\section{Records}
\label{sec:records}
\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 basic Isabelle record has a fixed set of fields, with access
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 fields 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 field \cdx{more}, which is
present in every record type. Generic methods, or operations that
work on all possible extensions of a given record, can be expressed by
definitions involving \isa{more}, but the details are complicated.
\subsection{Record Basics}
Record types are not primitive in Isabelle and have a complex internal
representation. A \commdx{record} declaration
introduces a new record type:
\begin{isabelle}
\isacommand{record}\ point\ =\isanewline
\ \ Xcoord\ ::\ int\isanewline
\ \ Ycoord\ ::\ int
\end{isabelle}
Records of type \isa{point} have two fields named \isa{Xcoord} and \isa{Ycoord},
both of type~\isa{int}. We now declare a constant of type
\isa{point}:
\begin{isabelle}
\isacommand{constdefs}\ \ \ pt1\ ::\ point\isanewline
\ \ \ \ \ \ \ \ \ \ \ \ "pt1\ ==\ (|\ Xcoord\ =\ 999,\ Ycoord\ =\ 23\ |)"
\end{isabelle}
We see above the ASCII notation for record brackets. You can also use
the symbolic brackets \isa{\isasymlparr} and \isa{\isasymrparr}.
Record types can be written directly, rather than referring to
previously declared names:
\begin{isabelle}
\isacommand{constdefs}\ \ \ pt2\ ::\ "(|\ Xcoord\ ::\ int,\ Ycoord\ ::\ int\
|)"\ \isanewline
\ \ \ \ \ \ \ \ \ \ \ \ "pt2\ ==\ (|\ Xcoord\ =\ -45,\ Ycoord\ =\ 97\ |)"
\end{isabelle}
For each field, there is a \emph{selector} 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 are simplified automatically:
\begin{isabelle}
\isacommand{lemma}\ "Xcoord\ (|\ Xcoord\ =\ a,\ Ycoord\ =\ b\ |)\ =\ a"\isanewline
\isacommand{by}\ simp
\end{isabelle}
The \emph{update} operation is functional. For example,
\isa{p\isasymlparr Xcoord:=0\isasymrparr} is a record whose \isa{Xcoord} value
is zero and whose
\isa{Ycoord} value is copied from~\isa{p}. Updates are also simplified
automatically:
\begin{isabelle}
\isacommand{lemma}\ "(|\ Xcoord\ =\ a,\ Ycoord\ =\ b\ |)\ (|\ Xcoord:=\ 0\ |)\
=\isanewline
\ \ \ \ \ \ \ \ (|\ Xcoord\ =\ 0,\ Ycoord\ =\ b\ |)"\isanewline
\isacommand{by}\ simp
\end{isabelle}
\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}. Each record
declaration introduces a constant \cdx{more}.
\end{warn}
\subsection{Extensible Records and Generic Operations}
\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}:
\begin{isabelle}
\isacommand{datatype}\ colour\ =\ Red\ |\ Green\ |\
Blue\isanewline
\isanewline
\isacommand{record}\ cpoint\ =\ point\ +\isanewline
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ col\ ::\ colour
\end{isabelle}
The fields of this new type are \isa{Xcoord}, \isa{Ycoord} and \isa{col}, in that
order:
\begin{isabelle}
\isacommand{constdefs}\ \ \ cpt1\ ::\ cpoint\isanewline
\ \ \ \ \ \ \ \ \ \ \ \ "cpt1\ ==\ (|\ Xcoord\ =\ 999,\ Ycoord\ =\ 23,\ col\
=\ Green\ |)"
\end{isabelle}
Unfortunately, there are no built-in conversions between types
\isa{point} and \isa{cpoint}: to add a colour to
a point, or to convert a \isa{cpoint} to a \isa{point} by forgetting
its colour, we must define operations that copy across the other
fields. However, we can define generic operations
that work on type
\isa{point} and all extensions of it.
