\documentclass{article}
\newtheorem{axiom}{Axiom}
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\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\_2Ebinary\_\_ieee\_2Eulp (TMLNeMHfuiQ1Lg8A9kYArfJcU9FxvvsDMGs)}
\maketitle
Let $ ty\_2Ehreal\_2Ehreal : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Ehreal\_2Ehreal}}\label{hyp:ogfgpfehjhpfcdfeigchfgbgmgpfcdfeigchfgbgmg}
\end{equation}
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 $ ty\_2Erealax\_2Ereal : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Erealax\_2Ereal}}\label{hyp:ogfgpfehjhpfcdfechfgbgmgbgihpfcdfechfgbgmg}
\end{equation}
Let $ c\_2Erealax\_2Ereal\_\_REP\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Ereal\_\_REP\_\_CLASS \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})^{{ty\_2Erealax\_2Ereal}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfcffeafpfpfdemebedfdf}
\end{equation}
\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: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:dgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfcffeaf}
We define $\mname{c\_2Erealax\_2Ereal\_\_REP}$ to be ${{\uniddcgcg} V0a \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})}}~{({{{ap}~{c\_2Erealax\_2Ereal\_\_REP\_\_CLASS}}~{V0a}})}})}}$.
\end{definition}
Let $ c\_2Erealax\_2Etreal\_\_add : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Etreal\_\_add \in (((ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~\\
ty\_2Ehreal\_2Ehreal)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfeehchfgbgmgpfpfbgegeg}
\end{equation}
Let $ c\_2Erealax\_2Etreal\_\_eq : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Etreal\_\_eq \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfeehchfgbgmgpfpffgbh}
\end{equation}
Let $ c\_2Erealax\_2Ereal\_\_ABS\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Ereal\_\_ABS\_\_CLASS \in (ty\_2Erealax\_2Ereal^{{({{2}^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfbecedfpfpfdemebedfdf}
\end{equation}
\begin{definition}\label{def:dgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfbecedf}
We define $\mname{c\_2Erealax\_2Ereal\_\_ABS}$ to be ${{\uniddcgcg} V0r \in {({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})} .  {({{{ap}~{c\_2Erealax\_2Ereal\_\_ABS\_\_CLASS}}~{({{{ap}~{c\_2Erealax\_2Etreal\_\_eq}}~{V0r}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfbgegeg}
We define $\mname{c\_2Erealax\_2Ereal\_\_add}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V1T2 \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_ABS}}}~{({{{ap}~{({{{ap}~{c\_2Erealax\_2Etreal\_\_add}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V1T2}})}})}})}}}$.
\end{definition}
Let $ ty\_2Efcp\_2Ecart : {{\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\_2Efcp\_2Ecart~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfeggdgahpfcdfedgbgcheh}
\end{equation}
Let $ ty\_2Ebinary\_\_ieee\_2Efloat : {{\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\_2Ebinary\_\_ieee\_2Efloat~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgeh}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand~\\
A\_27t~A\_27w \in ((ty\_2Efcp\_2Ecart~2~A\_27t)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfdfjghgogjgggjgdgbgogeg}
\end{equation}
Let $ ty\_2Eone\_2Eone : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Eone\_2Eone}}\label{hyp:ogfgpfehjhpfcdfepgogfgpfcdfepgogfg}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign~\\
A\_27t~A\_27w \in ((ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfdfjghgog}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent~\\
A\_27t~A\_27w \in ((ty\_2Efcp\_2Ecart~2~A\_27w)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpffeihahpgogfgogeh}
\end{equation}
Let $ ty\_2Ebool\_2Eitself : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Ebool\_2Eitself}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfecgpgpgmgpfcdfejgehdhfgmggg}
\end{equation}
Let $ c\_2Ebool\_2Ethe\_\_value : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ebool\_2Ethe\_\_value~A\_27a \in (\\
ty\_2Ebool\_2Eitself~A\_27a)\end{array}
\label{hyp:ngfgngpfdgpfcdfecgpgpgmgpfcdfeehigfgpfpfghbgmgfhfg}
\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\_2Efcp\_2Edimindex : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Efcp\_2Edimindex~A\_27a \in (ty\_2Enum\_2Enum^{{({{ty\_2Ebool\_2Eitself}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggdgahpfcdfeegjgngjgogegfgih}
\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: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 $ c\_2Ereal\_2Ereal\_\_of\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Ereal\_2Ereal\_\_of\_\_num} \in {({{ty\_2Erealax\_2Ereal}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfechfgbgmgpfcdfechfgbgmgpfpfpgggpfpfogfhng}
\end{equation}
Let $ c\_2Ereal\_2Epow : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Ereal\_2Epow \in ((ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}})^{{ty\_2Erealax\_2Ereal}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgpfcdfeahpghh}
\end{equation}
Let $ ty\_2Efcp\_2Efinite\_\_image : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Efcp\_2Efinite\_\_image}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfeggdgahpfcdfeggjgogjgehfgpfpfjgngbghgfg}
\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: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}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfddgepfcdbd}
We define $\mname{c\_2Ebool\_2E\_3F\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{V0P}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V0P}}~{V1x}})}})}}~{({{{ap}~{V0P}}~{V2y}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V1x}})}}~{V2y}})}})}})}})}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggdgahpfcdfeggjgogjgehfgpfpfjgogegfgih}
We define $\mname{c\_2Efcp\_2Efinite\_\_index}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{A\_27a}^{{ty\_2Enum\_2Enum}}})}})}}~{({{\uniddcgcg} V0f \in {({{A\_27a}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V2n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V2n}})}}~{({{{ap}~{({{c\_2Efcp\_2Edimindex}~{A\_27a}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27a}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{({{{ap}~{V0f}}~{V2n}})}})}}~{V1x}})}})}})}})}})}})}})}})}}$.
