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xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>814</fr:anchor><fr:addr
type="user">gsm-0001</fr:addr><fr:route>gsm-0001.xml</fr:route><fr:title
text="Yue Niu">Yue Niu</fr:title><fr:authors><fr:author><fr:link
type="local"
href="smaren.xml"
addr="smaren"
title="Yue Niu">Yue Niu</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>I am a postdoctoral researcher at the <fr:link
type="external"
href="https://www.nii.ac.jp/en/research/projects/jst-crest/">National Institute of Informatics</fr:link> in Tokyo, Japan working with <fr:link
type="local"
href="sekiyama-taro.xml"
addr="sekiyama-taro"
title="関山太朗(Sekiyama Taro)">Sekiyama Taro</fr:link>. I graduated from the programming languages group <fr:link
type="external"
href="https://www.cs.cmu.edu/~pop/">(PoP)</fr:link> at Carnegie Mellon University, where I was advised by <fr:link
type="local"
href="robertharper.xml"
addr="robertharper"
title="Robert Harper">Robert Harper</fr:link>.</fr:p><fr:p>My research is centered around the application of denotational semantics to program verification, guided by type theory, domain theory, and category theory.</fr:p><fr:p>Please <fr:strong>contact</fr:strong> me at yue_niu[at]nii.ac.jp.</fr:p><fr:tree
toc="true"
numbered="true"
show-heading="true"
show-metadata="false"
expanded="true"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr
type="machine">#237</fr:addr><fr:route>unstable-237.xml</fr:route><fr:title
text="Publications">Publications</fr:title><fr:authors><fr:author><fr:link
type="local"
href="smaren.xml"
addr="smaren"
title="Yue Niu">Yue Niu</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
toc="false"
numbered="false"
show-heading="true"
show-metadata="true"
expanded="false"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr
type="user">niu-sterling-harper-2024</fr:addr><fr:route>niu-sterling-harper-2024.xml</fr:route><fr:title
text="Cost-sensitive computational adequacy of higher-order recursion in synthetic domain theory">Cost-sensitive computational adequacy of higher-order recursion in synthetic domain theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>3</fr:month></fr:date><fr:authors><fr:author><fr:link
type="local"
href="smaren.xml"
addr="smaren"
title="Yue Niu">Yue Niu</fr:link></fr:author><fr:author><fr:link
type="local"
href="jonsterling.xml"
addr="jonsterling"
title="Jon Sterling">Jon Sterling</fr:link></fr:author><fr:author>Robert Harper</fr:author></fr:authors><fr:meta
name="venue">MFPS '24</fr:meta><fr:meta
name="slides">https://yuesforest.com/talks/cost-adequacy-sdt.pdf</fr:meta><fr:meta
name="external">https://arxiv.org/abs/2404.00212</fr:meta></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
toc="false"
numbered="false"
show-heading="true"
show-metadata="true"
expanded="false"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr
type="user">grodin-harper-niu-sterling-2023</fr:addr><fr:route>grodin-harper-niu-sterling-2023.xml</fr:route><fr:title
text="Decalf: A Directed, Effectful Cost-Aware Logical Framework">Decalf: A Directed, Effectful Cost-Aware Logical Framework</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2023</fr:year><fr:month>7</fr:month><fr:day>12</fr:day></fr:date><fr:authors><fr:author><fr:link
type="local"
href="harrisongrodin.xml"
addr="harrisongrodin"
title="Harrison Grodin">Harrison Grodin</fr:link></fr:author><fr:author><fr:link
type="local"
href="robertharper.xml"
addr="robertharper"
title="Robert Harper">Robert Harper</fr:link></fr:author><fr:author><fr:link
type="local"
href="smaren.xml"
addr="smaren"
title="Yue Niu">Yue Niu</fr:link></fr:author><fr:author><fr:link
type="local"
href="jonsterling.xml"
addr="jonsterling"
title="Jon Sterling">Jon Sterling</fr:link></fr:author></fr:authors><fr:meta
name="external">https://arxiv.org/abs/2307.05938</fr:meta><fr:meta
name="venue">POPL '24</fr:meta><fr:meta
name="bibtex"><![CDATA[%
@article{10.1145/3632852,
author = {Grodin, Harrison and Niu, Yue and Sterling, Jonathan and Harper, Robert},
title = {Decalf: A Directed, Effectful Cost-Aware Logical Framework},
year = {2024},
issue_date = {January 2024},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
volume = {8},
number = {POPL},
url = {https://doi.org/10.1145/3632852},
doi = {10.1145/3632852},
abstract = {We present decalf, a directed, effectful cost-aware logical framework for studying quantitative aspects of functional programs with effects. Like calf, the language is based on a formal phase distinction between the extension and the intension of a program, its pure behavior as distinct from its cost measured by an effectful step-counting primitive. The type theory ensures that the behavior is unaffected by the cost accounting. Unlike calf, the present language takes account of effects, such as probabilistic choice and mutable state. This extension requires a reformulation of calf’s approach to cost accounting: rather than rely on a ”separable” notion of cost, here a cost bound is simply another program. To make this formal, we equip every type with an intrinsic preorder, relaxing the precise cost accounting intrinsic to a program to a looser but nevertheless informative estimate. For example, the cost bound of a probabilistic program is itself a probabilistic program that specifies the distribution of costs. This approach serves as a streamlined alternative to the standard method of isolating a cost recurrence and readily extends to higher-order, effectful programs. The development proceeds by first introducing the decalf type system, which is based on an intrinsic ordering among terms that restricts in the extensional phase to extensional equality, but in the intensional phase reflects an approximation of the cost of a program of interest. This formulation is then applied to a number of illustrative examples, including pure and effectful sorting algorithms, simple probabilistic programs, and higher-order functions. Finally, we justify decalf via a model in the topos of augmented simplicial sets.},
journal = {Proc. ACM Program. Lang.},
month = {jan},
articleno = {10},
numpages = {29},
keywords = {algorithm analysis, amortized analysis, behavioral verification, cost models, equational reasoning, intensional property, mechanized proof, modal type theory, noninterference, parallel algorithms, phase distinction, proof assistants, recurrence relations}
}]]></fr:meta></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
toc="false"
numbered="false"
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expanded="false"
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xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>303</fr:anchor><fr:addr
type="user">niu-harper-2023</fr:addr><fr:route>niu-harper-2023.xml</fr:route><fr:title
text="A Metalanguage for Cost-Aware Denotational Semantics">A Metalanguage for Cost-Aware Denotational Semantics</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2023</fr:year><fr:month>6</fr:month></fr:date><fr:authors><fr:author><fr:link
type="local"
href="smaren.xml"
addr="smaren"
title="Yue Niu">Yue Niu</fr:link></fr:author><fr:author><fr:link
type="local"
href="robertharper.xml"
addr="robertharper"
title="Robert Harper">Robert Harper</fr:link></fr:author></fr:authors><fr:meta
name="doi">10.1109/LICS56636.2023.10175777</fr:meta><fr:meta
name="external">https://yuesforest.com/papers/lics-23.pdf</fr:meta><fr:meta
name="slides">https://yuesforest.com/talks/lics-23.pdf</fr:meta><fr:meta
name="venue">LICS '23</fr:meta><fr:meta
name="bibtex"><![CDATA[%
@INPROCEEDINGS{niu-harper-2023,
  author={Niu, Yue and Harper, Robert},
  booktitle={2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)}, 
  title={A Metalanguage for Cost-Aware Denotational Semantics}, 
  year={2023},
  volume={},
  number={},
  pages={1-14},
  doi={10.1109/LICS56636.2023.10175777}}]]></fr:meta></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
toc="false"
numbered="false"
show-heading="true"
show-metadata="true"
expanded="false"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr
type="user">niu-sterling-grodin-harper-2022</fr:addr><fr:route>niu-sterling-grodin-harper-2022.xml</fr:route><fr:title
text="A Cost-Aware Logical Framework">A Cost-Aware Logical Framework</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2022</fr:year><fr:month>1</fr:month></fr:date><fr:authors><fr:author><fr:link
type="local"
href="smaren.xml"
addr="smaren"
title="Yue Niu">Yue Niu</fr:link></fr:author><fr:author><fr:link
type="local"
href="jonsterling.xml"
addr="jonsterling"
title="Jon Sterling">Jon Sterling</fr:link></fr:author><fr:author><fr:link
type="local"
href="harrisongrodin.xml"
addr="harrisongrodin"
title="Harrison Grodin">Harrison Grodin</fr:link></fr:author><fr:author><fr:link
type="local"
href="robertharper.xml"
addr="robertharper"
title="Robert Harper">Robert Harper</fr:link></fr:author></fr:authors><fr:meta
name="doi">10.1145/3498670</fr:meta><fr:meta
name="venue">POPL '22</fr:meta><fr:meta
name="bibtex"><![CDATA[%
@article{niu-sterling-grodin-harper-2022,
author = {Niu, Yue and Sterling, Jonathan and Grodin, Harrison and Harper, Robert},
title = {A Cost-Aware Logical Framework},
year = {2022},
issue_date = {January 2022},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
volume = {6},
number = {POPL},
url = {https://doi.org/10.1145/3498670},
doi = {10.1145/3498670},
abstract = {We present calf, a cost-aware logical framework for studying quantitative aspects of functional programs. Taking inspiration from recent work that reconstructs traditional aspects of programming languages in terms of a modal account of phase distinctions, we argue that the cost structure of programs motivates a phase distinction between intension and extension. Armed with this technology, we contribute a synthetic account of cost structure as a computational effect in which cost-aware programs enjoy an internal noninterference property: input/output behavior cannot depend on cost. As a full-spectrum dependent type theory, calf presents a unified language for programming and specification of both cost and behavior that can be integrated smoothly with existing mathematical libraries available in type theoretic proof assistants. We evaluate calf as a general framework for cost analysis by implementing two fundamental techniques for algorithm analysis: the method of recurrence relations and physicist’s method for amortized analysis. We deploy these techniques on a variety of case studies: we prove a tight, closed bound for Euclid’s algorithm, verify the amortized complexity of batched queues, and derive tight, closed bounds for the sequential and parallel complexity of merge sort, all fully mechanized in the Agda proof assistant. Lastly we substantiate the soundness of quantitative reasoning in calf by means of a model construction.},
journal = {Proc. ACM Program. Lang.},
month = {jan},
articleno = {9},
numpages = {31},
keywords = {noninterference, behavioral verification, recurrence relations, cost models, intensional property, mechanized proof, algorithm analysis, parallel algorithms, proof assistants, phase distinction, equational reasoning, amortized analysis, modal type theory}
}]]></fr:meta></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
toc="false"
numbered="false"
show-heading="true"
show-metadata="true"
expanded="false"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>311</fr:anchor><fr:addr
type="user">niu-hoffmann</fr:addr><fr:route>niu-hoffmann.xml</fr:route><fr:title
text="Automatic Space Bound Analysis for Functional Programs with Garbage Collection">Automatic Space Bound Analysis for Functional Programs with Garbage Collection</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2018</fr:year><fr:month>11</fr:month></fr:date><fr:authors><fr:author><fr:link
type="local"
href="smaren.xml"
addr="smaren"
title="Yue Niu">Yue Niu</fr:link></fr:author><fr:author><fr:link
type="local"
href="janhoffmann.xml"
addr="janhoffmann"
title="Jan Hoffmann">Jan Hoffmann</fr:link></fr:author></fr:authors><fr:meta
name="doi">10.29007/xkwx</fr:meta><fr:meta
name="venue">LPAR '18</fr:meta></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
toc="true"
numbered="true"
show-heading="true"
show-metadata="false"
expanded="true"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:addr
type="machine">#238</fr:addr><fr:route>unstable-238.xml</fr:route><fr:title
text="Manuscripts">Manuscripts</fr:title><fr:authors><fr:author><fr:link
type="local"
href="smaren.xml"
addr="smaren"
title="Yue Niu">Yue Niu</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
toc="false"
numbered="false"
show-heading="true"
show-metadata="true"
expanded="false"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:addr
type="user">niu-harper-2020</fr:addr><fr:route>niu-harper-2020.xml</fr:route><fr:title
text="Cost-Aware Type Theory">Cost-Aware Type Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2020</fr:year><fr:month>11</fr:month><fr:day>7</fr:day></fr:date><fr:authors><fr:author><fr:link
type="local"
href="smaren.xml"
addr="smaren"
title="Yue Niu">Yue Niu</fr:link></fr:author><fr:author><fr:link
type="local"
href="robertharper.xml"
addr="robertharper"
title="Robert Harper">Robert Harper</fr:link></fr:author></fr:authors><fr:meta
name="external">https://arxiv.org/abs/2011.03660</fr:meta></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
toc="true"
numbered="true"
show-heading="true"
show-metadata="false"
expanded="true"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>319</fr:anchor><fr:addr
type="machine">#239</fr:addr><fr:route>unstable-239.xml</fr:route><fr:title
text="Doctoral work">Doctoral work</fr:title><fr:authors><fr:author><fr:link
type="local"
href="smaren.xml"
addr="smaren"
title="Yue Niu">Yue Niu</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
  
