Problem
Research questions and friction points this paper is trying to address.
exploratory mathematics
equational reasoning
proof interfaces
formal methods
mathematics education
Proof Interfaces for Exploratory Mathematics
🤖 AI Summary
This study addresses the challenges of error-prone transcription, cumbersome notation, and high barriers to entry in formalizing mathematical derivations by proposing an equational reasoning approach that supports multi-level automation. Building upon a rewrite-search architecture, we extend the Hazel live programming environment to enable seamless exportation from interactive exploration to the Rocq theorem prover, thereby accommodating both novice practice and expert-level research needs. Case studies demonstrate that the proposed system effectively bridges the accessibility gap for formal methods in exploratory mathematics, exhibiting strong usability and efficacy across contexts ranging from foundational education to professional research.
📝 Abstract
This paper describes a mathematics interface intended for both educational applications and exploratory mathematics. Both learners and mathematics practitioners often use pen and paper or a whiteboard to perform equational reasoning, manipulate expressions, or construct proofs. These workflows lead to a variety of problems: transcription errors, tedious writing and notation, and/or unclear standards for proof justification. We extend the Hazel Prover, an equational-reasoning interface in the Hazel live programming environment, to add capabilities for exploratory mathematics, with varying levels of verbosity and automation aimed at both students and expert users. Students require more deliberate practice when learning mathematical concepts and correspondingly more verbose justifications, while experts may benefit from significant mathematical automation. With this in mind, our interface supports multiple levels of mathematical automation and simplification, grounded in a rewrite search architecture. For expert users, motivated by a gap in the accessibility of formal methods, we support proof export to the Rocq theorem prover, extending this rewrite search to proof tactics. This paper closes with several case studies covering elementary- to college-level mathematics.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Exploratory Mathematics
Equational Reasoning
Rewrite Search
Proof Export
Theorem Proving
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