src/HOL/Imperative_HOL/Array.thy
 author haftmann Fri Jul 23 10:58:13 2010 +0200 (2010-07-23) changeset 37947 844977c7abeb parent 37845 b70d7a347964 child 37964 0a1ae22df1f1 permissions -rw-r--r--
```     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 get :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a list" where
```
```    17   "get h a = map from_nat (arrays h (TYPEREP('a)) (addr_of_array a))"
```
```    18
```
```    19 definition set :: "'a\<Colon>heap array \<Rightarrow> 'a list \<Rightarrow> heap \<Rightarrow> heap" where
```
```    20   "set a x = arrays_update (\<lambda>h. h(TYPEREP('a) := ((h(TYPEREP('a))) (addr_of_array a:=map to_nat x))))"
```
```    21
```
```    22 definition alloc :: "'a list \<Rightarrow> heap \<Rightarrow> 'a\<Colon>heap array \<times> heap" where
```
```    23   "alloc xs h = (let
```
```    24      l = lim h;
```
```    25      r = Array l;
```
```    26      h'' = set r xs (h\<lparr>lim := l + 1\<rparr>)
```
```    27    in (r, h''))"
```
```    28
```
```    29 definition length :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> nat" where
```
```    30   "length h a = List.length (get h a)"
```
```    31
```
```    32 definition update :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> heap \<Rightarrow> heap" where
```
```    33   "update a i x h = set a ((get h a)[i:=x]) h"
```
```    34
```
```    35 definition noteq :: "'a\<Colon>heap array \<Rightarrow> 'b\<Colon>heap array \<Rightarrow> bool" (infix "=!!=" 70) where
```
```    36   "r =!!= s \<longleftrightarrow> TYPEREP('a) \<noteq> TYPEREP('b) \<or> addr_of_array r \<noteq> addr_of_array s"
```
```    37
```
```    38
```
```    39 subsection {* Monad operations *}
```
```    40
```
```    41 definition new :: "nat \<Rightarrow> 'a\<Colon>heap \<Rightarrow> 'a array Heap" where
```
```    42   [code del]: "new n x = Heap_Monad.heap (alloc (replicate n x))"
```
```    43
```
```    44 definition of_list :: "'a\<Colon>heap list \<Rightarrow> 'a array Heap" where
```
```    45   [code del]: "of_list xs = Heap_Monad.heap (alloc xs)"
```
```    46
```
```    47 definition make :: "nat \<Rightarrow> (nat \<Rightarrow> 'a\<Colon>heap) \<Rightarrow> 'a array Heap" where
```
```    48   [code del]: "make n f = Heap_Monad.heap (alloc (map f [0 ..< n]))"
```
```    49
```
```    50 definition len :: "'a\<Colon>heap array \<Rightarrow> nat Heap" where
```
```    51   [code del]: "len a = Heap_Monad.tap (\<lambda>h. length h a)"
```
```    52
```
```    53 definition nth :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a Heap" where
```
```    54   [code del]: "nth a i = Heap_Monad.guard (\<lambda>h. i < length h a)
```
```    55     (\<lambda>h. (get h a ! i, h))"
```
```    56
```
```    57 definition upd :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a\<Colon>heap array Heap" where
```
```    58   [code del]: "upd i x a = Heap_Monad.guard (\<lambda>h. i < length h a)
```
```    59     (\<lambda>h. (a, update a i x h))"
```
```    60
```
```    61 definition map_entry :: "nat \<Rightarrow> ('a\<Colon>heap \<Rightarrow> 'a) \<Rightarrow> 'a array \<Rightarrow> 'a array Heap" where
```
```    62   [code del]: "map_entry i f a = Heap_Monad.guard (\<lambda>h. i < length h a)
```
```    63     (\<lambda>h. (a, update a i (f (get h a ! i)) h))"
```
```    64
```
```    65 definition swap :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a Heap" where
```