  <fr:tree
toc="true"
numbered="true"
show-heading="true"
show-metadata="true"
expanded="true"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr
type="user">niu-thesis</fr:addr><fr:route>niu-thesis.xml</fr:route><fr:title
text="Cost-sensitive programming, verification, and semantics">Cost-sensitive programming, verification, and semantics</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>9</fr:month></fr:date><fr:authors><fr:author><fr:link
type="local"
href="smaren.xml"
addr="smaren"
title="Yue Niu">Yue Niu</fr:link></fr:author></fr:authors><fr:meta
name="slides">https://yuesforest.com/talks/defense.pdf</fr:meta><fr:meta
name="external">https://yuesforest.com/papers/thesis.pdf</fr:meta></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree>
  <fr:tree
toc="true"
numbered="true"
show-heading="true"
show-metadata="true"
expanded="true"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr
type="user">niu-proposal</fr:addr><fr:route>niu-proposal.xml</fr:route><fr:title
text="A cost-aware logical framework">A cost-aware logical framework</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2022</fr:year><fr:month>12</fr:month></fr:date><fr:authors><fr:author><fr:link
type="local"
href="smaren.xml"
addr="smaren"
title="Yue Niu">Yue Niu</fr:link></fr:author></fr:authors><fr:meta
name="external">https://yuesforest.com/papers/proposal.pdf</fr:meta></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree>
  </fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
toc="true"
numbered="true"
show-heading="true"
show-metadata="false"
expanded="true"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr
type="user">gsm-000L</fr:addr><fr:route>gsm-000L.xml</fr:route><fr:title
text="Notes">Notes</fr:title><fr:date><fr:year>2023</fr:year><fr:month>9</fr:month><fr:day>20</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter>
  
