Rules Standalone
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Andreas Stadelmeier
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vortragKRRUpSeminar/latexImages/rules.tex
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267
vortragKRRUpSeminar/latexImages/rules.tex
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% This is samplepaper.tex, a sample chapter demonstrating the
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% LLNCS macro package for Springer Computer Science proceedings;
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% Version 2.21 of 2022/01/12
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%
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\documentclass{standalone}
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\usepackage{underscore}
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\usepackage[T1]{fontenc}
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% T1 fonts will be used to generate the final print and online PDFs,
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% so please use T1 fonts in your manuscript whenever possible.
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% Other font encondings may result in incorrect characters.
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%
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\usepackage{graphicx}
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% Used for displaying a sample figure. If possible, figure files should
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% be included in EPS format.
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%
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% If you use the hyperref package, please uncomment the following two lines
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% to display URLs in blue roman font according to Springer's eBook style:
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%\usepackage{color}
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%\renewcommand\UrlFont{\color{blue}\rmfamily}
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%\urlstyle{rm}
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%
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\include{prolog}
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\usepackage{mathpartir}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{amsthm}
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\usepackage{enumitem}
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\usepackage{xcolor}
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%\usepackage{amsthm}
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\newtheorem{theorem}{Theorem}
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\newtheorem{lemma}[theorem]{Lemma}
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\begin{document}
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\begin{tabular}{lcll}
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$c $ &$::=$ & $\type{T} \lessdot \type{T} \mid \type{T} \doteq \type{T}$ & Constraint \\
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$\type{T} $ & $::=$ & $\tv{a} \mid \type{N}$ & Type placeholder or Type \\
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$\ntype{N}$ & $::=$ & $\exptype{C}{\ol{T}}$ & Class Type containing type placeholders\\
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$\gtype{G}$ & $::=$ & $\exptype{C}{\ol{G}}$ & Class Type not containing type placeholders \\
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\end{tabular}
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%\subsection{Subtyping}
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\begin{mathpar}
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\inferrule[S-Refl]{}{
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\type{T} <: \type{T}
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}
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\and
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\inferrule[S-Trans]{\type{T}_1 <: \type{T}_2 \\ \type{T}_2 <: \type{T}_3}{
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\type{T}_1 <: \type{T}_3
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}
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\and
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\inferrule[S-Var]{}{\type{A} <: \Delta(\type{A})}
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\and
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\inferrule[S-Class]{\texttt{class}\ \exptype{C}{\ol{X}} \triangleleft \type{N}}{
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\exptype{C}{\ol{T}} <: [\ol{T}/\ol{X}]\type{N}
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}
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\end{mathpar}
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\begin{mathpar}
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\inferrule[N-Refl]{}{
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\type{C} << \type{C}
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}
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\and
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\inferrule[N-Trans]{\type{C}_1 << \type{C}_2 \\ \type{C}_2 << \type{C}_3}{
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\type{C}_1 << \type{C}_3
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}
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\and
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\inferrule[N-Class]{\texttt{class}\ \exptype{C}{\ldots} \triangleleft \exptype{D}{\ldots}}{
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\type{C} << \type{D}
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}
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\end{mathpar}
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\label{sec:implicationRules}
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\begin{mathpar}
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\inferrule[Subst-L]{
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\tv{a} \doteq \type{T}_1 \\
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\tv{a} \lessdot \type{T}_2
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}{
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\type{T}_1 \lessdot \type{T}_2
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}
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\and
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\inferrule[Subst-R]{
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\tv{a} \doteq \type{T}_1 \\
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\type{T}_2 \lessdot \tv{a}
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}{
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\type{T}_2 \lessdot \type{T}_1
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}
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\and
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\inferrule[Subst-Equal]{
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\tv{a} \doteq \type{T}_1 \\
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\tv{a} \doteq \type{T}_2
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}{
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\type{T}_1 \doteq \type{T}_2
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}
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\and
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\inferrule[Swap]{
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\type{T}_1 \doteq \type{T}_2
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}{
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\type{T}_2 \doteq \type{T}_1
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}
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\and
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\inferrule[Unfold]{
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\tv{b} \doteq \exptype{C}{\type{T}_1 \ldots \type{T}_n}
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}{
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\type{T}_i \doteq \type{T}_i
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}
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\and
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\inferrule[Unfold']{
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\tv{b} \lessdot \exptype{C}{\type{T}_1 \ldots \type{T}_n}
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}{
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\type{T}_i \doteq \type{T}_i
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}
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\and
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\inferrule[Subst-Param]{
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\type{T}' \doteq \type{S} \\
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\type{T} \doteq \exptype{C}{\type{T}_1 \ldots, \type{T}', \ldots \type{T}_n} \\
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}{
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\type{T} \doteq \exptype{C}{\type{T}_1, \ldots \type{S}, \ldots \type{T}_n}
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}
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\and
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\inferrule[Subst-Param']{
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\type{T}' \doteq \type{S} \\
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\tv{b} \lessdot \exptype{C}{\type{T}_1 \ldots, \type{T}', \ldots \type{T}_n} \\
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}{
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\tv{b} \lessdot \exptype{C}{\type{T}_1, \ldots \type{S}, \ldots \type{T}_n}
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}
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\and
