811 lines
25 KiB
TeX
811 lines
25 KiB
TeX
\documentclass[%
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pdftex,
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t, % Folieninhalt am oberen Rand beginnen.
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10pt, % Schriftgroesse
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]{beamer}
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% Einstellungen laden
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\usetheme{DHBW}
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\usepackage{xstring}
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\usepackage[utf8]{inputenc}
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%\usepackage[latin1]{inputenc}
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\usepackage[T1]{fontenc}
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%\usepackage{amsmath}
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%\usepackage{amsfonts}
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%\usepackage{amssymb}
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\usepackage{alltt}
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\usepackage{nameref}
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\newcommand{\einstellung}[1]{%
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\expandafter\newcommand\csname #1\endcsname{}
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\expandafter\newcommand\csname setze#1\endcsname[1]{\expandafter\renewcommand\csname#1\endcsname{##1}}
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}
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\input{einstellungen_liste} % verfügbare Einstellungen
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%\input{einstellungen} % lese Einstellungen
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\setzetitel{Typeinferenz für Java mit Wildcards}
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\setzeuntertitel{Bad Honnef 2024}
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\setzedatum{30. April 2024}
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\setzestudiengang{Stuidengang Informatik}
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\setzedhbw{DHBW Stuttgart Campus Horb}
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\setzeinternetadresse{www.dhbw-stuttgart.de/horb}
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\setzeautor{Andreas Stadelmeier}%\\Andreas Stadelmeier}
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\setzeschriftart{palatino} % oder goudysans, lmodern, libertine
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%% Farben (Angabe in HTML-Notation mit großen Buchstaben)
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\newcommand{\ladefarben}{%
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\definecolor{LinkColor}{HTML}{00007A}
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\definecolor{ListingBackground}{HTML}{FCF7DE}
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}
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%% Mathematikpakete benutzen (Pakete aktivieren)
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%\usepackage{amsmath}
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%\usepackage{amssymb}
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%% Programmiersprachen Highlighting (Listings)
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\newcommand{\listingsettings}{%
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\lstset{%
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language=Java, % Standardsprache des Quellcodes
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%numbers=left, % Zeilennummern links
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stepnumber=1, % Jede Zeile nummerieren.
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numbersep=5pt, % 5pt Abstand zum Quellcode
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numberstyle=\tiny, % Zeichengrösse 'tiny' für die Nummern.
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breaklines=false, % Zeilen umbrechen wenn notwendig.
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breakautoindent=true, % Nach dem Zeilenumbruch Zeile einrücken.
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postbreak=\space, % Bei Leerzeichen umbrechen.
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tabsize=2, % Tabulatorgrösse 2
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basicstyle=\ttfamily\small, % Nichtproportionale Schrift, klein für den Quellcode
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showspaces=false, % Leerzeichen nicht anzeigen.
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showstringspaces=false, % Leerzeichen auch in Strings ('') nicht anzeigen.
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extendedchars=true, % Alle Zeichen vom Latin1 Zeichensatz anzeigen.
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%captionpos=b, % sets the caption-position to bottom
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%backgroundcolor=\color{ListingBackground}, % Hintergrundfarbe des Quellcodes setzen.
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xleftmargin=0pt, % Rand links
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xrightmargin=0pt, % Rand rechts
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%frame=single, % Rahmen an
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%frameround=ffff,
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rulecolor=\color{darkgray}, % Rahmenfarbe
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fillcolor=\color{ListingBackground}
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showtabs=false,
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%breaklines=true,
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%breakatwhitespace=true,
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%basicstyle=\ttfamily\fontsize{8}{9.6}\selectfont, %\footnotesize
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escapeinside={(*@}{@*)},
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captionpos=t,float,abovecaptionskip=-\medskipamount
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}
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}
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\usepackage{prolog_slide}
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%\input{prolog}
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\usepackage[english, ngerman]{babel}
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{\selectlanguage{ngerman}} % Paket babel benutzt neue deutsche Rechtschreibung
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%{\selectlanguage{english}} % Paket babel benutzt Englisch
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%%%%%%% Package Includes %%%%%%%
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\usepackage[overlay, absolute]{textpos}
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\usepackage{longtable}
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\usepackage{pgffor} % für automatische Kapiteldateieinbindung
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\usepackage{listings}
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\usepackage{calc}
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%%%%%% Configuration %%%%%
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%% Anwenden der Einstellungen
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\usepackage{\schriftart}
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\ladefarben{}
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% Titel, Autor und Datum
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\title{\titel}
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\subtitle{\untertitel}
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\author{\autor}
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\date{\datum}
