Here are a bunch of pre-tested operations for the untyped lambda calculus. In some cases multiple versions are offered. ; booleans let true = \y n. y in ; aka K let false = \y n. n in ; aka K I let and = \p q. p q false in ; or let and = \p q. p q p in ; aka S C I let or = \p q. p true q in ; or let or = \p q. p p q in ; aka M let not = \p. p false true in ; or let not = \p y n. p n y in ; aka C let xor = \p q. p (not q) q in let iff = \p q. not (xor p q) in ; or let iff = \p q. p q (not q) in ; pairs let make_pair = \x y f. f x y in let get_1st = \x y. x in ; aka true let get_2nd = \x y. y in ; aka false ; triples let make_triple = \x y z f. f x y z in ; version 3 lists let empty = \f z. z in let make_list = \h t f z. f h (t f z) in let isempty = \lst. lst (\h sofar. false) true in let head = \lst. lst (\h sofar. h) junk in let tail = \lst. (\shift lst. lst shift (make_pair empty junk) get_2nd) ; where shift is (\h p. p (\t y. make_pair (make-list h t) t)) in let length = \lst. lst (\h sofar. succ sofar) zero in let map = \f lst. lst (\h sofar. make_list (f h) sofar) empty in let filter = \f lst. lst (\h sofar. f h (make_list h sofar) sofar) empty in ; or let filter = \f lst. lst (\h. f h (make_list h) I) empty in ; version 1 lists let empty = make_pair true junk in let make_list = \h t. make_pair false (make_pair h t) in let isempty = \lst. lst get_1st in let head = \lst. isempty lst error (lst get_2nd get_1st) in let tail_empty = empty in let tail = \lst. isempty lst tail_empty (lst get_2nd get_2nd) in ; Church numerals: easy operations let zero = \s z. z in ; aka false let one = \s z. s z in ; aka I let succ = \n s z. s (n s z) in ; for any Church numeral n > zero : n (K y) z ~~> y let iszero = \n. n (\x. false) true in let add = \m n. m succ n in ; or let add = \m n s z. m s (n s z) in let mul = \m n. m (\z. add n z) zero in ; or let mul = \m n s. m (n s) in let pow = \b exp. exp (mul b) one in ; or ; b succ : adds b ; b (b succ) ; adds b b times, ie adds b^2 ; b (b (b succ)) ; adds b^2 b times, ie adds b^3 ; exp b succ ; adds b^exp let pow = \b exp s z. exp b s z in ; three strategies for predecessor let pred_zero = zero in let pred = (\shift n. n shift (make_pair zero pred_zero) get_2nd) ; where shift is (\p. p (\x y. make_pair (succ x) x)) in ; or ; from Oleg; observe that for any Church numeral n: n I ~~> I let pred = \n. iszero n zero ; else (n (\x. x I ; when x is the base term, this will be K zero ; when x is a Church numeral, it will be I (succ x)) ; base term (K (K zero)) ) in ; from Bunder/Urbanek let pred = \n s z. n (\u v. v (u s)) (K z) I in ; or ; inefficient but simple comparisons let leq = \m n. iszero (n pred m) in let lt = \m n. not (leq n m) in let eq = \m n. and (leq m n) (leq n m) in ; or ; more efficient comparisons, Oleg's gt provided some simplifications let leq = (\base build consume. \m n. n consume (m build base) get_1st) ; where base is (make_pair true junk) ; and build is (\p. make_pair false p) ; and consume is (\p. p get_1st p (p get_2nd)) in let lt = \m n. not (leq n m) in let eq = (\base build consume. \m n. n consume (m build base) get_1st) ; 2nd element of a pair will now be of the form (K sthg) or I ; we supply the pair being consumed itself as an argument ; getting back either sthg or the pair we just consumed ; base is (make_pair true (K (make_pair false I))) ; and build is (\p. make_pair false (K p)) ; and consume is (\p. p get_2nd p) in ; -n is a fixedpoint of \x. add (add n x) x ; but unfortunately Y that_function doesn't normalize ; instead: let sub = \m n. n pred m in ; or ; how many times we can succ n until m <= result let sub = \m n. (\base build. m build base (\cur fin sofar. sofar)) ; where base is (make_triple n false zero) ; and build is (\t. t (\cur fin sofar. or fin (leq m cur) (make_triple cur true sofar) ; enough (make_triple (succ cur) false (succ sofar)) ; continue )) in ; or let sub = (\base build consume. \m n. n consume (m build base) get_1st) ; where base is (make_pair zero I) ; see second defn of eq for explanation of 2nd element ; and build is (\p. p (\x y. make_pair (succ x) (K p))) ; and consume is (\p. p get_2nd p) in let min = \m n. sub m (sub m n) in let max = \m n. add n (sub m n) in ; (m/n) is a fixedpoint of \x. add (sub (mul n x) m) x ; but unfortunately Y that_function doesn't normalize ; instead: ; how many times we can sub n from m while n <= result let div = \m n. (\base build. m build base (\cur go sofar. sofar)) ; where base is (make_triple m true zero) ; and build is (\t. t (\cur go sofar. and go (leq n cur) (make_triple (sub cur n) true (succ sofar)) ; continue (make_triple cur false sofar) ; enough )) in ; what's left after sub n from m while n <= result let mod = \m n. (\base build. m build base (\cur go. cur)) ; where base is (make_pair m true) ; and build is (\p. p (\cur go. and go (leq n cur) (make_pair (sub cur n) true) ; continue (make_pair cur false) ; enough )) in ; or let divmod = (\base build mtail. \m n. (\dhead. m (mtail dhead) (\sel. dhead (sel 0 0))) (n build base (\x y z. z junk)) (\t u x y z. make_pair t u) ) ; where base is (make_triple succ (K 0) I) ; see second defn of eq for explanation of 3rd element ; and build is (\t. make_triple I succ (K t)) ; and mtail is (\dhead d. d (\dz mz df mf drest sel. drest dhead (sel (df dz) (mf mz)))) in let div = \n d. divmod n d get_1st in let mod = \n d. divmod n d get_2nd in ; sqrt n is a fixedpoint of \x. div (div (add n (mul x x)) 2) x ; but unfortunately Y that_function doesn't normalize ; (log base b of m) is a fixedpoint of \x. add (sub (pow b x) m) x ; but unfortunately Y that_function doesn't normalize ; instead: ; how many times we can mul b by b while result <= m let log = \m b. (\base build. m build base (\cur go sofar. sofar)) ; where base is (make_triple b true 0) ; and build is (\t. t (\cur go sofar. and go (leq cur m) (make_triple (mul cur b) true (succ sofar)) ; continue (make_triple cur false sofar) ; enough )) in ; Curry's fixed point combinator let Y = \f. (\h. f (h h)) (\h. f (h h)) in ; Turing's fixed point combinator let Z = (\u f. f (u u f)) (\u f. f (u u f)) in ; length for version 1 lists let length = Y (\self lst. isempty lst 0 (succ (self (tail lst)))) in ; numhelper 0 f z ~~> z ; when n > 0: numhelper n f z ~~> f (pred n) ; compare Bunder/Urbanek pred let numhelper = \n. n (\u v. v (u succ)) (K 0) (\p f z. f p) in ; accepts fixed point combinator as a parameter, so you can use different ones let fact = \y. y (\self n. numhelper n (\p. mul n (self p)) 1) in fact Z 3 ; returns 6