--- a/doc-src/TutorialI/Advanced/document/simp.tex Mon Dec 04 23:36:16 2000 +0100
+++ b/doc-src/TutorialI/Advanced/document/simp.tex Mon Dec 04 23:38:19 2000 +0100
@@ -28,7 +28,7 @@
controlled by so-called \bfindex{congruence rules}. This is the one for
\isa{{\isasymlongrightarrow}}:
\begin{isabelle}%
-\ \ \ \ \ {\isasymlbrakk}P\ {\isacharequal}\ P{\isacharprime}{\isacharsemicolon}\ P{\isacharprime}\ {\isasymLongrightarrow}\ Q\ {\isacharequal}\ Q{\isacharprime}{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}P\ {\isasymlongrightarrow}\ Q{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}P{\isacharprime}\ {\isasymlongrightarrow}\ Q{\isacharprime}{\isacharparenright}%
+\ \ \ \ \ P\ {\isacharequal}\ P{\isacharprime}\ {\isasymLongrightarrow}\ {\isacharparenleft}P{\isacharprime}\ {\isasymLongrightarrow}\ Q\ {\isacharequal}\ Q{\isacharprime}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}P\ {\isasymlongrightarrow}\ Q{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}P{\isacharprime}\ {\isasymlongrightarrow}\ Q{\isacharprime}{\isacharparenright}%
\end{isabelle}
It should be read as follows:
In order to simplify \isa{P\ {\isasymlongrightarrow}\ Q} to \isa{P{\isacharprime}\ {\isasymlongrightarrow}\ Q{\isacharprime}},
@@ -38,14 +38,15 @@
Here are some more examples. The congruence rules for bounded
quantifiers supply contextual information about the bound variable:
\begin{isabelle}%
-\ \ \ \ \ {\isasymlbrakk}A\ {\isacharequal}\ B{\isacharsemicolon}\ {\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ B\ {\isasymLongrightarrow}\ P\ x\ {\isacharequal}\ Q\ x{\isasymrbrakk}\isanewline
-\ \ \ \ \ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymforall}x{\isasymin}A{\isachardot}\ P\ x{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isasymforall}x{\isasymin}B{\isachardot}\ Q\ x{\isacharparenright}%
+\ \ \ \ \ A\ {\isacharequal}\ B\ {\isasymLongrightarrow}\isanewline
+\ \ \ \ \ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ B\ {\isasymLongrightarrow}\ P\ x\ {\isacharequal}\ Q\ x{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymforall}x{\isasymin}A{\isachardot}\ P\ x{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isasymforall}x{\isasymin}B{\isachardot}\ Q\ x{\isacharparenright}%
\end{isabelle}
The congruence rule for conditional expressions supply contextual
information for simplifying the arms:
\begin{isabelle}%
-\ \ \ \ \ {\isasymlbrakk}b\ {\isacharequal}\ c{\isacharsemicolon}\ c\ {\isasymLongrightarrow}\ x\ {\isacharequal}\ u{\isacharsemicolon}\ {\isasymnot}\ c\ {\isasymLongrightarrow}\ y\ {\isacharequal}\ v{\isasymrbrakk}\isanewline
-\ \ \ \ \ {\isasymLongrightarrow}\ {\isacharparenleft}if\ b\ then\ x\ else\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}if\ c\ then\ u\ else\ v{\isacharparenright}%
+\ \ \ \ \ b\ {\isacharequal}\ c\ {\isasymLongrightarrow}\isanewline
+\ \ \ \ \ {\isacharparenleft}c\ {\isasymLongrightarrow}\ x\ {\isacharequal}\ u{\isacharparenright}\ {\isasymLongrightarrow}\isanewline
+\ \ \ \ \ {\isacharparenleft}{\isasymnot}\ c\ {\isasymLongrightarrow}\ y\ {\isacharequal}\ v{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}if\ b\ then\ x\ else\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}if\ c\ then\ u\ else\ v{\isacharparenright}%
\end{isabelle}
A congruence rule can also \emph{prevent} simplification of some arguments.
Here is an alternative congruence rule for conditional expressions:
@@ -72,7 +73,7 @@
\begin{warn}
The congruence rule \isa{conj{\isacharunderscore}cong}
\begin{isabelle}%
-\ \ \ \ \ {\isasymlbrakk}P\ {\isacharequal}\ P{\isacharprime}{\isacharsemicolon}\ P{\isacharprime}\ {\isasymLongrightarrow}\ Q\ {\isacharequal}\ Q{\isacharprime}{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}P\ {\isasymand}\ Q{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}P{\isacharprime}\ {\isasymand}\ Q{\isacharprime}{\isacharparenright}%
+\ \ \ \ \ P\ {\isacharequal}\ P{\isacharprime}\ {\isasymLongrightarrow}\ {\isacharparenleft}P{\isacharprime}\ {\isasymLongrightarrow}\ Q\ {\isacharequal}\ Q{\isacharprime}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}P\ {\isasymand}\ Q{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}P{\isacharprime}\ {\isasymand}\ Q{\isacharprime}{\isacharparenright}%
\end{isabelle}
is occasionally useful but not a default rule; you have to use it explicitly.
\end{warn}%
--- a/doc-src/TutorialI/CTL/document/CTL.tex Mon Dec 04 23:36:16 2000 +0100
+++ b/doc-src/TutorialI/CTL/document/CTL.tex Mon Dec 04 23:38:19 2000 +0100
@@ -73,12 +73,11 @@
\isacommand{apply}{\isacharparenleft}clarsimp\ simp\ add{\isacharcolon}\ af{\isacharunderscore}def\ Paths{\isacharunderscore}def{\isacharparenright}%
\begin{isamarkuptxt}%
\begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}p{\isachardot}\ {\isasymlbrakk}p\ {\isadigit{0}}\ {\isasymin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ {\isacharparenleft}{\isasymforall}t{\isachardot}\ {\isacharparenleft}p\ {\isadigit{0}}{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\isasymforall}p{\isachardot}\ t\ {\isacharequal}\ p\ {\isadigit{0}}\ {\isasymand}\ {\isacharparenleft}{\isasymforall}i{\isachardot}\ {\isacharparenleft}p\ i{\isacharcomma}\ p\ {\isacharparenleft}Suc\ i{\isacharparenright}{\isacharparenright}\ {\isasymin}\ M{\isacharparenright}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\isasymexists}i{\isachardot}\ p\ i\ {\isasymin}\ A{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isacharsemicolon}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isasymforall}i{\isachardot}\ {\isacharparenleft}p\ i{\isacharcomma}\ p\ {\isacharparenleft}Suc\ i{\isacharparenright}{\isacharparenright}\ {\isasymin}\ M{\isasymrbrakk}\isanewline
-\ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isasymexists}i{\isachardot}\ p\ i\ {\isasymin}\ A%
+\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}p{\isachardot}\ p\ {\isadigit{0}}\ {\isasymin}\ A\ {\isasymor}\isanewline
+\ \ \ \ \ \ \ \ {\isacharparenleft}{\isasymforall}t{\isachardot}\ {\isacharparenleft}p\ {\isadigit{0}}{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\ {\isasymlongrightarrow}\isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\isasymforall}p{\isachardot}\ t\ {\isacharequal}\ p\ {\isadigit{0}}\ {\isasymand}\ {\isacharparenleft}{\isasymforall}i{\isachardot}\ {\isacharparenleft}p\ i{\isacharcomma}\ p\ {\isacharparenleft}Suc\ i{\isacharparenright}{\isacharparenright}\ {\isasymin}\ M{\isacharparenright}\ {\isasymlongrightarrow}\isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\isasymexists}i{\isachardot}\ p\ i\ {\isasymin}\ A{\isacharparenright}{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\isanewline
+\ \ \ \ \ \ \ \ {\isasymforall}i{\isachardot}\ {\isacharparenleft}p\ i{\isacharcomma}\ p\ {\isacharparenleft}Suc\ i{\isacharparenright}{\isacharparenright}\ {\isasymin}\ M\ {\isasymLongrightarrow}\ {\isasymexists}i{\isachardot}\ p\ i\ {\isasymin}\ A%
\end{isabelle}
Now we eliminate the disjunction. The case \isa{p\ {\isadigit{0}}\ {\isasymin}\ A} is trivial:%
\end{isamarkuptxt}%
@@ -92,10 +91,10 @@
\isacommand{apply}{\isacharparenleft}clarsimp{\isacharparenright}%
\begin{isamarkuptxt}%
\begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}p{\isachardot}\ {\isasymlbrakk}{\isasymforall}i{\isachardot}\ {\isacharparenleft}p\ i{\isacharcomma}\ p\ {\isacharparenleft}Suc\ i{\isacharparenright}{\isacharparenright}\ {\isasymin}\ M{\isacharsemicolon}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isasymforall}pa{\isachardot}\ p\ {\isadigit{1}}\ {\isacharequal}\ pa\ {\isadigit{0}}\ {\isasymand}\ {\isacharparenleft}{\isasymforall}i{\isachardot}\ {\isacharparenleft}pa\ i{\isacharcomma}\ pa\ {\isacharparenleft}Suc\ i{\isacharparenright}{\isacharparenright}\ {\isasymin}\ M{\isacharparenright}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\isasymexists}i{\isachardot}\ pa\ i\ {\isasymin}\ A{\isacharparenright}{\isasymrbrakk}\isanewline
-\ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isasymexists}i{\isachardot}\ p\ i\ {\isasymin}\ A%
+\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}p{\isachardot}\ {\isasymforall}i{\isachardot}\ {\isacharparenleft}p\ i{\isacharcomma}\ p\ {\isacharparenleft}Suc\ i{\isacharparenright}{\isacharparenright}\ {\isasymin}\ M\ {\isasymLongrightarrow}\isanewline
+\ \ \ \ \ \ \ \ {\isasymforall}pa{\isachardot}\ p\ {\isadigit{1}}\ {\isacharequal}\ pa\ {\isadigit{0}}\ {\isasymand}\ {\isacharparenleft}{\isasymforall}i{\isachardot}\ {\isacharparenleft}pa\ i{\isacharcomma}\ pa\ {\isacharparenleft}Suc\ i{\isacharparenright}{\isacharparenright}\ {\isasymin}\ M{\isacharparenright}\ {\isasymlongrightarrow}\isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\isasymexists}i{\isachardot}\ pa\ i\ {\isasymin}\ A{\isacharparenright}\ {\isasymLongrightarrow}\isanewline
