author | blanchet |
Thu, 18 Feb 2010 18:48:07 +0100 | |
changeset 35220 | 2bcdae5f4fdb |
parent 35194 | a6c573d13385 |
child 36109 | 1028cf8c0d1b |
permissions | -rw-r--r-- |
22803 | 1 |
(* Title: HOL/Library/AssocList.thy |
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Author: Norbert Schirmer, Tobias Nipkow, Martin Wildmoser, TU Muenchen |
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*) |
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header {* Map operations implemented on association lists*} |
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theory AssocList |
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imports Main Mapping |
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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: expand_fun_eq) |
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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' expand_fun_eq) |
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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 al = foldl (\<lambda>al (k, v). update k v al) al (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 "foldl (\<lambda>f (k, v). f(k \<mapsto> v)) (map_of al) (zip ks vs) = |
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map_of (foldl (\<lambda>al (k, v). update k v al) al (zip ks vs))" |
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by (rule foldl_apply) (auto simp add: expand_fun_eq update_conv') |
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then show ?thesis |
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by (simp add: updates_def map_upds_fold_map_upd) |
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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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from assms have "distinct (foldl |
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(\<lambda>al (k, v). if k \<in> set al then al else al @ [k]) |
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(map fst al) (zip ks vs))" |
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by (rule foldl_invariant) auto |
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moreover have "foldl (\<lambda>al (k, v). if k \<in> set al then al else al @ [k]) |
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(map fst al) (zip ks vs) |
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= map fst (foldl (\<lambda>al (k, v). update k v al) al (zip ks vs))" |
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by (rule foldl_apply) (simp add: update_keys split_def comp_def) |
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ultimately show ?thesis by (simp add: updates_def) |
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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: expand_fun_eq) |
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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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190 |
lemma delete_notin_dom: "k \<notin> fst ` set (delete k al)" |
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191 |
by (auto simp add: delete_eq) |
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192 |
|
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lemma dom_delete_subset: "fst ` set (delete k al) \<subseteq> fst ` set al" |
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194 |
by (auto simp add: delete_eq) |
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195 |
|
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lemma delete_update_same: |
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197 |
"delete k (update k v al) = delete k al" |
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198 |
by (induct al) simp_all |
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199 |
|
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200 |
lemma delete_update: |
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201 |
"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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203 |
|
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204 |
lemma delete_twist: "delete x (delete y al) = delete y (delete x al)" |
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205 |
by (simp add: delete_eq conj_commute) |
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|
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207 |
lemma length_delete_le: "length (delete k al) \<le> length al" |
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208 |
by (simp add: delete_eq) |
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209 |
|
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210 |
|
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211 |
subsection {* @{text restrict} *} |
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212 |
|
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213 |
definition restrict :: "'key set \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where |
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214 |
restrict_eq: "restrict A = filter (\<lambda>(k, v). k \<in> A)" |
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215 |
|
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216 |
lemma restr_simps [simp]: |
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217 |
"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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219 |
