\documentclass [fleqn]{article}

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\setlength{\parskip}{1ex}

\usepackage{graphicx}

\usepackage[dvipdfm=true]{hyperref}

\hypersetup{pdfpagemode=none}

\hypersetup{pdfstartpage=1}

\hypersetup{pdfstartview=FitBH}

\hypersetup{pdfpagescrop={120 130 490 730}}

\hypersetup{pdftitle=Logiweb codex of problemtwo}

\hypersetup{colorlinks=true}

\everymath{\rm}

\begin {document}\raggedright

\makebox [0mm][l]{\makebox [\linewidth ][r]{\includegraphics [0mm,18.5mm][16.5mm,19mm]{/home/disk07/grue/logiweb/20060417/html/logiweb/doc/logiweb.eps}}}%
{\bf {\Large Logiweb codex of $\rm 
\mbox{problemtwo}$}}

\vspace {12.5mm}

\href {../index.html}{Up}
\href {http://www.diku.dk/~grue/logiweb/20060417/logiweb/doc/browser/codex/index.html}{Help}


\begin {flushleft}\hyperlink {url:1808490373749049856838801749754674032079431248934152926926024270046521601:102}{$\rm \ast
\Rightarrow \ast$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:0}{$\rm 
\mbox{problemtwo}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:1}{$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:2}{$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:3}{$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:4}{$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:5}{$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:6}{$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:7}{$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:8}{$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:9}{$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:10}{$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:11}{$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:12}{$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:13}{$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:14}{$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:15}{$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:16}{$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:17}{$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:18}{$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:19}{$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:20}{$\rm 
\mbox{pred calc}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:21}{$\rm 
\mbox{pc1}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:22}{$\rm 
\mbox{pc2}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:23}{$\rm 
\mbox{pc3}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:24}{$\rm 
\mbox{pc4}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:25}{$\rm 
\mbox{pc5}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:26}{$\rm 
\mbox{pc6}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:27}{$\rm 
\mbox{pc7}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:28}{$\rm 
\mbox{pc8}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:29}{$\rm 
\mbox{pc9}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:30}{$\rm 
\mbox{pc10}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:31}{$\rm 
\mbox{pc11}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:32}{$\rm 
\mbox{pc12}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:33}{$\rm 
\mbox{pcmp}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:34}{$\rm 
\mbox{pcunsound}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:35}{$\rm 
\mbox{pcded}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:36}{$\rm 
\mbox{pcia}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:37}{$\rm 
\mbox{pcie}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:38}{$\rm 
\mbox{pcdeduction}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:39}{$\rm 
\mbox{trivia}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:40}{$\rm 
\mbox{trivia2}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:41}{$\rm 
\mbox{iatest}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:42}{$\rm 
\mbox{andintro}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:43}{$\rm 
\mbox{andelim1}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:44}{$\rm 
\mbox{andelim2}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:45}{$\rm 
\mbox{orintro1}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:46}{$\rm 
\mbox{orintro2}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:47}{$\rm 
\mbox{orelim}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:48}{$\rm 
\mbox{notintro}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:49}{$\rm 
\mbox{notnotintro}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:50}{$\rm 
\mbox{notnotelim}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:51}{$\rm 
\mbox{mt}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:52}{$\rm 
\mbox{pbc}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:53}{$\rm 
\mbox{repeat}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:54}{$\rm 
\mbox{lem}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:55}{$\rm 
\mbox{forallintro}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:56}{$\rm 
\mbox{forallelim}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:57}{$\rm 
\mbox{existsintro}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:58}{$\rm 
\mbox{existselim}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:59}{$\rm 
\mbox{bottomelim}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:60}{$\rm 
\mbox{lemnotintro}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:61}{$\rm 
\mbox{nontriv0}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:62}{$\rm 
\mbox{nontriv1}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:63}{$\rm 
\mbox{nontriv2}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:64}{$\rm 
\mbox{nontriv3}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:65}{$\rm 
\mbox{nontriv4}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:66}{$\rm 
\mbox{nontriv5}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:67}{$\rm 
\mbox{nontriv6}$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:68}{$\rm \ast \equiv \ast$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:69}{$\rm \ast = \ast$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:70}{$\rm \neg \ast$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:71}{$\rm \ast \wedge \ast$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:72}{$\rm \ast \vee \ast$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:73}{$\rm \forall \ast . \left(\ast\right)$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:74}{$\rm \exists \ast . \left(\ast\right)$},
\hyperlink {url:1808490463983504709587517268929939939143332479535486097762496273530505985:75}{$\rm \ast \in \ast$},
\end {flushleft}

\hypertarget {url:1808490373749049856838801749754674032079431248934152926926024270046521601:102}{}\section* {$\rm \ast
\Rightarrow \ast$}

$\rm
[
\mathsf{x}
\Rightarrow 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}Rightarrow
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:0}{}\section* {$\rm 
\mbox{problemtwo}$}

