src/HOL/Library/Multiset.thy
author wenzelm
Mon Dec 28 21:47:32 2015 +0100 (2015-12-28)
changeset 61955 e96292f32c3c
parent 61890 f6ded81f5690
child 62082 614ef6d7a6b6
permissions -rw-r--r--
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
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(*  Title:      HOL/Library/Multiset.thy
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    Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
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    Author:     Andrei Popescu, TU Muenchen
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    Author:     Jasmin Blanchette, Inria, LORIA, MPII
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    Author:     Dmitriy Traytel, TU Muenchen
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    Author:     Mathias Fleury, MPII
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*)
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section \<open>(Finite) multisets\<close>
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theory Multiset
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imports Main
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begin
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subsection \<open>The type of multisets\<close>
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definition "multiset = {f :: 'a \<Rightarrow> nat. finite {x. f x > 0}}"
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typedef 'a multiset = "multiset :: ('a \<Rightarrow> nat) set"
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  morphisms count Abs_multiset
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  unfolding multiset_def
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proof
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  show "(\<lambda>x. 0::nat) \<in> {f. finite {x. f x > 0}}" by simp
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qed
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setup_lifting type_definition_multiset
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abbreviation Melem :: "'a \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "\<in>#" 50)
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  where "a \<in># M \<equiv> 0 < count M a"
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notation (ASCII)
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  Melem  ("(_/ :# _)" [50, 51] 50)  (* FIXME !? *)
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lemma multiset_eq_iff: "M = N \<longleftrightarrow> (\<forall>a. count M a = count N a)"
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  by (simp only: count_inject [symmetric] fun_eq_iff)
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lemma multiset_eqI: "(\<And>x. count A x = count B x) \<Longrightarrow> A = B"
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  using multiset_eq_iff by auto
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text \<open>Preservation of the representing set @{term multiset}.\<close>
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lemma const0_in_multiset: "(\<lambda>a. 0) \<in> multiset"
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  by (simp add: multiset_def)
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lemma only1_in_multiset: "(\<lambda>b. if b = a then n else 0) \<in> multiset"
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  by (simp add: multiset_def)
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lemma union_preserves_multiset: "M \<in> multiset \<Longrightarrow> N \<in> multiset \<Longrightarrow> (\<lambda>a. M a + N a) \<in> multiset"
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  by (simp add: multiset_def)
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lemma diff_preserves_multiset:
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  assumes "M \<in> multiset"
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  shows "(\<lambda>a. M a - N a) \<in> multiset"
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proof -
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  have "{x. N x < M x} \<subseteq> {x. 0 < M x}"
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    by auto
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  with assms show ?thesis
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    by (auto simp add: multiset_def intro: finite_subset)
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qed
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lemma filter_preserves_multiset:
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  assumes "M \<in> multiset"
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  shows "(\<lambda>x. if P x then M x else 0) \<in> multiset"
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proof -
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  have "{x. (P x \<longrightarrow> 0 < M x) \<and> P x} \<subseteq> {x. 0 < M x}"
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    by auto
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  with assms show ?thesis
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    by (auto simp add: multiset_def intro: finite_subset)
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qed
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lemmas in_multiset = const0_in_multiset only1_in_multiset
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  union_preserves_multiset diff_preserves_multiset filter_preserves_multiset
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subsection \<open>Representing multisets\<close>
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text \<open>Multiset enumeration\<close>
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instantiation multiset :: (type) cancel_comm_monoid_add
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begin
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lift_definition zero_multiset :: "'a multiset" is "\<lambda>a. 0"
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by (rule const0_in_multiset)
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abbreviation Mempty :: "'a multiset" ("{#}") where
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  "Mempty \<equiv> 0"
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lift_definition plus_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" is "\<lambda>M N. (\<lambda>a. M a + N a)"
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by (rule union_preserves_multiset)
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lift_definition minus_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" is "\<lambda> M N. \<lambda>a. M a - N a"
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by (rule diff_preserves_multiset)
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instance
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  by (standard; transfer; simp add: fun_eq_iff)
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end
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lift_definition single :: "'a \<Rightarrow> 'a multiset" is "\<lambda>a b. if b = a then 1 else 0"
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by (rule only1_in_multiset)
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syntax
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  "_multiset" :: "args \<Rightarrow> 'a multiset"    ("{#(_)#}")
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translations
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  "{#x, xs#}" == "{#x#} + {#xs#}"
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  "{#x#}" == "CONST single x"
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lemma count_empty [simp]: "count {#} a = 0"
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  by (simp add: zero_multiset.rep_eq)
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lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
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  by (simp add: single.rep_eq)
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subsection \<open>Basic operations\<close>
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subsubsection \<open>Union\<close>
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lemma count_union [simp]: "count (M + N) a = count M a + count N a"
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  by (simp add: plus_multiset.rep_eq)
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subsubsection \<open>Difference\<close>
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instantiation multiset :: (type) comm_monoid_diff
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begin
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instance
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  by (standard; transfer; simp add: fun_eq_iff)
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end
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lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
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  by (simp add: minus_multiset.rep_eq)
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lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
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  by rule (fact Groups.diff_zero, fact Groups.zero_diff)
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lemma diff_cancel[simp]: "A - A = {#}"
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  by (fact Groups.diff_cancel)
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lemma diff_union_cancelR [simp]: "M + N - N = (M::'a multiset)"
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  by (fact add_diff_cancel_right')
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lemma diff_union_cancelL [simp]: "N + M - N = (M::'a multiset)"
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  by (fact add_diff_cancel_left')
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lemma diff_right_commute:
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  fixes M N Q :: "'a multiset"
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  shows "M - N - Q = M - Q - N"
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  by (fact diff_right_commute)
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lemma diff_add:
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  fixes M N Q :: "'a multiset"
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  shows "M - (N + Q) = M - N - Q"
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  by (rule sym) (fact diff_diff_add)
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lemma insert_DiffM: "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
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  by (clarsimp simp: multiset_eq_iff)
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lemma insert_DiffM2 [simp]: "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
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  by (clarsimp simp: multiset_eq_iff)
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lemma diff_union_swap: "a \<noteq> b \<Longrightarrow> M - {#a#} + {#b#} = M + {#b#} - {#a#}"
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  by (auto simp add: multiset_eq_iff)
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lemma diff_union_single_conv: "a \<in># J \<Longrightarrow> I + J - {#a#} = I + (J - {#a#})"
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  by (simp add: multiset_eq_iff)
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subsubsection \<open>Equality of multisets\<close>
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lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
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  by (simp add: multiset_eq_iff)
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lemma single_eq_single [simp]: "{#a#} = {#b#} \<longleftrightarrow> a = b"
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  by (auto simp add: multiset_eq_iff)
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lemma union_eq_empty [iff]: "M + N = {#} \<longleftrightarrow> M = {#} \<and> N = {#}"
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  by (auto simp add: multiset_eq_iff)
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lemma empty_eq_union [iff]: "{#} = M + N \<longleftrightarrow> M = {#} \<and> N = {#}"
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  by (auto simp add: multiset_eq_iff)
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lemma multi_self_add_other_not_self [simp]: "M = M + {#x#} \<longleftrightarrow> False"
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  by (auto simp add: multiset_eq_iff)
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lemma diff_single_trivial: "\<not> x \<in># M \<Longrightarrow> M - {#x#} = M"
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  by (auto simp add: multiset_eq_iff)
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lemma diff_single_eq_union: "x \<in># M \<Longrightarrow> M - {#x#} = N \<longleftrightarrow> M = N + {#x#}"
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  by auto
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lemma union_single_eq_diff: "M + {#x#} = N \<Longrightarrow> M = N - {#x#}"
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  by (auto dest: sym)
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lemma union_single_eq_member: "M + {#x#} = N \<Longrightarrow> x \<in># N"
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  by auto
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lemma union_is_single: "M + N = {#a#} \<longleftrightarrow> M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#}"
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  (is "?lhs = ?rhs")
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proof
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  show ?lhs if ?rhs using that by auto
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  show ?rhs if ?lhs
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    using that by (simp add: multiset_eq_iff split: if_splits) (metis add_is_1)
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qed
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lemma single_is_union: "{#a#} = M + N \<longleftrightarrow> {#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N"
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  by (auto simp add: eq_commute [of "{#a#}" "M + N"] union_is_single)
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lemma add_eq_conv_diff:
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  "M + {#a#} = N + {#b#} \<longleftrightarrow> M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#}"
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  (is "?lhs \<longleftrightarrow> ?rhs")
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(* shorter: by (simp add: multiset_eq_iff) fastforce *)
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proof
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  show ?lhs if ?rhs
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    using that
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    by (auto simp add: add.assoc add.commute [of "{#b#}"])
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      (drule sym, simp add: add.assoc [symmetric])
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  show ?rhs if ?lhs
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  proof (cases "a = b")
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    case True with \<open>?lhs\<close> show ?thesis by simp
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  next
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    case False
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    from \<open>?lhs\<close> have "a \<in># N + {#b#}" by (rule union_single_eq_member)
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    with False have "a \<in># N" by auto
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    moreover from \<open>?lhs\<close> have "M = N + {#b#} - {#a#}" by (rule union_single_eq_diff)
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    moreover note False
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    ultimately show ?thesis by (auto simp add: diff_right_commute [of _ "{#a#}"] diff_union_swap)
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  qed
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qed
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lemma insert_noteq_member:
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  assumes BC: "B + {#b#} = C + {#c#}"
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   and bnotc: "b \<noteq> c"
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  shows "c \<in># B"
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proof -
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  have "c \<in># C + {#c#}" by simp
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  have nc: "\<not> c \<in># {#b#}" using bnotc by simp
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  then have "c \<in># B + {#b#}" using BC by simp
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  then show "c \<in># B" using nc by simp
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qed
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lemma add_eq_conv_ex:
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  "(M + {#a#} = N + {#b#}) =
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    (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
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  by (auto simp add: add_eq_conv_diff)
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lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
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  by (rule exI [where x = "M - {#x#}"]) simp
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lemma multiset_add_sub_el_shuffle:
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  assumes "c \<in># B"
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    and "b \<noteq> c"
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  shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
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proof -
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  from \<open>c \<in># B\<close> obtain A where B: "B = A + {#c#}"
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    by (blast dest: multi_member_split)
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  have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
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  then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}"
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    by (simp add: ac_simps)
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  then show ?thesis using B by simp
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qed
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subsubsection \<open>Pointwise ordering induced by count\<close>
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definition subseteq_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "\<subseteq>#" 50)
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  where "A \<subseteq># B = (\<forall>a. count A a \<le> count B a)"
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definition subset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "\<subset>#" 50)
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  where "A \<subset># B = (A \<subseteq># B \<and> A \<noteq> B)"
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notation (input)
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  subseteq_mset  (infix "\<le>#" 50)
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notation (ASCII)
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  subseteq_mset  (infix "<=#" 50) and
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  subset_mset  (infix "<#" 50)
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interpretation subset_mset: ordered_ab_semigroup_add_imp_le "op +" "op -" "op \<subseteq>#" "op \<subset>#"
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  by standard (auto simp add: subset_mset_def subseteq_mset_def multiset_eq_iff intro: order_trans antisym)
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lemma mset_less_eqI: "(\<And>x. count A x \<le> count B x) \<Longrightarrow> A \<le># B"
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  by (simp add: subseteq_mset_def)
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lemma mset_le_exists_conv: "(A::'a multiset) \<le># B \<longleftrightarrow> (\<exists>C. B = A + C)"
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  unfolding subseteq_mset_def
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  apply (rule iffI)
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   apply (rule exI [where x = "B - A"])
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   apply (auto intro: multiset_eq_iff [THEN iffD2])
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  done
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interpretation subset_mset: ordered_cancel_comm_monoid_diff  "op +" "op -" 0 "op \<le>#" "op <#"
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  by standard (simp, fact mset_le_exists_conv)
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lemma mset_le_mono_add_right_cancel [simp]: "(A::'a multiset) + C \<le># B + C \<longleftrightarrow> A \<le># B"
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  by (fact subset_mset.add_le_cancel_right)
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lemma mset_le_mono_add_left_cancel [simp]: "C + (A::'a multiset) \<le># C + B \<longleftrightarrow> A \<le># B"
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  by (fact subset_mset.add_le_cancel_left)
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lemma mset_le_mono_add: "(A::'a multiset) \<le># B \<Longrightarrow> C \<le># D \<Longrightarrow> A + C \<le># B + D"
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  by (fact subset_mset.add_mono)
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lemma mset_le_add_left [simp]: "(A::'a multiset) \<le># A + B"
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  unfolding subseteq_mset_def by auto
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lemma mset_le_add_right [simp]: "B \<le># (A::'a multiset) + B"
Mathias@60397
   310
  unfolding subseteq_mset_def by auto
haftmann@34943
   311
wenzelm@60607
   312
lemma mset_le_single: "a \<in># B \<Longrightarrow> {#a#} \<le># B"
Mathias@60397
   313
  by (simp add: subseteq_mset_def)
haftmann@34943
   314
haftmann@35268
   315
lemma multiset_diff_union_assoc:
wenzelm@60606
   316
  fixes A B C D :: "'a multiset"
wenzelm@60606
   317
  shows "C \<le># B \<Longrightarrow> A + B - C = A + (B - C)"
Mathias@60397
   318
  by (simp add: subset_mset.diff_add_assoc)
haftmann@34943
   319
haftmann@34943
   320
lemma mset_le_multiset_union_diff_commute:
wenzelm@60606
   321
  fixes A B C D :: "'a multiset"
wenzelm@60606
   322
  shows "B \<le># A \<Longrightarrow> A - B + C = A + C - B"
Mathias@60397
   323
by (simp add: subset_mset.add_diff_assoc2)
Mathias@60397
   324
Mathias@60397
   325
lemma diff_le_self[simp]: "(M::'a multiset) - N \<le># M"
Mathias@60397
   326
by(simp add: subseteq_mset_def)
Mathias@60397
   327
Mathias@60397
   328
lemma mset_lessD: "A <# B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
Mathias@60397
   329
apply (clarsimp simp: subset_mset_def subseteq_mset_def)
wenzelm@60678
   330
apply (erule allE [where x = x])
haftmann@34943
   331
apply auto
haftmann@34943
   332
done
haftmann@34943
   333
Mathias@60397
   334
lemma mset_leD: "A \<le># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
Mathias@60397
   335
apply (clarsimp simp: subset_mset_def subseteq_mset_def)
wenzelm@60678
   336
apply (erule allE [where x = x])
haftmann@34943
   337
apply auto
haftmann@34943
   338
done
blanchet@58425
   339
Mathias@60397
   340
lemma mset_less_insertD: "(A + {#x#} <# B) \<Longrightarrow> (x \<in># B \<and> A <# B)"
haftmann@34943
   341
apply (rule conjI)
haftmann@34943
   342
 apply (simp add: mset_lessD)
Mathias@60397
   343
apply (clarsimp simp: subset_mset_def subseteq_mset_def)
haftmann@34943
   344
apply safe
haftmann@34943
   345
 apply (erule_tac x = a in allE)
haftmann@34943
   346
 apply (auto split: split_if_asm)
haftmann@34943
   347
done
haftmann@34943
   348
Mathias@60397
   349
lemma mset_le_insertD: "(A + {#x#} \<le># B) \<Longrightarrow> (x \<in># B \<and> A \<le># B)"
haftmann@34943
   350
apply (rule conjI)
haftmann@34943
   351
 apply (simp add: mset_leD)
Mathias@60397
   352
apply (force simp: subset_mset_def subseteq_mset_def split: split_if_asm)
haftmann@34943
   353
done
haftmann@34943
   354
Mathias@60397
   355
lemma mset_less_of_empty[simp]: "A <# {#} \<longleftrightarrow> False"
Mathias@60397
   356
  by (auto simp add: subseteq_mset_def subset_mset_def multiset_eq_iff)
Mathias@60397
   357
Mathias@60397
   358
lemma empty_le[simp]: "{#} \<le># A"
nipkow@55808
   359
  unfolding mset_le_exists_conv by auto
nipkow@55808
   360
Mathias@60397
   361
lemma le_empty[simp]: "(M \<le># {#}) = (M = {#})"
nipkow@55808
   362
  unfolding mset_le_exists_conv by auto
nipkow@55808
   363
Mathias@60397
   364
lemma multi_psub_of_add_self[simp]: "A <# A + {#x#}"
Mathias@60397
   365
  by (auto simp: subset_mset_def subseteq_mset_def)
Mathias@60397
   366
Mathias@60397
   367
lemma multi_psub_self[simp]: "(A::'a multiset) <# A = False"
haftmann@35268
   368
  by simp
haftmann@34943
   369
Mathias@60397
   370
lemma mset_less_add_bothsides: "N + {#x#} <# M + {#x#} \<Longrightarrow> N <# M"
Mathias@60397
   371
  by (fact subset_mset.add_less_imp_less_right)
haftmann@35268
   372
wenzelm@60606
   373
lemma mset_less_empty_nonempty: "{#} <# S \<longleftrightarrow> S \<noteq> {#}"
Mathias@60397
   374
  by (auto simp: subset_mset_def subseteq_mset_def)
haftmann@35268
   375
wenzelm@60606
   376
lemma mset_less_diff_self: "c \<in># B \<Longrightarrow> B - {#c#} <# B"
Mathias@60397
   377
  by (auto simp: subset_mset_def subseteq_mset_def multiset_eq_iff)
haftmann@35268
   378
haftmann@35268
   379
wenzelm@60500
   380
subsubsection \<open>Intersection\<close>
haftmann@35268
   381
Mathias@60397
   382
definition inf_subset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where
Mathias@60397
   383
  multiset_inter_def: "inf_subset_mset A B = A - (A - B)"
Mathias@60397
   384
Mathias@60397
   385
interpretation subset_mset: semilattice_inf inf_subset_mset "op \<le>#" "op <#"
wenzelm@46921
   386
proof -
wenzelm@60678
   387
  have [simp]: "m \<le> n \<Longrightarrow> m \<le> q \<Longrightarrow> m \<le> n - (n - q)" for m n q :: nat
wenzelm@60678
   388
    by arith
wenzelm@60678
   389
  show "class.semilattice_inf op #\<inter> op \<le># op <#"
wenzelm@60678
   390
    by standard (auto simp add: multiset_inter_def subseteq_mset_def)
haftmann@35268
   391
qed
haftmann@35268
   392
haftmann@34943
   393
haftmann@41069
   394
lemma multiset_inter_count [simp]:
wenzelm@60606
   395
  fixes A B :: "'a multiset"
wenzelm@60606
   396
  shows "count (A #\<inter> B) x = min (count A x) (count B x)"
bulwahn@47429
   397
  by (simp add: multiset_inter_def)
haftmann@35268
   398
haftmann@35268
   399
lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
wenzelm@46730
   400
  by (rule multiset_eqI) auto
haftmann@34943
   401
haftmann@35268
   402
lemma multiset_union_diff_commute:
haftmann@35268
   403
  assumes "B #\<inter> C = {#}"
haftmann@35268
   404
  shows "A + B - C = A - C + B"
nipkow@39302
   405
proof (rule multiset_eqI)
haftmann@35268
   406
  fix x
haftmann@35268
   407
  from assms have "min (count B x) (count C x) = 0"
wenzelm@46730
   408
    by (auto simp add: multiset_eq_iff)
haftmann@35268
   409
  then have "count B x = 0 \<or> count C x = 0"
haftmann@35268
   410
    by auto
haftmann@35268
   411
  then show "count (A + B - C) x = count (A - C + B) x"
haftmann@35268
   412
    by auto
haftmann@35268
   413
qed
haftmann@35268
   414
wenzelm@60606
   415
lemma empty_inter [simp]: "{#} #\<inter> M = {#}"
haftmann@51600
   416
  by (simp add: multiset_eq_iff)
haftmann@51600
   417
wenzelm@60606
   418
lemma inter_empty [simp]: "M #\<inter> {#} = {#}"
haftmann@51600
   419
  by (simp add: multiset_eq_iff)
haftmann@51600
   420
wenzelm@60606
   421
lemma inter_add_left1: "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = M #\<inter> N"
haftmann@51600
   422
  by (simp add: multiset_eq_iff)
haftmann@51600
   423
wenzelm@60606
   424
lemma inter_add_left2: "x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = (M #\<inter> (N - {#x#})) + {#x#}"
haftmann@51600
   425
  by (simp add: multiset_eq_iff)
haftmann@51600
   426
wenzelm@60606
   427
lemma inter_add_right1: "\<not> x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = N #\<inter> M"
haftmann@51600
   428
  by (simp add: multiset_eq_iff)
haftmann@51600
   429
wenzelm@60606
   430
lemma inter_add_right2: "x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = ((N - {#x#}) #\<inter> M) + {#x#}"
haftmann@51600
   431
  by (simp add: multiset_eq_iff)
haftmann@51600
   432
haftmann@35268
   433
wenzelm@60500
   434
subsubsection \<open>Bounded union\<close>
wenzelm@60678
   435
wenzelm@60678
   436
definition sup_subset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset"(infixl "#\<union>" 70)
wenzelm@60678
   437
  where "sup_subset_mset A B = A + (B - A)"
Mathias@60397
   438
Mathias@60397
   439
interpretation subset_mset: semilattice_sup sup_subset_mset "op \<le>#" "op <#"
haftmann@51623
   440
proof -
wenzelm@60678
   441
  have [simp]: "m \<le> n \<Longrightarrow> q \<le> n \<Longrightarrow> m + (q - m) \<le> n" for m n q :: nat
wenzelm@60678
   442
    by arith
Mathias@60397
   443
  show "class.semilattice_sup op #\<union> op \<le># op <#"
wenzelm@60678
   444
    by standard (auto simp add: sup_subset_mset_def subseteq_mset_def)
haftmann@51623
   445
qed
haftmann@51623
   446
wenzelm@60606
   447
lemma sup_subset_mset_count [simp]: "count (A #\<union> B) x = max (count A x) (count B x)"
Mathias@60397
   448
  by (simp add: sup_subset_mset_def)
haftmann@51623
   449
wenzelm@60606
   450
lemma empty_sup [simp]: "{#} #\<union> M = M"
haftmann@51623
   451
  by (simp add: multiset_eq_iff)
haftmann@51623
   452
wenzelm@60606
   453
lemma sup_empty [simp]: "M #\<union> {#} = M"
haftmann@51623
   454
  by (simp add: multiset_eq_iff)
haftmann@51623
   455
wenzelm@60606
   456
lemma sup_add_left1: "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> N) + {#x#}"
haftmann@51623
   457
  by (simp add: multiset_eq_iff)
haftmann@51623
   458
wenzelm@60606
   459
lemma sup_add_left2: "x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> (N - {#x#})) + {#x#}"
haftmann@51623
   460
  by (simp add: multiset_eq_iff)
haftmann@51623
   461
wenzelm@60606
   462
lemma sup_add_right1: "\<not> x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = (N #\<union> M) + {#x#}"
haftmann@51623
   463
  by (simp add: multiset_eq_iff)
haftmann@51623
   464
wenzelm@60606
   465
lemma sup_add_right2: "x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = ((N - {#x#}) #\<union> M) + {#x#}"
haftmann@51623
   466
  by (simp add: multiset_eq_iff)
haftmann@51623
   467
wenzelm@60500
   468
subsubsection \<open>Subset is an order\<close>
Mathias@60397
   469
interpretation subset_mset: order "op \<le>#" "op <#" by unfold_locales auto
haftmann@51623
   470
wenzelm@60500
   471
subsubsection \<open>Filter (with comprehension syntax)\<close>
wenzelm@60500
   472
wenzelm@60500
   473
text \<open>Multiset comprehension\<close>
haftmann@41069
   474
nipkow@59998
   475
lift_definition filter_mset :: "('a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset"
nipkow@59998
   476
is "\<lambda>P M. \<lambda>x. if P x then M x else 0"
bulwahn@47429
   477
by (rule filter_preserves_multiset)
haftmann@35268
   478
wenzelm@60606
   479
lemma count_filter_mset [simp]: "count (filter_mset P M) a = (if P a then count M a else 0)"
nipkow@59998
   480
  by (simp add: filter_mset.rep_eq)
nipkow@59998
   481
wenzelm@60606
   482
lemma filter_empty_mset [simp]: "filter_mset P {#} = {#}"
nipkow@59998
   483
  by (rule multiset_eqI) simp
nipkow@59998
   484
wenzelm@60606
   485
lemma filter_single_mset [simp]: "filter_mset P {#x#} = (if P x then {#x#} else {#})"
nipkow@39302
   486
  by (rule multiset_eqI) simp
haftmann@35268
   487
wenzelm@60606
   488
lemma filter_union_mset [simp]: "filter_mset P (M + N) = filter_mset P M + filter_mset P N"
haftmann@41069
   489
  by (rule multiset_eqI) simp
haftmann@41069
   490
wenzelm@60606
   491
lemma filter_diff_mset [simp]: "filter_mset P (M - N) = filter_mset P M - filter_mset P N"
nipkow@39302
   492
  by (rule multiset_eqI) simp
haftmann@35268
   493
wenzelm@60606
   494
lemma filter_inter_mset [simp]: "filter_mset P (M #\<inter> N) = filter_mset P M #\<inter> filter_mset P N"
haftmann@41069
   495
  by (rule multiset_eqI) simp
haftmann@41069
   496
Mathias@60397
   497
lemma multiset_filter_subset[simp]: "filter_mset f M \<le># M"
Mathias@60397
   498
  by (simp add: mset_less_eqI)
Mathias@60397
   499
wenzelm@60606
   500
lemma multiset_filter_mono:
wenzelm@60606
   501
  assumes "A \<le># B"
Mathias@60397
   502
  shows "filter_mset f A \<le># filter_mset f B"
blanchet@58035
   503
proof -
blanchet@58035
   504
  from assms[unfolded mset_le_exists_conv]
blanchet@58035
   505
  obtain C where B: "B = A + C" by auto
blanchet@58035
   506
  show ?thesis unfolding B by auto
blanchet@58035
   507
qed
blanchet@58035
   508
wenzelm@61955
   509
syntax (ASCII)
haftmann@41069
   510
  "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ :# _./ _#})")
wenzelm@61955
   511
syntax
haftmann@41069
   512
  "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ \<in># _./ _#})")
haftmann@41069
   513
translations
nipkow@59998
   514
  "{#x \<in># M. P#}" == "CONST filter_mset (\<lambda>x. P) M"
haftmann@41069
   515
wenzelm@10249
   516
wenzelm@60500
   517
subsubsection \<open>Set of elements\<close>
wenzelm@10249
   518
wenzelm@60606
   519
definition set_mset :: "'a multiset \<Rightarrow> 'a set"
wenzelm@60607
   520
  where "set_mset M = {x. x \<in># M}"
nipkow@60495
   521
nipkow@60495
   522
lemma set_mset_empty [simp]: "set_mset {#} = {}"
nipkow@60495
   523
by (simp add: set_mset_def)
nipkow@60495
   524
nipkow@60495
   525
lemma set_mset_single [simp]: "set_mset {#b#} = {b}"
nipkow@60495
   526
by (simp add: set_mset_def)
nipkow@60495
   527
nipkow@60495
   528
lemma set_mset_union [simp]: "set_mset (M + N) = set_mset M \<union> set_mset N"
nipkow@60495
   529
by (auto simp add: set_mset_def)
nipkow@60495
   530
nipkow@60495
   531
lemma set_mset_eq_empty_iff [simp]: "(set_mset M = {}) = (M = {#})"
nipkow@60495
   532
by (auto simp add: set_mset_def multiset_eq_iff)
nipkow@60495
   533
wenzelm@60607
   534
lemma mem_set_mset_iff [simp]: "(x \<in> set_mset M) = (x \<in># M)"
nipkow@60495
   535
by (auto simp add: set_mset_def)
nipkow@60495
   536
wenzelm@60607
   537
lemma set_mset_filter [simp]: "set_mset {# x\<in>#M. P x #} = set_mset M \<inter> {x. P x}"
nipkow@60495
   538
by (auto simp add: set_mset_def)
nipkow@60495
   539
nipkow@60495
   540
lemma finite_set_mset [iff]: "finite (set_mset M)"
nipkow@60495
   541
  using count [of M] by (simp add: multiset_def set_mset_def)
haftmann@34943
   542
wenzelm@60607
   543
lemma finite_Collect_mem [iff]: "finite {x. x \<in># M}"
nipkow@60495
   544
  unfolding set_mset_def[symmetric] by simp
nipkow@60495
   545
nipkow@60495
   546
lemma set_mset_mono: "A \<le># B \<Longrightarrow> set_mset A \<subseteq> set_mset B"
nipkow@60495
   547
  by (metis mset_leD subsetI mem_set_mset_iff)
nipkow@60495
   548
nipkow@60495
   549
lemma ball_set_mset_iff: "(\<forall>x \<in> set_mset M. P x) \<longleftrightarrow> (\<forall>x. x \<in># M \<longrightarrow> P x)"
blanchet@59813
   550
  by auto
blanchet@59813
   551
blanchet@59813
   552
wenzelm@60500
   553
subsubsection \<open>Size\<close>
wenzelm@10249
   554
blanchet@56656
   555
definition wcount where "wcount f M = (\<lambda>x. count M x * Suc (f x))"
blanchet@56656
   556
blanchet@56656
   557
lemma wcount_union: "wcount f (M + N) a = wcount f M a + wcount f N a"
blanchet@56656
   558
  by (auto simp: wcount_def add_mult_distrib)
blanchet@56656
   559
blanchet@56656
   560
definition size_multiset :: "('a \<Rightarrow> nat) \<Rightarrow> 'a multiset \<Rightarrow> nat" where
nipkow@60495
   561
  "size_multiset f M = setsum (wcount f M) (set_mset M)"
blanchet@56656
   562
blanchet@56656
   563
lemmas size_multiset_eq = size_multiset_def[unfolded wcount_def]
blanchet@56656
   564
wenzelm@60606
   565
instantiation multiset :: (type) size
wenzelm@60606
   566
begin
wenzelm@60606
   567
blanchet@56656
   568
definition size_multiset where
blanchet@56656
   569
  size_multiset_overloaded_def: "size_multiset = Multiset.size_multiset (\<lambda>_. 0)"
haftmann@34943
   570
instance ..
wenzelm@60606
   571
haftmann@34943
   572
end
haftmann@34943
   573
blanchet@56656
   574
lemmas size_multiset_overloaded_eq =
blanchet@56656
   575
  size_multiset_overloaded_def[THEN fun_cong, unfolded size_multiset_eq, simplified]
blanchet@56656
   576
blanchet@56656
   577
lemma size_multiset_empty [simp]: "size_multiset f {#} = 0"
blanchet@56656
   578
by (simp add: size_multiset_def)
blanchet@56656
   579
haftmann@28708
   580
lemma size_empty [simp]: "size {#} = 0"
blanchet@56656
   581
by (simp add: size_multiset_overloaded_def)
blanchet@56656
   582
blanchet@56656
   583
lemma size_multiset_single [simp]: "size_multiset f {#b#} = Suc (f b)"
blanchet@56656
   584
by (simp add: size_multiset_eq)
wenzelm@10249
   585
haftmann@28708
   586
lemma size_single [simp]: "size {#b#} = 1"
blanchet@56656
   587
by (simp add: size_multiset_overloaded_def)
blanchet@56656
   588
blanchet@56656
   589
lemma setsum_wcount_Int:
nipkow@60495
   590
  "finite A \<Longrightarrow> setsum (wcount f N) (A \<inter> set_mset N) = setsum (wcount f N) A"
nipkow@26178
   591
apply (induct rule: finite_induct)
nipkow@26178
   592
 apply simp
nipkow@60495
   593
apply (simp add: Int_insert_left set_mset_def wcount_def)
blanchet@56656
   594
done
blanchet@56656
   595
blanchet@56656
   596
lemma size_multiset_union [simp]:
blanchet@56656
   597
  "size_multiset f (M + N::'a multiset) = size_multiset f M + size_multiset f N"
haftmann@57418
   598
apply (simp add: size_multiset_def setsum_Un_nat setsum.distrib setsum_wcount_Int wcount_union)
blanchet@56656
   599
apply (subst Int_commute)
blanchet@56656
   600
apply (simp add: setsum_wcount_Int)
nipkow@26178
   601
done
wenzelm@10249
   602
haftmann@28708
   603
lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
blanchet@56656
   604
by (auto simp add: size_multiset_overloaded_def)
blanchet@56656
   605
blanchet@56656
   606
lemma size_multiset_eq_0_iff_empty [iff]: "(size_multiset f M = 0) = (M = {#})"
blanchet@56656
   607
by (auto simp add: size_multiset_eq multiset_eq_iff)
wenzelm@10249
   608
wenzelm@17161
   609
lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
blanchet@56656
   610
by (auto simp add: size_multiset_overloaded_def)
nipkow@26016
   611
nipkow@26016
   612
lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
nipkow@26178
   613
by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
wenzelm@10249
   614
wenzelm@60607
   615
lemma size_eq_Suc_imp_elem: "size M = Suc n \<Longrightarrow> \<exists>a. a \<in># M"
blanchet@56656
   616
apply (unfold size_multiset_overloaded_eq)
nipkow@26178
   617
apply (drule setsum_SucD)
nipkow@26178
   618
apply auto
nipkow@26178
   619
done
wenzelm@10249
   620
haftmann@34943
   621
lemma size_eq_Suc_imp_eq_union:
haftmann@34943
   622
  assumes "size M = Suc n"
haftmann@34943
   623
  shows "\<exists>a N. M = N + {#a#}"
haftmann@34943
   624
proof -
haftmann@34943
   625
  from assms obtain a where "a \<in># M"
haftmann@34943
   626
    by (erule size_eq_Suc_imp_elem [THEN exE])
haftmann@34943
   627
  then have "M = M - {#a#} + {#a#}" by simp
haftmann@34943
   628
  then show ?thesis by blast
nipkow@23611
   629
qed
kleing@15869
   630
wenzelm@60606
   631
lemma size_mset_mono:
wenzelm@60606
   632
  fixes A B :: "'a multiset"
wenzelm@60606
   633
  assumes "A \<le># B"
wenzelm@60606
   634
  shows "size A \<le> size B"
nipkow@59949
   635
proof -
nipkow@59949
   636
  from assms[unfolded mset_le_exists_conv]
nipkow@59949
   637
  obtain C where B: "B = A + C" by auto
wenzelm@60606
   638
  show ?thesis unfolding B by (induct C) auto
nipkow@59949
   639
qed
nipkow@59949
   640
nipkow@59998
   641
lemma size_filter_mset_lesseq[simp]: "size (filter_mset f M) \<le> size M"
nipkow@59949
   642
by (rule size_mset_mono[OF multiset_filter_subset])
nipkow@59949
   643
nipkow@59949
   644
lemma size_Diff_submset:
Mathias@60397
   645
  "M \<le># M' \<Longrightarrow> size (M' - M) = size M' - size(M::'a multiset)"
nipkow@59949
   646
by (metis add_diff_cancel_left' size_union mset_le_exists_conv)
nipkow@26016
   647
wenzelm@60500
   648
subsection \<open>Induction and case splits\<close>
wenzelm@10249
   649
wenzelm@18258
   650
theorem multiset_induct [case_names empty add, induct type: multiset]:
huffman@48009
   651
  assumes empty: "P {#}"
huffman@48009
   652
  assumes add: "\<And>M x. P M \<Longrightarrow> P (M + {#x#})"
huffman@48009
   653
  shows "P M"
huffman@48009
   654
proof (induct n \<equiv> "size M" arbitrary: M)
huffman@48009
   655
  case 0 thus "P M" by (simp add: empty)
huffman@48009
   656
next
huffman@48009
   657
  case (Suc k)
huffman@48009
   658
  obtain N x where "M = N + {#x#}"
wenzelm@60500
   659
    using \<open>Suc k = size M\<close> [symmetric]
huffman@48009
   660
    using size_eq_Suc_imp_eq_union by fast
huffman@48009
   661
  with Suc add show "P M" by simp
wenzelm@10249
   662
qed
wenzelm@10249
   663
kleing@25610
   664
lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
nipkow@26178
   665
by (induct M) auto
kleing@25610
   666
wenzelm@55913
   667
lemma multiset_cases [cases type]:
wenzelm@55913
   668
  obtains (empty) "M = {#}"
wenzelm@55913
   669
    | (add) N x where "M = N + {#x#}"
wenzelm@55913
   670
  using assms by (induct M) simp_all
kleing@25610
   671
haftmann@34943
   672
lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
haftmann@34943
   673
by (cases "B = {#}") (auto dest: multi_member_split)
haftmann@34943
   674
wenzelm@60607
   675
lemma multiset_partition: "M = {# x\<in>#M. P x #} + {# x\<in>#M. \<not> P x #}"
nipkow@39302
   676
apply (subst multiset_eq_iff)
nipkow@26178
   677
apply auto
nipkow@26178
   678
done
wenzelm@10249
   679
Mathias@60397
   680
lemma mset_less_size: "(A::'a multiset) <# B \<Longrightarrow> size A < size B"
haftmann@34943
   681
proof (induct A arbitrary: B)
haftmann@34943
   682
  case (empty M)
haftmann@34943
   683
  then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
blanchet@58425
   684
  then obtain M' x where "M = M' + {#x#}"
haftmann@34943
   685
    by (blast dest: multi_nonempty_split)
haftmann@34943
   686
  then show ?case by simp
haftmann@34943
   687
next
haftmann@34943
   688
  case (add S x T)
Mathias@60397
   689
  have IH: "\<And>B. S <# B \<Longrightarrow> size S < size B" by fact
Mathias@60397
   690
  have SxsubT: "S + {#x#} <# T" by fact
Mathias@60397
   691
  then have "x \<in># T" and "S <# T" by (auto dest: mset_less_insertD)
blanchet@58425
   692
  then obtain T' where T: "T = T' + {#x#}"
haftmann@34943
   693
    by (blast dest: multi_member_split)
Mathias@60397
   694
  then have "S <# T'" using SxsubT
haftmann@34943
   695
    by (blast intro: mset_less_add_bothsides)
haftmann@34943
   696
  then have "size S < size T'" using IH by simp
haftmann@34943
   697
  then show ?case using T by simp
haftmann@34943
   698
qed
haftmann@34943
   699
haftmann@34943
   700
nipkow@59949
   701
lemma size_1_singleton_mset: "size M = 1 \<Longrightarrow> \<exists>a. M = {#a#}"
nipkow@59949
   702
by (cases M) auto
nipkow@59949
   703
wenzelm@60500
   704
subsubsection \<open>Strong induction and subset induction for multisets\<close>
wenzelm@60500
   705
wenzelm@60500
   706
text \<open>Well-foundedness of strict subset relation\<close>
haftmann@58098
   707
Mathias@60397
   708
lemma wf_less_mset_rel: "wf {(M, N :: 'a multiset). M <# N}"
haftmann@34943
   709
apply (rule wf_measure [THEN wf_subset, where f1=size])
haftmann@34943
   710
apply (clarsimp simp: measure_def inv_image_def mset_less_size)
haftmann@34943
   711
done
haftmann@34943
   712
haftmann@34943
   713
lemma full_multiset_induct [case_names less]:
Mathias@60397
   714
assumes ih: "\<And>B. \<forall>(A::'a multiset). A <# B \<longrightarrow> P A \<Longrightarrow> P B"
haftmann@34943
   715
shows "P B"
haftmann@58098
   716
apply (rule wf_less_mset_rel [THEN wf_induct])
haftmann@58098
   717
apply (rule ih, auto)
haftmann@34943
   718
done
haftmann@34943
   719
haftmann@34943
   720
lemma multi_subset_induct [consumes 2, case_names empty add]:
wenzelm@60606
   721
  assumes "F \<le># A"
wenzelm@60606
   722
    and empty: "P {#}"
wenzelm@60606
   723
    and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
wenzelm@60606
   724
  shows "P F"
haftmann@34943
   725
proof -
wenzelm@60500
   726
  from \<open>F \<le># A\<close>
haftmann@34943
   727
  show ?thesis
haftmann@34943
   728
  proof (induct F)
haftmann@34943
   729
    show "P {#}" by fact
haftmann@34943
   730
  next
haftmann@34943
   731
    fix x F
Mathias@60397
   732
    assume P: "F \<le># A \<Longrightarrow> P F" and i: "F + {#x#} \<le># A"
haftmann@34943
   733
    show "P (F + {#x#})"
haftmann@34943
   734
    proof (rule insert)
haftmann@34943
   735
      from i show "x \<in># A" by (auto dest: mset_le_insertD)
Mathias@60397
   736
      from i have "F \<le># A" by (auto dest: mset_le_insertD)
haftmann@34943
   737
      with P show "P F" .