+\ \ \ \ \ \ \ \ {\isasymexists}i{\isachardot}\ p\ i\ {\isasymin}\ A%
\end{isabelle}
It merely remains to set \isa{pa} to \isa{{\isasymlambda}i{\isachardot}\ p\ {\isacharparenleft}i\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}}, i.e.\ \isa{p} without its
first element. The rest is practically automatic:%
@@ -171,9 +170,10 @@
\noindent
After simplification and clarification the subgoal has the following compact form
\begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}i{\isachardot}\ {\isasymlbrakk}P\ s{\isacharsemicolon}\ {\isasymforall}s{\isachardot}\ P\ s\ {\isasymlongrightarrow}\ {\isacharparenleft}{\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ P\ t{\isacharparenright}{\isasymrbrakk}\isanewline
-\ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isacharparenleft}path\ s\ P\ i{\isacharcomma}\ SOME\ t{\isachardot}\ {\isacharparenleft}path\ s\ P\ i{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ P\ t{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ P\ {\isacharparenleft}path\ s\ P\ i{\isacharparenright}%
+\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}i{\isachardot}\ P\ s\ {\isasymLongrightarrow}\isanewline
+\ \ \ \ \ \ \ \ {\isasymforall}s{\isachardot}\ P\ s\ {\isasymlongrightarrow}\ {\isacharparenleft}{\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ P\ t{\isacharparenright}\ {\isasymLongrightarrow}\isanewline
+\ \ \ \ \ \ \ \ {\isacharparenleft}path\ s\ P\ i{\isacharcomma}\ SOME\ t{\isachardot}\ {\isacharparenleft}path\ s\ P\ i{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ P\ t{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\isanewline
+\ \ \ \ \ \ \ \ P\ {\isacharparenleft}path\ s\ P\ i{\isacharparenright}%
\end{isabelle}
and invites a proof by induction on \isa{i}:%
\end{isamarkuptxt}%
@@ -183,14 +183,15 @@
\noindent
After simplification the base case boils down to
\begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ {\isasymlbrakk}P\ s{\isacharsemicolon}\ {\isasymforall}s{\isachardot}\ P\ s\ {\isasymlongrightarrow}\ {\isacharparenleft}{\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ P\ t{\isacharparenright}{\isasymrbrakk}\isanewline
-\ \ \ \ {\isasymLongrightarrow}\ {\isacharparenleft}s{\isacharcomma}\ SOME\ t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ P\ t{\isacharparenright}\ {\isasymin}\ M%
+\ {\isadigit{1}}{\isachardot}\ P\ s\ {\isasymLongrightarrow}\isanewline
+\ \ \ \ {\isasymforall}s{\isachardot}\ P\ s\ {\isasymlongrightarrow}\ {\isacharparenleft}{\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ P\ t{\isacharparenright}\ {\isasymLongrightarrow}\isanewline
+\ \ \ \ {\isacharparenleft}s{\isacharcomma}\ SOME\ t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ P\ t{\isacharparenright}\ {\isasymin}\ M%
\end{isabelle}
The conclusion looks exceedingly trivial: after all, \isa{t} is chosen such that \isa{{\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M}
holds. However, we first have to show that such a \isa{t} actually exists! This reasoning
is embodied in the theorem \isa{someI{\isadigit{2}}{\isacharunderscore}ex}:
\begin{isabelle}%
-\ \ \ \ \ {\isasymlbrakk}{\isasymexists}a{\isachardot}\ {\isacharquery}P\ a{\isacharsemicolon}\ {\isasymAnd}x{\isachardot}\ {\isacharquery}P\ x\ {\isasymLongrightarrow}\ {\isacharquery}Q\ x{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharquery}Q\ {\isacharparenleft}SOME\ x{\isachardot}\ {\isacharquery}P\ x{\isacharparenright}%
+\ \ \ \ \ {\isasymexists}a{\isachardot}\ {\isacharquery}P\ a\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ {\isacharquery}P\ x\ {\isasymLongrightarrow}\ {\isacharquery}Q\ x{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharquery}Q\ {\isacharparenleft}SOME\ x{\isachardot}\ {\isacharquery}P\ x{\isacharparenright}%
\end{isabelle}
When we apply this theorem as an introduction rule, \isa{{\isacharquery}P\ x} becomes
\isa{{\isacharparenleft}s{\isacharcomma}\ x{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ P\ x} and \isa{{\isacharquery}Q\ x} becomes \isa{{\isacharparenleft}s{\isacharcomma}\ x{\isacharparenright}\ {\isasymin}\ M} and we have to prove
--- a/doc-src/TutorialI/CTL/document/CTLind.tex Mon Dec 04 23:36:16 2000 +0100
+++ b/doc-src/TutorialI/CTL/document/CTLind.tex Mon Dec 04 23:38:19 2000 +0100
@@ -121,7 +121,7 @@
into a \isa{{\isasymAnd}p}, which would complicate matters below. As it is,
\isa{Avoid{\isacharunderscore}in{\isacharunderscore}lfp} is now
\begin{isabelle}%
-\ \ \ \ \ {\isasymlbrakk}{\isasymforall}p{\isasymin}Paths\ s{\isachardot}\ {\isasymexists}i{\isachardot}\ p\ i\ {\isasymin}\ A{\isacharsemicolon}\ t\ {\isasymin}\ Avoid\ s\ A{\isasymrbrakk}\ {\isasymLongrightarrow}\ t\ {\isasymin}\ lfp\ {\isacharparenleft}af\ A{\isacharparenright}%
+\ \ \ \ \ {\isasymforall}p{\isasymin}Paths\ s{\isachardot}\ {\isasymexists}i{\isachardot}\ p\ i\ {\isasymin}\ A\ {\isasymLongrightarrow}\ t\ {\isasymin}\ Avoid\ s\ A\ {\isasymLongrightarrow}\ t\ {\isasymin}\ lfp\ {\isacharparenleft}af\ A{\isacharparenright}%
\end{isabelle}
The main theorem is simply the corollary where \isa{t\ {\isacharequal}\ s},
in which case the assumption \isa{t\ {\isasymin}\ Avoid\ s\ A} is trivially true
--- a/doc-src/TutorialI/CTL/document/PDL.tex Mon Dec 04 23:36:16 2000 +0100
+++ b/doc-src/TutorialI/CTL/document/PDL.tex Mon Dec 04 23:38:19 2000 +0100
@@ -127,7 +127,7 @@
\noindent
After simplification and clarification we are left with
\begin{isabelle}%
-\ {\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}%
+\ {\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}%
\end{isabelle}
This goal is proved by induction on \isa{{\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}}. But since the model
checker works backwards (from \isa{t} to \isa{s}), we cannot use the
@@ -135,9 +135,9 @@
forward direction. Fortunately the converse induction theorem
\isa{converse{\isacharunderscore}rtrancl{\isacharunderscore}induct} already exists:
\begin{isabelle}%
-\ \ \ \ \ {\isasymlbrakk}{\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isasymin}\ r\isactrlsup {\isacharasterisk}{\isacharsemicolon}\ P\ b{\isacharsemicolon}\isanewline
-\ \ \ \ \ \ \ \ {\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
-\ \ \ \ \ {\isasymLongrightarrow}\ P\ a%
+\ \ \ \ \ {\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isasymin}\ r\isactrlsup {\isacharasterisk}\ {\isasymLongrightarrow}\isanewline
+\ \ \ \ \ P\ b\ {\isasymLongrightarrow}\isanewline
+\ \ \ \ \ {\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%
\end{isabelle}
It says that if \isa{{\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isasymin}\ r\isactrlsup {\isacharasterisk}} and we know \isa{P\ b} then we can infer
\isa{P\ a} provided each step backwards from a predecessor \isa{z} of
--- a/doc-src/TutorialI/Inductive/document/AB.tex Mon Dec 04 23:36:16 2000 +0100
+++ b/doc-src/TutorialI/Inductive/document/AB.tex Mon Dec 04 23:38:19 2000 +0100
@@ -96,8 +96,8 @@
1 on our way from 0 to 2. Formally, we appeal to the following discrete
intermediate value theorem \isa{nat{\isadigit{0}}{\isacharunderscore}intermed{\isacharunderscore}int{\isacharunderscore}val}
\begin{isabelle}%
-\ \ \ \ \ {\isasymlbrakk}{\isasymforall}i{\isachardot}\ i\ {\isacharless}\ n\ {\isasymlongrightarrow}\ {\isasymbar}f\ {\isacharparenleft}i\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}\ {\isacharminus}\ f\ i{\isasymbar}\ {\isasymle}\ {\isacharhash}{\isadigit{1}}{\isacharsemicolon}\ f\ {\isadigit{0}}\ {\isasymle}\ k{\isacharsemicolon}\ k\ {\isasymle}\ f\ n{\isasymrbrakk}\isanewline
-\ \ \ \ \ {\isasymLongrightarrow}\ {\isasymexists}i{\isachardot}\ i\ {\isasymle}\ n\ {\isasymand}\ f\ i\ {\isacharequal}\ k%
+\ \ \ \ \ {\isasymforall}i{\isachardot}\ i\ {\isacharless}\ n\ {\isasymlongrightarrow}\ {\isasymbar}f\ {\isacharparenleft}i\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}\ {\isacharminus}\ f\ i{\isasymbar}\ {\isasymle}\ {\isacharhash}{\isadigit{1}}\ {\isasymLongrightarrow}\isanewline
+\ \ \ \ \ f\ {\isadigit{0}}\ {\isasymle}\ k\ {\isasymLongrightarrow}\ k\ {\isasymle}\ f\ n\ {\isasymLongrightarrow}\ {\isasymexists}i{\isachardot}\ i\ {\isasymle}\ n\ {\isasymand}\ f\ i\ {\isacharequal}\ k%
\end{isabelle}
where \isa{f} is of type \isa{nat\ {\isasymRightarrow}\ int}, \isa{int} are the integers,
\isa{abs} is the absolute value function, and \isa{{\isacharhash}{\isadigit{1}}} is the
--- a/doc-src/TutorialI/Inductive/document/Advanced.tex Mon Dec 04 23:36:16 2000 +0100
+++ b/doc-src/TutorialI/Inductive/document/Advanced.tex Mon Dec 04 23:38:19 2000 +0100
@@ -36,7 +36,7 @@
We completely forgot about "rule inversion".