$\rm
[
\mbox{problemtwo}
\stackrel{
\mathrm{prio}
}{\rightarrow}
\newline \mathbf {Preassociative} \newline 
[
\mbox{problemtwo}
] , \linebreak [0] 
[
\mbox{base}
] , \linebreak [0] 
[
\mbox{bracket $
\ast $ end bracket}
] , \linebreak [0] 
[
\mbox{big bracket $
\ast $ end bracket}
] , \linebreak [0] 
[
\ \$
\ast \$\linebreak[0]\ 
] , \linebreak [0] 
[
\mathbf{flush\ left\ }[ 
\ast 
]
] , \linebreak [0] 
[
\mathsf{x}
] , \linebreak [0] 
[
\mathsf{y}
] , \linebreak [0] 
[
\mathsf{z}
] , \linebreak [0] 
[
[
\ast 
\bowtie
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\stackrel{
\ast 
}{\rightarrow}
\ast 
]
] , \linebreak [0] 
[
\mathrm{pyk}
] , \linebreak [0] 
[
\mathrm{tex}
] , \linebreak [0] 
[
\mathrm{name}
] , \linebreak [0] 
[
\mathrm{prio}
] , \linebreak [0] 
[
\ast 
] , \linebreak [0] 
[
\mathsf {T}
] , \linebreak [0] 
[
\mathrm {if}(
\ast 
,\linebreak [0]
\ast 
,\linebreak [0]
\ast 
)
] , \linebreak [0] 
[
[
\ast 
\stackrel {
\ast 
}{\Rightarrow }
\ast 
]
] , \linebreak [0] 
[
\mathrm {val}
] , \linebreak [0] 
[
\mathrm {claim}
] , \linebreak [0] 
[
\bot 
] , \linebreak [0] 
[
f(
\ast 
)
] , \linebreak [0] 
[
(
\ast 
){}^{I} 
] , \linebreak [0] 
[
\mathsf {F}
] , \linebreak [0] 
[
\underline {0}
] , \linebreak [0] 
[
\underline {1}
] , \linebreak [0] 
[
\underline {2}
] , \linebreak [0] 
[
\underline {3}
] , \linebreak [0] 
[
\underline {4}
] , \linebreak [0] 
[
\underline {5}
] , \linebreak [0] 
[
\underline {6}
] , \linebreak [0] 
[
\underline {7}
] , \linebreak [0] 
[
\underline {8}
] , \linebreak [0] 
[
\underline {9}
] , \linebreak [0] 
[
0
] , \linebreak [0] 
[
1
] , \linebreak [0] 
[
2
] , \linebreak [0] 
[
3
] , \linebreak [0] 
[
4
] , \linebreak [0] 
[
5
] , \linebreak [0] 
[
6
] , \linebreak [0] 
[
7
] , \linebreak [0] 
[
8
] , \linebreak [0] 
[
9
] , \linebreak [0] 
[
\mathsf {a}
] , \linebreak [0] 
[
\mathsf {b}
] , \linebreak [0] 
[
\mathsf {c}
] , \linebreak [0] 
[
\mathsf {d}
] , \linebreak [0] 
[
\mathsf {e}
] , \linebreak [0] 
[
\mathsf {f}
] , \linebreak [0] 
[
\mathsf {g}
] , \linebreak [0] 
[
\mathsf {h}
] , \linebreak [0] 
[
\mathsf {i}
] , \linebreak [0] 
[
\mathsf {j}
] , \linebreak [0] 
[
\mathsf {k}
] , \linebreak [0] 
[
\mathsf {l}
] , \linebreak [0] 
[
\mathsf {m}
] , \linebreak [0] 
[
\mathsf {n}
] , \linebreak [0] 
[
\mathsf {o}
] , \linebreak [0] 
[
\mathsf {p}
] , \linebreak [0] 
[
\mathsf {q}
] , \linebreak [0] 
[
\mathsf {r}
] , \linebreak [0] 
[
\mathsf {s}
] , \linebreak [0] 
[
\mathsf {t}
] , \linebreak [0] 
[
\mathsf {u}
] , \linebreak [0] 
[
\mathsf {v}
] , \linebreak [0] 
[
\mathsf {w}
] , \linebreak [0] 
[
(
\ast 
)^M
] , \linebreak [0] 
[
\mathrm {If}(
\ast 
,\linebreak [0]
\ast 
,\linebreak [0]
\ast 
)
] , \linebreak [0] 
[\mathrm {array}\{
\ast 
\}
\ast 
\mathrm {end\ array}
] , \linebreak [0] 
[
l
] , \linebreak [0] 
[
c
] , \linebreak [0] 
[
r
] , \linebreak [0] 
[
\mathrm {empty}
] , \linebreak [0] 
[
\langle 
\ast 
\,{|}
\ast 
{:=}\,
\ast 
\rangle 
] , \linebreak [0] 
[
{\cal M}(
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal U}}(
\ast 
)
] , \linebreak [0] 
[
{\cal U}(
\ast 
)
] , \linebreak [0] 
[
{\cal U}^M(
\ast 
)
] , \linebreak [0] 
[
\mathbf {apply}(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathbf {apply}_1(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
identifier( 
\ast 
)
] , \linebreak [0] 
[
identifier_{1}( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
array\mbox{-}\linebreak[0]plus( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
array\mbox{-}\linebreak[0]remove( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
array\mbox{-}\linebreak[0]put( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
array\mbox{-}\linebreak[0]add( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
bit( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
bit_1( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
rack
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {vector}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {bibliography}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {dictionary}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {body}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {codex}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {expansion}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {code}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {cache}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {diagnose}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {pyk}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {tex}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {texname}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {value}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {message}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {macro}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {definition}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {unpack}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {claim}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {priority}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {lambda}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {apply}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {true}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {if}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {quote}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {proclaim}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {define}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {introduce}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {hide}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {pre}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {post}\mbox {\tt \char34}
] , \linebreak [0] 
[
{\cal E}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal E}_2(
\ast 
,
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal E}_3(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal E}_4(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathbf {lookup}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathbf {abstract}(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\lceil 
\ast 
\rceil 
] , \linebreak [0] 
[
{\cal M}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal M}_2(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal M}^*(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathrm {macro}
] , \linebreak [0] 
[
s_0
] , \linebreak [0] 
[
\mathbf {zip}(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathbf {assoc}_1(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[(
\ast 
)^{\mathbf {p}}
] , \linebreak [0] 
[
\mbox{self}
] , \linebreak [0] 
[
[
\ast 
\mathrel {\ddot {=}}
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\mathrel {\dot {=}}
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\mathrel {\acute {=}}
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\stackrel {\mathrm {pyk}}{=} 
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\stackrel {\mathrm {tex}}{=} 
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\stackrel {\mathrm {name}}{=} 
\ast 
]
] , \linebreak [0] 
[
\mathbf{Priority\ table} [ 
\ast 
]
] , \linebreak [0] 
[
\tilde {{\cal M}}_1
] , \linebreak [0] 
[
\tilde {{\cal M}}_2(
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal M}}_3(
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal M}}_4(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal M}}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal Q}}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal Q}}_2(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal Q}}_3(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal Q}}^*(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
(
\ast 
)
] , \linebreak [0] 
[
\left( 
\ast 
\right) 
] , \linebreak [0] 
[
display(
\ast 
)
] , \linebreak [0] 
[
statement(
\ast 
)
] , \linebreak [0] 
[
[
\ast 
]^{\cdot}
] , \linebreak [0] 
[
[
\ast 
]^-
] , \linebreak [0] 
[
\mathbf{aspect}( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{aspect}( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\langle 
\ast 
\rangle 
] , \linebreak [0] 
[
\mathbf {tuple}_1( 
\ast 
) 
] , \linebreak [0] 
[
\mathbf {tuple}_2( 
\ast 
) 
] , \linebreak [0] 
[
let_2( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
let_1( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
[
\ast 
\stackrel {claim}{=}
\ast 
]
] , \linebreak [0] 
[
\mbox{checker}
] , \linebreak [0] 
[
\mathbf{check}( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{check}_2( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{check}_3( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{check}^*( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{check}^*_2( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\relax [ 
\ast 
\relax ]^{\cdot} 
] , \linebreak [0] 
[
\relax [ 
\ast 
\relax ]^{-} 
] , \linebreak [0] 
[
\relax [ 
\ast 
\relax ]^{\circ} 
] , \linebreak [0] 
[
msg
] , \linebreak [0] 
[
[
\ast 
\stackrel {msg}{=} 
\ast 
]
] , \linebreak [0] 
[
{<}stmt{>}
] , \linebreak [0] 
[
stmt
] , \linebreak [0] 
[
[
\ast 
\stackrel {stmt}{=}
\ast 
]
] , \linebreak [0] 
[
HeadNil'
] , \linebreak [0] 
[
HeadPair'
] , \linebreak [0] 
[
Transitivity'
] , \linebreak [0] 
[
{\makebox [0mm][l]{$\bot $}\,{\bot }}
] , \linebreak [0] 
[
Contra'
] , \linebreak [0] 
[
T'_{E}
] , \linebreak [0] 
[
L_{1}
] , \linebreak [0] 
[\underline{
\ast }
] , \linebreak [0] 
[{\cal A}
] , \linebreak [0] 
[{\cal B}
] , \linebreak [0] 
[{\cal C}
] , \linebreak [0] 
[{\cal D}
] , \linebreak [0] 
[{\cal E}
] , \linebreak [0] 
[{\cal F}
] , \linebreak [0] 
[{\cal G}
] , \linebreak [0] 
[{\cal H}
] , \linebreak [0] 
[{\cal I}
] , \linebreak [0] 
[{\cal J}
] , \linebreak [0] 
[{\cal K}
] , \linebreak [0] 
[{\cal L}
] , \linebreak [0] 
[{\cal M}
] , \linebreak [0] 
[{\cal N}
] , \linebreak [0] 
[{\cal O}
] , \linebreak [0] 
[{\cal P}
] , \linebreak [0] 
[{\cal Q}
] , \linebreak [0] 
[{\cal R}
] , \linebreak [0] 
[{\cal S}
] , \linebreak [0] 
[{\cal T}
] , \linebreak [0] 
[{\cal U}
] , \linebreak [0] 
[{\cal V}
] , \linebreak [0] 
[{\cal W}
] , \linebreak [0] 
[{\cal X}
] , \linebreak [0] 
[{\cal Y}
] , \linebreak [0] 
[{\cal Z}
] , \linebreak [0] 
[
\langle 
\ast 
\,{\protect\vert}
\ast 
{:=}\,
\ast 
\rangle 
] , \linebreak [0] 
[
\langle ^ { \ast } 
\ast 
\,{\protect\vert}
\ast 
{:=}\,
\ast 
\rangle 
] , \linebreak [0] 
[
\emptyset 
] , \linebreak [0] 
[
Remainder
] , \linebreak [0] 
[
( 
\ast 
)^{\bf v}
] , \linebreak [0] 
[
intro(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
intro(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
error(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
error_{2}(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
proof( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
proof_{2}( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{I}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\rhd}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\rhd}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{E}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{E}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{+}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{+}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{-}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{-}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\ast}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\ast}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{2}^{\ast}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\char64}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\char64}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\vdash}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\vdash}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\makebox [0mm][l]{\scriptsize $\vdash $}\,{\vdash }}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\makebox [0mm][l]{\scriptsize $\vdash $}\,{\vdash }}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{i.e.}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{i.e.}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{2}^{i.e.}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\forall}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\forall}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{;}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{;}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{2}^{;}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal T}( 
\ast 
)
] , \linebreak [0] 
[
claims( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
claims_2( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{<}proof{>}
] , \linebreak [0] 
[
proof
] , \linebreak [0] 
[
[ \mathbf{Lemma\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ \mathbf{Proof\ of\ }
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ 
\ast 
\mathbf{\ lemma\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ 
\ast 
\mathbf{\ antilemma\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ 
\ast 
\mathbf{\ rule\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ 
\ast 
\mathbf{\ antirule\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
\mbox{verifier}
] , \linebreak [0] 
[
{\cal V}_1( 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_2( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_3( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_4( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_5( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_6( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_7( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
Cut( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
Head_{\oplus} ( 
\ast 
)
] , \linebreak [0] 
[
Tail_{\oplus} ( 
\ast 
)
] , \linebreak [0] 
[
rule_1( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
rule( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
Rule\ tactic
] , \linebreak [0] 
[
Plus( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
[ \mathbf{Theory\ } 
\ast 
]
] , \linebreak [0] 
[
theory_2( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
theory_3( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
theory_4( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
HeadNil''
] , \linebreak [0] 
[
HeadPair''
] , \linebreak [0] 
[
Transitivity''
] , \linebreak [0] 
[
Contra''
] , \linebreak [0] 
[
HeadNil
] , \linebreak [0] 
[
HeadPair
] , \linebreak [0] 
[
Transitivity
] , \linebreak [0] 
[
Contra
] , \linebreak [0] 
[
T_E
] , \linebreak [0] 
[
ragged\ right
] , \linebreak [0] 
[
ragged\ right\ expansion\ 
] , \linebreak [0] 
[
parm(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
parm^*(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
inst(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
inst^*(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
occur(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
occur^*(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
unify(
\ast 
=
\ast 
,
\ast 
)
] , \linebreak [0] 
[
unify^*(
\ast 
=
\ast 
,
\ast 
)
] , \linebreak [0] 
[
unify_2(
\ast 
=
\ast 
,
\ast 
)
] , \linebreak [0] 
[L_a
] , \linebreak [0] 
[L_b
] , \linebreak [0] 
[L_c
] , \linebreak [0] 
[L_d
] , \linebreak [0] 
[L_e
] , \linebreak [0] 
[L_f
] , \linebreak [0] 
[L_g
] , \linebreak [0] 
[L_h
] , \linebreak [0] 
[L_i
] , \linebreak [0] 
[L_j
] , \linebreak [0] 
[L_k
] , \linebreak [0] 
[L_l
] , \linebreak [0] 
[L_m
] , \linebreak [0] 
[L_n
] , \linebreak [0] 
[L_o
] , \linebreak [0] 
[L_p
] , \linebreak [0] 
[L_q
] , \linebreak [0] 
[L_r
] , \linebreak [0] 
[L_s
] , \linebreak [0] 
[L_t
] , \linebreak [0] 
[L_u
] , \linebreak [0] 
[L_v
] , \linebreak [0] 
[L_w
] , \linebreak [0] 
[L_x
] , \linebreak [0] 
[L_y
] , \linebreak [0] 
[L_z
] , \linebreak [0] 
[L_A
] , \linebreak [0] 
[L_B
] , \linebreak [0] 
[L_C
] , \linebreak [0] 
[L_D
] , \linebreak [0] 
[L_E
] , \linebreak [0] 
[L_F
] , \linebreak [0] 
[L_G
] , \linebreak [0] 
[L_H
] , \linebreak [0] 
[L_I
] , \linebreak [0] 
[L_J
] , \linebreak [0] 
[L_K
] , \linebreak [0] 
[L_L
] , \linebreak [0] 
[L_M
] , \linebreak [0] 
[L_N
] , \linebreak [0] 
[L_O
] , \linebreak [0] 
[L_P
] , \linebreak [0] 
[L_Q
] , \linebreak [0] 
[L_R
] , \linebreak [0] 
[L_S
] , \linebreak [0] 
[L_T
] , \linebreak [0] 
[L_U
] , \linebreak [0] 
[L_V
] , \linebreak [0] 
[L_W
] , \linebreak [0] 
[L_X
] , \linebreak [0] 
[L_Y
] , \linebreak [0] 
[L_Z
] , \linebreak [0] 
[L_?
] , \linebreak [0] 
[
Reflexivity
] , \linebreak [0] 
[
Reflexivity_1
] , \linebreak [0] 
[
Commutativity
] , \linebreak [0] 
[
Commutativity_1
] , \linebreak [0] 
[
{<}tactic{>}
] , \linebreak [0] 
[
tactic
] , \linebreak [0] 
[
[
\ast 
\stackrel {tactic}{=}
\ast 
]
] , \linebreak [0] 
[
{\cal P}( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal P}^*( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
p_0
] , \linebreak [0] 
[
conclude_1 ( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
conclude_2 ( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
conclude_3 ( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
conclude_4 ( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
] , \linebreak [0] 
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
] , \linebreak [0] 
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
] , \linebreak [0] 
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
] , \linebreak [0] 
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
] , \linebreak [0] 
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
] , \linebreak [0] 
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
] , \linebreak [0] 
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
] , \linebreak [0] 
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
] , \linebreak [0] 
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
] , \linebreak [0] 
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
] , \linebreak [0] 
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
] , \linebreak [0] 
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
] , \linebreak [0] 
