A Novice-Friendly Induction Tactic for Lean

TL;DR

This paper introduces a new, user-friendly induction tactic for Lean 3 that improves ergonomics, supports inductive predicates, and simplifies proof development, especially for educational purposes.

Contribution

It presents a novel induction tactic for Lean 3 that enhances usability and addresses limitations of existing tactics, serving as a case study for metaprogramming in Lean.

Findings

The new tactic reliably supports inductive predicates.

It produces less complex and more appropriate induction hypotheses.

It demonstrates the metaprogramming capabilities of Lean 3.

Abstract

In theorem provers based on dependent type theory such as Coq and Lean, induction is a fundamental proof method and induction tactics are omnipresent in proof scripts. Yet the ergonomics of existing induction tactics are not ideal: they do not reliably support inductive predicates and relations; they sometimes generate overly specific or unnecessarily complex induction hypotheses; and they occasionally choose confusing names for the hypotheses they introduce. This paper describes a new induction tactic, implemented in Lean 3, which addresses these issues. The tactic is particularly suitable for educational use, but experts should also find it more convenient than existing induction tactics. In addition, the tactic serves as a moderately complex case study for the metaprogramming framework of Lean 3. The paper describes some difficulties encountered during the implementation and suggests improvements to the framework.