Cryptol.TypeCheck.Solver.InfNat

This module defines natural numbers with an additional infinity element, and various arithmetic operators on them.

Documentation

data Nat'

Natural numbers with an infinity element

Constructors

Nat Integer

Inf

Instances

Eq Nat'

Methods

(==) :: Nat' -> Nat' -> Bool

(/=) :: Nat' -> Nat' -> Bool

Ord Nat'

Methods

compare :: Nat' -> Nat' -> Ordering

(<) :: Nat' -> Nat' -> Bool

(<=) :: Nat' -> Nat' -> Bool

(>) :: Nat' -> Nat' -> Bool

(>=) :: Nat' -> Nat' -> Bool

max :: Nat' -> Nat' -> Nat'

min :: Nat' -> Nat' -> Nat'

Show Nat'

Methods

showsPrec :: Int -> Nat' -> ShowS

show :: Nat' -> String

showList :: [Nat'] -> ShowS

Generic Nat'

Associated Types

type Rep Nat' :: * -> *

Methods

from :: Nat' -> Rep Nat' x

to :: Rep Nat' x -> Nat'

NFData Nat'

Methods

rnf :: Nat' -> ()

type Rep Nat'

type Rep Nat' = D1 (MetaData "Nat'" "Cryptol.TypeCheck.Solver.InfNat" "cryptol-2.4.0-AtabUoGsZJn8kSvO8P84NP" False) ((:+:) (C1 (MetaCons "Nat" PrefixI False) (S1 (MetaSel (Nothing Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 Integer))) (C1 (MetaCons "Inf" PrefixI False) U1))

fromNat :: Nat' -> Maybe Integer

nEq :: Maybe Nat' -> Maybe Nat' -> Bool

nGt :: Maybe Nat' -> Maybe Nat' -> Bool

nFin :: Maybe Nat' -> Bool

nAdd :: Nat' -> Nat' -> Nat'

nMul :: Nat' -> Nat' -> Nat'

Some algebraic properties of interest:

1 * x = x
x * (y * z) = (x * y) * z
0 * x = 0
x * y = y * x
x * (a + b) = x * a + x * b

nExp :: Nat' -> Nat' -> Nat'

Some algebraic properties of interest:

x ^ 0        = 1
x ^ (n + 1)  = x * (x ^ n)
x ^ (m + n)  = (x ^ m) * (x ^ n)
x ^ (m * n)  = (x ^ m) ^ n

nMin :: Nat' -> Nat' -> Nat'

nMax :: Nat' -> Nat' -> Nat'

nSub :: Nat' -> Nat' -> Maybe Nat'

nSub x y = Just z iff z is the unique value such that Add y z = Just x.

nDiv :: Nat' -> Nat' -> Maybe Nat'

Rounds down.

y * q + r = x
x / y     = q with remainder r
0 <= r && r < y

We don't allow Inf in the first argument for two reasons: 1. It matches the behavior of nMod, 2. The well-formedness constraints can be expressed as a conjunction.

nMod :: Nat' -> Nat' -> Maybe Nat'

nLg2 :: Nat' -> Nat'

Rounds up. lg2 x = y, iff y is the smallest number such that x <= 2 ^ y

nWidth :: Nat' -> Nat'

nWidth n is number of bits needed to represent all numbers from 0 to n, inclusive. nWidth x = nLg2 (x + 1).

nLenFromThen :: Nat' -> Nat' -> Nat' -> Maybe Nat'

length ([ x, y .. ] : [_][w]) We don't check that the second element fits in w many bits as the second element may not be part of the list. For example, the length of [ 0 .. ] : [_][0] is nLenFromThen 0 1 0, which should evaluate to 1.

nLenFromThenTo :: Nat' -> Nat' -> Nat' -> Maybe Nat'

length [ x, y .. z ]

genLog :: Integer -> Integer -> Maybe (Integer, Bool)

Compute the logarithm of a number in the given base, rounded down to the closest integer. The boolean indicates if we the result is exact (i.e., True means no rounding happened, False means we rounded down). The logarithm base is the second argument.

widthInteger :: Integer -> Integer

Compute the number of bits required to represent the given integer.

rootExact :: Integer -> Integer -> Maybe Integer

Compute the exact root of a natural number. The second argument specifies which root we are computing.

genRoot :: Integer -> Integer -> Maybe (Integer, Bool)

Compute the the n-th root of a natural number, rounded down to the closest natural number. The boolean indicates if the result is exact (i.e., True means no rounding was done, False means rounded down). The second argument specifies which root we are computing.