```    66   [code del]: "swap i x a = Heap_Monad.guard (\<lambda>h. i < length h a)
```
```    67     (\<lambda>h. (get h a ! i, update a i x h))"
```
```    68
```
```    69 definition freeze :: "'a\<Colon>heap array \<Rightarrow> 'a list Heap" where
```
```    70   [code del]: "freeze a = Heap_Monad.tap (\<lambda>h. get h a)"
```
```    71
```
```    72
```
```    73 subsection {* Properties *}
```
```    74
```
```    75 text {* FIXME: Does there exist a "canonical" array axiomatisation in
```
```    76 the literature?  *}
```
```    77
```
```    78 text {* Primitives *}
```
```    79
```
```    80 lemma noteq_sym: "a =!!= b \<Longrightarrow> b =!!= a"
```
```    81   and unequal [simp]: "a \<noteq> a' \<longleftrightarrow> a =!!= a'"
```
```    82   unfolding noteq_def by auto
```
```    83
```
```    84 lemma noteq_irrefl: "r =!!= r \<Longrightarrow> False"
```
```    85   unfolding noteq_def by auto
```
```    86
```
```    87 lemma present_alloc_noteq: "present h a \<Longrightarrow> a =!!= fst (alloc xs h)"
```
```    88   by (simp add: present_def noteq_def alloc_def Let_def)
```
```    89
```
```    90 lemma get_set_eq [simp]: "get (set r x h) r = x"
```
```    91   by (simp add: get_def set_def o_def)
```
```    92
```
```    93 lemma get_set_neq [simp]: "r =!!= s \<Longrightarrow> get (set s x h) r = get h r"
```
```    94   by (simp add: noteq_def get_def set_def)
```
```    95
```
```    96 lemma set_same [simp]:
```
```    97   "set r x (set r y h) = set r x h"
```
```    98   by (simp add: set_def)
```
```    99
```
```   100 lemma set_set_swap:
```
```   101   "r =!!= r' \<Longrightarrow> set r x (set r' x' h) = set r' x' (set r x h)"
```
```   102   by (simp add: Let_def expand_fun_eq noteq_def set_def)
```
```   103
```
```   104 lemma get_update_eq [simp]:
```
```   105   "get (update a i v h) a = (get h a) [i := v]"
```
```   106   by (simp add: update_def)
```
```   107
```
```   108 lemma nth_update_neq [simp]:
```
```   109   "a =!!= b \<Longrightarrow> get (update b j v h) a ! i = get h a ! i"
```
```   110   by (simp add: update_def noteq_def)
```
```   111
```
```   112 lemma get_update_elem_neqIndex [simp]:
```
```   113   "i \<noteq> j \<Longrightarrow> get (update a j v h) a ! i = get h a ! i"
```
```   114   by simp
```
```   115
```
```   116 lemma length_update [simp]:
```
```   117   "length (update b i v h) = length h"
```
```   118   by (simp add: update_def length_def set_def get_def expand_fun_eq)
```
```   119
```
```   120 lemma update_swap_neq:
```
```   121   "a =!!= a' \<Longrightarrow>
```
```   122   update a i v (update a' i' v' h)
```
```   123   = update a' i' v' (update a i v h)"
```
```   124 apply (unfold update_def)
```
```   125 apply simp
```
```   126 apply (subst set_set_swap, assumption)
```
```   127 apply (subst get_set_neq)
```
```   128 apply (erule noteq_sym)
```
```   129 apply simp
```
```   130 done
```
```   131
```
```   132 lemma update_swap_neqIndex:
```
```   133   "\<lbrakk> i \<noteq> i' \<rbrakk> \<Longrightarrow> update a i v (update a i' v' h) = update a i' v' (update a i v h)"
```
```   134   by (auto simp add: update_def set_set_swap list_update_swap)
```
```   135
```
```   136 lemma get_alloc:
```
```   137   "get (snd (alloc ls' h)) (fst (alloc ls h)) = ls'"
```
```   138   by (simp add: Let_def split_def alloc_def)
```
```   139
```
```   140 lemma set:
```
```   141   "set (fst (alloc ls h))
```
```   142      new_ls (snd (alloc ls h))
```
