{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@curveto{0.46236pt}{0.0pt}{0.69353pt}{1.1381pt}{1.15588pt}{1.1381pt}\pgfsys@curveto{1.49062pt}{1.1381pt}{1.77913pt}{0.58272pt}{2.0806pt}{0.0pt}\pgfsys@curveto{2.38205pt}{-0.5827pt}{2.67056pt}{-1.1381pt}{3.00531pt}{-1.1381pt}\pgfsys@curveto{3.34004pt}{-1.1381pt}{3.62856pt}{-0.58272pt}{3.93002pt}{0.0pt}\pgfsys@curveto{4.23148pt}{0.5827pt}{4.51999pt}{1.1381pt}{4.85474pt}{1.1381pt}\pgfsys@curveto{5.18947pt}{1.1381pt}{5.47798pt}{0.58272pt}{5.77945pt}{0.0pt}\pgfsys@curveto{6.0809pt}{-0.5827pt}{6.36942pt}{-1.1381pt}{6.70416pt}{-1.1381pt}\pgfsys@lineto{6.70416pt}{0.0pt}\pgfsys@lineto{9.38113pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.38113pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}^{*}(q_{w},\star,\gamma_{w})\\}$
$\varPhi_{w}(q,\gamma)=\\{\gamma_{w}\in\Gamma^{+}\cup\\{\bot\\}\mid\exists
q_{w}\in Q,(q,w\star,\gamma)\mathrel{\raisebox{0.0pt}[0.0pt][0.0pt]{
\leavevmode\hbox to11.78pt{\vbox to2.68pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-1.3381pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{} {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}}{{}}} {{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}
{{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@curveto{0.46236pt}{0.0pt}{0.69353pt}{1.1381pt}{1.15588pt}{1.1381pt}\pgfsys@curveto{1.49062pt}{1.1381pt}{1.77913pt}{0.58272pt}{2.0806pt}{0.0pt}\pgfsys@curveto{2.38205pt}{-0.5827pt}{2.67056pt}{-1.1381pt}{3.00531pt}{-1.1381pt}\pgfsys@curveto{3.34004pt}{-1.1381pt}{3.62856pt}{-0.58272pt}{3.93002pt}{0.0pt}\pgfsys@curveto{4.23148pt}{0.5827pt}{4.51999pt}{1.1381pt}{4.85474pt}{1.1381pt}\pgfsys@curveto{5.18947pt}{1.1381pt}{5.47798pt}{0.58272pt}{5.77945pt}{0.0pt}\pgfsys@curveto{6.0809pt}{-0.5827pt}{6.36942pt}{-1.1381pt}{6.70416pt}{-1.1381pt}\pgfsys@lineto{6.70416pt}{0.0pt}\pgfsys@lineto{9.38113pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.38113pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}^{*}(q_{w},\star,\gamma_{w})\\}$
$g_{w}(q,q^{\prime},\gamma)=\\{p_{w}\in
Q\mid\exists\gamma_{w}\in\varPhi_{w}(q,\gamma),(q^{\prime},\varepsilon,\gamma_{w})\mathrel{\raisebox{0.0pt}[0.0pt][0.0pt]{
\leavevmode\hbox to11.78pt{\vbox to2.68pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-1.3381pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{} {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}}{{}}} {{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}
{{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@curveto{0.46236pt}{0.0pt}{0.69353pt}{1.1381pt}{1.15588pt}{1.1381pt}\pgfsys@curveto{1.49062pt}{1.1381pt}{1.77913pt}{0.58272pt}{2.0806pt}{0.0pt}\pgfsys@curveto{2.38205pt}{-0.5827pt}{2.67056pt}{-1.1381pt}{3.00531pt}{-1.1381pt}\pgfsys@curveto{3.34004pt}{-1.1381pt}{3.62856pt}{-0.58272pt}{3.93002pt}{0.0pt}\pgfsys@curveto{4.23148pt}{0.5827pt}{4.51999pt}{1.1381pt}{4.85474pt}{1.1381pt}\pgfsys@curveto{5.18947pt}{1.1381pt}{5.47798pt}{0.58272pt}{5.77945pt}{0.0pt}\pgfsys@curveto{6.0809pt}{-0.5827pt}{6.36942pt}{-1.1381pt}{6.70416pt}{-1.1381pt}\pgfsys@lineto{6.70416pt}{0.0pt}\pgfsys@lineto{9.38113pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.38113pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}^{*}(p_{w},\varepsilon,\bot)\\}$

