src/HOL/Imperative_HOL/Array.thy
 author haftmann Tue Jul 13 11:38:03 2010 +0200 (2010-07-13 ago) changeset 37787 30dc3abf4a58 parent 37771 1bec64044b5e child 37792 ba0bc31b90d7 child 37796 08bd610b2583 permissions -rw-r--r--
theorem collections do not contain default rules any longer
```     1 (*  Title:      HOL/Imperative_HOL/Array.thy
```
```     2     Author:     John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* Monadic arrays *}
```
```     6
```
```     7 theory Array
```
```     8 imports Heap_Monad
```
```     9 begin
```
```    10
```
```    11 subsection {* Primitives *}
```
```    12
```
```    13 definition (*FIXME present :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> bool" where*)
```
```    14   array_present :: "'a\<Colon>heap array \<Rightarrow> heap \<Rightarrow> bool" where
```
```    15   "array_present a h \<longleftrightarrow> addr_of_array a < lim h"
```
```    16
```
```    17 definition (*FIXME get :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a list" where*)
```
```    18   get_array :: "'a\<Colon>heap array \<Rightarrow> heap \<Rightarrow> 'a list" where
```
```    19   "get_array a h = map from_nat (arrays h (TYPEREP('a)) (addr_of_array a))"
```
```    20
```
```    21 definition (*FIXME set*)
```
```    22   set_array :: "'a\<Colon>heap array \<Rightarrow> 'a list \<Rightarrow> heap \<Rightarrow> heap" where
```
```    23   "set_array a x =
```
```    24   arrays_update (\<lambda>h. h(TYPEREP('a) := ((h(TYPEREP('a))) (addr_of_array a:=map to_nat x))))"
```
```    25
```
```    26 definition (*FIXME alloc*)
```
```    27   array :: "'a list \<Rightarrow> heap \<Rightarrow> 'a\<Colon>heap array \<times> heap" where
```
```    28   "array xs h = (let
```
```    29      l = lim h;
```
```    30      r = Array l;
```
```    31      h'' = set_array r xs (h\<lparr>lim := l + 1\<rparr>)
```
```    32    in (r, h''))"
```
```    33
```
```    34 definition (*FIXME length :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> nat" where*)
```
```    35   length :: "'a\<Colon>heap array \<Rightarrow> heap \<Rightarrow> nat" where
```
```    36   "length a h = List.length (get_array a h)"
```
```    37
```
```    38 definition (*FIXME update*)
```
```    39   change :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> heap \<Rightarrow> heap" where
```
```    40   "change a i x h = set_array a ((get_array a h)[i:=x]) h"
```
```    41
```
```    42 definition (*FIXME noteq*)
```
```    43   noteq_arrs :: "'a\<Colon>heap array \<Rightarrow> 'b\<Colon>heap array \<Rightarrow> bool" (infix "=!!=" 70) where
```
```    44   "r =!!= s \<longleftrightarrow> TYPEREP('a) \<noteq> TYPEREP('b) \<or> addr_of_array r \<noteq> addr_of_array s"
```
```    45
```
```    46
```
```    47 subsection {* Monad operations *}
```
```    48
```
```    49 definition new :: "nat \<Rightarrow> 'a\<Colon>heap \<Rightarrow> 'a array Heap" where
```
```    50   [code del]: "new n x = Heap_Monad.heap (array (replicate n x))"
```
```    51
```
```    52 definition of_list :: "'a\<Colon>heap list \<Rightarrow> 'a array Heap" where
```
```    53   [code del]: "of_list xs = Heap_Monad.heap (array xs)"
```
```    54
```
```    55 definition make :: "nat \<Rightarrow> (nat \<Rightarrow> 'a\<Colon>heap) \<Rightarrow> 'a array Heap" where
```
```    56   [code del]: "make n f = Heap_Monad.heap (array (map f [0 ..< n]))"
```
```    57
```
```    58 definition len :: "'a\<Colon>heap array \<Rightarrow> nat Heap" where
```
```    59   [code del]: "len a = Heap_Monad.tap (\<lambda>h. length a h)"
```
```    60
```
