Download the skeleton files `LazyListPlus.elm`

and `Deque.elm`

, and use them as a starting point for the following problems. Look for all occurrences of `TODO`

in comments, which point out where you should implement your solutions. Once you are done, follow the submission instructions below.

In this problem, you will implement `LazyList`

analogues of a couple more standard `List`

functions. You will need to download `LazyList.elm`

and `elm-package.json`

and place them in the directory where you are working.

First, implement the following functions.

```
map : (a -> b) -> LazyList a -> LazyList b
concat : LazyList (LazyList a) -> LazyList a
```

Your implementations should be as lazy as possible. In particular, for any `f`

and `xs`

, the expression `map f xs`

should evaluate very quickly even if, for example, `head (map f xs)`

does not.

Next, use `map`

and `concat`

to define the following.

`concatMap : (a -> LazyList b) -> LazyList a -> LazyList b`

Finally, implement the following function that computes the Cartesian product of two `LazyList`

s and transforms each its pairs using the input function.

`cartProdWith : (a -> b -> c) -> LazyList a -> LazyList b -> LazyList c`

A *double-ended queue*, or *deque* (pronounced “deck”), allows adding and removing elements from both ends of the data structure. In this problem, you will adapt the `FastQueue`

strategy from Chapter 5 to implement the `Deque`

abstraction. In particular, use the representation

`type Deque a = D { front : List a, back : List a }`

and maintain the invariant that `front`

and `back`

are both non-empty whenever there are at least two elements in the `Deque`

.

Implement the following three operations (which we referred to in the `Queue`

context as `enqueue`

, `dequeue`

, and `peek`

, respectively):

```
addBack : a -> Deque a -> Deque a
removeFront : Deque a -> Maybe (Deque a)
peekFront : Deque a -> Maybe a
```

In addition, implement the following three analogous operators:

```
addFront : a -> Deque a -> Deque a
removeBack : Deque a -> Maybe (Deque a)
peekBack : Deque a -> Maybe a
```

Implement and use a helper function

`check : List a -> List a -> Deque a`

that enforces the invariant by checking whether either list is empty and, if so, splitting the other in half and reversing one of the halves.

All operations should run in *O(1)* amortized time assuming, as in Chapter 5, that values are not used persistently (but you are not asked to prove it).

**Hint**: The frontmost and backmost element of a `Deque`

may be the same.

Start by navigating to the folder where you checked out your repo. Next, create a subfolder for this assignment and populate it with the skeleton code:

```
% svn mkdir hw6
% cd hw6
% wget http://www.classes.cs.uchicago.edu/current/22300-1/assignments/hw6/Deque.elm
% wget http://www.classes.cs.uchicago.edu/current/22300-1/assignments/hw6/LazyListPlus.elm
% wget http://www.classes.cs.uchicago.edu/current/22300-1/assignments/hw6/elm-package.json
% wget http://www.classes.cs.uchicago.edu/current/22300-1/lectures/LazyLists/LazyList.elm
```

If `wget`

or a similar tool (such as `curl`

) is not available on your machine, download and save the skeleton files provided above in some other way. Then add *only* these files to your repo:

```
% svn add LazyListPlus.elm
% svn add Deque.elm
```

Make sure you choose the *same exact* names for directories and files that you create. Once you are ready to submit:

`% svn commit -m "hw6 submission"`

You can resubmit as many times as you wish, and we will grade the most recent versions submitted. Late days, if any, will be computed based on your submission times.

As a sanity check, you can visit the Web-based frontend for your repository to make sure that you have submitted your files as intended:

`https://phoenixforge.cs.uchicago.edu/projects/USER-cs223-spr-17/repository`