# Theory HOL-Imperative_HOL.Array

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

theory Array
begin

subsection ‹Primitives›

definition present :: "heap ⇒ 'a::heap array ⇒ bool" where
"present h a ⟷ addr_of_array a < lim h"

definition get :: "heap ⇒ 'a::heap array ⇒ 'a list" where
"get h a = map from_nat (arrays h (TYPEREP('a)) (addr_of_array a))"

definition set :: "'a::heap array ⇒ 'a list ⇒ heap ⇒ heap" where
"set a x = arrays_update (λh. h(TYPEREP('a) := ((h(TYPEREP('a))) (addr_of_array a:=map to_nat x))))"

definition alloc :: "'a list ⇒ heap ⇒ 'a::heap array × heap" where
"alloc xs h = (let
l = lim h;
r = Array l;
h'' = set r xs (h⦇lim := l + 1⦈)
in (r, h''))"

definition length :: "heap ⇒ 'a::heap array ⇒ nat" where
"length h a = List.length (get h a)"

definition update :: "'a::heap array ⇒ nat ⇒ 'a ⇒ heap ⇒ heap" where
"update a i x h = set a ((get h a)[i:=x]) h"

definition noteq :: "'a::heap array ⇒ 'b::heap array ⇒ bool" (infix "=!!=" 70) where
"r =!!= s ⟷ TYPEREP('a) ≠ TYPEREP('b) ∨ addr_of_array r ≠ addr_of_array s"

definition new :: "nat ⇒ 'a::heap ⇒ 'a array Heap" where
[code del]: "new n x = Heap_Monad.heap (alloc (replicate n x))"

definition of_list :: "'a::heap list ⇒ 'a array Heap" where
[code del]: "of_list xs = Heap_Monad.heap (alloc xs)"

definition make :: "nat ⇒ (nat ⇒ 'a::heap) ⇒ 'a array Heap" where
[code del]: "make n f = Heap_Monad.heap (alloc (map f [0 ..< n]))"

definition len :: "'a::heap array ⇒ nat Heap" where
[code del]: "len a = Heap_Monad.tap (λh. length h a)"

definition nth :: "'a::heap array ⇒ nat ⇒ 'a Heap" where
[code del]: "nth a i = Heap_Monad.guard (λh. i < length h a)
(λh. (get h a ! i, h))"

definition upd :: "nat ⇒ 'a ⇒ 'a::heap array ⇒ 'a::heap array Heap" where
[code del]: "upd i x a = Heap_Monad.guard (λh. i < length h a)
(λh. (a, update a i x h))"

definition map_entry :: "nat ⇒ ('a::heap ⇒ 'a) ⇒ 'a array ⇒ 'a array Heap" where
[code del]: "map_entry i f a = Heap_Monad.guard (λh. i < length h a)
(λh. (a, update a i (f (get h a ! i)) h))"

definition swap :: "nat ⇒ 'a ⇒ 'a::heap array ⇒ 'a Heap" where
[code del]: "swap i x a = Heap_Monad.guard (λh. i < length h a)
(λh. (get h a ! i, update a i x h))"

definition freeze :: "'a::heap array ⇒ 'a list Heap" where
[code del]: "freeze a = Heap_Monad.tap (λh. get h a)"

subsection ‹Properties›

text ‹FIXME: Does there exist a "canonical" array axiomatisation in
the literature?›

text ‹Primitives›

lemma noteq_sym: "a =!!= b ⟹ b =!!= a"
and unequal [simp]: "a ≠ a' ⟷ a =!!= a'"
unfolding noteq_def by auto

lemma noteq_irrefl: "r =!!= r ⟹ False"
unfolding noteq_def by auto

lemma present_alloc_noteq: "present h a ⟹ 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 ⟹ 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"

lemma set_set_swap:
"r =!!= r' ⟹ 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]"

lemma nth_update_neq [simp]:
"a =!!= b ⟹ get (update b j v h) a ! i = get h a ! i"

lemma get_update_elem_neqIndex [simp]:
"i ≠ j ⟹ get (update a j v h) a ! i = get h a ! i"
by simp

lemma 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' ⟹
update a i v (update a' i' v' h)
= update a' i' v' (update a i v h)"
apply (unfold update_def)
apply simp
apply (subst set_set_swap, assumption)
apply (subst get_set_neq)
apply (erule noteq_sym)
apply simp
done

lemma update_swap_neqIndex:
"⟦ i ≠ i' ⟧ ⟹ 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"

