Coalgebraic Infinite Traces and Kleisli Simulations
Natsuki Urabe
Ichiro Hasuo
%Natsuki Urabe %
%Ichiro Hasuo %
% 4th ver. Kleisli simulation is a categorical notion introduced by Hasuo to verify finite trace inclusion. They allow us to give definitions of forward and backward simulation for various types of systems. A generic categorical theory behind Kleisli simulation has been developed and it guarantees the soundness of those simulations wrt.\ finite trace semantics. Moreover, those simulations can be aided by forward partial execution (FPE)---a categorical transformation of systems previously introduced by the authors. In this paper, we give Kleisli simulation a theoretical foundation that assures its soundness also wrt.\ infinite trace. %show that Kleisli simulation can be used also to verify infinite trace inclusion. There, following Jacobs' work, infinite trace semantics is characterized as the ``largest homomorphism.'' It turns out that soundness of forward simulations is rather straightforward; that of backward simulation holds too, although it requires certain additional conditions and its proof is more involved. % It is proved using categorical characterization of the infinite trace introduced by Jacobs intensively for nondeterministic systems. % There, infinite trace semantics is captured by the ``largest homomorphism''. %to the ``lifted'' final coalgebra. % For a nondeterministic system that outputs trees, we show that their automata-theoretic infinite language % can be captured with this framework. %by the largest homomorphism. %Then we show the soundness of Kleisli simulation wrt.\ finite trace. %the inclusion between the coalgebraic infinite trace semantics. We also show that FPE can be successfully employed in the infinite trace setting to enhance the applicability of Kleisli simulations as witnesses of trace inclusion. Our framework is parameterized in the monad for branching as well as in the functor for linear-time behaviors; for the former we use the powerset monad (for nondeterminism) as well as the sub-Giry monad (for probability). %The same process is done for probabilistic systems. % % %We give a way to construction %Soundness of forward Kleisli simulation wrt.\ infinite trace holds while soundness of backward Kleisli simulation fails. % %Their is proved wrt.\ coalgebraically characterized infinite trace semantics, namely, %as the largest homomorphism to the lifted final coalgebra. %The idea to capture infinite trace semantics as the largest homomorphism is introduced by Jacobs for nondeterministic systems. %We %We also show that FPE can aid Kleisli simulation also in infinite setting. %We do the same thing for probabilistic systems. % % % 3rd ver. %Forward and backward Kleisli simulation are categorical notions introduced by Hasuo %that satisfy soundness wrt.\ finite trace: %namely, existence of fwd.\ or bwd.\ Kleisli simulation %imply inclusion between coalgebraically captured finite trace semantics. %They allow us to define a notion of simulation for concrete systems whose soundness is assured by %general theory of category. %In this paper, we show that Kleisli simulation satisfy soundness wrt.\ infinite trace, too. % %Then we % % % 2nd ver. %We use Kleisli simulation to verify infinite trace inclusion. %Kleisli simulation is a categorical notion introduced by Hasuo that allows us to define two types of simulations---forward and backward---for various concrete systems. %By the general theory of coalgebra, the resulting simulations are assured to satisfy soundness wrt.\ finite trace, i.e.\ they witness finite trace inclusion. %inclusion between finite traces of systems. %In this paper, we show that forward Kleisli simulation is also sound wrt.\ infinite trace. %Moreover, we also show that backward Kleisli simulation is sound wrt.\ infinite trace %if we require the simulation to satisfy suitable conditions like ``image finiteness'' or ``nonemptiness''. %These properties are proved for categorical infinite trace semantics introduced by Jacobs where it is captured as the largest homomorphism. %We also show that FPE---transformation of systems introduced by the authors in the past %paper that aids Kleisli simulation to witness finite trace inclusion---aids to witness infinite trace inclusion, too. % % % 1st ver. %Kleisli simulation is a categorical notion introduced by Hasuo. It has two types---forward and backward---and they allow us to define simulations for various concrete systems. It is assured by general theory that derived simulation satisfies soundness wrt.\ finite trace. %Namely, existence of simulation between two systems implies inclusion between finite-length languages of the systems. %In this paper, we claim that Kleisli simulation can be also used to prove infinite trace inclusion. %More concretely, we show that forward Kleisli simulation is sound wrt.\ infinite trace. %Moreover, we also show that backward Kleisli simulation is sound wrt.\ infinite trace if we require the simulation to satisfy suitable conditions like ``image finiteness'' %or ``nonemptiness''. %% satisfy some conditions. %A transformation of systems that aids Kleisli simulation to witness finite trace inclusion was introduced by the authors in the past paper. %We also show that this transformation aids to witness infinite trace inclusion, too.
infiniteTraceExtended.tex
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% Author macros::begin %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title{Coalgebraic Infinite Traces and Kleisli Simulations}
%\titlerunning{} %optional, in case that the title is too long; the running title should fit into the top page column

\author{Natsuki Urabe}
\author{Ichiro Hasuo}

\affil{Dept.\ Computer Science, The University of Tokyo\\ Hongo 7-3-1, Tokyo 113-8656, 
  Japan\\
\texttt{\{urabenatsuki,ichiro\}@is.s.u-tokyo.ac.jp}
  }
%\affil[2]{University of Tokyo\\
%  Japan
%%  \texttt{urabenatsuki@is.s.u-tokyo.ac.jp}
%  }

\authorrunning{N. Urabe and I. Hasuo} %mandatory. First: Use abbreviated first/middle names. Second (only in severe cases): Use first author plus 'et. al.'

\Copyright{Natsuki Urabe and Ichiro Hasuo} %mandatory, please use full first names. LIPIcs license is "CC-BY";  http://creativecommons.org/licenses/by/3.0/

\subjclass{F.1.1 Models of Computation}% mandatory: Please choose ACM 1998 classifications from http://www.acm.org/about/class/ccs98-html . E.g., cite as "F.1.1 Models of Computation". 

\keywords{category theory, coalgebra, simulation, verification, trace semantics} % mandatory: Please provide 1-5 keywords
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\begin{document}
%
%\begin{frontmatter}              % start of the contributions
%\mainmatter
%


%
%\titlerunning{Hamiltonian Mechanics}  % abbreviated title (for running head)
%                                     also used for the TOC unless
%                                     \toctitle is used
%
%\author[ut]{Natsuki Urabe\corref{cor1}}
%\ead{urabenatsuki@is.s.u-tokyo.ac.jp}
%
%\author[ut]{Ichiro Hasuo}
%\ead{ichiro@is.s.u-tokyo.ac.jp}
%
%\cortext[cor1]{Corresponding author}
%
%\address[ut]{Department of Computer Science, Graduate School of Information Science and Technology, the University of Tokyo, 
%7-3-1, Hongo, Bunkyo-ku, Tokyo, Japan}

%\author{
%Natsuki Urabe
%\and
%Ichiro Hasuo
%}%\inst{1}}
%%
%\authorrunning{Natsuki Urabe et al.} % abbreviated author list (for running head)
%%
%%%%% list of authors for the TOC (use if author list has to be modified)
%\tocauthor{Natsuki Urabe}
%%
%\institute{University of Tokyo, Japan
%% ,\\
%% \email{urabenatsuki@is.s.u-tokyo.ac.jp}
%}



% 


\maketitle              % typeset the title of the contribution

\begin{abstract}
% 4th ver.
Kleisli simulation is a categorical notion introduced by Hasuo 
to verify finite trace inclusion.
They allow us to give  definitions of \emph{forward and backward simulation} for various types of systems.
A generic categorical theory behind Kleisli simulation has been
 developed and it guarantees the
soundness of those simulations wrt.\ \emph{finite} trace semantics.
Moreover, those simulations can be aided by \emph{forward partial execution} (FPE)---a 
categorical transformation of systems previously introduced by the authors.

In this paper, we give Kleisli simulation a theoretical foundation that assures 
its soundness also wrt.\ \emph{infinite} trace.
%show that Kleisli simulation can be used also to verify \emph{infinite} trace inclusion.
There, following Jacobs' work, infinite trace semantics is characterized
 as the ``largest homomorphism.'' It turns out that soundness of forward
 simulations is rather straightforward; that of backward simulation
 holds too, although it requires certain additional conditions and its 
proof is more involved.
% It is proved using categorical characterization of the infinite trace introduced by Jacobs intensively for nondeterministic systems.
% There, infinite trace semantics is captured by the ``largest homomorphism''. %to the ``lifted'' final coalgebra.
% For a nondeterministic system that outputs trees, we show that their automata-theoretic infinite language
% can be captured with this framework. %by the largest homomorphism.
%Then we show the soundness of Kleisli simulation wrt.\ finite trace.
%the inclusion between the coalgebraic infinite trace semantics.
We also show that FPE can be successfully employed in the infinite trace
 setting
to enhance the applicability of Kleisli simulations as witnesses of
 trace inclusion.
Our framework is parameterized in the monad for branching as well as in
 the functor for linear-time behaviors; for the former we use the
 powerset monad (for nondeterminism) as well as the sub-Giry monad
(for probability).
%The same process is done for probabilistic systems. 
%
%
%We give a way to construction 
%Soundness of forward Kleisli simulation wrt.\ infinite trace holds while soundness of backward Kleisli simulation fails.
%
%Their  is proved wrt.\ coalgebraically characterized infinite trace semantics, namely, 
%as the largest homomorphism to the lifted final coalgebra.
%The idea to capture infinite trace semantics as the largest homomorphism is introduced by Jacobs for nondeterministic systems.
%We 
%We also show that FPE can aid Kleisli simulation also in infinite setting.
%We do the same thing for probabilistic systems.
%
%
% 3rd ver.
%Forward and backward Kleisli simulation are categorical notions introduced by Hasuo 
%that satisfy soundness wrt.\ \emph{finite} trace: 
%namely, existence of fwd.\ or bwd.\ Kleisli simulation 
%imply inclusion between coalgebraically captured finite trace semantics.
%They allow us to define a notion of simulation for concrete systems whose soundness is assured by 
%general theory of category.
%In this paper, we show that Kleisli simulation satisfy soundness wrt.\ \emph{infinite} trace, too.
%
%Then we 
%
%
% 2nd ver.
%We use Kleisli simulation to verify infinite trace inclusion.
%Kleisli simulation is a categorical notion introduced by Hasuo that allows us to define two types of simulations---forward and backward---for various concrete systems.
%By the general theory of coalgebra, the resulting simulations are assured to satisfy soundness wrt.\ finite trace, i.e.\ they witness finite trace inclusion.
%inclusion between finite traces of systems.
%In this paper, we show that forward Kleisli simulation is also sound wrt.\ infinite trace.
%Moreover, we also show that backward Kleisli simulation is sound wrt.\ infinite trace 
%if we require the simulation to satisfy suitable conditions like ``image finiteness'' or ``nonemptiness''.
%These properties are proved for categorical infinite trace semantics introduced by Jacobs where it is captured as the largest homomorphism.
%We also show that FPE---transformation of systems introduced by the authors in the past 
%paper that aids Kleisli simulation to witness finite trace inclusion---aids to witness infinite trace inclusion, too.
%
%
% 1st ver.
%Kleisli simulation is a categorical notion introduced by Hasuo. It has two types---forward and backward---and they allow us to define simulations for various concrete systems. It is assured by general theory that derived simulation satisfies soundness wrt.\ finite trace. 
%Namely, existence of simulation between two systems implies inclusion between finite-length languages of the systems.
%In this paper, we claim that Kleisli simulation can be also used to prove infinite trace inclusion.
%More concretely, we show that forward Kleisli simulation is sound wrt.\ infinite trace.
%Moreover, we also show that backward Kleisli simulation is sound wrt.\ infinite trace if we require the simulation to satisfy suitable conditions like ``image finiteness'' 
%or ``nonemptiness''.
%% satisfy some conditions.
%A transformation of systems that aids Kleisli simulation to witness finite trace inclusion was introduced by the authors in the past paper.
%We also show that this transformation aids to witness infinite trace inclusion, too.
\end{abstract}

%\end{frontmatter}

%\paragraph*{To do}
%\begin{itemize}
%\item Change letters for ordinals into $\mathfrak{a,b,c,\ldots}$.
%\end{itemize}

\section{Introduction}
\emph{Language inclusion} of transition systems is an important problem 
in both qualitative and quantitative verification.
In a qualitative setting the problem is concretely as follows:
for given two nondeterministic systems $\mathcal{X}$ and $\mathcal{Y}$,
check if $\lang(\mathcal{X})\subseteq\lang(\mathcal{Y})$---that is, if the set of words 
generated by $\mathcal{X}$ is included in the set of words generated by $\mathcal{Y}$.
In a typical usage scenario, $\mathcal{X}$ is a model of the \emph{implementation} in question while
$\mathcal{Y}$ is a model that represents the \emph{specification} of $\mathcal{X}$.
More concretely, $\mathcal{Y}$ is a system such that
$\lang(\mathcal{Y})$ is easily seen not to contain anything ``dangerous''---therefore the language inclusion $\lang(\mathcal{X})\subseteq\lang(\mathcal{Y})$ immediately implies that
$\lang(\mathcal{X})$ contains no dangerous output, either.
Such a situation can also arise in a \emph{quantitative setting} where 
a specification is about \emph{probability}, \emph{reward}, and so on.
%it is aimed to verify that the system satisfy specifications about
%\emph{probability} or \emph{rewards}.

\begin{myexample}\label{ex:firstExample}
%----------------------------------------------------------------
\begin{figure}
 \begin{tabular}{cccc}
\begin{xy}
(-5,8)*{\mathcal{X}}="",
(0,0)*{\circ} = "x",
(8,0)*{\circ} = "y",
(2,4)*{\circ} = "z",
(14,0)*{\checkmark} = "xc",
\ar _{} (-6,0);"x"
\ar ^{a} "x";"z"
\ar _{b} "x";"y"
\ar _{} "y";"xc"
%\ar @(ur,ul)_{a} "x";"x"
\ar @(ur,ul)_{b} "z";"z"
\end{xy}
&
\begin{xy}
(-5,8)*{\mathcal{Y}}="",
(0,0)*{\circ} = "x",
(8,0)*{\circ} = "y",
%(8,8)*{\circ} = "z",
(14,0)*{\checkmark} = "xc",
\ar _{} (-6,0);"x"
%\ar _{a} "x";"z"
\ar _{b} "x";"y"
\ar _{} "y";"xc"
\ar @(ur,ul)_{a} "x";"x"
\end{xy}
&
\begin{xy}
(-10,8)*{\mathcal{Z}}="",
(0,0)*{\circ} = "x",
(10,0)*{\circ} = "y",
(5,10)*{\circ} = "z",
(18,0)*{\checkmark} = "xc",
\ar _{} (-6,0);"x"
\ar ^(.3){a,\frac{2}{3}} "x";"z"
\ar _{b,\frac{1}{3}} "x";"y"
\ar _{\frac{1}{2}} "y";"xc"
%\ar @(ur,ul)_{a} "x";"x"
\ar @(ur,ul)_(.2){a,\frac{1}{2}} "y";"y"
\ar @(dr,ur)_{b,\frac{1}{2}} "z";"z"
\ar @(dl,ul)^{c,\frac{1}{2}} "z";"z"
\end{xy}
&
\begin{xy}
(-5,8)*{\mathcal{W}}="",
(0,0)*{\circ} = "x",
(10,0)*{\circ} = "y",
%(8,8)*{\circ} = "z",
(20,0)*{\checkmark} = "xc",
\ar _{} (-6,0);"x"
%\ar _{a} "x";"z"
\ar ^{b,1} "x";"y"
\ar ^{\frac{1}{2}} "y";"xc"
%\ar @(ur,ul)_{a} "x";"x"
\ar @(ur,ul)_{a,\frac{1}{2}} "y";"y"
\end{xy}
\end{tabular}
\caption{Examples of nondeterministic and probabilistic automata}
\label{fig:exampleautomataintro}
\end{figure}
In Fig.~\ref{fig:exampleautomataintro} are four examples of transition
 systems; $\mathcal{X}$ and $\mathcal{Y}$  are
 qualitative/nondeterministic; $\mathcal{Z}$ and $\mathcal{W}$ exhibit
 probabilistic
 branching. We shall denote the \emph{finite} language of a system
 $\mathcal{A}$ 
 by $\langfin(\mathcal{A})$ and the \emph{infinite} one by
 $\langinf(\mathcal{A})$. We define that a generated finite word is one
 with a run that ends with the termination symbol $\checkmark$.

 In the nondeterministic setting, languages are \emph{sets} of words. We
 have
 $\langfin(\mathcal{X})=\{b\}\subseteq
 \{b,ab,aab,\ldots\}=\langfin(\mathcal{Y})$, i.e.\ \emph{finite}
 language inclusion from $\mathcal{X}$ to $\mathcal{Y}$. However 
 $abb\ldots\in \langinf(\mathcal{X})$ while
 $abb\ldots\not\in \langinf(\mathcal{Y})$, hence \emph{infinite} 
language inclusion fails.

 In the probabilistic setting, languages are naturally \emph{probability
 distributions} over words; and language inclusion refers to the
 pointwise order between probabilities. For example 
$\langfin(\mathcal{Z})=[b\mapsto \frac{1}{6},ba\mapsto \frac{1}{12},baa\mapsto \frac{1}{24},\ldots]$ and
$\langfin(\mathcal{W})=[b\mapsto \frac{1}{2},ba\mapsto
 \frac{1}{4},baa\mapsto \frac{1}{8},\ldots]$; since the former assigns
 no greater probabilities to all the words, we say that the
 finite language of $\mathcal{Z}$ is \emph{included} in that of $\mathcal{W}$.
 This quantitative notion of trace inclusion is also useful in
 verification: it gives e.g.\ an upper bound for the probability for
 something bad.

 Finally, the \emph{infinite} languages for probabilistic systems call
 for measure-theoretic machinery since, in most of the cases, any
 infinite word gets assigned the probability $0$ (which is also the case
 in
 $\mathcal{Z}$ and $\mathcal{W}$). Here it is standard to assign
 probabilities to \emph{cylinder sets} rather than to individual
 words; see e.g.~\cite{baier08principlesof}. An example of a cylinder
 set is $\{aw\mid w\in\{b,c\}^{\omega}\}$. The language
 $\langinf(\mathcal{Z})$ assigns $\frac{2}{3}$ to it, while 
  $\langinf(\mathcal{W})$ assigns $0$; therefore we do not have
 \emph{infinite} language inclusion from $\mathcal{Z}$ to $\mathcal{W}$.
%
% Here are some examples that show the difference between finite and infinite languages.
%
% The nondeterministic automata $\mathcal{X}$ and $\mathcal{Y}$~in
% Fig.~\ref{fig:exampleautomataintro} give qualitative version of such an example.
% If we consider only finite trace, the language of the nondeterministic automaton is 
% a set of finite words collected along finite accepting paths on the automaton.
% Hence finite language of $\mathcal{X}$  is a set $\{b\}$ while that of $\mathcal{Y}$ is $\{b,ab,aab,\ldots\}$.
% Obviously the former set is included in the latter set,
% therefore we can say that finite trace inclusion holds between $\mathcal{X}$ and $\mathcal{Y}$.
%
% In contrast to the finite setting, when we are considering infinite language, infinite words along infinite paths are also collected.
% Therefore, infinite language of $\mathcal{X}$ is $\{b\}\cup\{abb\ldots\}$ while that of $\mathcal{Y}$ is 
% a set $\{b,ab,aab,\ldots\}\cup\{aaa\ldots\}$.
% Differently from the above, it is easy to see that the former is \emph{not} included in the latter.
% Therefore infinite language inclusion between $\mathcal{X}$ and $\mathcal{Y}$ fails.
%
% The probabilistic automata $\mathcal{Z}$ and $\mathcal{W}$~in
% Fig.~\ref{fig:exampleautomataintro} gives an quantitative example.
% The finite language of probabilistic automaton is given as a distribution over a set of finite words.
% For example, the finite language of $\mathcal{Z}$ is a distribution  
% $[b\mapsto \frac{1}{6},ba\mapsto \frac{1}{12},baa\mapsto \frac{1}{24},\ldots]$ while
% that of $\mathcal{W}$ is 
% $[b\mapsto \frac{1}{2},ba\mapsto \frac{1}{4},baa\mapsto \frac{1}{8},\ldots]$.
% The ``inclusion'' between the languages of probabilistic systems is defined by the pointwise comparison,
% and we can see that the former distribution is ``included'' than the latter in this sense.
% Therefore we can say that the finite language inclusion holds.
%
% The idea of infinite language in probabilistic systems is similar to that of nondeterministic systems:
% the probabilities to output infinite words are also witnessed.
% For example, for the automaton $\mathcal{Z}$ in Fig.~\ref{fig:exampleautomataintro},
% we want to the infinite language  to witness that 
% words such as $abbb\ldots$ or $abcbc\ldots$ can be generated.
% However, we can  see that if we use discrete distribution to define the language,
% the probabilities corresponding those words converge to $0$.
% Therefore we use measure-theoretic definition of the infinite language.
% Here, the language is defined as a probability measure 
% over a set of infinite words equipped with a suitable $\sigma$-algebra---the one generated from the \emph{cylinder sets}.
% For example, the infinite language of $\mathcal{Z}$ is given as a probability measure 
% $\lang(\mathcal{Z})$ 
% over the set of finite and infinite words 
% such that 
% $\lang(\mathcal{Z})(\{ba\})=\frac{1}{6}$ or
% $\lang(\mathcal{Z})(\{aw\mid w\in\{b,c\}^{\omega})=\frac{1}{3}$, for example.
% It is easy to see that this is not included in $\lang(\mathcal{W})$ where 
% $\lang(\mathcal{W})(\{aw\mid w\in\{b,c\}^{\omega})=0$
% in the above sense.
% Therefore infinite trace inclusion fails.
\end{myexample}

There are many known algorithms for checking language inclusion. A
well-known one for NFA is a complete one 
that reduces the problem to emptiness check; however it
involves complementation, hence determinization, that
incurs an exponential blowup.

{\par\sloppy
One of the alternative approaches to  language inclusion is by
\emph{simulation}. 
% It is  an alternative to the well-known algorithms, e.g.\
% for NFA, that reduce the problem to emptiness checking: these algorithms involve complementation, hence determinization, that
% results in an exponential blowup.
%Instead of directly checking language inclusion---that can be
%inefficient---
In the simulation-based verification we look for a simulation, that is, a
witness for \emph{stepwise} language inclusion.
The notion of simulation is commonly defined so that it implies
(proper, global) language inclusion---a property  called
\emph{soundness}. Although its converse (\emph{completeness}) fails 
in many settings, such simulation-based approaches tend to have an
advantage
in computational cost.
% It should be noted that the converse (\emph{completeness}) is not imposed and indeed fails in many cases.
One prototype example of such simulation notions is \emph{forward} and \emph{backward simulation}~\cite{lynch95forwardand}, by Lynch and Vaandrager, for nondeterministic automata.
They are shown in~\cite{lynch95forwardand} to satisfy soundness wrt.\
finite trace: explicitly, existence of a forward (or backward) simulation from $\mathcal{X}$ to $\mathcal{Y}$ 
implies $\lang(\mathcal{X})
\subseteq
\lang(\mathcal{Y})$, where the languages collects all the \emph{finite}
words generated.
\par}

% that all finite words that $\mathcal{X}$ outputs are also output by $\mathcal{Y}$.

\emph{Kleisli
simulation}~\cite{hasuo06genericforward,hasuo10genericforward,urabeH14genericforward}
is a categorical generalization of these notions of forward and backward simulation by Lynch and Vaandrager.
It builds upon the use of coalgebras in a \emph{Kleisli category},
in~\cite{hasuo07generictrace}, where they are used to characterize
finite traces.
% for characterizing finite traces.
 % coalgebraic characterization of \emph{finite} trace semantics
 % studied in~\cite{hasuo07generictrace}.
Specifically:
\begin{itemize}
 \item A branching system $\mathcal{X}$ is represented as  an
       \emph{$F$-coalgebra} $c:X\kto FX$ in the Kleisli category $\Kl(T)$, for
       a suitable choice of a functor $F$ and a monad $T$. 
Here $F$ and $T$ are  parameters that determine
the (linear-time) \emph{transition type} and the \emph{branching type},
       respectively, of the  system $\mathcal{X}$. Examples are:
       \begin{itemize}
	\item $F=1+\Sigma\times(\place)$ (terminate, or (output and
	      continue)) and the
\emph{powerset monad} $T=\pow$ on $\Sets$ (nondeterminism), if
	      $\mathcal{X}$ is
	      a nondeterministic automaton (with explicit termination); and
	\item the same functor $F=1+\Sigma\times(\place)$ and the
\emph{sub-Giry monad} 
$T=\giry$~\cite{giry82categoricalapproach} on the category  $\Meas$  of
measurable spaces and measurable functions, for their probabilistic variant.
       \end{itemize}
 \item In~\cite{hasuo07generictrace}, under certain conditions
on $F$ and $T$,  it is shown that a \emph{final
       $F$-coalgebra} in $\Kl(T)$ arises as a lifting of an initial 
       $F$-algebra in $\Sets$. Moreover, it is observed that the natural
       notion of
       ``finite trace semantics'' or ``(finite) languages''  is  captured
       by a unique homomorphism via finality. This works uniformly for 
       a wide variety of systems, by changing $F$ and $T$.
%  from a system $\mathcal{X}$---modelled as a
%        coalgebra $c$ in $\Kl(T)$---to the final coalgebra thus obtained,
%        is obs
% the coalgebraic modeling $c$  of
% a system $\mathcal{X}$, to $J\iota^{-1}$, gives the \emph{finite} trace semantics of $\mathcal{X}$.
% , 
%  the final $F$-coalgebra $J\iota^{-1}:A\kto FA$ in $\Kl(T)$ is obtained
%  as a lifting of
%  the $F$-initial algebra $\iota:FA\to A$ in $\mathbb{C}$.
% %and reversing it.
% Moreover, the unique homomorphism from the coalgebraic modeling $c$  of
% a system $\mathcal{X}$, to $J\iota^{-1}$, gives the \emph{finite} trace semantics of $\mathcal{X}$.
\end{itemize}

% , for a functor $F$ and a monad $T$ on a category $\mathbb{C}$,
% a system $\mathcal{X}$ is categorically represented as an $F$-coalgebra $c:X\kto FX$ in the Kleisli category $\Kl(T)$.
% Here, $F$ and $T$ are  parameters that determine
% the (linear-time) transition type and the branching type, respectively, of the represented system.
% For example, the combination of $F=1+\Sigma\times(\place)$ and the
% \emph{powerset monad} $T=\pow$  on $\Sets$ is used to represent
% nondeterministic $\Sigma$-labeled automata; and another combination of $F=1+\Sigma\times(\place)$ and the \emph{sub-Giry monad} 
% $T=\giry$~\cite{giry82categoricalapproach} on $\Meas$ (the category of
% measurable spaces and measurable functions) is used for
% probabilistic $\Sigma$-labeled automata.

It is shown in~\cite{hasuo06genericforward} that,
with respect to this categorical characterization of finite trace~\cite{hasuo07generictrace},
both forward and backward Kleisli simulation are indeed sound.
This categorical background allows us to instantiate Kleisli simulation
for various concrete systems---including both qualitative and
quantitative ones---and obtain simulation notions whose soundness wrt.\ \emph{finite} traces comes for free~\cite{hasuo06genericforward,hasuo10genericforward}.
%define simulation for various concrete systems---including both qualitative and quantitative ones---where
%its soundness wrt.\ \emph{finite} trace is assured while completeness is not.
%By It allows us to define a notion of forward and backward simulation for various concrete systems 
%including both qualitative and quantitative systems.
%the defined simulation is assured to be sound wrt.\ \emph{finite} trace.
%In contrast its completeness is not assured and fails for many systems.
Like many other notions of simulation, the resulting simulation sometimes fails to be complete.
This drawback of incompleteness 
%feature of forward and backward Kleisli simulation
wrt.\ finite trace can be partly mended by 
%to some extent by 
\emph{forward partial execution} (FPE), a transformation of coalgebraic
systems introduced in~\cite{urabeH14genericforward} that potentially
increases the likelihood of existence of simulations.
% The property of FPE that it aids Kleisli simulation 
% can be expressed using two terms---\emph{soundness} and \emph{adequacy}.

\subparagraph*{Contributions}
In this paper we continue our series of
work~\cite{hasuo06genericforward,hasuo10genericforward,urabeH14genericforward}
and study the relationship between Kleisli simulations and
\emph{infinite} traces. This turns out to be more complicated than 
we had expected, a principal reason being that \emph{infinite} traces
are less well-behaved than finite traces (that are characterized simply
by finality).
% Using Kleisli simulation to check \emph{infinite} trace inclusion is
% the main aim of this paper.

For a suitable coalgebraic characterization of infinite traces we
principally follow~\cite{jacobs04tracesemantics}---also relying on 
observations
in~\cite{cirstea10genericinfinite,kerstan13coalgebraictrace}---and
characterize infinite traces in terms of \emph{largest homomorphisms}.
% To this end, we need a categorical characterization of infinite trace;
% we follow the framework introduced in~\cite{jacobs04tracesemantics}.
%where nondeterministic systems are intensively studied.
%With respect to \emph{infinite} trace,
%similar study is done in~\cite{jacobs04tracesemantics} 
%intensively for nondeterministic systems.
More specifically, we lift 
a final $F$-coalgebra
% $\zeta:Z\to FZ$ 
in $\Sets$ to
%  the coalgebra
% $J\zeta:Z\kto FZ$ in
the Kleisli category $\Kl(T)$ 
and exhibit that the latter admits a largest homomorphism. 
In this paper
we (principally) work with: the powerset monad $\pow$ (on $\Sets$) and the
sub-Giry monad $\giry$ (on $\Meas$), as a monad $T$ for branching; and
a polynomial functor $F$ for linear-time behaviors.

Here are our concrete contributions.
For each of the above combinations of $T$ and $F$:
\begin{itemize}
 \item We show that
       forward Kleisli simulations are sound with respect to inclusion of
       \emph{infinite}
       languages. The proof of this general result is not hard,
       exploiting the above
       coalgebraic characterization of infinite languages as largest homomorphisms.
 \item We show that backward simulations are sound too, although here we have
       to impose 
       suitable restrictions, like \emph{totality} and
       \emph{image-finiteness}. The soundness
       proofs are much more involved, too, and calls for careful inspection of the 
       construction of infinite trace semantics. The proofs are
       separately for $T=\pow$ and for $\giry$.
 \item We show that \emph{forward partial execution} (FPE)---a
       transformation from~\cite{urabeH14genericforward} that aids 
       discovery of fwd./bwd.\ simulations---is applicable also to
       the current setting of \emph{infinite} trace inclusion. More
       specifically we prove: \emph{soundness} of FPE (discovery of a
       simulation after FPE indeed witnesses infinite language
       inclusion); and its \emph{adequacy} (FPE does not destroy simulations that
       are already there).       
\end{itemize}
% In the framework, for a concrete setting where $F$ is a polynomial
% functor on $\Sets$ and $T$ is a powerset monad $\pow$,
% it is shown that the \emph{final} $F$-coalgebra $\zeta:Z\to FZ$ in $\Sets$ is lifted to 
% a weakly final $F$-coalgebra $J\zeta:Z\kto FZ$ in $\Kl(T)$ that admits the largest homomorphism.
% Then the largest homomorphism from $c$ to $J\zeta$ gives the \emph{infinite} trace semantics of $\mathcal{X}$.



%\subparagraph*{Contributions}
%In this paper, we show that Kleisli simulation and FPE can also work in 
%the infinite trace framework in~\cite{jacobs04tracesemantics}. More concretely,
%Our concrete contributions are as follows.
% \begin{itemize}
% \item For the powerset monad $\pow$ and a polynomial functor $F$ on $\Sets$, we do the followings.
% \begin{enumerate}
% \item 
% We give an alternative proof for the result in~\cite{jacobs04tracesemantics}
%  that $J\zeta$
% % ---lifted final coalgebra---
%  admits the largest homomorphism.
% %We prove that $J\zeta$---lifted final coalgebra---admits the largest homomorphism.
% %It is a known result, but we give another proof.
% %It is easy to see that $g:X\kto Z$ is a homomorphism from $c$ to $J\zeta$ iff 
% %$g$ is a fixed point of $\Phi_X:\Kl(\pow)(X,Z)\to\Kl(\pow)(X,Z)$ defined by 
% %$\Phi_X(g)=J\zeta^{-1}\odot \overline{F}
% More specifically, the largest homomorphism from $c:X\kto FX$ to $J\zeta:Z\kto FZ$ can be given as the greatest lower bound of 
% a certain decreasing transfinite sequence in $\Kl(\pow)(X,Z)$ 
% starting from the largest element $\top_{X,Z}$.
% %
% %for a general monad $T$ and a general functor $F$ that satisfy some conditions,
% %we present a way to construct the largest homomorphism to $J\zeta$ in $\Kl(T)$.
% %Then we show that $\pow$ and a polynomial functor on $\Sets$ satisfy the conditions.
%  This characterization of the largest homomorphism helps us to come up with the notion of 
% \emph{total and image-finite backward simulation} and prove that it is sound wrt.\ infinite trace.
% \label{item:intropow1}
% %Going thorough the general construction allows us to define a notion of restricted backward simulation that is sound wrt.\ infinite trace.
% \item 
% %For $\pow$ and $\FSigma$, 
% We show that the coalgebraic infinite trace semantics (by maximality)
% indeed captures (the usual definition of) infinite trace semantics.
% %via the largest homomorphism 
% %coincides with the automata-theoretic semantics.
% %---that is defined by regarding the system as
% %an automaton that nondeterministically outputs trees.
% %We characterize the largest homomorphism to $J\zeta$ using automata-theoretic terms.
% %(This justify notions of simulation that witness the inclusion between the largest homomorphism).
% \label{item:intropow2}
% \item 
% %For a general monad $T$ and a general functor $F$ where $J\zeta$ admits the largest homomorphism, 
% We prove the soundness of fwd.\ Kleisli simulation wrt.\ coalgebraic infinite trace 
% %(Namely, existence of a forward Kleisli simulation witnesses inclusion between the largest homomorphism to $J\zeta$.
% %Moreover, 
% %This implies that we can use forward Kleisli simulation to verify inclusion between nondeterministic systems.
% %\item 
% %For $\pow$ and $\FSigma$, 
% and moreover,
% along the construction in~\ref{item:intropow1}, %of the largest homomorphism, 
% we prove that backward Kleisli simulation is sound wrt.\ coalgebraic
%       infinite trace if the simulation is \emph{total} and \emph{image-finite}.
% % Moreover, we give a set-theoretic sufficient  conditions for the simulation to satisfy the 
% % conditions---they are ``image nonemptiness'' and ``image finiteness''. 
% \label{item:intropow3}
% %For a general monad $T$ and a general functor $F$ where the largest homomorphism can be constructed via the general construction,
% %we prove that backward Kleisli simulation is sound wrt.\ coalgebraic infinite trace if the simulation satisfy some conditions.
% \item 
% %For a general monad $T$ and a general functor $F$ where $J\zeta$ admits the largest homomorphism, 
% %We show that the soundness and the adequacy of FPE for forward simulation wrt.\ coalgebraic infinite trace.
% %\item
% %For $\pow$ and $\FSigma$, 
% In the context of infinite trace inclusion,
% we show \emph{soundness} and \emph{adequacy} of FPE wrt.\ forward
%       simulation---two properties that justifies use of FPE.
% For \emph{backward} simulation,
%  soundness of FPE holds but adequacy 
% %The remaining one---adequacy of FPE wrt.\ backward simulation---
% needs some care:
% the current proof only works for a restricted class of systems.
% %We also show that FPE for backward simulation satisfies soundness wrt.\ infinite trace while
% %adequacy is assured only for systems satisfying some conditions.
% %%We show that the soundness of FPE for backward simulation wrt.\ coalgebraic infinite trace.
% %%We also show that FPE for backward simulation is adequate wrt.\ coalgebraic infinite trace if 
% %%the simulating system satisfy some conditions.
% %We give a sufficient set-theoretic condition 
% %for the system to satisfy the conditions.
% \label{item:intropow4}
% %
% %For a monad $T$ and a functor $F$ where the largest homomorphism can be constructed via the general construction,
% %we first show that the soundness of FPE wrt.\ coalgebraic infinite trace and backward simulation.
% %We also show that FPE is adequate wrt.\ coalgebraic infinite trace and backward simulation if 
% %the simulating system satisfy some conditions.
% %Moreover, we give a sufficient condition for the system to satisfy the conditions when $T=\pow$ and $F$ is polynomial.
% \end{enumerate}
% \item We follow the same line for the (sub-)Giry monad $\giry$ and a polynomial functor $F$ on $\Meas$.
% The difference from the above $\pow$ case are as follows.
% %The differences from the above are as follows.
% \begin{enumerate}
% \item 
% The construction of the largest homomorphism becomes more complicated.
% Specifically, we cannot take $\top_{X,Z}\in\Kl(\giry)(X,Z)$ from which we 
% would be starting the iteration to construct the largest homomorphism.
% Instead, starting from the largest element $J!_{X}$ in $\Kl(\giry)(X,1)$,
% we calculate the glb.\ of a certain decreasing sequence along the final $F$-sequence in $\Meas$ lifted to the Kleisli category.
% This allows us to obtain the largest homomorphism as the mediating arrow to $Z$---a limit of the lifted final sequence. 
% %Instead, we calculate the glb.\ 
% %%of a certain decreasing sequence 
% %in $\Kl(\giry)(X,1)$,
% %along the final $F$-sequence in $\Meas$ lifted to the Kleisli category,
% %starting from its largest element $J!_{X}$.
% %%the top element $\top_{X,1}$.
% %Using the resulting glb., we can construct a cone over a lifted final sequence 
% %%$1\overset{J!_{F1}}{\kleftarrow} F1\overset{JF!_{F1}}{\kleftarrow} F^2 1\overset{JF^2!_{F1}}{\kleftarrow} \ldots$,
% %that allows us to obtain an arrow $l_{\max}:X\kto Z$ from which we can start the iteration. %to obtain the largest homomorphism.
% %
% %Specifically, we cannot take $\top_{X,Z}\in\Kl(\giry)(X,Z)$ from which we 
% %would be starting the iteration to construct the largest homomorphism.
% %Instead, we calculate the glb.\ of a certain decreasing transfinite
% %      sequence in $\Kl(\giry)(X,1)$,
% %along the final $F$-sequence  in $\Meas$ lifted to the Kleisli category,
% %starting from its largest element $J!_{X}$.
% %%the top element $\top_{X,1}$.
% %Using the resulting glb., we can construct a cone over a lifted final sequence 
% %%$1\overset{J!_{F1}}{\kleftarrow} F1\overset{JF!_{F1}}{\kleftarrow} F^2 1\overset{JF^2!_{F1}}{\kleftarrow} \ldots$,
% %that allows us to obtain an arrow $l_{\max}:X\kto Z$ from which we can start the iteration. %to obtain the largest homomorphism.
% \label{item:introgiry1}
% %\item
% %Naturally, automata-theoretic definition of infinite trace semantics is different.
% %\label{item:introgiry1}
% \item 
% Soundness of Kleisli backward simulation is satisfied if 
% This changes restriction imposed on the systems so that FPE for backward simulation becomes adequate.
% %As the construction of the largest homomorphism is complicated,
% %the condition for backward simulation to be sound should be complicated, too.
% \label{item:introgiry3}
% %\item 
% %As the construction of the largest homomorphism is complicated,
% %the condition for FPE for backward simulation to be adequate should be complicated, too.
% %\label{item:introgiry4}
% \end{enumerate}
% \end{itemize}


%\begin{itemize}
%\item For $\pow$ and $\FSigma$ on $\Sets$, 
%\begin{itemize}
%\item 
%We prove that $J\zeta$ admits the largest homomorphism.
%It is a known result, but we give another proof.
%%It is easy to see that $g:X\kto Z$ is a homomorphism from $c$ to $J\zeta$ iff 
%%$g$ is a fixed point of $\Phi_X:\Kl(\pow)(X,Z)\to\Kl(\pow)(X,Z)$ defined by 
%%$\Phi_X(g)=J\zeta^{-1}\odot \overline{F}
%Namely, the largest homomorphism can be given as the greatest lower bound of 
%a certain decreasing transfinite sequence in $\Kl(\pow)(X,Z)$ 
%starting from the largest element $\top_{X,Z}$.
%%
%%for a general monad $T$ and a general functor $F$ that satisfy some conditions,
%%we present a way to construct the largest homomorphism to $J\zeta$ in $\Kl(T)$.
%%Then we show that $\pow$ and a polynomial functor on $\Sets$ satisfy the conditions.
%This construction of the largest homomorphism allows us to define a notion of 
%\emph{GFP-abstracting backward simulation} that is sound wrt.\ infinite trace in later sections.
%%Going thorough the general construction allows us to define a notion of restricted backward simulation that is sound wrt.\ infinite trace.
%\item 
%%For $\pow$ and $\FSigma$, 
%We show that the coalgebraic infinite trace semantics via the largest homomorphism coincides with
%the automata-theoretic semantics.
%%We characterize the largest homomorphism to $J\zeta$ using automata-theoretic terms.
%(This justify notions of simulation that witness the inclusion between the largest homomorphism).
%\item 
%%For a general monad $T$ and a general functor $F$ where $J\zeta$ admits the largest homomorphism, 
%We prove the soundness of fwd.\ Kleisli simulation wrt.\ coalgebraic infinite trace.
%(Namely, existence of a forward Kleisli simulation witnesses inclusion between the largest homomorphism to $J\zeta$.
%Moreover, this implies that we can use fwd.\ Kleisli simulation to verify inclusion between nondeterministic systems.)
%\item 
%%For $\pow$ and $\FSigma$, 
%Along the construction of the largest homomorphism, 
%we prove that backward Kleisli simulation is sound wrt.\ coalgebraic infinite trace if the simulation satisfy some conditions.
%Moreover, we give a set-theoretic sufficient  conditions for the simulation to satisfy the 
%conditions---they are ``image nonemptiness'' and ``image finiteness''. 
%%For a general monad $T$ and a general functor $F$ where the largest homomorphism can be constructed via the general construction,
%%we prove that backward Kleisli simulation is sound wrt.\ coalgebraic infinite trace if the simulation satisfy some conditions.
%\item 
%%For a general monad $T$ and a general functor $F$ where $J\zeta$ admits the largest homomorphism, 
%We show the soundness and the adequacy of FPE for forward simulation wrt.\ coalgebraic infinite trace.
%\item
%%For $\pow$ and $\FSigma$, 
%We show that the soundness of FPE for backward simulation wrt.\ coalgebraic infinite trace.
%We also show that FPE is adequate wrt.\ coalgebraic infinite trace and backward simulation if 
%the simulating system satisfy some conditions.
%Moreover, we give a sufficient and set-theoretic condition 
%for the system to satisfy the conditions.
%%
%%For a monad $T$ and a functor $F$ where the largest homomorphism can be constructed via the general construction,
%%we first show that the soundness of FPE wrt.\ coalgebraic infinite trace and backward simulation.
%%We also show that FPE is adequate wrt.\ coalgebraic infinite trace and backward simulation if 
%%the simulating system satisfy some conditions.
%%Moreover, we give a sufficient condition for the system to satisfy the conditions when $T=\pow$ and $F$ is polynomial.
%\end{itemize}
%\item We do a similar thing for the sub-Giry monad $\giry$ and a polynomial functor on $\Meas$.
%The differences from the above are as follows.
%\begin{itemize}
%\item 
%The construction of the largest homomorphism becomes more complicated.
%Namely, we cannot take $\top_{X,Z}\in\Kl(\giry)(X,Z)$ which we start the iteration to construct the largest homomorphism from.
%Instead, we calculate the greatest lower bound of a certain decreasing transfinite sequence in $\Kl(\giry)(X,1)$
%starting from 
%the largest element $J!_{X}$.
%%the top element $\top_{X,1}$.
%Using the resulting glb., we can construct a cone 
%over a lifted final sequence 
%$1\overset{J!_{F1}}{\kleftarrow} \overline{F}1\overset{JF!_{F1}}{\kleftarrow} \overline{F}^2 1\overset{JF^2!_{F1}}{\kleftarrow} \ldots$,
%and it allows us to obtain an arrow $l_{\max}:X\kto Z$ from which we can start an iteration to obtain the largest homomorphism.
%\item 
%As the construction of the largest homomorphism is complicated,
%the condition for backward simulation to be sound is complicated, too.
%\item 
%As the construction of the largest homomorphism is complicated,
%the condition for FPE for backward simulation to be adequate is complicated, too.
%\end{itemize}
%\end{itemize}




\subparagraph*{Organization}
 \S\ref{sec:preliminaries} is devoted to categorical preliminaries; we
 fix notations there.
In \S\ref{sec:weaklyfinalcoalg} we review the previous works that we
 rely on, namely coalgebraic infinite trace
 semantics~\cite{jacobs04tracesemantics}, Kleisli
 simulation~\cite{hasuo06genericforward,hasuo10genericforward,urabeH14genericforward},
and FPE~\cite{urabeH14genericforward}.
Our technical contributions are in the subsequent sections: in
\S\ref{sec:powersetmonad} we study the nondeterministic setting (i.e.\
 the powerset monad $\pow$ on $\Sets$ and a polynomial functor $F$); 
 \S\ref{sec:girymonad} is for the probabilistic setting (where the monad
 $T$ is the sub-Giry monad $\giry$). In \S\ref{sec:othermonads} we
 briefly
discuss other monads like the \emph{lift monad} $\lift$ (for divergence) and the
\emph{subdistribution monad} $\dist$ on $\Sets$ (for discrete probabilities).
% In \S\ref{sec:powersetmonad}, 
% we investigate systems represented with $\pow$ and $\FSigma$.
% In \S\ref{sec:girymonad}, 
% we investigate systems represented with $\giry$ and $\FXi$.
% In \S\ref{sec:othermonads}, we investigate other pairs of a functor and a monad.
% In \S\ref{sec:conclusion}, we conclude and give future work.

Some definitions and results
in~\S{}\ref{sec:powersetmonad}--\ref{sec:girymonad} are marked with
$\dagger$. Those marked ones are essentially proofs of the results for specific
settings (namely $T=\pow$ and $T=\giry$) but formulated in general
terms with a general $T$. We do so in the hope that the axioms thus
identified
will help to discover new instances. 

Most proofs are deferred to the appendices, where auxiliary definitions
and examples are also found.
%Most proofs are deferred to the appendices,
%%that are available at the first author's webpage, 
%where auxiliary definitions
%and examples are also found.
%Omitted proofs, definitions and examples are found in the appendices.

%\hrulefill
%
%For example, suppose that we want to verify that a implementation $\mathcal{I}$ 
%satisfy a certain specification $\mathcal{S}$. 
%Let $\mathcal{A}_{\mathcal{I}}$ and $\mathcal{A}_{\mathcal{S}}$ be nondeterministic automata
%that represent the implementation and the specification respectively.
%More concretely, $\mathcal{A}_{\mathcal{I}}$ is an automaton 
%
%
%Similar situation can arise in \emph{quantitative} setting. 
%
%
%
%
%
%suppose that we are given two nondeterministic 
%automata $\mathcal{A}$ and $\mathcal{B}$ that nondeterministically output a finite word respectively.
%Here, 
%$\mathcal{A}$ models the implementation of a certain system while
%$\mathcal{B}$ models the specification of the system. 
%More concretely, $\mathcal{B}$ never outputs a word 
%
%that nondeterministically output a finite word and 
%a specification $\mathcal{S}$ that 
%
%
%The notion of language inclusion can be defined for various types of transition systems.
%For example, for two nondeterministic automata $\mathcal{X}$ and $\mathcal{Y}$ 
%that nondeterministically output a finite word respectively, 
%language inclusion $\lang(\mathcal{X})\subseteq\lang(\mathcal{Y})$ means 
%all the output of $\mathcal{X}$ can be output by $\mathcal{Y}$.
%
%
%For example, suppose that we are given two nondeterministic automata $\mathcal{A}$ and $\mathcal{B}$
%that nondeterministically output a finite word respectively.
%The automaton $\mathcal{A}$ is 
%
%and a specification $\mathcal{S}$.
%that nondeterministically output a finite word.
%We 
%
%
%
%%\subparagraph*{Contributions of the paper}
%
%
%%\hrulefill
%%
%%Checking \emph{language inclusion} of transition systems is a significant problem in model checking.
%%For example, suppose that we are given a nondeterministic automaton $\mathcal{A}$ 
%%that nondeterministically output a finite word $w\in\Sigma^*$ on an alphabet $\Sigma$.
%%We are also given a set $W\subseteq\Sigma^*$ of finite words.
%%Our aim is to prove that $\mathcal{A}$ never output word $w$ in $W$.
%%Practically, $\mathcal{A}$ is a model for an implementation of a real system 
%%while $W$ is 
%%
%%Similar situation can arise for \emph{quantitative} setting.
%%
%%
%%
%%One of the verification techniques to check such language inclusion is \emph{simulation}.
%%Intuitively, simulation can be 
%%
%%
%%Finite ...
%%
%%
%%To infinite ...
%
%\hrulefill
%
%Simulation is a notion that is used for a model checking.
%Suppose that we are given an implementation $\mathcal{I}$  and a specification $\mathcal{S}$ of a system,
% and desire to verify that the implementation satisfies the specification.
%One of the way to achieve this goal is to check language inclusion. % between systems.
%Namely, from the specification $\mathcal{S}$, we first construct a system $\mathcal{I}_{\mathcal{S}}$.
%Intuitively, $\mathcal{I}_{\mathcal{S}}$ is a system that behave without violating the specification $\mathcal{S}$.
%Then we check if the language of the implementation $\mathcal{I}$ is included in that of $\mathcal{I}_{\mathcal{S}}$.
%For example, if $\mathcal{I}$ is a nondeterministic automaton,  its language is a set $\lang(\mathcal{I})$ of words output by $\mathcal{I}$.
%Then the language inclusion is  set inclusion $\lang(\mathcal{I})\subseteq\lang(\mathcal{I}_{\mathcal{S}})$,
%that means $\mathcal{I}$ never output something dangerous defined by $\mathcal{S}$.
%%For another example, if $\mathcal{I}$ is a probabilistic automaton, its language is a distribution $\lang(\mathcal{I})$ that gives 
%%the probability of output for each word.
%%Then the language inclusion between $\lang(\mathcal{I})$ and $\mathcal{I}_{\mathcal{S}}$ means pointwise 
%
%Therefore our new goal is to check language inclusion of the systems.
%However, here is a problem: the language of a system is infinite in many cases.
%For instance, in the above example, the output $\lang(\mathcal{I})$ of a nondeterministic automaton is usually an infinite set.
%This makes it very difficult (or sometimes impossible because it becomes undecidable) to check language inclusion directly.
%%
%This problem can be resolved as follows: instead of checking entire language inclusion, we check stepwise language inclusion---\emph{simulation}.
%%This stepwise language inclusion is called \emph{simulation}.
%Here, it is required that the simulation satisfy \emph{soundness}; namely,
%the stepwise language inclusion (simulation) imply entire language inclusion.
%%This property is called \emph{soundness} of simulation.
%One of the example of simulation is forward and backward simulation for nondeterministic automaton systematically studied in~\cite{lynch95forwardand}.
%
%Forward and backward \emph{Kleisli simulation}---a categorical generalization 
%%of the simulation in
%of~\cite{lynch95forwardand}---is introduced in~\cite{hasuo06genericforward}.
%%To apply the framework, a system should be represented as a coalgebra of a functor $F$ in a Kleisli category of a monad $T$.
%Kleisli simulation is defined for systems that is represented as a coalgebra of a functor $F$ in a Kleisli category of a monad $T$.
%Here, $F$ 
%%is a endofunctor on some category that 
%represents the branching type of the system,
%while $T$ %is a monad that 
%represents the transition type.
%Using various $F$ and $T$, we can represent various state-based systems.
%For such an $F$-coalgebra, it is known that its finite trace semantics can be defined  coalgebraically~\cite{hasuo07generictrace}.
%With respect to this categorical finite trace semantics,  Kleisli simulation satisfies desired property---soundness.
%
%%It is known that simulation satisfies soundness---the stepwise language inclusion imply the entire language inclusion.
%While simulation satisfy the soundness, we should note that the converse does not necessarily hold.
%Namely, %it is possible that 
%there might exists a pair of systems where language inclusion holds while a simulation does not exist.
%In~\cite{urabeH14genericforward}, a transformation of systems called forward partial execution (FPE) is introduced.
%By applying FPE to such a pair of systems, we can modify the situation; namely, a simulation may exist between the systems
%after the transformation.
%
%
%%for a given endofunctor $F$ and a monad $T$ on a category $\mathbb{C}$ where $F$ can be lifted to an endofunctor $\overline{F}$ on $\Kl(T)$, 
%%a general construction of a $\overline{F}$-final coalgebra in the Kleisli category $\Kl(T)$ was presented.
%%More concretely, if $T$ and $F$ satisfy some conditions, 
%%then the inversion of the $F$-initial algebra in $\mathbb{C}$ is lifted to $\overline{F}$-final coalgebra in $\Kl(T)$.
%%For a system that is represented as a $\overline{F}$-coalgebra $c$ in $\Kl(T)$, 
%%the unique homomorphism from $c$ to the constructed $\overline{F}$-final coalgebra gives 
%%a finite trace semantics of the represented system.
%%For a pair of systems represented as $\overline{F}$-coalgebras,  notions of forward and backward Kleisli simulation are introduced in~\cite{hasuo07generictrace},.
%%They are defined as Kleisli arrows that satisfy some diagrams about the coalgebras.
%%Existence of a forward or backward Kleisli simulation implies finite trace inclusion between the systems.
%%In~\cite{urabeH14genericforward}, a transformation of system called FPE is introduced.
%%It increases Kleisli simulation and helps us to find a simulation.
%
%
%
%\subparagraph*{Contributions of the Paper}
%In this paper, we show that 
%%the notion of 
%forward and backward Kleisli simulation introduced in~\cite{hasuo07generictrace}
%can be also used to witness \emph{infinite} trace inclusion.
%To this end, for a system represented as a $\overline{F}$-coalgebra in $\Kl(T)$,
%we first have to capture its infinite trace semantics coalgebraically.
%We follow the framework introduced in~\cite{jacobs04tracesemantics} where 
%infinite trace semantics is captured as the largest homomorphism to a lifted final $F$-coalgebra.
%For two pairs of a functor and a monad, 
%we characterize the coalgebraic infinite trace semantics via the largest homomorphism using automata theoretic terms.
%Moreover, we also give a general construction of the largest homomorphism, that is applicable to another pair of a functor and a monad.
%Then we show that Kleisli simulation in~\cite{hasuo07generictrace} and FPE in~\cite{urabeH14genericforward} can be 
%used to check  trace inclusion with respect to this infinite trace via the largest homomorphism.
%
%%The general construction is based on~\cite{cirstea10genericinfinite}.
%%There, for a given $\overline{F}$-coalgebra, they construct a cone over a lifted final sequence in $\mathbb{C}$ using the coalgebra.
%%To assure that the constructed family of Kleisli arrows indeed constitute a cone, 
%%it is assumed that the final object in $\mathbb{C}$ is lifted to a final object in $\Kl(T)$.
%%Moreover, it is assumed that the limit of the final sequence is lifted to a weak limit.
%%Therefore we have the largest mediating arrow, that can be regardezd as a infinite trace semantics.
%
%%In this paper, we mainly use one type of functor---polynomial functor on $\Sets$ and $\Meas$---and 
%%three types of monads---powerset monad $\pow$, lift monad $\lift$, and sub-Giry monad $\giry$.
%%In~\cite{jacobs04tracesemantics}, it is already shown that a polynomial functor $F$ on $\Sets$ and 
%%the powerset monad $\pow$ constitute an infinite trace situation (i.e.\ final $F$-final coalgebra in $\Sets$ is lifted a weakly final $\overline{F}$-coalgebra in $\Kl(\pow)$).
%%Moreover, set-theoretic characterization of the coalgebraic infinite trace semantics was given there.
%%In this paper, we give another proof for $F$ and $\pow$ to constitute an infinite trace situation using general construction.
%%Then we give a bit more automata-theoretic characterization of  the coalgebraic infinite trace semantics.
%
%%\begin{itemize}
%%\item We give a sufficient condition for a functor and a monad where a final coalgebra is lifted to a weakly final coalgebra in the Kleisli category.
%%\item For three types of systems, we characterize the infinite trace semantics via weakly final coalgebra using automata-theoretic terms.
%%%We show that not only semantics of nondeterministic automata but also that of automata with exception and 
%%%probabilistic automata can be captured via the largest homomorphism to the weakly final coalgebra.
%%\item We show soundness of forward Kleisli simulation with respect to infinite trace.
%%\item We show soundness of restricted backward Kleisli simulation with respect to infinite trace.
%%\item We show soundness and adequacy of forward partial execution for forward Kleisli simulation with respect to infinite trace.
%%\item We show soundness and adequacy of forward partial execution for backward Kleisli simulation with respect to infinite trace under some conditions.
%%\end{itemize}
%
%%\subparagraph*{Related Works}
%%The construction of the largest homomorphism given in this paper is based on~\cite{cirstea10genericinfinite}.
%%There, for a given $\overline{F}$-coalgebra, they construct a cone over a lifted final sequence in $\mathbb{C}$ using the coalgebra. 
%%In~\cite{cirstea10genericinfinite}, it is assumed that the limit of the final sequence is lifted to a weak limit.
%%Therefore we have the largest mediating arrow, that is regarded as a infinite trace semantics in the paper.
%%Our general construction is different from~\cite{cirstea10genericinfinite} in  two points.
%%First, in~\cite{cirstea10genericinfinite} to assure that the constructed family of Kleisli arrows indeed constitute a cone, 
%%it is assumed that the final object in $\mathbb{C}$ is lifted to a final object in $\Kl(T)$.
%%In contrast, we do not assume that the final object in $\mathbb{C}$ is lifted to a final object in $\Kl(T)$.
%%Instead, we construct a cone over a lifted final sequence by taking the greatest lower bound of the family of Kleisli arrows.
%%Second, we do not quit at obtaining the largest mediating arrow to the weak limit.
%%We go furthermore, namely, we again take the greatest lower bound and construct a homomorphism.
%%There exists a concrete system whose infinite trace semantics is captured differently by the largest homomorphism and by the largest mediating arrow.
%%
%%In~\cite{kerstan13coalgebraictrace}, an explicit description of a $\overline{F}$-(not weakly) final coalgebra is given 
%%for four combinations of a functor $F$ and a monad $T$ on 
%%$\Meas$---$F=\Sigma\times(\place)$ and $F=1+\Sigma\times(\place)$, and $T=\giry$ (sub-Giry monad) and $T=\Geq$ (strict-Giry monad). 
%%However, we do not use final $\overline{F}$-coalgebra to capture trace semantics in this paper.
%%This is because if we choose sub-Giry monad $\giry$ for a monad, then the carrier of $F$-final coalgebra becomes an empty set or $\Sigma^*$,
%%and the unique homomorphism to the final $\overline{F}$-coalgebra does not capture infinite trace.
%%If we choose strict-Giry monad $\Gleq$, then  the carrier of $F$-final coalgebra becomes an $\Sigma^{\omega}$ or $\Sigma^{\infty}$,
%%and the unique homomorphism to the final $\overline{F}$-coalgebra does  capture infinite trace as we desire.
%%However, in this setting, infinite trace inclusion coincides with infinite trace equivalence.
%%As it is often more difficult to prove trace inclusion than to prove trace equivalence 
%%(e.g.\ finite trace inclusion is undecidable~\cite{blondel03undecidableproblems} while finite trace equivalence is decidable~\cite{kiefer11languageequivalence}),
%%it seems to be unsuitable to use simulation to prove trace equivalence.
%%
%%In~\cite{schubert09terminalcoalgebras}, it is shown that 
%%polynomial functor on a category $\Meas$ and sub-Giry monad $\giry$ preserves 
%%a limit of a $\omega^{\text{op}}$-sequence consisting of standard Borel sets and surjective measurable functions.
%%Moreover, polynomial functor and Giry monad preserves standard Borel sets.
%%These facts imply that there exists final $\giry F$-coalgebra in $\Meas$ for every polynomial functor $F$ on $\Meas$.
%
%
%%\begin{center}
%%\begin{tabular}{c | c | c || c | c | c | c | c}
%%$\mathbb{C}$ & $T$ & $F$ & \shortstack{inf. trace\\situation} & \shortstack{general\\construction} & \shortstack{existence\\of $\top$} & \shortstack{continuity of\\composition} & \shortstack{limit of final\\seq. lifted to} \\ \hline
%%$\Sets$ & $\pow$ & {\scriptsize\shortstack{polynomial\\functor}} & \checkmark & \checkmark & \checkmark & \times & \shortstack{weak\\2-limit} \\ \hline
%%$\Sets$ & $\lift$ & {\scriptsize\shortstack{polynomial\\functor}} & \checkmark & \checkmark & \times & \checkmark & 2-limit \\ \hline
%%$\Sets$ & $\dist$ & {\scriptsize\shortstack{polynomial\\functor}} & \checkmark & \times & \times & \checkmark & \times \\ \hline
%%$\Meas$ & $\giry$ & {\scriptsize\shortstack{polynomial\\functor}} & \checkmark & \checkmark & \times & \checkmark & 2-limit 
%%\end{tabular}
%%\end{center}
%
%\subparagraph*{Organization of the paper}
%In \S\ref{sec:preliminaries}, we give preliminaries about  categorical notions and about infinite trees.
%In \S\ref{sec:weaklyfinalcoalg}, following~\cite{jacobs04tracesemantics}, 
%we give a definition of a coalgebraic infinite trace semantics via the largest homomorphism. % to the lifted final coalgebra.
%In \S\ref{sec:powersetmonad}, for a polynomial functor on $\Sets$ and the powerset monad, we show that they 
%constitute an infinite trace situation---namely, final coalgebra admits the largest homomorphism if it is lifted to the Kleisli category---by giving a general construction of the largest homomorphism that is applicable to other pairs of a functor and a monad.
%Moreover, we characterize the obtained semantics using automata-theoretic terms.
%Then we show that Kleisli simulation and FPE can be used to verify the inclusion between the coalgebraic infinite trace semantics
%defined via the largest homomorphism.
%In \S\ref{sec:girymonad}, we do the same  for a polynomial functor on $\Meas$ and the Giry monad.
%In \S\ref{sec:othermonads}, we investigate other two pairs of a functor and a monad---a polynomial functor on $\Sets$ and the lift monad,
%and the same functor and the subdistribution monad.
%In \S\ref{sec:conclusion}, we conclude and give  future work.
%Omitted proofs and examples can be found in the appendices.
%%In \S\ref{sec:genconstruction}, we give a sufficient condition for a final coalgebra to be lifted to a weakly final coalgebra in the Kleisli category.
%%In \S\ref{sec:coincidencesemantics}, 
%%for three types of systems, 
%%we characterize their coalgebraic infinite trace semantics using automata-theoretic terms.
%%%we show that their ``automata-theoretic'' semantics  can be captured via weakly final coalgebra.
%%In \S\ref{sec:Kleislisim}, we review the definitions of forward and backward Kleisli simulation, 
%%and prove the soundness of forward simulation and restricted backward  simulation with respect to infinite trace.
%%In \S\ref{sec:fpe}, we review the definitions of forward partial execution, and prove their soundness and adequacy 
%%with respect to infinite trace.