```    61 definition nth :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a Heap" where
```
```    62   [code del]: "nth a i = Heap_Monad.guard (\<lambda>h. i < length a h)
```
```    63     (\<lambda>h. (get_array a h ! i, h))"
```
```    64
```
```    65 definition upd :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a\<Colon>heap array Heap" where
```
```    66   [code del]: "upd i x a = Heap_Monad.guard (\<lambda>h. i < length a h)
```
```    67     (\<lambda>h. (a, change a i x h))"
```
```    68
```
```    69 definition map_entry :: "nat \<Rightarrow> ('a\<Colon>heap \<Rightarrow> 'a) \<Rightarrow> 'a array \<Rightarrow> 'a array Heap" where
```
```    70   [code del]: "map_entry i f a = Heap_Monad.guard (\<lambda>h. i < length a h)
```
```    71     (\<lambda>h. (a, change a i (f (get_array a h ! i)) h))"
```
```    72
```
```    73 definition swap :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a Heap" where
```
```    74   [code del]: "swap i x a = Heap_Monad.guard (\<lambda>h. i < length a h)
```
```    75     (\<lambda>h. (get_array a h ! i, change a i x h))"
```
```    76
```
```    77 definition freeze :: "'a\<Colon>heap array \<Rightarrow> 'a list Heap" where
```
```    78   [code del]: "freeze a = Heap_Monad.tap (\<lambda>h. get_array a h)"
```
```    79
```
```    80
```
```    81 subsection {* Properties *}
```
```    82
```
```    83 text {* FIXME: Does there exist a "canonical" array axiomatisation in
```
```    84 the literature?  *}
```
```    85
```
```    86 text {* Primitives *}
```
```    87
```
```    88 lemma noteq_arrs_sym: "a =!!= b \<Longrightarrow> b =!!= a"
```
```    89   and unequal_arrs [simp]: "a \<noteq> a' \<longleftrightarrow> a =!!= a'"
```
```    90   unfolding noteq_arrs_def by auto
```
```    91
```
```    92 lemma noteq_arrs_irrefl: "r =!!= r \<Longrightarrow> False"
```
```    93   unfolding noteq_arrs_def by auto
```
```    94
```
```    95 lemma present_new_arr: "array_present a h \<Longrightarrow> a =!!= fst (array xs h)"
```
```    96   by (simp add: array_present_def noteq_arrs_def array_def Let_def)
```
```    97
```
```    98 lemma array_get_set_eq [simp]: "get_array r (set_array r x h) = x"
```
```    99   by (simp add: get_array_def set_array_def o_def)
```
```   100
```
```   101 lemma array_get_set_neq [simp]: "r =!!= s \<Longrightarrow> get_array r (set_array s x h) = get_array r h"
```
```   102   by (simp add: noteq_arrs_def get_array_def set_array_def)
```
```   103
```
```   104 lemma set_array_same [simp]:
```
```   105   "set_array r x (set_array r y h) = set_array r x h"
```
```   106   by (simp add: set_array_def)
```
```   107
```
```   108 lemma array_set_set_swap:
```
```   109   "r =!!= r' \<Longrightarrow> set_array r x (set_array r' x' h) = set_array r' x' (set_array r x h)"
```
```   110   by (simp add: Let_def expand_fun_eq noteq_arrs_def set_array_def)
```
```   111
```
```   112 lemma get_array_change_eq [simp]:
```
```   113   "get_array a (change a i v h) = (get_array a h) [i := v]"
```
```   114   by (simp add: change_def)
```
```   115
```
```   116 lemma nth_change_array_neq_array [simp]:
```
```   117   "a =!!= b \<Longrightarrow> get_array a (change b j v h) ! i = get_array a h ! i"
```
```   118   by (simp add: change_def noteq_arrs_def)
```
```   119
```
```   120 lemma get_arry_array_change_elem_neqIndex [simp]:
```
```   121   "i \<noteq> j \<Longrightarrow> get_array a (change a j v h) ! i = get_array a h ! i"
```
```   122   by simp
```
```   123
```
```   124 lemma length_change [simp]:
```
```   125   "length a (change b i v h) = length a h"
