#Reversing a list#
- How would you define an operation to reverse a list? (Don't peek at the
[[lambda_library]]! Try to figure it out on your own.) Choose whichever
implementation of list you like. Even then, there are various strategies you
can use.
(See [[hints/Assignment 4 hint 1]] if you need some hints.)

#Comparing lists for equality#
- Suppose you have two lists of integers, `left` and `right`. You want to determine whether those lists are equal: that is, whether they have all the same members in the same order. (Equality for the lists we're working with is *extensional*, or parasitic on the equality of their members, and the list structure. Later in the course we'll see lists which aren't extensional in this way.)
How would you implement such a list comparison?
(See [[hints/Assignment 4 hint 2]] if you need some hints.)

#Enumerating the fringe of a leaf-labeled tree#
First, read this: [[Implementing trees]]
- blah
(See [[hints/Assignment 4 hint 3]] if you need some hints.)

#Mutually-recursive functions#
- (Challenging.) One way to define the function `even` is to have it hand off part of the work to another function `odd`:
let even = \x. iszero x
; if x == 0 then result is
true
; else result turns on whether x's pred is odd
(odd (pred x))
At the same tme, though, it's natural to define `odd` in such a way that it hands off part of the work to `even`:
let odd = \x. iszero x
; if x == 0 then result is
false
; else result turns on whether x's pred is even
(even (pred x))
Such a definition of `even` and `odd` is called **mutually recursive**. If you trace through the evaluation of some sample numerical arguments, you can see that eventually we'll always reach a base step. So the recursion should be perfectly well-grounded:
even 3
~~> iszero 3 true (odd (pred 3))
~~> odd 2
~~> iszero 2 false (even (pred 2))
~~> even 1
~~> iszero 1 true (odd (pred 1))
~~> odd 0
~~> iszero 0 false (even (pred 0))
~~> false
But we don't yet know how to implement this kind of recursion in the lambda calculus.
The fixed point operators we've been working with so far worked like this:
let X = Y T in
X <~~> T X
Suppose we had a pair of fixed point operators, `Y1` and `Y2`, that operated on a *pair* of functions `T1` and `T2`, as follows:
let X1 = Y1 T1 T2 in
let X2 = Y2 T1 T2 in
X1 <~~> T1 X1 X2 and
X2 <~~> T2 X1 X2
If we gave you such a `Y1` and `Y2`, how would you implement the above definitions of `even` and `odd`?
- (More challenging.) Using our derivation of Y from the [[Week2]] notes as a model, construct a pair `Y1` and `Y2` that behave in the way described.
(See [[hints/Assignment 4 hint 4]] if you need some hints.)