt;-</span> operationThatReturnsStuff</span>
 <span id="cb123-4"><a href="#cb123-4" aria-hidden="true" tabindex="-1"></a>   <span class="kw">let</span> var2 <span class="ot">=</span> expression</span>
 <span id="cb123-5"><a href="#cb123-5" aria-hidden="true" tabindex="-1"></a>   operationThatProducesTheResult var2</span></code></pre></div>
-<section id="examples" data-number="3.2.1">
-<h3 data-number="11.2.1"><span class="header-section-number">11.2.1</span> Examples</h3>
-<p>You can find useful IO operations in the standard library modules <a href="https://hackage.haskell.org/package/base-4.16.4.0/docs/Prelude.html">Prelude</a> and <a href="https://hackage.haskell.org/package/base-4.16.4.0/docs/System-IO.html">System.IO</a></p>
-<p>Here’s an IO operation that asks the user for a string, and prints out the length of the string.</p>
-<div class="sourceCode" id="cb124"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb124-1"><a href="#cb124-1" aria-hidden="true" tabindex="-1"></a><span class="ot">query ::</span> <span class="dt">IO</span> ()</span>
+<h3 data-number="11.2.1" id="examples"><span
+class="header-section-number">11.2.1</span> Examples</h3>
+<p>You can find useful IO operations in the standard library modules <a
+href="https://hackage.haskell.org/package/base-4.16.4.0/docs/Prelude.html">Prelude</a>
+and <a
+href="https://hackage.haskell.org/package/base-4.16.4.0/docs/System-IO.html">System.IO</a></p>
+<p>Here’s an IO operation that asks the user for a string, and prints
+out the length of the string.</p>
+<div class="sourceCode" id="cb124"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb124-1"><a href="#cb124-1" aria-hidden="true" tabindex="-1"></a><span class="ot">query ::</span> <span class="dt">IO</span> ()</span>
 <span id="cb124-2"><a href="#cb124-2" aria-hidden="true" tabindex="-1"></a>query <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb124-3"><a href="#cb124-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">putStrLn</span> <span class="st">&quot;Write something!&quot;</span>                    <span class="co">-- run an operation, ignore produced value</span></span>
 <span id="cb124-4"><a href="#cb124-4" aria-hidden="true" tabindex="-1"></a>  s <span class="ot">&lt;-</span> <span class="fu">getLine</span>                                   <span class="co">-- run an operation, capture produced value</span></span>
 <span id="cb124-5"><a href="#cb124-5" aria-hidden="true" tabindex="-1"></a>  <span class="kw">let</span> n <span class="ot">=</span> <span class="fu">length</span> s                               <span class="co">-- run a pure function</span></span>
 <span id="cb124-6"><a href="#cb124-6" aria-hidden="true" tabindex="-1"></a>  <span class="fu">putStrLn</span> (<span class="st">&quot;You wrote &quot;</span><span class="op">++</span><span class="fu">show</span> n<span class="op">++</span><span class="st">&quot; characters&quot;</span>) <span class="co">-- run an operation, passing on the produced value</span></span></code></pre></div>
-<div class="sourceCode" id="cb125"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb125-1"><a href="#cb125-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> query</span>
+<div class="sourceCode" id="cb125"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb125-1"><a href="#cb125-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> query</span>
 <span id="cb125-2"><a href="#cb125-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Write</span> something<span class="op">!</span></span>
 <span id="cb125-3"><a href="#cb125-3" aria-hidden="true" tabindex="-1"></a>lorem ipsum</span>
 <span id="cb125-4"><a href="#cb125-4" aria-hidden="true" tabindex="-1"></a><span class="dt">You</span> wrote <span class="dv">11</span> characters</span></code></pre></div>
-<p>The value produced by the last line of a <code>do</code> block is the value produced by the whole block. Note how <code>askForALine</code> has the same type as <code>getLine</code>, <code>IO String</code>:</p>
-<div class="sourceCode" id="cb126"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb126-1"><a href="#cb126-1" aria-hidden="true" tabindex="-1"></a><span class="ot">askForALine ::</span> <span class="dt">IO</span> <span class="dt">String</span></span>
+<p>The value produced by the last line of a <code>do</code> block is the
+value produced by the whole block. Note how <code>askForALine</code> has
+the same type as <code>getLine</code>, <code>IO String</code>:</p>
+<div class="sourceCode" id="cb126"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb126-1"><a href="#cb126-1" aria-hidden="true" tabindex="-1"></a><span class="ot">askForALine ::</span> <span class="dt">IO</span> <span class="dt">String</span></span>
 <span id="cb126-2"><a href="#cb126-2" aria-hidden="true" tabindex="-1"></a>askForALine <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb126-3"><a href="#cb126-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">putStrLn</span> <span class="st">&quot;Please give me a line&quot;</span></span>
 <span id="cb126-4"><a href="#cb126-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">getLine</span></span></code></pre></div>
-<p>In addition to IO operations like <code>query</code> you can also run IO operations that produce values, like <code>askForALine</code>, in GHCi. You can use <code>&lt;-</code> to capture the result of the operation into a variable if you want.</p>
-<div class="sourceCode" id="cb127"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb127-1"><a href="#cb127-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> askForALine</span>
+<p>In addition to IO operations like <code>query</code> you can also run
+IO operations that produce values, like <code>askForALine</code>, in
+GHCi. You can use <code>&lt;-</code> to capture the result of the
+operation into a variable if you want.</p>
+<div class="sourceCode" id="cb127"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb127-1"><a href="#cb127-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> askForALine</span>
 <span id="cb127-2"><a href="#cb127-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Please</span> give me a line</span>
 <span id="cb127-3"><a href="#cb127-3" aria-hidden="true" tabindex="-1"></a>this is a line</span>
 <span id="cb127-4"><a href="#cb127-4" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;this is a line&quot;</span></span>
@@ -1778,12 +2476,18 @@ word: ever</code></pre>
 <span id="cb127-9"><a href="#cb127-9" aria-hidden="true" tabindex="-1"></a><span class="ot">line ::</span> <span class="dt">String</span></span>
 <span id="cb127-10"><a href="#cb127-10" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> line</span>
 <span id="cb127-11"><a href="#cb127-11" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;this is a line&quot;</span></span></code></pre></div>
-<p>If you need to give your operation parameters, you can just make <em>a function that returns an operation</em>. Note how <code>ask</code> has a function type with a <code>-&gt;</code>, just like a normal function. We also use normal function definition syntax to give the parameter the name <code>question</code>.</p>
-<div class="sourceCode" id="cb128"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb128-1"><a href="#cb128-1" aria-hidden="true" tabindex="-1"></a><span class="ot">ask ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> <span class="dt">String</span></span>
+<p>If you need to give your operation parameters, you can just make
+<em>a function that returns an operation</em>. Note how <code>ask</code>
+has a function type with a <code>-&gt;</code>, just like a normal
+function. We also use normal function definition syntax to give the
+parameter the name <code>question</code>.</p>
+<div class="sourceCode" id="cb128"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb128-1"><a href="#cb128-1" aria-hidden="true" tabindex="-1"></a><span class="ot">ask ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> <span class="dt">String</span></span>
 <span id="cb128-2"><a href="#cb128-2" aria-hidden="true" tabindex="-1"></a>ask question <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb128-3"><a href="#cb128-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">putStrLn</span> question</span>
 <span id="cb128-4"><a href="#cb128-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">getLine</span></span></code></pre></div>
-<div class="sourceCode" id="cb129"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb129-1"><a href="#cb129-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> ask <span class="st">&quot;What is love?&quot;</span></span>
+<div class="sourceCode" id="cb129"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb129-1"><a href="#cb129-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> ask <span class="st">&quot;What is love?&quot;</span></span>
 <span id="cb129-2"><a href="#cb129-2" aria-hidden="true" tabindex="-1"></a><span class="dt">What</span> is love<span class="op">?</span></span>
 <span id="cb129-3"><a href="#cb129-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Baby</span> don&#39;t hurt me<span class="op">!</span></span>
 <span id="cb129-4"><a href="#cb129-4" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;Baby don&#39;t hurt me!&quot;</span></span>
@@ -1798,25 +2502,33 @@ word: ever</code></pre>
 <span id="cb129-13"><a href="#cb129-13" aria-hidden="true" tabindex="-1"></a><span class="ot">ask ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> <span class="dt">String</span></span>
 <span id="cb129-14"><a href="#cb129-14" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t ask <span class="st">&quot;Who are you?&quot;</span></span>
 <span id="cb129-15"><a href="#cb129-15" aria-hidden="true" tabindex="-1"></a>ask <span class="st">&quot;Who are you?&quot;</span><span class="ot"> ::</span> <span class="dt">IO</span> <span class="dt">String</span></span></code></pre></div>
-</section>
-</section>
-<section id="the-subtle-return" data-number="3.3">
-<h2 data-number="11.3"><span class="header-section-number">11.3</span> The Subtle <code>return</code></h2>
-<p>The Haskell function <code>return</code> is named a bit misleadingly. In other languages <code>return</code> is a built-in keyword, but in Haskell it’s just a function. The <code>return :: a -&gt; IO a</code> function takes a value and turns it into an <em>operation, that produces the value</em>.</p>
-<div class="sourceCode" id="cb130"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb130-1"><a href="#cb130-1" aria-hidden="true" tabindex="-1"></a><span class="ot">produceThree ::</span> <span class="dt">IO</span> <span class="dt">Int</span></span>
+<h2 data-number="11.3" id="the-subtle-return"><span
+class="header-section-number">11.3</span> The Subtle
+<code>return</code></h2>
+<p>The Haskell function <code>return</code> is named a bit misleadingly.
+In other languages <code>return</code> is a built-in keyword, but in
+Haskell it’s just a function. The <code>return :: a -&gt; IO a</code>
+function takes a value and turns it into an <em>operation, that produces
+the value</em>.</p>
+<div class="sourceCode" id="cb130"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb130-1"><a href="#cb130-1" aria-hidden="true" tabindex="-1"></a><span class="ot">produceThree ::</span> <span class="dt">IO</span> <span class="dt">Int</span></span>
 <span id="cb130-2"><a href="#cb130-2" aria-hidden="true" tabindex="-1"></a>produceThree <span class="ot">=</span> <span class="fu">return</span> <span class="dv">3</span></span>
 <span id="cb130-3"><a href="#cb130-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb130-4"><a href="#cb130-4" aria-hidden="true" tabindex="-1"></a><span class="ot">printThree ::</span> <span class="dt">IO</span> ()</span>
 <span id="cb130-5"><a href="#cb130-5" aria-hidden="true" tabindex="-1"></a>printThree <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb130-6"><a href="#cb130-6" aria-hidden="true" tabindex="-1"></a>  three <span class="ot">&lt;-</span> produceThree</span>
 <span id="cb130-7"><a href="#cb130-7" aria-hidden="true" tabindex="-1"></a>  <span class="fu">putStrLn</span> (<span class="fu">show</span> three)</span></code></pre></div>
-<p>That doesn’t sound very useful does it? Combined with do-notation it is. Here we return a boolean according to whether the user answered <code>Y</code> or <code>N</code>:</p>
-<div class="sourceCode" id="cb131"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb131-1"><a href="#cb131-1" aria-hidden="true" tabindex="-1"></a><span class="ot">yesNoQuestion ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> <span class="dt">Bool</span></span>
+<p>That doesn’t sound very useful does it? Combined with do-notation it
+is. Here we return a boolean according to whether the user answered
+<code>Y</code> or <code>N</code>:</p>
+<div class="sourceCode" id="cb131"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb131-1"><a href="#cb131-1" aria-hidden="true" tabindex="-1"></a><span class="ot">yesNoQuestion ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> <span class="dt">Bool</span></span>
 <span id="cb131-2"><a href="#cb131-2" aria-hidden="true" tabindex="-1"></a>yesNoQuestion question <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb131-3"><a href="#cb131-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">putStrLn</span> question</span>
 <span id="cb131-4"><a href="#cb131-4" aria-hidden="true" tabindex="-1"></a>  s <span class="ot">&lt;-</span> <span class="fu">getLine</span></span>
 <span id="cb131-5"><a href="#cb131-5" aria-hidden="true" tabindex="-1"></a>  <span class="fu">return</span> (s <span class="op">==</span> <span class="st">&quot;Y&quot;</span>)</span></code></pre></div>
-<div class="sourceCode" id="cb132"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb132-1"><a href="#cb132-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> yesNoQuestion <span class="st">&quot;Fire the missiles?&quot;</span></span>
+<div class="sourceCode" id="cb132"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb132-1"><a href="#cb132-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> yesNoQuestion <span class="st">&quot;Fire the missiles?&quot;</span></span>
 <span id="cb132-2"><a href="#cb132-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Fire</span> the missiles<span class="op">?</span></span>
 <span id="cb132-3"><a href="#cb132-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Y</span></span>
 <span id="cb132-4"><a href="#cb132-4" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span>
@@ -1827,13 +2539,20 @@ word: ever</code></pre>
 <span id="cb132-9"><a href="#cb132-9" aria-hidden="true" tabindex="-1"></a><span class="ot">answer ::</span> <span class="dt">Bool</span></span>
 <span id="cb132-10"><a href="#cb132-10" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> answer</span>
 <span id="cb132-11"><a href="#cb132-11" aria-hidden="true" tabindex="-1"></a><span class="dt">False</span></span></code></pre></div>
-<p><strong>Note!</strong> This means that return <em>does not stop execution</em> of an operation (unlike return in Java or C). Remember that in do-blocks, the last line decides which value to produce. This means that this operation produces <code>2</code>:</p>
-<div class="sourceCode" id="cb133"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb133-1"><a href="#cb133-1" aria-hidden="true" tabindex="-1"></a><span class="ot">produceTwo ::</span> <span class="dt">IO</span> <span class="dt">Int</span></span>
+<p><strong>Note!</strong> This means that return <em>does not stop
+execution</em> of an operation (unlike return in Java or C). Remember
+that in do-blocks, the last line decides which value to produce. This
+means that this operation produces <code>2</code>:</p>
+<div class="sourceCode" id="cb133"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb133-1"><a href="#cb133-1" aria-hidden="true" tabindex="-1"></a><span class="ot">produceTwo ::</span> <span class="dt">IO</span> <span class="dt">Int</span></span>
 <span id="cb133-2"><a href="#cb133-2" aria-hidden="true" tabindex="-1"></a>produceTwo <span class="ot">=</span> <span class="kw">do</span> <span class="fu">return</span> <span class="dv">1</span></span>
 <span id="cb133-3"><a href="#cb133-3" aria-hidden="true" tabindex="-1"></a>                <span class="fu">return</span> <span class="dv">2</span></span></code></pre></div>
-<div class="sourceCode" id="cb134"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb134-1"><a href="#cb134-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> produceTwo</span>
+<div class="sourceCode" id="cb134"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb134-1"><a href="#cb134-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> produceTwo</span>
 <span id="cb134-2"><a href="#cb134-2" aria-hidden="true" tabindex="-1"></a><span class="dv">2</span></span></code></pre></div>
-<p>Let’s look at this another way. The <code>do</code> notation allows us to cause a sequence of side-effects, and finally to produce a value.</p>
+<p>Let’s look at this another way. The <code>do</code> notation allows
+us to cause a sequence of side-effects, and finally to produce a
+value.</p>
 <pre><code>produceThree = do putStrLn &quot;1&quot;   -- side effect, produces (), which is ignored
                   return    2    -- no side effect, produces 2, which is ignored
                   getLine        -- side effect, produces a String, which is ignored
@@ -1844,36 +2563,60 @@ this line is ignored
 Prelude&gt; final
 3</code></pre>
 <p>Also note that these are the same operation:</p>
-<div class="sourceCode" id="cb137"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb137-1"><a href="#cb137-1" aria-hidden="true" tabindex="-1"></a><span class="kw">do</span> <span class="op">...</span></span>
+<div class="sourceCode" id="cb137"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb137-1"><a href="#cb137-1" aria-hidden="true" tabindex="-1"></a><span class="kw">do</span> <span class="op">...</span></span>
 <span id="cb137-2"><a href="#cb137-2" aria-hidden="true" tabindex="-1"></a>   x <span class="ot">&lt;-</span> op</span>
 <span id="cb137-3"><a href="#cb137-3" aria-hidden="true" tabindex="-1"></a>   <span class="fu">return</span> x</span></code></pre></div>
-<div class="sourceCode" id="cb138"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb138-1"><a href="#cb138-1" aria-hidden="true" tabindex="-1"></a><span class="kw">do</span> <span class="op">...</span></span>
+<div class="sourceCode" id="cb138"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb138-1"><a href="#cb138-1" aria-hidden="true" tabindex="-1"></a><span class="kw">do</span> <span class="op">...</span></span>
 <span id="cb138-2"><a href="#cb138-2" aria-hidden="true" tabindex="-1"></a>   op</span></code></pre></div>
-<p>Since <code>return</code> is a function, you should remember to parenthesize any complex expressions:</p>
-<div class="sourceCode" id="cb139"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb139-1"><a href="#cb139-1" aria-hidden="true" tabindex="-1"></a><span class="fu">return</span> (f x <span class="op">:</span> xs)</span>
+<p>Since <code>return</code> is a function, you should remember to
+parenthesize any complex expressions:</p>
+<div class="sourceCode" id="cb139"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb139-1"><a href="#cb139-1" aria-hidden="true" tabindex="-1"></a><span class="fu">return</span> (f x <span class="op">:</span> xs)</span>
 <span id="cb139-2"><a href="#cb139-2" aria-hidden="true" tabindex="-1"></a><span class="co">-- alternatively:</span></span>
 <span id="cb139-3"><a href="#cb139-3" aria-hidden="true" tabindex="-1"></a><span class="fu">return</span> <span class="op">$</span> f x <span class="op">:</span> xs</span></code></pre></div>
-</section>
-<section id="do-and-types" data-number="3.4">
-<h2 data-number="11.4"><span class="header-section-number">11.4</span> <code>do</code> and Types</h2>
-<p>Let’s look at the typing of do-notation in more detail. A do-block builds a value of type <code>IO &lt;something&gt;</code>. For example in</p>
-<div class="sourceCode" id="cb140"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb140-1"><a href="#cb140-1" aria-hidden="true" tabindex="-1"></a>foo <span class="ot">=</span> <span class="kw">do</span></span>
+<h2 data-number="11.4" id="do-and-types"><span
+class="header-section-number">11.4</span> <code>do</code> and Types</h2>
+<p>Let’s look at the typing of do-notation in more detail. A do-block
+builds a value of type <code>IO &lt;something&gt;</code>. For example
+in</p>
+<div class="sourceCode" id="cb140"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb140-1"><a href="#cb140-1" aria-hidden="true" tabindex="-1"></a>foo <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb140-2"><a href="#cb140-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">...</span></span>
 <span id="cb140-3"><a href="#cb140-3" aria-hidden="true" tabindex="-1"></a>  lastOp</span></code></pre></div>
-<p>The <code>lastOp</code> must be of type <code>IO X</code> (for some <code>X</code>). The type of <code>foo</code> will also be <code>IO X</code>. Let’s look at an example with parameters next:</p>
-<div class="sourceCode" id="cb141"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb141-1"><a href="#cb141-1" aria-hidden="true" tabindex="-1"></a>bar x y <span class="ot">=</span> <span class="kw">do</span></span>
+<p>The <code>lastOp</code> must be of type <code>IO X</code> (for some
+<code>X</code>). The type of <code>foo</code> will also be
+<code>IO X</code>. Let’s look at an example with parameters next:</p>
+<div class="sourceCode" id="cb141"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb141-1"><a href="#cb141-1" aria-hidden="true" tabindex="-1"></a>bar x y <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb141-2"><a href="#cb141-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">...</span></span>
 <span id="cb141-3"><a href="#cb141-3" aria-hidden="true" tabindex="-1"></a>  lastOp arg</span></code></pre></div>
-<p>The <code>lastOp</code> must be of type <code>Y -&gt; IO X</code> (so that <code>lastOp arg</code> has type <code>IO X</code>). The type of <code>bar</code> will be <code>A -&gt; B -&gt; IO X</code> (and inside <code>bar</code> we’ll have <code>x :: A</code> and <code>y :: B</code>).</p>
+<p>The <code>lastOp</code> must be of type <code>Y -&gt; IO X</code> (so
+that <code>lastOp arg</code> has type <code>IO X</code>). The type of
+<code>bar</code> will be <code>A -&gt; B -&gt; IO X</code> (and inside
+<code>bar</code> we’ll have <code>x :: A</code> and
+<code>y :: B</code>).</p>
 <p>If we use <code>return</code>:</p>
-<div class="sourceCode" id="cb142"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb142-1"><a href="#cb142-1" aria-hidden="true" tabindex="-1"></a>quux x <span class="ot">=</span> <span class="kw">do</span></span>
+<div class="sourceCode" id="cb142"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb142-1"><a href="#cb142-1" aria-hidden="true" tabindex="-1"></a>quux x <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb142-2"><a href="#cb142-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">...</span></span>
 <span id="cb142-3"><a href="#cb142-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">return</span> value</span></code></pre></div>
-<p>The function <code>quux</code> will have type <code>A -&gt; IO B</code>, where <code>x :: A</code> and <code>value :: B</code>.</p>
-<p>Let’s look at the typing of <code>&lt;-</code> next. If <code>op :: IO X</code> and you have <code>var &lt;- op</code>, <code>var</code> will have type <code>X</code>. We’ve seen this in many GHCi examples.</p>
-<p>The last line of a <code>do</code> cannot be <code>foo &lt;- bar</code>. It can’t be <code>let foo = bar</code> either. The last line determines what the whole operation produces, so it must be an operation (for example, <code>return something</code>).</p>
+<p>The function <code>quux</code> will have type
+<code>A -&gt; IO B</code>, where <code>x :: A</code> and
+<code>value :: B</code>.</p>
+<p>Let’s look at the typing of <code>&lt;-</code> next. If
+<code>op :: IO X</code> and you have <code>var &lt;- op</code>,
+<code>var</code> will have type <code>X</code>. We’ve seen this in many
+GHCi examples.</p>
+<p>The last line of a <code>do</code> cannot be
+<code>foo &lt;- bar</code>. It can’t be <code>let foo = bar</code>
+either. The last line determines what the whole operation produces, so
+it must be an operation (for example,
+<code>return something</code>).</p>
 <p>Here’s a worked example:</p>
-<div class="sourceCode" id="cb143"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb143-1"><a href="#cb143-1" aria-hidden="true" tabindex="-1"></a><span class="ot">alwaysFine ::</span> <span class="dt">IO</span> <span class="dt">Bool</span></span>
+<div class="sourceCode" id="cb143"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb143-1"><a href="#cb143-1" aria-hidden="true" tabindex="-1"></a><span class="ot">alwaysFine ::</span> <span class="dt">IO</span> <span class="dt">Bool</span></span>
 <span id="cb143-2"><a href="#cb143-2" aria-hidden="true" tabindex="-1"></a>alwaysFine <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb143-3"><a href="#cb143-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">putStrLn</span> <span class="st">&quot;What?&quot;</span> <span class="co">-- :: IO ()</span></span>
 <span id="cb143-4"><a href="#cb143-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">return</span> <span class="dv">2</span>         <span class="co">-- :: IO Int, produced value is discarded</span></span>
@@ -1882,49 +2625,73 @@ Prelude&gt; final
 <span id="cb143-7"><a href="#cb143-7" aria-hidden="true" tabindex="-1"></a>  <span class="kw">let</span> b <span class="ot">=</span> <span class="dt">True</span>     <span class="co">-- b :: Bool</span></span>
 <span id="cb143-8"><a href="#cb143-8" aria-hidden="true" tabindex="-1"></a>  <span class="fu">return</span> b         <span class="co">-- :: IO Bool</span></span>
 <span id="cb143-9"><a href="#cb143-9" aria-hidden="true" tabindex="-1"></a>                   <span class="co">-- Thus, alwaysFine :: IO Bool</span></span></code></pre></div>
-<p>The typing rules guarantee that you can not “escape” <code>IO</code>. Even though <code>&lt;-</code> gives you an <code>X</code> from an <code>IO X</code>, you can only use <code>&lt;-</code> inside <code>do</code>. However a <code>do</code> always means a value of type <code>IO Y</code>. In other words: you can temporarily open the <code>IO</code> box, but you must return into it. <em>“What happens in IO, stays in IO.”</em></p>
-<p>We’ll talk more about what this means later. For now, it’s enough to know that if you have a function with a non-IO type, like for example <code>myFunction :: Int -&gt; [String] -&gt; String</code>, the function can not have IO happening inside it. It is a pure function.</p>
-</section>
-<section id="control-structures" data-number="3.5">
-<h2 data-number="11.5"><span class="header-section-number">11.5</span> Control Structures</h2>
+<p>The typing rules guarantee that you can not “escape” <code>IO</code>.
+Even though <code>&lt;-</code> gives you an <code>X</code> from an
+<code>IO X</code>, you can only use <code>&lt;-</code> inside
+<code>do</code>. However a <code>do</code> always means a value of type
+<code>IO Y</code>. In other words: you can temporarily open the
+<code>IO</code> box, but you must return into it. <em>“What happens in
+IO, stays in IO.”</em></p>
+<p>We’ll talk more about what this means later. For now, it’s enough to
+know that if you have a function with a non-IO type, like for example
+<code>myFunction :: Int -&gt; [String] -&gt; String</code>, the function
+can not have IO happening inside it. It is a pure function.</p>
+<h2 data-number="11.5" id="control-structures"><span
+class="header-section-number">11.5</span> Control Structures</h2>
 <p>For the following examples, we’ll need two new operations.</p>
-<div class="sourceCode" id="cb144"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb144-1"><a href="#cb144-1" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span><span class="ot"> ::</span> <span class="dt">Show</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">IO</span> ()   <span class="co">-- print a value using the show function</span></span>
+<div class="sourceCode" id="cb144"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb144-1"><a href="#cb144-1" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span><span class="ot"> ::</span> <span class="dt">Show</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">IO</span> ()   <span class="co">-- print a value using the show function</span></span>
 <span id="cb144-2"><a href="#cb144-2" aria-hidden="true" tabindex="-1"></a><span class="fu">readLn</span><span class="ot"> ::</span> <span class="dt">Read</span> a <span class="ot">=&gt;</span> <span class="dt">IO</span> a        <span class="co">-- get a line and convert it to a value using the read function</span></span></code></pre></div>
-<p>The usual tools of recursion, guards and if-then-else also work in the <code>IO</code> world. Here’s an IO operation that’s defined using a guard:</p>
-<div class="sourceCode" id="cb145"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb145-1"><a href="#cb145-1" aria-hidden="true" tabindex="-1"></a><span class="ot">printDescription ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span>
+<p>The usual tools of recursion, guards and if-then-else also work in
+the <code>IO</code> world. Here’s an IO operation that’s defined using a
+guard:</p>
+<div class="sourceCode" id="cb145"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb145-1"><a href="#cb145-1" aria-hidden="true" tabindex="-1"></a><span class="ot">printDescription ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span>
 <span id="cb145-2"><a href="#cb145-2" aria-hidden="true" tabindex="-1"></a>printDescription n</span>
 <span id="cb145-3"><a href="#cb145-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> <span class="fu">even</span> n    <span class="ot">=</span> <span class="fu">putStrLn</span> <span class="st">&quot;even&quot;</span></span>
 <span id="cb145-4"><a href="#cb145-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> n<span class="op">==</span><span class="dv">3</span>      <span class="ot">=</span> <span class="fu">putStrLn</span> <span class="st">&quot;three&quot;</span></span>
 <span id="cb145-5"><a href="#cb145-5" aria-hidden="true" tabindex="-1"></a>  <span class="op">|</span> <span class="fu">otherwise</span> <span class="ot">=</span> <span class="fu">print</span> n</span></code></pre></div>
-<div class="sourceCode" id="cb146"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb146-1"><a href="#cb146-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> printDescription <span class="dv">2</span></span>
+<div class="sourceCode" id="cb146"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb146-1"><a href="#cb146-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> printDescription <span class="dv">2</span></span>
 <span id="cb146-2"><a href="#cb146-2" aria-hidden="true" tabindex="-1"></a><span class="fu">even</span></span>
 <span id="cb146-3"><a href="#cb146-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> printDescription <span class="dv">3</span></span>
 <span id="cb146-4"><a href="#cb146-4" aria-hidden="true" tabindex="-1"></a>three</span>
 <span id="cb146-5"><a href="#cb146-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> printDescription <span class="dv">5</span></span>
 <span id="cb146-6"><a href="#cb146-6" aria-hidden="true" tabindex="-1"></a><span class="dv">5</span></span></code></pre></div>
-<p>Here’s an operation that prints all numbers in a list using recursion and pattern matching:</p>
-<div class="sourceCode" id="cb147"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb147-1"><a href="#cb147-1" aria-hidden="true" tabindex="-1"></a><span class="ot">printList ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span>
+<p>Here’s an operation that prints all numbers in a list using recursion
+and pattern matching:</p>
+<div class="sourceCode" id="cb147"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb147-1"><a href="#cb147-1" aria-hidden="true" tabindex="-1"></a><span class="ot">printList ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span>
 <span id="cb147-2"><a href="#cb147-2" aria-hidden="true" tabindex="-1"></a>printList [] <span class="ot">=</span> <span class="fu">return</span> () <span class="co">-- do nothing</span></span>
 <span id="cb147-3"><a href="#cb147-3" aria-hidden="true" tabindex="-1"></a>printList (x<span class="op">:</span>xs) <span class="ot">=</span> <span class="kw">do</span> <span class="fu">print</span> x</span>
 <span id="cb147-4"><a href="#cb147-4" aria-hidden="true" tabindex="-1"></a>                      printList xs <span class="co">-- recursion</span></span></code></pre></div>
-<div class="sourceCode" id="cb148"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb148-1"><a href="#cb148-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> printList [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
+<div class="sourceCode" id="cb148"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb148-1"><a href="#cb148-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> printList [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb148-2"><a href="#cb148-2" aria-hidden="true" tabindex="-1"></a><span class="dv">1</span></span>
 <span id="cb148-3"><a href="#cb148-3" aria-hidden="true" tabindex="-1"></a><span class="dv">2</span></span>
 <span id="cb148-4"><a href="#cb148-4" aria-hidden="true" tabindex="-1"></a><span class="dv">3</span></span></code></pre></div>
-<p>Here are two slightly more complicated examples of recursive IO operations. They use the value produced by the recursive call. The operation <code>readAndSum n</code> reads <code>n</code> numbers from the user and prints their sum. The operation <code>ask questions</code> shows each string in <code>questions</code> to the user, reads a response, and returns a list of all the responses.</p>
-<div class="sourceCode" id="cb149"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb149-1"><a href="#cb149-1" aria-hidden="true" tabindex="-1"></a><span class="ot">readAndSum ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> <span class="dt">Int</span></span>
+<p>Here are two slightly more complicated examples of recursive IO
+operations. They use the value produced by the recursive call. The
+operation <code>readAndSum n</code> reads <code>n</code> numbers from
+the user and prints their sum. The operation <code>ask questions</code>
+shows each string in <code>questions</code> to the user, reads a
+response, and returns a list of all the responses.</p>
+<div class="sourceCode" id="cb149"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb149-1"><a href="#cb149-1" aria-hidden="true" tabindex="-1"></a><span class="ot">readAndSum ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> <span class="dt">Int</span></span>
 <span id="cb149-2"><a href="#cb149-2" aria-hidden="true" tabindex="-1"></a>readAndSum <span class="dv">0</span> <span class="ot">=</span> <span class="fu">return</span> <span class="dv">0</span></span>
 <span id="cb149-3"><a href="#cb149-3" aria-hidden="true" tabindex="-1"></a>readAndSum n <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb149-4"><a href="#cb149-4" aria-hidden="true" tabindex="-1"></a>  i <span class="ot">&lt;-</span> <span class="fu">readLn</span>            <span class="co">-- read one number</span></span>
 <span id="cb149-5"><a href="#cb149-5" aria-hidden="true" tabindex="-1"></a>  s <span class="ot">&lt;-</span> readAndSum (n<span class="op">-</span><span class="dv">1</span>)  <span class="co">-- recursion: read and sum rest of numbers</span></span>
 <span id="cb149-6"><a href="#cb149-6" aria-hidden="true" tabindex="-1"></a>  <span class="fu">return</span> (i<span class="op">+</span>s)           <span class="co">-- produce result</span></span></code></pre></div>
-<div class="sourceCode" id="cb150"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb150-1"><a href="#cb150-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> s <span class="ot">&lt;-</span> readAndSum <span class="dv">3</span></span>
+<div class="sourceCode" id="cb150"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb150-1"><a href="#cb150-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> s <span class="ot">&lt;-</span> readAndSum <span class="dv">3</span></span>
 <span id="cb150-2"><a href="#cb150-2" aria-hidden="true" tabindex="-1"></a><span class="dv">2</span></span>
 <span id="cb150-3"><a href="#cb150-3" aria-hidden="true" tabindex="-1"></a><span class="dv">4</span></span>
 <span id="cb150-4"><a href="#cb150-4" aria-hidden="true" tabindex="-1"></a><span class="dv">5</span></span>
 <span id="cb150-5"><a href="#cb150-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> s</span>
 <span id="cb150-6"><a href="#cb150-6" aria-hidden="true" tabindex="-1"></a><span class="dv">11</span></span></code></pre></div>
-<div class="sourceCode" id="cb151"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb151-1"><a href="#cb151-1" aria-hidden="true" tabindex="-1"></a><span class="ot">ask ::</span> [<span class="dt">String</span>] <span class="ot">-&gt;</span> <span class="dt">IO</span> [<span class="dt">String</span>]</span>
+<div class="sourceCode" id="cb151"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb151-1"><a href="#cb151-1" aria-hidden="true" tabindex="-1"></a><span class="ot">ask ::</span> [<span class="dt">String</span>] <span class="ot">-&gt;</span> <span class="dt">IO</span> [<span class="dt">String</span>]</span>
 <span id="cb151-2"><a href="#cb151-2" aria-hidden="true" tabindex="-1"></a>ask [] <span class="ot">=</span> <span class="fu">return</span> []</span>
 <span id="cb151-3"><a href="#cb151-3" aria-hidden="true" tabindex="-1"></a>ask (question<span class="op">:</span>questions) <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb151-4"><a href="#cb151-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">putStr</span> question</span>
@@ -1932,15 +2699,19 @@ Prelude&gt; final
 <span id="cb151-6"><a href="#cb151-6" aria-hidden="true" tabindex="-1"></a>  answer <span class="ot">&lt;-</span> <span class="fu">getLine</span>         <span class="co">-- get one answer</span></span>
 <span id="cb151-7"><a href="#cb151-7" aria-hidden="true" tabindex="-1"></a>  answers <span class="ot">&lt;-</span> ask questions  <span class="co">-- recursion: get rest of answers</span></span>
 <span id="cb151-8"><a href="#cb151-8" aria-hidden="true" tabindex="-1"></a>  <span class="fu">return</span> (answer<span class="op">:</span>answers)   <span class="co">-- produce result</span></span></code></pre></div>
-<div class="sourceCode" id="cb152"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb152-1"><a href="#cb152-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> replies <span class="ot">&lt;-</span> ask [<span class="st">&quot;What is your name&quot;</span>,<span class="st">&quot;How old are you&quot;</span>]</span>
+<div class="sourceCode" id="cb152"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb152-1"><a href="#cb152-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> replies <span class="ot">&lt;-</span> ask [<span class="st">&quot;What is your name&quot;</span>,<span class="st">&quot;How old are you&quot;</span>]</span>
 <span id="cb152-2"><a href="#cb152-2" aria-hidden="true" tabindex="-1"></a><span class="dt">What</span> is your name<span class="op">?</span></span>
 <span id="cb152-3"><a href="#cb152-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Yog</span><span class="op">-</span><span class="dt">Sothoth</span></span>
 <span id="cb152-4"><a href="#cb152-4" aria-hidden="true" tabindex="-1"></a><span class="dt">How</span> old are you<span class="op">?</span></span>
 <span id="cb152-5"><a href="#cb152-5" aria-hidden="true" tabindex="-1"></a><span class="dt">The</span> question is meaningless</span>
 <span id="cb152-6"><a href="#cb152-6" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> replies</span>
 <span id="cb152-7"><a href="#cb152-7" aria-hidden="true" tabindex="-1"></a>[<span class="st">&quot;Yog-Sothoth&quot;</span>,<span class="st">&quot;The question is meaningless&quot;</span>]</span></code></pre></div>
-<p>Additionally, we have some <code>IO</code>-specific control structures, or rather, functions. These come from the module <code>Control.Monad</code>.</p>
-<div class="sourceCode" id="cb153"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb153-1"><a href="#cb153-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- when b op performs op if b is true</span></span>
+<p>Additionally, we have some <code>IO</code>-specific control
+structures, or rather, functions. These come from the module
+<code>Control.Monad</code>.</p>
+<div class="sourceCode" id="cb153"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb153-1"><a href="#cb153-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- when b op performs op if b is true</span></span>
 <span id="cb153-2"><a href="#cb153-2" aria-hidden="true" tabindex="-1"></a><span class="ot">when ::</span> <span class="dt">Bool</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> () <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span>
 <span id="cb153-3"><a href="#cb153-3" aria-hidden="true" tabindex="-1"></a><span class="co">-- unless b op performs op if b is false</span></span>
 <span id="cb153-4"><a href="#cb153-4" aria-hidden="true" tabindex="-1"></a><span class="ot">unless ::</span> <span class="dt">Bool</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> () <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span>
@@ -1956,12 +2727,15 @@ Prelude&gt; final
 <span id="cb153-14"><a href="#cb153-14" aria-hidden="true" tabindex="-1"></a><span class="ot">forM  ::</span> [a] <span class="ot">-&gt;</span> (a <span class="ot">-&gt;</span> <span class="dt">IO</span> b) <span class="ot">-&gt;</span> <span class="dt">IO</span> [b]</span>
 <span id="cb153-15"><a href="#cb153-15" aria-hidden="true" tabindex="-1"></a><span class="ot">forM_ ::</span> [a] <span class="ot">-&gt;</span> (a <span class="ot">-&gt;</span> <span class="dt">IO</span> b) <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span></code></pre></div>
 <p>Using these, we can rewrite our earlier examples:</p>
-<div class="sourceCode" id="cb154"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb154-1"><a href="#cb154-1" aria-hidden="true" tabindex="-1"></a><span class="ot">printList ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span>
+<div class="sourceCode" id="cb154"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb154-1"><a href="#cb154-1" aria-hidden="true" tabindex="-1"></a><span class="ot">printList ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span>
 <span id="cb154-2"><a href="#cb154-2" aria-hidden="true" tabindex="-1"></a>printList xs <span class="ot">=</span> <span class="fu">mapM_</span> <span class="fu">print</span> xs</span></code></pre></div>
-<div class="sourceCode" id="cb155"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb155-1"><a href="#cb155-1" aria-hidden="true" tabindex="-1"></a>readAndSum n <span class="ot">=</span> <span class="kw">do</span></span>
+<div class="sourceCode" id="cb155"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb155-1"><a href="#cb155-1" aria-hidden="true" tabindex="-1"></a>readAndSum n <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb155-2"><a href="#cb155-2" aria-hidden="true" tabindex="-1"></a>  numbers <span class="ot">&lt;-</span> replicateM n <span class="fu">readLn</span></span>
 <span id="cb155-3"><a href="#cb155-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">return</span> (<span class="fu">sum</span> numbers)</span></code></pre></div>
-<div class="sourceCode" id="cb156"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb156-1"><a href="#cb156-1" aria-hidden="true" tabindex="-1"></a><span class="ot">ask ::</span> [<span class="dt">String</span>] <span class="ot">-&gt;</span> <span class="dt">IO</span> [<span class="dt">String</span>]</span>
+<div class="sourceCode" id="cb156"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb156-1"><a href="#cb156-1" aria-hidden="true" tabindex="-1"></a><span class="ot">ask ::</span> [<span class="dt">String</span>] <span class="ot">-&gt;</span> <span class="dt">IO</span> [<span class="dt">String</span>]</span>
 <span id="cb156-2"><a href="#cb156-2" aria-hidden="true" tabindex="-1"></a>ask questions <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb156-3"><a href="#cb156-3" aria-hidden="true" tabindex="-1"></a>  forM questions askOne</span>
 <span id="cb156-4"><a href="#cb156-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -1970,13 +2744,16 @@ Prelude&gt; final
 <span id="cb156-7"><a href="#cb156-7" aria-hidden="true" tabindex="-1"></a>  <span class="fu">putStr</span> question</span>
 <span id="cb156-8"><a href="#cb156-8" aria-hidden="true" tabindex="-1"></a>  <span class="fu">putStrLn</span> <span class="st">&quot;?&quot;</span></span>
 <span id="cb156-9"><a href="#cb156-9" aria-hidden="true" tabindex="-1"></a>  <span class="fu">getLine</span></span></code></pre></div>
-</section>
-<section id="a-word-about-do-and-indentation" data-number="3.6">
-<h2 data-number="11.6"><span class="header-section-number">11.6</span> A Word About <code>do</code> and Indentation</h2>
-<p>It’s easy to run into weird indentation problems when using do-notation. Here are some rules of thumb to help you get it right.</p>
-<p>The most important rule of do and indentation is <em>all operations in a do-block must start in the same column</em>.</p>
+<h2 data-number="11.6" id="a-word-about-do-and-indentation"><span
+class="header-section-number">11.6</span> A Word About <code>do</code>
+and Indentation</h2>
+<p>It’s easy to run into weird indentation problems when using
+do-notation. Here are some rules of thumb to help you get it right.</p>
+<p>The most important rule of do and indentation is <em>all operations
+in a do-block must start in the same column</em>.</p>
 <p>Some examples of this rule:</p>
-<div class="sourceCode" id="cb157"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb157-1"><a href="#cb157-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- This is not OK, putStrLn is way too left</span></span>
+<div class="sourceCode" id="cb157"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb157-1"><a href="#cb157-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- This is not OK, putStrLn is way too left</span></span>
 <span id="cb157-2"><a href="#cb157-2" aria-hidden="true" tabindex="-1"></a>foo <span class="ot">=</span> <span class="kw">do</span> y <span class="ot">&lt;-</span> <span class="fu">getLine</span></span>
 <span id="cb157-3"><a href="#cb157-3" aria-hidden="true" tabindex="-1"></a>   <span class="fu">putStrLn</span> y</span>
 <span id="cb157-4"><a href="#cb157-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -1992,8 +2769,11 @@ Prelude&gt; final
 <span id="cb157-14"><a href="#cb157-14" aria-hidden="true" tabindex="-1"></a>foo <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb157-15"><a href="#cb157-15" aria-hidden="true" tabindex="-1"></a>  y <span class="ot">&lt;-</span> <span class="fu">getLine</span></span>
 <span id="cb157-16"><a href="#cb157-16" aria-hidden="true" tabindex="-1"></a>  <span class="fu">putStrLn</span> y</span></code></pre></div>
-<p>A related rule is <em>when an operation goes over multiple lines, indent the follow-up lines</em>. If you don’t indent, it’ll look like a new operation!</p>
-<div class="sourceCode" id="cb158"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb158-1"><a href="#cb158-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- This is not OK, the string starts a new operation</span></span>
+<p>A related rule is <em>when an operation goes over multiple lines,
+indent the follow-up lines</em>. If you don’t indent, it’ll look like a
+new operation!</p>
+<div class="sourceCode" id="cb158"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb158-1"><a href="#cb158-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- This is not OK, the string starts a new operation</span></span>
 <span id="cb158-2"><a href="#cb158-2" aria-hidden="true" tabindex="-1"></a>quux <span class="ot">=</span> <span class="kw">do</span> <span class="fu">putStrLn</span></span>
 <span id="cb158-3"><a href="#cb158-3" aria-hidden="true" tabindex="-1"></a>          <span class="st">&quot;this long string&quot;</span></span>
 <span id="cb158-4"><a href="#cb158-4" aria-hidden="true" tabindex="-1"></a>          <span class="fu">print</span> <span class="dv">1</span></span>
@@ -2002,8 +2782,10 @@ Prelude&gt; final
 <span id="cb158-7"><a href="#cb158-7" aria-hidden="true" tabindex="-1"></a>quux <span class="ot">=</span> <span class="kw">do</span> <span class="fu">putStrLn</span></span>
 <span id="cb158-8"><a href="#cb158-8" aria-hidden="true" tabindex="-1"></a>            <span class="st">&quot;this long string&quot;</span></span>
 <span id="cb158-9"><a href="#cb158-9" aria-hidden="true" tabindex="-1"></a>          <span class="fu">print</span> <span class="dv">1</span></span></code></pre></div>
-<p>Here’s one more example, with nested do-blocks, and two different valid indentations.</p>
-<div class="sourceCode" id="cb159"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb159-1"><a href="#cb159-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- This is OK</span></span>
+<p>Here’s one more example, with nested do-blocks, and two different
+valid indentations.</p>
+<div class="sourceCode" id="cb159"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb159-1"><a href="#cb159-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- This is OK</span></span>
 <span id="cb159-2"><a href="#cb159-2" aria-hidden="true" tabindex="-1"></a>foo x <span class="ot">=</span> <span class="kw">do</span> quux</span>
 <span id="cb159-3"><a href="#cb159-3" aria-hidden="true" tabindex="-1"></a>           y <span class="ot">&lt;-</span> blorg</span>
 <span id="cb159-4"><a href="#cb159-4" aria-hidden="true" tabindex="-1"></a>           when y (<span class="kw">do</span> thing</span>
@@ -2018,11 +2800,17 @@ Prelude&gt; final
 <span id="cb159-13"><a href="#cb159-13" aria-hidden="true" tabindex="-1"></a>    thing</span>
 <span id="cb159-14"><a href="#cb159-14" aria-hidden="true" tabindex="-1"></a>    otherThing</span>
 <span id="cb159-15"><a href="#cb159-15" aria-hidden="true" tabindex="-1"></a>  <span class="fu">return</span> <span class="dv">3</span></span></code></pre></div>
-</section>
-<section id="lets-write-a-program" data-number="3.7">
-<h2 data-number="11.7"><span class="header-section-number">11.7</span> Let’s Write a Program</h2>
-<p>After all these short one-off examples, let’s turn to something a bit longer. Let’s write a program to fetch all type annotations from all <code>.hs</code> files. We use IO operations like <code>readFile</code> and <code>listDirectory</code> to read and find files, but also pure code like <code>map</code> and <code>filter</code> to do the actual processing. First off, here’s a recap of the library operations we’re using:</p>
-<div class="sourceCode" id="cb160"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb160-1"><a href="#cb160-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- split string into lines</span></span>
+<h2 data-number="11.7" id="lets-write-a-program"><span
+class="header-section-number">11.7</span> Let’s Write a Program</h2>
+<p>After all these short one-off examples, let’s turn to something a bit
+longer. Let’s write a program to fetch all type annotations from all
+<code>.hs</code> files. We use IO operations like <code>readFile</code>
+and <code>listDirectory</code> to read and find files, but also pure
+code like <code>map</code> and <code>filter</code> to do the actual
+processing. First off, here’s a recap of the library operations we’re
+using:</p>
+<div class="sourceCode" id="cb160"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb160-1"><a href="#cb160-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- split string into lines</span></span>
 <span id="cb160-2"><a href="#cb160-2" aria-hidden="true" tabindex="-1"></a><span class="fu">lines</span><span class="ot"> ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> [<span class="dt">String</span>]</span>
 <span id="cb160-3"><a href="#cb160-3" aria-hidden="true" tabindex="-1"></a><span class="co">-- `isSuffixOf suf list` is true if list ends in suf</span></span>
 <span id="cb160-4"><a href="#cb160-4" aria-hidden="true" tabindex="-1"></a>Data.List.isSuffixOf<span class="ot"> ::</span> <span class="dt">Eq</span> a <span class="ot">=&gt;</span> [a] <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
@@ -2036,8 +2824,11 @@ Prelude&gt; final
 <span id="cb160-12"><a href="#cb160-12" aria-hidden="true" tabindex="-1"></a>System.Directory.listDirectory<span class="ot"> ::</span> <span class="dt">FilePath</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> [<span class="dt">FilePath</span>]</span>
 <span id="cb160-13"><a href="#cb160-13" aria-hidden="true" tabindex="-1"></a><span class="co">-- is the given file a directory?</span></span>
 <span id="cb160-14"><a href="#cb160-14" aria-hidden="true" tabindex="-1"></a>System.Directory.doesDirectoryExist<span class="ot"> ::</span> <span class="dt">FilePath</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> <span class="dt">Bool</span></span></code></pre></div>
-<p>And here’s the program itself. You can also find it in the course repository as <a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Examples/ReadTypes.hs"><code>exercises/Examples/ReadTypes.hs</code></a>.</p>
-<div class="sourceCode" id="cb161"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb161-1"><a href="#cb161-1" aria-hidden="true" tabindex="-1"></a><span class="kw">module</span> <span class="dt">Examples.ReadTypes</span> <span class="kw">where</span></span>
+<p>And here’s the program itself. You can also find it in the course
+repository as <a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Examples/ReadTypes.hs"><code>exercises/Examples/ReadTypes.hs</code></a>.</p>
+<div class="sourceCode" id="cb161"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb161-1"><a href="#cb161-1" aria-hidden="true" tabindex="-1"></a><span class="kw">module</span> <span class="dt">Examples.ReadTypes</span> <span class="kw">where</span></span>
 <span id="cb161-2"><a href="#cb161-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb161-3"><a href="#cb161-3" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="dt">Control.Monad</span> (forM)</span>
 <span id="cb161-4"><a href="#cb161-4" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="dt">Data.List</span> (isInfixOf, isSuffixOf)</span>
@@ -2077,25 +2868,43 @@ Prelude&gt; final
 <span id="cb161-38"><a href="#cb161-38" aria-hidden="true" tabindex="-1"></a><span class="ot">main ::</span> <span class="dt">IO</span> ()</span>
 <span id="cb161-39"><a href="#cb161-39" aria-hidden="true" tabindex="-1"></a>main <span class="ot">=</span> <span class="kw">do</span> ts <span class="ot">&lt;-</span> readTypes <span class="st">&quot;.&quot;</span></span>
 <span id="cb161-40"><a href="#cb161-40" aria-hidden="true" tabindex="-1"></a>          <span class="fu">mapM_</span> <span class="fu">putStrLn</span> ts</span></code></pre></div>
-<p>We can run this program by going to the directory <code>exercises/Examples</code> and running:</p>
-<div class="sourceCode" id="cb162"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb162-1"><a href="#cb162-1" aria-hidden="true" tabindex="-1"></a><span class="op">$</span> stack runhaskell ReadTypes.hs</span>
+<p>We can run this program by going to the directory
+<code>exercises/Examples</code> and running:</p>
+<div class="sourceCode" id="cb162"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb162-1"><a href="#cb162-1" aria-hidden="true" tabindex="-1"></a><span class="op">$</span> stack runhaskell ReadTypes.hs</span>
 <span id="cb162-2"><a href="#cb162-2" aria-hidden="true" tabindex="-1"></a><span class="ot">deposit ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Bank</span> <span class="ot">-&gt;</span> <span class="dt">Bank</span></span>
 <span id="cb162-3"><a href="#cb162-3" aria-hidden="true" tabindex="-1"></a><span class="ot">withdraw ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Bank</span> <span class="ot">-&gt;</span> (<span class="dt">Int</span>,<span class="dt">Bank</span>)</span>
 <span id="cb162-4"><a href="#cb162-4" aria-hidden="true" tabindex="-1"></a><span class="ot">runBankOp ::</span> <span class="dt">BankOp</span> a <span class="ot">-&gt;</span> <span class="dt">Bank</span> <span class="ot">-&gt;</span> (a,<span class="dt">Bank</span>)</span>
 <span id="cb162-5"><a href="#cb162-5" aria-hidden="true" tabindex="-1"></a><span class="op">...</span> <span class="fu">and</span> so on</span></code></pre></div>
-<p>The exact output will vary according to the contents of the directory, of course.</p>
-</section>
-<section id="what-does-it-all-mean" data-number="3.8">
-<h2 data-number="11.8"><span class="header-section-number">11.8</span> What Does It All Mean?</h2>
-<p>Let’s return to the functional world. How can we reconcile IO operations with Haskell being a <em>pure</em> and <em>lazy</em> language? Something like <code>putStrLn :: String -&gt; IO ()</code> is a <em>pure</em> function that returns an operation. How is it pure? <code>putStrLn x</code> is the same when <code>x</code> is the same. In other words: an operation is a <em>pure description</em> of a chain of side effects. Only <em>executing</em> the operation causes those side effects. When a Haskell program is run, only one operation is executed - it’s called <code>main :: IO ()</code>. Other operations can be run only by linking them up to <code>main</code>.</p>
-<p>When in GHCi, if an expression you type in evaluates to an operation, GHCi runs that operation for you. Here’s a demonstration of the purity of <code>print</code>:</p>
-<div class="sourceCode" id="cb163"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb163-1"><a href="#cb163-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> x <span class="ot">=</span> <span class="fu">print</span> <span class="dv">1</span>   <span class="co">-- creates operation, doesn&#39;t run it</span></span>
+<p>The exact output will vary according to the contents of the
+directory, of course.</p>
+<h2 data-number="11.8" id="what-does-it-all-mean"><span
+class="header-section-number">11.8</span> What Does It All Mean?</h2>
+<p>Let’s return to the functional world. How can we reconcile IO
+operations with Haskell being a <em>pure</em> and <em>lazy</em>
+language? Something like <code>putStrLn :: String -&gt; IO ()</code> is
+a <em>pure</em> function that returns an operation. How is it pure?
+<code>putStrLn x</code> is the same when <code>x</code> is the same. In
+other words: an operation is a <em>pure description</em> of a chain of
+side effects. Only <em>executing</em> the operation causes those side
+effects. When a Haskell program is run, only one operation is executed -
+it’s called <code>main :: IO ()</code>. Other operations can be run only
+by linking them up to <code>main</code>.</p>
+<p>When in GHCi, if an expression you type in evaluates to an operation,
+GHCi runs that operation for you. Here’s a demonstration of the purity
+of <code>print</code>:</p>
+<div class="sourceCode" id="cb163"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb163-1"><a href="#cb163-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> x <span class="ot">=</span> <span class="fu">print</span> <span class="dv">1</span>   <span class="co">-- creates operation, doesn&#39;t run it</span></span>
 <span id="cb163-2"><a href="#cb163-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> x             <span class="co">-- runs the operation</span></span>
 <span id="cb163-3"><a href="#cb163-3" aria-hidden="true" tabindex="-1"></a><span class="dv">1</span></span>
 <span id="cb163-4"><a href="#cb163-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> x             <span class="co">-- runs it again!</span></span>
 <span id="cb163-5"><a href="#cb163-5" aria-hidden="true" tabindex="-1"></a><span class="dv">1</span></span></code></pre></div>
-<p><em>Operations are values</em> just like numbers, lists and functions. We can write code that operates on operations. This function takes two operations, <code>a</code> and <code>b</code>, and returns an operation that asks the user which one he’d like to run.</p>
-<div class="sourceCode" id="cb164"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb164-1"><a href="#cb164-1" aria-hidden="true" tabindex="-1"></a><span class="ot">choice ::</span> <span class="dt">IO</span> x <span class="ot">-&gt;</span> <span class="dt">IO</span> x <span class="ot">-&gt;</span> <span class="dt">IO</span> x</span>
+<p><em>Operations are values</em> just like numbers, lists and
+functions. We can write code that operates on operations. This function
+takes two operations, <code>a</code> and <code>b</code>, and returns an
+operation that asks the user which one he’d like to run.</p>
+<div class="sourceCode" id="cb164"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb164-1"><a href="#cb164-1" aria-hidden="true" tabindex="-1"></a><span class="ot">choice ::</span> <span class="dt">IO</span> x <span class="ot">-&gt;</span> <span class="dt">IO</span> x <span class="ot">-&gt;</span> <span class="dt">IO</span> x</span>
 <span id="cb164-2"><a href="#cb164-2" aria-hidden="true" tabindex="-1"></a>choice a b <span class="ot">=</span></span>
 <span id="cb164-3"><a href="#cb164-3" aria-hidden="true" tabindex="-1"></a>  <span class="kw">do</span> <span class="fu">putStr</span> <span class="st">&quot;a or b? &quot;</span></span>
 <span id="cb164-4"><a href="#cb164-4" aria-hidden="true" tabindex="-1"></a>     x <span class="ot">&lt;-</span> <span class="fu">getLine</span></span>
@@ -2103,31 +2912,47 @@ Prelude&gt; final
 <span id="cb164-6"><a href="#cb164-6" aria-hidden="true" tabindex="-1"></a>               <span class="st">&quot;b&quot;</span> <span class="ot">-&gt;</span> b</span>
 <span id="cb164-7"><a href="#cb164-7" aria-hidden="true" tabindex="-1"></a>               _ <span class="ot">-&gt;</span> <span class="kw">do</span> <span class="fu">putStrLn</span> <span class="st">&quot;Wrong!&quot;</span></span>
 <span id="cb164-8"><a href="#cb164-8" aria-hidden="true" tabindex="-1"></a>                       choice a b</span></code></pre></div>
-<div class="sourceCode" id="cb165"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb165-1"><a href="#cb165-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> choice (<span class="fu">putStrLn</span> <span class="st">&quot;A!!!!&quot;</span>) (<span class="fu">putStrLn</span> <span class="st">&quot;B!!!!&quot;</span>)</span>
+<div class="sourceCode" id="cb165"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb165-1"><a href="#cb165-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> choice (<span class="fu">putStrLn</span> <span class="st">&quot;A!!!!&quot;</span>) (<span class="fu">putStrLn</span> <span class="st">&quot;B!!!!&quot;</span>)</span>
 <span id="cb165-2"><a href="#cb165-2" aria-hidden="true" tabindex="-1"></a>a <span class="fu">or</span> b<span class="op">?</span> z</span>
 <span id="cb165-3"><a href="#cb165-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Wrong</span><span class="op">!</span></span>
 <span id="cb165-4"><a href="#cb165-4" aria-hidden="true" tabindex="-1"></a>a <span class="fu">or</span> b<span class="op">?</span> a</span>
 <span id="cb165-5"><a href="#cb165-5" aria-hidden="true" tabindex="-1"></a><span class="dt">A</span><span class="op">!!!!</span></span></code></pre></div>
-<p>Using operations specified as parameters lets us write functions like <code>mapM_</code>, which we met earlier. The implementation is a recursive IO operation that takes another IO operation as a parameter. Conceptually complicated, but simple when you read the code:</p>
-<div class="sourceCode" id="cb166"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb166-1"><a href="#cb166-1" aria-hidden="true" tabindex="-1"></a><span class="fu">mapM_</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> <span class="dt">IO</span> b) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span>
+<p>Using operations specified as parameters lets us write functions like
+<code>mapM_</code>, which we met earlier. The implementation is a
+recursive IO operation that takes another IO operation as a parameter.
+Conceptually complicated, but simple when you read the code:</p>
+<div class="sourceCode" id="cb166"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb166-1"><a href="#cb166-1" aria-hidden="true" tabindex="-1"></a><span class="fu">mapM_</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> <span class="dt">IO</span> b) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span>
 <span id="cb166-2"><a href="#cb166-2" aria-hidden="true" tabindex="-1"></a><span class="fu">mapM_</span> op     [] <span class="ot">=</span> <span class="fu">return</span> ()       <span class="co">-- do nothing for an empty list</span></span>
 <span id="cb166-3"><a href="#cb166-3" aria-hidden="true" tabindex="-1"></a><span class="fu">mapM_</span> op (x<span class="op">:</span>xs) <span class="ot">=</span> <span class="kw">do</span> op x         <span class="co">-- run operation on first element</span></span>
 <span id="cb166-4"><a href="#cb166-4" aria-hidden="true" tabindex="-1"></a>                     <span class="fu">mapM_</span> op xs  <span class="co">-- run operation on rest of list, recursively</span></span></code></pre></div>
-<div class="sourceCode" id="cb167"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb167-1"><a href="#cb167-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">mapM_</span> <span class="fu">print</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
+<div class="sourceCode" id="cb167"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb167-1"><a href="#cb167-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">mapM_</span> <span class="fu">print</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb167-2"><a href="#cb167-2" aria-hidden="true" tabindex="-1"></a><span class="dv">1</span></span>
 <span id="cb167-3"><a href="#cb167-3" aria-hidden="true" tabindex="-1"></a><span class="dv">2</span></span>
 <span id="cb167-4"><a href="#cb167-4" aria-hidden="true" tabindex="-1"></a><span class="dv">3</span></span></code></pre></div>
-</section>
-<section id="one-more-thing-ioref" data-number="3.9">
-<h2 data-number="11.9"><span class="header-section-number">11.9</span> One More Thing: IORef</h2>
-<p>So far the only side-effects we’ve been able to produce in IO have been terminal (<code>getLine</code>, <code>print</code>) and file (<code>readFile</code>, <code>listDirectory</code>) IO. Imperative programs written in Java, Python or C have other types of side effects too that we can’t express in pure Haskell. One of these is <em>mutable (i.e. changeable) state</em>. A pure function can not read mutable state, because otherwise two invocations of the same function might not return the same value.</p>
-<p>The Haskell type <code>IORef a</code> from the module <code>Data.IORef</code> is a mutable reference to a value of type <code>a</code></p>
-<div class="sourceCode" id="cb168"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb168-1"><a href="#cb168-1" aria-hidden="true" tabindex="-1"></a><span class="ot">newIORef ::</span> a <span class="ot">-&gt;</span> <span class="dt">IO</span> (<span class="dt">IORef</span> a)                <span class="co">-- create a new IORef containing a value</span></span>
+<h2 data-number="11.9" id="one-more-thing-ioref"><span
+class="header-section-number">11.9</span> One More Thing: IORef</h2>
+<p>So far the only side-effects we’ve been able to produce in IO have
+been terminal (<code>getLine</code>, <code>print</code>) and file
+(<code>readFile</code>, <code>listDirectory</code>) IO. Imperative
+programs written in Java, Python or C have other types of side effects
+too that we can’t express in pure Haskell. One of these is <em>mutable
+(i.e. changeable) state</em>. A pure function can not read mutable
+state, because otherwise two invocations of the same function might not
+return the same value.</p>
+<p>The Haskell type <code>IORef a</code> from the module
+<code>Data.IORef</code> is a mutable reference to a value of type
+<code>a</code></p>
+<div class="sourceCode" id="cb168"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb168-1"><a href="#cb168-1" aria-hidden="true" tabindex="-1"></a><span class="ot">newIORef ::</span> a <span class="ot">-&gt;</span> <span class="dt">IO</span> (<span class="dt">IORef</span> a)                <span class="co">-- create a new IORef containing a value</span></span>
 <span id="cb168-2"><a href="#cb168-2" aria-hidden="true" tabindex="-1"></a><span class="ot">readIORef ::</span> <span class="dt">IORef</span> a <span class="ot">-&gt;</span> <span class="dt">IO</span> a                 <span class="co">-- produce value contained in IORef</span></span>
 <span id="cb168-3"><a href="#cb168-3" aria-hidden="true" tabindex="-1"></a><span class="ot">writeIORef ::</span> <span class="dt">IORef</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">IO</span> ()          <span class="co">-- set value in IORef</span></span>
 <span id="cb168-4"><a href="#cb168-4" aria-hidden="true" tabindex="-1"></a><span class="ot">modifyIORef ::</span> <span class="dt">IORef</span> a <span class="ot">-&gt;</span> (a <span class="ot">-&gt;</span> a) <span class="ot">-&gt;</span> <span class="dt">IO</span> ()  <span class="co">-- modify value contained in IORef with a pure function</span></span></code></pre></div>
 <p>Here are some examples of using an IORef in GHCi:</p>
-<div class="sourceCode" id="cb169"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb169-1"><a href="#cb169-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>m <span class="op">+</span><span class="dt">Data.IORef</span></span>
+<div class="sourceCode" id="cb169"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb169-1"><a href="#cb169-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>m <span class="op">+</span><span class="dt">Data.IORef</span></span>
 <span id="cb169-2"><a href="#cb169-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.IORef</span><span class="op">&gt;</span> myRef <span class="ot">&lt;-</span> newIORef <span class="st">&quot;banana&quot;</span></span>
 <span id="cb169-3"><a href="#cb169-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.IORef</span><span class="op">&gt;</span> readIORef myRef</span>
 <span id="cb169-4"><a href="#cb169-4" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;banana&quot;</span></span>
@@ -2137,26 +2962,37 @@ Prelude&gt; final
 <span id="cb169-8"><a href="#cb169-8" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.IORef</span><span class="op">&gt;</span> modifyIORef myRef <span class="fu">reverse</span></span>
 <span id="cb169-9"><a href="#cb169-9" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.IORef</span><span class="op">&gt;</span> readIORef myRef</span>
 <span id="cb169-10"><a href="#cb169-10" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;elppa&quot;</span></span></code></pre></div>
-<p>Here’s an example of using an <code>IORef</code> to sum the values in a list. Note the similarity with an imperative loop.</p>
-<div class="sourceCode" id="cb170"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb170-1"><a href="#cb170-1" aria-hidden="true" tabindex="-1"></a><span class="ot">sumList ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">IO</span> <span class="dt">Int</span></span>
+<p>Here’s an example of using an <code>IORef</code> to sum the values in
+a list. Note the similarity with an imperative loop.</p>
+<div class="sourceCode" id="cb170"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb170-1"><a href="#cb170-1" aria-hidden="true" tabindex="-1"></a><span class="ot">sumList ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">IO</span> <span class="dt">Int</span></span>
 <span id="cb170-2"><a href="#cb170-2" aria-hidden="true" tabindex="-1"></a>sumList xs <span class="ot">=</span> <span class="kw">do</span> r <span class="ot">&lt;-</span> newIORef <span class="dv">0</span>                       <span class="co">-- initialize r to 0</span></span>
 <span id="cb170-3"><a href="#cb170-3" aria-hidden="true" tabindex="-1"></a>                forM_ xs (\x <span class="ot">-&gt;</span> modifyIORef r (x<span class="op">+</span>))   <span class="co">-- for every xs, add it to r</span></span>
 <span id="cb170-4"><a href="#cb170-4" aria-hidden="true" tabindex="-1"></a>                readIORef r                           <span class="co">-- get last value of r</span></span></code></pre></div>
-<p>Using <code>IORef</code> isn’t necessary most of the time. Haskell style prefers recursion, arguments and return values. However real world programs might need one or two IORefs occasionally.</p>
-</section>
-<section id="summary-of-io" data-number="3.10">
-<h2 data-number="11.10"><span class="header-section-number">11.10</span> Summary of IO</h2>
-<p>A value of type <code>IO X</code> is an <em>IO operation</em> that <em>produces</em> a value of type X <em>when run</em>. Operations are pure values. Only <em>running</em> the operation causes the side effects.</p>
-<p>IO operations can combined together using <code>do</code>-notation:</p>
-<div class="sourceCode" id="cb171"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb171-1"><a href="#cb171-1" aria-hidden="true" tabindex="-1"></a><span class="ot">op ::</span> <span class="dt">X</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> <span class="dt">Y</span></span>
+<p>Using <code>IORef</code> isn’t necessary most of the time. Haskell
+style prefers recursion, arguments and return values. However real world
+programs might need one or two IORefs occasionally.</p>
+<h2 data-number="11.10" id="summary-of-io"><span
+class="header-section-number">11.10</span> Summary of IO</h2>
+<p>A value of type <code>IO X</code> is an <em>IO operation</em> that
+<em>produces</em> a value of type X <em>when run</em>. Operations are
+pure values. Only <em>running</em> the operation causes the side
+effects.</p>
+<p>IO operations can combined together using
+<code>do</code>-notation:</p>
+<div class="sourceCode" id="cb171"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb171-1"><a href="#cb171-1" aria-hidden="true" tabindex="-1"></a><span class="ot">op ::</span> <span class="dt">X</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> <span class="dt">Y</span></span>
 <span id="cb171-2"><a href="#cb171-2" aria-hidden="true" tabindex="-1"></a>op arg <span class="ot">=</span> <span class="kw">do</span> operation                 <span class="co">-- run operation</span></span>
 <span id="cb171-3"><a href="#cb171-3" aria-hidden="true" tabindex="-1"></a>            operation2 arg            <span class="co">-- run operation with argument</span></span>
 <span id="cb171-4"><a href="#cb171-4" aria-hidden="true" tabindex="-1"></a>            result <span class="ot">&lt;-</span> operation3 arg  <span class="co">-- run operation with argument, store result</span></span>
 <span id="cb171-5"><a href="#cb171-5" aria-hidden="true" tabindex="-1"></a>            <span class="kw">let</span> something <span class="ot">=</span> f result  <span class="co">-- run a pure function f, store result</span></span>
 <span id="cb171-6"><a href="#cb171-6" aria-hidden="true" tabindex="-1"></a>            finalOperation            <span class="co">-- last operation produces the the return value</span></span></code></pre></div>
-<p>The <code>return x</code> operation is an operation that always produces value <code>x</code>. When <code>x :: a</code>, <code>return x :: IO a</code>.</p>
+<p>The <code>return x</code> operation is an operation that always
+produces value <code>x</code>. When <code>x :: a</code>,
+<code>return x :: IO a</code>.</p>
 <p>Useful IO operations:</p>
-<div class="sourceCode" id="cb172"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb172-1"><a href="#cb172-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- printing &amp; reading</span></span>
+<div class="sourceCode" id="cb172"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb172-1"><a href="#cb172-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- printing &amp; reading</span></span>
 <span id="cb172-2"><a href="#cb172-2" aria-hidden="true" tabindex="-1"></a><span class="fu">putStr</span><span class="ot"> ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span>
 <span id="cb172-3"><a href="#cb172-3" aria-hidden="true" tabindex="-1"></a><span class="fu">putStrLn</span><span class="ot"> ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span>
 <span id="cb172-4"><a href="#cb172-4" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span><span class="ot"> ::</span> <span class="dt">Show</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span>
@@ -2175,11 +3011,11 @@ Prelude&gt; final
 <span id="cb172-17"><a href="#cb172-17" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb172-18"><a href="#cb172-18" aria-hidden="true" tabindex="-1"></a><span class="co">-- files</span></span>
 <span id="cb172-19"><a href="#cb172-19" aria-hidden="true" tabindex="-1"></a><span class="fu">readFile</span><span class="ot"> ::</span> <span class="dt">FilePath</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> <span class="dt">String</span></span></code></pre></div>
-</section>
-<section id="quiz-2" data-number="3.11">
-<h2 data-number="11.11"><span class="header-section-number">11.11</span> Quiz</h2>
+<h2 data-number="11.11" id="quiz-2"><span
+class="header-section-number">11.11</span> Quiz</h2>
 <p>What is the type of this IO operation?</p>
-<div class="sourceCode" id="cb173"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb173-1"><a href="#cb173-1" aria-hidden="true" tabindex="-1"></a>foo x <span class="ot">=</span> <span class="kw">do</span> <span class="fu">putStrLn</span> x</span>
+<div class="sourceCode" id="cb173"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb173-1"><a href="#cb173-1" aria-hidden="true" tabindex="-1"></a>foo x <span class="ot">=</span> <span class="kw">do</span> <span class="fu">putStrLn</span> x</span>
 <span id="cb173-2"><a href="#cb173-2" aria-hidden="true" tabindex="-1"></a>           y <span class="ot">&lt;-</span> <span class="fu">getLine</span></span>
 <span id="cb173-3"><a href="#cb173-3" aria-hidden="true" tabindex="-1"></a>           <span class="fu">return</span> (<span class="fu">length</span> y)</span></code></pre></div>
 <ol class="quiz">
@@ -2197,7 +3033,8 @@ Prelude&gt; final
 </li>
 </ol>
 <p>Which of these lines could be used in place of <code>????</code></p>
-<div class="sourceCode" id="cb174"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb174-1"><a href="#cb174-1" aria-hidden="true" tabindex="-1"></a><span class="ot">quux ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> [<span class="dt">String</span>]</span>
+<div class="sourceCode" id="cb174"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb174-1"><a href="#cb174-1" aria-hidden="true" tabindex="-1"></a><span class="ot">quux ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> [<span class="dt">String</span>]</span>
 <span id="cb174-2"><a href="#cb174-2" aria-hidden="true" tabindex="-1"></a>quux q <span class="ot">=</span> <span class="kw">do</span> y <span class="ot">&lt;-</span> <span class="fu">getLine</span></span>
 <span id="cb174-3"><a href="#cb174-3" aria-hidden="true" tabindex="-1"></a>            z <span class="ot">&lt;-</span> <span class="fu">getLine</span></span>
 <span id="cb174-4"><a href="#cb174-4" aria-hidden="true" tabindex="-1"></a>            <span class="fu">putStrLn</span> (y<span class="op">++</span>z)</span>
@@ -2216,8 +3053,11 @@ Prelude&gt; final
 <code>ans &lt;- return [y,z]</code>
 </li>
 </ol>
-<p>What values does <code>blorg [1,2,3]</code> print? That is, what values <code>x</code> does it call <code>print x</code> for. The value produced by <code>blorg</code> doesn’t count.</p>
-<div class="sourceCode" id="cb175"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb175-1"><a href="#cb175-1" aria-hidden="true" tabindex="-1"></a>blorg [] <span class="ot">=</span> <span class="fu">return</span> <span class="dv">0</span></span>
+<p>What values does <code>blorg [1,2,3]</code> print? That is, what
+values <code>x</code> does it call <code>print x</code> for. The value
+produced by <code>blorg</code> doesn’t count.</p>
+<div class="sourceCode" id="cb175"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb175-1"><a href="#cb175-1" aria-hidden="true" tabindex="-1"></a>blorg [] <span class="ot">=</span> <span class="fu">return</span> <span class="dv">0</span></span>
 <span id="cb175-2"><a href="#cb175-2" aria-hidden="true" tabindex="-1"></a>blorg (x<span class="op">:</span>xs) <span class="ot">=</span> <span class="kw">do</span> m <span class="ot">&lt;-</span> blorg xs</span>
 <span id="cb175-3"><a href="#cb175-3" aria-hidden="true" tabindex="-1"></a>                  <span class="fu">print</span> x</span>
 <span id="cb175-4"><a href="#cb175-4" aria-hidden="true" tabindex="-1"></a>                  <span class="fu">return</span> (m<span class="op">+</span>x)</span></code></pre></div>
@@ -2235,7 +3075,8 @@ It prints <code>3</code>, <code>2</code>, <code>1</code>
 It prints <code>3</code>, <code>2</code>, <code>1</code>, <code>6</code>
 </li>
 </ol>
-<p>Which of these can a function of type <code>Int -&gt; IO Int</code> do?</p>
+<p>Which of these can a function of type <code>Int -&gt; IO Int</code>
+do?</p>
 <ol class="quiz">
 <li>
 No function of this type can be defined.
