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