haftmann@34943
   738
    qed
haftmann@34943
   739
  qed
haftmann@34943
   740
qed
wenzelm@26145
   741
wenzelm@17161
   742
wenzelm@60500
   743
subsection \<open>The fold combinator\<close>
huffman@48023
   744
nipkow@59998
   745
definition fold_mset :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b"
huffman@48023
   746
where
nipkow@60495
   747
  "fold_mset f s M = Finite_Set.fold (\<lambda>x. f x ^^ count M x) s (set_mset M)"
huffman@48023
   748
wenzelm@60606
   749
lemma fold_mset_empty [simp]: "fold_mset f s {#} = s"
nipkow@59998
   750
  by (simp add: fold_mset_def)
huffman@48023
   751
huffman@48023
   752
context comp_fun_commute
huffman@48023
   753
begin
huffman@48023
   754
wenzelm@60606
   755
lemma fold_mset_insert: "fold_mset f s (M + {#x#}) = f x (fold_mset f s M)"
haftmann@49822
   756
proof -
haftmann@49822
   757
  interpret mset: comp_fun_commute "\<lambda>y. f y ^^ count M y"
haftmann@49822
   758
    by (fact comp_fun_commute_funpow)
haftmann@49822
   759
  interpret mset_union: comp_fun_commute "\<lambda>y. f y ^^ count (M + {#x#}) y"
haftmann@49822
   760
    by (fact comp_fun_commute_funpow)
haftmann@49822
   761
  show ?thesis
nipkow@60495
   762
  proof (cases "x \<in> set_mset M")
haftmann@49822
   763
    case False
haftmann@49822
   764
    then have *: "count (M + {#x#}) x = 1" by simp
nipkow@60495
   765
    from False have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s (set_mset M) =
nipkow@60495
   766
      Finite_Set.fold (\<lambda>y. f y ^^ count M y) s (set_mset M)"
haftmann@49822
   767
      by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
haftmann@49822
   768
    with False * show ?thesis
nipkow@59998
   769
      by (simp add: fold_mset_def del: count_union)
huffman@48023
   770
  next
haftmann@49822
   771
    case True
nipkow@60495
   772
    def N \<equiv> "set_mset M - {x}"
nipkow@60495
   773
    from N_def True have *: "set_mset M = insert x N" "x \<notin> N" "finite N" by auto
haftmann@49822
   774
    then have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s N =
haftmann@49822
   775
      Finite_Set.fold (\<lambda>y. f y ^^ count M y) s N"
haftmann@49822
   776
      by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
nipkow@59998
   777
    with * show ?thesis by (simp add: fold_mset_def del: count_union) simp
huffman@48023
   778
  qed
huffman@48023
   779
qed
huffman@48023
   780
wenzelm@60606
   781
corollary fold_mset_single [simp]: "fold_mset f s {#x#} = f x s"
haftmann@49822
   782
proof -
nipkow@59998
   783
  have "fold_mset f s ({#} + {#x#}) = f x s" by (simp only: fold_mset_insert) simp
haftmann@49822
   784
  then show ?thesis by simp
haftmann@49822
   785
qed
huffman@48023
   786
wenzelm@60606
   787
lemma fold_mset_fun_left_comm: "f x (fold_mset f s M) = fold_mset f (f x s) M"
haftmann@49822
   788
  by (induct M) (simp_all add: fold_mset_insert fun_left_comm)
huffman@48023
   789
wenzelm@60606
   790
lemma fold_mset_union [simp]: "fold_mset f s (M + N) = fold_mset f (fold_mset f s M) N"
haftmann@49822
   791
proof (induct M)
huffman@48023
   792
  case empty then show ?case by simp
huffman@48023
   793
next
haftmann@49822
   794
  case (add M x)
haftmann@49822
   795
  have "M + {#x#} + N = (M + N) + {#x#}"
haftmann@57514
   796
    by (simp add: ac_simps)
haftmann@51548
   797
  with add show ?case by (simp add: fold_mset_insert fold_mset_fun_left_comm)
huffman@48023
   798
qed
huffman@48023
   799
huffman@48023
   800
lemma fold_mset_fusion:
huffman@48023
   801
  assumes "comp_fun_commute g"
wenzelm@60606
   802
    and *: "\<And>x y. h (g x y) = f x (h y)"
wenzelm@60606
   803
  shows "h (fold_mset g w A) = fold_mset f (h w) A"
huffman@48023
   804
proof -
huffman@48023
   805
  interpret comp_fun_commute g by (fact assms)
wenzelm@60606
   806
  from * show ?thesis by (induct A) auto
huffman@48023
   807
qed
huffman@48023
   808
huffman@48023
   809
end
huffman@48023
   810
wenzelm@60500
   811
text \<open>
huffman@48023
   812
  A note on code generation: When defining some function containing a
nipkow@59998
   813
  subterm @{term "fold_mset F"}, code generation is not automatic. When
wenzelm@61585
   814
  interpreting locale \<open>left_commutative\<close> with \<open>F\<close>, the
nipkow@59998
   815
  would be code thms for @{const fold_mset} become thms like
wenzelm@61585
   816
  @{term "fold_mset F z {#} = z"} where \<open>F\<close> is not a pattern but
huffman@48023
   817
  contains defined symbols, i.e.\ is not a code thm. Hence a separate
wenzelm@61585
   818
  constant with its own code thms needs to be introduced for \<open>F\<close>. See the image operator below.
wenzelm@60500
   819
\<close>
wenzelm@60500
   820
wenzelm@60500
   821
wenzelm@60500
   822
subsection \<open>Image\<close>
huffman@48023
   823
huffman@48023
   824
definition image_mset :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" where
wenzelm@60607
   825
  "image_mset f = fold_mset (plus \<circ> single \<circ> f) {#}"
wenzelm@60607
   826
wenzelm@60607
   827
lemma comp_fun_commute_mset_image: "comp_fun_commute (plus \<circ> single \<circ> f)"
haftmann@49823
   828
proof
haftmann@57514
   829
qed (simp add: ac_simps fun_eq_iff)
huffman@48023
   830
huffman@48023
   831
lemma image_mset_empty [simp]: "image_mset f {#} = {#}"
haftmann@49823
   832
  by (simp add: image_mset_def)
huffman@48023
   833
huffman@48023
   834
lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}"
haftmann@49823
   835
proof -
wenzelm@60607
   836
  interpret comp_fun_commute "plus \<circ> single \<circ> f"
haftmann@49823
   837
    by (fact comp_fun_commute_mset_image)
haftmann@49823
   838
  show ?thesis by (simp add: image_mset_def)
haftmann@49823
   839
qed
huffman@48023
   840
wenzelm@60606
   841
lemma image_mset_union [simp]: "image_mset f (M + N) = image_mset f M + image_mset f N"
haftmann@49823
   842
proof -
wenzelm@60607
   843
  interpret comp_fun_commute "plus \<circ> single \<circ> f"
haftmann@49823
   844
    by (fact comp_fun_commute_mset_image)
haftmann@57514
   845
  show ?thesis by (induct N) (simp_all add: image_mset_def ac_simps)
haftmann@49823
   846
qed
haftmann@49823
   847
wenzelm@60606
   848
corollary image_mset_insert: "image_mset f (M + {#a#}) = image_mset f M + {#f a#}"
haftmann@49823
   849
  by simp
huffman@48023
   850
wenzelm@60606
   851
lemma set_image_mset [simp]: "set_mset (image_mset f M) = image f (set_mset M)"
haftmann@49823
   852
  by (induct M) simp_all
huffman@48040
   853
wenzelm@60606
   854
lemma size_image_mset [simp]: "size (image_mset f M) = size M"
haftmann@49823
   855
  by (induct M) simp_all
huffman@48023
   856
wenzelm@60606
   857
lemma image_mset_is_empty_iff [simp]: "image_mset f M = {#} \<longleftrightarrow> M = {#}"
haftmann@49823
   858
  by (cases M) auto
huffman@48023
   859
wenzelm@61955
   860
syntax (ASCII)
wenzelm@61955
   861
  "_comprehension_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"  ("({#_/. _ :# _#})")
huffman@48023
   862
syntax
wenzelm@61955
   863
  "_comprehension_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"  ("({#_/. _ \<in># _#})")
blanchet@59813
   864
translations
wenzelm@61955
   865
  "{#e. x \<in># M#}" \<rightleftharpoons> "CONST image_mset (\<lambda>x. e) M"
wenzelm@61955
   866
wenzelm@61955
   867
syntax (ASCII)
wenzelm@61955
   868
  "_comprehension_mset'" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"  ("({#_/ | _ :# _./ _#})")
huffman@48023
   869
syntax
wenzelm@61955
   870
  "_comprehension_mset'" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"  ("({#_/ | _ \<in># _./ _#})")
blanchet@59813
   871
translations
wenzelm@60606
   872
  "{#e | x\<in>#M. P#}" \<rightharpoonup> "{#e. x \<in># {# x\<in>#M. P#}#}"
blanchet@59813
   873
wenzelm@60500
   874
text \<open>
wenzelm@60607
   875
  This allows to write not just filters like @{term "{#x\<in>#M. x<c#}"}
wenzelm@60607
   876
  but also images like @{term "{#x+x. x\<in>#M #}"} and @{term [source]
wenzelm@60607
   877
  "{#x+x|x\<in>#M. x<c#}"}, where the latter is currently displayed as
wenzelm@60607
   878
  @{term "{#x+x|x\<in>#M. x<c#}"}.
wenzelm@60500
   879
\<close>
huffman@48023
   880
nipkow@60495
   881
lemma in_image_mset: "y \<in># {#f x. x \<in># M#} \<longleftrightarrow> y \<in> f ` set_mset M"
nipkow@60498
   882
by (metis mem_set_mset_iff set_image_mset)
blanchet@59813
   883
blanchet@55467
   884
functor image_mset: image_mset
huffman@48023
   885
proof -
huffman@48023
   886
  fix f g show "image_mset f \<circ> image_mset g = image_mset (f \<circ> g)"
huffman@48023
   887
  proof
huffman@48023
   888
    fix A
huffman@48023
   889
    show "(image_mset f \<circ> image_mset g) A = image_mset (f \<circ> g) A"
huffman@48023
   890
      by (induct A) simp_all
huffman@48023
   891
  qed
huffman@48023
   892
  show "image_mset id = id"
huffman@48023
   893
  proof
huffman@48023
   894
    fix A
huffman@48023
   895
    show "image_mset id A = id A"
huffman@48023
   896
      by (induct A) simp_all
huffman@48023
   897
  qed
huffman@48023
   898
qed
huffman@48023
   899
blanchet@59813
   900
declare
blanchet@59813
   901
  image_mset.id [simp]
blanchet@59813
   902
  image_mset.identity [simp]
blanchet@59813
   903
blanchet@59813
   904
lemma image_mset_id[simp]: "image_mset id x = x"
blanchet@59813
   905
  unfolding id_def by auto
blanchet@59813
   906
blanchet@59813
   907
lemma image_mset_cong: "(\<And>x. x \<in># M \<Longrightarrow> f x = g x) \<Longrightarrow> {#f x. x \<in># M#} = {#g x. x \<in># M#}"
blanchet@59813
   908
  by (induct M) auto
blanchet@59813
   909
blanchet@59813
   910
lemma image_mset_cong_pair:
blanchet@59813
   911
  "(\<forall>x y. (x, y) \<in># M \<longrightarrow> f x y = g x y) \<Longrightarrow> {#f x y. (x, y) \<in># M#} = {#g x y. (x, y) \<in># M#}"
blanchet@59813
   912
  by (metis image_mset_cong split_cong)
haftmann@49717
   913
huffman@48023
   914
wenzelm@60500
   915
subsection \<open>Further conversions\<close>
haftmann@34943
   916
nipkow@60515
   917
primrec mset :: "'a list \<Rightarrow> 'a multiset" where
nipkow@60515
   918
  "mset [] = {#}" |
nipkow@60515
   919
  "mset (a # x) = mset x + {# a #}"
haftmann@34943
   920
haftmann@37107
   921
lemma in_multiset_in_set:
nipkow@60515
   922
  "x \<in># mset xs \<longleftrightarrow> x \<in> set xs"
haftmann@37107
   923
  by (induct xs) simp_all
haftmann@37107
   924
nipkow@60515
   925
lemma count_mset:
nipkow@60515
   926
  "count (mset xs) x = length (filter (\<lambda>y. x = y) xs)"
haftmann@37107
   927
  by (induct xs) simp_all
haftmann@37107
   928
nipkow@60515
   929
lemma mset_zero_iff[simp]: "(mset x = {#}) = (x = [])"
blanchet@59813
   930
  by (induct x) auto
haftmann@34943
   931
nipkow@60515
   932
lemma mset_zero_iff_right[simp]: "({#} = mset x) = (x = [])"
haftmann@34943
   933
by (induct x) auto
haftmann@34943
   934
nipkow@60515
   935
lemma set_mset_mset[simp]: "set_mset (mset x) = set x"
haftmann@34943
   936
by (induct x) auto
haftmann@34943
   937
wenzelm@60607
   938
lemma mem_set_multiset_eq: "x \<in> set xs = (x \<in># mset xs)"
haftmann@34943
   939
by (induct xs) auto
haftmann@34943
   940
nipkow@60515
   941
lemma size_mset [simp]: "size (mset xs) = length xs"
huffman@48012
   942
  by (induct xs) simp_all
huffman@48012
   943
wenzelm@60606
   944
lemma mset_append [simp]: "mset (xs @ ys) = mset xs + mset ys"
haftmann@57514
   945
  by (induct xs arbitrary: ys) (auto simp: ac_simps)
haftmann@34943
   946
wenzelm@60607
   947
lemma mset_filter: "mset (filter P xs) = {#x \<in># mset xs. P x #}"
haftmann@40303
   948
  by (induct xs) simp_all
haftmann@40303
   949
nipkow@60515
   950
lemma mset_rev [simp]:
nipkow@60515
   951
  "mset (rev xs) = mset xs"
haftmann@40950
   952
  by (induct xs) simp_all
haftmann@40950
   953
nipkow@60515
   954
lemma surj_mset: "surj mset"
haftmann@34943
   955
apply (unfold surj_def)
haftmann@34943
   956
apply (rule allI)
haftmann@34943
   957
apply (rule_tac M = y in multiset_induct)
haftmann@34943
   958
 apply auto
haftmann@34943
   959
apply (rule_tac x = "x # xa" in exI)
haftmann@34943
   960
apply auto
haftmann@34943
   961
done
haftmann@34943
   962
nipkow@60515
   963
lemma set_count_greater_0: "set x = {a. count (mset x) a > 0}"
haftmann@34943
   964
by (induct x) auto
haftmann@34943
   965
haftmann@34943
   966
lemma distinct_count_atmost_1:
wenzelm@60606
   967
  "distinct x = (\<forall>a. count (mset x) a = (if a \<in> set x then 1 else 0))"
wenzelm@60678
   968
  apply (induct x, simp, rule iffI, simp_all)
wenzelm@60678
   969
  subgoal for a b
wenzelm@60678
   970
    apply (rule conjI)
wenzelm@60678
   971
    apply (simp_all add: set_mset_mset [symmetric] del: set_mset_mset)
wenzelm@60678
   972
    apply (erule_tac x = a in allE, simp)
wenzelm@60678
   973
    apply clarify
wenzelm@60678
   974
    apply (erule_tac x = aa in allE, simp)
wenzelm@60678
   975
    done
wenzelm@60678
   976
  done
haftmann@34943
   977
wenzelm@60606
   978
lemma mset_eq_setD: "mset xs = mset ys \<Longrightarrow> set xs = set ys"
nipkow@39302
   979
by (rule) (auto simp add:multiset_eq_iff set_count_greater_0)
haftmann@34943
   980
nipkow@60515
   981
lemma set_eq_iff_mset_eq_distinct:
haftmann@34943
   982
  "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
nipkow@60515
   983
    (set x = set y) = (mset x = mset y)"
nipkow@39302
   984
by (auto simp: multiset_eq_iff distinct_count_atmost_1)
haftmann@34943
   985
nipkow@60515
   986
lemma set_eq_iff_mset_remdups_eq:
nipkow@60515
   987
   "(set x = set y) = (mset (remdups x) = mset (remdups y))"
haftmann@34943
   988
apply (rule iffI)
nipkow@60515
   989
apply (simp add: set_eq_iff_mset_eq_distinct[THEN iffD1])
haftmann@34943
   990
apply (drule distinct_remdups [THEN distinct_remdups
nipkow@60515
   991
      [THEN set_eq_iff_mset_eq_distinct [THEN iffD2]]])
haftmann@34943
   992
apply simp
haftmann@34943
   993
done
haftmann@34943
   994
wenzelm@60606
   995
lemma mset_compl_union [simp]: "mset [x\<leftarrow>xs. P x] + mset [x\<leftarrow>xs. \<not>P x] = mset xs"
haftmann@57514
   996
  by (induct xs) (auto simp: ac_simps)
haftmann@34943
   997
wenzelm@60607
   998
lemma nth_mem_mset: "i < length ls \<Longrightarrow> (ls ! i) \<in># mset ls"
wenzelm@60678
   999
proof (induct ls arbitrary: i)
wenzelm@60678
  1000
  case Nil
wenzelm@60678
  1001
  then show ?case by simp
wenzelm@60678
  1002
next
wenzelm@60678
  1003
  case Cons
wenzelm@60678
  1004
  then show ?case by (cases i) auto
wenzelm@60678
  1005
qed
haftmann@34943
  1006
wenzelm@60606
  1007
lemma mset_remove1[simp]: "mset (remove1 a xs) = mset xs - {#a#}"
wenzelm@60678
  1008
  by (induct xs) (auto simp add: multiset_eq_iff)
haftmann@34943
  1009
nipkow@60515
  1010
lemma mset_eq_length:
nipkow@60515
  1011
  assumes "mset xs = mset ys"
haftmann@37107
  1012
  shows "length xs = length ys"
nipkow@60515
  1013
  using assms by (metis size_mset)
nipkow@60515
  1014
nipkow@60515
  1015
lemma mset_eq_length_filter:
nipkow@60515
  1016
  assumes "mset xs = mset ys"
haftmann@39533
  1017
  shows "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) ys)"
nipkow@60515
  1018
  using assms by (metis count_mset)
haftmann@39533
  1019
haftmann@45989
  1020
lemma fold_multiset_equiv:
haftmann@45989
  1021
  assumes f: "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
nipkow@60515
  1022
    and equiv: "mset xs = mset ys"
haftmann@49822
  1023
  shows "List.fold f xs = List.fold f ys"
wenzelm@60606
  1024
  using f equiv [symmetric]
wenzelm@46921
  1025
proof (induct xs arbitrary: ys)
wenzelm@60678
  1026
