Superposition as a logical glue

The typical mathematical language systematically exploits notational and logical abuses whose resolution requires not just the knowledge of domain specific notation and conventions, but not trivial skills in the given mathematical discipline. A large part of this background knowledge is expressed in form of equalities and isomorphisms, allowing mathematicians to freely move between different incarnations of the same entity without even mentioning the transformation. Providing ITP-systems with similar capabilities seems to be a major way to improve their intelligence, and to ease the communication between the user and the machine. The present paper discusses our experience of integration of a superposition calculus within the Matita interactive prover. Superposition provides the key ingredient for a “smart” application tactic that allows the user to disregard many tedious details otherwise needed to convince the system that his reasoning step is indeed correct. We also show how this kind of automation, named small scale, can serve as the building block for the more general, large scale, case, allowing a smooth integration of equational reasoning with backward-based proof searching procedures.