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
haftmann@31870
     1
(*  Title:      HOL/Imperative_HOL/Array.thy
haftmann@26170
     2
    Author:     John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen
haftmann@26170
     3
*)
haftmann@26170
     4
haftmann@26170
     5
header {* Monadic arrays *}
haftmann@26170
     6
haftmann@26170
     7
theory Array
haftmann@31203
     8
imports Heap_Monad
haftmann@26170
     9
begin
haftmann@26170
    10
haftmann@37752
    11
subsection {* Primitives *}
haftmann@37719
    12
haftmann@37804
    13
definition present :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> bool" where
haftmann@37804
    14
  "present h a \<longleftrightarrow> addr_of_array a < lim h"
haftmann@37719
    15
haftmann@37805
    16
definition (*FIXME get *)
haftmann@37805
    17
  get_array :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a list" where
haftmann@37805
    18
  "get_array h a = map from_nat (arrays h (TYPEREP('a)) (addr_of_array a))"
haftmann@37719
    19
haftmann@37804
    20
definition set :: "'a\<Colon>heap array \<Rightarrow> 'a list \<Rightarrow> heap \<Rightarrow> heap" where
haftmann@37804
    21
  "set a x = arrays_update (\<lambda>h. h(TYPEREP('a) := ((h(TYPEREP('a))) (addr_of_array a:=map to_nat x))))"
haftmann@37719
    22
haftmann@37804
    23
definition alloc :: "'a list \<Rightarrow> heap \<Rightarrow> 'a\<Colon>heap array \<times> heap" where
haftmann@37804
    24
  "alloc xs h = (let
haftmann@37719
    25
     l = lim h;
haftmann@37719
    26
     r = Array l;
haftmann@37804
    27
     h'' = set r xs (h\<lparr>lim := l + 1\<rparr>)
haftmann@37719
    28
   in (r, h''))"
haftmann@37719
    29
haftmann@37802
    30
definition length :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> nat" where
haftmann@37805
    31
  "length h a = List.length (get_array h a)"
haftmann@37719
    32
  
haftmann@37798
    33
definition update :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> heap \<Rightarrow> heap" where
haftmann@37805
    34
  "update a i x h = set a ((get_array h a)[i:=x]) h"
haftmann@37719
    35
haftmann@37804
    36
definition noteq :: "'a\<Colon>heap array \<Rightarrow> 'b\<Colon>heap array \<Rightarrow> bool" (infix "=!!=" 70) where
haftmann@37752
    37
  "r =!!= s \<longleftrightarrow> TYPEREP('a) \<noteq> TYPEREP('b) \<or> addr_of_array r \<noteq> addr_of_array s"
haftmann@37752
    38
haftmann@37752
    39
haftmann@37752
    40
subsection {* Monad operations *}
haftmann@37752
    41
haftmann@37752
    42
definition new :: "nat \<Rightarrow> 'a\<Colon>heap \<Rightarrow> 'a array Heap" where
haftmann@37804
    43
  [code del]: "new n x = Heap_Monad.heap (alloc (replicate n x))"
haftmann@37752
    44
haftmann@37752
    45
definition of_list :: "'a\<Colon>heap list \<Rightarrow> 'a array Heap" where
haftmann@37804
    46
  [code del]: "of_list xs = Heap_Monad.heap (alloc xs)"
haftmann@37752
    47
haftmann@37752
    48
definition make :: "nat \<Rightarrow> (nat \<Rightarrow> 'a\<Colon>heap) \<Rightarrow> 'a array Heap" where
haftmann@37804
    49
  [code del]: "make n f = Heap_Monad.heap (alloc (map f [0 ..< n]))"
haftmann@37752
    50
haftmann@37752
    51
definition len :: "'a\<Colon>heap array \<Rightarrow> nat Heap" where
haftmann@37802
    52
  [code del]: "len a = Heap_Monad.tap (\<lambda>h. length h a)"
haftmann@37752
    53
haftmann@37752
    54
definition nth :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a Heap" where
haftmann@37802
    55
