src/HOL/Imperative_HOL/Array.thy
 author haftmann Tue Jul 13 16:12:40 2010 +0200 (2010-07-13 ago) changeset 37805 0f797d586ce5 parent 37804 0145e59c1f6c child 37806 a7679be14442 permissions -rw-r--r--
canonical argument order for get
```     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 present :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> bool" where
```
```    14   "present h a \<longleftrightarrow> addr_of_array a < lim h"
```
```    15
```
```    16 definition (*FIXME get *)
```
```    17   get_array :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a list" where
```
```    18   "get_array h a = map from_nat (arrays h (TYPEREP('a)) (addr_of_array a))"
```
```    19
```
```    20 definition set :: "'a\<Colon>heap array \<Rightarrow> 'a list \<Rightarrow> heap \<Rightarrow> heap" where
```
```    21   "set a x = arrays_update (\<lambda>h. h(TYPEREP('a) := ((h(TYPEREP('a))) (addr_of_array a:=map to_nat x))))"
```
```    22
```
```    23 definition alloc :: "'a list \<Rightarrow> heap \<Rightarrow> 'a\<Colon>heap array \<times> heap" where
```
```    24   "alloc xs h = (let
```
```    25      l = lim h;
```
```    26      r = Array l;
```
```    27      h'' = set r xs (h\<lparr>lim := l + 1\<rparr>)
```
```    28    in (r, h''))"
```
```    29
```
```    30 definition length :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> nat" where
```
```    31   "length h a = List.length (get_array h a)"
```
```    32
```
```    33 definition update :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> heap \<Rightarrow> heap" where
```
```    34   "update a i x h = set a ((get_array h a)[i:=x]) h"
```
```    35
```
```    36 definition noteq :: "'a\<Colon>heap array \<Rightarrow> 'b\<Colon>heap array \<Rightarrow> bool" (infix "=!!=" 70) where
```
```    37   "r =!!= s \<longleftrightarrow> TYPEREP('a) \<noteq> TYPEREP('b) \<or> addr_of_array r \<noteq> addr_of_array s"
```
```    38
```
```    39
```
```    40 subsection {* Monad operations *}
```
```    41
```
```    42 definition new :: "nat \<Rightarrow> 'a\<Colon>heap \<Rightarrow> 'a array Heap" where
```
```    43   [code del]: "new n x = Heap_Monad.heap (alloc (replicate n x))"
```
```    44
```
```    45 definition of_list :: "'a\<Colon>heap list \<Rightarrow> 'a array Heap" where
```
```    46   [code del]: "of_list xs = Heap_Monad.heap (alloc xs)"
```
```    47
```
```    48 definition make :: "nat \<Rightarrow> (nat \<Rightarrow> 'a\<Colon>heap) \<Rightarrow> 'a array Heap" where
```
```    49   [code del]: "make n f = Heap_Monad.heap (alloc (map f [0 ..< n]))"
```
```    50
```
```    51 definition len :: "'a\<Colon>heap array \<Rightarrow> nat Heap" where
```
```    52   [code del]: "len a = Heap_Monad.tap (\<lambda>h. length h a)"
```
```    53
```
```    54 definition nth :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a Heap" where
```
```    55   [code del]: "nth a i = Heap_Monad.guard (\<lambda>h. i < length h a)
```
```    56     (\<lambda>h. (get_array h a ! i, h))"
```
```    57
```
```    58 definition upd :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a\<Colon>heap array Heap" where
```
```    59   [code del]: "upd i x a = Heap_Monad.guard (\<lambda>h. i < length h a)
```
```    60     (\<lambda>h. (a, update a i x h))"
```
```    61
```
```    62 definition map_entry :: "nat \<Rightarrow> ('a\<Colon>heap \<Rightarrow> 'a) \<Rightarrow> 'a array \<Rightarrow> 'a array Heap" where
```
