<html xmlns:o="urn:schemas-microsoft-com:office:office" xmlns:w="urn:schemas-microsoft-com:office:word" xmlns:m="http://schemas.microsoft.com/office/2004/12/omml" xmlns="http://www.w3.org/TR/REC-html40">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<meta name="Generator" content="Microsoft Word 15 (filtered medium)">
<style><!--
/* Font Definitions */
@font-face
{font-family:"Cambria Math";
panose-1:2 4 5 3 5 4 6 3 2 4;}
@font-face
{font-family:Calibri;
panose-1:2 15 5 2 2 2 4 3 2 4;}
/* Style Definitions */
p.MsoNormal, li.MsoNormal, div.MsoNormal
{margin:0cm;
margin-bottom:.0001pt;
font-size:11.0pt;
font-family:"Calibri",sans-serif;}
a:link, span.MsoHyperlink
{mso-style-priority:99;
color:blue;
text-decoration:underline;}
a:visited, span.MsoHyperlinkFollowed
{mso-style-priority:99;
color:purple;
text-decoration:underline;}
p.msonormal0, li.msonormal0, div.msonormal0
{mso-style-name:msonormal;
mso-margin-top-alt:auto;
margin-right:0cm;
mso-margin-bottom-alt:auto;
margin-left:0cm;
font-size:11.0pt;
font-family:"Calibri",sans-serif;}
span.EmailStyle18
{mso-style-type:personal-reply;
font-family:"Calibri",sans-serif;
color:windowtext;}
.MsoChpDefault
{mso-style-type:export-only;
font-size:10.0pt;}
@page WordSection1
{size:612.0pt 792.0pt;
margin:72.0pt 72.0pt 72.0pt 72.0pt;}
div.WordSection1
{page:WordSection1;}
--></style>
</head>
<body lang="EN-GB" link="blue" vlink="purple">
<div class="WordSection1">
<p class="MsoNormal"><span style="mso-fareast-language:EN-US">Hi,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="mso-fareast-language:EN-US">I didn’t see Wouter’s comment but I guess I can add another unhelpful comment along the same lines. Really a strictly positive operator should be a container and we do need some syntactic sugar for
this so that we just have to write something similar as what you do (didn’t even know there were strict positivity annotations). The same criticism applies to the whole sized type story: a reasonable implementation of sized types should be just syntactic sugar
for programs we can define without them.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="mso-fareast-language:EN-US">But I realize you want to use agda as it is and my principled complaints are not helping. However, they are the reason that I haven’t taken these adhoc extensions serious. I am still waiting that
somebody provides a reasonable explanation (i.e. semantics / translation) for them.
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="mso-fareast-language:EN-US">Thorsten<o:p></o:p></span></p>
<p class="MsoNormal"><span style="mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<div style="border:none;border-top:solid #B5C4DF 1.0pt;padding:3.0pt 0cm 0cm 0cm">
<p class="MsoNormal"><b><span style="font-size:12.0pt;color:black">From: </span></b><span style="font-size:12.0pt;color:black">Agda <agda-bounces@lists.chalmers.se> on behalf of Andrew Pitts <andrew.pitts@cl.cam.ac.uk><br>
<b>Date: </b>Thursday, 7 November 2019 at 11:01<br>
<b>To: </b>agda list <agda@lists.chalmers.se><br>
<b>Subject: </b>[Agda] initial T-algebra for an endofunctor T, via sized types<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
<p class="MsoNormal">Recently I have been more or less forced into using Agda’s sized types to get a certain development to go through and it has set me thinking about the use of sized types with inductive definitions. The current documentation stub says<br>
<br>
"Sizes help the termination checker by tracking the depth of data structures across definition boundaries”
<o:p></o:p></p>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal">and then goes on to give an extended example to do with coinduction; but of course sizes are useful for induction too. <o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal">To get a better feel for what one can achieve, I tried using sizes to construct the (provably) initial algebra for a postulated strictly positive endofunctor on Set and eventually succeeded; but the construction was much more elaborate
than I expected (and involved the use of quotient types). You can see it here:<o:p></o:p></p>
</div>
<div>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal"><<a href="https://www.cl.cam.ac.uk/~amp12/agda/initial-T-algebras/">https://www.cl.cam.ac.uk/~amp12/agda/initial-T-algebras/</a>><o:p></o:p></p>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal">Did I somehow over-complicate things? Am I re-inventing a construction that already exists?<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal">Andy Pitts<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
</div>
</div>
</div>
<PRE>
This message and any attachment are intended solely for the addressee
and may contain confidential information. If you have received this
message in error, please contact the sender and delete the email and
attachment.
Any views or opinions expressed by the author of this email do not
necessarily reflect the views of the University of Nottingham. Email
communications with the University of Nottingham may be monitored
where permitted by law.
</PRE></body>
</html>