author | blanchet |
Wed, 24 Sep 2014 15:45:55 +0200 | |
changeset 58425 | 246985c6b20b |
parent 56327 | 3e62e68fb342 |
child 58881 | b9556a055632 |
permissions | -rw-r--r-- |
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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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header {* Implementation of Association Lists *} |
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theory AList |
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imports Main |
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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" |
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where |
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"update k v [] = [(k, v)]" |
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| "update k v (p # ps) = (if fst p = k then (k, v) # ps else p # update k v ps)" |
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lemma update_conv': "map_of (update k v al) = (map_of al)(k\<mapsto>v)" |
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by (induct al) (auto simp add: fun_eq_iff) |
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corollary update_conv: "map_of (update k v al) k' = ((map_of al)(k\<mapsto>v)) k'" |
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by (simp add: update_conv') |
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lemma dom_update: "fst ` set (update k v al) = {k} \<union> fst ` set al" |
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by (induct al) auto |
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lemma update_keys: |
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"map fst (update k v al) = |
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(if k \<in> set (map fst al) then map fst al else map fst al @ [k])" |
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by (induct al) simp_all |
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lemma distinct_update: |
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assumes "distinct (map fst al)" |
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shows "distinct (map fst (update k v al))" |
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using assms by (simp add: update_keys) |
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lemma update_filter: |
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"a \<noteq> k \<Longrightarrow> update k v [q\<leftarrow>ps. fst q \<noteq> a] = [q\<leftarrow>update k v ps. fst q \<noteq> a]" |
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by (induct ps) auto |
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lemma update_triv: "map_of al k = Some v \<Longrightarrow> update k v al = al" |
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by (induct al) auto |
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lemma update_nonempty [simp]: "update k v al \<noteq> []" |
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by (induct al) auto |
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lemma update_eqD: "update k v al = update k v' al' \<Longrightarrow> v = v'" |
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proof (induct al arbitrary: al') |
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case Nil |
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then show ?case |
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by (cases al') (auto split: split_if_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: 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: |
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"k \<noteq> k' \<Longrightarrow> |
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map_of (update k v (update k' v' al)) = map_of (update k' v' (update k v al))" |
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by (simp add: update_conv' fun_eq_iff) |
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lemma update_Some_unfold: |
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"map_of (update k v al) x = Some y \<longleftrightarrow> |
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x = k \<and> v = y \<or> x \<noteq> k \<and> map_of al x = Some y" |
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by (simp add: update_conv' map_upd_Some_unfold) |
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lemma image_update [simp]: |
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"x \<notin> A \<Longrightarrow> map_of (update x y al) ` A = map_of al ` A" |
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by (simp add: update_conv') |