by (auto simp add: restrict_eq) |
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220 |
|
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221 |
lemma restr_conv': "map_of (restrict A al) = ((map_of al)|` A)" |
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222 |
proof |
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223 |
fix k |
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224 |
show "map_of (restrict A al) k = ((map_of al)|` A) k" |
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225 |
by (induct al) (simp, cases "k \<in> A", auto) |
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226 |
qed |
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227 |
|
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228 |
corollary restr_conv: "map_of (restrict A al) k = ((map_of al)|` A) k" |
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229 |
by (simp add: restr_conv') |
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230 |
|
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231 |
lemma distinct_restr: |
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232 |
"distinct (map fst al) \<Longrightarrow> distinct (map fst (restrict A al))" |
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233 |
by (induct al) (auto simp add: restrict_eq) |
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234 |
|
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235 |
lemma restr_empty [simp]: |
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236 |
"restrict {} al = []" |
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237 |
"restrict A [] = []" |
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238 |
by (induct al) (auto simp add: restrict_eq) |
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239 |
|
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240 |
lemma restr_in [simp]: "x \<in> A \<Longrightarrow> map_of (restrict A al) x = map_of al x" |
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241 |
by (simp add: restr_conv') |
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242 |
|
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243 |
lemma restr_out [simp]: "x \<notin> A \<Longrightarrow> map_of (restrict A al) x = None" |
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244 |
by (simp add: restr_conv') |
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245 |
|
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246 |
lemma dom_restr [simp]: "fst ` set (restrict A al) = fst ` set al \<inter> A" |
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247 |
by (induct al) (auto simp add: restrict_eq) |
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248 |
|
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249 |
lemma restr_upd_same [simp]: "restrict (-{x}) (update x y al) = restrict (-{x}) al" |
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250 |
by (induct al) (auto simp add: restrict_eq) |
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251 |
|
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252 |
lemma restr_restr [simp]: "restrict A (restrict B al) = restrict (A\<inter>B) al" |
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253 |
by (induct al) (auto simp add: restrict_eq) |
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254 |
|
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255 |
lemma restr_update[simp]: |
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256 |
"map_of (restrict D (update x y al)) = |
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257 |
map_of ((if x \<in> D then (update x y (restrict (D-{x}) al)) else restrict D al))" |
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258 |
by (simp add: restr_conv' update_conv') |
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259 |
|
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260 |
lemma restr_delete [simp]: |
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|
261 |
"(delete x (restrict D al)) = |
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262 |
(if x \<in> D then restrict (D - {x}) al else restrict D al)" |
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263 |
apply (simp add: delete_eq restrict_eq) |
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264 |
apply (auto simp add: split_def) |
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|
265 |
proof - |
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266 |
have "\<And>y. y \<noteq> x \<longleftrightarrow> x \<noteq> y" by auto |
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|
267 |
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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|
268 |
by simp |
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269 |
assume "x \<notin> D" |
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|
270 |
then have "\<And>y. y \<in> D \<longleftrightarrow> y \<in> D \<and> x \<noteq> y" by auto |
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271 |
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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|
272 |
by simp |
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|
273 |
qed |
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274 |
|
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275 |
lemma update_restr: |
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|
276 |
