# HG changeset patch # User wenzelm # Date 971365727 -7200 # Node ID c7c64cd26fc9e22862297bdf34be5ca899e6edfe # Parent d48a2e324abfbccc3d30a6a1b47d27627a4c79d1 updated; diff -r d48a2e324abf -r c7c64cd26fc9 doc-src/AxClass/generated/Group.tex --- a/doc-src/AxClass/generated/Group.tex Thu Oct 12 17:48:06 2000 +0200 +++ b/doc-src/AxClass/generated/Group.tex Thu Oct 12 17:48:47 2000 +0200 @@ -21,8 +21,8 @@ signature parts of our structures.% \end{isamarkuptext}% \isacommand{consts}\isanewline -\ \ times\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymodot}{\isachardoublequote}\ \isadigit{7}\isadigit{0}{\isacharparenright}\isanewline -\ \ inverse\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}{\isachardoublequote}{\isacharparenleft}{\isacharunderscore}{\isasyminv}{\isacharparenright}{\isachardoublequote}\ {\isacharbrackleft}\isadigit{1}\isadigit{0}\isadigit{0}\isadigit{0}{\isacharbrackright}\ \isadigit{9}\isadigit{9}\isadigit{9}{\isacharparenright}\isanewline +\ \ times\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymodot}{\isachardoublequote}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline +\ \ inverse\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}{\isachardoublequote}{\isacharparenleft}{\isacharunderscore}{\isasyminv}{\isacharparenright}{\isachardoublequote}\ {\isacharbrackleft}{\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharbrackright}\ {\isadigit{9}}{\isadigit{9}}{\isadigit{9}}{\isacharparenright}\isanewline \ \ one\ {\isacharcolon}{\isacharcolon}\ {\isacharprime}a\ \ \ \ {\isacharparenleft}{\isachardoublequote}{\isasymunit}{\isachardoublequote}{\isacharparenright}% \begin{isamarkuptext}% \noindent Next we define class \isa{monoid} of monoids with diff -r d48a2e324abf -r c7c64cd26fc9 doc-src/AxClass/generated/NatClass.tex --- a/doc-src/AxClass/generated/NatClass.tex Thu Oct 12 17:48:06 2000 +0200 +++ b/doc-src/AxClass/generated/NatClass.tex Thu Oct 12 17:48:47 2000 +0200 @@ -22,12 +22,12 @@ \isacommand{axclass}\ nat\ {\isacharless}\ {\isachardoublequote}term{\isachardoublequote}\isanewline \ \ induct{\isacharcolon}\ {\isachardoublequote}P{\isacharparenleft}{\isasymzero}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ P{\isacharparenleft}x{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharparenleft}Suc{\isacharparenleft}x{\isacharparenright}{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharparenleft}n{\isacharparenright}{\isachardoublequote}\isanewline \ \ Suc{\isacharunderscore}inject{\isacharcolon}\ {\isachardoublequote}Suc{\isacharparenleft}m{\isacharparenright}\ {\isacharequal}\ Suc{\isacharparenleft}n{\isacharparenright}\ {\isasymLongrightarrow}\ m\ {\isacharequal}\ n{\isachardoublequote}\isanewline -\ \ Suc{\isacharunderscore}neq{\isacharunderscore}\isadigit{0}{\isacharcolon}\ {\isachardoublequote}Suc{\isacharparenleft}m{\isacharparenright}\ {\isacharequal}\ {\isasymzero}\ {\isasymLongrightarrow}\ R{\isachardoublequote}\isanewline -\ \ rec{\isacharunderscore}\isadigit{0}{\isacharcolon}\ {\isachardoublequote}rec{\isacharparenleft}{\isasymzero}{\isacharcomma}\ a{\isacharcomma}\ f{\isacharparenright}\ {\isacharequal}\ a{\isachardoublequote}\isanewline +\ \ Suc{\isacharunderscore}neq{\isacharunderscore}{\isadigit{0}}{\isacharcolon}\ {\isachardoublequote}Suc{\isacharparenleft}m{\isacharparenright}\ {\isacharequal}\ {\isasymzero}\ {\isasymLongrightarrow}\ R{\isachardoublequote}\isanewline +\ \ rec{\isacharunderscore}{\isadigit{0}}{\isacharcolon}\ {\isachardoublequote}rec{\isacharparenleft}{\isasymzero}{\isacharcomma}\ a{\isacharcomma}\ f{\isacharparenright}\ {\isacharequal}\ a{\isachardoublequote}\isanewline \ \ rec{\isacharunderscore}Suc{\isacharcolon}\ {\isachardoublequote}rec{\isacharparenleft}Suc{\isacharparenleft}m{\isacharparenright}{\isacharcomma}\ a{\isacharcomma}\ f{\isacharparenright}\ {\isacharequal}\ f{\isacharparenleft}m{\isacharcomma}\ rec{\isacharparenleft}m{\isacharcomma}\ a{\isacharcomma}\ f{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline \isanewline \isacommand{constdefs}\isanewline -\ \ add\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isacharcolon}{\isacharcolon}nat\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isacharplus}{\isachardoublequote}\ \isadigit{6}\isadigit{0}{\isacharparenright}\isanewline +\ \ add\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isacharcolon}{\isacharcolon}nat\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isacharplus}{\isachardoublequote}\ {\isadigit{6}}{\isadigit{0}}{\isacharparenright}\isanewline \ \ {\isachardoublequote}m\ {\isacharplus}\ n\ {\isasymequiv}\ rec{\isacharparenleft}m{\isacharcomma}\ n{\isacharcomma}\ {\isasymlambda}x\ y{\isachardot}\ Suc{\isacharparenleft}y{\isacharparenright}{\isacharparenright}{\isachardoublequote}% \begin{isamarkuptext}% This is an abstract version of the plain \isa{Nat} theory in diff -r d48a2e324abf -r c7c64cd26fc9 doc-src/AxClass/generated/Product.tex --- a/doc-src/AxClass/generated/Product.tex Thu Oct 12 17:48:06 2000 +0200 +++ b/doc-src/AxClass/generated/Product.tex Thu Oct 12 17:48:47 2000 +0200 @@ -20,7 +20,7 @@ \isacommand{axclass}\isanewline \ \ product\ {\isacharless}\ {\isachardoublequote}term{\isachardoublequote}\isanewline \isacommand{consts}\isanewline -\ \ product\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isasymColon}product\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymodot}{\isachardoublequote}\ \isadigit{7}\isadigit{0}{\isacharparenright}% +\ \ product\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isasymColon}product\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymodot}{\isachardoublequote}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}% \begin{isamarkuptext}% Here class \isa{product} is defined as subclass of \isa{term} without any additional axioms. This effects in logical equivalence diff -r d48a2e324abf -r c7c64cd26fc9 doc-src/AxClass/generated/Semigroups.tex --- a/doc-src/AxClass/generated/Semigroups.tex Thu Oct 12 17:48:06 2000 +0200 +++ b/doc-src/AxClass/generated/Semigroups.tex Thu Oct 12 17:48:47 2000 +0200 @@ -17,7 +17,7 @@ semigroups.% \end{isamarkuptext}% \isacommand{consts}\isanewline -\ \ times\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymodot}{\isachardoublequote}\ \isadigit{7}\isadigit{0}{\isacharparenright}\isanewline +\ \ times\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymodot}{\isachardoublequote}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline \isacommand{axclass}\ semigroup\ {\isacharless}\ {\isachardoublequote}term{\isachardoublequote}\isanewline \ \ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymodot}\ y{\isacharparenright}\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequote}% \begin{isamarkuptext}% @@ -36,7 +36,7 @@ correspond to semigroups \isa{{\isacharparenleft}{\isasymtau}{\isacharcomma}\ {\isasymodot}{\isacharparenright}}.% \end{isamarkuptext}% \isacommand{consts}\isanewline -\ \ plus\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymoplus}{\isachardoublequote}\ \isadigit{7}\isadigit{0}{\isacharparenright}\isanewline +\ \ plus\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymoplus}{\isachardoublequote}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline \isacommand{axclass}\ plus{\isacharunderscore}semigroup\ {\isacharless}\ {\isachardoublequote}term{\isachardoublequote}\isanewline \ \ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymoplus}\ y{\isacharparenright}\ {\isasymoplus}\ z\ {\isacharequal}\ x\ {\isasymoplus}\ {\isacharparenleft}y\ {\isasymoplus}\ z{\isacharparenright}{\isachardoublequote}% \begin{isamarkuptext}% diff -r d48a2e324abf -r c7c64cd26fc9 