  [code del]: "nth a i = Heap_Monad.guard (\<lambda>h. i < length h a)
haftmann@37805
    56
    (\<lambda>h. (get_array h a ! i, h))"
haftmann@37752
    57
haftmann@37752
    58
definition upd :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a\<Colon>heap array Heap" where
haftmann@37802
    59
  [code del]: "upd i x a = Heap_Monad.guard (\<lambda>h. i < length h a)
haftmann@37798
    60
    (\<lambda>h. (a, update a i x h))"
haftmann@37752
    61
haftmann@37752
    62
definition map_entry :: "nat \<Rightarrow> ('a\<Colon>heap \<Rightarrow> 'a) \<Rightarrow> 'a array \<Rightarrow> 'a array Heap" where
haftmann@37802
    63
  [code del]: "map_entry i f a = Heap_Monad.guard (\<lambda>h. i < length h a)
haftmann@37805
    64
    (\<lambda>h. (a, update a i (f (get_array h a ! i)) h))"
haftmann@37752
    65
haftmann@37752
    66
definition swap :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a Heap" where
haftmann@37802
    67
  [code del]: "swap i x a = Heap_Monad.guard (\<lambda>h. i < length h a)
haftmann@37805
    68
    (\<lambda>h. (get_array h a ! i, update a i x h))"
haftmann@37752
    69
haftmann@37752
    70
definition freeze :: "'a\<Colon>heap array \<Rightarrow> 'a list Heap" where
haftmann@37805
    71
  [code del]: "freeze a = Heap_Monad.tap (\<lambda>h. get_array h a)"
haftmann@37752
    72
haftmann@37752
    73
haftmann@37752
    74
subsection {* Properties *}
haftmann@37719
    75
haftmann@37719
    76
text {* FIXME: Does there exist a "canonical" array axiomatisation in
haftmann@37719
    77
the literature?  *}
haftmann@37719
    78
haftmann@37758
    79
text {* Primitives *}
haftmann@37758
    80
haftmann@37719
    81
lemma noteq_arrs_sym: "a =!!= b \<Longrightarrow> b =!!= a"
haftmann@37719
    82
  and unequal_arrs [simp]: "a \<noteq> a' \<longleftrightarrow> a =!!= a'"
haftmann@37804
    83
  unfolding noteq_def by auto
haftmann@37719
    84
haftmann@37719
    85
lemma noteq_arrs_irrefl: "r =!!= r \<Longrightarrow> False"
haftmann@37804
    86
  unfolding noteq_def by auto
haftmann@37719
    87
haftmann@37804
    88
lemma present_new_arr: "present h a \<Longrightarrow> a =!!= fst (alloc xs h)"
haftmann@37804
    89
  by (simp add: present_def noteq_def alloc_def Let_def)
haftmann@37719
    90
haftmann@37805
    91
lemma array_get_set_eq [simp]: "get_array (set r x h) r = x"
haftmann@37804
    92
  by (simp add: get_array_def set_def o_def)
haftmann@37719
    93
haftmann@37805
    94
lemma array_get_set_neq [simp]: "r =!!= s \<Longrightarrow> get_array (set s x h) r = get_array h r"
haftmann@37804
    95
  by (simp add: noteq_def get_array_def set_def)
haftmann@37719
    96
haftmann@37719
    97
lemma set_array_same [simp]:
haftmann@37804
    98
  "set r x (set r y h) = set r x h"
haftmann@37804
    99
  by (simp add: set_def)
haftmann@37719
   100
haftmann@37719
   101
lemma array_set_set_swap:
haftmann@37804
   102
  "r =!!= r' \<Longrightarrow> set r x (set r' x' h) = set r' x' (set r x h)"
haftmann@37804
   103
  by (simp add: Let_def expand_fun_eq noteq_def set_def)
haftmann@37719
   104
haftmann@37798
   105
lemma get_array_update_eq [simp]:
haftmann@37805
   106
  "get_array (update a i v h) a = (get_array h a) [i := v]"
haftmann@37798
   107
  by (simp add: update_def)
haftmann@37719
   108
haftmann@37798
   109
lemma nth_update_array_neq_array [simp]:
haftmann@37805
   110
  "a =!!= b \<Longrightarrow> get_array (update b j v h) a ! i = get_array h a ! i"
haftmann@37804
   111
  by (simp add: update_def noteq_def)
haftmann@37719
   112
haftmann@37798
   113
