Program extraction in exact real arithmetic

The importance of an abstract approach to a computation theory over general data types has been stressed by Tucker in many of his papers. Berger and Seisenberger recently elaborated the idea for extraction out of proofs involving (only) abstract reals. They considered a proof involving coinduction of the proposition that any two reals in [−1, 1] have their average in the same interval, and informally extract a Haskell program from this proof, which works with stream representations of reals. Here we formalize the proof, and machine extract its computational content using the Minlog proof assistant. This required an extension of this system to also take coinduction into account.

[1]  Ulrich Kohlenbach,et al.  Applied Proof Theory - Proof Interpretations and their Use in Mathematics , 2008, Springer Monographs in Mathematics.

[2]  J. V. Tucker,et al.  Theory of Computation over Stream Algebras, and its Applications , 1992, MFCS.

[3]  Christine Paulin-Mohring,et al.  The coq proof assistant reference manual , 2000 .

[4]  Pierre Letouzey,et al.  A New Extraction for Coq , 2002, TYPES.

[5]  Edwin Wiedmer Exaktes Rechnen mit reellen Zahlen und anderen unendlichen Objekten , 1977 .

[6]  E. Wiedmer,et al.  Computing with Infinite Objects , 1980, Theor. Comput. Sci..

[7]  Alberto Ciaffaglione,et al.  A certified, corecursive implementation of exact real numbers , 2006, Theor. Comput. Sci..

[8]  D. Dalen Review: Georg Kreisel, Godel's Intepretation of Heyting's Arithmetic; G. Kreisel, Relations Between Classes of Constructive Functionals; Georg Kreisel, A. Heyting, Interpretation of Analysis by Means of Constructive Functionals of Finite Types , 1971 .

[9]  Stefan Berghofer,et al.  Proofs, programs and executable specifications in higher order logic , 2003 .

[10]  Ulrich Berger,et al.  From Coinductive Proofs to Exact Real Arithmetic , 2009, CSL.

[11]  Chi Ming Chuang,et al.  Extraction of Programs for Exact Real Number Computation Using Agda , 2011 .

[12]  Ulrich Berger,et al.  Term rewriting for normalization by evaluation , 2003, Inf. Comput..

[13]  R.S.S. O'Connor,et al.  Incompleteness & completeness : formalizing logic and analysis in type theory , 2005 .

[14]  Ulrich Berger,et al.  Program Extraction from Normalization Proofs , 2006, Stud Logica.

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

[16]  D. Plume A calculator for exact real number computation , 1998 .

[17]  Ulrich Berger,et al.  Proofs, Programs, Processes , 2011, Theory of Computing Systems.

[18]  Helmut Schwichtenberg,et al.  Realizability interpretation of proofs in constructive analysis , 2008, Theory of Computing Systems.

[19]  Douglas S. Bridges,et al.  Constructivity in Mathematics , 2004 .