Mercurial > cpdt > repo
changeset 101:bc12662ae895
Exercise added at end of MoreDep
author | Adam Chlipala <adamc@hcoop.net> |
---|---|
date | Wed, 08 Oct 2008 13:55:20 -0400 |
parents | 070f4de92311 |
children | d829cc24faee |
files | src/MoreDep.v |
diffstat | 1 files changed, 17 insertions(+), 0 deletions(-) [+] |
line wrap: on
line diff
--- a/src/MoreDep.v Wed Oct 08 10:01:26 2008 -0400 +++ b/src/MoreDep.v Wed Oct 08 13:55:20 2008 -0400 @@ -1078,3 +1078,20 @@ 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