\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\_2Emergesort\_2Esort2\_\_perm (TMVWZ37gmvbxvNvrR7G5sViXkrxuMUdsAwN)}
\maketitle
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddeepfddeepfddfe}
We define $\mname{c\_2Emin\_2E\_3D\_3D\_3E}$ to be ${{\uniddcgcg} P \in {2} .  {{\uniddcgcg} Q \in {2} .  {{inj\_\_o}~{({{{p}~{P}}{\unicdbdjdcd}{{p}~{Q}}})}}}}$ of type ${\iota}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddee}
We define $\mname{c\_2Emin\_2E\_3D}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} x \in {A} .  {{\uniddcgcg} y \in {A} .  {{inj\_\_o}~{({{x}={y}})}}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfeef}
We define $\mname{c\_2Ebool\_2ET}$ to be ${({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{2}}})}})}}~{({{\uniddcgcg} V0x \in {2} .  {V0x}})}})}}~{({{\uniddcgcg} V1x \in {2} .  {V1x}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdbd}
We define $\mname{c\_2Ebool\_2E\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V0P}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepffddepfcdge}
We define $\mname{c\_2Ebool\_2E\_5C\_2F}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{V2t}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}}~{V2t}})}})}})}})}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfege}
We define $\mname{c\_2Ebool\_2EF}$ to be ${({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V0t \in {2} .  {V0t}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfhdfe}
We define $\mname{c\_2Ebool\_2E\_7E}$ to be ${({{\uniddcgcg} V0t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdgepffdde}
We define $\mname{c\_2Ebool\_2E\_2F\_5C}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}})}}~{V2t}})}})}})}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfedad}
We define $\mname{c\_2Emin\_2E\_40}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} P \in {{2}^{{A}}} .  {{\mathbf{if}}~{({{\unicdcdaddd} x \in {A} .  {{p}~{({{{ap}~{P}}~{x}})}}})}~{\mathbf{then}}~{({{the}~{({{\uniddcgcg} x .  {{{x} \in {A}}{\unicdcdcdhd}{{p}~{({{{ap}~{P}}~{x}})}}}})}})}~{\mathbf{else}}~{({{the}~{({{\uniddcgcg} x .  {{x} \in {A}}})}})}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfedepeoeee}
We define $\mname{c\_2Ebool\_2ECOND}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0t \in {2} .  {({{\uniddcgcg} V1t1 \in {A\_27a} .  {({{\uniddcgcg} V2t2 \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{({{\uniddcgcg} V3x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2ET}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V1t1}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V2t2}})}})}})}})}})}})}})}})}}$.
\end{definition}
Let $ ty\_2Elist\_2Elist : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Elist\_2Elist}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfemgjgdhehpfcdfemgjgdheh}
\end{equation}
Let $ c\_2Elist\_2ENIL : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ENIL~A\_27a \in (ty\_2Elist\_2Elist~\\
A\_27a)\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfeoejeme}
\end{equation}
Let $ 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}
\begin{definition}\label{def:dgpfcdfengfgchhgfgdhpgchehpfcdfedhpgchehcd}
We define $\mname{c\_2Emergesort\_2Esort2}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1x \in {A\_27a} .  {{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{V0R}}~{V1x}})}}~{V2y}})}})}}~{({{{ap}~{({{{ap}~{({{c\_2Elist\_2ECONS}~{A\_27a}})}}~{V1x}})}}~{({{{ap}~{({{{ap}~{({{c\_2Elist\_2ECONS}~{A\_27a}})}}~{V2y}})}}~{({{c\_2Elist\_2ENIL}~{A\_27a}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{c\_2Elist\_2ECONS}~{A\_27a}})}}~{V2y}})}}~{({{{ap}~{({{{ap}~{({{c\_2Elist\_2ECONS}~{A\_27a}})}}~{V1x}})}}~{({{c\_2Elist\_2ENIL}~{A\_27a}})}})}})}})}}}}}$.
\end{definition}
Let $ c\_2Elist\_2EAPPEND : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2EAPPEND~A\_27a \in (((ty\_2Elist\_2Elist~\\
A\_27a)^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfebeafaffeoeee}
\end{equation}
Let $ c\_2Elist\_2EFILTER : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2EFILTER~A\_27a \in (((ty\_2Elist\_2Elist~\\
A\_27a)^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{({{2}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfegejemeeffecf}
\end{equation}
\begin{definition}\label{def:dgpfcdfedhpgchehjgoghgpfcdfeaffecfne}
We define $\mname{c\_2Esorting\_2EPERM}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0L1 \in {({{ty\_2Elist\_2Elist}~{A\_27a}})} .  {{\uniddcgcg} V1L2 \in {({{ty\_2Elist\_2Elist}~{A\_27a}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{c\_2Elist\_2EFILTER}~{A\_27a}})}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2x}})}})}}~{V0L1}})}})}}~{({{{ap}~{({{{ap}~{({{c\_2Elist\_2EFILTER}~{A\_27a}})}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2x}})}})}}~{V1L2}})}})}})}})}}}}$.
