\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\_2Epatricia\_2EBRANCH\_\_def (TML5nL3vmv9VMBL2S359RNDN9n9ESKRBRoP)}
\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: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: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: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\_2Epair\_2Eprod : {{\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\_2Epair\_2Eprod~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfeahbgjgchpfcdfeahchpgeg}
\end{equation}
Let $ c\_2Epair\_2EABS\_\_prod : {{\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\_2Epair\_2EABS\_\_prod~\\
A\_27a~A\_27b \in ((ty\_2Epair\_2Eprod~A\_27a~A\_27b)^{{({{({{2}^{{A\_27b}}})}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfebecedfpfpfahchpgeg}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgjgchpfcdfepfcdde}
We define $\mname{c\_2Epair\_2E\_2C}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {{\uniddcgcg} V1y \in {A\_27b} .  {({{{ap}~{({{{c\_2Epair\_2EABS\_\_prod}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V2a \in {A\_27a} .  {({{\uniddcgcg} V3b \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2a}})}}~{V0x}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{V3b}})}}~{V1y}})}})}})}})}})}}}}}$.
\end{definition}
Let $ ty\_2Epatricia\_2Eptree : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Epatricia\_2Eptree}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfeahbgehchjgdgjgbgpfcdfeahehchfgfg}
\end{equation}
Let $ c\_2Epatricia\_2EEmpty : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Epatricia\_2EEmpty~A\_27a \in (ty\_2Epatricia\_2Eptree~\\
A\_27a)\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgehchjgdgjgbgpfcdfefengahehjh}
\end{equation}
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
Let $ c\_2Epatricia\_2EBranch : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Epatricia\_2EBranch~A\_27a \in (\\
((((ty\_2Epatricia\_2Eptree~A\_27a)^{{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})^{{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgehchjgdgjgbgpfcdfecechbgogdgig}
\end{equation}
Let $ c\_2Epatricia\_2ELeaf : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Epatricia\_2ELeaf~A\_27a \in (((\\
ty\_2Epatricia\_2Eptree~A\_27a)^{{A\_27a}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgehchjgdgjgbgpfcdfemefgbggg}
\end{equation}
\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\_2Epatricia\_2Eptree\_\_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\_2Epatricia\_2Eptree\_\_CASE~\\
A\_27a~A\_27b \in ((((A\_27b^{{({{({{({{({{A\_27b}^{{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}}})}^{{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}}})}^{{ty\_2Enum\_2Enum}}})}^{{ty\_2Enum\_2Enum}}})}})^{{({{({{A\_27b}^{{A\_27a}}})}^{{ty\_2Enum\_2Enum}}})}})^{{A\_27b}})^{{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgehchjgdgjgbgpfcdfeahehchfgfgpfpfdebedffe}
\end{equation}
Let $ c\_2Epair\_2ESND : {{\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\_2Epair\_2ESND~\\
A\_27a~A\_27b \in (A\_27b^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfedfoeee}
\end{equation}
Let $ c\_2Epair\_2EFST : {{\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\_2Epair\_2EFST~\\
A\_27a~A\_27b \in (A\_27a^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfegedfef}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgjgchpfcdfeahbgjgchpfpfdebedffe}
We define $\mname{c\_2Epair\_2Epair\_\_CASE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0p \in {({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27c}})} .  {{\uniddcgcg} V1f \in {({{({{A\_27a}^{{A\_27c}}})}^{{A\_27b}}})} .  {({{{ap}~{({{{ap}~{V1f}}~{({{{ap}~{({{{c\_2Epair\_2EFST}~{A\_27b}}~{A\_27c}})}}~{V0p}})}})}}~{({{{ap}~{({{{c\_2Epair\_2ESND}~{A\_27b}}~{A\_27c}})}}~{V0p}})}})}}}}}}$.
\end{definition}
\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: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: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: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}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfehfge}
We define $\mname{c\_2Erelation\_2EWF}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V1B \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V2w \in {A\_27a} .  {({{{ap}~{V1B}}~{V2w}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V3min \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V1B}}~{V3min}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4b \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{V0R}}~{V4b}})}}~{V3min}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{V1B}}~{V4b}})}})}})}})}})}})}})}})}})}})}})}}}$.
