# Theory Mapping

theory Mapping
imports Main
```(*  Title:      HOL/Library/Mapping.thy
Author:     Florian Haftmann and Ondrej Kuncar
*)

section ‹An abstract view on maps for code generation.›

theory Mapping
imports Main
begin

subsection ‹Parametricity transfer rules›

lemma map_of_foldr: "map_of xs = foldr (λ(k, v) m. m(k ↦ v)) xs Map.empty"  (* FIXME move *)
using map_add_map_of_foldr [of Map.empty] by auto

context includes lifting_syntax
begin

lemma empty_parametric: "(A ===> rel_option B) Map.empty Map.empty"
by transfer_prover

lemma lookup_parametric: "((A ===> B) ===> A ===> B) (λm k. m k) (λm k. m k)"
by transfer_prover

lemma update_parametric:
assumes [transfer_rule]: "bi_unique A"
shows "(A ===> B ===> (A ===> rel_option B) ===> A ===> rel_option B)
(λk v m. m(k ↦ v)) (λk v m. m(k ↦ v))"
by transfer_prover

lemma delete_parametric:
assumes [transfer_rule]: "bi_unique A"
shows "(A ===> (A ===> rel_option B) ===> A ===> rel_option B)
(λk m. m(k := None)) (λk m. m(k := None))"
by transfer_prover

lemma is_none_parametric [transfer_rule]:
"(rel_option A ===> HOL.eq) Option.is_none Option.is_none"
by (auto simp add: Option.is_none_def rel_fun_def rel_option_iff split: option.split)

lemma dom_parametric:
assumes [transfer_rule]: "bi_total A"
shows "((A ===> rel_option B) ===> rel_set A) dom dom"
unfolding dom_def [abs_def] Option.is_none_def [symmetric] by transfer_prover

lemma map_of_parametric [transfer_rule]:
assumes [transfer_rule]: "bi_unique R1"
shows "(list_all2 (rel_prod R1 R2) ===> R1 ===> rel_option R2) map_of map_of"
unfolding map_of_def by transfer_prover

lemma map_entry_parametric [transfer_rule]:
assumes [transfer_rule]: "bi_unique A"
shows "(A ===> (B ===> B) ===> (A ===> rel_option B) ===> A ===> rel_option B)
(λk f m. (case m k of None ⇒ m
| Some v ⇒ m (k ↦ (f v)))) (λk f m. (case m k of None ⇒ m
| Some v ⇒ m (k ↦ (f v))))"
by transfer_prover

lemma tabulate_parametric:
assumes [transfer_rule]: "bi_unique A"
shows "(list_all2 A ===> (A ===> B) ===> A ===> rel_option B)
(λks f. (map_of (map (λk. (k, f k)) ks))) (λks f. (map_of (map (λk. (k, f k)) ks)))"
by transfer_prover

lemma bulkload_parametric:
"(list_all2 A ===> HOL.eq ===> rel_option A)
(λxs k. if k < length xs then Some (xs ! k) else None)
(λxs k. if k < length xs then Some (xs ! k) else None)"
proof
fix xs ys
assume "list_all2 A xs ys"
then show
"(HOL.eq ===> rel_option A)
(λk. if k < length xs then Some (xs ! k) else None)
(λk. if k < length ys then Some (ys ! k) else None)"
apply induct
apply auto
unfolding rel_fun_def
apply clarsimp
apply (case_tac xa)
apply (auto dest: list_all2_lengthD list_all2_nthD)
done
qed

lemma map_parametric:
"((A ===> B) ===> (C ===> D) ===> (B ===> rel_option C) ===> A ===> rel_option D)
(λf g m. (map_option g ∘ m ∘ f)) (λf g m. (map_option g ∘ m ∘ f))"
by transfer_prover

lemma combine_with_key_parametric:
"((A ===> B ===> B ===> B) ===> (A ===> rel_option B) ===> (A ===> rel_option B) ===>
(A ===> rel_option B)) (λf m1 m2 x. combine_options (f x) (m1 x) (m2 x))
(λf m1 m2 x. combine_options (f x) (m1 x) (m2 x))"
unfolding combine_options_def by transfer_prover

lemma combine_parametric:
"((B ===> B ===> B) ===> (A ===> rel_option B) ===> (A ===> rel_option B) ===>
(A ===> rel_option B)) (λf m1 m2 x. combine_options f (m1 x) (m2 x))
(λf m1 m2 x. combine_options f (m1 x) (m2 x))"
unfolding combine_options_def by transfer_prover

end

subsection ‹Type definition and primitive operations›

typedef ('a, 'b) mapping = "UNIV :: ('a ⇀ 'b) set"
morphisms rep Mapping ..