"map_of (update x y (restrict D al)) = map_of (update x y (restrict (D-{x}) al))" |
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277 |
by (simp add: update_conv' restr_conv') (rule fun_upd_restrict) |
19234 | 278 |
|
34975
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279 |
lemma upate_restr_conv [simp]: |
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|
280 |
"x \<in> D \<Longrightarrow> |
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|
281 |
map_of (update x y (restrict D al)) = map_of (update x y (restrict (D-{x}) al))" |
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|
282 |
by (simp add: update_conv' restr_conv') |
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283 |
|
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284 |
lemma restr_updates [simp]: " |
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285 |
\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk> |
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286 |
\<Longrightarrow> map_of (restrict D (updates xs ys al)) = |
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|
287 |
map_of (updates xs ys (restrict (D - set xs) al))" |
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|
288 |
by (simp add: updates_conv' restr_conv') |
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|
289 |
|
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|
290 |
lemma restr_delete_twist: "(restrict A (delete a ps)) = delete a (restrict A ps)" |
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|
291 |
by (induct ps) auto |
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|
292 |
|
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293 |
|
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|
294 |
subsection {* @{text clearjunk} *} |
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295 |
|
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296 |
function clearjunk :: "('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where |
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297 |
"clearjunk [] = []" |
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298 |
| "clearjunk (p#ps) = p # clearjunk (delete (fst p) ps)" |
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299 |
by pat_completeness auto |
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300 |
termination by (relation "measure length") |
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301 |
(simp_all add: less_Suc_eq_le length_delete_le) |
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|
302 |
|
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303 |
lemma map_of_clearjunk: |
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304 |
"map_of (clearjunk al) = map_of al" |
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305 |
by (induct al rule: clearjunk.induct) |
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|
306 |
(simp_all add: expand_fun_eq) |
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|
307 |
|
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|
308 |
lemma clearjunk_keys_set: |
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|
309 |
"set (map fst (clearjunk al)) = set (map fst al)" |
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310 |
by (induct al rule: clearjunk.induct) |
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311 |
(simp_all add: delete_keys) |
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|
312 |
|
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313 |
lemma dom_clearjunk: |
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|
314 |
"fst ` set (clearjunk al) = fst ` set al" |
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|
315 |
using clearjunk_keys_set by simp |
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|
316 |
|
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|
317 |
lemma distinct_clearjunk [simp]: |
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|
318 |
"distinct (map fst (clearjunk al))" |
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|
319 |
by (induct al rule: clearjunk.induct) |
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|
320 |
(simp_all del: set_map add: clearjunk_keys_set delete_keys) |
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changeset
|
321 |
|
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|
322 |
lemma ran_clearjunk: |
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|
323 |
"ran (map_of (clearjunk al)) = ran (map_of al)" |
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|
324 |
by (simp add: map_of_clearjunk) |
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changeset
|
325 |
|
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|
326 |
lemma ran_map_of: |
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|
327 |
"ran (map_of al) = snd ` set (clearjunk al)" |
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|
328 |
proof - |
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|
329 |
have "ran (map_of al) = ran (map_of (clearjunk al))" |
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|
330 |
by (simp add: ran_clearjunk) |
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|
331 |
also have "\<dots> = snd ` set (clearjunk al)" |
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|
332 |
by (simp add: ran_distinct) |
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|
333 |
finally show ?thesis . |
