# HG changeset patch # User wenzelm # Date 966877438 -7200 # Node ID 2c208c98f541e58c0b29bc06e9cc2f646cd80f63 # Parent 8741740ea6d6cf33178c5eb67d4b90a5d2d44264 updated; diff -r 8741740ea6d6 -r 2c208c98f541 doc-src/AxClass/generated/Group.tex --- a/doc-src/AxClass/generated/Group.tex Mon Aug 21 18:49:38 2000 +0200 +++ b/doc-src/AxClass/generated/Group.tex Mon Aug 21 19:03:58 2000 +0200 @@ -1,7 +1,7 @@ \begin{isabelle}% % \isamarkupheader{Basic group theory} -\isacommand{theory}\ Group\ =\ Main:% +\isacommand{theory}\ Group\ {\isacharequal}\ Main{\isacharcolon}% \begin{isamarkuptext}% \medskip\noindent The meta-type system of Isabelle supports \emph{intersections} and \emph{inclusions} of type classes. These @@ -19,9 +19,9 @@ signature parts of our structures.% \end{isamarkuptext}% \isacommand{consts}\isanewline -\ \ times\ ::\ {\isachardoublequote}{\isacharprime}a\ ={\isachargreater}\ {\isacharprime}a\ ={\isachargreater}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymOtimes}{\isachardoublequote}\ 70{\isacharparenright}\isanewline -\ \ inverse\ ::\ {\isachardoublequote}{\isacharprime}a\ ={\isachargreater}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ \ \ \ \ {\isacharparenleft}{\isachardoublequote}{\isacharparenleft}{\isacharunderscore}{\isasyminv}{\isacharparenright}{\isachardoublequote}\ {\isacharbrackleft}1000{\isacharbrackright}\ 999{\isacharparenright}\isanewline -\ \ one\ ::\ {\isacharprime}a\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\isachardoublequote}{\isasymunit}{\isachardoublequote}{\isacharparenright}% +\ \ times\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isacharequal}{\isachargreater}\ {\isacharprime}a\ {\isacharequal}{\isachargreater}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymOtimes}{\isachardoublequote}\ \isadigit{7}\isadigit{0}{\isacharparenright}\isanewline +\ \ inverse\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isacharequal}{\isachargreater}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ \ \ \ \ {\isacharparenleft}{\isachardoublequote}{\isacharparenleft}{\isacharunderscore}{\isasyminv}{\isacharparenright}{\isachardoublequote}\ {\isacharbrackleft}\isadigit{1}\isadigit{0}\isadigit{0}\isadigit{0}{\isacharbrackright}\ \isadigit{9}\isadigit{9}\isadigit{9}{\isacharparenright}\isanewline +\ \ one\ {\isacharcolon}{\isacharcolon}\ {\isacharprime}a\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\isachardoublequote}{\isasymunit}{\isachardoublequote}{\isacharparenright}% \begin{isamarkuptext}% \noindent Next we define class $monoid$ of monoids with operations $\TIMES$ and $1$. Note that multiple class axioms are allowed for @@ -30,9 +30,9 @@ \end{isamarkuptext}% \isacommand{axclass}\isanewline \ \ monoid\ {\isacharless}\ {\isachardoublequote}term{\isachardoublequote}\isanewline -\ \ assoc:\ \ \ \ \ \ {\isachardoublequote}{\isacharparenleft}x\ {\isasymOtimes}\ y{\isacharparenright}\ {\isasymOtimes}\ z\ =\ x\ {\isasymOtimes}\ {\isacharparenleft}y\ {\isasymOtimes}\ z{\isacharparenright}{\isachardoublequote}\isanewline -\ \ left{\isacharunderscore}unit:\ \ {\isachardoublequote}{\isasymunit}\ {\isasymOtimes}\ x\ =\ x{\isachardoublequote}\isanewline -\ \ right{\isacharunderscore}unit:\ {\isachardoublequote}x\ {\isasymOtimes}\ {\isasymunit}\ =\ x{\isachardoublequote}% +\ \ assoc{\isacharcolon}\ \ \ \ \ \ {\isachardoublequote}{\isacharparenleft}x\ {\isasymOtimes}\ y{\isacharparenright}\ {\isasymOtimes}\ z\ {\isacharequal}\ x\ {\isasymOtimes}\ {\isacharparenleft}y\ {\isasymOtimes}\ z{\isacharparenright}{\isachardoublequote}\isanewline +\ \ left{\isacharunderscore}unit{\isacharcolon}\ \ {\isachardoublequote}{\isasymunit}\ {\isasymOtimes}\ x\ {\isacharequal}\ x{\isachardoublequote}\isanewline +\ \ right{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}x\ {\isasymOtimes}\ {\isasymunit}\ {\isacharequal}\ x{\isachardoublequote}% \begin{isamarkuptext}% \noindent So class $monoid$ contains exactly those types $\tau$ where $\TIMES :: \tau \To \tau \To \tau$ and $1 :: \tau$ are specified @@ -48,16 +48,16 @@ \end{isamarkuptext}% \isacommand{axclass}\isanewline \ \ semigroup\ {\isacharless}\ {\isachardoublequote}term{\isachardoublequote}\isanewline -\ \ assoc:\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymOtimes}\ y{\isacharparenright}\ {\isasymOtimes}\ z\ =\ x\ {\isasymOtimes}\ {\isacharparenleft}y\ {\isasymOtimes}\ z{\isacharparenright}{\isachardoublequote}\isanewline +\ \ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymOtimes}\ y{\isacharparenright}\ {\isasymOtimes}\ z\ {\isacharequal}\ x\ {\isasymOtimes}\ {\isacharparenleft}y\ {\isasymOtimes}\ z{\isacharparenright}{\isachardoublequote}\isanewline \isanewline \isacommand{axclass}\isanewline \ \ group\ {\isacharless}\ semigroup\isanewline -\ \ left{\isacharunderscore}unit:\ \ \ \ {\isachardoublequote}{\isasymunit}\ {\isasymOtimes}\ x\ =\ x{\isachardoublequote}\isanewline -\ \ left{\isacharunderscore}inverse:\ {\isachardoublequote}x{\isasyminv}\ {\isasymOtimes}\ x\ =\ {\isasymunit}{\isachardoublequote}\isanewline +\ \ left{\isacharunderscore}unit{\isacharcolon}\ \ \ \ {\isachardoublequote}{\isasymunit}\ {\isasymOtimes}\ x\ {\isacharequal}\ x{\isachardoublequote}\isanewline +\ \ left{\isacharunderscore}inverse{\isacharcolon}\ {\isachardoublequote}x{\isasyminv}\ {\isasymOtimes}\ x\ {\isacharequal}\ {\isasymunit}{\isachardoublequote}\isanewline \isanewline \isacommand{axclass}\isanewline \ \ agroup\ {\isacharless}\ group\isanewline -\ \ commute:\ {\isachardoublequote}x\ {\isasymOtimes}\ y\ =\ y\ {\isasymOtimes}\ x{\isachardoublequote}% +\ \ commute{\isacharcolon}\ {\isachardoublequote}x\ {\isasymOtimes}\ y\ {\isacharequal}\ y\ {\isasymOtimes}\ x{\isachardoublequote}% \begin{isamarkuptext}% \noindent Class $group$ inherits associativity of $\TIMES$ from $semigroup$ and adds two further group axioms. Similarly, $agroup$ @@ -85,42 +85,42 @@ ``abstract theorems''. For example, we may now derive the following well-known laws of general groups.% \end{isamarkuptext}% -\isacommand{theorem}\ group{\isacharunderscore}right{\isacharunderscore}inverse:\ {\isachardoublequote}x\ {\isasymOtimes}\ x{\isasyminv}\ =\ {\isacharparenleft}{\isasymunit}{\isasymColon}{\isacharprime}a{\isasymColon}group{\isacharparenright}{\isachardoublequote}\isanewline +\isacommand{theorem}\ group{\isacharunderscore}right{\isacharunderscore}inverse{\isacharcolon}\ {\isachardoublequote}x\ {\isasymOtimes}\ x{\isasyminv}\ {\isacharequal}\ {\isacharparenleft}{\isasymunit}{\isasymColon}{\isacharprime}a{\isasymColon}group{\isacharparenright}{\isachardoublequote}\isanewline \isacommand{proof}\ {\isacharminus}\isanewline -\ \ \isacommand{have}\ {\isachardoublequote}x\ {\isasymOtimes}\ x{\isasyminv}\ =\ {\isasymunit}\ {\isasymOtimes}\ {\isacharparenleft}x\ {\isasymOtimes}\ x{\isasyminv}{\isacharparenright}{\isachardoublequote}\isanewline -\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only:\ group.left{\isacharunderscore}unit{\isacharparenright}\isanewline -\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}...\ =\ {\isasymunit}\ {\isasymOtimes}\ x\ {\isasymOtimes}\ x{\isasyminv}{\isachardoublequote}\isanewline -\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only:\ semigroup.assoc{\isacharparenright}\isanewline -\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}...\ =\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymOtimes}\ x{\isasyminv}\ {\isasymOtimes}\ x\ {\isasymOtimes}\ x{\isasyminv}{\isachardoublequote}\isanewline -\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only:\ group.left{\isacharunderscore}inverse{\isacharparenright}\isanewline -\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}...\ =\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymOtimes}\ {\isacharparenleft}x{\isasyminv}\ {\isasymOtimes}\ x{\isacharparenright}\ {\isasymOtimes}\ x{\isasyminv}{\isachardoublequote}\isanewline -\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only:\ semigroup.assoc{\isacharparenright}\isanewline -\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}...\ =\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymOtimes}\ {\isasymunit}\ {\isasymOtimes}\ x{\isasyminv}{\isachardoublequote}\isanewline -\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only:\ group.left{\isacharunderscore}inverse{\isacharparenright}\isanewline -\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}...\ =\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymOtimes}\ {\isacharparenleft}{\isasymunit}\ {\isasymOtimes}\ x{\isasyminv}{\isacharparenright}{\isachardoublequote}\isanewline -\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only:\ semigroup.assoc{\isacharparenright}\isanewline -\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}...\ =\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymOtimes}\ x{\isasyminv}{\isachardoublequote}\isanewline -\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only:\ group.left{\isacharunderscore}unit{\isacharparenright}\isanewline -\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}...\ =\ {\isasymunit}{\isachardoublequote}\isanewline -\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only:\ group.left{\isacharunderscore}inverse{\isacharparenright}\isanewline -\ \ \isacommand{finally}\ \isacommand{show}\ ?thesis\ \isacommand{.