\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\_2EEMPTY\_\_IS\_\_PTREE (TMFLkMzpBh5HDxMDpBwFupXV4h2u5hxPreS)}
\maketitle
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: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: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: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: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}
\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}
Let $ c\_2Earithmetic\_2EEXP : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2EEXP \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfefeifaf}
\end{equation}
Let $ c\_2Earithmetic\_2EDIV : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2EDIV \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfeeejegf}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgjgehpfcdfeeejegfpfpfcdfeifaf}
We define $\mname{c\_2Ebit\_2EDIV\_\_2EXP}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2EDIV}}~{V1n}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2EEXP}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}}~{V0x}})}})}}}$.
\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 $ c\_2Earithmetic\_2EMOD : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2EMOD \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfenepeee}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgjgehpfcdfenepeeepfpfcdfeifaf}
We define $\mname{c\_2Ebit\_2EMOD\_\_2EXP}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2EMOD}}~{V1n}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2EEXP}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}}~{V0x}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgehpfcdfecejeefdf}
We define $\mname{c\_2Ebit\_2EBITS}$ to be ${{\uniddcgcg} V0h \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1l \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V2n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebit\_2EMOD\_\_2EXP}}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2D}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V0h}})}})}}~{V1l}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebit\_2EDIV\_\_2EXP}}}~{V1l}})}}~{V2n}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgehpfcdfecejeef}
We define $\mname{c\_2Ebit\_2EBIT}$ to be ${{\uniddcgcg} V0b \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebit\_2EBITS}}}~{V0b}})}}~{V0b}})}}~{V1n}})}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgehpfcdfenepeeepfpfcdfeifafpfpffebf}
We define $\mname{c\_2Ebit\_2EMOD\_\_2EXP\_\_EQ}$ to be ${{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1a \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V2b \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebit\_2EMOD\_\_2EXP}}}~{V0n}})}}~{V1a}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebit\_2EMOD\_\_2EXP}}}~{V0n}})}}~{V2b}})}})}}}}$.
\end{definition}
Let $ ty\_2Epatricia\_2Eptree : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Epatricia\_2Eptree}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfeahbgehchjgdgjgbgpfcdfeahehchfgfg}
\end{equation}
Let $ c\_2Epatricia\_2EEVERY\_\_LEAF : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Epatricia\_2EEVERY\_\_LEAF~A\_27a \in \\
((2^{{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})^{{({{({{2}^{{A\_27a}}})}^{{ty\_2Enum\_2Enum}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgehchjgdgjgbgpfcdfefegffecfjfpfpfmefebege}
\end{equation}
\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}
\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: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:dgpfcdfeahchjgngpfpfchfgdgpfcdfepfddde}
We define $\mname{c\_2Eprim\_\_rec\_2E\_3C}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{2}^{{ty\_2Enum\_2Enum}}})}})}}~{({{\uniddcgcg} V2P \in {({{2}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V3n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{V2P}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V3n}})}})}})}}~{({{{ap}~{V2P}}~{V3n}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V2P}}~{V0m}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{V2P}}~{V1n}})}})}})}})}})}})}}}$.