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% Literaturverweise (sowohl deutsch als auch englisch)
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% \usepackage[
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% %backend=biber, % empfohlen. Falls biber Probleme macht: bibtex
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% %backend=bibtex, % empfohlen. Falls biber Probleme macht: bibtex
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% bibwarn=true,
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% bibencoding=utf8, % wenn .bib in utf8, sonst ascii
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% sortlocale=de_DE, % oder en_US
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% style=alphabetic %Zitierstil. Siehe http://ctan.mirrorcatalogs.com/macros/latex/contrib/biblatex/doc/biblatex.pdf
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% ]{biblatex}
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%%%%%% Additional settings %%%%%%
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% Hurenkinder und Schusterjungen verhindern
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% http://projekte.dante.de/DanteFAQ/Silbentrennung
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\clubpenalty=10000
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\widowpenalty=10000
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\displaywidowpenalty=10000
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% Bildpfad
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\graphicspath{{images/}}
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% Einige häufig verwendete Sprachen
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\lstloadlanguages{PHP,Python,Java,C,C++,bash}
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\listingsettings{}
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%\includeonly{content/01kapitel}
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%\includeonly{content/01kapitel,content/02kapitel}
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%\includeonly{content/02kapitel}
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%\includeonly{content/03kapitel}
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%\includeonly{content/04kapitel}
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%\includeonly{content/02kapitel,content/03kapitel}
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%\includeonly{content/04kapitel,content/05kapitel}
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%\includeonly{content/05kapitel}
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%\includeonly{content/04kapitel,content/05kapitel,content/06kapitel}
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%\includeonly{content/06kapitel}
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%\includeonly{content/07kapitel}
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%\includeonly{content/08kapitel}
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%\includeonly{content/07kapitel, content/08kapitel}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\newcommand\ddfrac[2]{\frac{\displaystyle #1}{\displaystyle #2}}
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\newcommand{\tv}[1]{\mathit{ #1 }}
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\newcommand{\tto}{\texttt{ -> }}
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\newcommand{\funcType}{\ensuremath{\exptype{Function}{\type{T}, \type{R}}}}
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%\newcommand{\fjtypeInsert}{\textbf{FJTypeInsert}}
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\newcommand{\unifyGenerics}{\ensuremath{\gamma}}% {\ensuremath{\overline{\type{G}\triangleleft \type{H}}}}
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\newcommand{\fjtype}{\textbf{FJType}}
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\newcommand{\unify}{\textbf{Unify}}
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\newcommand{\typeMethod}{\textbf{TYPEMethod}}
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\newcommand{\typeExpr}{\textbf{TYPEExpr}}
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\newcommand{\constraint}{\ensuremath{\mathit{c}}}%{\ensuremath{\mathtt{C}}}
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\newcommand{\consSet}{C}%{\ensuremath{\overline{\mathtt{C}}}}
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\newcommand{\orCons}{\textit{oc}}%{\ensuremath{\textbf{C}_{||}}}
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\newcommand{\simpleCons}{\textit{sc}}
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\newcommand{\overridesFunc}{\textit{overrides}}
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\newcommand{\typeAssumptionsSymbol}{\ensuremath{\Theta}}
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\newcommand{\typeAssumptions}{\ensuremath{(\mv{\Pi} ; \overline{\localVarAssumption})}}%{\ensuremath{(\overline{\methodAssumption} ; \overline{\fieldAssumption}; \overline{\localVarAssumption})}}
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\newcommand{\constraints}{\ensuremath{\mathit{\overline{c}}}}
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\newcommand{\itype}[1]{\ensuremath{\mathit{#1}}}
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\newcommand{\il}[1]{\ensuremath{\overline{\itype{#1}}}}
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\newcommand{\gType}[1]{\texttt{#1}}
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\newcommand{\mtypeEnvironment}{\ensuremath{\Pi}}
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\newcommand{\methodAssumption}{\ensuremath{\mathtt{\lambda}}}
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\newcommand{\localVarAssumption}{\ensuremath{\mathtt{\eta}}}
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\newcommand{\expandLB}{\textit{expandLB}}
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\newcommand{\type}[1]{\texttt{#1}}
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\newcommand{\environmentvdash}{\Pi;\Delta;\Gamma \vdash}
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\newcommand\subeq{\mathbin{\texttt{<:}}}
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\newcommand\mv[1]{{\tt #1}}
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\def\exptypett#1#2{\texttt{#1}\textrm{{\tt <#2}}\textrm{{\tt >}}\xspace}
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\def\exp#1#2{#1(\,#2\,)\xspace}
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\begin{document}
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\bibliographystyle{unsrt}
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% Titelfolie
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\IfFileExists{content/00titelfolie}{%
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\input{content/00titelfolie}
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}
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{%
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%file does not exist
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}
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% Inhalt
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\IfFileExists{content/01inhalt}{%
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\input{content/01inhalt}
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}
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{%
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%file does not exist
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}
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% Kapitel
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\foreach \i in {01,02,03,04,05,06,07,08,09,10,...,99} {%
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\edef\FileName{content/\i kapitel}%
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\IfFileExists{\FileName}{%
