feat(TOC): Table of Contents
post(typoclassopedia): alternative formulations for Applicative
This commit is contained in:
Mahdi Dibaiee
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ae6b5b2be7
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Resume.pdf
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Resume.pdf
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@ -32,3 +32,6 @@ collections:
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output: true
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permalink: /travel/:path/
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path: /travel
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contentsLabel: "Table of Contents"
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showToggleButton: true
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@ -10,7 +10,11 @@
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<div class="page-content">
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<div class="wrapper">
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<h1 class="page-heading"></h1>
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{% if page.toc %}
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{{ content | toc_generate }}
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{% else %}
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{{ content }}
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{% endif %}
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</div>
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</div>
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132
_plugins/tocGenerator.rb
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132
_plugins/tocGenerator.rb
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@ -0,0 +1,132 @@
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require 'nokogiri'
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module Jekyll
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module TOCGenerator
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TOGGLE_HTML = '<h2>%1</h2>'
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TOC_CONTAINER_HTML = '<div id="toc-container"><table class="toc" id="toc"><tbody><tr><td>%1%2<ul>%3</ul></td></tr></tbody></table></div>'
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HIDE_HTML = '<input type="checkbox" id="toctogglelink" href="#" checked></input><span></span>'
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def toc_generate(html)
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# No Toc can be specified on every single page
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# For example the index page has no table of contents
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return html if (@context.environments.first["page"]["noToc"] || false)
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config = @context.registers[:site].config
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# Minimum number of items needed to show TOC, default 0 (0 means no minimum)
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min_items_to_show_toc = config["minItemsToShowToc"] || 0
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anchor_prefix = config["anchorPrefix"] || 'tocAnchor-'
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# better for traditional page seo, commonlly use h1 as title
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toc_top_tag = config["tocTopTag"] || 'h1'
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toc_top_tag = toc_top_tag.gsub(/h/, '').to_i
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toc_top_tag = 5 if toc_top_tag > 5
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toc_sec_tag = toc_top_tag + 1
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toc_top_tag = "h#{toc_top_tag}"
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toc_sec_tag = "h#{toc_sec_tag}"
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# Text labels
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contents_label = config["contentsLabel"] || 'Contents'
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hide_label = config["hideLabel"] || 'hide'
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# show_label = config["showLabel"] || 'show' # unused
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show_toggle_button = config["showToggleButton"]
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toc_html = ''
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toc_level = 1
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toc_section = 1
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item_number = 1
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level_html = ''
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doc = Nokogiri::HTML(html)
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# Find H1 tag and all its H2 siblings until next H1
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doc.css(toc_top_tag).each do |tag|
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# TODO This XPATH expression can greatly improved
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ct = tag.xpath("count(following-sibling::#{toc_top_tag})")
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sects = tag.xpath("following-sibling::#{toc_sec_tag}[count(following-sibling::#{toc_top_tag})=#{ct}]")
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level_html = ''
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inner_section = 0
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sects.each do |sect|
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inner_section += 1
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anchor_id = [
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anchor_prefix, toc_level, '-', toc_section, '-',
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inner_section
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].map(&:to_s).join ''
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sect['id'] = "#{anchor_id}"
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level_html += create_level_html(anchor_id,
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toc_level + 1,
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toc_section + inner_section,
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item_number.to_s + '.' + inner_section.to_s,
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sect.text,
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'')
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end
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level_html = '<ul>' + level_html + '</ul>' if level_html.length > 0
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anchor_id = anchor_prefix + toc_level.to_s + '-' + toc_section.to_s
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tag['id'] = "#{anchor_id}"
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toc_html += create_level_html(anchor_id,
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toc_level,
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toc_section,
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item_number,
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tag.text,
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level_html)
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toc_section += 1 + inner_section
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item_number += 1
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end
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# for convenience item_number starts from 1
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# so we decrement it to obtain the index count
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toc_index_count = item_number - 1
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return html unless toc_html.length > 0
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hide_html = ''
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hide_html = HIDE_HTML.gsub('%1', hide_label) if (show_toggle_button)
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if min_items_to_show_toc <= toc_index_count
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replaced_toggle_html = TOGGLE_HTML
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.gsub('%1', contents_label)
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toc_table = TOC_CONTAINER_HTML
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.gsub('%1', replaced_toggle_html)
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.gsub('%2', hide_html)
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.gsub('%3', toc_html)
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doc.css('.post-header').after(toc_table)
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end
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doc.css('body').children.to_xhtml
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end
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private
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def create_level_html(anchor_id, toc_level, toc_section, tocNumber, tocText, tocInner)
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link = '<a href="#%1"><span class="tocnumber">%2</span> <span class="toctext">%3</span></a>%4'
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.gsub('%1', anchor_id.to_s)
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.gsub('%2', tocNumber.to_s)
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.gsub('%3', tocText)
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.gsub('%4', tocInner ? tocInner : '')
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'<li class="toc_level-%1 toc_section-%2">%3</li>'
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.gsub('%1', toc_level.to_s)
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.gsub('%2', toc_section.to_s)
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.gsub('%3', link)
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end
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end
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end
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Liquid::Template.register_filter(Jekyll::TOCGenerator)
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@ -5,9 +5,10 @@ date: 2017-09-27
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permalink: typoclassopedia-exercise-solutions/
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categories: programming
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math: true
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toc: true
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---
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I wanted to get proficient in Haskell so I decided to follow [An [Essential] Haskell Reading List](http://www.stephendiehl.com/posts/essential_haskell.html), there I stumbled upon [Typoclassopedia](https://wiki.haskell.org/Typeclassopedia), while the material is great, I couldn't find solutions for the exercises to check against, so I decided I would write my own and hopefully the solutions would get fixed in case I have gone wrong by others. So if you think a solution is wrong, let me know in the comments!
