latexify

astro.config.ts

@@ -8,6 +8,9 @@ import emoji from "remark-emoji";
 import remarkMermaid from "astro-diagram/remark-mermaid";
 import remarkDescription from "astro-remark-description";
 import remarkAdmonitions from "./plugin/remark-admonitions";
+import remarkMath from "remark-math";
+
+import rehypeKatex from "rehype-katex";
 
 // https://astro.build/config
 export default defineConfig({
@@ -18,9 +21,11 @@ export default defineConfig({
     remarkPlugins: [
       remarkAdmonitions,
       remarkReadingTime,
+      remarkMath,
       remarkMermaid,
       emoji,
       [remarkDescription, { name: "excerpt" }],
     ],
+    rehypePlugins: [rehypeKatex],
   },
 });

package-lock.json

@@ -16,11 +16,14 @@
         "astro-imagetools": "^0.9.0",
         "astro-remark-description": "^1.0.1",
         "fork-awesome": "^1.2.0",
+        "katex": "^0.16.8",
         "lodash-es": "^4.17.21",
         "mdast-util-to-string": "^4.0.0",
         "reading-time": "^1.5.0",
+        "rehype-katex": "^6.0.3",
         "remark-emoji": "^4.0.0",
         "remark-github-beta-blockquote-admonitions": "^2.1.0",
+        "remark-math": "^5.1.1",
         "remark-parse": "^10.0.2"
       },
       "devDependencies": {
@@ -1574,6 +1577,11 @@
       "resolved": "https://registry.npmjs.org/@types/json5/-/json5-0.0.30.tgz",
       "integrity": "sha512-sqm9g7mHlPY/43fcSNrCYfOeX9zkTTK+euO5E6+CVijSMm5tTjkVdwdqRkY3ljjIAf8679vps5jKUoJBCLsMDA=="
     },
+    "node_modules/@types/katex": {
+      "version": "0.14.0",
+      "resolved": "https://registry.npmjs.org/@types/katex/-/katex-0.14.0.tgz",
+      "integrity": "sha512-+2FW2CcT0K3P+JMR8YG846bmDwplKUTsWgT2ENwdQ1UdVfRk3GQrh6Mi4sTopy30gI8Uau5CEqHTDZ6YvWIUPA=="
+    },
     "node_modules/@types/lodash": {
       "version": "4.14.197",
       "resolved": "https://registry.npmjs.org/@types/lodash/-/lodash-4.14.197.tgz",
@@ -3148,6 +3156,17 @@
         "once": "^1.4.0"
       }
     },
+    "node_modules/entities": {
+      "version": "4.5.0",
+      "resolved": "https://registry.npmjs.org/entities/-/entities-4.5.0.tgz",
+      "integrity": "sha512-V0hjH4dGPh9Ao5p0MoRY6BVqtwCjhz6vI5LT8AJ55H+4g9/4vbHx1I54fS0XuclLhDHArPQCiMjDxjaL8fPxhw==",
+      "engines": {
+        "node": ">=0.12"
+      },
+      "funding": {
+        "url": "https://github.com/fb55/entities?sponsor=1"
+      }
+    },
     "node_modules/es-module-lexer": {
       "version": "1.3.0",
       "resolved": "https://registry.npmjs.org/es-module-lexer/-/es-module-lexer-1.3.0.tgz",
@@ -3694,6 +3713,61 @@
         "node": ">=4"
       }
     },
+    "node_modules/hast-util-from-dom": {
+      "version": "4.2.0",
+      "resolved": "https://registry.npmjs.org/hast-util-from-dom/-/hast-util-from-dom-4.2.0.tgz",
+      "integrity": "sha512-t1RJW/OpJbCAJQeKi3Qrj1cAOLA0+av/iPFori112+0X7R3wng+jxLA+kXec8K4szqPRGI8vPxbbpEYvvpwaeQ==",
+      "dependencies": {
+        "hastscript": "^7.0.0",
+        "web-namespaces": "^2.0.0"
+      },
+      "funding": {
+        "type": "opencollective",
+        "url": "https://opencollective.com/unified"
+      }
+    },
+    "node_modules/hast-util-from-html": {
+      "version": "1.0.2",
+      "resolved": "https://registry.npmjs.org/hast-util-from-html/-/hast-util-from-html-1.0.2.tgz",
+      "integrity": "sha512-LhrTA2gfCbLOGJq2u/asp4kwuG0y6NhWTXiPKP+n0qNukKy7hc10whqqCFfyvIA1Q5U5d0sp9HhNim9gglEH4A==",
+      "dependencies": {
+        "@types/hast": "^2.0.0",
