Theory TypeRel

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theory TypeRel
imports Decl
begin

(*  Title:      HOL/MicroJava/J/TypeRel.thy
    ID:         $Id: TypeRel.thy,v 1.27 2005/06/17 14:13:09 haftmann Exp $
    Author:     David von Oheimb
    Copyright   1999 Technische Universitaet Muenchen
*)

header {* \isaheader{Relations between Java Types} *}

theory TypeRel imports Decl begin

consts
  subcls1 :: "'c prog => (cname × cname) set"  -- "subclass"
  widen   :: "'c prog => (ty    × ty   ) set"  -- "widening"
  cast    :: "'c prog => (ty    × ty   ) set"  -- "casting"

syntax (xsymbols)
  subcls1 :: "'c prog => [cname, cname] => bool" ("_ \<turnstile> _ \<prec>C1 _" [71,71,71] 70)
  subcls  :: "'c prog => [cname, cname] => bool" ("_ \<turnstile> _ \<preceq>C _"  [71,71,71] 70)
  widen   :: "'c prog => [ty   , ty   ] => bool" ("_ \<turnstile> _ \<preceq> _"   [71,71,71] 70)
  cast    :: "'c prog => [ty   , ty   ] => bool" ("_ \<turnstile> _ \<preceq>? _"  [71,71,71] 70)

syntax
  subcls1 :: "'c prog => [cname, cname] => bool" ("_ |- _ <=C1 _" [71,71,71] 70)
  subcls  :: "'c prog => [cname, cname] => bool" ("_ |- _ <=C _"  [71,71,71] 70)
  widen   :: "'c prog => [ty   , ty   ] => bool" ("_ |- _ <= _"   [71,71,71] 70)
  cast    :: "'c prog => [ty   , ty   ] => bool" ("_ |- _ <=? _"  [71,71,71] 70)

translations
  "G\<turnstile>C \<prec>C1 D" == "(C,D) ∈ subcls1 G"
  "G\<turnstile>C \<preceq>C  D" == "(C,D) ∈ (subcls1 G)^*"
  "G\<turnstile>S \<preceq>   T" == "(S,T) ∈ widen   G"
  "G\<turnstile>C \<preceq>?  D" == "(C,D) ∈ cast    G"

-- "direct subclass, cf. 8.1.3"
inductive "subcls1 G" intros
  subcls1I: "[|class G C = Some (D,rest); C ≠ Object|] ==> G\<turnstile>C\<prec>C1D"
  
lemma subcls1D: 
  "G\<turnstile>C\<prec>C1D ==> C ≠ Object ∧ (∃fs ms. class G C = Some (D,fs,ms))"
apply (erule subcls1.elims)
apply auto
done

lemma subcls1_def2: 
  "subcls1 G = 
     (SIGMA C: {C. is_class G C} . {D. C≠Object ∧ fst (the (class G C))=D})"
  by (auto simp add: is_class_def dest: subcls1D intro: subcls1I)

lemma finite_subcls1: "finite (subcls1 G)"
apply(subst subcls1_def2)
apply(rule finite_SigmaI [OF finite_is_class])
apply(rule_tac B = "{fst (the (class G C))}" in finite_subset)
apply  auto
done

lemma subcls_is_class: "(C,D) ∈ (subcls1 G)^+ ==> is_class G C"
apply (unfold is_class_def)
apply(erule trancl_trans_induct)
apply (auto dest!: subcls1D)
done

lemma subcls_is_class2 [rule_format (no_asm)]: 
  "G\<turnstile>C\<preceq>C D ==> is_class G D --> is_class G C"
apply (unfold is_class_def)
apply (erule rtrancl_induct)
apply  (drule_tac [2] subcls1D)
apply  auto
done

constdefs
  class_rec :: "'c prog => cname => 'a =>
    (cname => fdecl list => 'c mdecl list => 'a => 'a) => 'a"
  "class_rec G == wfrec ((subcls1 G)^-1)
    (λr C t f. case class G C of
         None => arbitrary
       | Some (D,fs,ms) => 
           f C fs ms (if C = Object then t else r D t f))"

lemma class_rec_lemma: "wf ((subcls1 G)^-1) ==> class G C = Some (D,fs,ms) ==>
 class_rec G C t f = f C fs ms (if C=Object then t else class_rec G D t f)"
  by (simp add: class_rec_def wfrec cut_apply [OF converseI [OF subcls1I]])

consts

  method :: "'c prog × cname => ( sig   \<rightharpoonup> cname × ty × 'c)" (* ###curry *)
  field  :: "'c prog × cname => ( vname \<rightharpoonup> cname × ty     )" (* ###curry *)
  fields :: "'c prog × cname => ((vname × cname) × ty) list" (* ###curry *)

