author  nipkow 
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parent 69661  a03a63b81f44 
child 73680  50437744eb1c 
permissions  rwrr 
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(* Title: HOL/Library/AList.thy 
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Author: Norbert Schirmer, Tobias Nipkow, Martin Wildmoser, TU Muenchen 
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*) 
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section \<open>Implementation of Association Lists\<close> 
19234  6 

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theory AList 
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imports Main 
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begin 
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context 
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begin 
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text \<open> 
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The operations preserve distinctness of keys and 
69593  16 
function \<^term>\<open>clearjunk\<close> distributes over them. Since 
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\<^term>\<open>clearjunk\<close> 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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\<close> 
19234  20 

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subsection \<open>\<open>update\<close> and \<open>updates\<close>\<close> 
19323  22 

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qualified primrec update :: "'key \<Rightarrow> 'val \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" 
63462  24 
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)" 

19234  27 

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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 

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then show ?case 

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by (cases al') (auto split: if_split_asm) 
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next 
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case Cons 
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then show ?case 

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by (cases al') (auto split: if_split_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 \<open>Note that the lists are not necessarily the same: 
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\<^term>\<open>update k v (update k' v' []) = [(k', v'), (k, v)]\<close> and 
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\<^term>\<open>update k' v' (update k v []) = [(k, v), (k', v')]\<close>.\<close> 

56327  74 

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lemma update_swap: 

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"k \<noteq> k' \<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]: "x \<notin> A \<Longrightarrow> map_of (update x y al) ` A = map_of al ` A" 
69661  85 
by (auto simp add: update_conv') 
19234  86 

63476  87 
qualified definition updates :: 
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"'key list \<Rightarrow> 'val list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" 

56327  89 
where "updates ks vs = fold (case_prod update) (zip ks vs)" 
19234  90 

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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> fold (case_prod update) (zip ks vs) = 
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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 
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by (auto simp add: updates_def fun_eq_iff map_upds_fold_map_upd foldl_conv_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 (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> fold (case_prod update) (zip ks vs) = 
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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 case_prod_beta comp_def) 
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ultimately show ?thesis 
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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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"size ks \<le> i \<Longrightarrow> i < size vs \<Longrightarrow> 
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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 

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(if x \<in> set (take (length ys) xs) 

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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') 
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lemma updates_apply_notin [simp]: 
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"k \<notin> set ks \<Longrightarrow> 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 \<open>\<open>delete\<close>\<close> 
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qualified definition delete :: "'key \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" 
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where 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: "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]: "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: "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: "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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61585  216 
subsection \<open>\<open>update_with_aux\<close> and \<open>delete_aux\<close>\<close> 
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qualified primrec update_with_aux :: 
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"'val \<Rightarrow> 'key \<Rightarrow> ('val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" 

63462  220 
where 
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"update_with_aux v k f [] = [(k, f v)]" 

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 "update_with_aux v k f (p # ps) = 

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(if (fst p = k) then (k, f (snd p)) # ps else p # update_with_aux v k f ps)" 

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60500  225 
text \<open> 
69593  226 
The above \<^term>\<open>delete\<close> traverses all the list even if it has found the key. 
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This one does not have to keep going because is assumes the invariant that keys are distinct. 
60500  228 
\<close> 
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qualified fun delete_aux :: "'key \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" 
63462  230 
where 
231 
"delete_aux k [] = []" 

232 
 "delete_aux k ((k', v) # xs) = (if k = k' then xs else (k', v) # delete_aux k xs)" 

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233 

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lemma map_of_update_with_aux': 
63462  235 
"map_of (update_with_aux v k f ps) k' = 
236 
((map_of ps)(k \<mapsto> (case map_of ps k of None \<Rightarrow> f v  Some v \<Rightarrow> f v))) k'" 

237 
by (induct ps) auto 

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238 

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lemma map_of_update_with_aux: 
63462  240 
"map_of (update_with_aux v k f ps) = 
241 
(map_of ps)(k \<mapsto> (case map_of ps k of None \<Rightarrow> f v  Some v \<Rightarrow> f v))" 

242 
by (simp add: fun_eq_iff map_of_update_with_aux') 

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lemma dom_update_with_aux: "fst ` set (update_with_aux v k f ps) = {k} \<union> fst ` set ps" 
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by (induct ps) auto 
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246 

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lemma distinct_update_with_aux [simp]: 
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"distinct (map fst (update_with_aux v k f ps)) = distinct (map fst ps)" 
63462  249 
by (induct ps) (auto simp add: dom_update_with_aux) 
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250 

