From 6b73d5ec8318b4832c2132a9a06c6d500b2084d6 Mon Sep 17 00:00:00 2001 From: Ulya Trofimovich Date: Mon, 2 Jul 2018 23:06:40 +0100 Subject: [PATCH] Paper: another example of traces computation. --- re2c/doc/tdfa_v2/img/Makefile | 2 +- re2c/doc/tdfa_v2/img/pe.tex | 58 ++--- re2c/doc/tdfa_v2/img/pe2.tex | 422 +++++++++++++++++++++++++++++++ re2c/doc/tdfa_v2/part_1_tnfa.tex | 11 + 4 files changed, 463 insertions(+), 30 deletions(-) create mode 100644 re2c/doc/tdfa_v2/img/pe2.tex diff --git a/re2c/doc/tdfa_v2/img/Makefile b/re2c/doc/tdfa_v2/img/Makefile index 120d8c27..0656219e 100644 --- a/re2c/doc/tdfa_v2/img/Makefile +++ b/re2c/doc/tdfa_v2/img/Makefile @@ -1,7 +1,7 @@ %.pdf : %.tex lualatex -shell-escape $< $<.build_log -all : mark_enum.pdf pe.pdf trees.pdf tnfa.pdf gor1.pdf +all : mark_enum.pdf pe.pdf pe2.pdf trees.pdf tnfa.pdf gor1.pdf clean: rm *.log *.build_log *.aux *.pdf *.pag diff --git a/re2c/doc/tdfa_v2/img/pe.tex b/re2c/doc/tdfa_v2/img/pe.tex index e4b69712..8980859b 100644 --- a/re2c/doc/tdfa_v2/img/pe.tex +++ b/re2c/doc/tdfa_v2/img/pe.tex @@ -23,14 +23,14 @@ \begin{tikzpicture}[>=stealth, ->, auto, node distance=0.2in] -\tikzstyle{every node}=[draw=none] +\tikzstyle{every node}=[draw=none, shape=rectangle] % $(\epsilon|a^{0,\infty})(a|\epsilon)^{0,\infty}$ \begin{scope}[xshift=0in, yshift=0in] \node (a) {{ $\begin{aligned} - \quad + \quad\quad & \Phi_0 \big( {T}^{1} ( {T}^{2} ( @@ -53,7 +53,7 @@ && \overbracket { \Xr_1 \Xl_2 \Xm_2 \Xr_1 \Xr_0 }%^{\alpha_1} % &&= \alpha_0 a \alpha_1 &&= \alpha - \\[-0.3em] + \\[-0.4em] & \Phi_0 \big( {T}^{1} ( {T}^{2}( @@ -78,7 +78,7 @@ && \overbracket { \Xr_2 \Xr_1 \Xr_0 }%^{\beta_1} % &&= \beta_0 a \beta_1 &&= \beta - \\[-0.3em] + \\[-0.4em] & \Phi_0 \big( {T}^{1} ( {T}^{2}( @@ -106,54 +106,54 @@ && \overbracket { \Xr_2 \Xl_3 \Xr_2 \Xr_1 \Xr_0 }%^{\gamma_1} % &&= \gamma_0 a \gamma_1 &&= \gamma - \quad + \quad\quad \end{aligned}$ }}; \end{scope} -\begin{scope}[xshift=0in, yshift=-2.1in] +\begin{scope}[xshift=0in, yshift=-1.7in] \node (a) { $\begin{aligned} &\begin{tabular}{c|ll} $traces (\alpha, \beta)$ & 0 & 1 \\ - \hline \\[-0.5em] - $\rho^{\alpha/\beta}$ & 2 & 0 \\ - $\rho^{\beta/\alpha}$ & 1 & 0 \\ + \hline \\[-1em] + $\rho$ & 2 & 0 \\ + $\rho'$ & 1 & 0 \\ \end{tabular} &&\quad \left[\begin{aligned} - \rho^{\alpha/\beta}_0 &= min (lasth (\Xl_1 \Xl_2), minh (\epsilon)) = min (2, \infty) = 2 \\[-0.3em] - \rho^{\beta/\alpha}_0 &= min (lasth (\Xl_1 \Xl_2), minh (\Xr_1 \Xl_2 \Xl_3)) = min (2, 1) = 1 \\[-0.3em] - \rho^{\alpha/\beta}_1 &= min (\rho^{\alpha/\beta}_0, minh (\Xr_1 \Xl_2 \Xm_2 \Xr_1 \Xr_0)) = min (2,0) = 0 \\[-0.3em] - \rho^{\beta/\alpha}_1 &= min (\rho^{\beta/\alpha}_0, minh (\Xr_2 \Xr_1 \Xr_0)) = min (1,0) = 0 + \rho_0 &= min (lasth (\Xl_1 \Xl_2), minh (\epsilon)) = min (2, \infty) = 2 \\[-0.3em] + \rho'_0 &= min (lasth (\Xl_1 \Xl_2), minh (\Xr_1 \Xl_2 \Xl_3)) = min (2, 1) = 1 \\[-0.3em] + \rho_1 &= min (\rho_0, minh (\Xr_1 \Xl_2 \Xm_2 \Xr_1 \Xr_0)) = min (2,0) = 0 \\[-0.3em] + \rho'_1 &= min (\rho'_0, minh (\Xr_2 \Xr_1 \Xr_0)) = min (1,0) = 0 \end{aligned}\right. - \\[1em] + \\[0.5em] &\begin{tabular}{c|ll} $traces (\beta, \gamma)$ & 0 & 1 \\ - \hline \\[-0.5em] - $\rho^{\beta/\gamma}$ & -1 & 0 \\ - $\rho^{\gamma/\beta}$ & -1 & 0 \\ + \hline \\[-1em] + $\rho$ & \!-1 & 0 \\ + $\rho'$ & \!-1 & 0 \\ \end{tabular} &&\quad \left[\begin{aligned} - \rho^{\beta/\gamma}_0 &= -1 \\[-0.3em] - \rho^{\gamma/\beta}_0 &= -1 \\[-0.3em] - \rho^{\beta/\gamma}_1 &= min (lasth (\Xr_2), minh (\Xr_1 \Xr_0)) = min (2,0) = 0 \\[-0.3em] - \rho^{\gamma/\beta}_1 &= min (lasth (\Xr_2), minh (\Xl_3 \Xr_2 \Xr_1 \Xr_0)) = min (2,0) = 0 + \rho_0 &= -1 \\[-0.3em] + \rho'_0 &= -1 \\[-0.3em] + \rho_1 &= min (lasth (\Xr_2), minh (\Xr_1 \Xr_0)) = min (2,0) = 0 \\[-0.3em] + \rho'_1 &= min (lasth (\Xr_2), minh (\Xl_3 \Xr_2 \Xr_1 \Xr_0)) = min (2,0) = 0 \end{aligned}\right. - \\[1em] + \\[0.5em] &\begin{tabular}{c|ll} $traces (\alpha, \gamma)$ & 0 & 1 \\ - \hline \\[-0.5em] - $\rho^{\alpha/\gamma}$ & 2 & 0 \\ - $\rho^{\gamma/\alpha}$ & 1 & 0 \\ + \hline \\[-1em] + $\rho$ & 2 & 0 \\ + $\rho'$ & 1 & 0 \\ \end{tabular} &&\quad \left[\begin{aligned} - \rho^{\alpha/\gamma}_0 &= min (lasth (\Xl_1 \Xl_2), minh (\epsilon)) = min (2, \infty) = 2 \\[-0.3em] - \rho^{\gamma/\alpha}_0 &= min (lasth (\Xl_1 \Xl_2), minh (\Xr_1 \Xl_2 \Xl_3)) = min (2, 1) = 1 \\[-0.3em] - \rho^{\alpha/\gamma}_1 &= min (\rho^{\alpha/\gamma}_0, minh (\Xr_1 \Xl_2 \Xm_2 \Xr_1 \Xr_0)) = min (2,0) = 0 \\[-0.3em] - \rho^{\gamma/\alpha}_1 &= min (\rho^{\gamma/\alpha}_0, minh (\Xr_2 \Xl_3 \Xr_2 \Xr_1 \Xr_0)) = min (1,0) = 0 + \rho_0 &= min (lasth (\Xl_1 \Xl_2), minh (\epsilon)) = min (2, \infty) = 2 \\[-0.3em] + \rho'_0 &= min (lasth (\Xl_1 \Xl_2), minh (\Xr_1 \Xl_2 \Xl_3)) = min (2, 1) = 1 \\[-0.3em] + \rho_1 &= min (\rho_0, minh (\Xr_1 \Xl_2 \Xm_2 \Xr_1 \Xr_0)) = min (2,0) = 0 \\[-0.3em] + \rho'_1 &= min (\rho'_0, minh (\Xr_2 \Xl_3 \Xr_2 \Xr_1 \Xr_0)) = min (1,0) = 0 \end{aligned}\right. \end{aligned}$ }; diff --git a/re2c/doc/tdfa_v2/img/pe2.tex b/re2c/doc/tdfa_v2/img/pe2.tex new file mode 100644 index 00000000..9be68fcc --- /dev/null +++ b/re2c/doc/tdfa_v2/img/pe2.tex @@ -0,0 +1,422 @@ + +\documentclass[tikz,border=10pt]{standalone} + + +\RequirePackage{luatex85} +\usepackage[utf8]{inputenc} +\usepackage{amsmath, amssymb, amsfonts, accents} +\usetikzlibrary{graphdrawing, graphs, arrows, shapes, automata, calc} +\usegdlibrary{trees, layered} +\usepackage{stix} + + +%\newcommand{\Xund}{\rule{.4em}{.4pt}} +%\newcommand{\IRE}{I\!RE} + +\newcommand{\Xund}{\rule{.4em}{.4pt}} +\newcommand{\Xl}{\langle} +\newcommand{\Xr}{\rangle} +\newcommand{\Xm}{\langle\!