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__Sign Up__- sum (take (s (s (s (s (s z))))) (cycle (take (s (s (s z))) (filter (leq (s z)) nats)))) where
- id = \x. x;
- force = \thunk. thunk _;
- fix = \f. (\x. f (\y. x x y)) (\x. f (\y. x x y));
- true = \p _. p;
- false = \_ q. q;
- or = \p q. p p q;
- if = id;
- z = \_ z. z;
- s = \n s z. s (n s z);
- eq0 = \n. n (\_. false) true;
- plus = \m n s z. m s (n s z);
- pred = \n s z. n (\g h. h (g s)) (\_. z) (\x. x);
- sub = \n m. m pred n;
- leq = \n m. eq0 (sub n m);
- nil = \ _ z. z;
- cons = \x xs f z. f (\_. x) (\f' _. xs f' z);
- consC = \x xs f z. f x (\f' _. xs f' z);
- tail = \xs f z. xs (\_ r. force (r f)) z;
- foldr = \f z xs. xs (fix (\rec f x r. f x (r (rec f))) f) z;
- defhead = foldr (\x _. force x);
- head = defhead _;
- null = foldr (\_ _. false) true;
- ifNull = \xs def f. if (null xs) def (\_. force (f xs));
- ifNullNil = \xs. ifNull xs nil;
- append = fix (\append xs1 xs2. ifNull xs1 xs2
- (\xs1. consC (\_. head xs1) (append (tail xs1) xs2)));
- map = fix (\map f xs. ifNullNil xs
- (\xs. consC (\_. f (head xs)) (map f (tail xs))));
- take = fix (\take n xs. if (eq0 n) nil (ifNullNil xs
- (\xs. consC (\_. head xs) (take (pred n) (tail xs)))));
- filter = fix (\filter p xs. ifNullNil xs
- (\xs. if (p x)
- (cons x (filter p (tail xs)))
- (filter p (tail xs))
- where x = head xs.));
- cycle = fix (\cycle xs. \_. force (append xs (cycle xs)));
- s1foldr = \f. foldr (\x acc. f x (force acc));
- s2foldr = \f. foldr (\x acc. f (force x) acc );
- sfoldr = \f. foldr (\x acc. f (force x) (force acc));
- nats = fix (\rec n. \_. force (cons n (rec (s n)))) z;
- sum = sfoldr plus z.
- result: \s z. s (s (s (s (s (s (s (s (s z))))))))
- equals to haskell "sum $ take 5 $ cycle $ take 3 $ filter (1 <=) [0..]" --> 1 + 2 + 3 + 1 + 2 = 9

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