An Interactive SMT Tactic in Coq using Abductive Reasoning

Abstract

A well-known challenge in leveraging automatic theorem provers, such as satisfiability modulo theories (SMT) solvers, to discharge proof obligations from interactive theorem provers (ITPs) is determining which axioms to send to the solver together with the con- jecture to be proven. Too many axioms may confuse or clog the solver, while too few may make a theorem unprovable. When a solver fails to prove a conjecture, it is unclear to the user which case transpired. In this paper we enhance SMTCoq — an integration between the Coq ITP and the cvc5 SMT solver — with a tactic called abduce aimed at mitigating the uncertainty above. When the solver fails to prove the goal, the user may invoke abduce which will use abductive reasoning to provide facts that will allow the solver to prove the goal, if any.

Automated Completion of Statements and Proofs in Synthetic Geometry: an Approach based on Constraint Solving

“…For instance, an automated system may help a student or a researcher who tries to prove (or formalize) a theorem with a missing assumption. Barbosa et al have proposed such goal (although not for geometry) in the context of interactive proof assistants where conjectures are sent to an SMT solver [2].…”