Due by 11:59pm on Monday, 7/27

## Instructions

Download hw08.zip. Inside the archive, you will find a file called hw08.py, along with a copy of the OK autograder.

Submission: When you are done, submit with ```python3 ok --submit```. You may submit more than once before the deadline; only the final submission will be scored. See Lab 1 for instructions on submitting assignments.

Using OK: If you have any questions about using OK, please refer to this guide.

Readings: You might find the following references useful:

## Required questions

### Question 1

Implement `__contains__` for the `Link` class, which allows us to use the `in` operator to check if a `value` is contained in a linked list.

``````class Link:
empty = ()

def __init__(self, first, rest=empty):
self.first = first
self.rest = rest

def __contains__(self, value):

Use OK to test your code:

``python3 ok -q contains``

### Generating natural numbers

The following questions use the `naturals` generator function, which yields an infinite sequence of integers starting at 1.

``````def naturals():
"""A generator function that yields the infinite sequence of natural
numbers, starting at 1.

>>> m = naturals()
>>> type(m)
<class 'generator'>
>>> [next(m) for _ in range(10)]
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
"""
i = 1
while True:
yield i
i += 1``````

### Question 2

Implement an iterator class called `ScaleIterator` that scales elements in an iterable `s` by a number `k`.

``````class ScaleIterator:
"""An iterator the scales elements of the iterable s by a number k.

>>> s = ScaleIterator([1, 5, 2], 5)
>>> list(s)
[5, 25, 10]

>>> m = ScaleIterator(naturals(), 2)
>>> [next(m) for _ in range(5)]
[2, 4, 6, 8, 10]
"""
def __init__(self, s, k):

def __iter__(self):
return self

def __next__(self):

Use OK to test your code:

``python3 ok -q ScaleIterator``

### Question 3

Implement the generator function `scale(s, k)`, which yields elements of the given iterable `s`, scaled by `k`.

``````def scale(s, k):
"""Yield elements of the iterable s scaled by a number k.

>>> s = scale([1, 5, 2], 5)
>>> type(s)
<class 'generator'>
>>> list(s)
[5, 25, 10]

>>> m = scale(naturals(), 2)
>>> [next(m) for _ in range(5)]
[2, 4, 6, 8, 10]
"""

Use OK to test your code:

``python3 ok -q scale``

### Question 4

Implement `merge(s1, s2)`, which takes two iterables `s1` and `s2` whose elements are ordered. `merge` yields elements from `s1` and `s2` in sorted order, elimnating repetition. You may assume `s0` and `s1` themselves do not contain repeats. You may also assume `s0` and `s1` represent infinite sequences; that is, their iterators never raise `StopIteration`.

See the doctests for example behavior.

``````def merge(s0, s1):
"""Yield the elements of strictly increasing iterables s0 and s1, removing
repeats. Assume that s0 and s1 have no repeats. You can also assume that s0
and s1 represent infinite sequences.

>>> twos = scale(naturals(), 2)
>>> threes = scale(naturals(), 3)
>>> m = merge(twos, threes)
>>> type(m)
<class 'generator'>
>>> [next(m) for _ in range(10)]
[2, 3, 4, 6, 8, 9, 10, 12, 14, 15]
"""
i0, i1 = iter(s0), iter(s1)
e0, e1 = next(i0), next(i1)

Use OK to test your code:

``python3 ok -q merge``

### Question 5

A famous problem, first raised by Richard Hamming, is to enumerate, in ascending order with no repetitions, all positive integers with no prime factors other than 2, 3, or 5. These are called regular numbers. One obvious way to do this is to simply test each integer in turn to see whether it has any factors other than 2, 3, and 5. But this is very inefficient, since, as the integers get larger, fewer and fewer of them fit the requirement.

As an alternative, we can write a generator function of such numbers. Let us call the sequence of numbers `s` and notice the following facts about it.

• `s` begins with `1`.
• The elements of `scale(s, 2)` are also elements of `s`.
• The same is true for `scale(s, 3)` and `scale(s, 5)`.
• These are all of the elements of `s`.

Now all we have to do is combine elements from these sources. Use the `merge` function you defined previously to fill in the definition of `make_s`:

``````def make_s():
"""A generator function that yields all positive integers with only factors
2, 3, and 5.

>>> s = make_s()
>>> type(s)
<class 'generator'>
>>> [next(s) for _ in range(20)]
[1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36]
"""
``python3 ok -q make_s``