Logic Gates and Logical Equivalence of Adders

This is an experimental article which shows that logical correctness of logic circuits can be easily proven by the Mizar system. First, we define the notion of logic gates. Then we prove that an MSB carry of ’4 Bit Carry Skip Adder’ is equivalent to an MSB carry of a normal 4 bit adder. In the last theorem, we show that outputs of the ’4 Bit Carry Look Ahead Adder’ are equivalent to the corresponding outputs of the normal 4 bits adder. The policy here is as follows: when the functional (semantic) correctness of a system is already proven, and the correspondence of the system to a (normal) logic circuit is given, it is enough to prove the correctness of the new circuit if we only prove the logical equivalence between them. Although the article is very fundamental (it contains few environment files), it can be applied to real problems. The key of the method introduced here is to put the specification of the logic circuit into the Mizar propositional formulae, and to use the strong inference ability of the Mizar checker. The proof is done formally so that the automation of the proof writing is possible. Even in the 5.3.07 version of Mizar, it can handle a formulae of more than 100 lines, and a formula which contains more than 100 variables. This means that the Mizar system is enough to prove logical correctness of middle scaled logic circuits.