```
```   126   by (simp add: change_def length_def set_array_def get_array_def)
```
```   127
```
```   128 lemma change_swap_neqArray:
```
```   129   "a =!!= a' \<Longrightarrow>
```
```   130   change a i v (change a' i' v' h)
```
```   131   = change a' i' v' (change a i v h)"
```
```   132 apply (unfold change_def)
```
```   133 apply simp
```
```   134 apply (subst array_set_set_swap, assumption)
```
```   135 apply (subst array_get_set_neq)
```
```   136 apply (erule noteq_arrs_sym)
```
```   137 apply (simp)
```
```   138 done
```
```   139
```
```   140 lemma change_swap_neqIndex:
```
```   141   "\<lbrakk> i \<noteq> i' \<rbrakk> \<Longrightarrow> change a i v (change a i' v' h) = change a i' v' (change a i v h)"
```
```   142   by (auto simp add: change_def array_set_set_swap list_update_swap)
```
```   143
```
```   144 lemma get_array_init_array_list:
```
```   145   "get_array (fst (array ls h)) (snd (array ls' h)) = ls'"
```
```   146   by (simp add: Let_def split_def array_def)
```
```   147
```
```   148 lemma set_array:
```
```   149   "set_array (fst (array ls h))
```
```   150      new_ls (snd (array ls h))
```
```   151        = snd (array new_ls h)"
```
```   152   by (simp add: Let_def split_def array_def)
```
```   153
```
```   154 lemma array_present_change [simp]:
```
```   155   "array_present a (change b i v h) = array_present a h"
```
```   156   by (simp add: change_def array_present_def set_array_def get_array_def)
```
```   157
```
```   158 lemma array_present_array [simp]:
```
```   159   "array_present (fst (array xs h)) (snd (array xs h))"
```
```   160   by (simp add: array_present_def array_def set_array_def Let_def)
```
```   161
```
```   162 lemma not_array_present_array [simp]:
```
```   163   "\<not> array_present (fst (array xs h)) h"
```
```   164   by (simp add: array_present_def array_def Let_def)
```
```   165
```
```   166
```
```   167 text {* Monad operations *}
```
```   168
```
```   169 lemma execute_new [execute_simps]:
```
```   170   "execute (new n x) h = Some (array (replicate n x) h)"
```
```   171   by (simp add: new_def execute_simps)
```
```   172
```
```   173 lemma success_newI [success_intros]:
```
```   174   "success (new n x) h"
```
```   175   by (auto intro: success_intros simp add: new_def)
```
```   176
```
```   177 lemma crel_newI [crel_intros]:
```
```   178   assumes "(a, h') = array (replicate n x) h"
```
```   179   shows "crel (new n x) h h' a"
```
```   180   by (rule crelI) (simp add: assms execute_simps)
```
```   181
```
```   182 lemma crel_newE [crel_elims]:
```
```   183   assumes "crel (new n x) h h' r"
```
```   184   obtains "r = fst (array (replicate n x) h)" "h' = snd (array (replicate n x) h)"
```
```   185     "get_array r h' = replicate n x" "array_present r h'" "\<not> array_present r h"
```
```   186   using assms by (rule crelE) (simp add: get_array_init_array_list execute_simps)
```
```   187
```
```   188 lemma execute_of_list [execute_simps]:
```
```   189   "execute (of_list xs) h = Some (array xs h)"
```
```   190   by (simp add: of_list_def execute_simps)
```
```   191
```
```   192 lemma success_of_listI [success_intros]:
```
```   193   "success (of_list xs) h"
```
```   194   by (auto intro: success_intros simp add: of_list_def)
```
```   195
```
```   196 lemma crel_of_listI [crel_intros]:
```
```   197   assumes "(a, h') = array xs h"
```
```   198   shows "crel (of_list xs) h h' a"
```
```   199   by (rule crelI) (simp add: assms execute_simps)
```
```   200
```
```   201 lemma crel_of_listE [crel_elims]:
