Theory Map

theory Map
imports List
```(*  Title:      HOL/Map.thy
Author:     Tobias Nipkow, based on a theory by David von Oheimb

The datatype of "maps"; strongly resembles maps in VDM.
*)

section ‹Maps›

theory Map
imports List
abbrevs "(=" = "⊆⇩m"
begin

type_synonym ('a, 'b) "map" = "'a ⇒ 'b option" (infixr "⇀" 0)

abbreviation
empty :: "'a ⇀ 'b" where
"empty ≡ λx. None"

definition
map_comp :: "('b ⇀ 'c) ⇒ ('a ⇀ 'b) ⇒ ('a ⇀ 'c)"  (infixl "∘⇩m" 55) where
"f ∘⇩m g = (λk. case g k of None ⇒ None | Some v ⇒ f v)"

definition
map_add :: "('a ⇀ 'b) ⇒ ('a ⇀ 'b) ⇒ ('a ⇀ 'b)"  (infixl "++" 100) where
"m1 ++ m2 = (λx. case m2 x of None ⇒ m1 x | Some y ⇒ Some y)"

definition
restrict_map :: "('a ⇀ 'b) ⇒ 'a set ⇒ ('a ⇀ 'b)"  (infixl "|`"  110) where
"m|`A = (λx. if x ∈ A then m x else None)"

notation (latex output)
restrict_map  ("_↾⇘_⇙" [111,110] 110)

definition
dom :: "('a ⇀ 'b) ⇒ 'a set" where
"dom m = {a. m a ≠ None}"

definition
ran :: "('a ⇀ 'b) ⇒ 'b set" where
"ran m = {b. ∃a. m a = Some b}"

definition
map_le :: "('a ⇀ 'b) ⇒ ('a ⇀ 'b) ⇒ bool"  (infix "⊆⇩m" 50) where
"(m⇩1 ⊆⇩m m⇩2) ⟷ (∀a ∈ dom m⇩1. m⇩1 a = m⇩2 a)"

nonterminal maplets and maplet

syntax
"_maplet"  :: "['a, 'a] ⇒ maplet"             ("_ /↦/ _")
"_maplets" :: "['a, 'a] ⇒ maplet"             ("_ /[↦]/ _")
""         :: "maplet ⇒ maplets"             ("_")
"_Maplets" :: "[maplet, maplets] ⇒ maplets" ("_,/ _")
"_MapUpd"  :: "['a ⇀ 'b, maplets] ⇒ 'a ⇀ 'b" ("_/'(_')" [900, 0] 900)
"_Map"     :: "maplets ⇒ 'a ⇀ 'b"            ("(1[_])")

syntax (ASCII)
"_maplet"  :: "['a, 'a] ⇒ maplet"             ("_ /|->/ _")
"_maplets" :: "['a, 'a] ⇒ maplet"             ("_ /[|->]/ _")

translations
"_MapUpd m (_Maplets xy ms)"  ⇌ "_MapUpd (_MapUpd m xy) ms"
"_MapUpd m (_maplet  x y)"    ⇌ "m(x := CONST Some y)"
"_Map ms"                     ⇌ "_MapUpd (CONST empty) ms"
"_Map (_Maplets ms1 ms2)"     ↽ "_MapUpd (_Map ms1) ms2"
"_Maplets ms1 (_Maplets ms2 ms3)" ↽ "_Maplets (_Maplets ms1 ms2) ms3"

primrec map_of :: "('a × 'b) list ⇒ 'a ⇀ 'b"
where
"map_of [] = empty"
| "map_of (p # ps) = (map_of ps)(fst p ↦ snd p)"

definition map_upds :: "('a ⇀ 'b) ⇒ 'a list ⇒ 'b list ⇒ 'a ⇀ 'b"
where "map_upds m xs ys = m ++ map_of (rev (zip xs ys))"
translations
"_MapUpd m (_maplets x y)" ⇌ "CONST map_upds m x y"

lemma map_of_Cons_code [code]:
"map_of [] k = None"
"map_of ((l, v) # ps) k = (if l = k then Some v else map_of ps k)"
by simp_all

subsection ‹@{term [source] empty}›

lemma empty_upd_none [simp]: "empty(x := None) = empty"
by (rule ext) simp

subsection ‹@{term [source] map_upd}›

lemma map_upd_triv: "t k = Some x ⟹ t(k↦x) = t"
by (rule ext) simp

lemma map_upd_nonempty [simp]: "t(k↦x) ≠ empty"
proof
assume "t(k ↦ x) = empty"