lemma get_arry_array_update_elem_neqIndex [simp]:
haftmann@37805
   114
  "i \<noteq> j \<Longrightarrow> get_array (update a j v h) a ! i = get_array h a ! i"
haftmann@37719
   115
  by simp
haftmann@37719
   116
haftmann@37798
   117
lemma length_update [simp]: 
haftmann@37802
   118
  "length (update b i v h) = length h"
haftmann@37804
   119
  by (simp add: update_def length_def set_def get_array_def expand_fun_eq)
haftmann@37719
   120
haftmann@37798
   121
lemma update_swap_neqArray:
haftmann@37719
   122
  "a =!!= a' \<Longrightarrow> 
haftmann@37798
   123
  update a i v (update a' i' v' h) 
haftmann@37798
   124
  = update a' i' v' (update a i v h)"
haftmann@37798
   125
apply (unfold update_def)
haftmann@37719
   126
apply simp
haftmann@37719
   127
apply (subst array_set_set_swap, assumption)
haftmann@37719
   128
apply (subst array_get_set_neq)
haftmann@37719
   129
apply (erule noteq_arrs_sym)
haftmann@37719
   130
apply (simp)
haftmann@37719
   131
done
haftmann@37719
   132
haftmann@37798
   133
lemma update_swap_neqIndex:
haftmann@37798
   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)"
haftmann@37798
   135
  by (auto simp add: update_def array_set_set_swap list_update_swap)
haftmann@37719
   136
haftmann@37719
   137
lemma get_array_init_array_list:
haftmann@37805
   138
  "get_array (snd (alloc ls' h)) (fst (alloc ls h)) = ls'"
haftmann@37804
   139
  by (simp add: Let_def split_def alloc_def)
haftmann@37719
   140
haftmann@37719
   141
lemma set_array:
haftmann@37804
   142
  "set (fst (alloc ls h))
haftmann@37804
   143
     new_ls (snd (alloc ls h))
haftmann@37804
   144
       = snd (alloc new_ls h)"
haftmann@37804
   145
  by (simp add: Let_def split_def alloc_def)
haftmann@37719
   146
haftmann@37798
   147
lemma array_present_update [simp]: 
haftmann@37804
   148
  "present (update b i v h) = present h"
haftmann@37804
   149
  by (simp add: update_def present_def set_def get_array_def expand_fun_eq)
haftmann@37719
   150
haftmann@37771
   151
lemma array_present_array [simp]:
haftmann@37804
   152
  "present (snd (alloc xs h)) (fst (alloc xs h))"
haftmann@37804
   153
  by (simp add: present_def alloc_def set_def Let_def)
haftmann@37771
   154
haftmann@37771
   155
lemma not_array_present_array [simp]:
haftmann@37804
   156
  "\<not> present h (fst (alloc xs h))"
haftmann@37804
   157
  by (simp add: present_def alloc_def Let_def)
haftmann@37771
   158
haftmann@37758
   159
haftmann@37758
   160
text {* Monad operations *}
haftmann@37758
   161
haftmann@37787
   162
lemma execute_new [execute_simps]:
haftmann@37804
   163
  "execute (new n x) h = Some (alloc (replicate n x) h)"
haftmann@37787
   164
  by (simp add: new_def execute_simps)
haftmann@37758
   165
haftmann@37787
   166
lemma success_newI [success_intros]:
haftmann@37758
   167
  "success (new n x) h"
haftmann@37787
   168
  by (auto intro: success_intros simp add: new_def)
haftmann@26170
   169
haftmann@37771
   170
lemma crel_newI [crel_intros]:
haftmann@37804
   171
  assumes "(a, h') = alloc (replicate n x) h"
haftmann@37771
   172
  shows "crel (new n x) h h' a"
haftmann@37787
   173
  by (rule crelI) (simp add: assms execute_simps)
haftmann@37771
   174
haftmann@37771
   175
lemma crel_newE [crel_elims]:
haftmann@37771
   176
  assumes "crel (new n x) h h' r"
haftmann@37804
   177
  obtains "r = fst (alloc (replicate n x) h)" "h' = snd (alloc (replicate n x) h)" 
haftmann@37805
   178
    "get_array h' r = replicate n x" "present h' r" "\<not> present h r"
haftmann@37787
