\NeedsTeXFormat{LaTeX2e} \newcommand*{\@curpack}{\@currname} %rec-thy.sty \newcommand*{\@packinfo}[1]{\PackageInfo{\@curpack}{#1}} \newcommand*{\recthy@strip}[1]{%\expandafter\newcommand{\csname #1\endcsname}{}% Makes sure arg isn't already a command \expandafter\edef\csname #1\endcsname{\expandafter\noexpand\csname @recthy@#1\endcsname}} %leaves us a command \arg defined to do \@recthy@arg \ProvidesPackage{\@curpack} [2010/10/14 v1.0 Provides commands for writing up recursion theory papers] %file name is package name \RequirePackage{suffix} \RequirePackage{ifmtarg} \RequirePackage{xifthen} \RequirePackage{xkeyval} \RequirePackage{undertilde} \RequirePackage{stmaryrd} \RequirePackage{amsmath} \RequirePackage{amssymb} \RequirePackage{marvosym} \DeclareMathAlphabet{\mathbrush}{T1}{pbsi}{xl}{n} \newcommand*{\PMG@arginit}{\let\@PMG@parenarg\@PMG@undefined\let\@PMG@braketarg\@PMG@undefined} \newcommand*{\@recthy@family}{\@currname.\@currext} %The default family for this package...filename.ext %Define boolean options true if option passed false otherwise \define@boolkeys{\@recthy@family}[recthy@]{nohyphenation,nomath,nodelim,nosets,noquants,nospaces,nostrings,notrees,nopair,nonames,nosetrels,noordinalnotations,nosyntax,nodegclasses,noforcing} %of form \ifrecthy@nosets etc... \AtEndOfPackage{ \ifrecthy@nohyphenation \@packinfo{No hyphenation loading.} \else \hyphenation{ra-tion-als ra-tion-al Harrington Slaman Cholak Knight Gerdes Lange} \fi } \newcommand*{\@recthy@abs}[1]{\lvert#1\rvert} \newcommand*{\@recthy@ensuretext}[1]{\ensuremath{\text{#1}}} \AtEndOfPackage{ \ifrecthy@nomath \@packinfo{No math loading.} \else \@packinfo{math loading.} \newcommand*{\eqdef}{\stackrel{\text{\tiny def}}{=}} %\newcommand*{\eqdef}{\ensuremath{=\limits_{\text{\tiny def}}}} \newcommand*{\iffdef}{\stackrel{\text{\tiny def}}{\iff}} \DeclareMathOperator{\aut}{Aut} \DeclareMathOperator{\Ord}{Ord} \newcommand*{\meet}{\mathbin{\wedge}} \newcommand*{\join}{\mathbin{\vee}} \newcommand*{\Meet}{\mathop{\bigwedge}} \let\MnJoin=\Join \renewcommand*{\Join}{\mathop{\bigvee}} \recthy@strip{abs} \DeclareMathOperator{\dom}{dom} \DeclareMathOperator{\rng}{rng} \newcommand*{\restr}[1]{\mathpunct{\restriction_{#1}}} \newcommand*{\ordpair}[2]{\left( #1, #2 \right) } %%%%%%%%Function commands \newcommand*{\map}[2]{:#1\mapsto #2} \newcommand*{\functo}[3]{#1\map{#2}{#3}} \newcommand*{\compfunc}{\mathbin{\circ}} \let\compose=\compfunc \let\funcomp=\compfunc \recthy@strip{ensuretext} \fi } % corner quotes From Sam Buss http://www.phil.cam.ac.uk/teaching_staff/Smith/LaTeX/resources/godelquotes.txt \newbox\@recthy@gnBoxA \newdimen\@recthy@gnCornerHgt \setbox\@recthy@gnBoxA=\hbox{$\ulcorner$} \global\@recthy@gnCornerHgt=\ht\@recthy@gnBoxA \newdimen\@recthy@gnArgHgt \newcommand{\@recthy@godelnum}[1]{% \setbox\@recthy@gnBoxA=\hbox{$#1$}% \@recthy@gnArgHgt=\ht\@recthy@gnBoxA% \ifnum \@recthy@gnArgHgt<\@recthy@gnCornerHgt \@recthy@gnArgHgt=0pt% \else \advance \@recthy@gnArgHgt by -\@recthy@gnCornerHgt% \fi \raise\@recthy@gnArgHgt\hbox{$\ulcorner$} \box\@recthy@gnBoxA % \raise\@recthy@gnArgHgt\hbox{$\urcorner$}} \newcommand*{\@recthy@llangle}{\mathopen{\langle\!\langle}} \newcommand*{\@recthy@rrangle}{\mathclose{\rangle\!\rangle}} \AtEndOfPackage{ \ifrecthy@nodelim \@packinfo{No delimiters loading.} \else \newcommand*{\godelnum}[1]{\@recthy@godelnum{#1}} \let\cornerquote=\godelnum \let\gcode=\godelnum \recthy@strip{llangle} \recthy@strip{rrangle} \fi } \newcommand*{\@recthy@re}{\@recthy@ensuretext{r.e.}} \newcommand*{\@recthy@ce}{\@recthy@ensuretext{c.e.