then have "(t(k ↦ x)) k = None" by simp
then show False by simp
qed

lemma map_upd_eqD1:
assumes "m(a↦x) = n(a↦y)"
shows "x = y"
proof -
from assms have "(m(a↦x)) a = (n(a↦y)) a" by simp
then show ?thesis by simp
qed

lemma map_upd_Some_unfold:
"((m(a↦b)) x = Some y) = (x = a ∧ b = y ∨ x ≠ a ∧ m x = Some y)"
by auto

lemma image_map_upd [simp]: "x ∉ A ⟹ m(x ↦ y) ` A = m ` A"
by auto

lemma finite_range_updI: "finite (range f) ⟹ finite (range (f(a↦b)))"
unfolding image_def
apply (rule finite_subset)
prefer 2 apply assumption
apply (auto)
done

subsection ‹@{term [source] map_of}›

lemma map_of_eq_empty_iff [simp]:
"map_of xys = empty ⟷ xys = []"
proof
show "map_of xys = empty ⟹ xys = []"
by (induction xys) simp_all
qed simp

lemma empty_eq_map_of_iff [simp]:
"empty = map_of xys ⟷ xys = []"
by(subst eq_commute) simp

lemma map_of_eq_None_iff:
"(map_of xys x = None) = (x ∉ fst ` (set xys))"
by (induct xys) simp_all

lemma map_of_eq_Some_iff [simp]:
"distinct(map fst xys) ⟹ (map_of xys x = Some y) = ((x,y) ∈ set xys)"
apply (induct xys)
apply simp
apply (auto simp: map_of_eq_None_iff [symmetric])
done

lemma Some_eq_map_of_iff [simp]:
"distinct(map fst xys) ⟹ (Some y = map_of xys x) = ((x,y) ∈ set xys)"
by (auto simp del: map_of_eq_Some_iff simp: map_of_eq_Some_iff [symmetric])

lemma map_of_is_SomeI [simp]: "⟦ distinct(map fst xys); (x,y) ∈ set xys ⟧
⟹ map_of xys x = Some y"
apply (induct xys)
apply simp
apply force
done

lemma map_of_zip_is_None [simp]:
"length xs = length ys ⟹ (map_of (zip xs ys) x = None) = (x ∉ set xs)"
by (induct rule: list_induct2) simp_all

lemma map_of_zip_is_Some:
assumes "length xs = length ys"
shows "x ∈ set xs ⟷ (∃y. map_of (zip xs ys) x = Some y)"
using assms by (induct rule: list_induct2) simp_all

lemma map_of_zip_upd:
fixes x :: 'a and xs :: "'a list" and ys zs :: "'b list"
assumes "length ys = length xs"
and "length zs = length xs"
and "x ∉ set xs"
and "map_of (zip xs ys)(x ↦ y) = map_of (zip xs zs)(x ↦ z)"
shows "map_of (zip xs ys) = map_of (zip xs zs)"
proof
fix x' :: 'a
show "map_of (zip xs ys) x' = map_of (zip xs zs) x'"
proof (cases "x = x'")
case True
from assms True map_of_zip_is_None [of xs ys x']
have "map_of (zip xs ys) x' = None" by simp
moreover from assms True map_of_zip_is_None [of xs zs x']
have "map_of (zip xs zs) x' = None" by simp
ultimately show ?thesis by simp
next
case False from assms
have "(map_of (zip xs ys)(x ↦ y)) x' = (map_of (zip xs zs)(x ↦ z)) x'" by auto
with False show ?thesis by simp
qed
qed

lemma map_of_zip_inject:
assumes "length ys = length xs"
and "length zs = length xs"
and dist: "distinct xs"
and map_of: "map_of (zip xs ys) = map_of (zip xs zs)"
shows "ys = zs"
using assms(1) assms(2)[symmetric]
