some support for inital command markup;
tuned signature;
(* Title: HOL/Imperative_HOL/Array.thy Author: John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen*)header {* Monadic arrays *}theory Arrayimports Heap_Monadbeginsubsection {* Primitives *}definition present :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> bool" where "present h a \<longleftrightarrow> addr_of_array a < lim h"definition get :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a list" where "get 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))))"definition alloc :: "'a list \<Rightarrow> heap \<Rightarrow> 'a\<Colon>heap array \<times> heap" where "alloc xs h = (let l = lim h; r = Array l; h'' = set r xs (h\<lparr>lim := l + 1\<rparr>) in (r, h''))"definition length :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> nat" where "length h a = List.length (get 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 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"subsection {* Monad operations *}definition new :: "nat \<Rightarrow> 'a\<Colon>heap \<Rightarrow> 'a array Heap" where [code del]: "new n x = Heap_Monad.heap (alloc (replicate n x))"definition of_list :: "'a\<Colon>heap list \<Rightarrow> 'a array Heap" where [code del]: "of_list xs = Heap_Monad.heap (alloc xs)"definition make :: "nat \<Rightarrow> (nat \<Rightarrow> 'a\<Colon>heap) \<Rightarrow> 'a array Heap" where [code del]: "make n f = Heap_Monad.heap (alloc (map f [0 ..< n]))"definition len :: "'a\<Colon>heap array \<Rightarrow> nat Heap" where [code del]: "len a = Heap_Monad.tap (\<lambda>h. length h a)"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 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) (\<lambda>h. (a, update 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 h a) (\<lambda>h. (a, update a i (f (get 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 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 h a)"subsection {* Properties *}text {* FIXME: Does there exist a "canonical" array axiomatisation inthe literature? *}text {* Primitives *}lemma noteq_sym: "a =!!= b \<Longrightarrow> b =!!= a" and unequal [simp]: "a \<noteq> a' \<longleftrightarrow> a =!!= a'" unfolding noteq_def by autolemma noteq_irrefl: "r =!!= r \<Longrightarrow> False" unfolding noteq_def by autolemma present_alloc_noteq: "present h a \<Longrightarrow> a =!!= fst (alloc xs h)" by (simp add: present_def noteq_def alloc_def Let_def)lemma get_set_eq [simp]: "get (set r x h) r = x" by (simp add: get_def set_def o_def)lemma get_set_neq [simp]: "r =!!= s \<Longrightarrow> get (set s x h) r = get h r" by (simp add: noteq_def get_def set_def)lemma set_same [simp]: "set r x (set r y h) = set r x h" by (simp add: set_def)lemma set_set_swap: "r =!!= r' \<Longrightarrow> set r x (set r' x' h) = set r' x' (set r x h)" by (simp add: Let_def fun_eq_iff noteq_def set_def)lemma get_update_eq [simp]: "get (update a i v h) a = (get h a) [i := v]" by (simp add: update_def)lemma nth_update_neq [simp]: "a =!!= b \<Longrightarrow> get (update b j v h) a ! i = get h a ! i" by (simp add: update_def noteq_def)lemma get_update_elem_neqIndex [simp]: "i \<noteq> j \<Longrightarrow> get (update a j v h) a ! i = get h a ! i" by simplemma length_update [simp]: "length (update b i v h) = length h" by (simp add: update_def length_def set_def get_def fun_eq_iff)lemma update_swap_neq: "a =!!