\section{Preliminaries}\label{sec:preliminaries}
%\subsection{Categorical Notions}
\begin{mydefinition}
 A  \emph{polynomial functor} $F$ on $\Sets$  is defined by the following BNF notation:
 $F::=\id\mid A\mid F_1\times F_2\mid \coprod_{i\in I} F_i$. 
 Here $A\in\Sets$ and $I$ is a countable set.
\end{mydefinition}
The notion of polynomial functor can be also defined  for $\Meas$---the category 
of measurable spaces and measurable functions between them. 
%For the sake of later sections,
%we require the constant to be a \emph{standard Borel space} (see~\cite{doob94measuretheory}, for example).
%In this paper, we use two types of system---a \emph{nondeterministic tree automaton} and \emph{probabilistic tree automaton}.
%
%To model a certain type of transition systems,
%we first have to fix three parameters---a category $\mathbb{C}$,
%a functor $F$, and a monad $T$.
%In this paper, 
%
%A functor $F$ represents the branching type of the system while $T$ represents the transition type.
%In this paper, we use two combinations:
%the first is a category $\Sets$ of sets and functions, a polynomial functor $F $
%systems of two transition 
%types---nondeterministic systems and probabilistic systems.
%A transition type is modelled by transition 
%Both of them are represented 
%
%%\begin{mydefinition}[polynomial functor on $\Sets$]
%%\end{mydefinition}
%%
%In this paper, we mainly use two pairs of a functor and a monad that are defined on the different categories.
%One of the categories is $\Sets$---a category whose objects are sets and arrows are functions between them.
%The other is $\Meas$---a category whose whose objects are measurable spaces and arrows are measurable functions between them.
%%\begin{mydefinition}[category $\Meas$]
%%A \emph{category of measurable spaces} is a category $\Meas$ 
%%whose objects are measurable spaces and arrows are measurable functions between them.
%%\end{mydefinition}
%%If confusion is unlike, we denote $X$ for a measurable space $(X,\sigalg_X)$.
%
%For each of the categories, we use the following functor.
%For the category $\Sets$, we use a polynomial functor---a functor constructed from identity, a constant,
%a finite product, and a countable coproduct.
%%, that is defined by the following BNF notation:
%%$F\;::=\; \id\,\mid\, A \,\mid\, F_1\times F_2 \,\mid\, \coprod_{i\in I} F_i$
%%where $A$ is a set and $I$ is a countable set.
%%For its semantics, see~\cite{} for example.
%For the category $\Meas$, we also use a \emph{polynomial functor}, that is defined as follows.
%Next, we define a polynomial functor on $\Meas$.
%Here, we use a restricted version used in~\cite{cirstea10genericinfinite} where the constant has to be countable space with discrete $\sigma$-algebra.
\begin{mydefinition}\label{def:polyfuncmeas}
A \emph{(standard Borel) polynomial functor} $F$ on $\Meas$ is defined by the following BNF notation:
$%\begin{equation*}
F\;::=\; \id\,\mid\, (A,\sigalg_A) \,\mid\, F_1\times F_2 \,\mid\, \coprod_{i\in I} F_i\, .
$ %\end{equation*} 
%Here, $A\in\Meas$.
%Here, $A$ is a countable set.
Here
 $I$ is a countable set; and
 we require that $(A,\sigalg_A)\in\Meas$  is a \emph{standard Borel
 space}
 (see e.g.~\cite{doob94measuretheory}). 
The  $\sigma$-algebra $\sigalg_{FX}$ associated to $FX$ is defined in the obvious
 manner. Namely: 
%We have to specify $\sigma$-algebras.
% For $(X,\sigalg_X)\in\Meas$, $F(X,\sigalg_X)=(FX,\sigalg_{FX})$ is given as follows:
% the underling set $FX$ is given in the same manner as that for the  polynomial functors on $\Sets$.
% The $\sigma$-algebra $\sigalg_{FX}$ is defined inductively as follows.
for $F=\id$, $\sigalg_{FX}=\sigalg_X$; for $F=(A,\sigalg_A)$, $\sigalg_{FX}=\sigalg_A$;
for $F=F_1\times F_2$, $\sigalg_{FX}$ is the smallest $\sigma$-algebra that contains $A_1\times A_2$ for 
all $A_1\in\sigalg_{F_1X}$ and $A_2\in\sigalg_{F_2X}$;
for for $F=\coprod_{i\in I} F_i$, 
$\sigalg_{FX}=\{\coprod_{i\in I} A_i\mid A_i\in \sigalg_{F_iX}\}$.

For arrows, $F$ works in the same manner as a polynomial functor on
 $\Sets$.

In what follows, a standard Borel polynomial functor is often called
 simply a \emph{polynomial functor}. 
\end{mydefinition}
%We only give their action on objects $(X,\sigalg_X)$ in $\Meas$.
 The technical  requirement of being standard Borel in the above will be used in the probabilistic
setting of~\S{}\ref{sec:girymonad} (it is also exploited in~\cite{cirstea10genericinfinite,schubert09terminalcoalgebras}).  A standard Borel space is a measurable
space induced by a Polish space; for further details see
e.g.~\cite{doob94measuretheory}.


There is a natural correspondence between polynomial functors and
\emph{ranked alphabets}. In this paper  a functor $F$ for the (linear-time)
transition type is restricted to a polynomial one; this means that we are dealing with
($T$-branching) systems that generate \emph{trees}
over some ranked alphabet. We collect some standard notions and notations
for such trees in Appendix~\ref{subsec:rankedAlphabetAndInfiniteTrees};
they will be used later in showing that our coalgebraic infinite traces indeed
capture
infinite tree languages of such systems.



 We go on to introduce monads $T$ for branching.
We principally use two monads---the \emph{powerset monad} $\pow$ on
$\Sets$ and the \emph{sub-Giry} monad $\giry$ on $\Meas$. The latter is
an adaptation of the \emph{Giry monad}~\cite{giry82categoricalapproach}
and inherits most of its structure from the Giry monad;
see Rem.~\ref{rem:giryAndSubGiry}.
%Giry monad is defined in \cite{giry82categoricalapproach}
\begin{mydefinition}[monads $\pow$ and $\giry$]\label{def:variousmonads}
The \emph{powerset monad} is the monad $(\pow,\eta^{\pow},\mu^{\pow})$ on $\Sets$ such that 
$\pow X=\{A\subseteq X\}$ and $\pow f(A)=\{f(x) \mid x\in A\}$.
Its unit is given by the singleton set $\eta^{\pow}_{X}(x)=\{x\}$ and its multiplication is given by $\mu^{\pow}_{X}(M)=\bigcup_{A\in M}A$.
%\begin{itemize}
%\item $\pow X=\{A\subseteq X\}$,
%\item $\pow f(A)=\{f(x) \mid x\in A\}$,
%\item $\eta^{\pow}_{X}(x)=\{x\}$, and 
%\item $\mu^{\pow}_{X}(M)=\bigcup_{A\in M}P$.
%\end{itemize}
%
%A \emph{lift monad} is a monad $(\lift,\eta^{\lift},\mu^{\lift})$ on $\Sets$  such that
%\begin{itemize}
%\item $\lift X=\{\bot\}+X$,
%\item $\lift f(y)=\begin{cases} f(y) & (y\neq \bot) \\ \bot & (\text{otherwise}),\end{cases} $
%\item $\eta^{\lift}_{X}(x)=x$, and 
%\item $\mu^{\pow}_{X}(z)=z$.
%\end{itemize}
%
%A \emph{Giry monad} is a monad $(\giry,\eta^{\giry},\mu^{\giry})$ on $\SB$ that is defined by

The \emph{sub-Giry monad} is the monad $(\giry,\eta^{\giry},\mu^{\giry})$ on $\Meas$ such that
\begin{itemize}
\item $\giry(X,\sigalg_X)=(\giry X, \sigalg_{\giry X})$, where 
%\begin{itemize}
%\item 
the underling set $\giry X$ is the set of all \emph{subprobability
      measures} on $(X,\sigalg_X)$. The latter means those measures which assign to the whole
      space $X$ a value in the unit interval $[0,1]$.
 \item 
%\item 
The $\sigma$-algebra $\sigalg_{\giry X}$ on $\giry X$ is the smallest $\sigma$-algebra such that, for all $S\in\sigalg_X$, 
the function $\text{ev}_S:\giry X\to[0,1]$ defined by $\text{ev}_S(P)=P(S)$ is measurable.
%\end{itemize}
\item $\giry f(\nu)(S)=\nu(f^{-1}(S))$ where $f:(X,\sigalg_X)\to(Y,\sigalg_Y)$ is measurable, $\nu\in\giry X$, and $S\in\sigalg_Y$.
\item $\eta^{\giry}_{(X,\sigalg_X)(x)}$ is given by the \emph{Dirac measure}: $\eta^{\giry}_{(X,\sigalg_X)}(x)(S)$ is $1$ if $x\in S$ and $0$ otherwise.
%$\eta^{\giry}_{(X,\sigalg_X)}(x)(S)=\begin{cases} 1 & (x\in S) \\ 0 & (x\notin S)\end{cases}$ 
%where $x\in X$ and $S\in\sigalg_X$, and
\item $\mu^{\giry}_{(X,\sigalg_X)}(\Psi)(S)=\int_{\giry (X,\sigalg_X)} \text{ev}_S \,d\Psi$ where 
$\Psi\in\giry^2 X$, $S\in\sigalg_X$ and $\text{ev}_S$ is defined as above.
\end{itemize}
\end{mydefinition}


A monad gives rise to a category called its \emph{Kleisli category} (see
e.g.~\cite{mac98categoriesfor}).

\begin{mydefinition}[Kleisli category $\Kl(T)$]
Given
%a category $\mathbb{C}$ and 
a monad $(T,\eta,\mu)$ on a category $\mathbb{C}$,
the \emph{Kleisli category} for $T$ is the category $\Kl(T)$ whose objects are the same as $\mathbb{C}$, 
and for each pair of objects $X,Y$, the homset $\Kl(T)(X,Y)$ is given by 
$\mathbb{C}(X,TY)$. 
An arrow  in $\Kl(T)$ is referred to as a \emph{Kleisli arrow}, and
 depicted by $X\kto Y$ for distinction. Note that it is nothing but
 an arrow $X\to TY$ in the base category $\mathbb{C}$.
% a set of \emph{Kleisli arrows}: namely,  
% $\Kl(T)(X,Y)=\mathbb{C}(X,TY)$.
% An arrow $f$ from $X$ to $Y$ in the Kleisli category is denoted by
 % $f:X\kto Y$ for distinction.

Moreover, for two sequential Kleisli arrows $f:X\kto Y$ and $g:Y\kto Z$,
their composition is given by $\mu_Z\circ Tg\circ f$ and denoted by $g\odot f$.
The \emph{Kleisli inclusion functor} is the functor $J:\mathbb{C}\to\Kl(T)$ 
%that is defined by
such that
$JX=X$ and $Jf=\eta_Y\circ f$ for $f:X\to Y$ in $\mathbb{C}$.
\end{mydefinition}

%Let $F$ be an endofunctor and $T$ be a monad on a category $\mathbb{C}$.
%For a monad $T$, 
It is known that a functor $F:\mathbb{C}\to\mathbb{C}$ canonically lifts
to a functor $\overline{F}:\Kl(T)\to\Kl(T)$, given that 
there exists a natural transformation $\lambda:FT\Rightarrow TF$ that is compatible 
with the unit and the multiplication of $T$.
Such a natural transformation is called a \emph{distributive law}. 
%Namely, for $X\in\Kl(T)$ (hence $X\in\mathbb{C}$), $\overline{F}X$ is given by $\overline{F}X=FX$ and
%for $f\in\Kl(T)(X,Y)=\mathbb{C}(X,TY)$, $\overline{F}f\in\Kl(T)(\overline{F}X,\overline{F}Y)=\mathbb(C)(FX,TFY)$ is given by 
%$\overline{F}f=\lambda_{Y}\circ Ff$.
For more details, see~\cite{mulry93liftingtheorems}.

%We  define orders on each of the homsets in $\Kl(\pow)$ and $\Kl(\giry)$ as follows.
Throughout this paper, we fix the orders on the homsets of $\Kl(\pow)$ and $\Kl(\giry)$ as follows.
\begin{mydefinition}[order enrichment of $\Kl(\pow)$ and $\Kl(\giry)$]\label{def:orderenrichment}
We define an order on $\Kl(\pow)(X,Y)$ by $f\sqsubseteq g \;\defarrow\; \forall x\in X.\, f(x)\subseteq g(x)$.
We define an order on $\Kl(\giry)(X,Y)$ by $f\sqsubseteq g \;\defarrow\;
 \forall x\in X.\,\forall A\in \sigalg_Y.\, f(x)(A)\leq g(x)(A)$. Here
 the last $\le$ is the usual order in the unit interval $[0,1]$.
%
%For $T=\pow$ and $\lift$, we define an order on $\Kl(T)(X,Y)$  as follows.
%\begin{itemize}
%\item For $T=\pow$, $f\sqsubseteq g \;\defarrow\; \forall x\in X.\, f(x)\subseteq g(x)$.
%\item For $T=\lift$, $f\sqsubseteq g \;\defarrow\; \forall x\in X.\, g(x)=\bot \Rightarrow f(x)=\bot$.
%\end{itemize}
%For $T=\giry$, we define an order on $\Kl(T)((X,\sigalg_X),(Y,\sigalg_Y))$ as follows.
%\begin{itemize}
%\item $f\sqsubseteq g \;\defarrow\; \forall x\in X.\,\forall A\in \sigalg_Y.\, f(x)(A)\leq g(x)(A)$.
%\end{itemize}
\end{mydefinition}

\begin{myremark}\label{rem:giryAndSubGiry}
 The sub-Giry monad $\giry$ is an adaptation of the \emph{Giry monad}
from~\cite{giry82categoricalapproach}; in the original Giry monad we only allow
(proper) \emph{probability measures}, i.e.\  measures that map the
 whole space to $1$. We work with the sub-Giry monad because, without
 this relaxation from probability to subprobability, the order
 structure in Def.~\ref{def:orderenrichment} is reduced to the equality.
\end{myremark}









\section{Infinite Traces, Kleisli Simulations and Coalgebras in $\Kl(T)$}\label{sec:weaklyfinalcoalg}
%\begin{mydefinition}[weakly final coalgebra]
%Let $\mathbb{D}$ be a category where each homset carries a partial order $\sqsubseteq$.
%Let $F$ be an endofunctor on $\mathbb{D}$.
%An $F$-coalgebra $z:Z\to FZ$ is \emph{weakly final} if for every $F$-coalgebra $c:X\to FX$ ,
%there exists the largest homomorphism $\trinf(c):X\to Z$.
%That is, $\trinf(c)$ is a homomorphism from $c$ to $z$ and moreover, for every homomorphism $f:X\to Z$ 
%from $c$ to $z$, $f\sqsubseteq \trinf(c)$ holds.
%\end{mydefinition}
In this section we review the categorical constructs, the relationship
among which lies at the heart of this paper. They are namely: coalgebraic
infinite trace semantics~\cite{jacobs04tracesemantics}, Kleisli
simulation~\cite{hasuo06genericforward,hasuo10genericforward,urabeH14genericforward}
and forward partial execution (FPE)~\cite{urabeH14genericforward}.

% In this section, 
% we introduce a categorical way to capture infinite trace semantics,
% that is originally introduced in~\cite{jacobs04tracesemantics} to capture 
% infinite trace semantics of nondeterministic systems.
% Moreover, we review categorical definitions of simulations~\cite{hasuo06genericforward}
% and transformations of systems that aids the simulation~\cite{urabeH14genericforward}.
%review categorical ways to 
%capture infinite trace semantics~\cite{jacobs04tracesemantics},
%define simulation~\cite{hasuo06genericforward}, and
%aid the simulation~\cite{urabeH14genericforward}.

%The following situation is identified in~\cite{jacobs04tracesemantics}.
The following situation is identified in~\cite{jacobs04tracesemantics}, (see also \S\ref{subsec:automchara}
and \S\ref{subsubsec:automcharaexception}):
the largest homomorphism 
to a certain coalgebra that we describe below
%to an $\overline{F}$-coalgebra $J\zeta$ in $\Kl(T)$ 
%where $\zeta$ is a final $F$-coalgebra in $\mathbb{C}$
is observed to coincide with the standard, conventionally defined notion of infinite language, for a variety of systems.
%This situation allows us to characterize the infinite trace semantics of a system as 
%the largest homomorphism.
% to an $\overline{F}$-coalgebra in $\Kl(T)$ that is obtained by
%lifting a final $F$-coalgebra in $\mathbb{C}$.
An instance of it is shown to arise, in~\cite{jacobs04tracesemantics}, when $\mathbb{C}=\Sets$, $T=\pow$ and $F$ is a
polynomial functor.
In~\S\ref{sec:powersetmonad} we will give another proof for this fact;
the new proof will
 serve our goal of showing soundness of backward simulations.
\begin{mydefinition}[infinite trace situation]\label{def:infiniteTraceSituation}
%Let $\mathbb{C}$ be a category, let $F$ be an endofunctor on $\mathbb{C}$ and $T$ be a monad on $\mathbb{C}$.
Let $F$ be an endofunctor and $T$ be a monad on a category $\mathbb{C}$.
We assume that each homset of the Kleisli category $\Kl(T)$ carries an order $\sqsubseteq$.
A functor $F$ and a monad $T$ constitute an \emph{infinite trace situation} with respect to $\sqsubseteq$ if they satisfy the following conditions.
\begin{itemize}
\item There exists a final $F$-coalgebra $\zeta:Z\to FZ$ in $\mathbb{C}$.
\item There exists a distributive law $\lambda:FT\Rightarrow TF$,
      yielding a lifting $\overline{F}$ on $\Kl(T)$ of $F$.
%, and hence $F$ can be lifted to an endofunctor $\overline{F}$ on $\Kl(T)$ (see~\cite{mulry93liftingtheorems} for more details).
\item For each coalgebra $c:X\kto \overline{F}X$ in $\Kl(T)$, the
      lifting $J\zeta:Z\kto \overline{F}Z$ of $\zeta$ admits the largest homomorphism.
That is, there exists a homomorphism $\trinf(c):X\kto Z$ from $c$ to
      $J\zeta$ such that, for any homomorphism $f$ from $c$ to $J\zeta$, $f\sqsubseteq\trinf(c)$ holds.
%(i.e.\ $\trinf(c)$ is a A coalgebra $J\zeta:Z\kto \overline{F}Z$ is weakly final.
\end{itemize}
%In this setting, for an $\overline{F}$-coalgebra $c:X\kto\overline{F}X$, 
%its \emph{coalgebraic infinite trace semantics} is the largest homomorphism  from $c$ to $J\zeta$ and denoted by $\trinf(c)$.
%The largest homomorphism $\trinf(c)$ is 
%Namely, for every homomorphism $f:X\kto Z$ from $c$ to $J\zeta$, $f\sqsubseteq \trinf(c)$ holds.
\end{mydefinition}



In~\cite{hasuo06genericforward,hasuo10genericforward,urabeH14genericforward}
we augment a coalgebra with an explicit arrow for initial states. The
resulting notion is called a $(T,F)$-system.

\vspace*{1mm}
\noindent\begin{minipage}{0.75\hsize}
\begin{mydefinition}[infinite trace semantics for $(T,F)$-systems~\cite{hasuo07generictrace,jacobs04tracesemantics}]
Let $\mathbb{C}$ be a category with a final object $1\in\mathbb{C}$.
A \emph{$(T,F)$-system} is a triple $\mathcal{X}=(X,s,c)$ consisting of
a \emph{state space} $X\in\mathbb{C}$,
a Kleisli arrow $s:1\kto X$ for  \emph{initial states}, and
%a Kleisli arrow 
$c :X\kto \overline{F}X$ for \emph{transition}.
%
%We further assume that each homset of the Kleisli category $\Kl(T)$ carries an order $\sqsubseteq$
%and an endofunctor $F$ and a monad $T$ on $\mathbb{C}$ constitute an
 %infinite trace situation with respect to the order.

Let us assume that the endofunctor $F$ and the monad $T$ on $\mathbb{C}$
 constitute an infinite trace situation.
%  with respect to 
% an order $\sqsubseteq$.
The \emph{coalgebraic infinite trace semantics} of a $(T,F)$-system
 $\mathcal{X}=(X,s,c)$ is the Kleisli arrow $\trinf(c)\odot s:1\kto Z$
 (see the diagram, in $\Kl(T)$, on the right).
\end{mydefinition}
\end{minipage}
%\begin{wrapfigure}{r}{3cm}
%\vspace*{-\intextsep}
\begin{minipage}{0.25\hsize}
\vspace{-5mm}
\begin{xy}
 \xymatrix@R=2.4em@C=1.8em{
 {\overline{F} X} \ar@{}[drr]|{=} \kar@{~>}[rr]^{\overline{F}(\trinf(c))} & & {\overline{F}Z}  \\
 {X} \kar[u]_{c}  \kar@{~>}^{\trinf(c)}[rr]  & & {Z} \kar[u]_{J\zeta}  \\
1 \kar[u]_{s} &   & }
\end{xy}
\end{minipage}\vspace{3mm}
%\end{wrapfigure}

Suppose that we are given two $(T,F)$-systems $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$.
Let us say we aim to prove the inclusion between infinite trace
semantics, that is, to show
% that the coalgebraic infinite trace semantics of $\mathcal{X}$ is included in that of $\mathcal{Y}$.
%Namely, we want to verify that
 $\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$
with respect to the order in the homset of $\Kl(T)$.
%This situation is called \emph{(coalgebraic) infinite trace inclusion}.
Our goal in this paper is to offer \emph{Kleisli simulations} as a sound means
to do so.

The notions of \emph{forward} and \emph{backward Kleisli simulation} are introduced in~\cite{hasuo06genericforward}
as a categorical generalization of fwd./bwd.\ simulations in~\cite{lynch95forwardand}.
They are defined as Kleisli arrows between (the state spaces of) two
$(T,F)$-system that are subject to certain inequalities---in short they
are \emph{lax/oplax coalgebra
homomorphisms}. In~\cite{hasuo06genericforward} they are
shown to be sound with respect to \emph{finite} trace semantics---the
languages of finite words, concretely; and the unique arrow to a
lifted initial algebra (that is a final coalgebra,
see~\cite{hasuo07generictrace} and the introduction), abstractly. In this
paper we are interested in their relation to \emph{infinite} trace semantics.
% They satisfy soundness, namely, 
% the existence of forward or backward Kleisli simulation witness coalgebraic \emph{finite} trace inclusion.
% In the later sections, we will see that forward and backward Kleisli simulation can be also used to witness infinite trace inclusion defined above.

\vspace{5mm}\noindent\begin{minipage}{0.8\hsize}
\begin{mydefinition}[fwd./bwd.\ Kleisli simulation~\cite{hasuo06genericforward}]\label{def:fwdbwdsim}
Let $F$ be an endofunctor and $T$ be a monad on $\mathbb{C}$ 
such that each homset of $\Kl(T)$ carries an order $\sqsubseteq$.
%that constitute an infinite trace situation with respect to an order $\sqsubseteq$.
Let $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$ be $(T,F)$-systems.

A \emph{forward Kleisli simulation} from $\mathcal{X}$ to $\mathcal{Y}$
 is a Kleisli arrow $f:Y\kto X$ that satisfies the following conditions
 (see the diagram).
\begin{equation*}
s\sqsubseteq f\odot t, \;\;\;\text{and}\;\;\;
c\odot f \sqsubseteq \overline{F}f\odot d.
\end{equation*}
We write $\mathcal{X}\fwd\mathcal{Y}$ if there exists a forward simulation from $\mathcal{X}$ to $\mathcal{Y}$.

A \emph{backward Kleisli simulation} from $\mathcal{X}$ to $\mathcal{Y}$
 is a Kleisli arrow 
 $b:X\kto Y$ that satisfies the following conditions
 (see the diagram).
\begin{equation*}
b\odot s\sqsubseteq t, \;\;\;\text{and}\;\;\;
\overline{F}b\odot c \sqsubseteq d\odot f.
\end{equation*}
We write $\mathcal{X}\bwd\mathcal{Y}$ if there exists a backward simulation from $\mathcal{X}$ to $\mathcal{Y}$.
\end{mydefinition}
\end{minipage}
\begin{minipage}{0.2\hsize}
\vspace*{-\intextsep}
 \begin{xy}
 \xymatrix@R=1.6em@C=1.2em{
 {\overline{F} X} \ar@{}[drr]|{\sqsubseteq}  & & {\overline{F} Y} \kar[ll]_{\overline{F} f } \\
 {X} \kar[u]_{c} \ar@{}[drr]|{\sqsubseteq} & & {Y} \kar[u]_{d} \ar_(.3){f}|-*\dir{|}[ll]  \\
 {} \ar@{}[ru]|(.10)*\dir{-} & {1} \ar`l[lu][lu]^(0){s}  \ar`r[ru][ru]_(0){t}  & \ar@{}[lu]|(.10)*\dir{-} }
 \end{xy}
 \vspace{5mm}
 \begin{xy}
 \xymatrix@R=1.6em@C=1.2em{
 {\overline{F} X} \ar@{}[drr]|{\sqsubseteq} \kar[rr]^{\overline{F} b } & & {\overline{F} Y}  \\
 {X} \kar[u]_{c} \ar@{}[drr]|{\sqsubseteq} \ar^(.3){b}|-*\dir{|}[rr]  & & {Y} \kar[u]_{d}  \\
 {} \ar@{}[ru]|(.10)*\dir{-} & {1} \ar`l[lu][lu]^(0)s \ar`r[ru][ru]_(0)t  & \ar@{}[lu]|(.10)*\dir{-} }
 \end{xy}
\end{minipage}
\vspace{1mm}


\emph{Forward partial execution} (FPE) is a transformation of a
$(T,F)$-system introduced in~\cite{urabeH14genericforward} for the
purpose of aiding discovery of Kleisli simulations.
Intuitively, it ``executes'' the given system by one step.

\begin{mydefinition}[FPE~\cite{urabeH14genericforward}]\label{def:fpe}
Let $F$ be an endofunctor and $T$ be a monad on $\mathbb{C}$.
\emph{Forward partial execution} (FPE) is a transformation that takes 
a $(T,F)$-system $\mathcal{X}=(X,s,c)$ 
%and $X_1\in X$ such that there exists $X_2\in X$ satisfying $X=X_1+X_2$
as an input
and returns a $(T,F)$-system 
$\mathcal{X}_{\FPE}=(\overline{F}X,c\odot s,\overline{F}c)$ 
as an output.
%Here, $c_i:X_i\kto \overline{F}X$ ($i\in\{1,2\}$) is defined by 
%$c_i=c\odot \kappa_i$ where $\kappa_i:X_i\kto X=X_1+X_2$ is an injection.
\end{mydefinition}
\noindent
It is shown in~\cite{urabeH14genericforward} that FPE is a valid technique for
establishing  inclusion of \emph{finite} trace semantics, in the
 technical senses of \emph{soundness} and \emph{adequacy}. Soundness 
asserts that  discovery of a Kleisli simulation after applying FPE
indeed witnesses trace inclusion between the original systems; adequacy
asserts that if there is a Kleisli simulation between the original
systems,
then there is too between the transformed systems. In this paper, naturally, we wish to 
establish the same results for \emph{infinite} trace semantics.

%  we can conclude finite trace inclusion between systems by checking finite trace inclusion between 
% the transformed systems.
%  ``increases'' forward and backward Kleisli simulations for finite trace in two sense---soundness and adequacy.
% %Formally, we can prove that FPE satisfies \emph{soundness} and \emph{adequacy}.
% Soundness means  we can conclude finite trace inclusion between systems by checking finite trace inclusion between 
% the transformed systems.
% Adequacy means  FPE does not destroy already existing Kleisli simulation.
% %In the later sections, 
% Later,
% we will see that FPE can be also used to increase Kleisli simulation for infinite trace.


\section{Systems with Nondeterministic Branching}\label{sec:powersetmonad}
%We call a \emph{nondeterministic tree automaton} for a
%$(\pow,\FSigma)$-system.
In the rest of the paper we develop a coalgebraic theory of infinite
traces and (Kleisli) simulations---the main contribution of the paper. We do so separately for the
nondeterministic
setting ($T=\pow$) and for the probabilistic one ($T=\giry$). This is
because of the difference in the constructions of infinite traces, and
consequently in the soundness proofs. 

In this section we focus on the
nondeterministic setting; we assume that $F$ is a polynomial functor.




\subsection{Construction of Infinite Traces
%the Largest Homomorphism for  $(\pow,\FSigma)$-systems
}\label{subsec:constructlargesthompow}
%It is already known that $\FSigma$ on $\Sets$ and the powerset  monad $\pow$ constitute an infinite trace situation~\cite{jacobs04tracesemantics}.
%%
%\begin{myproposition}\label{prop:satassumptionnondet}
%For a ranked alphabet $\Sigma$,
%the functor $F_{\Sigma}$ on $\Sets$ and the powerset monad $\pow$ satisfy assumptions~\ref{asm:generalconstructiontopFinalCoalg},
%\ref{asm:generalconstructiontopDistLaw}, \ref{asm:existenceWeaklyFinalCoalgFMonotone}, \ref{asm:generalconstructiontopGFPtransfinite}, and
%\ref{asm:generalconstructiontopTop} in Prop.~\ref{prop:generalconstructiontop}
%with respect to the order in Def.~\ref{def:orderenrichment}. 
%Therefore, $\FSigma$ and $\pow$ constitute an infinite trace situation. 
%\qed
%\end{myproposition}
%%
%In this paper, 
%
The following is already known from~\cite{jacobs04tracesemantics}. 
% Let $\Sigma$ be a ranked alphabet.
% It is already known that $\FSigma$ on $\Sets$ and the powerset  monad $\pow$ constitute an infinite trace situation~\cite{jacobs04tracesemantics}.
\begin{mytheorem}\label{thm:inftrsituationpow}
The combination of  polynomial $F$  and
$T=\pow$  constitute an
infinite trace situation (Def.~\ref{def:infiniteTraceSituation}).
% The functor $\FSigma$ and $\pow$ constitute an infinite trace situation.
%with respect to the order in Def.~\ref{def:orderenrichment}. 
\qed
\end{mytheorem}
The proof in~\cite{jacobs04tracesemantics} combines
fibrational intuitions with some constructions
that are
specific to $\Sets$. Here we present a
different proof. It exploits an order-theoretic structure of the
Kleisli category $\Kl(\pow)$; this will be useful later in showing
soundness of (restricted) backward simulations.
Our proof also paves the way to the probabilistic case in~\S\ref{sec:girymonad}.

% \begin{myproof}
% This holds because $\FSigma$ and $\pow$ satisfy assumptions in Prop.~\ref{prop:generalconstructiontop} below
% (Lem.~\ref{lem:satassumptionnondet}).
% \qed
% \end{myproof}

% In~\cite{jacobs04tracesemantics}, it is proved by giving an explicit set-theoretic definition of the largest homomorphism from each $\overline{\FSigma}$-coalgebra $c$ 
% to $J\zeta$ where $\zeta$ is a final $\FSigma$-coalgebra.
% In this section, we prove it in another way:
% we give a general construction of the largest homomorphism that may be applicable to another functors and monads.
% %by constructing the largest homomorphism categorically.
% This is because the soundness of \emph{total and image-finite backward simulation}---a kind of simulation introduced in 
% \S\ref{subsec:Klsimpow}---is proved along this general construction.
% Moreover, the construction of the largest homomorphism for the giry monad discussed in \S\ref{sec:girymonad}
% can be given by developing the construction in this section. 
%Let $F$ be a polynomial functor on $\Sets$ and $\pow$ be a powerset monad. 
%It is known that polynomial functor $F$ on $\Sets$ has $F$-final coalgebra $\zeta:X\to FX$~\cite{adamekK79leastfixed}.
%In this section, we show that $F$ and $\pow$ exhibit infinite trace situation.
%Namely, we show that for an arbitrary $\overline{F}$-coalgebra $c:X\kto \overline{F}X$ in $\Kl(\pow)$,
%there exists the largest homomorphism $\trinf(c):X\kto Z$ from $c$ to $J\zeta$.
%To this end, for a $\overline{F}$-coalgebra on $\Kl(\pow)$, we have to construct the largest homomorphism from $c:X\kto \overline{F}X$ to $J\zeta$.

\auxproof{
Let $F$ be an endofunctor and $T$ be a monad on a category $\mathbb{C}$.
Let $\zeta:Z\to FZ$ be a final $F$-coalgebra in $\mathbb{C}$.
We assume that there exists a distributive law $\lambda:FT\Rightarrow TF$, and hence $F$ can be lifted to 
an endofunctor $\overline{F}$ on $\Kl(T)$.
Moreover, we also assume that each homset of $\Kl(T)$ carries a partial order $\sqsubseteq$.
In general, in order to prove that $F$ and $T$ constitute an infinite trace situation, for each $\overline{F}$-coalgebra $c:X\kto \overline{F}X$,
we have to construct the largest homomorphism $\trinf(c):X\kto Z$ from $c$ to $J\zeta:Z\kto\overline{F}Z$.

Before giving the construction, 
we start with giving the construction of the \emph{smallest} homomorphism $\tr(c):X\kto Z$ from $c:X\kto\overline{F}X$ to $J\zeta$ in order to describe the basic idea. 
We assume the following conditions.
\begin{enumerate}
\item The homset $\Kl(T)(X,Z)$ has the smallest element $\bot_{X,Z}$.
\label{asm:constleast1}
\item Composition of arrows in $\Kl(T)$ and $\overline{F}$ are both monotone.
\label{asm:constleast2}
\item Each increasing sequence $f_0\sqsubseteq f_1\sqsubseteq \ldots$ in $\Kl(T)(X,Z)$ has the least upper bound $\bigsqcup_{i\in\omega}f_i$.
\label{asm:constleast3}
\item Composition of arrows in $\Kl(T)$ and $\overline{F}$'s action on
      arrows are both $\omega$-continuous  (i.e.\ preserves the least upper bound of an increasing sequence).
\label{asm:constleast4}
\end{enumerate}
\begin{floatingfigure}[r]{0pt}
\begin{center}
\begin{xy}
 \xymatrix@R=2.4em@C=1.7em{
 {\overline{F}X} \ar@{}[dr]|{\sqsubseteq} \kar[r]^{\overline{F}\bot_{X,Z}} & \overline{F}Z \\
 {X} \kar[u]^{c} \kar[r]_(.47){\bot_{X,Z}} & {Z} \kar[u]^{J\zeta}_{\cong} }
\end{xy}
\end{center}
\end{floatingfigure}
We define a function $\Phi_X:\Kl(T)(X,Z)\to\Kl(T)(X,Z)$ by $\Phi_X(f)=J\zeta^{-1}\odot\overline{F}f\odot c$ 
(recall that $\zeta$ is the final $F$-coalgebra hence it is isomorphism).
As it is assumed that composition and $\overline{F}$ are monotone (assumption~\ref{asm:constleast2}), $\Phi_X$ is also monotone.
Therefore, starting with $\bot\in\Kl(T)$ (see assumption~\ref{asm:constleast1}), we can construct an increasing sequence 
$\bot\sqsubseteq \Phi_X\bot\sqsubseteq\Phi_X^2\bot\ldots$.
By assumption~\ref{asm:constleast3}, this increasing sequence has the least upper bound $\Phi_X^{\omega}\bot=\bigsqcup_{i\in\omega}\Phi_X^i\bot$.
As it is assumed that composition and $\overline{F}$'s action on arrows are  continuous (assumption~\ref{asm:constleast4}), $\Phi_X$ is also continuous.
 Therefore $\Phi_X^{\omega}\bot$ is a fixed point of $\Phi_X$.
 By the construction, it is easy to see that it is the least fixed point.
 This immediately implies that $\Phi_X^{\omega}\bot$ is the smallest homomorphism $\tr(c):X\kto Z$ from $c$ to $J\zeta$.
  
The conditions~\ref{asm:constleast1} to \ref{asm:constleast4} are satisfied by $\FSigma$ and a powerset monad.
Hence we can construct the smallest homomorphism from an arbitrary $\overline{\FSigma}$-coalgebra to $J\zeta$ as above.
The obtained smallest homomorphism captures \emph{finite} trace semantics of the nondeterministic system

This construction of the smallest homomorphism gives us an idea about the construction of the largest homomorphism.


The largest homomorphism can be constructed by starting from the top element, and going downward.
More concretely, we assume the following conditions.
\begin{enumerate}
%\setcounter{enumi}{4}
\item The homset $\Kl(T)(X,Z)$ has the largest element $\top_{X,Z}$.
\label{asm:constleast5}
\item Composition of arrows in $\Kl(T)$ and $\overline{F}$'s action on arrows are monotone.
\label{asm:constleast6}
\item %Each decreasing sequence $f_0\sqsupseteq f_1\sqsupseteq \ldots$ in $\Kl(T)(X,Z)$ has the greatest lower bound $\bigsqcap_{i\in\omega}f_i$.
\label{asm:constleast7}
%\item Composition of arrows in $\Kl(T)$ and $\overline{F}$ are locally $\omega^{\text{op}}$-continuous respectively (i.e.\ preserves the greatest lower bound of a decreasing sequence).
%\label{asm:constleast8}
\end{enumerate}
%
\begin{floatingfigure}[r]{0pt}
\begin{center}
\begin{xy}
 \xymatrix@R=2.4em@C=1.7em{
 {\overline{F}X} \ar@{}[dr]|{\rotatebox{270}{\sqsubseteq}} \kar[r]^{\overline{F}\top_{X,Z}} & \overline{F}Z \\
 {X} \kar[u]^{c} \kar[r]_(.47){\top_{X,Z}} & {Z} \kar[u]^{J\zeta}_{\rotatebox{270}{$\cong$}} }
\end{xy}
\end{center}
\end{floatingfigure}
%Then in the similar manner to the construction of the smallest homomorphism, we can construct the largest homomorphism $\trinf(c):X\kto Z$ from $c$ to $J\zeta$.
As we will see later, these assumptions are satisfied by $\FSigma$ and $\pow$.
%
We define a function $\Phi_X:\Kl(T)(X,Z)\to\Kl(T)(X,Z)$ by $\Phi_X(f)=J\zeta^{-1}\odot\overline{F}f\odot c$ 
(recall that $\zeta$ is the final $F$-coalgebra hence it is isomorphism).
It is easy to see that $f:X\kto Z$ is a homomorphism from $c$ to $J\zeta$ iff it is a fixed point of $\Phi_X$.
Therefore our goal is to construct the greatest fixed point of $\Phi_X$.
As it is assumed that composition and $\overline{F}$ are monotone (assumption~\ref{asm:constleast6}), $\Phi_X$ is also monotone.
Therefore, starting with $\top_{X,Z}\in\Kl(T)$ (see assumption~\ref{asm:constleast5}), we can construct an decreasing sequence 
$\top_{X,Z}\sqsubseteq \Phi_X\top_{X,Z}\sqsubseteq\Phi_X^2\top_{X,Z}\ldots$.
By assumption~\ref{asm:constleast7}, this decreasing sequence has the least upper bound $\Phi_X^{\omega}\top_{X,Z}=\bigsqcup_{i\in\omega}\Phi_X^i\top_{X,Z}$.
Here, we should note that it is not assumed that composition in $\Kl(T)$ and $\overline{F}$ are  $\omega^{\text{op}}$-continuous.
This implies that $\Phi_X^{\omega}\top_{X,Z}$ is not necessarily a fixed point of $\Phi_X$.
We do not assume the local $\omega^{\text{op}}$-continuity of composition because it does not hold in $\Kl(\pow)$.
Indeed, for $f:X\kto Y$ and a decreasing sequence $(g_i:Y\kto Z)_{i\in\omega}$ of Kleisli arrows, 
$\bigl(\bigsqcap_{i\in\omega}g_i\bigr)\odot f(x)=\bigcup_{y\in f(x)}\bigcap_{i\in\omega} g_i(y)$
while $\bigsqcap_{i\in\omega}(g_i\odot f)(x)=\bigcap_{i\in\omega}\bigcup_{y\in f(x)} g_i(y)$. 
Therefore, instead of assuming local continuity of composition, we follow the construction of fixed point in~\cite{cousotC79constructiveversions}.
Namely, in the calculation of the greatest fixed point, we do not stop at $\Phi_X^{\omega}\top_{X,Z}$, but go further using transfinite induction.
As a result, we have the following proposition.
}
%As it is assumed that composition and $\overline{F}$ are locally continuous (assumption~\ref{asm:constleast4}), $\Phi_X$ is also continuous.
%Therefore $\Phi_X^{\omega}\bot$ is a fixed point of $\Phi_X$.
%By the construction, it is easy to see that it is the least fixed point.
%This immediately implies that $\Phi_X^{\omega}\bot$ is the smallest homomorphism $\tr(c):X\kto Z$ from $c$ to $J\zeta$.

\auxproof{
However, here is a problem: powerset monad does not satisfy assumption~\ref{asm:constleast8}.
Namely, composition of arrows in $\Kl(\pow)$ does not preserve the greatest lower bound.
Indeed, for $f:X\kto Y$ and a decreasing sequence $(g_i:Y\kto Z)_{i\in\omega}$ of Kleisli arrows, 
$\bigl(\bigsqcap_{i\in\omega}g_i\bigr)\odot f(x)=\bigcup_{y\in f(x)}\bigcap_{i\in\omega} g_i(y)$
while $\bigsqcap_{i\in\omega}(g_i\odot f)(x)=\bigcap_{i\in\omega}\bigcup_{y\in f(x)} g_i(y)$. 
%It is well known that the former is always included in the latter, but the latter is not always included in the former.
This implies that $\Phi^{\omega}(\top_{X,Z})$ is not a fixed point of $\Phi$.
To resolve this problem, we follow~\cite{cousotC79constructiveversions}.
Namely, in the calculation of the greatest fixed point, we do not stop at $\Phi_X^{\omega}\top_{X,Z}$, but go further using transfinite induction.
}%\end auxproof

In fact, our proof of Thm.~\ref{thm:inftrsituationpow} is stated
axiomatically, in the form of the following proposition. 
This is potentially useful in identifying new examples other than the
combination of polynomial $F$ and $T=\pow$ (although we have not yet
managed to do so). 
% (We note,
% however, that we are yet to find examples other than polynomial $F$ and $T=\pow$.) 
It is essentially  the construction of a greatest fixed
point by transfinite induction~\cite{cousotC79constructiveversions}.
 % we start from the largest element in $\Kl(T)(X,Z)$, and calculate the largest homomorphism by iteration. 
\begin{myproposition}\label{prop:generalconstructiontop}$\!\!\!{}^\dagger$
Let $\mathbb{C}$ be a category, $F$ be an endofunctor on $\mathbb{C}$, 
and $T$ be a monad on $\mathbb{C}$.
%such that each homset of $\Kl(T)$ carries an order $\sqsubseteq$. 
%Let $J:\mathbb{C}\to\Kl(T)$ be a Kleisli inclusion functor.
 Assume the following conditions.
%
\begin{enumerate}
\item There exists a final $F$-coalgebra $\zeta:Z\to FZ$ in $\mathbb{C}$.
 \label{asm:generalconstructiontopFinalCoalg}
\item There exists a distributive law $\lambda:FT\Rightarrow TF$,
      yielding a lifting $\overline{F}$ on $\Kl(T)$ of $F$.
%, and hence $F$ can be lifted to an endofunctor $\overline{F}$ on $\Kl(T)$. 
\label{asm:generalconstructiontopDistLaw}
%
\item For each 
%pair of objects $X$ and $Y$ in $\Kl(T)$, 
$X,Y\in\Kl(T)$,
the homset $\Kl(T)(X,Y)$ carries a partial order $\sqsubseteq$.
Moreover, $\overline{F}$'s action on arrows, as well as composition of
      arrows in $\Kl(T)$, is monotone with respect to this order.
%and continuous (i.e.\ preserves the greatest lower bound). 
\label{asm:generalconstructiontopMonotone}
%\item One of the following  is satisfied.
%\begin{enumerate}
%\item  For each pair of objects $X$ and $Y$ in $\Kl(T)$,  
%every decreasing sequence $f_0\sqsupseteq f_1\sqsubseteq \ldots$ in $\Kl(T)(X,Y)$ has the greatest lower bound $\bigsqcap_{i\in\omega}f_i$.
%%The Kleisli category $\Kl(T)$ is $\omega\mathbf{Cpo}$-enriched in downward manner
%Moreover, composition in $\Kl(T)$ and a lifted functor $\overline{F}$ are  locally $\omega^{\text{op}}$-continuous, respectively.
%(i.e.\ for each $g:Z\kto X$ and $h:Y\kto W$, we have 
%$g\odot (\bigsqcap_{i\in\omega}f_i)=\bigsqcap_{i\in\omega}(g\odot f_i)$, 
%$(\bigsqcap_{i\in\omega}f_i)\odot h=\bigsqcap_{i\in\omega}(f_i\odot h)$, and
%$F(\bigsqcap_{i\in\omega}f_i)=\bigsqcap_{i\in\omega}(F f_i)$).
%\label{asm:generalconstructiontopGFPcont}
\item For each 
$X,Y\in \Kl(T)$,
%pair of objects $X$ and $Y$ in $\Kl(T)$,  
every  (possibly transfinite) decreasing sequence in $\Kl(T)(X,Y)$ has the greatest lower bound. 
That is: let $\mathfrak{a}$ be a limit ordinal and
      $(g_{\mathfrak{i}}:X\kto Y)_{\mathfrak{i}<\mathfrak{a}}$  be 
a family of arrows such that $\mathfrak{i}\leq \mathfrak{j}$ implies
      $g_{\mathfrak{i}}\sqsupseteq g_{\mathfrak{j}}$. Then
$\bigsqcap_{\mathfrak{i}<\mathfrak{a}}g_{\mathfrak{i}}$ exists.
 
%  given a family of arrows $(g_{\mathfrak{i}}:X\kto Y)_{\mathfrak{i}<\mathfrak{a}}$ where $\mathfrak{a}$ is any limit ordinal and 
% $\mathfrak{i}\leq \mathfrak{j}$ implies $g_{\mathfrak{i}}\sqsupseteq g_{\mathfrak{j}}$,
% the greatest lower bound $\bigsqcap_{\mathfrak{i}<\mathfrak{a}}g_{\mathfrak{i}}$ exists.
%The Kleisli category $\Kl(T)$ is $\mathbf{Dcpo}$-enriched in downward manner
\label{asm:generalconstructiontopGFPtransfinite}
%\end{enumerate}
%\label{asm:generalconstructiontopGFP}
%(We denote this order by $\sqsubseteq$.)
\item For each  $X\in\mathbb{C}$, 
the homset $\Kl(T)(X,Z)$ has the largest element $\top_{X,Z}$.
%the lifting $J(!_X)$ of the unique arrow to the final object is the largest element of $\Kl(T)(X,1)$.
 \label{asm:generalconstructiontopTop}
%
\end{enumerate}
Then $T$ and $F$ constitute an infinite trace situation with respect to $\sqsubseteq$.%$J\zeta:X\kto\overline{F}Z$ is a weakly final $\overline{F}$-coalgebra.
\end{myproposition}


%\begin{myproof}
\proofmark
Let $c:X\kto \overline{F}X$ be an $\overline{F}$-coalgebra in $\Kl(T)$.
We shall construct the largest homomorphism $\trinf(c):X\kto Z$ from $c$
to $J\zeta$,
by transfinite induction.

\begin{wrapfigure}[4]{r}{0pt}
\vspace*{-\intextsep}
\begin{xy}
 \xymatrix@R=1.5em@C=3.2em{
 {\overline{F}X} \ar@{}[dr]|{\sqsubseteq} \kar[r]^{\overline{F}\top_{X,Z}} & \overline{F}Z \\
 {X} \kar[u]^{c} \kar[r]_(.47){\top_{X,Z}} & {Z} \kar[u]^{J\zeta}_{\cong} }
\end{xy}
\end{wrapfigure}
%\vspace{1mm}
%\noindent\begin{minipage}{0.8\hsize}
We define an endofunction $\Phi_X:\Kl(T)(X,Z)\to\Kl(T)(X,Z)$ by $\Phi_X(f)=J\zeta^{-1}\odot \overline{F}f\odot c$.
By the monotonicity of $\odot$ and $\overline{F}$ (Assumption~\ref{asm:generalconstructiontopMonotone}), 
$\Phi_X$ is also monotone.
%Moreover, 
For each ordinal $\mathfrak{a}$, we define
$\Phi_X^{\mathfrak{a}}(\top_{X,Z})\in\Kl(T)(X,Z)$ by the following transfinite induction.
% on $\mathfrak{a}$ 
%as follows:
\begin{itemize}
\item $\Phi_X^0(\top_{X,Z})= \top_{X,Z}$.
\item For a successor ordinal $\mathfrak{a}$, $\Phi_X^{\mathfrak{a}}(\top_{X,Z})=\Phi_X(\Phi_X^{\mathfrak{a}-1}(\top_{X,Z}))$.
\item For a limit ordinal $\mathfrak{a}$,
      $\Phi_X^{\mathfrak{a}}(\top_{X,Z})=\bigsqcap_{\mathfrak{i}<\mathfrak{a}}\Phi^{\mathfrak{i}}_X(\top_{X,Z})$. (cf.\
      Assumption~\ref{asm:generalconstructiontopGFPtransfinite})
\end{itemize}
%\end{minipage}
%\begin{minipage}{0.2\hsize}
%
%\end{minipage}\vspace{1mm}
%We define an ordinal $\mathfrak{l}$ as follows.
%
%If Assumption~\ref{asm:generalconstructiontopGFPcont} is satisfied, then $\Phi_X^{\mathfrak{a}}$ is well-defined for all $\mathfrak{a}\leq\omega$.
%We define $\mathfrak{l}$ by $\mathfrak{l}=\omega$.
%By the $\omega^{\text{op}}$-continuity of composition in $\Kl(T)$ and $\overline{F}$, $\Phi_X$ is also $\omega$-continuous.
%Therefore $\Phi_X^{\mathfrak{l}}(f)$ is the greatest fixed point of $\Phi_X$.
%
%If Assumption~\ref{asm:generalconstructiontopGFPtransfinite} is satisfied, then $\Phi_X^{\mathfrak{a}}$ is well-defined for all ordinal $\mathfrak{a}$.
%
We define $\mathfrak{l}$ to be the smallest ordinal such that the cardinality of $\mathfrak{l}$ is greater than that of $\Kl(T)(X,Z)$.
Then from~\cite{cousotC79constructiveversions}, $\Phi_X^{\mathfrak{l}}(\top_{X,Z})$ is the greatest fixed point of $\Phi_X$.
%
%As a consequence, $\Phi_X^{\mathfrak{l}}(f)$ is the greatest fixed point of $\Phi_X$ in both cases.
This immediately implies that $\Phi_X^{\mathfrak{l}}(\top_{X,Z})$ is the largest homomorphism from $c$ to $J\zeta$.
\qed
%\end{myproof}
\vspace{2mm}

Note that the local continuity of  composition in $\Kl(T)$ is not assumed.
This is because $\pow$---our choice for $T$ in this section---does not satisfy it.
Indeed,  consider $f:X\kto Y$ and a decreasing sequence $(g_i:Y\kto
Z)_{i\in\omega}$, both
in $\Kl(\pow)$.
Then we have
$\bigl(\bigsqcap_{i\in\omega}g_i\bigr)\odot f(x)=\bigcup_{y\in f(x)}\bigcap_{i\in\omega} g_i(y)$
while $\bigsqcap_{i\in\omega}(g_i\odot
f)(x)=\bigcap_{i\in\omega}\bigcup_{y\in f(x)} g_i(y)$, and these two are
not equal in general (e.g.\ Example~\ref{example:needtransfinite}).
 This failure of continuity prevents us from applying the (simpler)
 \emph{Kleene fixed-point theorem}, in which induction terminates after
 $\omega$ steps.
% This restriction prevents us from stopping the iteration at $\mathfrak{i}=\omega$ and require going further using transfinite induction.
There does exist a nondeterministic automaton for which the largest
homomorphism is obtained after steps bigger than $\omega$; see
Example~\ref{example:needtransfinite}.
% where we need transfinite induction to obtain the largest homomorphism (e.g.\ Example~\ref{example:needtransfinite}).

%As a result of employing transfinite induction,
%The functor and the monad we use in this section---$\FSigma$ and $\pow$---satisfy the Assumption of the above proposition.
%The assumption of the above proposition are satisfied by $\FSigma$ and $\pow$.

It is easy to check that all the assumptions in
Prop.~\ref{prop:generalconstructiontop} are satisfied by polynomial $F$
and $T=\pow$. This yields Thm.~\ref{thm:inftrsituationpow}.
We can also show that the resulting coalgebraic infinite trace semantics coincides
with 
the usual definition of (infinite) tree languages for nondeterministic
systems. See \S\ref{subsec:automcharapow} for  details.
%automata-theoretically defined infinite semantics (see~\S\ref{subsec:automcharapow} for more details).






\subsection{Kleisli Simulations for Nondeterministic Systems}\label{subsec:Klsimpow}
% In this section, we show that forward and backward Kleisli simulations (Def.~\ref{def:fwdbwdsim}) can be used to witness infinite trace inclusion.
%In~\cite{hasuo06genericforward}, notions of forward and backward Kleisli simulation were defined for $(T,F)$-system where 
%each homset of $\Kl(T)$ carries an order.
%Then it is proved that Kleisli simulation satisfy soundness---existence
%of forward or backward Kleisli simulation implies finite trace inclusion that is coalgebraically defined using (not-weakly) final $\overline{F}$-coalgebra.
%As we also model a system as a $(T,F)$-system and assume that homsets of $\Kl(T)$ is ordered,
%we can define forward and backward simulation exactly in the same manner as~\cite{hasuo06genericforward}.
%
%
%We have to check that these two simulations satisfy a desired property---soundness---with respect to infinite trace semantics.
%It is easy to prove the soundness of forward simulation with respect to
%infinite trace semantics.
\subsubsection{Forward Simulations}
Soundness of forward simulation is not hard; we do not have to go
into the  construction in Prop.~\ref{prop:generalconstructiontop}.
% because it can be proved without going through a general construction in Prop.~\ref{prop:generalconstructiontop}.

\begin{mytheorem}\label{thm:soundnessFwdpow}
Given  two $(\pow,F)$-systems $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$,
$\mathcal{X}\fwd\mathcal{Y}$ implies $\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$.
\qed
\end{mytheorem}

The proof, again, is formulated as a general result, singling out some
sufficient axioms. 