\rangle} + + +\begin{document} + +\begin{tikzpicture}[>=stealth, ->, auto, sibling distance = 0.3in, inner sep = 1.5pt] + +\tikzstyle{every node}=[draw, shape = circle] + +\begin{scope}[xshift=0in, yshift=0in] + \graph [tree layout, grow=down, fresh nodes] { + "$1$"[draw] -- { + ""[draw] -- { + ""[draw] -- { "$a$", "$a$", "$a$" } + } + } + , "$2$"[draw] -- { + ""[draw] -- { + ""[draw] -- { "$a$", "$a$" }, + ""[draw] -- { "$a$" } + } + } + , "$3$"[draw] -- { + ""[draw] -- { + ""[draw] -- { "$a$" }, + ""[draw] -- { "$a$", "$a$" } + } + } + , "$4$"[draw] -- { + ""[draw] -- { + ""[draw] -- { "$a$" }, + ""[draw] -- { "$a$" }, + ""[draw] -- { "$a$" } + } + } + , "$5$"[draw] -- { + ""[draw] -- { + ""[draw] -- { "$a$", "$a$" } + }, + ""[draw] -- { + ""[draw] -- { "$a$" } + } + } + , "$6$"[draw] -- { + ""[draw] -- { + ""[draw] -- { "$a$" } + }, + ""[draw] -- { + ""[draw] -- { "$a$", "$a$" } + } + } + , "$7$"[draw] -- { + ""[draw] -- { + ""[draw] -- { "$a$"}, + ""[draw] -- { "$a$"} + }, + ""[draw] -- { + ""[draw] -- { "$a$" } + } + } + , "$8$"[draw] -- { + ""[draw] -- { + ""[draw] -- { "$a$"} + }, + ""[draw] -- { + ""[draw] -- { "$a$"}, + ""[draw] -- { "$a$" } + } + } + , "$9$"[draw] -- { + ""[draw] -- { + ""[draw] -- { "$a$" } + }, + ""[draw] -- { + ""[draw] -- { "$a$" } + }, + ""[draw] -- { + ""[draw] -- { "$a$" } + } + } + }; +\end{scope} + +\begin{scope}[xshift=1in, yshift=0in] + \draw [dash pattern = on 2pt off 2pt, rounded corners] (2,-0.3) -- (2,-1) -- (1.45,-1.9) -- (1.45,-2.75); + \draw [dash pattern = on 2pt off 2pt, rounded corners] (1.75,-2.75) -- (1.75,-2) -- (2.2,-1.3) -- (2.6,-2) -- (2.3,-2.75); + \draw [dash pattern = on 2pt off 2pt, rounded corners] (2.55,-2.8) -- (2.75,-2.3) -- (2.9,-2.8); + \draw [dash pattern = on 2pt off 2pt, rounded corners] (3.2,-2.75) -- (2.9,-1.9) -- (2.35,-1) -- (2.35,-0.3); +\end{scope} + +%\begin{scope}[xshift=0in, yshift=0.1in] +% \node (a1) {1}; +% \node [right of = a1, xshift = 0.5in] (a2) {2}; +% \node [right of = a2, xshift = 0.35in] (a3) {3}; +% \node [right of = a3, xshift = 0.45in] (a4) {4}; +% \node [right of = a4, xshift = 0.45in] (a5) {5}; +%\end{scope} + +\begin{scope}[xshift=6.5in, yshift=-3.1in] + \node [shape=rectangle, draw = none] (a) { + $\begin{aligned} + &1 &&\overbracket {\Xl_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xl_4} a \overbracket {\Xr_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xr_0} \\ + &2 &&\overbracket {\Xl_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xr_0} \\ + &3 &&\overbracket {\Xl_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xr_0} \\ + &4 &&\overbracket {\Xl_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xr_0} \\ + &5 &&\overbracket {\Xl_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xr_0} \\ + &6 &&\overbracket {\Xl_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xr_0} \\ + &7 &&\overbracket {\Xl_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xr_0} \\ + &8 &&\overbracket {\Xl_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xr_0} \\ + &9 &&\overbracket {\Xl_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xr_0} + \end{aligned}$ + }; +\end{scope} + +\begin{scope}[xshift=2.2in, yshift=-3.1in] + \node [shape=rectangle, draw = none] (a) { + \setlength\tabcolsep{2pt} + \renewcommand{\arraystretch}{1.1} + $\begin{aligned} + &\begin{tabular}{c ccccccccc} + 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ + % + \begin{tabular}{|cccc|} + \hline + -1 & \!-1 & 3 & 0 \\[-3pt] + -1 & \!-1 & 2 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & 3 & 3 & 0 \\[-3pt] + -1 & 2 & 2 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & 3 & 3 & 0 \\[-3pt] + -1 & 2 & 2 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & \!-1 & 3 & 0 \\[-3pt] + -1 & \!-1 & 1 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & 3 & 3 & 0 \\[-3pt] + -1 & 1 & 1 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & 3 & 3 & 0 \\[-3pt] + -1 & 2 & 1 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & 3 & 3 & 0 \\[-3pt] + -1 & 1 & 1 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & 3 & 3 & 0 \\[-3pt] + -1 & 1 & 1 & 0 \\ + \hline + \end{tabular} + & 1 + \\[1em] +% + & + \begin{tabular}{|cccc|} + \hline + -1 & 3 & 2 & 0 \\[-3pt] + -1 & 2 & 2 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & 3 & 2 & 0 \\[-3pt] + -1 & 2 & 2 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & \!-1 & 2 & 0 \\[-3pt] + -1 & \!-1 & 1 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & 3 & 2 & 0 \\[-3pt] + -1 & 1 & 1 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & 3 & 2 & 0 \\[-3pt] + -1 & 2 & 1 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & 3 & 2 & 0 \\[-3pt] + -1 & 1 & 1 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & 3 & 2 & 0 \\[-3pt] + -1 & 1 & 1 & 0 \\ + \hline + \end{tabular} + & 2 + \\[1em] +% + & & + \begin{tabular}{|cccc|} + \hline + -1 & \!-1 & 3 & 0 \\[-3pt] + -1 & \!-1 & 2 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & 2 & 2 & 0 \\[-3pt] + -1 & 3 & 1 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & 2 & 2 & 0 \\[-3pt] + -1 & 1 & 1 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & \!-1 & 3 & 0 \\[-3pt] + -1 & \!