
haftmann@31870: (*  Title:      HOL/Imperative_HOL/Array.thy
haftmann@26170:     Author:     John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen
haftmann@26170: *)
haftmann@26170: 
haftmann@26170: header {* Monadic arrays *}
haftmann@26170: 
haftmann@26170: theory Array
haftmann@31203: imports Heap_Monad
haftmann@26170: begin
haftmann@26170: 
haftmann@37752: subsection {* Primitives *}
haftmann@37719: 
haftmann@37752: definition (*FIXME present :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> bool" where*)
haftmann@37719:   array_present :: "'a\<Colon>heap array \<Rightarrow> heap \<Rightarrow> bool" where
haftmann@37719:   "array_present a h \<longleftrightarrow> addr_of_array a < lim h"
haftmann@37719: 
haftmann@37752: definition (*FIXME get :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a list" where*)
haftmann@37719:   get_array :: "'a\<Colon>heap array \<Rightarrow> heap \<Rightarrow> 'a list" where
haftmann@37719:   "get_array a h = map from_nat (arrays h (TYPEREP('a)) (addr_of_array a))"
haftmann@37719: 
haftmann@37752: definition (*FIXME set*)
haftmann@37719:   set_array :: "'a\<Colon>heap array \<Rightarrow> 'a list \<Rightarrow> heap \<Rightarrow> heap" where
haftmann@37719:   "set_array a x = 
haftmann@37719:   arrays_update (\<lambda>h. h(TYPEREP('a) := ((h(TYPEREP('a))) (addr_of_array a:=map to_nat x))))"
haftmann@37719: 
haftmann@37752: definition (*FIXME alloc*)
haftmann@37752:   array :: "'a list \<Rightarrow> heap \<Rightarrow> 'a\<Colon>heap array \<times> heap" where
haftmann@37719:   "array xs h = (let
haftmann@37719:      l = lim h;
haftmann@37719:      r = Array l;
haftmann@37719:      h'' = set_array r xs (h\<lparr>lim := l + 1\<rparr>)
haftmann@37719:    in (r, h''))"
haftmann@37719: 
haftmann@37752: definition (*FIXME length :: "heap \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> nat" where*)
haftmann@37752:   length :: "'a\<Colon>heap array \<Rightarrow> heap \<Rightarrow> nat" where
haftmann@37719:   "length a h = List.length (get_array a h)"
haftmann@37719:   
haftmann@37752: definition (*FIXME update*)
haftmann@37752:   change :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> heap \<Rightarrow> heap" where
haftmann@37719:   "change a i x h = set_array a ((get_array a h)[i:=x]) h"
haftmann@37719: 
haftmann@37752: definition (*FIXME noteq*)
haftmann@37752:   noteq_arrs :: "'a\<Colon>heap array \<Rightarrow> 'b\<Colon>heap array \<Rightarrow> bool" (infix "=!!=" 70) where
haftmann@37752:   "r =!!= s \<longleftrightarrow> TYPEREP('a) \<noteq> TYPEREP('b) \<or> addr_of_array r \<noteq> addr_of_array s"
haftmann@37752: 
haftmann@37752: 
haftmann@37752: subsection {* Monad operations *}
haftmann@37752: 
haftmann@37752: definition new :: "nat \<Rightarrow> 'a\<Colon>heap \<Rightarrow> 'a array Heap" where
haftmann@37752:   [code del]: "new n x = Heap_Monad.heap (array (replicate n x))"
haftmann@37752: 
haftmann@37752: definition of_list :: "'a\<Colon>heap list \<Rightarrow> 'a array Heap" where
haftmann@37752:   [code del]: "of_list xs = Heap_Monad.heap (array xs)"
haftmann@37752: 
haftmann@37752: definition make :: "nat \<Rightarrow> (nat \<Rightarrow> 'a\<Colon>heap) \<Rightarrow> 'a array Heap" where
haftmann@37752:   [code del]: "make n f = Heap_Monad.heap (array (map f [0 ..< n]))"
haftmann@37752: 
haftmann@37752: definition len :: "'a\<Colon>heap array \<Rightarrow> nat Heap" where
haftmann@37752:   [code del]: "len a = Heap_Monad.heap (\<lambda>h. (length a h, h))"
haftmann@37752: 
haftmann@37752: definition nth :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a Heap" where