```    63   [code del]: "map_entry i f a = Heap_Monad.guard (\<lambda>h. i < length h a)
```
```    64     (\<lambda>h. (a, update a i (f (get_array h a ! i)) h))"
```
```    65
```
```    66 definition swap :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a Heap" where
```
```    67   [code del]: "swap i x a = Heap_Monad.guard (\<lambda>h. i < length h a)
```
```    68     (\<lambda>h. (get_array h a ! i, update a i x h))"
```
```    69
```
```    70 definition freeze :: "'a\<Colon>heap array \<Rightarrow> 'a list Heap" where
```
```    71   [code del]: "freeze a = Heap_Monad.tap (\<lambda>h. get_array h a)"
```
```    72
```
```    73
```
```    74 subsection {* Properties *}
```
```    75
```
```    76 text {* FIXME: Does there exist a "canonical" array axiomatisation in
```
```    77 the literature?  *}
```
```    78
```
```    79 text {* Primitives *}
```
```    80
```
```    81 lemma noteq_arrs_sym: "a =!!= b \<Longrightarrow> b =!!= a"
```
```    82   and unequal_arrs [simp]: "a \<noteq> a' \<longleftrightarrow> a =!!= a'"
```
```    83   unfolding noteq_def by auto
```
```    84
```
```    85 lemma noteq_arrs_irrefl: "r =!!= r \<Longrightarrow> False"
```
```    86   unfolding noteq_def by auto
```
```    87
```
```    88 lemma present_new_arr: "present h a \<Longrightarrow> a =!!= fst (alloc xs h)"
```
```    89   by (simp add: present_def noteq_def alloc_def Let_def)
```
```    90
```
```    91 lemma array_get_set_eq [simp]: "get_array (set r x h) r = x"
```
```    92   by (simp add: get_array_def set_def o_def)
```
```    93
```
```    94 lemma array_get_set_neq [simp]: "r =!!= s \<Longrightarrow> get_array (set s x h) r = get_array h r"
```
```    95   by (simp add: noteq_def get_array_def set_def)
```
```    96
```
```    97 lemma set_array_same [simp]:
```
```    98   "set r x (set r y h) = set r x h"
```
```    99   by (simp add: set_def)
```
```   100
```
```   101 lemma array_set_set_swap:
```
```   102   "r =!!= r' \<Longrightarrow> set r x (set r' x' h) = set r' x' (set r x h)"
```
```   103   by (simp add: Let_def expand_fun_eq noteq_def set_def)
```
```   104
```
```   105 lemma get_array_update_eq [simp]:
```
```   106   "get_array (update a i v h) a = (get_array h a) [i := v]"
```
```   107   by (simp add: update_def)
```
```   108
```
```   109 lemma nth_update_array_neq_array [simp]:
```
```   110   "a =!!= b \<Longrightarrow> get_array (update b j v h) a ! i = get_array h a ! i"
```
```   111   by (simp add: update_def noteq_def)
```
```   112
```
```   113 lemma get_arry_array_update_elem_neqIndex [simp]:
```
```   114   "i \<noteq> j \<Longrightarrow> get_array (update a j v h) a ! i = get_array h a ! i"
```
```   115   by simp
```
```   116
```
```   117 lemma length_update [simp]:
```
```   118   "length (update b i v h) = length h"
```
```   119   by (simp add: update_def length_def set_def get_array_def expand_fun_eq)
```
```   120
```
```   121 lemma update_swap_neqArray:
```
```   122   "a =!!= a' \<Longrightarrow>
```
```   123   update a i v (update a' i' v' h)
```
```   124   = update a' i' v' (update a i v h)"
```
```   125 apply (unfold update_def)
```
```   126 apply simp
```
```   127 apply (subst array_set_set_swap, assumption)
```
```   128 apply (subst array_get_set_neq)
```
```   129 apply (erule noteq_arrs_sym)
```
```   130 apply (simp)
```
```   131 done
```
```   132
```
```   133 lemma update_swap_neqIndex:
```