@@ -2250,7 +3091,8 @@ Run the IO operation it has been given and return its value.
 Query the user for a number and return it.
 </li>
 </ol>
-<p>Which of these can a function of type <code>IO Int -&gt; Int</code> do?</p>
+<p>Which of these can a function of type <code>IO Int -&gt; Int</code>
+do?</p>
 <ol class="quiz">
 <li>
 No function of this type can be defined.
@@ -2265,34 +3107,50 @@ Run the IO operation it has been given and return its value.
 Query the user for a number and return it.
 </li>
 </ol>
-</section>
-<section id="exercises-2" data-number="3.12">
-<h2 data-number="11.12"><span class="header-section-number">11.12</span> Exercises</h2>
+<h2 data-number="11.12" id="exercises-2"><span
+class="header-section-number">11.12</span> Exercises</h2>
 <ul>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set11a.hs">Set11a</a> - basic IO exercises</li>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set11b.hs">Set11b</a> - advanced IO exercises</li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set11a.hs">Set11a</a>
+- basic IO exercises</li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set11b.hs">Set11b</a>
+- advanced IO exercises</li>
 </ul>
-</section>
-</section>
-<section id="lecture-12-fmap-fmap-fmap" data-number="4">
-<h1 data-number="12"><span class="header-section-number">12</span> Lecture 12: fmap fmap fmap</h1>
-<section id="contents-1" data-number="4.1">
-<h2 data-number="12.1"><span class="header-section-number">12.1</span> Contents</h2>
+<h1 data-number="12" id="lecture-12-fmap-fmap-fmap"><span
+class="header-section-number">12</span> Lecture 12: fmap fmap fmap</h1>
+<h2 data-number="12.1" id="contents-1"><span
+class="header-section-number">12.1</span> Contents</h2>
 <ul>
 <li>Functors</li>
 </ul>
-</section>
-<section id="functors" data-number="4.2">
-<h2 data-number="12.2"><span class="header-section-number">12.2</span> Functors</h2>
-<section id="preserving-structure" data-number="4.2.1">
-<h3 data-number="12.2.1"><span class="header-section-number">12.2.1</span> Preserving Structure</h3>
-<p>Remember the <code>map</code> function for lists? Here’s the definition again:</p>
-<div class="sourceCode" id="cb176"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb176-1"><a href="#cb176-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [b]</span>
+<h2 data-number="12.2" id="functors"><span
+class="header-section-number">12.2</span> Functors</h2>
+<h3 data-number="12.2.1" id="preserving-structure"><span
+class="header-section-number">12.2.1</span> Preserving Structure</h3>
+<p>Remember the <code>map</code> function for lists? Here’s the
+definition again:</p>
+<div class="sourceCode" id="cb176"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb176-1"><a href="#cb176-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span><span class="ot"> ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [b]</span>
 <span id="cb176-2"><a href="#cb176-2" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> _ []     <span class="ot">=</span> []</span>
 <span id="cb176-3"><a href="#cb176-3" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> g (x<span class="op">:</span>xs) <span class="ot">=</span> g x <span class="op">:</span> <span class="fu">map</span> g xs</span></code></pre></div>
-<p>It applies a function <code>g :: a -&gt; b</code> to each element of a list of type <code>[a]</code>, returning a list of type <code>[b]</code>. Another way to express the type of <code>map</code> would be <code>(a -&gt; b) -&gt; ([a] -&gt; [b])</code>. This is the same type because <code>-&gt;</code> associates to right. The extra parentheses emphasize the fact that <code>map</code> converts the function <code>g :: a -&gt; b</code> into a function <code>map g :: [a] -&gt; [b]</code>. This means that <code>map</code> is a <em>higher-order function</em> that transforms functions to functions.</p>
-<p>As <code>map</code> is parametrically polymorphic, its definition doesn’t depend on the type of values stored in the list. Thus, every function of type <code>a -&gt; b</code> is converted into a function of type <code>[a] -&gt; [b]</code> using exactly the same logic. Using the definition above, we can see that:</p>
-<div class="sourceCode" id="cb177"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb177-1"><a href="#cb177-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> (<span class="op">||</span> <span class="dt">True</span>) [<span class="dt">True</span>, <span class="dt">True</span>, <span class="dt">False</span>]</span>
+<p>It applies a function <code>g :: a -&gt; b</code> to each element of
+a list of type <code>[a]</code>, returning a list of type
+<code>[b]</code>. Another way to express the type of <code>map</code>
+would be <code>(a -&gt; b) -&gt; ([a] -&gt; [b])</code>. This is the
+same type because <code>-&gt;</code> associates to right. The extra
+parentheses emphasize the fact that <code>map</code> converts the
+function <code>g :: a -&gt; b</code> into a function
+<code>map g :: [a] -&gt; [b]</code>. This means that <code>map</code> is
+a <em>higher-order function</em> that transforms functions to
+functions.</p>
+<p>As <code>map</code> is parametrically polymorphic, its definition
+doesn’t depend on the type of values stored in the list. Thus, every
+function of type <code>a -&gt; b</code> is converted into a function of
+type <code>[a] -&gt; [b]</code> using exactly the same logic. Using the
+definition above, we can see that:</p>
+<div class="sourceCode" id="cb177"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb177-1"><a href="#cb177-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> (<span class="op">||</span> <span class="dt">True</span>) [<span class="dt">True</span>, <span class="dt">True</span>, <span class="dt">False</span>]</span>
 <span id="cb177-2"><a href="#cb177-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="dt">True</span> <span class="op">||</span> <span class="dt">True</span>, <span class="dt">True</span> <span class="op">||</span> <span class="dt">True</span>, <span class="dt">False</span> <span class="op">||</span> <span class="dt">True</span>]</span>
 <span id="cb177-3"><a href="#cb177-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="dt">True</span>, <span class="dt">True</span>, <span class="dt">True</span>]</span>
 <span id="cb177-4"><a href="#cb177-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -2303,83 +3161,154 @@ Query the user for a number and return it.
 <span id="cb177-9"><a href="#cb177-9" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span> (<span class="op">++</span><span class="st">&quot;1&quot;</span>) [<span class="st">&quot;1&quot;</span>, <span class="st">&quot;2&quot;</span>, <span class="st">&quot;3&quot;</span>]</span>
 <span id="cb177-10"><a href="#cb177-10" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="st">&quot;1&quot;</span> <span class="op">++</span> <span class="st">&quot;1&quot;</span>, <span class="st">&quot;2&quot;</span> <span class="op">++</span> <span class="st">&quot;1&quot;</span>, <span class="st">&quot;3&quot;</span> <span class="op">++</span> <span class="st">&quot;1&quot;</span>]</span>
 <span id="cb177-11"><a href="#cb177-11" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="st">&quot;11&quot;</span>, <span class="st">&quot;21&quot;</span>, <span class="st">&quot;31&quot;</span>]</span></code></pre></div>
-<p>What’s notable here is that <code>map</code> <em>preserves the structure</em> of a list. The length of the list and the relative positions of the elements are the same. The general idea is demonstrated in the picture below.</p>
+<p>What’s notable here is that <code>map</code> <em>preserves the
+structure</em> of a list. The length of the list and the relative
+positions of the elements are the same. The general idea is demonstrated
+in the picture below.</p>
 <figure>
-<img src="img/map.svg" alt="Mapping a function g to a list" /><figcaption aria-hidden="true">Mapping a function <code>g</code> to a list</figcaption>
+<img src="img/map.svg" alt="Mapping a function g to a list" />
+<figcaption aria-hidden="true">Mapping a function <code>g</code> to a
+list</figcaption>
 </figure>
-<p>Let’s see if we can find other similar functions. A value of type <code>Maybe a</code> is kind of like a list of length at most 1. Let’s map over a <code>Maybe</code>! Can you see the similarity with the definition of <code>map</code>?</p>
-<div class="sourceCode" id="cb178"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb178-1"><a href="#cb178-1" aria-hidden="true" tabindex="-1"></a><span class="ot">mapMaybe ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> <span class="dt">Maybe</span> a <span class="ot">-&gt;</span> <span class="dt">Maybe</span> b</span>
+<p>Let’s see if we can find other similar functions. A value of type
+<code>Maybe a</code> is kind of like a list of length at most 1. Let’s
+map over a <code>Maybe</code>! Can you see the similarity with the
+definition of <code>map</code>?</p>
+<div class="sourceCode" id="cb178"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb178-1"><a href="#cb178-1" aria-hidden="true" tabindex="-1"></a><span class="ot">mapMaybe ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> <span class="dt">Maybe</span> a <span class="ot">-&gt;</span> <span class="dt">Maybe</span> b</span>
 <span id="cb178-2"><a href="#cb178-2" aria-hidden="true" tabindex="-1"></a>mapMaybe f <span class="dt">Nothing</span> <span class="ot">=</span> <span class="dt">Nothing</span></span>
 <span id="cb178-3"><a href="#cb178-3" aria-hidden="true" tabindex="-1"></a>mapMaybe f (<span class="dt">Just</span> x) <span class="ot">=</span> <span class="dt">Just</span> (f x)</span></code></pre></div>
-<p>Here too, the structure of the value is preserved. A <code>Nothing</code> turns into a <code>Nothing</code>, and a <code>Just</code> turns into a <code>Just</code>. Here too, we can think of the type as <code>(a -&gt; b) -&gt; (Maybe a -&gt; Maybe b)</code>, converting (or “lifting”) a normal function into a function that works on Maybes.</p>
+<p>Here too, the structure of the value is preserved. A
+<code>Nothing</code> turns into a <code>Nothing</code>, and a
+<code>Just</code> turns into a <code>Just</code>. Here too, we can think
+of the type as <code>(a -&gt; b) -&gt; (Maybe a -&gt; Maybe b)</code>,
+converting (or “lifting”) a normal function into a function that works
+on Maybes.</p>
 <p>One more example: consider binary trees.</p>
-<div class="sourceCode" id="cb179"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb179-1"><a href="#cb179-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Tree</span> a <span class="ot">=</span> <span class="dt">Leaf</span> <span class="op">|</span> <span class="dt">Node</span> a (<span class="dt">Tree</span> a) (<span class="dt">Tree</span> a)</span>
+<div class="sourceCode" id="cb179"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb179-1"><a href="#cb179-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Tree</span> a <span class="ot">=</span> <span class="dt">Leaf</span> <span class="op">|</span> <span class="dt">Node</span> a (<span class="dt">Tree</span> a) (<span class="dt">Tree</span> a)</span>
 <span id="cb179-2"><a href="#cb179-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb179-3"><a href="#cb179-3" aria-hidden="true" tabindex="-1"></a><span class="ot">mapTree ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> <span class="dt">Tree</span> a <span class="ot">-&gt;</span> <span class="dt">Tree</span> b</span>
 <span id="cb179-4"><a href="#cb179-4" aria-hidden="true" tabindex="-1"></a>mapTree f <span class="dt">Leaf</span> <span class="ot">=</span> <span class="dt">Leaf</span></span>
 <span id="cb179-5"><a href="#cb179-5" aria-hidden="true" tabindex="-1"></a>mapTree f (<span class="dt">Node</span> val left right) <span class="ot">=</span> <span class="dt">Node</span> (f val) (mapTree f left) (mapTree f right)</span></code></pre></div>
 <p>A binary tree might look like this:</p>
 <figure>
-<img src="img/mapTreeBefore.svg" alt="A binary tree" /><figcaption aria-hidden="true">A binary tree</figcaption>
+<img src="img/mapTreeBefore.svg" alt="A binary tree" />
+<figcaption aria-hidden="true">A binary tree</figcaption>
 </figure>
 <p>After <code>mapTree g</code> the tree would looke like this:</p>
 <figure>
-<img src="img/mapTreeAfter.svg" alt="A binary tree after mapping g into its nodes" /><figcaption aria-hidden="true">A binary tree after mapping <code>g</code> into its nodes</figcaption>
+<img src="img/mapTreeAfter.svg"
+alt="A binary tree after mapping g into its nodes" />
+<figcaption aria-hidden="true">A binary tree after mapping
+<code>g</code> into its nodes</figcaption>
 </figure>
-</section>
-<section id="the-functor-class" data-number="4.2.2">
-<h3 data-number="12.2.2"><span class="header-section-number">12.2.2</span> The <code>Functor</code> Class</h3>
-<p>Now we have three different structure-preserving mapping functions. Three similar operations over different types. Could we write a type class to capture this similarity?</p>
-<div class="sourceCode" id="cb180"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb180-1"><a href="#cb180-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span><span class="ot">      ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span>      [a] <span class="ot">-&gt;</span>      [b]</span>
+<h3 data-number="12.2.2" id="the-functor-class"><span
+class="header-section-number">12.2.2</span> The <code>Functor</code>
+Class</h3>
+<p>Now we have three different structure-preserving mapping functions.
+Three similar operations over different types. Could we write a type
+class to capture this similarity?</p>
+<div class="sourceCode" id="cb180"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb180-1"><a href="#cb180-1" aria-hidden="true" tabindex="-1"></a><span class="fu">map</span><span class="ot">      ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span>      [a] <span class="ot">-&gt;</span>      [b]</span>
 <span id="cb180-2"><a href="#cb180-2" aria-hidden="true" tabindex="-1"></a><span class="ot">mapMaybe ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> <span class="dt">Maybe</span> a  <span class="ot">-&gt;</span> <span class="dt">Maybe</span> b</span>
 <span id="cb180-3"><a href="#cb180-3" aria-hidden="true" tabindex="-1"></a><span class="ot">mapTree  ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> <span class="dt">Tree</span>  a  <span class="ot">-&gt;</span> <span class="dt">Tree</span>  b</span></code></pre></div>
-<p>A naive attempt at writing a type class runs into problems. If we try to abstract over <code>Maybe c</code>, it seems we can’t write the right type for the map operation. We’d need to be able to change the type parameter <code>c</code> somehow.</p>
-<div class="sourceCode" id="cb181"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb181-1"><a href="#cb181-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Mappable</span> m <span class="kw">where</span></span>
+<p>A naive attempt at writing a type class runs into problems. If we try
+to abstract over <code>Maybe c</code>, it seems we can’t write the right
+type for the map operation. We’d need to be able to change the type
+parameter <code>c</code> somehow.</p>
+<div class="sourceCode" id="cb181"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb181-1"><a href="#cb181-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Mappable</span> m <span class="kw">where</span></span>
 <span id="cb181-2"><a href="#cb181-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  mapThing ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> m <span class="ot">-&gt;</span> m</span>
 <span id="cb181-3"><a href="#cb181-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb181-4"><a href="#cb181-4" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Mappable</span> (<span class="dt">Maybe</span> c) <span class="kw">where</span></span>
 <span id="cb181-5"><a href="#cb181-5" aria-hidden="true" tabindex="-1"></a><span class="ot">  mapThing ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> <span class="dt">Maybe</span> c <span class="ot">-&gt;</span> <span class="dt">Maybe</span> c</span>
 <span id="cb181-6"><a href="#cb181-6" aria-hidden="true" tabindex="-1"></a>  mapThing <span class="ot">=</span> <span class="op">...</span></span></code></pre></div>
-<p>Luckily Haskell type classes have a feature we haven’t covered before. You can write classes for <em>type constructors</em> in addition to types. What does this mean? Let’s just have a look at the standard type class <code>Functor</code> that does what we tried to do with our <code>Mappable</code>.</p>
-<div class="sourceCode" id="cb182"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb182-1"><a href="#cb182-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Functor</span> f <span class="kw">where</span></span>
+<p>Luckily Haskell type classes have a feature we haven’t covered
+before. You can write classes for <em>type constructors</em> in addition
+to types. What does this mean? Let’s just have a look at the standard
+type class <code>Functor</code> that does what we tried to do with our
+<code>Mappable</code>.</p>
+<div class="sourceCode" id="cb182"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb182-1"><a href="#cb182-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Functor</span> f <span class="kw">where</span></span>
 <span id="cb182-2"><a href="#cb182-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  fmap ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> f a <span class="ot">-&gt;</span> f b</span></code></pre></div>
-<p>Note how the type parameter <code>f</code> is a type constructor: it’s being passed <code>a</code> and <code>b</code> arguments in different parts of the type of <code>fmap</code>. Now let’s see the instance for <code>Maybe</code>.</p>
-<div class="sourceCode" id="cb183"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb183-1"><a href="#cb183-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Functor</span> <span class="dt">Maybe</span> <span class="kw">where</span></span>
+<p>Note how the type parameter <code>f</code> is a type constructor:
+it’s being passed <code>a</code> and <code>b</code> arguments in
+different parts of the type of <code>fmap</code>. Now let’s see the
+instance for <code>Maybe</code>.</p>
+<div class="sourceCode" id="cb183"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb183-1"><a href="#cb183-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Functor</span> <span class="dt">Maybe</span> <span class="kw">where</span></span>
 <span id="cb183-2"><a href="#cb183-2" aria-hidden="true" tabindex="-1"></a>  <span class="co">-- In this instance, the type of fmap is:</span></span>
 <span id="cb183-3"><a href="#cb183-3" aria-hidden="true" tabindex="-1"></a>  <span class="co">-- fmap :: (a -&gt; b) -&gt; Maybe a -&gt; Maybe b</span></span>
 <span id="cb183-4"><a href="#cb183-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">fmap</span> f <span class="dt">Nothing</span> <span class="ot">=</span> <span class="dt">Nothing</span></span>
 <span id="cb183-5"><a href="#cb183-5" aria-hidden="true" tabindex="-1"></a>  <span class="fu">fmap</span> f (<span class="dt">Just</span> x) <span class="ot">=</span> <span class="dt">Just</span> (f x)</span></code></pre></div>
-<p>Now <code>fmap</code> has the right type and we can implement it like <code>mapMaybe</code>! Note how we’ve declared <code>instance Functor Maybe</code> instead of <code>instance Functor (Maybe a)</code>. The type <code>Maybe a</code> isn’t a functor, the type constructor <code>Maybe</code> is.</p>
-<p>The type constructor for lists is written <code>[]</code>. It’s special syntax, just like other list syntax. However if the type <code>[a]</code> was written <code>List a</code>, the type constructor <code>[]</code> would mean <code>List</code>.</p>
-<div class="sourceCode" id="cb184"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb184-1"><a href="#cb184-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Functor</span> [] <span class="kw">where</span></span>
+<p>Now <code>fmap</code> has the right type and we can implement it like
+<code>mapMaybe</code>! Note how we’ve declared
+<code>instance Functor Maybe</code> instead of
+<code>instance Functor (Maybe a)</code>. The type <code>Maybe a</code>
+isn’t a functor, the type constructor <code>Maybe</code> is.</p>
+<p>The type constructor for lists is written <code>[]</code>. It’s
+special syntax, just like other list syntax. However if the type
+<code>[a]</code> was written <code>List a</code>, the type constructor
+<code>[]</code> would mean <code>List</code>.</p>
+<div class="sourceCode" id="cb184"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb184-1"><a href="#cb184-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Functor</span> [] <span class="kw">where</span></span>
 <span id="cb184-2"><a href="#cb184-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">fmap</span> <span class="ot">=</span> <span class="fu">map</span></span></code></pre></div>
-<p>Here’s the final of our examples, as a <code>Functor</code> instance.</p>
-<div class="sourceCode" id="cb185"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb185-1"><a href="#cb185-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Tree</span> a <span class="ot">=</span> <span class="dt">Leaf</span> <span class="op">|</span> <span class="dt">Node</span> a (<span class="dt">Tree</span> a) (<span class="dt">Tree</span> a)</span>
+<p>Here’s the final of our examples, as a <code>Functor</code>
+instance.</p>
+<div class="sourceCode" id="cb185"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb185-1"><a href="#cb185-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Tree</span> a <span class="ot">=</span> <span class="dt">Leaf</span> <span class="op">|</span> <span class="dt">Node</span> a (<span class="dt">Tree</span> a) (<span class="dt">Tree</span> a)</span>
 <span id="cb185-2"><a href="#cb185-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb185-3"><a href="#cb185-3" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Functor</span> <span class="dt">Tree</span> <span class="kw">where</span></span>
 <span id="cb185-4"><a href="#cb185-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">fmap</span> _ <span class="dt">Leaf</span> <span class="ot">=</span> <span class="dt">Leaf</span></span>
 <span id="cb185-5"><a href="#cb185-5" aria-hidden="true" tabindex="-1"></a>  <span class="fu">fmap</span> f (<span class="dt">Node</span> val left right) <span class="ot">=</span> <span class="dt">Node</span> (f val) (<span class="fu">fmap</span> f left) (<span class="fu">fmap</span> f right)</span></code></pre></div>
-<p>Sidenote: the term functor comes originally from a branch of mathematics called <a href="https://en.wikipedia.org/wiki/Category_theory">category theory</a>. However, to work with Haskell you don’t need to know any category theory. As you progress in learning Haskell, you might get interested in category theory, and it can be a valuable source of new ideas for programming. Category theory can feel intimidating, so it’s good to know that you can get on fine without it. For now, when you see functor, you can just think “something I can map over”, or perhaps “a container”.</p>
-<p>Let’s zoom out a bit. When we have an instance <code>Functor MyFun</code> we know that we can map a type <code>X</code> into a new type <code>MyFun X</code> (since <code>MyFun</code> is a type constructor), but also that we can lift a function <code>f</code> that takes an <code>X</code> argument into a function <code>fmap f</code> that takes a <code>MyFun X</code> argument! So you could say we’re mapping both on the type level and the value level.</p>
-<p>Oh right, one more thing. Once you’ve gotten the hang of <code>fmap</code> you might find yourself using it quite a bit. For code that uses <code>fmap</code> heavily it can be nice to use its infix alias, <code>&lt;$&gt;</code>. Consider the symmetry between <code>$</code> and <code>&lt;$&gt;</code> in these examples:</p>
-<div class="sourceCode" id="cb186"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb186-1"><a href="#cb186-1" aria-hidden="true" tabindex="-1"></a>(<span class="op">+</span><span class="dv">1</span>) <span class="op">&lt;$&gt;</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]    <span class="op">==&gt;</span>  [<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>]</span>
+<p>Sidenote: the term functor comes originally from a branch of
+mathematics called <a
+href="https://en.wikipedia.org/wiki/Category_theory">category
+theory</a>. However, to work with Haskell you don’t need to know any
+category theory. As you progress in learning Haskell, you might get
+interested in category theory, and it can be a valuable source of new
+ideas for programming. Category theory can feel intimidating, so it’s
+good to know that you can get on fine without it. For now, when you see
+functor, you can just think “something I can map over”, or perhaps “a
+container”.</p>
+<p>Let’s zoom out a bit. When we have an instance
+<code>Functor MyFun</code> we know that we can map a type <code>X</code>
+into a new type <code>MyFun X</code> (since <code>MyFun</code> is a type
+constructor), but also that we can lift a function <code>f</code> that
+takes an <code>X</code> argument into a function <code>fmap f</code>
+that takes a <code>MyFun X</code> argument! So you could say we’re
+mapping both on the type level and the value level.</p>
+<p>Oh right, one more thing. Once you’ve gotten the hang of
+<code>fmap</code> you might find yourself using it quite a bit. For code
+that uses <code>fmap</code> heavily it can be nice to use its infix
+alias, <code>&lt;$&gt;</code>. Consider the symmetry between
+<code>$</code> and <code>&lt;$&gt;</code> in these examples:</p>
+<div class="sourceCode" id="cb186"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb186-1"><a href="#cb186-1" aria-hidden="true" tabindex="-1"></a>(<span class="op">+</span><span class="dv">1</span>) <span class="op">&lt;$&gt;</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]    <span class="op">==&gt;</span>  [<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>]</span>
 <span id="cb186-2"><a href="#cb186-2" aria-hidden="true" tabindex="-1"></a><span class="fu">not</span> <span class="op">&lt;$&gt;</span> <span class="dt">Just</span> <span class="dt">False</span>  <span class="op">==&gt;</span>  <span class="dt">Just</span> <span class="dt">True</span></span>
 <span id="cb186-3"><a href="#cb186-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb186-4"><a href="#cb186-4" aria-hidden="true" tabindex="-1"></a><span class="fu">reverse</span> <span class="op">.</span> <span class="fu">tail</span>  <span class="op">$</span>       <span class="st">&quot;hello&quot;</span>       <span class="op">==&gt;</span>  <span class="st">&quot;olle&quot;</span></span>
 <span id="cb186-5"><a href="#cb186-5" aria-hidden="true" tabindex="-1"></a><span class="fu">reverse</span> <span class="op">.</span> <span class="fu">tail</span> <span class="op">&lt;$&gt;</span> <span class="dt">Just</span> <span class="st">&quot;hello&quot;</span>       <span class="op">==&gt;</span>  <span class="dt">Just</span> <span class="st">&quot;olle&quot;</span></span>
 <span id="cb186-6"><a href="#cb186-6" aria-hidden="true" tabindex="-1"></a><span class="co">-- which is the same as</span></span>
 <span id="cb186-7"><a href="#cb186-7" aria-hidden="true" tabindex="-1"></a><span class="fu">fmap</span> (<span class="fu">reverse</span> <span class="op">.</span> <span class="fu">tail</span>) (<span class="dt">Just</span> <span class="st">&quot;hello&quot;</span>)  <span class="op">==&gt;</span>  <span class="dt">Just</span> <span class="st">&quot;olle&quot;</span></span></code></pre></div>
-</section>
-</section>
-<section id="lawful-instances" data-number="4.3">
-<h2 data-number="12.3"><span class="header-section-number">12.3</span> Lawful Instances</h2>
-<p>What is this “preserving of the structure” that was mentioned above exactly? The following two <em>functor laws</em> are expected to hold for any <code>Functor</code> instance <code>f</code> (though unfortunately Haskell compilers can’t enforce them):</p>
+<h2 data-number="12.3" id="lawful-instances"><span
+class="header-section-number">12.3</span> Lawful Instances</h2>
+<p>What is this “preserving of the structure” that was mentioned above
+exactly? The following two <em>functor laws</em> are expected to hold
+for any <code>Functor</code> instance <code>f</code> (though
+unfortunately Haskell compilers can’t enforce them):</p>
 <ol type="1">
 <li><code>fmap id      === id</code></li>
 <li><code>fmap (f . g) === fmap f . fmap g</code></li>
 </ol>
-<p>Don’t worry if that sounded abstract! The first law says that a functor maps <code>id :: a -&gt; a</code> into <code>id :: f a -&gt; f a</code>. (<code>id</code> is the identity function, meaning that <code>id x = x</code>.) Let’s be concrete and see how it works for the list <code>[1,2,3]</code>:</p>
-<div class="sourceCode" id="cb187"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb187-1"><a href="#cb187-1" aria-hidden="true" tabindex="-1"></a><span class="fu">fmap</span> <span class="fu">id</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] <span class="op">==&gt;</span> <span class="fu">map</span> <span class="fu">id</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
+<p>Don’t worry if that sounded abstract! The first law says that a
+functor maps <code>id :: a -&gt; a</code> into
+<code>id :: f a -&gt; f a</code>. (<code>id</code> is the identity
+function, meaning that <code>id x = x</code>.) Let’s be concrete and see
+how it works for the list <code>[1,2,3]</code>:</p>
+<div class="sourceCode" id="cb187"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb187-1"><a href="#cb187-1" aria-hidden="true" tabindex="-1"></a><span class="fu">fmap</span> <span class="fu">id</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] <span class="op">==&gt;</span> <span class="fu">map</span> <span class="fu">id</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb187-2"><a href="#cb187-2" aria-hidden="true" tabindex="-1"></a>                <span class="op">==&gt;</span> <span class="fu">map</span> <span class="fu">id</span> (<span class="dv">1</span><span class="op">:</span>[<span class="dv">2</span>,<span class="dv">3</span>])</span>
 <span id="cb187-3"><a href="#cb187-3" aria-hidden="true" tabindex="-1"></a>                <span class="op">==&gt;</span> <span class="fu">id</span> <span class="dv">1</span> <span class="op">:</span> <span class="fu">map</span> <span class="fu">id</span> [<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb187-4"><a href="#cb187-4" aria-hidden="true" tabindex="-1"></a>                <span class="op">==&gt;</span> <span class="dv">1</span> <span class="op">:</span> <span class="fu">map</span> <span class="fu">id</span> [<span class="dv">2</span>,<span class="dv">3</span>]</span>
@@ -2388,11 +3317,23 @@ Query the user for a number and return it.
 <span id="cb187-7"><a href="#cb187-7" aria-hidden="true" tabindex="-1"></a>                <span class="op">==&gt;</span> <span class="dv">1</span> <span class="op">:</span> <span class="dv">2</span> <span class="op">:</span> <span class="dv">3</span> <span class="op">:</span> []</span>
 <span id="cb187-8"><a href="#cb187-8" aria-hidden="true" tabindex="-1"></a>                <span class="op">===</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span></code></pre></div>
 <p>On the other hand,</p>
-<div class="sourceCode" id="cb188"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb188-1"><a href="#cb188-1" aria-hidden="true" tabindex="-1"></a><span class="fu">id</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] <span class="op">==&gt;</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span></code></pre></div>
-<p>Hence, the result of <code>fmap id [1,2,3]</code> was the same as the result of <code>id [1,2,3]</code>, so the first functor law holds in this case. It’s not hard to show that the first functor law holds for any list whatsoever.</p>
-<p>The first functor law is really a very simple proposition if you think about it. It just says that if we apply <code>fmap</code> to a function that changes nothing (<code>id</code>) the resulting function (<code>fmap id</code>) again changes nothing. Thus, the act of applying <code>fmap</code> itself preserves the structure of the functor.</p>
-<p>How about the second functor law? For lists, consider what happens if we <code>fmap</code> the function <code>negate.(*2)</code> (remember, <code>negate</code> maps <code>x</code> to <code>-x</code> and <code>(*2)</code> multiplies its argument by <code>2</code>):</p>
-<div class="sourceCode" id="cb189"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb189-1"><a href="#cb189-1" aria-hidden="true" tabindex="-1"></a><span class="fu">fmap</span> (<span class="fu">negate</span><span class="op">.</span>(<span class="op">*</span><span class="dv">2</span>)) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] <span class="op">==&gt;</span> <span class="fu">map</span> (<span class="fu">negate</span><span class="op">.</span>(<span class="op">*</span><span class="dv">2</span>)) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
+<div class="sourceCode" id="cb188"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb188-1"><a href="#cb188-1" aria-hidden="true" tabindex="-1"></a><span class="fu">id</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] <span class="op">==&gt;</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span></code></pre></div>
+<p>Hence, the result of <code>fmap id [1,2,3]</code> was the same as the
+result of <code>id [1,2,3]</code>, so the first functor law holds in
+this case. It’s not hard to show that the first functor law holds for
+any list whatsoever.</p>
+<p>The first functor law is really a very simple proposition if you
+think about it. It just says that if we apply <code>fmap</code> to a
+function that changes nothing (<code>id</code>) the resulting function
+(<code>fmap id</code>) again changes nothing. Thus, the act of applying
+<code>fmap</code> itself preserves the structure of the functor.</p>
+<p>How about the second functor law? For lists, consider what happens if
+we <code>fmap</code> the function <code>negate.(*2)</code> (remember,
+<code>negate</code> maps <code>x</code> to <code>-x</code> and
+<code>(*2)</code> multiplies its argument by <code>2</code>):</p>
+<div class="sourceCode" id="cb189"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb189-1"><a href="#cb189-1" aria-hidden="true" tabindex="-1"></a><span class="fu">fmap</span> (<span class="fu">negate</span><span class="op">.</span>(<span class="op">*</span><span class="dv">2</span>)) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] <span class="op">==&gt;</span> <span class="fu">map</span> (<span class="fu">negate</span><span class="op">.</span>(<span class="op">*</span><span class="dv">2</span>)) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb189-2"><a href="#cb189-2" aria-hidden="true" tabindex="-1"></a>                           <span class="op">==&gt;</span> (<span class="fu">negate</span><span class="op">.</span>(<span class="op">*</span><span class="dv">2</span>)) <span class="dv">1</span> <span class="op">:</span> <span class="fu">map</span> (<span class="fu">negate</span><span class="op">.</span>(<span class="op">*</span><span class="dv">2</span>)) [<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb189-3"><a href="#cb189-3" aria-hidden="true" tabindex="-1"></a>                           <span class="op">==&gt;</span> <span class="fu">negate</span> (<span class="dv">1</span> <span class="op">*</span> <span class="dv">2</span>)  <span class="op">:</span> <span class="fu">map</span> (<span class="fu">negate</span><span class="op">.</span>(<span class="op">*</span><span class="dv">2</span>)) [<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb189-4"><a href="#cb189-4" aria-hidden="true" tabindex="-1"></a>                           <span class="op">==&gt;</span> <span class="op">-</span><span class="dv">2</span> <span class="op">:</span> <span class="fu">map</span> (<span class="fu">negate</span><span class="op">.</span>(<span class="op">*</span><span class="dv">2</span>)) [<span class="dv">2</span>,<span class="dv">3</span>]</span>
@@ -2401,13 +3342,21 @@ Query the user for a number and return it.
 <span id="cb189-7"><a href="#cb189-7" aria-hidden="true" tabindex="-1"></a>                           <span class="op">==&gt;</span> <span class="op">-</span><span class="dv">2</span> <span class="op">:</span> <span class="op">-</span><span class="dv">4</span> <span class="op">:</span> (<span class="fu">negate</span><span class="op">.</span>(<span class="op">*</span><span class="dv">2</span>)) <span class="dv">3</span> <span class="op">:</span> <span class="fu">map</span> (<span class="fu">negate</span><span class="op">.</span>(<span class="op">*</span><span class="dv">2</span>)) []</span>
 <span id="cb189-8"><a href="#cb189-8" aria-hidden="true" tabindex="-1"></a>                           <span class="op">==&gt;</span> <span class="op">-</span><span class="dv">2</span> <span class="op">:</span> <span class="op">-</span><span class="dv">4</span> <span class="op">:</span> <span class="op">-</span><span class="dv">6</span> <span class="op">:</span> []</span>
 <span id="cb189-9"><a href="#cb189-9" aria-hidden="true" tabindex="-1"></a>                           <span class="op">==&gt;</span> [<span class="op">-</span><span class="dv">2</span>,<span class="op">-</span><span class="dv">4</span>,<span class="op">-</span><span class="dv">6</span>]</span></code></pre></div>
-<p>Let’s consider the right-hand side of the second functor law in this case:</p>
-<div class="sourceCode" id="cb190"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb190-1"><a href="#cb190-1" aria-hidden="true" tabindex="-1"></a>(<span class="fu">fmap</span> <span class="fu">negate</span> <span class="op">.</span> <span class="fu">fmap</span> (<span class="op">*</span><span class="dv">2</span>)) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] <span class="op">==&gt;</span> (<span class="fu">map</span> <span class="fu">negate</span> <span class="op">.</span> <span class="fu">map</span> (<span class="op">*</span><span class="dv">2</span>)) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
+<p>Let’s consider the right-hand side of the second functor law in this
+case:</p>
+<div class="sourceCode" id="cb190"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb190-1"><a href="#cb190-1" aria-hidden="true" tabindex="-1"></a>(<span class="fu">fmap</span> <span class="fu">negate</span> <span class="op">.</span> <span class="fu">fmap</span> (<span class="op">*</span><span class="dv">2</span>)) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] <span class="op">==&gt;</span> (<span class="fu">map</span> <span class="fu">negate</span> <span class="op">.</span> <span class="fu">map</span> (<span class="op">*</span><span class="dv">2</span>)) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb190-2"><a href="#cb190-2" aria-hidden="true" tabindex="-1"></a>                                  <span class="op">==&gt;</span> <span class="fu">map</span> <span class="fu">negate</span> (<span class="fu">map</span> (<span class="op">*</span><span class="dv">2</span>) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>])</span>
 <span id="cb190-3"><a href="#cb190-3" aria-hidden="true" tabindex="-1"></a>                                  <span class="op">==&gt;</span> <span class="fu">map</span> <span class="fu">negate</span> [<span class="dv">2</span>,<span class="dv">4</span>,<span class="dv">6</span>]</span>
 <span id="cb190-4"><a href="#cb190-4" aria-hidden="true" tabindex="-1"></a>                                  <span class="op">==&gt;</span> [<span class="op">-</span><span class="dv">2</span>,<span class="op">-</span><span class="dv">4</span>,<span class="op">-</span><span class="dv">6</span>]</span></code></pre></div>
-<p>The second functor law turns out to hold in this particular case. In fact, it holds in all cases (exercise!).</p>
-<p>In general, the second functor law says that composing two functions first and then applying <code>fmap</code> must yield the same result as performing <code>fmap</code> on those functions and then composing the resulting function. In other words, the order of applying <code>fmap</code> and composing doesn’t matter. (These two operations are said to <em>commute</em>.)</p>
+<p>The second functor law turns out to hold in this particular case. In
+fact, it holds in all cases (exercise!).</p>
+<p>In general, the second functor law says that composing two functions
+first and then applying <code>fmap</code> must yield the same result as
+performing <code>fmap</code> on those functions and then composing the
+resulting function. In other words, the order of applying
+<code>fmap</code> and composing doesn’t matter. (These two operations
+are said to <em>commute</em>.)</p>
 <!-- I don't know if this is helpful or more confusing:
 You might remember the distributivity law from algebra:
 ```
@@ -2421,46 +3370,71 @@ which we perform `fmap` and `.` and expect to get the same result. Purity is
 necessary for having a concept like functor laws in a programming language so
 this topic connects to equational reasoning discussed in chapter 10.
 -->
-<p>There are also higher-order functions that fail to satisfy functor laws. Consider the function <code>badMap</code>:</p>
-<div class="sourceCode" id="cb191"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb191-1"><a href="#cb191-1" aria-hidden="true" tabindex="-1"></a><span class="ot">badMap ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [b]</span>
+<p>There are also higher-order functions that fail to satisfy functor
+laws. Consider the function <code>badMap</code>:</p>
+<div class="sourceCode" id="cb191"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb191-1"><a href="#cb191-1" aria-hidden="true" tabindex="-1"></a><span class="ot">badMap ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [b]</span>
 <span id="cb191-2"><a href="#cb191-2" aria-hidden="true" tabindex="-1"></a>badMap f [] <span class="ot">=</span> []</span>
 <span id="cb191-3"><a href="#cb191-3" aria-hidden="true" tabindex="-1"></a>badMap f (x<span class="op">:</span>y<span class="op">:</span>xs) <span class="ot">=</span> f x <span class="op">:</span> badMap f xs</span>
 <span id="cb191-4"><a href="#cb191-4" aria-hidden="true" tabindex="-1"></a>badMap f (x<span class="op">:</span>xs) <span class="ot">=</span> f x <span class="op">:</span> badMap f xs</span></code></pre></div>
 <p>This function violates the first functor law. For instance:</p>
-<div class="sourceCode" id="cb192"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb192-1"><a href="#cb192-1" aria-hidden="true" tabindex="-1"></a>badMap <span class="fu">id</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] <span class="op">==&gt;</span> badMap <span class="fu">id</span> (<span class="dv">1</span><span class="op">:</span><span class="dv">2</span><span class="op">:</span>[<span class="dv">3</span>])</span>
+<div class="sourceCode" id="cb192"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb192-1"><a href="#cb192-1" aria-hidden="true" tabindex="-1"></a>badMap <span class="fu">id</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] <span class="op">==&gt;</span> badMap <span class="fu">id</span> (<span class="dv">1</span><span class="op">:</span><span class="dv">2</span><span class="op">:</span>[<span class="dv">3</span>])</span>
 <span id="cb192-2"><a href="#cb192-2" aria-hidden="true" tabindex="-1"></a>                  <span class="op">==&gt;</span> <span class="fu">id</span> <span class="dv">1</span> <span class="op">:</span> badMap <span class="fu">id</span> [<span class="dv">3</span>]</span>
 <span id="cb192-3"><a href="#cb192-3" aria-hidden="true" tabindex="-1"></a>                  <span class="op">==&gt;</span> <span class="dv">1</span> <span class="op">:</span> badMap <span class="fu">id</span> [<span class="dv">3</span>]</span>
 <span id="cb192-4"><a href="#cb192-4" aria-hidden="true" tabindex="-1"></a>                  <span class="op">==&gt;</span> <span class="dv">1</span> <span class="op">:</span> badMap <span class="fu">id</span> (<span class="dv">3</span><span class="op">:</span>[])</span>
 <span id="cb192-5"><a href="#cb192-5" aria-hidden="true" tabindex="-1"></a>                  <span class="op">==&gt;</span> <span class="dv">1</span> <span class="op">:</span> <span class="fu">id</span> <span class="dv">3</span> <span class="op">:</span> badMap <span class="fu">id</span> []</span>
 <span id="cb192-6"><a href="#cb192-6" aria-hidden="true" tabindex="-1"></a>                  <span class="op">==&gt;</span> <span class="dv">1</span> <span class="op">:</span> <span class="dv">3</span> <span class="op">:</span> []</span>
 <span id="cb192-7"><a href="#cb192-7" aria-hidden="true" tabindex="-1"></a>                  <span class="op">==&gt;</span> [<span class="dv">1</span>,<span class="dv">3</span>]</span></code></pre></div>
-<p>Applying <code>badMap id</code> to the list <code>[1,2,3]</code> changed the list as the element <code>2</code> was dropped.</p>
-<p>As mentioned, Haskell compilers can’t detect if a functor obeys its laws or not. A Haskell compiler would happily accept an instance <code>Functor []</code> that used <code>badMap</code> instead of <code>map</code> as the <code>fmap</code> implementation. This is a limitation of the type system of Haskell. There are technologies such as LiquidHaskell or dependently typed languages such as Agda, Idris, Coq, or Lean that could actually enforce functor laws so that unlawful functor instances wouldn’t compile. These technologies are outside the scope of this course, however.</p>
-</section>
-<section id="sidenote-kinds" data-number="4.4">
-<h2 data-number="12.4"><span class="header-section-number">12.4</span> Sidenote: Kinds</h2>
-<p>Remember that <code>Functor</code> was a class for type constructors. If we try to define an instance of <code>Functor</code> for a type, we get an error:</p>
-<div class="sourceCode" id="cb193"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb193-1"><a href="#cb193-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">instance</span> <span class="dt">Functor</span> <span class="dt">Int</span> <span class="kw">where</span></span>
+<p>Applying <code>badMap id</code> to the list <code>[1,2,3]</code>
+changed the list as the element <code>2</code> was dropped.</p>
+<p>As mentioned, Haskell compilers can’t detect if a functor obeys its
+laws or not. A Haskell compiler would happily accept an instance
+<code>Functor []</code> that used <code>badMap</code> instead of
+<code>map</code> as the <code>fmap</code> implementation. This is a
+limitation of the type system of Haskell. There are technologies such as
+LiquidHaskell or dependently typed languages such as Agda, Idris, Coq,
+or Lean that could actually enforce functor laws so that unlawful
+functor instances wouldn’t compile. These technologies are outside the
+scope of this course, however.</p>
+<h2 data-number="12.4" id="sidenote-kinds"><span
+class="header-section-number">12.4</span> Sidenote: Kinds</h2>
+<p>Remember that <code>Functor</code> was a class for type constructors.
+If we try to define an instance of <code>Functor</code> for a type, we
+get an error:</p>
+<div class="sourceCode" id="cb193"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb193-1"><a href="#cb193-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">instance</span> <span class="dt">Functor</span> <span class="dt">Int</span> <span class="kw">where</span></span>
 <span id="cb193-2"><a href="#cb193-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb193-3"><a href="#cb193-3" aria-hidden="true" tabindex="-1"></a><span class="op">&lt;</span>interactive<span class="op">&gt;:</span><span class="dv">1</span><span class="op">:</span><span class="dv">18</span><span class="op">:</span> <span class="fu">error</span><span class="op">:</span></span>
 <span id="cb193-4"><a href="#cb193-4" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">Expected</span> kind ‘<span class="op">*</span> <span class="ot">-&gt;</span> <span class="op">*</span>’, but ‘<span class="dt">Int</span>’ has kind ‘<span class="op">*</span>’</span>
 <span id="cb193-5"><a href="#cb193-5" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">In</span> the first argument <span class="kw">of</span> ‘<span class="dt">Functor</span>’, namely ‘<span class="dt">Int</span>’</span>
 <span id="cb193-6"><a href="#cb193-6" aria-hidden="true" tabindex="-1"></a>      <span class="dt">In</span> the <span class="kw">instance</span> declaration for ‘<span class="dt">Functor</span> <span class="dt">Int</span>’</span></code></pre></div>
-<p>The error message talks about <em>kinds</em>. Kinds are <em>types of types</em>. A type like <code>Int</code>, <code>Bool</code> or <code>Maybe Int</code> that can contain values has kind <code>*</code>. A type constructor has a kind that looks like a function, for example, <code>Maybe</code> has kind <code>* -&gt; *</code>. This means that the <code>Maybe</code> type constructor must be applied to a type of kind <code>*</code> to get a type of kind <code>*</code>.</p>
+<p>The error message talks about <em>kinds</em>. Kinds are <em>types of
+types</em>. A type like <code>Int</code>, <code>Bool</code> or
+<code>Maybe Int</code> that can contain values has kind <code>*</code>.
+A type constructor has a kind that looks like a function, for example,
+<code>Maybe</code> has kind <code>* -&gt; *</code>. This means that the
+<code>Maybe</code> type constructor must be applied to a type of kind
+<code>*</code> to get a type of kind <code>*</code>.</p>
 <p>We can ask GHCi for the kinds of types:</p>
-<div class="sourceCode" id="cb194"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb194-1"><a href="#cb194-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>kind <span class="dt">Int</span></span>
+<div class="sourceCode" id="cb194"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb194-1"><a href="#cb194-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>kind <span class="dt">Int</span></span>
 <span id="cb194-2"><a href="#cb194-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Int</span><span class="ot"> ::</span> <span class="op">*</span></span>
 <span id="cb194-3"><a href="#cb194-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>kind <span class="dt">Maybe</span></span>
 <span id="cb194-4"><a href="#cb194-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Maybe</span><span class="ot"> ::</span> <span class="op">*</span> <span class="ot">-&gt;</span> <span class="op">*</span></span>
 <span id="cb194-5"><a href="#cb194-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>kind <span class="dt">Maybe</span> <span class="dt">Int</span></span>
 <span id="cb194-6"><a href="#cb194-6" aria-hidden="true" tabindex="-1"></a><span class="dt">Maybe</span> <span class="dt">Int</span><span class="ot"> ::</span> <span class="op">*</span></span></code></pre></div>
-<p>If we ask GHCi for info about the <code>Functor</code> class, it tells us that instances of <code>Functor</code> must have kind <code>* -&gt; *</code>:</p>
-<div class="sourceCode" id="cb195"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb195-1"><a href="#cb195-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>info <span class="dt">Functor</span></span>
+<p>If we ask GHCi for info about the <code>Functor</code> class, it
+tells us that instances of <code>Functor</code> must have kind
+<code>* -&gt; *</code>:</p>
+<div class="sourceCode" id="cb195"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb195-1"><a href="#cb195-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>info <span class="dt">Functor</span></span>
 <span id="cb195-2"><a href="#cb195-2" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Functor</span> (<span class="ot">f ::</span> <span class="op">*</span> <span class="ot">-&gt;</span> <span class="op">*</span>) <span class="kw">where</span></span>
 <span id="cb195-3"><a href="#cb195-3" aria-hidden="true" tabindex="-1"></a><span class="ot">  fmap ::</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> f a <span class="ot">-&gt;</span> f b</span>
 <span id="cb195-4"><a href="#cb195-4" aria-hidden="true" tabindex="-1"></a><span class="op">...</span></span></code></pre></div>
 <p>Here are some examples of even more complex kinds.</p>
-<div class="sourceCode" id="cb196"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb196-1"><a href="#cb196-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- multiple type parameters</span></span>
+<div class="sourceCode" id="cb196"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb196-1"><a href="#cb196-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- multiple type parameters</span></span>
 <span id="cb196-2"><a href="#cb196-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>kind <span class="dt">Either</span></span>
 <span id="cb196-3"><a href="#cb196-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Either</span><span class="ot"> ::</span> <span class="op">*</span> <span class="ot">-&gt;</span> <span class="op">*</span> <span class="ot">-&gt;</span> <span class="op">*</span></span>
 <span id="cb196-4"><a href="#cb196-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">data</span> <span class="dt">Either3</span> a b c <span class="ot">=</span> <span class="dt">Left</span> a <span class="op">|</span> <span class="dt">Middle</span> b <span class="op">|</span> <span class="dt">Right</span> c</span>
@@ -2470,22 +3444,39 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb196-8"><a href="#cb196-8" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">data</span> <span class="dt">IntInside</span> f <span class="ot">=</span> <span class="dt">IntInside</span> (f <span class="dt">Int</span>)</span>
 <span id="cb196-9"><a href="#cb196-9" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>kind <span class="dt">IntInside</span></span>
 <span id="cb196-10"><a href="#cb196-10" aria-hidden="true" tabindex="-1"></a><span class="dt">IntInside</span><span class="ot"> ::</span> (<span class="op">*</span> <span class="ot">-&gt;</span> <span class="op">*</span>) <span class="ot">-&gt;</span> <span class="op">*</span></span></code></pre></div>
-<p>You won’t bump into kinds that much in Haskell programming, but sometimes you’ll see error messages that talk about kinds, so it’s good to know what they are.</p>
-</section>
-<section id="foldable-again" data-number="4.5">
-<h2 data-number="12.5"><span class="header-section-number">12.5</span> <code>Foldable</code>, Again</h2>
-<p>We briefly covered the class <code>Foldable</code>, which occurs in many type signatures of basic functions, in part 1. For example:</p>
-<div class="sourceCode" id="cb197"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb197-1"><a href="#cb197-1" aria-hidden="true" tabindex="-1"></a><span class="fu">length</span><span class="ot"> ::</span> <span class="dt">Foldable</span> t <span class="ot">=&gt;</span> t a <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<p>You won’t bump into kinds that much in Haskell programming, but
+sometimes you’ll see error messages that talk about kinds, so it’s good
+to know what they are.</p>
+<h2 data-number="12.5" id="foldable-again"><span
+class="header-section-number">12.5</span> <code>Foldable</code>,
+Again</h2>
+<p>We briefly covered the class <code>Foldable</code>, which occurs in
+many type signatures of basic functions, in part 1. For example:</p>
+<div class="sourceCode" id="cb197"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb197-1"><a href="#cb197-1" aria-hidden="true" tabindex="-1"></a><span class="fu">length</span><span class="ot"> ::</span> <span class="dt">Foldable</span> t <span class="ot">=&gt;</span> t a <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb197-2"><a href="#cb197-2" aria-hidden="true" tabindex="-1"></a><span class="fu">sum</span><span class="ot"> ::</span> (<span class="dt">Foldable</span> t, <span class="dt">Num</span> a) <span class="ot">=&gt;</span> t a <span class="ot">-&gt;</span> a</span>
 <span id="cb197-3"><a href="#cb197-3" aria-hidden="true" tabindex="-1"></a><span class="fu">minimum</span><span class="ot"> ::</span> (<span class="dt">Foldable</span> t, <span class="dt">Ord</span> a) <span class="ot">=&gt;</span> t a <span class="ot">-&gt;</span> a</span>
 <span id="cb197-4"><a href="#cb197-4" aria-hidden="true" tabindex="-1"></a><span class="fu">foldMap</span><span class="ot"> ::</span> (<span class="dt">Foldable</span> t, <span class="dt">Monoid</span> m) <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> m) <span class="ot">-&gt;</span> t a <span class="ot">-&gt;</span> m</span></code></pre></div>
-<p>From these type signatures we can see that <code>Foldable</code>, just like <code>Functor</code>, is a class for type constructors (things of kind <code>* -&gt; *</code>). The essence of <code>Foldable</code> is to be a class for <em>things you can fold over</em>. The class definition could be as simple as</p>
-<div class="sourceCode" id="cb198"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb198-1"><a href="#cb198-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Foldable</span> (<span class="ot">t ::</span> <span class="op">*-&gt;*</span>) <span class="kw">where</span></span>
+<p>From these type signatures we can see that <code>Foldable</code>,
+just like <code>Functor</code>, is a class for type constructors (things
+of kind <code>* -&gt; *</code>). The essence of <code>Foldable</code> is
+to be a class for <em>things you can fold over</em>. The class
+definition could be as simple as</p>
+<div class="sourceCode" id="cb198"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb198-1"><a href="#cb198-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Foldable</span> (<span class="ot">t ::</span> <span class="op">*-&gt;*</span>) <span class="kw">where</span></span>
 <span id="cb198-2"><a href="#cb198-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  foldr ::</span> (a <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> t a <span class="ot">-&gt;</span> b</span></code></pre></div>
-<p>However, for performance reasons, the class contains many methods (you can see them yourself by checking <code>:info Foldable</code> in GHCi!), but when we’re defining an instance for <code>Foldable</code> it’s enought to define just <code>foldr</code>.</p>
-<p>Another way of thinking of the <code>Foldable</code> class is processing elements <em>left-to-right</em>, in other words, if <code>Functor</code> was the class for containers, <code>Foldable</code> is the class for <em>ordered containers</em>.</p>
-<p>As an example, let’s implement <code>Functor</code> and <code>Foldable</code> for our own pair type.</p>
-<div class="sourceCode" id="cb199"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb199-1"><a href="#cb199-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Pair</span> a <span class="ot">=</span> <span class="dt">Pair</span> a a</span>
+<p>However, for performance reasons, the class contains many methods
+(you can see them yourself by checking <code>:info Foldable</code> in
+GHCi!), but when we’re defining an instance for <code>Foldable</code>
+it’s enought to define just <code>foldr</code>.</p>
+<p>Another way of thinking of the <code>Foldable</code> class is
+processing elements <em>left-to-right</em>, in other words, if
+<code>Functor</code> was the class for containers, <code>Foldable</code>
+is the class for <em>ordered containers</em>.</p>
+<p>As an example, let’s implement <code>Functor</code> and
+<code>Foldable</code> for our own pair type.</p>
+<div class="sourceCode" id="cb199"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb199-1"><a href="#cb199-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Pair</span> a <span class="ot">=</span> <span class="dt">Pair</span> a a</span>
 <span id="cb199-2"><a href="#cb199-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">deriving</span> <span class="dt">Show</span></span>
 <span id="cb199-3"><a href="#cb199-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb199-4"><a href="#cb199-4" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Functor</span> <span class="dt">Pair</span> <span class="kw">where</span></span>
@@ -2499,8 +3490,10 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb199-12"><a href="#cb199-12" aria-hidden="true" tabindex="-1"></a><span class="co">-- an example function that uses both instances</span></span>
 <span id="cb199-13"><a href="#cb199-13" aria-hidden="true" tabindex="-1"></a><span class="ot">doubleAndCount ::</span> (<span class="dt">Functor</span> f, <span class="dt">Foldable</span> f) <span class="ot">=&gt;</span> f <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb199-14"><a href="#cb199-14" aria-hidden="true" tabindex="-1"></a>doubleAndCount <span class="ot">=</span> <span class="fu">sum</span> <span class="op">.</span> <span class="fu">fmap</span> (<span class="op">*</span><span class="dv">2</span>)</span></code></pre></div>
-<p>Now, we can use <code>Pair</code> almost wherever we can use a list:</p>
-<div class="sourceCode" id="cb200"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb200-1"><a href="#cb200-1" aria-hidden="true" tabindex="-1"></a><span class="fu">fmap</span> (<span class="op">+</span><span class="dv">1</span>) (<span class="dt">Pair</span> <span class="dv">3</span> <span class="dv">6</span>)   <span class="op">==&gt;</span> <span class="dt">Pair</span> <span class="dv">4</span> <span class="dv">7</span></span>
+<p>Now, we can use <code>Pair</code> almost wherever we can use a
+list:</p>
+<div class="sourceCode" id="cb200"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb200-1"><a href="#cb200-1" aria-hidden="true" tabindex="-1"></a><span class="fu">fmap</span> (<span class="op">+</span><span class="dv">1</span>) (<span class="dt">Pair</span> <span class="dv">3</span> <span class="dv">6</span>)   <span class="op">==&gt;</span> <span class="dt">Pair</span> <span class="dv">4</span> <span class="dv">7</span></span>
 <span id="cb200-2"><a href="#cb200-2" aria-hidden="true" tabindex="-1"></a><span class="fu">fmap</span> (<span class="op">+</span><span class="dv">1</span>) [<span class="dv">3</span>,<span class="dv">6</span>]        <span class="op">==&gt;</span> [<span class="dv">4</span>,<span class="dv">7</span>]</span>
 <span id="cb200-3"><a href="#cb200-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb200-4"><a href="#cb200-4" aria-hidden="true" tabindex="-1"></a><span class="fu">foldr</span> (<span class="op">*</span>) <span class="dv">1</span> (<span class="dt">Pair</span> <span class="dv">3</span> <span class="dv">6</span>) <span class="op">==&gt;</span> <span class="dv">18</span></span>
@@ -2514,15 +3507,21 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb200-12"><a href="#cb200-12" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb200-13"><a href="#cb200-13" aria-hidden="true" tabindex="-1"></a>doubleAndCount (<span class="dt">Pair</span> <span class="dv">3</span> <span class="dv">6</span>)  <span class="op">==&gt;</span> <span class="dv">18</span></span>
 <span id="cb200-14"><a href="#cb200-14" aria-hidden="true" tabindex="-1"></a>doubleAndCount [<span class="dv">3</span>,<span class="dv">6</span>]       <span class="op">==&gt;</span> <span class="dv">18</span></span></code></pre></div>
-<p>Other types we’ve met that are <code>Foldable</code> include <code>Data.Map</code> and <code>Data.Array</code>.</p>
-</section>
-<section id="recap" data-number="4.6">
-<h2 data-number="12.6"><span class="header-section-number">12.6</span> Recap</h2>
-<p>So, to summarize, a functor is a type constructor <code>f</code> and the corresponding <code>Functor f</code> instance such that <code>fmap</code> satisfies the two functor laws. These laws assert that <code>fmap</code> must preserve the identity function and distribute over function composition. More informally, <code>fmap</code> lifts a function <code>g :: a -&gt; b</code> operating on values to one operating on containers: <code>fmap g :: f a -&gt; f b</code>. Basically all well-behaving data structures in Haskell are functors.</p>
+<p>Other types we’ve met that are <code>Foldable</code> include
+<code>Data.Map</code> and <code>Data.Array</code>.</p>
+<h2 data-number="12.6" id="recap"><span
+class="header-section-number">12.6</span> Recap</h2>
+<p>So, to summarize, a functor is a type constructor <code>f</code> and
+the corresponding <code>Functor f</code> instance such that
+<code>fmap</code> satisfies the two functor laws. These laws assert that
+<code>fmap</code> must preserve the identity function and distribute
+over function composition. More informally, <code>fmap</code> lifts a
+function <code>g :: a -&gt; b</code> operating on values to one
+operating on containers: <code>fmap g :: f a -&gt; f b</code>. Basically
+all well-behaving data structures in Haskell are functors.</p>
 <!-- ## TODO: something more? move some stuff from lecture 13 here -->
-</section>
-<section id="quiz-3" data-number="4.7">
-<h2 data-number="12.7"><span class="header-section-number">12.7</span> Quiz</h2>
+<h2 data-number="12.7" id="quiz-3"><span
+class="header-section-number">12.7</span> Quiz</h2>
 <p>What’s the type of <code>fmap</code>?</p>
 <ol class="quiz">
 <li>
@@ -2538,8 +3537,10 @@ this topic connects to equational reasoning discussed in chapter 10.
 <code>Functor f =&gt; (a -&gt; b) -&gt; f a -&gt; f b</code>
 </li>
 </ol>
-<p>Which code snippet completes the next <code>Functor</code> instance?</p>
-<div class="sourceCode" id="cb201"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb201-1"><a href="#cb201-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Container</span> x <span class="ot">=</span> <span class="dt">Things</span> x [x]</span>
+<p>Which code snippet completes the next <code>Functor</code>
+instance?</p>
+<div class="sourceCode" id="cb201"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb201-1"><a href="#cb201-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Container</span> x <span class="ot">=</span> <span class="dt">Things</span> x [x]</span>
 <span id="cb201-2"><a href="#cb201-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb201-3"><a href="#cb201-3" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Functor</span> <span class="dt">Container</span> <span class="kw">where</span></span>
 <span id="cb201-4"><a href="#cb201-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">????</span></span></code></pre></div>
@@ -2570,7 +3571,8 @@ this topic connects to equational reasoning discussed in chapter 10.
 </li>
 </ol>
 <p>What’s the kind of <code>Foo</code>?</p>
-<div class="sourceCode" id="cb202"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb202-1"><a href="#cb202-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Foo</span> x <span class="ot">=</span> <span class="dt">FooConst</span></span></code></pre></div>
+<div class="sourceCode" id="cb202"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb202-1"><a href="#cb202-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Foo</span> x <span class="ot">=</span> <span class="dt">FooConst</span></span></code></pre></div>
 <ol class="quiz">
 <li>
 <code>*</code>
@@ -2583,7 +3585,8 @@ this topic connects to equational reasoning discussed in chapter 10.
 </li>
 </ol>
 <p>What’s the kind of <code>Bar</code>?</p>
-<div class="sourceCode" id="cb203"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb203-1"><a href="#cb203-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Bar</span> <span class="ot">=</span> <span class="dt">Baz</span> <span class="op">|</span> <span class="dt">Qux</span> <span class="dt">Int</span></span></code></pre></div>
+<div class="sourceCode" id="cb203"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb203-1"><a href="#cb203-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Bar</span> <span class="ot">=</span> <span class="dt">Baz</span> <span class="op">|</span> <span class="dt">Qux</span> <span class="dt">Int</span></span></code></pre></div>
 <ol class="quiz">
 <li class="correct">
 <code>*</code>
@@ -2610,8 +3613,10 @@ this topic connects to equational reasoning discussed in chapter 10.
 <code>Just 1</code>
 </li>
 </ol>
-<p>Which code snippet completes the next <code>Foldable</code> instance?</p>
-<div class="sourceCode" id="cb204"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb204-1"><a href="#cb204-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Container</span> x <span class="ot">=</span> <span class="dt">Things</span> x [x]</span>
+<p>Which code snippet completes the next <code>Foldable</code>
+instance?</p>
+<div class="sourceCode" id="cb204"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb204-1"><a href="#cb204-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Container</span> x <span class="ot">=</span> <span class="dt">Things</span> x [x]</span>
 <span id="cb204-2"><a href="#cb204-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb204-3"><a href="#cb204-3" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Foldable</span> <span class="dt">Container</span> <span class="kw">where</span></span>
 <span id="cb204-4"><a href="#cb204-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">????</span></span></code></pre></div>
@@ -2629,27 +3634,49 @@ this topic connects to equational reasoning discussed in chapter 10.
