# Theory AList

theory AList
imports Main
```(*  Title:      HOL/Library/AList.thy
Author:     Norbert Schirmer, Tobias Nipkow, Martin Wildmoser, TU Muenchen
*)

section ‹Implementation of Association Lists›

theory AList
imports Main
begin

context
begin

text ‹
The operations preserve distinctness of keys and
function @{term "clearjunk"} distributes over them. Since
@{term clearjunk} enforces distinctness of keys it can be used
to establish the invariant, e.g. for inductive proofs.
›

qualified primrec update :: "'key ⇒ 'val ⇒ ('key × 'val) list ⇒ ('key × 'val) list"
where
"update k v [] = [(k, v)]"
| "update k v (p # ps) = (if fst p = k then (k, v) # ps else p # update k v ps)"

lemma update_conv': "map_of (update k v al)  = (map_of al)(k↦v)"
by (induct al) (auto simp add: fun_eq_iff)

corollary update_conv: "map_of (update k v al) k' = ((map_of al)(k↦v)) k'"

lemma dom_update: "fst ` set (update k v al) = {k} ∪ fst ` set al"
by (induct al) auto

lemma update_keys:
"map fst (update k v al) =
(if k ∈ set (map fst al) then map fst al else map fst al @ [k])"
by (induct al) simp_all

lemma distinct_update:
assumes "distinct (map fst al)"
shows "distinct (map fst (update k v al))"
using assms by (simp add: update_keys)

lemma update_filter:
"a ≠ k ⟹ update k v [q←ps. fst q ≠ a] = [q←update k v ps. fst q ≠ a]"
by (induct ps) auto

lemma update_triv: "map_of al k = Some v ⟹ update k v al = al"
by (induct al) auto

lemma update_nonempty [simp]: "update k v al ≠ []"
by (induct al) auto

lemma update_eqD: "update k v al = update k v' al' ⟹ v = v'"
proof (induct al arbitrary: al')
case Nil
then show ?case
by (cases al') (auto split: split_if_asm)
next
case Cons
then show ?case
by (cases al') (auto split: split_if_asm)
qed

lemma update_last [simp]: "update k v (update k v' al) = update k v al"
by (induct al) auto

text ‹Note that the lists are not necessarily the same:
@{term "update k v (update k' v' []) = [(k', v'), (k, v)]"} and
@{term "update k' v' (update k v []) = [(k, v), (k', v')]"}.›

lemma update_swap:
"k ≠ k' ⟹
map_of (update k v (update k' v' al)) = map_of (update k' v' (update k v al))"

lemma update_Some_unfold:
"map_of (update k v al) x = Some y ⟷
x = k ∧ v = y ∨ x ≠ k ∧ map_of al x = Some y"

lemma image_update [simp]:
"x ∉ A ⟹ map_of (update x y al) ` A = map_of al ` A"

qualified definition
updates :: "'key list ⇒ 'val list ⇒ ('key × 'val) list ⇒ ('key × 'val) list"
where "updates ks vs = fold (case_prod update) (zip ks vs)"

"updates [] vs ps = ps"
"updates ks [] ps = ps"
"updates (k#ks) (v#vs) ps = updates ks vs (update k v ps)"

"updates (k # ks) vs ps =
(case vs of [] ⇒ ps | v # vs ⇒ updates ks vs (update k v ps))"
by (cases vs) simp_all

proof -
have "map_of ∘ fold (case_prod update) (zip ks vs) =
fold (λ(k, v) f. f(k ↦ v)) (zip ks vs) ∘ map_of"
by (rule fold_commute) (auto simp add: fun_eq_iff update_conv')
then show ?thesis
qed

lemma updates_conv: "map_of (updates ks vs al) k = ((map_of al)(ks[↦]vs)) k"

assumes "distinct (map fst al)"
shows "distinct (map fst (updates ks vs al))"
proof -
have "distinct (fold
(λ(k, v) al. if k ∈ set al then al else al @ [k])
