Theory AList

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


header {* Implementation of Association Lists *}

theory AList
imports Main
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.
*}


subsection {* @{text update} and @{text updates} *}

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\<mapsto>v)"
by (induct al) (auto simp add: fun_eq_iff)

corollary update_conv: "map_of (update k v al) k' = ((map_of al)(k\<mapsto>v)) k'"
by (simp add: update_conv')

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\<leftarrow>ps . fst q ≠ a] = [q\<leftarrow>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 thus ?case
by (cases al') (auto split: split_if_asm)
next
case Cons thus ?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))"

by (simp add: update_conv' fun_eq_iff)

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"

by (simp add: update_conv' map_upd_Some_unfold)

lemma image_update [simp]:
"x ∉ A ==> map_of (update x y al) ` A = map_of al ` A"
by (simp add: update_conv')

definition updates :: "'key list => 'val list => ('key × 'val) list => ('key × 'val) list" where
"updates ks vs = fold (prod_case update) (zip ks vs)"

lemma updates_simps [simp]:
"updates [] vs ps = ps"
"updates ks [] ps = ps"
"updates (k#ks) (v#vs) ps = updates ks vs (update k v ps)"
by (simp_all add: updates_def)

lemma updates_key_simp [simp]:
"updates (k # ks) vs ps =
(case vs of [] => ps | v # vs => updates ks vs (update k v ps))"

by (cases vs) simp_all

lemma updates_conv': "map_of (updates ks vs al) = (map_of al)(ks[\<mapsto>]vs)"
proof -
have "map_of o fold (prod_case update) (zip ks vs) =
fold (λ(k, v) f. f(k \<mapsto> v)) (zip ks vs) o map_of"

by (rule fold_commute) (auto simp add: fun_eq_iff update_conv')
then show ?thesis by (auto simp add: updates_def fun_eq_iff map_upds_fold_map_upd foldl_conv_fold split_def)
qed

lemma updates_conv: "map_of (updates ks vs al) k = ((map_of al)(ks[\<mapsto>]vs)) k"
by (simp add: updates_conv')

lemma distinct_updates:
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 o fold (prod_case update) (zip ks vs) =
fold (λ(k, v) al. if k ∈ set al then al else al @ [k]) (zip ks vs) o map fst"

by (rule fold_commute) (simp add: update_keys split_def prod_case_beta comp_def)
ultimately show ?thesis by (simp add: updates_def fun_eq_iff)
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)

lemma updates_list_update_drop[simp]:
"[|size ks ≤ i; i < size vs|]
==> updates ks (vs[i:=v]) al = updates ks vs al"

by (induct ks arbitrary: al vs i) (auto split:list.splits nat.splits)

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

by (simp add: updates_conv' update_conv' map_upd_upds_conv_if)

lemma updates_twist [simp]:
"k ∉ set ks ==>
map_of (updates ks vs (update k v al)) = map_of (update k v (updates ks vs al))"

by (simp add: updates_conv' update_conv')

lemma updates_apply_notin[simp]:
"k ∉ set ks ==> map_of (updates ks vs al) k = map_of al k"
by (simp add: updates_conv)

lemma updates_append_drop[simp]:
"size xs = size ys ==> updates (xs@zs) ys al = updates xs ys al"
by (induct xs arbitrary: ys al) (auto split: list.splits)

lemma updates_append2_drop[simp]:
"size xs = size ys ==> updates xs (ys@zs) al = updates xs ys al"
by (induct xs arbitrary: ys al) (auto split: list.splits)


subsection {* @{text delete} *}

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)"
by (auto simp add: delete_eq)

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'"
by (simp add: delete_conv')

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"
by (simp add: delete_eq)

lemma map_of_delete [simp]:
"k' ≠ k ==> map_of (delete k al) k' = map_of al k'"
by (simp add: delete_conv')

lemma delete_notin_dom: "k ∉ fst ` set (delete k al)"
by (auto simp add: delete_eq)

lemma dom_delete_subset: "fst ` set (delete k al) ⊆ fst ` set al"
by (auto simp add: delete_eq)

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)"
by (simp add: delete_eq conj_commute)

lemma length_delete_le: "length (delete k al) ≤ length al"
by (simp add: delete_eq)


subsection {* @{text restrict} *}

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)"
by (auto simp add: restrict_eq)

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"
by (simp add: restr_conv')

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"
by (simp add: restr_conv')

lemma restr_out [simp]: "x ∉ A ==> map_of (restrict A al) x = None"
by (simp add: restr_conv')