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definition updates :: "'key list \<Rightarrow> 'val list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" |
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where "updates ks vs = fold (case_prod update) (zip ks vs)" |
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lemma updates_simps [simp]: |
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"updates [] vs ps = ps" |
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"updates ks [] ps = ps" |
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"updates (k#ks) (v#vs) ps = updates ks vs (update k v ps)" |
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by (simp_all add: updates_def) |
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lemma updates_key_simp [simp]: |
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"updates (k # ks) vs ps = |
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(case vs of [] \<Rightarrow> ps | v # vs \<Rightarrow> updates ks vs (update k v ps))" |
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by (cases vs) simp_all |
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lemma updates_conv': "map_of (updates ks vs al) = (map_of al)(ks[\<mapsto>]vs)" |
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proof - |
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have "map_of \<circ> 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 {* @{text delete} *} |
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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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202 |
by (induct al) simp_all |
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203 |
|
56327 | 204 |
lemma delete_update: "k \<noteq> l \<Longrightarrow> delete l (update k v al) = update k v (delete l al)" |
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205 |
by (induct al) simp_all |
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206 |
|
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207 |
lemma delete_twist: "delete x (delete y al) = delete y (delete x al)" |
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208 |
by (simp add: delete_eq conj_commute) |
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209 |
|
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210 |
lemma length_delete_le: "length (delete k al) \<le> length al" |
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211 |
by (simp add: delete_eq) |
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212 |
|
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213 |
|
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214 |
subsection {* @{text restrict} *} |
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215 |
|
56327 | 216 |
definition restrict :: "'key set \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" |
217 |
where restrict_eq: "restrict A = filter (\<lambda>(k, v). k \<in> A)" |
|
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218 |
|
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219 |
lemma restr_simps [simp]: |
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220 |
"restrict A [] = []" |
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221 |
"restrict A (p#ps) = (if fst p \<in> A then p # restrict A ps else restrict A ps)" |
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222 |
by (auto simp add: restrict_eq) |
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223 |
|
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224 |
lemma restr_conv': "map_of (restrict A al) = ((map_of al)|` A)" |
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225 |
proof |
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226 |
fix k |
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227 |
show "map_of (restrict A al) k = ((map_of al)|` A) k" |
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228 |
by (induct al) (simp, cases "k \<in> A", auto) |
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229 |
qed |
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230 |
|
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231 |
corollary restr_conv: "map_of (restrict A al) k = ((map_of al)|` A) k" |
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232 |
by (simp add: restr_conv') |
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233 |
|
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234 |
lemma distinct_restr: |
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235 |
"distinct (map fst al) \<Longrightarrow> distinct (map fst (restrict A al))" |
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236 |
