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<p class="MsoNormal"><span style="mso-fareast-language:EN-US">First of all I don’t like the word “dependent type theory”. Dependent types are one important feature of modern Type Theory but hardly the only one.<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">To me the most important feature of Type Theory is the support of abstraction in Mathematics and computer science. Using types instead of sets means that you can hide implementation choices which is essential if you want to build towers
of abstraction. Set theory fails here badly. Just as a very simple example: in set theory you have the notion of union, so for example
<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal" style="text-indent:36.0pt">{0,1} \cup {0,1,2,3} = {0,1,2,3}<o:p></o:p></p>
<p class="MsoNormal" style="text-indent:36.0pt"><o:p> </o:p></p>
<p class="MsoNormal">However, if we change the representation of the first set and use lets say {true,false} we get a different result:<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal" style="text-indent:36.0pt">{true , false} \cup {0,1,2,3} = {true,false,0,1,2,3}<o:p></o:p></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">This means that \cup exposes implementation details because the results are not equivalent upto renaming. In Type Theory we have the notion of sum, sometimes called disjoint union, which is well
behaved<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" style="text-indent:36.0pt"><span lang="DE">{0,1} + {0,1,2,3} = {in1 0,in1 1,in2 0,in2 1,in2 2,in2 3}<o:p></o:p></span></p>
<p class="MsoNormal" style="text-indent:36.0pt"><span lang="DE"><o:p> </o:p></span></p>
<p class="MsoNormal" style="text-indent:36.0pt"><span lang="DE">{true , false} + {0,1,2,3} = {in1 true,in1 false ,in2 0,in2 1,in2 2,in2 3}<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="DE"><o:p> </o:p></span></p>
<p class="MsoNormal">Unlike \cup, + doesn’t reveal any implementation details it is a purely structural operation. Having only structural operations means that everything you do is stable under equivalence, that is you can replace one object with another one
that behaves the same. This is the essence of Voevodsky’s univalence principle.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">There are other nice aspects of Type Theory. From a constructive point of view (which should come naturally to a computer scientists) the proporsitions as types explanation provides a very natural way to obtain “logic for free” and paedagogically
helpful since it reduces logical reasoning to programming.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">There are performance issues with implementations of Type Theory, however, in my experience (mainly agda) the execution of functions at compile time isn’t one of them. In my experience the main problem is to deal with a loss of sharing
when handling equational constraints which can blow up the time needed for type checking. I think this is an engineering problem and there are some suggestions how to fix this.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">Thorsten<o:p></o:p></p>
<p class="MsoNormal" style="text-indent:36.0pt"><o:p> </o:p></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>
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<p class="MsoNormal" style="margin-left:36.0pt"><b><span style="font-size:12.0pt;color:black">From:
</span></b><span style="font-size:12.0pt;color:black">"coq-club-request@inria.fr" <coq-club-request@inria.fr> on behalf of Jason Gross <jasongross9@gmail.com><br>
<b>Reply to: </b>"coq-club@inria.fr" <coq-club@inria.fr><br>
<b>Date: </b>Tuesday, 3 March 2020 at 19:44<br>
<b>To: </b>coq-club <coq-club@inria.fr>, agda-list <agda@lists.chalmers.se>, "coq+miscellaneous@discoursemail.com" <coq+miscellaneous@discoursemail.com>, lean-user <lean-user@googlegroups.com><br>
<b>Subject: </b>[Coq-Club] Why dependent type theory?<o:p></o:p></span></p>
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<p class="MsoNormal" style="margin-left:36.0pt"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt">I'm in the process of writing my thesis on proof assistant performance bottlenecks (with a focus on Coq), and there's a large class of performance bottlenecks that come from (mis)using the power of dependent types.
So in writing the introduction, I want to provide some justification for the design decision of using dependent types, rather than, say, set theory or classical logic (as in, e.g., Isabelle/HOL). And the only reasons I can come up with are "it's fun" and
"lots of people do it" <o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt">So I'm asking these mailing lists: why do we base proof assistants on dependent type theory? What are the trade-offs involved?<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt">I'm interested both in explanations and arguments given on list, as well as in references to papers that discuss these sorts of choices.<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt">Thanks,<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt">Jason<o:p></o:p></p>
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