\documentclass{article}
\newtheorem{axiom}{Axiom}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{conjecture}{Conjecture}
\newcommand\unicdcdgdad{\not=}
\newcommand\unicdcdcdbg{\cup}
\newcommand\unicdcdbdgd{\setminus}
\newcommand\unicdcdcdjd{\cap}
\newcommand\unicdcdadad{\forall}
\newcommand\unicdbdjdcd{\Rightarrow}
\newcommand\unicdcdadjd{\notin}
\newcommand\unicdbdjded{\Leftrightarrow}
\newcommand\unicdcdcdid{\lor}
\newcommand\unicdcdcdhd{\land}
\newcommand\uniddcgcg{\lambda}
\newcommand\unicdcdaddd{\exists}
\newcommand\unicdcdidid{\not\subseteq}
\newcommand\preoh[1]{\lnot #1}
\newcommand\mname[1]{{\mathsf{#1}}}
\begin{document}
\title{t3\_gate\_4 (TMHiUzygrbFKFbrZaeqo6XUXpS9YMBnzm3r)}
\maketitle
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k4\_gate\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k2\_gate\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(\preoh v1\_xboole\_0~X1)){\unicdbdjdcd}\\
(v1\_xboole\_0~(k4\_gate\_1~X0~X1))\end{array}
\label{hyp:ggdgjdpfhgbgehfgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xboole\_0~X0){\unicdcdcdhd}(v1\_xboole\_0~X1)){\unicdbdjdcd}(\\
v1\_xboole\_0~(k4\_gate\_1~X0~X1))\end{array}
\label{hyp:ggdgidpfhgbgehfgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xboole\_0~X0){\unicdcdcdhd}(\preoh v1\_xboole\_0~X1)){\unicdbdjdcd}\\
(\preoh v1\_xboole\_0~(k4\_gate\_1~X0~X1))\end{array}
\label{hyp:ggdghdpfhgbgehfgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (v1\_xboole\_0~X0){\unicdbdjdcd}(v1\_xboole\_0~(k2\_gate\_1~\\
X0~X1))\end{array}
\label{hyp:ggdgedpfhgbgehfgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(\preoh v1\_xboole\_0~X1)){\unicdbdjdcd}\\
(\preoh v1\_xboole\_0~(k2\_gate\_1~X0~X1))\end{array}
\label{hyp:ggdgddpfhgbgehfgpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{\unicdcdadad} X1 .  {{{{k4\_gate\_1}~{X0}}~{X1}}={{{k4\_gate\_1}~{X1}}~{X0}}}}}\label{hyp:dgpgngngfhehbgehjgghjgehjhpflgedpfhgbgehfgpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{\unicdcdadad} X1 .  {{{{k2\_gate\_1}~{X0}}~{X1}}={{{k2\_gate\_1}~{X1}}~{X0}}}}}\label{hyp:dgpgngngfhehbgehjgghjgehjhpflgcdpfhgbgehfgpfbd}
\end{equation}
\begin{theorem}\label{thm:ehddpfhgbgehfgpfed}
$$\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  {\unicdcdadad} X4 .  {\unicdcdadad} X5 .  \\
{\unicdcdadad} X6 .  {\unicdcdadad} X7 .  {\unicdcdadad} X8 .  {\unicdcdadad} X9 .  {\unicdcdadad} X10 .  {\unicdcdadad} X11 .  \\
{\unicdcdadad} X12 .  {\unicdcdadad} X13 .  {\unicdcdadad} X14 .  {\unicdcdadad} X15 .  {\unicdcdadad} X16 .  \\