% \begin{myproof}
% It is known that $\FSigma$ and $\pow$ satisfy the assumptions in Lem.~\ref{lem:soundnessFwd}~\cite{hasuo07generictrace} below.
% Therefore immediate from Thm.~\ref{thm:soundnessFwd}.
% \qed
% \end{myproof}

%
\begin{mylemma}\label{lem:soundnessFwd}$\!\!\!{}^\dagger$
Let $F$ be an endofunctor and $T$ be a monad on $\mathbb{C}$; assume
 further that they constitute an infinite trace situation (with respect to $\sqsubseteq$).
We assume the following conditions.
\begin{enumerate}
\item Each homset of $\Kl(T)$ is $\omega$-complete, that is, each
      increasing $\omega$-sequence in it has the lub. 
\label{asm:soundnessFwd1}
% \item Composition of arrows in $\Kl(T)$ and $\overline{F}$ are respectively monotone. %(i.e.\ composition of Kleisli arrows and $\overline{F}$ preserve the order).
% \label{asm:soundnessFwd2}
\item Composition $\odot$ of arrows in $\Kl(T)$ and $\overline{F}$'s
      action on arrows are both  $\omega$-continuous (i.e.\ they
      preserve the lub.\ of an increasing $\omega$-sequence). It follows
      that they are both  monotone.
\label{asm:soundnessFwd3}
\end{enumerate}
For two $(T,F)$-systems  $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$,
if $f:Y\kto X$ is a forward simulation from $\mathcal{X}$ to
 $\mathcal{Y}$, then $\trinf(c)\odot f\sqsubseteq \trinf(d)$.
As a consequence we have
% We assume the conditions in Lem.~\ref{lem:soundnessFwd}.
% For  two $(T,F)$-system $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$,
% $\mathcal{X}\fwd\mathcal{Y}$ implies
 $\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$.
\end{mylemma}



%\noindent\begin{minipage}{0.68\hsize}
%\begin{myproof}


\proofmark
Let $\zeta:Z\to FZ$ be a final $F$-coalgebra in $\mathbb{C}$.
%Let $f:Y\kto X$ be a forward Kleisli simulation from $\mathcal{X}$ to $\mathcal{Y}$. %(see the above diagram). 
We define a function $\Phi_Y:\Kl(T)(Y,Z)\to\Kl(T)(Y,Z)$ by
$\Phi_Y(g)=J\zeta^{-1}\odot \overline{F}g\odot d$; note that
$\zeta$ is a final coalgebra and hence an isomorphism.
% (recall that weakly final coalgebra $J\zeta$ is an isomorphism).
Then 
%\end{myproof}\end{minipage}
%\begin{minipage}{0.32\hsize}
%\vspace*{-6mm}
%{\small
%\begin{xy}
% \xymatrix@R=2.0em@C=1.3em{
% {\overline{F}Y} \ar@{}[drr]|{\sqsupseteq} \kar[rr]_{\overline{F} f } \kar@/^4mm/[rrrr]^{\overline{F}(\trinf(d))} & & {\overline{F} X} \ar@{}[drr]|{=} \kar[rr]_{\overline{F}(\trinf(c))} & & {\overline{F} Z} \\
% {Y} \kar@/_4mm/[rrrr]_(.8){\trinf(d)} \kar[u]_{d} \ar@{}[drr]|{\sqsupseteq} \ar^{f}|-*\dir{|}[rr]  & & {X} \kar[u]_{c} \kar[rr]^{\trinf(c)} & & {Z} \kar[u]_{J\zeta} \\
% {} \ar@{}[ru]|(.10)*\dir{-} & {1} \ar`l[lu][lu]^(0)t \ar`r[ru][ru]_(0)s  & \ar@{}[lu]|(.10)*\dir{-} & & }
% \end{xy}}
%\end{minipage}
\begin{align*}
\trinf(c)\odot f 
%&= J\zeta^{-1}\odot J\zeta \odot \trinf(c)\odot f & (J\zeta\text{ is an isomorphism}) \\
&=J\zeta^{-1}\odot \overline{F}(\trinf(c))\odot c \odot f & (\text{$\trinf(c)$ is a homomorphism}) \\ 
&\sqsubseteq \Phi_Y(\trinf(c)\odot f) & (\text{$f$ is a fwd.\
 sim., and the definition of $\Phi_{Y}$}).
%&\sqsubseteq (J\zeta)^{-1}\odot \overline{F}(\trinf(c))\odot \overline{F}f \odot f & (\trinf(c)\text{ is a homomorphism}) \\ 
%&= \Phi_Y(\trinf(c)\odot f) & (\text{by definition}).
\end{align*}

\begin{wrapfigure}[8]{r}{0pt}
\vspace*{-\intextsep}
{\small
\begin{xy}
 \xymatrix@R=2.0em@C=1.3em{
 {\overline{F}Y} \ar@{}[drr]|{\sqsupseteq} \kar[rr]_{\overline{F} f } \kar@/^4mm/[rrrr]^{\overline{F}(\trinf(d))} & & {\overline{F} X} \ar@{}[drr]|{=} \kar[rr]_{\overline{F}(\trinf(c))} & & {\overline{F} Z} \\
 {Y} \kar@/_4mm/[rrrr]_(.8){\trinf(d)} \kar[u]_{d} \ar@{}[drr]|{\sqsupseteq} \ar^{f}|-*\dir{|}[rr]  & & {X} \kar[u]_{c} \kar[rr]^{\trinf(c)} & & {Z} \kar[u]_{J\zeta} \\
 {} \ar@{}[ru]|(.10)*\dir{-} & {1} \ar`l[lu][lu]^(0)t \ar`r[ru][ru]_(0)s  & \ar@{}[lu]|(.10)*\dir{-} & & }
 \end{xy}}
%\hbox{}
% \vspace{-30pt}
\end{wrapfigure}
%
By the assumption that $\overline{F}$ and the composition are   monotone,
$\Phi_Y$ is also monotone.
Therefore by repeatedly applying  $\Phi_Y$ to the both sides of the above inequality, we obtain an increasing sequence
$\trinf(c)\odot f\sqsubseteq\Phi_Y(\trinf(c)\odot f)\sqsubseteq\Phi_Y^2(\trinf(c)\odot f)\sqsubseteq\cdots$ in $\Kl(T)(Y,Z)$.

%By the assumption that each homset of $\Kl(T)$ is $\omega$-complete, % in upward manner,
As $\Kl(T)(Y,Z)$ is $\omega$-complete,
the least upper bound $\bigsqcup_{i<\omega}\Phi^i(\trinf(c)\odot f)$ exists.
By the assumption that $\overline{F}$ and $\odot$ are both locally $\omega$-continuous,
$\Phi_Y$ is also $\omega$-continuous.
Therefore
%$\Phi(\bigsqcup_{i<\omega}\Phi^i(\trinf(c)\odot f))=\bigsqcup_{i<\omega}\Phi^{i+1}(\trinf(c)\odot f)=\bigsqcup_{i<\omega}\Phi^i(\tr^{\infty}_{z}(c)\odot f)$ 
%holds.
%This means that 
%$\Phi_Y^{\omega}(\trinf(c)\odot f)$ 
$\bigsqcup_{i<\omega}\Phi^i(\trinf(c)\odot f)$ is a fixed point of $\Phi_Y$, and hence a homomorphism from $d$ to $J\zeta$.
%
As $\trinf(d)$ is the largest homomorphism from $d$ to $J\zeta$, this implies 
$\trinf(c)\odot f \sqsubseteq \bigsqcup_{i<\omega}\Phi^i(\trinf(c)\odot f)\sqsubseteq\trinf(d)$.
Combining with the assumption that $f$ is a forward simulation (its
condition on initial states), 
we have $\trinf(c)\odot s \sqsubseteq \trinf(c)\odot f\odot t \sqsubseteq \trinf(d)\odot t$.
%Hence
%\begin{align*}
%\trinf(c)\odot s
%&\sqsubseteq \trinf(c)\odot f\odot t &(s \text{ is a fwd. simulation})\\
%&\sqsubseteq \trinf(d)\odot t &(\text{above result})
%\end{align*}
%This concludes the proof.
\qed
\vspace{3mm}
%\end{myproof}


% This lemma immediately implies the following---soundness of forward Kleisli simulation.
% %
% \begin{mytheorem}\label{thm:soundnessFwd}
% We assume the conditions in Lem.~\ref{lem:soundnessFwd}.
% For  two $(T,F)$-system $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$,
% $\mathcal{X}\fwd\mathcal{Y}$ implies $\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$.
% \qed
% \end{mytheorem}
%

It is known from~\cite{hasuo07generictrace} that the combination of polynomial $F$ and
$T=\pow$ satisfy the conditions of Lem.~\ref{lem:soundnessFwd}. Hence we
obtain Thm.~\ref{thm:soundnessFwdpow}, i.e.\ soundness of fwd.\
simulation in the nondeterministic setting.


% Therefore existence of a forward Kleisli simulation witnesses coalgebraic infinite trace inclusion.
%We can apply these result to $\FSigma$ and $\pow$.
%
%\begin{myproposition}
%Moreover, it is known that 
%%a polynomial functor 
%$F_{\Sigma}$ 
%%on $\Sets$ 
%and 
%%a powerset monad 
%$\pow$ satisfy assumptions in Lem.~\ref{lem:soundnessFwd}~\cite{hasuo06genericforward}.
%
%It is known that polynomial functors on $\Sets$ and $\pow$ satisfy the assumptions in 
%Lem.~\ref{lem:soundnessFwd}~\cite{hasuo07generictrace}.
%Therefore, we can use forward Kleisli simulation to verify infinite language inclusion between nondeterministic tree automata.
%%
%\begin{mylemma}\label{prop:satassumptionsfwdpow}
%The functor $\FSigma$  and $\pow$ satisfy assumptions~\ref{asm:soundnessFwd1}--\ref{asm:soundnessFwd3}
%in Lem.~\ref{lem:soundnessFwd}.
%\qed
%\end{mylemma}
%\end{myproposition}
%
%Hence we can use forward Kleisli simulation to verify infinite language inclusion between nondeterministic tree automata.



\auxproof{
Next we have to show the soundness of backward Kleisli simulation with respect to infinite trace. 
Before that, we start with showing the soundness of \emph{forward} Kleisli simulation with respect to finite trace that is captured via the smallest homomorphism.
Let $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$ be $(T,F)$-systems and $f:Y\kto X$ be a forward simulation from $\mathcal{X}$ to $\mathcal{Y}$.
Moreover, let $\zeta:Z\to FZ$ be a final $F$-coalgebra.
We assume the conditions \ref{asm:constleast1}--\ref{asm:constleast4} in \S\ref{subsec:constructlargesthompow}.
Recall that then the smallest homomorphisms $\tr(c):X\kto Z$ from $c$ to $J\zeta$ and $\tr(d):Y\kto Z$ from $d$ to $J\zeta$ can be constructed 
by repeatedly applying $\Phi_X$ and $\Phi_Y$ to $\bot_{X,Z}$ and $\bot_{Y,Z}$.
We further assume the following additional conditions.
\begin{enumerate}
\setcounter{enumi}{8}
\item A composition in $\Kl(T)$ is left-strict with respect to $\bot$. (i.e.\ For each $g:Y\kto X$, $\bot_{X,Z}\odot g=\bot_{Y,Z}$.)
\label{asm:forwardleast9}
\end{enumerate}
%
\begin{floatingfigure}[r]{0pt}
\begin{center}
\begin{xy}
 \xymatrix@R=2.4em@C=1.7em{
{\overline{F}Y} \ar@{}[dr]|{\sqsupseteq} \kar[r]_{\overline{F}f} \kar@/^5mm/[rr]^{\overline{F}\bot_{Y,Z}} & {\overline{F}X} \ar@{}[dr]|{\sqsubseteq} \kar[r]_{\overline{F}\bot_{X,Z}} & \overline{F}Z \\
 {Y} \kar[u]^{d} \kar[r]^{f} \kar@/_5mm/[rr]_{\bot_{Y,Z}} &  {X} \kar[u]^{c} \kar[r]^(.47){\bot_{X,Z}}  & {Z} \kar[u]^{J\zeta}_{\cong} }
\end{xy}
\end{center}
\end{floatingfigure}
From the additional assumption~\ref{asm:forwardleast9}, $\bot_{X,Z}\odot f=\bot_{Y,Z}$ holds.
Repeatedly applying $\Phi_Y$ to the both sides, %we have $\Phi_Y^{i}(\bot_{X,Z}\odot f)= \Phi_Y^{i}(\bot_{Y,Z})$ for all $i\in\omega$.
%Moreover, 
by the assumption that $f:Y\kto X$ is a forward simulation, we can inductively prove $\Phi_X^i(\bot_{X,Z})\odot f\sqsubseteq\Phi_Y^{i}(\bot_{Y,Z})$.
We calculate the least upper bound of the both sides. Then by the assumption that composition is locally continuous, we have 
\begin{equation*}
\tr(c)\odot f=\Phi_X^{\omega}(\bot_{X,Z})\odot f\sqsubseteq\Phi_Y^{\omega}(\bot_{Y,Z})=\tr(d)\,.
\end{equation*}
This equation immediately implies $\tr(c)\odot s\sqsubseteq\tr(d)\odot t$---coalgebraic finite trace inclusion.


By reversing the orders appearing in the above discussion, we can prove the soundness of backward simulation with respect to infinite trace
that is captured via the largest homomorphism.
Namely, in addition to  Assumptions~\ref{asm:constleast5}--\ref{asm:constleast8}, we assume the following extra assumption.
\begin{enumerate}
\setcounter{enumi}{9}
\item A composition in $\Kl(T)$ is left-strict with respect to $\top$. (i.e.\ For each $g:Y\kto X$, $\top_{X,Z}\odot g=\top_{Y,Z}$.)
\label{asm:backwardleast10}
\end{enumerate}
%
\begin{floatingfigure}[r]{0pt}
\begin{center}
\begin{xy}
 \xymatrix@R=2.4em@C=1.7em{
{\overline{F}X} \ar@{}[dr]|{\sqsubseteq} \kar[r]_{\overline{F}b} \kar@/^5mm/[rr]^{\overline{F}\top_{X,Z}} & {\overline{F}Y} \ar@{}[dr]|{\sqsubseteq} \kar[r]_{\overline{F}\bot_{Y,Z}} & \overline{F}Z \\
 {X} \kar[u]^{c} \kar[r]^{b} \kar@/_5mm/[rr]_{\top_{X,Z}} &  {Y} \kar[u]^{d} \kar[r]^(.47){\bot_{Y,Z}}  & {Z} \kar[u]^{J\zeta}_{\cong} }
\end{xy}
\end{center}
\end{floatingfigure}
Then in the similar manner to the above discussion, we can prove that
the existence of backward simulation $b:X\kto Y$ from $\mathcal{X}$ to $\mathcal{Y}$ implies 
$\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$---infinite trace inclusion.
%
However, here is a problem: $\FSigma$ and $\pow$ do not satisfy the assumptions.
%More concretely, we cannot prove the soundness of backward simulation for $\FSigma$ and $\pow$ because of the following reasons.
First, while composition is left-strict with respect to $\bot$ in $\Kl(\pow)$, it is not left-strict with respect to $\top$.
For example, let $X=Y=Z=\{\*\}$ and define $b:X\kto Y$ by $b(*)=\emptyset$.
Then $\top_{Y,Z}\odot b\neq\top_{X,Z}$ because $\top_{Y,Z}\odot b(*)=\emptyset\neq \{*\}=\top_{X,Z}(*)$.
Second, as we have seen in \S\ref{subsec:constructlargesthompow}, composition in $\Kl(\pow)$ does not preserve the greatest lower bound.
Hence we cannot derive $\trinf(c)=\bigsqcup_{\mathfrak{i}<\mathfrak{a}}\Phi_X^{\mathfrak{i}}(\top_{X,Z}) \sqsubseteq \left(\bigsqcup_{\mathfrak{i}<\mathfrak{a}}\Phi_X^{\mathfrak{i}}(\top_{Y,Z})\right)\odot b=\trinf(d)\odot b$.
there exists a pair of nondeterministic automata where backward simulation exists but infinite language inclusion does not hold
(see Example~\ref{example:bwdsimnotsound} for example).
To recover this situation, we put some restriction on the definition of backward simulation.
}

\subsubsection{Backward Simulations}\label{subsubsec:powbwd}
Next we 
%focus on
%want to show the soundness of 
wish to establish soundness of
\emph{backward} Kleisli simulations
with respect to \emph{infinite} traces (for finite traces it is shown in~\cite{hasuo06genericforward}). 
In fact, the desired soundness fails in general---a counterexample is in
Example~\ref{example:bwdsimnotsound}. It turns out that we can impose
certain restrictions on backward Kleisli simulations and ensure soundness.

% counterexamples show that soundness of backward Kleisli simulation with respect to infinite trace 
% cannot be proved as long as we put an assumption on a functor and a monad so that $\FSigma$ and $\pow$ are allowed.
% More concretely, there exist pairs of nondeterministic automata where a backward simulation exists but infinite language inclusion does not hold (see Example~\ref{example:bwdsimnotsound}).
% To recover this situation, we put some restriction on the definition of backward simulation.


\begin{mydefinition}[totality, image-finiteness, TIF-backward simulation]\label{def:restrictedbwdsimpow}
%Let $F$ be an endofunctor and $T$ be a monad on $\mathbb{C}$ that 
%satisfy the conditions in Prop.~\ref{prop:generalconstructiontop} wrt.\ $\sqsubseteq$. 
%constitute an infinite trace situation with respect to an order $\sqsubseteq$.
%
Let $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$ be
 $(\pow,F)$-systems. 
A backward simulation $b:X\kto Y$ 
from $\mathcal{X}$ to $\mathcal{Y}$ is \emph{total} if
$b(x)\neq\emptyset$ for all $x\in X$; it is \emph{image-finite} if
 $b(x)$ is finite for all $x\in X$.
If $b$ satisfies both of the two conditions, it is called
a \emph{TIF-backward simulation}. 
% a \emph{total and image finite backward Kleisli simulation} (\emph{TIF-backward simulation} for short) is 
% a backward simulation $b:X\kto Y$ 
% from $\mathcal{X}$ to $\mathcal{Y}$ 
% that satisfies the following conditions.
% \begin{enumerate}
% \item $b(x)\neq\emptyset$ for all $x\in X$.
% \label{item:restrictedbwdsim2pow}
% \item $b(x)$ is finite for all $x\in X$.
% %Assumption~\ref{asm:generalconstructiontopGFPtransfinite} in Prop.~\ref{prop:generalconstructiontop} is satisfied.
% %Precomposing $b:X\kto Y$ preserves the greatest lower bound of a decreasing transfinite sequence.
% %Namely, for $A\in\Kl(T)$ and a family of Kleisli arrows $(g_{\mathfrak{i}}:Y\kto A)_{\mathfrak{i}<\mathfrak{a}}$ where $\mathfrak{a}$ is a limit ordinal and $\mathfrak{i}\leq \mathfrak{j}$ implies $g_{\mathfrak{i}}\sqsupseteq g_{\mathfrak{j}}$,
% %%$g_0\sqsupseteq g_1 \sqsupseteq g_2\ldots :Y\kto A$, 
% %we have $\bigsqcap_{i\in\mathfrak{a}}(g_i\odot b)=(\bigsqcap_{i\in\mathfrak{a}}g_i)\odot b$.
% %\item Precomposing $b:X\kto Y$ preserves the greatest fixed point of $\Psi$. 
% %Namely, if $f:Y\kto 1$ is the greatest fixed point of $\Psi_Y$, then $f\odot b:X\kto 1$ is the greatest fixed point of $\Psi_X$.
% \label{item:restrictedbwdsim3pow}
% %\end{enumerate}
% %\label{item:restrictedbwdsim3}
% %\item Precomposing $b:X\kto Y$ preserves the largest mediating arrow in assumption~\ref{asm:existenceWeaklyFinalCoalgWeak2Lim} in 
% %Prop.~\ref{prop:generalconstructionnotop}.
% %Namely, if $l_{\max}:Y\kto Z$ is the largest mediating arrow from a cone $(Y,(\delta_i:Y\kto \overline{F}^i1))$ to 
% %a weak 2-limit $(Z,(J\gamma_i:Z\kto \overline{F}^i 1))$, then 
% %$l_{\max}\odot b:X\kto Z$ is the largest mediating arrow from a cone $(X,(\delta_i\odot b:X\kto \overline{F}^i1))$ to 
% %the weak 2-limit. 
% %\label{item:restrictedbwdsim4}
% \end{enumerate}
We write $\mathcal{X}\tifbwd\mathcal{Y}$ if there exists a TIF-backward simulation from $\mathcal{X}$ to $\mathcal{Y}$.
\end{mydefinition}


\begin{mytheorem}[soundness of $\tifbwd$]\label{thm:soundnesstifbwdpow}
For  two $(\pow,F)$-systems $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$,
$\mathcal{X}\tifbwd\mathcal{Y}$ implies $\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$.
\qed
\end{mytheorem}

The proof of Thm.~\ref{thm:soundnesstifbwdpow} is, yet again, via the following axiomatic development.


%
\begin{mydefinition}[TIF-backward simulation, generally]\label{def:restrictedbwdsimtop}$\!\!\!{}^\dagger$
Let $F$ be an endofunctor and $T$ be a monad on $\mathbb{C}$ that 
satisfy the conditions in Prop.~\ref{prop:generalconstructiontop} wrt.\ $\sqsubseteq$. 
%constitute an infinite trace situation with respect to an order $\sqsubseteq$.
%
For  two $(T,F)$-systems $\mathcal{X}=(X,s,c)$ and
 $\mathcal{Y}=(Y,t,d)$,
a \emph{TIF-backward simulation} 
from $\mathcal{X}$ to $\mathcal{Y}$ 
% a \emph{total and image finite backward Kleisli simulation} (\emph{TIF-backward simulation} for short) is 
is a backward simulation $b:X\kto Y$ 
that satisfies the following conditions.
\begin{enumerate}
\item The arrow $b:X\kto Y$ satisfies $\top_{Y,Z}\odot b = \top_{X,Z}$. \label{item:restrictedbwdsim2top}
%\item One of the following is satisfied.
%\begin{enumerate}
%\item Assumption~\ref{asm:generalconstructiontopGFPcont} in Prop.~\ref{prop:generalconstructiontop} is satisfied. \label{item:restrictedbwdsim3cont}
\item
\label{item:restrictedbwdsim3top}  
Precomposing $b:X\kto Y$ preserves the greatest lower bound of any decreasing transfinite sequence.
That is, let   $A\in\Kl(T)$, $\mathfrak{a}$ be a limit ordinal, and
       $(g_{\mathfrak{i}}:Y\kto A)_{\mathfrak{i}<\mathfrak{a}}$ be 
a family of Kleisli arrows such that $\mathfrak{i}\leq \mathfrak{j}$
       implies $g_{\mathfrak{i}}\sqsupseteq g_{\mathfrak{j}}$. Then
%$g_0\sqsupseteq g_1 \sqsupseteq g_2\ldots :Y\kto A$, 
we have $\bigsqcap_{\mathfrak{i}\in\mathfrak{a}}(g_\mathfrak{i}\odot
       b)=(\bigsqcap_{\mathfrak{i}\in\mathfrak{a}}g_\mathfrak{i})\odot
       b$.
\end{enumerate}
We write $\mathcal{X}\tifbwd\mathcal{Y}$ if there exists a TIF-backward simulation from $\mathcal{X}$ to $\mathcal{Y}$.
\end{mydefinition}

Assumption~\ref{item:restrictedbwdsim3top} of
Def.~\ref{def:restrictedbwdsimtop} resembles how ``finiteness'' is
formulated in category theory, e.g.\ in the definition of \emph{finitary} objects.

This general TIF-backward simulation satisfies soundness. For its proof
we have to look into the inductive 
% It can be proved along the
 construction of the largest homomorphism in \S\ref{subsec:constructlargesthompow}.

\begin{mylemma}\label{lem:soundnessRBwdtop}$\!\!\!{}^\dagger$
Let $F$ and $T$ be
%We assume that $F$ and $T$ 
as in Prop.~\ref{prop:generalconstructiontop}.
% wrt.\ $\sqsubseteq$. 
% that constitute an infinite trace situation with respect to an order $\sqsubseteq$.
%We assume that the weakly final coalgebra is obtained by using Prop.~\ref{prop:generalconstructiontop}. 
For two $(T,F)$-systems $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$, 
$\mathcal{X}\tifbwd\mathcal{Y}$ (in the sense of Def.~\ref{def:restrictedbwdsimtop}) implies $\trinf(c)\sqsubseteq
 \trinf(d)\odot b$.
 Furthermore it follows that  $\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$.
\end{mylemma}
%
%\begin{minipage}{0.7\hsize}

\begin{wrapfigure}[8]{r}{0pt}
\vspace*{-\intextsep}
\begin{xy}
 \xymatrix@R=2.3em@C=2.8em{
{\overline{F}X} \ar@{}[dr]|{\sqsubseteq} \kar[r]_{\overline{F}b} \kar@/^5mm/[rr]^{\overline{F}\top_{X,Z}} & {\overline{F}Y} \ar@{}[dr]|{\sqsubseteq} \kar[r]_{\overline{F}\top_{Y,Z}} & \overline{F}Z \\
 {X} \kar[u]^{c} \kar[r]^{b} \kar@/_5mm/[rr]_{\top_{X,Z}} &  {Y} \kar[u]^{d} \kar[r]^(.47){\top_{Y,Z}}  & {Z} \kar[u]^{J\zeta}_{\cong} }
\end{xy}
%\hbox{}
\end{wrapfigure}
%
\proofmark%\begin{myproof}
Let $\zeta:Z\to FZ$ be a final $F$-coalgebra in $\mathbb{C}$.
%and $b:X\kto Y$ be a restricted backward simulation from $\mathcal{X}$ to $\mathcal{Y}$.
We define  $\Phi_X:\Kl(T)(X,Z)\to\Kl(T)(X,Z)$ and $\Phi_Y:\Kl(T)(Y,Z)\to\Kl(T)(Y,Z)$ as in the proof of Prop.~\ref{prop:generalconstructiontop}. 
Moreover, 
in the same manner as in
%similarly to
%in the same manner as 
the proof of Prop.~\ref{prop:generalconstructiontop},
for each 
%$g:X\kto Z$ and  
ordinal $\mathfrak{a}$, we define $\Phi_X^{\mathfrak{a}}(\top_{X,Z}):X\kto Z$ and $\Phi_Y^{\mathfrak{a}}(\top_{Y,Z}):Y\kto Z$ by the transfinite induction on $\mathfrak{a}$.
As we have seen in the proof of Prop.~\ref{prop:generalconstructiontop}, 
there exist ordinals $\mathfrak{l}_X$ and $\mathfrak{l}_Y$ s.t.\
$\trinf(c)=\Phi_X^{\mathfrak{l}_X}(\top_{X,Z})$ and $\trinf(d)=\Phi_Y^{\mathfrak{l}_Y}(\top_{Y,Z})$. Let $\mathfrak{l}=\max(\mathfrak{l}_X,\mathfrak{l}_Y)$.
%\end{myproof}
%\end{minipage}
%\begin{minipage}{0.3\hsize}{\small
%\begin{xy}
% \xymatrix@R=2.3em@C=2.8em{
%{\overline{F}X} \ar@{}[dr]|{\sqsubseteq} \kar[r]_{\overline{F}b} \kar@/^5mm/[rr]^{\overline{F}\top_{X,Z}} & {\overline{F}Y} \ar@{}[dr]|{\rotatebox{270}{\sqsubseteq}} \kar[r]_{\overline{F}\top_{Y,Z}} & \overline{F}Z \\
% {X} \kar[u]^{c} \kar[r]^{b} \kar@/_5mm/[rr]_{\top_{X,Z}} &  {Y} \kar[u]^{d} \kar[r]^(.47){\top_{Y,Z}}  & {Z} \kar[u]^{J\zeta}_{\rotatebox{270}{$\cong$}} }
%\end{xy}}
%\end{minipage}\vspace{1mm}
%
%From assumption~\ref{item:restrictedbwdsim2}, we have $\top_{X,Z}=\top_{Y,Z}\odot b$.
%Then by transfinite induction, we can prove $\trinf(c)=\Phi_X^{\mathfrak{l}}(\top_{X,Z})=\Phi_X^{\mathfrak{l}}(\top_{Y,Z}\odot b)$.
%
We shall now prove by 
transfinite induction that, 
for each 
%ordinal 
$\mathfrak{a}$, we have
$\Phi_X^{\mathfrak{a}}(\top_{X,Z})\sqsubseteq\Phi_Y^{\mathfrak{a}}(\top_{Y,Z})\odot
b$; this will yield our goal by taking $\mathfrak{a}=\mathfrak{l}$.

% by the
% transfinite induction on $\mathfrak{a}$ as follows.

For $\mathfrak{a}=0$, from Assumption~\ref{item:restrictedbwdsim2top} of
Def.~\ref{def:restrictedbwdsimtop}, we have 
$\Phi_X^{\mathfrak{a}}(\top_{X,Z})=\top_{X,Z} = \top_{Y,Z}\odot b=\Phi_Y^{\mathfrak{a}}(\top_{Y,Z})\odot b$.

Assume that $\mathfrak{a}$ is a successor ordinal and $\Phi_X^{\mathfrak{a-1}}(\top_{X,Z})\sqsubseteq\Phi_Y^{\mathfrak{a-1}}(\top_{Y,Z})\odot b$.
Then %we have
\begin{align*}
\Phi^{\mathfrak{a}}_X(\top_{X,Z})
%&= J\zeta^{-1}\odot \overline{F}(\Phi^{\mathfrak{a}-1}_X(\top_{X,Z})) \odot d & (\text{by definition of $\Phi_X$ }) \\
&\sqsubseteq J\zeta^{-1}\odot \overline{F}(\Phi^{\mathfrak{a}-1}_Y(\top_{Y,Z}))\odot \overline{F}b \odot c  & (\text{by induction hypothesis}) \\
&\sqsubseteq  \Phi^{\mathfrak{a}}_Y(\top_{Y,Z})\odot b & (\text{$b$ is a bwd.\ simulation})\,.
%&\sqsubseteq J\zeta^{-1}\odot \overline{F}(\Psi^{\mathfrak{a}-1}_Y(\top_{Y,Z}))\odot d\odot b & (\text{$b$ is a bwd.\ simulation}) \\
%&=  \Phi^{\mathfrak{a}}_Y(\top_{Y,Z})\odot b & (\text{by definition of $\Phi_Y$})\,.
\end{align*}

Let $\mathfrak{a}$ be a limit ordinal and assume that $\Phi^{\mathfrak{i}}_X(\top_{X,Z})\sqsubseteq \Psi^{\mathfrak{i}}_Y(\top_{Y,Z})\odot b$ for all $\mathfrak{i}<\mathfrak{a}$.
Then 
\begin{align*}
\Phi_X^{\mathfrak{a}}(\top_{X,Z})
%&= \bigsqcap_{\mathfrak{i}<\mathfrak{a}}\Phi_X^{\mathfrak{i}}(\top_{X,Z}) & (\text{by definition}) \\
&\sqsubseteq\textstyle \bigsqcap_{\mathfrak{i}<\mathfrak{a}}\left(\Phi_Y^{\mathfrak{i}}(\top_{Y,Z}) \odot b\right) & (\text{by induction hypothesis}) \\
&= \Phi^{\mathfrak{a}}_Y(\top_{Y,Z})\odot b & (\text{by
 Assumption~\ref{item:restrictedbwdsim3top} of Def.~\ref{def:restrictedbwdsimtop}})\,.
%&=\left(\bigsqcap_{\mathfrak{i}<\mathfrak{a}}\Phi_Y^{\mathfrak{i}}(\top_{Y,Z})\right) \odot b & (\text{by assumption~\ref{item:restrictedbwdsim3top}}) \\
%&= \Phi^{\mathfrak{a}}_Y(\top_{Y,Z})\odot b & (\text{by definition})\,.
\end{align*} 
%Therefore we have $\Phi^{\mathfrak{a}}_X(\top_{X,Z})\sqsubseteq \Phi^{\mathfrak{a}}_Y(\top_{Y,Z})\odot b$ for each ordinal $\mathfrak{a}$.
%Letting $\mathfrak{a}=\mathfrak{l}$, 
Thus $\trinf(c)\sqsubseteq
\trinf(d)\odot b$. The last claim follows from $b$'s condition on
initial states.
\qed\vspace{3mm}

%Note that analogical assumption to Lem.~\ref{lem:soundnessFwd} (e.g.\ $\omega^{\text{op}}$-completeness of homset or monotonicity of composition and $\overline{F}$)
%is included in the assumption of Prop.~\ref{prop:generalconstructiontop}. 
%This lemma immediately implies the following theorem---soundness of TIF-backward Kleisli simulation.
% The following theorem is immediately implied by the above lemma.
% %
% \begin{mytheorem}\label{thm:soundnessRBwdtop}
% We assume the conditions in Lem.~\ref{lem:soundnessRBwdtop}.
% For  two $(T,F)$-system $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$,
% $\mathcal{X}\tifbwd\mathcal{Y}$ implies $\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$.
% \qed
% \end{mytheorem}

%For a $(\pow,\FSigma)$-system, the assumptions required in the definition of restricted backward simulation can be rephrased as follows.
%For a $(\pow,\FSigma)$-system, 
%the sufficient conditions for a backward simulation to satisfy
%the assumptions in Def.~\ref{def:restrictedbwdsimtop} can be given as follows.
%

\newtheorem*{proof:soundnesstifbwdpow}{Proof of Thm.\ \ref{thm:soundnesstifbwdpow}}{\itshape}{\rmfamily}
\begin{proof:soundnesstifbwdpow}
%\begin{myproof} (Of Thm.~\ref{thm:soundnesstifbwdpow})
In Lem.~\ref{lem:whenrestrictedpow} we prove that a TIF-backward
 simulation
in the specific sense of Def.~\ref{def:restrictedbwdsimpow} is also 
a TIF-backward
 simulation in the general sense of Def.~\ref{def:restrictedbwdsimtop}.
 Therefore Lem.~\ref{lem:soundnessRBwdtop} yields trace inclusion.
% It is immediate from  Lem.~\ref{lem:soundnessRBwdtop} and that
% if $T=\pow$ and $F=\FSigma$, for a backward simulatio $b:X\kto Y$,
% Assumption~\ref{item:restrictedbwdsim2} In Def.~\ref{def:restrictedbwdsimtop} is satisfied if $b(x)\neq\emptyset$ for each $x\in X$, 
% while Assumption~\ref{item:restrictedbwdsim3} is satisfied if $b(x)$ is finite for each $x\in X$ (Lem.~\ref{lem:whenrestrictedpow}).
\qed
\end{proof:soundnesstifbwdpow}
%\end{myproof}

Even with the additional constraints of totality and image-finiteness,
backward Kleisli simulations are a viable method for establishing
infinite trace inclusion. An example is in
Example~\ref{example:bwdsimnecessary}, where a fwd.\ simulation does not
exist but a TIF-bwd.\ simulation does.

% The TIF-backward simulations increases the number of pairs of automata where we can check infinite trace inclusion.
% Namely, there exists a pair of %nondeterministic 
% automata where forward  simulation does not exist but TIF-backward  simulation exists 
% (e.g.\ Example~\ref{example:bwdsimnecessary}).


\subsection{Forward Partial Execution for Nondeterministic
Systems}\label{subsec:FPEpow}
We now apply \emph{forward partial execution} (FPE)~\cite{urabeH14genericforward}---a transformation of coalgebraic
systems that potentially
increases the likelihood of existence of simulations---in the current
setting of nondeterminism and infinite traces. We follow the setting
in~\cite{urabeH14genericforward} for the \emph{finite} traces, and
formulate FPE's ``correctness'' in the following theorem.

% In this section, we discuss about forward partial execution (FPE, see Def.~\ref{def:fpe}).
% %In~\cite{urabeH14genericforward}, 
% %it is shown that forward partial execution (FPE, see Def.~\ref{def:fpe}) satisfy soundness and adequacy with respect to finite trace.
% FPE can increase a forward simulation, namely, by searching for a forward simulation from $\mathcal{X}_{\FPE}$ to $\mathcal{Y}$ instead of $\mathcal{X}$ to $\mathcal{Y}$,
% we can increase the possibility to find a forward simulation without losing soundness.
% Similarly, FPE can also increase a backward simulation, namely, 
% by searching for a backward simulation from $\mathcal{X}$ to $\mathcal{Y}_{\FPE}$
%  instead of $\mathcal{X}$ to $\mathcal{Y}$, we can increase the possibility to find a backward simulation without losing soundness.
% %we apply FPE to the simulated system to increase forward simulation,
% %or the simulating system to increase backward simulation. 
% The theorem below states that 
% the former situation holds
% %FPE satisfy soundness and adequacy 
% with respect to also infinite trace. %when it is used to increase forward simulation.

\begin{mytheorem}\label{thm:propertiesFPEpow}
Let $F$ be a polynomial functor on $\Sets$.
For $(\pow,F)$-systems $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$, the following  hold.
\begin{enumerate}
\item
\begin{enumerate}
\item (soundness of FPE for fwd.\ sim.)  $\mathcal{X}_{\FPE}\fwd\mathcal{Y}$ implies $\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$.
\label{item:soundnessfpefwdpow}
\item (adequacy of FPE for fwd.\ sim.) $\mathcal{X}\fwd\mathcal{Y}$ implies $\mathcal{X}_{\FPE}\fwd\mathcal{Y}$.
\label{item:adequacyfpefwdpow}
\end{enumerate}
\label{item:safpefwdpow}
\item
\begin{enumerate}
\item (soundness of FPE for bwd.\ sim.)  $\mathcal{X}\tifbwd\mathcal{Y}_{\FPE}$ implies $\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$.
\label{item:soundnessfpebwdtoppow}
\item (adequacy of FPE for bwd.\ sim.) 
$\mathcal{X}\tifbwd\mathcal{Y}$ implies
      $\mathcal{X}\tifbwd\mathcal{Y}_{\FPE}$, assuming that the following hold.
\begin{enumerate}
\item 
$d(y)\neq\emptyset$ for all $y\in Y$.
\label{item:restrictedfpe2toppow}
\item 
$d(y)$ is finite for all $y\in Y$.
\label{item:restrictedfpe3toppow}
\qed
\end{enumerate}
\label{item:adequacyfpebwdtoppow}
\end{enumerate}
\label{item:safpebwdpow}
\end{enumerate}
\end{mytheorem}
Informally: \emph{soundness} means that discovery after applying FPE
still witnesses the trace inclusion between the original systems; and
\emph{adequacy} means that the relationship $\fwd$ (or $\tifbwd$) is not
destroyed by application of FPE. The theorem also implies that FPE must
be applied to the ``correct side'' of the desired trace inclusion: 
$\mathcal{X}$ in the search for a fwd.\ simulation; and 
$\mathcal{Y}$ in the search for a bwd.\ one.

Note that the adequacy property is independent from the choice of trace
semantics (finite or infinite). Therefore the
statement~\ref{item:adequacyfpefwdpow} of
Thm.~\ref{thm:propertiesFPEpow} is the same as its counterpart
in~\cite{urabeH14genericforward}. For the
statement~\ref{item:adequacyfpebwdtoppow}, however, we have to 
check that the TIF restriction (that is absent in~\cite{urabeH14genericforward}) is indeed carried over. 


In~\cite{urabeH14genericforward} it is shown that FPE can indeed create
a simulation that does not exist between the original systems. Its
practical
use is witnessed by experimental results
in~\cite{urabeH14genericforward}, too. It would not be hard to observe
the same in the current setting for \emph{infinite} traces.


For the proof of Thm.~\ref{thm:propertiesFPEpow}, once again, we turn 
to an axiomatic development.
%
\begin{mytheorem}[FPE and fwd.\ sim.]%[correctness of FPE for fwd.\ sim.]
\label{thm:soundnessFPEfwd}$\!\!\!{}^\dagger$
Let $F$ be an endofunctor and $T$ be a monad on $\mathbb{C}$, as in
Lem.~\ref{lem:soundnessFwd} (that is, they constitute an infinite trace
 situation and satisfy the two additional assumptions.)
% that
%  constitute an infinite trace situation with respect to $\sqsubseteq$ (Def.~\ref{def:infiniteTraceSituation}).
Let $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$ be $(T,F)$-systems. % and $X=X_1+X_2$.
%Assume that $F$ and $T$ satisfy the assumptions in  Lem.~\ref{lem:soundnessFwd}.
Then we have:
\begin{enumerate}
%\item (monotonicity) If $X_1=X_{1,1}+X_{1,2}$, then $\mathcal{X}_{\FPE,X_{1,1}}\fwd\mathcal{Y}$ implies $\mathcal{X}_{\FPE,X_1}\fwd\mathcal{Y}$.
%\label{item:monotonicityfpefwd}
\item (soundness for fwd.\ sim.)  $\mathcal{X}_{\FPE}\fwd\mathcal{Y}$ implies $\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$.
\label{item:soundnessfpefwd}
\item (adequacy for fwd.\ sim.) $\mathcal{X}\fwd\mathcal{Y}$ implies $\mathcal{X}_{\FPE}\fwd\mathcal{Y}$.
\qed
\label{item:adequacyfpefwd}
\end{enumerate}
\end{mytheorem}

%The next aim is %we want 
%to show that FPE satisfy soundness and adequacy with respect to  infinite trace when it is used to increase restricted backward simulation.



% Next we focus on using FPE to increase TIF-backward simulation.
% It is easy to prove the soundness.
% However, adequacy requires some care. %additional assumptions to prove.
% It is easy to prove that if there exists a backward simulation from $\mathcal{X}$ to $\mathcal{Y}$, 
% then there also exists backward simulation from $\mathcal{X}$ to $\mathcal{Y}_{\FPE}$.
% However, to assure that the resulting backward simulation is a total and image-finite one, we have to require the simulating system to satisfy some conditions.%pu
%t some assumption on the simulated system.
%
\begin{mytheorem}[FPE and bwd.\ sim.]%[correctness of FPE for restricted bwd.\ sim.]
\label{thm:soundnessFPEbwdtop}$\!\!\!{}^\dagger$
Let $F$ be an endofunctor and $T$ be a monad on $\mathbb{C}$
that satisfy the conditions in Prop.~\ref{prop:generalconstructiontop}
 (hence those in Lem.~\ref{lem:soundnessRBwdtop}). 
%Let $F$ be an endofunctor and $T$ be a monad on $\mathbb{C}$ that constitute an infinite trace situation with respect to 
%%an order 
%$\sqsubseteq$.
%We assume that the largest homomorphism is obtained by using Prop.~\ref{prop:generalconstructiontop}. 
%satisfies the assumptions \ref{item:restrictedbwdsima} and \ref{item:restrictedbwdsimb} in Def.~\ref{def:restrictedbwdsim}.
Let $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$ be $(T,F)$-systems. % and $Y=Y_1+Y_2$.
%Moreover, we define $d_i:Y_i\kto \overline{F}Y$ ($i\in\{1,2\}$) by 
%$d_i=d\odot \kappa_i$.
\begin{enumerate}
%\item (monotonicity) If $Y_1=Y_{1,1}+Y_{1,2}$, then $\mathcal{X}\tifbwd\mathcal{Y}_{\FPE,Y_{1,1}}$ implies $\mathcal{X}\bwd\mathcal{Y}_{\FPE,Y_1}$.
%\label{item:monotonicityfpebwd}
\item (soundness for bwd.\ sim.)  $\mathcal{X}\tifbwd\mathcal{Y}_{\FPE}$ implies $\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$.
\label{item:soundnessfpebwdtop}
\item (adequacy for bwd.\ sim.) 
$\mathcal{X}\tifbwd\mathcal{Y}$ implies $\mathcal{X}\tifbwd\mathcal{Y}_{\FPE}$ if the following conditions are satisfied.
%Assume the following conditions.
\begin{enumerate}
\item 
The coalgebra $d:Y\kto \overline{F}Y$ satisfies 
$\top_{\overline{F}Y,Z}\odot d = \top_{Y,Z}$. 
%$J!_{FY}\odot d =      J!_Y$. 
%The arrow $(d_1+\id_{X_2}):Y\kto \overline{F}Y+Y_2$ satisfies $J!_{\overline{F}Y+Y_2}\odot (c_1+\id_{X_2}) = J!_Y$. 
\label{item:restrictedfpe2top}
%\item One of the following is satisfied.
%\begin{enumerate}
%\item Assumption~\ref{asm:existenceWeaklyFinalCoalgGFPcont} in Prop.~\ref{prop:generalconstructionnotop} is satisfied. \label{item:restrictedfpe3cont}
\item Precomposing $d$ preserves the glb.\ of a decreasing transfinite sequence.
%Namely, for $A\in\Kl(T)$ and a family of Kleisli arrows $(g_{\mathfrak{i}}:\overline{F}Y\kto A)_{\mathfrak{i}\in\mathfrak{a}}$ where 
%$\mathfrak{a}$ is a limit ordinal and $\mathfrak{i}\leq \mathfrak{j}$ implies $g_{\mathfrak{i}}\sqsupseteq g_{\mathfrak{j}}$,
%%$g_0\sqsupseteq g_1 \sqsupseteq g_2\ldots :Y\kto A$,
%$\bigsqcap_{\mathfrak{i}<\mathfrak{a}}(g_{\mathfrak{i}}\odot d)=(\bigsqcap_{\mathfrak{i}<\mathfrak{a}}g_{\mathfrak{i}})\odot d$ holds.
%\item Precomposing $b:X\kto Y$ preserves the greatest fixed point of $\Psi$. 
%Namely, if $f:Y\kto 1$ is the greatest fixed point of $\Psi_Y$, then $f\odot b:X\kto 1$ is the greatest fixed point of $\Psi_X$.
\label{item:restrictedfpe3top}
%\end{enumerate}
%
%\item Precomposing $d:Y\kto \overline{F}Y$ preserves the greatest fixed point of a decreasing sequence. % in $\Kl(T)(\overline{F}Y,1)$.
%Namely, for $A\in\Kl(T)$ and a decreasing sequence $g_0\sqsupseteq g_1 \sqsupseteq g_2\ldots :\overline{F}Y\kto A$ of Kleisli arrows, 
%$\bigsqcap_{i\in\omega}(g_i\odot d)=(\bigsqcap_{i\in\omega}g_i)\odot d$ holds.
%%\item Precomposing $(d_1+\id_{X_2}):Y\kto \overline{F}Y+Y_2$ preserves the greatest fixed point of a decreasing sequence. % in $\Kl(T)(\overline{F}Y,1)$.
%%Namely, for $A\in\Kl(T)$ and a decreasing sequence $g_0\sqsupseteq g_1 \sqsupseteq g_2\ldots :\overline{F}Y+Y_2\kto A$ of Kleisli arrows, 
%%$\bigsqcap_{i\in\omega}(g_i\odot b)=(\bigsqcap_{i\in\omega}g_i)\odot b$ holds.
%%\item Precomposing $d:Y\kto \overline{F}Y$ preserves the greatest fixed point of $\Psi$. 
%%Namely, if $f:\overline{F}Y\kto 1$ is the greatest fixed point of $\Psi_{\overline{F}Y}$, then $f\odot d:Y\kto 1$ is the greatest fixed point of $\Psi_{\overline{F}Y}$.
%\label{item:restrictedfpe3}
%\item Precomposing $d:Y\kto \overline{F}Y$ preserves the largest mediating arrow  
%in assumption~\ref{asm:existenceWeaklyFinalCoalgWeak2Lim} in Prop.~\ref{prop:generalconstructionnotop}.
%Namely, if $l_{\max}:\overline{F}Y\kto Z$ is the largest mediating arrow from a cone $(\overline{F}Y,(\delta_i:\overline{F}Y\kto \overline{F}^i1))$ to 
%a weak 2-limit $(Z,(J\gamma_i:Z\kto \overline{F}^i 1))$, then 
%$l_{\max}\odot d:Y\kto Z$ is the largest mediating arrow from a cone $(X,(\delta_i\odot b:X\kto \overline{F}^i1))$ to 
%the weak 2-limit. 
%\label{item:restrictedfpe4}
\qed
\end{enumerate}
%Then $\mathcal{X}\tifbwd\mathcal{Y}$ implies $\mathcal{X}\tifbwd\mathcal{Y}_{\FPE}$.
\label{item:adequacyfpebwdtop}
\end{enumerate}
\end{mytheorem}


\newtheorem*{proof:propertiesFPEpow}{Proof of Thm.\ \ref{thm:propertiesFPEpow}}{\itshape}{\rmfamily}
\begin{proof:propertiesFPEpow}
%\begin{myproof} (Of Thm.~\ref{thm:propertiesFPEpow})
% NATSUKI: proof environment
\ref{item:safpefwdpow} is immediate from Thm.~\ref{thm:soundnessFPEfwd}.
In a similar manner to Lem.~\ref{lem:whenrestrictedpow},
we can prove \ref{item:safpebwdpow} using Thm.~\ref{thm:soundnessFPEbwdtop}.
%\end{myproof}
\qed
\end{proof:propertiesFPEpow}


%For a $(\pow,\FSigma)$-system, 
%a sufficient condition
%the additional assumptions imposed 
%%on the simulating automaton 
%in the above theorem can be rephrased as follows.
%For a $(\pow,\FSigma)$-system, 
%the sufficient conditions for the simulating system to satisfy
%the assumptions in the above theorem can be given as follows.
%It can be proved in the similar manner to 
%Prop.~\ref{lem:whenrestrictedpow}.
%\begin{myproposition}\label{lem:whenrestrictedfpepow}
%In Thm.~\ref{thm:soundnessFPEbwdtop}, if $T=\pow$ and $F=\FSigma$, 
%%\begin{enumerate}
%%\item 
%assumption~\ref{item:restrictedfpe2top} is satisfied if $d(y)\neq\emptyset$ for each $y\in Y$ while
%%\item 
%assumption~\ref{item:restrictedfpe3top} is satisfied if $d(y)$ is finite for each $y\in Y$. 
%\qed
%%\end{enumerate}
%\end{myproposition}

\auxproof{
In~\cite{urabeH14genericforward}, for a $(T,F)$-system  on $\Sets$,
FPE is defined so that it takes not only a $(T,F)$-system $\mathcal{X}=(X,s,c)$ but also
a subset $X_1\in X$ as inputs, and output a $(T,F)$-system $\mathcal{X}_{\FPE,X_1}$.
The second input $X_1$ is a set of states that is ``executed forward''.
In other words, it was allowed to apply FPE to only a part of states.
%
It is not so difficult to define this ``partial'' FPE for $(T,F)$-systems on general categories
as long as a state space $X$ can be decomposed into a coproduct $X=X_1+X_2$. 
We can prove that this partial FPE also satisfies soundness and adequacy with respect to infinite trace.
Therefore it can be used in the similar manner to the ``total'' FPE, 
that is used in this section. % and can be denoted by $\mathcal{X}_{\FPE,X}$.
However, it is known that FPE satisfies \emph{monotonicity} i.e.\ $X_1\subseteq X_1'$ implies 
$\mathcal{X}_{\FPE,X_1}\fwd \mathcal{Y} \,\Rightarrow\,\mathcal{X}_{\FPE,X_1'}\fwd \mathcal{Y}$.
This means that the more states we apply FPE to, the more simulations we can found.
Therefore theoretically, there is no reason to apply FPE to only a part of states.

However, FPE usually increases the number of states.
Therefore, practically,  by applying FPE to only a part of states we can avoid state explosion.
Moreover, in an infinite trace setting, there is another reason to use ``partial'' FPE.
When we apply FPE to increase TIF-backward Kleisli simulation, 
it is required that $d(y)$ is finite for all states $y$ of the simulating automaton $\mathcal{Y}$ 
in order to assure adequacy.
However, when FPE is applied to a part of states, 
it suffices to require that $d(y)$ is finite for each state $y$ that is applied FPE.
Therefore applying FPE only to the states having only a finite number of successors, 
we can increase TIF-backward simulation without losing adequacy.
}

\section{Systems with  Probabilistic Branching}\label{sec:girymonad}
We now turn to probabilistic systems. They are modeled as $(\giry,
F)$-systems in the category $\Meas$. 
Here we establish largely the same statements as
in~\S{}\ref{sec:powersetmonad}, but many constructions and proofs are different.
Throughout this section $F$ is
assumed to be a (standard Borel) polynomial functor on $\Meas$
(Def.~\ref{def:polyfuncmeas}).


\subsection{Construction of Infinite Traces}\label{subsec:constructlargesthomgiry}
\begin{mytheorem}\label{thm:inftrsitgiry}
The combination of polynomial $F$ and $T=\giry$ constitute an infinite
 trace situation (Def.~\ref{def:infiniteTraceSituation}).
%A functor $\FXi$ and $\giry$ constitute an infinite trace situation.
%with respect to the order in Def.~\ref{def:orderenrichment}. 
 %\qed
\end{mytheorem}

Our basic idea of the construction is similar to that for $\pow$
(\S\ref{subsec:constructlargesthompow}). Our goal is to construct
the largest homomorphism from an $\overline{F}$-coalgebra $c$ in to 
the lifted final coalgebra $J\zeta:Z\kto \overline{F}Z$; we do so
inductively, starting from the top element and going down along a
decreasing 
sequence. Compared to the nondeterministic case ($T=\pow$), major
differences are as follows.
\begin{itemize}
 \item  Composition of Kleisli arrows is $\omega^{\text{op}}$-continuous
	in $\Kl(\giry)$. This is an advantage, because we can appeal to
	the Kleene fixed point theorem and we only need inductive
	construction 
	up-to $\omega$ steps (while, for $\pow$, we needed transfinite
	induction).
 \item  A big disadvantage, however, is the absence of the top
 element $\top_{X,Z}$ in $\Kl(T)(X,Z)$. One can imagine a top element
	$\top_{X,Z}$ to 
	assign $1$ to every event---this is however not a (probability)
	measure. 
\end{itemize}


\begin{wrapfigure}[6]{r}{50pt}
\vspace{-\intextsep}
\vspace{-5pt}
{\tiny
\begin{xy}
(0,0)*+{1} = "F01",
(10,0)*+{\overline{F}1} = "F11",
(22,0)*+{\overline{F}^21} = "F21",
%(30,0)*+{\overline{F}^31} = "F31",
%
(-6,10)*+{X} = "F0X",
%
(-6,-10)*+{Z} = "F0Z",
%
(24,6)*+{\dots}= "",
(24,-6)*+{\dots}= "",
(31,0)*+{\dots}= "",
%
(-16,10)*+{\overline{F}X} ="F1X",
(-16,-10)*+{\overline{F}Z} ="F1Z",
(-11.5,0)*+{=} ="",
%
\kar _{J!}"F11";"F01"
\kar _{JF!}"F21";"F11"
%\kar _{JF^2!_{F1}}"F31";"F21"
\kar _(.1){JF^2!} (29,0);"F21"
%\ar @{.} (31,0);(34,0)
%
\kar ^{\pi_0} "F0X";"F01"
\kar @/^2mm/ ^(.7){\pi_1} "F0X";"F11"
\kar @/^4mm/ ^(.7){\pi_2} "F0X";"F21"
%
\kar _(.7){J\gamma_0} "F0Z";"F01"
\kar @/_2mm/ _(.7){J\gamma_1} "F0Z";"F11"
\kar @/_4mm/ _(.7){J\gamma_2} "F0Z";"F21"
%
\kar @{-->} _{\rotatebox{90}{\raisebox{1ex}{$\trinf(c)$}}} "F0X";"F0Z"
%
\kar _{c} "F0X";"F1X"
\kar _{\rotatebox{90}{\raisebox{1ex}{$\overline{F}(\trinf(c))$}}} "F1X";"F1Z"
\kar _{J\zeta}^{\cong} "F0Z";"F1Z"
\end{xy}}
\end{wrapfigure}
%Our alternative idea is as follows: the greatest element might not exists in $\Kl(T)(X,Z)$, 
%however, it exists in $\Kl(T)(X,1)$ in many cases. 
%The alternative idea is as follows: 
To cope with the latter challenge, we turn to the \emph{final
$F$-sequence} 
in $\Meas$ that yields a final $F$-coalgebra as its limit. 
Instead of using a sequence like $\top\sqsupseteq\Phi(\top)\sqsupseteq\cdots$ in
$\Kl(T)(X,Z)$ (where the largest element $\top$ does not exist anyway), 
we use a decreasing sequence that goes along the final sequence. 

% Concretely:
% % The alternative idea is as follows:
% % %Alternatively, 
% % we start from the greatest element $J!_X$ in $\Kl(T)(X,1)$,
% %  that is the greatest element for many monads $T$.
% % This idea results in the following construction of the largest homomorphism, that 
% % is applicable to both $\giry$ and $\pow$ (c.f.\ Prop.~\ref{prop:satassumptionnondet2}).
% %
% %By calculating the greatest fixed point of a certain function 
% %$\Psi_X:\Kl(T)(X,1)\to\Kl(T)(X,1)$ 
% %on $\Kl(T)(X,1)$
% %from the greatest element,
% %Then 
% starting from $J!_X: X\kto 1$,
% %instead of a homomorphism,
% we  construct a cone $(X,(\pi_i:X\kto \overline{F}^i1))$ over a sequence 
% $1\overset{J!_{F1}}{\kleftarrow} \overline{F}1\overset{JF!_{F1}}{\kleftarrow} \overline{F}^2 1\overset{JF^2!_{F1}}{\kleftarrow} \ldots$.
% From this cone, next we want to obtain a Kleisli arrow from $X$ to $Z$. 
% We use the framework in~\cite{cirstea10genericinfinite}.
% %Here, we note 
% It is known that if the functor $F$ is $\omega^{\text{op}}$-continuous (i.e.\ preserves $\omega^{\text{op}}$-limit),
%  the carrier $Z$ of the final $F$-coalgebra can be obtained
% as the limit $(Z,(\gamma_i:Z\to F^i1)_{i\in\omega})$ of a final sequence 
% $1\overset{!_{F1}}{\leftarrow}F1\overset{F!_{F1}}{\leftarrow}F^2 1\overset{F^2!_{F1}}{\leftarrow} \ldots$~\cite{adamekK79leastfixed}.
% Hence we put an additional assumption: the limit $(Z,(\gamma_i:Z\to F^i1)_{i\in\omega})$
%  %over $1\overset{!_{F1}}{\leftarrow}F1\overset{F!_{F1}}{\leftarrow}F^2 1\overset{F^2!_{F1}}{\leftarrow} \ldots$ 
%  in $\mathbb{C}$ is lifted to a weak limit over the sequence $1\overset{J!_{F1}}{\kleftarrow} \overline{F}1\overset{JF!_{F1}}{\kleftarrow} \overline{F}^2 1\overset{JF^2!_{F1}}{\kleftarrow} \ldots$ in $\Kl(T)$ that admits the largest mediating arrow. 
% %In the proof of Prop.~\ref{prop:generalconstructionnotop}, 
% %
% Then we can obtain  the largest mediating arrow $l_{\max}:X\kto Z$, from which we can start the iteration in Prop.~\ref{prop:generalconstructiontop}.