```   143        = snd (alloc new_ls h)"
```
```   144   by (simp add: Let_def split_def alloc_def)
```
```   145
```
```   146 lemma present_update [simp]:
```
```   147   "present (update b i v h) = present h"
```
```   148   by (simp add: update_def present_def set_def get_def expand_fun_eq)
```
```   149
```
```   150 lemma present_alloc [simp]:
```
```   151   "present (snd (alloc xs h)) (fst (alloc xs h))"
```
```   152   by (simp add: present_def alloc_def set_def Let_def)
```
```   153
```
```   154 lemma not_present_alloc [simp]:
```
```   155   "\<not> present h (fst (alloc xs h))"
```
```   156   by (simp add: present_def alloc_def Let_def)
```
```   157
```
```   158
```
```   159 text {* Monad operations *}
```
```   160
```
```   161 lemma execute_new [execute_simps]:
```
```   162   "execute (new n x) h = Some (alloc (replicate n x) h)"
```
```   163   by (simp add: new_def execute_simps)
```
```   164
```
```   165 lemma success_newI [success_intros]:
```
```   166   "success (new n x) h"
```
```   167   by (auto intro: success_intros simp add: new_def)
```
```   168
```
```   169 lemma crel_newI [crel_intros]:
```
```   170   assumes "(a, h') = alloc (replicate n x) h"
```
```   171   shows "crel (new n x) h h' a"
```
```   172   by (rule crelI) (simp add: assms execute_simps)
```
```   173
```
```   174 lemma crel_newE [crel_elims]:
```
```   175   assumes "crel (new n x) h h' r"
```
```   176   obtains "r = fst (alloc (replicate n x) h)" "h' = snd (alloc (replicate n x) h)"
```
```   177     "get h' r = replicate n x" "present h' r" "\<not> present h r"
```
```   178   using assms by (rule crelE) (simp add: get_alloc execute_simps)
```
```   179
```
```   180 lemma execute_of_list [execute_simps]:
```
```   181   "execute (of_list xs) h = Some (alloc xs h)"
```
```   182   by (simp add: of_list_def execute_simps)
```
```   183
```
```   184 lemma success_of_listI [success_intros]:
```
```   185   "success (of_list xs) h"
```
```   186   by (auto intro: success_intros simp add: of_list_def)
```
```   187
```
```   188 lemma crel_of_listI [crel_intros]:
```
```   189   assumes "(a, h') = alloc xs h"
```
```   190   shows "crel (of_list xs) h h' a"
```
```   191   by (rule crelI) (simp add: assms execute_simps)
```
```   192
```
```   193 lemma crel_of_listE [crel_elims]:
```
```   194   assumes "crel (of_list xs) h h' r"
```
```   195   obtains "r = fst (alloc xs h)" "h' = snd (alloc xs h)"
```
```   196     "get h' r = xs" "present h' r" "\<not> present h r"
```
```   197   using assms by (rule crelE) (simp add: get_alloc execute_simps)
```
```   198
```
```   199 lemma execute_make [execute_simps]:
```
```   200   "execute (make n f) h = Some (alloc (map f [0 ..< n]) h)"
```
```   201   by (simp add: make_def execute_simps)
```
```   202
```
```   203 lemma success_makeI [success_intros]:
```
```   204   "success (make n f) h"
```
```   205   by (auto intro: success_intros simp add: make_def)
```
```   206
```
```   207 lemma crel_makeI [crel_intros]:
```
```   208   assumes "(a, h') = alloc (map f [0 ..< n]) h"
```
```   209   shows "crel (make n f) h h' a"
```
```   210   by (rule crelI) (simp add: assms execute_simps)
```
```   211
```
```   212 lemma crel_makeE [crel_elims]:
```
```   213   assumes "crel (make n f) h h' r"
```
```   214   obtains "r = fst (alloc (map f [0 ..< n]) h)" "h' = snd (alloc (map f [0 ..< n]) h)"
```
```   215     "get h' r = map f [0 ..< n]" "present h' r" "\<not> present h r"
