Formal proof of prefix adders

The paper presents an algebraic analysis for the correctness of prefix-based adders. In contrast to using higher-order functions and rewriting systems previously, we harness first-order recursive equations for correctness proof. A new carry operator is defined in terms of a semi-group with the set of binary bits. Both sequential and parallel addition algorithms are formalized and analyzed. The formal analysis on some special prefix adder circuits demonstrates the effectiveness of our algebraic approach. This study lays an underpinning for further understanding on computer arithmetic systems.