Towards an Interactive Mathematical Proof Mode

Abstract

Formalizing mathematical proofs has as aim to represent arbitrary mathematical notions and proofs on a computer in order to construct a database of certified results useful to learn and develop the subject. At present it is mathematically not appealing to construct formal proofs. To make formalzing more mathematician-friendly one should have a good interface for proofs, definitions and computations. The proof-assistant Mizar does have a good interface for proofs, but not for making computations. Other assistants, like Coq based on type theory, do have a good interface for computations, but not for proofs. This paper sketches ways in which proofs are represented in a mathematical way. Although the underlying formalized statements come from the system Coq, this is not essential Mainly the paper has as aim to convince implementers of mathematical assistants to make systems in such a way that formalizing proofs becomes natural. Much further developed is the work on Isar providing a mathematical proof language for the assistant Isabelle. The approach in this paper is to approximate a proof language by writing proof-sketches, a notion by Wiedijk, with the aim that they should eventually be verifiable by a proof-checker. [Nederpelt, 2002] has a different approach: there the emphasis is on the ease of providing formalizations of mathematical definitions.