src/HOL/Imperative_HOL/Array.thy
 author haftmann Tue Jul 13 15:34:02 2010 +0200 (2010-07-13 ago) changeset 37802 f2e9c104cebd parent 37798 0b0570445a2a child 37803 582d0fbd201e permissions -rw-r--r--
canonical argument order for length
```     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 length :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> nat" where
```
```    35   "length h a = List.length (get_array a h)"
```
```    36
```
```    37 definition update :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> heap \<Rightarrow> heap" where
```
```    38   "update a i x h = set_array a ((get_array a h)[i:=x]) h"
```
```    39
```
```    40 definition (*FIXME noteq*)
```
```    41   noteq_arrs :: "'a\<Colon>heap array \<Rightarrow> 'b\<Colon>heap array \<Rightarrow> bool" (infix "=!!=" 70) where
```
```    42   "r =!!= s \<longleftrightarrow> TYPEREP('a) \<noteq> TYPEREP('b) \<or> addr_of_array r \<noteq> addr_of_array s"
```
```    43
```
```    44
```
```    45 subsection {* Monad operations *}
```
```    46
```
```    47 definition new :: "nat \<Rightarrow> 'a\<Colon>heap \<Rightarrow> 'a array Heap" where
```
```    48   [code del]: "new n x = Heap_Monad.heap (array (replicate n x))"
```
```    49
```
```    50 definition of_list :: "'a\<Colon>heap list \<Rightarrow> 'a array Heap" where
```
```    51   [code del]: "of_list xs = Heap_Monad.heap (array xs)"
```
```    52
```
```    53 definition make :: "nat \<Rightarrow> (nat \<Rightarrow> 'a\<Colon>heap) \<Rightarrow> 'a array Heap" where
```
```    54   [code del]: "make n f = Heap_Monad.heap (array (map f [0 ..< n]))"
```
```    55
```
```    56 definition len :: "'a\<Colon>heap array \<Rightarrow> nat Heap" where
```
```    57   [code del]: "len a = Heap_Monad.tap (\<lambda>h. length h a)"
```
```    58
```
```    59 definition nth :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a Heap" where
```
```    60   [code del]: "nth a i = Heap_Monad.guard (\<lambda>h. i < length h a)
```
```    61     (\<lambda>h. (get_array a h ! i, h))"
```
```    62
```
```    63 definition upd :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a\<Colon>heap array Heap" where
```
```    64   [code del]: "upd i x a = Heap_Monad.guard (\<lambda>h. i < length h a)
```
```    65     (\<lambda>h. (a, update a i x h))"
```
```    66
```
```    67 definition map_entry :: "nat \<Rightarrow> ('a\<Colon>heap \<Rightarrow> 'a) \<Rightarrow> 'a array \<Rightarrow> 'a array Heap" where
```
```    68   [code del]: "map_entry i f a = Heap_Monad.guard (\<lambda>h. i < length h a)
```
```    69     (\<lambda>h. (a, update a i (f (get_array a h ! i)) h))"
```
```    70
```
```    71 definition swap :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a Heap" where
```
```    72   [code del]: "swap i x a = Heap_Monad.guard (\<lambda>h. i < length h a)
```
```    73     (\<lambda>h. (get_array a h ! i, update a i x h))"
```
```    74
```
```    75 definition freeze :: "'a\<Colon>heap array \<Rightarrow> 'a list Heap" where
```
```    76   [code del]: "freeze a = Heap_Monad.tap (\<lambda>h. get_array a h)"
```
```    77
```
```    78
```
```    79 subsection {* Properties *}
```
```    80
```
```    81 text {* FIXME: Does there exist a "canonical" array axiomatisation in
```
```    82 the literature?  *}
```
```    83
```
```    84 text {* Primitives *}
