Superposition Theorem Proving for Abelian Groups Represented as Integer Modules

We define a superposition calculus specialized for abelian groups represented as integer modules, and show its refutational completeness. This allows to substantially reduce the number of inferences compared to a standard superposition prover which applies the axioms directly. Specifically, equational literals are simplified, so that only the maximal term of the sums is on the left-hand side. Only certain minimal superpositions need to be considered; other superpositions which a standard prover would consider become redundant. This not only reduces the number of inferences, but also reduces the size of the AC-unification problems which are generated. That is, AC-unification is not necessary at the top of a term, only below some non-AC-symbol. Further, we consider situations where the axioms give rise to variable overlaps and develop techniques to avoid these explosive cases where possible