Add Dave's proof macros

docs/iris/locallabel.sty

+../locallabel.sty
\ No newline at end of file

docs/iris/model.tex

@@ -82,7 +82,7 @@ action on morphisms).
 \begin{array}[t]{rcl}
 %  \protStatus &::=& \enabled \ALT \disabled \\[0.4em]
 \textdom{Res} &\eqdef&
-\{\, \res = (\pres, \ghostRes) \mid
+\{\, \rs = (\pres, \ghostRes) \mid
 \pres \in \textdom{State} \uplus \{\munit\} \land \ghostRes \in \mcarp{\monoid} \,\} \\[0.5em]
 (\pres, \ghostRes) \rsplit
 (\pres', \ghostRes') &\eqdef&
@@ -92,17 +92,17 @@ action on morphisms).
 \end{cases}
 \\[0.5em]
 %
-\res \leq \res' & \eqdef &
-\Exists \res''. \res' = \res \rsplit \res''\\[1em]
+\rs \leq \rs' & \eqdef &
+\Exists \rs''. \rs' = \rs \rsplit \rs''\\[1em]
 %
 \UPred(\textdom{Res}) &\eqdef& 
 \{\, p \subseteq \mathbb{N} \times \textdom{Res} \mid
-\All (k,\res) \in p.
+\All (k,\rs) \in p.
 \All j\leq k.
-\All \res' \geq \res.
-(j,\res')\in p \,\}\\[0.5em]
+\All \rs' \geq \rs.
+(j,\rs')\in p \,\}\\[0.5em]
 \restr{p}{k} &\eqdef& 
-\{\, (j, \res) \in p \mid j < k \,\}\\[0.5em]
+\{\, (j, \rs) \in p \mid j < k \,\}\\[0.5em]
 p \nequiv{n} q & \eqdef & \restr{p}{n} = \restr{q}{n}\\[1em]
 %
 \textdom{PreProp} & \cong  &
@@ -187,7 +187,7 @@ $\rho\nequiv{n} \rho' \iff n=0 \lor \bigl(\dom(\rho)=\dom(\rho') \land
 %\typedsection{Validity}{valid : \pset{\textdom{Prop}} \in Sets}
 %
 %\begin{align*}
-%valid(p) &\iff \All n \in \mathbb{N}. \All \res \in \textdom{Res}. \All W \in \textdom{World}. (n, \res) \in p(W)
+%valid(p) &\iff \All n \in \mathbb{N}. \All \rs \in \textdom{Res}. \All W \in \textdom{World}. (n, \rs) \in p(W)
 %\end{align*}
 
 \typedsection{Later modality}{\later : \textdom{Prop} \to \textdom{Prop} \in {\cal U}}
@@ -213,7 +213,7 @@ $\rho\nequiv{n} \rho' \iff n=0 \lor \bigl(\dom(\rho)=\dom(\rho') \land
 %\begin{align*}
 %&\forall p, q \in \textdom{Prop}.~\\
 %&\qquad
-%  (\forall n \in \mathbb{N}.~\forall \res \in \textdom{Res}.~\forall W \in \textdom{World}.~(n, \res) \in p(W) \Rightarrow (n, \res) \in q(W)) \Leftrightarrow~valid(\always{(p \Rightarrow q)})
+%  (\forall n \in \mathbb{N}.~\forall \rs \in \textdom{Res}.~\forall W \in \textdom{World}.~(n, \rs) \in p(W) \Rightarrow (n, \rs) \in q(W)) \Leftrightarrow~valid(\always{(p \Rightarrow q)})
 %\end{align*}
 %\end{lem}
 
