75 lines
3.9 KiB
TeX
75 lines
3.9 KiB
TeX
\chapter{Perspectives}
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\section{Discussion}
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In the previous chapter the practical aspects of proving things in Cubical Agda
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were highlighted. I also demonstrated the usefulness of separating ``laws'' from
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``data''. One of the reasons for this is that dependencies within types can lead
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to very complicated goals. One technique for alleviating this was to prove that
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certain types are mere propositions.
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\subsection{Computational properties}
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Another aspect (\TODO{That I actually did not highlight very well in the
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previous chapter}) is the computational nature of paths. Say we have
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formalized this common result about monads:
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\TODO{Some equation\ldots}
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By transporting this to the Kleisli formulation we get a result that we can use
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to compute with. This is particularly useful because the Kleisli formulation
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will be more familiar to programmers e.g. those coming from a background in
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Haskell. Whereas the theory usually talks about monoidal monads.
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\TODO{Mention that with postulates we cannot do this}
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\subsection{Reusability of proofs}
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The previous example also illustrate how univalence unifies two otherwise
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disparate areas: The category-theoretic study of monads; and monads as in
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functional programming. Univalence thus allows one to reuse proofs. You could
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say that univalence gives the developer two proofs for the price of one.
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The introduction (section \S\ref{sec:context}) mentioned an often
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employed-technique for enabling extensional equalities is to use the
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setoid-interpretation. Nowhere in this formalization has this been necessary,
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$\Path$ has been used globally in the project as propositional equality. One
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interesting place where this becomes apparent is in interfacing with the Agda
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standard library. Multiple definitions in the Agda standard library have been
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designed with the setoid-interpretation in mind. E.g. the notion of ``unique
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existential'' is indexed by a relation that should play the role of
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propositional equality. Likewise for equivalence relations, they are indexed,
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not only by the actual equivalence relation, but also by another relation that
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serve as propositional equality.
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%% Unfortunately we cannot use the definition of equivalences found in the
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%% standard library to do equational reasoning directly. The reason for this is
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%% that the equivalence relation defined there must be a homogenous relation,
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%% but paths are heterogeneous relations.
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In the formalization at present a significant amount of energy has been put
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towards proving things that would not have been needed in classical Agda. The
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proofs that some given type is a proposition were provided as a strategy to
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simplify some otherwise very complicated proofs (e.g.
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\ref{eq:proof-prop-IsPreCategory} and \label{eq:productPath}). Often these
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proofs would not be this complicated. If the J-rule holds definitionally the
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proof-assistant can help simplify these goals considerably. The lack of the
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J-rule has a significant impact on the complexity of these kinds of proofs.
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\TODO{Universe levels.}
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\section{Future work}
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\subsection{Agda \texttt{Prop}}
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Jesper Cockx' work extending the universe-level-laws for Agda and the
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\texttt{Prop}-type.
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\subsection{Compiling Cubical Agda}
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\label{sec:compiling-cubical-agda}
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Compilation of program written in Cubical Agda is currently not supported. One
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issue here is that the backends does not provide an implementation for the
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cubical primitives (such as the path-type). This means that even though the
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path-type gives us a computational interpretation of functional extensionality,
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univalence, transport, etc., we do not have a way of actually using this to
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compile our programs that use these primitives. It would be interesting to see
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practical applications of this. The path between monads that this library
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exposes could provide one particularly interesting case-study.
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\subsection{Higher inductive types}
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This library has not explored the usefulness of higher inductive types in the
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context of Category Theory.
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