Inverting Monotone Continuous Functions in Constructive Analysis

We prove constructively (in the style of Bishop) that every monotone continuous function with a uniform modulus of increase has a continuous inverse. The proof is formalized, and a realizing term extracted. This term can be applied to concrete continuous functions and arguments, and then normalized to a rational approximation of say a zero of a given function. It turns out that even in the logical term language “normalization by evaluation” is reasonably efficient.

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

[2]  Patrik Andersson.  Exact Real Arithmetic with Automatic Error Estimates in a Computer Algebra System , 2001 .

[3]  Ulrich Berger,et al.  Uniform Heyting arithmetic , 2005, Ann. Pure Appl. Log..

[4]  Witold Hurewicz,et al.  Lectures on Ordinary Differential Equations , 1959 .

[5]  Herman Geuvers,et al.  A Constructive Proof of the Fundamental Theorem of Algebra without Using the Rationals , 2000, TYPES.

[6]  Maribel Fernández,et al.  Curry-Style Types for Nominal Terms , 2006, TYPES.

[7]  Von Kurt Gödel,et al.  ÜBER EINE BISHER NOCH NICHT BENÜTZTE ERWEITERUNG DES FINITEN STANDPUNKTES , 1958 .

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

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

[10]  Sam Lindley,et al.  Extensional Rewriting with Sums , 2007, TLCA.

[11]  M. Mandelkern.  Continuity of monotone functions. , 1982 .

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

[13]  Josef Berger.  Exact calculation of inverse functions , 2005, Math. Log. Q..

[14]  Cruz Filipe,et al.  Constructive real analysis : a type-theoretical formalization and applications , 2004 .

[15]  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 .