Question

For the while loop below, use the loop invariant given to show that if the pre-condition is the loop.In each step, clearly state true before the loop then the post-condition is true after what facts are assumed and what facts will be proven. (computes the sum of a list of numbers.) fa ,an While (<n) sum := sum + am j j+1 End-while ° Pre-condition: j- 1, sum a, n is a positive integer, a,,.. a, is a list of n numbers. . Post-condition: sum-a n, and sum-Σ ak Loop invariant: j is an integer, j

Answer 1

What is assumed: The pre-condition i.e.
00i
and the loop invariant
Ci sum -
.

To prove: after the execution of all these loops .

sum -

First, note that because of the pre-condition, when the code first enters the loop, the invariant is satisfied. Now, the loop invariant ensures that it is satisfied in each subsequent loop. Finally, when the code comes out of the loop, the value of j is exactly n. As the loop invariant was still satisfied, the value of sum must be exactly, $\sum_{i=1}^{n} a_i$ . This proves the post condition.

Comment in case of any doubts.