diff -r 619531d87ce4 -r 4e2ee88276d2 doc-src/TutorialI/document/Axioms.tex --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/doc-src/TutorialI/document/Axioms.tex Thu Jul 26 19:59:06 2012 +0200 @@ -0,0 +1,487 @@ +% +\begin{isabellebody}% +\def\isabellecontext{Axioms}% +% +\isadelimtheory +% +\endisadelimtheory +% +\isatagtheory +% +\endisatagtheory +{\isafoldtheory}% +% +\isadelimtheory +% +\endisadelimtheory +% +\isamarkupsubsection{Axioms% +} +\isamarkuptrue% +% +\begin{isamarkuptext}% +Attaching axioms to our classes lets us reason on the level of +classes. The results will be applicable to all types in a class, just +as in axiomatic mathematics. + +\begin{warn} +Proofs in this section use structured \emph{Isar} proofs, which are not +covered in this tutorial; but see \cite{Nipkow-TYPES02}.% +\end{warn}% +\end{isamarkuptext}% +\isamarkuptrue% +% +\isamarkupsubsubsection{Semigroups% +} +\isamarkuptrue% +% +\begin{isamarkuptext}% +We specify \emph{semigroups} as subclass of \isa{plus}:% +\end{isamarkuptext}% +\isamarkuptrue% +\isacommand{class}\isamarkupfalse% +\ semigroup\ {\isaliteral{3D}{\isacharequal}}\ plus\ {\isaliteral{2B}{\isacharplus}}\isanewline +\ \ \isakeyword{assumes}\ assoc{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ z\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}% +\begin{isamarkuptext}% +\noindent This \hyperlink{command.class}{\mbox{\isa{\isacommand{class}}}} specification requires that +all instances of \isa{semigroup} obey \hyperlink{fact.assoc:}{\mbox{\isa{assoc{\isaliteral{3A}{\isacharcolon}}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x\ y\ z\ {\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}\ {\isaliteral{27}{\isacharprime}}a{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}semigroup{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ z\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}. + +We can use this class axiom to derive further abstract theorems +relative to class \isa{semigroup}:% +\end{isamarkuptext}% +\isamarkuptrue% +\isacommand{lemma}\isamarkupfalse% +\ assoc{\isaliteral{5F}{\isacharunderscore}}left{\isaliteral{3A}{\isacharcolon}}\isanewline +\ \ \isakeyword{fixes}\ x\ y\ z\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}semigroup{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\ \ \isakeyword{shows}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ z{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +% +\isadelimproof +\ \ % +\endisadelimproof +% +\isatagproof +\isacommand{using}\isamarkupfalse% +\ assoc\ \isacommand{by}\isamarkupfalse% +\ {\isaliteral{28}{\isacharparenleft}}rule\ sym{\isaliteral{29}{\isacharparenright}}% +\endisatagproof +{\isafoldproof}% +% +\isadelimproof +% +\endisadelimproof +% +\begin{isamarkuptext}% +\noindent The \isa{semigroup} constraint on type \isa{{\isaliteral{27}{\isacharprime}}a} restricts instantiations of \isa{{\isaliteral{27}{\isacharprime}}a} to types of class +\isa{semigroup} and during the proof enables us to use the fact +\hyperlink{fact.assoc}{\mbox{\isa{assoc}}} whose type parameter is itself constrained to class +\isa{semigroup}. The main advantage of classes is that theorems +can be proved in the abstract and freely reused for each instance. + +On instantiation, we have to give a proof that the given operations +obey the class axioms:% +\end{isamarkuptext}% +\isamarkuptrue% +\isacommand{instantiation}\isamarkupfalse% +\ nat\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ semigroup\isanewline +\isakeyword{begin}\isanewline +\isanewline +\isacommand{instance}\isamarkupfalse% +% +\isadelimproof +\ % +\endisadelimproof +% +\isatagproof +\isacommand{proof}\isamarkupfalse% +% +\begin{isamarkuptxt}% +\noindent The proof opens with a default proof step, which for +instance judgements invokes method \hyperlink{method.intro-classes}{\mbox{\isa{intro{\isaliteral{5F}{\isacharunderscore}}classes}}}.