diff -r b05331869f79 -r 7237c6497cb1 doc-src/AxClass/generated/Group.tex --- a/doc-src/AxClass/generated/Group.tex Mon Dec 03 20:59:29 2001 +0100 +++ b/doc-src/AxClass/generated/Group.tex Mon Dec 03 20:59:57 2001 +0100 @@ -30,25 +30,25 @@ \isacommand{consts}\isanewline \ \ times\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymodot}{\isachardoublequote}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline \ \ invers\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\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}\isamarkupfalse% +\ \ one\ {\isacharcolon}{\isacharcolon}\ {\isacharprime}a\ \ \ \ {\isacharparenleft}{\isachardoublequote}{\isasymone}{\isachardoublequote}{\isacharparenright}\isamarkupfalse% % \begin{isamarkuptext}% \noindent Next we define class \isa{monoid} of monoids with - operations \isa{{\isasymodot}} and \isa{{\isasymunit}}. Note that multiple class + operations \isa{{\isasymodot}} and \isa{{\isasymone}}. Note that multiple class axioms are allowed for user convenience --- they simply represent the conjunction of their respective universal closures.% \end{isamarkuptext}% \isamarkuptrue% \isacommand{axclass}\ monoid\ {\isasymsubseteq}\ type\isanewline \ \ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymodot}\ y{\isacharparenright}\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequote}\isanewline -\ \ left{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}{\isasymunit}\ {\isasymodot}\ x\ {\isacharequal}\ x{\isachardoublequote}\isanewline -\ \ right{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymunit}\ {\isacharequal}\ x{\isachardoublequote}\isamarkupfalse% +\ \ left{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}{\isasymone}\ {\isasymodot}\ x\ {\isacharequal}\ x{\isachardoublequote}\isanewline +\ \ right{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymone}\ {\isacharequal}\ x{\isachardoublequote}\isamarkupfalse% % \begin{isamarkuptext}% \noindent So class \isa{monoid} contains exactly those types - \isa{{\isasymtau}} where \isa{{\isasymodot}\ {\isasymColon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}} and \isa{{\isasymunit}\ {\isasymColon}\ {\isasymtau}} + \isa{{\isasymtau}} where \isa{{\isasymodot}\ {\isasymColon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}} and \isa{{\isasymone}\ {\isasymColon}\ {\isasymtau}} are specified appropriately, such that \isa{{\isasymodot}} is associative and - \isa{{\isasymunit}} is a left and right unit element for the \isa{{\isasymodot}} + \isa{{\isasymone}} is a left and right unit element for the \isa{{\isasymodot}} operation.% \end{isamarkuptext}% \isamarkuptrue% @@ -65,8 +65,8 @@ \isanewline \isamarkupfalse% \isacommand{axclass}\ group\ {\isasymsubseteq}\ semigroup\isanewline -\ \ left{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}{\isasymunit}\ {\isasymodot}\ x\ {\isacharequal}\ x{\isachardoublequote}\isanewline -\ \ left{\isacharunderscore}inverse{\isacharcolon}\ {\isachardoublequote}x{\isasyminv}\ {\isasymodot}\ x\ {\isacharequal}\ {\isasymunit}{\isachardoublequote}\isanewline +\ \ left{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}{\isasymone}\ {\isasymodot}\ x\ {\isacharequal}\ x{\isachardoublequote}\isanewline +\ \ left{\isacharunderscore}inverse{\isacharcolon}\ {\isachardoublequote}x{\isasyminv}\ {\isasymodot}\ x\ {\isacharequal}\ {\isasymone}{\isachardoublequote}\isanewline \isanewline \isamarkupfalse% \isacommand{axclass}\ agroup\ {\isasymsubseteq}\ group\isanewline @@ -98,16 +98,16 @@ well-known laws of general groups.% \end{isamarkuptext}% \isamarkuptrue% -\isacommand{theorem}\ group{\isacharunderscore}right{\isacharunderscore}inverse{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ x{\isasyminv}\ {\isacharequal}\ {\isacharparenleft}{\isasymunit}{\isasymColon}{\isacharprime}a{\isasymColon}group{\isacharparenright}{\isachardoublequote}\isanewline +\isacommand{theorem}\ group{\isacharunderscore}right{\isacharunderscore}inverse{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ x{\isasyminv}\ {\isacharequal}\ {\isacharparenleft}{\isasymone}{\isasymColon}{\isacharprime}a{\isasymColon}group{\isacharparenright}{\isachardoublequote}\isanewline \isamarkupfalse% \isacommand{proof}\ {\isacharminus}\isanewline \ \ \isamarkupfalse% -\isacommand{have}\ {\isachardoublequote}x\ {\isasymodot}\ x{\isasyminv}\ {\isacharequal}\ {\isasymunit}\ {\isasymodot}\ {\isacharparenleft}x\ {\isasymodot}\ x{\isasyminv}{\isacharparenright}{\isachardoublequote}\isanewline +\isacommand{have}\ {\isachardoublequote}x\ {\isasymodot}\ x{\isasyminv}\ {\isacharequal}\ {\isasymone}\ {\isasymodot}\ {\isacharparenleft}x\ {\isasymodot}\ x{\isasyminv}{\isacharparenright}{\isachardoublequote}\isanewline \ \ \ \ \isamarkupfalse% \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline \ \ \isamarkupfalse% \isacommand{also}\ \isamarkupfalse% -\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymunit}\ {\isasymodot}\ x\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline +\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymone}\ {\isasymodot}\ x\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline \ \ \ \ \isamarkupfalse% \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline \ \ \isamarkupfalse% @@ -122,12 +122,12 @@ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline \ \ \isamarkupfalse% \isacommand{also}\ \isamarkupfalse% -\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ {\isasymunit}\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline +\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ {\isasymone}\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline \ \ \ \ \isamarkupfalse% \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline \ \ \isamarkupfalse% \isacommand{also}\ \isamarkupfalse% -\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ {\isacharparenleft}{\isasymunit}\ {\isasymodot}\ x{\isasyminv}{\isacharparenright}{\isachardoublequote}\isanewline +\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ {\isacharparenleft}{\isasymone}\ {\isasymodot}\ x{\isasyminv}{\isacharparenright}{\isachardoublequote}\isanewline \ \ \ \ \isamarkupfalse% \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline \ \ \isamarkupfalse% @@ -137,7 +137,7 @@ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline \ \ \isamarkupfalse% \isacommand{also}\ \isamarkupfalse% -\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymunit}{\isachardoublequote}\isanewline +\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymone}{\isachardoublequote}\isanewline \ \ \ \ \isamarkupfalse% \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline \ \ \isamarkupfalse% @@ -152,11 +152,11 @@ much easier.% \end{isamarkuptext}% \isamarkuptrue% -\isacommand{theorem}\ group{\isacharunderscore}right{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymunit}\ {\isacharequal}\ {\isacharparenleft}x{\isasymColon}{\isacharprime}a{\isasymColon}group{\isacharparenright}{\isachardoublequote}\isanewline +\isacommand{theorem}\ group{\isacharunderscore}right{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymone}\ {\isacharequal}\ {\isacharparenleft}x{\isasymColon}{\isacharprime}a{\isasymColon}group{\isacharparenright}{\isachardoublequote}\isanewline \isamarkupfalse% \isacommand{proof}\ {\isacharminus}\isanewline \ \ \isamarkupfalse% -\isacommand{have}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymunit}\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}x{\isasyminv}\ {\isasymodot}\ x{\isacharparenright}{\isachardoublequote}\isanewline +\isacommand{have}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymone}\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}x{\isasyminv}\ {\isasymodot}\ x{\isacharparenright}{\isachardoublequote}\isanewline \ \ \ \ \isamarkupfalse% \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline \ \ \isamarkupfalse% @@ -166,7 +166,7 @@ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline \ \ \isamarkupfalse% \isacommand{also}\ \isamarkupfalse% -\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymunit}\ {\isasymodot}\ x{\isachardoublequote}\isanewline +\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymone}\ {\isasymodot}\ x{\isachardoublequote}\isanewline \ \ \ \ \isamarkupfalse% \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isacharunderscore}right{\isacharunderscore}inverse{\isacharparenright}\isanewline \ \ \isamarkupfalse% @@ -251,11 +251,11 @@ \ \ \ \ \isamarkupfalse% \isacommand{by}\ {\isacharparenleft}rule\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline \ \ \isamarkupfalse% -\isacommand{show}\ {\isachardoublequote}{\isasymunit}\ {\isasymodot}\ x\ {\isacharequal}\ x{\isachardoublequote}\isanewline +\isacommand{show}\ {\isachardoublequote}{\isasymone}\ {\isasymodot}\ x\ {\isacharequal}\ x{\isachardoublequote}\isanewline \ \ \ \ \isamarkupfalse% \isacommand{by}\ {\isacharparenleft}rule\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline \ \ \isamarkupfalse% -\isacommand{show}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymunit}\ {\isacharequal}\ x{\isachardoublequote}\isanewline +\isacommand{show}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymone}\ {\isacharequal}\ x{\isachardoublequote}\isanewline \ \ \ \ \isamarkupfalse% \isacommand{by}\ {\isacharparenleft}rule\ group{\isacharunderscore}right{\isacharunderscore}unit{\isacharparenright}\isanewline \isamarkupfalse% @@ -289,13 +289,13 @@ \medskip As a typical example, we show that type \isa{bool} with exclusive-or as \isa{{\isasymodot}} operation, identity as \isa{{\isasyminv}}, and - \isa{False} as \isa{{\isasymunit}} forms an Abelian group.% + \isa{False} as \isa{{\isasymone}} forms an Abelian group.% \end{isamarkuptext}% \isamarkuptrue% \isacommand{defs}\ {\isacharparenleft}\isakeyword{overloaded}{\isacharparenright}\isanewline \ \ times{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ 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}\isamarkupfalse% +\ \ unit{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}{\isasymone}\ {\isasymequiv}\ False{\isachardoublequote}\isamarkupfalse% % \begin{isamarkuptext}% \medskip It is important to note that above $\DEFS$ are just @@ -342,9 +342,9 @@ \medskip We could now also instantiate our group theory classes to many other concrete types. For example, \isa{int\ {\isasymColon}\ agroup} (e.g.\ by defining \isa{{\isasymodot}} as addition, \isa{{\isasyminv}} as negation - and \isa{{\isasymunit}} as zero) or \isa{list\ {\isasymColon}\ {\isacharparenleft}type{\isacharparenright}\ semigroup} + and \isa{{\isasymone}} as zero) or \isa{list\ {\isasymColon}\ {\isacharparenleft}type{\isacharparenright}\ semigroup} (e.g.\ if \isa{{\isasymodot}} is defined as list append). Thus, the - characteristic constants \isa{{\isasymodot}}, \isa{{\isasyminv}}, \isa{{\isasymunit}} + characteristic constants \isa{{\isasymodot}}, \isa{{\isasyminv}}, \isa{{\isasymone}} really become overloaded, i.e.\ have different meanings on different types.% \end{isamarkuptext}%