src/HOL/MicroJava/J/TypeRel.thy
author wenzelm
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explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions; tuned;
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(*  Title:      HOL/MicroJava/J/TypeRel.thy
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    Author:     David von Oheimb, Technische Universitaet Muenchen
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*)
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header {* \isaheader{Relations between Java Types} *}
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theory TypeRel
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imports Decl "~~/src/HOL/Library/Wfrec"
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begin
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-- "direct subclass, cf. 8.1.3"
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inductive_set
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  subcls1 :: "'c prog => (cname \<times> cname) set"
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  and subcls1' :: "'c prog => cname \<Rightarrow> cname => bool" ("_ \<turnstile> _ \<prec>C1 _" [71,71,71] 70)
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  for G :: "'c prog"
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where
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  "G \<turnstile> C \<prec>C1 D \<equiv> (C, D) \<in> subcls1 G"
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  | subcls1I: "\<lbrakk>class G C = Some (D,rest); C \<noteq> Object\<rbrakk> \<Longrightarrow> G \<turnstile> C \<prec>C1 D"
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abbreviation
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  subcls  :: "'c prog => cname \<Rightarrow> cname => bool" ("_ \<turnstile> _ \<preceq>C _"  [71,71,71] 70)
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  where "G \<turnstile> C \<preceq>C D \<equiv> (C, D) \<in> (subcls1 G)^*"
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lemma subcls1D: 
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  "G\<turnstile>C\<prec>C1D \<Longrightarrow> C \<noteq> Object \<and> (\<exists>fs ms. class G C = Some (D,fs,ms))"
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apply (erule subcls1.cases)
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apply auto
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done
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lemma subcls1_def2:
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  "subcls1 P =
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     (SIGMA C:{C. is_class P C}. {D. C\<noteq>Object \<and> fst (the (class P C))=D})"
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  by (auto simp add: is_class_def dest: subcls1D intro: subcls1I)
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lemma finite_subcls1: "finite (subcls1 G)"
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apply(simp add: subcls1_def2 del: mem_Sigma_iff)
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apply(rule finite_SigmaI [OF finite_is_class])
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apply(rule_tac B = "{fst (the (class G C))}" in finite_subset)
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apply  auto
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done
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lemma subcls_is_class: "(C, D) \<in> (subcls1 G)^+  ==> is_class G C"
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apply (unfold is_class_def)
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apply(erule trancl_trans_induct)
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apply (auto dest!: subcls1D)
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done
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lemma subcls_is_class2 [rule_format (no_asm)]: 
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  "G\<turnstile>C\<preceq>C D \<Longrightarrow> is_class G D \<longrightarrow> is_class G C"
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apply (unfold is_class_def)
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apply (erule rtrancl_induct)
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apply  (drule_tac [2] subcls1D)
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apply  auto
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done
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definition class_rec :: "'c prog \<Rightarrow> cname \<Rightarrow> 'a \<Rightarrow>
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    (cname \<Rightarrow> fdecl list \<Rightarrow> 'c mdecl list \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a" where
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  "class_rec G == wfrec ((subcls1 G)^-1)
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    (\<lambda>r C t f. case class G C of
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         None \<Rightarrow> undefined
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       | Some (D,fs,ms) \<Rightarrow> 
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           f C fs ms (if C = Object then t else r D t f))"
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lemma class_rec_lemma:
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  assumes wf: "wf ((subcls1 G)^-1)"
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    and cls: "class G C = Some (D, fs, ms)"
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  shows "class_rec G C t f = f C fs ms (if C=Object then t else class_rec G D t f)"
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proof -
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  from wf have step: "\<And>H a. wfrec ((subcls1 G)\<inverse>) H a =
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    H (cut (wfrec ((subcls1 G)\<inverse>) H) ((subcls1 G)\<inverse>) a) a"
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    by (rule wfrec)
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  have cut: "\<And>f. C \<noteq> Object \<Longrightarrow> cut f ((subcls1 G)\<inverse>) C D = f D"
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    by (rule cut_apply [where r="(subcls1 G)^-1", simplified, OF subcls1I, OF cls])
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  from cls show ?thesis by (simp add: step cut class_rec_def)
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qed
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definition
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  "wf_class G = wf ((subcls1 G)^-1)"
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text {* Code generator setup *}
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code_pred 
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  (modes: i \<Rightarrow> i \<Rightarrow> o \<Rightarrow> bool, i \<Rightarrow> i \<Rightarrow> i \<Rightarrow> bool)
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  subcls1p 
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  .
