src/HOL/Library/Array.thy
author haftmann
Mon Jul 07 08:47:17 2008 +0200 (2008-07-07)
changeset 27487 c8a6ce181805
parent 26752 6b276119139b
child 27596 bc47d30747e6
permissions -rw-r--r--
absolute imports of HOL/*.thy theories
     1 (*  Title:      HOL/Library/Array.thy
     2     ID:         $Id$
     3     Author:     John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen
     4 *)
     5 
     6 header {* Monadic arrays *}
     7 
     8 theory Array
     9 imports Heap_Monad Code_Index
    10 begin
    11 
    12 subsection {* Primitives *}
    13 
    14 definition
    15   new :: "nat \<Rightarrow> 'a\<Colon>heap \<Rightarrow> 'a array Heap" where
    16   [code del]: "new n x = Heap_Monad.heap (Heap.array n x)"
    17 
    18 definition
    19   of_list :: "'a\<Colon>heap list \<Rightarrow> 'a array Heap" where
    20   [code del]: "of_list xs = Heap_Monad.heap (Heap.array_of_list xs)"
    21 
    22 definition
    23   length :: "'a\<Colon>heap array \<Rightarrow> nat Heap" where
    24   [code del]: "length arr = Heap_Monad.heap (\<lambda>h. (Heap.length arr h, h))"
    25 
    26 definition
    27   nth :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a Heap"
    28 where
    29   [code del]: "nth a i = (do len \<leftarrow> length a;
    30                  (if i < len
    31                      then Heap_Monad.heap (\<lambda>h. (get_array a h ! i, h))
    32                      else raise (''array lookup: index out of range''))
    33               done)"
    34 
    35 -- {* FIXME adjustion for List theory *}
    36 no_syntax
    37   nth  :: "'a list \<Rightarrow> nat \<Rightarrow> 'a" (infixl "!" 100)
    38 
    39 abbreviation
    40   nth_list :: "'a list \<Rightarrow> nat \<Rightarrow> 'a" (infixl "!" 100)
    41 where
    42   "nth_list \<equiv> List.nth"
    43 
    44 definition
    45   upd :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a\<Colon>heap array Heap"
    46 where
    47   [code del]: "upd i x a = (do len \<leftarrow> length a;
    48                       (if i < len
    49                            then Heap_Monad.heap (\<lambda>h. (a, Heap.upd a i x h))
    50                            else raise (''array update: index out of range''))
    51                    done)" 
    52 
    53 lemma upd_return:
    54   "upd i x a \<guillemotright> return a = upd i x a"
    55 proof (rule Heap_eqI)
    56   fix h
    57   obtain len h' where "Heap_Monad.execute (Array.length a) h = (len, h')"
    58     by (cases "Heap_Monad.execute (Array.length a) h")
    59   then show "Heap_Monad.execute (upd i x a \<guillemotright> return a) h = Heap_Monad.execute (upd i x a) h"
    60     by (auto simp add: upd_def bindM_def run_drop split: sum.split)
    61 qed
    62 
    63 
    64 subsection {* Derivates *}
    65 
    66 definition
    67   map_entry :: "nat \<Rightarrow> ('a\<Colon>heap \<Rightarrow> 'a) \<Rightarrow> 'a array \<Rightarrow> 'a array Heap"
    68 where
    69   "map_entry i f a = (do
    70      x \<leftarrow> nth a i;
    71      upd i (f x) a
    72    done)"
    73 
    74 definition
    75   swap :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a Heap"
    76 where
    77   "swap i x a = (do
    78      y \<leftarrow> nth a i;
    79      upd i x a;
    80      return x
    81    done)"
    82 
    83 definition
    84   make :: "nat \<Rightarrow> (nat \<Rightarrow> 'a\<Colon>heap) \<Rightarrow> 'a array Heap"
    85 where
    86   "make n f = of_list (map f [0 ..< n])"
    87 
    88 definition
    89   freeze :: "'a\<Colon>heap array \<Rightarrow> 'a list Heap"
    90 where
    91   "freeze a = (do
    92      n \<leftarrow> length a;
    93      mapM (nth a) [0..<n]
    94    done)"
    95 
    96 definition
    97   map :: "('a\<Colon>heap \<Rightarrow> 'a) \<Rightarrow> 'a array \<Rightarrow> 'a array Heap"
    98 where
    99   "map f a = (do
   100      n \<leftarrow> length a;
   101      foldM (\<lambda>n. map_entry n f) [0..<n] a
   102    done)"
   103 
   104 hide (open) const new map -- {* avoid clashed with some popular names *}
   105 
   106 
   107 subsection {* Properties *}
   108 
   109 lemma array_make [code func]:
   110   "Array.new n x = make n (\<lambda>_. x)"
   111   by (induct n) (simp_all add: make_def new_def Heap_Monad.heap_def
   112     monad_simp array_of_list_replicate [symmetric]
   113     map_replicate_trivial replicate_append_same
   114     of_list_def)
   115 
   116 lemma array_of_list_make [code func]:
   117   "of_list xs = make (List.length xs) (\<lambda>n. xs ! n)"
   118   unfolding make_def map_nth ..
