author  haftmann 
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permissions  rwrr 
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(* Title: HOL/Imperative_HOL/Array.thy 
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Author: John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen 
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*) 

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header {* Monadic arrays *} 

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theory Array 

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imports Heap_Monad 
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begin 
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subsection {* Primitive layer *} 
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definition 
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array_present :: "'a\<Colon>heap array \<Rightarrow> heap \<Rightarrow> bool" where 
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"array_present a h \<longleftrightarrow> addr_of_array a < lim h" 
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definition 
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get_array :: "'a\<Colon>heap array \<Rightarrow> heap \<Rightarrow> 'a list" where 
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"get_array a h = map from_nat (arrays h (TYPEREP('a)) (addr_of_array a))" 
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definition 
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set_array :: "'a\<Colon>heap array \<Rightarrow> 'a list \<Rightarrow> heap \<Rightarrow> heap" where 
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"set_array a x = 
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arrays_update (\<lambda>h. h(TYPEREP('a) := ((h(TYPEREP('a))) (addr_of_array a:=map to_nat x))))" 
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definition array :: "'a list \<Rightarrow> heap \<Rightarrow> 'a\<Colon>heap array \<times> heap" where 
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"array xs h = (let 
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l = lim h; 
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r = Array l; 
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h'' = set_array r xs (h\<lparr>lim := l + 1\<rparr>) 
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in (r, h''))" 
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definition length :: "'a\<Colon>heap array \<Rightarrow> heap \<Rightarrow> nat" where 
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"length a h = List.length (get_array a h)" 
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definition change :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> heap \<Rightarrow> heap" where 
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"change a i x h = set_array a ((get_array a h)[i:=x]) h" 
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text {* Properties of imperative arrays *} 
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text {* FIXME: Does there exist a "canonical" array axiomatisation in 
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the literature? *} 
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definition noteq_arrs :: "('a\<Colon>heap) array \<Rightarrow> ('b\<Colon>heap) array \<Rightarrow> bool" (infix "=!!=" 70) where 
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"r =!!= s \<longleftrightarrow> TYPEREP('a) \<noteq> TYPEREP('b) \<or> addr_of_array r \<noteq> addr_of_array s" 
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lemma noteq_arrs_sym: "a =!!= b \<Longrightarrow> b =!!= a" 
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and unequal_arrs [simp]: "a \<noteq> a' \<longleftrightarrow> a =!!= a'" 
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unfolding noteq_arrs_def by auto 
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lemma noteq_arrs_irrefl: "r =!!= r \<Longrightarrow> False" 
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unfolding noteq_arrs_def by auto 
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lemma present_new_arr: "array_present a h \<Longrightarrow> a =!!= fst (array xs h)" 
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by (simp add: array_present_def noteq_arrs_def array_def Let_def) 
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lemma array_get_set_eq [simp]: "get_array r (set_array r x h) = x" 
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by (simp add: get_array_def set_array_def o_def) 
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lemma array_get_set_neq [simp]: "r =!!= s \<Longrightarrow> get_array r (set_array s x h) = get_array r h" 
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by (simp add: noteq_arrs_def get_array_def set_array_def) 
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lemma set_array_same [simp]: 
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"set_array r x (set_array r y h) = set_array r x h" 
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by (simp add: set_array_def) 
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lemma array_set_set_swap: 
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"r =!!= r' \<Longrightarrow> set_array r x (set_array r' x' h) = set_array r' x' (set_array r x h)" 
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by (simp add: Let_def expand_fun_eq noteq_arrs_def set_array_def) 
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lemma get_array_change_eq [simp]: 
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"get_array a (change a i v h) = (get_array a h) [i := v]" 
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by (simp add: change_def) 
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lemma nth_change_array_neq_array [simp]: 
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"a =!!= b \<Longrightarrow> get_array a (change b j v h) ! i = get_array a h ! i" 
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by (simp add: change_def noteq_arrs_def) 
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lemma get_arry_array_change_elem_neqIndex [simp]: 
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"i \<noteq> j \<Longrightarrow> get_array a (change a j v h) ! i = get_array a h ! i" 
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by simp 
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lemma length_change [simp]: 
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"length a (change b i v h) = length a h" 
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by (simp add: change_def length_def set_array_def get_array_def) 
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lemma change_swap_neqArray: 
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"a =!!= a' \<Longrightarrow> 
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change a i v (change a' i' v' h) 
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= change a' i' v' (change a i v h)" 
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apply (unfold change_def) 
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apply simp 
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apply (subst array_set_set_swap, assumption) 
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apply (subst array_get_set_neq) 
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apply (erule noteq_arrs_sym) 
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apply (simp) 
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done 
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lemma change_swap_neqIndex: 
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"\<lbrakk> i \<noteq> i' \<rbrakk> \<Longrightarrow> change a i v (change a i' v' h) = change a i' v' (change a i v h)" 
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by (auto simp add: change_def array_set_set_swap list_update_swap) 
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lemma get_array_init_array_list: 
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"get_array (fst (array ls h)) (snd (array ls' h)) = ls'" 
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by (simp add: Let_def split_def array_def) 
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lemma set_array: 
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"set_array (fst (array ls h)) 
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new_ls (snd (array ls h)) 
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= snd (array new_ls h)" 
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by (simp add: Let_def split_def array_def) 
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lemma array_present_change [simp]: 
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"array_present a (change b i v h) = array_present a h" 
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subsection {* Primitives *} 
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definition 

