Commit 5535d6ec authored by Adam Chlipala's avatar Adam Chlipala

Exercise added at end of MoreDep

parent 56efc518
......@@ -1078,3 +1078,20 @@ Eval simpl in matches a_star "a".
Eval simpl in matches a_star "b".
Eval simpl in matches a_star "aa".
(* end hide *)
(** * Exercises *)
(** %\begin{enumerate}%#<ol>#
%\item%#<li># Define a kind of dependently-typed lists, where a list's type index gives a lower bound on how many of its elements satisfy a particular predicate. In particular, for an arbitrary set [A] and a predicate [P] over it:
%\begin{enumerate}%#<ol>#
%\item%#<li># Define a type [plist : nat -> Set]. Each [plist n] should be a list of [A]s, where it is guaranteed that at least [n] distinct elements satisfy [P]. There is wide latitude in choosing how to encode this. You should try to avoid using subset types or any other mechanism based on annotating non-dependent types with propositions after-the-fact.#</li>#
%\item%#<li># Define a version of list concatenation that works on [plist]s. The type of this new function should express as much information as possible about the outpit [plist].#</li>#
%\item%#<li># Define a function [plistOut] for translating [plist]s to normal [list]s.#</li>#
%\item%#<li># Define a function [plistIn] for translating [list]s to [plist]s. The type of [plistIn] should make it clear that the best bound on [P]-matching elements is chosen. You may assume that you are given a dependently-typed function for deciding instances of [P].#</li>#
%\item%#<li># Prove that, for any list [ls], [plistOut (plistIn ls) = ls]. This should be the only part of the exercise where you use tactic-based proving.#</li>#
%\item%#<li># Define a function [grab : forall n (ls : plist (S n)), sig P]. That is, when given a [plist] guaranteed to contain at least one element satisfying [P], [grab] produces such an element. [sig] is the type family of sigma types, and [sig P] is extensionally equivalent to [{x : A | P x}], though the latter form uses an eta-expansion of [P] instead of [P] itself as the predicate.#</li>#
#</ol>#%\end{enumerate}% #</li>#
#</ol>#%\end{enumerate}% *)
\ No newline at end of file
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