Ilya Beylin and Peter Dybjer. Extracting a proof of coherence for monoidal categories from a proof of normalization for monoids. In Stefano Berardi and Mario Coppo, editors, Types for Proofs and Programs, International Workshop TYPES'95, number 1158 in Lecture Notes in Computer Science, pages 47--61, Torino, Italy, June 1995. SpringerVerlag.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....that the empty set, the singleton set and the set of natural numbers all are CI sets. He also shows that the class of CI sets is closed under the formation of identity set and indexed sum. 1.12 Monoidal coherence and the discrete category. In the work on monoidal coherence by Beylin and Dybjer [3], the notion of the set of natural numbers seen as a category plays a central part. On the informal level this category may be thought of as the discrete category defined to have exactly one or no arrow from m to n depending on whether or not m and n are the same natural number. In their ....

I. Beylin and P. Dybjer. Extracting a proof of coherence for monoidal categories from a proof of normalization for monoids. In S. Berardi and M. Coppo, editors, TYPES '95, LNCS 1158, pages 47--61, 1996.

.... bicategorical analysis of E categories Yoshiki Kinoshita Electrotechnical Laboratory January 10, 1997 1 Introduction Beylin and Dybjer introduced a reductionless approach to normalisation in [2]. Dybjer in his talk [9] pointed out an essential use of Yoneda lemma in using presheaf models in this approach to normalisation. They introduced E categories whose homsets are equipped with equivalence relations and axioms for categories hold only up to equivalence, and developed enough theory ....

....between pseudo (bicategory) and strict (2 category) notions. This paper is organized as follows. We first give the basic definitions on the notion of E category in Section 2 including the E Yoneda lemma. We also introduce the constructive proof of normalisation proof given by Beylin and Dybjer [2] at the end of the section. Then we introduce the basic notion of bicategories in Section 3, pointing out the relationship to the corresponding notion in E categories. Since a detailed account of bicategorical Yoneda lemma cannot be found in the literature we are aware of, we gave a fairly ....

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Ilya Beylin and Peter Dybjer. Extracting a proof of coherence for monoidal categories from a formal proof of normalization for monoids. To appear in TYPES '95, Lecture Notes in Computer Science, 1995.

....: 18 7.2 A Higher Order Conversion for Associativity : 20 7.3 From Conversions to Tactics : 22 7. 4 Proving Side Conditions : 23 8 Conclusion 23 1 Introduction In [2], Beylin and Dybjer show how a Curry Howard interpretation of a formal proof of normalization for monoids almost directly yield a coherence proof for monoidal categories. They then formalize the proof in the ALF theorem prover [8] which supports a version of Martin Lof type theory. The ....

.... can be computed more easily (simply by rewriting) The structural induction yields nine cases which are all proved by diagram chasing (well, in a paper and pencil proof at least) Three examples of the more interesting diagrams are listed in Figure 1 and Figure 2 (the diagrams are borrowed from [2]) The diagrams may look frightening, certainly the LaTeX source does, but they are not as complicated as they seem. To read a diagram, start in the upper left corner (the source ) and follow the arrows towards the lower right corner (the target ) The goal is to prove that the diagram commutes, ....

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I. Beylin and P. Dybjer. Extracting a proof of coherence for monoidal categories from a formal proof of normalization for monoids. Draft, Chalmers University of Technology, September 1995.

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I. Beylin and P. Dybjer. Extracting a proof of coherence for monoidal categories from a proof of normalization for monoids. This volume.

....necessary categories and functors are constructed, and the main lemma (nat) is proved for all cases, except mult and inverse arrows. The case mult was already proved in the version 3a (which used dioeerent ddenitions incompatible with version 4) 1 Introduction Throughout the text, the paper means [1] or its earlier version [2] Whenever we refer to the classical Mac Lane s book [4] we mean the chapter iMonoidal Categoriesj, pp. 158 159, and the formula numbers in brackets refer to diagrams (5. 9) in that chapter. All les mentioned in this paper can be found in the ftp directory ....

I. Beylin and P. Dybjer. Extracting a proof of coherence for monoidal categories from a formal proof of normalisation for monoids. Submitted to TYPES-CLICS proceedings, September 1995.

....development was greatly facilitated by an implementation of tool support for equational reasoning in Standard ML. 1 Introduction We compare the two proof assistants ALF and HOL by using them for implementing a proof in elementary category theory 3 . This proof was presented by Beylin and Dybjer [3] and shows how a Curry Howard interpretation of a formal proof of normalization for monoids almost directly yields a coherence proof for monoidal categories. It is an interesting example of an application of intuitionistic type theory and can be viewed as part of the larger enterprise of ....

....the reader to keep the interleaved developments apart. Only some key parts of the developments are shown. The full proofs can be retrieved by ftp from ftp.cs.chalmers.se pub users ilya FMC and from ftp.ifad.dk pub users sten FMC respectively. We would also like to refer to Beylin and Dybjer [3] for a more complete presentation of the proof and for more background and motivation. Foe a more complete presentation of the tool support for diagram chasing, see Agerholm [1] The paper is organized as follows. In Section 2 we give a brief introduction to the systems. In the remaining sections ....

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Ilya Beylin and Peter Dybjer. Extracting a proof of coherence for monoidal categories from a formal proof of normalization for monoids. In Stefano Berardi and Mario Coppo, editors, TYPES '95, LNCS, 1996. To appear.

....and categories with attributes. Cwfs have also been used in recent unpublished work on Tarski Semantics for Type Theory by Per Martin Lof (lecture at the meeting Twenty Five Years of Constructive Type Theory , Venice, October, 1995) The reader is also referred to the paper by Beylin and Dybjer [4] which shows how related phenomena appear in another proof of coherence in type theory. Acknowledgements. The author is grateful for support from the ESPRIT BRA s TYPES and CLICS II, from TFR (the Swedish Technical Research Council) and from The Isaac Newton Institute for Mathematical ....

I. Beylin and P. Dybjer. Extracting a proof of coherence for monoidal categories from a proof of normalization for monoids. This volume.

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Ilya Beylin and Peter Dybjer. Extracting a proof of coherence for monoidal categories from a proof of normalization for monoids. In Stefano Berardi and Mario Coppo, editors, Types for Proofs and Programs, International Workshop TYPES'95, number 1158 in Lecture Notes in Computer Science, pages 47--61, Torino, Italy, June 1995. SpringerVerlag.

No context found.

Ilya Beylin and Peter Dybjer. Extracting a proof of coherence for monoidal categories from a proof of normalization for monoids. In Stefano Berardi and Mario Coppo, editors, Types for Proofs and Programs, International Workshop TYPES'95, number 1158 in Lecture Notes in Computer Science, pages 47--61, Torino, Italy, June 1995. Springer-Verlag.

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Ilya Beylin and Peter Dybjer. Extracting a proof of coherence for monoidal categories from a proof of normalization for monoids. In TYPES, pages 47--61, 1995.

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