% In the above construction, we calculated the greatest fixed point 
% twice---once for constructing a cone and once for constructing the largest homomorphism.
% The second iteration is necessary for $\pow$ because there exists a pair of nondeterministic automata that construct exactly the same cone in the first iteration
%  (e.g.\ Example~\ref{example:sameconediffhom}).


% % Eventually, we obtain the following proposition. Its formal proof is in the appendix.

% The following proposition yields Thm.~\ref{thm:inftrsitgiry} as an
% instance. In fact Thm.~\ref{thm:inftrsituationpow} is its instance, too,
% by using the
% condition~\ref{asm:existenceWeaklyFinalCoalgGFPtransfinite} (this is
% proved
% in Prop.~\ref{prop:satassumptionnondet2}). The axioms in the following
% proposition are possibly overly general; but we hope we find 
% an example of a monad $T$
% where neither of the two useful properties
% (namely: existence of $\top_{X,Z}$ for $\pow$, and $\omega$-continuity
% of $\odot$ for $\giry$) fail. For such an example $T$ the following
% proposition will be useful.


% is applicable to both $\giry$ and $\pow$ (c.f.\ Prop.~\ref{prop:satassumptionnondet2}).

The precise construction is found in the proof of the following
proposition (the proof is in
Appendix~\ref{subsec:omittedProofsForGiryMonad}). 
\begin{myproposition}\label{prop:generalconstructionnotop}$\!\!\!{}^\dagger$
Let $\mathbb{C}$ be a category, $F$ be an endofunctor on $\mathbb{C}$, 
and $T$ be a monad on $\mathbb{C}$ where each homset of $\Kl(T)$ carries an order $\sqsubseteq$. 
%Let $J:\mathbb{C}\to\Kl(T)$ be a Kleisli inclusion functor.
We assume the following conditions.
%
\begin{enumerate}
\item The category $\mathbb{C}$ has a final object $1$;
the final sequence $1\overset{!_{F1}}{\leftarrow}F1\overset{F!_{F1}}{\leftarrow}F^2 1\overset{F^2!_{F1}}{\leftarrow} \ldots$ 
has a limit 
$(Z,(\gamma_i:Z\to F^i1)_{i\in\omega})$; and moreover,
$F$ preserves this limit. (Hence the limit carries a final $F$-coalgebra~\cite{adamekK79leastfixed}.)
%(Then this limit yields a final $F$-coalgebra $\zeta:Z\to FZ$.) 
\label{asm:existenceWeaklyFinalCoalgFinalSeq}
%
\item There exists a distributive law $\lambda:FT\Rightarrow TF$,
      yielding a lifting $\overline{F}$ on $\Kl(T)$ of $F$.
%and hence $F$ can be lifted to an endofunctor $\overline{F}$ on $\Kl(T)$. 
\label{asm:existenceWeaklyFinalCoalgDistLaw}
%
%\item For each 
%%pair of objects $X$ and $Y$ in $\Kl(T)$, 
%$X,Y\in\Kl(T)$,
%the homset $\Kl(T)(X,Y)$ carries a partial order $\sqsubseteq$. 
%Moreover, $\overline{F}$ and the composition of arrows in $\Kl(T)$ 
%%and the functor  $\overline{F}$ 
%are  monotone  wrt.\ this order.
%%composition is locally monotone (i.e.\ preserves order) with respect to this order.
%\label{asm:existenceWeaklyFinalCoalgFMonotone}
\item 
\label{asm:existenceWeaklyFinalCoalgGFPcont}
For 
%each 
$X,Y\in\Kl(T)$,
%pair of objects $X$ and $Y$ in $\Kl(T)$,  
every decreasing $\omega$-sequence $f_0\sqsupseteq f_1\sqsupseteq
      \ldots$ in $\Kl(T)(X,Y)$ has the greatest lower bound $\bigsqcap_{i\in\omega}f_i$.
Moreover,  composition of arrows in $\Kl(T)$ and $\overline{F}$'s action on arrows  are both $\omega^{\text{op}}$-continuous. %, respectively.
That is, for each $g:Z\kto X$ and $h:Y\kto W$, we have 
$g\odot (\bigsqcap_{i\in\omega}f_i)=\bigsqcap_{i\in\omega}(g\odot f_i)$, 
$(\bigsqcap_{i\in\omega}f_i)\odot h=\bigsqcap_{i\in\omega}(f_i\odot h)$, and
$\overline{F}(\bigsqcap_{i\in\omega}f_i)=\bigsqcap_{i\in\omega}(\overline{F} f_i)$.
%(i.e.\ composition of arrows and applying $\overline{F}$ preserves the greatest lower bound).
%(We denote this order by $\sqsubseteq$.)
\item 
%For each object $X\in\mathbb{C}$, 
The lifting $J(!_X)$ of the unique arrow to 
$1$ %the final object 
is the largest element of $\Kl(T)(X,1)$.\!\!
 \label{asm:existenceWeaklyFinalCoalgTop}
%
%\item The functor  $\overline{F}$ is locally monotone. 
%%and continuous (i.e.\ preserves the greatest lower bound). 
%\label{asm:existenceWeaklyFinalCoalgFMonotone}
%
\item 
The functor $J$ lifts the limit in Assumption~\ref{asm:existenceWeaklyFinalCoalgFinalSeq} to a 2-limit.
% $(Z,(J\gamma_i:Z\kto F^i1)_{i\in\omega})$.
Namely, for any cone $(X,(\pi_i:X\kto F^i1)_{i\in\omega})$ over the sequence 
$1\overset{J!_{F1}}{\kleftarrow} \overline{F}1\overset{JF!_{F1}}{\kleftarrow} \overline{F}^2 1\overset{JF^2!_{F1}}{\kleftarrow} \cdots$,
there uniquely exists 
$l:X\kto Z$ s.t.\ $\pi_i=J\gamma_i\odot l$ holds for each $i\in\omega$.
Moreover, if $l':X\kto Z$ satisfies $J\gamma_i\odot l'\sqsubseteq J\gamma_i\odot l$ for each $i\in\omega$, then 
$l'\sqsubseteq l$ holds.
\label{asm:existenceWeaklyFinalCoalgWeak2Lim}
\end{enumerate}
Then $F$ and $T$ constitute an infinite trace situation with respect to $\sqsubseteq$.
%Then $J\zeta:X\kto\overline{F}Z$ is a weakly final $\overline{F}$-coalgebra.
\qed
\end{myproposition}
In more elementary terms,
Assumption~\ref{asm:existenceWeaklyFinalCoalgWeak2Lim} asserts that: $J$
lifts the limit $Z$; and the lifted limit satisfies a stronger condition
of ``carrying over'' the order between cones to the order between
mediating maps. 


\newtheorem*{proof:inftrsitgiry}{Proof of Thm.\ \ref{thm:inftrsitgiry}}{\itshape}{\rmfamily}
\begin{proof:inftrsitgiry}
%\begin{myproof}
We have to check that
 polynomial $F$ and $T=\giry$ satisfy the assumptions in
 Prop.~\ref{prop:generalconstructionnotop}. The most nontrivial is
 Assumption~\ref{asm:existenceWeaklyFinalCoalgWeak2Lim}; there we rely
 on Kolmogorov's consistency theorem, for the fact that a limit is
 lifted to a limit. That the latter is indeed a 2-limit is not hard,
 exploiting suitable monotonicity. Details are found in Lem.~\ref{lem:satassumptionprob}.
\qed
%\end{myproof}
\end{proof:inftrsitgiry}

%We can construct the largest homomorphism to $J\zeta$ when the functor is $\FXi$ on $\Meas$ and the monad is a sub-Giry monad $\giry$.
%The conditions in Prop.~\ref{prop:generalconstructionnotop} are satisfied by $\FXi$ on $\Meas$ and $\giry$.



We can also show that the resulting coalgebraic infinite trace semantics coincides
with 
the usual definition of (infinite) tree languages for probabilistic
systems. See \S\ref{subsec:automcharagiry} for  details.



%We can show that this coalgebraic infinite trace semantics describes
%automata-theoretic infinite language (see \S\ref{subsec:automcharagiry} for more details).





\subsection{Kleisli Simulations for Probabilistic Systems}\label{subsec:Klsimgiry}
\subsubsection{Forward Simulations}
Soundness of forward simulation, in the current probabilistic setting, 
follows immediately from the 
 the axiomatic development in Lem.~\ref{lem:soundnessFwd}.

\begin{mytheorem}\label{thm:soundnessFwdgiry}
Given  two $(\giry,F)$-systems $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$,
$\mathcal{X}\fwd\mathcal{Y}$ implies $\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$.
\qed
\end{mytheorem}


%In this section, we show that forward and backward Kleisli simulations can be used to
%verify infinite trace inclusion of $(\giry,\FXi)$-systems. 
%%
%Soundness of forward simulation 
%always holds
%because $\giry$ and $\FXi$ satisfy assumptions in Lem.~\ref{lem:soundnessFwd}.
%%can be proved in the same way as \S\ref{subsec:Klsimpow} 
%% 
%\begin{mytheorem}\label{thm:soundnessFwdgiry}
%For  two $(\giry,\FXi)$-systems $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$,
%$\mathcal{X}\fwd\mathcal{Y}$ implies $\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$.
%\qed
%\end{mytheorem}
%
%\begin{myproposition}\label{prop:satassumptionsfwdgiry}
%The functor $\FXi$ and $\giry$  satisfy assumptions~\ref{asm:soundnessFwd1}--\ref{asm:soundnessFwd3}
%in Lem.~\ref{lem:soundnessFwd}.
%\qed
%\end{myproposition}
%we can prove that $\giry$ and $\FXi$ also satisfy the assumptions of Lem.~\ref{lem:soundnessFwd}.


\subsubsection{Backward Simulations}
Next we 
turn to
%want to prove the soundness of 
backward simulations.
%Recall that 
Similarly to nondeterministic setting (\S{}\ref{subsubsec:powbwd}),
we have to impose a certain restriction on backward Kleisli simulations to ensure soundness.
By the feature of $\giry$ that composition in $\Kl(\giry)$ is $\omega$-continuous,
the  image-finiteness condition is no longer needed.


\begin{mydefinition}[totality, T-backward simulation]\label{def:restrictedbwdsimgiry}
Let $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$ be
 $(\giry,F)$-systems. 
A backward simulation $b:X\kto Y$ 
from $\mathcal{X}$ to $\mathcal{Y}$ is \emph{total} if
$b(x)(Y)=1$ for all $x\in X$.
If $b$ is total, it is called
a \emph{T-backward simulation}. 
We write $\mathcal{X}\tbwd\mathcal{Y}$ if there exists a T-backward simulation from $\mathcal{X}$ to $\mathcal{Y}$.
\end{mydefinition}

\begin{mytheorem}[soundness of $\tbwd$]\label{thm:soundnesstifbwdgiry}
For  two $(\giry,F)$-systems $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$,
$\mathcal{X}\tbwd\mathcal{Y}$ implies $\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$.
\qed
\end{mytheorem}

The proof of Thm.~\ref{thm:soundnesstifbwdgiry} is via the following axiomatic development.


\begin{mydefinition}[T-backward simulation, generally]\label{def:restrictedbwdsimnotop}$\!\!\!{}^\dagger$
Let $F$ be an endofunctor and $T$ be a monad on $\mathbb{C}$ that 
satisfy the conditions in Prop.~\ref{prop:generalconstructionnotop} wrt.\ $\sqsubseteq$. 
%constitute an infinite trace situation with respect to an order $\sqsubseteq$.
%
For  two $(T,F)$-systems $\mathcal{X}=(X,s,c)$ and
 $\mathcal{Y}=(Y,t,d)$,
a \emph{T-backward simulation} 
from $\mathcal{X}$ to $\mathcal{Y}$ 
% a \emph{total and image finite backward Kleisli simulation} (\emph{TIF-backward simulation} for short) is 
is a backward simulation $b:X\kto Y$ 
that satisfies the following condition:
\begin{enumerate}
\item 
The arrow $b:X\kto Y$ satisfies $J!_Y\odot b = J!_X$. Here $!_{Y}\colon
      Y\to 1$ is the unique function.
\label{item:restrictedbwdsim2notop}
%\item One of the following is satisfied.
%\begin{enumerate}
%\item Assumption~\ref{asm:generalconstructiontopGFPcont} in Prop.~\ref{prop:generalconstructiontop} is satisfied. \label{item:restrictedbwdsim3cont}
\end{enumerate}
We write $\mathcal{X}\tbwd\mathcal{Y}$ if there exists a T-backward simulation from $\mathcal{X}$ to $\mathcal{Y}$.
\end{mydefinition}

This general T-backward simulation satisfies soundness. For its proof
we have to look into the inductive 
 construction of the largest homomorphism in
 \S\ref{subsec:constructlargesthomgiry} (Prop.~\ref{prop:generalconstructionnotop}).
 
 \begin{mylemma}\label{lem:soundnessRBwdnotop}$\!\!\!{}^\dagger$
Let $F$ and $T$ be
%We assume that $F$ and $T$ 
as in Prop.~\ref{prop:generalconstructionnotop}.
% wrt.\ $\sqsubseteq$. 
% that constitute an infinite trace situation with respect to an order $\sqsubseteq$.
%We assume that the weakly final coalgebra is obtained by using Prop.~\ref{prop:generalconstructiontop}. 
For two $(T,F)$-systems $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$, 
$\mathcal{X}\tbwd\mathcal{Y}$ (in the sense of Def.~\ref{def:restrictedbwdsimnotop}) implies $\trinf(c)\sqsubseteq
 \trinf(d)\odot b$.
 Furthermore it follows that  $\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$.
 \qed
\end{mylemma}
 

\newtheorem*{proof:soundnesstifbwdgiry}{Proof of Thm.\ \ref{thm:soundnesstifbwdgiry}}{\itshape}{\rmfamily}
\begin{proof:soundnesstifbwdgiry}
In Lem.~\ref{lem:whenrestrictedgiry} we prove that a T-backward
 simulation
in the specific sense of Def.~\ref{def:restrictedbwdsimgiry} is also 
a T-backward
 simulation in the general sense of Def.~\ref{def:restrictedbwdsimgiry}.
 Therefore Lem.~\ref{lem:soundnessRBwdnotop} yields trace inclusion.
% It is immediate from  Lem.~\ref{lem:soundnessRBwdtop} and that
% if $T=\pow$ and $F=\FSigma$, for a backward simulatio $b:X\kto Y$,
% Assumption~\ref{item:restrictedbwdsim2} In Def.~\ref{def:restrictedbwdsimtop} is satisfied if $b(x)\neq\emptyset$ for each $x\in X$, 
% while Assumption~\ref{item:restrictedbwdsim3} is satisfied if $b(x)$ is finite for each $x\in X$ (Lem.~\ref{lem:whenrestrictedpow}).
\qed
\end{proof:soundnesstifbwdgiry}


%
%In \S\ref{subsec:Klsimpow}, we proved the soundness of a total and image-finite backward simulation 
%%for $(\pow,\FSigma)$-system 
%along the construction of the largest homomorphism 
%%presented 
%in Prop.~\ref{prop:generalconstructiontop}.
%%In the current setting,
%Now,
%as the construction of the largest homomorphism % in Prop.~\ref{prop:generalconstructionnotop} 
%is more complicated (Prop.~\ref{prop:generalconstructionnotop}),
%the definition of a restricted backward simulation is complicated, too. %should be more complicated, too.
%%we have to put additional restriction on the definition of restricted backward simulation.

%\begin{mydefinition}\label{def:restrictedbwdsimnotop}$\!\!\!{}^\dagger$
%Let $F$ be an endofunctor and $T$ be a monad on $\mathbb{C}$
%%We assume that $F$ and $T$ 
%that
%satisfy the conditions of Prop.~\ref{prop:generalconstructionnotop}. 
%%that constitute an infinite trace situation with respect to an order $\sqsubseteq$.
%%Let $F$ be an endofunctor on $\mathbb{C}$ and $T$ be a monad on $\mathbb{C}$.
%%We assume that each homset of $\Kl(T)$ is enriched with an order $\sqsubseteq$.
%%We assume that the weakly final coalgebra is obtained by using Prop.~\ref{prop:generalconstructionnotop}. 
%%
%For  two $(T,F)$-systems $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$,
%a \emph{restricted backward Kleisli simulation} is 
%a backward simulation $b:X\kto Y$ 
%from $\mathcal{X}$ to $\mathcal{Y}$ 
%that satisfies the following conditions.
%\begin{enumerate}
%\item The arrow $b:X\kto Y$ satisfies $J!_Y\odot b = J!_X$. \label{item:restrictedbwdsim2}
%%\item 
%%%One of the following conditions is satisfied.
%%%\begin{enumerate}
%%%\item 
%%a) Assumption~\ref{asm:existenceWeaklyFinalCoalgGFPcont} of Prop.~\ref{prop:generalconstructionnotop} is satisfied,
%%or
%%% \label{item:restrictedbwdsim3cont}
%%%\item 
%%b) Assumption~\ref{asm:existenceWeaklyFinalCoalgGFPtransfinite} of Prop.~\ref{prop:generalconstructionnotop} is satisfied
%%and precomposing $b$ preserves the glb.\ of a decreasing transfinite sequence.
%%%Namely, for $A\in\Kl(T)$ and a family of Kleisli arrows $(g_{\mathfrak{i}}:Y\kto A)_{\mathfrak{i}<\mathfrak{a}}$ where $\mathfrak{a}$ is a limit ordinal and $\mathfrak{i}\leq \mathfrak{j}$ implies $g_{\mathfrak{i}}\sqsupseteq g_{\mathfrak{j}}$,
%%%%$g_0\sqsupseteq g_1 \sqsupseteq g_2\ldots :Y\kto A$,
%%%we have $\bigsqcap_{i\in\mathfrak{a}}(g_i\odot b)=(\bigsqcap_{i\in\mathfrak{a}}g_i)\odot b$.
%%%\item Precomposing $b:X\kto Y$ preserves the greatest fixed point of $\Psi$. 
%%%Namely, if $f:Y\kto 1$ is the greatest fixed point of $\Psi_Y$, then $f\odot b:X\kto 1$ is the greatest fixed point of $\Psi_X$.
%%%\label{item:restrictedbwdsim3transfinite}
%%%\end{enumerate}
%%\label{item:restrictedbwdsim3}
%%\item Precomposing $b$ preserves the largest mediating arrow in Assumption~\ref{asm:existenceWeaklyFinalCoalgWeak2Lim} of 
%%Prop.~\ref{prop:generalconstructionnotop}.
%%Namely, if $l_{\max}:Y\kto Z$ is the largest mediating arrow from a cone $(Y,(\delta_i:Y\kto \overline{F}^i1))$ to 
%%a weak 2-limit $(Z,(J\gamma_i:Z\kto \overline{F}^i 1))$, then 
%%$l_{\max}\odot b:X\kto Z$ is the largest mediating arrow from a cone $(X,(\delta_i\odot b:X\kto \overline{F}^i1))$ to 
%%the weak 2-limit. 
%%\label{item:restrictedbwdsim4}
%\end{enumerate}
%We write $\mathcal{X}\infbwd\mathcal{Y}$ if there exists a restricted backward simulation from $\mathcal{X}$ to $\mathcal{Y}$.
%\end{mydefinition}

%The following theorem states the 
%The soundness of this restricted backward simulation
%%It is 
%can be
%proved in the similar manner to Lem.~\ref{lem:soundnessRBwdtop},
%%and Thm.~\ref{thm:soundnessFwd}, 
%%namely, 
%i.e.\
%along the  construction of the largest homomorphism.
%The soundness of this restricted backward simulation can be proved 

%\begin{mytheorem}\label{thm:soundnessRBwd}$\!\!\!{}^\dagger$
%Let $F$ be an endofunctor and $T$ be a monad on $\mathbb{C}$ 
%%that constitute an infinite trace situation with respect to an order $\sqsubseteq$.
%%We assume that $T$ and $F$ 
%that satisfy the conditions of Prop.~\ref{prop:generalconstructionnotop}
%wrt.\ an order $\sqsubseteq$. 
%%We assume the conditions in Lem.~\ref{lem:soundnessRBwd}.
%For  two $(T,F)$-systems $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$,
%$\mathcal{X}\infbwd\mathcal{Y}$ implies $\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$.
%\qed
%\end{mytheorem}

%For a $(\pow,\FSigma)$-system and a $(\giry,\FXi)$-system, we give sufficient conditions for 
%a backward simulation to satisfy the assumptions in Def.~\ref{def:restrictedbwdsimnotop}.
%For a $(\pow,\FSigma)$-system, sufficient conditions to satisfy Assumptions~\ref{item:restrictedbwdsim2} and
%\ref{item:restrictedbwdsim3}b are already presented in Lem.~\ref{lem:whenrestrictedpow}. 
%%Hence we present the remaining conditions.
%
%\begin{myproposition}\label{prop:whenrestrictedgiry}
%In Def.~\ref{def:restrictedbwdsimnotop}, 
%%\begin{enumerate}
%%\item %if $T=\pow$ and $F=\FSigma$, 
%%\begin{enumerate}
%%\item Assumption \ref{item:restrictedbwdsim2} is satisfied if $b(x)\neq\emptyset$ for each $x\in X$. \label{item:whenrestrictedpow2}
%%\item Assumption \ref{item:restrictedbwdsim3} is satisfied if $b(x)$ is finite for each $x\in X$. \label{item:whenrestrictedpow3}
%%\item 
%if $T=\pow$ and $F=\FSigma$, 
%Assumption~\ref{item:restrictedbwdsim4} is satisfied if $b(x)$ is finite for each $x\in X$. \label{item:whenrestrictedpow4}
%%\end{enumerate}
%%\label{item:whenrestrictedpow}
%%\item if $T=\lift$ and $F=\FSigma$, 
%%\begin{enumerate}
%%\item Assumption \ref{item:restrictedbwdsim2} is satisfied if $b(x)\neq\bot$ for each $x\in X$. \label{item:whenrestrictedlift2}
%%\item Assumption \ref{item:restrictedbwdsim3} is always satisfied. \label{item:whenrestrictedlift3}
%%\item Assumption \ref{item:restrictedbwdsim4} is always satisfied.\label{item:whenrestrictedlift4}
%%\end{enumerate}
%%\label{item:whenrestrictedlift}
%%\item 
%Moreover,
%if $T=\giry$ and $F=\FXi$, 
%%\begin{enumerate}
%%\item 
%Assumption~\ref{item:restrictedbwdsim2} is satisfied if $b(x)(Y)=1$ for each $x\in X$ \label{item:whenrestrictedgiry2} while
%%\item 
%Assumptions~\ref{item:restrictedbwdsim3}a %is always satisfied. \label{item:whenrestrictedgiry3}
%%\item 
%and \ref{item:restrictedbwdsim4} are always satisfied. \label{item:whenrestrictedgiry4}
%\qed
%%\end{enumerate}
%\label{item:whenrestrictedgiry}
%%\end{enumerate}
%\end{myproposition}







\subsection{Forward Partial Execution for Probabilistic Systems}\label{subsec:FPEgiry}
We show that FPE can be used to aid discovery of forward and backward
 simulations,
also in the current probabilistic setting.
%


\begin{mytheorem}\label{thm:propertiesFPEgiry}
Let $F$ be a polynomial functor on $\Meas$.
For $(\giry,F)$-systems $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$, the following  hold.
\begin{enumerate}
\item
\begin{enumerate}
\item (soundness of FPE for fwd.\ sim.)  $\mathcal{X}_{\FPE}\fwd\mathcal{Y}$ implies $\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$.
\label{item:soundnessfpefwdgiry}
\item (adequacy of FPE for fwd.\ sim.) $\mathcal{X}\fwd\mathcal{Y}$ implies $\mathcal{X}_{\FPE}\fwd\mathcal{Y}$.
\label{item:adequacyfpefwdgiry}
\end{enumerate}
\label{item:safpefwdgiry}
\item
\begin{enumerate}
\item (soundness of FPE for bwd.\ sim.)  $\mathcal{X}\tbwd\mathcal{Y}_{\FPE}$ implies $\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$.
\label{item:soundnessfpebwdtopgiry}
\item (adequacy of FPE for bwd.\ sim.) 
$\mathcal{X}\tbwd\mathcal{Y}$ implies
      $\mathcal{X}\tbwd\mathcal{Y}_{\FPE}$, assuming that:
$d(y)(FY)=1$ for all $y\in Y$.
%  the following hold.
% \begin{enumerate}
% \item 
% $d(y)\neq\emptyset$ for all $y\in Y$.
% \label{item:restrictedfpe2topgiry}
% \qed
% \end{enumerate}
\label{item:adequacyfpebwdtopgiry}
\end{enumerate}
\label{item:safpebwdgiry}
\end{enumerate}
\end{mytheorem}

The item~\ref{item:safpefwdgiry} for forward simulations follows
immediately from Thm.~\ref{thm:soundnessFPEfwd}. For the relationship to
backward simulations, we develop another general result.

% The proof of soundness and adequacy of FPE for forward simulation can be 
% proved using axiomatic result in Thm.~\ref{thm:soundnessFPEfwd}.
% Those of FPE for backward simulation is derived from the following axiomatic result.


\begin{mytheorem}[FPE and bwd.\ sim.]%[correctness of FPE for restricted bwd.\ sim.]
\label{thm:soundnessFPEbwdnotop}$\!\!\!{}^\dagger$
Let $F$ be an endofunctor and $T$ be a monad on $\mathbb{C}$
that satisfy the conditions in Prop.~\ref{prop:generalconstructionnotop}
 (hence those in Lem.~\ref{lem:soundnessRBwdnotop}). 
Let $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$ be $(T,F)$-systems. % and $Y=Y_1+Y_2$.
\begin{enumerate}
\item (soundness for bwd.\ sim.)  $\mathcal{X}\tbwd\mathcal{Y}_{\FPE}$ implies $\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$.
\label{item:soundnessfpebwdnotop}
\item (adequacy for bwd.\ sim.) 
$\mathcal{X}\tbwd\mathcal{Y}$ implies
      $\mathcal{X}\tbwd\mathcal{Y}_{\FPE}$, 
assuming that:
the coalgebra $d:Y\kto \overline{F}Y$ satisfies 
$J!_{FY}\odot d = J!_Y$. 
% if the following conditions are satisfied.
% \begin{enumerate}
% \item 
% The coalgebra $d:Y\kto \overline{F}Y$ satisfies 
% $J!_{FY}\odot d = J!_Y$. 
% \label{item:restrictedfpe2notop}
% \qed
% \end{enumerate}
\label{item:adequacyfpebwdnotop}
\end{enumerate}
\end{mytheorem}


\newtheorem*{proof:propertiesFPEgiry}{Proof of Thm.\ \ref{thm:propertiesFPEgiry}}{\itshape}{\rmfamily}
\begin{proof:propertiesFPEgiry}
The item \ref{item:safpefwdgiry} is immediate from Thm.~\ref{thm:soundnessFPEfwd}.
In a similar manner to Lem.~\ref{lem:whenrestrictedgiry},
we can prove the item \ref{item:safpebwdgiry} using Thm.~\ref{thm:soundnessFPEbwdnotop}.
\qed
\end{proof:propertiesFPEgiry}







%
%Soundness and adequacy of FPE for forward simulation can be proved using Thm.~\ref{thm:soundnessFPEfwd}.
%
%In contrast, as 
%conditions required for a backward simulation to satisfy soundness
%%the definition of a restricted backward simulation 
%are different from that in \S\ref{subsec:FPEpow},
%the condition imposed on the simulating system to assure the adequacy of FPE for backward simulation is also different. 
%%should be modified.
%%
%\begin{mytheorem}\label{thm:soundnessFPEbwdnotop}$\!\!\!{}^\dagger$
%Let $F$ be an endofunctor and $T$ be a monad on $\mathbb{C}$ that 
%%constitute an infinite trace situation with respect to an order $\sqsubseteq$.
%%We assume that $T$ and $F$ 
%satisfy the assumptions in Prop.~\ref{prop:generalconstructionnotop}. 
%%satisfies the assumptions \ref{item:restrictedbwdsima} and \ref{item:restrictedbwdsimb} in Def.~\ref{def:restrictedbwdsim}.
%Let $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$ be $(T,F)$-systems. % and $Y=Y_1+Y_2$.
%%Moreover, we define $d_i:Y_i\kto \overline{F}Y$ ($i\in\{1,2\}$) by 
%%$d_i=d\odot \kappa_i$.
%\begin{enumerate}
%%\item (monotonicity) If $Y_1=Y_{1,1}+Y_{1,2}$, then $\mathcal{X}\infbwd\mathcal{Y}_{\FPE,Y_{1,1}}$ implies $\mathcal{X}\bwd\mathcal{Y}_{\FPE,Y_1}$.
%%\label{item:monotonicityfpebwd}
%\item (soundness)  $\mathcal{X}\infbwd\mathcal{Y}_{\FPE}$ implies $\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$.
%\label{item:soundnessfpebwd}
%\item (adequacy) 
%$\mathcal{X}\infbwd\mathcal{Y}$ implies $\mathcal{X}\infbwd\mathcal{Y}_{\FPE}$ if the following conditions are satisfied.
%%Assume the following conditions.
%\begin{enumerate}
%\item 
%The arrow $d:Y\kto \overline{F}Y$ satisfies $J!_{FY}\odot d = J!_Y$. 
%%The arrow $(d_1+\id_{X_2}):Y\kto \overline{F}Y+Y_2$ satisfies $J!_{\overline{F}Y+Y_2}\odot (c_1+\id_{X_2}) = J!_Y$. 
%\label{item:restrictedfpe2}
%\item 
%%One of the followings is satisfied.
%%\begin{enumerate}
%%\item 
%i) Assumption~\ref{asm:existenceWeaklyFinalCoalgGFPcont} in Prop.~\ref{prop:generalconstructionnotop} is satisfied, or,  
%%\label{item:restrictedfpe3cont}
%%\item 
%ii) Assumption~\ref{asm:existenceWeaklyFinalCoalgGFPtransfinite} in Prop.~\ref{prop:generalconstructionnotop} is satisfied and
%precomposing $d$ preserves the glb.\ of a decreasing transfinite sequence.
%%Namely, for $A\in\Kl(T)$ and a family of Kleisli arrows $(g_{\mathfrak{i}}:\overline{F}Y\kto A)_{\mathfrak{i}\in\mathfrak{a}}$ where 
%%$\mathfrak{a}$ is a limit ordinal and $\mathfrak{i}\leq \mathfrak{j}$ implies $g_{\mathfrak{i}}\sqsupseteq g_{\mathfrak{j}}$,
%%%$g_0\sqsupseteq g_1 \sqsupseteq g_2\ldots :Y\kto A$,
%%$\bigsqcap_{\mathfrak{i}<\mathfrak{a}}(g_{\mathfrak{i}}\odot d)=(\bigsqcap_{\mathfrak{i}<\mathfrak{a}}g_{\mathfrak{i}})\odot d$ holds.
%%
%%\item Precomposing $b:X\kto Y$ preserves the greatest fixed point of $\Psi$. 
%%Namely, if $f:Y\kto 1$ is the greatest fixed point of $\Psi_Y$, then $f\odot b:X\kto 1$ is the greatest fixed point of $\Psi_X$.
%%\label{item:restrictedfpe3transfinite}
%%\end{enumerate}
%%
%%\item Precomposing $d:Y\kto \overline{F}Y$ preserves the greatest fixed point of a decreasing sequence. % in $\Kl(T)(\overline{F}Y,1)$.
%%Namely, for $A\in\Kl(T)$ and a decreasing sequence $g_0\sqsupseteq g_1 \sqsupseteq g_2\ldots :\overline{F}Y\kto A$ of Kleisli arrows, 
%%$\bigsqcap_{i\in\omega}(g_i\odot d)=(\bigsqcap_{i\in\omega}g_i)\odot d$ holds.
%%%\item Precomposing $(d_1+\id_{X_2}):Y\kto \overline{F}Y+Y_2$ preserves the greatest fixed point of a decreasing sequence. % in $\Kl(T)(\overline{F}Y,1)$.
%%%Namely, for $A\in\Kl(T)$ and a decreasing sequence $g_0\sqsupseteq g_1 \sqsupseteq g_2\ldots :\overline{F}Y+Y_2\kto A$ of Kleisli arrows, 
%%%$\bigsqcap_{i\in\omega}(g_i\odot b)=(\bigsqcap_{i\in\omega}g_i)\odot b$ holds.
%%%\item Precomposing $d:Y\kto \overline{F}Y$ preserves the greatest fixed point of $\Psi$. 
%%%Namely, if $f:\overline{F}Y\kto 1$ is the greatest fixed point of $\Psi_{\overline{F}Y}$, then $f\odot d:Y\kto 1$ is the greatest fixed point of $\Psi_{\overline{F}Y}$.
%\label{item:restrictedfpe3}
%\item Precomposing $d$ preserves the largest mediating arrow  
%in Assumption~\ref{asm:existenceWeaklyFinalCoalgWeak2Lim} of Prop.~\ref{prop:generalconstructionnotop}.
%%Namely, if $l_{\max}:\overline{F}Y\kto Z$ is the largest mediating arrow from a cone $(\overline{F}Y,(\delta_i:\overline{F}Y\kto \overline{F}^i1))$ to 
%%a weak 2-limit $(Z,(J\gamma_i:Z\kto \overline{F}^i 1))$, then 
%%$l_{\max}\odot d:Y\kto Z$ is the largest mediating arrow from a cone $(X,(\delta_i\odot b:X\kto \overline{F}^i1))$ to 
%%the weak 2-limit. 
%\label{item:restrictedfpe4}
%\qed
%\end{enumerate}
%%Then $\mathcal{X}\infbwd\mathcal{Y}$ implies $\mathcal{X}\infbwd\mathcal{Y}_{\FPE}$.
%\label{item:adequacyfpebwd}
%\end{enumerate}
%\end{mytheorem}
%
%For $(\pow,\FSigma)$-systems and $(\giry,\FXi)$-systems, 
%sufficient conditions for the simulating system to satisfy the assumptions in Thm.~\ref{thm:soundnessFPEbwdnotop} are as follows.
%They can be proved in the similar manner to a proof for Prop.~\ref{prop:whenrestrictedgiry}.
%The conditions for a $(\pow,\FSigma)$-system corresponding to Assumptions~\ref{item:restrictedfpe2} and \ref{item:restrictedfpe3}(ii) are already presented in Prop.~\ref{thm:soundnessFPEbwdtop}.
%Interestingly, the conditions for a $(\pow,\FSigma)$-system corresponding to Assumption~\ref{item:restrictedfpe4} is
%again \emph{image-finiteness}.
%%hence we present the remaining ones. 
%
%
%\begin{myproposition}\label{prop:whenrestrictedfpegiry}
%In Thm.~\ref{thm:soundnessFPEbwdnotop}, 
%%\begin{enumerate}
%%\item % 
%%\begin{enumerate}
%%\item Assumption \ref{item:restrictedfpe2} is satisfied if $d(y)\neq\emptyset$ for each $y\in Y$.
%%\item Assumption \ref{item:restrictedfpe3} is satisfied if $d(y)$ is finite for each $y\in Y$. 
%%\item 
%if $T=\pow$ and $F=\FSigma$,
%Assumption~\ref{item:restrictedfpe4} is satisfied if $d(y)$ is finite for each $y\in Y$. 
%%\item Assumption \ref{item:restrictedfpe4} is satisfied if $d(y)$ is finite for each $y\in Y$.
%%\end{enumerate}
%%\item if $T=\lift$ and $F=\FSigma$,
%%\begin{enumerate}
%%\item Assumption \ref{item:restrictedfpe2} is satisfied if $d(y)\neq\bot$ for each $y\in Y$.
%%\item Assumption \ref{item:restrictedfpe3} is always satisfied.
%%\item Assumption \ref{item:restrictedfpe4} is always satisfied.
%%\end{enumerate}
%%\item 
%Moreover,
%if $T=\giry$ and $F=\FXi$, 
%%\begin{enumerate}
%%\item 
%Assumption~\ref{item:restrictedfpe2} is satisfied if $d(y)(FY)=1$ for each $y\in X$ while
%%\item 
%Assumptions~\ref{item:restrictedfpe3}(i) %is always satisfied.
%%\item 
%and \ref{item:restrictedfpe4} are always satisfied.
%\qed
%%\end{enumerate}
%%\end{enumerate}
%\end{myproposition}




\section{ Systems with Other Branching Types}\label{sec:othermonads}
In this section we briefly discuss two more pairs of $F$ and $T$ that constitute  infinite trace situations.

The first pair is a polynomial functor $F$ on $\Sets$ and the \emph{lift monad} $\lift$.
For a given set $X\in\Sets$, $\lift X$ is given by $\{\bot\}+X$.
The added element $\bot$ represents the aborting or non-termination of the program,
and hence an $(\lift,F)$-system can be regarded as a \emph{tree
automaton with exception}. To show that  $F$ and $\lift$ constitute an
infinite trace situation, we rely on
Prop.~\ref{prop:generalconstructionnotop}
(but not Prop.~\ref{prop:generalconstructiontop}, since $\lift X$ does
not have the greatest element).	
%   by constructing 
% the largest homomorphism using Prop.~\ref{prop:generalconstructionnotop}.
Therefore, much like for $\giry$,  we can check trace inclusion by forward or T-backward simulations (see \S\ref{subsec:Klsimgiry}).
 More details are found in \S\ref{subsec:liftmonad}.

The second pair is that of polynomial $F$ on $\Sets$ and the \emph{subdistribution monad} $\dist$.
For a given set $X\in\Sets$, $\dist X$ is the set 
$\{d\colon X\to [0,1]\mid \sum_{x\in X}d(x)\le 1\}$ of (discrete) subdistributions over $X$.
The subdistribution monad $\dist$ is similar to the sub-Giry monad $\giry$, and
a $(\dist,F)$-system can be also regarded as a  probabilistic tree automaton.
We can prove that $F$ and $\dist$ constitute an infinite trace situation. 
The resulting infinite trace semantics has limited use, however, due to
the discrete nature of an arrow $X\kto \dist Z$ (it assigns a
probability  to  a single tree and the probability is most of the time
$0$; see Example~\ref{ex:firstExample}).
%  is different from that of a $(\giry,F)$-system:
% the former is given as a \emph{distribution} over $\Treeinf(\Sigma)$ 
% while the latter is given as a \emph{measure} on
% $(\Treeinf(\Sigma),\sigalg_{\infty})$.
Another difficulty is that  infinite traces for $T=\dist$ does not
follow from either of our general results  (Prop.~\ref{prop:generalconstructiontop} or
Prop.~\ref{prop:generalconstructionnotop})---in~\S\ref{subsec:subdistmonad}
we construct infinite traces for $T=\dist$ in concrete terms. 
% In fact, we cannot construct the largest homomorphism neither by using the method 
% in Prop.~\ref{prop:generalconstructiontop} nor Prop.~\ref{prop:generalconstructionnotop}.
%prove that $\FSigma$ and  $\dist$ constitute an infinite trace situation by constructing 
%the largest homomorphism using Prop.~\ref{prop:generalconstructiontop} nor Prop.~\ref{prop:generalconstructionnotop}.
%This means that the conditions of  these propositions are only  a sufficient condition and 
%not a necessary condition for a functor and a monad to constitute an infinite trace situation.
%Furthermore, 
This prevents us from applying the general theories for Kleisli
simulations in~\S{}\ref{sec:powersetmonad}--\ref{sec:girymonad}.
% introducing a notion of backward simulation that is sound for $(\dist,F)$-systems.
 For more details, see \S\ref{subsec:subdistmonad}.


%For the definition of subdistribution monad $\dist$, see~\cite{hasuo07generictrace} for example.
%We define an endofunctor $F$ on $\Sets$ by $F=\Sigma\times(\place)$ where $\Sigma=\{p,q\}$.
%
%For each pair of objects $X,Y\in\Sets$, we can define an order $\sqsubseteq$ on $\Kl(\dist)(X,Y)$ by 
%$f\sqsubseteq g \;\Leftrightarrow\; \forall x\in X, y\in Y.\, f(x)(y)\leq g(x)(y)$.
%The $F$-final coalgebra on $\Sets$ exists and is given by $\zeta:\Sigma^{\omega}\to F\Sigma^{\omega}$ where $\zeta(aw)=(a,w)$ for $a\in\Sigma$ and $w\in\Sigma^{\omega}$. 
%It is known that distributive law $\lambda:F\dist\Rightarrow \dist F$ 
%for $F$ and $\dist$ exists~\cite{hasuo07generictrace}. Hence $F$ can be lifted to an endofunctor $\overline{F}$ on $\Kl(\dist)$.
%Moreover, for each $\overline{F}$-coalgebra $c:X\kto \overline{F}X$, we can define the largest homomorphism from $c$ to $J\zeta$ as follows.
%For each $x\in X$, $w=a_0a_1a_2\ldots\in\Sigma^{\omega}$ and $i\in\omega$, 
%we define a distribution $p_{c,x,w,i}\in\dist X$ by the induction on $i$ as follows:
%$p_{c,x,w,0}(y)=[y\mapsto 1]$ and $p_{c,x,w,i+1}(y)=\sum_{z\in X}p_{c,x,w,i}(z)\cdot c(z)(a_i,y)$.
%Moreover, we define $r_{c,x,w,i}\in[0,1]$ by $r_{c,x,w,i}=\sum_{y\in X}p_{c,x,w,i}(y)$.
%As $\sum_{y\in X} c(x)(a,y)\leq 1$ for all $x\in X$ and $a\in\Sigma$, $r_{c,x,w,i}$ is decreasing with respect to $i$.
%Therefore we can define an arrow $f_c:X\kto \Sigma^{\omega}$ by $f_c(x)(w)=\lim_{i\to\infty}r_{c,x,w,i}$.
%It is not so difficult to see that $f_c$ is the largest homomorphism from $c$ to $J\zeta$.
%Therefore $F$ and $\dist$ satisfy infinite trace situation. 
%
%\begin{floatingfigure}[r]{0pt}
%%\begin{center}
%\begin{xy}
%%(0,-20)*{} = "",
%(-8,8)*{\mathcal{X}} = "",
%(0,0)*+{\circ} = "x0",
%\ar (0,-5);"x0"
%\ar @(dl,ul)^{p,\frac{1}{2}} "x0";"x0"
%\ar @(dr,ur)_{q,\frac{1}{2}} "x0";"x0"
%\end{xy}
%\vspace{5mm}
%%\end{center}
%\end{floatingfigure}
%However, $F$ and $\dist$ do not satisfy assumptions in Prop.~\ref{prop:generalconstructionnotop}.
%More concretely, assumption~\ref{asm:existenceWeaklyFinalCoalgWeak2Lim} is not satisfied.
%We define a cone $(X,(\pi_i:X\kto F^i1)_{i\in\omega})$ over a sequence 
%$1\overset{J!_{F1}}{\kleftarrow} \overline{F}1\overset{JF!_{F1}}{\kleftarrow} \overline{F}^2 1\overset{JF^2!_{F1}}{\kleftarrow} \ldots$ in $\Kl(\dist)$ as follows:
%$X=1$ and $\pi_i(*)(a_0a_1\ldots a_{i-1})=\frac{1}{2^i}$ for each $a_0a_1\ldots a_{i-1}\in F^i 1=\{p,q\}^i$.
%(Such a cone is indeed constructed in the proof of Prop.~\ref{prop:generalconstructionnotop} if we define $c:X\kto \overline{F}1$ by 
%$c(*)(p,*)=c(*)(q,*)=\frac{1}{2}$. See an automaton $\mathcal{X}$ in the right.)
%It is easy to see that there exists no $l:X\kto Z$ such that $J\gamma_i\odot l=\pi_i$ for all $i\in\omega$.
%This means that $J:\Sets\to\Kl(\dist)$ does not lift the limit of the final sequence to a weak limit.

\section{Related Work}
The construction of the largest homomorphism given in
Prop.~\ref{prop:generalconstructionnotop} is based on the one
in~\cite{cirstea10genericinfinite}.
The latter imposes some technical conditions  on a monad $T$, including
 a ``totality'' condition that excludes $T=\pow$ from its instances (the
 nonempty powerset monad is an instance). Our assumption of lifting to 
a 2-limit (Assumption~\ref{asm:existenceWeaklyFinalCoalgWeak2Lim} in
Prop.~\ref{prop:generalconstructionnotop}) is inspired by a condition
in~\cite{cirstea10genericinfinite}, namely that the limit $Z$ is lifted
to a \emph{weak} limit in $\Kl(T)$. It is not the case that
Prop.~\ref{prop:generalconstructionnotop} subsumes the construction 
in~\cite{cirstea10genericinfinite}: the former does not apply to the
nonempty powerset monad (but our Prop.~\ref{prop:generalconstructiontop}
does apply to it).

% There, for a given $\overline{F}$-coalgebra $c$, 
% a cone over a sequence 
% $1\overset{J!_{F1}}{\kleftarrow} \overline{F}1\overset{JF!_{F1}}{\kleftarrow} \overline{F}^2 1\overset{JF^2!_{F1}}{\kleftarrow} \ldots$
% %lifted final sequence in $\mathbb{C}$ using the coalgebra. 
% is constructed using $c$.
% %In~\cite{cirstea10genericinfinite}, 
% It is assumed that the limit of the final sequence is lifted to a weak limit
% that admits the largest mediating arrow. 
% %Therefore we have the largest mediating arrow, that 
% Then the largest mediating arrow
% is regarded as a infinite trace semantics. % in~\cite{cirstea10genericinfinite}.
% The construction in Prop.~\ref{prop:generalconstructionnotop} is different from it 
% %~\cite{cirstea10genericinfinite} 
% in two points.
% First, in~\cite{cirstea10genericinfinite}, 
% calculation of the greatest lower bound is not 
% to assure that the constructed family of Kleisli arrows  constitute a cone, 
% it is assumed that the final object in $\mathbb{C}$ is lifted to a final object in $\Kl(T)$.
% In contrast, we do not assume it %that  the final object in $\mathbb{C}$ is lifted to a final object in $\Kl(T)$.
% and instead, 
% we construct a cone 
% %over the sequence 
% %$1\overset{J!_{F1}}{\kleftarrow} \overline{F}1\overset{JF!_{F1}}{\kleftarrow} \overline{F}^2 1\overset{JF^2!_{F1}}{\kleftarrow} \ldots$
%  by taking the greatest lower bound of a family of Kleisli arrows.
% Second, instead of stopping the construction when we obtain the largest mediating arrow to the weak limit, we go furthermore.
% Then we take the greatest lower bound again and construct a homomorphism, that is regarded as the infinite trace semantics in this paper.
% There exists a a pair of concrete systems that constitute exactly the same cone over the sequence but 
% whose infinite trace semantics is captured differently via the largest homomorphism (e.g.\ Example~\ref{example:sameconediffhom}).
% %concrete system whose infinite trace semantics is captured differently by the largest homomorphism and by the largest mediating arrow.
% %There exists a concrete system whose infinite trace semantics is captured differently by the largest homomorphism and by the largest mediating arrow.

In~\cite{kerstan13coalgebraictrace}, an explicit description of a
(proper, not weakly) final $\overline{F}$-coalgebra is given 
for
$F\in\bigl\{\,\Sigma\times(\place),\,1+\Sigma\times(\place)\,\bigr\}$
and 
$T\in\{\giry,\Geq\}$. Here $\Geq$ is the \emph{Giry monad} and restricts
$\giry$
to proper, not sub-, distributions.
%four combinations of a 
% two functors $F$ and two monads $T$ on 
% $\Meas$---$F=\Sigma\times(\place)$ and $F=1+\Sigma\times(\place)$, and
% $T=\giry$ (sub-Giry monad) and $T=\Geq$ (strict-Giry monad). 
We do not use their results (proper finality) for characterization of
infinite traces, because: 1) if $T=\giry$ then the final coalgebras do
not coincide with the set of possibly infinite words; and 2) if $T=\Geq$
then language inclusion is reduced to the equality. We doubt about the value
of
developing simulation-based methods for the latter degenerate case, one
reason being that trace inclusion is often a more difficult problem than
trace equivalence.
For example, finite trace inclusion for probabilistic systems is undecidable~\cite{blondel03undecidableproblems} while trace equivalence is decidable~\cite{kiefer11languageequivalence}.
% As trace inclusion is often more difficult to prove than trace equivalence 
% (e.g.\ finite trace inclusion for probabilistic systems is undecidable~\cite{blondel03undecidableproblems} while trace equivalence is decidable~\cite{kiefer11languageequivalence}),
% it seems to be unsuitable to use simulation to prove trace equivalence.


% However, we do not use the final $\overline{F}$-coalgebra to capture infinite trace semantics in this paper because of the following reason.
% If we choose  $\giry$ for a monad, then the carrier of final $F$-coalgebra becomes 
% an empty set (if $F=\Sigma\times(\place)$) or $\Sigma^*$ (if $F=1+\Sigma\times(\place)$),
% and the unique homomorphism to the final $\overline{F}$-coalgebra does not capture infinite trace.
% In contrast, if we choose $\Geq$ for a monad, then  the carrier of final $F$-coalgebra 
% becomes $\Sigma^{\omega}$ (if $F=\Sigma\times(\place)$) or $\Sigma^{\infty}$ (if $F=1+\Sigma\times(\place)$),
% and the unique homomorphism to the final $\overline{F}$-coalgebra does  capture infinite trace as we desire.
% However, in this setting, infinite trace inclusion coincides with infinite trace equivalence.
% As trace inclusion is often more difficult to prove than trace equivalence 
% (e.g.\ finite trace inclusion for probabilistic systems is undecidable~\cite{blondel03undecidableproblems} while trace equivalence is decidable~\cite{kiefer11languageequivalence}),
% it seems to be unsuitable to use simulation to prove trace equivalence.

In~\cite{schubert09terminalcoalgebras}, it is shown that:
a limit of a $\omega^{\text{op}}$-sequence consisting of standard Borel
spaces and surjective measurable functions is preserved by
a polynomial functor $F$ 
(where  constants are restricted to  standard Borel spaces), and also by $\giry$.
It is also shown there that such a polynomial functor $F$
preserves standard Borel spaces, and so does $\giry$.
These facts imply the existence of a final $\giry F$-coalgebra in
$\Meas$ for every polynomial functor $F$. Note however that this final
$\giry F$-coalgebra captures (probabilistic) bisimilarity, not trace semantics.

\section{Conclusions and Future Work}\label{sec:conclusion}
We have shown that the technique
 forward and backward Kleisli simulations~\cite{hasuo06genericforward}
 and that of FPE~\cite{urabeH14genericforward}---techniques originally 
developed for witnessing \emph{finite} trace inclusion---are also
 applicable to \emph{infinite} trace semantics.
%  that are introduced in~\cite{hasuo06genericforward} 
% to witness finite trace inclusion and FPE that was introduced in~\cite{urabeH14genericforward} to aid the simulation can be also used %to witness 
% for
% infinite trace setting.
 We followed~\cite{jacobs04tracesemantics} (and
 also~\cite{cirstea10genericinfinite,kerstan13coalgebraictrace}) to 
characterize infinite trace semantics in coalgebraic terms, on which we
 established properties of Kleisli simulations such as soundness.
 We developed our theory for two classes of instances: nondeterministic
 systems and probabilistic ones.

% Their correctness with respect to infinite trace was proved using coalgebraic 
% framework introduced in~\cite{jacobs04tracesemantics} where
% the infinite trace semantics is captured as the largest homomorphism to the lifted final coalgebra.
% For two types of systems---a nondeterministic tree automaton and a probabilistic tree automaton,
% we have 
% characterized their coalgebraic infinite trace semantics using automata-theoretic terms.
%shown that their automata-theoretic infinite languages can be captured as the coalgebraic infinite trace semantics.

There are some directions for a future work.
In~\cite{urabeH14genericforward}, in addition to FPE, a transformation called \emph{backward partial execution} (BPE) is introduced.
Similarly to FPE, BPE can also aid forward and backward Kleisli simulation for finite trace in the sense that it satisfy soundness and adequacy.
However, BPE is only defined for word automata (with $T$-branching) and
not generally for $(T,F)$-systems.
Defining BPE categorically and proving its soundness and adequacy with
respect to infinite trace, possibly restricting to word automata,  is one of the future work.

Another direction is implementation and experiments.
As forward and backward Kleisli simulations in this paper are defined in almost the same way as~\cite{urabeH14genericforward},
we can use the implementation already developed
there
%in~\cite{urabeH14genericforward} 
to check  infinite trace inclusion.
%A problem is that we have few examples where the implemented programs are tested.
% Finding a good working example is 
% a challenge to overcome. 
%another direction of the future work.

%For two types of systems---a nondeterministic tree automaton and a probabilistic tree automaton,
%we have shown that their infinite trace semantics can be captured via the framework introduced in~\cite{jacobs04tracesemantics}.
%In the framework, the infinite trace semantics is 


%Future work:
%\begin{itemize}
%\item BPE?
%\end{itemize}

\subparagraph*{Acknowledgments}
The authors  are supported by Grants-in-Aid
No.\ 24680001 \& 15K11984, JSPS.

%
%\section{Generic Construction of Weakly Final Coalgebra in Kleisli Category}\label{sec:genconstruction}
%In this section, we give a sufficient condition for a category $\mathbb{C}$, a functor $F$, and a monad $T$ to satisfy infinite trace situation.
%Under some conditions, for a given $\overline{F}$-coalgebra $c$, 
%we can construct the coalgebraic infinite trace semantics $\trinf(c)$---the largest homomorphism from $c$ to $J\zeta$.

%We note that the assumptions given below is only a sufficient condition and not a necessary condition for satisfying infinite trace situation.
%There exist $\mathbb{C}$, $F$ and $T$ where
%infinite trace situation holds but the assumptions given below are not satisfied.
%(e.g.\ $\mathbb{C}=\Sets$, $F=\{a,b\}\times(\place)$, and $T$ is a \emph{subdistribution monad} $\dist$. See Example~\ref{example:nogeneralconstruction} for more details.)





%\section{Coalgebraic Infinite Trace Semantics and Automata-theoretic Semantics}\label{sec:coincidencesemantics}
%In \S\ref{sec:weaklyfinalcoalg}, we gave a definition of coalgebraic infinite trace semantics of a system in a general way where 
%a functor $F$ and a monad $T$ are arbitrary as long as they satisfy 
%%a condition called 
%infinite trace situation.
%In \S\ref{sec:genconstruction}, we gave a sufficient condition for a functor and a monad to satisfy infinite trace situation.
%%
%In this section, we fix $F$ and $T$ to three pairs of concrete functors and monads.
%For each pair of $F$ and $T$, we
%\begin{itemize}
%\item show that $F$ and $T$ satisfy infinite trace situation using result in \S\ref{sec:genconstruction},
%\item define the semantics of $(F,T)$-system using automata-theoretic terms, and
%\item show that the automata-theoretic semantics coincides with the coalgebraic infinite trace semantics defined using a weakly final coalgebra.
%\end{itemize}
%We prove that these combinations satisfy the condition given in \S\ref{sec:genconstruction} and therefore satisfy infinite trace situation.
%Then we see concrete systems represented by these $(T,F)$-systems. 
%Moreover, we investigate what kind of semantics do their coalgebraic infinite trace semantics captures.

%\subsection{Example {I}: Nondeterministic Tree Automaton} \label{subsec:examplenondet}
%In this section, we let $\mathbb{C}$ be $\Sets$---a category of small sets and functions between them.
%A functor $F$ is fixed as follows. 
%\begin{mydefinition}\label{def:FSigmaSets}
%For a ranked alphabet $\Sigma=(\Sigma_n)_{n\in\omega}$,
%we define a polynomial functor $F_{\Sigma}:\Sets\to\Sets$ by 
%$F_{\Sigma}=\coprod_{n\in\omega} \Sigma_n\times (\place)^n$.
%\end{mydefinition}
%A monad $T$ is fixed to a powerset monad $\pow$. 
%
%
%It is already known that  $\FSigma$ and $\pow$ satisfy infinite trace situation \cite{jacobs04tracesemantics}.
%Alternative proof that they satisfy infinite trace situation can be given using Prop.~\ref{prop:generalconstructionnotop}. %framework in \S\ref{sec:genconstruction}.
%\begin{myproposition}\label{prop:satassumptionnondet}
%A polynomial functor $F_{\Sigma}$ on $\Sets$ and a powerset monad $\pow$ satisfy assumptions in Prop.~\ref{prop:generalconstructionnotop}
%with respect to the order in Def.~\ref{def:orderenrichment}. 
%Therefore, $\FSigma$ and $\pow$ satisfy infinite trace situation. 
%\qed
%\end{myproposition}
%
%
%%
%%\begin{myproposition}
%%%Let $F$ be a polynomial functor on $\Sets$ and $\pow$ be a powerset monad. 
%%%For each homset $\Kl(\pow)(X,Y)$, we define an order $\sqsubseteq$ as follows:
%%%\begin{equation*}
%%%f\sqsubseteq g \;\Leftrightarrow\; \forall x\in X.\, f(x)\subseteq g(x),.
%%%\end{equation*}
%%The functor $F_{\Sigma}$ and the monad $\pow$ satisfy infinite trace situation with respect to $\sqsubseteq$ in Def.~\ref{def:orderenrichment}.
%%\end{myproposition}
%
%In \cite{jacobs04tracesemantics}, an explicit description of the coalgebraic infinite trace semantics is given using set-theoretic terms.
%Here, we characterize coalgebraic infinite trace semantics using more automata-theoretic terms.
%%We call a \emph{nondeterministic tree automaton} for a $(\pow,\FSigma)$-system.
%%
%%A $(\pow,\FSigma)$-system can be regarded as a nondeterministic automaton that generates an infinite trees.
%%Its language can be formally defined as follows.
%%The formal definition of the language of a $(\pow,\FSigma)$-system can be given as follows.
%%For a $(\pow,\FSigma)$-system, we can define its  infinite language as follows.
%\begin{mydefinition}\label{def:nondettreeautomaton}
%A $(\pow,\FSigma)$-system is called a \emph{nondeterministic tree automaton}.
%
%Let $\mathcal{X}=(X,s,c)$ be a nondeterministic tree automaton. 
%For a $\Sigma$-labeled  infinite tree $t=(D,l)$ and a state $x\in X$, 
%a $(X)_{n\in\omega}$-labeled  infinite tree $t_r=(D,l_r)$ that has the same domain as $t$ is 
%called a \emph{run of $\mathcal{X}$ from $x$ that generates $t$} if  
%\begin{itemize}
%\item $l_r(\empseq)=x$, and
%\item for each $\alpha\in D$ such that $l(\alpha)=a\in\Sigma_n$, $l_r(\alpha)=y$ and 
%$l_r(\alpha i)=y_i$ for each $0\leq i \leq n-1$, we have $(a,y_0,\ldots,y_{n-1}) \in c(y)$.
%\end{itemize}
%
%For a state $x\in X$, 
%an \emph{infinite language of $\mathcal{X}$ from $x$} is a set 
%$\lang(\mathcal{X},x)\subseteq \Treeinf(\Sigma)$ that is defined by
%$%\begin{equation*}
%\lang(\mathcal{X},x)=\{
%T\in \Treeinf(\Sigma)
%\mid
%\text{$\exists$  run of $\mathcal{X}$ from $x$ that generates $T$}
%\}
%$. %\end{equation*}
%%
%An \emph{infinite language of $\mathcal{X}$} is a set $\lang(\mathcal{X})\subseteq \Treeinf(\Sigma)$ that is defined by
%$\lang(\mathcal{X})=\bigcup_{X\in s(*)} \lang(\mathcal{X},x)$.
%\end{mydefinition}
%%
%%In the below, we show the coincidence between  infinite language of nondeterministic automaton and its infinite trace semantics.
%
%
%In the rest of this section, we show that this automata-theoretic semantics of nondeterministic tree automaton coincides with coalgebraic infinite trace semantics.
%
%The carrier of $F_{\Sigma}$-final coalgebra in $\Sets$ is given by a set of  infinite trees.
%\begin{mylemma}\label{lem:finalcoalgsets}
%%Let $F_{\Sigma}$ be an endofunctor on $\Sets$ that is defined as Def.~\ref{def:FSigmaSets}.
%For the functor $\FSigma$ in Def.~\ref{def:FSigmaSets}, we define an arrow $\zeta:\Treeinf(\Sigma)\to F_{\Sigma}\Treeinf(\Sigma)$ in $\Sets$ by 
%$%\begin{equation*}
%\zeta(t)=(a,(t_0,\ldots,t_{n-1}))
%$. %\end{equation*}
%Here, $t=(D,l)$, $a=l(\empseq)\in\Sigma_n$, and $t_i$ is the $i$'th subtree of $t$.
%%and for each $i\in\{0,\ldots,n-1\}$, $D_i=\{\alpha\in\mathbb{N}^*\mid i\alpha\in D\}$ and $l_i(\alpha)=l(i\alpha)$.
%%
%Then $\zeta$ is a final $F_{\Sigma}$-coalgebra.
%\qed
%\end{mylemma}
%
%The following theorem states that the infinite language of a nondeterministic tree automaton can be captured as coalgebraic infinite trace semantics.
%\begin{mytheorem}\label{thm:largestsemanticspow}
%Let $\mathcal{X}=(X,s,c)$ be a nondeterministic tree automaton.
%With respect to the order in Def.~\ref{def:orderenrichment}, for each $x\in X$, we have $\trinf(c)(x)=\lang(\mathcal{X},x)$.
%Moreover, $\trinf(c)\odot s(*)=\lang(\mathcal{X})$.
%\qed
%\end{mytheorem}

%\begin{mycorollary}
%Let $\mathcal{X}=(X,s,c)$ be a nondeterministic tree automaton.
%In Thm.~\ref{thm:largestsemanticspow}, $\trinf(c)\odot s:1\kto \Treeinf(\Sigma)$ gives the  infinite nondeterministic tree language of $\mathcal{A}$,
%namely, $\trinf(c)\odot s(*)=\lang(\mathcal{A})$.
%\qed
%\end{mycorollary}

%In the above, we reviewed that $\FSigma$ and $\pow$ satisfy infinite trace situation,
%and for a given $\overline{\FSigma}$-coalgebra $c$, 
%we constructed  the largest homomorphism from $c$ to $J\zeta$.


