author | haftmann |
Tue, 03 Feb 2009 19:37:00 +0100 | |
changeset 29793 | 86cac1fab613 |
parent 29399 | ebcd69a00872 |
child 30738 | 0842e906300c |
permissions | -rw-r--r-- |
26170 | 1 |
(* Title: HOL/Library/Heap.thy |
2 |
ID: $Id$ |
|
3 |
Author: John Matthews, Galois Connections; Alexander Krauss, TU Muenchen |
|
4 |
*) |
|
5 |
||
6 |
header {* A polymorphic heap based on cantor encodings *} |
|
7 |
||
8 |
theory Heap |
|
28952
15a4b2cf8c34
made repository layout more coherent with logical distribution structure; stripped some $Id$s
haftmann
parents:
28524
diff
changeset
|
9 |
imports Plain "~~/src/HOL/List" Countable Typerep |
26170 | 10 |
begin |
11 |
||
12 |
subsection {* Representable types *} |
|
13 |
||
14 |
text {* The type class of representable types *} |
|
15 |
||
28335 | 16 |
class heap = typerep + countable |
26170 | 17 |
|
18 |
text {* Instances for common HOL types *} |
|
19 |
||
20 |
instance nat :: heap .. |
|
21 |
||
22 |
instance "*" :: (heap, heap) heap .. |
|
23 |
||
24 |
instance "+" :: (heap, heap) heap .. |
|
25 |
||
26 |
instance list :: (heap) heap .. |
|
27 |
||
28 |
instance option :: (heap) heap .. |
|
29 |
||
30 |
instance int :: heap .. |
|
31 |
||
32 |
instance message_string :: countable |
|
33 |
by (rule countable_classI [of "message_string_case to_nat"]) |
|
34 |
(auto split: message_string.splits) |
|
35 |
||
36 |
instance message_string :: heap .. |
|
37 |
||
38 |
text {* Reflected types themselves are heap-representable *} |
|
39 |
||
28335 | 40 |
instantiation typerep :: countable |
26170 | 41 |
begin |
42 |
||
28335 | 43 |
fun to_nat_typerep :: "typerep \<Rightarrow> nat" where |
28952
15a4b2cf8c34
made repository layout more coherent with logical distribution structure; stripped some $Id$s
haftmann
parents:
28524
diff
changeset
|
44 |
"to_nat_typerep (Typerep.Typerep c ts) = to_nat (to_nat c, to_nat (map to_nat_typerep ts))" |
26170 | 45 |
|
26932 | 46 |
instance |
47 |
proof (rule countable_classI) |
|
28335 | 48 |
fix t t' :: typerep and ts |
49 |
have "(\<forall>t'. to_nat_typerep t = to_nat_typerep t' \<longrightarrow> t = t') |
|
50 |
\<and> (\<forall>ts'. map to_nat_typerep ts = map to_nat_typerep ts' \<longrightarrow> ts = ts')" |
|
51 |
proof (induct rule: typerep.induct) |
|
52 |
case (Typerep c ts) show ?case |
|
26170 | 53 |
proof (rule allI, rule impI) |
54 |
fix t' |
|
28952
15a4b2cf8c34
made repository layout more coherent with logical distribution structure; stripped some $Id$s
haftmann
parents:
28524
diff
changeset
|
55 |
assume hyp: "to_nat_typerep (Typerep.Typerep c ts) = to_nat_typerep t'" |
15a4b2cf8c34
made repository layout more coherent with logical distribution structure; stripped some $Id$s
haftmann
parents:
28524
diff
changeset
|
56 |
then obtain c' ts' where t': "t' = (Typerep.Typerep c' ts')" |
26170 | 57 |
by (cases t') auto |
28335 | 58 |
with Typerep hyp have "c = c'" and "ts = ts'" by simp_all |
28952
15a4b2cf8c34
made repository layout more coherent with logical distribution structure; stripped some $Id$s
haftmann
parents:
28524
diff
changeset
|
59 |
with t' show "Typerep.Typerep c ts = t'" by simp |
26170 | 60 |
qed |
61 |
next |
|
28335 | 62 |
case Nil_typerep then show ?case by simp |
26170 | 63 |
next |
28335 | 64 |
case (Cons_typerep t ts) then show ?case by auto |
