\documentclass{article}
\newtheorem{axiom}{Axiom}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{conjecture}{Conjecture}
\newcommand\unicdcdgdad{\not=}
\newcommand\unicdcdcdbg{\cup}
\newcommand\unicdcdbdgd{\setminus}
\newcommand\unicdcdcdjd{\cap}
\newcommand\unicdcdadad{\forall}
\newcommand\unicdbdjdcd{\Rightarrow}
\newcommand\unicdcdadjd{\notin}
\newcommand\unicdbdjded{\Leftrightarrow}
\newcommand\unicdcdcdid{\lor}
\newcommand\unicdcdcdhd{\land}
\newcommand\uniddcgcg{\lambda}
\newcommand\unicdcdaddd{\exists}
\newcommand\unicdcdidid{\not\subseteq}
\newcommand\preoh[1]{\lnot #1}
\newcommand\mname[1]{{\mathsf{#1}}}
\begin{document}
\title{thm\_2Epatricia\_\_casts\_2Enum\_\_to\_\_string\_\_string\_\_to\_\_num (TMbwnP7DahcHByJKhpPcC6FFUNH5JRTxTNw)}
\maketitle
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddee}
We define $\mname{c\_2Emin\_2E\_3D}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} x \in {A} .  {{\uniddcgcg} y \in {A} .  {{inj\_\_o}~{({{x}={y}})}}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfeef}
We define $\mname{c\_2Ebool\_2ET}$ to be ${({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{2}}})}})}}~{({{\uniddcgcg} V0x \in {2} .  {V0x}})}})}}~{({{\uniddcgcg} V1x \in {2} .  {V1x}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdbd}
We define $\mname{c\_2Ebool\_2E\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V0P}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfege}
We define $\mname{c\_2Ebool\_2EF}$ to be ${({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V0t \in {2} .  {V0t}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddeepfddeepfddfe}
We define $\mname{c\_2Emin\_2E\_3D\_3D\_3E}$ to be ${{\uniddcgcg} P \in {2} .  {{\uniddcgcg} Q \in {2} .  {{inj\_\_o}~{({{{p}~{P}}{\unicdbdjdcd}{{p}~{Q}}})}}}}$ of type ${\iota}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfhdfe}
We define $\mname{c\_2Ebool\_2E\_7E}$ to be ${({{\uniddcgcg} V0t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}$.
\end{definition}
\begin{definition}\label{def: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\_2Estring\_2Echar : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Estring\_2Echar}}\label{hyp:ogfgpfehjhpfcdfedhehchjgoghgpfcdfedgigbgch}
\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\_2Estring\_2ECHR : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Estring\_2ECHR} \in {({{ty\_2Estring\_2Echar}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfedhehchjgoghgpfcdfedeiecf}
\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}
\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: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:dgpfcdfebgchjgehigngfgehjgdgpfcdfeoeffnefecfbeme}
We define $\mname{c\_2Earithmetic\_2ENUMERAL}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {V0x}}$.
\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\_2Enumposrep\_2En2l : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Enumposrep\_2En2l \in (((ty\_2Elist\_2Elist~ty\_2Enum\_2Enum)^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeogfhngahpgdhchfgahpfcdfeogcdmg}
\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\_2EREVERSE : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2EREVERSE~A\_27a \in ((ty\_2Elist\_2Elist~\\
A\_27a)^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfecffegffecfdffe}
\end{equation}
\begin{definition}\label{def:dgpfcdfebedfdejejeogfhngcgfgchdhpfcdfeogcddh}
We define $\mname{c\_2EASCIInumbers\_2En2s}$ to be ${{\uniddcgcg} V0b \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1f \in {({{ty\_2Estring\_2Echar}^{{ty\_2Enum\_2Enum}}})} .  {{\uniddcgcg} V2n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{c\_2Elist\_2EREVERSE}~{ty\_2Estring\_2Echar}})}}~{({{{ap}~{({{{ap}~{({{{c\_2Elist\_2EMAP}~{ty\_2Enum\_2Enum}}~{ty\_2Estring\_2Echar}})}}~{V1f}})}}~{({{{ap}~{({{{ap}~{c\_2Enumposrep\_2En2l}}~{V0b}})}}~{V2n}})}})}})}}}}$.