Now, let $x,y,u,v,u_{0},v_{0}\in\widehat{\Sigma}^{*}$ such that
$u_{0}x^{\triangleleft},u_{0}y^{\triangleleft}\in\widehat{\Sigma}^{*}_{\mathrel{\mathchoice{\makebox[5.60002pt]{{\makebox[0.0pt]{$\geq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\gtrdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{$\geq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\gtrdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\scriptsize$\geq$}}{\makebox[0.0pt]{\raisebox{0.8pt}{\scriptsize$\gtrdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\tiny$\geq$}}{\makebox[0.0pt]{\raisebox{0.6pt}{\tiny$\gtrdot$}}}}}}}$,
$x^{\triangleright}v_{0},y^{\triangleright}v_{0}\in\widehat{\Sigma}^{*}_{\mathrel{\mathchoice{\makebox[5.60002pt]{{\makebox[0.0pt]{$\leq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\lessdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{$\leq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\lessdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\scriptsize$\leq$}}{\makebox[0.0pt]{\raisebox{0.8pt}{\scriptsize$\lessdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\tiny$\leq$}}{\makebox[0.0pt]{\raisebox{0.6pt}{\tiny$\lessdot$}}}}}}}$,
${{}^{u}[u_{0}xv_{0}]^{v}}$, ${{}^{u}[u_{0}yv_{0}]^{v}}$, and
$x\leqslant_{\mathcal{A}}y$. Since $x\leqslant_{\mathcal{A}}y$ implies
$x\approx y$, which implies $x^{\triangleleft}=y^{\triangleleft}$ and
$x^{\triangleright}=y^{\triangleright}$, it is clear that from the definitions
that $f_{uu_{0}xv_{0}v}(q,\gamma)\subseteq f_{uu_{0}yv_{0}v}(q,\gamma)$ and
$\varPhi_{uu_{0}xv_{0}v}(q,\gamma)\subseteq\varPhi_{uu_{0}yv_{0}v}(q,\gamma)$
for all $q\in Q$ and $\gamma\in\Gamma\cup\\{\bot\\}$. Intuitively, the
reasoning for this is as follows: $\mathcal{A}$ reaches a set of
configurations after processing $uu_{0}$. For every configuration in this set,
consider the state and the top stack symbol given as inputs to $f_{x}$ and
$f_{y}$, as well as $\varPhi_{x}$ and $\varPhi_{y}$, for whose outputs we know
the inclusion relation above. It means that after reading $uu_{0}x$ and
$uu_{0}y$ from any state and top stack symbol we have the same relations as
well. Now, consider this time the states and top stack symbols of the
configurations reached after $uu_{0}x$ and $uu_{0}y$. Proceeding the
computation from these configurations with the suffix $v_{0}v$ clearly
preserves the inclusion relation for the sets of states reached. For the sets
of stacks, note that ${{}^{u}[u_{0}xv_{0}]^{v}}$ and
${{}^{u}[u_{0}yv_{0}]^{v}}$, therefore the suffix $v_{0}v$ pops the stack
beyond what has been pushed while reading $x$ and $y$, which is the same for
both words. Finally, for all $q,q^{\prime}\in Q$ and all
$\gamma\in\Gamma\cup\\{\bot\\}$, we have that
$g_{uu_{0}xv_{0}v}(q,q^{\prime},\gamma)\subseteq
g_{uu_{0}yv_{0}v}(q,q^{\prime},\gamma)$ from
$\varPhi_{uu_{0}xv_{0}v}(q,\gamma)\subseteq\varPhi_{uu_{0}yv_{0}v}(q,\gamma)$
which implies that
$\\{(q^{\prime},\varepsilon,\gamma_{w})\mid\gamma_{w}\in\varPhi_{uu_{0}xv_{0}v}(q,\gamma)\\}\subseteq\\{(q^{\prime},\varepsilon,\gamma_{w})\mid\gamma_{w}\in\varPhi_{uu_{0}yv_{0}v}(q,\gamma)\\}$.
Therefore, $uu_{0}xv_{0}v\ll uu_{0}yv_{0}v$, and thus
$uu_{0}xv_{0}v\leqslant_{\mathcal{A}}uu_{0}yv_{0}v$, implying that
$\leqslant_{\mathcal{A}}$ is chain-monotonic.

#### Lemma 4.7

*Statement. For every OPA $\mathcal{A}$ and $w_{1},w_{2}\in\widehat{\Sigma}^{*}$, if $w_{1}\leqslant_{\mathcal{A}}w_{2}$ and $w_{1}\in L(\mathcal{A})$ then $w_{2}\in L(\mathcal{A})$.

######  Proof A.7.

Let $\mathcal{A}=(Q,I,F,\Delta)$. If $w_{1}\in L(\mathcal{A})$ then
$(q_{I},w_{1},\bot)\mathrel{\raisebox{0.0pt}[0.0pt][0.0pt]{ \leavevmode\hbox
to11.78pt{\vbox to2.68pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-1.3381pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{} {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}}{{}}} {{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}
{{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@curveto{0.46236pt}{0.0pt}{0.69353pt}{1.1381pt}{1.15588pt}{1.1381pt}\pgfsys@curveto{1.49062pt}{1.1381pt}{1.77913pt}{0.58272pt}{2.0806pt}{0.0pt}\pgfsys@curveto{2.38205pt}{-0.5827pt}{2.67056pt}{-1.1381pt}{3.00531pt}{-1.1381pt}\pgfsys@curveto{3.34004pt}{-1.1381pt}{3.62856pt}{-0.58272pt}{3.93002pt}{0.0pt}\pgfsys@curveto{4.23148pt}{0.5827pt}{4.51999pt}{1.1381pt}{4.85474pt}{1.1381pt}\pgfsys@curveto{5.18947pt}{1.1381pt}{5.47798pt}{0.58272pt}{5.77945pt}{0.0pt}\pgfsys@curveto{6.0809pt}{-0.5827pt}{6.36942pt}{-1.1381pt}{6.70416pt}{-1.1381pt}\pgfsys@lineto{6.70416pt}{0.0pt}\pgfsys@lineto{9.38113pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.38113pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}^{*}(q_{F},\varepsilon,\bot)$
for some $q_{I}\in I$ and $q_{F}\in F$. Since
$w_{1}\leqslant_{\mathcal{A}}w_{2}$, we also have that
$(q_{I},w_{1},\bot)\mathrel{\raisebox{0.0pt}[0.0pt][0.0pt]{ \leavevmode\hbox
to11.78pt{\vbox to2.68pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-1.3381pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{} {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}}{{}}} {{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}
{{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@curveto{0.46236pt}{0.0pt}{0.69353pt}{1.1381pt}{1.15588pt}{1.1381pt}\pgfsys@curveto{1.49062pt}{1.1381pt}{1.77913pt}{0.58272pt}{2.0806pt}{0.0pt}\pgfsys@curveto{2.38205pt}{-0.5827pt}{2.67056pt}{-1.1381pt}{3.00531pt}{-1.1381pt}\pgfsys@curveto{3.34004pt}{-1.1381pt}{3.62856pt}{-0.58272pt}{3.93002pt}{0.0pt}\pgfsys@curveto{4.23148pt}{0.5827pt}{4.51999pt}{1.1381pt}{4.85474pt}{1.1381pt}\pgfsys@curveto{5.18947pt}{1.1381pt}{5.47798pt}{0.58272pt}{5.77945pt}{0.0pt}\pgfsys@curveto{6.0809pt}{-0.5827pt}{6.36942pt}{-1.1381pt}{6.70416pt}{-1.1381pt}\pgfsys@lineto{6.70416pt}{0.0pt}\pgfsys@lineto{9.38113pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.38113pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}^{*}(q_{F},\varepsilon,\bot)$
implying that $w_{2}\in L(\mathcal{A})$.

