(*  Title:      HOL/Library/Array.thy

     ID:         $Id$

     Author:     John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen

 *)

 header {* Monadic arrays *}

 theory Array

 imports Heap_Monad Code_Index

 begin

 subsection {* Primitives *}

 definition

   new :: "nat \<Rightarrow> 'a\<Colon>heap \<Rightarrow> 'a array Heap" where

   [code del]: "new n x = Heap_Monad.heap (Heap.array n x)"

 definition

   of_list :: "'a\<Colon>heap list \<Rightarrow> 'a array Heap" where

   [code del]: "of_list xs = Heap_Monad.heap (Heap.array_of_list xs)"

 definition

   length :: "'a\<Colon>heap array \<Rightarrow> nat Heap" where

   [code del]: "length arr = Heap_Monad.heap (\<lambda>h. (Heap.length arr h, h))"

 definition

   nth :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a Heap"

 where

   [code del]: "nth a i = (do len \<leftarrow> length a;

                  (if i < len

                      then Heap_Monad.heap (\<lambda>h. (get_array a h ! i, h))

                      else raise (''array lookup: index out of range''))

               done)"

 -- {* FIXME adjustion for List theory *}

 no_syntax

   nth  :: "'a list \<Rightarrow> nat \<Rightarrow> 'a" (infixl "!" 100)

 abbreviation

   nth_list :: "'a list \<Rightarrow> nat \<Rightarrow> 'a" (infixl "!" 100)

 where

   "nth_list \<equiv> List.nth"

 definition

   upd :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a\<Colon>heap array Heap"

 where

   [code del]: "upd i x a = (do len \<leftarrow> length a;

                       (if i < len

                            then Heap_Monad.heap (\<lambda>h. (a, Heap.upd a i x h))

                            else raise (''array update: index out of range''))

                    done)" 

 lemma upd_return:

   "upd i x a \<guillemotright> return a = upd i x a"

 proof (rule Heap_eqI)

   fix h

   obtain len h' where "Heap_Monad.execute (Array.length a) h = (len, h')"

     by (cases "Heap_Monad.execute (Array.length a) h")

   then show "Heap_Monad.execute (upd i x a \<guillemotright> return a) h = Heap_Monad.execute (upd i x a) h"

     by (auto simp add: upd_def bindM_def split: sum.split)

 qed

 subsection {* Derivates *}

 definition

   map_entry :: "nat \<Rightarrow> ('a\<Colon>heap \<Rightarrow> 'a) \<Rightarrow> 'a array \<Rightarrow> 'a array Heap"

 where

   "map_entry i f a = (do

      x \<leftarrow> nth a i;

      upd i (f x) a

    done)"

 definition

   swap :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a Heap"

 where

   "swap i x a = (do

      y \<leftarrow> nth a i;

      upd i x a;

      return y

    done)"

 definition

   make :: "nat \<Rightarrow> (nat \<Rightarrow> 'a\<Colon>heap) \<Rightarrow> 'a array Heap"

 where

   "make n f = of_list (map f [0 ..< n])"

 definition

   freeze :: "'a\<Colon>heap array \<Rightarrow> 'a list Heap"

 where

   "freeze a = (do

      n \<leftarrow> length a;

      mapM (nth a) [0..<n]

    done)"

 definition

    map :: "('a\<Colon>heap \<Rightarrow> 'a) \<Rightarrow> 'a array \<Rightarrow> 'a array Heap"

 where

   "map f a = (do

      n \<leftarrow> length a;

      mapM (\<lambda>n. map_entry n f a) [0..<n];

      return a

    done)"

 hide (open) const new map -- {* avoid clashed with some popular names *}

 subsection {* Properties *}

 lemma array_make [code]:

   "Array.new n x = make n (\<lambda>_. x)"

   by (induct n) (simp_all add: make_def new_def Heap_Monad.heap_def

     monad_simp array_of_list_replicate [symmetric]

     map_replicate_trivial replicate_append_same

     of_list_def)

 lemma array_of_list_make [code]:

   "of_list xs = make (List.length xs) (\<lambda>n. xs ! n)"

   unfolding make_def map_nth ..

