# Church Numerals

Church Numerals
You are encouraged to solve this task according to the task description, using any language you may know.

In the Church encoding of natural numbers, the number N is encoded by a function that applies its first argument N times to its second argument.

• Church zero always returns the identity function, regardless of its first argument. In other words, the first argument is not applied to the second argument at all.
• Church one applies its first argument f just once to its second argument x, yielding f(x)
• Church two applies its first argument f twice to its second argument x, yielding f(f(x))
• and each successive Church numeral applies its first argument one additional time to its second argument, f(f(f(x))), f(f(f(f(x)))) ... The Church numeral 4, for example, returns a quadruple composition of the function supplied as its first argument.

Arithmetic operations on natural numbers can be similarly represented as functions on Church numerals.

• Church Zero,
• a Church successor function (a function on a Church numeral which returns the next Church numeral in the series),
• functions for Addition, Multiplication and Exponentiation over Church numerals,
• a function to convert integers to corresponding Church numerals,
• and a function to convert Church numerals to corresponding integers.

You should:

• Derive Church numerals three and four in terms of Church zero and a Church successor function.
• use Church numeral arithmetic to obtain the the sum and the product of Church 3 and Church 4,
• similarly obtain 4^3 and 3^4 in terms of Church numerals, using a Church numeral exponentiation function,
• convert each result back to an integer, and return it or print it to the console.

## AppleScript

Implementing churchFromInt as a fold seems to protect Applescript from overflowing its (famously shallow) stack with even quite low Church numerals.

on run    set cThree to churchFromInt(3)    set cFour to churchFromInt(4)     map(intFromChurch, ¬        {churchAdd(cThree, cFour), churchMult(cThree, cFour), ¬            churchExp(cFour, cThree), churchExp(cThree, cFour)})end run -- churchZero :: (a -> a) -> a -> aon churchZero(f, x)    xend churchZero -- churchSucc :: ((a -> a) -> a -> a) -> (a -> a) -> a -> aon churchSucc(n)    script        on |λ|(f)            script                property mf : mReturn(f)                on |λ|(x)                    mf's |λ|(mReturn(n)'s |λ|(mf)'s |λ|(x))                end |λ|            end script        end |λ|    end scriptend churchSucc -- churchFromInt(n) :: Int -> (b -> b) -> b -> bon churchFromInt(n)    script        on |λ|(f)            foldr(my compose, my |id|, replicate(n, f))        end |λ|    end scriptend churchFromInt -- intFromChurch :: ((Int -> Int) -> Int -> Int) -> Inton intFromChurch(cn)    mReturn(cn)'s |λ|(my succ)'s |λ|(0)end intFromChurch on churchAdd(m, n)    script        on |λ|(f)            script                property mf : mReturn(m)                property nf : mReturn(n)                on |λ|(x)                    nf's |λ|(f)'s |λ|(mf's |λ|(f)'s |λ|(x))                end |λ|            end script        end |λ|    end scriptend churchAdd on churchMult(m, n)    script        on |λ|(f)            script                property mf : mReturn(m)                property nf : mReturn(n)                on |λ|(x)                    mf's |λ|(nf's |λ|(f))'s |λ|(x)                end |λ|            end script        end |λ|    end scriptend churchMult on churchExp(m, n)    n's |λ|(m)end churchExp  -- GENERIC ----------------------------------------------------------- -- compose (<<<) :: (b -> c) -> (a -> b) -> a -> con compose(f, g)    script        property mf : mReturn(f)        property mg : mReturn(g)        on |λ|(x)            mf's |λ|(mg's |λ|(x))        end |λ|    end scriptend compose -- id :: a -> aon |id|(x)    xend |id| -- foldr :: (a -> b -> b) -> b -> [a] -> bon foldr(f, startValue, xs)    tell mReturn(f)        set v to startValue        set lng to length of xs        repeat with i from lng to 1 by -1            set v to |λ|(item i of xs, v, i, xs)        end repeat        return v    end tellend foldr -- map :: (a -> b) -> [a] -> [b]on map(f, xs)    tell mReturn(f)        set lng to length of xs        set lst to {}        repeat with i from 1 to lng            set end of lst to |λ|(item i of xs, i, xs)        end repeat        return lst    end tellend map -- Lift 2nd class handler function into 1st class script wrapper -- mReturn :: First-class m => (a -> b) -> m (a -> b)on mReturn(f)    if class of f is script then        f    else        script            property |λ| : f        end script    end ifend mReturn -- Egyptian multiplication - progressively doubling a list, appending-- stages of doubling to an accumulator where needed for binary -- assembly of a target length-- replicate :: Int -> a -> [a]on replicate(n, a)    set out to {}    if n < 1 then return out    set dbl to {a}     repeat while (n > 1)        if (n mod 2) > 0 then set out to out & dbl        set n to (n div 2)        set dbl to (dbl & dbl)    end repeat    return out & dblend replicate -- succ :: Int -> Inton succ(x)    1 + xend succ
Output:
{7, 12, 64, 81}

