scholarly journals Homotopy limits in type theory

2015 ◽  
Vol 25 (5) ◽  
pp. 1040-1070 ◽  
Author(s):  
JEREMY AVIGAD ◽  
KRZYSZTOF KAPULKIN ◽  
PETER LEFANU LUMSDAINE
Keyword(s):  
Homotopy Type ◽  
Systematic Study ◽  
Type Theory ◽  
Proof Assistant ◽  
Classical Approach ◽  
Homotopy Type Theory ◽  
The Coq Proof Assistant ◽  
Homotopy Limits ◽  
Coq Proof Assistant

Working in homotopy type theory, we provide a systematic study of homotopy limits of diagrams over graphs, formalized in the Coq proof assistant. We discuss some of the challenges posed by this approach to the formalizing homotopy-theoretic material. We also compare our constructions with the more classical approach to homotopy limits via fibration categories.

scholarly journals An experimental library of formalized Mathematics based on the univalent foundations

2015 ◽  
Vol 25 (5) ◽  
pp. 1278-1294 ◽  
Author(s):  
VLADIMIR VOEVODSKY
Keyword(s):  
Type Theory ◽  
Proof Assistant ◽  
Formalization Of Mathematics ◽  
Constructive Type Theory ◽  
Short Overview ◽  
Formalized Mathematics ◽  
The Coq Proof Assistant ◽  
Constructive Type ◽  
Coq Proof Assistant ◽  
Univalent Foundations

This is a short overview of an experimental library of Mathematics formalized in the Coq proof assistant using the univalent interpretation of the underlying type theory of Coq. I started to work on this library in February 2010 in order to gain experience with formalization of Mathematics in a constructive type theory based on the intuition gained from the univalent models (see Kapulkin et al. 2012).

Verification of non-functional programs using interpretations in type theory

2003 ◽  
Vol 13 (4) ◽  
pp. 709-745 ◽  
Author(s):  
JEAN-CHRISTOPHE FILLIÂTRE
Keyword(s):  
Static Analysis ◽  
Type Theory ◽  
General Framework ◽  
Specification Language ◽  
Proof Assistant ◽  
Logical Interpretation ◽  
Total Correctness ◽  
The Coq Proof Assistant ◽  
Coq Proof Assistant ◽  
Functional Features

We study the problem of certifying programs combining imperative and functional features within the general framework of type theory. Type theory is a powerful specification language which is naturally suited for the proof of purely functional programs. To deal with imperative programs, we propose a logical interpretation of an annotated program as a partial proof of its specification. The construction of the corresponding partial proof term is based on a static analysis of the effects of the program which excludes aliases. The missing subterms in the partial proof term are seen as proof obligations, whose actual proofs are left to the user. We show that the validity of those proof obligations implies the total correctness of the program. This work has been implemented in the Coq proof assistant. It appears as a tactic taking an annotated program as argument and generating a set of proof obligations. Several nontrivial algorithms have been certified using this tactic.

scholarly journals Homotopy type-theoretic interpretations of constructive set theories

2019 ◽  
pp. 1-32
Author(s):  
Cesare Gallozzi
Keyword(s):  
Comparative Analysis ◽  
Set Theory ◽  
Homotopy Type ◽  
Type Theory ◽  
Axiom Of Choice ◽  
Homotopy Type Theory ◽  
Constructive Set Theory

Abstract We introduce a family of (k, h)-interpretations for 2 ≤ k ≤ ∞ and 1 ≤ h ≤ ∞ of constructive set theory into type theory, in which sets and formulas are interpreted as types of homotopy level k and h, respectively. Depending on the values of the parameters k and h, we are able to interpret different theories, like Aczel’s CZF and Myhill’s CST. We also define a proposition-as-hproposition interpretation in the context of logic-enriched type theories. The rest of the paper is devoted to characterising and analysing the interpretations considered. The formulas valid in the prop-as-hprop interpretation are characterised in terms of the axiom of unique choice. We also analyse the interpretations of CST into homotopy type theory, providing a comparative analysis with Aczel’s interpretation. This is done by formulating in a logic-enriched type theory the key principles used in the proofs of the two interpretations. Finally, we characterise a class of sentences valid in the (k, ∞)-interpretations in terms of the ΠΣ axiom of choice.

scholarly journals Reviving the Philosophy of Geometry

2018 ◽  
Author(s):  
David Corfield
Keyword(s):  
Homotopy Type ◽  
Type Theory ◽  
Great Part ◽  
Mathematical Physics ◽  
Relevant Today ◽  
Homotopy Type Theory ◽  
Philosophical Account ◽  
Hard Times ◽  
Anglophone World

In the Anglophone world, the philosophical treatment of geometry has fallen on hard times. This chapter argues that philosophy can come to a better understanding of mathematics by providing an account of modern geometry, including its development of new forms of space, both for mathematical physics and for arithmetic. It returns to the discussions of Weyl and Cassirer on geometry whose concerns are very much relevant today. A way of encompassing a great part of modern geometry via homotopy toposes is discussed, along with the `cohesive’ variant of their internal language, known as `homotopy type theory’. With these tools in place, we can now start to see what an adequate philosophical account of current geometry might look like.

Homotopy Types

2020 ◽  
pp. 77-106
Author(s):  
David Corfield
Keyword(s):  
Homotopy Type ◽  
Type Theory ◽  
Definite Descriptions ◽  
Homotopy Types ◽  
Homotopy Type Theory

A further innovation is the introduction of an intensional type theory. Here one need not reduce equivalence to mere identity. How two entities are the same tells us more than whether they are the same. This is explained by the homotopical aspect of HoTT, where types are taken to resemble spaces of points, paths, paths between paths, and so on. This allows us to rethink Russell’s definite descriptions. Mathematicians use a ‘generalized the’ in situations where it appears that they might be talking about a multiplicity of instances, but there is essentially a unique instance. Taken together with the ‘univalence axiom’ there results a language in which anything that can be said of a type can be said of an equivalent type. This allows homotopy type theory to become the language of choice for a structuralist, avoiding the need for any kind of abstraction away from multiple instantiations.

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