\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{t2\_gate\_3 (TMdbYR1usLT5AxiuTVa3M4eGGRkd9CVTEf6)}
\maketitle
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k2\_gate\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k1\_gate\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_gate\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k8\_gate\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (\preoh (\preoh v1\_xboole\_0~(k3\_gate\_1~X0~X1)){\unicdcdcdhd}((v1\_xboole\_0~\\
X0){\unicdcdcdhd}(v1\_xboole\_0~X1))){\unicdcdcdhd}(\preoh (\preoh (v1\_xboole\_0~X0){\unicdcdcdhd}(v1\_xboole\_0~X1)){\unicdcdcdhd}\\
(v1\_xboole\_0~(k3\_gate\_1~X0~X1)))\end{array}
\label{hyp:ehfdpfhgbgehfgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (\preoh v1\_xboole\_0~(k2\_gate\_1~X0~X1)){\unicdbdjded}((\preoh v1\_xboole\_0~\\
X0){\unicdcdcdhd}(\preoh v1\_xboole\_0~X1))\end{array}
\label{hyp:ehedpfhgbgehfgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  (\preoh (v1\_xboole\_0~(\\
k2\_gate\_1~X0~X1)){\unicdcdcdhd}(v1\_xboole\_0~(k3\_gate\_1~(k1\_gate\_1~X0)~(k1\_gate\_1~\\
X1)))){\unicdcdcdhd}(((\preoh v1\_xboole\_0~(k3\_gate\_1~(k1\_gate\_1~X0)~(k1\_gate\_1~\\
X1))){\unicdbdjdcd}(v1\_xboole\_0~(k2\_gate\_1~X0~X1))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k3\_gate\_1~\\
X0~X1)){\unicdcdcdhd}((\preoh v1\_xboole\_0~(k3\_gate\_1~X2~X1)){\unicdcdcdhd}(v1\_xboole\_0~(k3\_gate\_1~\\
(k2\_gate\_1~X0~X2)~X1)))){\unicdcdcdhd}(((\preoh v1\_xboole\_0~(k3\_gate\_1~(k2\_gate\_1~\\
X0~X2)~X1)){\unicdbdjdcd}((\preoh v1\_xboole\_0~(k3\_gate\_1~X0~X1)){\unicdcdcdhd}(\preoh v1\_xboole\_0~(\\
k3\_gate\_1~X2~X1)))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k3\_gate\_1~X0~X1)){\unicdcdcdhd}((\preoh v1\_xboole\_0~\\
(k3\_gate\_1~X2~X1)){\unicdcdcdhd}((\preoh v1\_xboole\_0~(k3\_gate\_1~X3~X1)){\unicdcdcdhd}(v1\_xboole\_0~\\
(k3\_gate\_1~(k8\_gate\_1~X0~X2~X3)~X1))))){\unicdcdcdhd}(((\preoh v1\_xboole\_0~(k3\_gate\_1~\\
(k8\_gate\_1~X0~X2~X3)~X1)){\unicdbdjdcd}((\preoh v1\_xboole\_0~(k3\_gate\_1~X0~X1)){\unicdcdcdhd}((\\
\preoh v1\_xboole\_0~(k3\_gate\_1~X2~X1)){\unicdcdcdhd}(\preoh v1\_xboole\_0~(k3\_gate\_1~X3~X1))))){\unicdcdcdhd}\\
(\preoh (\preoh v1\_xboole\_0~(k3\_gate\_1~X0~X1)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v1\_xboole\_0~\\
X2)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X2){\unicdcdcdhd}(v1\_xboole\_0~X0)){\unicdcdcdhd}(v1\_xboole\_0~(\\
k3\_gate\_1~X2~X1))))))))))\end{array}
\label{hyp:ehddpfhgbgehfgpfcd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({\preoh{{{v1\_xboole\_0}~{X0}}}})}{\unicdbdjdcd}{({{v1\_xboole\_0}~{({{k1\_gate\_1}~{X0}})}})}}}\label{hyp:ggdgcdpfhgbgehfgpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xboole\_0}~{X0}})}{\unicdbdjdcd}{({\preoh{{{v1\_xboole\_0}~{({{k1\_gate\_1}~{X0}})}}}})}}}\label{hyp:ggdgbdpfhgbgehfgpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{\unicdcdadad} X1 .  {{{{k3\_gate\_1}~{X0}}~{X1}}={{{k3\_gate\_1}~{X1}}~{X0}}}}}\label{hyp:dgpgngngfhehbgehjgghjgehjhpflgddpfhgbgehfgpfbd}
\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:ehcdpfhgbgehfgpfdd}
$$\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 .  \preoh (\preoh (\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v1\_xboole\_0~(k2\_gate\_1~(k1\_gate\_1~\\