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lemma set_update_with_aux: 
63462  252 
"distinct (map fst xs) \<Longrightarrow> 
253 
set (update_with_aux v k f xs) = 

254 
(set xs  {k} \<times> UNIV \<union> {(k, f (case map_of xs k of None \<Rightarrow> v  Some v \<Rightarrow> v))})" 

255 
by (induct xs) (auto intro: rev_image_eqI) 

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256 

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lemma set_delete_aux: "distinct (map fst xs) \<Longrightarrow> set (delete_aux k xs) = set xs  {k} \<times> UNIV" 
63462  258 
apply (induct xs) 
63476  259 
apply simp_all 
63462  260 
apply clarsimp 
261 
apply (fastforce intro: rev_image_eqI) 

262 
done 

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lemma dom_delete_aux: "distinct (map fst ps) \<Longrightarrow> fst ` set (delete_aux k ps) = fst ` set ps  {k}" 
63462  265 
by (auto simp add: set_delete_aux) 
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63462  267 
lemma distinct_delete_aux [simp]: "distinct (map fst ps) \<Longrightarrow> distinct (map fst (delete_aux k ps))" 
268 
proof (induct ps) 

269 
case Nil 

270 
then show ?case by simp 

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next 
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case (Cons a ps) 
63462  273 
obtain k' v where a: "a = (k', v)" 
274 
by (cases a) 

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show ?case 
63462  276 
proof (cases "k' = k") 
277 
case True 

278 
with Cons a show ?thesis by simp 

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next 
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case False 
63462  281 
with Cons a have "k' \<notin> fst ` set ps" "distinct (map fst ps)" 
282 
by simp_all 

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with False a have "k' \<notin> fst ` set (delete_aux k ps)" 
63462  284 
by (auto dest!: dom_delete_aux[where k=k]) 
285 
with Cons a show ?thesis 

286 
by simp 

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qed 
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qed 
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289 

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lemma map_of_delete_aux': 
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"distinct (map fst xs) \<Longrightarrow> map_of (delete_aux k xs) = (map_of xs)(k := None)" 
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apply (induct xs) 
63476  293 
apply (fastforce simp add: map_of_eq_None_iff fun_upd_twist) 
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apply (auto intro!: ext) 
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apply (simp add: map_of_eq_None_iff) 
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done 
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297 

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lemma map_of_delete_aux: 
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"distinct (map fst xs) \<Longrightarrow> map_of (delete_aux k xs) k' = ((map_of xs)(k := None)) k'" 
63462  300 
by (simp add: map_of_delete_aux') 
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301 

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lemma delete_aux_eq_Nil_conv: "delete_aux k ts = [] \<longleftrightarrow> ts = [] \<or> (\<exists>v. ts = [(k, v)])" 
63462  303 
by (cases ts) (auto split: if_split_asm) 
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304 

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61585  306 
subsection \<open>\<open>restrict\<close>\<close> 
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307 

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qualified definition restrict :: "'key set \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" 
56327  309 
where restrict_eq: "restrict A = filter (\<lambda>(k, v). k \<in> A)" 
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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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by (auto simp add: restrict_eq) 
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lemma restr_conv': "map_of (restrict A al) = ((map_of al)` A)" 
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317 
proof 
63462  318 
show "map_of (restrict A al) k = ((map_of al)` A) k" for k 
319 
apply (induct al) 

63476  320 
apply simp 
63462  321 
apply (cases "k \<in> A") 
63476  322 
apply auto 
63462  323 
done 
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qed 
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corollary restr_conv: "map_of (restrict A al) k = ((map_of al)` A) k" 
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by (simp add: restr_conv') 
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328 

63462  329 
lemma distinct_restr: "distinct (map fst al) \<Longrightarrow> distinct (map fst (restrict A al))" 
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by (induct al) (auto simp add: restrict_eq) 
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56327  332 
lemma restr_empty [simp]: 
333 
"restrict {} al = []" 

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"restrict A [] = []" 
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by (induct al) (auto simp add: restrict_eq) 
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336 

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lemma restr_in [simp]: "x \<in> A \<Longrightarrow> map_of (restrict A al) x = map_of al x" 
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by (simp add: restr_conv') 
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339 

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lemma restr_out [simp]: "x \<notin> A \<Longrightarrow> map_of (restrict A al) x = None" 
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by (simp add: restr_conv') 
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342 

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lemma dom_restr [simp]: "fst ` set (restrict A al) = fst ` set al \<inter> A" 
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by (induct al) (auto simp add: restrict_eq) 
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345 