26170 | 65 |
qed |
28335 | 66 |
then have "to_nat_typerep t = to_nat_typerep t' \<Longrightarrow> t = t'" by auto |
67 |
moreover assume "to_nat_typerep t = to_nat_typerep t'" |
|
26170 | 68 |
ultimately show "t = t'" by simp |
69 |
qed |
|
70 |
||
71 |
end |
|
72 |
||
28335 | 73 |
instance typerep :: heap .. |
26170 | 74 |
|
75 |
||
76 |
subsection {* A polymorphic heap with dynamic arrays and references *} |
|
77 |
||
78 |
types addr = nat -- "untyped heap references" |
|
79 |
||
80 |
datatype 'a array = Array addr |
|
81 |
datatype 'a ref = Ref addr -- "note the phantom type 'a " |
|
82 |
||
83 |
primrec addr_of_array :: "'a array \<Rightarrow> addr" where |
|
84 |
"addr_of_array (Array x) = x" |
|
85 |
||
86 |
primrec addr_of_ref :: "'a ref \<Rightarrow> addr" where |
|
87 |
"addr_of_ref (Ref x) = x" |
|
88 |
||
89 |
lemma addr_of_array_inj [simp]: |
|
90 |
"addr_of_array a = addr_of_array a' \<longleftrightarrow> a = a'" |
|
91 |
by (cases a, cases a') simp_all |
|
92 |
||
93 |
lemma addr_of_ref_inj [simp]: |
|
94 |
"addr_of_ref r = addr_of_ref r' \<longleftrightarrow> r = r'" |
|
95 |
by (cases r, cases r') simp_all |
|
96 |
||
97 |
instance array :: (type) countable |
|
98 |
by (rule countable_classI [of addr_of_array]) simp |
|
99 |
||
100 |
instance ref :: (type) countable |
|
101 |
by (rule countable_classI [of addr_of_ref]) simp |
|
102 |
||
103 |
setup {* |
|
104 |
Sign.add_const_constraint (@{const_name Array}, SOME @{typ "nat \<Rightarrow> 'a\<Colon>heap array"}) |
|
105 |
#> Sign.add_const_constraint (@{const_name Ref}, SOME @{typ "nat \<Rightarrow> 'a\<Colon>heap ref"}) |
|
106 |
#> Sign.add_const_constraint (@{const_name addr_of_array}, SOME @{typ "'a\<Colon>heap array \<Rightarrow> nat"}) |
|
107 |
#> Sign.add_const_constraint (@{const_name addr_of_ref}, SOME @{typ "'a\<Colon>heap ref \<Rightarrow> nat"}) |
|
108 |
*} |
|
109 |
||
110 |
types heap_rep = nat -- "representable values" |
|
111 |
||
112 |
record heap = |
|
28335 | 113 |
arrays :: "typerep \<Rightarrow> addr \<Rightarrow> heap_rep list" |
114 |
refs :: "typerep \<Rightarrow> addr \<Rightarrow> heap_rep" |
|
26170 | 115 |
lim :: addr |
116 |
||
117 |
definition empty :: heap where |
|
28524 | 118 |
"empty = \<lparr>arrays = (\<lambda>_. undefined), refs = (\<lambda>_. undefined), lim = 0\<rparr>" -- "why undefined?" |
26170 | 119 |
|
120 |
||
121 |
subsection {* Imperative references and arrays *} |
|
122 |
||
123 |
text {* |
|
124 |
References and arrays are developed in parallel, |
|
26586 | 125 |
but keeping them separate makes some later proofs simpler. |
26170 | 126 |
*} |
127 |
||
128 |
subsubsection {* Primitive operations *} |
|
129 |
||
130 |
definition |
|
131 |
new_ref :: "heap \<Rightarrow> ('a\<Colon>heap) ref \<times> heap" where |
|
132 |
"new_ref h = (let l = lim h in (Ref l, h\<lparr>lim := l + 1\<rparr>))" |
|
133 |
||
134 |
definition |
|
135 |
new_array :: "heap \<Rightarrow> ('a\<Colon>heap) array \<times> heap" where |
|
136 |
"new_array h = (let l = lim h in (Array l, h\<lparr>lim := l + 1\<rparr>))" |
|
137 |
||
138 |
definition |
|
139 |
ref_present :: "'a\<Colon>heap ref \<Rightarrow> heap \<Rightarrow> bool" where |
|
140 |
"ref_present r h \<longleftrightarrow> addr_of_ref r < lim h" |
|
141 |
||
142 |
definition |
|
143 |