\end{definition}
Let $ c\_2Epatricia\_2EBranch : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Epatricia\_2EBranch~A\_27a \in (\\
((((ty\_2Epatricia\_2Eptree~A\_27a)^{{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})^{{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgehchjgdgjgbgpfcdfecechbgogdgig}
\end{equation}
Let $ c\_2Epatricia\_2ELeaf : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Epatricia\_2ELeaf~A\_27a \in (((\\
ty\_2Epatricia\_2Eptree~A\_27a)^{{A\_27a}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgehchjgdgjgbgpfcdfemefgbggg}
\end{equation}
Let $ c\_2Epatricia\_2EEmpty : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Epatricia\_2EEmpty~A\_27a \in (ty\_2Epatricia\_2Eptree~\\
A\_27a)\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgehchjgdgjgbgpfcdfefengahehjh}
\end{equation}
Let $ c\_2Epatricia\_2EIS\_\_PTREE : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Epatricia\_2EIS\_\_PTREE~A\_27a \in \\
(2^{{({{ty\_2Epatricia\_2Eptree}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgehchjgdgjgbgpfcdfejedfpfpfafefcffefe}
\end{equation}
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}(((p~(ap~(c\_2Epatricia\_2EIS\_\_PTREE~\\
A\_27a)~(c\_2Epatricia\_2EEmpty~A\_27a))){\unicdbdjded}True){\unicdcdcdhd}(({\unicdcdadad} V0k \in \\
ty\_2Enum\_2Enum .  ({\unicdcdadad} V1d \in A\_27a .  ((p~(ap~(c\_2Epatricia\_2EIS\_\_PTREE~\\
A\_27a)~(ap~(ap~(c\_2Epatricia\_2ELeaf~A\_27a)~V0k)~V1d))){\unicdbdjded}True))){\unicdcdcdhd}\\
({\unicdcdadad} V2p \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V3m \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V4l \in (ty\_2Epatricia\_2Eptree~A\_27a) .  ({\unicdcdadad} V5r \in (ty\_2Epatricia\_2Eptree~\\
A\_27a) .  ((p~(ap~(c\_2Epatricia\_2EIS\_\_PTREE~A\_27a)~(ap~(ap~(ap~(ap~\\
(c\_2Epatricia\_2EBranch~A\_27a)~V2p)~V3m)~V4l)~V5r))){\unicdbdjded}((p~(ap~(ap~\\
\mname{c\_2Eprim\_\_rec\_2E\_3C}~V2p)~(ap~(ap~c\_2Earithmetic\_2EEXP~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~\mname{c\_2Earithmetic\_2EZERO})))~V3m))){\unicdcdcdhd}\\
((p~(ap~(c\_2Epatricia\_2EIS\_\_PTREE~A\_27a)~V4l)){\unicdcdcdhd}((p~(ap~(c\_2Epatricia\_2EIS\_\_PTREE~\\
A\_27a)~V5r)){\unicdcdcdhd}((\preoh (V4l=(c\_2Epatricia\_2EEmpty~A\_27a))){\unicdcdcdhd}((\preoh (V5r=\\
(c\_2Epatricia\_2EEmpty~A\_27a))){\unicdcdcdhd}((p~(ap~(ap~(c\_2Epatricia\_2EEVERY\_\_LEAF~\\
A\_27a)~({\uniddcgcg} V6k \in ty\_2Enum\_2Enum .  ({\uniddcgcg} V7d \in A\_27a .  (ap~(ap~\mname{c\_2Ebool\_2E\_2F\_5C}~\\
(ap~(ap~(ap~\mname{c\_2Ebit\_2EMOD\_\_2EXP\_\_EQ}~V3m)~V6k)~V2p))~(ap~(ap~\mname{c\_2Ebit\_2EBIT}~\\
V3m)~V6k)))))~V4l)){\unicdcdcdhd}(p~(ap~(ap~(c\_2Epatricia\_2EEVERY\_\_LEAF~A\_27a)~\\
({\uniddcgcg} V8k \in ty\_2Enum\_2Enum .  ({\uniddcgcg} V9d \in A\_27a .  (ap~(ap~\mname{c\_2Ebool\_2E\_2F\_5C}~\\
(ap~(ap~(ap~\mname{c\_2Ebit\_2EMOD\_\_2EXP\_\_EQ}~V3m)~V8k)~V2p))~(ap~\mname{c\_2Ebool\_2E\_7E}~\\
(ap~(ap~\mname{c\_2Ebit\_2EBIT}~V3m)~V8k))))))~V5r)))))))))))))))\end{array}
\label{hyp:ehigngpfcdfeahbgehchjgdgjgbgpfcdfejedfpfpfafefcffefepfpfegfggg}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfeahbgehchjgdgjgbgpfcdfefeneafefjfpfpfjedfpfpfafefcffefe}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(p~(ap~(c\_2Epatricia\_2EIS\_\_PTREE~\\
A\_27a)~(c\_2Epatricia\_2EEmpty~A\_27a)))\end{array}$$
\end{theorem}
\end{document}