@@ -232,11 +232,11 @@ $\rho\nequiv{n} \rho' \iff n=0 \lor \bigl(\dom(\rho)=\dom(\rho') \land
 	\textdom{World} \to \psetdown{\mathbb{N}} \in {\cal U}}
 \ralf{Make this Dave-compatible: Explicitly compose all the things in $s$}
 \begin{align*}
-	\fullSat{\state}{\mask}{\res}{W} &=
+	\fullSat{\state}{\mask}{\rs}{W} &=
 	\begin{aligned}[t]
-		\{\, n + 1 \in \mathbb{N} \mid &\Exists  \resB:\mathbb{N} \fpfn \textdom{Res}. (\res \rsplit \resB).\pres = \state \land{}\\
-		&\quad \All \iota \in \dom(W). \iota \in \dom(W) \leftrightarrow \iota \in \dom(\resB) \land {}\\
-		&\quad\quad \iota \in \mask \ra (n, \resB(\iota)) \in \wIso^{-1}(W(\iota))(W) \,\} \cup \{ 0 \}
+		\{\, n + 1 \in \mathbb{N} \mid &\Exists  \rsB:\mathbb{N} \fpfn \textdom{Res}. (\rs \rsplit \rsB).\pres = \state \land{}\\
+		&\quad \All \iota \in \dom(W). \iota \in \dom(W) \leftrightarrow \iota \in \dom(\rsB) \land {}\\
+		&\quad\quad \iota \in \mask \ra (n, \rsB(\iota)) \in \wIso^{-1}(W(\iota))(W) \,\} \cup \{ 0 \}
 	\end{aligned}
 \end{align*}
 \begin{lem}\label{lem:fullsat-nonexpansive}
@@ -248,9 +248,9 @@ $\rho\nequiv{n} \rho' \iff n=0 \lor \bigl(\dom(\rho)=\dom(\rho') \land
 		\MoveEqLeft
 		\All \state \in \Delta(\textdom{State}).
 		\All \mask_1, \mask_2 \in \Delta(\pset{\mathbb{N}}).
-		\All \res, \resB \in \Delta(\textdom{Res}).
+		\All \rs, \rsB \in \Delta(\textdom{Res}).
 		\All W \in \textdom{World}. \\&
-		\mask_1 \subseteq \mask_2 \implies (\fullSat{\state}{\mask_2}{\res}{W}) \subseteq (\fullSat{\state}{\mask_1}{\res}{W})
+		\mask_1 \subseteq \mask_2 \implies (\fullSat{\state}{\mask_2}{\rs}{W}) \subseteq (\fullSat{\state}{\mask_1}{\rs}{W})
 	\end{align*}
 \end{lem}
 
@@ -293,11 +293,11 @@ $\rho\nequiv{n} \rho' \iff n=0 \lor \bigl(\dom(\rho)=\dom(\rho') \land
 \begin{align*}
 	\mathit{vs}_{\mask_1}^{\mask_2}(q) &= \Lam W.
 	\begin{aligned}[t]
-		\{\, (n, \res) &\mid \All W_F \geq W. \All \res_F, \mask_F, \state. \All k \leq n.\\
+		\{\, (n, \rs) &\mid \All W_F \geq W. \All \rs_F, \mask_F, \state. \All k \leq n.\\
 		&\qquad 
-		k \in (\fullSat{\state}{\mask_1 \cup \mask_F}{\res \rsplit \res_F}{W_F}) \land k > 0 \land \mask_F \sep (\mask_1 \cup \mask_2) \implies{} \\
+		k \in (\fullSat{\state}{\mask_1 \cup \mask_F}{\rs \rsplit \rs_F}{W_F}) \land k > 0 \land \mask_F \sep (\mask_1 \cup \mask_2) \implies{} \\
 		&\qquad
-		\Exists W' \geq W_F. \Exists \res'. k \in (\fullSat{\state}{\mask_2 \cup \mask_F}{\res' \rsplit \res_F}{W'}) \land (k, \res') \in q(W')
+		\Exists W' \geq W_F. \Exists \rs'. k \in (\fullSat{\state}{\mask_2 \cup \mask_F}{\rs' \rsplit \rs_F}{W'}) \land (k, \rs') \in q(W')
 		\,\}
 	\end{aligned}
 \end{align*}
@@ -338,10 +338,10 @@ $\rho\nequiv{n} \rho' \iff n=0 \lor \bigl(\dom(\rho)=\dom(\rho') \land
 %  \All \iota \in \mathbb{N}.
 %  \All p \in \textdom{Prop}.
 %  \All W \in \textdom{World}.
-%  \All \res \in \textdom{Res}.
+%  \All \rs \in \textdom{Res}.
 %  \All n \in \mathbb{N}. \\
 %&
-%  (n, \res) \in inv(\iota, p)(W) \implies (n, \res) \in vs_{\{ \iota \}}^{\emptyset}(\later p)(W)
+%  (n, \rs) \in inv(\iota, p)(W) \implies (n, \rs) \in vs_{\{ \iota \}}^{\emptyset}(\later p)(W)
 %\end{align*}
 %\end{lem}
 
