Coinductive Correctness of Homographic and Quadratic Algorithms for Exact Real Numbers

Abstract

In this article we present a method for formally proving the correctness of the lazy algorithms for computing homographic and quadratic transformations — of which field operations are special cases— on a representation of real numbers by coinductive streams. The algorithms work on coinductive stream of Möbius maps and form the basis of Edalat–Potts exact real arithmetic. We build upon our earlier work of formalising the homographic and quadratic algorithms in constructive type theory via general corecursion. Based on the notion of cofixed point equations for general corecursive definitions we prove by coinduction the correctness of the algorithms. We use the machinery of the Coq proof assistant for coinductive types to present the formalisation. The material in this article is fully formalised in the Coq proof assistant.KeywordsPoint EquationProof ObligationCorrectness ProofExact ArithmeticGuardedness ConditionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.