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
] , \linebreak [0] 
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
] , \linebreak [0] 
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
] , \linebreak [0] 
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
] , \linebreak [0] 
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
] , \linebreak [0] 
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
] , \linebreak [0] 
[
\mbox{check}
] , \linebreak [0] 
[
[
\ast 
\stackrel {\circ}{=}
\ast 
]
] , \linebreak [0] 
[
RootVisible(
\ast 
)
] , \linebreak [0] 
[
A
] , \linebreak [0] 
[
R
] , \linebreak [0] 
[
C
] , \linebreak [0] 
[
T
] , \linebreak [0] 
[
L
] , \linebreak [0] 
[
\{
\ast 
\}
] , \linebreak [0] 
[\overline{
\ast }
] , \linebreak [0] 
[
\mathit{a}
] , \linebreak [0] 
[
\mathit{b}
] , \linebreak [0] 
[
\mathit{c}
] , \linebreak [0] 
[
\mathit{d}
] , \linebreak [0] 
[
\mathit{e}
] , \linebreak [0] 
[
\mathit{f}
] , \linebreak [0] 
[
\mathit{g}
] , \linebreak [0] 
[
\mathit{h}
] , \linebreak [0] 
[
\mathit{i}
] , \linebreak [0] 
[
\mathit{j}
] , \linebreak [0] 
[
\mathit{k}
] , \linebreak [0] 
[
\mathit{l}
] , \linebreak [0] 
[
\mathit{m}
] , \linebreak [0] 
[
\mathit{n}
] , \linebreak [0] 
[
\mathit{o}
] , \linebreak [0] 
[
\mathit{p}
] , \linebreak [0] 
[
\mathit{q}
] , \linebreak [0] 
[
\mathit{r}
] , \linebreak [0] 
[
\mathit{s}
] , \linebreak [0] 
[
\mathit{t}
] , \linebreak [0] 
[
\mathit{u}
] , \linebreak [0] 
[
\mathit{v}
] , \linebreak [0] 
[
\mathit{w}
] , \linebreak [0] 
[
\mathit{x}
] , \linebreak [0] 
[
\mathit{y}
] , \linebreak [0] 
[
\mathit{z}
] , \linebreak [0] 
[
\langle 
\ast 
{\equiv} 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
] , \linebreak [0] 
[
\langle 
\ast 
{\equiv}^0 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
] , \linebreak [0] 
[
\langle 
\ast 
{\equiv}^1 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
] , \linebreak [0] 
[
\langle 
\ast 
{\equiv}^* 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
] , \linebreak [0] 
[
Ded(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_0(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_1(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_2(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_3(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_4(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_4^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_5(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_6(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_6^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_7(
\ast 
)
] , \linebreak [0] 
[
Ded_8(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_8^*(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
S
] , \linebreak [0] 
[
Neg
] , \linebreak [0] 
[
MP
] , \linebreak [0] 
[
Gen
] , \linebreak [0] 
[
Ded
] , \linebreak [0] 
[
S1
] , \linebreak [0] 
[
S2
] , \linebreak [0] 
[
S3
] , \linebreak [0] 
[
S4
] , \linebreak [0] 
[
S5
] , \linebreak [0] 
[
S6
] , \linebreak [0] 
[
S7
] , \linebreak [0] 
[
S8
] , \linebreak [0] 
[
S9
] , \linebreak [0] 
[
Repetition
] , \linebreak [0] 
[
A1'
] , \linebreak [0] 
[
A2'
] , \linebreak [0] 
[
A4'
] , \linebreak [0] 
[
A5'
] , \linebreak [0] 
[
Prop\ 3.2a
] , \linebreak [0] 
[
Prop\ 3.2b
] , \linebreak [0] 
[
Prop\ 3.2c
] , \linebreak [0] 
[
Prop\ 3.2d
] , \linebreak [0] 
[
Prop\ 3.2e_1
] , \linebreak [0] 
[
Prop\ 3.2e_2
] , \linebreak [0] 
[
Prop\ 3.2e
] , \linebreak [0] 
[
Prop\ 3.2f_1
] , \linebreak [0] 
[
Prop\ 3.2f_2
] , \linebreak [0] 
[
Prop\ 3.2f
] , \linebreak [0] 
[
Prop\ 3.2g_1
] , \linebreak [0] 
[
Prop\ 3.2g_2
] , \linebreak [0] 
[
Prop\ 3.2g
] , \linebreak [0] 
[
Prop\ 3.2h_1
] , \linebreak [0] 
[
Prop\ 3.2h_2
] , \linebreak [0] 
[
Prop\ 3.2h
] , \linebreak [0] 
[
Block_1(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Block_2(
\ast 
)
] , \linebreak [0] 
[
\mbox{pred calc}
] , \linebreak [0] 
[
\mbox{pc1}
] , \linebreak [0] 
[
\mbox{pc2}
] , \linebreak [0] 
[
\mbox{pc3}
] , \linebreak [0] 
[
\mbox{pc4}
] , \linebreak [0] 
[
\mbox{pc5}
] , \linebreak [0] 
[
\mbox{pc6}
] , \linebreak [0] 
[
\mbox{pc7}
] , \linebreak [0] 
[
\mbox{pc8}
] , \linebreak [0] 
[
\mbox{pc9}
] , \linebreak [0] 
[
\mbox{pc10}
] , \linebreak [0] 
[
\mbox{pc11}
] , \linebreak [0] 
[
\mbox{pc12}
] , \linebreak [0] 
[
\mbox{pcmp}
] , \linebreak [0] 
[
\mbox{pcunsound}
] , \linebreak [0] 
[
\mbox{pcded}
] , \linebreak [0] 
[
\mbox{pcia}
] , \linebreak [0] 
[
\mbox{pcie}
] , \linebreak [0] 
[
\mbox{pcdeduction}
] , \linebreak [0] 
[
\mbox{trivia}
] , \linebreak [0] 
[
\mbox{trivia2}
] , \linebreak [0] 
[
\mbox{iatest}
] , \linebreak [0] 
[
\mbox{andintro}
] , \linebreak [0] 
[
\mbox{andelim1}
] , \linebreak [0] 
[
\mbox{andelim2}
] , \linebreak [0] 
[
\mbox{orintro1}
] , \linebreak [0] 
[
\mbox{orintro2}
] , \linebreak [0] 
[
\mbox{orelim}
] , \linebreak [0] 
[
\mbox{notintro}
] , \linebreak [0] 
[
\mbox{notnotintro}
] , \linebreak [0] 
[
\mbox{notnotelim}
] , \linebreak [0] 
[
\mbox{mt}
] , \linebreak [0] 
[
\mbox{pbc}
] , \linebreak [0] 
[
\mbox{repeat}
] , \linebreak [0] 
[
\mbox{lem}
] , \linebreak [0] 
[
\mbox{forallintro}
] , \linebreak [0] 
[
\mbox{forallelim}
] , \linebreak [0] 
[
\mbox{existsintro}
] , \linebreak [0] 
[
\mbox{existselim}
] , \linebreak [0] 
[
\mbox{bottomelim}
] , \linebreak [0] 
[
\mbox{lemnotintro}
] , \linebreak [0] 
[
\mbox{nontriv0}
] , \linebreak [0] 
[
\mbox{nontriv1}
] , \linebreak [0] 
[
\mbox{nontriv2}
] , \linebreak [0] 
[
\mbox{nontriv3}
] , \linebreak [0] 
[
\mbox{nontriv4}
] , \linebreak [0] 
[
\mbox{nontriv5}
] , \linebreak [0] 
[
\mbox{nontriv6}
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\_\{
\ast 
\}
] , \linebreak [0] 
[
\ast 
/indexintro(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast 
/intro(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast 
/bothintro(
\ast 
,
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast 
/nameintro(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast '
] , \linebreak [0] 
[
\ast 
{[}
\ast 
{]}
] , \linebreak [0] 
[
\ast 
[ 
\ast 
{\rightarrow} 
\ast 
]
] , \linebreak [0] 
[
\ast 
[ 
\ast 
{\Rightarrow} 
\ast 
]
] , \linebreak [0] 
[
\ast 
0
] , \linebreak [0] 
[
\ast 
1
] , \linebreak [0] 
[
0 \mathrm {b}
] , \linebreak [0] 
[
\ast 
\mbox {-color}( 
\ast 
)
] , \linebreak [0] 
[
\ast 
\mbox {-color}^{\ast}( 
\ast 
)
] , \linebreak [0] 
[
\ast 
{}^H
] , \linebreak [0] 
[
\ast 
{}^T
] , \linebreak [0] 
[
\ast 
{}^U
] , \linebreak [0] 
[
\ast 
{}^h
] , \linebreak [0] 
[
\ast 
{}^t
] , \linebreak [0] 
[
\ast 
{}^s
] , \linebreak [0] 
[
\ast 
{}^c
] , \linebreak [0] 
[
\ast 
{}^d
] , \linebreak [0] 
[
\ast 
{}^a
] , \linebreak [0] 
[
\ast 
{}^C
] , \linebreak [0] 
[
\ast 
{}^M
] , \linebreak [0] 
[
\ast 
{}^B
] , \linebreak [0] 
[
\ast 
{}^{r}
] , \linebreak [0] 
[
\ast 
{}^{i}
] , \linebreak [0] 
[
\ast 
{}^{d}
] , \linebreak [0] 
[
\ast 
{}^{R}
] , \linebreak [0] 
[
\ast 
{}^{0}
] , \linebreak [0] 
[
\ast 
{}^{1}
] , \linebreak [0] 
[
\ast 
{}^{2}
] , \linebreak [0] 
[
\ast 
{}^{3}
] , \linebreak [0] 
[
\ast 
{}^{4}
] , \linebreak [0] 
[
\ast 
{}^{5}
] , \linebreak [0] 
[
\ast 
{}^{6}
] , \linebreak [0] 
[
\ast 
{}^{7}
] , \linebreak [0] 
[
\ast 
{}^{8}
] , \linebreak [0] 
[
\ast 
{}^{9}
] , \linebreak [0] 
[
\ast 
{} ^ { E }
] , \linebreak [0] 
[
\ast 
{} ^ {\cal V}
] , \linebreak [0] 
[
\ast 
{} ^ {\cal C}
] , \linebreak [0] 
[
\ast 
{} ^ {{\cal C} ^ { \ast }}
] , \linebreak [0] 
[
\ast 
{}^{hide}
]
; 
\newline \mathbf {Preassociative} \newline 
[
\mbox{``}
\ast 
\mbox{''}
] , \linebreak [0] 
[
] , \linebreak [0] 
[
(
\ast 
)^{\bf t}
] , \linebreak [0] 
[
\mbox{string}(
\ast 
)+
\ast 
] , \linebreak [0] 
[\mbox{string}(
\ast 
)\mathrel{++}
\ast 
] , \linebreak [0] 
[
\newline 
\ast 
] , \linebreak [0] 
[
\linebreak [0]\ \hskip0em plus2.0em{}
\ast 
] , \linebreak [0] 
[!
\ast 
] , \linebreak [0] 
[
\mbox {\tt \char34}
\ast 
] , \linebreak [0] 
[
\#
\ast 
] , \linebreak [0] 
[
\$
\ast 
] , \linebreak [0] 
[
\%
\ast 
] , \linebreak [0] 
[
\&
\ast 
] , \linebreak [0] 
[
\mbox {'}
\ast 
] , \linebreak [0] 
[(
\ast 
] , \linebreak [0] 
[)
\ast 
] , \linebreak [0] 
[
{*}
\ast 
] , \linebreak [0] 
[
{+}
\ast 
] , \linebreak [0] 
[,
\ast 
] , \linebreak [0] 
[
\mbox{-}
\ast 
] , \linebreak [0] 
[.
\ast 
] , \linebreak [0] 
[/
\ast 
] , \linebreak [0] 
[0
\ast 
] , \linebreak [0] 
[1
\ast 
] , \linebreak [0] 
[2
\ast 
] , \linebreak [0] 
[3
\ast 
] , \linebreak [0] 
[4
\ast 
] , \linebreak [0] 
[5
\ast 
] , \linebreak [0] 
[6
\ast 
] , \linebreak [0] 
[7
\ast 
] , \linebreak [0] 
[8
\ast 
] , \linebreak [0] 
[9
\ast 
] , \linebreak [0] 
[
{:}
\ast 
] , \linebreak [0] 
[;
\ast 
] , \linebreak [0] 
[
{<}
\ast 
] , \linebreak [0] 
[
{=}
\ast 
] , \linebreak [0] 
[
{>}
\ast 
] , \linebreak [0] 
[?
\ast 
] , \linebreak [0] 
[@
\ast 
] , \linebreak [0] 
[A
\ast 
] , \linebreak [0] 
[B
\ast 
] , \linebreak [0] 
[C
\ast 
] , \linebreak [0] 
[D
\ast 
] , \linebreak [0] 
[E
\ast 
] , \linebreak [0] 
[F
\ast 
] , \linebreak [0] 
[G
\ast 
] , \linebreak [0] 
[H
\ast 
] , \linebreak [0] 
[I
\ast 
] , \linebreak [0] 
[J
\ast 
] , \linebreak [0] 
[K
\ast 
] , \linebreak [0] 
[L
\ast 
] , \linebreak [0] 
[M
\ast 
] , \linebreak [0] 
[N
\ast 
] , \linebreak [0] 
[O
\ast 
] , \linebreak [0] 
[P
\ast 
] , \linebreak [0] 
[Q
\ast 
] , \linebreak [0] 
[R
\ast 
] , \linebreak [0] 
[S
\ast 
] , \linebreak [0] 
[T
\ast 
] , \linebreak [0] 
[U
\ast 
] , \linebreak [0] 
[V
\ast 
] , \linebreak [0] 
[W
\ast 
] , \linebreak [0] 
[X
\ast 
] , \linebreak [0] 
[Y
\ast 
] , \linebreak [0] 
[Z
\ast 
] , \linebreak [0] 
[[
\ast 
] , \linebreak [0] 
[
\mbox {$\backslash $}
\ast 
] , \linebreak [0] 
[]
\ast 
] , \linebreak [0] 
[
{\char94}
\ast 
] , \linebreak [0] 
[
\_
\ast 
] , \linebreak [0] 
[
\mbox {`}
\ast 
] , \linebreak [0] 
[a
\ast 
] , \linebreak [0] 
[b
\ast 
] , \linebreak [0] 
[c
\ast 
] , \linebreak [0] 
[d
\ast 
] , \linebreak [0] 
[e
\ast 
] , \linebreak [0] 
[f
\ast 
] , \linebreak [0] 
[g
\ast 
] , \linebreak [0] 
[h
\ast 
] , \linebreak [0] 
[i
\ast 
] , \linebreak [0] 
[j
\ast 
] , \linebreak [0] 
[k
\ast 
] , \linebreak [0] 
[l
\ast 
] , \linebreak [0] 
[m
\ast 
] , \linebreak [0] 
[n
\ast 
] , \linebreak [0] 
[o
\ast 
] , \linebreak [0] 
[p
\ast 
] , \linebreak [0] 
[q
\ast 
] , \linebreak [0] 
[r
\ast 
] , \linebreak [0] 
[s
\ast 
] , \linebreak [0] 
[t
\ast 
] , \linebreak [0] 
[u
\ast 
] , \linebreak [0] 
[v
\ast 
] , \linebreak [0] 
[w
\ast 
] , \linebreak [0] 
[x
\ast 
] , \linebreak [0] 
[y
\ast 
] , \linebreak [0] 
[z
\ast 
] , \linebreak [0] 
[
\{
\ast 
] , \linebreak [0] 
[|
\ast 
] , \linebreak [0] 
[
\}
\ast 
] , \linebreak [0] 
[
\char126 
\ast 
] , \linebreak [0] 
[
\mathbf{Preassociative}\,
\ast 
; 
\ast 
] , \linebreak [0] 
[
\mathbf{Postassociative}\,
\ast 
; 
\ast 
] , \linebreak [0] 
[
[
\ast 
] , \linebreak [0] 
\ast 
] , \linebreak [0] 
[
\mathrm{priority} \,
\ast 
\, \mathrm{end}
] , \linebreak [0] 
[
newline\ 
\ast 
] , \linebreak [0] 
[
macro\ newline\ 
\ast 
] , \linebreak [0] 
[
MacroIndent(
\ast 
)
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\mathbin {\mbox {'}}
\ast 
] , \linebreak [0] 
[
\ast 
\mathbin {\mbox {`}}
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
{}'
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\mathbin {\mbox {'}}
\ast 
] , \linebreak [0] 
[
\ast 
\mathbin {\mbox {`}}
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\cdot 
\ast 
] , \linebreak [0] 
[
\ast 
\cdot_0 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
+ 
\ast 
] , \linebreak [0] 
[
\ast 
\mathop{+_0} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathop{+_1} 
\ast 
] , \linebreak [0] 
[
\ast 
- 
\ast 
] , \linebreak [0] 
[
\ast 
\mathop{-_0} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathop{-_1} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\cup \{ 
\ast 
\}
] , \linebreak [0] 
[
\ast 
\cup 
\ast 
] , \linebreak [0] 
[
\ast 
\backslash \{ 
\ast 
\}
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel { \dot { . \, . } }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { \underline { \dot { . \, . } } }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { \underline { : \, : } }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { \underline { {+} 2 \ast } }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { : \, : }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { {+} 2 \ast }
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
,\linebreak [0] 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\stackrel {B}{\approx }
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {D}{\approx }
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {C}{\approx }
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {P}{\approx }
\ast 
] , \linebreak [0] 
[
\ast 
\approx 
\ast 
] , \linebreak [0] 
[
\ast 
=
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {+}{\rightarrow }
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {t}{=}
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {t^*}{=}
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel{r}{=} 
\ast 
] , \linebreak [0] 
[
\ast 
\in_t 
\ast 
] , \linebreak [0] 
[
\ast 
\subseteq_T 
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel{T}{=} 
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel{s}{=} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {free\ in} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {free\ in}^{\ast} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {free\ for} 
\ast 
\mathrel {in} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {free\ for}^{\ast} 
\ast 
\mathrel {in} 
\ast 
] , \linebreak [0] 
[
\ast 
\in_c 
\ast 
] , \linebreak [0] 
[
\ast 
< 
\ast 
] , \linebreak [0] 
[
\ast 
<' 
\ast 
] , \linebreak [0] 
[
\ast 
\le' 
\ast 
] , \linebreak [0] 
[
\ast 
= 
\ast 
] , \linebreak [0] 
[
\ast 
\neq 
\ast 
] , \linebreak [0] 
[
\ast 
{}^{var}
] , \linebreak [0] 
[
\ast 
\#^0
\ast 
] , \linebreak [0] 
[
\ast 
\#^1
\ast 
] , \linebreak [0] 
[
\ast 
\#^*
\ast 
] , \linebreak [0] 
[
\ast  \equiv 
\ast 
] , \linebreak [0] 
[
\ast  = 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
{\neg }
\ast 
] , \linebreak [0] 
[\neg 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\wedge 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\ddot{\wedge}} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\tilde{\wedge}} 
\ast 
] , \linebreak [0] 
[
\ast 
\wedge_c 
\ast 
] , \linebreak [0] 
[
\ast  \wedge 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\vee 
\ast 
] , \linebreak [0] 
[
\ast 
\parallel 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\ddot{\vee}} 
\ast 
] , \linebreak [0] 
[
\ast  \vee 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\exists 
\ast 
\colon 
\ast 
] , \linebreak [0] 
[
\forall 
\ast 
\colon 
\ast 
] , \linebreak [0] 
[
\forall_{obj} 
\ast 
\colon 
\ast 
] , \linebreak [0] 
[\forall 
\ast  . \left(
\ast \right)
] , \linebreak [0] 
[\exists 
\ast  . \left(
\ast \right)
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel{\ddot{\Rightarrow}} 
\ast 
] , \linebreak [0] 
[
\ast 
\Rightarrow 
\ast 
] , \linebreak [0] 
[
\ast 
\Leftrightarrow 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
:
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{spy}
\ast 
] , \linebreak [0] 
[
\ast 
!
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\left\{\protect \begin {array}{l}
\ast 
\\
\ast 
\protect \end {array}\right.
]
; 
\newline \mathbf {Preassociative} \newline 
[
\lambda 
\ast 
.
\ast 
] , \linebreak [0] 
[
\Lambda 
\ast 
. 
\ast 
] , \linebreak [0] 
[
\Lambda 
\ast 
] , \linebreak [0] 
[
{\bf if} \ 
\ast 
\ {\bf then} \ 
\ast 
\ {\bf else} \ 
\ast 
] , \linebreak [0] 
[
\mathbf{let\ } 
\ast 
= 
\ast 
\mathbf{\ in\ } 
\ast 
] , \linebreak [0] 
[
\mathbf{let\ } 
\ast 
\mathrel{\ddot{=}} 
\ast 
\mathrel{\ in\ } 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\#
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
{} ^ { I }
] , \linebreak [0] 
[
\ast 
{} ^ { \rhd }
] , \linebreak [0] 
[
\ast 
{} ^ { V } 
] , \linebreak [0] 
[
\ast 
{} ^ { + } 
] , \linebreak [0] 
[
\ast 
{} ^ { - } 
] , \linebreak [0] 
[
\ast 
{} ^ { \ast } 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\mathop {\char64} 
\ast 
] , \linebreak [0] 
[
\ast 
\rhd 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\ast 
] , \linebreak [0] 
[
\ast 
\gg 
\ast 
] , \linebreak [0] 
[
\ast 
\unrhd 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\vdash 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {i.e.} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\forall 
\ast 
\colon 
\ast 
] , \linebreak [0] 
[
\Pi 
\ast 
\colon 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel {\oplus} 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
; 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\ proves\ 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\mathbf{\ proof\ of\ } 
\ast 
: 
\ast 
] , \linebreak [0] 
[
Line \, 
\ast 
: 
\ast 
\gg 
\ast 
; 
\ast 
] , \linebreak [0] 
[
Last\ line \, 
\ast 
\gg 
\ast 
\,\Box
] , \linebreak [0] 
[
Line \, 
\ast 
: Premise \gg 
\ast 
;
\ast 
] , \linebreak [0] 
[
Line \, 
\ast 
: \mbox{Side-condition} \gg 
\ast 
; 
\ast 
] , \linebreak [0] 
[
Arbitrary \gg 
\ast 
;
\ast 
] , \linebreak [0] 
[
Local \gg 
\ast 
= 
\ast 
; 
\ast 
] , \linebreak [0] 
[
Begin \, 
\ast 
 ; 
\ast 
 : End ; 
\ast 
] , \linebreak [0] 
[
Last\ block\ line \, 
\ast 
\gg 
\ast 
\,;
] , \linebreak [0] 
[
Arbitrary \gg 
\ast 
;
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel{|} 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel{\mathrm{,}}
\ast 
] , \linebreak [0] 
[
\ast 
{[}
\ast 
{]}
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\& 
\ast 
] , \linebreak [0] 
[
 \rightarrow 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\backslash \backslash 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{linebreak[4]} 
\ast 
] , \linebreak [0] 
[
\ast 
\backslash \backslash 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast  \in 
\ast 
]
; 
]$

