author  bulwahn 
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child 45884  58a10da12812 
permissions  rwrr 
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(* Title: HOL/Library/AList_Impl.thy 
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Author: Norbert Schirmer, Tobias Nipkow, Martin Wildmoser, TU Muenchen 
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*) 
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header {* Implementation of Association Lists *} 
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theory AList_Impl 
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imports Main More_List 
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begin 
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text {* 
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The operations preserve distinctness of keys and 

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function @{term "clearjunk"} distributes over them. Since 

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@{term clearjunk} enforces distinctness of keys it can be used 

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to establish the invariant, e.g. for inductive proofs. 

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*} 

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subsection {* @{text update} and @{text updates} *} 
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primrec update :: "'key \<Rightarrow> 'val \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where 
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"update k v [] = [(k, v)]" 
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 "update k v (p#ps) = (if fst p = k then (k, v) # ps else p # update k v ps)" 

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lemma update_conv': "map_of (update k v al) = (map_of al)(k\<mapsto>v)" 
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by (induct al) (auto simp add: fun_eq_iff) 
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corollary update_conv: "map_of (update k v al) k' = ((map_of al)(k\<mapsto>v)) k'" 
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by (simp add: update_conv') 
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lemma dom_update: "fst ` set (update k v al) = {k} \<union> fst ` set al" 

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by (induct al) auto 

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lemma update_keys: 
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"map fst (update k v al) = 
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(if k \<in> set (map fst al) then map fst al else map fst al @ [k])" 
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by (induct al) simp_all 
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lemma distinct_update: 
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assumes "distinct (map fst al)" 

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shows "distinct (map fst (update k v al))" 

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using assms by (simp add: update_keys) 
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lemma update_filter: 

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"a\<noteq>k \<Longrightarrow> update k v [q\<leftarrow>ps . fst q \<noteq> a] = [q\<leftarrow>update k v ps . fst q \<noteq> a]" 
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by (induct ps) auto 
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lemma update_triv: "map_of al k = Some v \<Longrightarrow> update k v al = al" 

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by (induct al) auto 

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lemma update_nonempty [simp]: "update k v al \<noteq> []" 

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by (induct al) auto 

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lemma update_eqD: "update k v al = update k v' al' \<Longrightarrow> v = v'" 
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proof (induct al arbitrary: al') 
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case Nil thus ?case 
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by (cases al') (auto split: split_if_asm) 

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next 

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case Cons thus ?case 

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by (cases al') (auto split: split_if_asm) 

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qed 

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lemma update_last [simp]: "update k v (update k v' al) = update k v al" 

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by (induct al) auto 

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text {* Note that the lists are not necessarily the same: 

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@{term "update k v (update k' v' []) = [(k', v'), (k, v)]"} and 
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@{term "update k' v' (update k v []) = [(k, v), (k', v')]"}.*} 
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lemma update_swap: "k\<noteq>k' 
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\<Longrightarrow> map_of (update k v (update k' v' al)) = map_of (update k' v' (update k v al))" 

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by (simp add: update_conv' fun_eq_iff) 
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lemma update_Some_unfold: 

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"map_of (update k v al) x = Some y \<longleftrightarrow> 
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x = k \<and> v = y \<or> x \<noteq> k \<and> map_of al x = Some y" 
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by (simp add: update_conv' map_upd_Some_unfold) 
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lemma image_update [simp]: 
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"x \<notin> A \<Longrightarrow> map_of (update x y al) ` A = map_of al ` A" 
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by (simp add: update_conv' image_map_upd) 
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definition updates :: "'key list \<Rightarrow> 'val list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where 
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"updates ks vs = More_List.fold (prod_case update) (zip ks vs)" 
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lemma updates_simps [simp]: 
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"updates [] vs ps = ps" 
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"updates ks [] ps = ps" 
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"updates (k#ks) (v#vs) ps = updates ks vs (update k v ps)" 
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by (simp_all add: updates_def) 
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lemma updates_key_simp [simp]: 
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"updates (k # ks) vs ps = 
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(case vs of [] \<Rightarrow> ps  v # vs \<Rightarrow> updates ks vs (update k v ps))" 
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by (cases vs) simp_all 
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lemma updates_conv': "map_of (updates ks vs al) = (map_of al)(ks[\<mapsto>]vs)" 
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proof  
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have "map_of \<circ> More_List.fold (prod_case update) (zip ks vs) = 
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More_List.fold (\<lambda>(k, v) f. f(k \<mapsto> v)) (zip ks vs) \<circ> map_of" 

