Here are the definitions pre-loaded for working on assignment 3:
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+; booleans
let true = \x y. x in
let false = \x y. y in
let and = \l r. l (r true false) false in
-let makePair = \f s g. g f s in
-let fst = true in
-let snd = false in
-let nil = makePair true meh in
-let isNil = \x. x fst in
-let makeList = \h t. makePair false (makePair h t) in
-let head = \l. isNil l err (l snd fst) in
-let tail = \l. isNil l err (l snd snd) in
-let mylist = makeList 1 (makeList 2 (makeList 3 nil)) in
-let Y = \f. (\h. f (h h)) (\h. f (h h)) in
-let isZero = \n. n (\x. false) true in
+let or = \l r. l true r 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 mult = \m n s. m (n s) in
-let length = Y (\length l. isNil l 0 (succ (length (tail l)))) in
-let pred = \n. isZero n 0 (length (tail (n (\p. makeList meh p) nil))) in
-let leq = \m n. isZero(n pred m) 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
;
-length (tail mylist)
+; 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
+;
+; synonyms
+let makePair = make\_pair in
+let fst = get\_fst in
+let snd = get\_snd in
+let nil = empty in
+let isNil = isempty in
+let makeList = make\_list in
+let isZero = iszero in
+let mult = mul in
+;
+let t1 = (make\_list 1 empty) in
+let t2 = (make\_list 2 empty) in
+let t3 = (make\_list 3 empty) in
+let t12 = (make\_list t1 (make\_list t2 empty)) in
+let t23 = (make\_list t2 (make\_list t3 empty)) in
+let ta = (make\_list t1 t23) in
+let tb = (make\_list t12 (make\_list t3 empty)) in
+let tc = (make\_list t1 (make\_list t23 empty)) in
+;
+;sum-leaves t1 ; ~~> 1
+;sum-leaves t2 ; ~~> 2
+;sum-leaves t3 ; ~~> 3
+;sum-leaves t12 ; ~~> 3
+;sum-leaves t23 ; ~~> 5
+;sum-leaves ta ; ~~> 6
+;sum-leaves tb ; ~~> 6
+;sum-leaves tc ; ~~> 6
+;
+; updated: added add, and fold for v1 lists; and defn of tb fixed
+; hint:
+fold mylist add 0
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+
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