using dist map_of
proof (induct ys xs zs rule: list_induct3)
case Nil show ?case by simp
next
case (Cons y ys x xs z zs)
from ‹map_of (zip (x#xs) (y#ys)) = map_of (zip (x#xs) (z#zs))›
have map_of: "map_of (zip xs ys)(x ↦ y) = map_of (zip xs zs)(x ↦ z)" by simp
from Cons have "length ys = length xs" and "length zs = length xs"
and "x ∉ set xs" by simp_all
then have "map_of (zip xs ys) = map_of (zip xs zs)" using map_of by (rule map_of_zip_upd)
with Cons.hyps ‹distinct (x # xs)› have "ys = zs" by simp
moreover from map_of have "y = z" by (rule map_upd_eqD1)
ultimately show ?case by simp
qed

lemma map_of_zip_nth:
assumes "length xs = length ys"
assumes "distinct xs"
assumes "i < length ys"
shows "map_of (zip xs ys) (xs ! i) = Some (ys ! i)"
using assms proof (induct arbitrary: i rule: list_induct2)
case Nil
then show ?case by simp
next
case (Cons x xs y ys)
then show ?case
using less_Suc_eq_0_disj by auto
qed

lemma map_of_zip_map:
"map_of (zip xs (map f xs)) = (λx. if x ∈ set xs then Some (f x) else None)"
by (induct xs) (simp_all add: fun_eq_iff)

lemma finite_range_map_of: "finite (range (map_of xys))"
apply (induct xys)
apply (rule finite_subset)
prefer 2 apply assumption
apply auto
done

lemma map_of_SomeD: "map_of xs k = Some y ⟹ (k, y) ∈ set xs"
by (induct xs) (auto split: if_splits)

lemma map_of_mapk_SomeI:
"inj f ⟹ map_of t k = Some x ⟹
map_of (map (case_prod (λk. Pair (f k))) t) (f k) = Some x"
by (induct t) (auto simp: inj_eq)

lemma weak_map_of_SomeI: "(k, x) ∈ set l ⟹ ∃x. map_of l k = Some x"
by (induct l) auto

lemma map_of_filter_in:
"map_of xs k = Some z ⟹ P k z ⟹ map_of (filter (case_prod P) xs) k = Some z"
by (induct xs) auto

lemma map_of_map:
"map_of (map (λ(k, v). (k, f v)) xs) = map_option f ∘ map_of xs"
by (induct xs) (auto simp: fun_eq_iff)

lemma dom_map_option:
"dom (λk. map_option (f k) (m k)) = dom m"

lemma dom_map_option_comp [simp]:
"dom (map_option g ∘ m) = dom m"
using dom_map_option [of "λ_. g" m] by (simp add: comp_def)

subsection ‹\<^const>‹map_option› related›

lemma map_option_o_empty [simp]: "map_option f ∘ empty = empty"
by (rule ext) simp

lemma map_option_o_map_upd [simp]:
"map_option f ∘ m(a↦b) = (map_option f ∘ m)(a↦f b)"
by (rule ext) simp

subsection ‹@{term [source] map_comp} related›

lemma map_comp_empty [simp]:
"m ∘⇩m empty = empty"
"empty ∘⇩m m = empty"
by (auto simp: map_comp_def split: option.splits)

lemma map_comp_simps [simp]:
"m2 k = None ⟹ (m1 ∘⇩m m2) k = None"
"m2 k = Some k' ⟹ (m1 ∘⇩m m2) k = m1 k'"
by (auto simp: map_comp_def)

lemma map_comp_Some_iff:
"((m1 ∘⇩m m2) k = Some v) = (∃k'. m2 k = Some k' ∧ m1 k' = Some v)"
by (auto simp: map_comp_def split: option.splits)

lemma map_comp_None_iff:
"((m1 ∘⇩m m2) k = None) = (m2 k = None ∨ (∃k'. m2 k = Some k' ∧ m1 k' = None)) "
by (auto simp: map_comp_def split: option.splits)

subsection ‹‹++››

lemma map_add_empty[simp]: "m ++ empty = m"

lemma empty_map_add[simp]: "empty ++ m = m"

lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3"

"((m ++ n) k = Some x) = (n k = Some x ∨ n k = None ∧ m k = Some x)"

"(m ++ n) k = Some x ⟹ n k = Some x ∨ n k = None ∧ m k = Some x"

lemma map_add_find_right [simp]: "n k = Some xx ⟹ (m ++ n) k = Some xx"

lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None ∧ m k = None)"

lemma map_add_upd[simp]: "f ++ g(x↦y) = (f ++ g)(x↦y)"

lemma map_add_upds[simp]: "m1 ++ (m2(xs[↦]ys)) = (m1++m2)(xs[↦]ys)"

lemma map_add_upd_left: "m∉dom e2 ⟹ e1(m ↦ u1) ++ e2 = (e1 ++ e2)(m ↦ u1)"
by (rule ext) (auto simp: map_add_def dom_def split: option.split)

lemma map_of_append[simp]: "map_of (xs @ ys) = map_of ys ++ map_of xs"
apply (induct xs)
apply simp
apply (rule ext)
apply (simp split: option.split)
done

"finite (range f) ⟹ finite (range (f ++ map_of l))"
apply (induct l)
apply (auto simp del: fun_upd_apply)
apply (erule finite_range_updI)
done

"inj_on (m ++ m') (dom m') = inj_on m' (dom m')"
by (fastforce simp: map_add_def dom_def inj_on_def split: option.splits)

lemma map_upds_fold_map_upd:
"m(ks[↦]vs) = foldl (λm (k, v). m(k ↦ v)) m (zip ks vs)"
unfolding map_upds_def proof (rule sym, rule zip_obtain_same_length)
fix ks :: "'a list" and vs :: "'b list"
assume "length ks = length vs"
then show "foldl (λm (k, v). m(k↦v)) m (zip ks vs) = m ++ map_of (rev (zip ks vs))"
by(induct arbitrary: m rule: list_induct2) simp_all
qed

"m ++ map_of ps = foldr (λ(k, v) m. m(k ↦ v)) ps m"
by (induct ps) (auto simp: fun_eq_iff map_add_def)

subsection ‹@{term [source] restrict_map}›

lemma restrict_map_to_empty [simp]: "m|`{} = empty"

lemma restrict_map_insert: "f |` (insert a A) = (f |` A)(a := f a)"
by (auto simp: restrict_map_def)

lemma restrict_map_empty [simp]: "empty|`D = empty"

lemma restrict_in [simp]: "x ∈ A ⟹ (m|`A) x = m x"

lemma restrict_out [simp]: "x ∉ A ⟹ (m|`A) x = None"

lemma ran_restrictD: "y ∈ ran (m|`A) ⟹ ∃x∈A. m x = Some y"
by (auto simp: restrict_map_def ran_def split: if_split_asm)

lemma dom_restrict [simp]: "dom (m|`A) = dom m ∩ A"
by (auto simp: restrict_map_def dom_def split: if_split_asm)

lemma restrict_upd_same [simp]: "m(x↦y)|`(-{x}) = m|`(-{x})"
by (rule ext) (auto simp: restrict_map_def)

lemma restrict_restrict [simp]: "m|`A|`B = m|`(A∩B)"
by (rule ext) (auto simp: restrict_map_def)

lemma restrict_fun_upd [simp]:
"m(x := y)|`D = (if x ∈ D then (m|`(D-{x}))(x := y) else m|`D)"

lemma fun_upd_None_restrict [simp]:
"(m|`D)(x := None) = (if x ∈ D then m|`(D - {x}) else m|`D)"

lemma fun_upd_restrict: "(m|`D)(x := y) = (m|`(D-{x}))(x := y)"

lemma fun_upd_restrict_conv [simp]:
"x ∈ D ⟹ (m|`D)(x := y) = (m|`(D-{x}))(x := y)"

lemma map_of_map_restrict:
"map_of (map (λk. (k, f k)) ks) = (Some ∘ f) |` set ks"
by (induct ks) (simp_all add: fun_eq_iff restrict_map_insert)

lemma restrict_complement_singleton_eq:
"f |` (- {x}) = f(x := None)"

subsection ‹@{term [source] map_upds}›

lemma map_upds_Nil1 [simp]: "m([] [↦] bs) = m"

lemma map_upds_Nil2 [simp]: "m(as [↦] []) = m"

lemma map_upds_Cons [simp]: "m(a#as [↦] b#bs) = (m(a↦b))(as[↦]bs)"

lemma map_upds_append1 [simp]: "size xs < size ys ⟹
m(xs@[x] [↦] ys) = m(xs [↦] ys)(x ↦ ys!size xs)"
apply(induct xs arbitrary: ys m)
apply (case_tac ys)
apply simp
apply simp
done

lemma map_upds_list_update2_drop [simp]:
"size xs ≤ i ⟹ m(xs[↦]ys[i:=y]) = m(xs[↦]ys)"
apply (induct xs arbitrary: m ys i)
apply simp
apply (case_tac ys)
apply simp
apply (simp split: nat.split)
done

lemma map_upd_upds_conv_if:
"(f(x↦y))(xs [↦] ys) =
(if x ∈ set(take (length ys) xs) then f(xs [↦] ys)
else (f(xs [↦] ys))(x↦y))"
apply (induct xs arbitrary: x y ys f)
apply simp
apply (case_tac ys)
apply (auto split: if_split simp: fun_upd_twist)
done

lemma map_upds_twist [simp]:
"a ∉ set as ⟹ m(a↦b)(as[↦]bs) = m(as[↦]bs)(a↦b)"
using set_take_subset by (fastforce simp add: map_upd_upds_conv_if)

lemma map_upds_apply_nontin [simp]:
"x ∉ set xs ⟹ (f(xs[↦]ys)) x = f x"
apply (induct xs arbitrary: ys)
apply simp
apply (case_tac ys)
apply (auto simp: map_upd_upds_conv_if)
done

lemma fun_upds_append_drop [simp]:
"size xs = size ys ⟹ m(xs@zs[↦]ys) = m(xs[↦]ys)"
apply (induct xs arbitrary: m ys)
apply simp
apply (case_tac ys)
apply simp_all
done

lemma fun_upds_append2_drop [simp]:
"size xs = size ys ⟹ m(xs[↦]ys@zs) = m(xs[↦]ys)"
apply (induct xs arbitrary: m ys)
apply simp
apply (case_tac ys)
apply simp_all
done

lemma restrict_map_upds[simp]:
"⟦ length xs = length ys; set xs ⊆ D ⟧
⟹ m(xs [↦] ys)|`D = (m|`(D - set xs))(xs [↦] ys)"
apply (induct xs arbitrary: m ys)
apply simp
apply (case_tac ys)
apply simp
apply (simp add: Diff_insert [symmetric] insert_absorb)
done

subsection ‹@{term [source] dom}›

lemma dom_eq_empty_conv [simp]: "dom f = {} ⟷ f = empty"
by (auto simp: dom_def)

lemma domI: "m a = Some b ⟹ a ∈ dom m"
(* declare domI [intro]? *)

lemma domD: "a ∈ dom m ⟹ ∃b. m a = Some b"
by (cases "m a") (auto simp add: dom_def)

lemma domIff [iff, simp del, code_unfold]: "a ∈ dom m ⟷ m a ≠ None"

lemma dom_empty [simp]: "dom empty = {}"

lemma dom_fun_upd [simp]:
"dom(f(x := y)) = (if y = None then dom f - {x} else insert x (dom f))"
by (auto simp: dom_def)

lemma dom_if:
"dom (λx. if P x then f x else g x) = dom f ∩ {x. P x} ∪ dom g ∩ {x. ¬ P x}"
by (auto split: if_splits)

lemma dom_map_of_conv_image_fst:
"dom (map_of xys) = fst ` set xys"
by (induct xys) (auto simp add: dom_if)

lemma dom_map_of_zip [simp]: "length xs = length ys ⟹ dom (map_of (zip xs ys)) = set xs"
by (induct rule: list_induct2) (auto simp: dom_if)

lemma finite_dom_map_of: "finite (dom (map_of l))"
by (induct l) (auto simp: dom_def insert_Collect [symmetric])

lemma dom_map_upds [simp]:
"dom(m(xs[↦]ys)) = set(take (length ys) xs) ∪ dom m"
apply (induct xs arbitrary: m ys)
apply simp
apply (case_tac ys)
apply auto
done

lemma dom_map_add [simp]: "dom (m ++ n) = dom n ∪ dom m"
by (auto simp: dom_def)

lemma dom_override_on [simp]:
"dom (override_on f g A) =
(dom f  - {a. a ∈ A - dom g}) ∪ {a. a ∈ A ∩ dom g}"
by (auto simp: dom_def override_on_def)

lemma map_add_comm: "dom m1 ∩ dom m2 = {} ⟹ m1 ++ m2 = m2 ++ m1"
by (rule ext) (force simp: map_add_def dom_def split: option.split)

"m ∈ dom l2 ⟹ (l1 ++ l2) m = l2 m"
"m ∉ dom l1 ⟹ (l1 ++ l2) m = l2 m"
"m ∉ dom l2 ⟹ (l1 ++ l2) m = l1 m"

lemma dom_const [simp]:
"dom (λx. Some (f x)) = UNIV"
by auto

(* Due to John Matthews - could be rephrased with dom *)
lemma finite_map_freshness:
"finite (dom (f :: 'a ⇀ 'b)) ⟹ ¬ finite (UNIV :: 'a set) ⟹
∃x. f x = None"
by (bestsimp dest: ex_new_if_finite)

lemma dom_minus:
"f x = None ⟹ dom f - insert x A = dom f - A"
unfolding dom_def by simp

lemma insert_dom:
"f x = Some y ⟹ insert x (dom f) = dom f"
unfolding dom_def by auto

lemma map_of_map_keys:
"set xs = dom m ⟹ map_of (map (λk. (k, the (m k))) xs) = m"
by (rule ext) (auto simp add: map_of_map_restrict restrict_map_def)

lemma map_of_eqI:
assumes set_eq: "set (map fst xs) = set (map fst ys)"
assumes map_eq: "∀k∈set (map fst xs). map_of xs k = map_of ys k"
shows "map_of xs = map_of ys"
proof (rule ext)
fix k show "map_of xs k = map_of ys k"
proof (cases "map_of xs k")
case None
then have "k ∉ set (map fst xs)" by (simp add: map_of_eq_None_iff)
with set_eq have "k ∉ set (map fst ys)" by simp
then have "map_of ys k = None" by (simp add: map_of_eq_None_iff)
with None show ?thesis by simp
next
case (Some v)
then have "k ∈ set (map fst xs)" by (auto simp add: dom_map_of_conv_image_fst [symmetric])
with map_eq show ?thesis by auto
qed
qed

lemma map_of_eq_dom:
assumes "map_of xs = map_of ys"
shows "fst ` set xs = fst ` set ys"
proof -
from assms have "dom (map_of xs) = dom (map_of ys)" by simp
then show ?thesis by (simp add: dom_map_of_conv_image_fst)
qed

lemma finite_set_of_finite_maps:
assumes "finite A" "finite B"
shows "finite {m. dom m = A ∧ ran m ⊆ B}" (is "finite ?S")
proof -
let ?S' = "{m. ∀x. (x ∈ A ⟶ m x ∈ Some ` B) ∧ (x ∉ A ⟶ m x = None)}"