array_present :: "'a\<Colon>heap array \<Rightarrow> heap \<Rightarrow> bool" where |
|
144 |
"array_present a h \<longleftrightarrow> addr_of_array a < lim h" |
|
145 |
||
146 |
definition |
|
147 |
get_ref :: "'a\<Colon>heap ref \<Rightarrow> heap \<Rightarrow> 'a" where |
|
28335 | 148 |
"get_ref r h = from_nat (refs h (TYPEREP('a)) (addr_of_ref r))" |
26170 | 149 |
|
150 |
definition |
|
151 |
get_array :: "'a\<Colon>heap array \<Rightarrow> heap \<Rightarrow> 'a list" where |
|
28335 | 152 |
"get_array a h = map from_nat (arrays h (TYPEREP('a)) (addr_of_array a))" |
26170 | 153 |
|
154 |
definition |
|
155 |
set_ref :: "'a\<Colon>heap ref \<Rightarrow> 'a \<Rightarrow> heap \<Rightarrow> heap" where |
|
156 |
"set_ref r x = |
|
28335 | 157 |
refs_update (\<lambda>h. h(TYPEREP('a) := ((h (TYPEREP('a))) (addr_of_ref r:=to_nat x))))" |
26170 | 158 |
|
159 |
definition |
|
160 |
set_array :: "'a\<Colon>heap array \<Rightarrow> 'a list \<Rightarrow> heap \<Rightarrow> heap" where |
|
161 |
"set_array a x = |
|
28335 | 162 |
arrays_update (\<lambda>h. h(TYPEREP('a) := ((h(TYPEREP('a))) (addr_of_array a:=map to_nat x))))" |
26170 | 163 |
|
164 |
subsubsection {* Interface operations *} |
|
165 |
||
166 |
definition |
|
167 |
ref :: "'a \<Rightarrow> heap \<Rightarrow> 'a\<Colon>heap ref \<times> heap" where |
|
168 |
"ref x h = (let (r, h') = new_ref h; |
|
169 |
h'' = set_ref r x h' |
|
170 |
in (r, h''))" |
|
171 |
||
172 |
definition |
|
173 |
array :: "nat \<Rightarrow> 'a \<Rightarrow> heap \<Rightarrow> 'a\<Colon>heap array \<times> heap" where |
|
174 |
"array n x h = (let (r, h') = new_array h; |
|
175 |
h'' = set_array r (replicate n x) h' |
|
176 |
in (r, h''))" |
|
177 |
||
178 |
definition |
|
179 |
array_of_list :: "'a list \<Rightarrow> heap \<Rightarrow> 'a\<Colon>heap array \<times> heap" where |
|
180 |
"array_of_list xs h = (let (r, h') = new_array h; |
|
181 |
h'' = set_array r xs h' |
|
182 |
in (r, h''))" |
|
183 |
||
184 |
definition |
|
185 |
upd :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> heap \<Rightarrow> heap" where |
|
186 |
"upd a i x h = set_array a ((get_array a h)[i:=x]) h" |
|
187 |
||
188 |
definition |
|
189 |
length :: "'a\<Colon>heap array \<Rightarrow> heap \<Rightarrow> nat" where |
|
190 |
"length a h = size (get_array a h)" |
|
191 |
||
192 |
definition |
|
193 |
array_ran :: "('a\<Colon>heap) option array \<Rightarrow> heap \<Rightarrow> 'a set" where |
|
194 |
"array_ran a h = {e. Some e \<in> set (get_array a h)}" |
|
195 |
-- {*FIXME*} |
|
196 |
||
197 |
||
198 |
subsubsection {* Reference equality *} |
|
199 |
||
200 |
text {* |
|
201 |
The following relations are useful for comparing arrays and references. |
|
202 |
*} |
|
203 |
||
204 |
definition |
|
205 |
noteq_refs :: "('a\<Colon>heap) ref \<Rightarrow> ('b\<Colon>heap) ref \<Rightarrow> bool" (infix "=!=" 70) |
|
206 |
where |
|
28335 | 207 |
"r =!= s \<longleftrightarrow> TYPEREP('a) \<noteq> TYPEREP('b) \<or> addr_of_ref r \<noteq> addr_of_ref s" |
26170 | 208 |
|
209 |
definition |
|
210 |
noteq_arrs :: "('a\<Colon>heap) array \<Rightarrow> ('b\<Colon>heap) array \<Rightarrow> bool" (infix "=!!=" 70) |
|
211 |
where |
|
28335 | 212 |
"r =!!= s \<longleftrightarrow> TYPEREP('a) \<noteq> TYPEREP('b) \<or> addr_of_array r \<noteq> addr_of_array s" |
26170 | 213 |