+        "hast-util-from-parse5": "^7.0.0",
+        "parse5": "^7.0.0",
+        "vfile": "^5.0.0",
+        "vfile-message": "^3.0.0"
+      },
+      "funding": {
+        "type": "opencollective",
+        "url": "https://opencollective.com/unified"
+      }
+    },
+    "node_modules/hast-util-from-html-isomorphic": {
+      "version": "1.0.0",
+      "resolved": "https://registry.npmjs.org/hast-util-from-html-isomorphic/-/hast-util-from-html-isomorphic-1.0.0.tgz",
+      "integrity": "sha512-Yu480AKeOEN/+l5LA674a+7BmIvtDj24GvOt7MtQWuhzUwlaaRWdEPXAh3Qm5vhuthpAipFb2vTetKXWOjmTvw==",
+      "dependencies": {
+        "@types/hast": "^2.0.0",
+        "hast-util-from-dom": "^4.0.0",
+        "hast-util-from-html": "^1.0.0",
+        "unist-util-remove-position": "^4.0.0"
+      },
+      "funding": {
+        "type": "opencollective",
+        "url": "https://opencollective.com/unified"
+      }
+    },
+    "node_modules/hast-util-from-html/node_modules/parse5": {
+      "version": "7.1.2",
+      "resolved": "https://registry.npmjs.org/parse5/-/parse5-7.1.2.tgz",
+      "integrity": "sha512-Czj1WaSVpaoj0wbhMzLmWD69anp2WH7FXMB9n1Sy8/ZFF9jolSQVMu1Ij5WIyGmcBmhk7EOndpO4mIpihVqAXw==",
+      "dependencies": {
+        "entities": "^4.4.0"
+      },
+      "funding": {
+        "url": "https://github.com/inikulin/parse5?sponsor=1"
+      }
+    },
     "node_modules/hast-util-from-parse5": {
       "version": "7.1.2",
       "resolved": "https://registry.npmjs.org/hast-util-from-parse5/-/hast-util-from-parse5-7.1.2.tgz",
@@ -3712,6 +3786,19 @@
         "url": "https://opencollective.com/unified"
       }
     },
+    "node_modules/hast-util-is-element": {
+      "version": "2.1.3",
+      "resolved": "https://registry.npmjs.org/hast-util-is-element/-/hast-util-is-element-2.1.3.tgz",
+      "integrity": "sha512-O1bKah6mhgEq2WtVMk+Ta5K7pPMqsBBlmzysLdcwKVrqzZQ0CHqUPiIVspNhAG1rvxpvJjtGee17XfauZYKqVA==",
+      "dependencies": {
+        "@types/hast": "^2.0.0",
+        "@types/unist": "^2.0.0"
+      },
+      "funding": {
+        "type": "opencollective",
+        "url": "https://opencollective.com/unified"
+      }
+    },
     "node_modules/hast-util-parse-selector": {
       "version": "3.1.1",
       "resolved": "https://registry.npmjs.org/hast-util-parse-selector/-/hast-util-parse-selector-3.1.1.tgz",
@@ -3825,6 +3912,21 @@
         "url": "https://opencollective.com/unified"
       }
     },
+    "node_modules/hast-util-to-text": {
+      "version": "3.1.2",
+      "resolved": "https://registry.npmjs.org/hast-util-to-text/-/hast-util-to-text-3.1.2.tgz",
+      "integrity": "sha512-tcllLfp23dJJ+ju5wCCZHVpzsQQ43+moJbqVX3jNWPB7z/KFC4FyZD6R7y94cHL6MQ33YtMZL8Z0aIXXI4XFTw==",
+      "dependencies": {
+        "@types/hast": "^2.0.0",
+        "@types/unist": "^2.0.0",
+        "hast-util-is-element": "^2.0.0",
+        "unist-util-find-after": "^4.0.0"
+      },
+      "funding": {
+        "type": "opencollective",
+        "url": "https://opencollective.com/unified"
+      }
+    },
     "node_modules/hast-util-whitespace": {
       "version": "2.0.1",
       "resolved": "https://registry.npmjs.org/hast-util-whitespace/-/hast-util-whitespace-2.0.1.tgz",
@@ -4382,6 +4484,29 @@
       "resolved": "https://registry.npmjs.org/jsonc-parser/-/jsonc-parser-3.2.0.tgz",
       "integrity": "sha512-gfFQZrcTc8CnKXp6Y4/CBT3fTc0OVuDofpre4aEeEpSBPV5X5v4+Vmx+8snU7RLPrNHPKSgLxGo9YuQzz20o+w=="
