12857
|
1 |
(* Title: HOL/Bali/Conform.thy
|
12854
|
2 |
ID: $Id$
|
|
3 |
Author: David von Oheimb
|
12858
|
4 |
License: GPL (GNU GENERAL PUBLIC LICENSE)
|
12854
|
5 |
*)
|
|
6 |
|
|
7 |
header {* Conformance notions for the type soundness proof for Java *}
|
|
8 |
|
|
9 |
theory Conform = State:
|
|
10 |
|
|
11 |
text {*
|
|
12 |
design issues:
|
|
13 |
\begin{itemize}
|
|
14 |
\item lconf allows for (arbitrary) inaccessible values
|
|
15 |
\item ''conforms'' does not directly imply that the dynamic types of all
|
|
16 |
objects on the heap are indeed existing classes. Yet this can be
|
|
17 |
inferred for all referenced objs.
|
|
18 |
\end{itemize}
|
|
19 |
*}
|
|
20 |
|
|
21 |
types env_ = "prog \<times> (lname, ty) table" (* same as env of WellType.thy *)
|
|
22 |
|
|
23 |
|
|
24 |
section "extension of global store"
|
|
25 |
|
|
26 |
constdefs
|
|
27 |
|
|
28 |
gext :: "st \<Rightarrow> st \<Rightarrow> bool" ("_\<le>|_" [71,71] 70)
|
|
29 |
"s\<le>|s' \<equiv> \<forall>r. \<forall>obj\<in>globs s r: \<exists>obj'\<in>globs s' r: tag obj'= tag obj"
|
|
30 |
|
|
31 |
lemma gext_objD:
|
|
32 |
"\<lbrakk>s\<le>|s'; globs s r = Some obj\<rbrakk>
|
|
33 |
\<Longrightarrow> \<exists>obj'. globs s' r = Some obj' \<and> tag obj' = tag obj"
|
|
34 |
apply (simp only: gext_def)
|
|
35 |
by force
|
|
36 |
|
|
37 |
lemma rev_gext_objD:
|
|
38 |
"\<lbrakk>globs s r = Some obj; s\<le>|s'\<rbrakk>
|
|
39 |
\<Longrightarrow> \<exists>obj'. globs s' r = Some obj' \<and> tag obj' = tag obj"
|
|
40 |
by (auto elim: gext_objD)
|
|
41 |
|
|
42 |
lemma init_class_obj_inited:
|
|
43 |
"init_class_obj G C s1\<le>|s2 \<Longrightarrow> inited C (globs s2)"
|
|
44 |
apply (unfold inited_def init_obj_def)
|
|
45 |
apply (auto dest!: gext_objD)
|
|
46 |
done
|
|
47 |
|
|
48 |
lemma gext_refl [intro!, simp]: "s\<le>|s"
|
|
49 |
apply (unfold gext_def)
|
|
50 |
apply (fast del: fst_splitE)
|
|
51 |
done
|
|
52 |
|
|
53 |
lemma gext_gupd [simp, elim!]: "\<And>s. globs s r = None \<Longrightarrow> s\<le>|gupd(r\<mapsto>x)s"
|
|
54 |
by (auto simp: gext_def)
|
|
55 |
|
|
56 |
lemma gext_new [simp, elim!]: "\<And>s. globs s r = None \<Longrightarrow> s\<le>|init_obj G oi r s"
|
|
57 |
apply (simp only: init_obj_def)
|
|
58 |
apply (erule_tac gext_gupd)
|
|
59 |
done
|
|
60 |
|
|
61 |
lemma gext_trans [elim]: "\<And>X. \<lbrakk>s\<le>|s'; s'\<le>|s''\<rbrakk> \<Longrightarrow> s\<le>|s''"
|
|
62 |
by (force simp: gext_def)
|
|
63 |
|
|
64 |
lemma gext_upd_gobj [intro!]: "s\<le>|upd_gobj r n v s"
|
|
65 |
apply (simp only: gext_def)
|
|
66 |
apply auto
|
|
67 |
apply (case_tac "ra = r")
|
|
68 |
apply auto
|
|
69 |
apply (case_tac "globs s r = None")
|
|
70 |
apply auto
|
|
