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From type theory to setoids and back

Published online by Cambridge University Press:  12 April 2023

Erik Palmgren Open the ORCID record for Erik Palmgren [Opens in a new window]
Erik Palmgren*
Affiliation:
Department of Mathematics, Stockholm University, Stockholm, Sweden
*
Email: palmgren@math.su.se
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Abstract

A model of Martin-Löf extensional type theory with universes is formalized in Agda, an interactive proof system based on Martin-Löf intensional type theory. This may be understood, we claim, as a solution to the old problem of modeling the full extensional theory in the intensional theory. Types are interpreted as setoids, and the model is therefore a setoid model.We solve the problem of interpreting type universes by utilizing Aczel’s type of iterative sets and show how it can be made into a setoid of small setoids containing the necessary setoid constructions. In addition, we interpret the bracket types of Awodey and Bauer. Further quotient types should be interpretable.

Keywords

Martin-Löf type theoryextensionalityconstructive set theorysetoids
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 32 , Issue 10 , November 2022 , pp. 1283 - 1312
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Copyright
© The Estate of Erik Palmgren, 2023. Published by Cambridge University Press

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