```
```   216   using assms by (rule crelE) (simp add: get_alloc execute_simps)
```
```   217
```
```   218 lemma execute_len [execute_simps]:
```
```   219   "execute (len a) h = Some (length h a, h)"
```
```   220   by (simp add: len_def execute_simps)
```
```   221
```
```   222 lemma success_lenI [success_intros]:
```
```   223   "success (len a) h"
```
```   224   by (auto intro: success_intros simp add: len_def)
```
```   225
```
```   226 lemma crel_lengthI [crel_intros]:
```
```   227   assumes "h' = h" "r = length h a"
```
```   228   shows "crel (len a) h h' r"
```
```   229   by (rule crelI) (simp add: assms execute_simps)
```
```   230
```
```   231 lemma crel_lengthE [crel_elims]:
```
```   232   assumes "crel (len a) h h' r"
```
```   233   obtains "r = length h' a" "h' = h"
```
```   234   using assms by (rule crelE) (simp add: execute_simps)
```
```   235
```
```   236 lemma execute_nth [execute_simps]:
```
```   237   "i < length h a \<Longrightarrow>
```
```   238     execute (nth a i) h = Some (get h a ! i, h)"
```
```   239   "i \<ge> length h a \<Longrightarrow> execute (nth a i) h = None"
```
```   240   by (simp_all add: nth_def execute_simps)
```
```   241
```
```   242 lemma success_nthI [success_intros]:
```
```   243   "i < length h a \<Longrightarrow> success (nth a i) h"
```
```   244   by (auto intro: success_intros simp add: nth_def)
```
```   245
```
```   246 lemma crel_nthI [crel_intros]:
```
```   247   assumes "i < length h a" "h' = h" "r = get h a ! i"
```
```   248   shows "crel (nth a i) h h' r"
```
```   249   by (rule crelI) (insert assms, simp add: execute_simps)
```
```   250
```
```   251 lemma crel_nthE [crel_elims]:
```
```   252   assumes "crel (nth a i) h h' r"
```
```   253   obtains "i < length h a" "r = get h a ! i" "h' = h"
```
```   254   using assms by (rule crelE)
```
```   255     (erule successE, cases "i < length h a", simp_all add: execute_simps)
```
```   256
```
```   257 lemma execute_upd [execute_simps]:
```
```   258   "i < length h a \<Longrightarrow>
```
```   259     execute (upd i x a) h = Some (a, update a i x h)"
```
```   260   "i \<ge> length h a \<Longrightarrow> execute (upd i x a) h = None"
```
```   261   by (simp_all add: upd_def execute_simps)
```
```   262
```
```   263 lemma success_updI [success_intros]:
```
```   264   "i < length h a \<Longrightarrow> success (upd i x a) h"
```
```   265   by (auto intro: success_intros simp add: upd_def)
```
```   266
```
```   267 lemma crel_updI [crel_intros]:
```
```   268   assumes "i < length h a" "h' = update a i v h"
```
```   269   shows "crel (upd i v a) h h' a"
```
```   270   by (rule crelI) (insert assms, simp add: execute_simps)
```
```   271
```
```   272 lemma crel_updE [crel_elims]:
```
```   273   assumes "crel (upd i v a) h h' r"
```
```   274   obtains "r = a" "h' = update a i v h" "i < length h a"
```
```   275   using assms by (rule crelE)
```
```   276     (erule successE, cases "i < length h a", simp_all add: execute_simps)
```
```   277
```
```   278 lemma execute_map_entry [execute_simps]:
```
```   279   "i < length h a \<Longrightarrow>
```
```   280    execute (map_entry i f a) h =
```
```   281       Some (a, update a i (f (get h a ! i)) h)"
```
```   282   "i \<ge> length h a \<Longrightarrow> execute (map_entry i f a) h = None"
```
```   283   by (simp_all add: map_entry_def execute_simps)
```
```   284
```
```   285 lemma success_map_entryI [success_intros]:
```
```   286   "i < length h a \<Longrightarrow> success (map_entry i f a) h"
```
```   287   by (auto intro: success_intros simp add: map_entry_def)
```
```   288
```
```   289 lemma crel_map_entryI [crel_intros]:
```
```   290   assumes "i < length h a" "h' = update a i (f (get h a ! i)) h" "r = a"
```
```   291   shows "crel (map_entry i f a) h h' r"
```
```   292   by (rule crelI) (insert assms, simp add: execute_simps)
```
```   293
```
```   294 lemma crel_map_entryE [crel_elims]:
```
```   295   assumes "crel (map_entry i f a) h h' r"
```
```   296   obtains "r = a" "h' = update a i (f (get h a ! i)) h" "i < length h a"
```
```   297   using assms by (rule crelE)
```
```   298     (erule successE, cases "i < length h a", simp_all add: execute_simps)
```
```   299
```
```   300 lemma execute_swap [execute_simps]:
```
```   301   "i < length h a \<Longrightarrow>
```
```   302    execute (swap i x a) h =
```
```   303       Some (get h a ! i, update a i x h)"
```
```   304   "i \<ge> length h a \<Longrightarrow> execute (swap i x a) h = None"
```
```   305   by (simp_all add: swap_def execute_simps)
```
```   306
```
```   307 lemma success_swapI [success_intros]:
```
```   308   "i < length h a \<Longrightarrow> success (swap i x a) h"
```
```   309   by (auto intro: success_intros simp add: swap_def)
```
```   310
```
```   311 lemma crel_swapI [crel_intros]:
```
```   312   assumes "i < length h a" "h' = update a i x h" "r = get h a ! i"
```
```   313   shows "crel (swap i x a) h h' r"
```
```   314   by (rule crelI) (insert assms, simp add: execute_simps)
```
```   315
```
```   316 lemma crel_swapE [crel_elims]:
```
```   317   assumes "crel (swap i x a) h h' r"
```
```   318   obtains "r = get h a ! i" "h' = update a i x h" "i < length h a"
```
```   319   using assms by (rule crelE)
```
```   320     (erule successE, cases "i < length h a", simp_all add: execute_simps)
```
```   321
```
```   322 lemma execute_freeze [execute_simps]:
```
```   323   "execute (freeze a) h = Some (get h a, h)"
```
```   324   by (simp add: freeze_def execute_simps)
```
```   325
```
```   326 lemma success_freezeI [success_intros]:
```
```   327   "success (freeze a) h"
```
```   328   by (auto intro: success_intros simp add: freeze_def)
```
```   329
```
```   330 lemma crel_freezeI [crel_intros]:
```
```   331   assumes "h' = h" "r = get h a"
```
```   332   shows "crel (freeze a) h h' r"
```
```   333   by (rule crelI) (insert assms, simp add: execute_simps)
```
```   334
```
```   335 lemma crel_freezeE [crel_elims]:
```
```   336   assumes "crel (freeze a) h h' r"
```
```   337   obtains "h' = h" "r = get h a"
```
```   338   using assms by (rule crelE) (simp add: execute_simps)
```
```   339
```
```   340 lemma upd_return:
```
```   341   "upd i x a \<guillemotright> return a = upd i x a"
```
```   342   by (rule Heap_eqI) (simp add: bind_def guard_def upd_def execute_simps)
```
```   343
```
```   344 lemma array_make:
```
```   345   "new n x = make n (\<lambda>_. x)"
```
```   346   by (rule Heap_eqI) (simp add: map_replicate_trivial execute_simps)
```
```   347
```
```   348 lemma array_of_list_make [code]:
```
```   349   "of_list xs = make (List.length xs) (\<lambda>n. xs ! n)"
```
```   350   by (rule Heap_eqI) (simp add: map_nth execute_simps)
```
```   351
```
```   352 hide_const (open) present get set alloc length update noteq new of_list make len nth upd map_entry swap freeze
```
```   353
