--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/TutorialI/document/Records.tex Thu Jul 26 19:59:06 2012 +0200
@@ -0,0 +1,665 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{Records}%
+%
+\isamarkupheader{Records \label{sec:records}%
+}
+\isamarkuptrue%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\begin{isamarkuptext}%
+\index{records|(}%
+ Records are familiar from programming languages. A record of $n$
+ fields is essentially an $n$-tuple, but the record's components have
+ names, which can make expressions easier to read and reduces the
+ risk of confusing one field for another.
+
+ A record of Isabelle/HOL covers a collection of fields, with select
+ and update operations. Each field has a specified type, which may
+ be polymorphic. The field names are part of the record type, and
+ the order of the fields is significant --- as it is in Pascal but
+ not in Standard ML. If two different record types have field names
+ in common, then the ambiguity is resolved in the usual way, by
+ qualified names.
+
+ Record types can also be defined by extending other record types.
+ Extensible records make use of the reserved pseudo-field \cdx{more},
+ which is present in every record type. Generic record operations
+ work on all possible extensions of a given type scheme; polymorphism
+ takes care of structural sub-typing behind the scenes. There are
+ also explicit coercion functions between fixed record types.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Record Basics%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Record types are not primitive in Isabelle and have a delicate
+ internal representation \cite{NaraschewskiW-TPHOLs98}, based on
+ nested copies of the primitive product type. A \commdx{record}
+ declaration introduces a new record type scheme by specifying its
+ fields, which are packaged internally to hold up the perception of
+ the record as a distinguished entity. Here is a simple example:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{record}\isamarkupfalse%
+\ point\ {\isaliteral{3D}{\isacharequal}}\isanewline
+\ \ Xcoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int\isanewline
+\ \ Ycoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int%
+\begin{isamarkuptext}%
+\noindent
+ Records of type \isa{point} have two fields named \isa{Xcoord}
+ and \isa{Ycoord}, both of type~\isa{int}. We now define a
+ constant of type \isa{point}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ pt{\isadigit{1}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ point\ \isakeyword{where}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}pt{\isadigit{1}}\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{7C}{\isacharbar}}\ Xcoord\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{9}}{\isadigit{9}}{\isadigit{9}}{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{2}}{\isadigit{3}}\ {\isaliteral{7C}{\isacharbar}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+\noindent
+ We see above the ASCII notation for record brackets. You can also
+ use the symbolic brackets \isa{{\isaliteral{5C3C6C706172723E}{\isasymlparr}}} and \isa{{\isaliteral{5C3C72706172723E}{\isasymrparr}}}. Record type
+ expressions can be also written directly with individual fields.
+ The type name above is merely an abbreviation.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ pt{\isadigit{2}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}pt{\isadigit{2}}\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{2D}{\isacharminus}}{\isadigit{4}}{\isadigit{5}}{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{9}}{\isadigit{7}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+For each field, there is a \emph{selector}\index{selector!record}
+ function of the same name. For example, if \isa{p} has type \isa{point} then \isa{Xcoord\ p} denotes the value of the \isa{Xcoord} field of~\isa{p}. Expressions involving field selection
+ of explicit records are simplified automatically:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}Xcoord\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ simp%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+The \emph{update}\index{update!record} operation is functional. For
+ example, \isa{p{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}} is a record whose \isa{Xcoord}
+ value is zero and whose \isa{Ycoord} value is copied from~\isa{p}. Updates of explicit records are also simplified automatically:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{3D}{\isacharequal}}\isanewline
+\ \ \ \ \ \ \ \ \ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ simp%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+\begin{warn}
+ Field names are declared as constants and can no longer be used as
+ variables. It would be unwise, for example, to call the fields of
+ type \isa{point} simply \isa{x} and~\isa{y}.
