author  Andreas Lochbihler 
Fri, 05 Aug 2011 14:16:44 +0200  
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permissions  rwrr 
8011  1 
(* 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 imports Decl "~~/src/HOL/Library/Wfrec" 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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11266  47 
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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20970  76 
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[unfolded Collect_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_inline]: 
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"(subcls1 G)^* = (\<lambda>(C, D). subcls' G C D)" 
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by(simp add: subcls'_def subcls1_def rtrancl_def)(simp add: Collect_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" 
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hence wf: "wf (((subcls1 G)^+)^1)" unfolding wf_class_def by(rule wf_converse_trancl) 
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hence acyc: "acyclic ((subcls1 G)^+)" by(auto dest: wf_acyclic) 
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show "\<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(safe) 
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fix C D fs ms 
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assume "(C, D, fs, ms) \<in> set G" 
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and "C \<noteq> Object" 
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and subcls: "G \<turnstile> fst (the (class G C)) \<preceq>C C" 
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from `(C, D, fs, ms) \<in> set G` obtain D' fs' ms' 
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where "class": "class G C = Some (D', fs', ms')" 
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unfolding class_def by(auto dest!: weak_map_of_SomeI) 
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hence "G \<turnstile> C \<prec>C1 D'" using `C \<noteq> Object` .. 
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hence "(C, D') \<in> (subcls1 G)^+" .. 
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also with acyc have "C \<noteq> D'" by(auto simp add: acyclic_def) 
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with subcls "class" have "(D', C) \<in> (subcls1 G)^+" by(auto dest: rtranclD) 
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finally show False using acyc by(auto simp add: acyclic_def) 
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qed 
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next 
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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" 
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have "acyclic (subcls1 G)" 
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proof(intro acyclicI strip notI) 
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fix C 
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assume "(C, C) \<in> (subcls1 G)\<^sup>+" 
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thus False 
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proof(cases) 
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case base 
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then obtain rest where "class G C = Some (C, rest)" 
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and "C \<noteq> Object" by cases 
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from `class G C = Some (C, rest)` have "(C, C, rest) \<in> set G" 
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158 
unfolding class_def by(rule map_of_SomeD) 
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159 
with `C \<noteq> Object` `class G C = Some (C, rest)` 
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have "\<not> G \<turnstile> C \<preceq>C C" by(auto dest: rhs) 
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thus False by simp 
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next 
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case (step D) 
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from `G \<turnstile> D \<prec>C1 C` obtain rest where "class G D = Some (C, rest)" 
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165 
and "D \<noteq> Object" by cases 
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166 
from `class G D = Some (C, rest)` have "(D, C, rest) \<in> set G" 
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167 
unfolding class_def by(rule map_of_SomeD) 
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168 
with `D \<noteq> Object` `class G D = Some (C, rest)` 
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have "\<not> G \<turnstile> C \<preceq>C D" by(auto dest: rhs) 
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moreover from `(C, D) \<in> (subcls1 G)\<^sup>+` 
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171 
have "G \<turnstile> C \<preceq>C D" by(rule trancl_into_rtrancl) 
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172 
ultimately show False by contradiction 
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173 
qed 
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174 
qed 
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175 
thus "wf_class G" unfolding wf_class_def 
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176 
by(rule finite_acyclic_wf_converse[OF finite_subcls1]) 
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177 
qed 
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178 

8011  179 
consts 
14134  180 
method :: "'c prog \<times> cname => ( sig \<rightharpoonup> cname \<times> ty \<times> 'c)" (* ###curry *) 
181 
field :: "'c prog \<times> cname => ( vname \<rightharpoonup> cname \<times> ty )" (* ###curry *) 

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fields :: "'c prog \<times> cname => ((vname \<times> cname) \<times> ty) list" (* ###curry *) 
8011  183 

12517  184 
 "methods of a class, with inheritance, overriding and hiding, cf. 8.4.6" 
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defs method_def [code]: "method \<equiv> \<lambda>(G,C). class_rec G C empty (\<lambda>C fs ms ts. 
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ts ++ map_of (map (\<lambda>(s,m). (s,(C,m))) ms))" 
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187 

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lemma method_rec_lemma: "[class G C = Some (D,fs,ms); wf ((subcls1 G)^1)] ==> 
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method (G,C) = (if C = Object then empty else method (G,D)) ++ 
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190 
map_of (map (\<lambda>(s,m). (s,(C,m))) ms)" 
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191 
apply (unfold method_def) 
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192 
apply (simp split del: split_if) 
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193 
apply (erule (1) class_rec_lemma [THEN trans]); 
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194 
apply auto 
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195 
done 
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196 

8011  197 

12517  198 
 "list of fields of a class, including inherited and hidden ones" 
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defs fields_def [code]: "fields \<equiv> \<lambda>(G,C). class_rec G C [] (\<lambda>C fs ms ts. 
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200 
map (\<lambda>(fn,ft). ((fn,C),ft)) fs @ ts)" 
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201 

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lemma fields_rec_lemma: "[class G C = Some (D,fs,ms); wf ((subcls1 G)^1)] ==> 
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fields (G,C) = 
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map (\<lambda>(fn,ft). ((fn,C),ft)) fs @ (if C = Object then [] else fields (G,D))" 
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205 
apply (unfold fields_def) 
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206 
apply (simp split del: split_if) 
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207 
apply (erule (1) class_rec_lemma [THEN trans]); 
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208 
apply auto 
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209 
done 
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210 

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211 

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defs field_def [code]: "field == map_of o (map (\<lambda>((fn,fd),ft). (fn,(fd,ft)))) o fields" 
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213 

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214 
lemma field_fields: 
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"field (G,C) fn = Some (fd, fT) \<Longrightarrow> map_of (fields (G,C)) (fn, fd) = Some fT" 
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216 
apply (unfold field_def) 
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217 
apply (rule table_of_remap_SomeD) 
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218 
apply simp 
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219 
done 
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220 

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221 

12517  222 
 "widening, viz. method invocation conversion,cf. 5.3 i.e. sort of syntactic subtyping" 
23757  223 
inductive 
22271  224 
widen :: "'c prog => [ty , ty ] => bool" ("_ \<turnstile> _ \<preceq> _" [71,71,71] 70) 
225 
for G :: "'c prog" 

226 
where 

12517  227 
refl [intro!, simp]: "G\<turnstile> T \<preceq> T"  "identity conv., cf. 5.1.1" 
22271  228 
 subcls : "G\<turnstile>C\<preceq>C D ==> G\<turnstile>Class C \<preceq> Class D" 
229 
 null [intro!]: "G\<turnstile> NT \<preceq> RefT R" 

8011  230 

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code_pred widen . 
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232 

22597  233 
lemmas refl = HOL.refl 
234 

12517  235 
 "casting conversion, cf. 5.5 / 5.1.5" 
236 
 "left out casts on primitve types" 

23757  237 
inductive 
22271  238 
cast :: "'c prog => [ty , ty ] => bool" ("_ \<turnstile> _ \<preceq>? _" [71,71,71] 70) 
239 
for G :: "'c prog" 

240 
where 

14045  241 
widen: "G\<turnstile> C\<preceq> D ==> G\<turnstile>C \<preceq>? D" 
22271  242 
 subcls: "G\<turnstile> D\<preceq>C C ==> G\<turnstile>Class C \<preceq>? Class D" 
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243 

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244 
lemma widen_PrimT_RefT [iff]: "(G\<turnstile>PrimT pT\<preceq>RefT rT) = False" 
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245 
apply (rule iffI) 
22271  246 
apply (erule widen.cases) 
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247 
apply auto 
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248 
done 
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249 

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250 
lemma widen_RefT: "G\<turnstile>RefT R\<preceq>T ==> \<exists>t. T=RefT t" 
23757  251 
apply (ind_cases "G\<turnstile>RefT R\<preceq>T") 
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252 
apply auto 
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253 
done 
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254 

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255 
lemma widen_RefT2: "G\<turnstile>S\<preceq>RefT R ==> \<exists>t. S=RefT t" 
23757  256 
apply (ind_cases "G\<turnstile>S\<preceq>RefT R") 
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257 
apply auto 
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258 
done 
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259 

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260 
lemma widen_Class: "G\<turnstile>Class C\<preceq>T ==> \<exists>D. T=Class D" 
23757  261 
apply (ind_cases "G\<turnstile>Class C\<preceq>T") 
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262 
apply auto 
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263 
done 
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264 

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265 
lemma widen_Class_NullT [iff]: "(G\<turnstile>Class C\<preceq>NT) = False" 
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266 
apply (rule iffI) 
23757  267 
apply (ind_cases "G\<turnstile>Class C\<preceq>NT") 
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268 
apply auto 
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269 
done 
8011  270 

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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) 
23757  273 
apply (ind_cases "G\<turnstile>Class C \<preceq> Class D") 
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274 
apply (auto elim: widen.subcls) 
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275 
done 
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276 

14045  277 
lemma widen_NT_Class [simp]: "G \<turnstile> T \<preceq> NT \<Longrightarrow> G \<turnstile> T \<preceq> Class D" 
23757  278 
by (ind_cases "G \<turnstile> T \<preceq> NT", auto) 
14045  279 

280 
lemma cast_PrimT_RefT [iff]: "(G\<turnstile>PrimT pT\<preceq>? RefT rT) = False" 

281 
apply (rule iffI) 

22271  282 
apply (erule cast.cases) 
14045  283 
apply auto 
284 
done 

285 

286 
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 