71 |
done
|
|
72 |
|
|
73 |
lemma gext_cong1 [simp]: "set_locals l s1\<le>|s2 = s1\<le>|s2"
|
|
74 |
by (auto simp: gext_def)
|
|
75 |
|
|
76 |
lemma gext_cong2 [simp]: "s1\<le>|set_locals l s2 = s1\<le>|s2"
|
|
77 |
by (auto simp: gext_def)
|
|
78 |
|
|
79 |
|
|
80 |
lemma gext_lupd1 [simp]: "lupd(vn\<mapsto>v)s1\<le>|s2 = s1\<le>|s2"
|
|
81 |
by (auto simp: gext_def)
|
|
82 |
|
|
83 |
lemma gext_lupd2 [simp]: "s1\<le>|lupd(vn\<mapsto>v)s2 = s1\<le>|s2"
|
|
84 |
by (auto simp: gext_def)
|
|
85 |
|
|
86 |
|
|
87 |
lemma inited_gext: "\<lbrakk>inited C (globs s); s\<le>|s'\<rbrakk> \<Longrightarrow> inited C (globs s')"
|
|
88 |
apply (unfold inited_def)
|
|
89 |
apply (auto dest: gext_objD)
|
|
90 |
done
|
|
91 |
|
|
92 |
|
|
93 |
section "value conformance"
|
|
94 |
|
|
95 |
constdefs
|
|
96 |
|
|
97 |
conf :: "prog \<Rightarrow> st \<Rightarrow> val \<Rightarrow> ty \<Rightarrow> bool" ("_,_\<turnstile>_\<Colon>\<preceq>_" [71,71,71,71] 70)
|
|
98 |
"G,s\<turnstile>v\<Colon>\<preceq>T \<equiv> \<exists>T'\<in>typeof (\<lambda>a. option_map obj_ty (heap s a)) v:G\<turnstile>T'\<preceq>T"
|
|
99 |
|
|
100 |
lemma conf_cong [simp]: "G,set_locals l s\<turnstile>v\<Colon>\<preceq>T = G,s\<turnstile>v\<Colon>\<preceq>T"
|
|
101 |
by (auto simp: conf_def)
|
|
102 |
|
|
103 |
lemma conf_lupd [simp]: "G,lupd(vn\<mapsto>va)s\<turnstile>v\<Colon>\<preceq>T = G,s\<turnstile>v\<Colon>\<preceq>T"
|
|
104 |
by (auto simp: conf_def)
|
|
105 |
|
|
106 |
lemma conf_PrimT [simp]: "\<forall>dt. typeof dt v = Some (PrimT t) \<Longrightarrow> G,s\<turnstile>v\<Colon>\<preceq>PrimT t"
|
|
107 |
apply (simp add: conf_def)
|
|
108 |
done
|
|
109 |
|
|
110 |
lemma conf_litval [rule_format (no_asm)]:
|
|
111 |
"typeof (\<lambda>a. None) v = Some T \<longrightarrow> G,s\<turnstile>v\<Colon>\<preceq>T"
|
|
112 |
apply (unfold conf_def)
|
|
113 |
apply (rule val.induct)
|
|
114 |
apply auto
|
|
115 |
done
|
|
116 |
|
|
117 |
lemma conf_Null [simp]: "G,s\<turnstile>Null\<Colon>\<preceq>T = G\<turnstile>NT\<preceq>T"
|
|
118 |
by (simp add: conf_def)
|
|
119 |
|
|
120 |
lemma conf_Addr:
|
|
121 |
"G,s\<turnstile>Addr a\<Colon>\<preceq>T = (\<exists>obj. heap s a = Some obj \<and> G\<turnstile>obj_ty obj\<preceq>T)"
|
|
122 |
by (auto simp: conf_def)
|
|
123 |
|
|
124 |
lemma conf_AddrI:"\<lbrakk>heap s a = Some obj; G\<turnstile>obj_ty obj\<preceq>T\<rbrakk> \<Longrightarrow> G,s\<turnstile>Addr a\<Colon>\<preceq>T"
|
|
125 |
apply (rule conf_Addr [THEN iffD2])
|
|
126 |
by fast
|
|
127 |
|
|
128 |
lemma defval_conf [rule_format (no_asm), elim]:
|
|
129 |
"is_type G T \<longrightarrow> G,s\<turnstile>default_val T\<Colon>\<preceq>T"
|
|
130 |
apply (unfold conf_def)
|
|
131 |
