author  streckem 
Mon, 26 May 2003 18:36:15 +0200  
changeset 14045  a34d89ce6097 
parent 13090  4fb7a2f2c1df 
child 14134  0fdf5708c7a8 
permissions  rwrr 
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(* Title: HOL/MicroJava/J/TypeRel.thy 
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ID: $Id$ 

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Author: David von Oheimb 

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Copyright 1999 Technische Universitaet Muenchen 

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*) 
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header {* \isaheader{Relations between Java Types} *} 
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theory TypeRel = Decl: 
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consts 

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subcls1 :: "'c prog => (cname \<times> cname) set"  "subclass" 
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widen :: "'c prog => (ty \<times> ty ) set"  "widening" 

14045  14 
cast :: "'c prog => (ty \<times> ty ) set"  "casting" 
8011  15 

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syntax (xsymbols) 
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subcls1 :: "'c prog => [cname, cname] => bool" ("_ \<turnstile> _ \<prec>C1 _" [71,71,71] 70) 
11372  18 
subcls :: "'c prog => [cname, cname] => bool" ("_ \<turnstile> _ \<preceq>C _" [71,71,71] 70) 
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widen :: "'c prog => [ty , ty ] => bool" ("_ \<turnstile> _ \<preceq> _" [71,71,71] 70) 

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cast :: "'c prog => [ty , ty ] => bool" ("_ \<turnstile> _ \<preceq>? _" [71,71,71] 70) 
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11372  22 
syntax 
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subcls1 :: "'c prog => [cname, cname] => bool" ("_  _ <=C1 _" [71,71,71] 70) 
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subcls :: "'c prog => [cname, cname] => bool" ("_  _ <=C _" [71,71,71] 70) 
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widen :: "'c prog => [ty , ty ] => bool" ("_  _ <= _" [71,71,71] 70) 

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cast :: "'c prog => [ty , ty ] => bool" ("_  _ <=? _" [71,71,71] 70) 
8011  27 

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translations 

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"G\<turnstile>C \<prec>C1 D" == "(C,D) \<in> subcls1 G" 
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"G\<turnstile>C \<preceq>C D" == "(C,D) \<in> (subcls1 G)^*" 
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"G\<turnstile>S \<preceq> T" == "(S,T) \<in> widen G" 
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"G\<turnstile>C \<preceq>? D" == "(C,D) \<in> cast G" 
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 "direct subclass, cf. 8.1.3" 
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inductive "subcls1 G" intros 
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subcls1I: "\<lbrakk>class G C = Some (D,rest); C \<noteq> Object\<rbrakk> \<Longrightarrow> G\<turnstile>C\<prec>C1D" 
8011  37 

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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.elims) 
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apply auto 
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done 
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lemma subcls1_def2: 
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"subcls1 G = (\<Sigma>C\<in>{C. is_class G C} . {D. C\<noteq>Object \<and> fst (the (class G 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(subst subcls1_def2) 
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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  61 
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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constdefs 
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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" 
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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> arbitrary 
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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))" 
11284  77 

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lemma class_rec_lemma: "wf ((subcls1 G)^1) \<Longrightarrow> class G C = Some (D,fs,ms) \<Longrightarrow> 
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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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by (simp add: class_rec_def wfrec cut_apply [OF converseI [OF subcls1I]]) 
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8011  82 
consts 
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method :: "'c prog \<times> cname => ( sig \<leadsto> cname \<times> ty \<times> 'c)" (* ###curry *) 
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field :: "'c prog \<times> cname => ( vname \<leadsto> cname \<times> ty )" (* ###curry *) 
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fields :: "'c prog \<times> cname => ((vname \<times> cname) \<times> ty) list" (* ###curry *) 
8011  87 

12517  88 
 "methods of a class, with inheritance, overriding and hiding, cf. 8.4.6" 
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defs method_def: "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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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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map_of (map (\<lambda>(s,m). (s,(C,m))) ms)" 
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apply (unfold method_def) 
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apply (simp split del: split_if) 
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apply (erule (1) class_rec_lemma [THEN trans]); 
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apply auto 
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done 
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8011  101 

12517  102 
 "list of fields of a class, including inherited and hidden ones" 
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defs fields_def: "fields \<equiv> \<lambda>(G,C). class_rec G C [] (\<lambda>C fs ms ts. 
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map (\<lambda>(fn,ft). ((fn,C),ft)) fs @ ts)" 
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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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apply (unfold fields_def) 
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apply (simp split del: split_if) 
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apply (erule (1) class_rec_lemma [THEN trans]); 
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apply auto 
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done 
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defs field_def: "field == map_of o (map (\<lambda>((fn,fd),ft). (fn,(fd,ft)))) o fields" 
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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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apply (unfold field_def) 
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apply (rule table_of_remap_SomeD) 
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apply simp 
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done 
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12517  126 
 "widening, viz. method invocation conversion,cf. 5.3 i.e. sort of syntactic subtyping" 
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inductive "widen G" intros 

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refl [intro!, simp]: "G\<turnstile> T \<preceq> T"  "identity conv., cf. 5.1.1" 

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subcls : "G\<turnstile>C\<preceq>C D ==> G\<turnstile>Class C \<preceq> Class D" 
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null [intro!]: "G\<turnstile> NT \<preceq> RefT R" 
8011  131 

12517  132 
 "casting conversion, cf. 5.5 / 5.1.5" 
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 "left out casts on primitve types" 

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inductive "cast G" intros 

14045  135 
widen: "G\<turnstile> C\<preceq> D ==> G\<turnstile>C \<preceq>? D" 
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subcls: "G\<turnstile> D\<preceq>C C ==> G\<turnstile>Class C \<preceq>? Class D" 

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lemma widen_PrimT_RefT [iff]: "(G\<turnstile>PrimT pT\<preceq>RefT rT) = False" 
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apply (rule iffI) 
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apply (erule widen.elims) 
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apply auto 
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done 
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lemma widen_RefT: "G\<turnstile>RefT R\<preceq>T ==> \<exists>t. T=RefT t" 
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apply (ind_cases "G\<turnstile>S\<preceq>T") 
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apply auto 
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done 
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lemma widen_RefT2: "G\<turnstile>S\<preceq>RefT R ==> \<exists>t. S=RefT t" 
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apply (ind_cases "G\<turnstile>S\<preceq>T") 
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apply auto 
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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>S\<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>S\<preceq>T") 
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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>S\<preceq>T") 
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apply (auto elim: widen.subcls) 
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done 
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14045  171 
lemma widen_NT_Class [simp]: "G \<turnstile> T \<preceq> NT \<Longrightarrow> G \<turnstile> T \<preceq> Class D" 
172 
by (ind_cases "G \<turnstile> T \<preceq> NT", auto) 

173 

174 
lemma cast_PrimT_RefT [iff]: "(G\<turnstile>PrimT pT\<preceq>? RefT rT) = False" 

175 
apply (rule iffI) 

176 
apply (erule cast.elims) 

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apply auto 

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done 

179 

180 
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 

184 
done 

185 

12517  186 
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  
12517  188 
assume "G\<turnstile>S\<preceq>U" thus "\<And>T. G\<turnstile>U\<preceq>T \<Longrightarrow> G\<turnstile>S\<preceq>T" 
11987  189 
proof induct 
12517  190 
case (refl T T') thus "G\<turnstile>T\<preceq>T'" . 
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next 
11987  192 
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 elim: rtrancl_trans) 
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next 
11987  196 
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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8011  202 
end 