  case Nil
wenzelm@60678
  1027
  then show ?case by simp
haftmann@45989
  1028
next
haftmann@45989
  1029
  case (Cons x xs)
wenzelm@60678
  1030
  then have *: "set ys = set (x # xs)"
wenzelm@60678
  1031
    by (blast dest: mset_eq_setD)
blanchet@58425
  1032
  have "\<And>x y. x \<in> set ys \<Longrightarrow> y \<in> set ys \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
haftmann@45989
  1033
    by (rule Cons.prems(1)) (simp_all add: *)
wenzelm@60678
  1034
  moreover from * have "x \<in> set ys"
wenzelm@60678
  1035
    by simp
wenzelm@60678
  1036
  ultimately have "List.fold f ys = List.fold f (remove1 x ys) \<circ> f x"
wenzelm@60678
  1037
    by (fact fold_remove1_split)
wenzelm@60678
  1038
  moreover from Cons.prems have "List.fold f xs = List.fold f (remove1 x ys)"
wenzelm@60678
  1039
    by (auto intro: Cons.hyps)
haftmann@45989
  1040
  ultimately show ?case by simp
haftmann@45989
  1041
qed
haftmann@45989
  1042
wenzelm@60606
  1043
lemma mset_insort [simp]: "mset (insort x xs) = mset xs + {#x#}"
haftmann@51548
  1044
  by (induct xs) (simp_all add: ac_simps)
haftmann@51548
  1045
wenzelm@60606
  1046
lemma mset_map: "mset (map f xs) = image_mset f (mset xs)"
haftmann@51600
  1047
  by (induct xs) simp_all
haftmann@51600
  1048
haftmann@61890
  1049
global_interpretation mset_set: folding "\<lambda>x M. {#x#} + M" "{#}"
haftmann@61832
  1050
  defines mset_set = "folding.F (\<lambda>x M. {#x#} + M) {#}"
haftmann@61832
  1051
  by standard (simp add: fun_eq_iff ac_simps)
haftmann@51548
  1052
nipkow@60513
  1053
lemma count_mset_set [simp]:
nipkow@60513
  1054
  "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> count (mset_set A) x = 1" (is "PROP ?P")
nipkow@60513
  1055
  "\<not> finite A \<Longrightarrow> count (mset_set A) x = 0" (is "PROP ?Q")
nipkow@60513
  1056
  "x \<notin> A \<Longrightarrow> count (mset_set A) x = 0" (is "PROP ?R")
haftmann@51600
  1057
proof -
wenzelm@60606
  1058
  have *: "count (mset_set A) x = 0" if "x \<notin> A" for A
wenzelm@60606
  1059
  proof (cases "finite A")
wenzelm@60606
  1060
    case False then show ?thesis by simp
wenzelm@60606
  1061
  next
wenzelm@60606
  1062
    case True from True \<open>x \<notin> A\<close> show ?thesis by (induct A) auto
wenzelm@60606
  1063
  qed
haftmann@51600
  1064
  then show "PROP ?P" "PROP ?Q" "PROP ?R"
haftmann@51600
  1065
  by (auto elim!: Set.set_insert)
wenzelm@61585
  1066
qed \<comment> \<open>TODO: maybe define @{const mset_set} also in terms of @{const Abs_multiset}\<close>
nipkow@60513
  1067
nipkow@60513
  1068
lemma elem_mset_set[simp, intro]: "finite A \<Longrightarrow> x \<in># mset_set A \<longleftrightarrow> x \<in> A"
blanchet@59813
  1069
  by (induct A rule: finite_induct) simp_all
blanchet@59813
  1070
haftmann@51548
  1071
context linorder
haftmann@51548
  1072
begin
haftmann@51548
  1073
haftmann@51548
  1074
definition sorted_list_of_multiset :: "'a multiset \<Rightarrow> 'a list"
haftmann@51548
  1075
where
nipkow@59998
  1076
  "sorted_list_of_multiset M = fold_mset insort [] M"
haftmann@51548
  1077
haftmann@51548
  1078
lemma sorted_list_of_multiset_empty [simp]:
haftmann@51548
  1079
  "sorted_list_of_multiset {#} = []"
haftmann@51548
  1080
  by (simp add: sorted_list_of_multiset_def)
haftmann@51548
  1081
haftmann@51548
  1082
lemma sorted_list_of_multiset_singleton [simp]:
haftmann@51548
  1083
  "sorted_list_of_multiset {#x#} = [x]"
haftmann@51548
  1084
proof -
haftmann@51548
  1085
  interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
haftmann@51548
  1086
  show ?thesis by (simp add: sorted_list_of_multiset_def)
haftmann@51548
  1087
qed
haftmann@51548
  1088
haftmann@51548
  1089
lemma sorted_list_of_multiset_insert [simp]:
haftmann@51548
  1090
  "sorted_list_of_multiset (M + {#x#}) = List.insort x (sorted_list_of_multiset M)"
haftmann@51548
  1091
proof -
haftmann@51548
  1092
  interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
haftmann@51548
  1093
  show ?thesis by (simp add: sorted_list_of_multiset_def)
haftmann@51548
  1094
qed
haftmann@51548
  1095
haftmann@51548
  1096
end
haftmann@51548
  1097
nipkow@60515
  1098
lemma mset_sorted_list_of_multiset [simp]:
nipkow@60515
  1099
  "mset (sorted_list_of_multiset M) = M"
nipkow@60513
  1100
by (induct M) simp_all
haftmann@51548
  1101
nipkow@60515
  1102
lemma sorted_list_of_multiset_mset [simp]:
nipkow@60515
  1103
  "sorted_list_of_multiset (mset xs) = sort xs"
nipkow@60513
  1104
by (induct xs) simp_all
nipkow@60513
  1105
nipkow@60513
  1106
lemma finite_set_mset_mset_set[simp]:
nipkow@60513
  1107
  "finite A \<Longrightarrow> set_mset (mset_set A) = A"
nipkow@60513
  1108
by (induct A rule: finite_induct) simp_all
nipkow@60513
  1109
nipkow@60513
  1110
lemma infinite_set_mset_mset_set:
nipkow@60513
  1111
  "\<not> finite A \<Longrightarrow> set_mset (mset_set A) = {}"
nipkow@60513
  1112
by simp
haftmann@51548
  1113
haftmann@51548
  1114
lemma set_sorted_list_of_multiset [simp]:
nipkow@60495
  1115
  "set (sorted_list_of_multiset M) = set_mset M"
nipkow@60513
  1116
by (induct M) (simp_all add: set_insort)
nipkow@60513
  1117
nipkow@60513
  1118
lemma sorted_list_of_mset_set [simp]:
nipkow@60513
  1119
  "sorted_list_of_multiset (mset_set A) = sorted_list_of_set A"
nipkow@60513
  1120
by (cases "finite A") (induct A rule: finite_induct, simp_all add: ac_simps)
haftmann@51548
  1121
haftmann@51548
  1122
haftmann@60804
  1123
subsection \<open>Replicate operation\<close>
haftmann@60804
  1124
haftmann@60804
  1125
definition replicate_mset :: "nat \<Rightarrow> 'a \<Rightarrow> 'a multiset" where
haftmann@60804
  1126
  "replicate_mset n x = ((op + {#x#}) ^^ n) {#}"
haftmann@60804
  1127
haftmann@60804
  1128
lemma replicate_mset_0[simp]: "replicate_mset 0 x = {#}"
haftmann@60804
  1129
  unfolding replicate_mset_def by simp
haftmann@60804
  1130
haftmann@60804
  1131
lemma replicate_mset_Suc[simp]: "replicate_mset (Suc n) x = replicate_mset n x + {#x#}"
haftmann@60804
  1132
  unfolding replicate_mset_def by (induct n) (auto intro: add.commute)
haftmann@60804
  1133
haftmann@60804
  1134
lemma in_replicate_mset[simp]: "x \<in># replicate_mset n y \<longleftrightarrow> n > 0 \<and> x = y"
haftmann@60804
  1135
  unfolding replicate_mset_def by (induct n) simp_all
haftmann@60804
  1136
haftmann@60804
  1137
lemma count_replicate_mset[simp]: "count (replicate_mset n x) y = (if y = x then n else 0)"
haftmann@60804
  1138
  unfolding replicate_mset_def by (induct n) simp_all
haftmann@60804
  1139
haftmann@60804
  1140
lemma set_mset_replicate_mset_subset[simp]: "set_mset (replicate_mset n x) = (if n = 0 then {} else {x})"
haftmann@60804
  1141
  by (auto split: if_splits)
haftmann@60804
  1142
haftmann@60804
  1143
lemma size_replicate_mset[simp]: "size (replicate_mset n M) = n"
haftmann@60804
  1144
  by (induct n, simp_all)
haftmann@60804
  1145
haftmann@60804
  1146
lemma count_le_replicate_mset_le: "n \<le> count M x \<longleftrightarrow> replicate_mset n x \<le># M"
haftmann@60804
  1147
  by (auto simp add: assms mset_less_eqI) (metis count_replicate_mset subseteq_mset_def)
haftmann@60804
  1148
haftmann@60804
  1149
lemma filter_eq_replicate_mset: "{#y \<in># D. y = x#} = replicate_mset (count D x) x"
haftmann@60804
  1150
  by (induct D) simp_all
haftmann@60804
  1151
haftmann@61031
  1152
lemma replicate_count_mset_eq_filter_eq:
haftmann@61031
  1153
  "replicate (count (mset xs) k) k = filter (HOL.eq k) xs"
haftmann@61031
  1154
  by (induct xs) auto
haftmann@61031
  1155
haftmann@60804
  1156
wenzelm@60500
  1157
subsection \<open>Big operators\<close>
haftmann@51548
  1158
haftmann@51548
  1159
no_notation times (infixl "*" 70)
haftmann@51548
  1160
no_notation Groups.one ("1")
haftmann@51548
  1161
haftmann@51548
  1162
locale comm_monoid_mset = comm_monoid
haftmann@51548
  1163
begin
haftmann@51548
  1164
haftmann@51548
  1165
definition F :: "'a multiset \<Rightarrow> 'a"
wenzelm@60606
  1166
  where eq_fold: "F M = fold_mset f 1 M"
wenzelm@60606
  1167
wenzelm@60606
  1168
lemma empty [simp]: "F {#} = 1"
haftmann@51548
  1169
  by (simp add: eq_fold)
haftmann@51548
  1170
wenzelm@60678
  1171
lemma singleton [simp]: "F {#x#} = x"
haftmann@51548
  1172
proof -
haftmann@51548
  1173
  interpret comp_fun_commute
wenzelm@60678
  1174
    by standard (simp add: fun_eq_iff left_commute)
haftmann@51548
  1175
  show ?thesis by (simp add: eq_fold)
haftmann@51548
  1176
qed
haftmann@51548
  1177
wenzelm@60606
  1178
lemma union [simp]: "F (M + N) = F M * F N"
haftmann@51548
  1179
proof -
haftmann@51548
  1180
  interpret comp_fun_commute f
wenzelm@60678
  1181
    by standard (simp add: fun_eq_iff left_commute)
wenzelm@60678
  1182
  show ?thesis
wenzelm@60678
  1183
    by (induct N) (simp_all add: left_commute eq_fold)
haftmann@51548
  1184
qed
haftmann@51548
  1185
haftmann@51548
  1186
end
haftmann@51548
  1187
wenzelm@61076
  1188
lemma comp_fun_commute_plus_mset[simp]: "comp_fun_commute (op + :: 'a multiset \<Rightarrow> _ \<Rightarrow> _)"
wenzelm@60678
  1189
  by standard (simp add: add_ac comp_def)
blanchet@59813
  1190
blanchet@59813
  1191
declare comp_fun_commute.fold_mset_insert[OF comp_fun_commute_plus_mset, simp]
blanchet@59813
  1192
nipkow@59998
  1193
lemma in_mset_fold_plus_iff[iff]: "x \<in># fold_mset (op +) M NN \<longleftrightarrow> x \<in># M \<or> (\<exists>N. N \<in># NN \<and> x \<in># N)"
blanchet@59813
  1194
  by (induct NN) auto
blanchet@59813
  1195
haftmann@51548
  1196
notation times (infixl "*" 70)
haftmann@51548
  1197
notation Groups.one ("1")
haftmann@51548
  1198
haftmann@54868
  1199
context comm_monoid_add
haftmann@54868
  1200
begin
haftmann@54868
  1201
wenzelm@61605
  1202
sublocale msetsum: comm_monoid_mset plus 0
haftmann@61832
  1203
  defines msetsum = msetsum.F ..
haftmann@51548
  1204
haftmann@60804
  1205
lemma (in semiring_1) msetsum_replicate_mset [simp]:
haftmann@60804
  1206
  "msetsum (replicate_mset n a) = of_nat n * a"
haftmann@60804
  1207
  by (induct n) (simp_all add: algebra_simps)
haftmann@60804
  1208
haftmann@51548
  1209
lemma setsum_unfold_msetsum:
nipkow@60513
  1210
  "setsum f A = msetsum (image_mset f (mset_set A))"
haftmann@51548
  1211
  by (cases "finite A") (induct A rule: finite_induct, simp_all)
haftmann@51548
  1212
haftmann@51548
  1213
end
haftmann@51548
  1214
blanchet@59813
  1215
lemma msetsum_diff:
wenzelm@61076
  1216
  fixes M N :: "('a :: ordered_cancel_comm_monoid_diff) multiset"
Mathias@60397
  1217
  shows "N \<le># M \<Longrightarrow> msetsum (M - N) = msetsum M - msetsum N"
Mathias@60397
  1218
  by (metis add_diff_cancel_right' msetsum.union subset_mset.diff_add)
blanchet@59813
  1219
nipkow@59949
  1220
lemma size_eq_msetsum: "size M = msetsum (image_mset (\<lambda>_. 1) M)"
nipkow@59949
  1221
proof (induct M)
nipkow@59949
  1222
  case empty then show ?case by simp
nipkow@59949
  1223
next
nipkow@59949
  1224
  case (add M x) then show ?case
nipkow@60495
  1225
    by (cases "x \<in> set_mset M")
nipkow@60495
  1226
      (simp_all del: mem_set_mset_iff add: size_multiset_overloaded_eq setsum.distrib setsum.delta' insert_absorb, simp)
nipkow@59949
  1227
qed
nipkow@59949
  1228
nipkow@59949
  1229
wenzelm@61955
  1230
abbreviation Union_mset :: "'a multiset multiset \<Rightarrow> 'a multiset"  ("\<Union>#_" [900] 900)
wenzelm@61955
  1231
  where "\<Union># MM \<equiv> msetsum MM"
blanchet@59813
  1232
nipkow@60495
  1233
lemma set_mset_Union_mset[simp]: "set_mset (\<Union># MM) = (\<Union>M \<in> set_mset MM. set_mset M)"
blanchet@59813
  1234
  by (induct MM) auto
blanchet@59813
  1235
blanchet@59813
  1236
lemma in_Union_mset_iff[iff]: "x \<in># \<Union># MM \<longleftrightarrow> (\<exists>M. M \<in># MM \<and> x \<in># M)"
blanchet@59813
  1237
  by (induct MM) auto
blanchet@59813
  1238
wenzelm@61955
  1239
syntax (ASCII)
wenzelm@61955
  1240
  "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"  ("(3SUM _:#_. _)" [0, 51, 10] 10)
haftmann@51548
  1241
syntax
wenzelm@61955
  1242
  "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"  ("(3\<Sum>_\<in>#_. _)" [0, 51, 10] 10)
haftmann@51548
  1243
translations
wenzelm@61955
  1244
  "\<Sum>i \<in># A. b" \<rightleftharpoons> "CONST msetsum (CONST image_mset (\<lambda>i. b) A)"
haftmann@51548
  1245
haftmann@54868
  1246
context comm_monoid_mult
haftmann@54868
  1247
begin
haftmann@54868
  1248
wenzelm@61605
  1249
sublocale msetprod: comm_monoid_mset times 1
haftmann@61832
  1250
  defines msetprod = msetprod.F ..
haftmann@51548
  1251
haftmann@51548
  1252
lemma msetprod_empty:
haftmann@51548
  1253
  "msetprod {#} = 1"
haftmann@51548
  1254
  by (fact msetprod.empty)
haftmann@51548
  1255
haftmann@51548
  1256
lemma msetprod_singleton:
haftmann@51548
  1257
  "msetprod {#x#} = x"
haftmann@51548
  1258
  by (fact msetprod.singleton)
haftmann@51548
  1259
haftmann@51548
  1260
lemma msetprod_Un:
blanchet@58425
  1261
  "msetprod (A + B) = msetprod A * msetprod B"
haftmann@51548
  1262
  by (fact msetprod.union)
haftmann@51548
  1263
haftmann@60804
  1264
lemma msetprod_replicate_mset [simp]:
haftmann@60804
  1265
  "msetprod (replicate_mset n a) = a ^ n"
haftmann@60804
  1266
  by (induct n) (simp_all add: ac_simps)
haftmann@60804
  1267
haftmann@51548
  1268
lemma setprod_unfold_msetprod:
nipkow@60513
  1269
  "setprod f A = msetprod (image_mset f (mset_set A))"
haftmann@51548
  1270
  by (cases "finite A") (induct A rule: finite_induct, simp_all)
haftmann@51548
  1271
haftmann@51548
  1272
lemma msetprod_multiplicity:
nipkow@60495
  1273
  "msetprod M = setprod (\<lambda>x. x ^ count M x) (set_mset M)"
nipkow@59998
  1274
  by (simp add: fold_mset_def setprod.eq_fold msetprod.eq_fold funpow_times_power comp_def)
haftmann@51548
  1275
haftmann@51548
  1276
end
haftmann@51548
  1277
wenzelm@61955
  1278
syntax (ASCII)
wenzelm@61955
  1279
  "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"  ("(3PROD _:#_. _)" [0, 51, 10] 10)
haftmann@51548
  1280
syntax
wenzelm@61955
  1281
  "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"  ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
haftmann@51548
  1282
translations
wenzelm@61955
  1283
  "\<Prod>i \<in># A. b" \<rightleftharpoons> "CONST msetprod (CONST image_mset (\<lambda>i. b) A)"
haftmann@51548
  1284
haftmann@51548
  1285
lemma (in comm_semiring_1) dvd_msetprod:
haftmann@51548
  1286
  assumes "x \<in># A"
haftmann@51548
  1287
  shows "x dvd msetprod A"
haftmann@51548
  1288
proof -
haftmann@51548
  1289
  from assms have "A = (A - {#x#}) + {#x#}" by simp
haftmann@51548
  1290
  then obtain B where "A = B + {#x#}" ..
haftmann@51548
  1291
  then show ?thesis by simp
haftmann@51548
  1292
qed
haftmann@51548
  1293
haftmann@51548
  1294
wenzelm@60500
  1295
subsection \<open>Alternative representations\<close>
wenzelm@60500
  1296
wenzelm@60500
  1297
subsubsection \<open>Lists\<close>
haftmann@51548
  1298
haftmann@39533
  1299
context linorder
haftmann@39533
  1300
begin
haftmann@39533
  1301
nipkow@60515
  1302
lemma mset_insort [simp]:
nipkow@60515
  1303
  "mset (insort_key k x xs) = {#x#} + mset xs"
haftmann@37107
  1304
  by (induct xs) (simp_all add: ac_simps)
blanchet@58425
  1305
nipkow@60515
  1306
lemma mset_sort [simp]:
nipkow@60515
  1307
  "mset (sort_key k xs) = mset xs"
haftmann@37107
  1308
  by (induct xs) (simp_all add: ac_simps)
haftmann@37107
  1309
wenzelm@60500
  1310
text \<open>
haftmann@34943
  1311
  This lemma shows which properties suffice to show that a function
wenzelm@61585
  1312
  \<open>f\<close> with \<open>f xs = ys\<close> behaves like sort.
