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Equivalences for free: univalent parametricity for effective transport

Nicolas Tabareau, Éric Tanter, Matthieu Sozeau
2018 Proceedings of the ACM on Programming Languages  
We study how to maximize the effectiveness of these transports in terms of computational behavior, and identify a fragment useful for certified programming on which univalent transport is guaranteed to  ...  In particular, univalent parametricity strengthens parametricity to ensure preservation of type equivalences.  ...  ACKNOWLEDGMENTS We thank Pierre-Évariste Dagand, Eric Finster and the anonymous reviewers for useful feedback, as well as Simon Boulier and Gaëtan Gilbert for contributing parts of the Coq development.  ... 
doi:10.1145/3236787 dblp:journals/pacmpl/TabareauTS18 fatcat:5v7ewz4nxrahrbwpu6k3zr37yy

Refinements for Free! [chapter]

Cyril Cohen, Maxime Dénès, Anders Mörtberg
2013 Lecture Notes in Computer Science  
We relieve the user of these tedious refinements by introducing a framework where correctness is established in a proof-oriented context and automatically transported to computation-oriented data structures  ...  formalism of the Coq proof assistant enables us to develop this within Coq, without having to maintain an external tool. 1. relate a proof-oriented data representation with a more efficient one (Sect. 2), 2. parametrize  ...  We also thank Bassel Mannaa and Dan Rosén for proof reading the final version of this paper.  ... 
doi:10.1007/978-3-319-03545-1_10 fatcat:atas3io5mjfiloi5njvrnxpwmm

Equivalence relations for homology cylinders and the core of the Casson invariant [article]

Gwenael Massuyeau, Jean-Baptiste Meilhan
2012 arXiv   pre-print
Similarly, we show that the J_3-equivalence is classified by (1) and (2).  ...  The J_k-equivalence relation on IC is defined in a similar way using the k-th term of the Johnson filtration.  ...  In [22, Proposition 2.7], Habiro gives a list of 12 moves on claspers which give equivalent claspers, that is claspers with homeomorphic surgery effect.  ... 
arXiv:1104.2763v3 fatcat:irqhgkperrb4bc4ijj24lih3bq

The Marriage of Univalence and Parametricity [article]

Nicolas Tabareau, Éric Tanter, Matthieu Sozeau
2020 arXiv   pre-print
The combination of parametricity and univalence supports transport \'a la carte: basic univalent transport, which stems from a type equivalence, can be complemented with additional proofs of equivalences  ...  Parametricity and univalence are two major concepts that have been explored to transport programs and proofs across type equivalences, but they fall short of achieving seamless, automatic transport.  ...  We are thankful as well to Assia Mahboubi for suggesting concrete applications for validating the use of our approach, which likewise  ... 
arXiv:1909.05027v3 fatcat:bhzydmmuc5euxd2wz4cdnrmymu

The Marriage of Univalence and Parametricity

Nicolas Tabareau, Éric Tanter, Matthieu Sozeau
2021 Journal of the ACM  
The combination of parametricity and univalence supports transport à la carte : basic univalent transport, which stems from a type equivalence, can be complemented with additional proofs of equivalences  ...  The resulting concept, called univalent parametricity , is an extension of parametricity strengthened with univalence that fully realizes programming and proving modulo equivalences.  ...  We are thankful as well to Assia Mahboubi for suggesting concrete applications for validating the use of our approach, which likewise pushed us to extend and enhance this work on univalent parametricity  ... 
doi:10.1145/3429979 fatcat:xrcsi5wjnbdunbm4djsqxfdbwq

Equivalence relations for homology cylinders and the core of the Casson invariant

Gwénaël Massuyeau, Jean–Baptiste Meilhan
2013 Transactions of the American Mathematical Society  
EQUIVALENCE RELATIONS FOR HOMOLOGY CYLINDERS 5435 but using the Johnson filtration of the Torelli group instead of its lower central series.  ...  It follows from these definitions that J 1 = Y 1 and that Y k implies J k for every k ≥ 1. But the converse is not true for k ≥ 2, as illustrated by the following statements. Theorem B.  ...  Acknowledgement The authors would like to thank Kazuo Habiro for stimulating discussions about the diagrammatic description of the group IC/Y 3 , and for his comments on the first version of this manuscript  ... 
doi:10.1090/s0002-9947-2013-05818-7 fatcat:xwfizdm5hjbglpixzvypjypua4

Internalizing Representation Independence with Univalence [article]

Carlo Angiuli, Evan Cavallo, Anders Mörtberg, Max Zeuner
2020 arXiv   pre-print
The correspondence becomes an isomorphism between the quotiented types, thereby allowing us to obtain an equality of implementations by univalence.  ...  If our programming language is dependently-typed, however, we would like to appeal to such invariance results within the language itself, in order to obtain correctness theorems for complex implementations  ...  ACKNOWLEDGMENTS We thank Robert Harper and Jonathan Sterling for helpful conversations about this work.  ... 
arXiv:2009.05547v2 fatcat:zrmwjkuul5cgdcrcpmqu34wasu

