Quotienting the Delay Monad by Weak Bisimilarity
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The delay datatype was introduced by Capretta  as a means to deal with partial functions (as in computability theory) in Martin-Löf type theory. It is a monad and it constitutes a constructive alternative to the maybe monad. It is often desirable to consider two delayed computations equal, if they terminate with equal values, whenever one of them terminates. The equivalence relation underlying this identification is called weak bisimilarity. In type theory, one commonly replaces quotients with setoids. In this approach, the delay monad quotiented by weak bisimilarity is still a monad. In this paper, we consider Hofmann’s alternative approach  of extending type theory with inductive-like quotient types. In this setting, it is difficult to define the intended monad multiplication for the quotiented datatype. We give a solution where we postulate some principles, crucially proposition extensionality and the (semi-classical) axiom of countable choice. We have fully formalized our results in the Agda dependently typed programming language.
We thank Thorsten Altenkirch, Andrej Bauer, Bas Spitters and our anonymous referees for comments.
This research was supported by the ERDF funded Estonian CoE project EXCS and ICT national programme project “Coinduction”, the Estonian Science Foundation grants No. 9219 and 9475 and the Estonian Ministry of Education and Research institutional research grant IUT33-13.
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