\documentclass [fleqn]{article}

\setlength {\overfullrule }{1mm}

\usepackage{latexsym}

\setlength{\parindent}{0em}

\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 check}

\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 $\rm 
\mbox{check}$}}

\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:1808490373737845726312426323955533139748491690065847855319167193584235521:319}{$\rm {\cal X}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:0}{$\rm 
\mbox{check}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:1}{$\rm 
[\ast
\stackrel {\circ}{=}\ast
]$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:2}{$\rm 
RootVisible(\ast
)$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:3}{$\rm 
A$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:4}{$\rm 
R$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:5}{$\rm 
C$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:6}{$\rm 
T$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:7}{$\rm 
L$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:8}{$\rm 
\{\ast
\}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:9}{$\rm \overline{\ast}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:10}{$\rm 
\mathit{a}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:11}{$\rm 
\mathit{b}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:12}{$\rm 
\mathit{c}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:13}{$\rm 
\mathit{d}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:14}{$\rm 
\mathit{e}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:15}{$\rm 
\mathit{f}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:16}{$\rm 
\mathit{g}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:17}{$\rm 
\mathit{h}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:18}{$\rm 
\mathit{i}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:19}{$\rm 
\mathit{j}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:20}{$\rm 
\mathit{k}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:21}{$\rm 
\mathit{l}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:22}{$\rm 
\mathit{m}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:23}{$\rm 
\mathit{n}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:24}{$\rm 
\mathit{o}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:25}{$\rm 
\mathit{p}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:26}{$\rm 
\mathit{q}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:27}{$\rm 
\mathit{r}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:28}{$\rm 
\mathit{s}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:29}{$\rm 
\mathit{t}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:30}{$\rm 
\mathit{u}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:31}{$\rm 
\mathit{v}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:32}{$\rm 
\mathit{w}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:33}{$\rm 
\mathit{x}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:34}{$\rm 
\mathit{y}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:35}{$\rm 
\mathit{z}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:36}{$\rm 
\langle \ast
{\equiv} \ast
| \ast
{:=} \ast
\rangle $},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:37}{$\rm 
\langle \ast
{\equiv}^0 \ast
| \ast
{:=} \ast
\rangle $},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:38}{$\rm 
\langle \ast
{\equiv}^1 \ast
| \ast
{:=} \ast
\rangle $},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:39}{$\rm 
\langle \ast
{\equiv}^* \ast
| \ast
{:=} \ast
\rangle $},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:40}{$\rm 
Ded(\ast
,\ast
)$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:41}{$\rm 
Ded_0(\ast
,\ast
)$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:42}{$\rm 
Ded_1(\ast
,\ast
,\ast
)$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:43}{$\rm 
Ded_2(\ast
,\ast
,\ast
)$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:44}{$\rm 
Ded_3(\ast
,\ast
,\ast
,\ast
)$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:45}{$\rm 
Ded_4(\ast
,\ast
,\ast
,\ast
)$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:46}{$\rm 
Ded_4^*(\ast
,\ast
,\ast
,\ast
)$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:47}{$\rm 
Ded_5(\ast
,\ast
,\ast
)$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:48}{$\rm 
Ded_6(\ast
,\ast
,\ast
,\ast
)$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:49}{$\rm 
Ded_6^*(\ast
,\ast
,\ast
,\ast
)$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:50}{$\rm 
Ded_7(\ast
)$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:51}{$\rm 
Ded_8(\ast
,\ast
)$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:52}{$\rm 
Ded_8^*(\ast
,\ast
)$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:53}{$\rm 
S$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:54}{$\rm 
Neg$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:55}{$\rm 
MP$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:56}{$\rm 
Gen$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:57}{$\rm 
Ded$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:58}{$\rm 
S1$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:59}{$\rm 
S2$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:60}{$\rm 
S3$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:61}{$\rm 
S4$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:62}{$\rm 
S5$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:63}{$\rm 
S6$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:64}{$\rm 
S7$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:65}{$\rm 
S8$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:66}{$\rm 
S9$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:67}{$\rm 
Repetition$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:68}{$\rm 
A1'$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:69}{$\rm 
A2'$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:70}{$\rm 
A4'$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:71}{$\rm 
A5'$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:72}{$\rm 
Prop\ 3.2a$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:73}{$\rm 
Prop\ 3.2b$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:74}{$\rm 
Prop\ 3.2c$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:75}{$\rm 
Prop\ 3.2d$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:76}{$\rm 
Prop\ 3.2e_1$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:77}{$\rm 
Prop\ 3.2e_2$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:78}{$\rm 
Prop\ 3.2e$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:79}{$\rm 
Prop\ 3.2f_1$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:80}{$\rm 
Prop\ 3.2f_2$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:81}{$\rm 
Prop\ 3.2f$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:82}{$\rm 
Prop\ 3.2g_1$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:83}{$\rm 
Prop\ 3.2g_2$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:84}{$\rm 
Prop\ 3.2g$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:85}{$\rm 
Prop\ 3.2h_1$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:86}{$\rm 
Prop\ 3.2h_2$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:87}{$\rm 
Prop\ 3.2h$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:88}{$\rm 
Block_1(\ast
,\ast
,\ast
)$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:89}{$\rm 
Block_2(\ast
)$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:90}{$\rm \ast
{}^{hide}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:91}{$\rm 
MacroIndent(\ast
)$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:92}{$\rm \ast
{}'$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:93}{$\rm \ast
= \ast$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:94}{$\rm \ast
\neq \ast$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:95}{$\rm \ast
{}^{var}$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:96}{$\rm \ast
\#^0\ast$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:97}{$\rm \ast
\#^1\ast$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:98}{$\rm \ast
\#^*\ast$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:99}{$\rm 
\exists \ast
\colon \ast$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:100}{$\rm 
\forall \ast
\colon \ast$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:101}{$\rm 
\forall_{obj} \ast
\colon \ast$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:102}{$\rm \ast
\Rightarrow \ast$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:103}{$\rm \ast
\Leftrightarrow \ast$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:104}{$\rm \ast
\#\ast$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:105}{$\rm \ast
\unrhd \ast$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:106}{$\rm 
\Pi \ast
\colon \ast$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:107}{$\rm 
Begin \, \ast
 ; \ast
 : End ; \ast$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:108}{$\rm 
Last\ block\ line \, \ast
\gg \ast
\,;$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:109}{$\rm 
Arbitrary \gg \ast
;\ast$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:110}{$\rm \ast
\mathrel{|} \ast$},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:111}{$\rm 
 \rightarrow $},
\hyperlink {url:1808490384217020148094711118359520484602074145217451012073151273782664705:112}{$\rm \ast
\backslash \backslash \ast$},
\end {flushleft}

\hypertarget {url:1808490373737845726312426323955533139748491690065847855319167193584235521:319}{}\section* {$\rm {\cal X}$}

$\rm
[{\cal X}
\stackrel{
stmt
}{\rightarrow}
\mathsf{x}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:0}{}\section* {$\rm 
\mbox{check}$}

$\rm
[
\mbox{check}
\stackrel{
\mathrm{prio}
}{\rightarrow}
\newline \mathbf {Preassociative} \newline 
[
\mbox{check}
] , \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] 
[
[
\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 
)
]
; 
\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 
\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 
]
; 
\newline \mathbf {Preassociative} \newline 
[
{\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 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\vee 
\ast 
] , \linebreak [0] 
[
\ast 
\parallel 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\ddot{\vee}} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\exists 
\ast 
\colon 
\ast 
] , \linebreak [0] 
[
\forall 
\ast 
\colon 
\ast 
] , \linebreak [0] 
[
\forall_{obj} 
\ast 
\colon 
\ast 
]
; 
\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 $}\,{\vdash }} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {i.e.} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\forall 
\ast 
\colon 
\ast 
] , \linebreak [0] 
[
\Pi 
\ast 
\colon 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel {\oplus} 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
; 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\ proves\ 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\mathbf{\ proof\ of\ } 
\ast 
: 
\ast 
] , \linebreak [0] 
[
Line \, 
\ast 
: 
\ast 
\gg 
\ast 
; 
\ast 
] , \linebreak [0] 
[
Last\ line \, 
\ast 
\gg 
\ast 
\,\Box
] , \linebreak [0] 
[
Line \, 
\ast 
: Premise \gg 
\ast 
;
\ast 
] , \linebreak [0] 
[
Line \, 
\ast 
: \mbox{Side-condition} \gg 
\ast 
; 
\ast 
] , \linebreak [0] 
[
Arbitrary \gg 
\ast 
;
\ast 
] , \linebreak [0] 
[
Local \gg 
\ast 
= 
\ast 
; 
\ast 
] , \linebreak [0] 
[
Begin \, 
\ast 
 ; 
\ast 
 : End ; 
\ast 
] , \linebreak [0] 
[
Last\ block\ line \, 
\ast 
\gg 
\ast 
\,;
] , \linebreak [0] 
[
Arbitrary \gg 
\ast 
;
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel{|} 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel{\mathrm{,}}
\ast 
] , \linebreak [0] 
[
\ast 
{[}
\ast 
{]}
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\& 
\ast 
] , \linebreak [0] 
[
 \rightarrow 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\backslash \backslash 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{linebreak[4]} 
\ast 
] , \linebreak [0] 
[
\ast 
\backslash \backslash 
\ast 
]
; 
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:1}{}\section* {$\rm 
[\ast
\stackrel {\circ}{=}\ast
]$}

$\rm
[
[
\mathsf{x}
\stackrel {\circ}{=}
\mathsf{y}
]
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
[
\mathsf{x}
\stackrel {\circ}{=}
\mathsf{y}
]
\mathrel {\ddot {=}}
[(
\mathsf{x}
)^{\mathbf {p}}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\mathsf{y}
]
]
\rceil 
)
]$

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

$\rm
[
[
\ast 
\stackrel {\circ}{=}
\ast 
]
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}general
\linebreak [0]\ \hskip0em plus2.0em{}macro
\linebreak [0]\ \hskip0em plus2.0em{}define
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}as
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}define
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:2}{}\section* {$\rm 
RootVisible(\ast
)$}

$\rm
[
RootVisible(
\mathsf{x}
)
\stackrel{
\mathrm{name}
}{\rightarrow}
\mbox{``}
\newline RootVisible(
\#1.
\newline )
\mbox{''}
]$

$\rm
[
RootVisible(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1/tex
\linebreak [0]\ \hskip0em plus2.0em{}name/tex.
\mbox{''}
]$

$\rm
[
RootVisible(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}make
\linebreak [0]\ \hskip0em plus2.0em{}root
\linebreak [0]\ \hskip0em plus2.0em{}visible
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}visible
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:3}{}\section* {$\rm 
A$}

$\rm
[
A
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline A
\mbox{''}
]$

$\rm
[
A
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}sequent
\linebreak [0]\ \hskip0em plus2.0em{}example
\linebreak [0]\ \hskip0em plus2.0em{}axiom
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:4}{}\section* {$\rm 
R$}

$\rm
[
R
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline R
\mbox{''}
]$

$\rm
[
R
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}sequent
\linebreak [0]\ \hskip0em plus2.0em{}example
\linebreak [0]\ \hskip0em plus2.0em{}rule
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:5}{}\section* {$\rm 
C$}

$\rm
[
C
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline C
\mbox{''}
]$

$\rm
[
C
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}sequent
\linebreak [0]\ \hskip0em plus2.0em{}example
\linebreak [0]\ \hskip0em plus2.0em{}contradiction
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:6}{}\section* {$\rm 
T$}

$\rm
[
T
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline T
\mbox{''}
]$

$\rm
[
T
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}sequent
\linebreak [0]\ \hskip0em plus2.0em{}example
\linebreak [0]\ \hskip0em plus2.0em{}theory
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:7}{}\section* {$\rm 
L$}

$\rm
[
L
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline L
\mbox{''}
]$

$\rm
[
L
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}sequent
\linebreak [0]\ \hskip0em plus2.0em{}example
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:8}{}\section* {$\rm 
\{\ast
\}$}

$\rm
[
\{
\mathsf{x}
\}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}
\{
\#1.
\newline 
\mbox {$\backslash $}
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:9}{}\section* {$\rm \overline{\ast}$}

$\rm
[\overline{
\mathsf{x}}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}overline
\{
\#1.
\}
\mbox{''}
]$

$\rm
[\overline{
\ast }
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}object
\linebreak [0]\ \hskip0em plus2.0em{}var
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}var
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:10}{}\section* {$\rm 
\mathit{a}$}

$\rm
[
\mathit{a}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{a}
\mathrel {\ddot {=}}\overline{
\mathsf {a}}
]
\rceil 
)
]$

$\rm
[
\mathit{a}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{a
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:11}{}\section* {$\rm 
\mathit{b}$}

$\rm
[
\mathit{b}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{b}
\mathrel {\ddot {=}}\overline{
\mathsf {b}}
]
\rceil 
)
]$

$\rm
[
\mathit{b}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{b
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:12}{}\section* {$\rm 
\mathit{c}$}

$\rm
[
\mathit{c}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{c}
\mathrel {\ddot {=}}\overline{
\mathsf {c}}
]
\rceil 
)
]$

$\rm
[
\mathit{c}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{c
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:13}{}\section* {$\rm 
\mathit{d}$}

$\rm
[
\mathit{d}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{d}
\mathrel {\ddot {=}}\overline{
\mathsf {d}}
]
\rceil 
)
]$

