\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\_2EPMATCH\_\_EQUIV\_\_ROWS\_\_CONS\_\_NONE (TMdy18fzwV1vepoinjLYposU4xcyFWzRnY7)}
\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}
\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:dgpfcdfecgpgpgmgpfcdfepfhdfe}
We define $\mname{c\_2Ebool\_2E\_7E}$ to be ${({{\uniddcgcg} V0t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepffddepfcdge}
We define $\mname{c\_2Ebool\_2E\_5C\_2F}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{V2t}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}}~{V2t}})}})}})}})}})}})}$.
\end{definition}
Let $ ty\_2Eone\_2Eone : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Eone\_2Eone}}\label{hyp:ogfgpfehjhpfcdfepgogfgpfcdfepgogfg}
\end{equation}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfedad}
We define $\mname{c\_2Emin\_2E\_40}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} P \in {{2}^{{A}}} .  {{\mathbf{if}}~{({{\unicdcdaddd} x \in {A} .  {{p}~{({{{ap}~{P}}~{x}})}}})}~{\mathbf{then}}~{({{the}~{({{\uniddcgcg} x .  {{{x} \in {A}}{\unicdcdcdhd}{{p}~{({{{ap}~{P}}~{x}})}}}})}})}~{\mathbf{else}}~{({{the}~{({{\uniddcgcg} x .  {{x} \in {A}}})}})}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfepgogfgpfcdfepgogfg}
We define $\mname{c\_2Eone\_2Eone}$ to be ${({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{ty\_2Eone\_2Eone}})}}~{({{\uniddcgcg} V0x \in {ty\_2Eone\_2Eone} .  {\mname{c\_2Ebool\_2ET}}})}})}$.
\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 $ ty\_2Esum\_2Esum : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A0 .  nonempty~A0{\unicdbdjdcd}{\unicdcdadad} A1 .  nonempty~A1{\unicdbdjdcd}nonempty~(ty\_2Esum\_2Esum~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfedhfhngpfcdfedhfhng}
\end{equation}
Let $ c\_2Esum\_2EABS\_\_sum : {{\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}c\_2Esum\_2EABS\_\_sum~\\
A\_27a~A\_27b \in ((ty\_2Esum\_2Esum~A\_27a~A\_27b)^{{({{({{({{2}^{{A\_27b}}})}^{{A\_27a}}})}^{{2}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhfhngpfcdfebecedfpfpfdhfhng}
\end{equation}
\begin{definition}\label{def:dgpfcdfedhfhngpfcdfejeoecf}
We define $\mname{c\_2Esum\_2EINR}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0e \in {A\_27b} .  {({{{ap}~{({{{c\_2Esum\_2EABS\_\_sum}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V1b \in {2} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{\uniddcgcg} V3y \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{V3y}})}}~{V0e}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{V1b}})}})}})}})}})}})}}}}$.
\end{definition}
Let $ ty\_2Eoption\_2Eoption : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Eoption\_2Eoption}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfepgahehjgpgogpfcdfepgahehjgpgog}
\end{equation}
Let $ c\_2Eoption\_2Eoption\_\_ABS : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Eoption\_2Eoption\_\_ABS~A\_27a \in \\
((ty\_2Eoption\_2Eoption~A\_27a)^{{({{{ty\_2Esum\_2Esum}~{A\_27a}}~{ty\_2Eone\_2Eone}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfepgahehjgpgogpfcdfepgahehjgpgogpfpfbecedf}
\end{equation}
\begin{definition}\label{def:dgpfcdfepgahehjgpgogpfcdfeoepeoefe}
We define $\mname{c\_2Eoption\_2ENONE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{c\_2Eoption\_2Eoption\_\_ABS}~{A\_27a}})}}~{({{{ap}~{({{{\mname{c\_2Esum\_2EINR}}~{A\_27a}}~{ty\_2Eone\_2Eone}})}}~{\mname{c\_2Eone\_2Eone}}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfhngpfcdfejeoeme}
We define $\mname{c\_2Esum\_2EINL}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0e \in {A\_27a} .  {({{{ap}~{({{{c\_2Esum\_2EABS\_\_sum}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V1b \in {2} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{\uniddcgcg} V3y \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2x}})}}~{V0e}})}})}}~{V1b}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfepgahehjgpgogpfcdfedfpenefe}
We define $\mname{c\_2Eoption\_2ESOME}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {({{{ap}~{({{c\_2Eoption\_2Eoption\_\_ABS}~{A\_27a}})}}~{({{{ap}~{({{{\mname{c\_2Esum\_2EINL}}~{A\_27a}}~{ty\_2Eone\_2Eone}})}}~{V0x}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfele}
We define $\mname{c\_2Ecombin\_2EK}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {({{\uniddcgcg} V1y \in {A\_27b} .  {V0x}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfedf}
We define $\mname{c\_2Ecombin\_2ES}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {({{\uniddcgcg} V0f \in {({{({{A\_27c}^{{A\_27b}}})}^{{A\_27a}}})} .  {({{\uniddcgcg} V1g \in {({{A\_27b}^{{A\_27a}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{V0f}}~{V2x}})}}~{({{{ap}~{V1g}}~{V2x}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfeje}
We define $\mname{c\_2Ecombin\_2EI}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{{ap}~{({{{{\mname{c\_2Ecombin\_2ES}}~{A\_27a}}~{({{A\_27a}^{{A\_27a}}})}}~{A\_27a}})}}~{({{{\mname{c\_2Ecombin\_2EK}}~{A\_27a}}~{({{A\_27a}^{{A\_27a}}})}})}})}}~{({{{\mname{c\_2Ecombin\_2EK}}~{A\_27a}}~{A\_27a}})}})}}$.
\end{definition}
Let $ c\_2Eoption\_2Eoption\_\_CASE : {{\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}c\_2Eoption\_2Eoption\_\_CASE~\\
A\_27a~A\_27b \in (((A\_27b^{{({{A\_27b}^{{A\_27a}}})}})^{{A\_27b}})^{{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfepgahehjgpgogpfcdfepgahehjgpgogpfpfdebedffe}
\end{equation}
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\_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}
Let $ c\_2Ebool\_2EARB : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A\_27a .  {{{nonempty}~{A\_27a}}{\unicdbdjdcd}{{{c\_2Ebool\_2EARB}~{A\_27a}} \in {A\_27a}}}}\label{hyp:ngfgngpfdgpfcdfecgpgpgmgpfcdfebecfce}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgehehfgchognebgehdgigfgdhpfcdfeafnebeefdeiepfpfjeoedepeneafmefeeffe}
We define $\mname{c\_2EpatternMatches\_2EPMATCH\_\_INCOMPLETE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{c\_2Ebool\_2EARB}~{A\_27a}})}}$.
\end{definition}
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\_2Eoption\_2EIS\_\_SOME : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Eoption\_2EIS\_\_SOME~A\_27a \in (\\
2^{{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfepgahehjgpgogpfcdfejedfpfpfdfpenefe}
\end{equation}
Let $ c\_2Elist\_2ELIST\_\_TO\_\_SET : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ELIST\_\_TO\_\_SET~A\_27a \in \\
((2^{{A\_27a}})^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfemejedfefpfpfefpepfpfdffeef}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfejeoe}
We define $\mname{c\_2Ebool\_2EIN}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {({{\uniddcgcg} V1f \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{V1f}}~{V0x}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfddge}
We define $\mname{c\_2Ebool\_2E\_3F}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{V0P}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{V0P}})}})}})}}$.
