--- a/src/HOL/Imperative_HOL/Array.thy Fri Jul 09 08:11:10 2010 +0200
+++ b/src/HOL/Imperative_HOL/Array.thy Fri Jul 09 09:48:52 2010 +0200
@@ -8,42 +8,81 @@
imports Heap_Monad
begin
-subsection {* Primitive layer *}
+subsection {* Primitives *}
-definition
+definition (*FIXME present :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> bool" where*)
array_present :: "'a\<Colon>heap array \<Rightarrow> heap \<Rightarrow> bool" where
"array_present a h \<longleftrightarrow> addr_of_array a < lim h"
-definition
+definition (*FIXME get :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a list" where*)
get_array :: "'a\<Colon>heap array \<Rightarrow> heap \<Rightarrow> 'a list" where
"get_array a h = map from_nat (arrays h (TYPEREP('a)) (addr_of_array a))"
-definition
+definition (*FIXME set*)
set_array :: "'a\<Colon>heap array \<Rightarrow> 'a list \<Rightarrow> heap \<Rightarrow> heap" where
"set_array a x =
arrays_update (\<lambda>h. h(TYPEREP('a) := ((h(TYPEREP('a))) (addr_of_array a:=map to_nat x))))"
-definition array :: "'a list \<Rightarrow> heap \<Rightarrow> 'a\<Colon>heap array \<times> heap" where
+definition (*FIXME alloc*)
+ array :: "'a list \<Rightarrow> heap \<Rightarrow> 'a\<Colon>heap array \<times> heap" where
"array xs h = (let
l = lim h;
r = Array l;
h'' = set_array r xs (h\<lparr>lim := l + 1\<rparr>)
in (r, h''))"
-definition length :: "'a\<Colon>heap array \<Rightarrow> heap \<Rightarrow> nat" where
+definition (*FIXME length :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> nat" where*)
+ length :: "'a\<Colon>heap array \<Rightarrow> heap \<Rightarrow> nat" where
"length a h = List.length (get_array a h)"
-definition change :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> heap \<Rightarrow> heap" where
+definition (*FIXME update*)
+ change :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> heap \<Rightarrow> heap" where
"change a i x h = set_array a ((get_array a h)[i:=x]) h"
-text {* Properties of imperative arrays *}
+definition (*FIXME noteq*)
+ noteq_arrs :: "'a\<Colon>heap array \<Rightarrow> 'b\<Colon>heap array \<Rightarrow> bool" (infix "=!!=" 70) where
+ "r =!!= s \<longleftrightarrow> TYPEREP('a) \<noteq> TYPEREP('b) \<or> addr_of_array r \<noteq> addr_of_array s"
+
+
+subsection {* Monad operations *}
+
+definition new :: "nat \<Rightarrow> 'a\<Colon>heap \<Rightarrow> 'a array Heap" where
+ [code del]: "new n x = Heap_Monad.heap (array (replicate n x))"
+
+definition of_list :: "'a\<Colon>heap list \<Rightarrow> 'a array Heap" where
+ [code del]: "of_list xs = Heap_Monad.heap (array xs)"
+
+definition make :: "nat \<Rightarrow> (nat \<Rightarrow> 'a\<Colon>heap) \<Rightarrow> 'a array Heap" where
+ [code del]: "make n f = Heap_Monad.heap (array (map f [0 ..< n]))"
+
+definition len :: "'a\<Colon>heap array \<Rightarrow> nat Heap" where
+ [code del]: "len a = Heap_Monad.heap (\<lambda>h. (length a h, h))"
+
+definition nth :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a Heap" where
+ [code del]: "nth a i = Heap_Monad.guard (\<lambda>h. i < length a h)
+ (\<lambda>h. (get_array a h ! i, h))"
+