(zip ks vs) (map fst al))"
by (rule fold_invariant [of "zip ks vs" "λ_. True"]) (auto intro: assms)
moreover have "map fst ∘ fold (case_prod update) (zip ks vs) =
fold (λ(k, v) al. if k ∈ set al then al else al @ [k]) (zip ks vs) ∘ map fst"
by (rule fold_commute) (simp add: update_keys split_def case_prod_beta comp_def)
ultimately show ?thesis
qed

lemma updates_append1[simp]: "size ks < size vs ⟹
updates (ks@[k]) vs al = update k (vs!size ks) (updates ks vs al)"
by (induct ks arbitrary: vs al) (auto split: list.splits)

"size ks ≤ i ⟹ i < size vs ⟹
by (induct ks arbitrary: al vs i) (auto split: list.splits nat.splits)

"map_of (updates xs ys (update x y al)) =
map_of
(if x ∈ set (take (length ys) xs)
else (update x y (updates xs ys al)))"

"k ∉ set ks ⟹
map_of (updates ks vs (update k v al)) = map_of (update k v (updates ks vs al))"

"k ∉ set ks ⟹ map_of (updates ks vs al) k = map_of al k"

"size xs = size ys ⟹ updates (xs @ zs) ys al = updates xs ys al"
by (induct xs arbitrary: ys al) (auto split: list.splits)

"size xs = size ys ⟹ updates xs (ys @ zs) al = updates xs ys al"
by (induct xs arbitrary: ys al) (auto split: list.splits)

subsection ‹‹delete››

qualified definition delete :: "'key ⇒ ('key × 'val) list ⇒ ('key × 'val) list"
where delete_eq: "delete k = filter (λ(k', _). k ≠ k')"

lemma delete_simps [simp]:
"delete k [] = []"
"delete k (p # ps) = (if fst p = k then delete k ps else p # delete k ps)"

lemma delete_conv': "map_of (delete k al) = (map_of al)(k := None)"
by (induct al) (auto simp add: fun_eq_iff)

corollary delete_conv: "map_of (delete k al) k' = ((map_of al)(k := None)) k'"

lemma delete_keys: "map fst (delete k al) = removeAll k (map fst al)"
by (simp add: delete_eq removeAll_filter_not_eq filter_map split_def comp_def)

lemma distinct_delete:
assumes "distinct (map fst al)"
shows "distinct (map fst (delete k al))"
using assms by (simp add: delete_keys distinct_removeAll)

lemma delete_id [simp]: "k ∉ fst ` set al ⟹ delete k al = al"
by (auto simp add: image_iff delete_eq filter_id_conv)

lemma delete_idem: "delete k (delete k al) = delete k al"

lemma map_of_delete [simp]: "k' ≠ k ⟹ map_of (delete k al) k' = map_of al k'"

lemma delete_notin_dom: "k ∉ fst ` set (delete k al)"

lemma dom_delete_subset: "fst ` set (delete k al) ⊆ fst ` set al"

lemma delete_update_same: "delete k (update k v al) = delete k al"
by (induct al) simp_all

lemma delete_update: "k ≠ l ⟹ delete l (update k v al) = update k v (delete l al)"
by (induct al) simp_all

lemma delete_twist: "delete x (delete y al) = delete y (delete x al)"

lemma length_delete_le: "length (delete k al) ≤ length al"

subsection ‹‹update_with_aux› and ‹delete_aux››

qualified primrec update_with_aux :: "'val ⇒ 'key ⇒ ('val ⇒ 'val) ⇒ ('key × 'val) list ⇒ ('key × 'val) list"
where
"update_with_aux v k f [] = [(k, f v)]"
| "update_with_aux v k f (p # ps) = (if (fst p = k) then (k, f (snd p)) # ps else p # update_with_aux v k f ps)"

text ‹
The above @{term "delete"} traverses all the list even if it has found the key.
This one does not have to keep going because is assumes the invariant that keys are distinct.