  
  <fr:tree
toc="true"
numbered="false"
show-heading="true"
show-metadata="false"
expanded="false"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr
type="user">gsm-000H</fr:addr><fr:route>gsm-000H.xml</fr:route><fr:title
text="The intrinsic vertical {}cpo structure of complete -spaces">The intrinsic vertical <fr:tex
display="inline"><![CDATA[\mathsf {\omega }]]></fr:tex>cpo structure of complete <fr:tex
display="inline"><![CDATA[\Sigma ]]></fr:tex>-spaces</fr:title><fr:date><fr:year>2023</fr:year><fr:month>9</fr:month><fr:day>20</fr:day></fr:date><fr:authors><fr:author><fr:link
type="local"
href="smaren.xml"
addr="smaren"
title="Yue Niu">Yue Niu</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>Every <fr:link
type="local"
href="gsm-0003.xml"
addr="gsm-0003"
title="Complete types in synthetic domain theory">complete</fr:link> <fr:tex
display="inline"><![CDATA[\Sigma ]]></fr:tex>-space in a model of synthetic domain theory has an <fr:link
type="local"
href="gsm-0004.xml"
addr="gsm-0004"
title="The instrinsic preorder in synthetic domain theory">intrinsic preorder order structure</fr:link> that is cocomplete with respect to <fr:tex
display="inline"><![CDATA[\omega ]]></fr:tex>-indexed families.</fr:p><fr:tree
toc="true"
numbered="true"
show-heading="true"
show-metadata="false"
expanded="true"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>328</fr:anchor><fr:addr
type="user">gsm-0002</fr:addr><fr:route>gsm-0002.xml</fr:route><fr:title
text="Every complete type has least upper bounds for synthetic -chains">Every complete type has least upper bounds for synthetic <fr:tex
display="inline"><![CDATA[\omega ]]></fr:tex>-chains</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2023</fr:year><fr:month>9</fr:month><fr:day>11</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>For every map <fr:tex
display="inline"><![CDATA[f:\omega  \to  A]]></fr:tex> into a <fr:link
type="local"
href="gsm-0003.xml"
addr="gsm-0003"
title="Complete types in synthetic domain theory">complete type</fr:link> <fr:tex
display="inline"><![CDATA[A]]></fr:tex>, there exists an element <fr:tex
display="inline"><![CDATA[a_\infty  : A]]></fr:tex> such that <fr:tex
display="inline"><![CDATA[a_\infty ]]></fr:tex> is a least upper bound of <fr:tex
display="inline"><![CDATA[f]]></fr:tex> with respect to the <fr:link
type="local"
href="gsm-0004.xml"
addr="gsm-0004"
title="The instrinsic preorder in synthetic domain theory">intrinsic preorder order structure</fr:link>.</fr:p>
 