@@ -352,10 +352,10 @@ $\rho\nequiv{n} \rho' \iff n=0 \lor \bigl(\dom(\rho)=\dom(\rho') \land
 %  \forall \iota \in \mathbb{N}.~
 %  \forall p \in \textdom{Prop}.~
 %  \forall W \in \textdom{World}.~
-%  \forall \res \in \textdom{Res}.~
+%  \forall \rs \in \textdom{Res}.~
 %  \forall n \in \mathbb{N}.~\\
 %&\qquad
-%  (n, \res) \in (inv(\iota, p) * \later p)(W) \Rightarrow (n, \res) \in vs^{\{ \iota \}}_{\emptyset}(\top)(W)
+%  (n, \rs) \in (inv(\iota, p) * \later p)(W) \Rightarrow (n, \rs) \in vs^{\{ \iota \}}_{\emptyset}(\top)(W)
 %\end{align*}
 %\end{lem}
 
@@ -392,22 +392,22 @@ $\rho\nequiv{n} \rho' \iff n=0 \lor \bigl(\dom(\rho)=\dom(\rho') \land
 \begin{align*}
 	\mathit{wp}_\mask(\expr, q) &\eqdef \Lam W.
 	\begin{aligned}[t]
-		\{\, (n, \res) &\mid \All W_F \geq W; k \leq n; \res_F; \state; \mask_F \sep \mask. k > 0 \land k \in (\fullSat{\state}{\mask \cup \mask_F}{\res \rsplit \res_F}{W_F}) \implies{}\\
+		\{\, (n, \rs) &\mid \All W_F \geq W; k \leq n; \rs_F; \state; \mask_F \sep \mask. k > 0 \land k \in (\fullSat{\state}{\mask \cup \mask_F}{\rs \rsplit \rs_F}{W_F}) \implies{}\\
 		&\qquad
-		(\expr \in \textdom{Val} \implies \Exists W' \geq W_F. \Exists \res'. \\
+		(\expr \in \textdom{Val} \implies \Exists W' \geq W_F. \Exists \rs'. \\
 		&\qquad\qquad
-		k \in (\fullSat{\state}{\mask \cup \mask_F}{\res' \rsplit \res_F}{W'}) \land (k, \res') \in q(\expr)(W'))~\land \\
+		k \in (\fullSat{\state}{\mask \cup \mask_F}{\rs' \rsplit \rs_F}{W'}) \land (k, \rs') \in q(\expr)(W'))~\land \\
 		&\qquad
-		(\All\ectx,\expr_0,\expr'_0,\state'. \expr = \ectx[\expr_0] \land \cfg{\state}{\expr_0} \step \cfg{\state'}{\expr'_0} \implies \Exists W' \geq W_F. \Exists \res'. \\
+		(\All\ectx,\expr_0,\expr'_0,\state'. \expr = \ectx[\expr_0] \land \cfg{\state}{\expr_0} \step \cfg{\state'}{\expr'_0} \implies \Exists W' \geq W_F. \Exists \rs'. \\
 		&\qquad\qquad
-		k - 1 \in (\fullSat{\state'}{\mask \cup \mask_F}{\res' \rsplit \res_F}{W'}) \land (k-1, \res') \in wp_\mask(\ectx[\expr_0'], q)(W'))~\land \\
+		k - 1 \in (\fullSat{\state'}{\mask \cup \mask_F}{\rs' \rsplit \rs_F}{W'}) \land (k-1, \rs') \in wp_\mask(\ectx[\expr_0'], q)(W'))~\land \\
 		&\qquad
-		(\All\ectx,\expr'. \expr = \ectx[\fork{\expr'}] \implies \Exists W' \geq W_F. \Exists \res', \res_1', \res_2'. \\
