12854
|
1 |
(* Title: isabelle/Bali/TypeRel.thy
|
|
2 |
ID: $Id$
|
|
3 |
Author: David von Oheimb
|
|
4 |
Copyright 1997 Technische Universitaet Muenchen
|
|
5 |
*)
|
|
6 |
header {* The relations between Java types *}
|
|
7 |
|
|
8 |
theory TypeRel = Decl:
|
|
9 |
|
|
10 |
text {*
|
|
11 |
simplifications:
|
|
12 |
\begin{itemize}
|
|
13 |
\item subinterface, subclass and widening relation includes identity
|
|
14 |
\end{itemize}
|
|
15 |
improvements over Java Specification 1.0:
|
|
16 |
\begin{itemize}
|
|
17 |
\item narrowing reference conversion also in cases where the return types of a
|
|
18 |
pair of methods common to both types are in widening (rather identity)
|
|
19 |
relation
|
|
20 |
\item one could add similar constraints also for other cases
|
|
21 |
\end{itemize}
|
|
22 |
design issues:
|
|
23 |
\begin{itemize}
|
|
24 |
\item the type relations do not require @{text is_type} for their arguments
|
|
25 |
\item the subint1 and subcls1 relations imply @{text is_iface}/@{text is_class}
|
|
26 |
for their first arguments, which is required for their finiteness
|
|
27 |
\end{itemize}
|
|
28 |
*}
|
|
29 |
|
|
30 |
consts
|
|
31 |
|
|
32 |
(*subint1, in Decl.thy*) (* direct subinterface *)
|
|
33 |
(*subint , by translation*) (* subinterface (+ identity) *)
|
|
34 |
(*subcls1, in Decl.thy*) (* direct subclass *)
|
|
35 |
(*subcls , by translation*) (* subclass *)
|
|
36 |
(*subclseq, by translation*) (* subclass + identity *)
|
|
37 |
implmt1 :: "prog \<Rightarrow> (qtname \<times> qtname) set" (* direct implementation *)
|
|
38 |
implmt :: "prog \<Rightarrow> (qtname \<times> qtname) set" (* implementation *)
|
|
39 |
widen :: "prog \<Rightarrow> (ty \<times> ty ) set" (* widening *)
|
|
40 |
narrow :: "prog \<Rightarrow> (ty \<times> ty ) set" (* narrowing *)
|
|
41 |
cast :: "prog \<Rightarrow> (ty \<times> ty ) set" (* casting *)
|
|
42 |
|
|
43 |
syntax
|
|
44 |
|
|
45 |
"@subint1" :: "prog => [qtname, qtname] => bool" ("_|-_<:I1_" [71,71,71] 70)
|
|
46 |
"@subint" :: "prog => [qtname, qtname] => bool" ("_|-_<=:I _"[71,71,71] 70)
|
|
47 |
(* Defined in Decl.thy:
|
|
48 |
"@subcls1" :: "prog => [qtname, qtname] => bool" ("_|-_<:C1_" [71,71,71] 70)
|
|
49 |
"@subclseq":: "prog => [qtname, qtname] => bool" ("_|-_<=:C _"[71,71,71] 70)
|
|
50 |
"@subcls" :: "prog => [qtname, qtname] => bool" ("_|-_<:C _"[71,71,71] 70)
|
|
51 |
*)
|
|
52 |
"@implmt1" :: "prog => [qtname, qtname] => bool" ("_|-_~>1_" [71,71,71] 70)
|
|
53 |
"@implmt" :: "prog => [qtname, qtname] => bool" ("_|-_~>_" [71,71,71] 70)
|
|
54 |
"@widen" :: "prog => [ty , ty ] => bool" ("_|-_<=:_" [71,71,71] 70)
|
|
55 |
"@narrow" :: "prog => [ty , ty ] => bool" ("_|-_:>_" [71,71,71] 70)
|
|
56 |
"@cast" :: "prog => [ty , ty ] => bool" ("_|-_<=:? _"[71,71,71] 70)
|
|
57 |
|
|
58 |
syntax (symbols)
|
|
59 |
|
|
60 |
"@subint1" :: "prog \<Rightarrow> [qtname, qtname] \<Rightarrow> bool" ("_\<turnstile>_\<prec>I1_" [71,71,71] 70)
|
|
61 |
"@subint" :: "prog \<Rightarrow> [qtname, qtname] \<Rightarrow> bool" ("_\<turnstile>_\<preceq>I _" [71,71,71] 70)
|
|
62 |
(* Defined in Decl.thy:
|
|
63 |
\ "@subcls1" :: "prog \<Rightarrow> [qtname, qtname] \<Rightarrow> bool" ("_\<turnstile>_\<prec>\<^sub>C\<^sub>1_" [71,71,71] 70)
|
|
64 |
"@subclseq":: "prog \<Rightarrow> [qtname, qtname] \<Rightarrow> bool" ("_\<turnstile>_\<preceq>\<^sub>C _" [71,71,71] 70)
|
|
65 |
"@subcls" :: "prog \<Rightarrow> [qtname, qtname] \<Rightarrow> bool" ("_\<turnstile>_\<prec>\<^sub>C _" [71,71,71] 70)
|
|
66 |
*)
|
|
67 |
"@implmt1" :: "prog \<Rightarrow> [qtname, qtname] \<Rightarrow> bool" ("_\<turnstile>_\<leadsto>1_" [71,71,71] 70)
|
|
68 |
"@implmt" :: "prog \<Rightarrow> [qtname, qtname] \<Rightarrow> bool" ("_\<turnstile>_\<leadsto>_" [71,71,71] 70)
|
|
69 |
