🤖 AI Summary
This work investigates the goal-directed generation of mathematically meaningful theorems—or lemmas suitable for automated proof—from a given set of axioms. To this end, it introduces a novel approach grounded in the propositions-as-types paradigm, which systematically partitions the space of proof terms according to inductive levels and integrates proof-term enumeration with compression techniques, including separation-based reduction, DAG compression, and combinatory logic. This framework enables the efficient construction and compact representation of proof structures. Experimental evaluation on a fragment of Metamath’s set.mm library demonstrates that the method successfully produces nontrivial and semantically relevant theorems, thereby confirming its feasibility and advantages in the context of automated theorem discovery.
📝 Abstract
We address generating theorems from a given set of axioms, without proof goal, aiming at value from a mathematical point of view or as lemmas for automated proving. As benchmark, we convert a fragment of the Metamath database set.mm. Our techniques are centered on proof terms and condensed detachment, which ties in with approaches to automated first-order proving by proof structure enumeration, and links to Metamath as well as to formulas-as-types. Our methods for generating theorems are based on partitioning the set of proof terms into inductively characterized levels. We study two ideas for improvement: Lemma synthesis by DAG compression of proof term sets and incorporating combinators into proof term construction.