   119 
   120 
   121 subsection {* Code generator setup *}
   122 
   123 subsubsection {* Logical intermediate layer *}
   124 
   125 definition new' where
   126   [code del]: "new' = Array.new o nat_of_index"
   127 hide (open) const new'
   128 lemma [code func]:
   129   "Array.new = Array.new' o index_of_nat"
   130   by (simp add: new'_def o_def)
   131 
   132 definition of_list' where
   133   [code del]: "of_list' i xs = Array.of_list (take (nat_of_index i) xs)"
   134 hide (open) const of_list'
   135 lemma [code func]:
   136   "Array.of_list xs = Array.of_list' (index_of_nat (List.length xs)) xs"
   137   by (simp add: of_list'_def)
   138 
   139 definition make' where
   140   [code del]: "make' i f = Array.make (nat_of_index i) (f o index_of_nat)"
   141 hide (open) const make'
   142 lemma [code func]:
   143   "Array.make n f = Array.make' (index_of_nat n) (f o nat_of_index)"
   144   by (simp add: make'_def o_def)
   145 
   146 definition length' where
   147   [code del]: "length' = Array.length \<guillemotright>== liftM index_of_nat"
   148 hide (open) const length'
   149 lemma [code func]:
   150   "Array.length = Array.length' \<guillemotright>== liftM nat_of_index"
   151   by (simp add: length'_def monad_simp',
   152     simp add: liftM_def comp_def monad_simp,
   153     simp add: monad_simp')
   154 
   155 definition nth' where
   156   [code del]: "nth' a = Array.nth a o nat_of_index"
   157 hide (open) const nth'
   158 lemma [code func]:
   159   "Array.nth a n = Array.nth' a (index_of_nat n)"
   160   by (simp add: nth'_def)
   161 
   162 definition upd' where
   163   [code del]: "upd' a i x = Array.upd (nat_of_index i) x a \<guillemotright> return ()"
   164 hide (open) const upd'
   165 lemma [code func]:
   166   "Array.upd i x a = Array.upd' a (index_of_nat i) x \<guillemotright> return a"
   167   by (simp add: upd'_def monad_simp upd_return)
   168 
   169 
   170 subsubsection {* SML *}
   171 
   172 code_type array (SML "_/ array")
   173 code_const Array (SML "raise/ (Fail/ \"bare Array\")")
   174 code_const Array.new' (SML "(fn/ ()/ =>/ Array.array/ ((_),/ (_)))")
   175 code_const Array.of_list (SML "(fn/ ()/ =>/ Array.fromList/ _)")
   176 code_const Array.make' (SML "(fn/ ()/ =>/ Array.tabulate/ ((_),/ (_)))")
   177 code_const Array.length' (SML "(fn/ ()/ =>/ Array.length/ _)")
   178 code_const Array.nth' (SML "(fn/ ()/ =>/ Array.sub/ ((_),/ (_)))")
   179 code_const Array.upd' (SML "(fn/ ()/ =>/ Array.update/ ((_),/ (_),/ (_)))")
   180 
   181 code_reserved SML Array
   182 
   183 
   184 subsubsection {* OCaml *}
   185 
   186 code_type array (OCaml "_/ array")
   187 code_const Array (OCaml "failwith/ \"bare Array\"")
   188 code_const Array.new' (OCaml "(fn/ ()/ =>/ Array.make/ _/ _)")
   189 code_const Array.of_list (OCaml "(fn/ ()/ =>/ Array.of'_list/ _)")
   190 code_const Array.make' (OCaml "(fn/ ()/ =>/ Array.init/ _/ _)")
   191 code_const Array.length' (OCaml "(fn/ ()/ =>/ Array.length/ _)")
   192 code_const Array.nth' (OCaml "(fn/ ()/ =>/ Array.get/ _/ _)")
   193 code_const Array.upd' (OCaml "(fn/ ()/ =>/ Array.set/ _/ _/ _)")
   194 
   195 code_reserved OCaml Array
   196 
   197 
   198 subsubsection {* Haskell *}
   199 
   200 code_type array (Haskell "STArray '_s _")
   201 code_const Array (Haskell "error/ \"bare Array\"")
   202 code_const Array.new' (Haskell "newArray/ (0,/ _)")
   203 code_const Array.of_list' (Haskell "newListArray/ (0,/ _)")
   204 code_const Array.length' (Haskell "length")
   205 code_const Array.nth' (Haskell "readArray")
   206 code_const Array.upd' (Haskell "writeArray")
   207 
   208 end