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@@ -1,2 +1,3 @@
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.build/
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graphs/*.tex
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svg-inkscape
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+89
-45
@@ -1,54 +1,98 @@
|
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@InProceedings{Fiore2008,
|
||||
|
||||
author={Fiore, Marcelo},
|
||||
|
||||
booktitle={2008 23rd Annual IEEE Symposium on Logic in Computer Science},
|
||||
|
||||
title={Second-Order and Dependently-Sorted Abstract Syntax},
|
||||
|
||||
year={2008},
|
||||
|
||||
volume={},
|
||||
|
||||
number={},
|
||||
|
||||
pages={57-68},
|
||||
|
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keywords={Algebra;Computer science;Mathematical model;Logic functions;Laboratories;MONOS devices;Sorting;abstract syntax;second-order syntax;dependently-sorted syntax;alpha-equivalence;variable binding;substitution;metavariable;meta-substitution;categorical algebra},
|
||||
|
||||
doi={10.1109/LICS.2008.38}
|
||||
@inproceedings{Fiore2008,
|
||||
author = {Marcelo P. Fiore},
|
||||
title = {Second-Order and Dependently-Sorted Abstract Syntax},
|
||||
booktitle = {Proceedings of the Twenty-Third Annual {IEEE} Symposium on Logic in
|
||||
Computer Science, {LICS} 2008, 24-27 June 2008, Pittsburgh, PA, {USA}},
|
||||
pages = {57--68},
|
||||
publisher = {{IEEE} Computer Society},
|
||||
year = {2008},
|
||||
url = {https://doi.org/10.1109/LICS.2008.38},
|
||||
doi = {10.1109/LICS.2008.38},
|
||||
timestamp = {Fri, 24 Mar 2023 00:01:50 +0100},
|
||||
biburl = {https://dblp.org/rec/conf/lics/Fiore08.bib},
|
||||
bibsource = {dblp computer science bibliography, https://dblp.org}
|
||||
}
|
||||
|
||||
@InProceedings{Altenkirch2018,
|
||||
author={"Altenkirch, Thorsten and Capriotti, Paolo and Dijkstra, Gabe and Kraus, Nicolai and Nordvall Forsberg, Fredrik"},
|
||||
editor={"Baier, Christel and Dal Lago, Ugo"},
|
||||
title={"Quotient Inductive-Inductive Types"},
|
||||
booktitle={"Foundations of Software Science and Computation Structures"},
|
||||
year={"2018"},
|
||||
publisher={"Springer International Publishing"},
|
||||
address={"Cham"},
|
||||
pages={"293--310"},
|
||||
abstract={"Higher inductive types (HITs) in Homotopy Type Theory allow the definition of datatypes which have constructors for equalities over the defined type. HITs generalise quotient types, and allow to define types with non-trivial higher equality types, such as spheres, suspensions and the torus. However, there are also interesting uses of HITs to define types satisfying uniqueness of equality proofs, such as the Cauchy reals, the partiality monad, and the well-typed syntax of type theory. In each of these examples we define several types that depend on each other mutually, i.e. they are inductive-inductive definitions. We call those HITs quotient inductive-inductive types (QIITs). Although there has been recent progress on a general theory of HITs, there is not yet a theoretical foundation for the combination of equality constructors and induction-induction, despite many interesting applications. In the present paper we present a first step towards a semantic definition of QIITs. In particular, we give an initial-algebra semantics. We further derive a section induction principle, stating that every algebra morphism into the algebra in question has a section, which is close to the intuitively expected elimination rules."},
|
||||
isbn={"978-3-319-89366-2"}
|
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@inproceedings{Altenkirch2018,
|
||||
author = {Thorsten Altenkirch and
|
||||
Paolo Capriotti and
|
||||
Gabe Dijkstra and
|
||||
Nicolai Kraus and
|
||||
Fredrik Nordvall Forsberg},
|
||||
editor = {Christel Baier and
|
||||
Ugo Dal Lago},
|
||||
title = {Quotient Inductive-Inductive Types},
|
||||
booktitle = {Foundations of Software Science and Computation Structures - 21st
|
||||
International Conference, {FOSSACS} 2018, Held as Part of the European
|
||||
Joint Conferences on Theory and Practice of Software, {ETAPS} 2018,
|
||||
Thessaloniki, Greece, April 14-20, 2018, Proceedings},
|
||||
series = {Lecture Notes in Computer Science},
|
||||
volume = {10803},
|
||||
pages = {293--310},
|
||||
publisher = {Springer},
|
||||
year = {2018},
|
||||
url = {https://doi.org/10.1007/978-3-319-89366-2\_16},
|
||||
doi = {10.1007/978-3-319-89366-2\_16},
|
||||
timestamp = {Sun, 04 Aug 2024 19:40:23 +0200},
|
||||
biburl = {https://dblp.org/rec/conf/fossacs/AltenkirchCDKF18.bib},
|
||||
bibsource = {dblp computer science bibliography, https://dblp.org}
|
||||
}
|
||||
|
||||
@InProceedings{Munchhausen,
|
||||
author = {Altenkirch, Thorsten and Kaposi, Ambrus and \v{S}inkarovs, Artjoms and V\'{e}gh, Tam\'{a}s},
|
||||
title = {{The M\"{u}nchhausen Method in Type Theory}},
|
||||
booktitle = {28th International Conference on Types for Proofs and Programs (TYPES 2022)},