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lemma restr_upd_same [simp]: "restrict ({x}) (update x y al) = restrict ({x}) al" 
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by (induct al) (auto simp add: restrict_eq) 
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348 

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lemma restr_restr [simp]: "restrict A (restrict B al) = restrict (A\<inter>B) al" 
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by (induct al) (auto simp add: restrict_eq) 
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351 

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lemma restr_update[simp]: 
63462  353 
"map_of (restrict D (update x y al)) = 
354 
map_of ((if x \<in> D then (update x y (restrict (D{x}) al)) else restrict D al))" 

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by (simp add: restr_conv' update_conv') 
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356 

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lemma restr_delete [simp]: 
56327  358 
"delete x (restrict D al) = (if x \<in> D then restrict (D  {x}) al else restrict D al)" 
359 
apply (simp add: delete_eq restrict_eq) 

360 
apply (auto simp add: split_def) 

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361 
proof  
63462  362 
have "y \<noteq> x \<longleftrightarrow> x \<noteq> y" for y 
56327  363 
by auto 
68386  364 
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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365 
by simp 
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366 
assume "x \<notin> D" 
63462  367 
then have "y \<in> D \<longleftrightarrow> y \<in> D \<and> x \<noteq> y" for y 
56327  368 
by auto 
68386  369 
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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370 
by simp 
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qed 
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372 

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373 
lemma update_restr: 
56327  374 
"map_of (update x y (restrict D al)) = map_of (update x y (restrict (D  {x}) al))" 
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375 
by (simp add: update_conv' restr_conv') (rule fun_upd_restrict) 
19234  376 

45867  377 
lemma update_restr_conv [simp]: 
56327  378 
"x \<in> D \<Longrightarrow> 
379 
map_of (update x y (restrict D al)) = map_of (update x y (restrict (D  {x}) al))" 

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380 
by (simp add: update_conv' restr_conv') 
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381 

56327  382 
lemma restr_updates [simp]: 
383 
"length xs = length ys \<Longrightarrow> set xs \<subseteq> D \<Longrightarrow> 

384 
map_of (restrict D (updates xs ys al)) = 

385 
map_of (updates xs ys (restrict (D  set xs) al))" 

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386 
by (simp add: updates_conv' restr_conv') 
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387 

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388 
lemma restr_delete_twist: "(restrict A (delete a ps)) = delete a (restrict A ps)" 
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389 
by (induct ps) auto 
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390 

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391 

61585  392 
subsection \<open>\<open>clearjunk\<close>\<close> 
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393 

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394 
qualified function clearjunk :: "('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" 
63462  395 
where 
396 
"clearjunk [] = []" 

397 
 "clearjunk (p#ps) = p # clearjunk (delete (fst p) ps)" 

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by pat_completeness auto 
56327  399 
termination 
400 
by (relation "measure length") (simp_all add: less_Suc_eq_le length_delete_le) 

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401 

56327  402 
lemma map_of_clearjunk: "map_of (clearjunk al) = map_of al" 
403 
by (induct al rule: clearjunk.induct) (simp_all add: fun_eq_iff) 

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404 

56327  405 
lemma clearjunk_keys_set: "set (map fst (clearjunk al)) = set (map fst al)" 
406 
by (induct al rule: clearjunk.induct) (simp_all add: delete_keys) 

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407 

56327  408 
lemma dom_clearjunk: "fst ` set (clearjunk al) = fst ` set al" 
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409 
using clearjunk_keys_set by simp 
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410 

56327  411 
lemma distinct_clearjunk [simp]: "distinct (map fst (clearjunk al))" 
412 
by (induct al rule: clearjunk.induct) (simp_all del: set_map add: clearjunk_keys_set delete_keys) 

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413 

56327  414 
lemma ran_clearjunk: "ran (map_of (clearjunk al)) = ran (map_of al)" 
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415 
by (simp add: map_of_clearjunk) 
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changeset

416 

56327  417 
lemma ran_map_of: "ran (map_of al) = snd ` set (clearjunk al)" 
34975
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diff
changeset

418 
proof  
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changeset

419 
have "ran (map_of al) = ran (map_of (clearjunk al))" 
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diff
changeset

420 
by (simp add: ran_clearjunk) 
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parents:
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diff
changeset

421 
also have "\<dots> = snd ` set (clearjunk al)" 
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parents:
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diff
changeset

422 
by (simp add: ran_distinct) 
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diff
changeset

423 
finally show ?thesis . 
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parents:
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diff
changeset

