Proving and rewriting

This paper presents some ways to prove theorems in first and second order logic, such that rewriting does the routine work automatically, and partially successful proofs often return information that suggests what to try next. The theoretical framework makes extensive use of general algebra, and main results include an extension of many-sorted equational logic to universal quantification over functions, some techniques for handling first order logic, and some structural induction principles. The OBJ language is used for illustration, and initiality is a recurrent theme.