by (induct al) (auto simp add: restrict_eq) |
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237 |
|
56327 | 238 |
lemma restr_empty [simp]: |
239 |
"restrict {} al = []" |
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240 |
"restrict A [] = []" |
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241 |
by (induct al) (auto simp add: restrict_eq) |
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242 |
|
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243 |
lemma restr_in [simp]: "x \<in> A \<Longrightarrow> map_of (restrict A al) x = map_of al x" |
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244 |
by (simp add: restr_conv') |
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245 |
|
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246 |
lemma restr_out [simp]: "x \<notin> A \<Longrightarrow> map_of (restrict A al) x = None" |
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247 |
by (simp add: restr_conv') |
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248 |
|
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249 |
lemma dom_restr [simp]: "fst ` set (restrict A al) = fst ` set al \<inter> A" |
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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_upd_same [simp]: "restrict (-{x}) (update x y al) = restrict (-{x}) 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_restr [simp]: "restrict A (restrict B al) = restrict (A\<inter>B) al" |
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256 |
by (induct al) (auto simp add: restrict_eq) |
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257 |
|
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258 |
lemma restr_update[simp]: |
56327 | 259 |
"map_of (restrict D (update x y al)) = |
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260 |
map_of ((if x \<in> D then (update x y (restrict (D-{x}) al)) else restrict D al))" |
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|
261 |
by (simp add: restr_conv' update_conv') |
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262 |
|
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263 |
lemma restr_delete [simp]: |
56327 | 264 |
"delete x (restrict D al) = (if x \<in> D then restrict (D - {x}) al else restrict D al)" |
265 |
apply (simp add: delete_eq restrict_eq) |
|
266 |
apply (auto simp add: split_def) |
|
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267 |
proof - |
56327 | 268 |
have "\<And>y. y \<noteq> x \<longleftrightarrow> x \<noteq> y" |
269 |
by auto |
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270 |
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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|
271 |
by simp |
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|
272 |
assume "x \<notin> D" |
56327 | 273 |
then have "\<And>y. y \<in> D \<longleftrightarrow> y \<in> D \<and> x \<noteq> y" |
274 |
by auto |
|
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|
275 |
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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|
276 |
by simp |
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|
277 |
qed |
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|
278 |
|
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|
279 |
lemma update_restr: |
56327 | 280 |
"map_of (update x y (restrict D al)) = map_of (update x y (restrict (D - {x}) al))" |
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|
281 |
by (simp add: update_conv' restr_conv') (rule fun_upd_restrict) |
19234 | 282 |
|
45867 | 283 |
lemma update_restr_conv [simp]: |
56327 | 284 |
"x \<in> D \<Longrightarrow> |
285 |
map_of (update x y (restrict D al)) = map_of (update x y (restrict (D - {x}) al))" |
|
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|
286 |
by (simp add: update_conv' restr_conv') |
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|
287 |
|
56327 | 288 |
lemma restr_updates [simp]: |
289 |
"length xs = length ys \<Longrightarrow> set xs \<subseteq> D \<Longrightarrow> |
|
290 |
map_of (restrict D (updates xs ys al)) = |
|
291 |
map_of (updates xs ys (restrict (D - set xs) al))" |
|
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|
292 |
by (simp add: updates_conv' restr_conv') |
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|
293 |
|
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|
294 |
lemma restr_delete_twist: "(restrict A (delete a ps)) = delete a (restrict A ps)" |
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|
295 |
by (induct ps) auto |
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|
296 |
|