424 
qed 
f099b0b20646
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parents:
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diff
changeset

425 

56327  426 
lemma clearjunk_update: "clearjunk (update k v al) = update k v (clearjunk al)" 
427 
by (induct al rule: clearjunk.induct) (simp_all add: delete_update) 

19234  428 

56327  429 
lemma clearjunk_updates: "clearjunk (updates ks vs al) = updates ks vs (clearjunk al)" 
34975
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changeset

430 
proof  
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
47397
diff
changeset

431 
have "clearjunk \<circ> fold (case_prod update) (zip ks vs) = 
63462  432 
fold (case_prod update) (zip ks vs) \<circ> clearjunk" 
55414
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blanchet
parents:
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diff
changeset

433 
by (rule fold_commute) (simp add: clearjunk_update case_prod_beta o_def) 
56327  434 
then show ?thesis 
435 
by (simp add: updates_def fun_eq_iff) 

34975
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diff
changeset

436 
qed 
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changeset

437 

56327  438 
lemma clearjunk_delete: "clearjunk (delete x al) = delete x (clearjunk al)" 
34975
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haftmann
parents:
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changeset

439 
by (induct al rule: clearjunk.induct) (auto simp add: delete_idem delete_twist) 
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more correspondence lemmas between related operations; tuned some proofs
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parents:
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diff
changeset

440 

56327  441 
lemma clearjunk_restrict: "clearjunk (restrict A al) = restrict A (clearjunk al)" 
34975
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parents:
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diff
changeset

442 
by (induct al rule: clearjunk.induct) (auto simp add: restr_delete_twist) 
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more correspondence lemmas between related operations; tuned some proofs
haftmann
parents:
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diff
changeset

443 

56327  444 
lemma distinct_clearjunk_id [simp]: "distinct (map fst al) \<Longrightarrow> clearjunk al = al" 
34975
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diff
changeset

445 
by (induct al rule: clearjunk.induct) auto 
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diff
changeset

446 

56327  447 
lemma clearjunk_idem: "clearjunk (clearjunk al) = clearjunk al" 
34975
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parents:
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diff
changeset

448 
by simp 
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haftmann
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changeset

449 

56327  450 
lemma length_clearjunk: "length (clearjunk al) \<le> length al" 
34975
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haftmann
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changeset

451 
proof (induct al rule: clearjunk.induct [case_names Nil Cons]) 
56327  452 
case Nil 
453 
then show ?case by simp 

34975
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diff
changeset

454 
next 
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changeset

455 
case (Cons kv al) 
56327  456 
moreover have "length (delete (fst kv) al) \<le> length al" 
457 
by (fact length_delete_le) 

458 
ultimately have "length (clearjunk (delete (fst kv) al)) \<le> length al" 

459 
by (rule order_trans) 

460 
then show ?case 

461 
by simp 

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462 
qed 
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more correspondence lemmas between related operations; tuned some proofs
haftmann
parents:
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diff
changeset

463 

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diff
changeset

464 
lemma delete_map: 
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parents:
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diff
changeset

465 
assumes "\<And>kv. fst (f kv) = fst kv" 
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parents:
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diff
changeset

466 
shows "delete k (map f ps) = map f (delete k ps)" 
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more correspondence lemmas between related operations; tuned some proofs
haftmann
parents:
32960
diff
changeset

467 
by (simp add: delete_eq filter_map comp_def split_def assms) 
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more correspondence lemmas between related operations; tuned some proofs
haftmann
parents:
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diff
changeset

468 

f099b0b20646
more correspondence lemmas between related operations; tuned some proofs
haftmann
parents:
32960
diff
changeset

469 
lemma clearjunk_map: 
f099b0b20646
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parents:
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diff
changeset

470 
assumes "\<And>kv. fst (f kv) = fst kv" 
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more correspondence lemmas between related operations; tuned some proofs
haftmann
parents:
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diff
changeset

471 
shows "clearjunk (map f ps) = map f (clearjunk ps)" 
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haftmann
parents:
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diff
changeset

472 
by (induct ps rule: clearjunk.induct [case_names Nil Cons]) 
f099b0b20646
more correspondence lemmas between related operations; tuned some proofs
haftmann
parents:
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diff
changeset

473 
(simp_all add: clearjunk_delete delete_map assms) 
f099b0b20646
more correspondence lemmas between related operations; tuned some proofs
haftmann
parents:
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diff
changeset

474 

f099b0b20646
more correspondence lemmas between related operations; tuned some proofs
haftmann
parents:
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diff
changeset