+definition upd :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a\<Colon>heap array Heap" where
+ [code del]: "upd i x a = Heap_Monad.guard (\<lambda>h. i < length a h)
+ (\<lambda>h. (a, change a i x h))"
+
+definition map_entry :: "nat \<Rightarrow> ('a\<Colon>heap \<Rightarrow> 'a) \<Rightarrow> 'a array \<Rightarrow> 'a array Heap" where
+ [code del]: "map_entry i f a = Heap_Monad.guard (\<lambda>h. i < length a h)
+ (\<lambda>h. (a, change a i (f (get_array a h ! i)) h))"
+
+definition swap :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a Heap" where
+ [code del]: "swap i x a = Heap_Monad.guard (\<lambda>h. i < length a h)
+ (\<lambda>h. (get_array a h ! i, change a i x h))"
+
+definition freeze :: "'a\<Colon>heap array \<Rightarrow> 'a list Heap" where
+ [code del]: "freeze a = Heap_Monad.heap (\<lambda>h. (get_array a h, h))"
+
+
+subsection {* Properties *}
text {* FIXME: Does there exist a "canonical" array axiomatisation in
the literature? *}
-definition noteq_arrs :: "('a\<Colon>heap) array \<Rightarrow> ('b\<Colon>heap) array \<Rightarrow> bool" (infix "=!!=" 70) where
- "r =!!= s \<longleftrightarrow> TYPEREP('a) \<noteq> TYPEREP('b) \<or> addr_of_array r \<noteq> addr_of_array s"
-
lemma noteq_arrs_sym: "a =!!= b \<Longrightarrow> b =!!= a"
and unequal_arrs [simp]: "a \<noteq> a' \<longleftrightarrow> a =!!= a'"
unfolding noteq_arrs_def by auto
@@ -114,97 +153,65 @@
"array_present a (change b i v h) = array_present a h"
by (simp add: change_def array_present_def set_array_def get_array_def)
-
-
-subsection {* Primitives *}
+lemma execute_new [simp]:
+ "Heap_Monad.execute (new n x) h = Some (array (replicate n x) h)"
+ by (simp add: new_def)
-definition
- new :: "nat \<Rightarrow> 'a\<Colon>heap \<Rightarrow> 'a array Heap" where
- [code del]: "new n x = Heap_Monad.heap (Array.array (replicate n x))"
+lemma execute_of_list [simp]:
+ "Heap_Monad.execute (of_list xs) h = Some (array xs h)"
+ by (simp add: of_list_def)
-definition
- of_list :: "'a\<Colon>heap list \<Rightarrow> 'a array Heap" where
- [code del]: "of_list xs = Heap_Monad.heap (Array.array xs)"
+lemma execute_make [simp]:
+ "Heap_Monad.execute (make n f) h = Some (array (map f [0 ..< n]) h)"
+ by (simp add: make_def)
-definition
- len :: "'a\<Colon>heap array \<Rightarrow> nat Heap" where
- [code del]: "len arr = Heap_Monad.heap (\<lambda>h. (Array.length arr h, h))"
+lemma execute_len [simp]:
+ "Heap_Monad.execute (len a) h = Some (length a h, h)"
+ by (simp add: len_def)
+
+lemma execute_nth [simp]:
+ "i < length a h \<Longrightarrow>
+ Heap_Monad.execute (nth a i) h = Some (get_array a h ! i, h)"
+ "i \<ge> length a h \<Longrightarrow> Heap_Monad.execute (nth a i) h = None"
+ by (simp_all add: nth_def)
-definition
- nth :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a Heap"
-where
- [code del]: "nth a i = (do len \<leftarrow> len 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)"
+lemma execute_upd [simp]:
+ "i < length a h \<Longrightarrow>
+ Heap_Monad.execute (upd i x a) h = Some (a, change a i x h)"
+ "i \<ge> length a h \<Longrightarrow> Heap_Monad.execute (nth a i) h = None"
