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Cubical Type Theory: a constructive interpretation of the univalence axiom [article]

Cyril Cohen, Thierry Coquand, Simon Huber, Anders Mörtberg
2016 arXiv   pre-print
This 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  ...  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.  ...  The authors would like to thank the referees and Martín Escardó, Georges Gonthier, Dan Grayson, Peter Hancock, Dan Licata, Peter LeFanu Lumsdaine, Christian Sattler, Andrew Swan, Vladimir Voevodsky for  ... 
arXiv:1611.02108v1 fatcat:ewgjnfayxrcobcp3hw7sieeh6y

Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom

Cyril Cohen, Thierry Coquand, Simon Huber, Anders Mörtberg, Marc Herbstritt
2018 Types for Proofs and Programs  
The authors would like to thank the referees and Martín Escardó, Georges Gonthier, Dan Grayson, Peter Hancock, Dan Licata, Peter LeFanu Lumsdaine, Christian Sattler, Andrew Swan, Vladimir Voevodsky for  ...  Introduction This work is a continuation of the program started in [6, 13] to provide a constructive justification of Voevodsky's univalence axiom [27] .  ...  This operation will then be used to define the composition operation for the universe and to prove the univalence axiom. 5:12 Cubical Type Theory Contractible Types We define isContr A = (x : A) ×  ... 
doi:10.4230/lipics.types.2015.5 dblp:conf/types/CohenCHM15 fatcat:v5zcyb7cvvfl5lofulmfixg5oa

Guarded Cubical Type Theory

Lars Birkedal, Aleš Bizjak, Ranald Clouston, Hans Bugge Grathwohl, Bas Spitters, Andrea Vezzosi
2018 Journal of automated reasoning  
Cubical type theory [2] is an extension of Martin-Löf type theory with the goal of obtaining a computational interpretation of the univalence axiom.  ...  The important novel idea in cubical type theory is that the identity type is not inductively defined. Instead the identity type 1 Id A (x, y) is defined to be the type of paths starting from x to y.  ...  Cubical type theory [2] is an extension of Martin-Löf type theory with the goal of obtaining a computational interpretation of the univalence axiom.  ... 
doi:10.1007/s10817-018-9471-7 fatcat:dnv5azxrnrawtjolai4o6bhfsm

Internal Parametricity for Cubical Type Theory [article]

Evan Cavallo, Robert Harper
2021 arXiv   pre-print
We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives.  ...  We demonstrate the use of the theory by analyzing polymorphic functions between higher inductive types, observe how cubical equality regularizes parametric type theory, and examine the similarities and  ...  This was finally addressed by the development of cubical type theories [CCHM15, AFH18, OP18, ABC + 19, CMS20], a family of univalent type theories (with constructive models) where the univalence axiom  ... 
arXiv:2005.11290v5 fatcat:rnbnor535jcbbeb74726r7mltm

Axioms for Modelling Cubical Type Theory in a Topos [article]

Ian Orton, Andrew M. Pitts
2018 arXiv   pre-print
This clarifies the definition and properties of the notion of uniform Kan filling that lies at the heart of their constructive interpretation of Voevodsky's univalence axiom.  ...  We investigate the extent to which their model construction can be expressed in the internal type theory of any topos and identify a collection of quite weak axioms for this purpose.  ...  This paper is a revised and expanded version of a paper of the same name that appeared in the proceedings of the 25th EACSL Annual Conference on Computer Science Logic (CSL 2016).  ... 
arXiv:1712.04864v3 fatcat:st6xdwahnfcnrd7jb2oicjw3lq

Normalization for Cubical Type Theory [article]

Jonathan Sterling, Carlo Angiuli
2021 arXiv   pre-print
We prove normalization for (univalent, Cartesian) cubical type theory, closing the last major open problem in the syntactic metatheory of cubical type theory.  ...  Our normalization result is reduction-free, in the sense of yielding a bijection between equivalence classes of terms in context and a tractable language of β/η-normal forms.  ...  This can be seen by a model construction in which T is interpreted into cubical sets, where the interpretation of cofibrations is standard but each type is interpreted as an inhabited type.  ... 
arXiv:2101.11479v2 fatcat:ubf2xtqncjf4xd4hmfso47xqty

Unifying Cubical Models of Univalent Type Theory

Evan Cavallo, Anders Mörtberg, Andrew W Swan, Michael Wagner
2020 Annual Conference for Computer Science Logic  
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.  ...  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.  ...  Licata, Andrew Pitts and Jon Sterling for helpful comments and remarks on earlier versions of this work.  ... 
doi:10.4230/lipics.csl.2020.14 dblp:conf/csl/CavalloMS20 fatcat:rgzcvxrphbf67djmhglropid4q

Parametric Cubical Type Theory [article]