|
||||
pages = {10:1--10:20},
|
||||
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
|
||||
ISBN = {978-3-95977-285-3},
|
||||
ISSN = {1868-8969},
|
||||
year = {2023},
|
||||
author = {Thorsten Altenkirch and
|
||||
Ambrus Kaposi and
|
||||
Artjoms Sinkarovs and
|
||||
Tam{\'{a}}s V{\'{e}}gh},
|
||||
editor = {Delia Kesner and
|
||||
Pierre{-}Marie P{\'{e}}drot},
|
||||
title = {The M{\"{u}}nchhausen Method in Type Theory},
|
||||
booktitle = {28th International Conference on Types for Proofs and Programs, {TYPES}
|
||||
2022, June 20-25, 2022, LS2N, University of Nantes, France},
|
||||
series = {LIPIcs},
|
||||
volume = {269},
|
||||
editor = {Kesner, Delia and P\'{e}drot, Pierre-Marie},
|
||||
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
|
||||
address = {Dagstuhl, Germany},
|
||||
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2022.10},
|
||||
URN = {urn:nbn:de:0030-drops-184534},
|
||||
doi = {10.4230/LIPIcs.TYPES.2022.10},
|
||||
annote = {Keywords: type theory, proof assistants, very dependent types}
|
||||
pages = {10:1--10:20},
|
||||
publisher = {Schloss Dagstuhl - Leibniz-Zentrum f{\"{u}}r Informatik},
|
||||
year = {2022},
|
||||
url = {https://doi.org/10.4230/LIPIcs.TYPES.2022.10},
|
||||
doi = {10.4230/LIPICS.TYPES.2022.10},
|
||||
timestamp = {Mon, 31 Jul 2023 17:17:51 +0200},
|
||||
biburl = {https://dblp.org/rec/conf/types/AltenkirchKSV22.bib},
|
||||
bibsource = {dblp computer science bibliography, https://dblp.org}
|
||||
}
|
||||
@article{CartmellGATs,
|
||||
author = {John Cartmell},
|
||||
title = {Generalised algebraic theories and contextual categories},
|
||||
journal = {Ann. Pure Appl. Log.},
|
||||
volume = {32},
|
||||
pages = {209--243},
|
||||
year = {1986},
|
||||
url = {https://doi.org/10.1016/0168-0072(86)90053-9},
|
||||
doi = {10.1016/0168-0072(86)90053-9},
|
||||
timestamp = {Fri, 21 Feb 2020 21:18:13 +0100},
|
||||
biburl = {https://dblp.org/rec/journals/apal/Cartmell86.bib},
|
||||
bibsource = {dblp computer science bibliography, https://dblp.org}
|
||||
}
|
||||
|
||||
@phdthesis{SestiniPhD,
|
||||
author = {Filippo Sestini},
|
||||
title = {Bootstrapping Extensionality},
|
||||
school = {University of Nottingham},
|
||||
year = 2023,
|
||||
month = mar
|
||||
}
|
||||
|
||||
@misc{AmbrusSzumiXie2sort,
|
||||
author = {Ambrus Kaposi},
|
||||
title = {Message to the Agda mailing list},
|
||||
howpublished = {\url{https://lists.chalmers.se/pipermail/agda/2019/011176.html}},
|
||||
year = 2019
|
||||
}
|
||||
|
||||
@misc{nlab:reflective_subcategory,
|
||||
author = {{nLab authors}},
|
||||
title = {reflective subcategory},
|
||||
howpublished = {\url{https://ncatlab.org/nlab/show/reflective+subcategory}},
|
||||
note = {\href{https://ncatlab.org/nlab/revision/reflective+subcategory/116}{Revision 116}},
|
||||
month = jul,
|
||||
year = 2024
|
||||
}
|
||||
@@ -0,0 +1,530 @@
|
||||
% !TeX spellcheck = en_US
|
||||
\DocumentMetadata{}
|
||||
\documentclass[12pt,xcolor={dvipsnames},aspectratio=169]{beamer}
|
||||
|
||||
|
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\input{./headerDiapo.tex}
|
||||
|
||||
\title[Semantics of 2-sortification]{Categorical semantics of the reduction of GATs to two-sorted GATs.
|
||||
\\[1ex] \large Notes on my 4.5-month internship at the LIX}
|
||||
\hypersetup{pdftitle={Categorical semantics of the reduction of GATs to two-sorted GATs}}
|
||||
\author[Samy Avrillon]{Samy Avrillon, supervised by
|
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\\[1ex] Ambroise Lafont (LIX, Palaiseau, France)}
|
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\date{}
|
||||
|
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\begin{document}
|
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|
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\begin{frame}[plain]
|
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\maketitle
|
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\end{frame}
|
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|
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\section{GATs and 2-sortification}
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\begin{frame}{What is a GAT ?}
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\begin{itemize}
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\item A \textbf{list} of declarations
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\item[\ding{220}] Enables to declare sorts, objects of those sorts and equalities between those objects
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\item A \textbf{syntactic object}
|
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\item[\ding{220}] Type judgements are defined with induction rules
|
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\item Describes \textbf{models}
|
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\item[\ding{220}] Defines a category of models
