Abstract
The delay datatype was introduced by Capretta [3] 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 [6] 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.
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Notes
- 1.
Working in homotopy type theory [15], we would assume this principle only for 0-types, i.e., sets, and that would also be enough for our purposes.
- 2.
Propositions are (\(-1\))-types and proposition extensionality is univalence for (\(-1\))-types.
- 3.
Notice that \(\mathsf {AC}\) is fundamentally different from the type-theoretic axiom of choice:
$$\begin{aligned} \prod _{\{X, Y: \mathcal {U}\}}\,\prod _{P : X \rightarrow Y \rightarrow \mathcal {U}}\,\left( \prod _{x : X} \, \sum _{y : Y} \, P\,x\,y \right) \rightarrow \sum _{f : X \rightarrow Y} \, \prod _{x : X} \, P\,x\,(f\,x) \end{aligned}$$which is provable in type theory.
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Acknowledgement
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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Chapman, J., Uustalu, T., Veltri, N. (2015). Quotienting the Delay Monad by Weak Bisimilarity. In: Leucker, M., Rueda, C., Valencia, F. (eds) Theoretical Aspects of Computing - ICTAC 2015. ICTAC 2015. Lecture Notes in Computer Science(), vol 9399. Springer, Cham. https://doi.org/10.1007/978-3-319-25150-9_8
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