author haftmann Mon Jul 26 11:09:44 2010 +0200 (2010-07-26) changeset 37964 0a1ae22df1f1 parent 37947 844977c7abeb child 38057 5ac79735cfef permissions -rw-r--r--
use Natural as index type for Haskell and Scala
```     1 (*  Title:      HOL/Imperative_HOL/Heap_Monad.thy
```
```     2     Author:     John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* A monad with a polymorphic heap and primitive reasoning infrastructure *}
```
```     6
```
```     7 theory Heap_Monad
```
```     8 imports Heap Monad_Syntax Code_Natural
```
```     9 begin
```
```    10
```
```    11 subsection {* The monad *}
```
```    12
```
```    13 subsubsection {* Monad construction *}
```
```    14
```
```    15 text {* Monadic heap actions either produce values
```
```    16   and transform the heap, or fail *}
```
```    17 datatype 'a Heap = Heap "heap \<Rightarrow> ('a \<times> heap) option"
```
```    18
```
```    19 primrec execute :: "'a Heap \<Rightarrow> heap \<Rightarrow> ('a \<times> heap) option" where
```
```    20   [code del]: "execute (Heap f) = f"
```
```    21
```
```    22 lemma Heap_cases [case_names succeed fail]:
```
```    23   fixes f and h
```
```    24   assumes succeed: "\<And>x h'. execute f h = Some (x, h') \<Longrightarrow> P"
```
```    25   assumes fail: "execute f h = None \<Longrightarrow> P"
```
```    26   shows P
```
```    27   using assms by (cases "execute f h") auto
```
```    28
```
```    29 lemma Heap_execute [simp]:
```
```    30   "Heap (execute f) = f" by (cases f) simp_all
```
```    31
```
```    32 lemma Heap_eqI:
```
```    33   "(\<And>h. execute f h = execute g h) \<Longrightarrow> f = g"
```
```    34     by (cases f, cases g) (auto simp: expand_fun_eq)
```
```    35
```
```    36 ML {* structure Execute_Simps = Named_Thms(
```
```    37   val name = "execute_simps"
```
```    38   val description = "simplification rules for execute"
```
```    39 ) *}
```
```    40
```
```    41 setup Execute_Simps.setup
```
```    42
```
```    43 lemma execute_Let [execute_simps]:
```
```    44   "execute (let x = t in f x) = (let x = t in execute (f x))"
```
```    45   by (simp add: Let_def)
```
```    46
```
```    47
```
```    48 subsubsection {* Specialised lifters *}
```
```    49
```
```    50 definition tap :: "(heap \<Rightarrow> 'a) \<Rightarrow> 'a Heap" where
```
```    51   [code del]: "tap f = Heap (\<lambda>h. Some (f h, h))"
```
```    52
```
```    53 lemma execute_tap [execute_simps]:
```
```    54   "execute (tap f) h = Some (f h, h)"
```
```    55   by (simp add: tap_def)
```
```    56
```
```    57 definition heap :: "(heap \<Rightarrow> 'a \<times> heap) \<Rightarrow> 'a Heap" where
```
```    58   [code del]: "heap f = Heap (Some \<circ> f)"
```
```    59
```
```    60 lemma execute_heap [execute_simps]:
```
```    61   "execute (heap f) = Some \<circ> f"
```
```    62   by (simp add: heap_def)
```
```    63
```
```    64 definition guard :: "(heap \<Rightarrow> bool) \<Rightarrow> (heap \<Rightarrow> 'a \<times> heap) \<Rightarrow> 'a Heap" where
```
```    65   [code del]: "guard P f = Heap (\<lambda>h. if P h then Some (f h) else None)"
```
```    66
```
```    67 lemma execute_guard [execute_simps]:
```
```    68   "\<not> P h \<Longrightarrow> execute (guard P f) h = None"
```
```    69   "P h \<Longrightarrow> execute (guard P f) h = Some (f h)"
```
```    70   by (simp_all add: guard_def)
```
```    71
```
```    72
```
