src/HOL/MicroJava/J/WellType.thy

corrected xsymbol/HTML syntax

(*  Title:      HOL/MicroJava/J/WellType.thy
    ID:         $Id$
    Author:     David von Oheimb
    Copyright   1999 Technische Universitaet Muenchen
*)

header "Well-typedness Constraints"

theory WellType = Term + WellForm:

text {*
the formulation of well-typedness of method calls given below (as well as
the Java Specification 1.0) is a little too restrictive: Is does not allow
methods of class Object to be called upon references of interface type.

\begin{description}
\item[simplifications:]\ \\
\begin{itemize}
\item the type rules include all static checks on expressions and statements, 
  e.g.\ definedness of names (of parameters, locals, fields, methods)
\end{itemize}
\end{description}
*}
types	lenv (* local variables, including method parameters and This *)
	= "vname \<leadsto> ty"
        'c env
	= "'c prog \<times> lenv"

syntax
  prg    :: "'c env => 'c prog"
  localT :: "'c env => (vname \<leadsto> ty)"

translations	
  "prg"    => "fst"
  "localT" => "snd"

consts
  more_spec :: "'c prog => (ty \<times> 'x) \<times> ty list =>
                (ty \<times> 'x) \<times> ty list => bool"
  appl_methds :: "'c prog =>  cname => sig => ((ty \<times> ty) \<times> ty list) set"
  max_spec :: "'c prog =>  cname => sig => ((ty \<times> ty) \<times> ty list) set"

defs
  more_spec_def: "more_spec G == \<lambda>((d,h),pTs). \<lambda>((d',h'),pTs'). G\<turnstile>d\<preceq>d' \<and>
		                            list_all2 (\<lambda>T T'. G\<turnstile>T\<preceq>T') pTs pTs'"
  
  (* applicable methods, cf. 15.11.2.1 *)
  appl_methds_def: "appl_methds G C == \<lambda>(mn, pTs).
		                 {((Class md,rT),pTs') |md rT mb pTs'.
		                  method (G,C)  (mn, pTs') = Some (md,rT,mb) \<and>
		                  list_all2 (\<lambda>T T'. G\<turnstile>T\<preceq>T') pTs pTs'}"

  (* maximally specific methods, cf. 15.11.2.2 *)
  max_spec_def: "max_spec G C sig == {m. m \<in>appl_methds G C sig \<and> 
                                       (\<forall>m'\<in>appl_methds G C sig.
                                         more_spec G m' m --> m' = m)}"

lemma max_spec2appl_meths: 
  "x \<in> max_spec G C sig ==> x \<in> appl_methds G C sig"
apply (unfold max_spec_def)
apply (fast)
done

lemma appl_methsD: 
"((md,rT),pTs')\<in>appl_methds G C (mn, pTs) ==>  
  \<exists>D b. md = Class D \<and> method (G,C) (mn, pTs') = Some (D,rT,b)  
  \<and> list_all2 (\<lambda>T T'. G\<turnstile>T\<preceq>T') pTs pTs'"
apply (unfold appl_methds_def)
apply (fast)
done

lemmas max_spec2mheads = insertI1 [THEN [2] equalityD2 [THEN subsetD], 
                         THEN max_spec2appl_meths, THEN appl_methsD]


consts
  typeof :: "(loc => ty option) => val => ty option"

primrec
	"typeof dt  Unit    = Some (PrimT Void)"
	"typeof dt  Null    = Some NT"
	"typeof dt (Bool b) = Some (PrimT Boolean)"
	"typeof dt (Intg i) = Some (PrimT Integer)"
	"typeof dt (Addr a) = dt a"

lemma is_type_typeof [rule_format (no_asm), simp]: "(\<forall>a. v \<noteq> Addr a) --> (\<exists>T. typeof t v = Some T \<and> is_type G T)"
apply (rule val.induct)
apply     auto
done

lemma typeof_empty_is_type [rule_format (no_asm)]: 
  "typeof (\<lambda>a. None) v = Some T \<longrightarrow> is_type G T"
apply (rule val.induct)
apply     auto
done

types
	java_mb = "vname list \<times> (vname \<times> ty) list \<times> stmt \<times> expr"
	(* method body with parameter names, local variables, block, result expression *)

consts
  ty_expr :: "java_mb env => (expr      \<times> ty     ) set"
  ty_exprs:: "java_mb env => (expr list \<times> ty list) set"
  wt_stmt :: "java_mb env =>  stmt                 set"

syntax (xsymbols)
  ty_expr :: "java_mb env => [expr     , ty     ] => bool" ("_ \<turnstile> _ :: _"   [51,51,51]50)
  ty_exprs:: "java_mb env => [expr list, ty list] => bool" ("_ \<turnstile> _ [::] _" [51,51,51]50)
  wt_stmt :: "java_mb env =>  stmt                => bool" ("_ \<turnstile> _ \<surd>"      [51,51   ]50)

syntax
  ty_expr :: "java_mb env => [expr     , ty     ] => bool" ("_ |- _ :: _"   [51,51,51]50)
  ty_exprs:: "java_mb env => [expr list, ty list] => bool" ("_ |- _ [::] _" [51,51,51]50)
  wt_stmt :: "java_mb env =>  stmt                => bool" ("_ |- _ [ok]"   [51,51   ]50)


translations
	"E\<turnstile>e :: T" == "(e,T) \<in> ty_expr  E"
	"E\<turnstile>e[::]T" == "(e,T) \<in> ty_exprs E"
	"E\<turnstile>c \<surd>"    == "c     \<in> wt_stmt  E"
  
inductive "ty_expr E" "ty_exprs E" "wt_stmt E" intros

(* well-typed expressions *)

  (* cf. 15.8 *)
  NewC:	"[| is_class (prg E) C |] ==>
         E\<turnstile>NewC C::Class C"

  (* cf. 15.15 *)
  Cast:	"[| E\<turnstile>e::Class C; is_class (prg E) D;
            prg E\<turnstile>C\<preceq>? D |] ==>
         E\<turnstile>Cast D e::Class D"

  (* cf. 15.7.1 *)
  Lit:	  "[| typeof (\<lambda>v. None) x = Some T |] ==>
         E\<turnstile>Lit x::T"

  
  (* cf. 15.13.1 *)
  LAcc:	"[| localT E v = Some T; is_type (prg E) T |] ==>
         E\<turnstile>LAcc v::T"

  BinOp:"[| E\<turnstile>e1::T;
            E\<turnstile>e2::T;
            if bop = Eq then T' = PrimT Boolean
                        else T' = T \<and> T = PrimT Integer|] ==>
         E\<turnstile>BinOp bop e1 e2::T'"

  (* cf. 15.25, 15.25.1 *)
  LAss: "[| E\<turnstile>LAcc v::T;
	          E\<turnstile>e::T';
            prg E\<turnstile>T'\<preceq>T |] ==>
         E\<turnstile>v::=e::T'"

  (* cf. 15.10.1 *)
  FAcc: "[| E\<turnstile>a::Class C; 
            field (prg E,C) fn = Some (fd,fT) |] ==>
         E\<turnstile>{fd}a..fn::fT"

  (* cf. 15.25, 15.25.1 *)
  FAss: "[| E\<turnstile>{fd}a..fn::T;
            E\<turnstile>v        ::T';
            prg E\<turnstile>T'\<preceq>T |] ==>
         E\<turnstile>{fd}a..fn:=v::T'"


  (* cf. 15.11.1, 15.11.2, 15.11.3 *)
  Call: "[| E\<turnstile>a::Class C;
            E\<turnstile>ps[::]pTs;
            max_spec (prg E) C (mn, pTs) = {((md,rT),pTs')} |] ==>
         E\<turnstile>{C}a..mn({pTs'}ps)::rT"

(* well-typed expression lists *)

  (* cf. 15.11.??? *)
  Nil: "E\<turnstile>[][::][]"

  (* cf. 15.11.??? *)
  Cons:"[| E\<turnstile>e::T;
           E\<turnstile>es[::]Ts |] ==>
        E\<turnstile>e#es[::]T#Ts"

(* well-typed statements *)

  Skip:"E\<turnstile>Skip\<surd>"

  Expr:"[| E\<turnstile>e::T |] ==>
        E\<turnstile>Expr e\<surd>"

  Comp:"[| E\<turnstile>s1\<surd>; 
           E\<turnstile>s2\<surd> |] ==>
        E\<turnstile>s1;; s2\<surd>"

  (* cf. 14.8 *)
  Cond:"[| E\<turnstile>e::PrimT Boolean;
           E\<turnstile>s1\<surd>;
           E\<turnstile>s2\<surd> |] ==>
         E\<turnstile>If(e) s1 Else s2\<surd>"

  (* cf. 14.10 *)
  Loop:"[| E\<turnstile>e::PrimT Boolean;
           E\<turnstile>s\<surd> |] ==>
        E\<turnstile>While(e) s\<surd>"

constdefs

 wf_java_mdecl :: "java_mb prog => cname => java_mb mdecl => bool"
"wf_java_mdecl G C == \<lambda>((mn,pTs),rT,(pns,lvars,blk,res)).
	length pTs = length pns \<and>
	nodups pns \<and>
	unique lvars \<and>
	(\<forall>pn\<in>set pns. map_of lvars pn = None) \<and>
	(\<forall>(vn,T)\<in>set lvars. is_type G T) &
	(let E = (G,map_of lvars(pns[\<mapsto>]pTs)(This\<mapsto>Class C)) in
	 E\<turnstile>blk\<surd> \<and> (\<exists>T. E\<turnstile>res::T \<and> G\<turnstile>T\<preceq>rT))"

 wf_java_prog :: "java_mb prog => bool"
"wf_java_prog G == wf_prog wf_java_mdecl G"


lemma wt_is_type: "wf_prog wf_mb G \<Longrightarrow> ((G,L)\<turnstile>e::T \<longrightarrow> is_type G T) \<and>  
       ((G,L)\<turnstile>es[::]Ts \<longrightarrow> Ball (set Ts) (is_type G)) \<and> ((G,L)\<turnstile>c \<surd> \<longrightarrow> True)"
apply (rule ty_expr_ty_exprs_wt_stmt.induct)
apply auto
apply (   erule typeof_empty_is_type)
apply (  simp split add: split_if_asm)
apply ( drule field_fields)
apply ( drule (1) fields_is_type)
apply (  simp (no_asm_simp))
apply  (assumption)
apply (auto dest!: max_spec2mheads method_wf_mdecl is_type_rTI simp add: wf_mdecl_def)
done

lemmas ty_expr_is_type = wt_is_type [THEN conjunct1,THEN mp, COMP swap_prems_rl]

end