   
   <fr:tree
toc="false"
numbered="true"
show-heading="true"
show-metadata="false"
expanded="true"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr
type="machine">#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2023</fr:year><fr:month>9</fr:month><fr:day>11</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter>Define <fr:tex
display="inline"><![CDATA[f_\infty ]]></fr:tex> be the element determined by the unique extension <fr:tex
display="inline"><![CDATA[\overline {f} : \overline {\omega } \to  A]]></fr:tex> evaluated at the invariant point <fr:tex
display="inline"><![CDATA[\infty ]]></fr:tex>. 
  <fr:ol><fr:li>First we show that <fr:tex
display="inline"><![CDATA[f_\infty ]]></fr:tex> is an upper bound for <fr:tex
display="inline"><![CDATA[f]]></fr:tex>. Fixing <fr:tex
display="inline"><![CDATA[i: \omega ]]></fr:tex>, we need to show that <fr:tex
display="inline"><![CDATA[f~i \sqsubseteq ^\circ _A f_\infty ]]></fr:tex>. Because <fr:tex
display="inline"><![CDATA[\overline {f}]]></fr:tex> extends <fr:tex
display="inline"><![CDATA[f]]></fr:tex>, it suffices to show <fr:tex
display="inline"><![CDATA[\overline {f}~i \sqsubseteq ^\circ _A f_\infty ]]></fr:tex>. Using the fact that every map is monotone with respect to the specialization order, the result holds because <fr:link
type="local"
href="gsm-0006.xml"
addr="gsm-0006"
title="The invariant point is an upper bound for the initial lift algebra"><fr:tex
display="inline"><![CDATA[i \sqsubseteq ^\circ _{\overline {\omega }} \infty ]]></fr:tex></fr:link>.</fr:li>