```   134   "\<lbrakk> i \<noteq> i' \<rbrakk> \<Longrightarrow> update a i v (update a i' v' h) = update a i' v' (update a i v h)"
```
```   135   by (auto simp add: update_def array_set_set_swap list_update_swap)
```
```   136
```
```   137 lemma get_array_init_array_list:
```
```   138   "get_array (snd (alloc ls' h)) (fst (alloc ls h)) = ls'"
```
```   139   by (simp add: Let_def split_def alloc_def)
```
```   140
```
```   141 lemma set_array:
```
```   142   "set (fst (alloc ls h))
```
```   143      new_ls (snd (alloc ls h))
```
```   144        = snd (alloc new_ls h)"
```
```   145   by (simp add: Let_def split_def alloc_def)
```
```   146
```
```   147 lemma array_present_update [simp]:
```
```   148   "present (update b i v h) = present h"
```
```   149   by (simp add: update_def present_def set_def get_array_def expand_fun_eq)
```
```   150
```
```   151 lemma array_present_array [simp]:
```
```   152   "present (snd (alloc xs h)) (fst (alloc xs h))"
```
```   153   by (simp add: present_def alloc_def set_def Let_def)
```
```   154
```
```   155 lemma not_array_present_array [simp]:
```
```   156   "\<not> present h (fst (alloc xs h))"
```
```   157   by (simp add: present_def alloc_def Let_def)
```
```   158
```
```   159
```
```   160 text {* Monad operations *}
```
```   161
```
```   162 lemma execute_new [execute_simps]:
```
```   163   "execute (new n x) h = Some (alloc (replicate n x) h)"
```
```   164   by (simp add: new_def execute_simps)
```
```   165
```
```   166 lemma success_newI [success_intros]:
```
```   167   "success (new n x) h"
```
```   168   by (auto intro: success_intros simp add: new_def)
```
```   169
```
```   170 lemma crel_newI [crel_intros]:
```
```   171   assumes "(a, h') = alloc (replicate n x) h"
```
```   172   shows "crel (new n x) h h' a"
```
```   173   by (rule crelI) (simp add: assms execute_simps)
```
```   174
```
```   175 lemma crel_newE [crel_elims]:
```
```   176   assumes "crel (new n x) h h' r"
```
```   177   obtains "r = fst (alloc (replicate n x) h)" "h' = snd (alloc (replicate n x) h)"
```
```   178     "get_array h' r = replicate n x" "present h' r" "\<not> present h r"
```
```   179   using assms by (rule crelE) (simp add: get_array_init_array_list execute_simps)
```
```   180
```
```   181 lemma execute_of_list [execute_simps]:
```
```   182   "execute (of_list xs) h = Some (alloc xs h)"
```
```   183   by (simp add: of_list_def execute_simps)
```
```   184
```
```   185 lemma success_of_listI [success_intros]:
```
```   186   "success (of_list xs) h"
```
```   187   by (auto intro: success_intros simp add: of_list_def)
```
```   188
```
```   189 lemma crel_of_listI [crel_intros]:
```
```   190   assumes "(a, h') = alloc xs h"
```
```   191   shows "crel (of_list xs) h h' a"
```
```   192   by (rule crelI) (simp add: assms execute_simps)
```
```   193
```
```   194 lemma crel_of_listE [crel_elims]:
```
```   195   assumes "crel (of_list xs) h h' r"
```
```   196   obtains "r = fst (alloc xs h)" "h' = snd (alloc xs h)"
```
```   197     "get_array h' r = xs" "present h' r" "\<not> present h r"
```
```   198   using assms by (rule crelE) (simp add: get_array_init_array_list execute_simps)
```
```   199
```
```   200 lemma execute_make [execute_simps]:
```
```   201   "execute (make n f) h = Some (alloc (map f [0 ..< n]) h)"
```
```   202   by (simp add: make_def execute_simps)
```
```   203
```
```   204 lemma success_makeI [success_intros]:
```
```   205   "success (make n f) h"
```