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|
334 |
qed |
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changeset
|
335 |
|
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|
336 |
lemma clearjunk_update: |
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|
337 |
"clearjunk (update k v al) = update k v (clearjunk al)" |
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|
338 |
by (induct al rule: clearjunk.induct) |
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|
339 |
(simp_all add: delete_update) |
19234 | 340 |
|
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|
341 |
lemma clearjunk_updates: |
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|
342 |
"clearjunk (updates ks vs al) = updates ks vs (clearjunk al)" |
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|
343 |
proof - |
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|
344 |
have "foldl (\<lambda>al (k, v). update k v al) (clearjunk al) (zip ks vs) = |
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|
345 |
clearjunk (foldl (\<lambda>al (k, v). update k v al) al (zip ks vs))" |
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|
346 |
by (rule foldl_apply) (simp add: clearjunk_update expand_fun_eq split_def) |
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|
347 |
then show ?thesis by (simp add: updates_def) |
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|
348 |
qed |
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changeset
|
349 |
|
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|
350 |
lemma clearjunk_delete: |
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|
351 |
"clearjunk (delete x al) = delete x (clearjunk al)" |
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|
352 |
by (induct al rule: clearjunk.induct) (auto simp add: delete_idem delete_twist) |
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changeset
|
353 |
|
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|
354 |
lemma clearjunk_restrict: |
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|
355 |
"clearjunk (restrict A al) = restrict A (clearjunk al)" |
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|
356 |
by (induct al rule: clearjunk.induct) (auto simp add: restr_delete_twist) |
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changeset
|
357 |
|
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changeset
|
358 |
lemma distinct_clearjunk_id [simp]: |
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|
359 |
"distinct (map fst al) \<Longrightarrow> clearjunk al = al" |
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|
360 |
by (induct al rule: clearjunk.induct) auto |
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changeset
|
361 |
|
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changeset
|
362 |
lemma clearjunk_idem: |
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|
363 |
"clearjunk (clearjunk al) = clearjunk al" |
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changeset
|
364 |
by simp |
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changeset
|
365 |
|
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|
366 |
lemma length_clearjunk: |
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|
367 |
"length (clearjunk al) \<le> length al" |
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|
368 |
proof (induct al rule: clearjunk.induct [case_names Nil Cons]) |
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|
369 |
case Nil then show ?case by simp |
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|
370 |
next |
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|
371 |
case (Cons kv al) |
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|
372 |
moreover have "length (delete (fst kv) al) \<le> length al" by (fact length_delete_le) |
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|
373 |
ultimately have "length (clearjunk (delete (fst kv) al)) \<le> length al" by (rule order_trans) |
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parents:
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changeset
|
374 |
then show ?case by simp |
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parents:
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|
375 |
qed |
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parents:
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changeset
|
376 |
|
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|
377 |
lemma delete_map: |
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parents:
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changeset
|
378 |
assumes "\<And>kv. fst (f kv) = fst kv" |
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changeset
|
379 |
shows "delete k (map f ps) = map f (delete k ps)" |
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parents:
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changeset
|
380 |
by (simp add: delete_eq filter_map comp_def split_def assms) |
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parents:
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changeset
|
381 |
|
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parents:
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changeset
|
382 |
lemma clearjunk_map: |
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changeset
|
383 |
assumes "\<And>kv. fst (f kv) = fst kv" |
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changeset