```
```   354
```
```   355 subsection {* Code generator setup *}
```
```   356
```
```   357 subsubsection {* Logical intermediate layer *}
```
```   358
```
```   359 definition new' where
```
```   360   [code del]: "new' = Array.new o Code_Numeral.nat_of"
```
```   361
```
```   362 lemma [code]:
```
```   363   "Array.new = new' o Code_Numeral.of_nat"
```
```   364   by (simp add: new'_def o_def)
```
```   365
```
```   366 definition make' where
```
```   367   [code del]: "make' i f = Array.make (Code_Numeral.nat_of i) (f o Code_Numeral.of_nat)"
```
```   368
```
```   369 lemma [code]:
```
```   370   "Array.make n f = make' (Code_Numeral.of_nat n) (f o Code_Numeral.nat_of)"
```
```   371   by (simp add: make'_def o_def)
```
```   372
```
```   373 definition len' where
```
```   374   [code del]: "len' a = Array.len a \<guillemotright>= (\<lambda>n. return (Code_Numeral.of_nat n))"
```
```   375
```
```   376 lemma [code]:
```
```   377   "Array.len a = len' a \<guillemotright>= (\<lambda>i. return (Code_Numeral.nat_of i))"
```
```   378   by (simp add: len'_def)
```
```   379
```
```   380 definition nth' where
```
```   381   [code del]: "nth' a = Array.nth a o Code_Numeral.nat_of"
```
```   382
```
```   383 lemma [code]:
```
```   384   "Array.nth a n = nth' a (Code_Numeral.of_nat n)"
```
```   385   by (simp add: nth'_def)
```
```   386
```
```   387 definition upd' where
```
```   388   [code del]: "upd' a i x = Array.upd (Code_Numeral.nat_of i) x a \<guillemotright> return ()"
```
```   389
```
```   390 lemma [code]:
```
```   391   "Array.upd i x a = upd' a (Code_Numeral.of_nat i) x \<guillemotright> return a"
```
```   392   by (simp add: upd'_def upd_return)
```
```   393
```
```   394 lemma [code]:
```
```   395   "Array.map_entry i f a = do {
```
```   396      x \<leftarrow> Array.nth a i;
```
```   397      Array.upd i (f x) a
```
```   398    }"
```
```   399   by (rule Heap_eqI) (simp add: bind_def guard_def map_entry_def execute_simps)
```
```   400
```
```   401 lemma [code]:
```
```   402   "Array.swap i x a = do {
```
```   403      y \<leftarrow> Array.nth a i;
```
```   404      Array.upd i x a;
```
```   405      return y
```
```   406    }"
```
```   407   by (rule Heap_eqI) (simp add: bind_def guard_def swap_def execute_simps)
```
```   408
```
```   409 lemma [code]:
```
```   410   "Array.freeze a = do {
```
```   411      n \<leftarrow> Array.len a;
```
```   412      Heap_Monad.fold_map (\<lambda>i. Array.nth a i) [0..<n]
```
```   413    }"
```
```   414 proof (rule Heap_eqI)
```
```   415   fix h
```
```   416   have *: "List.map
```
```   417      (\<lambda>x. fst (the (if x < Array.length h a
```
```   418                     then Some (Array.get h a ! x, h) else None)))
```
```   419      [0..<Array.length h a] =
```
```   420        List.map (List.nth (Array.get h a)) [0..<Array.length h a]"
```
```   421     by simp
```
```   422   have "execute (Heap_Monad.fold_map (Array.nth a) [0..<Array.length h a]) h =
```
```   423     Some (Array.get h a, h)"
```
```   424     apply (subst execute_fold_map_unchanged_heap)
```
```   425     apply (simp_all add: nth_def guard_def *)
```
```   426     apply (simp add: length_def map_nth)
```
```   427     done
```
```   428   then have "execute (do {
```
```   429       n \<leftarrow> Array.len a;
```
```   430       Heap_Monad.fold_map (Array.nth a) [0..<n]
```
```   431     }) h = Some (Array.get h a, h)"
```
