--- a/src/HOL/Imperative_HOL/Array.thy Tue Jul 13 16:00:56 2010 +0200
+++ b/src/HOL/Imperative_HOL/Array.thy Tue Jul 13 16:12:40 2010 +0200
@@ -13,9 +13,9 @@
definition present :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> bool" where
"present h a \<longleftrightarrow> addr_of_array a < lim h"
-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 (*FIXME get *)
+ get_array :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a list" where
+ "get_array h a = map from_nat (arrays h (TYPEREP('a)) (addr_of_array a))"
definition set :: "'a\<Colon>heap array \<Rightarrow> 'a list \<Rightarrow> heap \<Rightarrow> heap" where
"set a x = arrays_update (\<lambda>h. h(TYPEREP('a) := ((h(TYPEREP('a))) (addr_of_array a:=map to_nat x))))"
@@ -28,10 +28,10 @@
in (r, h''))"
definition length :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> nat" where
- "length h a = List.length (get_array a h)"
+ "length h a = List.length (get_array h a)"
definition update :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> heap \<Rightarrow> heap" where
- "update a i x h = set a ((get_array a h)[i:=x]) h"
+ "update a i x h = set a ((get_array h a)[i:=x]) h"
definition noteq :: "'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"
@@ -53,7 +53,7 @@
definition nth :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a Heap" where
[code del]: "nth a i = Heap_Monad.guard (\<lambda>h. i < length h a)
- (\<lambda>h. (get_array a h ! i, h))"
+ (\<lambda>h. (get_array h a ! 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 h a)
@@ -61,14 +61,14 @@
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 h a)
- (\<lambda>h. (a, update a i (f (get_array a h ! i)) h))"
+ (\<lambda>h. (a, update a i (f (get_array h a ! 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 h a)
- (\<lambda>h. (get_array a h ! i, update a i x h))"
+ (\<lambda>h. (get_array h a ! i, update a i x h))"
definition freeze :: "'a\<Colon>heap array \<Rightarrow> 'a list Heap" where
- [code del]: "freeze a = Heap_Monad.tap (\<lambda>h. get_array a h)"
+ [code del]: "freeze a = Heap_Monad.tap (\<lambda>h. get_array h a)"
subsection {* Properties *}
@@ -88,10 +88,10 @@
lemma present_new_arr: "present h a \<Longrightarrow> a =!!= fst (alloc xs h)"
by (simp add: present_def noteq_def alloc_def Let_def)
-lemma array_get_set_eq [simp]: "get_array r (set r x h) = x"
+lemma array_get_set_eq [simp]: "get_array (set r x h) r = x"
by (simp add: get_array_def set_def o_def)
-lemma array_get_set_neq [simp]: "r =!!= s \<Longrightarrow> get_array r (set s x h) = get_array r h"
+lemma array_get_set_neq [simp]: "r =!!= s \<Longrightarrow> get_array (set s x h) r = get_array h r"
by (simp add: noteq_def get_array_def set_def)
lemma set_array_same [simp]:
@@ -103,15 +103,15 @@
by (simp add: Let_def expand_fun_eq noteq_def set_def)
lemma get_array_update_eq [simp]:
- "get_array a (update a i v h) = (get_array a h) [i := v]"
+ "get_array (update a i v h) a = (get_array h a) [i := v]"
by (simp add: update_def)
lemma nth_update_array_neq_array [simp]:
- "a =!!= b \<Longrightarrow> get_array a (update b j v h) ! i = get_array a h ! i"
+ "a =!!= b \<Longrightarrow> get_array (update b j v h) a ! i = get_array h a ! i"