\begin{isabelle}%
-\ \ \ \ \ {\isasymlbrakk}a\ {\isasymin}\ even{\isacharsemicolon}\ a\ {\isacharequal}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ P{\isacharsemicolon}\ {\isasymAnd}n{\isachardot}\ {\isasymlbrakk}a\ {\isacharequal}\ Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isacharsemicolon}\ n\ {\isasymin}\ even{\isasymrbrakk}\ {\isasymLongrightarrow}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ P%
+\ \ \ \ \ a\ {\isasymin}\ even\ {\isasymLongrightarrow}\ {\isacharparenleft}a\ {\isacharequal}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ P{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymAnd}n{\isachardot}\ a\ {\isacharequal}\ Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isasymLongrightarrow}\ n\ {\isasymin}\ even\ {\isasymLongrightarrow}\ P{\isacharparenright}\ {\isasymLongrightarrow}\ P%
\end{isabelle}
\rulename{even.cases}
@@ -50,7 +50,7 @@
\isacommand{thm}\ Suc{\isacharunderscore}Suc{\isacharunderscore}cases%
\begin{isamarkuptext}%
\begin{isabelle}%
-\ \ \ \ \ {\isasymlbrakk}Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isasymin}\ even{\isacharsemicolon}\ n\ {\isasymin}\ even\ {\isasymLongrightarrow}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ P%
+\ \ \ \ \ Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isasymin}\ even\ {\isasymLongrightarrow}\ {\isacharparenleft}n\ {\isasymin}\ even\ {\isasymLongrightarrow}\ P{\isacharparenright}\ {\isasymLongrightarrow}\ P%
\end{isabelle}
\rulename{Suc_Suc_cases}
@@ -65,7 +65,7 @@
this is what we get:
\begin{isabelle}%
-\ \ \ \ \ {\isasymlbrakk}Apply\ f\ args\ {\isasymin}\ gterms\ F{\isacharsemicolon}\ {\isasymlbrakk}{\isasymforall}t{\isasymin}set\ args{\isachardot}\ t\ {\isasymin}\ gterms\ F{\isacharsemicolon}\ f\ {\isasymin}\ F{\isasymrbrakk}\ {\isasymLongrightarrow}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ P%
+\ \ \ \ \ Apply\ f\ args\ {\isasymin}\ gterms\ F\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymforall}t{\isasymin}set\ args{\isachardot}\ t\ {\isasymin}\ gterms\ F\ {\isasymLongrightarrow}\ f\ {\isasymin}\ F\ {\isasymLongrightarrow}\ P{\isacharparenright}\ {\isasymLongrightarrow}\ P%
\end{isabelle}
\rulename{gterm_Apply_elim}%
\end{isamarkuptext}%
--- a/doc-src/TutorialI/Inductive/document/Even.tex Mon Dec 04 23:36:16 2000 +0100
+++ b/doc-src/TutorialI/Inductive/document/Even.tex Mon Dec 04 23:38:19 2000 +0100
@@ -31,7 +31,7 @@
\rulename{even.step}
\begin{isabelle}%
-\ \ \ \ \ {\isasymlbrakk}xa\ {\isasymin}\ even{\isacharsemicolon}\ P\ {\isadigit{0}}{\isacharsemicolon}\ {\isasymAnd}n{\isachardot}\ {\isasymlbrakk}n\ {\isasymin}\ even{\isacharsemicolon}\ P\ n{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isacharparenright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ xa%
+\ \ \ \ \ xa\ {\isasymin}\ even\ {\isasymLongrightarrow}\ P\ {\isadigit{0}}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymAnd}n{\isachardot}\ n\ {\isasymin}\ even\ {\isasymLongrightarrow}\ P\ n\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\ P\ xa%
\end{isabelle}
\rulename{even.induct}
--- a/doc-src/TutorialI/Inductive/document/Star.tex Mon Dec 04 23:36:16 2000 +0100
+++ b/doc-src/TutorialI/Inductive/document/Star.tex Mon Dec 04 23:38:19 2000 +0100
@@ -51,9 +51,9 @@
To prove transitivity, we need rule induction, i.e.\ theorem
\isa{rtc{\isachardot}induct}:
\begin{isabelle}%
-\ \ \ \ \ {\isasymlbrakk}{\isacharparenleft}{\isacharquery}xb{\isacharcomma}\ {\isacharquery}xa{\isacharparenright}\ {\isasymin}\ {\isacharquery}r{\isacharasterisk}{\isacharsemicolon}\ {\isasymAnd}x{\isachardot}\ {\isacharquery}P\ x\ x{\isacharsemicolon}\isanewline
-\ \ \ \ \ \ \ \ {\isasymAnd}x\ y\ z{\isachardot}\ {\isasymlbrakk}{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ {\isacharquery}r{\isacharsemicolon}\ {\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ {\isacharquery}r{\isacharasterisk}{\isacharsemicolon}\ {\isacharquery}P\ y\ z{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharquery}P\ x\ z{\isasymrbrakk}\isanewline
-\ \ \ \ \ {\isasymLongrightarrow}\ {\isacharquery}P\ {\isacharquery}xb\ {\isacharquery}xa%
+\ \ \ \ \ {\isacharparenleft}{\isacharquery}xb{\isacharcomma}\ {\isacharquery}xa{\isacharparenright}\ {\isasymin}\ {\isacharquery}r{\isacharasterisk}\ {\isasymLongrightarrow}\isanewline
+\ \ \ \ \ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ {\isacharquery}P\ x\ x{\isacharparenright}\ {\isasymLongrightarrow}\isanewline
+\ \ \ \ \ {\isacharparenleft}{\isasymAnd}x\ y\ z{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ {\isacharquery}r\ {\isasymLongrightarrow}\ {\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ {\isacharquery}r{\isacharasterisk}\ {\isasymLongrightarrow}\ {\isacharquery}P\ y\ z\ {\isasymLongrightarrow}\ {\isacharquery}P\ x\ z{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharquery}P\ {\isacharquery}xb\ {\isacharquery}xa%
\end{isabelle}
It says that \isa{{\isacharquery}P} holds for an arbitrary pair \isa{{\isacharparenleft}{\isacharquery}xb{\isacharcomma}{\isacharquery}xa{\isacharparenright}\ {\isasymin}\ {\isacharquery}r{\isacharasterisk}} if \isa{{\isacharquery}P} is preserved by all rules of the inductive definition,
i.e.\ if \isa{{\isacharquery}P} holds for the conclusion provided it holds for the
@@ -110,8 +110,9 @@
\begin{isabelle}%
\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymlongrightarrow}\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\isanewline
\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}x\ y\ za{\isachardot}\isanewline
-\ \ \ \ \ \ \ {\isasymlbrakk}{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ r{\isacharsemicolon}\ {\isacharparenleft}y{\isacharcomma}\ za{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isacharsemicolon}\ {\isacharparenleft}za{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymlongrightarrow}\ {\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isasymrbrakk}\isanewline
-\ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isacharparenleft}za{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymlongrightarrow}\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}%
+\ \ \ \ \ \ \ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ r\ {\isasymLongrightarrow}\isanewline
+\ \ \ \ \ \ \ {\isacharparenleft}y{\isacharcomma}\ za{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymLongrightarrow}\isanewline
+\ \ \ \ \ \ \ {\isacharparenleft}za{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymlongrightarrow}\ {\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymLongrightarrow}\ {\isacharparenleft}za{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymlongrightarrow}\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}%
\end{isabelle}%
\end{isamarkuptxt}%
\ \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
@@ -156,7 +157,7 @@
\begin{exercise}\label{ex:converse-rtc-step}
Show that the converse of \isa{rtc{\isacharunderscore}step} also holds:
\begin{isabelle}%
-\ \ \ \ \ {\isasymlbrakk}{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isacharsemicolon}\ {\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}%
+\ \ \ \ \ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymLongrightarrow}\ {\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}%
\end{isabelle}
\end{exercise}
\begin{exercise}
--- a/doc-src/TutorialI/Misc/document/AdvancedInd.tex Mon Dec 04 23:36:16 2000 +0100
+++ b/doc-src/TutorialI/Misc/document/AdvancedInd.tex Mon Dec 04 23:38:19 2000 +0100
@@ -95,7 +95,7 @@
\isacommand{lemmas}\ myrule\ {\isacharequal}\ simple{\isacharbrackleft}rule{\isacharunderscore}format{\isacharbrackright}%
\begin{isamarkuptext}%
\noindent
-yielding \isa{{\isasymlbrakk}A\ y{\isacharsemicolon}\ B\ y{\isasymrbrakk}\ {\isasymLongrightarrow}\ B\ y\ {\isasymand}\ A\ y}.