}} \newcommand*{\@recthy@recursive}{\@recthy@ensuretext{recursive}} \newcommand*{\@recthy@computable}{\@recthy@ensuretext{computable}} \newcommand*{\@recthy@recursivelyEnumerable}{\@recthy@ensuretext{recursively enumerable}} \newcommand*{\@recthy@computablyEnumerable}{\@recthy@ensuretext{computably enumerable}} \newcommand*{\@recthy@Recursive}{\@recthy@ensuretext{Recursive}} \newcommand*{\@recthy@Computable}{\@recthy@ensuretext{Computable}} \newcommand*{\@recthy@RecursivelyEnumerable}{\@recthy@ensuretext{Recursively enumerable}} \newcommand*{\@recthy@ComputablyEnumerable}{\@recthy@ensuretext{Computably enumerable}} \newcommand*{\@recthy@REA}{\ensuremath{\text{REA}}} \WithSuffix\def\@recthy@REA[#1]{#1-\REA\relax} \newcommand*{\@recthy@CEA}{\ensuremath{\text{CEA}}} \WithSuffix\def\@recthy@CEA[#1]{#1-\CEA\relax} \AtEndOfPackage{ \ifrecthy@nonames \@packinfo{No names loading.} \else \@packinfo{Names loading.} \recthy@strip{re} \recthy@strip{ce} \recthy@strip{REA} \recthy@strip{CEA} \recthy@strip{recursive} \recthy@strip{computable} \recthy@strip{recursivelyEnumerable} \recthy@strip{computablyEnumerable} \recthy@strip{Recursive} \recthy@strip{Computable} \recthy@strip{RecursivelyEnumerable} \recthy@strip{ComputablyEnumerable} \fi } \DeclareOptionX<\@recthy@family>{re}{% \let\@recthy@ce=\@recthy@re \let\@recthy@computable=\@recthy@recursive \let\@recthy@computablyEnumerable=\@recthy@recursivelyEnumerable \let\@recthy@ComputablyEnumerable=\@recthy@RecursivelyEnumerable } %Make all references to ce change to re \DeclareOptionX<\@recthy@family>{rea}{% \let\@recthy@CEA=\@recthy@REA } \DeclareOptionX<\@recthy@family>{ce}{% \let\@recthy@re=\@recthy@ce \let\@recthy@recursive=\@recthy@computable \let\@recthy@recursivelyEnumerable=\@recthy@computablyEnumerable \let\@recthy@RecursivelyEnumerable=\@recthy@ComputablyEnumerable } %Make all references to re change to ce \DeclareOptionX<\@recthy@family>{cea}{% \let\@recthy@REA=\@recthy@CEA } % Set stuff \newcommand*{\@recthy@set}[2]{\ifthenelse{\isempty{#2}}{\left \{ #1 \right \}}{\left \{ #1 \middle | #2\right \}} } \newcommand*{\@recthy@card}[1]{\lvert#1\rvert} \newcommand*{\@recthy@union}{\mathbin{\cup}} \newcommand*{\@recthy@Union}{\mathop{\bigcup}} \newcommand*{\@recthy@isect}{\mathbin{\cap}} \newcommand*{\@recthy@Isect}{\mathop{\bigcap}} \newcommand*{\@recthy@powset}[1]{\mathcal{P}\left(#1\right)} \newcommand*{\@recthy@eset}{\emptyset} \newcommand*{\@recthy@nin}{\mathrel{\not\in}} \newcommand*{\@recthy@setcmpminusSYM}{\backsim} \newcommand*{\@recthy@setminuscmp}[1]{\mathop{\@recthy@setcmpminusSYM} #1} \newcommand*{\@recthy@setovercmp}[1]{\overline{#1}} \let\@recthy@setcmp=\@recthy@setovercmp \def\@recthy@setminusSYM{-} \DeclareOptionX<\@recthy@family>{setminussym}{\def\@recthy@setminusSYM{#1}} \DeclareOptionX<\@recthy@family>{setcmpminussym}{\def\@recthy@setminusSYM{#1}} \DeclareOptionX<\@recthy@family>{minussetcmp}{\let\@recthy@setcmp=\@recthy@setminuscmp} \newcommand*{\@recthy@setminus}{\mathbin{\@recthy@setminusSYM}} \AtEndOfPackage{ \WithSuffix\def\@recthy@setcmp[#1]#2{#1\@recthy@setminus#2} \ifrecthy@nosets \@packinfo{No set cmds loading.} \else \@packinfo{Set cmds loading.} \recthy@strip{set} \recthy@strip{card} \recthy@strip{union} \recthy@strip{Union} \recthy@strip{isect} \recthy@strip{Isect} \recthy@strip{powset} \recthy@strip{eset} \recthy@strip{nin} \recthy@strip{setcmp} \recthy@strip{setovercmp} \recthy@strip{setminuscmp} \recthy@strip{setminus} \newcommand*{\cross}{\mathbin{\times}} \let\CrossOrig=\Cross \def\Cross{} \renewcommand*{\Cross}{\mathop{\Pi}} \newcommand*{\symdiff}{\mathbin{\Delta}} \DeclareMathOperator{\interior}{int} \DeclareMathOperator{\closure}{cl} \fi } \AtEndOfPackage{ \ifrecthy@noquants \@packinfo{No quants loading.} \else \@packinfo{Quants loading.} \WithSuffix\def\exists(#1){\left(\exists\, #1 \right)\!} \WithSuffix\def\forall(#1){\left(\forall\, #1 \right)\!} \WithSuffix\def\nexists(#1){\left(\nexists\, #1 \right)\!} \WithSuffix\def\exists[#1]{\left[\exists\, #1 \right]\!} \WithSuffix\def\forall[#1]{\left[\forall\, #1 \right]\!} \WithSuffix\def\nexists[#1]{\left[\nexists\, #1 \right]\!