using System; public delegate Numeral Numeral(Numeral f); public static class ChurchNumeral{    public static readonly Numeral Zero = f => x => x;     public static Numeral Successor(this Numeral n) => f => x => f(n(f)(x));    public static Numeral Add(this Numeral m, Numeral n) => f => x => m(f)(n(f)(x));    public static Numeral Multiply(this Numeral m, Numeral n) => f => m(n(f));    public static Numeral Pow(this Numeral m, Numeral n) => n(m);     public static Numeral FromInt(int i) => i < 0 ? throw new ArgumentException("Negative church numeral.")        : i == 0 ? Zero : Successor(FromInt(i - 1));     public static int ToInt(this Numeral f) {        int count = 0;        f(x => { count++; return x; })(null);        return count;    }     public static void Main2() {        Numeral c3 = FromInt(3);        Numeral c4 = c3.Successor();        int sum = c3.Add(c4).ToInt();        int product = c3.Multiply(c4).ToInt();        int exp43 = c4.Pow(c3).ToInt();        int exp34 = c3.Pow(c4).ToInt();        Console.WriteLine($"{sum} {product} {exp43} {exp34}"); } } Output: 7 12 64 81  ## Fōrmulæ In this page you can see the solution of this task. Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text (more info). Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for transportation effects more than visualization and edition. The option to show Fōrmulæ programs and their results is showing images. Unfortunately images cannot be uploaded in Rosetta Code. ## F# type INumeral = abstract Apply : ('a -> 'a) -> 'a -> 'a let zero = {new INumeral with override __.Apply _ x = x}let successor (n: INumeral) = {new INumeral with override __.Apply f x = f (n.Apply f x)}let addition (m: INumeral) (n: INumeral) = {new INumeral with override __.Apply f x = m.Apply f (n.Apply f x)}let multiplication (m: INumeral) (n: INumeral) = {new INumeral with override __.Apply f x = m.Apply (n.Apply f) x}let exponential (m: INumeral) (n: INumeral) = {new INumeral with override __.Apply f x = n.Apply m.Apply f x} let ntoi (n: INumeral) = n.Apply ((+) 1) 0let iton i = List.fold (>>) id (List.replicate i successor) zero let c3 = iton 3let c4 = successor c3 [addition c3 c4;multiplication c3 c4;exponential c4 c3;exponential c3 c4]|> List.map ntoi|> printfn "%A"  Output: [7; 12; 64; 81] ## Go package main import "fmt" type any = interface{} type fn func(any) any type church func(fn) fn func zero(f fn) fn { return func(x any) any { return x }} func (c church) succ() church { return func(f fn) fn { return func(x any) any { return f(c(f)(x)) } }} func (c church) add(d church) church { return func(f fn) fn { return func(x any) any { return c(f)(d(f)(x)) } }} func (c church) mul(d church) church { return func(f fn) fn { return func(x any) any { return c(d(f))(x) } }} func (c church) pow(d church) church { di := d.toInt() prod := c for i := 1; i < di; i++ { prod = prod.mul(c) } return prod} func (c church) toInt() int { return c(incr)(0).(int)} func intToChurch(i int) church { if i == 0 { return zero } else { return intToChurch(i - 1).succ() }} func incr(i any) any { return i.