\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{thm\_2EpatternMatches\_2EREDUNDANT\_\_ROWS\_\_INFOS\_\_CONJ\_\_REWRITE (TMdJUk3Q5XSiP8BD5krj6eVEXZos2fX8HWg)}
\maketitle
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddee}
We define $\mname{c\_2Emin\_2E\_3D}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} x \in {A} .  {{\uniddcgcg} y \in {A} .  {{inj\_\_o}~{({{x}={y}})}}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfeef}
We define $\mname{c\_2Ebool\_2ET}$ to be ${({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{2}}})}})}}~{({{\uniddcgcg} V0x \in {2} .  {V0x}})}})}}~{({{\uniddcgcg} V1x \in {2} .  {V1x}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdbd}
We define $\mname{c\_2Ebool\_2E\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V0P}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfege}
We define $\mname{c\_2Ebool\_2EF}$ to be ${({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V0t \in {2} .  {V0t}})}})}$.
\end{definition}
Let $ ty\_2Elist\_2Elist : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Elist\_2Elist}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfemgjgdhehpfcdfemgjgdheh}
\end{equation}
Let $ c\_2Elist\_2ENIL : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ENIL~A\_27a \in (ty\_2Elist\_2Elist~\\
A\_27a)\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfeoejeme}
\end{equation}
Let $ c\_2Elist\_2ECONS : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ECONS~A\_27a \in (((ty\_2Elist\_2Elist~\\
A\_27a)^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{A\_27a}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfedepeoedf}
\end{equation}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddeepfddeepfddfe}
We define $\mname{c\_2Emin\_2E\_3D\_3D\_3E}$ to be ${{\uniddcgcg} P \in {2} .  {{\uniddcgcg} Q \in {2} .  {{inj\_\_o}~{({{{p}~{P}}{\unicdbdjdcd}{{p}~{Q}}})}}}}$ of type ${\iota}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdgepffdde}
We define $\mname{c\_2Ebool\_2E\_2F\_5C}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}})}}~{V2t}})}})}})}})}})}$.
\end{definition}
Let $ c\_2Elist\_2EMAP2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}c\_2Elist\_2EMAP2~A\_27a~A\_27b~A\_27c \in ((((ty\_2Elist\_2Elist~\\
A\_27a)^{{({{ty\_2Elist\_2Elist}~{A\_27c}})}})^{{({{ty\_2Elist\_2Elist}~{A\_27b}})}})^{{({{({{A\_27a}^{{A\_27c}}})}^{{A\_27b}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfenebeafcd}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgehehfgchognebgehdgigfgdhpfcdfecffeeeffoeeebeoeefpfpfcfpehfdfpfpfjeoegepedfpfpfdepeoeke}
We define $\mname{c\_2EpatternMatches\_2EREDUNDANT\_\_ROWS\_\_INFOS\_\_CONJ}$ to be ${{\uniddcgcg} V0ip1 \in {({{ty\_2Elist\_2Elist}~{2}})} .  {{\uniddcgcg} V1ip2 \in {({{ty\_2Elist\_2Elist}~{2}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{{c\_2Elist\_2EMAP2}~{2}}~{2}}~{2}})}}~{({{\uniddcgcg} V2i1 \in {2} .  {({{\uniddcgcg} V3i2 \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{V2i1}})}}~{V3i2}})}})}})}})}}~{V0ip1}})}}~{V1ip2}})}}}$.