›
qualified fun delete_aux :: "'key ⇒ ('key × 'val) list ⇒ ('key × 'val) list"
where
"delete_aux k [] = []"
| "delete_aux k ((k', v) # xs) = (if k = k' then xs else (k', v) # delete_aux k xs)"

lemma map_of_update_with_aux':
"map_of (update_with_aux v k f ps) k' = ((map_of ps)(k ↦ (case map_of ps k of None ⇒ f v | Some v ⇒ f v))) k'"
by(induct ps) auto

lemma map_of_update_with_aux:
"map_of (update_with_aux v k f ps) = (map_of ps)(k ↦ (case map_of ps k of None ⇒ f v | Some v ⇒ f v))"

lemma dom_update_with_aux: "fst ` set (update_with_aux v k f ps) = {k} ∪ fst ` set ps"
by (induct ps) auto

lemma distinct_update_with_aux [simp]:
"distinct (map fst (update_with_aux v k f ps)) = distinct (map fst ps)"

lemma set_update_with_aux:
"distinct (map fst xs)
⟹ set (update_with_aux v k f xs) = (set xs - {k} × UNIV ∪ {(k, f (case map_of xs k of None ⇒ v | Some v ⇒ v))})"
by(induct xs)(auto intro: rev_image_eqI)

lemma set_delete_aux: "distinct (map fst xs) ⟹ set (delete_aux k xs) = set xs - {k} × UNIV"
apply(induct xs)
apply simp_all
apply clarsimp
apply(fastforce intro: rev_image_eqI)
done

lemma dom_delete_aux: "distinct (map fst ps) ⟹ fst ` set (delete_aux k ps) = fst ` set ps - {k}"

lemma distinct_delete_aux [simp]:
"distinct (map fst ps) ⟹ distinct (map fst (delete_aux k ps))"
proof(induct ps)
case Nil thus ?case by simp
next
case (Cons a ps)
obtain k' v where a: "a = (k', v)" by(cases a)
show ?case
proof(cases "k' = k")
case True with Cons a show ?thesis by simp
next
case False
with Cons a have "k' ∉ fst ` set ps" "distinct (map fst ps)" by simp_all
with False a have "k' ∉ fst ` set (delete_aux k ps)"
by(auto dest!: dom_delete_aux[where k=k])
with Cons a show ?thesis by simp
qed
qed

lemma map_of_delete_aux':
"distinct (map fst xs) ⟹ map_of (delete_aux k xs) = (map_of xs)(k := None)"
apply (induct xs)
apply (fastforce simp add: map_of_eq_None_iff fun_upd_twist)
apply (auto intro!: ext)
done

lemma map_of_delete_aux:
"distinct (map fst xs) ⟹ map_of (delete_aux k xs) k' = ((map_of xs)(k := None)) k'"

lemma delete_aux_eq_Nil_conv: "delete_aux k ts = [] ⟷ ts = [] ∨ (∃v. ts = [(k, v)])"
by(cases ts)(auto split: split_if_asm)

subsection ‹‹restrict››

qualified definition restrict :: "'key set ⇒ ('key × 'val) list ⇒ ('key × 'val) list"
where restrict_eq: "restrict A = filter (λ(k, v). k ∈ A)"

lemma restr_simps [simp]:
"restrict A [] = []"
"restrict A (p#ps) = (if fst p ∈ A then p # restrict A ps else restrict A ps)"

lemma restr_conv': "map_of (restrict A al) = ((map_of al)|` A)"
proof
fix k
show "map_of (restrict A al) k = ((map_of al)|` A) k"
by (induct al) (simp, cases "k ∈ A", auto)
qed

corollary restr_conv: "map_of (restrict A al) k = ((map_of al)|` A) k"

lemma distinct_restr:
"distinct (map fst al) ⟹ distinct (map fst (restrict A al))"