-1 & 1 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & 2 & 2 & 0 \\[-3pt] + -1 & 1 & 1 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & 2 & 2 & 0 \\[-3pt] + -1 & 1 & 1 & 0 \\ + \hline + \end{tabular} + & 3 + \\[1em] +% + & & & + \begin{tabular}{|cccc|} + \hline + -1 & 2 & 2 & 0 \\[-3pt] + -1 & 3 & 1 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & 2 & 2 & 0 \\[-3pt] + -1 & 1 & 1 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & \!-1 & 2 & 0 \\[-3pt] + -1 & \!-1 & 1 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & 2 & 2 & 0 \\[-3pt] + -1 & 1 & 1 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & 2 & 2 & 0 \\[-3pt] + -1 & 1 & 1 & 0 \\ + \hline + \end{tabular} + & 4 + \\[1em] +% + & & & & + \begin{tabular}{|cccc|} + \hline + -1 & 3 & 1 & 0 \\[-3pt] + -1 & 1 & 1 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & 3 & 1 & 0 \\[-3pt] + -1 & 2 & 1 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & 3 & 1 & 0 \\[-3pt] + -1 & 1 & 1 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & 3 & 1 & 0 \\[-3pt] + -1 & 1 & 1 & 0 \\ + \hline + \end{tabular} + & 5 + \\[1em] +% + & & & & & + \begin{tabular}{|cccc|} + \hline + -1 & 1 & 1 & 0 \\[-3pt] + -1 & 2 & 1 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & \!-1 & 3 & 0 \\[-3pt] + -1 & \!-1 & 2 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & \!-1 & 3 & 0 \\[-3pt] + -1 & \!-1 & 1 & 0 \\ + \hline + \end{tabular} + & 6 + \\[1em] +% + & & & & & & + \begin{tabular}{|cccc|} + \hline + -1 & 2 & 1 & 0 \\[-3pt] + -1 & 1 & 1 & 0 \\ + \hline + \end{tabular} + & + \begin{tabular}{|cccc|} + \hline + -1 & 2 & 1 & 0 \\[-3pt] + -1 & 1 & 1 & 0 \\ + \hline + \end{tabular} + & 7 + \\[1em] +% + & & & & & & & + \begin{tabular}{|cccc|} + \hline + -1 & \!-1 & 2 & 0 \\[-3pt] + -1 & \!-1 & 1 & 0 \\ + \hline + \end{tabular} + & 8 + \end{tabular} + \end{aligned}$ + }; +\end{scope} + +\end{tikzpicture} + +\end{document} + diff --git a/re2c/doc/tdfa_v2/part_1_tnfa.tex b/re2c/doc/tdfa_v2/part_1_tnfa.tex index 3bbf0ad1..b1df341d 100644 --- a/re2c/doc/tdfa_v2/part_1_tnfa.tex +++ b/re2c/doc/tdfa_v2/part_1_tnfa.tex @@ -730,6 +730,17 @@ $\beta \sim \gamma$ and $\beta \subset \gamma$, because $first (\beta \backslash \gamma) = \Xr < \Xl = first (\gamma \backslash \beta)$. Therefore $\alpha < \beta < \gamma$. } +\end{figure} + +\begin{figure}\label{fig_pe2} +\includegraphics[width=\linewidth]{img/pe2.pdf} +\caption{ +An example of PEs for IPTs from figure \ref{fig_mark_enum} and the computation of $traces$ for each pair of PEs.\\ +Here $\alpha \sqsubset \beta$ and $\alpha \sqsubset \gamma$, while +$\beta \sim \gamma$ and $\beta \subset \gamma$, +because $first (\beta \backslash \gamma) = \Xr < \Xl = first (\gamma \backslash \beta)$. +Therefore $\alpha < \beta < \gamma$. +} \end{figure} \begin{Xdef}\label{prec1} -- 2.40.0