```    73 subsubsection {* Predicate classifying successful computations *}
```
```    74
```
```    75 definition success :: "'a Heap \<Rightarrow> heap \<Rightarrow> bool" where
```
```    76   "success f h \<longleftrightarrow> execute f h \<noteq> None"
```
```    77
```
```    78 lemma successI:
```
```    79   "execute f h \<noteq> None \<Longrightarrow> success f h"
```
```    80   by (simp add: success_def)
```
```    81
```
```    82 lemma successE:
```
```    83   assumes "success f h"
```
```    84   obtains r h' where "r = fst (the (execute c h))"
```
```    85     and "h' = snd (the (execute c h))"
```
```    86     and "execute f h \<noteq> None"
```
```    87   using assms by (simp add: success_def)
```
```    88
```
```    89 ML {* structure Success_Intros = Named_Thms(
```
```    90   val name = "success_intros"
```
```    91   val description = "introduction rules for success"
```
```    92 ) *}
```
```    93
```
```    94 setup Success_Intros.setup
```
```    95
```
```    96 lemma success_tapI [success_intros]:
```
```    97   "success (tap f) h"
```
```    98   by (rule successI) (simp add: execute_simps)
```
```    99
```
```   100 lemma success_heapI [success_intros]:
```
```   101   "success (heap f) h"
```
```   102   by (rule successI) (simp add: execute_simps)
```
```   103
```
```   104 lemma success_guardI [success_intros]:
```
```   105   "P h \<Longrightarrow> success (guard P f) h"
```
```   106   by (rule successI) (simp add: execute_guard)
```
```   107
```
```   108 lemma success_LetI [success_intros]:
```
```   109   "x = t \<Longrightarrow> success (f x) h \<Longrightarrow> success (let x = t in f x) h"
```
```   110   by (simp add: Let_def)
```
```   111
```
```   112 lemma success_ifI:
```
```   113   "(c \<Longrightarrow> success t h) \<Longrightarrow> (\<not> c \<Longrightarrow> success e h) \<Longrightarrow>
```
```   114     success (if c then t else e) h"
```
```   115   by (simp add: success_def)
```
```   116
```
```   117
```
```   118 subsubsection {* Predicate for a simple relational calculus *}
```
```   119
```
```   120 text {*
```
```   121   The @{text crel} predicate states that when a computation @{text c}
```
```   122   runs with the heap @{text h} will result in return value @{text r}
```
```   123   and a heap @{text "h'"}, i.e.~no exception occurs.
```
```   124 *}
```
```   125
```
```   126 definition crel :: "'a Heap \<Rightarrow> heap \<Rightarrow> heap \<Rightarrow> 'a \<Rightarrow> bool" where
```
```   127   crel_def: "crel c h h' r \<longleftrightarrow> execute c h = Some (r, h')"
```
```   128
```
```   129 lemma crelI:
```
```   130   "execute c h = Some (r, h') \<Longrightarrow> crel c h h' r"
```
```   131   by (simp add: crel_def)
```
```   132
```
```   133 lemma crelE:
```
```   134   assumes "crel c h h' r"
```
```   135   obtains "r = fst (the (execute c h))"
```
```   136     and "h' = snd (the (execute c h))"
```
```   137     and "success c h"
```
```   138 proof (rule that)
```
```   139   from assms have *: "execute c h = Some (r, h')" by (simp add: crel_def)
```
```   140   then show "success c h" by (simp add: success_def)
```
```   141   from * have "fst (the (execute c h)) = r" and "snd (the (execute c h)) = h'"
```
```   142     by simp_all
```
```   143   then show "r = fst (the (execute c h))"
```
```   144     and "h' = snd (the (execute c h))" by simp_all
```
```   145 qed
```
```   146
```
```   147 lemma crel_success:
```
```   148   "crel c h h' r \<Longrightarrow> success c h"
```
```   149   by (simp add: crel_def success_def)
```
```   150
```
```   151 lemma success_crelE:
```
```   152   assumes "success c h"
```
```   153   obtains r h' where "crel c h h' r"
```
```   154   using assms by (auto simp add: crel_def success_def)
```
```   155
```
```   156 lemma crel_deterministic:
```
```   157   assumes "crel f h h' a"
```
```   158     and "crel f h h'' b"
```
```   159   shows "a = b" and "h' = h''"
```
```   160   using assms unfolding crel_def by auto
```
```   161
```
```   162 ML {* structure Crel_Intros = Named_Thms(
```
```   163   val name = "crel_intros"
```
```   164   val description = "introduction rules for crel"
```
```   165 ) *}
```
```   166
```
```   167 ML {* structure Crel_Elims = Named_Thms(
```
```   168   val name = "crel_elims"
```
```   169   val description = "elimination rules for crel"
```
```   170 ) *}
```
```   171
```
```   172 setup "Crel_Intros.setup #> Crel_Elims.setup"
```
```   173
```
```   174 lemma crel_LetI [crel_intros]:
```
```   175   assumes "x = t" "crel (f x) h h' r"
```
```   176   shows "crel (let x = t in f x) h h' r"
```
```   177   using assms by simp
```
```   178
```
```   179 lemma crel_LetE [crel_elims]:
```
```   180   assumes "crel (let x = t in f x) h h' r"
```
```   181   obtains "crel (f t) h h' r"
```
```   182   using assms by simp
```
```   183
```
```   184 lemma crel_ifI:
```
```   185   assumes "c \<Longrightarrow> crel t h h' r"
```
```   186     and "\<not> c \<Longrightarrow> crel e h h' r"
```
```   187   shows "crel (if c then t else e) h h' r"
```
```   188   by (cases c) (simp_all add: assms)
```
```   189
```
```   190 lemma crel_ifE:
```
```   191   assumes "crel (if c then t else e) h h' r"
```
```   192   obtains "c" "crel t h h' r"
```
```   193     | "\<not> c" "crel e h h' r"
```
```   194   using assms by (cases c) simp_all
```
```   195
```
```   196 lemma crel_tapI [crel_intros]:
```
```   197   assumes "h' = h" "r = f h"
```
```   198   shows "crel (tap f) h h' r"
```
```   199   by (rule crelI) (simp add: assms execute_simps)
```
```   200
```
```   201 lemma crel_tapE [crel_elims]:
```
```   202   assumes "crel (tap f) h h' r"
```
```   203   obtains "h' = h" and "r = f h"
```
```   204   using assms by (rule crelE) (auto simp add: execute_simps)
```
```   205
```
```   206 lemma crel_heapI [crel_intros]:
```
```   207   assumes "h' = snd (f h)" "r = fst (f h)"
```
```   208   shows "crel (heap f) h h' r"
```
```   209   by (rule crelI) (simp add: assms execute_simps)
```
```   210
```
```   211 lemma crel_heapE [crel_elims]:
```
```   212   assumes "crel (heap f) h h' r"
```
```   213   obtains "h' = snd (f h)" and "r = fst (f h)"
```
```   214   using assms by (rule crelE) (simp add: execute_simps)
```
```   215
```
```   216 lemma crel_guardI [crel_intros]:
```
```   217   assumes "P h" "h' = snd (f h)" "r = fst (f h)"
```
```   218   shows "crel (guard P f) h h' r"
```
```   219   by (rule crelI) (simp add: assms execute_simps)
```
```   220
```
```   221 lemma crel_guardE [crel_elims]:
```
```   222   assumes "crel (guard P f) h h' r"
```
```   223   obtains "h' = snd (f h)" "r = fst (f h)" "P h"
```
```   224   using assms by (rule crelE)
```
```   225     (auto simp add: execute_simps elim!: successE, cases "P h", auto simp add: execute_simps)
```
```   226
```
```   227
```
```   228 subsubsection {* Monad combinators *}
```
```   229
```
```   230 definition return :: "'a \<Rightarrow> 'a Heap" where
```
```   231   [code del]: "return x = heap (Pair x)"
```
```   232
```
```   233 lemma execute_return [execute_simps]:
```
```   234   "execute (return x) = Some \<circ> Pair x"
```
```   235   by (simp add: return_def execute_simps)
```
```   236
```
```   237 lemma success_returnI [success_intros]:
```
```   238   "success (return x) h"
```
```   239   by (rule successI) (simp add: execute_simps)
```
```   240
```
```   241 lemma crel_returnI [crel_intros]:
```
```   242   "h = h' \<Longrightarrow> crel (return x) h h' x"
```
```   243   by (rule crelI) (simp add: execute_simps)
```
```   244
```
```   245 lemma crel_returnE [crel_elims]:
```
```   246   assumes "crel (return x) h h' r"
```
```   247   obtains "r = x" "h' = h"
```
```   248   using assms by (rule crelE) (simp add: execute_simps)
```
```   249
```
```   250 definition raise :: "string \<Rightarrow> 'a Heap" where -- {* the string is just decoration *}
```
```   251   [code del]: "raise s = Heap (\<lambda>_. None)"
```
```   252
```
```   253 lemma execute_raise [execute_simps]:
```
```   254   "execute (raise s) = (\<lambda>_. None)"
```
```   255   by (simp add: raise_def)
```
```   256
```
```   257 lemma crel_raiseE [crel_elims]:
```
```   258   assumes "crel (raise x) h h' r"
```
```   259   obtains "False"
```
```   260   using assms by (rule crelE) (simp add: success_def execute_simps)
```
```   261
```
```   262 definition bind :: "'a Heap \<Rightarrow> ('a \<Rightarrow> 'b Heap) \<Rightarrow> 'b Heap" where
```
```   263   [code del]: "bind f g = Heap (\<lambda>h. case execute f h of
```
```   264                   Some (x, h') \<Rightarrow> execute (g x) h'
```
```   265                 | None \<Rightarrow> None)"
```
```   266
```
```   267 setup {*
```
```   268   Adhoc_Overloading.add_variant
```
```   269     @{const_name Monad_Syntax.bind} @{const_name Heap_Monad.bind}
```
```   270 *}
```
```   271
```
```   272 lemma execute_bind [execute_simps]:
```
```   273   "execute f h = Some (x, h') \<Longrightarrow> execute (f \<guillemotright>= g) h = execute (g x) h'"
```
```   274   "execute f h = None \<Longrightarrow> execute (f \<guillemotright>= g) h = None"
```
```   275   by (simp_all add: bind_def)
```
```   276
```
```   277 lemma execute_bind_success:
```
```   278   "success f h \<Longrightarrow> execute (f \<guillemotright>= g) h = execute (g (fst (the (execute f h)))) (snd (the (execute f h)))"
```
```   279   by (cases f h rule: Heap_cases) (auto elim!: successE simp add: bind_def)
```
```   280
```
```   281 lemma success_bind_executeI:
```
```   282   "execute f h = Some (x, h') \<Longrightarrow> success (g x) h' \<Longrightarrow> success (f \<guillemotright>= g) h"
```
```   283   by (auto intro!: successI elim!: successE simp add: bind_def)
```
```   284
```
```   285 lemma success_bind_crelI [success_intros]:
```
```   286   "crel f h h' x \<Longrightarrow> success (g x) h' \<Longrightarrow> success (f \<guillemotright>= g) h"
```
```   287   by (auto simp add: crel_def success_def bind_def)
```
```   288
```
```   289 lemma crel_bindI [crel_intros]:
```
```   290   assumes "crel f h h' r" "crel (g r) h' h'' r'"
```
```   291   shows "crel (f \<guillemotright>= g) h h'' r'"
```
```   292   using assms
```
```   293   apply (auto intro!: crelI elim!: crelE successE)
```
```   294   apply (subst execute_bind, simp_all)
