Foundational aspects of multiscale digitization

Abstract

International audienceIn this article, we describe the theoretical foundations of the Ω-arithmetization. This method provides a multi-scale discretization of a continuous function that is a solution of a differential equation. This discretization process is based on the Harthong-Reeb line HRω. The Harthong-Reeb line is a linear space that is both discrete and continuous. This strange line HRω stems from a nonstandard point of view on arithmetic based, in this paper, on the concept of Ω-numbers introduced by Laugwitz and Schmieden. After a full description of this nonstandard background and of the first properties of HRω, we introduce the Ω-arithmetization and we apply it to some significant examples. An important point is that the constructive properties of our approach leads to algorithms which can be exactly translated into functional computer programs without uncontrolled numerical error. Afterwards, we investigate to what extent HRω fits Bridges's axioms of the constructive continuum. Finally, we give an overview of a formalization of the Harthong-Reeb line with the Coq proof assistant

Formalizing a discrete model of the continuum in Coq from a discrete geometry perspective

“…In our formal developement in Coq, we follow the layout of previous papers on the Harthong-Reeb line [8,7] and we consider a minimal axiomatic form of nonstandard analysis, related to Nelson' s Internal Set Theory (IST) [28]. We also intend to capture the inheritely constructive nature of the model of the Harthong-Reeb line in our formalization, using Ω-numbers, introduced by Laugwitz and Schmieden in [22].…”

“…As what matters is only that P holds for the considered element, we use the principle of proof irrelevance to show all proofs of the same property (e.g. P (x)) are equal 7 . This principle is expressed with the following axiom in Coq:…”

“…Such a construction, based on the Ω-numbers of Laugwitz and Schmieden [21], was made by some of the authors with others in [9,7].…”

“…This model was at the origin of important developments in the Discrete Geometry field [31]. The formalization is based on the work presented in [8,7] where it was shown that the Harthong-Reeb line satisfies the axioms for constructive real numbers introduced by Bridges [4]. Laugwitz-Schmieden numbers [20] are then introduced and their limitations with respect to being a model of the Harthong-Reeb line is investigated [7].…”

“…The formalization is based on the work presented in [8,7] where it was shown that the Harthong-Reeb line satisfies the axioms for constructive real numbers introduced by Bridges [4]. Laugwitz-Schmieden numbers [20] are then introduced and their limitations with respect to being a model of the Harthong-Reeb line is investigated [7]. In this paper, we transpose all these definitions and properties into a formal description using the Coq proof assistant.…”