author | Lars Hupel <lars.hupel@mytum.de> |
Mon, 18 Jun 2018 10:50:24 +0200 | |
changeset 68462 | 8d1bf38c6fe6 |
parent 67408 | 4a4c14b24800 |
child 69064 | 5840724b1d71 |
permissions | -rw-r--r-- |
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(* Title: HOL/Library/DAList_Multiset.thy |
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Author: Lukas Bulwahn, TU Muenchen |
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*) |
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section \<open>Multisets partially implemented by association lists\<close> |
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theory DAList_Multiset |
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imports Multiset DAList |
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begin |
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text \<open>Delete prexisting code equations\<close> |
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declare [[code drop: "{#}" Multiset.is_empty add_mset |
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"plus :: 'a multiset \<Rightarrow> _" "minus :: 'a multiset \<Rightarrow> _" |
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inf_subset_mset sup_subset_mset image_mset filter_mset count |
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"size :: _ multiset \<Rightarrow> nat" sum_mset prod_mset |
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set_mset sorted_list_of_multiset subset_mset subseteq_mset |
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equal_multiset_inst.equal_multiset]] |
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text \<open>Raw operations on lists\<close> |
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definition join_raw :: |
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"('key \<Rightarrow> 'val \<times> 'val \<Rightarrow> 'val) \<Rightarrow> |
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('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" |
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where "join_raw f xs ys = foldr (\<lambda>(k, v). map_default k v (\<lambda>v'. f k (v', v))) ys xs" |
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lemma join_raw_Nil [simp]: "join_raw f xs [] = xs" |
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by (simp add: join_raw_def) |
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lemma join_raw_Cons [simp]: |
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"join_raw f xs ((k, v) # ys) = map_default k v (\<lambda>v'. f k (v', v)) (join_raw f xs ys)" |
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by (simp add: join_raw_def) |
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lemma map_of_join_raw: |
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assumes "distinct (map fst ys)" |
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shows "map_of (join_raw f xs ys) x = |
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(case map_of xs x of |
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None \<Rightarrow> map_of ys x |
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| Some v \<Rightarrow> (case map_of ys x of None \<Rightarrow> Some v | Some v' \<Rightarrow> Some (f x (v, v'))))" |
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using assms |
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apply (induct ys) |
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apply (auto simp add: map_of_map_default split: option.split) |
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apply (metis map_of_eq_None_iff option.simps(2) weak_map_of_SomeI) |
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apply (metis Some_eq_map_of_iff map_of_eq_None_iff option.simps(2)) |
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done |
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lemma distinct_join_raw: |
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assumes "distinct (map fst xs)" |
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shows "distinct (map fst (join_raw f xs ys))" |
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using assms |
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proof (induct ys) |
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case Nil |
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then show ?case by simp |
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next |
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case (Cons y ys) |
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then show ?case by (cases y) (simp add: distinct_map_default) |
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qed |
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definition "subtract_entries_raw xs ys = foldr (\<lambda>(k, v). AList.map_entry k (\<lambda>v'. v' - v)) ys xs" |
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lemma map_of_subtract_entries_raw: |
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assumes "distinct (map fst ys)" |
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shows "map_of (subtract_entries_raw xs ys) x = |
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(case map_of xs x of |