setup_lifting type_definition_mapping

lift_definition empty :: "('a, 'b) mapping"
is Map.empty parametric empty_parametric .

lift_definition lookup :: "('a, 'b) mapping ⇒ 'a ⇒ 'b option"
is "λm k. m k" parametric lookup_parametric .

definition "lookup_default d m k = (case Mapping.lookup m k of None ⇒ d | Some v ⇒ v)"

lemma [code abstract]:
"lookup (Mapping f) = f"
by (fact Mapping.lookup.abs_eq) (* FIXME lifting *)

lift_definition update :: "'a ⇒ 'b ⇒ ('a, 'b) mapping ⇒ ('a, 'b) mapping"
is "λk v m. m(k ↦ v)" parametric update_parametric .

lift_definition delete :: "'a ⇒ ('a, 'b) mapping ⇒ ('a, 'b) mapping"
is "λk m. m(k := None)" parametric delete_parametric .

lift_definition filter :: "('a ⇒ 'b ⇒ bool) ⇒ ('a, 'b) mapping ⇒ ('a, 'b) mapping"
is "λP m k. case m k of None ⇒ None | Some v ⇒ if P k v then Some v else None" .

lift_definition keys :: "('a, 'b) mapping ⇒ 'a set"
is dom parametric dom_parametric .

lift_definition tabulate :: "'a list ⇒ ('a ⇒ 'b) ⇒ ('a, 'b) mapping"
is "λks f. (map_of (List.map (λk. (k, f k)) ks))" parametric tabulate_parametric .

lift_definition bulkload :: "'a list ⇒ (nat, 'a) mapping"
is "λxs k. if k < length xs then Some (xs ! k) else None" parametric bulkload_parametric .

lift_definition map :: "('c ⇒ 'a) ⇒ ('b ⇒ 'd) ⇒ ('a, 'b) mapping ⇒ ('c, 'd) mapping"
is "λf g m. (map_option g ∘ m ∘ f)" parametric map_parametric .

lift_definition map_values :: "('c ⇒ 'a ⇒ 'b) ⇒ ('c, 'a) mapping ⇒ ('c, 'b) mapping"
is "λf m x. map_option (f x) (m x)" .

lift_definition combine_with_key ::
"('a ⇒ 'b ⇒ 'b ⇒ 'b) ⇒ ('a,'b) mapping ⇒ ('a,'b) mapping ⇒ ('a,'b) mapping"
is "λf m1 m2 x. combine_options (f x) (m1 x) (m2 x)" parametric combine_with_key_parametric .

lift_definition combine ::
"('b ⇒ 'b ⇒ 'b) ⇒ ('a,'b) mapping ⇒ ('a,'b) mapping ⇒ ('a,'b) mapping"
is "λf m1 m2 x. combine_options f (m1 x) (m2 x)" parametric combine_parametric .

definition "All_mapping m P ⟷
(∀x. case Mapping.lookup m x of None ⇒ True | Some y ⇒ P x y)"

declare [[code drop: map]]

subsection ‹Functorial structure›

functor map: map
by (transfer, auto simp add: fun_eq_iff option.map_comp option.map_id)+

subsection ‹Derived operations›

definition ordered_keys :: "('a::linorder, 'b) mapping ⇒ 'a list"
where "ordered_keys m = (if finite (keys m) then sorted_list_of_set (keys m) else [])"

definition is_empty :: "('a, 'b) mapping ⇒ bool"
where "is_empty m ⟷ keys m = {}"

definition size :: "('a, 'b) mapping ⇒ nat"
where "size m = (if finite (keys m) then card (keys m) else 0)"

definition replace :: "'a ⇒ 'b ⇒ ('a, 'b) mapping ⇒ ('a, 'b) mapping"
where "replace k v m = (if k ∈ keys m then update k v m else m)"

definition default :: "'a ⇒ 'b ⇒ ('a, 'b) mapping ⇒ ('a, 'b) mapping"
where "default k v m = (if k ∈ keys m then m else update k v m)"

text ‹Manual derivation of transfer rule is non-trivial›

lift_definition map_entry :: "'a ⇒ ('b ⇒ 'b) ⇒ ('a, 'b) mapping ⇒ ('a, 'b) mapping" is
"λk f m.
