author | blanchet |
Sun, 16 Feb 2014 21:33:28 +0100 | |
changeset 55524 | f41ef840f09d |
parent 42150 | b0c0638c4aad |
child 56073 | 29e308b56d23 |
permissions | -rw-r--r-- |
42150 | 1 |
(* Title: HOL/MicroJava/BV/EffectMono.thy |
12516 | 2 |
Author: Gerwin Klein |
3 |
Copyright 2000 Technische Universitaet Muenchen |
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4 |
*) |
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||
12911 | 6 |
header {* \isaheader{Monotonicity of eff and app} *} |
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|
33954
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
25362
diff
changeset
|
8 |
theory EffectMono |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
25362
diff
changeset
|
9 |
imports Effect |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
25362
diff
changeset
|
10 |
begin |
12516 | 11 |
|
12 |
lemma PrimT_PrimT: "(G \<turnstile> xb \<preceq> PrimT p) = (xb = PrimT p)" |
|
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by (auto elim: widen.cases) |
12516 | 14 |
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15 |
||
16 |
lemma sup_loc_some [rule_format]: |
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"\<forall>y n. (G \<turnstile> b <=l y) \<longrightarrow> n < length y \<longrightarrow> y!n = OK t \<longrightarrow> |
34915 | 18 |
(\<exists>t. b!n = OK t \<and> (G \<turnstile> (b!n) <=o (y!n)))" |
19 |
proof (induct b) |
|
20 |
case Nil |
|
21 |
show ?case by simp |
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25362 | 22 |
next |
23 |
case (Cons a list) |
|
34915 | 24 |
show ?case |
12516 | 25 |
proof (clarsimp simp add: list_all2_Cons1 sup_loc_def Listn.le_def lesub_def) |
26 |
fix z zs n |
|
25362 | 27 |
assume *: |
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"G \<turnstile> a <=o z" "list_all2 (sup_ty_opt G) list zs" |
29 |
"n < Suc (length list)" "(z # zs) ! n = OK t" |
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30 |
||
31 |
show "(\<exists>t. (a # list) ! n = OK t) \<and> G \<turnstile>(a # list) ! n <=o OK t" |
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32 |
proof (cases n) |
|
33 |
case 0 |
|
34 |
with * show ?thesis by (simp add: sup_ty_opt_OK) |
|
35 |
next |
|
36 |
case Suc |
|
37 |
with Cons * |
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38 |
show ?thesis by (simp add: sup_loc_def Listn.le_def lesub_def) |
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39 |
qed |
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40 |
qed |
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41 |
qed |
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42 |
||
43 |
||
44 |
lemma all_widen_is_sup_loc: |
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"\<forall>b. length a = length b \<longrightarrow> |
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(\<forall>(x, y)\<in>set (zip a b). G \<turnstile> x \<preceq> y) = (G \<turnstile> (map OK a) <=l (map OK b))" |
13006 | 47 |
(is "\<forall>b. length a = length b \<longrightarrow> ?Q a b" is "?P a") |
12516 | 48 |
proof (induct "a") |
49 |
show "?P []" by simp |
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50 |
||
51 |
fix l ls assume Cons: "?P ls" |
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52 |
||
53 |
show "?P (l#ls)" |
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54 |
proof (intro allI impI) |
|
55 |
fix b |
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56 |
assume "length (l # ls) = length (b::ty list)" |
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57 |
with Cons |
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58 |
show "?Q (l # ls) b" by - (cases b, auto) |
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59 |
qed |
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60 |
qed |
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61 |
||
62 |
||
63 |
lemma append_length_n [rule_format]: |
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34915 | 64 |
"\<forall>n. n \<le> length x \<longrightarrow> (\<exists>a b. x = a@b \<and> length a = n)" |
65 |
proof (induct x) |
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case Nil |
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show ?case by simp |
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next |
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case (Cons l ls) |