apply (induct "T")
|
|
132 |
apply (auto intro: prim_ty.induct)
|
|
133 |
done
|
|
134 |
|
|
135 |
lemma conf_widen [rule_format (no_asm), elim]:
|
|
136 |
"G\<turnstile>T\<preceq>T' \<Longrightarrow> G,s\<turnstile>x\<Colon>\<preceq>T \<longrightarrow> ws_prog G \<longrightarrow> G,s\<turnstile>x\<Colon>\<preceq>T'"
|
|
137 |
apply (unfold conf_def)
|
|
138 |
apply (rule val.induct)
|
|
139 |
apply (auto elim: ws_widen_trans)
|
|
140 |
done
|
|
141 |
|
|
142 |
lemma conf_gext [rule_format (no_asm), elim]:
|
|
143 |
"G,s\<turnstile>v\<Colon>\<preceq>T \<longrightarrow> s\<le>|s' \<longrightarrow> G,s'\<turnstile>v\<Colon>\<preceq>T"
|
|
144 |
apply (unfold gext_def conf_def)
|
|
145 |
apply (rule val.induct)
|
|
146 |
apply force+
|
|
147 |
done
|
|
148 |
|
|
149 |
|
|
150 |
lemma conf_list_widen [rule_format (no_asm)]:
|
|
151 |
"ws_prog G \<Longrightarrow>
|
|
152 |
\<forall>Ts Ts'. list_all2 (conf G s) vs Ts
|
|
153 |
\<longrightarrow> G\<turnstile>Ts[\<preceq>] Ts' \<longrightarrow> list_all2 (conf G s) vs Ts'"
|
|
154 |
apply (unfold widens_def)
|
|
155 |
apply (rule list_all2_trans)
|
|
156 |
apply auto
|
|
157 |
done
|
|
158 |
|
|
159 |
lemma conf_RefTD [rule_format (no_asm)]:
|
|
160 |
"G,s\<turnstile>a'\<Colon>\<preceq>RefT T
|
|
161 |
\<longrightarrow> a' = Null \<or> (\<exists>a obj T'. a' = Addr a \<and> heap s a = Some obj \<and>
|
|
162 |
obj_ty obj = T' \<and> G\<turnstile>T'\<preceq>RefT T)"
|
|
163 |
apply (unfold conf_def)
|
|
164 |
apply (induct_tac "a'")
|
|
165 |
apply (auto dest: widen_PrimT)
|
|
166 |
done
|
|
167 |
|
|
168 |
|
|
169 |
section "value list conformance"
|
|
170 |
|
|
171 |
constdefs
|
|
172 |
|
|
173 |
lconf :: "prog \<Rightarrow> st \<Rightarrow> ('a, val) table \<Rightarrow> ('a, ty) table \<Rightarrow> bool"
|
|
174 |
("_,_\<turnstile>_[\<Colon>\<preceq>]_" [71,71,71,71] 70)
|
|
175 |
"G,s\<turnstile>vs[\<Colon>\<preceq>]Ts \<equiv> \<forall>n. \<forall>T\<in>Ts n: \<exists>v\<in>vs n: G,s\<turnstile>v\<Colon>\<preceq>T"
|
|
176 |
|
|
177 |
lemma lconfD: "\<lbrakk>G,s\<turnstile>vs[\<Colon>\<preceq>]Ts; Ts n = Some T\<rbrakk> \<Longrightarrow> G,s\<turnstile>(the (vs n))\<Colon>\<preceq>T"
|
|
178 |
by (force simp: lconf_def)
|
|
179 |
|
|
180 |
|
|
181 |
lemma lconf_cong [simp]: "\<And>s. G,set_locals x s\<turnstile>l[\<Colon>\<preceq>]L = G,s\<turnstile>l[\<Colon>\<preceq>]L"
|
|
182 |
by (auto simp: lconf_def)
|
|
183 |
|
|
184 |
lemma lconf_lupd [simp]: "G,lupd(vn\<mapsto>v)s\<turnstile>l[\<Colon>\<preceq>]L = G,s\<turnstile>l[\<Colon>\<preceq>]L"
|
|
185 |
by (auto simp: lconf_def)
|
|
186 |
|
|
187 |
(* unused *)
|
|
188 |