(case m k of
None ⇒ m
| Some v ⇒ m (k ↦ (f v)))" parametric map_entry_parametric .

lemma map_entry_code [code]:
"map_entry k f m =
(case lookup m k of
None ⇒ m
| Some v ⇒ update k (f v) m)"
by transfer rule

definition map_default :: "'a ⇒ 'b ⇒ ('b ⇒ 'b) ⇒ ('a, 'b) mapping ⇒ ('a, 'b) mapping"
where "map_default k v f m = map_entry k f (default k v m)"

definition of_alist :: "('k × 'v) list ⇒ ('k, 'v) mapping"
where "of_alist xs = foldr (λ(k, v) m. update k v m) xs empty"

instantiation mapping :: (type, type) equal
begin

definition "HOL.equal m1 m2 ⟷ (∀k. lookup m1 k = lookup m2 k)"

instance
apply standard
unfolding equal_mapping_def
apply transfer
apply auto
done

end

context includes lifting_syntax
begin

lemma [transfer_rule]:
assumes [transfer_rule]: "bi_total A"
and [transfer_rule]: "bi_unique B"
shows "(pcr_mapping A B ===> pcr_mapping A B ===> (=)) HOL.eq HOL.equal"
unfolding equal by transfer_prover

lemma of_alist_transfer [transfer_rule]:
assumes [transfer_rule]: "bi_unique R1"
shows "(list_all2 (rel_prod R1 R2) ===> pcr_mapping R1 R2) map_of of_alist"
unfolding of_alist_def [abs_def] map_of_foldr [abs_def] by transfer_prover

end

subsection ‹Properties›

lemma mapping_eqI: "(⋀x. lookup m x = lookup m' x) ⟹ m = m'"
by transfer (simp add: fun_eq_iff)

lemma mapping_eqI':
assumes "⋀x. x ∈ Mapping.keys m ⟹ Mapping.lookup_default d m x = Mapping.lookup_default d m' x"
and "Mapping.keys m = Mapping.keys m'"
shows "m = m'"
proof (intro mapping_eqI)
show "Mapping.lookup m x = Mapping.lookup m' x" for x
proof (cases "Mapping.lookup m x")
case None
then have "x ∉ Mapping.keys m"
by transfer (simp add: dom_def)
then have "x ∉ Mapping.keys m'"
by (simp add: assms)
then have "Mapping.lookup m' x = None"
by transfer (simp add: dom_def)
with None show ?thesis
by simp
next
case (Some y)
then have A: "x ∈ Mapping.keys m"
by transfer (simp add: dom_def)
then have "x ∈ Mapping.keys m'"
by (simp add: assms)
then have "∃y'. Mapping.lookup m' x = Some y'"
by transfer (simp add: dom_def)
with Some assms(1)[OF A] show ?thesis
by (auto simp add: lookup_default_def)
qed
qed

lemma lookup_update: "lookup (update k v m) k = Some v"
by transfer simp

lemma lookup_update_neq: "k ≠ k' ⟹ lookup (update k v m) k' = lookup m k'"
by transfer simp

lemma lookup_update': "Mapping.lookup (update k v m) k' = (if k = k' then Some v else lookup m k')"
by (auto simp: lookup_update lookup_update_neq)

lemma lookup_empty: "lookup empty k = None"
by transfer simp

lemma lookup_filter:
"lookup (filter P m) k =
(case lookup m k of
None ⇒ None
| Some v ⇒ if P k v then Some v else None)"
by transfer simp_all

lemma lookup_map_values: "lookup (map_values f m) k = map_option (f k) (lookup m k)"
by transfer simp_all

lemma lookup_default_empty: "lookup_default d empty k = d"
by (simp add: lookup_default_def lookup_empty)

lemma lookup_default_update: "lookup_default d (update k v m) k = v"
by (simp add: lookup_default_def lookup_update)

lemma lookup_default_update_neq:
"k ≠ k' ⟹ lookup_default d (update k v m) k' = lookup_default d m k'"
by (simp add: lookup_default_def lookup_update_neq)

lemma lookup_default_update':
"lookup_default d (update k v m) k' = (if k = k' then v else lookup_default d m k')"
by (auto simp: lookup_default_update lookup_default_update_neq)

lemma lookup_default_filter:
"lookup_default d (filter P m) k =