```   206   by (auto intro: success_intros simp add: make_def)
```
```   207
```
```   208 lemma crel_makeI [crel_intros]:
```
```   209   assumes "(a, h') = alloc (map f [0 ..< n]) h"
```
```   210   shows "crel (make n f) h h' a"
```
```   211   by (rule crelI) (simp add: assms execute_simps)
```
```   212
```
```   213 lemma crel_makeE [crel_elims]:
```
```   214   assumes "crel (make n f) h h' r"
```
```   215   obtains "r = fst (alloc (map f [0 ..< n]) h)" "h' = snd (alloc (map f [0 ..< n]) h)"
```
```   216     "get_array h' r = map f [0 ..< n]" "present h' r" "\<not> present h r"
```
```   217   using assms by (rule crelE) (simp add: get_array_init_array_list execute_simps)
```
```   218
```
```   219 lemma execute_len [execute_simps]:
```
```   220   "execute (len a) h = Some (length h a, h)"
```
```   221   by (simp add: len_def execute_simps)
```
```   222
```
```   223 lemma success_lenI [success_intros]:
```
```   224   "success (len a) h"
```
```   225   by (auto intro: success_intros simp add: len_def)
```
```   226
```
```   227 lemma crel_lengthI [crel_intros]:
```
```   228   assumes "h' = h" "r = length h a"
```
```   229   shows "crel (len a) h h' r"
```
```   230   by (rule crelI) (simp add: assms execute_simps)
```
```   231
```
```   232 lemma crel_lengthE [crel_elims]:
```
```   233   assumes "crel (len a) h h' r"
```
```   234   obtains "r = length h' a" "h' = h"
```
```   235   using assms by (rule crelE) (simp add: execute_simps)
```
```   236
```
```   237 lemma execute_nth [execute_simps]:
```
```   238   "i < length h a \<Longrightarrow>
```
```   239     execute (nth a i) h = Some (get_array h a ! i, h)"
```
```   240   "i \<ge> length h a \<Longrightarrow> execute (nth a i) h = None"
```
```   241   by (simp_all add: nth_def execute_simps)
```
```   242
```
```   243 lemma success_nthI [success_intros]:
```
```   244   "i < length h a \<Longrightarrow> success (nth a i) h"
```
```   245   by (auto intro: success_intros simp add: nth_def)
```
```   246
```
```   247 lemma crel_nthI [crel_intros]:
```
```   248   assumes "i < length h a" "h' = h" "r = get_array h a ! i"
```
```   249   shows "crel (nth a i) h h' r"
```
```   250   by (rule crelI) (insert assms, simp add: execute_simps)
```
```   251
```
```   252 lemma crel_nthE [crel_elims]:
```
```   253   assumes "crel (nth a i) h h' r"
```
```   254   obtains "i < length h a" "r = get_array h a ! i" "h' = h"
```
```   255   using assms by (rule crelE)
```
```   256     (erule successE, cases "i < length h a", simp_all add: execute_simps)
```
```   257
```
```   258 lemma execute_upd [execute_simps]:
```
```   259   "i < length h a \<Longrightarrow>
```
```   260     execute (upd i x a) h = Some (a, update a i x h)"
```
```   261   "i \<ge> length h a \<Longrightarrow> execute (upd i x a) h = None"
```
```   262   by (simp_all add: upd_def execute_simps)
```
```   263
```
```   264 lemma success_updI [success_intros]:
```
```   265   "i < length h a \<Longrightarrow> success (upd i x a) h"
```
```   266   by (auto intro: success_intros simp add: upd_def)
```
```   267
```
```   268 lemma crel_updI [crel_intros]:
```
```   269   assumes "i < length h a" "h' = update a i v h"
```
```   270   shows "crel (upd i v a) h h' a"
```
```   271   by (rule crelI) (insert assms, simp add: execute_simps)
```
```   272
```
```   273 lemma crel_updE [crel_elims]:
```
```   274   assumes "crel (upd i v a) h h' r"
```