\end{definition}
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:dgpfcdfecgpgpgmgpfcdfedepeoeee}
We define $\mname{c\_2Ebool\_2ECOND}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0t \in {2} .  {({{\uniddcgcg} V1t1 \in {A\_27a} .  {({{\uniddcgcg} V2t2 \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{({{\uniddcgcg} V3x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2ET}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V1t1}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V2t2}})}})}})}})}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfecffedfefcfjedeef}
We define $\mname{c\_2Erelation\_2ERESTRICT}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V2x \in {A\_27a} .  {({{\uniddcgcg} V3y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{A\_27b}})}}~{({{{ap}~{({{{ap}~{V1R}}~{V3y}})}}~{V2x}})}})}}~{({{{ap}~{V0f}}~{V3y}})}})}}~{({{c\_2Ebool\_2EARB}~{A\_27b}})}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfeefde}
We define $\mname{c\_2Erelation\_2ETC}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1a \in {A\_27a} .  {{\uniddcgcg} V2b \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{({{2}^{{A\_27a}}})}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V3P \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V5y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{V0R}}~{V4x}})}}~{V5y}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V4x}})}}~{V5y}})}})}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V6x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V7y \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V8z \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{V3P}}~{V6x}})}}~{V7y}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V7y}})}}~{V8z}})}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V6x}})}}~{V8z}})}})}})}})}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V1a}})}}~{V2b}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfebgahahchpgih}
We define $\mname{c\_2Erelation\_2Eapprox}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1M \in {({{({{A\_27b}^{{A\_27a}}})}^{{({{A\_27b}^{{A\_27a}}})}}})} .  {{\uniddcgcg} V2x \in {A\_27a} .  {{\uniddcgcg} V3f \in {({{A\_27b}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{A\_27b}^{{A\_27a}}})}})}}~{V3f}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V4y \in {A\_27a} .  {({{{ap}~{({{{ap}~{V1M}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{V3f}})}}~{V0R}})}}~{V4y}})}})}}~{V4y}})}})}})}}~{V0R}})}}~{V2x}})}})}}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfeehigfgpfpfggfhog}
We define $\mname{c\_2Erelation\_2Ethe\_\_fun}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1M \in {({{({{A\_27b}^{{A\_27a}}})}^{{({{A\_27b}^{{A\_27a}}})}}})} .  {{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{A\_27b}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V3f \in {({{A\_27b}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2Eapprox}}~{A\_27a}}~{A\_27b}})}}~{V0R}})}}~{V1M}})}}~{V2x}})}}~{V3f}})}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfehfgecffede}
We define $\mname{c\_2Erelation\_2EWFREC}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1M \in {({{({{A\_27b}^{{A\_27a}}})}^{{({{A\_27b}^{{A\_27a}}})}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{V1M}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2Ethe\_\_fun}}~{A\_27a}}~{A\_27b}})}}~{({{{ap}~{({{\mname{c\_2Erelation\_2ETC}}~{A\_27a}})}}~{V0R}})}})}}~{({{\uniddcgcg} V3f \in {({{A\_27b}^{{A\_27a}}})} .  {({{\uniddcgcg} V4v \in {A\_27a} .  {({{{ap}~{({{{ap}~{V1M}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{V3f}})}}~{V0R}})}}~{V4v}})}})}}~{V4v}})}})}})}})}}~{V2x}})}})}}~{V0R}})}}~{V2x}})}})}}~{V2x}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahbgehchjgdgjgbgpfcdfececfbeoedeie}