\end{definition}
Let $ c\_2Efcp\_2Edest\_\_cart : {{\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\_2Efcp\_2Edest\_\_cart~\\
A\_27a~A\_27b \in ((A\_27a^{{({{ty\_2Efcp\_2Efinite\_\_image}~{A\_27b}})}})^{{({{{ty\_2Efcp\_2Ecart}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggdgahpfcdfeegfgdhehpfpfdgbgcheh}
\end{equation}
\begin{definition}\label{def:dgpfcdfeggdgahpfcdfeggdgahpfpfjgogegfgih}
We define $\mname{c\_2Efcp\_2Efcp\_\_index}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0x \in {({{{ty\_2Efcp\_2Ecart}~{A\_27a}}~{A\_27b}})} .  {{\uniddcgcg} V1i \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{c\_2Efcp\_2Edest\_\_cart}~{A\_27a}}~{A\_27b}})}}~{V0x}})}}~{({{{ap}~{({{\mname{c\_2Efcp\_2Efinite\_\_index}}~{({{ty\_2Efcp\_2Efinite\_\_image}~{A\_27b}})}})}}~{V1i}})}})}}}}}$.
\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}
\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}
\begin{definition}\label{def:dgpfcdfecgjgehpfcdfedfcejeef}
We define $\mname{c\_2Ebit\_2ESBIT}$ to be ${{\uniddcgcg} V0b \in {2} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Enum\_2Enum}})}}~{V0b}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2EEXP}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}}~{V1n}})}})}}~{\mname{c\_2Enum\_2E0}}})}}}$.
\end{definition}
Let $ c\_2Esum\_\_num\_2ESUM : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Esum\_\_num\_2ESUM \in ((ty\_2Enum\_2Enum^{{({{ty\_2Enum\_2Enum}^{{ty\_2Enum\_2Enum}}})}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhfhngpfpfogfhngpfcdfedfffne}
\end{equation}
\begin{definition}\label{def:dgpfcdfehhpgchegdhpfcdfehhcdog}
We define $\mname{c\_2Ewords\_2Ew2n}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0w \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{c\_2Esum\_\_num\_2ESUM}}~{({{{ap}~{({{c\_2Efcp\_2Edimindex}~{A\_27a}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27a}})}})}})}}~{({{\uniddcgcg} V1i \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebit\_2ESBIT}}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efcp\_2Efcp\_\_index}}~{2}}~{A\_27a}})}}~{V0w}})}}~{V1i}})}})}}~{V1i}})}})}})}}}$.
\end{definition}
Let $ c\_2Erealax\_2Etreal\_\_inv : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Etreal\_\_inv \in ((ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~\\
ty\_2Ehreal\_2Ehreal)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfeehchfgbgmgpfpfjgoggh}
\end{equation}
\begin{definition}\label{def:dgpfcdfechfgbgmgbgihpfcdfejgoggh}
We define $\mname{c\_2Erealax\_2Einv}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_ABS}}}~{({{{ap}~{c\_2Erealax\_2Etreal\_\_inv}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V0T1}})}})}})}}$.
\end{definition}
Let $ c\_2Erealax\_2Etreal\_\_mul : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Etreal\_\_mul \in (((ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~\\
ty\_2Ehreal\_2Ehreal)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfeehchfgbgmgpfpfngfhmg}
\end{equation}
\begin{definition}\label{def:dgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfngfhmg}
We define $\mname{c\_2Erealax\_2Ereal\_\_mul}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V1T2 \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_ABS}}}~{({{{ap}~{({{{ap}~{c\_2Erealax\_2Etreal\_\_mul}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V1T2}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgbgmgpfcdfepfcdge}
We define $\mname{c\_2Ereal\_2E\_2F}$ to be ${{\uniddcgcg} V0x \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V1y \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_mul}}}~{V0x}})}}~{({{{ap}~{\mname{c\_2Erealax\_2Einv}}}~{V1y}})}})}}}$.
\end{definition}
\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}
Let $ c\_2Ewords\_2EINT\_\_MAX : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ewords\_2EINT\_\_MAX~A\_27a \in (ty\_2Enum\_2Enum^{{({{ty\_2Ebool\_2Eitself}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfehhpgchegdhpfcdfejeoeefpfpfnebeif}
\end{equation}
Let $ c\_2Erealax\_2Etreal\_\_neg : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Etreal\_\_neg \in ((ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~\\
ty\_2Ehreal\_2Ehreal)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfeehchfgbgmgpfpfogfghg}
\end{equation}
\begin{definition}\label{def:dgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfogfghg}
We define $\mname{c\_2Erealax\_2Ereal\_\_neg}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_ABS}}}~{({{{ap}~{c\_2Erealax\_2Etreal\_\_neg}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V0T1}})}})}})}}$.
