Abstract
AbstractThe Isabelle Higher-order Tarski–Grothendieck object logic includes in its foundations both higher-order logic and set theory, which allows importing the libraries of Isabelle/HOL and Isabelle/Mizar. The two libraries, however, define all the basic concepts independently, which means that the results in the two are disconnected. In this paper, we align significant parts of these two libraries, by defining isomorphisms between their concepts, including the real numbers and algebraic structures. The isomorphisms allow us to transport theorems between the foundations and use the results from the libraries simultaneously.
Funder
European Research Council
Publisher
Springer Science and Business Media LLC
Subject
Artificial Intelligence,Computational Theory and Mathematics,Software
Reference55 articles.
1. Assaf, A., Cauderlier, R.: Mixing HOL and Coq in Dedukti. In: Kaliszyk, C., Paskevich, A. (eds.) Proof eXchange for Theorem Proving (PxTP 2015), vol. 186 of EPTCS, pp. 89–96 (2015)
2. Assaf, A.: A framework for defining computational higher-order logics. (Un cadre de définition de logiques calculatoires d’ordre supérieur). PhD thesis, École Polytechnique, Palaiseau, France (2015)
3. Awodey, S: Type theory and homotopy. In: Dybjer, P., Lindström, S., Palmgren, E., Sundholm, G. (eds.) Epistemology versus Ontology - Essays on the Philosophy and Foundations of Mathematics in Honour of Per Martin-Löf, vol. 27 of Logic, Epistemology, and the Unity of Science, pp. 183–201. Springer (2012)
4. Bancerek, G., Byliński, C., Grabowski, A., Korniłowicz, A., Matuszewski, R., Naumowicz, A., Pąk, K.: The role of the Mizar Mathematical Library for interactive proof development in Mizar. J. Automat. Reason. 61, 9–32 (2017)
5. Bezem, M., Coquand, T., Huber, S.: The univalence axiom in cubical sets. J. Autom. Reason. 63(2), 159–171 (2019)