+		(\All\ectx,\expr'. \expr = \ectx[\fork{\expr'}] \implies \Exists W' \geq W_F. \Exists \rs', \rs_1', \rs_2'. \\
 		&\qquad\qquad
-		k - 1 \in (\fullSat{\state}{\mask \cup \mask_F}{\res' \rsplit \res_F}{W'}) \land \res' = \res_1' \rsplit \res_2'~\land \\
+		k - 1 \in (\fullSat{\state}{\mask \cup \mask_F}{\rs' \rsplit \rs_F}{W'}) \land \rs' = \rs_1' \rsplit \rs_2'~\land \\
 		&\qquad\qquad
-		(k-1, \res_1') \in \mathit{wp}_\mask(\ectx[\textsf{fRet}], q)(W') \land
-		(k-1, \res_2') \in \mathit{wp}_\top(\expr', \Lam\any. \top)(W'))
+		(k-1, \rs_1') \in \mathit{wp}_\mask(\ectx[\textsf{fRet}], q)(W') \land
+		(k-1, \rs_2') \in \mathit{wp}_\top(\expr', \Lam\any. \top)(W'))
 		\,\}
 	\end{aligned}
 \end{align*}
@@ -491,9 +491,9 @@ $\rho\nequiv{n} \rho' \iff n=0 \lor \bigl(\dom(\rho)=\dom(\rho') \land
 	\semTerm{\vctx \proves \knowInv{\iname}{\prop} : \Prop}_\gamma &=
 	inv(\semTerm{\vctx \proves \iname : \textsort{InvName}}_\gamma, \semTerm{\vctx \proves \prop : \Prop}_\gamma) \\
 	\semTerm{\vctx \proves \ownGGhost{\melt} : \Prop}_\gamma &=
-	\Lam W. \{\, (n, \res) \mid \res.\ghostRes \geq \semTerm{\vctx \proves \melt : \textsort{Monoid}}_\gamma \,\} \\
+	\Lam W. \{\, (n, \rs) \mid \rs.\ghostRes \geq \semTerm{\vctx \proves \melt : \textsort{Monoid}}_\gamma \,\} \\
 	\semTerm{\vctx \proves \ownPhys{\state} : \Prop}_\gamma &=
-	\Lam W. \{\, (n, \res) \mid \res.\pres = \semTerm{\vctx \proves \state : \textsort{State}}_\gamma \,\}
+	\Lam W. \{\, (n, \rs) \mid \rs.\pres = \semTerm{\vctx \proves \state : \textsort{State}}_\gamma \,\}
 \end{align*}
 %
 \begin{align*}
@@ -513,10 +513,10 @@ $\rho\nequiv{n} \rho' \iff n=0 \lor \bigl(\dom(\rho)=\dom(\rho') \land
 \MoveEqLeft
 \forall n \in \mathbb{N}.\;
 \forall W \in \textdom{World}.\;
-\forall \res \in \textdom{Res}.\; 
+\forall \rs \in \textdom{Res}.\; 
 \forall \gamma \in \semSort{\vctx},\;
 \\&
-\bigl(\All \propB \in \pfctx. (n, \res) \in \semTerm{\vctx \proves \propB : \Prop}_\gamma(W)\bigr)
-\implies (n, \res) \in \semTerm{\vctx \proves \prop : \Prop}_\gamma(W)
+\bigl(\All \propB \in \pfctx. (n, \rs) \in \semTerm{\vctx \proves \propB : \Prop}_\gamma(W)\bigr)
+\implies (n, \rs) \in \semTerm{\vctx \proves \prop : \Prop}_\gamma(W)
 \end{aligned}
 \]

