Discussion of Homotopy Type Theory and Univalent Foundations
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From: Michael Shulman <shulman@sandiego.edu>
To: "HomotopyTypeTheory@googlegroups.com"
	<homotopytypetheory@googlegroups.com>
Subject: [HoTT] two's complement integers
Date: Thu, 4 Mar 2021 12:43:28 -0800
Message-ID: <CAOvivQw4h46DtKATiwUvm=L=jEGng2XdYGAhKbi7fhis+y_TPw@mail.gmail.com> (raw)

Has anyone considered the following HIT definition of the integers?

data ℤ : Type where
  zero : ℤ                              -- 0
  negOne : ℤ                            -- -1
  dbl : ℤ → ℤ                           -- n ↦ 2n
  sucDbl : ℤ → ℤ                        -- n ↦ 2n+1
  dblZero : dbl zero ≡ zero             -- 2·0 = 0
  sucDblNegOne : sucDbl negOne ≡ negOne -- 2·(-1)+1 = -1

The idea is that it's an arbitrary-precision version of little-endian
two's-complement, with sucDbl and dbl representing 1 and 0
respectively, and negOne and zero representing the highest-order sign
bit 1 and 0 respectively.  Thus for instance

sucDbl (dbl (sucDbl (sucDbl zero))) = 01101 = 13
dbl (sucDbl (dbl (dbl negOne))) = 10010 = -14

The two path-constructors enable arbitrary expansion of the precision, e.g.

01101 = 001101 = 0001101 = ...
10010 = 110010 = 1110010 = ...

This seems like a fairly nice definition: it should already be a set
without any truncation, it's binary rather than unary, and the
arithmetic operations can be defined in the usual digit-by-digit way
without splitting into cases by sign.  Mathematically speaking, it
represents integers by their images in the 2-adic integers, with zero
and negOne representing infinite tails of 0s and 1s respectively.  (An
arbitrary 2-adic integer, of course, is just a stream of bits.)

Best,
Mike

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             reply	other threads:[~2021-03-04 20:43 UTC|newest]

Thread overview: 8+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2021-03-04 20:43 Michael Shulman [this message]
2021-03-04 21:11 ` Martin Escardo
2021-03-04 22:05   ` Michael Shulman
2021-03-04 22:42     ` Martin Escardo
2021-03-04 23:16     ` Nicolai Kraus
2021-03-05  2:27       ` Michael Shulman
2021-03-05  3:02         ` Jason Gross
2021-03-05  4:41           ` Michael Shulman

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Discussion of Homotopy Type Theory and Univalent Foundations

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