\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\_2Ellist\_2ELZIP\_\_LUNZIP (TMJLjGZcynxZ4JYy1455NmUXx1sSyCs9r3q)}
\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:dgpfcdfecgpgpgmgpfcdfecepeffoeeefeee}
We define $\mname{c\_2Ebool\_2EBOUNDED}$ to be ${({{\uniddcgcg} V0v \in {2} .  {\mname{c\_2Ebool\_2ET}}})}$.
\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\_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 $ ty\_2Ellist\_2Ellist : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Ellist\_2Ellist}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfemgmgjgdhehpfcdfemgmgjgdheh}
\end{equation}
Let $ c\_2Ellist\_2ELZIP : {{\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\_2Ellist\_2ELZIP~\\
A\_27a~A\_27b \in ((ty\_2Ellist\_2Ellist~(ty\_2Epair\_2Eprod~A\_27a~A\_27b))^{{({{{ty\_2Epair\_2Eprod}~{({{ty\_2Ellist\_2Ellist}~{A\_27a}})}}~{({{ty\_2Ellist\_2Ellist}~{A\_27b}})}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgmgjgdhehpfcdfemekfjeaf}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfemefeef}
We define $\mname{c\_2Ebool\_2ELET}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27a}}})} .  {({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{V0f}}~{V1x}})}})}})}}}$.
\end{definition}
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 $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\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}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfekffecfpe}
We define $\mname{c\_2Earithmetic\_2EZERO}$ to be ${\mname{c\_2Enum\_2E0}}$.
\end{definition}
Let $ c\_2Enum\_2EREP\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EREP\_\_num} \in {({{omega}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfecffeafpfpfogfhng}
\end{equation}
Let $ c\_2Enum\_2ESUC\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2ESUC\_\_REP} \in {({{omega}^{{omega}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfedfffdepfpfcffeaf}
\end{equation}
\begin{definition}\label{def:dgpfcdfeogfhngpfcdfedfffde}
We define $\mname{c\_2Enum\_2ESUC}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{({{{ap}~{c\_2Enum\_2ESUC\_\_REP}}~{({{{ap}~{c\_2Enum\_2EREP\_\_num}}~{V0m}})}})}})}}$.
\end{definition}
Let $ c\_2Earithmetic\_2E\_2B : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2E\_2B \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepfcdce}
\end{equation}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfecejeefbd}
We define $\mname{c\_2Earithmetic\_2EBIT1}$ to be ${{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{\mname{c\_2Enum\_2E0}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfeoeffnefecfbeme}
We define $\mname{c\_2Earithmetic\_2ENUMERAL}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {V0x}}$.
\end{definition}
Let $ c\_2Earithmetic\_2E\_2D : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2E\_2D \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepfcdee}
\end{equation}
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\_2Ellist\_2Ellist\_\_rep : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ellist\_2Ellist\_\_rep~A\_27a \in \\
(((ty\_2Eoption\_2Eoption~A\_27a)^{{ty\_2Enum\_2Enum}})^{{({{ty\_2Ellist\_2Ellist}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgmgjgdhehpfcdfemgmgjgdhehpfpfchfgah}
\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: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: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}
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: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: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 $ c\_2Ellist\_2Ellist\_\_abs : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ellist\_2Ellist\_\_abs~A\_27a \in \\
((ty\_2Ellist\_2Ellist~A\_27a)^{{({{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}^{{ty\_2Enum\_2Enum}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgmgjgdhehpfcdfemgmgjgdhehpfpfbgcgdh}
\end{equation}
\begin{definition}\label{def:dgpfcdfemgmgjgdhehpfcdfemedepeoedf}
We define $\mname{c\_2Ellist\_2ELCONS}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0h \in {A\_27a} .  {{\uniddcgcg} V1t \in {({{ty\_2Ellist\_2Ellist}~{A\_27a}})} .  {({{{ap}~{({{c\_2Ellist\_2Ellist\_\_abs}~{A\_27a}})}}~{({{\uniddcgcg} V2n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V2n}})}}~{\mname{c\_2Enum\_2E0}}})}})}}~{({{{ap}~{({{\mname{c\_2Eoption\_2ESOME}}~{A\_27a}})}}~{V0h}})}})}}~{({{{ap}~{({{{ap}~{({{c\_2Ellist\_2Ellist\_\_rep}~{A\_27a}})}}~{V1t}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2D}}~{V2n}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}})}})}}}}$.