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\end{itemize}
|
||||
\end{frame}
|
||||
|
||||
|
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\begin{frame}{A GAT for a function in Set}
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\begin{tcolorbox}
|
||||
\begin{columns}
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\begin{column}{.6\textwidth}
|
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\renewcommand\arraystretch{1.5}
|
||||
\begin{tabular}{ll}
|
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$A : \Set$&\multirow{2}{*}{$\left.\begin{array}{c}\\\\\end{array}\right\}\text{sorts}$}\\
|
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$B : \Set$&\\\hdashline
|
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$\operatorname{exec} : A \to B$&$\}\text{constructors}$
|
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\end{tabular}
|
||||
\end{column}
|
||||
\begin{column}{.4\textwidth}
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Models:
|
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triples (A,B,f)
|
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\end{column}
|
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\end{columns}
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\end{tcolorbox}
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\end{frame}
|
||||
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\begin{frame}{A GAT for an bijective function in Set}
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\begin{tcolorbox}
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\begin{columns}
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\begin{column}{.7\textwidth}
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\renewcommand\arraystretch{1.5}
|
||||
\begin{tabular}{ll}
|
||||
$A : \Set$ & \multirow{2}{*}{$\left.\begin{array}{c}\\\\\end{array}\right\}\text{sorts}$}\\
|
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$B : \Set$ &\\\hdashline
|
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$\operatorname{exec} : A \to B$ &\multirow{2}{*}{$\left.\begin{array}{c}\\\\\end{array}\right\}\text{constructors}$}\\
|
||||
$\operatorname{invexec} : B \to A$&\\\hdashline
|
||||
$\operatorname{isol} : (x : A) \to \operatorname{invexec}(\operatorname{exec}\;x) = x$&
|
||||
\multirow{2}{*}{$\left.\begin{array}{c}\\\\\end{array}\right\}\text{equalities}$}\\
|
||||
$\operatorname{isor} : (y : B) \to \operatorname{exec}(\operatorname{invexec}\;y) = y$&\\
|
||||
\end{tabular}
|
||||
\end{column}
|
||||
\begin{column}{.3\textwidth}
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||||
Models:
|
||||
triples (A,B,f)
|
||||
|
||||
s.t. f is bijective
|
||||
\end{column}
|
||||
\end{columns}
|
||||
\end{tcolorbox}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{A GAT for a small category}
|
||||
\begin{tcolorbox}
|
||||
\begin{columns}
|
||||
\begin{column}{0.8\textwidth}
|
||||
\renewcommand\arraystretch{1.5}
|
||||
\begin{tabular}{l}
|
||||
$\Obj : \Set$ \\
|
||||
$\Hom : \Obj \to \Obj \to \Set$ \\\hdashline
|
||||
$\id : (A : \Obj) \to \Hom\;A\;A$ \\
|
||||
$\circ : (A\;B\;C:\Obj) \to \Hom\;B\;C \to\Hom\;A\;B \to \Hom\;A\;C$ \\\hdashline
|
||||
$\operatorname{idl}: (A B : \Obj) \to (\sigma : \Hom\;A\;B) \to \circ\;(\id\;B)\;\sigma = \sigma$\\
|
||||
$\operatorname{idr}: (A B : \Obj) \to (\sigma : \Hom\;A\;B) \to \circ\;\sigma\;(\id\;A) = \sigma$\\
|
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$\operatorname{\circ-trans}: \dots$\\
|
||||
\end{tabular}
|
||||
\end{column}
|
||||
\begin{column}{0.2\textwidth}
|
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Models:
|
||||
|
||||
small categories
|
||||
\end{column}
|
||||
\end{columns}
|
||||
\end{tcolorbox}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{A GAT for Type Theory}
|
||||
\begin{tcolorbox}
|
||||
\begin{columns}
|
||||
\begin{column}{0.7\textwidth}
|
||||
\renewcommand\arraystretch{1.5}
|
||||
\begin{tabular}{l}
|
||||
$\Con : \Set$ \\
|
||||
$\Ty : \Con \to \Set$ \\
|
||||
$\Tm : (\Gamma : \Con) \to \Ty\;\Gamma \to \Set$ \\\hdashline
|
||||
$\operatorname{empty}: \Con$ \\
|
||||
$\operatorname{ext}: (\Gamma:\Con)\to(A:\Ty\;\Gamma)\to\Con$ \\
|
||||
$\operatorname{implies} : (\Gamma:\Con) \to \Ty\;\Gamma \to \Ty\;\Gamma \to \Ty\;\Gamma$ \\
|
||||
$\operatorname{app} : (\Gamma : \Con) \to (A\;B :\Ty\;\Gamma) \to$\\
|
||||
\qquad$\Tm\;\Gamma\;(\operatorname{implies}\;A\;B)\to\Tm\;\Gamma\; A \to \Tm\;\Gamma\; B$
|
||||
\end{tabular}
|
||||
\end{column}
|
||||
|
||||
\begin{column}{0.3\textwidth}
|
||||
Models : Triples
|
||||
|
||||
$(X_\Con,X_\Ty,X_\Tm)$
|
||||
|
||||
with constructors
|
||||
|
||||
$\operatorname{empty} \in X_\Con$
|
||||
|
||||
\qquad\vdots
|
||||
\end{column}
|
||||
\end{columns}
|
||||
\end{tcolorbox}
|
||||
|
||||
\end{frame}
|
||||
\begin{frame}{Subject of my internship}{2-sortification of GATs}
|
||||
\begin{center}
|
||||
{\large Transform a GAT into a GAT with only two sorts}
|
||||
\end{center}
|
||||