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declare subcls1_def [code_pred_def]
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code_pred 
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  (modes: i \<Rightarrow> i \<times> o \<Rightarrow> bool, i \<Rightarrow> i \<times> i \<Rightarrow> bool)
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  [inductify]
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  subcls1 
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  .
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definition subcls' where "subcls' G = (subcls1p G)^**"
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code_pred
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  (modes: i \<Rightarrow> i \<Rightarrow> i \<Rightarrow> bool, i \<Rightarrow> i \<Rightarrow> o \<Rightarrow> bool)
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  [inductify]
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  subcls'
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  .
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lemma subcls_conv_subcls' [code_unfold]:
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  "(subcls1 G)^* = {(C, D). subcls' G C D}"
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by(simp add: subcls'_def subcls1_def rtrancl_def)
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lemma class_rec_code [code]:
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  "class_rec G C t f = 
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  (if wf_class G then 
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    (case class G C of
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       None \<Rightarrow> class_rec G C t f
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     | Some (D, fs, ms) \<Rightarrow> 
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       if C = Object then f Object fs ms t else f C fs ms (class_rec G D t f))
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   else class_rec G C t f)"
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apply(cases "wf_class G")
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 apply(unfold class_rec_def wf_class_def)
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 apply(subst wfrec, assumption)
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 apply(cases "class G C")
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  apply(simp add: wfrec)
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 apply clarsimp
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 apply(rename_tac D fs ms)
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 apply(rule_tac f="f C fs ms" in arg_cong)
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 apply(clarsimp simp add: cut_def)
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 apply(blast intro: subcls1I)
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apply simp
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done
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lemma wf_class_code [code]:
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  "wf_class G \<longleftrightarrow> (\<forall>(C, rest) \<in> set G. C \<noteq> Object \<longrightarrow> \<not> G \<turnstile> fst (the (class G C)) \<preceq>C C)"
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proof
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  assume "wf_class G"
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
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   135
  hence wf: "wf (((subcls1 G)^+)^-1)" unfolding wf_class_def by(rule wf_converse_trancl)
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
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diff changeset
   136
  hence acyc: "acyclic ((subcls1 G)^+)" by(auto dest: wf_acyclic)
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
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parents: 44014
diff changeset
   137
  show "\<forall>(C, rest) \<in> set G. C \<noteq> Object \<longrightarrow> \<not> G \<turnstile> fst (the (class G C)) \<preceq>C C"
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   138
  proof(safe)
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   139
    fix C D fs ms
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
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diff changeset
   140
    assume "(C, D, fs, ms) \<in> set G"
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
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diff changeset
   141
      and "C \<noteq> Object"
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diff changeset
   142
      and subcls: "G \<turnstile> fst (the (class G C)) \<preceq>C C"
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   143
    from `(C, D, fs, ms) \<in> set G` obtain D' fs' ms'
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
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diff changeset
   144
      where "class": "class G C = Some (D', fs', ms')"
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
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diff changeset
   145
      unfolding class_def by(auto dest!: weak_map_of_SomeI)
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   146
    hence "G \<turnstile> C \<prec>C1 D'" using `C \<noteq> Object` ..
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
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diff changeset
   147
    hence "(C, D') \<in> (subcls1 G)^+" ..
