--- a/src/HOL/Library/AList_Impl.thy Tue Sep 13 07:13:49 2011 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,656 +0,0 @@
-(* 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 upate_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, standard]
-
-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)
-
-end