wenzelm@60500
  1313
\<close>
haftmann@37074
  1314
haftmann@39533
  1315
lemma properties_for_sort_key:
nipkow@60515
  1316
  assumes "mset ys = mset xs"
wenzelm@60606
  1317
    and "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>x. f k = f x) ys = filter (\<lambda>x. f k = f x) xs"
wenzelm@60606
  1318
    and "sorted (map f ys)"
haftmann@39533
  1319
  shows "sort_key f xs = ys"
wenzelm@60606
  1320
  using assms
wenzelm@46921
  1321
proof (induct xs arbitrary: ys)
haftmann@34943
  1322
  case Nil then show ?case by simp
haftmann@34943
  1323
next
haftmann@34943
  1324
  case (Cons x xs)
haftmann@39533
  1325
  from Cons.prems(2) have
haftmann@40305
  1326
    "\<forall>k \<in> set ys. filter (\<lambda>x. f k = f x) (remove1 x ys) = filter (\<lambda>x. f k = f x) xs"
haftmann@39533
  1327
    by (simp add: filter_remove1)
haftmann@39533
  1328
  with Cons.prems have "sort_key f xs = remove1 x ys"
haftmann@39533
  1329
    by (auto intro!: Cons.hyps simp add: sorted_map_remove1)
haftmann@39533
  1330
  moreover from Cons.prems have "x \<in> set ys"
haftmann@39533
  1331
    by (auto simp add: mem_set_multiset_eq intro!: ccontr)
haftmann@39533
  1332
  ultimately show ?case using Cons.prems by (simp add: insort_key_remove1)
haftmann@34943
  1333
qed
haftmann@34943
  1334
haftmann@39533
  1335
lemma properties_for_sort:
nipkow@60515
  1336
  assumes multiset: "mset ys = mset xs"
wenzelm@60606
  1337
    and "sorted ys"
haftmann@39533
  1338
  shows "sort xs = ys"
haftmann@39533
  1339
proof (rule properties_for_sort_key)
nipkow@60515
  1340
  from multiset show "mset ys = mset xs" .
wenzelm@60500
  1341
  from \<open>sorted ys\<close> show "sorted (map (\<lambda>x. x) ys)" by simp
wenzelm@60678
  1342
  from multiset have "length (filter (\<lambda>y. k = y) ys) = length (filter (\<lambda>x. k = x) xs)" for k
nipkow@60515
  1343
    by (rule mset_eq_length_filter)
wenzelm@60678
  1344
  then have "replicate (length (filter (\<lambda>y. k = y) ys)) k =
wenzelm@60678
  1345
    replicate (length (filter (\<lambda>x. k = x) xs)) k" for k
haftmann@39533
  1346
    by simp
wenzelm@60678
  1347
  then show "k \<in> set ys \<Longrightarrow> filter (\<lambda>y. k = y) ys = filter (\<lambda>x. k = x) xs" for k
haftmann@39533
  1348
    by (simp add: replicate_length_filter)
haftmann@39533
  1349
qed
haftmann@39533
  1350
haftmann@61031
  1351
lemma sort_key_inj_key_eq:
haftmann@61031
  1352
  assumes mset_equal: "mset xs = mset ys"
haftmann@61031
  1353
    and "inj_on f (set xs)"
haftmann@61031
  1354
    and "sorted (map f ys)"
haftmann@61031
  1355
  shows "sort_key f xs = ys"
haftmann@61031
  1356
proof (rule properties_for_sort_key)
haftmann@61031
  1357
  from mset_equal
haftmann@61031
  1358
  show "mset ys = mset xs" by simp
wenzelm@61188
  1359
  from \<open>sorted (map f ys)\<close>
haftmann@61031
  1360
  show "sorted (map f ys)" .
haftmann@61031
  1361
  show "[x\<leftarrow>ys . f k = f x] = [x\<leftarrow>xs . f k = f x]" if "k \<in> set ys" for k
haftmann@61031
  1362
  proof -
haftmann@61031
  1363
    from mset_equal
haftmann@61031
  1364
    have set_equal: "set xs = set ys" by (rule mset_eq_setD)
haftmann@61031
  1365
    with that have "insert k (set ys) = set ys" by auto
wenzelm@61188
  1366
    with \<open>inj_on f (set xs)\<close> have inj: "inj_on f (insert k (set ys))"
haftmann@61031
  1367
      by (simp add: set_equal)
haftmann@61031
  1368
    from inj have "[x\<leftarrow>ys . f k = f x] = filter (HOL.eq k) ys"
haftmann@61031
  1369
      by (auto intro!: inj_on_filter_key_eq)
haftmann@61031
  1370
    also have "\<dots> = replicate (count (mset ys) k) k"
haftmann@61031
  1371
      by (simp add: replicate_count_mset_eq_filter_eq)
haftmann@61031
  1372
    also have "\<dots> = replicate (count (mset xs) k) k"
haftmann@61031
  1373
      using mset_equal by simp
haftmann@61031
  1374
    also have "\<dots> = filter (HOL.eq k) xs"
haftmann@61031
  1375
      by (simp add: replicate_count_mset_eq_filter_eq)
haftmann@61031
  1376
    also have "\<dots> = [x\<leftarrow>xs . f k = f x]"
haftmann@61031
  1377
      using inj by (auto intro!: inj_on_filter_key_eq [symmetric] simp add: set_equal)
haftmann@61031
  1378
    finally show ?thesis .
haftmann@61031
  1379
  qed
haftmann@61031
  1380
qed
haftmann@61031
  1381
haftmann@61031
  1382
lemma sort_key_eq_sort_key:
haftmann@61031
  1383
  assumes "mset xs = mset ys"
haftmann@61031
  1384
    and "inj_on f (set xs)"
haftmann@61031
  1385
  shows "sort_key f xs = sort_key f ys"
haftmann@61031
  1386
  by (rule sort_key_inj_key_eq) (simp_all add: assms)
haftmann@61031
  1387
haftmann@40303
  1388
lemma sort_key_by_quicksort:
haftmann@40303
  1389
  "sort_key f xs = sort_key f [x\<leftarrow>xs. f x < f (xs ! (length xs div 2))]
haftmann@40303
  1390
    @ [x\<leftarrow>xs. f x = f (xs ! (length xs div 2))]
haftmann@40303
  1391
    @ sort_key f [x\<leftarrow>xs. f x > f (xs ! (length xs div 2))]" (is "sort_key f ?lhs = ?rhs")
haftmann@40303
  1392
proof (rule properties_for_sort_key)
nipkow@60515
  1393
  show "mset ?rhs = mset ?lhs"
nipkow@60515
  1394
    by (rule multiset_eqI) (auto simp add: mset_filter)
haftmann@40303
  1395
  show "sorted (map f ?rhs)"
haftmann@40303
  1396
    by (auto simp add: sorted_append intro: sorted_map_same)
haftmann@40303
  1397
next
haftmann@40305
  1398
  fix l
haftmann@40305
  1399
  assume "l \<in> set ?rhs"
haftmann@40346
  1400
  let ?pivot = "f (xs ! (length xs div 2))"
haftmann@40346
  1401
  have *: "\<And>x. f l = f x \<longleftrightarrow> f x = f l" by auto
haftmann@40306
  1402
  have "[x \<leftarrow> sort_key f xs . f x = f l] = [x \<leftarrow> xs. f x = f l]"
haftmann@40305
  1403
    unfolding filter_sort by (rule properties_for_sort_key) (auto intro: sorted_map_same)
haftmann@40346
  1404
  with * have **: "[x \<leftarrow> sort_key f xs . f l = f x] = [x \<leftarrow> xs. f l = f x]" by simp
haftmann@40346
  1405
  have "\<And>x P. P (f x) ?pivot \<and> f l = f x \<longleftrightarrow> P (f l) ?pivot \<and> f l = f x" by auto
haftmann@40346
  1406
  then have "\<And>P. [x \<leftarrow> sort_key f xs . P (f x) ?pivot \<and> f l = f x] =
haftmann@40346
  1407
    [x \<leftarrow> sort_key f xs. P (f l) ?pivot \<and> f l = f x]" by simp
haftmann@40346
  1408
  note *** = this [of "op <"] this [of "op >"] this [of "op ="]
haftmann@40306
  1409
  show "[x \<leftarrow> ?rhs. f l = f x] = [x \<leftarrow> ?lhs. f l = f x]"
haftmann@40305
  1410
  proof (cases "f l" ?pivot rule: linorder_cases)
wenzelm@46730
  1411
    case less
wenzelm@46730
  1412
    then have "f l \<noteq> ?pivot" and "\<not> f l > ?pivot" by auto
wenzelm@46730
  1413
    with less show ?thesis
haftmann@40346
  1414
      by (simp add: filter_sort [symmetric] ** ***)
haftmann@40305
  1415
  next
haftmann@40306
  1416
    case equal then show ?thesis
haftmann@40346
  1417
      by (simp add: * less_le)
haftmann@40305
  1418
  next
wenzelm@46730
  1419
    case greater
wenzelm@46730
  1420
    then have "f l \<noteq> ?pivot" and "\<not> f l < ?pivot" by auto
wenzelm@46730
  1421
    with greater show ?thesis
haftmann@40346
  1422
      by (simp add: filter_sort [symmetric] ** ***)
haftmann@40306
  1423
  qed
haftmann@40303
  1424
qed
haftmann@40303
  1425
haftmann@40303
  1426
lemma sort_by_quicksort:
haftmann@40303
  1427
  "sort xs = sort [x\<leftarrow>xs. x < xs ! (length xs div 2)]
haftmann@40303
  1428
    @ [x\<leftarrow>xs. x = xs ! (length xs div 2)]
haftmann@40303
  1429
    @ sort [x\<leftarrow>xs. x > xs ! (length xs div 2)]" (is "sort ?lhs = ?rhs")
haftmann@40303
  1430
  using sort_key_by_quicksort [of "\<lambda>x. x", symmetric] by simp
haftmann@40303
  1431
wenzelm@60500
  1432
text \<open>A stable parametrized quicksort\<close>
haftmann@40347
  1433
haftmann@40347
  1434
definition part :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'b list \<times> 'b list \<times> 'b list" where
haftmann@40347
  1435
  "part f pivot xs = ([x \<leftarrow> xs. f x < pivot], [x \<leftarrow> xs. f x = pivot], [x \<leftarrow> xs. pivot < f x])"
haftmann@40347
  1436
haftmann@40347
  1437
lemma part_code [code]:
haftmann@40347
  1438
  "part f pivot [] = ([], [], [])"
haftmann@40347
  1439
  "part f pivot (x # xs) = (let (lts, eqs, gts) = part f pivot xs; x' = f x in
haftmann@40347
  1440
     if x' < pivot then (x # lts, eqs, gts)
haftmann@40347
  1441
     else if x' > pivot then (lts, eqs, x # gts)
haftmann@40347
  1442
     else (lts, x # eqs, gts))"
haftmann@40347
  1443
  by (auto simp add: part_def Let_def split_def)
haftmann@40347
  1444
haftmann@40347
  1445
lemma sort_key_by_quicksort_code [code]:
wenzelm@60606
  1446
  "sort_key f xs =
wenzelm@60606
  1447
    (case xs of
wenzelm@60606
  1448
      [] \<Rightarrow> []
haftmann@40347
  1449
    | [x] \<Rightarrow> xs
haftmann@40347
  1450
    | [x, y] \<Rightarrow> (if f x \<le> f y then xs else [y, x])
wenzelm@60606
  1451
    | _ \<Rightarrow>
wenzelm@60606
  1452
        let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
wenzelm@60606
  1453
        in sort_key f lts @ eqs @ sort_key f gts)"
haftmann@40347
  1454
proof (cases xs)
haftmann@40347
  1455
  case Nil then show ?thesis by simp
haftmann@40347
  1456
next
wenzelm@46921
  1457
  case (Cons _ ys) note hyps = Cons show ?thesis
wenzelm@46921
  1458
  proof (cases ys)
haftmann@40347
  1459
    case Nil with hyps show ?thesis by simp
haftmann@40347
  1460
  next
wenzelm@46921
  1461
    case (Cons _ zs) note hyps = hyps Cons show ?thesis
wenzelm@46921
  1462
    proof (cases zs)
haftmann@40347
  1463
      case Nil with hyps show ?thesis by auto
haftmann@40347
  1464
    next
blanchet@58425
  1465
      case Cons
haftmann@40347
  1466
      from sort_key_by_quicksort [of f xs]
haftmann@40347
  1467
      have "sort_key f xs = (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
haftmann@40347
  1468
        in sort_key f lts @ eqs @ sort_key f gts)"
haftmann@40347
  1469
      by (simp only: split_def Let_def part_def fst_conv snd_conv)
haftmann@40347
  1470
      with hyps Cons show ?thesis by (simp only: list.cases)
haftmann@40347
  1471
    qed
haftmann@40347
  1472
  qed
haftmann@40347
  1473
qed
haftmann@40347
  1474
haftmann@39533
  1475
end
haftmann@39533
  1476
haftmann@40347
  1477
hide_const (open) part
haftmann@40347
  1478
nipkow@60515
  1479
lemma mset_remdups_le: "mset (remdups xs) \<le># mset xs"
Mathias@60397
  1480
  by (induct xs) (auto intro: subset_mset.order_trans)
haftmann@34943
  1481
nipkow@60515
  1482
lemma mset_update:
nipkow@60515
  1483
  "i < length ls \<Longrightarrow> mset (ls[i := v]) = mset ls - {#ls ! i#} + {#v#}"
haftmann@34943
  1484
proof (induct ls arbitrary: i)
haftmann@34943
  1485
  case Nil then show ?case by simp
haftmann@34943
  1486
next
haftmann@34943
  1487
  case (Cons x xs)
haftmann@34943
  1488
  show ?case
haftmann@34943
  1489
  proof (cases i)
haftmann@34943
  1490
    case 0 then show ?thesis by simp
haftmann@34943
  1491
  next
haftmann@34943
  1492
    case (Suc i')
haftmann@34943
  1493
    with Cons show ?thesis
haftmann@34943
  1494
      apply simp
haftmann@57512
  1495
      apply (subst add.assoc)
haftmann@57512
  1496
      apply (subst add.commute [of "{#v#}" "{#x#}"])
haftmann@57512
  1497
      apply (subst add.assoc [symmetric])
haftmann@34943
  1498
      apply simp
haftmann@34943
  1499
      apply (rule mset_le_multiset_union_diff_commute)
nipkow@60515
  1500
      apply (simp add: mset_le_single nth_mem_mset)
haftmann@34943
  1501
      done
haftmann@34943
  1502
  qed
haftmann@34943
  1503
qed
haftmann@34943
  1504
nipkow@60515
  1505
lemma mset_swap:
haftmann@34943
  1506
  "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
nipkow@60515
  1507
    mset (ls[j := ls ! i, i := ls ! j]) = mset ls"
nipkow@60515
  1508
  by (cases "i = j") (simp_all add: mset_update nth_mem_mset)
haftmann@34943
  1509
haftmann@34943
  1510
wenzelm@60500
  1511
subsection \<open>The multiset order\<close>
wenzelm@60500
  1512
wenzelm@60500
  1513
subsubsection \<open>Well-foundedness\<close>
wenzelm@10249
  1514
wenzelm@60606
  1515
definition mult1 :: "('a \<times> 'a) set \<Rightarrow> ('a multiset \<times> 'a multiset) set" where
haftmann@37765
  1516
  "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
wenzelm@60607
  1517
      (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r)}"
wenzelm@60606
  1518
wenzelm@60606
  1519
definition mult :: "('a \<times> 'a) set \<Rightarrow> ('a multiset \<times> 'a multiset) set" where
haftmann@37765
  1520
  "mult r = (mult1 r)\<^sup>+"
wenzelm@10249
  1521
berghofe@23751
  1522
lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
nipkow@26178
  1523
by (simp add: mult1_def)
wenzelm@10249
  1524
wenzelm@60608
  1525
lemma less_add:
wenzelm@60608
  1526
  assumes mult1: "(N, M0 + {#a#}) \<in> mult1 r"
wenzelm@60608
  1527
  shows
wenzelm@60608
  1528
    "(\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
wenzelm@60608
  1529
     (\<exists>K. (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r) \<and> N = M0 + K)"
wenzelm@60608
  1530
proof -
wenzelm@60607
  1531
  let ?r = "\<lambda>K a. \<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r"
nipkow@11464
  1532
  let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
wenzelm@60608
  1533
  obtain a' M0' K where M0: "M0 + {#a#} = M0' + {#a'#}"
wenzelm@60608
  1534
    and N: "N = M0' + K"
wenzelm@60608
  1535
    and r: "?r K a'"
wenzelm@60608
  1536
    using mult1 unfolding mult1_def by auto
wenzelm@60608
  1537
  show ?thesis (is "?case1 \<or> ?case2")
wenzelm@60606
  1538
  proof -
wenzelm@60606
  1539
    from M0 consider "M0 = M0'" "a = a'"
wenzelm@60606
  1540
      | K' where "M0 = K' + {#a'#}" "M0' = K' + {#a#}"
wenzelm@60606
  1541
      by atomize_elim (simp only: add_eq_conv_ex)
wenzelm@18258
  1542
    then show ?thesis
wenzelm@60606
  1543
    proof cases
wenzelm@60606
  1544
      case 1
nipkow@11464
  1545
      with N r have "?r K a \<and> N = M0 + K" by simp
wenzelm@60606
  1546
      then have ?case2 ..
wenzelm@60606
  1547
      then show ?thesis ..
wenzelm@10249
  1548
    next
wenzelm@60606
  1549
      case 2
wenzelm@60606
  1550
      from N 2(2) have n: "N = K' + K + {#a#}" by (simp add: ac_simps)
wenzelm@60606
  1551
      with r 2(1) have "?R (K' + K) M0" by blast
wenzelm@60608
  1552
      with n have ?case1 by (simp add: mult1_def)
wenzelm@60606
  1553
      then show ?thesis ..
wenzelm@10249
  1554
    qed
wenzelm@10249
  1555
  qed
wenzelm@10249
  1556
qed
wenzelm@10249
  1557
wenzelm@60608
  1558
lemma all_accessible:
wenzelm@60608
  1559
  assumes "wf r"
wenzelm@60608
  1560
  shows "\<forall>M. M \<in> Wellfounded.acc (mult1 r)"
wenzelm@10249
  1561
proof
wenzelm@10249
  1562
  let ?R = "mult1 r"
haftmann@54295
  1563
  let ?W = "Wellfounded.acc ?R"
wenzelm@10249
  1564
  {
wenzelm@10249
  1565
    fix M M0 a
berghofe@23751
  1566
    assume M0: "M0 \<in> ?W"
wenzelm@60606
  1567
      and wf_hyp: "\<And>b. (b, a) \<in> r \<Longrightarrow> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
wenzelm@60606
  1568
      and acc_hyp: "\<forall>M. (M, M0) \<in> ?R \<longrightarrow> M + {#a#} \<in> ?W"
berghofe@23751
  1569
    have "M0 + {#a#} \<in> ?W"
berghofe@23751
  1570
    proof (rule accI [of "M0 + {#a#}"])
wenzelm@10249
  1571
      fix N
berghofe@23751
  1572
      assume "(N, M0 + {#a#}) \<in> ?R"
wenzelm@60608
  1573
      then consider M where "(M, M0) \<in> ?R" "N = M + {#a#}"
wenzelm@60608
  1574
        | K where "\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r" "N = M0 + K"
wenzelm@60608
  1575
        by atomize_elim (rule less_add)
berghofe@23751
  1576
      then show "N \<in> ?W"
wenzelm@60608
  1577
      proof cases
wenzelm@60608
  1578
        case 1
wenzelm@60606
  1579
        from acc_hyp have "(M, M0) \<in> ?R \<longrightarrow> M + {#a#} \<in> ?W" ..