-- "methods of a class, with inheritance, overriding and hiding, cf. 8.4.6"
defs method_def: "method ≡ λ(G,C). class_rec G C empty (λC fs ms ts.
                           ts ++ map_of (map (λ(s,m). (s,(C,m))) ms))"

lemma method_rec_lemma: "[|class G C = Some (D,fs,ms); wf ((subcls1 G)^-1)|] ==>
  method (G,C) = (if C = Object then empty else method (G,D)) ++  
  map_of (map (λ(s,m). (s,(C,m))) ms)"
apply (unfold method_def)
apply (simp split del: split_if)
apply (erule (1) class_rec_lemma [THEN trans]);
apply auto
done


-- "list of fields of a class, including inherited and hidden ones"
defs fields_def: "fields ≡ λ(G,C). class_rec G C []    (λC fs ms ts.
                           map (λ(fn,ft). ((fn,C),ft)) fs @ ts)"

lemma fields_rec_lemma: "[|class G C = Some (D,fs,ms); wf ((subcls1 G)^-1)|] ==>
 fields (G,C) = 
  map (λ(fn,ft). ((fn,C),ft)) fs @ (if C = Object then [] else fields (G,D))"
apply (unfold fields_def)
apply (simp split del: split_if)
apply (erule (1) class_rec_lemma [THEN trans]);
apply auto
done


defs field_def: "field == map_of o (map (λ((fn,fd),ft). (fn,(fd,ft)))) o fields"

lemma field_fields: 
"field (G,C) fn = Some (fd, fT) ==> map_of (fields (G,C)) (fn, fd) = Some fT"
apply (unfold field_def)
apply (rule table_of_remap_SomeD)
apply simp
done


-- "widening, viz. method invocation conversion,cf. 5.3 i.e. sort of syntactic subtyping"
inductive "widen G" intros 
  refl   [intro!, simp]:       "G\<turnstile>      T \<preceq> T"   -- "identity conv., cf. 5.1.1"
  subcls         : "G\<turnstile>C\<preceq>C D ==> G\<turnstile>Class C \<preceq> Class D"
  null   [intro!]:             "G\<turnstile>     NT \<preceq> RefT R"

-- "casting conversion, cf. 5.5 / 5.1.5"
-- "left out casts on primitve types"
inductive "cast G" intros
  widen:  "G\<turnstile> C\<preceq> D ==> G\<turnstile>C \<preceq>? D"
  subcls: "G\<turnstile> D\<preceq>C C ==> G\<turnstile>Class C \<preceq>? Class D"

lemma widen_PrimT_RefT [iff]: "(G\<turnstile>PrimT pT\<preceq>RefT rT) = False"
apply (rule iffI)
apply (erule widen.elims)
apply auto
done

lemma widen_RefT: "G\<turnstile>RefT R\<preceq>T ==> ∃t. T=RefT t"
apply (ind_cases "G\<turnstile>S\<preceq>T")
apply auto
done

lemma widen_RefT2: "G\<turnstile>S\<preceq>RefT R ==> ∃t. S=RefT t"
apply (ind_cases "G\<turnstile>S\<preceq>T")
apply auto
done

lemma widen_Class: "G\<turnstile>Class C\<preceq>T ==> ∃D. T=Class D"
apply (ind_cases "G\<turnstile>S\<preceq>T")
apply auto
done

lemma widen_Class_NullT [iff]: "(G\<turnstile>Class C\<preceq>NT) = False"
apply (rule iffI)
apply (ind_cases "G\<turnstile>S\<preceq>T")
apply auto
done