\end{definition}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdcdcdhd}(p~V0t)){\unicdbdjded}False){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}False){\unicdbdjded}False){\unicdcdcdhd}\\
(((p~V0t){\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ((V0x=V0x){\unicdbdjded}\\
True))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecffegemepfpfdemebeffdffe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}{\unicdcdadad} A\_27d .  nonempty~A\_27d{\unicdbdjdcd}{\unicdcdadad} A\_27e .  nonempty~\\
A\_27e{\unicdbdjdcd}{\unicdcdadad} A\_27f .  nonempty~A\_27f{\unicdbdjdcd}(({\unicdcdadad} V0f \in ((A\_27c^{{A\_27b}})^{{A\_27a}}) .  \\
((ap~(ap~(ap~(c\_2Elist\_2EMAP2~A\_27c~A\_27a~A\_27b)~V0f)~(c\_2Elist\_2ENIL~\\
A\_27a))~(c\_2Elist\_2ENIL~A\_27b))=(c\_2Elist\_2ENIL~A\_27c))){\unicdcdcdhd}(\\
{\unicdcdadad} V1f \in ((A\_27f^{{A\_27e}})^{{A\_27d}}) .  ({\unicdcdadad} V2h1 \in A\_27d .  ({\unicdcdadad} V3t1 \in \\
(ty\_2Elist\_2Elist~A\_27d) .  ({\unicdcdadad} V4h2 \in A\_27e .  ({\unicdcdadad} V5t2 \in \\
(ty\_2Elist\_2Elist~A\_27e) .  ((ap~(ap~(ap~(c\_2Elist\_2EMAP2~A\_27f~\\
A\_27d~A\_27e)~V1f)~(ap~(ap~(c\_2Elist\_2ECONS~A\_27d)~V2h1)~V3t1))~(\\
ap~(ap~(c\_2Elist\_2ECONS~A\_27e)~V4h2)~V5t2))=(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27f)~(ap~(ap~V1f~V2h1)~V4h2))~(ap~(ap~(ap~(c\_2Elist\_2EMAP2~A\_27f~\\
A\_27d~A\_27e)~V1f)~V3t1)~V5t2)))))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfenebeafcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0a0 \in A\_27a .  ({\unicdcdadad} V1a1 \in \\
(ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V2a0\_27 \in A\_27a .  ({\unicdcdadad} V3a1\_27 \in \\
(ty\_2Elist\_2Elist~A\_27a) .  (((ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V0a0)~\\
V1a1)=(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V2a0\_27)~V3a1\_27)){\unicdbdjded}((V0a0=\\
V2a0\_27){\unicdcdcdhd}(V1a1=V3a1\_27)))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfedepeoedfpfpfbdbd}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfeahbgehehfgchognebgehdgigfgdhpfcdfecffeeeffoeeebeoeefpfpfcfpehfdfpfpfjeoegepedfpfpfdepeoekepfpfcffehfcfjeeffe}
$$\begin{array}{c}({\unicdcdadad} V0i1 \in 2 .  ({\unicdcdadad} V1is1 \in (ty\_2Elist\_2Elist~2) .  ({\unicdcdadad} V2i2 \in \\
2 .  ({\unicdcdadad} V3is2 \in (ty\_2Elist\_2Elist~2) .  (((ap~(ap~\mname{c\_2EpatternMatches\_2EREDUNDANT\_\_ROWS\_\_INFOS\_\_CONJ}~\\
(c\_2Elist\_2ENIL~2))~(c\_2Elist\_2ENIL~2))=(c\_2Elist\_2ENIL~2)){\unicdcdcdhd}\\
((ap~(ap~\mname{c\_2EpatternMatches\_2EREDUNDANT\_\_ROWS\_\_INFOS\_\_CONJ}~\\
(ap~(ap~(c\_2Elist\_2ECONS~2)~V0i1)~V1is1))~(ap~(ap~(c\_2Elist\_2ECONS~\\
2)~V2i2)~V3is2))=(ap~(ap~(c\_2Elist\_2ECONS~2)~(ap~(ap~\mname{c\_2Ebool\_2E\_2F\_5C}~\\
V0i1)~V2i2))~(ap~(ap~\mname{c\_2EpatternMatches\_2EREDUNDANT\_\_ROWS\_\_INFOS\_\_CONJ}~\\
V1is1)~V3is2))))))))\end{array}$$
\end{theorem}
\end{document}