     },
+    "node_modules/katex": {
+      "version": "0.16.8",
+      "resolved": "https://registry.npmjs.org/katex/-/katex-0.16.8.tgz",
+      "integrity": "sha512-ftuDnJbcbOckGY11OO+zg3OofESlbR5DRl2cmN8HeWeeFIV7wTXvAOx8kEjZjobhA+9wh2fbKeO6cdcA9Mnovg==",
+      "funding": [
+        "https://opencollective.com/katex",
+        "https://github.com/sponsors/katex"
+      ],
+      "dependencies": {
+        "commander": "^8.3.0"
+      },
+      "bin": {
+        "katex": "cli.js"
+      }
+    },
+    "node_modules/katex/node_modules/commander": {
+      "version": "8.3.0",
+      "resolved": "https://registry.npmjs.org/commander/-/commander-8.3.0.tgz",
+      "integrity": "sha512-OkTL9umf+He2DZkUq8f8J9of7yL6RJKI24dVITBmNfZBmri9zYZQrKkuXiKhyfPSu8tUhnVBB1iKXevvnlR4Ww==",
+      "engines": {
+        "node": ">= 12"
+      }
+    },
     "node_modules/khroma": {
       "version": "2.0.0",
       "resolved": "https://registry.npmjs.org/khroma/-/khroma-2.0.0.tgz",
@@ -4791,6 +4916,20 @@
         "url": "https://opencollective.com/unified"
       }
     },
+    "node_modules/mdast-util-math": {
+      "version": "2.0.2",
+      "resolved": "https://registry.npmjs.org/mdast-util-math/-/mdast-util-math-2.0.2.tgz",
+      "integrity": "sha512-8gmkKVp9v6+Tgjtq6SYx9kGPpTf6FVYRa53/DLh479aldR9AyP48qeVOgNZ5X7QUK7nOy4yw7vg6mbiGcs9jWQ==",
+      "dependencies": {
+        "@types/mdast": "^3.0.0",
+        "longest-streak": "^3.0.0",
+        "mdast-util-to-markdown": "^1.3.0"
+      },
+      "funding": {
+        "type": "opencollective",
+        "url": "https://opencollective.com/unified"
+      }
+    },
     "node_modules/mdast-util-mdx": {
       "version": "2.0.1",
       "resolved": "https://registry.npmjs.org/mdast-util-mdx/-/mdast-util-mdx-2.0.1.tgz",
@@ -5205,6 +5344,29 @@
         "url": "https://opencollective.com/unified"
       }
     },
+    "node_modules/micromark-extension-math": {
+      "version": "2.1.2",
+      "resolved": "https://registry.npmjs.org/micromark-extension-math/-/micromark-extension-math-2.1.2.tgz",
+      "integrity": "sha512-es0CcOV89VNS9wFmyn+wyFTKweXGW4CEvdaAca6SWRWPyYCbBisnjaHLjWO4Nszuiud84jCpkHsqAJoa768Pvg==",
+      "dependencies": {
+        "@types/katex": "^0.16.0",
+        "katex": "^0.16.0",
+        "micromark-factory-space": "^1.0.0",
+        "micromark-util-character": "^1.0.0",
+        "micromark-util-symbol": "^1.0.0",
+        "micromark-util-types": "^1.0.0",
+        "uvu": "^0.5.0"
+      },
+      "funding": {
+        "type": "opencollective",
+        "url": "https://opencollective.com/unified"
+      }
+    },
+    "node_modules/micromark-extension-math/node_modules/@types/katex": {
+      "version": "0.16.2",
+      "resolved": "https://registry.npmjs.org/@types/katex/-/katex-0.16.2.tgz",
+      "integrity": "sha512-dHsSjSlU/EWEEbeNADr3FtZZOAXPkFPUO457QCnoNqcZQXNqNEu/svQd0Nritvd3wNff4vvC/f4e6xgX3Llt8A=="
+    },
     "node_modules/micromark-extension-mdx-expression": {
       "version": "1.0.8",
       "resolved": "https://registry.npmjs.org/micromark-extension-mdx-expression/-/micromark-extension-mdx-expression-1.0.8.tgz",
@@ -6738,6 +6900,37 @@
         "url": "https://opencollective.com/unified"
       }
     },
+    "node_modules/rehype-katex": {
+      "version": "6.0.3",
+      "resolved": "https://registry.npmjs.org/rehype-katex/-/rehype-katex-6.0.3.tgz",
+      "integrity": "sha512-ByZlRwRUcWegNbF70CVRm2h/7xy7jQ3R9LaY4VVSvjnoVWwWVhNL60DiZsBpC5tSzYQOCvDbzncIpIjPZWodZA==",