```
```    85
```
```    86 lemma noteq_arrs_sym: "a =!!= b \<Longrightarrow> b =!!= a"
```
```    87   and unequal_arrs [simp]: "a \<noteq> a' \<longleftrightarrow> a =!!= a'"
```
```    88   unfolding noteq_arrs_def by auto
```
```    89
```
```    90 lemma noteq_arrs_irrefl: "r =!!= r \<Longrightarrow> False"
```
```    91   unfolding noteq_arrs_def by auto
```
```    92
```
```    93 lemma present_new_arr: "array_present a h \<Longrightarrow> a =!!= fst (array xs h)"
```
```    94   by (simp add: array_present_def noteq_arrs_def array_def Let_def)
```
```    95
```
```    96 lemma array_get_set_eq [simp]: "get_array r (set_array r x h) = x"
```
```    97   by (simp add: get_array_def set_array_def o_def)
```
```    98
```
```    99 lemma array_get_set_neq [simp]: "r =!!= s \<Longrightarrow> get_array r (set_array s x h) = get_array r h"
```
```   100   by (simp add: noteq_arrs_def get_array_def set_array_def)
```
```   101
```
```   102 lemma set_array_same [simp]:
```
```   103   "set_array r x (set_array r y h) = set_array r x h"
```
```   104   by (simp add: set_array_def)
```
```   105
```
```   106 lemma array_set_set_swap:
```
```   107   "r =!!= r' \<Longrightarrow> set_array r x (set_array r' x' h) = set_array r' x' (set_array r x h)"
```
```   108   by (simp add: Let_def expand_fun_eq noteq_arrs_def set_array_def)
```
```   109
```
```   110 lemma get_array_update_eq [simp]:
```
```   111   "get_array a (update a i v h) = (get_array a h) [i := v]"
```
```   112   by (simp add: update_def)
```
```   113
```
```   114 lemma nth_update_array_neq_array [simp]:
```
```   115   "a =!!= b \<Longrightarrow> get_array a (update b j v h) ! i = get_array a h ! i"
```
```   116   by (simp add: update_def noteq_arrs_def)
```
```   117
```
```   118 lemma get_arry_array_update_elem_neqIndex [simp]:
```
```   119   "i \<noteq> j \<Longrightarrow> get_array a (update a j v h) ! i = get_array a h ! i"
```
```   120   by simp
```
```   121
```
```   122 lemma length_update [simp]:
```
```   123   "length (update b i v h) = length h"
```
```   124   by (simp add: update_def length_def set_array_def get_array_def expand_fun_eq)
```
```   125
```
```   126 lemma update_swap_neqArray:
```
```   127   "a =!!= a' \<Longrightarrow>
```
```   128   update a i v (update a' i' v' h)
```
```   129   = update a' i' v' (update a i v h)"
```
```   130 apply (unfold update_def)
```
```   131 apply simp
```
```   132 apply (subst array_set_set_swap, assumption)
```
```   133 apply (subst array_get_set_neq)
```
```   134 apply (erule noteq_arrs_sym)
```
```   135 apply (simp)
```
```   136 done
```
```   137
```
```   138 lemma update_swap_neqIndex:
```
```   139   "\<lbrakk> i \<noteq> i' \<rbrakk> \<Longrightarrow> update a i v (update a i' v' h) = update a i' v' (update a i v h)"
```
```   140   by (auto simp add: update_def array_set_set_swap list_update_swap)
```
```   141
```
```   142 lemma get_array_init_array_list:
```
```   143   "get_array (fst (array ls h)) (snd (array ls' h)) = ls'"
```
```   144   by (simp add: Let_def split_def array_def)
```
```   145
```
```   146 lemma set_array:
```
```   147   "set_array (fst (array ls h))
```
```   148      new_ls (snd (array ls h))
```
```   149        = snd (array new_ls h)"
```
```   150   by (simp add: Let_def split_def array_def)
```
```   151
```