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\include{\FileName}
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}
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{%
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%file does not exist
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}
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}
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%\small
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%\renewcommand{\pause}{}
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\bibliographystyle{alpha}
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%\centerslidesfalse
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\begin{frame}
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\maketitle
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\end{frame}
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\newcommand{\rulenameAfter}[1]{\begin{array}[b]{l}\rulename{#1}\\[-0.5em] \ \end{array}}
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\begin{frame}[fragile]
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\frametitle{Java Method Call}
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\rulename{T-Call} \\[1em]
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\begin{center}
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$\begin{array}{l}
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\begin{array}{@{}c}
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\textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T} \quad \quad
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\ol{v} : [\ol{T'}/\ol{X}]\ol{T}
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\\
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\hline
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\vspace*{-0.3cm}\\
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\generics{\ol{T'}}\texttt{m}(\ol{v}) : [\ol{T'}/\ol{X}]\type{T}
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\end{array}
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\end{array}$
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\end{center}
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\textbf{Example:}
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$\generics{\type{String}}\texttt{emptyList}() : \exptype{List}{\type{String}}$
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\\[2em]
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\rulename{TI-Call} \\[1em]
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\begin{center}
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$\begin{array}{l}
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\begin{array}{@{}c}
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\textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T} \quad \quad
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\ol{v} : [\ol{T'}/\ol{X}]\ol{T}
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\\
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\hline
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\vspace*{-0.3cm}\\
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\texttt{m}(\ol{v}) : [\ol{T'}/\ol{X}]\type{T}
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\end{array}
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\end{array}$
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\end{center}
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\textbf{Example:}
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$\texttt{emptyList}() : \exptype{List}{\type{String}}$
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\end{frame}
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% \begin{frame}[fragile]
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% \frametitle{Java Method Call}
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% \rulename{M-Class} \\[1em]
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% \begin{center}
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% $\begin{array}{l}
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% \begin{array}{@{}c}
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% \generics{\ol{X}}\ \type{T} \ \texttt{m}(\ol{T\ x}) \set{ \ldots }
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% \\
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% \hline
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% \vspace*{-0.3cm}\\
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% \textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T}
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% \end{array}
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% \end{array}$
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% \end{center}
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% \textbf{Beispiele:}\\
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% \begin{verbatim}
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% X head(List<X> l) { ... }
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% \end{verbatim}
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% \begin{itemize}
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% \item
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% $\textit{mtype}(\texttt{head}) = \generics{\type{X}}\ \exptype{List}{\type{X}} \to \type{X}$
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% \end{itemize}
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% \begin{verbatim}
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% <X> List<X> emptyList() { ... }
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% \end{verbatim}
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% \begin{itemize}
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% \item
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% $\textit{mtype}(\texttt{emptyList}) = \generics{\type{X}}\ [] \to \exptype{List}{\type{X}}$
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% \end{itemize}
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% \end{frame}
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\begin{frame}[fragile]
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\frametitle{Java Method Call}
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\rulename{T-Call} \\[1em]
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\begin{center}
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$\begin{array}{l}
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\begin{array}{@{}c}
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\textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T} \quad \quad
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\Delta \vdash \expr{v}, \ol{v} : [\ol{T'}/\ol{X}]\ol{T}
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\\
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\hline
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\vspace*{-0.3cm}\\
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\Delta \vdash \expr{v}.\generics{\ol{T'}}\texttt{m}(\ol{v}) : [\ol{T'}/\ol{X}]\type{T}
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\end{array}
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\end{array}$
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\end{center}
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\textbf{Beispiel:}\\
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\begin{verbatim}
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<X> X head(List<X> l){ ... }
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head(new List<String>());
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\end{verbatim}
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\pause
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$\begin{array}{l}
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\begin{array}{@{}c}
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\textit{mtype}(\texttt{head}) = \generics{\ol{X}}\ \exptype{List}{\type{X}} \to \type{X} \\