doc-src/AxClass/generated/isabelle.sty --- a/doc-src/AxClass/generated/isabelle.sty Thu Oct 12 17:48:06 2000 +0200 +++ b/doc-src/AxClass/generated/isabelle.sty Thu Oct 12 17:48:47 2000 +0200 @@ -98,7 +98,7 @@ \newcommand{\isabellestylett}{% \renewcommand{\isastyle}{\small\tt}% -\renewcommand{\isastyleminor}{\tt}% +\renewcommand{\isastyleminor}{\small\tt}% } \newcommand{\isabellestyleit}{% \renewcommand{\isastyle}{\small\it}% diff -r d48a2e324abf -r c7c64cd26fc9 doc-src/AxClass/generated/isabellesym.sty --- a/doc-src/AxClass/generated/isabellesym.sty Thu Oct 12 17:48:06 2000 +0200 +++ b/doc-src/AxClass/generated/isabellesym.sty Thu Oct 12 17:48:47 2000 +0200 @@ -18,219 +18,225 @@ % automatically inserts italic corrections around symbols wherever % appropriate. -\newcommand{\isasymspacespace}{~~} -\newcommand{\isasymGamma}{$\Gamma$} -\newcommand{\isasymDelta}{$\Delta$} -\newcommand{\isasymTheta}{$\Theta$} -\newcommand{\isasymLambda}{$\Lambda$} -\newcommand{\isasymXi}{$\Xi$} -\newcommand{\isasymPi}{$\Pi$} -\newcommand{\isasymSigma}{$\Sigma$} -\newcommand{\isasymUpsilon}{$\Upsilon$} -\newcommand{\isasymPhi}{$\Phi$} -\newcommand{\isasymPsi}{$\Psi$} -\newcommand{\isasymOmega}{$\Omega$} -\newcommand{\isasymalpha}{$\alpha$} -\newcommand{\isasymbeta}{$\beta$} -\newcommand{\isasymgamma}{$\gamma$} -\newcommand{\isasymdelta}{$\delta$} -\newcommand{\isasymepsilon}{$\varepsilon$} -\newcommand{\isasymzeta}{$\zeta$} -\newcommand{\isasymeta}{$\eta$} -\newcommand{\isasymtheta}{$\vartheta$} -\newcommand{\isasymiota}{$\iota$} -\newcommand{\isasymkappa}{$\kappa$} -\newcommand{\isasymlambda}{$\lambda$} -\newcommand{\isasymmu}{$\mu$} -\newcommand{\isasymnu}{$\nu$} -\newcommand{\isasymxi}{$\xi$} -\newcommand{\isasympi}{$\pi$} -\newcommand{\isasymrho}{$\varrho$} -\newcommand{\isasymsigma}{$\sigma$} -\newcommand{\isasymtau}{$\tau$} -\newcommand{\isasymupsilon}{$\upsilon$} -\newcommand{\isasymphi}{$\varphi$} -\newcommand{\isasymchi}{$\chi$} -\newcommand{\isasympsi}{$\psi$} -\newcommand{\isasymomega}{$\omega$} -\newcommand{\isasymnot}{\emph{$\neg$}} -\newcommand{\isasymand}{\emph{$\wedge$}} -\newcommand{\isasymor}{\emph{$\vee$}} -\newcommand{\isasymforall}{\emph{$\forall\,$}} -\newcommand{\isasymexists}{\emph{$\exists\,$}} -\newcommand{\isasymAnd}{\emph{$\bigwedge\,$}} -\newcommand{\isasymlceil}{\emph{$\lceil$}} -\newcommand{\isasymrceil}{\emph{$\rceil$}} -\newcommand{\isasymlfloor}{\emph{$\lfloor$}} -\newcommand{\isasymrfloor}{\emph{$\rfloor$}} -\newcommand{\isasymturnstile}{\emph{$\vdash$}} -\newcommand{\isasymTurnstile}{\emph{$\models$}} -\newcommand{\isasymlbrakk}{\emph{$\mathopen{\lbrack\mkern-3mu\lbrack}$}} -\newcommand{\isasymrbrakk}{\emph{$\mathclose{\rbrack\mkern-3mu\rbrack}$}} -\newcommand{\isasymcdot}{\emph{$\cdot$}} -\newcommand{\isasymin}{\emph{$\in$}} -\newcommand{\isasymsubseteq}{\emph{$\subseteq$}} -\newcommand{\isasyminter}{\emph{$\cap$}} -\newcommand{\isasymunion}{\emph{$\cup$}} -\newcommand{\isasymInter}{\emph{$\bigcap\,$}} -\newcommand{\isasymUnion}{\emph{$\bigcup\,$}} -\newcommand{\isasymsqinter}{\emph{$\sqcap$}} -\newcommand{\isasymsqunion}{\emph{$\sqcup$}} -\newcommand{\isasymSqinter}{\emph{$\bigsqcap\,$}} %requires stmaryrd -\newcommand{\isasymSqunion}{\emph{$\bigsqcup\,$}} -\newcommand{\isasymbottom}{\emph{$\bot$}} -\newcommand{\isasymdoteq}{\emph{$\doteq$}} -\newcommand{\isasymequiv}{\emph{$\equiv$}} -\newcommand{\isasymnoteq}{\emph{$\not=$}} -\newcommand{\isasymsqsubset}{\emph{$\sqsubset$}} -\newcommand{\isasymsqsubseteq}{\emph{$\sqsubseteq$}} -\newcommand{\isasymprec}{\emph{$\prec$}} -\newcommand{\isasympreceq}{\emph{$\preceq$}} -\newcommand{\isasymsucc}{\emph{$\succ$}} -\newcommand{\isasymapprox}{\emph{$\approx$}} -\newcommand{\isasymsim}{\emph{$\sim$}} -\newcommand{\isasymsimeq}{\emph{$\simeq$}} -\newcommand{\isasymle}{\emph{$\le$}} -\newcommand{\isasymColon}{\emph{$\mathrel{::}$}} -\newcommand{\isasymleftarrow}{\emph{$\leftarrow$}} -\newcommand{\isasymmidarrow}{\emph{$\relbar$}} -\newcommand{\isasymrightarrow}{\emph{$\rightarrow$}} -\newcommand{\isasymLeftarrow}{\emph{$\Leftarrow$}} -\newcommand{\isasymMidarrow}{\emph{$\Relbar$}} -\newcommand{\isasymRightarrow}{\emph{$\Rightarrow$}} -\newcommand{\isasymfrown}{\emph{$\frown$}} -\newcommand{\isasymmapsto}{\emph{$\mapsto$}} -\newcommand{\isasymleadsto}{\emph{$\leadsto$}} %requires