\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\_2 (TMcLFzD7Nk1v7a1uh23srERqSGHrDQp4xT2)}
\maketitle
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k2\_gate\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k3\_gate\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k1\_gate\_1 : {{\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 .  (\preoh v1\_xboole\_0~(k8\_gate\_1~X0~\\
X1~X2)){\unicdbdjded}((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}(\preoh v1\_xboole\_0~\\
X2)))\end{array}
\label{hyp:ehbdedpfhgbgehfgpfbd}
\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:ehddpfhgbgehfgpfcd}
$$\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  (\preoh (v1\_xboole\_0~(\\
k2\_gate\_1~X0~X1)){\unicdcdcdhd}(v1\_xboole\_0~(k3\_gate\_1~(k1\_gate\_1~X0)~(k1\_gate\_1~\\
X1)))){\unicdcdcdhd}(((\preoh v1\_xboole\_0~(k3\_gate\_1~(k1\_gate\_1~X0)~(k1\_gate\_1~\\
X1))){\unicdbdjdcd}(v1\_xboole\_0~(k2\_gate\_1~X0~X1))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k3\_gate\_1~\\
X0~X1)){\unicdcdcdhd}((\preoh v1\_xboole\_0~(k3\_gate\_1~X2~X1)){\unicdcdcdhd}(v1\_xboole\_0~(k3\_gate\_1~\\
(k2\_gate\_1~X0~X2)~X1)))){\unicdcdcdhd}(((\preoh v1\_xboole\_0~(k3\_gate\_1~(k2\_gate\_1~\\
X0~X2)~X1)){\unicdbdjdcd}((\preoh v1\_xboole\_0~(k3\_gate\_1~X0~X1)){\unicdcdcdhd}(\preoh v1\_xboole\_0~(\\
k3\_gate\_1~X2~X1)))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k3\_gate\_1~X0~X1)){\unicdcdcdhd}((\preoh v1\_xboole\_0~\\
(k3\_gate\_1~X2~X1)){\unicdcdcdhd}((\preoh v1\_xboole\_0~(k3\_gate\_1~X3~X1)){\unicdcdcdhd}(v1\_xboole\_0~\\
(k3\_gate\_1~(k8\_gate\_1~X0~X2~X3)~X1))))){\unicdcdcdhd}(((\preoh v1\_xboole\_0~(k3\_gate\_1~\\
(k8\_gate\_1~X0~X2~X3)~X1)){\unicdbdjdcd}((\preoh v1\_xboole\_0~(k3\_gate\_1~X0~X1)){\unicdcdcdhd}((\\
\preoh v1\_xboole\_0~(k3\_gate\_1~X2~X1)){\unicdcdcdhd}(\preoh v1\_xboole\_0~(k3\_gate\_1~X3~X1))))){\unicdcdcdhd}\\
(\preoh (\preoh v1\_xboole\_0~(k3\_gate\_1~X0~X1)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v1\_xboole\_0~\\
X2)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X2){\unicdcdcdhd}(v1\_xboole\_0~X0)){\unicdcdcdhd}(v1\_xboole\_0~(\\
k3\_gate\_1~X2~X1))))))))))\end{array}$$
\end{theorem}
\end{document}