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by (rule fold_commute) (auto simp add: fun_eq_iff update_conv') 
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then show ?thesis by (auto simp add: updates_def fun_eq_iff map_upds_fold_map_upd foldl_fold split_def) 
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qed 
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lemma updates_conv: "map_of (updates ks vs al) k = ((map_of al)(ks[\<mapsto>]vs)) k" 

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by (simp add: updates_conv') 
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lemma distinct_updates: 

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assumes "distinct (map fst al)" 

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shows "distinct (map fst (updates ks vs al))" 

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proof  
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have "distinct (More_List.fold 
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(\<lambda>(k, v) al. if k \<in> set al then al else al @ [k]) 

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(zip ks vs) (map fst al))" 

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by (rule fold_invariant [of "zip ks vs" "\<lambda>_. True"]) (auto intro: assms) 

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moreover have "map fst \<circ> More_List.fold (prod_case update) (zip ks vs) = 

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More_List.fold (\<lambda>(k, v) al. if k \<in> set al then al else al @ [k]) (zip ks vs) \<circ> map fst" 

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by (rule fold_commute) (simp add: update_keys split_def prod_case_beta comp_def) 
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ultimately show ?thesis by (simp add: updates_def fun_eq_iff) 
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qed 
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lemma updates_append1[simp]: "size ks < size vs \<Longrightarrow> 

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updates (ks@[k]) vs al = update k (vs!size ks) (updates ks vs al)" 

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by (induct ks arbitrary: vs al) (auto split: list.splits) 
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lemma updates_list_update_drop[simp]: 

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"\<lbrakk>size ks \<le> i; i < size vs\<rbrakk> 

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\<Longrightarrow> updates ks (vs[i:=v]) al = updates ks vs al" 

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by (induct ks arbitrary: al vs i) (auto split:list.splits nat.splits) 
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lemma update_updates_conv_if: " 

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map_of (updates xs ys (update x y al)) = 

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map_of (if x \<in> set(take (length ys) xs) then updates xs ys al 

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else (update x y (updates xs ys al)))" 

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by (simp add: updates_conv' update_conv' map_upd_upds_conv_if) 

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lemma updates_twist [simp]: 

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"k \<notin> set ks \<Longrightarrow> 

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map_of (updates ks vs (update k v al)) = map_of (update k v (updates ks vs al))" 

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by (simp add: updates_conv' update_conv' map_upds_twist) 

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lemma updates_apply_notin[simp]: 

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"k \<notin> set ks ==> map_of (updates ks vs al) k = map_of al k" 

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by (simp add: updates_conv) 

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lemma updates_append_drop[simp]: 

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"size xs = size ys \<Longrightarrow> updates (xs@zs) ys al = updates xs ys al" 

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by (induct xs arbitrary: ys al) (auto split: list.splits) 
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lemma updates_append2_drop[simp]: 

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"size xs = size ys \<Longrightarrow> updates xs (ys@zs) al = updates xs ys al" 