   179
  using assms by (rule crelE) (simp add: get_array_init_array_list execute_simps)
haftmann@37771
   180
haftmann@37787
   181
lemma execute_of_list [execute_simps]:
haftmann@37804
   182
  "execute (of_list xs) h = Some (alloc xs h)"
haftmann@37787
   183
  by (simp add: of_list_def execute_simps)
haftmann@37758
   184
haftmann@37787
   185
lemma success_of_listI [success_intros]:
haftmann@37758
   186
  "success (of_list xs) h"
haftmann@37787
   187
  by (auto intro: success_intros simp add: of_list_def)
haftmann@26170
   188
haftmann@37771
   189
lemma crel_of_listI [crel_intros]:
haftmann@37804
   190
  assumes "(a, h') = alloc xs h"
haftmann@37771
   191
  shows "crel (of_list xs) h h' a"
haftmann@37787
   192
  by (rule crelI) (simp add: assms execute_simps)
haftmann@37771
   193
haftmann@37771
   194
lemma crel_of_listE [crel_elims]:
haftmann@37771
   195
  assumes "crel (of_list xs) h h' r"
haftmann@37804
   196
  obtains "r = fst (alloc xs h)" "h' = snd (alloc xs h)" 
haftmann@37805
   197
    "get_array h' r = xs" "present h' r" "\<not> present h r"
haftmann@37787
   198
  using assms by (rule crelE) (simp add: get_array_init_array_list execute_simps)
haftmann@37771
   199
haftmann@37787
   200
lemma execute_make [execute_simps]:
haftmann@37804
   201
  "execute (make n f) h = Some (alloc (map f [0 ..< n]) h)"
haftmann@37787
   202
  by (simp add: make_def execute_simps)
haftmann@26170
   203
haftmann@37787
   204
lemma success_makeI [success_intros]:
haftmann@37758
   205
  "success (make n f) h"
haftmann@37787
   206
  by (auto intro: success_intros simp add: make_def)
haftmann@37758
   207
haftmann@37771
   208
lemma crel_makeI [crel_intros]:
haftmann@37804
   209
  assumes "(a, h') = alloc (map f [0 ..< n]) h"
haftmann@37771
   210
  shows "crel (make n f) h h' a"
haftmann@37787
   211
  by (rule crelI) (simp add: assms execute_simps)
haftmann@37771
   212
haftmann@37771
   213
lemma crel_makeE [crel_elims]:
haftmann@37771
   214
  assumes "crel (make n f) h h' r"
haftmann@37804
   215
  obtains "r = fst (alloc (map f [0 ..< n]) h)" "h' = snd (alloc (map f [0 ..< n]) h)" 
haftmann@37805
   216
    "get_array h' r = map f [0 ..< n]" "present h' r" "\<not> present h r"
haftmann@37787
   217
  using assms by (rule crelE) (simp add: get_array_init_array_list execute_simps)
haftmann@37771
   218
haftmann@37787
   219
lemma execute_len [execute_simps]:
haftmann@37802
   220
  "execute (len a) h = Some (length h a, h)"
haftmann@37787
   221
  by (simp add: len_def execute_simps)
haftmann@37758
   222
haftmann@37787
   223
lemma success_lenI [success_intros]:
haftmann@37758
   224
  "success (len a) h"
haftmann@37787
   225
  by (auto intro: success_intros simp add: len_def)
haftmann@37752
   226
haftmann@37771
   227
lemma crel_lengthI [crel_intros]:
haftmann@37802
   228
  assumes "h' = h" "r = length h a"
haftmann@37771
   229
  shows "crel (len a) h h' r"
haftmann@37787
   230
  by (rule crelI) (simp add: assms execute_simps)
haftmann@37771
   231
haftmann@37771
   232
lemma crel_lengthE [crel_elims]:
haftmann@37771
   233
  assumes "crel (len a) h h' r"
haftmann@37802
   234
  obtains "r = length h' a" "h' = h" 
haftmann@37787
   235
  using assms by (rule crelE) (simp add: execute_simps)
haftmann@37771
   236
haftmann@37758
   237
lemma execute_nth [execute_simps]:
haftmann@37802
   238
  "i < length h a \<Longrightarrow>
haftmann@37805
   239
    execute (nth a i) h = Some (get_array h a ! i, h)"
haftmann@37802
   240
  "i \<ge> length h a \<Longrightarrow> execute (nth a i) h = None"
