From informal to formal proofs in Euclidean geometry

• Published in: Annals of mathematics and artificial intelligence Vol. 85; no. 2-4; pp. 89 - 117
• Main Author: Stojanović-Ðurđević, Sana
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.04.2019; Springer; Springer Nature B.V
• Subjects: Artificial Intelligence; Automation; Complex Systems; Computer Science; Euclidean geometry; Geometry; Language; Logic; Mathematics; Natural language; Secondary schools; Textbooks; Theorem proving; Theorems; Euclidean geometry; 68T15; Informal proofs; Coherent logic; Interactive theorem proving; Automated theorem proving; 03B35
• ISSN: 1012-2443; 1573-7470
• DOI: 10.1007/s10472-018-9597-7

In this paper, we propose a new approach for automated verification of informal proofs in Euclidean geometry using a fragment of first-order logic called coherent logic and a corresponding proof representation. We use a TPTP inspired language to write a semi-formal proof of a theorem, that fairly accurately depicts a proof that can be found in mathematical textbooks. The semi-formal proof is verified by generating more detailed proof objects expressed in the coherent logic vernacular. Those proof objects can be easily transformed to Isabelle and Coq proof objects, and also in natural language proofs written in English and Serbian. This approach is tested on two sets of theorem proofs using classical axiomatic system for Euclidean geometry created by David Hilbert, and a modern axiomatic system E created by Jeremy Avigad, Edward Dean, and John Mumma.