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In questions $1$ and $2,$ we will create a program that calculates the factorial of natural number $$ and stores the result

as $=().$ Here, function $()$ is the function that gives the factorial of $$:

$()=\{$

$1,$

if

$$

$=$

$0$

$$

$$

$$

$($

$$

$1$

$)$

$,$

if

$$

$>$

$0$

$1.$ Answer the following questions.

a$.$ What are the pre$-$ and post$-$ conditions needed for this program?

b$.$ Create a loop invariant $$ by replacing the named constant $$ by a variable $$ in the postcondition $($what can

be the range of variable $?)$ and create the corresponding loop condition $.$

c$.$ Find a bound expression $$ for this program. Show your work.

$2.$ Create a full proof outline under the total correctness for this program. Don$$t forget to initialize variables $$ and

$$ before the loop starts, find reasonable values for them. Justify each logical implications used in the proof

outline.