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|
297 |
|
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|
298 |
subsection {* @{text clearjunk} *} |
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|
299 |
|
56327 | 300 |
function clearjunk :: "('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" |
301 |
where |
|
302 |
"clearjunk [] = []" |
|
303 |
| "clearjunk (p#ps) = p # clearjunk (delete (fst p) ps)" |
|
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304 |
by pat_completeness auto |
56327 | 305 |
termination |
306 |
by (relation "measure length") (simp_all add: less_Suc_eq_le length_delete_le) |
|
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|
307 |
|
56327 | 308 |
lemma map_of_clearjunk: "map_of (clearjunk al) = map_of al" |
309 |
by (induct al rule: clearjunk.induct) (simp_all add: fun_eq_iff) |
|
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|
310 |
|
56327 | 311 |
lemma clearjunk_keys_set: "set (map fst (clearjunk al)) = set (map fst al)" |
312 |
by (induct al rule: clearjunk.induct) (simp_all add: delete_keys) |
|
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313 |
|
56327 | 314 |
lemma dom_clearjunk: "fst ` set (clearjunk al) = fst ` set al" |
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|
315 |
using clearjunk_keys_set by simp |
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|
316 |
|
56327 | 317 |
lemma distinct_clearjunk [simp]: "distinct (map fst (clearjunk al))" |
318 |
by (induct al rule: clearjunk.induct) (simp_all del: set_map add: clearjunk_keys_set delete_keys) |
|
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|
319 |
|
56327 | 320 |
lemma ran_clearjunk: "ran (map_of (clearjunk al)) = ran (map_of al)" |
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|
321 |
by (simp add: map_of_clearjunk) |
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|
322 |
|
56327 | 323 |
lemma ran_map_of: "ran (map_of al) = snd ` set (clearjunk al)" |
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|
324 |
proof - |
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|
325 |
have "ran (map_of al) = ran (map_of (clearjunk al))" |
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|
326 |
by (simp add: ran_clearjunk) |
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|
327 |
also have "\<dots> = snd ` set (clearjunk al)" |
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changeset
|
328 |
by (simp add: ran_distinct) |
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|
329 |
finally show ?thesis . |
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|
330 |
qed |
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|
331 |
|
56327 | 332 |
lemma clearjunk_update: "clearjunk (update k v al) = update k v (clearjunk al)" |
333 |
by (induct al rule: clearjunk.induct) (simp_all add: delete_update) |
|
19234 | 334 |
|
56327 | 335 |
lemma clearjunk_updates: "clearjunk (updates ks vs al) = updates ks vs (clearjunk al)" |
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|
336 |
proof - |
55414
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changeset
|
337 |
have "clearjunk \<circ> fold (case_prod update) (zip ks vs) = |
eab03e9cee8a
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parents:
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changeset
|
338 |
fold (case_prod update) (zip ks vs) \<circ> clearjunk" |
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|
339 |
by (rule fold_commute) (simp add: clearjunk_update case_prod_beta o_def) |
56327 | 340 |
then show ?thesis |
341 |
by (simp add: updates_def fun_eq_iff) |
|
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|
342 |
qed |
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|
343 |
|
56327 | 344 |
lemma clearjunk_delete: "clearjunk (delete x al) = delete x (clearjunk al)" |
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|
345 |
by (induct al rule: clearjunk.induct) (auto simp add: delete_idem delete_twist) |
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|
346 |
|
56327 | 347 |
lemma clearjunk_restrict: "clearjunk (restrict A al) = restrict A (clearjunk al)" |
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|
348 |
by (induct al rule: clearjunk.induct) (auto simp add: restr_delete_twist) |
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|
349 |
|
56327 | 350 |
lemma distinct_clearjunk_id [simp]: "distinct (map fst al) \<Longrightarrow> clearjunk al = al" |
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|
351 |
by (induct al rule: clearjunk.induct) auto |
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changeset
|
352 |
|
56327 | 353 |