\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: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}
\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:dgpfcdfemgmgjgdhehpfcdfemeoejeme}
We define $\mname{c\_2Ellist\_2ELNIL}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{c\_2Ellist\_2Ellist\_\_abs}~{A\_27a}})}}~{({{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {({{\mname{c\_2Eoption\_2ENONE}}~{A\_27a}})}})}})}}$.
\end{definition}
Let $ c\_2Ellist\_2ELUNZIP : {{\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\_2Ellist\_2ELUNZIP~\\
A\_27a~A\_27b \in ((ty\_2Epair\_2Eprod~(ty\_2Ellist\_2Ellist~A\_27a)~(\\
ty\_2Ellist\_2Ellist~A\_27b))^{{({{ty\_2Ellist\_2Ellist}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgmgjgdhehpfcdfemeffoekfjeaf}
\end{equation}
\begin{definition}\label{def:dgpfcdfengbgchlgfgchpfcdfebecgcgchfggh}
We define $\mname{c\_2Emarker\_2EAbbrev}$ to be ${{\uniddcgcg} V0x \in {2} .  {V0x}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengbgchlgfgchpfcdfedepgoghg}
We define $\mname{c\_2Emarker\_2ECong}$ to be ${{\uniddcgcg} V0x \in {2} .  {V0x}}$.
\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}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfede}
We define $\mname{c\_2Ecombin\_2EC}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {({{\uniddcgcg} V0f \in {({{({{A\_27c}^{{A\_27b}}})}^{{A\_27a}}})} .  {({{\uniddcgcg} V1x \in {A\_27b} .  {({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{V0f}}~{V2y}})}}~{V1x}})}})}})}})}}}}$.
\end{definition}
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}
\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:dgpfcdfedgpgngcgjgogpfcdfepg}
We define $\mname{c\_2Ecombin\_2Eo}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27c}}})} .  {{\uniddcgcg} V1g \in {({{A\_27c}^{{A\_27a}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{V0f}}~{({{{ap}~{V1g}}~{V2x}})}})}})}}}}}}$.
\end{definition}
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:dgpfcdfeahbgjgchpfcdfeffoedeffcfcfjf}
We define $\mname{c\_2Epair\_2EUNCURRY}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0f \in {({{({{A\_27c}^{{A\_27b}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1v \in {({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})} .  {({{{ap}~{({{{ap}~{V0f}}~{({{{ap}~{({{{c\_2Epair\_2EFST}~{A\_27a}}~{A\_27b}})}}~{V1v}})}})}}~{({{{ap}~{({{{c\_2Epair\_2ESND}~{A\_27a}}~{A\_27b}})}}~{V1v}})}})}}}}}}$.
\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}
{({{\unicdcdadad} V0t \in {2} .  {({{False}{\unicdbdjdcd}{({{p}~{V0t}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegebemedfjeefjf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0t \in 2 .  (({\unicdcdaddd} V1x \in \\