lemma lconf_new: "\<lbrakk>L vn = None; G,s\<turnstile>l[\<Colon>\<preceq>]L\<rbrakk> \<Longrightarrow> G,s\<turnstile>l(vn\<mapsto>v)[\<Colon>\<preceq>]L"
|
|
189 |
by (auto simp: lconf_def)
|
|
190 |
|
|
191 |
lemma lconf_upd: "\<lbrakk>G,s\<turnstile>l[\<Colon>\<preceq>]L; G,s\<turnstile>v\<Colon>\<preceq>T; L vn = Some T\<rbrakk> \<Longrightarrow>
|
|
192 |
G,s\<turnstile>l(vn\<mapsto>v)[\<Colon>\<preceq>]L"
|
|
193 |
by (auto simp: lconf_def)
|
|
194 |
|
|
195 |
lemma lconf_ext: "\<lbrakk>G,s\<turnstile>l[\<Colon>\<preceq>]L; G,s\<turnstile>v\<Colon>\<preceq>T\<rbrakk> \<Longrightarrow> G,s\<turnstile>l(vn\<mapsto>v)[\<Colon>\<preceq>]L(vn\<mapsto>T)"
|
|
196 |
by (auto simp: lconf_def)
|
|
197 |
|
|
198 |
lemma lconf_map_sum [simp]:
|
|
199 |
"G,s\<turnstile>l1 (+) l2[\<Colon>\<preceq>]L1 (+) L2 = (G,s\<turnstile>l1[\<Colon>\<preceq>]L1 \<and> G,s\<turnstile>l2[\<Colon>\<preceq>]L2)"
|
|
200 |
apply (unfold lconf_def)
|
|
201 |
apply safe
|
|
202 |
apply (case_tac [3] "n")
|
|
203 |
apply (force split add: sum.split)+
|
|
204 |
done
|
|
205 |
|
|
206 |
lemma lconf_ext_list [rule_format (no_asm)]: "
|
|
207 |
\<And>X. \<lbrakk>G,s\<turnstile>l[\<Colon>\<preceq>]L\<rbrakk> \<Longrightarrow>
|
12893
|
208 |
\<forall>vs Ts. distinct vns \<longrightarrow> length Ts = length vns
|
12854
|
209 |
\<longrightarrow> list_all2 (conf G s) vs Ts \<longrightarrow> G,s\<turnstile>l(vns[\<mapsto>]vs)[\<Colon>\<preceq>]L(vns[\<mapsto>]Ts)"
|
|
210 |
apply (unfold lconf_def)
|
|
211 |
apply (induct_tac "vns")
|
|
212 |
apply clarsimp
|
|
213 |
apply clarsimp
|
|
214 |
apply (frule list_all2_lengthD)
|
|
215 |
apply clarsimp
|
|
216 |
done
|
|
217 |
|
|
218 |
|
|
219 |
lemma lconf_deallocL: "\<lbrakk>G,s\<turnstile>l[\<Colon>\<preceq>]L(vn\<mapsto>T); L vn = None\<rbrakk> \<Longrightarrow> G,s\<turnstile>l[\<Colon>\<preceq>]L"
|
|
220 |
apply (simp only: lconf_def)
|
|
221 |
apply safe
|
|
222 |
apply (drule spec)
|
|
223 |
apply (drule ospec)
|
|
224 |
apply auto
|
|
225 |
done
|
|
226 |
|
|
227 |
|
|
228 |
lemma lconf_gext [elim]: "\<lbrakk>G,s\<turnstile>l[\<Colon>\<preceq>]L; s\<le>|s'\<rbrakk> \<Longrightarrow> G,s'\<turnstile>l[\<Colon>\<preceq>]L"
|
|
229 |
apply (simp only: lconf_def)
|
|
230 |
apply fast
|
|
231 |
done
|
|
232 |
|
|
233 |
lemma lconf_empty [simp, intro!]: "G,s\<turnstile>vs[\<Colon>\<preceq>]empty"
|
|
234 |
apply (unfold lconf_def)
|
|
235 |
apply force
|
|
236 |
done
|
|
237 |
|
|
238 |
lemma lconf_init_vals [intro!]:
|
|
239 |
" \<forall>n. \<forall>T\<in>fs n:is_type G T \<Longrightarrow> G,s\<turnstile>init_vals fs[\<Colon>\<preceq>]fs"
|
|
240 |
apply (unfold lconf_def)
|
|
241 |
apply force
|
|
242 |
done
|
|
243 |
|
|
244 |
|
|
245 |
section "object conformance"
|
|
246 |
|
|
247 |
constdefs
|
|
248 |
