Simplifying Boolean Expression
No, you cannot simplify to $(A&lt;B)_i = B_i + (A&lt;B)_{i-1}$, Take for example $A=11$ and $B = 10$. 

Then $(A&lt;B)_2$ = False, $B_2$ = True, and $(A&lt;B)_1$ = False. If you substitute the values above then the equation is not satisfied. The original equation(s) is correct.

$(A&lt;B)_i = \neg A_i B_i + (\neg A_i+B_i)(A&lt;B)_{i-1}$
$(A&lt;B)_1 = \neg A_1 B_1$

In your derivation of simpler formula, $B_i + (A&lt;B)_{i-1} = Q$ but you are incorrectly substituting $B_i (A&lt;B)_{i-1} = Q$ in rightmost term.