by (simp add: update_def noteq_def)
lemma get_arry_array_update_elem_neqIndex [simp]:
- "i \<noteq> j \<Longrightarrow> get_array a (update a j v h) ! i = get_array a h ! i"
+ "i \<noteq> j \<Longrightarrow> get_array (update a j v h) a ! i = get_array h a ! i"
by simp
lemma length_update [simp]:
@@ -135,7 +135,7 @@
by (auto simp add: update_def array_set_set_swap list_update_swap)
lemma get_array_init_array_list:
- "get_array (fst (alloc ls h)) (snd (alloc ls' h)) = ls'"
+ "get_array (snd (alloc ls' h)) (fst (alloc ls h)) = ls'"
by (simp add: Let_def split_def alloc_def)
lemma set_array:
@@ -175,7 +175,7 @@
lemma crel_newE [crel_elims]:
assumes "crel (new n x) h h' r"
obtains "r = fst (alloc (replicate n x) h)" "h' = snd (alloc (replicate n x) h)"
- "get_array r h' = replicate n x" "present h' r" "\<not> present h r"
+ "get_array h' r = replicate n x" "present h' r" "\<not> present h r"
using assms by (rule crelE) (simp add: get_array_init_array_list execute_simps)
lemma execute_of_list [execute_simps]:
@@ -194,7 +194,7 @@
lemma crel_of_listE [crel_elims]:
assumes "crel (of_list xs) h h' r"
obtains "r = fst (alloc xs h)" "h' = snd (alloc xs h)"
- "get_array r h' = xs" "present h' r" "\<not> present h r"
+ "get_array h' r = xs" "present h' r" "\<not> present h r"
using assms by (rule crelE) (simp add: get_array_init_array_list execute_simps)
lemma execute_make [execute_simps]:
@@ -213,7 +213,7 @@
lemma crel_makeE [crel_elims]:
assumes "crel (make n f) h h' r"
obtains "r = fst (alloc (map f [0 ..< n]) h)" "h' = snd (alloc (map f [0 ..< n]) h)"
- "get_array r h' = map f [0 ..< n]" "present h' r" "\<not> present h r"
+ "get_array h' r = map f [0 ..< n]" "present h' r" "\<not> present h r"
using assms by (rule crelE) (simp add: get_array_init_array_list execute_simps)
lemma execute_len [execute_simps]:
@@ -236,7 +236,7 @@
lemma execute_nth [execute_simps]:
"i < length h a \<Longrightarrow>
- execute (nth a i) h = Some (get_array a h ! i, h)"
+ execute (nth a i) h = Some (get_array h a ! i, h)"
"i \<ge> length h a \<Longrightarrow> execute (nth a i) h = None"
by (simp_all add: nth_def execute_simps)
@@ -245,13 +245,13 @@
by (auto intro: success_intros simp add: nth_def)
lemma crel_nthI [crel_intros]:
- assumes "i < length h a" "h' = h" "r = get_array a h ! i"
+ assumes "i < length h a" "h' = h" "r = get_array h a ! i"
shows "crel (nth a i) h h' r"
by (rule crelI) (insert assms, simp add: execute_simps)
lemma crel_nthE [crel_elims]:
assumes "crel (nth a i) h h' r"
- obtains "i < length h a" "r = get_array a h ! i" "h' = h"
+ obtains "i < length h a" "r = get_array h a ! i" "h' = h"
using assms by (rule crelE)
(erule successE, cases "i < length h a", simp_all add: execute_simps)
@@ -279,7 +279,7 @@
lemma execute_map_entry [execute_simps]:
"i < length h a \<Longrightarrow>
execute (map_entry i f a) h =
- Some (a, update a i (f (get_array a h ! i)) h)"
+ Some (a, update a i (f (get_array h a ! i)) h)"
"i \<ge> length h a \<Longrightarrow> execute (map_entry i f a) h = None"
by (simp_all add: map_entry_def execute_simps)
@@ -288,20 +288,20 @@
by (auto intro: success_intros simp add: map_entry_def)
