Exercise 1 (The Ackermann function). The Ackermann function is a very important function in theoretical computer science and logic. It is a two-variable function A : N2 + N defined recursively by the following conditions: (i) A(0,y) = y +1 for every y EN, (ii) An +1,0) = An, 1) for every n EN, (iii) A(n +1, y + 1) = An(n, A(n +1, y) for every n, Y EN. The theoretical importance of the Ackermann function is the following: For any primitive recursive function F:N + N, there is a fixed nf E N such that F(x) A(n,x) + y. Hint: do induction first on n, then on y. (4) For every n, y EN, if n > 1, then An + 1,y) > An, y) + y. Hint: do a similar type of double-induction as in (3). (5) For every n,y EN, A(n+1,y) > An, y + 1). Hint: do induction on y. (6) For every n, y EN, 2A(n,y) 0 use (3), (4), and (5) above. (7) For every n, x,y EN, if x A(n,x) + y. Hint: do induction first on n, then on y. (4) For every n, y EN, if n > 1, then An + 1,y) > An, y) + y. Hint: do a similar type of double-induction as in (3). (5) For every n,y EN, A(n+1,y) > An, y + 1). Hint: do induction on y. (6) For every n, y EN, 2A(n,y) 0 use (3), (4), and (5) above. (7) For every n, x,y EN, if x