```
```   202   assumes "crel (of_list xs) h h' r"
```
```   203   obtains "r = fst (array xs h)" "h' = snd (array xs h)"
```
```   204     "get_array r h' = xs" "array_present r h'" "\<not> array_present r h"
```
```   205   using assms by (rule crelE) (simp add: get_array_init_array_list execute_simps)
```
```   206
```
```   207 lemma execute_make [execute_simps]:
```
```   208   "execute (make n f) h = Some (array (map f [0 ..< n]) h)"
```
```   209   by (simp add: make_def execute_simps)
```
```   210
```
```   211 lemma success_makeI [success_intros]:
```
```   212   "success (make n f) h"
```
```   213   by (auto intro: success_intros simp add: make_def)
```
```   214
```
```   215 lemma crel_makeI [crel_intros]:
```
```   216   assumes "(a, h') = array (map f [0 ..< n]) h"
```
```   217   shows "crel (make n f) h h' a"
```
```   218   by (rule crelI) (simp add: assms execute_simps)
```
```   219
```
```   220 lemma crel_makeE [crel_elims]:
```
```   221   assumes "crel (make n f) h h' r"
```
```   222   obtains "r = fst (array (map f [0 ..< n]) h)" "h' = snd (array (map f [0 ..< n]) h)"
```
```   223     "get_array r h' = map f [0 ..< n]" "array_present r h'" "\<not> array_present r h"
```
```   224   using assms by (rule crelE) (simp add: get_array_init_array_list execute_simps)
```
```   225
```
```   226 lemma execute_len [execute_simps]:
```
```   227   "execute (len a) h = Some (length a h, h)"
```
```   228   by (simp add: len_def execute_simps)
```
```   229
```
```   230 lemma success_lenI [success_intros]:
```
```   231   "success (len a) h"
```
```   232   by (auto intro: success_intros simp add: len_def)
```
```   233
```
```   234 lemma crel_lengthI [crel_intros]:
```
```   235   assumes "h' = h" "r = length a h"
```
```   236   shows "crel (len a) h h' r"
```
```   237   by (rule crelI) (simp add: assms execute_simps)
```
```   238
```
```   239 lemma crel_lengthE [crel_elims]:
```
```   240   assumes "crel (len a) h h' r"
```
```   241   obtains "r = length a h'" "h' = h"
```
```   242   using assms by (rule crelE) (simp add: execute_simps)
```
```   243
```
```   244 lemma execute_nth [execute_simps]:
```
```   245   "i < length a h \<Longrightarrow>
```
```   246     execute (nth a i) h = Some (get_array a h ! i, h)"
```
```   247   "i \<ge> length a h \<Longrightarrow> execute (nth a i) h = None"
```
```   248   by (simp_all add: nth_def execute_simps)
```
```   249
```
```   250 lemma success_nthI [success_intros]:
```
```   251   "i < length a h \<Longrightarrow> success (nth a i) h"
```
```   252   by (auto intro: success_intros simp add: nth_def)
```
```   253
```
```   254 lemma crel_nthI [crel_intros]:
```
```   255   assumes "i < length a h" "h' = h" "r = get_array a h ! i"
```
```   256   shows "crel (nth a i) h h' r"
```
```   257   by (rule crelI) (insert assms, simp add: execute_simps)
```
```   258
```
```   259 lemma crel_nthE [crel_elims]:
```
```   260   assumes "crel (nth a i) h h' r"
```
```   261   obtains "i < length a h" "r = get_array a h ! i" "h' = h"
```
```   262   using assms by (rule crelE)
```
```   263     (erule successE, cases "i < length a h", simp_all add: execute_simps)
```
```   264
```
```   265 lemma execute_upd [execute_simps]:
```
```   266   "i < length a h \<Longrightarrow>
```
```   267     execute (upd i x a) h = Some (a, change a i x h)"
```
```   268   "i \<ge> length a h \<Longrightarrow> execute (upd i x a) h = None"
```
```   269   by (simp_all add: upd_def execute_simps)
```
```   270
```
```   271 lemma success_updI [success_intros]:
```
```   272   "i < length a h \<Longrightarrow> success (upd i x a) h"
```
```   273   by (auto intro: success_intros simp add: upd_def)
```
```   274
```
```   275 lemma crel_updI [crel_intros]:
```
```   276   assumes "i < length a h" "h' = change a i v h"
```
```   277   shows "crel (upd i v a) h h' a"
```
```   278   by (rule crelI) (insert assms, simp add: execute_simps)
```
```   279
```
```   280 lemma crel_updE [crel_elims]:
```
```   281   assumes "crel (upd i v a) h h' r"
```
```   282   obtains "r = a" "h' = change a i v h" "i < length a h"
```
```   283   using assms by (rule crelE)
```
```   284     (erule successE, cases "i < length a h", simp_all add: execute_simps)
```
```   285
```
```   286 lemma execute_map_entry [execute_simps]:
```
```   287   "i < length a h \<Longrightarrow>
```
```   288    execute (map_entry i f a) h =
```
```   289       Some (a, change a i (f (get_array a h ! i)) h)"
```
```   290   "i \<ge> length a h \<Longrightarrow> execute (map_entry i f a) h = None"
```
```   291   by (simp_all add: map_entry_def execute_simps)
```
```   292
```
```   293 lemma success_map_entryI [success_intros]:
```
```   294   "i < length a h \<Longrightarrow> success (map_entry i f a) h"
```
```   295   by (auto intro: success_intros simp add: map_entry_def)
```
```   296
```
```   297 lemma crel_map_entryI [crel_intros]:
```
```   298   assumes "i < length a h" "h' = change a i (f (get_array a h ! i)) h" "r = a"
```
```   299   shows "crel (map_entry i f a) h h' r"
```
```   300   by (rule crelI) (insert assms, simp add: execute_simps)
```
```   301
```
```   302 lemma crel_map_entryE [crel_elims]:
```
```   303   assumes "crel (map_entry i f a) h h' r"
```
```   304   obtains "r = a" "h' = change a i (f (get_array a h ! i)) h" "i < length a h"
```
```   305   using assms by (rule crelE)
```
```   306     (erule successE, cases "i < length a h", simp_all add: execute_simps)
```
```   307
```
```   308 lemma execute_swap [execute_simps]:
```
```   309   "i < length a h \<Longrightarrow>
```
```   310    execute (swap i x a) h =
```
```   311       Some (get_array a h ! i, change a i x h)"
```
```   312   "i \<ge> length a h \<Longrightarrow> execute (swap i x a) h = None"
```
```   313   by (simp_all add: swap_def execute_simps)
```
```   314
```
```   315 lemma success_swapI [success_intros]:
```
```   316   "i < length a h \<Longrightarrow> success (swap i x a) h"
```
```   317   by (auto intro: success_intros simp add: swap_def)
```
```   318
```
```   319 lemma crel_swapI [crel_intros]:
```
```   320   assumes "i < length a h" "h' = change a i x h" "r = get_array a h ! i"
```
```   321   shows "crel (swap i x a) h h' r"
```
```   322   by (rule crelI) (insert assms, simp add: execute_simps)
```
```   323
```
```   324 lemma crel_swapE [crel_elims]:
```
```   325   assumes "crel (swap i x a) h h' r"
```
```   326   obtains "r = get_array a h ! i" "h' = change a i x h" "i < length a h"
```
```   327   using assms by (rule crelE)
```
```   328     (erule successE, cases "i < length a h", simp_all add: execute_simps)
```
```   329
```
```   330 lemma execute_freeze [execute_simps]:
```
```   331   "execute (freeze a) h = Some (get_array a h, h)"
```
```   332   by (simp add: freeze_def execute_simps)
```
```   333
```
```   334 lemma success_freezeI [success_intros]:
```
```   335   "success (freeze a) h"
```
```   336   by (auto intro: success_intros simp add: freeze_def)
```
```   337
```
```   338 lemma crel_freezeI [crel_intros]:
```
```   339   assumes "h' = h" "r = get_array a h"
```