lemma clearjunk_idem: "clearjunk (clearjunk al) = clearjunk al" |
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|
354 |
by simp |
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|
355 |
|
56327 | 356 |
lemma length_clearjunk: "length (clearjunk al) \<le> length al" |
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|
357 |
proof (induct al rule: clearjunk.induct [case_names Nil Cons]) |
56327 | 358 |
case Nil |
359 |
then show ?case by simp |
|
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|
360 |
next |
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|
361 |
case (Cons kv al) |
56327 | 362 |
moreover have "length (delete (fst kv) al) \<le> length al" |
363 |
by (fact length_delete_le) |
|
364 |
ultimately have "length (clearjunk (delete (fst kv) al)) \<le> length al" |
|
365 |
by (rule order_trans) |
|
366 |
then show ?case |
|
367 |
by simp |
|
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|
368 |
qed |
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changeset
|
369 |
|
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|
370 |
lemma delete_map: |
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|
371 |
assumes "\<And>kv. fst (f kv) = fst kv" |
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changeset
|
372 |
shows "delete k (map f ps) = map f (delete k ps)" |
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changeset
|
373 |
by (simp add: delete_eq filter_map comp_def split_def assms) |
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changeset
|
374 |
|
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|
375 |
lemma clearjunk_map: |
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|
376 |
assumes "\<And>kv. fst (f kv) = fst kv" |
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changeset
|
377 |
shows "clearjunk (map f ps) = map f (clearjunk ps)" |
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changeset
|
378 |
by (induct ps rule: clearjunk.induct [case_names Nil Cons]) |
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changeset
|
379 |
(simp_all add: clearjunk_delete delete_map assms) |
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changeset
|
380 |
|
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changeset
|
381 |
|
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changeset
|
382 |
subsection {* @{text map_ran} *} |
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|
383 |
|
56327 | 384 |
definition map_ran :: "('key \<Rightarrow> 'val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" |
385 |
where "map_ran f = map (\<lambda>(k, v). (k, f k v))" |
|
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changeset
|
386 |
|
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|
387 |
lemma map_ran_simps [simp]: |
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|
388 |
"map_ran f [] = []" |
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|
389 |
"map_ran f ((k, v) # ps) = (k, f k v) # map_ran f ps" |
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changeset
|
390 |
by (simp_all add: map_ran_def) |
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changeset
|
391 |
|
56327 | 392 |
lemma dom_map_ran: "fst ` set (map_ran f al) = fst ` set al" |
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|
393 |
by (simp add: map_ran_def image_image split_def) |
56327 | 394 |
|
395 |
lemma map_ran_conv: "map_of (map_ran f al) k = map_option (f k) (map_of al k)" |
|
19234 | 396 |
by (induct al) auto |
397 |
||
56327 | 398 |
lemma distinct_map_ran: "distinct (map fst al) \<Longrightarrow> distinct (map fst (map_ran f al))" |
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changeset
|
399 |
by (simp add: map_ran_def split_def comp_def) |
19234 | 400 |
|
56327 | 401 |
lemma map_ran_filter: "map_ran f [p\<leftarrow>ps. fst p \<noteq> a] = [p\<leftarrow>map_ran f ps. fst p \<noteq> a]" |
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changeset
|
402 |
by (simp add: map_ran_def filter_map split_def comp_def) |
19234 | 403 |
|
56327 | 404 |
lemma clearjunk_map_ran: "clearjunk (map_ran f al) = map_ran f (clearjunk al)" |
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changeset
|
405 |
by (simp add: map_ran_def split_def clearjunk_map) |
19234 | 406 |
|
23373 | 407 |
|
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|
408 |
subsection {* @{text merge} *} |
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|
409 |
|
56327 | 410 |
definition merge :: "('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" |
411 |
where "merge qs ps = foldr (\<lambda>(k, v). update k v) ps qs" |
|
34975
