Theory TypeRel

theory TypeRel
imports Decl
(*  Title:      HOL/MicroJava/J/TypeRel.thy
    Author:     David von Oheimb, Technische Universitaet Muenchen
*)

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

theory TypeRel
imports Decl
begin

-- "direct subclass, cf. 8.1.3"

inductive_set
  subcls1 :: "'c prog => (cname × cname) set"
  and subcls1' :: "'c prog => cname => cname => bool" ("_ \<turnstile> _ \<prec>C1 _" [71,71,71] 70)
  for G :: "'c prog"
where
  "G \<turnstile> C \<prec>C1 D ≡ (C, D) ∈ subcls1 G"
  | subcls1I: "[|class G C = Some (D,rest); C ≠ Object|] ==> G \<turnstile> C \<prec>C1 D"

abbreviation
  subcls  :: "'c prog => cname => cname => bool" ("_ \<turnstile> _ \<preceq>C _"  [71,71,71] 70)
  where "G \<turnstile> C \<preceq>C D ≡ (C, D) ∈ (subcls1 G)^*"

lemma subcls1D: 
  "G\<turnstile>C\<prec>C1D ==> C ≠ Object ∧ (∃fs ms. class G C = Some (D,fs,ms))"
apply (erule subcls1.cases)
apply auto
done

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

lemma finite_subcls1: "finite (subcls1 G)"
apply(simp add: subcls1_def2 del: mem_Sigma_iff)
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

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

lemma class_rec_lemma:
  assumes wf: "wf ((subcls1 G)^-1)"
    and cls: "class G C = Some (D, fs, ms)"
  shows "class_rec G C t f = f C fs ms (if C=Object then t else class_rec G D t f)"
proof -
  from wf have step: "!!H a. wfrec ((subcls1 G)¯) H a =
    H (cut (wfrec ((subcls1 G)¯) H) ((subcls1 G)¯) a) a"
    by (rule wfrec)
  have cut: "!!f. C ≠ Object ==> cut f ((subcls1 G)¯) C D = f D"
    by (rule cut_apply [where r="(subcls1 G)^-1", simplified, OF subcls1I, OF cls])
  from cls show ?thesis by (simp add: step cut class_rec_def)
qed

definition
  "wf_class G = wf ((subcls1 G)^-1)"



text {* Code generator setup *}

code_pred 
  (modes: i => i => o => bool, i => i => i => bool)
  subcls1p 
  .

declare subcls1_def [code_pred_def]

code_pred 
  (modes: i => i × o => bool, i => i × i => bool)
  [inductify]
  subcls1 
  .

definition subcls' where "subcls' G = (subcls1p G)^**"

code_pred
  (modes: i => i => i => bool, i => i => o => bool)
  [inductify]
  subcls'
  .

lemma subcls_conv_subcls' [code_unfold]:
  "(subcls1 G)^* = {(C, D). subcls' G C D}"
by(simp add: subcls'_def subcls1_def rtrancl_def)

lemma class_rec_code [code]:
  "class_rec G C t f = 
  (if wf_class G then 
    (case class G C of
       None => class_rec G C t f
     | Some (D, fs, ms) => 
       if C = Object then f Object fs ms t else f C fs ms (class_rec G D t f))
   else class_rec G C t f)"
apply(cases "wf_class G")
 apply(unfold class_rec_def wf_class_def)
 apply(subst wfrec, assumption)
 apply(cases "class G C")
  apply(simp add: wfrec)
 apply clarsimp
 apply(rename_tac D fs ms)
 apply(rule_tac f="f C fs ms" in arg_cong)
 apply(clarsimp simp add: cut_def)
 apply(blast intro: subcls1I)
apply simp
done

lemma wf_class_code [code]:
  "wf_class G <-> (∀(C, rest) ∈ set G. C ≠ Object --> ¬ G \<turnstile> fst (the (class G C)) \<preceq>C C)"
proof
  assume "wf_class G"
  hence wf: "wf (((subcls1 G)^+)^-1)" unfolding wf_class_def by(rule wf_converse_trancl)
  hence acyc: "acyclic ((subcls1 G)^+)" by(auto dest: wf_acyclic)
  show "∀(C, rest) ∈ set G. C ≠ Object --> ¬ G \<turnstile> fst (the (class G C)) \<preceq>C C"
  proof(safe)
    fix C D fs ms
    assume "(C, D, fs, ms) ∈ set G"
      and "C ≠ Object"
      and subcls: "G \<turnstile> fst (the (class G C)) \<preceq>C C"
    from `(C, D, fs, ms) ∈ set G` obtain D' fs' ms'
      where "class": "class G C = Some (D', fs', ms')"
      unfolding class_def by(auto dest!: weak_map_of_SomeI)
    hence "G \<turnstile> C \<prec>C1 D'" using `C ≠ Object` ..
    hence *: "(C, D') ∈ (subcls1 G)^+" ..
    also from * acyc have "C ≠ D'" by(auto simp add: acyclic_def)
    with subcls "class" have "(D', C) ∈ (subcls1 G)^+" by(auto dest: rtranclD)
    finally show False using acyc by(auto simp add: acyclic_def)
  qed
next
  assume rhs[rule_format]: "∀(C, rest) ∈ set G. C ≠ Object --> ¬ G \<turnstile> fst (the (class G C)) \<preceq>C C"
  have "acyclic (subcls1 G)"
  proof(intro acyclicI strip notI)
    fix C
    assume "(C, C) ∈ (subcls1 G)+"
    thus False
    proof(cases)
      case base
      then obtain rest where "class G C = Some (C, rest)"
        and "C ≠ Object" by cases
      from `class G C = Some (C, rest)` have "(C, C, rest) ∈ set G"
        unfolding class_def by(rule map_of_SomeD)
      with `C ≠ Object` `class G C = Some (C, rest)`
      have "¬ G \<turnstile> C \<preceq>C C" by(auto dest: rhs)
      thus False by simp
    next
      case (step D)
      from `G \<turnstile> D \<prec>C1 C` obtain rest where "class G D = Some (C, rest)"
        and "D ≠ Object" by cases
      from `class G D = Some (C, rest)` have "(D, C, rest) ∈ set G"
        unfolding class_def by(rule map_of_SomeD)
      with `D ≠ Object` `class G D = Some (C, rest)`
      have "¬ G \<turnstile> C \<preceq>C D" by(auto dest: rhs)
      moreover from `(C, D) ∈ (subcls1 G)+`
      have "G \<turnstile> C \<preceq>C D" by(rule trancl_into_rtrancl)
      ultimately show False by contradiction
    qed
  qed
  thus "wf_class G" unfolding wf_class_def
    by(rule finite_acyclic_wf_converse[OF finite_subcls1])
qed

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 [code]: "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 [code]: "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 [code]: "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   :: "'c prog => [ty   , ty   ] => bool" ("_ \<turnstile> _ \<preceq> _"   [71,71,71] 70)
  for G :: "'c prog"
where
  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"

code_pred widen .

lemmas refl = HOL.refl

-- "casting conversion, cf. 5.5 / 5.1.5"
-- "left out casts on primitve types"
inductive
  cast    :: "'c prog => [ty   , ty   ] => bool" ("_ \<turnstile> _ \<preceq>? _"  [71,71,71] 70)
  for G :: "'c prog"
where
  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.cases)
apply auto
done

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

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