Every record structure has an implicit field, \cdx{more}, to allow
extension. Its type is completely polymorphic:~\isa{'a}. When a
record value is expressed using just its standard fields, the value of
\isa{more} is implicitly set to \isa{()}, the empty tuple, which has
type \isa{unit}. Within the record brackets, you can refer to the
\isa{more} field by writing \isa{...} (three periods):
\begin{isabelle}
\isacommand{lemma}\ "Xcoord\ (|\ Xcoord\ =\ a,\ Ycoord\ =\ b,\ ...\ =\ p\ |)\ =\ a"
\end{isabelle}
This lemma (trivially proved using \isa{simp}) applies to any
record whose first two fields are \isa{Xcoord} and~\isa{Ycoord}. Field
\isa{more} can be selected in the usual way, but as all records share
this field, the identifier must be qualified:
\begin{isabelle}
\isacommand{lemma}\ "point.more\ cpt1\ =\ \isasymlparr col\ =\ Green\isasymrparr "\isanewline
\isacommand{by}\ (simp\ add:\ cpt1_def)
\end{isabelle}
%
We see that the colour attached to this \isa{point} is a record in its
own right, namely
\isa{\isasymlparr col\ =\ Green\isasymrparr}.
To define generic operations, we need to know a bit more about records.
Our declaration of \isa{point} above generated two type
abbreviations:
\begin{isabelle}
\ \ \ \ point\ =\ (|\ Xcoord\ ::\ int,\ Ycoord\ ::\ int\ |)\isanewline
\ \ \ \ 'a\ point_scheme\ =\ (|\ Xcoord\ ::\ int,\ Ycoord\ ::\ int,\ ...\ ::\ 'a\ |)
\end{isabelle}
%
Type \isa{point} is for rigid records having the two fields
\isa{Xcoord} and~\isa{Ycoord}, while the polymorphic type \isa{'a\ point_scheme}
comprises all possible extensions to those two fields. For example,
let us define two operations --- methods, if we regard records as
objects --- to get and set any point's
\isa{Xcoord} field.
\begin{isabelle}
\ \ getX\ ::\ "'a\ point_scheme\ \isasymRightarrow \ int"\isanewline
\ \ \ "getX\ r\ ==\ Xcoord\ r"\isanewline
\ \ setX\ ::\ "['a\ point_scheme,\ int]\ \isasymRightarrow \ 'a\ point_scheme"\isanewline
\ \ \ "setX\ r\ a\ ==\ r\ (|\ Xcoord\ :=\ a\ |)"
\end{isabelle}
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 type \isa{point} and any of its
extensions, such as \isa{cpoint}:
\begin{isabelle}
\isacommand{constdefs}\isanewline
\ \ incX\ ::\ "'a\ point_scheme\ \isasymRightarrow \ 'a\ point_scheme"\isanewline
\ \ "incX\ r\ ==\ \isasymlparr Xcoord\ =\ (Xcoord\ r)\ +\ 1,\isanewline
\ \ \ \ \ \ \ \ \ \ \ \ \ \ Ycoord\ =\ Ycoord\ r,\isanewline
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isasymdots \ =\ point.more\
r\isasymrparr"
\end{isabelle}
Generic theorems can be proved about generic methods. This trivial
lemma relates \isa{incX} to \isa{getX} and \isa{setX}:
\begin{isabelle}
\isacommand{lemma}\ "incX\ r\ =\ setX\ r\ ((getX\ r)\ +\ 1)"\isanewline
\isacommand{by}\ (simp\ add:\ getX_def\ setX_def\ incX_def)
\end{isabelle}
\begin{warn}
If you use the symbolic record brackets \isa{\isasymlparr} and
\isa{\isasymrparr}, then you must also use the symbolic ellipsis,
\isa{\isasymdots}, rather than three consecutive periods,
\isa{...}. 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|)}
\subsection{Record Equality}
Two records are equal\index{equality!of records}
if all pairs of corresponding fields are equal.
Record equalities are simplified automatically:
\begin{isabelle}
\isacommand{lemma}\ "(\isasymlparr Xcoord\ =\ a,\ Ycoord\ =\ b\isasymrparr \ =\ \isasymlparr Xcoord\ =\ a',\ Ycoord\ =\ b'\isasymrparr )\
=\isanewline
\ \ \ \ \ \ \ \ (a\ =\ a'\ \&\ b\ =\ b')"\isanewline
\isacommand{by}\ simp
\end{isabelle}
The following equality is similar, but generic, in that \isa{r} can
be any instance of \isa{point_scheme}:
\begin{isabelle}
\isacommand{lemma}\ "r\ \isasymlparr Xcoord\ :=\ a,\ Ycoord\ :=\
b\isasymrparr \ =\ r\ \isasymlparr Ycoord\ :=\ b,\ Xcoord\ :=\
a\isasymrparr "\isanewline
\isacommand{by}\ simp
\end{isabelle}
We see above the syntax for iterated updates. We could equivalently
have written the left-hand side as
\isa{r\ \isasymlparr Xcoord\ :=\ a\isasymrparr \ \isasymlparr
Ycoord\ :=\ b\isasymrparr}.