$\rm
[
\mathit{d}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{d
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:14}{}\section* {$\rm 
\mathit{e}$}

$\rm
[
\mathit{e}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{e}
\mathrel {\ddot {=}}\overline{
\mathsf {e}}
]
\rceil 
)
]$

$\rm
[
\mathit{e}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{e
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:15}{}\section* {$\rm 
\mathit{f}$}

$\rm
[
\mathit{f}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{f}
\mathrel {\ddot {=}}\overline{
\mathsf {f}}
]
\rceil 
)
]$

$\rm
[
\mathit{f}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{f
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:16}{}\section* {$\rm 
\mathit{g}$}

$\rm
[
\mathit{g}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{g}
\mathrel {\ddot {=}}\overline{
\mathsf {g}}
]
\rceil 
)
]$

$\rm
[
\mathit{g}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{g
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:17}{}\section* {$\rm 
\mathit{h}$}

$\rm
[
\mathit{h}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{h}
\mathrel {\ddot {=}}\overline{
\mathsf {h}}
]
\rceil 
)
]$

$\rm
[
\mathit{h}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{h
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:18}{}\section* {$\rm 
\mathit{i}$}

$\rm
[
\mathit{i}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{i}
\mathrel {\ddot {=}}\overline{
\mathsf {i}}
]
\rceil 
)
]$

$\rm
[
\mathit{i}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{i
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:19}{}\section* {$\rm 
\mathit{j}$}

$\rm
[
\mathit{j}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{j}
\mathrel {\ddot {=}}\overline{
\mathsf {j}}
]
\rceil 
)
]$

$\rm
[
\mathit{j}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{j
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:20}{}\section* {$\rm 
\mathit{k}$}

$\rm
[
\mathit{k}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{k}
\mathrel {\ddot {=}}\overline{
\mathsf {k}}
]
\rceil 
)
]$

$\rm
[
\mathit{k}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{k
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:21}{}\section* {$\rm 
\mathit{l}$}

$\rm
[
\mathit{l}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{l}
\mathrel {\ddot {=}}\overline{
\mathsf {l}}
]
\rceil 
)
]$

$\rm
[
\mathit{l}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{l
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:22}{}\section* {$\rm 
\mathit{m}$}

$\rm
[
\mathit{m}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{m}
\mathrel {\ddot {=}}\overline{
\mathsf {m}}
]
\rceil 
)
]$

$\rm
[
\mathit{m}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{m
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:23}{}\section* {$\rm 
\mathit{n}$}

$\rm
[
\mathit{n}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{n}
\mathrel {\ddot {=}}\overline{
\mathsf {n}}
]
\rceil 
)
]$

$\rm
[
\mathit{n}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{n
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:24}{}\section* {$\rm 
\mathit{o}$}

$\rm
[
\mathit{o}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{o}
\mathrel {\ddot {=}}\overline{
\mathsf {o}}
]
\rceil 
)
]$

$\rm
[
\mathit{o}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{o
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:25}{}\section* {$\rm 
\mathit{p}$}

$\rm
[
\mathit{p}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{p}
\mathrel {\ddot {=}}\overline{
\mathsf {p}}
]
\rceil 
)
]$

$\rm
[
\mathit{p}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{p
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:26}{}\section* {$\rm 
\mathit{q}$}

$\rm
[
\mathit{q}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{q}
\mathrel {\ddot {=}}\overline{
\mathsf {q}}
]
\rceil 
)
]$

$\rm
[
\mathit{q}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{q
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:27}{}\section* {$\rm 
\mathit{r}$}

$\rm
[
\mathit{r}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{r}
\mathrel {\ddot {=}}\overline{
\mathsf {r}}
]
\rceil 
)
]$

$\rm
[
\mathit{r}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{r
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:28}{}\section* {$\rm 
\mathit{s}$}

$\rm
[
\mathit{s}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{s}
\mathrel {\ddot {=}}\overline{
\mathsf {s}}
]
\rceil 
)
]$

$\rm
[
\mathit{s}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{s
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:29}{}\section* {$\rm 
\mathit{t}$}

$\rm
[
\mathit{t}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{t}
\mathrel {\ddot {=}}\overline{
\mathsf {t}}
]
\rceil 
)
]$

$\rm
[
\mathit{t}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{t
\}
\mbox{''}
]$

$\rm
[
\mathit{t}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}object
\linebreak [0]\ \hskip0em plus2.0em{}t
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:30}{}\section* {$\rm 
\mathit{u}$}

$\rm
[
\mathit{u}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{u}
\mathrel {\ddot {=}}\overline{
\mathsf {u}}
]
\rceil 
)
]$

$\rm
[
\mathit{u}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{u
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:31}{}\section* {$\rm 
\mathit{v}$}

$\rm
[
\mathit{v}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{v}
\mathrel {\ddot {=}}\overline{
\mathsf {v}}
]
\rceil 
)
]$

$\rm
[
\mathit{v}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{v
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:32}{}\section* {$\rm 
\mathit{w}$}

$\rm
[
\mathit{w}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{w}
\mathrel {\ddot {=}}\overline{
\mathsf {w}}
]
\rceil 
)
]$

$\rm
[
\mathit{w}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{w
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:33}{}\section* {$\rm 
\mathit{x}$}

$\rm
[
\mathit{x}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{x}
\mathrel {\ddot {=}}\overline{
\mathsf{x}}
]
\rceil 
)
]$

$\rm
[
\mathit{x}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{x
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:34}{}\section* {$\rm 
\mathit{y}$}

$\rm
[
\mathit{y}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{y}
\mathrel {\ddot {=}}\overline{
\mathsf{y}}
]
\rceil 
)
]$

$\rm
[
\mathit{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{y
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:35}{}\section* {$\rm 
\mathit{z}$}

$\rm
[
\mathit{z}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathit{z}
\mathrel {\ddot {=}}\overline{
\mathsf{z}}
]
\rceil 
)
]$

$\rm
[
\mathit{z}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}mathit
\{z
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:36}{}\section* {$\rm 
\langle \ast
{\equiv} \ast
| \ast
{:=} \ast
\rangle $}

$\rm
[
\langle 
\mathsf {a}
{\equiv} 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle 
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\langle 
\mathsf {a}
{\equiv} 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle 
\mathrel {\ddot {=}}
\langle 
\lceil 
\mathsf {a}
\rceil 
{\equiv}^0 
\lceil 
\mathsf {b}
\rceil 
| 
\lceil 
\mathsf{x}
\rceil 
{:=} 
\lceil 
\mathsf {t}
\rceil 
\rangle 
]
\rceil 
)
]$

$\rm
[
\langle 
\mathsf{x}
{\equiv} 
\mathsf{y}
| 
\mathsf{z}
{:=} 
\mathsf {u}
\rangle 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\{
\mbox {$\backslash $}equiv
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline |
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\newline 
\{
{:}
{=}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#4.
\newline 
\mbox {$\backslash $}rangle
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[
\langle 
\ast 
{\equiv} 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}sub
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}where
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}sub
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:37}{}\section* {$\rm 
\langle \ast
{\equiv}^0 \ast
| \ast
{:=} \ast
\rangle $}

$\rm
[
\langle 
\mathsf {a}
{\equiv}^0 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle 
\stackrel{
\mathrm {val}
}{\rightarrow}
\lambda 
\mathsf {c}
.
\mathsf{x}
{}^{var}
\wedge 
\langle 
\mathsf {a}
{\equiv}^1 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle 
]$

$\rm
[
\langle 
\mathsf{x}
{\equiv}^0 
\mathsf{y}
| 
\mathsf{z}
{:=} 
\mathsf {u}
\rangle 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\{
\mbox {$\backslash $}equiv
\}
{\char94}0
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline |
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\newline 
\{
{:}
{=}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#4.
\newline 
\mbox {$\backslash $}rangle
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[
\langle 
\ast 
{\equiv}^0 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}sub
\linebreak [0]\ \hskip0em plus2.0em{}zero
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}where
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}sub
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:38}{}\section* {$\rm 
\langle \ast
{\equiv}^1 \ast
| \ast
{:=} \ast
\rangle $}

$\rm
[
\langle 
\mathsf {a}
{\equiv}^1 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle 
\stackrel{
\mathrm {val}
}{\rightarrow}
\mathsf {a}
!
\mathsf{x}
!
\mathsf {t}
!
\newline 
\mathrm {If}(
\mathrm {If}(
\mathsf {b}
\stackrel{r}{=} 
\lceil 
\forall_{obj} 
\mathsf {u}
\colon 
\mathsf {v}
\rceil 
,\linebreak [0]
\mathsf {b}
{}^{1}
\stackrel {t}{=}
\mathsf{x}
,\linebreak [0]
\mathsf {F}
)
,\linebreak [0]
\mathsf {a}
\stackrel {t}{=}
\mathsf {b}
,\linebreak [0]
\newline 
\mathrm {If}(
\mathsf {b}
{}^{var}
\wedge 
\mathsf {b}
\stackrel {t}{=}
\mathsf{x}
,\linebreak [0]
\mathsf {a}
\stackrel {t}{=}
\mathsf {t}
,\linebreak [0]
\mathrm {If}(
\newline 
\mathsf {a}
\stackrel{r}{=} 
\mathsf {b}
,\linebreak [0]
\langle 
\mathsf {a}
{}^t
{\equiv}^* 
\mathsf {b}
{}^t
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle 
,\linebreak [0]
\mathsf {F}
)
)
)
]$

$\rm
[
\langle 
\mathsf{x}
{\equiv}^1 
\mathsf{y}
| 
\mathsf{z}
{:=} 
\mathsf {u}
\rangle 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\{
\mbox {$\backslash $}equiv
\}
{\char94}1
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline |
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\newline 
\{
{:}
{=}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#4.
\newline 
\mbox {$\backslash $}rangle
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[
\langle 
\ast 
{\equiv}^1 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}sub
\linebreak [0]\ \hskip0em plus2.0em{}one
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}where
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}sub
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:39}{}\section* {$\rm 
\langle \ast
{\equiv}^* \ast
| \ast
{:=} \ast
\rangle $}

$\rm
[
\langle 
\mathsf {a}
{\equiv}^* 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle 
\stackrel{
\mathrm {val}
}{\rightarrow}
\mathsf {b}
!
\mathsf{x}
!
\mathsf {t}
!
\mathrm {If}(
\mathsf {a}
,\linebreak [0]
\mathsf {T}
,\linebreak [0]
\mathrm {If}(
\langle 
\mathsf {a}
{}^h
{\equiv}^1 
\mathsf {b}
{}^h
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle 
,\linebreak [0]
\langle 
\mathsf {a}
{}^t
{\equiv}^* 
\mathsf {b}
{}^t
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle 
,\linebreak [0]
\mathsf {F}
)
)
]$

$\rm
[
\langle 
\mathsf{x}
{\equiv}^* 
\mathsf{y}
| 
\mathsf{z}
{:=} 
\mathsf {u}
\rangle 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\{
\mbox {$\backslash $}equiv
\}
{\char94}
{*}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline |
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\newline 
\{
{:}
{=}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#4.
\newline 
\mbox {$\backslash $}rangle
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[
\langle 
\ast 
{\equiv}^* 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}sub
\linebreak [0]\ \hskip0em plus2.0em{}star
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}where
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}sub
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:40}{}\section* {$\rm 
Ded(\ast
,\ast
)$}

$\rm
[
Ded(
\mathsf {p}
,
\mathsf {c}
)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
Ded(
\mathsf {p}
,
\mathsf {c}
)
\mathrel {\ddot {=}}
\lambda 
\mathsf{x}
.
Ded_0(
\lceil 
\mathsf {p}
\rceil 
,
\lceil 
\mathsf {c}
\rceil 
)
]
\rceil 
)
]$

$\rm
[
Ded(
\mathsf{x}
,
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Ded(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$

$\rm
[
Ded(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}deduction
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}deduction
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:41}{}\section* {$\rm 
Ded_0(\ast
,\ast
)$}

$\rm
[
Ded_0(
\mathsf {p}
,
\mathsf {c}
)
\stackrel{
\mathrm {val}
}{\rightarrow}
\mathsf {c}
!
\mathrm {If}(
Ded_8(
\mathsf {p}
,
\mathsf {T}
)
,\linebreak [0]
Ded_1(
Ded_7(
\mathsf {p}
)
,
\mathsf {c}
,
\mathsf {T}
)
,\linebreak [0]
\mathsf {F}
)
]$

$\rm
[
Ded_0(
\mathsf{x}
,
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Ded
\_0(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$

$\rm
[
Ded_0(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}deduction
\linebreak [0]\ \hskip0em plus2.0em{}zero
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}deduction
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:42}{}\section* {$\rm 
Ded_1(\ast
,\ast
,\ast
)$}

$\rm
[
Ded_1(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
)
\stackrel{
\mathrm {val}
}{\rightarrow}
\mathrm {If}(
\mathsf {c}
\stackrel{r}{=} 
\lceil 
\mathsf{x}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathsf{y}
\rceil 
,\linebreak [0]
Ded_1(
\mathsf {p}
,
\mathsf {c}
{}^{2}
,
\mathsf {c}
{}^{1}
\mathrel { : \, : }
\mathsf {s}
)
,\linebreak [0]
Ded_2(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
)
)
]$

$\rm
[
Ded_1(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Ded
\_1(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline )
\mbox{''}
]$

$\rm
[
Ded_1(
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}deduction
\linebreak [0]\ \hskip0em plus2.0em{}one
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}condition
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}deduction
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:43}{}\section* {$\rm 
Ded_2(\ast
,\ast
,\ast
)$}