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by (induct xs arbitrary: ys al) (auto split: list.splits) 
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subsection {* @{text delete} *} 
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definition delete :: "'key \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where 
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delete_eq: "delete k = filter (\<lambda>(k', _). k \<noteq> k')" 
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lemma delete_simps [simp]: 
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"delete k [] = []" 
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"delete k (p#ps) = (if fst p = k then delete k ps else p # delete k ps)" 
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by (auto simp add: delete_eq) 
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lemma delete_conv': "map_of (delete k al) = (map_of al)(k := None)" 
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by (induct al) (auto simp add: fun_eq_iff) 
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corollary delete_conv: "map_of (delete k al) k' = ((map_of al)(k := None)) k'" 
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by (simp add: delete_conv') 
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lemma delete_keys: 
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"map fst (delete k al) = removeAll k (map fst al)" 
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by (simp add: delete_eq removeAll_filter_not_eq filter_map split_def comp_def) 
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lemma distinct_delete: 
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assumes "distinct (map fst al)" 
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shows "distinct (map fst (delete k al))" 
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using assms by (simp add: delete_keys distinct_removeAll) 
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lemma delete_id [simp]: "k \<notin> fst ` set al \<Longrightarrow> delete k al = al" 
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by (auto simp add: image_iff delete_eq filter_id_conv) 
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lemma delete_idem: "delete k (delete k al) = delete k al" 
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by (simp add: delete_eq) 
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lemma map_of_delete [simp]: 
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"k' \<noteq> k \<Longrightarrow> map_of (delete k al) k' = map_of al k'" 
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by (simp add: delete_conv') 
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lemma delete_notin_dom: "k \<notin> fst ` set (delete k al)" 
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by (auto simp add: delete_eq) 
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lemma dom_delete_subset: "fst ` set (delete k al) \<subseteq> fst ` set al" 
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by (auto simp add: delete_eq) 
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lemma delete_update_same: 
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"delete k (update k v al) = delete k al" 
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by (induct al) simp_all 
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lemma delete_update: 
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"k \<noteq> l \<Longrightarrow> delete l (update k v al) = update k v (delete l al)" 
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by (induct al) simp_all 
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lemma delete_twist: "delete x (delete y al) = delete y (delete x al)" 
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by (simp add: delete_eq conj_commute) 
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lemma length_delete_le: "length (delete k al) \<le> length al" 
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by (simp add: delete_eq) 
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subsection {* @{text restrict} *} 
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definition restrict :: "'key set \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where 
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restrict_eq: "restrict A = filter (\<lambda>(k, v). k \<in> A)" 
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213 

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lemma restr_simps [simp]: 
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"restrict A [] = []" 
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"restrict A (p#ps) = (if fst p \<in> A then p # restrict A ps else restrict A ps)" 
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217 
by (auto simp add: restrict_eq) 
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218 

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lemma restr_conv': "map_of (restrict A al) = ((map_of al)` A)" 
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proof 
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221 
fix k 
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show "map_of (restrict A al) k = ((map_of al)` A) k" 
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223 
by (induct al) (simp, cases "k \<in> A", auto) 
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224 
qed 
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225 

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corollary restr_conv: "map_of (restrict A al) k = ((map_of al)` A) k" 
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227 
by (simp add: restr_conv') 
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228 

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229 
lemma distinct_restr: 
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"distinct (map fst al) \<Longrightarrow> distinct (map fst (restrict A al))" 
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231 
by (induct al) (auto simp add: restrict_eq) 
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232 

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lemma restr_empty [simp]: 
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234 
"restrict {} al = []" 
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235 
"restrict A [] = []" 
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236 
by (induct al) (auto simp add: restrict_eq) 
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237 

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238 
lemma restr_in [simp]: "x \<in> A \<Longrightarrow> map_of (restrict A al) x = map_of al x" 
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239 
by (simp add: restr_conv') 
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240 

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241 
lemma restr_out [simp]: "x \<notin> A \<Longrightarrow> map_of (restrict A al) x = None" 
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242 
by (simp add: restr_conv') 
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243 

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lemma dom_restr [simp]: "fst ` set (restrict A al) = fst ` set al \<inter> A" 
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245 
by (induct al) (auto simp add: restrict_eq) 
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246 

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247 
lemma restr_upd_same [simp]: "restrict ({x}) (update x y al) = restrict ({x}) al" 
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248 
by (induct al) (auto simp add: restrict_eq) 
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249 

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250 
lemma restr_restr [simp]: "restrict A (restrict B al) = restrict (A\<inter>B) al" 
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251 
by (induct al) (auto simp add: restrict_eq) 
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252 

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253 
lemma restr_update[simp]: 
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"map_of (restrict D (update x y al)) = 
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255 
map_of ((if x \<in> D then (update x y (restrict (D{x}) al)) else restrict D al))" 
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256 
by (simp add: restr_conv' update_conv') 
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257 