|
384 |
shows "clearjunk (map f ps) = map f (clearjunk ps)" |
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changeset
|
385 |
by (induct ps rule: clearjunk.induct [case_names Nil Cons]) |
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changeset
|
386 |
(simp_all add: clearjunk_delete delete_map assms) |
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parents:
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diff
changeset
|
387 |
|
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parents:
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diff
changeset
|
388 |
|
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parents:
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changeset
|
389 |
subsection {* @{text map_ran} *} |
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parents:
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changeset
|
390 |
|
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|
391 |
definition map_ran :: "('key \<Rightarrow> 'val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where |
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|
392 |
"map_ran f = map (\<lambda>(k, v). (k, f k v))" |
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parents:
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changeset
|
393 |
|
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changeset
|
394 |
lemma map_ran_simps [simp]: |
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|
395 |
"map_ran f [] = []" |
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|
396 |
"map_ran f ((k, v) # ps) = (k, f k v) # map_ran f ps" |
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changeset
|
397 |
by (simp_all add: map_ran_def) |
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parents:
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diff
changeset
|
398 |
|
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parents:
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changeset
|
399 |
lemma dom_map_ran: |
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changeset
|
400 |
"fst ` set (map_ran f al) = fst ` set al" |
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changeset
|
401 |
by (simp add: map_ran_def image_image split_def) |
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parents:
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diff
changeset
|
402 |
|
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parents:
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changeset
|
403 |
lemma map_ran_conv: |
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changeset
|
404 |
"map_of (map_ran f al) k = Option.map (f k) (map_of al k)" |
19234 | 405 |
by (induct al) auto |
406 |
||
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|
407 |
lemma distinct_map_ran: |
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changeset
|
408 |
"distinct (map fst al) \<Longrightarrow> distinct (map fst (map_ran f al))" |
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changeset
|
409 |
by (simp add: map_ran_def split_def comp_def) |
19234 | 410 |
|
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|
411 |
lemma map_ran_filter: |
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parents:
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changeset
|
412 |
"map_ran f [p\<leftarrow>ps. fst p \<noteq> a] = [p\<leftarrow>map_ran f ps. fst p \<noteq> a]" |
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parents:
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changeset
|
413 |
by (simp add: map_ran_def filter_map split_def comp_def) |
19234 | 414 |
|
34975
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changeset
|
415 |
lemma clearjunk_map_ran: |
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parents:
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diff
changeset
|
416 |
"clearjunk (map_ran f al) = map_ran f (clearjunk al)" |
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haftmann
parents:
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diff
changeset
|
417 |
by (simp add: map_ran_def split_def clearjunk_map) |
19234 | 418 |
|
23373 | 419 |
|
34975
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changeset
|
420 |
subsection {* @{text merge} *} |
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parents:
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changeset
|
421 |
|
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parents:
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changeset
|
422 |
definition merge :: "('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where |
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parents:
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changeset
|
423 |
"merge qs ps = foldr (\<lambda>(k, v). update k v) ps qs" |
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parents:
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diff
changeset
|
424 |
|
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parents:
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diff
changeset
|
425 |
lemma merge_simps [simp]: |
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haftmann
parents:
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diff
changeset
|
426 |
"merge qs [] = qs" |
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parents:
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diff