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I wanted to get proficient in Haskell so I decided to follow [An [Essential] Haskell Reading List](http://www.stephendiehl.com/posts/essential_haskell.html). There I stumbled upon [Typoclassopedia](https://wiki.haskell.org/Typeclassopedia), while the material is great, I couldn't find solutions for the exercises to check against, so I decided I would write my own and hopefully the solutions would get fixed in case I have gone wrong by others. So if you think a solution is wrong, let me know in the comments!
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In each section below, I left some reference material for the exercises and then the solutions.
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@ -347,3 +348,229 @@ You can check the type of `(flip ($) f) . (flip ($))` is something like this (de
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```
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Also see [this question on Stack Overflow](https://stackoverflow.com/questions/46503793/applicative-prove-pure-f-x-pure-flip-x-pure-f/46505868#46505868) which includes alternative proofs.
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## Instances
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Applicative instance of lists as a collection of values:
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```haskell
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newtype ZipList a = ZipList { getZipList :: [a] }
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instance Applicative ZipList where
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pure :: a -> ZipList a
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pure = undefined -- exercise
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(<*>) :: ZipList (a -> b) -> ZipList a -> ZipList b
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(ZipList gs) <*> (ZipList xs) = ZipList (zipWith ($) gs xs)
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```
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Applicative instance of lists as a non-deterministic computation context:
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```haskell
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instance Applicative [] where
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pure :: a -> [a]
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pure x = [x]
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(<*>) :: [a -> b] -> [a] -> [b]
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gs <*> xs = [ g x | g <- gs, x <- xs ]
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```
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### Exercises
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1. Implement an instance of `Applicative` for `Maybe`.
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**Solution**:
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```haskell
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instance Applicative (Maybe a) where
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pure :: a -> Maybe a
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pure x = Just x
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(<*>) :: Maybe (a -> b) -> Maybe a -> Maybe b
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_ <*> Nothing = Nothing
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Nothing <*> _ = Nothing
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(Just f) <*> (Just x) = Just (f x)
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```
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2. Determine the correct definition of `pure` for the `ZipList` instance of `Applicative`—there is only one implementation that satisfies the law relating `pure` and `(<*>)`.
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**Solution**:
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```haskell
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newtype ZipList a = ZipList { getZipList :: [a] }
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instance Functor ZipList where
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fmap f (ZipList list) = ZipList { getZipList = fmap f list }
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instance Applicative ZipList where
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pure = ZipList . pure
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(ZipList gs) <*> (ZipList xs) = ZipList (zipWith ($) gs xs)
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```
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You can check the Applicative laws for this implementation.
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## Utility functions
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### Exercises
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1. Implement a function
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`sequenceAL :: Applicative f => [f a] -> f [a]`
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There is a generalized version of this, `sequenceA`, which works for any `Traversable` (see the later section on `Traversable`), but implementing this version specialized to lists is a good exercise.
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**Solution**:
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```haskell
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createList = replicate 1
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sequenceAL :: Applicative f => [f a] -> f [a]
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sequenceAL = foldr (\x b -> ((++) . createList <$> x) <*> b) (pure [])
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```
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Explanation:
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First, `createList` is a simple function for creating a list of a single element, e.g. `createList 2 == [2]`.
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Now let's take `sequenceAL` apart, first, it does a fold over the list `[f a]`, and `b` is initialized to `pure []`, which results in `f [a]` as required by the function's output.