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258 
lemma restr_delete [simp]: 
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259 
"(delete x (restrict D al)) = 
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(if x \<in> D then restrict (D  {x}) al else restrict D al)" 
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261 
apply (simp add: delete_eq restrict_eq) 
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262 
apply (auto simp add: split_def) 
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263 
proof  
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264 
have "\<And>y. y \<noteq> x \<longleftrightarrow> x \<noteq> y" by auto 
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265 
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]" 
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266 
by simp 
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267 
assume "x \<notin> D" 
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then have "\<And>y. y \<in> D \<longleftrightarrow> y \<in> D \<and> x \<noteq> y" by auto 
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269 
then show "[p \<leftarrow> al . fst p \<in> D \<and> x \<noteq> fst p] = [p \<leftarrow> al . fst p \<in> D]" 
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270 
by simp 
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271 
qed 
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272 

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273 
lemma update_restr: 
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274 
"map_of (update x y (restrict D al)) = map_of (update x y (restrict (D{x}) al))" 
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275 
by (simp add: update_conv' restr_conv') (rule fun_upd_restrict) 
19234  276 

45867  277 
lemma update_restr_conv [simp]: 
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278 
"x \<in> D \<Longrightarrow> 
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279 
map_of (update x y (restrict D al)) = map_of (update x y (restrict (D{x}) al))" 
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280 
by (simp add: update_conv' restr_conv') 
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281 

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282 
lemma restr_updates [simp]: " 
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283 
\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk> 
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284 
\<Longrightarrow> map_of (restrict D (updates xs ys al)) = 
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285 
map_of (updates xs ys (restrict (D  set xs) al))" 
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286 
by (simp add: updates_conv' restr_conv') 
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287 

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288 
lemma restr_delete_twist: "(restrict A (delete a ps)) = delete a (restrict A ps)" 
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289 
by (induct ps) auto 
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290 

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291 

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292 
subsection {* @{text clearjunk} *} 
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293 

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294 
function clearjunk :: "('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where 
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295 
"clearjunk [] = []" 
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296 
 "clearjunk (p#ps) = p # clearjunk (delete (fst p) ps)" 
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297 
by pat_completeness auto 
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298 
termination by (relation "measure length") 
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299 
(simp_all add: less_Suc_eq_le length_delete_le) 
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300 

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301 
lemma map_of_clearjunk: 
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302 
"map_of (clearjunk al) = map_of al" 
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303 
by (induct al rule: clearjunk.induct) 
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304 
(simp_all add: fun_eq_iff) 
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305 

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306 
lemma clearjunk_keys_set: 
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307 
"set (map fst (clearjunk al)) = set (map fst al)" 
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308 
by (induct al rule: clearjunk.induct) 
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309 
(simp_all add: delete_keys) 
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310 

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311 
lemma dom_clearjunk: 
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312 
"fst ` set (clearjunk al) = fst ` set al" 
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313 
using clearjunk_keys_set by simp 
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314 

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315 
lemma distinct_clearjunk [simp]: 
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316 
"distinct (map fst (clearjunk al))" 
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317 
by (induct al rule: clearjunk.induct) 
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318 
(simp_all del: set_map add: clearjunk_keys_set delete_keys) 
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319 

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320 
lemma ran_clearjunk: 
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321 
"ran (map_of (clearjunk al)) = ran (map_of al)" 
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322 
by (simp add: map_of_clearjunk) 
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323 

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324 
lemma ran_map_of: 
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325 
"ran (map_of al) = snd ` set (clearjunk al)" 
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326 
proof  
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327 
have "ran (map_of al) = ran (map_of (clearjunk al))" 
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328 
by (simp add: ran_clearjunk) 
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329 
also have "\<dots> = snd ` set (clearjunk al)" 
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330 
by (simp add: ran_distinct) 
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331 
finally show ?thesis . 
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332 
qed 
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333 

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334 
lemma clearjunk_update: 
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335 
"clearjunk (update k v al) = update k v (clearjunk al)" 
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336 
by (induct al rule: clearjunk.induct) 
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337 
(simp_all add: delete_update) 
19234  338 

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339 
lemma clearjunk_updates: 
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340 
"clearjunk (updates ks vs al) = updates ks vs (clearjunk al)" 
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341 
proof  
37458  342 
have "clearjunk \<circ> More_List.fold (prod_case update) (zip ks vs) = 
343 
More_List.fold (prod_case update) (zip ks vs) \<circ> clearjunk" 

39921  344 
by (rule fold_commute) (simp add: clearjunk_update prod_case_beta o_def) 
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345 
then show ?thesis by (simp add: updates_def fun_eq_iff) 
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346 
qed 
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347 