```   275   obtains "r = a" "h' = update a i v h" "i < length h a"
```
```   276   using assms by (rule crelE)
```
```   277     (erule successE, cases "i < length h a", simp_all add: execute_simps)
```
```   278
```
```   279 lemma execute_map_entry [execute_simps]:
```
```   280   "i < length h a \<Longrightarrow>
```
```   281    execute (map_entry i f a) h =
```
```   282       Some (a, update a i (f (get_array h a ! i)) h)"
```
```   283   "i \<ge> length h a \<Longrightarrow> execute (map_entry i f a) h = None"
```
```   284   by (simp_all add: map_entry_def execute_simps)
```
```   285
```
```   286 lemma success_map_entryI [success_intros]:
```
```   287   "i < length h a \<Longrightarrow> success (map_entry i f a) h"
```
```   288   by (auto intro: success_intros simp add: map_entry_def)
```
```   289
```
```   290 lemma crel_map_entryI [crel_intros]:
```
```   291   assumes "i < length h a" "h' = update a i (f (get_array h a ! i)) h" "r = a"
```
```   292   shows "crel (map_entry i f a) h h' r"
```
```   293   by (rule crelI) (insert assms, simp add: execute_simps)
```
```   294
```
```   295 lemma crel_map_entryE [crel_elims]:
```
```   296   assumes "crel (map_entry i f a) h h' r"
```
```   297   obtains "r = a" "h' = update a i (f (get_array h a ! i)) h" "i < length h a"
```
```   298   using assms by (rule crelE)
```
```   299     (erule successE, cases "i < length h a", simp_all add: execute_simps)
```
```   300
```
```   301 lemma execute_swap [execute_simps]:
```
```   302   "i < length h a \<Longrightarrow>
```
```   303    execute (swap i x a) h =
```
```   304       Some (get_array h a ! i, update a i x h)"
```
```   305   "i \<ge> length h a \<Longrightarrow> execute (swap i x a) h = None"
```
```   306   by (simp_all add: swap_def execute_simps)
```
```   307
```
```   308 lemma success_swapI [success_intros]:
```
```   309   "i < length h a \<Longrightarrow> success (swap i x a) h"
```
```   310   by (auto intro: success_intros simp add: swap_def)
```
```   311
```
```   312 lemma crel_swapI [crel_intros]:
```
```   313   assumes "i < length h a" "h' = update a i x h" "r = get_array h a ! i"
```
```   314   shows "crel (swap i x a) h h' r"
```
```   315   by (rule crelI) (insert assms, simp add: execute_simps)
```
```   316
```
```   317 lemma crel_swapE [crel_elims]:
```
```   318   assumes "crel (swap i x a) h h' r"
```
```   319   obtains "r = get_array h a ! i" "h' = update a i x h" "i < length h a"
```
```   320   using assms by (rule crelE)
```
```   321     (erule successE, cases "i < length h a", simp_all add: execute_simps)
```
```   322
```
```   323 lemma execute_freeze [execute_simps]:
```
```   324   "execute (freeze a) h = Some (get_array h a, h)"
```
```   325   by (simp add: freeze_def execute_simps)
```
```   326
```
```   327 lemma success_freezeI [success_intros]:
```
```   328   "success (freeze a) h"
```
```   329   by (auto intro: success_intros simp add: freeze_def)
```
```   330
```
```   331 lemma crel_freezeI [crel_intros]:
```
```   332   assumes "h' = h" "r = get_array h a"
```
```   333   shows "crel (freeze a) h h' r"
```
```   334   by (rule crelI) (insert assms, simp add: execute_simps)
```
```   335
```
```   336 lemma crel_freezeE [crel_elims]:
```
```   337   assumes "crel (freeze a) h h' r"
```
```   338   obtains "h' = h" "r = get_array h a"
```
```   339   using assms by (rule crelE) (simp add: execute_simps)
```
```   340
```
```   341 lemma upd_return:
```
