\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\_2EDecode\_2Edec\_\_bnum\_\_def\_\_compute (TMRhRuoK31EXB5hKmWZVRuD87NxCtVnfVqk)}
\maketitle
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddee}
We define $\mname{c\_2Emin\_2E\_3D}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} x \in {A} .  {{\uniddcgcg} y \in {A} .  {{inj\_\_o}~{({{x}={y}})}}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfeef}
We define $\mname{c\_2Ebool\_2ET}$ to be ${({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{2}}})}})}}~{({{\uniddcgcg} V0x \in {2} .  {V0x}})}})}}~{({{\uniddcgcg} V1x \in {2} .  {V1x}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdbd}
We define $\mname{c\_2Ebool\_2E\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V0P}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfege}
We define $\mname{c\_2Ebool\_2EF}$ to be ${({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V0t \in {2} .  {V0t}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddeepfddeepfddfe}
We define $\mname{c\_2Emin\_2E\_3D\_3D\_3E}$ to be ${{\uniddcgcg} P \in {2} .  {{\uniddcgcg} Q \in {2} .  {{inj\_\_o}~{({{{p}~{P}}{\unicdbdjdcd}{{p}~{Q}}})}}}}$ of type ${\iota}$.
\end{definition}
\begin{definition}\label{def: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\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
Let $ c\_2Earithmetic\_2E\_2A : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2E\_2A \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepfcdbe}
\end{equation}
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}
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\_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}
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\_\_CASE : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}c\_2Eoption\_2Eoption\_\_CASE~\\
A\_27a~A\_27b \in (((A\_27b^{{({{A\_27b}^{{A\_27a}}})}})^{{A\_27b}})^{{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfepgahehjgpgogpfcdfepgahehjgpgogpfpfdebedffe}
\end{equation}
Let $ ty\_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}
\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}
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 $ 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}
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\_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}
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: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 $ 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 $ c\_2EDecode\_2Edec\_\_bnum : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2EDecode\_2Edec\_\_bnum \in (((ty\_2Eoption\_2Eoption~(ty\_2Epair\_2Eprod~\\
ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2)))^{{({{ty\_2Elist\_2Elist}~{2}})}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeeefgdgpgegfgpfcdfeegfgdgpfpfcgogfhng}
\end{equation}
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}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfecejeefcd}
We define $\mname{c\_2Earithmetic\_2EBIT2}$ to be ${{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{\mname{c\_2Enum\_2E0}}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfekffecfpe}
We define $\mname{c\_2Earithmetic\_2EZERO}$ to be ${\mname{c\_2Enum\_2E0}}$.
\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}
\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}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0l \in (ty\_2Elist\_2Elist~2) .  ((ap~(ap~c\_2EDecode\_2Edec\_\_bnum~\\
\mname{c\_2Enum\_2E0})~V0l)=(ap~(\mname{c\_2Eoption\_2ESOME}~(ty\_2Epair\_2Eprod~\\
ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2)))~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2))~\mname{c\_2Enum\_2E0})~V0l)))){\unicdcdcdhd}\\