```   152 lemma array_present_update [simp]:
```
```   153   "array_present a (update b i v h) = array_present a h"
```
```   154   by (simp add: update_def array_present_def set_array_def get_array_def)
```
```   155
```
```   156 lemma array_present_array [simp]:
```
```   157   "array_present (fst (array xs h)) (snd (array xs h))"
```
```   158   by (simp add: array_present_def array_def set_array_def Let_def)
```
```   159
```
```   160 lemma not_array_present_array [simp]:
```
```   161   "\<not> array_present (fst (array xs h)) h"
```
```   162   by (simp add: array_present_def array_def Let_def)
```
```   163
```
```   164
```
```   165 text {* Monad operations *}
```
```   166
```
```   167 lemma execute_new [execute_simps]:
```
```   168   "execute (new n x) h = Some (array (replicate n x) h)"
```
```   169   by (simp add: new_def execute_simps)
```
```   170
```
```   171 lemma success_newI [success_intros]:
```
```   172   "success (new n x) h"
```
```   173   by (auto intro: success_intros simp add: new_def)
```
```   174
```
```   175 lemma crel_newI [crel_intros]:
```
```   176   assumes "(a, h') = array (replicate n x) h"
```
```   177   shows "crel (new n x) h h' a"
```
```   178   by (rule crelI) (simp add: assms execute_simps)
```
```   179
```
```   180 lemma crel_newE [crel_elims]:
```
```   181   assumes "crel (new n x) h h' r"
```
```   182   obtains "r = fst (array (replicate n x) h)" "h' = snd (array (replicate n x) h)"
```
```   183     "get_array r h' = replicate n x" "array_present r h'" "\<not> array_present r h"
```
```   184   using assms by (rule crelE) (simp add: get_array_init_array_list execute_simps)
```
```   185
```
```   186 lemma execute_of_list [execute_simps]:
```
```   187   "execute (of_list xs) h = Some (array xs h)"
```
```   188   by (simp add: of_list_def execute_simps)
```
```   189
```
```   190 lemma success_of_listI [success_intros]:
```
```   191   "success (of_list xs) h"
```
```   192   by (auto intro: success_intros simp add: of_list_def)
```
```   193
```
```   194 lemma crel_of_listI [crel_intros]:
```
```   195   assumes "(a, h') = array xs h"
```
```   196   shows "crel (of_list xs) h h' a"
```
```   197   by (rule crelI) (simp add: assms execute_simps)
```
```   198
```
```   199 lemma crel_of_listE [crel_elims]:
```
```   200   assumes "crel (of_list xs) h h' r"
```
```   201   obtains "r = fst (array xs h)" "h' = snd (array xs h)"
```
```   202     "get_array r h' = xs" "array_present r h'" "\<not> array_present r h"
```
```   203   using assms by (rule crelE) (simp add: get_array_init_array_list execute_simps)
```
```   204
```
```   205 lemma execute_make [execute_simps]:
```
```   206   "execute (make n f) h = Some (array (map f [0 ..< n]) h)"
```
```   207   by (simp add: make_def execute_simps)
```
```   208
```
```   209 lemma success_makeI [success_intros]:
```
```   210   "success (make n f) h"
```
```   211   by (auto intro: success_intros simp add: make_def)
```
```   212
```
```   213 lemma crel_makeI [crel_intros]:
```
```   214   assumes "(a, h') = array (map f [0 ..< n]) h"
```
```   215   shows "crel (make n f) h h' a"
```
```   216   by (rule crelI) (simp add: assms execute_simps)
```
```   217
```
```   218 lemma crel_makeE [crel_elims]:
```
```   219   assumes "crel (make n f) h h' r"
```