We define $\mname{c\_2Epatricia\_2EBRANCH}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2EWFREC}}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{({{{ty\_2Epair\_2Eprod}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})}})}})}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{({{2}^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{({{{ty\_2Epair\_2Eprod}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})}})}})}}})}^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{({{{ty\_2Epair\_2Eprod}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})}})}})}}})}})}}~{({{\uniddcgcg} V0R \in {({{({{2}^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{({{{ty\_2Epair\_2Eprod}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})}})}})}}})}^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{({{{ty\_2Epair\_2Eprod}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})}})}})}}})} .  {({{{ap}~{({{\mname{c\_2Erelation\_2EWF}}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{({{{ty\_2Epair\_2Eprod}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})}})}})}})}}~{V0R}})}})}})}})}}~{({{\uniddcgcg} V1BRANCH \in {({{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{({{{ty\_2Epair\_2Eprod}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})}})}})}}})} .  {({{\uniddcgcg} V2a \in {({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{({{{ty\_2Epair\_2Eprod}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})}})}})} .  {({{{ap}~{({{{ap}~{({{{{\mname{c\_2Epair\_2Epair\_\_CASE}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}}~{ty\_2Enum\_2Enum}}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{({{{ty\_2Epair\_2Eprod}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})}})}})}}~{V2a}})}}~{({{\uniddcgcg} V3p \in {ty\_2Enum\_2Enum} .  {({{\uniddcgcg} V4v1 \in {({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{({{{ty\_2Epair\_2Eprod}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})}})} .  {({{{ap}~{({{{ap}~{({{{{\mname{c\_2Epair\_2Epair\_\_CASE}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}}~{ty\_2Enum\_2Enum}}~{({{{ty\_2Epair\_2Eprod}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})}})}}~{V4v1}})}}~{({{\uniddcgcg} V5m \in {ty\_2Enum\_2Enum} .  {({{\uniddcgcg} V6v3 \in {({{{ty\_2Epair\_2Eprod}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})} .  {({{{ap}~{({{{ap}~{({{{{\mname{c\_2Epair\_2Epair\_\_CASE}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})}}~{V6v3}})}}~{({{\uniddcgcg} V7v4 \in {({{ty\_2Epatricia\_2Eptree}~{A\_27a}})} .  {({{\uniddcgcg} V8t \in {({{ty\_2Epatricia\_2Eptree}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{c\_2Epatricia\_2Eptree\_\_CASE}~{A\_27a}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})}}~{V7v4}})}}~{({{{ap}~{({{\mname{c\_2Ecombin\_2EI}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})}}~{V8t}})}})}}~{({{\uniddcgcg} V9v18 \in {ty\_2Enum\_2Enum} .  {({{\uniddcgcg} V10v19 \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{c\_2Epatricia\_2Eptree\_\_CASE}~{A\_27a}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})}}~{V8t}})}}~{({{{ap}~{({{\mname{c\_2Ecombin\_2EI}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{c\_2Epatricia\_2ELeaf}~{A\_27a}})}}~{V9v18}})}}~{V10v19}})}})}})}}~{({{\uniddcgcg} V11v30 \in {ty\_2Enum\_2Enum} .  {({{\uniddcgcg} V12v31 \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ecombin\_2EI}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Epatricia\_2EBranch}~{A\_27a}})}}~{V3p}})}}~{V5m}})}}~{({{{ap}~{({{{ap}~{({{c\_2Epatricia\_2ELeaf}~{A\_27a}})}}~{V9v18}})}}~{V10v19}})}})}}~{({{{ap}~{({{{ap}~{({{c\_2Epatricia\_2ELeaf}~{A\_27a}})}}~{V11v30}})}}~{V12v31}})}})}})}})}})}})}}~{({{\uniddcgcg} V13v32 \in {ty\_2Enum\_2Enum} .  {({{\uniddcgcg} V14v33 \in {ty\_2Enum\_2Enum} .  {({{\uniddcgcg} V15v34 \in {({{ty\_2Epatricia\_2Eptree}~{A\_27a}})} .  {({{\uniddcgcg} V16v35 \in {({{ty\_2Epatricia\_2Eptree}~{A\_27a}})} .  {({{{ap}~{({{\mname{c\_2Ecombin\_2EI}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Epatricia\_2EBranch}~{A\_27a}})}}~{V3p}})}}~{V5m}})}}~{({{{ap}~{({{{ap}~{({{c\_2Epatricia\_2ELeaf}~{A\_27a}})}}~{V9v18}})}}~{V10v19}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Epatricia\_2EBranch}~{A\_27a}})}}~{V13v32}})}}~{V14v33}})}}~{V15v34}})}}~{V16v35}})}})}})}})}})}})}})}})}})}})}})}}~{({{\uniddcgcg} V17v20 \in {ty\_2Enum\_2Enum} .  {({{\uniddcgcg} V18v21 \in {ty\_2Enum\_2Enum} .  {({{\uniddcgcg} V19v22 \in {({{ty\_2Epatricia\_2Eptree}~{A\_27a}})} .  {({{\uniddcgcg} V20v23 \in {({{ty\_2Epatricia\_2Eptree}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{c\_2Epatricia\_2Eptree\_\_CASE}~{A\_27a}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})}}~{V8t}})}}~{({{{ap}~{({{\mname{c\_2Ecombin\_2EI}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Epatricia\_2EBranch}~{A\_27a}})}}~{V17v20}})}}~{V18v21}})}}~{V19v22}})}}~{V20v23}})}})}})}}~{({{\uniddcgcg} V21v42 \in {ty\_2Enum\_2Enum} .  {({{\uniddcgcg} V22v43 \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ecombin\_2EI}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Epatricia\_2EBranch}~{A\_27a}})}}~{V3p}})}}~{V5m}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Epatricia\_2EBranch}~{A\_27a}})}}~{V17v20}})}}~{V18v21}})}}~{V19v22}})}}~{V20v23}})}})}}~{({{{ap}~{({{{ap}~{({{c\_2Epatricia\_2ELeaf}~{A\_27a}})}}~{V21v42}})}}~{V22v43}})}})}})}})}})}})}}~{({{\uniddcgcg} V23v44 \in {ty\_2Enum\_2Enum} .  {({{\uniddcgcg} V24v45 \in {ty\_2Enum\_2Enum} .  {({{\uniddcgcg} V25v46 \in {({{ty\_2Epatricia\_2Eptree}~{A\_27a}})} .  {({{\uniddcgcg} V26v47 \in {({{ty\_2Epatricia\_2Eptree}~{A\_27a}})} .  {({{{ap}~{({{\mname{c\_2Ecombin\_2EI}}~{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Epatricia\_2EBranch}~{A\_27a}})}}~{V3p}})}}~{V5m}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Epatricia\_2EBranch}~{A\_27a}})}}~{V17v20}})}}~{V18v21}})}}~{V19v22}})}}~{V20v23}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Epatricia\_2EBranch}~{A\_27a}})}}~{V23v44}})}}~{V24v45}})}}~{V25v46}})}}~{V26v47}})}})}})}})}})}})}})}})}})}})}})}})}})}})}})}})}})}})}})}})}})}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfefeneafefjfpfpfcffeme}
We define $\mname{c\_2Erelation\_2EEMPTY\_\_REL}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {{\uniddcgcg} V1y \in {A\_27a} .  {\mname{c\_2Ebool\_2EF}}}}}$.
\end{definition}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ((ap~(\mname{c\_2Ecombin\_2EI}~\\
A\_27a)~V0x)=V0x))\end{array}
\label{hyp:ehigngpfcdfedgpgngcgjgogpfcdfejepfpfefiene}
\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} V0x \in A\_27b .  ({\unicdcdadad} V1y \in A\_27c .  ({\unicdcdadad} V2f \in \\
((A\_27a^{{A\_27c}})^{{A\_27b}}) .  ((ap~(ap~(\mname{c\_2Epair\_2Epair\_\_CASE}~A\_27a~A\_27b~\\
A\_27c)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27b~A\_27c)~V0x)~V1y))~V2f)=(ap~\\
(ap~V2f~V0x)~V1y)))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfeahbgjgchpfpfdgbgdhfgpfpfehigng}
\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}})^{{ty\_2Enum\_2Enum}}) .  \\
({\unicdcdadad} V2f1 \in ((((A\_27b^{{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})^{{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}}) .  \\
((ap~(ap~(ap~(ap~(c\_2Epatricia\_2Eptree\_\_CASE~A\_27a~A\_27b)~(c\_2Epatricia\_2EEmpty~\\
A\_27a))~V0v)~V1f)~V2f1)=V0v)))){\unicdcdcdhd}(({\unicdcdadad} V3a0 \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V4a1 \in A\_27a .  ({\unicdcdadad} V5v \in A\_27b .  ({\unicdcdadad} V6f \in ((A\_27b^{{A\_27a}})^{{ty\_2Enum\_2Enum}}) .  \\
({\unicdcdadad} V7f1 \in ((((A\_27b^{{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})^{{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}}) .  \\
((ap~(ap~(ap~(ap~(c\_2Epatricia\_2Eptree\_\_CASE~A\_27a~A\_27b)~(ap~(\\
ap~(c\_2Epatricia\_2ELeaf~A\_27a)~V3a0)~V4a1))~V5v)~V6f)~V7f1)=(ap~\\
(ap~V6f~V3a0)~V4a1))))))){\unicdcdcdhd}({\unicdcdadad} V8a0 \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V9a1 \in \\
ty\_2Enum\_2Enum .  ({\unicdcdadad} V10a2 \in (ty\_2Epatricia\_2Eptree~A\_27a) .  \\
({\unicdcdadad} V11a3 \in (ty\_2Epatricia\_2Eptree~A\_27a) .  ({\unicdcdadad} V12v \in \\
A\_27b .  ({\unicdcdadad} V13f \in ((A\_27b^{{A\_27a}})^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V14f1 \in \\
((((A\_27b^{{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})^{{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}}) .  \\
((ap~(ap~(ap~(ap~(c\_2Epatricia\_2Eptree\_\_CASE~A\_27a~A\_27b)~(ap~(\\
ap~(ap~(ap~(c\_2Epatricia\_2EBranch~A\_27a)~V8a0)~V9a1)~V10a2)~V11a3))~\\
V12v)~V13f)~V14f1)=(ap~(ap~(ap~(ap~V14f1~V8a0)~V9a1)~V10a2)~V11a3)))))))))))\end{array}
\label{hyp:ehigngpfcdfeahbgehchjgdgjgbgpfcdfeahehchfgfgpfpfdgbgdhfgpfpfegfggg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(p~(ap~(\mname{c\_2Erelation\_2EWF}~A\_27a)~\\
(\mname{c\_2Erelation\_2EEMPTY\_\_REL}~A\_27a)))\end{array}
\label{hyp:ehigngpfcdfechfgmgbgehjgpgogpfcdfehfgepfpffeneafefjfpfpfcffeme}
\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} V0M \in ((A\_27b^{{A\_27a}})^{{({{A\_27b}^{{A\_27a}}})}}) .  ({\unicdcdadad} V1R \in ((2^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V2f \in (A\_27b^{{A\_27a}}) .  ((V2f=(ap~(ap~(\mname{c\_2Erelation\_2EWFREC}~\\
A\_27a~A\_27b)~V1R)~V0M)){\unicdbdjdcd}((p~(ap~(\mname{c\_2Erelation\_2EWF}~A\_27a)~V1R)){\unicdbdjdcd}\\
({\unicdcdadad} V3x \in A\_27a .  ((ap~V2f~V3x)=(ap~(ap~V0M~(ap~(ap~(ap~(\mname{c\_2Erelation\_2ERESTRICT}~\\
A\_27a~A\_27b)~V2f)~V1R)~V3x))~V3x))))))))\end{array}
\label{hyp:ehigngpfcdfechfgmgbgehjgpgogpfcdfehfgecffedepfpfdepecfpememebecfjf}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfeahbgehchjgdgjgbgpfcdfececfbeoedeiepfpfegfggg}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0p \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V1m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V2t \in (ty\_2Epatricia\_2Eptree~\\
A\_27a) .  ({\unicdcdadad} V3v6 \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V4v7 \in A\_27a .  (\\
{\unicdcdadad} V5v8 \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V6v9 \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V7v10 \in (ty\_2Epatricia\_2Eptree~A\_27a) .  ({\unicdcdadad} V8v11 \in \\