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|
34915 | 71 |
show ?case |
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proof (intro allI impI) |
73 |
fix n |
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assume l: "n \<le> length (l # ls)" |
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75 |
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76 |
show "\<exists>a b. l # ls = a @ b \<and> length a = n" |
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proof (cases n) |
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assume "n=0" thus ?thesis by simp |
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79 |
next |
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80 |
fix n' assume s: "n = Suc n'" |
|
81 |
with l have "n' \<le> length ls" by simp |
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hence "\<exists>a b. ls = a @ b \<and> length a = n'" by (rule Cons [rule_format]) |
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then obtain a b where "ls = a @ b" "length a = n'" by iprover |
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with s have "l # ls = (l#a) @ b \<and> length (l#a) = n" by simp |
85 |
thus ?thesis by blast |
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qed |
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87 |
qed |
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qed |
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||
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lemma rev_append_cons: |
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"n < length x \<Longrightarrow> \<exists>a b c. x = (rev a) @ b # c \<and> length a = n" |
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proof - |
93 |
assume n: "n < length x" |
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94 |
hence "n \<le> length x" by simp |
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hence "\<exists>a b. x = a @ b \<and> length a = n" by (rule append_length_n) |
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17589 | 96 |
then obtain r d where x: "x = r@d" "length r = n" by iprover |
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with n have "\<exists>b c. d = b#c" by (simp add: neq_Nil_conv) |
17589 | 98 |
then obtain b c where "d = b#c" by iprover |
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with x have "x = (rev (rev r)) @ b # c \<and> length (rev r) = n" by simp |
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thus ?thesis by blast |
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qed |
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||
103 |
lemma sup_loc_length_map: |
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"G \<turnstile> map f a <=l map g b \<Longrightarrow> length a = length b" |
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proof - |
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assume "G \<turnstile> map f a <=l map g b" |
|
107 |
hence "length (map f a) = length (map g b)" by (rule sup_loc_length) |
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thus ?thesis by simp |
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109 |
qed |
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110 |
||
111 |
lemmas [iff] = not_Err_eq |
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112 |
||
113 |
lemma app_mono: |
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"\<lbrakk>G \<turnstile> s <=' s'; app i G m rT pc et s'\<rbrakk> \<Longrightarrow> app i G m rT pc et s" |
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proof - |
116 |
||
117 |
{ fix s1 s2 |
|
118 |
assume G: "G \<turnstile> s2 <=s s1" |
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assume app: "app i G m rT pc et (Some s1)" |
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120 |
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121 |
note [simp] = sup_loc_length sup_loc_length_map |
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122 |
||
123 |
have "app i G m rT pc et (Some s2)" |
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25362 | 124 |
proof (cases i) |
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case Load |
126 |
||
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from G Load app |
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have "G \<turnstile> snd s2 <=l snd s1" by (auto simp add: sup_state_conv) |
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129 |
