Problem $3[5+5$ Points$].$ Consider a function ite $($for if then else$)$ which takes three arguments: a boolean, an expression to evaluate if the boolean is true, and a second expression to evaluate if the boolean is false.

a$)$ Given the lambda expressions for boolean constants and operators presented in class, write the lambda expression for this ite function.

b$)$ Write the lambda expression for a function iteite $($for if then elseif then else$)$ which takes five arguments: a boolean, an expression to evaluate if the boolean is true, another boolean, an expression to evaluate if the second boolean is true, and a third expression to evaluate if both booleans are false. You may assume that a correct implementation of function ite from $($a$)$ is available to you, and use it if you like.

Problem $4[10$ Points$].$ Computing the exponent of a number $($i$.$e$.,$ nk$)$ can take O$($n$)$ time if done by simply multiplying by n$,$ k number of times. A logarithmic way for computing exponents is by using the idea of successive squaring. For instance, rather than computing n$8$ as: n $*$ n $*$ n $*$ n $*$ n $*$ n $*$ n $*$ n$,$ we can compute it by repeatedly squaring, beginning with n$,$ computing n$2,$ n$4$ and finally n$8.$ In general, the algorithm would do the following:

nk $=($n$($k$/2))2$ if k is even

nk $=$ n $*$ n$($k$-1)$ if k is odd

Write a lambda expression for this function. You may assume that the function required in Problem $3($a$)$ is already available to you, and you may use it if you like. You may also assume that if you use the function from Problem $3($a$),$ you are simply able to state the condition as a logical statement.

Because this is a recursive function, you will have to nest the lambda expression taking argument k within another lambda taking f $$ the function $$ which you will then recursively call within the body. Plus, you will have to precede this definition with the letter Y$,$ which represents a special kind of lambda expression called the Y combinator, which actually makes the recursion happen. So$,$ your solution will look something like:

Y lambda f $.[$rest of your solution$]$