have "?S = ?S'"
proof
show "?S ⊆ ?S'" by (auto simp: dom_def ran_def image_def)
show "?S' ⊆ ?S"
proof
fix m assume "m ∈ ?S'"
hence 1: "dom m = A" by force
hence 2: "ran m ⊆ B" using ‹m ∈ ?S'› by (auto simp: dom_def ran_def)
from 1 2 show "m ∈ ?S" by blast
qed
qed
with assms show ?thesis by(simp add: finite_set_of_finite_funs)
qed

subsection ‹@{term [source] ran}›

lemma ranI: "m a = Some b ⟹ b ∈ ran m"
by (auto simp: ran_def)
(* declare ranI [intro]? *)

lemma ran_empty [simp]: "ran empty = {}"
by (auto simp: ran_def)

lemma ran_map_upd [simp]: "m a = None ⟹ ran(m(a↦b)) = insert b (ran m)"
unfolding ran_def
apply auto
apply (subgoal_tac "aa ≠ a")
apply auto
done

assumes "dom m1 ∩ dom m2 = {}"
shows "ran (m1 ++ m2) = ran m1 ∪ ran m2"
proof
show "ran (m1 ++ m2) ⊆ ran m1 ∪ ran m2"
unfolding ran_def by auto
next
show "ran m1 ∪ ran m2 ⊆ ran (m1 ++ m2)"
proof -
have "(m1 ++ m2) x = Some y" if "m1 x = Some y" for x y
using assms map_add_comm that by fastforce
moreover have "(m1 ++ m2) x = Some y" if "m2 x = Some y" for x y
using assms that by auto
ultimately show ?thesis
unfolding ran_def by blast
qed
qed

lemma finite_ran:
assumes "finite (dom p)"
shows "finite (ran p)"
proof -
have "ran p = (λx. the (p x)) ` dom p"
unfolding ran_def by force
from this ‹finite (dom p)› show ?thesis by auto
qed

lemma ran_distinct:
assumes dist: "distinct (map fst al)"
shows "ran (map_of al) = snd ` set al"
using assms
proof (induct al)
case Nil
then show ?case by simp
next
case (Cons kv al)
then have "ran (map_of al) = snd ` set al" by simp
moreover from Cons.prems have "map_of al (fst kv) = None"
ultimately show ?case by (simp only: map_of.simps ran_map_upd) simp
qed

lemma ran_map_of_zip:
assumes "length xs = length ys" "distinct xs"
shows "ran (map_of (zip xs ys)) = set ys"
using assms by (simp add: ran_distinct set_map[symmetric])

lemma ran_map_option: "ran (λx. map_option f (m x)) = f ` ran m"

subsection ‹‹map_le››

lemma map_le_empty [simp]: "empty ⊆⇩m g"

lemma upd_None_map_le [simp]: "f(x := None) ⊆⇩m f"

lemma map_le_upd[simp]: "f ⊆⇩m g ==> f(a := b) ⊆⇩m g(a := b)"

lemma map_le_imp_upd_le [simp]: "m1 ⊆⇩m m2 ⟹ m1(x := None) ⊆⇩m m2(x ↦ y)"

lemma map_le_upds [simp]:
"f ⊆⇩m g ⟹ f(as [↦] bs) ⊆⇩m g(as [↦] bs)"
apply (induct as arbitrary: f g bs)
apply simp
apply (case_tac bs)
apply auto
done

lemma map_le_implies_dom_le: "(f ⊆⇩m g) ⟹ (dom f ⊆ dom g)"
by (fastforce simp add: map_le_def dom_def)

lemma map_le_refl [simp]: "f ⊆⇩m f"

lemma map_le_trans[trans]: "⟦ m1 ⊆⇩m m2; m2 ⊆⇩m m3⟧ ⟹ m1 ⊆⇩m m3"
by (auto simp add: map_le_def dom_def)

lemma map_le_antisym: "⟦ f ⊆⇩m g; g ⊆⇩m f ⟧ ⟹ f = g"
unfolding map_le_def
apply (rule ext)
apply (case_tac "x ∈ dom f", simp)
