%%
%% This is file `amsthm.sty',
%% generated with the docstrip utility.
%%
%% The original source files were:
%%
%% amsclass.dtx  (with options: `amsthm')
%% This is a generated file.
%% 
%% Copyright 1995, 1999, 2004, 2009-2020 American Mathematical Society.
%% 
%% American Mathematical Society
%% Technical Support
%% Publications Technical Group
%% 201 Charles Street
%% Providence, RI 02904
%% USA
%% tel: (401) 455-4080
%%      (800) 321-4267 (USA and Canada only)
%% fax: (401) 331-3842
%% email: tech-support@ams.org
%% 
%% This work may be distributed and/or modified under the
%% conditions of the LaTeX Project Public License, either version 1.3c
%% of this license or (at your option) any later version.
%% The latest version of this license is in
%%   http://www.latex-project.org/lppl.txt
%% and version 1.3c or later is part of all distributions of LaTeX
%% version 2005/12/01 or later.
%% 
%% This work has the LPPL maintenance status `maintained'.
%% 
%% The Current Maintainer of this work is the American Mathematical
%% Society.
%% 
%% ========================================================================
%% 
\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX)
[1995/06/01]% LaTeX date must be June 1995 or later
\ProvidesPackage{amsthm}[2020/05/29 v2.20.6]
\let\@xp=\expandafter
\let\@nx=\noexpand
\def\@oparg#1[#2]{\@ifnextchar[{#1}{#1[#2]}}
\long\def\@ifempty#1{\@xifempty#1@@..\@nil}
\long\def\@xifempty#1#2@#3#4#5\@nil{%
  \ifx#3#4\@xp\@firstoftwo\else\@xp\@secondoftwo\fi}
\long\def\@ifnotempty#1{\@ifempty{#1}{}}
\def\setboxz@h{\setbox\z@\hbox}
\def\@addpunct#1{%
  \relax\ifhmode
    \ifnum\spacefactor>\@m \else#1\fi
  \fi}
\def\nopunct{\spacefactor 1007 }
\def\frenchspacing{\sfcode`\.1006\sfcode`\?1005\sfcode`\!1004%
  \sfcode`\:1003\sfcode`\;1002\sfcode`\,1001 }
\DeclareOption*{\input{\CurrentOption .thm}}
\ProcessOptions

\newcommand{\theoremstyle}[1]{%
  \@ifundefined{th@#1}{%
    \PackageWarning{amsthm}{Unknown theoremstyle `#1'}%
    \thm@style{plain}%
  }{%
    \thm@style{#1}%
  }%
}
\newtoks\thm@style
\thm@style{plain}
\newtoks\thm@bodyfont  \thm@bodyfont{\itshape}
\newtoks\thm@headfont  \thm@headfont{\bfseries}
\newtoks\thm@notefont  \thm@notefont{}
\newtoks\thm@headpunct \thm@headpunct{.}
\newskip\thm@preskip \newskip\thm@postskip
\def\thm@space@setup{%
  \thm@preskip=\topsep \thm@postskip=\thm@preskip
}
\renewcommand{\newtheorem}{\@ifstar{\@xnthm *}{\@xnthm \relax}}
\def\@xnthm#1#2{%
  \let\@tempa\relax
  \@xp\@ifdefinable\csname #2\endcsname{%
    \global\@xp\let\csname end#2\endcsname\@endtheorem
    \ifx *#1% unnumbered, need to get one more mandatory arg
      \edef\@tempa##1{%
        \gdef\@xp\@nx\csname#2\endcsname{%
          \@nx\@thm{\@xp\@nx\csname th@\the\thm@style\endcsname}%
            {}{##1}}}%
    \else % numbered theorem, need to check for optional arg
      \def\@tempa{\@oparg{\@ynthm{#2}}[]}%
    \fi
  }%
  \@tempa
}
\def\@ynthm#1[#2]#3{%
  \ifx\relax#2\relax
    \def\@tempa{\@oparg{\@xthm{#1}{#3}}[]}%
  \else
    \@ifundefined{c@#2}{%
      \def\@tempa{\@nocounterr{#2}}%
    }{%
      \@xp\xdef\csname the#1\endcsname{\@xp\@nx\csname the#2\endcsname}%
