author haftmann Tue Jul 13 16:12:40 2010 +0200 (2010-07-13 ago) changeset 37805 0f797d586ce5 parent 37804 0145e59c1f6c child 37806 a7679be14442
canonical argument order for get
```     1.1 --- a/src/HOL/Imperative_HOL/Array.thy	Tue Jul 13 16:00:56 2010 +0200
1.2 +++ b/src/HOL/Imperative_HOL/Array.thy	Tue Jul 13 16:12:40 2010 +0200
1.3 @@ -13,9 +13,9 @@
1.4  definition present :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> bool" where
1.5    "present h a \<longleftrightarrow> addr_of_array a < lim h"
1.6
1.7 -definition (*FIXME get :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a list" where*)
1.8 -  get_array :: "'a\<Colon>heap array \<Rightarrow> heap \<Rightarrow> 'a list" where
1.9 -  "get_array a h = map from_nat (arrays h (TYPEREP('a)) (addr_of_array a))"
1.10 +definition (*FIXME get *)
1.11 +  get_array :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a list" where
1.12 +  "get_array h a = map from_nat (arrays h (TYPEREP('a)) (addr_of_array a))"
1.13
1.14  definition set :: "'a\<Colon>heap array \<Rightarrow> 'a list \<Rightarrow> heap \<Rightarrow> heap" where
1.15    "set a x = arrays_update (\<lambda>h. h(TYPEREP('a) := ((h(TYPEREP('a))) (addr_of_array a:=map to_nat x))))"
1.16 @@ -28,10 +28,10 @@
1.17     in (r, h''))"
1.18
1.19  definition length :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> nat" where
1.20 -  "length h a = List.length (get_array a h)"
1.21 +  "length h a = List.length (get_array h a)"
1.22
1.23  definition update :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> heap \<Rightarrow> heap" where
1.24 -  "update a i x h = set a ((get_array a h)[i:=x]) h"
1.25 +  "update a i x h = set a ((get_array h a)[i:=x]) h"
1.26
1.27  definition noteq :: "'a\<Colon>heap array \<Rightarrow> 'b\<Colon>heap array \<Rightarrow> bool" (infix "=!!=" 70) where
1.28    "r =!!= s \<longleftrightarrow> TYPEREP('a) \<noteq> TYPEREP('b) \<or> addr_of_array r \<noteq> addr_of_array s"
1.29 @@ -53,7 +53,7 @@
1.30
1.31  definition nth :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a Heap" where
1.32    [code del]: "nth a i = Heap_Monad.guard (\<lambda>h. i < length h a)
1.33 -    (\<lambda>h. (get_array a h ! i, h))"
1.34 +    (\<lambda>h. (get_array h a ! i, h))"
1.35
1.36  definition upd :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a\<Colon>heap array Heap" where
1.37    [code del]: "upd i x a = Heap_Monad.guard (\<lambda>h. i < length h a)
1.38 @@ -61,14 +61,14 @@
1.39
1.40  definition map_entry :: "nat \<Rightarrow> ('a\<Colon>heap \<Rightarrow> 'a) \<Rightarrow> 'a array \<Rightarrow> 'a array Heap" where
1.41    [code del]: "map_entry i f a = Heap_Monad.guard (\<lambda>h. i < length h a)
1.42 -    (\<lambda>h. (a, update a i (f (get_array a h ! i)) h))"
1.43 +    (\<lambda>h. (a, update a i (f (get_array h a ! i)) h))"
1.44
1.45  definition swap :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a Heap" where
1.46    [code del]: "swap i x a = Heap_Monad.guard (\<lambda>h. i < length h a)
1.47 -    (\<lambda>h. (get_array a h ! i, update a i x h))"
1.48 +    (\<lambda>h. (get_array h a ! i, update a i x h))"
1.49
1.50  definition freeze :: "'a\<Colon>heap array \<Rightarrow> 'a list Heap" where
1.51 -  [code del]: "freeze a = Heap_Monad.tap (\<lambda>h. get_array a h)"
1.52 +  [code del]: "freeze a = Heap_Monad.tap (\<lambda>h. get_array h a)"
1.53
1.54
1.55  subsection {* Properties *}
1.56 @@ -88,10 +88,10 @@
1.57  lemma present_new_arr: "present h a \<Longrightarrow> a =!!= fst (alloc xs h)"
1.58    by (simp add: present_def noteq_def alloc_def Let_def)
1.59
1.60 -lemma array_get_set_eq [simp]: "get_array r (set r x h) = x"
1.61 +lemma array_get_set_eq [simp]: "get_array (set r x h) r = x"
1.62    by (simp add: get_array_def set_def o_def)
1.63
1.64 -lemma array_get_set_neq [simp]: "r =!!= s \<Longrightarrow> get_array r (set s x h) = get_array r h"
1.65 +lemma array_get_set_neq [simp]: "r =!!= s \<Longrightarrow> get_array (set s x h) r = get_array h r"
1.66    by (simp add: noteq_def get_array_def set_def)
1.67
1.68  lemma set_array_same [simp]:
1.69 @@ -103,15 +103,15 @@
1.70    by (simp add: Let_def expand_fun_eq noteq_def set_def)
1.71
1.72  lemma get_array_update_eq [simp]:
1.73 -  "get_array a (update a i v h) = (get_array a h) [i := v]"
1.74 +  "get_array (update a i v h) a = (get_array h a) [i := v]"
1.76
1.77  lemma nth_update_array_neq_array [simp]:
1.78 -  "a =!!= b \<Longrightarrow> get_array a (update b j v h) ! i = get_array a h ! i"
1.79 +  "a =!!= b \<Longrightarrow> get_array (update b j v h) a ! i = get_array h a ! i"
1.80    by (simp add: update_def noteq_def)
1.81
1.82  lemma get_arry_array_update_elem_neqIndex [simp]:
1.83 -  "i \<noteq> j \<Longrightarrow> get_array a (update a j v h) ! i = get_array a h ! i"
1.84 +  "i \<noteq> j \<Longrightarrow> get_array (update a j v h) a ! i = get_array h a ! i"
1.85    by simp
1.86
1.87  lemma length_update [simp]:
1.88 @@ -135,7 +135,7 @@
1.89    by (auto simp add: update_def array_set_set_swap list_update_swap)
1.90
1.91  lemma get_array_init_array_list:
1.92 -  "get_array (fst (alloc ls h)) (snd (alloc ls' h)) = ls'"
1.93 +  "get_array (snd (alloc ls' h)) (fst (alloc ls h)) = ls'"
1.94    by (simp add: Let_def split_def alloc_def)
1.95
1.96  lemma set_array:
1.97 @@ -175,7 +175,7 @@
1.98  lemma crel_newE [crel_elims]:
1.99    assumes "crel (new n x) h h' r"
1.100    obtains "r = fst (alloc (replicate n x) h)" "h' = snd (alloc (replicate n x) h)"
1.101 -    "get_array r h' = replicate n x" "present h' r" "\<not> present h r"
1.102 +    "get_array h' r = replicate n x" "present h' r" "\<not> present h r"
1.103    using assms by (rule crelE) (simp add: get_array_init_array_list execute_simps)
1.104