wenzelm@60500
  1580
        from this and \<open>(M, M0) \<in> ?R\<close> have "M + {#a#} \<in> ?W" ..
wenzelm@60608
  1581
        then show "N \<in> ?W" by (simp only: \<open>N = M + {#a#}\<close>)
wenzelm@10249
  1582
      next
wenzelm@60608
  1583
        case 2
wenzelm@60608
  1584
        from this(1) have "M0 + K \<in> ?W"
wenzelm@10249
  1585
        proof (induct K)
wenzelm@18730
  1586
          case empty
berghofe@23751
  1587
          from M0 show "M0 + {#} \<in> ?W" by simp
wenzelm@18730
  1588
        next
wenzelm@18730
  1589
          case (add K x)
berghofe@23751
  1590
          from add.prems have "(x, a) \<in> r" by simp
berghofe@23751
  1591
          with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
berghofe@23751
  1592
          moreover from add have "M0 + K \<in> ?W" by simp
berghofe@23751
  1593
          ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
haftmann@57512
  1594
          then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: add.assoc)
wenzelm@10249
  1595
        qed
wenzelm@60608
  1596
        then show "N \<in> ?W" by (simp only: 2(2))
wenzelm@10249
  1597
      qed
wenzelm@10249
  1598
    qed
wenzelm@10249
  1599
  } note tedious_reasoning = this
wenzelm@10249
  1600
wenzelm@60608
  1601
  show "M \<in> ?W" for M
wenzelm@10249
  1602
  proof (induct M)
berghofe@23751
  1603
    show "{#} \<in> ?W"
wenzelm@10249
  1604
    proof (rule accI)
berghofe@23751
  1605
      fix b assume "(b, {#}) \<in> ?R"
berghofe@23751
  1606
      with not_less_empty show "b \<in> ?W" by contradiction
wenzelm@10249
  1607
    qed
wenzelm@10249
  1608
berghofe@23751
  1609
    fix M a assume "M \<in> ?W"
wenzelm@60608
  1610
    from \<open>wf r\<close> have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
wenzelm@10249
  1611
    proof induct
wenzelm@10249
  1612
      fix a
wenzelm@60606
  1613
      assume r: "\<And>b. (b, a) \<in> r \<Longrightarrow> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
berghofe@23751
  1614
      show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
wenzelm@10249
  1615
      proof
berghofe@23751
  1616
        fix M assume "M \<in> ?W"
berghofe@23751
  1617
        then show "M + {#a#} \<in> ?W"
wenzelm@23373
  1618
          by (rule acc_induct) (rule tedious_reasoning [OF _ r])
wenzelm@10249
  1619
      qed
wenzelm@10249
  1620
    qed
wenzelm@60500
  1621
    from this and \<open>M \<in> ?W\<close> show "M + {#a#} \<in> ?W" ..
wenzelm@10249
  1622
  qed
wenzelm@10249
  1623
qed
wenzelm@10249
  1624
wenzelm@60606
  1625
theorem wf_mult1: "wf r \<Longrightarrow> wf (mult1 r)"
nipkow@26178
  1626
by (rule acc_wfI) (rule all_accessible)
wenzelm@10249
  1627
wenzelm@60606
  1628
theorem wf_mult: "wf r \<Longrightarrow> wf (mult r)"
nipkow@26178
  1629
unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
wenzelm@10249
  1630
wenzelm@10249
  1631
wenzelm@60500
  1632
subsubsection \<open>Closure-free presentation\<close>
wenzelm@60500
  1633
wenzelm@60500
  1634
text \<open>One direction.\<close>
wenzelm@10249
  1635
wenzelm@10249
  1636
lemma mult_implies_one_step:
wenzelm@60606
  1637
  "trans r \<Longrightarrow> (M, N) \<in> mult r \<Longrightarrow>
nipkow@11464
  1638
    \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
nipkow@60495
  1639
    (\<forall>k \<in> set_mset K. \<exists>j \<in> set_mset J. (k, j) \<in> r)"
nipkow@60495
  1640
apply (unfold mult_def mult1_def set_mset_def)
nipkow@26178
  1641
apply (erule converse_trancl_induct, clarify)
nipkow@26178
  1642
 apply (rule_tac x = M0 in exI, simp, clarify)
wenzelm@60607
  1643
apply (case_tac "a \<in># K")
nipkow@26178
  1644
 apply (rule_tac x = I in exI)
nipkow@26178
  1645
 apply (simp (no_asm))
nipkow@26178
  1646
 apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
haftmann@57512
  1647
 apply (simp (no_asm_simp) add: add.assoc [symmetric])
wenzelm@59807
  1648
 apply (drule_tac f = "\<lambda>M. M - {#a#}" and x="S + T" for S T in arg_cong)
nipkow@26178
  1649
 apply (simp add: diff_union_single_conv)
nipkow@26178
  1650
 apply (simp (no_asm_use) add: trans_def)
nipkow@26178
  1651
 apply blast
wenzelm@60607
  1652
apply (subgoal_tac "a \<in># I")
nipkow@26178
  1653
 apply (rule_tac x = "I - {#a#}" in exI)
nipkow@26178
  1654
 apply (rule_tac x = "J + {#a#}" in exI)
nipkow@26178
  1655
 apply (rule_tac x = "K + Ka" in exI)
nipkow@26178
  1656
 apply (rule conjI)
nipkow@39302
  1657
  apply (simp add: multiset_eq_iff split: nat_diff_split)
nipkow@26178
  1658
 apply (rule conjI)
wenzelm@59807
  1659
  apply (drule_tac f = "\<lambda>M. M - {#a#}" and x="S + T" for S T in arg_cong, simp)
nipkow@39302
  1660
  apply (simp add: multiset_eq_iff split: nat_diff_split)
nipkow@26178
  1661
 apply (simp (no_asm_use) add: trans_def)
nipkow@26178
  1662
 apply blast
wenzelm@60607
  1663
apply (subgoal_tac "a \<in># (M0 + {#a#})")
nipkow@26178
  1664
 apply simp
nipkow@26178
  1665
apply (simp (no_asm))
nipkow@26178
  1666
done
wenzelm@10249
  1667
wenzelm@10249
  1668
lemma one_step_implies_mult_aux:
wenzelm@60678
  1669
  "\<forall>I J K. size J = n \<and> J \<noteq> {#} \<and> (\<forall>k \<in> set_mset K. \<exists>j \<in> set_mset J. (k, j) \<in> r)
wenzelm@60678
  1670
    \<longrightarrow> (I + K, I + J) \<in> mult r"
wenzelm@60678
  1671
apply (induct n)
wenzelm@60678
  1672
 apply auto
nipkow@26178
  1673
apply (frule size_eq_Suc_imp_eq_union, clarify)
nipkow@26178
  1674
apply (rename_tac "J'", simp)
nipkow@26178
  1675
apply (erule notE, auto)
nipkow@26178
  1676
apply (case_tac "J' = {#}")
nipkow@26178
  1677
 apply (simp add: mult_def)
nipkow@26178
  1678
 apply (rule r_into_trancl)
nipkow@60495
  1679
 apply (simp add: mult1_def set_mset_def, blast)
wenzelm@60500
  1680
txt \<open>Now we know @{term "J' \<noteq> {#}"}.\<close>
nipkow@26178
  1681
apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
nipkow@60495
  1682
apply (erule_tac P = "\<forall>k \<in> set_mset K. P k" for P in rev_mp)
nipkow@26178
  1683
apply (erule ssubst)
nipkow@26178
  1684
apply (simp add: Ball_def, auto)
nipkow@26178
  1685
apply (subgoal_tac
wenzelm@60607
  1686
  "((I + {# x \<in># K. (x, a) \<in> r #}) + {# x \<in># K. (x, a) \<notin> r #},
wenzelm@60607
  1687
    (I + {# x \<in># K. (x, a) \<in> r #}) + J') \<in> mult r")
nipkow@26178
  1688
 prefer 2
nipkow@26178
  1689
 apply force
haftmann@57512
  1690
apply (simp (no_asm_use) add: add.assoc [symmetric] mult_def)
nipkow@26178
  1691
apply (erule trancl_trans)
nipkow@26178
  1692
apply (rule r_into_trancl)
nipkow@60495
  1693
apply (simp add: mult1_def set_mset_def)
nipkow@26178
  1694
apply (rule_tac x = a in exI)
nipkow@26178
  1695
apply (rule_tac x = "I + J'" in exI)
haftmann@57514
  1696
apply (simp add: ac_simps)
nipkow@26178
  1697
done
wenzelm@10249
  1698
wenzelm@17161
  1699
lemma one_step_implies_mult:
wenzelm@60606
  1700
  "trans r \<Longrightarrow> J \<noteq> {#} \<Longrightarrow> \<forall>k \<in> set_mset K. \<exists>j \<in> set_mset J. (k, j) \<in> r
wenzelm@60606
  1701
    \<Longrightarrow> (I + K, I + J) \<in> mult r"
nipkow@26178
  1702
using one_step_implies_mult_aux by blast
wenzelm@10249
  1703
wenzelm@10249
  1704
wenzelm@60500
  1705
subsubsection \<open>Partial-order properties\<close>
wenzelm@10249
  1706
wenzelm@61955
  1707
definition less_multiset :: "'a::order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "#\<subset>#" 50)
wenzelm@61955
  1708
  where "M' #\<subset># M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"
wenzelm@61955
  1709
wenzelm@61955
  1710
definition le_multiset :: "'a::order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "#\<subseteq>#" 50)
wenzelm@61955
  1711
  where "M' #\<subseteq># M \<longleftrightarrow> M' #\<subset># M \<or> M' = M"
wenzelm@61955
  1712
wenzelm@61955
  1713
notation (ASCII)
wenzelm@61955
  1714
  less_multiset (infix "#<#" 50) and
wenzelm@61955
  1715
  le_multiset (infix "#<=#" 50)
wenzelm@10249
  1716
haftmann@35268
  1717
interpretation multiset_order: order le_multiset less_multiset
haftmann@35268
  1718
proof -
wenzelm@60606
  1719
  have irrefl: "\<not> M #\<subset># M" for M :: "'a multiset"
haftmann@35268
  1720
  proof
blanchet@59958
  1721
    assume "M #\<subset># M"
haftmann@35268
  1722
    then have MM: "(M, M) \<in> mult {(x, y). x < y}" by (simp add: less_multiset_def)
haftmann@35268
  1723
    have "trans {(x'::'a, x). x' < x}"
haftmann@35268
  1724
      by (rule transI) simp
haftmann@35268
  1725
    moreover note MM
haftmann@35268
  1726
    ultimately have "\<exists>I J K. M = I + J \<and> M = I + K
nipkow@60495
  1727
      \<and> J \<noteq> {#} \<and> (\<forall>k\<in>set_mset K. \<exists>j\<in>set_mset J. (k, j) \<in> {(x, y). x < y})"
haftmann@35268
  1728
      by (rule mult_implies_one_step)
haftmann@35268
  1729
    then obtain I J K where "M = I + J" and "M = I + K"
nipkow@60495
  1730
      and "J \<noteq> {#}" and "(\<forall>k\<in>set_mset K. \<exists>j\<in>set_mset J. (k, j) \<in> {(x, y). x < y})" by blast
wenzelm@60678
  1731
    then have *: "K \<noteq> {#}" and **: "\<forall>k\<in>set_mset K. \<exists>j\<in>set_mset K. k < j" by auto
nipkow@60495
  1732
    have "finite (set_mset K)" by simp
wenzelm@60678
  1733
    moreover note **
nipkow@60495
  1734
    ultimately have "set_mset K = {}"
haftmann@35268
  1735
      by (induct rule: finite_induct) (auto intro: order_less_trans)
wenzelm@60678
  1736
    with * show False by simp
haftmann@35268
  1737
  qed
wenzelm@60678
  1738
  have trans: "K #\<subset># M \<Longrightarrow> M #\<subset># N \<Longrightarrow> K #\<subset># N" for K M N :: "'a multiset"
haftmann@35268
  1739
    unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
wenzelm@46921
  1740
  show "class.order (le_multiset :: 'a multiset \<Rightarrow> _) less_multiset"
wenzelm@60678
  1741
    by standard (auto simp add: le_multiset_def irrefl dest: trans)
haftmann@35268
  1742
qed
wenzelm@10249
  1743
wenzelm@60678
  1744
lemma mult_less_irrefl [elim!]:
wenzelm@60678
  1745
  fixes M :: "'a::order multiset"
wenzelm@60678
  1746
  shows "M #\<subset># M \<Longrightarrow> R"
wenzelm@46730
  1747
  by simp
haftmann@26567
  1748
wenzelm@10249
  1749
wenzelm@60500
  1750
subsubsection \<open>Monotonicity of multiset union\<close>
wenzelm@10249
  1751
wenzelm@60606
  1752
lemma mult1_union: "(B, D) \<in> mult1 r \<Longrightarrow> (C + B, C + D) \<in> mult1 r"
nipkow@26178
  1753
apply (unfold mult1_def)
nipkow@26178
  1754
apply auto
nipkow@26178
  1755
apply (rule_tac x = a in exI)
nipkow@26178
  1756
apply (rule_tac x = "C + M0" in exI)
haftmann@57512
  1757
apply (simp add: add.assoc)
nipkow@26178
  1758
done
wenzelm@10249
  1759
wenzelm@60606
  1760
lemma union_less_mono2: "B #\<subset># D \<Longrightarrow> C + B #\<subset># C + (D::'a::order multiset)"
nipkow@26178
  1761
apply (unfold less_multiset_def mult_def)
nipkow@26178
  1762
apply (erule trancl_induct)
noschinl@40249
  1763
 apply (blast intro: mult1_union)
noschinl@40249
  1764
apply (blast intro: mult1_union trancl_trans)
nipkow@26178
  1765
done
wenzelm@10249
  1766
wenzelm@60606
  1767
lemma union_less_mono1: "B #\<subset># D \<Longrightarrow> B + C #\<subset># D + (C::'a::order multiset)"
haftmann@57512
  1768
apply (subst add.commute [of B C])
haftmann@57512
  1769
apply (subst add.commute [of D C])
nipkow@26178
  1770
apply (erule union_less_mono2)
nipkow@26178
  1771
done
wenzelm@10249
  1772
wenzelm@17161
  1773
lemma union_less_mono:
wenzelm@60606
  1774
  fixes A B C D :: "'a::order multiset"
wenzelm@60606
  1775
  shows "A #\<subset># C \<Longrightarrow> B #\<subset># D \<Longrightarrow> A + B #\<subset># C + D"
haftmann@35268
  1776
  by (blast intro!: union_less_mono1 union_less_mono2 multiset_order.less_trans)
wenzelm@10249
  1777
haftmann@35268
  1778
interpretation multiset_order: ordered_ab_semigroup_add plus le_multiset less_multiset
wenzelm@60678
  1779
  by standard (auto simp add: le_multiset_def intro: union_less_mono2)
wenzelm@26145
  1780
paulson@15072
  1781
wenzelm@60500
  1782
subsubsection \<open>Termination proofs with multiset orders\<close>
krauss@29125
  1783
krauss@29125
  1784
lemma multi_member_skip: "x \<in># XS \<Longrightarrow> x \<in># {# y #} + XS"
krauss@29125
  1785
  and multi_member_this: "x \<in># {# x #} + XS"
krauss@29125
  1786
  and multi_member_last: "x \<in># {# x #}"
krauss@29125
  1787
  by auto
krauss@29125
  1788
krauss@29125
  1789
definition "ms_strict = mult pair_less"
haftmann@37765
  1790
definition "ms_weak = ms_strict \<union> Id"
krauss@29125
  1791
krauss@29125
  1792
lemma ms_reduction_pair: "reduction_pair (ms_strict, ms_weak)"
krauss@29125
  1793
unfolding reduction_pair_def ms_strict_def ms_weak_def pair_less_def
krauss@29125
  1794
by (auto intro: wf_mult1 wf_trancl simp: mult_def)
krauss@29125
  1795
krauss@29125
  1796
lemma smsI:
nipkow@60495
  1797
  "(set_mset A, set_mset B) \<in> max_strict \<Longrightarrow> (Z + A, Z + B) \<in> ms_strict"
krauss@29125
  1798
  unfolding ms_strict_def
krauss@29125
  1799
by (rule one_step_implies_mult) (auto simp add: max_strict_def pair_less_def elim!:max_ext.cases)
krauss@29125
  1800
krauss@29125
  1801
lemma wmsI:
nipkow@60495
  1802
  "(set_mset A, set_mset B) \<in> max_strict \<or> A = {#} \<and> B = {#}
krauss@29125
  1803
  \<Longrightarrow> (Z + A, Z + B) \<in> ms_weak"
krauss@29125
  1804
unfolding ms_weak_def ms_strict_def
krauss@29125
  1805
by (auto simp add: pair_less_def max_strict_def elim!:max_ext.cases intro: one_step_implies_mult)
krauss@29125
  1806
krauss@29125
  1807
inductive pw_leq
krauss@29125
  1808
where
krauss@29125
  1809
  pw_leq_empty: "pw_leq {#} {#}"
krauss@29125
  1810
| pw_leq_step:  "\<lbrakk>(x,y) \<in> pair_leq; pw_leq X Y \<rbrakk> \<Longrightarrow> pw_leq ({#x#} + X) ({#y#} + Y)"
krauss@29125
  1811
krauss@29125
  1812
lemma pw_leq_lstep:
krauss@29125
  1813
  "(x, y) \<in> pair_leq \<Longrightarrow> pw_leq {#x#} {#y#}"
krauss@29125
  1814
by (drule pw_leq_step) (rule pw_leq_empty, simp)
krauss@29125
  1815
krauss@29125
  1816
lemma pw_leq_split:
krauss@29125
  1817
  assumes "pw_leq X Y"
nipkow@60495
  1818
  shows "\<exists>A B Z. X = A + Z \<and> Y = B + Z \<and> ((set_mset A, set_mset B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
krauss@29125
  1819
  using assms
wenzelm@60606
  1820
proof induct
krauss@29125
  1821
  case pw_leq_empty thus ?case by auto
krauss@29125
  1822
next
krauss@29125
  1823
  case (pw_leq_step x y X Y)
krauss@29125
  1824
  then obtain A B Z where
blanchet@58425
  1825
    [simp]: "X = A + Z" "Y = B + Z"
nipkow@60495
  1826
      and 1[simp]: "(set_mset A, set_mset B) \<in> max_strict \<or> (B = {#} \<and> A = {#})"
krauss@29125
  1827
    by auto
wenzelm@60606
  1828
  from pw_leq_step consider "x = y" | "(x, y) \<in> pair_less"
krauss@29125
  1829
    unfolding pair_leq_def by auto
krauss@29125
  1830
  thus ?case
wenzelm@60606
  1831
  proof cases
wenzelm@60606
  1832
    case [simp]: 1
wenzelm@60606
  1833
    have "{#x#} + X = A + ({#y#}+Z) \<and> {#y#} + Y = B + ({#y#}+Z) \<and>
wenzelm@60606
  1834
      ((set_mset A, set_mset B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
haftmann@57514
  1835
      by (auto simp: ac_simps)
wenzelm@60606
  1836
    thus ?thesis by blast
krauss@29125
  1837
  next
wenzelm@60606
  1838
    case 2
krauss@29125
  1839
    let ?A' = "{#x#} + A" and ?B' = "{#y#} + B"