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348 
lemma clearjunk_delete: 
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349 
"clearjunk (delete x al) = delete x (clearjunk al)" 
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350 
by (induct al rule: clearjunk.induct) (auto simp add: delete_idem delete_twist) 
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351 

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352 
lemma clearjunk_restrict: 
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353 
"clearjunk (restrict A al) = restrict A (clearjunk al)" 
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354 
by (induct al rule: clearjunk.induct) (auto simp add: restr_delete_twist) 
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355 

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356 
lemma distinct_clearjunk_id [simp]: 
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357 
"distinct (map fst al) \<Longrightarrow> clearjunk al = al" 
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358 
by (induct al rule: clearjunk.induct) auto 
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359 

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360 
lemma clearjunk_idem: 
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361 
"clearjunk (clearjunk al) = clearjunk al" 
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362 
by simp 
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363 

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364 
lemma length_clearjunk: 
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365 
"length (clearjunk al) \<le> length al" 
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366 
proof (induct al rule: clearjunk.induct [case_names Nil Cons]) 
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367 
case Nil then show ?case by simp 
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368 
next 
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369 
case (Cons kv al) 
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370 
moreover have "length (delete (fst kv) al) \<le> length al" by (fact length_delete_le) 
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371 
ultimately have "length (clearjunk (delete (fst kv) al)) \<le> length al" by (rule order_trans) 
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372 
then show ?case by simp 
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373 
qed 
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374 

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375 
lemma delete_map: 
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376 
assumes "\<And>kv. fst (f kv) = fst kv" 
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377 
shows "delete k (map f ps) = map f (delete k ps)" 
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378 
by (simp add: delete_eq filter_map comp_def split_def assms) 
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379 

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380 
lemma clearjunk_map: 
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381 
assumes "\<And>kv. fst (f kv) = fst kv" 
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382 
shows "clearjunk (map f ps) = map f (clearjunk ps)" 
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383 
by (induct ps rule: clearjunk.induct [case_names Nil Cons]) 
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384 
(simp_all add: clearjunk_delete delete_map assms) 
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385 

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386 

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387 
subsection {* @{text map_ran} *} 
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388 

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389 
definition map_ran :: "('key \<Rightarrow> 'val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where 
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390 
"map_ran f = map (\<lambda>(k, v). (k, f k v))" 
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391 

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392 
lemma map_ran_simps [simp]: 
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393 
"map_ran f [] = []" 
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394 
"map_ran f ((k, v) # ps) = (k, f k v) # map_ran f ps" 
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395 
by (simp_all add: map_ran_def) 
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396 

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397 
lemma dom_map_ran: 
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398 
"fst ` set (map_ran f al) = fst ` set al" 
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399 
by (simp add: map_ran_def image_image split_def) 
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400 

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401 
lemma map_ran_conv: 
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402 
"map_of (map_ran f al) k = Option.map (f k) (map_of al k)" 
19234  403 
by (induct al) auto 
404 

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405 
lemma distinct_map_ran: 
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406 
"distinct (map fst al) \<Longrightarrow> distinct (map fst (map_ran f al))" 
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407 
by (simp add: map_ran_def split_def comp_def) 
19234  408 

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409 
lemma map_ran_filter: 
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410 
"map_ran f [p\<leftarrow>ps. fst p \<noteq> a] = [p\<leftarrow>map_ran f ps. fst p \<noteq> a]" 
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411 
by (simp add: map_ran_def filter_map split_def comp_def) 
19234  412 

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413 
lemma clearjunk_map_ran: 
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414 
"clearjunk (map_ran f al) = map_ran f (clearjunk al)" 
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415 
by (simp add: map_ran_def split_def clearjunk_map) 
19234  416 

23373  417 

34975
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418 
subsection {* @{text merge} *} 
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419 

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420 
definition merge :: "('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where 
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421 
"merge qs ps = foldr (\<lambda>(k, v). update k v) ps qs" 
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422 

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423 
lemma merge_simps [simp]: 
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424 
"merge qs [] = qs" 
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425 
"merge qs (p#ps) = update (fst p) (snd p) (merge qs ps)" 
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426 
by (simp_all add: merge_def split_def) 
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427 