    <fr:li>Let <fr:tex
display="inline"><![CDATA[\alpha ]]></fr:tex> be an upper bound for <fr:tex
display="inline"><![CDATA[f]]></fr:tex>. We need to show that <fr:tex
display="inline"><![CDATA[f_\infty  \sqsubseteq ^\circ  \alpha ]]></fr:tex>. If the principal lower set <fr:tex
display="inline"><![CDATA[{\downarrow }(\alpha )]]></fr:tex> is complete, we have the following lifting situation:

      
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type="latex"
part="preamble"><![CDATA[
    
		\usepackage{tikz, tikz-cd, mathtools, amssymb, stmaryrd}
	\usetikzlibrary{matrix,arrows}
	\usetikzlibrary{fit,positioning,calc,shapes}
	\usetikzlibrary{decorations.pathreplacing}
	\usetikzlibrary{decorations.pathmorphing}
	\usetikzlibrary{decorations.markings}

	\tikzset{
	  desc/.style={sloped, fill=white,inner sep=2pt},
	  upright desc/.style={fill=white,inner sep=2pt},
	  pullback/.style = {
	    append after command={
	      \pgfextra{
	        \draw ($(\tikzlastnode) + (.2cm,-.5cm)$) -- ++(0.3cm,0) -- ++(0,0.3cm);
	      }
	    }
	  },
	  pullback 45/.style = {
	    append after command={
	      \pgfextra{
	        \draw[rotate = 45] ($(\tikzlastnode) + (.2cm,-.5cm)$) -- ++(0.3cm,0) -- ++(0,0.3cm);
	      }
	    }
	  },
	  ne pullback/.style = {
	    append after command={
	      \pgfextra{
	        \draw ($(\tikzlastnode) + (-.2cm,-.5cm)$) -- ++(-0.3cm,0) -- ++(0,0.3cm);
	      }
	    }
	  },
	  sw pullback/.style = {
	    append after command={
	      \pgfextra{
	        \draw ($(\tikzlastnode) + (.2cm,.5cm)$) -- ++(0.3cm,0) -- ++(0,-0.3cm);
	      }
	    }
	  },
	  dotted pullback/.style = {
	    append after command={
	      \pgfextra{
	        \draw [densely dotted] ($(\tikzlastnode) + (.2cm,-.5cm)$) -- ++(0.3cm,0) -- ++(0,0.3cm);
	      }
	    }
	  },
	  muted pullback/.style = {
	    append after command={
	      \pgfextra{
	        \draw [gray] ($(\tikzlastnode) + (.2cm,-.5cm)$) -- ++(0.3cm,0) -- ++(0,0.3cm);
	      }
	    }
	  },
	  pushout/.style = {
	    append after command={
	      \pgfextra{
	        \draw ($(\tikzlastnode) + (-.2cm,.5cm)$) -- ++(-0.3cm,0) -- ++(0,-0.3cm);
	      }
	    }
	  },
	  between/.style args={#1 and #2}{
	    at = ($(#1)!0.5!(#2)$)
	  },
	  diagram/.style = {
	    on grid,
	    node distance=2cm,
	    commutative diagrams/every diagram,
	    line width = .5pt,
	    every node/.append style = {
	      commutative diagrams/every cell,
	    }
	  },
	  fibration/.style = {
	    -{Triangle[open]}
	  },
	  etale/.style = {
	    -{Triangle[open]}
	  },
	  etale cover/.style= {
	    >={Triangle[open]},->.>
	  },
	  opfibration/.style = {
	    -{Triangle}
	  },
	  lies over/.style = {
	    |-{Triangle[open]}
	  },
	  op lies over/.style = {
	    |-{Triangle}
	  },
	  embedding/.style = {
	    {right hook}->
	  },
	  open immersion/.style = {
	    {right hook}-{Triangle[open]}
	  },
	  closed immersion/.style = {
	    {right hook}-{Triangle}
	  },
	  closed immersion*/.style = {
	    {left hook}-{Triangle}
	  },
	  embedding*/.style = {
	    {left hook}->
	  },
	  open immersion*/.style = {
	    {left hook}-{Triangle[open]}
	  },
	  exists/.style = {
	    densely dashed
	  },
	}

