# Theory HOL-Library.Mapping

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

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

theory Mapping
imports Main AList
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 graph_parametric:
assumes "bi_total A"
shows "((A ===> rel_option B) ===> rel_set (rel_prod A B)) Map.graph Map.graph"
proof
fix f g assume "(A ===> rel_option B) f g"
with assms[unfolded bi_total_def] show "rel_set (rel_prod A B) (Map.graph f) (Map.graph g)"
unfolding graph_def rel_set_def rel_fun_def
by auto (metis option_rel_Some1 option_rel_Some2)+
qed

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

"(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)"

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 entries :: "('a, 'b) mapping ⇒ ('a × 'b) set"
is Map.graph parametric graph_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 ordered_entries :: "('a::linorder, 'b) mapping ⇒ ('a × 'b) list"
where "ordered_entries m = (if finite (entries m) then sorted_key_list_of_set fst (entries m)
else [])"

definition fold :: "('a::linorder ⇒ 'b ⇒ 'c ⇒ 'c) ⇒ ('a, 'b) mapping ⇒ 'c ⇒ 'c"
where "fold f m a = List.fold (case_prod f) (ordered_entries m) a"

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'"

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"
then have "x ∉ Mapping.keys m'"
then have "Mapping.lookup m' x = None"
with None show ?thesis
by simp
next
case (Some y)
then have A: "x ∈ Mapping.keys m"
then have "x ∈ Mapping.keys m'"
then have "∃y'. Mapping.lookup m' x = Some y'"
with Some assms(1)[OF A] show ?thesis
qed
qed

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

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

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

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

lemma lookup_delete[simp]: "lookup (delete k m) k = None"
by transfer simp

lemma lookup_delete_neq[simp]: "k ≠ k' ⟹ lookup (delete k m) k' = lookup m k'"
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"

lemma lookup_default_update: "lookup_default d (update k v m) k = v"

lemma lookup_default_update_neq:
"k ≠ k' ⟹ lookup_default d (update k v m) k' = lookup_default d m k'"

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: "lookup (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)"

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)"

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

lemma Mapping_delete_if_notin_keys[simp]:
"k ∉ keys m ⟹ delete k m = m"
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)"

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)

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)"

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

lemma keys_empty [simp]: "keys empty = {}"
by transfer (fact dom_empty)

lemma in_keysD: "k ∈ keys m ⟹ ∃v. lookup m k = Some v"
by transfer (fact domD)

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)"

lemma keys_map_values [simp]: "keys (map_values f m) = keys m"

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"

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)"

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

lemma set_ordered_keys[simp]:
"finite (Mapping.keys m) ⟹ set (Mapping.ordered_keys m) = Mapping.keys m"
unfolding ordered_keys_def by transfer auto

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

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

lemma ordered_keys_empty [simp]: "ordered_keys empty = []"

lemma sorted_ordered_keys[simp]: "sorted (ordered_keys m)"
unfolding ordered_keys_def by simp

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)"
(auto simp only: sorted_list_of_set_insert_remove[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
next
case True
with fin show ?thesis
qed
qed

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

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)"

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

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)"

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

lemma tabulate_fold: "tabulate xs f = List.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 = List.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) = List.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"
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"

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

subsubsection ‹@{term [source] entries}, @{term [source] ordered_entries},
and @{term [source] fold}›

context linorder
begin

sublocale folding_Map_graph: folding_insort_key "(≤)" "(<)" "Map.graph m" fst for m
by unfold_locales (fact inj_on_fst_graph)

end

lemma sorted_fst_list_of_set_insort_Map_graph[simp]:
assumes "finite (dom m)" "fst x ∉ dom m"
shows "sorted_key_list_of_set fst (insert x (Map.graph m))
= insort_key fst x (sorted_key_list_of_set fst (Map.graph m))"
proof(cases x)
case (Pair k v)
with ‹fst x ∉ dom m› have "Map.graph m ⊆ Map.graph (m(k ↦ v))"
by(auto simp: graph_def)
moreover from Pair ‹fst x ∉ dom m› have "(k, v) ∉ Map.graph m"
using graph_domD by fastforce
ultimately show ?thesis
using Pair assms folding_Map_graph.sorted_key_list_of_set_insert[where ?m="m(k ↦ v)"]
by auto
qed