+ by (simp_all add: upd_def)
-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> len a;
- (if i < len
- then Heap_Monad.heap (\<lambda>h. (a, change a i x h))
- else raise ''array update: index out of range'')
- done)"
+lemma execute_map_entry [simp]:
+ "i < length a h \<Longrightarrow>
+ Heap_Monad.execute (map_entry i f a) h =
+ Some (a, change a i (f (get_array a h ! i)) h)"
+ "i \<ge> length a h \<Longrightarrow> Heap_Monad.execute (nth a i) h = None"
+ by (simp_all add: map_entry_def)
+
+lemma execute_swap [simp]:
+ "i < length a h \<Longrightarrow>
+ Heap_Monad.execute (swap i x a) h =
+ Some (get_array a h ! i, change a i x h)"
+ "i \<ge> length a h \<Longrightarrow> Heap_Monad.execute (nth a i) h = None"
+ by (simp_all add: swap_def)
+
+lemma execute_freeze [simp]:
+ "Heap_Monad.execute (freeze a) h = Some (get_array a h, h)"
+ by (simp add: freeze_def)
lemma upd_return:
"upd i x a \<guillemotright> return a = upd i x a"
- by (rule Heap_eqI) (simp add: upd_def bindM_def split: option.split)
-
-
-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)"
+ by (rule Heap_eqI) (simp add: bindM_def guard_def upd_def)
-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 y
- done)"
-
-definition
- make :: "nat \<Rightarrow> (nat \<Rightarrow> 'a\<Colon>heap) \<Rightarrow> 'a array Heap"
-where
- "make n f = of_list (map f [0 ..< n])"
+lemma array_make:
+ "new n x = make n (\<lambda>_. x)"
+ by (rule Heap_eqI) (simp add: map_replicate_trivial)
-definition
- freeze :: "'a\<Colon>heap array \<Rightarrow> 'a list Heap"
-where
- "freeze a = (do
- n \<leftarrow> len a;
- mapM (nth a) [0..<n]
- done)"
+lemma array_of_list_make:
+ "of_list xs = make (List.length xs) (\<lambda>n. xs ! n)"
+ by (rule Heap_eqI) (simp add: map_nth)
-definition
- map :: "('a\<Colon>heap \<Rightarrow> 'a) \<Rightarrow> 'a array \<Rightarrow> 'a array Heap"
-where
- "map f a = (do
- n \<leftarrow> len a;
- mapM (\<lambda>n. map_entry n f a) [0..<n];
- return a
- done)"
-
-
-
-subsection {* Properties *}
-
-lemma array_make [code]:
- "Array.new n x = make n (\<lambda>_. x)"
- by (rule Heap_eqI) (simp add: make_def new_def map_replicate_trivial of_list_def)
-
-lemma array_of_list_make [code]:
- "of_list xs = make (List.length xs) (\<lambda>n. xs ! n)"
- by (rule Heap_eqI) (simp add: make_def map_nth)
+hide_const (open) new map
subsection {* Code generator setup *}
@@ -213,48 +220,95 @@
definition new' where
[code del]: "new' = Array.new o Code_Numeral.nat_of"
-hide_const (open) new'
+
lemma [code]:
- "Array.new = Array.new' o Code_Numeral.of_nat"
+ "Array.new = new' o Code_Numeral.of_nat"
by (simp add: new'_def o_def)
definition of_list' where
[code del]: "of_list' i xs = Array.of_list (take (Code_Numeral.nat_of i) xs)"
-hide_const (open) of_list'
+
lemma [code]:
- "Array.of_list xs = Array.of_list' (Code_Numeral.of_nat (List.length xs)) xs"
+ "Array.of_list xs = of_list' (Code_Numeral.of_nat (List.length xs)) xs"
by (simp add: of_list'_def)
definition make' where
[code del]: "make' i f = Array.make (Code_Numeral.nat_of i) (f o Code_Numeral.of_nat)"
-hide_const (open) make'
+
lemma [code]:
- "Array.make n f = Array.make' (Code_Numeral.of_nat n) (f o Code_Numeral.nat_of)"
+ "Array.make n f = make' (Code_Numeral.of_nat n) (f o Code_Numeral.nat_of)"
by (simp add: make'_def o_def)
definition len' where
[code del]: "len' a = Array.len a \<guillemotright>= (\<lambda>n. return (Code_Numeral.of_nat n))"
-hide_const (open) len'
+
lemma [code]:
- "Array.len a = Array.len' a \<guillemotright>= (\<lambda>i. return (Code_Numeral.nat_of i))"
+ "Array.len a = len' a \<guillemotright>= (\<lambda>i. return (Code_Numeral.nat_of i))"
by (simp add: len'_def)
definition nth' where
[code del]: "nth' a = Array.nth a o Code_Numeral.nat_of"
-hide_const (open) nth'
+
lemma [code]:
- "Array.nth a n = Array.nth' a (Code_Numeral.of_nat n)"
+ "Array.nth a n = nth' a (Code_Numeral.of_nat n)"
by (simp add: nth'_def)
definition upd' where
[code del]: "upd' a i x = Array.upd (Code_Numeral.nat_of i) x a \<guillemotright> return ()"
-hide_const (open) upd'
+
lemma [code]:
- "Array.upd i x a = Array.upd' a (Code_Numeral.of_nat i) x \<guillemotright> return a"
+ "Array.upd i x a = upd' a (Code_Numeral.of_nat i) x \<guillemotright> return a"
by (simp add: upd'_def upd_return)
+lemma [code]:
+ "map_entry i f a = (do
+ x \<leftarrow> nth a i;
+ upd i (f x) a
+ done)"
+ by (rule Heap_eqI) (simp add: bindM_def guard_def map_entry_def)
-subsubsection {* SML *}
+lemma [code]:
+ "swap i x a = (do
+ y \<leftarrow> nth a i;
+ upd i x a;
+ return y
+ done)"
+ by (rule Heap_eqI) (simp add: bindM_def guard_def swap_def)
+
+lemma [code]:
+ "freeze a = (do
+ n \<leftarrow> len a;
+ mapM (\<lambda>i. nth a i) [0..<n]
+ done)"
+proof (rule Heap_eqI)
+ fix h
+ have *: "List.map
+ (\<lambda>x. fst (the (if x < length a h
+ then Some (get_array a h ! x, h) else None)))
+ [0..<length a h] =
+ List.map (List.nth (get_array a h)) [0..<length a h]"
+ by simp
+ have "Heap_Monad.execute (mapM (Array.nth a) [0..<length a h]) h =
+ Some (get_array a h, h)"
+ apply (subst execute_mapM_unchanged_heap)
+ apply (simp_all add: nth_def guard_def *)
+ apply (simp add: length_def map_nth)
+ done
+ then have "Heap_Monad.execute (do
+ n \<leftarrow> len a;
+ mapM (Array.nth a) [0..<n]
+ done) h = Some (get_array a h, h)"
+ by (auto intro: execute_eq_SomeI)
+ then show "Heap_Monad.execute (freeze a) h = Heap_Monad.execute (do
+ n \<leftarrow> len a;
+ mapM (Array.nth a) [0..<n]
+ done) h" by simp
+qed
+
+hide_const (open) new' of_list' make' len' nth' upd'
+
+
+text {* SML *}
code_type array (SML "_/ array")
code_const Array (SML "raise/ (Fail/ \"bare Array\")")
@@ -268,7 +322,7 @@
code_reserved SML Array
-subsubsection {* OCaml *}
+text {* OCaml *}
code_type array (OCaml "_/ array")
code_const Array (OCaml "failwith/ \"bare Array\"")
@@ -281,7 +335,7 @@
code_reserved OCaml Array
-subsubsection {* Haskell *}
+text {* Haskell *}
code_type array (Haskell "Heap.STArray/ Heap.RealWorld/ _")
code_const Array (Haskell "error/ \"bare Array\"")
@@ -291,6 +345,4 @@
code_const Array.nth' (Haskell "Heap.readArray")
code_const Array.upd' (Haskell "Heap.writeArray")
-hide_const (open) new map -- {* avoid clashed with some popular names *}
-
end