Theory AList

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theory AList
imports Main
`(*  Title:      HOL/Library/AList.thy    Author:     Norbert Schirmer, Tobias Nipkow, Martin Wildmoser, TU Muenchen*)header {* Implementation of Association Lists *}theory AListimports Mainbegintext {*  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) autolemma 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_alllemma 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) autolemma update_triv: "map_of al k = Some v ==> update k v al = al"  by (induct al) autolemma update_nonempty [simp]: "update k v al ≠ []"  by (induct al) autolemma 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)qedlemma update_last [simp]: "update k v (update k v' al) = update k v al"  by (induct al) autotext {* 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_alllemma 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)qedlemma 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)qedlemma 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_alllemma delete_update:  "k ≠ l ==> delete l (update k v al) = update k v (delete l al)"  by (induct al) simp_alllemma 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)qedcorollary 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 simpqedlemma 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) autosubsection {* @{text clearjunk} *}function clearjunk  :: "('key × 'val) list => ('key × 'val) list" where    "clearjunk [] = []"    | "clearjunk (p#ps) = p # clearjunk (delete (fst p) ps)"  by pat_completeness autotermination 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 simplemma 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 .qedlemma 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)qedlemma 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) autolemma clearjunk_idem:  "clearjunk (clearjunk al) = clearjunk al"  by simplemma length_clearjunk:  "length (clearjunk al) ≤ length al"proof (induct al rule: clearjunk.induct [case_names Nil Cons])  case Nil then show ?case by simpnext  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 simpqedlemma 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) autolemma 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)qedcorollary 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 autotermination 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 simpnext  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)  qedqed   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 simpnext  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  qedqedlemma distinct_compose: assumes "distinct (map fst xs)" shows "distinct (map fst (compose xs ys))"using assmsproof (induct xs ys rule: compose.induct)  case 1 thus ?case by simpnext  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)  qedqedlemma 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 simpnext  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  qedqedlemma 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) autolemma dom_map_entry:  "fst ` set (map_entry k f xs) = fst ` set xs"by (induct xs) autolemma 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) autolemma dom_map_default:  "fst ` set (map_default k v f xs) = insert k (fst ` set xs)" by (induct xs) autolemma 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_entryend`