(int) + 1} func main() { z := church(zero) three := z.succ().succ().succ() four := three.succ() fmt.Println("three ->", three.toInt()) fmt.Println("four ->", four.toInt()) fmt.Println("three + four ->", three.add(four).toInt()) fmt.Println("three * four ->", three.mul(four).toInt()) fmt.Println("three ^ four ->", three.pow(four).toInt()) fmt.Println("four ^ three ->", four.pow(three).toInt()) fmt.Println("5 -> five ->", intToChurch(5).toInt())} Output: three -> 3 four -> 4 three + four -> 7 three * four -> 12 three ^ four -> 81 four ^ three -> 64 5 -> five -> 5  ## Haskell churchZero = const id churchSucc = (<*>) (.) churchAdd = (<*>) . fmap (.) churchMult = (.) churchExp = flip id churchFromInt :: Int -> ((a -> a) -> a -> a)churchFromInt 0 = churchZerochurchFromInt n = churchSucc$ churchFromInt (n - 1) -- Or as a fold:-- churchFromInt n = foldr (.) id . replicate n -- Or as an iterate:-- churchFromInt n = iterate churchSucc churchZero !! n intFromChurch :: ((Int -> Int) -> Int -> Int) -> IntintFromChurch cn = cn succ 0 -- TEST --------------------------------------------[cThree, cFour] = churchFromInt <$> [3, 4] main :: IO ()main = print$  intFromChurch <$> [ churchAdd cThree cFour , churchMult cThree cFour , churchExp cFour cThree , churchExp cThree cFour ] Output: [7,12,64,81] ## JavaScript (() => { 'use strict'; const main = () => { const churchZero = f => x => x; const churchSucc = n => f => x => f(n(f)(x)); const churchAdd = m => n => f => x => n(f)(m(f)(x)); const churchMult = m => n => f => x => n(m(f))(x); const churchExp = m => n => n(m); const intFromChurch = n => n(succ)(0); const churchFromInt = n => f => foldl(composeR, id, replicate(n, f)); // Or, recursively ... // const churchFromInt = x => { // const go = i => // 0 === i ? ( // churchZero // ) : churchSucc(go(i - 1)); // return go(x); // }; // TEST ------------------------------------------- const [cThree, cFour] = map(churchFromInt, [3, 4]); return map( intFromChurch, [ churchAdd(cThree)(cFour), churchMult(cThree)(cFour), churchExp(cFour)(cThree), churchExp(cThree)(cFour), ] ); }; // GENERIC FUNCTIONS ------------------------------ // composeR (>>>) :: (a -> b) -> (b -> c) -> a -> c const composeR = (f, g) => x => f(g(x)); // foldl :: (a -> b -> a) -> a -> [b] -> a const foldl = (f, a, xs) => xs.reduce(f, a); // id :: a -> a const id = x => x; // map :: (a -> b) -> [a] -> [b] const map = (f, xs) => xs.map(f); // replicate :: Int -> a -> [a] const replicate = (n, x) => Array.from({ length: n }, () => x); // succ :: Enum a => a -> a const succ = x => 1 + x; // MAIN --------------------- return JSON.stringify(main());})(); Output: [7,12,64,81] ## Julia We could overload the Base operators, but that is not needed here.  id(x) = x -> xzero() = x -> id(x)add(m) = n -> (f -> (x -> n(f)(m(f)(x))))mult(m) = n -> (f -> (x -> n(m(f))(x)))exp(m) = n -> n(m)succ(i::Int) = i + 1succ(cn) = f -> (x -> f(cn(f)(x)))church2int(cn) = cn(succ)(0)int2church(n) = n < 0 ? throw("negative Church numeral") : (n == 0 ? zero() : succ(int2church(n - 1))) function runtests() church3 = int2church(3) church4 = int2church(4) println("Church 3 + Church 4 = ", church2int(add(church3)(church4))) println("Church 3 * Church 4 = ", church2int(mult(church3)(church4))) println("Church 4 ^ Church 3 = ", church2int(exp(church4)(church3))) println("Church 3 ^ Church 4 = ", church2int(exp(church3)(church4)))end runtests()  Output:  Church 3 + Church 4 = 7 Church 3 * Church 4 = 12 Church 4 ^ Church 3 = 64 Church 3 ^ Church 4 = 81  ## Lambdatalk  {def succ {lambda {:n :f :x} {:f {:n :f :x}}}}{def add {lambda {:n :m :f :x} {{:n :f} {:m :f :x}}}} {def mul {lambda {:n :m :f} {:m {:n :f}}}}{def power {lambda {:n :m} {:m :n}}} {def church {lambda {:n} {{:n {+ {lambda {:x} {+ :x 1}}}} 0}}} {def zero {lambda {:f :x} :x}}{def three {succ {succ {succ zero}}}}{def four {succ {succ {succ {succ zero}}}}} 3+4 = {church {add {three} {four}}} -> 73*4 = {church {mul {three} {four}}} -> 123^4 = {church {power {three} {four}}} -> 814^3 = {church {power {four} {three}}} -> 64  ## Lua  function churchZero() return function(x) return x end end function churchSucc(c) return function(f) return function(x) return f(c(f)(x)) end end end function churchAdd(c, d) return function(f) return function(x) return c(f)(d(f)(x)) end end end function churchMul(c, d) return function(f) return c(d(f)) end end function churchExp(c, e) return e(c)end function numToChurch(n) local ret = churchZero for i = 1, n do ret = succ(ret) end return ret end function churchToNum(c) return c(function(x) return x + 1 end)(0) end three = churchSucc(churchSucc(churchSucc(churchZero)))four = churchSucc(churchSucc(churchSucc(churchSucc(churchZero)))) print("'three'\t=", churchToNum(three))print("'four' \t=", churchToNum(four))print("'three' * 'four' =", churchToNum(churchMul(three, four)))print("'three' + 'four' =", churchToNum(churchAdd(three, four)))print("'three' ^ 'four' =", churchToNum(churchExp(three, four)))print("'four' ^ 'three' =", churchToNum(churchExp(four, three))) Output: 'three' = 3 'four' = 4 'three' * 'four' = 12 'three' + 'four' = 7 'three' ^ 'four' = 81 'four' ^ 'three' = 64 ## Perl Translation of: Perl 6 use 5.020;use feature qw<signatures>;no warnings qw<experimental::signatures>; use constant zero => sub ($f) {                      sub ($x) {$x }}; use constant succ  => sub ($n) { sub ($f) {                      sub ($x) {$f->($n->($f)($x)) }}}; use constant add => sub ($n) {                      sub ($m) { sub ($f) {                      sub ($x) {$m->($f)($n->($f)($x)) }}}}; use constant mult  => sub ($n) { sub ($m) {                      sub ($f) { sub ($x) { $m->($n->($f))($x) }}}}; use constant power => sub ($b) { sub ($e) { $e->($b) }}; use constant countup   => sub ($i) {$i + 1 };use constant countdown => sub ($i) {$i == 0 ? zero : succ->( __SUB__->($i - 1) ) };use constant to_int => sub ($f) { $f->(countup)->(0) };use constant from_int => sub ($x) { countdown->($x) }; use constant three => succ->(succ->(succ->(zero)));use constant four => from_int->(4); say join ' ', map { to_int->($_) } (    add  ->( three )->( four  ),    mult ->( three )->( four  ),    power->( four  )->( three ),    power->( three )->( four  ),);
Output:
7 12 64 81

