+#Writing recursive functions on version 1 style lists#
+
+Recall that version 1 style lists are constructed like this (see
+[[lists and numbers]]):
+
+ ; booleans
+ let true = \x y. x in
+ let false = \x y. y in
+ let and = \l r. l (r true false) false in
+
+ let make_pair = \f s g. g f s in
+ let get_fst = true in
+ let get_snd = false in
+ let empty = make_pair true junk in
+ let isempty = \x. x get_fst in
+ let make_list = \h t. make_pair false (make_pair h t) in
+ let head = \l. isempty l err (l get_snd get_fst) in
+ let tail = \l. isempty l err (l get_snd get_snd) in
+
+ ; a list of numbers to experiment on
+ let mylist = make_list 1 (make_list 2 (make_list 3 empty)) in
+
+ ; church numerals
+ let iszero = \n. n (\x. false) true in
+ let succ = \n s z. s (n s z) in
+ let add = \l r. l succ r in
+ let mul = \m n s. m (n s) in
+ let pred = (\shift n. n shift (make\_pair 0 0) get\_snd) (\p. p (\x y. make\_pair (succ x) x)) in
+ let leq = \m n. iszero(n pred m) in
+ let eq = \m n. and (leq m n)(leq n m) in
+
+ ; a fixed-point combinator for defining recursive functions
+ let Y = \f. (\h. f (h h)) (\h. f (h h)) in
+ let length = Y (\length l. isempty l 0 (succ (length (tail l)))) in
+ let fold = Y (\f l g z. isempty l z (g (head l)(f (tail l) g z))) in
+
+ eq 2 2 yes no
+
+
+Then `length mylist` evaluates to 3.
+
+1. What does `head (tail (tail mylist))` evaluate to?
+
+2. Using the `length` function as a model, and using the predecessor
+function, write a function that computes factorials. (Recall that n!,
+the factorial of n, is n times the factorial of n-1.)
+
+ Warning: it takes a long time for my browser to compute factorials larger than 4!
+
+3. (Easy) Write a function `equal_length` that returns true just in case
+two lists have the same length. That is,
+
+ equal_length mylist (make_list junk (make_list junk (make_list junk empty))) ~~> true