```
```   295   done
```
```   296
```
```   297 lemma crel_bindE [crel_elims]:
```
```   298   assumes "crel (f \<guillemotright>= g) h h'' r'"
```
```   299   obtains h' r where "crel f h h' r" "crel (g r) h' h'' r'"
```
```   300   using assms by (auto simp add: crel_def bind_def split: option.split_asm)
```
```   301
```
```   302 lemma execute_bind_eq_SomeI:
```
```   303   assumes "execute f h = Some (x, h')"
```
```   304     and "execute (g x) h' = Some (y, h'')"
```
```   305   shows "execute (f \<guillemotright>= g) h = Some (y, h'')"
```
```   306   using assms by (simp add: bind_def)
```
```   307
```
```   308 lemma return_bind [simp]: "return x \<guillemotright>= f = f x"
```
```   309   by (rule Heap_eqI) (simp add: execute_bind execute_simps)
```
```   310
```
```   311 lemma bind_return [simp]: "f \<guillemotright>= return = f"
```
```   312   by (rule Heap_eqI) (simp add: bind_def execute_simps split: option.splits)
```
```   313
```
```   314 lemma bind_bind [simp]: "(f \<guillemotright>= g) \<guillemotright>= k = (f :: 'a Heap) \<guillemotright>= (\<lambda>x. g x \<guillemotright>= k)"
```
```   315   by (rule Heap_eqI) (simp add: bind_def execute_simps split: option.splits)
```
```   316
```
```   317 lemma raise_bind [simp]: "raise e \<guillemotright>= f = raise e"
```
```   318   by (rule Heap_eqI) (simp add: execute_simps)
```
```   319
```
```   320
```
```   321 subsection {* Generic combinators *}
```
```   322
```
```   323 subsubsection {* Assertions *}
```
```   324
```
```   325 definition assert :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a Heap" where
```
```   326   "assert P x = (if P x then return x else raise ''assert'')"
```
```   327
```
```   328 lemma execute_assert [execute_simps]:
```
```   329   "P x \<Longrightarrow> execute (assert P x) h = Some (x, h)"
```
```   330   "\<not> P x \<Longrightarrow> execute (assert P x) h = None"
```
```   331   by (simp_all add: assert_def execute_simps)
```
```   332
```
```   333 lemma success_assertI [success_intros]:
```
```   334   "P x \<Longrightarrow> success (assert P x) h"
```
```   335   by (rule successI) (simp add: execute_assert)
```
```   336
```
```   337 lemma crel_assertI [crel_intros]:
```
```   338   "P x \<Longrightarrow> h' = h \<Longrightarrow> r = x \<Longrightarrow> crel (assert P x) h h' r"
```
```   339   by (rule crelI) (simp add: execute_assert)
```
```   340
```
```   341 lemma crel_assertE [crel_elims]:
```
```   342   assumes "crel (assert P x) h h' r"
```
```   343   obtains "P x" "r = x" "h' = h"
```
```   344   using assms by (rule crelE) (cases "P x", simp_all add: execute_assert success_def)
```
```   345
```
```   346 lemma assert_cong [fundef_cong]:
```
```   347   assumes "P = P'"
```
```   348   assumes "\<And>x. P' x \<Longrightarrow> f x = f' x"
```
```   349   shows "(assert P x >>= f) = (assert P' x >>= f')"
```
```   350   by (rule Heap_eqI) (insert assms, simp add: assert_def)
```
```   351
```
```   352
```
```   353 subsubsection {* Plain lifting *}
```
```   354
```
```   355 definition lift :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b Heap" where
```
```   356   "lift f = return o f"
```
```   357
```
```   358 lemma lift_collapse [simp]:
```
```   359   "lift f x = return (f x)"
```
```   360   by (simp add: lift_def)
```
```   361
```
```   362 lemma bind_lift:
```
```   363   "(f \<guillemotright>= lift g) = (f \<guillemotright>= (\<lambda>x. return (g x)))"
