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-#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]]
-
-
-- Write an implementation of leaf-labeled trees. You can do something v3-like, or use the Y combinator, as you prefer.
-
-You'll need an operation `make_leaf` that turns a label into a new leaf. You'll
-need an operation `make_node` that takes two subtrees (perhaps leaves, perhaps
-other nodes) and joins them into a new tree. You'll need an operation `isleaf`
-that tells you whether a given tree is a leaf. And an operation `extract_label`
-that tells you what value is associated with a given leaf. And an operation
-`extract_left` that tells you what the left subtree is of a tree that isn't a
-leaf. (Presumably, `extract_right` will work similarly.)
-
-
- The **fringe** of a leaf-labeled tree is the list of values at its leaves,
-ordered from left to right. For example, the fringe of this tree:
-
- .
- / \
- . 3
- / \
- 1 2
-
-is `[1;2;3]`. And that is also the fringe of this tree:
-
- .
- / \
- 1 .
- / \
- 2 3
-
-The two trees are different, but they have the same fringe. We're going to
-return later in the term to the problem of determining when two trees have the
-same fringe. For now, one straightforward way to determine this would be:
-enumerate the fringe of the first tree. That gives you a list. Enumerate the
-fringe of the second tree. That also gives you a list. Then compare the two
-lists to see if they're equal. (You just programmed this above.)
-
-Write the fringe-enumeration function. It should work on the implementation of
-trees you designed in the previous step.
-
-
-
-
-#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 [Week3
-notes](/week3/#index4h2) as a model, construct a pair `Y1` and `Y2` that behave
-in the way described.
-
-(See [[hints/Assignment 4 hint 3]] if you need some hints.)
-
-
-