+ \end{warn}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Extensible Records and Generic Operations%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\index{records!extensible|(}%
+
+ Now, let us define coloured points (type \isa{cpoint}) to be
+ points extended with a field \isa{col} of type \isa{colour}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{datatype}\isamarkupfalse%
+\ colour\ {\isaliteral{3D}{\isacharequal}}\ Red\ {\isaliteral{7C}{\isacharbar}}\ Green\ {\isaliteral{7C}{\isacharbar}}\ Blue\isanewline
+\isanewline
+\isacommand{record}\isamarkupfalse%
+\ cpoint\ {\isaliteral{3D}{\isacharequal}}\ point\ {\isaliteral{2B}{\isacharplus}}\isanewline
+\ \ col\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ colour%
+\begin{isamarkuptext}%
+\noindent
+ The fields of this new type are \isa{Xcoord}, \isa{Ycoord} and
+ \isa{col}, in that order.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ cpt{\isadigit{1}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ cpoint\ \isakeyword{where}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}cpt{\isadigit{1}}\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{9}}{\isadigit{9}}{\isadigit{9}}{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{2}}{\isadigit{3}}{\isaliteral{2C}{\isacharcomma}}\ col\ {\isaliteral{3D}{\isacharequal}}\ Green{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+We can define generic operations that work on arbitrary
+ instances of a record scheme, e.g.\ covering \isa{point}, \isa{cpoint}, and any further extensions. Every record structure has an
+ implicit pseudo-field, \cdx{more}, that keeps the extension as an
+ explicit value. Its type is declared as completely
+ polymorphic:~\isa{{\isaliteral{27}{\isacharprime}}a}. When a fixed record value is expressed
+ using just its standard fields, the value of \isa{more} is
+ implicitly set to \isa{{\isaliteral{28}{\isacharparenleft}}{\isaliteral{29}{\isacharparenright}}}, the empty tuple, which has type
+ \isa{unit}. Within the record brackets, you can refer to the
+ \isa{more} field by writing ``\isa{{\isaliteral{5C3C646F74733E}{\isasymdots}}}'' (three dots):%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}Xcoord\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ p{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ simp%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+This lemma applies to any record whose first two fields are \isa{Xcoord} and~\isa{Ycoord}. Note that \isa{{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}} is exactly the same as \isa{{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{5C3C72706172723E}{\isasymrparr}}}. Selectors and updates are always polymorphic wrt.\ the
+ \isa{more} part of a record scheme, its value is just ignored (for
+ select) or copied (for update).
+
+ The \isa{more} pseudo-field may be manipulated directly as well,
+ but the identifier needs to be qualified:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}point{\isaliteral{2E}{\isachardot}}more\ cpt{\isadigit{1}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}col\ {\isaliteral{3D}{\isacharequal}}\ Green{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ cpt{\isadigit{1}}{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+\noindent
+ We see that the colour part attached to this \isa{point} is a
+ rudimentary record in its own right, namely \isa{{\isaliteral{5C3C6C706172723E}{\isasymlparr}}col\ {\isaliteral{3D}{\isacharequal}}\ Green{\isaliteral{5C3C72706172723E}{\isasymrparr}}}. In order to select or update \isa{col}, this fragment
+ needs to be put back into the context of the parent type scheme, say
+ as \isa{more} part of another \isa{point}.
+
+ To define generic operations, we need to know a bit more about
+ records. Our definition of \isa{point} above has generated two
+ type abbreviations:
+
+ \medskip
+ \begin{tabular}{l}
+ \isa{point}~\isa{{\isaliteral{3D}{\isacharequal}}}~\isa{{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int{\isaliteral{5C3C72706172723E}{\isasymrparr}}} \\
+ \isa{{\isaliteral{27}{\isacharprime}}a\ point{\isaliteral{5F}{\isacharunderscore}}scheme}~\isa{{\isaliteral{3D}{\isacharequal}}}~\isa{{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a{\isaliteral{5C3C72706172723E}{\isasymrparr}}} \\
+ \end{tabular}
+ \medskip
+
+\noindent
+ Type \isa{point} is for fixed records having exactly the two fields
+ \isa{Xcoord} and~\isa{Ycoord}, while the polymorphic type \isa{{\isaliteral{27}{\isacharprime}}a\ point{\isaliteral{5F}{\isacharunderscore}}scheme} comprises all possible extensions to those two
+ fields. Note that \isa{unit\ point{\isaliteral{5F}{\isacharunderscore}}scheme} coincides with \isa{point}, and \isa{{\isaliteral{5C3C6C706172723E}{\isasymlparr}}col\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ colour{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ point{\isaliteral{5F}{\isacharunderscore}}scheme} with \isa{cpoint}.