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None \<Rightarrow> None |
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| Some v \<Rightarrow> (case map_of ys x of None \<Rightarrow> Some v | Some v' \<Rightarrow> Some (v - v')))" |
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using assms |
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unfolding subtract_entries_raw_def |
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apply (induct ys) |
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apply auto |
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apply (simp split: option.split) |
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apply (simp add: map_of_map_entry) |
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apply (auto split: option.split) |
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apply (metis map_of_eq_None_iff option.simps(3) option.simps(4)) |
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apply (metis map_of_eq_None_iff option.simps(4) option.simps(5)) |
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done |
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lemma distinct_subtract_entries_raw: |
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assumes "distinct (map fst xs)" |
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shows "distinct (map fst (subtract_entries_raw xs ys))" |
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using assms |
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unfolding subtract_entries_raw_def |
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by (induct ys) (auto simp add: distinct_map_entry) |
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text \<open>Operations on alists with distinct keys\<close> |
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lift_definition join :: "('a \<Rightarrow> 'b \<times> 'b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist" |
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is join_raw |
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by (simp add: distinct_join_raw) |
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lift_definition subtract_entries :: "('a, ('b :: minus)) alist \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist" |
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is subtract_entries_raw |
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by (simp add: distinct_subtract_entries_raw) |
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text \<open>Implementing multisets by means of association lists\<close> |
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definition count_of :: "('a \<times> nat) list \<Rightarrow> 'a \<Rightarrow> nat" |
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where "count_of xs x = (case map_of xs x of None \<Rightarrow> 0 | Some n \<Rightarrow> n)" |
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lemma count_of_multiset: "count_of xs \<in> multiset" |
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proof - |
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let ?A = "{x::'a. 0 < (case map_of xs x of None \<Rightarrow> 0::nat | Some n \<Rightarrow> n)}" |
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have "?A \<subseteq> dom (map_of xs)" |
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proof |
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fix x |
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assume "x \<in> ?A" |
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then have "0 < (case map_of xs x of None \<Rightarrow> 0::nat | Some n \<Rightarrow> n)" |
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by simp |
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then have "map_of xs x \<noteq> None" |
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by (cases "map_of xs x") auto |
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then show "x \<in> dom (map_of xs)" |
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by auto |
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qed |
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with finite_dom_map_of [of xs] have "finite ?A" |
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by (auto intro: finite_subset) |
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then show ?thesis |
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by (simp add: count_of_def fun_eq_iff multiset_def) |
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qed |
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lemma count_simps [simp]: |
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"count_of [] = (\<lambda>_. 0)" |
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"count_of ((x, n) # xs) = (\<lambda>y. if x = y then n else count_of xs y)" |
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by (simp_all add: count_of_def fun_eq_iff) |
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lemma count_of_empty: "x \<notin> fst ` set xs \<Longrightarrow> count_of xs x = 0" |
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by (induct xs) (simp_all add: count_of_def) |
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lemma count_of_filter: "count_of (List.filter (P \<circ> fst) xs) x = (if P x then count_of xs x else 0)" |
