Copyright | (c) 2013-2016 Galois Inc. |
---|---|

License | BSD3 |

Maintainer | cryptol@galois.com |

Stability | provisional |

Portability | portable |

Safe Haskell | Safe |

Language | Haskell2010 |

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

## Synopsis

- data Nat'
- fromNat :: Nat' -> Maybe Integer
- nAdd :: Nat' -> Nat' -> Nat'
- nMul :: Nat' -> Nat' -> Nat'
- nExp :: Nat' -> Nat' -> Nat'
- nMin :: Nat' -> Nat' -> Nat'
- nMax :: Nat' -> Nat' -> Nat'
- nSub :: Nat' -> Nat' -> Maybe Nat'
- nDiv :: Nat' -> Nat' -> Maybe Nat'
- nMod :: Nat' -> Nat' -> Maybe Nat'
- nCeilDiv :: Nat' -> Nat' -> Maybe Nat'
- nCeilMod :: Nat' -> Nat' -> Maybe Nat'
- nLg2 :: Nat' -> Nat'
- nWidth :: Nat' -> Nat'
- nLenFromThen :: Nat' -> Nat' -> Nat' -> Maybe Nat'
- nLenFromThenTo :: Nat' -> Nat' -> Nat' -> Maybe Nat'
- genLog :: Integer -> Integer -> Maybe (Integer, Bool)
- widthInteger :: Integer -> Integer
- rootExact :: Integer -> Integer -> Maybe Integer
- genRoot :: Integer -> Integer -> Maybe (Integer, Bool)

# Documentation

Natural numbers with an infinity element

## Instances

Eq Nat' Source # | |

Ord Nat' Source # | |

Show Nat' Source # | |

Generic Nat' Source # | |

NFData Nat' Source # | |

Defined in Cryptol.TypeCheck.Solver.InfNat | |

type Rep Nat' Source # | |

Defined in Cryptol.TypeCheck.Solver.InfNat type Rep Nat' = D1 (MetaData "Nat'" "Cryptol.TypeCheck.Solver.InfNat" "cryptol-2.6.0-24w5HMDd2znGLrodkM4xJM" False) (C1 (MetaCons "Nat" PrefixI False) (S1 (MetaSel (Nothing :: Maybe 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 :: Nat' -> Nat' -> Maybe Nat' Source #

`nSub x y = Just z`

iff `z`

is the unique value
such that `Add y z = Just x`

.

nCeilDiv :: Nat' -> Nat' -> Maybe Nat' Source #

`nCeilDiv msgLen blockSize`

computes the least `n`

such that
`msgLen <= blockSize * n`

. It is undefined when `blockSize = 0`

.
It is also undefined when either input is infinite; perhaps this
could be relaxed later.

nCeilMod :: Nat' -> Nat' -> Maybe Nat' Source #

`nCeilMod msgLen blockSize`

computes the least `k`

such that
`blockSize`

divides `msgLen + k`

. It is undefined when `blockSize = 0`

.
It is also undefined when either input is infinite; perhaps this
could be relaxed later.

Rounds up.
`lg2 x = y`

, iff `y`

is the smallest number such that `x <= 2 ^ y`

nWidth :: Nat' -> Nat' Source #

`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' Source #

`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.

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

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 Source #

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

rootExact :: Integer -> Integer -> Maybe Integer Source #

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

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

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.