1.105  lemma execute_of_list [execute_simps]:
1.106 @@ -194,7 +194,7 @@
1.107  lemma crel_of_listE [crel_elims]:
1.108    assumes "crel (of_list xs) h h' r"
1.109    obtains "r = fst (alloc xs h)" "h' = snd (alloc xs h)"
1.110 -    "get_array r h' = xs" "present h' r" "\<not> present h r"
1.111 +    "get_array h' r = xs" "present h' r" "\<not> present h r"
1.112    using assms by (rule crelE) (simp add: get_array_init_array_list execute_simps)
1.113
1.114  lemma execute_make [execute_simps]:
1.115 @@ -213,7 +213,7 @@
1.116  lemma crel_makeE [crel_elims]:
1.117    assumes "crel (make n f) h h' r"
1.118    obtains "r = fst (alloc (map f [0 ..< n]) h)" "h' = snd (alloc (map f [0 ..< n]) h)"
1.119 -    "get_array r h' = map f [0 ..< n]" "present h' r" "\<not> present h r"
1.120 +    "get_array h' r = map f [0 ..< n]" "present h' r" "\<not> present h r"
1.121    using assms by (rule crelE) (simp add: get_array_init_array_list execute_simps)
1.122
1.123  lemma execute_len [execute_simps]:
1.124 @@ -236,7 +236,7 @@
1.125
1.126  lemma execute_nth [execute_simps]:
1.127    "i < length h a \<Longrightarrow>
1.128 -    execute (nth a i) h = Some (get_array a h ! i, h)"
1.129 +    execute (nth a i) h = Some (get_array h a ! i, h)"
1.130    "i \<ge> length h a \<Longrightarrow> execute (nth a i) h = None"
1.131    by (simp_all add: nth_def execute_simps)
1.132
1.133 @@ -245,13 +245,13 @@
1.134    by (auto intro: success_intros simp add: nth_def)
1.135
1.136  lemma crel_nthI [crel_intros]:
1.137 -  assumes "i < length h a" "h' = h" "r = get_array a h ! i"
1.138 +  assumes "i < length h a" "h' = h" "r = get_array h a ! i"
1.139    shows "crel (nth a i) h h' r"
1.140    by (rule crelI) (insert assms, simp add: execute_simps)
1.141
1.142  lemma crel_nthE [crel_elims]:
1.143    assumes "crel (nth a i) h h' r"
1.144 -  obtains "i < length h a" "r = get_array a h ! i" "h' = h"
1.145 +  obtains "i < length h a" "r = get_array h a ! i" "h' = h"
1.146    using assms by (rule crelE)
1.147      (erule successE, cases "i < length h a", simp_all add: execute_simps)
1.148
1.149 @@ -279,7 +279,7 @@
1.150  lemma execute_map_entry [execute_simps]:
1.151    "i < length h a \<Longrightarrow>
1.152     execute (map_entry i f a) h =
1.153 -      Some (a, update a i (f (get_array a h ! i)) h)"
1.154 +      Some (a, update a i (f (get_array h a ! i)) h)"
1.155    "i \<ge> length h a \<Longrightarrow> execute (map_entry i f a) h = None"
1.156    by (simp_all add: map_entry_def execute_simps)
1.157
1.158 @@ -288,20 +288,20 @@
1.159    by (auto intro: success_intros simp add: map_entry_def)
1.160
1.161  lemma crel_map_entryI [crel_intros]:
1.162 -  assumes "i < length h a" "h' = update a i (f (get_array a h ! i)) h" "r = a"
1.163 +  assumes "i < length h a" "h' = update a i (f (get_array h a ! i)) h" "r = a"
1.164    shows "crel (map_entry i f a) h h' r"
1.165    by (rule crelI) (insert assms, simp add: execute_simps)
1.166
1.167  lemma crel_map_entryE [crel_elims]:
1.168    assumes "crel (map_entry i f a) h h' r"
1.169 -  obtains "r = a" "h' = update a i (f (get_array a h ! i)) h" "i < length h a"
1.170 +  obtains "r = a" "h' = update a i (f (get_array h a ! i)) h" "i < length h a"
1.171    using assms by (rule crelE)
1.172      (erule successE, cases "i < length h a", simp_all add: execute_simps)
1.173
1.174  lemma execute_swap [execute_simps]:
1.175    "i < length h a \<Longrightarrow>
1.176     execute (swap i x a) h =
1.177 -      Some (get_array a h ! i, update a i x h)"
1.178 +      Some (get_array h a ! i, update a i x h)"
1.179    "i \<ge> length h a \<Longrightarrow> execute (swap i x a) h = None"
1.180    by (simp_all add: swap_def execute_simps)
1.181
1.182 @@ -310,18 +310,18 @@
1.183    by (auto intro: success_intros simp add: swap_def)
1.184
1.185  lemma crel_swapI [crel_intros]:
1.186 -  assumes "i < length h a" "h' = update a i x h" "r = get_array a h ! i"
1.187 +  assumes "i < length h a" "h' = update a i x h" "r = get_array h a ! i"
1.188    shows "crel (swap i x a) h h' r"
1.189    by (rule crelI) (insert assms, simp add: execute_simps)
1.190
1.191  lemma crel_swapE [crel_elims]:
1.192    assumes "crel (swap i x a) h h' r"
1.193 -  obtains "r = get_array a h ! i" "h' = update a i x h" "i < length h a"
1.194 +  obtains "r = get_array h a ! i" "h' = update a i x h" "i < length h a"
1.195    using assms by (rule crelE)
1.196      (erule successE, cases "i < length h a", simp_all add: execute_simps)
1.197
1.198  lemma execute_freeze [execute_simps]:
1.199 -  "execute (freeze a) h = Some (get_array a h, h)"
1.200 +  "execute (freeze a) h = Some (get_array h a, h)"
1.201    by (simp add: freeze_def execute_simps)
1.202
1.203  lemma success_freezeI [success_intros]:
1.204 @@ -329,13 +329,13 @@
1.205    by (auto intro: success_intros simp add: freeze_def)
1.206
1.207  lemma crel_freezeI [crel_intros]:
1.208 -  assumes "h' = h" "r = get_array a h"
1.209 +  assumes "h' = h" "r = get_array h a"
1.210    shows "crel (freeze a) h h' r"
1.211    by (rule crelI) (insert assms, simp add: execute_simps)
1.212
1.213  lemma crel_freezeE [crel_elims]:
1.214    assumes "crel (freeze a) h h' r"
1.215 -  obtains "h' = h" "r = get_array a h"
1.216 +  obtains "h' = h" "r = get_array h a"
1.217    using assms by (rule crelE) (simp add: execute_simps)
1.218
1.219  lemma upd_return:
1.220 @@ -423,12 +423,12 @@
1.221    fix h
1.222    have *: "List.map
1.223       (\<lambda>x. fst (the (if x < Array.length h a
1.224 -                    then Some (get_array a h ! x, h) else None)))
1.225 +                    then Some (get_array h a ! x, h) else None)))
1.226       [0..<Array.length h a] =
1.227 -       List.map (List.nth (get_array a h)) [0..<Array.length h a]"
1.228 +       List.map (List.nth (get_array h a)) [0..<Array.length h a]"
