The Python function mergesort is shown below. It recursively sorts a list of numbers U using a recursive divide-and-conquer algorithm, then returns a sorted list S. The function head returns the first element of a non-empty list Q, and the function tail returns all but the first element of a non-empty list Q. Lines 0607 detect if U is trivially sorted. Lines 0916 split U into two halves, L and R, of approximately equal lengths. Lines 1718 recursively sort L and R. Lines 1928 merge the sorted L and R back into a sorted list S.

01 def head(Q): 02 return Q[0] 03 def tail(Q): 04 return Q[1:] 05 def mergesort(U): 06 if U == [] or tail(U) == []: 07 return U 08 else: 09 L = [] 10 R = [] 11 while U != [] and tail(U) != []: 12 L = L + [head(U)] 13 U = tail(U) 14 R = R + [head(U)] 15 U = tail(U) 16 L = L + U 17 L = mergesort(L) 18 R = mergesort(R) 19 S = [] 20 while L != [] and R != []: 21 if head(L) <= head(R): 22 S = S + [head(L)] 23 L = tail(L) 24 else: 25 S = S + [head(R)] 26 R = tail(R) 27 S = S + L + R 28 return S

Prove that mergesorts splitting loop is correct (lines 0916). Do not prove that the rest of mergesort is correct. Like the proofs in Cormens text, you must use a loop invariant. Your proof must have three parts: initialization, maintenance, and termination.

1a.		Find a loop invariant for the splitting loop.
1b.		Use your loop invariant to prove the initialization part.
1c.		Use your loop invariant to prove the maintenance part.
1d.		Use your loop invariant to prove the termination part.