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with G Load app show ?thesis |
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by (cases s2) (auto simp add: sup_state_conv dest: sup_loc_some) |
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132 |
next |
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133 |
case Store |
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with G app show ?thesis |
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25362 | 135 |
by (cases s2) (auto simp add: sup_loc_Cons2 sup_state_conv) |
12516 | 136 |
next |
137 |
case LitPush |
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25362 | 138 |
with G app show ?thesis by (cases s2) (auto simp add: sup_state_conv) |
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next |
140 |
case New |
|
25362 | 141 |
with G app show ?thesis by (cases s2) (auto simp add: sup_state_conv) |
12516 | 142 |
next |
143 |
case Getfield |
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with app G show ?thesis |
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by (cases s2) (clarsimp simp add: sup_state_Cons2, rule widen_trans) |
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146 |
next |
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case (Putfield vname cname) |
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149 |
with app |
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obtain vT oT ST LT b |
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where s1: "s1 = (vT # oT # ST, LT)" and |
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"field (G, cname) vname = Some (cname, b)" |
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"is_class G cname" and |
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oT: "G\<turnstile> oT\<preceq> (Class cname)" and |
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vT: "G\<turnstile> vT\<preceq> b" and |
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xc: "Ball (set (match G NullPointer pc et)) (is_class G)" |
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by force |
158 |
moreover |
|
159 |
from s1 G |
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obtain vT' oT' ST' LT' |
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where s2: "s2 = (vT' # oT' # ST', LT')" and |
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oT': "G\<turnstile> oT' \<preceq> oT" and |
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vT': "G\<turnstile> vT' \<preceq> vT" |
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by - (cases s2, simp add: sup_state_Cons2, elim exE conjE, simp, rule that) |
|
165 |
moreover |
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from vT' vT |
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have "G \<turnstile> vT' \<preceq> b" by (rule widen_trans) |
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168 |
moreover |
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169 |
from oT' oT |
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170 |
have "G\<turnstile> oT' \<preceq> (Class cname)" by (rule widen_trans) |
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171 |
ultimately |
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172 |
show ?thesis by (auto simp add: Putfield xc) |
|
173 |
next |
|
174 |
case Checkcast |
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175 |
with app G show ?thesis |
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25362 | 176 |
by (cases s2) (auto intro!: widen_RefT2 simp add: sup_state_Cons2) |
12516 | 177 |
next |
178 |
case Return |
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with app G show ?thesis |
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180 |
by (cases s2) (auto simp add: sup_state_Cons2, rule widen_trans) |
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181 |
next |
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182 |
case Pop |
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with app G show ?thesis |
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25362 | 184 |
by (cases s2) (clarsimp simp add: sup_state_Cons2) |
12516 | 185 |
next |
186 |
case Dup |
|
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with app G show ?thesis |
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25362 | 188 |
by (cases s2) (clarsimp simp add: sup_state_Cons2, |
12516 | 189 |