1.229      by simp
1.230    have "execute (Heap_Monad.fold_map (Array.nth a) [0..<Array.length h a]) h =
1.231 -    Some (get_array a h, h)"
1.232 +    Some (get_array h a, h)"
1.233      apply (subst execute_fold_map_unchanged_heap)
1.234      apply (simp_all add: nth_def guard_def *)
1.235      apply (simp add: length_def map_nth)
1.236 @@ -436,7 +436,7 @@
1.237    then have "execute (do {
1.238        n \<leftarrow> Array.len a;
1.240 -    }) h = Some (get_array a h, h)"
1.241 +    }) h = Some (get_array h a, h)"
1.242      by (auto intro: execute_bind_eq_SomeI simp add: execute_simps)
1.243    then show "execute (Array.freeze a) h = execute (do {
1.244        n \<leftarrow> Array.len a;
```
```     2.1 --- a/src/HOL/Imperative_HOL/Ref.thy	Tue Jul 13 16:00:56 2010 +0200
2.2 +++ b/src/HOL/Imperative_HOL/Ref.thy	Tue Jul 13 16:12:40 2010 +0200
2.3 @@ -219,15 +219,11 @@
2.4  text {* Non-interaction between imperative array and imperative references *}
2.5
2.6  lemma get_array_set [simp]:
2.7 -  "get_array a (set r v h) = get_array a h"
2.8 -  by (simp add: get_array_def set_def)
2.9 -
2.10 -lemma nth_set [simp]:
2.11 -  "get_array a (set r v h) ! i = get_array a h ! i"
2.12 -  by simp
2.13 +  "get_array (set r v h) = get_array h"
2.14 +  by (simp add: get_array_def set_def expand_fun_eq)
2.15
2.16  lemma get_update [simp]:
2.17 -  "get (Array.update a i v h) r  = get h r"
2.18 +  "get (Array.update a i v h) r = get h r"
2.19    by (simp add: get_def Array.update_def Array.set_def)
2.20
2.21  lemma alloc_update:
2.22 @@ -243,8 +239,8 @@
2.23    by (simp add: Array.length_def get_array_def alloc_def set_def Let_def)
2.24
2.25  lemma get_array_alloc [simp]:
2.26 -  "get_array a (snd (alloc v h)) = get_array a h"
2.27 -  by (simp add: get_array_def alloc_def set_def Let_def)
2.28 +  "get_array (snd (alloc v h)) = get_array h"
2.29 +  by (simp add: get_array_def alloc_def set_def Let_def expand_fun_eq)
2.30
2.31  lemma present_update [simp]:
2.32    "present (Array.update a i v h) = present h"
```
```     3.1 --- a/src/HOL/Imperative_HOL/ex/Imperative_Quicksort.thy	Tue Jul 13 16:00:56 2010 +0200
3.2 +++ b/src/HOL/Imperative_HOL/ex/Imperative_Quicksort.thy	Tue Jul 13 16:12:40 2010 +0200
3.3 @@ -23,15 +23,15 @@
3.4
3.5  lemma crel_swapI [crel_intros]:
3.6    assumes "i < Array.length h a" "j < Array.length h a"
3.7 -    "x = get_array a h ! i" "y = get_array a h ! j"
3.8 +    "x = get_array h a ! i" "y = get_array h a ! j"
3.9      "h' = Array.update a j x (Array.update a i y h)"
3.10    shows "crel (swap a i j) h h' r"
3.11    unfolding swap_def using assms by (auto intro!: crel_intros)
3.12
3.13  lemma swap_permutes:
3.14    assumes "crel (swap a i j) h h' rs"
3.15 -  shows "multiset_of (get_array a h')
3.16 -  = multiset_of (get_array a h)"
3.17 +  shows "multiset_of (get_array h' a)
3.18 +  = multiset_of (get_array h a)"
3.19    using assms
3.20    unfolding swap_def
3.21    by (auto simp add: Array.length_def multiset_of_swap dest: sym [of _ "h'"] elim!: crel_bindE crel_nthE crel_returnE crel_updE)
3.22 @@ -55,8 +55,8 @@
3.23
3.24  lemma part_permutes:
3.25    assumes "crel (part1 a l r p) h h' rs"
3.26 -  shows "multiset_of (get_array a h')
3.27 -  = multiset_of (get_array a h)"
3.28 +  shows "multiset_of (get_array h' a)
3.29 +  = multiset_of (get_array h a)"
3.30    using assms
3.31  proof (induct a l r p arbitrary: h h' rs rule:part1.induct)
3.32    case (1 a l r p h h' rs)
3.33 @@ -82,7 +82,7 @@
3.34    next
3.35      case False (* recursive case *)
3.36      note rec_condition = this
3.37 -    let ?v = "get_array a h ! l"
3.38 +    let ?v = "get_array h a ! l"
3.39      show ?thesis
3.40      proof (cases "?v \<le> p")
3.41        case True
3.42 @@ -130,7 +130,7 @@
3.43
3.44  lemma part_outer_remains:
3.45    assumes "crel (part1 a l r p) h h' rs"
3.46 -  shows "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array (a::nat array) h ! i = get_array a h' ! i"
3.47 +  shows "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array h (a::nat array) ! i = get_array h' a ! i"
3.48    using assms
3.49  proof (induct a l r p arbitrary: h h' rs rule:part1.induct)
3.50    case (1 a l r p h h' rs)
3.51 @@ -145,7 +145,7 @@
3.52    next
3.53      case False (* recursive case *)
3.54      note rec_condition = this
3.55 -    let ?v = "get_array a h ! l"
3.56 +    let ?v = "get_array h a ! l"
3.57      show ?thesis
3.58      proof (cases "?v \<le> p")
3.59        case True
3.60 @@ -163,7 +163,7 @@
3.61          unfolding part1.simps[of a l r p]
3.62          by (elim crel_bindE crel_nthE crel_ifE crel_returnE) auto
3.63        from swp rec_condition have
3.64 -        "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array a h ! i = get_array a h1 ! i"
3.65 +        "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array h a ! i = get_array h1 a ! i"
3.66          unfolding swap_def
3.67          by (elim crel_bindE crel_nthE crel_updE crel_returnE) auto
3.68        with 1(2) [OF rec_condition False rec2] show ?thesis by fastsimp
3.69 @@ -174,8 +174,8 @@
3.70
3.71  lemma part_partitions:
3.72    assumes "crel (part1 a l r p) h h' rs"
3.73 -  shows "(\<forall>i. l \<le> i \<and> i < rs \<longrightarrow> get_array (a::nat array) h' ! i \<le> p)
3.74 -  \<and> (\<forall>i. rs < i \<and> i \<le> r \<longrightarrow> get_array a h' ! i \<ge> p)"
3.75 +  shows "(\<forall>i. l \<le> i \<and> i < rs \<longrightarrow> get_array h' (a::nat array) ! i \<le> p)
3.76 +  \<and> (\<forall>i. rs < i \<and> i \<le> r \<longrightarrow> get_array h' a ! i \<ge> p)"
3.77    using assms
3.78  proof (induct a l r p arbitrary: h h' rs rule:part1.induct)
3.79    case (1 a l r p h h' rs)
3.80 @@ -192,7 +192,7 @@
3.81    next
