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
|
|
9 |
imports Main Countable RType
|
|
10 |
begin
|
|
11 |
|
|
12 |
subsection {* Representable types *}
|
|
13 |
|
|
14 |
text {* The type class of representable types *}
|
|
15 |
|
|
16 |
class heap = rtype + countable
|
|
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 set :: ("{heap, finite}") heap ..
|
|
33 |
|
|
34 |
instance message_string :: countable
|
|
35 |
by (rule countable_classI [of "message_string_case to_nat"])
|
|
36 |
(auto split: message_string.splits)
|
|
37 |
|
|
38 |
instance message_string :: heap ..
|
|
39 |
|
|
40 |
text {* Reflected types themselves are heap-representable *}
|
|
41 |
|
|
42 |
instantiation rtype :: countable
|
|
43 |
begin
|
|
44 |
|
|
45 |
lemma list_size_size_append:
|
|
46 |
"list_size size (xs @ ys) = list_size size xs + list_size size ys"
|
|
47 |
by (induct xs, auto)
|
|
48 |
|
|
49 |
lemma rtype_size: "t = RType.RType c ts \<Longrightarrow> t' \<in> set ts \<Longrightarrow> size t' < size t"
|
|
50 |
by (frule split_list) (auto simp add: list_size_size_append)
|
|
51 |
|
|
52 |
function to_nat_rtype :: "rtype \<Rightarrow> nat" where
|
|
53 |
"to_nat_rtype (RType.RType c ts) = to_nat (to_nat c, to_nat (map to_nat_rtype ts))"
|
|
54 |
by pat_completeness auto
|
|
55 |
|
|
56 |
termination by (relation "measure (\<lambda>x. size x)")
|
|
57 |
(simp, simp only: in_measure rtype_size)
|
|
58 |
|
|
59 |
instance proof (rule countable_classI)
|
|
60 |
fix t t' :: rtype
|
|
61 |
have "(\<forall>t'. to_nat_rtype t = to_nat_rtype t' \<longrightarrow> t = t')
|
|
62 |
\<and> (\<forall>ts'. map to_nat_rtype ts = map to_nat_rtype ts' \<longrightarrow> ts = ts')"
|
|
63 |
proof (induct rule: rtype.induct)
|
|
64 |
case (RType c ts) show ?case
|
|
65 |
proof (rule allI, rule impI)
|
|
66 |
fix t'
|
|
67 |
assume hyp: "to_nat_rtype (rtype.RType c ts) = to_nat_rtype t'"
|
|
68 |
then obtain c' ts' where t': "t' = (rtype.RType c' ts')"
|
|
69 |
by (cases t') auto
|
|
70 |
with RType hyp have "c = c'" and "ts = ts'" by simp_all
|
|
71 |
with t' show "rtype.RType c ts = t'" by simp
|
|
72 |
qed
|
|
73 |
next
|
|
74 |
case Nil_rtype then show ?case by simp
|
|
75 |
next
|
|
76 |
case (Cons_rtype t ts) then show ?case by auto
|
|
77 |
qed
|
|
78 |
then have "to_nat_rtype t = to_nat_rtype t' \<Longrightarrow> t = t'" by auto
|
|
79 |
moreover assume "to_nat_rtype t = to_nat_rtype t'"
|
|
80 |
ultimately show "t = t'" by simp
|
|
81 |
qed
|
|
82 |
|
|
83 |
end
|
|
84 |
|
|
85 |
instance rtype :: heap ..
