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src/HOL/Library/AList_Impl.thy

author | bulwahn |

Wed, 14 Dec 2011 16:30:30 +0100 | |

changeset 45872 | 3759fb8a02b8 |

parent 45871 | 1fec5b365f9b |

child 45884 | 58a10da12812 |

permissions | -rw-r--r-- |

tuned header after renaming

(* Title: HOL/Library/AList_Impl.thy Author: Norbert Schirmer, Tobias Nipkow, Martin Wildmoser, TU Muenchen *) header {* Implementation of Association Lists *} theory AList_Impl imports Main More_List 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 \<Rightarrow> 'val \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> '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} \<union> fst ` set al" by (induct al) auto lemma update_keys: "map fst (update k v al) = (if k \<in> 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\<noteq>k \<Longrightarrow> update k v [q\<leftarrow>ps . fst q \<noteq> a] = [q\<leftarrow>update k v ps . fst q \<noteq> a]" by (induct ps) auto lemma update_triv: "map_of al k = Some v \<Longrightarrow> update k v al = al" by (induct al) auto lemma update_nonempty [simp]: "update k v al \<noteq> []" by (induct al) auto lemma update_eqD: "update k v al = update k v' al' \<Longrightarrow> 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\<noteq>k' \<Longrightarrow> 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 \<longleftrightarrow> x = k \<and> v = y \<or> x \<noteq> k \<and> map_of al x = Some y" by (simp add: update_conv' map_upd_Some_unfold) lemma image_update [simp]: "x \<notin> A \<Longrightarrow> map_of (update x y al) ` A = map_of al ` A" by (simp add: update_conv' image_map_upd) definition updates :: "'key list \<Rightarrow> 'val list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where "updates ks vs = More_List.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 [] \<Rightarrow> ps | v # vs \<Rightarrow> 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 \<circ> More_List.fold (prod_case update) (zip ks vs) = More_List.fold (\<lambda>(k, v) f. f(k \<mapsto> v)) (zip ks vs) \<circ> 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_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 (More_List.fold (\<lambda>(k, v) al. if k \<in> set al then al else al @ [k]) (zip ks vs) (map fst al))" by (rule fold_invariant [of "zip ks vs" "\<lambda>_. True"]) (auto intro: assms) moreover have "map fst \<circ> More_List.fold (prod_case update) (zip ks vs) = More_List.fold (\<lambda>(k, v) al. if k \<in> set al then al else al @ [k]) (zip ks vs) \<circ> 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 \<Longrightarrow> 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]: "\<lbrakk>size ks \<le> i; i < size vs\<rbrakk> \<Longrightarrow> 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 \<in> 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 \<notin> set ks \<Longrightarrow> 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' map_upds_twist) lemma updates_apply_notin[simp]: "k \<notin> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where delete_eq: "delete k = filter (\<lambda>(k', _). k \<noteq> 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 \<notin> fst ` set al \<Longrightarrow> 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' \<noteq> k \<Longrightarrow> map_of (delete k al) k' = map_of al k'" by (simp add: delete_conv') lemma delete_notin_dom: "k \<notin> fst ` set (delete k al)" by (auto simp add: delete_eq) lemma dom_delete_subset: "fst ` set (delete k al) \<subseteq> 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 \<noteq> l \<Longrightarrow> 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) \<le> length al" by (simp add: delete_eq) subsection {* @{text restrict} *} definition restrict :: "'key set \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where restrict_eq: "restrict A = filter (\<lambda>(k, v). k \<in> A)" lemma restr_simps [simp]: "restrict A [] = []" "restrict A (p#ps) = (if fst p \<in> 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 \<in> 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) \<Longrightarrow> 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 \<in> A \<Longrightarrow> map_of (restrict A al) x = map_of al x" by (simp add: restr_conv') lemma restr_out [simp]: "x \<notin> A \<Longrightarrow> map_of (restrict A al) x = None" by (simp add: restr_conv') lemma dom_restr [simp]: "fst ` set (restrict A al) = fst ` set al \<inter> 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\<inter>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 \<in> 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 \<in> 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 "\<And>y. y \<noteq> x \<longleftrightarrow> x \<noteq> y" by auto then show "[p \<leftarrow> al. fst p \<in> D \<and> x \<noteq> fst p] = [p \<leftarrow> al. fst p \<in> D \<and> fst p \<noteq> x]" by simp assume "x \<notin> D" then have "\<And>y. y \<in> D \<longleftrightarrow> y \<in> D \<and> x \<noteq> y" by auto then show "[p \<leftarrow> al . fst p \<in> D \<and> x \<noteq> fst p] = [p \<leftarrow> al . fst p \<in> 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 \<in> D \<Longrightarrow> 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]: " \<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk> \<Longrightarrow> 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 \<times> 'val) list \<Rightarrow> ('key \<times> '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 "\<dots> = 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 \<circ> More_List.fold (prod_case update) (zip ks vs) = More_List.fold (prod_case update) (zip ks vs) \<circ> 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) \<Longrightarrow> 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) \<le> 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) \<le> length al" by (fact length_delete_le) ultimately have "length (clearjunk (delete (fst kv) al)) \<le> length al" by (rule order_trans) then show ?case by simp qed lemma delete_map: assumes "\<And>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 "\<And>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 \<Rightarrow> 'val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where "map_ran f = map (\<lambda>(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) \<Longrightarrow> 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 \<noteq> a] = [p\<leftarrow>map_ran f ps. fst p \<noteq> 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 \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where "merge qs ps = foldr (\<lambda>(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_fold_rev zip_rev zip_map_fst_snd) lemma dom_merge: "fst ` set (merge xs ys) = fst ` set xs \<union> 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 \<circ> More_List.fold (prod_case update) (rev ys) = More_List.fold (\<lambda>(k, v) m. m(k \<mapsto> v)) (rev ys) \<circ> 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_fold_rev 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 \<or> map_of n k = None \<and> 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 \<Longrightarrow> 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 \<and> 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 \<times> 'a) list \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('key \<times> 'b) list" where "compose [] ys = []" | "compose (x#xs) ys = (case map_of ys (snd x) of None \<Rightarrow> compose (delete (fst x) xs) ys | Some v \<Rightarrow> (fst x, v) # compose xs ys)" by pat_completeness auto termination by (relation "measure (length \<circ> 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 \<circ>\<^sub>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 \<circ>\<^sub>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 \<circ>\<^sub>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 \<circ>\<^sub>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) \<subseteq> 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) \<subseteq> fst ` set (delete (fst x) xs)" by simp also have "\<dots> \<subseteq> 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) \<subseteq> 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) = (\<exists>k'. map_of xs k = Some k' \<and> 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 \<or> (\<exists>k'. map_of xs k = Some k' \<and> map_of ys k' = None)) " by (simp add: compose_conv map_comp_None_iff) subsection {* @{text map_entry} *} fun map_entry :: "'key \<Rightarrow> ('val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> '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 \<Rightarrow> 'val \<Rightarrow> ('val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> '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