$\rm
[
\mbox{problemtwo}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}problemtwo
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:1}{}\section* {$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $}

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwprooflinep
\mbox {$\backslash $}endcsname
\linebreak [0]\ \hskip0em plus2.0em{}L
\_o
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}else
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwello
\mbox {$\backslash $}endcsname
\newline 
\mbox {$\backslash $}global
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}advance
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}by
\linebreak [0]\ \hskip0em plus2.0em{}1
\newline 
\mbox {$\backslash $}xdef
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\{L
\mbox {$\backslash $}ifnum
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
{<}10
\linebreak [0]\ \hskip0em plus2.0em{}0
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}number
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\newline 
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ell
\linebreak [0]\ \hskip0em plus2.0em{}aa
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:2}{}\section* {$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $}

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwprooflinep
\mbox {$\backslash $}endcsname
\linebreak [0]\ \hskip0em plus2.0em{}L
\_o
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}else
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwello
\mbox {$\backslash $}endcsname
\newline 
\mbox {$\backslash $}global
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}advance
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}by
\linebreak [0]\ \hskip0em plus2.0em{}1
\newline 
\mbox {$\backslash $}xdef
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\{L
\mbox {$\backslash $}ifnum
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
{<}10
\linebreak [0]\ \hskip0em plus2.0em{}0
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}number
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\newline 
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ell
\linebreak [0]\ \hskip0em plus2.0em{}ab
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:3}{}\section* {$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $}

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwprooflinep
\mbox {$\backslash $}endcsname
\linebreak [0]\ \hskip0em plus2.0em{}L
\_o
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}else
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwello
\mbox {$\backslash $}endcsname
\newline 
\mbox {$\backslash $}global
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}advance
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}by
\linebreak [0]\ \hskip0em plus2.0em{}1
\newline 
\mbox {$\backslash $}xdef
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\{L
\mbox {$\backslash $}ifnum
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
{<}10
\linebreak [0]\ \hskip0em plus2.0em{}0
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}number
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\newline 
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ell
\linebreak [0]\ \hskip0em plus2.0em{}ac
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:4}{}\section* {$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $}

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwprooflinep
\mbox {$\backslash $}endcsname
\linebreak [0]\ \hskip0em plus2.0em{}L
\_o
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}else
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwello
\mbox {$\backslash $}endcsname
\newline 
\mbox {$\backslash $}global
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}advance
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}by
\linebreak [0]\ \hskip0em plus2.0em{}1
\newline 
\mbox {$\backslash $}xdef
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\{L
\mbox {$\backslash $}ifnum
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
{<}10
\linebreak [0]\ \hskip0em plus2.0em{}0
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}number
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\newline 
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ell
\linebreak [0]\ \hskip0em plus2.0em{}ad
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:5}{}\section* {$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $}

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwprooflinep
\mbox {$\backslash $}endcsname
\linebreak [0]\ \hskip0em plus2.0em{}L
\_o
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}else
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwello
\mbox {$\backslash $}endcsname
\newline 
\mbox {$\backslash $}global
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}advance
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}by
\linebreak [0]\ \hskip0em plus2.0em{}1
\newline 
\mbox {$\backslash $}xdef
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\{L
\mbox {$\backslash $}ifnum
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
{<}10
\linebreak [0]\ \hskip0em plus2.0em{}0
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}number
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\newline 
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ell
\linebreak [0]\ \hskip0em plus2.0em{}ae
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:6}{}\section* {$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $}

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwprooflinep
\mbox {$\backslash $}endcsname
\linebreak [0]\ \hskip0em plus2.0em{}L
\_o
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}else
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwello
\mbox {$\backslash $}endcsname
\newline 
\mbox {$\backslash $}global
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}advance
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}by
\linebreak [0]\ \hskip0em plus2.0em{}1
\newline 
\mbox {$\backslash $}xdef
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\{L
\mbox {$\backslash $}ifnum
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
{<}10
\linebreak [0]\ \hskip0em plus2.0em{}0
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}number
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\newline 
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ell
\linebreak [0]\ \hskip0em plus2.0em{}af
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:7}{}\section* {$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $}

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwprooflinep
\mbox {$\backslash $}endcsname
\linebreak [0]\ \hskip0em plus2.0em{}L
\_o
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}else
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwello
\mbox {$\backslash $}endcsname
\newline 
\mbox {$\backslash $}global
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}advance
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}by
\linebreak [0]\ \hskip0em plus2.0em{}1
\newline 
\mbox {$\backslash $}xdef
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\{L
\mbox {$\backslash $}ifnum
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
{<}10
\linebreak [0]\ \hskip0em plus2.0em{}0
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}number
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\newline 
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ell
\linebreak [0]\ \hskip0em plus2.0em{}ag
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:8}{}\section* {$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $}

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwprooflinep
\mbox {$\backslash $}endcsname
\linebreak [0]\ \hskip0em plus2.0em{}L
\_o
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}else
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwello
\mbox {$\backslash $}endcsname
\newline 
\mbox {$\backslash $}global
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}advance
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}by
\linebreak [0]\ \hskip0em plus2.0em{}1
\newline 
\mbox {$\backslash $}xdef
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\{L
\mbox {$\backslash $}ifnum
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
{<}10
\linebreak [0]\ \hskip0em plus2.0em{}0
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}number
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\newline 
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ell
\linebreak [0]\ \hskip0em plus2.0em{}ah
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:9}{}\section* {$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $}

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwprooflinep
\mbox {$\backslash $}endcsname
\linebreak [0]\ \hskip0em plus2.0em{}L
\_o
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}else
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwello
\mbox {$\backslash $}endcsname
\newline 
\mbox {$\backslash $}global
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}advance
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}by
\linebreak [0]\ \hskip0em plus2.0em{}1
\newline 
\mbox {$\backslash $}xdef
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\{L
\mbox {$\backslash $}ifnum
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
{<}10
\linebreak [0]\ \hskip0em plus2.0em{}0
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}number
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\newline 
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ell
\linebreak [0]\ \hskip0em plus2.0em{}ai
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:10}{}\section* {$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $}

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwprooflinep
\mbox {$\backslash $}endcsname
\linebreak [0]\ \hskip0em plus2.0em{}L
\_o
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}else
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwello
\mbox {$\backslash $}endcsname
\newline 
\mbox {$\backslash $}global
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}advance
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}by
\linebreak [0]\ \hskip0em plus2.0em{}1
\newline 
\mbox {$\backslash $}xdef
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\{L
\mbox {$\backslash $}ifnum
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
{<}10
\linebreak [0]\ \hskip0em plus2.0em{}0
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}number
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\newline 
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ell
\linebreak [0]\ \hskip0em plus2.0em{}aj
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:11}{}\section* {$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $}