changeset
|
427 |
"merge qs (p#ps) = update (fst p) (snd p) (merge qs ps)" |
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haftmann
parents:
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diff
changeset
|
428 |
by (simp_all add: merge_def split_def) |
f099b0b20646
more correspondence lemmas between related operations; tuned some proofs
haftmann
parents:
32960
diff
changeset
|
429 |
|
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haftmann
parents:
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diff
changeset
|
430 |
lemma merge_updates: |
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parents:
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diff
changeset
|
431 |
"merge qs ps = updates (rev (map fst ps)) (rev (map snd ps)) qs" |
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haftmann
parents:
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diff
changeset
|
432 |
by (simp add: merge_def updates_def split_def |
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haftmann
parents:
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diff
changeset
|
433 |
foldr_foldl zip_rev zip_map_fst_snd) |
19234 | 434 |
|
435 |
lemma dom_merge: "fst ` set (merge xs ys) = fst ` set xs \<union> fst ` set ys" |
|
20503 | 436 |
by (induct ys arbitrary: xs) (auto simp add: dom_update) |
19234 | 437 |
|
438 |
lemma distinct_merge: |
|
439 |
assumes "distinct (map fst xs)" |
|
440 |
shows "distinct (map fst (merge xs ys))" |
|
34975
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parents:
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diff
changeset
|
441 |
using assms by (simp add: merge_updates distinct_updates) |
19234 | 442 |
|
443 |
lemma clearjunk_merge: |
|
34975
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parents:
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diff
changeset
|
444 |
"clearjunk (merge xs ys) = merge (clearjunk xs) ys" |
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parents:
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diff
changeset
|
445 |
by (simp add: merge_updates clearjunk_updates) |
19234 | 446 |
|
34975
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parents:
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diff
changeset
|
447 |
lemma merge_conv': |
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parents:
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diff
changeset
|
448 |
"map_of (merge xs ys) = map_of xs ++ map_of ys" |
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haftmann
parents:
32960
diff
changeset
|
449 |
proof - |
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haftmann
parents:
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diff
changeset
|
450 |
have "foldl (\<lambda>m (k, v). m(k \<mapsto> v)) (map_of xs) (rev ys) = |
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haftmann
parents:
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diff
changeset
|
451 |
map_of (foldl (\<lambda>xs (k, v). update k v xs) xs (rev ys)) " |
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parents:
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diff
changeset
|
452 |
by (rule foldl_apply) (simp add: expand_fun_eq split_def update_conv') |
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haftmann
parents:
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diff
changeset
|
453 |
then show ?thesis |
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haftmann
parents:
32960
diff
changeset
|
454 |
by (simp add: merge_def map_add_map_of_foldr foldr_foldl split_def) |
19234 | 455 |
qed |
456 |
||
34975
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diff
changeset
|
457 |
corollary merge_conv: |
f099b0b20646
more correspondence lemmas between related operations; tuned some proofs
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parents:
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diff
changeset
|
458 |
"map_of (merge xs ys) k = (map_of xs ++ map_of ys) k" |
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parents:
32960
diff
changeset
|
459 |
by (simp add: merge_conv') |
19234 | 460 |
|
34975
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parents:
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diff
changeset
|
461 |
lemma merge_empty: "map_of (merge [] ys) = map_of ys" |
19234 | 462 |
by (simp add: merge_conv') |
463 |
||
464 |
lemma merge_assoc[simp]: "map_of (merge m1 (merge m2 m3)) = |
|
465 |
map_of (merge (merge m1 m2) m3)" |
|
466 |
by (simp add: merge_conv') |
|
467 |
||
468 |
lemma merge_Some_iff: |
|
469 |
"(map_of (merge m n) k = Some x) = |
|
470 |
(map_of n k = Some x \<or> map_of n k = None \<and> map_of m k = Some x)" |
|
471 |
by (simp add: merge_conv' map_add_Some_iff) |
|
472 |
||
34975
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parents:
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diff
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|
473 |
lemmas merge_SomeD [dest!] = merge_Some_iff [THEN iffD1, standard] |
19234 | 474 |
|
475 |
lemma merge_find_right[simp]: "map_of n k = Some v \<Longrightarrow> map_of (merge m n) k = Some v" |
|
476 |
by (simp add: merge_conv') |
|
477 |
||
478 |
lemma merge_None [iff]: |
|
479 |
"(map_of (merge m n) k = None) = (map_of n k = None \<and> map_of m k = None)" |
|
480 |
by (simp add: merge_conv') |
|
481 |
||
482 |
lemma merge_upd[simp]: |
|
483 |
