\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{t1\_gate\_2 (TMQDLbVYBWmrSh7r62PoiEGxY4JcSDMXYKS)}
\maketitle
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k8\_gate\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k1\_gate\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k4\_gate\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k3\_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 (\preoh v1\_xboole\_0~(k4\_gate\_1~X0~X1)){\unicdcdcdhd}((\preoh \\
(\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v1\_xboole\_0~X1)){\unicdcdcdhd}(\preoh (v1\_xboole\_0~X0){\unicdcdcdhd}(\preoh v1\_xboole\_0~\\
X1)))){\unicdcdcdhd}(\preoh (((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v1\_xboole\_0~X1)){\unicdcdcdid}((v1\_xboole\_0~\\
X0){\unicdcdcdhd}(\preoh v1\_xboole\_0~X1))){\unicdcdcdhd}(v1\_xboole\_0~(k4\_gate\_1~X0~X1)))\end{array}
\label{hyp:ehgdpfhgbgehfgpfbd}
\end{equation}
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}
\begin{theorem}\label{thm:ehbdpfhgbgehfgpfcd}
$$\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 .  \\
\preoh (\preoh (\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v1\_xboole\_0~(k8\_gate\_1~(k1\_gate\_1~X18)~\\
(k1\_gate\_1~X17)~(k1\_gate\_1~X16)))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k8\_gate\_1~\\
(k1\_gate\_1~X18)~(k1\_gate\_1~X17)~(k1\_gate\_1~X16))){\unicdcdcdhd}(v1\_xboole\_0~\\
X0)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X1){\unicdcdcdhd}(v1\_xboole\_0~(k8\_gate\_1~(k1\_gate\_1~\\
X18)~(k1\_gate\_1~X17)~X16))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k8\_gate\_1~(k1\_gate\_1~\\
X18)~(k1\_gate\_1~X17)~X16)){\unicdcdcdhd}(v1\_xboole\_0~X1)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~\\
X2){\unicdcdcdhd}(v1\_xboole\_0~(k8\_gate\_1~(k1\_gate\_1~X18)~X17~(k1\_gate\_1~X16)))){\unicdcdcdhd}\\
((\preoh (\preoh v1\_xboole\_0~(k8\_gate\_1~(k1\_gate\_1~X18)~X17~(k1\_gate\_1~X16))){\unicdcdcdhd}\\
(v1\_xboole\_0~X2)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X3){\unicdcdcdhd}(v1\_xboole\_0~(k8\_gate\_1~\\
(k1\_gate\_1~X18)~X17~X16))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k8\_gate\_1~(k1\_gate\_1~\\
X18)~X17~X16)){\unicdcdcdhd}(v1\_xboole\_0~X3)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X4){\unicdcdcdhd}(v1\_xboole\_0~\\
(k8\_gate\_1~X18~(k1\_gate\_1~X17)~(k1\_gate\_1~X16)))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~\\
(k8\_gate\_1~X18~(k1\_gate\_1~X17)~(k1\_gate\_1~X16))){\unicdcdcdhd}(v1\_xboole\_0~\\
X4)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X5){\unicdcdcdhd}(v1\_xboole\_0~(k8\_gate\_1~X18~(k1\_gate\_1~\\
X17)~X16))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k8\_gate\_1~X18~(k1\_gate\_1~X17)~X16)){\unicdcdcdhd}\\
(v1\_xboole\_0~X5)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X6){\unicdcdcdhd}(v1\_xboole\_0~(k8\_gate\_1~\\
X18~X17~(k1\_gate\_1~X16)))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k8\_gate\_1~X18~X17~\\
(k1\_gate\_1~X16))){\unicdcdcdhd}(v1\_xboole\_0~X6)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X7){\unicdcdcdhd}(\\
v1\_xboole\_0~(k8\_gate\_1~X18~X17~X16))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k8\_gate\_1~\\
X18~X17~X16)){\unicdcdcdhd}(v1\_xboole\_0~X7)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X8){\unicdcdcdhd}(v1\_xboole\_0~\\
(k8\_gate\_1~(k1\_gate\_1~X21)~(k1\_gate\_1~X20)~(k1\_gate\_1~X19)))){\unicdcdcdhd}\\
((\preoh (\preoh v1\_xboole\_0~(k8\_gate\_1~(k1\_gate\_1~X21)~(k1\_gate\_1~X20)~(\\
k1\_gate\_1~X19))){\unicdcdcdhd}(v1\_xboole\_0~X8)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X9){\unicdcdcdhd}(v1\_xboole\_0~\\
(k8\_gate\_1~(k1\_gate\_1~X21)~(k1\_gate\_1~X20)~X19))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~\\
(k8\_gate\_1~(k1\_gate\_1~X21)~(k1\_gate\_1~X20)~X19)){\unicdcdcdhd}(v1\_xboole\_0~\\