haftmann@37752:   [code del]: "nth a i = Heap_Monad.guard (\<lambda>h. i < length a h)
haftmann@37752:     (\<lambda>h. (get_array a h ! i, h))"
haftmann@37752: 
haftmann@37752: definition upd :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a\<Colon>heap array Heap" where
haftmann@37752:   [code del]: "upd i x a = Heap_Monad.guard (\<lambda>h. i < length a h)
haftmann@37752:     (\<lambda>h. (a, change a i x h))"
haftmann@37752: 
haftmann@37752: definition map_entry :: "nat \<Rightarrow> ('a\<Colon>heap \<Rightarrow> 'a) \<Rightarrow> 'a array \<Rightarrow> 'a array Heap" where
haftmann@37752:   [code del]: "map_entry i f a = Heap_Monad.guard (\<lambda>h. i < length a h)
haftmann@37752:     (\<lambda>h. (a, change a i (f (get_array a h ! i)) h))"
haftmann@37752: 
haftmann@37752: definition swap :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a Heap" where
haftmann@37752:   [code del]: "swap i x a = Heap_Monad.guard (\<lambda>h. i < length a h)
haftmann@37752:     (\<lambda>h. (get_array a h ! i, change a i x h))"
haftmann@37752: 
haftmann@37752: definition freeze :: "'a\<Colon>heap array \<Rightarrow> 'a list Heap" where
haftmann@37752:   [code del]: "freeze a = Heap_Monad.heap (\<lambda>h. (get_array a h, h))"
haftmann@37752: 
haftmann@37752: 
haftmann@37752: subsection {* Properties *}
haftmann@37719: 
haftmann@37719: text {* FIXME: Does there exist a "canonical" array axiomatisation in
haftmann@37719: the literature?  *}
haftmann@37719: 
haftmann@37719: lemma noteq_arrs_sym: "a =!!= b \<Longrightarrow> b =!!= a"
haftmann@37719:   and unequal_arrs [simp]: "a \<noteq> a' \<longleftrightarrow> a =!!= a'"
haftmann@37719:   unfolding noteq_arrs_def by auto
haftmann@37719: 
haftmann@37719: lemma noteq_arrs_irrefl: "r =!!= r \<Longrightarrow> False"
haftmann@37719:   unfolding noteq_arrs_def by auto
haftmann@37719: 
haftmann@37719: lemma present_new_arr: "array_present a h \<Longrightarrow> a =!!= fst (array xs h)"
haftmann@37719:   by (simp add: array_present_def noteq_arrs_def array_def Let_def)
haftmann@37719: 
haftmann@37719: lemma array_get_set_eq [simp]: "get_array r (set_array r x h) = x"
haftmann@37719:   by (simp add: get_array_def set_array_def o_def)
haftmann@37719: 
haftmann@37719: lemma array_get_set_neq [simp]: "r =!!= s \<Longrightarrow> get_array r (set_array s x h) = get_array r h"
haftmann@37719:   by (simp add: noteq_arrs_def get_array_def set_array_def)
haftmann@37719: 
haftmann@37719: lemma set_array_same [simp]:
haftmann@37719:   "set_array r x (set_array r y h) = set_array r x h"
haftmann@37719:   by (simp add: set_array_def)
haftmann@37719: 
haftmann@37719: lemma array_set_set_swap:
haftmann@37719:   "r =!!= r' \<Longrightarrow> set_array r x (set_array r' x' h) = set_array r' x' (set_array r x h)"
haftmann@37719:   by (simp add: Let_def expand_fun_eq noteq_arrs_def set_array_def)
haftmann@37719: 
haftmann@37719: lemma get_array_change_eq [simp]:
haftmann@37719:   "get_array a (change a i v h) = (get_array a h) [i := v]"
haftmann@37719:   by (simp add: change_def)
haftmann@37719: 
haftmann@37719: lemma nth_change_array_neq_array [simp]:
haftmann@37719:   "a =!!= b \<Longrightarrow> get_array a (change b j v h) ! i = get_array a h ! i"
haftmann@37719:   by (simp add: change_def noteq_arrs_def)
haftmann@37719: 
haftmann@37719: lemma get_arry_array_change_elem_neqIndex [simp]:
haftmann@37719:   "i \<noteq> j \<Longrightarrow> get_array a (change a j v h) ! i = get_array a h ! i"
haftmann@37719:   by simp
haftmann@37719: 
haftmann@37719: lemma length_change [simp]: 
haftmann@37719:   "length a (change b i v h) = length a h"