lemma widen_Class_Class [iff]: "(G\<turnstile>Class C\<preceq> Class D) = (G\<turnstile>C\<preceq>C D)"
apply (rule iffI)
apply (ind_cases "G\<turnstile>S\<preceq>T")
apply (auto elim: widen.subcls)
done

lemma widen_NT_Class [simp]: "G \<turnstile> T \<preceq> NT ==> G \<turnstile> T \<preceq> Class D"
by (ind_cases "G \<turnstile> T \<preceq> NT",  auto)

lemma cast_PrimT_RefT [iff]: "(G\<turnstile>PrimT pT\<preceq>? RefT rT) = False"
apply (rule iffI)
apply (erule cast.elims)
apply auto
done

lemma cast_RefT: "G \<turnstile> C \<preceq>? Class D ==> ∃ rT. C = RefT rT"
apply (erule cast.cases)
apply simp apply (erule widen.cases) 
apply auto
done

theorem widen_trans[trans]: "[|G\<turnstile>S\<preceq>U; G\<turnstile>U\<preceq>T|] ==> G\<turnstile>S\<preceq>T"
proof -
  assume "G\<turnstile>S\<preceq>U" thus "!!T. G\<turnstile>U\<preceq>T ==> G\<turnstile>S\<preceq>T"
  proof induct
    case (refl T T') thus "G\<turnstile>T\<preceq>T'" .
  next
    case (subcls C D T)
    then obtain E where "T = Class E" by (blast dest: widen_Class)
    with subcls show "G\<turnstile>Class C\<preceq>T" by (auto elim: rtrancl_trans)
  next
    case (null R RT)
    then obtain rt where "RT = RefT rt" by (blast dest: widen_RefT)
    thus "G\<turnstile>NT\<preceq>RT" by auto
  qed
qed

end

lemma subcls1D:

  G |- C <=C1 D ==> C ≠ Object ∧ (∃fs ms. class G C = Some (D, fs, ms))

lemma subcls1_def2:

  subcls1 G =
  (SIGMA C:{C. is_class G C}. {D. C ≠ Object ∧ fst (the (class G C)) = D})

lemma finite_subcls1:

  finite (subcls1 G)

lemma subcls_is_class:

  (C, D) ∈ (subcls1 G)+ ==> is_class G C

lemma subcls_is_class2:

  [| G |- C <=C D; is_class G D |] ==> is_class G C

lemma class_rec_lemma:

  [| wf ((subcls1 G)^-1); class G C = Some (D, fs, ms) |]
  ==> class_rec G C t f = f C fs ms (if C = Object then t else class_rec G D t f)

lemma method_rec_lemma:

  [| class G C = Some (D, fs, ms); wf ((subcls1 G)^-1) |]
  ==> method (G, C) =
      (if C = Object then empty else method (G, D)) ++
      map_of (map (%(s, m). (s, C, m)) ms)

lemma fields_rec_lemma:

  [| class G C = Some (D, fs, ms); wf ((subcls1 G)^-1) |]
  ==> fields (G, C) =
      map (%(fn, ft). ((fn, C), ft)) fs @
      (if C = Object then [] else fields (G, D))

lemma field_fields:

  field (G, C) fn = Some (fd, fT) ==> map_of (fields (G, C)) (fn, fd) = Some fT

lemma widen_PrimT_RefT:

  G |- PrimT pT <= RefT rT = False

lemma widen_RefT:

  G |- RefT R <= T ==> ∃t. T = RefT t

lemma widen_RefT2:

  G |- S <= RefT R ==> ∃t. S = RefT t

lemma widen_Class:

  G |- Class C <= T ==> ∃D. T = Class D

lemma widen_Class_NullT:

  G |- Class C <= NT = False

lemma widen_Class_Class:

  G |- Class C <= Class D = G |- C <=C D

lemma widen_NT_Class:

  G |- T <= NT ==> G |- T <= Class D

lemma cast_PrimT_RefT:

  G |- PrimT pT <=? RefT rT = False

lemma cast_RefT:

  G |- C <=? Class D ==> ∃rT. C = RefT rT

theorem widen_trans:

  [| G |- S <= U; G |- U <= T |] ==> G |- S <= T