auto dest: sup_state_length) |
190 |
next |
|
191 |
case Dup_x1 |
|
192 |
with app G show ?thesis |
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25362 | 193 |
by (cases s2) (clarsimp simp add: sup_state_Cons2, |
12516 | 194 |
auto dest: sup_state_length) |
195 |
next |
|
196 |
case Dup_x2 |
|
197 |
with app G show ?thesis |
|
25362 | 198 |
by (cases s2) (clarsimp simp add: sup_state_Cons2, |
12516 | 199 |
auto dest: sup_state_length) |
200 |
next |
|
201 |
case Swap |
|
202 |
with app G show ?thesis |
|
25362 | 203 |
by (cases s2) (auto simp add: sup_state_Cons2) |
12516 | 204 |
next |
205 |
case IAdd |
|
206 |
with app G show ?thesis |
|
25362 | 207 |
by (cases s2) (auto simp add: sup_state_Cons2 PrimT_PrimT) |
12516 | 208 |
next |
209 |
case Goto |
|
210 |
with app show ?thesis by simp |
|
211 |
next |
|
212 |
case Ifcmpeq |
|
213 |
with app G show ?thesis |
|
25362 | 214 |
by (cases s2) (auto simp add: sup_state_Cons2 PrimT_PrimT widen_RefT2) |
12516 | 215 |
next |
25362 | 216 |
case (Invoke cname mname list) |
12516 | 217 |
|
218 |
with app |
|
219 |
obtain apTs X ST LT mD' rT' b' where |
|
220 |
s1: "s1 = (rev apTs @ X # ST, LT)" and |
|
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l: "length apTs = length list" and |
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222 |
c: "is_class G cname" and |
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C: "G \<turnstile> X \<preceq> Class cname" and |
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22271 | 224 |
w: "\<forall>(x, y) \<in> set (zip apTs list). G \<turnstile> x \<preceq> y" and |
12516 | 225 |
m: "method (G, cname) (mname, list) = Some (mD', rT', b')" and |
226 |
x: "\<forall>C \<in> set (match_any G pc et). is_class G C" |
|
227 |
by (simp del: not_None_eq, elim exE conjE) (rule that) |
|
228 |
||
229 |
obtain apTs' X' ST' LT' where |
|
230 |
s2: "s2 = (rev apTs' @ X' # ST', LT')" and |
|
231 |
l': "length apTs' = length list" |
|
232 |
proof - |
|
233 |
from l s1 G |
|
234 |
have "length list < length (fst s2)" |
|
235 |
by simp |
|
236 |
hence "\<exists>a b c. (fst s2) = rev a @ b # c \<and> length a = length list" |
|
237 |
by (rule rev_append_cons [rule_format]) |
|
238 |
thus ?thesis |
|
25362 | 239 |
by (cases s2) (elim exE conjE, simp, rule that) |
12516 | 240 |
qed |
241 |
||
242 |
from l l' |
|
243 |
have "length (rev apTs') = length (rev apTs)" by simp |
|
244 |
||
245 |
from this s1 s2 G |
|
246 |
obtain |
|
247 |
G': "G \<turnstile> (apTs',LT') <=s (apTs,LT)" and |
|
248 |
X : "G \<turnstile> X' \<preceq> X" and "G \<turnstile> (ST',LT') <=s (ST,LT)" |
|
249 |
by (simp add: sup_state_rev_fst sup_state_append_fst sup_state_Cons1) |
|
250 |
||
251 |
with C |
|
252 |
have C': "G \<turnstile> X' \<preceq> Class cname" |
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253 |
by - (rule widen_trans, auto) |
|
254 |
||
255 |
from G' |
|
256 |
have "G \<turnstile> map OK apTs' <=l map OK apTs" |
|
257 |
by (simp add: sup_state_conv) |
|
258 |
also |
|
259 |
from l w |
|
260 |
have "G \<turnstile> map OK apTs <=l map OK list" |
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261 |
by (simp add: all_widen_is_sup_loc) |
|
262 |
finally |
|
263 |
have "G \<turnstile> map OK apTs' <=l map OK list" . |
|
264 |
||
265 |
with l' |
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22271 | 266 |
have w': "\<forall>(x, y) \<in> set (zip apTs' list). G \<turnstile> x \<preceq> y" |
12516 | 267 |
by (simp add: all_widen_is_sup_loc) |
268 |
||
269 |
from Invoke s2 l' w' C' m c x |
|
270 |
show ?thesis |
|
271 |
by (simp del: split_paired_Ex) blast |
|
272 |
next |
|
273 |
case Throw |
|
274 |
with app G show ?thesis |
|
275 |
by (cases s2, clarsimp simp add: sup_state_Cons2 widen_RefT2) |
|
276 |
qed |
|
277 |
} note this [simp] |
|
278 |
||
279 |
assume "G \<turnstile> s <=' s'" "app i G m rT pc et s'" |
|
280 |
thus ?thesis by (cases s, cases s', auto) |
|
281 |
qed |
|
282 |
||
283 |
lemmas [simp del] = split_paired_Ex |
|
284 |
||
285 |
lemma eff'_mono: |
|
13006 | 286 |
"\<lbrakk> app i G m rT pc et (Some s2); G \<turnstile> s1 <=s s2 \<rbrakk> \<Longrightarrow> |
12516 | 287 |
G \<turnstile> eff' (i,G,s1) <=s eff' (i,G,s2)" |
288 |
proof (cases s1, cases s2) |
|
289 |
fix a1 b1 a2 b2 |
|
290 |
assume s: "s1 = (a1,b1)" "s2 = (a2,b2)" |
|
291 |
assume app2: "app i G m rT pc et (Some s2)" |
|
292 |
