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Homotopy Type Theory in Isabelle
[article]
2021
arXiv
pre-print
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This paper introduces Isabelle/HoTT, the first development of homotopy type theory in the Isabelle proof assistant. ...
The infrastructure developed is then used to formalize foundational results from the Homotopy Type Theory book. ...
This paper aims to do this by introducing Isabelle/HoTT, the first development of homotopy type theory in Isabelle. ...
arXiv:2002.09282v2
fatcat:cgaeh2o2jrhlrb37ofqayl4pre
An Implementation of Homotopy Type Theory in Isabelle/Pure
[article]
2019
arXiv
pre-print
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dependent type theory with universes inside a simple type-theoretic logical foundation. ...
In this Masters thesis we present an implementation of a fragment of "book HoTT" as an object logic for the interactive proof assistant Isabelle. ...
Introduction This thesis presents an implementation of an initial segment of a homotopy type theory in the proof assistant Isabelle. ...
arXiv:1911.00399v1
fatcat:vweng3xbezf6fecadvoy54l54a
Homotopy Type Theory in Isabelle
2021
Preserved Fulltext
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This paper introduces Isabelle/HoTT, the first development of homotopy type theory in the Isabelle proof assistant. ...
The infrastructure developed is then used to formalize foundational results from the Homotopy Type Theory book. ...
This paper aims to do this by introducing Isabelle/HoTT, the first development of homotopy type theory in Isabelle. ...
doi:10.4230/lipics.itp.2021.12
fatcat:oujlc4k7j5ewvcdtoymlyhzyia
Formalization of the Fundamental Group in Untyped Set Theory Using Auto2
[chapter]
2017
Lecture Notes in Computer Science
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2017 pre-print arXiv:1707.04757v1 fatcat:fzaqavihtjd2jgfdbuk7gxkrka
Using this framework, we formalize in Isabelle/FOL the entire chain of development from the axioms of set theory to the definition of the fundamental group for an arbitrary topological space. ...
We present a new framework for formalizing mathematics in untyped set theory using auto2. ...
In homotopy type theory, one can work with fundamental groups (and higherhomotopy groups) using synthetic definitions. ...
doi:10.1007/978-3-319-66107-0_32
fatcat:kaxjxbfrdfcwbduwjoacugsj3m
Hybrid and alternative logics in Isabelle
2019
International Conference on Intelligent Computer Mathematics
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. 3 Dependent types for HOL Initial work with implementing homotopy type theory as an Isabelle object logic [Che19] has indicated the potential of the soft type approach to adding dependent types on ...
Soft type infrastructure for Isabelle The soft type paradigm [Kra10; LP99] is an approach that views types in terms of their defining logical predicates, as opposed to the type-theoretic approach in which ...
dblp:conf/mkm/Chen19
fatcat:jkdaxc4vtfcibkijxnczgbo5sa
Page 3508 of Mathematical Reviews Vol. , Issue 2000e
[page]
2000
Mathematical Reviews
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In proper shape theory there are various approaches which at this time seem to fall into two classes: the Ball-Sher type, and all others [B. J. Ball, in Studies in topology (Proc. Conf., Univ. ...
With such homotopy classes of maps one obtains the proper n-homotopy category of spaces and an associated proper n-shape theory. ...
A new foundational crisis in mathematics, is it really happening?
[article]
2019
arXiv
pre-print
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Discussion that this discovery has generated in the community of mathematicians, philosophers and computer scientists might indicate a new crisis in the foundation of mathematics. ...
Acknowledgements Many thanks to the organisers of the FOMUS conference in July 2016 for their invitation to give a talk and to participate in the panel discussion. ...
My thanks equally go to the audiences in Paris, Nancy, Oxford, Teheran and Mexico City who have listened and contributed by their comments to the talk that accompanied the development of this paper. ...
arXiv:1802.06221v4
fatcat:jedzv3mrhrht7pgfbkmj35fjdq
A New Foundational Crisis in Mathematics, Is It Really Happening?
[chapter]
2019
Reflections on the Foundations of Mathematics
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Discussion that this discovery has generated in the community of mathematicians, philosophers and computer scientists might indicate a new crisis in the foundation of mathematics. ...
Acknowledgements Many thanks to the organisers of the FOMUS conference in July 2016 for their invitation to give a talk and to participate in the panel discussion. ...