(if P k (lookup_default d m k) then lookup_default d m k else d)"
by (simp add: lookup_default_def lookup_filter split: option.splits)

lemma lookup_default_map_values:
"lookup_default (f k d) (map_values f m) k = f k (lookup_default d m k)"
by (simp add: lookup_default_def lookup_map_values split: option.splits)

lemma lookup_combine_with_key:
"Mapping.lookup (combine_with_key f m1 m2) x =
combine_options (f x) (Mapping.lookup m1 x) (Mapping.lookup m2 x)"
by transfer (auto split: option.splits)

lemma combine_altdef: "combine f m1 m2 = combine_with_key (λ_. f) m1 m2"
by transfer' (rule refl)

lemma lookup_combine:
"Mapping.lookup (combine f m1 m2) x =
combine_options f (Mapping.lookup m1 x) (Mapping.lookup m2 x)"
by transfer (auto split: option.splits)

lemma lookup_default_neutral_combine_with_key:
assumes "⋀x. f k d x = x" "⋀x. f k x d = x"
shows "Mapping.lookup_default d (combine_with_key f m1 m2) k =
f k (Mapping.lookup_default d m1 k) (Mapping.lookup_default d m2 k)"
by (auto simp: lookup_default_def lookup_combine_with_key assms split: option.splits)

lemma lookup_default_neutral_combine:
assumes "⋀x. f d x = x" "⋀x. f x d = x"
shows "Mapping.lookup_default d (combine f m1 m2) x =
f (Mapping.lookup_default d m1 x) (Mapping.lookup_default d m2 x)"
by (auto simp: lookup_default_def lookup_combine assms split: option.splits)

lemma lookup_map_entry: "lookup (map_entry x f m) x = map_option f (lookup m x)"
by transfer (auto split: option.splits)

lemma lookup_map_entry_neq: "x ≠ y ⟹ lookup (map_entry x f m) y = lookup m y"
by transfer (auto split: option.splits)

lemma lookup_map_entry':
"lookup (map_entry x f m) y =
(if x = y then map_option f (lookup m y) else lookup m y)"
by transfer (auto split: option.splits)

lemma lookup_default: "lookup (default x d m) x = Some (lookup_default d m x)"
unfolding lookup_default_def default_def
by transfer (auto split: option.splits)

lemma lookup_default_neq: "x ≠ y ⟹ lookup (default x d m) y = lookup m y"
unfolding lookup_default_def default_def
by transfer (auto split: option.splits)

lemma lookup_default':
"lookup (default x d m) y =
(if x = y then Some (lookup_default d m x) else lookup m y)"
unfolding lookup_default_def default_def
by transfer (auto split: option.splits)

lemma lookup_map_default: "lookup (map_default x d f m) x = Some (f (lookup_default d m x))"
unfolding lookup_default_def default_def
by (simp add: map_default_def lookup_map_entry lookup_default lookup_default_def)

lemma lookup_map_default_neq: "x ≠ y ⟹ lookup (map_default x d f m) y = lookup m y"
unfolding lookup_default_def default_def
by (simp add: map_default_def lookup_map_entry_neq lookup_default_neq)

lemma lookup_map_default':
"lookup (map_default x d f m) y =
(if x = y then Some (f (lookup_default d m x)) else lookup m y)"
unfolding lookup_default_def default_def
by (simp add: map_default_def lookup_map_entry' lookup_default' lookup_default_def)

lemma lookup_tabulate:
assumes "distinct xs"
shows "Mapping.lookup (Mapping.tabulate xs f) x = (if x ∈ set xs then Some (f x) else None)"
using assms by transfer (auto simp: map_of_eq_None_iff o_def dest!: map_of_SomeD)

lemma lookup_of_alist: "Mapping.lookup (Mapping.of_alist xs) k = map_of xs k"
by transfer simp_all

lemma keys_is_none_rep [code_unfold]: "k ∈ keys m ⟷ ¬ (Option.is_none (lookup m k))"
by transfer (auto simp add: Option.is_none_def)

