Suppose a is an array of numbers of length n, and x holds a number. (a)[4] Write a contract (specification) for computing i{0,n}a(i)xi into y. (b)[4] Use the technique of replacing a constant by a variable to obtain an invariant. (c)[4] Write a correct proof outline that solves the problem. (You may assume that computing xi for any integer i is an operation in the programming language, although an expensive one.) Don't worry about efficiency at this point. (d)[4] What is the variant? (e)[4] Introduce a tracking variable to improve the efficiency of the algorithm. State the linking invariant that relates the new variable to the rest of the state. (f)[4] Rewrite the proof outline from part (c) to use the tracking variable