Documentation

Natural numbers with an infinity element

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

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.

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 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 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 .. ] : [_][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.