\end{definition}
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:dgpfcdfeggdgahpfcdfegedeaf}
We define $\mname{c\_2Efcp\_2EFCP}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{\uniddcgcg} V0g \in {({{A\_27a}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{{ty\_2Efcp\_2Ecart}~{A\_27a}}~{A\_27b}})}})}}~{({{\uniddcgcg} V1f \in {({{{ty\_2Efcp\_2Ecart}~{A\_27a}}~{A\_27b}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V2i \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V2i}})}}~{({{{ap}~{({{c\_2Efcp\_2Edimindex}~{A\_27b}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27b}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efcp\_2Efcp\_\_index}}~{A\_27a}}~{A\_27b}})}}~{V1f}})}}~{V2i}})}})}}~{({{{ap}~{V0g}}~{V2i}})}})}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfehhpgchegdhpfcdfeogcdhh}
We define $\mname{c\_2Ewords\_2En2w}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{\mname{c\_2Efcp\_2EFCP}}~{2}}~{A\_27a}})}}~{({{\uniddcgcg} V1i \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebit\_2EBIT}}}~{V1i}})}}~{V0n}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfehpgpfpfchfgbgmg}
We define $\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_to\_\_real}$ to be ${{\uniddcgcg} A\_27t : {\iota} .  {{\uniddcgcg} A\_27w : {\iota} .  {{\uniddcgcg} V0x \in {({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Erealax\_2Ereal}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}}~{({{{ap}~{({{\mname{c\_2Ewords\_2En2w}}~{A\_27w}})}}~{\mname{c\_2Enum\_2E0}}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_mul}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_mul}}}~{({{{ap}~{({{{ap}~{c\_2Ereal\_2Epow}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_neg}}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ewords\_2Ew2n}}~{ty\_2Eone\_2Eone}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ereal\_2E\_2F}}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{({{{ap}~{({{{ap}~{c\_2Ereal\_2Epow}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{({{{ap}~{({{c\_2Ewords\_2EINT\_\_MAX}~{A\_27w}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27w}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ereal\_2E\_2F}}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{({{\mname{c\_2Ewords\_2Ew2n}}~{A\_27t}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}})}})}}~{({{{ap}~{({{{ap}~{c\_2Ereal\_2Epow}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{({{{ap}~{({{c\_2Efcp\_2Edimindex}~{A\_27t}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27t}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_mul}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_mul}}}~{({{{ap}~{({{{ap}~{c\_2Ereal\_2Epow}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_neg}}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ewords\_2Ew2n}}~{ty\_2Eone\_2Eone}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ereal\_2E\_2F}}}~{({{{ap}~{({{{ap}~{c\_2Ereal\_2Epow}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ewords\_2Ew2n}}~{A\_27w}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}})}})}}~{({{{ap}~{({{{ap}~{c\_2Ereal\_2Epow}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{({{{ap}~{({{c\_2Ewords\_2EINT\_\_MAX}~{A\_27w}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27w}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_add}}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ereal\_2E\_2F}}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{({{\mname{c\_2Ewords\_2Ew2n}}~{A\_27t}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}})}})}}~{({{{ap}~{({{{ap}~{c\_2Ereal\_2Epow}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{({{{ap}~{({{c\_2Efcp\_2Edimindex}~{A\_27t}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27t}})}})}})}})}})}})}})}}}}$.
\end{definition}
Let $ c\_2Ebool\_2EARB : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A\_27a .  {{{nonempty}~{A\_27a}}{\unicdbdjdcd}{{{c\_2Ebool\_2EARB}~{A\_27a}} \in {A\_27a}}}}\label{hyp:ngfgngpfdgpfcdfecgpgpgmgpfcdfebecfce}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27u .  nonempty~A\_27u{\unicdbdjdcd}{\unicdcdadad} A\_27w .  \\
nonempty~A\_27w{\unicdbdjdcd}c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd~\\
A\_27t~A\_27u~A\_27w \in (((ty\_2Ebinary\_\_ieee\_2Efloat~A\_27u~A\_27w)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{({{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27u}})}^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27t}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfdfjghgogjgggjgdgbgogegpfpfggfhaheg}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}{\unicdcdadad} A\_27x .  \\
nonempty~A\_27x{\unicdbdjdcd}c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd~A\_27t~\\
A\_27w~A\_27x \in (((ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27x)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{({{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27x}})}^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpffeihahpgogfgogehpfpfggfhaheg}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd~\\
A\_27t~A\_27w \in (((ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{({{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfdfjghgogpfpfggfhaheg}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_plus\_\_min : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}c\_2Ebinary\_\_ieee\_2Efloat\_\_plus\_\_min~\\
A\_27t~A\_27w \in ((ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w)^{{({{ty\_2Ebool\_2Eitself}~{({{{ty\_2Epair\_2Eprod}~{A\_27t}}~{A\_27w}})}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfahmgfhdhpfpfngjgog}
\end{equation}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfele}
We define $\mname{c\_2Ecombin\_2EK}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {({{\uniddcgcg} V1y \in {A\_27b} .  {V0x}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfedf}
We define $\mname{c\_2Ecombin\_2ES}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {({{\uniddcgcg} V0f \in {({{({{A\_27c}^{{A\_27b}}})}^{{A\_27a}}})} .  {({{\uniddcgcg} V1g \in {({{A\_27b}^{{A\_27a}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{V0f}}~{V2x}})}}~{({{{ap}~{V1g}}~{V2x}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfeje}
We define $\mname{c\_2Ecombin\_2EI}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{{ap}~{({{{{\mname{c\_2Ecombin\_2ES}}~{A\_27a}}~{({{A\_27a}^{{A\_27a}}})}}~{A\_27a}})}}~{({{{\mname{c\_2Ecombin\_2EK}}~{A\_27a}}~{({{A\_27a}^{{A\_27a}}})}})}})}}~{({{{\mname{c\_2Ecombin\_2EK}}~{A\_27a}}~{A\_27a}})}})}}$.
