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% -*-Latex-*-

%if False

> %default total
> %auto_implicits off
> %access public export

%endif

\section{Conclusions, outlook}
\label{section:conclusions}

In dependently typed programming in the context of Martin-Löf type
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theories \citep{martinlof1984, nordstrom1990programming}, the problem of
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how to specify abstract data types for verified generic programming is
still not well understood.
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In this work, we have shown that requiring functors to preserve
extensional equality of arrows yields abstract data types that are
strong enough to support the verification of non-trivial monadic laws
and of generic results in domain specific languages for dynamical system
and control theory.

We have shown that such a minimalist approach can be exploited to derive
results that otherwise would require enforcing the relationships between
monadic operators -- |pure|, bind, join, Kleisli composition, etc. --
through intensional equalities or, even worse, postulating function
extensionality or similar \emph{impossible} specifications.

As a consequence we have proposed to carefully distinguish between
functors whose associated |map| can be shown to preserve extensional
equality (and identity arrows and arrow composition) and functors for
which this is not the case.
%
%(We are working on a characterization theorem relating extensional
%equality preservation to traversable functors.)

Current work by two of the authors shows that preservation
of extensional equality is useful in designing a verified ADT
for Applicative functors \citep{mcbride2008applicative} and that
all Traversable functors satisfy |mapPresEE|.

We conjecture that carefully distinguishing between higher order
functions that can be shown to preserve extensional equality and higher
order functions for which this is not the case can pay high dividends
(in terms of concise and correct generic implementations and avoidance
of boilerplate code) also for other abstract data types.