haftmann@37758
   241
  by (simp_all add: nth_def execute_simps)
haftmann@37758
   242
haftmann@37758
   243
lemma success_nthI [success_intros]:
haftmann@37802
   244
  "i < length h a \<Longrightarrow> success (nth a i) h"
haftmann@37758
   245
  by (auto intro: success_intros simp add: nth_def)
haftmann@26170
   246
haftmann@37771
   247
lemma crel_nthI [crel_intros]:
haftmann@37805
   248
  assumes "i < length h a" "h' = h" "r = get_array h a ! i"
haftmann@37771
   249
  shows "crel (nth a i) h h' r"
haftmann@37771
   250
  by (rule crelI) (insert assms, simp add: execute_simps)
haftmann@37771
   251
haftmann@37771
   252
lemma crel_nthE [crel_elims]:
haftmann@37771
   253
  assumes "crel (nth a i) h h' r"
haftmann@37805
   254
  obtains "i < length h a" "r = get_array h a ! i" "h' = h"
haftmann@37771
   255
  using assms by (rule crelE)
haftmann@37802
   256
    (erule successE, cases "i < length h a", simp_all add: execute_simps)
haftmann@37771
   257
haftmann@37758
   258
lemma execute_upd [execute_simps]:
haftmann@37802
   259
  "i < length h a \<Longrightarrow>
haftmann@37798
   260
    execute (upd i x a) h = Some (a, update a i x h)"
haftmann@37802
   261
  "i \<ge> length h a \<Longrightarrow> execute (upd i x a) h = None"
haftmann@37758
   262
  by (simp_all add: upd_def execute_simps)
haftmann@26170
   263
haftmann@37758
   264
lemma success_updI [success_intros]:
haftmann@37802
   265
  "i < length h a \<Longrightarrow> success (upd i x a) h"
haftmann@37758
   266
  by (auto intro: success_intros simp add: upd_def)
haftmann@37758
   267
haftmann@37771
   268
lemma crel_updI [crel_intros]:
haftmann@37802
   269
  assumes "i < length h a" "h' = update a i v h"
haftmann@37771
   270
  shows "crel (upd i v a) h h' a"
haftmann@37771
   271
  by (rule crelI) (insert assms, simp add: execute_simps)
haftmann@37771
   272
haftmann@37771
   273
lemma crel_updE [crel_elims]:
haftmann@37771
   274
  assumes "crel (upd i v a) h h' r"
haftmann@37802
   275
  obtains "r = a" "h' = update a i v h" "i < length h a"
haftmann@37771
   276
  using assms by (rule crelE)
haftmann@37802
   277
    (erule successE, cases "i < length h a", simp_all add: execute_simps)
haftmann@37771
   278
haftmann@37758
   279
lemma execute_map_entry [execute_simps]:
haftmann@37802
   280
  "i < length h a \<Longrightarrow>
haftmann@37758
   281
   execute (map_entry i f a) h =
haftmann@37805
   282
      Some (a, update a i (f (get_array h a ! i)) h)"
haftmann@37802
   283
  "i \<ge> length h a \<Longrightarrow> execute (map_entry i f a) h = None"
haftmann@37758
   284
  by (simp_all add: map_entry_def execute_simps)
haftmann@37752
   285
haftmann@37758
   286
lemma success_map_entryI [success_intros]:
haftmann@37802
   287
  "i < length h a \<Longrightarrow> success (map_entry i f a) h"
haftmann@37758
   288
  by (auto intro: success_intros simp add: map_entry_def)
haftmann@37758
   289
haftmann@37771
   290
lemma crel_map_entryI [crel_intros]:
haftmann@37805
   291
  assumes "i < length h a" "h' = update a i (f (get_array h a ! i)) h" "r = a"
haftmann@37771
   292
  shows "crel (map_entry i f a) h h' r"
haftmann@37771
   293
  by (rule crelI) (insert assms, simp add: execute_simps)
haftmann@37771
   294
haftmann@37771
   295
lemma crel_map_entryE [crel_elims]:
haftmann@37771
   296
  assumes "crel (map_entry i f a) h h' r"
haftmann@37805
   297
  obtains "r = a" "h' = update a i (f (get_array h a ! i)) h" "i < length h a"
haftmann@37771
   298
  using assms by (rule crelE)