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changeset
|
412 |
|
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|
413 |
lemma merge_simps [simp]: |
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changeset
|
414 |
"merge qs [] = qs" |
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changeset
|
415 |
"merge qs (p#ps) = update (fst p) (snd p) (merge qs ps)" |
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changeset
|
416 |
by (simp_all add: merge_def split_def) |
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changeset
|
417 |
|
56327 | 418 |
lemma merge_updates: "merge qs ps = updates (rev (map fst ps)) (rev (map snd ps)) qs" |
47397
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parents:
46507
diff
changeset
|
419 |
by (simp add: merge_def updates_def foldr_conv_fold zip_rev zip_map_fst_snd) |
19234 | 420 |
|
421 |
lemma dom_merge: "fst ` set (merge xs ys) = fst ` set xs \<union> fst ` set ys" |
|
20503 | 422 |
by (induct ys arbitrary: xs) (auto simp add: dom_update) |
19234 | 423 |
|
424 |
lemma distinct_merge: |
|
425 |
assumes "distinct (map fst xs)" |
|
426 |
shows "distinct (map fst (merge xs ys))" |
|
56327 | 427 |
using assms by (simp add: merge_updates distinct_updates) |
19234 | 428 |
|
56327 | 429 |
lemma clearjunk_merge: "clearjunk (merge xs ys) = merge (clearjunk xs) ys" |
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changeset
|
430 |
by (simp add: merge_updates clearjunk_updates) |
19234 | 431 |
|
56327 | 432 |
lemma merge_conv': "map_of (merge xs ys) = map_of xs ++ map_of ys" |
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changeset
|
433 |
proof - |
55414
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blanchet
parents:
47397
diff
changeset
|
434 |
have "map_of \<circ> fold (case_prod update) (rev ys) = |
56327 | 435 |
fold (\<lambda>(k, v) m. m(k \<mapsto> v)) (rev ys) \<circ> map_of" |
55414
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blanchet
parents:
47397
diff
changeset
|
436 |
by (rule fold_commute) (simp add: update_conv' case_prod_beta split_def fun_eq_iff) |
34975
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changeset
|
437 |
then show ?thesis |
47397
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no preference wrt. fold(l/r); prefer fold rather than foldr for iterating over lists in generated code
haftmann
parents:
46507
diff
changeset
|
438 |
by (simp add: merge_def map_add_map_of_foldr foldr_conv_fold fun_eq_iff) |
19234 | 439 |
qed |
440 |
||
56327 | 441 |
corollary merge_conv: "map_of (merge xs ys) k = (map_of xs ++ map_of ys) k" |
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changeset
|
442 |
by (simp add: merge_conv') |
19234 | 443 |
|
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|
444 |
lemma merge_empty: "map_of (merge [] ys) = map_of ys" |
19234 | 445 |
by (simp add: merge_conv') |
446 |
||
56327 | 447 |
lemma merge_assoc [simp]: "map_of (merge m1 (merge m2 m3)) = map_of (merge (merge m1 m2) m3)" |
19234 | 448 |
by (simp add: merge_conv') |
449 |
||
56327 | 450 |
lemma merge_Some_iff: |
451 |
"map_of (merge m n) k = Some x \<longleftrightarrow> |
|
452 |
map_of n k = Some x \<or> map_of n k = None \<and> map_of m k = Some x" |
|
19234 | 453 |
by (simp add: merge_conv' map_add_Some_iff) |
454 |
||
45605 | 455 |
lemmas merge_SomeD [dest!] = merge_Some_iff [THEN iffD1] |
19234 | 456 |
|
56327 | 457 |
lemma merge_find_right [simp]: "map_of n k = Some v \<Longrightarrow> map_of (merge m n) k = Some v" |
19234 | 458 |
by (simp add: merge_conv') |
459 |
||
56327 | 460 |
lemma merge_None [iff]: |
19234 | 461 |
"(map_of (merge m n) k = None) = (map_of n k = None \<and> map_of m k = None)" |
462 |
by (simp add: merge_conv') |
|
463 |
||
56327 | 464 |
lemma merge_upd [simp]: |
19234 | 465 |
"map_of (merge m (update k v n)) = map_of (update k v (merge m n))" |
466 |
by (simp add: update_conv' merge_conv') |
|
467 |
||
56327 | 468 |
lemma merge_updatess [simp]: |
19234 | 469 |
"map_of (merge m (updates xs ys n)) = map_of (updates xs ys (merge m n))" |
470 |
by (simp add: updates_conv' merge_conv') |
|
471 |
||
56327 | 472 |
lemma merge_append: "map_of (xs @ ys) = map_of (merge ys xs)" |
19234 | 473 |
by (simp add: merge_conv') |
474 |
||
23373 | 475 |
|
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|
476 |
subsection {* @{text compose} *} |
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parents:
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changeset
|
477 |
|
56327 | 478 |
function compose :: "('key \<times> 'a) list \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('key \<times> 'b) list" |
479 |
