more on induction + implement language switcher

assets/sass/_content.scss

@@ -338,6 +338,73 @@ table.table {
     }
 }
 
+// Tabbing
+.language-switcher-choice {
+    display: none;
+}
+
+.tabbed {
+    border: 1px solid #eee;
+
+    ul.tabs {
+        margin-top: 0;
+        margin-bottom: 0;
+        list-style: none;
+        padding-inline-start: 0;
+        border-bottom: 1px solid lightgray;
+
+        .tab {
+            display: inline-block;
+            list-style: none;
+
+            label {
+                display: inline-block;
+                padding: 4px 8px;
+                font-size: 0.9rem;
+            }
+        }
+    }
+
+    .contents {
+        .tab-content {
+            padding: 0 12px;
+
+            &:not(:last-child) {
+                border-bottom: 1px solid gray;
+            }
+        }
+    }
+}
+
+@for $i from 1 through 5 {
+    $colors: [green, red, blue, yellow, purple];
+
+    .language-switcher-choice:nth-of-type(#{$i}):checked {
+        outline: 2px solid nth($colors, $i);
+    }
+
+    body:has(.language-switcher-choice:nth-of-type(#{$i}):checked) .tabbed {
+        .tabs .tab:nth-child(#{$i}) {
+            border-bottom: 3px solid gray;
+        }
+
+        .tabs .tab:not(:nth-child(#{$i})) {
+            border-bottom: 3px solid transparent;
+        }
+
+        .contents .tab-content:nth-child(#{$i}) {
+            border-bottom: none;
+        }
+
+        .contents .tab-content:not(:nth-child(#{$i})) {
+            // text-shadow: 2px 2px nth($colors, $i);
+            display: none;
+        }
+    }
+}
+
+// Post container
+
 .post-container {
     display: flex;
 
@@ -375,7 +442,7 @@ table.table {
     */
 
     .post-content {
-        ul {
+        ul:not(.tabs) {
             padding-left: 1.5rem;
         }
 
@@ -403,6 +470,12 @@ table.table {
                 padding-left: 12px;
             }
         }
+
+        .highlight,
+        details {
+            margin-top: 16px;
+            margin-bottom: 16px;
+        }
     }
 
     &.logseq-post {

content/posts/2023-03-26-inductive-types.md

@@ -1,15 +1,17 @@
 +++
 title = "Inductive Types"
-slug = "inductive-types"
 date = 2023-03-26
 tags = ["type-theory"]
 math = true
 draft = true
+
+language_switcher_languages = ["ocaml", "python"]
 +++
 
 There's a feature common to many functional languages, the ability to have
-algebraic data types. It might look something like this (OCaml syntax):
+algebraic data types. It might look something like this:
 
