Equational Theories in CERES

Cut-elimination is the most important proof transformation in logic. Equality is a central paradigm in mathematics and plays a key role in automated deduction. Therefore its importance awakes the necessity of integrating equality into existing cut-elimination methods. In this paper we extend the resolution-based method of cut-elimination CERES to CERES-e by adding equality (and paramodulation to LK), where all the advantages of CERES are preserved; in particular CERES-e is superior to Gentzen type methods, is exible with respect to resolution, paramodulation and its re nements and admits a semantical use of cut. We go even further and combine CERES-e with equational theories yielding a system which adds simplicity of proof notation to the advantages gained from equality.