where |
|
480 |
"compose [] ys = []" |
|
481 |
| "compose (x # xs) ys = |
|
482 |
(case map_of ys (snd x) of |
|
483 |
None \<Rightarrow> compose (delete (fst x) xs) ys |
|
484 |
| Some v \<Rightarrow> (fst x, v) # compose xs ys)" |
|
34975
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parents:
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diff
changeset
|
485 |
by pat_completeness auto |
56327 | 486 |
termination |
487 |
by (relation "measure (length \<circ> fst)") (simp_all add: less_Suc_eq_le length_delete_le) |
|
19234 | 488 |
|
56327 | 489 |
lemma compose_first_None [simp]: |
490 |
assumes "map_of xs k = None" |
|
19234 | 491 |
shows "map_of (compose xs ys) k = None" |
56327 | 492 |
using assms by (induct xs ys rule: compose.induct) (auto split: option.splits split_if_asm) |
19234 | 493 |
|
56327 | 494 |
lemma compose_conv: "map_of (compose xs ys) k = (map_of ys \<circ>\<^sub>m map_of xs) k" |
22916 | 495 |
proof (induct xs ys rule: compose.induct) |
56327 | 496 |
case 1 |
497 |
then show ?case by simp |
|
19234 | 498 |
next |
56327 | 499 |
case (2 x xs ys) |
500 |
show ?case |
|
19234 | 501 |
proof (cases "map_of ys (snd x)") |
56327 | 502 |
case None |
503 |
with 2 have hyp: "map_of (compose (delete (fst x) xs) ys) k = |
|
504 |
(map_of ys \<circ>\<^sub>m map_of (delete (fst x) xs)) k" |
|
19234 | 505 |
by simp |
506 |
show ?thesis |
|
507 |
proof (cases "fst x = k") |
|
508 |
case True |
|
509 |
from True delete_notin_dom [of k xs] |
|
510 |
have "map_of (delete (fst x) xs) k = None" |
|
32960
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wenzelm
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30663
diff
changeset
|
511 |
by (simp add: map_of_eq_None_iff) |
19234 | 512 |
with hyp show ?thesis |
32960
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eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
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30663
diff
changeset
|
513 |
using True None |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
30663
diff
changeset
|
514 |
by simp |
19234 | 515 |
next |
516 |
case False |
|
517 |
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 tab-width;
wenzelm
parents:
30663
diff
changeset
|
518 |
by simp |
19234 | 519 |
with hyp show ?thesis |
56327 | 520 |
using False None by (simp add: map_comp_def) |
19234 | 521 |
qed |
522 |
next |
|
523 |
case (Some v) |
|
22916 | 524 |
with 2 |
19234 | 525 |
have "map_of (compose xs ys) k = (map_of ys \<circ>\<^sub>m map_of xs) k" |
526 |
by simp |
|
527 |
with Some show ?thesis |
|
528 |
by (auto simp add: map_comp_def) |
|
529 |
qed |
|
530 |
qed |
|
56327 | 531 |
|
532 |
lemma compose_conv': "map_of (compose xs ys) = (map_of ys \<circ>\<^sub>m map_of xs)" |
|
19234 | 533 |
by (rule ext) (rule compose_conv) |
534 |
||
535 |
lemma compose_first_Some [simp]: |
|
56327 | 536 |
assumes "map_of xs k = Some v" |
19234 | 537 |
shows "map_of (compose xs ys) k = map_of ys v" |
56327 | 538 |
using assms by (simp add: compose_conv) |
19234 | 539 |
|
540 |
lemma dom_compose: "fst ` set (compose xs ys) \<subseteq> fst ` set xs" |
|
22916 | 541 |
proof (induct xs ys rule: compose.induct) |
56327 | 542 |
case 1 |
543 |
then show ?case by simp |
|
19234 | 544 |
next |
22916 | 545 |
case (2 x xs ys) |
19234 | 546 |
show ?case |
547 |
proof (cases "map_of ys (snd x)") |
|
548 |
case None |
|
22916 | 549 |
with "2.hyps" |
19234 | 550 |
have "fst ` set (compose (delete (fst x) xs) ys) \<subseteq> fst ` set (delete (fst x) xs)" |
551 |
by simp |
|
552 |
also |
|
553 |
have "\<dots> \<subseteq> fst ` set xs" |
|
554 |
by (rule dom_delete_subset) |
|
555 |
finally show ?thesis |
|
556 |
using None |
|
557 |
by auto |
|
558 |
next |
|
559 |
case (Some v) |
|
22916 | 560 |
with "2.hyps" |
19234 | 561 |
have "fst ` set (compose xs ys) \<subseteq> fst ` set xs" |
562 |
by simp |
|
563 |
with Some show ?thesis |
|
564 |
by auto |
|
565 |
qed |
|
566 |
qed |
|
567 |
||
568 |
lemma distinct_compose: |
|
56327 | 569 |
assumes "distinct (map fst xs)" |
570 |
shows "distinct (map fst (compose xs ys))" |
|
571 |
using assms |
|
22916 | 572 |
proof (induct xs ys rule: compose.induct) |
56327 | 573 |
case 1 |
574 |
then show ?case by simp |
|
19234 | 575 |
next |
22916 | 576 |
case (2 x xs ys) |
19234 | 577 |
show ?case |
578 |
proof (cases "map_of ys (snd x)") |
|
579 |
case None |
|
22916 | 580 |
with 2 show ?thesis by simp |
19234 | 581 |
next |
582 |
case (Some v) |
|
56327 | 583 |
with 2 dom_compose [of xs ys] show ?thesis |
584 |
by auto |
|
19234 | 585 |
qed |
586 |
qed |
|
587 |
||
56327 | 588 |