apply (case_tac "x ∈ dom g", simp, fastforce)
done

lemma map_le_map_add [simp]: "f ⊆⇩m g ++ f"
by (fastforce simp: map_le_def)

lemma map_le_iff_map_add_commute: "f ⊆⇩m f ++ g ⟷ f ++ g = g ++ f"
by (fastforce simp: map_add_def map_le_def fun_eq_iff split: option.splits)

lemma map_add_le_mapE: "f ++ g ⊆⇩m h ⟹ g ⊆⇩m h"
by (fastforce simp: map_le_def map_add_def dom_def)

lemma map_add_le_mapI: "⟦ f ⊆⇩m h; g ⊆⇩m h ⟧ ⟹ f ++ g ⊆⇩m h"
by (auto simp: map_le_def map_add_def dom_def split: option.splits)

lemma map_add_subsumed1: "f ⊆⇩m g ⟹ f++g = g"

lemma map_add_subsumed2: "f ⊆⇩m g ⟹ g++f = g"

lemma dom_eq_singleton_conv: "dom f = {x} ⟷ (∃v. f = [x ↦ v])"
(is "?lhs ⟷ ?rhs")
proof
assume ?rhs
then show ?lhs by (auto split: if_split_asm)
next
assume ?lhs
then obtain v where v: "f x = Some v" by auto
show ?rhs
proof
show "f = [x ↦ v]"
proof (rule map_le_antisym)
show "[x ↦ v] ⊆⇩m f"
using v by (auto simp add: map_le_def)
show "f ⊆⇩m [x ↦ v]"
using ‹dom f = {x}› ‹f x = Some v› by (auto simp add: map_le_def)
qed
qed
qed

"(f++g = empty) ⟷ f = empty ∧ g = empty"

"(empty = f++g) ⟷ f = empty ∧ g = empty"

subsection ‹Various›

lemma set_map_of_compr:
assumes distinct: "distinct (map fst xs)"
shows "set xs = {(k, v). map_of xs k = Some v}"
using assms
proof (induct xs)
case Nil
then show ?case by simp
next
case (Cons x xs)
obtain k v where "x = (k, v)" by (cases x) blast
with Cons.prems have "k ∉ dom (map_of xs)"
then have *: "insert (k, v) {(k, v). map_of xs k = Some v} =
{(k', v'). (map_of xs(k ↦ v)) k' = Some v'}"
by (auto split: if_splits)
from Cons have "set xs = {(k, v). map_of xs k = Some v}" by simp
with * ‹x = (k, v)› show ?case by simp
qed

lemma eq_key_imp_eq_value:
"v1 = v2"
if "distinct (map fst xs)" "(k, v1) ∈ set xs" "(k, v2) ∈ set xs"
proof -
from that have "inj_on fst (set xs)"
moreover have "fst (k, v1) = fst (k, v2)"
by simp
ultimately have "(k, v1) = (k, v2)"
by (rule inj_onD) (fact that)+
then show ?thesis
by simp
qed

lemma map_of_inject_set:
assumes distinct: "distinct (map fst xs)" "distinct (map fst ys)"
shows "map_of xs = map_of ys ⟷ set xs = set ys" (is "?lhs ⟷ ?rhs")
proof
assume ?lhs
moreover from ‹distinct (map fst xs)› have "set xs = {(k, v). map_of xs k = Some v}"
by (rule set_map_of_compr)
moreover from ‹distinct (map fst ys)› have "set ys = {(k, v). map_of ys k = Some v}"
by (rule set_map_of_compr)
ultimately show ?rhs by simp
next
assume ?rhs show ?lhs
proof
fix k
show "map_of xs k = map_of ys k"
proof (cases "map_of xs k")
case None
with ‹?rhs› have "map_of ys k = None"
with None show ?thesis by simp
next
case (Some v)
with distinct ‹?rhs› have "map_of ys k = Some v"
by simp
with Some show ?thesis by simp
qed
qed
qed

hide_const (open) Map.empty

end
```