(* 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 o⇩_{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 o⇩_{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"

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 o⇩_{m}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 o⇩_{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 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