 <code>foldr f z (Things x ys) = foldr f z (x:ys)</code>
 </li>
 </ol>
-</section>
-<section id="exercises-3" data-number="4.8">
-<h2 data-number="12.8"><span class="header-section-number">12.8</span> Exercises</h2>
+<h2 data-number="12.8" id="exercises-3"><span
+class="header-section-number">12.8</span> Exercises</h2>
 <ul>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set12.hs">Set12</a></li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set12.hs">Set12</a></li>
 </ul>
-</section>
-</section>
-<section id="lecture-13-a-monoid-in-the-category-of-problems" data-number="5">
-<h1 data-number="13"><span class="header-section-number">13</span> Lecture 13: A Monoid in the Category of Problems</h1>
+<h1 data-number="13"
+id="lecture-13-a-monoid-in-the-category-of-problems"><span
+class="header-section-number">13</span> Lecture 13: A Monoid in the
+Category of Problems</h1>
 <ul>
 <li>Monads</li>
 </ul>
-<p>In this lecture we’ll build up to the concept of a <em>monad</em> using a number of examples. By now you should be familiar with all the Haskell features needed for understanding monads.</p>
-<p>Monads are a famously hard topic in programming, which is partly due to weird terminology, partly due to bad tutorials, and partly due to trying to understand monads too early when learning Haskell. Monads are introduced this late in the course in an attempt to make understanding them easier.</p>
-<p>If you find this lecture hard, don’t despair, many others have found the topic hard as well. There are many many productive Haskell programmers who have managed to understand monads, so the task is not hopeless.</p>
-<p>One final word of caution: monads, like functors, are a concept originally from a branch of mathematics called Category Theory. However, and I can’t stress this enough, <em>you do not need to know anything or even care about category theory to understand monads in Haskell programming.</em> Just like one can work with object-oriented programming or functional programming without knowing the theory of objects or functions, one can work with monads without understanding the math associated with them. Category theory can be a rewarding topic for a functional programmer, but it’s not a mandatory one.</p>
-<section id="example-1-maybes" data-number="5.1">
-<h2 data-number="13.1"><span class="header-section-number">13.1</span> Example 1: Maybes</h2>
-<p>When working with many <code>Maybe</code> values, the code tends to become a bit messy. Let’s look at some examples. First, we combine some functions returning <code>Maybe String</code>. Note the nested <code>case</code> we need in <code>stealSecret</code>: It’s not fun to write.</p>
-<div class="sourceCode" id="cb205"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb205-1"><a href="#cb205-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Try to login with a password.</span></span>
+<p>In this lecture we’ll build up to the concept of a <em>monad</em>
+using a number of examples. By now you should be familiar with all the
+Haskell features needed for understanding monads.</p>
+<p>Monads are a famously hard topic in programming, which is partly due
+to weird terminology, partly due to bad tutorials, and partly due to
+trying to understand monads too early when learning Haskell. Monads are
+introduced this late in the course in an attempt to make understanding
+them easier.</p>
+<p>If you find this lecture hard, don’t despair, many others have found
+the topic hard as well. There are many many productive Haskell
+programmers who have managed to understand monads, so the task is not
+hopeless.</p>
+<p>One final word of caution: monads, like functors, are a concept
+originally from a branch of mathematics called Category Theory. However,
+and I can’t stress this enough, <em>you do not need to know anything or
+even care about category theory to understand monads in Haskell
+programming.</em> Just like one can work with object-oriented
+programming or functional programming without knowing the theory of
+objects or functions, one can work with monads without understanding the
+math associated with them. Category theory can be a rewarding topic for
+a functional programmer, but it’s not a mandatory one.</p>
+<h2 data-number="13.1" id="example-1-maybes"><span
+class="header-section-number">13.1</span> Example 1: Maybes</h2>
+<p>When working with many <code>Maybe</code> values, the code tends to
+become a bit messy. Let’s look at some examples. First, we combine some
+functions returning <code>Maybe String</code>. Note the nested
+<code>case</code> we need in <code>stealSecret</code>: It’s not fun to
+write.</p>
+<div class="sourceCode" id="cb205"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb205-1"><a href="#cb205-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Try to login with a password.</span></span>
 <span id="cb205-2"><a href="#cb205-2" aria-hidden="true" tabindex="-1"></a><span class="co">-- `Just username` on success, `Nothing` otherwise.</span></span>
 <span id="cb205-3"><a href="#cb205-3" aria-hidden="true" tabindex="-1"></a><span class="ot">login ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">String</span></span>
 <span id="cb205-4"><a href="#cb205-4" aria-hidden="true" tabindex="-1"></a>login <span class="st">&quot;f4bulous!&quot;</span> <span class="ot">=</span> <span class="dt">Just</span> <span class="st">&quot;unicorn73&quot;</span></span>
@@ -2670,13 +3697,18 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb205-19"><a href="#cb205-19" aria-hidden="true" tabindex="-1"></a>    <span class="dt">Just</span> user <span class="ot">-&gt;</span> <span class="kw">case</span> secret user <span class="kw">of</span></span>
 <span id="cb205-20"><a href="#cb205-20" aria-hidden="true" tabindex="-1"></a>                   <span class="dt">Nothing</span> <span class="ot">-&gt;</span> <span class="dt">Nothing</span></span>
 <span id="cb205-21"><a href="#cb205-21" aria-hidden="true" tabindex="-1"></a>                   <span class="dt">Just</span> s <span class="ot">-&gt;</span> <span class="dt">Just</span> (<span class="st">&quot;Stole secret: &quot;</span><span class="op">++</span>s)</span></code></pre></div>
-<div class="sourceCode" id="cb206"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb206-1"><a href="#cb206-1" aria-hidden="true" tabindex="-1"></a>stealSecret <span class="st">&quot;swordfish&quot;</span>  <span class="op">==&gt;</span>  <span class="dt">Just</span> <span class="st">&quot;Stole secret: I like roses&quot;</span></span>
+<div class="sourceCode" id="cb206"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb206-1"><a href="#cb206-1" aria-hidden="true" tabindex="-1"></a>stealSecret <span class="st">&quot;swordfish&quot;</span>  <span class="op">==&gt;</span>  <span class="dt">Just</span> <span class="st">&quot;Stole secret: I like roses&quot;</span></span>
 <span id="cb206-2"><a href="#cb206-2" aria-hidden="true" tabindex="-1"></a>stealSecret <span class="st">&quot;f4bulous!&quot;</span>  <span class="op">==&gt;</span>  <span class="dt">Nothing</span></span>
 <span id="cb206-3"><a href="#cb206-3" aria-hidden="true" tabindex="-1"></a>stealSecret <span class="st">&quot;wrong_password&quot;</span>  <span class="op">==&gt;</span>  <span class="dt">Nothing</span></span></code></pre></div>
-<p>Next up, we modify a list of pairs. We use the <code>Maybe</code>-returning function <code>lookup</code> from the Prelude. Here we have an if inside a case instead of a nested case.</p>
-<div class="sourceCode" id="cb207"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb207-1"><a href="#cb207-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Get the value corresponding to a key from a key-value list.</span></span>
+<p>Next up, we modify a list of pairs. We use the
+<code>Maybe</code>-returning function <code>lookup</code> from the
+Prelude. Here we have an if inside a case instead of a nested case.</p>
+<div class="sourceCode" id="cb207"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb207-1"><a href="#cb207-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Get the value corresponding to a key from a key-value list.</span></span>
 <span id="cb207-2"><a href="#cb207-2" aria-hidden="true" tabindex="-1"></a><span class="fu">lookup</span><span class="ot"> ::</span> (<span class="dt">Eq</span> a) <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> [(a, b)] <span class="ot">-&gt;</span> <span class="dt">Maybe</span> b</span></code></pre></div>
-<div class="sourceCode" id="cb208"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb208-1"><a href="#cb208-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Set the value of key to val in the given key-value list,</span></span>
+<div class="sourceCode" id="cb208"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb208-1"><a href="#cb208-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Set the value of key to val in the given key-value list,</span></span>
 <span id="cb208-2"><a href="#cb208-2" aria-hidden="true" tabindex="-1"></a><span class="co">-- but only if val is larger than the current value!</span></span>
 <span id="cb208-3"><a href="#cb208-3" aria-hidden="true" tabindex="-1"></a><span class="ot">increase ::</span> <span class="dt">Eq</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> [(a,<span class="dt">Int</span>)] <span class="ot">-&gt;</span> <span class="dt">Maybe</span> [(a,<span class="dt">Int</span>)]</span>
 <span id="cb208-4"><a href="#cb208-4" aria-hidden="true" tabindex="-1"></a>increase key val assocs <span class="ot">=</span></span>
@@ -2685,20 +3717,30 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb208-7"><a href="#cb208-7" aria-hidden="true" tabindex="-1"></a>     <span class="dt">Just</span> x <span class="ot">-&gt;</span> <span class="kw">if</span> (val <span class="op">&lt;</span> x)</span>
 <span id="cb208-8"><a href="#cb208-8" aria-hidden="true" tabindex="-1"></a>                <span class="kw">then</span> <span class="dt">Nothing</span></span>
 <span id="cb208-9"><a href="#cb208-9" aria-hidden="true" tabindex="-1"></a>                <span class="kw">else</span> <span class="dt">Just</span> ((key,val) <span class="op">:</span> delete (key,x) assocs)</span></code></pre></div>
-<p>This type of code is pretty common, and usually repeats the same pattern: if any intermediate result is <code>Nothing</code>, the whole result is <code>Nothing</code>. Let’s try to make writing code like this easier by defining a <em>chaining operator</em> <code>?&gt;</code>. The chaining operator takes a result and the next step of computation, and only runs the next step if the result was a <code>Just</code> value.</p>
-<div class="sourceCode" id="cb209"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb209-1"><a href="#cb209-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(?&gt;) ::</span> <span class="dt">Maybe</span> a <span class="ot">-&gt;</span> (a <span class="ot">-&gt;</span> <span class="dt">Maybe</span> b) <span class="ot">-&gt;</span> <span class="dt">Maybe</span> b</span>
+<p>This type of code is pretty common, and usually repeats the same
+pattern: if any intermediate result is <code>Nothing</code>, the whole
+result is <code>Nothing</code>. Let’s try to make writing code like this
+easier by defining a <em>chaining operator</em> <code>?&gt;</code>. The
+chaining operator takes a result and the next step of computation, and
+only runs the next step if the result was a <code>Just</code> value.</p>
+<div class="sourceCode" id="cb209"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb209-1"><a href="#cb209-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(?&gt;) ::</span> <span class="dt">Maybe</span> a <span class="ot">-&gt;</span> (a <span class="ot">-&gt;</span> <span class="dt">Maybe</span> b) <span class="ot">-&gt;</span> <span class="dt">Maybe</span> b</span>
 <span id="cb209-2"><a href="#cb209-2" aria-hidden="true" tabindex="-1"></a><span class="co">-- if we failed, don&#39;t even bother running the next step:</span></span>
 <span id="cb209-3"><a href="#cb209-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Nothing</span> <span class="op">?&gt;</span> _ <span class="ot">=</span> <span class="dt">Nothing</span></span>
 <span id="cb209-4"><a href="#cb209-4" aria-hidden="true" tabindex="-1"></a><span class="co">-- otherwise run the next step:</span></span>
 <span id="cb209-5"><a href="#cb209-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Just</span> x  <span class="op">?&gt;</span> f <span class="ot">=</span> f x</span></code></pre></div>
-<p>The chaining operator streamlines our examples nicely. Note how we can define simple helper functions that take care of one step of the computation instead of writing one big expression.</p>
-<div class="sourceCode" id="cb210"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb210-1"><a href="#cb210-1" aria-hidden="true" tabindex="-1"></a><span class="ot">stealSecret ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">String</span></span>
+<p>The chaining operator streamlines our examples nicely. Note how we
+can define simple helper functions that take care of one step of the
+computation instead of writing one big expression.</p>
+<div class="sourceCode" id="cb210"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb210-1"><a href="#cb210-1" aria-hidden="true" tabindex="-1"></a><span class="ot">stealSecret ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">String</span></span>
 <span id="cb210-2"><a href="#cb210-2" aria-hidden="true" tabindex="-1"></a>stealSecret password <span class="ot">=</span></span>
 <span id="cb210-3"><a href="#cb210-3" aria-hidden="true" tabindex="-1"></a>    login password <span class="op">?&gt;</span></span>
 <span id="cb210-4"><a href="#cb210-4" aria-hidden="true" tabindex="-1"></a>    secret <span class="op">?&gt;</span></span>
 <span id="cb210-5"><a href="#cb210-5" aria-hidden="true" tabindex="-1"></a>    decorate</span>
 <span id="cb210-6"><a href="#cb210-6" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> decorate s <span class="ot">=</span> <span class="dt">Just</span> (<span class="st">&quot;Stole secret: &quot;</span><span class="op">++</span>s)</span></code></pre></div>
-<div class="sourceCode" id="cb211"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb211-1"><a href="#cb211-1" aria-hidden="true" tabindex="-1"></a><span class="ot">increase ::</span> <span class="dt">Eq</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> [(a,<span class="dt">Int</span>)] <span class="ot">-&gt;</span> <span class="dt">Maybe</span> [(a,<span class="dt">Int</span>)]</span>
+<div class="sourceCode" id="cb211"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb211-1"><a href="#cb211-1" aria-hidden="true" tabindex="-1"></a><span class="ot">increase ::</span> <span class="dt">Eq</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> [(a,<span class="dt">Int</span>)] <span class="ot">-&gt;</span> <span class="dt">Maybe</span> [(a,<span class="dt">Int</span>)]</span>
 <span id="cb211-2"><a href="#cb211-2" aria-hidden="true" tabindex="-1"></a>increase key val assocs <span class="ot">=</span></span>
 <span id="cb211-3"><a href="#cb211-3" aria-hidden="true" tabindex="-1"></a>    <span class="fu">lookup</span> key assocs <span class="op">?&gt;</span></span>
 <span id="cb211-4"><a href="#cb211-4" aria-hidden="true" tabindex="-1"></a>    check <span class="op">?&gt;</span></span>
@@ -2707,8 +3749,10 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb211-7"><a href="#cb211-7" aria-hidden="true" tabindex="-1"></a>           <span class="op">|</span> val <span class="op">&lt;</span> x   <span class="ot">=</span> <span class="dt">Nothing</span></span>
 <span id="cb211-8"><a href="#cb211-8" aria-hidden="true" tabindex="-1"></a>           <span class="op">|</span> <span class="fu">otherwise</span> <span class="ot">=</span> <span class="dt">Just</span> x</span>
 <span id="cb211-9"><a href="#cb211-9" aria-hidden="true" tabindex="-1"></a>        buildResult x <span class="ot">=</span> <span class="dt">Just</span> ((key,val) <span class="op">:</span> delete (key,x) assocs)</span></code></pre></div>
-<p>Here’s another example: safe list indexing built from <code>safeHead</code> and <code>safeTail</code>:</p>
-<div class="sourceCode" id="cb212"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb212-1"><a href="#cb212-1" aria-hidden="true" tabindex="-1"></a><span class="ot">safeHead ::</span> [a] <span class="ot">-&gt;</span> <span class="dt">Maybe</span> a</span>
+<p>Here’s another example: safe list indexing built from
+<code>safeHead</code> and <code>safeTail</code>:</p>
+<div class="sourceCode" id="cb212"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb212-1"><a href="#cb212-1" aria-hidden="true" tabindex="-1"></a><span class="ot">safeHead ::</span> [a] <span class="ot">-&gt;</span> <span class="dt">Maybe</span> a</span>
 <span id="cb212-2"><a href="#cb212-2" aria-hidden="true" tabindex="-1"></a>safeHead [] <span class="ot">=</span> <span class="dt">Nothing</span></span>
 <span id="cb212-3"><a href="#cb212-3" aria-hidden="true" tabindex="-1"></a>safeHead (x<span class="op">:</span>xs) <span class="ot">=</span> <span class="dt">Just</span> x</span>
 <span id="cb212-4"><a href="#cb212-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -2722,7 +3766,8 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb212-12"><a href="#cb212-12" aria-hidden="true" tabindex="-1"></a><span class="ot">safeNth ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> <span class="dt">Maybe</span> a</span>
 <span id="cb212-13"><a href="#cb212-13" aria-hidden="true" tabindex="-1"></a>safeNth <span class="dv">0</span> xs <span class="ot">=</span> safeHead xs</span>
 <span id="cb212-14"><a href="#cb212-14" aria-hidden="true" tabindex="-1"></a>safeNth n xs <span class="ot">=</span> safeTail xs <span class="op">?&gt;</span> safeNth (n<span class="op">-</span><span class="dv">1</span>)</span></code></pre></div>
-<div class="sourceCode" id="cb213"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb213-1"><a href="#cb213-1" aria-hidden="true" tabindex="-1"></a>safeThird [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>]</span>
+<div class="sourceCode" id="cb213"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb213-1"><a href="#cb213-1" aria-hidden="true" tabindex="-1"></a>safeThird [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>]</span>
 <span id="cb213-2"><a href="#cb213-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Just</span> <span class="dv">3</span></span>
 <span id="cb213-3"><a href="#cb213-3" aria-hidden="true" tabindex="-1"></a>safeThird [<span class="dv">1</span>,<span class="dv">2</span>]</span>
 <span id="cb213-4"><a href="#cb213-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Nothing</span></span>
@@ -2730,23 +3775,38 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb213-6"><a href="#cb213-6" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Just</span> <span class="dv">6</span></span>
 <span id="cb213-7"><a href="#cb213-7" aria-hidden="true" tabindex="-1"></a>safeNth <span class="dv">11</span> [<span class="dv">1</span><span class="op">..</span><span class="dv">10</span>]</span>
 <span id="cb213-8"><a href="#cb213-8" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Nothing</span></span></code></pre></div>
-<p>PS. note that <code>?&gt;</code> <em>associates to the left</em> as is the default in Haskell. That means that <code>op ?&gt; f ?&gt; g</code> means <code>(op ?&gt; f) ?&gt; g</code>. The alternative, <code>op ?&gt; (f ?&gt; g)</code> would not even type check!</p>
-<p>Sidenote: this <code>?&gt;</code> operator expresses the if-result pattern that’s very common in other languages. Here is how one would write <code>op val ?&gt; f</code> in Python and Java.</p>
-<div class="sourceCode" id="cb214"><pre class="sourceCode python"><code class="sourceCode python"><span id="cb214-1"><a href="#cb214-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Python</span></span>
+<p>PS. note that <code>?&gt;</code> <em>associates to the left</em> as
+is the default in Haskell. That means that
+<code>op ?&gt; f ?&gt; g</code> means <code>(op ?&gt; f) ?&gt; g</code>.
+The alternative, <code>op ?&gt; (f ?&gt; g)</code> would not even type
+check!</p>
+<p>Sidenote: this <code>?&gt;</code> operator expresses the if-result
+pattern that’s very common in other languages. Here is how one would
+write <code>op val ?&gt; f</code> in Python and Java.</p>
+<div class="sourceCode" id="cb214"><pre
+class="sourceCode python"><code class="sourceCode python"><span id="cb214-1"><a href="#cb214-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Python</span></span>
 <span id="cb214-2"><a href="#cb214-2" aria-hidden="true" tabindex="-1"></a>x <span class="op">=</span> op(val)</span>
 <span id="cb214-3"><a href="#cb214-3" aria-hidden="true" tabindex="-1"></a><span class="cf">if</span> x:</span>
 <span id="cb214-4"><a href="#cb214-4" aria-hidden="true" tabindex="-1"></a>  f(x)</span></code></pre></div>
-<div class="sourceCode" id="cb215"><pre class="sourceCode java"><code class="sourceCode java"><span id="cb215-1"><a href="#cb215-1" aria-hidden="true" tabindex="-1"></a><span class="co">// Java</span></span>
+<div class="sourceCode" id="cb215"><pre
+class="sourceCode java"><code class="sourceCode java"><span id="cb215-1"><a href="#cb215-1" aria-hidden="true" tabindex="-1"></a><span class="co">// Java</span></span>
 <span id="cb215-2"><a href="#cb215-2" aria-hidden="true" tabindex="-1"></a><span class="bu">Object</span> x <span class="op">=</span> <span class="fu">op</span><span class="op">(</span>val<span class="op">);</span></span>
 <span id="cb215-3"><a href="#cb215-3" aria-hidden="true" tabindex="-1"></a><span class="cf">if</span> <span class="op">(</span>x <span class="op">!=</span> <span class="kw">null</span><span class="op">)</span> <span class="op">{</span></span>
 <span id="cb215-4"><a href="#cb215-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">f</span><span class="op">(</span>x<span class="op">);</span></span>
 <span id="cb215-5"><a href="#cb215-5" aria-hidden="true" tabindex="-1"></a><span class="op">}</span></span></code></pre></div>
-<p>The difference between the if-result pattern and our <code>?&gt;</code> is that we use the <code>Nothing</code> value to explicitly signal failure, instead of relying on the fact that any variable can be <code>None</code> (or <code>False</code>) in Python, or that any <code>Object</code> reference can be <code>null</code> in Java.</p>
-</section>
-<section id="example-2-logging" data-number="5.2">
-<h2 data-number="13.2"><span class="header-section-number">13.2</span> Example 2: Logging</h2>
-<p>Let’s explore the concept of chaining with another example: logging. The type <code>Logger</code> represents a value plus a list of log messages (produced by the computation that produced the value).</p>
-<div class="sourceCode" id="cb216"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb216-1"><a href="#cb216-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Logger definition</span></span>
+<p>The difference between the if-result pattern and our
+<code>?&gt;</code> is that we use the <code>Nothing</code> value to
+explicitly signal failure, instead of relying on the fact that any
+variable can be <code>None</code> (or <code>False</code>) in Python, or
+that any <code>Object</code> reference can be <code>null</code> in
+Java.</p>
+<h2 data-number="13.2" id="example-2-logging"><span
+class="header-section-number">13.2</span> Example 2: Logging</h2>
+<p>Let’s explore the concept of chaining with another example: logging.
+The type <code>Logger</code> represents a value plus a list of log
+messages (produced by the computation that produced the value).</p>
+<div class="sourceCode" id="cb216"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb216-1"><a href="#cb216-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Logger definition</span></span>
 <span id="cb216-2"><a href="#cb216-2" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Logger</span> a <span class="ot">=</span> <span class="dt">Logger</span> [<span class="dt">String</span>] a  <span class="kw">deriving</span> <span class="dt">Show</span></span>
 <span id="cb216-3"><a href="#cb216-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb216-4"><a href="#cb216-4" aria-hidden="true" tabindex="-1"></a><span class="ot">getVal ::</span> <span class="dt">Logger</span> a <span class="ot">-&gt;</span> a</span>
@@ -2763,8 +3823,12 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb216-15"><a href="#cb216-15" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb216-16"><a href="#cb216-16" aria-hidden="true" tabindex="-1"></a><span class="ot">msg ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Logger</span> ()</span>
 <span id="cb216-17"><a href="#cb216-17" aria-hidden="true" tabindex="-1"></a>msg s <span class="ot">=</span> <span class="dt">Logger</span> [s] ()        <span class="co">-- just a message</span></span></code></pre></div>
-<p>Here’s a <code>login</code> function that logs some details about the usernames and passwords it processes. Note how we run into complicated code in <code>login</code> when we need to handle multiple <code>Logger</code> values.</p>
-<div class="sourceCode" id="cb217"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb217-1"><a href="#cb217-1" aria-hidden="true" tabindex="-1"></a><span class="ot">validateUser ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Logger</span> <span class="dt">Bool</span></span>
+<p>Here’s a <code>login</code> function that logs some details about the
+usernames and passwords it processes. Note how we run into complicated
+code in <code>login</code> when we need to handle multiple
+<code>Logger</code> values.</p>
+<div class="sourceCode" id="cb217"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb217-1"><a href="#cb217-1" aria-hidden="true" tabindex="-1"></a><span class="ot">validateUser ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Logger</span> <span class="dt">Bool</span></span>
 <span id="cb217-2"><a href="#cb217-2" aria-hidden="true" tabindex="-1"></a>validateUser <span class="st">&quot;paul.atreides&quot;</span> <span class="ot">=</span> annotate <span class="st">&quot;Valid user&quot;</span> <span class="dt">True</span></span>
 <span id="cb217-3"><a href="#cb217-3" aria-hidden="true" tabindex="-1"></a>validateUser <span class="st">&quot;ninja&quot;</span> <span class="ot">=</span> nomsg <span class="dt">True</span></span>
 <span id="cb217-4"><a href="#cb217-4" aria-hidden="true" tabindex="-1"></a>validateUser u <span class="ot">=</span> annotate (<span class="st">&quot;Invalid user: &quot;</span><span class="op">++</span>u) <span class="dt">False</span></span>
@@ -2781,7 +3845,8 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb217-15"><a href="#cb217-15" aria-hidden="true" tabindex="-1"></a>       <span class="kw">then</span> <span class="kw">let</span> check <span class="ot">=</span> checkPassword user password</span>
 <span id="cb217-16"><a href="#cb217-16" aria-hidden="true" tabindex="-1"></a>            <span class="kw">in</span> <span class="dt">Logger</span> (getLog validation <span class="op">++</span> getLog check) (getVal check)</span>
 <span id="cb217-17"><a href="#cb217-17" aria-hidden="true" tabindex="-1"></a>       <span class="kw">else</span> validation</span></code></pre></div>
-<div class="sourceCode" id="cb218"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb218-1"><a href="#cb218-1" aria-hidden="true" tabindex="-1"></a>login <span class="st">&quot;paul.atreides&quot;</span> <span class="st">&quot;muad&#39;dib&quot;</span></span>
+<div class="sourceCode" id="cb218"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb218-1"><a href="#cb218-1" aria-hidden="true" tabindex="-1"></a>login <span class="st">&quot;paul.atreides&quot;</span> <span class="st">&quot;muad&#39;dib&quot;</span></span>
 <span id="cb218-2"><a href="#cb218-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Logger</span> [<span class="st">&quot;Valid user&quot;</span>,<span class="st">&quot;Password ok&quot;</span>] <span class="dt">True</span></span>
 <span id="cb218-3"><a href="#cb218-3" aria-hidden="true" tabindex="-1"></a>login <span class="st">&quot;paul.atreides&quot;</span> <span class="st">&quot;arrakis&quot;</span></span>
 <span id="cb218-4"><a href="#cb218-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Logger</span> [<span class="st">&quot;Valid user&quot;</span>,<span class="st">&quot;Password wrong: arrakis&quot;</span>] <span class="dt">False</span></span>
@@ -2789,11 +3854,17 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb218-6"><a href="#cb218-6" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Logger</span> [<span class="st">&quot;Password ok&quot;</span>] <span class="dt">True</span></span>
 <span id="cb218-7"><a href="#cb218-7" aria-hidden="true" tabindex="-1"></a>login <span class="st">&quot;leto.atreides&quot;</span> <span class="st">&quot;paul&quot;</span></span>
 <span id="cb218-8"><a href="#cb218-8" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Logger</span> [<span class="st">&quot;Invalid user: leto.atreides&quot;</span>] <span class="dt">False</span></span></code></pre></div>
-<p>Let’s try to streamline this code by defining a chaining operator for <code>Logger</code>. The important thing when doing multiple <code>Logger</code> operations is to preserve all the logs. Here’s a chaining operator, <code>#&gt;</code>, and an example of how it can be used to log some arithmetic computations.</p>
-<div class="sourceCode" id="cb219"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb219-1"><a href="#cb219-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(#&gt;) ::</span> <span class="dt">Logger</span> a <span class="ot">-&gt;</span> (a <span class="ot">-&gt;</span> <span class="dt">Logger</span> b) <span class="ot">-&gt;</span> <span class="dt">Logger</span> b</span>
+<p>Let’s try to streamline this code by defining a chaining operator for
+<code>Logger</code>. The important thing when doing multiple
+<code>Logger</code> operations is to preserve all the logs. Here’s a
+chaining operator, <code>#&gt;</code>, and an example of how it can be
+used to log some arithmetic computations.</p>
+<div class="sourceCode" id="cb219"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb219-1"><a href="#cb219-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(#&gt;) ::</span> <span class="dt">Logger</span> a <span class="ot">-&gt;</span> (a <span class="ot">-&gt;</span> <span class="dt">Logger</span> b) <span class="ot">-&gt;</span> <span class="dt">Logger</span> b</span>
 <span id="cb219-2"><a href="#cb219-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Logger</span> la a <span class="op">#&gt;</span> f <span class="ot">=</span> <span class="kw">let</span> <span class="dt">Logger</span> lb b <span class="ot">=</span> f a  <span class="co">-- feed value to next step</span></span>
 <span id="cb219-3"><a href="#cb219-3" aria-hidden="true" tabindex="-1"></a>                   <span class="kw">in</span> <span class="dt">Logger</span> (la<span class="op">++</span>lb) b   <span class="co">-- bundle result with all messages</span></span></code></pre></div>
-<div class="sourceCode" id="cb220"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb220-1"><a href="#cb220-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- square a number and log a message about it</span></span>
+<div class="sourceCode" id="cb220"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb220-1"><a href="#cb220-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- square a number and log a message about it</span></span>
 <span id="cb220-2"><a href="#cb220-2" aria-hidden="true" tabindex="-1"></a><span class="ot">square ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Logger</span> <span class="dt">Int</span></span>
 <span id="cb220-3"><a href="#cb220-3" aria-hidden="true" tabindex="-1"></a>square val <span class="ot">=</span> annotate (<span class="fu">show</span> val <span class="op">++</span> <span class="st">&quot;^2&quot;</span>) (val<span class="op">^</span><span class="dv">2</span>)</span>
 <span id="cb220-4"><a href="#cb220-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -2811,17 +3882,26 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb220-16"><a href="#cb220-16" aria-hidden="true" tabindex="-1"></a>    square x</span>
 <span id="cb220-17"><a href="#cb220-17" aria-hidden="true" tabindex="-1"></a>    <span class="op">#&gt;</span> add</span>
 <span id="cb220-18"><a href="#cb220-18" aria-hidden="true" tabindex="-1"></a>    <span class="op">#&gt;</span> double</span></code></pre></div>
-<div class="sourceCode" id="cb221"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb221-1"><a href="#cb221-1" aria-hidden="true" tabindex="-1"></a>compute <span class="dv">3</span></span>
+<div class="sourceCode" id="cb221"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb221-1"><a href="#cb221-1" aria-hidden="true" tabindex="-1"></a>compute <span class="dv">3</span></span>
 <span id="cb221-2"><a href="#cb221-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Logger</span> [<span class="st">&quot;3^2&quot;</span>,<span class="st">&quot;9+1&quot;</span>,<span class="st">&quot;10*2&quot;</span>] <span class="dv">20</span></span></code></pre></div>
-<p>We can streamline <code>login</code> quite a bit by using <code>#&gt;</code>. Note how we don’t need to worry about combining logs together. Also note how we use a lambda expression instead of defining a helper function.</p>
-<div class="sourceCode" id="cb222"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb222-1"><a href="#cb222-1" aria-hidden="true" tabindex="-1"></a><span class="ot">login ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Logger</span> <span class="dt">Bool</span></span>
+<p>We can streamline <code>login</code> quite a bit by using
+<code>#&gt;</code>. Note how we don’t need to worry about combining logs
+together. Also note how we use a lambda expression instead of defining a
+helper function.</p>
+<div class="sourceCode" id="cb222"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb222-1"><a href="#cb222-1" aria-hidden="true" tabindex="-1"></a><span class="ot">login ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Logger</span> <span class="dt">Bool</span></span>
 <span id="cb222-2"><a href="#cb222-2" aria-hidden="true" tabindex="-1"></a>login user password <span class="ot">=</span></span>
 <span id="cb222-3"><a href="#cb222-3" aria-hidden="true" tabindex="-1"></a>  validateUser user</span>
 <span id="cb222-4"><a href="#cb222-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">#&gt;</span></span>
 <span id="cb222-5"><a href="#cb222-5" aria-hidden="true" tabindex="-1"></a>  \valid <span class="ot">-&gt;</span> <span class="kw">if</span> valid <span class="kw">then</span> checkPassword user password</span>
 <span id="cb222-6"><a href="#cb222-6" aria-hidden="true" tabindex="-1"></a>                     <span class="kw">else</span> nomsg <span class="dt">False</span></span></code></pre></div>
-<p>To ramp things up a bit, let’s use <code>Logger</code> in a recursive list processing function. Here’s a logging version of <code>filter</code>. Note how the code chains a log message before the recursive call in order to keep the order of log entries nice.</p>
-<div class="sourceCode" id="cb223"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb223-1"><a href="#cb223-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- sometimes you don&#39;t need the previous value:</span></span>
+<p>To ramp things up a bit, let’s use <code>Logger</code> in a recursive
+list processing function. Here’s a logging version of
+<code>filter</code>. Note how the code chains a log message before the
+recursive call in order to keep the order of log entries nice.</p>
+<div class="sourceCode" id="cb223"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb223-1"><a href="#cb223-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- sometimes you don&#39;t need the previous value:</span></span>
 <span id="cb223-2"><a href="#cb223-2" aria-hidden="true" tabindex="-1"></a><span class="ot">(##&gt;) ::</span> <span class="dt">Logger</span> a <span class="ot">-&gt;</span> <span class="dt">Logger</span> b <span class="ot">-&gt;</span> <span class="dt">Logger</span> b</span>
 <span id="cb223-3"><a href="#cb223-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Logger</span> la _ <span class="op">##&gt;</span> <span class="dt">Logger</span> lb b <span class="ot">=</span> <span class="dt">Logger</span> (la<span class="op">++</span>lb) b</span>
 <span id="cb223-4"><a href="#cb223-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -2830,27 +3910,41 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb223-7"><a href="#cb223-7" aria-hidden="true" tabindex="-1"></a>filterLog f (x<span class="op">:</span>xs)</span>
 <span id="cb223-8"><a href="#cb223-8" aria-hidden="true" tabindex="-1"></a>   <span class="op">|</span> f x       <span class="ot">=</span> msg (<span class="st">&quot;keeping &quot;</span><span class="op">++</span><span class="fu">show</span> x) <span class="op">##&gt;</span> filterLog f xs <span class="op">#&gt;</span> (\xs&#39; <span class="ot">-&gt;</span> nomsg (x<span class="op">:</span>xs&#39;))</span>
 <span id="cb223-9"><a href="#cb223-9" aria-hidden="true" tabindex="-1"></a>   <span class="op">|</span> <span class="fu">otherwise</span> <span class="ot">=</span> msg (<span class="st">&quot;dropping &quot;</span><span class="op">++</span><span class="fu">show</span> x) <span class="op">##&gt;</span> filterLog f xs</span></code></pre></div>
-<div class="sourceCode" id="cb224"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb224-1"><a href="#cb224-1" aria-hidden="true" tabindex="-1"></a>filterLog (<span class="op">&gt;</span><span class="dv">0</span>) [<span class="dv">1</span>,<span class="op">-</span><span class="dv">2</span>,<span class="dv">3</span>,<span class="op">-</span><span class="dv">4</span>,<span class="dv">0</span>]</span>
+<div class="sourceCode" id="cb224"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb224-1"><a href="#cb224-1" aria-hidden="true" tabindex="-1"></a>filterLog (<span class="op">&gt;</span><span class="dv">0</span>) [<span class="dv">1</span>,<span class="op">-</span><span class="dv">2</span>,<span class="dv">3</span>,<span class="op">-</span><span class="dv">4</span>,<span class="dv">0</span>]</span>
 <span id="cb224-2"><a href="#cb224-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Logger</span> [<span class="st">&quot;keeping 1&quot;</span>,<span class="st">&quot;dropping -2&quot;</span>,<span class="st">&quot;keeping 3&quot;</span>,<span class="st">&quot;dropping -4&quot;</span>,<span class="st">&quot;dropping 0&quot;</span>] [<span class="dv">1</span>,<span class="dv">3</span>]</span></code></pre></div>
-</section>
-<section id="example-3-keeping-state" data-number="5.3">
-<h2 data-number="13.3"><span class="header-section-number">13.3</span> Example 3: Keeping State</h2>
-<p>In the previous example we just wrote some state (the log). Sometimes we need computations that change some sort of shared state. Let’s look at accounts in a small bank. We’ll first define a datatype for the state of the bank: the balances of all accounts, as a map from account name to balance.</p>
-<div class="sourceCode" id="cb225"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb225-1"><a href="#cb225-1" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="kw">qualified</span> <span class="dt">Data.Map</span> <span class="kw">as</span> <span class="dt">Map</span></span>
+<h2 data-number="13.3" id="example-3-keeping-state"><span
+class="header-section-number">13.3</span> Example 3: Keeping State</h2>
+<p>In the previous example we just wrote some state (the log). Sometimes
+we need computations that change some sort of shared state. Let’s look
+at accounts in a small bank. We’ll first define a datatype for the state
+of the bank: the balances of all accounts, as a map from account name to
+balance.</p>
+<div class="sourceCode" id="cb225"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb225-1"><a href="#cb225-1" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="kw">qualified</span> <span class="dt">Data.Map</span> <span class="kw">as</span> <span class="dt">Map</span></span>
 <span id="cb225-2"><a href="#cb225-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb225-3"><a href="#cb225-3" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Bank</span> <span class="ot">=</span> <span class="dt">Bank</span> (<span class="dt">Map.Map</span> <span class="dt">String</span> <span class="dt">Int</span>)</span>
 <span id="cb225-4"><a href="#cb225-4" aria-hidden="true" tabindex="-1"></a>  <span class="kw">deriving</span> <span class="dt">Show</span></span></code></pre></div>
-<p>Here’s how we can deposit some money to an account. We use the function <code>adjust</code> from <code>Data.Map</code> to modify the map.</p>
-<div class="sourceCode" id="cb226"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb226-1"><a href="#cb226-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Apply a function to one value in a map</span></span>
+<p>Here’s how we can deposit some money to an account. We use the
+function <code>adjust</code> from <code>Data.Map</code> to modify the
+map.</p>
+<div class="sourceCode" id="cb226"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb226-1"><a href="#cb226-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Apply a function to one value in a map</span></span>
 <span id="cb226-2"><a href="#cb226-2" aria-hidden="true" tabindex="-1"></a>Map.adjust<span class="ot"> ::</span> <span class="dt">Ord</span> k <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> a) <span class="ot">-&gt;</span> k <span class="ot">-&gt;</span> <span class="dt">Map.Map</span> k a <span class="ot">-&gt;</span> <span class="dt">Map.Map</span> k a</span></code></pre></div>
-<div class="sourceCode" id="cb227"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb227-1"><a href="#cb227-1" aria-hidden="true" tabindex="-1"></a><span class="ot">deposit ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Bank</span> <span class="ot">-&gt;</span> <span class="dt">Bank</span></span>
+<div class="sourceCode" id="cb227"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb227-1"><a href="#cb227-1" aria-hidden="true" tabindex="-1"></a><span class="ot">deposit ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Bank</span> <span class="ot">-&gt;</span> <span class="dt">Bank</span></span>
 <span id="cb227-2"><a href="#cb227-2" aria-hidden="true" tabindex="-1"></a>deposit accountName amount (<span class="dt">Bank</span> accounts) <span class="ot">=</span></span>
 <span id="cb227-3"><a href="#cb227-3" aria-hidden="true" tabindex="-1"></a>  <span class="dt">Bank</span> (Map.adjust (\x <span class="ot">-&gt;</span> x<span class="op">+</span>amount) accountName accounts)</span></code></pre></div>
-<p>Withdrawing money is a bit more complicated, since we want to handle some special cases like the account not existing, or the account not having enough money. We use the library function <code>findWithDefault</code> to help us along.</p>
-<div class="sourceCode" id="cb228"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb228-1"><a href="#cb228-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Fetch the value corresponding to a key from a map,</span></span>
+<p>Withdrawing money is a bit more complicated, since we want to handle
+some special cases like the account not existing, or the account not
+having enough money. We use the library function
+<code>findWithDefault</code> to help us along.</p>
+<div class="sourceCode" id="cb228"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb228-1"><a href="#cb228-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Fetch the value corresponding to a key from a map,</span></span>
 <span id="cb228-2"><a href="#cb228-2" aria-hidden="true" tabindex="-1"></a><span class="co">-- or a default value in case the key does not exist</span></span>
 <span id="cb228-3"><a href="#cb228-3" aria-hidden="true" tabindex="-1"></a>Map.findWithDefault<span class="ot"> ::</span> <span class="dt">Ord</span> k <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> k <span class="ot">-&gt;</span> <span class="dt">Map.Map</span> k a <span class="ot">-&gt;</span> a</span></code></pre></div>
-<div class="sourceCode" id="cb229"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb229-1"><a href="#cb229-1" aria-hidden="true" tabindex="-1"></a><span class="ot">withdraw ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Bank</span> <span class="ot">-&gt;</span> (<span class="dt">Int</span>,<span class="dt">Bank</span>)</span>
+<div class="sourceCode" id="cb229"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb229-1"><a href="#cb229-1" aria-hidden="true" tabindex="-1"></a><span class="ot">withdraw ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Bank</span> <span class="ot">-&gt;</span> (<span class="dt">Int</span>,<span class="dt">Bank</span>)</span>
 <span id="cb229-2"><a href="#cb229-2" aria-hidden="true" tabindex="-1"></a>withdraw accountName amount (<span class="dt">Bank</span> accounts) <span class="ot">=</span></span>
 <span id="cb229-3"><a href="#cb229-3" aria-hidden="true" tabindex="-1"></a>  <span class="kw">let</span> <span class="co">-- balance is 0 for a nonexistant account</span></span>
 <span id="cb229-4"><a href="#cb229-4" aria-hidden="true" tabindex="-1"></a>      balance <span class="ot">=</span> Map.findWithDefault <span class="dv">0</span> accountName accounts</span>
@@ -2858,8 +3952,14 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb229-6"><a href="#cb229-6" aria-hidden="true" tabindex="-1"></a>      withdrawal <span class="ot">=</span> <span class="fu">min</span> amount balance</span>
 <span id="cb229-7"><a href="#cb229-7" aria-hidden="true" tabindex="-1"></a>      newAccounts <span class="ot">=</span> Map.adjust (\x <span class="ot">-&gt;</span> x<span class="op">-</span>withdrawal) accountName accounts</span>
 <span id="cb229-8"><a href="#cb229-8" aria-hidden="true" tabindex="-1"></a>  <span class="kw">in</span> (withdrawal, <span class="dt">Bank</span> newAccounts)</span></code></pre></div>
-<p>Finally, let’s write a function that takes at most 100 money from one account, splits the money in half, and deposits it in two accounts. Pay attention to how we need to carefully thread the different versions of the bank, <code>bank</code>, <code>bank1</code>, <code>bank2</code> and <code>bank3</code> to make sure all transactions happen in the right order.</p>
-<div class="sourceCode" id="cb230"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb230-1"><a href="#cb230-1" aria-hidden="true" tabindex="-1"></a><span class="ot">share ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Bank</span> <span class="ot">-&gt;</span> <span class="dt">Bank</span></span>
+<p>Finally, let’s write a function that takes at most 100 money from one
+account, splits the money in half, and deposits it in two accounts. Pay
+attention to how we need to carefully thread the different versions of
+the bank, <code>bank</code>, <code>bank1</code>, <code>bank2</code> and
+<code>bank3</code> to make sure all transactions happen in the right
+order.</p>
+<div class="sourceCode" id="cb230"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb230-1"><a href="#cb230-1" aria-hidden="true" tabindex="-1"></a><span class="ot">share ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Bank</span> <span class="ot">-&gt;</span> <span class="dt">Bank</span></span>
 <span id="cb230-2"><a href="#cb230-2" aria-hidden="true" tabindex="-1"></a>share from to1 to2 bank <span class="ot">=</span></span>
 <span id="cb230-3"><a href="#cb230-3" aria-hidden="true" tabindex="-1"></a>  <span class="kw">let</span> (amount,bank1) <span class="ot">=</span> withdraw from <span class="dv">100</span> bank</span>
 <span id="cb230-4"><a href="#cb230-4" aria-hidden="true" tabindex="-1"></a>      half <span class="ot">=</span> <span class="fu">div</span> amount <span class="dv">2</span></span>
@@ -2868,15 +3968,22 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb230-7"><a href="#cb230-7" aria-hidden="true" tabindex="-1"></a>      bank2 <span class="ot">=</span> deposit to1 half bank1</span>
 <span id="cb230-8"><a href="#cb230-8" aria-hidden="true" tabindex="-1"></a>      bank3 <span class="ot">=</span> deposit to2 rest bank2</span>
 <span id="cb230-9"><a href="#cb230-9" aria-hidden="true" tabindex="-1"></a>  <span class="kw">in</span> bank3</span></code></pre></div>
-<div class="sourceCode" id="cb231"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb231-1"><a href="#cb231-1" aria-hidden="true" tabindex="-1"></a>share <span class="st">&quot;wotan&quot;</span> <span class="st">&quot;siegfried&quot;</span> <span class="st">&quot;brunhilde&quot;</span></span>
+<div class="sourceCode" id="cb231"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb231-1"><a href="#cb231-1" aria-hidden="true" tabindex="-1"></a>share <span class="st">&quot;wotan&quot;</span> <span class="st">&quot;siegfried&quot;</span> <span class="st">&quot;brunhilde&quot;</span></span>
 <span id="cb231-2"><a href="#cb231-2" aria-hidden="true" tabindex="-1"></a>      (<span class="dt">Bank</span> (Map.fromList [(<span class="st">&quot;brunhilde&quot;</span>,<span class="dv">0</span>),(<span class="st">&quot;siegfried&quot;</span>,<span class="dv">0</span>),(<span class="st">&quot;wotan&quot;</span>,<span class="dv">1000</span>)]))</span>
 <span id="cb231-3"><a href="#cb231-3" aria-hidden="true" tabindex="-1"></a>   <span class="op">==&gt;</span> <span class="dt">Bank</span> (Map.fromList [(<span class="st">&quot;brunhilde&quot;</span>,<span class="dv">50</span>),(<span class="st">&quot;siegfried&quot;</span>,<span class="dv">50</span>),(<span class="st">&quot;wotan&quot;</span>,<span class="dv">900</span>)])</span>
 <span id="cb231-4"><a href="#cb231-4" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb231-5"><a href="#cb231-5" aria-hidden="true" tabindex="-1"></a>share <span class="st">&quot;wotan&quot;</span> <span class="st">&quot;siegfried&quot;</span> <span class="st">&quot;brunhilde&quot;</span></span>
 <span id="cb231-6"><a href="#cb231-6" aria-hidden="true" tabindex="-1"></a>      (<span class="dt">Bank</span> (Map.fromList [(<span class="st">&quot;brunhilde&quot;</span>,<span class="dv">0</span>),(<span class="st">&quot;siegfried&quot;</span>,<span class="dv">0</span>),(<span class="st">&quot;wotan&quot;</span>,<span class="dv">91</span>)]))</span>
 <span id="cb231-7"><a href="#cb231-7" aria-hidden="true" tabindex="-1"></a>   <span class="op">==&gt;</span> <span class="dt">Bank</span> (Map.fromList [(<span class="st">&quot;brunhilde&quot;</span>,<span class="dv">46</span>),(<span class="st">&quot;siegfried&quot;</span>,<span class="dv">45</span>),(<span class="st">&quot;wotan&quot;</span>,<span class="dv">0</span>)])</span></code></pre></div>
-<p>Code like this turns up often in Haskell when you’re doing serial updates to one value, while also performing some other computations on the side. It’s easy to make a mistake, and the type system won’t help you if you e.g. reuse the <code>bank1</code> value. Let’s rewrite <code>share</code> so that we don’t need to refer to the bank itself. We can again use the same chaining idea to accomplish this.</p>
-<div class="sourceCode" id="cb232"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb232-1"><a href="#cb232-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- `BankOp a` is an operation that transforms a Bank value,</span></span>
+<p>Code like this turns up often in Haskell when you’re doing serial
+updates to one value, while also performing some other computations on
+the side. It’s easy to make a mistake, and the type system won’t help
+you if you e.g. reuse the <code>bank1</code> value. Let’s rewrite
+<code>share</code> so that we don’t need to refer to the bank itself. We
+can again use the same chaining idea to accomplish this.</p>
+<div class="sourceCode" id="cb232"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb232-1"><a href="#cb232-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- `BankOp a` is an operation that transforms a Bank value,</span></span>
 <span id="cb232-2"><a href="#cb232-2" aria-hidden="true" tabindex="-1"></a><span class="co">-- while returning a value of type `a`</span></span>
 <span id="cb232-3"><a href="#cb232-3" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">BankOp</span> a <span class="ot">=</span> <span class="dt">BankOp</span> (<span class="dt">Bank</span> <span class="ot">-&gt;</span> (a,<span class="dt">Bank</span>))</span>
 <span id="cb232-4"><a href="#cb232-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -2910,7 +4017,8 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb232-32"><a href="#cb232-32" aria-hidden="true" tabindex="-1"></a><span class="ot">withdrawOp ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">BankOp</span> <span class="dt">Int</span></span>
 <span id="cb232-33"><a href="#cb232-33" aria-hidden="true" tabindex="-1"></a>withdrawOp accountName amount <span class="ot">=</span> <span class="dt">BankOp</span> (withdraw accountName amount)</span></code></pre></div>
 <p>Let’s see how chaining works with these bank operations.</p>
-<div class="sourceCode" id="cb233"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb233-1"><a href="#cb233-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> bank <span class="ot">=</span> <span class="dt">Bank</span> (Map.fromList [(<span class="st">&quot;edsger&quot;</span>,<span class="dv">10</span>),(<span class="st">&quot;grace&quot;</span>,<span class="dv">50</span>)])</span>
+<div class="sourceCode" id="cb233"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb233-1"><a href="#cb233-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> bank <span class="ot">=</span> <span class="dt">Bank</span> (Map.fromList [(<span class="st">&quot;edsger&quot;</span>,<span class="dv">10</span>),(<span class="st">&quot;grace&quot;</span>,<span class="dv">50</span>)])</span>
 <span id="cb233-2"><a href="#cb233-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb233-3"><a href="#cb233-3" aria-hidden="true" tabindex="-1"></a><span class="co">-- Running a number of operations using +&gt;&gt;</span></span>
 <span id="cb233-4"><a href="#cb233-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -2930,9 +4038,19 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb233-18"><a href="#cb233-18" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb233-19"><a href="#cb233-19" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> runBankOp (withdrawOp <span class="st">&quot;edsger&quot;</span> <span class="dv">100</span> <span class="op">+&gt;</span> depositOp <span class="st">&quot;grace&quot;</span>) bank</span>
 <span id="cb233-20"><a href="#cb233-20" aria-hidden="true" tabindex="-1"></a>((),<span class="dt">Bank</span> (fromList [(<span class="st">&quot;edsger&quot;</span>,<span class="dv">0</span>),(<span class="st">&quot;grace&quot;</span>,<span class="dv">60</span>)]))</span></code></pre></div>
-<p>Note how a value of type <code>BankOp</code> represents a process that transforms the bank. The initial state of the bank must be supplied using <code>runBankOp</code>. This makes sense because <code>BankOp</code> transformations can be composed, unlike <code>Bank</code> states. Having to use <code>runBankOp</code> makes the distinction between <em>defining</em> operations and <em>executing them</em> clearer.</p>
-<p>Now that we’re familiar with manipulating <code>BankOp</code> values, we can implement <code>share</code> as a <code>BankOp</code>. We implement a helper <code>distributeOp</code> to make the code a bit neater.</p>
-<div class="sourceCode" id="cb234"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb234-1"><a href="#cb234-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- distribute amount to two accounts</span></span>
+<p>Note how a value of type <code>BankOp</code> represents a process
+that transforms the bank. The initial state of the bank must be supplied
+using <code>runBankOp</code>. This makes sense because
+<code>BankOp</code> transformations can be composed, unlike
+<code>Bank</code> states. Having to use <code>runBankOp</code> makes the
+distinction between <em>defining</em> operations and <em>executing
+them</em> clearer.</p>
+<p>Now that we’re familiar with manipulating <code>BankOp</code> values,
+we can implement <code>share</code> as a <code>BankOp</code>. We
+implement a helper <code>distributeOp</code> to make the code a bit
+neater.</p>
+<div class="sourceCode" id="cb234"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb234-1"><a href="#cb234-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- distribute amount to two accounts</span></span>
 <span id="cb234-2"><a href="#cb234-2" aria-hidden="true" tabindex="-1"></a><span class="ot">distributeOp ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">BankOp</span> ()</span>
 <span id="cb234-3"><a href="#cb234-3" aria-hidden="true" tabindex="-1"></a>distributeOp to1 to2 amount <span class="ot">=</span></span>
 <span id="cb234-4"><a href="#cb234-4" aria-hidden="true" tabindex="-1"></a>  depositOp to1 half</span>
@@ -2946,45 +4064,64 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb234-12"><a href="#cb234-12" aria-hidden="true" tabindex="-1"></a>  withdrawOp from <span class="dv">100</span></span>
 <span id="cb234-13"><a href="#cb234-13" aria-hidden="true" tabindex="-1"></a>  <span class="op">+&gt;</span></span>
 <span id="cb234-14"><a href="#cb234-14" aria-hidden="true" tabindex="-1"></a>  distributeOp to1 to2</span></code></pre></div>
-<div class="sourceCode" id="cb235"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb235-1"><a href="#cb235-1" aria-hidden="true" tabindex="-1"></a>runBankOp (shareOp <span class="st">&quot;wotan&quot;</span> <span class="st">&quot;siegfried&quot;</span> <span class="st">&quot;brunhilde&quot;</span>)</span>
+<div class="sourceCode" id="cb235"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb235-1"><a href="#cb235-1" aria-hidden="true" tabindex="-1"></a>runBankOp (shareOp <span class="st">&quot;wotan&quot;</span> <span class="st">&quot;siegfried&quot;</span> <span class="st">&quot;brunhilde&quot;</span>)</span>
 <span id="cb235-2"><a href="#cb235-2" aria-hidden="true" tabindex="-1"></a>          (<span class="dt">Bank</span> (Map.fromList [(<span class="st">&quot;brunhilde&quot;</span>,<span class="dv">0</span>),(<span class="st">&quot;siegfried&quot;</span>,<span class="dv">0</span>),(<span class="st">&quot;wotan&quot;</span>,<span class="dv">1000</span>)]))</span>
 <span id="cb235-3"><a href="#cb235-3" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> ((),<span class="dt">Bank</span> (Map.fromList [(<span class="st">&quot;brunhilde&quot;</span>,<span class="dv">50</span>),(<span class="st">&quot;siegfried&quot;</span>,<span class="dv">50</span>),(<span class="st">&quot;wotan&quot;</span>,<span class="dv">900</span>)]))</span>
 <span id="cb235-4"><a href="#cb235-4" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb235-5"><a href="#cb235-5" aria-hidden="true" tabindex="-1"></a>runBankOp (shareOp <span class="st">&quot;wotan&quot;</span> <span class="st">&quot;siegfried&quot;</span> <span class="st">&quot;brunhilde&quot;</span>)</span>
 <span id="cb235-6"><a href="#cb235-6" aria-hidden="true" tabindex="-1"></a>          (<span class="dt">Bank</span> (Map.fromList [(<span class="st">&quot;brunhilde&quot;</span>,<span class="dv">0</span>),(<span class="st">&quot;siegfried&quot;</span>,<span class="dv">0</span>),(<span class="st">&quot;wotan&quot;</span>,<span class="dv">91</span>)]))</span>
 <span id="cb235-7"><a href="#cb235-7" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> ((),<span class="dt">Bank</span> (Map.fromList [(<span class="st">&quot;brunhilde&quot;</span>,<span class="dv">46</span>),(<span class="st">&quot;siegfried&quot;</span>,<span class="dv">45</span>),(<span class="st">&quot;wotan&quot;</span>,<span class="dv">0</span>)]))</span></code></pre></div>
-<p>That was pretty clean wasn’t it? We don’t need to mention the bank at all, we can almost program as if in an imperative language while staying completely pure.</p>
-<p>You can find all of this code in the course repository under <a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Examples/Bank.hs"><code>exercises/Examples/Bank.hs</code></a>.</p>
-</section>
-<section id="finally-the-monad-type-class" data-number="5.4">
-<h2 data-number="13.4"><span class="header-section-number">13.4</span> Finally: The Monad Type Class</h2>
+<p>That was pretty clean wasn’t it? We don’t need to mention the bank at
+all, we can almost program as if in an imperative language while staying
+completely pure.</p>
+<p>You can find all of this code in the course repository under <a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Examples/Bank.hs"><code>exercises/Examples/Bank.hs</code></a>.</p>
+<h2 data-number="13.4" id="finally-the-monad-type-class"><span
+class="header-section-number">13.4</span> Finally: The Monad Type
+Class</h2>
 <p>We’ve now seen three different types with a chaining operation:</p>
-<div class="sourceCode" id="cb236"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb236-1"><a href="#cb236-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(?&gt;) ::</span> <span class="dt">Maybe</span> a <span class="ot">-&gt;</span> (a <span class="ot">-&gt;</span> <span class="dt">Maybe</span> b) <span class="ot">-&gt;</span> <span class="dt">Maybe</span> b</span>
+<div class="sourceCode" id="cb236"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb236-1"><a href="#cb236-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(?&gt;) ::</span> <span class="dt">Maybe</span> a <span class="ot">-&gt;</span> (a <span class="ot">-&gt;</span> <span class="dt">Maybe</span> b) <span class="ot">-&gt;</span> <span class="dt">Maybe</span> b</span>
 <span id="cb236-2"><a href="#cb236-2" aria-hidden="true" tabindex="-1"></a><span class="ot">(#&gt;) ::</span> <span class="dt">Logger</span> a <span class="ot">-&gt;</span> (a <span class="ot">-&gt;</span> <span class="dt">Logger</span> b) <span class="ot">-&gt;</span> <span class="dt">Logger</span> b</span>
 <span id="cb236-3"><a href="#cb236-3" aria-hidden="true" tabindex="-1"></a><span class="ot">(+&gt;) ::</span> <span class="dt">BankOp</span> a <span class="ot">-&gt;</span> (a <span class="ot">-&gt;</span> <span class="dt">BankOp</span> b) <span class="ot">-&gt;</span> <span class="dt">BankOp</span> b</span></code></pre></div>
-<p>Just like previously with <code>map</code> and <code>Functor</code>, there is a type class that captures this pattern. Note that <code>Monad</code> is a class for <em>type constructors</em>, just like <code>Functor</code>.</p>
-<div class="sourceCode" id="cb237"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb237-1"><a href="#cb237-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Monad</span> m <span class="kw">where</span></span>
+<p>Just like previously with <code>map</code> and <code>Functor</code>,
+there is a type class that captures this pattern. Note that
+<code>Monad</code> is a class for <em>type constructors</em>, just like
+<code>Functor</code>.</p>
+<div class="sourceCode" id="cb237"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb237-1"><a href="#cb237-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Monad</span> m <span class="kw">where</span></span>
 <span id="cb237-2"><a href="#cb237-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  (&gt;&gt;=) ::</span> m a <span class="ot">-&gt;</span> (a <span class="ot">-&gt;</span> m b) <span class="ot">-&gt;</span> m b</span></code></pre></div>
 <p>There are some additional operations in <code>Monad</code> too:</p>
-<div class="sourceCode" id="cb238"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb238-1"><a href="#cb238-1" aria-hidden="true" tabindex="-1"></a>  <span class="co">-- lift a normal value into the monad</span></span>
+<div class="sourceCode" id="cb238"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb238-1"><a href="#cb238-1" aria-hidden="true" tabindex="-1"></a>  <span class="co">-- lift a normal value into the monad</span></span>
 <span id="cb238-2"><a href="#cb238-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  return ::</span> a <span class="ot">-&gt;</span> m a</span>
 <span id="cb238-3"><a href="#cb238-3" aria-hidden="true" tabindex="-1"></a>  <span class="co">-- simpler chaining (like our ##&gt;)</span></span>
 <span id="cb238-4"><a href="#cb238-4" aria-hidden="true" tabindex="-1"></a><span class="ot">  (&gt;&gt;) ::</span> m a <span class="ot">-&gt;</span> m b <span class="ot">-&gt;</span> m b</span>
 <span id="cb238-5"><a href="#cb238-5" aria-hidden="true" tabindex="-1"></a>  a <span class="op">&gt;&gt;</span> b  <span class="ot">=</span>  a <span class="op">&gt;&gt;=</span> \_ <span class="ot">-&gt;</span> b     <span class="co">-- remember: _ means ignored argument</span></span></code></pre></div>
-<p>Recall that the <code>Functor</code> class was about a generic <code>map</code> operation. Similarly, the <code>Monad</code> class is just about a generic chaining operation.</p>
-<div class="sourceCode" id="cb239"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb239-1"><a href="#cb239-1" aria-hidden="true" tabindex="-1"></a><span class="fu">fmap</span><span class="ot"> ::</span> <span class="dt">Functor</span> f <span class="ot">=&gt;</span> (a<span class="ot">-&gt;</span>b) <span class="ot">-&gt;</span> f a <span class="ot">-&gt;</span> f b</span>
+<p>Recall that the <code>Functor</code> class was about a generic
+<code>map</code> operation. Similarly, the <code>Monad</code> class is
+just about a generic chaining operation.</p>
+<div class="sourceCode" id="cb239"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb239-1"><a href="#cb239-1" aria-hidden="true" tabindex="-1"></a><span class="fu">fmap</span><span class="ot"> ::</span> <span class="dt">Functor</span> f <span class="ot">=&gt;</span> (a<span class="ot">-&gt;</span>b) <span class="ot">-&gt;</span> f a <span class="ot">-&gt;</span> f b</span>
 <span id="cb239-2"><a href="#cb239-2" aria-hidden="true" tabindex="-1"></a><span class="ot">(&gt;&gt;=) ::</span> <span class="dt">Monad</span> m <span class="ot">=&gt;</span> m a <span class="ot">-&gt;</span> (a <span class="ot">-&gt;</span> m b) <span class="ot">-&gt;</span> m b</span></code></pre></div>
-<p>The expression <code>operation &gt;&gt;= next</code> takes a monadic operation <code>operation :: m a</code>, and does some further computation with the value that it produces using <code>next :: a -&gt; m b</code>. If this feels too abstract, just recall how chaining works for <code>Maybe</code>:</p>
-<div class="sourceCode" id="cb240"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb240-1"><a href="#cb240-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(&gt;&gt;=) ::</span> <span class="dt">Maybe</span> a <span class="ot">-&gt;</span> (a <span class="ot">-&gt;</span> <span class="dt">Maybe</span> b) <span class="ot">-&gt;</span> <span class="dt">Maybe</span> b</span>
+<p>The expression <code>operation &gt;&gt;= next</code> takes a monadic
+operation <code>operation :: m a</code>, and does some further
+computation with the value that it produces using
+<code>next :: a -&gt; m b</code>. If this feels too abstract, just
+recall how chaining works for <code>Maybe</code>:</p>
+<div class="sourceCode" id="cb240"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb240-1"><a href="#cb240-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(&gt;&gt;=) ::</span> <span class="dt">Maybe</span> a <span class="ot">-&gt;</span> (a <span class="ot">-&gt;</span> <span class="dt">Maybe</span> b) <span class="ot">-&gt;</span> <span class="dt">Maybe</span> b</span>
 <span id="cb240-2"><a href="#cb240-2" aria-hidden="true" tabindex="-1"></a><span class="co">-- if we failed, don&#39;t even bother running the next step</span></span>
 <span id="cb240-3"><a href="#cb240-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Nothing</span> <span class="op">&gt;&gt;=</span> _ <span class="ot">=</span> <span class="dt">Nothing</span></span>
 <span id="cb240-4"><a href="#cb240-4" aria-hidden="true" tabindex="-1"></a><span class="co">-- otherwise run the next step</span></span>
 <span id="cb240-5"><a href="#cb240-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Just</span> x  <span class="op">&gt;&gt;=</span> f <span class="ot">=</span> f x</span></code></pre></div>
-</section>
-<section id="maybe-is-a-monad" data-number="5.5">
-<h2 data-number="13.5"><span class="header-section-number">13.5</span> Maybe is a Monad!</h2>
-<p>Here’s the full <code>Monad</code> instance for <code>Maybe</code> and some examples.</p>
-<div class="sourceCode" id="cb241"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb241-1"><a href="#cb241-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span>  <span class="dt">Monad</span> <span class="dt">Maybe</span>  <span class="kw">where</span></span>
+<h2 data-number="13.5" id="maybe-is-a-monad"><span
+class="header-section-number">13.5</span> Maybe is a Monad!</h2>
+<p>Here’s the full <code>Monad</code> instance for <code>Maybe</code>
+and some examples.</p>
+<div class="sourceCode" id="cb241"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb241-1"><a href="#cb241-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span>  <span class="dt">Monad</span> <span class="dt">Maybe</span>  <span class="kw">where</span></span>
 <span id="cb241-2"><a href="#cb241-2" aria-hidden="true" tabindex="-1"></a>    (<span class="dt">Just</span> x) <span class="op">&gt;&gt;=</span> k      <span class="ot">=</span> k x</span>
 <span id="cb241-3"><a href="#cb241-3" aria-hidden="true" tabindex="-1"></a>    <span class="dt">Nothing</span>  <span class="op">&gt;&gt;=</span> _      <span class="ot">=</span> <span class="dt">Nothing</span></span>
 <span id="cb241-4"><a href="#cb241-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -2992,7 +4129,8 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb241-6"><a href="#cb241-6" aria-hidden="true" tabindex="-1"></a>    <span class="dt">Nothing</span>  <span class="op">&gt;&gt;</span>  _      <span class="ot">=</span> <span class="dt">Nothing</span></span>
 <span id="cb241-7"><a href="#cb241-7" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb241-8"><a href="#cb241-8" aria-hidden="true" tabindex="-1"></a>    <span class="fu">return</span> x            <span class="ot">=</span> <span class="dt">Just</span> x</span></code></pre></div>
-<div class="sourceCode" id="cb242"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb242-1"><a href="#cb242-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Just</span> <span class="dv">1</span> <span class="op">&gt;&gt;=</span> \x <span class="ot">-&gt;</span> <span class="fu">return</span> (x<span class="op">+</span><span class="dv">1</span>)</span>
+<div class="sourceCode" id="cb242"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb242-1"><a href="#cb242-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Just</span> <span class="dv">1</span> <span class="op">&gt;&gt;=</span> \x <span class="ot">-&gt;</span> <span class="fu">return</span> (x<span class="op">+</span><span class="dv">1</span>)</span>
 <span id="cb242-2"><a href="#cb242-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Just</span> <span class="dv">2</span></span>
 <span id="cb242-3"><a href="#cb242-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Just</span> <span class="st">&quot;HELLO&quot;</span> <span class="op">&gt;&gt;=</span> (\x <span class="ot">-&gt;</span> <span class="fu">return</span> (<span class="fu">length</span> x)) <span class="op">&gt;&gt;=</span> (\x <span class="ot">-&gt;</span> <span class="fu">return</span> (x<span class="op">+</span><span class="dv">1</span>))</span>
 <span id="cb242-4"><a href="#cb242-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Just</span> <span class="dv">6</span></span>
@@ -3002,14 +4140,19 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb242-8"><a href="#cb242-8" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Just</span> <span class="dv">2</span></span>
 <span id="cb242-9"><a href="#cb242-9" aria-hidden="true" tabindex="-1"></a><span class="dt">Just</span> <span class="dv">2</span> <span class="op">&gt;&gt;</span> <span class="dt">Nothing</span></span>
 <span id="cb242-10"><a href="#cb242-10" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Nothing</span></span></code></pre></div>
-<p>Here are the <code>stealSecret</code> and <code>increase</code> examples rewritten with monad operations. The changes are <code>?&gt;</code> to <code>&gt;&gt;=</code> and <code>Just</code> to <code>return</code>.</p>
-<div class="sourceCode" id="cb243"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb243-1"><a href="#cb243-1" aria-hidden="true" tabindex="-1"></a><span class="ot">stealSecret ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">String</span></span>
+<p>Here are the <code>stealSecret</code> and <code>increase</code>
+examples rewritten with monad operations. The changes are
+<code>?&gt;</code> to <code>&gt;&gt;=</code> and <code>Just</code> to
+<code>return</code>.</p>
+<div class="sourceCode" id="cb243"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb243-1"><a href="#cb243-1" aria-hidden="true" tabindex="-1"></a><span class="ot">stealSecret ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">String</span></span>
 <span id="cb243-2"><a href="#cb243-2" aria-hidden="true" tabindex="-1"></a>stealSecret password <span class="ot">=</span></span>
 <span id="cb243-3"><a href="#cb243-3" aria-hidden="true" tabindex="-1"></a>    login password <span class="op">&gt;&gt;=</span></span>
 <span id="cb243-4"><a href="#cb243-4" aria-hidden="true" tabindex="-1"></a>    secret <span class="op">&gt;&gt;=</span></span>
 <span id="cb243-5"><a href="#cb243-5" aria-hidden="true" tabindex="-1"></a>    decorate</span>
 <span id="cb243-6"><a href="#cb243-6" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> decorate s <span class="ot">=</span> <span class="fu">return</span> (<span class="st">&quot;Stole secret: &quot;</span><span class="op">++</span>s)</span></code></pre></div>
-<div class="sourceCode" id="cb244"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb244-1"><a href="#cb244-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Set the value of key to val in the given key-value list,</span></span>
+<div class="sourceCode" id="cb244"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb244-1"><a href="#cb244-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Set the value of key to val in the given key-value list,</span></span>
 <span id="cb244-2"><a href="#cb244-2" aria-hidden="true" tabindex="-1"></a><span class="co">-- but only if val is larger than the current value!</span></span>
 <span id="cb244-3"><a href="#cb244-3" aria-hidden="true" tabindex="-1"></a><span class="ot">increase ::</span> <span class="dt">Eq</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> [(a,<span class="dt">Int</span>)] <span class="ot">-&gt;</span> <span class="dt">Maybe</span> [(a,<span class="dt">Int</span>)]</span>
 <span id="cb244-4"><a href="#cb244-4" aria-hidden="true" tabindex="-1"></a>increase key val assocs <span class="ot">=</span></span>
@@ -3020,41 +4163,57 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb244-9"><a href="#cb244-9" aria-hidden="true" tabindex="-1"></a>           <span class="op">|</span> val <span class="op">&lt;</span> x   <span class="ot">=</span> <span class="dt">Nothing</span></span>
 <span id="cb244-10"><a href="#cb244-10" aria-hidden="true" tabindex="-1"></a>           <span class="op">|</span> <span class="fu">otherwise</span> <span class="ot">=</span> <span class="fu">return</span> x</span>
 <span id="cb244-11"><a href="#cb244-11" aria-hidden="true" tabindex="-1"></a>        buildResult x <span class="ot">=</span> <span class="fu">return</span> ((key,val) <span class="op">:</span> delete (key,x) assocs)</span></code></pre></div>
-</section>
-<section id="the-return-of-do" data-number="5.6">
-<h2 data-number="13.6"><span class="header-section-number">13.6</span> The Return of <code>do</code></h2>
-<p>Here’s an example of what a complex monad operation might look like.</p>
-<div class="sourceCode" id="cb245"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb245-1"><a href="#cb245-1" aria-hidden="true" tabindex="-1"></a>f <span class="ot">=</span> op1 <span class="op">&gt;&gt;=</span> continue</span>
+<h2 data-number="13.6" id="the-return-of-do"><span
+class="header-section-number">13.6</span> The Return of
+<code>do</code></h2>
+<p>Here’s an example of what a complex monad operation might look
+like.</p>
+<div class="sourceCode" id="cb245"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb245-1"><a href="#cb245-1" aria-hidden="true" tabindex="-1"></a>f <span class="ot">=</span> op1 <span class="op">&gt;&gt;=</span> continue</span>
 <span id="cb245-2"><a href="#cb245-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> continue  x   <span class="ot">=</span> op2 <span class="op">&gt;&gt;</span> op3 <span class="op">&gt;&gt;=</span> continue2 x</span>
 <span id="cb245-3"><a href="#cb245-3" aria-hidden="true" tabindex="-1"></a>        continue2 x y <span class="ot">=</span> op4 <span class="op">&gt;&gt;</span> op5 x y</span></code></pre></div>
-<p>Let’s see what happens when we transform this code a bit. First off, let’s inline the definitions.</p>
-<div class="sourceCode" id="cb246"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb246-1"><a href="#cb246-1" aria-hidden="true" tabindex="-1"></a>f <span class="ot">=</span> op1 <span class="op">&gt;&gt;=</span> (\x <span class="ot">-&gt;</span></span>
+<p>Let’s see what happens when we transform this code a bit. First off,
+let’s inline the definitions.</p>
+<div class="sourceCode" id="cb246"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb246-1"><a href="#cb246-1" aria-hidden="true" tabindex="-1"></a>f <span class="ot">=</span> op1 <span class="op">&gt;&gt;=</span> (\x <span class="ot">-&gt;</span></span>
 <span id="cb246-2"><a href="#cb246-2" aria-hidden="true" tabindex="-1"></a>               op2 <span class="op">&gt;&gt;</span></span>
 <span id="cb246-3"><a href="#cb246-3" aria-hidden="true" tabindex="-1"></a>               op3 <span class="op">&gt;&gt;=</span> (\y <span class="ot">-&gt;</span></span>
 <span id="cb246-4"><a href="#cb246-4" aria-hidden="true" tabindex="-1"></a>                          op4 <span class="op">&gt;&gt;</span></span>
 <span id="cb246-5"><a href="#cb246-5" aria-hidden="true" tabindex="-1"></a>                          op5 x y))</span></code></pre></div>
-<p>Due to lambda expressions continuing to the end of the expression, we can omit the parentheses. Let’s also indent differently.</p>
-<div class="sourceCode" id="cb247"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb247-1"><a href="#cb247-1" aria-hidden="true" tabindex="-1"></a>f <span class="ot">=</span> op1 <span class="op">&gt;&gt;=</span> \x <span class="ot">-&gt;</span></span>
+<p>Due to lambda expressions continuing to the end of the expression, we
+can omit the parentheses. Let’s also indent differently.</p>
+<div class="sourceCode" id="cb247"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb247-1"><a href="#cb247-1" aria-hidden="true" tabindex="-1"></a>f <span class="ot">=</span> op1 <span class="op">&gt;&gt;=</span> \x <span class="ot">-&gt;</span></span>
 <span id="cb247-2"><a href="#cb247-2" aria-hidden="true" tabindex="-1"></a>    op2 <span class="op">&gt;&gt;</span></span>
 <span id="cb247-3"><a href="#cb247-3" aria-hidden="true" tabindex="-1"></a>    op3 <span class="op">&gt;&gt;=</span> \y <span class="ot">-&gt;</span></span>
 <span id="cb247-4"><a href="#cb247-4" aria-hidden="true" tabindex="-1"></a>    op4 <span class="op">&gt;&gt;</span></span>
 <span id="cb247-5"><a href="#cb247-5" aria-hidden="true" tabindex="-1"></a>    op5 x y</span></code></pre></div>
-<p>Now we can notice the similarity with <code>do</code> notation. The <code>do</code> block below is actually the same code!</p>
-<div class="sourceCode" id="cb248"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb248-1"><a href="#cb248-1" aria-hidden="true" tabindex="-1"></a>f <span class="ot">=</span> <span class="kw">do</span> x <span class="ot">&lt;-</span> op1</span>
+<p>Now we can notice the similarity with <code>do</code> notation. The
+<code>do</code> block below is actually the same code!</p>
+<div class="sourceCode" id="cb248"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb248-1"><a href="#cb248-1" aria-hidden="true" tabindex="-1"></a>f <span class="ot">=</span> <span class="kw">do</span> x <span class="ot">&lt;-</span> op1</span>
 <span id="cb248-2"><a href="#cb248-2" aria-hidden="true" tabindex="-1"></a>       op2</span>
 <span id="cb248-3"><a href="#cb248-3" aria-hidden="true" tabindex="-1"></a>       y <span class="ot">&lt;-</span> op3</span>
 <span id="cb248-4"><a href="#cb248-4" aria-hidden="true" tabindex="-1"></a>       op4</span>
 <span id="cb248-5"><a href="#cb248-5" aria-hidden="true" tabindex="-1"></a>       op5 x y</span></code></pre></div>
-<p>To clarify, <code>do</code> notation is just a nicer syntax for the monad operations (<code>&gt;&gt;=</code> and <code>&gt;&gt;</code>) and lambdas. Here’s how do notation gets transformed into monad operations. Note! the definition is recursive.</p>
-<div class="sourceCode" id="cb249"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb249-1"><a href="#cb249-1" aria-hidden="true" tabindex="-1"></a><span class="kw">do</span> x <span class="ot">&lt;-</span> op a       <span class="op">~~~&gt;</span>       op a <span class="op">&gt;&gt;=</span> \x <span class="ot">-&gt;</span> <span class="kw">do</span> <span class="op">...</span></span>
+<p>To clarify, <code>do</code> notation is just a nicer syntax for the
+monad operations (<code>&gt;&gt;=</code> and <code>&gt;&gt;</code>) and
+lambdas. Here’s how do notation gets transformed into monad operations.
+Note! the definition is recursive.</p>
+<div class="sourceCode" id="cb249"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb249-1"><a href="#cb249-1" aria-hidden="true" tabindex="-1"></a><span class="kw">do</span> x <span class="ot">&lt;-</span> op a       <span class="op">~~~&gt;</span>       op a <span class="op">&gt;&gt;=</span> \x <span class="ot">-&gt;</span> <span class="kw">do</span> <span class="op">...</span></span>
 <span id="cb249-2"><a href="#cb249-2" aria-hidden="true" tabindex="-1"></a>   <span class="op">...</span></span></code></pre></div>
-<div class="sourceCode" id="cb250"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb250-1"><a href="#cb250-1" aria-hidden="true" tabindex="-1"></a><span class="kw">do</span> op a            <span class="op">~~~&gt;</span>       op a <span class="op">&gt;&gt;</span> <span class="kw">do</span> <span class="op">...</span></span>
+<div class="sourceCode" id="cb250"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb250-1"><a href="#cb250-1" aria-hidden="true" tabindex="-1"></a><span class="kw">do</span> op a            <span class="op">~~~&gt;</span>       op a <span class="op">&gt;&gt;</span> <span class="kw">do</span> <span class="op">...</span></span>
 <span id="cb250-2"><a href="#cb250-2" aria-hidden="true" tabindex="-1"></a>   <span class="op">...</span></span></code></pre></div>
-<div class="sourceCode" id="cb251"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb251-1"><a href="#cb251-1" aria-hidden="true" tabindex="-1"></a><span class="kw">do</span> <span class="kw">let</span> x <span class="ot">=</span> expr    <span class="op">~~~&gt;</span>       <span class="kw">let</span> x <span class="ot">=</span> expr <span class="kw">in</span> <span class="kw">do</span> <span class="op">...</span></span>
+<div class="sourceCode" id="cb251"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb251-1"><a href="#cb251-1" aria-hidden="true" tabindex="-1"></a><span class="kw">do</span> <span class="kw">let</span> x <span class="ot">=</span> expr    <span class="op">~~~&gt;</span>       <span class="kw">let</span> x <span class="ot">=</span> expr <span class="kw">in</span> <span class="kw">do</span> <span class="op">...</span></span>
 <span id="cb251-2"><a href="#cb251-2" aria-hidden="true" tabindex="-1"></a>   <span class="op">...</span></span></code></pre></div>
-<div class="sourceCode" id="cb252"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb252-1"><a href="#cb252-1" aria-hidden="true" tabindex="-1"></a><span class="kw">do</span> finalOp         <span class="op">~~~&gt;</span>       finalOp</span></code></pre></div>
+<div class="sourceCode" id="cb252"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb252-1"><a href="#cb252-1" aria-hidden="true" tabindex="-1"></a><span class="kw">do</span> finalOp         <span class="op">~~~&gt;</span>       finalOp</span></code></pre></div>
 <p>Here’s <code>safeNth</code> using do notation:</p>
-<div class="sourceCode" id="cb253"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb253-1"><a href="#cb253-1" aria-hidden="true" tabindex="-1"></a><span class="ot">safeHead ::</span> [a] <span class="ot">-&gt;</span> <span class="dt">Maybe</span> a</span>
+<div class="sourceCode" id="cb253"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb253-1"><a href="#cb253-1" aria-hidden="true" tabindex="-1"></a><span class="ot">safeHead ::</span> [a] <span class="ot">-&gt;</span> <span class="dt">Maybe</span> a</span>
 <span id="cb253-2"><a href="#cb253-2" aria-hidden="true" tabindex="-1"></a>safeHead [] <span class="ot">=</span> <span class="dt">Nothing</span></span>
 <span id="cb253-3"><a href="#cb253-3" aria-hidden="true" tabindex="-1"></a>safeHead (x<span class="op">:</span>xs) <span class="ot">=</span> <span class="dt">Just</span> x</span>
 <span id="cb253-4"><a href="#cb253-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -3067,7 +4226,8 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb253-11"><a href="#cb253-11" aria-hidden="true" tabindex="-1"></a>safeNth n xs <span class="ot">=</span> <span class="kw">do</span> t <span class="ot">&lt;-</span> safeTail xs</span>
 <span id="cb253-12"><a href="#cb253-12" aria-hidden="true" tabindex="-1"></a>                  safeNth (n<span class="op">-</span><span class="dv">1</span>) t</span></code></pre></div>
 <p>Here is <code>increase</code> one last time, now with do notation</p>
-<div class="sourceCode" id="cb254"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb254-1"><a href="#cb254-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Set the value of key to val in the given key-value list,</span></span>
+<div class="sourceCode" id="cb254"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb254-1"><a href="#cb254-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Set the value of key to val in the given key-value list,</span></span>
 <span id="cb254-2"><a href="#cb254-2" aria-hidden="true" tabindex="-1"></a><span class="co">-- but only if val is larger than the current value!</span></span>
 <span id="cb254-3"><a href="#cb254-3" aria-hidden="true" tabindex="-1"></a><span class="ot">increase ::</span> <span class="dt">Eq</span> a <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> [(a,<span class="dt">Int</span>)] <span class="ot">-&gt;</span> <span class="dt">Maybe</span> [(a,<span class="dt">Int</span>)]</span>
 <span id="cb254-4"><a href="#cb254-4" aria-hidden="true" tabindex="-1"></a>increase key val assocs <span class="ot">=</span></span>
@@ -3077,12 +4237,21 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb254-8"><a href="#cb254-8" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> check x</span>
 <span id="cb254-9"><a href="#cb254-9" aria-hidden="true" tabindex="-1"></a>           <span class="op">|</span> val <span class="op">&lt;</span> x   <span class="ot">=</span> <span class="dt">Nothing</span></span>
 <span id="cb254-10"><a href="#cb254-10" aria-hidden="true" tabindex="-1"></a>           <span class="op">|</span> <span class="fu">otherwise</span> <span class="ot">=</span> <span class="fu">return</span> x</span></code></pre></div>
-</section>
-<section id="logger-is-a-monad" data-number="5.7">
-<h2 data-number="13.7"><span class="header-section-number">13.7</span> Logger is a Monad!</h2>
-<p>We should be able to write a <code>Monad</code> instance for <code>Logger</code> ourselves, by setting <code>&gt;&gt;=</code> to <code>#&gt;</code>. However, due to <a href="https://wiki.haskell.org/Functor-Applicative-Monad_Proposal">recent changes in the Haskell language</a> we must implement <code>Functor</code> and <code>Applicative</code> instances to be allowed to implement the <code>Monad</code> instance. <code>Functor</code> we’ve already met, but what’s <code>Applicative</code>? We’ll find out later. Let’s implement the instances:</p>
+<h2 data-number="13.7" id="logger-is-a-monad"><span
+class="header-section-number">13.7</span> Logger is a Monad!</h2>
+<p>We should be able to write a <code>Monad</code> instance for
+<code>Logger</code> ourselves, by setting <code>&gt;&gt;=</code> to
+<code>#&gt;</code>. However, due to <a
+href="https://wiki.haskell.org/Functor-Applicative-Monad_Proposal">recent
+changes in the Haskell language</a> we must implement
+<code>Functor</code> and <code>Applicative</code> instances to be
+allowed to implement the <code>Monad</code> instance.