$\rm
[
Ded_2(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
)
\stackrel{
\mathrm {val}
}{\rightarrow}
\mathsf {s}
!
\mathsf {p}
\stackrel{r}{=} 
\lceil 
\mathsf{x}
\vdash 
\mathsf{y}
\rceil 
\wedge 
\mathsf {c}
\stackrel{r}{=} 
\lceil 
\mathsf{x}
\Rightarrow 
\mathsf{y}
\rceil 
\left\{\protect \begin {array}{l}
Ded_3(
\mathsf {p}
{}^{1}
,
\mathsf {c}
{}^{1}
,
\mathsf {s}
,
\mathsf {T}
)
\wedge 
Ded_2(
\mathsf {p}
{}^{2}
,
\mathsf {c}
{}^{2}
,
\mathsf {s}
)
\\
Ded_4(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
,
Ded_6(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {T}
,
\mathsf {T}
)
)
\protect \end {array}\right.
]$

$\rm
[
Ded_2(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Ded
\_2(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline )
\mbox{''}
]$

$\rm
[
Ded_2(
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}deduction
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}condition
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}deduction
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:44}{}\section* {$\rm 
Ded_3(\ast
,\ast
,\ast
,\ast
)$}

$\rm
[
Ded_3(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
,
\mathsf {b}
)
\stackrel{
\mathrm {val}
}{\rightarrow}
\mathrm {If}(
{\neg }
\mathsf {c}
\stackrel{r}{=} 
\lceil 
\forall_{obj} 
\mathsf{x}
\colon 
\mathsf{y}
\rceil 
,\linebreak [0]
Ded_4(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
,
\mathsf {b}
)
,\linebreak [0]
\mathrm {If}(
\mathsf {p}
\stackrel{r}{=} 
\lceil 
\forall_{obj} 
\mathsf{x}
\colon 
\mathsf{y}
\rceil 
\wedge 
\mathsf {p}
{}^{1}
\stackrel {t}{=}
\mathsf {c}
{}^{1}
,\linebreak [0]
Ded_4(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
,
\mathsf {b}
)
,\linebreak [0]
Ded_3(
\mathsf {p}
,
\mathsf {c}
{}^{2}
,
\mathsf {s}
,
\mathsf {c}
{}^{1}
\mathrel { : \, : }
\mathsf {c}
{}^{1}
\mathrel { : \, : }
\mathsf {b}
)
)
)
]$

$\rm
[
Ded_3(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
,
\mathsf {u}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Ded
\_3(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline ,
\#4.
\newline )
\mbox{''}
]$

$\rm
[
Ded_3(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}deduction
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}condition
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}deduction
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:45}{}\section* {$\rm 
Ded_4(\ast
,\ast
,\ast
,\ast
)$}

$\rm
[
Ded_4(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
,
\mathsf {b}
)
\stackrel{
\mathrm {val}
}{\rightarrow}
\mathsf {s}
!
\mathsf {b}
!
\mathrm {If}(
\mathsf {p}
\stackrel{r}{=} 
\lceil \overline{
\mathsf{x}}
\rceil 
,\linebreak [0]
\mathbf {lookup}(
\mathsf {p}
,
\mathsf {b}
,
\mathsf {T}
)
\stackrel {t}{=}
\mathsf {c}
,\linebreak [0]
\mathrm {If}(
{\neg }
\mathsf {p}
\stackrel{r}{=} 
\mathsf {c}
,\linebreak [0]
\mathsf {F}
,\linebreak [0]
\mathrm {If}(
\mathsf {p}
\stackrel{r}{=} 
\lceil 
\forall_{obj} 
\mathsf{x}
\colon 
\mathsf{y}
\rceil 
,\linebreak [0]
\mathsf {p}
{}^{1}
\stackrel {t}{=}
\mathsf {c}
{}^{1}
\wedge 
Ded_4(
\mathsf {p}
{}^{2}
,
\mathsf {c}
{}^{2}
,
\mathsf {s}
,
\mathsf {p}
{}^{1}
\mathrel { : \, : }
\mathsf {p}
{}^{1}
\mathrel { : \, : }
\mathsf {b}
)
,\linebreak [0]
\mathrm {If}(
{\neg }
\mathsf {p}
\stackrel{r}{=} 
\lceil \underline{
\mathsf{x}}
\rceil 
,\linebreak [0]
Ded_4^*(
\mathsf {p}
{}^t
,
\mathsf {c}
{}^t
,
\mathsf {s}
,
\mathsf {b}
)
,\linebreak [0]
\mathsf {p}
{}^{1}
\stackrel {t}{=}
\mathsf {c}
{}^{1}
\wedge 
Ded_5(
\mathsf {p}
,
\mathsf {s}
,
\mathsf {b}
)
)
)
)
)
]$

$\rm
[
Ded_4(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
,
\mathsf {u}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Ded
\_4(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline ,
\#4.
\newline )
\mbox{''}
]$

$\rm
[
Ded_4(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}deduction
\linebreak [0]\ \hskip0em plus2.0em{}four
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}condition
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}deduction
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:46}{}\section* {$\rm 
Ded_4^*(\ast
,\ast
,\ast
,\ast
)$}

$\rm
[
Ded_4^*(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
,
\mathsf {b}
)
\stackrel{
\mathrm {val}
}{\rightarrow}
\mathsf {c}
!
\mathsf {s}
!
\mathsf {b}
!
\mathrm {If}(
\mathsf {p}
,\linebreak [0]
\mathsf {T}
,\linebreak [0]
Ded_4(
\mathsf {p}
{}^h
,
\mathsf {c}
{}^h
,
\mathsf {s}
,
\mathsf {b}
)
\wedge 
Ded_4^*(
\mathsf {p}
{}^t
,
\mathsf {c}
{}^t
,
\mathsf {s}
,
\mathsf {b}
)
)
]$

$\rm
[
Ded_4^*(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
,
\mathsf {u}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Ded
\_4
{\char94}
{*}(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline ,
\#4.
\newline )
\mbox{''}
]$

$\rm
[
Ded_4^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}deduction
\linebreak [0]\ \hskip0em plus2.0em{}four
\linebreak [0]\ \hskip0em plus2.0em{}star
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}condition
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}deduction
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:47}{}\section* {$\rm 
Ded_5(\ast
,\ast
,\ast
)$}

$\rm
[
Ded_5(
\mathsf {p}
,
\mathsf {s}
,
\mathsf {b}
)
\stackrel{
\mathrm {val}
}{\rightarrow}
\mathsf {p}
!
\mathsf {s}
!
\mathrm {If}(
\mathsf {b}
,\linebreak [0]
\mathsf {T}
,\linebreak [0]
\lceil 
\lceil 
\mathsf{x}
\rceil 
\#^0
\lceil 
\mathsf{y}
\rceil 
\rceil 
{}^h
\mathrel { : \, : }
\lceil 
\lceil 
\ast 
\rceil 
\rceil 
{}^h
\mathrel { : \, : }
\mathsf {b}
{}^h
{}^h
\mathrel { : \, : }
\mathsf {T}
\mathrel { : \, : }
\lceil 
\lceil 
\mathsf{x}
\rceil 
\rceil 
{}^h
\mathrel { : \, : }
\mathsf {p}
\mathrel { : \, : }
\mathsf {T}
\mathrel { : \, : }
\mathsf {T}
\in_t 
\mathsf {s}
\wedge 
Ded_5(
\mathsf {p}
,
\mathsf {s}
,
\mathsf {b}
{}^t
)
)
]$

$\rm
[
Ded_5(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Ded
\_5(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline )
\mbox{''}
]$

$\rm
[
Ded_5(
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}deduction
\linebreak [0]\ \hskip0em plus2.0em{}five
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}condition
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}deduction
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:48}{}\section* {$\rm 
Ded_6(\ast
,\ast
,\ast
,\ast
)$}

$\rm
[
Ded_6(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {e}
,
\mathsf {b}
)
\stackrel{
\mathrm {val}
}{\rightarrow}
\mathsf {p}
!
\mathsf {c}
!
\mathsf {b}
!
\mathsf {e}
!
\mathrm {If}(
\mathsf {p}
\stackrel{r}{=} 
\lceil \overline{
\mathsf{x}}
\rceil 
,\linebreak [0]
\mathsf {p}
\in_t 
\mathsf {e}
\left\{\protect \begin {array}{l}
\mathsf {b}
\\
\mathsf {p}
\mathrel { : \, : }
\mathsf {c}
\mathrel { : \, : }
\mathsf {b}
\protect \end {array}\right.
,\linebreak [0]
\mathrm {If}(
{\neg }
\mathsf {p}
\stackrel{r}{=} 
\mathsf {c}
,\linebreak [0]
\mathsf {T}
,\linebreak [0]
\mathrm {If}(
\mathsf {p}
\stackrel{r}{=} 
\lceil \underline{
\mathsf {a}}
\rceil 
,\linebreak [0]
\mathsf {b}
,\linebreak [0]
\mathrm {If}(
\mathsf {p}
\stackrel{r}{=} 
\lceil 
\forall_{obj} 
\mathsf{x}
\colon 
\mathsf{y}
\rceil 
,\linebreak [0]
Ded_6(
\mathsf {p}
{}^{2}
,
\mathsf {c}
{}^{2}
,
\mathsf {c}
{}^{1}
\mathrel { : \, : }
\mathsf {e}
,
\mathsf {b}
)
,\linebreak [0]
Ded_6^*(
\mathsf {p}
{}^t
,
\mathsf {c}
{}^t
,
\mathsf {e}
,
\mathsf {b}
)
)
)
)
)
]$

$\rm
[
Ded_6(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {e}
,
\mathsf {b}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Ded
\_6(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline ,
\#4.
\newline )
\mbox{''}
]$

$\rm
[
Ded_6(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}deduction
\linebreak [0]\ \hskip0em plus2.0em{}six
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}exception
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}deduction
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:49}{}\section* {$\rm 
Ded_6^*(\ast
,\ast
,\ast
,\ast
)$}

$\rm
[
Ded_6^*(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {e}
,
\mathsf {b}
)
\stackrel{
\mathrm {val}
}{\rightarrow}
\mathsf {p}
!
\mathsf {c}
!
\mathsf {b}
!
\mathsf {e}
!
\mathrm {If}(
\mathsf {p}
,\linebreak [0]
\mathsf {b}
,\linebreak [0]
Ded_6^*(
\mathsf {p}
{}^t
,
\mathsf {c}
{}^t
,
\mathsf {e}
,
Ded_6(
\mathsf {p}
{}^h
,
\mathsf {c}
{}^h
,
\mathsf {e}
,
\mathsf {b}
)
)
)
]$

$\rm
[
Ded_6^*(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {e}
,
\mathsf {b}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Ded
\_6
{\char94}
{*}(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline ,
\#4.
\newline )
\mbox{''}
]$

$\rm
[
Ded_6^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}deduction
\linebreak [0]\ \hskip0em plus2.0em{}six
\linebreak [0]\ \hskip0em plus2.0em{}star
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}exception
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}deduction
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:50}{}\section* {$\rm 
Ded_7(\ast
)$}

$\rm
[
Ded_7(
\mathsf {p}
)
\stackrel{
\mathrm {val}
}{\rightarrow}
\mathsf {p}
\stackrel{r}{=} 
\lceil 
\forall 
\mathsf{x}
\colon 
\mathsf{y}
\rceil 
\left\{\protect \begin {array}{l}
Ded_7(
\mathsf {p}
{}^{2}
)
\\
\mathsf {p}
\protect \end {array}\right.
]$

$\rm
[
Ded_7(
\mathsf {p}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Ded
\_7(
\#1.
\newline )
\mbox{''}
]$

$\rm
[
Ded_7(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}deduction
\linebreak [0]\ \hskip0em plus2.0em{}seven
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}deduction
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:51}{}\section* {$\rm 
Ded_8(\ast
,\ast
)$}

$\rm
[
Ded_8(
\mathsf {p}
,
\mathsf {b}
)
\stackrel{
\mathrm {val}
}{\rightarrow}
\mathrm {If}(
\mathsf {p}
\stackrel{r}{=} 
\lceil 
\forall 
\mathsf{x}
\colon 
\mathsf{y}
\rceil 
,\linebreak [0]
Ded_8(
\mathsf {p}
{}^{2}
,
\mathsf {p}
{}^{1}
\mathrel { : \, : }
\mathsf {b}
)
,\linebreak [0]
\mathrm {If}(
\mathsf {p}
\stackrel{r}{=} 
\lceil \underline{
\mathsf {a}}
\rceil 
,\linebreak [0]
\mathsf {p}
\in_t 
\mathsf {b}
,\linebreak [0]
Ded_8^*(
\mathsf {p}
{}^t
,
\mathsf {b}
)
)
)
]$

$\rm
[
Ded_8(
\mathsf {p}
,
\mathsf {b}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Ded
\_8(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$

$\rm
[
Ded_8(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}deduction
\linebreak [0]\ \hskip0em plus2.0em{}eight
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}deduction
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:52}{}\section* {$\rm 
Ded_8^*(\ast
,\ast
)$}