{\unicdcdadad} X17 .  {\unicdcdadad} X18 .  {\unicdcdadad} X19 .  {\unicdcdadad} X20 .  {\unicdcdadad} X21 .  \\
{\unicdcdadad} X22 .  {\unicdcdadad} X23 .  {\unicdcdadad} X24 .  {\unicdcdadad} X25 .  {\unicdcdadad} X26 .  \\
{\unicdcdadad} X27 .  {\unicdcdadad} X28 .  {\unicdcdadad} X29 .  {\unicdcdadad} X30 .  {\unicdcdadad} X31 .  \\
{\unicdcdadad} X32 .  {\unicdcdadad} X33 .  {\unicdcdadad} X34 .  {\unicdcdadad} X35 .  {\unicdcdadad} X36 .  \\
{\unicdcdadad} X37 .  {\unicdcdadad} X38 .  (v1\_xboole\_0~X37){\unicdbdjdcd}((v1\_xboole\_0~X0){\unicdcdcdid}\\
(((\preoh v1\_xboole\_0~X25){\unicdcdcdhd}(v1\_xboole\_0~(k4\_gate\_1~X38~X24))){\unicdcdcdid}(((\\
\preoh v1\_xboole\_0~(k4\_gate\_1~X38~X24)){\unicdcdcdhd}(v1\_xboole\_0~X25)){\unicdcdcdid}(((\preoh v1\_xboole\_0~\\
X26){\unicdcdcdhd}(v1\_xboole\_0~(k4\_gate\_1~X13~(k2\_gate\_1~X1~X25)))){\unicdcdcdid}(((\preoh v1\_xboole\_0~\\
(k4\_gate\_1~X13~(k2\_gate\_1~X1~X25))){\unicdcdcdhd}(v1\_xboole\_0~X26)){\unicdcdcdid}(((\preoh v1\_xboole\_0~\\
X27){\unicdcdcdhd}(v1\_xboole\_0~(k4\_gate\_1~X14~(k2\_gate\_1~X2~X25)))){\unicdcdcdid}(((\preoh v1\_xboole\_0~\\
(k4\_gate\_1~X14~(k2\_gate\_1~X2~X25))){\unicdcdcdhd}(v1\_xboole\_0~X27)){\unicdcdcdid}(((\preoh v1\_xboole\_0~\\
X28){\unicdcdcdhd}(v1\_xboole\_0~(k4\_gate\_1~X15~(k2\_gate\_1~X3~X25)))){\unicdcdcdid}(((\preoh v1\_xboole\_0~\\
(k4\_gate\_1~X15~(k2\_gate\_1~X3~X25))){\unicdcdcdhd}(v1\_xboole\_0~X28)){\unicdcdcdid}(((\preoh v1\_xboole\_0~\\
X29){\unicdcdcdhd}(v1\_xboole\_0~(k4\_gate\_1~X16~(k2\_gate\_1~X4~X25)))){\unicdcdcdid}(((\preoh v1\_xboole\_0~\\
(k4\_gate\_1~X16~(k2\_gate\_1~X4~X25))){\unicdcdcdhd}(v1\_xboole\_0~X29)){\unicdcdcdid}(((\preoh v1\_xboole\_0~\\
X30){\unicdcdcdhd}(v1\_xboole\_0~(k4\_gate\_1~X17~(k2\_gate\_1~X5~X25)))){\unicdcdcdid}(((\preoh v1\_xboole\_0~\\
(k4\_gate\_1~X17~(k2\_gate\_1~X5~X25))){\unicdcdcdhd}(v1\_xboole\_0~X30)){\unicdcdcdid}(((\preoh v1\_xboole\_0~\\
X31){\unicdcdcdhd}(v1\_xboole\_0~(k4\_gate\_1~X18~(k2\_gate\_1~X6~X25)))){\unicdcdcdid}(((\preoh v1\_xboole\_0~\\
(k4\_gate\_1~X18~(k2\_gate\_1~X6~X25))){\unicdcdcdhd}(v1\_xboole\_0~X31)){\unicdcdcdid}(((\preoh v1\_xboole\_0~\\
X32){\unicdcdcdhd}(v1\_xboole\_0~(k4\_gate\_1~X19~(k2\_gate\_1~X7~X25)))){\unicdcdcdid}(((\preoh v1\_xboole\_0~\\
(k4\_gate\_1~X19~(k2\_gate\_1~X7~X25))){\unicdcdcdhd}(v1\_xboole\_0~X32)){\unicdcdcdid}(((\preoh v1\_xboole\_0~\\
X33){\unicdcdcdhd}(v1\_xboole\_0~(k4\_gate\_1~X20~(k2\_gate\_1~X8~X25)))){\unicdcdcdid}(((\preoh v1\_xboole\_0~\\
(k4\_gate\_1~X20~(k2\_gate\_1~X8~X25))){\unicdcdcdhd}(v1\_xboole\_0~X33)){\unicdcdcdid}(((\preoh v1\_xboole\_0~\\