475 

61585  476 
subsection \<open>\<open>map_ran\<close>\<close> 
34975
f099b0b20646
more correspondence lemmas between related operations; tuned some proofs
haftmann
parents:
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diff
changeset

477 

73678  478 
definition map_ran :: "('key \<Rightarrow> 'val1 \<Rightarrow> 'val2) \<Rightarrow> ('key \<times> 'val1) list \<Rightarrow> ('key \<times> 'val2) list" 
56327  479 
where "map_ran f = map (\<lambda>(k, v). (k, f k v))" 
34975
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haftmann
parents:
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diff
changeset

480 

f099b0b20646
more correspondence lemmas between related operations; tuned some proofs
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diff
changeset

481 
lemma map_ran_simps [simp]: 
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diff
changeset

482 
"map_ran f [] = []" 
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diff
changeset

483 
"map_ran f ((k, v) # ps) = (k, f k v) # map_ran f ps" 
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diff
changeset

484 
by (simp_all add: map_ran_def) 
f099b0b20646
more correspondence lemmas between related operations; tuned some proofs
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diff
changeset

485 

56327  486 
lemma dom_map_ran: "fst ` set (map_ran f al) = fst ` set al" 
34975
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haftmann
parents:
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diff
changeset

487 
by (simp add: map_ran_def image_image split_def) 
56327  488 

489 
lemma map_ran_conv: "map_of (map_ran f al) k = map_option (f k) (map_of al k)" 

19234  490 
by (induct al) auto 
491 

56327  492 
lemma distinct_map_ran: "distinct (map fst al) \<Longrightarrow> distinct (map fst (map_ran f al))" 
34975
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haftmann
parents:
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diff
changeset

493 
by (simp add: map_ran_def split_def comp_def) 
19234  494 

68386  495 
lemma map_ran_filter: "map_ran f [p\<leftarrow>ps. fst p \<noteq> a] = [p\<leftarrow>map_ran f ps. fst p \<noteq> a]" 
34975
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diff
changeset

496 
by (simp add: map_ran_def filter_map split_def comp_def) 
19234  497 

56327  498 
lemma clearjunk_map_ran: "clearjunk (map_ran f al) = map_ran f (clearjunk al)" 
34975
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haftmann
parents:
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diff
changeset

499 
by (simp add: map_ran_def split_def clearjunk_map) 
19234  500 

23373  501 

61585  502 
subsection \<open>\<open>merge\<close>\<close> 
34975
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haftmann
parents:
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diff
changeset

503 

59990
a81dc82ecba3
clarified keyword 'qualified' in accordance to a similar keyword from Haskell (despite unrelated Binding.qualified in Isabelle/ML);
wenzelm
parents:
59943
diff
changeset

504 
qualified definition merge :: "('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" 
56327  505 
where "merge qs ps = foldr (\<lambda>(k, v). update k v) ps qs" 
34975
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more correspondence lemmas between related operations; tuned some proofs
haftmann
parents:
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diff
changeset

506 

f099b0b20646
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haftmann
parents:
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diff
changeset

507 
lemma merge_simps [simp]: 
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changeset

508 
"merge qs [] = qs" 
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diff
changeset

509 
"merge qs (p#ps) = update (fst p) (snd p) (merge qs ps)" 
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haftmann
parents:
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diff
changeset

510 
by (simp_all add: merge_def split_def) 
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haftmann
parents:
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diff
changeset

511 

56327  512 
lemma merge_updates: "merge qs ps = updates (rev (map fst ps)) (rev (map snd ps)) qs" 
47397
d654c73e4b12
no preference wrt. fold(l/r); prefer fold rather than foldr for iterating over lists in generated code
haftmann
parents:
46507
diff
changeset

513 
by (simp add: merge_def updates_def foldr_conv_fold zip_rev zip_map_fst_snd) 
19234  514 

515 
lemma dom_merge: "fst ` set (merge xs ys) = fst ` set xs \<union> fst ` set ys" 

20503  516 
by (induct ys arbitrary: xs) (auto simp add: dom_update) 
19234  517 

63462  518 
lemma distinct_merge: "distinct (map fst xs) \<Longrightarrow> distinct (map fst (merge xs ys))" 
519 
by (simp add: merge_updates distinct_updates) 

19234  520 

56327  521 
lemma clearjunk_merge: "clearjunk (merge xs ys) = merge (clearjunk xs) ys" 
34975
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haftmann
parents:
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diff
changeset