+<code>Functor</code> we’ve already met, but what’s
+<code>Applicative</code>? We’ll find out later. Let’s implement the
+instances:</p>
 <!-- TODO should this be in an Examples file as well? -->
-<div class="sourceCode" id="cb255"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb255-1"><a href="#cb255-1" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="dt">Control.Monad</span></span>
+<div class="sourceCode" id="cb255"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb255-1"><a href="#cb255-1" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="dt">Control.Monad</span></span>
 <span id="cb255-2"><a href="#cb255-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb255-3"><a href="#cb255-3" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Logger</span> a <span class="ot">=</span> <span class="dt">Logger</span> [<span class="dt">String</span>] a  <span class="kw">deriving</span> <span class="dt">Show</span></span>
 <span id="cb255-4"><a href="#cb255-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -3105,14 +4274,20 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb255-21"><a href="#cb255-21" aria-hidden="true" tabindex="-1"></a>  <span class="fu">return</span> x <span class="ot">=</span> <span class="dt">Logger</span> [] x</span>
 <span id="cb255-22"><a href="#cb255-22" aria-hidden="true" tabindex="-1"></a>  <span class="dt">Logger</span> la a <span class="op">&gt;&gt;=</span> f <span class="ot">=</span> <span class="dt">Logger</span> (la<span class="op">++</span>lb) b</span>
 <span id="cb255-23"><a href="#cb255-23" aria-hidden="true" tabindex="-1"></a>    <span class="kw">where</span> <span class="dt">Logger</span> lb b <span class="ot">=</span> f a</span></code></pre></div>
-<p>We don’t need the <code>nomsg</code> operation any more since it’s just <code>return</code>. We can also reimplement the <code>annotate</code> operation using monad operations.</p>
-<div class="sourceCode" id="cb256"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb256-1"><a href="#cb256-1" aria-hidden="true" tabindex="-1"></a><span class="ot">nomsg ::</span> a <span class="ot">-&gt;</span> <span class="dt">Logger</span> a</span>
+<p>We don’t need the <code>nomsg</code> operation any more since it’s
+just <code>return</code>. We can also reimplement the
+<code>annotate</code> operation using monad operations.</p>
+<div class="sourceCode" id="cb256"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb256-1"><a href="#cb256-1" aria-hidden="true" tabindex="-1"></a><span class="ot">nomsg ::</span> a <span class="ot">-&gt;</span> <span class="dt">Logger</span> a</span>
 <span id="cb256-2"><a href="#cb256-2" aria-hidden="true" tabindex="-1"></a>nomsg x <span class="ot">=</span> <span class="fu">return</span> x</span>
 <span id="cb256-3"><a href="#cb256-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb256-4"><a href="#cb256-4" aria-hidden="true" tabindex="-1"></a><span class="ot">annotate ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Logger</span> a</span>
 <span id="cb256-5"><a href="#cb256-5" aria-hidden="true" tabindex="-1"></a>annotate s x <span class="ot">=</span> msg s <span class="op">&gt;&gt;</span> <span class="fu">return</span> x</span></code></pre></div>
-<p>Here are the <code>compute</code> and <code>filterLog</code> examples rewritten using do-notation. Note how nice <code>filterLog</code> is with do-notation.</p>
-<div class="sourceCode" id="cb257"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb257-1"><a href="#cb257-1" aria-hidden="true" tabindex="-1"></a>compute x <span class="ot">=</span> <span class="kw">do</span></span>
+<p>Here are the <code>compute</code> and <code>filterLog</code> examples
+rewritten using do-notation. Note how nice <code>filterLog</code> is
+with do-notation.</p>
+<div class="sourceCode" id="cb257"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb257-1"><a href="#cb257-1" aria-hidden="true" tabindex="-1"></a>compute x <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb257-2"><a href="#cb257-2" aria-hidden="true" tabindex="-1"></a>  a <span class="ot">&lt;-</span> annotate <span class="st">&quot;^2&quot;</span> (x<span class="op">*</span>x)</span>
 <span id="cb257-3"><a href="#cb257-3" aria-hidden="true" tabindex="-1"></a>  b <span class="ot">&lt;-</span> annotate <span class="st">&quot;+1&quot;</span> (a<span class="op">+</span><span class="dv">1</span>)</span>
 <span id="cb257-4"><a href="#cb257-4" aria-hidden="true" tabindex="-1"></a>  annotate <span class="st">&quot;*2&quot;</span> (b<span class="op">*</span><span class="dv">2</span>)</span>
@@ -3125,15 +4300,25 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb257-11"><a href="#cb257-11" aria-hidden="true" tabindex="-1"></a>                    <span class="fu">return</span> (x<span class="op">:</span>xs&#39;)</span>
 <span id="cb257-12"><a href="#cb257-12" aria-hidden="true" tabindex="-1"></a>   <span class="op">|</span> <span class="fu">otherwise</span> <span class="ot">=</span> <span class="kw">do</span> msg (<span class="st">&quot;dropping &quot;</span><span class="op">++</span><span class="fu">show</span> x)</span>
 <span id="cb257-13"><a href="#cb257-13" aria-hidden="true" tabindex="-1"></a>                    filterLog f xs</span></code></pre></div>
-<div class="sourceCode" id="cb258"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb258-1"><a href="#cb258-1" aria-hidden="true" tabindex="-1"></a>compute <span class="dv">3</span></span>
+<div class="sourceCode" id="cb258"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb258-1"><a href="#cb258-1" aria-hidden="true" tabindex="-1"></a>compute <span class="dv">3</span></span>
 <span id="cb258-2"><a href="#cb258-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Logger</span> [<span class="st">&quot;^2&quot;</span>,<span class="st">&quot;+1&quot;</span>,<span class="st">&quot;*2&quot;</span>] <span class="dv">20</span></span>
 <span id="cb258-3"><a href="#cb258-3" aria-hidden="true" tabindex="-1"></a>filterLog (<span class="op">&gt;</span><span class="dv">0</span>) [<span class="dv">1</span>,<span class="op">-</span><span class="dv">2</span>,<span class="dv">3</span>,<span class="op">-</span><span class="dv">4</span>,<span class="dv">0</span>]</span>
 <span id="cb258-4"><a href="#cb258-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Logger</span> [<span class="st">&quot;keeping 1&quot;</span>,<span class="st">&quot;dropping -2&quot;</span>,<span class="st">&quot;keeping 3&quot;</span>,<span class="st">&quot;dropping -4&quot;</span>,<span class="st">&quot;dropping 0&quot;</span>] [<span class="dv">1</span>,<span class="dv">3</span>]</span></code></pre></div>
-</section>
-<section id="the-state-monad" data-number="5.8">
-<h2 data-number="13.8"><span class="header-section-number">13.8</span> The State Monad</h2>
-<p>Haskell’s <code>State</code> monad is a generalized version of our <code>BankOp</code> type. The <code>State</code> type is parameterized by two types, the first being the type of the state, and the second the type of the value produced. <code>State Bank a</code> would be equivalent to our <code>BankOp a</code>. You can find the <code>State</code> monad in the <a href="https://downloads.haskell.org/~ghc/latest/docs/html/libraries/transformers-0.5.6.2/Control-Monad-Trans-State.html">module <code>Control.Monad.Trans.State</code> of the <code>transformers</code> package</a>. Here’s a simplified implementation of <code>State</code>.</p>
-<div class="sourceCode" id="cb259"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb259-1"><a href="#cb259-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">State</span> s a <span class="ot">=</span> <span class="dt">State</span> (s <span class="ot">-&gt;</span> (a,s))</span>
+<h2 data-number="13.8" id="the-state-monad"><span
+class="header-section-number">13.8</span> The State Monad</h2>
+<p>Haskell’s <code>State</code> monad is a generalized version of our
+<code>BankOp</code> type. The <code>State</code> type is parameterized
+by two types, the first being the type of the state, and the second the
+type of the value produced. <code>State Bank a</code> would be
+equivalent to our <code>BankOp a</code>. You can find the
+<code>State</code> monad in the <a
+href="https://downloads.haskell.org/~ghc/latest/docs/html/libraries/transformers-0.5.6.2/Control-Monad-Trans-State.html">module
+<code>Control.Monad.Trans.State</code> of the <code>transformers</code>
+package</a>. Here’s a simplified implementation of
+<code>State</code>.</p>
+<div class="sourceCode" id="cb259"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb259-1"><a href="#cb259-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">State</span> s a <span class="ot">=</span> <span class="dt">State</span> (s <span class="ot">-&gt;</span> (a,s))</span>
 <span id="cb259-2"><a href="#cb259-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb259-3"><a href="#cb259-3" aria-hidden="true" tabindex="-1"></a>runState (<span class="dt">State</span> f) s <span class="ot">=</span> f s</span>
 <span id="cb259-4"><a href="#cb259-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -3158,8 +4343,14 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb259-23"><a href="#cb259-23" aria-hidden="true" tabindex="-1"></a>    <span class="kw">where</span> h state0 <span class="ot">=</span> <span class="kw">let</span> (val,state1) <span class="ot">=</span> runState op state0</span>
 <span id="cb259-24"><a href="#cb259-24" aria-hidden="true" tabindex="-1"></a>                         op2 <span class="ot">=</span> f val</span>
 <span id="cb259-25"><a href="#cb259-25" aria-hidden="true" tabindex="-1"></a>                     <span class="kw">in</span> runState op2 state1</span></code></pre></div>
-<p>Note how we declare an instance <code>Monad (State s)</code>. We’re using a <em>partially-applied type constructor</em> because instances of <code>Monad</code> can only be declared for type constructors that take one more type parameter. This might be a bit clearer if you look at how <code>m</code>, <code>Maybe</code> and <code>State</code> occur in the type of <code>&gt;&gt;=</code> below.</p>
-<div class="sourceCode" id="cb260"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb260-1"><a href="#cb260-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Monad</span> m <span class="kw">where</span></span>
+<p>Note how we declare an instance <code>Monad (State s)</code>. We’re
+using a <em>partially-applied type constructor</em> because instances of
+<code>Monad</code> can only be declared for type constructors that take
+one more type parameter. This might be a bit clearer if you look at how
+<code>m</code>, <code>Maybe</code> and <code>State</code> occur in the
+type of <code>&gt;&gt;=</code> below.</p>
+<div class="sourceCode" id="cb260"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb260-1"><a href="#cb260-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Monad</span> m <span class="kw">where</span></span>
 <span id="cb260-2"><a href="#cb260-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  (&gt;&gt;=) ::</span> m a <span class="ot">-&gt;</span> (a <span class="ot">-&gt;</span> m b) <span class="ot">-&gt;</span> m b</span>
 <span id="cb260-3"><a href="#cb260-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb260-4"><a href="#cb260-4" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Monad</span> <span class="dt">Maybe</span> <span class="kw">where</span></span>
@@ -3167,30 +4358,45 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb260-6"><a href="#cb260-6" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb260-7"><a href="#cb260-7" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Monad</span> (<span class="dt">State</span> s) <span class="kw">where</span></span>
 <span id="cb260-8"><a href="#cb260-8" aria-hidden="true" tabindex="-1"></a><span class="ot">  (&gt;&gt;=) ::</span> <span class="dt">State</span> s a <span class="ot">-&gt;</span> (a <span class="ot">-&gt;</span> <span class="dt">State</span> s b) <span class="ot">-&gt;</span> <span class="dt">State</span> s b</span></code></pre></div>
-<p>Let’s look at some examples of working with <code>State</code>. To start off, let’s consider computations of type <code>State Int a</code>, which represent working with a simple counter.</p>
-<div class="sourceCode" id="cb261"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb261-1"><a href="#cb261-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- adds i to the value of the counter</span></span>
+<p>Let’s look at some examples of working with <code>State</code>. To
+start off, let’s consider computations of type <code>State Int a</code>,
+which represent working with a simple counter.</p>
+<div class="sourceCode" id="cb261"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb261-1"><a href="#cb261-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- adds i to the value of the counter</span></span>
 <span id="cb261-2"><a href="#cb261-2" aria-hidden="true" tabindex="-1"></a><span class="ot">add ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">State</span> <span class="dt">Int</span> ()</span>
 <span id="cb261-3"><a href="#cb261-3" aria-hidden="true" tabindex="-1"></a>add i <span class="ot">=</span> <span class="kw">do</span> old <span class="ot">&lt;-</span> get</span>
 <span id="cb261-4"><a href="#cb261-4" aria-hidden="true" tabindex="-1"></a>           put (old<span class="op">+</span>i)</span></code></pre></div>
-<div class="sourceCode" id="cb262"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb262-1"><a href="#cb262-1" aria-hidden="true" tabindex="-1"></a>runState (add <span class="dv">1</span> <span class="op">&gt;&gt;</span> add <span class="dv">3</span> <span class="op">&gt;&gt;</span> add <span class="dv">5</span> <span class="op">&gt;&gt;</span> add <span class="dv">6</span>) <span class="dv">0</span></span>
+<div class="sourceCode" id="cb262"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb262-1"><a href="#cb262-1" aria-hidden="true" tabindex="-1"></a>runState (add <span class="dv">1</span> <span class="op">&gt;&gt;</span> add <span class="dv">3</span> <span class="op">&gt;&gt;</span> add <span class="dv">5</span> <span class="op">&gt;&gt;</span> add <span class="dv">6</span>) <span class="dv">0</span></span>
 <span id="cb262-2"><a href="#cb262-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> ((),<span class="dv">15</span>)</span></code></pre></div>
-<div class="sourceCode" id="cb263"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb263-1"><a href="#cb263-1" aria-hidden="true" tabindex="-1"></a><span class="ot">example ::</span> <span class="dt">State</span> <span class="dt">Int</span> <span class="dt">Int</span></span>
+<div class="sourceCode" id="cb263"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb263-1"><a href="#cb263-1" aria-hidden="true" tabindex="-1"></a><span class="ot">example ::</span> <span class="dt">State</span> <span class="dt">Int</span> <span class="dt">Int</span></span>
 <span id="cb263-2"><a href="#cb263-2" aria-hidden="true" tabindex="-1"></a>example <span class="ot">=</span> <span class="kw">do</span> add <span class="dv">3</span>           <span class="co">-- increment state by 3</span></span>
 <span id="cb263-3"><a href="#cb263-3" aria-hidden="true" tabindex="-1"></a>             value <span class="ot">&lt;-</span> get    <span class="co">-- value is current state, i.e. initial+3</span></span>
 <span id="cb263-4"><a href="#cb263-4" aria-hidden="true" tabindex="-1"></a>             add <span class="dv">1000</span>        <span class="co">-- increment state by 1000</span></span>
 <span id="cb263-5"><a href="#cb263-5" aria-hidden="true" tabindex="-1"></a>             put (value <span class="op">+</span> <span class="dv">1</span>) <span class="co">-- overwrite state with value+1, i.e. initial+4</span></span>
 <span id="cb263-6"><a href="#cb263-6" aria-hidden="true" tabindex="-1"></a>             <span class="fu">return</span> value    <span class="co">-- produce value, i.e. intial+3</span></span></code></pre></div>
-<div class="sourceCode" id="cb264"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb264-1"><a href="#cb264-1" aria-hidden="true" tabindex="-1"></a>runState example <span class="dv">1</span></span>
+<div class="sourceCode" id="cb264"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb264-1"><a href="#cb264-1" aria-hidden="true" tabindex="-1"></a>runState example <span class="dv">1</span></span>
 <span id="cb264-2"><a href="#cb264-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> (<span class="dv">4</span>,<span class="dv">5</span>)           <span class="co">-- initial is 1, state is initial+4=5, produces initial+3=4</span></span></code></pre></div>
-<p>Note how a value of type <code>State s a</code> represents a process that transforms the state (just like <code>BankOp</code>). The initial state must be supplied using <code>runState</code>. Again, having to use <code>runState</code> makes the distinction between <em>defining</em> operations and <em>executing them</em> clearer.</p>
-<p>A state can replace an accumulator parameter when processing a list. Here are two examples: finding the largest element of a list, and finding values in a list that occur directly after a <code>0</code>.</p>
-<div class="sourceCode" id="cb265"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb265-1"><a href="#cb265-1" aria-hidden="true" tabindex="-1"></a><span class="ot">findLargest ::</span> <span class="dt">Ord</span> a <span class="ot">=&gt;</span> [a] <span class="ot">-&gt;</span> <span class="dt">State</span> a ()</span>
+<p>Note how a value of type <code>State s a</code> represents a process
+that transforms the state (just like <code>BankOp</code>). The initial
+state must be supplied using <code>runState</code>. Again, having to use
+<code>runState</code> makes the distinction between <em>defining</em>
+operations and <em>executing them</em> clearer.</p>
+<p>A state can replace an accumulator parameter when processing a list.
+Here are two examples: finding the largest element of a list, and
+finding values in a list that occur directly after a <code>0</code>.</p>
+<div class="sourceCode" id="cb265"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb265-1"><a href="#cb265-1" aria-hidden="true" tabindex="-1"></a><span class="ot">findLargest ::</span> <span class="dt">Ord</span> a <span class="ot">=&gt;</span> [a] <span class="ot">-&gt;</span> <span class="dt">State</span> a ()</span>
 <span id="cb265-2"><a href="#cb265-2" aria-hidden="true" tabindex="-1"></a>findLargest [] <span class="ot">=</span> <span class="fu">return</span> ()</span>
 <span id="cb265-3"><a href="#cb265-3" aria-hidden="true" tabindex="-1"></a>findLargest (x<span class="op">:</span>xs) <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb265-4"><a href="#cb265-4" aria-hidden="true" tabindex="-1"></a>  modify (\y <span class="ot">-&gt;</span> <span class="fu">max</span> x y)  <span class="co">-- update state with max of current value and previous largest value</span></span>
 <span id="cb265-5"><a href="#cb265-5" aria-hidden="true" tabindex="-1"></a>  findLargest xs          <span class="co">-- process rest of list</span></span></code></pre></div>
-<div class="sourceCode" id="cb266"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb266-1"><a href="#cb266-1" aria-hidden="true" tabindex="-1"></a>runState (findLargest [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">7</span>,<span class="dv">3</span>]) <span class="dv">0</span>  <span class="op">==&gt;</span>  ((),<span class="dv">7</span>)</span></code></pre></div>
-<div class="sourceCode" id="cb267"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb267-1"><a href="#cb267-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- store the given value in the state list</span></span>
+<div class="sourceCode" id="cb266"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb266-1"><a href="#cb266-1" aria-hidden="true" tabindex="-1"></a>runState (findLargest [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">7</span>,<span class="dv">3</span>]) <span class="dv">0</span>  <span class="op">==&gt;</span>  ((),<span class="dv">7</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb267"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb267-1"><a href="#cb267-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- store the given value in the state list</span></span>
 <span id="cb267-2"><a href="#cb267-2" aria-hidden="true" tabindex="-1"></a><span class="ot">remember ::</span> a <span class="ot">-&gt;</span> <span class="dt">State</span> [a] ()</span>
 <span id="cb267-3"><a href="#cb267-3" aria-hidden="true" tabindex="-1"></a>remember x <span class="ot">=</span> modify (x<span class="op">:</span>)</span>
 <span id="cb267-4"><a href="#cb267-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -3201,15 +4407,21 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb267-9"><a href="#cb267-9" aria-hidden="true" tabindex="-1"></a>                         go (y<span class="op">:</span>xs)</span>
 <span id="cb267-10"><a href="#cb267-10" aria-hidden="true" tabindex="-1"></a>        go (x<span class="op">:</span>xs) <span class="ot">=</span> go xs</span>
 <span id="cb267-11"><a href="#cb267-11" aria-hidden="true" tabindex="-1"></a>        go [] <span class="ot">=</span> <span class="fu">return</span> ()</span></code></pre></div>
-<div class="sourceCode" id="cb268"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb268-1"><a href="#cb268-1" aria-hidden="true" tabindex="-1"></a>valuesAfterZero [<span class="dv">0</span>,<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">0</span>,<span class="dv">4</span>,<span class="dv">0</span>,<span class="dv">5</span>,<span class="dv">0</span>,<span class="dv">0</span>,<span class="dv">6</span>]</span>
+<div class="sourceCode" id="cb268"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb268-1"><a href="#cb268-1" aria-hidden="true" tabindex="-1"></a>valuesAfterZero [<span class="dv">0</span>,<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">0</span>,<span class="dv">4</span>,<span class="dv">0</span>,<span class="dv">5</span>,<span class="dv">0</span>,<span class="dv">0</span>,<span class="dv">6</span>]</span>
 <span id="cb268-2"><a href="#cb268-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> ((),[<span class="dv">6</span>,<span class="dv">0</span>,<span class="dv">5</span>,<span class="dv">4</span>,<span class="dv">1</span>])</span></code></pre></div>
-<p><strong>Note!</strong> By the way, the actual implementation of the <code>State</code> monad doesn’t have a <code>State</code> constructor like our simplified exmaple. If you want to wrap a function into a <code>State</code> operation, use this helper instead:</p>
+<p><strong>Note!</strong> By the way, the actual implementation of the
+<code>State</code> monad doesn’t have a <code>State</code> constructor
+like our simplified exmaple. If you want to wrap a function into a
+<code>State</code> operation, use this helper instead:</p>
 <pre><code>state :: (s -&gt; (a, s)) -&gt; State s a</code></pre>
-</section>
-<section id="the-return-of-mapm" data-number="5.9">
-<h2 data-number="13.9"><span class="header-section-number">13.9</span> The Return of <code>mapM</code></h2>
-<p>The control structures from the IO lecture work in <em>all monads</em>. Here are their real types.</p>
-<div class="sourceCode" id="cb270"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb270-1"><a href="#cb270-1" aria-hidden="true" tabindex="-1"></a><span class="ot">when ::</span> <span class="dt">Monad</span> m <span class="ot">=&gt;</span> <span class="dt">Bool</span> <span class="ot">-&gt;</span> m () <span class="ot">-&gt;</span> m ()        <span class="co">-- conditional operation</span></span>
+<h2 data-number="13.9" id="the-return-of-mapm"><span
+class="header-section-number">13.9</span> The Return of
+<code>mapM</code></h2>
+<p>The control structures from the IO lecture work in <em>all
+monads</em>. Here are their real types.</p>
+<div class="sourceCode" id="cb270"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb270-1"><a href="#cb270-1" aria-hidden="true" tabindex="-1"></a><span class="ot">when ::</span> <span class="dt">Monad</span> m <span class="ot">=&gt;</span> <span class="dt">Bool</span> <span class="ot">-&gt;</span> m () <span class="ot">-&gt;</span> m ()        <span class="co">-- conditional operation</span></span>
 <span id="cb270-2"><a href="#cb270-2" aria-hidden="true" tabindex="-1"></a><span class="ot">unless ::</span> <span class="dt">Monad</span> m <span class="ot">=&gt;</span> <span class="dt">Bool</span> <span class="ot">-&gt;</span> m () <span class="ot">-&gt;</span> m ()      <span class="co">-- same, but condition is flipped</span></span>
 <span id="cb270-3"><a href="#cb270-3" aria-hidden="true" tabindex="-1"></a><span class="ot">replicateM ::</span> <span class="dt">Monad</span> m <span class="ot">=&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> m a <span class="ot">-&gt;</span> m [a]   <span class="co">-- do something many times</span></span>
 <span id="cb270-4"><a href="#cb270-4" aria-hidden="true" tabindex="-1"></a><span class="ot">replicateM_ ::</span> <span class="dt">Monad</span> m <span class="ot">=&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> m a <span class="ot">-&gt;</span> m ()   <span class="co">-- same, but ignore the results</span></span>
@@ -3217,8 +4429,10 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb270-6"><a href="#cb270-6" aria-hidden="true" tabindex="-1"></a><span class="fu">mapM_</span><span class="ot"> ::</span> <span class="dt">Monad</span> m <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> m b) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> m ()  <span class="co">-- same, but ignore the results</span></span>
 <span id="cb270-7"><a href="#cb270-7" aria-hidden="true" tabindex="-1"></a><span class="ot">forM  ::</span> <span class="dt">Monad</span> m <span class="ot">=&gt;</span> [a] <span class="ot">-&gt;</span> (a <span class="ot">-&gt;</span> m b) <span class="ot">-&gt;</span> m [b] <span class="co">-- mapM but arguments reversed</span></span>
 <span id="cb270-8"><a href="#cb270-8" aria-hidden="true" tabindex="-1"></a><span class="ot">forM_ ::</span> <span class="dt">Monad</span> m <span class="ot">=&gt;</span> [a] <span class="ot">-&gt;</span> (a <span class="ot">-&gt;</span> m b) <span class="ot">-&gt;</span> m ()  <span class="co">-- same, but ignore the results</span></span></code></pre></div>
-<p>As we can see here, we can use <code>mapM</code> over all of the monads we’ve met so far:</p>
-<div class="sourceCode" id="cb271"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb271-1"><a href="#cb271-1" aria-hidden="true" tabindex="-1"></a><span class="fu">mapM</span> (\x <span class="ot">-&gt;</span> <span class="kw">if</span> (x<span class="op">&gt;</span><span class="dv">0</span>) <span class="kw">then</span> <span class="dt">Just</span> (x<span class="op">-</span><span class="dv">1</span>) <span class="kw">else</span> <span class="dt">Nothing</span>) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]  <span class="op">==&gt;</span>  <span class="dt">Just</span> [<span class="dv">0</span>,<span class="dv">1</span>,<span class="dv">2</span>]</span>
+<p>As we can see here, we can use <code>mapM</code> over all of the
+monads we’ve met so far:</p>
+<div class="sourceCode" id="cb271"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb271-1"><a href="#cb271-1" aria-hidden="true" tabindex="-1"></a><span class="fu">mapM</span> (\x <span class="ot">-&gt;</span> <span class="kw">if</span> (x<span class="op">&gt;</span><span class="dv">0</span>) <span class="kw">then</span> <span class="dt">Just</span> (x<span class="op">-</span><span class="dv">1</span>) <span class="kw">else</span> <span class="dt">Nothing</span>) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]  <span class="op">==&gt;</span>  <span class="dt">Just</span> [<span class="dv">0</span>,<span class="dv">1</span>,<span class="dv">2</span>]</span>
 <span id="cb271-2"><a href="#cb271-2" aria-hidden="true" tabindex="-1"></a><span class="fu">mapM</span> (\x <span class="ot">-&gt;</span> <span class="kw">if</span> (x<span class="op">&gt;</span><span class="dv">0</span>) <span class="kw">then</span> <span class="dt">Just</span> (x<span class="op">-</span><span class="dv">1</span>) <span class="kw">else</span> <span class="dt">Nothing</span>) [<span class="dv">1</span>,<span class="dv">0</span>,<span class="dv">3</span>]  <span class="op">==&gt;</span>  <span class="dt">Nothing</span></span>
 <span id="cb271-3"><a href="#cb271-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb271-4"><a href="#cb271-4" aria-hidden="true" tabindex="-1"></a><span class="fu">mapM</span> (\x <span class="ot">-&gt;</span> msg <span class="st">&quot;increment&quot;</span> <span class="op">&gt;&gt;</span> msg (<span class="fu">show</span> x) <span class="op">&gt;&gt;</span> <span class="fu">return</span> (x<span class="op">+</span><span class="dv">1</span>)) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
@@ -3227,66 +4441,89 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb271-7"><a href="#cb271-7" aria-hidden="true" tabindex="-1"></a>runState (<span class="fu">mapM</span> (\x <span class="ot">-&gt;</span> modify (x<span class="op">+</span>) <span class="op">&gt;&gt;</span> <span class="fu">return</span> (x<span class="op">+</span><span class="dv">1</span>)) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]) <span class="dv">0</span></span>
 <span id="cb271-8"><a href="#cb271-8" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> ([<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>],<span class="dv">6</span>)</span></code></pre></div>
 <p>Some more examples:</p>
-<div class="sourceCode" id="cb272"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb272-1"><a href="#cb272-1" aria-hidden="true" tabindex="-1"></a><span class="ot">safeHead ::</span> [a] <span class="ot">-&gt;</span> <span class="dt">Maybe</span> a</span>
+<div class="sourceCode" id="cb272"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb272-1"><a href="#cb272-1" aria-hidden="true" tabindex="-1"></a><span class="ot">safeHead ::</span> [a] <span class="ot">-&gt;</span> <span class="dt">Maybe</span> a</span>
 <span id="cb272-2"><a href="#cb272-2" aria-hidden="true" tabindex="-1"></a>safeHead [] <span class="ot">=</span> <span class="dt">Nothing</span></span>
 <span id="cb272-3"><a href="#cb272-3" aria-hidden="true" tabindex="-1"></a>safeHead (x<span class="op">:</span>xs) <span class="ot">=</span> <span class="dt">Just</span> x</span>
 <span id="cb272-4"><a href="#cb272-4" aria-hidden="true" tabindex="-1"></a><span class="ot">firsts ::</span> [[a]] <span class="ot">-&gt;</span> <span class="dt">Maybe</span> [a]</span>
 <span id="cb272-5"><a href="#cb272-5" aria-hidden="true" tabindex="-1"></a>firsts xs <span class="ot">=</span> forM xs safeHead</span></code></pre></div>
-<div class="sourceCode" id="cb273"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb273-1"><a href="#cb273-1" aria-hidden="true" tabindex="-1"></a>firsts [[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>],[<span class="dv">4</span>,<span class="dv">5</span>],[<span class="dv">6</span>]] <span class="op">==&gt;</span> <span class="dt">Just</span> [<span class="dv">1</span>,<span class="dv">4</span>,<span class="dv">6</span>]</span>
+<div class="sourceCode" id="cb273"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb273-1"><a href="#cb273-1" aria-hidden="true" tabindex="-1"></a>firsts [[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>],[<span class="dv">4</span>,<span class="dv">5</span>],[<span class="dv">6</span>]] <span class="op">==&gt;</span> <span class="dt">Just</span> [<span class="dv">1</span>,<span class="dv">4</span>,<span class="dv">6</span>]</span>
 <span id="cb273-2"><a href="#cb273-2" aria-hidden="true" tabindex="-1"></a>firsts [[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>],[],[<span class="dv">6</span>]]    <span class="op">==&gt;</span> <span class="dt">Nothing</span></span></code></pre></div>
-<div class="sourceCode" id="cb274"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb274-1"><a href="#cb274-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- an abbreviated version of an example from the last section</span></span>
+<div class="sourceCode" id="cb274"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb274-1"><a href="#cb274-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- an abbreviated version of an example from the last section</span></span>
 <span id="cb274-2"><a href="#cb274-2" aria-hidden="true" tabindex="-1"></a><span class="ot">findLargest ::</span> <span class="dt">Ord</span> a <span class="ot">=&gt;</span> [a] <span class="ot">-&gt;</span> <span class="dt">State</span> a ()</span>
 <span id="cb274-3"><a href="#cb274-3" aria-hidden="true" tabindex="-1"></a>findLargest xs <span class="ot">=</span> <span class="fu">mapM_</span> update xs</span>
 <span id="cb274-4"><a href="#cb274-4" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> update x <span class="ot">=</span> modify (\y <span class="ot">-&gt;</span> <span class="fu">max</span> x y)</span></code></pre></div>
-<div class="sourceCode" id="cb275"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb275-1"><a href="#cb275-1" aria-hidden="true" tabindex="-1"></a>runState (findLargest [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">7</span>,<span class="dv">3</span>]) <span class="dv">0</span>  <span class="op">==&gt;</span>  ((),<span class="dv">7</span>)</span></code></pre></div>
-<div class="sourceCode" id="cb276"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb276-1"><a href="#cb276-1" aria-hidden="true" tabindex="-1"></a><span class="kw">let</span> increment <span class="ot">=</span> modify (<span class="op">+</span><span class="dv">1</span>) <span class="op">&gt;&gt;</span> get</span>
+<div class="sourceCode" id="cb275"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb275-1"><a href="#cb275-1" aria-hidden="true" tabindex="-1"></a>runState (findLargest [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">7</span>,<span class="dv">3</span>]) <span class="dv">0</span>  <span class="op">==&gt;</span>  ((),<span class="dv">7</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb276"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb276-1"><a href="#cb276-1" aria-hidden="true" tabindex="-1"></a><span class="kw">let</span> increment <span class="ot">=</span> modify (<span class="op">+</span><span class="dv">1</span>) <span class="op">&gt;&gt;</span> get</span>
 <span id="cb276-2"><a href="#cb276-2" aria-hidden="true" tabindex="-1"></a>    ops <span class="ot">=</span> replicateM <span class="dv">4</span> increment</span>
 <span id="cb276-3"><a href="#cb276-3" aria-hidden="true" tabindex="-1"></a><span class="kw">in</span> runState ops <span class="dv">0</span></span>
 <span id="cb276-4"><a href="#cb276-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> ([<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>],<span class="dv">4</span>)</span></code></pre></div>
-<p>Here’s <code>filter</code> reimplemented using the <code>State</code> monad:</p>
-<div class="sourceCode" id="cb277"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb277-1"><a href="#cb277-1" aria-hidden="true" tabindex="-1"></a><span class="ot">rememberElements ::</span> (a <span class="ot">-&gt;</span> <span class="dt">Bool</span>) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> <span class="dt">State</span> [a] ()</span>
+<p>Here’s <code>filter</code> reimplemented using the <code>State</code>
+monad:</p>
+<div class="sourceCode" id="cb277"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb277-1"><a href="#cb277-1" aria-hidden="true" tabindex="-1"></a><span class="ot">rememberElements ::</span> (a <span class="ot">-&gt;</span> <span class="dt">Bool</span>) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> <span class="dt">State</span> [a] ()</span>
 <span id="cb277-2"><a href="#cb277-2" aria-hidden="true" tabindex="-1"></a>rememberElements f xs <span class="ot">=</span> <span class="fu">mapM_</span> maybePut xs</span>
 <span id="cb277-3"><a href="#cb277-3" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> maybePut x <span class="ot">=</span> when (f x) (modify (<span class="op">++</span>[x]))</span>
 <span id="cb277-4"><a href="#cb277-4" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb277-5"><a href="#cb277-5" aria-hidden="true" tabindex="-1"></a><span class="ot">sfilter ::</span> (a <span class="ot">-&gt;</span> <span class="dt">Bool</span>) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> [a]</span>
 <span id="cb277-6"><a href="#cb277-6" aria-hidden="true" tabindex="-1"></a>sfilter f xs <span class="ot">=</span> finalState</span>
 <span id="cb277-7"><a href="#cb277-7" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> (_, finalState) <span class="ot">=</span> runState (rememberElements f xs) []</span></code></pre></div>
-<div class="sourceCode" id="cb278"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb278-1"><a href="#cb278-1" aria-hidden="true" tabindex="-1"></a>sfilter <span class="fu">even</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>,<span class="dv">5</span>]</span>
+<div class="sourceCode" id="cb278"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb278-1"><a href="#cb278-1" aria-hidden="true" tabindex="-1"></a>sfilter <span class="fu">even</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>,<span class="dv">5</span>]</span>
 <span id="cb278-2"><a href="#cb278-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="dv">2</span>,<span class="dv">4</span>]</span></code></pre></div>
-<p>We can write our own operations that work for all monads. This is made possible by type classes, as we’ve seen before. If you only use monad operations like <code>return</code> and do-notation, the type system will infer a generic type for your function.</p>
-<div class="sourceCode" id="cb279"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb279-1"><a href="#cb279-1" aria-hidden="true" tabindex="-1"></a>mywhen b op <span class="ot">=</span> <span class="kw">if</span> b <span class="kw">then</span> op <span class="kw">else</span> <span class="fu">return</span> ()</span>
+<p>We can write our own operations that work for all monads. This is
+made possible by type classes, as we’ve seen before. If you only use
+monad operations like <code>return</code> and do-notation, the type
+system will infer a generic type for your function.</p>
+<div class="sourceCode" id="cb279"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb279-1"><a href="#cb279-1" aria-hidden="true" tabindex="-1"></a>mywhen b op <span class="ot">=</span> <span class="kw">if</span> b <span class="kw">then</span> op <span class="kw">else</span> <span class="fu">return</span> ()</span>
 <span id="cb279-2"><a href="#cb279-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb279-3"><a href="#cb279-3" aria-hidden="true" tabindex="-1"></a>mymapM_ op [] <span class="ot">=</span> <span class="fu">return</span> ()</span>
 <span id="cb279-4"><a href="#cb279-4" aria-hidden="true" tabindex="-1"></a>mymapM_ op (x<span class="op">:</span>xs) <span class="ot">=</span> <span class="kw">do</span> op x</span>
 <span id="cb279-5"><a href="#cb279-5" aria-hidden="true" tabindex="-1"></a>                       mymapM_ op xs</span></code></pre></div>
-<div class="sourceCode" id="cb280"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb280-1"><a href="#cb280-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Main</span><span class="op">&gt;</span> <span class="op">:</span>t mywhen</span>
+<div class="sourceCode" id="cb280"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb280-1"><a href="#cb280-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Main</span><span class="op">&gt;</span> <span class="op">:</span>t mywhen</span>
 <span id="cb280-2"><a href="#cb280-2" aria-hidden="true" tabindex="-1"></a><span class="ot">mywhen ::</span> (<span class="dt">Monad</span> m) <span class="ot">=&gt;</span> <span class="dt">Bool</span> <span class="ot">-&gt;</span> m () <span class="ot">-&gt;</span> m ()</span>
 <span id="cb280-3"><a href="#cb280-3" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Main</span><span class="op">&gt;</span> <span class="op">:</span>t mymapM_</span>
 <span id="cb280-4"><a href="#cb280-4" aria-hidden="true" tabindex="-1"></a><span class="ot">mymapM_ ::</span> (<span class="dt">Monad</span> m) <span class="ot">=&gt;</span> (t <span class="ot">-&gt;</span> m a) <span class="ot">-&gt;</span> [t] <span class="ot">-&gt;</span> m ()</span></code></pre></div>
-<p>We can use these generic operations in each of our example monads:</p>
-<div class="sourceCode" id="cb281"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb281-1"><a href="#cb281-1" aria-hidden="true" tabindex="-1"></a><span class="ot">perhapsDecrease ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">Int</span></span>
+<p>We can use these generic operations in each of our example
+monads:</p>
+<div class="sourceCode" id="cb281"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb281-1"><a href="#cb281-1" aria-hidden="true" tabindex="-1"></a><span class="ot">perhapsDecrease ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">Int</span></span>
 <span id="cb281-2"><a href="#cb281-2" aria-hidden="true" tabindex="-1"></a>perhapsDecrease x <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb281-3"><a href="#cb281-3" aria-hidden="true" tabindex="-1"></a>  mywhen (x<span class="op">&lt;=</span><span class="dv">0</span>) <span class="dt">Nothing</span></span>
 <span id="cb281-4"><a href="#cb281-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">return</span> (x<span class="op">-</span><span class="dv">1</span>)</span></code></pre></div>
-<div class="sourceCode" id="cb282"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb282-1"><a href="#cb282-1" aria-hidden="true" tabindex="-1"></a>perhapsDecrease <span class="dv">2</span>  <span class="op">==&gt;</span>  <span class="dt">Just</span> <span class="dv">1</span></span>
+<div class="sourceCode" id="cb282"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb282-1"><a href="#cb282-1" aria-hidden="true" tabindex="-1"></a>perhapsDecrease <span class="dv">2</span>  <span class="op">==&gt;</span>  <span class="dt">Just</span> <span class="dv">1</span></span>
 <span id="cb282-2"><a href="#cb282-2" aria-hidden="true" tabindex="-1"></a>perhapsDecrease <span class="dv">0</span>  <span class="op">==&gt;</span>  <span class="dt">Nothing</span></span></code></pre></div>
-<div class="sourceCode" id="cb283"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb283-1"><a href="#cb283-1" aria-hidden="true" tabindex="-1"></a><span class="ot">search ::</span> (<span class="dt">Show</span> a, <span class="dt">Eq</span> a) <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> <span class="dt">Logger</span> ()</span>
+<div class="sourceCode" id="cb283"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb283-1"><a href="#cb283-1" aria-hidden="true" tabindex="-1"></a><span class="ot">search ::</span> (<span class="dt">Show</span> a, <span class="dt">Eq</span> a) <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> <span class="dt">Logger</span> ()</span>
 <span id="cb283-2"><a href="#cb283-2" aria-hidden="true" tabindex="-1"></a>search x ys <span class="ot">=</span> mymapM_ look ys</span>
 <span id="cb283-3"><a href="#cb283-3" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> look y <span class="ot">=</span> mywhen (x<span class="op">==</span>y) (msg (<span class="st">&quot;Found &quot;</span><span class="op">++</span><span class="fu">show</span> y))</span></code></pre></div>
-<div class="sourceCode" id="cb284"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb284-1"><a href="#cb284-1" aria-hidden="true" tabindex="-1"></a>search <span class="dv">3</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>,<span class="dv">3</span>,<span class="dv">2</span>]  <span class="op">==&gt;</span>  <span class="dt">Logger</span> [<span class="st">&quot;Found 3&quot;</span>,<span class="st">&quot;Found 3&quot;</span>] ()</span></code></pre></div>
-<div class="sourceCode" id="cb285"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb285-1"><a href="#cb285-1" aria-hidden="true" tabindex="-1"></a><span class="ot">sumPositive ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">State</span> <span class="dt">Int</span> ()</span>
+<div class="sourceCode" id="cb284"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb284-1"><a href="#cb284-1" aria-hidden="true" tabindex="-1"></a>search <span class="dv">3</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>,<span class="dv">3</span>,<span class="dv">2</span>]  <span class="op">==&gt;</span>  <span class="dt">Logger</span> [<span class="st">&quot;Found 3&quot;</span>,<span class="st">&quot;Found 3&quot;</span>] ()</span></code></pre></div>
+<div class="sourceCode" id="cb285"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb285-1"><a href="#cb285-1" aria-hidden="true" tabindex="-1"></a><span class="ot">sumPositive ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">State</span> <span class="dt">Int</span> ()</span>
 <span id="cb285-2"><a href="#cb285-2" aria-hidden="true" tabindex="-1"></a>sumPositive xs <span class="ot">=</span> mymapM_ f xs</span>
 <span id="cb285-3"><a href="#cb285-3" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> f x <span class="ot">=</span> when (x<span class="op">&gt;</span><span class="dv">0</span>) (modify (x<span class="op">+</span>))</span></code></pre></div>
-<div class="sourceCode" id="cb286"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb286-1"><a href="#cb286-1" aria-hidden="true" tabindex="-1"></a>runState (sumPositive [<span class="dv">1</span>,<span class="op">-</span><span class="dv">4</span>,<span class="dv">2</span>,<span class="dv">3</span>]) <span class="dv">0</span>  <span class="op">==&gt;</span>  ((),<span class="dv">6</span>)</span></code></pre></div>
-</section>
-<section id="monads-are-functors" data-number="5.10">
-<h2 data-number="13.10"><span class="header-section-number">13.10</span> Monads are Functors</h2>
-<p>One useful operation hasn’t yet been introduced: <code>liftM</code>.</p>
-<div class="sourceCode" id="cb287"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb287-1"><a href="#cb287-1" aria-hidden="true" tabindex="-1"></a><span class="ot">liftM ::</span> <span class="dt">Monad</span> m <span class="ot">=&gt;</span> (a<span class="ot">-&gt;</span>b) <span class="ot">-&gt;</span> m a <span class="ot">-&gt;</span> m b</span>
+<div class="sourceCode" id="cb286"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb286-1"><a href="#cb286-1" aria-hidden="true" tabindex="-1"></a>runState (sumPositive [<span class="dv">1</span>,<span class="op">-</span><span class="dv">4</span>,<span class="dv">2</span>,<span class="dv">3</span>]) <span class="dv">0</span>  <span class="op">==&gt;</span>  ((),<span class="dv">6</span>)</span></code></pre></div>
+<h2 data-number="13.10" id="monads-are-functors"><span
+class="header-section-number">13.10</span> Monads are Functors</h2>
+<p>One useful operation hasn’t yet been introduced:
+<code>liftM</code>.</p>
+<div class="sourceCode" id="cb287"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb287-1"><a href="#cb287-1" aria-hidden="true" tabindex="-1"></a><span class="ot">liftM ::</span> <span class="dt">Monad</span> m <span class="ot">=&gt;</span> (a<span class="ot">-&gt;</span>b) <span class="ot">-&gt;</span> m a <span class="ot">-&gt;</span> m b</span>
 <span id="cb287-2"><a href="#cb287-2" aria-hidden="true" tabindex="-1"></a>liftM f op <span class="ot">=</span> <span class="kw">do</span> x <span class="ot">&lt;-</span> op</span>
 <span id="cb287-3"><a href="#cb287-3" aria-hidden="true" tabindex="-1"></a>                <span class="fu">return</span> (f x)</span></code></pre></div>
-<p>The <code>liftM</code> operation makes it easy to write code with pure and monadic parts.</p>
-<div class="sourceCode" id="cb288"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb288-1"><a href="#cb288-1" aria-hidden="true" tabindex="-1"></a>liftM <span class="fu">negate</span> (<span class="dt">Just</span> <span class="dv">3</span>)</span>
+<p>The <code>liftM</code> operation makes it easy to write code with
+pure and monadic parts.</p>
+<div class="sourceCode" id="cb288"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb288-1"><a href="#cb288-1" aria-hidden="true" tabindex="-1"></a>liftM <span class="fu">negate</span> (<span class="dt">Just</span> <span class="dv">3</span>)</span>
 <span id="cb288-2"><a href="#cb288-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Just</span> (<span class="op">-</span><span class="dv">3</span>)</span>
 <span id="cb288-3"><a href="#cb288-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb288-4"><a href="#cb288-4" aria-hidden="true" tabindex="-1"></a>liftM <span class="fu">sort</span> <span class="op">$</span> firsts [[<span class="dv">4</span>,<span class="dv">6</span>],[<span class="dv">2</span>,<span class="dv">1</span>,<span class="dv">0</span>],[<span class="dv">3</span>,<span class="dv">3</span>,<span class="dv">3</span>]]</span>
@@ -3294,9 +4531,15 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb288-6"><a href="#cb288-6" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb288-7"><a href="#cb288-7" aria-hidden="true" tabindex="-1"></a>runState (liftM <span class="fu">negate</span> get) <span class="dv">3</span></span>
 <span id="cb288-8"><a href="#cb288-8" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> (<span class="op">-</span><span class="dv">3</span>,<span class="dv">3</span>)</span></code></pre></div>
-<p>Does the type of <code>liftM</code> look familiar? It’s just like the type of <code>fmap</code>! In fact, it’s easy to define a functor instance for a monad: just set <code>fmap = liftM</code>. Since every <code>Monad</code> needs to be a <code>Functor</code> these days, modern Haskell style prefers <code>fmap</code> over <code>liftM</code>.</p>
-<div class="sourceCode" id="cb289"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb289-1"><a href="#cb289-1" aria-hidden="true" tabindex="-1"></a><span class="fu">fmap</span><span class="ot"> ::</span> <span class="dt">Functor</span> f <span class="ot">=&gt;</span> (a<span class="ot">-&gt;</span>b) <span class="ot">-&gt;</span> f a <span class="ot">-&gt;</span> f b</span></code></pre></div>
-<div class="sourceCode" id="cb290"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb290-1"><a href="#cb290-1" aria-hidden="true" tabindex="-1"></a><span class="fu">fmap</span> <span class="fu">negate</span> (<span class="dt">Just</span> <span class="dv">3</span>)</span>
+<p>Does the type of <code>liftM</code> look familiar? It’s just like the
+type of <code>fmap</code>! In fact, it’s easy to define a functor
+instance for a monad: just set <code>fmap = liftM</code>. Since every
+<code>Monad</code> needs to be a <code>Functor</code> these days, modern
+Haskell style prefers <code>fmap</code> over <code>liftM</code>.</p>
+<div class="sourceCode" id="cb289"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb289-1"><a href="#cb289-1" aria-hidden="true" tabindex="-1"></a><span class="fu">fmap</span><span class="ot"> ::</span> <span class="dt">Functor</span> f <span class="ot">=&gt;</span> (a<span class="ot">-&gt;</span>b) <span class="ot">-&gt;</span> f a <span class="ot">-&gt;</span> f b</span></code></pre></div>
+<div class="sourceCode" id="cb290"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb290-1"><a href="#cb290-1" aria-hidden="true" tabindex="-1"></a><span class="fu">fmap</span> <span class="fu">negate</span> (<span class="dt">Just</span> <span class="dv">3</span>)</span>
 <span id="cb290-2"><a href="#cb290-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Just</span> (<span class="op">-</span><span class="dv">3</span>)</span>
 <span id="cb290-3"><a href="#cb290-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb290-4"><a href="#cb290-4" aria-hidden="true" tabindex="-1"></a><span class="fu">fmap</span> <span class="fu">sort</span> <span class="op">$</span> firsts [[<span class="dv">4</span>,<span class="dv">6</span>],[<span class="dv">2</span>,<span class="dv">1</span>,<span class="dv">0</span>],[<span class="dv">3</span>,<span class="dv">3</span>,<span class="dv">3</span>]]</span>
@@ -3304,29 +4547,41 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb290-6"><a href="#cb290-6" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb290-7"><a href="#cb290-7" aria-hidden="true" tabindex="-1"></a>runState (<span class="fu">fmap</span> <span class="fu">negate</span> get) <span class="dv">3</span></span>
 <span id="cb290-8"><a href="#cb290-8" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> (<span class="op">-</span><span class="dv">3</span>,<span class="dv">3</span>)</span></code></pre></div>
-</section>
-<section id="one-more-monad" data-number="5.11">
-<h2 data-number="13.11"><span class="header-section-number">13.11</span> One More Monad</h2>
-<p><em>The list monad</em> (that is, the <code>Monad</code> instance for <code>[]</code>) represents computations with <em>multiple return values</em>. It’s useful for searching through alternatives. Here’s a first example. For every <code>x</code> we produce both <code>x</code> and <code>-x</code>:</p>
-<div class="sourceCode" id="cb291"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb291-1"><a href="#cb291-1" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] <span class="op">&gt;&gt;=</span> \x <span class="ot">-&gt;</span> [<span class="op">-</span>x,x]</span>
+<h2 data-number="13.11" id="one-more-monad"><span
+class="header-section-number">13.11</span> One More Monad</h2>
+<p><em>The list monad</em> (that is, the <code>Monad</code> instance for
+<code>[]</code>) represents computations with <em>multiple return
+values</em>. It’s useful for searching through alternatives. Here’s a
+first example. For every <code>x</code> we produce both <code>x</code>
+and <code>-x</code>:</p>
+<div class="sourceCode" id="cb291"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb291-1"><a href="#cb291-1" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] <span class="op">&gt;&gt;=</span> \x <span class="ot">-&gt;</span> [<span class="op">-</span>x,x]</span>
 <span id="cb291-2"><a href="#cb291-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="op">-</span><span class="dv">1</span>,<span class="dv">1</span>,<span class="op">-</span><span class="dv">2</span>,<span class="dv">2</span>,<span class="op">-</span><span class="dv">3</span>,<span class="dv">3</span>]</span></code></pre></div>
 <p>We can filter out unsuitable values by produing an empty list:</p>
-<div class="sourceCode" id="cb292"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb292-1"><a href="#cb292-1" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] <span class="op">&gt;&gt;=</span> \x <span class="ot">-&gt;</span> <span class="kw">if</span> x<span class="op">&gt;</span><span class="dv">1</span> <span class="kw">then</span> [x] <span class="kw">else</span> []</span>
+<div class="sourceCode" id="cb292"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb292-1"><a href="#cb292-1" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] <span class="op">&gt;&gt;=</span> \x <span class="ot">-&gt;</span> <span class="kw">if</span> x<span class="op">&gt;</span><span class="dv">1</span> <span class="kw">then</span> [x] <span class="kw">else</span> []</span>
 <span id="cb292-2"><a href="#cb292-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="dv">2</span>,<span class="dv">3</span>]</span></code></pre></div>
-<p>If we’re using do-notation the list monad starts to look more like a looping construct:</p>
-<div class="sourceCode" id="cb293"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb293-1"><a href="#cb293-1" aria-hidden="true" tabindex="-1"></a><span class="kw">do</span> word <span class="ot">&lt;-</span> [<span class="st">&quot;Blue&quot;</span>, <span class="st">&quot;Green&quot;</span>]</span>
+<p>If we’re using do-notation the list monad starts to look more like a
+looping construct:</p>
+<div class="sourceCode" id="cb293"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb293-1"><a href="#cb293-1" aria-hidden="true" tabindex="-1"></a><span class="kw">do</span> word <span class="ot">&lt;-</span> [<span class="st">&quot;Blue&quot;</span>, <span class="st">&quot;Green&quot;</span>]</span>
 <span id="cb293-2"><a href="#cb293-2" aria-hidden="true" tabindex="-1"></a>   number <span class="ot">&lt;-</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb293-3"><a href="#cb293-3" aria-hidden="true" tabindex="-1"></a>   <span class="fu">return</span> (word <span class="op">++</span> <span class="fu">show</span> number)</span>
 <span id="cb293-4"><a href="#cb293-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="st">&quot;Blue1&quot;</span>,<span class="st">&quot;Blue2&quot;</span>,<span class="st">&quot;Blue3&quot;</span>,<span class="st">&quot;Green1&quot;</span>,<span class="st">&quot;Green2&quot;</span>,<span class="st">&quot;Green3&quot;</span>]</span></code></pre></div>
-<p>More interesting example: find all the pairs in a list that sum to <code>k</code>. (The same element twice counts as a pair.)</p>
-<div class="sourceCode" id="cb294"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb294-1"><a href="#cb294-1" aria-hidden="true" tabindex="-1"></a><span class="ot">findSum ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> [(<span class="dt">Int</span>,<span class="dt">Int</span>)]</span>
+<p>More interesting example: find all the pairs in a list that sum to
+<code>k</code>. (The same element twice counts as a pair.)</p>
+<div class="sourceCode" id="cb294"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb294-1"><a href="#cb294-1" aria-hidden="true" tabindex="-1"></a><span class="ot">findSum ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> [(<span class="dt">Int</span>,<span class="dt">Int</span>)]</span>
 <span id="cb294-2"><a href="#cb294-2" aria-hidden="true" tabindex="-1"></a>findSum xs k <span class="ot">=</span> <span class="kw">do</span> a <span class="ot">&lt;-</span> xs</span>
 <span id="cb294-3"><a href="#cb294-3" aria-hidden="true" tabindex="-1"></a>                  b <span class="ot">&lt;-</span> xs</span>
 <span id="cb294-4"><a href="#cb294-4" aria-hidden="true" tabindex="-1"></a>                  <span class="kw">if</span> (a<span class="op">+</span>b<span class="op">==</span>k) <span class="kw">then</span> [(a,b)] <span class="kw">else</span> []</span></code></pre></div>
-<div class="sourceCode" id="cb295"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb295-1"><a href="#cb295-1" aria-hidden="true" tabindex="-1"></a>findSum [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>,<span class="dv">5</span>] <span class="dv">5</span></span>
+<div class="sourceCode" id="cb295"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb295-1"><a href="#cb295-1" aria-hidden="true" tabindex="-1"></a>findSum [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>,<span class="dv">5</span>] <span class="dv">5</span></span>
 <span id="cb295-2"><a href="#cb295-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [(<span class="dv">1</span>,<span class="dv">4</span>),(<span class="dv">2</span>,<span class="dv">3</span>),(<span class="dv">3</span>,<span class="dv">2</span>),(<span class="dv">4</span>,<span class="dv">1</span>)]</span></code></pre></div>
-<p>A final, more complex example. We find all palindromes from a string using the list monad, and then find the longest one.</p>
-<div class="sourceCode" id="cb296"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb296-1"><a href="#cb296-1" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="dt">Data.List</span> (sortBy)</span>
+<p>A final, more complex example. We find all palindromes from a string
+using the list monad, and then find the longest one.</p>
+<div class="sourceCode" id="cb296"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb296-1"><a href="#cb296-1" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="dt">Data.List</span> (sortBy)</span>
 <span id="cb296-2"><a href="#cb296-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb296-3"><a href="#cb296-3" aria-hidden="true" tabindex="-1"></a><span class="ot">substrings ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> [<span class="dt">String</span>]</span>
 <span id="cb296-4"><a href="#cb296-4" aria-hidden="true" tabindex="-1"></a>substrings xs <span class="ot">=</span> <span class="kw">do</span> start <span class="ot">&lt;-</span> [<span class="dv">0</span><span class="op">..</span><span class="fu">length</span> xs <span class="op">-</span> <span class="dv">1</span>]</span>
@@ -3339,24 +4594,39 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb296-11"><a href="#cb296-11" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb296-12"><a href="#cb296-12" aria-hidden="true" tabindex="-1"></a>longestPalindrome xs <span class="ot">=</span> <span class="fu">head</span> <span class="op">.</span> sortBy f <span class="op">$</span> palindromesIn xs</span>
 <span id="cb296-13"><a href="#cb296-13" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> f s s&#39; <span class="ot">=</span> <span class="fu">compare</span> (<span class="fu">length</span> s&#39;) (<span class="fu">length</span> s)  <span class="co">-- longer is smaller</span></span></code></pre></div>
-<div class="sourceCode" id="cb297"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb297-1"><a href="#cb297-1" aria-hidden="true" tabindex="-1"></a>palindromesIn <span class="st">&quot;aabbacddcaca&quot;</span></span>
+<div class="sourceCode" id="cb297"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb297-1"><a href="#cb297-1" aria-hidden="true" tabindex="-1"></a>palindromesIn <span class="st">&quot;aabbacddcaca&quot;</span></span>
 <span id="cb297-2"><a href="#cb297-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> [<span class="st">&quot;a&quot;</span>,<span class="st">&quot;aa&quot;</span>,<span class="st">&quot;a&quot;</span>,<span class="st">&quot;abba&quot;</span>,<span class="st">&quot;b&quot;</span>,<span class="st">&quot;bb&quot;</span>,<span class="st">&quot;b&quot;</span>,<span class="st">&quot;a&quot;</span>,<span class="st">&quot;acddca&quot;</span>,<span class="st">&quot;c&quot;</span>,<span class="st">&quot;cddc&quot;</span>,<span class="st">&quot;d&quot;</span>,<span class="st">&quot;dd&quot;</span>,<span class="st">&quot;d&quot;</span>,<span class="st">&quot;c&quot;</span>,<span class="st">&quot;cac&quot;</span>,<span class="st">&quot;a&quot;</span>,<span class="st">&quot;c&quot;</span>]</span>
 <span id="cb297-3"><a href="#cb297-3" aria-hidden="true" tabindex="-1"></a>longestPalindrome <span class="st">&quot;aabbacddcaca&quot;</span></span>
 <span id="cb297-4"><a href="#cb297-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="st">&quot;acddca&quot;</span></span></code></pre></div>
 <p>Here’s the surprisingly simple implementation of the list monad:</p>
-<div class="sourceCode" id="cb298"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb298-1"><a href="#cb298-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Monad</span> [] <span class="kw">where</span></span>
+<div class="sourceCode" id="cb298"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb298-1"><a href="#cb298-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Monad</span> [] <span class="kw">where</span></span>
 <span id="cb298-2"><a href="#cb298-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">return</span> x <span class="ot">=</span> [x]                  <span class="co">-- an operation that produces one value</span></span>
 <span id="cb298-3"><a href="#cb298-3" aria-hidden="true" tabindex="-1"></a>  lis <span class="op">&gt;&gt;=</span> f <span class="ot">=</span> <span class="fu">concat</span> (<span class="fu">map</span> f lis)  <span class="co">-- compute f for all values, combine the results</span></span></code></pre></div>
-<p>We’ve actually seen the list monad previously in the guise of list comprehensions. Compare this reimplementation of <code>findSum</code> to the earlier one that uses <code>do</code>-notation.</p>
-<div class="sourceCode" id="cb299"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb299-1"><a href="#cb299-1" aria-hidden="true" tabindex="-1"></a><span class="ot">findSum ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> [(<span class="dt">Int</span>,<span class="dt">Int</span>)]</span>
+<p>We’ve actually seen the list monad previously in the guise of list
+comprehensions. Compare this reimplementation of <code>findSum</code> to
+the earlier one that uses <code>do</code>-notation.</p>
+<div class="sourceCode" id="cb299"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb299-1"><a href="#cb299-1" aria-hidden="true" tabindex="-1"></a><span class="ot">findSum ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> [(<span class="dt">Int</span>,<span class="dt">Int</span>)]</span>
 <span id="cb299-2"><a href="#cb299-2" aria-hidden="true" tabindex="-1"></a>findSum xs k <span class="ot">=</span> [(a,b) <span class="op">|</span> a <span class="ot">&lt;-</span> xs, b <span class="ot">&lt;-</span> xs, a<span class="op">+</span>b<span class="op">==</span>k ]</span></code></pre></div>
-</section>
-<section id="oh-right-io" data-number="5.12">
-<h2 data-number="13.12"><span class="header-section-number">13.12</span> Oh Right, IO</h2>
-<p>As you’ve probably guessed by now, <code>IO</code> is a monad. However the implementations of the <code>IO</code> type and <code>instance Monad IO</code> are compiler built-ins. You couldn’t implement the IO monad just using standard Haskell, unlike <code>Maybe</code> monad, <code>State</code> monad and other monads we’ve seen.</p>
-<p>However, true side effects fit the monad pattern just like <code>State</code> and <code>Maybe</code>. Just like with other monads, we’re separating the <em>pure definitions of operations</em> from the process of <em>running the operations</em>. As a bonus, you can use all the generic monad operations (<code>mapM</code> and friends) with IO.</p>
+<h2 data-number="13.12" id="oh-right-io"><span
+class="header-section-number">13.12</span> Oh Right, IO</h2>
+<p>As you’ve probably guessed by now, <code>IO</code> is a monad.
+However the implementations of the <code>IO</code> type and
+<code>instance Monad IO</code> are compiler built-ins. You couldn’t
+implement the IO monad just using standard Haskell, unlike
+<code>Maybe</code> monad, <code>State</code> monad and other monads
+we’ve seen.</p>
+<p>However, true side effects fit the monad pattern just like
+<code>State</code> and <code>Maybe</code>. Just like with other monads,
+we’re separating the <em>pure definitions of operations</em> from the
+process of <em>running the operations</em>. As a bonus, you can use all
+the generic monad operations (<code>mapM</code> and friends) with
+IO.</p>
 <p>Here are some examples of writing IO using monad operations.</p>
-<div class="sourceCode" id="cb300"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb300-1"><a href="#cb300-1" aria-hidden="true" tabindex="-1"></a><span class="ot">printTwoThings ::</span> <span class="dt">IO</span> ()</span>
+<div class="sourceCode" id="cb300"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb300-1"><a href="#cb300-1" aria-hidden="true" tabindex="-1"></a><span class="ot">printTwoThings ::</span> <span class="dt">IO</span> ()</span>
 <span id="cb300-2"><a href="#cb300-2" aria-hidden="true" tabindex="-1"></a>printTwoThings <span class="ot">=</span> <span class="fu">putStrLn</span> <span class="st">&quot;One!&quot;</span> <span class="op">&gt;&gt;</span> <span class="fu">print</span> <span class="dv">2</span></span>
 <span id="cb300-3"><a href="#cb300-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb300-4"><a href="#cb300-4" aria-hidden="true" tabindex="-1"></a><span class="ot">echo ::</span> <span class="dt">IO</span> ()</span>
@@ -3372,7 +4642,8 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb300-14"><a href="#cb300-14" aria-hidden="true" tabindex="-1"></a>confirm question <span class="ot">=</span> <span class="fu">putStrLn</span> question <span class="op">&gt;&gt;</span> <span class="fu">fmap</span> interpret <span class="fu">getLine</span></span>
 <span id="cb300-15"><a href="#cb300-15" aria-hidden="true" tabindex="-1"></a>    <span class="kw">where</span> interpret <span class="st">&quot;Y&quot;</span> <span class="ot">=</span> <span class="dt">True</span></span>
 <span id="cb300-16"><a href="#cb300-16" aria-hidden="true" tabindex="-1"></a>          interpret _ <span class="ot">=</span> <span class="dt">False</span></span></code></pre></div>
-<div class="sourceCode" id="cb301"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb301-1"><a href="#cb301-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> printTwoThings</span>
+<div class="sourceCode" id="cb301"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb301-1"><a href="#cb301-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> printTwoThings</span>
 <span id="cb301-2"><a href="#cb301-2" aria-hidden="true" tabindex="-1"></a><span class="dt">One</span><span class="op">!</span></span>
 <span id="cb301-3"><a href="#cb301-3" aria-hidden="true" tabindex="-1"></a><span class="dv">2</span></span>
 <span id="cb301-4"><a href="#cb301-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -3396,60 +4667,108 @@ this topic connects to equational reasoning discussed in chapter 10.