latexsym -\newcommand{\isasymup}{\emph{$\uparrow$}} -\newcommand{\isasymdown}{\emph{$\downarrow$}} -\newcommand{\isasymnotin}{\emph{$\notin$}} -\newcommand{\isasymtimes}{\emph{$\times$}} -\newcommand{\isasymoplus}{\emph{$\oplus$}} -\newcommand{\isasymominus}{\emph{$\ominus$}} -\newcommand{\isasymotimes}{\emph{$\otimes$}} -\newcommand{\isasymoslash}{\emph{$\oslash$}} -\newcommand{\isasymsubset}{\emph{$\subset$}} -\newcommand{\isasyminfinity}{\emph{$\infty$}} -\newcommand{\isasymbox}{\emph{$\Box$}} %requires latexsym -\newcommand{\isasymdiamond}{\emph{$\Diamond$}} %requires latexsym -\newcommand{\isasymcirc}{\emph{$\circ$}} -\newcommand{\isasymbullet}{\emph{$\bullet$}} -\newcommand{\isasymparallel}{\emph{$\parallel$}} -\newcommand{\isasymsurd}{\emph{$\surd$}} -\newcommand{\isasymcopyright}{\emph{\copyright}} -\newcommand{\isasymplusminus}{\emph{$\pm$}} -\newcommand{\isasymdiv}{\emph{$\div$}} -\newcommand{\isasymlongrightarrow}{\emph{$\longrightarrow$}} -\newcommand{\isasymlongleftarrow}{\emph{$\longleftarrow$}} -\newcommand{\isasymlongleftrightarrow}{\emph{$\longleftrightarrow$}} -\newcommand{\isasymLongrightarrow}{\emph{$\Longrightarrow$}} -\newcommand{\isasymLongleftarrow}{\emph{$\Longleftarrow$}} -\newcommand{\isasymLongleftrightarrow}{\emph{$\Longleftrightarrow$}} -\newcommand{\isasymbar}{\emph{$\mid$}} -\newcommand{\isasymhyphen}{\emph{\rm-}} -\newcommand{\isasymmacron}{\emph{\rm\=\relax}} -\newcommand{\isasymexclamdown}{\emph{\rm\textexclamdown}} -\newcommand{\isasymquestiondown}{\emph{\rm\textquestiondown}} -\newcommand{\isasymguillemotleft}{\emph{\flqq}} %requires babel -\newcommand{\isasymguillemotright}{\emph{\frqq}} %requires babel -\newcommand{\isasymdegree}{\emph{\rm\textdegree}} %requires latin1 -\newcommand{\isasymonesuperior}{\emph{$\mathonesuperior$}} %requires latin1 -\newcommand{\isasymonequarter}{\emph{\rm\textonequarter}} %requires latin1 -\newcommand{\isasymtwosuperior}{\emph{$\mathtwosuperior$}} %requires latin1 -\newcommand{\isasymonehalf}{\emph{\rm\textonehalf}} %requires latin1 -\newcommand{\isasymthreesuperior}{\emph{$\maththreesuperior$}} %requires latin1 -\newcommand{\isasymthreequarters}{\emph{\rm\textthreequarters}} %requires latin1 -\newcommand{\isasymparagraph}{\emph{\P}} -\newcommand{\isasymregistered}{\emph{\rm\textregistered}} -\newcommand{\isasymordfeminine}{\emph{\rm\textordfeminine}} -\newcommand{\isasymordmasculine}{\emph{\rm\textordmasculine}} -\newcommand{\isasymsection}{\emph{\S}} -\newcommand{\isasympounds}{\emph{$\pounds$}} -\newcommand{\isasymyen}{\emph{\yen}} %requires amssymb -\newcommand{\isasymcent}{\emph{\cent}} %requires wasysym -\newcommand{\isasymcurrency}{\emph{\currency}} %requires wasysym -\newcommand{\isasymlbrace}{\emph{$\mathopen{\lbrace\mkern-4.5mu\mid}$}} -\newcommand{\isasymrbrace}{\emph{$\mathclose{\mid\mkern-4.5mu\rbrace}$}} -\newcommand{\isasymtop}{\emph{$\top$}} -\newcommand{\isasymcong}{\emph{$\cong$}} -\newcommand{\isasymclubsuit}{\emph{$\clubsuit$}} -\newcommand{\isasymdiamondsuit}{\emph{$\diamondsuit$}} -\newcommand{\isasymheartsuit}{\emph{$\heartsuit$}} -\newcommand{\isasymspadesuit}{\emph{$\spadesuit$}} -\newcommand{\isasymleftrightarrow}{\emph{$\leftrightarrow$}} -\newcommand{\isasymge}{\emph{$\ge$}} -\newcommand{\isasympropto}{\emph{$\propto$}} -\newcommand{\isasympartial}{\emph{$\partial$}} -\newcommand{\isasymdots}