|
|
249 |
oconf :: "prog \<Rightarrow> st \<Rightarrow> obj \<Rightarrow> oref \<Rightarrow> bool" ("_,_\<turnstile>_\<Colon>\<preceq>\<surd>_" [71,71,71,71] 70)
|
|
250 |
"G,s\<turnstile>obj\<Colon>\<preceq>\<surd>r \<equiv> G,s\<turnstile>values obj[\<Colon>\<preceq>]var_tys G (tag obj) r \<and>
|
|
251 |
(case r of
|
|
252 |
Heap a \<Rightarrow> is_type G (obj_ty obj)
|
|
253 |
| Stat C \<Rightarrow> True)"
|
|
254 |
(*
|
|
255 |
lemma oconf_def2: "G,s\<turnstile>\<lparr>tag=oi,values=fs\<rparr>\<Colon>\<preceq>\<surd>r =
|
|
256 |
(G,s\<turnstile>fs[\<Colon>\<preceq>]var_tys G oi r \<and>
|
|
257 |
(case r of Heap a \<Rightarrow> is_type G (obj_ty \<lparr>tag=oi,values=fs\<rparr>) | Stat C \<Rightarrow> True))"
|
|
258 |
by (simp add: oconf_def Let_def)
|
|
259 |
*)
|
|
260 |
(*
|
|
261 |
lemma oconf_def2: "G,s\<turnstile>obj\<Colon>\<preceq>\<surd>r =
|
|
262 |
(G,s\<turnstile>values obj[\<Colon>\<preceq>]var_tys G (tag obj) r \<and>
|
|
263 |
(case r of Heap a \<Rightarrow> is_type G (obj_ty obj) | Stat C \<Rightarrow> True))"
|
|
264 |
by (simp add: oconf_def Let_def)
|
|
265 |
*)
|
|
266 |
lemma oconf_is_type: "G,s\<turnstile>obj\<Colon>\<preceq>\<surd>Heap a \<Longrightarrow> is_type G (obj_ty obj)"
|
|
267 |
by (auto simp: oconf_def Let_def)
|
|
268 |
|
|
269 |
lemma oconf_lconf: "G,s\<turnstile>obj\<Colon>\<preceq>\<surd>r \<Longrightarrow> G,s\<turnstile>values obj[\<Colon>\<preceq>]var_tys G (tag obj) r"
|
|
270 |
by (simp add: oconf_def)
|
|
271 |
|
|
272 |
lemma oconf_cong [simp]: "G,set_locals l s\<turnstile>obj\<Colon>\<preceq>\<surd>r = G,s\<turnstile>obj\<Colon>\<preceq>\<surd>r"
|
|
273 |
by (auto simp: oconf_def Let_def)
|
|
274 |
|
|
275 |
lemma oconf_init_obj_lemma:
|
|
276 |
"\<lbrakk>\<And>C c. class G C = Some c \<Longrightarrow> unique (DeclConcepts.fields G C);
|
|
277 |
\<And>C c f fld. \<lbrakk>class G C = Some c;
|
|
278 |
table_of (DeclConcepts.fields G C) f = Some fld \<rbrakk>
|
|
279 |
\<Longrightarrow> is_type G (type fld);
|
|
280 |
(case r of
|
|
281 |
Heap a \<Rightarrow> is_type G (obj_ty obj)
|
|
282 |
| Stat C \<Rightarrow> is_class G C)
|
|
283 |
\<rbrakk> \<Longrightarrow> G,s\<turnstile>obj \<lparr>values:=init_vals (var_tys G (tag obj) r)\<rparr>\<Colon>\<preceq>\<surd>r"
|
|
284 |
apply (auto simp add: oconf_def)
|
|
285 |
apply (drule_tac var_tys_Some_eq [THEN iffD1])
|
|
286 |
defer
|
|
287 |
apply (subst obj_ty_cong)
|
|
288 |
apply(auto dest!: fields_table_SomeD obj_ty_CInst1 obj_ty_Arr1
|
|
289 |
split add: sum.split_asm obj_tag.split_asm)
|
|
290 |
done
|
|
291 |
|
|
292 |
(*
|
|
293 |
lemma oconf_init_obj_lemma:
|
|
294 |
"\<lbrakk>\<And>C c. class G C = Some c \<Longrightarrow> unique (fields G C);