+{{< language-switcher >}}
 ```ocaml
 type bool =
 | True
@@ -26,6 +28,20 @@ that has two _constructors_, or ways to create this type. The constructors are
 
 ---
 
+```python
+from typing import Literal
+MyBool = Literal[True] | Literal[False]
+```
+{{</ language-switcher >}}
+
+> **Note:** I'm using an experimental language switcher. It's implemented in
+> pure CSS using a feature called the [`:has` pseudo-class][has]. As of writing,
+> all major browsers _except_ Firefox has it implemented and enabled by default.
+> For Firefox there does exist a feature flag in about:config, but your mileage
+> may vary.
+
+---
+
 Many languages have this feature, under different names. Tagged unions, variant
 types, enumerations, but they all reflect a basic idea: a type with a limited
 set of variants.
@@ -41,12 +57,21 @@ unknown value of type `Boolean`, you know it can only take one of two values.
 There's actually nothing special about boolean itself. I could just as easily
 define a new type, like this:
 
+{{< language-switcher >}}
 ```ocaml
 type WeirdType =
 | Foo
 | Bar
 ```
 
+---
+
+```python
+from typing import Literal
+WeirdType = Literal['foo'] | Literal['bar']
+```
+{{</ language-switcher >}}
+
 Because this type can only have two values, it's _semantically_ equivalent to
 the `Boolean` type. I could use it anywhere I would typically use `Boolean`.
 
@@ -65,11 +90,20 @@ You can make any _finite_ type like this: just create an algebraic data type
 with unit constructors, and the result is a type with a finite cardinality. If I
 wanted to make a unit type for example:
 
+{{< language-switcher >}}
 ```ocaml
 type unit =
 | Unit
 ```
 
+---
+
+```python
+from typing import Literal
+Unit = Literal[None]
+```
+{{</ language-switcher >}}
+
 There's only one way to ever construct something of this type, so the
 cardinality of this type would be 1.
 
@@ -83,6 +117,7 @@ called structural matching in some languages).
 Let's see an example. Suppose I have a type with three values, defined like
 this:
 
+{{< language-switcher >}}
 ```ocaml
 type direction =
 | Left
@@ -90,10 +125,19 @@ type direction =
 | Right
 ```
 
-If I was given a value with type `direction`, but I wanted to do different
+---
+
+```python
+from typing import Literal
+Direction = Literal['left'] | Literal['middle'] | Literal['right']
+```
+{{</ language-switcher >}}
+
+If I was given a value with a type of direction, but I wanted to do different
 things depending on exactly which direction it was, I could use _pattern
 matching_ like this:
 
+{{< language-switcher >}}
 ```ocaml
 let do_something_with (d : direction) =
   match d with
@@ -102,6 +146,22 @@ let do_something_with (d : direction) =
   | Right -> do_this_if_right
 ```
 
+---
+
+```python
+def do_something_with(d : Direction) -> str:
+  match inp:
+    case 'left': return do_this_if_left
+    case 'middle': return do_this_if_middle
+    case 'right': return do_this_if_right
+    case _: assert_never(inp)
+```
+
+**Note:** the `assert_never` is a static check for exhaustiveness. If we missed
+a single one of the cases, a static type checker like [pyright] could catch it
+and tell us which of the remaining cases there are.
+{{</ language-switcher >}}
+
 This gives me a way to discriminate between the different variants of
 `direction`.
 
@@ -109,7 +169,7 @@ This gives me a way to discriminate between the different variants of
 > the `Boolean` type, called if-else. What would if-else look like if you wrote
 > it as a function in this pattern-matching form?
 
-## The Algebra of Types
+## Constructing larger types
 
 Finite-cardinality types like the ones we looked at just now are nice, but
 they're not super interesting. If you had a programming language with nothing
@@ -131,24 +191,106 @@ contain themselves as a type).
 
 You can see an example of this here:
 
+{{< language-switcher >}}
 ```ocaml
 type nat =
 | Suc of nat
 | Zero
 ```
+{{</ language-switcher >}}
 
 These are the natural numbers, which are defined inductively. Each number is
 just represented by a data type that wraps 0 that number of times. So 3 would be
 `Suc (Suc (Suc Zero))`.
 
-This data type is _inductive_ because the `Suc` case can contain arbitrarily
-many `nat`s inside of it. This also means that if we want to talk about writing
-any functions on `nat`, we just have to supply 2 cases instead of an infinite
-number of cases:
+At this point you can probably see why these have _infinite_ cardinality: with
+the Suc case, you can keep wrapping nats as many times as you want!
+
+One key observation here is that although the _cardinality_ of the entire type is
+infinite, it only uses _two_ constructors to build it. This also means that if
+we want to talk about writing any functions on `nat`, we just have to supply 2
+cases instead of an infinite number of cases:
 