\vspace{3ex}
|
||||
|
||||
\ding{229} Process observed since at least 2021
|
||||
|
||||
\ding{229} Never formally studied
|
||||
|
||||
\ding{229} Studying all GATs by only studying GATs with two sorts ?
|
||||
\end{frame}
|
||||
\begin{frame}{2-sortification of the Set Function GAT}
|
||||
\begin{tcolorbox}
|
||||
\renewcommand\arraystretch{1.5}
|
||||
\begin{tabular}{llrcl}
|
||||
\pause
|
||||
$\mathcal{O} : \Set$ & \color{RoyalBlue} \text{sorts} & $o:\mathcal{O}$ &$\leftrightarrow$&$o$ is a sort \\
|
||||
$\El : \mathcal{O} \to \Set$ & \color{RoyalBlue}\text{objects of that sort} & \qquad $x : \El\;o$ & $\leftrightarrow$ & $x : o$ \pause\\\hline
|
||||
$A : \mathcal{O}$ &&\multicolumn{1}{l}{\color{teal}$A : \Set$}&&\\
|
||||
$B : \mathcal{O}$ &&\multicolumn{1}{l}{\color{teal}$B : \Set$}&&\\
|
||||
$\operatorname{exec} : \El\;A \to \El\;B$&&\color{teal}$\operatorname{exec} : A \to B$&&
|
||||
\end{tabular}
|
||||
\end{tcolorbox}
|
||||
\end{frame}
|
||||
\begin{frame}{2-sortification of Type Theory GAT}
|
||||
\begin{tcolorbox}
|
||||
\renewcommand\arraystretch{1.4}
|
||||
\begin{tabular}{l}
|
||||
$\mathcal{O} : \Set$ \\
|
||||
$\El : \mathcal{O} \to \Set$ \pause\\\hline
|
||||
$\Con : \mathcal{O}$ \\
|
||||
$\Ty : \El\;\Con \to \mathcal{O}$ \\
|
||||
$\Tm : (\Gamma : \El\;\Con) \to \El\;(\Ty\;\Gamma) \to \mathcal{O}$ \\
|
||||
$\operatorname{empty}: \El\;\Con$ \\
|
||||
$\operatorname{ext}: (\Gamma:\El\;\Con)\to(A:\El\;(\Ty\;\Gamma))\to\El\;\Con$ \\
|
||||
$\operatorname{implies} : (\Gamma:\El\;\Con) \to \El\;(\Ty\;\Gamma) \to \El\;(\Ty\;\Gamma) \to \El\;(\Ty\;\Gamma)$
|
||||
\end{tabular}
|
||||
\end{tcolorbox}
|
||||
\end{frame}
|
||||
\begin{frame}{Goal of the internship}{}
|
||||
|
||||
{\large Is this transformation correct ?}
|
||||
|
||||
\ding{229} Can one study all GATs by studying only GATs with two sorts
|
||||
|
||||
\vspace{1ex}
|
||||
{\large How to state this fact ?}
|
||||
|
||||
\ding{229} Semantical proof
|
||||
|
||||
\begin{center}
|
||||
\includesvg[scale=.4]{graphs/diagrammeFG.svg}
|
||||
\end{center}
|
||||
|
||||
\ding{229} This adjunction proves that one can make the initial model of any GAT from the initial model of the transformed GAT
|
||||
|
||||
\end{frame}
|
||||
|
||||
\section{One Example}
|
||||
|
||||
\begin{frame}{Categories of models}
|
||||
\begin{center}
|
||||
\renewcommand{\ULthickness}{2.4pt}
|
||||
\only<1>{GAT = sorts + constructors + equalities}
|
||||
\only<2>{GAT = \textbf{sorts} + \sout{constructors} + \sout{equalities}}
|
||||
\end{center}
|
||||
\pause[2]
|
||||
\begin{center}
|
||||
\begin{tabular}{|l|l|}
|
||||
\hline
|
||||
$\begin{array}{l}
|
||||
\Con : \Set \\
|
||||
\Ty : \Con \to \Set \\
|
||||
\Tm : (\Gamma : \Con) \to \Ty\;\Gamma \to \Set
|
||||
\end{array}$
|
||||
&
|
||||
$\begin{array}{l}
|
||||
\mathcal{O} : \Set\\
|
||||
\El : \mathcal{O} \to \Set \\\hdashline
|
||||
\underline{\Con} : \mathcal{O} \\
|
||||
\underline{\Ty} : \El\;\underline{\Con} \to \mathcal{O} \\
|
||||
\underline{\Tm} : (\Gamma : \El\;\underline{\Con}) \to \El(\underline{\Ty}\;\Gamma) \to \mathcal{O}
|
||||
\end{array}$\\\hline\rule{0pt}{1.0\normalbaselineskip}
|
||||
\only<2>{$\CC \hookrightarrow (\Con,\Ty,\Tm)$}
|
||||
\only<3->{$\CC_0 \hookrightarrow ()$}
|
||||
&
|
||||
\only<2>{$\BB \hookrightarrow (\mathcal{O},\El,\underline{\Con},\underline{\Ty},\underline{\Tm})$}
|
||||
\only<3->{$\BB_0 \hookrightarrow (\mathcal{O},\El)$}\\
|
||||
\uncover<3->{$\CC_1 \hookrightarrow (\Con)$} &
|
||||
\uncover<3->{$\BB_1 \hookrightarrow (\mathcal{O},\El,\underline{\Con})$}\\
|
||||
\uncover<3->{$\CC_2 \hookrightarrow (\Con,\Ty)$} &
|
||||
\uncover<3->{$\BB_2 \hookrightarrow (\mathcal{O},\El,\underline{\Con},\underline{\Ty})$}\\
|
||||
\uncover<3->{$\CC_3 \hookrightarrow (\Con,\Ty,\Tm)$} &
|
||||
\uncover<3->{$\BB_3 \hookrightarrow (\mathcal{O},\El,\underline{\Con},\underline{\Ty},\underline{\Tm})$}\\\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{Categories of Models}{of the original GAT}
|
||||
\begin{tabular}{lcp{0.5\textwidth}}
|
||||
$\boxed{()}$ & $\CC_0 :=$ &
|
||||
$\one$ \\
|
||||
$\boxed{\Con : \Set}$ & $\CC_1 := $&
|
||||
\only<1>{$\left(X_\Con\right)$}
|
||||
\only<2>{$\left(X_\Con : \Set\right)$}
|
||||
\only<3>{$(\bullet : \CC_0) \times \left(\Set\right)$}
|
||||
\only<4>{$\left(X : \CC_0\right) \times \Set^{H_1(X)}$} \\
|
||||
$\boxed{\Ty : (\Gamma : \Con) \to \Set}$ & $\CC_2 := $&
|
||||
\only<1>{$\left(X_\Con, \left(X_\Ty(\Gamma)\right)_{\Gamma \in X_\Con}\right)$}
|
||||
\only<2>{$\left(X_\Con : \Set, X_\Ty : \Set^{X_\Con}\right)$}
|
||||
\only<3>{$\left(X_\Con : \CC_1\right) \times \left(\Set^{X_\Con}\right)$}
|
||||
\only<4>{$\left(X : \CC_1\right) \times \Set^{H_2(X)}$} \\
|
||||
$\boxed{\Tm : (\Delta : \Con) \to (A : \Ty\;\Delta) \to \Set}$ & $\CC_3 :=$&
|
||||
\only<1>{$\left(X_\Con, \left(X_\Ty(\Gamma)\right)_{\Gamma \in X_\Con},\right.$}
|
||||
\only<2>{$\left(X_\Con : \Set, X_\Ty : \Set^{X_\Con},\right.$}
|
||||