"@widen" :: "prog \<Rightarrow> [ty , ty ] \<Rightarrow> bool" ("_\<turnstile>_\<preceq>_" [71,71,71] 70)
|
|
70 |
"@narrow" :: "prog \<Rightarrow> [ty , ty ] \<Rightarrow> bool" ("_\<turnstile>_\<succ>_" [71,71,71] 70)
|
|
71 |
"@cast" :: "prog \<Rightarrow> [ty , ty ] \<Rightarrow> bool" ("_\<turnstile>_\<preceq>? _" [71,71,71] 70)
|
|
72 |
|
|
73 |
translations
|
|
74 |
|
|
75 |
"G\<turnstile>I \<prec>I1 J" == "(I,J) \<in> subint1 G"
|
|
76 |
"G\<turnstile>I \<preceq>I J" == "(I,J) \<in>(subint1 G)^*" (* cf. 9.1.3 *)
|
|
77 |
(* Defined in Decl.thy:
|
|
78 |
"G\<turnstile>C \<prec>\<^sub>C\<^sub>1 D" == "(C,D) \<in> subcls1 G"
|
|
79 |
"G\<turnstile>C \<preceq>\<^sub>C D" == "(C,D) \<in>(subcls1 G)^*"
|
|
80 |
*)
|
|
81 |
"G\<turnstile>C \<leadsto>1 I" == "(C,I) \<in> implmt1 G"
|
|
82 |
"G\<turnstile>C \<leadsto> I" == "(C,I) \<in> implmt G"
|
|
83 |
"G\<turnstile>S \<preceq> T" == "(S,T) \<in> widen G"
|
|
84 |
"G\<turnstile>S \<succ> T" == "(S,T) \<in> narrow G"
|
|
85 |
"G\<turnstile>S \<preceq>? T" == "(S,T) \<in> cast G"
|
|
86 |
|
|
87 |
|
|
88 |
section "subclass and subinterface relations"
|
|
89 |
|
|
90 |
(* direct subinterface in Decl.thy, cf. 9.1.3 *)
|
|
91 |
(* direct subclass in Decl.thy, cf. 8.1.3 *)
|
|
92 |
|
|
93 |
lemmas subcls_direct = subcls1I [THEN r_into_rtrancl, standard]
|
|
94 |
|
|
95 |
lemma subcls_direct1:
|
|
96 |
"\<lbrakk>class G C = Some c; C \<noteq> Object;D=super c\<rbrakk> \<Longrightarrow> G\<turnstile>C\<preceq>\<^sub>C D"
|
|
97 |
apply (auto dest: subcls_direct)
|
|
98 |
done
|
|
99 |
|
|
100 |
lemma subcls1I1:
|
|
101 |
"\<lbrakk>class G C = Some c; C \<noteq> Object;D=super c\<rbrakk> \<Longrightarrow> G\<turnstile>C\<prec>\<^sub>C\<^sub>1 D"
|
|
102 |
apply (auto dest: subcls1I)
|
|
103 |
done
|
|
104 |
|
|
105 |
lemma subcls_direct2:
|
|
106 |
"\<lbrakk>class G C = Some c; C \<noteq> Object;D=super c\<rbrakk> \<Longrightarrow> G\<turnstile>C\<prec>\<^sub>C D"
|
|
107 |
apply (auto dest: subcls1I1)
|
|
108 |
done
|
|
109 |
|
|
110 |
lemma subclseq_trans: "\<lbrakk>G\<turnstile>A \<preceq>\<^sub>C B; G\<turnstile>B \<preceq>\<^sub>C C\<rbrakk> \<Longrightarrow> G\<turnstile>A \<preceq>\<^sub>C C"
|
|
111 |
by (blast intro: rtrancl_trans)
|
|
112 |
|
|
113 |
lemma subcls_trans: "\<lbrakk>G\<turnstile>A \<prec>\<^sub>C B; G\<turnstile>B \<prec>\<^sub>C C\<rbrakk> \<Longrightarrow> G\<turnstile>A \<prec>\<^sub>C C"
|
|
114 |
by (blast intro: trancl_trans)
|
|
115 |
|
|
116 |
lemma SXcpt_subcls_Throwable_lemma:
|
|
117 |
"\<lbrakk>class G (SXcpt xn) = Some xc;
|
|
118 |
super xc = (if xn = Throwable then Object else SXcpt Throwable)\<rbrakk>
|
|
119 |
\<Longrightarrow> G\<turnstile>SXcpt xn\<preceq>\<^sub>C SXcpt Throwable"
|
|
120 |
apply (case_tac "xn = Throwable")
|
|
121 |
apply simp_all
|
|
122 |
apply (drule subcls_direct)
|
|
123 |
apply (auto dest: sym)
|
|
124 |
done
|
|
125 |
|
|
126 |
lemma subcls_ObjectI: "\<lbrakk>is_class G C; ws_prog G\<rbrakk> \<Longrightarrow> G\<turnstile>C\<preceq>\<^sub>C Object"
|
|
127 |
apply (erule ws_subcls1_induct)
|
|
128 |
apply clarsimp
|
|
129 |
apply (case_tac "C = Object")
|
|
130 |
apply (fast intro: r_into_rtrancl [THEN rtrancl_trans])+
|
|
131 |
done
|
|
132 |
|
|
133 |
lemma subclseq_ObjectD [dest!]: "G\<turnstile>Object\<preceq>\<^sub>C C \<Longrightarrow> C = Object"
|
|
134 |
apply (erule rtrancl_induct)
|
|
135 |
apply (auto dest: subcls1D)
|
|
136 |
done
|
|
137 |
|
|
138 |
lemma subcls_ObjectD [dest!]: "G\<turnstile>Object\<prec>\<^sub>C C \<Longrightarrow> False"
|
|
139 |
apply (erule trancl_induct)
|
|
140 |
apply (auto dest: subcls1D)
|
|
141 |
done
|
|
142 |
|
|
143 |
lemma subcls_ObjectI1 [intro!]:
|
|
144 |
"\<lbrakk>C \<noteq> Object;is_class G C;ws_prog G\<rbrakk> \<Longrightarrow> G\<turnstile>C \<prec>\<^sub>C Object"
|
|
145 |