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]:
"¬ present h (fst (alloc xs h))"
by (simp add: present_def alloc_def Let_def)

lemma execute_new [execute_simps]:
"execute (new n x) h = Some (alloc (replicate n x) h)"

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" "¬ 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)"

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" "¬ 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)"

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" "¬ 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)"

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 ⟹
execute (nth a i) h = Some (get h a ! i, h)"
"i ≥ length h a ⟹ execute (nth a i) h = None"

lemma success_nthI [success_intros]:
"i < length h a ⟹ 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) (cases "i < length h a", auto simp: execute_simps elim: successE)

lemma execute_upd [execute_simps]:
"i < length h a ⟹
execute (upd i x a) h = Some (a, update a i x h)"
"i ≥ length h a ⟹ execute (upd i x a) h = None"

lemma success_updI [success_intros]:
"i < length h a ⟹ 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) (cases "i < length h a", auto simp: execute_simps elim: successE)

lemma execute_map_entry [execute_simps]:
"i < length h a ⟹
execute (map_entry i f a) h =
Some (a, update a i (f (get h a ! i)) h)"
"i ≥ length h a ⟹ execute (map_entry i f a) h = None"

lemma success_map_entryI [success_intros]:
"i < length h a ⟹ 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) (cases "i < length h a", auto simp: execute_simps elim: successE)

lemma execute_swap [execute_simps]:
"i < length h a ⟹
execute (swap i x a) h =
Some (get h a ! i, update a i x h)"
"i ≥ length h a ⟹ execute (swap i x a) h = None"

lemma success_swapI [success_intros]:
"i < length h a ⟹ 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) (cases "i < length h a", auto simp: execute_simps elim: successE)

lemma execute_freeze [execute_simps]:
"execute (freeze a) h = Some (get h a, h)"

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 ⪢ 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 (λ_. 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) (λn. 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 freeze

subsection ‹Code generator setup›

subsubsection ‹Logical intermediate layer›

definition new' where
[code del]: "new' = Array.new o nat_of_integer"

lemma [code]:
"Array.new = new' o of_nat"

definition make' where
[code del]: "make' i f = Array.make (nat_of_integer i) (f o of_nat)"

lemma [code]:
"Array.make n f = make' (of_nat n) (f o nat_of_integer)"

definition len' where
[code del]: "len' a = Array.len a ⤜ (λn. return (of_nat n))"

lemma [code]:
"Array.len a = len' a ⤜ (λi. return (nat_of_integer i))"

definition nth' where
[code del]: "nth' a = Array.nth a o nat_of_integer"

lemma [code]:
"Array.nth a n = nth' a (of_nat n)"

definition upd' where
[code del]: "upd' a i x = Array.upd (nat_of_integer i) x a ⪢ return ()"

lemma [code]:
"Array.upd i x a = upd' a (of_nat i) x ⪢ return a"

lemma [code]:
"Array.map_entry i f a = do {
x ← 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 ← 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 ← Array.len a;
Heap_Monad.fold_map (λi. Array.nth a i) [0..<n]
}"
proof (rule Heap_eqI)
fix h
have *: "List.map
(λx. 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 *)
done
then have "execute (do {
n ← Array.len a;
}) 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 ← Array.len a;
}) h" by (simp add: execute_simps)
qed

hide_const (open) new' make' len' nth' upd'