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by (induct xs) auto |
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lemma count_of_map_default [simp]: |
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"count_of (map_default x b (\<lambda>x. x + b) xs) y = |
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(if x = y then count_of xs x + b else count_of xs y)" |
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unfolding count_of_def by (simp add: map_of_map_default split: option.split) |
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lemma count_of_join_raw: |
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"distinct (map fst ys) \<Longrightarrow> |
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count_of xs x + count_of ys x = count_of (join_raw (\<lambda>x (x, y). x + y) xs ys) x" |
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unfolding count_of_def by (simp add: map_of_join_raw split: option.split) |
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lemma count_of_subtract_entries_raw: |
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"distinct (map fst ys) \<Longrightarrow> |
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count_of xs x - count_of ys x = count_of (subtract_entries_raw xs ys) x" |
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unfolding count_of_def by (simp add: map_of_subtract_entries_raw split: option.split) |
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text \<open>Code equations for multiset operations\<close> |
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definition Bag :: "('a, nat) alist \<Rightarrow> 'a multiset" |
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where "Bag xs = Abs_multiset (count_of (DAList.impl_of xs))" |
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code_datatype Bag |
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lemma count_Bag [simp, code]: "count (Bag xs) = count_of (DAList.impl_of xs)" |
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by (simp add: Bag_def count_of_multiset) |
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lemma Mempty_Bag [code]: "{#} = Bag (DAList.empty)" |
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by (simp add: multiset_eq_iff alist.Alist_inverse DAList.empty_def) |
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lift_definition is_empty_Bag_impl :: "('a, nat) alist \<Rightarrow> bool" is |
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"\<lambda>xs. list_all (\<lambda>x. snd x = 0) xs" . |
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lemma is_empty_Bag [code]: "Multiset.is_empty (Bag xs) \<longleftrightarrow> is_empty_Bag_impl xs" |
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proof - |
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have "Multiset.is_empty (Bag xs) \<longleftrightarrow> (\<forall>x. count (Bag xs) x = 0)" |
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unfolding Multiset.is_empty_def multiset_eq_iff by simp |
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also have "\<dots> \<longleftrightarrow> (\<forall>x\<in>fst ` set (alist.impl_of xs). count (Bag xs) x = 0)" |
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proof (intro iffI allI ballI) |
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fix x assume A: "\<forall>x\<in>fst ` set (alist.impl_of xs). count (Bag xs) x = 0" |
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thus "count (Bag xs) x = 0" |
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proof (cases "x \<in> fst ` set (alist.impl_of xs)") |
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case False |
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thus ?thesis by (force simp: count_of_def split: option.splits) |
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qed (insert A, auto) |
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qed simp_all |
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also have "\<dots> \<longleftrightarrow> list_all (\<lambda>x. snd x = 0) (alist.impl_of xs)" |
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by (auto simp: count_of_def list_all_def) |
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finally show ?thesis by (simp add: is_empty_Bag_impl.rep_eq) |
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qed |
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lemma union_Bag [code]: "Bag xs + Bag ys = Bag (join (\<lambda>x (n1, n2). n1 + n2) xs ys)" |
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by (rule multiset_eqI) |
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(simp add: count_of_join_raw alist.Alist_inverse distinct_join_raw join_def) |
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lemma add_mset_Bag [code]: "add_mset x (Bag xs) = |
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Bag (join (\<lambda>x (n1, n2). n1 + n2) (DAList.update x 1 DAList.empty) xs)" |
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unfolding add_mset_add_single[of x "Bag xs"] union_Bag[symmetric] |
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by (simp add: multiset_eq_iff update.rep_eq empty.rep_eq) |