lemma crel_map_entryI [crel_intros]:
- assumes "i < length h a" "h' = update a i (f (get_array a h ! i)) h" "r = a"
+ assumes "i < length h a" "h' = update a i (f (get_array h a ! i)) h" "r = a"
shows "crel (map_entry i f a) h h' r"
by (rule crelI) (insert assms, simp add: execute_simps)
lemma crel_map_entryE [crel_elims]:
assumes "crel (map_entry i f a) h h' r"
- obtains "r = a" "h' = update a i (f (get_array a h ! i)) h" "i < length h a"
+ obtains "r = a" "h' = update a i (f (get_array h a ! i)) h" "i < length h a"
using assms by (rule crelE)
(erule successE, cases "i < length h a", simp_all add: execute_simps)
lemma execute_swap [execute_simps]:
"i < length h a \<Longrightarrow>
execute (swap i x a) h =
- Some (get_array a h ! i, update a i x h)"
+ Some (get_array h a ! i, update a i x h)"
"i \<ge> length h a \<Longrightarrow> execute (swap i x a) h = None"
by (simp_all add: swap_def execute_simps)
@@ -310,18 +310,18 @@
by (auto intro: success_intros simp add: swap_def)
lemma crel_swapI [crel_intros]:
- assumes "i < length h a" "h' = update a i x h" "r = get_array a h ! i"
+ assumes "i < length h a" "h' = update a i x h" "r = get_array h a ! i"
shows "crel (swap i x a) h h' r"
by (rule crelI) (insert assms, simp add: execute_simps)
lemma crel_swapE [crel_elims]:
assumes "crel (swap i x a) h h' r"
- obtains "r = get_array a h ! i" "h' = update a i x h" "i < length h a"
+ obtains "r = get_array h a ! i" "h' = update a i x h" "i < length h a"
using assms by (rule crelE)
(erule successE, cases "i < length h a", simp_all add: execute_simps)
lemma execute_freeze [execute_simps]:
- "execute (freeze a) h = Some (get_array a h, h)"
+ "execute (freeze a) h = Some (get_array h a, h)"
by (simp add: freeze_def execute_simps)
lemma success_freezeI [success_intros]:
@@ -329,13 +329,13 @@
by (auto intro: success_intros simp add: freeze_def)
lemma crel_freezeI [crel_intros]:
- assumes "h' = h" "r = get_array a h"
+ assumes "h' = h" "r = get_array h a"
shows "crel (freeze a) h h' r"
by (rule crelI) (insert assms, simp add: execute_simps)
lemma crel_freezeE [crel_elims]:
assumes "crel (freeze a) h h' r"
- obtains "h' = h" "r = get_array a h"
+ obtains "h' = h" "r = get_array h a"
using assms by (rule crelE) (simp add: execute_simps)
lemma upd_return:
@@ -423,12 +423,12 @@
fix h
have *: "List.map
(\<lambda>x. fst (the (if x < Array.length h a
- then Some (get_array a h ! x, h) else None)))
+ then Some (get_array h a ! x, h) else None)))
[0..<Array.length h a] =
- List.map (List.nth (get_array a h)) [0..<Array.length h a]"
+ List.map (List.nth (get_array h a)) [0..<Array.length h a]"
by simp
have "execute (Heap_Monad.fold_map (Array.nth a) [0..<Array.length h a]) h =
- Some (get_array a h, h)"
+ Some (get_array h a, h)"
apply (subst execute_fold_map_unchanged_heap)
apply (simp_all add: nth_def guard_def *)
apply (simp add: length_def map_nth)
@@ -436,7 +436,7 @@
then have "execute (do {
n \<leftarrow> Array.len a;
Heap_Monad.fold_map (Array.nth a) [0..<n]
- }) h = Some (get_array a h, h)"
+ }) h = Some (get_array h a, h)"
by (auto intro: execute_bind_eq_SomeI simp add: execute_simps)
then show "execute (Array.freeze a) h = execute (do {
n \<leftarrow> Array.len a;