apply (drule (1) subcls_ObjectI)
|
|
146 |
apply (auto intro: rtrancl_into_trancl3)
|
|
147 |
done
|
|
148 |
|
|
149 |
lemma subcls_is_class: "(C,D) \<in> (subcls1 G)^+ \<Longrightarrow> is_class G C"
|
|
150 |
apply (erule trancl_trans_induct)
|
|
151 |
apply (auto dest!: subcls1D)
|
|
152 |
done
|
|
153 |
|
|
154 |
lemma subcls_is_class2 [rule_format (no_asm)]:
|
|
155 |
"G\<turnstile>C\<preceq>\<^sub>C D \<Longrightarrow> is_class G D \<longrightarrow> is_class G C"
|
|
156 |
apply (erule rtrancl_induct)
|
|
157 |
apply (drule_tac [2] subcls1D)
|
|
158 |
apply auto
|
|
159 |
done
|
|
160 |
|
|
161 |
lemma single_inheritance:
|
|
162 |
"\<lbrakk>G\<turnstile>A \<prec>\<^sub>C\<^sub>1 B; G\<turnstile>A \<prec>\<^sub>C\<^sub>1 C\<rbrakk> \<Longrightarrow> B = C"
|
|
163 |
by (auto simp add: subcls1_def)
|
|
164 |
|
|
165 |
lemma subcls_compareable:
|
|
166 |
"\<lbrakk>G\<turnstile>A \<preceq>\<^sub>C X; G\<turnstile>A \<preceq>\<^sub>C Y
|
|
167 |
\<rbrakk> \<Longrightarrow> G\<turnstile>X \<preceq>\<^sub>C Y \<or> G\<turnstile>Y \<preceq>\<^sub>C X"
|
|
168 |
by (rule triangle_lemma) (auto intro: single_inheritance)
|
|
169 |
|
|
170 |
lemma subcls1_irrefl: "\<lbrakk>G\<turnstile>C \<prec>\<^sub>C\<^sub>1 D; ws_prog G \<rbrakk>
|
|
171 |
\<Longrightarrow> C \<noteq> D"
|
|
172 |
proof
|
|
173 |
assume ws: "ws_prog G" and
|
|
174 |
subcls1: "G\<turnstile>C \<prec>\<^sub>C\<^sub>1 D" and
|
|
175 |
eq_C_D: "C=D"
|
|
176 |
from subcls1 obtain c
|
|
177 |
where
|
|
178 |
neq_C_Object: "C\<noteq>Object" and
|
|
179 |
clsC: "class G C = Some c" and
|
|
180 |
super_c: "super c = D"
|
|
181 |
by (auto simp add: subcls1_def)
|
|
182 |
with super_c subcls1 eq_C_D
|
|
183 |
have subcls_super_c_C: "G\<turnstile>super c \<prec>\<^sub>C C"
|
|
184 |
by auto
|
|
185 |
from ws clsC neq_C_Object
|
|
186 |
have "\<not> G\<turnstile>super c \<prec>\<^sub>C C"
|
|
187 |
by (auto dest: ws_prog_cdeclD)
|
|
188 |
from this subcls_super_c_C
|
|
189 |
show "False"
|
|
190 |
by (rule notE)
|
|
191 |
qed
|
|
192 |
|
|
193 |
lemma no_subcls_Object: "G\<turnstile>C \<prec>\<^sub>C D \<Longrightarrow> C \<noteq> Object"
|
|
194 |
by (erule converse_trancl_induct) (auto dest: subcls1D)
|
|
195 |
|
|
196 |
lemma subcls_acyclic: "\<lbrakk>G\<turnstile>C \<prec>\<^sub>C D; ws_prog G\<rbrakk> \<Longrightarrow> \<not> G\<turnstile>D \<prec>\<^sub>C C"
|
|
197 |
proof -
|
|
198 |
assume ws: "ws_prog G"
|
|
199 |
assume subcls_C_D: "G\<turnstile>C \<prec>\<^sub>C D"
|
|
200 |
then show ?thesis
|
|
201 |
proof (induct rule: converse_trancl_induct)
|
|
202 |
fix C
|
|
203 |
assume subcls1_C_D: "G\<turnstile>C \<prec>\<^sub>C\<^sub>1 D"
|
|
204 |
then obtain c where
|
|
205 |
"C\<noteq>Object" and
|
|
206 |
"class G C = Some c" and
|
|
207 |
"super c = D"
|
|
208 |
by (auto simp add: subcls1_def)
|
|
209 |
with ws
|
|
210 |
show "\<not> G\<turnstile>D \<prec>\<^sub>C C"
|
|
211 |
by (auto dest: ws_prog_cdeclD)
|
|
212 |
next
|
|
213 |
fix C Z
|
|
214 |
assume subcls1_C_Z: "G\<turnstile>C \<prec>\<^sub>C\<^sub>1 Z" and
|
|
215 |
subcls_Z_D: "G\<turnstile>Z \<prec>\<^sub>C D" and
|
|
216 |
nsubcls_D_Z: "\<not> G\<turnstile>D \<prec>\<^sub>C Z"
|
|
217 |
show "\<not> G\<turnstile>D \<prec>\<^sub>C C"
|
|
218 |
proof
|
|
219 |
assume subcls_D_C: "G\<turnstile>D \<prec>\<^sub>C C"
|
|
220 |
show "False"
|
|
221 |
proof -
|
|
222 |
from subcls_D_C subcls1_C_Z
|
|
223 |
have "G\<turnstile>D \<prec>\<^sub>C Z"
|
|
224 |
by (auto dest: r_into_trancl trancl_trans)
|
|
225 |
with nsubcls_D_Z
|
|
226 |
show ?thesis
|
|
227 |
by (rule notE)
|
|
228 |
qed
|
|
229 |
qed
|
|
230 |
qed
|
|
231 |
qed
|
|
232 |
|
|
233 |
lemma subclseq_cases [consumes 1, case_names Eq Subcls]:
|
|
234 |
"\<lbrakk>G\<turnstile>C \<preceq>\<^sub>C D; C = D \<Longrightarrow> P; G\<turnstile>C \<prec>\<^sub>C D \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
