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

Authors:
Erik Palmgren
Affiliations:
Department of Mathematics, Stockholm University, Stockholm, Sweden 106 91
Venue:
Archive for Mathematical Logic
Year:
2012

Citing 3
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Abstract

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 proof-irrelevant if, and only if, the proof-objects of identity types are unique. A similar result is shown for fibre representations of families. The ubiquitous role of proof-irrelevant families is discussed.