%Package Identification \NeedsTeXFormat{LaTeX2e} \ProvidesPackage{principia}[2020/10/25 principia package version 1.1] %This is the principia package is for representing notations in Whitehead and Russell's ``Principia Mathematica" close to their appearance in the original. %Version 1.0 (superseded by Version 1.1): Covers typesetting of notation through Volume I. 2020/10/24 %Version 1.1 (updates): fixed the spacing of scope dots around parentheses; fixed spacing of theorem sign; fixed spacing around primitive proposition and definition signs. 2020/10/25 %Licensed under LaTeX Project Public License 1.3c. %Copyright Landon D. C. Elkind, 2020. (https://landondcelkind.com/contact/). %Principia package requirements \RequirePackage{amssymb} %This loads the relation domain and converse domain limitation symbols. \RequirePackage{amsmath} %This loads the circumflex, substitution into theorems, \text{}, \mathbf{}, \boldsymbol{}, \overleftarrow{}, \overrightarrow{}, etc. \RequirePackage{graphicx} %This loads commands that flip iota for definite descriptions, Lambda for the universal class, and so on. The (superseded) graphics package should also work here, but is not recommended. \RequirePackage{marvosym} %This loads the male and female symbol. \RequirePackage{pifont} %This loads the symbols such as the eight-pointed asterisk. %Meta-logical symbols \newcommand{\pmfd}{\begin{center} \rule{5cm}{.5pt} \end{center}} %Dividing line between introductory remarks in a starred number and the formal deductions. \newcommand{\pmdem}{\textit{Dem}.} %This notation begins a proof. \newcommand{\pmdemi}{\indent \pmdem} %This idents the notation that begins a proof. \newcommand{\pmhp}{\text{Hp}} %This typesets Hp (short for antecedent), which occurs at the beginning of a proof. \newcommand{\pmprop}{\text{Prop}} %This occurs at the end of a proof. \newcommand{\pmithm}{\pmimp\;\pmthm} %This occurs when a meta-theoretic implication is asserted. \newcommand{\pmbr}[1]{\bigg \lbrack \normalsize #1 \bigg \rbrack} %These are larger brackets for substitution. \newcommand{\pmsub}[2]{\bigg \lbrack \small \begin{array}{c} #1 \\ \hline #2 \end{array} \bigg \rbrack} %This is the substitution command. \newcommand{\pmsubb}[4]{\bigg \lbrack \small \begin{array}{c c} #1, & #3 \\ \hline #2, & #4 \end{array} \bigg \rbrack} %This is the substitution command. \newcommand{\pmsubbb}[6]{\bigg \lbrack \small \begin{array}{c c c} #1, & #3, & #5 \\ \hline #2, & #4, & #6 \end{array} \bigg \rbrack} %This is the substitution command. \newcommand{\pmsubbbb}[8]{\bigg \lbrack \small \begin{array}{c c c c} #1, & #3, & #5, & #7 \\ \hline #2, & #4, & #6, & #8 \end{array} \bigg \rbrack} %This is the substitution command. \newcommand{\pmSub}[3]{\bigg \lbrack \normalsize #1 \text{ } \small \begin{array}{c} #2 \\ \hline #3 \end{array} \bigg \rbrack} %This is the substitution command. \newcommand{\pmSubb}[5]{\bigg \lbrack \normalsize #1 \text{ } \small \begin{array}{c c} #2, & #4 \\ \hline #3, & #5 \end{array} \bigg \rbrack} %This is the substitution command. \newcommand{\pmSubbb}[7]{\bigg \lbrack \normalsize #1 \text{ } \small \begin{array}{c c c} #2, & #4, & #6 \\ \hline #3, & #5, & #7 \end{array} \bigg \rbrack} %This is the substitution command. \newcommand{\pmSubbbb}[9]{\bigg \lbrack \normalsize #1 \text{ } \small \begin{array}{c c c c} #1, & #3, & #5, & #7 \\ \hline #2, & #4, & #6, & #8 \end{array} \bigg \rbrack} %This is the substitution command. \newcommand{\pmsUb}[2]{\small \begin{array}{c} #1 \\ \hline #2 \end{array}} %This is the substitution command. \newcommand{\pmsUbb}[4]{\small \begin{array}{c c} #1, & #3 \\ \hline #2, & #4 \end{array}} %This is the substitution command. \newcommand{\pmsUbbb}[6]{\small \begin{array}{c c c} #1, & #3, & #5 \\ \hline #2, & #4, & #6 \end{array}} %This is the substitution command. \newcommand{\pmsUbbbb}[8]{\small \begin{array}{c c c c} #1, & #3, & #5, & #7 \\ \hline #2, & #4, & #6, & #8 \end{array}} %This is the substitution command. \newcommand{\pmSUb}[3]{\normalsize #1 \text{ } \small \begin{array}{c} #2 \\ \hline #3 \end{array}} %This is the substitution command. \newcommand{\pmSUbb}[5]{\normalsize #1 \text{ } \small \begin{array}{c c} #2, & #4 \\ \hline #3, & #5 \end{array}} %This is the substitution command. \newcommand{\pmSUbbb}[7]{\normalsize #1 \text{ } \small \begin{array}{c c c} #2, & #4, & #6 \\ \hline #3, & #5, & #7 \end{array}} %This is the substitution command. \newcommand{\pmSUbbbb}[9]{\normalsize #1 \text{ } \small \begin{array}{c c c c} #1, & #3, & #5, & #7 \\ \hline #2, & #4, & #6, & #8 \end{array}} %This is the substitution command. \newcommand{\pmthm}{\mathpunct{\text{\scalebox{.5}[1]{$\boldsymbol\vdash$}}}} %This is the theorem sign. \newcommand{\pmast}{\text{\resizebox{!}{.75\height}{\ding{107}}}} %This is the sign introducing a theorem number. \newcommand{\pmcdot}{\text{\raisebox{.05cm}{$\boldsymbol\cdot$}}} %This is a sign introducing a theorem sub-number. \newcommand{\pmiddf}{\mathbin{=}} \newcommand{\pmdf}{\quad \text{Df}} \newcommand{\pmpp}{\quad \text{Pp}} %Square dots for scope, defined for up to six dots \newcommand{\pmdot}{\mathrel{\hbox{\rule{.3ex}{.3ex}}}} \newcommand{\pmdott}{\mathrel{\overset{\pmdot}{\pmdot}}} \newcommand{\pmdottt}{\pmdott\hspace{.1em}\pmdot} \newcommand{\pmdotttt}{\pmdott\hspace{.1em}\pmdott} \newcommand{\pmdottttt}{\pmdott\hspace{.1em}\pmdott\hspace{.1em}\pmdot} \newcommand{\pmdotttttt}{\pmdott\hspace{.1em}\pmdott\hspace{.1em}\pmdott} %Logical connectives \newcommand{\pmnot}{\mathord{\ooalign{$\boldsymbol{\sim}\mkern.5mu$\hidewidth\cr$\boldsymbol{\sim}$\cr\hidewidth$\mkern.5mu\boldsymbol{\sim}$}}} \newcommand{\pmimp}{\mathbin{\ooalign{$\boldsymbol{\supset}\mkern.5mu$\hidewidth\cr$\boldsymbol{\supset}$\cr\hidewidth$\mkern.5mu\boldsymbol{\supset}$}}} \newcommand{\pmiff}{\mathbin{\ooalign{$\boldsymbol{\equiv}\mkern.5mu$\hidewidth\cr$\boldsymbol{\equiv}$\cr\hidewidth$\mkern.5mu\boldsymbol{\equiv}$}}} \newcommand{\pmor}{\mathbin{\ooalign{$\boldsymbol{\vee}\mkern.5mu$\hidewidth\cr$\boldsymbol{\vee}$\cr\hidewidth$\mkern.5mu\boldsymbol{\vee}$}}} \newcommand{\pmprod}{\mathbin{\ooalign{$\boldsymbol{\wedge}\mkern.5mu$\hidewidth\cr$\boldsymbol{\wedge}$\cr\hidewidth$\mkern.5mu\boldsymbol{\wedge}$}}} \newcommand{\pmall}[1]{(#1)} \newcommand{\pmsome}[1]{(\text{\raisebox{.5em}{\rotatebox{180}{\textbf{E}}}}#1)} \newcommand{\pmSome}{\text{\raisebox{.5em}{\rotatebox{180}{\textbf{E}}}}} \newcommand{\pmand}{\mathrel{\hbox{\rule{.3ex}{.3ex}}}} \newcommand{\pmandd}{\overset{\pmand}{\pmand}} \newcommand{\pmanddd}{\pmandd\hspace{.1em}\pmand} \newcommand{\pmandddd}{\pmandd\hspace{.1em}\pmandd} \newcommand{\pmanddddd}{\pmandd\hspace{.1em}\pmandd\hspace{.1em}\pmand} \newcommand{\pmandddddd}{\pmandd\hspace{.1em}\pmandd\hspace{.1em}\pmandd} %Additional defined logic signs \newcommand{\pmhat}[1]{\mathbf{\hat{\text{$#1$}}}} \newcommand{\pmpf}[2]{#1#2} %for propositional functions of one variable \newcommand{\pmpff}[3]{#1(#2, #3)} %for propositional functions of two variables \newcommand{\pmpfff}[4]{#1(#2, #3, #4)} %for propositional functions of three variables \newcommand{\pmshr}{\textbf{!}} %*12.1 and *12.11, used for predicative propositional functions \newcommand{\pmpred}[2]{#1\pmshr#2} %for predicates (``predicative functions'') of one variable \newcommand{\pmpredd}[3]{#1\pmshr(#2, #3)} %for predicates (``predicative functions'') of two variables \newcommand{\pmpreddd}[4]{#1\pmshr(#2, #3, #4)} %for predicates (``predicative functions'') of three variables \newcommand{\pmnid}{\mathrel{\ooalign{$=$\cr\hidewidth\footnotesize\rotatebox[origin=c]{210}{\textbf{/}}\hidewidth\cr}}} %*13.01 \newcommand{\pmiota}{\ooalign{\rotatebox[origin=c]{180}{$\mathbf{\iota}$}\cr\hidewidth\raisebox{.0125em}{\rotatebox[origin=c]{180}{$\mathbf{\iota}$}}\cr\hidewidth\raisebox{.025em}{\rotatebox[origin=c]{180}{$\mathbf{\iota}$}}\cr\hidewidth\raisebox{.0375em}{\rotatebox[origin=c]{180}{$\mathbf{\iota}$}}\cr\hidewidth\raisebox{.05em}{\rotatebox[origin=c]{180}{$\mathbf{\iota}$}}}} %the rotated Greek iota used in definite descriptions \newcommand{\pmdsc}[1]{(\pmiota#1)} %*14.01 \newcommand{\pmDsc}{\pmiota} %*14.01 \newcommand{\pmexists}{\textbf{E}\hspace{.1em}\pmshr} %*14.02 %Class signs \newcommand{\pmcuni}{\text{\rotatebox[origin=c]{180}{$\Lambda$}}} \newcommand{\pmcnull}{\Lambda} \newcommand{\pmcls}[2]{\pmhat{#1}(#2)} \newcommand{\pmCls}{\text{Cls}} \newcommand{\pmClsn}[1]{\text{Cls}^{#1}} \newcommand{\pmcexists}{\text{\raisebox{.5em}{\rotatebox{180}{\textbf{E}}}}\hspace{-.1em}\mathop{\pmshr}} \newcommand{\pmccmp}[1]{\boldsymbol{-}#1} \newcommand{\pmcmin}[2]{#1\boldsymbol{-}#2} \newcommand{\pmcin}{\mathop{\boldsymbol{\epsilon}}} \newcommand{\pmccup}{\mathop{\ooalign{\scalebox{1.3}[1.75]{$\put(3, 2.5){\oval(4,4)[b]}\phantom{\circ}$}\cr\hidewidth\hspace{.1em}\scalebox{1.3}[1.75]{$\put(3, 2.5){\oval(4,4)[b]}\phantom{\circ}$}\cr\hidewidth\hspace{.2em}\scalebox{1.3}[1.75]{$\put(3, 2.5){\oval(4,4)[b]}\phantom{\circ}$}\cr\hidewidth\hspace{.3em}\scalebox{1.3}[1.75]{$\put(3, 2.5){\oval(4,4)[b]}\phantom{\circ}$}\cr\hidewidth\hspace{.4em}\scalebox{1.3}[1.75]{$\put(3, 2.5){\oval(4,4)[b]}\phantom{\circ}$}\cr\hidewidth\hspace{.5em}\scalebox{1.3}[1.75]{$\put(3, 2.5){\oval(4,4)[b]}\phantom{\circ}$}\cr\hidewidth\hspace{.6em}\scalebox{1.3}[1.75]{$\put(3, 2.5){\oval(4,4)[b]}\phantom{\circ}$}}}} \newcommand{\pmccap}{\mathop{\ooalign{\scalebox{1.3}[1.75]{$\put(3, 2){\oval(4,1)[t]}\phantom{\circ}$}\cr\hidewidth\hspace{.1em}\scalebox{1.3}[1.75]{$\put(3, 2){\oval(4,1)[t]}\phantom{\circ}$}\cr\hidewidth\hspace{.2em}\scalebox{1.3}[1.75]{$\put(3, 2){\oval(4,1)[t]}\phantom{\circ}$}\cr\hidewidth\hspace{.3em}\scalebox{1.3}[1.75]{$\put(3, 2){\oval(4,1)[t]}\phantom{\circ}$}\cr\hidewidth\hspace{.4em}\scalebox{1.3}[1.75]{$\put(3, 2){\oval(4,1)[t]}\phantom{\circ}$}\cr\hidewidth\hspace{.5em}\scalebox{1.3}[1.75]{$\put(3, 2){\oval(4,1)[t]}\phantom{\circ}$}\cr\hidewidth\hspace{.6em}\scalebox{1.3}[1.75]{$\put(3, 2){\oval(4,1)[t]}\phantom{\circ}$}}}} \newcommand{\pmcinc}{\mathop{\ooalign{$\boldsymbol{\subset}$\cr\hidewidth$\hspace{.1em}\boldsymbol{\subset}$\cr\hidewidth$\hspace{.15em}\boldsymbol{\subset}$\cr\hidewidth$\hspace{.2em}\boldsymbol{\subset}$}}} %Relation signs \newcommand{\pmruni}{\dot{\text{\rotatebox[origin=c]{180}{$\Lambda$}}}} \newcommand{\pmrnull}{\dot{\Lambda}} \newcommand{\pmdscf}[2]{#1\textbf{`}#2} \newcommand{\pmdscff}[2]{#1\textbf{`}\textbf{`}#2} \newcommand{\pmdscfff}[2]{#1\textbf{`}\textbf{`}\textbf{`}#2} \newcommand{\pmdscfr}[2]{#1_{\pmcin}\textbf{`}#2} \newcommand{\pmdscfR}[1]{#1_{\pmcin}} \newcommand{\pmdscfe}[2]{\mathop{\text{E}}\mathop{\pmshr\pmshr}\pmdscff{#1}{#2}} \newcommand{\pmdm}[1]{\text{D}\textbf{`}#1} \newcommand{\pmDm}{\text{D}} \newcommand{\pmcdm}[1]{\text{\rotatebox[origin=c]{180}{D}}\textbf{`}#1} \newcommand{\pmCdm}{\text{\rotatebox[origin=c]{180}{D}}} \newcommand{\pmcmp}[1]{C\textbf{`}#1} \newcommand{\pmCmp}{C} \newcommand{\pmfld}[1]{F\textbf{`}#1} \newcommand{\pmFld}{F} \newcommand{\pmrel}[3]{\pmhat{#1}\pmhat{#2}#3} \newcommand{\pmrele}[5]{#1\{\pmhat{#2}\pmhat{#3}#4(#2, #3)\}#5} \newcommand{\pmrelep}[3]{#1\{#2\}#3} \newcommand{\pmrcmp}[1]{\ooalign{$\hidewidth\raisebox{.25em}{$\cdot$}\hidewidth$\cr$\mathbf{\pmccmp}$}#1} \newcommand{\pmrmin}[2]{#1\mathrel{\ooalign{$\hidewidth\raisebox{.25em}{$\cdot$}\hidewidth$\cr$\mathbf{\pmccmp}$}}#2} \newcommand{\pmrexists}{\dot{\mathop{\text{\raisebox{.5em}{\rotatebox{180}{E}}}}}\mathop{\pmshr}} \newcommand{\pmcrel}[1]{\breve{#1}} \newcommand{\pmCnv}{\text{Cnv}} \newcommand{\pmcnv}[1]{\breve{#1}} \newcommand{\pmcnvr}[1]{\text{Cnv}\textbf{`}#1} \newcommand{\pmrcup}{\mathrel{\ooalign{$\hidewidth\cdot\hidewidth$\cr$\mathbf{\pmccup}$}}} \newcommand{\pmrcap}{\mathrel{\ooalign{$\hidewidth\raisebox{.3em}{$\cdot$}\hidewidth$\cr$\mathbf{\pmccap}$}}} \newcommand{\pmrinc}{\mathrel{\ooalign{$\hidewidth\cdot\hidewidth$\cr$\mathbf{\pmcinc}$}}} \newcommand{\pmrrf}[2]{\overset{\boldsymbol{\rightarrow}}{#1\textbf{`}}#2} \newcommand{\pmRrf}[1]{\overset{\boldsymbol{\rightarrow}}{#1}} \newcommand{\pmrrl}[2]{\overset{\boldsymbol{\leftarrow}}{#1\textbf{`}}#2} \newcommand{\pmRrl}[1]{\overset{\boldsymbol{\leftarrow}}{#1}} \newcommand{\pmsg}[1]{\text{sg}\textbf{`}#1} \newcommand{\pmgs}[1]{\text{gs}\textbf{`}#1} \newcommand{\pmSg}{\text{sg}} \newcommand{\pmGs}{\text{gs}} \newcommand{\pmRprd}{\mathop{|}} \newcommand{\pmrprd}[2]{{#1}\mathop{|}{#2}} \newcommand{\pmrprdn}[2]{#1^{#2}} \newcommand{\pmrld}[2]{#1 \boldsymbol{\upharpoonleft} #2} \newcommand{\pmrlcd}[2]{#1 \boldsymbol{\upharpoonright} #2} \newcommand{\pmrlf}[3]{#1 \boldsymbol{\upharpoonleft} #2 \boldsymbol{\upharpoonright} #3} \newcommand{\pmrl}[2]{#1 \boldsymbol{\uparrow} #2} \newcommand{\pmrlF}[2]{#1 \mathbin{\ooalign{$\upharpoonright$\cr\hidewidth\rotatebox[origin=c]{180}{\text{$\upharpoonleft$}}\hidewidth\cr}} #2} \newcommand{\pmop}{\mathop{\text{\Female}}} \newcommand{\pmopc}[2]{#1 \mathop{\underset{\textbf{''}}{\text{\Female}}} #2} %Products and sums of classes of classes or relations \newcommand{\pmccsum}[1]{p\textbf{`}#1} \newcommand{\pmccprd}[1]{s\textbf{`}#1} \newcommand{\pmcrsum}[1]{\dot{p}\textbf{`}#1} \newcommand{\pmcrprd}[1]{\dot{s}\textbf{`}#1} \newcommand{\pmRprdd}{\mathop{||}} \newcommand{\pmrprdd}[2]{{#1}\mathop{||}{#2}} %Identity and Diversity \newcommand{\pmrid}{I} \newcommand{\pmrdiv}{J} \newcommand{\pmcunit}[1]{\iota\textbf{`}#1} \newcommand{\pmcUnit}{\iota} \newcommand{\pmcunits}[1]{\breve{\iota}\textbf{`}#1} %Cardinal numbers \newcommand{\pmcn}[1]{#1} %Ordinal numbers \newcommand{\pmrn}[1]{#1_r} \newcommand{\pmdn}[1]{\dot{#1}} \newcommand{\pmoc}[2]{#1 \boldsymbol{\downarrow} #2} %Subclasses and subrelations \newcommand{\pmscl}[1]{\text{Cl}\textbf{`}#1} \newcommand{\pmsCl}{\text{Cl}} \newcommand{\pmscle}[1]{\text{Cl ex}\textbf{`}#1} \newcommand{\pmsCle}{\text{Cl ex}} \newcommand{\pmscls}[1]{\text{Cls}\textbf{`}#1} \newcommand{\pmsrl}[1]{\text{Rl}\textbf{`}#1} \newcommand{\pmsRl}{\text{Rl}} \newcommand{\pmsrle}[1]{\text{Rl ex}\textbf{`}#1} \newcommand{\pmsRle}{\text{Rl ex}} \newcommand{\pmsrel}[1]{\text{Rel}\textbf{`}#1} \newcommand{\pmRel}{\text{Rel}} \newcommand{\pmReln}[1]{\text{Rel}^{#1}} \newcommand{\pmrin}{\mathop{\boldsymbol{\epsilon}}} %Relative