"map_of (merge m (update k v n)) = map_of (update k v (merge m n))" |
|
484 |
by (simp add: update_conv' merge_conv') |
|
485 |
||
486 |
lemma merge_updatess[simp]: |
|
487 |
"map_of (merge m (updates xs ys n)) = map_of (updates xs ys (merge m n))" |
|
488 |
by (simp add: updates_conv' merge_conv') |
|
489 |
||
490 |
lemma merge_append: "map_of (xs@ys) = map_of (merge ys xs)" |
|
491 |
by (simp add: merge_conv') |
|
492 |
||
23373 | 493 |
|
34975
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more correspondence lemmas between related operations; tuned some proofs
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parents:
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|
494 |
subsection {* @{text compose} *} |
f099b0b20646
more correspondence lemmas between related operations; tuned some proofs
haftmann
parents:
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diff
changeset
|
495 |
|
f099b0b20646
more correspondence lemmas between related operations; tuned some proofs
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parents:
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changeset
|
496 |
function compose :: "('key \<times> 'a) list \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('key \<times> 'b) list" where |
f099b0b20646
more correspondence lemmas between related operations; tuned some proofs
haftmann
parents:
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diff
changeset
|
497 |
"compose [] ys = []" |
f099b0b20646
more correspondence lemmas between related operations; tuned some proofs
haftmann
parents:
32960
diff
changeset
|
498 |
| "compose (x#xs) ys = (case map_of ys (snd x) |
f099b0b20646
more correspondence lemmas between related operations; tuned some proofs
haftmann
parents:
32960
diff
changeset
|
499 |
of None \<Rightarrow> compose (delete (fst x) xs) ys |
f099b0b20646
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haftmann
parents:
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diff
changeset
|
500 |
| Some v \<Rightarrow> (fst x, v) # compose xs ys)" |
f099b0b20646
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haftmann
parents:
32960
diff
changeset
|
501 |
by pat_completeness auto |
f099b0b20646
more correspondence lemmas between related operations; tuned some proofs
haftmann
parents:
32960
diff
changeset
|
502 |
termination by (relation "measure (length \<circ> fst)") |
f099b0b20646
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haftmann
parents:
32960
diff
changeset
|
503 |
(simp_all add: less_Suc_eq_le length_delete_le) |
19234 | 504 |
|
505 |
lemma compose_first_None [simp]: |
|
506 |
assumes "map_of xs k = None" |
|
507 |
shows "map_of (compose xs ys) k = None" |
|
23373 | 508 |
using assms by (induct xs ys rule: compose.induct) |
22916 | 509 |
(auto split: option.splits split_if_asm) |
19234 | 510 |
|
511 |
lemma compose_conv: |
|
512 |
shows "map_of (compose xs ys) k = (map_of ys \<circ>\<^sub>m map_of xs) k" |
|
22916 | 513 |
proof (induct xs ys rule: compose.induct) |
514 |
case 1 then show ?case by simp |
|
19234 | 515 |
next |
22916 | 516 |
case (2 x xs ys) show ?case |
19234 | 517 |
proof (cases "map_of ys (snd x)") |
22916 | 518 |
case None with 2 |
19234 | 519 |
have hyp: "map_of (compose (delete (fst x) xs) ys) k = |
520 |
(map_of ys \<circ>\<^sub>m map_of (delete (fst x) xs)) k" |
|
521 |
by simp |
|
522 |
show ?thesis |
|
523 |
proof (cases "fst x = k") |
|
524 |
case True |
|
525 |
from True delete_notin_dom [of k xs] |
|
526 |
have "map_of (delete (fst x) xs) k = None" |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
30663
diff
changeset
|
527 |
by (simp add: map_of_eq_None_iff) |
19234 | 528 |
with hyp show ?thesis |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
30663
diff
changeset
|
529 |
using True None |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
30663
diff
changeset
|
530 |
by simp |
19234 | 531 |
next |
532 |
case False |
|
533 |
from False have "map_of (delete (fst x) xs) k = map_of xs k" |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
30663
diff
changeset
|
534 |
by simp |
19234 | 535 |
with hyp show ?thesis |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
30663
diff
changeset
|
536 |
using False None |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
30663
diff
changeset
|
537 |
by (simp add: map_comp_def) |
19234 | 538 |
qed |
539 |
next |
|
540 |
case (Some v) |
|
22916 | 541 |
with 2 |
19234 | 542 |
have "map_of (compose xs ys) k = (map_of ys \<circ>\<^sub>m map_of xs) k" |
543 |
by simp |
|
544 |
with Some show ?thesis |
|
545 |
by (auto simp add: map_comp_def) |
|
546 |
qed |
|
547 |
qed |
|
548 |
||
549 |
lemma compose_conv': |
|
550 |
shows "map_of (compose xs ys) = (map_of ys \<circ>\<^sub>m map_of xs)" |
|
551 |
by (rule ext) (rule compose_conv) |
|
552 |
||
553 |
lemma compose_first_Some [simp]: |
|
554 |
assumes "map_of xs k = Some v" |
|
555 |
shows "map_of (compose xs ys) k = map_of ys v" |
|
23373 | 556 |
using assms by (simp add: compose_conv) |
19234 | 557 |
|
558 |
lemma dom_compose: "fst ` set (compose xs ys) \<subseteq> fst ` set xs" |
|
22916 | 559 |
proof (induct xs ys rule: compose.induct) |
560 |
case 1 thus ?case by simp |
|
19234 | 561 |
next |
22916 | 562 |
case (2 x xs ys) |
19234 | 563 |
show ?case |
564 |
proof (cases "map_of ys (snd x)") |
|
565 |
case None |
|
22916 | 566 |
with "2.hyps" |
19234 | 567 |