+      "dependencies": {
+        "@types/hast": "^2.0.0",
+        "@types/katex": "^0.14.0",
+        "hast-util-from-html-isomorphic": "^1.0.0",
+        "hast-util-to-text": "^3.1.0",
+        "katex": "^0.16.0",
+        "unist-util-visit": "^4.0.0"
+      },
+      "funding": {
+        "type": "opencollective",
+        "url": "https://opencollective.com/unified"
+      }
+    },
+    "node_modules/rehype-katex/node_modules/unist-util-visit": {
+      "version": "4.1.2",
+      "resolved": "https://registry.npmjs.org/unist-util-visit/-/unist-util-visit-4.1.2.tgz",
+      "integrity": "sha512-MSd8OUGISqHdVvfY9TPhyK2VdUrPgxkUtWSuMHF6XAAFuL4LokseigBnZtPnJMu+FbynTkFNnFlyjxpVKujMRg==",
+      "dependencies": {
+        "@types/unist": "^2.0.0",
+        "unist-util-is": "^5.0.0",
+        "unist-util-visit-parents": "^5.1.1"
+      },
+      "funding": {
+        "type": "opencollective",
+        "url": "https://opencollective.com/unified"
+      }
+    },
     "node_modules/rehype-parse": {
       "version": "8.0.5",
       "resolved": "https://registry.npmjs.org/rehype-parse/-/rehype-parse-8.0.5.tgz",
@@ -7018,6 +7211,39 @@
         "url": "https://opencollective.com/unified"
       }
     },
+    "node_modules/remark-math": {
+      "version": "5.1.1",
+      "resolved": "https://registry.npmjs.org/remark-math/-/remark-math-5.1.1.tgz",
+      "integrity": "sha512-cE5T2R/xLVtfFI4cCePtiRn+e6jKMtFDR3P8V3qpv8wpKjwvHoBA4eJzvX+nVrnlNy0911bdGmuspCSwetfYHw==",
+      "dependencies": {
+        "@types/mdast": "^3.0.0",
+        "mdast-util-math": "^2.0.0",
+        "micromark-extension-math": "^2.0.0",
+        "unified": "^10.0.0"
+      },
+      "funding": {
+        "type": "opencollective",
+        "url": "https://opencollective.com/unified"
+      }
+    },
+    "node_modules/remark-math/node_modules/unified": {
+      "version": "10.1.2",
+      "resolved": "https://registry.npmjs.org/unified/-/unified-10.1.2.tgz",
+      "integrity": "sha512-pUSWAi/RAnVy1Pif2kAoeWNBa3JVrx0MId2LASj8G+7AiHWoKZNTomq6LG326T68U7/e263X6fTdcXIy7XnF7Q==",
+      "dependencies": {
+        "@types/unist": "^2.0.0",
+        "bail": "^2.0.0",
+        "extend": "^3.0.0",
+        "is-buffer": "^2.0.0",
+        "is-plain-obj": "^4.0.0",
+        "trough": "^2.0.0",
+        "vfile": "^5.0.0"
+      },
+      "funding": {
+        "type": "opencollective",
+        "url": "https://opencollective.com/unified"
+      }
+    },
     "node_modules/remark-mdx": {
       "version": "2.3.0",
       "resolved": "https://registry.npmjs.org/remark-mdx/-/remark-mdx-2.3.0.tgz",
@@ -8152,6 +8378,19 @@
         "url": "https://opencollective.com/unified"
       }
     },
+    "node_modules/unist-util-find-after": {
+      "version": "4.0.1",
+      "resolved": "https://registry.npmjs.org/unist-util-find-after/-/unist-util-find-after-4.0.1.tgz",
+      "integrity": "sha512-QO/PuPMm2ERxC6vFXEPtmAutOopy5PknD+Oq64gGwxKtk4xwo9Z97t9Av1obPmGU0IyTa6EKYUfTrK2QJS3Ozw==",
+      "dependencies": {
+        "@types/unist": "^2.0.0",
+        "unist-util-is": "^5.0.0"
+      },
+      "funding": {
+        "type": "opencollective",
+        "url": "https://opencollective.com/unified"
+      }
+    },
     "node_modules/unist-util-generated": {
       "version": "2.0.1",
       "resolved": "https://registry.npmjs.org/unist-util-generated/-/unist-util-generated-2.0.1.tgz",