My thanks equally go to the audiences in Paris, Nancy, Oxford, Teheran and Mexico City who have listened and contributed by their comments to the talk that accompanied the development of this paper. ...
doi:10.1007/978-3-030-15655-8_11
fatcat:formjvpydvhele4z7fy3mn73y4
Social Network Processes in the Isabelle and Coq Theorem Proving Communities
[article]
2016
arXiv
pre-print
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We identify the main actors in the Isabelle and Coq communities and describe how they affect and influence their peers. ...
This work explores selected foundations of social networking analysis that we expect to be useful in the context of the ProofPeer project, which is developing a new model for interactive theorem proving ...
We examine some of these ideas in the next section. Coq: Homotopy Type Theory (HoTT) Figure 11 shows the graph of main actors for messages mentioning homotopy (type) theory. ...
arXiv:1609.07127v1
fatcat:gpmwsx6e7rdqrjwghh7zetk37a
A Type-based Formal Specification for Cryptographic Protocols
2018
Journal of Internet Services and Information Security
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Due to univalence and functoriality of mappings in homotopy type theory, the path structure will be preserved in the mapping and realized by equivalence in the universe. ...
It discusses specifying a cryptographic protocol using homotopy type theory which adds the notion of higher inductive type and univalence to Martin-Löf's intensional type theory. ...
of homotopy type theory. ...
doi:10.22667/jisis.2018.11.30.016
dblp:journals/jisis/Vivekanandan18
fatcat:e5j53njyyrazffj7wmzlzezxtm
Page 1841 of Mathematical Reviews Vol. , Issue 81E
[page]
1981
Mathematical Reviews
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Geometric applications of homotopy theory (Proc. Conf., Evanston,
Ill, 1977), I, pp. 303-352, Lecture Notes in Math., 657, Springer,
Berlin, 1978.
From recent work of the author, C. B. Thomas and C. ...
Next a suspension type of construction is performed to show that the inclusion map 2,(16r)~>Q,_,(16r) induces an iso- morphism of homotopy groups 2(Q,)—>7;,,(Q,_,). Also the sphere S***? ...
Computing in Coq with Infinite Algebraic Data Structures
[article]
2010
arXiv
pre-print
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Computational content encoded into constructive type theory proofs can be used to make computing experiments over concrete data structures. ...
In this paper, we explore this possibility when working in Coq with chain complexes of infinite type (that is to say, generated by infinite sets) as a part of the formalization of a hierarchy of homological ...
In order to deal with finite sets in a constructive type theory, more care is needed. For instance, several alternatives for defining finite sets in a constructive logic are included in [8] . ...
arXiv:1004.4998v1
fatcat:jm3tcxihizbvxpn5shfztydgym
Generating certified code from formal proofs: a case study in homological algebra
2009
Formal Aspects of Computing
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More concretely, based on a formal proof of the Basic Perturbation Lemma (a central result in homological algebra) in the prover Isabelle/HOL, we apply various code generation techniques, which lead to ...
This correspondence is established within the logical framework of Isabelle/HOL-that is, it is formally proved correct. The generated code is applied to and illustrated with a number of examples. ...
Nipkow and the members of the "Theorem Proving Group" at the Technische Universität München for hosting him during the elaboration of part of this work and patiently answering every question about the Isabelle ...
doi:10.1007/s00165-009-0120-0
fatcat:ehqm47d3zfc65hwr62cwcza3fi
On the Role of Formalization in Computational Mathematics
2013
Bulletin of the Belgian Mathematical Society Simon Stevin
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In this paper, we will report 1 on the developments carried out in Isabelle/HOL, ACL2 and Coq/SSReflect on Computational Algebraic Topology, in the frame of the ForMath European project 2 . ...
The aim is to illustrate, trough concrete examples, the role of formalization technologies in Computational Mathematics in general. ...
theory, theory underlying the Coq proof assistant). ...
doi:10.36045/bbms/1378314504
fatcat:bctdcuii2jhx3bkeg2mlbb77xy
Page 3535 of Mathematical Reviews Vol. , Issue 92g
[page]
1992
Mathematical Reviews
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theories of involution and polarity in the work of G. ...
in modular sequence spaces (Spanish) (pp. 702-707); Sergio Rodriguez-Machin, Addi- tive homotopy (Spanish) (pp. 708-713); Sergio Rodriguez-Machin, Postnikov systems in additive homotopy (Spanish) (pp. ...
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