+
+ In the following example we define two operations --- methods, if we
+ regard records as objects --- to get and set any point's \isa{Xcoord} field.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ getX\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ point{\isaliteral{5F}{\isacharunderscore}}scheme\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ int{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}getX\ r\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ Xcoord\ r{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isacommand{definition}\isamarkupfalse%
+\ setX\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ point{\isaliteral{5F}{\isacharunderscore}}scheme\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ int\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ point{\isaliteral{5F}{\isacharunderscore}}scheme{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}setX\ r\ a\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ r{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+Here is a generic method that modifies a point, incrementing its
+ \isa{Xcoord} field. The \isa{Ycoord} and \isa{more} fields
+ are copied across. It works for any record type scheme derived from
+ \isa{point} (including \isa{cpoint} etc.):%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ incX\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ point{\isaliteral{5F}{\isacharunderscore}}scheme\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ point{\isaliteral{5F}{\isacharunderscore}}scheme{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}incX\ r\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\isanewline
+\ \ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3D}{\isacharequal}}\ Xcoord\ r\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3D}{\isacharequal}}\ Ycoord\ r{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ point{\isaliteral{2E}{\isachardot}}more\ r{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+Generic theorems can be proved about generic methods. This trivial
+ lemma relates \isa{incX} to \isa{getX} and \isa{setX}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}incX\ r\ {\isaliteral{3D}{\isacharequal}}\ setX\ r\ {\isaliteral{28}{\isacharparenleft}}getX\ r\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ getX{\isaliteral{5F}{\isacharunderscore}}def\ setX{\isaliteral{5F}{\isacharunderscore}}def\ incX{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+\begin{warn}
+ If you use the symbolic record brackets \isa{{\isaliteral{5C3C6C706172723E}{\isasymlparr}}} and \isa{{\isaliteral{5C3C72706172723E}{\isasymrparr}}},
+ then you must also use the symbolic ellipsis, ``\isa{{\isaliteral{5C3C646F74733E}{\isasymdots}}}'', rather
+ than three consecutive periods, ``\isa{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}''. Mixing the ASCII
+ and symbolic versions causes a syntax error. (The two versions are
+ more distinct on screen than they are on paper.)
+ \end{warn}%
+ \index{records!extensible|)}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Record Equality%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Two records are equal\index{equality!of records} if all pairs of
+ corresponding fields are equal. Concrete record equalities are
+ simplified automatically:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{27}{\isacharprime}}{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{27}{\isacharprime}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\isanewline
+\ \ \ \ {\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ b\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ simp%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+The following equality is similar, but generic, in that \isa{r}
+ can be any instance of \isa{{\isaliteral{27}{\isacharprime}}a\ point{\isaliteral{5F}{\isacharunderscore}}scheme}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}r{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{3D}{\isacharequal}}\ r{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Ycoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{2C}{\isacharcomma}}\ Xcoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ simp%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+\noindent
+ We see above the syntax for iterated updates. We could equivalently
+ have written the left-hand side as \isa{r{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Ycoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{5C3C72706172723E}{\isasymrparr}}}.
