$2.$ Specifications $(10$p$,$ Questions $79)$ Recall that we often want to translate specifications written in English into predicate logic. For example, we may translate $$B$[1..$n$]$

contains duplicates$$ into $$i$,$ j in $1...$ n : i $6=$ j $$ B$[$i$]=$ B$[$j$].$ Now consider the specification:

Given a non$-$increasing array B$[1..$n$],$ and a value v$,$ return $0$ if v does not occur

in B$[1..$n$],$ but otherwise return the index of one of the occurrences.

Example: if B$[1.\mathrm{.}6]$ is $123456$

$211713131110$ then if v $=11$ then $5$ must be returned;

if v $=12$ then $0$ must be returned; if v $=13$ then either $3$ or $4$ must be returned.

For that specification, you must express in predicate logic

$7.(3$p$)$ the precondition, that is$,$ what is assumed about B$[1..$n$]$ and v$.$

and next the postcondition, that is$,$ what is required about r where r is the value returned.

We can write the postcondition as $($r $6=0\backslash$Phi $1)($r $=0\backslash$Phi $2).$

$8.(3$p$)$ What should $\backslash$Phi $1$ be$?$

$9.(4$p$)$ What should $\backslash$Phi $2$ be$?$

Whenever B occurs in your specification it must be as part