(* 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. › subsection ‹‹update› and ‹updates›› 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'" 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←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))" 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') qualified definition updates :: "'key list ⇒ 'val list ⇒ ('key × 'val) list ⇒ ('key × 'val) list" where "updates ks vs = fold (case_prod 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[↦]vs)" 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 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[↦]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 ∘ 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 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 ‹‹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)" 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 ‹‹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))" by(simp add: fun_eq_iff map_of_update_with_aux') 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)" by(induct ps)(auto simp add: dom_update_with_aux) 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}" by(auto simp add: set_delete_aux) 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) apply (simp add: map_of_eq_None_iff) done lemma map_of_delete_aux: "distinct (map fst xs) ⟹ map_of (delete_aux k xs) k' = ((map_of xs)(k := None)) k'" by(simp add: map_of_delete_aux') 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)" 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 ← 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))" 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 ‹‹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)" 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 ∘ 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 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 ‹‹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 = 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)" 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 ∘ 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 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 ‹‹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" 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 ∘⇩_{m}map_of xs) k" by simp with Some show ?thesis by (auto simp add: map_comp_def) 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 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 ‹‹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