\end{definition}
Let $ c\_2Epatricia\_\_casts\_2ESKIP1 : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Epatricia\_\_casts\_2ESKIP1 \in ((ty\_2Elist\_2Elist~ty\_2Estring\_2Echar)^{{({{ty\_2Elist\_2Elist}~{ty\_2Estring\_2Echar}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgehchjgdgjgbgpfpfdgbgdhehdhpfcdfedflejeafbd}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgehchjgdgjgbgpfpfdgbgdhehdhpfcdfeogfhngpfpfehpgpfpfdhehchjgoghg}
We define $\mname{c\_2Epatricia\_\_casts\_2Enum\_\_to\_\_string}$ to be ${{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{c\_2Epatricia\_\_casts\_2ESKIP1}}~{({{{ap}~{({{{ap}~{({{{ap}~{\mname{c\_2EASCIInumbers\_2En2s}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}})}})}})}})}})}}~{c\_2Estring\_2ECHR}})}}~{V0n}})}})}}$.
\end{definition}
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\_2Estring\_2EORD : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Estring\_2EORD} \in {({{ty\_2Enum\_2Enum}^{{ty\_2Estring\_2Echar}}})}}\label{hyp:ngfgngpfdgpfcdfedhehchjgoghgpfcdfepecfee}
\end{equation}
Let $ c\_2Enumposrep\_2El2n : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Enumposrep\_2El2n \in ((ty\_2Enum\_2Enum^{{({{ty\_2Elist\_2Elist}~{ty\_2Enum\_2Enum}})}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeogfhngahpgdhchfgahpfcdfemgcdog}
\end{equation}
\begin{definition}\label{def:dgpfcdfebedfdejejeogfhngcgfgchdhpfcdfedhcdog}
We define $\mname{c\_2EASCIInumbers\_2Es2n}$ to be ${{\uniddcgcg} V0b \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1f \in {({{ty\_2Enum\_2Enum}^{{ty\_2Estring\_2Echar}}})} .  {{\uniddcgcg} V2s \in {({{ty\_2Elist\_2Elist}~{ty\_2Estring\_2Echar}})} .  {({{{ap}~{({{{ap}~{c\_2Enumposrep\_2El2n}}~{V0b}})}}~{({{{ap}~{({{{ap}~{({{{c\_2Elist\_2EMAP}~{ty\_2Estring\_2Echar}}~{ty\_2Enum\_2Enum}})}}~{V1f}})}}~{({{{ap}~{({{c\_2Elist\_2EREVERSE}~{ty\_2Estring\_2Echar}})}}~{V2s}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahbgehchjgdgjgbgpfpfdgbgdhehdhpfcdfedhehchjgoghgpfpfehpgpfpfogfhng}
We define $\mname{c\_2Epatricia\_\_casts\_2Estring\_\_to\_\_num}$ to be ${{\uniddcgcg} V0s \in {({{ty\_2Elist\_2Elist}~{ty\_2Estring\_2Echar}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{\mname{c\_2EASCIInumbers\_2Es2n}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}})}})}})}})}})}}~{c\_2Estring\_2EORD}})}}~{({{{ap}~{({{{ap}~{({{c\_2Elist\_2ECONS}~{ty\_2Estring\_2Echar}})}}~{({{{ap}~{c\_2Estring\_2ECHR}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{V0s}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfeffoejegf}
We define $\mname{c\_2Epred\_\_set\_2EUNIV}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfejeoe}
We define $\mname{c\_2Ebool\_2EIN}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {({{\uniddcgcg} V1f \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{V1f}}~{V0x}})}})}})}}$.