```   220   obtains "r = fst (array (map f [0 ..< n]) h)" "h' = snd (array (map f [0 ..< n]) h)"
```
```   221     "get_array r h' = map f [0 ..< n]" "array_present r h'" "\<not> array_present r h"
```
```   222   using assms by (rule crelE) (simp add: get_array_init_array_list execute_simps)
```
```   223
```
```   224 lemma execute_len [execute_simps]:
```
```   225   "execute (len a) h = Some (length h a, h)"
```
```   226   by (simp add: len_def execute_simps)
```
```   227
```
```   228 lemma success_lenI [success_intros]:
```
```   229   "success (len a) h"
```
```   230   by (auto intro: success_intros simp add: len_def)
```
```   231
```
```   232 lemma crel_lengthI [crel_intros]:
```
```   233   assumes "h' = h" "r = length h a"
```
```   234   shows "crel (len a) h h' r"
```
```   235   by (rule crelI) (simp add: assms execute_simps)
```
```   236
```
```   237 lemma crel_lengthE [crel_elims]:
```
```   238   assumes "crel (len a) h h' r"
```
```   239   obtains "r = length h' a" "h' = h"
```
```   240   using assms by (rule crelE) (simp add: execute_simps)
```
```   241
```
```   242 lemma execute_nth [execute_simps]:
```
```   243   "i < length h a \<Longrightarrow>
```
```   244     execute (nth a i) h = Some (get_array a h ! i, h)"
```
```   245   "i \<ge> length h a \<Longrightarrow> execute (nth a i) h = None"
```
```   246   by (simp_all add: nth_def execute_simps)
```
```   247
```
```   248 lemma success_nthI [success_intros]:
```
```   249   "i < length h a \<Longrightarrow> success (nth a i) h"
```
```   250   by (auto intro: success_intros simp add: nth_def)
```
```   251
```
```   252 lemma crel_nthI [crel_intros]:
```
```   253   assumes "i < length h a" "h' = h" "r = get_array a h ! i"
```
```   254   shows "crel (nth a i) h h' r"
```
```   255   by (rule crelI) (insert assms, simp add: execute_simps)
```
```   256
```
```   257 lemma crel_nthE [crel_elims]:
```
```   258   assumes "crel (nth a i) h h' r"
```
```   259   obtains "i < length h a" "r = get_array a h ! i" "h' = h"
```
```   260   using assms by (rule crelE)
```
```   261     (erule successE, cases "i < length h a", simp_all add: execute_simps)
```
```   262
```
```   263 lemma execute_upd [execute_simps]:
```
```   264   "i < length h a \<Longrightarrow>
```
```   265     execute (upd i x a) h = Some (a, update a i x h)"
```
```   266   "i \<ge> length h a \<Longrightarrow> execute (upd i x a) h = None"
```
```   267   by (simp_all add: upd_def execute_simps)
```
```   268
```
```   269 lemma success_updI [success_intros]:
```
```   270   "i < length h a \<Longrightarrow> success (upd i x a) h"
```
```   271   by (auto intro: success_intros simp add: upd_def)
```
```   272
```
```   273 lemma crel_updI [crel_intros]:
```
```   274   assumes "i < length h a" "h' = update a i v h"
```
```   275   shows "crel (upd i v a) h h' a"
```
```   276   by (rule crelI) (insert assms, simp add: execute_simps)
```
```   277
```
```   278 lemma crel_updE [crel_elims]:
```
```   279   assumes "crel (upd i v a) h h' r"
```
```   280   obtains "r = a" "h' = update a i v h" "i < length h a"
```
```   281   using assms by (rule crelE)
```
```   282     (erule successE, cases "i < length h a", simp_all add: execute_simps)
```
```   283
```
```   284 lemma execute_map_entry [execute_simps]:
```
```   285   "i < length h a \<Longrightarrow>
```
