author | Andreas Lochbihler |
Fri, 05 Aug 2011 14:16:44 +0200 | |
changeset 44035 | 322d1657c40c |
parent 44014 | 88bd7d74a2c1 |
child 45231 | d85a2fdc586c |
permissions | -rw-r--r-- |
8011 | 1 |
(* Title: HOL/MicroJava/J/TypeRel.thy |
41589 | 2 |
Author: David von Oheimb, Technische Universitaet Muenchen |
11070 | 3 |
*) |
8011 | 4 |
|
12911 | 5 |
header {* \isaheader{Relations between Java Types} *} |
8011 | 6 |
|
44014
88bd7d74a2c1
moved recursion combinator to HOL/Library/Wfrec.thy -- it is so fundamental and well-known that it should survive recdef
krauss
parents:
44013
diff
changeset
|
7 |
theory TypeRel imports Decl "~~/src/HOL/Library/Wfrec" begin |
8011 | 8 |
|
22271 | 9 |
-- "direct subclass, cf. 8.1.3" |
33954
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
10 |
|
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
11 |
inductive_set |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
12 |
subcls1 :: "'c prog => (cname \<times> cname) set" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
13 |
and subcls1' :: "'c prog => cname \<Rightarrow> cname => bool" ("_ \<turnstile> _ \<prec>C1 _" [71,71,71] 70) |
22271 | 14 |
for G :: "'c prog" |
15 |
where |
|
33954
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
16 |
"G \<turnstile> C \<prec>C1 D \<equiv> (C, D) \<in> subcls1 G" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
17 |
| subcls1I: "\<lbrakk>class G C = Some (D,rest); C \<noteq> Object\<rbrakk> \<Longrightarrow> G \<turnstile> C \<prec>C1 D" |
10061
fe82134773dc
added HTML syntax; added spaces in normal syntax for better documents
kleing
parents:
10042
diff
changeset
|
18 |
|
22271 | 19 |
abbreviation |
33954
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
20 |
subcls :: "'c prog => cname \<Rightarrow> cname => bool" ("_ \<turnstile> _ \<preceq>C _" [71,71,71] 70) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
21 |
where "G \<turnstile> C \<preceq>C D \<equiv> (C, D) \<in> (subcls1 G)^*" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
22 |
|
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
23 |
lemma subcls1D: |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
24 |
"G\<turnstile>C\<prec>C1D \<Longrightarrow> C \<noteq> Object \<and> (\<exists>fs ms. class G C = Some (D,fs,ms))" |
22271 | 25 |
apply (erule subcls1.cases) |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
26 |
apply auto |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
27 |
done |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
28 |
|
33954
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
29 |
lemma subcls1_def2: |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
30 |
"subcls1 P = |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
31 |
(SIGMA C:{C. is_class P C}. {D. C\<noteq>Object \<and> fst (the (class P C))=D})" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
32 |
by (auto simp add: is_class_def dest: subcls1D intro: subcls1I) |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
33 |
|
33954
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
34 |
lemma finite_subcls1: "finite (subcls1 G)" |
23757 | 35 |
apply(simp add: subcls1_def2 del: mem_Sigma_iff) |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
36 |
apply(rule finite_SigmaI [OF finite_is_class]) |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
37 |
apply(rule_tac B = "{fst (the (class G C))}" in finite_subset) |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
38 |
apply auto |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
39 |
done |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
40 |
|
33954
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
41 |
lemma subcls_is_class: "(C, D) \<in> (subcls1 G)^+ ==> is_class G C" |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
42 |
apply (unfold is_class_def) |
33954
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
43 |
apply(erule trancl_trans_induct) |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
44 |
apply (auto dest!: subcls1D) |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
45 |
done |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
46 |
|
11266 | 47 |
lemma subcls_is_class2 [rule_format (no_asm)]: |
48 |
"G\<turnstile>C\<preceq>C D \<Longrightarrow> is_class G D \<longrightarrow> is_class G C" |
|
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
49 |
apply (unfold is_class_def) |
33954
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
50 |
apply (erule rtrancl_induct) |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
51 |
apply (drule_tac [2] subcls1D) |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
52 |
apply auto |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
53 |
done |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
54 |
|
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
33954
diff
changeset
|
55 |
definition class_rec :: "'c prog \<Rightarrow> cname \<Rightarrow> 'a \<Rightarrow> |
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
33954
diff
changeset
|
56 |
(cname \<Rightarrow> fdecl list \<Rightarrow> 'c mdecl list \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a" where |
33954
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
57 |
"class_rec G == wfrec ((subcls1 G)^-1) |
13090
4fb7a2f2c1df
Improved definition of class_rec: no longer mixes algorithm and
berghofe
parents:
12911
diff
changeset
|
58 |
(\<lambda>r C t f. case class G C of |
28524 | 59 |
None \<Rightarrow> undefined |
13090
4fb7a2f2c1df
Improved definition of class_rec: no longer mixes algorithm and
berghofe
parents:
12911
diff
changeset
|
60 |
| Some (D,fs,ms) \<Rightarrow> |
4fb7a2f2c1df
Improved definition of class_rec: no longer mixes algorithm and
berghofe
parents:
12911
diff
changeset
|
61 |
f C fs ms (if C = Object then t else r D t f))" |
11284 | 62 |
|
33954
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
63 |
lemma class_rec_lemma: |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
64 |
assumes wf: "wf ((subcls1 G)^-1)" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
65 |
and cls: "class G C = Some (D, fs, ms)" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
66 |
shows "class_rec G C t f = f C fs ms (if C=Object then t else class_rec G D t f)" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
67 |
proof - |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
68 |
from wf have step: "\<And>H a. wfrec ((subcls1 G)\<inverse>) H a = |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
69 |
H (cut (wfrec ((subcls1 G)\<inverse>) H) ((subcls1 G)\<inverse>) a) a" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
70 |
by (rule wfrec) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
71 |
have cut: "\<And>f. C \<noteq> Object \<Longrightarrow> cut f ((subcls1 G)\<inverse>) C D = f D" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
72 |
by (rule cut_apply [where r="(subcls1 G)^-1", simplified, OF subcls1I, OF cls]) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