 subsection {* Code generator setup *}

 subsubsection {* Logical intermediate layer *}

 definition new' where

   [code del]: "new' = Array.new o Code_Index.nat_of"

 hide (open) const new'

 lemma [code]:

   "Array.new = Array.new' o Code_Index.of_nat"

   by (simp add: new'_def o_def)

 definition of_list' where

   [code del]: "of_list' i xs = Array.of_list (take (Code_Index.nat_of i) xs)"

 hide (open) const of_list'

 lemma [code]:

   "Array.of_list xs = Array.of_list' (Code_Index.of_nat (List.length xs)) xs"

   by (simp add: of_list'_def)

 definition make' where

   [code del]: "make' i f = Array.make (Code_Index.nat_of i) (f o Code_Index.of_nat)"

 hide (open) const make'

 lemma [code]:

   "Array.make n f = Array.make' (Code_Index.of_nat n) (f o Code_Index.nat_of)"

   by (simp add: make'_def o_def)

 definition length' where

   [code del]: "length' = Array.length \<guillemotright>== liftM Code_Index.of_nat"

 hide (open) const length'

 lemma [code]:

   "Array.length = Array.length' \<guillemotright>== liftM Code_Index.nat_of"

   by (simp add: length'_def monad_simp',

     simp add: liftM_def comp_def monad_simp,

     simp add: monad_simp')

 definition nth' where

   [code del]: "nth' a = Array.nth a o Code_Index.nat_of"

 hide (open) const nth'

 lemma [code]:

   "Array.nth a n = Array.nth' a (Code_Index.of_nat n)"

   by (simp add: nth'_def)

 definition upd' where

   [code del]: "upd' a i x = Array.upd (Code_Index.nat_of i) x a \<guillemotright> return ()"

 hide (open) const upd'

 lemma [code]:

   "Array.upd i x a = Array.upd' a (Code_Index.of_nat i) x \<guillemotright> return a"

   by (simp add: upd'_def monad_simp upd_return)

 subsubsection {* SML *}

 code_type array (SML "_/ array")

 code_const Array (SML "raise/ (Fail/ \"bare Array\")")

 code_const Array.new' (SML "(fn/ ()/ =>/ Array.array/ ((_),/ (_)))")

 code_const Array.of_list (SML "(fn/ ()/ =>/ Array.fromList/ _)")

 code_const Array.make' (SML "(fn/ ()/ =>/ Array.tabulate/ ((_),/ (_)))")

 code_const Array.length' (SML "(fn/ ()/ =>/ Array.length/ _)")

 code_const Array.nth' (SML "(fn/ ()/ =>/ Array.sub/ ((_),/ (_)))")

 code_const Array.upd' (SML "(fn/ ()/ =>/ Array.update/ ((_),/ (_),/ (_)))")

 code_reserved SML Array

 subsubsection {* OCaml *}

 code_type array (OCaml "_/ array")

 code_const Array (OCaml "failwith/ \"bare Array\"")

 code_const Array.new' (OCaml "(fun/ ()/ ->/ Array.make/ _/ _)")

 code_const Array.of_list (OCaml "(fun/ ()/ ->/ Array.of'_list/ _)")

 code_const Array.make' (OCaml "(fun/ ()/ ->/ Array.init/ _/ _)")

 code_const Array.length' (OCaml "(fun/ ()/ ->/ Array.length/ _)")

 code_const Array.nth' (OCaml "(fun/ ()/ ->/ Array.get/ _/ _)")

 code_const Array.upd' (OCaml "(fun/ ()/ ->/ Array.set/ _/ _/ _)")

 code_reserved OCaml Array

 subsubsection {* Haskell *}

 code_type array (Haskell "Heap.STArray/ Heap.RealWorld/ _")

 code_const Array (Haskell "error/ \"bare Array\"")

 code_const Array.new' (Haskell "Heap.newArray/ (0,/ _)")

 code_const Array.of_list' (Haskell "Heap.newListArray/ (0,/ _)")

 code_const Array.length' (Haskell "Heap.lengthArray")

 code_const Array.nth' (Haskell "Heap.readArray")

 code_const Array.upd' (Haskell "Heap.writeArray")

 end