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428 
lemma merge_updates: 
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429 
"merge qs ps = updates (rev (map fst ps)) (rev (map snd ps)) qs" 
37591  430 
by (simp add: merge_def updates_def foldr_fold_rev zip_rev zip_map_fst_snd) 
19234  431 

432 
lemma dom_merge: "fst ` set (merge xs ys) = fst ` set xs \<union> fst ` set ys" 

20503  433 
by (induct ys arbitrary: xs) (auto simp add: dom_update) 
19234  434 

435 
lemma distinct_merge: 

436 
assumes "distinct (map fst xs)" 

437 
shows "distinct (map fst (merge xs ys))" 

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438 
using assms by (simp add: merge_updates distinct_updates) 
19234  439 

440 
lemma clearjunk_merge: 

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441 
"clearjunk (merge xs ys) = merge (clearjunk xs) ys" 
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442 
by (simp add: merge_updates clearjunk_updates) 
19234  443 

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444 
lemma merge_conv': 
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445 
"map_of (merge xs ys) = map_of xs ++ map_of ys" 
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446 
proof  
37458  447 
have "map_of \<circ> More_List.fold (prod_case update) (rev ys) = 
448 
More_List.fold (\<lambda>(k, v) m. m(k \<mapsto> v)) (rev ys) \<circ> map_of" 

39921  449 
by (rule fold_commute) (simp add: update_conv' prod_case_beta split_def fun_eq_iff) 
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450 
then show ?thesis 
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451 
by (simp add: merge_def map_add_map_of_foldr foldr_fold_rev fun_eq_iff) 
19234  452 
qed 
453 

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454 
corollary merge_conv: 
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455 
"map_of (merge xs ys) k = (map_of xs ++ map_of ys) k" 
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456 
by (simp add: merge_conv') 
19234  457 

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458 
lemma merge_empty: "map_of (merge [] ys) = map_of ys" 
19234  459 
by (simp add: merge_conv') 
460 

461 
lemma merge_assoc[simp]: "map_of (merge m1 (merge m2 m3)) = 

462 
map_of (merge (merge m1 m2) m3)" 

463 
by (simp add: merge_conv') 

464 

465 
lemma merge_Some_iff: 

466 
"(map_of (merge m n) k = Some x) = 

467 
(map_of n k = Some x \<or> map_of n k = None \<and> map_of m k = Some x)" 

468 
by (simp add: merge_conv' map_add_Some_iff) 

469 

45605  470 
lemmas merge_SomeD [dest!] = merge_Some_iff [THEN iffD1] 
19234  471 

472 
lemma merge_find_right[simp]: "map_of n k = Some v \<Longrightarrow> map_of (merge m n) k = Some v" 

473 
by (simp add: merge_conv') 

474 

475 
lemma merge_None [iff]: 

476 
"(map_of (merge m n) k = None) = (map_of n k = None \<and> map_of m k = None)" 

477 
by (simp add: merge_conv') 

478 

479 
lemma merge_upd[simp]: 

480 
"map_of (merge m (update k v n)) = map_of (update k v (merge m n))" 

481 
by (simp add: update_conv' merge_conv') 

482 

483 
lemma merge_updatess[simp]: 

484 
"map_of (merge m (updates xs ys n)) = map_of (updates xs ys (merge m n))" 

485 
by (simp add: updates_conv' merge_conv') 

486 

487 
lemma merge_append: "map_of (xs@ys) = map_of (merge ys xs)" 

488 
by (simp add: merge_conv') 

489 

23373  490 

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491 
subsection {* @{text compose} *} 
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492 

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493 
function compose :: "('key \<times> 'a) list \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('key \<times> 'b) list" where 
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494 
"compose [] ys = []" 
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495 
 "compose (x#xs) ys = (case map_of ys (snd x) 
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496 
of None \<Rightarrow> compose (delete (fst x) xs) ys 
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497 
 Some v \<Rightarrow> (fst x, v) # compose xs ys)" 
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498 
by pat_completeness auto 
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499 
termination by (relation "measure (length \<circ> fst)") 
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500 
(simp_all add: less_Suc_eq_le length_delete_le) 
19234  501 

502 
lemma compose_first_None [simp]: 