+
+ Record equality is \emph{extensional}:
+ \index{extensionality!for records} a record is determined entirely
+ by the values of its fields.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}r\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3D}{\isacharequal}}\ Xcoord\ r{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3D}{\isacharequal}}\ Ycoord\ r{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ simp%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+\noindent
+ The generic version of this equality includes the pseudo-field
+ \isa{more}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}r\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3D}{\isacharequal}}\ Xcoord\ r{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3D}{\isacharequal}}\ Ycoord\ r{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ point{\isaliteral{2E}{\isachardot}}more\ r{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ simp%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+The simplifier can prove many record equalities
+ automatically, but general equality reasoning can be tricky.
+ Consider proving this obvious fact:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}r{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{3D}{\isacharequal}}\ r{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{27}{\isacharprime}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ a\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{27}{\isacharprime}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+\ simp{\isaliteral{3F}{\isacharquery}}\isanewline
+\ \ \isacommand{oops}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+\noindent
+ Here the simplifier can do nothing, since general record equality is
+ not eliminated automatically. One way to proceed is by an explicit
+ forward step that applies the selector \isa{Xcoord} to both sides
+ of the assumed record equality:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}r{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{3D}{\isacharequal}}\ r{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{27}{\isacharprime}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ a\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{27}{\isacharprime}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}drule{\isaliteral{5F}{\isacharunderscore}}tac\ f\ {\isaliteral{3D}{\isacharequal}}\ Xcoord\ \isakeyword{in}\ arg{\isaliteral{5F}{\isacharunderscore}}cong{\isaliteral{29}{\isacharparenright}}%
+\begin{isamarkuptxt}%
+\begin{isabelle}%
+\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ Xcoord\ {\isaliteral{28}{\isacharparenleft}}r{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Xcoord\ {\isaliteral{28}{\isacharparenleft}}r{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{27}{\isacharprime}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ a\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{27}{\isacharprime}}%
+\end{isabelle}
+ Now, \isa{simp} will reduce the assumption to the desired
+ conclusion.%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\ \ \isacommand{apply}\isamarkupfalse%
+\ simp\isanewline
+\ \ \isacommand{done}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+The \isa{cases} method is preferable to such a forward proof. We
+ state the desired lemma again:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}r{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{3D}{\isacharequal}}\ r{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{27}{\isacharprime}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ a\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{27}{\isacharprime}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\begin{isamarkuptxt}%
+The \methdx{cases} method adds an equality to replace the
+ named record term by an explicit record expression, listing all
+ fields. It even includes the pseudo-field \isa{more}, since the
+ record equality stated here is generic for all extensions.%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\ \ \isacommand{apply}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}cases\ r{\isaliteral{29}{\isacharparenright}}%
+\begin{isamarkuptxt}%
+\begin{isabelle}%
+\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}Xcoord\ Ycoord\ more{\isaliteral{2E}{\isachardot}}\isanewline
+\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}r{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{3D}{\isacharequal}}\ r{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{27}{\isacharprime}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline
+\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ \ }r\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3D}{\isacharequal}}\ Xcoord{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3D}{\isacharequal}}\ Ycoord{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ more{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline
+\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ a\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{27}{\isacharprime}}%
+\end{isabelle} Again, \isa{simp} finishes the proof. Because \isa{r} is now represented as
+ an explicit record construction, the updates can be applied and the
+ record equality can be replaced by equality of the corresponding
+ fields (due to injectivity).%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\ \ \isacommand{apply}\isamarkupfalse%
+\ simp\isanewline
+\ \ \isacommand{done}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+The generic cases method does not admit references to locally bound
+ parameters of a goal. In longer proof scripts one might have to
+ fall back on the primitive \isa{rule{\isaliteral{5F}{\isacharunderscore}}tac} used together with the
+ internal field representation rules of records. The above use of
+ \isa{{\isaliteral{28}{\isacharparenleft}}cases\ r{\isaliteral{29}{\isacharparenright}}} would become \isa{{\isaliteral{28}{\isacharparenleft}}rule{\isaliteral{5F}{\isacharunderscore}}tac\ r\ {\isaliteral{3D}{\isacharequal}}\ r\ in\ point{\isaliteral{2E}{\isachardot}}cases{\isaliteral{5F}{\isacharunderscore}}scheme{\isaliteral{29}{\isacharparenright}}}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Extending and Truncating Records%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Each record declaration introduces a number of derived operations to
+ refer collectively to a record's fields and to convert between fixed
+ record types. They can, for instance, convert between types \isa{point} and \isa{cpoint}. We can add a colour to a point or convert
+ a \isa{cpoint} to a \isa{point} by forgetting its colour.