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
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diff changeset
   148
    also with acyc have "C \<noteq> D'" by(auto simp add: acyclic_def)
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
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diff changeset
   149
    with subcls "class" have "(D', C) \<in> (subcls1 G)^+" by(auto dest: rtranclD)
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
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   150
    finally show False using acyc by(auto simp add: acyclic_def)
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
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   151
  qed
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
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   152
next
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
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diff changeset
   153
  assume rhs[rule_format]: "\<forall>(C, rest) \<in> set G. C \<noteq> Object \<longrightarrow> \<not> G \<turnstile> fst (the (class G C)) \<preceq>C C"
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
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diff changeset
   154
  have "acyclic (subcls1 G)"
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   155
  proof(intro acyclicI strip notI)
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
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diff changeset
   156
    fix C
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
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parents: 44014
diff changeset
   157
    assume "(C, C) \<in> (subcls1 G)\<^sup>+"
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
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diff changeset
   158
    thus False
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   159
    proof(cases)
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
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diff changeset
   160
      case base
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
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diff changeset
   161
      then obtain rest where "class G C = Some (C, rest)"
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
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diff changeset
   162
        and "C \<noteq> Object" by cases
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
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parents: 44014
diff changeset
   163
      from `class G C = Some (C, rest)` have "(C, C, rest) \<in> set G"
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
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parents: 44014
diff changeset
   164
        unfolding class_def by(rule map_of_SomeD)
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
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parents: 44014
diff changeset
   165
      with `C \<noteq> Object` `class G C = Some (C, rest)`
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   166
      have "\<not> G \<turnstile> C \<preceq>C C" by(auto dest: rhs)
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   167
      thus False by simp
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   168
    next
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
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diff changeset
   169
      case (step D)
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
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parents: 44014
diff changeset
   170
      from `G \<turnstile> D \<prec>C1 C` obtain rest where "class G D = Some (C, rest)"
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   171
        and "D \<noteq> Object" by cases
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   172
      from `class G D = Some (C, rest)` have "(D, C, rest) \<in> set G"
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   173
        unfolding class_def by(rule map_of_SomeD)
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   174
      with `D \<noteq> Object` `class G D = Some (C, rest)`
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   175
      have "\<not> G \<turnstile> C \<preceq>C D" by(auto dest: rhs)
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   176
      moreover from `(C, D) \<in> (subcls1 G)\<^sup>+`
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   177
      have "G \<turnstile> C \<preceq>C D" by(rule trancl_into_rtrancl)
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   178
      ultimately show False by contradiction
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   179
    qed
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   180
  qed
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   181
  thus "wf_class G" unfolding wf_class_def
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   182
    by(rule finite_acyclic_wf_converse[OF finite_subcls1])
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   183
qed
32461
eee4fa79398f no consts_code for wfrec, as it violates the "code generation = equational reasoning" principle
krauss
parents: 28562
diff changeset
   184
8011
d14c4e9e9c8e *** empty log message ***
nipkow
parents:
diff changeset
   185
consts
14134
0fdf5708c7a8 Replaced \<leadsto> by \<rightharpoonup>
nipkow
parents: 14045
diff changeset
   186
  method :: "'c prog \<times> cname => ( sig   \<rightharpoonup> cname \<times> ty \<times> 'c)" (* ###curry *)
0fdf5708c7a8 Replaced \<leadsto> by \<rightharpoonup>
nipkow
parents: 14045
diff changeset
   187
  field  :: "'c prog \<times> cname => ( vname \<rightharpoonup> cname \<times> ty     )" (* ###curry *)
11026
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   188
  fields :: "'c prog \<times> cname => ((vname \<times> cname) \<times> ty) list" (* ###curry *)
8011
d14c4e9e9c8e *** empty log message ***
nipkow
parents:
diff changeset
   189
12517
360e3215f029 exception merge, cleanup, tuned
kleing
parents: 12443
diff changeset
   190
-- "methods of a class, with inheritance, overriding and hiding, cf. 8.4.6"
44035
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   191
defs method_def [code]: "method \<equiv> \<lambda>(G,C). class_rec G C empty (\<lambda>C fs ms ts.
11026
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   192
                           ts ++ map_of (map (\<lambda>(s,m). (s,(C,m))) ms))"
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   193
33954
1bc3b688548c backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents: 32461
diff changeset
   194
lemma method_rec_lemma: "[|class G C = Some (D,fs,ms); wf ((subcls1 G)^-1)|] ==>
11026
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   195
  method (G,C) = (if C = Object then empty else method (G,D)) ++  
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   196
  map_of (map (\<lambda>(s,m). (s,(C,m))) ms)"
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   197
apply (unfold method_def)
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   198
apply (simp split del: split_if)
45970
b6d0cff57d96 adjusted to set/pred distinction by means of type constructor `set`
haftmann
parents: 45231
diff changeset
   199
apply (erule (1) class_rec_lemma [THEN trans])
11026
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   200
apply auto
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   201
done
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   202
8011
d14c4e9e9c8e *** empty log message ***
nipkow
parents:
diff changeset
   203
12517
360e3215f029 exception merge, cleanup, tuned
kleing
parents: 12443
diff changeset
   204
-- "list of fields of a class, including inherited and hidden ones"
44035
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   205
defs fields_def [code]: "fields \<equiv> \<lambda>(G,C). class_rec G C []    (\<lambda>C fs ms ts.