 <span id="cb301-22"><a href="#cb301-22" aria-hidden="true" tabindex="-1"></a><span class="dt">Y</span></span>
 <span id="cb301-23"><a href="#cb301-23" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> b</span>
 <span id="cb301-24"><a href="#cb301-24" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span></code></pre></div>
-</section>
-<section id="monads-in-other-languages" data-number="5.13">
-<h2 data-number="13.13"><span class="header-section-number">13.13</span> Monads in Other Languages</h2>
-<p>Once you’ve gotten familiar with the concept of a monad, you’ll start seeing monadlike things in other languages too. The most well-known examples of this are <em>Option types</em>, <em>Java Streams</em> and <em>JavaScript promises</em> . If you know these languages or concepts from before, you might find this section illuminating. If you don’t, feel free to skip this.</p>
-<section id="optionals" data-number="5.13.1">
-<h3 data-number="13.13.1"><span class="header-section-number">13.13.1</span> Optionals</h3>
-<p>Many langages have an <a href="https://en.wikipedia.org/wiki/Option_type">option type</a>. This type is called <code>Optional&lt;T&gt;</code> in Java, <code>std::optional&lt;T&gt;</code> in C++, <code>Nullable&lt;T&gt;</code> in C#, and so on. These types often have behaviour resembling the Haskell <code>Maybe</code> monad, for example:</p>
+<h2 data-number="13.13" id="monads-in-other-languages"><span
+class="header-section-number">13.13</span> Monads in Other
+Languages</h2>
+<p>Once you’ve gotten familiar with the concept of a monad, you’ll start
+seeing monadlike things in other languages too. The most well-known
+examples of this are <em>Option types</em>, <em>Java Streams</em> and
+<em>JavaScript promises</em> . If you know these languages or concepts
+from before, you might find this section illuminating. If you don’t,
+feel free to skip this.</p>
+<h3 data-number="13.13.1" id="optionals"><span
+class="header-section-number">13.13.1</span> Optionals</h3>
+<p>Many langages have an <a
+href="https://en.wikipedia.org/wiki/Option_type">option type</a>. This
+type is called <code>Optional&lt;T&gt;</code> in Java,
+<code>std::optional&lt;T&gt;</code> in C++,
+<code>Nullable&lt;T&gt;</code> in C#, and so on. These types often have
+behaviour resembling the Haskell <code>Maybe</code> monad, for
+example:</p>
 <ul>
-<li>In Java, <a href="https://docs.oracle.com/en/java/javase/14/docs/api/java.base/java/util/Optional.html#flatMap(java.util.function.Function)"><code>Optional.flatMap</code></a> corresponds to <code>&gt;&gt;=</code>: it lets you apply a <code>Function&lt;T,Optional&lt;U&gt;&gt;</code> to an <code>Optional&lt;T&gt;</code> and get an <code>Optional&lt;U&gt;</code>.</li>
-<li>In C#, binary operations get automatically <a href="https://docs.microsoft.com/en-us/dotnet/csharp/language-reference/builtin-types/nullable-value-types#lifted-operators">lifted</a> to <code>Nullable</code> types. For example, <code>a + null</code> becomes <code>null</code>.</li>
+<li>In Java, <a
+href="https://docs.oracle.com/en/java/javase/14/docs/api/java.base/java/util/Optional.html#flatMap(java.util.function.Function)"><code>Optional.flatMap</code></a>
+corresponds to <code>&gt;&gt;=</code>: it lets you apply a
+<code>Function&lt;T,Optional&lt;U&gt;&gt;</code> to an
+<code>Optional&lt;T&gt;</code> and get an
+<code>Optional&lt;U&gt;</code>.</li>
+<li>In C#, binary operations get automatically <a
+href="https://docs.microsoft.com/en-us/dotnet/csharp/language-reference/builtin-types/nullable-value-types#lifted-operators">lifted</a>
+to <code>Nullable</code> types. For example, <code>a + null</code>
+becomes <code>null</code>.</li>
 </ul>
-</section>
-<section id="streams" data-number="5.13.2">
-<h3 data-number="13.13.2"><span class="header-section-number">13.13.2</span> Streams</h3>
-<p><a href="https://docs.oracle.com/en/java/javase/14/docs/api/java.base/java/util/stream/Stream.html">Java Streams</a> have a monadlike API too. Streams are about producing many values incrementally. Just like with Optional, the method <code>Stream.flatMap</code> lets us take a <code>Stream&lt;T&gt;</code>, combine it with a <code>Function&lt;T,Stream&lt;U&gt;&gt;</code> and get a <code>Stream&lt;U&gt;</code>.</p>
-<p>As an example, if <code>lines</code> is a <code>Stream&lt;String&gt;</code>, <code>words</code> takes a <code>String</code> and returns a <code>Stream&lt;String&gt;</code> and <code>readInt</code> takes a <code>String</code> and returns an <code>Integer</code>, we can write:</p>
-<div class="sourceCode" id="cb302"><pre class="sourceCode java"><code class="sourceCode java"><span id="cb302-1"><a href="#cb302-1" aria-hidden="true" tabindex="-1"></a>Stream<span class="op">&lt;</span><span class="bu">Integer</span><span class="op">&gt;</span> <span class="fu">parseNumbers</span><span class="op">(</span>Stream<span class="op">&lt;</span><span class="bu">String</span><span class="op">&gt;</span> lines<span class="op">)</span> <span class="op">{</span></span>
+<h3 data-number="13.13.2" id="streams"><span
+class="header-section-number">13.13.2</span> Streams</h3>
+<p><a
+href="https://docs.oracle.com/en/java/javase/14/docs/api/java.base/java/util/stream/Stream.html">Java
+Streams</a> have a monadlike API too. Streams are about producing many
+values incrementally. Just like with Optional, the method
+<code>Stream.flatMap</code> lets us take a <code>Stream&lt;T&gt;</code>,
+combine it with a <code>Function&lt;T,Stream&lt;U&gt;&gt;</code> and get
+a <code>Stream&lt;U&gt;</code>.</p>
+<p>As an example, if <code>lines</code> is a
+<code>Stream&lt;String&gt;</code>, <code>words</code> takes a
+<code>String</code> and returns a <code>Stream&lt;String&gt;</code> and
+<code>readInt</code> takes a <code>String</code> and returns an
+<code>Integer</code>, we can write:</p>
+<div class="sourceCode" id="cb302"><pre
+class="sourceCode java"><code class="sourceCode java"><span id="cb302-1"><a href="#cb302-1" aria-hidden="true" tabindex="-1"></a>Stream<span class="op">&lt;</span><span class="bu">Integer</span><span class="op">&gt;</span> <span class="fu">parseNumbers</span><span class="op">(</span>Stream<span class="op">&lt;</span><span class="bu">String</span><span class="op">&gt;</span> lines<span class="op">)</span> <span class="op">{</span></span>
 <span id="cb302-2"><a href="#cb302-2" aria-hidden="true" tabindex="-1"></a>    <span class="cf">return</span> lines<span class="op">.</span><span class="fu">flatMap</span><span class="op">(</span>words<span class="op">).</span><span class="fu">map</span><span class="op">(</span>read<span class="op">);</span></span>
 <span id="cb302-3"><a href="#cb302-3" aria-hidden="true" tabindex="-1"></a><span class="op">}</span></span></code></pre></div>
 <p>This corresponds to the following Haskell list monad code:</p>
-<div class="sourceCode" id="cb303"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb303-1"><a href="#cb303-1" aria-hidden="true" tabindex="-1"></a><span class="ot">parseNumbers ::</span> [<span class="dt">String</span>] <span class="ot">-&gt;</span> [<span class="dt">Int</span>]</span>
+<div class="sourceCode" id="cb303"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb303-1"><a href="#cb303-1" aria-hidden="true" tabindex="-1"></a><span class="ot">parseNumbers ::</span> [<span class="dt">String</span>] <span class="ot">-&gt;</span> [<span class="dt">Int</span>]</span>
 <span id="cb303-2"><a href="#cb303-2" aria-hidden="true" tabindex="-1"></a>parseNumbers strings <span class="ot">=</span> <span class="fu">fmap</span> <span class="fu">read</span> (strings <span class="op">&gt;&gt;=</span> <span class="fu">words</span>)</span></code></pre></div>
-<div class="sourceCode" id="cb304"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb304-1"><a href="#cb304-1" aria-hidden="true" tabindex="-1"></a>parseNumbers [<span class="st">&quot;123 456&quot;</span>,<span class="st">&quot;7 89&quot;</span>]  <span class="op">==&gt;</span>  [<span class="dv">123</span>,<span class="dv">456</span>,<span class="dv">7</span>,<span class="dv">89</span>]</span></code></pre></div>
-</section>
-<section id="promises" data-number="5.13.3">
-<h3 data-number="13.13.3"><span class="header-section-number">13.13.3</span> Promises</h3>
-<p>There is <a href="https://hackernoon.com/functional-javascript-functors-monads-and-promises-679ce2ab8abe">much</a> <a href="https://stackoverflow.com/questions/45712106/why-are-promises-monads">disagreement</a> about whether Promises in JavaScript <em>really</em> are monads or not. However, some similarities are obvious.</p>
-<p>First, consider the similarities between <code>Promise.then</code> and <code>&gt;&gt;=</code>. Both take an <em>operation</em> (promise or monadic operation), and combine it with a function that returns a new operation.</p>
-<div class="sourceCode" id="cb305"><pre class="sourceCode javascript"><code class="sourceCode javascript"><span id="cb305-1"><a href="#cb305-1" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">concatPromises</span>(promise1<span class="op">,</span> promise2) {</span>
+<div class="sourceCode" id="cb304"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb304-1"><a href="#cb304-1" aria-hidden="true" tabindex="-1"></a>parseNumbers [<span class="st">&quot;123 456&quot;</span>,<span class="st">&quot;7 89&quot;</span>]  <span class="op">==&gt;</span>  [<span class="dv">123</span>,<span class="dv">456</span>,<span class="dv">7</span>,<span class="dv">89</span>]</span></code></pre></div>
+<h3 data-number="13.13.3" id="promises"><span
+class="header-section-number">13.13.3</span> Promises</h3>
+<p>There is <a
+href="https://hackernoon.com/functional-javascript-functors-monads-and-promises-679ce2ab8abe">much</a>
+<a
+href="https://stackoverflow.com/questions/45712106/why-are-promises-monads">disagreement</a>
+about whether Promises in JavaScript <em>really</em> are monads or not.
+However, some similarities are obvious.</p>
+<p>First, consider the similarities between <code>Promise.then</code>
+and <code>&gt;&gt;=</code>. Both take an <em>operation</em> (promise or
+monadic operation), and combine it with a function that returns a new
+operation.</p>
+<div class="sourceCode" id="cb305"><pre
+class="sourceCode javascript"><code class="sourceCode javascript"><span id="cb305-1"><a href="#cb305-1" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">concatPromises</span>(promise1<span class="op">,</span> promise2) {</span>
 <span id="cb305-2"><a href="#cb305-2" aria-hidden="true" tabindex="-1"></a>  <span class="cf">return</span> promise1<span class="op">.</span><span class="fu">then</span>(value1 <span class="kw">=&gt;</span> promise2<span class="op">.</span><span class="fu">then</span>(value2 <span class="kw">=&gt;</span> value1<span class="op">+</span>value2))<span class="op">;</span></span>
 <span id="cb305-3"><a href="#cb305-3" aria-hidden="true" tabindex="-1"></a>}</span></code></pre></div>
-<div class="sourceCode" id="cb306"><pre class="sourceCode javascript"><code class="sourceCode javascript"><span id="cb306-1"><a href="#cb306-1" aria-hidden="true" tabindex="-1"></a><span class="op">&gt;&gt;</span> <span class="fu">concatPromises</span>(<span class="bu">Promise</span><span class="op">.</span><span class="fu">resolve</span>(<span class="st">&quot;abc&quot;</span>)<span class="op">,</span> <span class="bu">Promise</span><span class="op">.</span><span class="fu">resolve</span>(<span class="st">&quot;def&quot;</span>))<span class="op">.</span><span class="fu">then</span>(<span class="bu">console</span><span class="op">.</span><span class="fu">log</span>)</span>
+<div class="sourceCode" id="cb306"><pre
+class="sourceCode javascript"><code class="sourceCode javascript"><span id="cb306-1"><a href="#cb306-1" aria-hidden="true" tabindex="-1"></a><span class="op">&gt;&gt;</span> <span class="fu">concatPromises</span>(<span class="bu">Promise</span><span class="op">.</span><span class="fu">resolve</span>(<span class="st">&quot;abc&quot;</span>)<span class="op">,</span> <span class="bu">Promise</span><span class="op">.</span><span class="fu">resolve</span>(<span class="st">&quot;def&quot;</span>))<span class="op">.</span><span class="fu">then</span>(<span class="bu">console</span><span class="op">.</span><span class="fu">log</span>)</span>
 <span id="cb306-2"><a href="#cb306-2" aria-hidden="true" tabindex="-1"></a>abcdef</span></code></pre></div>
-<div class="sourceCode" id="cb307"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb307-1"><a href="#cb307-1" aria-hidden="true" tabindex="-1"></a><span class="ot">concatMonadic ::</span> <span class="dt">Monad</span> m <span class="ot">=&gt;</span> m <span class="dt">String</span> <span class="ot">-&gt;</span> m <span class="dt">String</span> <span class="ot">-&gt;</span> m <span class="dt">String</span></span>
+<div class="sourceCode" id="cb307"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb307-1"><a href="#cb307-1" aria-hidden="true" tabindex="-1"></a><span class="ot">concatMonadic ::</span> <span class="dt">Monad</span> m <span class="ot">=&gt;</span> m <span class="dt">String</span> <span class="ot">-&gt;</span> m <span class="dt">String</span> <span class="ot">-&gt;</span> m <span class="dt">String</span></span>
 <span id="cb307-2"><a href="#cb307-2" aria-hidden="true" tabindex="-1"></a>concatMonadic op1 op2 <span class="ot">=</span> op1 <span class="op">&gt;&gt;=</span> (\value1 <span class="ot">-&gt;</span> op2 <span class="op">&gt;&gt;=</span> (\value2 <span class="ot">-&gt;</span> <span class="fu">return</span> (value1<span class="op">++</span>value2)))</span></code></pre></div>
-<div class="sourceCode" id="cb308"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb308-1"><a href="#cb308-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> concatMonadic (<span class="dt">Just</span> <span class="st">&quot;abc&quot;</span>) (<span class="dt">Just</span> <span class="st">&quot;def&quot;</span>)</span>
+<div class="sourceCode" id="cb308"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb308-1"><a href="#cb308-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> concatMonadic (<span class="dt">Just</span> <span class="st">&quot;abc&quot;</span>) (<span class="dt">Just</span> <span class="st">&quot;def&quot;</span>)</span>
 <span id="cb308-2"><a href="#cb308-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Just</span> <span class="st">&quot;abcdef&quot;</span></span></code></pre></div>
-<p>Next, let’s consider the similarities between async/await and do-notation. Both are nicer syntaxes for working with the raw <code>Promise.then</code> or <code>&gt;&gt;=</code> mechanisms. We reimplement <code>concatPromises</code> using async/await, and <code>concatMonadic</code> using do-notation. Their behaviour stays the same.</p>
-<div class="sourceCode" id="cb309"><pre class="sourceCode javascript"><code class="sourceCode javascript"><span id="cb309-1"><a href="#cb309-1" aria-hidden="true" tabindex="-1"></a><span class="kw">async</span> <span class="kw">function</span> <span class="fu">concatPromises</span>(promise1<span class="op">,</span> promise2) {</span>
+<p>Next, let’s consider the similarities between async/await and
+do-notation. Both are nicer syntaxes for working with the raw
+<code>Promise.then</code> or <code>&gt;&gt;=</code> mechanisms. We
+reimplement <code>concatPromises</code> using async/await, and
+<code>concatMonadic</code> using do-notation. Their behaviour stays the
+same.</p>
+<div class="sourceCode" id="cb309"><pre
+class="sourceCode javascript"><code class="sourceCode javascript"><span id="cb309-1"><a href="#cb309-1" aria-hidden="true" tabindex="-1"></a><span class="kw">async</span> <span class="kw">function</span> <span class="fu">concatPromises</span>(promise1<span class="op">,</span> promise2) {</span>
 <span id="cb309-2"><a href="#cb309-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">let</span> value1 <span class="op">=</span> <span class="cf">await</span> promise1<span class="op">;</span></span>
 <span id="cb309-3"><a href="#cb309-3" aria-hidden="true" tabindex="-1"></a>  <span class="kw">let</span> value2 <span class="op">=</span> <span class="cf">await</span> promise2<span class="op">;</span></span>
 <span id="cb309-4"><a href="#cb309-4" aria-hidden="true" tabindex="-1"></a>  <span class="cf">return</span> value1<span class="op">+</span>value2<span class="op">;</span></span>
 <span id="cb309-5"><a href="#cb309-5" aria-hidden="true" tabindex="-1"></a>}</span></code></pre></div>
-<div class="sourceCode" id="cb310"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb310-1"><a href="#cb310-1" aria-hidden="true" tabindex="-1"></a><span class="ot">concatMonadic ::</span> <span class="dt">Monad</span> m <span class="ot">=&gt;</span> m <span class="dt">String</span> <span class="ot">-&gt;</span> m <span class="dt">String</span> <span class="ot">-&gt;</span> m <span class="dt">String</span></span>
+<div class="sourceCode" id="cb310"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb310-1"><a href="#cb310-1" aria-hidden="true" tabindex="-1"></a><span class="ot">concatMonadic ::</span> <span class="dt">Monad</span> m <span class="ot">=&gt;</span> m <span class="dt">String</span> <span class="ot">-&gt;</span> m <span class="dt">String</span> <span class="ot">-&gt;</span> m <span class="dt">String</span></span>
 <span id="cb310-2"><a href="#cb310-2" aria-hidden="true" tabindex="-1"></a>concatMonadic op1 op2 <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb310-3"><a href="#cb310-3" aria-hidden="true" tabindex="-1"></a>  value1 <span class="ot">&lt;-</span> op1</span>
 <span id="cb310-4"><a href="#cb310-4" aria-hidden="true" tabindex="-1"></a>  value2 <span class="ot">&lt;-</span> op2</span>
 <span id="cb310-5"><a href="#cb310-5" aria-hidden="true" tabindex="-1"></a>  <span class="fu">return</span> (value1<span class="op">++</span>value2)</span></code></pre></div>
-</section>
-</section>
-<section id="monads-wrap-up" data-number="5.14">
-<h2 data-number="13.14"><span class="header-section-number">13.14</span> Monads: Wrap-up</h2>
+<h2 data-number="13.14" id="monads-wrap-up"><span
+class="header-section-number">13.14</span> Monads: Wrap-up</h2>
 <ul>
-<li>The <code>Monad</code> type class is a way to represent different ways of <em>executing recipes</em>
+<li>The <code>Monad</code> type class is a way to represent different
+ways of <em>executing recipes</em>
 <ul>
 <li>failure (<code>Maybe</code>)</li>
 <li>logging</li>
@@ -3459,40 +4778,52 @@ this topic connects to equational reasoning discussed in chapter 10.
 </ul></li>
 <li>You can write monad code in two equivalent ways:
 <ul>
-<li>Using the <code>Monad</code> class operations (<code>&gt;&gt;=</code>, <code>&gt;&gt;</code>) directly</li>
+<li>Using the <code>Monad</code> class operations
+(<code>&gt;&gt;=</code>, <code>&gt;&gt;</code>) directly</li>
 <li>Using <code>do</code>-notation</li>
 </ul></li>
-<li>When <code>M</code> is a monad, values of type <code>M a</code> are <em>operations that produce a result of type <code>a</code></em></li>
-<li>Monads are a <em>design pattern</em> and a <em>library</em> (<code>mapM</code> etc)
+<li>When <code>M</code> is a monad, values of type <code>M a</code> are
+<em>operations that produce a result of type <code>a</code></em></li>
+<li>Monads are a <em>design pattern</em> and a <em>library</em>
+(<code>mapM</code> etc)
 <ul>
 <li>Using common abstractions makes it easier to understand code</li>
-<li>Reading a <code>State</code> operation is easier than deciphering a complicated recursion with state</li>
+<li>Reading a <code>State</code> operation is easier than deciphering a
+complicated recursion with state</li>
 </ul></li>
 <li>Everything you can do with monads, you can also do without them
 <ul>
 <li>Exception: IO</li>
 <li>Using a monad often simplifies code</li>
 </ul></li>
-<li><em>Warning</em>: the internet is full of tutorials trying to explain monads using a simple analogy
+<li><em>Warning</em>: the internet is full of tutorials trying to
+explain monads using a simple analogy
 <ul>
 <li>In my experience, this doesn’t work</li>
-<li>What works is using different monads and slowly getting used to the concept</li>
+<li>What works is using different monads and slowly getting used to the
+concept</li>
 </ul></li>
 </ul>
-</section>
-<section id="sidenote-standard-haskell" data-number="5.15">
-<h2 data-number="13.15"><span class="header-section-number">13.15</span> Sidenote: Standard Haskell</h2>
-<p>This and the previous lecture have covered many parts where the GHC version of Haskell differs from standard Haskell 2010. Here’s a short list of the changes GHC has made, just so you know:</p>
+<h2 data-number="13.15" id="sidenote-standard-haskell"><span
+class="header-section-number">13.15</span> Sidenote: Standard
+Haskell</h2>
+<p>This and the previous lecture have covered many parts where the GHC
+version of Haskell differs from standard Haskell 2010. Here’s a short
+list of the changes GHC has made, just so you know:</p>
 <ul>
-<li><code>length</code>, <code>sum</code>, <code>foldr</code> etc. generalized to work on <code>Foldable</code> instead of just lists</li>
-<li><code>Functor</code> and <code>Applicative</code> are superclasses of <code>Monad</code></li>
-<li>The <code>fail</code> method has been moved from the <code>Monad</code> type class to its own <code>MonadFail</code> class</li>
+<li><code>length</code>, <code>sum</code>, <code>foldr</code> etc.
+generalized to work on <code>Foldable</code> instead of just lists</li>
+<li><code>Functor</code> and <code>Applicative</code> are superclasses
+of <code>Monad</code></li>
+<li>The <code>fail</code> method has been moved from the
+<code>Monad</code> type class to its own <code>MonadFail</code>
+class</li>
 </ul>
-</section>
-<section id="quiz-4" data-number="5.16">
-<h2 data-number="13.16"><span class="header-section-number">13.16</span> Quiz</h2>
+<h2 data-number="13.16" id="quiz-4"><span
+class="header-section-number">13.16</span> Quiz</h2>
 <p>What is the expression equivalent to the following do block?</p>
-<div class="sourceCode" id="cb311"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb311-1"><a href="#cb311-1" aria-hidden="true" tabindex="-1"></a><span class="kw">do</span> y <span class="ot">&lt;-</span> z</span>
+<div class="sourceCode" id="cb311"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb311-1"><a href="#cb311-1" aria-hidden="true" tabindex="-1"></a><span class="kw">do</span> y <span class="ot">&lt;-</span> z</span>
 <span id="cb311-2"><a href="#cb311-2" aria-hidden="true" tabindex="-1"></a>   s y</span>
 <span id="cb311-3"><a href="#cb311-3" aria-hidden="true" tabindex="-1"></a>   <span class="fu">return</span> (f y)</span></code></pre></div>
 <ol class="quiz">
@@ -3551,36 +4882,80 @@ None of the above
 None of the above
 </li>
 </ol>
-</section>
-<section id="exercises-4" data-number="5.17">
-<h2 data-number="13.17"><span class="header-section-number">13.17</span> Exercises</h2>
+<h2 data-number="13.17" id="exercises-4"><span
+class="header-section-number">13.17</span> Exercises</h2>
 <ul>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set13a.hs">Set13a</a></li>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set13b.hs">Set13b</a></li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set13a.hs">Set13a</a></li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set13b.hs">Set13b</a></li>
 </ul>
-</section>
-</section>
-<section id="lecture-14-lets-use-some-libraries" data-number="6">
-<h1 data-number="14"><span class="header-section-number">14</span> Lecture 14: Let’s Use Some Libraries!</h1>
-<p>Now that you know monads, you pretty much know everything about Haskell to start writing real programs that use libraries to do useful things. This lecture will go over some examples of libraries that are commonly used in such real programs. Using these libraries is also a good way to practice using monads, reading docs, and understanding type errors.</p>
-<p><strong>Note!</strong> When reading the documentation for the libraries, remember to pay attention to the library version. You can see the versions used on the course in the <a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/tests.cabal"><code>tests.cabal</code> file</a>. The links in the course material always take you to the right version, as does the <code>stack haddock --open &lt;package&gt;</code> command. See also <a href="https://haskell.mooc.fi/part1#reading-docs">Reading Docs in Part 1</a>.</p>
-<section id="text-and-bytestring" data-number="6.1">
-<h2 data-number="14.1"><span class="header-section-number">14.1</span> <code>Text</code> and <code>ByteString</code></h2>
-<p>So far, we’ve been using the Haskell <code>String</code> type to work with strings. However <code>String</code> is just <code>[Char]</code>, a linked list of characters. This is horribly inefficient, both in terms of memory, and in terms of time. Once we move beyond processing short strings and start processing whole files or network requests, a more time efficient string type becomes a must.</p>
-<p>There are two types that are used as replacements for <code>String</code>, with slightly different semantics:</p>
+<h1 data-number="14" id="lecture-14-lets-use-some-libraries"><span
+class="header-section-number">14</span> Lecture 14: Let’s Use Some
+Libraries!</h1>
+<p>Now that you know monads, you pretty much know everything about
+Haskell to start writing real programs that use libraries to do useful
+things. This lecture will go over some examples of libraries that are
+commonly used in such real programs. Using these libraries is also a
+good way to practice using monads, reading docs, and understanding type
+errors.</p>
+<p><strong>Note!</strong> When reading the documentation for the
+libraries, remember to pay attention to the library version. You can see
+the versions used on the course in the <a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/tests.cabal"><code>tests.cabal</code>
+file</a>. The links in the course material always take you to the right
+version, as does the <code>stack haddock --open &lt;package&gt;</code>
+command. See also <a
+href="https://haskell.mooc.fi/part1#reading-docs">Reading Docs in Part
+1</a>.</p>
+<h2 data-number="14.1" id="text-and-bytestring"><span
+class="header-section-number">14.1</span> <code>Text</code> and
+<code>ByteString</code></h2>
+<p>So far, we’ve been using the Haskell <code>String</code> type to work
+with strings. However <code>String</code> is just <code>[Char]</code>, a
+linked list of characters. This is horribly inefficient, both in terms
+of memory, and in terms of time. Once we move beyond processing short
+strings and start processing whole files or network requests, a more
+time efficient string type becomes a must.</p>
+<p>There are two types that are used as replacements for
+<code>String</code>, with slightly different semantics:</p>
 <ul>
-<li><code>Data.Text</code> represents <em>a sequence of <a href="https://en.wikipedia.org/wiki/Unicode">Unicode characters</a></em>, just like <code>String</code>, only more efficient. Used when dealing with text.</li>
-<li><code>Data.ByteString</code> represents <em>a sequence of bytes</em>. Used when dealing with binary data.</li>
+<li><code>Data.Text</code> represents <em>a sequence of <a
+href="https://en.wikipedia.org/wiki/Unicode">Unicode
+characters</a></em>, just like <code>String</code>, only more efficient.
+Used when dealing with text.</li>
+<li><code>Data.ByteString</code> represents <em>a sequence of
+bytes</em>. Used when dealing with binary data.</li>
 </ul>
-<p>Additionally, both of these types come in <em>lazy</em> and <em>strict</em> variants. The <a href="https://hackage.haskell.org/package/text-1.2.5.0/docs/Data-Text.html">docs for <code>Data.Text</code></a> summarize the difference well:</p>
+<p>Additionally, both of these types come in <em>lazy</em> and
+<em>strict</em> variants. The <a
+href="https://hackage.haskell.org/package/text-1.2.5.0/docs/Data-Text.html">docs
+for <code>Data.Text</code></a> summarize the difference well:</p>
 <blockquote>
-<p>The strict <code>Text</code> type requires that an entire string fit into memory at once. The lazy <code>Text</code> type is capable of streaming strings that are larger than memory using a small memory footprint… Each module provides an almost identical API…</p>
+<p>The strict <code>Text</code> type requires that an entire string fit
+into memory at once. The lazy <code>Text</code> type is capable of
+streaming strings that are larger than memory using a small memory
+footprint… Each module provides an almost identical API…</p>
 </blockquote>
-<p>All of these types (<code>Text</code> and <code>ByteString</code>, strict and lazy) offer <code>pack</code> and <code>unpack</code> functions for converting from and to plain <code>String</code>s. The types also come with specialized versions of familiar list functions like <code>reverse</code>, <code>take</code>, <code>map</code> and so on.</p>
-<section id="examples-with-text" data-number="6.1.1">
-<h3 data-number="14.1.1"><span class="header-section-number">14.1.1</span> Examples with <code>Text</code></h3>
-<p>Let’s go through a short GHCi session demonstrating the use of <code>Data.Text</code>. As <a href="https://hackage.haskell.org/package/text-1.2.5.0/docs/Data-Text.html">the documentation</a> says, the <code>Data.Text</code> module is designed to be imported <em>qualified</em>. We can convert a <code>String</code> into a <code>Text</code> with the function <code>T.pack</code>. Note how a value of type <code>Text</code> gets printed just like a <code>String</code>.</p>
-<div class="sourceCode" id="cb312"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb312-1"><a href="#cb312-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="kw">qualified</span> <span class="dt">Data.Text</span> <span class="kw">as</span> <span class="dt">T</span></span>
+<p>All of these types (<code>Text</code> and <code>ByteString</code>,
+strict and lazy) offer <code>pack</code> and <code>unpack</code>
+functions for converting from and to plain <code>String</code>s. The
+types also come with specialized versions of familiar list functions
+like <code>reverse</code>, <code>take</code>, <code>map</code> and so
+on.</p>
+<h3 data-number="14.1.1" id="examples-with-text"><span
+class="header-section-number">14.1.1</span> Examples with
+<code>Text</code></h3>
+<p>Let’s go through a short GHCi session demonstrating the use of
+<code>Data.Text</code>. As <a
+href="https://hackage.haskell.org/package/text-1.2.5.0/docs/Data-Text.html">the
+documentation</a> says, the <code>Data.Text</code> module is designed to
+be imported <em>qualified</em>. We can convert a <code>String</code>
+into a <code>Text</code> with the function <code>T.pack</code>. Note how
+a value of type <code>Text</code> gets printed just like a
+<code>String</code>.</p>
+<div class="sourceCode" id="cb312"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb312-1"><a href="#cb312-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="kw">qualified</span> <span class="dt">Data.Text</span> <span class="kw">as</span> <span class="dt">T</span></span>
 <span id="cb312-2"><a href="#cb312-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span><span class="op">&gt;</span> <span class="op">:</span>t T.pack</span>
 <span id="cb312-3"><a href="#cb312-3" aria-hidden="true" tabindex="-1"></a>T.pack<span class="ot"> ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">T.Text</span></span>
 <span id="cb312-4"><a href="#cb312-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span><span class="op">&gt;</span> phrase <span class="ot">=</span> T.pack <span class="st">&quot;brevity is the soul of wit&quot;</span></span>
@@ -3588,8 +4963,12 @@ None of the above
 <span id="cb312-6"><a href="#cb312-6" aria-hidden="true" tabindex="-1"></a><span class="ot">phrase ::</span> <span class="dt">T.Text</span></span>
 <span id="cb312-7"><a href="#cb312-7" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span><span class="op">&gt;</span> phrase</span>
 <span id="cb312-8"><a href="#cb312-8" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;brevity is the soul of wit&quot;</span></span></code></pre></div>
-<p>We can use the functions from <code>Data.Text</code> to operate on values of <code>Text</code>. Many of these are named like their counterparts for <code>String</code>s or lists from <code>Prelude</code>.</p>
-<div class="sourceCode" id="cb313"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb313-1"><a href="#cb313-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span><span class="op">&gt;</span> <span class="op">:</span>t T.length</span>
+<p>We can use the functions from <code>Data.Text</code> to operate on
+values of <code>Text</code>. Many of these are named like their
+counterparts for <code>String</code>s or lists from
+<code>Prelude</code>.</p>
+<div class="sourceCode" id="cb313"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb313-1"><a href="#cb313-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span><span class="op">&gt;</span> <span class="op">:</span>t T.length</span>
 <span id="cb313-2"><a href="#cb313-2" aria-hidden="true" tabindex="-1"></a>T.length<span class="ot"> ::</span> <span class="dt">T.Text</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb313-3"><a href="#cb313-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span><span class="op">&gt;</span> T.length phrase</span>
 <span id="cb313-4"><a href="#cb313-4" aria-hidden="true" tabindex="-1"></a><span class="dv">26</span></span>
@@ -3605,31 +4984,49 @@ None of the above
 <span id="cb313-14"><a href="#cb313-14" aria-hidden="true" tabindex="-1"></a>T.map<span class="ot"> ::</span> (<span class="dt">Char</span> <span class="ot">-&gt;</span> <span class="dt">Char</span>) <span class="ot">-&gt;</span> <span class="dt">T.Text</span> <span class="ot">-&gt;</span> <span class="dt">T.Text</span></span>
 <span id="cb313-15"><a href="#cb313-15" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span><span class="op">&gt;</span> T.map (\c <span class="ot">-&gt;</span> <span class="kw">if</span> c<span class="op">==</span><span class="ch">&#39;o&#39;</span> <span class="kw">then</span> <span class="ch">&#39;0&#39;</span> <span class="kw">else</span> c) phrase</span>
 <span id="cb313-16"><a href="#cb313-16" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;brevity is the s0ul 0f wit&quot;</span></span></code></pre></div>
-<p>A useful detail is that <code>Text</code> has a <code>Monoid</code> instance that glues <code>Text</code> values together. You can also use the functions <code>T.append</code> and <code>T.concat</code>.</p>
-<div class="sourceCode" id="cb314"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb314-1"><a href="#cb314-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span><span class="op">&gt;</span> phrase <span class="op">&lt;&gt;</span> phrase</span>
+<p>A useful detail is that <code>Text</code> has a <code>Monoid</code>
+instance that glues <code>Text</code> values together. You can also use
+the functions <code>T.append</code> and <code>T.concat</code>.</p>
+<div class="sourceCode" id="cb314"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb314-1"><a href="#cb314-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span><span class="op">&gt;</span> phrase <span class="op">&lt;&gt;</span> phrase</span>
 <span id="cb314-2"><a href="#cb314-2" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;brevity is the soul of witbrevity is the soul of wit&quot;</span></span>
 <span id="cb314-3"><a href="#cb314-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span><span class="op">&gt;</span> T.append phrase phrase</span>
 <span id="cb314-4"><a href="#cb314-4" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;brevity is the soul of witbrevity is the soul of wit&quot;</span></span>
 <span id="cb314-5"><a href="#cb314-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span><span class="op">&gt;</span> T.concat [phrase,phrase,phrase]</span>
 <span id="cb314-6"><a href="#cb314-6" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;brevity is the soul of witbrevity is the soul of witbrevity is the soul of wit&quot;</span></span></code></pre></div>
-<p>If you want to write a recursive function that pattern matches on a <code>Text</code> like you would on a <code>String</code>, you can use the function <code>T.uncons :: T.Text -&gt; Maybe (Char, T.Text)</code> to split a <code>Text</code> into a head and a tail. Here’s a simple example:</p>
-<div class="sourceCode" id="cb315"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb315-1"><a href="#cb315-1" aria-hidden="true" tabindex="-1"></a><span class="ot">countLetter ::</span> <span class="dt">Char</span> <span class="ot">-&gt;</span> <span class="dt">T.Text</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
+<p>If you want to write a recursive function that pattern matches on a
+<code>Text</code> like you would on a <code>String</code>, you can use
+the function <code>T.uncons :: T.Text -&gt; Maybe (Char, T.Text)</code>
+to split a <code>Text</code> into a head and a tail. Here’s a simple
+example:</p>
+<div class="sourceCode" id="cb315"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb315-1"><a href="#cb315-1" aria-hidden="true" tabindex="-1"></a><span class="ot">countLetter ::</span> <span class="dt">Char</span> <span class="ot">-&gt;</span> <span class="dt">T.Text</span> <span class="ot">-&gt;</span> <span class="dt">Int</span></span>
 <span id="cb315-2"><a href="#cb315-2" aria-hidden="true" tabindex="-1"></a>countLetter c t <span class="ot">=</span></span>
 <span id="cb315-3"><a href="#cb315-3" aria-hidden="true" tabindex="-1"></a>  <span class="kw">case</span> T.uncons t <span class="kw">of</span></span>
 <span id="cb315-4"><a href="#cb315-4" aria-hidden="true" tabindex="-1"></a>    <span class="dt">Nothing</span> <span class="ot">-&gt;</span> <span class="dv">0</span></span>
 <span id="cb315-5"><a href="#cb315-5" aria-hidden="true" tabindex="-1"></a>    <span class="dt">Just</span> (x,rest) <span class="ot">-&gt;</span> (<span class="kw">if</span> x <span class="op">==</span> c <span class="kw">then</span> <span class="dv">1</span> <span class="kw">else</span> <span class="dv">0</span>) <span class="op">+</span> countLetter c rest</span></code></pre></div>
-<div class="sourceCode" id="cb316"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb316-1"><a href="#cb316-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span><span class="op">&gt;</span> countLetter <span class="ch">&#39;t&#39;</span> phrase</span>
+<div class="sourceCode" id="cb316"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb316-1"><a href="#cb316-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span><span class="op">&gt;</span> countLetter <span class="ch">&#39;t&#39;</span> phrase</span>
 <span id="cb316-2"><a href="#cb316-2" aria-hidden="true" tabindex="-1"></a><span class="dv">3</span></span></code></pre></div>
-<section id="strictness-and-laziness" data-number="6.1.1.1">
-<h4 data-number="14.1.1.1"><span class="header-section-number">14.1.1.1</span> Strictness and Laziness</h4>
-<p>Note that <code>Data.Text</code> implements the strict <code>Text</code> type. You need to use <code>Data.Text.Lazy</code> for the lazy variant. As mentioned earlier, one difference between these two types is that the strict type does not work for infinite strings:</p>
-<div class="sourceCode" id="cb317"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb317-1"><a href="#cb317-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span><span class="op">&gt;</span> T.head (T.pack (<span class="fu">repeat</span> <span class="ch">&#39;x&#39;</span>))</span>
+<h4 data-number="14.1.1.1" id="strictness-and-laziness"><span
+class="header-section-number">14.1.1.1</span> Strictness and
+Laziness</h4>
+<p>Note that <code>Data.Text</code> implements the strict
+<code>Text</code> type. You need to use <code>Data.Text.Lazy</code> for
+the lazy variant. As mentioned earlier, one difference between these two
+types is that the strict type does not work for infinite strings:</p>
+<div class="sourceCode" id="cb317"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb317-1"><a href="#cb317-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span><span class="op">&gt;</span> T.head (T.pack (<span class="fu">repeat</span> <span class="ch">&#39;x&#39;</span>))</span>
 <span id="cb317-2"><a href="#cb317-2" aria-hidden="true" tabindex="-1"></a><span class="co">-- never returns</span></span>
 <span id="cb317-3"><a href="#cb317-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="kw">qualified</span> <span class="dt">Data.Text.Lazy</span> <span class="kw">as</span> <span class="dt">TL</span></span>
 <span id="cb317-4"><a href="#cb317-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span> <span class="dt">TL</span><span class="op">&gt;</span> TL.head (TL.pack (<span class="fu">repeat</span> <span class="ch">&#39;x&#39;</span>))</span>
 <span id="cb317-5"><a href="#cb317-5" aria-hidden="true" tabindex="-1"></a><span class="ch">&#39;x&#39;</span></span></code></pre></div>
-<p>Another practical problem is that you can end up with a mismatch between strict and lazy <code>Text</code>s when using libraries. You can usually fix this by using <code>toStrict</code> or <code>fromStrict</code> as needed.</p>
-<div class="sourceCode" id="cb318"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb318-1"><a href="#cb318-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span> <span class="dt">TL</span><span class="op">&gt;</span> lazyPhrase <span class="ot">=</span> TL.pack <span class="st">&quot;brevity is the soul of wit&quot;</span></span>
+<p>Another practical problem is that you can end up with a mismatch
+between strict and lazy <code>Text</code>s when using libraries. You can
+usually fix this by using <code>toStrict</code> or
+<code>fromStrict</code> as needed.</p>
+<div class="sourceCode" id="cb318"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb318-1"><a href="#cb318-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span> <span class="dt">TL</span><span class="op">&gt;</span> lazyPhrase <span class="ot">=</span> TL.pack <span class="st">&quot;brevity is the soul of wit&quot;</span></span>
 <span id="cb318-2"><a href="#cb318-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span> <span class="dt">TL</span><span class="op">&gt;</span> <span class="op">:</span>t lazyPhrase</span>
 <span id="cb318-3"><a href="#cb318-3" aria-hidden="true" tabindex="-1"></a><span class="ot">lazyPhrase ::</span> <span class="dt">TL.Text</span></span>
 <span id="cb318-4"><a href="#cb318-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span> <span class="dt">TL</span><span class="op">&gt;</span> <span class="op">:</span>t phrase</span>
@@ -3651,12 +5048,19 @@ None of the above
 <span id="cb318-20"><a href="#cb318-20" aria-hidden="true" tabindex="-1"></a>TL.fromStrict<span class="ot"> ::</span> <span class="dt">T.Text</span> <span class="ot">-&gt;</span> <span class="dt">TL.Text</span></span>
 <span id="cb318-21"><a href="#cb318-21" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span> <span class="dt">TL</span><span class="op">&gt;</span> TL.toStrict lazyPhrase <span class="op">==</span> phrase</span>
 <span id="cb318-22"><a href="#cb318-22" aria-hidden="true" tabindex="-1"></a><span class="dt">True</span></span></code></pre></div>
-</section>
-</section>
-<section id="examples-with-bytestring" data-number="6.1.2">
-<h3 data-number="14.1.2"><span class="header-section-number">14.1.2</span> Examples with <code>ByteString</code></h3>
-<p>We can walk through pretty much the same GHCi session using <code>ByteString</code> instead of <code>Text</code>. However, note how the <code>ByteString</code> is built up from <code>Word8</code> values and not <code>Char</code> values. A <code>Char</code> can represent an arbitrary unicode codepoint like for a character like <code>'Å'</code>, but a <code>Word8</code> represents a byte: a number from 0 to 255. Unfortunately and somewhat confusingly, <code>ByteString</code> values get printed like <code>String</code>s.</p>
-<div class="sourceCode" id="cb319"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb319-1"><a href="#cb319-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="dt">Data.Word</span></span>
+<h3 data-number="14.1.2" id="examples-with-bytestring"><span
+class="header-section-number">14.1.2</span> Examples with
+<code>ByteString</code></h3>
+<p>We can walk through pretty much the same GHCi session using
+<code>ByteString</code> instead of <code>Text</code>. However, note how
+the <code>ByteString</code> is built up from <code>Word8</code> values
+and not <code>Char</code> values. A <code>Char</code> can represent an
+arbitrary unicode codepoint like for a character like <code>'Å'</code>,
+but a <code>Word8</code> represents a byte: a number from 0 to 255.
+Unfortunately and somewhat confusingly, <code>ByteString</code> values
+get printed like <code>String</code>s.</p>
+<div class="sourceCode" id="cb319"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb319-1"><a href="#cb319-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="dt">Data.Word</span></span>
 <span id="cb319-2"><a href="#cb319-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.Word</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="kw">qualified</span> <span class="dt">Data.ByteString</span> <span class="kw">as</span> <span class="dt">B</span></span>
 <span id="cb319-3"><a href="#cb319-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.Word</span> <span class="dt">B</span><span class="op">&gt;</span> binary <span class="ot">=</span> B.pack [<span class="dv">99</span>,<span class="dv">111</span>,<span class="dv">102</span>,<span class="dv">102</span>,<span class="dv">101</span>,<span class="dv">101</span>]</span>
 <span id="cb319-4"><a href="#cb319-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.Word</span> <span class="dt">B</span><span class="op">&gt;</span> <span class="op">:</span>t binary</span>
@@ -3679,8 +5083,10 @@ None of the above
 <span id="cb319-21"><a href="#cb319-21" aria-hidden="true" tabindex="-1"></a>B.map<span class="ot"> ::</span> (<span class="dt">Word8</span> <span class="ot">-&gt;</span> <span class="dt">Word8</span>) <span class="ot">-&gt;</span> <span class="dt">B.ByteString</span> <span class="ot">-&gt;</span> <span class="dt">B.ByteString</span></span>
 <span id="cb319-22"><a href="#cb319-22" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.Word</span> <span class="dt">B</span><span class="op">&gt;</span> B.map (<span class="op">+</span><span class="dv">1</span>) binary</span>
 <span id="cb319-23"><a href="#cb319-23" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;dpggff&quot;</span></span></code></pre></div>
-<p>The same caveats apply to the differences between strict and lazy <code>ByteString</code> as for <code>Text</code>:</p>
-<div class="sourceCode" id="cb320"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb320-1"><a href="#cb320-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">B</span> <span class="dt">Data.Char</span><span class="op">&gt;</span> B.head (B.pack (<span class="fu">repeat</span> <span class="dv">99</span>))</span>
+<p>The same caveats apply to the differences between strict and lazy
+<code>ByteString</code> as for <code>Text</code>:</p>
+<div class="sourceCode" id="cb320"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb320-1"><a href="#cb320-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">B</span> <span class="dt">Data.Char</span><span class="op">&gt;</span> B.head (B.pack (<span class="fu">repeat</span> <span class="dv">99</span>))</span>
 <span id="cb320-2"><a href="#cb320-2" aria-hidden="true" tabindex="-1"></a><span class="co">-- never returns</span></span>
 <span id="cb320-3"><a href="#cb320-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.Word</span> <span class="dt">B</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="kw">qualified</span> <span class="dt">Data.ByteString.Lazy</span> <span class="kw">as</span> <span class="dt">BL</span></span>
 <span id="cb320-4"><a href="#cb320-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.Word</span> <span class="dt">B</span> <span class="dt">BL</span><span class="op">&gt;</span> BL.head (BL.pack (<span class="fu">repeat</span> <span class="dv">99</span>))</span>
@@ -3702,18 +5108,51 @@ None of the above
 <span id="cb320-20"><a href="#cb320-20" aria-hidden="true" tabindex="-1"></a>BL.fromStrict<span class="ot"> ::</span> <span class="dt">B.ByteString</span> <span class="ot">-&gt;</span> <span class="dt">BL.ByteString</span></span>
 <span id="cb320-21"><a href="#cb320-21" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Data.Word</span> <span class="dt">B</span> <span class="dt">BL</span><span class="op">&gt;</span> binary <span class="op">==</span> BL.toStrict (BL.pack [<span class="dv">99</span>])</span>
 <span id="cb320-22"><a href="#cb320-22" aria-hidden="true" tabindex="-1"></a><span class="dt">False</span></span></code></pre></div>
-</section>
-<section id="sidenote-encodings" data-number="6.1.3">
-<h3 data-number="14.1.3"><span class="header-section-number">14.1.3</span> Sidenote: Encodings</h3>
-<p>You might be wondering why on earth do we have both Text and ByteString. The difference is subtle but real. When we operate on <code>Text</code> we operate character by character, regardless of what those characters are and how they are encoded. When we operate on <code>ByteString</code> we operate on bytes, regardless of what those bytes represent.</p>
-<p>Characters, numbers, and data structures are abstractions that help us humans to deal with complex programming tasks. The computer memory is essentially just an enormous sequence of bytes. The machine doesn’t care how we interpret those bytes. The essential difference between <code>Text</code> and <code>ByteString</code> is the way how the bytes are grouped and interpreted.</p>
-<p>To illustrate this difference, we’ll look at the <a href="https://en.wikipedia.org/wiki/UTF-8">UTF-8 text encoding</a>. A text encoding is a method for representing <em>characters</em> as <em>bytes</em>. UTF-8 can represent all the millions of characters defined by <a href="https://en.wikipedia.org/wiki/Unicode">Unicode</a>. Because bytes can only store values from 0 to 255, this means that a character can get encoded into multiple bytes. The bits and bytes of the UTF-8 string “Ha∫keλ!” can be interpreted in various ways:</p>
+<h3 data-number="14.1.3" id="sidenote-encodings"><span
+class="header-section-number">14.1.3</span> Sidenote: Encodings</h3>
+<p>You might be wondering why on earth do we have both Text and
+ByteString. The difference is subtle but real. When we operate on
+<code>Text</code> we operate character by character, regardless of what
+those characters are and how they are encoded. When we operate on
+<code>ByteString</code> we operate on bytes, regardless of what those
+bytes represent.</p>
+<p>Characters, numbers, and data structures are abstractions that help
+us humans to deal with complex programming tasks. The computer memory is
+essentially just an enormous sequence of bytes. The machine doesn’t care
+how we interpret those bytes. The essential difference between
+<code>Text</code> and <code>ByteString</code> is the way how the bytes
+are grouped and interpreted.</p>
+<p>To illustrate this difference, we’ll look at the <a
+href="https://en.wikipedia.org/wiki/UTF-8">UTF-8 text encoding</a>. A
+text encoding is a method for representing <em>characters</em> as
+<em>bytes</em>. UTF-8 can represent all the millions of characters
+defined by <a href="https://en.wikipedia.org/wiki/Unicode">Unicode</a>.
+Because bytes can only store values from 0 to 255, this means that a
+character can get encoded into multiple bytes. The bits and bytes of the
+UTF-8 string “Ha∫keλ!” can be interpreted in various ways:</p>
 <p><img src="img/string.svg" /></p>
-<p>(If you see different characters in the picture and in “Ha∫keλ!”, it means that your browser is either interpreting the encoding incorrectly or the font you’re using doesn’t support all of the characters.)</p>
-<p>If we read the same stream of bits using a different encoding, we’ll see other characters. For example the string above would be interpreted as “Ha∫keλ!” using the <a href="https://en.wikipedia.org/wiki/Latin-1">Latin-1 text encoding</a>.</p>
-<p>By the way, when dealing with raw binary data, it’s often convenient to use the hexadecimal number system which uses a single symbol <code>0</code>, <code>1</code>, …, <code>9</code>, <code>A</code>, <code>B</code>, …, <code>F</code> to represent all of the sixteen possible combinations of four bits. We don’t need hexadecimal in this course but you can check out <a href="https://en.wikipedia.org/wiki/Hexadecimal">Wikipedia</a> if you’re interested in learning more about hexadecimal.</p>
-<p>We can explore the same example using code. The function <code>Data.Text.Encoding.encodeUtf8 :: Text -&gt; ByteString</code> encodes the characters in a Text into bytes in a ByteString using UTF-8.</p>
-<div class="sourceCode" id="cb321"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb321-1"><a href="#cb321-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="kw">qualified</span> <span class="dt">Data.Text</span> <span class="kw">as</span> <span class="dt">T</span></span>
+<p>(If you see different characters in the picture and in “Ha∫keλ!”, it
+means that your browser is either interpreting the encoding incorrectly
+or the font you’re using doesn’t support all of the characters.)</p>
+<p>If we read the same stream of bits using a different encoding, we’ll
+see other characters. For example the string above would be interpreted
+as “Ha∫keλ!” using the <a
+href="https://en.wikipedia.org/wiki/Latin-1">Latin-1 text
+encoding</a>.</p>
+<p>By the way, when dealing with raw binary data, it’s often convenient
+to use the hexadecimal number system which uses a single symbol
+<code>0</code>, <code>1</code>, …, <code>9</code>, <code>A</code>,
+<code>B</code>, …, <code>F</code> to represent all of the sixteen
+possible combinations of four bits. We don’t need hexadecimal in this
+course but you can check out <a
+href="https://en.wikipedia.org/wiki/Hexadecimal">Wikipedia</a> if you’re
+interested in learning more about hexadecimal.</p>
+<p>We can explore the same example using code. The function
+<code>Data.Text.Encoding.encodeUtf8 :: Text -&gt; ByteString</code>
+encodes the characters in a Text into bytes in a ByteString using
+UTF-8.</p>
+<div class="sourceCode" id="cb321"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb321-1"><a href="#cb321-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="kw">qualified</span> <span class="dt">Data.Text</span> <span class="kw">as</span> <span class="dt">T</span></span>
 <span id="cb321-2"><a href="#cb321-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="kw">qualified</span> <span class="dt">Data.ByteString</span> <span class="kw">as</span> <span class="dt">B</span></span>
 <span id="cb321-3"><a href="#cb321-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span> <span class="dt">B</span><span class="op">&gt;</span> T.length (T.pack <span class="st">&quot;haskell&quot;</span>)</span>
 <span id="cb321-4"><a href="#cb321-4" aria-hidden="true" tabindex="-1"></a><span class="dv">7</span></span>
@@ -3728,8 +5167,18 @@ None of the above
 <span id="cb321-13"><a href="#cb321-13" aria-hidden="true" tabindex="-1"></a><span class="dv">7</span></span>
 <span id="cb321-14"><a href="#cb321-14" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span> <span class="dt">B</span> <span class="dt">Data.Text.Encoding</span><span class="op">&gt;</span> B.length (encodeUtf8 (T.pack <span class="st">&quot;Ha∫keλ!&quot;</span>))</span>
 <span id="cb321-15"><a href="#cb321-15" aria-hidden="true" tabindex="-1"></a><span class="dv">10</span></span></code></pre></div>
-<p>If we are processing <a href="https://en.wikipedia.org/wiki/ASCII">ASCII text</a>, that is, characters that can be represented using single bytes, we can use <code>Text</code> and <code>ByteString</code> interchangeably. There are namespaces <code>Data.ByteString.Char8</code> and <code>Data.ByteString.Lazy.Char8</code> that offer functions that operate on <code>ByteString</code>s using <code>Char</code> values instead of <code>Word8</code>. However, care must be taken to make sure all the characters really are plain ASCII characters, or surprising things will happen.</p>
-<div class="sourceCode" id="cb322"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb322-1"><a href="#cb322-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span> <span class="dt">B</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="kw">qualified</span> <span class="dt">Data.ByteString.Char8</span> <span class="kw">as</span> <span class="dt">B8</span></span>
+<p>If we are processing <a
+href="https://en.wikipedia.org/wiki/ASCII">ASCII text</a>, that is,
+characters that can be represented using single bytes, we can use
+<code>Text</code> and <code>ByteString</code> interchangeably. There are
+namespaces <code>Data.ByteString.Char8</code> and
+<code>Data.ByteString.Lazy.Char8</code> that offer functions that
+operate on <code>ByteString</code>s using <code>Char</code> values
+instead of <code>Word8</code>. However, care must be taken to make sure
+all the characters really are plain ASCII characters, or surprising
+things will happen.</p>
+<div class="sourceCode" id="cb322"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb322-1"><a href="#cb322-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span> <span class="dt">B</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="kw">qualified</span> <span class="dt">Data.ByteString.Char8</span> <span class="kw">as</span> <span class="dt">B8</span></span>
 <span id="cb322-2"><a href="#cb322-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span> <span class="dt">B</span> <span class="dt">B8</span><span class="op">&gt;</span> B8.pack <span class="st">&quot;abc&quot;</span></span>
 <span id="cb322-3"><a href="#cb322-3" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;abc&quot;</span></span>
 <span id="cb322-4"><a href="#cb322-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span> <span class="dt">B</span> <span class="dt">B8</span><span class="op">&gt;</span> <span class="op">:</span>t B8.pack</span>
@@ -3742,36 +5191,65 @@ None of the above
 <span id="cb322-11"><a href="#cb322-11" aria-hidden="true" tabindex="-1"></a>¬»È</span>
 <span id="cb322-12"><a href="#cb322-12" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">T</span> <span class="dt">B</span> <span class="dt">B8</span><span class="op">&gt;</span> <span class="fu">putStrLn</span> (T.unpack (T.pack <span class="st">&quot;€λ훈&quot;</span>))</span>
 <span id="cb322-13"><a href="#cb322-13" aria-hidden="true" tabindex="-1"></a>€λ훈</span></code></pre></div>
-</section>
-</section>
-<section id="monads-recap" data-number="6.2">
-<h2 data-number="14.2"><span class="header-section-number">14.2</span> Monads: Recap</h2>
-<p>The next libraries we’ll look at work inside the IO monad. Here’s a short recap of what we learned about monads in the last lecture.</p>
+<h2 data-number="14.2" id="monads-recap"><span
+class="header-section-number">14.2</span> Monads: Recap</h2>
+<p>The next libraries we’ll look at work inside the IO monad. Here’s a
+short recap of what we learned about monads in the last lecture.</p>
 <ul>
-<li>When <code>M</code> is a monad, values of type <code>M X</code> are <em>operations</em> that can be <em>executed</em> to <em>produce</em> a value of type <code>X</code>.</li>
+<li>When <code>M</code> is a monad, values of type <code>M X</code> are
+<em>operations</em> that can be <em>executed</em> to <em>produce</em> a
+value of type <code>X</code>.</li>
 <li>Monadic operations can be implemented using
 <ul>
-<li>the methods of the <code>Monad</code> type class (<code>return</code>, <code>&gt;&gt;=</code>, <code>&gt;&gt;</code>),</li>
+<li>the methods of the <code>Monad</code> type class
+(<code>return</code>, <code>&gt;&gt;=</code>,
+<code>&gt;&gt;</code>),</li>
 <li><code>do</code>-notation,</li>
 <li>and library functions like <code>mapM</code>.</li>
 </ul></li>
-<li>Unlike in other languages, <code>return</code> is not a keyword and does not cause an operation to stop executing. Instead, <code>return x</code> <em>is the operation that always produces x and does nothing else</em>.</li>
+<li>Unlike in other languages, <code>return</code> is not a keyword and
+does not cause an operation to stop executing. Instead,
+<code>return x</code> <em>is the operation that always produces x and
+does nothing else</em>.</li>
 <li>This is what <code>do</code>-notation looks like:</li>
 </ul>
-<div class="sourceCode" id="cb323"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb323-1"><a href="#cb323-1" aria-hidden="true" tabindex="-1"></a>foo y <span class="ot">=</span> <span class="kw">do</span></span>
+<div class="sourceCode" id="cb323"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb323-1"><a href="#cb323-1" aria-hidden="true" tabindex="-1"></a>foo y <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb323-2"><a href="#cb323-2" aria-hidden="true" tabindex="-1"></a>  operation1         <span class="co">-- run an operation</span></span>
 <span id="cb323-3"><a href="#cb323-3" aria-hidden="true" tabindex="-1"></a>  val <span class="ot">&lt;-</span> operation2  <span class="co">-- run an operation and keep the produced value</span></span>
 <span id="cb323-4"><a href="#cb323-4" aria-hidden="true" tabindex="-1"></a>  operation3 val y   <span class="co">-- run an operation with parameters</span></span>
 <span id="cb323-5"><a href="#cb323-5" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mapM_</span> (\x <span class="ot">-&gt;</span> operation4 val x) things  <span class="co">-- use a generic monad operation and a lambda</span></span>
 <span id="cb323-6"><a href="#cb323-6" aria-hidden="true" tabindex="-1"></a>  operation5 val     <span class="co">-- the final line of the do decides which value the whole block produces</span></span></code></pre></div>
 <!-- TODO more? -->
-</section>
-<section id="writing-a-http-server-wai-and-warp" data-number="6.3">
-<h2 data-number="14.3"><span class="header-section-number">14.3</span> Writing a HTTP Server: WAI and Warp</h2>
-<p>Sometimes it feels like everything in the world happens over <a href="https://en.wikipedia.org/wiki/Hypertext_Transfer_Protocol">HTTP</a> and <a href="https://en.wikipedia.org/wiki/Web_API">Web Apis</a>. Your web browser, your smartphone apps, <a href="https://developer.nordeaopenbanking.com/">your bank</a>, <a href="https://tools.ietf.org/html/rfc2324">your coffee pot</a>, <a href="https://support.ring.com/hc/en-us/articles/205385394-The-Protocols-and-Ports-Used-by-Ring-Devices">and even your doorbell</a>, all talk to servers using the HTTP protocol.</p>
-<p>Let’s look at how we can set up a simple HTTP server in Haskell. The standard low level components for this are called <a href="https://hackage.haskell.org/package/wai-3.2.3/docs/Network-Wai.html">WAI</a> and <a href="https://hackage.haskell.org/package/warp-3.3.23">Warp</a>. WAI, the Web Application Interface gives us a way to define how HTTP requests are handled. Warp is a simple HTTP server that runs the logic we have defined using WAI. That probably sounds a bit abstract right now, but a simple example will help.</p>
-<p>The file <a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Examples/HelloServer.hs"><code>exercises/Examples/HelloServer.hs</code></a> implements a HTTP server that always responds with “Hello World!”. You can try it out by going to the <code>exercises/Examples</code> directory and running it with <code>stack runhaskell HelloServer.hs</code>. After that you can visit <a href="http://localhost:3421" class="uri">http://localhost:3421</a> in your browser to see the response from the server.</p>
-<div class="sourceCode" id="cb324"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb324-1"><a href="#cb324-1" aria-hidden="true" tabindex="-1"></a><span class="kw">module</span> <span class="dt">Examples.HelloServer</span> <span class="kw">where</span></span>
+<h2 data-number="14.3" id="writing-a-http-server-wai-and-warp"><span
+class="header-section-number">14.3</span> Writing a HTTP Server: WAI and
+Warp</h2>
+<p>Sometimes it feels like everything in the world happens over <a
+href="https://en.wikipedia.org/wiki/Hypertext_Transfer_Protocol">HTTP</a>
+and <a href="https://en.wikipedia.org/wiki/Web_API">Web Apis</a>. Your
+web browser, your smartphone apps, <a
+href="https://developer.nordeaopenbanking.com/">your bank</a>, <a
+href="https://tools.ietf.org/html/rfc2324">your coffee pot</a>, <a
+href="https://support.ring.com/hc/en-us/articles/205385394-The-Protocols-and-Ports-Used-by-Ring-Devices">and
+even your doorbell</a>, all talk to servers using the HTTP protocol.</p>
+<p>Let’s look at how we can set up a simple HTTP server in Haskell. The
+standard low level components for this are called <a
+href="https://hackage.haskell.org/package/wai-3.2.3/docs/Network-Wai.html">WAI</a>
+and <a href="https://hackage.haskell.org/package/warp-3.3.23">Warp</a>.
+WAI, the Web Application Interface gives us a way to define how HTTP
+requests are handled. Warp is a simple HTTP server that runs the logic
+we have defined using WAI. That probably sounds a bit abstract right
+now, but a simple example will help.</p>
+<p>The file <a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Examples/HelloServer.hs"><code>exercises/Examples/HelloServer.hs</code></a>
+implements a HTTP server that always responds with “Hello World!”. You
+can try it out by going to the <code>exercises/Examples</code> directory
+and running it with <code>stack runhaskell HelloServer.hs</code>. After
+that you can visit <a href="http://localhost:3421"
+class="uri">http://localhost:3421</a> in your browser to see the
+response from the server.</p>
+<div class="sourceCode" id="cb324"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb324-1"><a href="#cb324-1" aria-hidden="true" tabindex="-1"></a><span class="kw">module</span> <span class="dt">Examples.HelloServer</span> <span class="kw">where</span></span>
 <span id="cb324-2"><a href="#cb324-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb324-3"><a href="#cb324-3" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="kw">qualified</span> <span class="dt">Data.ByteString.Lazy.Char8</span> <span class="kw">as</span> <span class="dt">BL</span></span>
 <span id="cb324-4"><a href="#cb324-4" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="dt">Network.HTTP.Types.Status</span> (status200)</span>
@@ -3788,40 +5266,128 @@ None of the above
 <span id="cb324-15"><a href="#cb324-15" aria-hidden="true" tabindex="-1"></a><span class="ot">application ::</span> <span class="dt">Application</span></span>
 <span id="cb324-16"><a href="#cb324-16" aria-hidden="true" tabindex="-1"></a>application request respond <span class="ot">=</span></span>
 <span id="cb324-17"><a href="#cb324-17" aria-hidden="true" tabindex="-1"></a>  respond (responseLBS status200 [] (BL.pack <span class="st">&quot;Hello World!&quot;</span>))</span></code></pre></div>
-<p>Let’s look at the types in this example. There’s a lot going on here. First off, <code>Application</code> is a <em>type alias</em> for a thing that implements the logic of a web server. The <a href="https://hackage.haskell.org/package/warp-3.3.23/docs/Network-Wai-Handler-Warp.html#v:run"><code>run</code> function from Warp</a> can run <code>Application</code>s:</p>
-<div class="sourceCode" id="cb325"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb325-1"><a href="#cb325-1" aria-hidden="true" tabindex="-1"></a><span class="ot">run ::</span> <span class="dt">Port</span> <span class="ot">-&gt;</span> <span class="dt">Application</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span>
+<p>Let’s look at the types in this example. There’s a lot going on here.
+First off, <code>Application</code> is a <em>type alias</em> for a thing
+that implements the logic of a web server. The <a
+href="https://hackage.haskell.org/package/warp-3.3.23/docs/Network-Wai-Handler-Warp.html#v:run"><code>run</code>
+function from Warp</a> can run <code>Application</code>s:</p>
+<div class="sourceCode" id="cb325"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb325-1"><a href="#cb325-1" aria-hidden="true" tabindex="-1"></a><span class="ot">run ::</span> <span class="dt">Port</span> <span class="ot">-&gt;</span> <span class="dt">Application</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span>
 <span id="cb325-2"><a href="#cb325-2" aria-hidden="true" tabindex="-1"></a><span class="kw">type</span> <span class="dt">Application</span> <span class="ot">=</span> <span class="dt">Request</span> <span class="ot">-&gt;</span> (<span class="dt">Response</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> <span class="dt">ResponseReceived</span>) <span class="ot">-&gt;</span> <span class="dt">IO</span> <span class="dt">ResponseReceived</span></span></code></pre></div>
-<p>We’ll talk about what <code>Request</code> and <code>Response</code> are soon, but from this type we can see that an <code>Application</code> is an IO operation that takes as arguments a request of type <code>Request</code>, and an IO operation <code>respond :: Response -&gt; IO ResponseReceived</code>. Arguments like <code>respond</code> are called <em>callbacks</em> in many contexts. They allow us to call back to the library who called the application. The <code>Application</code> operation has to produce the same special <code>ResponseReceived</code> type that <code>respond</code>. You can think of this type as a token proving that <code>respond</code> was called by the <code>Application</code>.</p>
-<p>That might sound intimidating: but looking at the code things are relatively simple: our <code>server</code> is an <code>Application</code> and takes two parameters: <code>request</code> and <code>respond</code>.</p>
-<p>WAI uses lots of types like <code>Port</code>, <code>Request</code>, <code>Response</code>, <code>Status</code> to represent HTTP concepts. It’s useful to look these up in the documentation when you bump into them. For example <a href="https://hackage.haskell.org/package/warp-3.3.23/docs/Network-Wai-Handler-Warp.html#t:Port"><code>Port</code> is just an alias for <code>Int</code></a>. As another example, we can see the <a href="https://hackage.haskell.org/package/wai-3.2.3/docs/Network-Wai.html#v:responseLBS"><code>responseLBS</code> function</a> has the type</p>
-<div class="sourceCode" id="cb326"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb326-1"><a href="#cb326-1" aria-hidden="true" tabindex="-1"></a><span class="ot">responseLBS ::</span> <span class="dt">Status</span> <span class="ot">-&gt;</span> <span class="dt">ResponseHeaders</span> <span class="ot">-&gt;</span> <span class="dt">ByteString</span> <span class="ot">-&gt;</span> <span class="dt">Response</span></span></code></pre></div>
-<p>where <a href="https://hackage.haskell.org/package/http-types-0.12.3/docs/Network-HTTP-Types-Status.html#t:Status"><code>Status</code> is defined in <code>Network.HTTP.Types.Status</code></a>, <a href="https://hackage.haskell.org/package/http-types-0.12.3/docs/Network-HTTP-Types-Header.html#t:ResponseHeaders"><code>ResponseHeaders</code> is a type alias for <code>[Header]</code> from <code>Network.HTTP.Types.Header</code></a>, <code>ByteString</code> is a lazy <code>ByteString</code>, and the result type <a href="https://hackage.haskell.org/package/wai-3.2.3/docs/Network-Wai.html#t:Response"><code>Response</code> is defined by <code>Network.WAI</code></a>.</p>
-<p>Finally, note that we take a shortcut and use the function <code>Data.ByteString.Lazy.Char8.pack</code> to convert a <code>String</code> into a <code>ByteString</code>. This only works for ASCII text.</p>
-<p>A web server that always responds with the same text isn’t that interesting. Let’s look at how we can give different responses to different requests next. There are many parts in a HTTP request, but for this lecture we’re going to focus on the <em>path</em>. In a URL like <code>http://example.com/abcd/ef/file</code>, the <code>/abcd/ef/file</code> part is the path. WAI has the function</p>
-<div class="sourceCode" id="cb327"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb327-1"><a href="#cb327-1" aria-hidden="true" tabindex="-1"></a><span class="ot">pathInfo ::</span> <span class="dt">Request</span> <span class="ot">-&gt;</span> [<span class="dt">Text</span>]</span></code></pre></div>
-<p>This gives us the path of the requested URL, split at the <code>/</code> characters.</p>
-<p>The file <a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Examples/PathServer.hs"><code>exercises/Examples/PathServer.hs</code></a> implements a web server that has three different pages:</p>
+<p>We’ll talk about what <code>Request</code> and <code>Response</code>
+are soon, but from this type we can see that an <code>Application</code>
+is an IO operation that takes as arguments a request of type
+<code>Request</code>, and an IO operation
+<code>respond :: Response -&gt; IO ResponseReceived</code>. Arguments
+like <code>respond</code> are called <em>callbacks</em> in many
+contexts. They allow us to call back to the library who called the
+application. The <code>Application</code> operation has to produce the
+same special <code>ResponseReceived</code> type that
+<code>respond</code>. You can think of this type as a token proving that
+<code>respond</code> was called by the <code>Application</code>.</p>
+<p>That might sound intimidating: but looking at the code things are
+relatively simple: our <code>server</code> is an
+<code>Application</code> and takes two parameters: <code>request</code>
+and <code>respond</code>.</p>
+<p>WAI uses lots of types like <code>Port</code>, <code>Request</code>,
+<code>Response</code>, <code>Status</code> to represent HTTP concepts.
+It’s useful to look these up in the documentation when you bump into
+them. For example <a
+href="https://hackage.haskell.org/package/warp-3.3.23/docs/Network-Wai-Handler-Warp.html#t:Port"><code>Port</code>
+is just an alias for <code>Int</code></a>. As another example, we can
+see the <a
+href="https://hackage.haskell.org/package/wai-3.2.3/docs/Network-Wai.html#v:responseLBS"><code>responseLBS</code>
+function</a> has the type</p>
+<div class="sourceCode" id="cb326"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb326-1"><a href="#cb326-1" aria-hidden="true" tabindex="-1"></a><span class="ot">responseLBS ::</span> <span class="dt">Status</span> <span class="ot">-&gt;</span> <span class="dt">ResponseHeaders</span> <span class="ot">-&gt;</span> <span class="dt">ByteString</span> <span class="ot">-&gt;</span> <span class="dt">Response</span></span></code></pre></div>
+<p>where <a
+href="https://hackage.haskell.org/package/http-types-0.12.3/docs/Network-HTTP-Types-Status.html#t:Status"><code>Status</code>
+is defined in <code>Network.HTTP.Types.Status</code></a>, <a
+href="https://hackage.haskell.org/package/http-types-0.12.3/docs/Network-HTTP-Types-Header.html#t:ResponseHeaders"><code>ResponseHeaders</code>
+is a type alias for <code>[Header]</code> from
+<code>Network.HTTP.Types.Header</code></a>, <code>ByteString</code> is a
+lazy <code>ByteString</code>, and the result type <a
+href="https://hackage.haskell.org/package/wai-3.2.3/docs/Network-Wai.html#t:Response"><code>Response</code>
+is defined by <code>Network.WAI</code></a>.</p>
+<p>Finally, note that we take a shortcut and use the function
+<code>Data.ByteString.Lazy.Char8.pack</code> to convert a
+<code>String</code> into a <code>ByteString</code>. This only works for
+ASCII text.</p>
+<p>A web server that always responds with the same text isn’t that
+interesting. Let’s look at how we can give different responses to
+different requests next. There are many parts in a HTTP request, but for
+this lecture we’re going to focus on the <em>path</em>. In a URL like
+<code>http://example.com/abcd/ef/file</code>, the
+<code>/abcd/ef/file</code> part is the path. WAI has the function</p>
+<div class="sourceCode" id="cb327"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb327-1"><a href="#cb327-1" aria-hidden="true" tabindex="-1"></a><span class="ot">pathInfo ::</span> <span class="dt">Request</span> <span class="ot">-&gt;</span> [<span class="dt">Text</span>]</span></code></pre></div>
+<p>This gives us the path of the requested URL, split at the
+<code>/</code> characters.</p>
+<p>The file <a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Examples/PathServer.hs"><code>exercises/Examples/PathServer.hs</code></a>
+implements a web server that has three different pages:</p>
 <ul>
-<li><a href="http://localhost:3421/source" class="uri">http://localhost:3421/source</a> is the source of the application itself, read from the file system</li>
-<li><a href="http://localhost:3421/secret/file" class="uri">http://localhost:3421/secret/file</a> is a secret string</li>
-<li><a href="http://localhost:3421/anything/else" class="uri">http://localhost:3421/anything/else</a> - for all other paths, a “Not found: anything/else” text is shown</li>
+<li><a href="http://localhost:3421/source"
+class="uri">http://localhost:3421/source</a> is the source of the
+application itself, read from the file system</li>
+<li><a href="http://localhost:3421/secret/file"
+class="uri">http://localhost:3421/secret/file</a> is a secret
+string</li>
+<li><a href="http://localhost:3421/anything/else"
+class="uri">http://localhost:3421/anything/else</a> - for all other
+paths, a “Not found: anything/else” text is shown</li>
 </ul>
-<p>As before, you can run the server by going into the <code>exercises/Examples</code> directory, and running <code>stack runhaskell PathServer.hs</code>.</p>
+<p>As before, you can run the server by going into the
+<code>exercises/Examples</code> directory, and running
+<code>stack runhaskell PathServer.hs</code>.</p>
 <!-- TODO walk through the code of the PathServer? -->
-</section>
-<section id="working-with-a-database-sqlite-simple" data-number="6.4">
-<h2 data-number="14.4"><span class="header-section-number">14.4</span> Working With a Database: sqlite-simple</h2>
-<p>After implementing a HTTP server we can participate in the global graph of applications talking to each other that’s called the internet. But what use is talking if we can’t remember? A real application needs to be able to <em>persist data</em> even when it is restarted. A common way to accomplish this is to use a database.</p>
-<p>There are many different kinds of databases, but arguably the most widely used simple database is <a href="https://www.sqlite.org/index.html">SQLite</a>. SQLite is a library that lets you store data in a file and process it using <a href="">SQL, the Structured Query Language</a>. With SQLite there is no need to run a separate database server a la <a href="https://www.postgresql.org/">PostgreSQL</a> or <a href="https://www.mysql.com/">MySQL</a>.</p>
-<p>In case you’re not familiar with SQL, don’t worry, you won’t need to write any queries of your own in the exercises. If you’d like to learn a bit of SQL now, there are lots of tutorials on the web. See <a href="https://www.w3schools.com/sql/">W3Schools</a>, <a href="https://sqlzoo.net/">SQL Zoo</a> or <a href="https://www.codecademy.com/learn/learn-sql">Codecademy</a>.</p>
-<p>There are many libraries for Haskell for using SQLite, but we’ll look at one called <a href="https://hackage.haskell.org/package/sqlite-simple-0.4.18.2">sqlite-simple</a> here. Let’s explore the library in GHCi for a bit.</p>
-<p>All the functions live inside <code>Database.SQLite.Simple</code>. You can open a database by giving a filename to <code>open</code>, which is an IO operation that produces a <code>Connection</code>.</p>
-<div class="sourceCode" id="cb328"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb328-1"><a href="#cb328-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="dt">Database.SQLite.Simple</span></span>
+<h2 data-number="14.4" id="working-with-a-database-sqlite-simple"><span
+class="header-section-number">14.4</span> Working With a Database:
+sqlite-simple</h2>
+<p>After implementing a HTTP server we can participate in the global
+graph of applications talking to each other that’s called the internet.
+But what use is talking if we can’t remember? A real application needs
+to be able to <em>persist data</em> even when it is restarted. A common
+way to accomplish this is to use a database.</p>
+<p>There are many different kinds of databases, but arguably the most
+widely used simple database is <a
+href="https://www.sqlite.org/index.html">SQLite</a>. SQLite is a library
+that lets you store data in a file and process it using <a href="">SQL,
+the Structured Query Language</a>. With SQLite there is no need to run a
+separate database server a la <a
+href="https://www.postgresql.org/">PostgreSQL</a> or <a
+href="https://www.mysql.com/">MySQL</a>.</p>
+<p>In case you’re not familiar with SQL, don’t worry, you won’t need to
+write any queries of your own in the exercises. If you’d like to learn a
+bit of SQL now, there are lots of tutorials on the web. See <a
+href="https://www.w3schools.com/sql/">W3Schools</a>, <a
+href="https://sqlzoo.net/">SQL Zoo</a> or <a
+href="https://www.codecademy.com/learn/learn-sql">Codecademy</a>.</p>
+<p>There are many libraries for Haskell for using SQLite, but we’ll look
+at one called <a
+href="https://hackage.haskell.org/package/sqlite-simple-0.4.18.2">sqlite-simple</a>
+here. Let’s explore the library in GHCi for a bit.</p>
+<p>All the functions live inside <code>Database.SQLite.Simple</code>.
