Semi-intelligible Isar Proofs from Machine-Generated Proofs

Abstract

Sledgehammer is a component of the Isabelle/HOL proof assistant that integrates external automatic theorem provers (ATPs) to discharge interactive proof obligations. As a safeguard against bugs, the proofs found by the external provers are reconstructed in Isabelle. Reconstructing complex arguments involves translating them to Isabelle's Isar format, supplying suitable justifications for each step. Sledgehammer transforms the proofs by contradiction into direct proofs; it iteratively tests and compresses the output, resulting in simpler and faster proofs; and it supports a wide range of ATPs, including E, LEO-II, Satallax, SPASS, Vampire, veriT, Waldmeister, and Z3. Keywords Automatic theorem provers • Proof assistants • Natural deduction 1 Introduction Sledgehammer [12,62] is a proof tool that connects the Isabelle/HOL proof assistant [53,54] with external automatic theorem provers (ATPs), including first-order superposition provers, higher-order provers, and solvers based on satisfiability modulo theories (SMT). Given an interactive proof goal, it heuristically selects hundreds of facts (lemmas, definitions, and axioms) from Isabelle's vast libraries, translates them to the external provers' logics, and invokes the provers in parallel (Section 2).

A Learning-Based Fact Selector for Isabelle/HOL

“…We use a collection of automatic provers for this. As we noted in a similar study [31], It is important to bear in mind that the evaluation is not a competition between the provers. Different provers are invoked with different problems and options, and The experiments were conducted on a 64-bit Linux server equipped with 12-core AMD Opteron 6174 processors running at 2.2 GHz.…”

“…Then it reconstructs the minimized proof in Isabelle by a suitable proof text-typically a single call to the built-in resolution prover Metis [30], but sometimes a detailed, structured proof [31,32].…”

Reconstructing veriT Proofs in Isabelle/HOL

“…While this experiment provides insight into the performance of the veriT-powered smt tactic relative to the Z3-powered variant, it does not tell us if the veriT-powered one is a useful and supplementary addition to the family of automated tactics provided by Isabelle. Towards that end, we try to generate new veriT-powered smt calls by using Sledgehammer [7], an Isabelle tool able to find proofs.…”

“…Sadly, the code to read the SMT-LIB input problems was never included in the standard Isabelle distribution and is now lost. Proof reconstruction has been heavily tested and succeeds in more than 90% of the cases according to Sledgehammer benchmark [7,Section 9], and is very efficient.…”

Language and Proofs for Higher-Order SMT (Work in Progress)

“…We focus on the proof format of the SMT solver veriT [9]. This solver is known for its very detailed proofs [2,6], which are reconstructed in the proof assistants Isabelle/HOL [7] and the GAPT system [10].…”