({\unicdcdadad} V1m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V2l \in (ty\_2Elist\_2Elist~\\
2) .  ((ap~(ap~c\_2EDecode\_2Edec\_\_bnum~(ap~\mname{c\_2Enum\_2ESUC}~V1m))~V2l)=\\
(ap~(ap~(ap~(c\_2Elist\_2Elist\_\_CASE~2~(ty\_2Eoption\_2Eoption~(\\
ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2))))~V2l)~\\
(\mname{c\_2Eoption\_2ENONE}~(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~\\
2))))~({\uniddcgcg} V3h \in 2 .  ({\uniddcgcg} V4t \in (ty\_2Elist\_2Elist~2) .  (ap~(ap~(\\
ap~(c\_2Eoption\_2Eoption\_\_CASE~(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~\\
(ty\_2Elist\_2Elist~2))~(ty\_2Eoption\_2Eoption~(ty\_2Epair\_2Eprod~\\
ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2))))~(ap~(ap~c\_2EDecode\_2Edec\_\_bnum~\\
V1m)~V4t))~(\mname{c\_2Eoption\_2ENONE}~(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~\\
(ty\_2Elist\_2Elist~2))))~({\uniddcgcg} V5v \in (ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~\\
(ty\_2Elist\_2Elist~2)) .  (ap~(ap~(\mname{c\_2Epair\_2Epair\_\_CASE}~(ty\_2Eoption\_2Eoption~\\
(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2)))~ty\_2Enum\_2Enum~\\
(ty\_2Elist\_2Elist~2))~V5v)~({\uniddcgcg} V6n \in ty\_2Enum\_2Enum .  ({\uniddcgcg} V7t\_27 \in \\
(ty\_2Elist\_2Elist~2) .  (ap~(\mname{c\_2Eoption\_2ESOME}~(ty\_2Epair\_2Eprod~\\
ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2)))~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2))~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
(ap~(ap~c\_2Earithmetic\_2E\_2A~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\\
\mname{c\_2Earithmetic\_2EBIT2}~\mname{c\_2Earithmetic\_2EZERO})))~V6n))~(ap~(ap~\\
(ap~(\mname{c\_2Ebool\_2ECOND}~ty\_2Enum\_2Enum)~V3h)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))~\mname{c\_2Enum\_2E0})))~\\
V7t\_27))))))))))))))\end{array}
\label{hyp:ehigngpfcdfeeefgdgpgegfgpfcdfeegfgdgpfpfcgogfhngpfpfegfggg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0f \in ((A\_27a^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}}) .  \\
({\unicdcdadad} V1g \in (A\_27a^{{ty\_2Enum\_2Enum}}) .  (({\unicdcdadad} V2n \in ty\_2Enum\_2Enum .  \\
((ap~V1g~(ap~\mname{c\_2Enum\_2ESUC}~V2n))=(ap~(ap~V0f~V2n)~(ap~\mname{c\_2Enum\_2ESUC}~\\
V2n)))){\unicdbdjded}(({\unicdcdadad} V3n \in ty\_2Enum\_2Enum .  ((ap~V1g~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~V3n)))=(ap~(ap~V0f~(ap~(ap~c\_2Earithmetic\_2E\_2D~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V3n)))~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V3n))))){\unicdcdcdhd}\\
({\unicdcdadad} V4n \in ty\_2Enum\_2Enum .  ((ap~V1g~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~V4n)))=(ap~(ap~V0f~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~V4n)))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~V4n)))))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfedfffdepfpffemejenepfpfoeffnefecfbemedf}
\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}
\begin{theorem}\label{thm:ehigngpfcdfeeefgdgpgegfgpfcdfeegfgdgpfpfcgogfhngpfpfegfgggpfpfdgpgngahfhehfg}
$$\begin{array}{c}(({\unicdcdadad} V0l \in (ty\_2Elist\_2Elist~2) .  ((ap~(ap~c\_2EDecode\_2Edec\_\_bnum~\\
\mname{c\_2Enum\_2E0})~V0l)=(ap~(\mname{c\_2Eoption\_2ESOME}~(ty\_2Epair\_2Eprod~\\
ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2)))~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2))~\mname{c\_2Enum\_2E0})~V0l)))){\unicdcdcdhd}\\
(({\unicdcdadad} V1m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V2l \in (ty\_2Elist\_2Elist~\\
2) .  ((ap~(ap~c\_2EDecode\_2Edec\_\_bnum~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~V1m)))~V2l)=(ap~(ap~(ap~(c\_2Elist\_2Elist\_\_CASE~\\