\end{definition}
Let $ c\_2Epair\_2ESND : {{\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\_2ESND~\\
A\_27a~A\_27b \in (A\_27b^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfedfoeee}
\end{equation}
Let $ c\_2Epair\_2EFST : {{\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\_2EFST~\\
A\_27a~A\_27b \in (A\_27a^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfegedfef}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgjgchpfcdfeahbgjgchpfpfdebedffe}
We define $\mname{c\_2Epair\_2Epair\_\_CASE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0p \in {({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27c}})} .  {{\uniddcgcg} V1f \in {({{({{A\_27a}^{{A\_27c}}})}^{{A\_27b}}})} .  {({{{ap}~{({{{ap}~{V1f}}~{({{{ap}~{({{{c\_2Epair\_2EFST}~{A\_27b}}~{A\_27c}})}}~{V0p}})}})}}~{({{{ap}~{({{{c\_2Epair\_2ESND}~{A\_27b}}~{A\_27c}})}}~{V0p}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfehfge}
We define $\mname{c\_2Erelation\_2EWF}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V1B \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V2w \in {A\_27a} .  {({{{ap}~{V1B}}~{V2w}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V3min \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V1B}}~{V3min}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4b \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{V0R}}~{V4b}})}}~{V3min}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{V1B}}~{V4b}})}})}})}})}})}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfecffedfefcfjedeef}
We define $\mname{c\_2Erelation\_2ERESTRICT}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V2x \in {A\_27a} .  {({{\uniddcgcg} V3y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{A\_27b}})}}~{({{{ap}~{({{{ap}~{V1R}}~{V3y}})}}~{V2x}})}})}}~{({{{ap}~{V0f}}~{V3y}})}})}}~{({{c\_2Ebool\_2EARB}~{A\_27b}})}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfeefde}
We define $\mname{c\_2Erelation\_2ETC}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1a \in {A\_27a} .  {{\uniddcgcg} V2b \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{({{2}^{{A\_27a}}})}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V3P \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V5y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{V0R}}~{V4x}})}}~{V5y}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V4x}})}}~{V5y}})}})}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V6x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V7y \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V8z \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{V3P}}~{V6x}})}}~{V7y}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V7y}})}}~{V8z}})}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V6x}})}}~{V8z}})}})}})}})}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V1a}})}}~{V2b}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfebgahahchpgih}
We define $\mname{c\_2Erelation\_2Eapprox}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1M \in {({{({{A\_27b}^{{A\_27a}}})}^{{({{A\_27b}^{{A\_27a}}})}}})} .  {{\uniddcgcg} V2x \in {A\_27a} .  {{\uniddcgcg} V3f \in {({{A\_27b}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{A\_27b}^{{A\_27a}}})}})}}~{V3f}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V4y \in {A\_27a} .  {({{{ap}~{({{{ap}~{V1M}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{V3f}})}}~{V0R}})}}~{V4y}})}})}}~{V4y}})}})}})}}~{V0R}})}}~{V2x}})}})}}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfeehigfgpfpfggfhog}
We define $\mname{c\_2Erelation\_2Ethe\_\_fun}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1M \in {({{({{A\_27b}^{{A\_27a}}})}^{{({{A\_27b}^{{A\_27a}}})}}})} .  {{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{A\_27b}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V3f \in {({{A\_27b}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2Eapprox}}~{A\_27a}}~{A\_27b}})}}~{V0R}})}}~{V1M}})}}~{V2x}})}}~{V3f}})}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfehfgecffede}
We define $\mname{c\_2Erelation\_2EWFREC}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1M \in {({{({{A\_27b}^{{A\_27a}}})}^{{({{A\_27b}^{{A\_27a}}})}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{V1M}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2Ethe\_\_fun}}~{A\_27a}}~{A\_27b}})}}~{({{{ap}~{({{\mname{c\_2Erelation\_2ETC}}~{A\_27a}})}}~{V0R}})}})}}~{({{\uniddcgcg} V3f \in {({{A\_27b}^{{A\_27a}}})} .  {({{\uniddcgcg} V4v \in {A\_27a} .  {({{{ap}~{({{{ap}~{V1M}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{V3f}})}}~{V0R}})}}~{V4v}})}})}}~{V4v}})}})}})}})}}~{V2x}})}})}}~{V0R}})}}~{V2x}})}})}}~{V2x}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeffmeaf}