	\newlength{\dontworryaboutit}

	\tikzset{
	  inline diagram/.style = {
	    commutative diagrams/every diagram,
	    commutative diagrams/cramped,
	    line width = .5pt,
	    every node/.append style = {
	      commutative diagrams/every cell,
	      anchor = base,
	      inner sep = 0pt
	    },
	    every path/.append style = {
	      outer xsep = 2pt
	    }
	  }
	}

	\tikzset{
	  square/nw/.style = {},
	  square/ne/.style = {},
	  square/se/.style = {},
	  square/sw/.style = {},
	  square/north/.style = {->},
	  square/south/.style = {->},
	  square/west/.style = {->},
	  square/east/.style = {->},
	  square/north/node/.style = {above},
	  square/south/node/.style = {below},
	  square/west/node/.style = {left},
	  square/east/node/.style = {right},
	}

	\ExplSyntaxOn

	\bool_new:N \l_jon_glue_west

	\keys_define:nn { jon-tikz/diagram } {
	  nw .tl_set:N = \l_jon_tikz_diagram_nw,
	  sw .tl_set:N = \l_jon_tikz_diagram_sw,
	  ne .tl_set:N = \l_jon_tikz_diagram_ne,
	  se .tl_set:N = \l_jon_tikz_diagram_se,

	  width .tl_set:N = \l_jon_tikz_diagram_width,
	  height .tl_set:N = \l_jon_tikz_diagram_height,

	  north .tl_set:N = \l_jon_tikz_diagram_north,
	  south .tl_set:N = \l_jon_tikz_diagram_south,
	  west .tl_set:N = \l_jon_tikz_diagram_west,
	  east .tl_set:N = \l_jon_tikz_diagram_east,

	  nw/style .code:n = {\tikzset{square/nw/.style = {#1}}},
	  sw/style .code:n = {\tikzset{square/sw/.style = {#1}}},
	  ne/style .code:n = {\tikzset{square/ne/.style = {#1}}},
	  se/style .code:n = {\tikzset{square/se/.style = {#1}}},

	  glue .choice:,
	  glue / west .code:n = {\bool_set:Nn \l_jon_glue_west \c_true_bool},

	  glue~target .tl_set:N = \l_jon_tikz_glue_target,

	  north/style .code:n = {\tikzset{square/north/.style = {#1}}},
	  north/node/style .code:n = {\tikzset{square/north/node/.style = {#1}}},
	  south/style .code:n = {\tikzset{square/south/.style = {#1}}},
	  south/node/style .code:n = {\tikzset{square/south/node/.style = {#1}}},
	  west/style .code:n = {\tikzset{square/west/.style = {#1}}},
	  west/node/style .code:n = {\tikzset{square/west/node/.style = {#1}}},
	  east/style .code:n = {\tikzset{square/east/.style = {#1}}},
	  east/node/style .code:n = {\tikzset{square/east/node/.style = {#1}}},

	  draft .meta:n = {
	    nw = {\__jon_tikz_diagram_fmt_placeholder:n {nw}},
	    sw = {\__jon_tikz_diagram_fmt_placeholder:n {sw}},
	    se = {\__jon_tikz_diagram_fmt_placeholder:n {se}},
	    ne = {\__jon_tikz_diagram_fmt_placeholder:n {ne}},
	    north = {\__jon_tikz_diagram_fmt_placeholder:n {north}},
	    south = {\__jon_tikz_diagram_fmt_placeholder:n {south}},
	    west = {\__jon_tikz_diagram_fmt_placeholder:n {west}},
	    east = {\__jon_tikz_diagram_fmt_placeholder:n {east}},
	  }
	}