|
214 |
lemma noteq_refs_sym: "r =!= s \<Longrightarrow> s =!= r" |
|
215 |
and noteq_arrs_sym: "a =!!= b \<Longrightarrow> b =!!= a" |
|
216 |
and unequal_refs [simp]: "r \<noteq> r' \<longleftrightarrow> r =!= r'" -- "same types!" |
|
217 |
and unequal_arrs [simp]: "a \<noteq> a' \<longleftrightarrow> a =!!= a'" |
|
218 |
unfolding noteq_refs_def noteq_arrs_def by auto |
|
219 |
||
220 |
lemma present_new_ref: "ref_present r h \<Longrightarrow> r =!= fst (ref v h)" |
|
221 |
by (simp add: ref_present_def new_ref_def ref_def Let_def noteq_refs_def) |
|
222 |
||
223 |
lemma present_new_arr: "array_present a h \<Longrightarrow> a =!!= fst (array v x h)" |
|
224 |
by (simp add: array_present_def noteq_arrs_def new_array_def array_def Let_def) |
|
225 |
||
226 |
||
227 |
subsubsection {* Properties of heap containers *} |
|
228 |
||
229 |
text {* Properties of imperative arrays *} |
|
230 |
||
231 |
text {* FIXME: Does there exist a "canonical" array axiomatisation in |
|
232 |
the literature? *} |
|
233 |
||
234 |
lemma array_get_set_eq [simp]: "get_array r (set_array r x h) = x" |
|
235 |
by (simp add: get_array_def set_array_def) |
|
236 |
||
237 |
lemma array_get_set_neq [simp]: "r =!!= s \<Longrightarrow> get_array r (set_array s x h) = get_array r h" |
|
238 |
by (simp add: noteq_arrs_def get_array_def set_array_def) |
|
239 |
||
240 |
lemma set_array_same [simp]: |
|
241 |
"set_array r x (set_array r y h) = set_array r x h" |
|
242 |
by (simp add: set_array_def) |
|
243 |
||
244 |
lemma array_set_set_swap: |
|
245 |
"r =!!= r' \<Longrightarrow> set_array r x (set_array r' x' h) = set_array r' x' (set_array r x h)" |
|
246 |
by (simp add: Let_def expand_fun_eq noteq_arrs_def set_array_def) |
|
247 |
||
248 |
lemma array_ref_set_set_swap: |
|
249 |
"set_array r x (set_ref r' x' h) = set_ref r' x' (set_array r x h)" |
|
250 |
by (simp add: Let_def expand_fun_eq set_array_def set_ref_def) |
|
251 |
||
252 |
lemma get_array_upd_eq [simp]: |
|
253 |
"get_array a (upd a i v h) = (get_array a h) [i := v]" |
|
254 |
by (simp add: upd_def) |
|
255 |
||
256 |
lemma nth_upd_array_neq_array [simp]: |
|
257 |
"a =!!= b \<Longrightarrow> get_array a (upd b j v h) ! i = get_array a h ! i" |
|
258 |
by (simp add: upd_def noteq_arrs_def) |
|
259 |
||
260 |
lemma get_arry_array_upd_elem_neqIndex [simp]: |
|
261 |
"i \<noteq> j \<Longrightarrow> get_array a (upd a j v h) ! i = get_array a h ! i" |
|
262 |
by simp |
|
263 |
||
264 |
lemma length_upd_eq [simp]: |
|
265 |
"length a (upd a i v h) = length a h" |
|
266 |
by (simp add: length_def upd_def) |
|
267 |
||
268 |
lemma length_upd_neq [simp]: |
|
269 |
"length a (upd b i v h) = length a h" |
|
270 |
by (simp add: upd_def length_def set_array_def get_array_def) |
|
271 |
||
272 |
lemma upd_swap_neqArray: |
|
273 |
"a =!!= a' \<Longrightarrow> |
|
274 |
upd a i v (upd a' i' v' h) |
|
275 |
= upd a' i' v' (upd a i v h)" |
|
276 |
apply (unfold upd_def) |
|
277 |
apply simp |
|
278 |
apply (subst array_set_set_swap, assumption) |
|
279 |
apply (subst array_get_set_neq) |
|
280 |
apply (erule noteq_arrs_sym) |
|
281 |
apply (simp) |
|
282 |
done |
|
283 |
||
284 |
lemma upd_swap_neqIndex: |
|
285 |
"\<lbrakk> i \<noteq> i' \<rbrakk> \<Longrightarrow> upd a i v (upd a i' v' h) = upd a i' v' (upd a i v h)" |
|