```
```   364   by (simp add: lift_def comp_def)
```
```   365
```
```   366
```
```   367 subsubsection {* Iteration -- warning: this is rarely useful! *}
```
```   368
```
```   369 primrec fold_map :: "('a \<Rightarrow> 'b Heap) \<Rightarrow> 'a list \<Rightarrow> 'b list Heap" where
```
```   370   "fold_map f [] = return []"
```
```   371 | "fold_map f (x # xs) = do {
```
```   372      y \<leftarrow> f x;
```
```   373      ys \<leftarrow> fold_map f xs;
```
```   374      return (y # ys)
```
```   375    }"
```
```   376
```
```   377 lemma fold_map_append:
```
```   378   "fold_map f (xs @ ys) = fold_map f xs \<guillemotright>= (\<lambda>xs. fold_map f ys \<guillemotright>= (\<lambda>ys. return (xs @ ys)))"
```
```   379   by (induct xs) simp_all
```
```   380
```
```   381 lemma execute_fold_map_unchanged_heap [execute_simps]:
```
```   382   assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<exists>y. execute (f x) h = Some (y, h)"
```
```   383   shows "execute (fold_map f xs) h =
```
```   384     Some (List.map (\<lambda>x. fst (the (execute (f x) h))) xs, h)"
```
```   385 using assms proof (induct xs)
```
```   386   case Nil show ?case by (simp add: execute_simps)
```
```   387 next
```
```   388   case (Cons x xs)
```
```   389   from Cons.prems obtain y
```
```   390     where y: "execute (f x) h = Some (y, h)" by auto
```
```   391   moreover from Cons.prems Cons.hyps have "execute (fold_map f xs) h =
```
```   392     Some (map (\<lambda>x. fst (the (execute (f x) h))) xs, h)" by auto
```
```   393   ultimately show ?case by (simp, simp only: execute_bind(1), simp add: execute_simps)
```
```   394 qed
```
```   395
```
```   396 subsection {* Code generator setup *}
```
```   397
```
```   398 subsubsection {* Logical intermediate layer *}
```
```   399
```
```   400 primrec raise' :: "String.literal \<Rightarrow> 'a Heap" where
```
```   401   [code del, code_post]: "raise' (STR s) = raise s"
```
```   402
```
```   403 lemma raise_raise' [code_inline]:
```
```   404   "raise s = raise' (STR s)"
```
```   405   by simp
```
```   406
```
```   407 code_datatype raise' -- {* avoid @{const "Heap"} formally *}
```
```   408
```
```   409
```
```   410 subsubsection {* SML and OCaml *}
```
```   411
```
```   412 code_type Heap (SML "unit/ ->/ _")
```
```   413 code_const bind (SML "!(fn/ f'_/ =>/ fn/ ()/ =>/ f'_/ (_/ ())/ ())")
```
```   414 code_const return (SML "!(fn/ ()/ =>/ _)")
```
```   415 code_const Heap_Monad.raise' (SML "!(raise/ Fail/ _)")
```
```   416
```
```   417 code_type Heap (OCaml "unit/ ->/ _")
```
```   418 code_const bind (OCaml "!(fun/ f'_/ ()/ ->/ f'_/ (_/ ())/ ())")
```
```   419 code_const return (OCaml "!(fun/ ()/ ->/ _)")
```
```   420 code_const Heap_Monad.raise' (OCaml "failwith")
```
```   421
```
```   422
```
```   423 subsubsection {* Haskell *}
```
```   424
```
```   425 text {* Adaption layer *}
```
```   426
```
```   427 code_include Haskell "Heap"
```
```   428 {*import qualified Control.Monad;
```
```   429 import qualified Control.Monad.ST;
```
```   430 import qualified Data.STRef;
```
```   431 import qualified Data.Array.ST;
```
```   432
```
```   433 import Natural;
```
```   434
```
```   435 type RealWorld = Control.Monad.ST.RealWorld;
```
```   436 type ST s a = Control.Monad.ST.ST s a;
```
```   437 type STRef s a = Data.STRef.STRef s a;
```
```   438 type STArray s a = Data.Array.ST.STArray s Natural a;