## Perl 6

Translation of: Python
constant $zero = sub (Code$f) {                  sub (     $x) {$x }} constant $succ = sub (Code$n) {                  sub (Code $f) { sub ($x) { $f($n($f)($x)) }}} constant $add = sub (Code$n) {                  sub (Code $m) { sub (Code$f) {                  sub (     $x) {$m($f)($n($f)($x)) }}}} constant $mult = sub (Code$n) {                  sub (Code $m) { sub (Code$f) {                  sub (     $x) {$m($n($f))($x) }}}} constant$power = sub (Code $b) { sub (Code$e) { $e($b) }} sub to_int (Code $f) { sub countup (Int$i) { $i + 1 } return$f(&countup).(0)} sub from_int (Int $x) { multi sub countdown ( 0) {$zero }    multi sub countdown (Int $i) {$succ( countdown($i - 1) ) } return countdown($x);} constant $three =$succ($succ($succ($zero)));constant$four  = from_int(4); say map &to_int,    $add($three )( $four ),$mult(  $three )($four  ),    $power($four  )( $three ),$power( $three )($four  ),;

### Arrow subs without sigils

Translation of: Julia
my \zero  = -> \f {                 -> \x { x               }}my \succ  = -> \n {         -> \f { -> \x { f.(n.(f)(x))    }}}my \add   = -> \n { -> \m { -> \f { -> \x { m.(f)(n.(f)(x)) }}}}my \mult  = -> \n { -> \m { -> \f { -> \x { m.(n.(f))(x)    }}}}my \power = -> \b {                 -> \e { e.(b)           }} my \to_int   = -> \f { f.( -> \i { i + 1 } ).(0) }my \from_int = -> \i { i == 0 ?? zero !! succ.( &?BLOCK(i - 1) ) } my \three = succ.(succ.(succ.(zero)));my \four  = from_int.(4); say map -> \f { to_int.(f) },    add.(   three )( four  ),    mult.(  three )( four  ),    power.( four  )( three ),    power.( three )( four  ),;
Output:
(7 12 64 81)

## Phix

Translation of: Go
type church(object c)-- eg {r_add,1,{a,b}}    return sequence(c) and length(c)=3        and integer(c[1]) and integer(c[2])        and sequence(c[3]) and length(c[3])=2end type function succ(church c)-- eg {r_add,1,{a,b}} => {r_add,2,{a,b}}  aka  a+b -> a+b+b    c[2] += 1    return cend function -- three normal integer-handling routines...function add(integer n, a, b)    for i=1 to n do        a += b    end for    return aend functionconstant r_add = routine_id("add") function mul(integer n, a, b)    for i=1 to n do        a *= b    end for    return aend functionconstant r_mul = routine_id("mul") function pow(integer n, a, b)    for i=1 to n do        a = power(a,b)    end for    return aend functionconstant r_pow = routine_id("pow") -- ...and three church constructors to match--    (no maths here, just pure static data)function addch(church c, d)    church res = {r_add,1,{c,d}}    return resend function function mulch(church c, d)    church res = {r_mul,1,{c,d}}    return resend function function powch(church c, d)    church res = {r_pow,1,{c,d}}    return resend function function tointch(church c)-- note this is where the bulk of any processing happens    {integer rid, integer n, object x} = c    for i=1 to length(x) do        if church(x[i]) then x[i] = tointch(x[i]) end if    end for    return call_func(rid,n&x)end function constant church zero = {r_add,0,{0,1}} function inttoch(integer i)    if i=0 then        return zero    else        return succ(inttoch(i-1))    end ifend function church three = succ(succ(succ(zero))),       four = succ(three)printf(1,"three        -> %d\n",tointch(three))printf(1,"four         -> %d\n",tointch(four))printf(1,"three + four -> %d\n",tointch(addch(three,four)))printf(1,"three * four -> %d\n",tointch(mulch(three,four)))printf(1,"three ^ four -> %d\n",tointch(powch(three,four)))printf(1,"four ^ three -> %d\n",tointch(powch(four,three)))printf(1,"5 -> five    -> %d\n",tointch(inttoch(5)))
Output:
three        -> 3
four         -> 4
three + four -> 7
three * four -> 12
three ^ four -> 81
four ^ three -> 64
5 -> five    -> 5


## Prolog

Prolog terms can be used to represent church numerals.

