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**Published in:** OASIcs, Volume 105, 4th International Workshop on Formal Methods for Blockchains (FMBC 2022)

We provide three detailed case studies of vulnerabilities in smart contracts, and show how property based testing would have found them: 1. the Dexter1 token exchange; 2. the iToken; 3. the ICO of Brave’s BAT token. The last example is, in fact, new, and was missed in the auditing process.
We have implemented this testing in ConCert, a general executable model/specification of smart contract execution in the Coq proof assistant. ConCert contracts can be used to generate verified smart contracts in Tezos' LIGO and Concordium’s rust language. We thus show the effectiveness of combining formal verification and property-based testing of smart contracts.

Mikkel Milo, Eske Hoy Nielsen, Danil Annenkov, and Bas Spitters. Finding Smart Contract Vulnerabilities with ConCert’s Property-Based Testing Framework. In 4th International Workshop on Formal Methods for Blockchains (FMBC 2022). Open Access Series in Informatics (OASIcs), Volume 105, pp. 2:1-2:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{milo_et_al:OASIcs.FMBC.2022.2, author = {Milo, Mikkel and Nielsen, Eske Hoy and Annenkov, Danil and Spitters, Bas}, title = {{Finding Smart Contract Vulnerabilities with ConCert’s Property-Based Testing Framework}}, booktitle = {4th International Workshop on Formal Methods for Blockchains (FMBC 2022)}, pages = {2:1--2:13}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-250-1}, ISSN = {2190-6807}, year = {2022}, volume = {105}, editor = {Dargaye, Zaynah and Schneidewind, Clara}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.FMBC.2022.2}, URN = {urn:nbn:de:0030-drops-171834}, doi = {10.4230/OASIcs.FMBC.2022.2}, annote = {Keywords: Smart Contracts, Formal Verification, Property-Based Testing, Coq} }

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**Published in:** LIPIcs, Volume 108, 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)

We begin by recalling the essentially global character of universes in various models of homotopy type theory, which prevents a straightforward axiomatization of their properties using the internal language of the presheaf toposes from which these model are constructed. We get around this problem by extending the internal language with a modal operator for expressing properties of global elements. In this setting we show how to construct a universe that classifies the Cohen-Coquand-Huber-Mörtberg (CCHM) notion of fibration from their cubical sets model, starting from the assumption that the interval is tiny - a property that the interval in cubical sets does indeed have. This leads to an elementary axiomatization of that and related models of homotopy type theory within what we call crisp type theory.

Daniel R. Licata, Ian Orton, Andrew M. Pitts, and Bas Spitters. Internal Universes in Models of Homotopy Type Theory. In 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 108, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{licata_et_al:LIPIcs.FSCD.2018.22, author = {Licata, Daniel R. and Orton, Ian and Pitts, Andrew M. and Spitters, Bas}, title = {{Internal Universes in Models of Homotopy Type Theory}}, booktitle = {3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)}, pages = {22:1--22:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-077-4}, ISSN = {1868-8969}, year = {2018}, volume = {108}, editor = {Kirchner, H\'{e}l\`{e}ne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2018.22}, URN = {urn:nbn:de:0030-drops-91929}, doi = {10.4230/LIPIcs.FSCD.2018.22}, annote = {Keywords: cubical sets, dependent type theory, homotopy type theory, internal language, modalities, univalent foundations, universes} }

Document

**Published in:** LIPIcs, Volume 62, 25th EACSL Annual Conference on Computer Science Logic (CSL 2016)

This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of program logics, and for programming and reasoning with coinductive types. We wish to implement GDTT with decidable type checking, while still supporting non-trivial equality proofs that reason about the extensions of guarded recursive constructions. CTT is a variation of Martin-Löf type theory in which the identity type is replaced by abstract paths between terms. CTT provides a computational interpretation of functional extensionality, is conjectured to have decidable type checking, and has an implemented type checker. Our new type theory, called guarded cubical type theory, provides a computational interpretation of extensionality for guarded recursive types. This further expands the foundations of CTT as a basis for formalisation in mathematics and computer science. We present examples to demonstrate the expressivity of our type theory, all of which have been checked using a prototype type-checker implementation, and present semantics in a presheaf category.