type symbols \newcommand{\pmrt}[1]{t\textbf{`}#1} \newcommand{\pmrti}[2]{t^{#1}\textbf{`}#2} \newcommand{\pmrtc}[2]{t_{#1}\textbf{`}#2} \newcommand{\pmrtri}[2]{t^{#1}\textbf{`}#2} \newcommand{\pmrtrc}[2]{t_{#1}\textbf{`}#2} \newcommand{\pmrtrci}[3]{t_{#1}^{\text{ }#2}\textbf{`}#3} \newcommand{\pmrtric}[3]{^{#1}t_{#2}\textbf{`}#3} \newcommand{\pmrtdi}[2]{#1_{#2}} \newcommand{\pmrtdc}[2]{#1(#2)} \newcommand{\pmrtdri}[2]{#1_{#2}} \newcommand{\pmrtdrc}[2]{#1(#2)} %Similarity relation signs \newcommand{\pmrdc}[2]{#1\boldsymbol{\to}#2} \newcommand{\pmsm}{\mathrel{\text{sm}}} \newcommand{\pmsmbar}{\mathrel{\overline{\text{sm}}}} \newcommand{\pmsmarr}{\overrightarrow{{\pmsm}}} \newcommand{\pmonemany}{1\boldsymbol{\to}\pmCls} \newcommand{\pmmanyone}{\pmCls\boldsymbol{\to}1} \newcommand{\pmoneone}{1\boldsymbol{\to}1} %Selections \newcommand{\pmselp}[1]{P_{\small\Delta}\mathbf{`}#1} \newcommand{\pmSelp}{P_{\Delta}} \newcommand{\pmsele}[1]{\pmcin_{\small\Delta}\mathbf{`}#1} \newcommand{\pmSele}{\pmcin_{\Delta}} \newcommand{\pmself}[1]{F_{\small\Delta}\mathbf{`}#1} \newcommand{\pmSelf}{F_{\Delta}} \newcommand{\pmex}[1]{\text{Cls} \mathop{\text{excl}}\mathbf{`}#1} \newcommand{\pmexc}{\text{Cls}^2 \mathop{\text{excl}}} \newcommand{\pmexcc}[1]{\text{Cl} \mathop{\text{excl}}\textbf{`}#1} \newcommand{\pmexcn}{\text{Cls} \mathop{\text{ex}^2} \mathop{\text{excl}}} \newcommand{\pmselc}[2]{#1 \mathrel{\rotatebox[origin=c]{270}{$\boldsymbol{\mapsto}$}} #2} \newcommand{\pmmultr}{\mathop{\text{Rel}} \mathop{\text{Mult}}} \newcommand{\pmmultc}{\mathop{\text{Cls}^2} \mathop{\text{Mult}}} \newcommand{\pmmultax}{\mathop{\text{Mult}} \mathop{\text{ax}}} %Inductive relations \newcommand{\pmanc}[1]{#1_\pmast} \newcommand{\pmancc}[1]{\pmcnv{#1}_\pmast} \newcommand{\pmrst}[1]{#1_\text{st}} \newcommand{\pmrts}[1]{#1_\text{ts}} \newcommand{\pmpot}[1]{\text{Pot}\mathbf{`}#1} \newcommand{\pmpotid}[1]{\text{Potid}\mathbf{`}#1} \newcommand{\pmpo}[1]{#1_\text{po}} \newcommand{\pmB}{B} \newcommand{\pmmin}[1]{\text{min}_{#1}} \newcommand{\pmmax}[1]{\text{max}_{#1}} \newcommand{\pmMin}{\text{min}} \newcommand{\pmMax}{\text{max}} \newcommand{\pmgen}[1]{\text{gen}\mathbf{`}#1} \newcommand{\pmGen}{\text{gen}} \newcommand{\pmefr}[2]{#1\pmast#2} \newcommand{\pmipr}[2]{I_{#1}\textbf{`}#2} \newcommand{\pmjpr}[2]{J_{#1}\textbf{`}#2} \newcommand{\pmfr}[2]{\overset{\boldsymbol{\leftrightarrow}}{#1}\textbf{`}#2} %Cardinality \newcommand{\pmnc}[1]{\text{Nc}\textbf{`}#1} \newcommand{\pmNc}{\text{Nc}} \newcommand{\pmNC}{\text{NC}} \newcommand{\pmnoc}[1]{\text{N}_0\text{c}\textbf{`}#1} \newcommand{\pmNoc}{\text{N}_0\text{c}} \newcommand{\pmnocind}[1]{\text{N}_0\text{Cinduct}\textbf{`}#1} \newcommand{\pmNocind}{\text{N}_0\text{Cinduct}} \newcommand{\pmclsrefl}{\text{Cls} \mathop{\text{refl}}} \newcommand{\pmncrefl}{\text{NC} \mathop{\text{refl}}}