% +\end{isamarkuptxt}% +\isamarkuptrue% +\ \ \isacommand{fix}\isamarkupfalse% +\ m\ n\ q\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ nat\isanewline +\ \ \isacommand{show}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}m\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ q\ {\isaliteral{3D}{\isacharequal}}\ m\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ q{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\ \ \ \ \isacommand{by}\isamarkupfalse% +\ {\isaliteral{28}{\isacharparenleft}}induct\ m{\isaliteral{29}{\isacharparenright}}\ simp{\isaliteral{5F}{\isacharunderscore}}all\isanewline +\isacommand{qed}\isamarkupfalse% +% +\endisatagproof +{\isafoldproof}% +% +\isadelimproof +% +\endisadelimproof +\isanewline +\isanewline +\isacommand{end}\isamarkupfalse% +% +\begin{isamarkuptext}% +\noindent Again, the interesting things enter the stage with +parametric types:% +\end{isamarkuptext}% +\isamarkuptrue% +\isacommand{instantiation}\isamarkupfalse% +\ prod\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{28}{\isacharparenleft}}semigroup{\isaliteral{2C}{\isacharcomma}}\ semigroup{\isaliteral{29}{\isacharparenright}}\ semigroup\isanewline +\isakeyword{begin}\isanewline +\isanewline +\isacommand{instance}\isamarkupfalse% +% +\isadelimproof +\ % +\endisadelimproof +% +\isatagproof +\isacommand{proof}\isamarkupfalse% +\isanewline +\ \ \isacommand{fix}\isamarkupfalse% +\ p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}\ p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{3}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}semigroup\ {\isaliteral{5C3C74696D65733E}{\isasymtimes}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}semigroup{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\ \ \isacommand{show}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{3}}\ {\isaliteral{3D}{\isacharequal}}\ p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ {\isaliteral{28}{\isacharparenleft}}p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{3}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\ \ \ \ \isacommand{by}\isamarkupfalse% +\ {\isaliteral{28}{\isacharparenleft}}cases\ p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ cases\ p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}{\isaliteral{2C}{\isacharcomma}}\ cases\ p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{3}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ assoc{\isaliteral{29}{\isacharparenright}}% +\begin{isamarkuptxt}% +\noindent Associativity of product semigroups is established +using the hypothetical associativity \hyperlink{fact.assoc}{\mbox{\isa{assoc}}} of the type +components, which holds due to the \isa{semigroup} constraints +imposed on the type components by the \hyperlink{command.instance}{\mbox{\isa{\isacommand{instance}}}} proposition. +Indeed, this pattern often occurs with parametric types and type +classes.% +\end{isamarkuptxt}% +\isamarkuptrue% +\isacommand{qed}\isamarkupfalse% +% +\endisatagproof +{\isafoldproof}% +% +\isadelimproof +% +\endisadelimproof +\isanewline +\isanewline +\isacommand{end}\isamarkupfalse% +% +\isamarkupsubsubsection{Monoids% +} +\isamarkuptrue% +% +\begin{isamarkuptext}% +We define a subclass \isa{monoidl} (a semigroup with a +left-hand neutral) by extending \isa{semigroup} with one additional +parameter \isa{neutral} together with its property:% +\end{isamarkuptext}% +\isamarkuptrue% +\isacommand{class}\isamarkupfalse% +\ monoidl\ {\isaliteral{3D}{\isacharequal}}\ semigroup\ {\isaliteral{2B}{\isacharplus}}\isanewline +\ \ \isakeyword{fixes}\ neutral\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C7A65726F3E}{\isasymzero}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}\isanewline +\ \ \isakeyword{assumes}\ neutl{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C7A65726F3E}{\isasymzero}}\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ x\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}% +\begin{isamarkuptext}% +\noindent Again, we prove some instances, by providing +suitable parameter definitions and proofs for the additional +specifications.% +\end{isamarkuptext}% +\isamarkuptrue% +\isacommand{instantiation}\isamarkupfalse% +\ nat\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ monoidl\isanewline +\isakeyword{begin}\isanewline +\isanewline +\isacommand{definition}\isamarkupfalse% +\isanewline +\ \ neutral{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C7A65726F3E}{\isasymzero}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}nat{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\isanewline +\isacommand{instance}\isamarkupfalse% +% +\isadelimproof +\ % +\endisadelimproof +% +\isatagproof +\isacommand{proof}\isamarkupfalse% +\isanewline +\ \ \isacommand{fix}\isamarkupfalse% +\ n\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ nat\isanewline +\ \ \isacommand{show}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C7A65726F3E}{\isasymzero}}\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ n\ {\isaliteral{3D}{\isacharequal}}\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\ \ \ \ \isacommand{unfolding}\isamarkupfalse% +\ neutral{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{5F}{\isacharunderscore}}def\ \isacommand{by}\isamarkupfalse% +\ simp\isanewline +\isacommand{qed}\isamarkupfalse% +% +\endisatagproof +{\isafoldproof}% +% +\isadelimproof +% +\endisadelimproof +\isanewline +\isanewline +\isacommand{end}\isamarkupfalse% +% +\begin{isamarkuptext}% +\noindent In contrast to the examples above, we here have both +specification of class operations and a non-trivial instance proof. + +This covers products as well:% +\end{isamarkuptext}% +\isamarkuptrue% +\isacommand{instantiation}\isamarkupfalse% +\ prod\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{28}{\isacharparenleft}}monoidl{\isaliteral{2C}{\isacharcomma}}\ monoidl{\isaliteral{29}{\isacharparenright}}\ monoidl\isanewline +\isakeyword{begin}\isanewline +\isanewline +\isacommand{definition}\isamarkupfalse% +\isanewline +\ \ neutral{\isaliteral{5F}{\isacharunderscore}}prod{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C7A65726F3E}{\isasymzero}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C7A65726F3E}{\isasymzero}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C7A65726F3E}{\isasymzero}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\isanewline +\isacommand{instance}\isamarkupfalse% +% +\isadelimproof +\ % +\endisadelimproof +% +\isatagproof +\isacommand{proof}\isamarkupfalse% +\isanewline +\ \ \isacommand{fix}\isamarkupfalse% +\ p\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}monoidl\ {\isaliteral{5C3C74696D65733E}{\isasymtimes}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}monoidl{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\ \ \isacommand{show}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C7A65726F3E}{\isasymzero}}\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ p\ {\isaliteral{3D}{\isacharequal}}\ p{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\ \ \ \ \isacommand{by}\isamarkupfalse% +\ {\isaliteral{28}{\isacharparenleft}}cases\ p{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ neutral{\isaliteral{5F}{\isacharunderscore}}prod{\isaliteral{5F}{\isacharunderscore}}def\ neutl{\isaliteral{29}{\isacharparenright}}\isanewline +\isacommand{qed}\isamarkupfalse% +% +\endisatagproof +{\isafoldproof}% +% +\isadelimproof +% +\endisadelimproof +\isanewline +\isanewline +\isacommand{end}\isamarkupfalse% +% +\begin{isamarkuptext}% +\noindent Fully-fledged monoids are modelled by another +subclass which does not add new parameters but tightens the +specification:% +\end{isamarkuptext}% +\isamarkuptrue% +\isacommand{class}\isamarkupfalse% +\ monoid\ {\isaliteral{3D}{\isacharequal}}\ monoidl\ {\isaliteral{2B}{\isacharplus}}\isanewline +\ \ \isakeyword{assumes}\ neutr{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ {\isaliteral{5C3C7A65726F3E}{\isasymzero}}\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}% +\begin{isamarkuptext}% +\noindent Corresponding instances for \isa{nat} and products +are left as an exercise to the reader.% +\end{isamarkuptext}% +\isamarkuptrue% +% +\isamarkupsubsubsection{Groups% +} +\isamarkuptrue% +% +\begin{isamarkuptext}% +\noindent To finish our small algebra example, we add a \isa{group} class:% +\end{isamarkuptext}% +\isamarkuptrue% +\isacommand{class}\isamarkupfalse% +\ group\ {\isaliteral{3D}{\isacharequal}}\ monoidl\ {\isaliteral{2B}{\isacharplus}}\isanewline +\ \ \isakeyword{fixes}\ inv\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6469763E}{\isasymdiv}}\ {\isaliteral{5F}{\isacharunderscore}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isadigit{8}}{\isadigit{1}}{\isaliteral{5D}{\isacharbrackright}}\ {\isadigit{8}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline +\ \ \isakeyword{assumes}\ invl{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6469763E}{\isasymdiv}}\ x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C7A65726F3E}{\isasymzero}}{\isaliteral{22}{\isachardoublequoteclose}}% +\begin{isamarkuptext}% +\noindent