lemma update_update:
"update k v (update k w m) = update k v m"
"k ≠ l ⟹ update k v (update l w m) = update l w (update k v m)"
by (transfer; simp add: fun_upd_twist)+

lemma update_delete [simp]: "update k v (delete k m) = update k v m"
by transfer simp

lemma delete_update:
"delete k (update k v m) = delete k m"
"k ≠ l ⟹ delete k (update l v m) = update l v (delete k m)"
by (transfer; simp add: fun_upd_twist)+

lemma delete_empty [simp]: "delete k empty = empty"
by transfer simp

lemma replace_update:
"k ∉ keys m ⟹ replace k v m = m"
"k ∈ keys m ⟹ replace k v m = update k v m"
by (transfer; auto simp add: replace_def fun_upd_twist)+

lemma map_values_update: "map_values f (update k v m) = update k (f k v) (map_values f m)"
by transfer (simp_all add: fun_eq_iff)

lemma size_mono: "finite (keys m') ⟹ keys m ⊆ keys m' ⟹ size m ≤ size m'"
unfolding size_def by (auto intro: card_mono)

lemma size_empty [simp]: "size empty = 0"
unfolding size_def by transfer simp

lemma size_update:
"finite (keys m) ⟹ size (update k v m) =
(if k ∈ keys m then size m else Suc (size m))"
unfolding size_def by transfer (auto simp add: insert_dom)

lemma size_delete: "size (delete k m) = (if k ∈ keys m then size m - 1 else size m)"
unfolding size_def by transfer simp

lemma size_tabulate [simp]: "size (tabulate ks f) = length (remdups ks)"
unfolding size_def by transfer (auto simp add: map_of_map_restrict card_set comp_def)

lemma keys_filter: "keys (filter P m) ⊆ keys m"
by transfer (auto split: option.splits)

lemma size_filter: "finite (keys m) ⟹ size (filter P m) ≤ size m"
by (intro size_mono keys_filter)

lemma bulkload_tabulate: "bulkload xs = tabulate [0..<length xs] (nth xs)"
by transfer (auto simp add: map_of_map_restrict)

lemma is_empty_empty [simp]: "is_empty empty"
unfolding is_empty_def by transfer simp

lemma is_empty_update [simp]: "¬ is_empty (update k v m)"
unfolding is_empty_def by transfer simp

lemma is_empty_delete: "is_empty (delete k m) ⟷ is_empty m ∨ keys m = {k}"
unfolding is_empty_def by transfer (auto simp del: dom_eq_empty_conv)

lemma is_empty_replace [simp]: "is_empty (replace k v m) ⟷ is_empty m"
unfolding is_empty_def replace_def by transfer auto

lemma is_empty_default [simp]: "¬ is_empty (default k v m)"
unfolding is_empty_def default_def by transfer auto

lemma is_empty_map_entry [simp]: "is_empty (map_entry k f m) ⟷ is_empty m"
unfolding is_empty_def by transfer (auto split: option.split)

lemma is_empty_map_values [simp]: "is_empty (map_values f m) ⟷ is_empty m"
unfolding is_empty_def by transfer (auto simp: fun_eq_iff)

lemma is_empty_map_default [simp]: "¬ is_empty (map_default k v f m)"
by (simp add: map_default_def)

lemma keys_dom_lookup: "keys m = dom (Mapping.lookup m)"
by transfer rule

lemma keys_empty [simp]: "keys empty = {}"
by transfer simp

lemma keys_update [simp]: "keys (update k v m) = insert k (keys m)"
by transfer simp

lemma keys_delete [simp]: "keys (delete k m) = keys m - {k}"
by transfer simp

lemma keys_replace [simp]: "keys (replace k v m) = keys m"
unfolding replace_def by transfer (simp add: insert_absorb)

lemma keys_default [simp]: "keys (default k v m) = insert k (keys m)"
unfolding default_def by transfer (simp add: insert_absorb)

lemma keys_map_entry [simp]: "keys (map_entry k f m) = keys m"
by transfer (auto split: option.split)