+yielding \isa{A\ y\ {\isasymLongrightarrow}\ B\ y\ {\isasymLongrightarrow}\ B\ y\ {\isasymand}\ A\ y}.
You can go one step further and include these derivations already in the
statement of your original lemma, thus avoiding the intermediate step:%
\end{isamarkuptext}%
@@ -182,8 +182,7 @@
\begin{isamarkuptxt}%
\begin{isabelle}%
\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}n\ i\ nat{\isachardot}\isanewline
-\ \ \ \ \ \ \ {\isasymlbrakk}{\isasymforall}m{\isachardot}\ m\ {\isacharless}\ n\ {\isasymlongrightarrow}\ {\isacharparenleft}{\isasymforall}i{\isachardot}\ m\ {\isacharequal}\ f\ i\ {\isasymlongrightarrow}\ i\ {\isasymle}\ f\ i{\isacharparenright}{\isacharsemicolon}\ i\ {\isacharequal}\ Suc\ nat{\isasymrbrakk}\isanewline
-\ \ \ \ \ \ \ {\isasymLongrightarrow}\ n\ {\isacharequal}\ f\ i\ {\isasymlongrightarrow}\ i\ {\isasymle}\ f\ i%
+\ \ \ \ \ \ \ {\isasymforall}m{\isachardot}\ m\ {\isacharless}\ n\ {\isasymlongrightarrow}\ {\isacharparenleft}{\isasymforall}i{\isachardot}\ m\ {\isacharequal}\ f\ i\ {\isasymlongrightarrow}\ i\ {\isasymle}\ f\ i{\isacharparenright}\ {\isasymLongrightarrow}\ i\ {\isacharequal}\ Suc\ nat\ {\isasymLongrightarrow}\ n\ {\isacharequal}\ f\ i\ {\isasymlongrightarrow}\ i\ {\isasymle}\ f\ i%
\end{isabelle}%
\end{isamarkuptxt}%
\isacommand{by}{\isacharparenleft}blast\ intro{\isacharbang}{\isacharcolon}\ f{\isacharunderscore}ax\ Suc{\isacharunderscore}leI\ intro{\isacharcolon}\ le{\isacharunderscore}less{\isacharunderscore}trans{\isacharparenright}%
@@ -196,7 +195,7 @@
proved as follows. From \isa{f{\isacharunderscore}ax} we have \isa{f\ {\isacharparenleft}f\ j{\isacharparenright}\ {\isacharless}\ f\ {\isacharparenleft}Suc\ j{\isacharparenright}}
(1) which implies \isa{f\ j\ {\isasymle}\ f\ {\isacharparenleft}f\ j{\isacharparenright}} (by the induction hypothesis).
Using (1) once more we obtain \isa{f\ j\ {\isacharless}\ f\ {\isacharparenleft}Suc\ j{\isacharparenright}} (2) by transitivity
-(\isa{le{\isacharunderscore}less{\isacharunderscore}trans}: \isa{{\isasymlbrakk}i\ {\isasymle}\ j{\isacharsemicolon}\ j\ {\isacharless}\ k{\isasymrbrakk}\ {\isasymLongrightarrow}\ i\ {\isacharless}\ k}).
+(\isa{le{\isacharunderscore}less{\isacharunderscore}trans}: \isa{i\ {\isasymle}\ j\ {\isasymLongrightarrow}\ j\ {\isacharless}\ k\ {\isasymLongrightarrow}\ i\ {\isacharless}\ k}).
Using the induction hypothesis once more we obtain \isa{j\ {\isasymle}\ f\ j}
which, together with (2) yields \isa{j\ {\isacharless}\ f\ {\isacharparenleft}Suc\ j{\isacharparenright}} (again by
\isa{le{\isacharunderscore}less{\isacharunderscore}trans}).
@@ -268,7 +267,7 @@
\noindent
The elimination rule \isa{less{\isacharunderscore}SucE} expresses the case distinction:
\begin{isabelle}%
-\ \ \ \ \ {\isasymlbrakk}m\ {\isacharless}\ Suc\ n{\isacharsemicolon}\ m\ {\isacharless}\ n\ {\isasymLongrightarrow}\ P{\isacharsemicolon}\ m\ {\isacharequal}\ n\ {\isasymLongrightarrow}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ P%
+\ \ \ \ \ m\ {\isacharless}\ Suc\ n\ {\isasymLongrightarrow}\ {\isacharparenleft}m\ {\isacharless}\ n\ {\isasymLongrightarrow}\ P{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}m\ {\isacharequal}\ n\ {\isasymLongrightarrow}\ P{\isacharparenright}\ {\isasymLongrightarrow}\ P%
\end{isabelle}
Now it is straightforward to derive the original version of
--- a/doc-src/TutorialI/Misc/document/simp.tex Mon Dec 04 23:36:16 2000 +0100
+++ b/doc-src/TutorialI/Misc/document/simp.tex Mon Dec 04 23:38:19 2000 +0100
@@ -299,8 +299,8 @@
In contrast to splitting the conclusion, this actually creates two
separate subgoals (which are solved by \isa{simp{\isacharunderscore}all}):
\begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ {\isasymlbrakk}xs\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}{\isacharsemicolon}\ ys\ {\isasymnoteq}\ {\isacharbrackleft}{\isacharbrackright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharat}\ ys\ {\isasymnoteq}\ {\isacharbrackleft}{\isacharbrackright}\isanewline
-\ {\isadigit{2}}{\isachardot}\ {\isasymlbrakk}xs\ {\isasymnoteq}\ {\isacharbrackleft}{\isacharbrackright}{\isacharsemicolon}\ ys\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ xs\ {\isacharat}\ {\isacharbrackleft}{\isacharbrackright}\ {\isasymnoteq}\ {\isacharbrackleft}{\isacharbrackright}%
+\ {\isadigit{1}}{\isachardot}\ xs\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}\ {\isasymLongrightarrow}\ ys\ {\isasymnoteq}\ {\isacharbrackleft}{\isacharbrackright}\ {\isasymLongrightarrow}\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharat}\ ys\ {\isasymnoteq}\ {\isacharbrackleft}{\isacharbrackright}\isanewline
+\ {\isadigit{2}}{\isachardot}\ xs\ {\isasymnoteq}\ {\isacharbrackleft}{\isacharbrackright}\ {\isasymLongrightarrow}\ ys\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}\ {\isasymLongrightarrow}\ xs\ {\isacharat}\ {\isacharbrackleft}{\isacharbrackright}\ {\isasymnoteq}\ {\isacharbrackleft}{\isacharbrackright}%
\end{isabelle}
If you need to split both in the assumptions and the conclusion,
use $t$\isa{{\isachardot}splits} which subsumes $t$\isa{{\isachardot}split} and
--- a/doc-src/TutorialI/Recdef/document/Nested2.tex Mon Dec 04 23:36:16 2000 +0100
+++ b/doc-src/TutorialI/Recdef/document/Nested2.tex Mon Dec 04 23:38:19 2000 +0100
@@ -61,8 +61,9 @@
\isacommand{recdef} has been supplied with the congruence theorem
\isa{map{\isacharunderscore}cong}:
\begin{isabelle}%
-\ \ \ \ \ {\isasymlbrakk}xs\ {\isacharequal}\ ys{\isacharsemicolon}\ {\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ set\ ys\ {\isasymLongrightarrow}\ f\ x\ {\isacharequal}\ g\ x{\isasymrbrakk}\isanewline
-\ \ \ \ \ {\isasymLongrightarrow}\ map\ f\ xs\ {\isacharequal}\ map\ g\ ys%
+\ \ \ \ \ xs\ {\isacharequal}\ ys\ {\isasymLongrightarrow}\isanewline
+\ \ \ \ \ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ set\ ys\ {\isasymLongrightarrow}\ f\ x\ {\isacharequal}\ g\ x{\isacharparenright}\ {\isasymLongrightarrow}\isanewline
+\ \ \ \ \ map\ f\ xs\ {\isacharequal}\ map\ g\ ys%
\end{isabelle}
Its second premise expresses (indirectly) that the second argument of
\isa{map} is only applied to elements of its third argument. Congruence
--- a/doc-src/TutorialI/Types/document/Axioms.tex Mon Dec 04 23:36:16 2000 +0100
+++ b/doc-src/TutorialI/Types/document/Axioms.tex Mon Dec 04 23:38:19 2000 +0100
@@ -68,8 +68,8 @@
specialized to type \isa{bool}, as subgoals:
\begin{isabelle}%
\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isasymColon}bool{\isachardot}\ x\ {\isacharless}{\isacharless}{\isacharequal}\ x\isanewline