 ```ocaml
-let is_even = fun (x : nat) ->
-  match x with
-  | Suc n -> not (is_even n)
+let rec is_even = fun (n : nat) ->
+  match n with
   | Zero -> true
+  | Suc n1 -> not (is_even n1)
 ```
+
+<details>
+  <summary>Try it for yourself</summary>
+
+  If you've got an OCaml interpreter handy, try a couple values for yourself and
+  convince yourself that this accurately represents the naturals and an even
+  testing function:
+
+  ```ocaml
+  utop # is_even Zero;;
+  - : bool = true
+  utop # is_even (Suc Zero);;
+  - : bool = false
+  ```
+
+  This is a good way of making sure the functions you write make sense!
+</details>
+
+## Induction principle
+
+Let's express this in the language of mathematical induction. If I have any
+natural number:
+
+- If the natural number is the base case of zero, then the `is_even` relation
+    automatically evaluates to true.
+- If the natural number is a successor, invert the induction hypothesis (which is
+    what `is_even` evaluates to for the previous step, a.k.a whether or not the
+    previous natural number is even), since every even number is succeeded by
+    an odd number and vice versa.
+
+Once these rules are followed, by induction we know that `is_even` can run on
+any natural number ever. In code, this looks like:
+
+```ocaml
+let is_even
+    (n_zero : bool)
+    (n_suc : nat -> bool)
+    (n : nat)
+  : bool =
+  match n with
+  | Zero -> n_zero
+  | Suc n1 -> n_suc n1
+```
+
+where n_zero defines what to do with the zero case, and n_suc defines what to do
+with the successor case.
+
+You might've noticed that this definition doesn't actually return any booleans.
+That's because this is not actually the is_even function! This is a general
+function that turns any natural into a boolean. In fact, we can go one step
+further and generalize this to all types:
+
+```ocaml
+let nat_transformer
+    (n_zero : 'a)
+    (n_suc : 'a -> 'a)
+    (n : nat)
+  : 'a =
+  match n with
+  | Zero -> n_zero
+  | Suc n1 -> n_suc n1
+```
+
+Let's say I wanted to write a function that converts from our custom-defined nat
+type into an OCaml integer. Using this constructor, that would look something
+like this:
+
+```ocaml
+let convert_nat = nat_transformer 0 (fun x -> x + 1)
+```
+
+
+
+TODO: Talk about https://counterexamples.org/currys-paradox.html
+
+[has]: https://developer.mozilla.org/en-US/docs/Web/CSS/:has
+[pyright]: https://github.com/microsoft/pyright

layouts/_default/baseof.html

@@ -14,7 +14,7 @@
         {{ end }}
 
         {{ $style := resources.Get "sass/main.scss" | resources.ToCSS }}
-        <link rel="stylesheet" href="{{ $style.RelPermalink }}" />
+        <link rel="stylesheet" href="{{ $style.RelPermalink }}" crossorigin="anonymous" />
     </head>
 
     <body>

layouts/posts/single.html

@@ -26,6 +26,27 @@
     - {{ .ReadingTime }} min read
 </small>
 
+{{ if .Params.language_switcher_languages }}
+<div class="has-warning">
+    warning! your browser does not support has
+</div>
+
+    <div class="language_switcher_choices">
+    {{ range $index, $lang := .Page.Params.language_switcher_languages }}
+        <input
+            type="radio"
+            name="language-switcher"
+            class="language-switcher-choice"
+            id="language-switcher-{{ $lang }}"
+            value="{{ $lang }}"
+            {{ if (eq $index 0) }}
+                checked
+            {{ end }}
+        >
+    {{ end }}
+    </div>
+{{ end }}
+
 <div class="post-container
     {{ if .Params.logseq }}logseq-post{{ end }}
     ">

layouts/shortcodes/language-switcher.html

@@ -0,0 +1,28 @@
+{{ $_hugo_config := `{ "version": 1 }` }}
+
+{{ $parts := split .Inner "---" }}
+{{ $page := .Page }}
+
+<div class="tabbed">
+
+<ul class="tabs">
+{{ range $index, $snippet := $parts }}
+	{{ $lang := index $page.Params.language_switcher_languages $index }}
+	<li class="tab">
+		<label for="language-switcher-{{ $lang }}">{{ $lang }}</label>
+	</li>
+{{ end }}
+</ul>
+
+
+<div class="contents">
+	{{ range $index, $snippet := $parts }}
+	<div class="tab-content">
+		{{ $snippet | markdownify }}
+	</div>
+	{{ end }}
+</div>
+
+</div>
+
+{{/* Thank you https://codepen.io/MPDoctor/pen/mpJdYe */}}