src/HOL/Library/Array.thy

+(*  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
+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. ((), Heap.upd a i x h))
+                           else raise (''array update: index out of range''));
+                      return a
+                   done)" 
+
+lemma upd_return:
+  "upd i x a \<guillemotright> return a = upd i x a"
+  unfolding upd_def by (simp add: monad_simp)
+
+
+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 x
+   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;
+     foldM (\<lambda>n. map_entry n f) [0..<n] a
+   done)"
+
+hide (open) const new map -- {* avoid clashed with some popular names *}
+
+
+subsection {* Converting arrays to lists *}
+
+primrec list_of_aux :: "nat \<Rightarrow> ('a\<Colon>heap) array \<Rightarrow> 'a list \<Rightarrow> 'a list Heap" where
+  "list_of_aux 0 a xs = return xs"
+  | "list_of_aux (Suc n) a xs = (do
+        x \<leftarrow> Array.nth a n;
+        list_of_aux n a (x#xs)
+     done)"
+
+definition list_of :: "('a\<Colon>heap) array \<Rightarrow> 'a list Heap" where
+  "list_of a = (do n \<leftarrow> Array.length a;
+                   list_of_aux n a []
+                done)"
+
+
+subsection {* Properties *}
+
+lemma array_make [code func]:
+  "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 func]:
+  "of_list xs = make (List.length xs) (\<lambda>n. xs ! n)"
+  unfolding make_def map_nth ..
+
+end