by (induct al) (auto simp add: restrict_eq)

lemma restr_empty [simp]:
"restrict {} al = []"
"restrict A [] = []"
by (induct al) (auto simp add: restrict_eq)

lemma restr_in [simp]: "x ∈ A ⟹ map_of (restrict A al) x = map_of al x"

lemma restr_out [simp]: "x ∉ A ⟹ map_of (restrict A al) x = None"

lemma dom_restr [simp]: "fst ` set (restrict A al) = fst ` set al ∩ A"
by (induct al) (auto simp add: restrict_eq)

lemma restr_upd_same [simp]: "restrict (-{x}) (update x y al) = restrict (-{x}) al"
by (induct al) (auto simp add: restrict_eq)

lemma restr_restr [simp]: "restrict A (restrict B al) = restrict (A∩B) al"
by (induct al) (auto simp add: restrict_eq)

lemma restr_update[simp]:
"map_of (restrict D (update x y al)) =
map_of ((if x ∈ D then (update x y (restrict (D-{x}) al)) else restrict D al))"

lemma restr_delete [simp]:
"delete x (restrict D al) = (if x ∈ D then restrict (D - {x}) al else restrict D al)"
proof -
have "⋀y. y ≠ x ⟷ x ≠ y"
by auto
then show "[p ← al. fst p ∈ D ∧ x ≠ fst p] = [p ← al. fst p ∈ D ∧ fst p ≠ x]"
by simp
assume "x ∉ D"
then have "⋀y. y ∈ D ⟷ y ∈ D ∧ x ≠ y"
by auto
then show "[p ← al . fst p ∈ D ∧ x ≠ fst p] = [p ← al . fst p ∈ D]"
by simp
qed

lemma update_restr:
"map_of (update x y (restrict D al)) = map_of (update x y (restrict (D - {x}) al))"
by (simp add: update_conv' restr_conv') (rule fun_upd_restrict)

lemma update_restr_conv [simp]:
"x ∈ D ⟹
map_of (update x y (restrict D al)) = map_of (update x y (restrict (D - {x}) al))"

"length xs = length ys ⟹ set xs ⊆ D ⟹
map_of (restrict D (updates xs ys al)) =
map_of (updates xs ys (restrict (D - set xs) al))"

lemma restr_delete_twist: "(restrict A (delete a ps)) = delete a (restrict A ps)"
by (induct ps) auto

subsection ‹‹clearjunk››

qualified function clearjunk  :: "('key × 'val) list ⇒ ('key × 'val) list"
where
"clearjunk [] = []"
| "clearjunk (p#ps) = p # clearjunk (delete (fst p) ps)"
by pat_completeness auto
termination
by (relation "measure length") (simp_all add: less_Suc_eq_le length_delete_le)

lemma map_of_clearjunk: "map_of (clearjunk al) = map_of al"
by (induct al rule: clearjunk.induct) (simp_all add: fun_eq_iff)

lemma clearjunk_keys_set: "set (map fst (clearjunk al)) = set (map fst al)"
by (induct al rule: clearjunk.induct) (simp_all add: delete_keys)

lemma dom_clearjunk: "fst ` set (clearjunk al) = fst ` set al"
using clearjunk_keys_set by simp

lemma distinct_clearjunk [simp]: "distinct (map fst (clearjunk al))"
by (induct al rule: clearjunk.induct) (simp_all del: set_map add: clearjunk_keys_set delete_keys)

lemma ran_clearjunk: "ran (map_of (clearjunk al)) = ran (map_of al)"

lemma ran_map_of: "ran (map_of al) = snd ` set (clearjunk al)"
proof -
have "ran (map_of al) = ran (map_of (clearjunk al))"
also have "… = snd ` set (clearjunk al)"
finally show ?thesis .
