\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\_2Esorting\_2EPERM\_\_lifts\_\_monotonicities (TMPk6G3PgrxC3eseoLv4awGwXHnKrMKVXHB)}
\maketitle
\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: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: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: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}
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\_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}
\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:dgpfcdfechfgmgbgehjgpgogpfcdfeehchbgogdhjgehjgghfg}
We define $\mname{c\_2Erelation\_2Etransitive}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3z \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{V0R}}~{V1x}})}}~{V2y}})}})}}~{({{{ap}~{({{{ap}~{V0R}}~{V2y}})}}~{V3z}})}})}})}}~{({{{ap}~{({{{ap}~{V0R}}~{V1x}})}}~{V3z}})}})}})}})}})}})}})}})}}}$.
\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}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\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 .  ({\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} V0l1 \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ({\unicdcdadad} V1l2 \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V2l3 \in \\
(ty\_2Elist\_2Elist~A\_27a) .  ((ap~(ap~(c\_2Elist\_2EAPPEND~A\_27a)~\\
V0l1)~(ap~(ap~(c\_2Elist\_2EAPPEND~A\_27a)~V1l2)~V2l3))=(ap~(ap~(c\_2Elist\_2EAPPEND~\\
A\_27a)~(ap~(ap~(c\_2Elist\_2EAPPEND~A\_27a)~V0l1)~V1l2))~V2l3)))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfebeafaffeoeeepfpfbedfdfpede}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(p~(ap~(\mname{c\_2Erelation\_2Etransitive}~\\
(ty\_2Elist\_2Elist~A\_27a))~(\mname{c\_2Esorting\_2EPERM}~A\_27a)))\end{array}
\label{hyp:ehigngpfcdfedhpgchehjgoghgpfcdfeaffecfnepfpfehchbgogdhjgehjgghfg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0l1 \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ({\unicdcdadad} V1l2 \in (ty\_2Elist\_2Elist~A\_27a) .  (p~(ap~(ap~(\mname{c\_2Esorting\_2EPERM}~\\
A\_27a)~(ap~(ap~(c\_2Elist\_2EAPPEND~A\_27a)~V0l1)~V1l2))~(ap~(ap~(c\_2Elist\_2EAPPEND~\\
A\_27a)~V1l2)~V0l1)))))\end{array}
\label{hyp:ehigngpfcdfedhpgchehjgoghgpfcdfeaffecfnepfpfbeafaffeoeee}
\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} V2l2 \in \\
(ty\_2Elist\_2Elist~A\_27a) .  ((p~(ap~(ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~\\
(ap~(ap~(c\_2Elist\_2EAPPEND~A\_27a)~V0l)~V1l1))~(ap~(ap~(c\_2Elist\_2EAPPEND~\\
A\_27a)~V0l)~V2l2))){\unicdbdjded}(p~(ap~(ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~V1l1)~\\
V2l2)))))){\unicdcdcdhd}({\unicdcdadad} V3l \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V4l1 \in \\
(ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V5l2 \in (ty\_2Elist\_2Elist~A\_27a) .  \\
((p~(ap~(ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~(ap~(ap~(c\_2Elist\_2EAPPEND~\\
A\_27a)~V4l1)~V3l))~(ap~(ap~(c\_2Elist\_2EAPPEND~A\_27a)~V5l2)~V3l))){\unicdbdjded}\\
(p~(ap~(ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~V4l1)~V5l2)))))))\end{array}
\label{hyp:ehigngpfcdfedhpgchehjgoghgpfcdfeaffecfnepfpfbeafaffeoeeepfpfjegege}
\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 (A\_27b^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}}) .  ({\unicdcdadad} V1Q \in (\\
(2^{{A\_27b}})^{{A\_27b}}) .  ((({\unicdcdadad} V2x1 \in (ty\_2Elist\_2Elist~A\_27a) .  \\
({\unicdcdadad} V3x2 \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V4x3 \in (ty\_2Elist\_2Elist~\\
A\_27a) .  (p~(ap~(ap~V1Q~(ap~V0f~(ap~(ap~(c\_2Elist\_2EAPPEND~A\_27a)~(\\
ap~(ap~(c\_2Elist\_2EAPPEND~A\_27a)~V2x1)~V3x2))~V4x3)))~(ap~V0f~(ap~\\
(ap~(c\_2Elist\_2EAPPEND~A\_27a)~(ap~(ap~(c\_2Elist\_2EAPPEND~A\_27a)~\\
V2x1)~V4x3))~V3x2))))))){\unicdcdcdhd}(p~(ap~(\mname{c\_2Erelation\_2Etransitive}~A\_27b)~\\
V1Q))){\unicdbdjdcd}({\unicdcdadad} V5x \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V6y \in \\
(ty\_2Elist\_2Elist~A\_27a) .  ((p~(ap~(ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~\\
V5x)~V6y)){\unicdbdjdcd}(p~(ap~(ap~V1Q~(ap~V0f~V5x))~(ap~V0f~V6y)))))))))\end{array}
\label{hyp:ehigngpfcdfedhpgchehjgoghgpfcdfeaffecfnepfpfmgjgggehdhpfpfehchbgogdhjgehjgghfgpfpfchfgmgbgehjgpgogdh}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfedhpgchehjgoghgpfcdfeaffecfnepfpfmgjgggehdhpfpfngpgogpgehpgogjgdgjgehjgfgdh}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0f \in ((ty\_2Elist\_2Elist~A\_27b)^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}}) .  \\
(({\unicdcdadad} V1x1 \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V2x2 \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ({\unicdcdadad} V3x3 \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdaddd} V4x1\_27 \in \\
(ty\_2Elist\_2Elist~A\_27b) .  ({\unicdcdaddd} V5x2\_27 \in (ty\_2Elist\_2Elist~\\
A\_27b) .  ({\unicdcdaddd} V6x3\_27 \in (ty\_2Elist\_2Elist~A\_27b) .  (((ap~V0f~\\
(ap~(ap~(c\_2Elist\_2EAPPEND~A\_27a)~(ap~(ap~(c\_2Elist\_2EAPPEND~A\_27a)~\\
V1x1)~V2x2))~V3x3))=(ap~(ap~(c\_2Elist\_2EAPPEND~A\_27b)~(ap~(ap~(\\
c\_2Elist\_2EAPPEND~A\_27b)~V4x1\_27)~V5x2\_27))~V6x3\_27)){\unicdcdcdhd}((ap~V0f~\\
(ap~(ap~(c\_2Elist\_2EAPPEND~A\_27a)~(ap~(ap~(c\_2Elist\_2EAPPEND~A\_27a)~\\
V1x1)~V3x3))~V2x2))=(ap~(ap~(c\_2Elist\_2EAPPEND~A\_27b)~(ap~(ap~(\\
c\_2Elist\_2EAPPEND~A\_27b)~V4x1\_27)~V6x3\_27))~V5x2\_27))))))))){\unicdbdjdcd}\\
({\unicdcdadad} V7x \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V8y \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ((p~(ap~(ap~(\mname{c\_2Esorting\_2EPERM}~A\_27a)~V7x)~V8y)){\unicdbdjdcd}(p~(ap~\\
(ap~(\mname{c\_2Esorting\_2EPERM}~A\_27b)~(ap~V0f~V7x))~(ap~V0f~V8y))))))))\end{array}$$
\end{theorem}
\end{document}