(ty\_2Epatricia\_2Eptree~A\_27a) .  ({\unicdcdadad} V9v12 \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V10v13 \in A\_27a .  ({\unicdcdadad} V11v24 \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V12v25 \in \\
A\_27a .  ({\unicdcdadad} V13v26 \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V14v27 \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V15v28 \in (ty\_2Epatricia\_2Eptree~A\_27a) .  ({\unicdcdadad} V16v29 \in \\
(ty\_2Epatricia\_2Eptree~A\_27a) .  ({\unicdcdadad} V17v14 \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V18v15 \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V19v16 \in (ty\_2Epatricia\_2Eptree~\\
A\_27a) .  ({\unicdcdadad} V20v17 \in (ty\_2Epatricia\_2Eptree~A\_27a) .  ({\unicdcdadad} V21v36 \in \\
ty\_2Enum\_2Enum .  ({\unicdcdadad} V22v37 \in A\_27a .  ({\unicdcdadad} V23v38 \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V24v39 \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V25v40 \in (ty\_2Epatricia\_2Eptree~\\
A\_27a) .  ({\unicdcdadad} V26v41 \in (ty\_2Epatricia\_2Eptree~A\_27a) .  (((ap~\\
(\mname{c\_2Epatricia\_2EBRANCH}~A\_27a)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Enum\_2Enum~\\
(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~(ty\_2Epair\_2Eprod~(ty\_2Epatricia\_2Eptree~\\
A\_27a)~(ty\_2Epatricia\_2Eptree~A\_27a))))~V0p)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
ty\_2Enum\_2Enum~(ty\_2Epair\_2Eprod~(ty\_2Epatricia\_2Eptree~A\_27a)~\\
(ty\_2Epatricia\_2Eptree~A\_27a)))~V1m)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
(ty\_2Epatricia\_2Eptree~A\_27a)~(ty\_2Epatricia\_2Eptree~A\_27a))~\\
(c\_2Epatricia\_2EEmpty~A\_27a))~V2t))))=V2t){\unicdcdcdhd}(((ap~(\mname{c\_2Epatricia\_2EBRANCH}~\\
A\_27a)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Enum\_2Enum~(ty\_2Epair\_2Eprod~\\
ty\_2Enum\_2Enum~(ty\_2Epair\_2Eprod~(ty\_2Epatricia\_2Eptree~A\_27a)~\\
(ty\_2Epatricia\_2Eptree~A\_27a))))~V0p)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
ty\_2Enum\_2Enum~(ty\_2Epair\_2Eprod~(ty\_2Epatricia\_2Eptree~A\_27a)~\\
(ty\_2Epatricia\_2Eptree~A\_27a)))~V1m)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
(ty\_2Epatricia\_2Eptree~A\_27a)~(ty\_2Epatricia\_2Eptree~A\_27a))~\\
(ap~(ap~(c\_2Epatricia\_2ELeaf~A\_27a)~V3v6)~V4v7))~(c\_2Epatricia\_2EEmpty~\\
A\_27a)))))=(ap~(ap~(c\_2Epatricia\_2ELeaf~A\_27a)~V3v6)~V4v7)){\unicdcdcdhd}\\
(((ap~(\mname{c\_2Epatricia\_2EBRANCH}~A\_27a)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Enum\_2Enum~\\
(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~(ty\_2Epair\_2Eprod~(ty\_2Epatricia\_2Eptree~\\
A\_27a)~(ty\_2Epatricia\_2Eptree~A\_27a))))~V0p)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
ty\_2Enum\_2Enum~(ty\_2Epair\_2Eprod~(ty\_2Epatricia\_2Eptree~A\_27a)~\\
(ty\_2Epatricia\_2Eptree~A\_27a)))~V1m)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
(ty\_2Epatricia\_2Eptree~A\_27a)~(ty\_2Epatricia\_2Eptree~A\_27a))~\\
(ap~(ap~(ap~(ap~(c\_2Epatricia\_2EBranch~A\_27a)~V5v8)~V6v9)~V7v10)~\\
V8v11))~(c\_2Epatricia\_2EEmpty~A\_27a)))))=(ap~(ap~(ap~(ap~(c\_2Epatricia\_2EBranch~\\
A\_27a)~V5v8)~V6v9)~V7v10)~V8v11)){\unicdcdcdhd}(((ap~(\mname{c\_2Epatricia\_2EBRANCH}~\\
A\_27a)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Enum\_2Enum~(ty\_2Epair\_2Eprod~\\
ty\_2Enum\_2Enum~(ty\_2Epair\_2Eprod~(ty\_2Epatricia\_2Eptree~A\_27a)~\\