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\inferrule[S-Object]{}{\tv{a} \lessdot \type{Object}}
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\and
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\inferrule[Match]{
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\tv{a} \lessdot \type{N}_1 \\
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\tv{a} \lessdot \type{N}_2 \\
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\type{N}_1 << \type{N}_2
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}{
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\type{N}_1 \lessdot \type{N}_2
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}
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\and
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\inferrule[Adopt]{
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\tv{a} \lessdot \tv{b} \\
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\tv{b} \lessdot \type{T}
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}{
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\tv{a} \lessdot \type{T}
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}
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\and
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% \inferrule[Subst-Param]{
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% \tv{a} \doteq \type{N} \\
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% \tv{a} = \type{T}_i \\
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% \exptype{C}{\type{T}_1 \ldots \type{T}_n} \lessdot \type{T} \\
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% }{
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% \type{T}_i \doteq \type{N} \\
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% }
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\and
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\inferrule[Adapt]{
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\type{N}_1 \lessdot \exptype{C}{\type{T}_1 \ldots \type{T}_n} \\
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\type{N}_1 <: \exptype{C}{\type{S}_1 \ldots \type{S}_n} \\
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}{
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\exptype{C}{\type{S}_1 \ldots \type{S}_n} \doteq \exptype{C}{\type{T}_1 \ldots \type{T}_n} \\
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}
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\and
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\inferrule[Reduce]{
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\exptype{C}{\type{S}_1 \ldots \type{S}_n} \doteq \exptype{C}{\type{T}_1 \ldots \type{T}_n} \\
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}{
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\type{S}_i \doteq \type{T}_i \\
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}
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\end{mathpar}
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\begin{mathpar}
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\text{Apply only once per constraint:}\quad
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\inferrule[Super]{
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\type{N} \lessdot \tv{a}\\
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\type{N} <: \type{N}'
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}{
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\tv{a} \doteq \type{N}'
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}
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\end{mathpar}
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\begin{mathpar}
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\inferrule[Split-L]{
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\tv{a} \lessdot \tv{b}\\
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\tv{a} \lessdot \type{N}\\
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}{
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\tv{b} \lessdot \type{N}
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}
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\quad \quad
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\vline
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\quad \quad
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\inferrule[Split-R]{
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\tv{a} \lessdot \tv{b}\\
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\tv{a} \lessdot \type{N}\\
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}{
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\type{N} \lessdot \tv{b}
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}
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\end{mathpar}
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Result:
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\begin{mathpar}
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\inferrule[Solution]{
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\tv{a} \doteq \type{G}
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}{
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\sigma(\tv{a}) = \type{G}
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}
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\and
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\inferrule[Solution-Gen]{
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\tv{a} \lessdot \type{C}_1, \ldots, \tv{a} \lessdot \type{C}_n \\
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\forall i: \type{C}_m << \type{C}_i \\
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}{
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\tv{a} \doteq \type{C}_m
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}
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% \and
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% \inferrule[Solution-Gen]{
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% \tv{a} \lessdot \type{C}\\
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% \sigma(\tv{a}) = \emptyset
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% }{
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% \tv{a} \doteq \type{A} \\ \sigma'(\tv{a}) = \type{A}
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% }
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% \and
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% \inferrule[Solution-Gen]{
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% \tv{a} \lessdot \type{G} \\
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% \tv{a} \lessdot \type{G}_1, \ldots, \tv{a} \lessdot \type{G}_n \\
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% \forall i: \type{G} << \type{G}_i \\
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% \sigma'(\tv{a}) = \type{A}
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% }{
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% \Delta(\type{A}) = \type{G}
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% }
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\end{mathpar}
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Fail:
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\begin{mathpar}
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% \inferrule[Fail]{
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% \type{T} \lessdot \type{N}\\
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% \type{T} \nless : \type{N}
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% }{
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% \emptyset
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% }
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% \and
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\inferrule[Fail]{
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\exptype{C}{\ldots} \doteq \exptype{D}{\ldots}\\
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\type{C} \neq \type{D}
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}{
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\emptyset
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}
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\and
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\inferrule[Fail-Generic]{
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\type{X} \doteq \type{T}\\
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\type{X} \neq \type{T}
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}{
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\emptyset
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}
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\and
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\inferrule[Fail-Sigma]{
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\tv{a} \doteq \type{N} \\
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\tv{a} \in \type{N}
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}{
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\emptyset
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}
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\and
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\inferrule[Fail]{
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\tv{a} \lessdot \type{N}_1 \\
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\tv{a} \lessdot \type{N}_2 \\
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\text{not}\ \type{N}_1 << \type{N}_2 \\
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\text{not}\ \type{N}_2 << \type{N}_1
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}{
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\emptyset
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}
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\end{mathpar}
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\end{document}
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