-\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}{\isacharparenleft}x{\isasymColon}bool{\isacharparenright}\ {\isacharparenleft}y{\isasymColon}bool{\isacharparenright}\ z{\isasymColon}bool{\isachardot}\ {\isasymlbrakk}x\ {\isacharless}{\isacharless}{\isacharequal}\ y{\isacharsemicolon}\ y\ {\isacharless}{\isacharless}{\isacharequal}\ z{\isasymrbrakk}\ {\isasymLongrightarrow}\ x\ {\isacharless}{\isacharless}{\isacharequal}\ z\isanewline
-\ {\isadigit{3}}{\isachardot}\ {\isasymAnd}{\isacharparenleft}x{\isasymColon}bool{\isacharparenright}\ y{\isasymColon}bool{\isachardot}\ {\isasymlbrakk}x\ {\isacharless}{\isacharless}{\isacharequal}\ y{\isacharsemicolon}\ y\ {\isacharless}{\isacharless}{\isacharequal}\ x{\isasymrbrakk}\ {\isasymLongrightarrow}\ x\ {\isacharequal}\ y\isanewline
+\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}{\isacharparenleft}x{\isasymColon}bool{\isacharparenright}\ {\isacharparenleft}y{\isasymColon}bool{\isacharparenright}\ z{\isasymColon}bool{\isachardot}\ x\ {\isacharless}{\isacharless}{\isacharequal}\ y\ {\isasymLongrightarrow}\ y\ {\isacharless}{\isacharless}{\isacharequal}\ z\ {\isasymLongrightarrow}\ x\ {\isacharless}{\isacharless}{\isacharequal}\ z\isanewline
+\ {\isadigit{3}}{\isachardot}\ {\isasymAnd}{\isacharparenleft}x{\isasymColon}bool{\isacharparenright}\ y{\isasymColon}bool{\isachardot}\ x\ {\isacharless}{\isacharless}{\isacharequal}\ y\ {\isasymLongrightarrow}\ y\ {\isacharless}{\isacharless}{\isacharequal}\ x\ {\isasymLongrightarrow}\ x\ {\isacharequal}\ y\isanewline
\ {\isadigit{4}}{\isachardot}\ {\isasymAnd}{\isacharparenleft}x{\isasymColon}bool{\isacharparenright}\ y{\isasymColon}bool{\isachardot}\ {\isacharparenleft}x\ {\isacharless}{\isacharless}\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isacharless}{\isacharless}{\isacharequal}\ y\ {\isasymand}\ x\ {\isasymnoteq}\ y{\isacharparenright}%
\end{isabelle}
Fortunately, the proof is easy for blast, once we have unfolded the definitions
--- a/doc-src/TutorialI/Types/document/Typedef.tex Mon Dec 04 23:36:16 2000 +0100
+++ b/doc-src/TutorialI/Types/document/Typedef.tex Mon Dec 04 23:38:19 2000 +0100
@@ -204,7 +204,7 @@
Expanding \isa{three{\isacharunderscore}def} yields the premise \isa{n\ {\isasymle}\ {\isadigit{2}}}. Repeated
elimination with \isa{le{\isacharunderscore}SucE}
\begin{isabelle}%
-\ \ \ \ \ {\isasymlbrakk}{\isacharquery}m\ {\isasymle}\ Suc\ {\isacharquery}n{\isacharsemicolon}\ {\isacharquery}m\ {\isasymle}\ {\isacharquery}n\ {\isasymLongrightarrow}\ {\isacharquery}R{\isacharsemicolon}\ {\isacharquery}m\ {\isacharequal}\ Suc\ {\isacharquery}n\ {\isasymLongrightarrow}\ {\isacharquery}R{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharquery}R%
+\ \ \ \ \ {\isacharquery}m\ {\isasymle}\ Suc\ {\isacharquery}n\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isacharquery}m\ {\isasymle}\ {\isacharquery}n\ {\isasymLongrightarrow}\ {\isacharquery}R{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isacharquery}m\ {\isacharequal}\ Suc\ {\isacharquery}n\ {\isasymLongrightarrow}\ {\isacharquery}R{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharquery}R%
\end{isabelle}
reduces \isa{n\ {\isasymle}\ {\isadigit{2}}} to the three cases \isa{n\ {\isasymle}\ {\isadigit{0}}}, \isa{n\ {\isacharequal}\ {\isadigit{1}}} and
\isa{n\ {\isacharequal}\ {\isadigit{2}}} which are trivial for simplification:%
@@ -231,10 +231,10 @@
\isacommand{apply}{\isacharparenleft}rule\ cases{\isacharunderscore}lemma{\isacharparenright}%
\begin{isamarkuptxt}%
\begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ {\isasymlbrakk}P\ A{\isacharsemicolon}\ P\ B{\isacharsemicolon}\ P\ C{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}Abs{\isacharunderscore}three\ {\isadigit{0}}{\isacharparenright}\isanewline
-\ {\isadigit{2}}{\isachardot}\ {\isasymlbrakk}P\ A{\isacharsemicolon}\ P\ B{\isacharsemicolon}\ P\ C{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}Abs{\isacharunderscore}three\ {\isadigit{1}}{\isacharparenright}\isanewline
-\ {\isadigit{3}}{\isachardot}\ {\isasymlbrakk}P\ A{\isacharsemicolon}\ P\ B{\isacharsemicolon}\ P\ C{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}Abs{\isacharunderscore}three\ {\isadigit{2}}{\isacharparenright}\isanewline
-\ {\isadigit{4}}{\isachardot}\ {\isasymlbrakk}P\ A{\isacharsemicolon}\ P\ B{\isacharsemicolon}\ P\ C{\isasymrbrakk}\ {\isasymLongrightarrow}\ Rep{\isacharunderscore}three\ x\ {\isasymin}\ three%
+\ {\isadigit{1}}{\isachardot}\ P\ A\ {\isasymLongrightarrow}\ P\ B\ {\isasymLongrightarrow}\ P\ C\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}Abs{\isacharunderscore}three\ {\isadigit{0}}{\isacharparenright}\isanewline
+\ {\isadigit{2}}{\isachardot}\ P\ A\ {\isasymLongrightarrow}\ P\ B\ {\isasymLongrightarrow}\ P\ C\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}Abs{\isacharunderscore}three\ {\isadigit{1}}{\isacharparenright}\isanewline
+\ {\isadigit{3}}{\isachardot}\ P\ A\ {\isasymLongrightarrow}\ P\ B\ {\isasymLongrightarrow}\ P\ C\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}Abs{\isacharunderscore}three\ {\isadigit{2}}{\isacharparenright}\isanewline
+\ {\isadigit{4}}{\isachardot}\ P\ A\ {\isasymLongrightarrow}\ P\ B\ {\isasymLongrightarrow}\ P\ C\ {\isasymLongrightarrow}\ Rep{\isacharunderscore}three\ x\ {\isasymin}\ three%
\end{isabelle}
The resulting subgoals are easily solved by simplification:%
\end{isamarkuptxt}%
--- a/doc-src/TutorialI/isabelle.sty Mon Dec 04 23:36:16 2000 +0100
+++ b/doc-src/TutorialI/isabelle.sty Mon Dec 04 23:38:19 2000 +0100
@@ -94,7 +94,9 @@
\newcommand{\isamarkupsubsect}[1]{\subsection{#1}}
\newcommand{\isamarkupsubsubsect}[1]{\subsubsection{#1}}
-\newenvironment{isapar}{\parindent\isa@parindent\parskip\isa@parskip\par\medskip}{\par\medskip}
+\newcommand{\isabeginpar}{\par\medskip}
+\newcommand{\isaendpar}{\par\medskip}
+\newenvironment{isapar}{\parindent\isa@parindent\parskip\isa@parskip\isabeginpar}{\isaendpar}
\newenvironment{isamarkuptext}{\isastyletext\begin{isapar}}{\end{isapar}}
\newenvironment{isamarkuptxt}{\isastyletxt\begin{isapar}}{\end{isapar}}
\newcommand{\isamarkupcmt}[1]{{\isastylecmt--- #1}}
--- a/doc-src/TutorialI/isabellesym.sty Mon Dec 04 23:36:16 2000 +0100
+++ b/doc-src/TutorialI/isabellesym.sty Mon Dec 04 23:38:19 2000 +0100
@@ -17,18 +17,32 @@
% symbol definitions
-\newcommand{\isasymspacespace}{\isamath{~~}}
-\newcommand{\isasymGamma}{\isamath{\Gamma}}
-\newcommand{\isasymDelta}{\isamath{\Delta}}
-\newcommand{\isasymTheta}{\isamath{\Theta}}