```   340   shows "crel (freeze a) h h' r"
```
```   341   by (rule crelI) (insert assms, simp add: execute_simps)
```
```   342
```
```   343 lemma crel_freezeE [crel_elims]:
```
```   344   assumes "crel (freeze a) h h' r"
```
```   345   obtains "h' = h" "r = get_array a h"
```
```   346   using assms by (rule crelE) (simp add: execute_simps)
```
```   347
```
```   348 lemma upd_return:
```
```   349   "upd i x a \<guillemotright> return a = upd i x a"
```
```   350   by (rule Heap_eqI) (simp add: bind_def guard_def upd_def execute_simps)
```
```   351
```
```   352 lemma array_make:
```
```   353   "new n x = make n (\<lambda>_. x)"
```
```   354   by (rule Heap_eqI) (simp add: map_replicate_trivial execute_simps)
```
```   355
```
```   356 lemma array_of_list_make:
```
```   357   "of_list xs = make (List.length xs) (\<lambda>n. xs ! n)"
```
```   358   by (rule Heap_eqI) (simp add: map_nth execute_simps)
```
```   359
```
```   360 hide_const (open) new
```
```   361
```
```   362
```
```   363 subsection {* Code generator setup *}
```
```   364
```
```   365 subsubsection {* Logical intermediate layer *}
```
```   366
```
```   367 definition new' where
```
```   368   [code del]: "new' = Array.new o Code_Numeral.nat_of"
```
```   369
```
```   370 lemma [code]:
```
```   371   "Array.new = new' o Code_Numeral.of_nat"
```
```   372   by (simp add: new'_def o_def)
```
```   373
```
```   374 definition of_list' where
```
```   375   [code del]: "of_list' i xs = Array.of_list (take (Code_Numeral.nat_of i) xs)"
```
```   376
```
```   377 lemma [code]:
```
```   378   "Array.of_list xs = of_list' (Code_Numeral.of_nat (List.length xs)) xs"
```
```   379   by (simp add: of_list'_def)
```
```   380
```
```   381 definition make' where
```
```   382   [code del]: "make' i f = Array.make (Code_Numeral.nat_of i) (f o Code_Numeral.of_nat)"
```
```   383
```
```   384 lemma [code]:
```
```   385   "Array.make n f = make' (Code_Numeral.of_nat n) (f o Code_Numeral.nat_of)"
```
```   386   by (simp add: make'_def o_def)
```
```   387
```
```   388 definition len' where
```
```   389   [code del]: "len' a = Array.len a \<guillemotright>= (\<lambda>n. return (Code_Numeral.of_nat n))"
```
```   390
```
```   391 lemma [code]:
```
```   392   "Array.len a = len' a \<guillemotright>= (\<lambda>i. return (Code_Numeral.nat_of i))"
```
```   393   by (simp add: len'_def)
```
```   394
```
```   395 definition nth' where
```
```   396   [code del]: "nth' a = Array.nth a o Code_Numeral.nat_of"
```
```   397
```
```   398 lemma [code]:
```
```   399   "Array.nth a n = nth' a (Code_Numeral.of_nat n)"
```
```   400   by (simp add: nth'_def)
```
```   401
```
```   402 definition upd' where
```
```   403   [code del]: "upd' a i x = Array.upd (Code_Numeral.nat_of i) x a \<guillemotright> return ()"
```
```   404
```
```   405 lemma [code]:
```
```   406   "Array.upd i x a = upd' a (Code_Numeral.of_nat i) x \<guillemotright> return a"
```
```   407   by (simp add: upd'_def upd_return)
```
```   408
```
```   409 lemma [code]:
```
```   410   "map_entry i f a = (do
```
```   411      x \<leftarrow> nth a i;
```
```   412      upd i (f x) a
```
```   413    done)"
```
```   414   by (rule Heap_eqI) (simp add: bind_def guard_def map_entry_def execute_simps)
```
```   415
```
```   416 lemma [code]:
```
```   417   "swap i x a = (do
```
```   418      y \<leftarrow> nth a i;
```
```   419      upd i x a;
```
```   420      return y
```
```   421    done)"
```
```   422   by (rule Heap_eqI) (simp add: bind_def guard_def swap_def execute_simps)
```