$\rm
[
Ded_8^*(
\mathsf {p}
,
\mathsf {b}
)
\stackrel{
\mathrm {val}
}{\rightarrow}
\mathsf {b}
!
\mathrm {If}(
\mathsf {p}
,\linebreak [0]
\mathsf {T}
,\linebreak [0]
\mathrm {If}(
Ded_8(
\mathsf {p}
{}^h
,
\mathsf {b}
)
,\linebreak [0]
Ded_8^*(
\mathsf {p}
{}^t
,
\mathsf {b}
)
,\linebreak [0]
\mathsf {F}
)
)
]$

$\rm
[
Ded_8^*(
\mathsf {p}
,
\mathsf {b}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Ded
\_8
{\char94}
{*}(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$

$\rm
[
Ded_8^*(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}deduction
\linebreak [0]\ \hskip0em plus2.0em{}eight
\linebreak [0]\ \hskip0em plus2.0em{}star
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}deduction
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:53}{}\section* {$\rm 
S$}

$\rm
[
S
\stackrel{
stmt
}{\rightarrow}
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
\mathrel {\oplus} 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\vdash \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {b}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\vdash \underline{
\mathsf {a}}
{}'
= \underline{
\mathsf {b}}
{}'
\mathrel {\oplus} 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
{}'
= \underline{
\mathsf {b}}
{}'
\vdash \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\lambda 
\mathsf{x}
.
Ded_0(
\lceil \underline{
\mathsf {a}}
\rceil 
,
\lceil \underline{
\mathsf {b}}
\rceil 
)
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {b}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
\cdot \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {a}}
\cdot \underline{
\mathsf {b}}
+ \underline{
\mathsf {a}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {a}}
\colon \underline{
\mathsf {a}}
+ 
0
= \underline{
\mathsf {a}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
{\neg }\underline{
\mathsf {b}}
\Rightarrow 
{\neg }\underline{
\mathsf {a}}
\vdash 
{\neg }\underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {b}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\vdash \underline{
\mathsf {a}}
= \underline{
\mathsf {c}}
\vdash \underline{
\mathsf {b}}
= \underline{
\mathsf {c}}
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\langle 
\lceil \underline{
\mathsf {b}}
\rceil 
{\equiv}^0 
\lceil \underline{
\mathsf {a}}
\rceil 
| 
\lceil \underline{
\mathsf{x}}
\rceil 
{:=} 
\lceil 
0
\rceil 
\rangle 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\langle 
\lceil \underline{
\mathsf {c}}
\rceil 
{\equiv}^0 
\lceil \underline{
\mathsf {a}}
\rceil 
| 
\lceil \underline{
\mathsf{x}}
\rceil 
{:=} 
\lceil \underline{
\mathsf{x}}
{}'
\rceil 
\rangle 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \underline{
\mathsf {b}}
\vdash \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {c}}
\vdash \underline{
\mathsf {a}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {a}}
\colon 
{\neg }
0
= \underline{
\mathsf {a}}
{}'
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf {a}}
\colon \underline{
\mathsf {a}}
\vdash 
\forall_{obj} \underline{
\mathsf{x}}
\colon \underline{
\mathsf {a}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {a}}
\colon \underline{
\mathsf {a}}
\cdot 
0
= 
0
]$

$\rm
[
S
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline S
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:54}{}\section* {$\rm 
Neg$}

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

$\rm
[
Neg
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
{\neg }\underline{
\mathsf {b}}
\Rightarrow 
{\neg }\underline{
\mathsf {a}}
\vdash 
{\neg }\underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {b}}
]$

$\rm
[
Neg
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Neg
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:55}{}\section* {$\rm 
MP$}

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

$\rm
[
MP
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\vdash \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {b}}
]$

$\rm
[
MP
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline MP
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:56}{}\section* {$\rm 
Gen$}

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

$\rm
[
Gen
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf {a}}
\colon \underline{
\mathsf {a}}
\vdash 
\forall_{obj} \underline{
\mathsf{x}}
\colon \underline{
\mathsf {a}}
]$

$\rm
[
Gen
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Gen
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:57}{}\section* {$\rm 
Ded$}

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

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

$\rm
[
Ded
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Ded
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:58}{}\section* {$\rm 
S1$}

$\rm
[
S1
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
S1
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\vdash \underline{
\mathsf {a}}
= \underline{
\mathsf {c}}
\vdash \underline{
\mathsf {b}}
= \underline{
\mathsf {c}}
]$

$\rm
[
S1
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline S1
\mbox{''}
]$

$\rm
[
S1
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}s
\linebreak [0]\ \hskip0em plus2.0em{}one
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:59}{}\section* {$\rm 
S2$}

$\rm
[
S2
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
S2
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\vdash \underline{
\mathsf {a}}
{}'
= \underline{
\mathsf {b}}
{}'
]$

$\rm
[
S2
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline S2
\mbox{''}
]$

$\rm
[
S2
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}s
\linebreak [0]\ \hskip0em plus2.0em{}two
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:60}{}\section* {$\rm 
S3$}

$\rm
[
S3
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
S3
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
{\neg }
0
= \underline{
\mathsf {a}}
{}'
]$

$\rm
[
S3
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline S3
\mbox{''}
]$

$\rm
[
S3
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}s
\linebreak [0]\ \hskip0em plus2.0em{}three
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:61}{}\section* {$\rm 
S4$}

$\rm
[
S4
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
S4
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
{}'
= \underline{
\mathsf {b}}
{}'
\vdash \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
]$

$\rm
[
S4
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline S4
\mbox{''}
]$

$\rm
[
S4
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}s
\linebreak [0]\ \hskip0em plus2.0em{}four
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:62}{}\section* {$\rm 
S5$}

$\rm
[
S5
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
S5
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon \underline{
\mathsf {a}}
+ 
0
= \underline{
\mathsf {a}}
]$

$\rm
[
S5
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline S5
\mbox{''}
]$

$\rm
[
S5
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}s
\linebreak [0]\ \hskip0em plus2.0em{}five
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:63}{}\section* {$\rm 
S6$}

$\rm
[
S6
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
S6
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
]$

$\rm
[
S6
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline S6
\mbox{''}
]$

$\rm
[
S6
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}s
\linebreak [0]\ \hskip0em plus2.0em{}six
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:64}{}\section* {$\rm 
S7$}

$\rm
[
S7
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
S7
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon \underline{
\mathsf {a}}
\cdot 
0
= 
0
]$

$\rm
[
S7
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline S7
\mbox{''}
]$

$\rm
[
S7
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}s
\linebreak [0]\ \hskip0em plus2.0em{}seven
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:65}{}\section* {$\rm 
S8$}

$\rm
[
S8
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
S8
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
\cdot \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {a}}
\cdot \underline{
\mathsf {b}}
+ \underline{
\mathsf {a}}
]$

$\rm
[
S8
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline S8
\mbox{''}
]$

$\rm
[
S8
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}s
\linebreak [0]\ \hskip0em plus2.0em{}eight
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:66}{}\section* {$\rm 
S9$}

$\rm
[
S9
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$\rm
[
S9
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\langle 
\lceil \underline{
\mathsf {b}}
\rceil 
{\equiv}^0 
\lceil \underline{
\mathsf {a}}
\rceil 
| 
\lceil \underline{
\mathsf{x}}
\rceil 
{:=} 
\lceil 
0
\rceil 
\rangle 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\langle 
\lceil \underline{
\mathsf {c}}
\rceil 
{\equiv}^0 
\lceil \underline{
\mathsf {a}}
\rceil 
| 
\lceil \underline{
\mathsf{x}}
\rceil 
{:=} 
\lceil \underline{
\mathsf{x}}
{}'
\rceil 
\rangle 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \underline{
\mathsf {b}}
\vdash \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {c}}
\vdash \underline{
\mathsf {a}}
]$

$\rm
[
S9
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline S9
\mbox{''}
]$

$\rm
[
S9
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}s
\linebreak [0]\ \hskip0em plus2.0em{}nine
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:67}{}\section* {$\rm 
Repetition$}

$\rm
[
Repetition
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
\lceil 
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon \underline{
\mathsf {a}}
{} ^ { I }
\rceil 
]$

$\rm
[
Repetition
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {a}}
]$

$\rm
[
Repetition
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Repetition
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:68}{}\section* {$\rm 
A1'$}

$\rm
[
A1'
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {b}}
\vdash 
Repetition
\rhd \underline{
\mathsf {a}}
\gg \underline{
\mathsf {a}}
; 
Ded
\rhd 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {b}}
\vdash \underline{
\mathsf {a}}
\gg \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {a}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
A1'
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {a}}
]$

$\rm
[
A1'
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline A1
\mbox {'}
\mbox{''}
]$

$\rm
[
A1'
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}a
\linebreak [0]\ \hskip0em plus2.0em{}one
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:69}{}\section* {$\rm 
A2'$}

$\rm
[
A2'
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\vdash \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\vdash \underline{
\mathsf {a}}
\vdash 
MP
\rhd \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\rhd \underline{
\mathsf {a}}
\gg \underline{
\mathsf {b}}
; 
MP
\rhd \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\rhd \underline{
\mathsf {a}}
\gg \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
; 
MP
\rhd \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\rhd \underline{
\mathsf {b}}
\gg \underline{
\mathsf {c}}
; 
Ded
\rhd 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\vdash \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\vdash \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {c}}
\gg \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {c}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
A2'
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {c}}
]$

$\rm
[
A2'
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline A2
\mbox {'}
\mbox{''}
]$

$\rm
[
A2'
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}a
\linebreak [0]\ \hskip0em plus2.0em{}two
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:70}{}\section* {$\rm 
A4'$}

$\rm
[
A4'
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
S
\vdash \overline{
\mathsf{x}}
+ \overline{
\mathsf{y}}
= \overline{
\mathsf{y}}
+ \overline{
\mathsf{x}}
\vdash 
Repetition
\rhd \overline{
\mathsf{x}}
+ \overline{
\mathsf{y}}
= \overline{
\mathsf{y}}
+ \overline{
\mathsf{x}}
\gg \overline{
\mathsf{x}}
+ \overline{
\mathsf{y}}
= \overline{
\mathsf{y}}
+ \overline{
\mathsf{x}}
; 
Ded
\rhd \overline{
\mathsf{x}}
+ \overline{
\mathsf{y}}
= \overline{
\mathsf{y}}
+ \overline{
\mathsf{x}}
\vdash \overline{
\mathsf{x}}
+ \overline{
\mathsf{y}}
= \overline{
\mathsf{y}}
+ \overline{
\mathsf{x}}
\gg 
\forall_{obj} \overline{
\mathsf{x}}
\colon 
\forall_{obj} \overline{
\mathsf{y}}
\colon \overline{
\mathsf{x}}
+ \overline{
\mathsf{y}}
= \overline{
\mathsf{y}}
+ \overline{
\mathsf{x}}
\Rightarrow 
2
+ 
3
= 
3
+ 
2
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
A4'
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall_{obj} \overline{
\mathsf{x}}
\colon 
\forall_{obj} \overline{
\mathsf{y}}
\colon \overline{
\mathsf{x}}
+ \overline{
\mathsf{y}}
= \overline{
\mathsf{y}}
+ \overline{
\mathsf{x}}
\Rightarrow 
2
+ 
3
= 
3
+ 
2
]$

$\rm
[
A4'
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline A4
\mbox {'}
\mbox{''}
]$

$\rm
[
A4'
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}a
\linebreak [0]\ \hskip0em plus2.0em{}four
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:71}{}\section* {$\rm 
A5'$}

$\rm
[
A5'
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
S
\vdash 
2
+ 
3
= 
5
\Rightarrow 
2
+ 
3
+ \overline{
\mathsf{x}}
= 
5
+ \overline{
\mathsf{x}}
\vdash 
2
+ 
3
= 
5
\vdash 
MP
\rhd 
2
+ 
3
= 
5
\Rightarrow 
2
+ 
3
+ \overline{
\mathsf{x}}
= 
5
+ \overline{
\mathsf{x}}
\rhd 
2
+ 
3
= 
5
\gg 
2
+ 
3
+ \overline{
\mathsf{x}}
= 
5
+ \overline{
\mathsf{x}}
; 
Gen
\rhd 
2
+ 
3
+ \overline{
\mathsf{x}}
= 
5
+ \overline{
\mathsf{x}}
\gg 
\forall_{obj} \overline{
\mathsf{x}}
\colon 
2
+ 
3
+ \overline{
\mathsf{x}}
= 
5
+ \overline{
\mathsf{x}}
; 
Ded
\rhd 
2
+ 
3
= 
5
\Rightarrow 
2
+ 
3
+ \overline{
\mathsf{x}}
= 
5
+ \overline{
\mathsf{x}}
\vdash 
2
+ 
3
= 
5
\vdash 
\forall_{obj} \overline{
\mathsf{x}}
\colon 
2
+ 
3
+ \overline{
\mathsf{x}}
= 
5
+ \overline{
\mathsf{x}}
\gg 
\forall_{obj} \overline{
\mathsf{x}}
\colon 
2
+ 
3
= 
5
\Rightarrow 
2
+ 
3
+ \overline{
\mathsf{x}}
= 
5
+ \overline{
\mathsf{x}}
\Rightarrow 
2
+ 
3
= 
5
\Rightarrow 
\forall_{obj} \overline{
\mathsf{x}}
\colon 
2
+ 
3
+ \overline{
\mathsf{x}}
= 
5
+ \overline{
\mathsf{x}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
A5'
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall_{obj} \overline{
\mathsf{x}}
\colon 
2
+ 
3
= 
5
\Rightarrow 
2
+ 
3
+ \overline{
\mathsf{x}}
= 
5
+ \overline{
\mathsf{x}}
\Rightarrow 
2
+ 
3
= 
5
\Rightarrow 
\forall_{obj} \overline{
\mathsf{x}}
\colon 
2
+ 
3
+ \overline{
\mathsf{x}}
= 
5
+ \overline{
\mathsf{x}}
]$