lemma sorted_fst_list_of_set_insort_insert_Map_graph[simp]:
assumes "finite (dom m)" "fst x ∉ dom m"
shows "sorted_key_list_of_set fst (insert x (Map.graph m))
= insort_insert_key fst x (sorted_key_list_of_set fst (Map.graph m))"
proof(cases x)
case (Pair k v)
with ‹fst x ∉ dom m› have "Map.graph m ⊆ Map.graph (m(k ↦ v))"
by(auto simp: graph_def)
with assms Pair show ?thesis
unfolding sorted_fst_list_of_set_insort_Map_graph[OF assms] insort_insert_key_def
using folding_Map_graph.set_sorted_key_list_of_set in_graphD by (fastforce split: if_splits)
qed

lemma linorder_finite_Map_induct[consumes 1, case_names empty update]:
fixes m :: "'a::linorder ⇀ 'b"
assumes "finite (dom m)"
assumes "P Map.empty"
assumes "⋀k v m. ⟦ finite (dom m); k ∉ dom m; (⋀k'. k' ∈ dom m ⟹ k' ≤ k); P m ⟧
⟹ P (m(k ↦ v))"
shows "P m"
proof -
let ?key_list = "λm. sorted_list_of_set (dom m)"
from assms(1,2) show ?thesis
proof(induction "length (?key_list m)" arbitrary: m)
case 0
then have "sorted_list_of_set (dom m) = []"
by auto
with ‹finite (dom m)› have "m = Map.empty"
by auto
with ‹P Map.empty› show ?case by simp
next
case (Suc n)
then obtain x xs where x_xs: "sorted_list_of_set (dom m) = xs @ [x]"
by (metis append_butlast_last_id length_greater_0_conv zero_less_Suc)
have "sorted_list_of_set (dom (m(x := None))) = xs"
proof -
have "distinct (xs @ [x])"
by (metis sorted_list_of_set.distinct_sorted_key_list_of_set x_xs)
then have "remove1 x (xs @ [x]) = xs"
with ‹finite (dom m)› x_xs show ?thesis
qed
moreover have "k ≤ x" if "k ∈ dom (m(x := None))" for k
proof -
from x_xs have "sorted (xs @ [x])"
by (metis sorted_list_of_set.sorted_sorted_key_list_of_set)
moreover from ‹k ∈ dom (m(x := None))› have "k ∈ set xs"
using ‹finite (dom m)› ‹sorted_list_of_set (dom (m(x := None))) = xs›
by auto
ultimately show "k ≤ x"
qed
moreover from ‹finite (dom m)› have "finite (dom (m(x := None)))" "x ∉ dom (m(x := None))"
by simp_all
moreover have "P (m(x := None))"
using Suc ‹sorted_list_of_set (dom (m(x := None))) = xs› x_xs by auto
ultimately show ?case
using assms(3)[where ?m="m(x := None)"] by (metis fun_upd_triv fun_upd_upd not_Some_eq)
qed
qed

lemma delete_insort_fst[simp]: "AList.delete k (insort_key fst (k, v) xs) = AList.delete k xs"
by (induction xs) simp_all

lemma insort_fst_delete: "⟦ fst x ≠ k2; sorted (List.map fst xs) ⟧
⟹ insort_key fst x (AList.delete k2 xs) = AList.delete k2 (insort_key fst x xs)"
by (induction xs) (fastforce simp add: insort_is_Cons order_trans)+

