header {* Records \label{sec:records} *}

(*<*)

theory Records imports Main begin

(*>*)

text {*

  \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.

*}

subsection {* Record Basics *}

text {*

  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:

*}

record point =

  Xcoord :: int

  Ycoord :: int

text {*\noindent

  Records of type @{typ point} have two fields named @{const Xcoord}

  and @{const Ycoord}, both of type~@{typ int}.  We now define a

  constant of type @{typ point}:

*}

definition pt1 :: point where

"pt1 \<equiv> (| Xcoord = 999, Ycoord = 23 |)"

text {*\noindent

  We see above the ASCII notation for record brackets.  You can also

  use the symbolic brackets @{text \<lparr>} and @{text \<rparr>}.  Record type

  expressions can be also written directly with individual fields.

  The type name above is merely an abbreviation.

*}

definition pt2 :: "\<lparr>Xcoord :: int, Ycoord :: int\<rparr>" where

"pt2 \<equiv> \<lparr>Xcoord = -45, Ycoord = 97\<rparr>"

text {*

  For each field, there is a \emph{selector}\index{selector!record}

  function of the same name.  For example, if @{text p} has type @{typ

  point} then @{text "Xcoord p"} denotes the value of the @{text

  Xcoord} field of~@{text p}.  Expressions involving field selection

  of explicit records are simplified automatically:

*}

lemma "Xcoord \<lparr>Xcoord = a, Ycoord = b\<rparr> = a"

  by simp

text {*

  The \emph{update}\index{update!record} operation is functional.  For

  example, @{term "p\<lparr>Xcoord := 0\<rparr>"} is a record whose @{const Xcoord}

  value is zero and whose @{const Ycoord} value is copied from~@{text

  p}.  Updates of explicit records are also simplified automatically:

*}

lemma "\<lparr>Xcoord = a, Ycoord = b\<rparr>\<lparr>Xcoord := 0\<rparr> =

         \<lparr>Xcoord = 0, Ycoord = b\<rparr>"

  by simp

text {*

  \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 @{typ point} simply @{text x} and~@{text y}.

  \end{warn}

*}

subsection {* Extensible Records and Generic Operations *}

text {*

  \index{records!extensible|(}%

  Now, let us define coloured points (type @{text cpoint}) to be

  points extended with a field @{text col} of type @{text colour}:

*}

datatype colour = Red | Green | Blue

record cpoint = point +

  col :: colour

text {*\noindent

  The fields of this new type are @{const Xcoord}, @{text Ycoord} and

  @{text col}, in that order.

*}

definition cpt1 :: cpoint where

"cpt1 \<equiv> \<lparr>Xcoord = 999, Ycoord = 23, col = Green\<rparr>"

text {*

  We can define generic operations that work on arbitrary

  instances of a record scheme, e.g.\ covering @{typ point}, @{typ

  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:~@{typ 'a}.  When a fixed record value is expressed

  using just its standard fields, the value of @{text more} is

  implicitly set to @{text "()"}, the empty tuple, which has type

  @{typ unit}.  Within the record brackets, you can refer to the

  @{text more} field by writing ``@{text "\<dots>"}'' (three dots):

*}

lemma "Xcoord \<lparr>Xcoord = a, Ycoord = b, \<dots> = p\<rparr> = a"

  by simp

text {*

  This lemma applies to any record whose first two fields are @{text

  Xcoord} and~@{const Ycoord}.  Note that @{text "\<lparr>Xcoord = a, Ycoord

  = b, \<dots> = ()\<rparr>"} is exactly the same as @{text "\<lparr>Xcoord = a, Ycoord

  = b\<rparr>"}.  Selectors and updates are always polymorphic wrt.\ the

  @{text more} part of a record scheme, its value is just ignored (for

  select) or copied (for update).

  The @{text more} pseudo-field may be manipulated directly as well,

  but the identifier needs to be qualified:

*}

lemma "point.more cpt1 = \<lparr>col = Green\<rparr>"

  by (simp add: cpt1_def)

text {*\noindent

  We see that the colour part attached to this @{typ point} is a

  rudimentary record in its own right, namely @{text "\<lparr>col =

  Green\<rparr>"}.  In order to select or update @{text col}, this fragment

  needs to be put back into the context of the parent type scheme, say

  as @{text more} part of another @{typ point}.

  To define generic operations, we need to know a bit more about

  records.  Our definition of @{typ point} above has generated two

  type abbreviations:

  \medskip

  \begin{tabular}{l}

  @{typ point}~@{text "="}~@{text "\<lparr>Xcoord :: int, Ycoord :: int\<rparr>"} \\

  @{typ "'a point_scheme"}~@{text "="}~@{text "\<lparr>Xcoord :: int, Ycoord :: int, \<dots> :: 'a\<rparr>"} \\

  \end{tabular}

  \medskip

\noindent

  Type @{typ point} is for fixed records having exactly the two fields

  @{const Xcoord} and~@{text Ycoord}, while the polymorphic type @{typ

  "'a point_scheme"} comprises all possible extensions to those two

  fields.  Note that @{typ "unit point_scheme"} coincides with @{typ

  point}, and @{typ "\<lparr>col :: colour\<rparr> point_scheme"} with @{typ

  cpoint}.

  In the following example we define two operations --- methods, if we

  regard records as objects --- to get and set any point's @{text

  Xcoord} field.

*}

definition getX :: "'a point_scheme \<Rightarrow> int" where

"getX r \<equiv> Xcoord r"

definition setX :: "'a point_scheme \<Rightarrow> int \<Rightarrow> 'a point_scheme" where

"setX r a \<equiv> r\<lparr>Xcoord := a\<rparr>"

text {*

  Here is a generic method that modifies a point, incrementing its

  @{const Xcoord} field.  The @{text Ycoord} and @{text more} fields

  are copied across.  It works for any record type scheme derived from

  @{typ point} (including @{typ cpoint} etc.):

*}

definition incX :: "'a point_scheme \<Rightarrow> 'a point_scheme" where

"incX r \<equiv>

  \<lparr>Xcoord = Xcoord r + 1, Ycoord = Ycoord r, \<dots> = point.more r\<rparr>"

text {*

  Generic theorems can be proved about generic methods.  This trivial

  lemma relates @{const incX} to @{text getX} and @{text setX}:

*}

lemma "incX r = setX r (getX r + 1)"

  by (simp add: getX_def setX_def incX_def)

text {*

  \begin{warn}

  If you use the symbolic record brackets @{text \<lparr>} and @{text \<rparr>},

  then you must also use the symbolic ellipsis, ``@{text \<dots>}'', rather

  than three consecutive periods, ``@{text "..."}''.  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 *}

text {*

  Two records are equal\index{equality!of records} if all pairs of

  corresponding fields are equal.  Concrete record equalities are

  simplified automatically:

*}

lemma "(\<lparr>Xcoord = a, Ycoord = b\<rparr> = \<lparr>Xcoord = a', Ycoord = b'\<rparr>) =

    (a = a' \<and> b = b')"

  by simp

text {*

  The following equality is similar, but generic, in that @{text r}

  can be any instance of @{typ "'a point_scheme"}:

*}

lemma "r\<lparr>Xcoord := a, Ycoord := b\<rparr> = r\<lparr>Ycoord := b, Xcoord := a\<rparr>"

  by simp

text {*\noindent

  We see above the syntax for iterated updates.  We could equivalently

  have written the left-hand side as @{text "r\<lparr>Xcoord := a\<rparr>\<lparr>Ycoord :=

  b\<rparr>"}.

  Record equality is \emph{extensional}:

  \index{extensionality!for records} a record is determined entirely

  by the values of its fields.

*}

lemma "r = \<lparr>Xcoord = Xcoord r, Ycoord = Ycoord r\<rparr>"

  by simp

text {*\noindent

  The generic version of this equality includes the pseudo-field

  @{text more}:

*}

lemma "r = \<lparr>Xcoord = Xcoord r, Ycoord = Ycoord r, \<dots> = point.more r\<rparr>"

  by simp

text {*

  The simplifier can prove many record equalities

  automatically, but general equality reasoning can be tricky.

  Consider proving this obvious fact:

*}

lemma "r\<lparr>Xcoord := a\<rparr> = r\<lparr>Xcoord := a'\<rparr> \<Longrightarrow> a = a'"

  apply simp?

  oops

text {*\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 @{const Xcoord} to both sides

  of the assumed record equality:

*}

lemma "r\<lparr>Xcoord := a\<rparr> = r\<lparr>Xcoord := a'\<rparr> \<Longrightarrow> a = a'"

  apply (drule_tac f = Xcoord in arg_cong)

  txt {* @{subgoals [display, indent = 0, margin = 65]}

    Now, @{text simp} will reduce the assumption to the desired

    conclusion. *}

  apply simp

  done

text {*

  The @{text cases} method is preferable to such a forward proof.  We

  state the desired lemma again:

*}

lemma "r\<lparr>Xcoord := a\<rparr> = r\<lparr>Xcoord := a'\<rparr> \<Longrightarrow> a = a'"

  txt {* 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 @{text more}, since the

  record equality stated here is generic for all extensions. *}

  apply (cases r)

  txt {* @{subgoals [display, indent = 0, margin = 65]} Again, @{text

  simp} finishes the proof.  Because @{text 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). *}

  apply simp

  done

text {*

  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 @{text rule_tac} used together with the

  internal field representation rules of records.  The above use of

  @{text "(cases r)"} would become @{text "(rule_tac r = r in

  point.cases_scheme)"}.

*}

subsection {* Extending and Truncating Records *}

text {*

  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 @{typ

  point} and @{typ cpoint}.  We can add a colour to a point or convert

  a @{typ cpoint} to a @{typ 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 @{text 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 @{text defs} (@{text

  point.defs}, @{text cpoint.defs}, etc.).

  For example, here are the versions of those functions generated for

  record @{typ point}.  We omit @{text point.fields}, which happens to

  be the same as @{text point.make}.

  @{thm [display, indent = 0, margin = 65] point.make_def [no_vars]

  point.extend_def [no_vars] point.truncate_def [no_vars]}

  Contrast those with the corresponding functions for record @{typ

  cpoint}.  Observe @{text cpoint.fields} in particular.

  @{thm [display, indent = 0, margin = 65] cpoint.make_def [no_vars]

  cpoint.fields_def [no_vars] cpoint.extend_def [no_vars]

  cpoint.truncate_def [no_vars]}

  To demonstrate these functions, we declare a new coloured point by

  extending an ordinary point.  Function @{text point.extend} augments

  @{text pt1} with a colour value, which is converted into an

  appropriate record fragment by @{text cpoint.fields}.

*}

definition cpt2 :: cpoint where

"cpt2 \<equiv> point.extend pt1 (cpoint.fields Green)"

text {*

  The coloured points @{const cpt1} and @{text cpt2} are equal.  The

  proof is trivial, by unfolding all the definitions.  We deliberately

  omit the definition of~@{text pt1} in order to reveal the underlying

  comparison on type @{typ point}.

*}

lemma "cpt1 = cpt2"

  apply (simp add: cpt1_def cpt2_def point.defs cpoint.defs)

  txt {* @{subgoals [display, indent = 0, margin = 65]} *}

  apply (simp add: pt1_def)

  done

text {*

  In the example below, a coloured point is truncated to leave a

  point.  We use the @{text truncate} function of the target record.

*}

lemma "point.truncate cpt2 = pt1"

  by (simp add: pt1_def cpt2_def point.defs)

text {*

  \begin{exercise}

  Extend record @{typ cpoint} to have a further field, @{text

  intensity}, of type~@{typ nat}.  Experiment with generic operations

  (using polymorphic selectors and updates) and explicit coercions

  (using @{text extend}, @{text 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

(*>*)