$($b$)$ This program puts c between a and b $,$ regardless of which of a and b is larger:

Prove that each of the following specifications is satisfied under partial correctness.

The domain of all the variables is the set of natural numbers.

i$.$ Give a full annotation of the specification, and

ii$.$ prove separately each implication arising from your annotation. The proofs of

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

$($a$)$ This program $($inefficiently$)$ computes $x+x$ :a $=$ x;

if $($a $=0)\{\}$ else $\{\}$

$($y$\backslash$approxx$+$x$)$