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))"

by (simp add: restr_conv' update_conv')

lemma restr_delete [simp]:
"(delete x (restrict D al)) =
(if x ∈ D then restrict (D - {x}) al else restrict D al)"

apply (simp add: delete_eq restrict_eq)
apply (auto simp add: split_def)
proof -
have "!!y. y ≠ x <-> x ≠ y" by auto
then show "[p \<leftarrow> al. fst p ∈ D ∧ x ≠ fst p] = [p \<leftarrow> 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 \<leftarrow> al . fst p ∈ D ∧ x ≠ fst p] = [p \<leftarrow> 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))"

by (simp add: update_conv' restr_conv')

lemma restr_updates [simp]: "
[| 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))"

by (simp add: updates_conv' restr_conv')

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


subsection {* @{text clearjunk} *}

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)"
by (simp add: map_of_clearjunk)

lemma ran_map_of:
"ran (map_of al) = snd ` set (clearjunk al)"
proof -
have "ran (map_of al) = ran (map_of (clearjunk al))"
by (simp add: ran_clearjunk)
also have "… = snd ` set (clearjunk al)"
by (simp add: ran_distinct)
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)

lemma clearjunk_updates:
"clearjunk (updates ks vs al) = updates ks vs (clearjunk al)"
proof -
have "clearjunk o fold (prod_case update) (zip ks vs) =
fold (prod_case update) (zip ks vs) o clearjunk"

by (rule fold_commute) (simp add: clearjunk_update prod_case_beta o_def)
then show ?thesis by (simp add: updates_def fun_eq_iff)
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])
(simp_all add: clearjunk_delete delete_map assms)


subsection {* @{text 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"
by (simp_all add: map_ran_def)

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 = Option.map (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\<leftarrow>ps. fst p ≠ a] = [p\<leftarrow>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 {* @{text merge} *}

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)"
by (simp_all add: merge_def split_def)

lemma merge_updates:
"merge qs ps = updates (rev (map fst ps)) (rev (map snd ps)) qs"
by (simp add: merge_def updates_def foldr_conv_fold zip_rev zip_map_fst_snd)

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))"
using assms by (simp add: merge_updates distinct_updates)

lemma clearjunk_merge:
"clearjunk (merge xs ys) = merge (clearjunk xs) ys"
by (simp add: merge_updates clearjunk_updates)

lemma merge_conv':
"map_of (merge xs ys) = map_of xs ++ map_of ys"
proof -
have "map_of o fold (prod_case update) (rev ys) =
fold (λ(k, v) m. m(k \<mapsto> v)) (rev ys) o map_of"

by (rule fold_commute) (simp add: update_conv' prod_case_beta split_def fun_eq_iff)
then show ?thesis
by (simp add: merge_def map_add_map_of_foldr foldr_conv_fold fun_eq_iff)
qed

corollary merge_conv:
"map_of (merge xs ys) k = (map_of xs ++ map_of ys) k"
by (simp add: merge_conv')

lemma merge_empty: "map_of (merge [] ys) = map_of ys"
by (simp add: merge_conv')

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

by (simp add: merge_conv')

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)"

by (simp add: merge_conv' map_add_Some_iff)

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"
by (simp add: merge_conv')

lemma merge_None [iff]:
"(map_of (merge m n) k = None) = (map_of n k = None ∧ map_of m k = None)"
by (simp add: merge_conv')

lemma merge_upd[simp]:
"map_of (merge m (update k v n)) = map_of (update k v (merge m n))"
by (simp add: update_conv' merge_conv')

lemma merge_updatess[simp]:
"map_of (merge m (updates xs ys n)) = map_of (updates xs ys (merge m n))"
by (simp add: updates_conv' merge_conv')

lemma merge_append: "map_of (xs@ys) = map_of (merge ys xs)"
by (simp add: merge_conv')


subsection {* @{text compose} *}

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 o 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:
shows "map_of (compose xs ys) k = (map_of ys om 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 om 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"
by (simp add: map_of_eq_None_iff)
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 om map_of xs) k"
by simp
with Some show ?thesis
by (auto simp add: map_comp_def)
qed
qed

lemma compose_conv':
shows "map_of (compose xs ys) = (map_of ys om 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 thus ?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 thus ?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 thus ?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
by (simp add: delete_idem)
next
case False
from None False hyp
show ?thesis
by (simp add: delete_twist)
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)"

by (simp add: compose_conv map_comp_Some_iff)

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)) "

by (simp add: compose_conv map_comp_None_iff)

subsection {* @{text map_entry} *}

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 {* @{text 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)

hide_const (open) update updates delete restrict clearjunk merge compose map_entry

end