11229
|
1 |
(* Title: HOL/MicroJava/BV/Typing_Framework_err.thy
|
|
2 |
ID: $Id$
|
|
3 |
Author: Gerwin Klein
|
|
4 |
Copyright 2000 TUM
|
|
5 |
|
|
6 |
*)
|
|
7 |
|
13224
|
8 |
header {* \isaheader{Lifting the Typing Framework to err, app, and eff} *}
|
11229
|
9 |
|
27680
|
10 |
theory Typing_Framework_err
|
|
11 |
imports Typing_Framework SemilatAlg
|
|
12 |
begin
|
11229
|
13 |
|
|
14 |
constdefs
|
|
15 |
|
13066
|
16 |
wt_err_step :: "'s ord \<Rightarrow> 's err step_type \<Rightarrow> 's err list \<Rightarrow> bool"
|
|
17 |
"wt_err_step r step ts \<equiv> wt_step (Err.le r) Err step ts"
|
12516
|
18 |
|
13066
|
19 |
wt_app_eff :: "'s ord \<Rightarrow> (nat \<Rightarrow> 's \<Rightarrow> bool) \<Rightarrow> 's step_type \<Rightarrow> 's list \<Rightarrow> bool"
|
|
20 |
"wt_app_eff r app step ts \<equiv>
|
12516
|
21 |
\<forall>p < size ts. app p (ts!p) \<and> (\<forall>(q,t) \<in> set (step p (ts!p)). t <=_r ts!q)"
|
|
22 |
|
|
23 |
map_snd :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('a \<times> 'c) list"
|
13066
|
24 |
"map_snd f \<equiv> map (\<lambda>(x,y). (x, f y))"
|
|
25 |
|
|
26 |
error :: "nat \<Rightarrow> (nat \<times> 'a err) list"
|
15425
|
27 |
"error n \<equiv> map (\<lambda>x. (x,Err)) [0..<n]"
|
11229
|
28 |
|
13066
|
29 |
err_step :: "nat \<Rightarrow> (nat \<Rightarrow> 's \<Rightarrow> bool) \<Rightarrow> 's step_type \<Rightarrow> 's err step_type"
|
|
30 |
"err_step n app step p t \<equiv>
|
|
31 |
case t of
|
|
32 |
Err \<Rightarrow> error n
|
|
33 |
| OK t' \<Rightarrow> if app p t' then map_snd OK (step p t') else error n"
|
11229
|
34 |
|
13224
|
35 |
app_mono :: "'s ord \<Rightarrow> (nat \<Rightarrow> 's \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> 's set \<Rightarrow> bool"
|
|
36 |
"app_mono r app n A \<equiv>
|
|
37 |
\<forall>s p t. s \<in> A \<and> p < n \<and> s <=_r t \<longrightarrow> app p t \<longrightarrow> app p s"
|
|
38 |
|
|
39 |
|
13066
|
40 |
lemmas err_step_defs = err_step_def map_snd_def error_def
|
12516
|
41 |
|
13224
|
42 |
|
13067
|
43 |
lemma bounded_err_stepD:
|
|
44 |
"bounded (err_step n app step) n \<Longrightarrow>
|
|
45 |
p < n \<Longrightarrow> app p a \<Longrightarrow> (q,b) \<in> set (step p a) \<Longrightarrow>
|
|
46 |
q < n"
|
|
47 |
apply (simp add: bounded_def err_step_def)
|
|
48 |
apply (erule allE, erule impE, assumption)
|
|
49 |
apply (erule_tac x = "OK a" in allE, drule bspec)
|
|
50 |
apply (simp add: map_snd_def)
|
|
51 |
apply fast
|
|
52 |
apply simp
|
|
53 |
done
|
|
54 |
|
|
55 |
|
|
56 |
lemma in_map_sndD: "(a,b) \<in> set (map_snd f xs) \<Longrightarrow> \<exists>b'. (a,b') \<in> set xs"
|
|
57 |
apply (induct xs)
|
|
58 |
apply (auto simp add: map_snd_def)
|
|
59 |
done
|
11229
|
60 |
|
12516
|
61 |
|
13067
|
62 |
lemma bounded_err_stepI:
|
|
63 |
"\<forall>p. p < n \<longrightarrow> (\<forall>s. ap p s \<longrightarrow> (\<forall>(q,s') \<in> set (step p s). q < n))
|
|
64 |
\<Longrightarrow> bounded (err_step n ap step) n"
|
|
65 |
apply (unfold bounded_def)
|
|
66 |
apply clarify
|
|
67 |
apply (simp add: err_step_def split: err.splits)
|
|
68 |
apply (simp add: error_def)
|
|
69 |
apply blast
|
|
70 |
apply (simp split: split_if_asm)
|
|
71 |
apply (blast dest: in_map_sndD)
|
|
72 |
apply (simp add: error_def)
|
|
73 |
apply blast
|
|
74 |
done
|
|
75 |
|
|
76 |
|
13224
|
77 |
lemma bounded_lift:
|
|
78 |
"bounded step n \<Longrightarrow> bounded (err_step n app step) n"
|
|
79 |
apply (unfold bounded_def err_step_def error_def)
|
|
80 |
apply clarify
|
|
81 |
apply (erule allE, erule impE, assumption)
|
|
82 |
apply (case_tac s)
|
|
83 |
apply (auto simp add: map_snd_def split: split_if_asm)
|
|
84 |
done
|
|
85 |
|
|
86 |
|
|
87 |
lemma le_list_map_OK [simp]:
|
|
88 |
"\<And>b. map OK a <=[Err.le r] map OK b = (a <=[r] b)"
|
|
89 |
apply (induct a)
|
|
90 |
apply simp
|
|
91 |
apply simp
|
|
92 |
apply (case_tac b)
|
|
93 |
apply simp
|
|
94 |
apply simp
|
|
95 |
done
|
|
96 |
|
|
97 |
|
|
98 |
lemma map_snd_lessI:
|
|
99 |
"x <=|r| y \<Longrightarrow> map_snd OK x <=|Err.le r| map_snd OK y"
|
|
100 |
apply (induct x)
|
|
101 |
apply (unfold lesubstep_type_def map_snd_def)
|
|
102 |
apply auto
|
|
103 |
done
|
|
104 |
|
|
105 |
|
|
106 |
lemma mono_lift:
|
|
107 |
"order r \<Longrightarrow> app_mono r app n A \<Longrightarrow> bounded (err_step n app step) n \<Longrightarrow>
|
|
108 |
\<forall>s p t. s \<in> A \<and> p < n \<and> s <=_r t \<longrightarrow> app p t \<longrightarrow> step p s <=|r| step p t \<Longrightarrow>
|
|
109 |
mono (Err.le r) (err_step n app step) n (err A)"
|
|
110 |
apply (unfold app_mono_def mono_def err_step_def)
|
|
111 |
apply clarify
|
|
112 |
apply (case_tac s)
|
|
113 |
apply simp
|
|
114 |
apply simp
|
|
115 |
apply (case_tac t)
|
|
116 |
apply simp
|
|
117 |
apply clarify
|
|
118 |
apply (simp add: lesubstep_type_def error_def)
|
|
119 |
apply clarify
|
|
120 |
apply (drule in_map_sndD)
|
|
121 |
apply clarify
|
|
122 |
apply (drule bounded_err_stepD, assumption+)
|
|
123 |
apply (rule exI [of _ Err])
|
|
124 |
apply simp
|
|
125 |
apply simp
|
|
126 |
apply (erule allE, erule allE, erule allE, erule impE)
|
|
127 |
apply (rule conjI, assumption)
|
|
128 |
apply (rule conjI, assumption)
|
|
129 |
apply assumption
|
|
130 |
apply (rule conjI)
|
|
131 |
apply clarify
|
|
132 |
apply (erule allE, erule allE, erule allE, erule impE)
|
|
133 |
apply (rule conjI, assumption)
|
|
134 |
apply (rule conjI, assumption)
|
|
135 |
apply assumption
|
|
136 |
apply (erule impE, assumption)
|
|
137 |
apply (rule map_snd_lessI, assumption)
|
|
138 |
apply clarify
|
|
139 |
apply (simp add: lesubstep_type_def error_def)
|
|
140 |
apply clarify
|
|
141 |
apply (drule in_map_sndD)
|
|
142 |