      \toks@{#3}%
      \@xp\xdef\csname#1\endcsname{%
        \@nx\@thm{%
          \let\@nx\thm@swap
            \if S\thm@swap\@nx\@firstoftwo\else\@nx\@gobble\fi
          \@xp\@nx\csname th@\the\thm@style\endcsname}%
            {#2}{\the\toks@}}%
      \let\@tempa\relax
    }%
  \fi
  \@tempa
}
\def\@xthm#1#2[#3]{%
  \ifx\relax#3\relax
    \newcounter{#1}%
  \else
    \newcounter{#1}[#3]%
    \@xp\xdef\csname the#1\endcsname{\@xp\@nx\csname the#3\endcsname
      \@thmcountersep\@thmcounter{#1}}%
  \fi
  \toks@{#2}%
  \@xp\xdef\csname#1\endcsname{%
    \@nx\@thm{%
      \let\@nx\thm@swap
        \if S\thm@swap\@nx\@firstoftwo\else\@nx\@gobble\fi
      \@xp\@nx\csname th@\the\thm@style\endcsname}%
      {#1}{\the\toks@}}%
}
\def\@thm#1#2#3{%
  \ifhmode\unskip\unskip\par\fi
  \normalfont
  \trivlist
  \let\thmheadnl\relax
  \let\thm@swap\@gobble
  \thm@notefont{\fontseries\mddefault\upshape}%
  \thm@headpunct{.}% add period after heading
  \thm@headsep 5\p@ plus\p@ minus\p@\relax
  \thm@space@setup
  #1% style overrides
  \@topsep \thm@preskip               % used by thm head
  \@topsepadd \thm@postskip           % used by \@endparenv
  \def\@tempa{#2}\ifx\@empty\@tempa
    \def\@tempa{\@oparg{\@begintheorem{#3}{}}[]}%
  \else
    \refstepcounter{#2}%
    \def\@tempa{\@oparg{\@begintheorem{#3}{\csname the#2\endcsname}}[]}%
  \fi
  \@tempa
}
\def\@restorelabelsep{\relax}
\let\@ythm\relax
\let\thmname\@iden \let\thmnote\@iden \let\thmnumber\@iden
\providecommand\@upn{\textup}
\def\thmhead@plain#1#2#3{%
  \thmname{#1}\thmnumber{\@ifnotempty{#1}{ }\@upn{#2}}%
  \thmnote{ {\the\thm@notefont(#3)}}}
\let\thmhead\thmhead@plain
\def\swappedhead#1#2#3{%
  \thmnumber{#2}%
  \thmname{\@ifnotempty{#2}{~}#1}%
  \thmnote{ {\the\thm@notefont(#3)}}}
\let\swappedhead@plain=\swappedhead
\let\thmheadnl\relax
\let\thm@indent\noindent
\let\thm@swap\@gobble
\def\@begintheorem#1#2[#3]{%
  \deferred@thm@head{\the\thm@headfont \thm@indent
    \@ifempty{#1}{\let\thmname\@gobble}{\let\thmname\@iden}%
    \@ifempty{#2}{\let\thmnumber\@gobble}{\let\thmnumber\@iden}%
    \@ifempty{#3}{\let\thmnote\@gobble}{\let\thmnote\@iden}%
    \thm@swap\swappedhead\thmhead{#1}{#2}{#3}%
    \the\thm@headpunct
    \thmheadnl % possibly a newline.
    \hskip\thm@headsep
  }%
  \ignorespaces}
\newskip\thm@headsep
\thm@headsep=5pt plus1pt minus1pt\relax
\let\adjust@parskip@nobreak=\@nbitem
\newtoks\dth@everypar
\dth@everypar={%
  \@minipagefalse \global\@newlistfalse
  \@noparitemfalse
  \if@inlabel
    \global\@inlabelfalse
    \begingroup \setbox\z@\lastbox
     \ifvoid\z@ \kern-\itemindent \fi
    \endgroup
    \unhbox\@labels
  \fi
  \if@nobreak \@nobreakfalse \clubpenalty\@M
  \else \clubpenalty\@clubpenalty \everypar{}%
  \fi
}%
\def\deferred@thm@head#1{%
  \if@inlabel \indent \par \fi % eject a section head if one is pending
  \if@nobreak
    \adjust@parskip@nobreak
  \else
    \addpenalty\@beginparpenalty
    \addvspace\@topsep
    \addvspace{-\parskip}%
  \fi
  \global\@inlabeltrue
  \everypar\dth@everypar
  \sbox\@labels{\normalfont#1}%
  \ignorespaces
}
\def\nonslanted{\relax
  \@xp\let\@xp\@tempa\csname\f@shape shape\endcsname