\only<3>{$\left((X_\Con,X_\Ty) : \CC_2\right) \times$}
|
||||
\only<4>{$\left(X : \CC_2\right) \times \Set^{H_3(X)}$} \\ &&
|
||||
\only<1>{$\left.\left(\left(X_\Tm(\Delta,A)\right)_{A \in \Ty(\Delta)}\right)_{\Delta \in X_\Con} \right)$}
|
||||
\only<2>{\quad$\left.X_\Tm : \Set^{\prod_{\Delta:X_\Con}X_\Ty(\Delta)}\right)$}
|
||||
\only<3>{\quad$\left(\Set^{\prod_{\Delta:X_\Con}X_\Ty(\Delta)}\right)$}
|
||||
\end{tabular}
|
||||
|
||||
\pause[4]
|
||||
\[\begin{array}{rcl}
|
||||
H_1(\bullet) &=& 1_\Set \\
|
||||
H_2(X_\Con) &=& X_\Con \\
|
||||
H_3(X_\Con,X_\Ty) &=& \displaystyle\prod_{\Delta:X_\Con}X_\Ty(\Delta)
|
||||
\end{array}\]
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{Category of Models}{of the transformed GAT}
|
||||
\[
|
||||
\begin{array}{|c|}
|
||||
\hline
|
||||
\mathcal{O} : \Set \\
|
||||
\El : \mathcal{O} \to \Set \\
|
||||
\hline
|
||||
\end{array}
|
||||
\]
|
||||
|
||||
\[
|
||||
\BB_0 := \left(X_\UU : \Set, X_\El : \Set^{X_\UU}\right)
|
||||
\]
|
||||
\pause
|
||||
\begin{center}
|
||||
\begin{tabular}{ll}
|
||||
$X_\UU$:& Sorts \\
|
||||
$X_\El(o)$:& Objects of sort $o$\\
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{Category of Models}{of the transformed GAT}
|
||||
|
||||
\begin{center}
|
||||
\color{blue}{
|
||||
\begin{tabular}{ll}
|
||||
$X_\UU$:& Sorts \\
|
||||
$X_\El(o)$:& Objects of sort $o$\\
|
||||
\end{tabular}}
|
||||
\end{center}
|
||||
\begin{center}
|
||||
\renewcommand\arraystretch{1.4}
|
||||
\begin{tabular}{lrp{0.4\textwidth}}
|
||||
$\boxed{\mathcal{O} : \Set\quad\El : \mathcal{O} \to \Set}$ & $\BB_0 =$ &
|
||||
$(X_\UU,X_\El)$ \\
|
||||
$\boxed{\Con : \mathcal{O}}$ & $\XCon_X :$ &
|
||||
\only<1>{$X_\UU$}
|
||||
\only<2>{$H_1F_0(X) \to X_\UU$} \\
|
||||
$\boxed{\Ty : (\Gamma : \underline{\Con}) \to \mathcal{O}}$ & $\XTy_X :$&
|
||||
\only<1>{$\left(\Gamma \in X_\El(\XCon_X)\right) \to X_\UU $}
|
||||
\only<2>{$H_2F_1(X,\XCon_X) \to X_\UU$} \\
|
||||
$\boxed{\Tm : (\Delta : \underline{\Con}) \to (A : \underline{\Ty\;\Delta}) \to \mathcal{O}}$ &
|
||||
$\XTm_X : $ &
|
||||
\only<1>{\rule[6ex]{0pt}{0pt}\renewcommand\arraystretch{0.9}$\begin{array}{c}\left(\Delta \in X_\El(\XCon_X)\right) \to\\ \left(A \in X_\El(\XTy_X(\Delta))\right) \to\\ X_\UU\end{array}$}
|
||||
\only<2>{$H_3F_2(X,\XCon_X,\XTy_X) \to X_\UU$}
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
\pause
|
||||
\begin{center}
|
||||
\includesvg[scale=.4]{graphs/diagrammeFGmini.svg}
|
||||
\end{center}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{Constructing $F$ and $G$}
|
||||
|
||||
\begin{center}
|
||||
\includesvg[scale=.4]{graphs/diagrammeFGmini.svg}
|
||||
\end{center}
|
||||
\[
|
||||
\begin{array}{l|l}
|
||||
Y : \mathbf{\CC_3} & X : \mathbf{\BB_3} \\\hline
|
||||
& X_\UU : \Set \quad X_\El : \Set^{X_\UU}\\
|
||||
Y_\Con : \Set & \XCon_X : X_\UU \\
|
||||
\left(Y_\Ty(\Gamma)\right)_{\Gamma \in Y_\Con} &
|
||||
\XTy_X : H_2 F_{1} (X,\XCon_X) \to X_\UU \\
|
||||
\left(\left(Y_\Tm(\Delta,A)\right)_{A \in Y_\Ty(\Delta)}\right)_{\Delta \in Y_\Con} &
|
||||
\XTm_X : H_3 F_{2} (X,\XCon_X,\XTy_X) \to X_\UU \\
|
||||
\end{array}
|
||||
\]
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{Constructing $G_3$}
|
||||
\vspace{-2ex}
|
||||
\begin{columns}
|
||||
\begin{column}{.5\textwidth}
|
||||
\begin{center}
|
||||
\includesvg[scale=.4]{graphs/diagrammeGmini.svg}
|
||||
\end{center}
|
||||
\end{column}
|
||||
\begin{column}{.5\textwidth}
|
||||
\[
|
||||
Y = \left(Y_\Con,Y_\Ty,Y_\Tm\right)
|
||||
\]
|
||||
\end{column}
|
||||
\end{columns}
|
||||
\vspace{.5ex}
|
||||
\only<1>{
|
||||
\[\begin{array}{ccl}
|
||||
X_\UU & = & \text{«sorts»}\\
|
||||
X_\El(o) & = & \text{«objects of sort $o$»}
|
||||
\end{array}\]
|
||||
}
|
||||
\pause
|
||||
\renewcommand{\arraystretch}{0.5}
|
||||
\[\begin{array}{ccccccccc}
|
||||
X_\UU & = &
|
||||
\{\star\} & \uplus &
|
||||
Y_\Con & \uplus&
|
||||
\displaystyle\coprod_{\Delta \in Y_\Con}Y_\Ty(\Delta) &
|
||||
\xrightarrow{X_\El}&\Set\\
|
||||
\pause
|
||||
&& \tikzmark{1}{\star} &&&&&&\tikzmark{2}{Y_\Con}\\
|
||||
X_\El & &
|
||||
&&\tikzmark{3}{\Gamma}&&&&\tikzmark{4}{Y_\Ty(\Gamma)}\\
|
||||
&&&&&& \tikzmark{5}{(\Delta,A)}&& \tikzmark{6}{Y_\Tm(\Delta,A)}\\
|
||||
|
||||
\end{array}\]
|
||||
\begin{tikzpicture}[overlay,remember picture]
|
||||
\draw[|->] (1) to (2);
|
||||
\draw[|->] (3) to (4);
|
||||
\draw[|->] (5) to (6);
|
||||
\end{tikzpicture}
|
||||
\pause
|
||||
\begin{columns}
|
||||
\begin{column}{.5\textwidth}
|
||||
\[\begin{array}{lcl}
|
||||
\XCon_X &=& \star \in \{\star\}\\
|
||||
\XTy_X(\Gamma) &=& \Gamma \in Y_\Con \\
|
||||
\XTm_X(\Delta,A) &=& (\Delta,A) \in \displaystyle\coprod_{\Delta \in Y_\Con}Y_\Ty(\Delta)
|
||||
\end{array}\]
|
||||
\end{column}
|
||||
\pause
|
||||
\begin{column}{.5\textwidth}
|
||||
\begin{tcolorbox}
|
||||
All sorts of $X_\UU$ are in the image of some constructor $(\XCon_X,\XTy_X,\XTm_X)$
|
||||
\end{tcolorbox}
|
||||
\end{column}
|
||||