#### Lemma 4.10

*Statement. Let $\mathcal{A}=(Q,I,F,\Delta)$ be an OPA, and let $\Gamma=\Sigma\times Q$. Considering $\vec{U}^{a,b,c}_{s,t}=\varepsilon$ when $a=b$ and $s=t$, otherwise $\vec{U}^{a,b,c}_{s,t}=\varnothing$. The following holds for all $n>0$:

$\texttt{Cat}^{n}(\vec{U}^{a,b,c}_{s,t}){=}\big{\\{}u\mid(s,uc,\alpha){\mathrel{\raisebox{0.0pt}[0.0pt][0.0pt]{
\leavevmode\hbox to11.78pt{\vbox to2.68pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-1.3381pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{} {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}}{{}}} {{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}
{{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@curveto{0.46236pt}{0.0pt}{0.69353pt}{1.1381pt}{1.15588pt}{1.1381pt}\pgfsys@curveto{1.49062pt}{1.1381pt}{1.77913pt}{0.58272pt}{2.0806pt}{0.0pt}\pgfsys@curveto{2.38205pt}{-0.5827pt}{2.67056pt}{-1.1381pt}{3.00531pt}{-1.1381pt}\pgfsys@curveto{3.34004pt}{-1.1381pt}{3.62856pt}{-0.58272pt}{3.93002pt}{0.0pt}\pgfsys@curveto{4.23148pt}{0.5827pt}{4.51999pt}{1.1381pt}{4.85474pt}{1.1381pt}\pgfsys@curveto{5.18947pt}{1.1381pt}{5.47798pt}{0.58272pt}{5.77945pt}{0.0pt}\pgfsys@curveto{6.0809pt}{-0.5827pt}{6.36942pt}{-1.1381pt}{6.70416pt}{-1.1381pt}\pgfsys@lineto{6.70416pt}{0.0pt}\pgfsys@lineto{9.38113pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.38113pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}^{*}}(t,c,\beta),|u|=n,\alpha\in\varTheta_{a},\beta\in\varTheta_{b},au\in\widehat{\Sigma}^{*}_{\mathrel{\mathchoice{\makebox[5.60002pt]{{\makebox[0.0pt]{$\leq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\lessdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{$\leq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\lessdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\scriptsize$\leq$}}{\makebox[0.0pt]{\raisebox{0.8pt}{\scriptsize$\lessdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\tiny$\leq$}}{\makebox[0.0pt]{\raisebox{0.6pt}{\tiny$\lessdot$}}}}}}},uc\in\widehat{\Sigma}^{*}_{\mathrel{\mathchoice{\makebox[5.60002pt]{{\makebox[0.0pt]{$\geq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\gtrdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{$\geq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\gtrdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\scriptsize$\geq$}}{\makebox[0.0pt]{\raisebox{0.8pt}{\scriptsize$\gtrdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\tiny$\geq$}}{\makebox[0.0pt]{\raisebox{0.6pt}{\tiny$\gtrdot$}}}}}}},u^{\triangleright}=b\big{\\}}$

where, for all $a\in\widehat{\Sigma}$, the set of stack symbols
$\varTheta_{a}\subseteq\Gamma\cup\\{\bot\\}$ is defined by
$\varTheta_{a}=\\{\bot\\}$ if $a=\varepsilon$, and $\varTheta_{a}=\\{\langle
a,q\rangle\mid q\in Q\\}$ otherwise.

######  Proof A.8.

For readability, we define the set of runs $\varOmega^{a,b,c}_{s,t,n}(w)$, for
all $a,b,c\in\widehat{\Sigma}$, $s,t\in Q$, $n\in\mathbb{N}$ and
$w\in\widehat{\Sigma}^{*}$ as follows.

$\varOmega^{a,b,c}_{s,t,m}(w)=\left\\{\begin{array}[]{c|c}\rho&\bigwedge\begin{array}[]{l}\rho=(s,wc,\alpha)\mathrel{\raisebox{0.0pt}[0.0pt][0.0pt]{
\leavevmode\hbox to11.78pt{\vbox to2.68pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-1.3381pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{} {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}}{{}}} {{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}
{{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@curveto{0.46236pt}{0.0pt}{0.69353pt}{1.1381pt}{1.15588pt}{1.1381pt}\pgfsys@curveto{1.49062pt}{1.1381pt}{1.77913pt}{0.58272pt}{2.0806pt}{0.0pt}\pgfsys@curveto{2.38205pt}{-0.5827pt}{2.67056pt}{-1.1381pt}{3.00531pt}{-1.1381pt}\pgfsys@curveto{3.34004pt}{-1.1381pt}{3.62856pt}{-0.58272pt}{3.93002pt}{0.0pt}\pgfsys@curveto{4.23148pt}{0.5827pt}{4.51999pt}{1.1381pt}{4.85474pt}{1.1381pt}\pgfsys@curveto{5.18947pt}{1.1381pt}{5.47798pt}{0.58272pt}{5.77945pt}{0.0pt}\pgfsys@curveto{6.0809pt}{-0.5827pt}{6.36942pt}{-1.1381pt}{6.70416pt}{-1.1381pt}\pgfsys@lineto{6.70416pt}{0.0pt}\pgfsys@lineto{9.38113pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.38113pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}^{m}(t,c,\beta)\land
m\leq 2|u|\\\ \alpha\in\varTheta_{a}\land\beta\in\varTheta_{b}\\\
aw\in\widehat{\Sigma}^{*}_{\mathrel{\mathchoice{\makebox[5.60002pt]{{\makebox[0.0pt]{$\leq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\lessdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{$\leq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\lessdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\scriptsize$\leq$}}{\makebox[0.0pt]{\raisebox{0.8pt}{\scriptsize$\lessdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\tiny$\leq$}}{\makebox[0.0pt]{\raisebox{0.6pt}{\tiny$\lessdot$}}}}}}}\land
wc\in\widehat{\Sigma}^{*}_{\mathrel{\mathchoice{\makebox[5.60002pt]{{\makebox[0.0pt]{$\geq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\gtrdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{$\geq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\gtrdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\scriptsize$\geq$}}{\makebox[0.0pt]{\raisebox{0.8pt}{\scriptsize$\gtrdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\tiny$\geq$}}{\makebox[0.0pt]{\raisebox{0.6pt}{\tiny$\gtrdot$}}}}}}}\land(aw)^{\triangleright}=b\end{array}\end{array}\right\\}$

