`λ`

s into the following abbreviated expressions:
1. `x x (x x x) x`
2. `v w (\x y. v x)`
3. `(\x y. x) u v`
4. `w (\x y z. x z (y z)) u v`
Mark all occurrences of `x y` in the following terms:
- `(\x y. x y) x y`
- `(\x y. x y) (x y)`
- `\x y. x y (x y)`

- `(\x. x (\y. y x)) (v w)`
- `(\x. x (\x. y x)) (v w)`
- `(\x. x (\y. y x)) (v x)`
- `(\x. x (\y. y x)) (v y)`
- `(\x y. x y y) u v`
- `(\x y. y x) (u v) z w`
- `(\x y. x) (\u u)`
- `(\x y z. x z (y z)) (\u v. u)`

- `Kxy`
- `KKxy`
- `KKKxy`
- `SKKxy`
- `SIII`
- `SII(SII)`
- Give Combinatory Logic combinators that behave like our boolean functions. You'll need combinators for `true`, `false`, `neg`, `and`, `or`, and `xor`.

- `\x.x`
- `\xy.x`
- `\xy.y`
- `\xy.yx`
- `\x.xx`
- `\xyz.x(yz)`
- For each translation, how many I's are there? Give a rule for describing what each I corresponds to in the original lambda term.

```
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)))))
- What would be the result of evaluating (see [[hints/Assignment 2 hint]] for a hint): LIST make-list empty
- Based on your answer to question 16, 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)))))
- Based on your answer to question 16, 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`.
- What goes wrong when we try to apply these techniques using the version 1 or version 2 implementation of lists?
- 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 essential to understand how the implementation we gave above works. You can treat it as a black box.