krauss@29125
  1840
    have "{#x#} + X = ?A' + Z"
krauss@29125
  1841
      "{#y#} + Y = ?B' + Z"
haftmann@57514
  1842
      by (auto simp add: ac_simps)
blanchet@58425
  1843
    moreover have
nipkow@60495
  1844
      "(set_mset ?A', set_mset ?B') \<in> max_strict"
wenzelm@60606
  1845
      using 1 2 unfolding max_strict_def
krauss@29125
  1846
      by (auto elim!: max_ext.cases)
krauss@29125
  1847
    ultimately show ?thesis by blast
krauss@29125
  1848
  qed
krauss@29125
  1849
qed
krauss@29125
  1850
blanchet@58425
  1851
lemma
krauss@29125
  1852
  assumes pwleq: "pw_leq Z Z'"
nipkow@60495
  1853
  shows ms_strictI: "(set_mset A, set_mset B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_strict"
wenzelm@60606
  1854
    and ms_weakI1:  "(set_mset A, set_mset B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_weak"
wenzelm@60606
  1855
    and ms_weakI2:  "(Z + {#}, Z' + {#}) \<in> ms_weak"
krauss@29125
  1856
proof -
blanchet@58425
  1857
  from pw_leq_split[OF pwleq]
krauss@29125
  1858
  obtain A' B' Z''
krauss@29125
  1859
    where [simp]: "Z = A' + Z''" "Z' = B' + Z''"
nipkow@60495
  1860
    and mx_or_empty: "(set_mset A', set_mset B') \<in> max_strict \<or> (A' = {#} \<and> B' = {#})"
krauss@29125
  1861
    by blast
krauss@29125
  1862
  {
nipkow@60495
  1863
    assume max: "(set_mset A, set_mset B) \<in> max_strict"
krauss@29125
  1864
    from mx_or_empty
krauss@29125
  1865
    have "(Z'' + (A + A'), Z'' + (B + B')) \<in> ms_strict"
krauss@29125
  1866
    proof
nipkow@60495
  1867
      assume max': "(set_mset A', set_mset B') \<in> max_strict"
nipkow@60495
  1868
      with max have "(set_mset (A + A'), set_mset (B + B')) \<in> max_strict"
krauss@29125
  1869
        by (auto simp: max_strict_def intro: max_ext_additive)
blanchet@58425
  1870
      thus ?thesis by (rule smsI)
krauss@29125
  1871
    next
krauss@29125
  1872
      assume [simp]: "A' = {#} \<and> B' = {#}"
krauss@29125
  1873
      show ?thesis by (rule smsI) (auto intro: max)
krauss@29125
  1874
    qed
wenzelm@60606
  1875
    thus "(Z + A, Z' + B) \<in> ms_strict" by (simp add: ac_simps)
krauss@29125
  1876
    thus "(Z + A, Z' + B) \<in> ms_weak" by (simp add: ms_weak_def)
krauss@29125
  1877
  }
krauss@29125
  1878
  from mx_or_empty
krauss@29125
  1879
  have "(Z'' + A', Z'' + B') \<in> ms_weak" by (rule wmsI)
haftmann@57514
  1880
  thus "(Z + {#}, Z' + {#}) \<in> ms_weak" by (simp add:ac_simps)
krauss@29125
  1881
qed
krauss@29125
  1882
nipkow@39301
  1883
lemma empty_neutral: "{#} + x = x" "x + {#} = x"
krauss@29125
  1884
and nonempty_plus: "{# x #} + rs \<noteq> {#}"
krauss@29125
  1885
and nonempty_single: "{# x #} \<noteq> {#}"
krauss@29125
  1886
by auto
krauss@29125
  1887
wenzelm@60500
  1888
setup \<open>
wenzelm@60606
  1889
  let
wenzelm@60606
  1890
    fun msetT T = Type (@{type_name multiset}, [T]);
wenzelm@60606
  1891
wenzelm@60606
  1892
    fun mk_mset T [] = Const (@{const_abbrev Mempty}, msetT T)
wenzelm@60606
  1893
      | mk_mset T [x] = Const (@{const_name single}, T --> msetT T) $ x
wenzelm@60606
  1894
      | mk_mset T (x :: xs) =
wenzelm@60606
  1895
            Const (@{const_name plus}, msetT T --> msetT T --> msetT T) $
wenzelm@60606
  1896
                  mk_mset T [x] $ mk_mset T xs
wenzelm@60606
  1897
wenzelm@60752
  1898
    fun mset_member_tac ctxt m i =
wenzelm@60606
  1899
      if m <= 0 then
wenzelm@60752
  1900
        resolve_tac ctxt @{thms multi_member_this} i ORELSE
wenzelm@60752
  1901
        resolve_tac ctxt @{thms multi_member_last} i
wenzelm@60606
  1902
      else
wenzelm@60752
  1903
        resolve_tac ctxt @{thms multi_member_skip} i THEN mset_member_tac ctxt (m - 1) i
wenzelm@60752
  1904
wenzelm@60752
  1905
    fun mset_nonempty_tac ctxt =
wenzelm@60752
  1906
      resolve_tac ctxt @{thms nonempty_plus} ORELSE'
wenzelm@60752
  1907
      resolve_tac ctxt @{thms nonempty_single}
krauss@29125
  1908
wenzelm@60606
  1909
    fun regroup_munion_conv ctxt =
wenzelm@60606
  1910
      Function_Lib.regroup_conv ctxt @{const_abbrev Mempty} @{const_name plus}
wenzelm@60606
  1911
        (map (fn t => t RS eq_reflection) (@{thms ac_simps} @ @{thms empty_neutral}))
wenzelm@60606
  1912
wenzelm@60752
  1913
    fun unfold_pwleq_tac ctxt i =
wenzelm@60752
  1914
      (resolve_tac ctxt @{thms pw_leq_step} i THEN (fn st => unfold_pwleq_tac ctxt (i + 1) st))
wenzelm@60752
  1915
        ORELSE (resolve_tac ctxt @{thms pw_leq_lstep} i)
wenzelm@60752
  1916
        ORELSE (resolve_tac ctxt @{thms pw_leq_empty} i)
wenzelm@60606
  1917
wenzelm@60606
  1918
    val set_mset_simps = [@{thm set_mset_empty}, @{thm set_mset_single}, @{thm set_mset_union},
wenzelm@60606
  1919
                        @{thm Un_insert_left}, @{thm Un_empty_left}]
wenzelm@60606
  1920
  in
wenzelm@60606
  1921
    ScnpReconstruct.multiset_setup (ScnpReconstruct.Multiset
wenzelm@60606
  1922
    {
wenzelm@60606
  1923
      msetT=msetT, mk_mset=mk_mset, mset_regroup_conv=regroup_munion_conv,
wenzelm@60606
  1924
      mset_member_tac=mset_member_tac, mset_nonempty_tac=mset_nonempty_tac,
wenzelm@60606
  1925
      mset_pwleq_tac=unfold_pwleq_tac, set_of_simps=set_mset_simps,
wenzelm@60606
  1926
      smsI'= @{thm ms_strictI}, wmsI2''= @{thm ms_weakI2}, wmsI1= @{thm ms_weakI1},
wenzelm@60752
  1927
      reduction_pair = @{thm ms_reduction_pair}
wenzelm@60606
  1928
    })
wenzelm@60606
  1929
  end
wenzelm@60500
  1930
\<close>
wenzelm@60500
  1931
wenzelm@60500
  1932
wenzelm@60500
  1933
subsection \<open>Legacy theorem bindings\<close>
haftmann@34943
  1934
nipkow@39302
  1935
lemmas multi_count_eq = multiset_eq_iff [symmetric]
haftmann@34943
  1936
haftmann@34943
  1937
lemma union_commute: "M + N = N + (M::'a multiset)"
haftmann@57512
  1938
  by (fact add.commute)
haftmann@34943
  1939
haftmann@34943
  1940
lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
haftmann@57512
  1941
  by (fact add.assoc)
haftmann@34943
  1942
haftmann@34943
  1943
lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
haftmann@57512
  1944
  by (fact add.left_commute)
haftmann@34943
  1945
haftmann@34943
  1946
lemmas union_ac = union_assoc union_commute union_lcomm
haftmann@34943
  1947
haftmann@34943
  1948
lemma union_right_cancel: "M + K = N + K \<longleftrightarrow> M = (N::'a multiset)"
haftmann@34943
  1949
  by (fact add_right_cancel)
haftmann@34943
  1950
haftmann@34943
  1951
lemma union_left_cancel: "K + M = K + N \<longleftrightarrow> M = (N::'a multiset)"
haftmann@34943
  1952
  by (fact add_left_cancel)
haftmann@34943
  1953
haftmann@34943
  1954
lemma multi_union_self_other_eq: "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"
haftmann@59557
  1955
  by (fact add_left_imp_eq)
haftmann@34943
  1956
Mathias@60397
  1957
lemma mset_less_trans: "(M::'a multiset) <# K \<Longrightarrow> K <# N \<Longrightarrow> M <# N"
Mathias@60397
  1958
  by (fact subset_mset.less_trans)
haftmann@35268
  1959
haftmann@35268
  1960
lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
Mathias@60397
  1961
  by (fact subset_mset.inf.commute)
haftmann@35268
  1962
haftmann@35268
  1963
lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
Mathias@60397
  1964
  by (fact subset_mset.inf.assoc [symmetric])
haftmann@35268
  1965
haftmann@35268
  1966
lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
Mathias@60397
  1967
  by (fact subset_mset.inf.left_commute)
haftmann@35268
  1968
haftmann@35268
  1969
lemmas multiset_inter_ac =
haftmann@35268
  1970
  multiset_inter_commute
haftmann@35268
  1971
  multiset_inter_assoc
haftmann@35268
  1972
  multiset_inter_left_commute
haftmann@35268
  1973
wenzelm@60606
  1974
lemma mult_less_not_refl: "\<not> M #\<subset># (M::'a::order multiset)"
haftmann@35268
  1975
  by (fact multiset_order.less_irrefl)
haftmann@35268
  1976
wenzelm@60606
  1977
lemma mult_less_trans: "K #\<subset># M \<Longrightarrow> M #\<subset># N \<Longrightarrow> K #\<subset># (N::'a::order multiset)"
haftmann@35268
  1978
  by (fact multiset_order.less_trans)
blanchet@58425
  1979
wenzelm@60606
  1980
lemma mult_less_not_sym: "M #\<subset># N \<Longrightarrow> \<not> N #\<subset># (M::'a::order multiset)"
haftmann@35268
  1981
  by (fact multiset_order.less_not_sym)
haftmann@35268
  1982
wenzelm@60606
  1983
lemma mult_less_asym: "M #\<subset># N \<Longrightarrow> (\<not> P \<Longrightarrow> N #\<subset># (M::'a::order multiset)) \<Longrightarrow> P"
haftmann@35268
  1984
  by (fact multiset_order.less_asym)
haftmann@34943
  1985
wenzelm@60500
  1986
declaration \<open>
wenzelm@60606
  1987
  let
wenzelm@60606
  1988
    fun multiset_postproc _ maybe_name all_values (T as Type (_, [elem_T])) (Const _ $ t') =
wenzelm@60606
  1989
          let
wenzelm@60606
  1990
            val (maybe_opt, ps) =
wenzelm@60606
  1991
              Nitpick_Model.dest_plain_fun t'
wenzelm@60606
  1992
              ||> op ~~
wenzelm@60606
  1993
              ||> map (apsnd (snd o HOLogic.dest_number))
wenzelm@60606
  1994
            fun elems_for t =
wenzelm@60606
  1995
              (case AList.lookup (op =) ps t of
wenzelm@60606
  1996
                SOME n => replicate n t
wenzelm@60606
  1997
              | NONE => [Const (maybe_name, elem_T --> elem_T) $ t])
wenzelm@60606
  1998
          in
wenzelm@60606
  1999
            (case maps elems_for (all_values elem_T) @
blanchet@61333
  2000
                 (if maybe_opt then [Const (Nitpick_Model.unrep_mixfix (), elem_T)] else []) of
wenzelm@60606
  2001
              [] => Const (@{const_name zero_class.zero}, T)
wenzelm@60606
  2002
            | ts =>
wenzelm@60606
  2003
                foldl1 (fn (t1, t2) =>
wenzelm@60606
  2004
                    Const (@{const_name plus_class.plus}, T --> T --> T) $ t1 $ t2)
wenzelm@60606
  2005
                  (map (curry (op $) (Const (@{const_name single}, elem_T --> T))) ts))
wenzelm@60606
  2006
          end
wenzelm@60606
  2007
      | multiset_postproc _ _ _ _ t = t
wenzelm@60606
  2008
  in Nitpick_Model.register_term_postprocessor @{typ "'a multiset"} multiset_postproc end
wenzelm@60500
  2009
\<close>
wenzelm@60500
  2010
wenzelm@60500
  2011
wenzelm@60500
  2012
subsection \<open>Naive implementation using lists\<close>
haftmann@51600
  2013
nipkow@60515
  2014
code_datatype mset
haftmann@51600
  2015
wenzelm@60606
  2016
lemma [code]: "{#} = mset []"
haftmann@51600
  2017
  by simp
haftmann@51600
  2018
wenzelm@60606
  2019
lemma [code]: "{#x#} = mset [x]"
haftmann@51600
  2020
  by simp
haftmann@51600
  2021
wenzelm@60606
  2022
lemma union_code [code]: "mset xs + mset ys = mset (xs @ ys)"
haftmann@51600
  2023
  by simp
haftmann@51600
  2024
wenzelm@60606
  2025
lemma [code]: "image_mset f (mset xs) = mset (map f xs)"
nipkow@60515
  2026
  by (simp add: mset_map)
haftmann@51600
  2027
wenzelm@60606
  2028
lemma [code]: "filter_mset f (mset xs) = mset (filter f xs)"
nipkow@60515
  2029
  by (simp add: mset_filter)
haftmann@51600
  2030
wenzelm@60606
  2031
lemma [code]: "mset xs - mset ys = mset (fold remove1 ys xs)"
haftmann@51600
  2032
  by (rule sym, induct ys arbitrary: xs) (simp_all add: diff_add diff_right_commute)
haftmann@51600
  2033
haftmann@51600
  2034
lemma [code]:
nipkow@60515
  2035
  "mset xs #\<inter> mset ys =
nipkow@60515
  2036
    mset (snd (fold (\<lambda>x (ys, zs).
haftmann@51600
  2037
      if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, [])))"
haftmann@51600
  2038
proof -
nipkow@60515
  2039
  have "\<And>zs. mset (snd (fold (\<lambda>x (ys, zs).
haftmann@51600
  2040
    if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, zs))) =
nipkow@60515
  2041
      (mset xs #\<inter> mset ys) + mset zs"
haftmann@51623
  2042
    by (induct xs arbitrary: ys)
haftmann@51623
  2043
      (auto simp add: mem_set_multiset_eq inter_add_right1 inter_add_right2 ac_simps)
haftmann@51623
  2044
  then show ?thesis by simp
haftmann@51623
  2045
qed
haftmann@51623
  2046
haftmann@51623
  2047
lemma [code]:
nipkow@60515
  2048
  "mset xs #\<union> mset ys =
haftmann@61424
  2049
    mset (case_prod append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, [])))"
haftmann@51623
  2050
proof -
haftmann@61424
  2051
  have "\<And>zs. mset (case_prod append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, zs))) =
nipkow@60515
  2052
      (mset xs #\<union> mset ys) + mset zs"
haftmann@51623
  2053
    by (induct xs arbitrary: ys) (simp_all add: multiset_eq_iff)
haftmann@51600
  2054
  then show ?thesis by simp
haftmann@51600
  2055
qed
haftmann@51600
  2056
blanchet@59813
  2057
declare in_multiset_in_set [code_unfold]
haftmann@51600
  2058
wenzelm@60606
  2059
lemma [code]: "count (mset xs) x = fold (\<lambda>y. if x = y then Suc else id) xs 0"
haftmann@51600
  2060
proof -
nipkow@60515
  2061
  have "\<And>n. fold (\<lambda>y. if x = y then Suc else id) xs n = count (mset xs) x + n"
haftmann@51600
  2062
    by (induct xs) simp_all
haftmann@51600
  2063
  then show ?thesis by simp
haftmann@51600
  2064
qed
haftmann@51600
  2065
nipkow@60515
  2066
declare set_mset_mset [code]
nipkow@60515
  2067
nipkow@60515
  2068
declare sorted_list_of_multiset_mset [code]
haftmann@51600
  2069
wenzelm@61585
  2070
lemma [code]: \<comment> \<open>not very efficient, but representation-ignorant!\<close>
nipkow@60515
  2071
  "mset_set A = mset (sorted_list_of_set A)"
haftmann@51600
  2072
  apply (cases "finite A")
haftmann@51600
  2073
  apply simp_all
haftmann@51600
  2074
  apply (induct A rule: finite_induct)
blanchet@59813
  2075
  apply (simp_all add: add.commute)
haftmann@51600
  2076
  done
haftmann@51600
  2077
nipkow@60515
  2078
declare size_mset [code]
haftmann@51600
  2079
blanchet@58425
  2080
fun ms_lesseq_impl :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool option" where
nipkow@55808
  2081
  "ms_lesseq_impl [] ys = Some (ys \<noteq> [])"
blanchet@58425
  2082
| "ms_lesseq_impl (Cons x xs) ys = (case List.extract (op = x) ys of
nipkow@55808
  2083
     None \<Rightarrow> None
nipkow@55808
  2084
   | Some (ys1,_,ys2) \<Rightarrow> ms_lesseq_impl xs (ys1 @ ys2))"
nipkow@55808
  2085
nipkow@60515
  2086
lemma ms_lesseq_impl: "(ms_lesseq_impl xs ys = None \<longleftrightarrow> \<not> mset xs \<le># mset ys) \<and>
nipkow@60515
  2087
  (ms_lesseq_impl xs ys = Some True \<longleftrightarrow> mset xs <# mset ys) \<and>
nipkow@60515
  2088
  (ms_lesseq_impl xs ys = Some False \<longrightarrow> mset xs = mset ys)"
nipkow@55808
  2089
proof (induct xs arbitrary: ys)
nipkow@55808
  2090
  case (Nil ys)
nipkow@55808
  2091
  show ?case by (auto simp: mset_less_empty_nonempty)
nipkow@55808
  2092
next
nipkow@55808
  2093
  case (Cons x xs ys)
nipkow@55808
  2094
  show ?case
nipkow@55808
  2095
  proof (cases "List.extract (op = x) ys")
nipkow@55808
  2096
    case None
nipkow@55808
  2097
    hence x: "x \<notin> set ys" by (simp add: extract_None_iff)
nipkow@55808
  2098
    {
nipkow@60515
  2099
      assume "mset (x # xs) \<le># mset ys"
nipkow@60495
  2100
      from set_mset_mono[OF this] x have False by simp
nipkow@55808
  2101
    } note nle = this
nipkow@55808
  2102
    moreover
nipkow@55808
  2103
    {
nipkow@60515
  2104
      assume "mset (x # xs) <# mset ys"
nipkow@60515
  2105
      hence "mset (x # xs) \<le># mset ys" by auto
nipkow@55808
  2106
      from nle[OF this] have False .