qed

lemma clearjunk_update: "clearjunk (update k v al) = update k v (clearjunk al)"
by (induct al rule: clearjunk.induct) (simp_all add: delete_update)

proof -
have "clearjunk ∘ fold (case_prod update) (zip ks vs) =
fold (case_prod update) (zip ks vs) ∘ clearjunk"
by (rule fold_commute) (simp add: clearjunk_update case_prod_beta o_def)
then show ?thesis
qed

lemma clearjunk_delete: "clearjunk (delete x al) = delete x (clearjunk al)"
by (induct al rule: clearjunk.induct) (auto simp add: delete_idem delete_twist)

lemma clearjunk_restrict: "clearjunk (restrict A al) = restrict A (clearjunk al)"
by (induct al rule: clearjunk.induct) (auto simp add: restr_delete_twist)

lemma distinct_clearjunk_id [simp]: "distinct (map fst al) ⟹ clearjunk al = al"
by (induct al rule: clearjunk.induct) auto

lemma clearjunk_idem: "clearjunk (clearjunk al) = clearjunk al"
by simp

lemma length_clearjunk: "length (clearjunk al) ≤ length al"
proof (induct al rule: clearjunk.induct [case_names Nil Cons])
case Nil
then show ?case by simp
next
case (Cons kv al)
moreover have "length (delete (fst kv) al) ≤ length al"
by (fact length_delete_le)
ultimately have "length (clearjunk (delete (fst kv) al)) ≤ length al"
by (rule order_trans)
then show ?case
by simp
qed

lemma delete_map:
assumes "⋀kv. fst (f kv) = fst kv"
shows "delete k (map f ps) = map f (delete k ps)"
by (simp add: delete_eq filter_map comp_def split_def assms)

lemma clearjunk_map:
assumes "⋀kv. fst (f kv) = fst kv"
shows "clearjunk (map f ps) = map f (clearjunk ps)"
by (induct ps rule: clearjunk.induct [case_names Nil Cons])

subsection ‹‹map_ran››

definition map_ran :: "('key ⇒ 'val ⇒ 'val) ⇒ ('key × 'val) list ⇒ ('key × 'val) list"
where "map_ran f = map (λ(k, v). (k, f k v))"

lemma map_ran_simps [simp]:
"map_ran f [] = []"
"map_ran f ((k, v) # ps) = (k, f k v) # map_ran f ps"

lemma dom_map_ran: "fst ` set (map_ran f al) = fst ` set al"
by (simp add: map_ran_def image_image split_def)

lemma map_ran_conv: "map_of (map_ran f al) k = map_option (f k) (map_of al k)"
by (induct al) auto

lemma distinct_map_ran: "distinct (map fst al) ⟹ distinct (map fst (map_ran f al))"
by (simp add: map_ran_def split_def comp_def)

lemma map_ran_filter: "map_ran f [p←ps. fst p ≠ a] = [p←map_ran f ps. fst p ≠ a]"
by (simp add: map_ran_def filter_map split_def comp_def)

lemma clearjunk_map_ran: "clearjunk (map_ran f al) = map_ran f (clearjunk al)"
by (simp add: map_ran_def split_def clearjunk_map)

subsection ‹‹merge››

qualified definition merge :: "('key × 'val) list ⇒ ('key × 'val) list ⇒ ('key × 'val) list"
where "merge qs ps = foldr (λ(k, v). update k v) ps qs"

lemma merge_simps [simp]:
"merge qs [] = qs"
"merge qs (p#ps) = update (fst p) (snd p) (merge qs ps)"

lemma merge_updates: "merge qs ps = updates (rev (map fst ps)) (rev (map snd ps)) qs"

lemma dom_merge: "fst ` set (merge xs ys) = fst ` set xs ∪ fst ` set ys"
by (induct ys arbitrary: xs) (auto simp add: dom_update)

lemma distinct_merge:
assumes "distinct (map fst xs)"
shows "distinct (map fst (merge xs ys))"

lemma clearjunk_merge: "clearjunk (merge xs ys) = merge (clearjunk xs) ys"