522 
by (simp add: merge_updates clearjunk_updates) 
19234  523 

56327  524 
lemma merge_conv': "map_of (merge xs ys) = map_of xs ++ map_of ys" 
34975
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haftmann
parents:
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diff
changeset

525 
proof  
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
47397
diff
changeset

526 
have "map_of \<circ> fold (case_prod update) (rev ys) = 
56327  527 
fold (\<lambda>(k, v) m. m(k \<mapsto> v)) (rev ys) \<circ> map_of" 
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
47397
diff
changeset

528 
by (rule fold_commute) (simp add: update_conv' case_prod_beta split_def fun_eq_iff) 
34975
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haftmann
parents:
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diff
changeset

529 
then show ?thesis 
47397
d654c73e4b12
no preference wrt. fold(l/r); prefer fold rather than foldr for iterating over lists in generated code
haftmann
parents:
46507
diff
changeset

530 
by (simp add: merge_def map_add_map_of_foldr foldr_conv_fold fun_eq_iff) 
19234  531 
qed 
532 

56327  533 
corollary merge_conv: "map_of (merge xs ys) k = (map_of xs ++ map_of ys) k" 
34975
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haftmann
parents:
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diff
changeset

534 
by (simp add: merge_conv') 
19234  535 

34975
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haftmann
parents:
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diff
changeset

536 
lemma merge_empty: "map_of (merge [] ys) = map_of ys" 
19234  537 
by (simp add: merge_conv') 
538 

56327  539 
lemma merge_assoc [simp]: "map_of (merge m1 (merge m2 m3)) = map_of (merge (merge m1 m2) m3)" 
19234  540 
by (simp add: merge_conv') 
541 

56327  542 
lemma merge_Some_iff: 
543 
"map_of (merge m n) k = Some x \<longleftrightarrow> 

544 
map_of n k = Some x \<or> map_of n k = None \<and> map_of m k = Some x" 

19234  545 
by (simp add: merge_conv' map_add_Some_iff) 
546 

45605  547 
lemmas merge_SomeD [dest!] = merge_Some_iff [THEN iffD1] 
19234  548 

56327  549 
lemma merge_find_right [simp]: "map_of n k = Some v \<Longrightarrow> map_of (merge m n) k = Some v" 
19234  550 
by (simp add: merge_conv') 
551 

63462  552 
lemma merge_None [iff]: "(map_of (merge m n) k = None) = (map_of n k = None \<and> map_of m k = None)" 
19234  553 
by (simp add: merge_conv') 
554 

63462  555 
lemma merge_upd [simp]: "map_of (merge m (update k v n)) = map_of (update k v (merge m n))" 
19234  556 
by (simp add: update_conv' merge_conv') 
557 

56327  558 
lemma merge_updatess [simp]: 
19234  559 
"map_of (merge m (updates xs ys n)) = map_of (updates xs ys (merge m n))" 
560 
by (simp add: updates_conv' merge_conv') 

561 

56327  562 
lemma merge_append: "map_of (xs @ ys) = map_of (merge ys xs)" 
19234  563 
by (simp add: merge_conv') 
564 

23373  565 

61585  566 
subsection \<open>\<open>compose\<close>\<close> 
34975
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haftmann
parents:
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diff
changeset

567 

59990
a81dc82ecba3
clarified keyword 'qualified' in accordance to a similar keyword from Haskell (despite unrelated Binding.qualified in Isabelle/ML);
wenzelm
parents:
59943
diff
changeset

568 
qualified function compose :: "('key \<times> 'a) list \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('key \<times> 'b) list" 
63462  569 
where 
570 
"compose [] ys = []" 

571 
 "compose (x # xs) ys = 

572 
(case map_of ys (snd x) of 

573 
None \<Rightarrow> compose (delete (fst x) xs) ys 

574 
 Some v \<Rightarrow> (fst x, v) # compose xs ys)" 

34975
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haftmann
parents:
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diff
changeset

575 
by pat_completeness auto 
56327  576 
termination 
577 
by (relation "measure (length \<circ> fst)") (simp_all add: less_Suc_eq_le length_delete_le) 

19234  578 

63462  579 
lemma compose_first_None [simp]: "map_of xs k = None \<Longrightarrow> map_of (compose xs ys) k = None" 
580 
by (induct xs ys rule: compose.induct) (auto split: option.splits if_split_asm) 

19234  581 

56327  582 
lemma compose_conv: "map_of (compose xs ys) k = (map_of ys \<circ>\<^sub>m map_of xs) k" 
22916  583 
proof (induct xs ys rule: compose.induct) 
56327  584 
case 1 
585 
then show ?case by simp 