```   342   "upd i x a \<guillemotright> return a = upd i x a"
```
```   343   by (rule Heap_eqI) (simp add: bind_def guard_def upd_def execute_simps)
```
```   344
```
```   345 lemma array_make:
```
```   346   "new n x = make n (\<lambda>_. x)"
```
```   347   by (rule Heap_eqI) (simp add: map_replicate_trivial execute_simps)
```
```   348
```
```   349 lemma array_of_list_make:
```
```   350   "of_list xs = make (List.length xs) (\<lambda>n. xs ! n)"
```
```   351   by (rule Heap_eqI) (simp add: map_nth execute_simps)
```
```   352
```
```   353 hide_const (open) present (*get*) set alloc length update noteq new of_list make len nth upd map_entry swap freeze
```
```   354
```
```   355
```
```   356 subsection {* Code generator setup *}
```
```   357
```
```   358 subsubsection {* Logical intermediate layer *}
```
```   359
```
```   360 definition new' where
```
```   361   [code del]: "new' = Array.new o Code_Numeral.nat_of"
```
```   362
```
```   363 lemma [code]:
```
```   364   "Array.new = new' o Code_Numeral.of_nat"
```
```   365   by (simp add: new'_def o_def)
```
```   366
```
```   367 definition of_list' where
```
```   368   [code del]: "of_list' i xs = Array.of_list (take (Code_Numeral.nat_of i) xs)"
```
```   369
```
```   370 lemma [code]:
```
```   371   "Array.of_list xs = of_list' (Code_Numeral.of_nat (List.length xs)) xs"
```
```   372   by (simp add: of_list'_def)
```
```   373
```
```   374 definition make' where
```
```   375   [code del]: "make' i f = Array.make (Code_Numeral.nat_of i) (f o Code_Numeral.of_nat)"
```
```   376
```
```   377 lemma [code]:
```
```   378   "Array.make n f = make' (Code_Numeral.of_nat n) (f o Code_Numeral.nat_of)"
```
```   379   by (simp add: make'_def o_def)
```
```   380
```
```   381 definition len' where
```
```   382   [code del]: "len' a = Array.len a \<guillemotright>= (\<lambda>n. return (Code_Numeral.of_nat n))"
```
```   383
```
```   384 lemma [code]:
```
```   385   "Array.len a = len' a \<guillemotright>= (\<lambda>i. return (Code_Numeral.nat_of i))"
```
```   386   by (simp add: len'_def)
```
```   387
```
```   388 definition nth' where
```
```   389   [code del]: "nth' a = Array.nth a o Code_Numeral.nat_of"
```
```   390
```
```   391 lemma [code]:
```
```   392   "Array.nth a n = nth' a (Code_Numeral.of_nat n)"
```
```   393   by (simp add: nth'_def)
```
```   394
```
```   395 definition upd' where
```
```   396   [code del]: "upd' a i x = Array.upd (Code_Numeral.nat_of i) x a \<guillemotright> return ()"
```
```   397
```
```   398 lemma [code]:
```
```   399   "Array.upd i x a = upd' a (Code_Numeral.of_nat i) x \<guillemotright> return a"
```
```   400   by (simp add: upd'_def upd_return)
```
```   401
```
```   402 lemma [code]:
```
```   403   "Array.map_entry i f a = do {
```
```   404      x \<leftarrow> Array.nth a i;
```
```   405      Array.upd i (f x) a
```
```   406    }"
```
```   407   by (rule Heap_eqI) (simp add: bind_def guard_def map_entry_def execute_simps)
```
```   408
```
```   409 lemma [code]:
```
```   410   "Array.swap i x a = do {
```
```   411      y \<leftarrow> Array.nth a i;
```
```   412      Array.upd i x a;
```
```   413      return y
```
```   414    }"
```
```   415   by (rule Heap_eqI) (simp add: bind_def guard_def swap_def execute_simps)
```
```   416
```
```   417 lemma [code]:
```
```   418   "Array.freeze a = do {
```
```   419      n \<leftarrow> Array.len a;
```
```   420      Heap_Monad.fold_map (\<lambda>i. Array.nth a i) [0..<n]