-\newcommand{\isasymLambda}{\isamath{\Lambda}}
-\newcommand{\isasymXi}{\isamath{\Xi}}
-\newcommand{\isasymPi}{\isamath{\Pi}}
-\newcommand{\isasymSigma}{\isamath{\Sigma}}
-\newcommand{\isasymUpsilon}{\isamath{\Upsilon}}
-\newcommand{\isasymPhi}{\isamath{\Phi}}
-\newcommand{\isasymPsi}{\isamath{\Psi}}
-\newcommand{\isasymOmega}{\isamath{\Omega}}
+\newcommand{\isasymA}{\isamath{\mathcal{A}}}
+\newcommand{\isasymB}{\isamath{\mathcal{B}}}
+\newcommand{\isasymC}{\isamath{\mathcal{C}}}
+\newcommand{\isasymD}{\isamath{\mathcal{D}}}
+\newcommand{\isasymE}{\isamath{\mathcal{E}}}
+\newcommand{\isasymF}{\isamath{\mathcal{F}}}
+\newcommand{\isasymG}{\isamath{\mathcal{G}}}
+\newcommand{\isasymH}{\isamath{\mathcal{H}}}
+\newcommand{\isasymI}{\isamath{\mathcal{I}}}
+\newcommand{\isasymJ}{\isamath{\mathcal{J}}}
+\newcommand{\isasymK}{\isamath{\mathcal{K}}}
+\newcommand{\isasymL}{\isamath{\mathcal{L}}}
+\newcommand{\isasymM}{\isamath{\mathcal{M}}}
+\newcommand{\isasymN}{\isamath{\mathcal{N}}}
+\newcommand{\isasymO}{\isamath{\mathcal{O}}}
+\newcommand{\isasymP}{\isamath{\mathcal{P}}}
+\newcommand{\isasymQ}{\isamath{\mathcal{Q}}}
+\newcommand{\isasymR}{\isamath{\mathcal{R}}}
+\newcommand{\isasymS}{\isamath{\mathcal{S}}}
+\newcommand{\isasymT}{\isamath{\mathcal{T}}}
+\newcommand{\isasymU}{\isamath{\mathcal{U}}}
+\newcommand{\isasymV}{\isamath{\mathcal{V}}}
+\newcommand{\isasymW}{\isamath{\mathcal{W}}}
+\newcommand{\isasymX}{\isamath{\mathcal{X}}}
+\newcommand{\isasymY}{\isamath{\mathcal{Y}}}
+\newcommand{\isasymZ}{\isamath{\mathcal{Z}}}
\newcommand{\isasymalpha}{\isamath{\alpha}}
\newcommand{\isasymbeta}{\isamath{\beta}}
\newcommand{\isasymgamma}{\isamath{\gamma}}
@@ -52,184 +66,196 @@
\newcommand{\isasymchi}{\isamath{\chi}}
\newcommand{\isasympsi}{\isamath{\psi}}
\newcommand{\isasymomega}{\isamath{\omega}}
-\newcommand{\isasymnot}{\isamath{\neg}}
-\newcommand{\isasymand}{\isamath{\wedge}}
-\newcommand{\isasymor}{\isamath{\vee}}
-\newcommand{\isasymforall}{\isamath{\forall\,}}
-\newcommand{\isasymexists}{\isamath{\exists\,}}
-\newcommand{\isasymAnd}{\isamath{\bigwedge\,}}
+\newcommand{\isasymGamma}{\isamath{\Gamma}}
+\newcommand{\isasymDelta}{\isamath{\Delta}}
+\newcommand{\isasymTheta}{\isamath{\Theta}}
+\newcommand{\isasymLambda}{\isamath{\Lambda}}
+\newcommand{\isasymXi}{\isamath{\Xi}}
+\newcommand{\isasymPi}{\isamath{\Pi}}
+\newcommand{\isasymSigma}{\isamath{\Sigma}}
+\newcommand{\isasymUpsilon}{\isamath{\Upsilon}}
+\newcommand{\isasymPhi}{\isamath{\Phi}}
+\newcommand{\isasymPsi}{\isamath{\Psi}}
+\newcommand{\isasymOmega}{\isamath{\Omega}}
+\newcommand{\isasymbool}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{B}}}
+\newcommand{\isasymcomplex}{\isamath{\mathrm{C}\mkern-15mu{\phantom{\mathrm{t}}\vrule}\mkern9mu}}
+\newcommand{\isasymnat}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{N}}}
+\newcommand{\isasymrat}{\isamath{\mathrm{Q}\mkern-16mu{\phantom{\mathrm{t}}\vrule}\mkern10mu}}
+\newcommand{\isasymreal}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{R}}}
+\newcommand{\isasymint}{\isamath{\mathsf{Z}\mkern-7.5mu\mathsf{Z}}}
+\newcommand{\isasymleftarrow}{\isamath{\leftarrow}}
+\newcommand{\isasymlongleftarrow}{\isamath{\longleftarrow}}
+\newcommand{\isasymrightarrow}{\isamath{\rightarrow}}
+\newcommand{\isasymlongrightarrow}{\isamath{\longrightarrow}}
+\newcommand{\isasymLeftarrow}{\isamath{\Leftarrow}}
+\newcommand{\isasymLongleftarrow}{\isamath{\Longleftarrow}}
+\newcommand{\isasymRightarrow}{\isamath{\Rightarrow}}
+\newcommand{\isasymLongrightarrow}{\isamath{\Longrightarrow}}
+\newcommand{\isasymleftrightarrow}{\isamath{\leftrightarrow}}
+\newcommand{\isasymlongleftrightarrow}{\isamath{\longleftrightarrow}}
+\newcommand{\isasymLeftrightarrow}{\isamath{\Leftrightarrow}}
+\newcommand{\isasymLongleftrightarrow}{\isamath{\Longleftrightarrow}}
+\newcommand{\isasymmapsto}{\isamath{\mapsto}}
+\newcommand{\isasymlongmapsto}{\isamath{\longmapsto}}
+\newcommand{\isasymmidarrow}{\isamath{\relbar}}
+\newcommand{\isasymMidarrow}{\isamath{\Relbar}}
+\newcommand{\isasymhookleftarrow}{\isamath{\hookleftarrow}}
+\newcommand{\isasymhookrightarrow}{\isamath{\hookrightarrow}}
+\newcommand{\isasymleftharpoondown}{\isamath{\leftharpoondown}}
+\newcommand{\isasymrightharpoondown}{\isamath{\rightharpoondown}}
+\newcommand{\isasymleftharpoonup}{\isamath{\leftharpoonup}}
+\newcommand{\isasymrightharpoonup}{\isamath{\rightharpoonup}}
+\newcommand{\isasymrightleftharpoons}{\isamath{\rightleftharpoons}}
+\newcommand{\isasymleadsto}{\isamath{\leadsto}} %requires latexsym
+\newcommand{\isasymup}{\isamath{\uparrow}}
+\newcommand{\isasymUparrow}{\isamath{\Uparrow}}
+\newcommand{\isasymdown}{\isamath{\downarrow}}
+\newcommand{\isasymDownarrow}{\isamath{\Downarrow}}
+\newcommand{\isasymupdownarrow}{\isamath{\updownarrow}}
+\newcommand{\isasymUpdownarrow}{\isamath{\Updownarrow}}
+\newcommand{\isasymlangle}{\isamath{\langle}}
+\newcommand{\isasymrangle}{\isamath{\rangle}}
\newcommand{\isasymlceil}{\isamath{\lceil}}
\newcommand{\isasymrceil}{\isamath{\rceil}}
\newcommand{\isasymlfloor}{\isamath{\lfloor}}
\newcommand{\isasymrfloor}{\isamath{\rfloor}}
-\newcommand{\isasymturnstile}{\isamath{\vdash}}
-\newcommand{\isasymTurnstile}{\isamath{\models}}
+\newcommand{\isasymlparr}{\isamath{\mathopen{(\mkern-3mu\mid}}}
+\newcommand{\isasymrparr}{\isamath{\mathclose{\mid\mkern-3mu)}}}
\newcommand{\isasymlbrakk}{\isamath{\mathopen{\lbrack\mkern-3mu\lbrack}}}
\newcommand{\isasymrbrakk}{\isamath{\mathclose{\rbrack\mkern-3mu\rbrack}}}
-\newcommand{\isasymcdot}{\isamath{\cdot}}
+\newcommand{\isasymlbrace}{\isamath{\mathopen{\lbrace\mkern-4.5mu\mid}}}
+\newcommand{\isasymrbrace}{\isamath{\mathclose{\mid\mkern-4.5mu\rbrace}}}
+\newcommand{\isasymguillemotleft}{\isatext{\flqq}} %requires babel
+\newcommand{\isasymguillemotright}{\isatext{\frqq}} %requires babel
+\newcommand{\isasymColon}{\isamath{\mathrel{::}}}
+\newcommand{\isasymnot}{\isamath{\neg}}
+\newcommand{\isasymbottom}{\isamath{\bot}}
+\newcommand{\isasymtop}{\isamath{\top}}
+\newcommand{\isasymand}{\isamath{\wedge}}
+\newcommand{\isasymor}{\isamath{\vee}}
+\newcommand{\isasymAnd}{\isamath{\bigwedge\,}}
+\newcommand{\isasymOr}{\isamath{\bigvee}}
+\newcommand{\isasymforall}{\isamath{\forall\,}}
+\newcommand{\isasymexists}{\isamath{\exists\,}}
+\newcommand{\isasymbox}{\isamath{\Box}} %requires latexsym
+\newcommand{\isasymdiamond}{\isamath{\Diamond}} %requires latexsym
+\newcommand{\isasymturnstile}{\isamath{\vdash}}
+\newcommand{\isasymTurnstile}{\isamath{\models}}