\end{columns}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{Constructing $F_3$}
|
||||
\begin{center}
|
||||
\includesvg[scale=.4]{graphs/diagrammeFmini.svg}
|
||||
\end{center}
|
||||
\vspace{-1ex}
|
||||
\renewcommand\arraystretch{1.5}
|
||||
\[
|
||||
\begin{array}{l|l}
|
||||
X : \mathbf{\BB_3} & Y = F_3(X): \mathbf{\CC_3}\\\hline
|
||||
X_\UU : \Set\quad X_\El : \Set^{X_\UU} &\\
|
||||
\XCon_X : X_\UU & Y_\Con =
|
||||
\uncover<2->{\ensuremath{X_\El(\XCon_X)}}\\
|
||||
\XTy_X : H_2 F_{1} (X,\XCon_X) \to X_\UU &
|
||||
Y_\Ty(\Gamma) =
|
||||
\uncover<3->{\ensuremath{X_\El(\XTy_X(\Gamma))}}\\
|
||||
\XTm_X : H_3 F_{2} (X,\XCon_X,\XTy_X) \to X_\UU &
|
||||
Y_\Tm(\Delta,A) =
|
||||
\uncover<4->{\ensuremath{X_\El(\XTm_X(\Delta,A))}} \\
|
||||
\end{array}
|
||||
\]
|
||||
\pause[5]
|
||||
\vspace{-.5ex}
|
||||
\begin{tcolorbox}
|
||||
The preimage by $X_\El$ of each object of $Y$ is a sort of $X_\UU$ constructed by some constructor
|
||||
\end{tcolorbox}
|
||||
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{Adjunction $F \vdash G$}
|
||||
\[
|
||||
\Hom_{\BB_3}\left(G_3Y,X\right) \simeq \Hom_{\CC_3}\left(Y,F_3X\right)
|
||||
\]
|
||||
\pause
|
||||
|
||||
\begin{remark}
|
||||
Morphisms of $\BB_3$ and $\CC_3$ both respect constructors
|
||||
\pause
|
||||
|
||||
All sorts of $G_3Y_\UU$ are in the image of some constructor $(\XCon_{G_3Y},\XTy_{G_3Y},\XTm_{G_3Y})$
|
||||
|
||||
\ding{220}$\Hom_{\BB_3}\left(G_3Y,X\right)$ morphisms are only defined on constructors
|
||||
|
||||
\pause
|
||||
|
||||
The preimage by $X_\El$ of each object of $F_3X$ is a sort of $X_\UU$ constructed by some constructor
|
||||
|
||||
\ding{220}$\Hom_{\CC_3}\left(Y,F_3X\right)$ morphisms only send to constructible terms
|
||||
\end{remark}
|
||||
\end{frame}
|
||||
|
||||
\section{Conclusion}
|
||||
|
||||
\begin{frame}{Conclusion}
|
||||
\begin{center}
|
||||
\hspace{9.2ex}$\CC$ \hspace{3.7cm} $\BB$
|
||||
\vspace{.5cm}
|
||||
|
||||
\includesvg[scale=.4]{graphs/diagrammeFG.svg}
|
||||
\end{center}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{Future work}
|
||||
\begin{itemize}
|
||||
\item Complete GAT (term constructors + equalities)
|
||||
\item Proof Assistant Formalization
|
||||
\item $S_i$ non-direct
|
||||
\end{itemize}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}
|
||||
\begin{center}
|
||||
\Large Thank you for your attention
|
||||
\end{center}
|
||||
\end{frame}
|
||||
|
||||
\appendix
|
||||
|
||||
\begin{frame}
|
||||
\[\begin{array}{lcl}
|
||||
F_3G_3(Y)_\Con &=& G_3(Y)_p^{-1}(\{\Cstr^{G_3(Y)}_\Con\})\\
|
||||
&=& G_3(Y)_p^{-1}(\{\inj_1 \star\}) \\
|
||||
&=& Y_\Con
|
||||
\end{array}\]
|
||||
and
|
||||
\[\begin{array}{lcl}
|
||||
F_3G_3(Y)_\Ty(\Gamma) &=& G_3(Y)_p^{-1}(\{\Cstr^{G_3(Y)}_\Ty(\Gamma)\})\\
|
||||
&=& G_3(Y)_p^{-1}(\{\inj_2 \Gamma\}) \\
|
||||
&=& \operatorname{proj}_1^{-1}(\Gamma) \\
|
||||
&=& \left\{(\Gamma',A) \in \coprod_{\Gamma' \in Y_\Con}Y_\Ty(\Gamma') \middle| \Gamma' = \Gamma\right\}\\
|
||||
&\simeq& Y_\Ty(\Gamma)
|
||||
\end{array}\]
|
||||
and finally, with the same method, we get that
|
||||
\[
|
||||
F_3G_3(Y)_\Tm(\Delta,A) \simeq Y_\Tm(\Delta,A)
|
||||
\]
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{Structure of the global proof}
|
||||
\begin{itemize}
|
||||
\item Categories $\CC_i$ \quad $\BB_i$
|
||||
\item Functors $F_i : \BB_i \to \CC_i : G_i$
|
||||
\item Adjunction $F_i \vdash G_i$
|
||||
\item Forgetful functor $R_{i-1}^i : \BB_i \to \BB_{i-1}$
|
||||
\item Operator $\tl^i : \BB_i \times \BB_0 \to \BB_i$ \quad
|
||||
$\inj_1^i : X \to X \tl^i Y$ \quad
|
||||
$\inj_2^i : Y \to R_0^i(X \tl^i Y)$
|
||||
\item Coreflection $F_iG_i \cong \Id_{\CC_i}$
|
||||
\item Isomorphism $F_i\inj_1^i$
|
||||
\item Isomorphism $(R_{i-1}^i X) \tl^{i-1} Y \to R_{i-1}^i (X \tl^i Y)$
|
||||
\end{itemize}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{Fibration of $\CC_i$}
|
||||
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{$S_i$ from syntax}
|
||||
|
||||
\end{frame}
|
||||
|
||||
\end{document}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
+1131
-390
File diff suppressed because it is too large
Load Diff
@@ -0,0 +1,71 @@
|
||||
# M2 Presentation
|
||||
|
||||
### Introduction slide
|
||||
Notes on my internship
|
||||
|
||||
### Plan of the presentation
|
||||
First present what is the subject of my study, and what i tried to achieve.
|
||||
|
||||
The whole proof is too formal and complex: Informally prove one specific example.
|
||||
|
||||
Then, i will present the «structure» of the global proof, and show some things discovered along the way
|
||||
|
||||
## GAT and 2-sortification
|
||||
### What is a GAT
|
||||
|
||||
### A GAT for a function in set
|
||||
A GAT can be composed of three kinds of lines:
|
||||
- Sort declarations
|
||||
- Objects costructors
|
||||
A model is the data of a function from a set to another set i.e. a triple the data of two sets, and a «mapping» between them
|
||||
We can spice up this GAT: Gat of isomorphic functions.