|
|
295 |
\<And>C c f fld. \<lbrakk>class G C = Some c; table_of (fields G C) f = Some fld \<rbrakk>
|
|
296 |
\<Longrightarrow> is_type G (type fld);
|
|
297 |
(case r of
|
|
298 |
Heap a \<Rightarrow> is_type G (obj_ty \<lparr>tag=oi,values=fs\<rparr>)
|
|
299 |
| Stat C \<Rightarrow> is_class G C)
|
|
300 |
\<rbrakk> \<Longrightarrow> G,s\<turnstile>\<lparr>tag=oi, values=init_vals (var_tys G oi r)\<rparr>\<Colon>\<preceq>\<surd>r"
|
|
301 |
apply (auto simp add: oconf_def)
|
|
302 |
apply (drule_tac var_tys_Some_eq [THEN iffD1])
|
|
303 |
defer
|
|
304 |
apply (subst obj_ty_eq)
|
|
305 |
apply(auto dest!: fields_table_SomeD split add: sum.split_asm obj_tag.split_asm)
|
|
306 |
done
|
|
307 |
*)
|
|
308 |
|
|
309 |
|
|
310 |
section "state conformance"
|
|
311 |
|
|
312 |
constdefs
|
|
313 |
|
|
314 |
conforms :: "state \<Rightarrow> env_ \<Rightarrow> bool" ( "_\<Colon>\<preceq>_" [71,71] 70)
|
|
315 |
"xs\<Colon>\<preceq>E \<equiv> let (G, L) = E; s = snd xs; l = locals s in
|
|
316 |
(\<forall>r. \<forall>obj\<in>globs s r: G,s\<turnstile>obj \<Colon>\<preceq>\<surd>r) \<and>
|
|
317 |
\<spacespace> G,s\<turnstile>l [\<Colon>\<preceq>]L\<spacespace> \<and>
|
|
318 |
(\<forall>a. fst xs=Some(Xcpt (Loc a)) \<longrightarrow> G,s\<turnstile>Addr a\<Colon>\<preceq> Class (SXcpt Throwable))"
|
|
319 |
|
|
320 |
section "conforms"
|
|
321 |
|
|
322 |
lemma conforms_globsD:
|
|
323 |
"\<lbrakk>(x, s)\<Colon>\<preceq>(G, L); globs s r = Some obj\<rbrakk> \<Longrightarrow> G,s\<turnstile>obj\<Colon>\<preceq>\<surd>r"
|
|
324 |
by (auto simp: conforms_def Let_def)
|
|
325 |
|
|
326 |
lemma conforms_localD: "(x, s)\<Colon>\<preceq>(G, L) \<Longrightarrow> G,s\<turnstile>locals s[\<Colon>\<preceq>]L"
|
|
327 |
by (auto simp: conforms_def Let_def)
|
|
328 |
|
|
329 |
lemma conforms_XcptLocD: "\<lbrakk>(x, s)\<Colon>\<preceq>(G, L); x = Some (Xcpt (Loc a))\<rbrakk> \<Longrightarrow>
|
|
330 |
G,s\<turnstile>Addr a\<Colon>\<preceq> Class (SXcpt Throwable)"
|
|
331 |
by (auto simp: conforms_def Let_def)
|
|
332 |
|
|
333 |
lemma conforms_RefTD:
|
|
334 |
"\<lbrakk>G,s\<turnstile>a'\<Colon>\<preceq>RefT t; a' \<noteq> Null; (x,s) \<Colon>\<preceq>(G, L)\<rbrakk> \<Longrightarrow>
|
|
335 |
\<exists>a obj. a' = Addr a \<and> globs s (Inl a) = Some obj \<and>
|
|
336 |
G\<turnstile>obj_ty obj\<preceq>RefT t \<and> is_type G (obj_ty obj)"
|
|
337 |
apply (drule_tac conf_RefTD)
|
|
338 |
apply clarsimp
|
|
339 |
apply (rule conforms_globsD [THEN oconf_is_type])
|
|
340 |
apply auto
|
|
341 |
done
|
|
342 |
|
|
343 |
lemma conforms_Jump [iff]:
|
|
344 |
"((Some (Jump j), s)\<Colon>\<preceq>(G, L)) = (Norm s\<Colon>\<preceq>(G, L))"
|
|
345 |
