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Joined 1 year ago
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Cake day: June 20th, 2023

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  • The recorder. It’s easy to get started, there is a lot of good information online (look up Sarah Jeffery), and professional plastic instruments are very cheap. It also requires no special care and doesn’t need maintenance, since it has no moving parts.

    The downside is that in order to play larger recorders your hands need to stretch a bit, so I’m no sure if that is a problem if you have joint issues.

    The other option is a hand ocarina, but it is very hard to even make a sound, even harder to make it clear and consistent. Also you need to play entirety by ear. It’s a fun challenge, though.

















  • As you already figured out the types are sets with a certain number of elements.

    Two types are isomorphic if you can write a function that converts all elements of the first one into the elements of the second one and a function which does the reverse. You can then use this as the equality.

    The types with the same number of elements are isomorphic, i.e True | False = Left | Right. For example, you can write a function that converts True to Left, False to Right, and a function that does the reverse.

    Therefore you essentially only need types 0, 1, 2, 3, …, where type 0 has 0 elements, type 1 has 1 element, etc. and all others are isomorphic to one of these.

    Let’s use (*) for the product and (+) for the sum, and letters for generic types. Then you can essentially manipulate types as natural numbers (the same laws hold, associativity, commutativity, identity elements, distributivity).

    For example:

    2 = 1 + 1 can be interpreted as Bool = True | False

    2 * 1 = 2 can be interpreted as (Bool, Unit) = Bool

    2 * x = x + x can be interpreted as (Bool, x) = This of x | That of x

    o(x) = x + 1 can be interpreted as Option x = Some of x | None

    l(x) = o(x * l(x)) = x * l(x) + 1 can be interpreted as List x = Option (x, List x)

    l(x) = x * l(x) + 1 = x * (x * l(x) + 1) + 1 = x * x * l(x) + x + 1 = x * x * (l(x) + 1) + x + 1 = x * x * l(x) + x * x + x + 1 so a list is either empty, has 1 element or 2 elements, … (if you keep substituting)

    For the expression problem, read this paper: doi:10.1007/BFb0019443