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwprooflinep
\mbox {$\backslash $}endcsname
\linebreak [0]\ \hskip0em plus2.0em{}L
\_o
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}else
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwello
\mbox {$\backslash $}endcsname
\newline 
\mbox {$\backslash $}global
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}advance
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}by
\linebreak [0]\ \hskip0em plus2.0em{}1
\newline 
\mbox {$\backslash $}xdef
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\{L
\mbox {$\backslash $}ifnum
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
{<}10
\linebreak [0]\ \hskip0em plus2.0em{}0
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}number
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\newline 
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ell
\linebreak [0]\ \hskip0em plus2.0em{}ak
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:12}{}\section* {$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $}

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwprooflinep
\mbox {$\backslash $}endcsname
\linebreak [0]\ \hskip0em plus2.0em{}L
\_o
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}else
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwello
\mbox {$\backslash $}endcsname
\newline 
\mbox {$\backslash $}global
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}advance
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}by
\linebreak [0]\ \hskip0em plus2.0em{}1
\newline 
\mbox {$\backslash $}xdef
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\{L
\mbox {$\backslash $}ifnum
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
{<}10
\linebreak [0]\ \hskip0em plus2.0em{}0
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}number
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\newline 
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ell
\linebreak [0]\ \hskip0em plus2.0em{}al
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:13}{}\section* {$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $}

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwprooflinep
\mbox {$\backslash $}endcsname
\linebreak [0]\ \hskip0em plus2.0em{}L
\_o
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}else
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwello
\mbox {$\backslash $}endcsname
\newline 
\mbox {$\backslash $}global
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}advance
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}by
\linebreak [0]\ \hskip0em plus2.0em{}1
\newline 
\mbox {$\backslash $}xdef
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\{L
\mbox {$\backslash $}ifnum
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
{<}10
\linebreak [0]\ \hskip0em plus2.0em{}0
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}number
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\newline 
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ell
\linebreak [0]\ \hskip0em plus2.0em{}am
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:14}{}\section* {$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $}

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwprooflinep
\mbox {$\backslash $}endcsname
\linebreak [0]\ \hskip0em plus2.0em{}L
\_o
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}else
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwello
\mbox {$\backslash $}endcsname
\newline 
\mbox {$\backslash $}global
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}advance
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}by
\linebreak [0]\ \hskip0em plus2.0em{}1
\newline 
\mbox {$\backslash $}xdef
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\{L
\mbox {$\backslash $}ifnum
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
{<}10
\linebreak [0]\ \hskip0em plus2.0em{}0
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}number
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\newline 
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ell
\linebreak [0]\ \hskip0em plus2.0em{}an
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:15}{}\section* {$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $}

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwprooflinep
\mbox {$\backslash $}endcsname
\linebreak [0]\ \hskip0em plus2.0em{}L
\_o
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}else
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwello
\mbox {$\backslash $}endcsname
\newline 
\mbox {$\backslash $}global
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}advance
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}by
\linebreak [0]\ \hskip0em plus2.0em{}1
\newline 
\mbox {$\backslash $}xdef
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\{L
\mbox {$\backslash $}ifnum
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
{<}10
\linebreak [0]\ \hskip0em plus2.0em{}0
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}number
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\newline 
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ell
\linebreak [0]\ \hskip0em plus2.0em{}ao
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:16}{}\section* {$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $}

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwprooflinep
\mbox {$\backslash $}endcsname
\linebreak [0]\ \hskip0em plus2.0em{}L
\_o
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}else
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwello
\mbox {$\backslash $}endcsname
\newline 
\mbox {$\backslash $}global
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}advance
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}by
\linebreak [0]\ \hskip0em plus2.0em{}1
\newline 
\mbox {$\backslash $}xdef
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\{L
\mbox {$\backslash $}ifnum
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
{<}10
\linebreak [0]\ \hskip0em plus2.0em{}0
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}number
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\newline 
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ell
\linebreak [0]\ \hskip0em plus2.0em{}ap
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:17}{}\section* {$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $}

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwprooflinep
\mbox {$\backslash $}endcsname
\linebreak [0]\ \hskip0em plus2.0em{}L
\_o
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}else
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwello
\mbox {$\backslash $}endcsname
\newline 
\mbox {$\backslash $}global
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}advance
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}by
\linebreak [0]\ \hskip0em plus2.0em{}1
\newline 
\mbox {$\backslash $}xdef
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\{L
\mbox {$\backslash $}ifnum
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
{<}10
\linebreak [0]\ \hskip0em plus2.0em{}0
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}number
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\newline 
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ell
\linebreak [0]\ \hskip0em plus2.0em{}aq
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:18}{}\section* {$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $}

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwprooflinep
\mbox {$\backslash $}endcsname
\linebreak [0]\ \hskip0em plus2.0em{}L
\_o
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}else
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwello
\mbox {$\backslash $}endcsname
\newline 
\mbox {$\backslash $}global
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}advance
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}by
\linebreak [0]\ \hskip0em plus2.0em{}1
\newline 
\mbox {$\backslash $}xdef
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\{L
\mbox {$\backslash $}ifnum
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
{<}10
\linebreak [0]\ \hskip0em plus2.0em{}0
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}number
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\newline 
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ell
\linebreak [0]\ \hskip0em plus2.0em{}ar
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:19}{}\section* {$\rm 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi $}

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwprooflinep
\mbox {$\backslash $}endcsname
\linebreak [0]\ \hskip0em plus2.0em{}L
\_o
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}else
\newline 
\mbox {$\backslash $}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}relax
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}csname
\linebreak [0]\ \hskip0em plus2.0em{}lgwello
\mbox {$\backslash $}endcsname
\newline 
\mbox {$\backslash $}global
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}advance
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}by
\linebreak [0]\ \hskip0em plus2.0em{}1
\newline 
\mbox {$\backslash $}xdef
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\{L
\mbox {$\backslash $}ifnum
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
{<}10
\linebreak [0]\ \hskip0em plus2.0em{}0
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}number
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\newline 
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwello
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ell
\linebreak [0]\ \hskip0em plus2.0em{}as
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:20}{}\section* {$\rm 
\mbox{pred calc}$}

$\rm
[
\mbox{pred calc}
\stackrel{
stmt
}{\rightarrow}
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf {r}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf {f}}
\colon 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {r}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {g}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\langle 
\lceil \underline{
\mathsf {g}}
\rceil 
{\equiv}^0 
\lceil \underline{
\mathsf {f}}
\rceil 
| 
\lceil \underline{
\mathsf{x}}
\rceil 
{:=} 
\lceil \underline{
\mathsf {r}}
\rceil 
\rangle 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \forall \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
\Rightarrow \underline{
\mathsf {g}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}} \wedge \underline{
\mathsf {g}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}} \wedge \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {g}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {g}}
\vdash \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \neg \underline{
\mathsf {g}}
\Rightarrow \neg \underline{
\mathsf {f}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf{x}}
\colon 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {g}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\vdash \exists \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
\Rightarrow \underline{
\mathsf {g}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}} \vee \underline{
\mathsf {f}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\vdash \underline{
\mathsf {g}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {f}}
\colon \neg \neg \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf {h}}
\colon \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {h}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {h}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\lambda 
\mathsf{x}
.
Ded_0(
\lceil \underline{
\mathsf {a}}
\rceil 
,
\lceil \underline{
\mathsf {b}}
\rceil 
)
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {b}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}} \wedge \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf {h}}
\colon \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {h}}
\Rightarrow \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}} \vee \underline{
\mathsf {h}}
\Rightarrow \underline{
\mathsf {g}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf{x}}
\colon 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {g}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {g}}
\Rightarrow \forall \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
\mathrel {\oplus} 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}} \vee \underline{
\mathsf {g}}
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf {r}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf {f}}
\colon 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {r}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {g}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\langle 
\lceil \underline{
\mathsf {g}}
\rceil 
{\equiv}^0 
\lceil \underline{
\mathsf {f}}
\rceil 
| 
\lceil \underline{
\mathsf{x}}
\rceil 
{:=} 
\lceil \underline{
\mathsf {r}}
\rceil 
\rangle 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \underline{
\mathsf {g}}
\Rightarrow \exists \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
]$

$\rm
[
\mbox{pred calc}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pred
\linebreak [0]\ \hskip0em plus2.0em{}calc
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:21}{}\section* {$\rm 
\mbox{pc1}$}

$\rm
[
\mbox{pc1}
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
\mbox{pc1}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}}
]$

$\rm
[
\mbox{pc1}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pc1
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:22}{}\section* {$\rm 
\mbox{pc2}$}

$\rm
[
\mbox{pc2}
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
\mbox{pc2}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf {h}}
\colon \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {h}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {h}}
]$

$\rm
[
\mbox{pc2}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pc2
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:23}{}\section* {$\rm 
\mbox{pc3}$}

$\rm
[
\mbox{pc3}
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
\mbox{pc3}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}} \wedge \underline{
\mathsf {g}}
]$

$\rm
[
\mbox{pc3}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pc3
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:24}{}\section* {$\rm 
\mbox{pc4}$}

$\rm
[
\mbox{pc4}
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
\mbox{pc4}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}} \vee \underline{
\mathsf {g}}
]$

$\rm
[
\mbox{pc4}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pc4
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:25}{}\section* {$\rm 
\mbox{pc5}$}

$\rm
[
\mbox{pc5}
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
\mbox{pc5}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}} \vee \underline{
\mathsf {f}}
]$

$\rm
[
\mbox{pc5}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pc5
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:26}{}\section* {$\rm 
\mbox{pc6}$}

$\rm
[
\mbox{pc6}
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
\mbox{pc6}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}} \wedge \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}}
]$

$\rm
[
\mbox{pc6}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pc6
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:27}{}\section* {$\rm 
\mbox{pc7}$}

$\rm
[
\mbox{pc7}
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
\mbox{pc7}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}} \wedge \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {g}}
]$

$\rm
[
\mbox{pc7}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pc7
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:28}{}\section* {$\rm 
\mbox{pc8}$}

$\rm
[
\mbox{pc8}
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
\mbox{pc8}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf {h}}
\colon \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {h}}
\Rightarrow \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}} \vee \underline{
\mathsf {h}}
\Rightarrow \underline{
\mathsf {g}}
]$

$\rm
[
\mbox{pc8}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pc8
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:29}{}\section* {$\rm 
\mbox{pc9}$}

$\rm
[
\mbox{pc9}
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
\mbox{pc9}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \neg \underline{
\mathsf {g}}
\Rightarrow \neg \underline{
\mathsf {f}}
]$

$\rm
[
\mbox{pc9}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pc9
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:30}{}\section* {$\rm 
\mbox{pc10}$}

$\rm
[
\mbox{pc10}
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
\mbox{pc10}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon \neg \neg \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
]$

$\rm
[
\mbox{pc10}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pc10
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:31}{}\section* {$\rm 
\mbox{pc11}$}

$\rm
[
\mbox{pc11}
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
\mbox{pc11}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf {r}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf {f}}
\colon 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {r}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {g}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\langle 
\lceil \underline{
\mathsf {g}}
\rceil 
{\equiv}^0 
\lceil \underline{
\mathsf {f}}
\rceil 
| 
\lceil \underline{
\mathsf{x}}
\rceil 
{:=} 
\lceil \underline{
\mathsf {r}}
\rceil 
\rangle 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \forall \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
\Rightarrow \underline{
\mathsf {g}}
]$

$\rm
[
\mbox{pc11}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pc11
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:32}{}\section* {$\rm 
\mbox{pc12}$}

$\rm
[
\mbox{pc12}
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
\mbox{pc12}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf {r}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf {f}}
\colon 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {r}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {g}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\langle 
\lceil \underline{
\mathsf {g}}
\rceil 
{\equiv}^0 
\lceil \underline{
\mathsf {f}}
\rceil 
| 
\lceil \underline{
\mathsf{x}}
\rceil 
{:=} 
\lceil \underline{
\mathsf {r}}
\rceil 
\rangle 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \underline{
\mathsf {g}}
\Rightarrow \exists \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
]$

$\rm
[
\mbox{pc12}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pc12
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:33}{}\section* {$\rm 
\mbox{pcmp}$}

$\rm
[
\mbox{pcmp}
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
\mbox{pcmp}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\vdash \underline{
\mathsf {g}}
]$

$\rm
[
\mbox{pcmp}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pcmp
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:34}{}\section* {$\rm 
\mbox{pcunsound}$}

$\rm
[
\mbox{pcunsound}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pcunsound
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:35}{}\section* {$\rm 
\mbox{pcded}$}

$\rm
[
\mbox{pcded}
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
\mbox{pcded}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {g}}
\vdash \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
]$