|
|
235 |
by (blast intro: rtrancl_cases)
|
|
236 |
|
|
237 |
lemma subclseq_acyclic:
|
|
238 |
"\<lbrakk>G\<turnstile>C \<preceq>\<^sub>C D; G\<turnstile>D \<preceq>\<^sub>C C; ws_prog G\<rbrakk> \<Longrightarrow> C=D"
|
|
239 |
by (auto elim: subclseq_cases dest: subcls_acyclic)
|
|
240 |
|
|
241 |
lemma subcls_irrefl: "\<lbrakk>G\<turnstile>C \<prec>\<^sub>C D; ws_prog G\<rbrakk>
|
|
242 |
\<Longrightarrow> C \<noteq> D"
|
|
243 |
proof -
|
|
244 |
assume ws: "ws_prog G"
|
|
245 |
assume subcls: "G\<turnstile>C \<prec>\<^sub>C D"
|
|
246 |
then show ?thesis
|
|
247 |
proof (induct rule: converse_trancl_induct)
|
|
248 |
fix C
|
|
249 |
assume "G\<turnstile>C \<prec>\<^sub>C\<^sub>1 D"
|
|
250 |
with ws
|
|
251 |
show "C\<noteq>D"
|
|
252 |
by (blast dest: subcls1_irrefl)
|
|
253 |
next
|
|
254 |
fix C Z
|
|
255 |
assume subcls1_C_Z: "G\<turnstile>C \<prec>\<^sub>C\<^sub>1 Z" and
|
|
256 |
subcls_Z_D: "G\<turnstile>Z \<prec>\<^sub>C D" and
|
|
257 |
neq_Z_D: "Z \<noteq> D"
|
|
258 |
show "C\<noteq>D"
|
|
259 |
proof
|
|
260 |
assume eq_C_D: "C=D"
|
|
261 |
show "False"
|
|
262 |
proof -
|
|
263 |
from subcls1_C_Z eq_C_D
|
|
264 |
have "G\<turnstile>D \<prec>\<^sub>C Z"
|
|
265 |
by (auto)
|
|
266 |
also
|
|
267 |
from subcls_Z_D ws
|
|
268 |
have "\<not> G\<turnstile>D \<prec>\<^sub>C Z"
|
|
269 |
by (rule subcls_acyclic)
|
|
270 |
ultimately
|
|
271 |
show ?thesis
|
|
272 |
by - (rule notE)
|
|
273 |
qed
|
|
274 |
qed
|
|
275 |
qed
|
|
276 |
qed
|
|
277 |
|
|
278 |
lemma invert_subclseq:
|
|
279 |
"\<lbrakk>G\<turnstile>C \<preceq>\<^sub>C D; ws_prog G\<rbrakk>
|
|
280 |
\<Longrightarrow> \<not> G\<turnstile>D \<prec>\<^sub>C C"
|
|
281 |
proof -
|
|
282 |
assume ws: "ws_prog G" and
|
|
283 |
subclseq_C_D: "G\<turnstile>C \<preceq>\<^sub>C D"
|
|
284 |
show ?thesis
|
|
285 |
proof (cases "D=C")
|
|
286 |
case True
|
|
287 |
with ws
|
|
288 |
show ?thesis
|
|
289 |
by (auto dest: subcls_irrefl)
|
|
290 |
next
|
|
291 |
case False
|
|
292 |
with subclseq_C_D
|
|
293 |
have "G\<turnstile>C \<prec>\<^sub>C D"
|
|
294 |
by (blast intro: rtrancl_into_trancl3)
|
|
295 |
with ws
|
|
296 |
show ?thesis
|
|
297 |
by (blast dest: subcls_acyclic)
|
|
298 |
qed
|
|
299 |
qed
|
|
300 |
|
|
301 |
lemma invert_subcls:
|
|
302 |
"\<lbrakk>G\<turnstile>C \<prec>\<^sub>C D; ws_prog G\<rbrakk>
|
|
303 |
\<Longrightarrow> \<not> G\<turnstile>D \<preceq>\<^sub>C C"
|
|
304 |
proof -
|
|
305 |
assume ws: "ws_prog G" and
|
|
306 |
subcls_C_D: "G\<turnstile>C \<prec>\<^sub>C D"
|
|
307 |
then
|
|
308 |
have nsubcls_D_C: "\<not> G\<turnstile>D \<prec>\<^sub>C C"
|
|
309 |
by (blast dest: subcls_acyclic)
|
|
310 |
show ?thesis
|
|
311 |
proof
|
|
312 |
assume "G\<turnstile>D \<preceq>\<^sub>C C"
|
|
313 |
then show "False"
|
|
314 |
proof (cases rule: subclseq_cases)
|
|
315 |
case Eq
|
|
316 |
with ws subcls_C_D
|
|
317 |
show ?thesis
|
|
318 |
by (auto dest: subcls_irrefl)
|
|
319 |
next
|
|
320 |
case Subcls
|
|
321 |
with nsubcls_D_C
|
|
322 |
show ?thesis
|
|
323 |
by blast
|
|
324 |
qed
|
|
325 |
qed
|
|
326 |
qed
|
|
327 |
|
|
328 |
lemma subcls_superD:
|
|
329 |
"\<lbrakk>G\<turnstile>C \<prec>\<^sub>C D; class G C = Some c\<rbrakk> \<Longrightarrow> G\<turnstile>(super c) \<preceq>\<^sub>C D"
|
|
330 |
proof -
|
|
331 |
assume clsC: "class G C = Some c"
|
|
332 |
assume subcls_C_C: "G\<turnstile>C \<prec>\<^sub>C D"
|
|
333 |
then obtain S where
|
|
334 |
"G\<turnstile>C \<prec>\<^sub>C\<^sub>1 S" and
|
|
335 |
subclseq_S_D: "G\<turnstile>S \<preceq>\<^sub>C D"
|
|
336 |
by (blast dest: tranclD)
|
|
337 |
with clsC
|
|
338 |
have "S=super c"
|
|
339 |
by (auto dest: subcls1D)
|
|
340 |
with subclseq_S_D show ?thesis by simp
|
|
341 |
qed
|
|
342 |
|
|
343 |
|
|
344 |
lemma subclseq_superD:
|
|
345 |
"\<lbrakk>G\<turnstile>C \<preceq>\<^sub>C D; C\<noteq>D;class G C = Some c\<rbrakk> \<Longrightarrow> G\<turnstile>(super c) \<preceq>\<^sub>C D"
|
|
346 |
proof -
|
|
347 |
assume neq_C_D: "C\<noteq>D"
|
|
348 |
assume clsC: "class G C = Some c"
|
|
349 |
assume subclseq_C_D: "G\<turnstile>C \<preceq>\<^sub>C D"
|
|
350 |
then show ?thesis
|
|
351 |
proof (cases rule: subclseq_cases)
|
|
352 |
case Eq with neq_C_D show ?thesis by contradiction
|
|
353 |
next
|
|
354 |
case Subcls
|
|
355 |
with clsC show ?thesis by (blast dest: subcls_superD)
|
|
356 |
qed
|
|
357 |
qed
|
|
358 |
|
|
359 |
section "implementation relation"
|
|
360 |
|
|
361 |
defs
|
|
362 |
(* direct implementation, cf. 8.1.3 *)
|
|
363 |
implmt1_def:"implmt1 G\<equiv>{(C,I). C\<noteq>Object \<and> (\<exists>c\<in>class G C: I\<in>set (superIfs c))}"
|
|
364 |
|
|
365 |
lemma implmt1D: "G\<turnstile>C\<leadsto>1I \<Longrightarrow> C\<noteq>Object \<and> (\<exists>c\<in>class G C: I\<in>set (superIfs c))"
|
|
366 |
apply (unfold implmt1_def)
|
|
367 |
apply auto
|
|
368 |
done
|
|
369 |
|
|
370 |
|
|
371 |
inductive "implmt G" intros (* cf. 8.1.4 *)
|
|
372 |
|
|
373 |
direct: "G\<turnstile>C\<leadsto>1J \<spacespace>\<spacespace> \<Longrightarrow> G\<turnstile>C\<leadsto>J"
|
|
374 |
subint: "\<lbrakk>G\<turnstile>C\<leadsto>1I; G\<turnstile>I\<preceq>I J\<rbrakk> \<Longrightarrow> G\<turnstile>C\<leadsto>J"
|
|
375 |
subcls1: "\<lbrakk>G\<turnstile>C\<prec>\<^sub>C\<^sub>1D; G\<turnstile>D\<leadsto>J \<rbrakk> \<Longrightarrow> G\<turnstile>C\<leadsto>J"
|
|
376 |
|
|
377 |
lemma implmtD: "G\<turnstile>C\<leadsto>J \<Longrightarrow> (\<exists>I. G\<turnstile>C\<leadsto>1I \<and> G\<turnstile>I\<preceq>I J) \<or> (\<exists>D. G\<turnstile>C\<prec>\<^sub>C\<^sub>1D \<and> G\<turnstile>D\<leadsto>J)"
|
|
378 |
apply (erule implmt.induct)
|
|
379 |
apply fast+
|
|
380 |
done
|
|
381 |
|
|
382 |
lemma implmt_ObjectE [elim!]: "G\<turnstile>Object\<leadsto>I \<Longrightarrow> R"
|
|
383 |
by (auto dest!: implmtD implmt1D subcls1D)
|
|
384 |
|
|
385 |
lemma subcls_implmt [rule_format (no_asm)]: "G\<turnstile>A\<preceq>\<^sub>C B \<Longrightarrow> G\<turnstile>B\<leadsto>K \<longrightarrow> G\<turnstile>A\<leadsto>K"
|
|
386 |
apply (erule rtrancl_induct)
|
|
387 |
apply (auto intro: implmt.subcls1)
|
|
388 |
done
|
|
389 |
|
|
390 |
lemma implmt_subint2: "\<lbrakk> G\<turnstile>A\<leadsto>J; G\<turnstile>J\<preceq>I K\<rbrakk> \<Longrightarrow> G\<turnstile>A\<leadsto>K"
|
|
391 |
apply (erule make_imp, erule implmt.induct)
|
|
392 |
apply (auto dest: implmt.subint rtrancl_trans implmt.subcls1)
|
|
393 |
done
|
|
394 |
|
|
395 |
lemma implmt_is_class: "G\<turnstile>C\<leadsto>I \<Longrightarrow> is_class G C"
|
|
396 |
apply (erule implmt.induct)
|
|
397 |
apply (blast dest: implmt1D subcls1D)+
|
|
398 |
done
|
|
399 |
|
|
400 |
|
|
401 |
section "widening relation"
|
|
402 |
|
|
403 |
inductive "widen G" intros(*widening, viz. method invocation conversion, cf. 5.3
|
|
404 |
i.e. kind of syntactic subtyping *)
|
|
405 |
refl: "G\<turnstile>T\<preceq>T"(*identity conversion, cf. 5.1.1 *)
|
|
406 |
subint: "G\<turnstile>I\<preceq>I J \<Longrightarrow> G\<turnstile>Iface I\<preceq> Iface J"(*wid.ref.conv.,cf. 5.1.4 *)
|
|
407 |
int_obj: "G\<turnstile>Iface I\<preceq> Class Object"
|
|
408 |
subcls: "G\<turnstile>C\<preceq>\<^sub>C D \<Longrightarrow> G\<turnstile>Class C\<preceq> Class D"
|
|
409 |
implmt: "G\<turnstile>C\<leadsto>I \<Longrightarrow> G\<turnstile>Class C\<preceq> Iface I"
|
|
410 |
null: "G\<turnstile>NT\<preceq> RefT R"
|
|
411 |
arr_obj: "G\<turnstile>T.[]\<preceq> Class Object"
|
|
412 |
array: "G\<turnstile>RefT S\<preceq>RefT T \<Longrightarrow> G\<turnstile>RefT S.[]\<preceq> RefT T.[]"
|
|
413 |
|
|
414 |
declare widen.refl [intro!]