{\emph{$\dots$}} -\newcommand{\isasymaleph}{\emph{$\aleph$}} -\newcommand{\isasymIm}{\emph{$\Im$}} -\newcommand{\isasymRe}{\emph{$\Re$}} -\newcommand{\isasymwp}{\emph{$\wp$}} -\newcommand{\isasymemptyset}{\emph{$\emptyset$}} -\newcommand{\isasymangle}{\emph{$\angle$}} -\newcommand{\isasymnabla}{\emph{$\nabla$}} -\newcommand{\isasymProd}{\emph{$\prod\,$}} -\newcommand{\isasymLeftrightarrow}{\emph{$\Leftrightarrow$}} -\newcommand{\isasymUparrow}{\emph{$\Uparrow$}} -\newcommand{\isasymDownarrow}{\emph{$\Downarrow$}} -\newcommand{\isasymlozenge}{\emph{$\lozenge$}} %requires amssym -\newcommand{\isasymlangle}{\emph{$\langle$}} -\newcommand{\isasymrangle}{\emph{$\rangle$}} -\newcommand{\isasymSum}{\emph{$\sum\,$}} -\newcommand{\isasymintegral}{\emph{$\int\,$}} -\newcommand{\isasymdagger}{\emph{$\dagger$}} -\newcommand{\isasymsharp}{\emph{$\sharp$}} -\newcommand{\isasymstar}{\emph{$\star$}} -\newcommand{\isasymtriangleright}{\emph{$\triangleright$}} -\newcommand{\isasymlhd}{\emph{$\lhd$}} -\newcommand{\isasymtriangle}{\emph{$\triangle$}} -\newcommand{\isasymrhd}{\emph{$\rhd$}} -\newcommand{\isasymunlhd}{\emph{$\unlhd$}} -\newcommand{\isasymunrhd}{\emph{$\unrhd$}} -\newcommand{\isasymtriangleleft}{\emph{$\triangleleft$}} -\newcommand{\isasymnatural}{\emph{$\natural$}} -\newcommand{\isasymflat}{\emph{$\flat$}} -\newcommand{\isasymamalg}{\emph{$\amalg$}} -\newcommand{\isasymmho}{\emph{$\mho$}} %requires latexsym -\newcommand{\isasymupdownarrow}{\emph{$\updownarrow$}} -\newcommand{\isasymlongmapsto}{\emph{$\longmapsto$}} -\newcommand{\isasymUpdownarrow}{\emph{$\Updownarrow$}} -\newcommand{\isasymhookleftarrow}{\emph{$\hookleftarrow$}} -\newcommand{\isasymhookrightarrow}{\emph{$\hookrightarrow$}} -\newcommand{\isasymrightleftharpoons}{\emph{$\rightleftharpoons$}} -\newcommand{\isasymleftharpoondown}{\emph{$\leftharpoondown$}} -\newcommand{\isasymrightharpoondown}{\emph{$\rightharpoondown$}} -\newcommand{\isasymleftharpoonup}{\emph{$\leftharpoonup$}} -\newcommand{\isasymrightharpoonup}{\emph{$\rightharpoonup$}} -\newcommand{\isasymasymp}{\emph{$\asymp$}} -\newcommand{\isasymminusplus}{\emph{$\mp$}} -\newcommand{\isasymbowtie}{\emph{$\bowtie$}} -\newcommand{\isasymcdots}{\emph{$\cdots$}} -\newcommand{\isasymodot}{\emph{$\odot$}} -\newcommand{\isasymsupset}{\emph{$\supset$}} -\newcommand{\isasymsupseteq}{\emph{$\supseteq$}} -\newcommand{\isasymsqsupset}{\emph{$\sqsupset$}} %requires latexsym -\newcommand{\isasymsqsupseteq}{\emph{$\sqsupseteq$}} -\newcommand{\isasymll}{\emph{$\ll$}} -\newcommand{\isasymgg}{\emph{$\gg$}} -\newcommand{\isasymuplus}{\emph{$\uplus$}} -\newcommand{\isasymsmile}{\emph{$\smile$}} -\newcommand{\isasymsucceq}{\emph{$\succeq$}} -\newcommand{\isasymstileturn}{\emph{$\dashv$}} -\newcommand{\isasymOr}{\emph{$\bigvee$}} -\newcommand{\isasymbiguplus}{\emph{$\biguplus$}} -\newcommand{\isasymddagger}{\emph{$\ddagger$}} -\newcommand{\isasymJoin}{\emph{$\Join$}} %requires latexsym -\newcommand{\isasymbool}{\emph{$\mathrm{I}\mkern-3.8mu\mathrm{B}$}} -\newcommand{\isasymcomplex}{\emph{$\mathrm{C}\mkern-15mu{\phantom{\mathrm{t}}\vrule}\mkern9mu$}} -\newcommand{\isasymnat}{\emph{$\mathrm{I}\mkern-3.8mu\mathrm{N}$}} -\newcommand{\isasymrat}{\emph{$\mathrm{Q}\mkern-16mu{\phantom{\mathrm{t}}\vrule}\mkern10mu$}} -\newcommand{\isasymreal}{\emph{$\mathrm{I}\mkern-3.8mu\mathrm{R}$}} -\newcommand{\isasymint}{\emph{$\mathsf{Z}\mkern-7.5mu\mathsf{Z}$}} -\newcommand{\isasymlesssim}{\emph{$\lesssim$}} %requires amssymb -\newcommand{\isasymgreatersim}{\emph{$\gtrsim$}} %requires amssymb -\newcommand{\isasymlessapprox}{\emph{$\lessapprox$}} %requires amssymb -\newcommand{\isasymgreaterapprox}{\emph{$\gtrapprox$}} %requires amssymb -\newcommand{\isasymtriangleq}{\emph{$\triangleq$}} %requires amssymb -\newcommand{\isasymlparr}{\emph{$\mathopen{(\mkern-3mu\mid}$}} -\newcommand{\isasymrparr}{\emph{$\mathclose{\mid\mkern-3mu)}$}} +\newcommand{\isamath}[1]{\emph{$#1$}} +\newcommand{\isatext}[1]{\emph{#1}} + + +% symbol definitions + +\newcommand{\isasymspacespace}{\isamath{~~}} +\newcommand{\isasymGamma}{\isamath{\Gamma}} +\newcommand{\isasymDelta}{\isamath{\Delta}} +\newcommand{\isasymTheta}{\isamath{\Theta}} +\newcommand{\isasymLambda}{\isamath{\Lambda}} +\newcommand{\isasymXi}{\isamath{\Xi}} +\newcommand{\isasymPi}{\isamath{\Pi}} +\newcommand{\isasymSigma}{\isamath{\Sigma}} +\newcommand{\isasymUpsilon}{\isamath{\Upsilon}} +\newcommand{\isasymPhi}{\isamath{\Phi}} +\newcommand{\isasymPsi}{\isamath{\Psi}} +\newcommand{\isasymOmega}{\isamath{\Omega}} +\newcommand{\isasymalpha}{\isamath{\alpha}} +\newcommand{\isasymbeta}{\isamath{\beta}} +\newcommand{\isasymgamma}{\isamath{\gamma}} +\newcommand{\isasymdelta}{\isamath{\delta}} +\newcommand{\isasymepsilon}{\isamath{\varepsilon}} +\newcommand{\isasymzeta}{\isamath{\zeta}} +\newcommand{\isasymeta}{\isamath{\eta}} +\newcommand{\isasymtheta}{\isamath{\vartheta}} +\newcommand{\isasymiota}{\isamath{\iota}} +\newcommand{\isasymkappa}{\isamath{\kappa}} +\newcommand{\isasymlambda}{\isamath{\lambda}} +\newcommand{\isasymmu}{\isamath{\mu}} +\newcommand{\isasymnu}{\isamath{\nu}} +\newcommand{\isasymxi}{\isamath{\xi}} +\newcommand{\isasympi}{\isamath{\pi}} +\newcommand{\isasymrho}{\isamath{\varrho}} +\newcommand{\isasymsigma}{\isamath{\sigma}} +\newcommand{\isasymtau}{\isamath{\tau}} +\newcommand{\isasymupsilon}{\isamath{\upsilon}} +\newcommand{\isasymphi}{\isamath{\varphi}} +\newcommand{\isasymchi}{\isamath{\chi}} +\newcommand{\isasympsi}{\isamath{\psi}} +\newcommand{\isasymomega}{\isamath{\omega}} +\newcommand{\isasymnot}{\isamath{\neg}} +\newcommand{\isasymand}{\isamath{\wedge}} +\newcommand{\isasymor}{\isamath{\vee}} +\newcommand{\isasymforall}{\isamath{\forall\,}} +\newcommand{\isasymexists}{\isamath{\exists\,}} +\newcommand{\isasymAnd}{\isamath{\bigwedge\,}} +\newcommand{\isasymlceil}{\isamath{\lceil}} +\newcommand{\isasymrceil}{\isamath{\rceil}} +\newcommand{\isasymlfloor}{\isamath{\lfloor}} +\newcommand{\isasymrfloor}{\isamath{\rfloor}} +\newcommand{\isasymturnstile}{\isamath{\vdash}} +\newcommand{\isasymTurnstile}{\isamath{\models}} +\newcommand{\isasymlbrakk}{\isamath{\mathopen{\lbrack\mkern-3mu\lbrack}}} +\newcommand{\isasymrbrakk}{\isamath{\mathclose{\rbrack\mkern-3mu\rbrack}}} +\newcommand{\isasymcdot}{\isamath{\cdot}} +\newcommand{\isasymin}{\isamath{\in}} +\newcommand{\isasymsubseteq}{\isamath{\subseteq}} +\newcommand{\isasyminter}{\isamath{\cap}} +\newcommand{\isasymunion}{\isamath{\cup}} +\newcommand{\isasymInter}{\isamath{\bigcap\,}} +\newcommand{\isasymUnion}{\isamath{\bigcup\,}} +\newcommand{\isasymsqinter}{\isamath{\sqcap}} +\newcommand{\isasymsqunion}{\isamath{\sqcup}} +\newcommand{\isasymSqinter}{\isamath{\bigsqcap\,}} %requires stmaryrd +\newcommand{\isasymSqunion}{\isamath{\bigsqcup\,}} +\newcommand{\isasymbottom}{\isamath{\bot}} +\newcommand{\isasymdoteq}{\isamath{\doteq}} +\newcommand{\isasymequiv}{\isamath{\equiv}} +\newcommand{\isasymnoteq}{\isamath{\not=}} +\newcommand{\isasymsqsubset}{\isamath{\sqsubset}} +\newcommand{\isasymsqsubseteq}{\isamath{\sqsubseteq}} +\newcommand{\isasymprec}{\isamath{\prec}} +\newcommand{\isasympreceq}{\isamath{\preceq}} +\newcommand{\isasymsucc}{\isamath{\succ}} +\newcommand{\isasymapprox}{\isamath{\approx}} +\newcommand{\isasymsim}{\isamath{\sim}} +\newcommand{\isasymsimeq}{\isamath{\simeq}} +\newcommand{\isasymle}{\isamath{\le}} +\newcommand{\isasymColon}{\isamath{\mathrel{::}}} +\newcommand{\isasymleftarrow}{\isamath{\leftarrow}} +\newcommand{\isasymmidarrow}{\isamath{\relbar}} +\newcommand{\isasymrightarrow}{\isamath{\rightarrow}} +\newcommand{\isasymLeftarrow}{\isamath{\Leftarrow}} +\newcommand{\isasymMidarrow}{\isamath{\Relbar}} +\newcommand{\isasymRightarrow}{\isamath{\Rightarrow}} +\newcommand{\isasymfrown}{\isamath{\frown}} +\newcommand{\isasymmapsto}{\isamath{\mapsto}} +\newcommand{\isasymleadsto}{\isamath{\leadsto}} %requires