}\isanewline +\ \ \isacommand{have}\ {\isachardoublequote}x\ {\isasymOtimes}\ x{\isasyminv}\ {\isacharequal}\ {\isasymunit}\ {\isasymOtimes}\ {\isacharparenleft}x\ {\isasymOtimes}\ x{\isasyminv}{\isacharparenright}{\isachardoublequote}\isanewline +\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline +\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymunit}\ {\isasymOtimes}\ x\ {\isasymOtimes}\ x{\isasyminv}{\isachardoublequote}\isanewline +\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline +\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymOtimes}\ x{\isasyminv}\ {\isasymOtimes}\ x\ {\isasymOtimes}\ x{\isasyminv}{\isachardoublequote}\isanewline +\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline +\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymOtimes}\ {\isacharparenleft}x{\isasyminv}\ {\isasymOtimes}\ x{\isacharparenright}\ {\isasymOtimes}\ x{\isasyminv}{\isachardoublequote}\isanewline +\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline +\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymOtimes}\ {\isasymunit}\ {\isasymOtimes}\ x{\isasyminv}{\isachardoublequote}\isanewline +\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline +\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymOtimes}\ {\isacharparenleft}{\isasymunit}\ {\isasymOtimes}\ x{\isasyminv}{\isacharparenright}{\isachardoublequote}\isanewline +\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline +\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymOtimes}\ x{\isasyminv}{\isachardoublequote}\isanewline +\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline +\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymunit}{\isachardoublequote}\isanewline +\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline +\ \ \isacommand{finally}\ \isacommand{show}\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isanewline \isacommand{qed}% \begin{isamarkuptext}% \noindent With $group_right_inverse$ already available, $group_right_unit$\label{thm:group-right-unit} is now established much easier.% \end{isamarkuptext}% -\isacommand{theorem}\ group{\isacharunderscore}right{\isacharunderscore}unit:\ {\isachardoublequote}x\ {\isasymOtimes}\ {\isasymunit}\ =\ {\isacharparenleft}x{\isasymColon}{\isacharprime}a{\isasymColon}group{\isacharparenright}{\isachardoublequote}\isanewline +\isacommand{theorem}\ group{\isacharunderscore}right{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}x\ {\isasymOtimes}\ {\isasymunit}\ {\isacharequal}\ {\isacharparenleft}x{\isasymColon}{\isacharprime}a{\isasymColon}group{\isacharparenright}{\isachardoublequote}\isanewline \isacommand{proof}\ {\isacharminus}\isanewline -\ \ \isacommand{have}\ {\isachardoublequote}x\ {\isasymOtimes}\ {\isasymunit}\ =\ x\ {\isasymOtimes}\ {\isacharparenleft}x{\isasyminv}\ {\isasymOtimes}\ x{\isacharparenright}{\isachardoublequote}\isanewline -\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only:\ group.left{\isacharunderscore}inverse{\isacharparenright}\isanewline -\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}...\ =\ x\ {\isasymOtimes}\ x{\isasyminv}\ {\isasymOtimes}\ x{\isachardoublequote}\isanewline -\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only:\ semigroup.assoc{\isacharparenright}\isanewline -\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}...\ =\ {\isasymunit}\ {\isasymOtimes}\ x{\isachardoublequote}\isanewline -\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only:\ group{\isacharunderscore}right{\isacharunderscore}inverse{\isacharparenright}\isanewline -\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}...\ =\ x{\isachardoublequote}\isanewline -\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only:\ group.left{\isacharunderscore}unit{\isacharparenright}\isanewline -\ \ \isacommand{finally}\ \isacommand{show}\ ?thesis\ \isacommand{.}\isanewline +\ \ \isacommand{have}\ {\isachardoublequote}x\ {\isasymOtimes}\ {\isasymunit}\ {\isacharequal}\ x\ {\isasymOtimes}\ {\isacharparenleft}x{\isasyminv}\ {\isasymOtimes}\ x{\isacharparenright}{\isachardoublequote}\isanewline +\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline +\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ x\ {\isasymOtimes}\ x{\isasyminv}\ {\isasymOtimes}\ x{\isachardoublequote}\isanewline +\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline +\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymunit}\ {\isasymOtimes}\ x{\isachardoublequote}\isanewline +\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isacharunderscore}right{\isacharunderscore}inverse{\isacharparenright}\isanewline +\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ x{\isachardoublequote}\isanewline +\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline +\ \ \isacommand{finally}\ \isacommand{show}\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isanewline \isacommand{qed}% \begin{isamarkuptext}% \medskip Abstract theorems may be instantiated to only those types @@ -171,19 +171,19 @@ \end{isamarkuptext}% \isacommand{instance}\ monoid\ {\isacharless}\ semigroup\isanewline \isacommand{proof}\ intro{\isacharunderscore}classes\isanewline -\ \ \isacommand{fix}\ x\ y\ z\ ::\ {\isachardoublequote}{\isacharprime}a{\isasymColon}monoid{\isachardoublequote}\isanewline -\ \ \isacommand{show}\ {\isachardoublequote}x\ {\isasymOtimes}\ y\ {\isasymOtimes}\ z\ =\ x\ {\isasymOtimes}\ {\isacharparenleft}y\ {\isasymOtimes}\ z{\isacharparenright}{\isachardoublequote}\isanewline -\ \ \ \ \isacommand{by}\ {\isacharparenleft}rule\ monoid.assoc{\isacharparenright}\isanewline +\ \ \isacommand{fix}\ x\ y\ z\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isasymColon}monoid{\isachardoublequote}\isanewline +\ \ \isacommand{show}\ {\isachardoublequote}x\ {\isasymOtimes}\ y\ {\isasymOtimes}\ z\ {\isacharequal}\ x\ {\isasymOtimes}\ {\isacharparenleft}y\ {\isasymOtimes}\ z{\isacharparenright}{\isachardoublequote}\isanewline +\ \ \ \ \isacommand{by}\ {\isacharparenleft}rule\ monoid{\isachardot}assoc{\isacharparenright}\isanewline \isacommand{qed}\isanewline \isanewline \isacommand{instance}\ group\ {\isacharless}\ monoid\isanewline \isacommand{proof}\ intro{\isacharunderscore}classes\isanewline -\ \ \isacommand{fix}\ x\ y\ z\ ::\ {\isachardoublequote}{\isacharprime}a{\isasymColon}group{\isachardoublequote}\isanewline -\ \ \isacommand{show}\ {\isachardoublequote}x\ {\isasymOtimes}\ y\ {\isasymOtimes}\ z\ =\ x\ {\isasymOtimes}\ {\isacharparenleft}y\ {\isasymOtimes}\ z{\isacharparenright}{\isachardoublequote}\isanewline -\ \ \ \ \isacommand{by}\ {\isacharparenleft}rule\ semigroup.assoc{\isacharparenright}\isanewline -\ \ \isacommand{show}\ {\isachardoublequote}{\isasymunit}\ {\isasymOtimes}\ x\ =\ x{\isachardoublequote}\isanewline -\ \ \ \ \isacommand{by}\ {\isacharparenleft}rule\ group.left{\isacharunderscore}unit{\isacharparenright}\isanewline -\ \ \isacommand{show}\ {\isachardoublequote}x\ {\isasymOtimes}\ {\isasymunit}\ =\ x{\isachardoublequote}\isanewline +\ \ \isacommand{fix}\ x\ y\ z\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isasymColon}group{\isachardoublequote}\isanewline +\ \ \isacommand{show}\ {\isachardoublequote}x\ {\isasymOtimes}\ y\ {\isasymOtimes}\ z\ {\isacharequal}\ x\ {\isasymOtimes}\ {\isacharparenleft}y\ {\isasymOtimes}\ z{\isacharparenright}{\isachardoublequote}\isanewline +\ \ \ \ \isacommand{by}\ {\isacharparenleft}rule\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline +\ \ \isacommand{show}\ {\isachardoublequote}{\isasymunit}\ {\isasymOtimes}\ x\ {\isacharequal}\ x{\isachardoublequote}\isanewline +\ \ \ \ \isacommand{by}\ {\isacharparenleft}rule\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline +\ \ \isacommand{show}\ {\isachardoublequote}x\ {\isasymOtimes}\ {\isasymunit}\ {\isacharequal}\ x{\isachardoublequote}\isanewline \ \ \ \ \isacommand{by}\ {\isacharparenleft}rule\ group{\isacharunderscore}right{\isacharunderscore}unit{\isacharparenright}\isanewline \isacommand{qed}% \begin{isamarkuptext}% @@ -215,9 +215,9 @@ $False$ as $1$ forms an Abelian group.% \end{isamarkuptext}% \isacommand{defs}\ {\isacharparenleft}\isakeyword{overloaded}{\isacharparenright}\isanewline -\ \ times{\isacharunderscore}bool{\isacharunderscore}def:\ \ \ {\isachardoublequote}x\ {\isasymOtimes}\ y\ {\isasymequiv}\ x\ {\isasymnoteq}\ {\isacharparenleft}y{\isasymColon}bool{\isacharparenright}{\isachardoublequote}\isanewline -\ \ inverse{\isacharunderscore}bool{\isacharunderscore}def:\ {\isachardoublequote}x{\isasyminv}\ {\isasymequiv}\ x{\isasymColon}bool{\isachardoublequote}\isanewline -\ \ unit{\isacharunderscore}bool{\isacharunderscore}def:\ \ \ \ {\isachardoublequote}{\isasymunit}\ {\isasymequiv}\ False{\isachardoublequote}% +\ \ times{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ \ \ {\isachardoublequote}x\ {\isasymOtimes}\ y\ {\isasymequiv}\ x\ {\isasymnoteq}\ {\isacharparenleft}y{\isasymColon}bool{\isacharparenright}{\isachardoublequote}\isanewline +\ \ inverse{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}x{\isasyminv}\ {\isasymequiv}\ x{\isasymColon}bool{\isachardoublequote}\isanewline +\ \ unit{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ \ \ \ {\isachardoublequote}{\isasymunit}\ {\isasymequiv}\ False{\isachardoublequote}% \begin{isamarkuptext}% \medskip It is important to note that above $\DEFS$ are just overloaded meta-level constant definitions, where type classes are @@ -232,14 +232,14 @@ operations on type $bool$ appropriately, the class membership $bool :: agroup$ may be now derived as follows.% \end{isamarkuptext}% -\isacommand{instance}\ bool\ ::\ agroup\isanewline -\isacommand{proof}\ {\isacharparenleft}intro{\isacharunderscore}classes,\isanewline +\isacommand{instance}\ bool\ {\isacharcolon}{\isacharcolon}\ agroup\isanewline +\isacommand{proof}\ {\isacharparenleft}intro{\isacharunderscore}classes{\isacharcomma}\isanewline \ \ \ \ unfold\ times{\isacharunderscore}bool{\isacharunderscore}def\ inverse{\isacharunderscore}bool{\isacharunderscore}def\ unit{\isacharunderscore}bool{\isacharunderscore}def{\isacharparenright}\isanewline \ \ \isacommand{fix}\ x\ y\ z\isanewline -\ \ \isacommand{show}\ {\isachardoublequote}{\isacharparenleft}{\isacharparenleft}x\ {\isasymnoteq}\ y{\isacharparenright}\ {\isasymnoteq}\ z{\isacharparenright}\ =\ {\isacharparenleft}x\ {\isasymnoteq}\ {\isacharparenleft}y\ {\isasymnoteq}\ z{\isacharparenright}{\isacharparenright}{\isachardoublequote}\ \isacommand{by}\ blast\isanewline -\ \ \isacommand{show}\ {\isachardoublequote}{\isacharparenleft}False\ {\isasymnoteq}\ x{\isacharparenright}\ =\ x{\isachardoublequote}\ \isacommand{by}\ blast\isanewline -\ \ \isacommand{show}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymnoteq}\ x{\isacharparenright}\ =\ False{\isachardoublequote}\ \isacommand{by}\ blast\isanewline -\ \ \isacommand{show}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymnoteq}\ y{\isacharparenright}\ =\ {\isacharparenleft}y\ {\isasymnoteq}\ x{\isacharparenright}{\isachardoublequote}\ \isacommand{by}\ blast\isanewline +\ \ \isacommand{show}\ {\isachardoublequote}{\isacharparenleft}{\isacharparenleft}x\ {\isasymnoteq}\ y{\isacharparenright}\ {\isasymnoteq}\ z{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymnoteq}\ {\isacharparenleft}y\ {\isasymnoteq}\ z{\isacharparenright}{\isacharparenright}{\isachardoublequote}\ \isacommand{by}\ blast\isanewline +\ \ \isacommand{show}\ {\isachardoublequote}{\isacharparenleft}False\ {\isasymnoteq}\ x{\isacharparenright}\ {\isacharequal}\ x{\isachardoublequote}\ \isacommand{by}\ blast\isanewline +\ \ \isacommand{show}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymnoteq}\ x{\isacharparenright}\ {\isacharequal}\ False{\isachardoublequote}\ \isacommand{by}\ blast\isanewline +\ \ \isacommand{show}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymnoteq}\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}y\ {\isasymnoteq}\ x{\isacharparenright}{\isachardoublequote}\ \isacommand{by}\ blast\isanewline \isacommand{qed}% \begin{isamarkuptext}% The result of an $\isakeyword{instance}$ statement is both expressed @@ -270,22 +270,22 @@ $\TIMES$ component-wise to binary products $\alpha \times \beta$.% \end{isamarkuptext}% \isacommand{defs}\ {\isacharparenleft}\isakeyword{overloaded}{\isacharparenright}\isanewline -\ \ times{\isacharunderscore}prod{\isacharunderscore}def:\ {\isachardoublequote}p\ {\isasymOtimes}\ q\ {\isasymequiv}\ {\isacharparenleft}fst\ p\ {\isasymOtimes}\ fst\ q,\ snd\ p\ {\isasymOtimes}\ snd\ q{\isacharparenright}{\isachardoublequote}% +\ \ times{\isacharunderscore}prod{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}p\ {\isasymOtimes}\ q\ {\isasymequiv}\ {\isacharparenleft}fst\ p\ {\isasymOtimes}\ fst\ q{\isacharcomma}\ snd\ p\ {\isasymOtimes}\ snd\ q{\isacharparenright}{\isachardoublequote}% \begin{isamarkuptext}% It is very easy to see that associativity of $\TIMES^\alpha$ and $\TIMES^\beta$ transfers to ${\TIMES}^{\alpha \times \beta}$. Hence the binary type constructor $\times$ maps semigroups to semigroups. This may be established formally as follows.% \end{isamarkuptext}% -\isacommand{instance}\ {\isacharasterisk}\ ::\ {\isacharparenleft}semigroup,\ semigroup{\isacharparenright}\ semigroup\isanewline -\isacommand{proof}\ {\isacharparenleft}intro{\isacharunderscore}classes,\ unfold\ times{\isacharunderscore}prod{\isacharunderscore}def{\isacharparenright}\isanewline -\ \ \isacommand{fix}\ p\ q\ r\ ::\ {\isachardoublequote}{\isacharprime}a{\isasymColon}semigroup\ {\isasymtimes}\ {\isacharprime}b{\isasymColon}semigroup{\isachardoublequote}\isanewline +\isacommand{instance}\ {\isacharasterisk}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}semigroup{\isacharcomma}\ semigroup{\isacharparenright}\ semigroup\isanewline +\isacommand{proof}\ {\isacharparenleft}intro{\isacharunderscore}classes{\isacharcomma}\ unfold\ times{\isacharunderscore}prod{\isacharunderscore}def{\isacharparenright}\isanewline +\ \ \isacommand{fix}\ p\ q\ r\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isasymColon}semigroup\ {\isasymtimes}\ {\isacharprime}b{\isasymColon}semigroup{\isachardoublequote}\isanewline \ \ \isacommand{show}\isanewline -\ \ \ \ {\isachardoublequote}{\isacharparenleft}fst\ {\isacharparenleft}fst\ p\ {\isasymOtimes}\ fst\ q,\ snd\ p\ {\isasymOtimes}\ snd\ q{\isacharparenright}\ {\isasymOtimes}\ fst\ r,\isanewline -\ \ \ \ \ \ snd\ {\isacharparenleft}fst\ p\ {\isasymOtimes}\ fst\ q,\ snd\ p\ {\isasymOtimes}\ snd\ q{\isacharparenright}\ {\isasymOtimes}\ snd\ r{\isacharparenright}\ =\isanewline -\ \ \ \ \ \ \ {\isacharparenleft}fst\ p\ {\isasymOtimes}\ fst\ {\isacharparenleft}fst\ q\ {\isasymOtimes}\ fst\ r,\ snd\ q\ {\isasymOtimes}\ snd\ r{\isacharparenright},\isanewline -\ \ \ \ \ \ \ \ snd\ p\ {\isasymOtimes}\ snd\ {\isacharparenleft}fst\ q\ {\isasymOtimes}\ fst\ r,\ snd\ q\ {\isasymOtimes}\ snd\ r{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline -\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ add:\ semigroup.assoc{\isacharparenright}\isanewline +\ \ \ \ {\isachardoublequote}{\isacharparenleft}fst\ {\isacharparenleft}fst\ p\ {\isasymOtimes}\ fst\ q{\isacharcomma}\ snd\ p\ {\isasymOtimes}\ snd\ q{\isacharparenright}\ {\isasymOtimes}\ fst\ r{\isacharcomma}\isanewline +\ \ \ \ \ \ snd\ {\isacharparenleft}fst\ p\ {\isasymOtimes}\ fst\ q{\isacharcomma}\ snd\ p\ {\isasymOtimes}\ snd\ q{\isacharparenright}\ {\isasymOtimes}\ snd\ r{\isacharparenright}\ {\isacharequal}\isanewline +\ \ \ \ \ \ \ {\isacharparenleft}fst\ p\ {\isasymOtimes}\ fst\ {\isacharparenleft}fst\ q\ {\isasymOtimes}\ fst\ r{\isacharcomma}\ snd\ q\ {\isasymOtimes}\ snd\ r{\isacharparenright}{\isacharcomma}\isanewline +\ \ \ \ \ \ \ \ snd\ p\ {\isasymOtimes}\ snd\ {\isacharparenleft}fst\ q\ {\isasymOtimes}\ fst\ r{\isacharcomma}\ snd\ q\ {\isasymOtimes}\ snd\ r{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline +\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ add{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline \isacommand{qed}% \begin{isamarkuptext}% Thus, if we view class instances as ``structures'', then overloaded diff -r 8741740ea6d6 -r 2c208c98f541 doc-src/AxClass/generated/NatClass.tex --- a/doc-src/AxClass/generated/NatClass.tex Mon Aug 21 18:49:38 2000 +0200 +++ b/doc-src/AxClass/generated/NatClass.tex Mon Aug 21 19:03:58 2000 +0200 @@ -1,7 +1,7 @@ \begin{isabelle}% % \isamarkupheader{Defining natural numbers in FOL \label{sec:ex-natclass}} -\isacommand{theory}\ NatClass\ =\ FOL:% +\isacommand{theory}\ NatClass\ {\isacharequal}\ FOL{\isacharcolon}% \begin{isamarkuptext}% \medskip\noindent Axiomatic type classes abstract over exactly one type argument. Thus, any \emph{axiomatic} theory extension where each @@ -13,21 +13,21 @@ \url{http://isabelle.in.tum.de/library/FOL/ex/NatClass.html}}% \end{isamarkuptext}% \isacommand{consts}\isanewline -\ \ zero\ ::\ {\isacharprime}a\ \ \ \ {\isacharparenleft}{\isachardoublequote}0{\isachardoublequote}{\isacharparenright}\isanewline -\ \ Suc\ ::\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\isanewline -\ \ rec\ ::\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\isanewline +\ \ zero\ {\isacharcolon}{\isacharcolon}\ {\isacharprime}a\ \ \ \ {\isacharparenleft}{\isachardoublequote}\isadigit{0}{\isachardoublequote}{\isacharparenright}\isanewline +\ \ Suc\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\isanewline +\ \ rec\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\isanewline \isanewline \isacommand{axclass}\isanewline \ \ nat\ {\isacharless}\ {\isachardoublequote}term{\isachardoublequote}\isanewline -\ \ induct:\ \ \ \ \ {\isachardoublequote}P{\isacharparenleft}0{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymAnd}x.\ P{\isacharparenleft}x{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharparenleft}Suc{\isacharparenleft}x{\isacharparenright}{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharparenleft}n{\isacharparenright}{\isachardoublequote}\isanewline -\ \ Suc{\isacharunderscore}inject:\ {\isachardoublequote}Suc{\isacharparenleft}m{\isacharparenright}\ =\ Suc{\isacharparenleft}n{\isacharparenright}\ {\isasymLongrightarrow}\ m\ =\ n{\isachardoublequote}\isanewline -\ \ Suc{\isacharunderscore}neq{\isacharunderscore}0:\ \ {\isachardoublequote}Suc{\isacharparenleft}m{\isacharparenright}\ =\ 0\ {\isasymLongrightarrow}\ R{\isachardoublequote}\isanewline -\ \ rec{\isacharunderscore}0:\ \ \ \ \ \ {\isachardoublequote}rec{\isacharparenleft}0,\ a,\ f{\isacharparenright}\ =\ a{\isachardoublequote}\isanewline -\ \ rec{\isacharunderscore}Suc:\ \ \ \ {\isachardoublequote}rec{\isacharparenleft}Suc{\isacharparenleft}m{\isacharparenright},\ a,\ f{\isacharparenright}\ =\ f{\isacharparenleft}m,\ rec{\isacharparenleft}m,\ a,\ f{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline +\ \ induct{\isacharcolon}\ \ \ \ \ {\isachardoublequote}P{\isacharparenleft}\isadigit{0}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ P{\isacharparenleft}x{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharparenleft}Suc{\isacharparenleft}x{\isacharparenright}{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharparenleft}n{\isacharparenright}{\isachardoublequote}\isanewline +\ \ Suc{\isacharunderscore}inject{\isacharcolon}\ {\isachardoublequote}Suc{\isacharparenleft}m{\isacharparenright}\ {\isacharequal}\ Suc{\isacharparenleft}n{\isacharparenright}\ {\isasymLongrightarrow}\ m\ {\isacharequal}\ n{\isachardoublequote}\isanewline +\ \ Suc{\isacharunderscore}neq{\isacharunderscore}\isadigit{0}{\isacharcolon}\ \ {\isachardoublequote}Suc{\isacharparenleft}m{\isacharparenright}\ {\isacharequal}\ \isadigit{0}\ {\isasymLongrightarrow}\ R{\isachardoublequote}\isanewline +\ \ rec{\isacharunderscore}\isadigit{0}{\isacharcolon}\ \ \ \ \ \ {\isachardoublequote}rec{\isacharparenleft}\isadigit{0}{\isacharcomma}\ a{\isacharcomma}\ f{\isacharparenright}\ {\isacharequal}\ a{\isachardoublequote}\isanewline +\ \ rec{\isacharunderscore}Suc{\isacharcolon}\ \ \ \ {\isachardoublequote}rec{\isacharparenleft}Suc{\isacharparenleft}m{\isacharparenright}{\isacharcomma}\ a{\isacharcomma}\ f{\isacharparenright}\ {\isacharequal}\ f{\isacharparenleft}m{\isacharcomma}\ rec{\isacharparenleft}m{\isacharcomma}\ a{\isacharcomma}\ f{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline \isanewline \isacommand{constdefs}\isanewline -\ \ add\ ::\ {\isachardoublequote}{\isacharprime}a::nat\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}+{\isachardoublequote}\ 60{\isacharparenright}\isanewline -\ \ {\isachardoublequote}m\ +\ n\ {\isasymequiv}\ rec{\isacharparenleft}m,\ n,\ {\isasymlambda}x\ y.