church_zero(z). church_successor(Z, c(Z)). church_add(z, Z, Z).church_add(c(X), Y, c(Z)) :-    church_add(X, Y, Z). church_multiply(z, _, z).church_multiply(c(X), Y, R) :-    church_add(Y, S, R),    church_multiply(X, Y, S). % N ^ Mchurch_power(z, z, z).church_power(N, c(z), N).church_power(N, c(c(Z)), R) :-    church_multiply(N, R1, R),    church_power(N, c(Z), R1). int_church(0, z).int_church(I, c(Z)) :-    int_church(Is, Z),    succ(Is, I). run :-    int_church(3, Three),    church_successor(Three, Four),    church_add(Three, Four, Sum),    church_multiply(Three, Four, Product),    church_power(Four, Three, Power43),    church_power(Three, Four, Power34),     int_church(ISum, Sum),    int_church(IProduct, Product),    int_church(IPower43, Power43),    int_church(IPower34, Power34),     !,    maplist(format('~w '), [ISum, IProduct, IPower43, IPower34]),    nl.
Output:
7 12 81 64


## Python

Works with: Python version 3.7
'''Church numerals''' import functoolsimport itertools  # CHURCH ENCODINGS OF NUMERALS AND OPERATIONS ------------- def churchZero():    '''The identity function.       No applications of any supplied f       to its argument.    '''    return lambda f: id_  def churchSucc(cn):    '''The successor of a given       Church numeral. One additional       application of f. Equivalent to       the arithmetic addition of one.    '''    return lambda f: lambda x: f(cn(f)(x))  def churchAdd(m):    '''The arithmetic sum of two Church numerals.'''    return lambda n: lambda f: lambda x: n(f)(m(f)(x))  def churchMult(m):    '''The arithmetic product of two Church numerals.'''    return lambda n: lambda f: lambda x: n(m(f))(x)  def churchExp(m):    '''Exponentiation of Church numerals. m^n'''    return lambda n: n(m)  def churchFromInt(n):    '''The Church numeral equivalent of       a given integer.    '''    return lambda f: (        foldl        (composeR)        (id_)        (replicate(n)(f))    ) # OR, recursively:# def churchFromInt(n):#    if 0 == n:#        return churchZero()#    else:#        return churchSucc(churchFromInt(n - 1))  def intFromChurch(cn):    '''The integer equivalent of a       given Church numeral.    '''    return cn(succ)(0)  # GENERIC FUNCTIONS ------------------------------- # composeR (>>>) :: (a -> b) -> (b -> c) -> a -> cdef composeR(f):    '''A left to right composition of two       functions f and g: f(g(x))'''    return lambda g: lambda x: f(g(x))  # foldl :: (a -> b -> a) -> a -> [b] -> adef foldl(f):    '''Left to right reduction of a list,       using the binary operator f, and       starting with an initial value a.    '''    reduce = functools.reduce     def go(a, xs):        return reduce(f, xs, a)    return lambda acc: lambda xs: go(acc, xs)  # id_ :: a -> adef id_(x):    '''The identity function.'''    return x  # replicate :: Int -> a -> [a]def replicate(n):    '''A list of length n in which every       element has the value x.    '''    return lambda x: list(itertools.repeat(x, n))  # succ :: Enum a => a -> adef succ(x):    '''The successor of a value.       For numeric types, (1 +).    '''    return 1 + x if isinstance(x, int) else (        chr(1 + ord(x))    )  # uncurry :: (a -> b -> c) -> ((a, b) -> c)def uncurry(f):    '''A function over a tuple,       derived from a default or       curried function.'''    return lambda xy: f(xy[0], xy[1])  # MAIN -------------------------------------------def main():    'Tests'     cThree = churchFromInt(3)    cFour = churchFromInt(4)     print(list(map(intFromChurch, [        churchAdd(cThree)(cFour),        churchMult(cThree)(cFour),        churchExp(cFour)(cThree),        churchExp(cThree)(cFour),    ])))  if __name__ == '__main__':    main()
Output:
[7, 12, 64, 81]