latexsym +\newcommand{\isasymup}{\isamath{\uparrow}} +\newcommand{\isasymdown}{\isamath{\downarrow}} +\newcommand{\isasymnotin}{\isamath{\notin}} +\newcommand{\isasymtimes}{\isamath{\times}} +\newcommand{\isasymoplus}{\isamath{\oplus}} +\newcommand{\isasymominus}{\isamath{\ominus}} +\newcommand{\isasymotimes}{\isamath{\otimes}} +\newcommand{\isasymoslash}{\isamath{\oslash}} +\newcommand{\isasymsubset}{\isamath{\subset}} +\newcommand{\isasyminfinity}{\isamath{\infty}} +\newcommand{\isasymbox}{\isamath{\Box}} %requires latexsym +\newcommand{\isasymdiamond}{\isamath{\Diamond}} %requires latexsym +\newcommand{\isasymcirc}{\isamath{\circ}} +\newcommand{\isasymbullet}{\isamath{\bullet}} +\newcommand{\isasymparallel}{\isamath{\parallel}} +\newcommand{\isasymsurd}{\isamath{\surd}} +\newcommand{\isasymcopyright}{\isatext{\copyright}} +\newcommand{\isasymplusminus}{\isamath{\pm}} +\newcommand{\isasymdiv}{\isamath{\div}} +\newcommand{\isasymlongrightarrow}{\isamath{\longrightarrow}} +\newcommand{\isasymlongleftarrow}{\isamath{\longleftarrow}} +\newcommand{\isasymlongleftrightarrow}{\isamath{\longleftrightarrow}} +\newcommand{\isasymLongrightarrow}{\isamath{\Longrightarrow}} +\newcommand{\isasymLongleftarrow}{\isamath{\Longleftarrow}} +\newcommand{\isasymLongleftrightarrow}{\isamath{\Longleftrightarrow}} +\newcommand{\isasymbar}{\isamath{\mid}} +\newcommand{\isasymhyphen}{\isatext{\rm-}} +\newcommand{\isasymmacron}{\isatext{\rm\=\relax}} +\newcommand{\isasymexclamdown}{\isatext{\rm\textexclamdown}} +\newcommand{\isasymquestiondown}{\isatext{\rm\textquestiondown}} +\newcommand{\isasymguillemotleft}{\isatext{\flqq}} %requires babel +\newcommand{\isasymguillemotright}{\isatext{\frqq}} %requires babel +\newcommand{\isasymdegree}{\isatext{\rm\textdegree}} %requires latin1 +\newcommand{\isasymonesuperior}{\isamath{\mathonesuperior}} %requires latin1 +\newcommand{\isasymonequarter}{\isatext{\rm\textonequarter}} %requires latin1 +\newcommand{\isasymtwosuperior}{\isamath{\mathtwosuperior}} %requires latin1 +\newcommand{\isasymonehalf}{\isatext{\rm\textonehalf}} %requires latin1 +\newcommand{\isasymthreesuperior}{\isamath{\maththreesuperior}} %requires latin1 +\newcommand{\isasymthreequarters}{\isatext{\rm\textthreequarters}} %requires latin1 +\newcommand{\isasymparagraph}{\isatext{\P}} +\newcommand{\isasymregistered}{\isatext{\rm\textregistered}} +\newcommand{\isasymordfeminine}{\isatext{\rm\textordfeminine}} +\newcommand{\isasymordmasculine}{\isatext{\rm\textordmasculine}} +\newcommand{\isasymsection}{\isatext{\S}} +\newcommand{\isasympounds}{\isamath{\pounds}} +\newcommand{\isasymyen}{\isatext{\yen}} %requires amssymb +\newcommand{\isasymcent}{\isatext{\cent}} %requires wasysym +\newcommand{\isasymcurrency}{\isatext{\currency}} %requires wasysym +\newcommand{\isasymlbrace}{\isamath{\mathopen{\lbrace\mkern-4.5mu\mid}}} +\newcommand{\isasymrbrace}{\isamath{\mathclose{\mid\mkern-4.5mu\rbrace}}} +\newcommand{\isasymtop}{\isamath{\top}} +\newcommand{\isasymcong}{\isamath{\cong}} +\newcommand{\isasymclubsuit}{\isamath{\clubsuit}} +\newcommand{\isasymdiamondsuit}{\isamath{\diamondsuit}} +\newcommand{\isasymheartsuit}{\isamath{\heartsuit}} +\newcommand{\isasymspadesuit}{\isamath{\spadesuit}} +\newcommand{\isasymleftrightarrow}{\isamath{\leftrightarrow}} +\newcommand{\isasymge}{\isamath{\ge}} +\newcommand{\isasympropto}{\isamath{\propto}} +\newcommand{\isasympartial}{\isamath{\partial}} +\newcommand{\isasymdots}{\isamath{\dots}} +\newcommand{\isasymaleph}{\isamath{\aleph}} +\newcommand{\isasymIm}{\isamath{\Im}} +\newcommand{\isasymRe}{\isamath{\Re}} +\newcommand{\isasymwp}{\isamath{\wp}} +\newcommand{\isasymemptyset}{\isamath{\emptyset}} +\newcommand{\isasymangle}{\isamath{\angle}} +\newcommand{\isasymnabla}{\isamath{\nabla}} +\newcommand{\isasymProd}{\isamath{\prod\,}} +\newcommand{\isasymLeftrightarrow}{\isamath{\Leftrightarrow}} +\newcommand{\isasymUparrow}{\isamath{\Uparrow}} +\newcommand{\isasymDownarrow}{\isamath{\Downarrow}} +\newcommand{\isasymlozenge}{\isamath{\lozenge}} %requires