src/HOL/Imperative_HOL/Array.thy
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(*  Title:      HOL/Imperative_HOL/Array.thy
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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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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 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 y
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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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     mapM (\<lambda>n. map_entry n f a) [0..<n];
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     return 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]:
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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]:
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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 Code_Numeral.nat_of"
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hide (open) const new'
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lemma [code]:
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  "Array.new = Array.new' o Code_Numeral.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 (Code_Numeral.nat_of i) xs)"
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hide (open) const of_list'
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lemma [code]:
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  "Array.of_list xs = Array.of_list' (Code_Numeral.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 (Code_Numeral.nat_of i) (f o Code_Numeral.of_nat)"
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hide (open) const make'
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lemma [code]:
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  "Array.make n f = Array.make' (Code_Numeral.of_nat n) (f o Code_Numeral.nat_of)"
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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 Code_Numeral.of_nat"
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hide (open) const length'
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lemma [code]:
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  "Array.length = Array.length' \<guillemotright>== liftM Code_Numeral.nat_of"
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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 Code_Numeral.nat_of"
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hide (open) const nth'
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lemma [code]:
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  "Array.nth a n = Array.nth' a (Code_Numeral.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 (Code_Numeral.nat_of i) x a \<guillemotright> return ()"
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hide (open) const upd'
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lemma [code]:
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  "Array.upd i x a = Array.upd' a (Code_Numeral.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 "(fn/ ()/ =>/ Array.array/ ((_),/ (_)))")
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code_const Array.of_list (SML "(fn/ ()/ =>/ Array.fromList/ _)")
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code_const Array.make' (SML "(fn/ ()/ =>/ Array.tabulate/ ((_),/ (_)))")
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code_const Array.length' (SML "(fn/ ()/ =>/ Array.length/ _)")
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code_const Array.nth' (SML "(fn/ ()/ =>/ Array.sub/ ((_),/ (_)))")
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code_const Array.upd' (SML "(fn/ ()/ =>/ 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 "(fun/ ()/ ->/ Array.make/ _/ _)")
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code_const Array.of_list (OCaml "(fun/ ()/ ->/ Array.of'_list/ _)")
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code_const Array.make' (OCaml "(fun/ ()/ ->/ Array.init/ _/ _)")
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code_const Array.length' (OCaml "(fun/ ()/ ->/ Array.length/ _)")
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code_const Array.nth' (OCaml "(fun/ ()/ ->/ Array.get/ _/ _)")
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code_const Array.upd' (OCaml "(fun/ ()/ ->/ Array.set/ _/ _/ _)")
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code_reserved OCaml Array
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subsubsection {* Haskell *}
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code_type array (Haskell "Heap.STArray/ Heap.RealWorld/ _")
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code_const Array (Haskell "error/ \"bare Array\"")
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code_const Array.new' (Haskell "Heap.newArray/ (0,/ _)")
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code_const Array.of_list' (Haskell "Heap.newListArray/ (0,/ _)")
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code_const Array.length' (Haskell "Heap.lengthArray")
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code_const Array.nth' (Haskell "Heap.readArray")
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code_const Array.upd' (Haskell "Heap.writeArray")
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end