We reformulate the statement as $u\in\texttt{Cat}^{n}(\vec{U}^{a,b,c}_{s,t})$
if and only if $\varOmega^{a,b,c}_{s,t,2n}(u)\neq\varnothing$. For all
$u\in\texttt{Cat}^{n}(\vec{U}^{a,b,c}_{s,t})$, it takes a simple induction on
$|u|$ to prove that $\varOmega^{a,b,c}_{s,t,2n}(u)\neq\varnothing$, because
Cat follows runs of $\mathcal{A}$ and preserves the invariants $m\leq 2|u|$,
$au\in\widehat{\Sigma}^{*}_{\mathrel{\mathchoice{\makebox[5.60002pt]{{\makebox[0.0pt]{$\leq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\lessdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{$\leq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\lessdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\scriptsize$\leq$}}{\makebox[0.0pt]{\raisebox{0.8pt}{\scriptsize$\lessdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\tiny$\leq$}}{\makebox[0.0pt]{\raisebox{0.6pt}{\tiny$\lessdot$}}}}}}}$,
$uc\in\widehat{\Sigma}^{*}_{\mathrel{\mathchoice{\makebox[5.60002pt]{{\makebox[0.0pt]{$\geq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\gtrdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{$\geq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\gtrdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\scriptsize$\geq$}}{\makebox[0.0pt]{\raisebox{0.8pt}{\scriptsize$\gtrdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\tiny$\geq$}}{\makebox[0.0pt]{\raisebox{0.6pt}{\tiny$\gtrdot$}}}}}}}$,
$(au)^{\triangleright}=b$ by definition. Next, we prove that
$\varOmega^{a,b,c}_{s,t,2n}(u)\neq\varnothing$ implies
$u\in\texttt{Cat}^{n}(\vec{U}^{a,b,c}_{s,t})$, where $n=|u|$.

The proof goes by induction on the structure of $w=auc$. In the base case, $w$
does not admit any subchains, i.e., $\lambda(w)=w$. If
$\varOmega^{a,b,c}_{s,t,2n}(u)\neq\varnothing$ holds, then,
$au\in\widehat{\Sigma}^{*}_{\mathrel{\mathchoice{\makebox[5.60002pt]{{\makebox[0.0pt]{$\leq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\lessdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{$\leq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\lessdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\scriptsize$\leq$}}{\makebox[0.0pt]{\raisebox{0.8pt}{\scriptsize$\lessdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\tiny$\leq$}}{\makebox[0.0pt]{\raisebox{0.6pt}{\tiny$\lessdot$}}}}}}}$
and
$uc\in\widehat{\Sigma}^{*}_{\mathrel{\mathchoice{\makebox[5.60002pt]{{\makebox[0.0pt]{$\geq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\gtrdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{$\geq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\gtrdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\scriptsize$\geq$}}{\makebox[0.0pt]{\raisebox{0.8pt}{\scriptsize$\gtrdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\tiny$\geq$}}{\makebox[0.0pt]{\raisebox{0.6pt}{\tiny$\gtrdot$}}}}}}}$.
Together with $\lambda(w)=w$, it implies that
$u\in\widehat{\Sigma}^{*}_{\mathrel{\mathchoice{\raisebox{1.0pt}{$\dot{=}$}}{\raisebox{1.0pt}{$\dot{=}$}}{\raisebox{0.8pt}{\scriptsize$\dot{=}$}}{\raisebox{0.6pt}{\tiny$\dot{=}$}}}}$.
Hence, any run witnessing $\varOmega^{a,b,c}_{s,t,2n}(u)\neq\varnothing$
performs exclusively shift-transitions. Actually, since that run performs
exclusively shift-transitions, it length must be $n=|u|$. The base case goes
by induction on such a witnessing run of the length $n$. Having $n=0$ implies
$u=\varepsilon$ and $(s,ua,\alpha)=(t,a,\beta)$. In fact $\vec{U}$ is defined
such that $\varepsilon\in\vec{U}^{a,b,c}_{s,t,a}$ exactly when $a=b$ and
$s=t$. By induction, we assume that there exists a run of
$\varOmega^{a,b,c}_{s,t,2n}(u)$ of length $n=|u|$ of the form
$(s,uc,\alpha)\mathrel{\raisebox{-0.4pt}[0.0pt][0.0pt]{ \leavevmode\hbox
to11.78pt{\vbox to11.17pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{9.38113pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.38113pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.3015pt}{3.533pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\normalsize{\scriptsize$a^{\prime}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}(q,vc,\theta)\mathrel{\raisebox{0.0pt}[0.0pt][0.0pt]{
\leavevmode\hbox to11.78pt{\vbox to2.68pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-1.3381pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{} {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}}{{}}} {{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}
{{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@curveto{0.46236pt}{0.0pt}{0.69353pt}{1.1381pt}{1.15588pt}{1.1381pt}\pgfsys@curveto{1.49062pt}{1.1381pt}{1.77913pt}{0.58272pt}{2.0806pt}{0.0pt}\pgfsys@curveto{2.38205pt}{-0.5827pt}{2.67056pt}{-1.1381pt}{3.00531pt}{-1.1381pt}\pgfsys@curveto{3.34004pt}{-1.1381pt}{3.62856pt}{-0.58272pt}{3.93002pt}{0.0pt}\pgfsys@curveto{4.23148pt}{0.5827pt}{4.51999pt}{1.1381pt}{4.85474pt}{1.1381pt}\pgfsys@curveto{5.18947pt}{1.1381pt}{5.47798pt}{0.58272pt}{5.77945pt}{0.0pt}\pgfsys@curveto{6.0809pt}{-0.5827pt}{6.36942pt}{-1.1381pt}{6.70416pt}{-1.1381pt}\pgfsys@lineto{6.70416pt}{0.0pt}\pgfsys@lineto{9.38113pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.38113pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}^{n-1}(t,c,\beta)$
where $u=a^{\prime}v$ and $a^{\prime}=\theta^{\top}$. In particular
$(q,vc,\theta)\mathrel{\raisebox{0.0pt}[0.0pt][0.0pt]{ \leavevmode\hbox
to11.78pt{\vbox to2.68pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-1.3381pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{} {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}}{{}}} {{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}
{{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@curveto{0.46236pt}{0.0pt}{0.69353pt}{1.1381pt}{1.15588pt}{1.1381pt}\pgfsys@curveto{1.49062pt}{1.1381pt}{1.77913pt}{0.58272pt}{2.0806pt}{0.0pt}\pgfsys@curveto{2.38205pt}{-0.5827pt}{2.67056pt}{-1.1381pt}{3.00531pt}{-1.1381pt}\pgfsys@curveto{3.34004pt}{-1.1381pt}{3.62856pt}{-0.58272pt}{3.93002pt}{0.0pt}\pgfsys@curveto{4.23148pt}{0.5827pt}{4.51999pt}{1.1381pt}{4.85474pt}{1.1381pt}\pgfsys@curveto{5.18947pt}{1.1381pt}{5.47798pt}{0.58272pt}{5.77945pt}{0.0pt}\pgfsys@curveto{6.0809pt}{-0.5827pt}{6.36942pt}{-1.1381pt}{6.70416pt}{-1.1381pt}\pgfsys@lineto{6.70416pt}{0.0pt}\pgfsys@lineto{9.38113pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.38113pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}^{n-1}(t,c,\beta)$
witnesses $\varOmega^{a^{\prime},b,c}_{q,t,2|v|}(v)\neq\varnothing$. So,
$v\in\texttt{Cat}^{|v|}(\vec{U}^{a^{\prime},b,c}_{q,t})$ by induction
hypothesis. Finally, $u\in\texttt{Cat}^{n}(\vec{U}^{a,b,c}_{s,t})$ by
definition of CatShift.