} \WithSuffix\def\forall*{\forallae} \WithSuffix\def\exists*{\existsinf} \WithSuffix\def\nexists*{\nexistsinf} \newcommand*{\forallae}{\forall^{*}} \WithSuffix\def\forallae(#1){\left(\forallae\, #1\right)\!} \WithSuffix\def\forallae[#1]{\left[\forallae\, #1\right]\!} \newcommand*{\existsinf}{\exists^{\infty}} \WithSuffix\def\existsinf(#1){\left(\existsinf\, #1\right)\!} \WithSuffix\def\existsinf[#1]{\left[\existsinf\, #1\right]\!} \newcommand*{\nexistsinf}{\nexists^{\infty}} \WithSuffix\def\nexistsinf(#1){\left(\nexistsinf\, #1\right)\!} \WithSuffix\def\nexistsinf[#1]{\left[\nexistsinf\, #1\right]\!} \newcommand*{\True}{\top} \newcommand*{\False}{\bot} \newcommand*{\Land}{\mathop{\bigwedge}} \newcommand*{\Lor}{\mathop{\bigvee}} \newcommand*{\LLand}{\mathop{\bigwedge\mkern-15mu\bigwedge}} \newcommand*{\LLor}{\mathop{\bigvee\mkern-15mu\bigvee}} \newcommand*{\liff}{\ensuremath{\leftrightarrow}} \newcommand*{\limplies}{\ensuremath{\rightarrow}} \fi } \AtEndOfPackage{ \ifrecthy@nospaces \@packinfo{No spaces loading.} \else \@packinfo{Spaces loading.} \newcommand*{\bstrs}{2^{<\omega}} \newcommand*{\wstrs}{\omega^{<\omega}} \newcommand*{\cantor}{2^{\omega}} % \WithSuffix\def\cantor*{\left(2\union \diverge \right)^{\omega}} \newcommand*{\baire}{\omega^{\omega}} % \WithSuffix\def\baire*{\left(\omega \union \diverge \right)^{\omega}} \newcommand*{\Baire}{\mathcal{N}} \fi } % \newcommand*{\recfnlSYM}{\Phi} % % \DeclareOptionX{recfn}{\def\recfnlSYM{#1}} %Strings \newcommand*{\@recthy@EmptyStr}{\str{}} \newcommand*{\@recthy@concatSYM}{\hat{}} \DeclareOptionX<\@recthy@family>{emptystr}[]{\def\@recthy@EmptyStr{#1}} \DeclareOptionX<\@recthy@family>{concat}[]{\def\@recthy@concatSYM{#1}} \AtEndOfPackage{ \ifrecthy@nostrings \@packinfo{No strings loading.} \else \@packinfo{Strings loading.} \newcommand*{\str}[1]{\mathopen{\@recthy@llangle}#1\mathclose{\@recthy@rrangle}} \newcommand*{\code}{\str} \newcommand*{\EmptyStr}{\@recthy@EmptyStr} \newcommand*{\estr}{\EmptyStr} \newcommand*{\decode}[2]{(#1)_{#2}} \newcommand*{\godelpair}[2]{\mathopen{\langle} #1, #2 \mathclose{\rangle}} \newcommand*{\gpair}{\godelpair} \newcommand*{\concat}{\mathbin{\@recthy@concatSYM}} \WithSuffix\def\concat[#1]{\concat\str{#1}} \newcommand*{\strpred}[1]{{#1}^{-}} \newcommand*{\lh}[2][]{\@recthy@abs{#2}_{#1}} %% Gives length of a string \newcommand*{\incompat}{\mathrel{\mid}} %incompatible stringes FIX SPACING \newcommand*{\incomp}{\incompat} \newcommand*{\compat}{\mathrel{\nmid}} \newcommand*{\setcol}[2]{{#1}^{[#2]}} \ifrecthy@nopair \@packinfo{No redefining pair.} \else \@packinfo{pair=godelpair.} \newcommand*{\pair}{\godelpair} \fi \fi } \AtEndOfPackage{ \ifrecthy@notrees \@packinfo{No trees loading.} \else \@packinfo{Trees loading.} \newcommand*{\CBderiv}[2][1]{#2^{\langle#1\rangle}} \newcommand*{\pruneTree}[1]{\CBderiv[\infty]{#1}} \newcommand*{\hgt}[1]{\lVert #1 \rVert} \fi } % Tree arguments % \newcommand*{\truepath}[2][]{\Gamma_{#1}\ifthenelse{\isempty{#2}}{}{(#2)}} % \newcommand*{\Tpath}{\truepath} %%%%%% \newcommand*{\murec}[2]{\ensuremath{\mu#1\left( #2 \right) }} %\newcommand*{\brecfSYM}{\phi} \newcommand*{\recfnlSYM}{\Phi} % \newcommand*{\recf}[3][]{\recfSYM^{#1}_{#2}\ifthenelse{\isempty{#3}}{}{\left(#3\right)}} %If give an empty arg just print function no parens %\newcommand*{\brecf}[3][]{\brecfSYM^{#1}_{#2}\ifthenelse{\isempty{#3}}{}{\left(#3\right)}} %\recfnl[d]{i}{Y}{x} gives i-th d-recursive functional operating on set Y and integer x \newcommand*{\recfnl}[4][]{\recfnlSYM_{#2\ifthenelse{\isempty{#1}}{}{,#1} }% \ifthenelse{\isempty{#4}}% {\ifthenelse{\isempty{#3}}% {}% No args then print no parents {(#3)} }%If no oracle we just refer to the functional without any arguments {\ifthenelse{\isempty{#3}}{(#4)}{(#3;#4)}% }} \newcommand*{\cequiv}{\mathrel{\backsimeq}} \newcommand*{\ncequiv}{\mathrel{\not\cequiv}} \newcommand*{\conv}[1][]{\mathpunct{\downarrow}_{#1}} %\newcommand*{\convs}[1]{\ensuremath{\conv_{#1}}} \newcommand*{\nconv}[1][]{\mathpunct{\mkern-4mu{\arrownot\mkern 4mu \downarrow}_{#1}}} \newcommand*{\diverge}{\mathpunct{\uparrow}} % \newcommand*{\use}[2]{\mathscr{u}(#1\ifthenelse{\isempty{#2}}{}{,#2})} \newcommand*{\use}[1]{\mathop{\mathbrush{u}}\!