+
+ \begin{itemize}
+
+ \item Function \cdx{make} takes as arguments all of the record's
+ fields (including those inherited from ancestors). It returns the
+ corresponding record.
+
+ \item Function \cdx{fields} takes the record's very own fields and
+ returns a record fragment consisting of just those fields. This may
+ be filled into the \isa{more} part of the parent record scheme.
+
+ \item Function \cdx{extend} takes two arguments: a record to be
+ extended and a record containing the new fields.
+
+ \item Function \cdx{truncate} takes a record (possibly an extension
+ of the original record type) and returns a fixed record, removing
+ any additional fields.
+
+ \end{itemize}
+ These functions provide useful abbreviations for standard
+ record expressions involving constructors and selectors. The
+ definitions, which are \emph{not} unfolded by default, are made
+ available by the collective name of \isa{defs} (\isa{point{\isaliteral{2E}{\isachardot}}defs}, \isa{cpoint{\isaliteral{2E}{\isachardot}}defs}, etc.).
+ For example, here are the versions of those functions generated for
+ record \isa{point}. We omit \isa{point{\isaliteral{2E}{\isachardot}}fields}, which happens to
+ be the same as \isa{point{\isaliteral{2E}{\isachardot}}make}.
+
+ \begin{isabelle}%
+point{\isaliteral{2E}{\isachardot}}make\ Xcoord\ Ycoord\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3D}{\isacharequal}}\ Xcoord{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3D}{\isacharequal}}\ Ycoord{\isaliteral{5C3C72706172723E}{\isasymrparr}}\isasep\isanewline%
+point{\isaliteral{2E}{\isachardot}}extend\ r\ more\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\isanewline
+{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3D}{\isacharequal}}\ Xcoord\ r{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3D}{\isacharequal}}\ Ycoord\ r{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ more{\isaliteral{5C3C72706172723E}{\isasymrparr}}\isasep\isanewline%
+point{\isaliteral{2E}{\isachardot}}truncate\ r\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3D}{\isacharequal}}\ Xcoord\ r{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3D}{\isacharequal}}\ Ycoord\ r{\isaliteral{5C3C72706172723E}{\isasymrparr}}%
+\end{isabelle}
+ Contrast those with the corresponding functions for record \isa{cpoint}. Observe \isa{cpoint{\isaliteral{2E}{\isachardot}}fields} in particular.