docs/iris/pfsteps.sty

-../pfsteps.sty
\ No newline at end of file

docs/iris/pftools.sty

+../pftools.sty
\ No newline at end of file

docs/locallabel.sty

+%  Locallabel
+%
+%  Copyright (C) 2001, 2002, 2003 Didier Rmy
+%
+%  Author         : Didier Remy 
+%  Version        : 1.1.1
+%  Bug Reports    : to author
+%  Web Site       : http://pauillac.inria.fr/~remy/latex/
+% 
+%  Locallabel is free software; you can redistribute it and/or modify
+%  it under the terms of the GNU General Public License as published by
+%  the Free Software Foundation; either version 2, or (at your option)
+%  any later version.
+%  
+%  Locallabel is distributed in the hope that it will be useful,
+%  but WITHOUT ANY WARRANTY; without even the implied warranty of
+%  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+%  GNU General Public License for more details 
+%  (http://pauillac.inria.fr/~remy/license/GPL).
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%  File locallabel.sty (LaTeX macros)
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%
+
+%% Identification
+\NeedsTeXFormat{LaTeX2e}
+\ProvidesPackage{locallabel}
+         [2001/23/02 v0.92 Locallabel]
+
+%% Preliminary declarations
+
+%% Options
+
+%% More declarations
+
+%% We use two counters: The global counter is incremented at each reset.
+%% Its value is the ``group'' of a local. 
+%% The local counter is the last numeric value of a bound label in the
+%% current group.  The value of a label #1 is globally set to 
+%% \csname llb@\the\c@llb@global-#1\endcsname
+%% The global command \csname llb@\the\c@llb@global-#1*\endcsname is 
+%% use to ensure that a label is only bound once. Usually a label is
+%% bound and declared at the same time with \llabel. It may also be bound in
+%% advance,  with \lbind, for instance so as to control the numbering.
+%% Then, another \llabel must be used to declare it in the text.
+%% If no \lbind has been used before, the \llabel calls \lbind implicitlt. 
+
+\newcounter{llb@global}
+\newcounter{llb@local}
+
+\newcommand \llb@find [1]
+  {\expandafter \ifx \csname llb@\the\c@llb@global-#1\endcsname \relax
+     \message {*** Local label #1 undefined in this context}%
+     \edef \llb@current {#1??}%
+   \else 
+     \edef \llb@current {\csname llb@\the\c@llb@global-#1\endcsname}%
+   \fi}
+
+\newcommand \llb@make [1]
+  {\expandafter \ifx \csname llb@\the\c@llb@global-#1\endcsname \relax
+     \stepcounter{llb@local}\relax \expandafter 
+     \xdef \csname llb@\the\c@llb@global-#1\endcsname {\the\c@llb@local}%
+     \edef \llb@current {\the\c@llb@local}%
+   \else
+     \expandafter \ifx \csname llb@\the\c@llb@global-#1*\endcsname \relax
+       \message {*** Local label #1 already defined in this countext!}%
+       \edef \llb@current {\csname llb@\the\c@llb@global-#1\endcsname ??}%
+     \else
+       \expandafter \global \expandafter \let
+          \csname llb@\the\c@llb@global-#1*\endcsname \relax
+     \edef \llb@current {\csname llb@\the\c@llb@global-#1\endcsname}
+     \fi 
+   \fi
+}
+
+%%% Redefine those macros to change typsetting
+
+\newcommand \thelocallabel {\the \c@llb@local}
+
+\newcommand \LlabelTypeset [1] {(\textrm {\bfseries #1})}
+\newcommand \LrefTypeset [1] {(\textrm {#1)}}
+\newcommand \glabel [1]{\LlabelTypeset{\softtarget {#1}{#1}}}
+\newcommand \gref [1]{\LrefTypeset{\softlink {#1}{#1}}}
+
+%%% To reset all local labels---which just increment a global prefix.
+\newcommand \locallabelreset[1][0]%
+  {\stepcounter {llb@global}\setcounter {llb@local}{#1}}
+
+%%% Make a new local label, typeset it, and bind to the given name
+\def \llb@relax {\relax}
+\newcommand {\llabel}[2][\relax]%
+  {\llb@make{#2}%
+   \def \@test {#1}\ifx \@test\llb@relax\else
+     \edef \@currentlabel {\the\c@llb@local}%
+     \def \@test {#1}\ifx \@test\empty \def \@test{#2}\fi
+     \label{\@test}%
+   \fi%
+   \LlabelTypeset {\softtarget{llb@\the\c@llb@global-#2}{\llb@current}}}
+
+%%% Retreive the local label of given name and type set it.
+\newcommand \lref [1]
+  {\llb@find {#1}%
+   \LrefTypeset {\softlink {llb@\the\c@llb@global-#1}{\llb@current}}}
+
+%%% Make a new local label and bind it to the given name but do not typeset
+%%% it. Typesetting may then be done with \llabel non locally. Useful to
+%%% control the order of numberring.
+\newcommand \lbind [1]
+  {\llb@make {#1}%
+   \expandafter \global \expandafter 
+      \let \csname llb@\the\c@llb@global-#1*\endcsname \empty}
+   
+\AtBeginDocument {%
+  \@ifundefined{softlink}{\let \softlink \@secondoftwo}{}%
+  \@ifundefined{softtarget}{\let \softtarget \@secondoftwo}{}%
+}  

docs/pfsteps.sty

-%%
-%% This is file `pfsteps.sty',
-%% generated with the docstrip utility.
-%%
-%% The original source files were:
-%%
-%% pfsteps.dtx  (with options: `package')
-%% 
-%% Copyright (C) 2011 by Jesse A. Tov
-%% 
-%% This file may be distributed and/or modified under the conditions of the
-%% LaTeX Project Public License, either version 1.2 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 1.2 or later is part of all distributions of LaTeX
-%% version 1999/12/01 or later.
-%% 
-\NeedsTeXFormat{LaTeX2e}[1999/12/01]
-\ProvidesPackage{pfsteps}
- [2011/04/04 v0.4 proof tools]
-\RequirePackage{listproc}
-\newcommand*\pfsteps@set[3][]{
-  \expandafter\let\csname #1pfsteps@#2\endcsname#3
-}
-\newcommand*\pfsteps@option[2][\iffalse]{
-  \pfsteps@set[if]{#2}#1
-  \pfsteps@set[if]{#2@set}\iffalse
-  \DeclareOption{#2}{
-    \pfsteps@set[if]{#2}\iftrue
-    \pfsteps@set[if]{#2@set}\iftrue
-  }
-  \DeclareOption{no#2}{
-    \pfsteps@set[if]{#2}\iffalse
-    \pfsteps@set[if]{#2@set}\iftrue
-  }
-}
-\pfsteps@option[\iftrue]{atsign}
-\pfsteps@option[\iftrue]{hyperref}
-\pfsteps@option[\iftrue]{loadunicode}
-\pfsteps@option[\iftrue]{mathpartir}
-\pfsteps@option{unicode}
-\ProcessOptions
-\ifpfsteps@unicode
-  \ifpfsteps@loadunicode
-    \RequirePackage{ucs}
-    \RequirePackage[utf8x]{inputenc}
-  \fi
-\fi
-\ifpfsteps@mathpartir
-  \ifpfsteps@mathpartir@set
-    \RequirePackage{mathpartir}
-  \fi
-\fi
-\ifpfsteps@hyperref
-  \ifpfsteps@hyperref@set
-    \RequirePackage{hyperref}
-  \fi
-\fi
-\newcommand{\pfcounteranchor}[1]{(#1)}
-\newcommand{\pfcounterref}[1]{(#1)}
-\newcounter{pfsteps@pfc@global}
-\newcounter{pfsteps@pfc@local}
-\newcommand{\resetpfcounter}[1][0]
-  {\stepcounter{pfsteps@pfc@global}\setcounter{pfsteps@pfc@local}{#1}}
-\newcommand{\thepfcounter}
-  {\the\value{pfsteps@pfc@local}}
-\newcommand{\thepfsectioncounter}
-  {\the\value{pfsteps@pfc@global}}
-\newcommand{\steppfcounter}[1][\relax]{%
-  \addtocounter{pfsteps@pfc@local}{1}%
-   \ifx\relax#1\relax\else
-     \pflabel{#1}%
-   \fi
-}
-\newcommand{\usepfcounter}[1][\relax]{%
-  \steppfcounter[#1]%
-  \pfsteps@hypertarget{pfc:\thepfsectioncounter:\thepfcounter}{%
-    \pfcounteranchor{\thepfcounter}%
-  }%
-}
-\newcommand{\pfsteps@pfc@cs}[1]
-  {\csname\pfsteps@pfc@{\pfsteps@strip#1 \@empty}\endcsname}
-\newcommand{\pfsteps@pfc@}[1]
-  {pfsteps@pfc@\pfsteps@strip#1 \@empty @\thepfsectioncounter}
-\def\pfsteps@strip#1 #2{%
-  #1%
-  \ifx#2\@empty\else\expandafter\pfsteps@strip\fi
-  #2}
-\newcommand{\pflabel}[1]
-  {\expandafter\ifx\csname\pfsteps@pfc@{#1}@thisrun\endcsname\relax
-     \expandafter\xdef\csname\pfsteps@pfc@{#1}\endcsname
-       {\thepfcounter}%
-     \expandafter\gdef\csname\pfsteps@pfc@{#1}@thisrun\endcsname
-       {}%
-     \immediate\write\@auxout{
-       \noexpand\pfsteps@def@label
-         {#1}{\thepfsectioncounter}{\thepfcounter}
-     }%
-   \else
-     \PackageWarning{pfsteps}
-        {Proof step (#1) already defined in this section}%
-   \fi}
-\newcommand*{\pfsteps@def@label}[3]{
-  \expandafter\gdef
-    \csname pfsteps@pfc@#1@#2\endcsname
-    {#3}
-}
-\newcommand*{\pfref}[1]
-{{\ListExprTo
-    {\Compress[\@apply@group\@firstoftwo]
-     {\Sort[\@apply@group\@firstoftwo]
-      {\Map
-       {%
-        {\@ifundefined{\pfsteps@pfc@{##1}}
-           {-1}
-           {\csname\pfsteps@pfc@{##1}\endcsname}}%
-        {\@ifundefined{\pfsteps@pfc@{##1}}
-           {\PackageWarning{pfsteps}
-              {Proof step (##1) not yet defined in this section}%
-            \textbf{?}}
-           {\pfsteps@hyperlink
-             {pfc:\thepfsectioncounter:\pfsteps@pfc@cs{##1}}
-             {\pfsteps@pfc@cs{##1}}}}}
-       {\List{#1}}}}}
-    \pfsteps@pfref@list
-  \let\listitem\pfsteps@pfref@listitem@first
-  \def\@single##1{\@secondoftwo##1}%
-  \def\@range##1##2{\@secondoftwo##1--\@secondoftwo##2}%
-  \pfcounterref{\pfsteps@pfref@list}%
-}}
-\newcommand\pfsteps@pfref@listitem@first[1]{%
-  #1\let\listitem\pfsteps@pfref@listitem@rest
-}
-\newcommand\pfsteps@pfref@listitem@rest[1]{%
-  , #1\let\listitem\pfsteps@pfref@listitem@rest
-}
-\newcommand\pfsteps@hypertarget[2]{#2}
-\newcommand\pfsteps@hyperlink[2]{#2}
-\ifpfsteps@hyperref
-  \AtBeginDocument{
-    \ifcsname hypertarget\endcsname
-      \let\pfsteps@hypertarget=\hypertarget
-      \let\pfsteps@hyperlink=\hyperlink
-    \fi
-  }
-\fi
-\newlength{\proofleftskip}
-\newlength{\proofrightwidth}
-\setlength{\proofleftskip}{2pc}
-\setlength{\proofrightwidth}{0.3\linewidth}
-\newenvironment{pfsteps}
-        {\begin{pfsteps@with}$}
-        {\end{pfsteps@with}}
-\newenvironment{pfsteps*}
-        {\begin{pfsteps@with}{}}
-        {\end{pfsteps@with}}
-\newenvironment{pfsteps@with}[1]
-{
-  \leavevmode\begingroup
-  \setlength{\parskip}{0pt}%
-  \trivlist
-  \raggedright
-  \setlength{\leftskip}{1.5\proofleftskip}
-  \let\pfstepsSavedItem\item
-  \let\pfstepsSavedLabel\label
-  \let\pfstepsSavedQedhere\qedhere
-  \newcommand\AND[1][and]{\mathrel{\mbox{##1}}}