+You can open a database by giving a filename to <code>open</code>, which
+is an IO operation that produces a <code>Connection</code>.</p>
+<div class="sourceCode" id="cb328"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb328-1"><a href="#cb328-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="dt">Database.SQLite.Simple</span></span>
 <span id="cb328-2"><a href="#cb328-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Database.SQLite.Simple</span><span class="op">&gt;</span> <span class="op">:</span>t open</span>
 <span id="cb328-3"><a href="#cb328-3" aria-hidden="true" tabindex="-1"></a><span class="ot">open ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> <span class="dt">Connection</span></span>
 <span id="cb328-4"><a href="#cb328-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Database.SQLite.Simple</span><span class="op">&gt;</span> db <span class="ot">&lt;-</span> open <span class="st">&quot;example.sqlite&quot;</span></span></code></pre></div>
-<p>To run an SQL query you can use IO operation <code>query_</code>, which takes a <code>Connection</code> and a <code>Query</code>, and produces a list of results. The <code>Query</code> type is just a simple <code>newtype</code> around <code>Text</code>. The result type of <code>query_</code>is polymorphic: you can read any type that satisfies the <code>FromRow</code> type class from the database. If this feels confusing, compare it to the type of <code>read</code>: <code>Read a =&gt; String -&gt; a</code>. The <code>FromRow</code> class is like <code>Read</code> for this database: it represents types that can be read out of the database. Anyway, let’s read the number <code>1</code> from the database:</p>
-<div class="sourceCode" id="cb329"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb329-1"><a href="#cb329-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Database.SQLite.Simple</span><span class="op">&gt;</span> <span class="op">:</span>t query_</span>
+<p>To run an SQL query you can use IO operation <code>query_</code>,
+which takes a <code>Connection</code> and a <code>Query</code>, and
+produces a list of results. The <code>Query</code> type is just a simple
+<code>newtype</code> around <code>Text</code>. The result type of
+<code>query_</code>is polymorphic: you can read any type that satisfies
+the <code>FromRow</code> type class from the database. If this feels
+confusing, compare it to the type of <code>read</code>:
+<code>Read a =&gt; String -&gt; a</code>. The <code>FromRow</code> class
+is like <code>Read</code> for this database: it represents types that
+can be read out of the database. Anyway, let’s read the number
+<code>1</code> from the database:</p>
+<div class="sourceCode" id="cb329"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb329-1"><a href="#cb329-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Database.SQLite.Simple</span><span class="op">&gt;</span> <span class="op">:</span>t query_</span>
 <span id="cb329-2"><a href="#cb329-2" aria-hidden="true" tabindex="-1"></a><span class="ot">query_ ::</span> <span class="dt">FromRow</span> r <span class="ot">=&gt;</span> <span class="dt">Connection</span> <span class="ot">-&gt;</span> <span class="dt">Query</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> [r]</span>
 <span id="cb329-3"><a href="#cb329-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Database.SQLite.Simple</span><span class="op">&gt;</span> <span class="op">:</span>info <span class="dt">Query</span></span>
 <span id="cb329-4"><a href="#cb329-4" aria-hidden="true" tabindex="-1"></a><span class="kw">newtype</span> <span class="dt">Query</span> <span class="ot">=</span> <span class="dt">Query</span> {<span class="ot">fromQuery ::</span> <span class="dt">Data.Text.Internal.Text</span>}</span>
@@ -3832,34 +5398,64 @@ None of the above
 <span id="cb329-9"><a href="#cb329-9" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Database.SQLite.Simple</span> <span class="dt">T</span><span class="op">&gt;</span> res <span class="ot">&lt;-</span> query_ db<span class="ot"> q ::</span> <span class="dt">IO</span> [[<span class="dt">Int</span>]]</span>
 <span id="cb329-10"><a href="#cb329-10" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Database.SQLite.Simple</span> <span class="dt">T</span><span class="op">&gt;</span> res</span>
 <span id="cb329-11"><a href="#cb329-11" aria-hidden="true" tabindex="-1"></a>[[<span class="dv">1</span>]]</span></code></pre></div>
-<p>By the way, all of these initial examples use simple <code>SELECT x, y, z;</code> queries that just return constant data. We’ll worry about actual tables in the database later.</p>
-<p>Without the type signature, we get an error from GHCi, which can’t decide which type we want to read out from the database:</p>
-<div class="sourceCode" id="cb330"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb330-1"><a href="#cb330-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Database.SQLite.Simple</span> <span class="dt">T</span><span class="op">&gt;</span> res <span class="ot">&lt;-</span> query_ db q</span>
+<p>By the way, all of these initial examples use simple
+<code>SELECT x, y, z;</code> queries that just return constant data.
+We’ll worry about actual tables in the database later.</p>
+<p>Without the type signature, we get an error from GHCi, which can’t
+decide which type we want to read out from the database:</p>
+<div class="sourceCode" id="cb330"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb330-1"><a href="#cb330-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Database.SQLite.Simple</span> <span class="dt">T</span><span class="op">&gt;</span> res <span class="ot">&lt;-</span> query_ db q</span>
 <span id="cb330-2"><a href="#cb330-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb330-3"><a href="#cb330-3" aria-hidden="true" tabindex="-1"></a><span class="op">&lt;</span>interactive<span class="op">&gt;:</span><span class="dv">17</span><span class="op">:</span><span class="dv">8</span><span class="op">:</span> <span class="fu">error</span><span class="op">:</span></span>
 <span id="cb330-4"><a href="#cb330-4" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">Ambiguous</span> <span class="kw">type</span> variable ‘r0’ arising from a use <span class="kw">of</span> ‘query_’</span>
 <span id="cb330-5"><a href="#cb330-5" aria-hidden="true" tabindex="-1"></a>      prevents the constraint ‘(<span class="dt">FromRow</span> r0)’ from being solved<span class="op">.</span></span>
 <span id="cb330-6"><a href="#cb330-6" aria-hidden="true" tabindex="-1"></a>      <span class="dt">Probable</span> fix<span class="op">:</span> use a <span class="kw">type</span> annotation to specify what ‘r0’ should be<span class="op">.</span></span>
 <span id="cb330-7"><a href="#cb330-7" aria-hidden="true" tabindex="-1"></a>      <span class="co">-- rest of error omitted</span></span></code></pre></div>
-<p>Before we go on, let’s have a closer look at <code>FromRow</code>. If you’ve bumped into SQL before, you know that an SQL query returns a number of <em>rows</em>, and each row consists of a number of <em>values</em> (also called <em>columns</em>). To be able to interpret the result of an SQL query into Haskell data, we need a way to interpret these values and rows. Thus sqlite-simple defines two classes, <code>FromField</code> and <code>FromRow</code>, and a bunch of instances like the following. (You can find these instances from <a href="https://hackage.haskell.org/package/sqlite-simple-0.4.18.2/docs/Database-SQLite-Simple.html#t:FromRow">the docs</a> or by asking GHCi with <code>:info FromRow</code> etc.)</p>
-<div class="sourceCode" id="cb331"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb331-1"><a href="#cb331-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">FromField</span> <span class="dt">Int</span></span>
+<p>Before we go on, let’s have a closer look at <code>FromRow</code>. If
+you’ve bumped into SQL before, you know that an SQL query returns a
+number of <em>rows</em>, and each row consists of a number of
+<em>values</em> (also called <em>columns</em>). To be able to interpret
+the result of an SQL query into Haskell data, we need a way to interpret
+these values and rows. Thus sqlite-simple defines two classes,
+<code>FromField</code> and <code>FromRow</code>, and a bunch of
+instances like the following. (You can find these instances from <a
+href="https://hackage.haskell.org/package/sqlite-simple-0.4.18.2/docs/Database-SQLite-Simple.html#t:FromRow">the
+docs</a> or by asking GHCi with <code>:info FromRow</code> etc.)</p>
+<div class="sourceCode" id="cb331"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb331-1"><a href="#cb331-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">FromField</span> <span class="dt">Int</span></span>
 <span id="cb331-2"><a href="#cb331-2" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">FromField</span> <span class="dt">Bool</span></span>
 <span id="cb331-3"><a href="#cb331-3" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">FromField</span> <span class="dt">String</span></span>
 <span id="cb331-4"><a href="#cb331-4" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">FromField</span> <span class="dt">Text</span></span>
 <span id="cb331-5"><a href="#cb331-5" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">FromField</span> a <span class="ot">=&gt;</span> <span class="dt">FromRow</span> [a]</span>
 <span id="cb331-6"><a href="#cb331-6" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> (<span class="dt">FromField</span> a, <span class="dt">FromField</span> b) <span class="ot">=&gt;</span> <span class="dt">FromRow</span> (a,b)</span>
 <span id="cb331-7"><a href="#cb331-7" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> (<span class="dt">FromField</span> a, <span class="dt">FromField</span> b, <span class="dt">FromField</span> c) <span class="ot">=&gt;</span> <span class="dt">FromRow</span> (a,b,c)</span></code></pre></div>
-<p>In essence, basic Haskell datatypes fulfill the <code>FromField</code> class, and various Haskell collections fulfill the <code>FromRow</code> class. Our earlier example was using the <code>FromRow [a]</code> and <code>FromField Int</code> instances to get a <code>[[Int]]</code> out of <code>query_</code>. Here’s a simple query that uses some other datatypes:</p>
-<div class="sourceCode" id="cb332"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb332-1"><a href="#cb332-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Database.SQLite.Simple</span> <span class="dt">T</span><span class="op">&gt;</span> q <span class="ot">=</span> <span class="dt">Query</span> (T.pack <span class="st">&quot;SELECT 1, true, &#39;string&#39;;&quot;</span>)</span>
+<p>In essence, basic Haskell datatypes fulfill the
+<code>FromField</code> class, and various Haskell collections fulfill
+the <code>FromRow</code> class. Our earlier example was using the
+<code>FromRow [a]</code> and <code>FromField Int</code> instances to get
+a <code>[[Int]]</code> out of <code>query_</code>. Here’s a simple query
+that uses some other datatypes:</p>
+<div class="sourceCode" id="cb332"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb332-1"><a href="#cb332-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Database.SQLite.Simple</span> <span class="dt">T</span><span class="op">&gt;</span> q <span class="ot">=</span> <span class="dt">Query</span> (T.pack <span class="st">&quot;SELECT 1, true, &#39;string&#39;;&quot;</span>)</span>
 <span id="cb332-2"><a href="#cb332-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Database.SQLite.Simple</span> <span class="dt">T</span><span class="op">&gt;</span> query_ db<span class="ot"> q ::</span> <span class="dt">IO</span> [(<span class="dt">Int</span>,<span class="dt">Bool</span>,<span class="dt">String</span>)]</span>
 <span id="cb332-3"><a href="#cb332-3" aria-hidden="true" tabindex="-1"></a>[(<span class="dv">1</span>,<span class="dt">True</span>,<span class="st">&quot;string&quot;</span>)]</span></code></pre></div>
-<p>What happens if the SQL and Haskell types don’t match? Well, you get a runtime error, just like if you try to invoke <code>read "True" :: Int</code>.</p>
-<div class="sourceCode" id="cb333"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb333-1"><a href="#cb333-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Database.SQLite.Simple</span> <span class="dt">T</span><span class="op">&gt;</span> query_ db<span class="ot"> q ::</span> <span class="dt">IO</span> [(<span class="dt">Int</span>,<span class="dt">Int</span>,<span class="dt">Int</span>)]</span>
+<p>What happens if the SQL and Haskell types don’t match? Well, you get
+a runtime error, just like if you try to invoke
+<code>read "True" :: Int</code>.</p>
+<div class="sourceCode" id="cb333"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb333-1"><a href="#cb333-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Database.SQLite.Simple</span> <span class="dt">T</span><span class="op">&gt;</span> query_ db<span class="ot"> q ::</span> <span class="dt">IO</span> [(<span class="dt">Int</span>,<span class="dt">Int</span>,<span class="dt">Int</span>)]</span>
 <span id="cb333-2"><a href="#cb333-2" aria-hidden="true" tabindex="-1"></a><span class="op">***</span> <span class="dt">Exception</span><span class="op">:</span> <span class="dt">ConversionFailed</span> {errSQLType <span class="ot">=</span> <span class="st">&quot;TEXT&quot;</span>, errHaskellType <span class="ot">=</span> <span class="st">&quot;Int&quot;</span>, errMessage <span class="ot">=</span> <span class="st">&quot;need an int&quot;</span>}</span></code></pre></div>
-<p>To mirror the <code>FromRow</code> and <code>FromField</code> classes, sqlite-simple also defines the <code>ToRow</code> and <code>ToField</code> classes for writing into the database. Here’s the type of the <code>query</code> function, which allows us to use <em>parameterized queries</em>.</p>
-<div class="sourceCode" id="cb334"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb334-1"><a href="#cb334-1" aria-hidden="true" tabindex="-1"></a><span class="ot">query ::</span> (<span class="dt">ToRow</span> q, <span class="dt">FromRow</span> r) <span class="ot">=&gt;</span> <span class="dt">Connection</span> <span class="ot">-&gt;</span> <span class="dt">Query</span> <span class="ot">-&gt;</span> q <span class="ot">-&gt;</span> <span class="dt">IO</span> [r]</span></code></pre></div>
-<p>And here are some instances for <code>ToRow</code> and <code>ToField</code>:</p>
-<div class="sourceCode" id="cb335"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb335-1"><a href="#cb335-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">ToField</span> <span class="dt">Int</span></span>
+<p>To mirror the <code>FromRow</code> and <code>FromField</code>
+classes, sqlite-simple also defines the <code>ToRow</code> and
+<code>ToField</code> classes for writing into the database. Here’s the
+type of the <code>query</code> function, which allows us to use
+<em>parameterized queries</em>.</p>
+<div class="sourceCode" id="cb334"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb334-1"><a href="#cb334-1" aria-hidden="true" tabindex="-1"></a><span class="ot">query ::</span> (<span class="dt">ToRow</span> q, <span class="dt">FromRow</span> r) <span class="ot">=&gt;</span> <span class="dt">Connection</span> <span class="ot">-&gt;</span> <span class="dt">Query</span> <span class="ot">-&gt;</span> q <span class="ot">-&gt;</span> <span class="dt">IO</span> [r]</span></code></pre></div>
+<p>And here are some instances for <code>ToRow</code> and
+<code>ToField</code>:</p>
+<div class="sourceCode" id="cb335"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb335-1"><a href="#cb335-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">ToField</span> <span class="dt">Int</span></span>
 <span id="cb335-2"><a href="#cb335-2" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">ToField</span> <span class="dt">Bool</span></span>
 <span id="cb335-3"><a href="#cb335-3" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">ToField</span> <span class="dt">String</span></span>
 <span id="cb335-4"><a href="#cb335-4" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">ToField</span> <span class="dt">Text</span></span>
@@ -3870,21 +5466,43 @@ None of the above
 <span id="cb335-9"><a href="#cb335-9" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> (<span class="dt">ToField</span> a, <span class="dt">ToField</span> b, <span class="dt">ToField</span> c) <span class="ot">=&gt;</span> <span class="dt">ToRow</span> (a, b, c)</span>
 <span id="cb335-10"><a href="#cb335-10" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">ToField</span> a <span class="ot">=&gt;</span> <span class="dt">ToRow</span> (<span class="dt">Only</span> a)</span></code></pre></div>
 <!-- instance ToField a => ToField (Maybe a)  -- TODO talk about sql null -->
-<p>Parameterized queries use a <code>?</code> character to denote slots where parameters can be passed in. Here’s a simple example:</p>
-<div class="sourceCode" id="cb336"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb336-1"><a href="#cb336-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Database.SQLite.Simple</span> <span class="dt">T</span><span class="op">&gt;</span> input <span class="ot">=</span> (<span class="dv">1</span>,<span class="st">&quot;hello&quot;</span>)<span class="ot"> ::</span> (<span class="dt">Int</span>,<span class="dt">String</span>)</span>
+<p>Parameterized queries use a <code>?</code> character to denote slots
+where parameters can be passed in. Here’s a simple example:</p>
+<div class="sourceCode" id="cb336"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb336-1"><a href="#cb336-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Database.SQLite.Simple</span> <span class="dt">T</span><span class="op">&gt;</span> input <span class="ot">=</span> (<span class="dv">1</span>,<span class="st">&quot;hello&quot;</span>)<span class="ot"> ::</span> (<span class="dt">Int</span>,<span class="dt">String</span>)</span>
 <span id="cb336-2"><a href="#cb336-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Database.SQLite.Simple</span> <span class="dt">T</span><span class="op">&gt;</span> parameterized <span class="ot">=</span> <span class="dt">Query</span> (T.pack <span class="st">&quot;SELECT ?+1, true, ?;&quot;</span>)</span>
 <span id="cb336-3"><a href="#cb336-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Database.SQLite.Simple</span> <span class="dt">T</span><span class="op">&gt;</span> query db parameterized<span class="ot"> input ::</span> <span class="dt">IO</span> [(<span class="dt">Int</span>,<span class="dt">Bool</span>,<span class="dt">String</span>)]</span>
 <span id="cb336-4"><a href="#cb336-4" aria-hidden="true" tabindex="-1"></a>[(<span class="dv">2</span>,<span class="dt">True</span>,<span class="st">&quot;hello&quot;</span>)]</span></code></pre></div>
-<p><strong>Note!</strong> When performing a query with only one parameter, there are two <code>ToRow</code> instances you can use: <code>ToField a =&gt; ToRow [a]</code> and <code>ToField a =&gt; Only a</code>. The <code>Only</code> datatype is defined in <code>Data.Tuple.Only</code> and is kind of a workaround for the fact that Haskell doesn’t have one-element tuples. Alternatively, a list of size 1 works as well. The same applies for queries that return rows with only one column: you can use either <code>[[X]]</code> or <code>[Only X]</code> as the return type. Here’s an example:</p>
-<div class="sourceCode" id="cb337"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb337-1"><a href="#cb337-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Database.SQLite.Simple</span> <span class="dt">T</span><span class="op">&gt;</span> q <span class="ot">=</span> <span class="dt">Query</span> (T.pack <span class="st">&quot;SELECT lower(?);&quot;</span>)</span>
+<p><strong>Note!</strong> When performing a query with only one
+parameter, there are two <code>ToRow</code> instances you can use:
+<code>ToField a =&gt; ToRow [a]</code> and
+<code>ToField a =&gt; Only a</code>. The <code>Only</code> datatype is
+defined in <code>Data.Tuple.Only</code> and is kind of a workaround for
+the fact that Haskell doesn’t have one-element tuples. Alternatively, a
+list of size 1 works as well. The same applies for queries that return
+rows with only one column: you can use either <code>[[X]]</code> or
+<code>[Only X]</code> as the return type. Here’s an example:</p>
+<div class="sourceCode" id="cb337"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb337-1"><a href="#cb337-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Database.SQLite.Simple</span> <span class="dt">T</span><span class="op">&gt;</span> q <span class="ot">=</span> <span class="dt">Query</span> (T.pack <span class="st">&quot;SELECT lower(?);&quot;</span>)</span>
 <span id="cb337-2"><a href="#cb337-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Database.SQLite.Simple</span> <span class="dt">T</span><span class="op">&gt;</span> query db q (<span class="dt">Only</span> <span class="st">&quot;HELLO&quot;</span>)<span class="ot"> ::</span> <span class="dt">IO</span> [<span class="dt">Only</span> <span class="dt">String</span>]</span>
 <span id="cb337-3"><a href="#cb337-3" aria-hidden="true" tabindex="-1"></a>[<span class="dt">Only</span> {fromOnly <span class="ot">=</span> <span class="st">&quot;hello&quot;</span>}]</span>
 <span id="cb337-4"><a href="#cb337-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Database.SQLite.Simple</span> <span class="dt">T</span><span class="op">&gt;</span> query db q [<span class="st">&quot;HELLO&quot;</span>]<span class="ot"> ::</span> <span class="dt">IO</span> [[<span class="dt">String</span>]]</span>
 <span id="cb337-5"><a href="#cb337-5" aria-hidden="true" tabindex="-1"></a>[[<span class="st">&quot;hello&quot;</span>]]</span></code></pre></div>
-<p>That’s pretty much all you need to know about sqlite-simple: <code>open</code>, <code>query_</code>, <code>query</code>, <code>FromRow</code>, <code>ToRow</code>. Oh right, one more thing. There are the functions <code>execute</code> and <code>execute_</code> if you don’t need the result of the query. They’re useful for inserting things into the database, for example.</p>
-<div class="sourceCode" id="cb338"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb338-1"><a href="#cb338-1" aria-hidden="true" tabindex="-1"></a><span class="ot">execute_ ::</span> <span class="dt">Connection</span> <span class="ot">-&gt;</span> <span class="dt">Query</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span>
+<p>That’s pretty much all you need to know about sqlite-simple:
+<code>open</code>, <code>query_</code>, <code>query</code>,
+<code>FromRow</code>, <code>ToRow</code>. Oh right, one more thing.
+There are the functions <code>execute</code> and <code>execute_</code>
+if you don’t need the result of the query. They’re useful for inserting
+things into the database, for example.</p>
+<div class="sourceCode" id="cb338"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb338-1"><a href="#cb338-1" aria-hidden="true" tabindex="-1"></a><span class="ot">execute_ ::</span> <span class="dt">Connection</span> <span class="ot">-&gt;</span> <span class="dt">Query</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span>
 <span id="cb338-2"><a href="#cb338-2" aria-hidden="true" tabindex="-1"></a><span class="ot">execute ::</span> <span class="dt">ToRow</span> q <span class="ot">=&gt;</span> <span class="dt">Connection</span> <span class="ot">-&gt;</span> <span class="dt">Query</span> <span class="ot">-&gt;</span> q <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span></code></pre></div>
-<p>You’ll find an example program that uses sqlite-simple to maintain a phone book under <a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Examples/Phonebook.hs"><code>exercises/Examples/Phonebook.hs</code></a>. The program keeps the phonebook in a file called <code>phonebook.db</code> and works like this (run from the <code>exercises/Examples</code> directory in the course repository):</p>
+<p>You’ll find an example program that uses sqlite-simple to maintain a
+phone book under <a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Examples/Phonebook.hs"><code>exercises/Examples/Phonebook.hs</code></a>.
+The program keeps the phonebook in a file called
+<code>phonebook.db</code> and works like this (run from the
+<code>exercises/Examples</code> directory in the course repository):</p>
 <pre><code>$ stack runhaskell Phonebook.hs
 (a)dd or (q)uery?
 a
@@ -3914,42 +5532,79 @@ bob
 2 numbers:
 [&quot;1234&quot;]
 [&quot;5678&quot;]</code></pre>
-<p>PS. If you’re devastated by the lack of compile-time type checking for SQL queries, you can have a look at some of the more advanced SQL libraries for Haskell like <a href="https://haskell-beam.github.io/beam/">Beam</a> or <a href="https://hackage.haskell.org/package/opaleye">Opaleye</a>. This course is using sqlite-simple for simplicity and to avoid dwelling on the details of SQL too much.</p>
+<p>PS. If you’re devastated by the lack of compile-time type checking
+for SQL queries, you can have a look at some of the more advanced SQL
+libraries for Haskell like <a
+href="https://haskell-beam.github.io/beam/">Beam</a> or <a
+href="https://hackage.haskell.org/package/opaleye">Opaleye</a>. This
+course is using sqlite-simple for simplicity and to avoid dwelling on
+the details of SQL too much.</p>
 <!-- ## TODO Quiz: this set isn't that well suited for quiz questions -->
-</section>
-<section id="exercises-5" data-number="6.5">
-<h2 data-number="14.5"><span class="header-section-number">14.5</span> Exercises</h2>
+<h2 data-number="14.5" id="exercises-5"><span
+class="header-section-number">14.5</span> Exercises</h2>
 <ul>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set14a.hs">Set14a</a>: Text &amp; ByteString</li>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set14b.hs">Set14b</a>: HTTP &amp; SQLite</li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set14a.hs">Set14a</a>:
+Text &amp; ByteString</li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set14b.hs">Set14b</a>:
+HTTP &amp; SQLite</li>
 </ul>
-</section>
-</section>
-<section id="lecture-15-youre-valid-even-without-monads" data-number="7">
-<h1 data-number="15"><span class="header-section-number">15</span> Lecture 15: You’re Valid Even Without Monads</h1>
-<section id="introduction-to-applicatives" data-number="7.1">
-<h2 data-number="15.1"><span class="header-section-number">15.1</span> Introduction to Applicatives</h2>
-<p>The <code>Applicative</code> type class is a middle ground between <code>Functor</code> (which you can’t do that much with) and <code>Monad</code> (which pretty much allows you to write arbitrary programs). Reasons to use <code>Applicative</code> instead of <code>Monad</code> include:</p>
+<h1 data-number="15"
+id="lecture-15-youre-valid-even-without-monads"><span
+class="header-section-number">15</span> Lecture 15: You’re Valid Even
+Without Monads</h1>
+<h2 data-number="15.1" id="introduction-to-applicatives"><span
+class="header-section-number">15.1</span> Introduction to
+Applicatives</h2>
+<p>The <code>Applicative</code> type class is a middle ground between
+<code>Functor</code> (which you can’t do that much with) and
+<code>Monad</code> (which pretty much allows you to write arbitrary
+programs). Reasons to use <code>Applicative</code> instead of
+<code>Monad</code> include:</p>
 <ul>
-<li>Performance: since <code>Applicative</code> allows less operations, it can be optimized better than <code>Monad</code>.</li>
-<li>Simplicity: an <code>Applicative</code> interface is easier to reason about.</li>
-<li>Necessity: there just is no way to define a <code>Monad</code> instance for your type, but there is an <code>Applicative</code> instance. This is rare.</li>
+<li>Performance: since <code>Applicative</code> allows less operations,
+it can be optimized better than <code>Monad</code>.</li>
+<li>Simplicity: an <code>Applicative</code> interface is easier to
+reason about.</li>
+<li>Necessity: there just is no way to define a <code>Monad</code>
+instance for your type, but there is an <code>Applicative</code>
+instance. This is rare.</li>
 </ul>
-<p>So what is an <code>Applicative</code>? Let’s look at a definition.</p>
-<div class="sourceCode" id="cb340"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb340-1"><a href="#cb340-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Functor</span> f <span class="ot">=&gt;</span> <span class="dt">Applicative</span> f <span class="kw">where</span></span>
+<p>So what is an <code>Applicative</code>? Let’s look at a
+definition.</p>
+<div class="sourceCode" id="cb340"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb340-1"><a href="#cb340-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Functor</span> f <span class="ot">=&gt;</span> <span class="dt">Applicative</span> f <span class="kw">where</span></span>
 <span id="cb340-2"><a href="#cb340-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  pure ::</span> a <span class="ot">-&gt;</span> f a</span>
 <span id="cb340-3"><a href="#cb340-3" aria-hidden="true" tabindex="-1"></a><span class="ot">  liftA2 ::</span> (a <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> c) <span class="ot">-&gt;</span> f a <span class="ot">-&gt;</span> f b <span class="ot">-&gt;</span> f c</span>
 <span id="cb340-4"><a href="#cb340-4" aria-hidden="true" tabindex="-1"></a>  <span class="co">-- other operations omitted for now</span></span></code></pre></div>
-<p>So an <code>Applicative</code> is a <code>Functor</code> that allows us to build singleton values via <code>pure</code>, and combine two values into one using <code>liftA2</code>. This adds a lot of power compared to a bare functor. Computations using functors are necessarily linear: <code>fmap :: (a -&gt; b) -&gt; f a -&gt; f b</code> takes in one functorial value, and outputs another. By contrast, <code>pure</code> takes in no functorial value and outputs one, and <code>liftA2</code> takes in two and returns one.</p>
-<p>Sidenote: the term Applicative comes from the term <a href="https://en.wikipedia.org/wiki/Applicative_functor">Applicative Functor</a>, which sounds like it comes from Category Theory, but was actually introduced in a programming paper.</p>
-<p>That’s enough abstract mumbo jumbo for now. Let’s look at what sort of computations we can express using Applicative operations (and <code>fmap</code>). We’ll start with the <code>Maybe</code> applicative. Here’s a streamlined definition:</p>
-<div class="sourceCode" id="cb341"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb341-1"><a href="#cb341-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Applicative</span> <span class="dt">Maybe</span> <span class="kw">where</span></span>
+<p>So an <code>Applicative</code> is a <code>Functor</code> that allows
+us to build singleton values via <code>pure</code>, and combine two
+values into one using <code>liftA2</code>. This adds a lot of power
+compared to a bare functor. Computations using functors are necessarily
+linear: <code>fmap :: (a -&gt; b) -&gt; f a -&gt; f b</code> takes in
+one functorial value, and outputs another. By contrast,
+<code>pure</code> takes in no functorial value and outputs one, and
+<code>liftA2</code> takes in two and returns one.</p>
+<p>Sidenote: the term Applicative comes from the term <a
+href="https://en.wikipedia.org/wiki/Applicative_functor">Applicative
+Functor</a>, which sounds like it comes from Category Theory, but was
+actually introduced in a programming paper.</p>
+<p>That’s enough abstract mumbo jumbo for now. Let’s look at what sort
+of computations we can express using Applicative operations (and
+<code>fmap</code>). We’ll start with the <code>Maybe</code> applicative.
+Here’s a streamlined definition:</p>
+<div class="sourceCode" id="cb341"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb341-1"><a href="#cb341-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Applicative</span> <span class="dt">Maybe</span> <span class="kw">where</span></span>
 <span id="cb341-2"><a href="#cb341-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">pure</span> x <span class="ot">=</span> <span class="dt">Just</span> x</span>
 <span id="cb341-3"><a href="#cb341-3" aria-hidden="true" tabindex="-1"></a>  liftA2 f (<span class="dt">Just</span> x) (<span class="dt">Just</span> y) <span class="ot">=</span> <span class="dt">Just</span> (f x y)</span>
 <span id="cb341-4"><a href="#cb341-4" aria-hidden="true" tabindex="-1"></a>  liftA2 f _        _        <span class="ot">=</span> <span class="dt">Nothing</span></span></code></pre></div>
 <!-- TODO some GHCi examples? -->
-<p>You’ll see the definition uses the same type of failure propagation as the <code>Monad Maybe</code> instance. Let’s use this when parsing monetary values:</p>
-<div class="sourceCode" id="cb342"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb342-1"><a href="#cb342-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Currency</span> <span class="ot">=</span> <span class="dt">EUR</span> <span class="op">|</span> <span class="dt">USD</span></span>
+<p>You’ll see the definition uses the same type of failure propagation
+as the <code>Monad Maybe</code> instance. Let’s use this when parsing
+monetary values:</p>
+<div class="sourceCode" id="cb342"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb342-1"><a href="#cb342-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Currency</span> <span class="ot">=</span> <span class="dt">EUR</span> <span class="op">|</span> <span class="dt">USD</span></span>
 <span id="cb342-2"><a href="#cb342-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">deriving</span> (<span class="dt">Show</span>, <span class="dt">Eq</span>)</span>
 <span id="cb342-3"><a href="#cb342-3" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Money</span> <span class="ot">=</span> <span class="dt">Money</span> <span class="dt">Int</span> <span class="dt">Currency</span></span>
 <span id="cb342-4"><a href="#cb342-4" aria-hidden="true" tabindex="-1"></a>  <span class="kw">deriving</span> (<span class="dt">Show</span>, <span class="dt">Eq</span>)</span>
@@ -3966,37 +5621,58 @@ bob
 <span id="cb342-15"><a href="#cb342-15" aria-hidden="true" tabindex="-1"></a><span class="ot">parseMoney ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">Money</span></span>
 <span id="cb342-16"><a href="#cb342-16" aria-hidden="true" tabindex="-1"></a>parseMoney amountString currencyString <span class="ot">=</span></span>
 <span id="cb342-17"><a href="#cb342-17" aria-hidden="true" tabindex="-1"></a>  liftA2 <span class="dt">Money</span> (parseAmount amountString) (parseCurrency currencyString)</span></code></pre></div>
-<div class="sourceCode" id="cb343"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb343-1"><a href="#cb343-1" aria-hidden="true" tabindex="-1"></a>parseMoney <span class="st">&quot;123&quot;</span> <span class="st">&quot;€&quot;</span>  <span class="op">==&gt;</span> <span class="dt">Just</span> (<span class="dt">Money</span> <span class="dv">123</span> <span class="dt">EUR</span>)</span>
+<div class="sourceCode" id="cb343"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb343-1"><a href="#cb343-1" aria-hidden="true" tabindex="-1"></a>parseMoney <span class="st">&quot;123&quot;</span> <span class="st">&quot;€&quot;</span>  <span class="op">==&gt;</span> <span class="dt">Just</span> (<span class="dt">Money</span> <span class="dv">123</span> <span class="dt">EUR</span>)</span>
 <span id="cb343-2"><a href="#cb343-2" aria-hidden="true" tabindex="-1"></a>parseMoney <span class="st">&quot;45&quot;</span> <span class="st">&quot;$&quot;</span>   <span class="op">==&gt;</span> <span class="dt">Just</span> (<span class="dt">Money</span> <span class="dv">45</span> <span class="dt">USD</span>)</span>
 <span id="cb343-3"><a href="#cb343-3" aria-hidden="true" tabindex="-1"></a>parseMoney <span class="st">&quot;4x&quot;</span> <span class="st">&quot;€&quot;</span>   <span class="op">==&gt;</span> <span class="dt">Nothing</span></span>
 <span id="cb343-4"><a href="#cb343-4" aria-hidden="true" tabindex="-1"></a>parseMoney <span class="st">&quot;45&quot;</span> <span class="st">&quot;£&quot;</span>   <span class="op">==&gt;</span> <span class="dt">Nothing</span></span></code></pre></div>
-<p>That worked out nicely. However, if we try to expand on this we soon run into the limits of <code>Applicative</code>. For example, consider this <code>sumMoney</code> function that sums <code>Money</code> values but fails if they are not in the same currency:</p>
-<div class="sourceCode" id="cb344"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb344-1"><a href="#cb344-1" aria-hidden="true" tabindex="-1"></a><span class="ot">sumMoney ::</span> <span class="dt">Money</span> <span class="ot">-&gt;</span> <span class="dt">Money</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">Money</span></span>
+<p>That worked out nicely. However, if we try to expand on this we soon
+run into the limits of <code>Applicative</code>. For example, consider
+this <code>sumMoney</code> function that sums <code>Money</code> values
+but fails if they are not in the same currency:</p>
+<div class="sourceCode" id="cb344"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb344-1"><a href="#cb344-1" aria-hidden="true" tabindex="-1"></a><span class="ot">sumMoney ::</span> <span class="dt">Money</span> <span class="ot">-&gt;</span> <span class="dt">Money</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">Money</span></span>
 <span id="cb344-2"><a href="#cb344-2" aria-hidden="true" tabindex="-1"></a>sumMoney (<span class="dt">Money</span> a c) (<span class="dt">Money</span> b c&#39;)</span>
 <span id="cb344-3"><a href="#cb344-3" aria-hidden="true" tabindex="-1"></a>    <span class="op">|</span> c <span class="op">==</span> c&#39;   <span class="ot">=</span> <span class="dt">Just</span> (<span class="dt">Money</span> (a<span class="op">+</span>b) c)</span>
 <span id="cb344-4"><a href="#cb344-4" aria-hidden="true" tabindex="-1"></a>    <span class="op">|</span> <span class="fu">otherwise</span> <span class="ot">=</span> <span class="dt">Nothing</span></span></code></pre></div>
-<p>We can’t apply it to two <code>Maybe Money</code> values using <code>Applicative</code> operations. We need the <code>Maybe</code> monad for that:</p>
-<div class="sourceCode" id="cb345"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb345-1"><a href="#cb345-1" aria-hidden="true" tabindex="-1"></a><span class="ot">example ::</span> <span class="dt">Maybe</span> <span class="dt">Money</span></span>
+<p>We can’t apply it to two <code>Maybe Money</code> values using
+<code>Applicative</code> operations. We need the <code>Maybe</code>
+monad for that:</p>
+<div class="sourceCode" id="cb345"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb345-1"><a href="#cb345-1" aria-hidden="true" tabindex="-1"></a><span class="ot">example ::</span> <span class="dt">Maybe</span> <span class="dt">Money</span></span>
 <span id="cb345-2"><a href="#cb345-2" aria-hidden="true" tabindex="-1"></a>example <span class="ot">=</span> <span class="kw">do</span> x <span class="ot">&lt;-</span> parseMoney <span class="st">&quot;123&quot;</span> <span class="st">&quot;€&quot;</span></span>
 <span id="cb345-3"><a href="#cb345-3" aria-hidden="true" tabindex="-1"></a>             y <span class="ot">&lt;-</span> parseMoney <span class="st">&quot;45&quot;</span> <span class="st">&quot;$&quot;</span></span>
 <span id="cb345-4"><a href="#cb345-4" aria-hidden="true" tabindex="-1"></a>             sumMoney x y</span></code></pre></div>
-<p>If we try to use <code>liftA2</code>, we’re stuck with a <code>Maybe (Maybe Money)</code> type. In addition, we can now get two different types of failures: <code>Nothing</code> and <code>Just Nothing</code>, depending on what level the error happens on. This would be a clear case for switching to a <code>Monad</code> instance.</p>
+<p>If we try to use <code>liftA2</code>, we’re stuck with a
+<code>Maybe (Maybe Money)</code> type. In addition, we can now get two
+different types of failures: <code>Nothing</code> and
+<code>Just Nothing</code>, depending on what level the error happens on.
+This would be a clear case for switching to a <code>Monad</code>
+instance.</p>
 <pre><code>liftA2 sumMoney (parseMoney &quot;123&quot; &quot;e&quot;) (parseMoney &quot;45&quot; &quot;€&quot;)
   ==&gt; Just (Just (Money 168 EUR))
 liftA2 sumMoney (parseMoney &quot;123&quot; &quot;e&quot;) (parseMoney &quot;45&quot; &quot;$&quot;)
   ==&gt; Just Nothing
 liftA2 sumMoney (parseMoney &quot;123&quot; &quot;e&quot;) (parseMoney &quot;xxx&quot; &quot;e&quot;)
   ==&gt; Nothing</code></pre>
-<p>Sidenote: the name <code>liftA2</code> sounds a bit cumbersome, but it’s an analogy with the <code>liftM</code>, <code>liftM2</code> and so on functions for monads. Recall that <code>liftM</code> was just <code>fmap</code>, so perhaps <code>liftA2</code> should’ve been called <code>fmap2</code>.</p>
-</section>
-<section id="the-list-applicative" data-number="7.2">
-<h2 data-number="15.2"><span class="header-section-number">15.2</span> The List Applicative</h2>
-<p>Let’s have a look at the applicative instances for another <code>Functor</code> we’ve seen. The <code>Applicative</code> instance for the list functor goes through all possible combinations of values (just like the list monad). Here’s the instance:</p>
-<div class="sourceCode" id="cb347"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb347-1"><a href="#cb347-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Applicative</span> [] <span class="kw">where</span></span>
+<p>Sidenote: the name <code>liftA2</code> sounds a bit cumbersome, but
+it’s an analogy with the <code>liftM</code>, <code>liftM2</code> and so
+on functions for monads. Recall that <code>liftM</code> was just
+<code>fmap</code>, so perhaps <code>liftA2</code> should’ve been called
+<code>fmap2</code>.</p>
+<h2 data-number="15.2" id="the-list-applicative"><span
+class="header-section-number">15.2</span> The List Applicative</h2>
+<p>Let’s have a look at the applicative instances for another
+<code>Functor</code> we’ve seen. The <code>Applicative</code> instance
+for the list functor goes through all possible combinations of values
+(just like the list monad). Here’s the instance:</p>
+<div class="sourceCode" id="cb347"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb347-1"><a href="#cb347-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Applicative</span> [] <span class="kw">where</span></span>
 <span id="cb347-2"><a href="#cb347-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">pure</span> x <span class="ot">=</span> [x]</span>
 <span id="cb347-3"><a href="#cb347-3" aria-hidden="true" tabindex="-1"></a>  liftA2 f xs ys <span class="ot">=</span> [f x y <span class="op">|</span> x <span class="ot">&lt;-</span> xs, y <span class="ot">&lt;-</span> ys]</span></code></pre></div>
 <p>And here’s an example: generating some phrases.</p>
-<div class="sourceCode" id="cb348"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb348-1"><a href="#cb348-1" aria-hidden="true" tabindex="-1"></a><span class="ot">things ::</span> [<span class="dt">String</span>]</span>
+<div class="sourceCode" id="cb348"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb348-1"><a href="#cb348-1" aria-hidden="true" tabindex="-1"></a><span class="ot">things ::</span> [<span class="dt">String</span>]</span>
 <span id="cb348-2"><a href="#cb348-2" aria-hidden="true" tabindex="-1"></a>things <span class="ot">=</span> [<span class="st">&quot;tangerine&quot;</span>,<span class="st">&quot;bandit&quot;</span>,<span class="st">&quot;diamond&quot;</span>]</span>
 <span id="cb348-3"><a href="#cb348-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb348-4"><a href="#cb348-4" aria-hidden="true" tabindex="-1"></a><span class="ot">fruits ::</span> [<span class="dt">String</span>]</span>
@@ -4008,7 +5684,8 @@ liftA2 sumMoney (parseMoney &quot;123&quot; &quot;e&quot;) (parseMoney &quot;xxx
 <span id="cb348-10"><a href="#cb348-10" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb348-11"><a href="#cb348-11" aria-hidden="true" tabindex="-1"></a>bunches <span class="ot">=</span> liftA2 copy [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] fruits</span>
 <span id="cb348-12"><a href="#cb348-12" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> copy n f <span class="ot">=</span> <span class="fu">unwords</span> (<span class="fu">replicate</span> n f)</span></code></pre></div>
-<div class="sourceCode" id="cb349"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb349-1"><a href="#cb349-1" aria-hidden="true" tabindex="-1"></a>phrases <span class="op">==&gt;</span> [<span class="st">&quot;a tangerine the size of a apple&quot;</span>,</span>
+<div class="sourceCode" id="cb349"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb349-1"><a href="#cb349-1" aria-hidden="true" tabindex="-1"></a>phrases <span class="op">==&gt;</span> [<span class="st">&quot;a tangerine the size of a apple&quot;</span>,</span>
 <span id="cb349-2"><a href="#cb349-2" aria-hidden="true" tabindex="-1"></a>             <span class="st">&quot;a tangerine the size of a tangerine&quot;</span>,</span>
 <span id="cb349-3"><a href="#cb349-3" aria-hidden="true" tabindex="-1"></a>             <span class="st">&quot;a bandit the size of a apple&quot;</span>,</span>
 <span id="cb349-4"><a href="#cb349-4" aria-hidden="true" tabindex="-1"></a>             <span class="st">&quot;a bandit the size of a tangerine&quot;</span>,</span>
@@ -4019,27 +5696,40 @@ liftA2 sumMoney (parseMoney &quot;123&quot; &quot;e&quot;) (parseMoney &quot;xxx
 <span id="cb349-9"><a href="#cb349-9" aria-hidden="true" tabindex="-1"></a>             <span class="st">&quot;apple apple&quot;</span>,<span class="st">&quot;tangerine tangerine&quot;</span>,</span>
 <span id="cb349-10"><a href="#cb349-10" aria-hidden="true" tabindex="-1"></a>             <span class="st">&quot;apple apple apple&quot;</span>,<span class="st">&quot;tangerine tangerine tangerine&quot;</span>]</span></code></pre></div>
 <!-- TODO more examples? -->
-</section>
-<section id="new-operators" data-number="7.3">
-<h2 data-number="15.3"><span class="header-section-number">15.3</span> New Operators</h2>
-<p>There are a handful of operators for Applicatives that are quite handy. They are <code>&lt;$&gt;</code>, <code>&lt;*&gt;</code>, <code>&lt;*</code> and <code>*&gt;</code>.</p>
-<p>Let’s start with <code>&lt;$&gt;</code>, which is just an infix version of <code>fmap</code>:</p>
-<div class="sourceCode" id="cb350"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb350-1"><a href="#cb350-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(&lt;$&gt;) ::</span> <span class="dt">Functor</span> f <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> f a <span class="ot">-&gt;</span> f b</span>
+<h2 data-number="15.3" id="new-operators"><span
+class="header-section-number">15.3</span> New Operators</h2>
+<p>There are a handful of operators for Applicatives that are quite
+handy. They are <code>&lt;$&gt;</code>, <code>&lt;*&gt;</code>,
+<code>&lt;*</code> and <code>*&gt;</code>.</p>
+<p>Let’s start with <code>&lt;$&gt;</code>, which is just an infix
+version of <code>fmap</code>:</p>
+<div class="sourceCode" id="cb350"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb350-1"><a href="#cb350-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(&lt;$&gt;) ::</span> <span class="dt">Functor</span> f <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> f a <span class="ot">-&gt;</span> f b</span>
 <span id="cb350-2"><a href="#cb350-2" aria-hidden="true" tabindex="-1"></a>f <span class="op">&lt;$&gt;</span> x <span class="ot">=</span> <span class="fu">fmap</span> f x</span></code></pre></div>
-<div class="sourceCode" id="cb351"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb351-1"><a href="#cb351-1" aria-hidden="true" tabindex="-1"></a><span class="fu">not</span> <span class="op">&lt;$&gt;</span> <span class="dt">Just</span> <span class="dt">True</span>   <span class="op">==&gt;</span> <span class="dt">Just</span> <span class="dt">False</span></span>
+<div class="sourceCode" id="cb351"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb351-1"><a href="#cb351-1" aria-hidden="true" tabindex="-1"></a><span class="fu">not</span> <span class="op">&lt;$&gt;</span> <span class="dt">Just</span> <span class="dt">True</span>   <span class="op">==&gt;</span> <span class="dt">Just</span> <span class="dt">False</span></span>
 <span id="cb351-2"><a href="#cb351-2" aria-hidden="true" tabindex="-1"></a><span class="fu">not</span> <span class="op">&lt;$&gt;</span> <span class="dt">Nothing</span>     <span class="op">==&gt;</span> <span class="dt">Nothing</span></span>
 <span id="cb351-3"><a href="#cb351-3" aria-hidden="true" tabindex="-1"></a><span class="fu">negate</span> <span class="op">&lt;$&gt;</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]  <span class="op">==&gt;</span> [<span class="op">-</span><span class="dv">1</span>,<span class="op">-</span><span class="dv">2</span>,<span class="op">-</span><span class="dv">3</span>]</span></code></pre></div>
-<p>That’s kinda nice on its own, but it really gets to shine when combined with this Applicative operator:</p>
-<div class="sourceCode" id="cb352"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb352-1"><a href="#cb352-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(&lt;*&gt;) ::</span> <span class="dt">Applicative</span> f <span class="ot">=&gt;</span> f (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> f a <span class="ot">-&gt;</span> f b</span></code></pre></div>
-<p>The type tells you what <code>&lt;*&gt;</code> does: it’s function application <em>lifted</em> to an applicative. Here are some standalone examples:</p>
-<div class="sourceCode" id="cb353"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb353-1"><a href="#cb353-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Just</span> <span class="fu">not</span> <span class="op">&lt;*&gt;</span> <span class="dt">Just</span> <span class="dt">True</span>    <span class="op">==&gt;</span> <span class="dt">Just</span> <span class="dt">False</span></span>
+<p>That’s kinda nice on its own, but it really gets to shine when
+combined with this Applicative operator:</p>
+<div class="sourceCode" id="cb352"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb352-1"><a href="#cb352-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(&lt;*&gt;) ::</span> <span class="dt">Applicative</span> f <span class="ot">=&gt;</span> f (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> f a <span class="ot">-&gt;</span> f b</span></code></pre></div>
+<p>The type tells you what <code>&lt;*&gt;</code> does: it’s function
+application <em>lifted</em> to an applicative. Here are some standalone
+examples:</p>
+<div class="sourceCode" id="cb353"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb353-1"><a href="#cb353-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Just</span> <span class="fu">not</span> <span class="op">&lt;*&gt;</span> <span class="dt">Just</span> <span class="dt">True</span>    <span class="op">==&gt;</span> <span class="dt">Just</span> <span class="dt">False</span></span>
 <span id="cb353-2"><a href="#cb353-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Nothing</span>  <span class="op">&lt;*&gt;</span> <span class="dt">Just</span> <span class="dt">True</span>    <span class="op">==&gt;</span> <span class="dt">Nothing</span></span>
 <span id="cb353-3"><a href="#cb353-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Just</span> <span class="fu">not</span> <span class="op">&lt;*&gt;</span> <span class="dt">Nothing</span>      <span class="op">==&gt;</span> <span class="dt">Nothing</span></span>
 <span id="cb353-4"><a href="#cb353-4" aria-hidden="true" tabindex="-1"></a>[(<span class="op">+</span><span class="dv">1</span>),(<span class="op">*</span><span class="dv">2</span>)] <span class="op">&lt;*&gt;</span> [<span class="dv">10</span>,<span class="dv">100</span>]  <span class="op">==&gt;</span> [<span class="dv">11</span>,<span class="dv">101</span>,<span class="dv">20</span>,<span class="dv">200</span>]</span></code></pre></div>
-<p>The real magic happens when we combine <code>&lt;$&gt;</code> and <code>&lt;*&gt;</code>: we can then lift functions of arbitrarily many arguments to an Applicative!</p>
-<div class="sourceCode" id="cb354"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb354-1"><a href="#cb354-1" aria-hidden="true" tabindex="-1"></a><span class="ot">say ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
+<p>The real magic happens when we combine <code>&lt;$&gt;</code> and
+<code>&lt;*&gt;</code>: we can then lift functions of arbitrarily many
+arguments to an Applicative!</p>
+<div class="sourceCode" id="cb354"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb354-1"><a href="#cb354-1" aria-hidden="true" tabindex="-1"></a><span class="ot">say ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span></span>
 <span id="cb354-2"><a href="#cb354-2" aria-hidden="true" tabindex="-1"></a>say x i y <span class="ot">=</span> x <span class="op">++</span> <span class="st">&quot; has &quot;</span> <span class="op">++</span> <span class="fu">show</span> i <span class="op">++</span> <span class="st">&quot; &quot;</span> <span class="op">++</span> y</span></code></pre></div>
-<div class="sourceCode" id="cb355"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb355-1"><a href="#cb355-1" aria-hidden="true" tabindex="-1"></a>say <span class="op">&lt;$&gt;</span> <span class="dt">Just</span> <span class="st">&quot;haskell&quot;</span> <span class="op">&lt;*&gt;</span> <span class="dt">Just</span> <span class="dv">99</span> <span class="op">&lt;*&gt;</span> <span class="dt">Just</span> <span class="st">&quot;operators&quot;</span></span>
+<div class="sourceCode" id="cb355"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb355-1"><a href="#cb355-1" aria-hidden="true" tabindex="-1"></a>say <span class="op">&lt;$&gt;</span> <span class="dt">Just</span> <span class="st">&quot;haskell&quot;</span> <span class="op">&lt;*&gt;</span> <span class="dt">Just</span> <span class="dv">99</span> <span class="op">&lt;*&gt;</span> <span class="dt">Just</span> <span class="st">&quot;operators&quot;</span></span>
 <span id="cb355-2"><a href="#cb355-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Just</span> <span class="st">&quot;haskell has 99 operators&quot;</span></span>
 <span id="cb355-3"><a href="#cb355-3" aria-hidden="true" tabindex="-1"></a>say <span class="op">&lt;$&gt;</span> <span class="dt">Nothing</span> <span class="op">&lt;*&gt;</span> <span class="dt">Just</span> <span class="dv">99</span> <span class="op">&lt;*&gt;</span> <span class="dt">Just</span> <span class="st">&quot;operators&quot;</span></span>
 <span id="cb355-4"><a href="#cb355-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Nothing</span></span>
@@ -4052,8 +5742,11 @@ liftA2 sumMoney (parseMoney &quot;123&quot; &quot;e&quot;) (parseMoney &quot;xxx
 <span id="cb355-11"><a href="#cb355-11" aria-hidden="true" tabindex="-1"></a>       <span class="st">&quot;jake has 2 cars&quot;</span>,</span>
 <span id="cb355-12"><a href="#cb355-12" aria-hidden="true" tabindex="-1"></a>       <span class="st">&quot;jake has 3 bananas&quot;</span>,</span>
 <span id="cb355-13"><a href="#cb355-13" aria-hidden="true" tabindex="-1"></a>       <span class="st">&quot;jake has 3 cars&quot;</span>]</span></code></pre></div>
-<p>What’s going on here? Let’s step through the evaluation. The key is that each <code>&lt;*&gt;</code> partially applies one more argument to the function.</p>
-<div class="sourceCode" id="cb356"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb356-1"><a href="#cb356-1" aria-hidden="true" tabindex="-1"></a>    say <span class="op">&lt;$&gt;</span> <span class="dt">Just</span> <span class="st">&quot;haskell&quot;</span> <span class="op">&lt;*&gt;</span> <span class="dt">Just</span> <span class="dv">99</span> <span class="op">&lt;*&gt;</span> <span class="dt">Just</span> <span class="st">&quot;operators&quot;</span></span>
+<p>What’s going on here? Let’s step through the evaluation. The key is
+that each <code>&lt;*&gt;</code> partially applies one more argument to
+the function.</p>
+<div class="sourceCode" id="cb356"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb356-1"><a href="#cb356-1" aria-hidden="true" tabindex="-1"></a>    say <span class="op">&lt;$&gt;</span> <span class="dt">Just</span> <span class="st">&quot;haskell&quot;</span> <span class="op">&lt;*&gt;</span> <span class="dt">Just</span> <span class="dv">99</span> <span class="op">&lt;*&gt;</span> <span class="dt">Just</span> <span class="st">&quot;operators&quot;</span></span>
 <span id="cb356-2"><a href="#cb356-2" aria-hidden="true" tabindex="-1"></a><span class="op">===</span> ((say <span class="op">&lt;$&gt;</span> <span class="dt">Just</span> <span class="st">&quot;haskell&quot;</span>) <span class="op">&lt;*&gt;</span> <span class="dt">Just</span> <span class="dv">99</span>) <span class="op">&lt;*&gt;</span> <span class="dt">Just</span> <span class="st">&quot;operators&quot;</span></span>
 <span id="cb356-3"><a href="#cb356-3" aria-hidden="true" tabindex="-1"></a><span class="op">===</span> (<span class="fu">fmap</span> say (<span class="dt">Just</span> <span class="st">&quot;haskell&quot;</span>) <span class="op">&lt;*&gt;</span> <span class="dt">Just</span> <span class="dv">99</span>) <span class="op">&lt;*&gt;</span> <span class="dt">Just</span> <span class="st">&quot;operators&quot;</span></span>
 <span id="cb356-4"><a href="#cb356-4" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> (<span class="dt">Just</span> (say <span class="st">&quot;haskell&quot;</span>) <span class="op">&lt;*&gt;</span> <span class="dt">Just</span> <span class="dv">99</span>) <span class="op">&lt;*&gt;</span> <span class="dt">Just</span> <span class="st">&quot;operators&quot;</span></span>
@@ -4061,24 +5754,32 @@ liftA2 sumMoney (parseMoney &quot;123&quot; &quot;e&quot;) (parseMoney &quot;xxx
 <span id="cb356-6"><a href="#cb356-6" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="dt">Just</span> (say <span class="st">&quot;haskell&quot;</span> <span class="dv">99</span> <span class="st">&quot;operators&quot;</span>)</span>
 <span id="cb356-7"><a href="#cb356-7" aria-hidden="true" tabindex="-1"></a><span class="op">==&gt;</span> <span class="dt">Just</span> <span class="st">&quot;haskell has 99 operators&quot;</span></span></code></pre></div>
 <p>Perhaps looking at the types will make it clearer:</p>
-<div class="sourceCode" id="cb357"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb357-1"><a href="#cb357-1" aria-hidden="true" tabindex="-1"></a>say <span class="op">&lt;$&gt;</span> <span class="dt">Just</span> <span class="st">&quot;haskell&quot;</span><span class="ot">                                  ::</span> <span class="dt">Maybe</span> (<span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span>)</span>
+<div class="sourceCode" id="cb357"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb357-1"><a href="#cb357-1" aria-hidden="true" tabindex="-1"></a>say <span class="op">&lt;$&gt;</span> <span class="dt">Just</span> <span class="st">&quot;haskell&quot;</span><span class="ot">                                  ::</span> <span class="dt">Maybe</span> (<span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span>)</span>
 <span id="cb357-2"><a href="#cb357-2" aria-hidden="true" tabindex="-1"></a>say <span class="op">&lt;$&gt;</span> <span class="dt">Just</span> <span class="st">&quot;haskell&quot;</span> <span class="op">&lt;*&gt;</span> <span class="dt">Just</span> <span class="dv">99</span><span class="ot">                      ::</span> <span class="dt">Maybe</span> (       <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">String</span>)</span>
 <span id="cb357-3"><a href="#cb357-3" aria-hidden="true" tabindex="-1"></a>say <span class="op">&lt;$&gt;</span> <span class="dt">Just</span> <span class="st">&quot;haskell&quot;</span> <span class="op">&lt;*&gt;</span> <span class="dt">Just</span> <span class="dv">99</span> <span class="op">&lt;*&gt;</span> <span class="dt">Just</span> <span class="st">&quot;operators&quot;</span><span class="ot"> ::</span> <span class="dt">Maybe</span> (                 <span class="dt">String</span>)</span></code></pre></div>
 <p>The next two operators are a bit simpler:</p>
-<div class="sourceCode" id="cb358"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb358-1"><a href="#cb358-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(*&gt;) ::</span> <span class="dt">Applicative</span> f <span class="ot">=&gt;</span> f a <span class="ot">-&gt;</span> f b <span class="ot">-&gt;</span> f b</span>
+<div class="sourceCode" id="cb358"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb358-1"><a href="#cb358-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(*&gt;) ::</span> <span class="dt">Applicative</span> f <span class="ot">=&gt;</span> f a <span class="ot">-&gt;</span> f b <span class="ot">-&gt;</span> f b</span>
 <span id="cb358-2"><a href="#cb358-2" aria-hidden="true" tabindex="-1"></a>x <span class="op">*&gt;</span> y <span class="ot">=</span> liftA2 (\a b <span class="ot">-&gt;</span> b) x y</span>
 <span id="cb358-3"><a href="#cb358-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb358-4"><a href="#cb358-4" aria-hidden="true" tabindex="-1"></a><span class="ot">(&lt;*) ::</span> <span class="dt">Applicative</span> f <span class="ot">=&gt;</span> f a <span class="ot">-&gt;</span> f b <span class="ot">-&gt;</span> f a</span>
 <span id="cb358-5"><a href="#cb358-5" aria-hidden="true" tabindex="-1"></a>x <span class="op">&lt;*</span> y <span class="ot">=</span> liftA2 (\a b <span class="ot">-&gt;</span> a) x y</span></code></pre></div>
 <p>You can compare the types to a more familiar operator:</p>
-<div class="sourceCode" id="cb359"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb359-1"><a href="#cb359-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(&gt;&gt;) ::</span> <span class="dt">Monad</span> m <span class="ot">=&gt;</span> m a <span class="ot">-&gt;</span> m b <span class="ot">-&gt;</span> m b</span></code></pre></div>
-<p>What the operators <code>&lt;*</code> and <code>*&gt;</code> mean is: run both of these operations, but only keep one result. The arrow points to the result that’s kept:</p>
-<div class="sourceCode" id="cb360"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb360-1"><a href="#cb360-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Just</span> <span class="dv">1</span> <span class="op">*&gt;</span> <span class="dt">Just</span> <span class="dv">2</span>  <span class="op">==&gt;</span> <span class="dt">Just</span> <span class="dv">2</span></span>
+<div class="sourceCode" id="cb359"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb359-1"><a href="#cb359-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(&gt;&gt;) ::</span> <span class="dt">Monad</span> m <span class="ot">=&gt;</span> m a <span class="ot">-&gt;</span> m b <span class="ot">-&gt;</span> m b</span></code></pre></div>
+<p>What the operators <code>&lt;*</code> and <code>*&gt;</code> mean is:
+run both of these operations, but only keep one result. The arrow points
+to the result that’s kept:</p>
+<div class="sourceCode" id="cb360"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb360-1"><a href="#cb360-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Just</span> <span class="dv">1</span> <span class="op">*&gt;</span> <span class="dt">Just</span> <span class="dv">2</span>  <span class="op">==&gt;</span> <span class="dt">Just</span> <span class="dv">2</span></span>
 <span id="cb360-2"><a href="#cb360-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Just</span> <span class="dv">1</span> <span class="op">&lt;*</span> <span class="dt">Just</span> <span class="dv">2</span>  <span class="op">==&gt;</span> <span class="dt">Just</span> <span class="dv">1</span></span>
 <span id="cb360-3"><a href="#cb360-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Just</span> <span class="dv">1</span> <span class="op">&lt;*</span> <span class="dt">Nothing</span> <span class="op">==&gt;</span> <span class="dt">Nothing</span></span>
 <span id="cb360-4"><a href="#cb360-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Nothing</span> <span class="op">&lt;*</span> <span class="dt">Just</span> <span class="dv">2</span> <span class="op">==&gt;</span> <span class="dt">Nothing</span></span></code></pre></div>
-<p>These operators might seem trivial, but they’re useful when combining checks. For example:</p>
-<div class="sourceCode" id="cb361"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb361-1"><a href="#cb361-1" aria-hidden="true" tabindex="-1"></a><span class="ot">decrease ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">Int</span></span>
+<p>These operators might seem trivial, but they’re useful when combining
+checks. For example:</p>
+<div class="sourceCode" id="cb361"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb361-1"><a href="#cb361-1" aria-hidden="true" tabindex="-1"></a><span class="ot">decrease ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">Int</span></span>
 <span id="cb361-2"><a href="#cb361-2" aria-hidden="true" tabindex="-1"></a>decrease i <span class="ot">=</span> <span class="kw">if</span> i<span class="op">&gt;</span><span class="dv">0</span> <span class="kw">then</span> <span class="dt">Just</span> (i<span class="op">-</span><span class="dv">1</span>) <span class="kw">else</span> <span class="dt">Nothing</span></span>
 <span id="cb361-3"><a href="#cb361-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb361-4"><a href="#cb361-4" aria-hidden="true" tabindex="-1"></a><span class="ot">small ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">Int</span></span>
@@ -4087,30 +5788,47 @@ liftA2 sumMoney (parseMoney &quot;123&quot; &quot;e&quot;) (parseMoney &quot;xxx
 <span id="cb361-7"><a href="#cb361-7" aria-hidden="true" tabindex="-1"></a><span class="ot">decreaseSmall ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">Int</span></span>
 <span id="cb361-8"><a href="#cb361-8" aria-hidden="true" tabindex="-1"></a><span class="co">-- do what decrease does, but fail if small fails</span></span>
 <span id="cb361-9"><a href="#cb361-9" aria-hidden="true" tabindex="-1"></a>decreaseSmall i <span class="ot">=</span> decrease i <span class="op">&lt;*</span> small i</span></code></pre></div>
-<div class="sourceCode" id="cb362"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb362-1"><a href="#cb362-1" aria-hidden="true" tabindex="-1"></a>decreaseSmall <span class="dv">4</span>   <span class="op">==&gt;</span> <span class="dt">Just</span> <span class="dv">3</span></span>
+<div class="sourceCode" id="cb362"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb362-1"><a href="#cb362-1" aria-hidden="true" tabindex="-1"></a>decreaseSmall <span class="dv">4</span>   <span class="op">==&gt;</span> <span class="dt">Just</span> <span class="dv">3</span></span>
 <span id="cb362-2"><a href="#cb362-2" aria-hidden="true" tabindex="-1"></a>decreaseSmall <span class="dv">0</span>   <span class="op">==&gt;</span> <span class="dt">Nothing</span></span>
 <span id="cb362-3"><a href="#cb362-3" aria-hidden="true" tabindex="-1"></a>decreaseSmall <span class="dv">11</span>  <span class="op">==&gt;</span> <span class="dt">Nothing</span></span></code></pre></div>
-<p>Now that we’ve seen all these operators, we can understand the full definition of <code>Applicative</code>. All the operators have definitions in terms of <code>liftA2</code>, so it’s enought to define <code>liftA2</code> and <code>pure</code> when implementing an <code>Applicative</code> instance.</p>
-<div class="sourceCode" id="cb363"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb363-1"><a href="#cb363-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Functor</span> f <span class="ot">=&gt;</span> <span class="dt">Applicative</span> f <span class="kw">where</span></span>
+<p>Now that we’ve seen all these operators, we can understand the full
+definition of <code>Applicative</code>. All the operators have
+definitions in terms of <code>liftA2</code>, so it’s enought to define
+<code>liftA2</code> and <code>pure</code> when implementing an
+<code>Applicative</code> instance.</p>
+<div class="sourceCode" id="cb363"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb363-1"><a href="#cb363-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Functor</span> f <span class="ot">=&gt;</span> <span class="dt">Applicative</span> f <span class="kw">where</span></span>
 <span id="cb363-2"><a href="#cb363-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  pure ::</span> a <span class="ot">-&gt;</span> f a</span>
 <span id="cb363-3"><a href="#cb363-3" aria-hidden="true" tabindex="-1"></a><span class="ot">  liftA2 ::</span> (a <span class="ot">-&gt;</span> b <span class="ot">-&gt;</span> c) <span class="ot">-&gt;</span> f a <span class="ot">-&gt;</span> f b <span class="ot">-&gt;</span> f c</span>
 <span id="cb363-4"><a href="#cb363-4" aria-hidden="true" tabindex="-1"></a><span class="ot">  (&lt;*&gt;) ::</span> f (a <span class="ot">-&gt;</span> b) <span class="ot">-&gt;</span> f a <span class="ot">-&gt;</span> f b</span>
 <span id="cb363-5"><a href="#cb363-5" aria-hidden="true" tabindex="-1"></a><span class="ot">  (*&gt;) ::</span> f a <span class="ot">-&gt;</span> f b <span class="ot">-&gt;</span> f b</span>
 <span id="cb363-6"><a href="#cb363-6" aria-hidden="true" tabindex="-1"></a><span class="ot">  (&lt;*) ::</span> f a <span class="ot">-&gt;</span> f b <span class="ot">-&gt;</span> f a</span></code></pre></div>
-</section>
-<section id="the-validation-applicative" data-number="7.4">
-<h2 data-number="15.4"><span class="header-section-number">15.4</span> The Validation Applicative</h2>
-<p>Let’s look at an Applicative that’s a bit more interesting than Maybe or lists. Often in programming we need to <em>validate</em> some inputs from the user. In these cases it’s useful to gather together all the errors that the input might have. The file <a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Examples/Validation.hs"><code>exercises/Examples/Validation.hs</code></a> implements the <code>Validation</code> datatype:</p>
-<div class="sourceCode" id="cb364"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb364-1"><a href="#cb364-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Validation</span> a <span class="ot">=</span> <span class="dt">Ok</span> a <span class="op">|</span> <span class="dt">Errors</span> [<span class="dt">String</span>]</span>
+<h2 data-number="15.4" id="the-validation-applicative"><span
+class="header-section-number">15.4</span> The Validation
+Applicative</h2>
+<p>Let’s look at an Applicative that’s a bit more interesting than Maybe
+or lists. Often in programming we need to <em>validate</em> some inputs
+from the user. In these cases it’s useful to gather together all the
+errors that the input might have. The file <a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Examples/Validation.hs"><code>exercises/Examples/Validation.hs</code></a>
+implements the <code>Validation</code> datatype:</p>
+<div class="sourceCode" id="cb364"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb364-1"><a href="#cb364-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Validation</span> a <span class="ot">=</span> <span class="dt">Ok</span> a <span class="op">|</span> <span class="dt">Errors</span> [<span class="dt">String</span>]</span>
 <span id="cb364-2"><a href="#cb364-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">deriving</span> (<span class="dt">Show</span>,<span class="dt">Eq</span>)</span></code></pre></div>
-<p>The <code>Applicative</code> instance for <code>Validation</code> works like this:</p>
-<div class="sourceCode" id="cb365"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb365-1"><a href="#cb365-1" aria-hidden="true" tabindex="-1"></a>liftA2 (<span class="op">+</span>) (<span class="dt">Ok</span> <span class="dv">1</span>) (<span class="dt">Ok</span> <span class="dv">2</span>)</span>
+<p>The <code>Applicative</code> instance for <code>Validation</code>
+works like this:</p>
+<div class="sourceCode" id="cb365"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb365-1"><a href="#cb365-1" aria-hidden="true" tabindex="-1"></a>liftA2 (<span class="op">+</span>) (<span class="dt">Ok</span> <span class="dv">1</span>) (<span class="dt">Ok</span> <span class="dv">2</span>)</span>
 <span id="cb365-2"><a href="#cb365-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Ok</span> <span class="dv">3</span></span>
 <span id="cb365-3"><a href="#cb365-3" aria-hidden="true" tabindex="-1"></a>liftA2 (<span class="op">+</span>) (<span class="dt">Errors</span> [<span class="st">&quot;oh no&quot;</span>]) (<span class="dt">Errors</span> [<span class="st">&quot;boom&quot;</span>])</span>
 <span id="cb365-4"><a href="#cb365-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Errors</span> [<span class="st">&quot;oh no&quot;</span>,<span class="st">&quot;boom&quot;</span>]</span></code></pre></div>
-<p>Note how in contrast to the <code>Maybe</code> Applicative, we have many different kinds of failures.</p>
-<p>Here’s a worked example that introduces some helpers and then uses them to congratulate somebody on their birthday:</p>
-<div class="sourceCode" id="cb366"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb366-1"><a href="#cb366-1" aria-hidden="true" tabindex="-1"></a><span class="ot">invalid ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Validation</span> a</span>
+<p>Note how in contrast to the <code>Maybe</code> Applicative, we have
+many different kinds of failures.</p>
+<p>Here’s a worked example that introduces some helpers and then uses
+them to congratulate somebody on their birthday:</p>
+<div class="sourceCode" id="cb366"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb366-1"><a href="#cb366-1" aria-hidden="true" tabindex="-1"></a><span class="ot">invalid ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Validation</span> a</span>
 <span id="cb366-2"><a href="#cb366-2" aria-hidden="true" tabindex="-1"></a>invalid err <span class="ot">=</span> <span class="dt">Errors</span> [err]</span>
 <span id="cb366-3"><a href="#cb366-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb366-4"><a href="#cb366-4" aria-hidden="true" tabindex="-1"></a><span class="ot">check ::</span> <span class="dt">Bool</span> <span class="ot">-&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Validation</span> a</span>
@@ -4123,14 +5841,17 @@ liftA2 sumMoney (parseMoney &quot;123&quot; &quot;e&quot;) (parseMoney &quot;xxx
 <span id="cb366-11"><a href="#cb366-11" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> checkedName <span class="ot">=</span> check (<span class="fu">length</span> name <span class="op">&lt;</span> <span class="dv">10</span>) <span class="st">&quot;Name too long&quot;</span> name</span>
 <span id="cb366-12"><a href="#cb366-12" aria-hidden="true" tabindex="-1"></a>        checkedAge <span class="ot">=</span> check (age <span class="op">&lt;</span> <span class="dv">99</span>) <span class="st">&quot;Too old&quot;</span> age</span>
 <span id="cb366-13"><a href="#cb366-13" aria-hidden="true" tabindex="-1"></a>        congratulate n a <span class="ot">=</span> <span class="st">&quot;Happy &quot;</span><span class="op">++</span><span class="fu">show</span> a<span class="op">++</span><span class="st">&quot;th birthday &quot;</span><span class="op">++</span>n<span class="op">++</span><span class="st">&quot;!&quot;</span></span></code></pre></div>
-<div class="sourceCode" id="cb367"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb367-1"><a href="#cb367-1" aria-hidden="true" tabindex="-1"></a>birthday <span class="st">&quot;Guy&quot;</span> <span class="dv">31</span></span>
+<div class="sourceCode" id="cb367"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb367-1"><a href="#cb367-1" aria-hidden="true" tabindex="-1"></a>birthday <span class="st">&quot;Guy&quot;</span> <span class="dv">31</span></span>
 <span id="cb367-2"><a href="#cb367-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Ok</span> <span class="st">&quot;Happy 31th birthday Guy!&quot;</span></span>
 <span id="cb367-3"><a href="#cb367-3" aria-hidden="true" tabindex="-1"></a>birthday <span class="st">&quot;Guybrush Threepwood&quot;</span> <span class="dv">31</span></span>
 <span id="cb367-4"><a href="#cb367-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Errors</span> [<span class="st">&quot;Name too long&quot;</span>]</span>
 <span id="cb367-5"><a href="#cb367-5" aria-hidden="true" tabindex="-1"></a>birthday <span class="st">&quot;Yog-sothoth&quot;</span> <span class="dv">10000</span></span>
 <span id="cb367-6"><a href="#cb367-6" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Errors</span> [<span class="st">&quot;Name too long&quot;</span>,<span class="st">&quot;Too old&quot;</span>]</span></code></pre></div>
-<p>Oh right, here are the <code>Functor</code> and <code>Applicative</code> instances for <code>Validation</code>:</p>
-<div class="sourceCode" id="cb368"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb368-1"><a href="#cb368-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Functor</span> <span class="dt">Validation</span> <span class="kw">where</span></span>
+<p>Oh right, here are the <code>Functor</code> and
+<code>Applicative</code> instances for <code>Validation</code>:</p>
+<div class="sourceCode" id="cb368"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb368-1"><a href="#cb368-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Functor</span> <span class="dt">Validation</span> <span class="kw">where</span></span>
 <span id="cb368-2"><a href="#cb368-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">fmap</span> f (<span class="dt">Ok</span> x) <span class="ot">=</span> <span class="dt">Ok</span> (f x)</span>
 <span id="cb368-3"><a href="#cb368-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">fmap</span> _ (<span class="dt">Errors</span> e) <span class="ot">=</span> <span class="dt">Errors</span> e</span>
 <span id="cb368-4"><a href="#cb368-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -4140,59 +5861,92 @@ liftA2 sumMoney (parseMoney &quot;123&quot; &quot;e&quot;) (parseMoney &quot;xxx
 <span id="cb368-8"><a href="#cb368-8" aria-hidden="true" tabindex="-1"></a>  liftA2 f (<span class="dt">Errors</span> e1) (<span class="dt">Ok</span> y)      <span class="ot">=</span> <span class="dt">Errors</span> e1</span>
 <span id="cb368-9"><a href="#cb368-9" aria-hidden="true" tabindex="-1"></a>  liftA2 f (<span class="dt">Ok</span> x)      (<span class="dt">Errors</span> e2) <span class="ot">=</span> <span class="dt">Errors</span> e2</span>
 <span id="cb368-10"><a href="#cb368-10" aria-hidden="true" tabindex="-1"></a>  liftA2 f (<span class="dt">Errors</span> e1) (<span class="dt">Errors</span> e2) <span class="ot">=</span> <span class="dt">Errors</span> (e1<span class="op">++</span>e2)</span></code></pre></div>
-<p>The definition of <code>liftA2</code> for <code>Validation</code> shows that the errors are collected together left-to-right. This can be seen in the example above where the expression <code>liftA2 congratulate checkedName checkedAge</code> outputs the error (<code>"Name too long"</code>) from <code>checkedName</code> first, and the error (<code>"Too old"</code>) from <code>checkedAge</code> last.</p>
+<p>The definition of <code>liftA2</code> for <code>Validation</code>
+shows that the errors are collected together left-to-right. This can be
+seen in the example above where the expression
+<code>liftA2 congratulate checkedName checkedAge</code> outputs the
+error (<code>"Name too long"</code>) from <code>checkedName</code>
+first, and the error (<code>"Too old"</code>) from
+<code>checkedAge</code> last.</p>
 <!-- TODO other applicatives: IO, ? -->
-</section>
-<section id="validating-lists-traverse" data-number="7.5">
-<h2 data-number="15.5"><span class="header-section-number">15.5</span> Validating Lists: <code>traverse</code></h2>
-<p>So far we’ve dealt with fixed-size things and Applicatives: we’ve applied a function of two or three arguments to some things. What if we have an arbitrary amount of inputs? What if we need to validate a list?</p>
+<h2 data-number="15.5" id="validating-lists-traverse"><span
+class="header-section-number">15.5</span> Validating Lists:
+<code>traverse</code></h2>
+<p>So far we’ve dealt with fixed-size things and Applicatives: we’ve
+applied a function of two or three arguments to some things. What if we
+have an arbitrary amount of inputs? What if we need to validate a
+list?</p>
 <p>Let’s look at some ways to implement a function like this:</p>
-<div class="sourceCode" id="cb369"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb369-1"><a href="#cb369-1" aria-hidden="true" tabindex="-1"></a>allPositive [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
+<div class="sourceCode" id="cb369"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb369-1"><a href="#cb369-1" aria-hidden="true" tabindex="-1"></a>allPositive [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb369-2"><a href="#cb369-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Ok</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb369-3"><a href="#cb369-3" aria-hidden="true" tabindex="-1"></a>allPositive [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="op">-</span><span class="dv">4</span>]</span>
 <span id="cb369-4"><a href="#cb369-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Errors</span> [<span class="st">&quot;Not positive: -4&quot;</span>]</span>
 <span id="cb369-5"><a href="#cb369-5" aria-hidden="true" tabindex="-1"></a>allPositive [<span class="dv">1</span>,<span class="op">-</span><span class="dv">2</span>,<span class="dv">3</span>,<span class="op">-</span><span class="dv">4</span>]</span>
 <span id="cb369-6"><a href="#cb369-6" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Errors</span> [<span class="st">&quot;Not positive: -2&quot;</span>,<span class="st">&quot;Not positive: -4&quot;</span>]</span></code></pre></div>
-<p>As always, when working with lists, pattern matching and recursion are usually the way to go. Here’s a recursive solution:</p>
-<div class="sourceCode" id="cb370"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb370-1"><a href="#cb370-1" aria-hidden="true" tabindex="-1"></a><span class="ot">allPositive ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Validation</span> [<span class="dt">Int</span>]</span>
+<p>As always, when working with lists, pattern matching and recursion
+are usually the way to go. Here’s a recursive solution:</p>
+<div class="sourceCode" id="cb370"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb370-1"><a href="#cb370-1" aria-hidden="true" tabindex="-1"></a><span class="ot">allPositive ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Validation</span> [<span class="dt">Int</span>]</span>
 <span id="cb370-2"><a href="#cb370-2" aria-hidden="true" tabindex="-1"></a>allPositive [] <span class="ot">=</span> <span class="dt">Ok</span> []</span>
 <span id="cb370-3"><a href="#cb370-3" aria-hidden="true" tabindex="-1"></a>allPositive (x<span class="op">:</span>xs) <span class="ot">=</span> liftA2 (<span class="op">:</span>) checkThis checkRest</span>
 <span id="cb370-4"><a href="#cb370-4" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> checkThis <span class="ot">=</span> check (x<span class="op">&gt;=</span><span class="dv">0</span>) (<span class="st">&quot;Not positive: &quot;</span><span class="op">++</span><span class="fu">show</span> x) x</span>
 <span id="cb370-5"><a href="#cb370-5" aria-hidden="true" tabindex="-1"></a>        checkRest <span class="ot">=</span> allPositive xs</span></code></pre></div>
-<p>It’s a bit of a chore to always spell out a recursion like this. If we were working in a <code>Monad</code> we could just use a helper like <code>mapM</code>:</p>
-<div class="sourceCode" id="cb371"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb371-1"><a href="#cb371-1" aria-hidden="true" tabindex="-1"></a><span class="fu">mapM</span> (\x <span class="ot">-&gt;</span> <span class="kw">if</span> x<span class="op">&gt;=</span><span class="dv">0</span> <span class="kw">then</span> <span class="dt">Just</span> x <span class="kw">else</span> <span class="dt">Nothing</span>) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
+<p>It’s a bit of a chore to always spell out a recursion like this. If
+we were working in a <code>Monad</code> we could just use a helper like
+<code>mapM</code>:</p>
+<div class="sourceCode" id="cb371"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb371-1"><a href="#cb371-1" aria-hidden="true" tabindex="-1"></a><span class="fu">mapM</span> (\x <span class="ot">-&gt;</span> <span class="kw">if</span> x<span class="op">&gt;=</span><span class="dv">0</span> <span class="kw">then</span> <span class="dt">Just</span> x <span class="kw">else</span> <span class="dt">Nothing</span>) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb371-2"><a href="#cb371-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Just</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb371-3"><a href="#cb371-3" aria-hidden="true" tabindex="-1"></a><span class="fu">mapM</span> (\x <span class="ot">-&gt;</span> <span class="kw">if</span> x<span class="op">&gt;=</span><span class="dv">0</span> <span class="kw">then</span> <span class="dt">Just</span> x <span class="kw">else</span> <span class="dt">Nothing</span>) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="op">-</span><span class="dv">4</span>]</span>
 <span id="cb371-4"><a href="#cb371-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Nothing</span></span></code></pre></div>
-<p>The equivalent of <code>mapM</code> for <code>Applicative</code> is called <code>traverse</code>. It’s a member of the type class <code>Traversable</code>:</p>
-<div class="sourceCode" id="cb372"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb372-1"><a href="#cb372-1" aria-hidden="true" tabindex="-1"></a><span class="fu">traverse</span><span class="ot"> ::</span> (<span class="dt">Traversable</span> t, <span class="dt">Applicative</span> f) <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> f b) <span class="ot">-&gt;</span> t a <span class="ot">-&gt;</span> f (t b)</span></code></pre></div>
-<p>That’s one heck of a type signature, so let’s simplify it a bit. Lists are <code>Traversable</code>, so we can specialize this type into:</p>
-<div class="sourceCode" id="cb373"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb373-1"><a href="#cb373-1" aria-hidden="true" tabindex="-1"></a><span class="fu">traverse</span><span class="ot"> ::</span> <span class="dt">Applicative</span> f <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> f b) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> f [b]</span></code></pre></div>
-<p>That looks like what we need! For Applicatives that are also Monads, <code>traverse</code> is just another name for <code>mapM</code>:</p>
-<div class="sourceCode" id="cb374"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb374-1"><a href="#cb374-1" aria-hidden="true" tabindex="-1"></a><span class="fu">traverse</span> (\x <span class="ot">-&gt;</span> <span class="kw">if</span> x<span class="op">&gt;=</span><span class="dv">0</span> <span class="kw">then</span> <span class="dt">Just</span> x <span class="kw">else</span> <span class="dt">Nothing</span>) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
+<p>The equivalent of <code>mapM</code> for <code>Applicative</code> is
+called <code>traverse</code>. It’s a member of the type class
+<code>Traversable</code>:</p>
+<div class="sourceCode" id="cb372"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb372-1"><a href="#cb372-1" aria-hidden="true" tabindex="-1"></a><span class="fu">traverse</span><span class="ot"> ::</span> (<span class="dt">Traversable</span> t, <span class="dt">Applicative</span> f) <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> f b) <span class="ot">-&gt;</span> t a <span class="ot">-&gt;</span> f (t b)</span></code></pre></div>
+<p>That’s one heck of a type signature, so let’s simplify it a bit.
