A case-study in algebraic manipulation using mechanized reasoning tools

In this article, two different mechanized reasoning tools (Coq and Isabelle/HOL) are used in order to prove some basic but significant properties extracted from a symbolic computation system called Kenzo. In particular, we focused on a property called ‘cancellation theorem’, which can be applied to the proof of the idempotence property of a differential morphism. This result is used as a case-study to compare the capabilities and styles of the two proof assistants. The conclusion of our comparison is that both tools are adequate to solve the case-study presented in this article in a rather similar way but depending on their specific styles. This research is part of a more general project devoted to increase the reliability of computer algebra systems for calculations in algebraic topology.

[1]  A. Zients.  Andy , 2003 .

[2]  Markus Wenzel,et al.  Context Aware Calculation and Deduction , 2007, Calculemus/MKM.

[3]  César Domínguez,et al.  Object oriented institutions to specify symbolic computation systems , 2007, RAIRO Theor. Informatics Appl..

[4]  Panagiotis Manolios,et al.  Computer-Aided Reasoning: An Approach , 2011 .

[5]  Samuel Eilenberg,et al.  On Products of Complexes , 1953 .

[6]  Makarius Wenzel,et al.  Context aware Calculation and Deduction Ring Equalities via Gröbner Bases in Isabelle , 2007 .

[7]  Lawrence Charles Paulson,et al.  Isabelle/HOL: A Proof Assistant for Higher-Order Logic , 2002 .

[8]  Jesús Aransay,et al.  A Mechanized Proof of the Basic Perturbation Lemma , 2008, Journal of Automated Reasoning.

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

[10]  Jon P. May.  Simplicial objects in algebraic topology , 1993 .

[11]  Alonzo Church,et al.  A formulation of the simple theory of types , 1940, Journal of Symbolic Logic.

[12]  Francis Sergeraert,et al.  Constructive algebraic topology , 1999, SIGS.

[13]  Clemens Ballarin,et al.  Towards a higher reasoning level in formalized Homological Algebra , 2002 .

[14]  Freek Wiedijk,et al.  The Seventeen Provers of the World, Foreword by Dana S. Scott , 2006, The Seventeen Provers of the World.

[15]  Thierry Coquand,et al.  The Calculus of Constructions , 1988, Inf. Comput..

[16]  Assia Mahboubi,et al.  Proving Formally the Implementation of an Efficient gcd Algorithm for Polynomials , 2006, IJCAR.

[17]  Jesús Aransay,et al.  Four Approaches to Automated Reasoning with Differential Algebraic Structures , 2004, AISC.

[18]  Pierre Castéran,et al.  Interactive Theorem Proving and Program Development , 2004, Texts in Theoretical Computer Science An EATCS Series.

[19]  Vico Pascual,et al.  An Object-oriented Interpretation of the EAT System , 2003, Applicable Algebra in Engineering, Communication and Computing.

[20]  César Domínguez.  Formalizing in Coq Hidden Algebras to Specify Symbolic Computation Systems , 2008, AISC/MKM/Calculemus.

[21]  Konrad Slind,et al.  Treating Partiality in a Logic of Total Functions , 1997, Comput. J..

[22]  Julio Rubio,et al.  Executing in Common Lisp, Proving in ACL2 , 2007, Calculemus/MKM.

[23]  Christine Paulin-Mohring,et al.  Inductive Definitions in the system Coq - Rules and Properties , 1993, TLCA.

[24]  Freek Wiedijk,et al.  The Seventeen Provers of the World , 2006 .

[25]  César Domínguez,et al.  Modeling Inheritance as Coercion in the Kenzo System , 2006, J. Univers. Comput. Sci..

[26]  Allen Newell,et al.  Report on a general problem-solving program , 1959, IFIP Congress.