From coinductive proofs to exact real arithmetic: theory and applications

Based on a new coinductive characterization of continuous functions we extract certified programs for exact real number computation from constructive proofs. The extracted programs construct and combine exact real number algorithms with respect to the binary signed digit representation of real numbers. The data type corresponding to the coinductive definition of continuous functions consists of finitely branching non-wellfounded trees describing when the algorithm writes and reads digits. We discuss several examples including the extraction of programs for polynomials up to degree two and the definite integral of continuous maps

Minlog - A Tool for Program Extraction Supporting Algebras and Coalgebras

Minlog is an interactive proof system which implements prooftheoreticmethods and applies them to verication and program extraction.We give an overview of the system and demonstrate how it can beused to exploit the computational content in (co)algebraic proofs and todevelop correct and ecient programs

Proofs, Programs, Processes

The objective of this paper is to provide a theoretical foundation for program extraction from proofs. We give a realizability interpretation for first-order proofs involving inductive and coinductive definitions and discuss its application to the synthesis of provably correct programs. We show that realizers, although per-se untyped, can be assigned polymorphic recursive types and hence represent valid programs in a lazy functional programming language such as Haskell. Programs extracted from proofs using coinduction can be understood as perpetual processes producing infinite streams of data. Typical applications of such processes are computations in exact real arithmetic. As an example we show how to extract a program computing the average of two real numbers w.r.t.\ to the binary signed digit representation

From Coinductive Proofs to Exact Real Arithmetic

Abstract. We give a coinductive characterisation of the set of continuous functions defined on a compact real interval, and extract certified programs that construct and combine exact real number algorithms with respect to the binary signed digit representation of real numbers. The data type corresponding to the coinductive definition of continuous functions consists of finitely branching non-wellfounded trees describing when the algorithm writes and reads digits. This is a pilot study in using proof-theoretic methods for obtaining certified algorithms in exact real arithmetic.