package.json

@@ -18,11 +18,14 @@
     "astro-imagetools": "^0.9.0",
     "astro-remark-description": "^1.0.1",
     "fork-awesome": "^1.2.0",
+    "katex": "^0.16.8",
     "lodash-es": "^4.17.21",
     "mdast-util-to-string": "^4.0.0",
     "reading-time": "^1.5.0",
+    "rehype-katex": "^6.0.3",
     "remark-emoji": "^4.0.0",
     "remark-github-beta-blockquote-admonitions": "^2.1.0",
+    "remark-math": "^5.1.1",
     "remark-parse": "^10.0.2"
   },
   "devDependencies": {

src/content/posts/2023-09-01-formal-cek-machine-in-agda/index.md

@@ -66,21 +66,21 @@ several constructors:
 
 [lambda calculus]: https://en.wikipedia.org/wiki/Lambda_calculus
 
-- **Var.** This is just a variable, like `x` or `y`. By itself it holds no
+- **Var.** This is just a variable, like $x$ or $y$. By itself it holds no
   meaning, but during evaluation, the evaluation _environment_ holds a mapping
-  from variable names to the values. If the environment says `{ x = 5 }`, then
+  from variable names to the values. If the environment says $\{ x = 5 \}$, then
-  evaluating `x` would result in 5.
+  evaluating $x$ would result in $5$.
 
-- **Abstraction, or lambda (λ).** An _abstraction_ is a term that describes some
+- **Abstraction, or lambda ($\lambda$).** An _abstraction_ is a term that describes some
   other computation. From an algebraic perspective, it can be thought of as a
-  function with a single argument (i.e f(x) = 2x is an abstraction, although
+  function with a single argument (i.e $f(x) = 2x$ is an abstraction, although
-  it would be written `(λx.2x)`)
+  it would be written using the notation $\lambda x.2x$)
 
 - **Application.** Application is sort of the opposite of abstraction, exposing
   the computation that was abstracted away. From an algebraic perspective,
-  this is just function application (i.e applying `f(x) = 2x` to 3 would
+  this is just function application (i.e applying $f(x) = 2x$ to $3$ would
-  result in 2\*3. Note that only a simple substitution has been done and
+  result in $2 \times 3 = 6$. Note that only a simple substitution has been done
-  further evaluation is required to reduce 2\*3)
+  and further evaluation is required to reduce $2\times 3$)
 
 ### Why?
 
@@ -99,20 +99,24 @@ evaluation.
 In fact, the lambda calculus is [Turing-complete][tc], so any computation can
 technically be reduced to those 3 constructs. I used numbers liberally in the
 examples above, but in a lambda calculus without numbers, you could define
-integers using a recursive strategy called [Church numerals]. It looks like this:
+integers using a recursive strategy called [Church numerals]. It works like this:
 
 [church numerals]: https://en.wikipedia.org/wiki/Church_encoding
 
-- Let `z` represent zero.
+- $z$ represents zero.
-- Let `s` represent a "successor", or increment function. `s(z)` represents 1,
+- $s$ represents a "successor", or increment function. So:
-  `s(s(z))` represents 2, and so on.
+  - $s(z)$ represents 1,
+  - $s(s(z))$ represents 2
+  - and so on.
 