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\texttt{new List<String>}() : [\type{String}/\type{X}]\exptype{List}{\type{X}}
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\\
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\hline
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\vspace*{-0.3cm}\\
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\generics{\type{String}}\texttt{head}(\texttt{new List<String>}()) : [\type{String}/\type{X}]\type{X}
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\end{array}
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\end{array}$\\[1em]
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\pause
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% $\begin{array}{l}
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% \begin{array}{@{}c}
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% \textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T} \quad \quad
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% \texttt{new List<String>}() : \exptype{List}{\type{String}}
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% \\
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% \hline
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% \vspace*{-0.3cm}\\
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% \generics{\type{String}}\texttt{head}(\texttt{new List<String>}()) : \type{String}
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% \end{array}
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% \end{array}$
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\begin{itemize}
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\item \texttt{head(new List<String>()) : String}
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\end{itemize}
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\end{frame}
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% \begin{frame}[fragile]
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% \frametitle{Java Method Call}
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% \begin{verbatim}
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% head(head(emptyList()))
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% \end{verbatim}
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% $
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% \begin{array}[b]{c}
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% \texttt{emptyList()} : \exptype{List}{\exptype{List}{\type{String}}} \\
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% \quad \textit{mtype}(\texttt{get}) = \generics{X}\ \exptype{List}{\type{X}} \to \type{X}
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% \\
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% \hline
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% \texttt{head(emptyList())} : \exptype{List}{\type{String}}
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% \end{array} \rulenameAfter{T-Call}
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% $
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% \end{frame}
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% \begin{frame}[fragile]
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% \frametitle{Java Method Call}
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% $
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% \begin{array}[b]{c}
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% \texttt{emptyList()} : \exptype{List}{\exptype{List}{\type{String}}} \\
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% \quad \textit{mtype}(\texttt{get}) = \generics{X}\ \exptype{List}{\type{X}} \to \type{X}
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% \\
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% \hline
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% \generics{\exptype{List}{\exptype{List}{\type{String}}}}\texttt{head(emptyList())} : \exptype{List}{\type{String}}
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% \end{array} \rulenameAfter{T-Call}
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% $
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% $
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% \begin{array}[b]{c}
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% \texttt{emptyList()} : \exptype{List}{\exptype{List}{\type{String}}} \\
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% \quad \textit{mtype}(\texttt{get}) = \generics{X}\ \exptype{List}{\type{X}} \to \type{X}
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% \\
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% \hline
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% \generics{\exptype{List}{\type{String}}}\texttt{head(emptyList())} : \exptype{List}{\type{String}}
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% \end{array} \rulenameAfter{T-Call}
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% $
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% \end{frame}
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% \begin{frame}[fragile]
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% \frametitle{Java Method Call}
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|
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% $
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% \begin{array}[b]{c}
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% \texttt{emptyList()} : \exptype{List}{\exptype{List}{\type{String}}} \\
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% \quad \textit{mtype}(\texttt{get}) = \generics{X}\ \exptype{List}{\type{X}} \to \type{X}
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% \\
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|
% \hline
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|
% \generics{\exptype{List}{\type{String}}}\texttt{head(emptyList())} : \exptype{List}{\type{String}}
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|
% \end{array} \rulenameAfter{T-Call}
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|
% $
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% \end{frame}
|
|
|
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% \begin{frame}[fragile]
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|
% \frametitle{Java Method Call}
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|
% $
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% \begin{array}[b]{c}
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% \begin{array}[b]{c}
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% \texttt{emptyList()} : \exptype{List}{\exptype{List}{\type{String}}}
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% \\
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|
% \hline
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|
% \texttt{emptyList().get()} : \exptype{List}{\type{String}}
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% \end{array} \rulenameAfter{T-Call}
|
|
% \\
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|
% \hline
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% \texttt{emptyList().get().get()} : \type{String}
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% \end{array} \rulenameAfter{T-Call}
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% $
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% \end{frame}
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|
|
|
% \begin{frame}[fragile]
|
|
% \frametitle{Method Call with Wildcards}
|
|
% Eigentlich kann man mit Wildcard Typen keine Methodenaufrufe durchführen.
|
|
% Der eigentliche Typ ist nicht bekannt.
|
|
% ? sind auch nicht reflexiv
|
|
% Es geht dennoch, weil sich hinter einem Wildcard Typ ein konkreter Typ verbirgt.
|
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% Dieser muss ausgepackt werden.
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% Die entstehende freie Typvariable kann nun als Typ benutzt werden.