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new :: "nat \<Rightarrow> 'a\<Colon>heap \<Rightarrow> 'a array Heap" where 

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[code del]: "new n x = Heap_Monad.heap (Array.array (replicate n x))" 
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definition 

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of_list :: "'a\<Colon>heap list \<Rightarrow> 'a array Heap" where 

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[code del]: "of_list xs = Heap_Monad.heap (Array.array xs)" 
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definition 

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len :: "'a\<Colon>heap array \<Rightarrow> nat Heap" where 
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[code del]: "len arr = Heap_Monad.heap (\<lambda>h. (Array.length arr h, h))" 
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definition 

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nth :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a Heap" 

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where 

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[code del]: "nth a i = (do len \<leftarrow> len a; 
26170  137 
(if i < len 
138 
then Heap_Monad.heap (\<lambda>h. (get_array a h ! i, h)) 

37709  139 
else raise ''array lookup: index out of range'') 
26170  140 
done)" 
141 

142 
definition 

143 
upd :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a\<Colon>heap array Heap" 

144 
where 

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[code del]: "upd i x a = (do len \<leftarrow> len a; 
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(if i < len 
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then Heap_Monad.heap (\<lambda>h. (a, change a i x h)) 
37709  148 
else raise ''array update: index out of range'') 
26170  149 
done)" 
150 

151 
lemma upd_return: 

152 
"upd i x a \<guillemotright> return a = upd i x a" 

37709  153 
by (rule Heap_eqI) (simp add: upd_def bindM_def split: option.split) 
26170  154 

155 

156 
subsection {* Derivates *} 

157 

158 
definition 

159 
map_entry :: "nat \<Rightarrow> ('a\<Colon>heap \<Rightarrow> 'a) \<Rightarrow> 'a array \<Rightarrow> 'a array Heap" 

160 
where 

161 
"map_entry i f a = (do 

162 
x \<leftarrow> nth a i; 

163 
upd i (f x) a 

164 
done)" 

165 

166 
definition 

167 
swap :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a Heap" 

168 
where 

169 
"swap i x a = (do 

170 
y \<leftarrow> nth a i; 

171 
upd i x a; 

27596  172 
return y 
26170  173 
done)" 
174 

175 
definition 

176 
make :: "nat \<Rightarrow> (nat \<Rightarrow> 'a\<Colon>heap) \<Rightarrow> 'a array Heap" 

177 
where 

178 
"make n f = of_list (map f [0 ..< n])" 

179 

180 
definition 

181 
freeze :: "'a\<Colon>heap array \<Rightarrow> 'a list Heap" 

182 
where 

183 
"freeze a = (do 

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n \<leftarrow> len a; 
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mapM (nth a) [0..<n] 
186 
done)" 

187 

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definition 
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map :: "('a\<Colon>heap \<Rightarrow> 'a) \<Rightarrow> 'a array \<Rightarrow> 'a array Heap" 
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where 
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"map f a = (do 
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n \<leftarrow> len a; 
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mapM (\<lambda>n. map_entry n f a) [0..<n]; 
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return a 
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done)" 
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196 

26170  197 

198 

199 
subsection {* Properties *} 

200 

28562  201 
lemma array_make [code]: 
26170  202 
"Array.new n x = make n (\<lambda>_. x)" 
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by (rule Heap_eqI) (simp add: make_def new_def map_replicate_trivial of_list_def) 
26170  204 

28562  205 
lemma array_of_list_make [code]: 
26170  206 
"of_list xs = make (List.length xs) (\<lambda>n. xs ! n)" 
37709  207 
by (rule Heap_eqI) (simp add: make_def map_nth) 
26170  208 

26182  209 

210 
subsection {* Code generator setup *} 

211 

212 
subsubsection {* Logical intermediate layer *} 

213 

214 
definition new' where 

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[code del]: "new' = Array.new o Code_Numeral.nat_of" 
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hide_const (open) new' 
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lemma [code]: 
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"Array.new = Array.new' o Code_Numeral.of_nat" 
26182  219 
by (simp add: new'_def o_def) 
220 