|
|
415 |
declare widen.intros [simp]
|
|
416 |
|
|
417 |
(* too strong in general:
|
|
418 |
lemma widen_PrimT: "G\<turnstile>PrimT x\<preceq>T \<Longrightarrow> T = PrimT x"
|
|
419 |
*)
|
|
420 |
lemma widen_PrimT: "G\<turnstile>PrimT x\<preceq>T \<Longrightarrow> (\<exists>y. T = PrimT y)"
|
|
421 |
apply (ind_cases "G\<turnstile>S\<preceq>T")
|
|
422 |
by auto
|
|
423 |
|
|
424 |
(* too strong in general:
|
|
425 |
lemma widen_PrimT2: "G\<turnstile>S\<preceq>PrimT x \<Longrightarrow> S = PrimT x"
|
|
426 |
*)
|
|
427 |
lemma widen_PrimT2: "G\<turnstile>S\<preceq>PrimT x \<Longrightarrow> \<exists>y. S = PrimT y"
|
|
428 |
apply (ind_cases "G\<turnstile>S\<preceq>T")
|
|
429 |
by auto
|
|
430 |
|
|
431 |
lemma widen_RefT: "G\<turnstile>RefT R\<preceq>T \<Longrightarrow> \<exists>t. T=RefT t"
|
|
432 |
apply (ind_cases "G\<turnstile>S\<preceq>T")
|
|
433 |
by auto
|
|
434 |
|
|
435 |
lemma widen_RefT2: "G\<turnstile>S\<preceq>RefT R \<Longrightarrow> \<exists>t. S=RefT t"
|
|
436 |
apply (ind_cases "G\<turnstile>S\<preceq>T")
|
|
437 |
by auto
|
|
438 |
|
|
439 |
lemma widen_Iface: "G\<turnstile>Iface I\<preceq>T \<Longrightarrow> T=Class Object \<or> (\<exists>J. T=Iface J)"
|
|
440 |
apply (ind_cases "G\<turnstile>S\<preceq>T")
|
|
441 |
by auto
|
|
442 |
|
|
443 |
lemma widen_Iface2: "G\<turnstile>S\<preceq> Iface J \<Longrightarrow> S = NT \<or> (\<exists>I. S = Iface I) \<or> (\<exists>D. S = Class D)"
|
|
444 |
apply (ind_cases "G\<turnstile>S\<preceq>T")
|
|
445 |
by auto
|
|
446 |
|
|
447 |
lemma widen_Iface_Iface: "G\<turnstile>Iface I\<preceq> Iface J \<Longrightarrow> G\<turnstile>I\<preceq>I J"
|
|
448 |
apply (ind_cases "G\<turnstile>S\<preceq>T")
|
|
449 |
by auto
|
|
450 |
|
|
451 |
lemma widen_Iface_Iface_eq [simp]: "G\<turnstile>Iface I\<preceq> Iface J = G\<turnstile>I\<preceq>I J"
|
|
452 |
apply (rule iffI)
|
|
453 |
apply (erule widen_Iface_Iface)
|
|
454 |
apply (erule widen.subint)
|
|
455 |
done
|
|
456 |
|
|
457 |
lemma widen_Class: "G\<turnstile>Class C\<preceq>T \<Longrightarrow> (\<exists>D. T=Class D) \<or> (\<exists>I. T=Iface I)"
|
|
458 |
apply (ind_cases "G\<turnstile>S\<preceq>T")
|
|
459 |
by auto
|
|
460 |
|
|
461 |
lemma widen_Class2: "G\<turnstile>S\<preceq> Class C \<Longrightarrow> C = Object \<or> S = NT \<or> (\<exists>D. S = Class D)"
|
|
462 |
apply (ind_cases "G\<turnstile>S\<preceq>T")
|
|
463 |
by auto
|
|
464 |
|
|
465 |
lemma widen_Class_Class: "G\<turnstile>Class C\<preceq> Class cm \<Longrightarrow> G\<turnstile>C\<preceq>\<^sub>C cm"
|
|
466 |
apply (ind_cases "G\<turnstile>S\<preceq>T")
|
|
467 |
by auto
|
|
468 |
|
|
469 |
lemma widen_Class_Class_eq [simp]: "G\<turnstile>Class C\<preceq> Class cm = G\<turnstile>C\<preceq>\<^sub>C cm"
|
|
470 |
apply (rule iffI)
|
|
471 |
apply (erule widen_Class_Class)
|
|
472 |
apply (erule widen.subcls)
|
|
473 |
done
|
|
474 |
|
|
475 |
lemma widen_Class_Iface: "G\<turnstile>Class C\<preceq> Iface I \<Longrightarrow> G\<turnstile>C\<leadsto>I"
|
|
476 |
apply (ind_cases "G\<turnstile>S\<preceq>T")
|
|
477 |
by auto
|
|
478 |
|
|
479 |
lemma widen_Class_Iface_eq [simp]: "G\<turnstile>Class C\<preceq> Iface I = G\<turnstile>C\<leadsto>I"
|
|
480 |
apply (rule iffI)
|
|
481 |
apply (erule widen_Class_Iface)
|
|
482 |
apply (erule widen.implmt)
|
|
483 |
done
|
|
484 |
|
|
485 |
lemma widen_Array: "G\<turnstile>S.[]\<preceq>T \<Longrightarrow> T=Class Object \<or> (\<exists>T'. T=T'.[] \<and> G\<turnstile>S\<preceq>T')"
|
|
486 |
apply (ind_cases "G\<turnstile>S\<preceq>T")
|
|
487 |
by auto
|
|
488 |
|
|
489 |
lemma widen_Array2: "G\<turnstile>S\<preceq>T.[] \<Longrightarrow> S = NT \<or> (\<exists>S'. S=S'.[] \<and> G\<turnstile>S'\<preceq>T)"
|
|
490 |
apply (ind_cases "G\<turnstile>S\<preceq>T")
|
|
491 |
by auto
|
|
492 |
|
|
493 |
|
|
494 |
lemma widen_ArrayPrimT: "G\<turnstile>PrimT t.[]\<preceq>T \<Longrightarrow> T=Class Object \<or> T=PrimT t.[]"
|
|
495 |
apply (ind_cases "G\<turnstile>S\<preceq>T")
|
|
496 |
by auto
|
|
497 |
|
|
498 |
lemma widen_ArrayRefT:
|
|
499 |
"G\<turnstile>RefT t.[]\<preceq>T \<Longrightarrow> T=Class Object \<or> (\<exists>s. T=RefT s.[] \<and> G\<turnstile>RefT t\<preceq>RefT s)"
|
|
500 |
apply (ind_cases "G\<turnstile>S\<preceq>T")