Record equality is \emph{extensional}:
\index{extensionality!for records}
a record is determined entirely by the values of its fields.
\begin{isabelle}
\isacommand{lemma}\ "r\ =\ \isasymlparr Xcoord\ =\ Xcoord\ r,\ Ycoord\ =\
Ycoord\ r\isasymrparr "\isanewline
\isacommand{by}\ simp
\end{isabelle}
The generic version of this equality includes the field \isa{more}:
\begin{isabelle}
\isacommand{lemma}\ "r\ =\ \isasymlparr Xcoord\ =\ Xcoord\ r,\ Ycoord\ =\ Ycoord\ r,\ \isasymdots \ =\ point.more\
r\isasymrparr"
\end{isabelle}
\medskip
The simplifier can prove many record equalities automatically,
but general equality reasoning can be tricky. Consider proving this
obvious fact:
\begin{isabelle}
\isacommand{lemma}\ "r\ \isasymlparr Xcoord\ :=\ a\isasymrparr \ =\ r\ \isasymlparr Xcoord\ :=\ a'\isasymrparr \ \isasymLongrightarrow \ a\ =\
a'"
\end{isabelle}
The simplifier can do nothing. One way to proceed is by an explicit
forward step that applies the selector \isa{Xcoord} to both sides
of the assumed record equality:
\begin{isabelle}
\isacommand{apply}\ (drule_tac\ f=Xcoord\ \isakeyword{in}\ arg_cong)\isanewline
\ 1.\ Xcoord\ (r\isasymlparr Xcoord\ :=\ a\isasymrparr )\ =\ Xcoord\
(r\isasymlparr Xcoord\ :=\ a'\isasymrparr )\ \isasymLongrightarrow \
a\ =\ a'
\end{isabelle}
Now, \isa{simp} will reduce the assumption to the desired
conclusion.
An alternative to such forward steps is record splitting. A record
variable can be split only if it is bound in the subgoal by the
meta-quantifier \isa{\isasymAnd}, or \isa{!!} in ASCII\@. So,
we enter the lemma again:
\begin{isabelle}
\isacommand{lemma}\ "!!r.\ r\ \isasymlparr Xcoord\ :=\ a\isasymrparr \ =\ r\ \isasymlparr Xcoord\ :=\ a'\isasymrparr \ \isasymLongrightarrow \ a\ =\
a'"\isanewline
\end{isabelle}
The \methdx{record_split} method replaces the record variable by an
explicit record, listing all fields. Even the field \isa{more} is
included, since the record equality is generic.
\begin{isabelle}
\isacommand{apply}\ record_split\isanewline
\ 1.\ \isasymAnd Xcoord\ Ycoord\ more.\isanewline
\isaindent{\ 1.\ \ \ \ }\isasymlparr Xcoord\ =\ Xcoord,\ Ycoord\ =\ Ycoord,\ \isasymdots \ =\ more\isasymrparr \isanewline
\isaindent{\ 1.\ \ \ \ }\isasymlparr Xcoord\ :=\ a\isasymrparr \ =\isanewline
\isaindent{\ 1.\ \ \ \ }\isasymlparr Xcoord\ =\ Xcoord,\ Ycoord\ =\ Ycoord,\ \isasymdots \ =\ more\isasymrparr \isanewline
\isaindent{\ 1.\ \ \ \ }\isasymlparr Xcoord\ :=\ a'\isasymrparr \ \isasymLongrightarrow \isanewline
\isaindent{\ 1.\ \ \ \ }a\ =\ a'
\end{isabelle}
Again, \isa{simp} finishes the proof. Because the records have
been split, the updates can be applied and the record equality can be
replaced by equality of the corresponding fields.
\begin{exercise}
\REMARK{There should be some, but I can't think of any.}
\end{exercise}
\index{records|)}
\endinput