\left[{#1}\right]} \newcommand*{\REset}{\let\@PMG@parenarg\@PMG@undefined\let\@PMG@braketarg\@PMG@undefined\@REsetbody} \newcommand*{\@REsetbody}[1]{W\ifdefined\@PMG@parenarg ^{\@PMG@parenarg}% \fi% \ifdefined\@PMG@braketarg _{#1, {\@PMG@braketarg}}% \else _{#1} \fi} \WithSuffix\def\@REsetbody(#1){\def\@PMG@parenarg{#1}\@REsetbody} \WithSuffix\def\@REsetbody[#1]{\def\@PMG@braketarg{#1}\@REsetbody} \newcommand*{\alphaREAop}[2][]{\mathcal{J}^{#2}_{#1}} \let\aREAop=\alphaREAop \let\reaop=\alphaREAop \newcommand*{\oneREAop}[1]{J_{#1}} \let\iREAop=\oneREAop \WithSuffix\def\reaop*#1{\oneREAop{#1}} %converges %%Computational Classes %%%%%%Turing Degrees \newcommand*{\Tdeg}[1]{\utilde{#1}} \newcommand*{\Tjump}[1]{#1'} \let\jump=\Tjump \newcommand*{\jumpn}[2]{#1^{(#2)}} \newcommand*{\jjump}[1]{#1''} % \newcommand*{\Tzerosym}{\MVZero} \newcommand*{\Tzerosym}{\MVZero} \newcommand*{\Tzero}{{\Tzerosym}} \newcommand*{\zeron}[1]{{\Tzerosym}^{(#1)}} \newcommand*{\zeroj}{\jump{{\Tzerosym}}} \let\Tzeroj=\zeroj \newcommand*{\zerojj}{\jump{\jump{{\Tzerosym}}}} \newcommand*{\zerojjj}{\jump{\jump{\jump{{\Tzerosym}}}}} \let\Tzerojj=\zerojj \let\Tzerojjj=\zerojjj %%Misc Recursion theory stuff %turing equivalent \newcommand*{\@recthy@TSYM}{\mathbf{T}} \newcommand*{\Tequiv}{\mathrel{\equiv_{\@recthy@TSYM}}} \newcommand*{\Teq}{\Tequiv} \newcommand*{\nTequiv}{\mathrel{\ncong_{\@recthy@TSYM}}} \newcommand*{\nTeq}{\nTequiv} %turing below \newcommand*{\Tlneq}{\lneq_{\@recthy@TSYM}} \newcommand*{\Tleq}{\leq_{\@recthy@TSYM}} \newcommand*{\Tgneq}{\gneq_{\@recthy@TSYM}} \newcommand*{\Tgeq}{\geq_{\@recthy@TSYM}} \newcommand*{\Tgtr}{>_{\@recthy@TSYM}} \newcommand*{\Tless}{<_{\@recthy@TSYM}} \newcommand*{\nTleq}{\nleq_{\@recthy@TSYM}} \newcommand*{\nTgeq}{\ngeq_{\@recthy@TSYM}} \newcommand*{\Tdegjoin}{\mathbin{\vee_{\@recthy@TSYM}}} \newcommand*{\Tdegmeet}{\mathbin{\wedge_{\@recthy@TSYM}}} \newcommand*{\Tmeet}{\Tdegmeet} \newcommand*{\Tsetjoin}{\mathbin{\oplus}} \newcommand*{\TsetJoin}{\mathop{\bigoplus}} \newcommand*{\Tplus}{\Tsetjoin} \newcommand*{\TPlus}{\TsetJoin} \newcommand*{\Tjoin}{\Tsetjoin} \newcommand*{\TJoin}{\TsetJoin} \DeclareOptionX<\@recthy@family>{degjoin}{\def\Tjoin{\Tdegjoin}} \newcommand*{\@recthy@ttSYM}{\ensuremath{\text{\tiny tt}}} \newcommand*{\ttSYM}{\@recthy@ttSYM} \newcommand*{\ttlneq}{\lneq_{\@recthy@ttSYM}} \newcommand*{\ttleq}{\leq_{\@recthy@ttSYM}} \newcommand*{\ttgneq}{\gneq_{\@recthy@ttSYM}} \newcommand*{\ttgeq}{\geq_{\@recthy@ttSYM}} \newcommand*{\ttgtr}{>_{\@recthy@ttSYM}} \newcommand*{\ttless}{<_{\@recthy@ttSYM}} \newcommand*{\ttTleq}{\nleq_{\@recthy@ttSYM}} \newcommand*{\ttTgeq}{\ngeq_{\@recthy@ttSYM}} %re set relations \AtEndOfPackage{ \ifrecthy@nosetrels \@packinfo{No setrels loading.} \else \@packinfo{setrels loading.} \WithSuffix\def\subset*{\subset^{*}} \WithSuffix\def\supset*{\supset^{*}} \WithSuffix\def\subseteq*{\subseteq^{*}} \WithSuffix\def\supseteq*{\supseteq^{*}} % \WithSuffix\def\subsetneq*{\subsetneq^{*}} % \WithSuffix\def\supsetneq*{\supsetneq^{*}} \providecommand*{\nsupset}{\not\supset} \providecommand*{\nsubset}{\not\subset} \newcommand*{\eq}{=} \WithSuffix\def\eq*{=^{*}} \newcommand*{\eqae}{=^{*}} \newcommand*{\infsubset}{\subset_{\infty}} \WithSuffix\def\infsubset*{\infsubset^{*}} \let\subsetnaeq=\infsubset \newcommand*{\infsupset}{\supset_{\infty}} \WithSuffix\def\infsupset*{\infsupset^{*}} \let\supsetnaeq=\infsupset \newcommand*{\majsubset}{\subset_{m}} \newcommand*{\majsupset}{\supset_{m}} \fi } %%% ORdinal notations % constructive/recursive ordinals \AtEndOfPackage{ \ifrecthy@noordinalnotations \@packinfo{No ordinal notations loading.} \else \@packinfo{ordinal notations loading.} \newcommand*{\wck}{\ensuremath{\omega^{ck}_1}} % \newcommand*{\ordtype}[1]{\ensuremath{\@recthy@abs{#1}_{\kleeneO}}} \newcommand*{\ordzero}{\ensuremath{0}} \newcommand*{\kleeneO}{\let\@PMG@parenarg\@PMG@undefined\let\@PMG@braketarg\@PMG@undefined\def\kleeneOSYM{\mathcal{O}}\@kleeneObody} \newcommand*{\@kleeneObody}{\kleeneOSYM\ifdefined\@PMG@parenarg ^{\@PMG@parenarg}% \fi% \ifdefined\@PMG@braketarg _{\abs{\@PMG@braketarg}}% \fi} \WithSuffix\def\@kleeneObody(#1){\def\@PMG@parenarg{#1}\@kleeneObody} \WithSuffix\def\@kleeneObody[#1]{\def\@PMG@braketarg{#1}\@kleeneObody} \WithSuffix\def\@kleeneObody*{\def\kleeneOSYM{\overline{\mathcal{O}}}\@kleeneObody} \newcommand*{\kleeneOuniq}{\kleeneO*} \let\ordNotations=\kleeneO \let\uniqOrdNotations=\kleeneOuniq \newcommand*{\kleeneless}[1][]{<_{\kleeneO\ifthenelse{\isempty{#1}}{}{,#1}}} \let\kleenel=\kleeneless \newcommand*{\kleeneleq}[1][]{\leq_{\kleeneO\ifthenelse{\isempty{#1}}{}{,#1}}} \newcommand*{\kleenenless}[1][]{\nless_{\kleeneO\ifthenelse{\isempty{#1}}{}{,#1}}} \let\kleenenl=\kleenenless \newcommand*{\kleenenleq}[1][]{\nleq_{\kleeneO\ifthenelse{\isempty{#1}}{}{,#1}}} \newcommand*{\kleenegtr}[1][]{>_{\kleeneO\ifthenelse{\isempty{#1}}{}{,#1}}} \let\kleeneg=\kleenegtr \newcommand*{\kleenegeq}[1][]{\geq_{\kleeneO\ifthenelse{\isempty{#1}}{}{,#1}}} \newcommand*{\kleenengtr}[1][]{\ngtr_{\kleeneO\ifthenelse{\isempty{#1}}{}{,#1}}} \let\kleeneng=\kleenengtr \newcommand*{\kleenengeq}[1][]{\ngeq_{\kleeneO\ifthenelse{\isempty{#1}}{}{,#1}}} \newcommand*{\kleenePlus}{\mathbin{+_{\kleeneO}}} \newcommand*{\kleeneMul}{\cdot_{\kleeneO}} \newcommand*{\kleenehgt}[1]{\lVert #1\rVert_{\kleeneO}} \let\hgtO=\kleenehgt \newcommand*{\kleenelim}[2]{{#1}_{[#2]}} \newcommand*{\kleenepred}[1]{{#1}^{-}} \fi } % \newcommand*{\axiom}{2}{#1} %alpha REA stuff % \newcommand*{\alphaREA}[1][\alpha]{\ensuremath{ #1 \text{-REA}} } \newcommand*{\@PMG@VdashStar}{{\Vdash\smash{\mkern-12mu\raise4\p@\hbox{\text{\tiny*}}}\mkern2mu}} % Forcing \AtEndOfPackage{ \ifrecthy@noforcing \@packinfo{No forcing loading.} \else \@packinfo{forcing loading.} \newcommand*{\forces}{\def\@PMG@frcparenarg{}\def\@PMG@frcbraketarg{}\def\@PMG@curfrcsym{\Vdash}\@forcesBody} \newcommand*{\@forcesBody}{\mathrel{{\@PMG@curfrcsym}^{\@PMG@frcparenarg}_{\@PMG@frcbraketarg}}} \WithSuffix\def\@forcesBody(#1){\def\@PMG@frcparenarg{#1}\@forcesBody} \WithSuffix\def\@forcesBody[#1]{\def\@PMG@frcbraketarg{#1}\@forcesBody} \WithSuffix\def\@forcesBody*{\let\@PMG@curfrcsym=\@PMG@VdashStar\@forcesBody} % \long\def\forces@[#1][#2]{\Vdash^{#2}_{#1}} % \newcommand{\forces}{\optparams{\forces@}{[][]}% % } \newcommand*{\frc}{\forces} \fi } %%%%%%%%% Formula Classes \newcommand*{\@recthy@CDelta}{\vphantom{\Delta}^{\mathcal{C}}\Delta} \newcommand*{\@recthy@CSigma}{\vphantom{\Sigma}^{\mathcal{C}}\Sigma} \newcommand*{\@recthy@CPi}{\vphantom{\Pi}^{\mathcal{C}}\Pi} \newcommand*{\@PMG@syntax}{\@PMG@synsym } \DeclareOptionX<\@recthy@family>{cdeltasym}{\def\@recthy@CDelta{#1}} \DeclareOptionX<\@recthy@family>{csigmasym}{\def\@recthy@CSigma{#1}} \DeclareOptionX<\@recthy@family>{cpisym}{\def\@recthy@CPi{#1}} \AtEndOfPackage{ \ifrecthy@nosyntax \@packinfo{No syntax loading.} \else \@packinfo{syntax loading.} \recthy@strip{CDelta} \recthy@strip{CSigma} \recthy@strip{CPi} \newcommand*{\deltan}[1]{\Delta_{#1}} \WithSuffix\def\deltan[#1]#2{\Delta^{#1}_{#2}} % \WithSuffix\def\deltan(#1)#2{\deltan{#2}\!\left(#1\right)} \newcommand*{\deltaZeroN}[1]{\Delta^{0}_{#1}} \WithSuffix\def\deltaZeroN[#1]#2{\Delta^{0,#1}_{#2}} % \WithSuffix\def\deltaZeroN(#1)#2{\deltaZeroN{#2}\!\left(#1\right)} \newcommand*{\deltaZeroTwo}{\deltaZeroN{2}} \WithSuffix\def\deltaZeroTwo[#1]{\deltaZeroN[#1]{2}} % \WithSuffix\def\deltaZeroTwo(#1){\deltaZeroN[#1]{2}} \newcommand*{\deltaZeroThree}{\deltaZeroN{3}} \WithSuffix\def\deltaZeroThree[#1]{\deltaZeroN[#1]{3}} % \WithSuffix\def\deltaZeroThree(#1){\deltaZeroN[#1]{3}} \newcommand*{\deltaZeroOne}{\deltaZeroN{1}} \WithSuffix\def\deltaZeroOne[#1]{\deltaZeroN[#1]{1}} % \WithSuffix\def\deltaZeroOne(#1){\deltaZeroN[#1]{1}} \newcommand*{\deltaZeroZero}{\deltaZeroN{0}} \WithSuffix\def\deltaZeroZero[#1]{\deltaZeroN[#1]{0}} % \WithSuffix\def\deltaZeroZero(#1){\deltaZeroN[#1]{0}} \newcommand*{\deltaOneN}[1]{\deltan{#1}^1} \WithSuffix\def\deltaOneN[#1]#2{\Delta^{1,#1}_{#2}} % \WithSuffix\def\deltaOneN(#1)#2{\deltaOneN{#2}\!\left(#1\right)} \newcommand*{\deltaOneOne}{\deltaOneN{1}} \WithSuffix\def\deltaOneOne[#1]{\deltaOneN[#1]{1}} % \WithSuffix\def\deltaOneOne(#1){\deltaOneN[#1]{1}} \newcommand*{\deltaOneTwo}{\deltaOneN{2}} \WithSuffix\def\deltaOneTwo[#1]{\deltaOneN[#1]{2}} % \WithSuffix\def\deltaOneTwo(#1){\deltaOneN[#1]{2}} \newcommand*{\deltaOneThree}{\deltaOneN{3}} \WithSuffix\def\deltaOneThree[#1]{\deltaOneN[#1]{3}} % \WithSuffix\def\deltaOneThree(#1){\deltaOneN[#1]{3}} \newcommand*{\sigman}[1]{\Sigma_{#1}} \WithSuffix\def\sigman[#1]#2{\Sigma^{#1}_{#2}} % \WithSuffix\def\sigman(#1)#2{\sigman{#2}\!\left(#1\right)} \newcommand*{\sigmaZeroN}[1]{\Sigma^{0}_{#1}} \WithSuffix\def\sigmaZeroN[#1]#2{\Sigma^{0,#1}_{#2}} % \WithSuffix\def\sigmaZeroN(#1)#2{\sigmaZeroN{#2}\!\left(#1\right)} \newcommand*{\sigmaZeroOne}{\sigmaZeroN{1}} \WithSuffix\def\sigmaZeroOne[#1]{\sigmaZeroN[#1]{1}} % \WithSuffix\def\sigmaZeroOne(#1){\sigmaZeroN[#1]{1}} \newcommand*{\sigmaZeroTwo}{\sigmaZeroN{2}} \WithSuffix\def\sigmaZeroTwo[#1]{\sigmaZeroN[#1]{2}} % \WithSuffix\def\sigmaZeroTwo(#1){\sigmaZeroN[#1]{2}} \newcommand*{\sigmaZeroThree}{\sigmaZeroN{3}} \WithSuffix\def\sigmaZeroThree[#1]{\sigmaZeroN[#1]{3}} % \WithSuffix\def\sigmaZeroThree(#1){\sigmaZeroN[#1]{3}} \newcommand*{\sigmaOneN}[1]{\Sigma^1_{#1}} \WithSuffix\def\sigmaOneN[#1]#2{\Sigma^{1,#1}_{#2}} % \WithSuffix\def\sigmaOneN(#1)#2{\sigmaOneN{#2}\!\left(#1\right)} \newcommand*{\sigmaOneOne}{\sigmaOneN{1}} \WithSuffix\def\sigmaOneOne[#1]{\sigmaOneN[#1]{1}} % \WithSuffix\def\sigmaOneOne(#1){\sigmaOneN[#1]{1}} \newcommand*{\sigmaOneTwo}{\sigmaOneN{2}} \WithSuffix\def\sigmaOneTwo[#1]{\sigmaOneN[#1]{2}} % \WithSuffix\def\sigmaOneTwo(#1){\sigmaOneN[#1]{2}} \newcommand*{\sigmaOneThree}{\sigmaOneN{3}} \WithSuffix\def\sigmaOneThree[#1]{\sigmaOneN[#1]{3}} % \WithSuffix\def\sigmaOneThree(#1){\sigmaOneN[#1]{3}} \newcommand*{\pin}[1]{\Pi_{#1}} \WithSuffix\def\pin[#1]#2{\Pi^{#1}_{#2}} % \WithSuffix\def\pin(#1)#2{\pin{#2}\!\left(#1\right)} \newcommand*{\piZeroN}[1]{\Pi^0_{#1}} \WithSuffix\def\piZeroN[#1]#2{\Pi^{0,#1}_{#2}} % \WithSuffix\def\piZeroN(#1)#2{\piZeroN{#2}\left(#1\right)} \newcommand*{\piZeroOne}{\piZeroN{1}} \WithSuffix\def\piZeroOne[#1]{\piZeroN[#1]{1}} % \WithSuffix\def\piZeroOne(#1){\piZeroN[#1]{1}} \newcommand*{\piZeroTwo}{\piZeroN{2}} \WithSuffix\def\piZeroTwo[#1]{\piZeroN[#1]{2}} % \WithSuffix\def\piZeroTwo(#1){\piZeroN[#1]{2}} \newcommand*{\piZeroThree}{\piZeroN{3}} \WithSuffix\def\piZeroThree[#1]{\piZeroN[#1]{3}} % \WithSuffix\def\piZeroThree(#1){\piZeroN[#1]{3}} \newcommand*{\piOneN}[1]{\Pi^1_{#1}} \WithSuffix\def\piOneN[#1]#2{\Pi^{1,#1}_{#2}} % \WithSuffix\def\piOneN(#1)#2{\piOneN{#2}\!\left(#1\right)} \newcommand*{\piOneOne}{\piOneN{1}} \WithSuffix\def\piOneOne[#1]{\piOneN[#1]{1}} % \WithSuffix\def\piOneOne(#1){\piOneN[#1]{1}} \newcommand*{\piOneTwo}{\piOneN{2}} \WithSuffix\def\piOneTwo[#1]{\piOneN[#1]{2}} % \WithSuffix\def\piOneTwo(#1){\piOneN[#1]{2}} \newcommand*{\piOneThree}{\piOneN{3}} \WithSuffix\def\piOneThree[#1]{\piOneN[#1]{3}} % \WithSuffix\def\piOneThree(#1){\piOneN[#1]{3}} %%%ABREVIATIONS \newcommand*{\deltaz}{\deltaN{0}} \newcommand*{\deltazn}{\deltaZeroN} \newcommand*{\deltazii}{\deltaZeroTwo} \newcommand*{\deltazi}{\deltaZeroOne} \newcommand*{\deltazz}{\deltaZeroZero} \newcommand*{\deltaziii}{\deltaZeroThree} \newcommand*{\deltaIn}{\deltaOneN} \newcommand*{\deltaIi}{\deltaOneOne} \newcommand*{\deltaoneone}{\deltaOneOne} \newcommand*{\deltaIii}{\deltaOneTwo} \newcommand*{\deltaIiii}{\deltaOneThree} \newcommand*{\sigmazn}{\sigmaZeroN} \newcommand*{\sigmazi}{\sigmaZeroOne} \newcommand*{\sigmazii}{\sigmaZeroTwo} \newcommand*{\sigmaziii}{\sigmaZeroThree} \newcommand*{\sigmaIn}{\sigmaOneN} \newcommand*{\sigmaIi}{\sigmaOneOne} \newcommand*{\sigmaIii}{\sigmaOneTwo} \newcommand*{\sigmaIiii}{\sigmaOneThree} \newcommand*{\pizn}{\piZeroN} \newcommand*{\pizi}{\piZeroOne} \newcommand*{\pizii}{\piZeroTwo} \newcommand*{\piziii}{\piZeroThree} \newcommand*{\piIn}{\piOneN} \newcommand*{\piIi}{\piOneOne} \newcommand*{\piIii}{\piOneTwo} \newcommand*{\piIiii}{\piOneThree} %%%%%%%%%%%%% % \newcommand*{\sigmapizn}[1]{\sigmazn{#1} \union \pizn{#1}} % \WithSuffix\def\sigmapizn[#1]#2{\sigmazn[#1]{#2} \union \pizn[#1]{#2}} % \WithSuffix\def\sigmapizn(#1)#2{\sigmazn[#1]{#2} \union \pizn[#1]{#2}} % \let\pisigmazn=\sigmapizn %%%%%%%%%%%%%% %%%%%%%% % Logics %\newcommand*{\equiv} \newcommand*{\logic}[2]{\mathcal{L}_{#1,#2}} % \newcommand*{\logicwiw}{\logic{\omega_{1}}{\omega}} % \newcommand*{\Clogic}[2]{\mathcal{CL}_{#1,#2}} % \newcommand*{\FOlogic}{\logic{\omega}{\omega}} % \newcommand*{\hyplogic}[1][]{\Clogic{\omega_{1}}{\omega}\ifthenelse{\isempty{#1}}{}{\left(#1\right)} } %May never use...specific notation for computable formulas...supplanted by \( \sigmazn{\alpha} \) notation. %%%%%%%%%%% \newcommand*{\Cdeltan}[1]{\CDelta_{#1}} \WithSuffix\def\Cdeltan[#1]#2{\Cdeltan{#2}^{#1}} % \WithSuffix\def\Cdeltan(#1)#2{\Cdeltan{#2}\left(#1\right)} \newcommand*{\Cpin}[1]{\CPi_{#1}} \WithSuffix\def\Cpin[#1]#2{\Cpin{#2}^{#1}} % \WithSuffix\def\Cpin(#1)#2{\Cpin{#2}\!\left(#1\right)} \newcommand*{\Csigman}[1]{\CSigma_{#1}} \WithSuffix\def\Csigman[#1]#2{\Csigman{#2}^{#1}} % \WithSuffix\def\Csigman(#1)#2{\Csigman{#2}\!\left(#1\right)} %%%BOLDFACE \newcommand*{\Deltan}[1]{\utilde{\mathbf{\Delta}}_{#1}} \WithSuffix\def\Deltan[#1]#2{\utilde{\mathbf{\Delta}}^{#1}_{#2}} % \WithSuffix\def\Deltan(#1)#2{\deltan{#2}\!\left(#1\right)} \newcommand*{\DeltaZeroN}[1]{\utilde{\mathbf{\Delta}}^{0}_{#1}} \WithSuffix\def\DeltaZeroN[#1]#2{\utilde{\mathbf{\Delta}}^{0,#1}_{#2}} % \WithSuffix\def\DeltaZeroN(#1)#2{\DeltaZeroN{#2}\!\left(#1\right)} \newcommand*{\DeltaZeroTwo}{\DeltaZeroN{2}} \WithSuffix\def\DeltaZeroTwo[#1]{\DeltaZeroN[#1]{2}} % \WithSuffix\def\DeltaZeroTwo(#1){\DeltaZeroN[#1]{2}} \newcommand*{\DeltaZeroThree}{\DeltaZeroN{3}} \WithSuffix\def\DeltaZeroThree[#1]{\DeltaZeroN[#1]{3}} % \WithSuffix\def\DeltaZeroThree(#1){\DeltaZeroN[#1]{3}} \newcommand*{\DeltaZeroOne}{\DeltaZeroN{1}} \WithSuffix\def\DeltaZeroOne[#1]{\DeltaZeroN[#1]{1}} % \WithSuffix\def\DeltaZeroOne(#1){\DeltaZeroN[#1]{1}} \newcommand*{\DeltaZeroZero}{\DeltaZeroN{0}} \WithSuffix\def\DeltaZeroZero[#1]{\DeltaZeroN[#1]{0}} % \WithSuffix\def\DeltaZeroZero(#1){\DeltaZeroN[#1]{0}} \newcommand*{\DeltaOneN}[1]{\Deltan{#1}^1} \WithSuffix\def\DeltaOneN[#1]#2{\utilde{\mathbf{\Delta}}^{1,#1}_{#2}} % \WithSuffix\def\DeltaOneN(#1)#2{\DeltaOneN{#2}\!\left(#1\right)} \newcommand*{\DeltaOneOne}{\DeltaOneN{1}} \WithSuffix\def\DeltaOneOne[#1]{\DeltaOneN[#1]{1}} % \WithSuffix\def\DeltaOneOne(#1){\DeltaOneN[#1]{1}} \newcommand*{\DeltaOneTwo}{\DeltaOneN{2}} \WithSuffix\def\DeltaOneTwo[#1]{\DeltaOneN[#1]{2}} % \WithSuffix\def\DeltaOneTwo(#1){\DeltaOneN[#1]{2}} \newcommand*{\DeltaOneThree}{\DeltaOneN{3}} \WithSuffix\def\DeltaOneThree[#1]{\DeltaOneN[#1]{3}} % \WithSuffix\def\DeltaOneThree(#1){\DeltaOneN[#1]{3}} \newcommand*{\Sigman}[1]{\utilde{\mathbf{\Sigma}}_{#1}} \WithSuffix\def\Sigman[#1]#2{\utilde{\mathbf{\Sigma}}^{#1}_{#2}} % \WithSuffix\def\Sigman(#1)#2{\sigman{#2}\!\left(#1\right)} \newcommand*{\SigmaZeroN}[1]{\utilde{\mathbf{\Sigma}}^{0}_{#1}} \WithSuffix\def\SigmaZeroN[#1]#2{\utilde{\mathbf{\Sigma}}^{0,#1}_{#2}} % \WithSuffix\def\SigmaZeroN(#1)#2{\SigmaZeroN{#2}\!