%\subsection{Example {II}: Tree Automaton with Exception} \label{subsec:exampleexception}
%In this section, we again let $\mathbb{C}$ be $\Sets$.
%Moreover, a functor is fixed to $\FSigma$ in Def.~\ref{def:FSigmaSets}, 
%and a monad is fixed to a lift monad $\lift$.
%%
%They satisfy infinite trace situation.
%\begin{myproposition}\label{prop:satassumptionexception}
%A functor $F_{\Sigma}$ on $\Sets$ and a lift monad $\lift$ satisfy assumptions in Prop.~\ref{prop:generalconstructionnotop}
%with respect to the order in Def.~\ref{def:orderenrichment}. 
%Therefore, $\FSigma$ and $\lift$ satisfy infinite trace situation.
%\qed
%\end{myproposition}


%In the above, we showed that $\FSigma$ and $\lift$ satisfy infinite trace situation
%by constructing the largest homomorphism from the given $\overline{\FSigma}$-coalgebra to $J\zeta$.
%Alternative proof of satisfaction of infinite trace situation can be given using framework in \S\ref{sec:genconstruction}.



%\subsection{Example {III}: Probabilistic Tree Automaton} \label{subsec:exampleprob}
%In this section, we fix a category $\mathbb{C}$ to $\Meas$---a category of measurable spaces and measurable functions between them.
%A functor $F$ is fixed as follows (c.f.\ Def.~\ref{def:polyfuncmeas}). 
%\begin{mydefinition}\label{def:FSigmaMeas}
%For a ranked alphabet $\Sigma=(\Sigma_n)_{n\in\omega}$,
%we define a restricted polynomial functor $F_{\Sigma}:\Meas\to\Meas$ by 
%$F_{\Sigma}=\coprod_{n\in\omega} \Sigma_n\times (\place)^n$.
%\end{mydefinition}
%%
%A monad $T$ is fixed to Giry monad $\giry$.
%%
%Then they satisfy infinite trace situation.









%\section{Kleisli Simulation for  Infinite Trace}\label{sec:Kleislisim}

%\begin{minipage}{0.8\hsize}
%\begin{mydefinition}\label{def:fwdsim}
%Let $F$ be an endofunctor and $T$ be a monad on $\mathbb{C}$ that satisfy infinite trace situation with respect to an order $\sqsubseteq$.
%%Let $F$ be an endofunctor on $\mathbb{C}$ and $T$ be a monad on $\mathbb{C}$.
%%We assume that $F$ and $T$ satisfy infinite trace situation with respect to an order $\sqsubseteq$.
%%We assume that each homset of $\Kl(T)$ is enriched with an order $\sqsubseteq$.
%For  two $(T,F)$-systems $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$,
%a \emph{forward Kleisli simulation} is a Kleisli arrow $f:Y\kto X$ that satisfies the following conditions.
%\begin{equation*}
%s\sqsubseteq f\odot t, \;\;\;\text{and}\;\;\;
%c\odot f \sqsubseteq \overline{F}f\odot d
%\end{equation*}
%We write $\mathcal{X}\fwd\mathcal{Y}$ if there exists a forward simulation from $\mathcal{X}$ to $\mathcal{Y}$.
%\end{mydefinition}
%\end{minipage}
%\begin{minipage}{0.2\hsize}
%\vspace*{-\intextsep}
% \begin{xy}
% \xymatrix@R=1.6em@C=1.2em{
% {F X} \ar@{}[drr]|{\sqsubseteq}  & & {F Y} \kar[ll]_{F f } \\
% {X} \kar[u]_{c} \ar@{}[drr]|{\sqsubseteq} & & {Y} \kar[u]_{d} \ar_(.3){f}|-*\dir{|}[ll]  \\
% {} \ar@{}[ru]|(.10)*\dir{-} & {1} \ar`l[lu][lu]^(0){s}  \ar`r[ru][ru]_(0){t}  & \ar@{}[lu]|(.10)*\dir{-} }
% \end{xy}
%\end{minipage}
%
%\begin{mylemma}\label{lem:soundnessFwd}
%Let $F$ be an endofunctor and $T$ be a monad on $\mathbb{C}$ that satisfy infinite trace situation with respect to an order $\sqsubseteq$.
%We assume the following conditions.
%\begin{itemize}
%\item Each homset of $\Kl(T)$ is $\omega$-complete in upward manner (i.e.\ each increasing sequence has the least upper bound).
%\item Composition in $\Kl(T)$ is locally continuous (i.e.\ composition of arrows preserves the least upper bound).
%\item The lifted functor $\overline{F}$ is locally monotone and continuous (i.e.\ it preserves the order and the least upper bound).
%\end{itemize}
%For two $(T,F)$-systems  $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$,
%if $f:Y\kto X$ is a forward simulation from $\mathcal{X}$ to $\mathcal{Y}$, then $\trinf(c)\odot f\sqsubseteq \trinf(d)$.
%\qed
%\end{mylemma}
%
%This lemma immediately implies the following theorem---soundness of forward Kleisli simulation.
%
%\begin{mytheorem}\label{thm:soundnessFwd}
%We assume the conditions in Lem.~\ref{lem:soundnessFwd}.
%For  two $(T,F)$-system $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$,
%$\mathcal{X}\fwd\mathcal{Y}$ implies $\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$.
%\qed
%\end{mytheorem}

%\begin{mydefinition}
%Let $F$ be an endofunctor on $\mathbb{C}$ and $T$ be a monad on $\mathbb{C}$.
%We assume that each homset of $\Kl(T)$ is enriched with an order $\sqsubseteq$.
%For  two $(T,F)$-systems $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$,
%a \emph{backward simulation} is a Kleisli arrow $b:X\kto Y$ that satisfies the following conditions.
%\begin{equation*}
%b\odot s\sqsubseteq t, \text{ and }
%\overline{F}b \odot c \sqsubseteq d \odot b
%\end{equation*}
%We write $\mathcal{X}\bwd\mathcal{Y}$ if there exists a backward simulation from $\mathcal{X}$ to $\mathcal{Y}$.
%\end{mydefinition}

%\noindent\begin{minipage}{0.8\hsize}
%\begin{mydefinition}\label{def:bwdsim}
%Let $F$ be an endofunctor and $T$ be a monad on $\mathbb{C}$ that satisfy infinite trace situation with respect to an order $\sqsubseteq$.
%For  two $(T,F)$-systems $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$,
%a \emph{backward Kleisli simulation} is a Kleisli arrow $b:X\kto Y$ that satisfies the following conditions.
%\begin{equation*}
%b\odot s\sqsubseteq t, \text{ and }
%\overline{F}b\odot c \sqsubseteq d\odot f
%\end{equation*}
%We write $\mathcal{X}\bwd\mathcal{Y}$ if there exists a backward simulation from $\mathcal{X}$ to $\mathcal{Y}$.
%\end{mydefinition}
%\end{minipage}
%\begin{minipage}{0.2\hsize}
%\vspace*{-\intextsep}
%\begin{xy}
% \xymatrix@R=1.6em@C=1.2em{
% {F X} \ar@{}[drr]|{\sqsubseteq} \kar[rr]^{F b } & & {F Y}  \\
% {X} \kar[u]_{c} \ar@{}[drr]|{\sqsubseteq} \ar^(.3){b}|-*\dir{|}[rr]  & & {Y} \kar[u]_{d}  \\
% {} \ar@{}[ru]|(.10)*\dir{-} & {1} \ar`l[lu][lu]^(0)s \ar`r[ru][ru]_(0)t  & \ar@{}[lu]|(.10)*\dir{-} }
% \end{xy}
%\end{minipage}

%There exists a pair of nondeterministic automata where backward simulation exists but infinite language inclusion does not hold
%(see Example~\ref{example:bwdsimnotsound} for example).
%However, if we put some restriction on the definition of backward simulation, then it becomes sound.



%
%\begin{mydefinition}\label{def:restrictedbwdsim}
%Let $F$ be an endofunctor on $\mathbb{C}$ and $T$ be a monad on $\mathbb{C}$.
%We assume that each homset of $\Kl(T)$ is enriched with an order $\sqsubseteq$.
%We assume the following conditions.
%\begin{enumerate}
%\renewcommand{\labelenumi}{\alph{enumi}.}
%\item The functor $F$ and a monad $T$ satisfy infinite trace situation with respect to an order $\sqsubseteq$.
%\label{item:restrictedbwdsima}
%\item The weakly final coalgebra is obtained by using Prop.~\ref{prop:generalconstructionnotop}. 
%\label{item:restrictedbwdsimb}
%\end{enumerate}
%For  two $(T,F)$-systems $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$,
%a \emph{restricted backward simulation} is 
%%a backward simulation 
%a Kleisli arrow $b:X\kto Y$ 
%%from $\mathcal{X}$ to $\mathcal{Y}$ 
%that satisfies the following conditions.
%\begin{enumerate}
%\item Inequalities $b\odot s\sqsubseteq t$, and $\overline{F}b \odot c \sqsubseteq d \odot b$ hold.
%\item The arrow $b:X\kto Y$ satisfies $J!_Y\odot b = J!_X$. \label{item:restrictedbwdsim2}
%\item Precomposing $b:X\kto Y$ preserves the greatest fixed point of $\Psi$. 
%Namely, if $f:Y\kto 1$ is the greatest fixed point of $\Psi_Y$, then $f\odot b:X\kto 1$ is the greatest fixed point of $\Psi_X$.
%\label{item:restrictedbwdsim3}
%\item Precomposing $b:X\kto Y$ preserves the largest mediating arrow.
%Namely, if $l_{\max}:Y\kto Z$ is the largest mediating arrow from a cone $(Y,(\delta_i:Y\kto \overline{F}^i1))$ to 
%a weak 2-limit $(Z,(J\pi_i:Z\kto \overline{F}^i 1))$, then 
%$l_{\max}\odot b:X\kto Z$ is the largest mediating arrow from a cone $(X,(\delta_i\odot b:X\kto \overline{F}^i1))$ to 
%the weak 2-limit. 
%\label{item:restrictedbwdsim4}
%\end{enumerate}
%We write $\mathcal{X}\infbwd\mathcal{Y}$ if there exists a restricted backward simulation from $\mathcal{X}$ to $\mathcal{Y}$.
%\end{mydefinition}



%This lemma immediately implies the following theorem---soundness of restricted backward Kleisli simulation.




%\begin{myproposition}\label{prop:whenrestricted}
%In Def.~\ref{def:restrictedbwdsim}, 
%\begin{enumerate}
%\item if $T=\pow$ and $F=\FSigma$, 
%\begin{enumerate}
%\item Assumption \ref{item:restrictedbwdsim2} is satisfied if $b(x)\neq\emptyset$ for each $x\in X$. \label{item:whenrestrictedpow2}
%\item Assumption \ref{item:restrictedbwdsim3} is satisfied if $b(x)$ is finite for each $x\in X$. \label{item:whenrestrictedpow3}
%\item Assumption \ref{item:restrictedbwdsim4} is satisfied if $b(x)$ is finite for each $x\in X$. \label{item:whenrestrictedpow4}
%%\item Assumption \ref{item:restrictedbwdsim4} is satisfied if $b(x)$ is finite for each $x\in X$.
%\end{enumerate}
%\label{item:whenrestrictedpow}
%\item if $T=\lift$ and $F=\FSigma$, 
%\begin{enumerate}
%\item Assumption \ref{item:restrictedbwdsim2} is satisfied if $b(x)\neq\bot$ for each $x\in X$. \label{item:whenrestrictedlift2}
%\item Assumption \ref{item:restrictedbwdsim3} is always satisfied. \label{item:whenrestrictedlift3}
%\item Assumption \ref{item:restrictedbwdsim4} is always satisfied.\label{item:whenrestrictedlift4}
%\end{enumerate}
%\label{item:whenrestrictedlift}
%\item if $T=\giry$ and $F=\FSigma$, 
%\begin{enumerate}
%\item Assumption \ref{item:restrictedbwdsim2} is satisfied if $b(x)(Y)=1$ for each $x\in X$. \label{item:whenrestrictedgiry2}
%\item Assumption \ref{item:restrictedbwdsim3} is always satisfied. \label{item:whenrestrictedgiry3}
%\item Assumption \ref{item:restrictedbwdsim4} is always satisfied. \label{item:whenrestrictedgiry4}
%\end{enumerate}
%\label{item:whenrestrictedgiry}
%\end{enumerate}
%\qed
%\end{myproposition}



%\section{Partial Execution for  Infinite Trace}\label{sec:fpe}
%\begin{mydefinition}\label{def:fpe}
%Let $F$ be an endofunctor and $T$ be a monad on $\mathbb{C}$.
%\emph{Forward Partial Execution} (FPE) is a transformation that takes 
%a $(T,F)$-system $\mathcal{X}=(X,s,c)$ 
%%and $X_1\in X$ such that there exists $X_2\in X$ satisfying $X=X_1+X_2$
%as an input
%and returns a $(T,F)$-system 
%$\mathcal{X}_{\FPE}=(\overline{F}X,c\odot s,\overline{F}c)$ 
%as an output.
%%Here, $c_i:X_i\kto \overline{F}X$ ($i\in\{1,2\}$) is defined by 
%%$c_i=c\odot \kappa_i$ where $\kappa_i:X_i\kto X=X_1+X_2$ is an injection.
%\end{mydefinition}

%\begin{mydefinition}\label{def:fpe}
%Let $F$ be an endofunctor and $T$ be a monad on $\mathbb{C}$.
%\emph{Forward Partial Execution} (FPE) is a transformation that takes 
%a $(T,F)$-system $\mathcal{X}=(X,s,c)$ 
%and $X_1\in X$ such that there exists $X_2\in X$ satisfying $X=X_1+X_2$
%as an input
%and returns a $(T,F)$-system 
%$\mathcal{X}_{\FPE,X_1}=(\overline{F}X+X_2,(c_1+\id_{X_2})\odot s,\overline{F}(c_1+\id_{X_2})\odot[\id_{\overline{F}X},c_2])$ 
%as an output.
%Here, $c_i:X_i\kto \overline{F}X$ ($i\in\{1,2\}$) is defined by 
%$c_i=c\odot \kappa_i$ where $\kappa_i:X_i\kto X=X_1+X_2$ is an injection.
%\end{mydefinition}

%\begin{mytheorem}[correctness of FPE for fwd.\ sim.]\label{thm:soundnessFPEfwd}
%Let $F$ be an endofunctor and $T$ be a monad on $\mathbb{C}$ that satisfy the assumptions in Def.~\ref{def:fwdsim}.
%Let $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$ be $(T,F)$-systems. % and $X=X_1+X_2$.
%Then the following statements hold.
%\begin{enumerate}
%%\item (monotonicity) If $X_1=X_{1,1}+X_{1,2}$, then $\mathcal{X}_{\FPE,X_{1,1}}\fwd\mathcal{Y}$ implies $\mathcal{X}_{\FPE,X_1}\fwd\mathcal{Y}$.
%%\label{item:monotonicityfpefwd}
%\item (soundness)  $\mathcal{X}_{\FPE}\fwd\mathcal{Y}$ implies $\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$.
%\label{item:soundnessfpefwd}
%\item (adequacy) $\mathcal{X}\fwd\mathcal{Y}$ implies $\mathcal{X}_{\FPE}\fwd\mathcal{Y}$.
%\label{item:adequacyfpefwd}
%\end{enumerate}
%\qed
%\end{mytheorem}
%
%\begin{mytheorem}[adequacy of FPE for fwd.\ simulation]\label{thm:adequacyFPEfwd}
%\qed
%\end{mytheorem}

%
%\begin{mytheorem}[adequacy of FPE for restricted bwd.\ simulation]\label{thm:adequacyFPEbwd}
%\qed
%\end{mytheorem}

%The following proposition gives a concrete condition for a $(T,F)$-systems for each monad $T$ and functor $F$ to satisfy the assumptions in 
%Thm.~\ref{thm:soundnessFPEbwd}.
%It can be proved in the similar manner to a proof for Prop.~\ref{prop:whenrestricted}.
%\begin{myproposition}\label{prop:whenrestrictedfpe}
%In Thm.~\ref{thm:soundnessFPEbwd}, 
%\begin{enumerate}
%\item if $T=\pow$ and $F=\FSigma$, 
%\begin{enumerate}
%\item Assumption \ref{item:restrictedfpe2} is satisfied if $d(y)\neq\emptyset$ for each $y\in Y$.
%\item Assumption \ref{item:restrictedfpe3} is satisfied if $d(y)$ is finite for each $y\in Y$. 
%\item Assumption \ref{item:restrictedfpe4} is satisfied if $d(y)$ is finite for each $y\in Y$. 
%%\item Assumption \ref{item:restrictedfpe4} is satisfied if $d(y)$ is finite for each $y\in Y$.
%\end{enumerate}
%\item if $T=\lift$ and $F=\FSigma$,
%\begin{enumerate}
%\item Assumption \ref{item:restrictedfpe2} is satisfied if $d(y)\neq\bot$ for each $y\in Y$.
%\item Assumption \ref{item:restrictedfpe3} is always satisfied.
%\item Assumption \ref{item:restrictedfpe4} is always satisfied.
%\end{enumerate}
%\item if $T=\giry$ and $F=\FSigma$, 
%\begin{enumerate}
%\item Assumption \ref{item:restrictedfpe2} is satisfied if $d(y)(FY)=1$ for each $y\in X$.
%\item Assumption \ref{item:restrictedfpe3} is always satisfied.
%\item Assumption \ref{item:restrictedfpe4} is always satisfied.
%\end{enumerate}
%\end{enumerate}
%\qed
%\end{myproposition}





%
% ---- Bibliography ----
%
%\section*{References}
% \bibliographystyle{splncs03}
 \bibliography{myref} %


%\end{document}

 
%
% ---- Appendix ----
%

\newpage
\appendix

\section{Appendix}
\counterwithin{theorem}{section}

\subsection{Ranked Alphabet and Infinite Trees}
\label{subsec:rankedAlphabetAndInfiniteTrees}
All  the systems that we will be using in this paper are those which
generate trees.
Here we collect some standard notions and notations on (the conventional
presentation of) such finite/infinite trees.

The generated trees are labeled with 
letters from an alphabet.
%a family of alphabets.
%Before focusing on specific functors and monads, we give a formal definition of trees.
\begin{mydefinition}
A \emph{ranked alphabet} is a family $\Sigma=(\Sigma_n)_{n\in\omega}$ of sets.
A \emph{standard Borel ranked alphabet} is a family $\Xi=\bigl((\Sigma_n,\sigalg_n)\bigr)_{n\in\omega}$ of
standard Borel spaces.
%In the above, 
The index $n\in\omega$ is called an \emph{arity}.
\end{mydefinition}

For the definition of infinite trees, we follow~\cite{courcelle83fundamentalproperties}.
%The defined tree is an infinite tree whose nodes are labeled with letters in a ranked alphabet.
Each node 
labeled with a letter of arity $n$
%whose arity is $n$ 
has $n$ children.
The idea of $k$-prefix trees is introduced in~\cite{ellis71probabilistictree}.
It can be regarded as a finite tree of depth $k$ 
that is obtained by extracting nodes of an infinite tree.
\begin{mydefinition}
Let $\Sigma=(\Sigma_n)_{n\in\omega}$ be a ranked alphabet.
A \emph{$\Sigma$-labeled  infinite tree} is a pair $(D,l)$ of the \emph{domain} $D\in \mathbb{N}^*$ and 
the \emph{labelling function} $l:D\to\bigcup_{n\in\omega}\Sigma_n$ such that: %satisfying the following conditions:
\begin{itemize}
\item the domain $D$ is
%\begin{itemize}
%\item 
prefix-closed 
(i.e.\ $\forall \alpha\in\mathbb{N}^*.\,\forall i\in\mathbb{N}.\; \alpha i\in D\,\Rightarrow\, \alpha \in D$),
%(i.e.\ for  $\alpha\in\mathbb{N}^*$ and $i\in\mathbb{N}$, $\alpha i\in D$ implies $\alpha \in D$),
%\item 
nonempty, and
%(equivalently, empty sequence $\empseq$ is in $D$), and
%\item 
downward-closed 
(i.e.\ $\forall \alpha\in\mathbb{N}^*.\,\forall i\in\mathbb{N}.\; \alpha
      i\in D\,\Rightarrow\, \forall j\leq i.\, \alpha j\in D$), and
%(i.e.\ for  $\alpha\in\mathbb{N}^*$ and $i\in\mathbb{N}$, $\alpha i\in D$ implies $\alpha j\in D$ for all $j\leq i$). 
%\end{itemize}
\item %The labelling function $l$ satisfies the arity condition: namely,
%\begin{itemize}
moreover, for each $\alpha\in D$ and $i\in\mathbb{N}$ such that $l(\alpha)\in \Sigma_n$, $\alpha i\in D$ iff $0\leq i\leq n-1$. 
%\end{itemize}
\end{itemize}
For 
%a ranked alphabet $\Sigma=(\Sigma_i)_{i\in\omega}$ and 
%a natural number 
$k\in\omega$,  
a \emph{$\Sigma$-labeled $k$-prefix tree} is 
a pair $(D,l)$ of the \emph{domain} $D$ and the \emph{labelling function} $l:D\to\bigcup_{n\in\omega}\Sigma_n$ 
that satisfy the following conditions.
If $k=0$, then $D$ is an empty set while
if $k>0$, then $D\subseteq \bigcup_{i\leq k} \mathbb{N}^{i}$ and it is prefix-closed, nonempty, and downward-closed.
Moreover, for $\alpha\in D$ and $i\in\mathbb{N}$ such that $|\alpha| <k$ and $l(\alpha)\in \Sigma_n$, $\alpha i\in D$ iff $0\leq i\leq n-1$. 
%defined in the similar manner to $\Sigma$-labeled  infinite tree, except for the following points.
%\begin{itemize}
%\item The domain $D$ is restricted to $D\subseteq \mathbb{N}^{k-1}$ if $k>0$ and $\emptyset$ if $k=0$.
%\item The empty sequence $\empseq$ does not have to be in  $D$.
%\item The condition for the labelling function is replaced by the following.
%\begin{itemize}
%\item for each $\alpha\in D$ such that $|\alpha| <n$ and $l(\alpha)\in \Sigma_i$, and each $n\in\mathbb{N}$, $\alpha n\in D$ iff $0\leq n\leq i-1$. 
%\end{itemize}
%\end{itemize}
Here,  $k$ is called the \emph{depth} of $(D,l)$.
%
%For a given ranked alphabet $\Sigma$, 
We write $\Treeinf(\Sigma)$ and $\Treeinf^k(\Sigma)$ for
the sets of all $\Sigma$-labeled  infinite trees and 
$\Sigma$-labeled  $k$-prefix trees, respectively.

A $\Sigma$-labeled $k$-prefix tree $t=(D,l)$ is said to be a \emph{prefix} of a
$\Sigma$-labeled  infinite (or $k'$-prefix for some $k'\geq k$) tree $t'(D',I')$ and denoted by $t\treeprefix t'$ if $D\subseteq D'$ and 
%restriction of $l'$ on $D$ coincides with $l$ (i.e.\ 
for each $\alpha\in D$, $l(\alpha)=l'(\alpha)$.
%
For a $\Sigma$-labeled  infinite $k$-prefix tree $t$, a \emph{cylinder over $t$} 
is the set 
%$\cyl(t)\subseteq\Treeinf(\Sigma)$ that is defined by
$%\begin{equation*}
\cyl(t)=\{t'\in\Treeinf(\Sigma)\mid \text{$t\treeprefix t'$}\}\,.
$%\end{equation*}

For a $\Sigma$-labeled  infinite tree $t=(D,l)$ and $\alpha\in D$, %such that $l(\empseq)\in\Sigma_n$ and $i\leq n-1$, %$\alpha\in D$,
\emph{$\alpha$'th subtree of $t$} is a $\Sigma$-labeled  infinite tree $t_{\alpha}=(D_{\alpha},l_{\alpha})$ where
$D_{\alpha}=\{\beta\in \mathbb{N}^* \mid \alpha\beta\in D\}$ and 
$l_{\alpha}(\beta)=l(\alpha\beta)$.

For $a\in\Sigma_n$ and $\Sigma$-labeled infinite trees $t_0=(D_0,l_0),\ldots,t_{n-1}=(D_{n-1},l_{n-1})$,
 %$(t_i=(D_i,l_i))_{0\leq i\leq n-1}$,
we write $(a,t_0,\ldots,t_{n-1})$ for a $\Sigma$-labeled infinite tree $t=(D,l)$ where
$D=\{i \alpha_i\mid 0\leq i\leq n-1, \alpha_i\in D_i\}$ and $l(i\alpha)=l_i(\alpha)$ for each $i\in\{0,\ldots,n-1\}$.
\end{mydefinition}
Later, we will use an \emph{infinite tree automaton}---an automaton that generate $\Sigma$-labeled trees.
When $\Sigma_0=\{\checkmark\}$, and $\Sigma_i=\emptyset$ for all $i\geq 2$, 
an infinite tree automaton can be regarded as an automaton that generates \emph{words} instead of trees. %---a special tree.
We call  such an automaton \emph{infinite automaton} (suppressing the word ``tree'').

A system that generates $\Sigma$-labeled infinite trees can be represented as an $\overline{F}$-coalgebra 
on the Kleisli category of some monad,
where $F$ is the polynomial functor defined as follows.
%A ranked alphabet and standard Borel ranked alphabet give rise to polynomial functors on $\Sets$ and $\Meas$ respectively.
%We will see later that they represent a branching type of systems that generate infinite trees.
\begin{mydefinition}\label{def:FSigma}
For a ranked alphabet $\Sigma=(\Sigma_n)_{n\in\omega}$,
we define 
%a polynomial functor 
$\FSigma:\Sets\to\Sets$ by $\FSigma=\coprod_{n\in\omega} \Sigma_n\times (\place)^n$.
%
For a standard Borel ranked alphabet $\Xi=\bigl((\Sigma_n,\sigalg_n)\bigr)_{n\in\omega}$,
we define 
%a polynomial functor 
$\FXi:\Meas\to\Meas$ by $\FXi=\coprod_{n\in\omega} (\Sigma_n,\sigalg_n)\times (\place)^n$.
%Let $\Sigma=(\Sigma_n)_{n\in\omega}$ be a ranked alphabet and
%$\mathbb{C}$ be a category where the notion of polynomial functors is defined.
%We define a polynomial functor $F_{\Sigma}:\mathbb{C}\to\mathbb{C}$ by 
%$F_{\Sigma}=\coprod_{n\in\omega} \Sigma_n\times (\place)^n$.
\end{mydefinition}


\subsection{Coincidence between Coalgebraic Infinite Trace Semantics and Automata-theoretic Semantics}\label{subsec:automchara}
\subsubsection{Nondeterministic Tree Automaton}\label{subsec:automcharapow}
%In the previous section, we proved that 
%%a functor 
%$\FSigma$ and 
%%a monad 
%$\pow$ constitute an infinite trace situation.
%This means that for an arbitrary $\overline{\FSigma}$-coalgebra $c:X\kto\overline{\FSigma}X$ in $\Kl(\pow)$,
%there exists the largest homomorphism $\trinf(c):X\kto Z$ from $c$ to $J\zeta$---the lifted final $\FSigma$-coalgebra.
%% where $\zeta:Z\to FZ$ is a final $F$-coalgebra in $\Sets$.
%Recall that we model a system as a triple $\mathcal{X}=(X,s,c)$ called a $(T,F)$-system, 
%and defined its infinite trace semantics as a Kleisli arrow $\trinf(c)\odot s:1\kto Z$.
In this section, we regard a $(\pow,\FSigma)$-system as an automaton that nondeterministically generate an infinite tree. 
Then we show that its automata-theoretic semantics coincides with coalgebraic infinite trace semantics.
%
\begin{mydefinition}\label{def:nondettreeautomaton}
Let $\Sigma$ be a ranked alphabet.
A $(\pow,\FSigma)$-system $\mathcal{X}=(X,s,c)$ is called a \emph{$\Sigma$-labeled nondeterministic tree automaton}.
%
%Let $\mathcal{X}=(X,s,c)$ be a nondeterministic tree automaton. 
For a $\Sigma$-labeled  infinite tree $t=(D,l)$ and a state $x\in X$, 
a $(X)_{n\in\omega}$-labeled  infinite tree $t_r=(D,l_r)$ that has the same domain as $t$ is 
called a \emph{run of $\mathcal{X}$ from $x$ that generates $t$} if  
\begin{itemize}
\item $l_r(\empseq)=x$, and
\item for each $\alpha\in D$ such that $l(\alpha)=a\in\Sigma_n$, $l_r(\alpha)=y$ and 
$l_r(\alpha i)=y_i$ for each $0\leq i \leq n-1$, we have $(a,y_0,\ldots,y_{n-1}) \in c(y)$.
\end{itemize}

For a state $x\in X$, 
an \emph{infinite language of $\mathcal{X}$ from $x$} is a set 
$\lang(\mathcal{X},x)\subseteq \Treeinf(\Sigma)$ that is defined by
$%\begin{equation*}
\lang(\mathcal{X},x)=\{
t\in \Treeinf(\Sigma)
\mid
\text{$\exists$  run of $\mathcal{X}$ from $x$ that generates $t$}
\}
$. %\end{equation*}
%
An \emph{infinite language of $\mathcal{X}$} is a set $\lang(\mathcal{X})\subseteq \Treeinf(\Sigma)$ that is defined by
$\lang(\mathcal{X})=\bigcup_{x\in s(*)} \lang(\mathcal{X},x)$.
\end{mydefinition}
%
%In the rest of this section, 
Then we can show that the 
the automata-theoretic semantics of $\Sigma$-labeled 
nondeterministic tree automaton coincides with coalgebraic infinite trace semantics.
\begin{mytheorem}\label{thm:largestsemanticspow}
Let $\Sigma$ be a ranked alphabet.
%We define an arrow $\zeta:\Treeinf(\Sigma)\to F_{\Sigma}\Treeinf(\Sigma)$ in $\Sets$ by 
%$\zeta(t)=(a,(t_0,\ldots,t_{n-1}))$.
%where $t=(D,l)$, $a=l(\empseq)\in\Sigma_n$, and $t_i$ is the $i$'th subtree of $t$.
%Then $\zeta$ is a final $F_{\Sigma}$-coalgebra.
The carrier of the final $\FSigma$-coalgebra is isomorphic to $\Treeinf(\Sigma)$ and
 for a $\Sigma$-labeled nondeterministic tree automaton $\mathcal{X}=(X,s,c)$,
%With respect to the order in Def.~\ref{def:orderenrichment}, 
 we have $\trinf(c)(x)=\lang(\mathcal{X},x)$ for all $x\in X$.
Moreover, $\trinf(c)\odot s(*)=\lang(\mathcal{X})$.
\end{mytheorem}
%
%
\auxproof{
We show that 
the automata-theoretic semantics of $\Sigma$-labeled 
nondeterministic tree automaton coincides with coalgebraic infinite trace semantics.
%
To this end, we first show that
the carrier of final $F_{\Sigma}$-coalgebra in $\Sets$ is given by a set of  infinite trees.
\begin{mylemma}\label{lem:finalcoalgsets}
%Let $F_{\Sigma}$ be an endofunctor on $\Sets$ that is defined as Def.~\ref{def:FSigmaSets}.
Let $\Sigma$ be a ranked alphabet.
%For the functor $\FSigma$ in Def.~\ref{def:FSigma}, 
We define an arrow $\zeta:\Treeinf(\Sigma)\to F_{\Sigma}\Treeinf(\Sigma)$ in $\Sets$ by 
$%\begin{equation*}
\zeta(t)=(a,(t_0,\ldots,t_{n-1}))
$. %\end{equation*}
Here, $t=(D,l)$, $a=l(\empseq)\in\Sigma_n$, and $t_i$ is the $i$'th subtree of $t$.
%and for each $i\in\{0,\ldots,n-1\}$, $D_i=\{\alpha\in\mathbb{N}^*\mid i\alpha\in D\}$ and $l_i(\alpha)=l(i\alpha)$.
%
Then $\zeta$ is a final $F_{\Sigma}$-coalgebra.
\qed
\end{mylemma}

%The following theorem states that the infinite language of a nondeterministic tree automaton can be captured as coalgebraic infinite trace semantics.
Then we have the following theorem.
\begin{mytheorem}\label{thm:largestsemanticspow}
Let $\mathcal{X}=(X,s,c)$ be a $\Sigma$-labeled nondeterministic tree automaton.
%With respect to the order in Def.~\ref{def:orderenrichment}, 
For all $x\in X$, we have $\trinf(c)(x)=\lang(\mathcal{X},x)$.
Moreover, $\trinf(c)\odot s(*)=\lang(\mathcal{X})$.
\qed
\end{mytheorem}
}


%\newtheorem*{proof:largestsemanticspow}{Proof of Thm.\ \ref{thm:largestsemanticspow}}{\itshape}{\rmfamily}
%\begin{proof:largestsemanticspow}
\begin{myproof}
%We define $f:X\kto \Treeinf(\Sigma)$ by $f(x)=\lang(\mathcal{X},x)$. 
We define an arrow $\zeta:\Treeinf(\Sigma)\to F_{\Sigma}\Treeinf(\Sigma)$ in $\Sets$ by 
$\zeta(t)=(a,(t_0,\ldots,t_{n-1}))$.
where $t=(D,l)$, $a=l(\empseq)\in\Sigma_n$, and $t_i$ is the $i$'th subtree of $t$.
Then it is easy to see that $\zeta$ is a final $F_{\Sigma}$-coalgebra.

We show that $\lang(\mathcal{X},\place):X\kto\Treeinf(\Sigma)$ is the largest homomorphism from $c$ to $J\zeta$.
% where $\zeta$ is a final coalgebra defined in Lem.~\ref{lem:finalcoalgsets}.

We first show that $\lang(\mathcal{X},\place)$ is a homomorphism.
For $x\in X$,
\allowdisplaybreaks[1]\begin{align*}
&(\overline{\FSigma}\lang(\mathcal{X},\place))\odot c (x) \\
%
&= (\overline{\FSigma}\lang(\mathcal{X},\place))\left(\left\{(a,x_0,\ldots,x_{n-1})\relmiddle|
\begin{array}{l}
 n\in\omega, a\in\Sigma_n, x_0,\ldots,x_{n-1}\in X, \\
 (a,x_0,\ldots,x_{n-1})\in c(x)
 \end{array}
 \right\}\right) \\
 %
&= \left\{(a,t_0,\ldots,t_{n-1})\relmiddle|
\begin{array}{l}
n\in\omega, \;a\in\Sigma_n,\; \\
\exists x_0,\ldots,x_{n-1}\in X.\, 
\left(\begin{array}{l}
(a,x_0,\ldots,x_{n-1})\in c(x), \\
t_i\in \lang(\mathcal{X},x_i)\text{ for each $i$}%$in\{0,\ldots,n-1\}$} 
\end{array}\right)
\end{array}
\right\}\\
%
&= J\zeta\left(\left\{(D,l)\in\Treeinf(\Sigma) \relmiddle|
\begin{array}{l}
n\in\omega, l(\empseq)\in\Sigma_n, \\
\exists x_0,\ldots,x_{n-1}\in X. 
\left(\begin{array}{l}(l(\empseq),x_0,\ldots,x_{n-1})\in c(x), \\
l(i)\in \lang(\mathcal{X},x_i) \text{ for each $i$}%\in\{0,\ldots,n-1\}$}
\end{array}\right)
\end{array}
\right\}\right) \\
%
&= J\zeta\left(\left\{(D,l)\in\Treeinf(\Sigma) \relmiddle|
\!\begin{array}{l}
n\in\omega, l(\empseq)\in\Sigma_n, \\
\exists x_0,\ldots,x_{n-1}\in X. \left(\begin{array}{l}
(l(\empseq),x_0,\ldots,x_{n-1})\in c(x), \\
\exists\text{run $t_{r,i}$ of $\mathcal{X}$ from $x_i$ that}\\
\;\;\;\;\;\text{generates $t_i$ for each $i$}%\in\{0,\ldots,n-1\}$}
\end{array}\right)
\end{array}\!
\right\}\right) \\
&= J\zeta(\{t\in\Treeinf(\Sigma) \mid \text{there exists a run $t_{r}$ of $\mathcal{X}$ from $x$ that generates $t$}\}) \\
&= J\zeta\odot \lang(\mathcal{X},\place)(x)\,.
\end{align*}
Therefore $\lang(\mathcal{X},\place)$ is a homomorphism from $c$ to $J\zeta$.

It remains to prove that $\lang(\mathcal{X},\place)$ is the largest homomorphism.
Let $g:X\kto\Treeinf(\Sigma)$ be a homomorphism from $c$ to $J\zeta$.
Assume that $t=(D,l)\in g(x)$ for $x\in X$. We show that $t\in\lang(\mathcal{X},x)$.
To prove this, it suffices to construct a run $t_r=(D,l_r)$ of $\mathcal{X}$ from $x$ that generates $t$.
For $\alpha\in D$, we denote $t_{\alpha}$ for the $\alpha$'th subtree of $t$.
For each $\alpha\in D$, we define a state $l_r(\alpha)\in X$ such that $t_{\alpha}\in g(l_r(\alpha))$
by the induction on the length of $\alpha$ as follows.

For $\alpha=\empseq$, we define it by $l_r(\empseq)=x$.
Then $t_{\alpha}=t\in g(x)=g(l_r(\empseq))$.

Assume that $l(\alpha)\in\Sigma_n$ and $t_{\alpha}\in g(l_r(\alpha))$. 
As $g$ is a homomorphism from $c$ to $J\zeta$, 
$t_{\alpha}\in g(l_r(\alpha))=J\zeta^{-1}\odot\overline{F}g\odot c(l_r(\alpha))$.
Therefore there exists $x_0,\ldots,x_{n-1}$ such that 
$(l(\alpha),x_0,\ldots,x_{n-1})\in c(l_r(\alpha))$ and $t_{\alpha i}\in g(x_i)$ for each $i\in\{0,\ldots, n-1\}$.
We define $l_r(\alpha i)$ by $l_r(\alpha i)=x_i$.
Then $t_{\alpha i}\in l_r(\alpha i)$.

By the axiom of dependent choice, this $l_r$ is well-defined.
Moreover by its construction, $(D,l_r)$ is a run of $\mathcal{X}$ from $x$ that generates $t$.
Therefore $t\in \lang(\mathcal{X},x)$  for each $x\in X$ and we have $g\sqsubseteq \lang(\mathcal{X},\place)$.

Hence $\trinf(c)=\lang(\mathcal{X},\place)$.
This immediately implies that $\trinf(c)\odot s(*)=\lang(\mathcal{X})$.
\qed
\end{myproof}
%\end{proof:largestsemanticspow}





\subsubsection{Coalgebraic Infinite Trace Semantics and Automata-theoretic Semantics of Probabilistic Tree Automata}\label{subsec:automcharagiry}
%Next, 
We give an automata-theoretic characterization of the coalgebraic infinite trace semantics of $(\giry,\FXi)$-systems
using the notion of \emph{branching process}~\cite{harris64theorybranching}---a kind of Markov process.
A $(\giry,\FXi)$-system can be regarded as a probabilistic automaton that generates trees.

In this section, for simplicity, we assume that 
the ranked alphabet is a family of countable sets $\Sigma$ equipped with a discrete $\sigma$-algebras.
However, it is not  difficult to generalize the results 
in this section 
%so that the automata-theoretic language is defined 
for 
%probabilistic tree 
automata 
labeled with general standard Borel ranked alphabet.
%whose alphabet is (standard Borel) uncountable sets.
%
Moreover, we also assume that the state space is finite because 
the result in~\cite{etessamiSY15greatestfixed} about branching processes that we will be using to show
the coincidence between the automata-theoretic language and the coalgebraic infinite trace semantics require the restriction.
%
%Moreover, we also assume that the state space is finite for the following reason. 
%To define a automata-theoretic language of a probabilistic tree automaton, we use a notion of a \emph{branching process}.
%Then the coincidence between the automata-theoretic language and the coalgebraic infinite trace semantics is shown using
%a result in~\cite{etessamiSY15greatestfixed}.
%---unreachable probability of a branching process 
%can be calculated as 
%%is given as
%the greatest fixed point of a certain function.
%However, in~\cite{etessamiSY15greatestfixed}, the state space of a branching process was restricted to finite one.
%Our restriction 
%%on the state space 
%is required to employ the result.
%This restriction is required in order to employ an existing theory of \emph{branching processes} to define
%automata-theoretic language of probabilistic tree automaton later. 
\begin{mydefinition}
For a ranked alphabet $\Sigma=(\Sigma_n)_{n\in\omega}$ where $\Sigma_n$ is countable for each $n\in\omega$, 
a \emph{$\Sigma$-labeled probabilistic tree automaton} is a $(\giry,\FSigma)$-system $\mathcal{X}=(X,s,c)$ 
s.t.\ $\FSigma:\Meas\to\Meas$ is defined by $\FSigma=\FXi$ where $\Xi=((\Sigma_n,\pow\Sigma_n))_{n\in\omega}$ and 
%such that
%$\Sigma=(\Sigma_n,\pow \Sigma_n)_{n\in\omega}$ 
%each the alphabet $\Sigma_n$ is a countable set with a discrete $\sigma$-algebra 
%and the state space $X$ is finite.
%$\Sigma$ is a standard Borel ranked alphabet and 
$X$ is a finite set.
\end{mydefinition}
%Here, differently from \S\ref{sec:powersetmonad},
%the state space is restricted to finite. 

To define the language of a $\Sigma$-labeled probabilistic tree automaton, we need some preparations.
For a given $\Sigma$-labeled probabilistic tree automaton $\mathcal{X}=(X,s,c)$ and a state $x\in X$, 
we will  be defining the \emph{infinite language of $\mathcal{X}$ from $x$} as a 
probability measure $\lang(\mathcal{X},x)$ on a set $\Treeinf(\Sigma)$ of infinite trees.
To this end, we first have to fix a  $\sigma$-algebra on $\Treeinf(\Sigma)$.
%It is generated from \emph{cylinders} over prefix trees.
\begin{mydefinition}
For a ranked alphabet $\Sigma$,  a \emph{cylinder set} $\mathcal{S}_{\Sigma}\subseteq \pow(\Treeinf(\Sigma))$ over $\Sigma$ is defined by
$%\begin{equation*}
\mathcal{S}_{\Sigma}=\{\cyl(t)\mid k\in\omega, t\in\Treeinf^k(\Sigma)\}
$. %\end{equation*}
A $\sigma$-algebra $\sigalg_{\infty}$ on $\Treeinf(\Sigma)$ is
%defined as
 the smallest $\sigma$-algebra that contains $\mathcal{S}_{\Sigma}$.
%$\{\cyl(t)\mid k\in\omega, t\in\Treeinf^k(\Sigma)\}\subseteq  \pow(\Treeinf(\Sigma))$.
%$\mathcal{S}_{\Sigma}$.
%follows.
%\begin{equation*}
%\sigalg_{\Treeinf(\Sigma)}=\sigma_{\Treeinf(\Sigma)}()
%\end{equation*}
%Here, for $A\subseteq X$, $\sigma_X(A)$ denotes the smallest $\sigma$-algebra that contains $A$.
\end{mydefinition}

%However, in order to do that, it suffices to define $\lang(\mathcal{X},x)$ only for all cylinders in $\mathcal{S}_{\Sigma}$.
%in a ``compatible'' manner.
%The following lemma is an analogy of \cite[Lemma 3.18]{kerstan13coalgebraictrace}.
%\begin{mylemma}\label{lem:uniqueextensionmeas}
%Let $\Sigma=(\Sigma_n)_{n\in\omega}$ be a ranked alphabet.
%Let $\nu:\mathcal{S}_{\Sigma}\to[0,\infty)$ be a function such that for all $k\in\omega$ and $t\in\Treeinf^k(\Sigma)$,
%\begin{equation*}
%\nu(\cyl(t))=\sum\{\nu(\cyl(t'))\mid t'\in\Treeinf^{k+1}(\Sigma), \text{$t$ is a prefix of $t'$}\} 
%\end{equation*}
%is satisfied.
%Then $\nu$ is uniquely extended to $\sigalg_{\Treeinf(\Sigma)}$.
%Namely, there exists a unique $\nu^*:\sigalg_{\Treeinf(\Sigma)}\to [0,\infty)$ such that 
%$\nu^*(\cyl(t))=\nu(\cyl(t))$ for all $k\in\omega$ and $t\in\Treeinf^k(\Sigma)$.
%\qed
%\end{mylemma}
%
%According to this lemma, especially, we have to define $\lang(\mathcal{X},x)$ for $\Treeinf(\Sigma)$---the entire set.

Next, we have to define a value $\lang(\mathcal{X},x)(A)$ for each measurable sets $A\in\sigalg_{\infty}$. 
By %According to 
Kolmogorov's consistency theorem (see~\cite{kechris95classicaldescriptive} for example), 
it suffices to define $\lang(\mathcal{X},x)$ only for each cylinder in $\mathcal{S}_{\Sigma}$ in a ``compatible'' manner. 
We start with defining $\lang(\mathcal{X},x)$ for $\Treeinf(\Sigma)\in\mathcal{S}_{\Sigma}$. %---the entire set.
Intuitively, the value $\lang(\mathcal{X},x)(\Treeinf(\Sigma))$ is a probability where the
 % probabilistic tree 
 automaton does not abort.
We characterize this value using the notion of \emph{branching processes}.%~\cite{harris64theorybranching}.
%Here, we follow the definition in~\cite{chen12modelchecking}.
%
\begin{mydefinition}%[branching process]
A \emph{branching process} is a triple $\Delta=(\Gamma,\tau)$ consisting of 
a finite set $\Gamma$ of \emph{types}, and
a \emph{transition function} $\tau: \Gamma\times \Gamma^*\to [0,1]$
such that $\sum_{\alpha\in  \Gamma^*} (x,\alpha)=1$ for all $x\in\Gamma$. %, and
%$\hookrightarrow\subseteq \Gamma\times\Gamma^*$ of \emph{transition rules}, 
%a \emph{transition distribution} $\text{Prob}:\hookrightarrow\to[0,1]$ such that $\sum_{\alpha\in \Gamma^*}\text{Prob}(x,\alpha)=1$ for all $x\in\Gamma$, and
%a initial process $x_0\in\Gamma$.
\end{mydefinition}
Here, we give an intuitive semantics of branching processes.
Formal definition can be found in~\cite{harris64theorybranching}, for example.
A branching process $\Delta=(\Gamma,\tau)$ and an \emph{initial process} $x_0\in\Gamma$ 
give rise to a discrete-time a Markov chain $\mathcal{M}_{\Delta,x_0}$.
Its state space is given by $\Gamma^*$ where $\alpha\in\Gamma^*$ can be regarded as a population of processes.
% where there are $\alpha(x)$ processes of type $x$ for each  $x \in\Gamma$.
We start from an initial population $\langle x_0\rangle$ in $\Gamma^*$, that means there is only one process of type $x_0$. 
In each transition, each process in the population gives birth to child processes randomly.
The probability that a process $x$ gives birth to children represented by a population $\alpha\in\Gamma^*$
is given by $\tau(x,\alpha)$.

For 
%a branching process $\Delta=(\Gamma,\tau)$ and 
a type $x\in \Gamma$, 
a \emph{probability of reaching $x$ from $x_0$} is a probability $\text{Reach}(\Delta,x_0,x)\in[0,1]$ where
a population 
%where a type
that has a type
 $x$ 
 %is appearing 
 is reached in $\mathcal{M}_{\Delta,x_0}$.

%A branching process $\Delta=(\Gamma,\hookrightarrow,\text{Prob},X_0)$ gives rise to a 
%discrete-time Markov chain $\mathcal{M}_{\Delta}=(S,\mathbf{P},s_0)$.
%Here, $S$ is a state space that is defined by $S=\mathbb{N}^{\Gamma}$,
%$\mathbf{P}\in[0,1]^{S\times S}$ is a transition matrix such that
% $\mathbf{P}$,
%and $s_0\in S$ is an initial state that is defined by 
%\begin{equation*}
%.
%\end{equation*}
%Here, ,


%For the semantics of a branching process, we follow the definition in~\cite{etessami13polynomialtime}.
%\begin{mydefinition}
%\end{mydefintion}
%

From a $\Sigma$-labeled probabilistic tree automaton $\mathcal{X}$, 
we can obtain a branching process $\Delta_{\mathcal{X}}$ by 
adding a new process $\bot$ that means aborting of the system and
``forgetting'' the labels on transitions.
%
%Formal definition can be found in Def.~\ref{def:skeleton}.

\begin{mydefinition}\label{def:skeleton}
For a $\Sigma$-labeled probabilistic tree automaton  $\mathcal{X}=(X,s,c)$,
its \emph{skeleton} is a branching process $\Delta_{\mathcal{X}}=(\Gamma_{\mathcal{X}},\tau_{\mathcal{X}})$.
Here 
%types is defined by 
$\Gamma_{\mathcal{X}}=X+\{\bot\}$
and
%A transition function 
$\tau_{\mathcal{X}}$ 
is defined as follows.
\begin{equation*}
\tau_{\mathcal{X}}(x,\alpha)=\begin{cases}
%s(*)(y) & (x=\bullet, \alpha=\langle y\rangle) \\
\sum_{a\in\Sigma_n} c(x)(\{(a,x_0,\ldots,x_{n-1})\})  & (x\in X, \alpha=\langle x_0,\ldots,x_{n-1}\rangle\in X^*) \\
%c(x)(\checkmark)  & (x\in X, \alpha=\langle\checkmark\rangle) \\
1-\sum_{\beta\in X^*} \tau(x,\beta)  & (x\in X, \alpha=\langle\bot\rangle) \\
1 & (x\in\{\bot\}, \alpha=\langle \bot\rangle) \\ 
0 & (\text{otherwise})\,
\end{cases}
\end{equation*}
%An initial state is given by $x_0=\bullet$.
\end{mydefinition}


%Intuitively, added process 
%$\bullet$ means an initial state and 
%$\bot$ means that the system aborts.
%
Using the skeleton $\Delta_{\mathcal{X}}$,
%Now 
we can define a value $\lang(\mathcal{X},x)(\Treeinf(\Sigma))$.
% as
Namely, it is defined as
the probability where $\bot$ is not reached  from $x$ in the induced skeleton.
It remains to define $\lang(\mathcal{X},x)$ for all cylinders $\cyl(t)\in \mathcal{S}_{\Sigma}$. % where $ t\in\Treeinf^k(\Sigma)$ and $k\in\omega$.
They are defined by the induction on the depth of $t$.
Then, 
%(see~\cite{kechris95classicaldescriptive} for example), 
a probability measure $\lang(\mathcal{X},x)$  on $(\Treeinf(\Sigma),\sigalg_{\infty})$
is uniquely determined.
%$\lang(\mathcal{X},x)$ 
%extended to a probability measure on $(\Treeinf(\Sigma),\sigalg_{\infty})$.

\begin{myproposition}\label{prop:existencesemantics}
Let $\mathcal{X}=(X,s,c)$ be a $\Sigma$-labeled probabilistic tree automaton. 
%
For each $x\in X$, $k\in\omega$ and $t\in \Treeinf^k(\Sigma)$, 
we define a value $\nu_x(t)\in[0,1]$ by the induction on $k$ as follows.
\begin{itemize}
\item If $k=0$, then 
$%\begin{equation}
\nu_x(t)
%=\nu(\Treeinf(\Sigma))
=1-\text{Reach}(\Delta_{\mathcal{X}},x,\bot) \,.
$ %\label{eq:def:probtreeautomaton1}%\end{equation}
\item If $t=(D,l)\in \Treeinf^{k+1}(\Sigma)$
where $l(\empseq)=a\in\Sigma_n$ and $t_i$ is the $i$'th subtree of $t$,
then 
\begin{equation*}
\nu_x(t)=\sum_{x_0,\ldots,x_{n-1}\in X} \bigl(c(x)(\{(a,x_0,\ldots,x_{n-1})\})\cdot \prod_{i=0}^{n-1} \nu_{x_i}(t_i)\bigr)\,.
%\nu_x(t)=\int_{(x_0,\ldots,x_{n-1})\in X^n} \bigl(\prod_{i=0}^{n-1} \nu_{x_i}(t_i)\bigr)d(c_a(x))\,.
%\label{eq:def:probtreeautomaton2}
\end{equation*}
%Here, $l(\empseq)=a\in\Sigma_n$, and
%$t_i$ is the $i$'th subtree of $t$.
%and $c_a(x)$ is a probability measure on $X^n$ that is defined by $c_a(x)(A)=c(x)(\{a\}\times A)$.
%, and $a_i=l_i(\empseq)$.
\end{itemize}
%
Then there exists 
a unique probability measure $\lang(\mathcal{X},x)$ on $(\Treeinf(\Sigma),\sigalg_{\infty})$ s.t.\ %such that 
$\lang(\mathcal{X},x)(\cyl(t))=\nu(t)$.
%The function $\lang(\mathcal{X},x)$ in Def.~\ref{def:probtreeautomaton} satisfies an equality in Lem.~\ref{lem:uniqueextensionmeas}.
%Therefore extension to $\sigalg_{\infty}$ uniquely exists.
%\qed
\end{myproposition}

This proposition can be proved using Kolmogorov's consistency theorem and the following lemma.

\begin{mylemma}\label{lem:existencesemanticslem}
In Prop.~\ref{prop:existencesemantics}, for all $k\in\omega$ and $t\in\Treeinf^k(\Sigma)$, we have
\begin{equation*}
\sum_{s\in \Treeinf^{k+1}(\Sigma), t\treeprefix s}\nu_x(s)=\nu_x(t)\,.
\hfill\qed
\end{equation*}
\end{mylemma}

%\newtheorem*{proof:existencesemanticslem}{Proof of Lem.\ \ref{lem:existencesemanticslem}}{\itshape}{\rmfamily}
%\begin{proof:existencesemanticslem}
\begin{myproof}
%By Kolmogorov's consistency theorem (see~\cite{kechris95classicaldescriptive}), 
%it suffices to prove that for all $k\in\omega$ and $t\in\Treeinf^k(\Sigma)$, $\sum_{s\in \Treeinf^{k+1}(\Sigma), t\treeprefix s}\nu_x(s)=\nu_x(t)$ holds.
For $s=(D_s,l_s)\in \Treeinf^{k+1}(\Sigma)$, we denote $a_{s}$ for $l_s(\empseq)$ and $n_s$ for the arity of $a_s$ (i.e.\ $a_s\in \Sigma_{n_s}$).
We prove the given equation by the induction on $k$. 

