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<p>Very nice paper, thanks for drawing my attention to it. You are
correct that I am looking for is "a parametric existential
quantifier, as exists in System F, Fω, Haskell, ...".</p>
<p>Unfortunately, for my application, I am not sure I can work in
Set 0. I could work in Set 1. But that's likely not small
enough... To be precise, I want to be able to hide a small type,
but I want to produce a non-small type that depends on it
parametrically.</p>
<p>Jacques<br>
</p>
<div class="moz-cite-prefix">On 2019-02-07 5:28 a.m., Andreas Nuyts
wrote:<br>
</div>
<blockquote type="cite"
cite="mid:3ef4c30a-7561-d969-9201-e7e73e38d3ad@cs.kuleuven.be">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
Dear Jacques,<br>
<br>
You seem to suggest that the first component of your dependent
pairs is going to be a type. Thus, it seems to me that what you
are looking for is a parametric existential quantifier, as exists
in System F, Fω, Haskell, ...<br>
<br>
Parametric quantification is certainly not available in vanilla
MLTT/Agda, but is implemented in an experimental branch of agda,
called agda-parametric:<br>
* github branch: <a class="moz-txt-link-freetext"
href="https://github.com/agda/agda/tree/parametric"
moz-do-not-send="true">https://github.com/agda/agda/tree/parametric</a><br>
* example code: <a class="moz-txt-link-freetext"
href="https://github.com/Saizan/parametric-demo"
moz-do-not-send="true">https://github.com/Saizan/parametric-demo</a><br>
* corresponding paper: "Parametric Quantifiers for Dependent Type
Theory", ICFP 2017, <a class="moz-txt-link-freetext"
href="https://doi.org/10.1145/3110276" moz-do-not-send="true">https://doi.org/10.1145/3110276</a><br>
I emphasize that - as far as I understand - the Agda
implementation (by Andrea Vezzosi) is purely for research and
there are currently no plans for continued support or integration
in the master branch.<br>
<br>
If you want to use vanilla MLTT/Agda, then the following theorem
may come in handy:<br>
<br>
<blockquote>Any function in MLTT with small codomain, is
parametric.<br>
</blockquote>
<br>
(Some papers claim that ANY function in MLTT is parametric; this
is true for a weaker definition of parametricity that is called
continuity in our ICFP paper cited above, and that seems
insufficient for your purposes.)<br>
<br>
What this boils down to is that any function of type<br>
(p : ∑[ X \in Set ] P X) -> ... -> T,<br>
where T : Set 0, will satisfy the property you are looking for:
its output does not depend on the first component of p.<br>
<br>
This is proven metatheoretically in:<br>
Atkey, Ghani, Johann, 2014, "<span style="left: 76.3802px; top:
615.668px; font-size: 13.2835px; font-family: sans-serif;
transform: scaleX(0.888169);">A Relationally Parametric Model of
Dependent Type Theory.</span>"<br>
Krishnaswami & Dreyer, 2013, "Internalizing Relational
Parametricity in the Extensional Calculusof Constructions",<br>
Takeuti, 2001, "The Theory of Parametricity in Lambda Cube."<br>
<br>
The theorem breaks down if you add axioms such as
(non-exhaustively):<br>
* choice / law of excluded middle: (X : Set) -> X + (X ->
False)<br>
* resizing axioms<br>
<br>
Best regards,<br>
Andreas<br>
<br>
<div class="moz-cite-prefix">On 6/02/19 22:30, Jacques Carette
wrote:<br>
</div>
<blockquote type="cite"
cite="mid:6bcd3784-b5d8-bc25-f042-09d96021f099@mcmaster.ca">Is
there a way to encode an existential in Agda where the "exists"
part of the dependent pair is abstract/private? ∃ from the
standard library just says that the first part is implicit (i.e.
can be uniquely inferred). <br>
<br>
I think I could use the same trick as in Haskell (i.e. use a
wrapper where I don't export the constructor but do provide
accessor functions that don't leak), but that somehow seems
heavy. Is there something simpler? <br>
<br>
I don't mean a fully non-constructive exists here: I mean
(informally!) "a dependent pair where the first projection is
abstract and cannot leak". So to build such a thing, a definite
type must be used, but once it is created, the 'clients' as it
were, cannot find out what that type is. <br>
<br>
Jacques <br>
<br>
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href="https://lists.chalmers.se/mailman/listinfo/agda"
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<br>
<br>
</blockquote>
<br>
</blockquote>
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