haftmann@37802
   299
    (erule successE, cases "i < length h a", simp_all add: execute_simps)
haftmann@37771
   300
haftmann@37758
   301
lemma execute_swap [execute_simps]:
haftmann@37802
   302
  "i < length h a \<Longrightarrow>
haftmann@37758
   303
   execute (swap i x a) h =
haftmann@37805
   304
      Some (get_array h a ! i, update a i x h)"
haftmann@37802
   305
  "i \<ge> length h a \<Longrightarrow> execute (swap i x a) h = None"
haftmann@37758
   306
  by (simp_all add: swap_def execute_simps)
haftmann@37758
   307
haftmann@37758
   308
lemma success_swapI [success_intros]:
haftmann@37802
   309
  "i < length h a \<Longrightarrow> success (swap i x a) h"
haftmann@37758
   310
  by (auto intro: success_intros simp add: swap_def)
haftmann@37752
   311
haftmann@37771
   312
lemma crel_swapI [crel_intros]:
haftmann@37805
   313
  assumes "i < length h a" "h' = update a i x h" "r = get_array h a ! i"
haftmann@37771
   314
  shows "crel (swap i x a) h h' r"
haftmann@37771
   315
  by (rule crelI) (insert assms, simp add: execute_simps)
haftmann@37771
   316
haftmann@37771
   317
lemma crel_swapE [crel_elims]:
haftmann@37771
   318
  assumes "crel (swap i x a) h h' r"
haftmann@37805
   319
  obtains "r = get_array h a ! i" "h' = update a i x h" "i < length h a"
haftmann@37771
   320
  using assms by (rule crelE)
haftmann@37802
   321
    (erule successE, cases "i < length h a", simp_all add: execute_simps)
haftmann@37771
   322
haftmann@37787
   323
lemma execute_freeze [execute_simps]:
haftmann@37805
   324
  "execute (freeze a) h = Some (get_array h a, h)"
haftmann@37787
   325
  by (simp add: freeze_def execute_simps)
haftmann@37758
   326
haftmann@37787
   327
lemma success_freezeI [success_intros]:
haftmann@37758
   328
  "success (freeze a) h"
haftmann@37787
   329
  by (auto intro: success_intros simp add: freeze_def)
haftmann@26170
   330
haftmann@37771
   331
lemma crel_freezeI [crel_intros]:
haftmann@37805
   332
  assumes "h' = h" "r = get_array h a"
haftmann@37771
   333
  shows "crel (freeze a) h h' r"
haftmann@37771
   334
  by (rule crelI) (insert assms, simp add: execute_simps)
haftmann@37771
   335
haftmann@37771
   336
lemma crel_freezeE [crel_elims]:
haftmann@37771
   337
  assumes "crel (freeze a) h h' r"
haftmann@37805
   338
  obtains "h' = h" "r = get_array h a"
haftmann@37787
   339
  using assms by (rule crelE) (simp add: execute_simps)
haftmann@37771
   340
haftmann@26170
   341
lemma upd_return:
haftmann@26170
   342
  "upd i x a \<guillemotright> return a = upd i x a"
haftmann@37787
   343
  by (rule Heap_eqI) (simp add: bind_def guard_def upd_def execute_simps)
haftmann@26170
   344
haftmann@37752
   345
lemma array_make:
haftmann@37752
   346
  "new n x = make n (\<lambda>_. x)"
haftmann@37787
   347
  by (rule Heap_eqI) (simp add: map_replicate_trivial execute_simps)
haftmann@26170
   348
haftmann@37752
   349
lemma array_of_list_make:
haftmann@37752
   350
  "of_list xs = make (List.length xs) (\<lambda>n. xs ! n)"
haftmann@37787
   351
  by (rule Heap_eqI) (simp add: map_nth execute_simps)
haftmann@26170
   352
haftmann@37804
   353
hide_const (open) present (*get*) set alloc length update noteq new of_list make len nth upd map_entry swap freeze
haftmann@26170
   354
haftmann@26182
   355
haftmann@26182
   356
subsection {* Code generator setup *}
haftmann@26182
   357
haftmann@26182
   358
subsubsection {* Logical intermediate layer *}
haftmann@26182
   359
haftmann@26182
   360
definition new' where
haftmann@31205