221 
definition of_list' where 

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[code del]: "of_list' i xs = Array.of_list (take (Code_Numeral.nat_of i) xs)" 
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hide_const (open) of_list' 
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lemma [code]: 
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"Array.of_list xs = Array.of_list' (Code_Numeral.of_nat (List.length xs)) xs" 
26182  226 
by (simp add: of_list'_def) 
227 

228 
definition make' where 

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[code del]: "make' i f = Array.make (Code_Numeral.nat_of i) (f o Code_Numeral.of_nat)" 
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hide_const (open) make' 
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lemma [code]: 
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"Array.make n f = Array.make' (Code_Numeral.of_nat n) (f o Code_Numeral.nat_of)" 
26182  233 
by (simp add: make'_def o_def) 
234 

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definition len' where 
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[code del]: "len' a = Array.len a \<guillemotright>= (\<lambda>n. return (Code_Numeral.of_nat n))" 
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hide_const (open) len' 
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lemma [code]: 
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"Array.len a = Array.len' a \<guillemotright>= (\<lambda>i. return (Code_Numeral.nat_of i))" 
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by (simp add: len'_def) 
26182  241 

242 
definition nth' where 

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[code del]: "nth' a = Array.nth a o Code_Numeral.nat_of" 
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hide_const (open) nth' 
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lemma [code]: 
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"Array.nth a n = Array.nth' a (Code_Numeral.of_nat n)" 
26182  247 
by (simp add: nth'_def) 
248 

249 
definition upd' where 

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[code del]: "upd' a i x = Array.upd (Code_Numeral.nat_of i) x a \<guillemotright> return ()" 
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hide_const (open) upd' 
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lemma [code]: 
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"Array.upd i x a = Array.upd' a (Code_Numeral.of_nat i) x \<guillemotright> return a" 
37709  254 
by (simp add: upd'_def upd_return) 
26182  255 

256 

257 
subsubsection {* SML *} 

258 

259 
code_type array (SML "_/ array") 

260 
code_const Array (SML "raise/ (Fail/ \"bare Array\")") 

26752  261 
code_const Array.new' (SML "(fn/ ()/ =>/ Array.array/ ((_),/ (_)))") 
35846  262 
code_const Array.of_list' (SML "(fn/ ()/ =>/ Array.fromList/ _)") 
26752  263 
code_const Array.make' (SML "(fn/ ()/ =>/ Array.tabulate/ ((_),/ (_)))") 
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code_const Array.len' (SML "(fn/ ()/ =>/ Array.length/ _)") 
26752  265 
code_const Array.nth' (SML "(fn/ ()/ =>/ Array.sub/ ((_),/ (_)))") 
266 
code_const Array.upd' (SML "(fn/ ()/ =>/ Array.update/ ((_),/ (_),/ (_)))") 

26182  267 

268 
code_reserved SML Array 

269 

270 

271 
subsubsection {* OCaml *} 

272 

273 
code_type array (OCaml "_/ array") 

274 
code_const Array (OCaml "failwith/ \"bare Array\"") 

32580  275 
code_const Array.new' (OCaml "(fun/ ()/ >/ Array.make/ (Big'_int.int'_of'_big'_int/ _)/ _)") 
35846  276 
code_const Array.of_list' (OCaml "(fun/ ()/ >/ Array.of'_list/ _)") 
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code_const Array.len' (OCaml "(fun/ ()/ >/ Big'_int.big'_int'_of'_int/ (Array.length/ _))") 
32580  278 
code_const Array.nth' (OCaml "(fun/ ()/ >/ Array.get/ _/ (Big'_int.int'_of'_big'_int/ _))") 
279 
code_const Array.upd' (OCaml "(fun/ ()/ >/ Array.set/ _/ (Big'_int.int'_of'_big'_int/ _)/ _)") 

26182  280 

281 
code_reserved OCaml Array 

282 

283 

284 
subsubsection {* Haskell *} 

285 

29793  286 
code_type array (Haskell "Heap.STArray/ Heap.RealWorld/ _") 
26182  287 
code_const Array (Haskell "error/ \"bare Array\"") 
29793  288 
code_const Array.new' (Haskell "Heap.newArray/ (0,/ _)") 
289 
code_const Array.of_list' (Haskell "Heap.newListArray/ (0,/ _)") 

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code_const Array.len' (Haskell "Heap.lengthArray") 
29793  291 
code_const Array.nth' (Haskell "Heap.readArray") 
292 
code_const Array.upd' (Haskell "Heap.writeArray") 

26182  293 

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hide_const (open) new map  {* avoid clashed with some popular names *} 
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295 

26170  296 
end 