|
|
86 |
|
|
87 |
|
|
88 |
subsection {* A polymorphic heap with dynamic arrays and references *}
|
|
89 |
|
|
90 |
types addr = nat -- "untyped heap references"
|
|
91 |
|
|
92 |
datatype 'a array = Array addr
|
|
93 |
datatype 'a ref = Ref addr -- "note the phantom type 'a "
|
|
94 |
|
|
95 |
primrec addr_of_array :: "'a array \<Rightarrow> addr" where
|
|
96 |
"addr_of_array (Array x) = x"
|
|
97 |
|
|
98 |
primrec addr_of_ref :: "'a ref \<Rightarrow> addr" where
|
|
99 |
"addr_of_ref (Ref x) = x"
|
|
100 |
|
|
101 |
lemma addr_of_array_inj [simp]:
|
|
102 |
"addr_of_array a = addr_of_array a' \<longleftrightarrow> a = a'"
|
|
103 |
by (cases a, cases a') simp_all
|
|
104 |
|
|
105 |
lemma addr_of_ref_inj [simp]:
|
|
106 |
"addr_of_ref r = addr_of_ref r' \<longleftrightarrow> r = r'"
|
|
107 |
by (cases r, cases r') simp_all
|
|
108 |
|
|
109 |
instance array :: (type) countable
|
|
110 |
by (rule countable_classI [of addr_of_array]) simp
|
|
111 |
|
|
112 |
instance ref :: (type) countable
|
|
113 |
by (rule countable_classI [of addr_of_ref]) simp
|
|
114 |
|
|
115 |
setup {*
|
|
116 |
Sign.add_const_constraint (@{const_name Array}, SOME @{typ "nat \<Rightarrow> 'a\<Colon>heap array"})
|
|
117 |
#> Sign.add_const_constraint (@{const_name Ref}, SOME @{typ "nat \<Rightarrow> 'a\<Colon>heap ref"})
|
|
118 |
#> Sign.add_const_constraint (@{const_name addr_of_array}, SOME @{typ "'a\<Colon>heap array \<Rightarrow> nat"})
|
|
119 |
#> Sign.add_const_constraint (@{const_name addr_of_ref}, SOME @{typ "'a\<Colon>heap ref \<Rightarrow> nat"})
|
|
120 |
*}
|
|
121 |
|
|
122 |
types heap_rep = nat -- "representable values"
|
|
123 |
|
|
124 |
record heap =
|
|
125 |
arrays :: "rtype \<Rightarrow> addr \<Rightarrow> heap_rep list"
|
|
126 |
refs :: "rtype \<Rightarrow> addr \<Rightarrow> heap_rep"
|
|
127 |
lim :: addr
|
|
128 |
|
|
129 |
definition empty :: heap where
|
|
130 |
"empty = \<lparr>arrays = (\<lambda>_. arbitrary), refs = (\<lambda>_. arbitrary), lim = 0\<rparr>" -- "why arbitrary?"
|
|
131 |
|
|
132 |
|
|
133 |
subsection {* Imperative references and arrays *}
|
|
134 |
|
|
135 |
text {*
|
|
136 |
References and arrays are developed in parallel,
|
|
137 |
but keeping them seperate makes some later proofs simpler.
|
|
138 |
*}
|
|
139 |
|
|
140 |
subsubsection {* Primitive operations *}
|
|
141 |
|
|
142 |
definition
|
|
143 |
new_ref :: "heap \<Rightarrow> ('a\<Colon>heap) ref \<times> heap" where
|
|
144 |
"new_ref h = (let l = lim h in (Ref l, h\<lparr>lim := l + 1\<rparr>))"
|
|
145 |
|
|
146 |
definition
|
|
147 |
new_array :: "heap \<Rightarrow> ('a\<Colon>heap) array \<times> heap" where
|
|
148 |
"new_array h = (let l = lim h in (Array l, h\<lparr>lim := l + 1\<rparr>))"
|
|
149 |
|
|
150 |
definition
|
|
151 |
ref_present :: "'a\<Colon>heap ref \<Rightarrow> heap \<Rightarrow> bool" where
|
|
152 |
"ref_present r h \<longleftrightarrow> addr_of_ref r < lim h"
|
|
153 |
|
|
154 |
definition
|
|
155 |
array_present :: "'a\<Colon>heap array \<Rightarrow> heap \<Rightarrow> bool" where
|
|
156 |
"array_present a h \<longleftrightarrow> addr_of_array a < lim h"
|
|
157 |
|
|
158 |
definition
|
|
159 |
get_ref :: "'a\<Colon>heap ref \<Rightarrow> heap \<Rightarrow> 'a" where