-  \newcommand\BY[2][by]
-    {\pfsteps@unmath{\penalty-1 \mbox{~}\hfill%
-     \begin{minipage}[t]{\proofrightwidth}%
-       \raggedright##1 ##2%
-     \end{minipage}}}
-  \def\pfstepsItem{%
-    \pfsteps@stopmath
-    \pfstepsSavedItem\mbox{}\kern-1.25\proofleftskip
-    \makebox[\proofleftskip]{\hfill\usepfcounter}\kern0.25\proofleftskip
-    #1\relax}
-  \def\pfstepsQedhere{\pfsteps@unmath{\pfstepsSavedQedhere}}
-  \let\item\pfstepsItem
-  \let\label\pflabel
-  \let\qedhere\pfstepsQedhere
-  \ifpfsteps@atsign
-    \pfsteps@setup@atsign
-  \fi
-  \relax
-}
-{
-  \pfsteps@stopmath
-  \endtrivlist\endgroup
-  \noindent\ignorespaces
-}
-\newcommand\pfsteps@stopmath{\ifmmode$\fi}
-\newcommand\pfsteps@unmath[1]{\ifmmode$\relax#1\relax$\else\relax#1\relax\fi}
-{
-  \def\atsign{@}
-  \catcode`\@=\active\relax
-  \expandafter\gdef\csname pfsteps\atsign setup\atsign atsign\endcsname{
-    \catcode`\@=\active\relax
-    \gdef@##1 {\pflabel{##1}}
-  }
-}
-\newcommand\pfstepsmathmode{\def\pfsteps@unicode@arg{$}}
-\newcommand\pfstepstextmode{\def\pfsteps@unicode@arg{\relax}}
-\newcommand\pfstepsSetupUnicode[3]{
-  \DeclareUnicodeCharacter{#1}{\pfsteps@unicode@startpfsteps}
-  \DeclareUnicodeCharacter{#3}{\pfsteps@unicode@item}
-  \def\pfsteps@unicode@startpfsteps
-    {\begingroup
-     \ifpfsteps@atsign\catcode`\@=\active\relax\fi
-     \pfsteps@unicode@startpfsteps@kont}
-  \def\pfsteps@unicode@startpfsteps@kont##1#2
-    {\begin{pfsteps@with}\pfsteps@unicode@arg\item##1\end{pfsteps@with}%
-     \endgroup}
-  \def\pfsteps@unicode@item{\item}
-  \pfstepsmathmode
-}
-\ifpfsteps@unicode
-  \pfstepsSetupUnicode{171}{»}{8226} % « » •
-\fi
-\newcommand\byCasesEveryCase{\resetpfcounter}
-\newcommand\byCasesEveryOtherwise{\byCasesEveryCase}
-\providecommand{\byCasesOtherwiseTemplate}{\textbf{Otherwise:}}
-\providecommand{\byCasesCaseTemplate}[1]{\textbf{Case\ \ \fbox{#1}}}
-\providecommand{\byCasesWhereTemplate}{\textbf{where}}
-\newenvironment{byCases}
-  {%
-    \begingroup
-    \let\case\byCases@case
-    \let\otherwise\byCases@otherwise
-    \ifpfsteps@mathpartir
-      \ifcsname inferrule\endcsname\let\icase\byCases@icase\fi
-    \fi
-    \list{}{\labelwidth\z@ \itemindent-\leftmargin
-            \let\makelabel\byCases@label}%
-  }
-  {%
-    \endlist
-    \endgroup
-  }
-\newcommand*\byCases@label[1]{%
-  \hspace\labelsep
-  \normalfont~\strut
-  \expandafter\ifx#1\relax\relax
-    \byCasesOtherwiseTemplate
-  \else
-    \byCasesCaseTemplate{\normalfont${#1}$}%