```
```   439
```
```   440 newSTRef = Data.STRef.newSTRef;
```
```   441 readSTRef = Data.STRef.readSTRef;
```
```   442 writeSTRef = Data.STRef.writeSTRef;
```
```   443
```
```   444 newArray :: Natural -> a -> ST s (STArray s a);
```
```   445 newArray k = Data.Array.ST.newArray (0, k);
```
```   446
```
```   447 newListArray :: [a] -> ST s (STArray s a);
```
```   448 newListArray xs = Data.Array.ST.newListArray (0, (fromInteger . toInteger . length) xs) xs;
```
```   449
```
```   450 newFunArray :: Natural -> (Natural -> a) -> ST s (STArray s a);
```
```   451 newFunArray k f = Data.Array.ST.newListArray (0, k) (map f [0..k-1]);
```
```   452
```
```   453 lengthArray :: STArray s a -> ST s Natural;
```
```   454 lengthArray a = Control.Monad.liftM snd (Data.Array.ST.getBounds a);
```
```   455
```
```   456 readArray :: STArray s a -> Natural -> ST s a;
```
```   457 readArray = Data.Array.ST.readArray;
```
```   458
```
```   459 writeArray :: STArray s a -> Natural -> a -> ST s ();
```
```   460 writeArray = Data.Array.ST.writeArray;*}
```
```   461
```
```   462 code_reserved Haskell Heap
```
```   463
```
```   464 text {* Monad *}
```
```   465
```
```   466 code_type Heap (Haskell "Heap.ST/ Heap.RealWorld/ _")
```
```   467 code_monad bind Haskell
```
```   468 code_const return (Haskell "return")
```
```   469 code_const Heap_Monad.raise' (Haskell "error")
```
```   470
```
```   471
```
```   472 subsubsection {* Scala *}
```
```   473
```
```   474 code_include Scala "Heap"
```
```   475 {*import collection.mutable.ArraySeq
```
```   476 import Natural._
```
```   477
```
```   478 def bind[A, B](f: Unit => A, g: A => Unit => B): Unit => B = (_: Unit) => g (f ()) ()
```
```   479
```
```   480 class Ref[A](x: A) {
```
```   481   var value = x
```
```   482 }
```
```   483
```
```   484 object Ref {
```
```   485   def apply[A](x: A): Ref[A] = new Ref[A](x)
```
```   486   def lookup[A](r: Ref[A]): A = r.value
```
```   487   def update[A](r: Ref[A], x: A): Unit = { r.value = x }
```
```   488 }
```
```   489
```
```   490 object Array {
```
```   491   def alloc[A](n: Natural)(x: A): ArraySeq[A] = ArraySeq.fill(n.as_Int)(x)
```
```   492   def make[A](n: Natural)(f: Natural => A): ArraySeq[A] = ArraySeq.tabulate(n.as_Int)((k: Int) => f(Natural(k)))
```
```   493   def len[A](a: ArraySeq[A]): Natural = Natural(a.length)
```
```   494   def nth[A](a: ArraySeq[A], n: Natural): A = a(n.as_Int)
```
```   495   def upd[A](a: ArraySeq[A], n: Natural, x: A): Unit = a.update(n.as_Int, x)
```
```   496   def freeze[A](a: ArraySeq[A]): List[A] = a.toList
```
```   497 }*}
```
```   498
```
```   499 code_reserved Scala bind Ref Array
```
```   500
```
```   501 code_type Heap (Scala "Unit/ =>/ _")
```
```   502 code_const bind (Scala "bind")
```
```   503 code_const return (Scala "('_: Unit)/ =>/ _")
```
```   504 code_const Heap_Monad.raise' (Scala "!error((_))")
```
```   505
```
```   506
```
```   507 subsubsection {* Target variants with less units *}
```
```   508
```
```   509 setup {*
```
```   510
```
```   511 let
```
```   512
```
```   513 open Code_Thingol;
```
```   514
```
```   515 fun imp_program naming =
```
```   516
```
```   517   let
```
```   518     fun is_const c = case lookup_const naming c
```
```   519      of SOME c' => (fn c'' => c' = c'')
```
```   520       | NONE => K false;
```
```   521     val is_bind = is_const @{const_name bind};
```