We continue with a further example for abstract +proofs relative to type classes:% +\end{isamarkuptext}% +\isamarkuptrue% +\isacommand{lemma}\isamarkupfalse% +\ left{\isaliteral{5F}{\isacharunderscore}}cancel{\isaliteral{3A}{\isacharcolon}}\isanewline +\ \ \isakeyword{fixes}\ x\ y\ z\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}group{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\ \ \isakeyword{shows}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ y\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ z\ {\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}\ y\ {\isaliteral{3D}{\isacharequal}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +% +\isadelimproof +% +\endisadelimproof +% +\isatagproof +\isacommand{proof}\isamarkupfalse% +\isanewline +\ \ \isacommand{assume}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ y\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\ \ \isacommand{then}\isamarkupfalse% +\ \isacommand{have}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6469763E}{\isasymdiv}}\ x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C6469763E}{\isasymdiv}}\ x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% +\ simp\isanewline +\ \ \isacommand{then}\isamarkupfalse% +\ \isacommand{have}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6469763E}{\isasymdiv}}\ x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ y\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6469763E}{\isasymdiv}}\ x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% +\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ assoc{\isaliteral{29}{\isacharparenright}}\isanewline +\ \ \isacommand{then}\isamarkupfalse% +\ \isacommand{show}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{3D}{\isacharequal}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% +\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ invl\ neutl{\isaliteral{29}{\isacharparenright}}\isanewline +\isacommand{next}\isamarkupfalse% +\isanewline +\ \ \isacommand{assume}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{3D}{\isacharequal}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\ \ \isacommand{then}\isamarkupfalse% +\ \isacommand{show}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ y\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% +\ simp\isanewline +\isacommand{qed}\isamarkupfalse% +% +\endisatagproof +{\isafoldproof}% +% +\isadelimproof +% +\endisadelimproof +% +\begin{isamarkuptext}% +\noindent Any \isa{group} is also a \isa{monoid}; this +can be made explicit by claiming an additional subclass relation, +together with a proof of the logical difference:% +\end{isamarkuptext}% +\isamarkuptrue% +\isacommand{instance}\isamarkupfalse% +\ group\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ monoid\isanewline +% +\isadelimproof +% +\endisadelimproof +% +\isatagproof +\isacommand{proof}\isamarkupfalse% +\isanewline +\ \ \isacommand{fix}\isamarkupfalse% +\ x\isanewline +\ \ \isacommand{from}\isamarkupfalse% +\ invl\ \isacommand{have}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6469763E}{\isasymdiv}}\ x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C7A65726F3E}{\isasymzero}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% +\isanewline +\ \ \isacommand{then}\isamarkupfalse% +\ \isacommand{have}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6469763E}{\isasymdiv}}\ x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ {\isaliteral{5C3C7A65726F3E}{\isasymzero}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C6469763E}{\isasymdiv}}\ x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\ \ \ \ \isacommand{by}\isamarkupfalse% +\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ neutl\ invl\ assoc\ {\isaliteral{5B}{\isacharbrackleft}}symmetric{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\isanewline +\ \ \isacommand{then}\isamarkupfalse% +\ \isacommand{show}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C6F706C75733E}{\isasymoplus}}\ {\isaliteral{5C3C7A65726F3E}{\isasymzero}}\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% +\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ left{\isaliteral{5F}{\isacharunderscore}}cancel{\isaliteral{29}{\isacharparenright}}\isanewline +\isacommand{qed}\isamarkupfalse% +% +\endisatagproof +{\isafoldproof}% +% +\isadelimproof +% +\endisadelimproof +% +\begin{isamarkuptext}% +\noindent The proof result is propagated to the type system, +making \isa{group} an instance of \isa{monoid} by adding an +additional edge to the graph of subclass relation; see also +Figure~\ref{fig:subclass}. + +\begin{figure}[htbp] + \begin{center} + \small + \unitlength 0.6mm + \begin{picture}(40,60)(0,0) + \put(20,60){\makebox(0,0){\isa{semigroup}}} + \put(20,40){\makebox(0,0){\isa{monoidl}}} + \put(00,20){\makebox(0,0){\isa{monoid}}} + \put(40,00){\makebox(0,0){\isa{group}}} + \put(20,55){\vector(0,-1){10}} + \put(15,35){\vector(-1,-1){10}} + \put(25,35){\vector(1,-3){10}} + \end{picture} + \hspace{8em} + \begin{picture}(40,60)(0,0) + \put(20,60){\makebox(0,0){\isa{semigroup}}} + \put(20,40){\makebox(0,0){\isa{monoidl}}} + \put(00,20){\makebox(0,0){\isa{monoid}}} + \put(40,00){\makebox(0,0){\isa{group}}} + \put(20,55){\vector(0,-1){10}} + \put(15,35){\vector(-1,-1){10}} + \put(05,15){\vector(3,-1){30}} + \end{picture} + \caption{Subclass relationship of monoids and groups: + before and after establishing the relationship + \isa{group\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ monoid}; transitive edges are left out.} + \label{fig:subclass} + \end{center} +\end{figure}% +\end{isamarkuptext}% +\isamarkuptrue% +% +\isamarkupsubsubsection{Inconsistencies% +} +\isamarkuptrue% +% +\begin{isamarkuptext}% +The reader may be wondering what happens if we attach an +inconsistent set of axioms to a class. So far we have always avoided +to add new axioms to HOL for fear of inconsistencies and suddenly it +seems that we are throwing all caution to the wind. So why is there no +problem? + +The point is that by construction, all type variables in the axioms of +a \isacommand{class} are automatically constrained with the class +being defined (as shown for axiom \isa{refl} above). These +constraints are always carried around and Isabelle takes care that +they are never lost, unless the type variable is instantiated with a +type that has been shown to belong to that class. Thus you may be able +to prove \isa{False} from your axioms, but Isabelle will remind you +that this theorem has the hidden hypothesis that the class is +non-empty. + +Even if each individual class is consistent, intersections of +(unrelated) classes readily become inconsistent in practice. Now we +know this need not worry us.% +\end{isamarkuptext}% +\isamarkuptrue% +% +\isamarkupsubsubsection{Syntactic Classes and Predefined Overloading% +} +\isamarkuptrue% +% +\begin{isamarkuptext}% +In our algebra example, we have started with a \emph{syntactic +class} \isa{plus} which only specifies operations but no axioms; it +would have been also possible to start immediately with class \isa{semigroup}, specifying the \isa{{\isaliteral{5C3C6F706C75733E}{\isasymoplus}}} operation and associativity at +the same time. + +Which approach is more appropriate depends. Usually it is more +convenient to introduce operations and axioms in the same class: then +the type checker will automatically insert the corresponding class +constraints whenever the operations occur, reducing the need of manual +annotations. However, when operations are decorated with popular +syntax, syntactic classes can be an option to re-use the syntax in +different contexts; this is indeed the way most overloaded constants +in HOL are introduced, of which the most important are listed in +Table~\ref{tab:overloading} in the appendix. Section +\ref{sec:numeric-classes} covers a range of corresponding classes +\emph{with} axioms. + +Further note that classes may contain axioms but \emph{no} operations. +An example is class \isa{finite} from theory \isa{Finite{\isaliteral{5F}{\isacharunderscore}}Set} +which specifies a type to be finite: \isa{{\isaliteral{22}{\isachardoublequote}}finite\ {\isaliteral{28}{\isacharparenleft}}UNIV\ {\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}\ {\isaliteral{27}{\isacharprime}}a{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}finite\ set{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}.% +\end{isamarkuptext}% +\isamarkuptrue% +% +\isadelimtheory +% +\endisadelimtheory +% +\isatagtheory +% +\endisatagtheory +{\isafoldtheory}% +% +\isadelimtheory +% +\endisadelimtheory +\end{isabellebody}% +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "root" +%%% End: