1.1 --- a/doc-src/TutorialI/CTL/document/PDL.tex	Mon Dec 18 16:11:53 2000 +0100
     1.2 +++ b/doc-src/TutorialI/CTL/document/PDL.tex	Mon Dec 18 16:45:17 2000 +0100
     1.3 @@ -127,7 +127,7 @@
     1.4  \noindent
     1.5  After simplification and clarification we are left with
     1.6  \begin{isabelle}%
     1.7 -\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ t{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}\ {\isasymLongrightarrow}\ t\ {\isasymin}\ A\ {\isasymLongrightarrow}\ x\ {\isasymin}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isasyminverse}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}%
     1.8 +\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ t{\isachardot}\ {\isasymlbrakk}{\isacharparenleft}x{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}{\isacharsemicolon}\ t\ {\isasymin}\ A{\isasymrbrakk}\ {\isasymLongrightarrow}\ x\ {\isasymin}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isasyminverse}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}%
     1.9  \end{isabelle}
    1.10  This goal is proved by induction on \isa{{\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}}. But since the model
    1.11  checker works backwards (from \isa{t} to \isa{s}), we cannot use the
    1.12 @@ -135,9 +135,9 @@
    1.13  forward direction. Fortunately the converse induction theorem
    1.14  \isa{converse{\isacharunderscore}rtrancl{\isacharunderscore}induct} already exists:
    1.15  \begin{isabelle}%
    1.16 -\ \ \ \ \ {\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isasymin}\ r\isactrlsup {\isacharasterisk}\ {\isasymLongrightarrow}\isanewline
    1.17 -\ \ \ \ \ P\ b\ {\isasymLongrightarrow}\isanewline
    1.18 -\ \ \ \ \ {\isacharparenleft}{\isasymAnd}y\ z{\isachardot}\ {\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r\ {\isasymLongrightarrow}\ {\isacharparenleft}z{\isacharcomma}\ b{\isacharparenright}\ {\isasymin}\ r\isactrlsup {\isacharasterisk}\ {\isasymLongrightarrow}\ P\ z\ {\isasymLongrightarrow}\ P\ y{\isacharparenright}\ {\isasymLongrightarrow}\ P\ a%
    1.19 +\ \ \ \ \ {\isasymlbrakk}{\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isasymin}\ r\isactrlsup {\isacharasterisk}{\isacharsemicolon}\ P\ b{\isacharsemicolon}\isanewline
    1.20 +\ \ \ \ \ \ \ \ {\isasymAnd}y\ z{\isachardot}\ {\isasymlbrakk}{\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharsemicolon}\ {\isacharparenleft}z{\isacharcomma}\ b{\isacharparenright}\ {\isasymin}\ r\isactrlsup {\isacharasterisk}{\isacharsemicolon}\ P\ z{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ y{\isasymrbrakk}\isanewline
    1.21 +\ \ \ \ \ {\isasymLongrightarrow}\ P\ a%
    1.22  \end{isabelle}
    1.23  It says that if \isa{{\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isasymin}\ r\isactrlsup {\isacharasterisk}} and we know \isa{P\ b} then we can infer
    1.24  \isa{P\ a} provided each step backwards from a predecessor \isa{z} of