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% \end{frame}
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\begin{frame}[fragile]
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\frametitle{Typeinference ohne Wildcards}
|
|
\rulename{TI-Call} \\[1em]
|
|
\begin{center}
|
|
$\begin{array}{l}
|
|
\begin{array}{@{}c}
|
|
\textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T} \quad \quad
|
|
\Delta \vdash \expr{v}, \ol{v} : [\ol{T'}/\ol{X}]\ol{T}
|
|
\\
|
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\hline
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\vspace*{-0.3cm}\\
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\Delta \vdash \expr{v}.\texttt{m}(\ol{v}) : [\ol{T'}/\ol{X}]\type{T}
|
|
\end{array}
|
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\end{array}$
|
|
\end{center}
|
|
\begin{verbatim}
|
|
untypedMethod(l){
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shuffle(l);
|
|
}
|
|
\end{verbatim}
|
|
\begin{itemize}
|
|
\item $\lessdot $
|
|
\end{itemize}
|
|
\end{frame}
|
|
|
|
\begin{frame}[fragile]{Capture Conversion}
|
|
\rulename{T-Call-Wildcards} \\[1em]
|
|
\begin{center}
|
|
$\begin{array}{l}
|
|
\begin{array}{@{}c}
|
|
\textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T} \quad \quad
|
|
\Delta \vdash \ol{v} : \ol{\wctype{\Delta}{C}{\ol{T}}} \\
|
|
\overline{\exptype{C}{\ol{T}}} <: [\ol{T'}/\ol{X}]\ol{T} \quad \quad [\ol{T'}/\ol{X}]\type{T} <: \type{R}
|
|
\\
|
|
\hline
|
|
\vspace*{-0.3cm}\\
|
|
\Delta \vdash \texttt{m}(\ol{v}) : \type{R}
|
|
\end{array}
|
|
\end{array}$
|
|
\end{center}
|
|
%Wildcard Capture
|
|
\begin{verbatim}
|
|
<X> List<X> shuffle(List<X> l){ ... }
|
|
|
|
List<?> l;
|
|
|
|
shuffle(l) : List<?>
|
|
\end{verbatim}
|
|
%why do we need capture constraints?
|
|
% List<List<?>> not a subtype of List<List<X>> for any X
|
|
|
|
\begin{center}
|
|
$\begin{array}{l}
|
|
\begin{array}{@{}c}
|
|
\textit{mtype}(\texttt{shuffle}) = \generics{\type{X}}\ \exptype{List}{\type{X}} \to \exptype{List}{\type{X}} \\
|
|
\texttt{l} : \wctype{\rwildcard{A}}{List}{\rwildcard{A}} \quad \quad
|
|
\exptype{List}{\type{A}} <: [\type{A}/\type{X}]\exptype{List}{\type{X}} \\
|
|
\ [\type{A}/\type{X}]\exptype{List}{\type{X}} <: \wctype{\rwildcard{A}}{List}{\rwildcard{A}}
|
|
\\
|
|
\hline
|
|
\vspace*{-0.3cm}\\
|
|
\texttt{shuffle}(\expr{l}) : \wctype{\rwildcard{A}}{List}{\rwildcard{A}}
|
|
\end{array}
|
|
\end{array}$
|
|
\end{center}
|
|
\end{frame}
|
|
|
|
\begin{frame}[fragile,t]
|
|
%Explain why ? cannot be used as a regular type
|
|
\begin{verbatim}
|
|
<X> List<X> concat(List<X> l1, List<X> l2){
|
|
return l1.addAll(l2);
|
|
}
|
|
|
|
List<?> l1;
|
|
List<?> l2;
|
|
|
|
concat(l1, l2); //Error
|
|
\end{verbatim}
|
|
\end{frame}
|
|
|
|
\begin{frame}[fragile,t]
|
|
\begin{verbatim}
|
|
<X> List<X> concat(List<X> l1, List<X> l2){
|
|
return l1.addAll(l2);
|
|
}
|
|
|
|
List<String> ls = new List<String>();
|
|
|
|
List<?> l1 = ls; //korrekt!
|
|
List<?> l2 = new List<Integer>(1);
|
|
|
|
concat(l1, l2); //Error (würde l2 an ls anfügen!)
|
|
// ls.getLast() -> 1 (ein Int obwohl ls : List<String>)
|
|
\end{verbatim}
|
|
\end{frame}
|
|
|
|
\begin{frame}[fragile]
|
|
%Wildcard Capture
|
|
\begin{verbatim}
|
|
<X> void shuffle2D(List<List<X>> l){
|
|
//randomly exchange every element
|
|
// in the two dimensional list l
|
|
}
|
|
|
|
List<List<?>> l;
|
|
|
|
shuffle2D(l); // Error
|
|
\end{verbatim}
|
|
|
|
$\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \nless: \exptype{List}{\rwildcard{A}}$
|
|
\end{frame}
|
|
|
|
\begin{frame}[fragile]
|
|
%Es gibt stellen wo capture conversion möglich ist und Stellen wo es nicht ist.