\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}
{({{\unicdcdadad} V0t \in {2} .  {({{({{p}~{V0t}})}{\unicdcdcdid}{({\preoh{{({{p}~{V0t}})}}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefeifdemeffeefeeepfpfnejeeeeemefe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdcdcdhd}(p~V0t)){\unicdbdjded}False){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}False){\unicdbdjded}False){\unicdcdcdhd}\\
(((p~V0t){\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ((V0x=V0x){\unicdbdjded}\\
True))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecffegemepfpfdemebeffdffe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in \\
A\_27a .  ((V0x=V1y){\unicdbdjded}(V1y=V0x))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdfjfnepfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjded}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjded}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdbdjded}(p~V0t)){\unicdbdjded}(\preoh (p~V0t))){\unicdcdcdhd}(((p~V0t){\unicdbdjded}False){\unicdbdjded}(\preoh (\\
p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0t1 \in A\_27a .  ({\unicdcdadad} V1t2 \in \\
A\_27a .  (((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\mname{c\_2Ebool\_2ET})~V0t1)~\\
V1t2)=V0t1){\unicdcdcdhd}((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\mname{c\_2Ebool\_2EF})~\\
V0t1)~V1t2)=V1t2))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfedepeoeeepfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(({\unicdcdadad} V0l \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ((ap~(ap~(c\_2Elist\_2EAPPEND~A\_27a)~(c\_2Elist\_2ENIL~A\_27a))~\\
V0l)=V0l)){\unicdcdcdhd}({\unicdcdadad} V1l1 \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V2l2 \in \\
(ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V3h \in A\_27a .  ((ap~(ap~(c\_2Elist\_2EAPPEND~\\
A\_27a)~(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V3h)~V1l1))~V2l2)=(ap~(ap~\\
(c\_2Elist\_2ECONS~A\_27a)~V3h)~(ap~(ap~(c\_2Elist\_2EAPPEND~A\_27a)~\\
V1l1)~V2l2)))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfebeafaffeoeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0a0 \in A\_27a .  ({\unicdcdadad} V1a1 \in \\
(ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V2a0\_27 \in A\_27a .  ({\unicdcdadad} V3a1\_27 \in \\
(ty\_2Elist\_2Elist~A\_27a) .  (((ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V0a0)~\\
V1a1)=(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V2a0\_27)~V3a1\_27)){\unicdbdjded}((V0a0=\\
V2a0\_27){\unicdcdcdhd}(V1a1=V3a1\_27)))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfedepeoedfpfpfbdbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1l2 \in \\
(ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V2l1 \in (ty\_2Elist\_2Elist~A\_27a) .  \\
((p~(ap~(ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V0x)~V2l1))~(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V0x)~V1l2))){\unicdbdjded}\\
(p~(ap~(ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~V2l1)~V1l2))))))\end{array}
\label{hyp:ehigngpfcdfedhpgchehjgoghgpfcdfeaffecfnepfpfdepeoedfpfpfjegege}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0L \in (ty\_2Elist\_2Elist~\\
A\_27a) .  (((p~(ap~(ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~V0L)~(c\_2Elist\_2ENIL~\\
A\_27a))){\unicdbdjded}(V0L=(c\_2Elist\_2ENIL~A\_27a))){\unicdcdcdhd}((p~(ap~(ap~(\mname{c\_2Esorting\_2EPERM}~\\
A\_27a)~(c\_2Elist\_2ENIL~A\_27a))~V0L)){\unicdbdjded}(V0L=(c\_2Elist\_2ENIL~A\_27a)))))\end{array}
\label{hyp:ehigngpfcdfedhpgchehjgoghgpfcdfeaffecfnepfpfoejeme}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0L \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ({\unicdcdadad} V1x \in A\_27a .  (((p~(ap~(ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~\\
V0L)~(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V1x)~(c\_2Elist\_2ENIL~A\_27a)))){\unicdbdjded}\\
(V0L=(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V1x)~(c\_2Elist\_2ENIL~A\_27a)))){\unicdcdcdhd}\\