3.82      case False (* recursive case *)
3.83      note lr = this
3.84 -    let ?v = "get_array a h ! l"
3.85 +    let ?v = "get_array h a ! l"
3.86      show ?thesis
3.87      proof (cases "?v \<le> p")
3.88        case True
3.89 @@ -200,7 +200,7 @@
3.90        have rec1: "crel (part1 a (l + 1) r p) h h' rs"
3.91          unfolding part1.simps[of a l r p]
3.92          by (elim crel_bindE crel_nthE crel_ifE crel_returnE) auto
3.93 -      from True part_outer_remains[OF rec1] have a_l: "get_array a h' ! l \<le> p"
3.94 +      from True part_outer_remains[OF rec1] have a_l: "get_array h' a ! l \<le> p"
3.95          by fastsimp
3.96        have "\<forall>i. (l \<le> i = (l = i \<or> Suc l \<le> i))" by arith
3.97        with 1(1)[OF False True rec1] a_l show ?thesis
3.98 @@ -212,10 +212,10 @@
3.99          and rec2: "crel (part1 a l (r - 1) p) h1 h' rs"
3.100          unfolding part1.simps[of a l r p]
3.101          by (elim crel_bindE crel_nthE crel_ifE crel_returnE) auto
3.102 -      from swp False have "get_array a h1 ! r \<ge> p"
3.103 +      from swp False have "get_array h1 a ! r \<ge> p"
3.104          unfolding swap_def
3.105          by (auto simp add: Array.length_def elim!: crel_bindE crel_nthE crel_updE crel_returnE)
3.106 -      with part_outer_remains [OF rec2] lr have a_r: "get_array a h' ! r \<ge> p"
3.107 +      with part_outer_remains [OF rec2] lr have a_r: "get_array h' a ! r \<ge> p"
3.108          by fastsimp
3.109        have "\<forall>i. (i \<le> r = (i = r \<or> i \<le> r - 1))" by arith
3.110        with 1(2)[OF lr False rec2] a_r show ?thesis
3.111 @@ -240,8 +240,8 @@
3.112
3.113  lemma partition_permutes:
3.114    assumes "crel (partition a l r) h h' rs"
3.115 -  shows "multiset_of (get_array a h')
3.116 -  = multiset_of (get_array a h)"
3.117 +  shows "multiset_of (get_array h' a)
3.118 +  = multiset_of (get_array h a)"
3.119  proof -
3.120      from assms part_permutes swap_permutes show ?thesis
3.121        unfolding partition.simps
3.122 @@ -260,7 +260,7 @@
3.123  lemma partition_outer_remains:
3.124    assumes "crel (partition a l r) h h' rs"
3.125    assumes "l < r"
3.126 -  shows "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array (a::nat array) h ! i = get_array a h' ! i"
3.127 +  shows "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array h (a::nat array) ! i = get_array h' a ! i"
3.128  proof -
3.129    from assms part_outer_remains part_returns_index_in_bounds show ?thesis
3.130      unfolding partition.simps swap_def
3.131 @@ -273,10 +273,10 @@
3.132    shows "l \<le> rs \<and> rs \<le> r"
3.133  proof -
3.134    from crel obtain middle h'' p where part: "crel (part1 a l (r - 1) p) h h'' middle"
3.135 -    and rs_equals: "rs = (if get_array a h'' ! middle \<le> get_array a h ! r then middle + 1
3.136 +    and rs_equals: "rs = (if get_array h'' a ! middle \<le> get_array h a ! r then middle + 1
3.137           else middle)"
3.138      unfolding partition.simps
3.139 -    by (elim crel_bindE crel_returnE crel_nthE crel_ifE crel_updE) simp
3.140 +    by (elim crel_bindE crel_returnE crel_nthE crel_ifE crel_updE) simp
3.141    from `l < r` have "l \<le> r - 1" by arith
3.142    from part_returns_index_in_bounds[OF part this] rs_equals `l < r` show ?thesis by auto
3.143  qed
3.144 @@ -284,18 +284,18 @@
3.145  lemma partition_partitions:
3.146    assumes crel: "crel (partition a l r) h h' rs"
3.147    assumes "l < r"
3.148 -  shows "(\<forall>i. l \<le> i \<and> i < rs \<longrightarrow> get_array (a::nat array) h' ! i \<le> get_array a h' ! rs) \<and>
3.149 -  (\<forall>i. rs < i \<and> i \<le> r \<longrightarrow> get_array a h' ! rs \<le> get_array a h' ! i)"
3.150 +  shows "(\<forall>i. l \<le> i \<and> i < rs \<longrightarrow> get_array h' (a::nat array) ! i \<le> get_array h' a ! rs) \<and>
3.151 +  (\<forall>i. rs < i \<and> i \<le> r \<longrightarrow> get_array h' a ! rs \<le> get_array h' a ! i)"
3.152  proof -
3.153 -  let ?pivot = "get_array a h ! r"
3.154 +  let ?pivot = "get_array h a ! r"
3.155    from crel obtain middle h1 where part: "crel (part1 a l (r - 1) ?pivot) h h1 middle"
3.156      and swap: "crel (swap a rs r) h1 h' ()"
3.157 -    and rs_equals: "rs = (if get_array a h1 ! middle \<le> ?pivot then middle + 1
3.158 +    and rs_equals: "rs = (if get_array h1 a ! middle \<le> ?pivot then middle + 1
3.159           else middle)"
3.160      unfolding partition.simps
3.161      by (elim crel_bindE crel_returnE crel_nthE crel_ifE crel_updE) simp
3.162 -  from swap have h'_def: "h' = Array.update a r (get_array a h1 ! rs)
3.163 -    (Array.update a rs (get_array a h1 ! r) h1)"
3.164 +  from swap have h'_def: "h' = Array.update a r (get_array h1 a ! rs)
3.165 +    (Array.update a rs (get_array h1 a ! r) h1)"
3.166      unfolding swap_def
3.167      by (elim crel_bindE crel_returnE crel_nthE crel_updE) simp
3.168    from swap have in_bounds: "r < Array.length h1 a \<and> rs < Array.length h1 a"
3.169 @@ -306,15 +306,15 @@
3.170    from `l < r` have "l \<le> r - 1" by simp
3.171    note middle_in_bounds = part_returns_index_in_bounds[OF part this]
3.172    from part_outer_remains[OF part] `l < r`
3.173 -  have "get_array a h ! r = get_array a h1 ! r"
3.174 +  have "get_array h a ! r = get_array h1 a ! r"
3.175      by fastsimp
3.176    with swap
3.177 -  have right_remains: "get_array a h ! r = get_array a h' ! rs"
3.178 +  have right_remains: "get_array h a ! r = get_array h' a ! rs"
3.179      unfolding swap_def
3.180      by (auto simp add: Array.length_def elim!: crel_bindE crel_returnE crel_nthE crel_updE) (cases "r = rs", auto)
3.181    from part_partitions [OF part]
3.182    show ?thesis
3.183 -  proof (cases "get_array a h1 ! middle \<le> ?pivot")
3.184 +  proof (cases "get_array h1 a ! middle \<le> ?pivot")
