Given:

$x=$ bit of number we want to add

$y=$ bit of number we want to add to $x$

cin $=$ carry$-$in to addition of $x$ and $y$

cout $=$ carry$-$out of addition of $x$ and $y$

$s=$ sum of $x$ and $y$ plus any carry$-$in$.$

The SOP for cout $=($from Module $35).$ The gate delay for this function is

Factoring out cin gives us cout $=$ The gate delay for this function is

Letting $g=xy$ and $p=x+y,$ we can functionally decompose cout $=$ The

gate delay for this function is

Slide $5$ of Module $43$ shows expanding this concept to the i$+1$ carry$-$out. The gate delay for $c_{i+1}$ as depicted on Slide $5$ is

Using this carry$-$lookahead formula, the gate delay for calculating the carry$-$out of a $4-$bit adder is

Using this formula, the gate delay for calculating the carry$-$in to the last stage of a $4-$bit