Prove this algorithm is correct

$1.$ Mutual exclusion is preserved

$2.$ The progress requirement is satisfied.

$3.$ The bounded waiting requirement is met

Please point out which part of the code meets the condition and explain it

do $\{$

waiting$[$i$]=$ true;

key $=$ true;

while$($waiting$[$i$]$ && key$)$

key $=$ test$\_$and$\_$set$($&lock$)$;

waiting$[$i$]=$ false;

$//$ critical section

j $=($i$+1)\%$ n;

while$(($j $!=$ i$)$ && $!$waiting$[$j$])$

j $=($j$+1)\%$ n;

if$($j $==$ i$)$

lock $=$ false;

else

waiting$[$j$]=$ false;

$//$remainder section

$\}$ while$($true$)$