(ty\_2Epatricia\_2Eptree~A\_27a))))~V0p)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
ty\_2Enum\_2Enum~(ty\_2Epair\_2Eprod~(ty\_2Epatricia\_2Eptree~A\_27a)~\\
(ty\_2Epatricia\_2Eptree~A\_27a)))~V1m)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
(ty\_2Epatricia\_2Eptree~A\_27a)~(ty\_2Epatricia\_2Eptree~A\_27a))~\\
(ap~(ap~(c\_2Epatricia\_2ELeaf~A\_27a)~V9v12)~V10v13))~(ap~(ap~(c\_2Epatricia\_2ELeaf~\\
A\_27a)~V11v24)~V12v25)))))=(ap~(ap~(ap~(ap~(c\_2Epatricia\_2EBranch~\\
A\_27a)~V0p)~V1m)~(ap~(ap~(c\_2Epatricia\_2ELeaf~A\_27a)~V9v12)~V10v13))~\\
(ap~(ap~(c\_2Epatricia\_2ELeaf~A\_27a)~V11v24)~V12v25))){\unicdcdcdhd}(((ap~(\\
\mname{c\_2Epatricia\_2EBRANCH}~A\_27a)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Enum\_2Enum~\\
(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~(ty\_2Epair\_2Eprod~(ty\_2Epatricia\_2Eptree~\\
A\_27a)~(ty\_2Epatricia\_2Eptree~A\_27a))))~V0p)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
ty\_2Enum\_2Enum~(ty\_2Epair\_2Eprod~(ty\_2Epatricia\_2Eptree~A\_27a)~\\
(ty\_2Epatricia\_2Eptree~A\_27a)))~V1m)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
(ty\_2Epatricia\_2Eptree~A\_27a)~(ty\_2Epatricia\_2Eptree~A\_27a))~\\
(ap~(ap~(c\_2Epatricia\_2ELeaf~A\_27a)~V9v12)~V10v13))~(ap~(ap~(ap~\\
(ap~(c\_2Epatricia\_2EBranch~A\_27a)~V13v26)~V14v27)~V15v28)~V16v29)))))=\\
(ap~(ap~(ap~(ap~(c\_2Epatricia\_2EBranch~A\_27a)~V0p)~V1m)~(ap~(ap~(\\
c\_2Epatricia\_2ELeaf~A\_27a)~V9v12)~V10v13))~(ap~(ap~(ap~(ap~(c\_2Epatricia\_2EBranch~\\
A\_27a)~V13v26)~V14v27)~V15v28)~V16v29))){\unicdcdcdhd}(((ap~(\mname{c\_2Epatricia\_2EBRANCH}~\\
A\_27a)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Enum\_2Enum~(ty\_2Epair\_2Eprod~\\
ty\_2Enum\_2Enum~(ty\_2Epair\_2Eprod~(ty\_2Epatricia\_2Eptree~A\_27a)~\\
(ty\_2Epatricia\_2Eptree~A\_27a))))~V0p)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
ty\_2Enum\_2Enum~(ty\_2Epair\_2Eprod~(ty\_2Epatricia\_2Eptree~A\_27a)~\\
(ty\_2Epatricia\_2Eptree~A\_27a)))~V1m)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
(ty\_2Epatricia\_2Eptree~A\_27a)~(ty\_2Epatricia\_2Eptree~A\_27a))~\\
(ap~(ap~(ap~(ap~(c\_2Epatricia\_2EBranch~A\_27a)~V17v14)~V18v15)~V19v16)~\\
V20v17))~(ap~(ap~(c\_2Epatricia\_2ELeaf~A\_27a)~V21v36)~V22v37)))))=\\
(ap~(ap~(ap~(ap~(c\_2Epatricia\_2EBranch~A\_27a)~V0p)~V1m)~(ap~(ap~(\\
ap~(ap~(c\_2Epatricia\_2EBranch~A\_27a)~V17v14)~V18v15)~V19v16)~V20v17))~\\
(ap~(ap~(c\_2Epatricia\_2ELeaf~A\_27a)~V21v36)~V22v37))){\unicdcdcdhd}((ap~(\mname{c\_2Epatricia\_2EBRANCH}~\\
A\_27a)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Enum\_2Enum~(ty\_2Epair\_2Eprod~\\
ty\_2Enum\_2Enum~(ty\_2Epair\_2Eprod~(ty\_2Epatricia\_2Eptree~A\_27a)~\\
(ty\_2Epatricia\_2Eptree~A\_27a))))~V0p)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
ty\_2Enum\_2Enum~(ty\_2Epair\_2Eprod~(ty\_2Epatricia\_2Eptree~A\_27a)~\\
(ty\_2Epatricia\_2Eptree~A\_27a)))~V1m)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
(ty\_2Epatricia\_2Eptree~A\_27a)~(ty\_2Epatricia\_2Eptree~A\_27a))~\\
(ap~(ap~(ap~(ap~(c\_2Epatricia\_2EBranch~A\_27a)~V17v14)~V18v15)~V19v16)~\\
V20v17))~(ap~(ap~(ap~(ap~(c\_2Epatricia\_2EBranch~A\_27a)~V23v38)~\\
V24v39)~V25v40)~V26v41)))))=(ap~(ap~(ap~(ap~(c\_2Epatricia\_2EBranch~\\
A\_27a)~V0p)~V1m)~(ap~(ap~(ap~(ap~(c\_2Epatricia\_2EBranch~A\_27a)~V17v14)~\\
V18v15)~V19v16)~V20v17))~(ap~(ap~(ap~(ap~(c\_2Epatricia\_2EBranch~\\
A\_27a)~V23v38)~V24v39)~V25v40)~V26v41))))))))))))))))))))))))))))))))))))\end{array}$$
\end{theorem}
\end{document}