We define $\mname{c\_2Ebinary\_\_ieee\_2EULP}$ to be ${{\uniddcgcg} A\_27t : {\iota} .  {{\uniddcgcg} A\_27w : {\iota} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2EWFREC}}~{({{{ty\_2Epair\_2Eprod}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}~{({{ty\_2Ebool\_2Eitself}~{A\_27t}})}})}}~{ty\_2Erealax\_2Ereal}})}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{({{2}^{{({{{ty\_2Epair\_2Eprod}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}~{({{ty\_2Ebool\_2Eitself}~{A\_27t}})}})}}})}^{{({{{ty\_2Epair\_2Eprod}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}~{({{ty\_2Ebool\_2Eitself}~{A\_27t}})}})}}})}})}}~{({{\uniddcgcg} V0R \in {({{({{2}^{{({{{ty\_2Epair\_2Eprod}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}~{({{ty\_2Ebool\_2Eitself}~{A\_27t}})}})}}})}^{{({{{ty\_2Epair\_2Eprod}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}~{({{ty\_2Ebool\_2Eitself}~{A\_27t}})}})}}})} .  {({{{ap}~{({{\mname{c\_2Erelation\_2EWF}}~{({{{ty\_2Epair\_2Eprod}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}~{({{ty\_2Ebool\_2Eitself}~{A\_27t}})}})}})}}~{V0R}})}})}})}})}}~{({{\uniddcgcg} V1ULP \in {({{ty\_2Erealax\_2Ereal}^{{({{{ty\_2Epair\_2Eprod}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}~{({{ty\_2Ebool\_2Eitself}~{A\_27t}})}})}}})} .  {({{\uniddcgcg} V2a \in {({{{ty\_2Epair\_2Eprod}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}~{({{ty\_2Ebool\_2Eitself}~{A\_27t}})}})} .  {({{{ap}~{({{{ap}~{({{{{\mname{c\_2Epair\_2Epair\_\_CASE}}~{ty\_2Erealax\_2Ereal}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}~{({{ty\_2Ebool\_2Eitself}~{A\_27t}})}})}}~{V2a}})}}~{({{\uniddcgcg} V3v \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})} .  {({{\uniddcgcg} V4v1 \in {({{ty\_2Ebool\_2Eitself}~{A\_27t}})} .  {({{{ap}~{({{\mname{c\_2Ecombin\_2EI}}~{ty\_2Erealax\_2Ereal}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ereal\_2E\_2F}}}~{({{{ap}~{({{{ap}~{c\_2Ereal\_2Epow}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Enum\_2Enum}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}})}}~{V3v}})}}~{({{{ap}~{({{\mname{c\_2Ewords\_2En2w}}~{A\_27w}})}}~{\mname{c\_2Enum\_2E0}}})}})}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ewords\_2Ew2n}}~{A\_27w}})}}~{V3v}})}})}})}})}}~{({{{ap}~{({{{ap}~{c\_2Ereal\_2Epow}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{({{{ap}~{({{c\_2Ewords\_2EINT\_\_MAX}~{A\_27w}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27w}})}})}})}}~{({{{ap}~{({{c\_2Efcp\_2Edimindex}~{A\_27t}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27t}})}})}})}})}})}})}})}})}})}})}})}})}}}$.
\end{definition}
Let $ c\_2Ebinary\_\_ieee\_2Eulp : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}c\_2Ebinary\_\_ieee\_2Eulp~\\
A\_27t~A\_27w \in (ty\_2Erealax\_2Ereal^{{({{ty\_2Ebool\_2Eitself}~{({{{ty\_2Epair\_2Eprod}~{A\_27t}}~{A\_27w}})}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfefhmgah}
\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}
\begin{definition}\label{def:dgpfcdfengbgchlgfgchpfcdfefhogjgogeh}
We define $\mname{c\_2Emarker\_2Eunint}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {V0x}}}$.
\end{definition}
Let $ c\_2Earithmetic\_2E\_2A : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2E\_2A \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepfcdbe}
\end{equation}
Let $ c\_2Earithmetic\_2EODD : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Earithmetic\_2EODD} \in {({{2}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepeeeee}
\end{equation}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfefeneafefjfpfpfcffeme}
We define $\mname{c\_2Erelation\_2EEMPTY\_\_REL}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {{\uniddcgcg} V1y \in {A\_27a} .  {\mname{c\_2Ebool\_2EF}}}}}$.
\end{definition}
Let $ c\_2Ewords\_2Edimword : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ewords\_2Edimword~A\_27a \in (ty\_2Enum\_2Enum^{{({{ty\_2Ebool\_2Eitself}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfehhpgchegdhpfcdfeegjgnghhpgcheg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2A~V0m)~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))=\\
V0m))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeneffmeefpfpfcfjeheieefpfpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (((ap~(ap~c\_2Earithmetic\_2EEXP~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))~\\
V0n)=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO}))){\unicdcdcdhd}((ap~(ap~c\_2Earithmetic\_2EEXP~V0n)~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))=\\
V0n)))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfefeifafpfpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27u .  nonempty~A\_27u{\unicdbdjdcd}{\unicdcdadad} A\_27w .  \\
nonempty~A\_27w{\unicdbdjdcd}{\unicdcdadad} A\_27x .  nonempty~A\_27x{\unicdbdjdcd}(({\unicdcdadad} V0f0 \in \\
((ty\_2Efcp\_2Ecart~2~A\_27x)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}) .  ({\unicdcdadad} V1f \in \\
(ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign~\\
A\_27t~A\_27x)~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd~\\
A\_27t~A\_27w~A\_27x)~V0f0)~V1f))=(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign~\\
A\_27t~A\_27w)~V1f)))){\unicdcdcdhd}(({\unicdcdadad} V2f0 \in ((ty\_2Efcp\_2Ecart~2~A\_27u)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27t}})}}) .  \\
({\unicdcdadad} V3f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~(\\
c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign~A\_27u~A\_27w)~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd~\\
A\_27t~A\_27u~A\_27w)~V2f0)~V3f))=(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign~\\