haftmann@37719:   by (simp add: change_def length_def set_array_def get_array_def)
haftmann@37719: 
haftmann@37719: lemma change_swap_neqArray:
haftmann@37719:   "a =!!= a' \<Longrightarrow> 
haftmann@37719:   change a i v (change a' i' v' h) 
haftmann@37719:   = change a' i' v' (change a i v h)"
haftmann@37719: apply (unfold change_def)
haftmann@37719: apply simp
haftmann@37719: apply (subst array_set_set_swap, assumption)
haftmann@37719: apply (subst array_get_set_neq)
haftmann@37719: apply (erule noteq_arrs_sym)
haftmann@37719: apply (simp)
haftmann@37719: done
haftmann@37719: 
haftmann@37719: lemma change_swap_neqIndex:
haftmann@37719:   "\<lbrakk> i \<noteq> i' \<rbrakk> \<Longrightarrow> change a i v (change a i' v' h) = change a i' v' (change a i v h)"
haftmann@37719:   by (auto simp add: change_def array_set_set_swap list_update_swap)
haftmann@37719: 
haftmann@37719: lemma get_array_init_array_list:
haftmann@37719:   "get_array (fst (array ls h)) (snd (array ls' h)) = ls'"
haftmann@37719:   by (simp add: Let_def split_def array_def)
haftmann@37719: 
haftmann@37719: lemma set_array:
haftmann@37719:   "set_array (fst (array ls h))
haftmann@37719:      new_ls (snd (array ls h))
haftmann@37719:        = snd (array new_ls h)"
haftmann@37719:   by (simp add: Let_def split_def array_def)
haftmann@37719: 
haftmann@37719: lemma array_present_change [simp]: 
haftmann@37719:   "array_present a (change b i v h) = array_present a h"
haftmann@37719:   by (simp add: change_def array_present_def set_array_def get_array_def)
haftmann@37719: 
haftmann@37752: lemma execute_new [simp]:
haftmann@37752:   "Heap_Monad.execute (new n x) h = Some (array (replicate n x) h)"
haftmann@37752:   by (simp add: new_def)
haftmann@26170: 
haftmann@37752: lemma execute_of_list [simp]:
haftmann@37752:   "Heap_Monad.execute (of_list xs) h = Some (array xs h)"
haftmann@37752:   by (simp add: of_list_def)
haftmann@26170: 
haftmann@37752: lemma execute_make [simp]:
haftmann@37752:   "Heap_Monad.execute (make n f) h = Some (array (map f [0 ..< n]) h)"
haftmann@37752:   by (simp add: make_def)
haftmann@26170: 
haftmann@37752: lemma execute_len [simp]:
haftmann@37752:   "Heap_Monad.execute (len a) h = Some (length a h, h)"
haftmann@37752:   by (simp add: len_def)
haftmann@37752: 
haftmann@37752: lemma execute_nth [simp]:
haftmann@37752:   "i < length a h \<Longrightarrow>
haftmann@37752:     Heap_Monad.execute (nth a i) h = Some (get_array a h ! i, h)"
haftmann@37752:   "i \<ge> length a h \<Longrightarrow> Heap_Monad.execute (nth a i) h = None"
haftmann@37752:   by (simp_all add: nth_def)
haftmann@26170: 
haftmann@37752: lemma execute_upd [simp]:
haftmann@37752:   "i < length a h \<Longrightarrow>
haftmann@37752:     Heap_Monad.execute (upd i x a) h = Some (a, change a i x h)"
haftmann@37752:   "i \<ge> length a h \<Longrightarrow> Heap_Monad.execute (nth a i) h = None"
haftmann@37752:   by (simp_all add: upd_def)
haftmann@26170: 
haftmann@37752: lemma execute_map_entry [simp]:
haftmann@37752:   "i < length a h \<Longrightarrow>
haftmann@37752:    Heap_Monad.execute (map_entry i f a) h =
haftmann@37752:       Some (a, change a i (f (get_array a h ! i)) h)"
haftmann@37752:   "i \<ge> length a h \<Longrightarrow> Heap_Monad.execute (nth a i) h = None"
haftmann@37752:   by (simp_all add: map_entry_def)
haftmann@37752: 
haftmann@37752: lemma execute_swap [simp]:
haftmann@37752:   "i < length a h \<Longrightarrow>
haftmann@37752:    Heap_Monad.execute (swap i x a) h =
haftmann@37752:       Some (get_array a h ! i, change a i x h)"
haftmann@37752:   "i \<ge> length a h \<Longrightarrow> Heap_Monad.execute (nth a i) h = None"