For $k=0$, as $t$ and subtrees of $s$ other than $s$ are $0$-prefix trees,
\allowdisplaybreaks[1]\begin{align*}
&\sum_{s\in \Treeinf^{k+1}(\Sigma), t\treeprefix s}\nu_x(s) \\
&= \sum_{s\in \Treeinf^{1}(\Sigma)}\sum_{x_0,\ldots,x_{n_s-1}\in X} c(x)(\{(a_s,x_0,\ldots,x_{n_s-1})\})\cdot \prod_{i=0}^{n_s-1} \nu_{x_i}(t) \\
& \pushright{\text{(by definition of $\nu_x(s)$)}} \\
&=\sum_{n=0}^{\infty}\sum_{a\in\Sigma_n}\sum_{x_0,\ldots,x_{n-1}\in X}c(x)(\{(a,x_0,\ldots,x_{n-1})\})\cdot \prod_{i=0}^{n-1} \nu_{x_i}(t)\\
&\pushright{(\text{as $s$ consists of one node})} \\
&= \sum_{n=0}^{\infty}\sum_{a\in\Sigma_n} \sum_{x_0,\ldots,x_{n-1}\in X}c(x)(\{(a,x_0,\ldots,x_{n-1})\})\cdot \prod_{i=0}^{n-1}(1-\Reach(\Delta_{\mathcal{X}},x_i,\bot)) \\
& \pushright{(\text{by definition of $\nu_x(t)$ where $t\in\Treeinf^0(\Sigma)$})} \\
&= \sum_{n=0}^{\infty}\sum_{x_0,\ldots,x_{n-1}\in X} \sum_{a\in\Sigma_n} c(x)(\{(a,x_0,\ldots,x_{n-1})\})\cdot \prod_{i=0}^{n-1}(1-\Reach(\Delta_{\mathcal{X}},x_i,\bot)) \\
&\pushright{(\text{as all elements are nonnegative})} \\
&=\sum_{n=0}^{\infty} \sum_{x_0,\ldots,x_{n-1}\in X} \tau_{\mathcal{X}}(x,\langle x_0,\ldots,x_{n-1}\rangle)\cdot \prod_{i=0}^{n-1}(1-\Reach(\Delta_{\mathcal{X}},x_i,\bot)) \\
&\pushright{(\text{by definition of $\Delta_{\mathcal{X}}$})} \\
&\pushleft{=1-\Reach(\Delta_{\mathcal{X}},x,\bot)} (\text{by definition of branching process}) \\
&\pushleft{=\nu_x(t)}  (\text{by definition of $\nu_x(t)$})\,.
\end{align*}
%
For $k>0$, we assume that  
$\sum_{v\in \Treeinf^{k}(\Sigma), u\treeprefix v}\nu_x(v)=\nu_x(u)$ holds for all $u\in\Treeinf^{k-1}(\Sigma)$ and $x\in X$.
Let $t=(D,l)\in\Treeinf^k(\Sigma)$, $a=l(\empseq)$ and assume that $a\in\Sigma_n$.
Moreover, let $t_i$ and $s_i$ be the $i$'th subtrees of $t\in\Treeinf^k(\Sigma)$ and $s\in\Treeinf^{k-1}(\Sigma)$, respectively.
Then
\begin{align*}
&\sum_{s\in \Treeinf^{k+1}(\Sigma), t\treeprefix s}\nu_x(s) \\
&= \sum_{s\in \Treeinf^{k+1}(\Sigma), t\treeprefix s}\sum_{x_0,\ldots,x_{n-1}\in X} \left( c(x)(\{(a,x_0,\ldots,x_{n-1})\})\cdot \prod_{i=0}^{n-1} \nu_{x_i}(s_i) \right) \\
& \pushright{(\text{by definition of $\nu_x(s)$})} \\
&=\sum_{x_0,\ldots,x_{n-1}\in X} \left(c(x)(\{(a,x_0,\ldots,x_{n-1})\})\cdot\left(\sum_{s\in \Treeinf^{k+1}(\Sigma), t\treeprefix s} \prod_{i=0}^{n-1} \nu_{x_i}(s_i)\right)\right) \\
&=\sum_{x_0,\ldots,x_{n-1}\in X}\left( c(x)(\{(a,x_0,\ldots,x_{n-1})\})\cdot\left(\sum_{\substack{s_0\in \Treeinf^{k}(\Sigma),\\ t_0\treeprefix s_0}}\cdots\sum_{\substack{s_{n-1}\in \Treeinf^{k}(\Sigma), \\t_{n-1}\treeprefix s_{n-1}}} \prod_{i=0}^{n-1} \nu_{x_i}(s_i)\right)\right) \\
&=\sum_{x_0,\ldots,x_{n-1}\in X} \left(c(x)(\{(a,x_0,\ldots,x_{n-1})\})\cdot\left(\prod_{i=0}^{n-1} \sum_{s_i\in \Treeinf^{k}(\Sigma), t_i\treeprefix s_i}\nu_{x_i}(s_i)\right)\right) \\
& \pushright{(\text{$s_i$ does not appear in $\nu_{x_j}(s_j)$ if $i\neq j$})} \\
&\pushleft{= \sum_{x_0,\ldots,x_{n-1}\in X} \bigl(c(x)(\{(a,x_0,\ldots,x_{n-1})\})\cdot \prod_{i=0}^{n-1} \nu_{x_i}(t_i)\bigr)} (\text{by induction hypothesis}) \\
&\pushleft{= \nu_x(t)}  (\text{by definition of $\nu_x(t)$}).
\end{align*}
%
Therefore $\sum_{s\in \Treeinf^{k+1}(\Sigma), t\treeprefix s}\nu_x(s)=\nu_x(t)$ holds for all $k\in\omega$ and $t\in\Treeinf^k(t)$. % and this concludes the proof.
\qed
%\end{proof:existencesemanticslem}
\end{myproof}

\newtheorem*{proof:existencesemantics}{Proof of Prop.\ \ref{prop:existencesemantics}}{\itshape}{\rmfamily}
\begin{proof:existencesemantics}
Immediate from Kolmogorov's consistency theorem~\cite{kechris95classicaldescriptive} and Lem.~\ref{lem:existencesemanticslem}.
\qed
\end{proof:existencesemantics}





\begin{mydefinition}\label{def:probtreeautomaton}
Let $\mathcal{X}=(X,s,c)$ be a $\Sigma$-labeled probabilistic tree automaton. 
%
%For each integer $k\in\omega$ and $k$-prefix tree $t\in \Treeinf^k(\Sigma)$, 
%we define a value $\nu(\cyl(t))\in[0,1]$ by an induction on $k$ as follows.
%\begin{itemize}
%\item If $k=0$, then 
%\begin{equation}
%\nu(\cyl(t))=\nu(\Treeinf(\Sigma))=\text{Reach}(\Delta_{\mathcal{X}},\bullet) \,.
%\label{eq:def:probtreeautomaton1}
%\end{equation}
%\item If $k=i+1$ and $t=(D,l)$ where , then 
%\begin{equation}
%\nu(\cyl(t))=\sum_{y_0,\ldots,y_{n-1}\in X} \bigl(c(x)(a,y_0,\ldots,y_{n-1})\cdot \prod_{i=0}^{n-1} \nu(\cyl(t_i))\bigr)\,.
%\label{eq:def:probtreeautomaton2}
%\end{equation}
%Here, $l(\empseq)=a\in\Sigma_n$, 
%$t_i=(D_i,l_i)$ is $\langle i \rangle$'th subtree of $t$, and
%$y_i=l_i(\empseq)$.
%\end{itemize}
%
For a state $x\in X$, an \emph{infinite language of $\mathcal{X}$ from $x$} is a 
%unique 
probability measure $\lang(\mathcal{X},x)$ on 
%a measurable space 
$(\Treeinf(\Sigma),\sigalg_{\infty})$ in Prop.~\ref{prop:existencesemantics}. %that is the unique extension of $\nu$ (i.e.\ $\lang(\mathcal{X},x)=\nu^*$).
%
An \emph{infinite language of $\mathcal{X}$} is a probability measure $\lang(\mathcal{X})$ on 
a $(\Treeinf(\Sigma),\sigalg_{\infty})$ that is defined by $\lang(\mathcal{X})(A)=\sum_{x\in X} \lang(\mathcal{X},x)(A)$ for each $A\in\sigalg_{\infty}$.
\end{mydefinition}

%The measure $\lang(\mathcal{X},x)$ defined above is well-defined.

%\begin{mylemma}\label{lem:existencesemantics}
%The function $\lang(\mathcal{X},x)$ in Def.~\ref{def:probtreeautomaton} satisfies an equality in Lem.~\ref{lem:uniqueextensionmeas}.
%Therefore extension to $\sigalg_{\infty}$ uniquely exists.
%\qed
%\end{mylemma}

Then we can show that 
the language defined in Def.~\ref{def:probtreeautomaton} 
coincides with coalgebraic infinite trace semantics.
\begin{mytheorem}\label{thm:largestsemanticsgiry}
The carrier of the final $\FSigma$-coalgebra in $\Meas$ is isomorphic to $(\Treeinf(\Sigma),\sigalg_{\infty})$,
and for a $\Sigma$-labeled probabilistic tree automaton $\mathcal{X}=(X,s,c)$,
%For the order in Def.~\ref{def:orderenrichment}, 
we have $\trinf(c)(x)=\lang(\mathcal{X},x)$ for all $x\in X$.
%wrt.\ the order in Def.~\ref{def:orderenrichment}.
Moreover, $\trinf\odot s(c)(*)=\lang(\mathcal{X})$.
%\qed
\end{mytheorem}

This theorem is proved using the result in~\cite{etessamiSY15greatestfixed}---the unreachable probability of a branching process
is calculated as the greatest fixed point of a certain function.


\begin{mylemma}[\cite{etessamiSY15greatestfixed}]\label{lem:BPunreachability}
Let $\Delta=(\Gamma,\tau)$ be a branching process and $y\in \Gamma$.
We define a function $P:[0,1]^{\Gamma}\to[0,1]^{\Gamma}$ as follows:
\begin{equation*}
(P(\mathbf{v}))_x=\sum_{\substack{\alpha\in\Gamma^*, \\ \text{$y$ does not appear in $\alpha$}}} \tau(x,\alpha)\cdot \prod_{i=0}^{|\alpha|-1} \mathbf{v}_{\alpha_i}\,.
\end{equation*}
Here, $|\alpha|$ denotes the length of $\alpha\in\Gamma^*$ and $\alpha_i$ denotes the $i$'th letter of $\alpha$.
As $P$ is a monotone function, $P$ has the greatest fixed point $\mathbf{v}^{y,\max}\in[0,1]^{\Gamma}$. 
Then we have $1-\Reach(\Delta,x,y)=(\mathbf{v}^{y,\max})_x$.
\qed
\end{mylemma}


\newtheorem*{proof:largestsemanticsgiry}{Proof of Thm.\ \ref{thm:largestsemanticsgiry}}{\itshape}{\rmfamily}
\begin{proof:largestsemanticsgiry}
We define an arrow $\zeta:(\Treeinf(\Sigma),\sigalg_{\infty})\to F_{\Sigma}(\Treeinf(\Sigma),\sigalg_{\infty})$ 
in $\Meas$ by $\zeta(t)=(a,(t_0,\ldots,t_{n-1}))$.
Here, $t=(D,l)$, $a=l(\empseq)\in\Sigma_n$, 
and for each $i\in\{0,\ldots,n-1\}$, $D_i=\{\alpha\in\mathbb{N}^*\mid i\alpha\in D\}$ and $l_i(\alpha)=l(i\alpha)$ holds.
Then it is easy to see that $\zeta$ is measurable and moreover, is a final $F_{\Sigma}$-coalgebra.

We show that a function $\lang(\mathcal{X},\place):X\to\giry \Treeinf(\Sigma)$ is the largest homomorphism from $c$ to $J\zeta$.
%where $\zeta$ is defined as in Lem.~\ref{lem:finalcoalgmeas}. 
As $X$ has a discrete $\sigma$-algebra, $\lang(\mathcal{X},\place)$ is indeed an arrow in $\Meas$.

Let $\mathbf{v}^{\max}\in[0,1]^X$ be the greatest fixed point of a function $P:[0,1]^X\to [0,1]^X$ that is defined as follows:
\begin{equation}
(P(\mathbf{v}))_x=\sum_{n=0}^{\infty}\sum_{x_0,\ldots,x_{n-1}\in X} \left(\sum_{a\in\Sigma_n} c(x)(\{(a,x_0,\ldots,x_{n-1}\}))\right) \cdot \prod_{i=0}^{n-1} \mathbf{v}_{x_i}\,.
\label{eq:largestsemanticsgiry1}
\end{equation}
As $\lang(\mathcal{X},x)(\Treeinf(\Sigma))$ is defined by $\lang(\mathcal{X},x)(\Treeinf(\Sigma))=1-\Reach(\Delta_{\mathcal{X}},x,\bot)$, by Lem.~\ref{lem:BPunreachability},
we have $\lang(\mathcal{X},x)(\Treeinf(\Sigma))=(\mathbf{v}^{\max})_x$.

We first show that $\lang(\mathcal{X},\place)$ is a homomorphism. 
By Kolmogorov's consistency theorem,  it suffices to prove the commutativity with respect to cylinder sets:
namely, we show that $(J\zeta^{-1}\odot \overline{F}\lang(\mathcal{X},\place)\odot c)(x)(\cyl(t))=\lang(\mathcal{X},x)(\cyl(t))$ 
for all $k\in\omega$ and $t\in\Treeinf^k(\Sigma)$.

If $k=0$, then because $\cyl(t)=\Treeinf(\Sigma)$,
\begin{align*}
&(J\zeta^{-1}\odot \overline{\FSigma}(\lang(\mathcal{X},\place))\odot c)(x)(\cyl(t)) \\
&= \sum_{n=0}^{\infty}\sum_{x_0,\ldots,x_{n-1}\in X} \sum_{a\in\Sigma_n} \left(c(x)(\{(a,x_0,\ldots,x_{n-1}\}))\cdot (J\zeta^{-1}\odot \overline{\FSigma}(\lang(\mathcal{X},\place)))(a,x_0,\ldots,x_{n-1})(\Treeinf(\Sigma))\right) \\
&= \sum_{n=0}^{\infty}\sum_{x_0,\ldots,x_{n-1}\in X} \sum_{a\in\Sigma_n} \left(c(x)(\{(a,x_0,\ldots,x_{n-1}\}))\cdot \prod_{i=0}^{n-1}\lang(\mathcal{X},x_i)(\Treeinf(\Sigma))\right) \\
&= \sum_{n=0}^{\infty}\sum_{x_0,\ldots,x_{n-1}\in X} \sum_{a\in\Sigma_n} \left(c(x)(\{(a,x_0,\ldots,x_{n-1}\}))\cdot \prod_{i=0}^{n-1}(\mathbf{v}^{\max})_{x_i}\right) \\
&= (P(\mathbf{v}^{\max}))_x \\
&=(\mathbf{v}^{\max})_x \\
\intertext{while} 
&\lang(\mathcal{X},\place)(x)(\cyl(t))=(\mathbf{v}^{\max})_x\,.
\end{align*}
Therefore we have $(J\zeta^{-1}\odot \overline{F}\lang(\mathcal{X},\place)\odot c)(x)(\cyl(t))=\lang(\mathcal{X},x)(\cyl(t))$. % for $k=0$.

Let $k>0$, $t=(D,l)$, $l(\empseq)=a$ and $a\in\Sigma_n$. 
Moreover, let $t_i$ be the $i$'th subtree of $t$ where $0\leq i \leq n-1$. 
Then
\begin{align*}
&(J\zeta^{-1}\odot \overline{\FSigma}(\lang(\mathcal{X},\place))\odot c)(x)(\cyl(t)) \\
&= \sum_{x_0,\ldots,x_{n-1}\in X} c(x)(\{(a,x_0,\ldots,x_{n-1})\})\cdot \prod_{i=0}^{n-1}\lang(\mathcal{X},x_i)(\cyl(t_i)) \\
&= \lang(\mathcal{X},x)(\cyl(t)) & (\text{by definition of $\lang(\mathcal{X},x)$})\,.
\end{align*}
%
Therefore we have $J\zeta^{-1}\odot \overline{F}\lang(\mathcal{X},\place)\odot c=\lang(\mathcal{X},\place)$.

Next we show that $\lang(\mathcal{X},\place)$ is the largest homomorphism.
Let $g:X\kto \Treeinf(\Sigma)$ be a homomorphism from $c$ to $J\zeta$.
By the monotonicity of the extension of measure on cylinder sets,
it suffices to show that $g(x)(\cyl(t))\leq\lang(\mathcal{X},x)(\cyl(t))$ for all $x\in X$, $k\in\omega$, and $t\in\Treeinf^k(\Sigma)$.
We prove it by the induction on $k$.

If $k=0$, then $\cyl(t)=\Treeinf(\Sigma)$. Then
\begin{align*}
&g(x)(\Treeinf(\Sigma)) \\
&\pushleft{=(J\zeta^{-1}\odot\overline{\FSigma}g\odot c)(x)(\Treeinf(\Sigma))}  \text{($g$ is a homomorphism)} \\ 
&= \sum_{n=0}^{\infty}\sum_{x_0,\ldots,x_{n-1}\in X} \left(\sum_{a\in\Sigma_n} c(x)(\{(a,x_0,\ldots,x_{n-1})\})\right) \cdot \prod_{i=0}^{n-1}g(x_i)(\Treeinf(\Sigma)) \\
& \pushright{\text{(by the definition of $\overline{\FSigma}$ on $\Kl(\giry)$)}} \,.
\end{align*}
Therefore a vector $\mathbf{w}\in[0,1]^{X}$ that is defined by $\mathbf{w}_x=g(x)(\Treeinf(\Sigma))$ is a fixed point of $P$ defined in (\ref{eq:largestsemanticsgiry1}).
As $\mathbf{v}^{\max}$ is the greatest fixed point of $P$, we have $g(x)(\Treeinf(\Sigma))=\mathbf{w}_x\leq\mathbf{v}_x=\lang(\mathcal{X},x)(\Treeinf(\Sigma))$.

Let $k>0$ and assume that $g(x)(\cyl(s))\leq\lang(\mathcal{X},x)(\cyl(s))$ for all $x\in X$ and $s\in\Treeinf^{k-1}(\Sigma)$.
Moreover, let $t=(D,l)$, $l(\empseq)=a$ and $a\in\Sigma_n$. 
We denote $t_i$ for the $i$'th subtree of $t$ where $0\leq i \leq n-1$.
Then %as $g$ is a homomorphism, by the definition of $\overline{\FSigma}$ on $\Kl(\giry)$, and by induction hypothesis, we have
\begin{align*}
g(x)(\cyl(t))
&\pushleft{=(J\zeta^{-1}\odot\overline{\FSigma}g\odot c)(x)(\cyl(t))}  \text{($g$ is a homomorphism)} \\
&= \sum_{x_0,\ldots,x_{n-1}} \left(c(x)(\{(a,x_0,\ldots,x_{n-1})\})\cdot  \prod_{i=0}^{n-1} g(x_i)(\cyl(t_i))\right) \\
&\pushright{\text{(by  definition of $\overline{\FSigma}$ on $\Kl(\giry)$)}} \\
&\leq \sum_{x_0,\ldots,x_{n-1}} \left(c(x)(\{(a,x_0,\ldots,x_{n-1})\})\cdot  \prod_{i=0}^{n-1} \lang(\mathcal{X},x_i)(\cyl(t_i))\right) \\
&\pushright{\text{(by induction hypothesis)}} \\
&\pushleft{= (J\zeta^{-1}\odot\overline{\FSigma}(\lang(\mathcal{X},\place)\odot c)(x)(\cyl(t))} \;\;\;\;\;\text{(by  definition of $\overline{\FSigma}$ on $\Kl(\giry)$)} \\
&\pushleft{= \lang(\mathcal{X},x)(\cyl(t))}  \text{($\lang(\mathcal{X},\place)$ is a homomorphism)}\,.
\end{align*}
%
Therefore $\trinf(c)(x)=\lang(\mathcal{X},x)$ for each $x\in X$.
This immediately implies  $\trinf(c)\odot s(*)=\lang(\mathcal{X})$
\qed
\end{proof:largestsemanticsgiry}


%
\auxproof{
We show that the language defined in Def.~\ref{def:probtreeautomaton} coincides with coalgebraic infinite trace semantics.
The carrier of the final $\FSigma$-coalgebra in $\Meas$ is given by a set of  infinite trees.
\begin{mylemma}\label{lem:finalcoalgmeas}
%For $\FSigma$ in Def.~\ref{def:FSigmaMeas}, 
We define an arrow $\zeta:(\Treeinf(\Sigma),\sigalg_{\infty})\to F_{\Sigma}(\Treeinf(\Sigma),\sigalg_{\infty})$ 
in $\Meas$ in the same way as Lem.~\ref{lem:finalcoalgsets}.
%\begin{equation*}
%\zeta(D,l)=(a,(t_0,\ldots,t_{n-1})).
%\end{equation*}
%Here, $a=l(\empseq)\in\Sigma_n$, 
%and for each $i\in\{0,\ldots,n-1\}$, $D_i=\{\alpha\in\mathbb{N}^*\mid i\alpha\in D\}$ and $l_i(\alpha)=l(i\alpha)$.
%
Then $\zeta$ is measurable and moreover,  is a final $F_{\Sigma}$-coalgebra.
\qed
\end{mylemma}

%The infinite language of a probabilistic tree automaton can be captured by the largest homomorphism.
Then we have the following theorem.
It is proved using the known result in~\cite{etessamiSY15greatestfixed}---the 
unreachable probability of a branching processes can be calculated as
the greatest fixed point of a certain function $P:[0,1]^{\Gamma}\to[0,1]^{\Gamma}$.

\begin{mytheorem}\label{thm:largestsemanticsgiry}
Let $\mathcal{X}=(X,s,c)$ be a $\Sigma$-labeled probabilistic tree automaton.
%For the order in Def.~\ref{def:orderenrichment}, 
For $x\in X$, we have $\trinf(c)(x)=\lang(\mathcal{X},x)$
wrt.\ the order in Def.~\ref{def:orderenrichment}.
Moreover, $\trinf\odot s(c)(*)=\lang(\mathcal{X})$.
\qed
\end{mytheorem}
}






\subsection{Omitted Proofs in Section~\ref{sec:powersetmonad}}
%\newtheorem*{proof:generalconstructiontop}{Proof of Prop.\ \ref{prop:generalconstructiontop}}{\itshape}{\rmfamily}
%\begin{proof:generalconstructiontop}
%\end{proof:generalconstructiontop}
%In this section, we assume that $F$ is a polynomial functor.

\begin{mylemma}\label{lem:satassumptionnondet}
%For a ranked alphabet $\Sigma$,
A polynomial functor $F$ and
%on $\Sets$ and the powerset monad 
$\pow$ satisfy Assumptions~\ref{asm:generalconstructiontopFinalCoalg},
\ref{asm:generalconstructiontopDistLaw}, \ref{asm:generalconstructiontopMonotone}, \ref{asm:generalconstructiontopGFPtransfinite}, and
\ref{asm:generalconstructiontopTop} in Prop.~\ref{prop:generalconstructiontop}.
%with respect to the order in Def.~\ref{def:orderenrichment}. 
%Therefore, $\FSigma$ and $\pow$ constitute an infinite trace situation. 
%\qed
\end{mylemma}


%\newtheorem*{proof:satassumptionnondet}{Proof of Lem.\ \ref{lem:satassumptionnondet}}{\itshape}{\rmfamily}
%\begin{proof:satassumptionnondet}
\begin{myproof}
It is known that Assumption~\ref{asm:generalconstructiontopFinalCoalg} is satisfied~\cite{adamekK79leastfixed}.
It is known that Assumption~\ref{asm:generalconstructiontopDistLaw} is satisfied~\cite[Lemma~2.4]{hasuo07generictrace}.
It is easy to see that $F$ and $\pow$ on $\Sets$ satisfy the Assumptions~\ref{asm:generalconstructiontopMonotone}, 
\ref{asm:generalconstructiontopGFPtransfinite}.
It is also easy to see that Assumption~\ref{asm:generalconstructiontopTop} is satisfied; $\top_{X,Z}:X\kto Z$ is given by $\top_{X,Z}(x)=Z$ for all $x\in X$.
\qed
\end{myproof}
%\end{proof:satassumptionnondet}



%\newtheorem*{proof:soundnessFwd}{Proof of Lem.\ \ref{lem:soundnessFwd}}{\itshape}{\rmfamily}
%\begin{proof:soundnessFwd}
%\end{proof:soundnessFwd}

\begin{mysublemma}\label{sublem:ordinalramsey}
Let $\mathfrak{a}$ be a limit ordinal, $C$ be a finite set, and $f:\mathfrak{a}\to C$.
Then there exists $c\in C$ such that for all $\mathfrak{i}<\mathfrak{a}$, there exists $\mathfrak{b}\geq\mathfrak{i}$ such that $f(\mathfrak{b})=c$
\end{mysublemma}

\begin{myproof}
For each $c\in C$, we define $A_c\subseteq \mathfrak{a}$ by $A_c=\{\mathfrak{d}<\mathfrak{a}\mid f(\mathfrak{d})=c\}$.
Assume that for all $c\in C$, there exists $\mathfrak{i}_c<\mathfrak{a}$ such that for all $\mathfrak{j}\in A_c$, $\mathfrak{j}\leq\mathfrak{i}_c$ holds.
Then $\bigsqcup_{\mathfrak{j}\in A_c}\mathfrak{j}\leq \mathfrak{i}_c<\mathfrak{a}$ for each $c\in C$.
As $C$ is finite, this implies 
$\mathfrak{a}=\bigsqcup_{\mathfrak{j}<\mathfrak{a}}\mathfrak{j}=
\bigsqcup_{c\in C}\bigsqcup_{\mathfrak{j}\in A_c}\mathfrak{j}\leq\bigsqcup_{c\in C}\mathfrak{i}_c<\mathfrak{a}$.
This contradicts and the statement is proved.
\qed
\end{myproof}

\begin{mylemma}\label{lem:whenrestrictedpow}
In Def.~\ref{def:restrictedbwdsimtop}, if $T=\pow$ and $F$ is a polynomial functor, 
%then
%\begin{enumerate}
%\item 
Assumption~\ref{item:restrictedbwdsim2top} is satisfied if $b(x)\neq\emptyset$ for each $x\in X$, 
%\label{item:whenrestrictedpow2}
%\item 
while Assumption~\ref{item:restrictedbwdsim3top} is satisfied if $b(x)$ is finite for each $x\in X$. 
%\label{item:whenrestrictedpow3}
%\qed
%\end{enumerate}
\end{mylemma}

%\newtheorem*{proof:whenrestrictedpow}{Proof of Prop.\ \ref{prop:whenrestrictedpow}}{\itshape}{\rmfamily}
%\begin{proof:whenrestrictedpow}
\begin{myproof}
%\noindent{\bf \ref{item:whenrestrictedpow2}.} 
Assume that $b(x)\neq\empty$ for all $x\in X$.
To prove that Assumption~\ref{item:restrictedbwdsim2top} in Def.~\ref{def:restrictedbwdsimtop} is satisfied,
it suffices to prove $z\in \top_{Y,Z}\odot b(x)$ for all $z\in Z$ and $x\in X$.
By the assumption, there exists $y\in Y$ such that $y\in b(x)$.
Therefore for all $z\in Z$, $z \in \top_{Y,Z}(y)\subseteq \top_{Y,Z}\odot b(x)$.

%\noindent{\bf \ref{item:whenrestrictedpow3}.}
Next we assume that $b(x)$ is finite for each $x\in X$ and 
prove that Assumption~\ref{item:restrictedbwdsim3top} in Def.~\ref{def:restrictedbwdsimtop} is satisfied.
Here, $\bigsqcap_{\mathfrak{i}<\mathfrak{a}}(g_{\mathfrak{a}}\odot b)(x)=\bigcap_{\mathfrak{i}<\mathfrak{a}}\bigcup_{y\in b(x)} g_{\mathfrak{i}}(y)$
while $\bigl(\bigsqcap_{\mathfrak{i}<\mathfrak{a}}g_{\mathfrak{i}}\bigr)\odot b(x)=\bigcup_{y\in b(x)}\bigcap_{\mathfrak{i}<\mathfrak{a}} g_{\mathfrak{i}}(y)$.
It is well-known that the latter is always included in the former.
Therefore 
it suffices to prove that $z\in \bigcap_{\mathfrak{i}<\mathfrak{a}}\bigcup_{y\in b(x)} g_{\mathfrak{i}}(y)$ implies 
 $z\in\bigcup_{y\in b(x)}\bigcap_{\mathfrak{i}<\mathfrak{a}} g_{\mathfrak{i}}(y)$.

Assume that $z\in \bigcap_{\mathfrak{i}<\mathfrak{a}}\bigcup_{y\in b(x)} g_{\mathfrak{i}}(y)$.
Then for each $\mathfrak{i}<\mathfrak{a}$, there exists $y_{\mathfrak{i}}\in b(x)$ such that $z\in g_{\mathfrak{i}}(y_{\mathfrak{i}})$.
As $b(x)$ is finite, from Sublem.~\ref{sublem:ordinalramsey}, for an arbitrary $\mathfrak{i}<\mathfrak{a}$, 
there exists an ordinal $\mathfrak{j}$ such that $\mathfrak{i}\leq \mathfrak{j}<\mathfrak{a}$ and 
$z\in g_{\mathfrak{j}}(y)$.
As $\mathfrak{i}\leq \mathfrak{j}$ implies $g_{\mathfrak{i}} \sqsupseteq g_{\mathfrak{j}}$, we have $z\in g_{\mathfrak{i}}(y)$.
Therefore $z\in\bigcup_{y\in b(x)}\bigcap_{\mathfrak{i}<\mathfrak{a}} g_{\mathfrak{i}}(y)$ holds.
\qed
\end{myproof}
%\end{proof:whenrestrictedpow}


\newtheorem*{proof:soundnessFPEfwd}{Proof of Thm.\ \ref{thm:soundnessFPEfwd}}{\itshape}{\rmfamily}
\begin{proof:soundnessFPEfwd}
\paragraph*{\ref{item:soundnessfpefwd}~(soundness).}
%\noindent{\bf \ref{item:soundnessfpefwd}(soundness).}
\begin{equation*}
\begin{xy}
 \xymatrix@R=2.4em@C=1.7em{
 {\overline{F}X} \ar@{}[drr]|{=} \kar[rr]^{\overline{F} c } \kar@/^6mm/[rrrr]^{\overline{F}(\trinf(c))} & & {\overline{F}^2 X} \ar@{}[drr]|{=} \kar[rr]_{\overline{F}(\trinf(\overline{F}c))} & & {\overline{F} Z} \ar@{}[drr]|{=} & & {\overline{F}Y} \kar[ll]_{ \overline{F}(\trinf(d))}\\  
 {X} \kar@/_4mm/[rrrr]_(.3){\trinf(c)} \kar[u]_{c} \ar@{}[drrr]|(.65){=} \kar[rr]^{c}  & & {\overline{F}X} \kar[u]_{\overline{F}c} \kar[rr]^{\trinf(\overline{F}c)} & & {Z} \kar[u]_{J\zeta} & &  {Y} \kar[ll]_{\trinf(d)} \kar[u]^{d}\\
 {} \ar@{}[ru]|(.10)*\dir{-} &   & & & {1} \ar@{}[ur]|(.5){\sqsubseteq} \ar`l[llllu][llllu]^(0){s} \kar[llu]^{c\odot s} \ar`r[rru][rru]_(0){t} & &  \ar@{}[lu]|(.10)*\dir{-} }
\end{xy}
\end{equation*}
%As $X\in\sigalg_X$, by letting $\mathcal{Y}=\mathcal{X}_{\FPE,X}$ in Prop.~\ref{prop:catmonotonicityOfFPEInf},
%from reflexivity of FPE, we obtain $\mathcal{X}_{\FPE,X}\fwd\mathcal{X}_{\FPE,X_1}$.
%Therefore by transitivity of FPE, we get $\mathcal{X}_{\FPE,X}\fwd\mathcal{Y}$.
%Hence it suffices to prove that  $\mathcal{X}_{\FPE,X}\fwd\mathcal{Y}$ implies $\tr(\mathcal{X})\sqsubseteq \tr(\mathcal{Y})$.
%
Let 
%$\mathcal{X}=((X,\sigalg_X),s,c)$, $\mathcal{Y}=((Y,\sigalg_Y),t,d)$ and 
$\zeta:Z\to FZ$ be a final $F$-coalgebra.
By definition, $\mathcal{X}_{\FPE}=(F(X),c\odot s, Fc)$.
By the assumption and the soundness of a forward Kleisli simulation, we have
\begin{equation}
\trinf(\overline{F}c)\odot (c\odot s)\sqsubseteq\trinf(d)\odot t \;. \label{eq:catsoundnessFPEinf1}
\end{equation}
As $\trinf(c)$ is a homomorphism from $c$ to $J\zeta$, 
%and as $J\zeta$ is an isomorphism, 
we have
\begin{equation}
\trinf(c)=(J\zeta)^{-1}\odot \overline{F}(\trinf(c))\odot c \;. \label{eq:catsoundnessFPEinf2}
\end{equation}
Here, $(J\zeta)^{-1}\odot \overline{F}(\trinf(c))$ is a homomorphism from $\overline{F}c$ to $\zeta$
because of the following equation.
\begin{align*}
J\zeta\odot \bigl((J\zeta)^{-1}\odot \overline{F}(\trinf(c))\bigr) &= \overline{F}(\trinf(c)) & \\
&= \overline{F}\bigl((J\zeta)^{-1}\odot \overline{F}(\trinf(c))\odot c\bigr) & (\text{by  (\ref{eq:catsoundnessFPEinf2})}) \\
&= \overline{F}\bigl((J\zeta)^{-1}\odot \overline{F}(\trinf(c))\bigr)\odot \overline{F}(c) &
\end{align*}
%$\zeta^{-1}\odot \overline{F}(\tr^{\infty}_{\zeta}(c))$ is a homomorphism from $\overline{F}c$ to $\zeta$.
As $\trinf(\overline{F}c)$ is the largest homomorphism from  $\overline{F}c$ to $J\zeta$, this implies
\begin{equation}
(J\zeta)^{-1}\odot \overline{F}(\trinf(c))\sqsubseteq\trinf(\overline{F}c) \;. \label{eq:catsoundnessFPEinf3}
\end{equation}
From the equations (\ref{eq:catsoundnessFPEinf1}--\ref{eq:catsoundnessFPEinf3}), % and the monotonicity of composition, 
$\trinf(d)\odot s\sqsubseteq\trinf(d)\odot t$ follows.

\vspace*{2mm}\noindent\begin{minipage}{0.65\hsize}
\noindent{\bf \ref{item:adequacyfpefwd}~(adequacy).} 
Let $f:Y\kto X$ be a forward Kleisli simulation from $\mathcal{X}$ to $\mathcal{Y}$.
Then as $f$ is a forward simulation,
\begin{equation*}
\overline{F}c\odot (c\odot f)
\sqsubseteq \overline{F}c\odot (\overline{F}f\odot d) 
\sqsubseteq \overline{F}(c\odot f)\odot d 
\end{equation*}
and
\begin{equation*}
c\odot s
\sqsubseteq (c\odot f)\odot t \,. %& (\text{$f$ is a fwd.\ simulation})\,.
\end{equation*}
Hence $c\odot f:Y\kto \overline{F}X$ is a forward simulation from $\mathcal{X}_{\FPE}$ to $\mathcal{Y}$.
\qed
\end{minipage}
\begin{minipage}{0.35\hsize}
\begin{xy}
 \xymatrix@R=2.4em@C=2.8em{
 {\overline{F}^2X} \ar@{}[dr]|{=} & {\overline{F}X} \ar@{}[dr]|{\sqsubseteq} \kar[l]_{\overline{F}c} & {\overline{F}Y} \kar[l]_{\overline{F}f} \\
  {\overline{F}X}  \kar[u]^{\overline{F}c} & {X} \ar@{}[dl]|(.45){=} \ar@{}[dr]|(.45){\sqsubseteq} \kar[l]_{c} \kar[u]^{c} & {Y} \kar[l]_{f} \kar[u]_{d} \\
  & {1} \kar@/^5mm/[ul]^{c\odot s}  \kar[u]^{s} \kar@/_5mm/[ur]_{t} & }
\end{xy}
\end{minipage}
\end{proof:soundnessFPEfwd}

\newtheorem*{proof:soundnessFPEbwdtop}{Proof of Thm.\ \ref{thm:soundnessFPEbwdtop}}{\itshape}{\rmfamily}
\begin{proof:soundnessFPEbwdtop}
\noindent{\bf \ref{item:soundnessfpebwdtop}~(soundness).} 
\begin{equation*}
\begin{xy}
 \xymatrix@R=2.4em@C=1.7em{
 {\overline{F}X} \ar@{}[drr]|{=} \kar[rr]^{\overline{F}(\trinf(c))} & & {\overline{F}Z} \ar@{}[drr]|{=}  & & {\overline{F}^2Y} \ar@{}[drr]|{=} \kar[ll]^{\overline{F}(\trinf(\overline{F}d))} & & {\overline{F}Y} \kar[ll]^{\overline{F}d} \kar@/_6mm/[llll]_{\overline{F}(\trinf(d))} \\  
 {X}  \kar[u]_{c}  \kar[rr]^{\trinf(c)}  & & {Z}\ar@{}[drrr]|(.65){=} \kar[u]_{J\zeta} & & {\overline{F}Y} \kar[u]_{\overline{F}d}  \kar[ll]_{\trinf(\overline{F}d)} & &  {Y} \kar[ll]_{d} \kar[u]^{d} \kar@/^4mm/[llll]^(.3){\trinf(d)}\\
 {} \ar@{}[ru]|(.10)*\dir{-} &   & {1} \ar@{}[ul]|(.5){\sqsubseteq} \ar`l[llu][llu]_(0){s} \ar`r[rrrru][rrrru]^(0){t} \kar[rru]^{d\odot t}  & &  & &  \ar@{}[lu]|(.10)*\dir{-} }
\end{xy}
\end{equation*}
Let $b:X\kto \overline{F}Y$ be a TIF-backward simulation from $\mathcal{X}$ to $\mathcal{Y}_{\FPE}$.
Then by the soundness of TIF-backward simulation (Lem.~\ref{lem:soundnessRBwdnotop}), 
we have
\begin{equation}
\trinf(c)\odot s\sqsubseteq \trinf(\overline{F}d)\odot (d\odot t)\,.
\label{eq:soundnessfpebwd1}
\end{equation}
It is easy to see that $d:Y\kto \overline{F}Y$ is a forward simulation from $\mathcal{Y}_{\FPE}$ to $\mathcal{Y}$.
Therefore by the soundness of forward simulation (Lem.~\ref{lem:soundnessFwd}), 
we have 
\begin{equation}
\trinf(\overline{F}d)\odot (d\odot t)\sqsubseteq \trinf(d)\odot t\,.
\label{eq:soundnessfpebwd2}
\end{equation}
From equations (\ref{eq:soundnessfpebwd1}) and (\ref{eq:soundnessfpebwd2}), we have 
$\trinf(c)\odot s\sqsubseteq \trinf(d)\odot t$.

\vspace{2mm}\noindent\begin{minipage}{0.65\hsize}
\noindent{\bf \ref{item:adequacyfpebwdtop}~(adequacy).} 
Let $b:X\kto Y$ be a TIF-backward simulation from $\mathcal{X}$ to $\mathcal{Y}$.
In the similar manner to the proof of Thm.~\ref{thm:soundnessFPEfwd}.\ref{item:soundnessfpefwd},
we can prove that $d\odot b:X\kto \overline{F}Y$ is a backward simulation from $\mathcal{X}$ to $\mathcal{Y}_{\FPE}$.
Moreover, the Assumptions~\ref{item:restrictedfpe2top} and \ref{item:restrictedfpe3top} imply that 
$d\odot b$ satisfy Assumptions~\ref{item:restrictedbwdsim2top} and \ref{item:restrictedbwdsim3top}  in 
the definition of a TIF-backward simulation (Def.~\ref{def:restrictedbwdsimtop}).
Therefore $d\odot b$ is a TIF-backward simulation from $\mathcal{X}$ to $\mathcal{Y}_{\FPE}$.
\qed
\end{minipage}
\begin{minipage}{0.35\hsize}
\begin{xy}
 \xymatrix@R=2.4em@C=2.8em{
 {\overline{F}X} \ar@{}[dr]|{\sqsubseteq} \kar[r]^{\overline{F}b} & {\overline{F}Y} \ar@{}[dr]|{=} \kar[r]^{\overline{F}d} & {\overline{F}^2Y}  \\
  {X}  \kar[u]^{c} \kar[r]^{b} & {Y} \ar@{}[dl]|(.45){\sqsubseteq} \ar@{}[dr]|(.45){=}  \kar[r]^{d} \kar[u]^{\overline{F}d} & {\overline{F}Y} \kar[u]_{\overline{F}d} \\
  & {1} \kar@/^5mm/[ul]^{s}  \kar[u]^{t} \kar@/_5mm/[ur]_{d\odot t} & }
\end{xy}
\end{minipage}
\end{proof:soundnessFPEbwdtop}

\subsection{Omitted Proofs in Section~\ref{sec:girymonad}}
\label{subsec:omittedProofsForGiryMonad}

\newtheorem*{proof:generalconstructionnotop}{Proof of Prop.\ \ref{prop:generalconstructionnotop}}{\itshape}{\rmfamily}
\begin{proof:generalconstructionnotop}
Let $c:X\kto\overline{F}X$ be a $\overline{F}$-coalgebra in $\Kl(T)$.
%We construct the largest homomorphism $\trinf(c):X\kto Z$ from $c$ to $J\zeta$.

We first construct a cone $(X,(\alpha_i:X\kto \overline{F}^i1)_{i\in\omega})$ over the sequence $1\overset{J!_{F1}}{\kleftarrow} \overline{F}1\overset{JF!_{F1}}{\kleftarrow} \overline{F}^2 1\overset{JF^2!_{F1}}{\kleftarrow} \ldots$.
We define a function $\Psi_X:\Kl(T)(X,1)\to\Kl(T)(X,1)$ by $\Psi_X(f)=J!_{F1}\odot \overline{F}f\odot c$.

As composition in $\Kl(T)$ and $\overline{F}$'s action on arrows are both monotone (by Assumption~\ref{asm:existenceWeaklyFinalCoalgGFPcont}), $\Psi_X$ is also monotone.
Moreover, as $J!_X$ is the largest element in $\Kl(T)(X,1)$ (Assumption~\ref{asm:existenceWeaklyFinalCoalgTop}), we have $J!_X\sqsupseteq \Psi_X(J!_X)$.
Therefore by repeatedly applying $\Psi_X$ to the both sides,
we can obtain a decreasing sequence $J!_X\sqsupseteq \Psi_X(J!_X)\sqsubseteq \Psi_X^2(J!_X)\cdots$.
By Assumption~\ref{asm:existenceWeaklyFinalCoalgGFPcont}, their greatest lower bound 
$\Psi_X^{\omega}(J!_X)=\bigsqcap_{i\in\omega}\Psi_X^i(J!_X)$ exists.
Here, as composition of arrows in $\Kl(T)$ and $\overline{F}$'s action on arrows are both $\omega^{\text{op}}$-continuous, 
$\Psi_X$ is also locally $\omega^{\text{op}}$-continuous.
Hence by Kleene fixed point theorem, $\Psi_X^{\omega}(J!_X)$ is the greatest fixed point of $\Psi_X$.
%
\begin{equation*}
\begin{xy}
(0,0)*+{1} = "F01",
(20,0)*+{\overline{F}1} = "F11",
(40,0)*+{\overline{F}^21} = "F21",
(60,0)*+{\overline{F}^31} = "F31",
%
(-5,15)*+{X} = "F0X",
(15,15)*+{\overline{F}X} = "F1X",
(35,15)*+{\overline{F}^2X} = "F2X",
(55,15)*+{\overline{F}^3X} = "F3X",
%
(8,7.5)*{\sqsupseteq} = "",
(28,7.5)*{\sqsupseteq} = "",
(48,7.5)*{\sqsupseteq} = "",
%
(-5,-15)*+{Z} = "F0Z",
(15,-15)*+{\overline{F}Z} = "F1Z",
(35,-15)*+{\overline{F}^2Z} = "F2Z",
(55,-15)*+{\overline{F}^3Z} = "F3Z",
%
(8,-7.5)*{=} = "",
(28,-7.5)*{=} = "",
(48,-7.5)*{=} = "",
%
(75,0)*{\cdots}="",
(70,15)*{\cdots}="",
(70,-15)*{\cdots}="",
%
\kar _{J!_{F1}}"F11";"F01"
\kar _{JF!_{F1}}"F21";"F11"
\kar _{JF^2!_{F1}}"F31";"F21"
\kar _(.1){JF^2!_{F1}} (70,0);"F31"
%\ar @{.} (75,0);(70,0)
%
\kar ^{c}"F0X";"F1X"
\kar ^{\overline{F}c}"F1X";"F2X"
\kar ^{\overline{F}^2c}"F2X";"F3X"
\kar ^(.9){\overline{F}^3c} "F3X";(65,15)
%\ar @{.} (65,15);(70,15)
%
\kar ^{J!_X} "F0X";"F01"
\kar ^{JF!_X} "F1X";"F11"
\kar ^{JF^2!_X} "F2X";"F21"
\kar ^{JF^3!_X} "F3X";"F31"
%
\kar _{J\zeta}"F0Z";"F1Z"
\kar _{JF\zeta}"F1Z";"F2Z"
\kar _{JF^2\zeta}"F2Z";"F3Z"
\kar _(.9){JF^3\zeta} "F3Z";(65,-15)
%\ar @{.} (65,-15);(70,-15)
%
\kar _{J!_Z} "F0Z";"F01"
\kar _{JF!_Z} "F1Z";"F11"
\kar _{JF^2!_Z} "F2Z";"F21"
\kar _{JF^3!_Z} "F3Z";"F31"
\end{xy}
\label{eq:diagramexistenceWeaklyFinalCoalg1}
\end{equation*}
%

%Therefore by applying $\Psi_X$ to the both side, 
%We define an ordinal $\mathfrak{l}$ as follows.
%
%If  Assumption~\ref{asm:existenceWeaklyFinalCoalgGFPcont} is satisfied, 
%then $\Psi_X^{\mathfrak{a}}(J!_X)$ is well-defined for all $\mathfrak{a}\leq\omega$,
%hence we define $\mathfrak{l}=\omega$.
%Moreover, as composition of arrows in $\Kl(T)$ and $\overline{F}$ are $\omega^{\text{op}}$-continuous, 
%$\Psi_X$ is also $\omega^{\text{op}}$-continuous.
%Hence by the Kleene fixed-point theorem,
%%for $\mathfrak{l}=\omega$, 
%$\Psi_X^{\mathfrak{l}}(J!_X)$ is the greatest fixed point of $\Psi_X$. 
%
%If the Assumption~\ref{asm:existenceWeaklyFinalCoalgGFPtransfinite} is satisfied, 
%then $\Psi_X^{\mathfrak{a}}(J!_X)$ are well-defined for all ordinals $\mathfrak{a}$, and
%we define $\mathfrak{l}$ as the smallest ordinal such that the cardinality of $\mathfrak{l}$ is greater than the cardinality of $\Kl(T)(X,1)$.
%%then the infimum $\bigsqcap_{\mathfrak{i}<\mathfrak{a}}\Psi_X^i(J!_X)$ in the above definition always exists 
%%because of the $\mathbf{Dcpo}$-enriched structure. 
%%$i<j$ implies $\Psi_X^i(J!_{F1})\sqsupseteq\Psi_X^j(J!_{F1})$ and by Assumption (\ref{asm:existenceWeaklyFinalCoalgGFP}).
%%Here, let $\mathfrak{l}\in\Ord$ be the smallest ordinal such that the cardinality of $\mathfrak{l}$ is greater than the cardinality of $\Kl(T)(X,1)$.
%Then from \cite{cousotC79constructiveversions}, $\Psi_X^{\mathfrak{l}}(J!_X)$ is the greatest fixed point of $\Psi_X$.
%
%In the end, $\Psi_X^{\mathfrak{l}}(J!_X)$ is the greatest fixed point of $\Psi_X$ in both cases.
Using this 
%$\Psi_X^{\mathfrak{l}}(J!_X)$, 
greatest fixed point,
for each $i<\omega$ we define an arrow $\alpha_i:X\kto \overline{F}^i1$ inductively as follows:
\begin{equation}
%\alpha_0=\bigsqcap_{i\in\omega} J!_{F^i 1}\odot \alpha'_i ,  \;\;\;\;\;
\alpha_0=\Psi_X^{\omega}(J!_X),  \;\;\;\text{and}\;\;\;
\alpha_{i+1}=\overline{F}\alpha_i\odot c\,.
\label{eq:existenceWeaklyFinalCoalgdefcone}
\end{equation}
%
Then for each $i\in\omega$, we can prove $\alpha_i=\overline{F}^iJ!_{F1}\odot \alpha_{i+1}$ inductively as follows. 

For $i=0$, we have:
\begin{align*}
J!_{F1}\odot\alpha_1
&=J!_{F1}\odot\overline{F}(\Psi_X^{\omega}(J!_X)) \odot c & (\text{by definition}) \\
&=\Psi_X(\Psi_X^{\omega}(J!_X)) & (\text{by definition}) \\
&= \Psi_X^{\omega}(J!_X) & (\text{$\Psi_X^{\omega}(J!_X)$ is a fixed point}) \\
&=\alpha_0 & (\text{by definition}).
\end{align*}

Assume that $\alpha_i=\overline{F}^iJ!_{F1}\odot \alpha_{i+1}$.
Applying $\overline{F}$ and composing $c$ from the right, we have $\alpha_{i+1}=\overline{F}^{i+1}J!_1\odot \alpha_{i+2}$.

Hence we have $\alpha_i=\overline{F}^iJ!_{F1}\odot \alpha_{i+1}$ for all $i\in\omega$. 
This means that $(X,(\alpha_i:X\kto \overline{F}^i1)_{i\in\omega})$ is a cone over the sequence 
$1\overset{J!_{F1}}{\kleftarrow} \overline{F}1\overset{JF!_{F1}}{\kleftarrow} \overline{F}^2 1\overset{JF^2!_{F1}}{\kleftarrow} \ldots$\,.
Therefore by Assumption~\ref{asm:existenceWeaklyFinalCoalgWeak2Lim}, 
there exists a unique mediating arrow $l:X\kto Z$ 
from the cone $(X,(\alpha_i)_{i\in\omega})$ to $(Z,(J\gamma_i)_{i\in\omega})$.
%that satisfies the conditions in Assumption~\ref{asm:existenceWeaklyFinalCoalgWeak2Lim}. 
%
\begin{equation*}
\begin{xy}
(0,0)*+{1} = "F01",
(20,0)*+{\overline{F}1} = "F11",
(40,0)*+{\overline{F}^21} = "F21",
(60,0)*+{\overline{F}^31} = "F31",
%
(-5,15)*+{X} = "F0X",
(15,15)*+{\overline{F}X} = "F1X",
(35,15)*+{\overline{F}^2X} = "F2X",
(55,15)*+{\overline{F}^3X} = "F3X",
%
(8,7.5)*{=} = "",
(28,7.5)*{=} = "",
(48,7.5)*{=} = "",
%
(-5,-15)*+{Z} = "F0Z",
(15,-15)*+{\overline{F}Z} = "F1Z",
(35,-15)*+{\overline{F}^2Z} = "F2Z",
(55,-15)*+{\overline{F}^3Z} = "F3Z",
%
(8,-7.5)*{=} = "",
(28,-7.5)*{=} = "",
(48,-7.5)*{=} = "",
%
(75,0)*{\cdots}="",
(70,15)*{\cdots}="",
(70,-15)*{\cdots}="",
%
\kar _{J!_{F1}}"F11";"F01"
\kar _{JF!_{F1}}"F21";"F11"
\kar _{JF^2!_{F1}}"F31";"F21"
\kar _(.1){JF^2!_{F1}} (70,0);"F31"
%\ar @{.} (75,0);(70,0)
%
\kar ^{c}"F0X";"F1X"
\kar ^{\overline{F}c}"F1X";"F2X"
\kar ^{\overline{F}^2c}"F2X";"F3X"
\kar ^(.9){\overline{F}^3c} "F3X";(65,15)
%\ar @{.} (65,15);(70,15)
%
\kar ^{\alpha_0} "F0X";"F01"
\kar ^{\overline{F}\alpha_0} "F1X";"F11"
\kar ^{\overline{F}^2\alpha_0} "F2X";"F21"
\kar ^{\overline{F}^3\alpha_0} "F3X";"F31"
%
\kar _{J\zeta}"F0Z";"F1Z"
\kar _{JF\zeta}"F1Z";"F2Z"
\kar _{JF^2\zeta}"F2Z";"F3Z"
\kar _(.9){JF^3\zeta} "F3Z";(65,-15)
%\ar @{.} (65,-15);(70,-15)
%
\kar _{J!_Z} "F0Z";"F01"
\kar _{JF!_Z} "F1Z";"F11"
\kar _{JF^2!_Z} "F2Z";"F21"
\kar _{JF^3!_Z} "F3Z";"F31"
%
\kar @{-->} _{l} "F0X";"F0Z"
\end{xy}
\label{eq:diagramexistenceWeaklyFinalCoalg1}
\end{equation*}
%
%We define a function $\Phi_X:\Kl(T)(X,Z)\to\Kl(T)(X,Z)$ by $\Phi_X(f)=(J\zeta)^{-1}\odot \overline{F}f \odot c$.
%Then for each $i\in\omega$, we have:
%\begin{align*}
%J\gamma_i\odot \Phi_X(l_{\max}) &= J\gamma_i\odot (J\zeta)^{-1}\odot \overline{F}l_{\max} \odot c  & (\text{by definition}) \\
%&= \overline{F}^{i+1}J!_{F1}\odot\overline{F}J\gamma_{i}\odot \overline{F}l_{\max} \odot c & (\text{$\zeta$ is a mediating arrow}) \\
%&= \overline{F}^{i+1}J!_{F1}\odot \overline{F}\alpha_{i} \odot c & (\text{$l_{\max}$ is a mediating arrow}) \\
%&= \overline{F}^{i+1}J!_{F1}\odot \alpha_{i+1}  & (\text{by definition of $\alpha_{i+1}$}) \\
%&= \alpha_{i}  & (\text{$(X,(\alpha_i)_{i\in\omega})$ is a cone}) \,.
%\end{align*}
%Hence $\Phi_X(l_{\max})$ is also a mediating arrow.
%%%%%%% To be modified
%Therefore by the maximality of $l_{\max}$, 
%we have $\Phi_X(l_{\max})\sqsubseteq l_{\max}$.
%
%
%For each an ordinal $\mathfrak{a}$, we define $\Phi_X^{\mathfrak{a}}(l_{\max})\in\Kl(T)(X,Z)$ by the transfinite induction on $\mathfrak{a}$ as follows:
%\begin{itemize}
%\item $\Phi_X^0(l_{\max})=l_{\max}$.
%\item For a successor ordinal $\mathfrak{a}$, $\Phi_X^{\mathfrak{a}}(l_{\max})=\Phi_X(\Phi_X^{\mathfrak{a}-1}(l_{\max}))$.
%\item For a limit ordinal $\mathfrak{a}$, $\Phi_X^{\mathfrak{a}}(l_{\max})=\bigsqcap_{\mathfrak{i}<\mathfrak{a}}\Phi_X^{\mathfrak{i}}(l_{\max})$.
%\end{itemize}
%%
%\begin{equation}
%\begin{xy}
%%(0,0)*+{1} = "F01",
%%(20,0)*+{\overline{F}1} = "F11",
%%(40,0)*+{\overline{F}^21} = "F21",
%%(60,0)*+{\overline{F}^31} = "F31",
%%
%(-5,15)*+{X} = "F0X",
%(15,15)*+{\overline{F}X} = "F1X",
%(35,15)*+{\overline{F}^2X} = "F2X",
%(55,15)*+{\overline{F}^3X} = "F3X",
%%
%%(8,7.5)*{=} = "",
%%(28,7.5)*{=} = "",
%%(48,7.5)*{=} = "",
%%
%(-5,-15)*+{Z} = "F0Z",
%(15,-15)*+{\overline{F}Z} = "F1Z",
%(35,-15)*+{\overline{F}^2Z} = "F2Z",
%(55,-15)*+{\overline{F}^3Z} = "F3Z",
%%
%%(8,-7.5)*{=} = "",
%%(28,-7.5)*{=} = "",
%%(48,-7.5)*{=} = "",
%%
%%\kar _{J!_{F1}}"F11";"F01"
%%\kar _{JF!_{F1}}"F21";"F11"
%%\kar _{JF^2!_{F1}}"F31";"F21"
%%\kar _(.1){JF^2!_{F1}} (70,0);"F31"
%%\ar @{.} (75,0);(70,0)
%%
%(5,0)*{\sqsupseteq} = "",
%(25,0)*{\sqsupseteq} = "",
%(45,0)*{\sqsupseteq} = "",
%%
%\kar ^{c}"F0X";"F1X"
%\kar ^{\overline{F}c}"F1X";"F2X"
%\kar ^{\overline{F}^2c}"F2X";"F3X"
%\kar ^(.9){\overline{F}^3c} "F3X";(65,15)
%\ar @{.} (65,15);(70,15)
%%
%%\kar ^{\alpha_0} "F0X";"F01"
%%\kar ^{\overline{F}\alpha_0} "F1X";"F11"
%%\kar ^{\overline{F}^2\alpha_0} "F2X";"F21"
%%\kar ^{\overline{F}^3\alpha_0} "F3X";"F31"
%%
%\kar _{J\zeta}"F0Z";"F1Z"
%\kar _{JF\zeta}"F1Z";"F2Z"
%\kar _{JF^2\zeta}"F2Z";"F3Z"
%\kar _(.9){JF^3\zeta} "F3Z";(65,-15)
%\ar @{.} (65,-15);(70,-15)
%%
%%\kar _{J!_Z} "F0Z";"F01"
%%\kar _{JF!_Z} "F1Z";"F11"
%%\kar _{JF^2!_Z} "F2Z";"F21"
%%\kar _{JF^3!_Z} "F3Z";"F31"
%%
%\kar @{~>} _{l_{\max}} "F0X";"F0Z"
%\kar  _(.3){\overline{F}l_{\max}} "F1X";"F1Z"
%\kar  _(.3){\overline{F}^2l_{\max}} "F2X";"F2Z"
%\kar  _(.3){\overline{F}^3l_{\max}} "F3X";"F3Z"
%\end{xy}
%\label{eq:diagramexistenceWeaklyFinalCoalg1}
%\end{equation}
%%
%By the monotonicity of $\overline{F}$ and the monotonicity of composition, $\Phi_X$ is also monotone.
%We define an ordinal $\mathfrak{k}$ as follows.
%
%If the Assumption~\ref{asm:existenceWeaklyFinalCoalgGFPcont} is satisfied, 
%then $\Phi_X^{\mathfrak{a}}(l_{\max})$ is well-defined for all $\mathfrak{a}\leq\omega$,
%hence we define $\mathfrak{k}=\omega$.
%Moreover, as composition of arrows in $\Kl(T)$ 
%and $\overline{F}$ are $\omega^{\text{op}}$-continuous,  $\Phi_X$ is also $\omega^{\text{op}}$-continuous.
%Hence by the Kleene fixed-point theorem,
%%for $\mathfrak{k}=\omega$, 
%$\Phi_X^{\mathfrak{k}}(l_{\max})$ is  a fixed point of $\Phi_X$. 
%
%If the Assumption~\ref{asm:existenceWeaklyFinalCoalgGFPtransfinite} is satisfied, 
%then $\Phi_X^{\mathfrak{a}}(l_{\max})$ is well-defined for all ordinals $\mathfrak{a}$, and
%we define $\mathfrak{k}$ as the smallest ordinal such that the cardinality of $\mathfrak{k}$ is greater than the cardinality of $\Kl(T)(X,Z)$.
%%
%%the infimum $\bigsqcap_{\mathfrak{i}<\mathfrak{a}}\Phi_X^{\mathfrak{i}}(l_{\max})$ always exists because of the $\mathbf{Dcpo}$-enriched structure.
%%Let $\mathfrak{l}\in\Ord$ be the smallest ordinal such that the cardinality of $\mathfrak{l}$ is greater than the cardinality of $\Kl(T)(X,Z)$.
%Then 
%%in the similar manner to 
%from~\cite{cousotC79constructiveversions}, 
%%we can prove that 
%this $\Phi_X^{\mathfrak{k}}(l_{\max})$ is a fixed point of $\Phi_X$.
%
%In the end, $\Phi_X^{\mathfrak{k}}(J!_{F1})$ is a fixed point of $\Phi_X$ in both cases.
%We denote $\Phi_X^{\mathfrak{k}}(J!_{F1})$ by $\trinf(c)$
%%%%%% End To be modified

Here, for each $i\in\omega$, we have
\begin{align*}
J\gamma_i\odot \bigl(J\zeta^{-1}\odot \overline{F}l \odot c\bigr)
&= \overline{F}^{i+1}J!_{F1}\odot\overline{F}J\gamma_{i}\odot \overline{F}l \odot c & (\text{$\zeta$ is a mediating arrow}) \\
&= \overline{F}^{i+1}J!_{F1}\odot \overline{F}\alpha_{i} \odot c & (\text{$l$ is a mediating arrow}) \\
&= \overline{F}^{i+1}J!_{F1}\odot \alpha_{i+1}  & (\text{by definition of $\alpha_{i+1}$}) \\
&= \alpha_{i}  & (\text{$(X,(\alpha_i)_{i\in\omega})$ is a cone}) \,.
\end{align*}
Therefore, $J\zeta^{-1}\odot \overline{F}l \odot c$ is a mediating arrow
from $(X,(\alpha_i)_{i\in\omega})$ to $(Z,(J\gamma_i)_{i\in\omega})$, too.
Hence by the uniqueness of the mediating arrow, $l$ is a homomorphism from $c$ to $j\zeta$.