X34){\unicdcdcdhd}(v1\_xboole\_0~(k4\_gate\_1~X21~(k2\_gate\_1~X9~X25)))){\unicdcdcdid}(((\preoh v1\_xboole\_0~\\
(k4\_gate\_1~X21~(k2\_gate\_1~X9~X25))){\unicdcdcdhd}(v1\_xboole\_0~X34)){\unicdcdcdid}(((\preoh v1\_xboole\_0~\\
X35){\unicdcdcdhd}(v1\_xboole\_0~(k4\_gate\_1~X22~(k2\_gate\_1~X10~X25)))){\unicdcdcdid}(((\preoh \\
v1\_xboole\_0~(k4\_gate\_1~X22~(k2\_gate\_1~X10~X25))){\unicdcdcdhd}(v1\_xboole\_0~\\
X35)){\unicdcdcdid}(((\preoh v1\_xboole\_0~X36){\unicdcdcdhd}(v1\_xboole\_0~(k4\_gate\_1~X23~(k2\_gate\_1~\\
X11~X25)))){\unicdcdcdid}(((\preoh v1\_xboole\_0~(k4\_gate\_1~X23~(k2\_gate\_1~X11~X25))){\unicdcdcdhd}\\
(v1\_xboole\_0~X36)){\unicdcdcdid}((\preoh (\preoh v1\_xboole\_0~X36){\unicdcdcdhd}(v1\_xboole\_0~(k4\_gate\_1~\\
(k4\_gate\_1~X23~(k2\_gate\_1~X11~X24))~(k4\_gate\_1~X37~(k2\_gate\_1~X11~\\
X38))))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k4\_gate\_1~(k4\_gate\_1~X23~(k2\_gate\_1~\\
X11~X24))~(k4\_gate\_1~X37~(k2\_gate\_1~X11~X38)))){\unicdcdcdhd}(v1\_xboole\_0~X36)){\unicdcdcdhd}\\
((\preoh (\preoh v1\_xboole\_0~X35){\unicdcdcdhd}(v1\_xboole\_0~(k4\_gate\_1~(k4\_gate\_1~X22~\\
(k2\_gate\_1~X10~X24))~(k4\_gate\_1~X37~(k2\_gate\_1~X10~X38))))){\unicdcdcdhd}((\\
\preoh (\preoh v1\_xboole\_0~(k4\_gate\_1~(k4\_gate\_1~X22~(k2\_gate\_1~X10~X24))~\\
(k4\_gate\_1~X37~(k2\_gate\_1~X10~X38)))){\unicdcdcdhd}(v1\_xboole\_0~X35)){\unicdcdcdhd}((\preoh \\
(\preoh v1\_xboole\_0~X34){\unicdcdcdhd}(v1\_xboole\_0~(k4\_gate\_1~(k4\_gate\_1~X21~(k2\_gate\_1~\\
X9~X24))~(k4\_gate\_1~X37~(k2\_gate\_1~X9~X38))))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~\\
(k4\_gate\_1~(k4\_gate\_1~X21~(k2\_gate\_1~X9~X24))~(k4\_gate\_1~X37~(k2\_gate\_1~\\
X9~X38)))){\unicdcdcdhd}(v1\_xboole\_0~X34)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X33){\unicdcdcdhd}(v1\_xboole\_0~\\
(k4\_gate\_1~(k4\_gate\_1~X20~(k2\_gate\_1~X8~X24))~(k4\_gate\_1~X37~(k2\_gate\_1~\\
X8~X38))))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k4\_gate\_1~(k4\_gate\_1~X20~(k2\_gate\_1~\\
X8~X24))~(k4\_gate\_1~X37~(k2\_gate\_1~X8~X38)))){\unicdcdcdhd}(v1\_xboole\_0~X33)){\unicdcdcdhd}\\
((\preoh (\preoh v1\_xboole\_0~X32){\unicdcdcdhd}(v1\_xboole\_0~(k4\_gate\_1~(k4\_gate\_1~X19~\\
(k2\_gate\_1~X7~X24))~(k4\_gate\_1~X37~(k2\_gate\_1~X7~X38))))){\unicdcdcdhd}((\preoh (\\
\preoh v1\_xboole\_0~(k4\_gate\_1~(k4\_gate\_1~X19~(k2\_gate\_1~X7~X24))~(k4\_gate\_1~\\
X37~(k2\_gate\_1~X7~X38)))){\unicdcdcdhd}(v1\_xboole\_0~X32)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~\\