|
|
160 |
"get_ref r h = from_nat (refs h (RTYPE('a)) (addr_of_ref r))"
|
|
161 |
|
|
162 |
definition
|
|
163 |
get_array :: "'a\<Colon>heap array \<Rightarrow> heap \<Rightarrow> 'a list" where
|
|
164 |
"get_array a h = map from_nat (arrays h (RTYPE('a)) (addr_of_array a))"
|
|
165 |
|
|
166 |
definition
|
|
167 |
set_ref :: "'a\<Colon>heap ref \<Rightarrow> 'a \<Rightarrow> heap \<Rightarrow> heap" where
|
|
168 |
"set_ref r x =
|
|
169 |
refs_update (\<lambda>h. h( RTYPE('a) := ((h (RTYPE('a))) (addr_of_ref r:=to_nat x))))"
|
|
170 |
|
|
171 |
definition
|
|
172 |
set_array :: "'a\<Colon>heap array \<Rightarrow> 'a list \<Rightarrow> heap \<Rightarrow> heap" where
|
|
173 |
"set_array a x =
|
|
174 |
arrays_update (\<lambda>h. h( RTYPE('a) := ((h (RTYPE('a))) (addr_of_array a:=map to_nat x))))"
|
|
175 |
|
|
176 |
subsubsection {* Interface operations *}
|
|
177 |
|
|
178 |
definition
|
|
179 |
ref :: "'a \<Rightarrow> heap \<Rightarrow> 'a\<Colon>heap ref \<times> heap" where
|
|
180 |
"ref x h = (let (r, h') = new_ref h;
|
|
181 |
h'' = set_ref r x h'
|
|
182 |
in (r, h''))"
|
|
183 |
|
|
184 |
definition
|
|
185 |
array :: "nat \<Rightarrow> 'a \<Rightarrow> heap \<Rightarrow> 'a\<Colon>heap array \<times> heap" where
|
|
186 |
"array n x h = (let (r, h') = new_array h;
|
|
187 |
h'' = set_array r (replicate n x) h'
|
|
188 |
in (r, h''))"
|
|
189 |
|
|
190 |
definition
|
|
191 |
array_of_list :: "'a list \<Rightarrow> heap \<Rightarrow> 'a\<Colon>heap array \<times> heap" where
|
|
192 |
"array_of_list xs h = (let (r, h') = new_array h;
|
|
193 |
h'' = set_array r xs h'
|
|
194 |
in (r, h''))"
|
|
195 |
|
|
196 |
definition
|
|
197 |
upd :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> heap \<Rightarrow> heap" where
|
|
198 |
"upd a i x h = set_array a ((get_array a h)[i:=x]) h"
|
|
199 |
|
|
200 |
definition
|
|
201 |
length :: "'a\<Colon>heap array \<Rightarrow> heap \<Rightarrow> nat" where
|
|
202 |
"length a h = size (get_array a h)"
|
|
203 |
|
|
204 |
definition
|
|
205 |
array_ran :: "('a\<Colon>heap) option array \<Rightarrow> heap \<Rightarrow> 'a set" where
|
|
206 |
"array_ran a h = {e. Some e \<in> set (get_array a h)}"
|
|
207 |
-- {*FIXME*}
|
|
208 |
|
|
209 |
|
|
210 |
subsubsection {* Reference equality *}
|
|
211 |
|
|
212 |
text {*
|
|
213 |
The following relations are useful for comparing arrays and references.
|
|
214 |
*}
|
|
215 |
|
|
216 |
definition
|
|
217 |
noteq_refs :: "('a\<Colon>heap) ref \<Rightarrow> ('b\<Colon>heap) ref \<Rightarrow> bool" (infix "=!=" 70)
|
|
218 |
where
|
|
219 |
"r =!= s \<longleftrightarrow> RTYPE('a) \<noteq> RTYPE('b) \<or> addr_of_ref r \<noteq> addr_of_ref s"
|
|
220 |
|
|
221 |
definition
|
|
222 |
noteq_arrs :: "('a\<Colon>heap) array \<Rightarrow> ('b\<Colon>heap) array \<Rightarrow> bool" (infix "=!!=" 70)
|
|
223 |
where
|
|
224 |
"r =!!= s \<longleftrightarrow> RTYPE('a) \<noteq> RTYPE('b) \<or> addr_of_array r \<noteq> addr_of_array s"
|
|
225 |
|
|
226 |
lemma noteq_refs_sym: "r =!= s \<Longrightarrow> s =!= r"
|
|
227 |
and noteq_arrs_sym: "a =!!= b \<Longrightarrow> b =!!= a"
|
|
228 |
and unequal_refs [simp]: "r \<noteq> r' \<longleftrightarrow> r =!= r'" -- "same types!"
