A Novice-Friendly Induction Tactic for Lean (2012.08990v1)

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.

• The paper introduces a refined induction tactic in Lean 3 that improves usability by generating more general hypotheses and intuitive names.
• It streamlines the treatment of indexed inductive types by avoiding redundant arguments, making complex proofs more accessible.
• The tactic’s integration into Lean's metaprogramming framework demonstrates significant educational and practical advancements in theorem proving.

A Novice-Friendly Induction Tactic for Lean

In theorem proving environments such as Lean and Coq, induction tactics are pivotal. Despite their importance, traditional tactics present usability issues, particularly for novices. Jannis Limperg's work introduces a refined induction tactic in Lean 3 that ameliorates these usability hurdles, broadening accessibility to both beginners and experts.

Key Improvements and Features

The proposed induction tactic in Lean 3 is primarily concerned with three major areas that affect usability: indexed inductive types, overly specific induction hypotheses, and naming conventions.

1. Indexed Inductive Types: Traditional induction tactics in Lean and Coq are often suboptimal when dealing with indexed inductive types. They can produce counterintuitive goals, complicating novice understanding. The paper proposes an enhanced tactic akin to dependent induction but with simplified induction hypotheses. This is pivotal in avoiding overly redundant arguments and simplifying goals, easing the learning curve for novice users.
2. Overly Specific Induction Hypotheses: Another significant issue with existent tactics is the generation of overly specific hypotheses. Limperg's tactic reverses this tendency by defaulting to more general hypotheses. This aids novices who may struggle to identify the limitations of specific induction hypotheses—enabling them to progress without getting bogged down in recognizing where generalization is needed.
3. Naming Conventions: Meaningful and intuitive naming of generated hypotheses is crucial, especially for educational purposes. The standard naming schemes in Lean are often too complex and obscure the connection to original hypotheses. By applying heuristics that align with common intuitions about induction, the new tactic generates names that make goals appear more natural and relatable.

Implementation and Usability in Education

The revised induction tactic explicitly addresses issues through metaprogramming within Lean 3’s framework. Crucially, this framework allows for a detailed case paper into improving Lean's usability in theorem proving education, particularly for students who may encounter unnecessary cognitive barriers due to complex tool workflows.

As part of the Logical Verification course at Vrije Universiteit Amsterdam, the new tactic was employed to replace Lean’s standard induction tactic, leading to a more streamlined educational process. The tactic is implemented in Lean’s mathlib, allowing seamless integration into the existing system.

Challenges and Insights from Lean's Metaprogramming Framework

Implementing the tactic surfaced various insights into Lean's metaprogramming framework. While the metaprogramming capabilities in Lean allowed exploration of complex tasks, some limitations required attention:

• Tracking and Naming: Unique approaches to tracking changes to hypothesis names and types were needed since many operations within Lean alter the context in a way that invalidates previous identifiers.
• Definitional Equality: The necessity to handle definitional equality with explicit tactics due to a lack of comprehensive interfaces posed challenges. This was further complicated by transparency levels and their management during expression construction and deconstruction.
• Elaboration Process: Constructing expressions required careful management of the elaboration process to appropriately fill in implicit arguments and maintain performance efficiency.
• Generalized Inductive Types: Lean's treatment of generalized inductive types unearthed challenges in representing them fully within tactics due to the translations applied internally by Lean.

Future Implications

The induction tactic described opens pathways for further development of user-friendly theorem proving tools. Future advancements could focus on integrating these improvements across other proof assistants, enhancing educational accessibility, and reducing the overhead associated with complex type theory and recursion learning curves.

The described enhancements mark an important stride in the evolution of Lean's toolset, potentially influencing broader metaprogramming frameworks. The lessons learned could inform system designers aiming to create robust, user-centric environments for formal verification and theorem proving.