73 |
from cls show ?thesis by (simp add: step cut class_rec_def) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
74 |
qed |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
75 |
|
20970 | 76 |
definition |
33954
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
77 |
"wf_class G = wf ((subcls1 G)^-1)" |
20970 | 78 |
|
79 |
||
44035
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
80 |
|
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
81 |
text {* Code generator setup *} |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
82 |
|
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
83 |
code_pred |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
84 |
(modes: i \<Rightarrow> i \<Rightarrow> o \<Rightarrow> bool, i \<Rightarrow> i \<Rightarrow> i \<Rightarrow> bool) |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
85 |
subcls1p |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
86 |
. |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
87 |
declare subcls1_def[unfolded Collect_def, code_pred_def] |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
88 |
code_pred |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
89 |
(modes: i \<Rightarrow> i \<times> o \<Rightarrow> bool, i \<Rightarrow> i \<times> i \<Rightarrow> bool) |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
90 |
[inductify] |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
91 |
subcls1 |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
92 |
. |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
93 |
|
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
94 |
definition subcls' where "subcls' G = (subcls1p G)^**" |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
95 |
code_pred |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
96 |
(modes: i \<Rightarrow> i \<Rightarrow> i \<Rightarrow> bool, i \<Rightarrow> i \<Rightarrow> o \<Rightarrow> bool) |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
97 |
[inductify] |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
98 |
subcls' |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
99 |
. |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
100 |
lemma subcls_conv_subcls' [code_inline]: |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
101 |
"(subcls1 G)^* = (\<lambda>(C, D). subcls' G C D)" |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
102 |
by(simp add: subcls'_def subcls1_def rtrancl_def)(simp add: Collect_def) |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
103 |
|
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
104 |
lemma class_rec_code [code]: |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
105 |
"class_rec G C t f = |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
106 |
(if wf_class G then |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
107 |
(case class G C of |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
108 |
None \<Rightarrow> class_rec G C t f |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
109 |
| Some (D, fs, ms) \<Rightarrow> |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
110 |
if C = Object then f Object fs ms t else f C fs ms (class_rec G D t f)) |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
111 |
else class_rec G C t f)" |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
112 |
apply(cases "wf_class G") |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
113 |
apply(unfold class_rec_def wf_class_def) |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
114 |
apply(subst wfrec, assumption) |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
115 |
apply(cases "class G C") |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
116 |
apply(simp add: wfrec) |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
117 |
apply clarsimp |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
118 |
apply(rename_tac D fs ms) |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
119 |
apply(rule_tac f="f C fs ms" in arg_cong) |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
120 |
apply(clarsimp simp add: cut_def) |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
121 |
apply(blast intro: subcls1I) |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
122 |
apply simp |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
123 |
done |
32461
eee4fa79398f
no consts_code for wfrec, as it violates the "code generation = equational reasoning" principle
krauss
parents:
28562
diff
changeset
|
124 |
|
44035
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
125 |
lemma wf_class_code [code]: |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
126 |
"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)" |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
127 |
proof |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
128 |
assume "wf_class G" |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
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" |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
137 |
from `(C, D, fs, ms) \<in> set G` obtain D' fs' ms' |
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) |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
140 |
hence "G \<turnstile> C \<prec>C1 D'" using `C \<noteq> Object` .. |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
141 |
hence "(C, D') \<in> (subcls1 G)^+" .. |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
142 |
also with acyc have "C \<noteq> D'" 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
|
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 |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
157 |
from `class G C = Some (C, rest)` have "(C, C, rest) \<in> set G" |
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) |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
159 |
with `C \<noteq> Object` `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
|
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) |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
164 |
from `G \<turnstile> D \<prec>C1 C` obtain rest where "class G D = Some (C, rest)" |
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 |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
166 |
from `class G D = Some (C, rest)` have "(D, C, rest) \<in> set G" |
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) |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
168 |
with `D \<noteq> Object` `class G D = Some (C, rest)` |
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) |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
170 |
moreover from `(C, D) \<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
|
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 |
|
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 *) |
|
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
182 |
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" |
44035
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