## Racket

#lang racket (define zero (λ (f) (λ (x) x)))(define zero* (const identity)) ; zero renamed (define one (λ (f) f))(define one* identity) ; one renamed (define succ (λ (n) (λ (f) (λ (x) (f ((n f) x))))))(define succ* (λ (n) (λ (f) (λ (x) ((n f) (f x)))))) ; different impl (define add (λ (n) (λ (m) (λ (f) (λ (x) ((m f) ((n f) x)))))))(define add* (λ (n) (n succ))) (define succ** (add one)) (define mult (λ (n) (λ (m) (λ (f) (m (n f))))))(define mult* (λ (n) (λ (m) ((m (add n)) zero)))) (define expt  (λ (n) (λ (m) (m n))))(define expt* (λ (n) (λ (m) ((m (mult n)) one)))) (define (nat->church n)  (cond    [(zero? n) zero]    [else (succ (nat->church (sub1 n)))])) (define (church->nat n) ((n add1) 0)) (define three (nat->church 3))(define four (nat->church 4)) (church->nat ((add three) four))(church->nat ((mult three) four))(church->nat ((expt three) four))(church->nat ((expt four) three))
Output:
7
12
81
64


## Rust

use std::rc::Rc;use std::ops::{Add, Mul}; #[derive(Clone)]struct Church<'a, T: 'a> {    runner: Rc<dyn Fn(Rc<dyn Fn(T) -> T + 'a>) -> Rc<dyn Fn(T) -> T + 'a> + 'a>,} impl<'a, T> Church<'a, T> {    fn zero() -> Self {        Church {            runner: Rc::new(|_f| {                Rc::new(|x| x)            })        }    }     fn succ(self) -> Self {        Church {            runner: Rc::new(move |f| {                let g = self.runner.clone();                Rc::new(move |x| f(g(f.clone())(x)))            })        }    }     fn run(&self, f: impl Fn(T) -> T + 'a) -> Rc<dyn Fn(T) -> T + 'a> {        (self.runner)(Rc::new(f))    }     fn exp(self, rhs: Church<'a, Rc<dyn Fn(T) -> T + 'a>>) -> Self    {        Church {            runner: (rhs.runner)(self.runner)        }    }} impl<'a, T> Add for Church<'a, T> {    type Output = Church<'a, T>;     fn add(self, rhs: Church<'a, T>) -> Church<T> {        Church {            runner: Rc::new(move |f| {                let self_runner = self.runner.clone();                let rhs_runner = rhs.runner.clone();                Rc::new(move |x| (self_runner)(f.clone())((rhs_runner)(f.clone())(x)))            })        }    }} impl<'a, T> Mul for Church<'a, T> {    type Output = Church<'a, T>;     fn mul(self, rhs: Church<'a, T>) -> Church<T> {        Church {            runner: Rc::new(move |f| {                (self.runner)((rhs.runner)(f))            })        }    }} impl<'a, T> From<i32> for Church<'a, T> {    fn from(n: i32) -> Church<'a, T> {        let mut ret = Church::zero();        for _ in 0..n {            ret = ret.succ();        }        ret    }} impl<'a> From<&Church<'a, i32>> for i32  {    fn from(c: &Church<'a, i32>) -> i32 {        c.run(|x| x + 1)(0)    }} fn three<'a, T>() -> Church<'a, T> {    Church::zero().succ().succ().succ()} fn four<'a, T>() -> Church<'a, T> {    Church::zero().succ().succ().succ().succ()} fn main() {    println!("three =\t{}", i32::from(&three()));    println!("four =\t{}", i32::from(&four()));     println!("three + four =\t{}", i32::from(&(three() + four())));    println!("three * four =\t{}", i32::from(&(three() * four())));     println!("three ^ four =\t{}", i32::from(&(three().exp(four()))));    println!("four ^ three =\t{}", i32::from(&(four().exp(three()))));}
Output:
three =	3
four =	4
three + four =	7
three * four =	12
three ^ four =	81
four ^ three =	64