$\rm
[
A5'
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline A5
\mbox {'}
\mbox{''}
]$

$\rm
[
A5'
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}a
\linebreak [0]\ \hskip0em plus2.0em{}five
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:72}{}\section* {$\rm 
Prop\ 3.2a$}

$\rm
[
Prop\ 3.2a
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
S5
\gg \underline{
\mathsf {a}}
+ 
0
= \underline{
\mathsf {a}}
; 
S1
\rhd \underline{
\mathsf {a}}
+ 
0
= \underline{
\mathsf {a}}
\rhd \underline{
\mathsf {a}}
+ 
0
= \underline{
\mathsf {a}}
\gg \underline{
\mathsf {a}}
= \underline{
\mathsf {a}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
Prop\ 3.2a
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon \underline{
\mathsf {a}}
= \underline{
\mathsf {a}}
]$

$\rm
[
Prop\ 3.2a
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Prop
\mbox {$\backslash $}
\linebreak [0]\ \hskip0em plus2.0em{}3.2a
\mbox{''}
]$

$\rm
[
Prop\ 3.2a
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}a
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:73}{}\section* {$\rm 
Prop\ 3.2b$}

$\rm
[
Prop\ 3.2b
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\vdash 
Prop\ 3.2a
\gg \underline{
\mathsf {a}}
= \underline{
\mathsf {a}}
; 
S1
\rhd \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\rhd \underline{
\mathsf {a}}
= \underline{
\mathsf {a}}
\gg \underline{
\mathsf {b}}
= \underline{
\mathsf {a}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
Prop\ 3.2b
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\vdash \underline{
\mathsf {b}}
= \underline{
\mathsf {a}}
]$

$\rm
[
Prop\ 3.2b
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Prop
\mbox {$\backslash $}
\linebreak [0]\ \hskip0em plus2.0em{}3.2b
\mbox{''}
]$

$\rm
[
Prop\ 3.2b
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}b
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:74}{}\section* {$\rm 
Prop\ 3.2c$}

$\rm
[
Prop\ 3.2c
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\vdash \underline{
\mathsf {b}}
= \underline{
\mathsf {c}}
\vdash 
Prop\ 3.2b
\rhd \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\gg \underline{
\mathsf {b}}
= \underline{
\mathsf {a}}
; 
S1
\rhd \underline{
\mathsf {b}}
= \underline{
\mathsf {a}}
\rhd \underline{
\mathsf {b}}
= \underline{
\mathsf {c}}
\gg \underline{
\mathsf {a}}
= \underline{
\mathsf {c}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
Prop\ 3.2c
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\vdash \underline{
\mathsf {b}}
= \underline{
\mathsf {c}}
\vdash \underline{
\mathsf {a}}
= \underline{
\mathsf {c}}
]$

$\rm
[
Prop\ 3.2c
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Prop
\mbox {$\backslash $}
\linebreak [0]\ \hskip0em plus2.0em{}3.2c
\mbox{''}
]$

$\rm
[
Prop\ 3.2c
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}c
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:75}{}\section* {$\rm 
Prop\ 3.2d$}

$\rm
[
Prop\ 3.2d
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon \underline{
\mathsf {a}}
= \underline{
\mathsf {c}}
\vdash \underline{
\mathsf {b}}
= \underline{
\mathsf {c}}
\vdash 
Prop\ 3.2b
\rhd \underline{
\mathsf {b}}
= \underline{
\mathsf {c}}
\gg \underline{
\mathsf {c}}
= \underline{
\mathsf {b}}
; 
Prop\ 3.2c
\rhd \underline{
\mathsf {a}}
= \underline{
\mathsf {c}}
\rhd \underline{
\mathsf {c}}
= \underline{
\mathsf {b}}
\gg \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
Prop\ 3.2d
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon \underline{
\mathsf {a}}
= \underline{
\mathsf {c}}
\vdash \underline{
\mathsf {b}}
= \underline{
\mathsf {c}}
\vdash \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
]$

$\rm
[
Prop\ 3.2d
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Prop
\mbox {$\backslash $}
\linebreak [0]\ \hskip0em plus2.0em{}3.2d
\mbox{''}
]$

$\rm
[
Prop\ 3.2d
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}d
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:76}{}\section* {$\rm 
Prop\ 3.2e_1$}

$\rm
[
Prop\ 3.2e_1
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\vdash 
S5
\gg \underline{
\mathsf {a}}
+ 
0
= \underline{
\mathsf {a}}
; 
Prop\ 3.2c
\rhd \underline{
\mathsf {a}}
+ 
0
= \underline{
\mathsf {a}}
\rhd \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\gg \underline{
\mathsf {a}}
+ 
0
= \underline{
\mathsf {b}}
; 
S5
\gg \underline{
\mathsf {b}}
+ 
0
= \underline{
\mathsf {b}}
; 
Prop\ 3.2d
\rhd \underline{
\mathsf {a}}
+ 
0
= \underline{
\mathsf {b}}
\rhd \underline{
\mathsf {b}}
+ 
0
= \underline{
\mathsf {b}}
\gg \underline{
\mathsf {a}}
+ 
0
= \underline{
\mathsf {b}}
+ 
0
; 
Ded
\rhd 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\vdash \underline{
\mathsf {a}}
+ 
0
= \underline{
\mathsf {b}}
+ 
0
\gg \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {a}}
+ 
0
= \underline{
\mathsf {b}}
+ 
0
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
Prop\ 3.2e_1
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {a}}
+ 
0
= \underline{
\mathsf {b}}
+ 
0
]$

$\rm
[
Prop\ 3.2e_1
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Prop
\mbox {$\backslash $}
\linebreak [0]\ \hskip0em plus2.0em{}3.2e
\_1
\mbox{''}
]$

$\rm
[
Prop\ 3.2e_1
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}e
\linebreak [0]\ \hskip0em plus2.0em{}one
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:77}{}\section* {$\rm 
Prop\ 3.2e_2$}

$\rm
[
Prop\ 3.2e_2
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {a}}
+ \underline{
\mathsf {c}}
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {c}}
\vdash \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\vdash 
MP
\rhd \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {a}}
+ \underline{
\mathsf {c}}
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {c}}
\rhd \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\gg \underline{
\mathsf {a}}
+ \underline{
\mathsf {c}}
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {c}}
; 
S2
\rhd \underline{
\mathsf {a}}
+ \underline{
\mathsf {c}}
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {c}}
\gg \underline{
\mathsf {a}}
+ \underline{
\mathsf {c}}
{}'
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {c}}
{}'
; 
S6
\gg \underline{
\mathsf {a}}
+ \underline{
\mathsf {c}}
{}'
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {c}}
{}'
; 
Prop\ 3.2c
\rhd \underline{
\mathsf {a}}
+ \underline{
\mathsf {c}}
{}'
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {c}}
{}'
\rhd \underline{
\mathsf {a}}
+ \underline{
\mathsf {c}}
{}'
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {c}}
{}'
\gg \underline{
\mathsf {a}}
+ \underline{
\mathsf {c}}
{}'
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {c}}
{}'
; 
S6
\gg \underline{
\mathsf {b}}
+ \underline{
\mathsf {c}}
{}'
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {c}}
{}'
; 
Prop\ 3.2d
\rhd \underline{
\mathsf {a}}
+ \underline{
\mathsf {c}}
{}'
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {c}}
{}'
\rhd \underline{
\mathsf {b}}
+ \underline{
\mathsf {c}}
{}'
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {c}}
{}'
\gg \underline{
\mathsf {a}}
+ \underline{
\mathsf {c}}
{}'
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {c}}
{}'
; 
Ded
\rhd 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {a}}
+ \underline{
\mathsf {c}}
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {c}}
\vdash \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\vdash \underline{
\mathsf {a}}
+ \underline{
\mathsf {c}}
{}'
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {c}}
{}'
\gg \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {a}}
+ \underline{
\mathsf {c}}
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {a}}
+ \underline{
\mathsf {c}}
{}'
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {c}}
{}'
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
Prop\ 3.2e_2
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {a}}
+ \underline{
\mathsf {c}}
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {a}}
+ \underline{
\mathsf {c}}
{}'
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {c}}
{}'
]$

$\rm
[
Prop\ 3.2e_2
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Prop
\mbox {$\backslash $}
\linebreak [0]\ \hskip0em plus2.0em{}3.2e
\_2
\mbox{''}
]$

$\rm
[
Prop\ 3.2e_2
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}e
\linebreak [0]\ \hskip0em plus2.0em{}two
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:78}{}\section* {$\rm 
Prop\ 3.2e$}

$\rm
[
Prop\ 3.2e
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\vdash 
Prop\ 3.2e_1
\gg \overline{
\mathsf{x}}
= \overline{
\mathsf{y}}
\Rightarrow \overline{
\mathsf{x}}
+ 
0
= \overline{
\mathsf{y}}
+ 
0
; 
Prop\ 3.2e_2
\gg \overline{
\mathsf{x}}
= \overline{
\mathsf{y}}
\Rightarrow \overline{
\mathsf{x}}
+ \overline{
\mathsf{z}}
= \overline{
\mathsf{y}}
+ \overline{
\mathsf{z}}
\Rightarrow \overline{
\mathsf{x}}
= \overline{
\mathsf{y}}
\Rightarrow \overline{
\mathsf{x}}
+ \overline{
\mathsf{z}}
{}'
= \overline{
\mathsf{y}}
+ \overline{
\mathsf{z}}
{}'
; 
S9
\mathop {\char64} \overline{
\mathsf{z}}
\rhd \overline{
\mathsf{x}}
= \overline{
\mathsf{y}}
\Rightarrow \overline{
\mathsf{x}}
+ 
0
= \overline{
\mathsf{y}}
+ 
0
\rhd \overline{
\mathsf{x}}
= \overline{
\mathsf{y}}
\Rightarrow \overline{
\mathsf{x}}
+ \overline{
\mathsf{z}}
= \overline{
\mathsf{y}}
+ \overline{
\mathsf{z}}
\Rightarrow \overline{
\mathsf{x}}
= \overline{
\mathsf{y}}
\Rightarrow \overline{
\mathsf{x}}
+ \overline{
\mathsf{z}}
{}'
= \overline{
\mathsf{y}}
+ \overline{
\mathsf{z}}
{}'
\gg \overline{
\mathsf{x}}
= \overline{
\mathsf{y}}
\Rightarrow \overline{
\mathsf{x}}
+ \overline{
\mathsf{z}}
= \overline{
\mathsf{y}}
+ \overline{
\mathsf{z}}
; 
Ded
\rhd \overline{
\mathsf{x}}
= \overline{
\mathsf{y}}
\Rightarrow \overline{
\mathsf{x}}
+ \overline{
\mathsf{z}}
= \overline{
\mathsf{y}}
+ \overline{
\mathsf{z}}
\gg \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {a}}
+ \underline{
\mathsf {c}}
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {c}}
; 
MP
\rhd \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {a}}
+ \underline{
\mathsf {c}}
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {c}}
\rhd \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\gg \underline{
\mathsf {a}}
+ \underline{
\mathsf {c}}
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {c}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
Prop\ 3.2e
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
\vdash \underline{
\mathsf {a}}
+ \underline{
\mathsf {c}}
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {c}}
]$

$\rm
[
Prop\ 3.2e
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Prop
\mbox {$\backslash $}
\linebreak [0]\ \hskip0em plus2.0em{}3.2e
\mbox{''}
]$

$\rm
[
Prop\ 3.2e
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}e
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:79}{}\section* {$\rm 
Prop\ 3.2f_1$}

$\rm
[
Prop\ 3.2f_1
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
S
\vdash 
S5
\gg 
0
+ 
0
= 
0
; 
Prop\ 3.2b
\rhd 
0
+ 
0
= 
0
\gg 
0
= 
0
+ 
0
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
Prop\ 3.2f_1
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
0
= 
0
+ 
0
]$

$\rm
[
Prop\ 3.2f_1
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Prop
\mbox {$\backslash $}
\linebreak [0]\ \hskip0em plus2.0em{}3.2f
\_1
\mbox{''}
]$

$\rm
[
Prop\ 3.2f_1
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}f
\linebreak [0]\ \hskip0em plus2.0em{}one
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:80}{}\section* {$\rm 
Prop\ 3.2f_2$}

$\rm
[
Prop\ 3.2f_2
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {a}}
\colon \underline{
\mathsf {a}}
= 
0
+ \underline{
\mathsf {a}}
\vdash 
S2
\rhd \underline{
\mathsf {a}}
= 
0
+ \underline{
\mathsf {a}}
\gg \underline{
\mathsf {a}}
{}'
= 
0
+ \underline{
\mathsf {a}}
{}'
; 
S6
\gg 
0
+ \underline{
\mathsf {a}}
{}'
= 
0
+ \underline{
\mathsf {a}}
{}'
; 
Prop\ 3.2d
\rhd \underline{
\mathsf {a}}
{}'
= 
0
+ \underline{
\mathsf {a}}
{}'
\rhd 
0
+ \underline{
\mathsf {a}}
{}'
= 
0
+ \underline{
\mathsf {a}}
{}'
\gg \underline{
\mathsf {a}}
{}'
= 
0
+ \underline{
\mathsf {a}}
{}'
; 
Ded
\rhd 
\forall \underline{
\mathsf {a}}
\colon \underline{
\mathsf {a}}
= 
0
+ \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {a}}
{}'
= 
0
+ \underline{
\mathsf {a}}
{}'
\gg \underline{
\mathsf {a}}
= 
0
+ \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {a}}
{}'
= 
0
+ \underline{
\mathsf {a}}
{}'
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
Prop\ 3.2f_2
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon \underline{
\mathsf {a}}
= 
0
+ \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {a}}
{}'
= 
0
+ \underline{
\mathsf {a}}
{}'
]$