A\_27t~A\_27w)~V3f)))){\unicdcdcdhd}(({\unicdcdadad} V4f0 \in ((ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}}) .  \\
({\unicdcdadad} V5f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~(\\
c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent~A\_27t~A\_27w)~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd~\\
A\_27t~A\_27w)~V4f0)~V5f))=(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent~\\
A\_27t~A\_27w)~V5f)))){\unicdcdcdhd}(({\unicdcdadad} V6f0 \in ((ty\_2Efcp\_2Ecart~2~A\_27u)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27t}})}}) .  \\
({\unicdcdadad} V7f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~(\\
c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent~A\_27u~A\_27w)~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd~\\
A\_27t~A\_27u~A\_27w)~V6f0)~V7f))=(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent~\\
A\_27t~A\_27w)~V7f)))){\unicdcdcdhd}(({\unicdcdadad} V8f0 \in ((ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}}) .  \\
({\unicdcdadad} V9f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~(\\
c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand~A\_27t~A\_27w)~(ap~(ap~(\\
c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd~A\_27t~A\_27w)~V8f0)~V9f))=\\
(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand~A\_27t~A\_27w)~V9f)))){\unicdcdcdhd}\\
(({\unicdcdadad} V10f0 \in ((ty\_2Efcp\_2Ecart~2~A\_27x)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}) .  \\
({\unicdcdadad} V11f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~\\
(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand~A\_27t~A\_27x)~(ap~(ap~\\
(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd~A\_27t~A\_27w~A\_27x)~\\
V10f0)~V11f))=(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand~A\_27t~\\
A\_27w)~V11f)))){\unicdcdcdhd}(({\unicdcdadad} V12f0 \in ((ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}}) .  \\
({\unicdcdadad} V13f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~\\
(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign~A\_27t~A\_27w)~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd~\\
A\_27t~A\_27w)~V12f0)~V13f))=(ap~V12f0~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign~\\
A\_27t~A\_27w)~V13f))))){\unicdcdcdhd}(({\unicdcdadad} V14f0 \in ((ty\_2Efcp\_2Ecart~2~\\
A\_27x)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}) .  ({\unicdcdadad} V15f \in (ty\_2Ebinary\_\_ieee\_2Efloat~\\
A\_27t~A\_27w) .  ((ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent~A\_27t~\\
A\_27x)~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd~A\_27t~\\
A\_27w~A\_27x)~V14f0)~V15f))=(ap~V14f0~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent~\\
A\_27t~A\_27w)~V15f))))){\unicdcdcdhd}({\unicdcdadad} V16f0 \in ((ty\_2Efcp\_2Ecart~2~A\_27u)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27t}})}}) .  \\
({\unicdcdadad} V17f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~\\
(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand~A\_27u~A\_27w)~(ap~(ap~\\
(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd~A\_27t~A\_27u~A\_27w)~\\
V16f0)~V17f))=(ap~V16f0~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand~\\
A\_27t~A\_27w)~V17f)))))))))))))\end{array}
\label{hyp:ehigngpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfbgdgdgggfhahegdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}(\\
(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_plus\_\_min~A\_27t~A\_27w)~(c\_2Ebool\_2Ethe\_\_value~\\
(ty\_2Epair\_2Eprod~A\_27t~A\_27w)))=(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd~\\
A\_27t~A\_27w)~(ap~(\mname{c\_2Ecombin\_2EK}~(ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone)~\\
(ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone))~(ap~(\mname{c\_2Ewords\_2En2w}~ty\_2Eone\_2Eone)~\\
\mname{c\_2Enum\_2E0})))~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd~\\
A\_27t~A\_27w~A\_27w)~(ap~(\mname{c\_2Ecombin\_2EK}~(ty\_2Efcp\_2Ecart~2~A\_27w)~\\
(ty\_2Efcp\_2Ecart~2~A\_27w))~(ap~(\mname{c\_2Ewords\_2En2w}~A\_27w)~\mname{c\_2Enum\_2E0})))~\\
(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd~A\_27t~\\
A\_27t~A\_27w)~(ap~(\mname{c\_2Ecombin\_2EK}~(ty\_2Efcp\_2Ecart~2~A\_27t)~(ty\_2Efcp\_2Ecart~\\
2~A\_27t))~(ap~(\mname{c\_2Ewords\_2En2w}~A\_27t)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))))~(c\_2Ebool\_2EARB~\\
(ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w))))))\end{array}
\label{hyp:ehigngpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfahmgfhdhpfpfngjgogpfpfegfggg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}(\\
(ap~(c\_2Ebinary\_\_ieee\_2Eulp~A\_27t~A\_27w)~(c\_2Ebool\_2Ethe\_\_value~\\
(ty\_2Epair\_2Eprod~A\_27t~A\_27w)))=(ap~(\mname{c\_2Ebinary\_\_ieee\_2EULP}~\\
A\_27t~A\_27w)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Efcp\_2Ecart~2~A\_27w)~\\
(ty\_2Ebool\_2Eitself~A\_27t))~(ap~(\mname{c\_2Ewords\_2En2w}~A\_27w)~\mname{c\_2Enum\_2E0}))~\\
(c\_2Ebool\_2Ethe\_\_value~A\_27t))))\end{array}
