Proof-relevance of families of setoids and identity in type theory

@article{Palmgren2012ProofrelevanceOF,
  title={Proof-relevance of families of setoids and identity in type theory},
  author={Erik Palmgren},
  journal={Arch. Math. Log.},
  year={2012},
  volume={51},
  pages={35-47}
}
Families of types are fundamental objects in Martin-Löf type theory. When extending the notion of setoid (type with an equivalence relation) to families of setoids, a choice between proof-relevant or proof-irrelevant indexing appears. It is shown that a family of types may be canonically extended to a proof-relevant family of setoids via the identity types, but that such a family is in general proofirrelevant if, and only if, the proof-objects of identity types are unique. A similar result is… CONTINUE READING