X9)~(k1\_gate\_1~X8)))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k2\_gate\_1~(k1\_gate\_1~\\
X9)~(k1\_gate\_1~X8))){\unicdcdcdhd}(v1\_xboole\_0~X0)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X1){\unicdcdcdhd}\\
(v1\_xboole\_0~(k2\_gate\_1~(k1\_gate\_1~X9)~X8))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~\\
(k2\_gate\_1~(k1\_gate\_1~X9)~X8)){\unicdcdcdhd}(v1\_xboole\_0~X1)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~\\
X2){\unicdcdcdhd}(v1\_xboole\_0~(k2\_gate\_1~X9~(k1\_gate\_1~X8)))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~\\
(k2\_gate\_1~X9~(k1\_gate\_1~X8))){\unicdcdcdhd}(v1\_xboole\_0~X2)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~\\
X3){\unicdcdcdhd}(v1\_xboole\_0~(k2\_gate\_1~X9~X8))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k2\_gate\_1~\\
X9~X8)){\unicdcdcdhd}(v1\_xboole\_0~X3)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X4){\unicdcdcdhd}(v1\_xboole\_0~\\
(k2\_gate\_1~(k1\_gate\_1~X11)~(k1\_gate\_1~X10)))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~\\
(k2\_gate\_1~(k1\_gate\_1~X11)~(k1\_gate\_1~X10))){\unicdcdcdhd}(v1\_xboole\_0~X4)){\unicdcdcdhd}\\
((\preoh (\preoh v1\_xboole\_0~X5){\unicdcdcdhd}(v1\_xboole\_0~(k2\_gate\_1~(k1\_gate\_1~X11)~\\
X10))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k2\_gate\_1~(k1\_gate\_1~X11)~X10)){\unicdcdcdhd}(v1\_xboole\_0~\\
X5)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X6){\unicdcdcdhd}(v1\_xboole\_0~(k2\_gate\_1~X11~(k1\_gate\_1~\\
X10)))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k2\_gate\_1~X11~(k1\_gate\_1~X10))){\unicdcdcdhd}(\\
v1\_xboole\_0~X6)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X7){\unicdcdcdhd}(v1\_xboole\_0~(k2\_gate\_1~\\
X11~X10))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k2\_gate\_1~X11~X10)){\unicdcdcdhd}(v1\_xboole\_0~\\
X7)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X10){\unicdcdcdhd}(v1\_xboole\_0~(k2\_gate\_1~(k1\_gate\_1~\\
X9)~X12))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k2\_gate\_1~(k1\_gate\_1~X9)~X12)){\unicdcdcdhd}\\
(v1\_xboole\_0~X10)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X11){\unicdcdcdhd}(v1\_xboole\_0~(k2\_gate\_1~\\
X8~X12))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k2\_gate\_1~X8~X12)){\unicdcdcdhd}(v1\_xboole\_0~\\
X11)){\unicdcdcdhd}(\preoh (\preoh (\preoh v1\_xboole\_0~X5){\unicdcdcdhd}(v1\_xboole\_0~(k2\_gate\_1~X0~X12))){\unicdcdcdhd}\\
((\preoh (\preoh v1\_xboole\_0~(k2\_gate\_1~X0~X12)){\unicdcdcdhd}(v1\_xboole\_0~X5)){\unicdcdcdhd}((\preoh (\\
\preoh v1\_xboole\_0~X7){\unicdcdcdhd}(v1\_xboole\_0~(k2\_gate\_1~X1~X12))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~\\
(k2\_gate\_1~X1~X12)){\unicdcdcdhd}(v1\_xboole\_0~X7)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X6){\unicdcdcdhd}\\
(v1\_xboole\_0~(k2\_gate\_1~X3~X12))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k2\_gate\_1~\\
X3~X12)){\unicdcdcdhd}(v1\_xboole\_0~X6)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X4){\unicdcdcdhd}(v1\_xboole\_0~\\
(k3\_gate\_1~(k2\_gate\_1~X2~X12)~(k1\_gate\_1~X12)))){\unicdcdcdhd}(\preoh (\preoh v1\_xboole\_0~\\
(k3\_gate\_1~(k2\_gate\_1~X2~X12)~(k1\_gate\_1~X12))){\unicdcdcdhd}(v1\_xboole\_0~\\
X4))))))))))))))))))))))))))))\end{array}$$
\end{theorem}
\end{document}