## Swift

func succ<A, B, C>(_ n: @escaping (@escaping (A) -> B) -> (C) -> A) -> (@escaping (A) -> B) -> (C) -> B {  return {f in    return {x in      return f(n(f)(x))    }  }} func zero<A, B>(_ a: A) -> (B) -> B {  return {b in    return b  }} func three<A>(_ f: @escaping (A) -> A) -> (A) -> A {  return {x in    return succ(succ(succ(zero)))(f)(x)  }} func four<A>(_ f: @escaping (A) -> A) -> (A) -> A {  return {x in    return succ(succ(succ(succ(zero))))(f)(x)  }} func add<A, B, C>(_ m: @escaping (B) -> (A) -> C) -> (@escaping (B) -> (C) -> A) -> (B) -> (C) -> C {  return {n in    return {f in      return {x in        return m(f)(n(f)(x))      }    }  }} func mult<A, B, C>(_ m: @escaping (A) -> B) -> (@escaping (C) -> A) -> (C) -> B {  return {n in    return {f in      return m(n(f))    }  }} func exp<A, B, C>(_ m: A) -> (@escaping (A) -> (B) -> (C) -> C) -> (B) -> (C) -> C {  return {n in    return {f in      return {x in        return n(m)(f)(x)      }    }  }} func church<A>(_ x: Int) -> (@escaping (A) -> A) -> (A) -> A {  guard x != 0 else { return zero }   return {f in    return {a in      return f(church(x - 1)(f)(a))    }  }} func unchurch<A>(_ f: (@escaping (Int) -> Int) -> (Int) -> A) -> A {  return f({i in    return i + 1  })(0)} let a = unchurch(add(three)(four))let b = unchurch(mult(three)(four))// We can even compose operationslet c = unchurch(exp(mult(four)(church(1)))(three))let d = unchurch(exp(mult(three)(church(1)))(four)) print(a, b, c, d)
Output:
7 12 64 81

## zkl

class Church{  // kinda heavy, just an int + fcn churchAdd(ca,cb) would also work   fcn init(N){ var n=N; }	// Church Zero is Church(0)   fcn toInt(f,x){ do(n){ x=f(x) } x } // c(3)(f,x) --> f(f(f(x)))   fcn succ{ self(n+1) }   fcn __opAdd(c){ self(n+c.n)      }   fcn __opMul(c){ self(n*c.n)      }   fcn pow(c)    { self(n.pow(c.n)) }   fcn toString{ String("Church(",n,")") }}
c3,c4 := Church(3),c3.succ();f,x := Op("+",1),0;println("f=",f,", x=",x);println("%s+%s=%d".fmt(c3,c4, (c3+c4).toInt(f,x)      ));println("%s*%s=%d".fmt(c3,c4, (c3*c4).toInt(f,x)      ));println("%s^%s=%d".fmt(c4,c3, (c4.pow(c3)).toInt(f,x) ));println("%s^%s=%d".fmt(c3,c4, (c3.pow(c4)).toInt(f,x) ));println();T(c3+c4,c3*c4,c4.pow(c3),c3.pow(c4)).apply("toInt",f,x).println();
Output:
f=Op(+1), x=0
Church(3)+Church(4)=7
Church(3)*Church(4)=12
Church(4)^Church(3)=64
Church(3)^Church(4)=81

L(7,12,64,81)


OK, that was the easy sleazy cheat around way to do it. The wad of nested functions way is as follows:

fcn churchZero{ return(fcn(x){ x }) } // or fcn churchZero{ self.fcn.idFcn }fcn churchSucc(c){ return('wrap(f){ return('wrap(x){ f(c(f)(x)) }) }) }fcn churchAdd(c1,c2){ return('wrap(f){ return('wrap(x){ c1(f)(c2(f)(x)) }) }) }fcn churchMul(c1,c2){ return('wrap(f){ c1(c2(f)) }) }fcn churchPow(c1,c2){ return('wrap(f){ c2(c1)(f) }) }fcn churchToInt(c,f,x){ c(f)(x) }fcn churchFromInt(n){ c:=churchZero; do(n){ c=churchSucc(c) } c }//fcn churchFromInt(n){ (0).reduce(n,churchSucc,churchZero) } // what ever
c3,c4 := churchFromInt(3),churchSucc(c3);f,x   := Op("+",1),0;	// x>=0, ie natural numberT(c3,c4,churchAdd(c3,c4),churchMul(c3,c4),churchPow(c4,c3),churchPow(c3,c4))   .apply(churchToInt,f,x).println();
Output:
L(3,4,7,12,64,81)