Remark:

$C_i=a_{i-1}b_{i-1} + \left(a_{i-1} \oplus b_{i-1} \right) C_{i-1}$ $,$ $i\geqslant1$ is equivalent to

$C_i=a_{i-1}b_{i-1} + \left(a_{i-1} + b_{i-1} \right) C_{i-1}$

To prove above equation, let’s take an example

For simplicity assume $C_1 = c_1$

Let $f(a_1,b_1,c_ 1)=a_1b_1 + \left(a_1 \oplus b_1 \right) c_ 1$

$\Rightarrow$ $f(a_1,b_1,c_ 1)=a_1b_1 + \overline {a_1}b_1c_1 + a_1\overline{b_1}c_1\\$

$\Rightarrow$ $f(a_1,b_1,c_ 1)=a_1b_1\overline{c_1} + a_1b_1c_1 + \overline {a_1}b_1c_1 + a_1\overline{b_1}c_1\\$

$\Rightarrow f(a_1,b_1,c_ 1)= \sum (3,5,6,7)$

After solving using K-Map you get $f=a_1b_1 + b_1c_1 + a_1c_1$