lemma sorted_fst_list_of_set_Map_graph_fun_upd_None[simp]:
"sorted_key_list_of_set fst (Map.graph (m(k := None)))
= AList.delete k (sorted_key_list_of_set fst (Map.graph m))"
proof(cases "finite (Map.graph m)")
assume "finite (Map.graph m)"
from this[unfolded finite_graph_iff_finite_dom] show ?thesis
proof(induction rule: finite_Map_induct)
let ?list_of="sorted_key_list_of_set fst"
case (update k2 v2 m)
note [simp] = ‹k2 ∉ dom m› ‹finite (dom m)›

have right_eq: "AList.delete k (?list_of (Map.graph (m(k2 ↦ v2))))
= AList.delete k (insort_key fst (k2, v2) (?list_of (Map.graph m)))"
by simp

show ?case
proof(cases "k = k2")
case True
then have "?list_of (Map.graph ((m(k2 ↦ v2))(k := None)))
= AList.delete k (insort_key fst (k2, v2) (?list_of (Map.graph m)))"
using fst_graph_eq_dom update.IH by auto
then show ?thesis
using right_eq by metis
next
case False
then have "AList.delete k (insort_key fst (k2, v2) (?list_of (Map.graph m)))
= insort_key fst (k2, v2) (?list_of (Map.graph (m(k := None))))"
by (auto simp add: insort_fst_delete update.IH
folding_Map_graph.sorted_sorted_key_list_of_set[OF subset_refl])
also have "… = ?list_of (insert (k2, v2) (Map.graph (m(k := None))))"
by auto
also from False ‹k2 ∉ dom m› have "… = ?list_of (Map.graph ((m(k2 ↦ v2))(k := None)))"
by (metis graph_map_upd domIff fun_upd_triv fun_upd_twist)
finally show ?thesis using right_eq by metis
qed
qed simp
qed simp

lemma entries_empty[simp]: "entries empty = {}"
by transfer (fact graph_empty)

lemma entries_lookup: "entries m = Map.graph (lookup m)"
by transfer rule

lemma in_entriesI: "lookup m k = Some v ⟹ (k, v) ∈ entries m"
by transfer (fact in_graphI)

lemma in_entriesD: "(k, v) ∈ entries m ⟹ lookup m k = Some v"
by transfer (fact in_graphD)

lemma fst_image_entries_eq_keys[simp]: "fst ` Mapping.entries m = Mapping.keys m"
by transfer (fact fst_graph_eq_dom)

lemma finite_entries_iff_finite_keys[simp]:
"finite (entries m) = finite (keys m)"
by transfer (fact finite_graph_iff_finite_dom)

lemma entries_update:
"entries (update k v m) = insert (k, v) (entries (delete k m))"
by transfer (fact graph_map_upd)

lemma entries_delete:
"entries (delete k m) = {e ∈ entries m. fst e ≠ k}"
by transfer (fact graph_fun_upd_None)

lemma entries_of_alist[simp]:
"distinct (List.map fst xs) ⟹ entries (of_alist xs) = set xs"
by transfer (fact graph_map_of_if_distinct_dom)

lemma entries_keysD:
"x ∈ entries m ⟹ fst x ∈ keys m"
by transfer (fact graph_domD)

lemma set_ordered_entries[simp]:
"finite (keys m) ⟹ set (ordered_entries m) = entries m"
unfolding ordered_entries_def
by transfer (auto simp: folding_Map_graph.set_sorted_key_list_of_set[OF subset_refl])

lemma distinct_ordered_entries[simp]: "distinct (List.map fst (ordered_entries m))"
unfolding ordered_entries_def
by transfer (simp add: folding_Map_graph.distinct_sorted_key_list_of_set[OF subset_refl])

lemma sorted_ordered_entries[simp]: "sorted (List.map fst (ordered_entries m))"
unfolding ordered_entries_def
by transfer (auto intro: folding_Map_graph.sorted_sorted_key_list_of_set)

lemma ordered_entries_infinite[simp]:
"¬ finite (Mapping.keys m) ⟹ ordered_entries m = []"

lemma ordered_entries_empty[simp]: "ordered_entries empty = []"