  \ifx\@tempa\itshape\upshape
  \else\ifx\@tempa\slshape\upshape\fi\fi}
\def\swapnumbers{\edef\thm@swap{\if S\thm@swap N\else S\fi}}
\def\thm@swap{N}%
\let\@opargbegintheorem\relax
\def\th@plain{%
%%  \let\thm@indent\noindent % no indent
%%  \thm@headfont{\bfseries}% heading font is bold
%%  \thm@notefont{}% same as heading font
%%  \thm@headpunct{.}% add period after heading
%%  \let\thm@swap\@gobble
%%  \thm@preskip\topsep
%%  \thm@postskip\theorempreskipamount
  \itshape % body font
}
\def\th@definition{%
  \normalfont % body font
}
\def\th@remark{%
  \thm@headfont{\itshape}%
  \normalfont % body font
  \thm@preskip\topsep \divide\thm@preskip\tw@
  \thm@postskip\thm@preskip
}
\def\@endtheorem{\endtrivlist\@endpefalse }
\newcommand{\newtheoremstyle}[9]{%
  \@ifempty{#5}{\dimen@\z@skip}{\dimen@#5\relax}%
  \ifdim\dimen@=\z@
    \toks@{#4\let\thm@indent\noindent}%
  \else
    \toks@{#4\def\thm@indent{\noindent\hbox to#5{}}}%
  \fi
  \def\@tempa{#8}\ifx\space\@tempa
    \toks@\@xp{\the\toks@ \thm@headsep\fontdimen\tw@\font\relax}%
  \else
    \def\@tempb{\newline}%
    \ifx\@tempb\@tempa
      \toks@\@xp{\the\toks@ \thm@headsep\z@skip
        \def\thmheadnl{\newline}}%
    \else
      \toks@\@xp{\the\toks@ \thm@headsep#8\relax}%
    \fi
  \fi
  \begingroup
  \thm@space@setup
  \@defaultunits\@tempskipa#2\thm@preskip\relax\@nnil
  \@defaultunits\@tempskipb#3\thm@postskip\relax\@nnil
  \xdef\@gtempa{\thm@preskip\the\@tempskipa
    \thm@postskip\the\@tempskipb\relax}%
  \endgroup
  \@temptokena\@xp{\@gtempa
    \thm@headfont{#6}\thm@headpunct{#7}%
  }%
  \@ifempty{#9}{%
    \let\thmhead\thmhead@plain
  }{%
    \@namedef{thmhead@#1}##1##2##3{#9}%
    \@temptokena\@xp{\the\@temptokena
      \@xp\let\@xp\thmhead\csname thmhead@#1\endcsname}%
  }%
  \@xp\xdef\csname th@#1\endcsname{\the\toks@ \the\@temptokena}%
}
\DeclareRobustCommand{\qed}{%
  \ifmmode \mathqed
  \else
    \leavevmode\unskip\penalty9999 \hbox{}\nobreak\hfill
    \quad\hbox{\qedsymbol}%
  \fi
}
\let\QED@stack\@empty
\let\qed@elt\relax
\newcommand{\pushQED}[1]{%
  \toks@{\qed@elt{#1}}\@temptokena\expandafter{\QED@stack}%
  \xdef\QED@stack{\the\toks@\the\@temptokena}%
}
\newcommand{\popQED}{%
  \begingroup\let\qed@elt\popQED@elt \QED@stack\relax\relax\endgroup
}
\def\popQED@elt#1#2\relax{#1\gdef\QED@stack{#2}}
\newcommand{\qedhere}{%
  \begingroup \let\mathqed\math@qedhere
    \let\qed@elt\setQED@elt \QED@stack\relax\relax \endgroup
}
\newif\ifmeasuring@
\newif\iffirstchoice@ \firstchoice@true
\def\setQED@elt#1#2\relax{%
  \ifmeasuring@
  \else \iffirstchoice@ \gdef\QED@stack{\qed@elt{}#2}\fi
  \fi
  #1%
}
\def\qed@warning{%
  \PackageWarning{amsthm}{The \@nx\qedhere command may not work
    correctly here}%
}
\newcommand{\mathqed}{\quad\hbox{\qedsymbol}}
\def\linebox@qed{\hfil\hbox{\qedsymbol}\hfilneg}
\@ifpackageloaded{amsmath}{%
  \def\math@qedhere{%
    \@ifundefined{\@currenvir @qed}{%
      \qed@warning\quad\hbox{\qedsymbol}%
    }{%
      \@xp\aftergroup\csname\@currenvir @qed\endcsname
    }%
  }
  \def\displaymath@qed{%
    \relax
    \ifmmode
      \ifinner \aftergroup\linebox@qed
      \else
        \eqno