|
||||
- Equalities
|
||||
|
||||
### A GAT for a category
|
||||
Another example of GATs:
|
||||
A category is the data of ...... with ...... such that ......
|
||||
|
||||
### A GAT for Type Theory
|
||||
Last example that of type theory
|
||||
-> a lot to add, contructors, equalities ..., just for the example.
|
||||
|
||||
### 2-sortification
|
||||
Known process
|
||||
Used by Sestini
|
||||
Can simplify a study of GATs to only GATs of two sorts
|
||||
|
||||
### 2-sortification of the Set function GAT
|
||||
Sort specification is always the same. O will describe «sorts» and El will describe «objects of that sort»
|
||||
|
||||
Then, we replace Set occuneces by O and we prefix everything else by El
|
||||
|
||||
### 2-sortification of the Type Theoyr GAT
|
||||
|
||||
### Goal of the internship
|
||||
|
||||
## One Example
|
||||
### Category of Models & Generalization
|
||||
|
||||
### Category of models of the Transformed GAT
|
||||
|
||||
### Adding transformed sort declarations
|
||||
|
||||
### Constructing the functors
|
||||
|
||||
### Contsructing G3
|
||||
|
||||
### Constructing F3
|
||||
|
||||
### Adjunction FG
|
||||
|
||||
## The complete proof & Discoveries
|
||||
### Structure of the proof
|
||||
|
||||
### Fibration of C
|
||||
|
||||
### S from syntax
|
||||
|
||||
## Conclusion
|
||||
### Conclusion
|
||||
|
||||
### Future work
|
||||
@@ -0,0 +1 @@
|
||||
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After Width: | Height: | Size: 18 KiB |
+58
-31
@@ -1,6 +1,9 @@
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% Loading packages
|
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\usepackage{autofe}
|
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\usepackage{hyperref}
|
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\usepackage{ae}
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\hypersetup{
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colorlinks=true,
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@@ -12,7 +15,6 @@
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\usepackage{amssymb}
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||||
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@@ -23,7 +25,6 @@
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\usepackage{tcolorbox}
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||||
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\usepackage{proof}
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@@ -39,11 +40,12 @@
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@@ -59,29 +61,16 @@
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% Création des environnement globaux
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\providecommand\ruleautorefname{règle}
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}
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% Commandes logiques globales
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\newcommand{\ifnullthenelse}[3]{
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@@ -99,24 +88,32 @@
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% Macros caractères spécifiques au document
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@@ -126,13 +123,13 @@
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\node (XEl) at (1,1) {\ensuremath{#1}};
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% Création des environnements spécifiques au document
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@@ -0,0 +1,199 @@
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% Loading packages
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% Macros caractères globales
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|
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|
||||
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|
||||
|
||||
% Création des environnement globaux
|
||||
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|
||||
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|
||||
|
||||
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|
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||||
% Commandes logiques globales
|
||||
\newcommand{\ifnullthenelse}[3]{
|
||||
\ifnum\value{#1}=0
|
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#2
|
||||
\else
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||||
#3
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||||
\fi
|
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}
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%%% Commande \newtag permettant de changer le label d'une equation
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||||
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% Macros caractères spécifiques au document
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||||
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||||
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|
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||||
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||||
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|
||||
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|
||||
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|
||||
\newcommand\XCon{{\ensuremath{\operatorname{\mathbf{Con}}}}}
|
||||
\newcommand\XTy{{\ensuremath{\operatorname{\mathbf{Ty}}}}}
|
||||
\newcommand\XTm{{\ensuremath{\operatorname{\mathbf{Tm}}}}}
|
||||
\newcommand\Cstr{{\ensuremath{\operatorname{\mathcal{C}str}}}}
|
||||
\newcommand\Rtsc{{\ensuremath{\operatorname{\mathcal{R}tsc}}}}
|
||||
\newcommand\Cat{{\ensuremath{\operatorname{\mathcal{C}at}}}}
|
||||
\newcommand\Set{{\ensuremath{\operatorname{\mathcal{S}et}}}}
|
||||
\newcommand\FamSet{{\ensuremath{\operatorname{\mathcal{F}am\mathcal{S}et}}}}