have "fst ` set (compose (delete (fst x) xs) ys) \<subseteq> fst ` set (delete (fst x) xs)" |
568 |
by simp |
|
569 |
also |
|
570 |
have "\<dots> \<subseteq> fst ` set xs" |
|
571 |
by (rule dom_delete_subset) |
|
572 |
finally show ?thesis |
|
573 |
using None |
|
574 |
by auto |
|
575 |
next |
|
576 |
case (Some v) |
|
22916 | 577 |
with "2.hyps" |
19234 | 578 |
have "fst ` set (compose xs ys) \<subseteq> fst ` set xs" |
579 |
by simp |
|
580 |
with Some show ?thesis |
|
581 |
by auto |
|
582 |
qed |
|
583 |
qed |
|
584 |
||
585 |
lemma distinct_compose: |
|
586 |
assumes "distinct (map fst xs)" |
|
587 |
shows "distinct (map fst (compose xs ys))" |
|
23373 | 588 |
using assms |
22916 | 589 |
proof (induct xs ys rule: compose.induct) |
590 |
case 1 thus ?case by simp |
|
19234 | 591 |
next |
22916 | 592 |
case (2 x xs ys) |
19234 | 593 |
show ?case |
594 |
proof (cases "map_of ys (snd x)") |
|
595 |
case None |
|
22916 | 596 |
with 2 show ?thesis by simp |
19234 | 597 |
next |
598 |
case (Some v) |
|
22916 | 599 |
with 2 dom_compose [of xs ys] show ?thesis |
19234 | 600 |
by (auto) |
601 |
qed |
|
602 |
qed |
|
603 |
||
604 |
lemma compose_delete_twist: "(compose (delete k xs) ys) = delete k (compose xs ys)" |
|
22916 | 605 |
proof (induct xs ys rule: compose.induct) |
606 |
case 1 thus ?case by simp |
|
19234 | 607 |
next |
22916 | 608 |
case (2 x xs ys) |
19234 | 609 |
show ?case |
610 |
proof (cases "map_of ys (snd x)") |
|
611 |
case None |
|
22916 | 612 |
with 2 have |
19234 | 613 |
hyp: "compose (delete k (delete (fst x) xs)) ys = |
614 |
delete k (compose (delete (fst x) xs) ys)" |
|
615 |
by simp |
|
616 |
show ?thesis |
|
617 |
proof (cases "fst x = k") |
|
618 |
case True |
|
619 |
with None hyp |
|
620 |
show ?thesis |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
30663
diff
changeset
|
621 |
by (simp add: delete_idem) |
19234 | 622 |
next |
623 |
case False |
|
624 |
from None False hyp |
|
625 |
show ?thesis |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
30663
diff
changeset
|
626 |
by (simp add: delete_twist) |
19234 | 627 |
qed |
628 |
next |
|
629 |
case (Some v) |
|
22916 | 630 |
with 2 have hyp: "compose (delete k xs) ys = delete k (compose xs ys)" by simp |
19234 | 631 |
with Some show ?thesis |
632 |
by simp |
|
633 |
qed |
|
634 |
qed |
|
635 |
||
636 |
lemma compose_clearjunk: "compose xs (clearjunk ys) = compose xs ys" |
|
22916 | 637 |
by (induct xs ys rule: compose.induct) |
19234 | 638 |
(auto simp add: map_of_clearjunk split: option.splits) |
639 |
||
640 |
lemma clearjunk_compose: "clearjunk (compose xs ys) = compose (clearjunk xs) ys" |
|
641 |
by (induct xs rule: clearjunk.induct) |
|
642 |
(auto split: option.splits simp add: clearjunk_delete delete_idem |
|
643 |
compose_delete_twist) |
|
644 |
||
645 |
lemma compose_empty [simp]: |
|
646 |
"compose xs [] = []" |
|
22916 | 647 |
by (induct xs) (auto simp add: compose_delete_twist) |
19234 | 648 |
|
649 |
lemma compose_Some_iff: |
|
650 |
"(map_of (compose xs ys) k = Some v) = |
|
651 |
(\<exists>k'. map_of xs k = Some k' \<and> map_of ys k' = Some v)" |
|
652 |
by (simp add: compose_conv map_comp_Some_iff) |
|
653 |
||
654 |
lemma map_comp_None_iff: |
|
655 |
"(map_of (compose xs ys) k = None) = |
|
656 |
(map_of xs k = None \<or> (\<exists>k'. map_of xs k = Some k' \<and> map_of ys k' = None)) " |
|
657 |
by (simp add: compose_conv map_comp_None_iff) |
|
658 |
||
35156 | 659 |
|
660 |
subsection {* Implementation of mappings *} |
|
661 |
||
662 |
definition AList :: "('a \<times> 'b) list \<Rightarrow> ('a, 'b) mapping" where |
|
663 |
"AList xs = Mapping (map_of xs)" |
|
664 |
||
665 |
code_datatype AList |
|
666 |
||
667 |
lemma lookup_AList [simp, code]: |
|
668 |
"Mapping.lookup (AList xs) = map_of xs" |
|
669 |
by (simp add: AList_def) |
|
670 |
||
671 |
lemma empty_AList [code]: |
|
672 |
"Mapping.empty = AList []" |
|
673 |
by (rule mapping_eqI) simp |
|
674 |
||
675 |
lemma is_empty_AList [code]: |
|
676 |
"Mapping.is_empty (AList xs) \<longleftrightarrow> null xs" |
|
677 |
by (cases xs) (simp_all add: is_empty_def) |
|
678 |
||
679 |
lemma update_AList [code]: |
|
680 |
"Mapping.update k v (AList xs) = AList (update k v xs)" |
|
681 |
by (rule mapping_eqI) (simp add: update_conv') |
|
682 |
||
683 |
lemma delete_AList [code]: |
|
684 |
"Mapping.delete k (AList xs) = AList (delete k xs)" |
|
685 |
by (rule mapping_eqI) (simp add: delete_conv') |
|
686 |
||
687 |
lemma keys_AList [code]: |
|
688 |
"Mapping.keys (AList xs) = set (map fst xs)" |
|
689 |
by (simp add: keys_def dom_map_of_conv_image_fst) |
|
690 |
||
35194 | 691 |
lemma ordered_keys_AList [code]: |
692 |
"Mapping.ordered_keys (AList xs) = sort (remdups (map fst xs))" |
|
693 |
by (simp only: ordered_keys_def keys_AList sorted_list_of_set_sort_remdups) |
|
694 |
||
35156 | 695 |
lemma size_AList [code]: |
696 |
"Mapping.size (AList xs) = length (remdups (map fst xs))" |
|
697 |
by (simp add: size_def length_remdups_card_conv dom_map_of_conv_image_fst) |
|
698 |
||
699 |
lemma tabulate_AList [code]: |
|
700 |
"Mapping.tabulate ks f = AList (map (\<lambda>k. (k, f k)) ks)" |
|
701 |
by (rule mapping_eqI) (simp add: map_of_map_restrict) |
|
702 |
||
703 |
lemma bulkload_AList [code]: |
|
704 |
"Mapping.bulkload vs = AList (map (\<lambda>n. (n, vs ! n)) [0..<length vs])" |
|
705 |
by (rule mapping_eqI) (simp add: map_of_map_restrict expand_fun_eq) |
|
706 |
||
19234 | 707 |
end |