Internalizing representation independence with univalence

Carlo Angiuli, Evan Cavallo, Anders Mörtberg, Max Zeuner
2021 Proceedings of the ACM on Programming Languages (PACMPL)  
The correspondence becomes an isomorphism between the quotiented types, thereby allowing us to obtain an equality of implementations by univalence.  ...  If our programming language is dependently-typed, however, we would like to appeal to such invariance results within the language itself, in order to obtain correctness theorems for complex implementations  ...  ACKNOWLEDGMENTS We thank Robert Harper and Jonathan Sterling for helpful conversations about this work.  ... 
doi:10.1145/3434293 fatcat:u6qhovi2crhobff3m4mhotqf4i

Circulation-induced shape deformations of drops and bubbles: Exact two-dimensional models

Darren Crowdy
1999 Physics of Fluids  
In this paper simple two-dimensional mathematical models for understanding the fluid dynamical problem of how circulation affects the free surface shapes of inviscid drops and bubbles with surface tension  ...  This theoretical paradigm is of interest in many areas of science including large-scale transport processes in chemical engineering.  ...  This would similarly provide a mechanism for enhanced transport rates.  ... 
doi:10.1063/1.870142 fatcat:gsu5srgdrffvxiocccipiyrhgy

The Definitional Side of the Forcing

Guilhem Jaber, Gabriel Lewertowski, Pierre-Marie Pédrot, Matthieu Sozeau, Nicolas Tabareau
2016 Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '16  
This is a novel example of a dependent theory with side-effects, clarifying how dependent elimination for inductive types must be restricted in a non-pure setting.  ...  Being implemented as a Coq plugin, this work gives the possibility to formalize easily consistency results, for instance the consistency of the negation of Voevodsky's univalence axiom.  ...  Preserving Univalence Axiom for Monotonous Types In the previous section, we have been able to negate the univalence axiom by using types that produce completely non-monotonous effects.  ... 
doi:10.1145/2933575.2935320 dblp:conf/lics/JaberLPST16 fatcat:nnumvblstngtrbfjrc5t74laxu

The Univalence Principle [article]

Benedikt Ahrens and Paige Randall North and Michael Shulman and Dimitris Tsementzis
2021 arXiv   pre-print
It follows that Univalent Foundations is a fully equivalence-invariant foundation for higher-categorical mathematics, as intended by Voevodsky.  ...  The Univalence Principle is the statement that equivalent mathematical structures are indistinguishable.  ...  This means that transport in these sorts is for free; an indiscernibility a ≍ b is exactly an isomorphism φ : a ∼ = b. π V T I E C T A p T m A A T O T m O O Here, (T O , In a univalent model of the above  ... 
arXiv:2102.06275v2 fatcat:4wcxpfzat5bg7c6gt7p6f6wge4

Internal Parametricity for Cubical Type Theory [article]

Evan Cavallo, Robert Harper
2021 arXiv   pre-print
The combined theory supports both univalence and its relational equivalent, which we call relativity.  ...  discrepancies between cubical and parametric type theory, which are closely related.  ...  Acknowledgments We thank Carlo Angiuli, Steve Awodey, Daniel Gratzer, Kuen-Bang Hou (Favonia), Dan Licata, Anders Mörtberg, Emily Riehl, Christian Sattler, Michael Shulman, Jonathan Sterling, and Andrew Swan for  ... 
arXiv:2005.11290v5 fatcat:rnbnor535jcbbeb74726r7mltm

An Implementation of Homotopy Type Theory in Isabelle/Pure [article]

Joshua Chen
2019 arXiv   pre-print
In this Masters thesis we present an implementation of a fragment of "book HoTT" as an object logic for the interactive proof assistant Isabelle.  ...  Lemma 1.5.5 (Transport lemma).  ...  Given a type A : U i , a type family P : A → U i , x, y : A and p : x = A y, we can define a function transport P x,y,p : P (x) → P (y) satisfying transport P x,x,refl(x) ≡ id P (x) for all x : A.  ... 
arXiv:1911.00399v1 fatcat:vweng3xbezf6fecadvoy54l54a

Construction of the Circle in UniMath [article]

Marc Bezem and Ulrik Buchholtz and Daniel R. Grayson and Michael Shulman
2020 arXiv   pre-print
We show that the type Tℤ of ℤ-torsors has the dependent universal property of the circle, which characterizes it up to a unique homotopy equivalence.  ...  The construction uses Voevodsky's Univalence Axiom and propositional truncation, yielding a stand-alone construction of the circle not using higher inductive types.  ...  For every p, q : a 1 = a 2 and 2-dimensional path α : p = q, transport along α induces an equivalence cp α : (b 1 = B p b 2 ) (b 1 = B q b 2 ) .  ... 
arXiv:1910.01856v2 fatcat:uz7ncjbd7bb7lg3unxbkdywuxy

A Dipole Model for Negative Steady-State Resistance in Excitable Membranes

Bernard B. Hamel, Irwin Zimmerman
1970 Biophysical Journal  
The theory, in addition, is in reasonable accord with nonequimolar ion flow, and provides an explanation for such effects as the following: the intercept of the voltage-current characteristic, the ion  ...  membrane concentrations inferred from electrodiffusion theories, and the effects of polyvalent cations  ...  The physics of the cooperative effect is modeled by means of an equivalent circuit.  ... 
doi:10.1016/s0006-3495(70)86351-2 pmid:5471696 pmcid:PMC1367981 fatcat:c4ilebgqujhy3dby6cafeh2zni
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