\end{definition}
Let $ ty\_2Epair\_2Eprod : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A0 .  nonempty~A0{\unicdbdjdcd}{\unicdcdadad} A1 .  nonempty~A1{\unicdbdjdcd}nonempty~(ty\_2Epair\_2Eprod~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfeahbgjgchpfcdfeahchpgeg}
\end{equation}
Let $ c\_2Epair\_2EABS\_\_prod : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}c\_2Epair\_2EABS\_\_prod~\\
A\_27a~A\_27b \in ((ty\_2Epair\_2Eprod~A\_27a~A\_27b)^{{({{({{2}^{{A\_27b}}})}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfebecedfpfpfahchpgeg}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgjgchpfcdfepfcdde}
We define $\mname{c\_2Epair\_2E\_2C}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {{\uniddcgcg} V1y \in {A\_27b} .  {({{{ap}~{({{{c\_2Epair\_2EABS\_\_prod}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V2a \in {A\_27a} .  {({{\uniddcgcg} V3b \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2a}})}}~{V0x}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{V3b}})}}~{V1y}})}})}})}})}})}}}}}$.
\end{definition}
Let $ c\_2Epred\_\_set\_2EGSPEC : {{\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\_2Epred\_\_set\_2EGSPEC~\\
A\_27a~A\_27b \in ((2^{{A\_27a}})^{{({{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{2}})}^{{A\_27b}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahchfgegpfpfdhfgehpfcdfehedfaffede}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfejenebehefe}
We define $\mname{c\_2Epred\_\_set\_2EIMAGE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27b}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27b}}~{2}})}}~{({{{ap}~{V0f}}~{V2x}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1s}})}})}})}})}}}}}$.
\end{definition}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\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} V0s \in (ty\_2Elist\_2Elist~ty\_2Estring\_2Echar) .  ({\unicdcdadad} V1t \in \\
(ty\_2Elist\_2Elist~ty\_2Estring\_2Echar) .  (((ap~\mname{c\_2Epatricia\_\_casts\_2Estring\_\_to\_\_num}~\\
V0s)=(ap~\mname{c\_2Epatricia\_\_casts\_2Estring\_\_to\_\_num}~V1t)){\unicdbdjded}(V0s=\\
V1t))))\end{array}
\label{hyp:ehigngpfcdfeahbgehchjgdgjgbgpfpfdgbgdhehdhpfcdfedhehchjgoghgpfpfehpgpfpfogfhngpfpfbdbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~ty\_2Enum\_2Enum)~\\
V0n)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EIMAGE}~(ty\_2Elist\_2Elist~ty\_2Estring\_2Echar)~\\
ty\_2Enum\_2Enum)~\mname{c\_2Epatricia\_\_casts\_2Estring\_\_to\_\_num})~(\mname{c\_2Epred\_\_set\_2EUNIV}~\\
(ty\_2Elist\_2Elist~ty\_2Estring\_2Echar))))){\unicdbdjdcd}((ap~\mname{c\_2Epatricia\_\_casts\_2Estring\_\_to\_\_num}~\\
(ap~\mname{c\_2Epatricia\_\_casts\_2Enum\_\_to\_\_string}~V0n))=V0n)))\end{array}
\label{hyp:ehigngpfcdfeahbgehchjgdgjgbgpfpfdgbgdhehdhpfcdfedhehchjgoghgpfpfehpgpfpfogfhngpfpfogfhngpfpfehpgpfpfdhehchjgoghg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  (p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V0x)~(\mname{c\_2Epred\_\_set\_2EUNIV}~A\_27a))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfffoejegf}
\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} V1s \in (2^{{A\_27a}}) .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V0x)~V1s)){\unicdbdjdcd}({\unicdcdadad} V2f \in (A\_27b^{{A\_27a}}) .  (p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27b)~(ap~V2f~V0x))~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EIMAGE}~A\_27a~A\_27b)~\\
V2f)~V1s)))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejenebehefepfpfjeoe}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfeahbgehchjgdgjgbgpfpfdgbgdhehdhpfcdfeogfhngpfpfehpgpfpfdhehchjgoghgpfpfdhehchjgoghgpfpfehpgpfpfogfhng}
$$\begin{array}{c}({\unicdcdadad} V0s \in (ty\_2Elist\_2Elist~ty\_2Estring\_2Echar) .  ((ap~\mname{c\_2Epatricia\_\_casts\_2Enum\_\_to\_\_string}~\\
(ap~\mname{c\_2Epatricia\_\_casts\_2Estring\_\_to\_\_num}~V0s))=V0s))\end{array}$$
\end{theorem}
\end{document}
