DFS$($G$)$

for each vertex $\}$u$\backslash$inG$.$

u$.$color $=$ WHITE

u$.\backslash$pi $=$ NIL

time $=0$

for each vertex $\}$u$\backslash$inG$.$

if u$.$color $==$ WHITE

DFS$-$Visit$($G$,$u$)$

DFS$-$VISIT $($G$,$u$)$

$1\}\backslash$mathrm$\{$ time $=$ time $+1$

u$.$d $=$ time

u$.$color $=$ GRAY

for each v$\}\backslash$inG$.$Adj$[$u

if v$.$color $==$ WHITE

v$.\backslash$pi $=$ u

DFS$-$VISIT $(\}$G$,$v

u$.$color $=$ BLACK

time $=$ time $+1$

u$.$f $=$ time

prove the correctness of the depth first search, use loop invariant to prove it mathematically, don't involve the discovery and finishing time. prove it without discovery, and finishing time