In the inductive step, we assume that $w$ is of the form
$a_{0}u_{0}a_{1}u_{1}\ldots a_{k}u_{k}a_{k+1}$ such that for all $0\leq i\leq
k$, either ${{}^{a_{i}}[u_{i}]^{a_{i+1}}}$ or $u_{i}=\varepsilon$, and
$\lambda(a_{0}a_{1}\dots a_{k+1})=a_{0}a_{1}\dots a_{k+1}$. It is worth
emphasizing that $a_{0}=a$, and $b=(au)^{\triangleright}$ and $a_{k+1}=c$ by
definition of $w$. Also $k>0$, since otherwise $\lambda(w)=w$, which is the
base case of this induction. As in the base case, if
$\varOmega^{a,b,c}_{s,t,2n}(u)\neq\varnothing$ then
$au\in\widehat{\Sigma}^{*}_{\mathrel{\mathchoice{\makebox[5.60002pt]{{\makebox[0.0pt]{$\leq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\lessdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{$\leq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\lessdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\scriptsize$\leq$}}{\makebox[0.0pt]{\raisebox{0.8pt}{\scriptsize$\lessdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\tiny$\leq$}}{\makebox[0.0pt]{\raisebox{0.6pt}{\tiny$\lessdot$}}}}}}}$
and
$uc\in\widehat{\Sigma}^{*}_{\mathrel{\mathchoice{\makebox[5.60002pt]{{\makebox[0.0pt]{$\geq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\gtrdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{$\geq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\gtrdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\scriptsize$\geq$}}{\makebox[0.0pt]{\raisebox{0.8pt}{\scriptsize$\gtrdot$}}}}}{\makebox[5.60002pt]{{\makebox[0.0pt]{\tiny$\geq$}}{\makebox[0.0pt]{\raisebox{0.6pt}{\tiny$\gtrdot$}}}}}}}$,
which implies $a_{0}a_{1}\dots
a_{k+1}\in\widehat{\Sigma}^{*}_{\mathrel{\mathchoice{\raisebox{1.0pt}{$\dot{=}$}}{\raisebox{1.0pt}{$\dot{=}$}}{\raisebox{0.8pt}{\scriptsize$\dot{=}$}}{\raisebox{0.6pt}{\tiny$\dot{=}$}}}}$
since $\lambda(a_{0}a_{1}\dots a_{k+1})=a_{0}a_{1}\dots a_{k+1}$. If there
exists a run $\rho$ witnessing
$\varOmega^{a_{0},(au)^{\triangleright},a_{k+1}}_{s,t,2n}(u)\neq\varnothing$
with $n=|u|$ then, due to $a_{0}a_{1}\dots
a_{k+1}\in\widehat{\Sigma}^{*}_{\mathrel{\mathchoice{\raisebox{1.0pt}{$\dot{=}$}}{\raisebox{1.0pt}{$\dot{=}$}}{\raisebox{0.8pt}{\scriptsize$\dot{=}$}}{\raisebox{0.6pt}{\tiny$\dot{=}$}}}}$,
for all $1\leq i\leq k$, there exists a run $\rho_{i}$ over $u_{i}$ such that
$\rho=\rho_{0}\mathrel{\raisebox{-0.4pt}[0.0pt][0.0pt]{ \leavevmode\hbox
to12.33pt{\vbox to11.34pt{\pgfpicture\makeatletter\hbox{\hskip
0.4725pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{9.38113pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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}\hbox{\normalsize{\scriptsize$a_{1}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\hbox{\normalsize{\scriptsize$a_{2}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>\leavevmode\hbox to12.53pt{\vbox
to11.44pt{\pgfpicture\makeatletter\hbox{\hskip 0.57458pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{9.38113pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.38113pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{2.75842pt}{4.89412pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\normalsize{\scriptsize$a_{k}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\rho_{k}$. Since