503 
assumes "map_of xs k = None" 

504 
shows "map_of (compose xs ys) k = None" 

23373  505 
using assms by (induct xs ys rule: compose.induct) 
22916  506 
(auto split: option.splits split_if_asm) 
19234  507 

508 
lemma compose_conv: 

509 
shows "map_of (compose xs ys) k = (map_of ys \<circ>\<^sub>m map_of xs) k" 

22916  510 
proof (induct xs ys rule: compose.induct) 
511 
case 1 then show ?case by simp 

19234  512 
next 
22916  513 
case (2 x xs ys) show ?case 
19234  514 
proof (cases "map_of ys (snd x)") 
22916  515 
case None with 2 
19234  516 
have hyp: "map_of (compose (delete (fst x) xs) ys) k = 
517 
(map_of ys \<circ>\<^sub>m map_of (delete (fst x) xs)) k" 

518 
by simp 

519 
show ?thesis 

520 
proof (cases "fst x = k") 

521 
case True 

522 
from True delete_notin_dom [of k xs] 

523 
have "map_of (delete (fst x) xs) k = None" 

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524 
by (simp add: map_of_eq_None_iff) 
19234  525 
with hyp show ?thesis 
32960
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526 
using True None 
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527 
by simp 
19234  528 
next 
529 
case False 

530 
from False have "map_of (delete (fst x) xs) k = map_of xs k" 

32960
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531 
by simp 
19234  532 
with hyp show ?thesis 
32960
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533 
using False None 
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534 
by (simp add: map_comp_def) 
19234  535 
qed 
536 
next 

537 
case (Some v) 

22916  538 
with 2 
19234  539 
have "map_of (compose xs ys) k = (map_of ys \<circ>\<^sub>m map_of xs) k" 
540 
by simp 

541 
with Some show ?thesis 

542 
by (auto simp add: map_comp_def) 

543 
qed 

544 
qed 

545 

546 
lemma compose_conv': 

547 
shows "map_of (compose xs ys) = (map_of ys \<circ>\<^sub>m map_of xs)" 

548 
by (rule ext) (rule compose_conv) 

549 

550 
lemma compose_first_Some [simp]: 

551 
assumes "map_of xs k = Some v" 

552 
shows "map_of (compose xs ys) k = map_of ys v" 

23373  553 
using assms by (simp add: compose_conv) 
19234  554 

555 
lemma dom_compose: "fst ` set (compose xs ys) \<subseteq> fst ` set xs" 

22916  556 
proof (induct xs ys rule: compose.induct) 
557 
case 1 thus ?case by simp 

19234  558 
next 
22916  559 
case (2 x xs ys) 
19234  560 
show ?case 
561 
proof (cases "map_of ys (snd x)") 

562 
case None 

22916  563 
with "2.hyps" 
19234  564 
have "fst ` set (compose (delete (fst x) xs) ys) \<subseteq> fst ` set (delete (fst x) xs)" 
565 
by simp 

566 
also 

567 
have "\<dots> \<subseteq> fst ` set xs" 

568 
by (rule dom_delete_subset) 

569 
finally show ?thesis 

570 
using None 

571 
by auto 

572 
next 

573 
case (Some v) 

22916  574 
with "2.hyps" 
19234  575 
have "fst ` set (compose xs ys) \<subseteq> fst ` set xs" 
576 
by simp 

577 
with Some show ?thesis 

578 
by auto 

579 
qed 

580 
qed 

581 

582 
lemma distinct_compose: 

583 
assumes "distinct (map fst xs)" 

584 
shows "distinct (map fst (compose xs ys))" 

23373  585 
using assms 
22916  586 
proof (induct xs ys rule: compose.induct) 
587 
case 1 thus ?case by simp 

19234  588 
next 
22916  589 
case (2 x xs ys) 
19234  590 
show ?case 
591 
proof (cases "map_of ys (snd x)") 

592 
case None 

22916  593 
with 2 show ?thesis by simp 
19234  594 
next 
595 
case (Some v) 

22916  596 
with 2 dom_compose [of xs ys] show ?thesis 
19234  597 
by (auto) 
598 
qed 

599 
qed 

600 

601 
lemma compose_delete_twist: "(compose (delete k xs) ys) = delete k (compose xs ys)" 