+\newcommand{\isasymstileturn}{\isamath{\dashv}}
+\newcommand{\isasymsurd}{\isamath{\surd}}
+\newcommand{\isasymle}{\isamath{\le}}
+\newcommand{\isasymge}{\isamath{\ge}}
+\newcommand{\isasymll}{\isamath{\ll}}
+\newcommand{\isasymgg}{\isamath{\gg}}
+\newcommand{\isasymlesssim}{\isamath{\lesssim}} %requires amssymb
+\newcommand{\isasymgreatersim}{\isamath{\gtrsim}} %requires amssymb
+\newcommand{\isasymlessapprox}{\isamath{\lessapprox}} %requires amssymb
+\newcommand{\isasymgreaterapprox}{\isamath{\gtrapprox}} %requires amssymb
\newcommand{\isasymin}{\isamath{\in}}
+\newcommand{\isasymnotin}{\isamath{\notin}}
+\newcommand{\isasymsubset}{\isamath{\subset}}
+\newcommand{\isasymsupset}{\isamath{\supset}}
\newcommand{\isasymsubseteq}{\isamath{\subseteq}}
+\newcommand{\isasymsupseteq}{\isamath{\supseteq}}
+\newcommand{\isasymsqsubset}{\isamath{\sqsubset}}
+\newcommand{\isasymsqsupset}{\isamath{\sqsupset}} %requires latexsym
+\newcommand{\isasymsqsubseteq}{\isamath{\sqsubseteq}}
+\newcommand{\isasymsqsupseteq}{\isamath{\sqsupseteq}}
\newcommand{\isasyminter}{\isamath{\cap}}
\newcommand{\isasymunion}{\isamath{\cup}}
\newcommand{\isasymInter}{\isamath{\bigcap\,}}
\newcommand{\isasymUnion}{\isamath{\bigcup\,}}
+\newcommand{\isasymsqunion}{\isamath{\sqcup}}
\newcommand{\isasymsqinter}{\isamath{\sqcap}}
-\newcommand{\isasymsqunion}{\isamath{\sqcup}}
-\newcommand{\isasymSqinter}{\isamath{\bigsqcap\,}} %requires stmaryrd
\newcommand{\isasymSqunion}{\isamath{\bigsqcup\,}}
-\newcommand{\isasymbottom}{\isamath{\bot}}
-\newcommand{\isasymdoteq}{\isamath{\doteq}}
-\newcommand{\isasymequiv}{\isamath{\equiv}}
+\newcommand{\isasymSqinter}{\isamath{\bigsqcap\,}} %requires stmaryrd
+\newcommand{\isasymuplus}{\isamath{\uplus}}
+\newcommand{\isasymbiguplus}{\isamath{\biguplus}}
\newcommand{\isasymnoteq}{\isamath{\not=}}
-\newcommand{\isasymsqsubset}{\isamath{\sqsubset}}
-\newcommand{\isasymsqsubseteq}{\isamath{\sqsubseteq}}
-\newcommand{\isasymprec}{\isamath{\prec}}
-\newcommand{\isasympreceq}{\isamath{\preceq}}
-\newcommand{\isasymsucc}{\isamath{\succ}}
-\newcommand{\isasymapprox}{\isamath{\approx}}
\newcommand{\isasymsim}{\isamath{\sim}}
+\newcommand{\isasymdoteq}{\isamath{\doteq}}
\newcommand{\isasymsimeq}{\isamath{\simeq}}
-\newcommand{\isasymle}{\isamath{\le}}
-\newcommand{\isasymColon}{\isamath{\mathrel{::}}}
-\newcommand{\isasymleftarrow}{\isamath{\leftarrow}}
-\newcommand{\isasymmidarrow}{\isamath{\relbar}}
-\newcommand{\isasymrightarrow}{\isamath{\rightarrow}}
-\newcommand{\isasymLeftarrow}{\isamath{\Leftarrow}}
-\newcommand{\isasymMidarrow}{\isamath{\Relbar}}
-\newcommand{\isasymRightarrow}{\isamath{\Rightarrow}}
+\newcommand{\isasymapprox}{\isamath{\approx}}
+\newcommand{\isasymasymp}{\isamath{\asymp}}
+\newcommand{\isasymcong}{\isamath{\cong}}
+\newcommand{\isasymsmile}{\isamath{\smile}}
+\newcommand{\isasymequiv}{\isamath{\equiv}}
\newcommand{\isasymfrown}{\isamath{\frown}}
-\newcommand{\isasymmapsto}{\isamath{\mapsto}}
-\newcommand{\isasymleadsto}{\isamath{\leadsto}} %requires latexsym
-\newcommand{\isasymup}{\isamath{\uparrow}}
-\newcommand{\isasymdown}{\isamath{\downarrow}}
-\newcommand{\isasymnotin}{\isamath{\notin}}
+\newcommand{\isasympropto}{\isamath{\propto}}
+\newcommand{\isasymbowtie}{\isamath{\bowtie}}
+\newcommand{\isasymprec}{\isamath{\prec}}
+\newcommand{\isasymsucc}{\isamath{\succ}}
+\newcommand{\isasympreceq}{\isamath{\preceq}}
+\newcommand{\isasymsucceq}{\isamath{\succeq}}
+\newcommand{\isasymparallel}{\isamath{\parallel}}
+\newcommand{\isasymbar}{\isamath{\mid}}
+\newcommand{\isasymplusminus}{\isamath{\pm}}
+\newcommand{\isasymminusplus}{\isamath{\mp}}
\newcommand{\isasymtimes}{\isamath{\times}}
+\newcommand{\isasymdiv}{\isamath{\div}}
+\newcommand{\isasymcdot}{\isamath{\cdot}}
+\newcommand{\isasymstar}{\isamath{\star}}
+\newcommand{\isasymdagger}{\isamath{\dagger}}
+\newcommand{\isasymddagger}{\isamath{\ddagger}}
+\newcommand{\isasymcirc}{\isamath{\circ}}
+\newcommand{\isasymbullet}{\isamath{\bullet}}
+\newcommand{\isasymlhd}{\isamath{\lhd}}
+\newcommand{\isasymrhd}{\isamath{\rhd}}
+\newcommand{\isasymunlhd}{\isamath{\unlhd}}
+\newcommand{\isasymunrhd}{\isamath{\unrhd}}
+\newcommand{\isasymtriangleleft}{\isamath{\triangleleft}}
+\newcommand{\isasymtriangleright}{\isamath{\triangleright}}
+\newcommand{\isasymtriangle}{\isamath{\triangle}}
+\newcommand{\isasymtriangleq}{\isamath{\triangleq}} %requires amssymb
\newcommand{\isasymoplus}{\isamath{\oplus}}
\newcommand{\isasymominus}{\isamath{\ominus}}
\newcommand{\isasymotimes}{\isamath{\otimes}}
\newcommand{\isasymoslash}{\isamath{\oslash}}
-\newcommand{\isasymsubset}{\isamath{\subset}}
+\newcommand{\isasymodot}{\isamath{\odot}}
\newcommand{\isasyminfinity}{\isamath{\infty}}
-\newcommand{\isasymbox}{\isamath{\Box}} %requires latexsym
-\newcommand{\isasymdiamond}{\isamath{\Diamond}} %requires latexsym
-\newcommand{\isasymcirc}{\isamath{\circ}}
-\newcommand{\isasymbullet}{\isamath{\bullet}}
-\newcommand{\isasymparallel}{\isamath{\parallel}}
-\newcommand{\isasymsurd}{\isamath{\surd}}
+\newcommand{\isasymdots}{\isamath{\dots}}
+\newcommand{\isasymcdots}{\isamath{\cdots}}
+\newcommand{\isasymSum}{\isamath{\sum\,}}
+\newcommand{\isasymProd}{\isamath{\prod\,}}
+\newcommand{\isasymintegral}{\isamath{\int\,}}
+\newcommand{\isasymJoin}{\isamath{\Join}} %requires latexsym
+\newcommand{\isasymclubsuit}{\isamath{\clubsuit}}
+\newcommand{\isasymdiamondsuit}{\isamath{\diamondsuit}}
+\newcommand{\isasymheartsuit}{\isamath{\heartsuit}}
+\newcommand{\isasymspadesuit}{\isamath{\spadesuit}}
+\newcommand{\isasymaleph}{\isamath{\aleph}}
+\newcommand{\isasymemptyset}{\isamath{\emptyset}}
+\newcommand{\isasymnabla}{\isamath{\nabla}}
+\newcommand{\isasympartial}{\isamath{\partial}}
+\newcommand{\isasymRe}{\isamath{\Re}}
+\newcommand{\isasymIm}{\isamath{\Im}}
+\newcommand{\isasymflat}{\isamath{\flat}}
+\newcommand{\isasymnatural}{\isamath{\natural}}
+\newcommand{\isasymsharp}{\isamath{\sharp}}
+\newcommand{\isasymangle}{\isamath{\angle}}
\newcommand{\isasymcopyright}{\isatext{\copyright}}
-\newcommand{\isasymplusminus}{\isamath{\pm}}
-\newcommand{\isasymdiv}{\isamath{\div}}
-\newcommand{\isasymlongrightarrow}{\isamath{\longrightarrow}}
-\newcommand{\isasymlongleftarrow}{\isamath{\longleftarrow}}
-\newcommand{\isasymlongleftrightarrow}{\isamath{\longleftrightarrow}}
-\newcommand{\isasymLongrightarrow}{\isamath{\Longrightarrow}}
-\newcommand{\isasymLongleftarrow}{\isamath{\Longleftarrow}}
-\newcommand{\isasymLongleftrightarrow}{\isamath{\Longleftrightarrow}}