 In lambda calculus terms, this would look like:
 
-- 0 = `λs.(λz.z)`
+| number | lambda calculus expression         |
-- 1 = `λs.(λz.s(z))`
+| ------ | ---------------------------------- |
-- 2 = `λs.(λz.s(s(z)))`
+| 0      | $\lambda s.(\lambda z.z)$          |
-- 3 = `λs.(λz.s(s(s(z))))`
+| 1      | $\lambda s.(\lambda z.s(z))$       |
+| 2      | $\lambda s.(\lambda z.s(s(z)))$    |
+| 3      | $\lambda s.(\lambda z.s(s(s(z))))$ |
 
 In practice, many lambda calculus incorporate higher level constructors such as
 numbers or lists to make it so we can avoid having to represent them using only
@@ -129,21 +133,31 @@ a fixed-point combinator:
 
 [tc]: https://en.wikipedia.org/wiki/Turing_completeness
 
-    Y = λf.(λx.f(x(x)))(λx.f(x(x)))
+$$
+Y = \lambda f.(\lambda x.f(x(x)))(\lambda x.f(x(x)))
+$$
 
-That's quite a mouthful. If you tried calling Y on some term, you will find that
+That's quite a mouthful. If you tried calling $Y$ on some term, you will find
-evaluation will quickly expand infinitely. That makes sense given its purpose:
+that evaluation will quickly expand infinitely. That makes sense given its
-to find a _fixed point_ of whatever function you pass in.
+purpose: to find a _fixed point_ of whatever function you pass in.
 
-> As an example, the fixed-point of the function f(x) = sqrt(x) is 1. That's
+> [!NOTE]
-> because f(1) = 1. The Y combinator attempts to find the fixed point by simply
+> As an example, the fixed-point of the function $f(x) = \sqrt{x}$ is $1$.
-> applying the function multiple times. In the untyped lambda calculus, this can
+> That's because $f(1) = 1$, and applying $f$ to any other number sort of
-> be used to implement simple (but possibly unbounded) recursion.
+> converges in on this value. If you took any number and applied $f$ infinitely
+> many times on it, you would get $1$.
+>
+> In this sense, the Y combinator can be seen as a sort of brute-force approach
+> of finding this fixed point by simply applying the function over and over until
+> the result stops changing. In the untyped lambda calculus, this can be used to
+> implement simple (but possibly unbounded) recursion.
 
 This actually proves disastrous for trying to reason about the logic of a
-program. If we don't even know for sure if something will halt, how can we know
+program. If something is able to recurse on itself without limit, we won't be
-that it'll produce the correct value? In fact, you can prove false statements
+able to tell what its result is, and we _definitely_ won't be able to know if
-using infinite recursion as a basis.
+the result is correct. This is why we typically ban unbounded recursion in
+proof systems. In fact, you can give proofs for false statements using infinite
+recursion.
 
 This is why we actually prefer _not_ to work with Turing-complete languages when
 doing logical reasoning on program evaluation. Instead, we always want to add
@@ -152,26 +166,29 @@ information about our program's behavior.
 
 ### Simply-typed lambda calculus
 
-The [simply-typed lambda calculus] (STLC) adds types to every term. Types are
+The [simply-typed lambda calculus] (STLC, or the notational variant
-crucial to any kind of static program analysis. Suppose I was trying to apply
+$\lambda^\rightarrow$) adds types to every term. Types are crucial to any kind
-the term 5 to 6 (in other words, call 5 with the argument 6 as if 5 was a
+of static program analysis. Suppose I was trying to apply the term $5$ to $6$ (in
-function). As humans we can look at that and instantly recognize that the
+other words, call $5$ with the argument $6$ as if $5$ was a function, like
+$5(6)$). As humans we can look at that and instantly recognize that the
 evaluation would be invalid, yet under the untyped lambda calculus, it would be
 completely representable.
 
 [simply-typed lambda calculus]: https://en.wikipedia.org/wiki/Simply_typed_lambda_calculus
 
 To solve this in STLC, we would make this term completely unrepresentable at
-all. To say you want to apply 5 to 6 would not be a legal STLC term. We do this
+all. To say you want to apply $5$ to $6$ would not be a legal STLC term. We do
-by requiring that all STLC terms are untyped lambda calculus terms accompanied
+this by requiring that all STLC terms are untyped lambda calculus terms
-by a _type_.
+accompanied by a _type_.
 