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lemma minus_Bag [code]: "Bag xs - Bag ys = Bag (subtract_entries xs ys)" |
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by (rule multiset_eqI) |
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(simp add: count_of_subtract_entries_raw alist.Alist_inverse |
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distinct_subtract_entries_raw subtract_entries_def) |
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lemma filter_Bag [code]: "filter_mset P (Bag xs) = Bag (DAList.filter (P \<circ> fst) xs)" |
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by (rule multiset_eqI) (simp add: count_of_filter DAList.filter.rep_eq) |
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lemma mset_eq [code]: "HOL.equal (m1::'a::equal multiset) m2 \<longleftrightarrow> m1 \<subseteq># m2 \<and> m2 \<subseteq># m1" |
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by (metis equal_multiset_def subset_mset.eq_iff) |
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text \<open>By default the code for \<open><\<close> is @{prop"xs < ys \<longleftrightarrow> xs \<le> ys \<and> \<not> xs = ys"}. |
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With equality implemented by \<open>\<le>\<close>, this leads to three calls of \<open>\<le>\<close>. |
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Here is a more efficient version:\<close> |
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lemma mset_less[code]: "xs \<subset># (ys :: 'a multiset) \<longleftrightarrow> xs \<subseteq># ys \<and> \<not> ys \<subseteq># xs" |
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by (rule subset_mset.less_le_not_le) |
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lemma mset_less_eq_Bag0: |
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"Bag xs \<subseteq># A \<longleftrightarrow> (\<forall>(x, n) \<in> set (DAList.impl_of xs). count_of (DAList.impl_of xs) x \<le> count A x)" |
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(is "?lhs \<longleftrightarrow> ?rhs") |
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proof |
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assume ?lhs |
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then show ?rhs by (auto simp add: subseteq_mset_def) |
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next |
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assume ?rhs |
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show ?lhs |
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proof (rule mset_subset_eqI) |
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fix x |
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from \<open>?rhs\<close> have "count_of (DAList.impl_of xs) x \<le> count A x" |
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by (cases "x \<in> fst ` set (DAList.impl_of xs)") (auto simp add: count_of_empty) |
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then show "count (Bag xs) x \<le> count A x" by (simp add: subset_mset_def) |
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qed |
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qed |
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lemma mset_less_eq_Bag [code]: |
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"Bag xs \<subseteq># (A :: 'a multiset) \<longleftrightarrow> (\<forall>(x, n) \<in> set (DAList.impl_of xs). n \<le> count A x)" |
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proof - |
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{ |
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fix x n |
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assume "(x,n) \<in> set (DAList.impl_of xs)" |
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then have "count_of (DAList.impl_of xs) x = n" |
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proof transfer |
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fix x n |
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fix xs :: "('a \<times> nat) list" |
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show "(distinct \<circ> map fst) xs \<Longrightarrow> (x, n) \<in> set xs \<Longrightarrow> count_of xs x = n" |
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proof (induct xs) |
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case Nil |
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then show ?case by simp |
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next |
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case (Cons ym ys) |
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obtain y m where ym: "ym = (y,m)" by force |
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note Cons = Cons[unfolded ym] |
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show ?case |
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proof (cases "x = y") |
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case False |
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with Cons show ?thesis |
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unfolding ym by auto |
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next |
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case True |