|
|
501 |
by auto
|
|
502 |
|
|
503 |
lemma widen_ArrayRefT_ArrayRefT_eq [simp]:
|
|
504 |
"G\<turnstile>RefT T.[]\<preceq>RefT T'.[] = G\<turnstile>RefT T\<preceq>RefT T'"
|
|
505 |
apply (rule iffI)
|
|
506 |
apply (drule widen_ArrayRefT)
|
|
507 |
apply simp
|
|
508 |
apply (erule widen.array)
|
|
509 |
done
|
|
510 |
|
|
511 |
lemma widen_Array_Array: "G\<turnstile>T.[]\<preceq>T'.[] \<Longrightarrow> G\<turnstile>T\<preceq>T'"
|
|
512 |
apply (drule widen_Array)
|
|
513 |
apply auto
|
|
514 |
done
|
|
515 |
|
|
516 |
lemma widen_Array_Class: "G\<turnstile>S.[] \<preceq> Class C \<Longrightarrow> C=Object"
|
|
517 |
by (auto dest: widen_Array)
|
|
518 |
|
|
519 |
(*
|
|
520 |
qed_typerel "widen_NT2" "G\<turnstile>S\<preceq>NT \<Longrightarrow> S = NT"
|
|
521 |
[prove_widen_lemma "G\<turnstile>S\<preceq>T \<Longrightarrow> T = NT \<longrightarrow> S = NT"]
|
|
522 |
*)
|
|
523 |
lemma widen_NT2: "G\<turnstile>S\<preceq>NT \<Longrightarrow> S = NT"
|
|
524 |
apply (ind_cases "G\<turnstile>S\<preceq>T")
|
|
525 |
by auto
|
|
526 |
|
|
527 |
lemma widen_Object:"\<lbrakk>isrtype G T;ws_prog G\<rbrakk> \<Longrightarrow> G\<turnstile>RefT T \<preceq> Class Object"
|
|
528 |
apply (case_tac T)
|
|
529 |
apply (auto)
|
|
530 |
apply (subgoal_tac "G\<turnstile>pid_field_type\<preceq>\<^sub>C Object")
|
|
531 |
apply (auto intro: subcls_ObjectI)
|
|
532 |
done
|
|
533 |
|
|
534 |
lemma widen_trans_lemma [rule_format (no_asm)]:
|
|
535 |
"\<lbrakk>G\<turnstile>S\<preceq>U; \<forall>C. is_class G C \<longrightarrow> G\<turnstile>C\<preceq>\<^sub>C Object\<rbrakk> \<Longrightarrow> \<forall>T. G\<turnstile>U\<preceq>T \<longrightarrow> G\<turnstile>S\<preceq>T"
|
|
536 |
apply (erule widen.induct)
|
|
537 |
apply safe
|
|
538 |
prefer 5 apply (drule widen_RefT) apply clarsimp
|
|
539 |
apply (frule_tac [1] widen_Iface)
|
|
540 |
apply (frule_tac [2] widen_Class)
|
|
541 |
apply (frule_tac [3] widen_Class)
|
|
542 |
apply (frule_tac [4] widen_Iface)
|
|
543 |
apply (frule_tac [5] widen_Class)
|
|
544 |
apply (frule_tac [6] widen_Array)
|
|
545 |
apply safe
|
|
546 |
apply (rule widen.int_obj)
|
|
547 |
prefer 6 apply (drule implmt_is_class) apply simp
|
|
548 |
apply (tactic "ALLGOALS (etac thin_rl)")
|
|
549 |
prefer 6 apply simp
|
|
550 |
apply (rule_tac [9] widen.arr_obj)
|
|
551 |
apply (rotate_tac [9] -1)
|
|
552 |
apply (frule_tac [9] widen_RefT)
|
|
553 |
apply (auto elim!: rtrancl_trans subcls_implmt implmt_subint2)
|
|
554 |
done
|
|
555 |
|
|
556 |
lemma ws_widen_trans: "\<lbrakk>G\<turnstile>S\<preceq>U; G\<turnstile>U\<preceq>T; ws_prog G\<rbrakk> \<Longrightarrow> G\<turnstile>S\<preceq>T"
|
|
557 |
by (auto intro: widen_trans_lemma subcls_ObjectI)
|
|
558 |
|
|
559 |
lemma widen_antisym_lemma [rule_format (no_asm)]: "\<lbrakk>G\<turnstile>S\<preceq>T;
|
|
560 |
\<forall>I J. G\<turnstile>I\<preceq>I J \<and> G\<turnstile>J\<preceq>I I \<longrightarrow> I = J;
|
|
561 |
\<forall>C D. G\<turnstile>C\<preceq>\<^sub>C D \<and> G\<turnstile>D\<preceq>\<^sub>C C \<longrightarrow> C = D;
|
|
562 |
\<forall>I . G\<turnstile>Object\<leadsto>I \<longrightarrow> False\<rbrakk> \<Longrightarrow> G\<turnstile>T\<preceq>S \<longrightarrow> S = T"
|
|
563 |
apply (erule widen.induct)
|
|
564 |
apply (auto dest: widen_Iface widen_NT2 widen_Class)
|
|
565 |
done
|
|
566 |
|
|
567 |
lemmas subint_antisym =
|
|
568 |
subint1_acyclic [THEN acyclic_impl_antisym_rtrancl, standard]
|
|
569 |
lemmas subcls_antisym =
|
|
570 |
subcls1_acyclic [THEN acyclic_impl_antisym_rtrancl, standard]
|
|
571 |
|
|
572 |
lemma widen_antisym: "\<lbrakk>G\<turnstile>S\<preceq>T; G\<turnstile>T\<preceq>S; ws_prog G\<rbrakk> \<Longrightarrow> S=T"
|
|
573 |
by (fast elim: widen_antisym_lemma subint_antisym [THEN antisymD]
|
|
574 |
subcls_antisym [THEN antisymD])
|
|
575 |
|
|
576 |
lemma widen_ObjectD [dest!]: "G\<turnstile>Class Object\<preceq>T \<Longrightarrow> T=Class Object"
|
|
577 |
apply (frule widen_Class)
|
|
578 |
apply (fast dest: widen_Class_Class widen_Class_Iface)
|
|
579 |
done
|
|
580 |
|
|
581 |
constdefs
|
|
582 |
widens :: "prog \<Rightarrow> [ty list, ty list] \<Rightarrow> bool" ("_\<turnstile>_[\<preceq>]_" [71,71,71] 70)
|
|
583 |