((p~(ap~(ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V1x)~(c\_2Elist\_2ENIL~A\_27a)))~V0L)){\unicdbdjded}(V0L=(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V1x)~(c\_2Elist\_2ENIL~A\_27a)))))))\end{array}
\label{hyp:ehigngpfcdfedhpgchehjgoghgpfcdfeaffecfnepfpfdfjeoehe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ({\unicdcdadad} V1y \in (ty\_2Elist\_2Elist~A\_27a) .  ((p~(ap~(ap~(\mname{c\_2Esorting\_2EPERM}~\\
A\_27a)~V0x)~V1y)){\unicdbdjded}((ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~V0x)=(ap~(\mname{c\_2Esorting\_2EPERM}~\\
A\_27a)~V1y)))))\end{array}
\label{hyp:ehigngpfcdfedhpgchehjgoghgpfcdfeaffecfnepfpffebfffjegfbemefeoedefepfpfbemeefpfpfeefege}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0y \in A\_27a .  ({\unicdcdadad} V1l1 \in \\
(ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V2x \in A\_27a .  ({\unicdcdadad} V3l2 \in \\
(ty\_2Elist\_2Elist~A\_27a) .  (((ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~V1l1)=\\
(ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~\\
V2x)~V3l2))){\unicdbdjdcd}((ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V0y)~V1l1))=(ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V2x)~(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V0y)~V3l2)))))))))\end{array}
\label{hyp:ehigngpfcdfedhpgchehjgoghgpfcdfeaffecfnepfpfgeffoepfpfdepeoedfpfpfbdbdpfpfdfhfbeafpfpfbeefpfpfgecfpeoeef}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0l \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ({\unicdcdadad} V1l1 \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V2x \in \\
A\_27a .  ({\unicdcdadad} V3l2 \in (ty\_2Elist\_2Elist~A\_27a) .  (((ap~(\mname{c\_2Esorting\_2EPERM}~\\
A\_27a)~V1l1)=(ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V2x)~V3l2))){\unicdbdjdcd}((ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~(ap~(ap~(c\_2Elist\_2EAPPEND~\\
A\_27a)~V0l)~V1l1))=(ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V2x)~(ap~(ap~(c\_2Elist\_2EAPPEND~A\_27a)~V0l)~V3l2)))))))))\end{array}
\label{hyp:ehigngpfcdfedhpgchehjgoghgpfcdfeaffecfnepfpfgeffoepfpfdepeoedfpfpfbeafaffeoeeepfpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1l1 \in \\
(ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V2l2 \in (ty\_2Elist\_2Elist~A\_27a) .  \\
(((ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~V1l1)=(ap~(\mname{c\_2Esorting\_2EPERM}~\\
A\_27a)~V2l2)){\unicdbdjdcd}((ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V0x)~V1l1))=(ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V0x)~V2l2)))))))\end{array}
\label{hyp:ehigngpfcdfedhpgchehjgoghgpfcdfeaffecfnepfpfgeffoepfpfdepeoedfpfpfjegege}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0l \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ({\unicdcdadad} V1l1 \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V2l1\_27 \in \\
(ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V3l2 \in (ty\_2Elist\_2Elist~A\_27a) .  \\
({\unicdcdadad} V4l2\_27 \in (ty\_2Elist\_2Elist~A\_27a) .  ((p~(ap~(ap~(\mname{c\_2Esorting\_2EPERM}~\\
A\_27a)~V1l1)~(ap~(ap~(c\_2Elist\_2EAPPEND~A\_27a)~V0l)~V2l1\_27))){\unicdbdjdcd}\\
((p~(ap~(ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~V3l2)~(ap~(ap~(c\_2Elist\_2EAPPEND~\\
A\_27a)~V0l)~V4l2\_27))){\unicdbdjdcd}((p~(ap~(ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~\\
V1l1)~V3l2)){\unicdbdjded}(p~(ap~(ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~V2l1\_27)~V4l2\_27))))))))))\end{array}
\label{hyp:ehigngpfcdfedhpgchehjgoghgpfcdfeaffecfnepfpfdepeoehepfpfbeafaffeoeeepfpfjegege}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfengfgchhgfgdhpgchehpfcdfedhpgchehcdpfpfahfgchng}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0R \in ((2^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V1x \in A\_27a .  ({\unicdcdadad} V2y \in A\_27a .  (p~(ap~(ap~(\mname{c\_2Esorting\_2EPERM}~\\
A\_27a)~(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V1x)~(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V2y)~(c\_2Elist\_2ENIL~A\_27a))))~(ap~(ap~(ap~(\mname{c\_2Emergesort\_2Esort2}~\\
A\_27a)~V0R)~V1x)~V2y))))))\end{array}$$
\end{theorem}
\end{document}
