theory Ind_Interface

imports "../Base" "../Parsing" Ind_Prelims Simple_Inductive_Package

begin

section {* Parsing and Typing the Specification *}

text {* 

  To be able to write down the specification in Isabelle, we have to introduce

  a new command (see Section~\ref{sec:newcommand}).  As the keyword for the

  new command we chose \simpleinductive{}. In the package we want to support

  some ``advanced'' features: First, we want that the package can cope with

  specifications inside locales. For example it should be possible to declare

*}

locale rel =

  fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"

text {*

  and then define the transitive closure and the accessible part as follows:

*}

simple_inductive (in rel) 

  trcl' 

where

  base: "trcl' x x"

| step: "trcl' x y \<Longrightarrow> R y z \<Longrightarrow> trcl' x z"

simple_inductive (in rel) 

  accpart'

where

  accpartI: "(\<And>y. R y x \<Longrightarrow> accpart' y) \<Longrightarrow> accpart' x"

text {* 

  Second, we want that the user can specify fixed parameters.

  Remember in the previous section we stated that the user can give the 

  specification for the transitive closure of a relation @{text R} as 

*}

simple_inductive

  trcl\<iota>\<iota> :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"

where

  base: "trcl\<iota>\<iota> R x x"

| step: "trcl\<iota>\<iota> R x y \<Longrightarrow> R y z \<Longrightarrow> trcl\<iota>\<iota> R x z"

text {*

  Note that there is no locale given in this specification---the parameter

  @{text "R"} therefore needs to be included explicitly in @{term trcl\<iota>\<iota>}, but

  stays fixed throughout the specification. The problem with this way of

  stating the specification for the transitive closure is that it derives the

  following induction principle.

  \begin{center}\small

  \mprset{flushleft}

  \mbox{\inferrule{

             @{thm_style prem1  trcl\<iota>\<iota>.induct[where P=P, where z=R, where za=x, where zb=y]}\\\\

             @{thm_style prem2  trcl\<iota>\<iota>.induct[where P=P, where z=R, where za=x, where zb=y]}\\\\

             @{thm_style prem3  trcl\<iota>\<iota>.induct[where P=P, where z=R, where za=x, where zb=y]}}

            {@{thm_style concl  trcl\<iota>\<iota>.induct[where P=P, where z=R, where za=x, where zb=y]}}}  

  \end{center}

  But this does not correspond to the induction principle we derived by hand, which

  was

  \begin{center}\small

  \mprset{flushleft}

  \mbox{\inferrule{

             @{thm_style prem1  trcl_induct[no_vars]}\\\\

             @{thm_style prem2  trcl_induct[no_vars]}\\\\

             @{thm_style prem3  trcl_induct[no_vars]}}

            {@{thm_style concl  trcl_induct[no_vars]}}}  

  \end{center}

  The difference is that in the one derived by hand the relation @{term R} is not

  a parameter of the proposition @{term P} to be proved and it is also not universally

  qunatified in the second and third premise. The point is that the parameter @{term R}

  stays fixed thoughout the definition and we do not want to regard it as an ``ordinary''

  argument of the transitive closure, but one that can be freely instantiated. 

  In order to recognise such parameters, we have to extend the specification

  to include a mechanism to state fixed parameters. The user should be able

  to write 

*}

simple_inductive

  trcl'' for R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"

where

  base: "trcl'' R x x"

| step: "trcl'' R x y \<Longrightarrow> R y z \<Longrightarrow> trcl'' R x z"

simple_inductive

  accpart'' for R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"

where

  accpartI: "(\<And>y. R y x \<Longrightarrow> accpart'' R y) \<Longrightarrow> accpart'' R x"

text {*

  \begin{figure}[t]

  \begin{isabelle}

  \railnontermfont{\rmfamily\itshape}

  \railterm{simpleinductive,where,for}

  \railalias{simpleinductive}{\simpleinductive{}}

  \railalias{where}{\isacommand{where}}

  \railalias{for}{\isacommand{for}}

  \begin{rail}

  simpleinductive target? fixes (for fixes)? \\

  (where (thmdecl? prop + '|'))?

  ;

  \end{rail}

  \end{isabelle}

  \caption{A railroad diagram describing the syntax of \simpleinductive{}. 

  The \emph{target} indicates an optional locale; the \emph{fixes} are an 

  \isacommand{and}-separated list of names for the inductive predicates (they

  can also contain typing- and syntax anotations); similarly the \emph{fixes} 

  after \isacommand{for} to indicate fixed parameters; \emph{prop} stands for a 

  introduction rule with an optional theorem declaration (\emph{thmdecl}).

  \label{fig:railroad}}

  \end{figure}

*}

text {*

  This leads directly to the railroad diagram shown in

  Figure~\ref{fig:railroad} for the syntax of \simpleinductive{}. This diagram

  more or less translates directly into the parser:

  @{ML_chunk [display,gray] parser}

  which we described in Section~\ref{sec:parsingspecs}. If we feed into the 

  parser the string (which corresponds to our definition of @{term even} and 

  @{term odd}):

  @{ML_response [display,gray]

"let

  val input = filtered_input

     (\"even and odd \" ^  

      \"where \" ^

      \"  even0[intro]: \\\"even 0\\\" \" ^ 

      \"| evenS[intro]: \\\"odd n \<Longrightarrow> even (Suc n)\\\" \" ^ 

      \"| oddS[intro]:  \\\"even n \<Longrightarrow> odd (Suc n)\\\"\")

in

  parse spec_parser input

end"

"((((NONE, [(even, NONE, NoSyn), (odd, NONE, NoSyn)]), []),

     [((even0,\<dots>), \"\\^E\\^Ftoken\\^Eeven 0\\^E\\^F\\^E\"),

      ((evenS,\<dots>), \"\\^E\\^Ftoken\\^Eodd n \<Longrightarrow> even (Suc n)\\^E\\^F\\^E\"),

      ((oddS,\<dots>), \"\\^E\\^Ftoken\\^Eeven n \<Longrightarrow> odd (Suc n)\\^E\\^F\\^E\")]), [])"}

*}

text {*

  then we get back a locale (in this case @{ML NONE}), the predicates (with type

  and syntax annotations), the parameters (similar as the predicates) and

  the specifications of the introduction rules. 

  This is all the information we

  need for calling the package and setting up the keyword. The latter is

  done in Lines 6 and 7 in the code below.

  @{ML_chunk [display,gray,linenos] syntax}

  We call @{ML OuterSyntax.command} with the kind-indicator @{ML

  OuterKeyword.thy_decl} since the package does not need to open up any goal

  state (see Section~\ref{sec:newcommand}). Note that the predicates and

  parameters are at the moment only some ``naked'' variables: they have no

  type yet (even if we annotate them with types) and they are also no defined

  constants yet (which the predicates will eventually be).  In Lines 1 to 4 we

  gather the information from the parser to be processed further. The locale

  is passed as argument to the function @{ML

  Toplevel.local_theory}.\footnote{FIXME Is this already described?} The other

  arguments, i.e.~the predicates, parameters and intro rule specifications,

  are passed to the function @{ML add_inductive in SimpleInductivePackage}

  (Line 4).

  We now come to the second subtask of the package, namely transforming the 

  parser output into some internal datastructures that can be processed further. 

  Remember that at the moment the introduction rules are just strings, and even

  if the predicates and parameters can contain some typing annotations, they

  are not yet in any way reflected in the introduction rules. So the task of

  @{ML add_inductive in SimpleInductivePackage} is to transform the strings

  into properly typed terms. For this it can use the function 

  @{ML read_specification in Specification}. This function takes some constants

  with possible typing annotations and some rule specifications and attempts to

  find a type according to the given type constraints and the type constraints

  by the surrounding (local theory). However this function is a bit

  too general for our purposes: we want that each introduction rule has only 

  name (for example @{text even0} or @{text evenS}), if a name is given at all.

  The function @{ML read_specification in Specification} however allows more

  than one rule. Since it is quite convenient to rely on this function (instead of

  building your own) we just quick ly write a wrapper function that translates

  between our specific format and the general format expected by 

  @{ML read_specification in Specification}. The code of this wrapper is as follows:

  @{ML_chunk [display,gray,linenos] read_specification}

  It takes a list of constants, a list of rule specifications and a local theory 

  as input. Does the transformation of the rule specifications in Line 3; calls

  the function and transforms the now typed rule specifications back into our

  format and returns the type parameter and typed rule specifications. 

   @{ML_chunk [display,gray,linenos] add_inductive}

  In order to add a new inductive predicate to a theory with the help of our

  package, the user must \emph{invoke} it. For every package, there are

  essentially two different ways of invoking it, which we will refer to as

  \emph{external} and \emph{internal}. By external invocation we mean that the

  package is called from within a theory document. In this case, the

  specification of the inductive predicate, including type annotations and

  introduction rules, are given as strings by the user. Before the package can

  actually make the definition, the type and introduction rules have to be

  parsed. In contrast, internal invocation means that the package is called by

  some other package. For example, the function definition package

  calls the inductive definition package to define the

  graph of the function. However, it is not a good idea for the function

  definition package to pass the introduction rules for the function graph to

  the inductive definition package as strings. In this case, it is better to

  directly pass the rules to the package as a list of terms, which is more

  robust than handling strings that are lacking the additional structure of

  terms. These two ways of invoking the package are reflected in its ML

  programming interface, which consists of two functions:

  @{ML_chunk [display,gray] SIMPLE_INDUCTIVE_PACKAGE}

*}

text {*

  (FIXME: explain Binding.binding; Attrib.binding somewhere else)

  The function for external invocation of the package is called @{ML

  add_inductive in SimpleInductivePackage}, whereas the one for internal

  invocation is called @{ML add_inductive_i in SimpleInductivePackage}. Both

  of these functions take as arguments the names and types of the inductive

  predicates, the names and types of their parameters, the actual introduction

  rules and a \emph{local theory}.  They return a local theory containing the

  definition and the induction principle as well introduction rules. 

  Note that @{ML add_inductive_i in SimpleInductivePackage} expects

  the types of the predicates and parameters to be specified using the

  datatype @{ML_type typ} of Isabelle's logical framework, whereas @{ML

  add_inductive in SimpleInductivePackage} expects them to be given as

  optional strings. If no string is given for a particular predicate or

  parameter, this means that the type should be inferred by the

  package. 

  Additional \emph{mixfix syntax} may be associated with the

  predicates and parameters as well. Note that @{ML add_inductive_i in

  SimpleInductivePackage} does not allow mixfix syntax to be associated with

  parameters, since it can only be used for parsing.\footnote{FIXME: why ist it there then?} 

  The names of the

  predicates, parameters and rules are represented by the type @{ML_type

  Binding.binding}. Strings can be turned into elements of the type @{ML_type

  Binding.binding} using the function @{ML [display] "Binding.name : string ->

  Binding.binding"} Each introduction rule is given as a tuple containing its

  name, a list of \emph{attributes} and a logical formula. Note that the type

  @{ML_type Attrib.binding} used in the list of introduction rules is just a

  shorthand for the type @{ML_type "Binding.binding * Attrib.src list"}.  The

  function @{ML add_inductive_i in SimpleInductivePackage} expects the formula

  to be specified using the datatype @{ML_type term}, whereas @{ML

  add_inductive in SimpleInductivePackage} expects it to be given as a string.

  An attribute specifies additional actions and transformations that should be

  applied to a theorem, such as storing it in the rule databases used by

  automatic tactics like the simplifier. The code of the package, which will

  be described in the following section, will mostly treat attributes as a

  black box and just forward them to other functions for storing theorems in

  local theories.  The implementation of the function @{ML add_inductive in

  SimpleInductivePackage} for external invocation of the package is quite

  simple. Essentially, it just parses the introduction rules and then passes

  them on to @{ML add_inductive_i in SimpleInductivePackage}:

  @{ML_chunk [display] add_inductive}

  For parsing and type checking the introduction rules, we use the function

  @{ML [display] "Specification.read_specification:

  (Binding.binding * string option * mixfix) list ->  (*{variables}*)

  (Attrib.binding * string list) list list ->  (*{rules}*)

  local_theory ->

  (((Binding.binding * typ) * mixfix) list *

   (Attrib.binding * term list) list) *

  local_theory"}

*}

text {*

  During parsing, both predicates and parameters are treated as variables, so

  the lists \verb!preds_syn! and \verb!params_syn! are just appended

  before being passed to @{ML read_specification in Specification}. Note that the format

  for rules supported by @{ML read_specification in Specification} is more general than

  what is required for our package. It allows several rules to be associated

  with one name, and the list of rules can be partitioned into several

  sublists. In order for the list \verb!intro_srcs! of introduction rules

  to be acceptable as an input for @{ML read_specification in Specification}, we first

  have to turn it into a list of singleton lists. This transformation

  has to be reversed later on by applying the function

  @{ML [display] "the_single: 'a list -> 'a"}

  to the list \verb!specs! containing the parsed introduction rules.

  The function @{ML read_specification in Specification} also returns the list \verb!vars!

  of predicates and parameters that contains the inferred types as well.

  This list has to be chopped into the two lists \verb!preds_syn'! and

  \verb!params_syn'! for predicates and parameters, respectively.

  All variables occurring in a rule but not in the list of variables passed to

  @{ML read_specification in Specification} will be bound by a meta-level universal

  quantifier.

*}

text {*

  Finally, @{ML read_specification in Specification} also returns another local theory,

  but we can safely discard it. As an example, let us look at how we can use this

  function to parse the introduction rules of the @{text trcl} predicate:

  @{ML_response [display]

"Specification.read_specification

  [(Binding.name \"trcl\", NONE, NoSyn),

   (Binding.name \"r\", SOME \"'a \<Rightarrow> 'a \<Rightarrow> bool\", NoSyn)]

  [[((Binding.name \"base\", []), [\"trcl r x x\"])],

   [((Binding.name \"step\", []), [\"trcl r x y \<Longrightarrow> r y z \<Longrightarrow> trcl r x z\"])]]

  @{context}"

"((\<dots>,

  [(\<dots>,

    [Const (\"all\", \<dots>) $ Abs (\"x\", TFree (\"'a\", \<dots>),

       Const (\"Trueprop\", \<dots>) $

         (Free (\"trcl\", \<dots>) $ Free (\"r\", \<dots>) $ Bound 0 $ Bound 0))]),

   (\<dots>,

    [Const (\"all\", \<dots>) $ Abs (\"x\", TFree (\"'a\", \<dots>),

       Const (\"all\", \<dots>) $ Abs (\"y\", TFree (\"'a\", \<dots>),

         Const (\"all\", \<dots>) $ Abs (\"z\", TFree (\"'a\", \<dots>),

           Const (\"==>\", \<dots>) $

             (Const (\"Trueprop\", \<dots>) $

               (Free (\"trcl\", \<dots>) $ Free (\"r\", \<dots>) $ Bound 2 $ Bound 1)) $

             (Const (\"==>\", \<dots>) $ \<dots> $ \<dots>))))])]),

 \<dots>)

: (((Binding.binding * typ) * mixfix) list *

   (Attrib.binding * term list) list) * local_theory"}

  In the list of variables passed to @{ML read_specification in Specification}, we have

  used the mixfix annotation @{ML NoSyn} to indicate that we do not want to associate any

  mixfix syntax with the variable. Moreover, we have only specified the type of \texttt{r},

  whereas the type of \texttt{trcl} is computed using type inference.

  The local variables \texttt{x}, \texttt{y} and \texttt{z} of the introduction rules

  are turned into bound variables with the de Bruijn indices,

  whereas \texttt{trcl} and \texttt{r} remain free variables.

*}

text {*

  \paragraph{Parsers for theory syntax}

  Although the function @{ML add_inductive in SimpleInductivePackage} parses terms and types, it still

  cannot be used to invoke the package directly from within a theory document.

  In order to do this, we have to write another parser. Before we describe

  the process of writing parsers for theory syntax in more detail, we first

  show some examples of how we would like to use the inductive definition

  package.

  The definition of the transitive closure should look as follows:

*}

text {*

  A proposition can be parsed using the function @{ML prop in OuterParse}.

  Essentially, a proposition is just a string or an identifier, but using the

  specific parser function @{ML prop in OuterParse} leads to more instructive

  error messages, since the parser will complain that a proposition was expected

  when something else than a string or identifier is found.

  An optional locale target specification of the form \isa{(\isacommand{in}\ $\ldots$)}

  can be parsed using @{ML opt_target in OuterParse}.

  The lists of names of the predicates and parameters, together with optional

  types and syntax, are parsed using the functions @{ML "fixes" in OuterParse}

  and @{ML for_fixes in OuterParse}, respectively.

  In addition, the following function from @{ML_struct SpecParse} for parsing

  an optional theorem name and attribute, followed by a delimiter, will be useful:

  \begin{table}

  @{ML "opt_thm_name:

  string -> token list -> Attrib.binding * token list" in SpecParse}

  \end{table}

  We now have all the necessary tools to write the parser for our

  \isa{\isacommand{simple{\isacharunderscore}inductive}} command:

  Once all arguments of the command have been parsed, we apply the function

  @{ML add_inductive in SimpleInductivePackage}, which yields a local theory

  transformer of type @{ML_type "local_theory -> local_theory"}. Commands in

  Isabelle/Isar are realized by transition transformers of type

  @{ML_type [display] "Toplevel.transition -> Toplevel.transition"}

  We can turn a local theory transformer into a transition transformer by using

  the function