3.185      case True
3.186      with rs_equals have rs_equals: "rs = middle + 1" by simp
3.187      {
3.188 @@ -324,8 +324,8 @@
3.189        have i_props: "i < Array.length h' a" "i \<noteq> r" "i \<noteq> rs" by auto
3.190        from i_is_left rs_equals have "l \<le> i \<and> i < middle \<or> i = middle" by arith
3.191        with part_partitions[OF part] right_remains True
3.192 -      have "get_array a h1 ! i \<le> get_array a h' ! rs" by fastsimp
3.193 -      with i_props h'_def in_bounds have "get_array a h' ! i \<le> get_array a h' ! rs"
3.194 +      have "get_array h1 a ! i \<le> get_array h' a ! rs" by fastsimp
3.195 +      with i_props h'_def in_bounds have "get_array h' a ! i \<le> get_array h' a ! rs"
3.196          unfolding Array.update_def Array.length_def by simp
3.197      }
3.198      moreover
3.199 @@ -334,13 +334,13 @@
3.200        assume "rs < i \<and> i \<le> r"
3.201
3.202        hence "(rs < i \<and> i \<le> r - 1) \<or> (rs < i \<and> i = r)" by arith
3.203 -      hence "get_array a h' ! rs \<le> get_array a h' ! i"
3.204 +      hence "get_array h' a ! rs \<le> get_array h' a ! i"
3.205        proof
3.206          assume i_is: "rs < i \<and> i \<le> r - 1"
3.207          with swap_length_remains in_bounds middle_in_bounds rs_equals
3.208          have i_props: "i < Array.length h' a" "i \<noteq> r" "i \<noteq> rs" by auto
3.209          from part_partitions[OF part] rs_equals right_remains i_is
3.210 -        have "get_array a h' ! rs \<le> get_array a h1 ! i"
3.211 +        have "get_array h' a ! rs \<le> get_array h1 a ! i"
3.212            by fastsimp
3.213          with i_props h'_def show ?thesis by fastsimp
3.214        next
3.215 @@ -348,7 +348,7 @@
3.216          with rs_equals have "Suc middle \<noteq> r" by arith
3.217          with middle_in_bounds `l < r` have "Suc middle \<le> r - 1" by arith
3.218          with part_partitions[OF part] right_remains
3.219 -        have "get_array a h' ! rs \<le> get_array a h1 ! (Suc middle)"
3.220 +        have "get_array h' a ! rs \<le> get_array h1 a ! (Suc middle)"
3.221            by fastsimp
3.222          with i_is True rs_equals right_remains h'_def
3.223          show ?thesis using in_bounds
3.224 @@ -366,8 +366,8 @@
3.225        with swap_length_remains in_bounds middle_in_bounds rs_equals
3.226        have i_props: "i < Array.length h' a" "i \<noteq> r" "i \<noteq> rs" by auto
3.227        from part_partitions[OF part] rs_equals right_remains i_is_left
3.228 -      have "get_array a h1 ! i \<le> get_array a h' ! rs" by fastsimp
3.229 -      with i_props h'_def have "get_array a h' ! i \<le> get_array a h' ! rs"
3.230 +      have "get_array h1 a ! i \<le> get_array h' a ! rs" by fastsimp
3.231 +      with i_props h'_def have "get_array h' a ! i \<le> get_array h' a ! rs"
3.232          unfolding Array.update_def by simp
3.233      }
3.234      moreover
3.235 @@ -375,13 +375,13 @@
3.236        fix i
3.237        assume "rs < i \<and> i \<le> r"
3.238        hence "(rs < i \<and> i \<le> r - 1) \<or> i = r" by arith
3.239 -      hence "get_array a h' ! rs \<le> get_array a h' ! i"
3.240 +      hence "get_array h' a ! rs \<le> get_array h' a ! i"
3.241        proof
3.242          assume i_is: "rs < i \<and> i \<le> r - 1"
3.243          with swap_length_remains in_bounds middle_in_bounds rs_equals
3.244          have i_props: "i < Array.length h' a" "i \<noteq> r" "i \<noteq> rs" by auto
3.245          from part_partitions[OF part] rs_equals right_remains i_is
3.246 -        have "get_array a h' ! rs \<le> get_array a h1 ! i"
3.247 +        have "get_array h' a ! rs \<le> get_array h1 a ! i"
3.248            by fastsimp
3.249          with i_props h'_def show ?thesis by fastsimp
3.250        next
3.251 @@ -420,8 +420,8 @@
3.252
3.253  lemma quicksort_permutes:
3.254    assumes "crel (quicksort a l r) h h' rs"
3.255 -  shows "multiset_of (get_array a h')
3.256 -  = multiset_of (get_array a h)"
3.257 +  shows "multiset_of (get_array h' a)
3.258 +  = multiset_of (get_array h a)"
3.259    using assms
3.260  proof (induct a l r arbitrary: h h' rs rule: quicksort.induct)
3.261    case (1 a l r h h' rs)
3.262 @@ -443,7 +443,7 @@
3.263
3.264  lemma quicksort_outer_remains:
3.265    assumes "crel (quicksort a l r) h h' rs"
3.266 -   shows "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array (a::nat array) h ! i = get_array a h' ! i"
3.267 +   shows "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array h (a::nat array) ! i = get_array h' a ! i"
3.268    using assms
3.269  proof (induct a l r arbitrary: h h' rs rule: quicksort.induct)
3.270    case (1 a l r h h' rs)
3.271 @@ -465,14 +465,14 @@
3.272        assume pivot: "l \<le> p \<and> p \<le> r"
3.273        assume i_outer: "i < l \<or> r < i"
3.274        from  partition_outer_remains [OF part True] i_outer
3.275 -      have "get_array a h !i = get_array a h1 ! i" by fastsimp
3.276 +      have "get_array h a !i = get_array h1 a ! i" by fastsimp
3.277        moreover
3.278        with 1(1) [OF True pivot qs1] pivot i_outer
3.279 -      have "get_array a h1 ! i = get_array a h2 ! i" by auto
3.280 +      have "get_array h1 a ! i = get_array h2 a ! i" by auto
3.281        moreover
3.282        with qs2 1(2) [of p h2 h' ret2] True pivot i_outer
3.283 -      have "get_array a h2 ! i = get_array a h' ! i" by auto
3.284 -      ultimately have "get_array a h ! i= get_array a h' ! i" by simp
3.285 +      have "get_array h2 a ! i = get_array h' a ! i" by auto
3.286 +      ultimately have "get_array h a ! i= get_array h' a ! i" by simp
3.287      }
3.288      with cr show ?thesis
3.289        unfolding quicksort.simps [of a l r]
3.290 @@ -512,7 +512,7 @@
3.291        have pivot: "l\<le> p \<and> p \<le> r" .
