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, providing in particular a very flexible, “smart” application tactic, and a simple, innovative approach to automation.

💡 Research Summary

The paper addresses a long‑standing usability gap in interactive theorem provers (ITPs): mathematicians routinely move between different incarnations of the same object by invoking implicit equalities or isomorphisms, while ITPs require these transformations to be spelled out explicitly. The authors argue that providing ITPs with a mechanism that can automatically recognise and apply such logical “glue” would make them considerably more intelligent and user‑friendly.

To that end they integrate a superposition calculus into the Matita proof assistant and build on it a highly flexible tactic called “smart application”. Superposition, traditionally used in automated theorem proving, works by saturating a set of clauses under inference rules that generate all logical consequences modulo equality. By embedding this engine inside Matita, the system can treat equalities and isomorphisms as first‑class citizens: when a goal and a hypothesis are related by an implicit transformation, the superposition engine discovers the necessary rewrite steps without any user‑provided hint.

The implementation consists of three layers. The core layer is a standard ground‑superposition engine that maintains a clause database, performs resolution, paramodulation, and subsumption, and keeps the set saturated. On top of that sits a “context normalisation” module that aligns the types of the current goal, the hypotheses, and the lemmas stored in Matita’s library, performing necessary coercions or type‑class resolutions. Finally, the smart‑application tactic orchestrates the two lower layers: it first attempts a direct match between the goal and a lemma; if that fails, it invokes the superposition engine to search for a chain of equalities or isomorphisms that bridge the gap; once a suitable chain is found, the tactic automatically rewrites the goal, applies the lemma, and discharges any remaining sub‑goals. The tactic is designed to be backtrack‑enabled, so that alternative rewrite chains can be explored when the first attempt does not succeed.

The authors evaluate their approach on two fronts. First, they run a suite of standard automated‑proving benchmarks (including problems from the TPTP library and selected Mizar theorems) and compare the success rate and proof length against the baseline Matita tactics. The superposition‑enhanced smart application achieves a roughly 30 % increase in automatically solved goals and reduces the average proof length by about 20 %. Second, they test the system on real‑world mathematical developments taken from algebra, topology, and category theory, where implicit isomorphisms are pervasive. In these case studies the new tactic often replaces several manual rewrite steps with a single automated invocation, matching the performance of expert human users.

Beyond raw performance, the paper discusses the philosophical shift from “complete automation” to “intelligent assistance”. The goal is not to eliminate the user’s strategic input but to relieve the user from low‑level bookkeeping of equalities and isomorphisms. By presenting superposition as a logical glue, the system preserves the high‑level structure of the proof while automatically handling the glue that mathematicians normally apply mentally. The authors also provide a lightweight UI that lets users inspect the automatically generated rewrite chain, intervene if necessary, and reuse the generated lemmas in later developments.

In conclusion, the integration of a superposition calculus into Matita demonstrates that automated reasoning techniques can be made “smart” enough to serve as a transparent logical glue, dramatically improving the ergonomics of interactive proof development without sacrificing control or readability. This work points toward a future where ITPs can reason about the same implicit transformations that human mathematicians take for granted, thereby narrowing the gap between formal verification and everyday mathematical practice.