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with Cons(2-3) have "m = n" by force |
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with True show ?thesis |
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unfolding ym by auto |
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qed |
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qed |
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qed |
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} |
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then show ?thesis |
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unfolding mset_less_eq_Bag0 by auto |
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qed |
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declare multiset_inter_def [code] |
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declare sup_subset_mset_def [code] |
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declare mset.simps [code] |
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fun fold_impl :: "('a \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> ('a \<times> nat) list \<Rightarrow> 'b" |
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where |
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"fold_impl fn e ((a,n) # ms) = (fold_impl fn ((fn a n) e) ms)" |
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| "fold_impl fn e [] = e" |
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context |
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begin |
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qualified definition fold :: "('a \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> ('a, nat) alist \<Rightarrow> 'b" |
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where "fold f e al = fold_impl f e (DAList.impl_of al)" |
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end |
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context comp_fun_commute |
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begin |
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lemma DAList_Multiset_fold: |
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assumes fn: "\<And>a n x. fn a n x = (f a ^^ n) x" |
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shows "fold_mset f e (Bag al) = DAList_Multiset.fold fn e al" |
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unfolding DAList_Multiset.fold_def |
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proof (induct al) |
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fix ys |
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let ?inv = "{xs :: ('a \<times> nat) list. (distinct \<circ> map fst) xs}" |
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note cs[simp del] = count_simps |
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have count[simp]: "\<And>x. count (Abs_multiset (count_of x)) = count_of x" |
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by (rule Abs_multiset_inverse[OF count_of_multiset]) |
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assume ys: "ys \<in> ?inv" |
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|
296 |
then show "fold_mset f e (Bag (Alist ys)) = fold_impl fn e (DAList.impl_of (Alist ys))" |
55887 | 297 |
unfolding Bag_def unfolding Alist_inverse[OF ys] |
298 |
proof (induct ys arbitrary: e rule: list.induct) |
|
299 |
case Nil |
|
300 |
show ?case |
|
59998
c54d36be22ef
renamed Multiset.fold -> fold_mset, Multiset.filter -> filter_mset
nipkow
parents:
59949
diff
changeset
|
301 |
by (rule trans[OF arg_cong[of _ "{#}" "fold_mset f e", OF multiset_eqI]]) |
55887 | 302 |
(auto, simp add: cs) |
303 |
next |
|
304 |
case (Cons pair ys e) |
|
58806 | 305 |
obtain a n where pair: "pair = (a,n)" |
306 |
by force |
|
307 |
from fn[of a n] have [simp]: "fn a n = (f a ^^ n)" |
|
308 |
by auto |
|
309 |
have inv: "ys \<in> ?inv" |
|
310 |
using Cons(2) by auto |
|
55887 | 311 |
note IH = Cons(1)[OF inv] |
63040 | 312 |
define Ys where "Ys = Abs_multiset (count_of ys)" |
67399 | 313 |
have id: "Abs_multiset (count_of ((a, n) # ys)) = (((+) {# a #}) ^^ n) Ys" |
55887 | 314 |
unfolding Ys_def |
315 |
proof (rule multiset_eqI, unfold count) |
|
58806 | 316 |
fix c |
317 |
show "count_of ((a, n) # ys) c = |
|
67399 | 318 |
count (((+) {#a#} ^^ n) (Abs_multiset (count_of ys))) c" (is "?l = ?r") |
55887 | 319 |
proof (cases "c = a") |
58806 | 320 |
case False |
321 |
then show ?thesis |
|
322 |
unfolding cs by (induct n) auto |
|
55887 | 323 |
next |
324 |
case True |
|
58806 | 325 |
then have "?l = n" by (simp add: cs) |
55887 | 326 |
also have "n = ?r" unfolding True |
327 |
proof (induct n) |
|
328 |
case 0 |
|
329 |
from Cons(2)[unfolded pair] have "a \<notin> fst ` set ys" by auto |
|
58806 | 330 |
then show ?case by (induct ys) (simp, auto simp: cs) |
331 |
next |
|
332 |
case Suc |
|
333 |
then show ?case by simp |
|
334 |
qed |
|
55887 | 335 |
finally show ?thesis . |
336 |
qed |
|
337 |
qed |
|
58806 | 338 |
show ?case |
339 |
unfolding pair |
|
340 |
apply (simp add: IH[symmetric]) |
|
341 |
unfolding id Ys_def[symmetric] |
|
342 |
apply (induct n) |
|
343 |
apply (auto simp: fold_mset_fun_left_comm[symmetric]) |
|
344 |
done |
|
