Problem $4$: Program Verification II

Consider the following program fragment:Where:

$P$:$n0(Precondition)$

$Q$:${\displaystyle \underset{?}{\overset{?}{}}}=\frac{n(n-1)}{2}(Postcondition)$

Define the Loop Invariant: Identify a suitable loop invariant I that holds: before entering the loop, at each iteration and after exiting the loop.

State the Verification Conditions: Using the identified loop invariant I, write the verification conditions for:

$($a$)$ Initialization: Show that I holds before the loop starts.

$($b$)$ Maintenance: Prove that I is preserved during each iteration of the loop.

$($c$)$ Termination: Show that upon exiting the loop, I combined with the loop's exit condition implies the postcondition Q$.$