19234  586 
next 
56327  587 
case (2 x xs ys) 
588 
show ?case 

19234  589 
proof (cases "map_of ys (snd x)") 
56327  590 
case None 
591 
with 2 have hyp: "map_of (compose (delete (fst x) xs) ys) k = 

592 
(map_of ys \<circ>\<^sub>m map_of (delete (fst x) xs)) k" 

19234  593 
by simp 
594 
show ?thesis 

595 
proof (cases "fst x = k") 

596 
case True 

597 
from True delete_notin_dom [of k xs] 

598 
have "map_of (delete (fst x) xs) k = None" 

32960
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wenzelm
parents:
30663
diff
changeset

599 
by (simp add: map_of_eq_None_iff) 
19234  600 
with hyp show ?thesis 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
30663
diff
changeset

601 
using True None 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
30663
diff
changeset

602 
by simp 
19234  603 
next 
604 
case False 

605 
from False have "map_of (delete (fst x) xs) k = map_of xs k" 

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

606 
by simp 
19234  607 
with hyp show ?thesis 
56327  608 
using False None by (simp add: map_comp_def) 
19234  609 
qed 
610 
next 

611 
case (Some v) 

22916  612 
with 2 
19234  613 
have "map_of (compose xs ys) k = (map_of ys \<circ>\<^sub>m map_of xs) k" 
614 
by simp 

615 
with Some show ?thesis 

616 
by (auto simp add: map_comp_def) 

617 
qed 

618 
qed 

56327  619 

620 
lemma compose_conv': "map_of (compose xs ys) = (map_of ys \<circ>\<^sub>m map_of xs)" 

19234  621 
by (rule ext) (rule compose_conv) 
622 

63462  623 
lemma compose_first_Some [simp]: "map_of xs k = Some v \<Longrightarrow> map_of (compose xs ys) k = map_of ys v" 
624 
by (simp add: compose_conv) 

19234  625 

626 
lemma dom_compose: "fst ` set (compose xs ys) \<subseteq> fst ` set xs" 

22916  627 
proof (induct xs ys rule: compose.induct) 
56327  628 
case 1 
629 
then show ?case by simp 

19234  630 
next 
22916  631 
case (2 x xs ys) 
19234  632 
show ?case 
633 
proof (cases "map_of ys (snd x)") 

634 
case None 

63462  635 
with "2.hyps" have "fst ` set (compose (delete (fst x) xs) ys) \<subseteq> fst ` set (delete (fst x) xs)" 
19234  636 
by simp 
63462  637 
also have "\<dots> \<subseteq> fst ` set xs" 
19234  638 
by (rule dom_delete_subset) 
639 
finally show ?thesis 

63462  640 
using None by auto 
19234  641 
next 
642 
case (Some v) 

63462  643 
with "2.hyps" have "fst ` set (compose xs ys) \<subseteq> fst ` set xs" 
19234  644 
by simp 
645 
with Some show ?thesis 

646 
by auto 

647 
qed 

648 
qed 

649 

650 
lemma distinct_compose: 

56327  651 
assumes "distinct (map fst xs)" 
652 
shows "distinct (map fst (compose xs ys))" 

653 
using assms 

22916  654 
proof (induct xs ys rule: compose.induct) 
56327  655 
case 1 
656 
then show ?case by simp 

19234  657 
next 
22916  658 
case (2 x xs ys) 
19234  659 
show ?case 
660 
proof (cases "map_of ys (snd x)") 

661 
case None 

22916  662 
with 2 show ?thesis by simp 
19234  663 
next 
664 
case (Some v) 

56327  665 
with 2 dom_compose [of xs ys] show ?thesis 
666 
by auto 

19234  667 
qed 
668 
qed 

669 

56327  670 
lemma compose_delete_twist: "compose (delete k xs) ys = delete k (compose xs ys)" 
22916  671 
proof (induct xs ys rule: compose.induct) 
56327  672 
case 1 
673 
then show ?case by simp 

19234  674 
next 
22916  675 
case (2 x xs ys) 
19234  676 
show ?case 
677 
proof (cases "map_of ys (snd x)") 

678 
case None 

56327  679 
with 2 have hyp: "compose (delete k (delete (fst x) xs)) ys = 
680 
delete k (compose (delete (fst x) xs) ys)" 

19234  681 
by simp 
682 
show ?thesis 

683 
proof (cases "fst x = k") 

684 
case True 

56327  685 
with None hyp show ?thesis 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
30663
diff
changeset