\end{definition}
Let $ c\_2EpatternMatches\_2EPMATCH : {{\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}c\_2EpatternMatches\_2EPMATCH~\\
A\_27a~A\_27b \in ((A\_27a^{{({{ty\_2Elist\_2Elist}~{({{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}^{{A\_27b}}})}})}})^{{A\_27b}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgehehfgchognebgehdgigfgdhpfcdfeafnebeefdeie}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgehehfgchognebgehdgigfgdhpfcdfeafnebeefdeiepfpffebfffjegfpfpfcfpehfdf}
We define $\mname{c\_2EpatternMatches\_2EPMATCH\_\_EQUIV\_\_ROWS}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0v \in {A\_27a} .  {{\uniddcgcg} V1rows1 \in {({{ty\_2Elist\_2Elist}~{({{({{ty\_2Eoption\_2Eoption}~{A\_27b}})}^{{A\_27a}}})}})} .  {{\uniddcgcg} V2rows2 \in {({{ty\_2Elist\_2Elist}~{({{({{ty\_2Eoption\_2Eoption}~{A\_27b}})}^{{A\_27a}}})}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{({{{ap}~{({{{ap}~{({{{c\_2EpatternMatches\_2EPMATCH}~{A\_27b}}~{A\_27a}})}}~{V0v}})}}~{V1rows1}})}})}}~{({{{ap}~{({{{ap}~{({{{c\_2EpatternMatches\_2EPMATCH}~{A\_27b}}~{A\_27a}})}}~{V0v}})}}~{V2rows2}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{({{ty\_2Eoption\_2Eoption}~{A\_27b}})}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V3r \in {({{({{ty\_2Eoption\_2Eoption}~{A\_27b}})}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{({{ty\_2Eoption\_2Eoption}~{A\_27b}})}^{{A\_27a}}})}})}}~{V3r}})}}~{({{{ap}~{({{c\_2Elist\_2ELIST\_\_TO\_\_SET}~{({{({{ty\_2Eoption\_2Eoption}~{A\_27b}})}^{{A\_27a}}})}})}}~{V1rows1}})}})}})}}~{({{{ap}~{({{c\_2Eoption\_2EIS\_\_SOME}~{A\_27b}})}}~{({{{ap}~{V3r}}~{V0v}})}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{({{ty\_2Eoption\_2Eoption}~{A\_27b}})}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V4r \in {({{({{ty\_2Eoption\_2Eoption}~{A\_27b}})}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{({{ty\_2Eoption\_2Eoption}~{A\_27b}})}^{{A\_27a}}})}})}}~{V4r}})}}~{({{{ap}~{({{c\_2Elist\_2ELIST\_\_TO\_\_SET}~{({{({{ty\_2Eoption\_2Eoption}~{A\_27b}})}^{{A\_27a}}})}})}}~{V2rows2}})}})}})}}~{({{{ap}~{({{c\_2Eoption\_2EIS\_\_SOME}~{A\_27b}})}}~{({{{ap}~{V4r}}~{V0v}})}})}})}})}})}})}})}}}}}}$.
\end{definition}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  (((p~V0t1){\unicdbdjdcd}(p~V1t2)){\unicdbdjdcd}(((p~\\
V1t2){\unicdbdjdcd}(p~V0t1)){\unicdbdjdcd}((p~V0t1){\unicdbdjded}(p~V1t2))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfbeoeefjedfjfnepfpfbeif}
\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} V0t \in 2 .  (((True{\unicdcdcdid}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}True){\unicdbdjded}True){\unicdcdcdhd}\\
(((False{\unicdcdcdid}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}False){\unicdbdjded}(p~V0t)){\unicdcdcdhd}(((p~V0t){\unicdcdcdid}\\
(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfepecfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjdcd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}True){\unicdbdjded}\\
True){\unicdcdcdhd}(((False{\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((\\