To conclude the proof, we have to show that this $l$ the largest homomorphism from $c$ to $J\zeta$.
%is the greatest fixed point of $\Phi_X$.
Let $g:X\kto Z$ be a homomorphism from $c$ to $J\zeta$.
We construct a cone $(X,(\beta_i:X\kto \overline{F}^i1)_{i\in\omega})$ over the sequence 
$1\overset{J!_{F1}}{\kleftarrow} \overline{F}1\overset{JF!_{F1}}{\kleftarrow} \overline{F}^2 1\overset{JF^2!_{F1}}{\kleftarrow} \ldots$
by $\beta_i=J\gamma_i\odot g$.
%
Then for each $i\in\omega$, we can prove $\beta_i\sqsubseteq \alpha_i$ by the induction on $i$ as follows.

For $i=0$, we have
\begin{align*}
\Psi_X(\beta_0)
&=J!_{F1}\odot \overline{F}J\gamma_0\odot \overline{F}g\odot c & (\text{by definition}) \\
%&=J!_{F1}\odot \overline{F}J\gamma_0\odot J\zeta\odot \Phi_X(g) & (\text{by definition}) \\
&=J!_{F1}\odot \overline{F}J\gamma_0\odot J\zeta\odot g & (\text{$g$ is a homomorphism}) \\
&=J\gamma_0\odot g & (\text{$1$ is a final object in $\mathbb{C}$}) \\
&=\beta_0 & (\text{by definition}).
\end{align*}
Therefore $\beta_0$ is a fixed point of $\Psi_X$.
As $\alpha_0=\Psi_X^{\omega}(J!_X)$ is the greatest fixed point of $\Psi_X$, we have $\beta_0\sqsubseteq \alpha_0$.

Assume $\beta_i\sqsubseteq \alpha_i$. Then we have:
\begin{align*}
\beta_{i+1}
&= J\gamma_{i+1}\odot g & (\text{by definition}) \\
&= \overline{F}\gamma_i\odot J\zeta \odot g & (\text{$\zeta$ is a mediating arrow}) \\
&= \overline{F}\gamma_i\odot \overline{F}g \odot c & (\text{$g$ is a homomorphism}) \\
&= \overline{F}\beta_i \odot c & (\text{$(X,(\beta_i)_{i\in\omega})$ is a cone}) \\
&\sqsubseteq \overline{F}\alpha_i \odot c & (\text{by inductive hypothesis and that $\overline{F}$ is monotone)}) \\
&= \alpha_{i+1} & (\text{by definition})\,.
\end{align*}

Hence $\beta_i\sqsubseteq \alpha_i$ holds for each $i \in\omega$.
This implies $J\gamma_i\odot g \sqsubseteq J\gamma_i\odot l$ for each $i\in\omega$.
As  $(Z,(J\gamma_i:Z\kto F^i1)_{i\in\omega})$ is a 2-limit (Assumption~\ref{asm:existenceWeaklyFinalCoalgWeak2Lim}), 
we have $g\sqsubseteq l$. Hence $l$ is the largest homomorphism from $c$ to $J\zeta$.
%
\begin{equation*}
\begin{xy}
(0,0)*+{1} = "F01",
(30,0)*+{\overline{F}1} = "F11",
(60,0)*+{\overline{F}^21} = "F21",
(90,0)*+{\overline{F}^31} = "F31",
%
(-20,20)*+{X} = "F0X",
(10,20)*+{\overline{F}X} = "F1X",
(40,20)*+{\overline{F}^2X} = "F2X",
(70,20)*+{\overline{F}^3X} = "F3X",
%
(-20,-15)*+{Z} = "F0Z",
(10,-15)*+{\overline{F}Z} = "F1Z",
(40,-15)*+{\overline{F}^2Z} = "F2Z",
(70,-15)*+{\overline{F}^3Z} = "F3Z",
%
(5,13)*+{Z} = "F0Z2",
(35,13)*+{\overline{F}Z} = "F1Z2",
(65,13)*+{\overline{F}^2Z} = "F2Z2",
(95,13)*+{\overline{F}^3Z} = "F3Z2",
%
(5,-7.5)*{=} = "",
(35,-7.5)*{=} = "",
(65,-7.5)*{=} = "",
%
(-2,10)*{\sqsupseteq} = "",
(28,10)*{\sqsupseteq} = "",
(58,10)*{\sqsupseteq} = "",
(88,10)*{\sqsupseteq} = "",
%
(-20,3)*{\sqsupseteq} = "",
%
(105,0)*{\cdots} = "",
(85,20)*{\cdots} = "",
(85,-15)*{\cdots} = "",
(102,13)*{\cdots} = "",
%
\kar _{J!_{F1}}"F11";"F01"
\kar _{JF!_{F1}}"F21";"F11"
\kar _{JF^2!_{F1}}"F31";"F21"
\kar _(.1){JF^2!_{F1}} (100,0);"F31"
%\ar @{.} (105,0);(100,0)
%
\kar ^{c}"F0X";"F1X"
\kar ^{\overline{F}c}"F1X";"F2X"
\kar ^{\overline{F}c}"F2X";"F3X"
\kar ^(.9){\overline{F}c} "F3X";(80,20)
%\ar @{.} (80,20);(85,20)
%
%\kar _{J!_X} "F0X";"F01"
%\kar _{JF!_X} "F1X";"F11"
%\kar _{JF^2!_X} "F2X";"F21"
%\kar _{JF^3!_X} "F3X";"F31"
%
\kar _{\alpha_0} "F0X";"F01"
\kar _{\overline{F}\alpha_0} "F1X";"F11"
\kar _{\overline{F}^2\alpha_0} "F2X";"F21"
\kar _{\overline{F}^3\alpha_0} "F3X";"F31"
%
\kar _{J\zeta}"F0Z";"F1Z"
\kar _{JF\zeta}"F1Z";"F2Z"
\kar _{JF^2\zeta}"F2Z";"F3Z"
\kar _(.9){JF^3\zeta} "F3Z";(80,-15)
%\ar @{.} (80,-15);(85,-15)
%
\kar ^(.2){J\zeta}"F0Z2";"F1Z2"
\kar ^(.2){JF\zeta}"F1Z2";"F2Z2"
\kar ^(.2){JF^2\zeta}"F2Z2";"F3Z2"
%\kar _(.9){JF^3\zeta} "F3Z2";(80,-15)
%\ar @{.} (80,-15);(85,-15)
%
\kar _{J!_Z} "F0Z";"F01"
\kar _{JF!_Z} "F1Z";"F11"
\kar _{JF^2!_Z} "F2Z";"F21"
\kar _{JF^3!_Z} "F3Z";"F31"
%
\kar ^{J!_Z} "F0Z2";"F01"
\kar ^{JF!_Z} "F1Z2";"F11"
\kar ^{JF^2!_Z} "F2Z2";"F21"
\kar ^{JF^3!_Z} "F3Z2";"F31"
%
\kar ^(.8){g} "F0X";"F0Z2"
\kar ^(.8){\overline{F}g} "F1X";"F1Z2"
\kar ^(.8){\overline{F}^2g} "F2X";"F2Z2"
\kar ^(.8){\overline{F}^3g} "F3X";"F3Z2"
%
\kar @/_4mm/@{-->} _{l} "F0X";"F0Z"
\kar  @/^4mm/^{g} "F0X";"F0Z"
\end{xy}
\label{eq:diagramexistenceWeaklyFinalCoalg1}
\end{equation*}
\qed
%
%From the monotonicity of $\Phi_X$, by the transfinite induction on ordinal $\mathfrak{a}$, we can prove $g=\Phi_X^{\mathfrak{a}}(g)\sqsubseteq \Phi_X^{\mathfrak{a}}(l_{\max})$ for each ordinal $\mathfrak{a}$ as long as $\Phi_X^{\mathfrak{a}}(l_{\max})$ is well-defined.
%Especially, by letting $\mathfrak{a}=\mathfrak{k}$, we have $g=\Phi_X^{\mathfrak{k}}(g)\sqsubseteq \Phi_X^{\mathfrak{k}}(l_{\max})=\trinf(c)$.
%Therefore $\trinf(c)$ is the largest fixed point of $\Phi_X$.
%This immediately implies that $\trinf(c)$ is the largest homomorphism from $c$ to $J\zeta$.
\end{proof:generalconstructionnotop}

%Here are some comments about the proof.

%Note that continuity of composition is not assumed.
%This is because continuity of composition fails in $\Kl(\pow)$.
%For $f:X\kto Y$ and $(g_i:Y\kto Z)_{i\in\omega}$ where $i\in \omega$ in $\Kl(\pow)$, 
%%\begin{equation*}
%$\bigl(\bigsqcap_{i\in\omega}g_i\bigr)\odot f(x)=\bigcup_{y\in f(x)}\bigcap_{i\in\omega} g_i(y)$
%%\end{equation*}
%while
%%\begin{equation*}
%$\bigsqcap_{i\in\omega}(g_i\odot f)(x)=\bigcap_{i\in\omega}\bigcup_{y\in f(x)} g_i(y)$
%\,.
%%\end{equation*}
%%
%There exists a a pair of nondeterministic automata where
%transfinite induction is indeed necessary (see Exmaple~\ref{example:needtransfinite}).

%The assumptions in Prop.~\ref{prop:generalconstructionnotop} is a sufficient but not a necessary condition for 
%a functor $F$ and a monad $T$ to satisfy infinite trace situation.
%Indeed, a functor $F=\{p,q\}\times (\place)$  and \emph{subdistribution monad} $\dist$ on $\Sets$ satisfy infinite trace situation.
%However, they do not satisfy assumption~\ref{asm:existenceWeaklyFinalCoalgWeak2Lim} of Prop.~\ref{prop:generalconstructionnotop}.
%but they satisfy infinite trace situation 
%(i.e.\ a final $F$-coalgebra in $\Sets$ lifts to a weakly final coalgebra in $\Kl(\dist)$.
%there exists a pair of a functor $F$ and a monad $T$ where
%the final coalgebra $\zeta$ lifts to weakly final coalgebra, but 
%Prop.~\ref{prop:generalconstructionnotop} is not applicable.
%\begin{myexample}\label{example:nogeneralconstruction}
%\end{myexample}

%We give an intuitive meaning of the cone $(X,(\alpha_i)_{i\in\omega})$ when $\mathbb{C}=\Meas$, $F=\Sigma\times(\place)$ and $T=\mathcal{G}$.
%Here, $\alpha_i:X\kto\Sigma^i$ where $\Sigma^i$ is obtained with discrete $\sigma$-algebra and $Z=\Sigma^{\omega}$ whose $\sigma$-algebra is generated from cylinder sets.
%For $w\in\Sigma^i$, $\alpha_i(x)(\{w\})$ gives the probability that 
%``in a system represented by a coalgebra $c$, execution is repeated infinitely and the first $i$ letters of the output are $w$''.
%Especially, $\alpha_0(x)(\{*\})$ is the probability that the represented system execute infinitely.

%In assumption \ref{asm:existenceWeaklyFinalCoalgTop}, we assumed that $J(!_X)$ is the largest element in $\Kl(T)(X,1)$.
%To prove that $J(\zeta)$ is a weakly final $\overline{F}$-coalgebra, 
%it suffices to assume that $\Kl(T)(X,1)$ has the largest element and its being $J(!_X)$ is not necessary. 
%However, to ensure that the resulting weakly final coalgebra is ``meaningful'' one, 
%it seems to be necessary to start with $J(!_X)$ in the construction of a cone $(X,(\alpha_i)_{i\in\omega})$.

%In the proof, we take the greatest lower bound twice: 
%the former is to construct a cone over a final sequence and the latter is to construct a homomorphism.
%The latter is necessary because there exists a pair of $(T,F)$-systems that 
%generates exactly the same cone over the final sequence but has different trace semantics (see Example~\ref{example:sameconediffhom}).


\begin{mylemma}\label{lem:satassumptionprob}
A polynomial functor $F$ 
%on $\Meas$ 
and 
%a sub-Giry monad 
$\giry$ on $\Meas$ satisfy Assumptions~\ref{asm:existenceWeaklyFinalCoalgFinalSeq}, 
\ref{asm:existenceWeaklyFinalCoalgDistLaw}, \ref{asm:existenceWeaklyFinalCoalgGFPcont}, 
 \ref{asm:existenceWeaklyFinalCoalgTop}, and \ref{asm:existenceWeaklyFinalCoalgWeak2Lim} in Prop.~\ref{prop:generalconstructionnotop}.
%with respect to the order in Def.~\ref{def:orderenrichment}. 
%Therefore, $\FXi$ and $\giry$ constitute an infinite trace situation.
%\qed
\end{mylemma}

%\newtheorem*{proof:satassumptionprob}{Proof of Lem.\ \ref{lem:satassumptionprob}}{\itshape}{\rmfamily}
%\begin{proof:satassumptionprob}
\begin{myproof}
It is known that Assumption~\ref{asm:existenceWeaklyFinalCoalgFinalSeq} is satisfied~\cite{schubert09terminalcoalgebras}.

It is also known that there exists distributive law $\lambda:F\giry\Rightarrow \giry F$ exists~\cite{cirstea10genericinfinite}.
Therefore Assumption~\ref{asm:existenceWeaklyFinalCoalgDistLaw} is satisfied.

Next, we prove that  Assumption~\ref{asm:existenceWeaklyFinalCoalgGFPcont} is satisfied.
Assume that a family of Kleisli arrows $(f_i:(X,\sigalg_X)\kto (Y,\sigalg_Y))_{i\in\omega}$ constitute a decreasing sequence.
We can define their greatest lower bound $\bigsqcap_{i\in\omega}f_i:X\kto Y$ in a pointwise manner:
namely, for $x\in X$ and $A\in\sigalg_Y$, 
\begin{equation*}
(\bigsqcap_{i\in\omega}f_i)(x)(A)=\lim_{i\to\infty}(f_i(x)(A))\,.
\end{equation*}
It is easy to see that polynomial $F$ preserves this pointwise greatest lower bound. 
It remains to prove that $\bigsqcap_{i\in\omega}f_i$ is a measurable function from $(X,\sigalg_X)$ to $\giry(Y,\sigalg_Y)$.
Moreover, we also have to prove the local continuity of composition. 
They can be proved in the similar manner to the proof of~\cite[Proposition~9]{brengosMP14behaviouralequivalences}.
Therefore Assumption~\ref{asm:existenceWeaklyFinalCoalgGFPcont} is satisfied.

It is easy to see that Assumption~\ref{asm:existenceWeaklyFinalCoalgTop} is satisfied.

We prove that Assumption~\ref{asm:existenceWeaklyFinalCoalgWeak2Lim} is satisfied.
If $F1$ is empty, then the limit $Z$ is also empty and Assumption~\ref{asm:existenceWeaklyFinalCoalgWeak2Lim} is satisfied.
Assume that $F1$ is not empty.
It is known that the sub-Giry monad $\giry$ preserves a limit over an $\omega^{\text{op}}$-sequence consisting of standard Borel spaces 
%(see~\cite{doob94measuretheory} for concrete definition) 
and surjective measurable functions~\cite{schubert09terminalcoalgebras}.
In our setting, for each $i\in\omega$, 
%as $\FSigma^i1$ is a countable set with a discrete $\sigma$-algebra, 
$F^i1$ is a standard Borel space because $1$ and all of $\Sigma_n$ are standard Borel spaces, and moreover, 
standard Borel space is preserved by countable coproducts and countable limits~\cite[12.B]{kechris95classicaldescriptive}.
Moreover, for each $i\in\omega$, $F^i!_{F1}$ is surjective.
Therefore the limit $(Z,(\gamma_i:Z\to F^i1)_{i\in\omega})$ over the final sequence 
$1\overset{!_{F1}}{\leftarrow}F1\overset{F!_{F1}}{\leftarrow}F^2 1\overset{F^2!_{F1}}{\leftarrow} \ldots$ is preserved by $\giry$.
This immediately implies that $J:\Meas\to\Kl(\giry)$ preserves the limit.
It is easy to see that the resulting limit is a $2$-limit.
%It is immediate from Lem.~\ref{lem:giry2lim} that assumption~\ref{asm:existenceWeaklyFinalCoalgWeak2Lim} is satisfied.
\qed
\end{myproof}
%\end{proof:satassumptionprob}

\auxproof{
\begin{myproposition}\label{prop:satassumptionnondet2}
A polynomial functor $F$
%For a ranked alphabet $\Sigma$, $\FSigma$ and 
%on $\Sets$ 
%where $\Sigma$ is a ranked alphabet 
and 
%a powerset monad 
$\pow$ satisfy 
Assumptions~\ref{asm:existenceWeaklyFinalCoalgFinalSeq}, 
\ref{asm:existenceWeaklyFinalCoalgDistLaw}, \ref{asm:existenceWeaklyFinalCoalgFMonotone},  \ref{asm:existenceWeaklyFinalCoalgGFPtransfinite},  \ref{asm:existenceWeaklyFinalCoalgTop}, and \ref{asm:existenceWeaklyFinalCoalgWeak2Lim} in Prop.~\ref{prop:generalconstructionnotop}.
%with respect to the order in Def.~\ref{def:orderenrichment}. 
%Therefore, $F$ and $\pow$ constitute an infinite trace situation.
\end{myproposition}

This proposition can be proved using the following two lemmas.

\begin{mylemma}[{\cite[Section 2.8]{borceux94handbookcategorical}}]\label{lem:limitinSets}
Let $(Z,(\gamma_i:Z\to A_i)_{i\in\omega})$ be a limit over a sequence 
%$1\overset{!_{F1}}{\leftarrow} F1\overset{F!_{F1}}{\leftarrow} F^21\overset{F^2!_{F1}}{\leftarrow}\ldots$ 
$A_0\overset{f_0}{\leftarrow} A_1\overset{f_1}{\leftarrow} A_2\overset{f_2}{\leftarrow}\ldots$ 
in $\Sets$.
Then the limit $(Z,(\gamma_i:Z\to A_i)_{i\in\omega})$ over the sequence is given as follows:
\begin{equation*}
%Z &= \{(a_j)_{j\in\omega}\,\mid\, \forall j\in\omega.\, a_j\in F^j1, F^j!_{F1}(a_{j+1})=a_j\} \,,\\
Z = \{(a_j)_{j\in\omega}\,\mid\, \forall j\in\omega.\, a_j\in A_j, f_j(a_{j+1})=a_j\} \,,\;\;\;\text{and}\;\;\;
\gamma_i((a_j)_{j\in\omega})=a_i\,.
\hfill\qed
\end{equation*}
\end{mylemma}

%We prove that this limit is lifted to a weak 2-limit in $\Kl(\pow)$ by
%giving an explicit definition of the largest mediating arrow for every cone in $\Kl(\pow)$.

\begin{mylemma}\label{lem:powweak2lim}
Let 
%$(X,(\pi_i:X\kto F^i1)_{i\in\omega})$ 
$(X,(\pi_i:X\kto A_i)_{i\in\omega})$ 
be a cone over the sequence 
%$1\overset{J!_{F1}}{\kleftarrow} F1\overset{JF!_{F1}}{\kleftarrow} F^21\overset{JF^2!_{F1}}{\kleftarrow}\ldots$
$A_0\overset{Jf_0}{\kleftarrow} A_1\overset{Jf_1}{\kleftarrow} A_2\overset{Jf_2}{\kleftarrow}\ldots$
in $\Kl(\pow)$.
We define $(Z,(\gamma_i)_{i\in\omega})$ as in Lem.~\ref{lem:limitinSets}.
We define a Kleisli arrow $l_{\max}:X\kto Z$ as follows.
\begin{equation*}
%l_{\max}(x)=\{(a_j)_{j\in\omega}\in Z\mid \forall j\in\omega.\, a_j\in \pi_j(x), F^j!_{F1}(a_{j+1})=a_j\}
l_{\max}(x)=\{(a_j)_{j\in\omega}\in Z\mid \forall j\in\omega.\, a_j\in \pi_j(x), f_j(a_{j+1})=a_j\}
\end{equation*}
Then
\begin{enumerate}
\item For each $i\in\omega$, $\pi_i=J\gamma_i\odot l_{\max}$.
\label{item:powweak2lim1}
\item If $l:X\kto Z$ satisfies $J\gamma_i\odot l\sqsubseteq J\gamma_i\odot l_{\max}$ for each $i\in\omega$, then $l\sqsubseteq l_{\max}$ holds.
\qed
\label{item:powweak2lim2}
\end{enumerate} 
\end{mylemma}

\begin{myproof}
%If $F1=\emptyset$, then $Z=\emptyset$ and it is easy to see that the statement is satisfied.
%
%Assume $F1\neq \emptyset$. Then, $F^i!_{F1}$ is surjective for each $i\in\omega$.
%We first show that $l_{\max}$ is a mediating arrow. 
To prove~\ref{item:powweak2lim1}, it suffices to prove that for all $x\in X$, $i\in\omega$, and $a\in A_i$,
$a\in (J\gamma_i\odot l_{\max})(x)$ iff $a\in\pi_i(x)$.

Assume $a\in (J\gamma_i\odot l_{\max})(x)$.
Then, by definition, there exists $(a_j)_{j\in\omega}\in Z$ such that 
$a_j\in\pi_j(x)$ and $f_j(a_{j+1})=a_j$ holds for all $j\in\omega$, and $a_i=a$.
Therefore, $a=a_i\in\pi_i(x)$

Conversely, assume that $a\in\pi_i(x)$.
As $(X,(\pi_j)_{j\in\omega})$ constitute a cone over a sequence $A_0\overset{Jf_0}{\kleftarrow} A_1\overset{Jf_1}{\kleftarrow} A_2\overset{Jf_2}{\kleftarrow}\ldots$,
for each $j<i$, we have $(f_{j}\circ\ldots\circ f_{i-1})(a)\in \pi_j(x)$.
Moreover, for each $j>i$, there exists $a_j\in\pi_j(x)$ such that $(f_{i}\circ\ldots\circ f_{j-1})(a_j)=a$.
Therefore by the axiom of dependent choice, there exists $(a_j)_{j\in\omega}$ such that 
$a_j\in\pi_j(x)$ and $f_j(a_{j+1})=a_j$ for all $j\in\omega$, and $a_i=a$.
Therefore, $a\in (J\gamma_i\odot l_{\max})(x)$.

Hence we have $\pi_i=J\gamma_i\odot l_{\max}$ for each $i\in\omega$.
To conclude the proof, we show that $l_{\max}$ is the largest mediating arrow.
Let $l:X\kto Z$ be a Kleisli arrow such that $J\gamma_i\odot l\sqsubseteq J\gamma_i\odot l_{\max}$ for each $i\in\omega$.
Assume that $x\in X$ and $(a_j)_{j\in\omega}\in l(x)$.
As $(a_j)_{j\in\omega}\in Z$, $f_i(a_{i+1})=a_i$ for each $i\in\omega$.
Moreover, from $J\gamma_i\odot l\sqsubseteq J\gamma_i\odot l_{\max}$, 
we have $a_i\in J\gamma_j\odot l_{\max}(x)=\pi_i(x)$ for each $i\in\omega$.
Therefore by definition of $l_{\max}$, $(a_j)_{j\in\omega}\in l_{\max}(x)$.
Hence we have $l\sqsubseteq l_{\max}$.
\qed
\end{myproof}



%\begin{myproof}
\newtheorem*{proof:satassumptionnondet2}{Proof of Prop.\ \ref{prop:satassumptionnondet2}}{\itshape}{\rmfamily}
\begin{proof:satassumptionnondet2}
It is already shown in Lem.~\ref{lem:satassumptionnondet} that Assumptions 
\ref{asm:existenceWeaklyFinalCoalgFinalSeq}, \ref{asm:existenceWeaklyFinalCoalgDistLaw}, 
\ref{asm:existenceWeaklyFinalCoalgFMonotone}, and \ref{asm:existenceWeaklyFinalCoalgGFPtransfinite} are satisfied.
It is easy to see that Assumption~\ref{asm:existenceWeaklyFinalCoalgTop} is satisfied.
It is immediate from Lem.~\ref{lem:powweak2lim} that Assumption~\ref{asm:existenceWeaklyFinalCoalgWeak2Lim} is satisfied.
\qed
\end{proof:satassumptionnondet2}
%\end{myproof}
}













%Let $F$ be a polynomial functor on $\Sets$ and 
%Let $(Z,(\gamma_i:Z\to A_i)_{i\in\omega})$ be a limit over a sequence 
%%$1\overset{!_{F1}}{\leftarrow} F1\overset{F!_{F1}}{\leftarrow} F^21\overset{F^2!_{F1}}{\leftarrow}\ldots$ 
%$A_0\overset{f_0}{\leftarrow} A_1\overset{f_1}{\leftarrow} A_2\overset{f_2}{\leftarrow}\ldots$ 
%in $\Sets$.
%It is well known that this limit can be given by
%a set of ``threads'' and projections (see~\cite[Section 2.8]{borceux94handbookcategorical}, for example).
%Namely, 
%\begin{equation*}
%%Z &= \{(a_j)_{j\in\omega}\,\mid\, \forall j\in\omega.\, a_j\in F^j1, F^j!_{F1}(a_{j+1})=a_j\} \,,\\
%Z = \{(a_j)_{j\in\omega}\,\mid\, \forall j\in\omega.\, a_j\in A_j, f_j(a_{j+1})=a_j\} \,,\;\;\;\text{and}\;\;\;
%\gamma_i((a_j)_{j\in\omega})=a_i\,.
%\end{equation*}



%thm:soundnessFwdgiry
\newtheorem*{proof:soundnessFwdgiry}{Proof of Thm.\ \ref{thm:soundnessFwdgiry}}{\itshape}{\rmfamily}
\begin{proof:soundnessFwdgiry}
%\begin{myproof}
In the similar manner to the proof of Lem.~\ref{lem:satassumptionprob},
we can show that $F$ and $\giry$ satisfy the assumptions in Lem.~\ref{lem:soundnessFwd}.
Therefore immediate from Lem.~\ref{lem:soundnessFwd}.
\qed
%\end{myproof}
\end{proof:soundnessFwdgiry}

%\newtheorem*{proof:satassumptionsfwdgiry}{Proof of Prop.\ \ref{prop:satassumptionsfwdgiry}}{\itshape}{\rmfamily}
%\begin{proof:satassumptionsfwdgiry}
%It can be proved in the similar manner to Lem.~\ref{lem:satassumptionprob}.
%\qed
%\end{proof:satassumptionsfwdgiry}

%\begin{mylemma}\label{lem:soundnessRBwd}
%Let $F$ be an endofunctor and $T$ be a monad on $\mathbb{C}$ 
%that satisfy the assumptions of Prop.~\ref{prop:generalconstructionnotop}  with respect to an order $\sqsubseteq$,
%and constitute an infinite trace situation.
%%We assume that $T$ and $F$ 
%Let  $\mathcal{X}=(X,s,c)$ and $\mathcal{Y}=(Y,t,d)$ be  $(T,F)$-systems.
%If $b:X\kto Y$ is a restricted backward simulation from $\mathcal{X}$ to $\mathcal{Y}$, then $\trinf(c)\sqsubseteq \trinf(d)\odot b$.
%\end{mylemma}



\newtheorem*{proof:soundnessRBwdnotop}{Proof of Lem.\ \ref{lem:soundnessRBwdnotop}}{\itshape}{\rmfamily}
\begin{proof:soundnessRBwdnotop}
We prove $\trinf(c)\sqsubseteq \trinf(d)\odot b$ along the construction of $\trinf(c)$ and $\trinf(d)$ 
in the proof of Prop.~\ref{prop:generalconstructionnotop}.

For each $i\in\omega$, we define $\Psi^{i}_X(J!_X):X\kto 1$ and $\Psi^{i}_Y(J!_Y):Y\kto 1$ 
as in the proof of Prop.~\ref{prop:generalconstructionnotop}.
We can prove that $\Psi^{i}_X(J!_X) \sqsubseteq \Psi^{i}_Y(J!_Y)\odot b$ 
for all $i\in\omega$ by the induction on $i$ as follows.
%It is proved by the  induction on $i$.

For $i=0$, by  Assumption~\ref{item:restrictedbwdsim2notop}, 
we have $\Psi^0_X(J!_X)=\Psi^0_Y(J!_Y)\odot b$.

Let $i>0$ and
assume that $\Psi^{i-1}_X(J!_X)\sqsubseteq \Psi^{i-1}_Y(J!_Y)\odot b$.
Then 
\begin{align*}
\Psi^{i}_X(J!_X)
&= J!_{F1}\odot \overline{F}(\Psi^{i-1}_X(J!_X)) \odot d & (\text{by definition of $\Psi_X$ }) \\
&\sqsubseteq J!_{F1}\odot \overline{F}(\Psi^{i-1}_Y(J!_Y))\odot \overline{F}b \odot c  & (\text{by induction hypothesis}) \\
&\sqsubseteq J!_{F1}\odot \overline{F}(\Psi^{i-1}_Y(J!_Y))\odot d\odot b & (\text{$b$ is a bwd.\ simulation}) \\
&=  \Psi^{i}_Y(J!_Y)\odot b & (\text{by definition of $\Psi_Y$})\,.
\end{align*}

%Let $\mathfrak{a}$ be a limit ordinal and assume that $\Psi^{\mathfrak{i}}_X(J!_X)\sqsubseteq \Psi^{\mathfrak{i}}_Y(J!_Y)\odot b$ for all $\mathfrak{i}<\mathfrak{a}$.
%Then, 
%\begin{align*}
%\Psi_X^{\mathfrak{a}}(J!_X)
%&= \bigsqcap_{\mathfrak{i}<\mathfrak{a}}\Psi_X^{\mathfrak{i}}(J!_X) & (\text{by definition}) \\
%&\sqsubseteq \bigsqcap_{\mathfrak{i}<\mathfrak{a}}\left(\Psi_Y^{\mathfrak{i}}(J!_Y) \odot b\right) & (\text{by induction hypothesis}) \\
%&=\left(\bigsqcap_{\mathfrak{i}<\mathfrak{a}}\Psi_Y^{\mathfrak{i}}(J!_Y)\right) \odot b & (\text{by Assumption~\ref{item:restrictedbwdsim3} of Def.~\ref{def:restrictedbwdsimnotop}}) \\
%&= \Psi^{\mathfrak{a}}_Y(J!_Y)\odot b & (\text{by definition})\,.
%\end{align*} 

Therefore we have $\Psi^{i}_X(J!_X)\sqsubseteq \Psi^{i}_Y(J!_Y)\odot b$ for all $i\in\omega$.
%
\begin{equation*}
\begin{xy}
(0,0)*+{1} = "F01",
(20,0)*+{\overline{F}1} = "F11",
(40,0)*+{\overline{F}^21} = "F21",
(60,0)*+{\overline{F}^31} = "F31",
%
(0,15)*+{Y} = "F0Y",
(20,15)*+{\overline{F}Y} = "F1Y",
(40,15)*+{\overline{F}^2Y} = "F2Y",
(60,15)*+{\overline{F}^3Y} = "F3Y",
%
(-15,30)*+{X} = "F0X",
(5,30)*+{\overline{F}X} = "F1X",
(25,30)*+{\overline{F}^2X} = "F2X",
(45,30)*+{\overline{F}^3X} = "F3X",
%
(12,7.5)*{\sqsupseteq} = "",
(32,7.5)*{\sqsupseteq} = "",
(52,7.5)*{\sqsupseteq} = "",
%
(3,22.5)*{\sqsupseteq} = "",
(23,22.5)*{\sqsupseteq} = "",
(43,22.5)*{\sqsupseteq} = "",
%
(-5,14)*{=} = "",
(13,18)*{{\scriptstyle=}} = "",
(33,18)*{{\scriptstyle=}} = "",
(53,18)*{{\scriptstyle=}} = "",
%
(-5,-15)*+{Z} = "F0Z",
(15,-15)*+{\overline{F}Z} = "F1Z",
(35,-15)*+{\overline{F}^2Z} = "F2Z",
(55,-15)*+{\overline{F}^3Z} = "F3Z",
%
(8,-7.5)*{=} = "",
(28,-7.5)*{=} = "",
(48,-7.5)*{=} = "",
%
(70,-15)*{\cdots} = "",
(75,0)*{\cdots} = "",
(75,15)*{\cdots} = "",
(60,30)*{\cdots} = "",
%
\kar _{J!_{F1}}"F11";"F01"
\kar _{JF!_{F1}}"F21";"F11"
\kar _{JF^2!_{F1}}"F31";"F21"
\kar _(.1){JF^2!_{F1}} (70,0);"F31"
%\ar @{.} (75,0);(70,0)
%
\kar _(.3){d}"F0Y";"F1Y"
\kar _(.3){\overline{F}d}"F1Y";"F2Y"
\kar _(.3){\overline{F}^2d}"F2Y";"F3Y"
\kar ^(.9){\overline{F}^3d} "F3Y";(70,15)
%\ar @{.} (70,15);(75,15)
%
\kar ^{c}"F0X";"F1X"
\kar ^{\overline{F}c}"F1X";"F2X"
\kar ^{\overline{F}^2c}"F2X";"F3X"
\kar ^(.9){\overline{F}^3c} "F3X";(55,30)
%\ar @{.} (55,30);(60,30)
%
\kar ^{J!_Y} "F0Y";"F01"
\kar ^{JF!_Y} "F1Y";"F11"
\kar ^{JF^2!_Y} "F2Y";"F21"
\kar ^{JF^3!_Y} "F3Y";"F31"
%
\kar ^{b} "F0X";"F0Y"
\kar ^{\overline{F}b} "F1X";"F1Y"
\kar ^{\overline{F}^2b} "F2X";"F2Y"
\kar ^{\overline{F}^3b} "F3X";"F3Y"
%
\kar @/_1mm/ _{J!_X} "F0X";"F01"
\kar @/_1mm/ _(.35){JF!_X} "F1X";"F11"
\kar @/_1mm/ _(.35){JF^2!_X} "F2X";"F21"
\kar @/_1mm/ _(.35){JF^3!_X} "F3X";"F31"
%
\kar _{J\zeta}"F0Z";"F1Z"
\kar _{JF\zeta}"F1Z";"F2Z"
\kar _{JF^2\zeta}"F2Z";"F3Z"
\kar _(.9){JF^3\zeta} "F3Z";(65,-15)
%\ar @{.} (65,-15);(70,-15)
%
\kar _{J!_Z} "F0Z";"F01"
\kar _{JF!_Z} "F1Z";"F11"
\kar _{JF^2!_Z} "F2Z";"F21"
\kar _{JF^3!_Z} "F3Z";"F31"
\end{xy}
%\label{eq:soundnessbwdsim1}
\end{equation*}
%
%Using $c$ and $d$, % respectively.
We define cones $(X,(\alpha^X_i:X\kto \overline{F}^i1)_{i\in\omega})$ and $(Y,(\alpha^Y_i:Y\kto \overline{F}^i1)_{i\in\omega})$
over the sequence
$1\overset{J!_{F1}}{\kleftarrow} \overline{F}1\overset{JF!_{F1}}{\kleftarrow} \overline{F}^2 1\overset{JF^2!_{F1}}{\kleftarrow} \ldots$
%  $1\kleftarrow \overline{F}1\kleftarrow \overline{F}^2 1\kleftarrow \ldots$ 
as equation (\ref{eq:existenceWeaklyFinalCoalgdefcone}) in the proof of Prop.~\ref{prop:generalconstructionnotop}.


As is shown in the proof of Prop.~\ref{prop:generalconstructionnotop}, 
 $\trinf(c):X\kto Z$ is the unique mediating arrow from a cone $(X,(\alpha^X_i:X\kto \overline{F}^i1)_{i\in\omega})$ 
to a  2-limit $(Z,(J\gamma_i:Z\kto \overline{F}^i1)_{i\in\omega})$.
Moreover, 
 $\trinf(d):Y\kto Z$ is the unique mediating arrow from a cone $(Y,(\alpha^Y_i:Y\kto \overline{F}^i1)_{i\in\omega})$ 
to $(Z,(J\gamma_i:Z\kto \overline{F}^i1)_{i\in\omega})$ and therefore 
$\trinf(d)\odot b:X\kto Z$ is the unique mediating arrow from a cone $(X,(\alpha^Y_i\odot b:X\kto \overline{F}^i1)_{i\in\omega})$ 
to $(Z,(J\gamma_i:Z\kto \overline{F}^i1)_{i\in\omega})$.
%, and therefore 
%$\trinf(d)\odot b$ is a mediating arrow.
% from a cone $(X,(\alpha^Y_i\odot b:X\kto \overline{F}^i1)_{i\in\omega})$
%to $(Z,(J\gamma_i:Z\kto \overline{F}^i1)_{i\in\omega})$.
%Hence by the uniqueness of the mediating arrow, we have $\trinf(c)=\trinf(d)\odot b$.
%By Assumption~\ref{item:restrictedbwdsim4}, $l^Y_{\max}\odot b:X\kto Z$ is the largest mediating arrow from a cone $(X,(\alpha^Y_i\odot b:X\kto \overline{F}^i1)_{i\in\omega})$ 
%to $(Z,(J\gamma_i:Z\kto \overline{F}^i1)_{i\in\omega})$.

For all $i\in\omega$, we can prove $\alpha^X_i\sqsubseteq\alpha^Y_i\odot b$ by the induction on $i$ as follows:

For $i=0$, 
\begin{align*}
\alpha^X_0&=\bigsqcap_{i\in\omega}\Psi^{i}_X(J!_X) & (\text{by definition}) \\
&\sqsubseteq \bigsqcap_{i\in\omega}(\Psi^{i}_Y(J!_Y)\odot b) & (\text{by the above result}) \\
&=\bigl(\bigsqcap_{i\in\omega}\Psi^{i}_Y(J!_Y)\bigr)\odot b & (\text{by Assumption~\ref{asm:existenceWeaklyFinalCoalgGFPcont} in Prop.~\ref{prop:generalconstructionnotop}}) \\
&= \alpha^Y_0\odot b & (\text{by definition})\,.
\end{align*}
%$\alpha^X_0=\Psi^{\omega}_X(J!_X)\sqsubseteq \Psi^{\omega}_Y(J!_Y)\odot b=\alpha^Y_0\odot b$

Let $i>0$ and assume that $\alpha^X_{i-1}\sqsubseteq\alpha^Y_{i-1}\odot b$. Then
\begin{align*}
\alpha^X_i 
&= \overline{F}\alpha^X_{i-1}\odot c & (\text{by definition}) \\
&\sqsubseteq \overline{F}\alpha^Y_{i-1}\odot\overline{F}b\odot c & (\text{by inductive hypothesis and the monotonicity of $\overline{F}$}) \\
&\sqsubseteq \overline{F}\alpha^Y_{i-1}\odot d  \odot b & (\text{$b$ is a backward simulation}) \\
&= \alpha^Y_i \odot b & (\text{by definition}).
\end{align*}

%Moreover, by the monotonicity of $\overline{F}$'s action on arrows and composition in $\Kl(T)$, 
%we can inductively prove that $\alpha^X_i\sqsubseteq\alpha^Y_i\odot b$ for all $i\in\omega$.
Therefore we have $\alpha^X_i\sqsubseteq\alpha^Y_i\odot b$ for all $i\in\omega$.
As $(Z,(J\gamma_i:Z\kto \overline{F}^i1)_{i\in\omega})$ is a  2-limit, this implies $\trinf(c)\sqsubseteq \trinf(d)\odot b$.

The last claim follows from $b$'s condition on initial states.
%
\begin{equation*}
\begin{xy}
(0,0)*+{1} = "F01",
(20,0)*+{\overline{F}1} = "F11",
(40,0)*+{\overline{F}^21} = "F21",
(60,0)*+{\overline{F}^31} = "F31",
%
(0,15)*+{Y} = "F0Y",
(20,15)*+{\overline{F}Y} = "F1Y",
(40,15)*+{\overline{F}^2Y} = "F2Y",
(60,15)*+{\overline{F}^3Y} = "F3Y",
%
(-15,30)*+{X} = "F0X",
(5,30)*+{\overline{F}X} = "F1X",
(25,30)*+{\overline{F}^2X} = "F2X",
(45,30)*+{\overline{F}^3X} = "F3X",
%
(12,7.5)*{=} = "",
(32,7.5)*{=} = "",
(52,7.5)*{=} = "",
%
(3,22.5)*{\sqsupseteq} = "",
(23,22.5)*{\sqsupseteq} = "",
(43,22.5)*{\sqsupseteq} = "",
%
(-5,14)*{\sqsubseteq} = "",
(13,18)*{{\scriptstyle\sqsubseteq}} = "",
(33,18)*{{\scriptstyle\sqsubseteq}} = "",
(53,18)*{{\scriptstyle\sqsubseteq}} = "",
%
(-7,-7)*{{\scriptstyle\sqsubseteq}} = "",
%
(-5,-15)*+{Z} = "F0Z",
(15,-15)*+{\overline{F}Z} = "F1Z",
(35,-15)*+{\overline{F}^2Z} = "F2Z",
(55,-15)*+{\overline{F}^3Z} = "F3Z",
%
(8,-7.5)*{=} = "",
(28,-7.5)*{=} = "",
(48,-7.5)*{=} = "",
%
(70,-15)*{\cdots} = "",
(75,0)*{\cdots} = "",
(75,15)*{\cdots} = "",
(60,30)*{\cdots} = "",
%
\kar _{J!_{F1}}"F11";"F01"
\kar _{JF!_{F1}}"F21";"F11"
\kar _{JF^2!_{F1}}"F31";"F21"
\kar _(.1){JF^2!_{F1}} (70,0);"F31"
%\ar @{.} (75,0);(70,0)
%
\kar _(.3){d}"F0Y";"F1Y"
\kar _(.3){\overline{F}d}"F1Y";"F2Y"
\kar _(.3){\overline{F}^2d}"F2Y";"F3Y"
\kar ^(.9){\overline{F}^3d} "F3Y";(70,15)
%\ar @{.} (70,15);(75,15)
%
\kar ^{c}"F0X";"F1X"
\kar ^{\overline{F}c}"F1X";"F2X"
\kar ^{\overline{F}^2c}"F2X";"F3X"
\kar ^(.9){\overline{F}^3c} "F3X";(55,30)
%\ar @{.} (55,30);(60,30)
%
\kar ^{\alpha^Y_0} "F0Y";"F01"
\kar ^{\overline{F}\alpha^Y_0} "F1Y";"F11"
\kar ^{\overline{F}^2\alpha^Y_0} "F2Y";"F21"
\kar ^{\overline{F}^3\alpha^Y_0} "F3Y";"F31"
%
\kar ^{b} "F0X";"F0Y"
\kar ^{\overline{F}b} "F1X";"F1Y"
\kar ^{\overline{F}^2b} "F2X";"F2Y"
\kar ^{\overline{F}^3b} "F3X";"F3Y"
%
\kar @/_1mm/ _{\alpha^X_0} "F0X";"F01"
\kar @/_1mm/ _(.35){\overline{F}\alpha^X_0} "F1X";"F11"
\kar @/_1mm/ _(.35){\overline{F}^2\alpha^X_0} "F2X";"F21"
\kar @/_1mm/ _(.35){\overline{F}^3\alpha^X_0} "F3X";"F31"
%
\kar _{J\zeta}"F0Z";"F1Z"
\kar _{JF\zeta}"F1Z";"F2Z"
\kar _{JF^2\zeta}"F2Z";"F3Z"
\kar _(.9){JF^3\zeta} "F3Z";(65,-15)
%\ar @{.} (65,-15);(70,-15)
%
\kar _{J!_Z} "F0Z";"F01"
\kar _{JF!_Z} "F1Z";"F11"
\kar _{JF^2!_Z} "F2Z";"F21"
\kar _{JF^3!_Z} "F3Z";"F31"
%
\kar @/_5mm/ @{-->} _{\trinf(c)} "F0X";"F0Z"
\kar @/_2mm/ @{-->} _{\trinf(d)} "F0Y";"F0Z"
\end{xy}
%\label{eq:soundnessbwdsim2}
\end{equation*}
\qed
%\end{myproof}
\end{proof:soundnessRBwdnotop}

%\newtheorem*{proof:soundnessRBwd}{Proof of Thm.\ \ref{thm:soundnessRBwd}}{\itshape}{\rmfamily}
%\begin{proof:soundnessRBwd}
%Immediate from Lem.~\ref{lem:soundnessRBwd}.
%\qed
%\end{proof:soundnessRBwd}


\begin{mylemma}\label{lem:whenrestrictedgiry}
In Def.~\ref{def:restrictedbwdsimnotop}, if $T=\giry$ and $F$ is a polynomial functor, 
Assumption~\ref{item:restrictedbwdsim2notop} is satisfied if $b(x)(Y)=1$ for each $x\in X$.
\end{mylemma}

%\newtheorem*{proof:whenrestrictedgiry}{Proof of Prop.\ \ref{prop:whenrestrictedgiry}}{\itshape}{\rmfamily}
%\begin{proof:whenrestrictedgiry}
\begin{myproof}
%In the below, 
%Let $*$ be the unique element of the final object $1$.
%
%\noindent{\bf \ref{item:whenrestrictedpow4}.}
%Let $T=\pow$ and 
%$F$ be a polynomial functor,
%%$F=\FSigma$, 
%and assume that $b(x)$ is finite for each $x\in X$.
%We prove that Assumption~\ref{item:restrictedbwdsim4} is satisfied.
%We define the limit $(Z,(\gamma_i:Z\to F^i1)_{i\in\omega})$  as in Lem.~\ref{lem:limitinSets}.
%We define $l^X_{\max}:X\kto Z$ and $l^Y_{\max}:Y\kto Z$ by
%\begin{align*}
%&l^X_{\max}(x)=\{(a_j)_{j\in\omega}\in Z\mid \forall j\in\omega.\, a_j\in \delta_j\odot b(x), F^j!_{F1}(a_{j+1})=a_j\},
%%\end{equation*}
%\;\;\;\text{and} \\
%%\begin{equation*}
%&l^Y_{\max}(y)=\{(a'_j)_{j\in\omega}\in Z\mid \forall j\in\omega.\, a'_j\in \delta_j(y), F^j!_{F1}(a'_{j+1})=a'_j\}\,.
%\end{align*}
%By Lem.~\ref{lem:powweak2lim}, it suffices to prove $l^X_{\max}(x)\subseteq l^Y_{\max}\odot b(x)$ for all $x\in X$.
%
%%We define
%%$l^X_{\max}(s)\subseteq l^Y_{\max}\odot b(x)$  for all $x\in X$
%%where $l^X_{\max}:X\kto Z$ is defined by
%%\begin{equation*}
%%l^X_{\max}(x)=\{(a_j)_{j\in\omega}\in Z\mid \forall j\in\omega.\, a_j\in \delta_j\odot b(x), F^j!_{F1}(a_{j+1})=a_j\}
%%\end{equation*}
%%while $l^Y_{\max}:Y\kto Z$ is defined by
%%\begin{equation*}
%%l^Y_{\max}(y)=\{(a'_j)_{j\in\omega}\in Z\mid \forall j\in\omega.\, a'_j\in \delta_j(y), F^j!_{F1}(a'_{j+1})=a'_j\}\,.
%%\end{equation*}
%%
%Let $x\in X$ and assume that $(a_j)_{j\in\omega}\in l^X_{\max}(x)$. 
%Then for each $i\in\omega$, there exists $y_i\in b(x)$ such that $a_i \in \delta_i(y_i)$.
%As it is assumed that $b(x)$ is finite, by the infinite pigeonhole principle,
%there exists $y\in b(x)$ such that $a_i\in \delta_j(y)$ for infinitely many $j\in\omega$.
%Then for an arbitrary $i\in\omega$, there exists $j\geq i$ such that $a_j\in \delta_j(y)$.
%As $(a_j)_{j\in\omega}\in Z$, we have 
%$a_i=F^i!_{F1}\circ F^{i+1}!_{F1}\circ\cdots\circ F^{j-1}!_{F1}(a_j)\in \delta_i(y)$.
%Therefore $(a_i)_{i\in\omega}\in l^Y_{\max}(y)=\{(a'_j)_{j\in\omega}\in Z\mid \forall j\in\omega.\, a'_j\in \delta_j(y), F^j!_{F1}(a'_{j+1})=a'_j\}$.


%%\noindent{\bf \ref{item:whenrestrictedgiry2}.}
%Next, let $T=\giry$ and $F=\FXi$.
We assume that $b(x)(Y)=1$ for each $x\in X$. %prove that Assumption~\ref{item:restrictedbwdsim2notop}.
By the definition of multiplication of the sub-Giry monad (see Def.~\ref{def:variousmonads}), for  $x\in X$,
\begin{equation*}
J!_Y\odot b(x)(1)=b(x)\bigl(!_Y^{-1}(1)\bigr)=b(x)(Y)=1=J!_X(x)(1)\,.
\end{equation*}
Therefore Assumption~\ref{item:restrictedbwdsim2notop} is satisfied.
%
%\noindent{\bf \ref{item:whenrestrictedgiry3}.}
%This holds 
%Assumption~\ref{item:restrictedbwdsim3}a is always satisfied
%because 
%$\FXi$ and $\giry$ satisfy Assumption~\ref{asm:existenceWeaklyFinalCoalgGFPcont} in Prop.~\ref{prop:generalconstructionnotop}.
%%in general, composition in $\Kl(\giry)$ is preserves the greatest lower bound of 
%%decreasing sequence of Kleisli arrows.
%%It can be proved  in the similar manner to the proof of~\cite[Proposition~9]{brengosMP14behaviouralequivalences}.
%%Therefore, Assumption~\ref{item:whenrestrictedgiry3} is always satisfied.
%%
%%\noindent{\bf \ref{item:whenrestrictedgiry4}.}
%Moreover, Assumption~\ref{item:restrictedbwdsim4} is satisfied
%%This holds 
%because for $T=\giry$ and $F=\FXi$, $(Z,(J\pi_i:Z\kto \overline{F}^i 1))$ is a (strong) 2-limit (see the proof of Lem.~\ref{lem:satassumptionprob}).
\qed
\end{myproof}
%\end{proof:whenrestrictedgiry}

\auxproof{
\subsection{Construction of the Largest Homomorphism for $(\pow,F)$-systems using Prop.~\ref{prop:generalconstructionnotop}}
In \S\ref{subsec:constructlargesthompow}, 
we have given a general construction of the largest homomorphism (Prop.~\ref{prop:generalconstructiontop}) 
that is applicable to a functor $\FSigma$ on $\Sets$ and a powerset monad $\pow$.
The largest homomorphism for them can be also constructed using Prop.~\ref{prop:generalconstructionnotop}.

In \S\ref{subsec:constructlargesthompow}, 
for a $\overline{F}$-coalgebra $c:X\kto\overline{F}X$ in $\Kl(\pow)$ where $F$ is a polynomial functor,
we have presented a method to construct the largest homomorphism $\trinf(c):X\kto Z$ 
from $c$ to $j\zeta:Z\kto\overline{F}Z$---the lifted final $F$-coalgebra.

The largest homomorphism $\trinf(c)$ can be obtained also by using the construction in Prop.~\ref{prop:generalconstructionnotop}.

}







%\subsection{Proof of Prop.~\ref{prop:generalconstructionnotop}}

\subsection{Lift Monad and a Tree Automaton with Exception}\label{subsec:liftmonad}
\subsubsection{Definitions}
In this section, we give a definition of the lift monad and the order on the homsets of the Kleisli category.

\begin{mydefinition}[lift monad]\label{def:liftmonad}
A \emph{lift monad} is a monad $(\lift,\eta^{\lift},\mu^{\lift})$ on $\Sets$  such that
\begin{itemize}
\item $\lift X=\{\bot\}+X$,
\item $\lift f(y)=\begin{cases} f(y) & (y\neq \bot) \\ \bot & (\text{otherwise}),\end{cases} $
\item $\eta^{\lift}_{X}(x)=x$, and 
\item $\mu^{\lift}_{X}(z)=z$.
\end{itemize}
\end{mydefinition}

\begin{mydefinition}[order enrichment of $\Kl(\lift)$]\label{def:orderenrichmentlift}
We define an order on $\Kl(\lift)(X,Y)$ by 
$f\sqsubseteq g \;\defarrow\; \forall x\in X.\, f(x)=\bot \,\text{or}\, f(x)=g(x)$.
\end{mydefinition}

\subsubsection{Construction of Infinite Traces}
In this section, we show that 
a polynomial functor $F$ on $\Sets$ 
%$\FSigma$ on $\Sets$ 
and the lift monad $\lift$ satisfy the assumptions in Prop.~\ref{prop:generalconstructionnotop},
hence constitute an infinite trace situation.

\begin{myproposition}\label{prop:satassumptionexception}
A polynomial functor $F$ 
% functor $F_{\Sigma}$ 
on $\Sets$ and a lift monad $\lift$ satisfy Assumptions~\ref{asm:existenceWeaklyFinalCoalgFinalSeq}, 
\ref{asm:existenceWeaklyFinalCoalgDistLaw}, \ref{asm:existenceWeaklyFinalCoalgGFPcont}, 
 \ref{asm:existenceWeaklyFinalCoalgTop}, and \ref{asm:existenceWeaklyFinalCoalgWeak2Lim} in Prop.~\ref{prop:generalconstructionnotop}
with respect to the order in Def.~\ref{def:orderenrichmentlift}. 
Therefore, $F$ and $\lift$ constitute an infinite trace situation.
\qed
\end{myproposition}

%\newtheorem*{proof:satassumptionexception}{Proof of Prop.\ \ref{prop:satassumptionexception}}{\itshape}{\rmfamily}
\begin{myproof}%\begin{proof:satassumptionexception}
It is easy to see that $F$ and $\lift$ on $\Sets$ satisfy the 
Assumptions~\ref{asm:existenceWeaklyFinalCoalgFinalSeq} and
%\ref{asm:existenceWeaklyFinalCoalgGFPcont}, 
\ref{asm:existenceWeaklyFinalCoalgTop}.

To prove that Assumption~\ref{asm:existenceWeaklyFinalCoalgGFPcont} is satisfied, it suffices to prove that for each $x\in X$, 
$\bigsqcap_{i\in\omega}(g_i\odot b)(x)=\bot$ iff $(\bigsqcap_{i\in\omega}g_i)\odot b(x)=\bot$.
If $b(x)=\bot$, then $\bigsqcap_{i\in\omega}(g_i\odot b)(x)=(\bigsqcap_{i\in\omega}g_i)\odot b(x)=\bot$.
Assume $b(x)\neq\bot$. Then
\begin{equation*}
\bigsqcap_{i\in\omega}(g_i\odot b)(x)=\bot
\Leftrightarrow \exists i\in\omega.\, g_i(b(x))=\bot 
\Leftrightarrow \bigsqcap_{i\in\omega}g_i(b(x))=\bot
\Leftrightarrow (\bigsqcap_{i\in\omega}g_i\odot b)(x)=\bot\,.
\end{equation*}

It is known that Assumption~\ref{asm:existenceWeaklyFinalCoalgDistLaw} is satisfied~\cite[Lemma~2.4]{hasuo07generictrace}.

As a connected limit and a coproduct commute in $\Sets$~\cite{adamekBLR02classificationaccessible}, 
the Kleisli inclusion functor $J:\Sets\to\Kl(T)$ preserves $\omega$-limit. % lifts to a limit in $\Kl(\lift)$.
It is easy to see that this limit is a 2-limit. 
Therefore Assumption~\ref{asm:existenceWeaklyFinalCoalgWeak2Lim} is also satisfied.
\qed
\end{myproof}%\end{proof:satassumptionexception}

\subsubsection{Coalgebraic Infinite Trace Semantics and Automata-theoretic Semantics of Tree Automata with Exception}
\label{subsubsec:automcharaexception}
Next we characterize the coalgebraic infinite trace semantics using automata-theoretic terms.
%we give automata-theoretic semantics of $(\lift,\FSigma)$-systems.
For a ranked alphabet $\Sigma$, $(\lift,\FSigma)$-system can be regarded as a automaton with exception that generates an infinite tree.
%Its language can be formally defined as follows.
%The formal definition of the language of a $(\lift,\FSigma)$-system can be given as follows.
\begin{mydefinition}
A $(\lift,\FSigma)$-system $\mathcal{X}=(X,s,c)$ is called a \emph{tree automaton with exception}.
%
%Let $\mathcal{X}=(X,s,c)$ be a tree automaton with exception. 
%Let $T=(D,l)$ be a $\Sigma$-labeled  infinite tree.
%Let $x\in X$. 
%For each $k\in\omega$, we define a $(X)_{n\in\omega}$-labeled  infinite $k$-prefix tree $T^x_k=(D^x_k,l^x_k)$ inductively as follows.
%\begin{itemize}
%\item For $k=0$, $D^x_k=\{\empseq\}$ and $l^x_k(\empseq)=x$.
%\item Assume that $T^x_k=(D^x_k,l^x_k)$ is defined. 
%\end{itemize}
For a $\Sigma$-labeled  infinite tree $t=(D,l)$ and a state $x\in X$,
a $(X)_{n\in\omega}$-labeled  infinite tree $t_r=(D,l_r)$  that has the same domain as $t$ is 
called a \emph{run of $\mathcal{X}$ from $x$ that generates $t$} if 
\begin{itemize}
\item $l_r(\empseq)=x$, and
\item for each $\alpha\in D$ where $l(\alpha)=a\in\Sigma_n$, $l_r(\alpha)=y$ and 
$l_r(\alpha i)=y_i$ for each $0\leq i \leq n-1$, we have $c(y)=(a,y_0,\ldots,y_{n-1})\neq\bot$.
\end{itemize}

For a state $x\in X$, 
an infinite tree $t$ is called an \emph{infinite output of $\mathcal{X}$ from $x$} and denoted by $\out(\mathcal{X},x)$ if 
there exists a  run of $\mathcal{X}$ from  $x$ that generates $t$.

