cryptol-2.4.0: Cryptol: The Language of Cryptography

Cryptol.TypeCheck.Solver.InfNat

Description

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

Synopsis

# Documentation

data Nat' Source #

Natural numbers with an infinity element

Constructors

 Nat Integer Inf

Instances

 Source # Methods(==) :: Nat' -> Nat' -> Bool #(/=) :: Nat' -> Nat' -> Bool # Source # Methodscompare :: Nat' -> Nat' -> Ordering #(<) :: Nat' -> Nat' -> Bool #(<=) :: Nat' -> Nat' -> Bool #(>) :: Nat' -> Nat' -> Bool #(>=) :: Nat' -> Nat' -> Bool #max :: Nat' -> Nat' -> Nat' #min :: Nat' -> Nat' -> Nat' # Source # MethodsshowsPrec :: Int -> Nat' -> ShowS #show :: Nat' -> String #showList :: [Nat'] -> ShowS # Source # Associated Typestype Rep Nat' :: * -> * # Methodsfrom :: Nat' -> Rep Nat' x #to :: Rep Nat' x -> Nat' # Source # Methodsrnf :: Nat' -> () # type Rep Nat' Source # 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))

nMul :: Nat' -> Nat' -> Nat' Source #

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' Source #

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

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

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.

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

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

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 .. ] : [_] is nLenFromThen 0 1 0, which should evaluate to 1.

length [ x, y .. z ]

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.

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

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

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.