Definitional Quantifiers Realise Semantic Reasoning for Proof by Induction

TL;DR

This paper introduces SeLFiE, a boolean query language with definitional quantifiers, to automate proof by induction in Isabelle/HOL, significantly improving the efficiency of inductive problem solving.

Contribution

We developed SeLFiE with definitional quantifiers to enhance automation in proof by induction, integrating syntactic and definitional analysis for better heuristic application.

Findings

Achieved 1,400% speedup over baseline prover within 1 second timeout.

Median speedup of 4.48x across 347 inductive problems.

Demonstrated effectiveness of definitional quantifiers in semantic reasoning.

Abstract

Proof assistants offer tactics to apply proof by induction, but these tactics rely on inputs given by human engineers. To automate this laborious process, we developed SeLFiE, a boolean query language to represent experienced users' knowledge on how to apply the induct tactic in Isabelle/HOL: when we apply an induction heuristic written in SeLFiE to an inductive problem and arguments to the induct tactic, the SeLFiE interpreter judges whether the arguments are plausible for that problem according to the heuristic by examining both the syntactic structure of the problem and definitions of the relevant constants. To examine the intricate interaction between syntactic analysis and analysis of constant definitions, we introduce definitional quantifiers. For evaluation we build an automatic induction prover using SeLFiE. Our evaluation based on 347 inductive problems shows that our new prover achieves 1.4 x 10^3% improvement over the corresponding baseline prover for 1.0 second of timeout and the median value of speedup is 4.48x.