Definitional Quantifiers Realise Semantic Reasoning for Proof by Induction

This year, I presented my domain-specific language, SeLFiE, and the concept it implements at TAP2022.

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⋅10^3% improvement over the corresponding baseline prover for 1.0 s of timeout and the median value of speedup is 4.48x.

The concept of definitional quantifiers is what allowed sem_ind to make accurate recommendations when proof by induction requires variable generalisation.