= a' \<Longrightarrow> update a i v (update a' i' v' h) = update a' i' v' (update a i v h)"apply (unfold update_def)apply simpapply (subst set_set_swap, assumption)apply (subst get_set_neq)apply (erule noteq_sym)apply simpdonelemma update_swap_neqIndex: "\<lbrakk> i \<noteq> i' \<rbrakk> \<Longrightarrow> update a i v (update a i' v' h) = update a i' v' (update a i v h)" by (auto simp add: update_def set_set_swap list_update_swap)lemma get_alloc: "get (snd (alloc xs h)) (fst (alloc ys h)) = xs" by (simp add: Let_def split_def alloc_def)lemma length_alloc: "length (snd (alloc (xs :: 'a::heap list) h)) (fst (alloc (ys :: 'a list) h)) = List.length xs" by (simp add: Array.length_def get_alloc)lemma set: "set (fst (alloc ls h)) new_ls (snd (alloc ls h)) = snd (alloc new_ls h)" by (simp add: Let_def split_def alloc_def)lemma present_update [simp]: "present (update b i v h) = present h" by (simp add: update_def present_def set_def get_def fun_eq_iff)lemma present_alloc [simp]: "present (snd (alloc xs h)) (fst (alloc xs h))" by (simp add: present_def alloc_def set_def Let_def)lemma not_present_alloc [simp]: "\<not> present h (fst (alloc xs h))" by (simp add: present_def alloc_def Let_def)text {* Monad operations *}lemma execute_new [execute_simps]: "execute (new n x) h = Some (alloc (replicate n x) h)" by (simp add: new_def execute_simps)lemma success_newI [success_intros]: "success (new n x) h" by (auto intro: success_intros simp add: new_def)lemma effect_newI [effect_intros]: assumes "(a, h') = alloc (replicate n x) h" shows "effect (new n x) h h' a" by (rule effectI) (simp add: assms execute_simps)lemma effect_newE [effect_elims]: assumes "effect (new n x) h h' r" obtains "r = fst (alloc (replicate n x) h)" "h' = snd (alloc (replicate n x) h)" "get h' r = replicate n x" "present h' r" "\<not> present h r" using assms by (rule effectE) (simp add: get_alloc execute_simps)lemma execute_of_list [execute_simps]: "execute (of_list xs) h = Some (alloc xs h)" by (simp add: of_list_def execute_simps)lemma success_of_listI [success_intros]: "success (of_list xs) h" by (auto intro: success_intros simp add: of_list_def)lemma effect_of_listI [effect_intros]: assumes "(a, h') = alloc xs h" shows "effect (of_list xs) h h' a" by (rule effectI) (simp add: assms execute_simps)lemma effect_of_listE [effect_elims]: assumes "effect (of_list xs) h h' r" obtains "r = fst (alloc xs h)" "h' = snd (alloc xs h)" "get h' r = xs" "present h' r" "\<not> present h r" using assms by (rule effectE) (simp add: get_alloc execute_simps)lemma execute_make [execute_simps]: "execute (make n f) h = Some (alloc (map f [0 ..< n]) h)" by (simp add: make_def execute_simps)lemma success_makeI [success_intros]: "success (make n f) h" by (auto intro: success_intros simp add: make_def)lemma effect_makeI [effect_intros]: assumes "(a, h') = alloc (map f [0 ..< n]) h" shows "effect (make n f) h h' a" by (rule effectI) (simp add: assms execute_simps)lemma effect_makeE [effect_elims]: assumes "effect (make n f) h h' r" obtains "r = fst (alloc (map f [0 ..< n]) h)" "h' = snd (alloc (map f [0 ..< n]) h)" "get h' r = map f [0 ..< n]" "present h' r" "\<not> present h r" using assms by (rule effectE) (simp add: get_alloc execute_simps)lemma execute_len [execute_simps]: "execute (len a) h = Some (length h a, h)" by (simp add: len_def execute_simps)lemma success_lenI [success_intros]: "success (len a) h" by (auto intro: success_intros simp add: len_def)lemma effect_lengthI [effect_intros]: assumes "h' = h" "r = length h a" shows "effect (len a) h h' r" by (rule effectI) (simp add: assms execute_simps)lemma effect_lengthE [effect_elims]: assumes "effect (len a) h h' r" obtains "r = length h' a" "h' = h" using assms by (rule effectE) (simp add: execute_simps)lemma execute_nth [execute_simps]: "i < length h a \<Longrightarrow> execute (nth a i) h = Some (get h a ! i, h)" "i \<ge> length h a \<Longrightarrow> execute (nth a i) h = None" by (simp_all add: nth_def execute_simps)lemma success_nthI [success_intros]: "i < length h a \<Longrightarrow> success (nth a i) h" by (auto intro: success_intros simp add: nth_def)lemma effect_nthI [effect_intros]: assumes "i < length h a" "h' = h" "r = get h a ! i" shows "effect (nth a i) h h' r" by (rule effectI) (insert assms, simp add: execute_simps)lemma effect_nthE [effect_elims]: assumes "effect (nth a i) h h' r" obtains "i < length h a" "r = get h a ! i" "h' = h" using assms by (rule effectE) (erule successE, cases "i < length h a", simp_all add: execute_simps)lemma execute_upd [execute_simps]: "i < length h a \<Longrightarrow> execute (upd i x a) h = Some (a, update a i x h)" "i \<ge> length h a \<Longrightarrow> execute (upd i x a) h = None" by (simp_all add: upd_def execute_simps)lemma success_updI [success_intros]: "i < length h a \<Longrightarrow> success (upd i x a) h" by (auto intro: success_intros simp add: upd_def)lemma effect_updI [effect_intros]: assumes "i < length h a" "h' = update a i v h" shows "effect (upd i v a) h h' a" by (rule effectI) (insert assms, simp add: execute_simps)lemma effect_updE [effect_elims]: assumes "effect (upd i v a) h h' r" obtains "r = a" "h' = update a i v h" "i < length h a" using assms by (rule effectE) (erule successE, cases "i < length h a", simp_all add: execute_simps)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 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)lemma success_map_entryI [success_intros]: "i < length h a \<Longrightarrow> success (map_entry i f a) h" by (auto intro: success_intros simp add: map_entry_def)lemma effect_map_entryI [effect_intros]: assumes "i < length h a" "h' = update a i (f (get h a ! i)) h" "r = a" shows "effect (map_entry i f a) h h' r" by (rule effectI) (insert assms, simp add: execute_simps)lemma effect_map_entryE [effect_elims]: assumes "effect (map_entry i f a) h h' r" obtains "r = a" "h' = update a i (f (get h a ! i)) h" "i < length h a" using assms by (rule effectE) (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 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)lemma success_swapI [success_intros]: "i < length h a \<Longrightarrow> success (swap i x a) h" by (auto intro: success_intros simp add: swap_def)lemma effect_swapI [effect_intros]: assumes "i < length h a" "h' = update a i x h" "r = get h a ! i" shows "effect (swap i x a) h h' r" by (rule effectI) (insert assms, simp add: execute_simps)lemma effect_swapE [effect_elims]: assumes "effect (swap i x a) h h' r" obtains "r = get h a ! i" "h' = update a i x h" "i < length h a" using assms by (rule effectE) (erule successE, cases "i < length h a", simp_all add: execute_simps)lemma execute_freeze [execute_simps]: "execute (freeze a) h = Some (get h a, h)" by (simp add: freeze_def execute_simps)lemma success_freezeI [success_intros]: "success (freeze a) h" by (auto intro: success_intros simp add: freeze_def)lemma effect_freezeI [effect_intros]: assumes "h' = h" "r = get h a" shows "effect (freeze a) h h' r" by (rule effectI) (insert assms, simp add: execute_simps)lemma effect_freezeE [effect_elims]: assumes "effect (freeze a) h h' r" obtains "h' = h" "r = get h a" using assms by (rule effectE) (simp add: execute_simps)lemma upd_return: "upd i x a \<guillemotright> return a = upd i x a" by (rule Heap_eqI) (simp add: bind_def guard_def upd_def execute_simps)lemma array_make: "new n x = make n (\<lambda>_. x)" by (rule Heap_eqI) (simp add: map_replicate_trivial execute_simps)lemma array_of_list_make [code]: "of_list xs = make (List.length xs) (\<lambda>n. xs ! n)" by (rule Heap_eqI) (simp add: map_nth execute_simps)hide_const (open) present get set alloc length update noteq new of_list make len nth upd map_entry swap freezesubsection {* Code generator setup *}subsubsection {* Logical intermediate layer *}definition new' where [code del]: "new' = Array.new o Code_Numeral.nat_of"lemma [code]: "Array.new = new' o Code_Numeral.of_nat" by (simp add: new'_def o_def)definition make' where [code del]: "make' i f = Array.make (Code_Numeral.nat_of i) (f o Code_Numeral.of_nat)"lemma [code]: "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))"lemma [code]: "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"lemma [code]: "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 ()"lemma [code]: "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]: "Array.map_entry i f a = do { x \<leftarrow> Array.nth a i; Array.upd i (f x) a }" by (rule Heap_eqI) (simp add: bind_def guard_def map_entry_def execute_simps)lemma [code]: "Array.swap i x a = do { y \<leftarrow> Array.nth a i; Array.upd i x a; return y }" by (rule Heap_eqI) (simp add: bind_def guard_def swap_def execute_simps)lemma [code]: "Array.freeze a = do { n \<leftarrow> Array.len a; Heap_Monad.fold_map (\<lambda>i. Array.nth a i) [0..<n] }"proof (rule Heap_eqI) fix h have *: "List.map (\<lambda>x. fst (the (if x < Array.length h a then Some (Array.get h a ! x, h) else None))) [0..<Array.length h a] = List.map (List.nth (Array.get 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 (Array.get 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) done then have "execute (do { n \<leftarrow> Array.len a; Heap_Monad.fold_map (Array.nth a) [0..<n] }) h = Some (Array.get 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; Heap_Monad.fold_map (Array.nth a) [0..<n] }) h" by (simp add: execute_simps)qedhide_const (open) new' make' len' nth' upd'text {* SML *}code_type array (SML "_/ array")code_const Array (SML "raise/ (Fail/ \"bare Array\")")code_const Array.new' (SML "(fn/ ()/ =>/ Array.array/ ((_),/ (_)))")code_const Array.of_list (SML "(fn/ ()/ =>/ Array.fromList/ _)")code_const Array.make' (SML "(fn/ ()/ =>/ Array.tabulate/ ((_),/ (_)))")code_const Array.len' (SML "(fn/ ()/ =>/ Array.length/ _)")code_const Array.nth' (SML "(fn/ ()/ =>/ Array.sub/ ((_),/ (_)))")code_const Array.upd' (SML "(fn/ ()/ =>/ Array.update/ ((_),/ (_),/ (_)))")code_const "HOL.equal :: 'a array \<Rightarrow> 'a array \<Rightarrow> bool" (SML infixl 6 "=")code_reserved SML Arraytext {* OCaml *}code_type array (OCaml "_/ array")code_const Array (OCaml "failwith/ \"bare Array\"")code_const Array.new' (OCaml "(fun/ ()/ ->/ Array.make/ (Big'_int.int'_of'_big'_int/ _)/ _)")code_const Array.of_list (OCaml "(fun/ ()/ ->/ Array.of'_list/ _)")code_const Array.make' (OCaml "(fun/ ()/ ->/ Array.init/ (Big'_int.int'_of'_big'_int/ _)/ (fun k'_ ->/ _/ (Big'_int.big'_int'_of'_int/ k'_)))")code_const Array.len' (OCaml "(fun/ ()/ ->/ Big'_int.big'_int'_of'_int/ (Array.length/ _))")code_const Array.nth' (OCaml "(fun/ ()/ ->/ Array.get/ _/ (Big'_int.int'_of'_big'_int/ _))")code_const Array.upd' (OCaml "(fun/ ()/ ->/ Array.set/ _/ (Big'_int.int'_of'_big'_int/ _)/ _)")code_const "HOL.equal :: 'a array \<Rightarrow> 'a array \<Rightarrow> bool" (OCaml infixl 4 "=")code_reserved OCaml Arraytext {* Haskell *}code_type array (Haskell "Heap.STArray/ Heap.RealWorld/ _")code_const Array (Haskell "error/ \"bare Array\"")code_const Array.new' (Haskell "Heap.newArray")code_const Array.of_list (Haskell "Heap.newListArray")code_const Array.make' (Haskell "Heap.newFunArray")code_const Array.len' (Haskell "Heap.lengthArray")code_const Array.nth' (Haskell "Heap.readArray")code_const Array.upd' (Haskell "Heap.writeArray")code_const "HOL.equal :: 'a array \<Rightarrow> 'a array \<Rightarrow> bool" (Haskell infix 4 "==")code_instance array :: HOL.equal (Haskell -)text {* Scala *}code_type array (Scala "!collection.mutable.ArraySeq[_]")code_const Array (Scala "!sys.error(\"bare Array\")")code_const Array.new' (Scala "('_: Unit)/ => / Array.alloc((_))((_))")code_const Array.make' (Scala "('_: Unit)/ =>/ Array.make((_))((_))")code_const Array.len' (Scala "('_: Unit)/ =>/ Array.len((_))")code_const Array.nth' (Scala "('_: Unit)/ =>/ Array.nth((_), (_))")code_const Array.upd' (Scala "('_: Unit)/ =>/ Array.upd((_), (_), (_))")code_const Array.freeze (Scala "('_: Unit)/ =>/ Array.freeze((_))")code_const "HOL.equal :: 'a array \<Rightarrow> 'a array \<Rightarrow> bool" (Scala infixl 5 "==")end