author | wenzelm |
Thu, 01 Sep 2016 16:05:22 +0200 | |
changeset 63750 | 9c8a366778e1 |
parent 62390 | 842917225d56 |
child 67443 | 3abf6a722518 |
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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section \<open>Relations between Java Types\<close> |
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theory TypeRel |
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imports Decl |
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begin |
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\<comment> "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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by (subst wfrec_def_adm[OF class_rec_def]) |
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(auto simp: assms adm_wf_def fun_eq_iff subcls1I split: option.split) |
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definition |
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"wf_class G = wf ((subcls1 G)^-1)" |
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text \<open>Code generator setup\<close> |
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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" |
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|
129 |
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
Andreas Lochbihler
parents:
44014
diff
changeset
|
130 |
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
Andreas Lochbihler
parents:
44014
diff
changeset
|
131 |
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
|
132 |
proof(safe) |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
133 |
fix C D fs ms |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
134 |
assume "(C, D, fs, ms) \<in> set G" |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
135 |
and "C \<noteq> Object" |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
136 |
and subcls: "G \<turnstile> fst (the (class G C)) \<preceq>C C" |
61361 | 137 |
from \<open>(C, D, fs, ms) \<in> set G\<close> obtain D' fs' ms' |
44035
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
138 |
where "class": "class G C = Some (D', fs', ms')" |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
139 |
unfolding class_def by(auto dest!: weak_map_of_SomeI) |
61361 | 140 |
hence "G \<turnstile> C \<prec>C1 D'" using \<open>C \<noteq> Object\<close> .. |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
45970
diff
changeset
|
141 |
hence *: "(C, D') \<in> (subcls1 G)^+" .. |
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
45970
diff
changeset
|
142 |
also from * acyc have "C \<noteq> D'" by(auto simp add: acyclic_def) |
44035
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
143 |
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
Andreas Lochbihler
parents:
44014
diff
changeset
|
144 |
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
Andreas Lochbihler
parents:
44014
diff
changeset
|
145 |
qed |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
146 |
next |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
147 |
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
parents:
44014
diff
changeset
|
148 |
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
|
149 |
proof(intro acyclicI strip notI) |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
150 |
fix C |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
151 |
assume "(C, C) \<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
|
152 |
thus False |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
153 |
proof(cases) |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
154 |
case base |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
155 |
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
Andreas Lochbihler
parents:
44014
diff
changeset
|
156 |
and "C \<noteq> Object" by cases |
61361 | 157 |
from \<open>class G C = Some (C, rest)\<close> have "(C, C, rest) \<in> set G" |
44035
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
158 |
unfolding class_def by(rule map_of_SomeD) |
61361 | 159 |
with \<open>C \<noteq> Object\<close> \<open>class G C = Some (C, rest)\<close> |
44035
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
160 |
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
|
161 |
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
|
162 |
next |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
163 |
case (step D) |
61361 | 164 |
from \<open>G \<turnstile> D \<prec>C1 C\<close> obtain rest where "class G D = Some (C, rest)" |
44035
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
165 |
and "D \<noteq> Object" by cases |
61361 | 166 |
from \<open>class G D = Some (C, rest)\<close> have "(D, C, rest) \<in> set G" |
44035
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
167 |
unfolding class_def by(rule map_of_SomeD) |
61361 | 168 |
with \<open>D \<noteq> Object\<close> \<open>class G D = Some (C, rest)\<close> |
44035
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
169 |
have "\<not> G \<turnstile> C \<preceq>C D" by(auto dest: rhs) |
61361 | 170 |
moreover from \<open>(C, D) \<in> (subcls1 G)\<^sup>+\<close> |
44035
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
171 |
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
|
172 |
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
|
173 |
qed |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
174 |
qed |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
175 |
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
|
176 |
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
|
177 |
qed |
32461
eee4fa79398f
no consts_code for wfrec, as it violates the "code generation = equational reasoning" principle
krauss
parents:
28562
diff
changeset
|
178 |
|
62145 | 179 |
definition "method" :: "'c prog \<times> cname => (sig \<rightharpoonup> cname \<times> ty \<times> 'c)" |
180 |
\<comment> "methods of a class, with inheritance, overriding and hiding, cf. 8.4.6" |
|
181 |
where [code]: "method \<equiv> \<lambda>(G,C). class_rec G C empty (\<lambda>C fs ms ts. |
|
182 |
ts ++ map_of (map (\<lambda>(s,m). (s,(C,m))) ms))" |
|
8011 | 183 |
|
62145 | 184 |
definition fields :: "'c prog \<times> cname => ((vname \<times> cname) \<times> ty) list" |
185 |
\<comment> "list of fields of a class, including inherited and hidden ones" |
|
186 |
where [code]: "fields \<equiv> \<lambda>(G,C). class_rec G C [] (\<lambda>C fs ms ts. |
|
187 |