X9)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X10){\unicdcdcdhd}(v1\_xboole\_0~(k8\_gate\_1~(k1\_gate\_1~\\
X21)~X20~(k1\_gate\_1~X19)))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k8\_gate\_1~(k1\_gate\_1~\\
X21)~X20~(k1\_gate\_1~X19))){\unicdcdcdhd}(v1\_xboole\_0~X10)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~\\
X11){\unicdcdcdhd}(v1\_xboole\_0~(k8\_gate\_1~(k1\_gate\_1~X21)~X20~X19))){\unicdcdcdhd}((\preoh (\\
\preoh v1\_xboole\_0~(k8\_gate\_1~(k1\_gate\_1~X21)~X20~X19)){\unicdcdcdhd}(v1\_xboole\_0~\\
X11)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X12){\unicdcdcdhd}(v1\_xboole\_0~(k8\_gate\_1~X21~(k1\_gate\_1~\\
X20)~(k1\_gate\_1~X19)))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k8\_gate\_1~X21~(k1\_gate\_1~\\
X20)~(k1\_gate\_1~X19))){\unicdcdcdhd}(v1\_xboole\_0~X12)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~\\
X13){\unicdcdcdhd}(v1\_xboole\_0~(k8\_gate\_1~X21~(k1\_gate\_1~X20)~X19))){\unicdcdcdhd}((\preoh (\\
\preoh v1\_xboole\_0~(k8\_gate\_1~X21~(k1\_gate\_1~X20)~X19)){\unicdcdcdhd}(v1\_xboole\_0~\\
X13)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X14){\unicdcdcdhd}(v1\_xboole\_0~(k8\_gate\_1~X21~X20~\\
(k1\_gate\_1~X19)))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k8\_gate\_1~X21~X20~(k1\_gate\_1~\\
X19))){\unicdcdcdhd}(v1\_xboole\_0~X14)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X15){\unicdcdcdhd}(v1\_xboole\_0~\\
(k8\_gate\_1~X21~X20~X19))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k8\_gate\_1~X21~X20~\\
X19)){\unicdcdcdhd}(v1\_xboole\_0~X15)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X19){\unicdcdcdhd}(v1\_xboole\_0~\\
(k1\_gate\_1~X16))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~(k1\_gate\_1~X16)){\unicdcdcdhd}(v1\_xboole\_0~\\
X19)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X20){\unicdcdcdhd}(v1\_xboole\_0~(k4\_gate\_1~X16~X17))){\unicdcdcdhd}\\
((\preoh (\preoh v1\_xboole\_0~(k4\_gate\_1~X16~X17)){\unicdcdcdhd}(v1\_xboole\_0~X20)){\unicdcdcdhd}((\\
\preoh (\preoh v1\_xboole\_0~X21){\unicdcdcdhd}(v1\_xboole\_0~(k3\_gate\_1~(k2\_gate\_1~X18~(\\
k1\_gate\_1~X16))~(k2\_gate\_1~X16~(k4\_gate\_1~X17~X18))))){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~\\
(k3\_gate\_1~(k2\_gate\_1~X18~(k1\_gate\_1~X16))~(k2\_gate\_1~X16~(k4\_gate\_1~\\
X17~X18)))){\unicdcdcdhd}(v1\_xboole\_0~X21)){\unicdcdcdhd}(\preoh (\preoh (\preoh v1\_xboole\_0~X9){\unicdcdcdhd}(v1\_xboole\_0~\\
X0)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v1\_xboole\_0~X9)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~\\
X10){\unicdcdcdhd}(v1\_xboole\_0~X1)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X1){\unicdcdcdhd}(v1\_xboole\_0~X10)){\unicdcdcdhd}\\
((\preoh (\preoh v1\_xboole\_0~X11){\unicdcdcdhd}(v1\_xboole\_0~X2)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X2){\unicdcdcdhd}\\
(v1\_xboole\_0~X11)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X12){\unicdcdcdhd}(v1\_xboole\_0~X3)){\unicdcdcdhd}\\
((\preoh (\preoh v1\_xboole\_0~X3){\unicdcdcdhd}(v1\_xboole\_0~X12)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X13){\unicdcdcdhd}\\
(v1\_xboole\_0~X4)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X4){\unicdcdcdhd}(v1\_xboole\_0~X13)){\unicdcdcdhd}\\
((\preoh (\preoh v1\_xboole\_0~X14){\unicdcdcdhd}(v1\_xboole\_0~X5)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X5){\unicdcdcdhd}\\
(v1\_xboole\_0~X14)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X15){\unicdcdcdhd}(v1\_xboole\_0~X6)){\unicdcdcdhd}\\
((\preoh (\preoh v1\_xboole\_0~X6){\unicdcdcdhd}(v1\_xboole\_0~X15)){\unicdcdcdhd}((\preoh (\preoh v1\_xboole\_0~X8){\unicdcdcdhd}\\
(v1\_xboole\_0~X7)){\unicdcdcdhd}(\preoh (\preoh v1\_xboole\_0~X7){\unicdcdcdhd}(v1\_xboole\_0~X8))))))))))))))))))))))))))))))))))))))))))))))))))))))\end{array}$$
\end{theorem}
\end{document}
