The groupoid model refutes uniqueness of identity proofs

Hofmann, Martin; Streicher, Thomas

doi:10.1109/lics.1994.316071

Cited by 64 publications

(66 citation statements)

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“…Using some earlier results, which I shall not treat here (see [230] for further references), Hofmann and Streicher [116], [117] homotopies. This geometrical viewpoint is not only intuitively appealing but also allows for a fruitful conceptual reconstruction of the initial groupoid model.…”

Section: Homotopy Type Theorymentioning

confidence: 99%

Axiomatic Method and Category Theory

Rodin

2014

Synthese Library

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PrefaceI first learned about category theory about 20 years ago from Yuri I. Manin's course on algebraic geometry [180] when I was preparing my dissertation on Euclid's Elements and was focused on studying Greek mathematics and classical Greek philosophy. Then I convinced myself that the mathematical category theory is philosophically relevant not only because of its name but also because of its content and because of its special role in the contemporary mathematics, which I privately compared to the role of the notion of figure in Euclid's geometry. Today I have more to say about these matters. The broad historical and philosophical context, in which I studied category theory, is made explicit throughout the present book. My interest to the Axiomatic Method stems from my work on Euclid and extends through Hilbert and axiomatic set theories to Lawvere's axiomatic topos theory to the Univalent Foundations of mathematics recently proposed by Vladimir Voevodsky. This explains what the two subjects appearing in the title of this book share in common.The next crucial biographical episode took place in 1999 when I was a young scholar visiting Columbia University on the Fulbright grant working on ontology of events under the supervision of Achille Varzi. As a part of my Fulbright program I had to make a presentation in a different American university, and I decided to use this opportunity for talking about the philosophical significance of category theory (I cannot now remember how exactly I married then this subject with the event ontology). Achille Varzi kindly arranged for me the invitation from Barry Smith to give a talk at his seminar on formal ontology in the SUNY in Buffalo. When I sent to Barry Smith my abstract he replied that nobody except probably Bill Lawvere will be able to understand my paper, and suggested to make the paper more accessible to the general audience.By that time I had already read some of Lawvere's papers but was wholly unaware about the fact that Lawvere worked in the same university and could attend my planned talk. So I took Smith's words for a joke. When I realized that this was not a joke I was very excited and, as Vladimir ArnoldThe main motivation of writing this book is to develop the view on mathematics described in the above epigraphs. Some 200 years ago this view used to be by far more common and easier to justify than today. It is sufficient to say that it made part of Kant's view on mathematics, and that Kant's view on mathematics remained extremely influential until the very end of the 19th century. When Cassirer defended this Kantian view in the beginning of the 20th century it was already polemical. When Arnold defended it in the end of the 20th century and in the beginning of this current century it already sounded as an intellectual provocation, and so his words sound today. Kant, Cassirer and Arnold do not speak about the same mathematics: each speaks about mathematics of his own time. So the growing polemical attitude to their shared view reflects not only a change of...

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Section: Homotopy Type Theorymentioning

confidence: 99%

Axiomatic Method and Category Theory

Rodin

2014

Synthese Library

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“…It is based on the unstated assumption that the equality type itself is proof-irrelevant: if there are several distinct proofs of x ≡ x , then x ∈ [x ] does not correspond directly to the positions of x in [x ]. However, in the absence of the K rule the equality type is not necessarily proof-irrelevant [9]. Fortunately, and maybe surprisingly, one can prove that the two definitions of bag equivalence above are equivalent even in the absence of proof-irrelevance (see Sect.…”

Section: Bag Equivalence For Listsmentioning

confidence: 96%

Bag Equivalence via a Proof-Relevant Membership Relation

Danielsson

2012

Interactive Theorem Proving

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Abstract. Two lists are bag equivalent if they are permutations of each other, i.e. if they contain the same elements, with the same multiplicity, but perhaps not in the same order. This paper describes how one can define bag equivalence as the presence of bijections between sets of membership proofs. This definition has some desirable properties:-Many bag equivalences can be proved using a flexible form of equational reasoning. -The definition generalises easily to arbitrary unary containers, including types with infinite values, such as streams. -By using a slight variation of the definition one gets set equivalence instead, i.e. equality up to order and multiplicity. Other variations give the subset and subbag preorders. -The definition works well in mechanised proofs.

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“…Axiom K and the elimination rule for Heq are not derivable in CIC [4], but it is no surprise that they are derivable with the new rule: Less-or-equal relation on natural numbers. We show two examples concerning the relation less-or-equal for natural numbers defined inductively as follows:…”

Section: Examplesmentioning

confidence: 99%

“…He already observed that the axiom K is derivable in his setting. Hofmann and Streicher [4] later proved that pattern matching is not a conservative extension of Type Theory, by showing that K is not derivable in Type Theory. Finally, Goguen et al [3] proved that pattern matching can be translated into a Type Theory with K as an axiom, showing that K is sufficient to support pattern matching -this result was already discovered by McBride [5].…”

Section: Metatheorymentioning

confidence: 99%

A New Elimination Rule for the Calculus of Inductive Constructions

Barras

Corbineau

Grégoire

et al. 2009

Lecture Notes in Computer Science

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Abstract. In Type Theory, definition by dependently-typed case analysis can be expressed by means of a set of equations -the semantic approach -or by an explicit pattern-matching construction -the syntactic approach. We aim at putting together the best of both approaches by extending the pattern-matching construction found in the Coq proof assistant in order to obtain the expressivity and flexibility of equationbased case analysis while remaining in a syntax-based setting, thus making dependently-typed programming more tractable in the Coq system. We provide a new rule that permits the omission of impossible cases, handles the propagation of inversion constraints, and allows to derive Streicher's K axiom. We show that subject reduction holds, and sketch a proof of relative consistency.

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The groupoid model refutes uniqueness of identity proofs

Cited by 64 publications

References 2 publications

Axiomatic Method and Category Theory

Axiomatic Method and Category Theory

Bag Equivalence via a Proof-Relevant Membership Relation

A New Elimination Rule for the Calculus of Inductive Constructions

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