Constructing a universe for the setoid model
Paper i proceeding, 2021

The setoid model is a model of intensional type theory that validates certain extensionality principles, like function extensionality and propositional extensionality, the latter being a limited form of univalence that equates logically equivalent propositions. The appeal of this model construction is that it can be constructed in a small, intensional, type theoretic metatheory, therefore giving a method to boostrap extensionality. The setoid model has been recently adapted into a formal system, namely Setoid Type Theory (SeTT). SeTT is an extension of intensional Martin-Löf type theory with constructs that give full access to the extensionality principles that hold in the setoid model. Although already a rich theory as currently defined, SeTT currently lacks a way to internalize the notion of type beyond propositions, hence we want to extend SeTT with a universe of setoids. To this aim, we present the construction of a (non-univalent) universe of setoids within the setoid model, first as an inductive-recursive definition, which is then translated to an inductive-inductive definition and finally to an inductive family. These translations from more powerful definition schemas to simpler ones ensure that our construction can still be defined in a relatively small metatheory which includes a proof-irrelevant identity type with a strong transport rule.



Setoid model


Type theory

Function extensionality


Thorsten Altenkirch

University of Nottingham

Simon Boulier

Institut National de Recherche en Informatique et en Automatique (INRIA)

Ambrus Kaposi

Eötvös Loránd University (ELTE)

Christian Sattler

Chalmers, Data- och informationsteknik, Datavetenskap

Filippo Sestini

University of Nottingham

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

03029743 (ISSN) 16113349 (eISSN)

Vol. 12650 1-21
9783030719944 (ISBN)

24th International Conference on Foundations of Software Science and Computation Structures, FOSSACS 2021 held as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021
Virtual, Online, ,


Algebra och logik

Annan fysik

Sannolikhetsteori och statistik



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