+ \begin{isabelle}%
+cpoint{\isaliteral{2E}{\isachardot}}make\ Xcoord\ Ycoord\ col\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\isanewline
+{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3D}{\isacharequal}}\ Xcoord{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3D}{\isacharequal}}\ Ycoord{\isaliteral{2C}{\isacharcomma}}\ col\ {\isaliteral{3D}{\isacharequal}}\ col{\isaliteral{5C3C72706172723E}{\isasymrparr}}\isasep\isanewline%
+cpoint{\isaliteral{2E}{\isachardot}}fields\ col\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}col\ {\isaliteral{3D}{\isacharequal}}\ col{\isaliteral{5C3C72706172723E}{\isasymrparr}}\isasep\isanewline%
+cpoint{\isaliteral{2E}{\isachardot}}extend\ r\ more\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\isanewline
+{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3D}{\isacharequal}}\ Xcoord\ r{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3D}{\isacharequal}}\ Ycoord\ r{\isaliteral{2C}{\isacharcomma}}\ col\ {\isaliteral{3D}{\isacharequal}}\ col\ r{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ more{\isaliteral{5C3C72706172723E}{\isasymrparr}}\isasep\isanewline%
+cpoint{\isaliteral{2E}{\isachardot}}truncate\ r\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\isanewline
+{\isaliteral{5C3C6C706172723E}{\isasymlparr}}Xcoord\ {\isaliteral{3D}{\isacharequal}}\ Xcoord\ r{\isaliteral{2C}{\isacharcomma}}\ Ycoord\ {\isaliteral{3D}{\isacharequal}}\ Ycoord\ r{\isaliteral{2C}{\isacharcomma}}\ col\ {\isaliteral{3D}{\isacharequal}}\ col\ r{\isaliteral{5C3C72706172723E}{\isasymrparr}}%
+\end{isabelle}
+
+ To demonstrate these functions, we declare a new coloured point by
+ extending an ordinary point. Function \isa{point{\isaliteral{2E}{\isachardot}}extend} augments
+ \isa{pt{\isadigit{1}}} with a colour value, which is converted into an
+ appropriate record fragment by \isa{cpoint{\isaliteral{2E}{\isachardot}}fields}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ cpt{\isadigit{2}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ cpoint\ \isakeyword{where}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}cpt{\isadigit{2}}\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ point{\isaliteral{2E}{\isachardot}}extend\ pt{\isadigit{1}}\ {\isaliteral{28}{\isacharparenleft}}cpoint{\isaliteral{2E}{\isachardot}}fields\ Green{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The coloured points \isa{cpt{\isadigit{1}}} and \isa{cpt{\isadigit{2}}} are equal. The
+ proof is trivial, by unfolding all the definitions. We deliberately
+ omit the definition of~\isa{pt{\isadigit{1}}} in order to reveal the underlying
+ comparison on type \isa{point}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}cpt{\isadigit{1}}\ {\isaliteral{3D}{\isacharequal}}\ cpt{\isadigit{2}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ cpt{\isadigit{1}}{\isaliteral{5F}{\isacharunderscore}}def\ cpt{\isadigit{2}}{\isaliteral{5F}{\isacharunderscore}}def\ point{\isaliteral{2E}{\isachardot}}defs\ cpoint{\isaliteral{2E}{\isachardot}}defs{\isaliteral{29}{\isacharparenright}}%
+\begin{isamarkuptxt}%
+\begin{isabelle}%
+\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ Xcoord\ pt{\isadigit{1}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{9}}{\isadigit{9}}{\isadigit{9}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ Ycoord\ pt{\isadigit{1}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{2}}{\isadigit{3}}%
+\end{isabelle}%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\ \ \isacommand{apply}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ pt{\isadigit{1}}{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{done}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+In the example below, a coloured point is truncated to leave a
+ point. We use the \isa{truncate} function of the target record.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}point{\isaliteral{2E}{\isachardot}}truncate\ cpt{\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ pt{\isadigit{1}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ pt{\isadigit{1}}{\isaliteral{5F}{\isacharunderscore}}def\ cpt{\isadigit{2}}{\isaliteral{5F}{\isacharunderscore}}def\ point{\isaliteral{2E}{\isachardot}}defs{\isaliteral{29}{\isacharparenright}}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+\begin{exercise}
+ Extend record \isa{cpoint} to have a further field, \isa{intensity}, of type~\isa{nat}. Experiment with generic operations
+ (using polymorphic selectors and updates) and explicit coercions
+ (using \isa{extend}, \isa{truncate} etc.) among the three record
+ types.
+ \end{exercise}
+
+ \begin{exercise}
+ (For Java programmers.)
+ Model a small class hierarchy using records.
+ \end{exercise}
+ \index{records|)}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End: