The following program computes a well-known function called Ackermann’s function. Note:...

The following program computes a well-known function called Ackermann’s function. Note: If you try out this function, don’t let x and y get too large.

f (x, y) = if x = 0 then y + 1

else if y = 0 then f (x − 1, 1)

else f (x − 1, f (x, y − 1)) .

Prove that f is defined for all pairs (x, y) in Hint: Use the lexicographic ordering on This gives the single basis element (0, 0). For the induction assumption, assume that such that Then show that f (x, y) is defined.