# 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}`

## Fōrmulæ

Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. 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.

## 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
```

`churchZero = const id churchSucc = (<*>) (.) churchAdd = (<*>) . (<\$>) (.) 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

```

## 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
```

## Python

`import functoolsimport itertools # CHURCH ENCODINGS ---------------------------------  def churchZero():    return lambda f: id  def churchSucc(cn):    return lambda f: lambda x: f(cn(f)(x))  def churchAdd(m):    return lambda n: lambda f: lambda x: n(f)(m(f)(x))  def churchMult(m):    return lambda n: lambda f: lambda x: n(m(f))(x)  def churchExp(m):    return lambda n: n(m)  def churchFromInt(n):    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):    return cn(succ)(0)  # GENERIC FUNCTIONS ------------------------------- # composeR (>>>) :: (a -> b) -> (b -> c) -> a -> cdef composeR(f):    return lambda g: lambda x: f(g(x))  # foldl :: (a -> b -> a) -> a -> [b] -> adef foldl(f):    return lambda a: lambda xs: (        functools.reduce(uncurry(f), xs, a)    )  # id :: a -> adef id(x):    return x  # replicate :: Int -> a -> [a]def replicate(n):    return lambda x: itertools.repeat(x, n)  # succ :: Int -> Intdef succ(x):    return 1 + x  # uncurry :: (a -> b -> c) -> ((a, b) -> c)def uncurry(f):    def g(x, y):        return f(x)(y)    return g  # MAIN -------------------------------------------def main():    cThree = churchFromInt(3)    cFour = churchFromInt(4)     print (list(map(intFromChurch, [        churchAdd(cThree)(cFour),        churchMult(cThree)(cFour),        churchExp(cFour)(cThree),        churchExp(cThree)(cFour),    ])))  main()`
Output:
`[7, 12, 64, 81]`

## 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)
```