$\rm
[
Prop\ 3.2f_2
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Prop
\mbox {$\backslash $}
\linebreak [0]\ \hskip0em plus2.0em{}3.2f
\_2
\mbox{''}
]$

$\rm
[
Prop\ 3.2f_2
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}f
\linebreak [0]\ \hskip0em plus2.0em{}two
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:81}{}\section* {$\rm 
Prop\ 3.2f$}

$\rm
[
Prop\ 3.2f
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
Prop\ 3.2f_1
\gg 
0
= 
0
+ 
0
; 
Prop\ 3.2f_2
\gg \overline{
\mathsf{x}}
= 
0
+ \overline{
\mathsf{x}}
\Rightarrow \overline{
\mathsf{x}}
{}'
= 
0
+ \overline{
\mathsf{x}}
{}'
; 
S9
\mathop {\char64} \overline{
\mathsf{x}}
\rhd 
0
= 
0
+ 
0
\rhd \overline{
\mathsf{x}}
= 
0
+ \overline{
\mathsf{x}}
\Rightarrow \overline{
\mathsf{x}}
{}'
= 
0
+ \overline{
\mathsf{x}}
{}'
\gg \overline{
\mathsf{x}}
= 
0
+ \overline{
\mathsf{x}}
; 
Ded
\rhd \overline{
\mathsf{x}}
= 
0
+ \overline{
\mathsf{x}}
\gg \underline{
\mathsf {a}}
= 
0
+ \underline{
\mathsf {a}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
Prop\ 3.2f
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon \underline{
\mathsf {a}}
= 
0
+ \underline{
\mathsf {a}}
]$

$\rm
[
Prop\ 3.2f
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Prop
\mbox {$\backslash $}
\linebreak [0]\ \hskip0em plus2.0em{}3.2f
\mbox{''}
]$

$\rm
[
Prop\ 3.2f
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}f
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:82}{}\section* {$\rm 
Prop\ 3.2g_1$}

$\rm
[
Prop\ 3.2g_1
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
S5
\gg \underline{
\mathsf {a}}
{}'
+ 
0
= \underline{
\mathsf {a}}
{}'
; 
S5
\gg \underline{
\mathsf {a}}
+ 
0
= \underline{
\mathsf {a}}
; 
S2
\rhd \underline{
\mathsf {a}}
+ 
0
= \underline{
\mathsf {a}}
\gg \underline{
\mathsf {a}}
+ 
0
{}'
= \underline{
\mathsf {a}}
{}'
; 
Prop\ 3.2d
\rhd \underline{
\mathsf {a}}
{}'
+ 
0
= \underline{
\mathsf {a}}
{}'
\rhd \underline{
\mathsf {a}}
+ 
0
{}'
= \underline{
\mathsf {a}}
{}'
\gg \underline{
\mathsf {a}}
{}'
+ 
0
= \underline{
\mathsf {a}}
+ 
0
{}'
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
Prop\ 3.2g_1
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon \underline{
\mathsf {a}}
{}'
+ 
0
= \underline{
\mathsf {a}}
+ 
0
{}'
]$

$\rm
[
Prop\ 3.2g_1
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Prop
\mbox {$\backslash $}
\linebreak [0]\ \hskip0em plus2.0em{}3.2g
\_1
\mbox{''}
]$

$\rm
[
Prop\ 3.2g_1
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}g
\linebreak [0]\ \hskip0em plus2.0em{}one
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:83}{}\section* {$\rm 
Prop\ 3.2g_2$}

$\rm
[
Prop\ 3.2g_2
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
{}'
+ \underline{
\mathsf {b}}
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
\vdash 
S2
\rhd \underline{
\mathsf {a}}
{}'
+ \underline{
\mathsf {b}}
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
\gg \underline{
\mathsf {a}}
{}'
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
{}'
; 
S6
\gg \underline{
\mathsf {a}}
{}'
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {a}}
{}'
+ \underline{
\mathsf {b}}
{}'
; 
Prop\ 3.2c
\rhd \underline{
\mathsf {a}}
{}'
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {a}}
{}'
+ \underline{
\mathsf {b}}
{}'
\rhd \underline{
\mathsf {a}}
{}'
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
{}'
\gg \underline{
\mathsf {a}}
{}'
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
{}'
; 
S6
\gg \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
; 
S2
\rhd \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
\gg \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
{}'
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
{}'
; 
Prop\ 3.2d
\rhd \underline{
\mathsf {a}}
{}'
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
{}'
\rhd \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
{}'
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
{}'
\gg \underline{
\mathsf {a}}
{}'
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
{}'
; 
Ded
\rhd 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
{}'
+ \underline{
\mathsf {b}}
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
\vdash \underline{
\mathsf {a}}
{}'
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
{}'
\gg \underline{
\mathsf {a}}
{}'
+ \underline{
\mathsf {b}}
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
\Rightarrow \underline{
\mathsf {a}}
{}'
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
{}'
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
Prop\ 3.2g_2
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
{}'
+ \underline{
\mathsf {b}}
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
\Rightarrow \underline{
\mathsf {a}}
{}'
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
{}'
]$

$\rm
[
Prop\ 3.2g_2
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Prop
\mbox {$\backslash $}
\linebreak [0]\ \hskip0em plus2.0em{}3.2g
\_2
\mbox{''}
]$

$\rm
[
Prop\ 3.2g_2
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}g
\linebreak [0]\ \hskip0em plus2.0em{}two
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:84}{}\section* {$\rm 
Prop\ 3.2g$}

$\rm
[
Prop\ 3.2g
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
Prop\ 3.2g_1
\gg \overline{
\mathsf{x}}
{}'
+ 
0
= \overline{
\mathsf{x}}
+ 
0
{}'
; 
Prop\ 3.2g_2
\gg \overline{
\mathsf{x}}
{}'
+ \overline{
\mathsf{y}}
= \overline{
\mathsf{x}}
+ \overline{
\mathsf{y}}
{}'
\Rightarrow \overline{
\mathsf{x}}
{}'
+ \overline{
\mathsf{y}}
{}'
= \overline{
\mathsf{x}}
+ \overline{
\mathsf{y}}
{}'
{}'
; 
S9
\mathop {\char64} \overline{
\mathsf{y}}
\rhd \overline{
\mathsf{x}}
{}'
+ 
0
= \overline{
\mathsf{x}}
+ 
0
{}'
\rhd \overline{
\mathsf{x}}
{}'
+ \overline{
\mathsf{y}}
= \overline{
\mathsf{x}}
+ \overline{
\mathsf{y}}
{}'
\Rightarrow \overline{
\mathsf{x}}
{}'
+ \overline{
\mathsf{y}}
{}'
= \overline{
\mathsf{x}}
+ \overline{
\mathsf{y}}
{}'
{}'
\gg \overline{
\mathsf{x}}
{}'
+ \overline{
\mathsf{y}}
= \overline{
\mathsf{x}}
+ \overline{
\mathsf{y}}
{}'
; 
Ded
\rhd \overline{
\mathsf{x}}
{}'
+ \overline{
\mathsf{y}}
= \overline{
\mathsf{x}}
+ \overline{
\mathsf{y}}
{}'
\gg \underline{
\mathsf {a}}
{}'
+ \underline{
\mathsf {b}}
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
Prop\ 3.2g
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
{}'
+ \underline{
\mathsf {b}}
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
]$

$\rm
[
Prop\ 3.2g
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Prop
\mbox {$\backslash $}
\linebreak [0]\ \hskip0em plus2.0em{}3.2g
\mbox{''}
]$

$\rm
[
Prop\ 3.2g
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}g
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:85}{}\section* {$\rm 
Prop\ 3.2h_1$}

$\rm
[
Prop\ 3.2h_1
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
S5
\gg \underline{
\mathsf {a}}
+ 
0
= \underline{
\mathsf {a}}
; 
Prop\ 3.2f
\gg \underline{
\mathsf {a}}
= 
0
+ \underline{
\mathsf {a}}
; 
Prop\ 3.2c
\rhd \underline{
\mathsf {a}}
+ 
0
= \underline{
\mathsf {a}}
\rhd \underline{
\mathsf {a}}
= 
0
+ \underline{
\mathsf {a}}
\gg \underline{
\mathsf {a}}
+ 
0
= 
0
+ \underline{
\mathsf {a}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
Prop\ 3.2h_1
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon \underline{
\mathsf {a}}
+ 
0
= 
0
+ \underline{
\mathsf {a}}
]$

$\rm
[
Prop\ 3.2h_1
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Prop
\mbox {$\backslash $}
\linebreak [0]\ \hskip0em plus2.0em{}3.2h
\_1
\mbox{''}
]$

$\rm
[
Prop\ 3.2h_1
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}h
\linebreak [0]\ \hskip0em plus2.0em{}one
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:86}{}\section* {$\rm 
Prop\ 3.2h_2$}

$\rm
[
Prop\ 3.2h_2
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {a}}
\vdash 
S2
\rhd \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {a}}
\gg \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {a}}
{}'
; 
S6
\gg \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
; 
Prop\ 3.2c
\rhd \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
\rhd \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {a}}
{}'
\gg \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {a}}
{}'
; 
Prop\ 3.2g
\gg \underline{
\mathsf {b}}
{}'
+ \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {a}}
{}'
; 
Prop\ 3.2d
\rhd \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {a}}
{}'
\rhd \underline{
\mathsf {b}}
{}'
+ \underline{
\mathsf {a}}
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {a}}
{}'
\gg \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {b}}
{}'
+ \underline{
\mathsf {a}}
; 
Ded
\rhd 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {b}}
{}'
+ \underline{
\mathsf {a}}
\gg \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {b}}
{}'
+ \underline{
\mathsf {a}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
Prop\ 3.2h_2
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
{}'
= \underline{
\mathsf {b}}
{}'
+ \underline{
\mathsf {a}}
]$

$\rm
[
Prop\ 3.2h_2
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Prop
\mbox {$\backslash $}
\linebreak [0]\ \hskip0em plus2.0em{}3.2h
\_2
\mbox{''}
]$

$\rm
[
Prop\ 3.2h_2
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}h
\linebreak [0]\ \hskip0em plus2.0em{}two
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:87}{}\section* {$\rm 
Prop\ 3.2h$}

$\rm
[
Prop\ 3.2h
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil 
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
Prop\ 3.2h_1
\gg \overline{
\mathsf{x}}
+ 
0
= 
0
+ \overline{
\mathsf{x}}
; 
Prop\ 3.2h_2
\gg \overline{
\mathsf{x}}
+ \overline{
\mathsf{y}}
= \overline{
\mathsf{y}}
+ \overline{
\mathsf{x}}
\Rightarrow \overline{
\mathsf{x}}
+ \overline{
\mathsf{y}}
{}'
= \overline{
\mathsf{y}}
{}'
+ \overline{
\mathsf{x}}
; 
S9
\mathop {\char64} \overline{
\mathsf{y}}
\rhd \overline{
\mathsf{x}}
+ 
0
= 
0
+ \overline{
\mathsf{x}}
\rhd \overline{
\mathsf{x}}
+ \overline{
\mathsf{y}}
= \overline{
\mathsf{y}}
+ \overline{
\mathsf{x}}
\Rightarrow \overline{
\mathsf{x}}
+ \overline{
\mathsf{y}}
{}'
= \overline{
\mathsf{y}}
{}'
+ \overline{
\mathsf{x}}
\gg \overline{
\mathsf{x}}
+ \overline{
\mathsf{y}}
= \overline{
\mathsf{y}}
+ \overline{
\mathsf{x}}
; 
Ded
\rhd \overline{
\mathsf{x}}
+ \overline{
\mathsf{y}}
= \overline{
\mathsf{y}}
+ \overline{
\mathsf{x}}
\gg \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {a}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[
Prop\ 3.2h
\stackrel{
stmt
}{\rightarrow}
S
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
+ \underline{
\mathsf {b}}
= \underline{
\mathsf {b}}
+ \underline{
\mathsf {a}}
]$

$\rm
[
Prop\ 3.2h
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Prop
\mbox {$\backslash $}
\linebreak [0]\ \hskip0em plus2.0em{}3.2h
\mbox{''}
]$

$\rm
[
Prop\ 3.2h
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}h
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:88}{}\section* {$\rm 
Block_1(\ast
,\ast
,\ast
)$}

$\rm
[
Block_1(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
)
\stackrel{
\mathrm {val}
}{\rightarrow}
\mathsf {t}
!
\mathsf {s}
!
\mathsf {c}
!
let_1( 
\lambda 
\mathsf {b}
.
let_1( 
\lambda 
\mathsf{x}
.
let_1( 
\lambda 
\mathsf {q}
.
let_1( 
\lambda 
\mathsf {q}
.
\tilde {{\cal Q}}(
\mathsf {t}
,
\lceil 
\mathsf {b}
; 
\mathsf {q}
\rceil 
,
\lceil 
\mathsf {b}
\rceil 
\mathrel { : \, : }
\mathsf {b}
\mathrel { : \, : }
\lceil 
\mathsf {q}
\rceil 
\mathrel { : \, : }
\mathsf {q}
\mathrel { : \, : }
\mathsf {T}
)
, 
\tilde {{\cal M}}(
\mathsf {q}
,
\mathsf {s}
,
\mathsf {c}
)
)
, 
\tilde {{\cal Q}}(
\mathsf {t}
,
\lceil 
\mathbf{let\ } 
\mathsf {l}
\mathrel{\ddot{=}} 
\mathsf{x}
\mathrel{\ in\ } 
\mathsf {p}
\rceil 
,
\lceil 
\mathsf {l}
\rceil 
\mathrel { : \, : }
\mathsf {t}
{}^{2}
\mathrel { : \, : }
\lceil 
\mathsf {p}
\rceil 
\mathrel { : \, : }
\mathsf {t}
{}^{3}
\mathrel { : \, : }
\lceil 
\mathsf{x}
\rceil 
\mathrel { : \, : }
\mathsf{x}
\mathrel { : \, : }
\mathsf {T}
)
)
, 
Block_2(
\mathsf {b}
)
)
, 
\tilde {{\cal M}}(
\mathsf {t}
{}^{1}
,
\mathsf {s}
,
\mathsf {c}
)
)
]$

$\rm
[
Block_1(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Block
\_1(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline )
\mbox{''}
]$

$\rm
[
Block_1(
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}block
\linebreak [0]\ \hskip0em plus2.0em{}one
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}state
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}cache
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}block
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:89}{}\section* {$\rm 
Block_2(\ast
)$}

$\rm
[
Block_2(
\mathsf {b}
)
\stackrel{
\mathrm {val}
}{\rightarrow}
\mathrm {If}(
\mathsf {b}
\stackrel{r}{=} 
\lceil 
\mathsf{x}
\vdash 
\mathsf{y}
\rceil 
,\linebreak [0]
\tilde {{\cal Q}}(
\mathsf {b}
,
\lceil 
\mathsf{x}
\vdash 
\mathsf{y}
\rceil 
,
\lceil 
\mathsf{x}
\rceil 
\mathrel { : \, : }
\mathsf {b}
{}^{1}
\mathrel { : \, : }
\lceil 
\mathsf{y}
\rceil 
\mathrel { : \, : }
Block_2(
\mathsf {b}
{}^{2}
)
\mathrel { : \, : }
\mathsf {T}
)
,\linebreak [0]
\mathrm {If}(
\mathsf {b}
\stackrel{r}{=} 
\lceil 
\mathsf{x}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathsf{y}
\rceil 
,\linebreak [0]
\tilde {{\cal Q}}(
\mathsf {b}
,
\lceil 
\mathsf{x}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathsf{y}
\rceil 
,
\lceil 
\mathsf{x}
\rceil 
\mathrel { : \, : }
\mathsf {b}
{}^{1}
\mathrel { : \, : }
\lceil 
\mathsf{y}
\rceil 
\mathrel { : \, : }
Block_2(
\mathsf {b}
{}^{2}
)
\mathrel { : \, : }
\mathsf {T}
)
,\linebreak [0]
\mathrm {If}(
\mathsf {b}
\stackrel{r}{=} 
\lceil 
\forall 
\mathsf{x}
\colon 
\mathsf{y}
\rceil 
,\linebreak [0]
\tilde {{\cal Q}}(
\mathsf {b}
,
\lceil 
\forall 
\mathsf{x}
\colon 
\mathsf{y}
\rceil 
,
\lceil 
\mathsf{x}
\rceil 
\mathrel { : \, : }
\mathsf {b}
{}^{1}
\mathrel { : \, : }
\lceil 
\mathsf{y}
\rceil 
\mathrel { : \, : }
Block_2(
\mathsf {b}
{}^{2}
)
\mathrel { : \, : }
\mathsf {T}
)
,\linebreak [0]
\mathrm {If}(
\mathsf {b}
\stackrel{r}{=} 
\lceil 
\mathsf{x}
; 
\mathsf{y}
\rceil 
,\linebreak [0]
Block_2(
\mathsf {b}
{}^{2}
)
,\linebreak [0]
\mathrm {If}(
\mathsf {b}
\stackrel{r}{=} 
\lceil 
\mathsf{x}
\gg 
\mathsf{y}
\rceil 
,\linebreak [0]
\mathsf {b}
{}^{2}
,\linebreak [0]
{\makebox [0mm][l]{$\bot $}\,{\bot }}
)
)
)
)
)
]$

$\rm
[
Block_2(
\mathsf {b}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Block
\_2(
\#1.
\newline )
\mbox{''}
]$

$\rm
[
Block_2(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}block
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}block
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:90}{}\section* {$\rm \ast
{}^{hide}$}


\mbox{Predef: hide}

$\rm
[
\mathsf{x}
{}^{hide}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\{
\}
{\char94}
\{hide
\}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:91}{}\section* {$\rm 
MacroIndent(\ast
)$}

$\rm
[
MacroIndent(
\mathsf{x}
)
\stackrel{
\mathrm{name}
}{\rightarrow}
\mbox{``}
\newline MacroIndent(
\#1.
\newline )
\mbox{''}
]$

$\rm
[
MacroIndent(
\mathsf{x}
)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
MacroIndent(
\mathsf{x}
)
\mathrel {\ddot {=}}
\mathsf{x}
]
\rceil 
)
]$

$\rm
[
MacroIndent(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\$
\%
\newline 
\mbox {$\backslash $}leftskip
{=}1em
\%
\newline 
\$
\#1.
\mbox{''}
]$

$\rm
[
MacroIndent(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}macro
\linebreak [0]\ \hskip0em plus2.0em{}indent
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:92}{}\section* {$\rm \ast
{}'$}

$\rm
[
\mathsf{x}
{}'
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\{
\}
\mbox {'}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:93}{}\section* {$\rm \ast
= \ast$}

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

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:94}{}\section* {$\rm \ast
\neq \ast$}

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

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:95}{}\section* {$\rm \ast
{}^{var}$}

$\rm
[
\mathsf{x}
{}^{var}
\stackrel{
\mathrm {val}
}{\rightarrow}
\mathsf{x}
\stackrel{r}{=} 
\lceil \overline{
\mathsf{x}}
\rceil 
]$

$\rm
[
\mathsf{x}
{}^{var}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\{
\}
{\char94}
\{var
\}
\mbox{''}
]$

$\rm
[
\ast 
{}^{var}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}object
\linebreak [0]\ \hskip0em plus2.0em{}var
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:96}{}\section* {$\rm \ast
\#^0\ast$}

$\rm
[
\mathsf{x}
\#^0
\mathsf{y}
\stackrel{
\mathrm {val}
}{\rightarrow}
\lambda 
\mathsf {c}
.
\mathsf{x}
{}^{var}
\wedge 
\mathsf{y}
{} ^ {\cal C}
\wedge 
\mathsf{x}
\#^1
\mathsf{y}
]$

$\rm
[
\mathsf{x}
\#^0
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\mbox {$\backslash $}
\#.
{\char94}0
\#2.
\mbox{''}
]$

$\rm
[
\ast 
\#^0
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}avoid
\linebreak [0]\ \hskip0em plus2.0em{}zero
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:97}{}\section* {$\rm \ast
\#^1\ast$}

$\rm
[
\mathsf{x}
\#^1
\mathsf{y}
\stackrel{
\mathrm {val}
}{\rightarrow}
\mathrm {If}(
\mathsf{y}
{}^{var}
,\linebreak [0]
{\neg }
\mathsf{x}
\stackrel {t}{=}
\mathsf{y}
,\linebreak [0]
\newline 
\mathrm {If}(
{\neg }
\mathsf{y}
\stackrel{r}{=} 
\lceil 
\forall_{obj} 
\mathsf{x}
\colon 
\mathsf{y}
\rceil 
,\linebreak [0]
\mathsf{x}
\#^*
\mathsf{y}
{}^t
,\linebreak [0]
\newline 
\mathrm {If}(
\mathsf{x}
\stackrel {t}{=}
\mathsf{y}
{}^{1}
,\linebreak [0]
\mathsf {T}
,\linebreak [0]
\mathsf{x}
\#^1
\mathsf{y}
{}^{2}
)
)
)
]$

$\rm
[
\mathsf{x}
\#^1
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\mbox {$\backslash $}
\#.
{\char94}1
\#2.
\mbox{''}
]$

$\rm
[
\ast 
\#^1
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}avoid
\linebreak [0]\ \hskip0em plus2.0em{}one
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:98}{}\section* {$\rm \ast
\#^*\ast$}

$\rm
[
\mathsf{x}
\#^*
\mathsf{y}
\stackrel{
\mathrm {val}
}{\rightarrow}
\mathsf{x}
!
\mathrm {If}(
\mathsf{y}
,\linebreak [0]
\mathsf {T}
,\linebreak [0]
\mathrm {If}(
\mathsf{x}
\#^1
\mathsf{y}
{}^h
,\linebreak [0]
\mathsf{x}
\#^*
\mathsf{y}
{}^t
,\linebreak [0]
\mathsf {F}
)
)
]$

$\rm
[
\mathsf{x}
\#^*
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\mbox {$\backslash $}
\#.
{\char94}
{*}
\#2.
\mbox{''}
]$

$\rm
[
\ast 
\#^*
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}avoid
\linebreak [0]\ \hskip0em plus2.0em{}star
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:99}{}\section* {$\rm 
\exists \ast
\colon \ast$}

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

$\rm
[
\exists 
\ast 
\colon 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}exist
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}indeed
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:100}{}\section* {$\rm 
\forall \ast
\colon \ast$}

$\rm
[
\forall 
\mathsf{x}
\colon 
\mathsf{y}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}(
\mathrm {If}(
{\neg }
\mathsf {t}
{}^{1}
\stackrel{r}{=} 
\lceil 
\mathsf{x}
,\linebreak [0] 
\mathsf{y}
\rceil 
,\linebreak [0]
\tilde {{\cal Q}}(
\mathsf {t}
,
\lceil 
\forall_{obj} 
\mathsf{x}
\colon 
\mathsf{y}
\rceil 
,
\lceil 
\mathsf{x}
\rceil 
\mathrel { : \, : }
\mathsf {t}
{}^{1}
\mathrel { : \, : }
\lceil 
\mathsf{y}
\rceil 
\mathrel { : \, : }
\mathsf {t}
{}^{2}
\mathrel { : \, : }
\mathsf {T}
)
,\linebreak [0]
\tilde {{\cal Q}}(
\mathsf {t}
,
\lceil 
\forall_{obj} 
\mathsf{x}
\colon 
\forall 
\mathsf{y}
\colon 
\mathsf{z}
\rceil 
,
\lceil 
\mathsf{x}
\rceil 
\mathrel { : \, : }
\mathsf {t}
{}^{1}
{}^{1}
\mathrel { : \, : }
\lceil 
\mathsf{y}
\rceil 
\mathrel { : \, : }
\mathsf {t}
{}^{1}
{}^{2}
\mathrel { : \, : }
\lceil 
\mathsf{z}
\rceil 
\mathrel { : \, : }
\mathsf {t}
{}^{2}
\mathrel { : \, : }
\mathsf {T}
)
)
,
\mathsf {s}
,
\mathsf {c}
)
]$

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

$\rm
[
\forall 
\ast 
\colon 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}for
\linebreak [0]\ \hskip0em plus2.0em{}all
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}indeed
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:101}{}\section* {$\rm 
\forall_{obj} \ast
\colon \ast$}

$\rm
[
\forall_{obj} 
\mathsf{x}
\colon 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}forall
\_
\{obj
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\mbox {$\backslash $}colon
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

$\rm
[
\forall_{obj} 
\ast 
\colon 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}for
\linebreak [0]\ \hskip0em plus2.0em{}all
\linebreak [0]\ \hskip0em plus2.0em{}objects
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}indeed
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:102}{}\section* {$\rm \ast
\Rightarrow \ast$}

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

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:103}{}\section* {$\rm \ast
\Leftrightarrow \ast$}

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

$\rm
[
\ast 
\Leftrightarrow 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}if
\linebreak [0]\ \hskip0em plus2.0em{}and
\linebreak [0]\ \hskip0em plus2.0em{}only
\linebreak [0]\ \hskip0em plus2.0em{}if
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:104}{}\section* {$\rm \ast
\#\ast$}

$\rm
[
\mathsf{x}
\#
\mathsf{y}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathsf{x}
\#
\mathsf{y}
\mathrel {\ddot {=}}
\lceil 
\mathsf{x}
\rceil 
\#^0
\lceil 
\mathsf{y}
\rceil 
]
\rceil 
)
]$

$\rm
[
\mathsf{x}
\#
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\mbox {$\backslash $}
\#.
\#2.
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:105}{}\section* {$\rm \ast
\unrhd \ast$}

$\rm
[
\mathsf{x}
\unrhd 
\mathsf{y}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathsf{x}
\unrhd 
\mathsf{y}
\mathrel {\ddot {=}}
MP
\rhd 
\mathsf{x}
\rhd 
\mathsf{y}
]
\rceil 
)
]$

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

$\rm
[
\ast 
\unrhd 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}object
\linebreak [0]\ \hskip0em plus2.0em{}modus
\linebreak [0]\ \hskip0em plus2.0em{}ponens
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:106}{}\section* {$\rm 
\Pi \ast
\colon \ast$}

