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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


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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. ((), Heap.upd a i x h))


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


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return a


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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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unfolding upd_def by (simp add: monad_simp)


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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 {* Converting arrays to lists *}


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primrec list_of_aux :: "nat \<Rightarrow> ('a\<Colon>heap) array \<Rightarrow> 'a list \<Rightarrow> 'a list Heap" where


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"list_of_aux 0 a xs = return xs"


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 "list_of_aux (Suc n) a xs = (do


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x \<leftarrow> Array.nth a n;


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list_of_aux n a (x#xs)


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done)"


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definition list_of :: "('a\<Colon>heap) array \<Rightarrow> 'a list Heap" where


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"list_of a = (do n \<leftarrow> Array.length a;


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list_of_aux n a []


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done)"


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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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end
