Reduction
---------

Find "normal forms" for the following---that is, reduce them until no more reductions are possible. As mentioned in the notes, we'll write <code>&lambda;x</code> as `\x`. If we ever say "reduce" without qualifications, we mean just "beta-reduce" (as opposed to "(beta + eta)-reduce").

1. `(\x \y. y x) z`
2. `(\x (x x)) z`
3. `(\x (\x x)) z`
4. `(\x (\z x)) z`
5. `(\x (x (\y y))) (\z (z z))`
6. `(\x (x x)) (\x (x x))`
7. `(\x (x x x)) (\x (x x x))`


Booleans
--------

Recall our definitions of true and false.

>   **true** is defined to be `\t f. t`  
>   **false** is defined to be `\t f. f`

In Racket, these functions can be defined like this:

    (define true (lambda (t) (lambda (f) t)))
    (define false (lambda (t) (lambda (f) f)))

(Note that they are different from Racket's *primitive* boolean values `#t` and `#f`.)

<small>(I know the numbering of the homework problems will restart instead of continuing as before. It's too much of a pain to fix it right now. We'll put in a better rendering engine later that will make this work right without laborious work-arounds on our part.)</small>

8.  Define a `neg` operator that negates `true` and `false`.

    Expected behavior: 

        (((neg true) 10) 20)

    evaluates to 20, and 

        (((neg false) 10) 20)

    evaluates to 10.

9.  Define an `or` operator.

10. Define an `xor` operator. If you haven't seen this term before, here's a truth table:

        true xor true == false
        true xor false == true
        false xor true == true
        false xor false == false



Triples
-------

Recall our definitions of ordered triples.

>   the triple **(**a**,**b**, **c**)** is defined to be `\f. f a b c`

To extract the first element of a triple t, you write:

    t (\fst snd trd. fst)

Here are some definitions in Racket:

    (define make-triple (lambda (fst) (lambda (snd) (lambda (trd) (lambda (f) (((f fst) snd) trd))))))
    (define fst_of_three (lambda (fst) (lambda (snd) (lambda (trd) fst))))
    (define snd_of_three (lambda (fst) (lambda (snd) (lambda (trd) snd))))

Now we can write:

    (define t (((make-triple 10) 20) 30))
    (t fst_of_three)  ; will evaluate to 10
    (t snd_of_three)  ; will evaluate to 20

If you're puzzled by having the triple to the left and the function that
operates on it come second, think about why it's being done this way: the pair
is a package that takes a function for operating on its elements *as an
argument*, and returns *the result of* operating on its elements with that
function. In other words, the triple is a higher-order function.


11. Define the `swap12` function that permutes the elements of a triple. Expected behavior:

        (define t (((make-triple 10) 20) 30))
        ((t swap12) fst_of_three) ; evaluates to 20
        ((t swap12) snd_of_three) ; evaluates to 10

    Write out the definition of `swap12` in Racket.


12. Define a `dup3` function that duplicates its argument to form a triple
whose elements are the same.  Expected behavior:

        ((dup3 10) fst_of_three) ; evaluates to 10
        ((dup3 10) snd_of_three) ; evaluates to 10

13. Define a `dup27` function that makes
twenty-seven copies of its argument (and stores them in a data structure of
your choice).