Computing in coq with infinite algebraic data structures

Computational content encoded into constructive type theory proofs can be used to make computing experiments over concrete data structures. In this paper, we explore this possibility when working in Coq with chain complexes of infinite type (that is to say, generated by infinite sets) as a part of the formalization of a hierarchy of homological algebra structures.

[1]  Thierry Coquand,et al.  Towards Constructive Homological Algebra in Type Theory , 2007, Calculemus/MKM.

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

[3]  Julio Rubio,et al.  Computing with locally effective matrices , 2005, Int. J. Comput. Math..

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

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

[6]  Julio Rubio,et al.  Constructive algebraic topology , 2002 .

[7]  Pierre Letouzey.  Extraction in Coq: An Overview , 2008, CiE.

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

[9]  Georges Gonthier,et al.  Formal Proof—The Four- Color Theorem , 2008 .

[10]  N. Jacobson,et al.  Basic Algebra II , 1989 .

[11]  César Domínguez,et al.  Effective homology of bicomplexes, formalized in Coq , 2011, Theor. Comput. Sci..

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

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

[14]  Koen Claessen,et al.  QuickCheck: a lightweight tool for random testing of Haskell programs , 2000, ICFP.

[15]  Jesús Aransay,et al.  Modelling Differential Structures in Proof Assistants: The Graded Case , 2009, EUROCAST.

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

[17]  Enrico Tassi,et al.  A Modular Formalisation of Finite Group Theory , 2007, TPHOLs.

[18]  Assia Mahboubi,et al.  Implementing the cylindrical algebraic decomposition within the Coq system , 2007, Mathematical Structures in Computer Science.

[19]  Herman Geuvers,et al.  A Constructive Algebraic Hierarchy in Coq , 2002, J. Symb. Comput..

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

[21]  Laureano Lambán Pardo,et al.  Contribuciones científicas en honor de Mirian Andrés Gómez , 2010 .

[22]  Jesús Aransay,et al.  Generating certified code from formal proofs: a case study in homological algebra , 2010, Formal Aspects of Computing.

[23]  Francis Sergeraert,et al.  The Computability Problem in Algebraic Topology , 1994 .

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