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(* Title: HOL/Library/Array.thy


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ID: $Id$


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


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*)


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header {* Monadic arrays *}


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theory Array

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imports Heap_Monad Code_Index

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begin


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subsection {* Primitives *}


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definition


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


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


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definition


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


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


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definition


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


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


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definition


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


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where


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


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(if i < len


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


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


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done)"


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 {* FIXME adjustion for List theory *}


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no_syntax


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


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abbreviation


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


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where


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"nth_list \<equiv> List.nth"


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definition


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


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where


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


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(if i < len

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


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

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done)"


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lemma upd_return:


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

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proof (rule Heap_eqI)


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fix h


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


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


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


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


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qed

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subsection {* Derivates *}


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definition


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


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where


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"map_entry i f a = (do


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x \<leftarrow> nth a i;


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upd i (f x) a


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done)"


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definition


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


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where


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"swap i x a = (do


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y \<leftarrow> nth a i;


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upd i x a;


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return x


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done)"


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definition


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


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where


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


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definition


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


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where


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"freeze a = (do


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n \<leftarrow> length a;


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


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done)"


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definition


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


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where


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"map f a = (do


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n \<leftarrow> length a;


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


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done)"


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


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subsection {* Properties *}


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lemma array_make [code func]:


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


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


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monad_simp array_of_list_replicate [symmetric]


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map_replicate_trivial replicate_append_same


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of_list_def)


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lemma array_of_list_make [code func]:


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


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unfolding make_def map_nth ..


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subsection {* Code generator setup *}


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subsubsection {* Logical intermediate layer *}


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definition new' where


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


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hide (open) const new'


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lemma [code func]:


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"Array.new = Array.new' o index_of_nat"


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by (simp add: new'_def o_def)


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definition of_list' where


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


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hide (open) const of_list'


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lemma [code func]:


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"Array.of_list xs = Array.of_list' (index_of_nat (List.length xs)) xs"


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by (simp add: of_list'_def)


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definition make' where


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[code del]: "make' i f = Array.make (nat_of_index i) (f o index_of_nat)"


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hide (open) const make'


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lemma [code func]:


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"Array.make n f = Array.make' (index_of_nat n) (f o nat_of_index)"


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by (simp add: make'_def o_def)


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definition length' where


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


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hide (open) const length'


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lemma [code func]:


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"Array.length = Array.length' \<guillemotright>== liftM nat_of_index"


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by (simp add: length'_def monad_simp',


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simp add: liftM_def comp_def monad_simp,


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simp add: monad_simp')


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definition nth' where


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


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hide (open) const nth'


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lemma [code func]:


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"Array.nth a n = Array.nth' a (index_of_nat n)"


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by (simp add: nth'_def)


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definition upd' where


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


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hide (open) const upd'


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lemma [code func]:


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

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by (simp add: upd'_def monad_simp upd_return)

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subsubsection {* SML *}


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code_type array (SML "_/ array")


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


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code_const Array.new' (SML "Array.array ((_), (_))")


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code_const Array.of_list (SML "Array.fromList")


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code_const Array.make' (SML "Array.tabulate ((_), (_))")


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code_const Array.length' (SML "Array.length")


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code_const Array.nth' (SML "Array.sub ((_), (_))")


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code_const Array.upd' (SML "Array.update ((_), (_), (_))")


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code_reserved SML Array


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subsubsection {* OCaml *}


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code_type array (OCaml "_/ array")


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


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code_const Array.new' (OCaml "Array.make")


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code_const Array.of_list (OCaml "Array.of_list")


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code_const Array.make' (OCaml "Array.init")


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code_const Array.length' (OCaml "Array.length")


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code_const Array.nth' (OCaml "Array.get")


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code_const Array.upd' (OCaml "Array.set")


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code_reserved OCaml Array


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subsubsection {* Haskell *}


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code_type array (Haskell "STArray '_s _")


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


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


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


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code_const Array.length' (Haskell "length")


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code_const Array.nth' (Haskell "readArray")


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code_const Array.upd' (Haskell "writeArray")


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end
