```
zero ≡ \s z. z
succ ≡ \n. \s z. s (n s z)
iszero ≡ \n. n (\x. false) true
add ≡ \m \n. m succ n
mul ≡ \m \n. \s. m (n s)
```

And:
```
empty ≡ \f z. z
make-list ≡ \hd tl. \f z. f hd (tl f z)
isempty ≡ \lst. lst (\hd sofar. false) true
extract-head ≡ \lst. lst (\hd sofar. hd) junk
```

The `junk` in `extract-head` is what you get back if you evaluate:
extract-head empty
As we said, the predecessor and the extract-tail functions are harder to define. We'll just give you one implementation of these, so that you'll be able to test and evaluate lambda-expressions using them in Scheme or OCaml.
```
predecesor ≡ (\shift n. n shift (make-pair zero junk) get-second) (\pair. pair (\fst snd. make-pair (successor fst) fst))
extract-tail ≡ (\shift lst. lst shift (make-pair empty junk) get-second) (\hd pair. pair (\fst snd. make-pair (make-list hd fst) fst))
```

The `junk` is what you get back if you evaluate:
predecessor zero
extract-tail empty
Alternatively, we might reasonably declare the predecessor of zero to be zero (this is a common construal of the predecessor function in discrete math), and the tail of the empty list to be the empty list.
For these exercises, assume that `LIST` is the result of evaluating:
(make-list a (make-list b (make-list c (make-list d (make-list e empty)))))
1. What would be the result of evaluating:
LIST make-list empty
2. Based on your answer to question 1, how might you implement the **map** function? Expected behavior:
`map f LIST <~~> (make-list (f a) (make-list (f b) (make-list (f c) (make-list (f d) (make-list (f e) empty)))))`

3. Based on your answer to question 1, how might you implement the **filter** function? The expected behavior is that:
filter f LIST
should evaluate to a list containing just those of `a`, `b`, `c`, `d`, and `e` such that `f` applied to them evaluates to `true`.
4. How would you implement map using the either the version 1 or the version 2 implementation of lists?
5. Our version 3 implementation of the numbers are usually called "Church numerals". If `m` is a Church numeral, then `m s z` applies the function `s` to the result of applying `s` to ... to `z`, for a total of *m* applications of `s`, where *m* is the number that `m` represents or encodes.
Given the primitive arithmetic functions above, how would you implement the less-than-or-equal function? Here is the expected behavior, where `one` abbreviates `succ zero`, and `two` abbreviates `succ (succ zero)`.
less-than-or-equal zero zero ~~> true
less-than-or-equal zero one ~~> true
less-than-or-equal zero two ~~> true
less-than-or-equal one zero ~~> false
less-than-or-equal one one ~~> true
less-than-or-equal one two ~~> true
less-than-or-equal two zero ~~> false
less-than-or-equal two one ~~> false
less-than-or-equal two two ~~> true
You'll need to make use of the predecessor function, but it's not important to understand how the implementation we gave above works. You can treat it as a black box.