|
|
229 |
and unequal_arrs [simp]: "a \<noteq> a' \<longleftrightarrow> a =!!= a'"
|
|
230 |
unfolding noteq_refs_def noteq_arrs_def by auto
|
|
231 |
|
|
232 |
lemma present_new_ref: "ref_present r h \<Longrightarrow> r =!= fst (ref v h)"
|
|
233 |
by (simp add: ref_present_def new_ref_def ref_def Let_def noteq_refs_def)
|
|
234 |
|
|
235 |
lemma present_new_arr: "array_present a h \<Longrightarrow> a =!!= fst (array v x h)"
|
|
236 |
by (simp add: array_present_def noteq_arrs_def new_array_def array_def Let_def)
|
|
237 |
|
|
238 |
|
|
239 |
subsubsection {* Properties of heap containers *}
|
|
240 |
|
|
241 |
text {* Properties of imperative arrays *}
|
|
242 |
|
|
243 |
text {* FIXME: Does there exist a "canonical" array axiomatisation in
|
|
244 |
the literature? *}
|
|
245 |
|
|
246 |
lemma array_get_set_eq [simp]: "get_array r (set_array r x h) = x"
|
|
247 |
by (simp add: get_array_def set_array_def)
|
|
248 |
|
|
249 |
lemma array_get_set_neq [simp]: "r =!!= s \<Longrightarrow> get_array r (set_array s x h) = get_array r h"
|
|
250 |
by (simp add: noteq_arrs_def get_array_def set_array_def)
|
|
251 |
|
|
252 |
lemma set_array_same [simp]:
|
|
253 |
"set_array r x (set_array r y h) = set_array r x h"
|
|
254 |
by (simp add: set_array_def)
|
|
255 |
|
|
256 |
lemma array_set_set_swap:
|
|
257 |
"r =!!= r' \<Longrightarrow> set_array r x (set_array r' x' h) = set_array r' x' (set_array r x h)"
|
|
258 |
by (simp add: Let_def expand_fun_eq noteq_arrs_def set_array_def)
|
|
259 |
|
|
260 |
lemma array_ref_set_set_swap:
|
|
261 |
"set_array r x (set_ref r' x' h) = set_ref r' x' (set_array r x h)"
|
|
262 |
by (simp add: Let_def expand_fun_eq set_array_def set_ref_def)
|
|
263 |
|
|
264 |
lemma get_array_upd_eq [simp]:
|
|
265 |
"get_array a (upd a i v h) = (get_array a h) [i := v]"
|
|
266 |
by (simp add: upd_def)
|
|
267 |
|
|
268 |
lemma nth_upd_array_neq_array [simp]:
|
|
269 |
"a =!!= b \<Longrightarrow> get_array a (upd b j v h) ! i = get_array a h ! i"
|
|
270 |
by (simp add: upd_def noteq_arrs_def)
|
|
271 |
|
|
272 |
lemma get_arry_array_upd_elem_neqIndex [simp]:
|
|
273 |
"i \<noteq> j \<Longrightarrow> get_array a (upd a j v h) ! i = get_array a h ! i"
|
|
274 |
by simp
|
|
275 |
|
|
276 |
lemma length_upd_eq [simp]:
|
|
277 |
"length a (upd a i v h) = length a h"
|
|
278 |
by (simp add: length_def upd_def)
|
|
279 |
|
|
280 |
lemma length_upd_neq [simp]:
|
|
281 |
"length a (upd b i v h) = length a h"
|
|
282 |
by (simp add: upd_def length_def set_array_def get_array_def)
|
|
283 |
|
|
284 |
lemma upd_swap_neqArray:
|
|
285 |
"a =!!= a' \<Longrightarrow>
|
|
286 |
upd a i v (upd a' i' v' h)
|
|
287 |
= upd a' i' v' (upd a i v h)"
|
|
288 |
apply (unfold upd_def)
|
|
289 |
apply simp
|
|
290 |
apply (subst array_set_set_swap, assumption)
|
|
291 |
apply (subst array_get_set_neq)
|
|
292 |
apply (erule noteq_arrs_sym)
|
|
293 |
apply (simp)
|
|
294 |
done
|
|
295 |
|
|
296 |
lemma upd_swap_neqIndex:
|
|
297 |
"\<lbrakk> i \<noteq> i' \<rbrakk> \<Longrightarrow> upd a i v (upd a i' v' h) = upd a i' v' (upd a i v h)"
|
|
298 |
by (auto simp add: upd_def array_set_set_swap list_update_swap)
|
|
299 |
|
|
300 |
lemma get_array_init_array_list:
|
|
301 |
"get_array (fst (array_of_list ls h)) (snd (array_of_list ls' h)) = ls'"
|
|
302 |