by (auto simp: conforms_def)
|
|
346 |
|
|
347 |
lemma conforms_StdXcpt [iff]:
|
|
348 |
"((Some (Xcpt (Std xn)), s)\<Colon>\<preceq>(G, L)) = (Norm s\<Colon>\<preceq>(G, L))"
|
|
349 |
by (auto simp: conforms_def)
|
|
350 |
|
|
351 |
lemma conforms_raise_if [iff]:
|
|
352 |
"((raise_if c xn x, s)\<Colon>\<preceq>(G, L)) = ((x, s)\<Colon>\<preceq>(G, L))"
|
|
353 |
by (auto simp: abrupt_if_def)
|
|
354 |
|
|
355 |
|
|
356 |
lemma conforms_NormI: "(x, s)\<Colon>\<preceq>(G, L) \<Longrightarrow> Norm s\<Colon>\<preceq>(G, L)"
|
|
357 |
by (auto simp: conforms_def Let_def)
|
|
358 |
|
|
359 |
|
|
360 |
lemma conforms_absorb [rule_format]:
|
|
361 |
"(a, b)\<Colon>\<preceq>(G, L) \<longrightarrow> (absorb j a, b)\<Colon>\<preceq>(G, L)"
|
|
362 |
apply (rule impI)
|
|
363 |
apply ( case_tac a)
|
|
364 |
apply (case_tac "absorb j a")
|
|
365 |
apply auto
|
|
366 |
apply (case_tac "absorb j (Some a)",auto)
|
|
367 |
apply (erule conforms_NormI)
|
|
368 |
done
|
|
369 |
|
|
370 |
lemma conformsI: "\<lbrakk>\<forall>r. \<forall>obj\<in>globs s r: G,s\<turnstile>obj\<Colon>\<preceq>\<surd>r;
|
|
371 |
G,s\<turnstile>locals s[\<Colon>\<preceq>]L;
|
|
372 |
\<forall>a. x = Some (Xcpt (Loc a)) \<longrightarrow> G,s\<turnstile>Addr a\<Colon>\<preceq> Class (SXcpt Throwable)\<rbrakk> \<Longrightarrow>
|
|
373 |
(x, s)\<Colon>\<preceq>(G, L)"
|
|
374 |
by (auto simp: conforms_def Let_def)
|
|
375 |
|
|
376 |
lemma conforms_xconf: "\<lbrakk>(x, s)\<Colon>\<preceq>(G,L);
|
|
377 |
\<forall>a. x' = Some (Xcpt (Loc a)) \<longrightarrow> G,s\<turnstile>Addr a\<Colon>\<preceq> Class (SXcpt Throwable)\<rbrakk> \<Longrightarrow>
|
|
378 |
(x',s)\<Colon>\<preceq>(G,L)"
|
|
379 |
by (fast intro: conformsI elim: conforms_globsD conforms_localD)
|
|
380 |
|
|
381 |
lemma conforms_lupd:
|
|
382 |
"\<lbrakk>(x, s)\<Colon>\<preceq>(G, L); L vn = Some T; G,s\<turnstile>v\<Colon>\<preceq>T\<rbrakk> \<Longrightarrow> (x, lupd(vn\<mapsto>v)s)\<Colon>\<preceq>(G, L)"
|
|
383 |
by (force intro: conformsI lconf_upd dest: conforms_globsD conforms_localD
|
|
384 |
conforms_XcptLocD simp: oconf_def)
|
|
385 |
|
|
386 |
|
|
387 |
lemmas conforms_allocL_aux = conforms_localD [THEN lconf_ext]
|
|
388 |
|
|
389 |
lemma conforms_allocL:
|
|
390 |
"\<lbrakk>(x, s)\<Colon>\<preceq>(G, L); G,s\<turnstile>v\<Colon>\<preceq>T\<rbrakk> \<Longrightarrow> (x, lupd(vn\<mapsto>v)s)\<Colon>\<preceq>(G, L(vn\<mapsto>T))"
|
|
391 |
by (force intro: conformsI dest: conforms_globsD
|
|
392 |
elim: conforms_XcptLocD conforms_allocL_aux simp: oconf_def)
|
|
393 |
|
|
394 |
lemmas conforms_deallocL_aux = conforms_localD [THEN lconf_deallocL]
|
|
395 |
|
|
396 |
