\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\_2Esptree\_2Emapi\_\_Alist (TMKKofc8XLPV4kK68s9t2J3meBwtH8MBGWZ)}
\maketitle
\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: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: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}
\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: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}
\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: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}
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\_2EMAP : {{\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\_2Elist\_2EMAP~\\
A\_27a~A\_27b \in (((ty\_2Elist\_2Elist~A\_27b)^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{({{A\_27b}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfenebeaf}
\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\_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 $ ty\_2Eone\_2Eone : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Eone\_2Eone}}\label{hyp:ogfgpfehjhpfcdfepgogfgpfcdfepgogfg}
\end{equation}
\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}
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: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: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}
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\_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}
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\_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 $ c\_2Eoption\_2EOPTION\_\_MAP : {{\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\_\_MAP~\\
A\_27a~A\_27b \in (((ty\_2Eoption\_2Eoption~A\_27b)^{{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}})^{{({{A\_27b}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfepgahehjgpgogpfcdfepeafefjepeoepfpfnebeaf}
\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\_2Enum\_2EZERO\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EZERO\_\_REP} \in {omega}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfekffecfpepfpfcffeaf}
\end{equation}
Let $ c\_2Enum\_2EABS\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EABS\_\_num} \in {({{ty\_2Enum\_2Enum}^{{omega}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfebecedfpfpfogfhng}
\end{equation}
\begin{definition}\label{def:dgpfcdfeogfhngpfcdfead}
We define $\mname{c\_2Enum\_2E0}$ to be ${({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{c\_2Enum\_2EZERO\_\_REP}})}$.
\end{definition}
Let $ ty\_2Esptree\_2Espt : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Esptree\_2Espt}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfedhahehchfgfgpfcdfedhaheh}
\end{equation}
Let $ c\_2Esptree\_2Efoldi : {{\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\_2Esptree\_2Efoldi~\\
A\_27a~A\_27b \in ((((A\_27a^{{({{ty\_2Esptree\_2Espt}~{A\_27b}})}})^{{A\_27a}})^{{ty\_2Enum\_2Enum}})^{{({{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27b}}})}^{{ty\_2Enum\_2Enum}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhahehchfgfgpfcdfeggpgmgegjg}
\end{equation}
\begin{definition}\label{def:dgpfcdfedhahehchfgfgpfcdfeehpgbemejgdheh}
We define $\mname{c\_2Esptree\_2EtoAList}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{c\_2Esptree\_2Efoldi}~{({{ty\_2Elist\_2Elist}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{A\_27a}})}})}}~{A\_27a}})}}~{({{\uniddcgcg} V0k \in {ty\_2Enum\_2Enum} .  {({{\uniddcgcg} V1v \in {A\_27a} .  {({{\uniddcgcg} V2a \in {({{ty\_2Elist\_2Elist}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{A\_27a}})}})} .  {({{{ap}~{({{{ap}~{({{c\_2Elist\_2ECONS}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Enum\_2Enum}}~{A\_27a}})}}~{V0k}})}}~{V1v}})}})}}~{V2a}})}})}})}})}})}}~{\mname{c\_2Enum\_2E0}}})}}~{({{c\_2Elist\_2ENIL}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{A\_27a}})}})}})}}$.
\end{definition}
Let $ c\_2Ealist\_2EALOOKUP : {{\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\_2Ealist\_2EALOOKUP~\\
A\_27a~A\_27b \in (((ty\_2Eoption\_2Eoption~A\_27a)^{{A\_27b}})^{{({{ty\_2Elist\_2Elist}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgmgjgdhehpfcdfebemepepeleffaf}
\end{equation}
Let $ c\_2Esptree\_2Elookup : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Esptree\_2Elookup~A\_27a \in (((\\
ty\_2Eoption\_2Eoption~A\_27a)^{{({{ty\_2Esptree\_2Espt}~{A\_27a}})}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhahehchfgfgpfcdfemgpgpglgfhah}
\end{equation}
Let $ c\_2Esptree\_2Einsert : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Esptree\_2Einsert~A\_27a \in (((\\
(ty\_2Esptree\_2Espt~A\_27a)^{{({{ty\_2Esptree\_2Espt}~{A\_27a}})}})^{{A\_27a}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhahehchfgfgpfcdfejgogdhfgcheh}
\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}
\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}
Let $ c\_2Esptree\_2ELN : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Esptree\_2ELN~A\_27a \in (ty\_2Esptree\_2Espt~\\
A\_27a)\end{array}
\label{hyp:ngfgngpfdgpfcdfedhahehchfgfgpfcdfemeoe}
\end{equation}
Let $ c\_2Elist\_2Elist\_\_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\_2Elist\_2Elist\_\_CASE~\\
A\_27a~A\_27b \in (((A\_27b^{{({{({{A\_27b}^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}}})}^{{A\_27a}}})}})^{{A\_27b}})^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfemgjgdhehpfpfdebedffe}
\end{equation}
\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: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:dgpfcdfedhahehchfgfgpfcdfeggchpgngbemejgdheh}
We define $\mname{c\_2Esptree\_2EfromAList}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2EWFREC}}~{({{ty\_2Elist\_2Elist}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{A\_27a}})}})}}~{({{ty\_2Esptree\_2Espt}~{A\_27a}})}})}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{({{2}^{{({{ty\_2Elist\_2Elist}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{A\_27a}})}})}}})}^{{({{ty\_2Elist\_2Elist}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{A\_27a}})}})}}})}})}}~{({{\uniddcgcg} V0R \in {({{({{2}^{{({{ty\_2Elist\_2Elist}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{A\_27a}})}})}}})}^{{({{ty\_2Elist\_2Elist}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{A\_27a}})}})}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Erelation\_2EWF}}~{({{ty\_2Elist\_2Elist}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{A\_27a}})}})}})}}~{V0R}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1y \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V2x \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{ty\_2Elist\_2Elist}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{A\_27a}})}})}})}}~{({{\uniddcgcg} V3xs \in {({{ty\_2Elist\_2Elist}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{A\_27a}})}})} .  {({{{ap}~{({{{ap}~{V0R}}~{V3xs}})}}~{({{{ap}~{({{{ap}~{({{c\_2Elist\_2ECONS}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Enum\_2Enum}}~{A\_27a}})}}~{V2x}})}}~{V1y}})}})}}~{V3xs}})}})}})}})}})}})}})}})}})}})}})}})}}~{({{\uniddcgcg} V4fromAList \in {({{({{ty\_2Esptree\_2Espt}~{A\_27a}})}^{{({{ty\_2Elist\_2Elist}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{A\_27a}})}})}}})} .  {({{\uniddcgcg} V5a \in {({{ty\_2Elist\_2Elist}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{A\_27a}})}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{c\_2Elist\_2Elist\_\_CASE}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{A\_27a}})}}~{({{ty\_2Esptree\_2Espt}~{A\_27a}})}})}}~{V5a}})}}~{({{{ap}~{({{\mname{c\_2Ecombin\_2EI}}~{({{ty\_2Esptree\_2Espt}~{A\_27a}})}})}}~{({{c\_2Esptree\_2ELN}~{A\_27a}})}})}})}}~{({{\uniddcgcg} V6v \in {({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{A\_27a}})} .  {({{\uniddcgcg} V7xs \in {({{ty\_2Elist\_2Elist}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{A\_27a}})}})} .  {({{{ap}~{({{{ap}~{({{{{\mname{c\_2Epair\_2Epair\_\_CASE}}~{({{ty\_2Esptree\_2Espt}~{A\_27a}})}}~{ty\_2Enum\_2Enum}}~{A\_27a}})}}~{V6v}})}}~{({{\uniddcgcg} V8x \in {ty\_2Enum\_2Enum} .  {({{\uniddcgcg} V9y \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ecombin\_2EI}}~{({{ty\_2Esptree\_2Espt}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Esptree\_2Einsert}~{A\_27a}})}}~{V8x}})}}~{V9y}})}}~{({{{ap}~{V4fromAList}}~{V7xs}})}})}})}})}})}})}})}})}})}})}})}})}}$.
\end{definition}
Let $ c\_2Esptree\_2Emapi0 : {{\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\_2Esptree\_2Emapi0~\\
A\_27a~A\_27b \in ((((ty\_2Esptree\_2Espt~A\_27a)^{{({{ty\_2Esptree\_2Espt}~{A\_27b}})}})^{{ty\_2Enum\_2Enum}})^{{({{({{A\_27a}^{{A\_27b}}})}^{{ty\_2Enum\_2Enum}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhahehchfgfgpfcdfengbgahjgad}
\end{equation}
\begin{definition}\label{def:dgpfcdfedhahehchfgfgpfcdfengbgahjg}
We define $\mname{c\_2Esptree\_2Emapi}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{({{A\_27a}^{{A\_27b}}})}^{{ty\_2Enum\_2Enum}}})} .  {{\uniddcgcg} V1pt \in {({{ty\_2Esptree\_2Espt}~{A\_27b}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{c\_2Esptree\_2Emapi0}~{A\_27a}}~{A\_27b}})}}~{V0f}})}}~{\mname{c\_2Enum\_2E0}}})}}~{V1pt}})}}}}}$.
\end{definition}
Let $ c\_2Esptree\_2Ewf : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Esptree\_2Ewf~A\_27a \in (2^{{({{ty\_2Esptree\_2Espt}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhahehchfgfgpfcdfehhgg}
\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} V0q \in A\_27b .  ((ap~(ap~(c\_2Ealist\_2EALOOKUP~A\_27a~A\_27b)~\\
(c\_2Elist\_2ENIL~(ty\_2Epair\_2Eprod~A\_27b~A\_27a)))~V0q)=(\mname{c\_2Eoption\_2ENONE}~\\
A\_27a))){\unicdcdcdhd}({\unicdcdadad} V1y \in A\_27a .  ({\unicdcdadad} V2x \in A\_27b .  ({\unicdcdadad} V3t \in \\
(ty\_2Elist\_2Elist~(ty\_2Epair\_2Eprod~A\_27b~A\_27a)) .  ({\unicdcdadad} V4q \in \\
A\_27b .  ((ap~(ap~(c\_2Ealist\_2EALOOKUP~A\_27a~A\_27b)~(ap~(ap~(c\_2Elist\_2ECONS~\\
(ty\_2Epair\_2Eprod~A\_27b~A\_27a))~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27b~\\
A\_27a)~V2x)~V1y))~V3t))~V4q)=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~(ty\_2Eoption\_2Eoption~\\
A\_27a))~(ap~(ap~(\mname{c\_2Emin\_2E\_3D}~A\_27b)~V2x)~V4q))~(ap~(\mname{c\_2Eoption\_2ESOME}~\\
A\_27a)~V1y))~(ap~(ap~(c\_2Ealist\_2EALOOKUP~A\_27a~A\_27b)~V3t)~V4q))))))))\end{array}
\label{hyp:ehigngpfcdfebgmgjgdhehpfcdfebemepepeleffafpfpfegfggg}
\end{equation}
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}
{({{\unicdcdadad} V0t \in {2} .  {({{False}{\unicdbdjdcd}{({{p}~{V0t}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegebemedfjeefjf}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0t \in {2} .  {({{({{p}~{V0t}})}{\unicdcdcdid}{({\preoh{{({{p}~{V0t}})}}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefeifdemeffeefeeepfpfnejeeeeemefe}
\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} 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} V0t \in 2 .  (((True{\unicdbdjded}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjded}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdbdjded}(p~V0t)){\unicdbdjded}(\preoh (p~V0t))){\unicdcdcdhd}(((p~V0t){\unicdbdjded}False){\unicdbdjded}(\preoh (\\
p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0t1 \in A\_27a .  ({\unicdcdadad} V1t2 \in \\
A\_27a .  (((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\mname{c\_2Ebool\_2ET})~V0t1)~\\
V1t2)=V0t1){\unicdcdcdhd}((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\mname{c\_2Ebool\_2EF})~\\
V0t1)~V1t2)=V1t2))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfedepeoeeepfpfdemebeffdffedf}
\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} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in 2 .  ({\unicdcdadad} V1Q \in 2 .  \\
({\unicdcdadad} V2x \in A\_27a .  ({\unicdcdadad} V3x\_27 \in A\_27a .  ({\unicdcdadad} V4y \in A\_27a .  \\
({\unicdcdadad} V5y\_27 \in A\_27a .  ((((p~V0P){\unicdbdjded}(p~V1Q)){\unicdcdcdhd}(((p~V1Q){\unicdbdjdcd}(V2x=V3x\_27)){\unicdcdcdhd}\\
((\preoh (p~V1Q)){\unicdbdjdcd}(V4y=V5y\_27)))){\unicdbdjdcd}((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\\
V0P)~V2x)~V4y)=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~V1Q)~V3x\_27)~\\
V5y\_27)))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfedepeoeeepfpfdepeoehe}
\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} V0t1 \in A\_27a .  ({\unicdcdadad} V1t2 \in \\
A\_27a .  ((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\mname{c\_2Ebool\_2ET})~V0t1)~\\
V1t2)=V0t1))){\unicdcdcdhd}({\unicdcdadad} V2t1 \in A\_27a .  ({\unicdcdadad} V3t2 \in A\_27a .  ((ap~\\
(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\mname{c\_2Ebool\_2EF})~V2t1)~V3t2)=V3t2))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecgpgpgmgpfpfdgbgdhfgpfpfehigng}
\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} V0f \in (A\_27b^{{A\_27a}}) .  ((ap~(ap~(c\_2Elist\_2EMAP~A\_27a~A\_27b)~\\
V0f)~(c\_2Elist\_2ENIL~A\_27a))=(c\_2Elist\_2ENIL~A\_27b))){\unicdcdcdhd}({\unicdcdadad} V1f \in \\
(A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V2h \in A\_27a .  ({\unicdcdadad} V3t \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ((ap~(ap~(c\_2Elist\_2EMAP~A\_27a~A\_27b)~V1f)~(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V2h)~V3t))=(ap~(ap~(c\_2Elist\_2ECONS~A\_27b)~(ap~V1f~V2h))~\\
(ap~(ap~(c\_2Elist\_2EMAP~A\_27a~A\_27b)~V1f)~V3t)))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfenebeaf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}}) .  \\
(((p~(ap~V0P~(c\_2Elist\_2ENIL~A\_27a))){\unicdcdcdhd}({\unicdcdadad} V1t \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ((p~(ap~V0P~V1t)){\unicdbdjdcd}({\unicdcdadad} V2h \in A\_27a .  (p~(ap~V0P~(ap~(ap~(\\
c\_2Elist\_2ECONS~A\_27a)~V2h)~V1t))))))){\unicdbdjdcd}({\unicdcdadad} V3l \in (ty\_2Elist\_2Elist~\\
A\_27a) .  (p~(ap~V0P~V3l)))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfemgjgdhehpfpfjgogegfhdgehjgpgog}
\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 .  (((ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~V0x)=(ap~(\mname{c\_2Eoption\_2ESOME}~\\
A\_27a)~V1y)){\unicdbdjded}(V0x=V1y))))\end{array}
\label{hyp:ehigngpfcdfepgahehjgpgogpfcdfedfpenefepfpfbdbd}
\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} V0f \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V1x \in A\_27a .  ((ap~(ap~(c\_2Eoption\_2EOPTION\_\_MAP~\\
A\_27a~A\_27b)~V0f)~(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~V1x))=(ap~(\mname{c\_2Eoption\_2ESOME}~\\
A\_27b)~(ap~V0f~V1x))))){\unicdcdcdhd}({\unicdcdadad} V2f \in (A\_27b^{{A\_27a}}) .  ((ap~(ap~(c\_2Eoption\_2EOPTION\_\_MAP~\\
A\_27a~A\_27b)~V2f)~(\mname{c\_2Eoption\_2ENONE}~A\_27a))=(\mname{c\_2Eoption\_2ENONE}~\\
A\_27b))))\end{array}
\label{hyp:ehigngpfcdfepgahehjgpgogpfcdfepeafefjepeoepfpfnebeafpfpfeefege}
\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} V0f \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V1x \in (ty\_2Eoption\_2Eoption~\\
A\_27a) .  ({\unicdcdadad} V2y \in A\_27b .  (((ap~(ap~(c\_2Eoption\_2EOPTION\_\_MAP~\\
A\_27a~A\_27b)~V0f)~V1x)=(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27b)~V2y)){\unicdbdjded}({\unicdcdaddd} V3z \in \\
A\_27a .  ((V1x=(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~V3z)){\unicdcdcdhd}(V2y=(ap~V0f~\\
V3z))))))))\end{array}
\label{hyp:ehigngpfcdfepgahehjgpgogpfcdfepeafefjepeoepfpfnebeafpfpffebfpfpfdfpenefe}
\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} V0x \in A\_27a .  ({\unicdcdadad} V1y \in A\_27b .  ((ap~(c\_2Epair\_2EFST~A\_27a~\\
A\_27b)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V0x)~V1y))=V0x)))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfegedfef}
\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} V0x \in A\_27a .  ({\unicdcdadad} V1y \in A\_27b .  ((ap~(c\_2Epair\_2ESND~A\_27a~\\
A\_27b)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V0x)~V1y))=V1y)))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfedfoeee}
\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} V0P \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}) .  (({\unicdcdadad} V1p \in \\
(ty\_2Epair\_2Eprod~A\_27a~A\_27b) .  (p~(ap~V0P~V1p))){\unicdbdjded}({\unicdcdadad} V2p\_\_1 \in \\
A\_27a .  ({\unicdcdadad} V3p\_\_2 \in A\_27b .  (p~(ap~V0P~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
A\_27a~A\_27b)~V2p\_\_1)~V3p\_\_2)))))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfegepecfbememepfpfafcfpeee}
\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} V0k \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1f \in ((A\_27a^{{A\_27b}})^{{ty\_2Enum\_2Enum}}) .  \\
({\unicdcdadad} V2pt \in (ty\_2Esptree\_2Espt~A\_27b) .  ((ap~(ap~(c\_2Esptree\_2Elookup~\\
A\_27a)~V0k)~(ap~(ap~(\mname{c\_2Esptree\_2Emapi}~A\_27a~A\_27b)~V1f)~V2pt))=\\
(ap~(ap~(c\_2Eoption\_2EOPTION\_\_MAP~A\_27b~A\_27a)~(ap~V1f~V0k))~(ap~\\
(ap~(c\_2Esptree\_2Elookup~A\_27b)~V0k)~V2pt))))))\end{array}
\label{hyp:ehigngpfcdfedhahehchfgfgpfcdfemgpgpglgfhahpfpfngbgahjg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0t \in (ty\_2Esptree\_2Espt~\\
A\_27a) .  ({\unicdcdadad} V1x \in ty\_2Enum\_2Enum .  ((ap~(ap~(c\_2Ealist\_2EALOOKUP~\\
A\_27a~ty\_2Enum\_2Enum)~(ap~(\mname{c\_2Esptree\_2EtoAList}~A\_27a)~V0t))~\\
V1x)=(ap~(ap~(c\_2Esptree\_2Elookup~A\_27a)~V1x)~V0t))))\end{array}
\label{hyp:ehigngpfcdfedhahehchfgfgpfcdfebemepepeleffafpfpfehpgbemejgdheh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0t1 \in (ty\_2Esptree\_2Espt~\\
A\_27a) .  ({\unicdcdadad} V1t2 \in (ty\_2Esptree\_2Espt~A\_27a) .  (((p~(ap~(c\_2Esptree\_2Ewf~\\
A\_27a)~V0t1)){\unicdcdcdhd}(p~(ap~(c\_2Esptree\_2Ewf~A\_27a)~V1t2))){\unicdbdjdcd}((V0t1=\\
V1t2){\unicdbdjded}({\unicdcdadad} V2n \in ty\_2Enum\_2Enum .  ((ap~(ap~(c\_2Esptree\_2Elookup~\\
A\_27a)~V2n)~V0t1)=(ap~(ap~(c\_2Esptree\_2Elookup~A\_27a)~V2n)~V1t2)))))))\end{array}
\label{hyp:ehigngpfcdfedhahehchfgfgpfcdfedhahehpfpffgbhpfpfehigng}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0ls \in (ty\_2Elist\_2Elist~\\
(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~A\_27a)) .  ({\unicdcdadad} V1x \in ty\_2Enum\_2Enum .  \\
((ap~(ap~(c\_2Esptree\_2Elookup~A\_27a)~V1x)~(ap~(\mname{c\_2Esptree\_2EfromAList}~\\
A\_27a)~V0ls))=(ap~(ap~(c\_2Ealist\_2EALOOKUP~A\_27a~ty\_2Enum\_2Enum)~\\
V0ls)~V1x))))\end{array}
\label{hyp:ehigngpfcdfedhahehchfgfgpfcdfemgpgpglgfhahpfpfggchpgngbemejgdheh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0ls \in (ty\_2Elist\_2Elist~\\
(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~A\_27a)) .  (p~(ap~(c\_2Esptree\_2Ewf~\\
A\_27a)~(ap~(\mname{c\_2Esptree\_2EfromAList}~A\_27a)~V0ls))))\end{array}
\label{hyp:ehigngpfcdfedhahehchfgfgpfcdfehhggpfpfggchpgngbemejgdheh}
\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} V0f \in ((A\_27a^{{A\_27b}})^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V1pt \in (ty\_2Esptree\_2Espt~\\
A\_27b) .  (p~(ap~(c\_2Esptree\_2Ewf~A\_27a)~(ap~(ap~(\mname{c\_2Esptree\_2Emapi}~\\
A\_27a~A\_27b)~V0f)~V1pt)))))\end{array}
\label{hyp:ehigngpfcdfedhahehchfgfgpfcdfehhggpfpfngbgahjg}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfedhahehchfgfgpfcdfengbgahjgpfpfbemgjgdheh}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0f \in ((A\_27a^{{A\_27b}})^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V1pt \in (ty\_2Esptree\_2Espt~\\
A\_27b) .  ((ap~(ap~(\mname{c\_2Esptree\_2Emapi}~A\_27a~A\_27b)~V0f)~V1pt)=(ap~\\
(\mname{c\_2Esptree\_2EfromAList}~A\_27a)~(ap~(ap~(c\_2Elist\_2EMAP~(ty\_2Epair\_2Eprod~\\
ty\_2Enum\_2Enum~A\_27b)~(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~A\_27a))~\\
({\uniddcgcg} V2kv \in (ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~A\_27b) .  (ap~(ap~(\\
\mname{c\_2Epair\_2E\_2C}~ty\_2Enum\_2Enum~A\_27a)~(ap~(c\_2Epair\_2EFST~ty\_2Enum\_2Enum~\\
A\_27b)~V2kv))~(ap~(ap~V0f~(ap~(c\_2Epair\_2EFST~ty\_2Enum\_2Enum~A\_27b)~\\
V2kv))~(ap~(c\_2Epair\_2ESND~ty\_2Enum\_2Enum~A\_27b)~V2kv)))))~(ap~\\
(\mname{c\_2Esptree\_2EtoAList}~A\_27b)~V1pt))))))\end{array}$$
\end{theorem}
\end{document}
