author | krauss |
Tue, 02 Aug 2011 10:36:50 +0200 | |
changeset 44013 | 5cfc1c36ae97 |
parent 41589 | bbd861837ebc |
child 44014 | 88bd7d74a2c1 |
permissions | -rw-r--r-- |
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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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|
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header {* \isaheader{Relations between Java Types} *} |
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theory TypeRel imports Decl "~~/src/HOL/Library/Old_Recdef" 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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18 |
|
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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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22 |
|
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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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|
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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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|
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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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|
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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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||
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text {* Code generator setup (FIXME!) *} |
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81 |
|
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consts_code |
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"wfrec" ("\<module>wfrec?") |
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attach {* |
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fun wfrec f x = f (wfrec f) x; |
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*} |
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87 |
|
8011 | 88 |
consts |
89 |
||
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method :: "'c prog \<times> cname => ( sig \<rightharpoonup> cname \<times> ty \<times> 'c)" (* ###curry *) |
91 |
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 *) |
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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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97 |
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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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100 |
map_of (map (\<lambda>(s,m). (s,(C,m))) ms)" |
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101 |
apply (unfold method_def) |
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102 |
apply (simp split del: split_if) |
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103 |
apply (erule (1) class_rec_lemma [THEN trans]); |
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104 |
apply auto |
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105 |
done |
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106 |
|
8011 | 107 |
|
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-- "list of fields of a class, including inherited and hidden ones" |
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109 |
defs fields_def: "fields \<equiv> \<lambda>(G,C). class_rec G C [] (\<lambda>C fs ms ts. |
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110 |
map (\<lambda>(fn,ft). ((fn,C),ft)) fs @ ts)" |
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111 |
|
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112 |
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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114 |
map (\<lambda>(fn,ft). ((fn,C),ft)) fs @ (if C = Object then [] else fields (G,D))" |
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115 |
apply (unfold fields_def) |
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116 |
apply (simp split del: split_if) |
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117 |
apply (erule (1) class_rec_lemma [THEN trans]); |
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118 |
apply auto |
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119 |
done |
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120 |
|
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121 |
|
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122 |
defs field_def: "field == map_of o (map (\<lambda>((fn,fd),ft). (fn,(fd,ft)))) o fields" |
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123 |
|
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124 |
lemma field_fields: |
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125 |
"field (G,C) fn = Some (fd, fT) \<Longrightarrow> map_of (fields (G,C)) (fn, fd) = Some fT" |
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126 |
apply (unfold field_def) |
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127 |
apply (rule table_of_remap_SomeD) |
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128 |
apply simp |
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129 |
done |
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130 |
|
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131 |
|
12517 | 132 |
-- "widening, viz. method invocation conversion,cf. 5.3 i.e. sort of syntactic subtyping" |
23757 | 133 |
inductive |
22271 | 134 |
widen :: "'c prog => [ty , ty ] => bool" ("_ \<turnstile> _ \<preceq> _" [71,71,71] 70) |
135 |
for G :: "'c prog" |
|
136 |
where |
|
12517 | 137 |
refl [intro!, simp]: "G\<turnstile> T \<preceq> T" -- "identity conv., cf. 5.1.1" |
22271 | 138 |
| subcls : "G\<turnstile>C\<preceq>C D ==> G\<turnstile>Class C \<preceq> Class D" |
139 |
| null [intro!]: "G\<turnstile> NT \<preceq> RefT R" |
|
8011 | 140 |
|
22597 | 141 |
lemmas refl = HOL.refl |
142 |
||
12517 | 143 |
-- "casting conversion, cf. 5.5 / 5.1.5" |
144 |
-- "left out casts on primitve types" |
|
23757 | 145 |
inductive |
22271 | 146 |
cast :: "'c prog => [ty , ty ] => bool" ("_ \<turnstile> _ \<preceq>? _" [71,71,71] 70) |
147 |
for G :: "'c prog" |
|
148 |
where |
|
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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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151 |
|
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152 |
lemma widen_PrimT_RefT [iff]: "(G\<turnstile>PrimT pT\<preceq>RefT rT) = False" |
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153 |
apply (rule iffI) |
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apply (erule widen.cases) |
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155 |
apply auto |
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156 |
done |
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157 |
|
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158 |
lemma widen_RefT: "G\<turnstile>RefT R\<preceq>T ==> \<exists>t. T=RefT t" |
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apply (ind_cases "G\<turnstile>RefT R\<preceq>T") |
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160 |
apply auto |
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161 |
done |
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162 |
|
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163 |
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>RefT R") |
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165 |
apply auto |
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166 |
done |
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167 |
|
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168 |
lemma widen_Class: "G\<turnstile>Class C\<preceq>T ==> \<exists>D. T=Class D" |
23757 | 169 |
apply (ind_cases "G\<turnstile>Class C\<preceq>T") |
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170 |
apply auto |
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171 |
done |
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172 |
|
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173 |
lemma widen_Class_NullT [iff]: "(G\<turnstile>Class C\<preceq>NT) = False" |
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174 |
apply (rule iffI) |
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apply (ind_cases "G\<turnstile>Class C\<preceq>NT") |
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176 |
apply auto |
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177 |
done |
8011 | 178 |
|
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179 |
lemma widen_Class_Class [iff]: "(G\<turnstile>Class C\<preceq> Class D) = (G\<turnstile>C\<preceq>C D)" |
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180 |
apply (rule iffI) |
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apply (ind_cases "G\<turnstile>Class C \<preceq> Class D") |
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182 |
apply (auto elim: widen.subcls) |
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183 |
done |
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184 |
|
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lemma widen_NT_Class [simp]: "G \<turnstile> T \<preceq> NT \<Longrightarrow> G \<turnstile> T \<preceq> Class D" |
23757 | 186 |
by (ind_cases "G \<turnstile> T \<preceq> NT", auto) |
14045 | 187 |
|
188 |
lemma cast_PrimT_RefT [iff]: "(G\<turnstile>PrimT pT\<preceq>? RefT rT) = False" |
|
189 |
apply (rule iffI) |
|
22271 | 190 |
apply (erule cast.cases) |
14045 | 191 |
apply auto |
192 |
done |
|
193 |
||
194 |
lemma cast_RefT: "G \<turnstile> C \<preceq>? Class D \<Longrightarrow> \<exists> rT. C = RefT rT" |
|
195 |
apply (erule cast.cases) |
|
196 |
apply simp apply (erule widen.cases) |
|
197 |
apply auto |
|
198 |
done |
|
199 |
||
12517 | 200 |
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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201 |
proof - |
12517 | 202 |
assume "G\<turnstile>S\<preceq>U" thus "\<And>T. G\<turnstile>U\<preceq>T \<Longrightarrow> G\<turnstile>S\<preceq>T" |
11987 | 203 |
proof induct |
12517 | 204 |
case (refl T T') thus "G\<turnstile>T\<preceq>T'" . |
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205 |
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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209 |
next |
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case (null R RT) |
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211 |
then obtain rt where "RT = RefT rt" by (blast dest: widen_RefT) |
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212 |
thus "G\<turnstile>NT\<preceq>RT" by auto |
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213 |
qed |
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214 |
qed |
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215 |
|
8011 | 216 |
end |