```   286    execute (map_entry i f a) h =
```
```   287       Some (a, update a i (f (get_array a h ! i)) h)"
```
```   288   "i \<ge> length h a \<Longrightarrow> execute (map_entry i f a) h = None"
```
```   289   by (simp_all add: map_entry_def execute_simps)
```
```   290
```
```   291 lemma success_map_entryI [success_intros]:
```
```   292   "i < length h a \<Longrightarrow> success (map_entry i f a) h"
```
```   293   by (auto intro: success_intros simp add: map_entry_def)
```
```   294
```
```   295 lemma crel_map_entryI [crel_intros]:
```
```   296   assumes "i < length h a" "h' = update a i (f (get_array a h ! i)) h" "r = a"
```
```   297   shows "crel (map_entry i f a) h h' r"
```
```   298   by (rule crelI) (insert assms, simp add: execute_simps)
```
```   299
```
```   300 lemma crel_map_entryE [crel_elims]:
```
```   301   assumes "crel (map_entry i f a) h h' r"
```
```   302   obtains "r = a" "h' = update a i (f (get_array a h ! i)) h" "i < length h a"
```
```   303   using assms by (rule crelE)
```
```   304     (erule successE, cases "i < length h a", simp_all add: execute_simps)
```
```   305
```
```   306 lemma execute_swap [execute_simps]:
```
```   307   "i < length h a \<Longrightarrow>
```
```   308    execute (swap i x a) h =
```
```   309       Some (get_array a h ! i, update a i x h)"
```
```   310   "i \<ge> length h a \<Longrightarrow> execute (swap i x a) h = None"
```
```   311   by (simp_all add: swap_def execute_simps)
```
```   312
```
```   313 lemma success_swapI [success_intros]:
```
```   314   "i < length h a \<Longrightarrow> success (swap i x a) h"
```
```   315   by (auto intro: success_intros simp add: swap_def)
```
```   316
```
```   317 lemma crel_swapI [crel_intros]:
```
```   318   assumes "i < length h a" "h' = update a i x h" "r = get_array a h ! i"
```
```   319   shows "crel (swap i x a) h h' r"
```
```   320   by (rule crelI) (insert assms, simp add: execute_simps)
```
```   321
```
```   322 lemma crel_swapE [crel_elims]:
```
```   323   assumes "crel (swap i x a) h h' r"
```
```   324   obtains "r = get_array a h ! i" "h' = update a i x h" "i < length h a"
```
```   325   using assms by (rule crelE)
```
```   326     (erule successE, cases "i < length h a", simp_all add: execute_simps)
```
```   327
```
```   328 lemma execute_freeze [execute_simps]:
```
```   329   "execute (freeze a) h = Some (get_array a h, h)"
```
```   330   by (simp add: freeze_def execute_simps)
```
```   331
```
```   332 lemma success_freezeI [success_intros]:
```
```   333   "success (freeze a) h"
```
```   334   by (auto intro: success_intros simp add: freeze_def)
```
```   335
```
```   336 lemma crel_freezeI [crel_intros]:
```
```   337   assumes "h' = h" "r = get_array a h"
```
```   338   shows "crel (freeze a) h h' r"
```
```   339   by (rule crelI) (insert assms, simp add: execute_simps)
```
```   340
```
```   341 lemma crel_freezeE [crel_elims]:
```
```   342   assumes "crel (freeze a) h h' r"
```
```   343   obtains "h' = h" "r = get_array a h"
```
```   344   using assms by (rule crelE) (simp add: execute_simps)
```
```   345
```
```   346 lemma upd_return:
```
```   347   "upd i x a \<guillemotright> return a = upd i x a"
```
```   348   by (rule Heap_eqI) (simp add: bind_def guard_def upd_def execute_simps)
```
```   349
```
```   350 lemma array_make:
```
```   351   "new n x = make n (\<lambda>_. x)"
```
```   352   by (rule Heap_eqI) (simp add: map_replicate_trivial execute_simps)
```
```   353
```
```   354 lemma array_of_list_make:
```
```   355   "of_list xs = make (List.length xs) (\<lambda>n. xs ! n)"
```
```   356   by (rule Heap_eqI) (simp add: map_nth execute_simps)
```
```   357
```
```   358 hide_const (open) update new of_list make len nth upd map_entry swap freeze
```
```   359
```
```   360
```
```   361 subsection {* Code generator setup *}
```
```   362
```
```   363 subsubsection {* Logical intermediate layer *}
```
```   364
```
```   365 definition new' where
```
```   366   [code del]: "new' = Array.new o Code_Numeral.nat_of"
```
```   367
```
```   368 lemma [code]:
```
```   369   "Array.new = new' o Code_Numeral.of_nat"
```
```   370   by (simp add: new'_def o_def)
```
```   371
```
```   372 definition of_list' where
```
```   373   [code del]: "of_list' i xs = Array.of_list (take (Code_Numeral.nat_of i) xs)"
```
```   374
```
```   375 lemma [code]:
```
```   376   "Array.of_list xs = of_list' (Code_Numeral.of_nat (List.length xs)) xs"
```
```   377   by (simp add: of_list'_def)
```
```   378
```
```   379 definition make' where
```
```   380   [code del]: "make' i f = Array.make (Code_Numeral.nat_of i) (f o Code_Numeral.of_nat)"
```
```   381
```
```   382 lemma [code]:
```
```   383   "Array.make n f = make' (Code_Numeral.of_nat n) (f o Code_Numeral.nat_of)"
```
```   384   by (simp add: make'_def o_def)
```
```   385
```
```   386 definition len' where
```
```   387   [code del]: "len' a = Array.len a \<guillemotright>= (\<lambda>n. return (Code_Numeral.of_nat n))"
```
```   388
```
```   389 lemma [code]:
```
```   390   "Array.len a = len' a \<guillemotright>= (\<lambda>i. return (Code_Numeral.nat_of i))"
```
```   391   by (simp add: len'_def)
```
```   392
```
```   393 definition nth' where
```
```   394   [code del]: "nth' a = Array.nth a o Code_Numeral.nat_of"
```
```   395
```
```   396 lemma [code]:
```
```   397   "Array.nth a n = nth' a (Code_Numeral.of_nat n)"
```
```   398   by (simp add: nth'_def)
```
```   399
```
```   400 definition upd' where
```
```   401   [code del]: "upd' a i x = Array.upd (Code_Numeral.nat_of i) x a \<guillemotright> return ()"
```
```   402
```
```   403 lemma [code]:
```
```   404   "Array.upd i x a = upd' a (Code_Numeral.of_nat i) x \<guillemotright> return a"
```
```   405   by (simp add: upd'_def upd_return)
```
```   406
```
```   407 lemma [code]:
```
```   408   "Array.map_entry i f a = do {
```
```   409      x \<leftarrow> Array.nth a i;
```
```   410      Array.upd i (f x) a
```
```   411    }"
```
```   412   by (rule Heap_eqI) (simp add: bind_def guard_def map_entry_def execute_simps)
```
```   413
```
```   414 lemma [code]:
```
```   415   "Array.swap i x a = do {
```
```   416      y \<leftarrow> Array.nth a i;
```
```   417      Array.upd i x a;
```
```   418      return y
```
```   419    }"
```
```   420   by (rule Heap_eqI) (simp add: bind_def guard_def swap_def execute_simps)
```
```   421
```
```   422 lemma [code]:
```
```   423   "Array.freeze a = do {
```
```   424      n \<leftarrow> Array.len a;
```
```   425      Heap_Monad.fold_map (\<lambda>i. Array.nth a i) [0..<n]
```
```   426    }"
```
```   427 proof (rule Heap_eqI)
```