text ‹SML›

code_printing type_constructor array ⇀ (SML) "_/ array"
code_printing constant Array ⇀ (SML) "raise/ (Fail/ \"bare Array\")"
code_printing constant Array.new' ⇀ (SML) "(fn/ ()/ =>/ Array.array/ (IntInf.toInt _,/ (_)))"
code_printing constant Array.of_list ⇀ (SML) "(fn/ ()/ =>/ Array.fromList/ _)"
code_printing constant Array.make' ⇀ (SML) "(fn/ ()/ =>/ Array.tabulate/ (IntInf.toInt _,/ _ o IntInf.fromInt))"
code_printing constant Array.len' ⇀ (SML) "(fn/ ()/ =>/ IntInf.fromInt (Array.length/ _))"
code_printing constant Array.nth' ⇀ (SML) "(fn/ ()/ =>/ Array.sub/ ((_),/ IntInf.toInt _))"
code_printing constant Array.upd' ⇀ (SML) "(fn/ ()/ =>/ Array.update/ ((_),/ IntInf.toInt _,/ (_)))"
code_printing constant "HOL.equal :: 'a array ⇒ 'a array ⇒ bool" ⇀ (SML) infixl 6 "="

code_reserved SML Array

text ‹OCaml›

code_printing type_constructor array ⇀ (OCaml) "_/ array"
code_printing constant Array ⇀ (OCaml) "failwith/ \"bare Array\""
code_printing constant Array.new' ⇀ (OCaml) "(fun/ ()/ ->/ Array.make/ (Z.to'_int/ _)/ _)"
code_printing constant Array.of_list ⇀ (OCaml) "(fun/ ()/ ->/ Array.of'_list/ _)"
code_printing constant Array.make' ⇀ (OCaml)
"(fun/ ()/ ->/ Array.init/ (Z.to'_int/ _)/ (fun k'_ ->/ _/ (Z.of'_int/ k'_)))"
code_printing constant Array.len' ⇀ (OCaml) "(fun/ ()/ ->/ Z.of'_int/ (Array.length/ _))"
code_printing constant Array.nth' ⇀ (OCaml) "(fun/ ()/ ->/ Array.get/ _/ (Z.to'_int/ _))"
code_printing constant Array.upd' ⇀ (OCaml) "(fun/ ()/ ->/ Array.set/ _/ (Z.to'_int/ _)/ _)"
code_printing constant "HOL.equal :: 'a array ⇒ 'a array ⇒ bool" ⇀ (OCaml) infixl 4 "="

code_reserved OCaml Array

code_printing type_constructor array ⇀ (Haskell) "Heap.STArray/ Heap.RealWorld/ _"
code_printing constant Array ⇀ (Haskell) "error/ \"bare Array\""
code_printing constant Array.new' ⇀ (Haskell) "Heap.newArray"
code_printing constant Array.of_list ⇀ (Haskell) "Heap.newListArray"
code_printing constant Array.make' ⇀ (Haskell) "Heap.newFunArray"
code_printing constant Array.len' ⇀ (Haskell) "Heap.lengthArray"
code_printing constant Array.upd' ⇀ (Haskell) "Heap.writeArray"
code_printing constant "HOL.equal :: 'a array ⇒ 'a array ⇒ bool" ⇀ (Haskell) infix 4 "=="
code_printing class_instance array :: HOL.equal ⇀ (Haskell) -

text ‹Scala›

code_printing type_constructor array ⇀ (Scala) "!Array.T[_]"
code_printing constant Array ⇀ (Scala) "!sys.error(\"bare Array\")"
code_printing constant Array.new' ⇀ (Scala) "('_: Unit)/ => / Array.alloc((_))((_))"
code_printing constant Array.make' ⇀ (Scala) "('_: Unit)/ =>/ Array.make((_))((_))"
code_printing constant Array.len' ⇀ (Scala) "('_: Unit)/ =>/ Array.len((_))"
code_printing constant Array.nth' ⇀ (Scala) "('_: Unit)/ =>/ Array.nth((_), (_))"
code_printing constant Array.upd' ⇀ (Scala) "('_: Unit)/ =>/ Array.upd((_), (_), (_))"
code_printing constant Array.freeze ⇀ (Scala) "('_: Unit)/ =>/ Array.freeze((_))"
code_printing constant "HOL.equal :: 'a array ⇒ 'a array ⇒ bool" ⇀ (Scala) infixl 5 "=="

end
```