${{}^{a_{i}}[u_{i}]^{a_{i+1}}}$ for all $0\leq i\leq k$, each $\rho_{i}$
witnesses
$\varOmega^{a_{i},(a_{i}u_{i})^{\triangleright},a_{i+1}}_{s_{i},t_{i},2n_{i}}(u_{i})\neq\varnothing$
with $n_{i}=|u_{i}|$ and $s_{i},t_{i}\in Q$. However, the induction hypothesis
cannot be apply on
$\varOmega^{a_{i},(a_{i}u_{i})^{\triangleright},a_{i+1}}_{s_{i},t_{i},2n_{i}}(u_{i})\neq\varnothing$,
as it may have the same chain structure as $w$. Hence, we proceed by cases to
prove that
$u_{i}\in\texttt{Cat}^{n_{i}}(\vec{U}^{a_{i},(a_{i}u_{i})^{\triangleright},a_{i+1}}_{s_{i},t_{i}})$.
In the case when $u_{i}=\varepsilon$ the run $\rho_{i}$ witnessing
$\varOmega^{a_{i},(a_{i}u_{i})^{\triangleright},a_{i+1}}_{s_{i},t_{i},2n_{i}}(u_{i})\neq\varnothing$,
or equivalently
$\varOmega^{a_{i},a_{i},a_{i+1}}_{s_{i},t_{i},2n_{i}}(u_{i})\neq\varnothing$
must be empty since
$a_{i}\mathrel{\mathchoice{\raisebox{1.0pt}{$\dot{=}$}}{\raisebox{1.0pt}{$\dot{=}$}}{\raisebox{0.8pt}{\scriptsize$\dot{=}$}}{\raisebox{0.6pt}{\tiny$\dot{=}$}}}a_{i+1}$.
This implies that $s_{i}=t_{i}$. In fact, we trivially have that
$\varepsilon\in\vec{U}^{a_{i},a_{i},a_{i+1}}_{s_{i},s_{i},2n_{i}}(u_{i})$ by
definition of $\vec{U}$. Equivalently,
$u_{i}\in\texttt{Cat}^{n_{i}}\vec{U}^{a_{i},(a_{i}u_{i})^{\triangleright},a_{i+1}}_{s_{i},t_{i},2n_{i}}(u_{i})\neq\varnothing$.
Otherwise $u_{i}\neq\varepsilon$. Let $b^{\prime}_{i}=u_{i}^{\triangleleft}$,
$b_{i}=u_{i}^{\triangleright}$, and $u^{\prime}_{i}$ be such that
$u_{i}=b^{\prime}_{i}u^{\prime}_{i}$. In this cases, the run $\rho_{i}$
witnessing
$\varOmega^{a_{i},(a_{i}u_{i})^{\triangleright},a_{i+1}}_{s_{i},t_{i},2n_{i}}(u_{i})\neq\varnothing$
must starts with a push-transition on $b^{\prime}_{i}$ since
$a_{i}\mathrel{\mathchoice{\raisebox{1.0pt}{$\lessdot$}}{\raisebox{1.0pt}{$\lessdot$}}{\raisebox{0.8pt}{\scriptsize$\lessdot$}}{\raisebox{0.6pt}{\tiny$\lessdot$}}}b^{\prime}_{i}$,
and must ends with a pop-transition on $b_{i}$ since
$b_{i}\mathrel{\mathchoice{\raisebox{1.0pt}{$\gtrdot$}}{\raisebox{1.0pt}{$\gtrdot$}}{\raisebox{0.8pt}{\scriptsize$\gtrdot$}}{\raisebox{0.6pt}{\tiny$\gtrdot$}}}a_{i+1}$.
In other words, there exists some run $\rho^{\prime}_{i}$ witnessing
$\varOmega^{b^{\prime}_{i},b_{i},a_{i+1}}_{s^{\prime}_{i},t^{\prime}_{i},2n^{\prime}_{i}}(u^{\prime}_{i})\neq\varnothing$
where $|u^{\prime}_{i}|=n^{\prime}_{i}$ and $s^{\prime}_{i},t^{\prime}_{i}\in
Q$. Now, we can apply the induction hypothesis on
$w^{\prime}=b^{\prime}_{i}u^{\prime}_{i}a_{i+1}$ which is a strict subchain of
$w$. Hence,
$u^{\prime}_{i}\in\texttt{Cat}^{n^{\prime}_{i}}(\vec{U}^{b^{\prime}_{i},b_{i},a_{i+1}}_{s^{\prime}_{i},t^{\prime}_{i}})$.
The rest of the proof is straightforward. For all $1\leq i\leq k$, we get
$u_{i}\in\texttt{Cat}^{n_{i}}\vec{U}^{a_{i},(a_{i}u_{i})^{\triangleright},a_{i+1}}_{s_{i},t_{i},2n_{i}}(u_{i})\neq\varnothing$
by definition of CatChain and since ${{}^{a_{i}}[u_{i}]^{a_{i+1}}}$. Finally,
we can prove that $u\in\texttt{Cat}^{n}(\vec{U}^{a,b,c}_{s,t})$ by induction
on $k$, by definition of CatShift and since $a_{0}a_{1}\dots
a_{k+1}\in\widehat{\Sigma}^{*}_{\mathrel{\mathchoice{\raisebox{1.0pt}{$\dot{=}$}}{\raisebox{1.0pt}{$\dot{=}$}}{\raisebox{0.8pt}{\scriptsize$\dot{=}$}}{\raisebox{0.6pt}{\tiny$\dot{=}$}}}}$.