```   428   fix h
```
```   429   have *: "List.map
```
```   430      (\<lambda>x. fst (the (if x < length h a
```
```   431                     then Some (get_array a h ! x, h) else None)))
```
```   432      [0..<length h a] =
```
```   433        List.map (List.nth (get_array a h)) [0..<length h a]"
```
```   434     by simp
```
```   435   have "execute (Heap_Monad.fold_map (Array.nth a) [0..<length h a]) h =
```
```   436     Some (get_array a h, h)"
```
```   437     apply (subst execute_fold_map_unchanged_heap)
```
```   438     apply (simp_all add: nth_def guard_def *)
```
```   439     apply (simp add: length_def map_nth)
```
```   440     done
```
```   441   then have "execute (do {
```
```   442       n \<leftarrow> Array.len a;
```
```   443       Heap_Monad.fold_map (Array.nth a) [0..<n]
```
```   444     }) h = Some (get_array a h, h)"
```
```   445     by (auto intro: execute_bind_eq_SomeI simp add: execute_simps)
```
```   446   then show "execute (Array.freeze a) h = execute (do {
```
```   447       n \<leftarrow> Array.len a;
```
```   448       Heap_Monad.fold_map (Array.nth a) [0..<n]
```
```   449     }) h" by (simp add: execute_simps)
```
```   450 qed
```
```   451
```
```   452 hide_const (open) new' of_list' make' len' nth' upd'
```
```   453
```
```   454
```
```   455 text {* SML *}
```
```   456
```
```   457 code_type array (SML "_/ array")
```
```   458 code_const Array (SML "raise/ (Fail/ \"bare Array\")")
```
```   459 code_const Array.new' (SML "(fn/ ()/ =>/ Array.array/ ((_),/ (_)))")
```
```   460 code_const Array.of_list' (SML "(fn/ ()/ =>/ Array.fromList/ _)")
```
```   461 code_const Array.make' (SML "(fn/ ()/ =>/ Array.tabulate/ ((_),/ (_)))")
```
```   462 code_const Array.len' (SML "(fn/ ()/ =>/ Array.length/ _)")
```
```   463 code_const Array.nth' (SML "(fn/ ()/ =>/ Array.sub/ ((_),/ (_)))")
```
```   464 code_const Array.upd' (SML "(fn/ ()/ =>/ Array.update/ ((_),/ (_),/ (_)))")
```
```   465
```
```   466 code_reserved SML Array
```
```   467
```
```   468
```
```   469 text {* OCaml *}
```
```   470
```
```   471 code_type array (OCaml "_/ array")
```
```   472 code_const Array (OCaml "failwith/ \"bare Array\"")
```
```   473 code_const Array.new' (OCaml "(fun/ ()/ ->/ Array.make/ (Big'_int.int'_of'_big'_int/ _)/ _)")
```
```   474 code_const Array.of_list' (OCaml "(fun/ ()/ ->/ Array.of'_list/ _)")
```
```   475 code_const Array.len' (OCaml "(fun/ ()/ ->/ Big'_int.big'_int'_of'_int/ (Array.length/ _))")
```
```   476 code_const Array.nth' (OCaml "(fun/ ()/ ->/ Array.get/ _/ (Big'_int.int'_of'_big'_int/ _))")
```
```   477 code_const Array.upd' (OCaml "(fun/ ()/ ->/ Array.set/ _/ (Big'_int.int'_of'_big'_int/ _)/ _)")
```
```   478
```
```   479 code_reserved OCaml Array
```
```   480
```
```   481
```
```   482 text {* Haskell *}
```
```   483
```
```   484 code_type array (Haskell "Heap.STArray/ Heap.RealWorld/ _")
```
```   485 code_const Array (Haskell "error/ \"bare Array\"")
```
```   486 code_const Array.new' (Haskell "Heap.newArray/ (0,/ _)")
```
```   487 code_const Array.of_list' (Haskell "Heap.newListArray/ (0,/ _)")
```
```   488 code_const Array.len' (Haskell "Heap.lengthArray")
```
```   489 code_const Array.nth' (Haskell "Heap.readArray")
```
```   490 code_const Array.upd' (Haskell "Heap.writeArray")
```
```   491
```
```   492 end
```