src/HOL/Library/DAList_Multiset.thy
author nipkow
Fri, 19 Jun 2015 15:55:22 +0200
changeset 60515 484559628038
parent 60495 d7ff0a1df90a
child 60679 ade12ef2773c
permissions -rw-r--r--
renamed multiset_of -> mset
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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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lemma [code, code del]: "{#} = {#}" ..
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lemma [code, code del]: "single = single" ..
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lemma [code, code del]: "plus = (plus :: 'a multiset \<Rightarrow> _)" ..
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lemma [code, code del]: "minus = (minus :: 'a multiset \<Rightarrow> _)" ..
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lemma [code, code del]: "inf_subset_mset = (inf_subset_mset :: 'a multiset \<Rightarrow> _)" ..
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lemma [code, code del]: "sup_subset_mset = (sup_subset_mset :: 'a multiset \<Rightarrow> _)" ..
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lemma [code, code del]: "image_mset = image_mset" ..
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lemma [code, code del]: "filter_mset = filter_mset" ..
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lemma [code, code del]: "count = count" ..
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lemma [code, code del]: "size = (size :: _ multiset \<Rightarrow> nat)" ..
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lemma [code, code del]: "msetsum = msetsum" ..
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lemma [code, code del]: "msetprod = msetprod" ..
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lemma [code, code del]: "set_mset = set_mset" ..
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lemma [code, code del]: "sorted_list_of_multiset = sorted_list_of_multiset" ..
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lemma [code, code del]: "subset_mset = subset_mset" ..
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lemma [code, code del]: "subseteq_mset = subseteq_mset" ..
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lemma [code, code del]: "equal_multiset_inst.equal_multiset = 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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lemma single_Bag [code]: "{#x#} = Bag (DAList.update x 1 DAList.empty)"
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  by (simp add: multiset_eq_iff alist.Alist_inverse update.rep_eq empty.rep_eq)
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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 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 \<le># m2 \<and> m2 \<le># 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 @{text "<"} is @{prop"xs < ys \<longleftrightarrow> xs \<le> ys \<and> \<not> xs = ys"}.
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With equality implemented by @{text"\<le>"}, this leads to three calls of  @{text"\<le>"}.
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Here is a more efficient version:\<close>
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lemma mset_less[code]: "xs <# (ys :: 'a multiset) \<longleftrightarrow> xs \<le># ys \<and> \<not> ys \<le># 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 \<le># 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_less_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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   232
lemma mset_less_eq_Bag [code]:
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  "Bag xs \<le># (A :: 'a multiset) \<longleftrightarrow> (\<forall>(x, n) \<in> set (DAList.impl_of xs). n \<le> count A x)"
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   234
proof -
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   235
  {
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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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   239
    proof transfer
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   240
      fix x n
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   241
      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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   243
      proof (induct xs)
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        case Nil
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        then show ?case by simp
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   246
      next
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   247
        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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   250
        show ?case
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   251
        proof (cases "x = y")
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          case False
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          with Cons show ?thesis
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   254
            unfolding ym by auto
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   255
        next
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   256
          case True
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          with Cons(2-3) have "m = n" by force
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   258
          with True show ?thesis
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   259
            unfolding ym by auto
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   260
        qed
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   261
      qed
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   262
    qed
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   263
  }
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   264
  then show ?thesis
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   265
    unfolding mset_less_eq_Bag0 by auto
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   266
qed
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   267
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declare multiset_inter_def [code]
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   269
declare sup_subset_mset_def [code]
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   270
declare mset.simps [code]
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   271
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   272
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   273
fun fold_impl :: "('a \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> ('a \<times> nat) list \<Rightarrow> 'b"
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   274
where
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   275
  "fold_impl fn e ((a,n) # ms) = (fold_impl fn ((fn a n) e) ms)"
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   276
| "fold_impl fn e [] = e"
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   277
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   278
definition fold :: "('a \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> ('a, nat) alist \<Rightarrow> 'b"
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   279
  where "fold f e al = fold_impl f e (DAList.impl_of al)"
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   280
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   281
hide_const (open) fold
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   282
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   283
context comp_fun_commute
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   284
begin
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   285
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   286
lemma DAList_Multiset_fold:
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   287
  assumes fn: "\<And>a n x. fn a n x = (f a ^^ n) x"
59998