apply clarify
|
|
143 |
apply (drule bounded_err_stepD, assumption+)
|
|
144 |
apply (rule exI [of _ Err])
|
|
145 |
apply simp
|
|
146 |
done
|
|
147 |
|
|
148 |
lemma in_errorD:
|
|
149 |
"(x,y) \<in> set (error n) \<Longrightarrow> y = Err"
|
|
150 |
by (auto simp add: error_def)
|
|
151 |
|
|
152 |
lemma pres_type_lift:
|
|
153 |
"\<forall>s\<in>A. \<forall>p. p < n \<longrightarrow> app p s \<longrightarrow> (\<forall>(q, s')\<in>set (step p s). s' \<in> A)
|
|
154 |
\<Longrightarrow> pres_type (err_step n app step) n (err A)"
|
|
155 |
apply (unfold pres_type_def err_step_def)
|
|
156 |
apply clarify
|
|
157 |
apply (case_tac b)
|
|
158 |
apply simp
|
|
159 |
apply (case_tac s)
|
|
160 |
apply simp
|
|
161 |
apply (drule in_errorD)
|
|
162 |
apply simp
|
|
163 |
apply (simp add: map_snd_def split: split_if_asm)
|
|
164 |
apply fast
|
|
165 |
apply (drule in_errorD)
|
|
166 |
apply simp
|
|
167 |
done
|
|
168 |
|
|
169 |
|
|
170 |
|
13067
|
171 |
text {*
|
|
172 |
There used to be a condition here that each instruction must have a
|
|
173 |
successor. This is not needed any more, because the definition of
|
|
174 |
@{term error} trivially ensures that there is a successor for
|
|
175 |
the critical case where @{term app} does not hold.
|
|
176 |
*}
|
13066
|
177 |
lemma wt_err_imp_wt_app_eff:
|
|
178 |
assumes wt: "wt_err_step r (err_step (size ts) app step) ts"
|
13067
|
179 |
assumes b: "bounded (err_step (size ts) app step) (size ts)"
|
13066
|
180 |
shows "wt_app_eff r app step (map ok_val ts)"
|
|
181 |
proof (unfold wt_app_eff_def, intro strip, rule conjI)
|
|
182 |
fix p assume "p < size (map ok_val ts)"
|
|
183 |
hence lp: "p < size ts" by simp
|
13067
|
184 |
hence ts: "0 < size ts" by (cases p) auto
|
|
185 |
hence err: "(0,Err) \<in> set (error (size ts))" by (simp add: error_def)
|
11229
|
186 |
|
|
187 |
from wt lp
|
13066
|
188 |
have [intro?]: "\<And>p. p < size ts \<Longrightarrow> ts ! p \<noteq> Err"
|
|
189 |
by (unfold wt_err_step_def wt_step_def) simp
|
11229
|
190 |
|
|
191 |
show app: "app p (map ok_val ts ! p)"
|
13067
|
192 |
proof (rule ccontr)
|
|
193 |
from wt lp obtain s where
|
12516
|
194 |
OKp: "ts ! p = OK s" and
|
13066
|
195 |
less: "\<forall>(q,t) \<in> set (err_step (size ts) app step p (ts!p)). t <=_(Err.le r) ts!q"
|
|
196 |
by (unfold wt_err_step_def wt_step_def stable_def)
|
12516
|
197 |
(auto iff: not_Err_eq)
|
13067
|
198 |
assume "\<not> app p (map ok_val ts ! p)"
|
|
199 |
with OKp lp have "\<not> app p s" by simp
|
|
200 |
with OKp have "err_step (size ts) app step p (ts!p) = error (size ts)"
|
|
201 |
by (simp add: err_step_def)
|
|
202 |
with err ts obtain q where
|
|
203 |
"(q,Err) \<in> set (err_step (size ts) app step p (ts!p))" and
|
|
204 |
q: "q < size ts" by auto
|
|
205 |
with less have "ts!q = Err" by auto
|
|
206 |
moreover from q have "ts!q \<noteq> Err" ..