lemma compose_delete_twist: "compose (delete k xs) ys = delete k (compose xs ys)" |
22916 | 589 |
proof (induct xs ys rule: compose.induct) |
56327 | 590 |
case 1 |
591 |
then show ?case by simp |
|
19234 | 592 |
next |
22916 | 593 |
case (2 x xs ys) |
19234 | 594 |
show ?case |
595 |
proof (cases "map_of ys (snd x)") |
|
596 |
case None |
|
56327 | 597 |
with 2 have hyp: "compose (delete k (delete (fst x) xs)) ys = |
598 |
delete k (compose (delete (fst x) xs) ys)" |
|
19234 | 599 |
by simp |
600 |
show ?thesis |
|
601 |
proof (cases "fst x = k") |
|
602 |
case True |
|
56327 | 603 |
with None hyp show ?thesis |
32960
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wenzelm
parents:
30663
diff
changeset
|
604 |
by (simp add: delete_idem) |
19234 | 605 |
next |
606 |
case False |
|
56327 | 607 |
from None False hyp show ?thesis |
32960
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eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
30663
diff
changeset
|
608 |
by (simp add: delete_twist) |
19234 | 609 |
qed |
610 |
next |
|
611 |
case (Some v) |
|
56327 | 612 |
with 2 have hyp: "compose (delete k xs) ys = delete k (compose xs ys)" |
613 |
by simp |
|
19234 | 614 |
with Some show ?thesis |
615 |
by simp |
|
616 |
qed |
|
617 |
qed |
|
618 |
||
619 |
lemma compose_clearjunk: "compose xs (clearjunk ys) = compose xs ys" |
|
56327 | 620 |
by (induct xs ys rule: compose.induct) |
621 |
(auto simp add: map_of_clearjunk split: option.splits) |
|
622 |
||
19234 | 623 |
lemma clearjunk_compose: "clearjunk (compose xs ys) = compose (clearjunk xs) ys" |
624 |
by (induct xs rule: clearjunk.induct) |
|
56327 | 625 |
(auto split: option.splits simp add: clearjunk_delete delete_idem compose_delete_twist) |
626 |
||
627 |
lemma compose_empty [simp]: "compose xs [] = []" |
|
22916 | 628 |
by (induct xs) (auto simp add: compose_delete_twist) |
19234 | 629 |
|
630 |
lemma compose_Some_iff: |
|
56327 | 631 |
"(map_of (compose xs ys) k = Some v) \<longleftrightarrow> |
632 |
(\<exists>k'. map_of xs k = Some k' \<and> map_of ys k' = Some v)" |
|
19234 | 633 |
by (simp add: compose_conv map_comp_Some_iff) |
634 |
||
635 |
lemma map_comp_None_iff: |
|
56327 | 636 |
"map_of (compose xs ys) k = None \<longleftrightarrow> |
637 |
(map_of xs k = None \<or> (\<exists>k'. map_of xs k = Some k' \<and> map_of ys k' = None))" |
|
19234 | 638 |
by (simp add: compose_conv map_comp_None_iff) |
639 |
||
56327 | 640 |
|
45869 | 641 |
subsection {* @{text map_entry} *} |
642 |
||
643 |
fun map_entry :: "'key \<Rightarrow> ('val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" |
|
644 |
where |
|
645 |
"map_entry k f [] = []" |
|
56327 | 646 |
| "map_entry k f (p # ps) = |
647 |
(if fst p = k then (k, f (snd p)) # ps else p # map_entry k f ps)" |
|
45869 | 648 |
|
649 |
lemma map_of_map_entry: |
|
56327 | 650 |
"map_of (map_entry k f xs) = |
651 |
(map_of xs)(k := case map_of xs k of None \<Rightarrow> None | Some v' \<Rightarrow> Some (f v'))" |
|
652 |
by (induct xs) auto |
|
45869 | 653 |
|
56327 | 654 |
lemma dom_map_entry: "fst ` set (map_entry k f xs) = fst ` set xs" |
655 |
by (induct xs) auto |
|
45869 | 656 |
|
657 |
lemma distinct_map_entry: |
|
658 |
assumes "distinct (map fst xs)" |
|
659 |
shows "distinct (map fst (map_entry k f xs))" |
|
56327 | 660 |
using assms by (induct xs) (auto simp add: dom_map_entry) |
661 |
||
45869 | 662 |
|
45868 | 663 |
subsection {* @{text map_default} *} |
664 |
||
665 |
fun map_default :: "'key \<Rightarrow> 'val \<Rightarrow> ('val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" |
|
666 |
where |
|
667 |
"map_default k v f [] = [(k, v)]" |
|
56327 | 668 |
| "map_default k v f (p # ps) = |
669 |
(if fst p = k then (k, f (snd p)) # ps else p # map_default k v f ps)" |
|
45868 | 670 |
|
671 |
lemma map_of_map_default: |
|
56327 | 672 |
"map_of (map_default k v f xs) = |
673 |
(map_of xs)(k := case map_of xs k of None \<Rightarrow> Some v | Some v' \<Rightarrow> Some (f v'))" |
|
674 |
by (induct xs) auto |
|
45868 | 675 |
|
56327 | 676 |
lemma dom_map_default: "fst ` set (map_default k v f xs) = insert k (fst ` set xs)" |
677 |
by (induct xs) auto |
|
45868 | 678 |
|
679 |
lemma distinct_map_default: |
|
680 |
assumes "distinct (map fst xs)" |
|
681 |
shows "distinct (map fst (map_default k v f xs))" |
|
56327 | 682 |
using assms by (induct xs) (auto simp add: dom_map_default) |
45868 | 683 |
|
46171
19f68d7671f0
proper hiding of facts and constants in AList_Impl and AList theory
bulwahn
parents:
46133
diff
changeset
|
684 |
hide_const (open) update updates delete restrict clearjunk merge compose map_entry |
45884 | 685 |
|
19234 | 686 |
end |