Quotients by Idempotent Functions in Cedille

Abstract

We present a simple characterization of definable quotient types as being induced by idempotent functions, and an encoding of this in Cedille (a dependently typed programming language) in which both equational constraints and the packaging that associates these with elements of the carrier type are irrelevant, facilitating equational reasoning in proofs. We provide several concrete examples of definable quotients using this encoding and give combinators for function lifting (with one variant having zero run-time cost).