A\_27a .  (p~V0t)){\unicdbdjded}(p~V0t)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefeifjedfefdfpfpfdfjeneaf}
\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} 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} 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}
{({{\unicdcdadad} V0v \in {2} .  {({{({{p}~{({{{ap}~{\mname{c\_2Ebool\_2EBOUNDED}}}~{V0v}})}})}{\unicdbdjded}{True}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecepeffoeeefeeepfpfefiene}
\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} V0f \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V1g \in (A\_27a^{{A\_27c}}) .  \\
({\unicdcdadad} V2x \in A\_27c .  ((ap~(ap~(ap~(\mname{c\_2Ecombin\_2Eo}~A\_27c~A\_27b~A\_27a)~\\
V0f)~V1g)~V2x)=(ap~V0f~(ap~V1g~V2x))))))\end{array}
\label{hyp:ehigngpfcdfedgpgngcgjgogpfcdfepgpfpfefiene}
\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} V0f \in (A\_27b^{{A\_27c}}) .  ({\unicdcdadad} V1g \in (A\_27c^{{A\_27a}}) .  \\
((ap~(ap~(\mname{c\_2Ecombin\_2Eo}~A\_27a~A\_27b~A\_27c)~V0f)~({\uniddcgcg} V2x \in A\_27a .  \\
(ap~V1g~V2x)))=({\uniddcgcg} V3x \in A\_27a .  (ap~V0f~(ap~V1g~V3x))))))\end{array}
\label{hyp:ehigngpfcdfedgpgngcgjgogpfcdfepgpfpfbecedfpfpfcf}
\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} V0f \in ((A\_27b^{{A\_27c}})^{{A\_27a}}) .  ({\unicdcdadad} V1g \in \\
(A\_27c^{{A\_27a}}) .  ((ap~(ap~(\mname{c\_2Ecombin\_2ES}~A\_27a~A\_27c~A\_27b)~V0f)~\\
({\uniddcgcg} V2x \in A\_27a .  (ap~V1g~V2x)))=({\uniddcgcg} V3x \in A\_27a .  (ap~(ap~V0f~V3x)~\\
(ap~V1g~V3x))))))\end{array}
\label{hyp:ehigngpfcdfedgpgngcgjgogpfcdfedfpfpfbecedfpfpfcf}
\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} V0f \in ((A\_27c^{{A\_27b}})^{{A\_27a}}) .  ({\unicdcdadad} V1x \in \\
A\_27b .  ({\unicdcdadad} V2y \in A\_27a .  ((ap~(ap~(ap~(\mname{c\_2Ecombin\_2EC}~A\_27a~A\_27b~\\
A\_27c)~V0f)~V1x)~V2y)=(ap~(ap~V0f~V2y)~V1x)))))\end{array}
\label{hyp:ehigngpfcdfedgpgngcgjgogpfcdfedepfpfefiene}
\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} V0f \in ((A\_27b^{{A\_27c}})^{{A\_27a}}) .  ({\unicdcdadad} V1y \in \\
A\_27c .  ((ap~(ap~(\mname{c\_2Ecombin\_2EC}~A\_27a~A\_27c~A\_27b)~({\uniddcgcg} V2x \in A\_27a .  \\
(ap~V0f~V2x)))~V1y)=({\uniddcgcg} V3x \in A\_27a .  (ap~(ap~V0f~V3x)~V1y)))))\end{array}
\label{hyp:ehigngpfcdfedgpgngcgjgogpfcdfedepfpfbecedfpfpfme}
\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} V0P \in (A\_27a^{{A\_27b}}) .  ({\unicdcdadad} V1f \in (A\_27b^{{A\_27c}}) .  \\
({\unicdcdadad} V2v \in A\_27c .  ((ap~V0P~(ap~(ap~(\mname{c\_2Ebool\_2ELET}~A\_27c~A\_27b)~\\
V1f)~V2v))=(ap~(ap~(\mname{c\_2Ebool\_2ELET}~A\_27c~A\_27a)~(ap~(ap~(\mname{c\_2Ecombin\_2Eo}~\\