```   432     by (auto intro: execute_bind_eq_SomeI simp add: execute_simps)
```
```   433   then show "execute (Array.freeze a) h = execute (do {
```
```   434       n \<leftarrow> Array.len a;
```
```   435       Heap_Monad.fold_map (Array.nth a) [0..<n]
```
```   436     }) h" by (simp add: execute_simps)
```
```   437 qed
```
```   438
```
```   439 hide_const (open) new' make' len' nth' upd'
```
```   440
```
```   441
```
```   442 text {* SML *}
```
```   443
```
```   444 code_type array (SML "_/ array")
```
```   445 code_const Array (SML "raise/ (Fail/ \"bare Array\")")
```
```   446 code_const Array.new' (SML "(fn/ ()/ =>/ Array.array/ ((_),/ (_)))")
```
```   447 code_const Array.of_list (SML "(fn/ ()/ =>/ Array.fromList/ _)")
```
```   448 code_const Array.make' (SML "(fn/ ()/ =>/ Array.tabulate/ ((_),/ (_)))")
```
```   449 code_const Array.len' (SML "(fn/ ()/ =>/ Array.length/ _)")
```
```   450 code_const Array.nth' (SML "(fn/ ()/ =>/ Array.sub/ ((_),/ (_)))")
```
```   451 code_const Array.upd' (SML "(fn/ ()/ =>/ Array.update/ ((_),/ (_),/ (_)))")
```
```   452
```
```   453 code_reserved SML Array
```
```   454
```
```   455
```
```   456 text {* OCaml *}
```
```   457
```
```   458 code_type array (OCaml "_/ array")
```
```   459 code_const Array (OCaml "failwith/ \"bare Array\"")
```
```   460 code_const Array.new' (OCaml "(fun/ ()/ ->/ Array.make/ (Big'_int.int'_of'_big'_int/ _)/ _)")
```
```   461 code_const Array.of_list (OCaml "(fun/ ()/ ->/ Array.of'_list/ _)")
```
```   462 code_const Array.make' (OCaml "(fun/ ()/ ->/ Array.init/ (Big'_int.int'_of'_big'_int/ _)/
```
```   463   (fun k'_ ->/ _/ (Big'_int.big'_int'_of'_int/ k'_)))")
```
```   464 code_const Array.len' (OCaml "(fun/ ()/ ->/ Big'_int.big'_int'_of'_int/ (Array.length/ _))")
```
```   465 code_const Array.nth' (OCaml "(fun/ ()/ ->/ Array.get/ _/ (Big'_int.int'_of'_big'_int/ _))")
```
```   466 code_const Array.upd' (OCaml "(fun/ ()/ ->/ Array.set/ _/ (Big'_int.int'_of'_big'_int/ _)/ _)")
```
```   467
```
```   468 code_reserved OCaml Array
```
```   469
```
```   470
```
```   471 text {* Haskell *}
```
```   472
```
```   473 code_type array (Haskell "Heap.STArray/ Heap.RealWorld/ _")
```
```   474 code_const Array (Haskell "error/ \"bare Array\"")
```
```   475 code_const Array.new' (Haskell "Heap.newArray")
```
```   476 code_const Array.of_list (Haskell "Heap.newListArray")
```
```   477 code_const Array.make' (Haskell "Heap.newFunArray")
```
```   478 code_const Array.len' (Haskell "Heap.lengthArray")
```
```   479 code_const Array.nth' (Haskell "Heap.readArray")
```
```   480 code_const Array.upd' (Haskell "Heap.writeArray")
```
```   481
```
```   482
```
```   483 text {* Scala *}
```
```   484
```
```   485 code_type array (Scala "!collection.mutable.ArraySeq[_]")
```
```   486 code_const Array (Scala "!error(\"bare Array\")")
```
```   487 code_const Array.new' (Scala "('_: Unit)/ => / collection.mutable.ArraySeq.fill((_))((_))")
```
```   488 code_const Array.make' (Scala "('_: Unit)/ =>/ collection.mutable.ArraySeq.tabulate((_))((_))")
```
```   489 code_const Array.len' (Scala "('_: Unit)/ =>/ _.length")
```
```   490 code_const Array.nth' (Scala "('_: Unit)/ =>/ _((_))")
```
```   491 code_const Array.upd' (Scala "('_: Unit)/ =>/ _.update((_),/ (_))")
```
```   492 code_const Array.freeze (Scala "('_: Unit)/ =>/ _.toList")
```
```   493
```
```   494 code_reserved Scala Array
```
```   495
```
```   496 end
```