686 
by (simp add: delete_idem) 
19234  687 
next 
688 
case False 

56327  689 
from None False hyp show ?thesis 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
30663
diff
changeset

690 
by (simp add: delete_twist) 
19234  691 
qed 
692 
next 

693 
case (Some v) 

56327  694 
with 2 have hyp: "compose (delete k xs) ys = delete k (compose xs ys)" 
695 
by simp 

19234  696 
with Some show ?thesis 
697 
by simp 

698 
qed 

699 
qed 

700 

701 
lemma compose_clearjunk: "compose xs (clearjunk ys) = compose xs ys" 

56327  702 
by (induct xs ys rule: compose.induct) 
703 
(auto simp add: map_of_clearjunk split: option.splits) 

704 

19234  705 
lemma clearjunk_compose: "clearjunk (compose xs ys) = compose (clearjunk xs) ys" 
706 
by (induct xs rule: clearjunk.induct) 

56327  707 
(auto split: option.splits simp add: clearjunk_delete delete_idem compose_delete_twist) 
708 

709 
lemma compose_empty [simp]: "compose xs [] = []" 

22916  710 
by (induct xs) (auto simp add: compose_delete_twist) 
19234  711 

712 
lemma compose_Some_iff: 

56327  713 
"(map_of (compose xs ys) k = Some v) \<longleftrightarrow> 
714 
(\<exists>k'. map_of xs k = Some k' \<and> map_of ys k' = Some v)" 

19234  715 
by (simp add: compose_conv map_comp_Some_iff) 
716 

717 
lemma map_comp_None_iff: 

56327  718 
"map_of (compose xs ys) k = None \<longleftrightarrow> 
719 
(map_of xs k = None \<or> (\<exists>k'. map_of xs k = Some k' \<and> map_of ys k' = None))" 

19234  720 
by (simp add: compose_conv map_comp_None_iff) 
721 

56327  722 

61585  723 
subsection \<open>\<open>map_entry\<close>\<close> 
45869  724 

59990
a81dc82ecba3
clarified keyword 'qualified' in accordance to a similar keyword from Haskell (despite unrelated Binding.qualified in Isabelle/ML);
wenzelm
parents:
59943
diff
changeset

725 
qualified fun map_entry :: "'key \<Rightarrow> ('val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" 
63462  726 
where 
727 
"map_entry k f [] = []" 

728 
 "map_entry k f (p # ps) = 

729 
(if fst p = k then (k, f (snd p)) # ps else p # map_entry k f ps)" 

45869  730 

731 
lemma map_of_map_entry: 

56327  732 
"map_of (map_entry k f xs) = 
733 
(map_of xs)(k := case map_of xs k of None \<Rightarrow> None  Some v' \<Rightarrow> Some (f v'))" 

734 
by (induct xs) auto 

45869  735 

56327  736 
lemma dom_map_entry: "fst ` set (map_entry k f xs) = fst ` set xs" 
737 
by (induct xs) auto 

45869  738 

739 
lemma distinct_map_entry: 

740 
assumes "distinct (map fst xs)" 

741 
shows "distinct (map fst (map_entry k f xs))" 

56327  742 
using assms by (induct xs) (auto simp add: dom_map_entry) 
743 

45869  744 

61585  745 
subsection \<open>\<open>map_default\<close>\<close> 
45868  746 

747 
fun map_default :: "'key \<Rightarrow> 'val \<Rightarrow> ('val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" 

63462  748 
where 
749 
"map_default k v f [] = [(k, v)]" 

750 
 "map_default k v f (p # ps) = 

751 
(if fst p = k then (k, f (snd p)) # ps else p # map_default k v f ps)" 

45868  752 

753 
lemma map_of_map_default: 

56327  754 
"map_of (map_default k v f xs) = 
755 
(map_of xs)(k := case map_of xs k of None \<Rightarrow> Some v  Some v' \<Rightarrow> Some (f v'))" 

756 
by (induct xs) auto 

45868  757 

56327  758 
lemma dom_map_default: "fst ` set (map_default k v f xs) = insert k (fst ` set xs)" 
759 
by (induct xs) auto 

45868  760 

761 
lemma distinct_map_default: 

762 
assumes "distinct (map fst xs)" 

763 
shows "distinct (map fst (map_default k v f xs))" 

56327  764 
using assms by (induct xs) (auto simp add: dom_map_default) 
45868  765 

59943
e83ecf0a0ee1
more qualified names  eliminated hide_const (open);
wenzelm
parents:
58881
diff
changeset

766 
end 
45884  767 

19234  768 
end 