$2)$ Following the construction of Theorem $7\mathrm{.}2,$ convert the following npda to an equivalent grammar. Show the initial results prior to simplification, and then the final results after simplification. You don't have to show each individual step of simplification.

states: $\{$q$0,$q$1,$q$2,$q$3\}$

input alphabet: $\{$a$,$b$\}$

stack alphabet: $\{$A$,$Z$\}$

initial state: q$0$

stack start symbol: Z

final states: $\{$q$3\}$

transitions:

$\backslash$delta $($q$2,\backslash$lambda $,$Z$)=\{($q$3,\backslash$lambda $)\}$

$\backslash$delta $($q$0,\backslash$lambda $,$Z$)=\{($q$1,$AZ$)\}$

$\backslash$delta $($q$2,$a$,$Z$)=\{($q$1,$AZ$)\}$

$\backslash$delta $($q$1,$b$,$A$)=\{($q$2,\backslash$lambda $)\}$