(p~V0t){\unicdbdjdcd}False){\unicdbdjded}(\preoh (p~V0t))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfdemebeffdffedf}
\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} V0x \in A\_27a .  ({\unicdcdadad} V1y \in \\
A\_27a .  ((V0x=V1y){\unicdbdjded}(V1y=V0x))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdfjfnepfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1Q \in \\
(2^{{A\_27a}}) .  (({\unicdcdaddd} V2x \in A\_27a .  ((p~(ap~V0P~V2x)){\unicdcdcdid}(p~(ap~V1Q~V2x)))){\unicdbdjded}\\
(({\unicdcdaddd} V3x \in A\_27a .  (p~(ap~V0P~V3x))){\unicdcdcdid}({\unicdcdaddd} V4x \in A\_27a .  (p~(\\
ap~V1Q~V4x)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefeifjedfefdfpfpfpecfpfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  ({\unicdcdadad} V2C \in 2 .  ((((p~V1B){\unicdcdcdid}\\
(p~V2C)){\unicdcdcdhd}(p~V0A)){\unicdbdjded}(((p~V1B){\unicdcdcdhd}(p~V0A)){\unicdcdcdid}((p~V2C){\unicdcdcdhd}(p~V0A)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecfjeheieefpfpfbeoeeepfpfpegffecfpfpfpecf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  ({\unicdcdadad} V2t3 \in 2 .  (((p~V0t1){\unicdbdjdcd}\\
((p~V1t2){\unicdbdjdcd}(p~V2t3))){\unicdbdjded}(((p~V0t1){\unicdcdcdhd}(p~V1t2)){\unicdbdjdcd}(p~V2t3))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfjeneafpfpfjeoeefcfpe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in 2 .  ({\unicdcdadad} V1x\_27 \in 2 .  ({\unicdcdadad} V2y \in 2 .  ({\unicdcdadad} V3y\_27 \in \\
2 .  ((((p~V0x){\unicdbdjded}(p~V1x\_27)){\unicdcdcdhd}((p~V1x\_27){\unicdbdjdcd}((p~V2y){\unicdbdjded}(p~V3y\_27)))){\unicdbdjdcd}\\
(((p~V0x){\unicdbdjdcd}(p~V2y)){\unicdbdjded}((p~V1x\_27){\unicdbdjdcd}(p~V3y\_27))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfdepeoehe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1a \in \\
A\_27a .  (({\unicdcdaddd} V2x \in A\_27a .  ((V2x=V1a){\unicdcdcdhd}(p~(ap~V0P~V2x)))){\unicdbdjded}(p~(\\
ap~V0P~V1a)))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeffoehfjeoeeepfpfefienecd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(({\unicdcdadad} V0x \in A\_27a .  ((p~(ap~(ap~\\
(\mname{c\_2Ebool\_2EIN}~A\_27a)~V0x)~(ap~(c\_2Elist\_2ELIST\_\_TO\_\_SET~A\_27a)~\\
(c\_2Elist\_2ENIL~A\_27a)))){\unicdbdjded}False)){\unicdcdcdhd}({\unicdcdadad} V1x \in A\_27a .  ({\unicdcdadad} V2h \in \\
A\_27a .  ({\unicdcdadad} V3t \in (ty\_2Elist\_2Elist~A\_27a) .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V1x)~(ap~(c\_2Elist\_2ELIST\_\_TO\_\_SET~A\_27a)~(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V2h)~V3t)))){\unicdbdjded}((V1x=V2h){\unicdcdcdid}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~\\
V1x)~(ap~(c\_2Elist\_2ELIST\_\_TO\_\_SET~A\_27a)~V3t)))))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfenefene}
\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} V0v \in A\_27b .  ({\unicdcdadad} V1f \in (A\_27b^{{A\_27a}}) .  ((ap~(ap~(ap~(c\_2Eoption\_2Eoption\_\_CASE~\\
A\_27a~A\_27b)~(\mname{c\_2Eoption\_2ENONE}~A\_27a))~V0v)~V1f)=V0v))){\unicdcdcdhd}({\unicdcdadad} V2x \in \\
A\_27a .  ({\unicdcdadad} V3v \in A\_27b .  ({\unicdcdadad} V4f \in (A\_27b^{{A\_27a}}) .  ((ap~(ap~\\