	\tl_set:Nn \l_jon_tikz_diagram_width { 2cm }
	\tl_set:Nn \l_jon_tikz_diagram_height { 2cm }


	\cs_new:Npn \__jon_tikz_diagram_fmt_placeholder:n #1 {
	  \texttt{\textcolor{red}{#1}}
	}

	\keys_set:nn { jon-tikz/diagram } {
	  glue~target = {},
	}


	\cs_new:Nn \__jon_tikz_render_square:nn {
	  \group_begin:
	  \keys_set:nn {jon-tikz/diagram} {#2}
	  \bool_if:nTF \l_jon_glue_west {
	    \node (#1ne) [right = \l_jon_tikz_diagram_width~of~\l_jon_tikz_glue_target ne,square/ne] {$\l_jon_tikz_diagram_ne$};
	    \node (#1se) [below = \l_jon_tikz_diagram_height~of~#1ne,square/se] {$\l_jon_tikz_diagram_se$};
	    \draw[square/north] (\l_jon_tikz_glue_target ne) to node [square/north/node] {$\l_jon_tikz_diagram_north$} (#1ne);
	    \draw[square/east] (#1ne) to node [square/east/node] {$\l_jon_tikz_diagram_east$} (#1se);
	    \draw[square/south] (\l_jon_tikz_glue_target se) to node [square/south/node] {$\l_jon_tikz_diagram_south$} (#1se);
	  } {
	    \node (#1nw) [square/nw] {$\l_jon_tikz_diagram_nw$};
	    \node (#1sw) [below = \l_jon_tikz_diagram_height~of~#1nw,square/sw] {$\l_jon_tikz_diagram_sw$};
	    \draw[square/west] (#1nw) to node [square/west/node] {$\l_jon_tikz_diagram_west$} (#1sw);

	    \node (#1ne) [right = \l_jon_tikz_diagram_width~of~#1nw,square/ne] {$\l_jon_tikz_diagram_ne$};
	    \node (#1se) [below = \l_jon_tikz_diagram_height~of~#1ne,square/se] {$\l_jon_tikz_diagram_se$};
	    \draw[square/north] (#1nw) to node [square/north/node] {$\l_jon_tikz_diagram_north$} (#1ne);
	    \draw[square/east] (#1ne) to node [square/east/node] {$\l_jon_tikz_diagram_east$} (#1se);
	    \draw[square/south] (#1sw) to node [square/south/node] {$\l_jon_tikz_diagram_south$} (#1se);
	  }
	  \group_end:
	}

	\NewDocumentCommand\SpliceDiagramSquare{D<>{}m}{
	  \__jon_tikz_render_square:nn {#1} {#2}
	}


	\NewDocumentCommand\DiagramSquare{D<>{}O{}m}{
	  \begin{tikzpicture}[diagram,#2,baseline=(#1sw.base)]
	    \__jon_tikz_render_square:nn {#1} {#3}
	  \end{tikzpicture}
	}

	\NewDocumentCommand{\Local}{oommmooo}{
		\begin{tikzpicture}[diagram]
		\SpliceDiagramSquare<sq/>{
			width = \IfValueT{#8}{#8}, 
			nw = #3,
			sw = #5,
			ne = #4,
			south/style = {opacity = 0},
			east/style = {opacity = 0},
			north = \IfValueT{#1}{#1}, 
			north/style = \IfValueTF{#2}{#2}{>->},
			west = \IfValueT{#6}{#6},
		}
		\draw[->,dotted] (sq/ne) to node[right] {$\IfValueT{#7}{#7}$} (sq/sw);
	\end{tikzpicture}}
	\ExplSyntaxOff

    \usepackage {mathpartir}
   ]]></fr:resource-source><fr:resource-source
type="latex"
part="body"><![CDATA[
        \begin {tikzpicture}[diagram]
        \path 
        (-1.5,0) node (I) {$\omega $}
        (1.5,0) node (F) {$\overline {\omega }$}
        (0,-2) node (A) [align=center] {${\downarrow }(\alpha ) = \{ a | a \sqsubseteq ^\circ  \alpha  \}$};
        \draw [embedding] (I) to (F); 
        \draw [->] (I) to node[left] {$f$} (A); 
        \draw [->,exists] (F) to node[right, shift={(0.1,0)}] {$\tilde {f}$} (A);
        \end {tikzpicture}
      ]]></fr:resource-source></fr:resource></fr:figure>
  

