# ECR: An equality conditional resolution proof procedure

## Abstract

This paper presents an equality conditional resolution proof procedure, ECR, that incorporates a user's knowledge concerning the different roles of input equations in a proof. The input equations are separated into different classes according to the roles they will play. Each such role has rule schema into which the corresponding equations are transformed. The conditions on the application of these rules control the inference, and prevent inappropriate use of the equations. The paper will introduce the concept of potential completeness to characterize the generality of a deduction method. ECR is potentially complete for proving the set of positive Horn theorems, that is, it can finally prove any of these theorems by first proving the needed lemmas, and then using them. This procedure has been used to prove a number of theorems of group theory, ring theory, boolean algebra and field theory. The paper will give a summary of these proofs, among which, the efficient proofs of the associativity law and De.Morgan's law of boolean algebra seem interesting.

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