\left(#1\right)} \newcommand*{\SigmaZeroOne}{\SigmaZeroN{1}} \WithSuffix\def\SigmaZeroOne[#1]{\SigmaZeroN[#1]{1}} % \WithSuffix\def\SigmaZeroOne(#1){\SigmaZeroN[#1]{1}} \newcommand*{\SigmaZeroTwo}{\SigmaZeroN{2}} \WithSuffix\def\SigmaZeroTwo[#1]{\SigmaZeroN[#1]{2}} % \WithSuffix\def\SigmaZeroTwo(#1){\SigmaZeroN[#1]{2}} \newcommand*{\SigmaZeroThree}{\SigmaZeroN{3}} \WithSuffix\def\SigmaZeroThree[#1]{\SigmaZeroN[#1]{3}} % \WithSuffix\def\SigmaZeroThree(#1){\SigmaZeroN[#1]{3}} \newcommand*{\SigmaOneN}[1]{\utilde{\mathbf{\Sigma}}^1_{#1}} \WithSuffix\def\SigmaOneN[#1]#2{\utilde{\mathbf{\Sigma}}^{1,#1}_{#2}} % \WithSuffix\def\SigmaOneN(#1)#2{\SigmaOneN{#2}\!\left(#1\right)} \newcommand*{\SigmaOneOne}{\SigmaOneN{1}} \WithSuffix\def\SigmaOneOne[#1]{\SigmaOneN[#1]{1}} % \WithSuffix\def\SigmaOneOne(#1){\SigmaOneN[#1]{1}} \newcommand*{\SigmaOneTwo}{\SigmaOneN{2}} \WithSuffix\def\SigmaOneTwo[#1]{\SigmaOneN[#1]{2}} % \WithSuffix\def\SigmaOneTwo(#1){\SigmaOneN[#1]{2}} \newcommand*{\SigmaOneThree}{\SigmaOneN{3}} \WithSuffix\def\SigmaOneThree[#1]{\SigmaOneN[#1]{3}} % \WithSuffix\def\SigmaOneThree(#1){\SigmaOneN[#1]{3}} \newcommand*{\Pin}[1]{\utilde{\mathbf{\Pi}}_{#1}} \WithSuffix\def\Pin[#1]#2{\utilde{\mathbf{\Pi}}^{#1}_{#2}} % \WithSuffix\def\Pin(#1)#2{\pin{#2}\!\left(#1\right)} \newcommand*{\PiZeroN}[1]{\utilde{\mathbf{\Pi}}^0_{#1}} \WithSuffix\def\PiZeroN[#1]#2{\utilde{\mathbf{\Pi}}^{0,#1}_{#2}} % \WithSuffix\def\PiZeroN(#1)#2{\PiZeroN{#2}\left(#1\right)} \newcommand*{\PiZeroOne}{\PiZeroN{1}} \WithSuffix\def\PiZeroOne[#1]{\PiZeroN[#1]{1}} % \WithSuffix\def\PiZeroOne(#1){\PiZeroN[#1]{1}} \newcommand*{\PiZeroTwo}{\PiZeroN{2}} \WithSuffix\def\PiZeroTwo[#1]{\PiZeroN[#1]{2}} % \WithSuffix\def\PiZeroTwo(#1){\PiZeroN[#1]{2}} \newcommand*{\PiZeroThree}{\PiZeroN{3}} \WithSuffix\def\PiZeroThree[#1]{\PiZeroN[#1]{3}} % \WithSuffix\def\PiZeroThree(#1){\PiZeroN[#1]{3}} \newcommand*{\PiOneN}[1]{\utilde{\mathbf{\Pi}}^1_{#1}} \WithSuffix\def\PiOneN[#1]#2{\utilde{\mathbf{\Pi}}^{1,#1}_{#2}} % \WithSuffix\def\PiOneN(#1)#2{\PiOneN{#2}\!\left(#1\right)} \newcommand*{\PiOneOne}{\PiOneN{1}} \WithSuffix\def\PiOneOne[#1]{\PiOneN[#1]{1}} % \WithSuffix\def\PiOneOne(#1){\PiOneN[#1]{1}} \newcommand*{\PiOneTwo}{\PiOneN{2}} \WithSuffix\def\PiOneTwo[#1]{\PiOneN[#1]{2}} % \WithSuffix\def\PiOneTwo(#1){\PiOneN[#1]{2}} \newcommand*{\PiOneThree}{\PiOneN{3}} \WithSuffix\def\PiOneThree[#1]{\PiOneN[#1]{3}} % \WithSuffix\def\PiOneThree(#1){\PiOneN[#1]{3}} %%%ABREVIATIONS \newcommand*{\Deltaz}{\deltaN{0}} \newcommand*{\Deltazn}{\DeltaZeroN} \newcommand*{\Deltazii}{\DeltaZeroTwo} \newcommand*{\Deltazi}{\DeltaZeroOne} \newcommand*{\Deltazz}{\DeltaZeroZero} \newcommand*{\Deltaziii}{\DeltaZeroThree} \newcommand*{\DeltaIn}{\DeltaOneN} \newcommand*{\DeltaIi}{\DeltaOneOne} \newcommand*{\Deltaoneone}{\DeltaOneOne} \newcommand*{\DeltaIii}{\DeltaOneTwo} \newcommand*{\DeltaIiii}{\DeltaOneThree} \newcommand*{\Sigmazn}{\SigmaZeroN} \newcommand*{\Sigmazi}{\SigmaZeroOne} \newcommand*{\Sigmazii}{\SigmaZeroTwo} \newcommand*{\Sigmaziii}{\SigmaZeroThree} \newcommand*{\SigmaIn}{\SigmaOneN} \newcommand*{\SigmaIi}{\SigmaOneOne} \newcommand*{\SigmaIii}{\SigmaOneTwo} \newcommand*{\SigmaIiii}{\SigmaOneThree} \newcommand*{\Pizn}{\PiZeroN} \newcommand*{\Pizi}{\PiZeroOne} \newcommand*{\Pizii}{\PiZeroTwo} \newcommand*{\Piziii}{\PiZeroThree} \newcommand*{\PiIn}{\PiOneN} \newcommand*{\PiIi}{\PiOneOne} \newcommand*{\PiIii}{\PiOneTwo} \newcommand*{\PiIiii}{\PiOneThree} \fi } %%%ABREVIATIONS %Degree Classes \AtEndOfPackage{ \ifrecthy@nodegclasses \@packinfo{No degree classes loading.} \else \@packinfo{Degree classes loading.} \newcommand{\Low}[1][]{\ensuremath{ \text{Low}^{#1} }} \newcommand{\LowN}[2][]{\ensuremath{ \text{Low}^{#1}_{#2} }} \newcommand{\lowN}[2][]{\ensuremath{ \mathcal{L}^{#1}_{#2} }} \newcommand{\req}[2]{\ensuremath{\mathcal{#1}_{#2}}} \fi } \ProcessOptionsX<\@recthy@family>