lemma keys_map_default [simp]: "keys (map_default k v f m) = insert k (keys m)"
by (simp add: map_default_def)

lemma keys_map_values [simp]: "keys (map_values f m) = keys m"
by transfer (simp_all add: dom_def)

lemma keys_combine_with_key [simp]:
"Mapping.keys (combine_with_key f m1 m2) = Mapping.keys m1 ∪ Mapping.keys m2"
by transfer (auto simp: dom_def combine_options_def split: option.splits)

lemma keys_combine [simp]: "Mapping.keys (combine f m1 m2) = Mapping.keys m1 ∪ Mapping.keys m2"
by (simp add: combine_altdef)

lemma keys_tabulate [simp]: "keys (tabulate ks f) = set ks"
by transfer (simp add: map_of_map_restrict o_def)

lemma keys_of_alist [simp]: "keys (of_alist xs) = set (List.map fst xs)"
by transfer (simp_all add: dom_map_of_conv_image_fst)

lemma keys_bulkload [simp]: "keys (bulkload xs) = {0..<length xs}"
by (simp add: bulkload_tabulate)

lemma distinct_ordered_keys [simp]: "distinct (ordered_keys m)"
by (simp add: ordered_keys_def)

lemma ordered_keys_infinite [simp]: "¬ finite (keys m) ⟹ ordered_keys m = []"
by (simp add: ordered_keys_def)

lemma ordered_keys_empty [simp]: "ordered_keys empty = []"
by (simp add: ordered_keys_def)

lemma ordered_keys_update [simp]:
"k ∈ keys m ⟹ ordered_keys (update k v m) = ordered_keys m"
"finite (keys m) ⟹ k ∉ keys m ⟹
ordered_keys (update k v m) = insort k (ordered_keys m)"
by (simp_all add: ordered_keys_def)
(auto simp only: sorted_list_of_set_insert [symmetric] insert_absorb)

lemma ordered_keys_delete [simp]: "ordered_keys (delete k m) = remove1 k (ordered_keys m)"
proof (cases "finite (keys m)")
case False
then show ?thesis by simp
next
case fin: True
show ?thesis
proof (cases "k ∈ keys m")
case False
with fin have "k ∉ set (sorted_list_of_set (keys m))"
by simp
with False show ?thesis
by (simp add: ordered_keys_def remove1_idem)
next
case True
with fin show ?thesis
by (simp add: ordered_keys_def sorted_list_of_set_remove)
qed
qed

lemma ordered_keys_replace [simp]: "ordered_keys (replace k v m) = ordered_keys m"
by (simp add: replace_def)

lemma ordered_keys_default [simp]:
"k ∈ keys m ⟹ ordered_keys (default k v m) = ordered_keys m"
"finite (keys m) ⟹ k ∉ keys m ⟹ ordered_keys (default k v m) = insort k (ordered_keys m)"
by (simp_all add: default_def)

lemma ordered_keys_map_entry [simp]: "ordered_keys (map_entry k f m) = ordered_keys m"
by (simp add: ordered_keys_def)

lemma ordered_keys_map_default [simp]:
"k ∈ keys m ⟹ ordered_keys (map_default k v f m) = ordered_keys m"
"finite (keys m) ⟹ k ∉ keys m ⟹ ordered_keys (map_default k v f m) = insort k (ordered_keys m)"
by (simp_all add: map_default_def)

lemma ordered_keys_tabulate [simp]: "ordered_keys (tabulate ks f) = sort (remdups ks)"
by (simp add: ordered_keys_def sorted_list_of_set_sort_remdups)

lemma ordered_keys_bulkload [simp]: "ordered_keys (bulkload ks) = [0..<length ks]"
by (simp add: ordered_keys_def)

lemma tabulate_fold: "tabulate xs f = fold (λk m. update k (f k) m) xs empty"
proof transfer
fix f :: "'a ⇒ 'b" and xs
have "map_of (List.map (λk. (k, f k)) xs) = foldr (λk m. m(k ↦ f k)) xs Map.empty"
by (simp add: foldr_map comp_def map_of_foldr)
also have "foldr (λk m. m(k ↦ f k)) xs = fold (λk m. m(k ↦ f k)) xs"
by (rule foldr_fold) (simp add: fun_eq_iff)
ultimately show "map_of (List.map (λk. (k, f k)) xs) = fold (λk m. m(k ↦ f k)) xs Map.empty"
by simp
qed

lemma All_mapping_mono:
"(⋀k v. k ∈ keys m ⟹ P k v ⟹ Q k v) ⟹ All_mapping m P ⟹ All_mapping m Q"
unfolding All_mapping_def by transfer (auto simp: All_mapping_def dom_def split: option.splits)

lemma All_mapping_empty [simp]: "All_mapping Mapping.empty P"
by (auto simp: All_mapping_def lookup_empty)

lemma All_mapping_update_iff:
"All_mapping (Mapping.update k v m) P ⟷ P k v ∧ All_mapping m (λk' v'. k = k' ∨ P k' v')"
unfolding All_mapping_def
proof safe
assume "∀x. case Mapping.lookup (Mapping.update k v m) x of None ⇒ True | Some y ⇒ P x y"
then have *: "case Mapping.lookup (Mapping.update k v m) x of None ⇒ True | Some y ⇒ P x y" for x
by blast
from *[of k] show "P k v"
by (simp add: lookup_update)
show "case Mapping.lookup m x of None ⇒ True | Some v' ⇒ k = x ∨ P x v'" for x
using *[of x] by (auto simp add: lookup_update' split: if_splits option.splits)
next
assume "P k v"
assume "∀x. case Mapping.lookup m x of None ⇒ True | Some v' ⇒ k = x ∨ P x v'"
then have A: "case Mapping.lookup m x of None ⇒ True | Some v' ⇒ k = x ∨ P x v'" for x
by blast
show "case Mapping.lookup (Mapping.update k v m) x of None ⇒ True | Some xa ⇒ P x xa" for x
using ‹P k v› A[of x] by (auto simp: lookup_update' split: option.splits)
qed

lemma All_mapping_update:
"P k v ⟹ All_mapping m (λk' v'. k = k' ∨ P k' v') ⟹ All_mapping (Mapping.update k v m) P"
by (simp add: All_mapping_update_iff)

lemma All_mapping_filter_iff: "All_mapping (filter P m) Q ⟷ All_mapping m (λk v. P k v ⟶ Q k v)"
by (auto simp: All_mapping_def lookup_filter split: option.splits)

lemma All_mapping_filter: "All_mapping m Q ⟹ All_mapping (filter P m) Q"
by (auto simp: All_mapping_filter_iff intro: All_mapping_mono)

lemma All_mapping_map_values: "All_mapping (map_values f m) P ⟷ All_mapping m (λk v. P k (f k v))"
by (auto simp: All_mapping_def lookup_map_values split: option.splits)

lemma All_mapping_tabulate: "(∀x∈set xs. P x (f x)) ⟹ All_mapping (Mapping.tabulate xs f) P"
unfolding All_mapping_def
apply (intro allI)
apply transfer
apply (auto split: option.split dest!: map_of_SomeD)
done

lemma All_mapping_alist:
"(⋀k v. (k, v) ∈ set xs ⟹ P k v) ⟹ All_mapping (Mapping.of_alist xs) P"
by (auto simp: All_mapping_def lookup_of_alist dest!: map_of_SomeD split: option.splits)

lemma combine_empty [simp]: "combine f Mapping.empty y = y" "combine f y Mapping.empty = y"
by (transfer; force)+

lemma (in abel_semigroup) comm_monoid_set_combine: "comm_monoid_set (combine f) Mapping.empty"
by standard (transfer fixing: f, simp add: combine_options_ac[of f] ac_simps)+

locale combine_mapping_abel_semigroup = abel_semigroup
begin

sublocale combine: comm_monoid_set "combine f" Mapping.empty
by (rule comm_monoid_set_combine)

lemma fold_combine_code:
"combine.F g (set xs) = foldr (λx. combine f (g x)) (remdups xs) Mapping.empty"
proof -
have "combine.F g (set xs) = foldr (λx. combine f (g x)) xs Mapping.empty"
if "distinct xs" for xs
using that by (induction xs) simp_all
from this[of "remdups xs"] show ?thesis by simp
qed

lemma keys_fold_combine: "finite A ⟹ Mapping.keys (combine.F g A) = (⋃x∈A. Mapping.keys (g x))"
by (induct A rule: finite_induct) simp_all

end

subsection ‹Code generator setup›

hide_const (open) empty is_empty rep lookup lookup_default filter update delete ordered_keys
keys size replace default map_entry map_default tabulate bulkload map map_values combine of_alist

end
```