Using small scale automation to improve both accessibility and readability of formal proofs in geometry

This paper describes some techniques to help building formal proofs in geometry and at the same time improving readability. Rather than trying to completely automate the proving process we provide symbolic manipulations which are useful to automate the parts of the formal proof that are usually implicit in a pen and paper proof. We test our framework using some well known theorems about triangles which are taught in high-school. We also highlight the proof steps which are usually overlooked in the informal proof and that we believe should be made explicit. Our framework is based on Tarski's geometry within the Coq proof assistant, but most of the ideas presented in this paper could be applied to other axiomatic systems or proof assistants.

[1]  Gilles Kahn,et al.  Extracting Text from Proofs , 1995, TLCA.

[2]  Pascal Schreck,et al.  Formalization of Wu's Simple Method in Coq , 2011, CPP.

[3]  Michael Beeson,et al.  A constructive version of Tarski's geometry , 2014, Ann. Pure Appl. Log..

[4]  Stéphane Lescuyer.  First-Class Containers in Coq , 2011, Stud. Inform. Univ..

[5]  Sana Stojanovic,et al.  A Coherent Logic Based Geometry Theorem Prover Capable of Producing Formal and Readable Proofs , 2010, Automated Deduction in Geometry.

[6]  Pascal Schreck,et al.  Formalizing Desargues' theorem in Coq using ranks , 2009, SAC '09.

[7]  Pedro Quaresma,et al.  The Area Method , 2010, Journal of Automated Reasoning.

[8]  Christoph Kreitz,et al.  The Nuprl Open Logical Environment , 2000, CADE.

[9]  Julien Narboux,et al.  A Graphical User Interface for Formal Proofs in Geometry , 2007, Journal of Automated Reasoning.

[10]  A. Tarski,et al.  Metamathematische Methoden in der Geometrie , 1983 .

[11]  Julien Narboux.  Formalisation et automatisation du raisonnement géométrique en Coq. (Formalisation and automation of geometric reasoning within Coq) , 2006 .

[12]  Julien Narboux,et al.  From Tarski to Hilbert , 2012, Automated Deduction in Geometry.

[13]  Pascal Schreck,et al.  Higher-Order Intuitionistic Formalization and Proofs in Hilbert's Elementary Geometry , 2000, Automated Deduction in Geometry.

[14]  Damien Pous,et al.  Tactics for Reasoning Modulo AC in Coq , 2011, CPP.

[15]  Julien Narboux,et al.  Mechanical Theorem Proving in Tarski's Geometry , 2006, Automated Deduction in Geometry.

[16]  Xiao-Shan Gao,et al.  Proving Geometry Statements of Constructive Type , 1992, CADE.

[17]  S. Chou.  Mechanical Geometry Theorem Proving , 1987 .

[18]  Tuan-Minh Pham,et al.  A Combination of a Dynamic Geometry Software With a Proof Assistant for Interactive Formal Proofs , 2012, Electron. Notes Theor. Comput. Sci..

[19]  Markus Wenzel,et al.  Isabelle, Isar - a versatile environment for human readable formal proof documents , 2002 .

[20]  Tobias Nipkow,et al.  Equational Reasoning in Isabelle , 1989, Sci. Comput. Program..

[21]  Pierre Corbineau,et al.  A Declarative Language for the Coq Proof Assistant , 2007, TYPES.

[22]  Yves Bertot,et al.  Interactive Theorem Proving and Program Development: Coq'Art The Calculus of Inductive Constructions , 2010 .

[23]  A. Tarski.  What is Elementary Geometry , 1959 .

[24]  Franz Baader,et al.  Unification theory , 1986, Decis. Support Syst..

[25]  Robert L. Constable,et al.  Verbalization of High-Level Formal Proofs , 1999, AAAI/IAAI.

[26]  Jürgen Avenhaus,et al.  Efficient Algorithms for Computing Modulo Permutation Theories , 2004, IJCAR.

[27]  B. Buchberger,et al.  Gröbner bases and applications , 1998 .

[28]  Enrico Tassi,et al.  A Small Scale Reflection Extension for the Coq system , 2008 .

[29]  Julien Narboux,et al.  A Decision Procedure for Geometry in Coq , 2004, TPHOLs.

[30]  S. Chou,et al.  A Class of Geometry Statements of Constructive Type and Geometry TheoremProving , 1989 .

[31]  W. Wu.  ON THE DECISION PROBLEM AND THE MECHANIZATION OF THEOREM-PROVING IN ELEMENTARY GEOMETRY , 2008 .

[32]  Samuel Boutin,et al.  Using Reflection to Build Efficient and Certified Decision Procedures , 1997, TACS.