 This gives us more information about what's allowed before we run the
-evaluation. For example, numbers may have their own type `Nat` (for "natural
+evaluation. For example, numbers may have their own type $\mathbb{N}$ (read
-number"), while functions have a special "arrow" type `_ -> _`, where the
+"nat", for "natural number") and booleans are $\mathrm{Bool}$, while functions
-underscores represent other types. A function that takes a number and returns a
+have a special "arrow" type $\_\rightarrow\_$, where the underscores represent
-boolean (like isEven) would have the type `Nat -> Bool`, while a function that
+other types. A function that takes a number and returns a boolean (like isEven)
-takes a boolean and returns another boolean would be `Bool -> Bool`.
+would have the type $\mathbb{N} \rightarrow \mathrm{Bool}$, while a function
+that takes a boolean and returns another boolean would be $\mathrm{Bool}
+\rightarrow \mathrm{Bool}$.
 
 With this, we have a framework for rejecting terms that would otherwise be legal
 in untyped lambda calculus, but would break when we tried to evaluate them. A
@@ -187,28 +204,33 @@ A semi-formal definition for STLC terms would look something like this:
 - **Var.** Same as before, it's a variable that can be looked up in the
   environment.
 
-- **Abstraction, or lambda (λ).** This is a function that carries three pieces
+- **Abstraction, or lambda ($\lambda$).** This is a function that carries three pieces
-  of information: (1) the name of the variable that its input will be substituted
+  of information:
-  for, (2) the _type_ of the input, and (3) the body in which the substitution
+
-  will happen.
+  1. the name of the variable that its input will be substituted for
+  2. the _type_ of the input, and
+  3. the body in which the substitution will happen.
 
 - **Application.** Same as before.
 
 It doesn't really seem like changing just one term changes the language all that
 much. But as a result of this tiny change, _every_ term now has a type:
 
-- `5 :: Nat`
+- $5 :: \mathbb{N}$
-- `λ(x:Nat).2x :: Nat -> Nat`
+- $λ(x:\mathbb{N}).2x :: \mathbb{N} \rightarrow \mathbb{N}$
-- `isEven(3) :: (Nat -> Bool) · Nat = Bool`
+- $isEven(3) :: (\mathbb{N} \rightarrow \mathrm{Bool}) · \mathbb{N} = \mathrm{Bool}$
 
-Notation: (`x :: T` means `x` has type `T`, and `f · x` means `f` applied to
+> [!NOTE]
-`x`)
+> Some notation:
+>
+> - $x :: T$ means $x$ has type $T$, and
+> - $f · x$ means $f$ applied to $x$
 
 This also means that some values are now unrepresentable:
 
-- `isEven(λx.2x) :: (Nat -> Bool) · (Nat -> Nat)` doesn't work because the type
+- $isEven(λx.2x)$ wouldn't work anymore because the type of the inner argument
-  of `λx.2x :: Nat -> Nat` can't be used as an input for `isEven`, which is
+  $λx.2x$ would be $\mathbb{N} \rightarrow \mathbb{N}$ can't be used as an input
-  expecting a `Nat`.
+  for $isEven$, which is expecting a $\mathbb{N}$.
 
 We have a good foundation for writing programs now, but this by itself can't
 qualify as a system for computation. We need an abstract machine of sorts that

src/layouts/BaseLayout.astro

@@ -2,6 +2,7 @@
 import Footer from "../components/Footer.astro";
 import LeftNav from "../components/LeftNav.astro";
 import "../styles/global.scss";
+import "katex/dist/katex.min.css";
 ---
 
 <!doctype html>

src/styles/post.scss

@@ -160,9 +160,12 @@
 
   table {
     border-collapse: collapse;
+    border: 1px solid var(--hr-color);
+    border-radius: 4px;
 
     thead {
-      background-color: black;
+      background-color: var(--hr-color);
+      // color: var(--background-color);
     }
 
     td,
@@ -239,10 +242,9 @@ hr.endline {
 
   .admonition-title {
     color: var(--color);
-    margin-bottom: 6px;
   }
 
   p {
-    margin: 0;
+    margin: 8px auto;
   }
 }