3.292       note length_remains = length_remains[OF qs2] length_remains[OF qs1] partition_length_remains[OF part]
3.293        from quicksort_outer_remains [OF qs2] quicksort_outer_remains [OF qs1] pivot quicksort_is_skip[OF qs1]
3.294 -      have pivot_unchanged: "get_array a h1 ! p = get_array a h' ! p" by (cases p, auto)
3.295 +      have pivot_unchanged: "get_array h1 a ! p = get_array h' a ! p" by (cases p, auto)
3.296          (*-- First of all, by induction hypothesis both sublists are sorted. *)
3.297        from 1(1)[OF True pivot qs1] length_remains pivot 1(5)
3.298        have IH1: "sorted (subarray l p a h2)"  by (cases p, auto simp add: subarray_Nil)
3.299 @@ -525,35 +525,35 @@
3.300          by (cases "Suc p \<le> r", auto simp add: subarray_Nil)
3.301             (* -- Secondly, both sublists remain partitioned. *)
3.302        from partition_partitions[OF part True]
3.303 -      have part_conds1: "\<forall>j. j \<in> set (subarray l p a h1) \<longrightarrow> j \<le> get_array a h1 ! p "
3.304 -        and part_conds2: "\<forall>j. j \<in> set (subarray (p + 1) (r + 1) a h1) \<longrightarrow> get_array a h1 ! p \<le> j"
3.305 +      have part_conds1: "\<forall>j. j \<in> set (subarray l p a h1) \<longrightarrow> j \<le> get_array h1 a ! p "
3.306 +        and part_conds2: "\<forall>j. j \<in> set (subarray (p + 1) (r + 1) a h1) \<longrightarrow> get_array h1 a ! p \<le> j"
3.307          by (auto simp add: all_in_set_subarray_conv)
3.308        from quicksort_outer_remains [OF qs1] quicksort_permutes [OF qs1] True
3.309 -        length_remains 1(5) pivot multiset_of_sublist [of l p "get_array a h1" "get_array a h2"]
3.310 +        length_remains 1(5) pivot multiset_of_sublist [of l p "get_array h1 a" "get_array h2 a"]
3.311        have multiset_partconds1: "multiset_of (subarray l p a h2) = multiset_of (subarray l p a h1)"
3.312          unfolding Array.length_def subarray_def by (cases p, auto)
3.313        with left_subarray_remains part_conds1 pivot_unchanged
3.314 -      have part_conds2': "\<forall>j. j \<in> set (subarray l p a h') \<longrightarrow> j \<le> get_array a h' ! p"
3.315 +      have part_conds2': "\<forall>j. j \<in> set (subarray l p a h') \<longrightarrow> j \<le> get_array h' a ! p"
3.316          by (simp, subst set_of_multiset_of[symmetric], simp)
3.317            (* -- These steps are the analogous for the right sublist \<dots> *)
3.318        from quicksort_outer_remains [OF qs1] length_remains
3.319        have right_subarray_remains: "subarray (p + 1) (r + 1) a h1 = subarray (p + 1) (r + 1) a h2"
3.320          by (auto simp add: subarray_eq_samelength_iff)
3.321        from quicksort_outer_remains [OF qs2] quicksort_permutes [OF qs2] True
3.322 -        length_remains 1(5) pivot multiset_of_sublist [of "p + 1" "r + 1" "get_array a h2" "get_array a h'"]
3.323 +        length_remains 1(5) pivot multiset_of_sublist [of "p + 1" "r + 1" "get_array h2 a" "get_array h' a"]
3.324        have multiset_partconds2: "multiset_of (subarray (p + 1) (r + 1) a h') = multiset_of (subarray (p + 1) (r + 1) a h2)"
3.325          unfolding Array.length_def subarray_def by auto
3.326        with right_subarray_remains part_conds2 pivot_unchanged
3.327 -      have part_conds1': "\<forall>j. j \<in> set (subarray (p + 1) (r + 1) a h') \<longrightarrow> get_array a h' ! p \<le> j"
3.328 +      have part_conds1': "\<forall>j. j \<in> set (subarray (p + 1) (r + 1) a h') \<longrightarrow> get_array h' a ! p \<le> j"
3.329          by (simp, subst set_of_multiset_of[symmetric], simp)
3.330            (* -- Thirdly and finally, we show that the array is sorted
3.331            following from the facts above. *)
3.332 -      from True pivot 1(5) length_remains have "subarray l (r + 1) a h' = subarray l p a h' @ [get_array a h' ! p] @ subarray (p + 1) (r + 1) a h'"
3.333 +      from True pivot 1(5) length_remains have "subarray l (r + 1) a h' = subarray l p a h' @ [get_array h' a ! p] @ subarray (p + 1) (r + 1) a h'"
3.334          by (simp add: subarray_nth_array_Cons, cases "l < p") (auto simp add: subarray_append subarray_Nil)
3.335        with IH1' IH2 part_conds1' part_conds2' pivot have ?thesis
3.336          unfolding subarray_def
3.337          apply (auto simp add: sorted_append sorted_Cons all_in_set_sublist'_conv)
3.338 -        by (auto simp add: set_sublist' dest: le_trans [of _ "get_array a h' ! p"])
3.339 +        by (auto simp add: set_sublist' dest: le_trans [of _ "get_array h' a ! p"])
3.340      }
3.341      with True cr show ?thesis
3.342        unfolding quicksort.simps [of a l r]
3.343 @@ -564,16 +564,16 @@
3.344
3.345  lemma quicksort_is_sort:
3.346    assumes crel: "crel (quicksort a 0 (Array.length h a - 1)) h h' rs"
3.347 -  shows "get_array a h' = sort (get_array a h)"
3.348 -proof (cases "get_array a h = []")
3.349 +  shows "get_array h' a = sort (get_array h a)"
3.350 +proof (cases "get_array h a = []")
3.351    case True
3.352    with quicksort_is_skip[OF crel] show ?thesis
3.353    unfolding Array.length_def by simp
3.354  next
3.355    case False
3.356 -  from quicksort_sorts [OF crel] False have "sorted (sublist' 0 (List.length (get_array a h)) (get_array a h'))"
3.357 +  from quicksort_sorts [OF crel] False have "sorted (sublist' 0 (List.length (get_array h a)) (get_array h' a))"
3.358      unfolding Array.length_def subarray_def by auto
3.359 -  with length_remains[OF crel] have "sorted (get_array a h')"
3.360 +  with length_remains[OF crel] have "sorted (get_array h' a)"
3.361      unfolding Array.length_def by simp
3.362    with quicksort_permutes [OF crel] properties_for_sort show ?thesis by fastsimp
3.363  qed
```
```     4.1 --- a/src/HOL/Imperative_HOL/ex/Imperative_Reverse.thy	Tue Jul 13 16:00:56 2010 +0200
4.2 +++ b/src/HOL/Imperative_HOL/ex/Imperative_Reverse.thy	Tue Jul 13 16:12:40 2010 +0200
4.3 @@ -27,17 +27,17 @@
4.4  declare swap.simps [simp del] rev.simps [simp del]
4.5
4.6  lemma swap_pointwise: assumes "crel (swap a i j) h h' r"
4.7 -  shows "get_array a h' ! k = (if k = i then get_array a h ! j
4.8 -      else if k = j then get_array a h ! i
4.9 -      else get_array a h ! k)"
4.10 +  shows "get_array h' a ! k = (if k = i then get_array h a ! j
4.11 +      else if k = j then get_array h a ! i
4.12 +      else get_array h a ! k)"
4.13  using assms unfolding swap.simps
4.14  by (elim crel_elims)
4.15   (auto simp: length_def)
4.16
4.17  lemma rev_pointwise: assumes "crel (rev a i j) h h' r"
4.18 -  shows "get_array a h' ! k = (if k < i then get_array a h ! k
4.19 -      else if j < k then get_array a h ! k
4.20 -      else get_array a h ! (j - (k - i)))" (is "?P a i j h h'")
4.21 +  shows "get_array h' a ! k = (if k < i then get_array h a ! k
4.22 +      else if j < k then get_array h a ! k
4.23 +      else get_array h a ! (j - (k - i)))" (is "?P a i j h h'")
4.24  using assms proof (induct a i j arbitrary: h h' rule: rev.induct)
4.25    case (1 a i j h h'')
4.26    thus ?case
4.27 @@ -94,7 +94,7 @@
4.28    {
4.29      fix k
4.30      assume "k < Suc j - i"
4.31 -    with rev_pointwise[OF assms(1)] have "get_array a h' ! (i + k) = get_array a h ! (j - k)"
4.32 +    with rev_pointwise[OF assms(1)] have "get_array h' a ! (i + k) = get_array h a ! (j - k)"
4.33        by auto
4.34    }
4.35    with assms(2) rev_length[OF assms(1)] show ?thesis
4.36 @@ -104,10 +104,10 @@
4.37
4.38  lemma rev2_rev:
4.39    assumes "crel (rev a 0 (Array.length h a - 1)) h h' u"
4.40 -  shows "get_array a h' = List.rev (get_array a h)"
4.41 +  shows "get_array h' a = List.rev (get_array h a)"
4.42    using rev2_rev'[OF assms] rev_length[OF assms] assms
4.43      by (cases "Array.length h a = 0", auto simp add: Array.length_def
4.44        subarray_def sublist'_all rev.simps[where j=0] elim!: crel_elims)
4.45 -  (drule sym[of "List.length (get_array a h)"], simp)
4.46 +  (drule sym[of "List.length (get_array h a)"], simp)
4.47
4.48  end
```
```     5.1 --- a/src/HOL/Imperative_HOL/ex/SatChecker.thy	Tue Jul 13 16:00:56 2010 +0200
5.2 +++ b/src/HOL/Imperative_HOL/ex/SatChecker.thy	Tue Jul 13 16:12:40 2010 +0200
5.3 @@ -120,17 +120,17 @@
5.4
5.5  definition
5.6    array_ran :: "('a\<Colon>heap) option array \<Rightarrow> heap \<Rightarrow> 'a set" where
5.7 -  "array_ran a h = {e. Some e \<in> set (get_array a h)}"
5.8 +  "array_ran a h = {e. Some e \<in> set (get_array h a)}"
5.9      -- {*FIXME*}
5.10
5.11 -lemma array_ranI: "\<lbrakk> Some b = get_array a h ! i; i < Array.length h a \<rbrakk> \<Longrightarrow> b \<in> array_ran a h"
5.12 +lemma array_ranI: "\<lbrakk> Some b = get_array h a ! i; i < Array.length h a \<rbrakk> \<Longrightarrow> b \<in> array_ran a h"
5.13  unfolding array_ran_def Array.length_def by simp
5.14
5.15  lemma array_ran_upd_array_Some:
5.16    assumes "cl \<in> array_ran a (Array.update a i (Some b) h)"
5.17    shows "cl \<in> array_ran a h \<or> cl = b"
5.18  proof -
5.19 -  have "set (get_array a h[i := Some b]) \<subseteq> insert (Some b) (set (get_array a h))" by (rule set_update_subset_insert)
5.20 +  have "set (get_array h a[i := Some b]) \<subseteq> insert (Some b) (set (get_array h a))" by (rule set_update_subset_insert)
5.21    with assms show ?thesis
5.22      unfolding array_ran_def Array.update_def by fastsimp
5.23  qed
5.24 @@ -139,7 +139,7 @@
5.25    assumes "cl \<in> array_ran a (Array.update a i None h)"
5.26    shows "cl \<in> array_ran a h"
5.27  proof -
5.28 -  have "set (get_array a h[i := None]) \<subseteq> insert None (set (get_array a h))" by (rule set_update_subset_insert)
5.29 +  have "set (get_array h a[i := None]) \<subseteq> insert None (set (get_array h a))" by (rule set_update_subset_insert)
5.30    with assms show ?thesis
5.31      unfolding array_ran_def Array.update_def by auto
5.32  qed
5.33 @@ -477,7 +477,7 @@
5.34      fix clj
5.35      let ?rs = "merge (remove l cli) (remove (compl l) clj)"
5.36      let ?rs' = "merge (remove (compl l) cli) (remove l clj)"
5.37 -    assume "h = h'" "Some clj = get_array a h' ! j" "j < Array.length h' a"
5.38 +    assume "h = h'" "Some clj = get_array h' a ! j" "j < Array.length h' a"
