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3. Using either Kapulet's or Haskell's list comprehension syntax, write an expression that transforms `[3, 1, 0, 2]` into `[3, 3, 3, 1, 2, 2]`. [[Here is a hint|assignment3 hint1]], if you need it.
-4. Last week you defined `head` in the Lambda Calculus, using our proposed encoding for lists. Now define `empty?` (It should require only small modifications to your solution for `head`.)
+4. Last week you defined `head` in terms of `fold_right`. Your solution should be straightforwardly translatable into one that uses our proposed right-fold encoding of lists in the Lambda Calculus. Now define `empty?` (It should require only small modifications to your solution for `head`.)
5. If we encode lists in terms of their *left*-folds, instead, `[a, b, c]` would be encoded as `\f z. f (f (f z a) b) c`. The empty list `[]` would still be encoded as `\f z. z`. What should `cons` be, for this encoding?
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10. In last week's homework, you gave a Lambda Calculus definition of `succ` for Church-encoded numbers. Can you now define `pred`? Let `pred 0` result in whatever `err` is bound to. This is challenging. For some time theorists weren't sure it could be done. (Here is [some interesting commentary](http://okmij.org/ftp/Computation/lambda-calc.html#predecessor).) However, in this week's notes we examined one strategy for defining `tail` for our chosen encodings of lists, and given the similarities we explained between lists and numbers, perhaps that will give you some guidance in defining `pred` for numbers.
+ (Want a further challenge? Define `map2` in the Lambda Calculus, using our right-fold encoding for lists, where `map2 g [a, b, c] [d, e, f]` should evaluate to `[g a d, g b e, g c f]`. Doing this will require drawing on a number of different tools we've developed, including that same strategy for defining `tail`. Purely extra credit.)
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11. Define `leq` for numbers (that is, ≤) in the Lambda Calculus. Here is the expected behavior,
where `one` abbreviates `succ zero`, and `two` abbreviates `succ (succ zero)`.