   361
  [code del]: "new' = Array.new o Code_Numeral.nat_of"
haftmann@37752
   362
haftmann@28562
   363
lemma [code]:
haftmann@37752
   364
  "Array.new = new' o Code_Numeral.of_nat"
haftmann@26182
   365
  by (simp add: new'_def o_def)
haftmann@26182
   366
haftmann@26182
   367
definition of_list' where
haftmann@31205
   368
  [code del]: "of_list' i xs = Array.of_list (take (Code_Numeral.nat_of i) xs)"
haftmann@37752
   369
haftmann@28562
   370
lemma [code]:
haftmann@37752
   371
  "Array.of_list xs = of_list' (Code_Numeral.of_nat (List.length xs)) xs"
haftmann@26182
   372
  by (simp add: of_list'_def)
haftmann@26182
   373
haftmann@26182
   374
definition make' where
haftmann@31205
   375
  [code del]: "make' i f = Array.make (Code_Numeral.nat_of i) (f o Code_Numeral.of_nat)"
haftmann@37752
   376
haftmann@28562
   377
lemma [code]:
haftmann@37752
   378
  "Array.make n f = make' (Code_Numeral.of_nat n) (f o Code_Numeral.nat_of)"
haftmann@26182
   379
  by (simp add: make'_def o_def)
haftmann@26182
   380
haftmann@37719
   381
definition len' where
haftmann@37719
   382
  [code del]: "len' a = Array.len a \<guillemotright>= (\<lambda>n. return (Code_Numeral.of_nat n))"
haftmann@37752
   383
haftmann@28562
   384
lemma [code]:
haftmann@37752
   385
  "Array.len a = len' a \<guillemotright>= (\<lambda>i. return (Code_Numeral.nat_of i))"
haftmann@37719
   386
  by (simp add: len'_def)
haftmann@26182
   387
haftmann@26182
   388
definition nth' where
haftmann@31205
   389
  [code del]: "nth' a = Array.nth a o Code_Numeral.nat_of"
haftmann@37752
   390
haftmann@28562
   391
lemma [code]:
haftmann@37752
   392
  "Array.nth a n = nth' a (Code_Numeral.of_nat n)"
haftmann@26182
   393
  by (simp add: nth'_def)
haftmann@26182
   394
haftmann@26182
   395
definition upd' where
haftmann@31205
   396
  [code del]: "upd' a i x = Array.upd (Code_Numeral.nat_of i) x a \<guillemotright> return ()"
haftmann@37752
   397
haftmann@28562
   398
lemma [code]:
haftmann@37752
   399
  "Array.upd i x a = upd' a (Code_Numeral.of_nat i) x \<guillemotright> return a"
haftmann@37709
   400
  by (simp add: upd'_def upd_return)
haftmann@26182
   401
haftmann@37752
   402
lemma [code]:
haftmann@37798
   403
  "Array.map_entry i f a = do {
haftmann@37798
   404
     x \<leftarrow> Array.nth a i;
haftmann@37798
   405
     Array.upd i (f x) a
krauss@37792
   406
   }"
haftmann@37758
   407
  by (rule Heap_eqI) (simp add: bind_def guard_def map_entry_def execute_simps)
haftmann@26182
   408
haftmann@37752
   409
lemma [code]:
haftmann@37798
   410
  "Array.swap i x a = do {
haftmann@37798
   411
     y \<leftarrow> Array.nth a i;
haftmann@37798
   412
     Array.upd i x a;
haftmann@37752
   413
     return y
krauss@37792
   414
   }"
haftmann@37758
   415
  by (rule Heap_eqI) (simp add: bind_def guard_def swap_def execute_simps)
haftmann@37752
   416
haftmann@37752
   417
lemma [code]:
haftmann@37798
   418
  "Array.freeze a = do {
haftmann@37798
   419
     n \<leftarrow> Array.len a;
haftmann@37798
   420
     Heap_Monad.fold_map (\<lambda>i. Array.nth a i) [0..<n]
krauss@37792
   421
   }"
haftmann@37752
   422
proof (rule Heap_eqI)
haftmann@37752
   423
  fix h
haftmann@37752
   424
  have *: "List.map
haftmann@37804
   425
     (\<lambda>x. fst (the (if x < Array.length h a
haftmann@37805
   426
                    then Some (get_array h a ! x, h) else None)))
haftmann@37804
   427
     [0..<Array.length h a] =
haftmann@37805
   428
       List.map (List.nth (get_array h a)) [0..<Array.length h a]"