\label{hyp:ehigngpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfefhmgahpfpfegfggg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (\preoh ((ap~(ap~c\_2Ereal\_2Epow~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\mname{c\_2Earithmetic\_2EZERO}))))~\\
V0n)=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))))\end{array}
\label{hyp:ehigngpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfekhfgchpgpfpfogfgbhpfpfehhhpgahpghh}
\end{equation}
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}
\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} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  ({\unicdcdadad} V2t3 \in 2 .  (((p~V0t1){\unicdbdjdcd}\\
((p~V1t2){\unicdbdjdcd}(p~V2t3))){\unicdbdjded}(((p~V0t1){\unicdcdcdhd}(p~V1t2)){\unicdbdjdcd}(p~V2t3))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfjeneafpfpfjeoeefcfpe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in 2 .  ({\unicdcdadad} V1Q \in 2 .  \\
({\unicdcdadad} V2x \in A\_27a .  ({\unicdcdadad} V3x\_27 \in A\_27a .  ({\unicdcdadad} V4y \in A\_27a .  \\
({\unicdcdadad} V5y\_27 \in A\_27a .  ((((p~V0P){\unicdbdjded}(p~V1Q)){\unicdcdcdhd}(((p~V1Q){\unicdbdjdcd}(V2x=V3x\_27)){\unicdcdcdhd}\\
((\preoh (p~V1Q)){\unicdbdjdcd}(V4y=V5y\_27)))){\unicdbdjdcd}((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\\
V0P)~V2x)~V4y)=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~V1Q)~V3x\_27)~\\
V5y\_27)))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfedepeoeeepfpfdepeoehe}
\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}({\unicdcdadad} V2t1 \in A\_27a .  ({\unicdcdadad} V3t2 \in A\_27a .  ((ap~\\
(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\mname{c\_2Ebool\_2EF})~V2t1)~V3t2)=V3t2))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecgpgpgmgpfpfdgbgdhfgpfpfehigng}
\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} V1y \in A\_27b .  ((ap~(ap~(\mname{c\_2Ecombin\_2EK}~\\
A\_27a~A\_27b)~V0x)~V1y)=V0x)))\end{array}
\label{hyp:ehigngpfcdfedgpgngcgjgogpfcdfelepfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ((ap~(\mname{c\_2Ecombin\_2EI}~\\
A\_27a)~V0x)=V0x))\end{array}
\label{hyp:ehigngpfcdfedgpgngcgjgogpfcdfejepfpfefiene}
\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} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27b .  ({\unicdcdadad} V1y \in A\_27c .  ({\unicdcdadad} V2f \in \\
((A\_27a^{{A\_27c}})^{{A\_27b}}) .  ((ap~(ap~(\mname{c\_2Epair\_2Epair\_\_CASE}~A\_27a~A\_27b~\\
A\_27c)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27b~A\_27c)~V0x)~V1y))~V2f)=(ap~\\
(ap~V2f~V0x)~V1y)))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfeahbgjgchpfpfdgbgdhfgpfpfehigng}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ((ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(\\
ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))~V0x)=V0x))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfneffmepfpfmejeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0c \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V1m \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V2n \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Ereal\_2Epow~V0c)~(ap~\\
(ap~c\_2Earithmetic\_2E\_2B~V1m)~V2n))=(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~\\
(ap~(ap~c\_2Ereal\_2Epow~V0c)~V1m))~(ap~(ap~c\_2Ereal\_2Epow~V0c)~V2n))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfeafpehfpfpfbeeeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V1y \in ty\_2Erealax\_2Ereal .  \\
({\unicdcdadad} V2u \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V3v \in ty\_2Erealax\_2Ereal .  \\
((ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~(ap~(ap~\mname{c\_2Ereal\_2E\_2F}~V0x)~V1y))~\\
(ap~(ap~\mname{c\_2Ereal\_2E\_2F}~V2u)~V3v))=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~\\
ty\_2Erealax\_2Ereal)~(ap~(ap~(\mname{c\_2Emin\_2E\_3D}~ty\_2Erealax\_2Ereal)~\\
V1y)~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0})))~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~\\
(ap~(\mname{c\_2Emarker\_2Eunint}~ty\_2Erealax\_2Ereal)~(ap~(ap~\mname{c\_2Ereal\_2E\_2F}~\\
V0x)~V1y)))~(ap~(ap~\mname{c\_2Ereal\_2E\_2F}~V2u)~V3v)))~(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~\\
ty\_2Erealax\_2Ereal)~(ap~(ap~(\mname{c\_2Emin\_2E\_3D}~ty\_2Erealax\_2Ereal)~\\
V3v)~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0})))~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~\\
(ap~(ap~\mname{c\_2Ereal\_2E\_2F}~V0x)~V1y))~(ap~(\mname{c\_2Emarker\_2Eunint}~ty\_2Erealax\_2Ereal)~\\
(ap~(ap~\mname{c\_2Ereal\_2E\_2F}~V2u)~V3v))))~(ap~(ap~\mname{c\_2Ereal\_2E\_2F}~(ap~(\\
ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~V0x)~V2u))~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~\\
V1y)~V3v)))))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfengfhmgehpfpfchbgeh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0a \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1b \in ty\_2Enum\_2Enum .  (\\
((ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
V0a))~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~V1b))=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
(ap~(ap~c\_2Earithmetic\_2E\_2A~V0a)~V1b))){\unicdcdcdhd}(((ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~\\
(ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~V0a)))~\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~V1b))=(ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~(ap~(ap~c\_2Earithmetic\_2E\_2A~V0a)~\\
V1b)))){\unicdcdcdhd}(((ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
V0a))~(ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
V1b)))=(ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
(ap~(ap~c\_2Earithmetic\_2E\_2A~V0a)~V1b)))){\unicdcdcdhd}((ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~\\
(ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~V0a)))~\\
(ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~V1b)))=\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~(ap~(ap~c\_2Earithmetic\_2E\_2A~V0a)~\\
V1b))))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfengfhmgehpfpfjgogehdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V2a \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V3y \in ty\_2Erealax\_2Ereal .  \\
(((ap~(ap~c\_2Ereal\_2Epow~V0x)~\mname{c\_2Enum\_2E0})=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))){\unicdcdcdhd}\\
(((ap~(ap~c\_2Ereal\_2Epow~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V1n)))=\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0})){\unicdcdcdhd}(((ap~(ap~c\_2Ereal\_2Epow~\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~V1n)))=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0})){\unicdcdcdhd}(((ap~(ap~c\_2Ereal\_2Epow~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V2a)))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V1n))=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~(ap~(ap~c\_2Earithmetic\_2EEXP~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V2a))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V1n)))){\unicdcdcdhd}(((ap~(ap~c\_2Ereal\_2Epow~(ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V2a))))~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V1n))=(ap~(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~\\
(ty\_2Erealax\_2Ereal^{{ty\_2Erealax\_2Ereal}}))~(ap~c\_2Earithmetic\_2EODD~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V1n)))~\mname{c\_2Erealax\_2Ereal\_\_neg})~\\
({\uniddcgcg} V4x \in ty\_2Erealax\_2Ereal .  V4x))~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
(ap~(ap~c\_2Earithmetic\_2EEXP~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V2a))~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V1n))))){\unicdcdcdhd}((ap~(ap~c\_2Ereal\_2Epow~\\
(ap~(ap~\mname{c\_2Ereal\_2E\_2F}~V0x)~V3y))~V1n)=(ap~(ap~\mname{c\_2Ereal\_2E\_2F}~(\\
ap~(ap~c\_2Ereal\_2Epow~V0x)~V1n))~(ap~(ap~c\_2Ereal\_2Epow~V3y)~V1n))))))))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfeahpghhpfpfchbgeh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(p~(ap~(\mname{c\_2Erelation\_2EWF}~A\_27a)~\\
(\mname{c\_2Erelation\_2EEMPTY\_\_REL}~A\_27a)))\end{array}
\label{hyp:ehigngpfcdfechfgmgbgehjgpgogpfcdfehfgepfpffeneafefjfpfpfcffeme}
\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} V0M \in ((A\_27b^{{A\_27a}})^{{({{A\_27b}^{{A\_27a}}})}}) .  ({\unicdcdadad} V1R \in ((2^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V2f \in (A\_27b^{{A\_27a}}) .  ((V2f=(ap~(ap~(\mname{c\_2Erelation\_2EWFREC}~\\
A\_27a~A\_27b)~V1R)~V0M)){\unicdbdjdcd}((p~(ap~(\mname{c\_2Erelation\_2EWF}~A\_27a)~V1R)){\unicdbdjdcd}\\
({\unicdcdadad} V3x \in A\_27a .  ((ap~V2f~V3x)=(ap~(ap~V0M~(ap~(ap~(ap~(\mname{c\_2Erelation\_2ERESTRICT}~\\
A\_27a~A\_27b)~V2f)~V1R)~V3x))~V3x))))))))\end{array}
\label{hyp:ehigngpfcdfechfgmgbgehjgpgogpfcdfehfgecffedepfpfdepecfpememebecfjf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (((ap~(\mname{c\_2Ewords\_2En2w}~A\_27a)~V0m)=\\
(ap~(\mname{c\_2Ewords\_2En2w}~A\_27a)~V1n)){\unicdbdjded}((ap~(ap~c\_2Earithmetic\_2EMOD~\\
V0m)~(ap~(c\_2Ewords\_2Edimword~A\_27a)~(c\_2Ebool\_2Ethe\_\_value~\\
A\_27a)))=(ap~(ap~c\_2Earithmetic\_2EMOD~V1n)~(ap~(c\_2Ewords\_2Edimword~\\
A\_27a)~(c\_2Ebool\_2Ethe\_\_value~A\_27a)))))))\end{array}
\label{hyp:ehigngpfcdfehhpgchegdhpfcdfeogcdhhpfpfbdbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}((ap~(\mname{c\_2Ewords\_2Ew2n}~A\_27a)~(ap~\\
(\mname{c\_2Ewords\_2En2w}~A\_27a)~\mname{c\_2Enum\_2E0}))=\mname{c\_2Enum\_2E0})\end{array}
\label{hyp:ehigngpfcdfehhpgchegdhpfcdfehhpgchegpfpfadpfpfogcdhh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}((ap~(\mname{c\_2Ewords\_2Ew2n}~A\_27a)~(ap~\\
(\mname{c\_2Ewords\_2En2w}~A\_27a)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO}))))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\\
\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))\end{array}
\label{hyp:ehigngpfcdfehhpgchegdhpfcdfehhpgchegpfpfbdpfpfogcdhh}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfefhmgah}
$$\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}(\\
(ap~(c\_2Ebinary\_\_ieee\_2Eulp~A\_27t~A\_27w)~(c\_2Ebool\_2Ethe\_\_value~\\
(ty\_2Epair\_2Eprod~A\_27t~A\_27w)))=(ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_to\_\_real}~\\
A\_27t~A\_27w)~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_plus\_\_min~A\_27t~\\
A\_27w)~(c\_2Ebool\_2Ethe\_\_value~(ty\_2Epair\_2Eprod~A\_27t~A\_27w)))))\end{array}$$
\end{theorem}
\end{document}
