\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\_2EquantHeuristics\_2EGUESSES\_\_NEG\_\_REWRITE (TMNjN94KkG4LiLSuwxgKhvgSGfak53q64S9)}
\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:dgpfcdfebhfhbgogehiefgfhchjgdhehjgdgdhpfcdfehefffedfdfpfpfgepecfbememepfpfafpejeoeef}
We define $\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL\_\_POINT}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0i \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1P \in {({{2}^{{A\_27b}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2fv \in {A\_27a} .  {({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{V1P}}~{({{{ap}~{V0i}}~{V2fv}})}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebhfhbgogehiefgfhchjgdhehjgdgdhpfcdfehefffedfdfpfpffeifjedfefdfpfpfafpejeoeef}
We define $\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS\_\_POINT}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0i \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1P \in {({{2}^{{A\_27b}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2fv \in {A\_27a} .  {({{{ap}~{V1P}}~{({{{ap}~{V0i}}~{V2fv}})}})}})}})}}}}}$.
\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:dgpfcdfebhfhbgogehiefgfhchjgdhehjgdgdhpfcdfehefffedfdfpfpfgepecfbememepfpfhebeaf}
We define $\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL\_\_GAP}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0i \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1P \in {({{2}^{{A\_27b}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27b}})}}~{({{\uniddcgcg} V2v \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{V1P}}~{V2v}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V3fv \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{V2v}})}}~{({{{ap}~{V0i}}~{V3fv}})}})}})}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebhfhbgogehiefgfhchjgdhehjgdgdhpfcdfehefffedfdfpfpffeifjedfefdfpfpfhebeaf}
We define $\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS\_\_GAP}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0i \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1P \in {({{2}^{{A\_27b}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27b}})}}~{({{\uniddcgcg} V2v \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{V1P}}~{V2v}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V3fv \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{V2v}})}}~{({{{ap}~{V0i}}~{V3fv}})}})}})}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebhfhbgogehiefgfhchjgdhehjgdgdhpfcdfehefffedfdfpfpfgepecfbememe}
We define $\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0i \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1P \in {({{2}^{{A\_27b}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27b}})}}~{({{\uniddcgcg} V2v \in {A\_27b} .  {({{{ap}~{V1P}}~{V2v}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3fv \in {A\_27a} .  {({{{ap}~{V1P}}~{({{{ap}~{V0i}}~{V3fv}})}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfepg}
We define $\mname{c\_2Ecombin\_2Eo}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27c}}})} .  {{\uniddcgcg} V1g \in {({{A\_27c}^{{A\_27a}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{V0f}}~{({{{ap}~{V1g}}~{V2x}})}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebhfhbgogehiefgfhchjgdhehjgdgdhpfcdfehefffedfdfpfpffeifjedfefdf}
We define $\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0i \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1P \in {({{2}^{{A\_27b}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27b}})}}~{({{\uniddcgcg} V2v \in {A\_27b} .  {({{{ap}~{V1P}}~{V2v}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V3fv \in {A\_27a} .  {({{{ap}~{V1P}}~{({{{ap}~{V0i}}~{V3fv}})}})}})}})}})}}}}}$.
\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}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0i \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V1P \in (2^{{A\_27b}}) .  (((p~(ap~(ap~\\
(\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS}~A\_27a~A\_27b)~V0i)~(ap~(\\
ap~(\mname{c\_2Ecombin\_2Eo}~A\_27b~2~2)~\mname{c\_2Ebool\_2E\_7E})~V1P))){\unicdbdjded}(p~(ap~(\\
ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL}~A\_27a~A\_27b)~V0i)~V1P))){\unicdcdcdhd}\\
(((p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL}~A\_27a~A\_27b)~\\
V0i)~(ap~(ap~(\mname{c\_2Ecombin\_2Eo}~A\_27b~2~2)~\mname{c\_2Ebool\_2E\_7E})~V1P))){\unicdbdjded}\\
(p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS}~A\_27a~A\_27b)~\\
V0i)~V1P))){\unicdcdcdhd}(((p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS\_\_GAP}~\\
A\_27a~A\_27b)~V0i)~(ap~(ap~(\mname{c\_2Ecombin\_2Eo}~A\_27b~2~2)~\mname{c\_2Ebool\_2E\_7E})~\\
V1P))){\unicdbdjded}(p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL\_\_GAP}~\\
A\_27a~A\_27b)~V0i)~V1P))){\unicdcdcdhd}(((p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL\_\_GAP}~\\
A\_27a~A\_27b)~V0i)~(ap~(ap~(\mname{c\_2Ecombin\_2Eo}~A\_27b~2~2)~\mname{c\_2Ebool\_2E\_7E})~\\
V1P))){\unicdbdjded}(p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS\_\_GAP}~\\
A\_27a~A\_27b)~V0i)~V1P))){\unicdcdcdhd}(((p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS\_\_POINT}~\\
A\_27a~A\_27b)~V0i)~(ap~(ap~(\mname{c\_2Ecombin\_2Eo}~A\_27b~2~2)~\mname{c\_2Ebool\_2E\_7E})~\\
V1P))){\unicdbdjded}(p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL\_\_POINT}~\\
A\_27a~A\_27b)~V0i)~V1P))){\unicdcdcdhd}((p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL\_\_POINT}~\\
A\_27a~A\_27b)~V0i)~(ap~(ap~(\mname{c\_2Ecombin\_2Eo}~A\_27b~2~2)~\mname{c\_2Ebool\_2E\_7E})~\\
V1P))){\unicdbdjded}(p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS\_\_POINT}~\\
A\_27a~A\_27b)~V0i)~V1P))))))))))\end{array}
\label{hyp:ehigngpfcdfebhfhbgogehiefgfhchjgdhehjgdgdhpfcdfehefffedfdffedfpfpfoefehepfpfeeffbemejeefjf}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfebhfhbgogehiefgfhchjgdhehjgdgdhpfcdfehefffedfdffedfpfpfoefehepfpfcffehfcfjeeffe}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0i \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V1P \in (2^{{A\_27b}}) .  (((p~(ap~(ap~\\
(\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS}~A\_27a~A\_27b)~V0i)~({\uniddcgcg} V2x \in \\
A\_27b .  (ap~\mname{c\_2Ebool\_2E\_7E}~(ap~V1P~V2x))))){\unicdbdjded}(p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL}~\\
A\_27a~A\_27b)~V0i)~({\uniddcgcg} V3x \in A\_27b .  (ap~V1P~V3x))))){\unicdcdcdhd}(((p~(ap~(ap~\\
(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL}~A\_27a~A\_27b)~V0i)~({\uniddcgcg} V4x \in \\
A\_27b .  (ap~\mname{c\_2Ebool\_2E\_7E}~(ap~V1P~V4x))))){\unicdbdjded}(p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS}~\\
A\_27a~A\_27b)~V0i)~({\uniddcgcg} V5x \in A\_27b .  (ap~V1P~V5x))))){\unicdcdcdhd}(((p~(ap~(ap~\\
(\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS\_\_GAP}~A\_27a~A\_27b)~V0i)~\\
({\uniddcgcg} V6x \in A\_27b .  (ap~\mname{c\_2Ebool\_2E\_7E}~(ap~V1P~V6x))))){\unicdbdjded}(p~(ap~(ap~\\
(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL\_\_GAP}~A\_27a~A\_27b)~V0i)~\\
({\uniddcgcg} V7x \in A\_27b .  (ap~V1P~V7x))))){\unicdcdcdhd}(((p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL\_\_GAP}~\\
A\_27a~A\_27b)~V0i)~({\uniddcgcg} V8x \in A\_27b .  (ap~\mname{c\_2Ebool\_2E\_7E}~(ap~V1P~V8x))))){\unicdbdjded}\\
(p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS\_\_GAP}~A\_27a~A\_27b)~\\
V0i)~({\uniddcgcg} V9x \in A\_27b .  (ap~V1P~V9x))))){\unicdcdcdhd}(((p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS\_\_POINT}~\\
A\_27a~A\_27b)~V0i)~({\uniddcgcg} V10x \in A\_27b .  (ap~\mname{c\_2Ebool\_2E\_7E}~(ap~V1P~V10x))))){\unicdbdjded}\\
(p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL\_\_POINT}~A\_27a~\\
A\_27b)~V0i)~({\uniddcgcg} V11x \in A\_27b .  (ap~V1P~V11x))))){\unicdcdcdhd}((p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL\_\_POINT}~\\
A\_27a~A\_27b)~V0i)~({\uniddcgcg} V12x \in A\_27b .  (ap~\mname{c\_2Ebool\_2E\_7E}~(ap~V1P~V12x))))){\unicdbdjded}\\
(p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS\_\_POINT}~A\_27a~\\
A\_27b)~V0i)~({\uniddcgcg} V13x \in A\_27b .  (ap~V1P~V13x))))))))))))\end{array}$$
\end{theorem}
\end{document}
