Theory TypeRel (Isabelle2005: October 2005)

# 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