55 lines
2 KiB
Markdown
55 lines
2 KiB
Markdown
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title = "Dependent Types from First Principles"
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slug = "dependent-types"
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date = 2022-10-27
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tags = ["type-theory"]
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toc = true
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math = true
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draft = true
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+++
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It's frequently said that computers are just made of 1s and 0s. As programmers,
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we induce structure into this sea of 1s and 0s to build a rich collection of
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abstractions that can produce the entire software ecosystem as we know it. But
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how does this work? Let's start from the beginning.
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### Bits and Bytes
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On the lowest level, your computer is actually a tiny network of many
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components, but let's pretend for a second it's just a giant list of **bits**.
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Something like this.
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```
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011101110110100001111001001000000110010001101001011001000010000001111001011011110111010100100000011001000110010101100011011011110110010001100101001000000111010001101000011010010111001100100000011011000110111101101100
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```
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This is often a mess, so we like to chunk these into groups of 8, called an
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_octet_, and then represent it as a 2-digit base 16 number. For example, the
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string above would look something like this:
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```
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77 68 79 20 64 69 64 20 79 6f 75 20 64 65 63 6f 64 65 20 74 68 69 73 20 6c 6f 6c
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```
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We've just turned our series of 1s and 0s into some numbers and letters. Great.
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Now what? How do we get programs, images, and video from this? Well, we have to
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induce some structure.
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### Building Blocks
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Suppose you had a pile of sugar and a pile of salt. They both look pretty
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similar, but if you were to put them into the same pile, cooking would be a
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disaster. You would not want to use salt where you wanted to use sugar, or vice
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versa.
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Same thing in computing. We have developed byte-level representations for
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several basic kinds of data, but without distinction, it's hard to tell them
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apart. So programmers developed **types**, ways of categorizing data so when
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we're just reading a series of 0s and 1s, we would know how to interpret it.
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|Raw|Interpreted as number|Interpreted as text|
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|---|---|---|
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|`68 65 6c 6c 6f`|448,378,203,247|`hello`|
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So now we know how to
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