11026
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   206
                           map (\<lambda>(fn,ft). ((fn,C),ft)) fs @ ts)"
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   207
33954
1bc3b688548c backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents: 32461
diff changeset
   208
lemma fields_rec_lemma: "[|class G C = Some (D,fs,ms); wf ((subcls1 G)^-1)|] ==>
11026
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   209
 fields (G,C) = 
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   210
  map (\<lambda>(fn,ft). ((fn,C),ft)) fs @ (if C = Object then [] else fields (G,D))"
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   211
apply (unfold fields_def)
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   212
apply (simp split del: split_if)
45970
b6d0cff57d96 adjusted to set/pred distinction by means of type constructor `set`
haftmann
parents: 45231
diff changeset
   213
apply (erule (1) class_rec_lemma [THEN trans])
11026
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   214
apply auto
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   215
done
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   216
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   217
44035
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   218
defs field_def [code]: "field == map_of o (map (\<lambda>((fn,fd),ft). (fn,(fd,ft)))) o fields"
11026
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   219
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   220
lemma field_fields: 
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   221
"field (G,C) fn = Some (fd, fT) \<Longrightarrow> map_of (fields (G,C)) (fn, fd) = Some fT"
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   222
apply (unfold field_def)
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   223
apply (rule table_of_remap_SomeD)
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   224
apply simp
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   225
done
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   226
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   227
12517
360e3215f029 exception merge, cleanup, tuned
kleing
parents: 12443
diff changeset
   228
-- "widening, viz. method invocation conversion,cf. 5.3 i.e. sort of syntactic subtyping"
23757
087b0a241557 - Renamed inductive2 to inductive
berghofe
parents: 22597
diff changeset
   229
inductive
22271
51a80e238b29 Adapted to new inductive definition package.
berghofe
parents: 20970
diff changeset
   230
  widen   :: "'c prog => [ty   , ty   ] => bool" ("_ \<turnstile> _ \<preceq> _"   [71,71,71] 70)
51a80e238b29 Adapted to new inductive definition package.
berghofe
parents: 20970
diff changeset
   231
  for G :: "'c prog"
51a80e238b29 Adapted to new inductive definition package.
berghofe
parents: 20970
diff changeset
   232
where
12517
360e3215f029 exception merge, cleanup, tuned
kleing
parents: 12443
diff changeset
   233
  refl   [intro!, simp]:       "G\<turnstile>      T \<preceq> T"   -- "identity conv., cf. 5.1.1"
22271
51a80e238b29 Adapted to new inductive definition package.
berghofe
parents: 20970
diff changeset
   234
| subcls         : "G\<turnstile>C\<preceq>C D ==> G\<turnstile>Class C \<preceq> Class D"
51a80e238b29 Adapted to new inductive definition package.
berghofe
parents: 20970
diff changeset
   235
| null   [intro!]:             "G\<turnstile>     NT \<preceq> RefT R"
8011
d14c4e9e9c8e *** empty log message ***
nipkow
parents:
diff changeset
   236
44035
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   237
code_pred widen .
322d1657c40c replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents: 44014
diff changeset
   238
22597
284b2183d070 rebind HOL.refl as refl (hidden by widen.refl);
wenzelm
parents: 22271
diff changeset
   239
lemmas refl = HOL.refl
284b2183d070 rebind HOL.refl as refl (hidden by widen.refl);
wenzelm
parents: 22271
diff changeset
   240
12517
360e3215f029 exception merge, cleanup, tuned
kleing
parents: 12443
diff changeset
   241
-- "casting conversion, cf. 5.5 / 5.1.5"
360e3215f029 exception merge, cleanup, tuned
kleing
parents: 12443
diff changeset
   242
-- "left out casts on primitve types"
23757
087b0a241557 - Renamed inductive2 to inductive
berghofe
parents: 22597
diff changeset
   243
inductive
22271
51a80e238b29 Adapted to new inductive definition package.
berghofe
parents: 20970
diff changeset
   244
  cast    :: "'c prog => [ty   , ty   ] => bool" ("_ \<turnstile> _ \<preceq>? _"  [71,71,71] 70)
51a80e238b29 Adapted to new inductive definition package.
