from operator import add, mul

def square(x):
    return x * x

def identity(x):
    return x

def triple(x):
    return 3 * x

def increment(x):
    return x + 1


SOURCE_FILE = __file__


def product(n, term):
    """Return the product of the first n terms in a sequence.

    n: a positive integer
    term: a function that takes an index as input and produces a term

    >>> product(3, identity)  # 1 * 2 * 3
    6
    >>> product(5, identity)  # 1 * 2 * 3 * 4 * 5
    120
    >>> product(3, square)    # 1^2 * 2^2 * 3^2
    36
    >>> product(5, square)    # 1^2 * 2^2 * 3^2 * 4^2 * 5^2
    14400
    >>> product(3, increment) # (1+1) * (2+1) * (3+1)
    24
    >>> product(3, triple)    # 1*3 * 2*3 * 3*3
    162
    """
    prod, k = 1, 1
    while k <= n:
        prod, k = term(k) * prod, k + 1
    return prod


def accumulate(fuse, start, n, term):
    """Return the result of fusing together the first n terms in a sequence 
    and start.  The terms to be fused are term(1), term(2), ..., term(n). 
    The function fuse is a two-argument commutative & associative function.

    >>> accumulate(add, 0, 5, identity)  # 0 + 1 + 2 + 3 + 4 + 5
    15
    >>> accumulate(add, 11, 5, identity) # 11 + 1 + 2 + 3 + 4 + 5
    26
    >>> accumulate(add, 11, 0, identity) # 11 (fuse is never used)
    11
    >>> accumulate(add, 11, 3, square)   # 11 + 1^2 + 2^2 + 3^2
    25
    >>> accumulate(mul, 2, 3, square)    # 2 * 1^2 * 2^2 * 3^2
    72
    >>> # 2 + (1^2 + 1) + (2^2 + 1) + (3^2 + 1)
    >>> accumulate(lambda x, y: x + y + 1, 2, 3, square)
    19
    """
    total, k = start, 1
    while k <= n:
        total, k = fuse(total, term(k)), k + 1
    return total

# Alternative solution
def accumulate_reverse(fuse, start, n, term):
    total, k = start, n
    while k >= 1:
        total, k = fuse(total, term(k)), k - 1
    return total


def summation_using_accumulate(n, term):
    """Returns the sum: term(1) + ... + term(n), using accumulate.

    >>> summation_using_accumulate(5, square) # square(1) + square(2) + ... + square(4) + square(5)
    55
    >>> summation_using_accumulate(5, triple) # triple(1) + triple(2) + ... + triple(4) + triple(5)
    45
    >>> # This test checks that the body of the function is just a return statement.
    >>> import inspect, ast
    >>> [type(x).__name__ for x in ast.parse(inspect.getsource(summation_using_accumulate)).body[0].body]
    ['Expr', 'Return']
    """
    return accumulate(add, 0, n, term)


def product_using_accumulate(n, term):
    """Returns the product: term(1) * ... * term(n), using accumulate.

    >>> product_using_accumulate(4, square) # square(1) * square(2) * square(3) * square()
    576
    >>> product_using_accumulate(6, triple) # triple(1) * triple(2) * ... * triple(5) * triple(6)
    524880
    >>> # This test checks that the body of the function is just a return statement.
    >>> import inspect, ast
    >>> [type(x).__name__ for x in ast.parse(inspect.getsource(product_using_accumulate)).body[0].body]
    ['Expr', 'Return']
    """
    return accumulate(mul, 1, n, term)


def make_repeater(f, n):
    """Returns the function that computes the nth application of f.

    >>> add_three = make_repeater(increment, 3)
    >>> add_three(5)
    8
    >>> make_repeater(triple, 5)(1) # 3 * (3 * (3 * (3 * (3 * 1))))
    243
    >>> make_repeater(square, 2)(5) # square(square(5))
    625
    >>> make_repeater(square, 3)(5) # square(square(square(5)))
    390625
    """
    def repeater(x):
        k = 0
        while k < n:
            x, k = f(x), k + 1
        return x
    return repeater