c54d36be22ef renamed Multiset.fold -> fold_mset, Multiset.filter -> filter_mset
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   288
  shows "fold_mset f e (Bag al) = DAList_Multiset.fold fn e al"
58806
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   289
  unfolding DAList_Multiset.fold_def
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   290
proof (induct al)
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   291
  fix ys
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   292
  let ?inv = "{xs :: ('a \<times> nat) list. (distinct \<circ> map fst) xs}"
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   293
  note cs[simp del] = count_simps
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   294
  have count[simp]: "\<And>x. count (Abs_multiset (count_of x)) = count_of x"
55887
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   295
    by (rule Abs_multiset_inverse[OF count_of_multiset])
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   296
  assume ys: "ys \<in> ?inv"
59998
c54d36be22ef renamed Multiset.fold -> fold_mset, Multiset.filter -> filter_mset
nipkow
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   297
  then show "fold_mset f e (Bag (Alist ys)) = fold_impl fn e (DAList.impl_of (Alist ys))"
55887
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   298
    unfolding Bag_def unfolding Alist_inverse[OF ys]
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   299
  proof (induct ys arbitrary: e rule: list.induct)
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   300
    case Nil
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   301
    show ?case
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   302
      by (rule trans[OF arg_cong[of _ "{#}" "fold_mset f e", OF multiset_eqI]])
55887
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   303
         (auto, simp add: cs)
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nipkow
parents: 55808
diff changeset
   304
  next
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   305
    case (Cons pair ys e)
58806
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   306
    obtain a n where pair: "pair = (a,n)"
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   307
      by force
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   308
    from fn[of a n] have [simp]: "fn a n = (f a ^^ n)"
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   309
      by auto
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   310
    have inv: "ys \<in> ?inv"
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   311
      using Cons(2) by auto
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   312
    note IH = Cons(1)[OF inv]
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   313
    def Ys \<equiv> "Abs_multiset (count_of ys)"
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   314
    have id: "Abs_multiset (count_of ((a, n) # ys)) = ((op + {# a #}) ^^ n) Ys"
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   315
      unfolding Ys_def
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   316
    proof (rule multiset_eqI, unfold count)
58806
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   317
      fix c
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   318
      show "count_of ((a, n) # ys) c =
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   319
        count ((op + {#a#} ^^ n) (Abs_multiset (count_of ys))) c" (is "?l = ?r")
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   320
      proof (cases "c = a")
58806
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   321
        case False
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   322
        then show ?thesis
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   323
          unfolding cs by (induct n) auto
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   324
      next
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   325
        case True
58806
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   326
        then have "?l = n" by (simp add: cs)
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   327
        also have "n = ?r" unfolding True
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   328
        proof (induct n)
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   329
          case 0
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   330
          from Cons(2)[unfolded pair] have "a \<notin> fst ` set ys" by auto
58806
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   331
          then show ?case by (induct ys) (simp, auto simp: cs)
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   332
        next
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   333
          case Suc
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   334
          then show ?case by simp
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   335
        qed
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   336
        finally show ?thesis .
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   337
      qed
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   338
    qed
58806
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   339
    show ?case
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   340
      unfolding pair
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   341
      apply (simp add: IH[symmetric])
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   342
      unfolding id Ys_def[symmetric]
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   343
      apply (induct n)
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   344
      apply (auto simp: fold_mset_fun_left_comm[symmetric])
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   345
      done
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   346
  qed
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   347
qed
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   348
58806
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   349
end
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   350
58806
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   351
lift_definition single_alist_entry :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) alist" is "\<lambda>a b. [(a, b)]"
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   352
  by auto
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   353
58806
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   354
lemma image_mset_Bag [code]:
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   355
  "image_mset f (Bag ms) =
58806
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   356
    DAList_Multiset.fold (\<lambda>a n m. Bag (single_alist_entry (f a) n) + m) {#} ms"
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   357
  unfolding image_mset_def
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   358
proof (rule comp_fun_commute.DAList_Multiset_fold, unfold_locales, (auto simp: ac_simps)[1])
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   359
  fix a n m
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   360
  show "Bag (single_alist_entry (f a) n) + m = ((op + \<circ> single \<circ> f) a ^^ n) m" (is "?l = ?r")
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   361
  proof (rule multiset_eqI)
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   362
    fix x
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   363
    have "count ?r x = (if x = f a then n + count m x else count m x)"
58806
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   364
      by (induct n) auto
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   365
    also have "\<dots> = count ?l x"
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   366
      by (simp add: single_alist_entry.rep_eq)
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   367
    finally show "count ?l x = count ?r x" ..
