56 lines
3.3 KiB
TeX
56 lines
3.3 KiB
TeX
\chapter{Conclusion}
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This thesis highlighted some issues with the standard inductive definition of
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propositional equality used in Agda. Functional extensionality and univalence
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are examples of two propositions not admissible in Intensional Type Theory
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(ITT). This has a big impact on what is provable and the reusability of proofs.
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This issue is overcome with an extension to Agda's type system called Cubical
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Agda. With Cubical Agda both functional extensionality and univalence are
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admissible. Cubical Agda is more expressive, but there are certain issues that
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arise that are not present in standard Agda. For one thing ITT and standard Agda
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enjoys Uniqueness of Identity Proofs (UIP). This is not the case in Cubical
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Agda. In stead there exists a hierarchy of types with increasing
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\nomen{homotopical structure}{homotopy levels}. It turns out to be useful to built the
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formalization with this hierarchy in mind as it can simplify proofs
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considerably. Another issue one must overcome in Cubical Agda is when a type has
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a field whose type depends on a previous field. In this case paths between such
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types will be heterogeneous paths. This problem is related to Cubical Agda not
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having the K-rule \TODO{Not mentioned anywhere in the report}. In practice it
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turns out to be considerably more difficult to work heterogeneous paths than
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with homogeneous paths. The thesis demonstrated some techniques to overcome
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these difficulties, such as based path-induction.
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This thesis formalized some of the core concepts from category theory including;
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categories, functors, products, exponentials, Cartesian closed categories,
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natural transformations, the yoneda embedding, monads and more. Category theory
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is an interesting case-study for the application of Cubical Agda for two reasons
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in particular: Because category theory is the study of abstract algebra of
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functions, meaning that functional extensionality is particularly relevant.
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Another reason is that in category theory it is commonplace to identify
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isomorphic structures and univalence allows for making this notion precise. This
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thesis also demonstrated another technique that is common in category theory;
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namely to define categories to prove properties of other structures.
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Specifically a category was defined to demonstrate that any two product objects
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in a category are isomorphic. Furthermore the thesis showed two formulations of
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monads and proved that they indeed are equivalent: Namely monads in the
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monoidal- and Kleisli- form. The monoidal formulation is more typical to
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category theoretic formulations and the Kleisli formulation will be more
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familiar to functional programmers. In the formulation we also saw how paths can
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be used to extract functions. A path between two types induce an isomorphism
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between the two types. This e.g. permits developers to write a monad instance
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for a given type using the Kleisli formulation. By transporting along the path
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between the monoidal- and Kleisli- formulation one can reuse all the operations
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and results shown for monoidal- monads in the context of kleisli monads.
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%%
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%% problem with inductive type
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%% overcome with cubical
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%% the path type
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%% homotopy levels
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%% depdendent paths
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%%
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%% category theory
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%% algebra of functions ~ funExt
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%% identify isomorphic types ~ univalence
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%% using categories to prove properties
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%% computational properties
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%% reusability, compositional
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