22916  602 
proof (induct xs ys rule: compose.induct) 
603 
case 1 thus ?case by simp 

19234  604 
next 
22916  605 
case (2 x xs ys) 
19234  606 
show ?case 
607 
proof (cases "map_of ys (snd x)") 

608 
case None 

22916  609 
with 2 have 
19234  610 
hyp: "compose (delete k (delete (fst x) xs)) ys = 
611 
delete k (compose (delete (fst x) xs) ys)" 

612 
by simp 

613 
show ?thesis 

614 
proof (cases "fst x = k") 

615 
case True 

616 
with None hyp 

617 
show ?thesis 

32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
30663
diff
changeset

618 
by (simp add: delete_idem) 
19234  619 
next 
620 
case False 

621 
from None False hyp 

622 
show ?thesis 

32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
30663
diff
changeset

623 
by (simp add: delete_twist) 
19234  624 
qed 
625 
next 

626 
case (Some v) 

22916  627 
with 2 have hyp: "compose (delete k xs) ys = delete k (compose xs ys)" by simp 
19234  628 
with Some show ?thesis 
629 
by simp 

630 
qed 

631 
qed 

632 

633 
lemma compose_clearjunk: "compose xs (clearjunk ys) = compose xs ys" 

22916  634 
by (induct xs ys rule: compose.induct) 
19234  635 
(auto simp add: map_of_clearjunk split: option.splits) 
636 

637 
lemma clearjunk_compose: "clearjunk (compose xs ys) = compose (clearjunk xs) ys" 

638 
by (induct xs rule: clearjunk.induct) 

639 
(auto split: option.splits simp add: clearjunk_delete delete_idem 

640 
compose_delete_twist) 

641 

642 
lemma compose_empty [simp]: 

643 
"compose xs [] = []" 

22916  644 
by (induct xs) (auto simp add: compose_delete_twist) 
19234  645 

646 
lemma compose_Some_iff: 

647 
"(map_of (compose xs ys) k = Some v) = 

648 
(\<exists>k'. map_of xs k = Some k' \<and> map_of ys k' = Some v)" 

649 
by (simp add: compose_conv map_comp_Some_iff) 

650 

651 
lemma map_comp_None_iff: 

652 
"(map_of (compose xs ys) k = None) = 

653 
(map_of xs k = None \<or> (\<exists>k'. map_of xs k = Some k' \<and> map_of ys k' = None)) " 

654 
by (simp add: compose_conv map_comp_None_iff) 

655 

45869  656 
subsection {* @{text map_entry} *} 
657 

658 
fun map_entry :: "'key \<Rightarrow> ('val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" 

659 
where 

660 
"map_entry k f [] = []" 

661 
 "map_entry k f (p # ps) = (if fst p = k then (k, f (snd p)) # ps else p # map_entry k f ps)" 

662 

663 
lemma map_of_map_entry: 

664 
"map_of (map_entry k f xs) = (map_of xs)(k := case map_of xs k of None => None  Some v' => Some (f v'))" 

665 
by (induct xs) auto 

666 

667 
lemma dom_map_entry: 

668 
"fst ` set (map_entry k f xs) = fst ` set xs" 

669 
by (induct xs) auto 

670 

671 
lemma distinct_map_entry: 

672 
assumes "distinct (map fst xs)" 

673 
shows "distinct (map fst (map_entry k f xs))" 

674 
using assms by (induct xs) (auto simp add: dom_map_entry) 

675 

45868  676 
subsection {* @{text map_default} *} 
677 

678 
fun map_default :: "'key \<Rightarrow> 'val \<Rightarrow> ('val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" 

679 
where 

680 
"map_default k v f [] = [(k, v)]" 

681 
 "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)" 

682 

683 
lemma map_of_map_default: 

684 
"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'))" 

685 
by (induct xs) auto 

686 

687 
lemma dom_map_default: 

688 
"fst ` set (map_default k v f xs) = insert k (fst ` set xs)" 

689 
by (induct xs) auto 

690 

691 
lemma distinct_map_default: 

692 
assumes "distinct (map fst xs)" 

693 
shows "distinct (map fst (map_default k v f xs))" 

694 
using assms by (induct xs) (auto simp add: dom_map_default) 

695 

19234  696 
end 