haftmann@37752:   by (simp_all add: swap_def)
haftmann@37752: 
haftmann@37752: lemma execute_freeze [simp]:
haftmann@37752:   "Heap_Monad.execute (freeze a) h = Some (get_array a h, h)"
haftmann@37752:   by (simp add: freeze_def)
haftmann@26170: 
haftmann@26170: lemma upd_return:
haftmann@26170:   "upd i x a \<guillemotright> return a = upd i x a"
haftmann@37752:   by (rule Heap_eqI) (simp add: bindM_def guard_def upd_def)
haftmann@26170: 
haftmann@37752: lemma array_make:
haftmann@37752:   "new n x = make n (\<lambda>_. x)"
haftmann@37752:   by (rule Heap_eqI) (simp add: map_replicate_trivial)
haftmann@26170: 
haftmann@37752: lemma array_of_list_make:
haftmann@37752:   "of_list xs = make (List.length xs) (\<lambda>n. xs ! n)"
haftmann@37752:   by (rule Heap_eqI) (simp add: map_nth)
haftmann@26170: 
haftmann@37752: hide_const (open) new map
haftmann@26170: 
haftmann@26182: 
haftmann@26182: subsection {* Code generator setup *}
haftmann@26182: 
haftmann@26182: subsubsection {* Logical intermediate layer *}
haftmann@26182: 
haftmann@26182: definition new' where
haftmann@31205:   [code del]: "new' = Array.new o Code_Numeral.nat_of"
haftmann@37752: 
haftmann@28562: lemma [code]:
haftmann@37752:   "Array.new = new' o Code_Numeral.of_nat"
haftmann@26182:   by (simp add: new'_def o_def)
haftmann@26182: 
haftmann@26182: definition of_list' where
haftmann@31205:   [code del]: "of_list' i xs = Array.of_list (take (Code_Numeral.nat_of i) xs)"
haftmann@37752: 
haftmann@28562: lemma [code]:
haftmann@37752:   "Array.of_list xs = of_list' (Code_Numeral.of_nat (List.length xs)) xs"
haftmann@26182:   by (simp add: of_list'_def)
haftmann@26182: 
haftmann@26182: definition make' where
haftmann@31205:   [code del]: "make' i f = Array.make (Code_Numeral.nat_of i) (f o Code_Numeral.of_nat)"
haftmann@37752: 
haftmann@28562: lemma [code]:
haftmann@37752:   "Array.make n f = make' (Code_Numeral.of_nat n) (f o Code_Numeral.nat_of)"
haftmann@26182:   by (simp add: make'_def o_def)
haftmann@26182: 
haftmann@37719: definition len' where
haftmann@37719:   [code del]: "len' a = Array.len a \<guillemotright>= (\<lambda>n. return (Code_Numeral.of_nat n))"
haftmann@37752: 
haftmann@28562: lemma [code]:
haftmann@37752:   "Array.len a = len' a \<guillemotright>= (\<lambda>i. return (Code_Numeral.nat_of i))"
haftmann@37719:   by (simp add: len'_def)
haftmann@26182: 
haftmann@26182: definition nth' where
haftmann@31205:   [code del]: "nth' a = Array.nth a o Code_Numeral.nat_of"
haftmann@37752: 
haftmann@28562: lemma [code]:
haftmann@37752:   "Array.nth a n = nth' a (Code_Numeral.of_nat n)"
haftmann@26182:   by (simp add: nth'_def)
haftmann@26182: 
haftmann@26182: definition upd' where
haftmann@31205:   [code del]: "upd' a i x = Array.upd (Code_Numeral.nat_of i) x a \<guillemotright> return ()"
haftmann@37752: 
haftmann@28562: lemma [code]:
haftmann@37752:   "Array.upd i x a = upd' a (Code_Numeral.of_nat i) x \<guillemotright> return a"
haftmann@37709:   by (simp add: upd'_def upd_return)
haftmann@26182: 
haftmann@37752: lemma [code]:
haftmann@37752:   "map_entry i f a = (do
haftmann@37752:      x \<leftarrow> nth a i;
haftmann@37752:      upd i (f x) a
haftmann@37752:    done)"
haftmann@37752:   by (rule Heap_eqI) (simp add: bindM_def guard_def map_entry_def)
haftmann@26182: 
haftmann@37752: lemma [code]:
haftmann@37752:   "swap i x a = (do
haftmann@37752:      y \<leftarrow> nth a i;
haftmann@37752:      upd i x a;
haftmann@37752:      return y
haftmann@37752:    done)"
haftmann@37752:   by (rule Heap_eqI) (simp add: bindM_def guard_def swap_def)
haftmann@37752: 
haftmann@37752: lemma [code]:
haftmann@37752:   "freeze a = (do
haftmann@37752:      n \<leftarrow> len a;
haftmann@37752:      mapM (\<lambda>i. nth a i) [0..<n]
haftmann@37752:    done)"
haftmann@37752: proof (rule Heap_eqI)
haftmann@37752:   fix h
haftmann@37752:   have *: "List.map
haftmann@37752:      (\<lambda>x. fst (the (if x < length a h
haftmann@37752:                     then Some (get_array a h ! x, h) else None)))
haftmann@37752:      [0..<length a h] =
haftmann@37752:        List.map (List.nth (get_array a h)) [0..<length a h]"
haftmann@37752:     by simp
haftmann@37752:   have "Heap_Monad.execute (mapM (Array.nth a) [0..<length a h]) h =
haftmann@37752:     Some (get_array a h, h)"
haftmann@37752:     apply (subst execute_mapM_unchanged_heap)
haftmann@37752:     apply (simp_all add: nth_def guard_def *)
haftmann@37752:     apply (simp add: length_def map_nth)
haftmann@37752:     done
haftmann@37752:   then have "Heap_Monad.execute (do
haftmann@37752:       n \<leftarrow> len a;
haftmann@37752:       mapM (Array.nth a) [0..<n]
haftmann@37752:     done) h = Some (get_array a h, h)"
haftmann@37752:     by (auto intro: execute_eq_SomeI)
haftmann@37752:   then show "Heap_Monad.execute (freeze a) h = Heap_Monad.execute (do
haftmann@37752:       n \<leftarrow> len a;
haftmann@37752:       mapM (Array.nth a) [0..<n]
haftmann@37752:     done) h" by simp
haftmann@37752: qed
haftmann@37752: 
haftmann@37752: hide_const (open) new' of_list' make' len' nth' upd'
haftmann@37752: 
haftmann@37752: 
haftmann@37752: text {* SML *}
haftmann@26182: 
haftmann@26182: code_type array (SML "_/ array")
haftmann@26182: code_const Array (SML "raise/ (Fail/ \"bare Array\")")
haftmann@26752: code_const Array.new' (SML "(fn/ ()/ =>/ Array.array/ ((_),/ (_)))")
haftmann@35846: code_const Array.of_list' (SML "(fn/ ()/ =>/ Array.fromList/ _)")
haftmann@26752: code_const Array.make' (SML "(fn/ ()/ =>/ Array.tabulate/ ((_),/ (_)))")
haftmann@37719: code_const Array.len' (SML "(fn/ ()/ =>/ Array.length/ _)")
haftmann@26752: code_const Array.nth' (SML "(fn/ ()/ =>/ Array.sub/ ((_),/ (_)))")
haftmann@26752: code_const Array.upd' (SML "(fn/ ()/ =>/ Array.update/ ((_),/ (_),/ (_)))")
haftmann@26182: 
haftmann@26182: code_reserved SML Array
haftmann@26182: 
haftmann@26182: 
haftmann@37752: text {* OCaml *}
haftmann@26182: 
haftmann@26182: code_type array (OCaml "_/ array")
haftmann@26182: code_const Array (OCaml "failwith/ \"bare Array\"")
haftmann@32580: code_const Array.new' (OCaml "(fun/ ()/ ->/ Array.make/ (Big'_int.int'_of'_big'_int/ _)/ _)")
haftmann@35846: code_const Array.of_list' (OCaml "(fun/ ()/ ->/ Array.of'_list/ _)")
haftmann@37719: code_const Array.len' (OCaml "(fun/ ()/ ->/ Big'_int.big'_int'_of'_int/ (Array.length/ _))")
haftmann@32580: code_const Array.nth' (OCaml "(fun/ ()/ ->/ Array.get/ _/ (Big'_int.int'_of'_big'_int/ _))")
haftmann@32580: code_const Array.upd' (OCaml "(fun/ ()/ ->/ Array.set/ _/ (Big'_int.int'_of'_big'_int/ _)/ _)")
haftmann@26182: 
haftmann@26182: code_reserved OCaml Array
haftmann@26182: 
haftmann@26182: 
haftmann@37752: text {* Haskell *}
haftmann@26182: 
haftmann@29793: code_type array (Haskell "Heap.STArray/ Heap.RealWorld/ _")
haftmann@26182: code_const Array (Haskell "error/ \"bare Array\"")
haftmann@29793: code_const Array.new' (Haskell "Heap.newArray/ (0,/ _)")
haftmann@29793: code_const Array.of_list' (Haskell "Heap.newListArray/ (0,/ _)")
haftmann@37719: code_const Array.len' (Haskell "Heap.lengthArray")
haftmann@29793: code_const Array.nth' (Haskell "Heap.readArray")
haftmann@29793: code_const Array.upd' (Haskell "Heap.writeArray")
haftmann@26182: 
haftmann@26170: end




