1,321 research outputs found
A Cubical Implementation of Homotopical Patch Theory
We consider theoretical models of version control systems based on Homotopy Type Theory (HoTT). The main contribution is an implementation of Angiuli et al.’s Homotopical Patch Theory in Cubical Agda. Additionally the first chapter contains an approachable introduction to HoTT and Cubical Agda aimed at an audience of interested computer science students covering dependent Martin-Löf-style type theory, propositions as types, univalent foundations, higher inductive types and CCHM cubical type theory. Finally, we discuss some other approaches to a theory of version control systems in Darcs’ “algebra of patches” and an unsuccessful attempt to model repositories in type theory as coequalizers.Master's Thesis in InformaticsINF399MAMN-INFMAMN-PRO
Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom
This paper presents a type theory in which it is possible to
directly manipulate -dimensional cubes (points, lines, squares,
cubes, etc.) based on an interpretation of dependent type theory in
a cubical set model. This enables new ways to reason about identity
types, for instance, function extensionality is directly provable in
the system. Further, Voevodsky\u27s univalence axiom is provable in
this system. We also explain an extension with some higher inductive
types like the circle and propositional truncation. Finally we
provide semantics for this cubical type theory in a constructive
meta-theory
Cubical Type Theory: a constructive interpretation of the univalence axiom
International audienceThis paper presents a type theory in which it is possible to directly manipulate n-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system. Further, Voevodsky's univalence axiom is provable in this system. We also explain an extension with some higher inductive types like the circle and propositional truncation. Finally we provide semantics for this cubical type theory in a constructive meta-theory
Internal Parametricity for Cubical Type Theory
We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives. The combined theory supports both univalence and its relational equivalent, which we call relativity. We demonstrate the use of the theory by analyzing polymorphic functions between higher inductive types, and we give an account of the identity extension lemma for internal parametricity
Homotopy Type Theory in Lean
We discuss the homotopy type theory library in the Lean proof assistant. The
library is especially geared toward synthetic homotopy theory. Of particular
interest is the use of just a few primitive notions of higher inductive types,
namely quotients and truncations, and the use of cubical methods.Comment: 17 pages, accepted for ITP 201
Unifying Cubical Models of Univalent Type Theory
We present a new constructive model of univalent type theory based on cubical sets. Unlike prior work on cubical models, ours depends neither on diagonal cofibrations nor connections. This is made possible by weakening the notion of fibration from the cartesian cubical set model, so that it is not necessary to assume that the diagonal on the interval is a cofibration. We have formally verified in Agda that these fibrations are closed under the type formers of cubical type theory and that the model satisfies the univalence axiom. By applying the construction in the presence of diagonal cofibrations or connections and reversals, we recover the existing cartesian and De Morgan cubical set models as special cases. Generalizing earlier work of Sattler for cubical sets with connections, we also obtain a Quillen model structure
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