X31){\unicdcdcdhd}(v1\_xboole\_0~(k4\_gate\_1~(k4\_gate\_1~X18~(k2\_gate\_1~X6~X24))~\\
(k4\_gate\_1~X37~(k2\_gate\_1~X6~X38))))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k4\_gate\_1~\\
(k4\_gate\_1~X18~(k2\_gate\_1~X6~X24))~(k4\_gate\_1~X37~(k2\_gate\_1~X6~\\
X38)))){\unicdcdcdhd}(v1\_xboole\_0~X31)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X30){\unicdcdcdhd}(v1\_xboole\_0~\\
(k4\_gate\_1~(k4\_gate\_1~X17~(k2\_gate\_1~X5~X24))~(k4\_gate\_1~X37~(k2\_gate\_1~\\
X5~X38))))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k4\_gate\_1~(k4\_gate\_1~X17~(k2\_gate\_1~\\
X5~X24))~(k4\_gate\_1~X37~(k2\_gate\_1~X5~X38)))){\unicdcdcdhd}(v1\_xboole\_0~X30)){\unicdcdcdhd}\\
((\preoh (\preoh v1\_xboole\_0~X29){\unicdcdcdhd}(v1\_xboole\_0~(k4\_gate\_1~(k4\_gate\_1~X16~\\
(k2\_gate\_1~X4~X24))~(k4\_gate\_1~X37~(k2\_gate\_1~X4~X38))))){\unicdcdcdhd}((\preoh (\\
\preoh v1\_xboole\_0~(k4\_gate\_1~(k4\_gate\_1~X16~(k2\_gate\_1~X4~X24))~(k4\_gate\_1~\\
X37~(k2\_gate\_1~X4~X38)))){\unicdcdcdhd}(v1\_xboole\_0~X29)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~\\
X28){\unicdcdcdhd}(v1\_xboole\_0~(k4\_gate\_1~(k4\_gate\_1~X15~(k2\_gate\_1~X3~X24))~\\
(k4\_gate\_1~X37~(k2\_gate\_1~X3~X38))))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k4\_gate\_1~\\
(k4\_gate\_1~X15~(k2\_gate\_1~X3~X24))~(k4\_gate\_1~X37~(k2\_gate\_1~X3~\\
X38)))){\unicdcdcdhd}(v1\_xboole\_0~X28)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X27){\unicdcdcdhd}(v1\_xboole\_0~\\
(k4\_gate\_1~(k4\_gate\_1~X14~(k2\_gate\_1~X2~X24))~(k4\_gate\_1~X37~(k2\_gate\_1~\\
X2~X38))))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k4\_gate\_1~(k4\_gate\_1~X14~(k2\_gate\_1~\\
X2~X24))~(k4\_gate\_1~X37~(k2\_gate\_1~X2~X38)))){\unicdcdcdhd}(v1\_xboole\_0~X27)){\unicdcdcdhd}\\
((\preoh (\preoh v1\_xboole\_0~X26){\unicdcdcdhd}(v1\_xboole\_0~(k4\_gate\_1~(k4\_gate\_1~X13~\\
(k2\_gate\_1~X1~X24))~(k4\_gate\_1~X37~(k2\_gate\_1~X1~X38))))){\unicdcdcdhd}((\preoh (\\
\preoh v1\_xboole\_0~(k4\_gate\_1~(k4\_gate\_1~X13~(k2\_gate\_1~X1~X24))~(k4\_gate\_1~\\
X37~(k2\_gate\_1~X1~X38)))){\unicdcdcdhd}(v1\_xboole\_0~X26)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~\\
X25){\unicdcdcdhd}(v1\_xboole\_0~(k4\_gate\_1~(k4\_gate\_1~X37~(k2\_gate\_1~X0~X24))~\\
(k4\_gate\_1~X37~(k2\_gate\_1~X0~X38))))){\unicdcdcdhd}(\preoh (\preoh v1\_xboole\_0~(k4\_gate\_1~\\
(k4\_gate\_1~X37~(k2\_gate\_1~X0~X24))~(k4\_gate\_1~X37~(k2\_gate\_1~X0~\\
X38)))){\unicdcdcdhd}(v1\_xboole\_0~X25))))))))))))))))))))))))))))))))))))))))))))))))))\end{array}$$
\end{theorem}
\end{document}
