Binary-3.2 -- a Pure Binary Natural Number Arithmetic library for Agda
is available here on GitHub and on
http://www.botik.ru/pub/local/Mechveliani/binNat/3.2
Binary-3.2 is a pure, regular-performance, complete, and certified binary arithmetic for natural numbers written in Agda.
It has been tested under Agda 2.5.4, MAlonzo, ghc-7.10.2.
`Pure'
means that Binary-3.2 never uses a built-in arithmetic (on Nat) to
essentially change performance.
The performance order is supposed to remain with replacing the Nat
arithmetic with the naive unary arithmetic.
`Regular performance' means that the arithmetic on Bin of Binary-3.2 has a regular performance cost order -- the one expected for the corresponding known naive operations with bit lists. At least this holds for <-cmp, +, -., *, divMod, gcd. Examples:
-
The comparison <-cmp x y on Bin costs O(|x| + |y|),
where |z| = bitLength z. -
Division with remainder divMod x y y≢0 on Bin is implemented as a loop of {s = shift n dr for certain n; dd := dd -. s} repeated O(|x|) times. So that it costs O( |x|^2 ).
`Complete'
means that all the items are implemented that are usually
required for standard arithmetic. There are provided the following items.
- DecTotalOrder instance for Bin, for x <= y defined on Bin as x < y or x == y.
- StrictTotalOrder instance for Bin, <.
- The bijection property for the map pair toNat, fromNat.
- Subtraction -. on Bin, with the needed properties proved.
- The proofs for isomorphism for +, *, -. for toNat, fromNat.
- The monotonicity proofs for toNat, fromNat for <= and Nat.<=. A similar monotonicity for < and Nat.<= are proved.
- Various kinds of monotonicity for +, *, -. for <= and <
are proved. - The CommutativeSemiring instance for Bin.
- Division with remainder and GCD for Bin.
- The demonstration/test programs for divMod and gcd.
`Certified' means that everything is proved in Agda which is regularly required of the above operations.
Bin0.agda copies the following items from Data.Bin of Standard library:
Bin⁺, data Bin, 2^, toBits, ⌊log₂⌋, 2,
*2+1, ⌊/2⌋, Carry, addBits, addCarry, +,
_, fromBits, addBitList,
The last three have a slight change in implementation.
Nothing else about Bin is taken from Standard library.
The definition for inequalities on Bin is totally changed in Bin0, Bin1.
More detailed explanations are given in README.agda.
Binary-3.2 has been tested under Agda 2.5.4, MAlozo, ghc-7.10.2
It also includes the module LtReasoning.agda -- a support for inequality reasoning by Ulf Norell.
Installation:
agda -c $agdaLibOpt GCDTest.agda
Comments and improvements are welcome.
Sergei D. Meshveliani mechvel@botik.ru