55887 | 345 |
qed |
346 |
qed |
|
347 |
||
58806 | 348 |
end |
55887 | 349 |
|
61115
3a4400985780
modernized name space management -- more uniform qualification;
wenzelm
parents:
60679
diff
changeset
|
350 |
context |
3a4400985780
modernized name space management -- more uniform qualification;
wenzelm
parents:
60679
diff
changeset
|
351 |
begin |
3a4400985780
modernized name space management -- more uniform qualification;
wenzelm
parents:
60679
diff
changeset
|
352 |
|
3a4400985780
modernized name space management -- more uniform qualification;
wenzelm
parents:
60679
diff
changeset
|
353 |
private lift_definition single_alist_entry :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) alist" is "\<lambda>a b. [(a, b)]" |
58806 | 354 |
by auto |
55887 | 355 |
|
58806 | 356 |
lemma image_mset_Bag [code]: |
55887 | 357 |
"image_mset f (Bag ms) = |
58806 | 358 |
DAList_Multiset.fold (\<lambda>a n m. Bag (single_alist_entry (f a) n) + m) {#} ms" |
359 |
unfolding image_mset_def |
|
55887 | 360 |
proof (rule comp_fun_commute.DAList_Multiset_fold, unfold_locales, (auto simp: ac_simps)[1]) |
361 |
fix a n m |
|
63793
e68a0b651eb5
add_mset constructor in multisets
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63310
diff
changeset
|
362 |
show "Bag (single_alist_entry (f a) n) + m = ((add_mset \<circ> f) a ^^ n) m" (is "?l = ?r") |
55887 | 363 |
proof (rule multiset_eqI) |
364 |
fix x |
|
365 |
have "count ?r x = (if x = f a then n + count m x else count m x)" |
|
58806 | 366 |
by (induct n) auto |
367 |
also have "\<dots> = count ?l x" |
|
368 |
by (simp add: single_alist_entry.rep_eq) |
|
55887 | 369 |
finally show "count ?l x = count ?r x" .. |
370 |
qed |
|
371 |
qed |
|
372 |
||
61115
3a4400985780
modernized name space management -- more uniform qualification;
wenzelm
parents:
60679
diff
changeset
|
373 |
end |
55887 | 374 |
|
67408 | 375 |
\<comment> \<open>we cannot use \<open>\<lambda>a n. (+) (a * n)\<close> for folding, since \<open>( * )\<close> is not defined in \<open>comm_monoid_add\<close>\<close> |
67399 | 376 |
lemma sum_mset_Bag[code]: "sum_mset (Bag ms) = DAList_Multiset.fold (\<lambda>a n. (((+) a) ^^ n)) 0 ms" |
63830 | 377 |
unfolding sum_mset.eq_fold |
58806 | 378 |
apply (rule comp_fun_commute.DAList_Multiset_fold) |
379 |
apply unfold_locales |
|
380 |
apply (auto simp: ac_simps) |
|
381 |
done |
|
55887 | 382 |
|
67408 | 383 |
\<comment> \<open>we cannot use \<open>\<lambda>a n. ( * ) (a ^ n)\<close> for folding, since \<open>(^)\<close> is not defined in \<open>comm_monoid_mult\<close>\<close> |
67399 | 384 |
lemma prod_mset_Bag[code]: "prod_mset (Bag ms) = DAList_Multiset.fold (\<lambda>a n. ((( * ) a) ^^ n)) 1 ms" |
63830 | 385 |
unfolding prod_mset.eq_fold |
58806 | 386 |
apply (rule comp_fun_commute.DAList_Multiset_fold) |
387 |
apply unfold_locales |
|
388 |
apply (auto simp: ac_simps) |
|
389 |
done |
|
55887 | 390 |
|
59998
c54d36be22ef
renamed Multiset.fold -> fold_mset, Multiset.filter -> filter_mset
nipkow
parents:
59949
diff
changeset
|
391 |
lemma size_fold: "size A = fold_mset (\<lambda>_. Suc) 0 A" (is "_ = fold_mset ?f _ _") |
55887 | 392 |
proof - |
60679 | 393 |
interpret comp_fun_commute ?f by standard auto |
55887 | 394 |
show ?thesis by (induct A) auto |
395 |
qed |
|
396 |
||
67399 | 397 |
lemma size_Bag[code]: "size (Bag ms) = DAList_Multiset.fold (\<lambda>a n. (+) n) 0 ms" |
59949 | 398 |
unfolding size_fold |
55887 | 399 |
proof (rule comp_fun_commute.DAList_Multiset_fold, unfold_locales, simp) |
400 |
fix a n x |
|
58806 | 401 |
show "n + x = (Suc ^^ n) x" |
402 |
by (induct n) auto |
|
55887 | 403 |
qed |
404 |
||
405 |
||
60495 | 406 |
lemma set_mset_fold: "set_mset A = fold_mset insert {} A" (is "_ = fold_mset ?f _ _") |
55887 | 407 |
proof - |
60679 | 408 |
interpret comp_fun_commute ?f by standard auto |
58806 | 409 |
show ?thesis by (induct A) auto |
55887 | 410 |
qed |
411 |
||
60495 | 412 |
lemma set_mset_Bag[code]: |
413 |
"set_mset (Bag ms) = DAList_Multiset.fold (\<lambda>a n. (if n = 0 then (\<lambda>m. m) else insert a)) {} ms" |
|
414 |
unfolding set_mset_fold |
|
55887 | 415 |
proof (rule comp_fun_commute.DAList_Multiset_fold, unfold_locales, (auto simp: ac_simps)[1]) |
416 |
fix a n x |
|
417 |
show "(if n = 0 then \<lambda>m. m else insert a) x = (insert a ^^ n) x" (is "?l n = ?r n") |
|
418 |
proof (cases n) |
|
58806 | 419 |
case 0 |
420 |
then show ?thesis by simp |
|
421 |
next |
|
55887 | 422 |
case (Suc m) |
58806 | 423 |
then have "?l n = insert a x" by simp |
55887 | 424 |
moreover have "?r n = insert a x" unfolding Suc by (induct m) auto |
425 |
ultimately show ?thesis by auto |
|
58806 | 426 |
qed |
55887 | 427 |
qed |
428 |
||
429 |
||
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
430 |
instantiation multiset :: (exhaustive) exhaustive |
51599 | 431 |
begin |
432 |
||
58806 | 433 |
definition exhaustive_multiset :: |
434 |
"('a multiset \<Rightarrow> (bool \<times> term list) option) \<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option" |
|
435 |
where "exhaustive_multiset f i = Quickcheck_Exhaustive.exhaustive (\<lambda>xs. f (Bag xs)) i" |
|
51599 | 436 |
|
437 |
instance .. |
|
438 |
||
439 |
end |
|
440 |
||
441 |
end |
|
442 |