haftmann@37752
   429
    by simp
haftmann@37804
   430
  have "execute (Heap_Monad.fold_map (Array.nth a) [0..<Array.length h a]) h =
haftmann@37805
   431
    Some (get_array h a, h)"
haftmann@37756
   432
    apply (subst execute_fold_map_unchanged_heap)
haftmann@37752
   433
    apply (simp_all add: nth_def guard_def *)
haftmann@37752
   434
    apply (simp add: length_def map_nth)
haftmann@37752
   435
    done
krauss@37792
   436
  then have "execute (do {
haftmann@37798
   437
      n \<leftarrow> Array.len a;
haftmann@37756
   438
      Heap_Monad.fold_map (Array.nth a) [0..<n]
haftmann@37805
   439
    }) h = Some (get_array h a, h)"
haftmann@37787
   440
    by (auto intro: execute_bind_eq_SomeI simp add: execute_simps)
haftmann@37798
   441
  then show "execute (Array.freeze a) h = execute (do {
haftmann@37798
   442
      n \<leftarrow> Array.len a;
haftmann@37756
   443
      Heap_Monad.fold_map (Array.nth a) [0..<n]
krauss@37792
   444
    }) h" by (simp add: execute_simps)
haftmann@37752
   445
qed
haftmann@37752
   446
haftmann@37752
   447
hide_const (open) new' of_list' make' len' nth' upd'
haftmann@37752
   448
haftmann@37752
   449
haftmann@37752
   450
text {* SML *}
haftmann@26182
   451
haftmann@26182
   452
code_type array (SML "_/ array")
haftmann@26182
   453
code_const Array (SML "raise/ (Fail/ \"bare Array\")")
haftmann@26752
   454
code_const Array.new' (SML "(fn/ ()/ =>/ Array.array/ ((_),/ (_)))")
haftmann@35846
   455
code_const Array.of_list' (SML "(fn/ ()/ =>/ Array.fromList/ _)")
haftmann@26752
   456
code_const Array.make' (SML "(fn/ ()/ =>/ Array.tabulate/ ((_),/ (_)))")
haftmann@37719
   457
code_const Array.len' (SML "(fn/ ()/ =>/ Array.length/ _)")
haftmann@26752
   458
code_const Array.nth' (SML "(fn/ ()/ =>/ Array.sub/ ((_),/ (_)))")
haftmann@26752
   459
code_const Array.upd' (SML "(fn/ ()/ =>/ Array.update/ ((_),/ (_),/ (_)))")
haftmann@26182
   460
haftmann@26182
   461
code_reserved SML Array
haftmann@26182
   462
haftmann@26182
   463
haftmann@37752
   464
text {* OCaml *}
haftmann@26182
   465
haftmann@26182
   466
code_type array (OCaml "_/ array")
haftmann@26182
   467
code_const Array (OCaml "failwith/ \"bare Array\"")
haftmann@32580
   468
code_const Array.new' (OCaml "(fun/ ()/ ->/ Array.make/ (Big'_int.int'_of'_big'_int/ _)/ _)")
haftmann@35846
   469
code_const Array.of_list' (OCaml "(fun/ ()/ ->/ Array.of'_list/ _)")
haftmann@37719
   470
code_const Array.len' (OCaml "(fun/ ()/ ->/ Big'_int.big'_int'_of'_int/ (Array.length/ _))")
haftmann@32580
   471
code_const Array.nth' (OCaml "(fun/ ()/ ->/ Array.get/ _/ (Big'_int.int'_of'_big'_int/ _))")
haftmann@32580
   472
code_const Array.upd' (OCaml "(fun/ ()/ ->/ Array.set/ _/ (Big'_int.int'_of'_big'_int/ _)/ _)")
haftmann@26182
   473
haftmann@26182
   474
code_reserved OCaml Array
haftmann@26182
   475
haftmann@26182
   476
haftmann@37752
   477
text {* Haskell *}
haftmann@26182
   478
haftmann@29793
   479
code_type array (Haskell "Heap.STArray/ Heap.RealWorld/ _")
haftmann@26182
   480
code_const Array (Haskell "error/ \"bare Array\"")
haftmann@29793
   481
code_const Array.new' (Haskell "Heap.newArray/ (0,/ _)")
haftmann@29793
   482
code_const Array.of_list' (Haskell "Heap.newListArray/ (0,/ _)")
haftmann@37719
   483
code_const Array.len' (Haskell "Heap.lengthArray")
haftmann@29793
   484
code_const Array.nth' (Haskell "Heap.readArray")
haftmann@29793
   485
code_const Array.upd' (Haskell "Heap.writeArray")
haftmann@26182
   486
haftmann@26170
   487
end