      <fr:p>In the above <fr:tex
display="inline"><![CDATA[\tilde {f}]]></fr:tex> is the unique extension of <fr:tex
display="inline"><![CDATA[f]]></fr:tex> considered as a map <fr:tex
display="inline"><![CDATA[\omega  \to  {\downarrow }(\alpha )]]></fr:tex>. By uniqueness of <fr:tex
display="inline"><![CDATA[\overline {f}]]></fr:tex> as the extension of <fr:tex
display="inline"><![CDATA[f : \omega  \to  A]]></fr:tex>, <fr:tex
display="inline"><![CDATA[\tilde {f}]]></fr:tex> is equal to <fr:tex
display="inline"><![CDATA[\overline {f}]]></fr:tex> considered as maps <fr:tex
display="inline"><![CDATA[\overline {\omega } \to  A]]></fr:tex>. Consequently we have that <fr:tex
display="inline"><![CDATA[f_\infty  = \overline {f}(\infty ) = \tilde {f}(\infty )]]></fr:tex>, so the result follows by observing that <fr:tex
display="inline"><![CDATA[\tilde {f}(\infty ) \in  {\downarrow }(\alpha )]]></fr:tex>.</fr:p> 
    
    
      <fr:p>It remains to show that <fr:tex
display="inline"><![CDATA[{\downarrow }(\alpha )]]></fr:tex> is complete. We can express the principal lower set as follows:</fr:p>

      <fr:tex
display="block"><![CDATA[\begin {aligned}           {\downarrow }(\alpha ) &= {\left \{a \mid  a \sqsubseteq ^\circ  \alpha \right \}}\\            &= {\left \{a \mid  \forall  f:A\to \Sigma ~ f(a) \to  f(\alpha )\right \}}\\            &= \bigcap _{f:A\to \Sigma } {\left \{a \mid  f(a) \to  f(\alpha )\right \}}       \end {aligned}]]></fr:tex> 

      <fr:p>Because complete types are internally complete, the result would follow if we can show that <fr:tex
display="inline"><![CDATA[S = {\left \{a \mid  f(a) \to  f(\alpha )\right \}}]]></fr:tex> is complete. Using similar reasoning to <fr:link
type="local"
href="gsm-0006.xml"
addr="gsm-0006"
title="The invariant point is an upper bound for the initial lift algebra">The invariant point is an upper bound for the initial lift algebra</fr:link>, we may show that <fr:tex
display="inline"><![CDATA[S]]></fr:tex> can be computed as a pullback of powers of <fr:tex
display="inline"><![CDATA[\Sigma ]]></fr:tex>, whence it follows that <fr:tex
display="inline"><![CDATA[S]]></fr:tex> is complete as well.</fr:p></fr:li></fr:ol>
</fr:mainmatter><fr:backmatter /></fr:tree>
 
</fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
toc="true"
numbered="true"
show-heading="true"
show-metadata="false"
expanded="true"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr
type="user">gsm-000G</fr:addr><fr:route>gsm-000G.xml</fr:route><fr:title
text="Complete -spaces are closed under -indexed joins">Complete <fr:tex
display="inline"><![CDATA[\Sigma ]]></fr:tex>-spaces are closed under <fr:tex
display="inline"><![CDATA[\omega ]]></fr:tex>-indexed joins</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2023</fr:year><fr:month>9</fr:month><fr:day>19</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
</fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter><fr:tree
toc="false"
numbered="false"
show-heading="true"
show-metadata="false"
expanded="true"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title
text="Related">Related</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
toc="true"
numbered="false"
show-heading="true"
show-metadata="true"
expanded="false"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>816</fr:anchor><fr:addr
type="user">robertharper</fr:addr><fr:route>robertharper.xml</fr:route><fr:title
text="Robert Harper">Robert Harper</fr:title><fr:taxon>Person</fr:taxon><fr:authors /><fr:meta
name="external">http://www.cs.cmu.edu/~rwh/</fr:meta></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
toc="true"
numbered="false"
show-heading="true"
show-metadata="true"
expanded="false"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>817</fr:anchor><fr:addr
type="user">sekiyama-taro</fr:addr><fr:route>sekiyama-taro.xml</fr:route><fr:title
text="関山太朗(Sekiyama Taro)">関山太朗(Sekiyama Taro)</fr:title><fr:taxon>Person</fr:taxon><fr:authors /><fr:meta
name="external">https://www.nii.ac.jp/en/faculty/architecture/sekiyama_taro/</fr:meta></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>