# bit-stream
Lazy, infinite, compact stream of `Bool` with O(1) indexing.
Most useful for memoization of predicates.
## Example 1
Consider following predicate:
```haskell
isOdd :: Word -> Bool
isOdd 0 = False
isOdd n = not (isOdd (n - 1))
```
Its computation is expensive, so we'd like to memoize its values into
`BitStream` using `tabulate` and access this stream via `index`
instead of recalculation of `isOdd`:
```haskell
isOddBS :: BitStream
isOddBS = tabulate isOdd
isOdd' :: Word -> Bool
isOdd' = index isOddBS
```
We can do even better by replacing part of recursive calls to `isOdd`
by indexing memoized values. Write `isOddF`
such that `isOdd = fix isOddF`:
```haskell
isOddF :: (Word -> Bool) -> Word -> Bool
isOddF _ 0 = False
isOddF f n = not (f (n - 1))
```
and use `tabulateFix`:
```haskell
isOddBS :: BitStream
isOddBS = tabulateFix isOddF
isOdd' :: Word -> Bool
isOdd' = index isOddBS
```
## Example 2
Define a predicate, which checks whether its argument is
a prime number by trial division.
```haskell
isPrime :: Word -> Bool
isPrime n
| n < 2 = False
| n < 4 = True
| even n = False
| otherwise = and [ n `rem` d /= 0 | d <- [3, 5 .. ceiling (sqrt (fromIntegral n))], isPrime d]
```
Convert it to unfixed form:
```haskell
isPrimeF :: (Word -> Bool) -> Word -> Bool
isPrimeF f n
| n < 2 = False
| n < 4 = True
| even n = False
| otherwise = and [ n `rem` d /= 0 | d <- [3, 5 .. ceiling (sqrt (fromIntegral n))], f d]
```
Create its memoized version for faster evaluation:
```haskell
isPrimeBS :: BitStream
isPrimeBS = tabulateFix isPrimeF
isPrime' :: Word -> Bool
isPrime' = index isPrimeBS
```