185 |
defs method_def [code]: "method \<equiv> \<lambda>(G,C). class_rec G C empty (\<lambda>C fs ms ts. |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
186 |
ts ++ map_of (map (\<lambda>(s,m). (s,(C,m))) ms))" |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
187 |
|
33954
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
188 |
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
|
189 |
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
|
190 |
map_of (map (\<lambda>(s,m). (s,(C,m))) ms)" |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
191 |
apply (unfold method_def) |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
192 |
apply (simp split del: split_if) |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
193 |
apply (erule (1) class_rec_lemma [THEN trans]); |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
194 |
apply auto |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
195 |
done |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
196 |
|
8011 | 197 |
|
12517 | 198 |
-- "list of fields of a class, including inherited and hidden ones" |
44035
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
199 |
defs fields_def [code]: "fields \<equiv> \<lambda>(G,C). class_rec G C [] (\<lambda>C fs ms ts. |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
200 |
map (\<lambda>(fn,ft). ((fn,C),ft)) fs @ ts)" |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
201 |
|
33954
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32461
diff
changeset
|
202 |
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
|
203 |
fields (G,C) = |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
204 |
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
|
205 |
apply (unfold fields_def) |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
206 |
apply (simp split del: split_if) |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
207 |
apply (erule (1) class_rec_lemma [THEN trans]); |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
208 |
apply auto |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
209 |
done |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
210 |
|
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
211 |
|
44035
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
212 |
defs field_def [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
|
213 |
|
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
214 |
lemma field_fields: |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
215 |
"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
|
216 |
apply (unfold field_def) |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
217 |
apply (rule table_of_remap_SomeD) |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
218 |
apply simp |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
219 |
done |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
220 |
|
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
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 |
|
44035
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
231 |
code_pred widen . |
322d1657c40c
replace old SML code generator by new code generator in MicroJava/JVM and /BV
Andreas Lochbihler
parents:
44014
diff
changeset
|
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" |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
243 |
|
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
244 |
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
|
245 |
apply (rule iffI) |
22271 | 246 |
apply (erule widen.cases) |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
247 |
apply auto |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
248 |
done |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
249 |
|
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
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") |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
252 |
apply auto |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
253 |
done |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
254 |
|
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
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") |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
257 |
apply auto |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
258 |
done |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
259 |
|
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
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") |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
262 |
apply auto |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
263 |
done |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
264 |
|
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
265 |
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
|
266 |
apply (rule iffI) |
23757 | 267 |
apply (ind_cases "G\<turnstile>Class C\<preceq>NT") |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
268 |
apply auto |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
269 |
done |
8011 | 270 |
|
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
271 |
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
|
272 |
apply (rule iffI) |
23757 | 273 |
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
|
274 |
apply (auto elim: widen.subcls) |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
275 |
done |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
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" |
|
287 |
apply (erule cast.cases) |
|
288 |
apply simp apply (erule widen.cases) |
|
289 |
apply auto |
|
290 |
done |
|
291 |
||
12517 | 292 |
theorem widen_trans[trans]: "\<lbrakk>G\<turnstile>S\<preceq>U; G\<turnstile>U\<preceq>T\<rbrakk> \<Longrightarrow> G\<turnstile>S\<preceq>T" |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
293 |
proof - |
12517 | 294 |
assume "G\<turnstile>S\<preceq>U" thus "\<And>T. G\<turnstile>U\<preceq>T \<Longrightarrow> G\<turnstile>S\<preceq>T" |
11987 | 295 |
proof induct |
12517 | 296 |
case (refl T T') thus "G\<turnstile>T\<preceq>T'" . |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
297 |
next |
11987 | 298 |
case (subcls C D T) |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
299 |
then obtain E where "T = Class E" by (blast dest: widen_Class) |
22271 | 300 |
with subcls show "G\<turnstile>Class C\<preceq>T" by auto |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
301 |
next |
11987 | 302 |
case (null R RT) |
11026
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
303 |
then obtain rt where "RT = RefT rt" by (blast dest: widen_RefT) |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
304 |
thus "G\<turnstile>NT\<preceq>RT" by auto |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
305 |
qed |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
306 |
qed |
a50365d21144
converted to Isar, simplifying recursion on class hierarchy
oheimb
parents:
10613
diff
changeset
|
307 |
|
8011 | 308 |
end |