$\rm
[
\mbox{pcded}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pcded
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:36}{}\section* {$\rm 
\mbox{pcia}$}

$\rm
[
\mbox{pcia}
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
\mbox{pcia}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf{x}}
\colon 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {g}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {g}}
\Rightarrow \forall \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
]$

$\rm
[
\mbox{pcia}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pcia
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:37}{}\section* {$\rm 
\mbox{pcie}$}

$\rm
[
\mbox{pcie}
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
\mbox{pcie}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf{x}}
\colon 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {g}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\vdash \exists \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
\Rightarrow \underline{
\mathsf {g}}
]$

$\rm
[
\mbox{pcie}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pcie
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:38}{}\section* {$\rm 
\mbox{pcdeduction}$}

$\rm
[
\mbox{pcdeduction}
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
\mbox{pcdeduction}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\lambda 
\mathsf{x}
.
Ded_0(
\lceil \underline{
\mathsf {a}}
\rceil 
,
\lceil \underline{
\mathsf {b}}
\rceil 
)
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {b}}
]$

$\rm
[
\mbox{pcdeduction}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pcdeduction
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:39}{}\section* {$\rm 
\mbox{trivia}$}

$\rm
[
\mbox{trivia}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\mbox{pc2}
\gg \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
; 
\mbox{pc1}
\gg \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
; 
\mbox{pcmp}
\rhd \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\rhd \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\gg \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
; 
\mbox{pc1}
\gg \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
; 
\mbox{pcmp}
\rhd \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\rhd \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\gg \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{trivia}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
]$

$\rm
[
\mbox{trivia}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}trivia
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:40}{}\section* {$\rm 
\mbox{trivia2}$}

$\rm
[
\mbox{trivia2}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {f}}
\colon \underline{
\mathsf {f}}
\vdash 
\mbox{repeat}
\rhd \underline{
\mathsf {f}}
\gg \underline{
\mathsf {f}}
; 
\mbox{pcdeduction}
\rhd 
\forall \underline{
\mathsf {f}}
\colon \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {f}}
\gg \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {f}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{trivia2}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {f}}
]$

$\rm
[
\mbox{trivia2}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}trivia2
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:41}{}\section* {$\rm 
\mbox{iatest}$}

$\rm
[
\mbox{iatest}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\lceil \underline{
\mathsf{y}}
\rceil 
\#^0
\lceil \underline{
\mathsf {g}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mbox{pc1}
\gg \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf{y}}
\Rightarrow \underline{
\mathsf {g}}
; 
\mbox{pcia}
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\lceil \underline{
\mathsf{y}}
\rceil 
\#^0
\lceil \underline{
\mathsf {g}}
\rceil 
\rhd \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf{y}}
\Rightarrow \underline{
\mathsf {g}}
\gg \underline{
\mathsf {g}}
\Rightarrow \forall \underline{
\mathsf{y}} . \left(\underline{
\mathsf{y}}
\Rightarrow \underline{
\mathsf {g}}\right)
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{iatest}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\lceil \underline{
\mathsf{y}}
\rceil 
\#^0
\lceil \underline{
\mathsf {g}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \underline{
\mathsf {g}}
\Rightarrow \forall \underline{
\mathsf{y}} . \left(\underline{
\mathsf{y}}
\Rightarrow \underline{
\mathsf {g}}\right)
]$

$\rm
[
\mbox{iatest}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}iatest
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:42}{}\section* {$\rm 
\mbox{andintro}$}

$\rm
[
\mbox{andintro}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {g}}
\vdash 
\mbox{pc3}
\gg \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}} \wedge \underline{
\mathsf {g}}
; 
\mbox{pcmp}
\rhd \underline{
\mathsf {f}}
\rhd \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}} \wedge \underline{
\mathsf {g}}
\gg \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}} \wedge \underline{
\mathsf {g}}
; 
\mbox{pcmp}
\rhd \underline{
\mathsf {g}}
\rhd \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}} \wedge \underline{
\mathsf {g}}
\gg \underline{
\mathsf {f}} \wedge \underline{
\mathsf {g}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{andintro}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {g}}
\vdash \underline{
\mathsf {f}} \wedge \underline{
\mathsf {g}}
]$

$\rm
[
\mbox{andintro}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}andintro
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:43}{}\section* {$\rm 
\mbox{andelim1}$}

$\rm
[
\mbox{andelim1}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}} \wedge \underline{
\mathsf {g}}
\vdash 
\mbox{pc6}
\gg \underline{
\mathsf {f}} \wedge \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}}
; 
\mbox{pcmp}
\rhd \underline{
\mathsf {f}} \wedge \underline{
\mathsf {g}}
\rhd \underline{
\mathsf {f}} \wedge \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}}
\gg \underline{
\mathsf {f}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{andelim1}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}} \wedge \underline{
\mathsf {g}}
\vdash \underline{
\mathsf {f}}
]$

$\rm
[
\mbox{andelim1}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}andelim1
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:44}{}\section* {$\rm 
\mbox{andelim2}$}

$\rm
[
\mbox{andelim2}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}} \wedge \underline{
\mathsf {g}}
\vdash 
\mbox{pc7}
\gg \underline{
\mathsf {f}} \wedge \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {g}}
; 
\mbox{pcmp}
\rhd \underline{
\mathsf {f}} \wedge \underline{
\mathsf {g}}
\rhd \underline{
\mathsf {f}} \wedge \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {g}}
\gg \underline{
\mathsf {g}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{andelim2}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}} \wedge \underline{
\mathsf {g}}
\vdash \underline{
\mathsf {g}}
]$

$\rm
[
\mbox{andelim2}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}andelim2
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:45}{}\section* {$\rm 
\mbox{orintro1}$}

$\rm
[
\mbox{orintro1}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\vdash 
\mbox{pc4}
\gg \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}} \vee \underline{
\mathsf {g}}
; 
\mbox{pcmp}
\rhd \underline{
\mathsf {f}}
\rhd \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}} \vee \underline{
\mathsf {g}}
\gg \underline{
\mathsf {f}} \vee \underline{
\mathsf {g}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{orintro1}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {f}} \vee \underline{
\mathsf {g}}
]$

$\rm
[
\mbox{orintro1}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}orintro1
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:46}{}\section* {$\rm 
\mbox{orintro2}$}

$\rm
[
\mbox{orintro2}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {g}}
\vdash 
\mbox{pc5}
\gg \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}} \vee \underline{
\mathsf {g}}
; 
\mbox{pcmp}
\rhd \underline{
\mathsf {g}}
\rhd \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}} \vee \underline{
\mathsf {g}}
\gg \underline{
\mathsf {f}} \vee \underline{
\mathsf {g}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{orintro2}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {g}}
\vdash \underline{
\mathsf {f}} \vee \underline{
\mathsf {g}}
]$

$\rm
[
\mbox{orintro2}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}orintro2
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:47}{}\section* {$\rm 
\mbox{orelim}$}

$\rm
[
\mbox{orelim}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf {h}}
\colon \underline{
\mathsf {f}} \vee \underline{
\mathsf {g}}
\vdash \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {h}}
\vdash \underline{
\mathsf {g}}
\vdash \underline{
\mathsf {h}}
\vdash 
\mbox{pcded}
\rhd \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {h}}
\gg \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {h}}
; 
\mbox{pcded}
\rhd \underline{
\mathsf {g}}
\vdash \underline{
\mathsf {h}}
\gg \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {h}}
; 
\mbox{pc8}
\gg \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {h}}
\Rightarrow \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {h}}
\Rightarrow \underline{
\mathsf {f}} \vee \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {h}}
; 
\mbox{pcmp}
\rhd \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {h}}
\rhd \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {h}}
\Rightarrow \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {h}}
\Rightarrow \underline{
\mathsf {f}} \vee \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {h}}
\gg \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {h}}
\Rightarrow \underline{
\mathsf {f}} \vee \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {h}}
; 
\mbox{pcmp}
\rhd \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {h}}
\rhd \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {h}}
\Rightarrow \underline{
\mathsf {f}} \vee \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {h}}
\gg \underline{
\mathsf {f}} \vee \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {h}}
; 
\mbox{pcmp}
\rhd \underline{
\mathsf {f}} \vee \underline{
\mathsf {g}}
\rhd \underline{
\mathsf {f}} \vee \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {h}}
\gg \underline{
\mathsf {h}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{orelim}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf {h}}
\colon \underline{
\mathsf {f}} \vee \underline{
\mathsf {g}}
\vdash \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {h}}
\vdash \underline{
\mathsf {g}}
\vdash \underline{
\mathsf {h}}
\vdash \underline{
\mathsf {h}}
]$

$\rm
[
\mbox{orelim}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}orelim
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:48}{}\section* {$\rm 
\mbox{notintro}$}

$\rm
[
\mbox{notintro}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {g}}
\vdash \underline{
\mathsf {f}}
\vdash \neg \underline{
\mathsf {g}}
\vdash 
\mbox{pcded}
\rhd \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {g}}
\gg \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
; 
\mbox{pcded}
\rhd \underline{
\mathsf {f}}
\vdash \neg \underline{
\mathsf {g}}
\gg \underline{
\mathsf {f}}
\Rightarrow \neg \underline{
\mathsf {g}}
; 
\mbox{pc9}
\gg \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \neg \underline{
\mathsf {g}}
\Rightarrow \neg \underline{
\mathsf {f}}
; 
\mbox{pcmp}
\rhd \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\rhd \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \neg \underline{
\mathsf {g}}
\Rightarrow \neg \underline{
\mathsf {f}}
\gg \underline{
\mathsf {f}}
\Rightarrow \neg \underline{
\mathsf {g}}
\Rightarrow \neg \underline{
\mathsf {f}}
; 
\mbox{pcmp}
\rhd \underline{
\mathsf {f}}
\Rightarrow \neg \underline{
\mathsf {g}}
\rhd \underline{
\mathsf {f}}
\Rightarrow \neg \underline{
\mathsf {g}}
\Rightarrow \neg \underline{
\mathsf {f}}
\gg \neg \underline{
\mathsf {f}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{notintro}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {g}}
\vdash \underline{
\mathsf {f}}
\vdash \neg \underline{
\mathsf {g}}
\vdash \neg \underline{
\mathsf {f}}
]$

$\rm
[
\mbox{notintro}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}notintro
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:49}{}\section* {$\rm 
\mbox{notnotintro}$}

$\rm
[
\mbox{notnotintro}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon \underline{
\mathsf {f}}
\vdash 
\forall \underline{
\mathsf {f}}
\colon \underline{
\mathsf {f}}
\vdash \neg \underline{
\mathsf {f}}
\vdash 
\mbox{repeat}
\rhd \underline{
\mathsf {f}}
\gg \underline{
\mathsf {f}}
; 
\mbox{pcdeduction}
\rhd 
\forall \underline{
\mathsf {f}}
\colon \underline{
\mathsf {f}}
\vdash \neg \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {f}}
\gg \underline{
\mathsf {f}}
\Rightarrow \neg \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
; 
\mbox{pcmp}
\rhd \underline{
\mathsf {f}}
\rhd \underline{
\mathsf {f}}
\Rightarrow \neg \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\gg \neg \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
; 
\mbox{trivia}
\gg \neg \underline{
\mathsf {f}}
\Rightarrow \neg \underline{
\mathsf {f}}
; 
\mbox{pc9}
\gg \neg \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \neg \underline{
\mathsf {f}}
\Rightarrow \neg \underline{
\mathsf {f}}
\Rightarrow \neg \neg \underline{
\mathsf {f}}
; 
\mbox{pcmp}
\rhd \neg \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\rhd \neg \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\Rightarrow \neg \underline{
\mathsf {f}}
\Rightarrow \neg \underline{
\mathsf {f}}
\Rightarrow \neg \neg \underline{
\mathsf {f}}
\gg \neg \underline{
\mathsf {f}}
\Rightarrow \neg \underline{
\mathsf {f}}
\Rightarrow \neg \neg \underline{
\mathsf {f}}
; 
\mbox{pcmp}
\rhd \neg \underline{
\mathsf {f}}
\Rightarrow \neg \underline{
\mathsf {f}}
\rhd \neg \underline{
\mathsf {f}}
\Rightarrow \neg \underline{
\mathsf {f}}
\Rightarrow \neg \neg \underline{
\mathsf {f}}
\gg \neg \neg \underline{
\mathsf {f}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{notnotintro}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon \underline{
\mathsf {f}}
\vdash \neg \neg \underline{
\mathsf {f}}
]$

$\rm
[
\mbox{notnotintro}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}notnotintro
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:50}{}\section* {$\rm 
\mbox{notnotelim}$}

$\rm
[
\mbox{notnotelim}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon \neg \neg \underline{
\mathsf {f}}
\vdash 
\mbox{pc10}
\gg \neg \neg \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
; 
\mbox{pcmp}
\rhd \neg \neg \underline{
\mathsf {f}}
\rhd \neg \neg \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\gg \underline{
\mathsf {f}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{notnotelim}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon \neg \neg \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {f}}
]$