berghofe
parents: 20970
diff changeset
   245
  for G :: "'c prog"
51a80e238b29 Adapted to new inductive definition package.
berghofe
parents: 20970
diff changeset
   246
where
14045
a34d89ce6097 Introduced distinction wf_prog vs. ws_prog
streckem
parents: 13090
diff changeset
   247
  widen:  "G\<turnstile> C\<preceq> D ==> G\<turnstile>C \<preceq>? D"
22271
51a80e238b29 Adapted to new inductive definition package.
berghofe
parents: 20970
diff changeset
   248
| subcls: "G\<turnstile> D\<preceq>C C ==> G\<turnstile>Class C \<preceq>? Class D"
11026
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oheimb
parents: 10613
diff changeset
   249
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   250
lemma widen_PrimT_RefT [iff]: "(G\<turnstile>PrimT pT\<preceq>RefT rT) = False"
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   251
apply (rule iffI)
22271
51a80e238b29 Adapted to new inductive definition package.
berghofe
parents: 20970
diff changeset
   252
apply (erule widen.cases)
11026
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   253
apply auto
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   254
done
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   255
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   256
lemma widen_RefT: "G\<turnstile>RefT R\<preceq>T ==> \<exists>t. T=RefT t"
23757
087b0a241557 - Renamed inductive2 to inductive
berghofe
parents: 22597
diff changeset
   257
apply (ind_cases "G\<turnstile>RefT R\<preceq>T")
11026
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   258
apply auto
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   259
done
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   260
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   261
lemma widen_RefT2: "G\<turnstile>S\<preceq>RefT R ==> \<exists>t. S=RefT t"
23757
087b0a241557 - Renamed inductive2 to inductive
berghofe
parents: 22597
diff changeset
   262
apply (ind_cases "G\<turnstile>S\<preceq>RefT R")
11026
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   263
apply auto
a50365d21144 converted to Isar, simplifying recursion on class hierarchy
oheimb
parents: 10613
diff changeset
   264
done
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lemma widen_Class: "G\<turnstile>Class C\<preceq>T ==> \<exists>D. T=Class D"
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apply (ind_cases "G\<turnstile>Class C\<preceq>T")
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apply auto
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done
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lemma widen_Class_NullT [iff]: "(G\<turnstile>Class C\<preceq>NT) = False"
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apply (rule iffI)
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apply (ind_cases "G\<turnstile>Class C\<preceq>NT")
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apply auto
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done
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lemma widen_Class_Class [iff]: "(G\<turnstile>Class C\<preceq> Class D) = (G\<turnstile>C\<preceq>C D)"
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apply (rule iffI)
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apply (ind_cases "G\<turnstile>Class C \<preceq> Class D")
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apply (auto elim: widen.subcls)
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done
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lemma widen_NT_Class [simp]: "G \<turnstile> T \<preceq> NT \<Longrightarrow> G \<turnstile> T \<preceq> Class D"
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by (ind_cases "G \<turnstile> T \<preceq> NT",  auto)
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lemma cast_PrimT_RefT [iff]: "(G\<turnstile>PrimT pT\<preceq>? RefT rT) = False"
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apply (rule iffI)
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apply (erule cast.cases)
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apply auto
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done
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lemma cast_RefT: "G \<turnstile> C \<preceq>? Class D \<Longrightarrow> \<exists> rT. C = RefT rT"
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apply (erule cast.cases)
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apply simp apply (erule widen.cases) 
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apply auto
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done
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theorem widen_trans[trans]: "\<lbrakk>G\<turnstile>S\<preceq>U; G\<turnstile>U\<preceq>T\<rbrakk> \<Longrightarrow> G\<turnstile>S\<preceq>T"
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proof -
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  assume "G\<turnstile>S\<preceq>U" thus "\<And>T. G\<turnstile>U\<preceq>T \<Longrightarrow> G\<turnstile>S\<preceq>T"
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  proof induct
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    case (refl T T') thus "G\<turnstile>T\<preceq>T'" .
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  next
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    case (subcls C D T)
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    then obtain E where "T = Class E" by (blast dest: widen_Class)
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    with subcls show "G\<turnstile>Class C\<preceq>T" by auto
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  next
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    case (null R RT)
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    then obtain rt where "RT = RefT rt" by (blast dest: widen_RefT)
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    thus "G\<turnstile>NT\<preceq>RT" by auto
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  qed
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qed
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end