author  kleing 
Sun, 16 Dec 2001 00:18:17 +0100  
changeset 12517  360e3215f029 
parent 12443  e56ab6134b41 
child 12911  704713ca07ea 
permissions  rwrr 
8011  1 
(* 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 "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" 

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cast :: "'c prog => (cname \<times> cname) 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) 
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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 => [cname, cname] => bool" ("_ \<turnstile> _ \<preceq>? _" [71,71,71] 70) 

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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 => [cname, cname] => bool" ("_  _ <=? _" [71,71,71] 70) 

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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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consts class_rec ::"'c prog \<times> cname \<Rightarrow> 
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'a \<Rightarrow> (cname \<Rightarrow> fdecl list \<Rightarrow> 'c mdecl list \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a" 
11266  71 

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recdef class_rec "same_fst (\<lambda>G. wf ((subcls1 G)^1)) (\<lambda>G. (subcls1 G)^1)" 
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"class_rec (G,C) = (\<lambda>t f. case class G C of None \<Rightarrow> arbitrary 
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 Some (D,fs,ms) \<Rightarrow> if wf ((subcls1 G)^1) then 
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f C fs ms (if C = Object then t else class_rec (G,D) t f) else arbitrary)" 
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(hints intro: subcls1I) 
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11266  78 
declare class_rec.simps [simp del] 
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11284  80 

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lemma class_rec_lemma: "\<lbrakk> wf ((subcls1 G)^1); class G C = Some (D,fs,ms)\<rbrakk> \<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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apply (rule class_rec.simps [THEN trans [THEN fun_cong [THEN fun_cong]]]) 
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apply simp 

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done 
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8011  87 
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 *) 
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 "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  106 

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 "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  131 
 "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" 
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12517  137 
 "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 

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widen: "G\<turnstile>C\<preceq>C D ==> G\<turnstile>C \<preceq>? D" 
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subcls: "G\<turnstile>D\<preceq>C C ==> G\<turnstile>C \<preceq>? 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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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 elim: rtrancl_trans) 
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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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8011  192 
end 