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Reconstruction of Euler’s proof published in the AFP

    Our development of “Euler’s Exponential Series as an Infinite Polynomial” has been accepted and published by the Archive of Formal Proofs.

    Abstract:
    In this formalisation, we reconstruct Euler’s derivation of the power series for the exponential function as expounded in his famous Introductio in analysin infinitorum, first published in 1748. Using nonstandard analysis, we mechanize his mixture of infinitesimal and infinite ‘algebraic’ reasoning in the proof assistant Isabelle. In so doing, we demonstrate that the gist of his arguments can be reconstructed formally, with Isabelle and nonstandard analysis shoring up crucial aspects of his reasoning that some historians have qualified as being “more a matter of faith than science”.

    The full formalization is available here:

    The Gelfand–Naimark–Segal Construction published in the AFP

      Our development of the “The Gelfand–Naimark–Segal Construction” has been accepted and published by the AFP.

      Abstract

      This entry formalises complete normed algebras equipped with an involution, so-called C*-algebras. We provide both a class definition, and a locale for C*-algebras on carrier sets in the spirit of existing developments of linear algebra and smooth manifolds. Bounded operators on a complex Hilbert space, with the operator norm and adjoints, form such an algebra. The main theorem of this entry is a result in the converse direction: the Gelfand–Naimark–Segal (GNS) construction, which starts with a single suitable functional on a C*-algebra in order to obtain both a Hilbert space and a representation of the algebra in terms of bounded operators on that space. This is implemented as a type construction in Isabelle/HOL, taking advantage of existing mechanisms for quotient types, and integrating with existing type classes for Hilbert spaces and Cauchy completions.

      The full formalisation is available here.

      Our formalisation of Linear Resources and Process Compositions has been published in the Archive of Formal Proof

        Abstract

        In this entry we formalise a framework for process composition based on actions that are specified by their input and output resources. We verify their correctness by translating compositions of process into deductions of intuitionistic linear logic. As part of the verification we derive simple conditions on the compositions which ensure well-formedness of the corresponding deduction.

        We describe an earlier version of this formalisation in our article Linear Resources in Isabelle/HOL, which also includes a formalisation of manufacturing processes in the simulation game Factorio.

        Paper accepted at CICM 2022

          Our paper, “Re-imagining the Isabelle Archive of Formal Proofs” (MacKenzie, Huch, Vaughan and Fleuriot), has been accepted at the 15th Conference on Intelligent Computer Mathematics (CICM 2022).

          New Archive of Formal Proof Website

            A project involving Carlin MacKenzie, James Vaughan and Jacques Fleuriot has resulted in a re-design and re-implementation of the Archive of Formal Proofs, which is a collection of proof libraries, examples, and larger scientific developments, mechanically checked in the theorem prover Isabelle. The revamped website (https://www.isa-afp.org) is now live and will serve the Isabelle community across the world. The Edinburgh team worked with Fabian Huch of TU Munich to integrate their work into the AFP infrastructure.