(* 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