$($b$)$ Prove that the specification in part $6$ a is satisfied under total correctness. You

may use the result of part $6,$ whether you solved it or not.This question is about a program that uses a while$-$loop to find a multiple of a which

lies betweeb $b$ and $2b.$ The domain of all the variables is the set of natural numbers.

$($a$)$ Prove that this specification is satisfied under partial correctness.

i$.$ Explicitly state your loop invariant,

ii$.$ give a full annotation of the specification, and

iii. prove separately each implication arising from your annotation. The proofs

of implications can be informal, like the proofs you wrote in MATH $135.$