by (simp add: Let_def split_def array_of_list_def)
|
|
303 |
|
|
304 |
lemma set_array:
|
|
305 |
"set_array (fst (array_of_list ls h))
|
|
306 |
new_ls (snd (array_of_list ls h))
|
|
307 |
= snd (array_of_list new_ls h)"
|
|
308 |
by (simp add: Let_def split_def array_of_list_def)
|
|
309 |
|
|
310 |
lemma array_present_upd [simp]:
|
|
311 |
"array_present a (upd b i v h) = array_present a h"
|
|
312 |
by (simp add: upd_def array_present_def set_array_def get_array_def)
|
|
313 |
|
|
314 |
lemma array_of_list_replicate:
|
|
315 |
"array_of_list (replicate n x) = array n x"
|
|
316 |
by (simp add: expand_fun_eq array_of_list_def array_def)
|
|
317 |
|
|
318 |
|
|
319 |
text {* Properties of imperative references *}
|
|
320 |
|
|
321 |
lemma next_ref_fresh [simp]:
|
|
322 |
assumes "(r, h') = new_ref h"
|
|
323 |
shows "\<not> ref_present r h"
|
|
324 |
using assms by (cases h) (auto simp add: new_ref_def ref_present_def Let_def)
|
|
325 |
|
|
326 |
lemma next_ref_present [simp]:
|
|
327 |
assumes "(r, h') = new_ref h"
|
|
328 |
shows "ref_present r h'"
|
|
329 |
using assms by (cases h) (auto simp add: new_ref_def ref_present_def Let_def)
|
|
330 |
|
|
331 |
lemma ref_get_set_eq [simp]: "get_ref r (set_ref r x h) = x"
|
|
332 |
by (simp add: get_ref_def set_ref_def)
|
|
333 |
|
|
334 |
lemma ref_get_set_neq [simp]: "r =!= s \<Longrightarrow> get_ref r (set_ref s x h) = get_ref r h"
|
|
335 |
by (simp add: noteq_refs_def get_ref_def set_ref_def)
|
|
336 |
|
|
337 |
(* FIXME: We need some infrastructure to infer that locally generated
|
|
338 |
new refs (by new_ref(_no_init), new_array(')) are distinct
|
|
339 |
from all existing refs.
|
|
340 |
*)
|
|
341 |
|
|
342 |
lemma ref_set_get: "set_ref r (get_ref r h) h = h"
|
|
343 |
apply (simp add: set_ref_def get_ref_def)
|
|
344 |
oops
|
|
345 |
|
|
346 |
lemma set_ref_same[simp]:
|
|
347 |
"set_ref r x (set_ref r y h) = set_ref r x h"
|
|
348 |
by (simp add: set_ref_def)
|
|
349 |
|
|
350 |
lemma ref_set_set_swap:
|
|
351 |
"r =!= r' \<Longrightarrow> set_ref r x (set_ref r' x' h) = set_ref r' x' (set_ref r x h)"
|
|
352 |
by (simp add: Let_def expand_fun_eq noteq_refs_def set_ref_def)
|
|
353 |
|
|
354 |
lemma ref_new_set: "fst (ref v (set_ref r v' h)) = fst (ref v h)"
|
|
355 |
by (simp add: ref_def new_ref_def set_ref_def Let_def)
|
|
356 |
|
|
357 |
lemma ref_get_new [simp]:
|
|
358 |
"get_ref (fst (ref v h)) (snd (ref v' h)) = v'"
|
|
359 |
by (simp add: ref_def Let_def split_def)
|
|
360 |
|
|
361 |
lemma ref_set_new [simp]:
|
|
362 |
"set_ref (fst (ref v h)) new_v (snd (ref v h)) = snd (ref new_v h)"
|
|
363 |
by (simp add: ref_def Let_def split_def)
|
|
364 |
|
|
365 |
lemma ref_get_new_neq: "r =!= (fst (ref v h)) \<Longrightarrow>
|
|
366 |
get_ref r (snd (ref v h)) = get_ref r h"
|
|
367 |
by (simp add: get_ref_def set_ref_def ref_def Let_def new_ref_def noteq_refs_def)
|
|
368 |
|
|
369 |
lemma lim_set_ref [simp]:
|
|
370 |
"lim (set_ref r v h) = lim h"
|
|
371 |
by (simp add: set_ref_def)
|
|
372 |
|
|
373 |
lemma ref_present_new_ref [simp]:
|
|
374 |
"ref_present r h \<Longrightarrow> ref_present r (snd (ref v h))"
|
|
375 |
by (simp add: new_ref_def ref_present_def ref_def Let_def)
|
|
376 |
|
|
377 |
lemma ref_present_set_ref [simp]:
|
|
378 |
"ref_present r (set_ref r' v h) = ref_present r h"
