From type theory to setoids and back
(Submitted on 3 Sep 2019)
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 modelling 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 intepreting 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.