'''Church numerals'''

from itertools import repeat
from functools import reduce




def churchZero():
    '''The identity function.
       No applications of any supplied f
       to its argument.
    '''
    return lambda f: identity


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: compose(f)(cn(f))


def churchAdd(m):
    '''The arithmetic sum of two Church numerals.'''
    return lambda n: lambda f: compose(m(f))(n(f))


def churchMult(m):
    '''The arithmetic product of two Church numerals.'''
    return lambda n: compose(m)(n)


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
        (compose)
        (identity)
        (replicate(n)(f))
    )



def churchFromInt_(n):
    '''The Church numeral equivalent of a given
       integer, by explicit recursion.
    '''
    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)




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),
    ])))





def compose(f):
    '''A left to right composition of two
       functions f and g'''
    return lambda g: lambda x: g(f(x))



def foldl(f):
    '''Left to right reduction of a list,
       using the binary operator f, and
       starting with an initial value a.
    '''
    def go(acc, xs):
        return reduce(lambda a, x: f(a)(x), xs, acc)
    return lambda acc: lambda xs: go(acc, xs)



def identity(x):
    '''The identity function.'''
    return x



def replicate(n):
    '''A list of length n in which every
       element has the value x.
    '''
    return lambda x: repeat(x, n)



def succ(x):
    '''The successor of a value.
       For numeric types, (1 +).
    '''
    return 1 + x if isinstance(x, int) else (
        chr(1 + ord(x))
    )


if __name__ == '__main__':
    main()