#### Lemma 4.12

*Statement. Let $\preccurlyeq$ be a chain-monotonic quasi-order over $\widehat{\Sigma}^{*}$. For every $A=(Q,I,F,\Delta)$ and $\vec{X},\vec{Y}$ such that $\mathfrak{B}(\vec{X}^{a,b,c}_{s,t}\preccurlyeq\vec{Y}^{a,b,c}_{s,t})$ holds for all $s,t\in Q$ and all $a,b,c\in\Sigma\cup\\{\varepsilon\\}$, we have $\mathfrak{B}(\texttt{Cat}(\vec{X}^{a,b,c}_{s,t})\preccurlyeq\texttt{Cat}(\vec{Y}^{a,b,c}_{s,t}))$ holds also for all $s,t\in Q$ and all $a,b,c\in\Sigma\cup\\{\varepsilon\\}$.

######  Proof A.9.

Assume that
$\mathfrak{B}(\vec{X}^{a,b,c}_{s,t}\preccurlyeq\vec{Y}^{a,b,c}_{s,t})$ holds
for all $s,t\in Q$ and all $a,b,c\in\Sigma\cup\\{\varepsilon\\}$. In
particular, for all $y_{0}\in\vec{Y}^{a,b,c}_{s,t}$, there exists
$x_{0}\in\vec{X}^{a,b,c}_{s,t}$ such that $x_{0}\preccurlyeq y_{0}$. Consider
$y\in\texttt{Cat}(\vec{Y}^{a,b,c}_{s,t})$, we show that there exists
$x\in\texttt{Cat}(\vec{X}^{a,b,c}_{s,t})$ such that $x\preccurlyeq y$. By
definition of Cat, there are three cases: Either (1)
$y\in\vec{Y}^{a,b,c}_{s,t}$ or, (2)
$y\in\texttt{CatShift}(\vec{Y}^{a,b,c}_{s,t})$, or (3)
$y\in\texttt{CatChain}(\vec{Y}^{a,b,c}_{s,t})$. We show (2) since (3) can be
prove similarly and (1) is trivial from
$\mathfrak{B}(\vec{X}^{a,b,c}_{s,t}\preccurlyeq\vec{Y}^{a,b,c}_{s,t})$.
Suppose that $y$ is of the form $y_{1}b^{\prime}y_{2}$ for some
$y_{1}\in\vec{Y}^{a,a^{\prime},b^{\prime}}_{s,s^{\prime}}$,
$b^{\prime}\in\Sigma$, and $y_{2}\in\vec{Y}^{b^{\prime},b,c}_{t^{\prime},t}$.
By hypothesis, there exist
$x_{1}\in\vec{X}^{a,a^{\prime},b^{\prime}}_{s,s^{\prime}}$ and
$x_{2}\in\vec{X}^{b^{\prime},b,c}_{t^{\prime},t}$ such that $x_{1}\preccurlyeq
y_{1}$ and $x_{2}\preccurlyeq y_{2}$. If $y_{1}=\varepsilon$ then
$x_{1}=\varepsilon$ and thus $x_{1}b^{\prime}\preccurlyeq y_{1}b^{\prime}$. If
$y_{1}\neq\varepsilon$ then
$y_{1}^{\triangleright}=a^{\prime}\mathrel{\mathchoice{\makebox[8.00003pt]{{\makebox[0.0pt]{$\leq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\lessdot$}}}}}{\makebox[8.00003pt]{{\makebox[0.0pt]{$\leq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\lessdot$}}}}}{\makebox[8.00003pt]{{\makebox[0.0pt]{\scriptsize$\leq$}}{\makebox[0.0pt]{\raisebox{0.8pt}{\scriptsize$\lessdot$}}}}}{\makebox[8.00003pt]{{\makebox[0.0pt]{\tiny$\leq$}}{\makebox[0.0pt]{\raisebox{0.6pt}{\tiny$\lessdot$}}}}}}b^{\prime}$
and
$x_{1}^{\triangleright}\mathrel{\mathchoice{\makebox[8.00003pt]{{\makebox[0.0pt]{$\leq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\lessdot$}}}}}{\makebox[8.00003pt]{{\makebox[0.0pt]{$\leq$}}{\makebox[0.0pt]{\raisebox{1.0pt}{$\lessdot$}}}}}{\makebox[8.00003pt]{{\makebox[0.0pt]{\scriptsize$\leq$}}{\makebox[0.0pt]{\raisebox{0.8pt}{\scriptsize$\lessdot$}}}}}{\makebox[8.00003pt]{{\makebox[0.0pt]{\tiny$\leq$}}{\makebox[0.0pt]{\raisebox{0.6pt}{\tiny$\lessdot$}}}}}}b^{\prime}$
since $y_{1}\approx x_{1}$. We have that
${{}^{\varepsilon}[x_{1}b^{\prime}]^{\varepsilon}}$ and
${{}^{\varepsilon}[y_{1}b^{\prime}]^{\varepsilon}}$. So,
$x_{1}b^{\prime}\preccurlyeq y_{1}b^{\prime}$. We have that
${{}^{\varepsilon}[x_{1}b^{\prime}y_{2}]^{\varepsilon}}$ and
${{}^{\varepsilon}[y_{1}b^{\prime}y_{2}]^{\varepsilon}}$. So,
$x_{1}b^{\prime}y_{2}\preccurlyeq y_{1}b^{\prime}y_{2}$. If
$b^{\prime}\mathrel{\mathchoice{\raisebox{1.0pt}{$\lessdot$}}{\raisebox{1.0pt}{$\lessdot$}}{\raisebox{0.8pt}{\scriptsize$\lessdot$}}{\raisebox{0.6pt}{\tiny$\lessdot$}}}y_{2}$
then
$b^{\prime}\mathrel{\mathchoice{\raisebox{1.0pt}{$\lessdot$}}{\raisebox{1.0pt}{$\lessdot$}}{\raisebox{0.8pt}{\scriptsize$\lessdot$}}{\raisebox{0.6pt}{\tiny$\lessdot$}}}x_{2}$.
We have that ${{}^{x_{1}b^{\prime}}[x_{2}]^{\varepsilon}}$ and
${{}^{x_{1}b^{\prime}}[y_{2}]^{\varepsilon}}$. So,
$x_{1}b^{\prime}x_{2}\preccurlyeq x_{1}b^{\prime}y_{2}$. By transitivity,
$x_{1}b^{\prime}x_{1}\preccurlyeq x_{1}b^{\prime}y_{2}$. If
$b^{\prime}\mathrel{\mathchoice{\raisebox{1.0pt}{$\dot{=}$}}{\raisebox{1.0pt}{$\dot{=}$}}{\raisebox{0.8pt}{\scriptsize$\dot{=}$}}{\raisebox{0.6pt}{\tiny$\dot{=}$}}}y_{2}$
then
$b^{\prime}\mathrel{\mathchoice{\raisebox{1.0pt}{$\dot{=}$}}{\raisebox{1.0pt}{$\dot{=}$}}{\raisebox{0.8pt}{\scriptsize$\dot{=}$}}{\raisebox{0.6pt}{\tiny$\dot{=}$}}}x_{2}$.
We have that ${{}^{\varepsilon}[x_{1}b^{\prime}x_{2}]^{\varepsilon}}$ and
${{}^{\varepsilon}[x_{1}b^{\prime}y_{2}]^{\varepsilon}}$. So,
$x_{1}b^{\prime}x_{2}\preccurlyeq x_{1}b^{\prime}y_{2}$. By transitivity,
$x_{1}b^{\prime}x_{1}\preccurlyeq y_{1}b^{\prime}y_{2}$.