\ Suc{\isacharparenleft}y{\isacharparenright}{\isacharparenright}{\isachardoublequote}% +\ \ add\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isacharcolon}{\isacharcolon}nat\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isacharplus}{\isachardoublequote}\ \isadigit{6}\isadigit{0}{\isacharparenright}\isanewline +\ \ {\isachardoublequote}m\ {\isacharplus}\ n\ {\isasymequiv}\ rec{\isacharparenleft}m{\isacharcomma}\ n{\isacharcomma}\ {\isasymlambda}x\ y{\isachardot}\ Suc{\isacharparenleft}y{\isacharparenright}{\isacharparenright}{\isachardoublequote}% \begin{isamarkuptext}% This is an abstract version of the plain $Nat$ theory in FOL.\footnote{See diff -r 8741740ea6d6 -r 2c208c98f541 doc-src/AxClass/generated/Product.tex --- a/doc-src/AxClass/generated/Product.tex Mon Aug 21 18:49:38 2000 +0200 +++ b/doc-src/AxClass/generated/Product.tex Mon Aug 21 19:03:58 2000 +0200 @@ -1,7 +1,7 @@ \begin{isabelle}% % \isamarkupheader{Syntactic classes} -\isacommand{theory}\ Product\ =\ Main:% +\isacommand{theory}\ Product\ {\isacharequal}\ Main{\isacharcolon}% \begin{isamarkuptext}% \medskip\noindent There is still a feature of Isabelle's type system left that we have not yet discussed. When declaring polymorphic @@ -19,7 +19,7 @@ \isacommand{axclass}\isanewline \ \ product\ {\isacharless}\ {\isachardoublequote}term{\isachardoublequote}\isanewline \isacommand{consts}\isanewline -\ \ product\ ::\ {\isachardoublequote}{\isacharprime}a::product\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymOtimes}{\isachardoublequote}\ 70{\isacharparenright}% +\ \ product\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isacharcolon}{\isacharcolon}product\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymOtimes}{\isachardoublequote}\ \isadigit{7}\isadigit{0}{\isacharparenright}% \begin{isamarkuptext}% Here class $product$ is defined as subclass of $term$ without any additional axioms. This effects in logical equivalence of $product$ @@ -48,10 +48,10 @@ This is done for class $product$ and type $bool$ as follows.% \end{isamarkuptext}% -\isacommand{instance}\ bool\ ::\ product\isanewline +\isacommand{instance}\ bool\ {\isacharcolon}{\isacharcolon}\ product\isanewline \ \ \isacommand{by}\ intro{\isacharunderscore}classes\isanewline \isacommand{defs}\ {\isacharparenleft}\isakeyword{overloaded}{\isacharparenright}\isanewline -\ \ product{\isacharunderscore}bool{\isacharunderscore}def:\ {\isachardoublequote}x\ {\isasymOtimes}\ y\ {\isasymequiv}\ x\ {\isasymand}\ y{\isachardoublequote}% +\ \ product{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}x\ {\isasymOtimes}\ y\ {\isasymequiv}\ x\ {\isasymand}\ y{\isachardoublequote}% \begin{isamarkuptext}% The definition $prod_bool_def$ becomes syntactically well-formed only after the arity $bool :: product$ is made known to the type checker. diff -r 8741740ea6d6 -r 2c208c98f541 doc-src/AxClass/generated/Semigroups.tex --- a/doc-src/AxClass/generated/Semigroups.tex Mon Aug 21 18:49:38 2000 +0200 +++ b/doc-src/AxClass/generated/Semigroups.tex Mon Aug 21 19:03:58 2000 +0200 @@ -1,7 +1,7 @@ \begin{isabelle}% % \isamarkupheader{Semigroups} -\isacommand{theory}\ Semigroups\ =\ Main:% +\isacommand{theory}\ Semigroups\ {\isacharequal}\ Main{\isacharcolon}% \begin{isamarkuptext}% \medskip\noindent An axiomatic type class is simply a class of types that all meet certain properties, which are also called \emph{class @@ -15,10 +15,10 @@ semigroups.% \end{isamarkuptext}% \isacommand{consts}\isanewline -\ \ times\ ::\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymOtimes}{\isachardoublequote}\ 70{\isacharparenright}\isanewline +\ \ times\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymOtimes}{\isachardoublequote}\ \isadigit{7}\isadigit{0}{\isacharparenright}\isanewline \isacommand{axclass}\isanewline \ \ semigroup\ {\isacharless}\ {\isachardoublequote}term{\isachardoublequote}\isanewline -\ \ assoc:\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymOtimes}\ y{\isacharparenright}\ {\isasymOtimes}\ z\ =\ x\ {\isasymOtimes}\ {\isacharparenleft}y\ {\isasymOtimes}\ z{\isacharparenright}{\isachardoublequote}% +\ \ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymOtimes}\ y{\isacharparenright}\ {\isasymOtimes}\ z\ {\isacharequal}\ x\ {\isasymOtimes}\ {\isacharparenleft}y\ {\isasymOtimes}\ z{\isacharparenright}{\isachardoublequote}% \begin{isamarkuptext}% \noindent Above we have first declared a polymorphic constant $\TIMES :: \alpha \To \alpha \To \alpha$ and then defined the class @@ -37,10 +37,10 @@ to semigroups $(\tau, \TIMES^\tau)$.