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Inside the function, `createList <$> x` applies `createList` to the value inside `f a`, resulting in `f [a]`, and then `(++)` is applied to the value again, so it becomes `f ((++) [a])`, now we can apply the function `(++) [a]` to `b` by `((++) . createList <$> x) <*> b`, which results in `f ([a] ++ b)`.
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## Alternative formulation
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### Definition
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```haskell
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class Functor f => Monoidal f where
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unit :: f ()
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(**) :: f a -> f b -> f (a,b)
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```
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### Laws:
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1. Left identity
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```haskell
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unit ** v ≅ v
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```
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2. Right identity
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```haskell
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u ** unit ≅ u
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```
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3. Associativity
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```haskell
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u ** (v ** w) ≅ (u ** v) ** w
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```
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4. Neutrality
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```haskell
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fmap (g *** h) (u ** v) = fmap g u ** fmap h v
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```
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### Isomorphism
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In the laws above, `≅` refers to isomorphism rather than equality. In particular we consider:
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```haskell
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(x,()) ≅ x ≅ ((),x)
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((x,y),z) ≅ (x,(y,z))
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```
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### Exercises
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instance Applicative [] where
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pure :: a -> [a]
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pure x = [x]
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(<*>) :: [a -> b] -> [a] -> [b]
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gs <*> xs = [ g x | g <- gs, x <- xs ]
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```haskell
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pure id <*> v = v
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```
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```haskell
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pure f <*> pure x = pure (f x)
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```
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```haskell
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u <*> pure y = pure ($ y) <*> u
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```
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```haskell
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u <*> (v <*> w) = pure (.) <*> u <*> v <*> w
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```
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1. Implement `pure` and `<*>` in terms of `unit` and `**`, and vice versa.
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```haskell
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unit :: f ()
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unit = pure ()
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(**) :: f a -> f b -> f (a, b)
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a ** b = fmap (,) a <*> b
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pure :: a -> f a
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pure x = unit ** x
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(<*>) :: f (a -> b) -> f a -> f b
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f <*> a = fmap (uncurry ($)) (f ** a) = fmap (\(f, a) -> f a) (f ** a)
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```
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2. Are there any `Applicative` instances for which there are also functions `f () -> ()` and `f (a,b) -> (f a, f b)`, satisfying some "reasonable" laws?
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The [`Arrow`](https://wiki.haskell.org/Typeclassopedia#Arrow) type class seems to satisfy these criteria.
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```haskell
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first unit = ()
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(id *** f) (a, b) = (f a, f b)
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```
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3. (Tricky) Prove that given your implementations from the first exercise, the usual Applicative laws and the Monoidal laws stated above are equivalent.
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1. Identity Law
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```haskell
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pure id <*> v
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= fmap (uncurry ($)) ((unit ** id) ** v)
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= fmap (uncurry ($)) (id ** v)
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= fmap id v
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= v
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```
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2. Homomorphism
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```haskell
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pure f <*> pure x
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= (unit ** f) <*> (unit ** x)
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= fmap (\(f, a) -> f a) (unit ** f) (unit ** x)
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= fmap (\(f, a) -> f a) (f ** x)
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= fmap f x
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= pure (f x)
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```
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3. Interchange
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```haskell
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u <*> pure y
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= fmap (uncurry ($)) (u ** (unit ** y))
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= fmap (uncurry ($)) (u ** y)
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= fmap (u $) y
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= fmap ($ y) u
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= pure ($ y) <*> u
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4. Composition
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```haskell
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u <*> (v <*> w)
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= fmap (uncurry ($)) (u ** (fmap (uncurry ($)) (v ** w)))
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= fmap (uncurry ($)) (u ** (fmap v w))
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= fmap u (fmap v w)
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= fmap (u . v) w
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= pure (.) <*> u <*> v <*> w =
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```
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29
_sass/toc.scss
Normal file
29
_sass/toc.scss
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@ -0,0 +1,29 @@
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#toc-container {
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h2 {
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display: inline-block;
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}
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input, input + span {
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margin-left: 1rem;
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}
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input {
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position: absolute;
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width: 50px;
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height: 20px;
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opacity: 0;
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margin-top: 10px;
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}
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input + span::before {
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content: '[hide]';
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}
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input:checked ~ ul {
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display: none;
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}
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input:checked + span::before {
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content: '[show]';
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}
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}
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@ -47,5 +47,6 @@ $on-laptop: 800px;
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@import
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"base",
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"layout",
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"syntax-highlighting"
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"syntax-highlighting",
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"toc"
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;
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