|
||||
\newcommand\Obj{{\ensuremath{\operatorname{\mathcal{O}bj}}}}
|
||||
\newcommand\Hom{{\ensuremath{\operatorname{\mathcal{H}om}}}}
|
||||
\newcommand\this{{\ensuremath{\operatorname{\texttt{this}}}}}
|
||||
\newcommand\one{{\ensuremath{\mathbf{1}}}}
|
||||
\newcommand\dash{{\;\textrm{---}\;}}
|
||||
\renewcommand\enquote[1]{``#1''}
|
||||
\newcommand\tl{{\triangleleft}}
|
||||
|
||||
\DeclareMathOperator{\inj}{inj}
|
||||
\DeclareMathOperator{\id}{id}
|
||||
\DeclareMathOperator{\Id}{\mathcal{I}d}
|
||||
|
||||
\newcommand\TSet{{\ensuremath{\left[\TT,\Set\right]}}}
|
||||
\newcommand\TSetObject[3]{{\ensuremath{
|
||||
\left[
|
||||
\begin{tikzpicture}[baseline=(base)]
|
||||
\node (U) at (0,0) {\UU};
|
||||
\node (El) at (0,1) {\El};
|
||||
\draw[->] (El) -- node[anchor=east] {\ensuremath{\scriptstyle p}} (U);
|
||||
|
||||
\node (XU) at (1.5,0) {\ensuremath{#3}};
|
||||
\node (XEl) at (1.5,1) {\ensuremath{#1}};
|
||||
\draw[->] (XEl) -- node[anchor=west] {\ensuremath{\scriptstyle #2}} (XU);
|
||||
|
||||
\draw[|->] (.3,0) -- (1,0);
|
||||
\draw[|->] (.3,0.5) -- (1,0.5);
|
||||
\draw[|->] (.3,1) -- (1,1);
|
||||
|
||||
\coordinate (base) at (.5,.4);
|
||||
\end{tikzpicture}
|
||||
\right]
|
||||
}}}
|
||||
|
||||
\newcommand\simpleArrow[3]{
|
||||
\begin{tikzpicture}
|
||||
\node (0) at (0,0) {$#1$};
|
||||
\node (1) at (6,0) {$#3$};
|
||||
\path[->] (0) edge["${\scriptstyle #2}$", pos=0.5, fore, black,=>, ] (1);
|
||||
\end{tikzpicture}
|
||||
}
|
||||
\newcommand\simpleUpDownArrow[3]{
|
||||
\begin{tikzpicture}
|
||||
\node (0) at (0,1) {$#1$};
|
||||
\node (1) at (0,0) {$#3$};
|
||||
\path[->] (0) edge["${\scriptstyle #2}$", pos=0.5, fore, black,=>, ] (1);
|
||||
\end{tikzpicture}
|
||||
}
|
||||
\newcommand\labeledupdownarrow[1]{
|
||||
\begin{tikzpicture}
|
||||
\path[->] (0,1) edge["${\scriptstyle #1}$", pos=0.5, fore, black,=>, ] (0,0);
|
||||
\end{tikzpicture}
|
||||
}
|
||||
|
||||
\newcommand\diagram[1]{\begin{tcolorbox}\begin{center}\vspace{1.5cm}\Huge \texttt{\textbf{#1}}\vspace{1.5cm}\end{center}\end{tcolorbox}}
|
||||
|
||||
\newcommand\todo[1]{\begin{tcolorbox}[colback=red!20!white,colframe=red!85!black,boxrule=4pt] #1 \end{tcolorbox}}
|
||||
\newcommand\inlinetodo[1]{\colorbox{orange}{#1}}
|
||||
|
||||
\newcommand{\backupbegin}{
|
||||
\newcounter{finalframe}
|
||||
\setcounter{finalframe}{\value{framenumber}}
|
||||
}
|
||||
\newcommand{\backupend}{
|
||||
\setcounter{framenumber}{\value{finalframe}}
|
||||
}
|
||||
|
||||
\addbibresource{Bilibibio.bib}
|
||||
|
||||
\usetheme{Madrid}
|
||||
|
||||
\makeatletter
|
||||
\setbeamertemplate{frametitle}{%
|
||||
\nointerlineskip
|
||||
\begin{beamercolorbox}[sep=1.6ex,wd=\paperwidth,leftskip=.5cm,rightskip=0cm]{frametitle}%
|
||||
\usebeamerfont{frametitle}\usebeamercolor[fg]{frametitle}\beamer@frametitle\\
|
||||
\usebeamerfont{framesubtitle}\usebeamercolor[fg]{framesubtitle}\insertframesubtitle
|
||||
\end{beamercolorbox}%
|
||||
}
|
||||
\makeatother
|
||||
|
||||
\hypersetup{pdfpagemode=FullScreen}
|
||||
% Transition en fade-in par défaut
|
||||
%\addtobeamertemplate{background canvas}{\transfade[duration=0.4]}{}
|
||||
|
||||
\addtobeamertemplate{frametitle}{
|
||||
\let\insertframetitle\insertsubsectionhead}{}
|
||||
\makeatletter
|
||||
\CheckCommand*\beamer@checkframetitle{\@ifnextchar\bgroup\beamer@inlineframetitle{}}
|
||||
\renewcommand*\beamer@checkframetitle{\global\let\beamer@frametitle\relax\@ifnextchar\bgroup\beamer@inlineframetitle{}}
|
||||
\makeatother
|
||||
|
||||
\setbeamertemplate{itemizeitem}{\scriptsize$\diamond$}
|
||||
\newcommand\sectocframe[1][]{
|
||||
\begin{frame}
|
||||
\pdfpcnote{#1}
|
||||
\tableofcontents[currentsection,hideothersubsections,sections=\value{section}]
|
||||
\end{frame}
|
||||
}
|
||||
|
||||
\newcommand{\tikzmark}[2]{\tikz[remember picture] \node (#1) {$#2$};}
|
||||
\makeatletter
|
||||
\newcommand{\mathleft}{\@fleqntrue\@mathmargin0pt}
|
||||
\newcommand{\mathcenter}{\@fleqnfalse}
|
||||
\makeatother
|
||||
@@ -0,0 +1,6 @@
|
||||
{
|
||||
"watchedFile": "M2Diapo.tex",
|
||||
"baseDir": "graphs",
|
||||
"externalOutput": true,
|
||||
"preambleFile": "yade-preamble.tex"
|
||||
}
|
||||
@@ -2,6 +2,7 @@
|
||||
\newcommand\BB{{\ensuremath{\mathcal{B}}}}
|
||||
\newcommand\TT{{\ensuremath{\mathcal{T}}}}
|
||||
\newcommand\UU{{\ensuremath{\mathcal{U}}}}
|
||||
\newcommand\CC{{\ensuremath{\mathcal{C}}}}
|
||||
\newcommand\El{{\ensuremath{\operatorname{\mathcal{E}l}}}}
|
||||
\newcommand\ii{{\ensuremath{\mathbf{i}}}}
|
||||
\newcommand\Cstr{{\ensuremath{\operatorname{\mathcal{C}str}}}}
|
||||
|
||||
+11
-2
@@ -47,8 +47,8 @@
|
||||
let \p1 = ($(\tikztostart) - (\tikztotarget)$) in
|
||||
(\tikztostart)
|
||||
.. controls
|
||||
($(\tikztostart)!\pv{pos}!(\tikztotarget)!{veclen(\x1,\y1)*\pv{ratio}*1pt}!270:(\tikztotarget)$)
|
||||
and ($(\tikztostart)!1-\pv{pos}!(\tikztotarget)!{veclen(\x1,\y1)*\pv{ratio}*1pt}!270:(\tikztotarget)$)
|
||||
($(\tikztostart)!\pv{pos}!(\tikztotarget)!{veclen(\x1,\y1)*\pv{ratio}*0.65pt}!270:(\tikztotarget)$)
|
||||
and ($(\tikztostart)!1-\pv{pos}!(\tikztotarget)!{veclen(\x1,\y1)*\pv{ratio}*0.65pt}!270:(\tikztotarget)$)
|
||||
.. (\tikztotarget)\tikztonodes}},
|
||||
settings/.code={\tikzset{yade/.cd,#1}
|
||||
\def\pv##1{\pgfkeysvalueof{/tikz/yade/##1}}},
|
||||
@@ -379,6 +379,15 @@ markings,
|
||||
mark=at position #2 with {\node[inner sep=0.1pt,fill=white] {$\scriptstyle #1$} ;}
|
||||
}}}}
|
||||
|
||||
\tikzset{
|
||||
labelonatsloped/.style 2 args={postaction={
|
||||
decorate,
|
||||
decoration={
|
||||
markings,
|
||||
mark=at position #2 with {\node[inner sep=0.1pt,fill=white,transform shape] {#1} ;}
|
||||
}}}}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
Reference in New Issue
Block a user