|
|
207 |
ultimately show False by blast
|
12516
|
208 |
qed
|
|
209 |
|
|
210 |
show "\<forall>(q,t)\<in>set(step p (map ok_val ts ! p)). t <=_r map ok_val ts ! q"
|
|
211 |
proof clarify
|
|
212 |
fix q t assume q: "(q,t) \<in> set (step p (map ok_val ts ! p))"
|
11229
|
213 |
|
|
214 |
from wt lp q
|
|
215 |
obtain s where
|
|
216 |
OKp: "ts ! p = OK s" and
|
13066
|
217 |
less: "\<forall>(q,t) \<in> set (err_step (size ts) app step p (ts!p)). t <=_(Err.le r) ts!q"
|
|
218 |
by (unfold wt_err_step_def wt_step_def stable_def)
|
11229
|
219 |
(auto iff: not_Err_eq)
|
|
220 |
|
13067
|
221 |
from b lp app q have lq: "q < size ts" by (rule bounded_err_stepD)
|
12516
|
222 |
hence "ts!q \<noteq> Err" ..
|
|
223 |
then obtain s' where OKq: "ts ! q = OK s'" by (auto iff: not_Err_eq)
|
11229
|
224 |
|
12516
|
225 |
from lp lq OKp OKq app less q
|
|
226 |
show "t <=_r map ok_val ts ! q"
|
|
227 |
by (auto simp add: err_step_def map_snd_def)
|
11229
|
228 |
qed
|
|
229 |
qed
|
|
230 |
|
|
231 |
|
13066
|
232 |
lemma wt_app_eff_imp_wt_err:
|
|
233 |
assumes app_eff: "wt_app_eff r app step ts"
|
|
234 |
assumes bounded: "bounded (err_step (size ts) app step) (size ts)"
|
|
235 |
shows "wt_err_step r (err_step (size ts) app step) (map OK ts)"
|
|
236 |
proof (unfold wt_err_step_def wt_step_def, intro strip, rule conjI)
|
|
237 |
fix p assume "p < size (map OK ts)"
|
|
238 |
hence p: "p < size ts" by simp
|
11229
|
239 |
thus "map OK ts ! p \<noteq> Err" by simp
|
12516
|
240 |
{ fix q t
|
13066
|
241 |
assume q: "(q,t) \<in> set (err_step (size ts) app step p (map OK ts ! p))"
|
|
242 |
with p app_eff obtain
|
12516
|
243 |
"app p (ts ! p)" "\<forall>(q,t) \<in> set (step p (ts!p)). t <=_r ts!q"
|
13066
|
244 |
by (unfold wt_app_eff_def) blast
|
11229
|
245 |
moreover
|
13066
|
246 |
from q p bounded have "q < size ts"
|
|
247 |
by - (rule boundedD)
|
11229
|
248 |
hence "map OK ts ! q = OK (ts!q)" by simp
|
|
249 |
moreover
|
|
250 |
have "p < size ts" by (rule p)
|
12516
|
251 |
moreover note q
|
11229
|
252 |
ultimately
|
12516
|
253 |
have "t <=_(Err.le r) map OK ts ! q"
|
|
254 |
by (auto simp add: err_step_def map_snd_def)
|
11229
|
255 |
}
|
13066
|
256 |
thus "stable (Err.le r) (err_step (size ts) app step) (map OK ts) p"
|
11229
|
257 |
by (unfold stable_def) blast
|
|
258 |
qed
|
|
259 |
|
|
260 |
end
|
13224
|
261 |
|