#### Theorem 4.13

*Statement. The algorithm from Figure 7 terminates and decides language inclusion.

######  Proof A.10.

First, we show that the inclusion algorithm from Figure 7 always terminates.
From the definition of Cat and the constant $\vec{U}$, we have that each
component of $\vec{X}$ holds a finite set of words after executing finitely
many instructions. The halting conditions of the repeat/until loop is
effectively computable. Indeed, deciding $\mathfrak{B}(X\preccurlyeq Y)$ where
$X$, $Y$ are finite sets and $\preccurlyeq$ a decidable quasi-order, can be
done by checking whether $X\subseteq Y$ and that for every $y\in Y$ there
exists $x\in X$ such that $x\preccurlyeq y$. Additionally, the quasi-order
$\preccurlyeq$ is a well-quasiorders and thus, there is no infinite sequence
$\\{X_{i}\\}_{i\in\mathbb{N}}$ such that
${{}_{\preccurlyeq}}{\upharpoonleft}X_{1}\subsetneq{{}_{\preccurlyeq}}{\upharpoonleft}X_{2}\subsetneq\dots$.
Since $\mathfrak{B}(X\preccurlyeq Y)$ is defined by $X\subseteq
Y\land{{}_{\preccurlyeq}}{\upharpoonleft}X={{}_{\preccurlyeq}}{\upharpoonleft}Y$
and since Cat only extends the upward closures of the components of $\vec{X}$,
we find that the repeat/until loop must terminate after finitely many
iterations.

Now, we show that the inclusion algorithm from Figure 7 is correct. Supposing
that the algorithm returns ko. Once $\vec{X}$ reached the fixpoint computed by
the repeat/until loop, all
$w\in\vec{X}^{\varepsilon,\varepsilon,\varepsilon}_{q_{I},q_{F}}$ where
$q_{I}\in I$ and $q_{F}\in F$ belong to $L_{1}$ by Lemma 4.10. Hence, when the
algorithm returns ko, then $L_{1}\not\subseteq L_{2}$. Conversely, supposing
that $L_{1}\not\subseteq L_{2}$, in particular let $w\in L_{2}\setminus
L_{1}$. In fact, $w$ belongs to
$\texttt{Cat}^{n}(\vec{U}_{q_{I},q_{F}}^{\varepsilon,\varepsilon,\varepsilon})$
for some $q_{I}\in I$, $q_{F}\in F$ and $n\in\mathbb{N}$, by Lemma 4.10.
Additionally, observe that once $\vec{X}$ reached the fixpoint computed by the
repeat/until loop,
$\vec{X}_{q_{I},q_{F}}^{\varepsilon,\varepsilon,\varepsilon}$ is a base of
$\texttt{Cat}^{n}(\vec{U}_{q_{I},q_{F}}^{\varepsilon,\varepsilon,\varepsilon})$,
i.e.
$\mathfrak{B}(\vec{X}_{q_{I},q_{F}}^{\varepsilon,\varepsilon,\varepsilon}\preccurlyeq\texttt{Cat}^{n}(\vec{U}_{q_{I},q_{F}}^{\varepsilon,\varepsilon,\varepsilon}))$.
This can be proved by induction thanks to Lemma 4.12. Hence, there exists
$w_{0}\in\vec{X}_{q_{I},q_{F}}^{\varepsilon,\varepsilon,\varepsilon}$ such
that $w_{0}\preccurlyeq w$. Since $\preccurlyeq$ satisfies that
$(w_{0}\preccurlyeq w\land w_{0}\in L_{2})\implies w\in L_{2}$, we get
$w_{0}\notin L_{2}$ and thus the algorithm returns ko.