lemma merge_conv': "map_of (merge xs ys) = map_of xs ++ map_of ys"
proof -
have "map_of ∘ fold (case_prod update) (rev ys) =
fold (λ(k, v) m. m(k ↦ v)) (rev ys) ∘ map_of"
by (rule fold_commute) (simp add: update_conv' case_prod_beta split_def fun_eq_iff)
then show ?thesis
qed

corollary merge_conv: "map_of (merge xs ys) k = (map_of xs ++ map_of ys) k"

lemma merge_empty: "map_of (merge [] ys) = map_of ys"

lemma merge_assoc [simp]: "map_of (merge m1 (merge m2 m3)) = map_of (merge (merge m1 m2) m3)"

lemma merge_Some_iff:
"map_of (merge m n) k = Some x ⟷
map_of n k = Some x ∨ map_of n k = None ∧ map_of m k = Some x"

lemmas merge_SomeD [dest!] = merge_Some_iff [THEN iffD1]

lemma merge_find_right [simp]: "map_of n k = Some v ⟹ map_of (merge m n) k = Some v"

lemma merge_None [iff]:
"(map_of (merge m n) k = None) = (map_of n k = None ∧ map_of m k = None)"

lemma merge_upd [simp]:
"map_of (merge m (update k v n)) = map_of (update k v (merge m n))"

"map_of (merge m (updates xs ys n)) = map_of (updates xs ys (merge m n))"

lemma merge_append: "map_of (xs @ ys) = map_of (merge ys xs)"

subsection ‹‹compose››

qualified function compose :: "('key × 'a) list ⇒ ('a × 'b) list ⇒ ('key × 'b) list"
where
"compose [] ys = []"
| "compose (x # xs) ys =
(case map_of ys (snd x) of
None ⇒ compose (delete (fst x) xs) ys
| Some v ⇒ (fst x, v) # compose xs ys)"
by pat_completeness auto
termination
by (relation "measure (length ∘ fst)") (simp_all add: less_Suc_eq_le length_delete_le)

lemma compose_first_None [simp]:
assumes "map_of xs k = None"
shows "map_of (compose xs ys) k = None"
using assms by (induct xs ys rule: compose.induct) (auto split: option.splits split_if_asm)

lemma compose_conv: "map_of (compose xs ys) k = (map_of ys ∘⇩m map_of xs) k"
proof (induct xs ys rule: compose.induct)
case 1
then show ?case by simp
next
case (2 x xs ys)
show ?case
proof (cases "map_of ys (snd x)")
case None
with 2 have hyp: "map_of (compose (delete (fst x) xs) ys) k =
(map_of ys ∘⇩m map_of (delete (fst x) xs)) k"
by simp
show ?thesis
proof (cases "fst x = k")
case True
from True delete_notin_dom [of k xs]
have "map_of (delete (fst x) xs) k = None"
with hyp show ?thesis
using True None
by simp
next
case False
from False have "map_of (delete (fst x) xs) k = map_of xs k"
by simp
with hyp show ?thesis
using False None by (simp add: map_comp_def)
qed
next
case (Some v)
with 2
have "map_of (compose xs ys) k = (map_of ys ∘⇩m map_of xs) k"
by simp
with Some show ?thesis
qed
qed

lemma compose_conv': "map_of (compose xs ys) = (map_of ys ∘⇩m map_of xs)"
by (rule ext) (rule compose_conv)

lemma compose_first_Some [simp]:
assumes "map_of xs k = Some v"
shows "map_of (compose xs ys) k = map_of ys v"
using assms by (simp add: compose_conv)

lemma dom_compose: "fst ` set (compose xs ys) ⊆ fst ` set xs"
proof (induct xs ys rule: compose.induct)
case 1
then show ?case by simp
next
case (2 x xs ys)
show ?case
proof (cases "map_of ys (snd x)")
case None
with "2.hyps"
have "fst ` set (compose (delete (fst x) xs) ys) ⊆ fst ` set (delete (fst x) xs)"
by simp
also
have "… ⊆ fst ` set xs"
by (rule dom_delete_subset)
finally show ?thesis
using None
by auto
next
case (Some v)
with "2.hyps"