|
|
\frametitle{Capture Constraint}
|
|
\begin{verbatim}
|
|
<X> List<List<X>> shuffle2D(List<? extends List<X>> l) {...}
|
|
|
|
List<List<?>> l = ...;
|
|
shuffle2D(l); // Fehler!
|
|
\end{verbatim}
|
|
\begin{itemize}
|
|
\item hier ergibt sich ein Subtype-Constraint der keine Capture Conversion zulässt:
|
|
\item $\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdot \exptype{List}{\tv{x}}$
|
|
\pause
|
|
\item $\unify{}(\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdot \exptype{List}{\tv{a}}) = \emptyset$
|
|
\end{itemize}
|
|
|
|
\end{frame}
|
|
|
|
\begin{frame}[fragile]
|
|
%Es gibt stellen wo capture conversion möglich ist und Stellen wo es nicht ist.
|
|
\frametitle{Capture Constraint}
|
|
\begin{verbatim}
|
|
<X> List<List<X>> shuffle2D(List<? extends List<X>> l) {...}
|
|
|
|
List<List<?>> l = ...;
|
|
shuffle2D(l); // Fehler!
|
|
\end{verbatim}
|
|
\begin{itemize}
|
|
\item hier ergibt sich ein Subtype-Constraint der keine Capture Conversion zulässt:
|
|
\item $\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdot \exptype{List}{\tv{x}}$
|
|
\pause
|
|
\item $\unify{}(\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdot \exptype{List}{\tv{a}}) = \emptyset$
|
|
\end{itemize}
|
|
|
|
\end{frame}
|
|
|
|
\begin{frame}[fragile]
|
|
\frametitle{Capture Constraint}
|
|
$\unify{}(\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdot \exptype{List}{\tv{a}}) = \emptyset$
|
|
$\unify{}(\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\tv{a}}) = \set{\tv{a} \to \rwildcard{A}}$
|
|
\end{frame}
|
|
|
|
% TODO:
|
|
% Folie zu aktueller Stand Unify: Es gibt <. Constraints. Unify löst diese
|
|
% Folie zu <? constraints: sie werden benötigt, weil unify während der Lösungsfindung Capture Conversion durchführen muss
|
|
%why do we need capture constraints?
|
|
% List<List<?>> not a subtype of List<List<X>> for any X
|
|
|
|
|
|
|
|
%TODO Unify erklären
|
|
% - es gibt die Capture Regel, welche Capture Conversion durchführt
|
|
% - die subst-Regel, welche keine Typen mit freien Variablen in normale Typvariablen einsetzt
|
|
\begin{frame}[fragile]{Reduce}
|
|
\rulename{Reduce} $
|
|
\begin{array}[c]{@{}ll}
|
|
\begin{array}[c]{l}
|
|
\wildcardEnv \vdash
|
|
C \cup \, \set{ \exptype{C}{\ol{S}} \lessdot
|
|
\wctype{\overline{\rwildcard{A}}}{C}{\ol{T}} } \\
|
|
\hline
|
|
\vspace*{-0.4cm}\\
|
|
\wildcardEnv
|
|
\vdash C \cup \, \set{
|
|
\ol{\type{S}} \doteq [\ol{\wtv{a}}/\overline{\rwildcard{A}}]\ol{\type{T}} }
|
|
\end{array}
|
|
%\quad \ol{Y} = \textit{fresh}(\ol{X})
|
|
\quad \begin{array}[c]{l}
|
|
\ol{\wtv{a}} \ \text{fresh}\\
|
|
%\text{fv}(\exptype{C}{\ol{S}}) \subseteq \text{dom}(\overline{\wildcard{B}{\type{U'}}{\type{L'}}})
|
|
%\text{dom}(\overline{\wildcard{A}{\type{U}}{\type{L}}}) \subseteq \text{fv}(\exptype{C}{\ol{T}}) \\
|
|
%\text{fv}(\wctype{\overline{\wildcard{A}{\type{U}}{\type{L}}}}{C}{\ol{T}}) = \emptyset
|
|
\end{array}
|
|
\end{array}
|
|
$
|
|
\begin{itemize}
|
|
\item \textbf{Example:}
|
|
$\begin{array}[t]{c}
|
|
\exptype{List}{\type{String}} \lessdot \wctype{\rwildcard{A}}{List}{\type{A}}\\
|
|
\hline
|
|
\type{String} \doteq \wtv{a}
|
|
\end{array}$
|
|
\item \textbf{Example:}
|
|
$\begin{array}[t]{c}
|
|
\wctype{\rwildcard{A}}{List}{\type{A}} \lessdot \exptype{List}{\type{String}}\\
|
|
\hline
|
|
\type{A} \doteq \type{String}
|
|
\end{array}$
|
|
\end{itemize}
|
|
\end{frame}
|
|
|
|
\begin{frame}[fragile]{Wildcard Creation}
|
|
\begin{lstlisting}
|
|
(*@\only<2>{$\color{red}\tv{r}$ }@*)someList(){
|
|
if(Math.random > 0.5){
|
|
return new List("String");