% \end{isamarkuptext}% \isacommand{consts}\isanewline -\ \ plus\ ::\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymOplus}{\isachardoublequote}\ 70{\isacharparenright}\isanewline +\ \ plus\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymOplus}{\isachardoublequote}\ \isadigit{7}\isadigit{0}{\isacharparenright}\isanewline \isacommand{axclass}\isanewline \ \ plus{\isacharunderscore}semigroup\ {\isacharless}\ {\isachardoublequote}term{\isachardoublequote}\isanewline -\ \ assoc:\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymOplus}\ y{\isacharparenright}\ {\isasymOplus}\ z\ =\ x\ {\isasymOplus}\ {\isacharparenleft}y\ {\isasymOplus}\ z{\isacharparenright}{\isachardoublequote}% +\ \ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymOplus}\ y{\isacharparenright}\ {\isasymOplus}\ z\ {\isacharequal}\ x\ {\isasymOplus}\ {\isacharparenleft}y\ {\isasymOplus}\ z{\isacharparenright}{\isachardoublequote}% \begin{isamarkuptext}% \noindent Even if classes $plus_semigroup$ and $semigroup$ both represent semigroups in a sense, they are certainly not quite the diff -r 8741740ea6d6 -r 2c208c98f541 doc-src/AxClass/generated/isabelle.sty --- a/doc-src/AxClass/generated/isabelle.sty Mon Aug 21 18:49:38 2000 +0200 +++ b/doc-src/AxClass/generated/isabelle.sty Mon Aug 21 19:03:58 2000 +0200 @@ -9,6 +9,7 @@ % isabelle environments \newcommand{\isastyle}{\small\tt\slshape} +\newcommand{\isastyleminor}{\small\tt\slshape} \newcommand{\isastyletext}{\normalsize\rm} \newcommand{\isastyletxt}{\rm} \newcommand{\isastylecmt}{\rm} @@ -20,10 +21,13 @@ \isa@parskip\parskip\parskip0pt% \isastyle}{} -\newcommand{\isa}[1]{\emph{\isastyle #1}} +\newcommand{\isa}[1]{\emph{\isastyleminor #1}} \newcommand{\isanewline}{\mbox{}\\\mbox{}} +\newcommand{\isadigit}[1]{#1} +\chardef\isacharbang=`\! +\chardef\isachardoublequote=`\" \chardef\isacharhash=`\# \chardef\isachardollar=`\$ \chardef\isacharpercent=`\% @@ -32,15 +36,24 @@ \chardef\isacharparenleft=`\( \chardef\isacharparenright=`\) \chardef\isacharasterisk=`\* +\chardef\isacharplus=`\+ +\chardef\isacharcomma=`\, \chardef\isacharminus=`\- +\chardef\isachardot=`\. +\chardef\isacharslash=`\/ +\chardef\isacharcolon=`\: +\chardef\isacharsemicolon=`\; \chardef\isacharless=`\< +\chardef\isacharequal=`\= \chardef\isachargreater=`\> +\chardef\isacharquery=`\? +\chardef\isacharat=`\@ \chardef\isacharbrackleft=`\[ -\chardef\isachardoublequote=`\" \chardef\isacharbackslash=`\\ \chardef\isacharbrackright=`\] \chardef\isacharcircum=`\^ \chardef\isacharunderscore=`\_ +\chardef\isacharbackquote=`\` \chardef\isacharbraceleft=`\{ \chardef\isacharbar=`\| \chardef\isacharbraceright=`\} @@ -49,10 +62,10 @@ % keyword and section markup -\newcommand{\isacommand}[1]{\emph{\bf #1}} -\newcommand{\isakeyword}[1]{\emph{\bf #1}} -\newcommand{\isabeginblock}{\isakeyword{\{}} -\newcommand{\isaendblock}{\isakeyword{\}}} +\newcommand{\isakeyword}[1] +{\emph{\bf\def\isachardot{.}\def\isacharunderscore{-}% +\def\isacharbraceleft{\{}\def\isacharbraceright{\}}#1}} +\newcommand{\isacommand}[1]{\isakeyword{#1}} \newcommand{\isamarkupheader}[1]{\section{#1}} \newcommand{\isamarkupchapter}[1]{\chapter{#1}} @@ -72,23 +85,34 @@ % alternative styles -- default is "tt" \newcommand{\isabellestyle}{} -\def\isabellestyle#1{\csname isasetup#1\endcsname} +\def\isabellestyle#1{\csname isabellestyle#1\endcsname} -\newcommand{\isasetupit}{% -\renewcommand{\isastyle}{\small\itshape}% -\renewcommand{\isastyletext}{\normalsize\rm}% -\renewcommand{\isastyletxt}{\rm}% -\renewcommand{\isastylecmt}{\rm}% -\renewcommand{\isachardollar}{\textsl{\$}}% -\renewcommand{\isacharampersand}{\textsl{\&}}% -\renewcommand{\isacharprime}{$'$}% +\newcommand{\isabellestyleit}{% +\renewcommand{\isastyle}{\small\it}% +\renewcommand{\isastyleminor}{\it}% +%\renewcommand{\isadigit}[1]{\emph{$##1$}} +\renewcommand{\isacharbang}{\emph{$!$}}% +\renewcommand{\isachardoublequote}{}% +\renewcommand{\isacharhash}{\emph{$\#$}}% +\renewcommand{\isachardollar}{\emph{$\$$}}% +\renewcommand{\isacharpercent}{\emph{$\%$}}% +\renewcommand{\isacharampersand}{\emph{$\&$}}% +\renewcommand{\isacharprime}{$\mskip2mu{'}\mskip-2mu$}% \renewcommand{\isacharparenleft}{\emph{$($}}% \renewcommand{\isacharparenright}{\emph{$)$}}% \renewcommand{\isacharasterisk}{\emph{$*$}}% +\renewcommand{\isacharplus}{\emph{$+$}}% +\renewcommand{\isacharcomma}{\emph{$\mathord,$}}% \renewcommand{\isacharminus}{\emph{$-$}}% +\renewcommand{\isachardot}{\emph{$\mathord.$}}% +\renewcommand{\isacharslash}{\emph{$/$}}% +\renewcommand{\isacharcolon}{\emph{$\mathord:$}}% +\renewcommand{\isacharsemicolon}{\emph{$\mathord;$}}% \renewcommand{\isacharless}{\emph{$<$}}% +\renewcommand{\isacharequal}{\emph{$=$}}% \renewcommand{\isachargreater}{\emph{$>$}}% -\renewcommand{\isachardoublequote}{}% +%\renewcommand{\isacharquery}{\emph{$\mathord?$}}% +\renewcommand{\isacharat}{\emph{$@$}}% \renewcommand{\isacharbrackleft}{\emph{$[$}}% \renewcommand{\isacharbrackright}{\emph{$]$}}% \renewcommand{\isacharunderscore}{-}% @@ -97,5 +121,4 @@ \renewcommand{\isacharbraceright}{\emph{$\}$}}% } -\newcommand{\isasetupsl}{\isasetupit\renewcommand{\isastyle}{\small\slshape}} - +\newcommand{\isabellestylesl}{\isabellestyleit\renewcommand{\isastyle}{\small\slshape}}