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   368
  qed
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   369
qed
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   370
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   371
hide_const single_alist_entry
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   372
58806
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   373
(* we cannot use (\<lambda>a n. op + (a * n)) for folding, since * is not defined
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   374
   in comm_monoid_add *)
58806
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   375
lemma msetsum_Bag[code]: "msetsum (Bag ms) = DAList_Multiset.fold (\<lambda>a n. ((op + a) ^^ n)) 0 ms"
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   376
  unfolding msetsum.eq_fold
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   377
  apply (rule comp_fun_commute.DAList_Multiset_fold)
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   378
  apply unfold_locales
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   379
  apply (auto simp: ac_simps)
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   380
  done
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   381
58806
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   382
(* we cannot use (\<lambda>a n. op * (a ^ n)) for folding, since ^ is not defined
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   383
   in comm_monoid_mult *)
58806
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   384
lemma msetprod_Bag[code]: "msetprod (Bag ms) = DAList_Multiset.fold (\<lambda>a n. ((op * a) ^^ n)) 1 ms"
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   385
  unfolding msetprod.eq_fold
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   386
  apply (rule comp_fun_commute.DAList_Multiset_fold)
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   387
  apply unfold_locales
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   388
  apply (auto simp: ac_simps)
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   389
  done
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   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
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   392
proof -
58806
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   393
  interpret comp_fun_commute ?f by default auto
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   394
  show ?thesis by (induct A) auto
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   395
qed
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   396
59949
fc4c896c8e74 Removed mcard because it is equal to size
nipkow
parents: 58881
diff changeset
   397
lemma size_Bag[code]: "size (Bag ms) = DAList_Multiset.fold (\<lambda>a n. op + n) 0 ms"
fc4c896c8e74 Removed mcard because it is equal to size
nipkow
parents: 58881
diff changeset
   398
  unfolding size_fold
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   399
proof (rule comp_fun_commute.DAList_Multiset_fold, unfold_locales, simp)
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   400
  fix a n x
58806
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   401
  show "n + x = (Suc ^^ n) x"
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   402
    by (induct n) auto
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   403
qed
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   404
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   405
60495
d7ff0a1df90a renamed Multiset.set_of to the canonical set_mset
nipkow
parents: 60397
diff changeset
   406
lemma set_mset_fold: "set_mset A = fold_mset insert {} A" (is "_ = fold_mset ?f _ _")
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   407
proof -
58806
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   408
  interpret comp_fun_commute ?f by default auto
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   409
  show ?thesis by (induct A) auto
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   410
qed
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   411
60495
d7ff0a1df90a renamed Multiset.set_of to the canonical set_mset
nipkow
parents: 60397
diff changeset
   412
lemma set_mset_Bag[code]:
d7ff0a1df90a renamed Multiset.set_of to the canonical set_mset
nipkow
parents: 60397
diff changeset
   413
  "set_mset (Bag ms) = DAList_Multiset.fold (\<lambda>a n. (if n = 0 then (\<lambda>m. m) else insert a)) {} ms"
d7ff0a1df90a renamed Multiset.set_of to the canonical set_mset
nipkow
parents: 60397
diff changeset
   414
  unfolding set_mset_fold
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   415
proof (rule comp_fun_commute.DAList_Multiset_fold, unfold_locales, (auto simp: ac_simps)[1])
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   416
  fix a n x
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   417
  show "(if n = 0 then \<lambda>m. m else insert a) x = (insert a ^^ n) x" (is "?l n = ?r n")
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   418
  proof (cases n)
58806
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   419
    case 0
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   420
    then show ?thesis by simp
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   421
  next
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   422
    case (Suc m)
58806
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   423
    then have "?l n = insert a x" by simp
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   424
    moreover have "?r n = insert a x" unfolding Suc by (induct m) auto
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   425
    ultimately show ?thesis by auto
58806
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   426
  qed
55887
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   427
qed
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   428
25bd4745ee38 more code lemmas by Rene Thiemann
nipkow
parents: 55808
diff changeset
   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
1559e9266280 optionalized very specific code setup for multisets
haftmann
parents:
diff changeset
   431
begin
1559e9266280 optionalized very specific code setup for multisets
haftmann
parents:
diff changeset
   432
58806
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   433
definition exhaustive_multiset ::
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   434
  "('a multiset \<Rightarrow> (bool \<times> term list) option) \<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option"
bb5ab5fce93a tuned proofs;
wenzelm
parents: 55887
diff changeset
   435
  where "exhaustive_multiset f i = Quickcheck_Exhaustive.exhaustive (\<lambda>xs. f (Bag xs)) i"
51599
1559e9266280 optionalized very specific code setup for multisets
haftmann
parents:
diff changeset
   436
1559e9266280 optionalized very specific code setup for multisets
haftmann
parents:
diff changeset
   437
instance ..
1559e9266280 optionalized very specific code setup for multisets
haftmann
parents:
diff changeset
   438
1559e9266280 optionalized very specific code setup for multisets
haftmann
parents:
diff changeset
   439
end
1559e9266280 optionalized very specific code setup for multisets
haftmann
parents:
diff changeset
   440
1559e9266280 optionalized very specific code setup for multisets
haftmann
parents:
diff changeset
   441
end
1559e9266280 optionalized very specific code setup for multisets
haftmann
parents:
diff changeset
   442