```
```   421    }"
```
```   422 proof (rule Heap_eqI)
```
```   423   fix h
```
```   424   have *: "List.map
```
```   425      (\<lambda>x. fst (the (if x < Array.length h a
```
```   426                     then Some (get_array h a ! x, h) else None)))
```
```   427      [0..<Array.length h a] =
```
```   428        List.map (List.nth (get_array h a)) [0..<Array.length h a]"
```
```   429     by simp
```
```   430   have "execute (Heap_Monad.fold_map (Array.nth a) [0..<Array.length h a]) h =
```
```   431     Some (get_array h a, h)"
```
```   432     apply (subst execute_fold_map_unchanged_heap)
```
```   433     apply (simp_all add: nth_def guard_def *)
```
```   434     apply (simp add: length_def map_nth)
```
```   435     done
```
```   436   then have "execute (do {
```
```   437       n \<leftarrow> Array.len a;
```
```   438       Heap_Monad.fold_map (Array.nth a) [0..<n]
```
```   439     }) h = Some (get_array h a, h)"
```
```   440     by (auto intro: execute_bind_eq_SomeI simp add: execute_simps)
```
```   441   then show "execute (Array.freeze a) h = execute (do {
```
```   442       n \<leftarrow> Array.len a;
```
```   443       Heap_Monad.fold_map (Array.nth a) [0..<n]
```
```   444     }) h" by (simp add: execute_simps)
```
```   445 qed
```
```   446
```
```   447 hide_const (open) new' of_list' make' len' nth' upd'
```
```   448
```
```   449
```
```   450 text {* SML *}
```
```   451
```
```   452 code_type array (SML "_/ array")
```
```   453 code_const Array (SML "raise/ (Fail/ \"bare Array\")")
```
```   454 code_const Array.new' (SML "(fn/ ()/ =>/ Array.array/ ((_),/ (_)))")
```
```   455 code_const Array.of_list' (SML "(fn/ ()/ =>/ Array.fromList/ _)")
```
```   456 code_const Array.make' (SML "(fn/ ()/ =>/ Array.tabulate/ ((_),/ (_)))")
```
```   457 code_const Array.len' (SML "(fn/ ()/ =>/ Array.length/ _)")
```
```   458 code_const Array.nth' (SML "(fn/ ()/ =>/ Array.sub/ ((_),/ (_)))")
```
```   459 code_const Array.upd' (SML "(fn/ ()/ =>/ Array.update/ ((_),/ (_),/ (_)))")
```
```   460
```
```   461 code_reserved SML Array
```
```   462
```
```   463
```
```   464 text {* OCaml *}
```
```   465
```
```   466 code_type array (OCaml "_/ array")
```
```   467 code_const Array (OCaml "failwith/ \"bare Array\"")
```
```   468 code_const Array.new' (OCaml "(fun/ ()/ ->/ Array.make/ (Big'_int.int'_of'_big'_int/ _)/ _)")
```
```   469 code_const Array.of_list' (OCaml "(fun/ ()/ ->/ Array.of'_list/ _)")
```
```   470 code_const Array.len' (OCaml "(fun/ ()/ ->/ Big'_int.big'_int'_of'_int/ (Array.length/ _))")
```
```   471 code_const Array.nth' (OCaml "(fun/ ()/ ->/ Array.get/ _/ (Big'_int.int'_of'_big'_int/ _))")
```
```   472 code_const Array.upd' (OCaml "(fun/ ()/ ->/ Array.set/ _/ (Big'_int.int'_of'_big'_int/ _)/ _)")
```
```   473
```
```   474 code_reserved OCaml Array
```
```   475
```
```   476
```
```   477 text {* Haskell *}
```
```   478
```
```   479 code_type array (Haskell "Heap.STArray/ Heap.RealWorld/ _")
```
```   480 code_const Array (Haskell "error/ \"bare Array\"")
```
```   481 code_const Array.new' (Haskell "Heap.newArray/ (0,/ _)")
```
```   482 code_const Array.of_list' (Haskell "Heap.newListArray/ (0,/ _)")
```
```   483 code_const Array.len' (Haskell "Heap.lengthArray")
```
```   484 code_const Array.nth' (Haskell "Heap.readArray")
```
```   485 code_const Array.upd' (Haskell "Heap.writeArray")
```
```   486
```
```   487 end
```