Re: [HoTT] two's complement integers

[Michael Shulman <shulman@sandiego.edu>]

I don't have a particular application in mind at the moment.  It just
struck me how 2-adic integers have such simple coinductive definitions
of the arithmetic operations, without any case distinctions at all,
and I wondered whether there was a definition of plain integers with a
similar property.  It seems annoying to have to constantly case-split
on a sign bit.

I guess a lower-inductive representation could take advantage of the
fact that the 2-adic representation of an integer other than 0 or -1
has a unique largest digit that differs from all larger digits, i.e.
it's either

...0000 (i.e. 0)
...1111 (i.e. -1)
...0001 + arbitrary bit string
...1110 +  arbitrary bit string

This involves more case splits when defining arithmetic, but it would
probably be an easy set to show that the HIT one is equivalent to.


On Thu, Mar 4, 2021 at 7:02 PM Jason Gross <jasongross9@gmail.com> wrote:
>
> Note that the Coq standard library Z is a binary representation of integers and Z.testbit gives access to the infinite twos-compliment representation.  Of course, it makes use of case-distinctions on sign to define it, but you go bit-by-bit; if the number is negative, you just invert the bit before returning it.  Is there something you're after by having the representation not encode sign bits separately?
>
> On Thu, Mar 4, 2021, 21:27 Michael Shulman <shulman@sandiego.edu> wrote:
>>
>> On Thu, Mar 4, 2021 at 3:16 PM Nicolai Kraus <nicolai.kraus@gmail.com> wrote:
>> > I'm not sure what the precise thing is that you're looking for because, without further specification, any standard definition of Z would qualify :-)
>>
>> Yes, that seems to be what Martin suggested too with ℕ + ℕ.  It seemed
>> to me as though the distance between ℕ + ℕ and my ℤ is greater than
>> the distance between his 𝔹 and 𝔹', but maybe not in any important
>> way.
>>
>> > The HIT is neat, but wouldn't it in practice behave pretty similar to a standard representation via binary lists? E.g. something like Unit + Bool * List(Bool), where inl(*) is zero, the first Bool is the sign, and you add a 1 in front of the list in order to get a positive integer. What's the advantage of the HIT - maybe one can avoid case distinctions?
>>
>> Is there a non-HIT binary representation that can be interpreted as
>> two's-complement (thereby avoiding case distinctions on sign)?  I
>> haven't been able to figure out a way to do that with mere lists of
>> booleans.
>>
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