```   423
```
```   424 lemma [code]:
```
```   425   "freeze a = (do
```
```   426      n \<leftarrow> len a;
```
```   427      Heap_Monad.fold_map (\<lambda>i. nth a i) [0..<n]
```
```   428    done)"
```
```   429 proof (rule Heap_eqI)
```
```   430   fix h
```
```   431   have *: "List.map
```
```   432      (\<lambda>x. fst (the (if x < length a h
```
```   433                     then Some (get_array a h ! x, h) else None)))
```
```   434      [0..<length a h] =
```
```   435        List.map (List.nth (get_array a h)) [0..<length a h]"
```
```   436     by simp
```
```   437   have "execute (Heap_Monad.fold_map (Array.nth a) [0..<length a h]) h =
```
```   438     Some (get_array a h, h)"
```
```   439     apply (subst execute_fold_map_unchanged_heap)
```
```   440     apply (simp_all add: nth_def guard_def *)
```
```   441     apply (simp add: length_def map_nth)
```
```   442     done
```
```   443   then have "execute (do
```
```   444       n \<leftarrow> len a;
```
```   445       Heap_Monad.fold_map (Array.nth a) [0..<n]
```
```   446     done) h = Some (get_array a h, h)"
```
```   447     by (auto intro: execute_bind_eq_SomeI simp add: execute_simps)
```
```   448   then show "execute (freeze a) h = execute (do
```
```   449       n \<leftarrow> len a;
```
```   450       Heap_Monad.fold_map (Array.nth a) [0..<n]
```
```   451     done) h" by (simp add: execute_simps)
```
```   452 qed
```
```   453
```
```   454 hide_const (open) new' of_list' make' len' nth' upd'
```
```   455
```
```   456
```
```   457 text {* SML *}
```
```   458
```
```   459 code_type array (SML "_/ array")
```
```   460 code_const Array (SML "raise/ (Fail/ \"bare Array\")")
```
```   461 code_const Array.new' (SML "(fn/ ()/ =>/ Array.array/ ((_),/ (_)))")
```
```   462 code_const Array.of_list' (SML "(fn/ ()/ =>/ Array.fromList/ _)")
```
```   463 code_const Array.make' (SML "(fn/ ()/ =>/ Array.tabulate/ ((_),/ (_)))")
```
```   464 code_const Array.len' (SML "(fn/ ()/ =>/ Array.length/ _)")
```
```   465 code_const Array.nth' (SML "(fn/ ()/ =>/ Array.sub/ ((_),/ (_)))")
```
```   466 code_const Array.upd' (SML "(fn/ ()/ =>/ Array.update/ ((_),/ (_),/ (_)))")
```
```   467
```
```   468 code_reserved SML Array
```
```   469
```
```   470
```
```   471 text {* OCaml *}
```
```   472
```
```   473 code_type array (OCaml "_/ array")
```
```   474 code_const Array (OCaml "failwith/ \"bare Array\"")
```
```   475 code_const Array.new' (OCaml "(fun/ ()/ ->/ Array.make/ (Big'_int.int'_of'_big'_int/ _)/ _)")
```
```   476 code_const Array.of_list' (OCaml "(fun/ ()/ ->/ Array.of'_list/ _)")
```
```   477 code_const Array.len' (OCaml "(fun/ ()/ ->/ Big'_int.big'_int'_of'_int/ (Array.length/ _))")
```
```   478 code_const Array.nth' (OCaml "(fun/ ()/ ->/ Array.get/ _/ (Big'_int.int'_of'_big'_int/ _))")
```
```   479 code_const Array.upd' (OCaml "(fun/ ()/ ->/ Array.set/ _/ (Big'_int.int'_of'_big'_int/ _)/ _)")
```
```   480
```
```   481 code_reserved OCaml Array
```
```   482
```
```   483
```
```   484 text {* Haskell *}
```
```   485
```
```   486 code_type array (Haskell "Heap.STArray/ Heap.RealWorld/ _")
```
```   487 code_const Array (Haskell "error/ \"bare Array\"")
```
```   488 code_const Array.new' (Haskell "Heap.newArray/ (0,/ _)")
```
```   489 code_const Array.of_list' (Haskell "Heap.newListArray/ (0,/ _)")
```
```   490 code_const Array.len' (Haskell "Heap.lengthArray")
```
```   491 code_const Array.nth' (Haskell "Heap.readArray")
```
```   492 code_const Array.upd' (Haskell "Heap.writeArray")
```
```   493
```
```   494 end
```