```   522     val is_return = is_const @{const_name return};
```
```   523     val dummy_name = "";
```
```   524     val dummy_type = ITyVar dummy_name;
```
```   525     val dummy_case_term = IVar NONE;
```
```   526     (*assumption: dummy values are not relevant for serialization*)
```
```   527     val unitt = case lookup_const naming @{const_name Unity}
```
```   528      of SOME unit' => IConst (unit', (([], []), []))
```
```   529       | NONE => error ("Must include " ^ @{const_name Unity} ^ " in generated constants.");
```
```   530     fun dest_abs ((v, ty) `|=> t, _) = ((v, ty), t)
```
```   531       | dest_abs (t, ty) =
```
```   532           let
```
```   533             val vs = fold_varnames cons t [];
```
```   534             val v = Name.variant vs "x";
```
```   535             val ty' = (hd o fst o unfold_fun) ty;
```
```   536           in ((SOME v, ty'), t `\$ IVar (SOME v)) end;
```
```   537     fun force (t as IConst (c, _) `\$ t') = if is_return c
```
```   538           then t' else t `\$ unitt
```
```   539       | force t = t `\$ unitt;
```
```   540     fun tr_bind' [(t1, _), (t2, ty2)] =
```
```   541       let
```
```   542         val ((v, ty), t) = dest_abs (t2, ty2);
```
```   543       in ICase (((force t1, ty), [(IVar v, tr_bind'' t)]), dummy_case_term) end
```
```   544     and tr_bind'' t = case unfold_app t
```
```   545          of (IConst (c, (_, ty1 :: ty2 :: _)), [x1, x2]) => if is_bind c
```
```   546               then tr_bind' [(x1, ty1), (x2, ty2)]
```
```   547               else force t
```
```   548           | _ => force t;
```
```   549     fun imp_monad_bind'' ts = (SOME dummy_name, dummy_type) `|=> ICase (((IVar (SOME dummy_name), dummy_type),
```
```   550       [(unitt, tr_bind' ts)]), dummy_case_term)
```
```   551     and imp_monad_bind' (const as (c, (_, tys))) ts = if is_bind c then case (ts, tys)
```
```   552        of ([t1, t2], ty1 :: ty2 :: _) => imp_monad_bind'' [(t1, ty1), (t2, ty2)]
```
```   553         | ([t1, t2, t3], ty1 :: ty2 :: _) => imp_monad_bind'' [(t1, ty1), (t2, ty2)] `\$ t3
```
```   554         | (ts, _) => imp_monad_bind (eta_expand 2 (const, ts))
```
```   555       else IConst const `\$\$ map imp_monad_bind ts
```
```   556     and imp_monad_bind (IConst const) = imp_monad_bind' const []
```
```   557       | imp_monad_bind (t as IVar _) = t
```
```   558       | imp_monad_bind (t as _ `\$ _) = (case unfold_app t
```
```   559          of (IConst const, ts) => imp_monad_bind' const ts
```
```   560           | (t, ts) => imp_monad_bind t `\$\$ map imp_monad_bind ts)
```
```   561       | imp_monad_bind (v_ty `|=> t) = v_ty `|=> imp_monad_bind t
```
```   562       | imp_monad_bind (ICase (((t, ty), pats), t0)) = ICase
```
```   563           (((imp_monad_bind t, ty),
```
```   564             (map o pairself) imp_monad_bind pats),
```
```   565               imp_monad_bind t0);
```
```   566
```
```   567   in (Graph.map_nodes o map_terms_stmt) imp_monad_bind end;
```
```   568
```
```   569 in
```
```   570
```
```   571 Code_Target.extend_target ("SML_imp", ("SML", imp_program))
```
```   572 #> Code_Target.extend_target ("OCaml_imp", ("OCaml", imp_program))
```
```   573 #> Code_Target.extend_target ("Scala_imp", ("Scala", imp_program))
```
```   574
```
```   575 end
```
```   576
```
```   577 *}
```
```   578
```
```   579
```
```   580 hide_const (open) Heap heap guard raise' fold_map
```
```   581
```
```   582 end
```