Evan Cavallo, Robert Harper
2019 arXiv   pre-print
We exhibit a computational type theory which combines the higher-dimensional structure of cartesian cubical type theory with the internal parametricity primitives of parametric type theory, drawing out  ...  interpretation of inductive types.  ...  Conversations with the directed type theory group at the 2017 Mathematics Research Communities workshop and with Emily Riehl and Christian Sattler afterwards were also instrumental to the first author's  ... 
arXiv:1901.00489v2 fatcat:rqbzj52wu5dujp4mchcxdxmm7y

Canonicity and homotopy canonicity for cubical type theory [article]

Thierry Coquand and Simon Huber and Christian Sattler
2022 arXiv   pre-print
Cubical type theory provides a constructive justification of homotopy type theory.  ...  A crucial ingredient of cubical type theory is a path lifting operation which is explained computationally by induction on the type involving several non-canonical choices.  ...  This model of univalence is a model of cubical type theory where each type has a filling operation. Univalence is then a theorem and not an axiom of cubical type theory.  ... 
arXiv:1902.06572v6 fatcat:ncv4mtdun5cbpct5wx7tajis7i

Naive cubical type theory [article]

Bruno Bentzen
2021 arXiv   pre-print
It can thus be viewed as a first step toward a cubical alternative to the program of informalization of type theory carried out in the homotopy type theory book for dependent type theory augmented with  ...  axioms for univalence and higher inductive types.  ...  Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the AFOSR.  ... 
arXiv:1911.05844v2 fatcat:jxqahdqi2zaz7iehsi5z5bi66m

Guarded Cubical Type Theory [article]

Lars Birkedal, Aleš Bizjak, Ranald Clouston, Hans Bugge Grathwohl, Bas Spitters, Andrea Vezzosi
2017 arXiv   pre-print
Our new type theory, guarded cubical type theory (GCTT), provides a computational interpretation of extensionality for guarded recursive types.  ...  This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT).  ...  This research was supported in part by the ModuRes Sapere Aude Advanced Grant from The Danish Council for Independent Research for the Natural Sciences (FNU), and in part by the Guarded homotopy type theory  ... 
arXiv:1611.09263v2 fatcat:n3vpcofribacdcmwdqhdhf2qii

Syntax and models of Cartesian cubical type theory

Carlo Angiuli, Guillaume Brunerie, Thierry Coquand, Robert Harper, Kuen-Bang Hou (Favonia), Daniel R. Licata
2021 Mathematical Structures in Computer Science  
Next, we describe a constructive model of this type theory in Cartesian cubical sets.  ...  An advantage of this formal approach is that our construction can also be interpreted in a range of other models, including cubical sets on the connections cube category and the De Morgan cube category  ...  We are very grateful to the late Vladimir Voevodsky for his vision and leadership, creating a rich research area that has captivated all of us over the past years and inspired the PhD theses of all four  ... 
doi:10.1017/s0960129521000347 dblp:journals/mscs/AngiuliBCHHL21 fatcat:kyppubqyxvepxj5pcq2um2hwhi

Constructing Inductive-Inductive Types in Cubical Type Theory [chapter]

Jasper Hugunin
2019 Green Chemistry and Sustainable Technology  
In this paper, we show that the existing construction requires Uniqueness of Identity Proofs, and present a new construction (which we conjecture generalizes) of one particular inductive-inductive type  ...  in cubical type theory, which is compatible with homotopy type theory.  ...  Some of this work was completed while studying at Tokyo Institute of Technology under Professor Ryo Kashima.  ... 
doi:10.1007/978-3-030-17127-8_17 dblp:conf/fossacs/Hugunin19 fatcat:j2lyqxbvtrhitp3zq3bul7aaay

Internal Parametricity for Cubical Type Theory

Evan Cavallo, Robert Harper, Michael Wagner
2020 Annual Conference for Computer Science Logic  
We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives.  ...  ACM Subject Classification Theory of computation → Type theory  ...  Reynolds captures this uniformity in the existence of a relational interpretation of type theory that expresses the invariance of type constructions under a broad class of relations.  ... 
doi:10.4230/lipics.csl.2020.13 dblp:conf/csl/Cavallo020 fatcat:qquvhque4rgnjdvf6izmcgvzz4

Homotopy Canonicity for Cubical Type Theory

Thierry Coquand, Simon Huber, Christian Sattler, Michael Wagner
2019 International Conference on Rewriting Techniques and Applications  
Cubical type theory provides a constructive justification of homotopy type theory and satisfies canonicity: every natural number is convertible to a numeral.  ...  A crucial ingredient of cubical type theory is a path lifting operation which is explained computationally by induction on the type involving several non-canonical choices.  ...  We can interpret univalent type theory in such any such cubical cwf as per Remark 1.  ... 
doi:10.4230/lipics.fscd.2019.11 dblp:conf/rta/CoquandHS19 fatcat:bxgmaj3xn5fk7piknzzojzazmm
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