map (\<lambda>(fn,ft). ((fn,C),ft)) fs @ ts)" |
|
188 |
||
189 |
definition field :: "'c prog \<times> cname => (vname \<rightharpoonup> cname \<times> ty)" |
|
190 |
where [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
|
191 |
|
33954
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
192 |
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
|
193 |
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
|
194 |
map_of (map (\<lambda>(s,m). (s,(C,m))) ms)" |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
195 |
apply (unfold method_def) |
62390 | 196 |
apply (simp split del: if_split) |
45970
b6d0cff57d96
adjusted to set/pred distinction by means of type constructor `set`
haftmann
parents:
45231
diff
changeset
|
197 |
apply (erule (1) class_rec_lemma [THEN trans]) |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
198 |
apply auto |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
199 |
done |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
200 |
|
33954
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
201 |
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
|
202 |
fields (G,C) = |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
203 |
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
|
204 |
apply (unfold fields_def) |
62390 | 205 |
apply (simp split del: if_split) |
45970
b6d0cff57d96
adjusted to set/pred distinction by means of type constructor `set`
haftmann
parents:
45231
diff
changeset
|
206 |
apply (erule (1) class_rec_lemma [THEN trans]) |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
207 |
apply auto |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
208 |
done |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
209 |
|
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
210 |
lemma field_fields: |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
211 |
"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
|
212 |
apply (unfold field_def) |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
213 |
apply (rule table_of_remap_SomeD) |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
214 |
apply simp |
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 |
|
62042 | 218 |
\<comment> "widening, viz. method invocation conversion,cf. 5.3 i.e. sort of syntactic subtyping" |
23757 | 219 |
inductive |
22271 | 220 |
widen :: "'c prog => [ty , ty ] => bool" ("_ \<turnstile> _ \<preceq> _" [71,71,71] 70) |
221 |
for G :: "'c prog" |
|
222 |
where |
|
62042 | 223 |
refl [intro!, simp]: "G\<turnstile> T \<preceq> T" \<comment> "identity conv., cf. 5.1.1" |
22271 | 224 |
| subcls : "G\<turnstile>C\<preceq>C D ==> G\<turnstile>Class C \<preceq> Class D" |
225 |
| null [intro!]: "G\<turnstile> NT \<preceq> RefT R" |
|
8011 | 226 |
|
44035
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
227 |
code_pred widen . |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
228 |
|
22597 | 229 |
lemmas refl = HOL.refl |
230 |
||
62042 | 231 |
\<comment> "casting conversion, cf. 5.5 / 5.1.5" |
232 |
\<comment> "left out casts on primitve types" |
|
23757 | 233 |
inductive |
22271 | 234 |
cast :: "'c prog => [ty , ty ] => bool" ("_ \<turnstile> _ \<preceq>? _" [71,71,71] 70) |
235 |
for G :: "'c prog" |
|
236 |
where |
|
14045 | 237 |
widen: "G\<turnstile> C\<preceq> D ==> G\<turnstile>C \<preceq>? D" |
22271 | 238 |
| subcls: "G\<turnstile> D\<preceq>C C ==> G\<turnstile>Class C \<preceq>? Class D" |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
239 |
|
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
240 |
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
|
241 |
apply (rule iffI) |
22271 | 242 |
apply (erule widen.cases) |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
243 |
apply auto |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
244 |
done |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
245 |
|
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
246 |
lemma widen_RefT: "G\<turnstile>RefT R\<preceq>T ==> \<exists>t. T=RefT t" |
23757 | 247 |
apply (ind_cases "G\<turnstile>RefT R\<preceq>T") |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
248 |
apply auto |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
249 |
done |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
250 |
|
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
251 |
lemma widen_RefT2: "G\<turnstile>S\<preceq>RefT R ==> \<exists>t. S=RefT t" |
23757 | 252 |
apply (ind_cases "G\<turnstile>S\<preceq>RefT R") |
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_Class: "G\<turnstile>Class C\<preceq>T ==> \<exists>D. T=Class D" |
23757 | 257 |
apply (ind_cases "G\<turnstile>Class C\<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_Class_NullT [iff]: "(G\<turnstile>Class C\<preceq>NT) = False" |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
262 |
apply (rule iffI) |
23757 | 263 |
apply (ind_cases "G\<turnstile>Class C\<preceq>NT") |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
264 |
apply auto |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
265 |
done |
8011 | 266 |
|
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
267 |
lemma widen_Class_Class [iff]: "(G\<turnstile>Class C\<preceq> Class D) = (G\<turnstile>C\<preceq>C D)" |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
268 |
apply (rule iffI) |
23757 | 269 |
apply (ind_cases "G\<turnstile>Class C \<preceq> Class D") |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
270 |
apply (auto elim: widen.subcls) |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
271 |
done |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
272 |
|
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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) |
14045 | 275 |
|
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lemma cast_PrimT_RefT [iff]: "(G\<turnstile>PrimT pT\<preceq>? RefT rT) = False" |
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apply (rule iffI) |
|
22271 | 278 |
apply (erule cast.cases) |
14045 | 279 |
apply auto |
280 |
done |
|
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||
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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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||
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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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303 |
|
8011 | 304 |
end |