lemma ordered_entries_update[simp]:
assumes "finite (keys m)"
shows "ordered_entries (update k v m)
= insort_insert_key fst (k, v) (AList.delete k (ordered_entries m))"
proof -
let ?list_of="sorted_key_list_of_set fst" and ?insort="insort_insert_key fst"

have *: "?list_of (insert (k, v) (Map.graph (m(k := None))))
= ?insort (k, v) (AList.delete k (?list_of (Map.graph m)))" if "finite (dom m)" for m
proof -
from ‹finite (dom m)› have "?list_of (insert (k, v) (Map.graph (m(k := None))))
= ?insort (k, v) (?list_of (Map.graph (m(k := None))))"
by (intro sorted_fst_list_of_set_insort_insert_Map_graph) (simp_all add: subset_insertI)
then show ?thesis by simp
qed
from assms show ?thesis
unfolding ordered_entries_def
apply (transfer fixing: k v) using "*" by auto
qed

lemma ordered_entries_delete[simp]:
"ordered_entries (delete k m) = AList.delete k (ordered_entries m)"
unfolding ordered_entries_def by transfer auto

lemma map_fst_ordered_entries[simp]:
"List.map fst (ordered_entries m) = ordered_keys m"
proof(cases "finite (Mapping.keys m)")
case True
then have "set (List.map fst (Mapping.ordered_entries m)) = set (Mapping.ordered_keys m)"
unfolding ordered_entries_def ordered_keys_def
by (transfer) (simp add: folding_Map_graph.set_sorted_key_list_of_set[OF subset_refl] fst_graph_eq_dom)
with True show "List.map fst (Mapping.ordered_entries m) = Mapping.ordered_keys m"
by (metis distinct_ordered_entries ordered_keys_def sorted_list_of_set.idem_if_sorted_distinct
sorted_list_of_set.set_sorted_key_list_of_set sorted_ordered_entries)
next
case False
then show ?thesis
unfolding ordered_entries_def ordered_keys_def by simp
qed

lemma fold_empty[simp]: "fold f empty a = a"
unfolding fold_def by simp

lemma insort_key_is_snoc_if_sorted_and_distinct:
assumes "sorted (List.map f xs)" "f y ∉ f ` set xs" "∀x ∈ set xs. f x ≤ f y"
shows "insort_key f y xs = xs @ [y]"
using assms by (induction xs) (auto dest!: insort_is_Cons)

lemma fold_update:
assumes "finite (keys m)"
assumes "k ∉ keys m" "⋀k'. k' ∈ keys m ⟹ k' ≤ k"
shows "fold f (update k v m) a = f k v (fold f m a)"
proof -
from assms have k_notin_entries: "k ∉ fst ` set (ordered_entries m)"
using entries_keysD by fastforce
with assms have "ordered_entries (update k v m)
= insort_insert_key fst (k, v) (ordered_entries m)"
by simp
also from k_notin_entries have "… = ordered_entries m @ [(k, v)]"
proof -
from assms have "∀x ∈ set (ordered_entries m). fst x ≤ fst (k, v)"
unfolding ordered_entries_def
by transfer (fastforce simp: folding_Map_graph.set_sorted_key_list_of_set[OF order_refl]
dest: graph_domD)
from insort_key_is_snoc_if_sorted_and_distinct[OF _ _ this] k_notin_entries ‹finite (keys m)›
show ?thesis
using sorted_ordered_keys
unfolding insort_insert_key_def by auto
qed
finally show ?thesis unfolding fold_def by simp
qed

lemma linorder_finite_Mapping_induct[consumes 1, case_names empty update]:
fixes m :: "('a::linorder, 'b) mapping"
assumes "finite (keys m)"
assumes "P empty"
assumes "⋀k v m.
⟦ finite (keys m); k ∉ keys m; (⋀k'. k' ∈ keys m ⟹ k' ≤ k); P m ⟧
⟹ P (update k v m)"
shows "P m"
using assms by transfer (simp add: linorder_finite_Map_induct)

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
entries ordered_entries fold

end
```