assume G: "G \<turnstile> s1 <=s s2" |
|
293 |
||
294 |
note [simp] = eff_def |
|
295 |
||
23467 | 296 |
with G have "G \<turnstile> (Some s1) <=' (Some s2)" by simp |
12516 | 297 |
from this app2 |
298 |
have app1: "app i G m rT pc et (Some s1)" by (rule app_mono) |
|
299 |
||
300 |
show ?thesis |
|
25362 | 301 |
proof (cases i) |
302 |
case (Load n) |
|
12516 | 303 |
|
304 |
with s app1 |
|
305 |
obtain y where |
|
25362 | 306 |
y: "n < length b1" "b1 ! n = OK y" by clarsimp |
12516 | 307 |
|
308 |
from Load s app2 |
|
309 |
obtain y' where |
|
25362 | 310 |
y': "n < length b2" "b2 ! n = OK y'" by clarsimp |
12516 | 311 |
|
312 |
from G s |
|
313 |
have "G \<turnstile> b1 <=l b2" by (simp add: sup_state_conv) |
|
314 |
||
315 |
with y y' |
|
316 |
have "G \<turnstile> y \<preceq> y'" |
|
317 |
by - (drule sup_loc_some, simp+) |
|
318 |
||
319 |
with Load G y y' s app1 app2 |
|
320 |
show ?thesis by (clarsimp simp add: sup_state_conv) |
|
321 |
next |
|
322 |
case Store |
|
323 |
with G s app1 app2 |
|
324 |
show ?thesis |
|
325 |
by (clarsimp simp add: sup_state_conv sup_loc_update) |
|
326 |
next |
|
327 |
case LitPush |
|
328 |
with G s app1 app2 |
|
329 |
show ?thesis |
|
330 |
by (clarsimp simp add: sup_state_Cons1) |
|
331 |
next |
|
332 |
case New |
|
333 |
with G s app1 app2 |
|
334 |
show ?thesis |
|
335 |
by (clarsimp simp add: sup_state_Cons1) |
|
336 |
next |
|
337 |
case Getfield |
|
338 |
with G s app1 app2 |
|
339 |
show ?thesis |
|
340 |
by (clarsimp simp add: sup_state_Cons1) |
|
341 |
next |
|
342 |
case Putfield |
|
343 |
with G s app1 app2 |
|
344 |
show ?thesis |
|
345 |
by (clarsimp simp add: sup_state_Cons1) |
|
346 |
next |
|
347 |
case Checkcast |
|
348 |
with G s app1 app2 |
|
349 |
show ?thesis |
|
350 |
by (clarsimp simp add: sup_state_Cons1) |
|
351 |
next |
|
25362 | 352 |
case (Invoke cname mname list) |
12516 | 353 |
|
354 |
with s app1 |
|
355 |
obtain a X ST where |
|
356 |
s1: "s1 = (a @ X # ST, b1)" and |
|
357 |
l: "length a = length list" |
|
13601 | 358 |
by (simp, elim exE conjE, simp (no_asm_simp)) |
12516 | 359 |
|
360 |
from Invoke s app2 |
|
361 |
obtain a' X' ST' where |
|
362 |
s2: "s2 = (a' @ X' # ST', b2)" and |
|
363 |
l': "length a' = length list" |
|
13601 | 364 |
by (simp, elim exE conjE, simp (no_asm_simp)) |
12516 | 365 |
|
366 |
from l l' |
|
367 |
have lr: "length a = length a'" by simp |
|
368 |
||
13601 | 369 |
from lr G s1 s2 |
12516 | 370 |
have "G \<turnstile> (ST, b1) <=s (ST', b2)" |
371 |
by (simp add: sup_state_append_fst sup_state_Cons1) |
|
372 |
||
373 |
moreover |
|
374 |
||
375 |
obtain b1' b2' where eff': |
|
376 |
"b1' = snd (eff' (i,G,s1))" |
|
377 |
"b2' = snd (eff' (i,G,s2))" by simp |
|
378 |
||
379 |
from Invoke G s eff' app1 app2 |
|
380 |
obtain "b1 = b1'" "b2 = b2'" by simp |
|
381 |
||
382 |
ultimately |
|
383 |
||
384 |
have "G \<turnstile> (ST, b1') <=s (ST', b2')" by simp |
|
385 |
||
386 |
with Invoke G s app1 app2 eff' s1 s2 l l' |
|
387 |
show ?thesis |
|
388 |
by (clarsimp simp add: sup_state_conv) |
|
389 |
next |
|
390 |
case Return |
|
391 |
with G |
|
392 |
show ?thesis |
|
393 |
by simp |
|
394 |
next |
|
395 |
case Pop |
|
396 |
with G s app1 app2 |
|
397 |
show ?thesis |
|
398 |
by (clarsimp simp add: sup_state_Cons1) |
|
399 |
next |
|
400 |
case Dup |
|
401 |
with G s app1 app2 |
|
402 |
show ?thesis |
|
403 |
by (clarsimp simp add: sup_state_Cons1) |
|
404 |
next |
|
405 |
case Dup_x1 |
|
406 |
with G s app1 app2 |
|
407 |
show ?thesis |
|
408 |
by (clarsimp simp add: sup_state_Cons1) |
|
409 |
next |
|
410 |
case Dup_x2 |
|
411 |
with G s app1 app2 |
|
412 |
show ?thesis |
|
413 |
by (clarsimp simp add: sup_state_Cons1) |
|
414 |
next |
|
415 |
case Swap |
|
416 |
with G s app1 app2 |
|
417 |
show ?thesis |
|
418 |
by (clarsimp simp add: sup_state_Cons1) |
|
419 |
next |
|
420 |
case IAdd |
|
421 |
with G s app1 app2 |
|
422 |
show ?thesis |
|
423 |
by (clarsimp simp add: sup_state_Cons1) |
|
424 |
next |
|
425 |
case Goto |
|
426 |
with G s app1 app2 |
|
427 |
show ?thesis by simp |
|
428 |
next |
|
429 |
case Ifcmpeq |
|
430 |
with G s app1 app2 |
|
431 |
show ?thesis |
|
432 |
by (clarsimp simp add: sup_state_Cons1) |
|
433 |
next |
|
434 |
case Throw |
|
435 |
with G |
|
436 |
show ?thesis |
|
437 |
by simp |
|
438 |
qed |
|
439 |
qed |
|
440 |
||
441 |
lemmas [iff del] = not_Err_eq |
|
442 |
||
443 |
end |