-\newcommand{\isasymbar}{\isamath{\mid}}
+\newcommand{\isasymregistered}{\isatext{\rm\textregistered}}
\newcommand{\isasymhyphen}{\isatext{\rm-}}
\newcommand{\isasyminverse}{\isamath{{}^{-1}}}
-\newcommand{\isasymexclamdown}{\isatext{\rm\textexclamdown}}
-\newcommand{\isasymquestiondown}{\isatext{\rm\textquestiondown}}
-\newcommand{\isasymguillemotleft}{\isatext{\flqq}} %requires babel
-\newcommand{\isasymguillemotright}{\isatext{\frqq}} %requires babel
-\newcommand{\isasymdegree}{\isatext{\rm\textdegree}} %requires latin1
\newcommand{\isasymonesuperior}{\isamath{\mathonesuperior}} %requires latin1
\newcommand{\isasymonequarter}{\isatext{\rm\textonequarter}} %requires latin1
\newcommand{\isasymtwosuperior}{\isamath{\mathtwosuperior}} %requires latin1
\newcommand{\isasymonehalf}{\isatext{\rm\textonehalf}} %requires latin1
\newcommand{\isasymthreesuperior}{\isamath{\maththreesuperior}} %requires latin1
\newcommand{\isasymthreequarters}{\isatext{\rm\textthreequarters}} %requires latin1
-\newcommand{\isasymparagraph}{\isatext{\P}}
-\newcommand{\isasymregistered}{\isatext{\rm\textregistered}}
\newcommand{\isasymordfeminine}{\isatext{\rm\textordfeminine}}
\newcommand{\isasymordmasculine}{\isatext{\rm\textordmasculine}}
\newcommand{\isasymsection}{\isatext{\S}}
+\newcommand{\isasymparagraph}{\isatext{\P}}
+\newcommand{\isasymexclamdown}{\isatext{\rm\textexclamdown}}
+\newcommand{\isasymquestiondown}{\isatext{\rm\textquestiondown}}
\newcommand{\isasympounds}{\isamath{\pounds}}
\newcommand{\isasymyen}{\isatext{\yen}} %requires amssymb
\newcommand{\isasymcent}{\isatext{\cent}} %requires wasysym
\newcommand{\isasymcurrency}{\isatext{\currency}} %requires wasysym
-\newcommand{\isasymlbrace}{\isamath{\mathopen{\lbrace\mkern-4.5mu\mid}}}
-\newcommand{\isasymrbrace}{\isamath{\mathclose{\mid\mkern-4.5mu\rbrace}}}
-\newcommand{\isasymtop}{\isamath{\top}}
-\newcommand{\isasymcong}{\isamath{\cong}}
-\newcommand{\isasymclubsuit}{\isamath{\clubsuit}}
-\newcommand{\isasymdiamondsuit}{\isamath{\diamondsuit}}
-\newcommand{\isasymheartsuit}{\isamath{\heartsuit}}
-\newcommand{\isasymspadesuit}{\isamath{\spadesuit}}
-\newcommand{\isasymleftrightarrow}{\isamath{\leftrightarrow}}
-\newcommand{\isasymge}{\isamath{\ge}}
-\newcommand{\isasympropto}{\isamath{\propto}}
-\newcommand{\isasympartial}{\isamath{\partial}}
-\newcommand{\isasymdots}{\isamath{\dots}}
-\newcommand{\isasymaleph}{\isamath{\aleph}}
-\newcommand{\isasymIm}{\isamath{\Im}}
-\newcommand{\isasymRe}{\isamath{\Re}}
+\newcommand{\isasymdegree}{\isatext{\rm\textdegree}} %requires latin1
\newcommand{\isasymwp}{\isamath{\wp}}
-\newcommand{\isasymemptyset}{\isamath{\emptyset}}
-\newcommand{\isasymangle}{\isamath{\angle}}
-\newcommand{\isasymnabla}{\isamath{\nabla}}
-\newcommand{\isasymProd}{\isamath{\prod\,}}
-\newcommand{\isasymLeftrightarrow}{\isamath{\Leftrightarrow}}
-\newcommand{\isasymUparrow}{\isamath{\Uparrow}}
-\newcommand{\isasymDownarrow}{\isamath{\Downarrow}}
-\newcommand{\isasymlozenge}{\isamath{\lozenge}} %requires amssym
-\newcommand{\isasymlangle}{\isamath{\langle}}
-\newcommand{\isasymrangle}{\isamath{\rangle}}
-\newcommand{\isasymSum}{\isamath{\sum\,}}
-\newcommand{\isasymintegral}{\isamath{\int\,}}
-\newcommand{\isasymdagger}{\isamath{\dagger}}
-\newcommand{\isasymsharp}{\isamath{\sharp}}
-\newcommand{\isasymstar}{\isamath{\star}}
-\newcommand{\isasymtriangleright}{\isamath{\triangleright}}
-\newcommand{\isasymlhd}{\isamath{\lhd}}
-\newcommand{\isasymtriangle}{\isamath{\triangle}}
-\newcommand{\isasymrhd}{\isamath{\rhd}}
-\newcommand{\isasymunlhd}{\isamath{\unlhd}}
-\newcommand{\isasymunrhd}{\isamath{\unrhd}}
-\newcommand{\isasymtriangleleft}{\isamath{\triangleleft}}
-\newcommand{\isasymnatural}{\isamath{\natural}}
-\newcommand{\isasymflat}{\isamath{\flat}}
\newcommand{\isasymamalg}{\isamath{\amalg}}
\newcommand{\isasymmho}{\isamath{\mho}} %requires latexsym
-\newcommand{\isasymupdownarrow}{\isamath{\updownarrow}}
-\newcommand{\isasymlongmapsto}{\isamath{\longmapsto}}
-\newcommand{\isasymUpdownarrow}{\isamath{\Updownarrow}}
-\newcommand{\isasymhookleftarrow}{\isamath{\hookleftarrow}}
-\newcommand{\isasymhookrightarrow}{\isamath{\hookrightarrow}}
-\newcommand{\isasymrightleftharpoons}{\isamath{\rightleftharpoons}}
-\newcommand{\isasymleftharpoondown}{\isamath{\leftharpoondown}}
-\newcommand{\isasymrightharpoondown}{\isamath{\rightharpoondown}}
-\newcommand{\isasymleftharpoonup}{\isamath{\leftharpoonup}}
-\newcommand{\isasymrightharpoonup}{\isamath{\rightharpoonup}}
-\newcommand{\isasymasymp}{\isamath{\asymp}}
-\newcommand{\isasymminusplus}{\isamath{\mp}}
-\newcommand{\isasymbowtie}{\isamath{\bowtie}}
-\newcommand{\isasymcdots}{\isamath{\cdots}}
-\newcommand{\isasymodot}{\isamath{\odot}}
-\newcommand{\isasymsupset}{\isamath{\supset}}
-\newcommand{\isasymsupseteq}{\isamath{\supseteq}}
-\newcommand{\isasymsqsupset}{\isamath{\sqsupset}} %requires latexsym
-\newcommand{\isasymsqsupseteq}{\isamath{\sqsupseteq}}
-\newcommand{\isasymll}{\isamath{\ll}}
-\newcommand{\isasymgg}{\isamath{\gg}}
-\newcommand{\isasymuplus}{\isamath{\uplus}}
-\newcommand{\isasymsmile}{\isamath{\smile}}
-\newcommand{\isasymsucceq}{\isamath{\succeq}}
-\newcommand{\isasymstileturn}{\isamath{\dashv}}
-\newcommand{\isasymOr}{\isamath{\bigvee}}
-\newcommand{\isasymbiguplus}{\isamath{\biguplus}}
-\newcommand{\isasymddagger}{\isamath{\ddagger}}
-\newcommand{\isasymJoin}{\isamath{\Join}} %requires latexsym
-\newcommand{\isasymbool}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{B}}}
-\newcommand{\isasymcomplex}{\isamath{\mathrm{C}\mkern-15mu{\phantom{\mathrm{t}}\vrule}\mkern9mu}}
-\newcommand{\isasymnat}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{N}}}
-\newcommand{\isasymrat}{\isamath{\mathrm{Q}\mkern-16mu{\phantom{\mathrm{t}}\vrule}\mkern10mu}}
-\newcommand{\isasymreal}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{R}}}
-\newcommand{\isasymint}{\isamath{\mathsf{Z}\mkern-7.5mu\mathsf{Z}}}
-\newcommand{\isasymlesssim}{\isamath{\lesssim}} %requires amssymb
-\newcommand{\isasymgreatersim}{\isamath{\gtrsim}} %requires amssymb
-\newcommand{\isasymlessapprox}{\isamath{\lessapprox}} %requires amssymb
-\newcommand{\isasymgreaterapprox}{\isamath{\gtrapprox}} %requires amssymb
-\newcommand{\isasymtriangleq}{\isamath{\triangleq}} %requires amssymb
-\newcommand{\isasymlparr}{\isamath{\mathopen{(\mkern-3mu\mid}}}
-\newcommand{\isasymrparr}{\isamath{\mathclose{\mid\mkern-3mu)}}}
+\newcommand{\isasymlozenge}{\isamath{\lozenge}} %requires amssym
+\newcommand{\isasymspacespace}{\isamath{~~}}