have "fst ` set (compose xs ys) ⊆ fst ` set xs"
by simp
with Some show ?thesis
by auto
qed
qed

lemma distinct_compose:
assumes "distinct (map fst xs)"
shows "distinct (map fst (compose xs ys))"
using assms
proof (induct xs ys rule: compose.induct)
case 1
then show ?case by simp
next
case (2 x xs ys)
show ?case
proof (cases "map_of ys (snd x)")
case None
with 2 show ?thesis by simp
next
case (Some v)
with 2 dom_compose [of xs ys] show ?thesis
by auto
qed
qed

lemma compose_delete_twist: "compose (delete k xs) ys = delete k (compose xs ys)"
proof (induct xs ys rule: compose.induct)
case 1
then show ?case by simp
next
case (2 x xs ys)
show ?case
proof (cases "map_of ys (snd x)")
case None
with 2 have hyp: "compose (delete k (delete (fst x) xs)) ys =
delete k (compose (delete (fst x) xs) ys)"
by simp
show ?thesis
proof (cases "fst x = k")
case True
with None hyp show ?thesis
next
case False
from None False hyp show ?thesis
qed
next
case (Some v)
with 2 have hyp: "compose (delete k xs) ys = delete k (compose xs ys)"
by simp
with Some show ?thesis
by simp
qed
qed

lemma compose_clearjunk: "compose xs (clearjunk ys) = compose xs ys"
by (induct xs ys rule: compose.induct)
(auto simp add: map_of_clearjunk split: option.splits)

lemma clearjunk_compose: "clearjunk (compose xs ys) = compose (clearjunk xs) ys"
by (induct xs rule: clearjunk.induct)
(auto split: option.splits simp add: clearjunk_delete delete_idem compose_delete_twist)

lemma compose_empty [simp]: "compose xs [] = []"
by (induct xs) (auto simp add: compose_delete_twist)

lemma compose_Some_iff:
"(map_of (compose xs ys) k = Some v) ⟷
(∃k'. map_of xs k = Some k' ∧ map_of ys k' = Some v)"

lemma map_comp_None_iff:
"map_of (compose xs ys) k = None ⟷
(map_of xs k = None ∨ (∃k'. map_of xs k = Some k' ∧ map_of ys k' = None))"

subsection ‹‹map_entry››

qualified fun map_entry :: "'key ⇒ ('val ⇒ 'val) ⇒ ('key × 'val) list ⇒ ('key × 'val) list"
where
"map_entry k f [] = []"
| "map_entry k f (p # ps) =
(if fst p = k then (k, f (snd p)) # ps else p # map_entry k f ps)"

lemma map_of_map_entry:
"map_of (map_entry k f xs) =
(map_of xs)(k := case map_of xs k of None ⇒ None | Some v' ⇒ Some (f v'))"
by (induct xs) auto

lemma dom_map_entry: "fst ` set (map_entry k f xs) = fst ` set xs"
by (induct xs) auto

lemma distinct_map_entry:
assumes "distinct (map fst xs)"
shows "distinct (map fst (map_entry k f xs))"
using assms by (induct xs) (auto simp add: dom_map_entry)

subsection ‹‹map_default››

fun map_default :: "'key ⇒ 'val ⇒ ('val ⇒ 'val) ⇒ ('key × 'val) list ⇒ ('key × 'val) list"
where
"map_default k v f [] = [(k, v)]"
| "map_default k v f (p # ps) =
(if fst p = k then (k, f (snd p)) # ps else p # map_default k v f ps)"

lemma map_of_map_default:
"map_of (map_default k v f xs) =
(map_of xs)(k := case map_of xs k of None ⇒ Some v | Some v' ⇒ Some (f v'))"
by (induct xs) auto

lemma dom_map_default: "fst ` set (map_default k v f xs) = insert k (fst ` set xs)"
by (induct xs) auto

lemma distinct_map_default:
assumes "distinct (map fst xs)"
shows "distinct (map fst (map_default k v f xs))"
using assms by (induct xs) (auto simp add: dom_map_default)

end

end
```