An  infinite tree $t$ is called an \emph{infinite output of $\mathcal{X}$} and denoted by $\out(\mathcal{X})$ if 
$s(*)\neq\bot$ and $t$ is an output of $\mathcal{X}$ from $s(*)$.
\end{mydefinition}
%
The notations $\out(\mathcal{X},x)$ and $\out(\mathcal{X})$ in the above definition are justified by the following lemma.
It is easy to prove, so we omit the proof.
\begin{mylemma}
Let $\mathcal{X}=(X,s,c)$ be a tree automaton with exception. 
For a state $x\in X$, if an infinite output of $\mathcal{X}$ from $x$ exists, then it is unique.
Moreover, an infinite output of $\mathcal{X}$ exists, then it is unique.
\qed
\end{mylemma}

%In the below, 
We show that for a tree automaton $\mathcal{X}=(X,s,c)$ with exception, 
the infinite output of $\mathcal{X}$ is the largest homomorphism from $c$ to $J\zeta$. 

\begin{mytheorem}\label{thm:largestsemanticslift}
%We assume that each homset of $\Kl(\lift)$ is ordered as Def.~\ref{def:orderenrichment}.
Let $\mathcal{X}=(X,s,c)$ be a tree automaton with exception. 

With respect to the order in Def.~\ref{def:orderenrichmentlift},
for each $x\in X$, we have
\begin{equation*}
\trinf(c)(x)=
\begin{cases}
\out(\mathcal{X},x) & (\text{an infinite output of $\mathcal{A}$ from $x$ exists}) \\
\bot & (\text{otherwise}).
\end{cases} 
\end{equation*}
Moreover, 
\begin{equation*}
\trinf(c)\odot s(*)=
\begin{cases}
\out(\mathcal{X}) & (\text{an infinite output of $\mathcal{A}$ exists}) \\
\bot & (\text{otherwise}). 
\end{cases} 
\end{equation*}
\end{mytheorem}

%\newtheorem*{proof:largestsemanticslift}{Proof of Thm.\ \ref{thm:largestsemanticslift}}{\itshape}{\rmfamily}
\begin{myproof}%\begin{proof:largestsemanticslift}
We define $h:X\kto \Treeinf(\Sigma)$ by 
\begin{equation*}
h(x)=
\begin{cases}
\out(\mathcal{X},x) & (\text{an infinite output of $\mathcal{A}$ from $x$ exists}) \\
\bot & (\text{otherwise}).
\end{cases} 
\end{equation*}
We show that $h$ is the largest homomorphism from $c$ to $J\zeta$ where
$\zeta$ is a $\FSigma$-coalgebra defined in the proof of Thm.~\ref{thm:largestsemanticsgiry}.
% Lem.~\ref{lem:finalcoalgsets}.

We first show that $h$ is a homomorphism.
Assume that $h(x)=t=(D,l)\neq \bot$.
Then there exists a run $t_r=(D,l_r)$ of $\mathcal{X}$ from $x$ that generates $t$.
By definition, we have $c(x)=c(l_r(\empseq))\neq\bot$.
Hence we define $n\in\omega$, $a\in\Sigma_n$ and 
$x_0,\ldots,x_{n-1}\in X$ by $(a,x_0,\ldots,x_{n-1})=c(x)$. %We assume that $a\in\Sigma_n$.
As $t_r=(D,l_r)$ is a run of $\mathcal{X}$ from $x$ that generates $t$,
%$t_{i,r}=(D_i,l_{i,r})$ where $D_i=\{\alpha\mid i\alpha\in D\}$ and $l_{i,r}(\alpha)=l(i\alpha)$ 
$i$'th subtree $t_{r,i}$ of $t_r$ is a run of $\mathcal{X}$ from $x_i$ that generates $i$'th subtree $t_i$ of $t$.
Therefore $h(x_i)=t_i\neq\bot$ for  each $i\in\{0,\ldots,n-1\}$.
Then 
\begin{equation*}
\overline{F}h\odot c(x)
=\overline{F}h(a,x_0,\ldots,x_{n-1}) %\\
=(a,t_0,\ldots,t_{n-1}) %\\
=J(\zeta)(t) %\\
= J(\zeta)\odot h(x).
\end{equation*}

Assume that $h(x)=\bot$. 
If $c(x)=\bot$, $\overline{F}h\odot c(x)=J\zeta\odot h(x)=\bot$ holds by definition.
Assume that $c(x)=(a,x_0,\ldots,x_{n-1})\neq\bot$ where $a\in\Sigma_n$.
If $h(x_i)=t_i\neq \bot$ for all $i\in\{0,\ldots,n-1\}$,
there exits  a run $t_{r,i}$ of $\mathcal{X}$ from $x_i$ that generates $t_i$ for each $i$.
Then a tree $(x,t_{r,0},\ldots,t_{r,n-1})$ is a run of $\mathcal{X}$ from $x$ that generates $t$,
and it contradicts to $h(x)=\bot$. Therefore there exists $i$
%\in\{0,\ldots,n-1\}$ 
such that $h(x_i)=\bot$ and
 we have $\overline{F}h\odot c(x)=J\zeta\odot h(x)=\bot$.

Hence $h$ is a homomorphism from $c$ to $J\zeta$.
It remains to prove that $h$ is the largest homomorphism.
Let $g:X\kto \Treeinf(\Sigma)$ be a homomorphism from $c$ to $J\zeta$.
It suffices to prove that $g(x)=t=(D,l)\neq \bot$, then $h(x)=t$, too.
To this end, we construct a run $t_r=(D,l_r)$ of $\mathcal{X}$ from $x$ that generates $t$.
For each $\alpha\in D$, we define $l_r(\alpha)\in X$ such that $g(l_r(\alpha))\neq\bot$ by the induction on the length of $\alpha$ as follows.

For $\alpha=\empseq$, we define it by $l_r(\empseq)=x$.
By assumption, $g(l_r(\empseq))=g(x)\neq\bot$.

Assume that $l(\alpha)\in\Sigma_n$ and $g(l(\alpha))\neq\bot$. 
As $g$ is a homomorphism from $c$ to $J\zeta$ and $g(l(\alpha))\neq\bot$, there exists $x_0,\ldots,x_{n-1}$ such that 
$c(l_r(\alpha))=(l(\beta),x_0,\ldots,x_{n-1})$ and $g(x_i)\neq\bot$ for each $i$.
We define $l_r(\alpha i)$ by $l_r(\alpha i)=x_i$. 

By axiom of dependent choice, this $l_r$ is well-defined and moreover,
by its construction, $(D,l_r)$ is a run of $\mathcal{X}$ from $x$ that generates $t$.
Therefore $t\in \lang(\mathcal{X},x)$  for each $x\in X$ and we have $g\sqsubseteq \lang(\mathcal{X},\place)$.

Hence $\trinf(c)=\lang(\mathcal{X},\place)$.
This immediately implies  $\trinf(c)\odot s(*)=\lang(\mathcal{X})$.
\qed
\end{myproof}%\end{proof:largestsemanticslift}


\subsubsection{Kleisli Simulation for Systems with Exception}
%It is not so difficult to see that $\FSigma$ and $\lift$ satisfy assumptions of Lem.~\ref{lem:soundnessFwd}.
It is known that a polynomial $F$ and $\lift$ satisfy assumptions of Lem.~\ref{lem:soundnessFwd}~\cite{hasuo06genericforward}.
Hence we can use  forward Kleisli simulation to check infinite trace inclusion between tree automata with exception. 

For a $(\lift,F)$-system, as we have seen in Prop.~\ref{prop:satassumptionexception}, 
the largest homomorphism can be constructed using Prop.~\ref{prop:generalconstructionnotop}.
Therefore from Lem.~\ref{lem:soundnessRBwdnotop}, we can use T-backward simulation in Def.~\ref{def:restrictedbwdsimnotop} to check infinite trace inclusion.
For a $(\lift,F)$-system, 
the sufficient condition for a backward simulation $b$ to satisfy 
the assumption in Def.~\ref{def:restrictedbwdsimnotop} can be given as follows.

\begin{myproposition}\label{prop:whenrestrictedlift}
In Def.~\ref{def:restrictedbwdsimnotop}, if $T=\lift$ and 
%$F=\FSigma$, then
$F$ is a polynomial functor, then
%\begin{enumerate}
%\item 
Assumption \ref{item:restrictedbwdsim2notop} is satisfied if $b(x)\neq\bot$ for each $x\in X$. 
%\label{item:whenrestrictedlift2}
%\item Assumption \ref{item:restrictedbwdsim3} is always satisfied. \label{item:whenrestrictedlift3}
%\item Assumption \ref{item:restrictedbwdsim4} is always satisfied. \label{item:whenrestrictedlift4}
%\end{enumerate}
\end{myproposition}

\begin{myproof}
%\noindent{\bf \ref{item:whenrestrictedlift2}.}
Let $*$ be the unique element in the final object $1$.
It suffices to prove that $J!_Y\odot b(x)=*$ for all $x\in X$.
By the assumption, $b(x)\neq \bot$.
Therefore $* = !_Y(b(x))= J!_Y\odot b(x)$.
%
%\noindent{\bf \ref{item:whenrestrictedlift3}.}
%This holds because $\FSigma$ and $\lift$ satisfy Assumption~\ref{asm:existenceWeaklyFinalCoalgGFPcont} in Prop.~\ref{prop:generalconstructionnotop}.
%
%\noindent{\bf \ref{item:whenrestrictedlift4}.}
%This holds because for $T=\lift$ and $F=\FSigma$, $(Z,(J\pi_i:Z\kto \overline{F}^i 1))$ is a (strong) 2-limit (see the proof of Prop.~\ref{prop:satassumptionexception}).
\qed
\end{myproof}


\subsubsection{Forward Partial Execution for Systems with Exception}
From Thm.~\ref{thm:soundnessFPEfwd}, soundness and adequacy of FPE for forward simulation hold.

By the construction of the largest homomorphism, soundness and adequacy of FPE for backward simulation hold if 
the simulating automaton satisfies the assumptions in Thm.~\ref{thm:soundnessFPEbwdnotop}.
For a $(\lift,F)$-system, the assumptions can be described as follows.

\begin{myproposition}\label{prop:whenrestrictedfpelift}
If $T=\lift$ and 
$F$ is a polynomial functor,
%$F=\FSigma$,
%\begin{enumerate}
%\item 
the assumption in Thm.~\ref{thm:soundnessFPEbwdnotop}
%Assumption \ref{item:restrictedfpe2notop}
 is satisfied if $d(y)\neq\bot$ for each $y\in Y$.
%\item Assumption \ref{item:restrictedfpe3} is always satisfied.
%\item Assumption \ref{item:restrictedfpe4} is always satisfied.
\qed
%\end{enumerate}
\end{myproposition}


\subsection{Subdistribution Monad and Infinite Trace Situation}\label{subsec:subdistmonad}
We first give definitions of the subdistribution monad and orders on the homsets of the Kleisli category.
%
\begin{mydefinition}[subdistribution monad]\label{def:subdistmonad}
A \emph{subdistribution monad} is a monad $(\dist,\eta^{\dist},\mu^{\dist})$ on $\Sets$  such that
\begin{itemize}
\item $\dist X=\{p:X\to [0,1]\mid \sum_{x\in X}p(x)\leq 1\}$,
\item $\dist f(p)(y)= \sum_{x\in f^{-1}(y)}p(x)$, 
\item $\eta^{\dist}_{X}(x)(y)=\begin{cases} 1 & (y=x) \\ 0 & (\text{otherwise}),\end{cases}$ and 
\item $\mu^{\dist}_{X}(\Phi)(x)=\sum_{p\in \dist X}\Phi(p)\cdot p(x)$. 
\end{itemize}
\end{mydefinition}

\begin{mydefinition}[order enrichment of $\Kl(\dist)$]\label{def:orderenrichmentdist}
We define an order on $\Kl(\dist)(X,Y)$ by $f\sqsubseteq g \;\Leftrightarrow\; \forall x\in X.\, \forall y\in Y.\, f(x)(y)\leq g(x)(y)$.
\end{mydefinition}

For a ranked alphabet $\Sigma$, we show that $\FSigma$ on $\Sets$ and the subdistribution monad $\dist$ constitute an infinite trace situation 
by giving an explicit definition of the largest homomorphism. % from each $\overline{\FSigma}$-coalgebra $c$ to $J\zeta$.

\begin{myproposition}\label{prop:satinftracesituationdist}
Let $\Sigma$ be a ranked alphabet and $\FSigma$ be a  functor on $\Sets$ defined in Def.~\ref{def:FSigma}.
Then $\FSigma$ and the subdistribution monad $\dist$ constitute an infinite trace situation.
\end{myproposition}

\begin{myproof}
We define $\zeta:\Treeinf(\Sigma)\to F_{\Sigma}\Treeinf(\Sigma)$ in $\Sets$ in the same way as 
the proof of Thm.~\ref{thm:largestsemanticsgiry}
%Lem.~\ref{lem:finalcoalgsets}.
Then $\zeta$ is a final $\FSigma$-coalgebra. 
For each $\overline{\FSigma}$-coalgebra $c:X\kto\overline{\FSigma}X$, we construct the largest homomorphism $h:X\kto \overline{\FSigma}\Treeinf(\Sigma)$ 
from $c$ to $J\zeta$.
%Let $t=(D,l)$ be a $\Sigma$-labeled infinite tree. 
%We define a value $h(c)(t)\in[0,1]$ as follows.

For each $x\in X$, an integer $k\in\omega$ and a $k$-prefix tree $t^k\in \Treeinf^k(\Sigma)$, 
we define a value $\xi_x(t^k)\in[0,1]$ by the induction on $k$ as follows:
\begin{itemize}
\item if $k=0$, then $\xi_x(t^k)=1$, and
\item if $t^k=(D^k,l^k)\in \Treeinf^{k}(\Sigma)$ 
such that $l^k(\empseq)=a\in\Sigma_n$ and
the $i$'th subtree of $t^k$ is $t^k_i$,
\begin{equation}
\xi_x(t^k)=\sum_{x_0,\ldots,x_{n-1}\in X} \bigl(c(x)(a,x_0,\ldots,x_{n-1})\cdot \prod_{i=0}^{n-1} \xi_{x_i}(t^k_i)\bigr)\,.
\label{eq:disttreeautomaton2}
\end{equation}
%Here, $a=l^k(\empseq)\in\Sigma_n$, and
%$t^k_i$ is the $i$'th subtree of $t^k$.
\end{itemize}

For 
%an infinite tree 
$t'\in\Treeinf(\Sigma)$ and $k\in\omega$, 
%we define a \emph{$k$-prefix of $t'$} as a $k$-prefix tree $\prefix_k(t')=(\prefix_k(D'),\prefix_k(l'))$ such that 
we denote $\prefix_k(t')=(\prefix_k(D'),\prefix_k(l'))$ for a unique $k$-prefix tree that is a prefix of $t'$.
%$\prefix_k(D')=\{\alpha\in \bigcup_{i<k}\mathbb{N}^i \;\mid\; \exists \beta\in\mathbb{N}^*.\, \alpha\beta\in D'\}$, and
%$\prefix_k(l')(\alpha)=l'(\alpha)$.
As $\sum_{n\in\omega}\sum_{a\in\Sigma_n}\sum_{x_0,\ldots,x_{n-1}\in X} c(x)(a,x_0,\ldots,x_{n-1})\leq 1$, 
the sequence $\bigl(\xi_x(\prefix_k(t))\bigr)_{k\in\omega}$ is decreasing with respect to $k$.
Therefore we define $h:X\kto \overline{\FSigma}\Treeinf(\Sigma)$ by $h(x)(t)=\lim_{k\to\infty}\xi_x(\prefix_k(t))$.

We first show that this $h$ is a homomorphism.
For each $x \in X$, $n\in\omega$, and $t=(D,l)\in\Treeinf(\Sigma)$ such that $l(\empseq)=a\in\Sigma_n$ and $i$'th subtree of $t$ is $t_i$,
%\allowdisplaybreaks[1]
\begin{align*}
&J\zeta^{-1}\odot \overline{\FSigma}h\odot c(x)(a,t_0,\ldots, t_{n-1}) \\
&= \sum_{x_0,\ldots,x_{n-1}\in X} c(x)(a,x_0,\ldots,x_{n-1})\cdot \prod_{i=0}^{n-1}h(x_i)(t_i) & (\text{by definition}) \\
&= \sum_{x_0,\ldots,x_{n-1}\in X} c(x)(a,x_0,\ldots,x_{n-1})\cdot \prod_{i=0}^{n-1}\lim_{k\to\infty}\xi_{x_i}(\prefix_k(t_i)) & (\text{by definition of $h$}) \\
%&= \sum_{x_0,\ldots,x_{n-1}\in X} \lim_{k\to\infty} c(x)(a,x_0,\ldots,x_n-1)\cdot \prod_{i=0}^{n-1}\xi_{x_i}(\prefix_k(t_i)) & \\
&= \lim_{k\to\infty} \sum_{x_0,\ldots,x_{n-1}\in X} c(x)(a,x_0,\ldots,x_{n-1})\cdot \prod_{i=0}^{n-1}\xi_{x_i}(\prefix_k(t_i)) & \\
&= \lim_{k\to\infty} \xi_x(t) & (\text{by definition of $\xi$}) \\
&= h(x)(t) & (\text{by definition of $h$})\,. 
\end{align*}

To conclude the proof, we show that $h$ is the largest homomorphism.
Let $g:X\kto \Treeinf(\Sigma)$ be a homomorphism from $c$ to $J\zeta$. 
We prove $g(x)(t)\leq h(x)(t)$ for all $x\in X$ and $t\in\Treeinf(\Sigma)$.
%To this end, we first prove that for all $k\in\omega$, for all $x\in X$ and $t\in\Treeinf(\Sigma)$,
%$g(x)(t)\leq \xi_x(\prefix_k(t))$ holds by the induction on $k$.
To this end, we first prove $g(x)(t)\leq \xi_x(\prefix_k(t))$ for all $k\in\omega$, $x\in X$ and $t\in\Treeinf(\Sigma)$,
by the induction on $k$.


If $k=0$ then for all $x$ and $t$, $g(x)(t)\leq 1 =\xi_x(\prefix_k(t))$.

Let $k>0$ and assume that $g(x)(t)\leq \xi_x(\prefix_{k-1}(t))$ for all $x$ and $t$.
Then 
\begin{align*}
&g(x)(t) \\
&= J\zeta^{-1}\odot \overline{\FSigma}g\odot c(x)(t) & (\text{$g$ is a homomorphism}) \\
&= \sum_{x_0,\ldots,x_{n-1}\in X} c(x)(a,x_0,\ldots,x_{n-1})\cdot \prod_{i=0}^{n-1}g(x_i)(t_i) & (\text{by definition}) \\
&\leq \sum_{x_0,\ldots,x_{n-1}\in X} c(x)(a,x_0,\ldots,x_{n-1})\cdot \prod_{i=0}^{n-1}\xi_{x_i}(\prefix_{k-1}(t_i)) & (\text{by induction hypothesis}) \\
&= \xi_x(t) & (\text{by definition of $\xi$}) 
\end{align*}
Hence for all $x$ and $t$, we have $g(x)(t)\leq\lim_{k\to\infty}\xi_x(\prefix_{k}(t))=h(x)(t)$.
\qed
\end{myproof}

We have shown that 
%a polynomial functor on $\Sets$ and the subdistribution monad 
$\FSigma$ and $\dist$
constitute an infinite trace situation.
However, the largest homomorphism to $J\zeta$ cannot be constructed in the general ways introduced in this paper.

It is easy to see that there exists $X$ and $Z$ in $\Sets$ such that $\Kl(\dist)(X,Z)$ does not have $\top_{X,Z}$.
Therefore we cannot construct the largest homomorphism by using Prop.~\ref{prop:generalconstructiontop}.

\begin{wrapfigure}[4]{r}{0pt}
%\begin{center}
\begin{xy}
%(0,-20)*{} = "",
(-8,6)*{\mathcal{X}} = "",
(0,0)*+{\circ} = "x0",
\ar (0,-5);"x0"
\ar @(dl,ul)^{p,\frac{1}{2}} "x0";"x0"
\ar @(dr,ur)_{q,\frac{1}{2}} "x0";"x0"
\end{xy}
\vspace{5mm}
%\end{center}
\end{wrapfigure}
%
Moreover, in fact, we cannot construct the largest homomorphism even by using Prop.~\ref{prop:generalconstructionnotop}.
Let $F$ be an endofunctor on $\Sets$ that is defined by $F(\place)=\{p,q\}\times(\place)$.
Then, the limit of the final sequence $1\overset{!_{F1}}{\leftarrow}F1\overset{F!_{F1}}{\leftarrow}F^2 1\overset{F^2!_{F1}}{\leftarrow} \ldots$
is given by $(Z,(\gamma_i:Z\to F^i1)_{i\in\omega})$ where $Z=\{p,q\}^{\omega}$ and $\gamma_i(a_0a_1\ldots)=a_0a_1\ldots a_{i-1}$.
We define $X\in\Kl(\dist)$ and $c:X\kto \overline{F}X$ by $X=\{*\}$ and $c(*)(a,*)=\frac{1}{2}$ where $a\in\{p,q\}$.
Moreover, for each $i\in\omega$, 
%we define $\alpha_i:X\kto\overline{F}^i1$ inductively as follows:
we inductively define $\alpha_i:X\kto\overline{F}^i1$ by $\alpha_0=J!_X$ and $\alpha_{i+1}=\overline{F}\alpha_i\odot c$.
%inductively as follows:
%\begin{equation*}
%\alpha_0=J!_X, \;\;\;\text{and}\;\;\; \alpha_{i+1}=\overline{F}\alpha_i\odot c\,.
%\end{equation*}
It is easy to see that $(X,(\alpha_i)_{i\in\omega})$ is a cone over a sequence 
$1\overset{J!_{F1}}{\kleftarrow} \overline{F}1\overset{JF!_{F1}}{\kleftarrow} \overline{F}^2 1\overset{JF^2!_{F1}}{\kleftarrow} \ldots$.
However, it is also easy to see that there does not exist $f:X\kto Z$ such that $J\gamma_i\odot f=\alpha_i$.
This means that Assumption~\ref{asm:existenceWeaklyFinalCoalgWeak2Lim} is not satisfied.

As a consequence, we can construct the largest homomorphism from $c$ to $J\zeta$ neither by using the construction in 
Prop.~\ref{prop:generalconstructiontop} nor Prop.~\ref{prop:generalconstructionnotop}.





%\subsection{Proof that Functor and Monad Satisfy Assumptions in Prop.~\ref{prop:generalconstructionnotop}}
%\subsubsection{Powerset Monad and Polynomial Functor}\label{subsubsec:powsatisfyasm}
%We fix $\FSigma$ and $\pow$ as  \S\ref{subsec:examplenondet}.



%\newtheorem*{proof:satassumptionnondet}{Proof of Prop.\ \ref{prop:satassumptionnondet}}{\itshape}{\rmfamily}
%\begin{proof:satassumptionnondet}
%It is easy to see that $\FSigma$ and $\pow$ on $\Sets$ satisfy the 
%assumptions~\ref{asm:existenceWeaklyFinalCoalgFinalSeq}, \ref{asm:existenceWeaklyFinalCoalgGFPtransfinite}, \ref{asm:existenceWeaklyFinalCoalgTop}, 
%and \ref{asm:existenceWeaklyFinalCoalgFMonotone}.
%
%It is known that assumption~\ref{asm:existenceWeaklyFinalCoalgDistLaw} is satisfied~\cite[Lemma~2.4]{hasuo07generictrace}.
%
%It is immediate from Lem.~\ref{lem:powweak2lim} that assumption \ref{asm:existenceWeaklyFinalCoalgWeak2Lim} is satisfied.
%\qed
%\end{proof:satassumptionnondet}
%
%\subsubsection{Lift Monad and Polynomial Functor}\label{subsubsec:liftsatisfyasm}
%We fix $\FSigma$ and $\lift$ as  \S\ref{subsec:exampleexception}.
%
%\subsubsection{sub-Giry Monad and Restricted Polynomial Functor}\label{subsubsec:girysatisfyasm}
%We fix $\FSigma$ and $\giry$ as  \S\ref{subsec:exampleprob}.
%\begin{mylemma}\label{lem:giry2lim}
%
%\end{mylemma}
%
%\begin{myproof}
%\qed
%\end{myproof}
%


%\subsection{Proof that Semantics of Probabilistic Tree Automaton is Well-defined}\label{subsec:semprobwelldefined}
%\newtheorem*{proof:uniqueextensionmeas}{Proof of Lem.\ \ref{lem:uniqueextensionmeas}}{\itshape}{\rmfamily}
%\begin{proof:uniqueextensionmeas}
%\qed
%\end{proof:uniqueextensionmeas}



%\subsection{Proof for Coincidence between Coalgebraic Semantics and Automata-theoretic Semantics}\label{subsec:coincidencesemantics}
%\subsubsection{nondeterministic Tree Automaton}\label{subsec:coincidencenondet}
%\newtheorem*{proof:largestsemanticspow}{Proof of Thm.\ \ref{thm:largestsemanticspow}}{\itshape}{\rmfamily}
%\begin{proof:largestsemanticspow}
%%We define $f:X\kto \Treeinf(\Sigma)$ by $f(x)=\lang(\mathcal{X},x)$. 
%We show that $\lang(\mathcal{X},\place):X\kto\Treeinf(\Sigma)$ is the largest homomorphism from $c$ to $J\zeta$ where
%$\zeta$ is a final coalgebra defined in Lem.~\ref{lem:finalcoalgsets}.
%
%We first show that $\lang(\mathcal{X},\place)$ is a homomorphism.
%For $x\in X$,
%\allowdisplaybreaks[1]\begin{align*}
%&(\overline{\FSigma}\lang(\mathcal{X},\place))\odot c (x) \\
%%
%&= (\overline{\FSigma}\lang(\mathcal{X},\place))\left(\left\{(a,x_0,\ldots,x_{n-1})\relmiddle|
%\begin{array}{l}
% n\in\omega, a\in\Sigma_n, x_0,\ldots,x_{n-1}\in X, \\
% (a,x_0,\ldots,x_{n-1})\in c(x)
% \end{array}
% \right\}\right) \\
% %
%&= \left\{(a,t_0,\ldots,t_{n-1})\relmiddle|
%\begin{array}{l}
%n\in\omega, \;a\in\Sigma_n,\; \\
%\exists x_0,\ldots,x_{n-1}\in X.\, 
%\left(\begin{array}{l}
%(a,x_0,\ldots,x_{n-1})\in c(x), \\
%t_i\in \lang(\mathcal{X},x_i)\text{ for each $i\in\{0,\ldots,n-1\}$} 
%\end{array}\right)
%\end{array}
%\right\}\\
%%
%&= J\zeta\left(\left\{(D,l)\in\Treeinf(\Sigma) \relmiddle|
%\begin{array}{l}
%n\in\omega, l(\empseq)\in\Sigma_n, \\
%\exists x_0,\ldots,x_{n-1}\in X. 
%\left(\begin{array}{l}(l(\empseq),x_0,\ldots,x_{n-1})\in c(x), \\
%l(i)\in \lang(\mathcal{X},x_i)\text{ for each $i\in\{0,\ldots,n-1\}$}
%\end{array}\right)
%\end{array}
%\right\}\right) \\
%%
%&= J\zeta\left(\left\{(D,l)\in\Treeinf(\Sigma) \relmiddle|
%\begin{array}{l}
%n\in\omega, l(\empseq)\in\Sigma_n, \\
%\exists x_0,\ldots,x_{n-1}\in X. \left(\begin{array}{l}
%(l(\empseq),x_0,\ldots,x_{n-1})\in c(x), \\
%\text{there exists a run $t_{r,i}$ of $\mathcal{X}$ from $x_i$ that}\\
%\text{generates $t_i$ for each $i\in\{0,\ldots,n-1\}$}
%\end{array}\right)
%\end{array}
%\right\}\right) \\
%&= J\zeta(\{t\in\Treeinf(\Sigma) \mid \text{there exists a run $t_{r}$ of $\mathcal{X}$ from $x$ that generates $t$}\}) \\
%&= J\zeta\odot \lang(\mathcal{X},\place)(x)\,.
%\end{align*}
%Therefore $\lang(\mathcal{X},\place)$ is a homomorphism from $c$ to $J\zeta$.
%
%It remains to prove that $\lang(\mathcal{X},\place)$ is the largest homomorphism.
%Let $g:X\kto\Treeinf(\Sigma)$ be a homomorphism from $c$ to $J\zeta$.
%Assume that $t=(D,l)\in g(x)$ for $x\in X$. We show that $t\in\lang(\mathcal{X},x)$.
%To prove this, it suffices to construct a run $t_r=(D,l_r)$ of $\mathcal{X}$ from $x$ that generates $t$.
%For $\alpha\in D$, we denote $t_{\alpha}$ for the $\alpha$'th subtree of $t$.
%For each $\alpha\in D$, we define $l_r(\alpha)\in X$ such that $t_{\alpha}\in g(l_r(\alpha))$
%by the induction on the length of $\alpha$ as follows.
%
%For $\alpha=\empseq$, we define it by $l_r(\empseq)=x$.
%Then $t_{\alpha}=t\in g(x)=g(l_r(\empseq))$.
%
%Assume that $l(\alpha)\in\Sigma_n$ and $t_{\alpha}\in g(l_r(\alpha))$. 
%As $g$ is a homomorphism from $c$ to $J\zeta$, 
%$t_{\alpha}\in g(l_r(\alpha))=J\zeta^{-1}\odot\overline{F}g\odot c(l_r(\alpha))$.
%Therefore there exists $x_0,\ldots,x_{n-1}$ such that 
%$(l(\alpha),x_0,\ldots,x_{n-1})\in c(l_r(\alpha))$ and $t_{\alpha i}\in g(x_i)$ for each $i\in\{0,\ldots, n-1\}$.
%We define $l_r(\alpha i)$ by $l_r(\alpha i)=x_i$.
%Then $t_{\alpha i}\in l_r(\alpha i)$.
%
%By the axiom of dependent choice, this $l_r$ is well-defined.
%Moreover by its construction, $(D,l_r)$ is a run of $\mathcal{X}$ from $x$ that generates $t$.
%Therefore $t\in \lang(\mathcal{X},x)$  for each $x\in X$ and we have $g\sqsubseteq \lang(\mathcal{X},\place)$.
%
%Hence $\trinf(c)=\lang(\mathcal{X},\place)$.
%This immediately implies that $\trinf(c)\odot s(*)=\lang(\mathcal{X})$.
%\qed
%\end{proof:largestsemanticspow}

%\subsubsection{Tree Automaton with Exception}\label{subsec:coincidenceexception}


%
%\subsection{Proofs for Section~\ref{sec:Kleislisim}}\label{subsec:proofKleislisim}
%\newtheorem*{proof:soundnessFwd}{Proof of Lem.\ \ref{lem:soundnessFwd}}{\itshape}{\rmfamily}
%\begin{proof:soundnessFwd}
%\begin{equation*}
%\begin{xy}
% \xymatrix@R=2.4em@C=1.5em{
% {\overline{F}Y} \ar@{}[drr]|{\sqsupseteq} \kar[rr]^{\overline{F} f } \kar@/^6mm/[rrrr]^{\overline{F}(\trinf(d))} & & {\overline{F} X} \ar@{}[drr]|{=} \kar[rr]^{\overline{F}(\trinf(c))} & & {\overline{F} Z} \\
% {Y} \kar@/_6mm/[rrrr]_(.8){\trinf(d)} \kar[u]_{d} \ar@{}[drr]|{\sqsupseteq} \ar^{f}|-*\dir{|}[rr]  & & {X} \kar[u]_{c} \kar[rr]^{\trinf(c)} & & {Z} \kar[u]_{J\zeta} \\
% {} \ar@{}[ru]|(.10)*\dir{-} & {1} \ar`l[lu][lu]^(0)t \ar`r[ru][ru]_(0)s  & \ar@{}[lu]|(.10)*\dir{-} & & }
% \end{xy}
% \end{equation*}
% 
%Let $\zeta:Z\to FZ$ be a final $F$-coalgebra in $\mathbb{C}$.
%Let $f:Y\kto X$ be a forward Kleisli simulation from $\mathcal{X}$ to $\mathcal{Y}$ (see the above diagram). 
%
%We define a function $\Phi:\Kl(T)(Y,Z)\to\Kl(T)(Y,Z)$ by $\Phi(f)=(J\zeta)^{-1}\odot \overline{F}f\odot d$ (recall that weakly final coalgebra $J\zeta$ is an isomorphism).
%
%From the assumption that $\overline{F}$ and composition are respectively locally monotone,
%$\Phi$ is  monotone.
%Moreover, from the assumption that $\overline{F}$ and composition are respectively locally continuous,
%$\Phi$ is  continuous.
%Therefore
%\begin{align*}
%\trinf(c)\odot f &= (J\zeta)^{-1}\odot J\zeta \odot \trinf(c)\odot f & (J\zeta\text{ is an isomorphism}) \\
%&=z^{-1}\odot \overline{F}(\tr^{\infty}_{z}(c))\odot c \odot f & (\text{$f$ is a fwd.\ simulation}) \\ 
%&\sqsubseteq (J\zeta)^{-1}\odot \overline{F}(\trinf(c))\odot \overline{F}f \odot f & (\trinf(c)\text{ is a homomorphism}) \\ 
%&= \Phi(\trinf(c)\odot f) & (\text{by definition}).
%\end{align*}
%As $\Phi$ is monotone, repeatedly applying $\Phi$ to the both sides of the above inequality, we can obtain an increasing sequence
%$\trinf(c)\odot f\sqsubseteq\Phi(\trinf(c)\odot f)\sqsubseteq\Phi^2(\trinf(c)\odot f)\sqsubseteq\ldots$ in $\Kl(T)(Y,Z)$.
%
%As it is assumed that each homset of $\Kl(T)$ is $\omega$-complete in upward manner,
%their least upper bound $\bigsqcup_{i<\omega}\Phi^i(\trinf(c)\odot f)$ exists.
%As $\Phi$ is continuous,
%%$\Phi(\bigsqcup_{i<\omega}\Phi^i(\trinf(c)\odot f))=\bigsqcup_{i<\omega}\Phi^{i+1}(\trinf(c)\odot f)=\bigsqcup_{i<\omega}\Phi^i(\tr^{\infty}_{z}(c)\odot f)$ 
%%holds.
%%This means that 
%$\bigsqcup_{i<\omega}\Phi^i(\trinf(c)\odot f)$ is the fixed point of $\Phi$, hence it is a homomorphism from $d$ to $J\zeta$.
%
%As $\trinf(d)$ is the largest homomorphism from $d$ to $J\zeta$, this implies 
%$\trinf(c)\odot f \sqsubseteq \bigsqcup_{i<\omega}\Phi^i(\trinf(c)\odot f)\sqsubseteq\trinf(d)$.
%Therefore, 
%\begin{align*}
%\trinf(c)\odot s
%&\sqsubseteq \tr^{\infty}_{z}(c)\odot f\odot t &(s \text{ is a fwd. simulation and composition is monotone})\\
%&\sqsubseteq \tr^{\infty}_{z}(d)\odot t &(\text{above result and composition is monotone})
%\end{align*}
%This concludes the proof.
%\qed
%\end{proof:soundnessFwd}



%\newtheorem*{proof:whenrestricted}{Proof of Prop.\ \ref{prop:whenrestricted}}{\itshape}{\rmfamily}
%\begin{proof:whenrestricted}
%In the below, let $*$ be the unique element of the final object $1$.
%
%\noindent{\bf \ref{item:whenrestrictedpow2}.} 
%It suffices to prove that $*\in J!_Y\odot b(x)$ for all $x\in X$.
%By the assumption, there exists $y\in Y$ such that $y\in b(x)$.
%Therefore $* = !_Y(y)\in J!_Y\odot b(x)$.
%
%\noindent{\bf \ref{item:whenrestrictedpow3}.}
%Here, $\bigsqcap_{\mathfrak{i}<\mathfrak{a}}(g_{\mathfrak{a}}\odot b)(x)=\bigcap_{\mathfrak{i}<\mathfrak{a}}\bigcup_{y\in b(x)} g_{\mathfrak{i}}(y)$
%while $\bigl(\bigsqcap_{\mathfrak{i}<\mathfrak{a}}g_{\mathfrak{i}}\bigr)\odot b(x)=\bigcup_{y\in b(x)}\bigcap_{\mathfrak{i}<\mathfrak{a}} g_{\mathfrak{i}}(y)$.
%It is well-known that the latter is always included in the former.
%Therefore it suffices to prove that $z\in \bigcap_{\mathfrak{i}<\mathfrak{a}}\bigcup_{y\in b(x)} g_{\mathfrak{i}}(y)$ implies 
% $z\in\bigcup_{y\in b(x)}\bigcap_{\mathfrak{i}<\mathfrak{a}} g_{\mathfrak{i}}(y)$.
%
%Assume that $z\in \bigcap_{\mathfrak{i}<\mathfrak{a}}\bigcup_{y\in b(x)} g_{\mathfrak{i}}(y)$.
%Then for each $\mathfrak{i}<\mathfrak{a}$, there exists $y_{\mathfrak{i}}\in b(x)$ such that $z\in g_{\mathfrak{i}}(y_{\mathfrak{i}})$.
%From Lem.~\ref{lem:ordinalramsey}, for arbitrary $\mathfrak{i}<\mathfrak{a}$, there exists $\mathfrak{j}<\mathfrak{a}$ such that $\mathfrak{i}\leq \mathfrak{j}$ and 
%$z\in g_{\mathfrak{j}}(y)$.
%As $\mathfrak{i}\leq \mathfrak{j}$ implies $g_{\mathfrak{i}} \sqsupseteq g_{\mathfrak{j}}$, we have $z\in g_{\mathfrak{i}}(y)$.
%Therefore $z\in\bigcup_{y\in b(x)}\bigcap_{\mathfrak{i}<\mathfrak{a}} g_{\mathfrak{i}}(y)$ holds.
%
%%Here, $\bigsqcap_{i\in\omega}(g_i\odot b)(x)=\bigcap_{i\in\omega}\bigcup_{y\in b(x)} g_i(y)$
%%while $\bigl(\bigsqcap_{i\in\omega}g_i\bigr)\odot b(x)=\bigcup_{y\in b(x)}\bigcap_{i\in\omega} g_i(y)$.
%%It is well-known that the latter is always included in the former.
%%Therefore it suffices to prove that $z\in \bigcap_{i\in\omega}\bigcup_{y\in b(x)} g_i(y)$ implies  $z\in\bigcup_{y\in b(x)}\bigcap_{i\in\omega} g_i(y)$.
%%
%%Assume that $z\in \bigcap_{i\in\omega}\bigcup_{y\in b(x)} g_i(y)$.
%%Then for each $i\in\omega$, there exists $y_i\in b(x)$ such that $z\in g_i(y_i)$.
%%As it is assumed that $b(x)$ is finite, by the infinite pigeonhole principle,
%%there exists $y\in b(x)$ such that $z\in g_j(y)$ for infinitely many $j\in\omega$.
%%Then for an arbitrary $i\in\omega$, there exists $j\geq i$ such that $z\in g_j(y)$.
%%As $g_i \sqsupseteq g_{i+1}\sqsupseteq\cdots\sqsupseteq g_j$, we have $z\in g_i(y)$.
%%Therefore $z\in\bigcup_{y\in b(x)}\bigcap_{i\in\omega} g_i(y)$ holds.
%
%\noindent{\bf \ref{item:whenrestrictedpow4}.}
%By Lem.~\ref{lem:powweak2lim}, 
%it suffices to prove that for all $x\in X$, 
%$l^X_{\max}\sqsubseteq l^Y_{\max}\odot b$ holds.
%Here, $l^X_{\max}:X\kto Z$ is defined by
%\begin{equation*}
%l^X_{\max}(x)=\{(a_j)_{j\in\omega}\in Z\mid \forall j\in\omega.\, a_j\in \delta_j\odot b(x), F^j!_{F1}(a_{j+1})=a_j\}
%\end{equation*}
%while $l^Y_{\max}:Y\kto Z$ is defined by
%\begin{equation*}
%l^Y_{\max}(y)=\{(a'_j)_{j\in\omega}\in Z\mid \forall j\in\omega.\, a'_j\in \delta_j(y), F^j!_{F1}(a'_{j+1})=a'_j\}\,.
%\end{equation*}
%%
%Let $x\in X$ and assume that $(a_j)_{j\in\omega}\in l^X_{\max}(x)$. 
%Then for each $i\in\omega$, there exists $y_i\in b(x)$ such that $a_i \in \delta_i(y_i)$.
%As it is assumed that $b(x)$ is finite, by the infinite pigeonhole principle,
%there exists $y\in b(x)$ such that $a_i\in \delta_j(y)$ for infinitely many $j\in\omega$.
%Then for an arbitrary $i\in\omega$, there exists $j\geq i$ such that $a_j\in \delta_j(y)$.
%As $(a_j)_{j\in\omega}\in Z$, we have 
%$a_i=F^i!_{F1}\circ F^{i+1}!_{F1}\circ\cdots\circ F^{j-1}!_{F1}(a_j)\in \delta_i(y)$.
%Therefore $(a_i)_{i\in\omega}\in l^Y_{\max}(y)=\{(a'_j)_{j\in\omega}\in Z\mid \forall j\in\omega.\, a'_j\in \delta_j(y), F^j!_{F1}(a'_{j+1})=a'_j\}$.
%
%
%\noindent{\bf \ref{item:whenrestrictedlift2}.}
%It suffices to prove that $J!_Y\odot b(x)=*$ for all $x\in X$.
%By assumption, $b(x)\neq \bot$.
%Therefore $* = !_Y(b(x))= J!_Y\odot b(x)$.
%
%\noindent{\bf \ref{item:whenrestrictedlift3}.}
%This holds because $\FSigma$ and $\lift$ satisfy assumption~\ref{asm:existenceWeaklyFinalCoalgGFPcont} in Prop.~\ref{prop:generalconstructionnotop}.
%
%%It suffices to prove that for each $x\in X$, 
%%$\bigsqcap_{i\in\omega}(g_i\odot b)(x)=\bot$ iff $(\bigsqcap_{i\in\omega}g_i)\odot b(x)=\bot$.
%%If $b(x)=\bot$, then $\bigsqcap_{i\in\omega}(g_i\odot b)(x)(\bigsqcap_{i\in\omega}g_i)\odot b(x)=\bot$.
%%Assume $b(x)\neq\bot$. Then
%%\begin{equation*}
%%\bigsqcap_{i\in\omega}(g_i\odot b)(x)=\bot
%%\Leftrightarrow \exists i\in\omega.\, g_i(b(x))=\bot 
%%\Leftrightarrow \bigsqcap_{i\in\omega}g_i(b(x))=\bot
%%\Leftrightarrow (\bigsqcap_{i\in\omega}g_i\odot b)(x)=\bot\,.
%%\end{equation*}
%
%\noindent{\bf \ref{item:whenrestrictedlift4}.}
%This holds because for $T=\lift$ and $F=\FSigma$, $(Z,(J\pi_i:Z\kto \overline{F}^i 1))$ is a (strong) 2-limit (see the proof of Prop.~\ref{prop:satassumptionexception}).
%
%\noindent{\bf \ref{item:whenrestrictedgiry2}.}
%By the definition of multiplication of sub-Giry monad (see Def.~\ref{def:variousmonads}), for each $x\in X$,
%\begin{equation*}
%J!_Y\odot b(x)(1)=b(x)\bigl(!_Y^{-1}(1)\bigr)=b(x)(Y)=1=J!_X(x)(1)\,.
%\end{equation*}
%Therefore assumption~\ref{item:restrictedbwdsim2} is satisfied.
%
%\noindent{\bf \ref{item:whenrestrictedgiry3}.}
%This holds because 
%$\FSigma$ and $\giry$ satisfy assumption~\ref{asm:existenceWeaklyFinalCoalgGFPcont} in Prop.~\ref{prop:generalconstructionnotop}.
%%in general, composition in $\Kl(\giry)$ is preserves the greatest lower bound of 
%%decreasing sequence of Kleisli arrows.
%%It can be proved  in the similar manner to the proof of~\cite[Proposition~9]{brengosMP14behaviouralequivalences}.
%%Therefore, assumption~\ref{item:whenrestrictedgiry3} is always satisfied.
%
%\noindent{\bf \ref{item:whenrestrictedgiry4}.}
%This holds because for $T=\giry$ and $F=\FSigma$, $(Z,(J\pi_i:Z\kto \overline{F}^i 1))$ is a (strong) 2-limit (see the proof of Prop.~\ref{prop:satassumptionprob}).
%\qed
%\end{proof:whenrestricted}


%\subsection{Proofs for Section~\ref{sec:fpe}}\label{subsec:prooffpe}

\subsection{Examples and Counterexamples}\label{subsec:examples}
%\begin{myexample}\label{example:nogeneralconstruction}
%
%\end{myexample}

\begin{myexample}\label{example:needtransfinite}
In the construction of the largest homomorphism in Prop.~\ref{prop:generalconstructiontop},
we need $\omega+1$ times of iterations 
%in order to construct the homomorphism 
for the nondeterministic automaton $\mathcal{X}$ on the left below. 
 We need $2\omega+1$ times of iterations
% in order to construct the homomorphism
%a homomorphism 
for $\mathcal{Y}$ on the right below. 
%In the proof of Prop.~\ref{prop:generalconstructionnotop}, we used transfinite induction 
%twice---once for constructing a cone over a lifted final sequence, and once for constructing a homomorphism.
%We need $\omega+1$ times of iterations in order to construct a cone for a nondeterministic automaton $\mathcal{X}$ on the left below. 
%We need $2\omega+1$ times of iterations in order to construct a homomorphism for a nondeterministic automaton $\mathcal{Y}$ on the right below. 
In a similar manner, for each ordinal $\mathfrak{a}$,
%by the transfinite induction on the ordinal $\mathfrak{a}$,
 we can construct an automaton for which we need $\mathfrak{a}$ times of
 iteration.
% or a homomorphism for it.

\begin{minipage}{0.5\hsize}
\begin{center}
\begin{xy}
(5,12)*{\mathcal{X}} = "",
%(10,0)*+{} = "xc",
(10,0)*+{\checkmark} = "x0",
(20,0)*+{\circ} = "x1",
(30,0)*+{\circ} = "x2",
(40,0)*+{\circ} = "x3",
(48,0)*{} = "x4",
(56,0)*{} = "x5",
(52,0)*{\cdots}="",
(34,7)*{\dots}="",
(15,10)*+{\circ} = "xo",
%\ar ^{} "x0";"xc"
\ar ^{} "x1";"x0"
\ar ^{a} "x2";"x1"
\ar ^{a} "x3";"x2"
\ar ^{a} "x4";"x3"
%\ar @{.}^{} "x5";"x4"
\ar _{} "xo";"x0"
\ar _{a} "xo";"x1"
\ar ^(.8){a} "xo";"x2"
\ar ^(.8){a} "xo";"x3"
\ar (15,15);"xo"
\end{xy}
\end{center}
\end{minipage}
%
\begin{minipage}{0.5\hsize}
\begin{center}
\begin{xy}
(5,22)*{\mathcal{Y}} = "",
%(0,0)*+{\checkmark} = "xc",
(10,0)*+{\checkmark} = "x0",
(20,0)*+{\circ} = "x1",
(30,0)*+{\circ} = "x2",
(40,0)*+{\circ} = "x3",
(48,0)*{} = "x4",
(56,0)*{} = "x5",
(15,10)*+{\circ} = "xo",
(25,10)*+{\circ} = "xo1",
(35,10)*+{\circ} = "xo2",
(45,10)*+{\circ} = "xo3",
(52,0)*{\cdots}="",
(56,10)*{\cdots}="",
(51,10)*{} = "xo4",
(61,10)*{} = "xo5",
(20,20)*+{\circ} = "x2o",
(45,4)*{\dots}="",
(46,17)*{\dots}="",
%\ar ^{} "x0";"xc"
\ar ^{} "x1";"x0"
\ar ^{a} "x2";"x1"
\ar ^{a} "x3";"x2"
\ar ^{a} "x4";"x3"
%\ar @{.}^{} "x5";"x4"
\ar _{} "xo";"x0"
\ar _{a} "xo";"x1"
\ar ^(.8){a} "xo";"x2"
\ar ^(.8){a} "xo";"x3"
%
\ar _{a} "xo1";"xo"
\ar ^{a} "xo2";"xo1"
\ar ^{a} "xo3";"xo2"
\ar ^{a} "xo4";"xo3"
%\ar @{.}^{} "xo5";"xo4"
\ar _{a} "x2o";"xo"
\ar _{a} "x2o";"xo1"
\ar ^(.8){a} "x2o";"xo2"
\ar ^(.8){a} "x2o";"xo3"
%
\ar (20,25);"x2o"
\end{xy}
\end{center}
\end{minipage}
%
\end{myexample}

\auxproof{
\begin{myexample}\label{example:sameconediffhom}
%In the proof of Prop.~\ref{prop:generalconstructionnotop}, we calculated a limit of a decreasing sequence 
%twice---once for constructing a cone and once for constructing the largest homomorphism.
The following two nondeterministic automata $\mathcal{X}$ and $\mathcal{Y}$  have the same state space
$X=\{x_0,x_1,x_2,\ldots\}\cup\{x_{\omega}\}$.
They have different infinite trace semantics
but 
construct exactly the same cone $(X,(\pi_i:X\kto \overline{F}^i1)_{i\in\omega})$ 
over the sequence $1\overset{J!_{F1}}{\kleftarrow} \overline{F}1\overset{JF!_{F1}}{\kleftarrow} \overline{F}^2 1\overset{JF^2!_{F1}}{\kleftarrow} \ldots$ in the construction in Prop.~\ref{prop:generalconstructionnotop}.
Here, $\pi_i$ is given by $\pi_i(x_j)=\{a^j\}$ for $j<i$, $\pi_i(x_j)=\{a^i\}$ for $i\leq j<\omega$ and $\pi_i(x_{\omega})=\{\empseq,a,a^2,\ldots,a^i\}$.
The resulting largest mediating arrow $l_{\max}:X\kto Z$ is given by $l_{\max}(x_i)=\{a^i\}$ for $i<\omega$ and $l_{\max}(x_{\omega})=\{\empseq,a,a^1,\ldots\}\cup\{a^{\omega}\}$.
%This Kleisli arrow seems to be giving a proper infinite trace semantics for the system $\mathcal{Y}$, but not for  $\mathcal{X}$.

\begin{minipage}{0.5\hsize}
\begin{center}
\begin{xy}
(5,12)*{\mathcal{X}} = "",
%(0,0)*+{\checkmark} = "xc",
(10,0)*+{\checkmark} = "x0",
(20,0)*+{x_0} = "x1",
(30,0)*+{x_1} = "x2",
(40,0)*+{x_2} = "x3",
(48,0)*{} = "x4",
(56,0)*{} = "x5",
(15,10)*+{x_{\omega}} = "xo",
(34,7)*{\dots}="",
(52,0)*{\cdots}="",
%\ar ^{} "x0";"xc"
\ar ^{} "x1";"x0"
\ar ^{a} "x2";"x1"
\ar ^{a} "x3";"x2"
\ar ^{a} "x4";"x3"
%\ar @{.}^{} "x5";"x4"
\ar _{} "xo";"x0"
\ar _{a} "xo";"x1"
\ar ^(.8){a} "xo";"x2"
\ar ^(.8){a} "xo";"x3"
\ar (15,15);"xo"
\end{xy}
\end{center}
\end{minipage}
%
\begin{minipage}{0.5\hsize}
\begin{center}
\begin{xy}
(5,12)*{\mathcal{Y}} = "",
%(0,0)*+{\checkmark} = "xc",
(10,0)*+{\checkmark} = "x0",
(20,0)*+{x_0} = "x1",
(30,0)*+{x_1} = "x2",
(40,0)*+{x_2} = "x3",
(48,0)*{} = "x4",
(56,0)*{} = "x5",
(15,10)*+{x_{\omega}} = "xo",
(52,0)*{\cdots}="",
%\ar ^{} "x0";"xc"
\ar ^{} "x1";"x0"
\ar ^{a} "x2";"x1"
\ar ^{a} "x3";"x2"
\ar ^{a} "x4";"x3"
%\ar @{.}^{} "x5";"x4"
\ar @(dr,ur)_{a} "xo";"xo"
\ar "xo";"x0"
%\ar _{a} "xo";"x1"
%\ar ^(.8){a} "xo";"x2"
%\ar ^(.8){a} "xo";"x3"
\ar (15,15);"xo"
\end{xy}
\end{center}
\end{minipage}
%Therefore the second calculation for the limit is necessary to distinguish them with respect to the largest homomorphism.
\end{myexample}
}


\begin{myexample}\label{example:bwdsimnotsound}
There exists a (not total) backward simulation from the nondeterministic automaton $\mathcal{X}$ to $\mathcal{Y}$ below.
However, the simulated automaton $\mathcal{X}$ outputs an infinite word $aaa\ldots$ while $\mathcal{Y}$ does not.
Therefore the infinite trace of $\mathcal{X}$ is not include in that of $\mathcal{Y}$.
%We can see that the backward simulation between them does not satisfy assumption~\ref{item:restrictedbwdsim2top} 
%in the definition of restricted backward simulation (
%of Def.~\ref{def:restrictedbwdsimtop}.

\begin{minipage}{0.5\hsize}
\begin{center}
\begin{xy}
(-25,0)*{} = "",
(0,10)*{\mathcal{X}} = "",
(0,0)*+{\checkmark} = "xc",
(10,0)*+{\circ} = "x0",
(20,0)*+{\circ} = "x1",
(13,7)*+{\circ} = "x2",
\ar (25,0);"x1"
\ar @(ul,ur)^{a} "x2";"x2"
\ar ^{a} "x1";"x0"
\ar _{a} "x1";"x2"
\ar ^{} "x0";"xc"
\end{xy}
\end{center}
\end{minipage}
%
\begin{minipage}{0.5\hsize}
\begin{center}
\begin{xy}
%(-25,0)*{} = "",
(0,10)*{\mathcal{Y}} = "",
(0,0)*+{\checkmark} = "xc",
(10,0)*+{\circ} = "x0",
(20,0)*+{\circ} = "x1",
%(13,7)*+{\circ} = "x2",
\ar (25,0);"x1"
%\ar @(ul,ur)^{a} "x2";"x2"
\ar ^{a} "x1";"x0"
%\ar _{a} "x1";"x2"
\ar ^{} "x0";"xc"
\end{xy}
\end{center}
\end{minipage}
%
%The following pair of nondeterministic automata $\mathcal{Z}$ and $\mathcal{W}$ 
There exists a (not image-finite) backward simulation from $\mathcal{Z}$ to $\mathcal{W}$ below,
while trace inclusion from $\mathcal{Z}$ to $\mathcal{W}$  does not hold.
%The automata $\mathcal{Z}$ and $\mathcal{W}$ below
%constitute another pair of such an example.
%The backward simulation from $\mathcal{Z}$ to $\mathcal{W}$ 
%does not satisfy assumption~\ref{item:restrictedbwdsim3top} 
%in the definition of restricted backward simulation (
%of Def.~\ref{def:restrictedbwdsimtop}.

\begin{minipage}{0.5\hsize}
\begin{center}
\begin{xy}
(-25,0)*{} = "",
(-5,8)*{\mathcal{Z}} = "",
(0,0)*+{\checkmark} = "xc",
(10,0)*+{\circ} = "xo",
(20,0)*+{\circ} = "xo2",
\ar (25,0);"xo2"
\ar ^{a} "xo2";"xo"
\ar @(ul,ur)^{a} "xo";"xo"
\ar ^{} "xo";"xc"
\end{xy}
\end{center}
\end{minipage}
%
\begin{minipage}{0.5\hsize}
\begin{center}
\begin{xy}
(5,12)*{\mathcal{W}} = "",
(0,0)*+{\checkmark} = "xc",
(10,0)*+{\circ} = "x0",
(20,0)*+{\circ} = "x1",
(30,0)*+{\circ} = "x2",
(40,0)*+{\circ} = "x3",
(48,0)*{} = "x4",
(56,0)*{} = "x5",
(15,10)*+{\circ} = "xo",
(34,7)*{\dots}="",
(52,0)*{\cdots}="",
\ar ^{} "x0";"xc"
\ar ^{a} "x1";"x0"
\ar ^{a} "x2";"x1"
\ar ^{a} "x3";"x2"
\ar ^{a} "x4";"x3"
%\ar @{.}^{} "x5";"x4"
\ar _{a} "xo";"x0"
\ar _{a} "xo";"x1"
\ar ^(.8){a} "xo";"x2"
\ar ^(.8){a} "xo";"x3"
\ar (15,15);"xo"
\end{xy}
\end{center}
\end{minipage}
\end{myexample}


\begin{myexample}\label{example:bwdsimnecessary}
The infinite trace of the nondeterministic automata $\mathcal{X}$ below is included in the infinite trace of $\mathcal{Y}$.
%The following pair of nondeterministic automata $\mathcal{X}$ and $\mathcal{Y}$ exhibits infinite trace inclusion.
There exists no forward simulation from $\mathcal{X}$ to $\mathcal{Y}$
 but a TIF-backward simulation does exist.

\begin{minipage}{0.5\hsize}
\begin{center}
\begin{xy}
(-25,0)*{} = "",
(-8,8)*{\mathcal{X}} = "",
(20,0)*+{\circ} = "x0",
(10,0)*+{\circ} = "x1",
(0,0)*+{\circ} = "x2",
\ar (25,0);"x0"
\ar _{a} "x0";"x1"
\ar @/^/ ^{b} "x1";"x2"
\ar @/_/ _{c} "x1";"x2"
\ar @(dl,ul)^{d} "x2";"x2"
\end{xy}
\end{center}
\end{minipage}
%
\begin{minipage}{0.5\hsize}
\begin{center}
\begin{xy}
%(-25,0)*{} = "",
(-8,8)*{\mathcal{Y}} = "",
(20,0)*+{\circ} = "x0",
(10,-5)*+{\circ} = "x11",
(10,5)*+{\circ} = "x12",
(0,0)*+{\circ} = "x2",
\ar (25,0);"x0"
\ar ^{a} "x0";"x11"
\ar _{a} "x0";"x12"
\ar ^{b} "x11";"x2"
\ar _{c} "x12";"x2"
\ar @(dl,ul)^{d} "x2";"x2"
\end{xy}
\end{center}
\end{minipage}
\end{myexample}



\end{document}

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