amssym +\newcommand{\isasymlangle}{\isamath{\langle}} +\newcommand{\isasymrangle}{\isamath{\rangle}} +\newcommand{\isasymSum}{\isamath{\sum\,}} +\newcommand{\isasymintegral}{\isamath{\int\,}} +\newcommand{\isasymdagger}{\isamath{\dagger}} +\newcommand{\isasymsharp}{\isamath{\sharp}} +\newcommand{\isasymstar}{\isamath{\star}} +\newcommand{\isasymtriangleright}{\isamath{\triangleright}} +\newcommand{\isasymlhd}{\isamath{\lhd}} +\newcommand{\isasymtriangle}{\isamath{\triangle}} +\newcommand{\isasymrhd}{\isamath{\rhd}} +\newcommand{\isasymunlhd}{\isamath{\unlhd}} +\newcommand{\isasymunrhd}{\isamath{\unrhd}} +\newcommand{\isasymtriangleleft}{\isamath{\triangleleft}} +\newcommand{\isasymnatural}{\isamath{\natural}} +\newcommand{\isasymflat}{\isamath{\flat}} +\newcommand{\isasymamalg}{\isamath{\amalg}} +\newcommand{\isasymmho}{\isamath{\mho}} %requires latexsym +\newcommand{\isasymupdownarrow}{\isamath{\updownarrow}} +\newcommand{\isasymlongmapsto}{\isamath{\longmapsto}} +\newcommand{\isasymUpdownarrow}{\isamath{\Updownarrow}} +\newcommand{\isasymhookleftarrow}{\isamath{\hookleftarrow}} +\newcommand{\isasymhookrightarrow}{\isamath{\hookrightarrow}} +\newcommand{\isasymrightleftharpoons}{\isamath{\rightleftharpoons}} +\newcommand{\isasymleftharpoondown}{\isamath{\leftharpoondown}} +\newcommand{\isasymrightharpoondown}{\isamath{\rightharpoondown}} +\newcommand{\isasymleftharpoonup}{\isamath{\leftharpoonup}} +\newcommand{\isasymrightharpoonup}{\isamath{\rightharpoonup}} +\newcommand{\isasymasymp}{\isamath{\asymp}} +\newcommand{\isasymminusplus}{\isamath{\mp}} +\newcommand{\isasymbowtie}{\isamath{\bowtie}} +\newcommand{\isasymcdots}{\isamath{\cdots}} +\newcommand{\isasymodot}{\isamath{\odot}} +\newcommand{\isasymsupset}{\isamath{\supset}} +\newcommand{\isasymsupseteq}{\isamath{\supseteq}} +\newcommand{\isasymsqsupset}{\isamath{\sqsupset}} %requires latexsym +\newcommand{\isasymsqsupseteq}{\isamath{\sqsupseteq}} +\newcommand{\isasymll}{\isamath{\ll}} +\newcommand{\isasymgg}{\isamath{\gg}} +\newcommand{\isasymuplus}{\isamath{\uplus}} +\newcommand{\isasymsmile}{\isamath{\smile}} +\newcommand{\isasymsucceq}{\isamath{\succeq}} +\newcommand{\isasymstileturn}{\isamath{\dashv}} +\newcommand{\isasymOr}{\isamath{\bigvee}} +\newcommand{\isasymbiguplus}{\isamath{\biguplus}} +\newcommand{\isasymddagger}{\isamath{\ddagger}} +\newcommand{\isasymJoin}{\isamath{\Join}} %requires latexsym +\newcommand{\isasymbool}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{B}}} +\newcommand{\isasymcomplex}{\isamath{\mathrm{C}\mkern-15mu{\phantom{\mathrm{t}}\vrule}\mkern9mu}} +\newcommand{\isasymnat}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{N}}} +\newcommand{\isasymrat}{\isamath{\mathrm{Q}\mkern-16mu{\phantom{\mathrm{t}}\vrule}\mkern10mu}} +\newcommand{\isasymreal}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{R}}} +\newcommand{\isasymint}{\isamath{\mathsf{Z}\mkern-7.5mu\mathsf{Z}}} +\newcommand{\isasymlesssim}{\isamath{\lesssim}} %requires amssymb +\newcommand{\isasymgreatersim}{\isamath{\gtrsim}} %requires amssymb +\newcommand{\isasymlessapprox}{\isamath{\lessapprox}} %requires amssymb +\newcommand{\isasymgreaterapprox}{\isamath{\gtrapprox}} %requires amssymb +\newcommand{\isasymtriangleq}{\isamath{\triangleq}} %requires amssymb +\newcommand{\isasymlparr}{\isamath{\mathopen{(\mkern-3mu\mid}}} +\newcommand{\isasymrparr}{\isamath{\mathclose{\mid\mkern-3mu)}}}