        \let\eqno\relax \let\leqno\relax \let\veqno\relax
        \hbox{\qedsymbol}%
      \fi
    \else
       \aftergroup\linebox@qed
    \fi
  }
  \@xp\let\csname equation*@qed\endcsname\displaymath@qed
  \def\equation@qed{%
    \iftagsleft@
      \hbox{\phantom{\quad\qedsymbol}}%
      \gdef\alt@tag{%
        \rlap{\hbox to\displaywidth{\hfil\qedsymbol}}%
        \global\let\alt@tag\@empty
      }%
    \else
      \gdef\alt@tag{%
        \global\let\alt@tag\@empty
        \vtop{\ialign{\hfil####\cr
                \tagform@\theequation\cr
                \qedsymbol\cr}}%
        \setbox\z@
      }%
    \fi
  }
  \def\qed@tag{%
    \global\tag@true \nonumber
    &\omit\setboxz@h {\strut@ \qedsymbol}\tagsleft@false
    \place@tag@gather
    \kern-\tabskip
    \ifst@rred \else \global\@eqnswtrue \fi \global\advance\row@\@ne \cr
  }
  \def\split@qed{%
    \def\endsplit{\crcr\egroup \egroup \ctagsplit@false \rendsplit@
      \aftergroup\align@qed
    }%
  }
  \def\align@qed{%
    \ifmeasuring@ \tag*{\qedsymbol}%
    \else \let\math@cr@@@\qed@tag
    \fi
  }
  \@xp\let\csname align*@qed\endcsname\align@qed
  \@xp\let\csname gather*@qed\endcsname\align@qed
%% Needs some patching up for amsmath 1.2
}{% end of amsmath branch, start plain LaTeX branch
  \def\math@qedhere{%
    \@ifundefined{\@currenvir @qed}{%
      \qed@warning \aftergroup\displaymath@qed
    }{%
      \@xp\aftergroup\csname\@currenvir @qed\endcsname
    }%
  }
  \def\displaymath@qed{%
    \relax
    \ifmmode
      \ifinner \aftergroup\aftergroup\aftergroup\linebox@qed
      \else
        \eqno \def\@badmath{$$}%
        \let\eqno\relax \let\leqno\relax \let\veqno\relax
        \hbox{\qedsymbol}%
      \fi
    \else
       \aftergroup\linebox@qed
    \fi
  }
  \@ifundefined{ver@leqno.clo}{%
    \def\equation@qed{\displaymath@qed \quad}%
  }{%
    \def\equation@qed{\displaymath@qed}%
  }
    \def\@tempa#1$#2#3\@nil#4{%
        \def#4{#1$#2\def\@currenvir{displaymath}#3}%
    }%
    \expandafter\ifx\csname[ \endcsname\relax
        \expandafter\@tempa\[\@nil\[%
    \else
        \expandafter\expandafter\expandafter\@tempa\csname[
        \expandafter\endcsname\expandafter\@nil
        \csname[ \endcsname
    \fi
}
\@ifpackageloaded{amstex}{%
  \def\@tempa{TT}%
}{%
  \@ifpackageloaded{amsmath}{%
    \def\@tempb#1 v#2.#3\@nil{#2}%
    \ifnum\@xp\@xp\@xp\@tempb\csname ver@amsmath.sty\endcsname v0.0\@nil
       <\tw@
      \def\@tempa{TT}%
    \else
      \def\@tempa{TF}%
    \fi
  }{%
    \def\@tempa{TF}
  }%
}
\if\@tempa
  \renewcommand{\math@qedhere}{\quad\hbox{\qedsymbol}}%
\fi
\newcommand{\openbox}{\leavevmode
  \hbox to.77778em{%
  \hfil\vrule
  \vbox to.675em{\hrule width.6em\vfil\hrule}%
  \vrule\hfil}}
\DeclareRobustCommand{\textsquare}{%
  \begingroup \usefont{U}{msa}{m}{n}\thr@@\endgroup
}
\providecommand{\qedsymbol}{\openbox}
\newenvironment{proof}[1][\proofname]{\par
  \pushQED{\qed}%
  \normalfont \topsep6\p@\@plus6\p@\relax
  \trivlist
  \item[\hskip\labelsep
        \itshape
    #1\@addpunct{.}]\ignorespaces
}{%
  \popQED\endtrivlist\@endpefalse
}
\providecommand{\proofname}{Proof}
\endinput
%%
%% End of file `amsthm.sty'.