|
|
} else {
|
|
return new List(42);
|
|
}
|
|
}
|
|
\end{lstlisting}
|
|
\pause
|
|
$
|
|
\exptype{List}{String} \lessdot {\only<2>{\color{red}}\tv{r}},
|
|
\exptype{List}{Integer} \lessdot {\only<2>{\color{red}}\tv{r}}
|
|
$\\[1em]
|
|
\pause
|
|
\rulename{General} $
|
|
\begin{array}[c]{l}
|
|
\wildcardEnv \vdash C \cup \wctype{\Delta}{C}{\ol{T}} \lessdot \ntv{a}\\
|
|
\hline
|
|
\wildcardEnv \vdash C \cup \set{\wctype{\Delta}{C}{\ol{T}} \lessdot \ntv{a},
|
|
\ntv{a} \doteq \wctype{\overline{\wildcard{A}{\tv{u}}{\tv{l}}}}{C}{\overline{\rwildcard{A}}}
|
|
}
|
|
\end{array} \quad \begin{array}[c]{l}
|
|
\texttt{class} \ \exptype{C}{\ol{X}} \set{\ldots }\\%\triangleleft \exptype{D}{\ol{N}} \\
|
|
\tv{u}, \tv{l}\ \text{fresh}
|
|
\end{array}
|
|
$
|
|
\\[1em]
|
|
\pause
|
|
$
|
|
\exptype{List}{String} \lessdot {\tv{r}},
|
|
\exptype{List}{Integer} \lessdot {\tv{r}},
|
|
\highlight{\tv{r} \doteq \wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\rwildcard{X}}}
|
|
$
|
|
\pause
|
|
\\[0.5em]
|
|
\textbf{Substitution:}
|
|
|
|
$
|
|
\exptype{List}{String} \lessdot \highlight{\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\rwildcard{X}}},
|
|
\exptype{List}{Integer} \lessdot \highlight{\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\rwildcard{X}}}
|
|
$
|
|
\end{frame}
|
|
|
|
\begin{frame}[fragile]
|
|
$
|
|
\begin{array}{l}
|
|
\exptype{List}{String} \lessdot \highlight{\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\rwildcard{X}}},
|
|
\exptype{List}{Integer} \lessdot \highlight{\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\rwildcard{X}}}
|
|
\\
|
|
\hline
|
|
\type{String} \doteq \wtv{x1}, \type{Integer} \doteq \wtv{x2}
|
|
\end{array}
|
|
\rulenameAfter{Reduce}
|
|
$
|
|
\end{frame}
|
|
|
|
\begin{frame}[fragile]{Wildcard Elminiation}
|
|
\begin{lstlisting}
|
|
<X> List<X> concat(List<X> l1, List<X> l2){ ... }
|
|
someList(){
|
|
if(Math.random > 0.5){
|
|
return new List("String");
|
|
} else {
|
|
return new List(42);
|
|
}
|
|
}
|
|
|
|
concat(someList(), someList())
|
|
\end{lstlisting}
|
|
|
|
\only<1>{\textbf{Input}:}
|
|
$\only<2>{\highlight{\rulename{General}}}$
|
|
$\only<3>{\highlight{\rulename{Reduce}}}$
|
|
|
|
$
|
|
\begin{array}{l}
|
|
\exptype{List}{String} \lessdot {\tv{r}},
|
|
\exptype{List}{Integer} \lessdot {\tv{r}},
|
|
\tv{r} \lessdotCC \exptype{List}{\wtv{x}},
|
|
\tv{r} \lessdotCC \exptype{List}{\wtv{x}}
|
|
\pause
|
|
\\
|
|
\hline
|
|
\exptype{List}{String} \lessdot \only<2>{\highlight}{\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\rwildcard{X}}},
|
|
\exptype{List}{Integer} \lessdot \only<2>{\highlight}{\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\rwildcard{X}}},\\
|
|
\only<2>{\highlight}{\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\rwildcard{X}}} \lessdotCC \exptype{List}{\wtv{x}},
|
|
\only<2>{\highlight}{\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\rwildcard{X}}} \lessdotCC \exptype{List}{\wtv{x}}
|
|
\pause
|
|
\\
|
|
\hline
|
|
\highlight{\type{String} \doteq \wtv{x1}, \wtv{x1} \lessdot \tv{u}, \tv{l} \lessdot \wtv{x1},
|
|
\type{Integer} \doteq \wtv{x2}, \wtv{x2} \lessdot \tv{u}, \tv{l} \lessdot \wtv{x2}}, \\
|
|
\only<2>{\highlight}{\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\rwildcard{X}}} \lessdotCC \exptype{List}{\wtv{x}},
|
|
\only<2>{\highlight}{\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\rwildcard{X}}} \lessdotCC \exptype{List}{\wtv{x}}
|
|
\end{array}$
|
|
|
|
\end{frame}
|
|
|
|
% wieso dürfen normale Typvariablen keine freien Variablen enthalten?