+Lists are <code>Traversable</code>, so we can specialize this type
+into:</p>
+<div class="sourceCode" id="cb373"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb373-1"><a href="#cb373-1" aria-hidden="true" tabindex="-1"></a><span class="fu">traverse</span><span class="ot"> ::</span> <span class="dt">Applicative</span> f <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> f b) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> f [b]</span></code></pre></div>
+<p>That looks like what we need! For Applicatives that are also Monads,
+<code>traverse</code> is just another name for <code>mapM</code>:</p>
+<div class="sourceCode" id="cb374"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb374-1"><a href="#cb374-1" aria-hidden="true" tabindex="-1"></a><span class="fu">traverse</span> (\x <span class="ot">-&gt;</span> <span class="kw">if</span> x<span class="op">&gt;=</span><span class="dv">0</span> <span class="kw">then</span> <span class="dt">Just</span> x <span class="kw">else</span> <span class="dt">Nothing</span>) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb374-2"><a href="#cb374-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Just</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb374-3"><a href="#cb374-3" aria-hidden="true" tabindex="-1"></a><span class="fu">traverse</span> (\x <span class="ot">-&gt;</span> <span class="kw">if</span> x<span class="op">&gt;=</span><span class="dv">0</span> <span class="kw">then</span> <span class="dt">Just</span> x <span class="kw">else</span> <span class="dt">Nothing</span>) [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="op">-</span><span class="dv">4</span>]</span>
 <span id="cb374-4"><a href="#cb374-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Nothing</span></span></code></pre></div>
-<p>But for our <code>Validation</code>, which isn’t a <code>Monad</code>, <code>traverse</code> is exactly what we want:</p>
-<div class="sourceCode" id="cb375"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb375-1"><a href="#cb375-1" aria-hidden="true" tabindex="-1"></a><span class="ot">allPositive ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Validation</span> [<span class="dt">Int</span>]</span>
+<p>But for our <code>Validation</code>, which isn’t a
+<code>Monad</code>, <code>traverse</code> is exactly what we want:</p>
+<div class="sourceCode" id="cb375"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb375-1"><a href="#cb375-1" aria-hidden="true" tabindex="-1"></a><span class="ot">allPositive ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Validation</span> [<span class="dt">Int</span>]</span>
 <span id="cb375-2"><a href="#cb375-2" aria-hidden="true" tabindex="-1"></a>allPositive xs <span class="ot">=</span> <span class="fu">traverse</span> checkNumber xs</span>
 <span id="cb375-3"><a href="#cb375-3" aria-hidden="true" tabindex="-1"></a>  <span class="kw">where</span> checkNumber x <span class="ot">=</span> check (x<span class="op">&gt;=</span><span class="dv">0</span>) (<span class="st">&quot;Not positive: &quot;</span><span class="op">++</span><span class="fu">show</span> x) x</span></code></pre></div>
-<div class="sourceCode" id="cb376"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb376-1"><a href="#cb376-1" aria-hidden="true" tabindex="-1"></a>allPositive [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
+<div class="sourceCode" id="cb376"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb376-1"><a href="#cb376-1" aria-hidden="true" tabindex="-1"></a>allPositive [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb376-2"><a href="#cb376-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Ok</span> [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>]</span>
 <span id="cb376-3"><a href="#cb376-3" aria-hidden="true" tabindex="-1"></a>allPositive [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="op">-</span><span class="dv">4</span>]</span>
 <span id="cb376-4"><a href="#cb376-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Errors</span> [<span class="st">&quot;Not positive: -4&quot;</span>]</span>
 <span id="cb376-5"><a href="#cb376-5" aria-hidden="true" tabindex="-1"></a>allPositive [<span class="dv">1</span>,<span class="op">-</span><span class="dv">2</span>,<span class="dv">3</span>,<span class="op">-</span><span class="dv">4</span>]</span>
 <span id="cb376-6"><a href="#cb376-6" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Errors</span> [<span class="st">&quot;Not positive: -2&quot;</span>,<span class="st">&quot;Not positive: -4&quot;</span>]</span></code></pre></div>
-<p>Note how <code>traverse</code> for <code>Validation</code> collects all the errors together, in the order they occur in the original list.</p>
-<p>PS. In fact, <code>Validation</code> is one of the few examples of an <code>Applicative</code> that can’t be a <code>Monad</code>. Can you figure out why?</p>
+<p>Note how <code>traverse</code> for <code>Validation</code> collects
+all the errors together, in the order they occur in the original
+list.</p>
+<p>PS. In fact, <code>Validation</code> is one of the few examples of an
+<code>Applicative</code> that can’t be a <code>Monad</code>. Can you
+figure out why?</p>
 <!-- TODO `traverse_`? -->
-</section>
-<section id="sidenote-traversable" data-number="7.6">
-<h2 data-number="15.6"><span class="header-section-number">15.6</span> Sidenote: <code>Traversable</code></h2>
-<p>So what things are <code>Traversable</code>? Many familiar structures. Here are some examples:</p>
-<div class="sourceCode" id="cb377"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb377-1"><a href="#cb377-1" aria-hidden="true" tabindex="-1"></a><span class="ot">decrease ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">Int</span></span>
+<h2 data-number="15.6" id="sidenote-traversable"><span
+class="header-section-number">15.6</span> Sidenote:
+<code>Traversable</code></h2>
+<p>So what things are <code>Traversable</code>? Many familiar
+structures. Here are some examples:</p>
+<div class="sourceCode" id="cb377"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb377-1"><a href="#cb377-1" aria-hidden="true" tabindex="-1"></a><span class="ot">decrease ::</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">Int</span></span>
 <span id="cb377-2"><a href="#cb377-2" aria-hidden="true" tabindex="-1"></a>decrease i <span class="ot">=</span> <span class="kw">if</span> i<span class="op">&gt;</span><span class="dv">0</span> <span class="kw">then</span> <span class="dt">Just</span> (i<span class="op">-</span><span class="dv">1</span>) <span class="kw">else</span> <span class="dt">Nothing</span></span></code></pre></div>
-<div class="sourceCode" id="cb378"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb378-1"><a href="#cb378-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Lists are Traversable</span></span>
+<div class="sourceCode" id="cb378"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb378-1"><a href="#cb378-1" aria-hidden="true" tabindex="-1"></a><span class="co">-- Lists are Traversable</span></span>
 <span id="cb378-2"><a href="#cb378-2" aria-hidden="true" tabindex="-1"></a><span class="fu">traverse</span> decrease [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>] <span class="op">==&gt;</span> <span class="dt">Just</span> [<span class="dv">0</span>,<span class="dv">1</span>,<span class="dv">2</span>]</span>
 <span id="cb378-3"><a href="#cb378-3" aria-hidden="true" tabindex="-1"></a><span class="fu">traverse</span> decrease [<span class="dv">1</span>,<span class="dv">0</span>,<span class="dv">3</span>] <span class="op">==&gt;</span> <span class="dt">Nothing</span></span>
 <span id="cb378-4"><a href="#cb378-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -4210,19 +5964,37 @@ liftA2 sumMoney (parseMoney &quot;123&quot; &quot;e&quot;) (parseMoney &quot;xxx
 <span id="cb378-16"><a href="#cb378-16" aria-hidden="true" tabindex="-1"></a><span class="fu">traverse</span> decrease (<span class="dt">Left</span> <span class="st">&quot;abc&quot;</span>) <span class="op">==&gt;</span> <span class="dt">Just</span> (<span class="dt">Left</span> <span class="st">&quot;abc&quot;</span>)</span>
 <span id="cb378-17"><a href="#cb378-17" aria-hidden="true" tabindex="-1"></a><span class="fu">traverse</span> decrease (<span class="dt">Right</span> <span class="dv">3</span>)    <span class="op">==&gt;</span> <span class="dt">Just</span> (<span class="dt">Right</span> <span class="dv">2</span>)</span>
 <span id="cb378-18"><a href="#cb378-18" aria-hidden="true" tabindex="-1"></a><span class="fu">traverse</span> decrease (<span class="dt">Right</span> <span class="dv">0</span>)    <span class="op">==&gt;</span> <span class="dt">Nothing</span></span></code></pre></div>
-<p>So <code>Traversable</code> is a type class for all sorts of containers, kind of like <code>Foldable</code>. Indeed, if you look at the definition, <code>Traversable</code> is a subclass of <code>Foldable</code>. It also turns out that <code>traverse</code> and <code>mapM</code> are methods of the class!</p>
-<div class="sourceCode" id="cb379"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb379-1"><a href="#cb379-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> (<span class="dt">Functor</span> t, <span class="dt">Foldable</span> t) <span class="ot">=&gt;</span> <span class="dt">Traversable</span> t <span class="kw">where</span></span>
+<p>So <code>Traversable</code> is a type class for all sorts of
+containers, kind of like <code>Foldable</code>. Indeed, if you look at
+the definition, <code>Traversable</code> is a subclass of
+<code>Foldable</code>. It also turns out that <code>traverse</code> and
+<code>mapM</code> are methods of the class!</p>
+<div class="sourceCode" id="cb379"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb379-1"><a href="#cb379-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> (<span class="dt">Functor</span> t, <span class="dt">Foldable</span> t) <span class="ot">=&gt;</span> <span class="dt">Traversable</span> t <span class="kw">where</span></span>
 <span id="cb379-2"><a href="#cb379-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  traverse ::</span> <span class="dt">Applicative</span> f <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> f b) <span class="ot">-&gt;</span> t a <span class="ot">-&gt;</span> f (t b)</span>
 <span id="cb379-3"><a href="#cb379-3" aria-hidden="true" tabindex="-1"></a><span class="ot">  mapM ::</span> <span class="dt">Monad</span> m <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> m b) <span class="ot">-&gt;</span> t a <span class="ot">-&gt;</span> m (t b)</span></code></pre></div>
-<p>It can be hard to keep the types straight here. Let’s go back to the type of <code>traverse</code>:</p>
-<div class="sourceCode" id="cb380"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb380-1"><a href="#cb380-1" aria-hidden="true" tabindex="-1"></a><span class="fu">traverse</span><span class="ot"> ::</span> (<span class="dt">Traversable</span> t, <span class="dt">Applicative</span> f) <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> f b) <span class="ot">-&gt;</span> t a <span class="ot">-&gt;</span> f (t b)</span></code></pre></div>
-<p>Here we have two functors: <code>t</code> and <code>f</code>. The <code>t</code> functor is also a <code>Foldable</code> and a <code>Traversable</code> and the <code>f</code> functor is also an <code>Applicative</code>. The <code>traverse</code> function lets us run <code>f</code>-operations inside a <code>t</code>-container.</p>
-<p>Don’t worry if that feels abstract. In practice, you pretty much always use <code>traverse</code> on lists.</p>
-</section>
-<section id="dealing-with-failure-alternative" data-number="7.7">
-<h2 data-number="15.7"><span class="header-section-number">15.7</span> Dealing with Failure: <code>Alternative</code></h2>
-<p>If you play around with applicatives a bit you’ll start to notice some limits to their power. For example, when writing parsers like we did in the <code>parseMoney</code> example, it would be nice to be able to try a couple of different parsers and take any non-failure result. This is easy enough to write for a concrete Applicative like <code>Maybe</code>, as can be seen below.</p>
-<div class="sourceCode" id="cb381"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb381-1"><a href="#cb381-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Answer</span> <span class="ot">=</span> <span class="dt">Yes</span> <span class="op">|</span> <span class="dt">No</span></span>
+<p>It can be hard to keep the types straight here. Let’s go back to the
+type of <code>traverse</code>:</p>
+<div class="sourceCode" id="cb380"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb380-1"><a href="#cb380-1" aria-hidden="true" tabindex="-1"></a><span class="fu">traverse</span><span class="ot"> ::</span> (<span class="dt">Traversable</span> t, <span class="dt">Applicative</span> f) <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> f b) <span class="ot">-&gt;</span> t a <span class="ot">-&gt;</span> f (t b)</span></code></pre></div>
+<p>Here we have two functors: <code>t</code> and <code>f</code>. The
+<code>t</code> functor is also a <code>Foldable</code> and a
+<code>Traversable</code> and the <code>f</code> functor is also an
+<code>Applicative</code>. The <code>traverse</code> function lets us run
+<code>f</code>-operations inside a <code>t</code>-container.</p>
+<p>Don’t worry if that feels abstract. In practice, you pretty much
+always use <code>traverse</code> on lists.</p>
+<h2 data-number="15.7" id="dealing-with-failure-alternative"><span
+class="header-section-number">15.7</span> Dealing with Failure:
+<code>Alternative</code></h2>
+<p>If you play around with applicatives a bit you’ll start to notice
+some limits to their power. For example, when writing parsers like we
+did in the <code>parseMoney</code> example, it would be nice to be able
+to try a couple of different parsers and take any non-failure result.
+This is easy enough to write for a concrete Applicative like
+<code>Maybe</code>, as can be seen below.</p>
+<div class="sourceCode" id="cb381"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb381-1"><a href="#cb381-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Answer</span> <span class="ot">=</span> <span class="dt">Yes</span> <span class="op">|</span> <span class="dt">No</span></span>
 <span id="cb381-2"><a href="#cb381-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">deriving</span> (<span class="dt">Show</span>, <span class="dt">Eq</span>)</span>
 <span id="cb381-3"><a href="#cb381-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb381-4"><a href="#cb381-4" aria-hidden="true" tabindex="-1"></a><span class="ot">parseYes ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">Answer</span></span>
@@ -4244,19 +6016,29 @@ liftA2 sumMoney (parseMoney &quot;123&quot; &quot;e&quot;) (parseMoney &quot;xxx
 <span id="cb381-20"><a href="#cb381-20" aria-hidden="true" tabindex="-1"></a><span class="ot">parseAnswer ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Maybe</span> <span class="dt">Answer</span></span>
 <span id="cb381-21"><a href="#cb381-21" aria-hidden="true" tabindex="-1"></a><span class="co">-- prefer positive answers!</span></span>
 <span id="cb381-22"><a href="#cb381-22" aria-hidden="true" tabindex="-1"></a>parseAnswer s <span class="ot">=</span> eitherOf (parseYes s) (parseNo s)</span></code></pre></div>
-<div class="sourceCode" id="cb382"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb382-1"><a href="#cb382-1" aria-hidden="true" tabindex="-1"></a>parseAnswer <span class="st">&quot;yes&quot;</span>    <span class="op">==&gt;</span> <span class="dt">Just</span> <span class="dt">Yes</span></span>
+<div class="sourceCode" id="cb382"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb382-1"><a href="#cb382-1" aria-hidden="true" tabindex="-1"></a>parseAnswer <span class="st">&quot;yes&quot;</span>    <span class="op">==&gt;</span> <span class="dt">Just</span> <span class="dt">Yes</span></span>
 <span id="cb382-2"><a href="#cb382-2" aria-hidden="true" tabindex="-1"></a>parseAnswer <span class="st">&quot;y&quot;</span>      <span class="op">==&gt;</span> <span class="dt">Just</span> <span class="dt">Yes</span></span>
 <span id="cb382-3"><a href="#cb382-3" aria-hidden="true" tabindex="-1"></a>parseAnswer <span class="st">&quot;n&quot;</span>      <span class="op">==&gt;</span> <span class="dt">Just</span> <span class="dt">No</span></span>
 <span id="cb382-4"><a href="#cb382-4" aria-hidden="true" tabindex="-1"></a>parseAnswer <span class="st">&quot;maybe&quot;</span>  <span class="op">==&gt;</span> <span class="dt">Just</span> <span class="dt">Yes</span></span>
 <span id="cb382-5"><a href="#cb382-5" aria-hidden="true" tabindex="-1"></a>parseAnswer <span class="st">&quot;x&quot;</span>      <span class="op">==&gt;</span> <span class="dt">Nothing</span></span></code></pre></div>
-<p>How could we generalize <code>eitherOf</code>? We can’t give it the type <code>Applicative f =&gt; f x -&gt; f x -&gt; f x</code> because then the implementation would need to be effectively something like <code>eitherOf a b = liftA2 something a b</code>, but then <code>eitherOf Nothing (Just x)</code> would be <code>Nothing</code> (since that’s how the Applicative instance works)!</p>
-<p>It turns out we need a new type class: <code>Alternative</code>. An Alternative adds two operations to Applicative: <code>empty</code> means no results, and <code>&lt;|&gt;</code> means combining results.</p>
-<div class="sourceCode" id="cb383"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb383-1"><a href="#cb383-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Applicative</span> f <span class="ot">=&gt;</span> <span class="dt">Alternative</span> f <span class="kw">where</span></span>
+<p>How could we generalize <code>eitherOf</code>? We can’t give it the
+type <code>Applicative f =&gt; f x -&gt; f x -&gt; f x</code> because
+then the implementation would need to be effectively something like
+<code>eitherOf a b = liftA2 something a b</code>, but then
+<code>eitherOf Nothing (Just x)</code> would be <code>Nothing</code>
+(since that’s how the Applicative instance works)!</p>
+<p>It turns out we need a new type class: <code>Alternative</code>. An
+Alternative adds two operations to Applicative: <code>empty</code> means
+no results, and <code>&lt;|&gt;</code> means combining results.</p>
+<div class="sourceCode" id="cb383"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb383-1"><a href="#cb383-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Applicative</span> f <span class="ot">=&gt;</span> <span class="dt">Alternative</span> f <span class="kw">where</span></span>
 <span id="cb383-2"><a href="#cb383-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  empty ::</span> f a</span>
 <span id="cb383-3"><a href="#cb383-3" aria-hidden="true" tabindex="-1"></a><span class="ot">  (&lt;|&gt;) ::</span> f a <span class="ot">-&gt;</span> f a <span class="ot">-&gt;</span> f a</span>
 <span id="cb383-4"><a href="#cb383-4" aria-hidden="true" tabindex="-1"></a>  <span class="co">-- some other operations omitted</span></span></code></pre></div>
 <p>Now we can rewrite our parsing code using general operations:</p>
-<div class="sourceCode" id="cb384"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb384-1"><a href="#cb384-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Answer</span> <span class="ot">=</span> <span class="dt">Yes</span> <span class="op">|</span> <span class="dt">No</span></span>
+<div class="sourceCode" id="cb384"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb384-1"><a href="#cb384-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Answer</span> <span class="ot">=</span> <span class="dt">Yes</span> <span class="op">|</span> <span class="dt">No</span></span>
 <span id="cb384-2"><a href="#cb384-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">deriving</span> (<span class="dt">Show</span>, <span class="dt">Eq</span>)</span>
 <span id="cb384-3"><a href="#cb384-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb384-4"><a href="#cb384-4" aria-hidden="true" tabindex="-1"></a><span class="ot">parseYes ::</span> <span class="dt">Alternative</span> f <span class="ot">=&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> f <span class="dt">Answer</span></span>
@@ -4273,8 +6055,11 @@ liftA2 sumMoney (parseMoney &quot;123&quot; &quot;e&quot;) (parseMoney &quot;xxx
 <span id="cb384-15"><a href="#cb384-15" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb384-16"><a href="#cb384-16" aria-hidden="true" tabindex="-1"></a><span class="ot">parseAnswer ::</span> <span class="dt">Alternative</span> f <span class="ot">=&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> f <span class="dt">Answer</span></span>
 <span id="cb384-17"><a href="#cb384-17" aria-hidden="true" tabindex="-1"></a>parseAnswer s <span class="ot">=</span> parseYes s <span class="op">&lt;|&gt;</span> parseNo s</span></code></pre></div>
-<p>We can also pick which <code>Alternative</code> we run our parser in to get different behaviours. <code>Maybe</code> gives us only one result, while <code>[]</code> gives us all possible results.</p>
-<div class="sourceCode" id="cb385"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb385-1"><a href="#cb385-1" aria-hidden="true" tabindex="-1"></a><span class="op">&gt;</span> parseAnswer <span class="st">&quot;yes&quot;</span><span class="ot"> ::</span> <span class="dt">Maybe</span> <span class="dt">Answer</span></span>
+<p>We can also pick which <code>Alternative</code> we run our parser in
+to get different behaviours. <code>Maybe</code> gives us only one
+result, while <code>[]</code> gives us all possible results.</p>
+<div class="sourceCode" id="cb385"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb385-1"><a href="#cb385-1" aria-hidden="true" tabindex="-1"></a><span class="op">&gt;</span> parseAnswer <span class="st">&quot;yes&quot;</span><span class="ot"> ::</span> <span class="dt">Maybe</span> <span class="dt">Answer</span></span>
 <span id="cb385-2"><a href="#cb385-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Just</span> <span class="dt">Yes</span></span>
 <span id="cb385-3"><a href="#cb385-3" aria-hidden="true" tabindex="-1"></a><span class="op">&gt;</span> parseAnswer <span class="st">&quot;maybe&quot;</span><span class="ot"> ::</span> <span class="dt">Maybe</span> <span class="dt">Answer</span></span>
 <span id="cb385-4"><a href="#cb385-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Just</span> <span class="dt">Yes</span></span>
@@ -4282,8 +6067,10 @@ liftA2 sumMoney (parseMoney &quot;123&quot; &quot;e&quot;) (parseMoney &quot;xxx
 <span id="cb385-6"><a href="#cb385-6" aria-hidden="true" tabindex="-1"></a>[<span class="dt">Yes</span>]</span>
 <span id="cb385-7"><a href="#cb385-7" aria-hidden="true" tabindex="-1"></a><span class="op">&gt;</span> parseAnswer <span class="st">&quot;maybe&quot;</span><span class="ot"> ::</span> [<span class="dt">Answer</span>]</span>
 <span id="cb385-8"><a href="#cb385-8" aria-hidden="true" tabindex="-1"></a>[<span class="dt">Yes</span>,<span class="dt">No</span>]</span></code></pre></div>
-<p>The <code>Alternative</code> instances for <code>[]</code> and <code>Maybe</code> are unsurprising:</p>
-<div class="sourceCode" id="cb386"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb386-1"><a href="#cb386-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Alternative</span> [] <span class="kw">where</span></span>
+<p>The <code>Alternative</code> instances for <code>[]</code> and
+<code>Maybe</code> are unsurprising:</p>
+<div class="sourceCode" id="cb386"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb386-1"><a href="#cb386-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Alternative</span> [] <span class="kw">where</span></span>
 <span id="cb386-2"><a href="#cb386-2" aria-hidden="true" tabindex="-1"></a>  empty <span class="ot">=</span> []</span>
 <span id="cb386-3"><a href="#cb386-3" aria-hidden="true" tabindex="-1"></a>  (<span class="op">&lt;|&gt;</span>) <span class="ot">=</span> (<span class="op">++</span>)</span>
 <span id="cb386-4"><a href="#cb386-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -4291,14 +6078,19 @@ liftA2 sumMoney (parseMoney &quot;123&quot; &quot;e&quot;) (parseMoney &quot;xxx
 <span id="cb386-6"><a href="#cb386-6" aria-hidden="true" tabindex="-1"></a>  empty <span class="ot">=</span> <span class="dt">Nothing</span></span>
 <span id="cb386-7"><a href="#cb386-7" aria-hidden="true" tabindex="-1"></a>  <span class="dt">Just</span> x  <span class="op">&lt;|&gt;</span> _  <span class="ot">=</span> <span class="dt">Just</span> x</span>
 <span id="cb386-8"><a href="#cb386-8" aria-hidden="true" tabindex="-1"></a>  <span class="dt">Nothing</span> <span class="op">&lt;|&gt;</span> mx <span class="ot">=</span> mx</span></code></pre></div>
-<p>The <code>Validation</code> type is also an <code>Alternative</code>. The instance collects together all error messages, just like the <code>Applicative</code> instance did.</p>
-<div class="sourceCode" id="cb387"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb387-1"><a href="#cb387-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Alternative</span> <span class="dt">Validation</span> <span class="kw">where</span></span>
+<p>The <code>Validation</code> type is also an <code>Alternative</code>.
+The instance collects together all error messages, just like the
+<code>Applicative</code> instance did.</p>
+<div class="sourceCode" id="cb387"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb387-1"><a href="#cb387-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Alternative</span> <span class="dt">Validation</span> <span class="kw">where</span></span>
 <span id="cb387-2"><a href="#cb387-2" aria-hidden="true" tabindex="-1"></a>  empty <span class="ot">=</span> <span class="dt">Errors</span> []</span>
 <span id="cb387-3"><a href="#cb387-3" aria-hidden="true" tabindex="-1"></a>  <span class="dt">Ok</span> x <span class="op">&lt;|&gt;</span> _ <span class="ot">=</span> <span class="dt">Ok</span> x</span>
 <span id="cb387-4"><a href="#cb387-4" aria-hidden="true" tabindex="-1"></a>  <span class="dt">Errors</span> e1 <span class="op">&lt;|&gt;</span> <span class="dt">Ok</span> y <span class="ot">=</span> <span class="dt">Ok</span> y</span>
 <span id="cb387-5"><a href="#cb387-5" aria-hidden="true" tabindex="-1"></a>  <span class="dt">Errors</span> e1 <span class="op">&lt;|&gt;</span> <span class="dt">Errors</span> e2 <span class="ot">=</span> <span class="dt">Errors</span> (e1<span class="op">++</span>e2)</span></code></pre></div>
-<p>Here’s a final example: validating contact information, which is either a phone number or an email address.</p>
-<div class="sourceCode" id="cb388"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb388-1"><a href="#cb388-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">ContactInfo</span> <span class="ot">=</span> <span class="dt">Email</span> <span class="dt">String</span> <span class="op">|</span> <span class="dt">Phone</span> <span class="dt">String</span></span>
+<p>Here’s a final example: validating contact information, which is
+either a phone number or an email address.</p>
+<div class="sourceCode" id="cb388"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb388-1"><a href="#cb388-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">ContactInfo</span> <span class="ot">=</span> <span class="dt">Email</span> <span class="dt">String</span> <span class="op">|</span> <span class="dt">Phone</span> <span class="dt">String</span></span>
 <span id="cb388-2"><a href="#cb388-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">deriving</span> <span class="dt">Show</span></span>
 <span id="cb388-3"><a href="#cb388-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb388-4"><a href="#cb388-4" aria-hidden="true" tabindex="-1"></a><span class="ot">validateEmail ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Validation</span> <span class="dt">ContactInfo</span></span>
@@ -4315,7 +6107,8 @@ liftA2 sumMoney (parseMoney &quot;123&quot; &quot;e&quot;) (parseMoney &quot;xxx
 <span id="cb388-15"><a href="#cb388-15" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb388-16"><a href="#cb388-16" aria-hidden="true" tabindex="-1"></a><span class="ot">validateContactInfo ::</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">Validation</span> <span class="dt">ContactInfo</span></span>
 <span id="cb388-17"><a href="#cb388-17" aria-hidden="true" tabindex="-1"></a>validateContactInfo s <span class="ot">=</span> validateEmail s <span class="op">&lt;|&gt;</span> validatePhone s</span></code></pre></div>
-<div class="sourceCode" id="cb389"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb389-1"><a href="#cb389-1" aria-hidden="true" tabindex="-1"></a>validateContactInfo <span class="st">&quot;user@example.com&quot;</span></span>
+<div class="sourceCode" id="cb389"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb389-1"><a href="#cb389-1" aria-hidden="true" tabindex="-1"></a>validateContactInfo <span class="st">&quot;user@example.com&quot;</span></span>
 <span id="cb389-2"><a href="#cb389-2" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Ok</span> (<span class="dt">Email</span> <span class="st">&quot;user@example.com&quot;</span>)</span>
 <span id="cb389-3"><a href="#cb389-3" aria-hidden="true" tabindex="-1"></a>validateContactInfo <span class="st">&quot;01234&quot;</span></span>
 <span id="cb389-4"><a href="#cb389-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Ok</span> (<span class="dt">Phone</span> <span class="st">&quot;01234&quot;</span>)</span>
@@ -4328,7 +6121,11 @@ liftA2 sumMoney (parseMoney &quot;123&quot; &quot;e&quot;) (parseMoney &quot;xxx
 <span id="cb389-11"><a href="#cb389-11" aria-hidden="true" tabindex="-1"></a>validateContactInfo <span class="st">&quot;x&quot;</span></span>
 <span id="cb389-12"><a href="#cb389-12" aria-hidden="true" tabindex="-1"></a>  <span class="op">==&gt;</span> <span class="dt">Errors</span> [<span class="st">&quot;Not an email: should contain a @&quot;</span>,</span>
 <span id="cb389-13"><a href="#cb389-13" aria-hidden="true" tabindex="-1"></a>              <span class="st">&quot;Not a phone number: should be all numbers&quot;</span>]</span></code></pre></div>
-<p>Note how here, just like in previous examples, the errors are collected left-to-right: the errors from <code>validateEmail</code> come before the errors from <code>validatePhone</code>. The error from <code>checkDigits</code> comes before the error from <code>checkLength</code>.</p>
+<p>Note how here, just like in previous examples, the errors are
+collected left-to-right: the errors from <code>validateEmail</code> come
+before the errors from <code>validatePhone</code>. The error from
+<code>checkDigits</code> comes before the error from
+<code>checkLength</code>.</p>
 <!--
 TODO compare to `Monoid`?
 
@@ -4336,26 +6133,61 @@ TODO another example
 
 TODO explain `some` and `many`
 -->
-</section>
-<section id="sidenote-applicatives-in-context" data-number="7.8">
-<h2 data-number="15.8"><span class="header-section-number">15.8</span> Sidenote: Applicatives in Context</h2>
-<section id="why-applicatives" data-number="7.8.1">
-<h3 data-number="15.8.1"><span class="header-section-number">15.8.1</span> Why Applicatives?</h3>
-<p>There are multiple reasons for learning Applicatives, even if they do not provide any additional power over Monads. First off, as discussed in Lecture 13, the GHC standard library nowadays makes Applicative instances for all Monads compulsory. Thus a working Haskell programmer is bound to see lots of Applicative instances.</p>
-<p>Secondly, you’ll bump into Applicative operators a lot even in Monadic code. Expressions like <code>f &lt;$&gt; x &lt;*&gt; y &lt;*&gt; z</code> can be useful in many Monadic contexts. Also, since the <code>Traversable</code> type class is built in terms of <code>Applicative</code>, you’ll often use applicative operations with it.</p>
-<p>Thirdly, Applicatives are an excellent exercise in understanding functional design patterns. They marry the Functor pattern with the Monoid pattern, and Alternative brings in yet another Monoid-like dimension. Being able to efficiently work with Applicatives will make working with further abstractions like <em>monad transformers</em> or <em>lenses</em> easier.</p>
-<p>Lastly, there are a couple of types that are Applicatives but not Monads. <code>Validation</code> is one example, and a very practical one at that. Without understanding of Applicative, we’d have no way to recognize and generalize operations over types like this. Another such type is <a href="https://hackage.haskell.org/package/base-4.16.4.0/docs/Control-Applicative.html#t:ZipList"><code>ZipList</code></a>.</p>
-</section>
-<section id="applicatives-in-the-wild" data-number="7.8.2">
-<h3 data-number="15.8.2"><span class="header-section-number">15.8.2</span> Applicatives in the Wild</h3>
-<p>Even though we only looked at some very simple and concrete Applicatives in this lecture, there are plenty of Haskell libraries that use Applicatives for big things. Here are some examples.</p>
-<p>The same sort of idea as our <code>Validation</code> Applicative has been implemented in the <a href="https://hackage.haskell.org/package/validation">validation</a> and <a href="https://hackage.haskell.org/package/either">either</a> libraries.</p>
-<p>There are multiple parser libraries that use Applicatives. For example, <a href="https://hackage.haskell.org/package/regex-applicative">regex-applicative</a>, <a href="https://hackage.haskell.org/package/optparse-applicative">optparse-applicative</a>, <a href="https://cs-syd.eu/posts/2020-06-28-yamlparse-applicative">yamlparse-applicative</a>, <a href="https://hackage.haskell.org/package/json-stream">json-stream</a>, and so on.</p>
-</section>
-<section id="monads-and-applicatives" data-number="7.8.3">
-<h3 data-number="15.8.3"><span class="header-section-number">15.8.3</span> Monads and Applicatives</h3>
-<p>So what’s the relationship between an Monad and an Applicative? If an Applicative is also a Monad, the following laws hold:</p>
-<div class="sourceCode" id="cb390"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb390-1"><a href="#cb390-1" aria-hidden="true" tabindex="-1"></a><span class="fu">pure</span>             <span class="op">===</span> <span class="fu">return</span></span>
+<h2 data-number="15.8" id="sidenote-applicatives-in-context"><span
+class="header-section-number">15.8</span> Sidenote: Applicatives in
+Context</h2>
+<h3 data-number="15.8.1" id="why-applicatives"><span
+class="header-section-number">15.8.1</span> Why Applicatives?</h3>
+<p>There are multiple reasons for learning Applicatives, even if they do
+not provide any additional power over Monads. First off, as discussed in
+Lecture 13, the GHC standard library nowadays makes Applicative
+instances for all Monads compulsory. Thus a working Haskell programmer
+is bound to see lots of Applicative instances.</p>
+<p>Secondly, you’ll bump into Applicative operators a lot even in
+Monadic code. Expressions like
+<code>f &lt;$&gt; x &lt;*&gt; y &lt;*&gt; z</code> can be useful in many
+Monadic contexts. Also, since the <code>Traversable</code> type class is
+built in terms of <code>Applicative</code>, you’ll often use applicative
+operations with it.</p>
+<p>Thirdly, Applicatives are an excellent exercise in understanding
+functional design patterns. They marry the Functor pattern with the
+Monoid pattern, and Alternative brings in yet another Monoid-like
+dimension. Being able to efficiently work with Applicatives will make
+working with further abstractions like <em>monad transformers</em> or
+<em>lenses</em> easier.</p>
+<p>Lastly, there are a couple of types that are Applicatives but not
+Monads. <code>Validation</code> is one example, and a very practical one
+at that. Without understanding of Applicative, we’d have no way to
+recognize and generalize operations over types like this. Another such
+type is <a
+href="https://hackage.haskell.org/package/base-4.16.4.0/docs/Control-Applicative.html#t:ZipList"><code>ZipList</code></a>.</p>
+<h3 data-number="15.8.2" id="applicatives-in-the-wild"><span
+class="header-section-number">15.8.2</span> Applicatives in the
+Wild</h3>
+<p>Even though we only looked at some very simple and concrete
+Applicatives in this lecture, there are plenty of Haskell libraries that
+use Applicatives for big things. Here are some examples.</p>
+<p>The same sort of idea as our <code>Validation</code> Applicative has
+been implemented in the <a
+href="https://hackage.haskell.org/package/validation">validation</a> and
+<a href="https://hackage.haskell.org/package/either">either</a>
+libraries.</p>
+<p>There are multiple parser libraries that use Applicatives. For
+example, <a
+href="https://hackage.haskell.org/package/regex-applicative">regex-applicative</a>,
+<a
+href="https://hackage.haskell.org/package/optparse-applicative">optparse-applicative</a>,
+<a
+href="https://cs-syd.eu/posts/2020-06-28-yamlparse-applicative">yamlparse-applicative</a>,
+<a
+href="https://hackage.haskell.org/package/json-stream">json-stream</a>,
+and so on.</p>
+<h3 data-number="15.8.3" id="monads-and-applicatives"><span
+class="header-section-number">15.8.3</span> Monads and Applicatives</h3>
+<p>So what’s the relationship between an Monad and an Applicative? If an
+Applicative is also a Monad, the following laws hold:</p>
+<div class="sourceCode" id="cb390"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb390-1"><a href="#cb390-1" aria-hidden="true" tabindex="-1"></a><span class="fu">pure</span>             <span class="op">===</span> <span class="fu">return</span></span>
 <span id="cb390-2"><a href="#cb390-2" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb390-3"><a href="#cb390-3" aria-hidden="true" tabindex="-1"></a><span class="fu">fmap</span>             <span class="op">===</span> liftM</span>
 <span id="cb390-4"><a href="#cb390-4" aria-hidden="true" tabindex="-1"></a></span>
@@ -4373,25 +6205,35 @@ TODO explain `some` and `many`
 <span id="cb390-16"><a href="#cb390-16" aria-hidden="true" tabindex="-1"></a>op1 <span class="op">&lt;*&gt;</span> op2      <span class="op">===</span> <span class="kw">do</span> f <span class="ot">&lt;-</span> op1</span>
 <span id="cb390-17"><a href="#cb390-17" aria-hidden="true" tabindex="-1"></a>                        x <span class="ot">&lt;-</span> op2</span>
 <span id="cb390-18"><a href="#cb390-18" aria-hidden="true" tabindex="-1"></a>                        <span class="fu">return</span> (f x)</span></code></pre></div>
-<p>When working in a <code>Monad</code>, you can freely mix <code>Applicative</code> and <code>Functor</code> with <code>Monad</code> operations. As an example, let’s rewrite <code>mapM</code> until it only uses applicative operations, and thus get an implementation for <code>traverse</code>. This is our starting point:</p>
-<div class="sourceCode" id="cb391"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb391-1"><a href="#cb391-1" aria-hidden="true" tabindex="-1"></a>myMapM op [] <span class="ot">=</span> <span class="fu">return</span> []</span>
+<p>When working in a <code>Monad</code>, you can freely mix
+<code>Applicative</code> and <code>Functor</code> with
+<code>Monad</code> operations. As an example, let’s rewrite
+<code>mapM</code> until it only uses applicative operations, and thus
+get an implementation for <code>traverse</code>. This is our starting
+point:</p>
+<div class="sourceCode" id="cb391"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb391-1"><a href="#cb391-1" aria-hidden="true" tabindex="-1"></a>myMapM op [] <span class="ot">=</span> <span class="fu">return</span> []</span>
 <span id="cb391-2"><a href="#cb391-2" aria-hidden="true" tabindex="-1"></a>myMapM op (x<span class="op">:</span>xs) <span class="ot">=</span> <span class="kw">do</span> y <span class="ot">&lt;-</span> op x</span>
 <span id="cb391-3"><a href="#cb391-3" aria-hidden="true" tabindex="-1"></a>                      ys <span class="ot">&lt;-</span> myMapM op xs</span>
 <span id="cb391-4"><a href="#cb391-4" aria-hidden="true" tabindex="-1"></a>                      <span class="fu">return</span> (y<span class="op">:</span>ys)</span></code></pre></div>
 <p>GHCi tells us it only works for Monads:</p>
-<div class="sourceCode" id="cb392"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb392-1"><a href="#cb392-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t myMapM</span>
+<div class="sourceCode" id="cb392"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb392-1"><a href="#cb392-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t myMapM</span>
 <span id="cb392-2"><a href="#cb392-2" aria-hidden="true" tabindex="-1"></a><span class="ot">myMapM ::</span> <span class="dt">Monad</span> m <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> m b) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> m [b]</span></code></pre></div>
-<p>Let’s apply the <code>pure === return</code> and the <code>liftA2</code> laws from above:</p>
-<div class="sourceCode" id="cb393"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb393-1"><a href="#cb393-1" aria-hidden="true" tabindex="-1"></a>myMapM op [] <span class="ot">=</span> <span class="fu">pure</span> []</span>
+<p>Let’s apply the <code>pure === return</code> and the
+<code>liftA2</code> laws from above:</p>
+<div class="sourceCode" id="cb393"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb393-1"><a href="#cb393-1" aria-hidden="true" tabindex="-1"></a>myMapM op [] <span class="ot">=</span> <span class="fu">pure</span> []</span>
 <span id="cb393-2"><a href="#cb393-2" aria-hidden="true" tabindex="-1"></a>myMapM op (x<span class="op">:</span>xs) <span class="ot">=</span> liftA2 (<span class="op">:</span>) (op x) (myMapM op xs)</span></code></pre></div>
-<p>Ta-da! Now <code>myMapM</code> works for any <code>Applicative</code>:</p>
-<div class="sourceCode" id="cb394"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb394-1"><a href="#cb394-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t myMapM</span>
+<p>Ta-da! Now <code>myMapM</code> works for any
+<code>Applicative</code>:</p>
+<div class="sourceCode" id="cb394"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb394-1"><a href="#cb394-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>t myMapM</span>
 <span id="cb394-2"><a href="#cb394-2" aria-hidden="true" tabindex="-1"></a><span class="ot">myMapM ::</span> <span class="dt">Applicative</span> f <span class="ot">=&gt;</span> (a <span class="ot">-&gt;</span> f b) <span class="ot">-&gt;</span> [a] <span class="ot">-&gt;</span> f [b]</span></code></pre></div>
-</section>
-</section>
-<section id="quiz-5" data-number="7.9">
-<h2 data-number="15.9"><span class="header-section-number">15.9</span> Quiz</h2>
-<p>What’s the type of <code>x</code> in <code>liftA2 (&amp;&amp;) Nothing x</code>?</p>
+<h2 data-number="15.9" id="quiz-5"><span
+class="header-section-number">15.9</span> Quiz</h2>
+<p>What’s the type of <code>x</code> in
+<code>liftA2 (&amp;&amp;) Nothing x</code>?</p>
 <ol class="quiz">
 <li>
 <code>Applicative f =&gt; f Bool</code>
@@ -4403,7 +6245,8 @@ TODO explain `some` and `many`
 <code>Maybe Bool</code>
 </li>
 </ol>
-<p>How many elements does <code>liftA2 f xs ys</code> have when <code>xs</code> and <code>ys</code> are lists?</p>
+<p>How many elements does <code>liftA2 f xs ys</code> have when
+<code>xs</code> and <code>ys</code> are lists?</p>
 <ol class="quiz">
 <li>
 <code>length xs + length ys</code>
@@ -4415,7 +6258,8 @@ TODO explain `some` and `many`
 <code>min (length xs) (length ys)</code>
 </li>
 </ol>
-<p>Which of these expressions is equivalent to <code>liftA2 f x y</code>? There might be multiple correct answers.</p>
+<p>Which of these expressions is equivalent to
+<code>liftA2 f x y</code>? There might be multiple correct answers.</p>
 <ol class="quiz">
 <li class="correct">
 <code>f &lt;$&gt; x &lt;*&gt; y</code>
@@ -4433,7 +6277,8 @@ TODO explain `some` and `many`
 <code>pure f &lt;*&gt; x &lt;*&gt; y</code>
 </li>
 </ol>
-<p>If <code>f :: a -&gt; Maybe b</code> and <code>xs :: [a]</code>, which of these expressions has a type different from the others?</p>
+<p>If <code>f :: a -&gt; Maybe b</code> and <code>xs :: [a]</code>,
+which of these expressions has a type different from the others?</p>
 <ol class="quiz">
 <li>
 <code>fmap f xs</code>
@@ -4445,7 +6290,9 @@ TODO explain `some` and `many`
 <code>map f xs</code>
 </li>
 </ol>
-<p>For which <code>Applicative</code> do the expressions <code>pure x &lt;* pure y</code> and <code>pure x &lt;|&gt; pure y</code> produce different results?</p>
+<p>For which <code>Applicative</code> do the expressions
+<code>pure x &lt;* pure y</code> and
+<code>pure x &lt;|&gt; pure y</code> produce different results?</p>
 <ol class="quiz">
 <li>
 <code>Maybe</code>
@@ -4457,36 +6304,61 @@ TODO explain `some` and `many`
 <code>Validation</code>
 </li>
 </ol>
-</section>
-<section id="exercises-6" data-number="7.10">
-<h2 data-number="15.10"><span class="header-section-number">15.10</span> Exercises</h2>
+<h2 data-number="15.10" id="exercises-6"><span
+class="header-section-number">15.10</span> Exercises</h2>
 <ul>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set15.hs">Set15</a></li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set15.hs">Set15</a></li>
 </ul>
-</section>
-</section>
-<section id="lecture-16-odds-and-ends" data-number="8">
-<h1 data-number="16"><span class="header-section-number">16</span> Lecture 16: Odds and Ends</h1>
-<p>This final lecture will go over some minor topics that didn’t fit in anywhere else. You’ve finished all the hard parts of the course. Now it’s time to sit back, relax, and enjoy some cool Haskell!</p>
-<section id="testing-with-quickcheck" data-number="8.1">
-<h2 data-number="16.1"><span class="header-section-number">16.1</span> Testing with QuickCheck</h2>
-<p>One of the benefits of purity is that pure functions are easy to test: you don’t need to set up any global state, you can just pass in arguments and check that the result is ok. In this section, we’ll take a quick tour of the <em>property-based testing</em> library QuickCheck, which has also been used for checking that your answers to exercises are right on this course!</p>
-<p>Let’s look at testing a (faulty) implementation of <code>reverse</code>. You can find this and the following examples in the file <a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Examples/QuickCheck.hs"><code>exercises/Examples/QuickCheck.hs</code></a>.</p>
-<div class="sourceCode" id="cb395"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb395-1"><a href="#cb395-1" aria-hidden="true" tabindex="-1"></a><span class="ot">rev ::</span> [a] <span class="ot">-&gt;</span> [a]</span>
+<h1 data-number="16" id="lecture-16-odds-and-ends"><span
+class="header-section-number">16</span> Lecture 16: Odds and Ends</h1>
+<p>This final lecture will go over some minor topics that didn’t fit in
+anywhere else. You’ve finished all the hard parts of the course. Now
+it’s time to sit back, relax, and enjoy some cool Haskell!</p>
+<h2 data-number="16.1" id="testing-with-quickcheck"><span
+class="header-section-number">16.1</span> Testing with QuickCheck</h2>
+<p>One of the benefits of purity is that pure functions are easy to
+test: you don’t need to set up any global state, you can just pass in
+arguments and check that the result is ok. In this section, we’ll take a
+quick tour of the <em>property-based testing</em> library QuickCheck,
+which has also been used for checking that your answers to exercises are
+right on this course!</p>
+<p>Let’s look at testing a (faulty) implementation of
+<code>reverse</code>. You can find this and the following examples in
+the file <a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Examples/QuickCheck.hs"><code>exercises/Examples/QuickCheck.hs</code></a>.</p>
+<div class="sourceCode" id="cb395"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb395-1"><a href="#cb395-1" aria-hidden="true" tabindex="-1"></a><span class="ot">rev ::</span> [a] <span class="ot">-&gt;</span> [a]</span>
 <span id="cb395-2"><a href="#cb395-2" aria-hidden="true" tabindex="-1"></a>rev [] <span class="ot">=</span> []</span>
 <span id="cb395-3"><a href="#cb395-3" aria-hidden="true" tabindex="-1"></a>rev (x<span class="op">:</span>xs) <span class="ot">=</span> xs <span class="op">++</span> [x]</span></code></pre></div>
-<p>We can write an individual test case using the <code>===</code> operator from QuickCheck:</p>
-<div class="sourceCode" id="cb396"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb396-1"><a href="#cb396-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(===) ::</span> (<span class="dt">Eq</span> a, <span class="dt">Show</span> a) <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Property</span></span></code></pre></div>
-<div class="sourceCode" id="cb397"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb397-1"><a href="#cb397-1" aria-hidden="true" tabindex="-1"></a><span class="ot">propRevSmall ::</span> <span class="dt">Property</span></span>
+<p>We can write an individual test case using the <code>===</code>
+operator from QuickCheck:</p>
+<div class="sourceCode" id="cb396"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb396-1"><a href="#cb396-1" aria-hidden="true" tabindex="-1"></a><span class="ot">(===) ::</span> (<span class="dt">Eq</span> a, <span class="dt">Show</span> a) <span class="ot">=&gt;</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">Property</span></span></code></pre></div>
+<div class="sourceCode" id="cb397"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb397-1"><a href="#cb397-1" aria-hidden="true" tabindex="-1"></a><span class="ot">propRevSmall ::</span> <span class="dt">Property</span></span>
 <span id="cb397-2"><a href="#cb397-2" aria-hidden="true" tabindex="-1"></a>propRevSmall <span class="ot">=</span> rev [<span class="dv">1</span>,<span class="dv">2</span>] <span class="op">===</span> [<span class="dv">2</span>,<span class="dv">1</span>]</span></code></pre></div>
 <p>We can ask QuickCheck to run them in GHCi:</p>
-<div class="sourceCode" id="cb398"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb398-1"><a href="#cb398-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck propRevSmall</span>
+<div class="sourceCode" id="cb398"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb398-1"><a href="#cb398-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck propRevSmall</span>
 <span id="cb398-2"><a href="#cb398-2" aria-hidden="true" tabindex="-1"></a><span class="op">+++</span> <span class="dt">OK</span>, passed <span class="dv">1</span> test<span class="op">.</span></span></code></pre></div>
-<p>So far so good. However this isn’t really what QuickCheck was made for. QuickCheck was designed for <em>property-based testing</em> where you can state a property your code should have, and QuickCheck runs your code with randomized inputs, checking the property every time. What would be a simple property that a correct implementation of <code>reverse</code> has? Reversing a list twice should certainly give us back the same list. Let’s write it out:</p>
-<div class="sourceCode" id="cb399"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb399-1"><a href="#cb399-1" aria-hidden="true" tabindex="-1"></a><span class="ot">propRevTwice ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Property</span></span>
+<p>So far so good. However this isn’t really what QuickCheck was made
+for. QuickCheck was designed for <em>property-based testing</em> where
+you can state a property your code should have, and QuickCheck runs your
+code with randomized inputs, checking the property every time. What
+would be a simple property that a correct implementation of
+<code>reverse</code> has? Reversing a list twice should certainly give
+us back the same list. Let’s write it out:</p>
+<div class="sourceCode" id="cb399"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb399-1"><a href="#cb399-1" aria-hidden="true" tabindex="-1"></a><span class="ot">propRevTwice ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Property</span></span>
 <span id="cb399-2"><a href="#cb399-2" aria-hidden="true" tabindex="-1"></a>propRevTwice xs <span class="ot">=</span> rev (rev xs) <span class="op">===</span> xs</span></code></pre></div>
-<p>Our <code>Property</code> has an argument, which means that QuickCheck will generate random values and run the test. We can use the <code>verboseCheck</code> function to see which values are run. We can also give a parameter to the test ourselves if we wish to check a specific value.</p>
-<div class="sourceCode" id="cb400"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb400-1"><a href="#cb400-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck propRevTwice</span>
+<p>Our <code>Property</code> has an argument, which means that
+QuickCheck will generate random values and run the test. We can use the
+<code>verboseCheck</code> function to see which values are run. We can
+also give a parameter to the test ourselves if we wish to check a
+specific value.</p>
+<div class="sourceCode" id="cb400"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb400-1"><a href="#cb400-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck propRevTwice</span>
 <span id="cb400-2"><a href="#cb400-2" aria-hidden="true" tabindex="-1"></a><span class="op">+++</span> <span class="dt">OK</span>, passed <span class="dv">100</span> tests<span class="op">.</span></span>
 <span id="cb400-3"><a href="#cb400-3" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> verboseCheck propRevTwice</span>
 <span id="cb400-4"><a href="#cb400-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Passed</span><span class="op">:</span></span>
@@ -4504,21 +6376,35 @@ TODO explain `some` and `many`
 <span id="cb400-16"><a href="#cb400-16" aria-hidden="true" tabindex="-1"></a><span class="op">+++</span> <span class="dt">OK</span>, passed <span class="dv">100</span> tests<span class="op">.</span></span>
 <span id="cb400-17"><a href="#cb400-17" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck (propRevTwice [<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>])</span>
 <span id="cb400-18"><a href="#cb400-18" aria-hidden="true" tabindex="-1"></a><span class="op">+++</span> <span class="dt">OK</span>, passed <span class="dv">1</span> test<span class="op">.</span></span></code></pre></div>
-<p>Even this property didn’t catch the bug in our implementation. Let’s try another one. Here’s a property about how <code>rev (xs ++ ys)</code> behaves. You might want to take a moment to convince yourself that it should hold for a correct <code>rev</code> function.</p>
-<div class="sourceCode" id="cb401"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb401-1"><a href="#cb401-1" aria-hidden="true" tabindex="-1"></a><span class="ot">propRevTwo ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Property</span></span>
+<p>Even this property didn’t catch the bug in our implementation. Let’s
+try another one. Here’s a property about how <code>rev (xs ++ ys)</code>
+behaves. You might want to take a moment to convince yourself that it
+should hold for a correct <code>rev</code> function.</p>
+<div class="sourceCode" id="cb401"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb401-1"><a href="#cb401-1" aria-hidden="true" tabindex="-1"></a><span class="ot">propRevTwo ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Property</span></span>
 <span id="cb401-2"><a href="#cb401-2" aria-hidden="true" tabindex="-1"></a>propRevTwo xs ys <span class="ot">=</span> rev (xs <span class="op">++</span> ys) <span class="op">===</span> rev ys <span class="op">++</span> rev xs</span></code></pre></div>
 <p>Let’s see if it holds for our implementation:</p>
-<div class="sourceCode" id="cb402"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb402-1"><a href="#cb402-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck propRevTwo</span>
+<div class="sourceCode" id="cb402"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb402-1"><a href="#cb402-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck propRevTwo</span>
 <span id="cb402-2"><a href="#cb402-2" aria-hidden="true" tabindex="-1"></a><span class="op">***</span> <span class="dt">Failed</span><span class="op">!</span> <span class="dt">Falsified</span> (after <span class="dv">5</span> tests <span class="fu">and</span> <span class="dv">3</span> shrinks)<span class="op">:</span></span>
 <span id="cb402-3"><a href="#cb402-3" aria-hidden="true" tabindex="-1"></a>[<span class="dv">0</span>,<span class="dv">0</span>]</span>
 <span id="cb402-4"><a href="#cb402-4" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span>]</span>
 <span id="cb402-5"><a href="#cb402-5" aria-hidden="true" tabindex="-1"></a>[<span class="dv">0</span>,<span class="dv">1</span>,<span class="dv">0</span>] <span class="op">/=</span> [<span class="dv">1</span>,<span class="dv">0</span>,<span class="dv">0</span>]</span></code></pre></div>
-<p>Finally, a failure! There’s a bit to unpack here. First off, QuickCheck tells us the arguments with which the property failed: they are <code>[0,0]</code> and <code>[1]</code>. We can check this ourselves:</p>
-<div class="sourceCode" id="cb403"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb403-1"><a href="#cb403-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck (propRevTwo [<span class="dv">0</span>,<span class="dv">0</span>] [<span class="dv">1</span>])</span>
+<p>Finally, a failure! There’s a bit to unpack here. First off,
+QuickCheck tells us the arguments with which the property failed: they
+are <code>[0,0]</code> and <code>[1]</code>. We can check this
+ourselves:</p>
+<div class="sourceCode" id="cb403"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb403-1"><a href="#cb403-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck (propRevTwo [<span class="dv">0</span>,<span class="dv">0</span>] [<span class="dv">1</span>])</span>
 <span id="cb403-2"><a href="#cb403-2" aria-hidden="true" tabindex="-1"></a><span class="op">***</span> <span class="dt">Failed</span><span class="op">!</span> <span class="dt">Falsified</span> (after <span class="dv">1</span> test)<span class="op">:</span></span>
 <span id="cb403-3"><a href="#cb403-3" aria-hidden="true" tabindex="-1"></a>[<span class="dv">0</span>,<span class="dv">1</span>,<span class="dv">0</span>] <span class="op">/=</span> [<span class="dv">1</span>,<span class="dv">0</span>,<span class="dv">0</span>]</span></code></pre></div>
-<p>Next up, what does “after 5 tests and 3 shrinks” mean? One of the cool things about QuickCheck is that when it finds a failure, it tries some related values in order to find a nicer, smaller failure. We can see this in action with <code>verboseShrinking</code>, which prints out all the failures QuickCheck goes through:</p>
-<div class="sourceCode" id="cb404"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb404-1"><a href="#cb404-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck (verboseShrinking propRevTwo)</span>
+<p>Next up, what does “after 5 tests and 3 shrinks” mean? One of the
+cool things about QuickCheck is that when it finds a failure, it tries
+some related values in order to find a nicer, smaller failure. We can
+see this in action with <code>verboseShrinking</code>, which prints out
+all the failures QuickCheck goes through:</p>
+<div class="sourceCode" id="cb404"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb404-1"><a href="#cb404-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck (verboseShrinking propRevTwo)</span>
 <span id="cb404-2"><a href="#cb404-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Failed</span><span class="op">:</span></span>
 <span id="cb404-3"><a href="#cb404-3" aria-hidden="true" tabindex="-1"></a>[<span class="dv">4</span>,<span class="op">-</span><span class="dv">1</span>,<span class="op">-</span><span class="dv">4</span>]</span>
 <span id="cb404-4"><a href="#cb404-4" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span>,<span class="dv">4</span>]</span>
@@ -4558,39 +6444,70 @@ TODO explain `some` and `many`
 <span id="cb404-38"><a href="#cb404-38" aria-hidden="true" tabindex="-1"></a>[<span class="dv">0</span>]</span>
 <span id="cb404-39"><a href="#cb404-39" aria-hidden="true" tabindex="-1"></a>[<span class="dv">0</span>,<span class="dv">1</span>]</span>
 <span id="cb404-40"><a href="#cb404-40" aria-hidden="true" tabindex="-1"></a>[<span class="dv">0</span>,<span class="dv">1</span>,<span class="dv">0</span>] <span class="op">/=</span> [<span class="dv">1</span>,<span class="dv">0</span>,<span class="dv">0</span>]</span></code></pre></div>
-<p>QuickCheck went down from a counterexample of <code>[4,1,-1,4,4]</code> all the way to <code>[1,0,0]</code>. Pretty sweet!</p>
-<section id="modifiers" data-number="8.1.1">
-<h3 data-number="16.1.1"><span class="header-section-number">16.1.1</span> Modifiers</h3>
-<p>Sometimes you need to limit the values QuickCheck generates. Your function might not work on all inputs, for instance? Let’s try writing a test for <code>last</code>.</p>
-<div class="sourceCode" id="cb405"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb405-1"><a href="#cb405-1" aria-hidden="true" tabindex="-1"></a><span class="ot">propLast ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Property</span></span>
+<p>QuickCheck went down from a counterexample of
+<code>[4,1,-1,4,4]</code> all the way to <code>[1,0,0]</code>. Pretty
+sweet!</p>
+<h3 data-number="16.1.1" id="modifiers"><span
+class="header-section-number">16.1.1</span> Modifiers</h3>
+<p>Sometimes you need to limit the values QuickCheck generates. Your
+function might not work on all inputs, for instance? Let’s try writing a
+test for <code>last</code>.</p>
+<div class="sourceCode" id="cb405"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb405-1"><a href="#cb405-1" aria-hidden="true" tabindex="-1"></a><span class="ot">propLast ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Property</span></span>
 <span id="cb405-2"><a href="#cb405-2" aria-hidden="true" tabindex="-1"></a>propLast xs <span class="ot">=</span> <span class="fu">last</span> xs <span class="op">===</span> <span class="fu">head</span> (<span class="fu">reverse</span> xs)</span></code></pre></div>
-<div class="sourceCode" id="cb406"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb406-1"><a href="#cb406-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck propLast</span>
+<div class="sourceCode" id="cb406"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb406-1"><a href="#cb406-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck propLast</span>
 <span id="cb406-2"><a href="#cb406-2" aria-hidden="true" tabindex="-1"></a><span class="op">***</span> <span class="dt">Failed</span><span class="op">!</span> <span class="dt">Exception</span><span class="op">:</span> <span class="dt">&#39;Prelude</span><span class="op">.</span><span class="fu">last</span><span class="op">:</span> empty list&#39; (after <span class="dv">1</span> test)<span class="op">:</span></span>
 <span id="cb406-3"><a href="#cb406-3" aria-hidden="true" tabindex="-1"></a>[]</span></code></pre></div>
-<p>In this case, we can fix the test just by switching to another input type. QuickCheck defines the <code>NonEmptyList</code> type (not to be confused with <code>Data.List.NonEmpty</code>!), which is just a wrapper for a normal list. However, when generating values of <code>NonEmptyList</code>, QuickCheck won’t generate empty lists.</p>
-<div class="sourceCode" id="cb407"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb407-1"><a href="#cb407-1" aria-hidden="true" tabindex="-1"></a><span class="kw">newtype</span> <span class="dt">NonEmptyList</span> a <span class="ot">=</span> <span class="dt">NonEmpty</span> [a]</span></code></pre></div>
-<div class="sourceCode" id="cb408"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb408-1"><a href="#cb408-1" aria-hidden="true" tabindex="-1"></a><span class="ot">propLastFixed ::</span> <span class="dt">NonEmptyList</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Property</span></span>
+<p>In this case, we can fix the test just by switching to another input
+type. QuickCheck defines the <code>NonEmptyList</code> type (not to be
+confused with <code>Data.List.NonEmpty</code>!), which is just a wrapper
+for a normal list. However, when generating values of
+<code>NonEmptyList</code>, QuickCheck won’t generate empty lists.</p>
+<div class="sourceCode" id="cb407"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb407-1"><a href="#cb407-1" aria-hidden="true" tabindex="-1"></a><span class="kw">newtype</span> <span class="dt">NonEmptyList</span> a <span class="ot">=</span> <span class="dt">NonEmpty</span> [a]</span></code></pre></div>
+<div class="sourceCode" id="cb408"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb408-1"><a href="#cb408-1" aria-hidden="true" tabindex="-1"></a><span class="ot">propLastFixed ::</span> <span class="dt">NonEmptyList</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Property</span></span>
 <span id="cb408-2"><a href="#cb408-2" aria-hidden="true" tabindex="-1"></a>propLastFixed (<span class="dt">NonEmpty</span> xs) <span class="ot">=</span> <span class="fu">last</span> xs <span class="op">===</span> <span class="fu">head</span> (<span class="fu">reverse</span> xs)</span></code></pre></div>
-<div class="sourceCode" id="cb409"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb409-1"><a href="#cb409-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck propLastFixed</span>
+<div class="sourceCode" id="cb409"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb409-1"><a href="#cb409-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck propLastFixed</span>
 <span id="cb409-2"><a href="#cb409-2" aria-hidden="true" tabindex="-1"></a><span class="op">+++</span> <span class="dt">OK</span>, passed <span class="dv">100</span> tests<span class="op">.</span></span></code></pre></div>
-<p>There are <a href="https://hackage.haskell.org/package/QuickCheck-2.14.3/docs/Test-QuickCheck.html#g:16">other modifiers</a> like this, for example <code>Positive</code> for positive numbers, <code>NonNegative</code> for non-negative numbers, or <code>SortedList</code> for a sorted list. Here’s an example of a more complex test. We check that the nth element of <code>cycle xs</code> is correct. Both of the modifiers are needed, since <code>!!</code> doesn’t work with negative inputs, and <code>cycle []</code> is an error.</p>
-<div class="sourceCode" id="cb410"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb410-1"><a href="#cb410-1" aria-hidden="true" tabindex="-1"></a><span class="ot">propCycle ::</span> <span class="dt">NonEmptyList</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">NonNegative</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Property</span></span>
+<p>There are <a
+href="https://hackage.haskell.org/package/QuickCheck-2.14.3/docs/Test-QuickCheck.html#g:16">other
+modifiers</a> like this, for example <code>Positive</code> for positive
+numbers, <code>NonNegative</code> for non-negative numbers, or
+<code>SortedList</code> for a sorted list. Here’s an example of a more
+complex test. We check that the nth element of <code>cycle xs</code> is
+correct. Both of the modifiers are needed, since <code>!!</code> doesn’t
+work with negative inputs, and <code>cycle []</code> is an error.</p>
+<div class="sourceCode" id="cb410"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb410-1"><a href="#cb410-1" aria-hidden="true" tabindex="-1"></a><span class="ot">propCycle ::</span> <span class="dt">NonEmptyList</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">NonNegative</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Property</span></span>
 <span id="cb410-2"><a href="#cb410-2" aria-hidden="true" tabindex="-1"></a>propCycle (<span class="dt">NonEmpty</span> xs) (<span class="dt">NonNegative</span> n) <span class="ot">=</span></span>
 <span id="cb410-3"><a href="#cb410-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">cycle</span> xs <span class="op">!!</span> n <span class="op">===</span> xs <span class="op">!!</span> (<span class="fu">mod</span> n (<span class="fu">length</span> xs))</span></code></pre></div>
-</section>
-<section id="generators-and-forall" data-number="8.1.2">
-<h3 data-number="16.1.2"><span class="header-section-number">16.1.2</span> Generators and <code>forAll</code></h3>
-<p>Sometimes we need to limit the range of inputs to a test even further. As a simple example, here’s a test that <code>Data.Char.toUpper</code> changes the character passed to it:</p>
-<div class="sourceCode" id="cb411"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb411-1"><a href="#cb411-1" aria-hidden="true" tabindex="-1"></a><span class="ot">propToUpperChanges ::</span> <span class="dt">Char</span> <span class="ot">-&gt;</span> <span class="dt">Property</span></span>
+<h3 data-number="16.1.2" id="generators-and-forall"><span
+class="header-section-number">16.1.2</span> Generators and
+<code>forAll</code></h3>
+<p>Sometimes we need to limit the range of inputs to a test even
+further. As a simple example, here’s a test that
+<code>Data.Char.toUpper</code> changes the character passed to it:</p>
+<div class="sourceCode" id="cb411"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb411-1"><a href="#cb411-1" aria-hidden="true" tabindex="-1"></a><span class="ot">propToUpperChanges ::</span> <span class="dt">Char</span> <span class="ot">-&gt;</span> <span class="dt">Property</span></span>
 <span id="cb411-2"><a href="#cb411-2" aria-hidden="true" tabindex="-1"></a>propToUpperChanges c <span class="ot">=</span> <span class="fu">toUpper</span> c <span class="op">=/=</span> c</span></code></pre></div>
-<div class="sourceCode" id="cb412"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb412-1"><a href="#cb412-1" aria-hidden="true" tabindex="-1"></a>quickCheck propToUpperChanges</span>
+<div class="sourceCode" id="cb412"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb412-1"><a href="#cb412-1" aria-hidden="true" tabindex="-1"></a>quickCheck propToUpperChanges</span>
 <span id="cb412-2"><a href="#cb412-2" aria-hidden="true" tabindex="-1"></a><span class="op">***</span> <span class="dt">Failed</span><span class="op">!</span> <span class="dt">Falsified</span> (after <span class="dv">1</span> test <span class="fu">and</span> <span class="dv">1</span> shrink)<span class="op">:</span></span>
 <span id="cb412-3"><a href="#cb412-3" aria-hidden="true" tabindex="-1"></a><span class="ch">&#39;A&#39;</span></span>
 <span id="cb412-4"><a href="#cb412-4" aria-hidden="true" tabindex="-1"></a><span class="ch">&#39;A&#39;</span> <span class="op">==</span> <span class="ch">&#39;A&#39;</span></span></code></pre></div>
-<p>Of course, it only changes <em>lowercase letters</em>. How can we write a test for that? There is no <code>Lowercase</code> modifier available that would work like <code>Positive</code> or <code>NonEmptyList</code>. We need to fall back to explicit generation of values using <code>forAll</code>:</p>
-<div class="sourceCode" id="cb413"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb413-1"><a href="#cb413-1" aria-hidden="true" tabindex="-1"></a><span class="ot">propToUpperChangesLetter ::</span> <span class="dt">Property</span></span>
+<p>Of course, it only changes <em>lowercase letters</em>. How can we
+write a test for that? There is no <code>Lowercase</code> modifier
+available that would work like <code>Positive</code> or
+<code>NonEmptyList</code>. We need to fall back to explicit generation
+of values using <code>forAll</code>:</p>
+<div class="sourceCode" id="cb413"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb413-1"><a href="#cb413-1" aria-hidden="true" tabindex="-1"></a><span class="ot">propToUpperChangesLetter ::</span> <span class="dt">Property</span></span>
 <span id="cb413-2"><a href="#cb413-2" aria-hidden="true" tabindex="-1"></a>propToUpperChangesLetter <span class="ot">=</span> forAll (elements [<span class="ch">&#39;a&#39;</span><span class="op">..</span><span class="ch">&#39;z&#39;</span>]) propToUpperChanges</span></code></pre></div>
-<div class="sourceCode" id="cb414"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb414-1"><a href="#cb414-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> verboseCheck propToUpperChangesLetter</span>
+<div class="sourceCode" id="cb414"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb414-1"><a href="#cb414-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> verboseCheck propToUpperChangesLetter</span>
 <span id="cb414-2"><a href="#cb414-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Passed</span><span class="op">:</span></span>
 <span id="cb414-3"><a href="#cb414-3" aria-hidden="true" tabindex="-1"></a><span class="ch">&#39;s&#39;</span></span>
 <span id="cb414-4"><a href="#cb414-4" aria-hidden="true" tabindex="-1"></a><span class="ch">&#39;S&#39;</span> <span class="op">/=</span> <span class="ch">&#39;s&#39;</span></span>
@@ -4601,50 +6518,83 @@ TODO explain `some` and `many`
 <span id="cb414-9"><a href="#cb414-9" aria-hidden="true" tabindex="-1"></a><span class="co">-- lots of output omitted</span></span>
 <span id="cb414-10"><a href="#cb414-10" aria-hidden="true" tabindex="-1"></a><span class="op">+++</span> <span class="dt">OK</span>, passed <span class="dv">100</span> tests<span class="op">.</span></span></code></pre></div>
 <p>Perfect! Let’s look at the types to see what’s going on here.</p>
-<div class="sourceCode" id="cb415"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb415-1"><a href="#cb415-1" aria-hidden="true" tabindex="-1"></a><span class="ot">elements ::</span> [a] <span class="ot">-&gt;</span> <span class="dt">Gen</span> a</span>
+<div class="sourceCode" id="cb415"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb415-1"><a href="#cb415-1" aria-hidden="true" tabindex="-1"></a><span class="ot">elements ::</span> [a] <span class="ot">-&gt;</span> <span class="dt">Gen</span> a</span>
 <span id="cb415-2"><a href="#cb415-2" aria-hidden="true" tabindex="-1"></a>elements [<span class="ch">&#39;a&#39;</span><span class="op">..</span><span class="ch">&#39;z&#39;</span>]<span class="ot"> ::</span> <span class="dt">Gen</span> <span class="dt">Char</span></span>
 <span id="cb415-3"><a href="#cb415-3" aria-hidden="true" tabindex="-1"></a><span class="ot">forAll ::</span> (<span class="dt">Show</span> a, <span class="dt">Testable</span> prop) <span class="ot">=&gt;</span> <span class="dt">Gen</span> a <span class="ot">-&gt;</span> (a <span class="ot">-&gt;</span> prop) <span class="ot">-&gt;</span> <span class="dt">Property</span></span>
 <span id="cb415-4"><a href="#cb415-4" aria-hidden="true" tabindex="-1"></a>forAll (elements [<span class="ch">&#39;a&#39;</span><span class="op">..</span><span class="ch">&#39;z&#39;</span>])<span class="ot"> ::</span> <span class="dt">Testable</span> prop <span class="ot">=&gt;</span> (<span class="dt">Char</span> <span class="ot">-&gt;</span> prop) <span class="ot">-&gt;</span> <span class="dt">Property</span></span></code></pre></div>
-<p>There are some new types here. A value of type <code>Gen a</code> is a generator for values of type <code>a</code>. We’ll talk a bit more about <code>Gen</code> in the next section, but in this section you’ll see a couple of functions that return <code>Gen</code>s so that we can use them with <code>forAll</code>. The <code>elements</code> function is a generator that returns one of the elements of the given list at random, as you might have guessed.</p>
-<p>The <code>Testable</code> type class is the same that the <code>quickCheck</code> function uses. It exists so that <code>quickCheck</code> can test types like <code>[Int] -&gt; Bool -&gt; Property</code> in addition to just plain <code>Property</code> values.</p>
-<div class="sourceCode" id="cb416"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb416-1"><a href="#cb416-1" aria-hidden="true" tabindex="-1"></a><span class="ot">quickCheck ::</span> <span class="dt">Testable</span> prop <span class="ot">=&gt;</span> prop <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span></code></pre></div>
-<p>In addition, some simple types like <code>Bool</code> have a <code>Testable</code> instance, so that you can write tests using normal Haskell predicates instead of <code>===</code>:</p>
-<div class="sourceCode" id="cb417"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb417-1"><a href="#cb417-1" aria-hidden="true" tabindex="-1"></a><span class="ot">listHasZero ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
+<p>There are some new types here. A value of type <code>Gen a</code> is
+a generator for values of type <code>a</code>. We’ll talk a bit more
+about <code>Gen</code> in the next section, but in this section you’ll
+see a couple of functions that return <code>Gen</code>s so that we can
+use them with <code>forAll</code>. The <code>elements</code> function is
+a generator that returns one of the elements of the given list at
+random, as you might have guessed.</p>
+<p>The <code>Testable</code> type class is the same that the
+<code>quickCheck</code> function uses. It exists so that
+<code>quickCheck</code> can test types like
+<code>[Int] -&gt; Bool -&gt; Property</code> in addition to just plain
+<code>Property</code> values.</p>
+<div class="sourceCode" id="cb416"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb416-1"><a href="#cb416-1" aria-hidden="true" tabindex="-1"></a><span class="ot">quickCheck ::</span> <span class="dt">Testable</span> prop <span class="ot">=&gt;</span> prop <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span></code></pre></div>
+<p>In addition, some simple types like <code>Bool</code> have a
+<code>Testable</code> instance, so that you can write tests using normal
+Haskell predicates instead of <code>===</code>:</p>
+<div class="sourceCode" id="cb417"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb417-1"><a href="#cb417-1" aria-hidden="true" tabindex="-1"></a><span class="ot">listHasZero ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Bool</span></span>
 <span id="cb417-2"><a href="#cb417-2" aria-hidden="true" tabindex="-1"></a>listHasZero xs <span class="ot">=</span> <span class="fu">elem</span> <span class="dv">0</span> xs</span></code></pre></div>
-<div class="sourceCode" id="cb418"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb418-1"><a href="#cb418-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck (listHasZero [<span class="dv">1</span>,<span class="dv">0</span>,<span class="dv">2</span>])</span>
+<div class="sourceCode" id="cb418"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb418-1"><a href="#cb418-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck (listHasZero [<span class="dv">1</span>,<span class="dv">0</span>,<span class="dv">2</span>])</span>
 <span id="cb418-2"><a href="#cb418-2" aria-hidden="true" tabindex="-1"></a><span class="op">+++</span> <span class="dt">OK</span>, passed <span class="dv">1</span> test<span class="op">.</span></span>
 <span id="cb418-3"><a href="#cb418-3" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck listHasZero</span>
 <span id="cb418-4"><a href="#cb418-4" aria-hidden="true" tabindex="-1"></a><span class="op">***</span> <span class="dt">Failed</span><span class="op">!</span> <span class="dt">Falsified</span> (after <span class="dv">1</span> test)<span class="op">:</span></span>
 <span id="cb418-5"><a href="#cb418-5" aria-hidden="true" tabindex="-1"></a>[]</span></code></pre></div>
-<p>Getting back to <code>forAll</code>, we can use <code>forAll</code> to write more complex tests. Here’s a test that checks that <code>sort xs</code> has the same elements as <code>xs</code>. Note how we use <code>NonEmptyList</code> to guarantee that the <code>forAll</code> has some elements to pick from.</p>
-<div class="sourceCode" id="cb419"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb419-1"><a href="#cb419-1" aria-hidden="true" tabindex="-1"></a><span class="ot">propSort ::</span> <span class="dt">NonEmptyList</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Property</span></span>
+<p>Getting back to <code>forAll</code>, we can use <code>forAll</code>
+to write more complex tests. Here’s a test that checks that
+<code>sort xs</code> has the same elements as <code>xs</code>. Note how
+we use <code>NonEmptyList</code> to guarantee that the
+<code>forAll</code> has some elements to pick from.</p>
+<div class="sourceCode" id="cb419"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb419-1"><a href="#cb419-1" aria-hidden="true" tabindex="-1"></a><span class="ot">propSort ::</span> <span class="dt">NonEmptyList</span> <span class="dt">Int</span> <span class="ot">-&gt;</span> <span class="dt">Property</span></span>
 <span id="cb419-2"><a href="#cb419-2" aria-hidden="true" tabindex="-1"></a>propSort (<span class="dt">NonEmpty</span> xs) <span class="ot">=</span></span>
 <span id="cb419-3"><a href="#cb419-3" aria-hidden="true" tabindex="-1"></a>  forAll (elements xs) (\x <span class="ot">-&gt;</span> <span class="fu">elem</span> x (<span class="fu">sort</span> xs))</span></code></pre></div>
-</section>
-<section id="further-with-quickcheck" data-number="8.1.3">
-<h3 data-number="16.1.3"><span class="header-section-number">16.1.3</span> Further with QuickCheck</h3>
-<p>We’ve only just scratched the surface of QuickCheck. Here are some pointers to some things you’ll find useful when you start writing larger QuickCheck tests.</p>
-<p>Sometimes the output from QuickCheck isn’t quite verbose enough. You can add your own lines to the output by using the <code>counterexample</code> combinator:</p>
-<div class="sourceCode" id="cb420"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb420-1"><a href="#cb420-1" aria-hidden="true" tabindex="-1"></a><span class="ot">counterexample ::</span> <span class="dt">Testable</span> prop <span class="ot">=&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> prop <span class="ot">-&gt;</span> <span class="dt">Property</span></span></code></pre></div>
-<p>As an example, let’s add logging of the input to <code>rev</code> to <code>propRevTwo</code>:</p>
-<div class="sourceCode" id="cb421"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb421-1"><a href="#cb421-1" aria-hidden="true" tabindex="-1"></a><span class="ot">propRevTwo&#39; ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Property</span></span>
+<h3 data-number="16.1.3" id="further-with-quickcheck"><span
+class="header-section-number">16.1.3</span> Further with QuickCheck</h3>
+<p>We’ve only just scratched the surface of QuickCheck. Here are some
+pointers to some things you’ll find useful when you start writing larger
+QuickCheck tests.</p>
+<p>Sometimes the output from QuickCheck isn’t quite verbose enough. You
+can add your own lines to the output by using the
+<code>counterexample</code> combinator:</p>
+<div class="sourceCode" id="cb420"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb420-1"><a href="#cb420-1" aria-hidden="true" tabindex="-1"></a><span class="ot">counterexample ::</span> <span class="dt">Testable</span> prop <span class="ot">=&gt;</span> <span class="dt">String</span> <span class="ot">-&gt;</span> prop <span class="ot">-&gt;</span> <span class="dt">Property</span></span></code></pre></div>
+<p>As an example, let’s add logging of the input to <code>rev</code> to
+<code>propRevTwo</code>:</p>
+<div class="sourceCode" id="cb421"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb421-1"><a href="#cb421-1" aria-hidden="true" tabindex="-1"></a><span class="ot">propRevTwo&#39; ::</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> [<span class="dt">Int</span>] <span class="ot">-&gt;</span> <span class="dt">Property</span></span>
 <span id="cb421-2"><a href="#cb421-2" aria-hidden="true" tabindex="-1"></a>propRevTwo&#39; xs ys <span class="ot">=</span></span>
 <span id="cb421-3"><a href="#cb421-3" aria-hidden="true" tabindex="-1"></a>  <span class="kw">let</span> input <span class="ot">=</span> xs <span class="op">++</span> ys</span>
 <span id="cb421-4"><a href="#cb421-4" aria-hidden="true" tabindex="-1"></a>  <span class="kw">in</span> counterexample (<span class="st">&quot;Input: &quot;</span> <span class="op">++</span> <span class="fu">show</span> input) <span class="op">$</span></span>
 <span id="cb421-5"><a href="#cb421-5" aria-hidden="true" tabindex="-1"></a>     rev input <span class="op">===</span> rev ys <span class="op">++</span> rev xs</span></code></pre></div>
-<div class="sourceCode" id="cb422"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb422-1"><a href="#cb422-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck propRevTwo&#39;</span>
+<div class="sourceCode" id="cb422"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb422-1"><a href="#cb422-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck propRevTwo&#39;</span>
 <span id="cb422-2"><a href="#cb422-2" aria-hidden="true" tabindex="-1"></a><span class="op">***</span> <span class="dt">Failed</span><span class="op">!</span> <span class="dt">Falsified</span> (after <span class="dv">4</span> tests <span class="fu">and</span> <span class="dv">5</span> shrinks)<span class="op">:</span></span>
 <span id="cb422-3"><a href="#cb422-3" aria-hidden="true" tabindex="-1"></a>[<span class="dv">0</span>]</span>
 <span id="cb422-4"><a href="#cb422-4" aria-hidden="true" tabindex="-1"></a>[<span class="dv">0</span>,<span class="dv">1</span>]</span>
 <span id="cb422-5"><a href="#cb422-5" aria-hidden="true" tabindex="-1"></a><span class="dt">Input</span><span class="op">:</span> [<span class="dv">0</span>,<span class="dv">0</span>,<span class="dv">1</span>]</span>
 <span id="cb422-6"><a href="#cb422-6" aria-hidden="true" tabindex="-1"></a>[<span class="dv">0</span>,<span class="dv">1</span>,<span class="dv">0</span>] <span class="op">/=</span> [<span class="dv">1</span>,<span class="dv">0</span>,<span class="dv">0</span>]</span></code></pre></div>
-<p>As you might have guessed, <code>Gen</code> is a <code>Monad</code>. You can write your own generators by combining the generators defined by QuickCheck. You can check what your generators output using <code>sample</code>.</p>
-<div class="sourceCode" id="cb423"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb423-1"><a href="#cb423-1" aria-hidden="true" tabindex="-1"></a><span class="ot">someLetters ::</span> <span class="dt">Gen</span> <span class="dt">String</span></span>
+<p>As you might have guessed, <code>Gen</code> is a <code>Monad</code>.
+You can write your own generators by combining the generators defined by
+QuickCheck. You can check what your generators output using
+<code>sample</code>.</p>
+<div class="sourceCode" id="cb423"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb423-1"><a href="#cb423-1" aria-hidden="true" tabindex="-1"></a><span class="ot">someLetters ::</span> <span class="dt">Gen</span> <span class="dt">String</span></span>
 <span id="cb423-2"><a href="#cb423-2" aria-hidden="true" tabindex="-1"></a>someLetters <span class="ot">=</span> <span class="kw">do</span></span>
 <span id="cb423-3"><a href="#cb423-3" aria-hidden="true" tabindex="-1"></a>  c <span class="ot">&lt;-</span> elements <span class="st">&quot;xyzw&quot;</span></span>
 <span id="cb423-4"><a href="#cb423-4" aria-hidden="true" tabindex="-1"></a>  n <span class="ot">&lt;-</span> choose (<span class="dv">1</span>,<span class="dv">10</span>)</span>
 <span id="cb423-5"><a href="#cb423-5" aria-hidden="true" tabindex="-1"></a>  <span class="fu">return</span> (<span class="fu">replicate</span> n c)</span></code></pre></div>
-<div class="sourceCode" id="cb424"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb424-1"><a href="#cb424-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> sample someLetters</span>
+<div class="sourceCode" id="cb424"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb424-1"><a href="#cb424-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> sample someLetters</span>
 <span id="cb424-2"><a href="#cb424-2" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;yyyyyyyy&quot;</span></span>
 <span id="cb424-3"><a href="#cb424-3" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;zzzzzzzzz&quot;</span></span>
 <span id="cb424-4"><a href="#cb424-4" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;xxxxxxxxx&quot;</span></span>
@@ -4656,12 +6606,18 @@ TODO explain `some` and `many`
 <span id="cb424-10"><a href="#cb424-10" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;yyyyyyy&quot;</span></span>
 <span id="cb424-11"><a href="#cb424-11" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;xxxxxxxxxx&quot;</span></span>
 <span id="cb424-12"><a href="#cb424-12" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;y&quot;</span></span></code></pre></div>
-<p>Closely related to generators is the <code>Arbitrary</code> type class. <code>Arbitrary</code> is how QuickCheck generates all those inputs automatically.</p>
-<div class="sourceCode" id="cb425"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb425-1"><a href="#cb425-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Arbitrary</span> a <span class="kw">where</span></span>
+<p>Closely related to generators is the <code>Arbitrary</code> type
+class. <code>Arbitrary</code> is how QuickCheck generates all those
+inputs automatically.</p>
+<div class="sourceCode" id="cb425"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb425-1"><a href="#cb425-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> <span class="dt">Arbitrary</span> a <span class="kw">where</span></span>
 <span id="cb425-2"><a href="#cb425-2" aria-hidden="true" tabindex="-1"></a><span class="ot">  arbitrary ::</span> <span class="dt">Gen</span> a</span>
 <span id="cb425-3"><a href="#cb425-3" aria-hidden="true" tabindex="-1"></a><span class="ot">  shrink ::</span> a <span class="ot">-&gt;</span> [a]</span></code></pre></div>
-<p>If you’re writing tests for custom types, you either need to use <code>forAll</code>, or implement an <code>Arbitrary</code> instance. Here’s what happens if you’re missing an instance:</p>
-<div class="sourceCode" id="cb426"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb426-1"><a href="#cb426-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Switch</span> <span class="ot">=</span> <span class="dt">On</span> <span class="op">|</span> <span class="dt">Off</span></span>
+<p>If you’re writing tests for custom types, you either need to use
+<code>forAll</code>, or implement an <code>Arbitrary</code> instance.
+Here’s what happens if you’re missing an instance:</p>
+<div class="sourceCode" id="cb426"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb426-1"><a href="#cb426-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Switch</span> <span class="ot">=</span> <span class="dt">On</span> <span class="op">|</span> <span class="dt">Off</span></span>
 <span id="cb426-2"><a href="#cb426-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">deriving</span> (<span class="dt">Show</span>, <span class="dt">Eq</span>)</span>
 <span id="cb426-3"><a href="#cb426-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb426-4"><a href="#cb426-4" aria-hidden="true" tabindex="-1"></a><span class="ot">toggle ::</span> <span class="dt">Switch</span> <span class="ot">-&gt;</span> <span class="dt">Switch</span></span>
@@ -4670,25 +6626,38 @@ TODO explain `some` and `many`
 <span id="cb426-7"><a href="#cb426-7" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb426-8"><a href="#cb426-8" aria-hidden="true" tabindex="-1"></a><span class="ot">propToggleTwice ::</span> <span class="dt">Switch</span> <span class="ot">-&gt;</span> <span class="dt">Property</span></span>
 <span id="cb426-9"><a href="#cb426-9" aria-hidden="true" tabindex="-1"></a>propToggleTwice s <span class="ot">=</span> s <span class="op">===</span> toggle (toggle s)</span></code></pre></div>
-<div class="sourceCode" id="cb427"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb427-1"><a href="#cb427-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck propToggleTwice</span>
+<div class="sourceCode" id="cb427"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb427-1"><a href="#cb427-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck propToggleTwice</span>
 <span id="cb427-2"><a href="#cb427-2" aria-hidden="true" tabindex="-1"></a><span class="fu">error</span><span class="op">:</span></span>
 <span id="cb427-3"><a href="#cb427-3" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">No</span> <span class="kw">instance</span> for (<span class="dt">Arbitrary</span> <span class="dt">Switch</span>)</span>
 <span id="cb427-4"><a href="#cb427-4" aria-hidden="true" tabindex="-1"></a>        arising from a use <span class="kw">of</span> ‘quickCheck’</span>
 <span id="cb427-5"><a href="#cb427-5" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">In</span> the expression<span class="op">:</span> quickCheck propToggleTwice</span></code></pre></div>
 <p>Here are the two ways to fix it:</p>
-<div class="sourceCode" id="cb428"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb428-1"><a href="#cb428-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck (forAll (elements [<span class="dt">On</span>,<span class="dt">Off</span>]) propToggleTwice)</span>
+<div class="sourceCode" id="cb428"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb428-1"><a href="#cb428-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.QuickCheck</span><span class="op">&gt;</span> quickCheck (forAll (elements [<span class="dt">On</span>,<span class="dt">Off</span>]) propToggleTwice)</span>
 <span id="cb428-2"><a href="#cb428-2" aria-hidden="true" tabindex="-1"></a><span class="op">+++</span> <span class="dt">OK</span>, passed <span class="dv">100</span> tests<span class="op">.</span></span></code></pre></div>
-<div class="sourceCode" id="cb429"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb429-1"><a href="#cb429-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Arbitrary</span> <span class="dt">Switch</span> <span class="kw">where</span></span>
+<div class="sourceCode" id="cb429"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb429-1"><a href="#cb429-1" aria-hidden="true" tabindex="-1"></a><span class="kw">instance</span> <span class="dt">Arbitrary</span> <span class="dt">Switch</span> <span class="kw">where</span></span>
 <span id="cb429-2"><a href="#cb429-2" aria-hidden="true" tabindex="-1"></a>  arbitrary <span class="ot">=</span> elements [<span class="dt">On</span>,<span class="dt">Off</span>]</span></code></pre></div>
 <!-- TODO (==>)? -->
-</section>
-</section>
-<section id="phantom-types" data-number="8.2">
-<h2 data-number="16.2"><span class="header-section-number">16.2</span> Phantom Types</h2>
-<p>Boo! There’s a ghost in the type system! Let’s have a look at what <em>phantom types</em> can do for you.</p>
-<p>Phantom types are types that don’t take any values. They are related to newtypes (see lecture 10) in that both are a way to add additional type checking without affecting the evaluation of the program at all.</p>
-<p>Let’s use phantom types to track which currency an amount of money is in. We define the phantom types <code>EUR</code> and <code>USD</code> (note how they don’t have any constructors!), and the parameterized type <code>Money a</code> that doesn’t use the type parameter <code>a</code> for anything. Then we can define two constants, one in euros and the other in dollars. You can find all of the code from this section in the file <a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Examples/Phantom.hs"><code>exercises/Examples/Phantom.hs</code></a>.</p>
-<div class="sourceCode" id="cb430"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb430-1"><a href="#cb430-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">EUR</span></span>
+<h2 data-number="16.2" id="phantom-types"><span
+class="header-section-number">16.2</span> Phantom Types</h2>
+<p>Boo! There’s a ghost in the type system! Let’s have a look at what
+<em>phantom types</em> can do for you.</p>
+<p>Phantom types are types that don’t take any values. They are related
+to newtypes (see lecture 10) in that both are a way to add additional
+type checking without affecting the evaluation of the program at
+all.</p>
+<p>Let’s use phantom types to track which currency an amount of money is
+in. We define the phantom types <code>EUR</code> and <code>USD</code>
+(note how they don’t have any constructors!), and the parameterized type
+<code>Money a</code> that doesn’t use the type parameter <code>a</code>
+for anything. Then we can define two constants, one in euros and the
+other in dollars. You can find all of the code from this section in the
+file <a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Examples/Phantom.hs"><code>exercises/Examples/Phantom.hs</code></a>.</p>
+<div class="sourceCode" id="cb430"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb430-1"><a href="#cb430-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">EUR</span></span>
 <span id="cb430-2"><a href="#cb430-2" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">USD</span></span>
 <span id="cb430-3"><a href="#cb430-3" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Money</span> currency <span class="ot">=</span> <span class="dt">Money</span> <span class="dt">Double</span></span>
 <span id="cb430-4"><a href="#cb430-4" aria-hidden="true" tabindex="-1"></a>  <span class="kw">deriving</span> <span class="dt">Show</span></span>
@@ -4698,22 +6667,41 @@ TODO explain `some` and `many`
 <span id="cb430-8"><a href="#cb430-8" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb430-9"><a href="#cb430-9" aria-hidden="true" tabindex="-1"></a><span class="ot">twoEuros ::</span> <span class="dt">Money</span> <span class="dt">EUR</span></span>
 <span id="cb430-10"><a href="#cb430-10" aria-hidden="true" tabindex="-1"></a>twoEuros <span class="ot">=</span> <span class="dt">Money</span> <span class="dv">2</span></span></code></pre></div>
-<p>Note how the type signatures for <code>dollar</code> and <code>twoEuros</code> are doing all the work here. An expression like <code>Money 1</code> has the polymorphic type <code>Money currency</code> if we don’t restrict it to a more specific type. We give <code>dollar</code> and <code>twoEuros</code> more limited types explicitly. This is analogous to defining something like <code>one :: Int; one = 1</code>, since the constant <code>1</code> has the polymorphic type <code>Num p =&gt; p</code> but we give it a more restricted type.</p>
-<div class="sourceCode" id="cb431"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb431-1"><a href="#cb431-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.Phantom</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="dt">Money</span></span>
+<p>Note how the type signatures for <code>dollar</code> and
+<code>twoEuros</code> are doing all the work here. An expression like
+<code>Money 1</code> has the polymorphic type
+<code>Money currency</code> if we don’t restrict it to a more specific
+type. We give <code>dollar</code> and <code>twoEuros</code> more limited
+types explicitly. This is analogous to defining something like
+<code>one :: Int; one = 1</code>, since the constant <code>1</code> has
+the polymorphic type <code>Num p =&gt; p</code> but we give it a more
+restricted type.</p>
+<div class="sourceCode" id="cb431"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb431-1"><a href="#cb431-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.Phantom</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="dt">Money</span></span>
 <span id="cb431-2"><a href="#cb431-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Money</span><span class="ot"> ::</span> <span class="dt">Double</span> <span class="ot">-&gt;</span> <span class="dt">Money</span> currency</span>
 <span id="cb431-3"><a href="#cb431-3" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.Phantom</span><span class="op">&gt;</span> <span class="op">:</span>t <span class="dt">Money</span> <span class="dv">1</span></span>
 <span id="cb431-4"><a href="#cb431-4" aria-hidden="true" tabindex="-1"></a><span class="dt">Money</span> <span class="dv">1</span><span class="ot"> ::</span> <span class="dt">Money</span> currency</span></code></pre></div>
-<p>Now that we have some constants, we can write functions that operate on them. Let’s start with a function <code>scaleMoney</code> that multiplies an amount of money by a number. The currency stays constant. Here too, the type signature is doing the work: without the type signature, Haskell would infer a type of <code>Double -&gt; Money a -&gt; Money b</code>.</p>
-<div class="sourceCode" id="cb432"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb432-1"><a href="#cb432-1" aria-hidden="true" tabindex="-1"></a><span class="ot">scaleMoney ::</span> <span class="dt">Double</span> <span class="ot">-&gt;</span> <span class="dt">Money</span> currency <span class="ot">-&gt;</span> <span class="dt">Money</span> currency</span>
+<p>Now that we have some constants, we can write functions that operate
+on them. Let’s start with a function <code>scaleMoney</code> that
+multiplies an amount of money by a number. The currency stays constant.
+Here too, the type signature is doing the work: without the type
+signature, Haskell would infer a type of
+<code>Double -&gt; Money a -&gt; Money b</code>.</p>
+<div class="sourceCode" id="cb432"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb432-1"><a href="#cb432-1" aria-hidden="true" tabindex="-1"></a><span class="ot">scaleMoney ::</span> <span class="dt">Double</span> <span class="ot">-&gt;</span> <span class="dt">Money</span> currency <span class="ot">-&gt;</span> <span class="dt">Money</span> currency</span>
 <span id="cb432-2"><a href="#cb432-2" aria-hidden="true" tabindex="-1"></a>scaleMoney factor (<span class="dt">Money</span> a) <span class="ot">=</span> <span class="dt">Money</span> (factor <span class="op">*</span> a)</span></code></pre></div>
-<div class="sourceCode" id="cb433"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb433-1"><a href="#cb433-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.Phantom</span><span class="op">&gt;</span> <span class="op">:</span>t scaleMoney <span class="dv">3</span> twoEuros</span>
+<div class="sourceCode" id="cb433"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb433-1"><a href="#cb433-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.Phantom</span><span class="op">&gt;</span> <span class="op">:</span>t scaleMoney <span class="dv">3</span> twoEuros</span>
 <span id="cb433-2"><a href="#cb433-2" aria-hidden="true" tabindex="-1"></a>scaleMoney <span class="dv">3</span><span class="ot"> twoEuros ::</span> <span class="dt">Money</span> <span class="dt">EUR</span></span>
 <span id="cb433-3"><a href="#cb433-3" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.Phantom</span><span class="op">&gt;</span> <span class="op">:</span>t scaleMoney <span class="dv">3</span> dollar</span>
 <span id="cb433-4"><a href="#cb433-4" aria-hidden="true" tabindex="-1"></a>scaleMoney <span class="dv">3</span><span class="ot"> dollar ::</span> <span class="dt">Money</span> <span class="dt">USD</span></span></code></pre></div>
-<p>Next up: adding two amounts that are in the same currency. We get a nice type error if we try to add values in two different currencies.</p>
-<div class="sourceCode" id="cb434"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb434-1"><a href="#cb434-1" aria-hidden="true" tabindex="-1"></a><span class="ot">addMoney ::</span> <span class="dt">Money</span> currency <span class="ot">-&gt;</span> <span class="dt">Money</span> currency <span class="ot">-&gt;</span> <span class="dt">Money</span> currency</span>
+<p>Next up: adding two amounts that are in the same currency. We get a
+nice type error if we try to add values in two different currencies.</p>
+<div class="sourceCode" id="cb434"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb434-1"><a href="#cb434-1" aria-hidden="true" tabindex="-1"></a><span class="ot">addMoney ::</span> <span class="dt">Money</span> currency <span class="ot">-&gt;</span> <span class="dt">Money</span> currency <span class="ot">-&gt;</span> <span class="dt">Money</span> currency</span>
 <span id="cb434-2"><a href="#cb434-2" aria-hidden="true" tabindex="-1"></a>addMoney (<span class="dt">Money</span> a) (<span class="dt">Money</span> b) <span class="ot">=</span> <span class="dt">Money</span> (a<span class="op">+</span>b)</span></code></pre></div>
-<div class="sourceCode" id="cb435"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb435-1"><a href="#cb435-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.Phantom</span><span class="op">&gt;</span> <span class="op">:</span>t addMoney dollar dollar</span>
+<div class="sourceCode" id="cb435"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb435-1"><a href="#cb435-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.Phantom</span><span class="op">&gt;</span> <span class="op">:</span>t addMoney dollar dollar</span>
 <span id="cb435-2"><a href="#cb435-2" aria-hidden="true" tabindex="-1"></a>addMoney dollar<span class="ot"> dollar ::</span> <span class="dt">Money</span> <span class="dt">USD</span></span>
 <span id="cb435-3"><a href="#cb435-3" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.Phantom</span><span class="op">&gt;</span> <span class="op">:</span>t addMoney twoEuros twoEuros</span>
 <span id="cb435-4"><a href="#cb435-4" aria-hidden="true" tabindex="-1"></a>addMoney twoEuros<span class="ot"> twoEuros ::</span> <span class="dt">Money</span> <span class="dt">EUR</span></span>
@@ -4724,13 +6712,23 @@ TODO explain `some` and `many`
 <span id="cb435-9"><a href="#cb435-9" aria-hidden="true" tabindex="-1"></a>        <span class="dt">Actual</span> <span class="kw">type</span><span class="op">:</span> <span class="dt">Money</span> <span class="dt">USD</span></span>
 <span id="cb435-10"><a href="#cb435-10" aria-hidden="true" tabindex="-1"></a>    • <span class="dt">In</span> the second argument <span class="kw">of</span> ‘addMoney’, namely ‘dollar’</span>
 <span id="cb435-11"><a href="#cb435-11" aria-hidden="true" tabindex="-1"></a>      <span class="dt">In</span> the expression<span class="op">:</span> addMoney twoEuros dollar</span></code></pre></div>
-<p>As before, the type signature is crucial. Here’s the same implementation with an unrestricted type. Now we can add anything to anything!</p>
-<div class="sourceCode" id="cb436"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb436-1"><a href="#cb436-1" aria-hidden="true" tabindex="-1"></a><span class="ot">addMoneyUnsafe ::</span> <span class="dt">Money</span> x <span class="ot">-&gt;</span> <span class="dt">Money</span> y <span class="ot">-&gt;</span> <span class="dt">Money</span> z</span>
+<p>As before, the type signature is crucial. Here’s the same
+implementation with an unrestricted type. Now we can add anything to
+anything!</p>
+<div class="sourceCode" id="cb436"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb436-1"><a href="#cb436-1" aria-hidden="true" tabindex="-1"></a><span class="ot">addMoneyUnsafe ::</span> <span class="dt">Money</span> x <span class="ot">-&gt;</span> <span class="dt">Money</span> y <span class="ot">-&gt;</span> <span class="dt">Money</span> z</span>
 <span id="cb436-2"><a href="#cb436-2" aria-hidden="true" tabindex="-1"></a>addMoneyUnsafe (<span class="dt">Money</span> a) (<span class="dt">Money</span> b) <span class="ot">=</span> <span class="dt">Money</span> (a<span class="op">+</span>b)</span></code></pre></div>
-<div class="sourceCode" id="cb437"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb437-1"><a href="#cb437-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.Phantom</span><span class="op">&gt;</span> addMoneyUnsafe twoEuros dollar</span>
+<div class="sourceCode" id="cb437"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb437-1"><a href="#cb437-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.Phantom</span><span class="op">&gt;</span> addMoneyUnsafe twoEuros dollar</span>
 <span id="cb437-2"><a href="#cb437-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Money</span> <span class="fl">3.0</span></span></code></pre></div>
-<p>We can keep going with this approach, and define currency conversions. We define the type <code>Rate</code> that uses phantom types to track the currencies it’s translating between. The types of <code>convert</code> and <code>invert</code> are restricted to have the properties we want. There’s also an unrestricted version of the convert function to let you compare the types.</p>
-<div class="sourceCode" id="cb438"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb438-1"><a href="#cb438-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Rate</span> from to <span class="ot">=</span> <span class="dt">Rate</span> <span class="dt">Double</span></span>
+<p>We can keep going with this approach, and define currency
+conversions. We define the type <code>Rate</code> that uses phantom
+types to track the currencies it’s translating between. The types of
+<code>convert</code> and <code>invert</code> are restricted to have the
+properties we want. There’s also an unrestricted version of the convert
+function to let you compare the types.</p>
+<div class="sourceCode" id="cb438"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb438-1"><a href="#cb438-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Rate</span> from to <span class="ot">=</span> <span class="dt">Rate</span> <span class="dt">Double</span></span>
 <span id="cb438-2"><a href="#cb438-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">deriving</span> <span class="dt">Show</span></span>
 <span id="cb438-3"><a href="#cb438-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb438-4"><a href="#cb438-4" aria-hidden="true" tabindex="-1"></a><span class="ot">eurToUsd ::</span> <span class="dt">Rate</span> <span class="dt">EUR</span> <span class="dt">USD</span></span>
@@ -4744,7 +6742,8 @@ TODO explain `some` and `many`
 <span id="cb438-12"><a href="#cb438-12" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb438-13"><a href="#cb438-13" aria-hidden="true" tabindex="-1"></a><span class="ot">convertUnsafe ::</span> <span class="dt">Rate</span> from to <span class="ot">-&gt;</span> <span class="dt">Money</span> x <span class="ot">-&gt;</span> <span class="dt">Money</span> y</span>
 <span id="cb438-14"><a href="#cb438-14" aria-hidden="true" tabindex="-1"></a>convertUnsafe (<span class="dt">Rate</span> r) (<span class="dt">Money</span> a) <span class="ot">=</span> <span class="dt">Money</span> (r<span class="op">*</span>a)</span></code></pre></div>
-<div class="sourceCode" id="cb439"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb439-1"><a href="#cb439-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.Phantom</span><span class="op">&gt;</span> convert eurToUsd twoEuros</span>
+<div class="sourceCode" id="cb439"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb439-1"><a href="#cb439-1" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.Phantom</span><span class="op">&gt;</span> convert eurToUsd twoEuros</span>
 <span id="cb439-2"><a href="#cb439-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Money</span> <span class="fl">2.44</span></span>
 <span id="cb439-3"><a href="#cb439-3" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.Phantom</span><span class="op">&gt;</span> convert eurToUsd dollar</span>
 <span id="cb439-4"><a href="#cb439-4" aria-hidden="true" tabindex="-1"></a><span class="fu">error</span><span class="op">:</span></span>
@@ -4758,12 +6757,49 @@ TODO explain `some` and `many`
 <span id="cb439-12"><a href="#cb439-12" aria-hidden="true" tabindex="-1"></a><span class="dt">Money</span> <span class="fl">0.819672131147541</span></span>
 <span id="cb439-13"><a href="#cb439-13" aria-hidden="true" tabindex="-1"></a><span class="op">*</span><span class="dt">Examples.Phantom</span><span class="op">&gt;</span> convertUnsafe eurToUsd dollar</span>
 <span id="cb439-14"><a href="#cb439-14" aria-hidden="true" tabindex="-1"></a><span class="dt">Money</span> <span class="fl">1.22</span></span></code></pre></div>
-<p>Note! The words <code>currency</code>, <code>from</code>, <code>to</code> and so forth in the previous examples are <em>just type variables</em>. There’s nothing special going on with them. We could’ve as well given <code>invert</code> a type like <code>Rate a b -&gt; Rate b a</code> without any change in type safety.</p>
-<p>This approach that uses phantom types has clear benefits: giving us type errors for invalid code. Also, compared to defining lots of concrete types like <code>data MoneyEur = MoneyEur Double</code>, with phantom types we need to implement functions like <code>scaleMoney</code> and <code>addMoney</code> only once. Also, we’re able to define polymorphic and reusable concepts like <code>Rate</code>. You can contrast this approach with the Boxing section of Lecture 7.</p>
-<p>However, phantom types also have downsides. Without advanced tricks we can’t really handle currencies that are defined at runtime (for example: reading an amount from a user). It’s also easy to end up in a place where you start needing language extensions like <a href="https://wiki.haskell.org/GADTs_for_dummies"><em>Generalized Algebraic Datatypes</em></a>, <a href="https://wiki.haskell.org/GHC/Type_families"><em>type families</em></a> and other <a href="https://aphyr.com/posts/342-typing-the-technical-interview"><em>type-level programming</em></a> constructs. Eventually you’re all the way in the world of <a href="https://mitpress.mit.edu/books/little-typer"><em>dependent typing</em></a>.</p>
-<p>So what are good applications for phantom types? When you need to track some simple, but crucial information, that is known at compile-time. An example that’s somewhat better than currencies is tracking whether inputs from the user have been sanitated to prevent attacks like <a href="https://en.wikipedia.org/wiki/SQL_injection">SQL injection</a> or <a href="https://en.wikipedia.org/wiki/Cross-site_scripting">cross-site scripting</a>.</p>
-<p>We can use the types <code>Input Safe</code> and <code>Input Unsafe</code> to track whether strings are safe for passing into the database or not. If our module only exports the <code>makeInput</code> function, and not the <code>Input</code> constructor, the type system ensures that any inputs must pass through the <code>escapeInput</code> function at some point before going into a database function like <code>addForumComment</code>.</p>
-<div class="sourceCode" id="cb440"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb440-1"><a href="#cb440-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Safe</span></span>
+<p>Note! The words <code>currency</code>, <code>from</code>,
+<code>to</code> and so forth in the previous examples are <em>just type
+variables</em>. There’s nothing special going on with them. We could’ve
+as well given <code>invert</code> a type like
+<code>Rate a b -&gt; Rate b a</code> without any change in type
+safety.</p>
+<p>This approach that uses phantom types has clear benefits: giving us
+type errors for invalid code. Also, compared to defining lots of
+concrete types like <code>data MoneyEur = MoneyEur Double</code>, with
+phantom types we need to implement functions like
+<code>scaleMoney</code> and <code>addMoney</code> only once. Also, we’re
+able to define polymorphic and reusable concepts like <code>Rate</code>.
+You can contrast this approach with the Boxing section of Lecture 7.</p>
+<p>However, phantom types also have downsides. Without advanced tricks
+we can’t really handle currencies that are defined at runtime (for
+example: reading an amount from a user). It’s also easy to end up in a
+place where you start needing language extensions like <a
+href="https://wiki.haskell.org/GADTs_for_dummies"><em>Generalized
+Algebraic Datatypes</em></a>, <a
+href="https://wiki.haskell.org/GHC/Type_families"><em>type
+families</em></a> and other <a
+href="https://aphyr.com/posts/342-typing-the-technical-interview"><em>type-level
+programming</em></a> constructs. Eventually you’re all the way in the
+world of <a
+href="https://mitpress.mit.edu/books/little-typer"><em>dependent
+typing</em></a>.</p>
+<p>So what are good applications for phantom types? When you need to
+track some simple, but crucial information, that is known at
+compile-time. An example that’s somewhat better than currencies is
+tracking whether inputs from the user have been sanitated to prevent
+attacks like <a href="https://en.wikipedia.org/wiki/SQL_injection">SQL
+injection</a> or <a
+href="https://en.wikipedia.org/wiki/Cross-site_scripting">cross-site
+scripting</a>.</p>
+<p>We can use the types <code>Input Safe</code> and
+<code>Input Unsafe</code> to track whether strings are safe for passing
+into the database or not. If our module only exports the
+<code>makeInput</code> function, and not the <code>Input</code>
+constructor, the type system ensures that any inputs must pass through
+the <code>escapeInput</code> function at some point before going into a
+database function like <code>addForumComment</code>.</p>
+<div class="sourceCode" id="cb440"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb440-1"><a href="#cb440-1" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Safe</span></span>
 <span id="cb440-2"><a href="#cb440-2" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Unsafe</span></span>
 <span id="cb440-3"><a href="#cb440-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb440-4"><a href="#cb440-4" aria-hidden="true" tabindex="-1"></a><span class="kw">data</span> <span class="dt">Input</span> a <span class="ot">=</span> <span class="dt">Input</span> <span class="dt">String</span></span>
@@ -4784,31 +6820,58 @@ TODO explain `some` and `many`
 <span id="cb440-19"><a href="#cb440-19" aria-hidden="true" tabindex="-1"></a><span class="co">-- Strip bad characters to turn an unsafe input safe</span></span>
 <span id="cb440-20"><a href="#cb440-20" aria-hidden="true" tabindex="-1"></a><span class="ot">escapeInput ::</span> <span class="dt">Input</span> <span class="dt">Unsafe</span> <span class="ot">-&gt;</span> <span class="dt">Input</span> <span class="dt">Safe</span></span>
 <span id="cb440-21"><a href="#cb440-21" aria-hidden="true" tabindex="-1"></a>escapeInput (<span class="dt">Input</span> xs) <span class="ot">=</span> <span class="dt">Input</span> (<span class="fu">filter</span> (\c <span class="ot">-&gt;</span> <span class="fu">isAlpha</span> c <span class="op">||</span> <span class="fu">isSpace</span> c) xs)</span></code></pre></div>
-</section>
-<section id="simultaneity" data-number="8.3">
-<h2 data-number="16.3"><span class="header-section-number">16.3</span> Simultaneity</h2>
-<section id="parallelism" data-number="8.3.1">
-<h3 data-number="16.3.1"><span class="header-section-number">16.3.1</span> Parallelism</h3>
-<p>One of the great things about purity is that it makes <em>parallelism</em>, computing many things at the same time, very easy. Let’s have a look at how we can do this in Haskell. First off, we need to start a new GHCi that has parallel execution enabled. The simplest way is:</p>
+<h2 data-number="16.3" id="simultaneity"><span
+class="header-section-number">16.3</span> Simultaneity</h2>
+<h3 data-number="16.3.1" id="parallelism"><span
+class="header-section-number">16.3.1</span> Parallelism</h3>
+<p>One of the great things about purity is that it makes
+<em>parallelism</em>, computing many things at the same time, very easy.
+Let’s have a look at how we can do this in Haskell. First off, we need
+to start a new GHCi that has parallel execution enabled. The simplest
+way is:</p>
 <pre><code>$ stack ghci --ghci-options &quot;+RTS -N&quot;</code></pre>
-<p>Next up, let’s define a very naive version of the Fibonacci function (remember Lecture 1?) , enable performance statistics with <code>:set +s</code>, and see how long it takes to compute five values of the function:</p>
-<div class="sourceCode" id="cb442"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb442-1"><a href="#cb442-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> fib <span class="dv">0</span> <span class="ot">=</span> <span class="dv">1</span>; fib <span class="dv">1</span> <span class="ot">=</span> <span class="dv">1</span>; fib n <span class="ot">=</span> fib (n<span class="op">-</span><span class="dv">1</span>) <span class="op">+</span> fib (n<span class="op">-</span><span class="dv">2</span>)</span>
+<p>Next up, let’s define a very naive version of the Fibonacci function
+(remember Lecture 1?) , enable performance statistics with
+<code>:set +s</code>, and see how long it takes to compute five values
+of the function:</p>
+<div class="sourceCode" id="cb442"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb442-1"><a href="#cb442-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> fib <span class="dv">0</span> <span class="ot">=</span> <span class="dv">1</span>; fib <span class="dv">1</span> <span class="ot">=</span> <span class="dv">1</span>; fib n <span class="ot">=</span> fib (n<span class="op">-</span><span class="dv">1</span>) <span class="op">+</span> fib (n<span class="op">-</span><span class="dv">2</span>)</span>
 <span id="cb442-2"><a href="#cb442-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="op">:</span>set <span class="op">+</span>s</span>
 <span id="cb442-3"><a href="#cb442-3" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="fu">map</span> fib [<span class="dv">29</span>,<span class="dv">29</span>,<span class="dv">29</span>,<span class="dv">29</span>,<span class="dv">29</span>]</span>
 <span id="cb442-4"><a href="#cb442-4" aria-hidden="true" tabindex="-1"></a>[<span class="dv">832040</span>,<span class="dv">832040</span>,<span class="dv">832040</span>,<span class="dv">832040</span>,<span class="dv">832040</span>]</span>
 <span id="cb442-5"><a href="#cb442-5" aria-hidden="true" tabindex="-1"></a>(<span class="fl">7.54</span> secs, <span class="dv">2</span>,<span class="dv">440</span>,<span class="dv">860</span>,<span class="dv">632</span> bytes)</span></code></pre></div>
-<p>Now let’s bring in the module <a href="https://hackage.haskell.org/package/parallel-3.2.2.0/docs/Control-Parallel-Strategies.html">Control.Parallel.Strategies</a> that defines ways to evaluate values in parallel. We’ll use the <code>parList rseq</code> strategy to evaluate all the elements of the list to WHNF, <em>in parallel</em>.</p>
-<div class="sourceCode" id="cb443"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb443-1"><a href="#cb443-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="dt">Control.Parallel.Strategies</span></span>
+<p>Now let’s bring in the module <a
+href="https://hackage.haskell.org/package/parallel-3.2.2.0/docs/Control-Parallel-Strategies.html">Control.Parallel.Strategies</a>
+that defines ways to evaluate values in parallel. We’ll use the
+<code>parList rseq</code> strategy to evaluate all the elements of the
+list to WHNF, <em>in parallel</em>.</p>
+<div class="sourceCode" id="cb443"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb443-1"><a href="#cb443-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span><span class="op">&gt;</span> <span class="kw">import</span> <span class="dt">Control.Parallel.Strategies</span></span>
 <span id="cb443-2"><a href="#cb443-2" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Control.Parallel.Strategies</span><span class="op">&gt;</span> withStrategy (parList rseq) (<span class="fu">map</span> fib [<span class="dv">29</span>,<span class="dv">29</span>,<span class="dv">29</span>,<span class="dv">29</span>,<span class="dv">29</span>])</span>
 <span id="cb443-3"><a href="#cb443-3" aria-hidden="true" tabindex="-1"></a>[<span class="dv">832040</span>,<span class="dv">832040</span>,<span class="dv">832040</span>,<span class="dv">832040</span>,<span class="dv">832040</span>]</span>
 <span id="cb443-4"><a href="#cb443-4" aria-hidden="true" tabindex="-1"></a>(<span class="fl">4.80</span> secs, <span class="dv">488</span>,<span class="dv">531</span>,<span class="dv">384</span> bytes)</span></code></pre></div>
-<p>That’s almost 2 times as fast, on the 2-core machine that this example is being run on. Pretty nice. The coolest thing here is that we were able to define the <em>computation</em> (<code>map fib ...</code>) completely separately from the <em>evaluation strategy</em> (<code>parList rseq</code>), separating the <em>what to compute</em> from the <em>how to compute</em>.</p>
-</section>
-<section id="concurrency" data-number="8.3.2">
-<h3 data-number="16.3.2"><span class="header-section-number">16.3.2</span> Concurrency</h3>
-<p>Computer science makes the distinction between parallel and <em>concurrent</em> computations. Parallel computations are those that just run separate independent computations in parallel (in other words, parallelism is <em>pure</em>). Concurrent computations are those where there are multiple interacting threads of computation. Concurrency usually involves threads, locks, messages and deadlocks.</p>
-<p>In addition to great tooling for parallelism, Haskell also has good tooling for concurrency via <em>threads</em>. Since concurrency is all about side-effects, concurrent computations happen in the <code>IO</code> Monad. The classic example of threading is two threads, one printing a stream of As and the other a stream of Bs. Here it is in Haskell:</p>
-<div class="sourceCode" id="cb444"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb444-1"><a href="#cb444-1" aria-hidden="true" tabindex="-1"></a><span class="ot">printA ::</span> <span class="dt">IO</span> ()</span>
+<p>That’s almost 2 times as fast, on the 2-core machine that this
+example is being run on. Pretty nice. The coolest thing here is that we
+were able to define the <em>computation</em> (<code>map fib ...</code>)
+completely separately from the <em>evaluation strategy</em>
+(<code>parList rseq</code>), separating the <em>what to compute</em>
+from the <em>how to compute</em>.</p>
+<h3 data-number="16.3.2" id="concurrency"><span
+class="header-section-number">16.3.2</span> Concurrency</h3>
+<p>Computer science makes the distinction between parallel and
+<em>concurrent</em> computations. Parallel computations are those that
+just run separate independent computations in parallel (in other words,
+parallelism is <em>pure</em>). Concurrent computations are those where
+there are multiple interacting threads of computation. Concurrency
+usually involves threads, locks, messages and deadlocks.</p>
+<p>In addition to great tooling for parallelism, Haskell also has good
+tooling for concurrency via <em>threads</em>. Since concurrency is all
+about side-effects, concurrent computations happen in the
+<code>IO</code> Monad. The classic example of threading is two threads,
+one printing a stream of As and the other a stream of Bs. Here it is in
+Haskell:</p>
+<div class="sourceCode" id="cb444"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb444-1"><a href="#cb444-1" aria-hidden="true" tabindex="-1"></a><span class="ot">printA ::</span> <span class="dt">IO</span> ()</span>
 <span id="cb444-2"><a href="#cb444-2" aria-hidden="true" tabindex="-1"></a>printA <span class="ot">=</span> <span class="fu">putStrLn</span> (<span class="fu">replicate</span> <span class="dv">40</span> <span class="ch">&#39;A&#39;</span>)</span>
 <span id="cb444-3"><a href="#cb444-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb444-4"><a href="#cb444-4" aria-hidden="true" tabindex="-1"></a><span class="ot">printB ::</span> <span class="dt">IO</span> ()</span>
@@ -4821,13 +6884,27 @@ TODO explain `some` and `many`
 <span id="cb444-11"><a href="#cb444-11" aria-hidden="true" tabindex="-1"></a>  <span class="fu">return</span> ()</span></code></pre></div>
 <pre><code>Prelude Control.Concurrent&gt; concurrency
 AABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABB</code></pre>
-<p>The operation <code>forkIO :: IO () -&gt; IO ThreadId</code> takes an IO operation and starts running it in the background. It produces a <code>ThreadId</code> that can be used to e.g. terminate the thread.</p>
-<p>If we want to add actual communication between threads, we can use abstractions like <a href="https://hackage.haskell.org/package/base-4.16.4.0/docs/Control-Concurrent-MVar.html"><code>MVar</code></a> (a mutable thread-safe variable) or <a href="https://hackage.haskell.org/package/base-4.16.4.0/docs/Control-Concurrent-Chan.html"><code>Chan</code></a> (a queue).</p>
-<p>Here’s a simple example where one thread writes a value to an <code>MVar</code> and another one waits for them and prints them. An <code>MVar</code> works like a mailbox: it is either empty or full. Calling <code>takeMVar</code> on an empty box waits for the box to get filled (with a <code>putMVar</code>). Symmetrically, trying to <code>putMVar</code> into a full box waits until the box is empty.</p>
-<div class="sourceCode" id="cb446"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb446-1"><a href="#cb446-1" aria-hidden="true" tabindex="-1"></a><span class="ot">takeMVar ::</span> <span class="dt">MVar</span> a <span class="ot">-&gt;</span> <span class="dt">IO</span> a</span>
+<p>The operation <code>forkIO :: IO () -&gt; IO ThreadId</code> takes an
+IO operation and starts running it in the background. It produces a
+<code>ThreadId</code> that can be used to e.g. terminate the thread.</p>
+<p>If we want to add actual communication between threads, we can use
+abstractions like <a
+href="https://hackage.haskell.org/package/base-4.16.4.0/docs/Control-Concurrent-MVar.html"><code>MVar</code></a>
+(a mutable thread-safe variable) or <a
+href="https://hackage.haskell.org/package/base-4.16.4.0/docs/Control-Concurrent-Chan.html"><code>Chan</code></a>
+(a queue).</p>
+<p>Here’s a simple example where one thread writes a value to an
+<code>MVar</code> and another one waits for them and prints them. An
+<code>MVar</code> works like a mailbox: it is either empty or full.
+Calling <code>takeMVar</code> on an empty box waits for the box to get
+filled (with a <code>putMVar</code>). Symmetrically, trying to
+<code>putMVar</code> into a full box waits until the box is empty.</p>
+<div class="sourceCode" id="cb446"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb446-1"><a href="#cb446-1" aria-hidden="true" tabindex="-1"></a><span class="ot">takeMVar ::</span> <span class="dt">MVar</span> a <span class="ot">-&gt;</span> <span class="dt">IO</span> a</span>
 <span id="cb446-2"><a href="#cb446-2" aria-hidden="true" tabindex="-1"></a><span class="ot">putMVar ::</span> <span class="dt">MVar</span> a <span class="ot">-&gt;</span> a <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span>
 <span id="cb446-3"><a href="#cb446-3" aria-hidden="true" tabindex="-1"></a><span class="ot">newEmptyMVar ::</span> <span class="dt">IO</span> (<span class="dt">MVar</span> a)</span></code></pre></div>
-<div class="sourceCode" id="cb447"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb447-1"><a href="#cb447-1" aria-hidden="true" tabindex="-1"></a><span class="ot">send ::</span> [<span class="dt">String</span>] <span class="ot">-&gt;</span> <span class="dt">MVar</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span>
+<div class="sourceCode" id="cb447"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb447-1"><a href="#cb447-1" aria-hidden="true" tabindex="-1"></a><span class="ot">send ::</span> [<span class="dt">String</span>] <span class="ot">-&gt;</span> <span class="dt">MVar</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span>
 <span id="cb447-2"><a href="#cb447-2" aria-hidden="true" tabindex="-1"></a>send values var <span class="ot">=</span> <span class="fu">mapM_</span> (putMVar var) values</span>
 <span id="cb447-3"><a href="#cb447-3" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb447-4"><a href="#cb447-4" aria-hidden="true" tabindex="-1"></a><span class="ot">receive ::</span> <span class="dt">MVar</span> <span class="dt">String</span> <span class="ot">-&gt;</span> <span class="dt">IO</span> ()</span>
@@ -4842,83 +6919,129 @@ AABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABB
 <span id="cb447-13"><a href="#cb447-13" aria-hidden="true" tabindex="-1"></a>  forkIO (send [<span class="st">&quot;hello&quot;</span>,<span class="st">&quot;world&quot;</span>,<span class="st">&quot;and&quot;</span>,<span class="st">&quot;goodbye&quot;</span>,<span class="st">&quot;end&quot;</span>] var)</span>
 <span id="cb447-14"><a href="#cb447-14" aria-hidden="true" tabindex="-1"></a>  forkIO (receive var)</span>
 <span id="cb447-15"><a href="#cb447-15" aria-hidden="true" tabindex="-1"></a>  <span class="fu">return</span> ()</span></code></pre></div>
-<div class="sourceCode" id="cb448"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb448-1"><a href="#cb448-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Control.Concurrent</span> <span class="dt">Control.Monad</span><span class="op">&gt;</span> concurrency2</span>
+<div class="sourceCode" id="cb448"><pre
+class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb448-1"><a href="#cb448-1" aria-hidden="true" tabindex="-1"></a><span class="dt">Prelude</span> <span class="dt">Control.Concurrent</span> <span class="dt">Control.Monad</span><span class="op">&gt;</span> concurrency2</span>
 <span id="cb448-2"><a href="#cb448-2" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;hello&quot;</span></span>
 <span id="cb448-3"><a href="#cb448-3" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;world&quot;</span></span>
 <span id="cb448-4"><a href="#cb448-4" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;and&quot;</span></span>
 <span id="cb448-5"><a href="#cb448-5" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;goodbye&quot;</span></span>
 <span id="cb448-6"><a href="#cb448-6" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;end&quot;</span></span></code></pre></div>
-</section>
-</section>
-<section id="exercises-7" data-number="8.4">
-<h2 data-number="16.4"><span class="header-section-number">16.4</span> Exercises</h2>
+<h2 data-number="16.4" id="exercises-7"><span
+class="header-section-number">16.4</span> Exercises</h2>
 <ul>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set16a.hs">Set16a</a>: QuickCheck</li>
-<li><a href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set16b.hs">Set16b</a>: Phantom types</li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set16a.hs">Set16a</a>:
+QuickCheck</li>
+<li><a
+href="https://github.com/moocfi/haskell-mooc/blob/master/exercises/Set16b.hs">Set16b</a>:
+Phantom types</li>
 <li>No exercises for parallel or concurrent Haskell, sorry!</li>
 </ul>
-</section>
-<section id="where-to-go-from-here" data-number="8.5">
-<h2 data-number="16.5"><span class="header-section-number">16.5</span> Where to Go From Here?</h2>
-<p>Congratulations! You’ve reached the end of this two part course on Functional Programming in Haskell. What next? You definitely know enough Haskell to keep learning on your own. The online Haskell community is very friendly and there are lots of blog posts and other content explaining advanced techniques and features. You can find lots of interesting stuff by following for example:</p>
+<h2 data-number="16.5" id="where-to-go-from-here"><span
+class="header-section-number">16.5</span> Where to Go From Here?</h2>
+<p>Congratulations! You’ve reached the end of this two part course on
+Functional Programming in Haskell. What next? You definitely know enough
+Haskell to keep learning on your own. The online Haskell community is
+very friendly and there are lots of blog posts and other content
+explaining advanced techniques and features. You can find lots of
+interesting stuff by following for example:</p>
 <ul>
-<li><a href="https://www.reddit.com/r/haskell/">/r/haskell</a> on reddit</li>
-<li><code>#haskell</code> on <a href="https://libera.chat">libera.chat</a></li>
+<li><a href="https://www.reddit.com/r/haskell/">/r/haskell</a> on
+reddit</li>
+<li><code>#haskell</code> on <a
+href="https://libera.chat">libera.chat</a></li>
 <li><a href="https://haskellweekly.news/">Haskell Weekly</a></li>
 <li>StackOverflow</li>
 </ul>
-<p>Just keep writing Haskell, studying things (like libraries and tools) when you bump into them, and slowly accumulate experience. Lots of the things you learn with Haskell will be transferable to other languages like <a href="https://www.typescriptlang.org/">TypeScript</a>, <a href="https://elm-lang.org/">Elm</a>, <a href="https://www.rust-lang.org/">Rust</a> or <a href="https://fsharp.org/">F#</a>.</p>
-<p>Finally, here is an incomplete list of things that got left out of this course, but are worth looking into:</p>
+<p>Just keep writing Haskell, studying things (like libraries and tools)
+when you bump into them, and slowly accumulate experience. Lots of the
+things you learn with Haskell will be transferable to other languages
+like <a href="https://www.typescriptlang.org/">TypeScript</a>, <a
+href="https://elm-lang.org/">Elm</a>, <a
+href="https://www.rust-lang.org/">Rust</a> or <a
+href="https://fsharp.org/">F#</a>.</p>
+<p>Finally, here is an incomplete list of things that got left out of
+this course, but are worth looking into:</p>
 <ul>
 <li>Language features
 <ul>
-<li><a href="https://en.wikibooks.org/wiki/Haskell/Modules">Modules</a></li>
-<li>Lazy patterns (<code>~</code> patterns) and <code>@</code> patterns. See e.g. <a href="https://www.haskell.org/tutorial/patterns.html">A Gentle Introduction to Haskell</a>.</li>
-<li>Language extensions like <code>MultiParamTypeClasses</code>, <code>ViewPatterns</code> etc. <a href="https://limperg.de/ghc-extensions/">This is one good guide</a></li>
+<li><a
+href="https://en.wikibooks.org/wiki/Haskell/Modules">Modules</a></li>
+<li>Lazy patterns (<code>~</code> patterns) and <code>@</code> patterns.
+See e.g. <a href="https://www.haskell.org/tutorial/patterns.html">A
+Gentle Introduction to Haskell</a>.</li>
+<li>Language extensions like <code>MultiParamTypeClasses</code>,
+<code>ViewPatterns</code> etc. <a
+href="https://limperg.de/ghc-extensions/">This is one good
+guide</a></li>
 <li>The <code>fix</code> function</li>
 <li>The Foreign Function Interface for calling C code from Haskell</li>
 </ul></li>
 <li>Abstractions
 <ul>
-<li>Monad transformers: <a href="https://book.realworldhaskell.org/read/monad-transformers.html">RWH</a>, <a href="https://en.wikibooks.org/wiki/Haskell/Monad_transformers">Wikibook</a></li>
-<li>Free monads (advanced topic): <a href="https://www.haskellforall.com/2012/06/you-could-have-invented-free-monads.html">blog</a></li>
-<li>Lenses (advanced topic): <a href="https://hackage.haskell.org/package/lens-tutorial-1.0.4/docs/Control-Lens-Tutorial.html">tutorial</a> <a href="https://oleg.fi/gists/posts/2017-04-18-glassery.html">glassery</a></li>
-<li><a href="https://bartoszmilewski.com/">Bartosz Milewski</a> covers lots of intermediate and advanced topics on his blog</li>
+<li>Monad transformers: <a
+href="https://book.realworldhaskell.org/read/monad-transformers.html">RWH</a>,
+<a
+href="https://en.wikibooks.org/wiki/Haskell/Monad_transformers">Wikibook</a></li>
+<li>Free monads (advanced topic): <a
+href="https://www.haskellforall.com/2012/06/you-could-have-invented-free-monads.html">blog</a></li>
+<li>Lenses (advanced topic): <a
+href="https://hackage.haskell.org/package/lens-tutorial-1.0.4/docs/Control-Lens-Tutorial.html">tutorial</a>
+<a
+href="https://oleg.fi/gists/posts/2017-04-18-glassery.html">glassery</a></li>
+<li><a href="https://bartoszmilewski.com/">Bartosz Milewski</a> covers
+lots of intermediate and advanced topics on his blog</li>
 </ul></li>
 <li>Tooling
 <ul>
-<li>Using <a href="https://www.haskell.org/cabal/">Cabal</a> and <a href="https://www.haskellstack.org">Stack</a> to build your own projects</li>
-<li>Profiling: <a href="https://book.realworldhaskell.org/read/profiling-and-optimization.html">RWH</a>, <a href="https://downloads.haskell.org/~ghc/latest/docs/html/users_guide/profiling.html">GHC</a></li>
+<li>Using <a href="https://www.haskell.org/cabal/">Cabal</a> and <a
+href="https://www.haskellstack.org">Stack</a> to build your own
+projects</li>
+<li>Profiling: <a
+href="https://book.realworldhaskell.org/read/profiling-and-optimization.html">RWH</a>,
+<a
+href="https://downloads.haskell.org/~ghc/latest/docs/html/users_guide/profiling.html">GHC</a></li>
 <li><a href="https://github.com/ndmitchell/hlint#readme">Hlint</a></li>
 </ul></li>
 <li>Libraries
 <ul>
-<li>Parsec (parsing): <a href="https://book.realworldhaskell.org/read/using-parsec.html">RWH</a></li>
-<li>Scotty (simple web framework), Aeson (json): <a href="https://seanhess.github.io/2015/08/19/practical-haskell-json-api.html">blog</a></li>
-<li><a href="https://haskell-servant.github.io/">Servant</a> (fancy web framework with phantom types)</li>
+<li>Parsec (parsing): <a
+href="https://book.realworldhaskell.org/read/using-parsec.html">RWH</a></li>
+<li>Scotty (simple web framework), Aeson (json): <a
+href="https://seanhess.github.io/2015/08/19/practical-haskell-json-api.html">blog</a></li>
+<li><a href="https://haskell-servant.github.io/">Servant</a> (fancy web
+framework with phantom types)</li>
 </ul></li>
 <li>Category theory
 <ul>
 <li>Many Haskell abstractions are based on Category theory</li>
-<li>Category theory can be a valuable source of new ideas for programming</li>
-<li>Category theory can feel intimidating, so it’s good to know that you can get on fine without it</li>
-<li>Bartosz Milewski has lots of good material, for example <a href="https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/">Category Theory for Programmers</a></li>
-<li>The <a href="https://wiki.haskell.org/Category_theory">Haskell Wiki</a> and <a href="https://en.wikibooks.org/wiki/Haskell/Category_theory">Wikibook</a> have sections on category theory</li>
+<li>Category theory can be a valuable source of new ideas for
+programming</li>
+<li>Category theory can feel intimidating, so it’s good to know that you
+can get on fine without it</li>
+<li>Bartosz Milewski has lots of good material, for example <a
+href="https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/">Category
+Theory for Programmers</a></li>
+<li>The <a href="https://wiki.haskell.org/Category_theory">Haskell
+Wiki</a> and <a
+href="https://en.wikibooks.org/wiki/Haskell/Category_theory">Wikibook</a>
+have sections on category theory</li>
 </ul></li>
 </ul>
-</section>
-<section id="acknowledgements" data-number="8.6">
-<h2 data-number="16.6"><span class="header-section-number">16.6</span> Acknowledgements</h2>
-<p>This course was made possible by <a href="https://nitor.com/en">Nitor</a> who donated hours and hours of Joel’s working time for this project. Thank you!</p>
+<h2 data-number="16.6" id="acknowledgements"><span
+class="header-section-number">16.6</span> Acknowledgements</h2>
+<p>This course was made possible by <a
+href="https://nitor.com/en">Nitor</a> who donated hours and hours of
+Joel’s working time for this project. Thank you!</p>
 <p>Thanks to the whole Haskell Mooc team, especially</p>
 <ul>
 <li>John Lång for help on the material</li>
-<li>Antti Laaksonen for setting up the course and helping with arrangements</li>
+<li>Antti Laaksonen for setting up the course and helping with
+arrangements</li>
 <li>Topi Talvitie for the exercise checking infrastructure</li>
 </ul>
-<p>Thanks to all the students who patiently waited for part 2 and reported errors in the material &amp; exercises!</p>
-</section>
-</section>
+<p>Thanks to all the students who patiently waited for part 2 and
+reported errors in the material &amp; exercises!</p>
 </div>
 </div>
 </body>