2~(ty\_2Eoption\_2Eoption~(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~(\\
ty\_2Elist\_2Elist~2))))~V2l)~(\mname{c\_2Eoption\_2ENONE}~(ty\_2Epair\_2Eprod~\\
ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2))))~({\uniddcgcg} V3h \in 2 .  ({\uniddcgcg} V4t \in \\
(ty\_2Elist\_2Elist~2) .  (ap~(ap~(ap~(c\_2Eoption\_2Eoption\_\_CASE~\\
(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2))~(ty\_2Eoption\_2Eoption~\\
(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2))))~(\\
ap~(ap~c\_2EDecode\_2Edec\_\_bnum~(ap~(ap~c\_2Earithmetic\_2E\_2D~(ap~\\
\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V1m)))~(\\
ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))~\\
V4t))~(\mname{c\_2Eoption\_2ENONE}~(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~(\\
ty\_2Elist\_2Elist~2))))~({\uniddcgcg} V5v \in (ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~\\
(ty\_2Elist\_2Elist~2)) .  (ap~(ap~(\mname{c\_2Epair\_2Epair\_\_CASE}~(ty\_2Eoption\_2Eoption~\\
(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2)))~ty\_2Enum\_2Enum~\\
(ty\_2Elist\_2Elist~2))~V5v)~({\uniddcgcg} V6n \in ty\_2Enum\_2Enum .  ({\uniddcgcg} V7t\_27 \in \\
(ty\_2Elist\_2Elist~2) .  (ap~(\mname{c\_2Eoption\_2ESOME}~(ty\_2Epair\_2Eprod~\\
ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2)))~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2))~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
(ap~(ap~c\_2Earithmetic\_2E\_2A~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\\
\mname{c\_2Earithmetic\_2EBIT2}~\mname{c\_2Earithmetic\_2EZERO})))~V6n))~(ap~(ap~\\
(ap~(\mname{c\_2Ebool\_2ECOND}~ty\_2Enum\_2Enum)~V3h)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))~\mname{c\_2Enum\_2E0})))~\\
V7t\_27))))))))))))){\unicdcdcdhd}({\unicdcdadad} V8m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V9l \in \\
(ty\_2Elist\_2Elist~2) .  ((ap~(ap~c\_2EDecode\_2Edec\_\_bnum~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~V8m)))~V9l)=(ap~(ap~(ap~(c\_2Elist\_2Elist\_\_CASE~\\
2~(ty\_2Eoption\_2Eoption~(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~(\\
ty\_2Elist\_2Elist~2))))~V9l)~(\mname{c\_2Eoption\_2ENONE}~(ty\_2Epair\_2Eprod~\\
ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2))))~({\uniddcgcg} V10h \in 2 .  ({\uniddcgcg} V11t \in \\
(ty\_2Elist\_2Elist~2) .  (ap~(ap~(ap~(c\_2Eoption\_2Eoption\_\_CASE~\\
(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2))~(ty\_2Eoption\_2Eoption~\\
(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2))))~(\\
ap~(ap~c\_2EDecode\_2Edec\_\_bnum~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(\\
ap~\mname{c\_2Earithmetic\_2EBIT1}~V8m)))~V11t))~(\mname{c\_2Eoption\_2ENONE}~(ty\_2Epair\_2Eprod~\\
ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2))))~({\uniddcgcg} V12v \in (ty\_2Epair\_2Eprod~\\
ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2)) .  (ap~(ap~(\mname{c\_2Epair\_2Epair\_\_CASE}~\\
(ty\_2Eoption\_2Eoption~(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~\\
2)))~ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2))~V12v)~({\uniddcgcg} V13n \in \\
ty\_2Enum\_2Enum .  ({\uniddcgcg} V14t\_27 \in (ty\_2Elist\_2Elist~2) .  (ap~(\mname{c\_2Eoption\_2ESOME}~\\
(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2)))~(ap~\\
(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Enum\_2Enum~(ty\_2Elist\_2Elist~2))~(ap~\\
(ap~c\_2Earithmetic\_2E\_2B~(ap~(ap~c\_2Earithmetic\_2E\_2A~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~\mname{c\_2Earithmetic\_2EZERO})))~V13n))~(\\
ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~ty\_2Enum\_2Enum)~V10h)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))~\mname{c\_2Enum\_2E0})))~\\
V14t\_27)))))))))))))))\end{array}$$
\end{theorem}
\end{document}