286 |
by (auto simp add: upd_def array_set_set_swap list_update_swap) |
|
287 |
||
288 |
lemma get_array_init_array_list: |
|
289 |
"get_array (fst (array_of_list ls h)) (snd (array_of_list ls' h)) = ls'" |
|
290 |
by (simp add: Let_def split_def array_of_list_def) |
|
291 |
||
292 |
lemma set_array: |
|
293 |
"set_array (fst (array_of_list ls h)) |
|
294 |
new_ls (snd (array_of_list ls h)) |
|
295 |
= snd (array_of_list new_ls h)" |
|
296 |
by (simp add: Let_def split_def array_of_list_def) |
|
297 |
||
298 |
lemma array_present_upd [simp]: |
|
299 |
"array_present a (upd b i v h) = array_present a h" |
|
300 |
by (simp add: upd_def array_present_def set_array_def get_array_def) |
|
301 |
||
302 |
lemma array_of_list_replicate: |
|
303 |
"array_of_list (replicate n x) = array n x" |
|
304 |
by (simp add: expand_fun_eq array_of_list_def array_def) |
|
305 |
||
306 |
||
307 |
text {* Properties of imperative references *} |
|
308 |
||
309 |
lemma next_ref_fresh [simp]: |
|
310 |
assumes "(r, h') = new_ref h" |
|
311 |
shows "\<not> ref_present r h" |
|
312 |
using assms by (cases h) (auto simp add: new_ref_def ref_present_def Let_def) |
|
313 |
||
314 |
lemma next_ref_present [simp]: |
|
315 |
assumes "(r, h') = new_ref h" |
|
316 |
shows "ref_present r h'" |
|
317 |
using assms by (cases h) (auto simp add: new_ref_def ref_present_def Let_def) |
|
318 |
||
319 |
lemma ref_get_set_eq [simp]: "get_ref r (set_ref r x h) = x" |
|
320 |
by (simp add: get_ref_def set_ref_def) |
|
321 |
||
322 |
lemma ref_get_set_neq [simp]: "r =!= s \<Longrightarrow> get_ref r (set_ref s x h) = get_ref r h" |
|
323 |
by (simp add: noteq_refs_def get_ref_def set_ref_def) |
|
324 |
||
325 |
(* FIXME: We need some infrastructure to infer that locally generated |
|
326 |
new refs (by new_ref(_no_init), new_array(')) are distinct |
|
327 |
from all existing refs. |
|
328 |
*) |
|
329 |
||
330 |
lemma ref_set_get: "set_ref r (get_ref r h) h = h" |
|
331 |
apply (simp add: set_ref_def get_ref_def) |
|
332 |
oops |
|
333 |
||
334 |
lemma set_ref_same[simp]: |
|
335 |
"set_ref r x (set_ref r y h) = set_ref r x h" |
|
336 |
by (simp add: set_ref_def) |
|
337 |
||
338 |
lemma ref_set_set_swap: |
|
339 |
"r =!= r' \<Longrightarrow> set_ref r x (set_ref r' x' h) = set_ref r' x' (set_ref r x h)" |
|
340 |
by (simp add: Let_def expand_fun_eq noteq_refs_def set_ref_def) |
|
341 |
||
342 |
lemma ref_new_set: "fst (ref v (set_ref r v' h)) = fst (ref v h)" |
|
343 |
by (simp add: ref_def new_ref_def set_ref_def Let_def) |
|
344 |
||
345 |
lemma ref_get_new [simp]: |
|
346 |
"get_ref (fst (ref v h)) (snd (ref v' h)) = v'" |
|
347 |
by (simp add: ref_def Let_def split_def) |
|
348 |
||
349 |
lemma ref_set_new [simp]: |
|
350 |
"set_ref (fst (ref v h)) new_v (snd (ref v h)) = snd (ref new_v h)" |
|
351 |
by (simp add: ref_def Let_def split_def) |
|
352 |
||
353 |
lemma ref_get_new_neq: "r =!= (fst (ref v h)) \<Longrightarrow> |
|
354 |
get_ref r (snd (ref v h)) = get_ref r h" |
|
355 |
by (simp add: get_ref_def set_ref_def ref_def Let_def new_ref_def noteq_refs_def) |
|
356 |
||
357 |
lemma lim_set_ref [simp]: |
|
358 |
"lim (set_ref r v h) = lim h" |
|
359 |
by (simp add: set_ref_def) |
|
360 |
||
361 |
lemma ref_present_new_ref [simp]: |
|
362 |
"ref_present r h \<Longrightarrow> ref_present r (snd (ref v h))" |
