*Due by 11:59pm on Monday, 9/15*

**Submission:** See
Lab 1 for
submission instructions. We have provided a hw2.py starter file for the questions below.

**Readings:** You might find the following references
useful:

Several doctests refer to these one-argument functions:

```
def square(x):
return x * x
def triple(x):
return 3 * x
def identity(x):
return x
def increment(x):
return x + 1
```

Implement `piecewise`

, which takes two one-argument functions, `f`

and `g`

,
along with a number `b`

. It returns a new function that takes a number `x`

and
returns either `f(x)`

if `x`

is less than `b`

, or `g(x)`

if `x`

is greater than
or equal to `b`

.

```
def piecewise(f, g, b):
"""Returns the piecewise function h where:
h(x) = f(x) if x < b,
g(x) otherwise
>>> def negate(x):
... return -x
>>> abs_value = piecewise(negate, identity, 0)
>>> abs_value(6)
6
>>> abs_value(-1)
1
"""
"*** YOUR CODE HERE ***"
```

Two functions intersect at an argument `x`

if they return equal values.
Implement `intersects`

, which takes a one-argument functions `f`

and a value `x`

.
It returns a function that takes another function `g`

and returns whether
`f`

and `g`

intersect at `x`

.

```
def intersects(f, x):
"""Returns a function that returns whether f intersects g at x.
>>> at_three = intersects(square, 3)
>>> at_three(triple) # triple(3) == square(3)
True
>>> at_three(increment)
False
>>> at_one = intersects(identity, 1)
>>> at_one(square)
True
>>> at_one(triple)
False
"""
"*** YOUR CODE HERE ***"
```

If `f`

is a numerical function and `n`

is a positive integer, then we
can form the nth repeated application of `f`

, which is defined to be
the function whose value at `x`

is `f(f(...(f(x))...))`

. For example,
if `f`

adds `1`

to its argument, then the nth repeated application of
`f`

adds `n`

. Write a function that takes as inputs a function `f`

and
a positive integer `n`

and returns the function that computes the nth
repeated application of f:

```
def repeated(f, n):
"""Return the function that computes the nth application of f.
>>> add_three = repeated(increment, 3)
>>> add_three(5)
8
>>> repeated(triple, 5)(1) # 3 * 3 * 3 * 3 * 3 * 1
243
>>> repeated(square, 2)(5) # square(square(5))
625
>>> repeated(square, 4)(5) # square(square(square(square(5))))
152587890625
"""
"*** YOUR CODE HERE ***"
```

*Hint*: You may find it convenient to use `compose1`

from the
textbook:

```
def compose1(f, g):
"""Return a function h, such that h(x) = f(g(x))."""
def h(x):
return f(g(x))
return h
```

The logician Alonzo Church invented a system of representing non-negative integers entirely using functions. The purpose was to show that functions are sufficient to describe all of number theory: if we have functions, we do not need to assume that numbers exist, but instead we can invent them.

Your goal in this problem is to rediscover this representation known as *Church
numerals*. Here are the definitions of `zero`

, as well as a function that
returns one more than its argument:

```
def zero(f):
return lambda x: x
def successor(n):
return lambda f: lambda x: f(n(f)(x))
```

First, define functions `one`

and `two`

such that they have the same behavior
as `successor(zero)`

and `successsor(successor(zero))`

respectively, but *do
not call successor in your implementation*.

Next, implement a function `church_to_int`

that converts a church numeral
argument to a regular Python integer.

Finally, implement functions `add_church`

, `mul_church`

, and `pow_church`

that
perform addition, multiplication, and exponentiation on church numerals.

```
def one(f):
"""Church numeral 1: same as successor(zero)"""
"*** YOUR CODE HERE ***"
def two(f):
"""Church numeral 2: same as successor(successor(zero))"""
"*** YOUR CODE HERE ***"
three = successor(two)
def church_to_int(n):
"""Convert the Church numeral n to a Python integer.
>>> church_to_int(zero)
0
>>> church_to_int(one)
1
>>> church_to_int(two)
2
>>> church_to_int(three)
3
"""
"*** YOUR CODE HERE ***"
def add_church(m, n):
"""Return the Church numeral for m + n, for Church numerals m and n.
>>> church_to_int(add_church(two, three))
5
"""
"*** YOUR CODE HERE ***"
def mul_church(m, n):
"""Return the Church numeral for m * n, for Church numerals m and n.
>>> four = successor(three)
>>> church_to_int(mul_church(two, three))
6
>>> church_to_int(mul_church(three, four))
12
"""
"*** YOUR CODE HERE ***"
def pow_church(m, n):
"""Return the Church numeral m ** n, for Church numerals m and n.
>>> church_to_int(pow_church(two, three))
8
>>> church_to_int(pow_church(three, two))
9
"""
"*** YOUR CODE HERE ***"
```