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-For these assignments, you'll probably want to use a "lambda calculator" to check your work. This accepts any grammatical lambda expression and reduces it to normal form, when possible. See the page on [[using the programming languages]] for instructions and links about setting this up.
-
-
-More Lambda Practice
---------------------
-
-Insert all the implicit `( )`s and 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:
-
-
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)))))
-
-
-