lemma conforms_deallocL: "\<And>s.\<lbrakk>s\<Colon>\<preceq>(G, L(vn\<mapsto>T)); L vn = None\<rbrakk> \<Longrightarrow> s\<Colon>\<preceq>(G,L)"
|
|
397 |
by (fast intro: conformsI dest: conforms_globsD
|
|
398 |
elim: conforms_XcptLocD conforms_deallocL_aux)
|
|
399 |
|
|
400 |
lemma conforms_gext: "\<lbrakk>(x, s)\<Colon>\<preceq>(G,L); s\<le>|s';
|
|
401 |
\<forall>r. \<forall>obj\<in>globs s' r: G,s'\<turnstile>obj\<Colon>\<preceq>\<surd>r;
|
|
402 |
locals s'=locals s\<rbrakk> \<Longrightarrow> (x,s')\<Colon>\<preceq>(G,L)"
|
|
403 |
by (force intro!: conformsI dest: conforms_localD conforms_XcptLocD)
|
|
404 |
|
|
405 |
|
|
406 |
lemma conforms_xgext:
|
|
407 |
"\<lbrakk>(x ,s)\<Colon>\<preceq>(G,L); (x', s')\<Colon>\<preceq>(G, L); s'\<le>|s\<rbrakk> \<Longrightarrow> (x',s)\<Colon>\<preceq>(G,L)"
|
|
408 |
apply (erule_tac conforms_xconf)
|
|
409 |
apply (fast dest: conforms_XcptLocD)
|
|
410 |
done
|
|
411 |
|
|
412 |
lemma conforms_gupd: "\<And>obj. \<lbrakk>(x, s)\<Colon>\<preceq>(G, L); G,s\<turnstile>obj\<Colon>\<preceq>\<surd>r; s\<le>|gupd(r\<mapsto>obj)s\<rbrakk>
|
|
413 |
\<Longrightarrow> (x, gupd(r\<mapsto>obj)s)\<Colon>\<preceq>(G, L)"
|
|
414 |
apply (rule conforms_gext)
|
|
415 |
apply auto
|
|
416 |
apply (force dest: conforms_globsD simp add: oconf_def)+
|
|
417 |
done
|
|
418 |
|
|
419 |
lemma conforms_upd_gobj: "\<lbrakk>(x,s)\<Colon>\<preceq>(G, L); globs s r = Some obj;
|
|
420 |
var_tys G (tag obj) r n = Some T; G,s\<turnstile>v\<Colon>\<preceq>T\<rbrakk> \<Longrightarrow> (x,upd_gobj r n v s)\<Colon>\<preceq>(G,L)"
|
|
421 |
apply (rule conforms_gext)
|
|
422 |
apply auto
|
|
423 |
apply (drule (1) conforms_globsD)
|
|
424 |
apply (simp add: oconf_def)
|
|
425 |
apply safe
|
|
426 |
apply (rule lconf_upd)
|
|
427 |
apply auto
|
|
428 |
apply (simp only: obj_ty_cong)
|
|
429 |
apply (force dest: conforms_globsD intro!: lconf_upd
|
|
430 |
simp add: oconf_def cong del: sum.weak_case_cong)
|
|
431 |
done
|
|
432 |
|
|
433 |
lemma conforms_set_locals:
|
|
434 |
"\<lbrakk>(x,s)\<Colon>\<preceq>(G, L'); G,s\<turnstile>l[\<Colon>\<preceq>]L\<rbrakk> \<Longrightarrow> (x,set_locals l s)\<Colon>\<preceq>(G,L)"
|
|
435 |
apply (auto intro!: conformsI dest: conforms_globsD
|
|
436 |
elim!: conforms_XcptLocD simp add: oconf_def)
|
|
437 |
done
|
|
438 |
|
|
439 |
lemma conforms_return: "\<And>s'. \<lbrakk>(x,s)\<Colon>\<preceq>(G, L); (x',s')\<Colon>\<preceq>(G, L'); s\<le>|s'\<rbrakk> \<Longrightarrow>
|
|
440 |
(x',set_locals (locals s) s')\<Colon>\<preceq>(G, L)"
|
|
441 |
apply (rule conforms_xconf)
|
|
442 |
prefer 2 apply (force dest: conforms_XcptLocD)
|
|
443 |
apply (erule conforms_gext)
|
|
444 |
apply (force dest: conforms_globsD)+
|
|
445 |
done
|
|
446 |
|
|
447 |
end
|