(ap~(c\_2Eoption\_2Eoption\_\_CASE~A\_27a~A\_27b)~(ap~(\mname{c\_2Eoption\_2ESOME}~\\
A\_27a)~V2x))~V3v)~V4f)=(ap~V4f~V2x))))))\end{array}
\label{hyp:ehigngpfcdfepgahehjgpgogpfcdfepgahehjgpgogpfpfdgbgdhfgpfpfegfggg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(({\unicdcdadad} V0x \in A\_27a .  ((p~(ap~(c\_2Eoption\_2EIS\_\_SOME~\\
A\_27a)~(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~V0x))){\unicdbdjded}True)){\unicdcdcdhd}((p~(ap~(c\_2Eoption\_2EIS\_\_SOME~\\
A\_27a)~(\mname{c\_2Eoption\_2ENONE}~A\_27a))){\unicdbdjded}False))\end{array}
\label{hyp:ehigngpfcdfepgahehjgpgogpfcdfejedfpfpfdfpenefepfpfeefege}
\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} V0v \in A\_27b .  ((ap~(ap~(c\_2EpatternMatches\_2EPMATCH~A\_27a~\\
A\_27b)~V0v)~(c\_2Elist\_2ENIL~((ty\_2Eoption\_2Eoption~A\_27a)^{{A\_27b}})))=\\
(\mname{c\_2EpatternMatches\_2EPMATCH\_\_INCOMPLETE}~A\_27a))){\unicdcdcdhd}({\unicdcdadad} V1v \in \\
A\_27b .  ({\unicdcdadad} V2r \in ((ty\_2Eoption\_2Eoption~A\_27a)^{{A\_27b}}) .  ({\unicdcdadad} V3rs \in \\
(ty\_2Elist\_2Elist~((ty\_2Eoption\_2Eoption~A\_27a)^{{A\_27b}})) .  ((\\
ap~(ap~(c\_2EpatternMatches\_2EPMATCH~A\_27a~A\_27b)~V1v)~(ap~(ap~(\\
c\_2Elist\_2ECONS~((ty\_2Eoption\_2Eoption~A\_27a)^{{A\_27b}}))~V2r)~V3rs))=\\
(ap~(ap~(ap~(c\_2Eoption\_2Eoption\_\_CASE~A\_27a~A\_27a)~(ap~V2r~V1v))~\\
(ap~(ap~(c\_2EpatternMatches\_2EPMATCH~A\_27a~A\_27b)~V1v)~V3rs))~\\
(\mname{c\_2Ecombin\_2EI}~A\_27a)))))))\end{array}
\label{hyp:ehigngpfcdfeahbgehehfgchognebgehdgigfgdhpfcdfeafnebeefdeiepfpfegfggg}
\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} V0v \in A\_27a .  ({\unicdcdadad} V1rows1 \in (ty\_2Elist\_2Elist~((ty\_2Eoption\_2Eoption~\\
A\_27b)^{{A\_27a}})) .  ({\unicdcdadad} V2rows2 \in (ty\_2Elist\_2Elist~((ty\_2Eoption\_2Eoption~\\
A\_27b)^{{A\_27a}})) .  ((p~(ap~(ap~(ap~(\mname{c\_2EpatternMatches\_2EPMATCH\_\_EQUIV\_\_ROWS}~\\
A\_27a~A\_27b)~V0v)~V1rows1)~V2rows2)){\unicdbdjded}((ap~(ap~(\mname{c\_2EpatternMatches\_2EPMATCH\_\_EQUIV\_\_ROWS}~\\
A\_27a~A\_27b)~V0v)~V1rows1)=(ap~(ap~(\mname{c\_2EpatternMatches\_2EPMATCH\_\_EQUIV\_\_ROWS}~\\
A\_27a~A\_27b)~V0v)~V2rows2))))))\end{array}
\label{hyp:ehigngpfcdfeahbgehehfgchognebgehdgigfgdhpfcdfeafnebeefdeiepfpffebfffjegfpfpfcfpehfdfpfpffebfffjegfpfpffeifafbeoeee}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfeahbgehehfgchognebgehdgigfgdhpfcdfeafnebeefdeiepfpffebfffjegfpfpfcfpehfdfpfpfdepeoedfpfpfoepeoefe}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0row \in ((ty\_2Eoption\_2Eoption~A\_27a)^{{A\_27b}}) .  ({\unicdcdadad} V1v \in \\
A\_27b .  ({\unicdcdadad} V2rows \in (ty\_2Elist\_2Elist~((ty\_2Eoption\_2Eoption~\\
A\_27a)^{{A\_27b}})) .  (((ap~V0row~V1v)=(\mname{c\_2Eoption\_2ENONE}~A\_27a)){\unicdbdjdcd}\\
((ap~(ap~(\mname{c\_2EpatternMatches\_2EPMATCH\_\_EQUIV\_\_ROWS}~A\_27b~A\_27a)~\\
V1v)~(ap~(ap~(c\_2Elist\_2ECONS~((ty\_2Eoption\_2Eoption~A\_27a)^{{A\_27b}}))~\\
V0row)~V2rows))=(ap~(ap~(\mname{c\_2EpatternMatches\_2EPMATCH\_\_EQUIV\_\_ROWS}~\\
A\_27b~A\_27a)~V1v)~V2rows))))))\end{array}$$
\end{theorem}
\end{document}