$\rm
[
\mbox{notnotelim}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}notnotelim
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:51}{}\section* {$\rm 
\mbox{mt}$}

$\rm
[
\mbox{mt}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\vdash \neg \underline{
\mathsf {g}}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\vdash 
\mbox{pcmp}
\rhd \underline{
\mathsf {f}}
\rhd \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\gg \underline{
\mathsf {g}}
; 
\mbox{pcdeduction}
\rhd 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {g}}
\gg \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {g}}
; 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\vdash 
\mbox{repeat}
\rhd \neg \underline{
\mathsf {g}}
\gg \neg \underline{
\mathsf {g}}
; 
\mbox{pcdeduction}
\rhd 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\vdash \neg \underline{
\mathsf {g}}
\gg \underline{
\mathsf {f}}
\vdash \neg \underline{
\mathsf {g}}
; 
\mbox{notintro}
\rhd \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {g}}
\rhd \underline{
\mathsf {f}}
\vdash \neg \underline{
\mathsf {g}}
\gg \neg \underline{
\mathsf {f}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{mt}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\vdash \neg \underline{
\mathsf {g}}
\vdash \neg \underline{
\mathsf {f}}
]$

$\rm
[
\mbox{mt}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}mt
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:52}{}\section* {$\rm 
\mbox{pbc}$}

$\rm
[
\mbox{pbc}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \neg \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {g}}
\vdash \neg \underline{
\mathsf {f}}
\vdash \neg \underline{
\mathsf {g}}
\vdash 
\mbox{notintro}
\rhd \neg \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {g}}
\rhd \neg \underline{
\mathsf {f}}
\vdash \neg \underline{
\mathsf {g}}
\gg \neg \neg \underline{
\mathsf {f}}
; 
\mbox{notnotelim}
\rhd \neg \neg \underline{
\mathsf {f}}
\gg \underline{
\mathsf {f}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{pbc}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \neg \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {g}}
\vdash \neg \underline{
\mathsf {f}}
\vdash \neg \underline{
\mathsf {g}}
\vdash \underline{
\mathsf {f}}
]$

$\rm
[
\mbox{pbc}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pbc
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:53}{}\section* {$\rm 
\mbox{repeat}$}

$\rm
[
\mbox{repeat}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon \underline{
\mathsf {f}}
\vdash 
\mbox{trivia}
\gg \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
; 
\mbox{pcmp}
\rhd \underline{
\mathsf {f}}
\rhd \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {f}}
\gg \underline{
\mathsf {f}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{repeat}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {f}}
]$

$\rm
[
\mbox{repeat}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}repeat
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:54}{}\section* {$\rm 
\mbox{lem}$}

$\rm
[
\mbox{lem}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {f}}
\colon \neg \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
\vdash 
\forall \underline{
\mathsf {f}}
\colon \underline{
\mathsf {f}}
\vdash 
\mbox{orintro1}
\rhd \underline{
\mathsf {f}}
\gg \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
; 
\mbox{pcdeduction}
\rhd 
\forall \underline{
\mathsf {f}}
\colon \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
\gg \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
; 
\forall \underline{
\mathsf {f}}
\colon \underline{
\mathsf {f}}
\vdash 
\mbox{repeat}
\rhd \neg \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
\gg \neg \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
; 
\mbox{pcdeduction}
\rhd 
\forall \underline{
\mathsf {f}}
\colon \underline{
\mathsf {f}}
\vdash \neg \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
\gg \underline{
\mathsf {f}}
\vdash \neg \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
; 
\mbox{notintro}
\rhd \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
\rhd \underline{
\mathsf {f}}
\vdash \neg \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
\gg \neg \underline{
\mathsf {f}}
; 
\mbox{orintro2}
\rhd \neg \underline{
\mathsf {f}}
\gg \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
; 
\mbox{pcdeduction}
\rhd 
\forall \underline{
\mathsf {f}}
\colon \neg \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
\gg \neg \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
; 
\forall \underline{
\mathsf {f}}
\colon \neg \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
\vdash 
\mbox{repeat}
\rhd \neg \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
\gg \neg \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
; 
\mbox{pcdeduction}
\rhd 
\forall \underline{
\mathsf {f}}
\colon \neg \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
\vdash \neg \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
\gg \neg \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
\vdash \neg \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
; 
\mbox{notintro}
\rhd \neg \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
\rhd \neg \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
\vdash \neg \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
\gg \neg \neg \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
; 
\mbox{notnotelim}
\rhd \neg \neg \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
\gg \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{lem}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon \underline{
\mathsf {f}} \vee \neg \underline{
\mathsf {f}}
]$

$\rm
[
\mbox{lem}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lem
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:55}{}\section* {$\rm 
\mbox{forallintro}$}

$\rm
[
\mbox{forallintro}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf{x}}
\colon 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {g}} \vee \neg \underline{
\mathsf {g}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \underline{
\mathsf {f}}
\vdash 
\mbox{lem}
\gg \underline{
\mathsf {g}} \vee \neg \underline{
\mathsf {g}}
; 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf {f}}
\colon \underline{
\mathsf {g}} \vee \neg \underline{
\mathsf {g}}
\vdash 
\mbox{repeat}
\rhd \underline{
\mathsf {f}}
\gg \underline{
\mathsf {f}}
; 
\mbox{pcdeduction}
\rhd 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf {f}}
\colon \underline{
\mathsf {g}} \vee \neg \underline{
\mathsf {g}}
\vdash \underline{
\mathsf {f}}
\gg \underline{
\mathsf {g}} \vee \neg \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}}
; 
\mbox{pcia}
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {g}} \vee \neg \underline{
\mathsf {g}}
\rceil 
\rhd \underline{
\mathsf {g}} \vee \neg \underline{
\mathsf {g}}
\Rightarrow \underline{
\mathsf {f}}
\gg \underline{
\mathsf {g}} \vee \neg \underline{
\mathsf {g}}
\Rightarrow \forall \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
; 
\mbox{pcmp}
\rhd \underline{
\mathsf {g}} \vee \neg \underline{
\mathsf {g}}
\rhd \underline{
\mathsf {g}} \vee \neg \underline{
\mathsf {g}}
\Rightarrow \forall \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
\gg \forall \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{forallintro}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf{x}}
\colon 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {g}} \vee \neg \underline{
\mathsf {g}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \underline{
\mathsf {f}}
\vdash \forall \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
]$

$\rm
[
\mbox{forallintro}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}forallintro
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:56}{}\section* {$\rm 
\mbox{forallelim}$}

$\rm
[
\mbox{forallelim}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf {r}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf {f}}
\colon 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {r}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {g}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\langle 
\lceil \underline{
\mathsf {g}}
\rceil 
{\equiv}^0 
\lceil \underline{
\mathsf {f}}
\rceil 
| 
\lceil \underline{
\mathsf{x}}
\rceil 
{:=} 
\lceil \underline{
\mathsf {r}}
\rceil 
\rangle 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mbox{pc11}
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {r}}
\rceil 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {g}}
\rceil 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\langle 
\lceil \underline{
\mathsf {g}}
\rceil 
{\equiv}^0 
\lceil \underline{
\mathsf {f}}
\rceil 
| 
\lceil \underline{
\mathsf{x}}
\rceil 
{:=} 
\lceil \underline{
\mathsf {r}}
\rceil 
\rangle 
\gg \forall \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
\Rightarrow \underline{
\mathsf {g}}
; 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf {f}}
\colon \forall \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
\vdash 
\mbox{pcmp}
\rhd \forall \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
\rhd \forall \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
\Rightarrow \underline{
\mathsf {g}}
\gg \underline{
\mathsf {g}}
; 
\mbox{pcdeduction}
\rhd 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf {f}}
\colon \forall \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
\vdash \underline{
\mathsf {g}}
\gg \forall \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
\vdash \underline{
\mathsf {g}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{forallelim}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf {r}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf {f}}
\colon 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {r}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {g}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\langle 
\lceil \underline{
\mathsf {g}}
\rceil 
{\equiv}^0 
\lceil \underline{
\mathsf {f}}
\rceil 
| 
\lceil \underline{
\mathsf{x}}
\rceil 
{:=} 
\lceil \underline{
\mathsf {r}}
\rceil 
\rangle 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \forall \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
\vdash \underline{
\mathsf {g}}
]$

$\rm
[
\mbox{forallelim}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}forallelim
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:57}{}\section* {$\rm 
\mbox{existsintro}$}

$\rm
[
\mbox{existsintro}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf {r}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf {f}}
\colon 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {r}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {g}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\langle 
\lceil \underline{
\mathsf {g}}
\rceil 
{\equiv}^0 
\lceil \underline{
\mathsf {f}}
\rceil 
| 
\lceil \underline{
\mathsf{x}}
\rceil 
{:=} 
\lceil \underline{
\mathsf {r}}
\rceil 
\rangle 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mbox{pc12}
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {r}}
\rceil 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {g}}
\rceil 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\langle 
\lceil \underline{
\mathsf {g}}
\rceil 
{\equiv}^0 
\lceil \underline{
\mathsf {f}}
\rceil 
| 
\lceil \underline{
\mathsf{x}}
\rceil 
{:=} 
\lceil \underline{
\mathsf {r}}
\rceil 
\rangle 
\gg \underline{
\mathsf {g}}
\Rightarrow \exists \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
; 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf {f}}
\colon \underline{
\mathsf {g}}
\vdash 
\mbox{pcmp}
\rhd \underline{
\mathsf {g}}
\rhd \underline{
\mathsf {g}}
\Rightarrow \exists \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
\gg \exists \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
; 
\mbox{pcdeduction}
\rhd 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf {f}}
\colon \underline{
\mathsf {g}}
\vdash \exists \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
\gg \underline{
\mathsf {g}}
\vdash \exists \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{existsintro}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf {r}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\forall \underline{
\mathsf {f}}
\colon 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {r}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {g}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\langle 
\lceil \underline{
\mathsf {g}}
\rceil 
{\equiv}^0 
\lceil \underline{
\mathsf {f}}
\rceil 
| 
\lceil \underline{
\mathsf{x}}
\rceil 
{:=} 
\lceil \underline{
\mathsf {r}}
\rceil 
\rangle 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \underline{
\mathsf {g}}
\vdash \exists \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
]$

$\rm
[
\mbox{existsintro}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}existsintro
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:58}{}\section* {$\rm 
\mbox{existselim}$}

$\rm
[
\mbox{existselim}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {g}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \exists \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
\vdash \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {g}}
\vdash 
\mbox{pcded}
\rhd \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {g}}
\gg \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
; 
\mbox{pcie}
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {g}}
\rceil 
\rhd \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}}
\gg \exists \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
\Rightarrow \underline{
\mathsf {g}}
; 
\mbox{pcmp}
\rhd \exists \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
\rhd \exists \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
\Rightarrow \underline{
\mathsf {g}}
\gg \underline{
\mathsf {g}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{existselim}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon 
\lceil \underline{
\mathsf{x}}
\rceil 
\#^0
\lceil \underline{
\mathsf {g}}
\rceil 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \exists \underline{
\mathsf{x}} . \left(\underline{
\mathsf {f}}\right)
\vdash \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {g}}
\vdash \underline{
\mathsf {g}}
]$

$\rm
[
\mbox{existselim}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}existselim
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:59}{}\section* {$\rm 
\mbox{bottomelim}$}

$\rm
[
\mbox{bottomelim}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}} \wedge \neg \underline{
\mathsf {f}}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \neg \underline{
\mathsf {g}}
\vdash 
\mbox{andelim1}
\rhd \underline{
\mathsf {f}} \wedge \neg \underline{
\mathsf {f}}
\gg \underline{
\mathsf {f}}
; 
\mbox{pcdeduction}
\rhd 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \neg \underline{
\mathsf {g}}
\vdash \underline{
\mathsf {f}}
\gg \neg \underline{
\mathsf {g}}
\vdash \underline{
\mathsf {f}}
; 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \neg \underline{
\mathsf {g}}
\vdash 
\mbox{andelim2}
\rhd \underline{
\mathsf {f}} \wedge \neg \underline{
\mathsf {f}}
\gg \neg \underline{
\mathsf {f}}
; 
\mbox{pcdeduction}
\rhd 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \neg \underline{
\mathsf {g}}
\vdash \neg \underline{
\mathsf {f}}
\gg \neg \underline{
\mathsf {g}}
\vdash \neg \underline{
\mathsf {f}}
; 
\mbox{notintro}
\rhd \neg \underline{
\mathsf {g}}
\vdash \underline{
\mathsf {f}}
\rhd \neg \underline{
\mathsf {g}}
\vdash \neg \underline{
\mathsf {f}}
\gg \neg \neg \underline{
\mathsf {g}}
; 
\mbox{notnotelim}
\rhd \neg \neg \underline{
\mathsf {g}}
\gg \underline{
\mathsf {g}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{bottomelim}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}} \wedge \neg \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {g}}
]$

$\rm
[
\mbox{bottomelim}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}bottomelim
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:60}{}\section* {$\rm 
\mbox{lemnotintro}$}

$\rm
[
\mbox{lemnotintro}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}} \wedge \neg \underline{
\mathsf {g}}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\vdash 
\mbox{pcmp}
\rhd \underline{
\mathsf {f}}
\rhd \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}} \wedge \neg \underline{
\mathsf {g}}
\gg \underline{
\mathsf {g}} \wedge \neg \underline{
\mathsf {g}}
; 
\mbox{andelim1}
\rhd \underline{
\mathsf {g}} \wedge \neg \underline{
\mathsf {g}}
\gg \underline{
\mathsf {g}}
; 
\mbox{pcdeduction}
\rhd 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {g}}
\gg \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {g}}
; 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\vdash 
\mbox{pcmp}
\rhd \underline{
\mathsf {f}}
\rhd \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}} \wedge \neg \underline{
\mathsf {g}}
\gg \underline{
\mathsf {g}} \wedge \neg \underline{
\mathsf {g}}
; 
\mbox{andelim2}
\rhd \underline{
\mathsf {g}} \wedge \neg \underline{
\mathsf {g}}
\gg \neg \underline{
\mathsf {g}}
; 
\mbox{pcdeduction}
\rhd 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\vdash \neg \underline{
\mathsf {g}}
\gg \underline{
\mathsf {f}}
\vdash \neg \underline{
\mathsf {g}}
; 
\mbox{notintro}
\rhd \underline{
\mathsf {f}}
\vdash \underline{
\mathsf {g}}
\rhd \underline{
\mathsf {f}}
\vdash \neg \underline{
\mathsf {g}}
\gg \neg \underline{
\mathsf {f}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{lemnotintro}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {f}}
\colon 
\forall \underline{
\mathsf {g}}
\colon \underline{
\mathsf {f}}
\Rightarrow \underline{
\mathsf {g}} \wedge \neg \underline{
\mathsf {g}}
\vdash \neg \underline{
\mathsf {f}}
]$

$\rm
[
\mbox{lemnotintro}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemnotintro
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:61}{}\section* {$\rm 
\mbox{nontriv0}$}

$\rm
[
\mbox{nontriv0}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {p}}
\colon 
\forall \underline{
\mathsf {q}}
\colon \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\Rightarrow \underline{
\mathsf {q}}
\vdash \underline{
\mathsf {q}}
\Rightarrow \underline{
\mathsf {p}}
\vdash \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\vdash 
\mbox{pcmp}
\rhd \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\rhd \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\Rightarrow \underline{
\mathsf {q}}
\gg \underline{
\mathsf {q}}
; 
\mbox{pcmp}
\rhd \underline{
\mathsf {q}}
\rhd \underline{
\mathsf {q}}
\Rightarrow \underline{
\mathsf {p}}
\gg \underline{
\mathsf {p}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{nontriv0}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {p}}
\colon 
\forall \underline{
\mathsf {q}}
\colon \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\Rightarrow \underline{
\mathsf {q}}
\vdash \underline{
\mathsf {q}}
\Rightarrow \underline{
\mathsf {p}}
\vdash \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\vdash \underline{
\mathsf {p}}
]$

$\rm
[
\mbox{nontriv0}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}nontriv0
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:62}{}\section* {$\rm 
\mbox{nontriv1}$}

$\rm
[
\mbox{nontriv1}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {p}}
\colon 
\forall \underline{
\mathsf {q}}
\colon \underline{
\mathsf {p}}
\vdash \neg \underline{
\mathsf {p}}
\vdash 
\mbox{andintro}
\rhd \underline{
\mathsf {p}}
\rhd \neg \underline{
\mathsf {p}}
\gg \underline{
\mathsf {p}} \wedge \neg \underline{
\mathsf {p}}
; 
\mbox{bottomelim}
\rhd \underline{
\mathsf {p}} \wedge \neg \underline{
\mathsf {p}}
\gg \underline{
\mathsf {q}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{nontriv1}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {p}}
\colon 
\forall \underline{
\mathsf {q}}
\colon \underline{
\mathsf {p}}
\vdash \neg \underline{
\mathsf {p}}
\vdash \underline{
\mathsf {q}}
]$

$\rm
[
\mbox{nontriv1}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}nontriv1
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:63}{}\section* {$\rm 
\mbox{nontriv2}$}

$\rm
[
\mbox{nontriv2}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {p}}
\colon 
\forall \underline{
\mathsf {q}}
\colon \neg \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\vdash \neg \underline{
\mathsf {p}}
\vdash 
\forall \underline{
\mathsf {p}}
\colon 
\forall \underline{
\mathsf {q}}
\colon \underline{
\mathsf {p}}
\vdash 
\mbox{nontriv1}
\rhd \underline{
\mathsf {p}}
\rhd \neg \underline{
\mathsf {p}}
\gg \underline{
\mathsf {q}}
; 
\mbox{pcdeduction}
\rhd 
\forall \underline{
\mathsf {p}}
\colon 
\forall \underline{
\mathsf {q}}
\colon \underline{
\mathsf {p}}
\vdash \underline{
\mathsf {q}}
\gg \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
; 
\mbox{andintro}
\rhd \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\rhd \neg \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\gg \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}} \wedge \neg \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{nontriv2}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {p}}
\colon 
\forall \underline{
\mathsf {q}}
\colon \neg \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\vdash \neg \underline{
\mathsf {p}}
\vdash \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}} \wedge \neg \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
]$

$\rm
[
\mbox{nontriv2}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}nontriv2
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:64}{}\section* {$\rm 
\mbox{nontriv3}$}

$\rm
[
\mbox{nontriv3}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {p}}
\colon 
\forall \underline{
\mathsf {q}}
\colon \neg \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\vdash 
\forall \underline{
\mathsf {p}}
\colon 
\forall \underline{
\mathsf {q}}
\colon \neg \underline{
\mathsf {p}}
\vdash 
\mbox{nontriv2}
\rhd \neg \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\rhd \neg \underline{
\mathsf {p}}
\gg \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}} \wedge \neg \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
; 
\mbox{pcdeduction}
\rhd 
\forall \underline{
\mathsf {p}}
\colon 
\forall \underline{
\mathsf {q}}
\colon \neg \underline{
\mathsf {p}}
\vdash \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}} \wedge \neg \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\gg \neg \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}} \wedge \neg \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
; 
\mbox{lemnotintro}
\rhd \neg \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}} \wedge \neg \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\gg \neg \neg \underline{
\mathsf {p}}
; 
\mbox{notnotelim}
\rhd \neg \neg \underline{
\mathsf {p}}
\gg \underline{
\mathsf {p}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{nontriv3}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {p}}
\colon 
\forall \underline{
\mathsf {q}}
\colon \neg \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\vdash \underline{
\mathsf {p}}
]$

$\rm
[
\mbox{nontriv3}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}nontriv3
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:65}{}\section* {$\rm 
\mbox{nontriv4}$}

$\rm
[
\mbox{nontriv4}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {p}}
\colon 
\forall \underline{
\mathsf {q}}
\colon \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\Rightarrow \underline{
\mathsf {q}}
\vdash \underline{
\mathsf {q}}
\Rightarrow \underline{
\mathsf {p}}
\vdash 
\forall \underline{
\mathsf {p}}
\colon 
\forall \underline{
\mathsf {q}}
\colon \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\vdash 
\mbox{nontriv0}
\rhd \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\Rightarrow \underline{
\mathsf {q}}
\rhd \underline{
\mathsf {q}}
\Rightarrow \underline{
\mathsf {p}}
\rhd \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\gg \underline{
\mathsf {p}}
; 
\mbox{pcdeduction}
\rhd 
\forall \underline{
\mathsf {p}}
\colon 
\forall \underline{
\mathsf {q}}
\colon \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\vdash \underline{
\mathsf {p}}
\gg \underline{
\mathsf {q}}
\Rightarrow \underline{
\mathsf {q}}
\vdash \underline{
\mathsf {p}}
; 
\forall \underline{
\mathsf {p}}
\colon 
\forall \underline{
\mathsf {q}}
\colon \neg \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\vdash 
\mbox{nontriv3}
\rhd \neg \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\gg \underline{
\mathsf {p}}
; 
\mbox{pcdeduction}
\rhd 
\forall \underline{
\mathsf {p}}
\colon 
\forall \underline{
\mathsf {q}}
\colon \neg \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\vdash \underline{
\mathsf {p}}
\gg \neg \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\vdash \underline{
\mathsf {p}}
; 
\mbox{orelim}
\rhd \underline{
\mathsf {q}}
\Rightarrow \underline{
\mathsf {q}}
\vdash \underline{
\mathsf {p}}
\rhd \neg \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\vdash \underline{
\mathsf {p}}
\gg \underline{
\mathsf {p}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
\mbox{nontriv4}
\stackrel{
stmt
}{\rightarrow}
\mbox{pred calc}
\vdash 
\forall \underline{
\mathsf {p}}
\colon 
\forall \underline{
\mathsf {q}}
\colon \underline{
\mathsf {p}}
\Rightarrow \underline{
\mathsf {q}}
\Rightarrow \underline{
\mathsf {q}}
\vdash \underline{
\mathsf {q}}
\Rightarrow \underline{
\mathsf {p}}
\vdash \underline{
\mathsf {p}}
]$

$\rm
[
\mbox{nontriv4}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}nontriv4
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:66}{}\section* {$\rm 
\mbox{nontriv5}$}

$\rm
[
\mbox{nontriv5}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}nontriv5
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:67}{}\section* {$\rm 
\mbox{nontriv6}$}

$\rm
[
\mbox{nontriv6}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}nontriv6
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:68}{}\section* {$\rm \ast \equiv \ast$}

$\rm
[
\mathsf{y} \equiv 
\mathsf {b}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}equiv
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

$\rm
[
\ast  \equiv 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}setequiv
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:69}{}\section* {$\rm \ast = \ast$}

$\rm
[
\mathsf{y} = 
\mathsf {b}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\linebreak [0]\ \hskip0em plus2.0em{}
{=}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

$\rm
[
\ast  = 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}setequals
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:70}{}\section* {$\rm \neg \ast$}

$\rm
[\neg 
\mathsf{x}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}neg
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\mbox{''}
]$

$\rm
[\neg 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lnot
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:71}{}\section* {$\rm \ast \wedge \ast$}

$\rm
[
\mathsf{x} \wedge 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}wedge
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

$\rm
[
\ast  \wedge 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}land
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:72}{}\section* {$\rm \ast \vee \ast$}

$\rm
[
\mathsf{x} \vee 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}vee
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

$\rm
[
\ast  \vee 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}lor
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:73}{}\section* {$\rm \forall \ast . \left(\ast\right)$}

$\rm
[\forall 
\mathsf{y} . \left(
\mathsf {b}\right)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}forall
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\linebreak [0]\ \hskip0em plus2.0em{}.
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}left(
\#2.
\mbox {$\backslash $}right)
\mbox{''}
]$

$\rm
[\forall 
\ast  . \left(
\ast \right)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}forall
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}dot
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}forall
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:74}{}\section* {$\rm \exists \ast . \left(\ast\right)$}

$\rm
[\exists 
\mathsf{y} . \left(
\mathsf {b}\right)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}exists
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\linebreak [0]\ \hskip0em plus2.0em{}.
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}left(
\#2.
\mbox {$\backslash $}right)
\mbox{''}
]$

$\rm
[\exists 
\ast  . \left(
\ast \right)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}exists
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}dot
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}exists
\mbox{''}
]$

\hypertarget {url:1808490463983504709587517268929939939143332479535486097762496273530505985:75}{}\section* {$\rm \ast \in \ast$}

$\rm
[
\mathsf{y} \in 
\mathsf {b}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}in
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

$\rm
[
\ast  \in 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}setin
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]$

\vspace{2ex}

{\em \href {http://www.diku.dk/~grue/logiweb/20060417/logiweb/doc/compiler/index.html}{The pyk compiler},
version 0.grue.20060417+
by
\href {http://www.diku.dk/~grue/index.html}{Klaus Grue}\\
\href {http://www.diku.dk/~grue/logiweb/20060417/logiweb/doc/misc/time.html}{GRD-2006-07-12.UTC:19:00:11.024866} = 
\href {http://www.diku.dk/~grue/logiweb/20060417/logiweb/doc/misc/time.html}{MJD-53928.TAI:19:00:44.024866} = 
\href {http://www.diku.dk/~grue/logiweb/20060417/logiweb/doc/misc/time.html}{LGT-4659447644024866e-6}
}

\end{document}