$\rm
[
\Pi 
\mathsf{x}
\colon 
\mathsf{y}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}(
\mathrm {If}(
{\neg }
\mathsf {t}
{}^{1}
\stackrel{r}{=} 
\lceil 
\mathsf{x}
,\linebreak [0] 
\mathsf{y}
\rceil 
,\linebreak [0]
\tilde {{\cal Q}}(
\mathsf {t}
,
\lceil 
\forall 
\mathsf{x}
\colon 
\mathsf{y}
\rceil 
,
\lceil 
\mathsf{x}
\rceil 
\mathrel { : \, : }
\mathsf {t}
{}^{1}
\mathrel { : \, : }
\lceil 
\mathsf{y}
\rceil 
\mathrel { : \, : }
\mathsf {t}
{}^{2}
\mathrel { : \, : }
\mathsf {T}
)
,\linebreak [0]
\tilde {{\cal Q}}(
\mathsf {t}
,
\lceil 
\forall 
\mathsf{x}
\colon 
\Pi 
\mathsf{y}
\colon 
\mathsf{z}
\rceil 
,
\lceil 
\mathsf{x}
\rceil 
\mathrel { : \, : }
\mathsf {t}
{}^{1}
{}^{1}
\mathrel { : \, : }
\lceil 
\mathsf{y}
\rceil 
\mathrel { : \, : }
\mathsf {t}
{}^{1}
{}^{2}
\mathrel { : \, : }
\lceil 
\mathsf{z}
\rceil 
\mathrel { : \, : }
\mathsf {t}
{}^{2}
\mathrel { : \, : }
\mathsf {T}
)
)
,
\mathsf {s}
,
\mathsf {c}
)
]$

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

$\rm
[
\Pi 
\ast 
\colon 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}for
\linebreak [0]\ \hskip0em plus2.0em{}all
\linebreak [0]\ \hskip0em plus2.0em{}terms
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}indeed
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:107}{}\section* {$\rm 
Begin \, \ast
 ; \ast
 : End ; \ast$}

$\rm
[
Begin \, 
\mathsf {b}
 ; 
\mathsf {l}
 : End ; 
\mathsf {p}
\stackrel{
\mathrm{name}
}{\rightarrow}
\mbox{``}
\newline Begin
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $},
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{};
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
{:}
\linebreak [0]\ \hskip0em plus2.0em{}End
\linebreak [0]\ \hskip0em plus2.0em{};
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\mbox{''}
]$

$\rm
[
Begin \, 
\mathsf {b}
 ; 
\mathsf {l}
 : End ; 
\mathsf {p}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
Block_1(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
)
]$

$\rm
[
Begin \, 
\mathsf {b}
 ; 
\mathsf {l}
 : End ; 
\mathsf {p}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}newline
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}makebox
\linebreak [0]\ \hskip0em plus2.0em{}[0.1
\mbox {$\backslash $}textwidth]
\{
\}
\%
\newline 
\mbox {$\backslash $}parbox
\linebreak [0]\ \hskip0em plus2.0em{}[b]
\{0.4
\mbox {$\backslash $}textwidth
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\{
\mbox {$\backslash $}raggedright
\newline 
\mbox {$\backslash $}setlength
\linebreak [0]\ \hskip0em plus2.0em{}
\{
\mbox {$\backslash $}parindent
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\{
\mbox{-}0.1
\mbox {$\backslash $}textwidth
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\%
\newline 
\mbox {$\backslash $}makebox
\linebreak [0]\ \hskip0em plus2.0em{}[0.1
\mbox {$\backslash $}textwidth
\linebreak [0]\ \hskip0em plus2.0em{}][l]
\{
\$
\newline 
\mbox {$\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 $}endcsname
\linebreak [0]\ \hskip0em plus2.0em{}L
\_?
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}else
\newline 
\mbox {$\backslash $}global
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}advance
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}by
\linebreak [0]\ \hskip0em plus2.0em{}1
\newline L
\mbox {$\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 $}lgwproofline
\newline 
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\newline 
\$
{:}
\}
\$Block
\linebreak [0]\ \hskip0em plus2.0em{}
\{
\}
\mbox {$\backslash $}gg
\linebreak [0]\ \hskip0em plus2.0em{}
\{
\}
\$
\}
\mbox {$\backslash $}quad
\newline 
\mbox {$\backslash $}parbox
\linebreak [0]\ \hskip0em plus2.0em{}[t]
\{0.4
\mbox {$\backslash $}textwidth
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\{
\$Begin
\newline 
\$
\mbox {$\backslash $}hfill
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}makebox
\linebreak [0]\ \hskip0em plus2.0em{}[0mm][l]
\{
\mbox {$\backslash $}quad
\linebreak [0]\ \hskip0em plus2.0em{};
\}
\}
\#1.
\newline 
\mbox {$\backslash $}newline
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}makebox
\linebreak [0]\ \hskip0em plus2.0em{}[0.1
\mbox {$\backslash $}textwidth]
\{
\}
\%
\newline 
\mbox {$\backslash $}parbox
\linebreak [0]\ \hskip0em plus2.0em{}[b]
\{0.4
\mbox {$\backslash $}textwidth
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\{
\mbox {$\backslash $}raggedright
\newline 
\mbox {$\backslash $}setlength
\linebreak [0]\ \hskip0em plus2.0em{}
\{
\mbox {$\backslash $}parindent
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\{
\mbox{-}0.1
\mbox {$\backslash $}textwidth
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\%
\newline 
\mbox {$\backslash $}makebox
\linebreak [0]\ \hskip0em plus2.0em{}[0.1
\mbox {$\backslash $}textwidth
\linebreak [0]\ \hskip0em plus2.0em{}][l]
\{
\$
\#2.
\newline 
\$
{:}
\}
\$Block
\linebreak [0]\ \hskip0em plus2.0em{}
\{
\}
\mbox {$\backslash $}gg
\linebreak [0]\ \hskip0em plus2.0em{}
\{
\}
\$
\}
\mbox {$\backslash $}quad
\newline 
\mbox {$\backslash $}parbox
\linebreak [0]\ \hskip0em plus2.0em{}[t]
\{0.4
\mbox {$\backslash $}textwidth
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\{
\$End
\newline 
\$
\mbox {$\backslash $}hfill
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}makebox
\linebreak [0]\ \hskip0em plus2.0em{}[0mm][l]
\{
\mbox {$\backslash $}quad
\linebreak [0]\ \hskip0em plus2.0em{};
\}
\}
\#3.
\mbox{''}
]$

$\rm
[
Begin \, 
\ast 
 ; 
\ast 
 : End ; 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}block
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}line
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}block
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:108}{}\section* {$\rm 
Last\ block\ line \, \ast
\gg \ast
\,;$}

$\rm
[
Last\ block\ line \, 
\mathsf {a}
\gg 
\mathsf {i}
\,;
\stackrel{
\mathrm{name}
}{\rightarrow}
\mbox{``}
\newline Last
\mbox {$\backslash $}
\linebreak [0]\ \hskip0em plus2.0em{}block
\mbox {$\backslash $}
\linebreak [0]\ \hskip0em plus2.0em{}line
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $},
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\mbox {$\backslash $}gg
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline 
\mbox {$\backslash $},;
\mbox{''}
]$

$\rm
[
Last\ block\ line \, 
\mathsf {a}
\gg 
\mathsf {i}
\,;
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
Last\ block\ line \, 
\mathsf {a}
\gg 
\mathsf {i}
\,;
\mathrel {\ddot {=}}
(
\mathsf {a}
\gg 
\mathsf {i}
)
]
\rceil 
)
]$

$\rm
[
Last\ block\ line \, 
\mathsf {a}
\gg 
\mathsf {i}
\,;
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}newline
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}makebox
\linebreak [0]\ \hskip0em plus2.0em{}[0.1
\mbox {$\backslash $}textwidth]
\{
\}
\%
\newline 
\mbox {$\backslash $}parbox
\linebreak [0]\ \hskip0em plus2.0em{}[b]
\{0.4
\mbox {$\backslash $}textwidth
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\{
\mbox {$\backslash $}raggedright
\newline 
\mbox {$\backslash $}setlength
\linebreak [0]\ \hskip0em plus2.0em{}
\{
\mbox {$\backslash $}parindent
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\{
\mbox{-}0.1
\mbox {$\backslash $}textwidth
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\%
\newline 
\mbox {$\backslash $}makebox
\linebreak [0]\ \hskip0em plus2.0em{}[0.1
\mbox {$\backslash $}textwidth
\linebreak [0]\ \hskip0em plus2.0em{}][l]
\{
\$
\newline 
\mbox {$\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 $}endcsname
\linebreak [0]\ \hskip0em plus2.0em{}L
\_?
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}else
\newline 
\mbox {$\backslash $}global
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}advance
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}by
\linebreak [0]\ \hskip0em plus2.0em{}1
\newline L
\mbox {$\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 $}lgwproofline
\newline 
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\newline 
\$
{:}
\}
\$
\#1.
\newline 
\{
\}
\mbox {$\backslash $}gg
\linebreak [0]\ \hskip0em plus2.0em{}
\{
\}
\$
\}
\mbox {$\backslash $}quad
\newline 
\mbox {$\backslash $}parbox
\linebreak [0]\ \hskip0em plus2.0em{}[t]
\{0.4
\mbox {$\backslash $}textwidth
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\{
\$
\#2.
\newline 
\$
\mbox {$\backslash $}hfill
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}makebox
\linebreak [0]\ \hskip0em plus2.0em{}[0mm][l]
\{
\mbox {$\backslash $}quad
\linebreak [0]\ \hskip0em plus2.0em{};
\}
\}
\mbox{''}
]$

$\rm
[
Last\ block\ line \, 
\ast 
\gg 
\ast 
\,;
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}because
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}indeed
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}line
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:109}{}\section* {$\rm 
Arbitrary \gg \ast
;\ast$}

$\rm
[
Arbitrary \gg 
\mathsf {i}
;
\mathsf {p}
\stackrel{
\mathrm{name}
}{\rightarrow}
\mbox{``}
\newline Arbitrary
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}gg
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline ;
\#2.
\mbox{''}
]$

$\rm
[
Arbitrary \gg 
\mathsf {i}
;
\mathsf {p}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
Arbitrary \gg 
\mathsf {i}
;
\mathsf {p}
\mathrel {\ddot {=}}
\Pi 
\mathsf {i}
\colon 
\mathsf {p}
]
\rceil 
)
]$

$\rm
[
Arbitrary \gg 
\mathsf {i}
;
\mathsf {p}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}newline
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}makebox
\linebreak [0]\ \hskip0em plus2.0em{}[0.1
\mbox {$\backslash $}textwidth
\linebreak [0]\ \hskip0em plus2.0em{}][l]
\{
\$
\newline 
\mbox {$\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 $}endcsname
\linebreak [0]\ \hskip0em plus2.0em{}L
\_?
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}else
\newline 
\mbox {$\backslash $}global
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}advance
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}lgwproofline
\linebreak [0]\ \hskip0em plus2.0em{}by
\linebreak [0]\ \hskip0em plus2.0em{}1
\newline L
\mbox {$\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 $}lgwproofline
\newline 
\mbox {$\backslash $}fi
\linebreak [0]\ \hskip0em plus2.0em{}
\newline 
\$
{:}
\}
\mbox {$\backslash $}makebox
\linebreak [0]\ \hskip0em plus2.0em{}[0.4
\mbox {$\backslash $}textwidth
\linebreak [0]\ \hskip0em plus2.0em{}][l]
\{
\$Arbitrary
\{
\}
\mbox {$\backslash $}gg
\{
\}
\$
\}
\mbox {$\backslash $}quad
\newline 
\mbox {$\backslash $}parbox
\linebreak [0]\ \hskip0em plus2.0em{}[t]
\{0.4
\mbox {$\backslash $}textwidth
\linebreak [0]\ \hskip0em plus2.0em{}
\}
\{
\$
\#1.
\newline 
\$
\mbox {$\backslash $}hfill
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}makebox
\linebreak [0]\ \hskip0em plus2.0em{}[0mm][l]
\{
\mbox {$\backslash $}quad
\linebreak [0]\ \hskip0em plus2.0em{};
\}
\}
\#2.
\mbox{''}
]$

$\rm
[
Arbitrary \gg 
\ast 
;
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}any
\linebreak [0]\ \hskip0em plus2.0em{}term
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}line
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]$

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:110}{}\section* {$\rm \ast
\mathrel{|} \ast$}

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

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:111}{}\section* {$\rm 
 \rightarrow $}

$\rm
[
 \rightarrow 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}rightarrow
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

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

\hypertarget {url:1808490384217020148094711118359520484602074145217451012073151273782664705:112}{}\section* {$\rm \ast
\backslash \backslash \ast$}

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

$\rm
[
\mathsf{x}
\backslash \backslash 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\mbox {$\backslash $}
\mbox {$\backslash $}
\{
\}
\#2.
\mbox{''}
]$

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

\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-2006-06-16.UTC:14:58:03.474266} = 
\href {http://www.diku.dk/~grue/logiweb/20060417/logiweb/doc/misc/time.html}{MJD-53902.TAI:14:58:36.474266} = 
\href {http://www.diku.dk/~grue/logiweb/20060417/logiweb/doc/misc/time.html}{LGT-4657186716474266e-6}
}

\end{document}