5.39      with correct_a have clj: "correctClause r clj" "sorted clj" "distinct clj"
5.40        unfolding correctArray_def by (auto intro: array_ranI)
5.41      with clj l_not_zero correct_cli
5.42 @@ -491,7 +491,7 @@
5.43    }
5.44    {
5.45      fix v clj
5.46 -    assume "Some clj = get_array a h ! j" "j < Array.length h a"
5.47 +    assume "Some clj = get_array h a ! j" "j < Array.length h a"
5.48      with correct_a have clj: "correctClause r clj \<and> sorted clj \<and> distinct clj"
5.49        unfolding correctArray_def by (auto intro: array_ranI)
5.50      assume "crel (res_thm' l cli clj) h h' rs"
```
```     6.1 --- a/src/HOL/Imperative_HOL/ex/Subarray.thy	Tue Jul 13 16:00:56 2010 +0200
6.2 +++ b/src/HOL/Imperative_HOL/ex/Subarray.thy	Tue Jul 13 16:12:40 2010 +0200
6.3 @@ -9,7 +9,7 @@
6.4  begin
6.5
6.6  definition subarray :: "nat \<Rightarrow> nat \<Rightarrow> ('a::heap) array \<Rightarrow> heap \<Rightarrow> 'a list" where
6.7 -  "subarray n m a h \<equiv> sublist' n m (get_array a h)"
6.8 +  "subarray n m a h \<equiv> sublist' n m (get_array h a)"
6.9
6.10  lemma subarray_upd: "i \<ge> m \<Longrightarrow> subarray n m a (Array.update a i v h) = subarray n m a h"
6.11  apply (simp add: subarray_def Array.update_def)
6.12 @@ -30,7 +30,7 @@
6.13  lemma subarray_Nil: "n \<ge> m \<Longrightarrow> subarray n m a h = []"
6.14  by (simp add: subarray_def sublist'_Nil')
6.15
6.16 -lemma subarray_single: "\<lbrakk> n < Array.length h a \<rbrakk> \<Longrightarrow> subarray n (Suc n) a h = [get_array a h ! n]"
6.17 +lemma subarray_single: "\<lbrakk> n < Array.length h a \<rbrakk> \<Longrightarrow> subarray n (Suc n) a h = [get_array h a ! n]"
6.18  by (simp add: subarray_def length_def sublist'_single)
6.19
6.20  lemma length_subarray: "m \<le> Array.length h a \<Longrightarrow> List.length (subarray n m a h) = m - n"
6.21 @@ -39,11 +39,11 @@
6.22  lemma length_subarray_0: "m \<le> Array.length h a \<Longrightarrow> List.length (subarray 0 m a h) = m"
6.24
6.25 -lemma subarray_nth_array_Cons: "\<lbrakk> i < Array.length h a; i < j \<rbrakk> \<Longrightarrow> (get_array a h ! i) # subarray (Suc i) j a h = subarray i j a h"
6.26 +lemma subarray_nth_array_Cons: "\<lbrakk> i < Array.length h a; i < j \<rbrakk> \<Longrightarrow> (get_array h a ! i) # subarray (Suc i) j a h = subarray i j a h"
6.27  unfolding Array.length_def subarray_def
6.29
6.30 -lemma subarray_nth_array_back: "\<lbrakk> i < j; j \<le> Array.length h a\<rbrakk> \<Longrightarrow> subarray i j a h = subarray i (j - 1) a h @ [get_array a h ! (j - 1)]"
6.31 +lemma subarray_nth_array_back: "\<lbrakk> i < j; j \<le> Array.length h a\<rbrakk> \<Longrightarrow> subarray i j a h = subarray i (j - 1) a h @ [get_array h a ! (j - 1)]"
6.32  unfolding Array.length_def subarray_def
6.34
6.35 @@ -51,21 +51,21 @@
6.36  unfolding subarray_def
6.38
6.39 -lemma subarray_all: "subarray 0 (Array.length h a) a h = get_array a h"
6.40 +lemma subarray_all: "subarray 0 (Array.length h a) a h = get_array h a"
6.41  unfolding Array.length_def subarray_def
6.43
6.44 -lemma nth_subarray: "\<lbrakk> k < j - i; j \<le> Array.length h a \<rbrakk> \<Longrightarrow> subarray i j a h ! k = get_array a h ! (i + k)"
6.45 +lemma nth_subarray: "\<lbrakk> k < j - i; j \<le> Array.length h a \<rbrakk> \<Longrightarrow> subarray i j a h ! k = get_array h a ! (i + k)"
6.46  unfolding Array.length_def subarray_def
6.48
6.49 -lemma subarray_eq_samelength_iff: "Array.length h a = Array.length h' a \<Longrightarrow> (subarray i j a h = subarray i j a h') = (\<forall>i'. i \<le> i' \<and> i' < j \<longrightarrow> get_array a h ! i' = get_array a h' ! i')"
6.50 +lemma subarray_eq_samelength_iff: "Array.length h a = Array.length h' a \<Longrightarrow> (subarray i j a h = subarray i j a h') = (\<forall>i'. i \<le> i' \<and> i' < j \<longrightarrow> get_array h a ! i' = get_array h' a ! i')"
6.51  unfolding Array.length_def subarray_def by (rule sublist'_eq_samelength_iff)
6.52
6.53 -lemma all_in_set_subarray_conv: "(\<forall>j. j \<in> set (subarray l r a h) \<longrightarrow> P j) = (\<forall>k. l \<le> k \<and> k < r \<and> k < Array.length h a \<longrightarrow> P (get_array a h ! k))"
6.54 +lemma all_in_set_subarray_conv: "(\<forall>j. j \<in> set (subarray l r a h) \<longrightarrow> P j) = (\<forall>k. l \<le> k \<and> k < r \<and> k < Array.length h a \<longrightarrow> P (get_array h a ! k))"
6.55  unfolding subarray_def Array.length_def by (rule all_in_set_sublist'_conv)
6.56
6.57 -lemma ball_in_set_subarray_conv: "(\<forall>j \<in> set (subarray l r a h). P j) = (\<forall>k. l \<le> k \<and> k < r \<and> k < Array.length h a \<longrightarrow> P (get_array a h ! k))"
6.58 +lemma ball_in_set_subarray_conv: "(\<forall>j \<in> set (subarray l r a h). P j) = (\<forall>k. l \<le> k \<and> k < r \<and> k < Array.length h a \<longrightarrow> P (get_array h a ! k))"
6.59  unfolding subarray_def Array.length_def by (rule ball_in_set_sublist'_conv)
6.60
6.61  end
6.62 \ No newline at end of file
```