|
363 |
by (simp add: new_ref_def ref_present_def ref_def Let_def) |
|
364 |
||
365 |
lemma ref_present_set_ref [simp]: |
|
366 |
"ref_present r (set_ref r' v h) = ref_present r h" |
|
367 |
by (simp add: set_ref_def ref_present_def) |
|
368 |
||
369 |
lemma array_ranI: "\<lbrakk> Some b = get_array a h ! i; i < Heap.length a h \<rbrakk> \<Longrightarrow> b \<in> array_ran a h" |
|
370 |
unfolding array_ran_def Heap.length_def by simp |
|
371 |
||
372 |
lemma array_ran_upd_array_Some: |
|
373 |
assumes "cl \<in> array_ran a (Heap.upd a i (Some b) h)" |
|
374 |
shows "cl \<in> array_ran a h \<or> cl = b" |
|
375 |
proof - |
|
376 |
have "set (get_array a h[i := Some b]) \<subseteq> insert (Some b) (set (get_array a h))" by (rule set_update_subset_insert) |
|
377 |
with assms show ?thesis |
|
378 |
unfolding array_ran_def Heap.upd_def by fastsimp |
|
379 |
qed |
|
380 |
||
381 |
lemma array_ran_upd_array_None: |
|
382 |
assumes "cl \<in> array_ran a (Heap.upd a i None h)" |
|
383 |
shows "cl \<in> array_ran a h" |
|
384 |
proof - |
|
385 |
have "set (get_array a h[i := None]) \<subseteq> insert None (set (get_array a h))" by (rule set_update_subset_insert) |
|
386 |
with assms show ?thesis |
|
387 |
unfolding array_ran_def Heap.upd_def by auto |
|
388 |
qed |
|
389 |
||
390 |
||
391 |
text {* Non-interaction between imperative array and imperative references *} |
|
392 |
||
393 |
lemma get_array_set_ref [simp]: "get_array a (set_ref r v h) = get_array a h" |
|
394 |
by (simp add: get_array_def set_ref_def) |
|
395 |
||
396 |
lemma nth_set_ref [simp]: "get_array a (set_ref r v h) ! i = get_array a h ! i" |
|
397 |
by simp |
|
398 |
||
399 |
lemma get_ref_upd [simp]: "get_ref r (upd a i v h) = get_ref r h" |
|
400 |
by (simp add: get_ref_def set_array_def upd_def) |
|
401 |
||
402 |
lemma new_ref_upd: "fst (ref v (upd a i v' h)) = fst (ref v h)" |
|
403 |
by (simp add: set_array_def get_array_def Let_def ref_new_set upd_def ref_def new_ref_def) |
|
404 |
||
26300 | 405 |
text {*not actually true ???*} |
26170 | 406 |
lemma upd_set_ref_swap: "upd a i v (set_ref r v' h) = set_ref r v' (upd a i v h)" |
407 |
apply (case_tac a) |
|
408 |
apply (simp add: Let_def upd_def) |
|
409 |
apply auto |
|
26300 | 410 |
oops |
26170 | 411 |
|
412 |
lemma length_new_ref[simp]: |
|
413 |
"length a (snd (ref v h)) = length a h" |
|
414 |
by (simp add: get_array_def set_ref_def length_def new_ref_def ref_def Let_def) |
|
415 |
||
416 |
lemma get_array_new_ref [simp]: |
|
417 |
"get_array a (snd (ref v h)) = get_array a h" |
|
418 |
by (simp add: new_ref_def ref_def set_ref_def get_array_def Let_def) |
|
419 |
||
420 |
lemma ref_present_upd [simp]: |
|
421 |
"ref_present r (upd a i v h) = ref_present r h" |
|
422 |
by (simp add: upd_def ref_present_def set_array_def get_array_def) |
|
423 |
||
424 |
lemma array_present_set_ref [simp]: |
|
425 |
"array_present a (set_ref r v h) = array_present a h" |
|
426 |
by (simp add: array_present_def set_ref_def) |
|
427 |
||
428 |
lemma array_present_new_ref [simp]: |
|
429 |
"array_present a h \<Longrightarrow> array_present a (snd (ref v h))" |
|
430 |
by (simp add: array_present_def new_ref_def ref_def Let_def) |
|
431 |
||
432 |
hide (open) const empty array array_of_list upd length ref |
|
433 |
||
434 |
end |