Lars Birkedal, Aleš Bizjak, Ranald Clouston, Hans Bugge Grathwohl, Bas Spitters, and Andrea Vezzosi. Guarded Cubical Type Theory: Path Equality for Guarded Recursion. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 23:1-23:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{birkedal_et_al:LIPIcs.CSL.2016.23, author = {Birkedal, Lars and Bizjak, Ale\v{s} and Clouston, Ranald and Grathwohl, Hans Bugge and Spitters, Bas and Vezzosi, Andrea}, title = {{Guarded Cubical Type Theory: Path Equality for Guarded Recursion}}, booktitle = {25th EACSL Annual Conference on Computer Science Logic (CSL 2016)}, pages = {23:1--23:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-022-4}, ISSN = {1868-8969}, year = {2016}, volume = {62}, editor = {Talbot, Jean-Marc and Regnier, Laurent}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2016.23}, URN = {urn:nbn:de:0030-drops-65638}, doi = {10.4230/LIPIcs.CSL.2016.23}, annote = {Keywords: Guarded Recursion, Dependent Type Theory, Cubical Type Theory, Denotational Semantics, Homotopy Type Theory} }

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Tutorial

**Published in:** OASIcs, Volume 11, 6th International Conference on Computability and Complexity in Analysis (CCA'09) (2009)

This tutorial will illustrate how to use the Coq proof assistant to implement effective and provably correct computation for analysis. Coq provides a dependently typed functional programming language that allows users to specify both programs and formal proofs.
We will introduce dependent type theory and show how it can be used to develop both mathematics and programming. We will show how to use dependent type theory to implement constructive analysis. Specifically we will cover how to implement effective real numbers and effective integration.
This work will be done using the Coq proof assistant. The tutorial will cover how to use the Coq proof assistant. Attendees are encouraged to download and install Coq 8.2 from {\tt http://coq.inria.fr/download} and also download and make the full system of C-CoRN from {\tt http://c-corn.cs.ru.nl/download.html} beforehand.

Bas Spitters and Russell O'Connor. Computer Verified Exact Analysis (Tutorial). In 6th International Conference on Computability and Complexity in Analysis (CCA'09). Open Access Series in Informatics (OASIcs), Volume 11, p. 23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{spitters_et_al:OASIcs.CCA.2009.2255, author = {Spitters, Bas and O'Connor, Russell}, title = {{Computer Verified Exact Analysis}}, booktitle = {6th International Conference on Computability and Complexity in Analysis (CCA'09)}, pages = {23--23}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-939897-12-5}, ISSN = {2190-6807}, year = {2009}, volume = {11}, editor = {Bauer, Andrej and Hertling, Peter and Ko, Ker-I}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.CCA.2009.2255}, URN = {urn:nbn:de:0030-drops-22554}, doi = {10.4230/OASIcs.CCA.2009.2255}, annote = {Keywords: Proof assistant, dependent type theory, constructive analysis} }

Document

**Published in:** Dagstuhl Seminar Proceedings, Volume 5021, Mathematics, Algorithms, Proofs (2006)

We present a constructive algebraic integration theory. The theory is constructive in the sense of Bishop, however we avoid the axiom of countable, or dependent, choice. Thus our results can be interpreted in any topos. Since we avoid impredicative methods the results may also be interpreted in Martin-LÃƒÂ¶f type theory or in a predicative topos in the sense of Moerdijk and Palmgren.
We outline how to develop most of Bishop's theorems on integration theory that do not mention points explicitly. Coquand's constructive version of the
Stone representation theorem is an important tool in this process. It is also used to give a new proof of Bishop's spectral theorem.

Bas Spitters. Constructive algebraic integration theory without choice. In Mathematics, Algorithms, Proofs. Dagstuhl Seminar Proceedings, Volume 5021, pp. 1-13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{spitters:DagSemProc.05021.9, author = {Spitters, Bas}, title = {{Constructive algebraic integration theory without choice}}, booktitle = {Mathematics, Algorithms, Proofs}, pages = {1--13}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2006}, volume = {5021}, editor = {Thierry Coquand and Henri Lombardi and Marie-Fran\c{c}oise Roy}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.05021.9}, URN = {urn:nbn:de:0030-drops-2905}, doi = {10.4230/DagSemProc.05021.9}, annote = {Keywords: Algebraic integration theory, spectral theorem, choiceless constructive mathematics, pointfree topology} }

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