"G\<turnstile>Ts[\<preceq>]Ts' \<equiv> list_all2 (\<lambda>T T'. G\<turnstile>T\<preceq>T') Ts Ts'"
|
|
584 |
|
|
585 |
lemma widens_Nil [simp]: "G\<turnstile>[][\<preceq>][]"
|
|
586 |
apply (unfold widens_def)
|
|
587 |
apply auto
|
|
588 |
done
|
|
589 |
|
|
590 |
lemma widens_Cons [simp]: "G\<turnstile>(S#Ss)[\<preceq>](T#Ts) = (G\<turnstile>S\<preceq>T \<and> G\<turnstile>Ss[\<preceq>]Ts)"
|
|
591 |
apply (unfold widens_def)
|
|
592 |
apply auto
|
|
593 |
done
|
|
594 |
|
|
595 |
|
|
596 |
section "narrowing relation"
|
|
597 |
|
|
598 |
(* all properties of narrowing and casting conversions we actually need *)
|
|
599 |
(* these can easily be proven from the definitions below *)
|
|
600 |
(*
|
|
601 |
rules
|
|
602 |
cast_RefT2 "G\<turnstile>S\<preceq>? RefT R \<Longrightarrow> \<exists>t. S=RefT t"
|
|
603 |
cast_PrimT2 "G\<turnstile>S\<preceq>? PrimT pt \<Longrightarrow> \<exists>t. S=PrimT t \<and> G\<turnstile>PrimT t\<preceq>PrimT pt"
|
|
604 |
*)
|
|
605 |
|
|
606 |
(* more detailed than necessary for type-safety, see above rules. *)
|
|
607 |
inductive "narrow G" intros (* narrowing reference conversion, cf. 5.1.5 *)
|
|
608 |
|
|
609 |
subcls: "G\<turnstile>C\<preceq>\<^sub>C D \<Longrightarrow> G\<turnstile> Class D\<succ>Class C"
|
|
610 |
implmt: "\<not>G\<turnstile>C\<leadsto>I \<Longrightarrow> G\<turnstile> Class C\<succ>Iface I"
|
|
611 |
obj_arr: "G\<turnstile>Class Object\<succ>T.[]"
|
|
612 |
int_cls: "G\<turnstile> Iface I\<succ>Class C"
|
|
613 |
subint: "imethds G I hidings imethds G J entails
|
|
614 |
(\<lambda>(md, mh ) (md',mh').\<spacespace>G\<turnstile>mrt mh\<preceq>mrt mh') \<Longrightarrow>
|
|
615 |
\<not>G\<turnstile>I\<preceq>I J \<spacespace>\<spacespace>\<spacespace>\<Longrightarrow> G\<turnstile> Iface I\<succ>Iface J"
|
|
616 |
array: "G\<turnstile>RefT S\<succ>RefT T \<spacespace>\<Longrightarrow> G\<turnstile> RefT S.[]\<succ>RefT T.[]"
|
|
617 |
|
|
618 |
(*unused*)
|
|
619 |
lemma narrow_RefT: "G\<turnstile>RefT R\<succ>T \<Longrightarrow> \<exists>t. T=RefT t"
|
|
620 |
apply (ind_cases "G\<turnstile>S\<succ>T")
|
|
621 |
by auto
|
|
622 |
|
|
623 |
lemma narrow_RefT2: "G\<turnstile>S\<succ>RefT R \<Longrightarrow> \<exists>t. S=RefT t"
|
|
624 |
apply (ind_cases "G\<turnstile>S\<succ>T")
|
|
625 |
by auto
|
|
626 |
|
|
627 |
(*unused*)
|
|
628 |
lemma narrow_PrimT: "G\<turnstile>PrimT pt\<succ>T \<Longrightarrow> \<exists>t. T=PrimT t"
|
|
629 |
apply (ind_cases "G\<turnstile>S\<succ>T")
|
|
630 |
by auto
|
|
631 |
|
|
632 |
lemma narrow_PrimT2: "G\<turnstile>S\<succ>PrimT pt \<Longrightarrow>
|
|
633 |
\<exists>t. S=PrimT t \<and> G\<turnstile>PrimT t\<preceq>PrimT pt"
|
|
634 |
apply (ind_cases "G\<turnstile>S\<succ>T")
|
|
635 |
by auto
|
|
636 |
|
|
637 |
|
|
638 |
section "casting relation"
|
|
639 |
|
|
640 |
inductive "cast G" intros (* casting conversion, cf. 5.5 *)
|
|
641 |
|
|
642 |
widen: "G\<turnstile>S\<preceq>T \<Longrightarrow> G\<turnstile>S\<preceq>? T"
|
|
643 |
narrow: "G\<turnstile>S\<succ>T \<Longrightarrow> G\<turnstile>S\<preceq>? T"
|
|
644 |
|
|
645 |
(*
|
|
646 |
lemma ??unknown??: "\<lbrakk>G\<turnstile>S\<preceq>T; G\<turnstile>S\<succ>T\<rbrakk> \<Longrightarrow> R"
|
|
647 |
deferred *)
|
|
648 |
|
|
649 |
(*unused*)
|
|
650 |
lemma cast_RefT: "G\<turnstile>RefT R\<preceq>? T \<Longrightarrow> \<exists>t. T=RefT t"
|
|
651 |
apply (ind_cases "G\<turnstile>S\<preceq>? T")
|
|
652 |
by (auto dest: widen_RefT narrow_RefT)
|
|
653 |
|
|
654 |
lemma cast_RefT2: "G\<turnstile>S\<preceq>? RefT R \<Longrightarrow> \<exists>t. S=RefT t"
|
|
655 |
apply (ind_cases "G\<turnstile>S\<preceq>? T")
|
|
656 |
by (auto dest: widen_RefT2 narrow_RefT2)
|
|
657 |
|
|
658 |
(*unused*)
|
|
659 |
lemma cast_PrimT: "G\<turnstile>PrimT pt\<preceq>? T \<Longrightarrow> \<exists>t. T=PrimT t"
|
|
660 |
apply (ind_cases "G\<turnstile>S\<preceq>? T")
|
|
661 |
by (auto dest: widen_PrimT narrow_PrimT)
|
|
662 |
|
|
663 |
lemma cast_PrimT2: "G\<turnstile>S\<preceq>? PrimT pt \<Longrightarrow> \<exists>t. S=PrimT t \<and> G\<turnstile>PrimT t\<preceq>PrimT pt"
|
|
664 |
apply (ind_cases "G\<turnstile>S\<preceq>? T")
|
|
665 |
by (auto dest: widen_PrimT2 narrow_PrimT2)
|
|
666 |
|
|
667 |
end
|