nipkow@55808
  2107
    }
nipkow@55808
  2108
    ultimately show ?thesis using None by auto
nipkow@55808
  2109
  next
nipkow@55808
  2110
    case (Some res)
nipkow@55808
  2111
    obtain ys1 y ys2 where res: "res = (ys1,y,ys2)" by (cases res, auto)
nipkow@55808
  2112
    note Some = Some[unfolded res]
nipkow@55808
  2113
    from extract_SomeE[OF Some] have "ys = ys1 @ x # ys2" by simp
nipkow@60515
  2114
    hence id: "mset ys = mset (ys1 @ ys2) + {#x#}"
nipkow@55808
  2115
      by (auto simp: ac_simps)
nipkow@55808
  2116
    show ?thesis unfolding ms_lesseq_impl.simps
nipkow@55808
  2117
      unfolding Some option.simps split
nipkow@55808
  2118
      unfolding id
nipkow@55808
  2119
      using Cons[of "ys1 @ ys2"]
Mathias@60397
  2120
      unfolding subset_mset_def subseteq_mset_def by auto
nipkow@55808
  2121
  qed
nipkow@55808
  2122
qed
nipkow@55808
  2123
nipkow@60515
  2124
lemma [code]: "mset xs \<le># mset ys \<longleftrightarrow> ms_lesseq_impl xs ys \<noteq> None"
nipkow@55808
  2125
  using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
nipkow@55808
  2126
nipkow@60515
  2127
lemma [code]: "mset xs <# mset ys \<longleftrightarrow> ms_lesseq_impl xs ys = Some True"
nipkow@55808
  2128
  using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
haftmann@51600
  2129
haftmann@51600
  2130
instantiation multiset :: (equal) equal
haftmann@51600
  2131
begin
haftmann@51600
  2132
haftmann@51600
  2133
definition
nipkow@55808
  2134
  [code del]: "HOL.equal A (B :: 'a multiset) \<longleftrightarrow> A = B"
nipkow@60515
  2135
lemma [code]: "HOL.equal (mset xs) (mset ys) \<longleftrightarrow> ms_lesseq_impl xs ys = Some False"
nipkow@55808
  2136
  unfolding equal_multiset_def
nipkow@55808
  2137
  using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
haftmann@51600
  2138
haftmann@51600
  2139
instance
wenzelm@60678
  2140
  by standard (simp add: equal_multiset_def)
wenzelm@60678
  2141
blanchet@37169
  2142
end
haftmann@49388
  2143
wenzelm@60606
  2144
lemma [code]: "msetsum (mset xs) = listsum xs"
haftmann@51600
  2145
  by (induct xs) (simp_all add: add.commute)
haftmann@51600
  2146
wenzelm@60606
  2147
lemma [code]: "msetprod (mset xs) = fold times xs 1"
haftmann@51600
  2148
proof -
nipkow@60515
  2149
  have "\<And>x. fold times xs x = msetprod (mset xs) * x"
haftmann@51600
  2150
    by (induct xs) (simp_all add: mult.assoc)
haftmann@51600
  2151
  then show ?thesis by simp
haftmann@51600
  2152
qed
haftmann@51600
  2153
wenzelm@60500
  2154
text \<open>
haftmann@51600
  2155
  Exercise for the casual reader: add implementations for @{const le_multiset}
haftmann@51600
  2156
  and @{const less_multiset} (multiset order).
wenzelm@60500
  2157
\<close>
wenzelm@60500
  2158
wenzelm@60500
  2159
text \<open>Quickcheck generators\<close>
haftmann@51600
  2160
haftmann@51600
  2161
definition (in term_syntax)
wenzelm@61076
  2162
  msetify :: "'a::typerep list \<times> (unit \<Rightarrow> Code_Evaluation.term)
haftmann@51600
  2163
    \<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
nipkow@60515
  2164
  [code_unfold]: "msetify xs = Code_Evaluation.valtermify mset {\<cdot>} xs"
haftmann@51600
  2165
haftmann@51600
  2166
notation fcomp (infixl "\<circ>>" 60)
haftmann@51600
  2167
notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@51600
  2168
haftmann@51600
  2169
instantiation multiset :: (random) random
haftmann@51600
  2170
begin
haftmann@51600
  2171
haftmann@51600
  2172
definition
haftmann@51600
  2173
  "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (msetify xs))"
haftmann@51600
  2174
haftmann@51600
  2175
instance ..
haftmann@51600
  2176
haftmann@51600
  2177
end
haftmann@51600
  2178
haftmann@51600
  2179
no_notation fcomp (infixl "\<circ>>" 60)
haftmann@51600
  2180
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@51600
  2181
haftmann@51600
  2182
instantiation multiset :: (full_exhaustive) full_exhaustive
haftmann@51600
  2183
begin
haftmann@51600
  2184
haftmann@51600
  2185
definition full_exhaustive_multiset :: "('a multiset \<times> (unit \<Rightarrow> term) \<Rightarrow> (bool \<times> term list) option) \<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option"
haftmann@51600
  2186
where
haftmann@51600
  2187
  "full_exhaustive_multiset f i = Quickcheck_Exhaustive.full_exhaustive (\<lambda>xs. f (msetify xs)) i"
haftmann@51600
  2188
haftmann@51600
  2189
instance ..
haftmann@51600
  2190
haftmann@51600
  2191
end
haftmann@51600
  2192
haftmann@51600
  2193
hide_const (open) msetify
haftmann@51600
  2194
blanchet@55129
  2195
wenzelm@60500
  2196
subsection \<open>BNF setup\<close>
blanchet@55129
  2197
blanchet@57966
  2198
definition rel_mset where
nipkow@60515
  2199
  "rel_mset R X Y \<longleftrightarrow> (\<exists>xs ys. mset xs = X \<and> mset ys = Y \<and> list_all2 R xs ys)"
nipkow@60515
  2200
nipkow@60515
  2201
lemma mset_zip_take_Cons_drop_twice:
blanchet@57966
  2202
  assumes "length xs = length ys" "j \<le> length xs"
nipkow@60515
  2203
  shows "mset (zip (take j xs @ x # drop j xs) (take j ys @ y # drop j ys)) =
nipkow@60515
  2204
    mset (zip xs ys) + {#(x, y)#}"
wenzelm@60606
  2205
  using assms
blanchet@57966
  2206
proof (induct xs ys arbitrary: x y j rule: list_induct2)
blanchet@57966
  2207
  case Nil
blanchet@57966
  2208
  thus ?case
blanchet@57966
  2209
    by simp
blanchet@55129
  2210
next
blanchet@57966
  2211
  case (Cons x xs y ys)
blanchet@57966
  2212
  thus ?case
blanchet@57966
  2213
  proof (cases "j = 0")
blanchet@57966
  2214
    case True
blanchet@57966
  2215
    thus ?thesis
blanchet@57966
  2216
      by simp
blanchet@55129
  2217
  next
blanchet@57966
  2218
    case False
blanchet@57966
  2219
    then obtain k where k: "j = Suc k"
wenzelm@60678
  2220
      by (cases j) simp
blanchet@57966
  2221
    hence "k \<le> length xs"
blanchet@57966
  2222
      using Cons.prems by auto
nipkow@60515
  2223
    hence "mset (zip (take k xs @ x # drop k xs) (take k ys @ y # drop k ys)) =
nipkow@60515
  2224
      mset (zip xs ys) + {#(x, y)#}"
blanchet@57966
  2225
      by (rule Cons.hyps(2))
blanchet@57966
  2226
    thus ?thesis
blanchet@57966
  2227
      unfolding k by (auto simp: add.commute union_lcomm)
blanchet@58425
  2228
  qed
blanchet@55129
  2229
qed
blanchet@55129
  2230
nipkow@60515
  2231
lemma ex_mset_zip_left:
nipkow@60515
  2232
  assumes "length xs = length ys" "mset xs' = mset xs"
nipkow@60515
  2233
  shows "\<exists>ys'. length ys' = length xs' \<and> mset (zip xs' ys') = mset (zip xs ys)"
blanchet@58425
  2234
using assms
blanchet@57966
  2235
proof (induct xs ys arbitrary: xs' rule: list_induct2)
blanchet@57966
  2236
  case Nil
blanchet@57966
  2237
  thus ?case
blanchet@57966
  2238
    by auto
blanchet@57966
  2239
next
blanchet@57966
  2240
  case (Cons x xs y ys xs')
blanchet@57966
  2241
  obtain j where j_len: "j < length xs'" and nth_j: "xs' ! j = x"
nipkow@60515
  2242
    by (metis Cons.prems in_set_conv_nth list.set_intros(1) mset_eq_setD)
blanchet@58425
  2243
blanchet@58425
  2244
  def xsa \<equiv> "take j xs' @ drop (Suc j) xs'"
nipkow@60515
  2245
  have "mset xs' = {#x#} + mset xsa"
blanchet@57966
  2246
    unfolding xsa_def using j_len nth_j
nipkow@58247
  2247
    by (metis (no_types) ab_semigroup_add_class.add_ac(1) append_take_drop_id Cons_nth_drop_Suc
nipkow@60515
  2248
      mset.simps(2) union_code add.commute)
nipkow@60515
  2249
  hence ms_x: "mset xsa = mset xs"
nipkow@60515
  2250
    by (metis Cons.prems add.commute add_right_imp_eq mset.simps(2))
blanchet@57966
  2251
  then obtain ysa where
nipkow@60515
  2252
    len_a: "length ysa = length xsa" and ms_a: "mset (zip xsa ysa) = mset (zip xs ys)"
blanchet@57966
  2253
    using Cons.hyps(2) by blast
blanchet@57966
  2254
blanchet@57966
  2255
  def ys' \<equiv> "take j ysa @ y # drop j ysa"
blanchet@57966
  2256
  have xs': "xs' = take j xsa @ x # drop j xsa"
blanchet@57966
  2257
    using ms_x j_len nth_j Cons.prems xsa_def
nipkow@58247
  2258
    by (metis append_eq_append_conv append_take_drop_id diff_Suc_Suc Cons_nth_drop_Suc length_Cons
nipkow@60515
  2259
      length_drop size_mset)
blanchet@57966
  2260
  have j_len': "j \<le> length xsa"
blanchet@57966
  2261
    using j_len xs' xsa_def
blanchet@57966
  2262
    by (metis add_Suc_right append_take_drop_id length_Cons length_append less_eq_Suc_le not_less)
blanchet@57966
  2263
  have "length ys' = length xs'"
blanchet@57966
  2264
    unfolding ys'_def using Cons.prems len_a ms_x
nipkow@60515
  2265
    by (metis add_Suc_right append_take_drop_id length_Cons length_append mset_eq_length)
nipkow@60515
  2266
  moreover have "mset (zip xs' ys') = mset (zip (x # xs) (y # ys))"
blanchet@57966
  2267
    unfolding xs' ys'_def
nipkow@60515
  2268
    by (rule trans[OF mset_zip_take_Cons_drop_twice])
blanchet@57966
  2269
      (auto simp: len_a ms_a j_len' add.commute)
blanchet@57966
  2270
  ultimately show ?case
blanchet@57966
  2271
    by blast
blanchet@55129
  2272
qed
blanchet@55129
  2273
blanchet@57966
  2274
lemma list_all2_reorder_left_invariance:
nipkow@60515
  2275
  assumes rel: "list_all2 R xs ys" and ms_x: "mset xs' = mset xs"
nipkow@60515
  2276
  shows "\<exists>ys'. list_all2 R xs' ys' \<and> mset ys' = mset ys"
blanchet@57966
  2277
proof -
blanchet@57966
  2278
  have len: "length xs = length ys"
blanchet@57966
  2279
    using rel list_all2_conv_all_nth by auto
blanchet@57966
  2280
  obtain ys' where
nipkow@60515
  2281
    len': "length xs' = length ys'" and ms_xy: "mset (zip xs' ys') = mset (zip xs ys)"
nipkow@60515
  2282
    using len ms_x by (metis ex_mset_zip_left)
blanchet@57966
  2283
  have "list_all2 R xs' ys'"
nipkow@60515
  2284
    using assms(1) len' ms_xy unfolding list_all2_iff by (blast dest: mset_eq_setD)
nipkow@60515
  2285
  moreover have "mset ys' = mset ys"
nipkow@60515
  2286
    using len len' ms_xy map_snd_zip mset_map by metis
blanchet@57966
  2287
  ultimately show ?thesis
blanchet@57966
  2288
    by blast
blanchet@55129
  2289
qed
blanchet@55129
  2290
nipkow@60515
  2291
lemma ex_mset: "\<exists>xs. mset xs = X"
nipkow@60515
  2292
  by (induct X) (simp, metis mset.simps(2))
blanchet@55129
  2293
blanchet@55129
  2294
bnf "'a multiset"
blanchet@57966
  2295
  map: image_mset
nipkow@60495
  2296
  sets: set_mset
blanchet@55129
  2297
  bd: natLeq
blanchet@55129
  2298
  wits: "{#}"
blanchet@57966
  2299
  rel: rel_mset
blanchet@57966
  2300
proof -
blanchet@57966
  2301
  show "image_mset id = id"
blanchet@57966
  2302
    by (rule image_mset.id)
wenzelm@60606
  2303
  show "image_mset (g \<circ> f) = image_mset g \<circ> image_mset f" for f g
blanchet@59813
  2304
    unfolding comp_def by (rule ext) (simp add: comp_def image_mset.compositionality)
wenzelm@60606
  2305
  show "(\<And>z. z \<in> set_mset X \<Longrightarrow> f z = g z) \<Longrightarrow> image_mset f X = image_mset g X" for f g X
wenzelm@60606
  2306
    by (induct X) (simp_all (no_asm),
nipkow@60495
  2307
      metis One_nat_def Un_iff count_single mem_set_mset_iff set_mset_union zero_less_Suc)
wenzelm@60606
  2308
  show "set_mset \<circ> image_mset f = op ` f \<circ> set_mset" for f
blanchet@57966
  2309
    by auto
blanchet@57966
  2310
  show "card_order natLeq"
blanchet@57966
  2311
    by (rule natLeq_card_order)
blanchet@57966
  2312
  show "BNF_Cardinal_Arithmetic.cinfinite natLeq"
blanchet@57966
  2313
    by (rule natLeq_cinfinite)
wenzelm@60606
  2314
  show "ordLeq3 (card_of (set_mset X)) natLeq" for X
blanchet@57966
  2315
    by transfer
blanchet@57966
  2316
      (auto intro!: ordLess_imp_ordLeq simp: finite_iff_ordLess_natLeq[symmetric] multiset_def)
wenzelm@60606
  2317
  show "rel_mset R OO rel_mset S \<le> rel_mset (R OO S)" for R S
blanchet@57966
  2318
    unfolding rel_mset_def[abs_def] OO_def
blanchet@57966
  2319
    apply clarify
wenzelm@60678
  2320
    subgoal for X Z Y xs ys' ys zs
wenzelm@60678
  2321
      apply (drule list_all2_reorder_left_invariance [where xs = ys' and ys = zs and xs' = ys])
wenzelm@60678
  2322
      apply (auto intro: list_all2_trans)
wenzelm@60678
  2323
      done
wenzelm@60606
  2324
    done
wenzelm@60606
  2325
  show "rel_mset R =
nipkow@60495
  2326
    (BNF_Def.Grp {x. set_mset x \<subseteq> {(x, y). R x y}} (image_mset fst))\<inverse>\<inverse> OO
wenzelm@60606
  2327
    BNF_Def.Grp {x. set_mset x \<subseteq> {(x, y). R x y}} (image_mset snd)" for R
blanchet@57966
  2328
    unfolding rel_mset_def[abs_def] BNF_Def.Grp_def OO_def
blanchet@57966
  2329
    apply (rule ext)+
blanchet@57966
  2330
    apply auto
nipkow@60515
  2331
     apply (rule_tac x = "mset (zip xs ys)" in exI; auto)
nipkow@60515
  2332
        apply (metis list_all2_lengthD map_fst_zip mset_map)
blanchet@57966
  2333
       apply (auto simp: list_all2_iff)[1]
nipkow@60515
  2334
      apply (metis list_all2_lengthD map_snd_zip mset_map)
blanchet@57966
  2335
     apply (auto simp: list_all2_iff)[1]
blanchet@57966
  2336
    apply (rename_tac XY)
nipkow@60515
  2337
    apply (cut_tac X = XY in ex_mset)
blanchet@57966
  2338
    apply (erule exE)
blanchet@57966
  2339
    apply (rename_tac xys)
blanchet@57966
  2340
    apply (rule_tac x = "map fst xys" in exI)
nipkow@60515
  2341
    apply (auto simp: mset_map)
blanchet@57966
  2342
    apply (rule_tac x = "map snd xys" in exI)
nipkow@60515
  2343
    apply (auto simp: mset_map list_all2I subset_eq zip_map_fst_snd)
wenzelm@59997
  2344
    done
wenzelm@60606
  2345
  show "z \<in> set_mset {#} \<Longrightarrow> False" for z
blanchet@57966
  2346
    by auto
blanchet@57966
  2347
qed
blanchet@57966
  2348
wenzelm@60606
  2349
inductive rel_mset'
wenzelm@60606
  2350
where
blanchet@57966
  2351
  Zero[intro]: "rel_mset' R {#} {#}"
blanchet@57966
  2352
| Plus[intro]: "\<lbrakk>R a b; rel_mset' R M N\<rbrakk> \<Longrightarrow> rel_mset' R (M + {#a#}) (N + {#b#})"
blanchet@57966
  2353
blanchet@57966
  2354
lemma rel_mset_Zero: "rel_mset R {#} {#}"
blanchet@57966
  2355
unfolding rel_mset_def Grp_def by auto
blanchet@55129
  2356
blanchet@55129
  2357
declare multiset.count[simp]
blanchet@55129
  2358
declare Abs_multiset_inverse[simp]
blanchet@55129
  2359
declare multiset.count_inverse[simp]
blanchet@55129
  2360
declare union_preserves_multiset[simp]
blanchet@55129
  2361
blanchet@57966
  2362
lemma rel_mset_Plus:
wenzelm@60606
  2363
  assumes ab: "R a b"
wenzelm@60606
  2364
    and MN: "rel_mset R M N"
wenzelm@60606
  2365
  shows "rel_mset R (M + {#a#}) (N + {#b#})"
wenzelm@60606
  2366
proof -
wenzelm@60606
  2367
  have "\<exists>ya. image_mset fst y + {#a#} = image_mset fst ya \<and>
wenzelm@60606
  2368
    image_mset snd y + {#b#} = image_mset snd ya \<and>
wenzelm@60606
  2369
    set_mset ya \<subseteq> {(x, y). R x y}"
wenzelm@60606
  2370
    if "R a b" and "set_mset y \<subseteq> {(x, y). R x y}" for y
wenzelm@60606
  2371
    using that by (intro exI[of _ "y + {#(a,b)#}"]) auto
blanchet@55129
  2372
  thus ?thesis
blanchet@55129
  2373
  using assms
blanchet@57966
  2374
  unfolding multiset.rel_compp_Grp Grp_def by blast
blanchet@55129
  2375
qed
blanchet@55129
  2376
wenzelm@60606
  2377
lemma rel_mset'_imp_rel_mset: "rel_mset' R M N \<Longrightarrow> rel_mset R M N"
wenzelm@60678
  2378
  by (induct rule: rel_mset'.induct) (auto simp: rel_mset_Zero rel_mset_Plus)
blanchet@57966
  2379
wenzelm@60606
  2380
lemma rel_mset_size: "rel_mset R M N \<Longrightarrow> size M = size N"
wenzelm@60678
  2381
  unfolding multiset.rel_compp_Grp Grp_def by auto
blanchet@55129
  2382
blanchet@55129
  2383
lemma multiset_induct2[case_names empty addL addR]:
wenzelm@60678
  2384
  assumes empty: "P {#} {#}"
wenzelm@60678
  2385
    and addL: "\<And>M N a. P M N \<Longrightarrow> P (M + {#a#}) N"
wenzelm@60678
  2386
    and addR: "\<And>M N a. P M N \<Longrightarrow> P M (N + {#a#})"
wenzelm@60678
  2387
  shows "P M N"
blanchet@55129
  2388
apply(induct N rule: multiset_induct)
blanchet@55129
  2389
  apply(induct M rule: multiset_induct, rule empty, erule addL)
blanchet@55129
  2390
  apply(induct M rule: multiset_induct, erule addR, erule addR)
blanchet@55129
  2391
done
blanchet@55129
  2392
nipkow@59949
  2393
lemma multiset_induct2_size[consumes 1, case_names empty add]:
wenzelm@60606
  2394
  assumes c: "size M = size N"
wenzelm@60606
  2395
    and empty: "P {#} {#}"
wenzelm@60606
  2396
    and add: "\<And>M N a b. P M N \<Longrightarrow> P (M + {#a#}) (N + {#b#})"
wenzelm@60606
  2397
  shows "P M N"
wenzelm@60678
  2398
  using c
wenzelm@60678
  2399
proof (induct M arbitrary: N rule: measure_induct_rule[of size])
wenzelm@60606
  2400
  case (less M)
wenzelm@60606
  2401
  show ?case