|
|
379 |
by (simp add: set_ref_def ref_present_def)
|
|
380 |
|
|
381 |
lemma array_ranI: "\<lbrakk> Some b = get_array a h ! i; i < Heap.length a h \<rbrakk> \<Longrightarrow> b \<in> array_ran a h"
|
|
382 |
unfolding array_ran_def Heap.length_def by simp
|
|
383 |
|
|
384 |
lemma array_ran_upd_array_Some:
|
|
385 |
assumes "cl \<in> array_ran a (Heap.upd a i (Some b) h)"
|
|
386 |
shows "cl \<in> array_ran a h \<or> cl = b"
|
|
387 |
proof -
|
|
388 |
have "set (get_array a h[i := Some b]) \<subseteq> insert (Some b) (set (get_array a h))" by (rule set_update_subset_insert)
|
|
389 |
with assms show ?thesis
|
|
390 |
unfolding array_ran_def Heap.upd_def by fastsimp
|
|
391 |
qed
|
|
392 |
|
|
393 |
lemma array_ran_upd_array_None:
|
|
394 |
assumes "cl \<in> array_ran a (Heap.upd a i None h)"
|
|
395 |
shows "cl \<in> array_ran a h"
|
|
396 |
proof -
|
|
397 |
have "set (get_array a h[i := None]) \<subseteq> insert None (set (get_array a h))" by (rule set_update_subset_insert)
|
|
398 |
with assms show ?thesis
|
|
399 |
unfolding array_ran_def Heap.upd_def by auto
|
|
400 |
qed
|
|
401 |
|
|
402 |
|
|
403 |
text {* Non-interaction between imperative array and imperative references *}
|
|
404 |
|
|
405 |
lemma get_array_set_ref [simp]: "get_array a (set_ref r v h) = get_array a h"
|
|
406 |
by (simp add: get_array_def set_ref_def)
|
|
407 |
|
|
408 |
lemma nth_set_ref [simp]: "get_array a (set_ref r v h) ! i = get_array a h ! i"
|
|
409 |
by simp
|
|
410 |
|
|
411 |
lemma get_ref_upd [simp]: "get_ref r (upd a i v h) = get_ref r h"
|
|
412 |
by (simp add: get_ref_def set_array_def upd_def)
|
|
413 |
|
|
414 |
lemma new_ref_upd: "fst (ref v (upd a i v' h)) = fst (ref v h)"
|
|
415 |
by (simp add: set_array_def get_array_def Let_def ref_new_set upd_def ref_def new_ref_def)
|
|
416 |
|
|
417 |
(*not actually true ???
|
|
418 |
lemma upd_set_ref_swap: "upd a i v (set_ref r v' h) = set_ref r v' (upd a i v h)"
|
|
419 |
apply (case_tac a)
|
|
420 |
apply (simp add: Let_def upd_def)
|
|
421 |
apply auto
|
|
422 |
done*)
|
|
423 |
|
|
424 |
lemma length_new_ref[simp]:
|
|
425 |
"length a (snd (ref v h)) = length a h"
|
|
426 |
by (simp add: get_array_def set_ref_def length_def new_ref_def ref_def Let_def)
|
|
427 |
|
|
428 |
lemma get_array_new_ref [simp]:
|
|
429 |
"get_array a (snd (ref v h)) = get_array a h"
|
|
430 |
by (simp add: new_ref_def ref_def set_ref_def get_array_def Let_def)
|
|
431 |
|
|
432 |
lemma get_array_new_ref [simp]:
|
|
433 |
"get_array a (snd (ref v h)) ! i = get_array a h ! i"
|
|
434 |
by (simp add: get_array_def new_ref_def ref_def set_ref_def Let_def)
|
|
435 |
|
|
436 |
lemma ref_present_upd [simp]:
|
|
437 |
"ref_present r (upd a i v h) = ref_present r h"
|
|
438 |
by (simp add: upd_def ref_present_def set_array_def get_array_def)
|
|
439 |
|
|
440 |
lemma array_present_set_ref [simp]:
|
|
441 |
"array_present a (set_ref r v h) = array_present a h"
|
|
442 |
by (simp add: array_present_def set_ref_def)
|
|
443 |
|
|
444 |
lemma array_present_new_ref [simp]:
|
|
445 |
"array_present a h \<Longrightarrow> array_present a (snd (ref v h))"
|
|
446 |
by (simp add: array_present_def new_ref_def ref_def Let_def)
|
|
447 |
|
|
448 |
hide (open) const empty array array_of_list upd length ref
|
|
449 |
|
|
450 |
end
|