|
|
% sonst könnte man der shuffle2D Method ein
|
|
\begin{frame}[fragile]
|
|
\begin{verbatim}
|
|
<A> List<A> shuffle(List<A> l){...}
|
|
<A,B> List<B> map(List<A> l, Function<A,B> f) { ... }
|
|
List<List<?>> l;
|
|
l.map((List<?> x) -> shuffle(x));
|
|
\end{verbatim}
|
|
$
|
|
\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\tv{x}},
|
|
\exptype{List}{\tv{x}} \lessdot \tv{r}
|
|
$
|
|
\end{frame}
|
|
\begin{frame}[fragile]
|
|
\begin{verbatim}
|
|
List<List<?>> l;
|
|
l2 = l.map(x -> shuffle(x));
|
|
shuffle2D(l2);
|
|
\end{verbatim}
|
|
|
|
$
|
|
\begin{array}{l}
|
|
\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\tv{x}}, \\
|
|
\exptype{List}{\exptype{List}{\tv{x}}} \lessdot \tv{l2}, \\
|
|
\tv{l2} \lessdotCC \exptype{List}{\exptype{List}{\tv{y}}}
|
|
\end{array}
|
|
$
|
|
|
|
\begin{itemize}
|
|
\item Falsche Lösung:
|
|
$
|
|
\begin{array}{l}
|
|
\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\highlight{\rwildcard{A}}}, \\
|
|
\exptype{List}{\exptype{List}{\tv{x}}} \lessdot \exptype{List}{\exptype{List}{\tv{x}}},
|
|
\tv{r} \lessdot \tv{l2},
|
|
\tv{l2} \lessdotCC \exptype{List}{\exptype{List}{\tv{y}}}
|
|
\end{array}
|
|
$
|
|
\end{itemize}
|
|
\end{frame}
|
|
|
|
\begin{frame}[fragile]{Lösung durch Unify}
|
|
$
|
|
\begin{array}{l}
|
|
\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\tv{x}}, \\
|
|
\exptype{List}{\exptype{List}{\tv{x}}} \lessdot \tv{l2}, \\
|
|
\tv{l2} \lessdotCC \exptype{List}{\exptype{List}{\tv{y}}}
|
|
\end{array}
|
|
$
|
|
|
|
\begin{itemize}
|
|
\item Falsche Lösung:
|
|
$
|
|
\begin{array}{l}
|
|
\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\highlight{\rwildcard{A}}}, \\
|
|
\exptype{List}{\exptype{List}{\tv{x}}} \lessdot \exptype{List}{\exptype{List}{\tv{x}}},
|
|
\tv{r} \lessdot \tv{l2},
|
|
\tv{l2} \lessdotCC \exptype{List}{\exptype{List}{\tv{y}}}
|
|
\end{array}
|
|
$
|
|
\end{itemize}
|
|
\end{frame}
|
|
|
|
\begin{frame}[fragile]
|
|
\begin{verbatim}
|
|
List<List<?>> l;
|
|
l2 = l.map(x -> shuffle(x));
|
|
shuffle2D(l2);
|
|
\end{verbatim}
|
|
|
|
$
|
|
\begin{array}{l}
|
|
\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\tv{x}}, \\
|
|
\exptype{List}{\exptype{List}{\tv{x}}} \lessdot \tv{r},
|
|
\tv{r} \lessdot \tv{l2},
|
|
\tv{l2} \lessdotCC \exptype{List}{\exptype{List}{\tv{y}}}
|
|
\end{array}
|
|
$
|
|
\end{frame}
|
|
|
|
% könnte es nicht well-formed typen geben, wenn man beliebige substitutionen zulässt?
|
|
% (brauchen wir die wildcard-TVs überhaupt? es wird eigentlich nur für den Soundness-Beweis gebraucht)
|
|
|
|
|
|
|
|
\end{document}
|