theory Ind_Interface

imports Ind_Intro Simple_Inductive_Package

keywords "simple_inductive2" :: thy_decl

begin

\begin{figure}[t]

\begin{boxedminipage}{\textwidth}

\begin{isabelle}

simple_inductive

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

  even and odd

where

  even0: "even 0"

| evenS: "odd n \<Longrightarrow> even (Suc n)"

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

simple_inductive

  accpart :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"

where

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

(*<*)

datatype trm =

  Var "string"

| App "trm" "trm"

| Lam "string" "trm"

(*>*)

simple_inductive 

  fresh :: "string \<Rightarrow> trm \<Rightarrow> bool" 

where

  fresh_var: "a\<noteq>b \<Longrightarrow> fresh a (Var b)"

| fresh_app: "\<lbrakk>fresh a t; fresh a s\<rbrakk> \<Longrightarrow> fresh a (App t s)"

| fresh_lam1: "fresh a (Lam a t)"

| fresh_lam2: "\<lbrakk>a\<noteq>b; fresh a t\<rbrakk> \<Longrightarrow> fresh a (Lam b t)"

\end{isabelle}

\end{boxedminipage}

\caption{Specification given by the user for the inductive predicates

@{term "trcl"}, @{term "even"} and @{term "odd"}, @{term "accpart"} and 

@{term "fresh"}.\label{fig:specs}}

\end{figure}  

  To be able to write down the specifications of inductive predicates, we have

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

  keyword for the new command we chose \simpleinductive{}. Examples of

  specifications from the previous section are shown in

  Figure~\ref{fig:specs}. The syntax used in these examples more or

  less translates directly into the parser:

   Parse.vars -- 

   Scan.optional 

     (Parse.$$$ "where" |--

        Parse.!!! 

          (Parse.enum1 "|" 

  which we explained in Section~\ref{sec:parsingspecs}.  There is no code included 

  for parsing the keyword and what is called a \emph{target}. The latter  can be given 

  optionally after the keyword. The target is an ``advanced'' feature which we will 

  inherit for ``free'' from the infrastructure on which we shall build the package. 

  The target stands for a locale and allows us to specify

locale rel =

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

  and then define the transitive closure and the accessible part of this 

  locale 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"

  filter Token.is_proper (Token.explode (Thy_Header.get_keywords' @{context}) Position.none str) 

  relation, is left implicit. For the moment we will ignore this kind

  of implicit parameters and rely on the fact that the infrastructure will

  deal with them. Later, however, we will come back to them.

  If we feed into the parser the string that 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"

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

     [((even0,\<dots>), \<dots>),

      ((evenS,\<dots>), \<dots>),

      ((oddS,\<dots>), \<dots>)]), [])"}

  then we get back the specifications of the predicates (with type and syntax annotations), 

  and 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 5 to 7 in the code below.

  spec_parser >>

    (fn (pred_specs, rule_specs) => add_inductive_cmd pred_specs rule_specs)

val _ = Outer_Syntax.local_theory @{command_keyword "simple_inductive2"}

          "definition of simple inductive predicates"

  We call @{ML_ind  local_theory in Outer_Syntax} with the kind-indicator 

  @{ML_ind  thy_decl in Keyword} since the package does not need to open 

  up any proof (see Section~\ref{sec:newcommand}). 

  gathers the information from the parser to be processed further

  Note that the predicates when they come out of the parser are just some

  ``naked'' strings: they have no type yet (even if we annotate them with

  types) and they are also not defined constants yet (which the predicates 

  eventually will be).  Also the introduction rules are just strings. What we have

  to do first is to transform the parser's output into some internal

  datastructures that can be processed further. For this we can use the

  function @{ML_ind  read_multi_specs in Specification}. This function takes some strings

  (with possible typing annotations) and some rule specifications, and attempts

  to find a typing according to the given type constraints given by the 

  user and the type constraints by the ``ambient'' theory. It returns 

  the type for the predicates and also returns typed terms for the

  introduction rules. So at the heart of the function 

let

  val ((pred_specs', rule_specs'), _) = 

         Specification.read_multi_specs pred_specs rule_specs lthy

in

  add_inductive pred_specs' rule_specs' lthy

  Once we have the input data as some internal datastructure, we call

  lifting in the package: it generates definitions for the 

  predicates and derives from them corresponding induction principles and 

  introduction rules. The description of this function will span over

  the next two sections.

(*<*)end(*>*)