A\_27c~A\_27a~A\_27b)~V0P)~V1f))~V2v)))))\end{array}
\label{hyp:ehigngpfcdfedgpgngcgjgogpfcdfehefeoepfpfmefeefpfpfcfbeoeee}
\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} V0f \in ((A\_27a^{{A\_27c}})^{{A\_27b}}) .  ({\unicdcdadad} V1v \in \\
A\_27b .  ({\unicdcdadad} V2x \in A\_27c .  ((ap~(ap~(ap~(\mname{c\_2Ebool\_2ELET}~A\_27b~(\\
A\_27a^{{A\_27c}}))~V0f)~V1v)~V2x)=(ap~(ap~(\mname{c\_2Ebool\_2ELET}~A\_27b~A\_27a)~\\
(ap~(ap~(\mname{c\_2Ecombin\_2EC}~A\_27b~A\_27c~A\_27a)~V0f)~V2x))~V1v)))))\end{array}
\label{hyp:ehigngpfcdfedgpgngcgjgogpfcdfehefeoepfpfmefeefpfpfcfbeefpecf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0f \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1v \in \\
A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2ELET}~A\_27a~2)~V0f)~V1v)){\unicdbdjded}(p~(ap~(\mname{c\_2Ebool\_2E\_21}~\\
A\_27a)~(ap~(ap~(\mname{c\_2Ecombin\_2ES}~A\_27a~2~2)~(ap~(ap~(\mname{c\_2Ecombin\_2Eo}~\\
A\_27a~(2^{{2}})~2)~\mname{c\_2Emin\_2E\_3D\_3D\_3E})~(ap~(ap~(\mname{c\_2Ecombin\_2Eo}~A\_27a~\\
2~2)~\mname{c\_2Emarker\_2EAbbrev})~(ap~(ap~(\mname{c\_2Ecombin\_2EC}~A\_27a~A\_27a~\\
2)~(\mname{c\_2Emin\_2E\_3D}~A\_27a))~V1v))))~V0f))))))\end{array}
\label{hyp:ehigngpfcdfedgpgngcgjgogpfcdfemefeefpfpfgepecfbememepfpffemejene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0l \in (ty\_2Ellist\_2Ellist~\\
A\_27a) .  ((V0l=(\mname{c\_2Ellist\_2ELNIL}~A\_27a)){\unicdcdcdid}({\unicdcdaddd} V1h \in A\_27a .  \\
({\unicdcdaddd} V2t \in (ty\_2Ellist\_2Ellist~A\_27a) .  (V0l=(ap~(ap~(\mname{c\_2Ellist\_2ELCONS}~\\
A\_27a)~V1h)~V2t))))))\end{array}
\label{hyp:ehigngpfcdfemgmgjgdhehpfcdfemgmgjgdhehpfpfdebedffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0h \in A\_27a .  ({\unicdcdadad} V1t \in \\
(ty\_2Ellist\_2Ellist~A\_27a) .  ((\preoh ((ap~(ap~(\mname{c\_2Ellist\_2ELCONS}~A\_27a)~\\
V0h)~V1t)=(\mname{c\_2Ellist\_2ELNIL}~A\_27a))){\unicdcdcdhd}(\preoh ((\mname{c\_2Ellist\_2ELNIL}~\\
A\_27a)=(ap~(ap~(\mname{c\_2Ellist\_2ELCONS}~A\_27a)~V0h)~V1t))))))\end{array}
\label{hyp:ehigngpfcdfemgmgjgdhehpfcdfemedepeoedfpfpfoepeefpfpfoejeme}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0h1 \in A\_27a .  ({\unicdcdadad} V1t1 \in \\
(ty\_2Ellist\_2Ellist~A\_27a) .  ({\unicdcdadad} V2h2 \in A\_27a .  ({\unicdcdadad} V3t2 \in \\
(ty\_2Ellist\_2Ellist~A\_27a) .  (((ap~(ap~(\mname{c\_2Ellist\_2ELCONS}~A\_27a)~\\
V0h1)~V1t1)=(ap~(ap~(\mname{c\_2Ellist\_2ELCONS}~A\_27a)~V2h2)~V3t2)){\unicdbdjded}((\\
V0h1=V2h2){\unicdcdcdhd}(V1t1=V3t2)))))))\end{array}
\label{hyp:ehigngpfcdfemgmgjgdhehpfcdfemedepeoedfpfpfbdbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0ll1 \in (ty\_2Ellist\_2Ellist~\\
A\_27a) .  ({\unicdcdadad} V1ll2 \in (ty\_2Ellist\_2Ellist~A\_27a) .  ((V0ll1=\\
V1ll2){\unicdbdjded}({\unicdcdaddd} V2R \in ((2^{{({{ty\_2Ellist\_2Ellist}~{A\_27a}})}})^{{({{ty\_2Ellist\_2Ellist}~{A\_27a}})}}) .  \\
((p~(ap~(ap~V2R~V0ll1)~V1ll2)){\unicdcdcdhd}({\unicdcdadad} V3ll3 \in (ty\_2Ellist\_2Ellist~\\
A\_27a) .  ({\unicdcdadad} V4ll4 \in (ty\_2Ellist\_2Ellist~A\_27a) .  ((p~(ap~(ap~\\
V2R~V3ll3)~V4ll4)){\unicdbdjdcd}((V3ll3=V4ll4){\unicdcdcdid}({\unicdcdaddd} V5h \in A\_27a .  ({\unicdcdaddd} V6t1 \in \\
(ty\_2Ellist\_2Ellist~A\_27a) .  ({\unicdcdaddd} V7t2 \in (ty\_2Ellist\_2Ellist~\\
A\_27a) .  ((V3ll3=(ap~(ap~(\mname{c\_2Ellist\_2ELCONS}~A\_27a)~V5h)~V6t1)){\unicdcdcdhd}\\
((V4ll4=(ap~(ap~(\mname{c\_2Ellist\_2ELCONS}~A\_27a)~V5h)~V7t2)){\unicdcdcdhd}(p~(ap~(\\
ap~V2R~V6t1)~V7t2))))))))))))))))\end{array}
\label{hyp:ehigngpfcdfemgmgjgdhehpfcdfememejedfefpfpfdfefcfpeoehepfpfcejedfjeneffmebeefjepeoe}
\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} V0l1 \in (ty\_2Ellist\_2Ellist~A\_27a) .  ((ap~(c\_2Ellist\_2ELZIP~\\
A\_27a~A\_27b)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Ellist\_2Ellist~A\_27a)~\\
(ty\_2Ellist\_2Ellist~A\_27b))~V0l1)~(\mname{c\_2Ellist\_2ELNIL}~A\_27b)))=\\
(\mname{c\_2Ellist\_2ELNIL}~(ty\_2Epair\_2Eprod~A\_27a~A\_27b)))){\unicdcdcdhd}(({\unicdcdadad} V1l2 \in \\
(ty\_2Ellist\_2Ellist~A\_27b) .  ((ap~(c\_2Ellist\_2ELZIP~A\_27a~A\_27b)~\\
(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Ellist\_2Ellist~A\_27a)~(ty\_2Ellist\_2Ellist~\\
A\_27b))~(\mname{c\_2Ellist\_2ELNIL}~A\_27a))~V1l2))=(\mname{c\_2Ellist\_2ELNIL}~\\
(ty\_2Epair\_2Eprod~A\_27a~A\_27b)))){\unicdcdcdhd}({\unicdcdadad} V2h1 \in A\_27a .  ({\unicdcdadad} V3h2 \in \\
A\_27b .  ({\unicdcdadad} V4t1 \in (ty\_2Ellist\_2Ellist~A\_27a) .  ({\unicdcdadad} V5t2 \in \\
(ty\_2Ellist\_2Ellist~A\_27b) .  ((ap~(c\_2Ellist\_2ELZIP~A\_27a~A\_27b)~\\
(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Ellist\_2Ellist~A\_27a)~(ty\_2Ellist\_2Ellist~\\
A\_27b))~(ap~(ap~(\mname{c\_2Ellist\_2ELCONS}~A\_27a)~V2h1)~V4t1))~(ap~(ap~(\\
\mname{c\_2Ellist\_2ELCONS}~A\_27b)~V3h2)~V5t2)))=(ap~(ap~(\mname{c\_2Ellist\_2ELCONS}~\\
(ty\_2Epair\_2Eprod~A\_27a~A\_27b))~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~\\
A\_27b)~V2h1)~V3h2))~(ap~(c\_2Ellist\_2ELZIP~A\_27a~A\_27b)~(ap~(ap~(\\
\mname{c\_2Epair\_2E\_2C}~(ty\_2Ellist\_2Ellist~A\_27a)~(ty\_2Ellist\_2Ellist~\\
A\_27b))~V4t1)~V5t2))))))))))\end{array}
\label{hyp:ehigngpfcdfemgmgjgdhehpfcdfemekfjeafpfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
((ap~(c\_2Ellist\_2ELUNZIP~A\_27a~A\_27b)~(\mname{c\_2Ellist\_2ELNIL}~(ty\_2Epair\_2Eprod~\\
A\_27a~A\_27b)))=(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Ellist\_2Ellist~A\_27a)~\\
(ty\_2Ellist\_2Ellist~A\_27b))~(\mname{c\_2Ellist\_2ELNIL}~A\_27a))~(\mname{c\_2Ellist\_2ELNIL}~\\
A\_27b))){\unicdcdcdhd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in A\_27b .  ({\unicdcdadad} V2t \in \\
(ty\_2Ellist\_2Ellist~(ty\_2Epair\_2Eprod~A\_27a~A\_27b)) .  ((ap~(c\_2Ellist\_2ELUNZIP~\\
A\_27a~A\_27b)~(ap~(ap~(\mname{c\_2Ellist\_2ELCONS}~(ty\_2Epair\_2Eprod~A\_27a~\\
A\_27b))~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V0x)~V1y))~V2t))=(\\
ap~(ap~(\mname{c\_2Ebool\_2ELET}~(ty\_2Epair\_2Eprod~(ty\_2Ellist\_2Ellist~\\
A\_27a)~(ty\_2Ellist\_2Ellist~A\_27b))~(ty\_2Epair\_2Eprod~(ty\_2Ellist\_2Ellist~\\
A\_27a)~(ty\_2Ellist\_2Ellist~A\_27b)))~(ap~(\mname{c\_2Epair\_2EUNCURRY}~\\
(ty\_2Ellist\_2Ellist~A\_27a)~(ty\_2Ellist\_2Ellist~A\_27b)~(ty\_2Epair\_2Eprod~\\
(ty\_2Ellist\_2Ellist~A\_27a)~(ty\_2Ellist\_2Ellist~A\_27b)))~({\uniddcgcg} V3ll1 \in \\
(ty\_2Ellist\_2Ellist~A\_27a) .  ({\uniddcgcg} V4ll2 \in (ty\_2Ellist\_2Ellist~\\
A\_27b) .  (ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Ellist\_2Ellist~A\_27a)~(ty\_2Ellist\_2Ellist~\\
A\_27b))~(ap~(ap~(\mname{c\_2Ellist\_2ELCONS}~A\_27a)~V0x)~V3ll1))~(ap~(ap~(\\
\mname{c\_2Ellist\_2ELCONS}~A\_27b)~V1y)~V4ll2))))))~(ap~(c\_2Ellist\_2ELUNZIP~\\
A\_27a~A\_27b)~V2t)))))))\end{array}
\label{hyp:ehigngpfcdfemgmgjgdhehpfcdfemeffoekfjeafpfpfefiene}
\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 .  ({\unicdcdadad} V2a \in A\_27a .  ({\unicdcdadad} V3b \in \\
A\_27b .  (((ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V0x)~V1y)=(ap~(ap~\\
(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V2a)~V3b)){\unicdbdjded}((V0x=V2a){\unicdcdcdhd}(V1y=V3b)))))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfedemepedffeeepfpfafbejecfpfpffebf}
\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 (ty\_2Epair\_2Eprod~A\_27a~A\_27b) .  ({\unicdcdaddd} V1q \in A\_27a .  \\
({\unicdcdaddd} V2r \in A\_27b .  (V0x=(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~\\
V1q)~V2r)))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfebecedfpfpfafbejecfpfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}{\unicdcdadad} A\_27d .  nonempty~A\_27d{\unicdbdjdcd}({\unicdcdadad} V0f \in (A\_27c^{{A\_27d}}) .  \\
({\unicdcdadad} V1g \in ((A\_27d^{{A\_27b}})^{{A\_27a}}) .  ((ap~(ap~(\mname{c\_2Ecombin\_2Eo}~(\\
ty\_2Epair\_2Eprod~A\_27a~A\_27b)~A\_27c~A\_27d)~V0f)~(ap~(\mname{c\_2Epair\_2EUNCURRY}~\\
A\_27a~A\_27b~A\_27d)~V1g))=(ap~(\mname{c\_2Epair\_2EUNCURRY}~A\_27a~A\_27b~A\_27c)~\\
(ap~(ap~(\mname{c\_2Ecombin\_2Eo}~A\_27a~(A\_27c^{{A\_27b}})~(A\_27d^{{A\_27b}}))~(ap~(\\
\mname{c\_2Ecombin\_2Eo}~A\_27b~A\_27c~A\_27d)~V0f))~V1g)))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfepgpfpfffoedeffcfcfjfpfpfcf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}{\unicdcdadad} A\_27d .  nonempty~A\_27d{\unicdbdjdcd}({\unicdcdadad} V0f \in ((\\
(A\_27c^{{A\_27d}})^{{A\_27b}})^{{A\_27a}}) .  ({\unicdcdadad} V1x \in A\_27d .  ((ap~(ap~(\mname{c\_2Ecombin\_2EC}~\\
(ty\_2Epair\_2Eprod~A\_27a~A\_27b)~A\_27d~A\_27c)~(ap~(\mname{c\_2Epair\_2EUNCURRY}~\\
A\_27a~A\_27b~(A\_27c^{{A\_27d}}))~V0f))~V1x)=(ap~(\mname{c\_2Epair\_2EUNCURRY}~\\
A\_27a~A\_27b~A\_27c)~(ap~(ap~(\mname{c\_2Ecombin\_2EC}~A\_27a~A\_27d~(A\_27c^{{A\_27b}}))~\\
(ap~(ap~(\mname{c\_2Ecombin\_2Eo}~A\_27a~((A\_27c^{{A\_27b}})^{{A\_27d}})~((A\_27c^{{A\_27d}})^{{A\_27b}}))~\\
(\mname{c\_2Ecombin\_2EC}~A\_27b~A\_27d~A\_27c))~V0f))~V1x)))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfedepfpfffoedeffcfcfjfpfpfme}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}{\unicdcdadad} A\_27d .  nonempty~A\_27d{\unicdbdjdcd}({\unicdcdadad} V0f \in ((\\
A\_27c^{{A\_27d}})^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}) .  ({\unicdcdadad} V1g \in ((\\
A\_27d^{{A\_27b}})^{{A\_27a}}) .  ((ap~(ap~(\mname{c\_2Ecombin\_2ES}~(ty\_2Epair\_2Eprod~\\
A\_27a~A\_27b)~A\_27d~A\_27c)~V0f)~(ap~(\mname{c\_2Epair\_2EUNCURRY}~A\_27a~A\_27b~\\
A\_27d)~V1g))=(ap~(\mname{c\_2Epair\_2EUNCURRY}~A\_27a~A\_27b~A\_27c)~(ap~(ap~\\
(\mname{c\_2Ecombin\_2ES}~A\_27a~(A\_27d^{{A\_27b}})~(A\_27c^{{A\_27b}}))~(ap~(ap~(\mname{c\_2Ecombin\_2Eo}~\\
A\_27a~((A\_27c^{{A\_27b}})^{{({{A\_27d}^{{A\_27b}}})}})~((A\_27c^{{A\_27d}})^{{A\_27b}}))~(\mname{c\_2Ecombin\_2ES}~\\
A\_27b~A\_27d~A\_27c))~(ap~(ap~(\mname{c\_2Ecombin\_2Eo}~A\_27a~((A\_27c^{{A\_27d}})^{{A\_27b}})~\\
((ty\_2Epair\_2Eprod~A\_27a~A\_27b)^{{A\_27b}}))~(ap~(\mname{c\_2Ecombin\_2Eo}~A\_27b~\\
(A\_27c^{{A\_27d}})~(ty\_2Epair\_2Eprod~A\_27a~A\_27b))~V0f))~(\mname{c\_2Epair\_2E\_2C}~\\
A\_27a~A\_27b))))~V1g)))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfedfpfpfffoedeffcfcfjfpfpfcf}
\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 ((2^{{A\_27b}})^{{A\_27a}}) .  ((p~(ap~(\mname{c\_2Ebool\_2E\_21}~(ty\_2Epair\_2Eprod~\\
A\_27a~A\_27b))~(ap~(\mname{c\_2Epair\_2EUNCURRY}~A\_27a~A\_27b~2)~V0f))){\unicdbdjded}(\\
p~(ap~(\mname{c\_2Ebool\_2E\_21}~A\_27a)~(ap~(ap~(\mname{c\_2Ecombin\_2Eo}~A\_27a~2~(2^{{A\_27b}}))~\\
(\mname{c\_2Ebool\_2E\_21}~A\_27b))~V0f)))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfegepecfbememepfpfffoedeffcfcfjf}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfemgmgjgdhehpfcdfemekfjeafpfpfmeffoekfjeaf}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0ll \in (ty\_2Ellist\_2Ellist~(ty\_2Epair\_2Eprod~A\_27a~A\_27b)) .  \\
((ap~(c\_2Ellist\_2ELZIP~A\_27a~A\_27b)~(ap~(c\_2Ellist\_2ELUNZIP~A\_27a~\\
A\_27b)~V0ll))=V0ll))\end{array}$$
\end{theorem}
\end{document}
